Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner
144 A. Bensoussan, J. L. Lions (Editors)
Analysis and Optimization of Systems Proceedings of the 9th International Conference Antibes, June 12-15, 1990
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong
Series Editors M. Thoma • A. Wyner
Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi
Editors A. Bensoussan INRIA - Universite Paris IX Dauphine J. L. Lions College de France - CNES, Paris INRIA Institut National de Recherche en Informatique et en Automatique Domaine de Voluceau, Rocquencourt, B.P.105 78153 Le Chesnay/France
ISBN 3-540-52630-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52630-7 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1990 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 2161/3020-543210 Printed on acid-free paper.
FOREWORD
These Proceedings of the 9th series of the I n t e r n a t i o n a l Conference "Analysis and Optimisation of Systems" gather more than 100 papers covering b o t h Theory and Applications in the field of Systems Analysis and Control (1 invited paper - 96 contributed papers - 10 papers presented at the two Invited Sessions).
This year again, we have noticed countries represented among the This vitality shows the complete continues to progress and allows promising results and applications
a large international participation through 18 speakers. maturity of this discipline which, regularly, scientists and engineers to expect the most for the future.
We would like to express our thanks to the Organizations which have sponsored this Conference : AFCET, CNRS, IEEE-CSS, IFAC, IIASA, INSERM, SEE, SIAM and SMAI.
We also would like to extend our gratitude to : the authors who have shown a real interest in this Conference the many reviewers who have accepted the hard task of selecting papers the chairpersons for having run with efficiency all the sessions of the Conference all the members of the Organization Committee the Scientific Secretaries who deserve a special mention the staff of the Public Relations Department for their excellent work i n making this Conference really happen. Professor M. THOMA and the Editor SPRINGER VERLAG who have accepted to publish this series in the Lecture Notes in Control and Information Sciences.
A. BENSOUSSAN
J.L. LIONS
THIS CONFERENCE IS UNDER THE SPONSORSHIP OF : AFCET
Association Franqaise pour la Cybern6tique Economiquc et Technique
CNRS
Centre National de la Recherche
IEEE-CSS
The Institute of Electrical and Electronics Enginecrs, Inc. Control System Society "Participating Societies"
IFAC
International Federation of
IIASA
International Institute for Applied
Scientifique
Automatic
Control
Systems
Analysis
INSERM
Institut National de la Sant6 et de la Recherche M6dicale
SEE SIAM
Soci6t6 des Electriciens et Electroniciens Society for Industrial and Applied Mathematics
SMAI
Socidt6 pour les Mathdmatiques Appliqu6es et Industrielles
ORGANIZATION COMMITTEE K.J. A. A. P. P. A. A. I.D. J.L. M. R. J.C.
ASTR6M BENSOUSSAN BENVENISTE BERNHARD FAURRE FOSSARD ISIDORI LANDAU LIONS THOMA VINTER WILLEMS
Lund Institute of Technology INRIA Universit6 Paris-Dauphine INRIA - IRISA - Rennes INRIA - Sophia-Antipolis SAGEM, Paris ENSAE, Toulouse Universita di Roma I.N.P.G., Grenoble Cotl6ge de France/CNES, Paris Technische Universit~t, Hannover Imperial College, London GrOningen U n i v e r s i t y
SCIENTIFIC SECRETARIES L. J.F. A.
BARATCHART BONNANS SULEM
INRIA INRIA INRIA
-
-
Sophia-Antipolis Rocqucncourt Rocquencourt
CONFERENCE SECRETARIAT Th. S. E.
BRICHETEAU GOSSET MANY
INRIA-France INRIA-France INRIA-France
AUTHORS
ABU EL ATA S AKIAN M. ANANTHARAM V. ANTOULAS A.C. AUBIN J.P. BAIRD C.R. BARATCHART L. BARBOT J.P. BARDI M. BEHAEGEL E. BENSOUSSAN D. BERGER W.A. BLANCHON G. BONGERS P.M.M. BONILLA E.M. BONNAN$ J.F. BOSE A.K. BRDYS M.A. BROGLIATO B. BULGAKOV A. YA. BYRNES C.I.
851 113 937 297 821 467 477 314 103 145 873 341 423 307 279 423 632 674 798 95 821-861
CANPONT D. CARVALHO M. TEIXEIRA. M. CHA I T.Y. CHANGYOU L. CHANG SHI-CHUNG CHANTRE P. CHAVENT G. CHEN CHUN-HUNG CHERNOUSKO F.L. COIC A. COMMAULT C.
37 900 684 985 133 851 452 133 580 85 t 361
DESCUSSE J. DE KEYSER R.M.C. DE LUCA A. DELFINI P. DI BENEDETrO M.D. DION J.M. DODU J.C. DONG CAO DULUC R.
288 77 833 623 843 361 423 755 549
EL-ANSARY M.
890
FALCONE M. FAYAZ M.A. FELIACHI A. FEVO"I'TE G. FLANDOLI F. FLIESS M. FLORCHINGER P . FONG I-KONG FRANKOWSKA H. GABASOV R. GAUBERT S. GAUVR1T M.
I 03 611 208 37 694 778-851 2 2 8 - 258 133 519 570 957 912
INDEX
GEORGES D. GOMES G. GOUZE J.L. GRIZZLE J.W. GRORUD A. GUGLIELMI M.
600 912 324 401-843 704 497
HAMAM Y. HAMMOURI H. HAN CHI-GEUN HANUS R. HARCAUT J.P HEJMO W. HEUBERGER P.S.C. HOANG NGOC MINH V. INSTALLE M. IOFFE A. ISIDORI A.
600 391 413 664 27 714 307 381 755 442 821-861
JACOB G. JODAR L.
381 87
KAMEN E.W. KARATZAS i. KEANE M.A. KINNAERT M. KIRILLOVA F.M. KOKOTOVIC P.V. KOWALEWSKI A. KOZA J.R. KULC"ZYCKIP.
655 3 47 880 570 861 174 47 714
LAKRORI M. LALL H.S. LAMNABHI-LAGARR1GUEF. LANARI L. LEBRET G. LEHOCZKY J.P. LEPSCH': A. LE GLAND F. LESCH K. LOBRY C. LOISEAU J.J.
37 123 611 833 37 1 3 218 228 590 623 37 1
MALABRE M. MARCHETTI C. MESSAGER F. MIAN G.A. MILHEIRO de OLIVEIRA P. MONACO S. MORAAL P.E. MORDUKHOVICH B.S. MYSLINSKI A.
279 187 778 218 198 788 401 539 164
NAVARRO E. NIKOUKHAH R. NORMAND-CYROT D.
87 67 788
OLIVI M. OTHMAN S.
477 391
OUSTALOUP A. OZVEREN C.M.
767 947
PARDALOS P.M, PENG S. PENG Y. PERCELL P.B. PEREZ A. PERRY R.J. POMET J.B. POMMARET J.F. POURTALLIER O. PRALY L.
413. 724 664 123 361 341 808 271 745 808
RENEKE J.A. ROCHA P. RUBIO J.E. RUNCHAL A.K.
632 332 154 145
SACHS G. SASTRY S.S. SHREVE S.E. SPIZZICHINO F.
590 861 3 238
TAKEUCHI Y. TALAY D. TAPIERO C.S. THEPOT J. TSIOTRAS G.
248 704 967 735 967
ULANICKI B. ULIVI G. URYAS'EV S.P.
674 833 432
VIARO U. VIGNERON C. V1NTER R.B.
218 549 529
WAGNEUR E. WAHNON E. WAYNE BEQUETTE B. WANG D. WANG P,K.C. WENBIN LIU WIELONSKY F. WILLEMS J.C. WILLSKY A.S. XU GAN-L1N XU HONG YAO.
977 487 57 600 145 154 477 297-332 947 3 467
YANG WE1LIN YE YINYU YONG J1ONGMIN YOU YUNCHENG
133-351-924 413 559 642
ZHANG SIY1NG ZI-QIANG LANG
351-924 507
TABLE OF CONTENTS
INVITED
CONFERENCE
Optimality conditions for utility maximization in an incomplete market I. K a r a t z a s , J.P. Lehoezky, S.E. Shreve, Gan-Lin
NUMERICAL
Xu
........................
ALGORITHMS
Nonlinear control of missiles through a geometric approach J.P. H a r c a u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Non linear control of a batch evaporative crystallization using an algorithm of "L/A" type G. Fevotte, D. Canpont, M. Lakrori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Genetic breeding of non-linear optimal control strategies for broom balancing J.R. Koza, M.A. Keane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Process control using nonlinear programming techniques B. W a y n e Bequette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Construction of autonomous boundary-value linear systems from acausal input-output functions R. Nikoukhah ...............................................................................
67
PC-TACT : personal computer tool for advanced control techniques R.M.C. De Keyser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Solving singular regular systems A2X" + A1X' + AoX = F(t) without increasing the dimension of the problem L. J o d a r , E. Navarro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
Matrix spectrum dichotomy and generalized Lyapunov matrix cquation A. Ya. Bulgakov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Discrete approximation of the minimal time function for systems with regular optimal trajectories M. Bardi, M. Faicone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
Analyse de l'algorithme multigrille FMGH de rdsolution d'6quations d'Hamilton-Jacobi-B ellman M. Akian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
A dynamic programming based gas pipeline optimizer H.S. Lali, P.B. Percell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Two effective approaches for hydroelectric generation scheduling
Shi-Chung I-Kong
Chang,
Fong,
Weilin
Chun-Hung
Yang, Chen
........................ . ..............................
133
VII
CONTROL OF DISTRIBUTED P A R A M E T E R S Y S T E M S Iterative solution of a free-boundary problem arising in microscopic particle manipulation inside a liquid layer P.K.C. W a n g , E. Behaegei, A.K. R u n c h a l .......................................... 145 Optimality conditions for elliptic variational inequalities Liu Wenbin, J.E. Rubio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
Minimax shape optimization problem for Von Karman system A. M y s l i n s k i ..............................................................................
164
Optimality conditions for a parabolic time delay system A. Kowalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
LINEAR
AND
NONLINEAR
FILTERING
A note about singular perturbations in the filtering of a Markov chain C. M a r e h e t t i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187
Filtres approch6s pour un probl~me de filtrage non lindaire discret avec petit bruit d'obscrvation P. M i l h e i r o De Oliveira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
198
;.................
Decentralized filtering for interconnected systems A. F e l i a c h i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
208
Split forms of z-domain algorithms for linear prediction and stability analysis A. Lepschy, G.A. Mian, U. Viaro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
218
Time-discretization of the Zakai equation for diffusion processes observed in correlated noise P. F l o r c h i n g e r , F. Le Gland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
228
Finite-dimensional stochastic filtering in discrete time : the role of convolution semigroups F. S p i z z i c h i n o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
238
On the decompositions of observations with non-Gaussian additive noise and their innovations processes Y. T a k e u c h i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
248
Nonlinear filtering with dependent noises. The case of unbounded coefficients P. F l o r c h i n g e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
VIII A L G E B R A I C AND
GEOMETRIC S Y S T E M T H E O R Y
Group theory and controllability of partial differential control systems J.F. Pommaret ..............................................................................
271
Algebraic characterization of invariant zeros at infinity for generalized systems E.M. Bonilla, M. Malabre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 The dynamic block decoupling problem : a minimal solution by precompensation J. Descusse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
288
Minimal rational interpolation and Prony's method A.C. Antoulas, J.C. Willems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
297
Discrete normalized coprime factorization P.M.M. Bongers, P.S.C. Heuberger
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
A forward accessibility algorithm for nonlinear discrete time systems J.P. B a r b o t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
314
Stabilisation globale de syst~mes non-lin6aires par un contr61e positif J.L. Gouz~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
Controllability of delay-differential systems P. Rocha, J.C. Wiilems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
332
Geometric approach to parametric sensitivity and gain suppression W.A. Berger, R.J. Perry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
341
A kind of nonlinear system and its reduction and structural control Weilin Yang, Siying Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
351
Rejet de perturbation dans les syst~mes structur6s C. Commault, J.M. Dion, A. Perez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
361
A new canonical form for descriptor systems with outputs J.J. Loiseau, G. Lebret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Evaluation transform and symbolic calculus for nonlinear control systems V. Hoang Ngoc Minh, G. Jacob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
381
Immersion in infinite dimension H. Hammouri, S. Othman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
391
On observers for smooth nonlinear digital systems J . W . Grizzle, P.E. Moraal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
401
NONLINEAR
PROGRAMMING
An interior-point algorithm for large-scale quadratic problems with box constraints P.M. P a r d a l o s , Yinyu Ye, Chi-Geun Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
413
IX Optimisation des r6seaux 61ectriques de grande taille G. Bianchon, J.C. Dodu, J.F. B o n n a n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
423
Adaptive variable metric methods for nondifferentiable optimization problems S.P. U r y a s ' e v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
432
Composite optimization : second order conditions, value functions and sensitivity A. Ioffe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
442
A new sufficient condition for the well-posedness of non-linear least square problems arising in identification and control G. Chavent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . .
452
SIGNAL PROCESSING An identification technique for adaptive systems in the case of poor excitation Hong Yao Xu, C.R. Baird . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
467
Asymptotic properties in rational L2-approximation L. Baratchart, M. Olivi~ F. Wielonsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 A min-max testing approach to failure detection and identification E. Wahnon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
487
On-line detection of minimal order for linear pieccwise stationary systems M. Guglielmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
497
A simple method for identification of linear dynamics with hysteresis nonlinear input Lang Zi-Qiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
507
DETERMINISTIC
CONTROL
A priori estimates for operational differential inclusions and necessary conditions for optimality H. F r a n k o w s k a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
519
Relaxed controls, for time delay systems R.B. Vinter ..................................................................................
529
Maximum principle for nonconvex finite difference control systems B.S. M o r d u k h o v i e h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
539
Functional R. Duluc,
viability constraints C. Vigneron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
549
Maximum principle of optimal controls for a nonsmooth semilinear evolution system J i o n g m i n Yong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
559
X New algorithms of solving extremal problems R. Gabasov, F.M. Kirillova .............................................................
570
Constrained controls in linear oscillating systems F.L. Chernousko ...........................................................................
580
Fuel savings by optimal aircraft cruise with singular and chattering control G. Sachs, K. Lesch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
590
Optimal path planning of manipulators with singular configurations, workspace and collision-free constraints D. Wang, D. Georges, Y. Hamam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
600
On singular output tracking in multivariable nonlinear systems M.A. F a y a z , F. L a m n a b h i - L a g a r r i g u e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
611
CONTROLLABILITY AND STABILIZATION DISTRIBUTED PARAMETER SYSTEMS
OF
Formal controllability and physical controllability of linear systems C. L o b r y , P. Deifini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
623
Conditions yielding weak controllability for a class of linear hereditary systems J.A. Reneke, A.K. Bose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
632
Nonlinear exponential stabilization of Boussinesq equations Y u n e h e n g You . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
642
STOCHASTIC
AND
ADAPTIVE
CONTROL
Direct adaptive control in a state-space setting E.W. K a m e n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
655
Pole placement via generalized predictive control Y. Peng, R. H a n u s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
664
Separation principle in optimizing control of state-constrained dynamical systems under bounded uncertainty M.A. Brdys, B. Ulanicki ................................................................
674
Unification of some MIMO adaptive control algorithms and global convergence analysis T.Y. Chai .....................................................................................
684
Boundary control of a stochastic parabolic equation with nonsmooth final cost F. Flandoli ..................................................................................
694
XI Approximation of Lyapunov exponents of stochastic differential systems on compact manifolds A. Grorud, D. Talay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 Synthesis of closed-loop system controlling a random object Hejmo, P. Kuiczycki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
W.
Maximum principle for stochastic optimal control with non convex control domain S. Peng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MULTIDECISION
CONTROL AND
724
GAMES
Nash vs Stackelberg strategies in a capital accumulation game J. Th~pot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
735
A two player dynamical game with imperfect information O. P o u r t a l l i e r .............................................................................
745
An interactive multiple criteria decision supporting tool with application to a simplified regional development problem Cao Dong, M. Installe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
755
CONTROL OF FINITE D I M E N S I O N A L S Y S T E M S CRONE control : principle synthesis, performances with non-linearities and robustness-input immunity dilemma A. O u s t a l o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vers une
767
stabilisation non-lingaire discontinue
M. Fliess, F. Messager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
778
A combinatorial approach of the nonlinear sampling problem S. Monaco, D. N o r m a n d - C y r o t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
788
Non-linear MRAS in robots motion control B. Brogliato .................................................................................
798
Adaptive control of feedback equivalent systems J.B. Pomet, L. Praly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
808
NONLINEAR
ZERO DYNAMICS
AND
OUTPUT R E G U L A T I O N
Viability kernels, controlled invariance and zero dynamics for nonlinear systems J.P. Aubin, C.I. Byrnes, A. Isidori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
821
Output regulation of a flexible robot arm A. De Luca, L. Lanari, G. Ulivi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
833
An analysis of regularity conditions in nonlinear synthesis M.D. Di Benedetto, J.W. Grizzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
843
XII Discontinuous predictive control, inversion and singularities. Application to a heat exchanger M. Fliess, P. C h a n t r e , S. Abu El Ata, A. Coic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
851
The analysis of singularly perturbed zero dynamics A. I s i d o r i , P.V. Kokotovic, S.S. Sastry, C.I. Byrnes,
861
........................
ROB U S T N E S S Elargissement des objectifs de robustesse des syst~mes de commande D. B e n s o u s s a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
873
Right half plane poles and zeros and robustness limitations in feedback systems M. Kinnaert .................................................................................
880
Stabilizing control of a singularly perturbed system driven by wide-band noises M. E I - A n s a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
890
Control o f uncertain dynamical systems using strictly positive real systems M. Carvalho Minhoto Teixeira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900 Parameter robust control design based on parametric identification quality M. Gauvrit, G. Gomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
912
Rotation symmetric structure in control systems and related stability analysis Weilin Yang, Siying Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
924
DISCRETE EVENTS SYSTEMS A hydrodynamic limit for a lattice caricature of dynamic routing in circuit switched networks V. A n a n t h a r a m ............................................................................
937
Aggregation and multi-level control in discrete event dynamic systems C.M. O z v e r e n , A.S. Willsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
947
An algebraic method for optimizing resources in timed event graphs S. G a u b e r t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
957
WIP and CSP-1 quality control in a two-stages queue like production C.S. T a p i e r o , G. Tsiot~ras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
967
Modulo'fds and pseudomodules : 3 - the lattice structure problem E. Wagneur ..................................................................................
977
Modelling and analysing a class of flexible manufacturing system L. C h a n g y o u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
985
I N V I T E D CONFERENCE
OPTIMALITY CONDITIONS F O I l UTILITY MAXIMIZATION IN AN INCOMPLETE MAltKET T Ioannis Karatzas Columbia University New York, NY 10027 **
John P. Lehoczky** Steven E. Shreve *#
Gan-Lin Xu Carnegie Mellon University Pittsburgh, PA 15213
ABSTRACT The problem of maximizing expected utility of final wealth in an incomplete market is investigatsd~ The incomplete market is modelled by a bond and a finite number of stocks, the latter being driven by a d--dimensional Brownian motion. The coefficients of the bond and stock price processes are adapted to this Brownian motion, and the number of stocks is less than or equal to the dimension of the driving Brownian motion. It is shown that there is a way to "fictitiously" complete this market so that the optimal po~tfoho for the resulting completed market coincides with the optimal portfolio for the original incomplete maxket. A number of equivalent characterizations of the fictitious completion are given, and examples are provided. SECTION 1. INTj~.ODUCTION This paper utudies the problem of an agent who receives a deterministic initial capital, and must invest it in an incomplete market so as to maximize the expected utihty from wealth at a prespecified terminal time. The market consists of a bond and m stocks, and stock prices are driven by a d---dimensional Brownian motion. Incompleteness in this market arises when m is strictly smaller than d. The market coefficients, i.e., the interest rate, the rates of stock appreciation, and the stock volatihty coefficients, are random processes adapted to the full
Research supported by the National Science Foundation under Grant DMS--87-2307g. Research supported by the National Science Foundation under Grant DMS-87-02537. Vpresented at the gth Intsr'natwnal Conference on Analysis and Optimization of Systems (Antibes, June 1990), and at the MS[ Workshop on the Mathematical Theory o/Modern Financial Markets (Cornell University, July 1989).
d---dimensionat Brownian motion. When m < d, it is impossible to construct a portfolio consisting of the bond and the m available stocks so as to completely hedge the risk associated with these coefficient processes. In Sections 2 through 5, we define the utility maximization problem faced by the agent. In Section 6 we present the solution when the market is complete (m = d), and complete hedging is possible. This solution proceeds in three steps. First, on the underlying probability space one determines a new measure P* which discounts the growth inherent in the market; under this measure, the expected value of the final wealth attained by any "reasonable" portfolio is equal to the initial capital. Secondly, among all random variables whose expectation under the new measure is equal to the initial endowment, a most desirable one is determined. Thirdly, it is shown that a portfolio can be constructed which obtains this most desirable random variable as terminal wealth; this portfolio is optimal. A complete market is one in which the agent can construct a portfolio which attains as terminal wealth any random variable whose P * - expectation is equal to the initial capital. Because such a construction is possible, it is said that the agent can hedge against the risk associated with this market. Mathematically, the construction of a portfolio uses the fact that any martingale with respect to a Brownian filtration can be represented as a stochastic integral with respect to the Brownian motion; the integrand in this representation leads directly to the portfolio that we are seeking, if m=d. However, if there are fewer than d stocks, (m < d) this line of argument fails. In Section 7 we introduce a convenient way of thinking about an incomplete market: fictitious completion. When m < d, then one introduces new, fictitious stocks so as to create a complete market. If these fictitious stocks have a high appreciation rate, then under an optimal portfolio the agent wiU hold a long position in them, but if they have a low (even negative) appreciation rate, then he will hold a short position. Thus one would expect to be able to adjust the appreciation rates of the fictitious stocks so that the agent, by optimal choice, does not invest in them at all. These judiciously chosen fictitious stocks allow us to write down the complete market solution for the utility maximization problem, but they are superfluous in the actual implementation of the optimal portfolio which must then also be optimal for the original incomplete market. The fictitious completion with the above property is the least favorable to the agent, because the portfolio which is optimal under this completion is available to him under every other fictitious completion. We thus have the notion of a ~inimax fictitious completion: for every fictitious completion we compute the portfolio which maximizes the expected utility of final wealth, and then choose the completion that makes this maximum expected utility as small as possible. As explained in Section 7, a convenient way to parametrize fictitious completions of an incomplete market is by a certain space of continuous local martingales, each of them being the Radon-Nikodym derivative process of the new measure P*, mentioned in the discussion of complete markets. One would like to be able to characterize the local martingale corresponding to the minimax fictitious completion, and to show that it gives rise to an optimal portfolio in the original incomplete problem. This program is carried out in Section 8. Section 9 provides two examples in which the minimax fictitious completion can be computed fairly explicitly. In the first example, the utility function is logarithmic, and it is discovered that the fictitious stocks in the minimax completion should have rates of appreciation
equal to the interest rate of the bond. This is a very general result, insensitive to the nature of the dependence of market coefficients on the driving Brownian motion. In the second example, it is assumed that the utility function is of the power form, and that the driving Brownian motion splits into two independent parts; the first part drives the stock price processes, whose coefficients are adapted solely to the second part. The minimax local martingale is exhibited as the solution to a martingale representation problem, and the optimal portfolio is found to be given by the formula already known to be correct for deterministic model coefficients. Our model for the financial market can be traced back to Merton (1969, 1971) and Samuelson (1969). The modern mathematical approach to portfolio management in complete markets, built around the ideas of equivalent martingale measures and the creation of portfolios from martingale representation theorems, began with Harrison & Kreps (1979) and was further developed by Harrison & Pliska (1981, 1983), in the context of option pricing. Phska (1986), Cox & Huang (1987a, 1987b) and Karatzas, Lehoczky & Shreve (1987) adapted the martingale ideas to problems of utility maximization. Much of this development appears in Section 5.8 of Karatzas & Shreve (1988), from which Section 6 of the present paper is drawn; see also the review article of Karatzas (1989) for a survey of financial economics problems in complete markets. A first step toward a martingale analysis of incomplete markets was taken by Pages (1987), who considered a Brownian model in which the number of stocks was strictly less than the dimension of the driving Brownian motion. However, the coefficients of the bond and stock prices in this model were allowed to depend on the underlying Brownian model only through the bond and stock prices themselves. Thus, the vector of bond and stock prices formed a Markov process. This specialization created an essentially complete market, and thus it avoided the more interesting case of a market with genuinely unhedgeable risk. However, Pages did characterize the class of equivalent martingale measures which could arise in an incomplete model, and this laid the groundwork for further developments. A more substantive step was taken by He & Pearson (1988) in a discrete-time, finite probability space model, where the authors proposed finding the optimal intermediate consumption and terminal wealth corresponding to each of the equivalent martingale measures, and then searching over those policies to find a pair yielding the minimum expected total utility. Using separating hyperplane arguments, they were able to show that the total utility obtained by this two-step "minimax" process is the optimal total value for the incomplete problem. He & Pearson have also studied the incomplete problem in a continuous-time, Brownian model. In an early version of He & Pearson (1989), the authors consider Pag~s's characterization of the family of equivalent martingales measures and search over this family for a "minimax" equivalent martingale measure which would lead them to the optimal consumption and portfolio processes just as in a complete market. The martingale associated with this measure would be the "Arrow-Debreu" state prices in the incomplete model. However, the continuous-time model is more subtle than one might expect, and although it is now clear that Azrow-Debreu state prices exist for the incomplete model under some assumptions, it is not clear that they are associated with a martingale. The present paper uses local martingales rather than martingales to address the issue of market incompleteness in continuous-time models. This work was motivated by the aforementioned previous version of He & Pearson (1989), and by the use of local martingale
methods introduced by Xu (1990) and Xu & Shreve (1990) in the study of incompleteness induced by a prohibition on the short--selling of stocks. Using the stochastic duality theory of Bismut (1973), Xu formulated a dual problem whose solution could be shown to exist and could then be used to obtain existence and characterization of the solution of the original problem. As this paper shows, such duality methods can also be used in the traditional incomplete Brownian market model. While we still do not know if the minimax equivalent martingale measure sought by He & Pearson exists in any generality, we show here that the solution to Bismut's dual problem is a "least favorable local martingale" which can be used to generate a sequence of equivalent measures. The existence of this least favorable local martingale is sufficient for the study of many models. A notable exception is the incomplete model in which the agent's endowment is a stochastic process; we do not know how to obtain existence and characterization of the optimal policy for such a model in terms of a least favorable local martingale, unless it is actually a martingale. tie & Pearson (1989) have incorporated Xu's local martingale techniques into their original work. He & Pearson (1989) report the existence of an optimal portfolio for the terminal wealth utility maximization problem when the index of relative risk aversion is everywhere greater than or equal to one, and they report similar results for the problem with intermediate consumption and consumption at the terminal time when the index of relative risk aversion is everywhere less than or equal to one. Our paper deals only with the case of terminal wealth utility maximization when the index of relative risk aversion is everywhere less than or equal to one; the generalization to also allow for intermediate consumption is straight-forward. Whereas He & Pearson (1989) assume that some augmentation of the market model will result in Markov prices, we allow general It5 price processes. He & Pearson (1989) do not provide the full set of equivalent conditions contained in our Theorem 8.4, nor do they use our assumption (8.2), which plays a fundamental role in our proof of Theorem 8.4. For a fuller account of this theory, and for proofs of the results which are not included hcre, we refer the interested reader to the paper Karatzas, Lehoczky, Shreve & Xu (1989). SECTION 2. 3?tIE MARKET MODEL We assume a model for the financial market consisting of one bond with price P0(t) given by (2.1)
dP0(t) = r(t)P0(t)dt,
P0(0) -- 1,
and m stocks with prices-per--share Pi(t), i = 1,...,m, satisfying the equations
(2.2)
d dPi(t) = Pi(t)[bi(t)dt + Z aij(t)dW(J)(t)],
j=1
i = 1,...,m.
ttere W = r~(W~U,...,W~dj)* _ t~ is a d-dimensional Brownian motion on a probabihty space ( f l , J , P ) , and we denote by {.~t} the P-augmentation of the filtration generated by W. We suppose that d > m, i.e., the number of stocks does not exceed the number of sources of uncertainty in the model.
7 The interest rate r(t), the vector bit ) = (bl(t),...,b~(t)) of stock appreciation rates, and the volatility matrix act ) = {crij(t)}l
(2.3)
T
[ J
J]b(t)J[dt <,~,
a.s.
0
The positive constant T is the terminal time for the problem. All processes are defined on [0,T]. $ We assume that the matrix a(t) has full rank for every t, so the matrix (a(t)g (t)) "l and the relative risk process (2.4)
0(t)~ ~ (t)(a(t)~ (t))-l[b(t) --r(t)l]
are defined. Throughout this paper, we denote by 1 a vector whose every component is 1 and whose dimension is appropriate for the context. We assume that T (2.5)
I
II0(t)II2dt < ~ '
a.s.P.
0 We shall have occasion to use the so---called discount process
(2.6)
fl(t)
=tr~-~=exp{--a 1 It0 r(s)ds},
as well as the process t
(2.7)
Wo(t) a W(t) + I
0(s)ds. 0
SECTION ~. PORTFOLIO AND WEALTH PROCESSES A portfolio process ~ t ) = (~h(t),...,~r~(t)) satisfying
is an ~-valued, {J(t)}-adapted process
T (3.1)
I
Ho'*(t)lr(t)]l~ at < ~, a . s . P . 0
We regard ~ri(t) as the proportion of an agent's wealth invested in stock i at time t, while the
8
remaining proportion 1 - ~" (t).l is invested in the bond. We do not constrain these proportions to be in [0,1]. For a given, no~andom, initial wealth x > 0, let X x'~r denote the wealth process corresponding to a portfolio ~r defined by Xxjr(0) = x and (3.2)
dXX'~r(t) = r(t)xX'Tr(t)dt + xX'~r(t)Tr*(t)[(b(t) - r(t)l)dt + a(t)dW(t)]
= r(t)Xx3r(t)dt + xx'~r(t)~r*(t)a(t)dW0(t). In other words,
(3.3)
~rCs)a(s)dWo(s)+ 0
I t ( s ) - 8 9 I1~ * (s)~s)ll2]ds} '
SECTION 4. UTILITYFUNCTIONS We introduce a utility function U: (0,~) -~ ~, which is strictly increasing, strictly concave, twice continuously differentiable, and satisfies (4.1)
U'(0)-A l i m U ' ( x ) = | x~ 0
U'(|
limU'(x)=0. x-~ |
We denote by I: (0,| -~ (0,~) the (continuous, strictly decreasing) inverse of U ' , and consistent with (4.1) we note that (4.2)
I(~) A lira I(y) = 0, I(0) ~ lira I(y) = ~. y-~| y~ 0
We introduce the Legendre transform of U: (4.3)
Off) _Am i n [xy - U(x)] = yI(y) - U(I(y)), x>0
0 < y < ~.
The function lJ is strictly increasing, strictly concave and satisfies (4.4)
U'(y) = I(y),
0 < y < ~.
It is easily verified that (4.5)
U ( x ) = m i n { x y - lJ(y)} = x U ' ( x ) - 0 ( U ' ( x ) ) ,
y>0 From (4.3) and (4.5) we have the useful inequalities
0 < x < ~.
9 (4.6) (4.T)
u(I(y)) _> u(~) + y[I(y) - x], v x > 0, y > 0, 0 ( u ' ( x ) ) _> 0(y) + x[U' (x) - y], v x > 0, y > 0.
SECTION 5. THE UTILITY MAXIMIZATION P R O B L E M The problem at hand is, for a given initial wealth x > 0, to choose a portfolio a" that maximizes E u ( x X ' r ( T ) ) . If U(0) > - | this quantity is defined (but may be | for all portfolios r, whereas it may be undefined if U(0) = - | and U(co) = |
Thus, we define ,/#(x) to be the set
of portfolios ~r for which E max{0,-u(xX'Tr(T))) < |
(5.1) and set
V(x) =a
(5.2)
sup n~(x)
E u(xX'rr(T)).
We assume throughout that
(5.3)
v(,:) < |
v x > 0.
A portfolio process rr which attains the supremum in (5.2) is called optimal. In Section 8 we give conditions which ensure the existence of an optimal portfolio, and provide several characterizations for it. SECTION 6. THE COMPLETE M A R K E T SOLUTION The market model of Section 2 is complete if and only if a(t) is invertible for all t, i.e., d = m. Under this condition, the solution of the utility maximization problem has a simple solution, which we present below. Define the nonnegative local martingale
(6.1)
it , 1 I t z0(t ) =A e~p { - 0 O (s)dW(s) - , ~ 0 IIO(s)ll2ds},
a~d note from (3.3) that for any portfolio process ~r, flCt)Z0(t)xX'~(t) = x e~p { f t
( / C s) ~(s) - 0* (s))dWCs)- 21 f t
0
0
II~ . (s)~Cs) - 0(s)ll 2 ds}
is a nonnegative local martingale, hence a supcrmartingale. It follows that
10 (8.2)
E [Z(T) Z0(T ) xX'r(T)] <_x. We now assume that
(6.3)
E[fl(T)Z0(T)I(yf~(T)Zo(T))] < ~, V y > 0,
so we may define a function ~o: (0,| -~ (0,~) by (6.4)
~o(Y) = E[~(T)Z0(T)I(y~(T)Z0(T))].
The function ~$0 inherits from I the property of being a continuous, strictlydecreasing mapping
of (0,| onto itself, and so ~'0 has a (continuous, strictly decreasing) inverse ]/o from (0,| itself. We define (6.5)
onto
~ a= I(]/o(x)fl(T)Z0(T)).
6.1 LSMMA. The random variable ~ satisfies (6.6)
E[fl(T)Z0(T)~] = x,
(6.7)
E ma~{0,-U(~)} <
~,
and for every portfolio ~r s ~4(x) we have (6.8)
Eu(xX'~r(T)) _<EU(~).
PIt00F: Equation (6.6) follows directly from the definitions of f~ and ]/o. From (4.6) we have U(~) > U(1)
+
]o(x)fl(T)Zo(T)[~
--
1]
_>- [U(1)[ - ~/o(x)/~(T)Zo(T). But $ is nonnegative and bounded a.s.,and Z0, being a nonnegative local martingale, is a
supermartingale. Therefore Ef~(T)Zo(T) < | and (6.7) follows. Now let ~r be a portfolio satisfying (5.1). From (4.6), (6.2) and (6.6), we have EU(~) > E{u(xX'~(T))
+
]/o(x)~(T)Zo(T)[f~- xX'~r(T)]}
> E u(xX'r(T)). From L e m m a 6.1 we see that if there exists a portfolio ~" that "finances" f~, i.e., {~ = xx'~'(T), then ~ is optimal. W e have so far not used the assumption of market
[]
]] completeness; this assumption will be employed now in the construction of the portfolio ~- which finances ~. We begin with the martingale
(6.9)
M(t) _a E[fl(T)Zo(T)4~I j(t)].
Being adapted to the Brownian filtration {~'t}, M has a representation as
~0 r (s)dW(s) t
(6.11)
M(t) = x +
,
T for some {Jr}--adapted process r satisfying f
llr < ~ a.s. (Karatzas and Shreve (1988), 0 Theorem 3.4.1 and Problem 3.4.16). From It6's lemma we obtain that
d( ~ )
=
z~
(~, *(t) + M(t)0 *(t))dW0(t),
and /?(T)~ = ~ = < +
I~ Z~
~b*(t) + M(t)#*(t))dW0(t).
We define 1
(6.11)
(6.12)
~(t) =A
1 Z(t)Z0(t)s
*
*
(o * (t))-l(C(t) § M(t)0(t)).
It is now easily verified that X(0) = x, ~:(T) = ~, and dX(t) = r(t):X(t)dt + ~:(t)~ (t)a(t)dW0(t) (cf. (3.2)). We have proved the following theorem. 5.2 T~s0Rr, M. Let an initial wealth x > 0 be given. In a complete market (d = m), under assumption (6.3), the portfolio ~" given by (6.12) is optimal. The resulting optimal terminal wealth is ~ a__I(~/o(x)/~(T)Zo(T)).
]2 6.3 Example. (Logarithmic utility function). Suppose U(x) = log x. Then ,~0(x) = ~1 and
(6.13)
T ~ = x exp{f 0 (r(t) +
T,
89IIo(t)ll2) dt + f 0 0 (t)dW(t)}.
From (3.3) we have xX'~r(T) = ~, where (6.14)
~r(t) _a_(a(t)~r (t)) "~ [b(t) - r(t) 11.
6.4 Example. (Power utility function and deterministic model coefficients.) Suppose that U(x) = 1 x,where 5 6< 1,6~0, and that the processes r and O are
6It * 62 deterministic. Then exP{1-~_ 0 0 (s)dW(s) - ~ - b ~
f0t []0(s)[[2ds} is a martingale with
expectation equal to one (Karatzas & Shreve (1988), p. 199, Corollary 5.13), and we have 1 exp {i-~_ ~o~0(y)= y]~"f 6 fT0 (r(s)+ 89lla(s)II=)ds).Eexp{l ~ ITo O,(s)dW(s)} 1
= y'~---fexp{i-~IT m(s)ds}, where
(6.15)
m(t) A=r(t) +
~
[la(t)H2.
It foUows that ~/o(x) -- x & l exp { 6 I~ m(s)ds}, a,nd
(6.16)
~ = x exp{ IT (r(t) + ~
1 IT0 O*(t)dW(t)}. I[0(t)ll2)dt + i-~
From (3.3) we have xXfi(T) = ~], where
(6.17)
~t) a__~ (a(t)q*(t))-'[b(t)- r(t) _1].
13 SECTION 7. FICTITIOUS COMPLETION OF AN INCOMPLETE MAI~KET The market model of Section 2 is incomplete if and only if In < d, i.e., the number of stocks is less than the dimension of the driving Brownian motion. We propose to study this model by introducing d - m additional stocks, so that a complete market solution to the utility maximization problem can be derived, and then determining appreciation rates for the additional stocks so that the complete market optimal portfolio does not invest in them. Following this program, we introduce an {~'t}-progressively measurable, uniformly bounded, (d-m) x d matrix-valued process p(t).whose rows, thought of as vectors in ~d, are orthonormal and in the kernel of cr(t), i.e., g(t)p (t) = 0. We also introduce an {Jt}-progressively measurable (d-m)---dimensional vector process a satisfying T
(r.1)
~ Ila(t)lldt < ~, a.s. J
0
We create fictitious stocks with prices Si(t) governed by d
(7.2)
dSi(t) = Si(t)[ai(t)dt+ Z Pij(t)dW(J)(t)], i = l,...,d-m. j=l
We will hold p fixed throughout the remainder of the paper, but we will consider the process a as a parameter. For the augmented stock appreciation rate vector b a_ [a~] and the augmented volatility matrix ~ ~ [;] we can define an augmented relative risk process ~ by (cf. (2.4))
(7.a)
0(t) = ~ (t)(5-(t)5" ( t ) ) - l [ b ( t ) - r(t) 1] = 0(t) + u(t),
where
(r.4)
~(t) =A p (t) [a(t)- r(t) _1].
Note that 0 (t)u(t) = 0, so
{{O(t)][~= [{O(t)l[~ + {[u(t)[[a. We assume that
(7.s)
T I
[]v(t)[[ ~dt < ~, a.s., 0
14 so we can define the nonnegative local martingale (7.6)
Z ( t ) a exp{-
(0 (s) + v*(s))dW(s)
2
0(s)l[2+ []"(s)lI~)ds}
by analogy with (6.1), and the function (7.6)
~(y)
__aE[~(T)Z (T)I(yfl(T)Z (T))], y > 0
by analogy with (6.4). If
(7.7)
~V(y) < |
we may define ~ v to be the inverse of ~ ,
v y > 0,
and set
~x =~ I(~/~(x)fl(T)Zv(T))
(7.8)
by analogy with (6.5) If the fictitious stocks introduced in this section were really available, then EU(~,) would be the maximal expected utility of final wealth (Theorem 6.2). Since these stocks are not available, we have
(7.0)
v(x) ~= s u p
EU(XX'~(T)) _<E u ( c ) ,
and equality holds if there exists a portfolio process r such that
(7.10)
Xx'~(T) = C'
i.e., if the terminal wealth ~
can be financed without investment in the fictitious stocks. In light
of (7.9), such a z would be optimal for the problem of utility maximization in the incomplete market. In the next section, we discuss properties which 7r and v must have in order that they be related by (7.10). SECTION 8. ~
T I
0
CONDITION~ FOR OPTIMALITY IN THE INCOMPLETE
Let us define L2[0,T] to be the set of {~'(t)}-adapted ~d--valued processes r satisfying H~t)[I2dt < ~, a.s. We decompose L2[fl,T] into the orthogonal subspaces
15 K(c) A={v E L2[0,Tll a(t)u(t) = 0 V t E [0,T], a.s. P},
K~(a) ~ {~ e 1,2[0,T]I ~(t) e l~nge(a*(t)) V t e [0,T], a.s. P}. Note that the process # of (2.4) belongs to K• whereas the process u of (7.4) belongs to K(r On the other hand, if v E K((~) is given, then (7.4) can be solved for the fictitious stock appreciation rate process a. Thus, K(c) provides a parametrization space for fictitious completious of the incomplete market. We define KI(a) A_ {g E K(cr); v satisfies (7.7)}, and we recall definition (7.8). Then
E[#(T)
(8.1)
Z (t)r
= x.
We shall assume on occasion that the function I(.) has the property
(8.~.)
I ( ~ ) _<# I(y) v y > o
for some a E (0,1) and fl E(1,m). One consequence of (8.2) is that if JFv(y ) < m for some ~, E K(r and some y > 0, then ,.~ (y) < | for a]l y > 0, so that v E Kl(g). Let x > 0 and ~r E ,A(x) be given, and consider the statement that ~- is optimal for the incomplete market utility maximization problem: (A) Ootimalitv of ~r: Eu(xX,~'(T)) < Eu(xX,~(T)), V ~r E ~4(x). We shall characterize condition (A) with the help of the following conditions. For A E KI(c), consider the statements: (B) Finaneiliby of ~
There exists a protfolio ~r E ~ ( x ) such that Xx'~(T) = ~ ~, a.s.;
(C) Minimality of A: E U ( ~ ) < EU(~)
V L, E gl((r);,
(D) Dual optimality of A: For all v E K~((~), E0(~A(x)fl(T)Z~,(T)) < E0(~/A(x)fl(T)ZA(T)); (E) Parsimony of A: E[~7(T)Zy(T)~] _<x,
V ~, E Kl(a).
The relations among these conditions are expounded in the following two theorems. In the interest of brevity, we omit the lengthy proof of Theorem 8.1. The examples in Section 9 use only Theorem 8.2, whose proof we provide. 8.1 THEOREM. Assume that U(0) > ---m. Then conditions (B)--(E) are equivalent and imply (A) with the same ~- as in (B). Conversely, suppose that ~"E ..4(x) satisfies (A) and (8.2) holds; then there exists
A E Kl(r for which (B)--(E) hold.
The assumption U(0) > --m in Theorem 8.1 rules out the important special case of a logarithmic utility function. In the absence of this assumption, the following weaker rcsult can be established.
16 8.2 Tazo~r.M. Conditions (B) and (E) are equivalent, and imply both (C) and (A) with the same ~r as in (B). PIt00F: (B) ~ (E): The inequality E[fl(T)Zv(T)XX'~(T)] < x follows from (6.2) applied to the fictiously completed market corresponding to v. (B) v CA): The argument for this is given in the last paragraph of Section 7. (B) ~ (C): Under (B), we have (A), which means that V(x) = E U ( ~ ) . Now use
(7.9). (E) ~ (B): If ~r can be found for which xX'~r(T) = ~ , then the proof of (6.7) can be applied to show that ~r c M(x). Assume (E), and consider the Brownian martingale M(T) A=E[fl(T)ZA(T)~[~t]. From (8.1) we have M(0) = x. Imitating the argument preceding Theorem 6.2, we produce r e L2[0,T] such that t
M(t) = x + fl(T)~ =
= x +
f0
. r (s)dW(s),
- ' 3 . + M t O*.t + A 0
Decompose r as r = r + r where r E K• A E K((r), we may simplify the above equation to r 1 /~(T)~ -- x + J0 " g ~
+ IT ~ 0
(t))]
9 [dW(t) + O(t)dt + ,~(t)dt].
and r E K(cr). Because O E K• r) and
* * (~b'(t) + M(t)0 (t))dWo(t)
[r .
+ M(t) A. (t)][dW(t) + 0(t)dt + A(t)dt].
We will show that (8.3)
r
+ M(t)A(t) = 0 V t E [0,T], a.s.
We will then define
s
~
~t 1 * [x + 0 Z ~ -o(r
* + M(s)0 (s))dW0(s)].
Because r + M(t)0(t) e Kx(c), this d-dimensional vector can be written as r = ~ (t)a(t) for some m-dimensional a. Indeed,
+ M(t)0(t)
17 (8.4)
a(t) -- (a(t)a (t))'la(t)(r
§ M(t)0(t)).
We can define
~(t) A
(8.5)
~(t) f~(t)zA(t)x(t)
and it is then easily verified that 2(0) = x, :Xi(T) = ~ ,
'
and d•
= r(t)~:(t)dt +
X(t)~*(t)a(t)dWo(t) (cf. (3.2)). In other words, once we prove (8.3), then (8.4), (8.5) will define ~" e ..4(x) such that (B) holds. In order to establish (8.3), consider an arbitrary u E K((r) and introduce the sequence of stopping times {vn}:: l given by
rn = T h inf {t E [O,T]; M(t) + I fi~*(s)dW(s)l +
I t (llr
~ + IIr
0
+ IIA(s)ll ~ + 114s)ll~)ds _>n}. Denote un(t) A y(t) l[0,vn](t). Then for e E (-1,1),
rthrn , (8.6) ZA%eun(t) -- ZA(t)exp{- eJ0
e2 [tA~'n
u (s)(dW(s) + A(s)ds) - - 2 - j0
e--nlel < ZA§
(8.7)
ZA(T)
H4s)l[2ds},
(T) < en] el -
9
Because of (8.7) and the fact that A e K,(cr), we have A + e Vn e K,(~). It follows from (8.6), (8.7), the Dominated Convergence Theorem, and condition (E) that
(..8)
o = ~ E I~(T) Z~§176
;-E[~(T)Za(T) ~i
e=0 = ~E~(T) ~ ~ Z~§176
~=0
I?",, (s)(dW(s) + a(s)ds)] =-E[M(T~) s?", (s)(dW(s) + a(s)ds)].
According to ItS's rule, (8.9) M(rn) 0 u (s)(dW(s) + A(s)ds) =
v (t)(M(t)A(t) + r
+
(t)dW(t)
18 gTn
+ J|0
Ft ,
*
M(t)[|0uj(s)(dW(s) + A(s)ds)](C~(t)
*
§ r
From the definition of rn, we see that the two stochastic integrals on the right-hand side of (8.9) have expectation zero, so (8.8) implies that E is arbitrary, [M(t)A(t) +
r
u (t)(M(t)A(t) + r
= 0. Since u e K(cr)
= o. Letting n -~ | we obtain (8.3). []
8.3 Remark. Using Theorem 8.2, one can prove Theorem 8.1 by establishing the implications CA) (B), (C) v (D) and (D) v (E). In each of these, the hypothesis is a statement of optimality, and in each case, the proof of the implication is a variational argument similar to that just used for the implication (E) ~ (B). We have the following result concerning the existence of A E Kl(g) which satisfies (D), and by inference, the existence of an optimal portfolio process. 8.4 TlfE01tBM. Assume that (8.2) and U(0) > - co hold, as well as T
(i)
E I llO(t)ll~dt <| 0
(ii)
~-(xg'(x))_>0,
V x>0,
(iii)
V y > 0, 3 y E K(a) such that E 0(yfl(T)Zu(T)) > - ~.
Then there exists A E Kl(a) satisfying condition (D). According to Theorem 8.1, this A also satisfies (B), (C) and (E), and condition (A) holds with the same ~ as in (B). SECTION 9. EXAMPLES IN AN INCOMPLETE MARKET In the examples of this section, we assume m < d and produce the optimal portfolio ~- and the process A e Kl(a) satisfying conditions (B)---(E). 9.1 Examnle. (Logarilhmic u~ili~yf~nc~ion). Suppose U(x) = log x. Then o~u(x) = ~ V u e K(a), and {u =
. The process A
-- 0 satisfies (E), because g u e K(~),
E[/~(T)Zu(T)5~] = x E exp{-
v*(s)dW(s) -89
) < x.
The last inequality follows from the fact that exp{- It0u*(s)dW(s ) - 5 l i t 0 H~(s)ll2ds}' being a nonnegative local martingale, must be a supermartingale. According to Theorem 8.2, the optimal
19
portfolio process ~r must satisfy xx'~r(T) = ~ , and this ~rwas determined in Example 6.3. From i7.4) we see that ,~ = 0 corresponds to completion of the market by stocks whose appreciation rate equals the interest rate. With a logarithmic utility function, the agent will not use such stocks even for hedging purposes. 9.2 Example (Power utility ~nction and totally unhedgeable market coefficients). Suppose U(x) = ~ x 6, where ~ < 1, ~ ~ 0. Suppose that the volatility matrix act ) has the form a(t) = [~rit),0], where ~r(t) is an m x m, nonsingulax matrix for all t e [0,T], almost surely. Decompose W into ~ r * ( t ) = (W(1)(t),...,w(m)(t))and
o*
W ( t ) = (w(m+l)(t),...,w(d)(t)), and
let {fi't} and { ~ t ) be the augmentations under P of the (independent) filtrations generated by o
o
~/ and W, respectively. Assume that the processes r, b and ~r are adapted to {Jr}; we refer to this situation as one with totally unhedgeable market coefficients, because the stock prices are driven solely by "v~r: m
dPi(t) = Pi(t)[bi(t)dt + Z ~ij(t)d'v~r(J)(t)], i = 1,...,m. j=l We show that under these conditions, the portfolio process given by (6.17) is optimal. In the o
present context, this process is random and {Jt}-adapted, rather than determimstic as in Example 6.4. v* To verify the above assertion, we note first that 0*(t) = [0 (t),0], where ~Tit) =a ~ (t)(~r(t)h (t)).,[bit)_r(t)l].
*
o*
o
We note also that the processes ,~ e Ki~r) axe of the form A (t) = [0,A it)I, where ACt) is (d-m)--dimensional. With m given by (6.15), we define A ~ E exp {SI~m(t)dt}.
o
The {Jt}-martingale Nit ) ~ E[exp{~
I: m(s)ds}l~t } has a representation as
N(t) = A +
where
o
t
i0
,
o
~ (s)dW(s), rT
is an {~t}-adapted,/d-m)-dimensional process satisfying ]
ll~(t)ll~dt <
0 surely. Since N is continuous and N(t) > 0 for all t e [0,T], we may define J
almost
20 ~(t) 1 ~(t) so that I = LIT 0 E K(a) and dN(t) = - N ( t ) io, (t)dW(t). o = -- N'(I'~ o Therefore,
(9.1)
o - 2 1 f T~ llA(t)[l~dt}. exp{ 6~m(t)dt} = N(T) = A exp{- f T oI. (t)dW(t) 0
We may assume without loss of generality that W is the coordinate mapping process VV(t/o) = ~ t ) defined on {l A C([0,T], ~), the space of continuous functions from [0,T] to ~ , o
o
and W is the coordinate mapping process on fl =~ C([0,T], ~d--). Then fl = flx fi, and P is the o
product of m-dimensional Wiener measure t' on (1 and (d-m)--dimensiona/Wiener measure P o
on ~1. Abusing notation slightly, we regard the {~t} adapted process 0 as a process on n. For almost every ~ E fl, we have * O (s,w)dW(s,w)
62 I t - 2(I:~
0
[]0 (s,~)[[2ds < | and thus the process ~ H exp{ 0
0
H0(s,~o)ll2ds} is a meamone {,~(t)}-martingale on ~ under ~'
(Karatzas & Skreve (1988), p. 199, Corollary 5.13). Consequently, (0.2)
E[exp {
=
-O (s)dW(s)}l . o) J~](w,w
=
Ifi exp{ ~6 f*T0 D (s,~o)d'v%r(s/o)}P(d~) =
exp {
0
[[D(s,D)[i2ds}.
From (9.2) and (9.1) we have 1 5~A(y) = y~---fE[exp{
(r(s) + 89]]0(s)l1' + 89]lA(s)ll2)ds + l--~J 0 A (sldW(s)}
9 E[exp { ~
6 IT., . o j O 8 (s)dW(s)} [ ~,~T]]
1 ~ ~ 6 ] T(m(s)d s + 21 II)'(s)ll2)ds + ~ 6 I T 0~ A (s)dW(s)}] = yI-~ E[exp{ 1-=-8 0
6 1 1 T = AI':~y~--rE[exp {6 IO m(slds}]= (~)~-'Y. It follows that A e Kl(a), ~ i ( x ) = Ax & l and
21 I ~ = (#g~)~CT)ZgT))~-r
T
+~
T O*(t)dW(t)}.
Just as in Example 6.4, we conclude from (3.3) that xX'Tr(T) = ~ , where ~" is given by (6.17). 9.3 Remark. An important unresolved question is whether there are simple, widely applicable conditions which guarantee that for the process A satisfying conditions (B)-(E) of Section 8, the nonnegative local martingale ZA is actually a martingale. In Example 9.1, we have ft
1 rt
*
zgt) - z0(t) = exp{- Jo o (s)dW(s) - ~ J0 II 0(s)H 2ds}, so we must at least assume that Z0 is a martingale in order to conclude that ZA is. In Example 9.2, a computation similar to (9.2) reveals that
E Z),(T) = E[exp{-J0 A ( t ) d W ( t ) -
IT.,
.
o
E[exp{- 0 o (t)dW(t)}
= E[exp{-
I (t)dW(t)-~
Ib(t)II2)dt}
~TI]
0 [IA(t)ll2dt}]-
Taking expectations in (9.1) and recalling the definition of A, we see that E ZA(T ) = 1. This is enough to ensure that ZA is a martingale (Karatzas & Shreve (1988), p. 198). We have so far been unable to produce an example in which Zo is a martingale but Z A is not.
SECTION
i0. REFERENCES
Bismut, J.M. (1973) Conjugate convex functions in optimal stochastic control, Y. Math. anal. Appl. 44, 384--404. Cox, J.C. & Huang, C. (1987a) A variational problem arising in financial economics, Sloan School of Managment, M.I.T., preprint.
22 Cox, J.C. & Huang, C. (1987b) Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ Theory, to appear. Harrison, J.M. & Kreps, D. (1979) Martingales and multiperiod securities markets, J. Econ. Theory 20, 381---408. Harrison, J.M. & Phska, S. (1981) Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl. 11, 215-260. Harrison, J.M. & Phska, S. (1983) A stochastic calculus model of continuous trading: complete markets, Stochastic Process. Appl. 15, 313--316. He, H. & Pearson, N.D. (1988) Consumption and portfoho pohcies with incomplete markets and short-sale constraints: The finite dimensional case. Working paper, Simon Graduate School of Business Administration, University of Rochester. He, H. & Pearson, N.D. (1989) Consumption and portfoho pohcies with incomplete markets and short-sale constraints: the infinite dimensional case. Working paper, University of Cahfornia, Berkeley. Karatzas, I. (1989) Optimization problems in the theory of continuous trading, SIAM J. Control Optimization 27, 1221-1259. Karatzas, I., Lehoczky, J. & Shreve, S. (1987) Optimal portfoho and consumption decisions for a "small investor" on a finite horizon. SIAM J. Control Optimization 25, 1557-1586. Karatzas, I., Lehoczky, J.P., Shreve, S.E. & Xu, G.-L. (1989) Martingale and duality methods for utility maximization in an incomplete market. Submitted for publication. Karatzas, I. & Shreve, S. (1988) Brownia~ Motion and Stochastic Calculus, Springer-Veflag, New York. Merton, R.C. (1969) Lifetime portfolio selection under uncertainty: the continuous-time case, t~ev. Econom. Stalist., 51,247-257. Merton, R.C. (1971) Optimum consumption and portfoho rules in a continuous-time model, d. Econom. Theory 3, 373-413. Erratum: ibid. 6(1973), 213-214. Pages, H. (1987) Optimal consumption and portfolio pohcies when markets are incomplete. MIT mimeo.
23 Pliska, S. (1986) A stochastic calculus model of continuous trading: optimal portfolios. Math. Operations Research 11, 371-382. Samuelsoa, P.A. (1969) Lifetime portfolio selection by dynamic stochastic programming. Rr Econom. Statist~ 51, 239-246. Xu, G.-L. (1990) Ph.D. dissertation, Department of Mathematics, Carnegie Mellon University. Xu, G.-L. & Shreve, S.E. (1990) A duality method for optimal consumption and investing under short-selling prohibition. Submitted for publication.
NONLINEAR CONTROL OF MISSILES THROUGH A GEOMETRIC APPROACH* Jean-Philippe HARCAUT CENTRE D'ETUDES ET DE RECHERCHES DE TOULOUSE D6partement d'Etudes et de Recherches en Autornatique 2 avenue Edouard Belin 31055 TOULOUSE Cedex
Abstract
Many theorical studies have been conducted about decoupling and linearization of nonlinear systems. This approach is a means to synthesize the control laws of single-input single-output systems. It is used here for the piloting of nonrolling missiles, of which transverse acceleration is to be controlled. In this paper, it is proved that for this kind of problem, it is possible to associate such a method to simplifications of the model on the one hand and separation of the system into multiple time scale cascade subsystems on the other. Thus, the dimension of the unobservable part can be minimized and the control law is simpler. I
Introduction
:
Nonlinear control via static state feedback has been extensively treated in the literature. In 1970, Porter [17] solved the problem of noninteracting control by inverting the system. If needed, this method also makes it easy to linearize the system. Singh et Rugh [18], and Freund [5] completed this work. Isidori et al. [12] then worked out the problem through a differential geometric approach. Later, Ha and Gilbert [7] characterized the decoupling laws and Isidori and Grizzle [11] proved that the decoupled system has fixed inner dynamics. Grizzle [6] , Monaco and Normand-Cyrot [16] studied discrete time nonlinear systems. Decoupling and linearizing via state feedback techniques leads, however, to serious difficulties in real cases. These difficulties are due on the one hand to the complexity of the control law and its computation, on the other to the possibility of part of the state space becoming unobservable because of noninteraction constraints. A compromise between these two problems is hardly realistic. In fact, trying to decrease the dimension of the unobservable manifold, which can be minimized (Marino [14]), leads to an increase of the characteristic numbers and thus usually to a much more complicated control law. This probably explains why, when dealing with aeronautics, the synthesis of control laws is often obtained from models that are linearized at different operating points, the laws thus obtained being afterwards concatenated or commuted. What is more, in order to simplify the synthesis, some couplings are eliminated. This happens traditionally in aeronautics, where pitch is usually decoupled from roll and yaw. However, the evolution of aircraft and missiles, as well as that of acceptable performance, makes it necessary to go further than this method. Furthermore, nonlinearities on the one hand and coupling on the other demand a nonlinear approach. In this paper, we intend to show that part of the above-mentioned difficulties can be solved by using decoupling and linearizing techniques together with time scale separation ~'
The author would like to thank thc DRET, which financially supportcd this work.
28 of the system into slow/fast subsystems. Thus, it is possible to find a solution in which the dimension of the unobservable subsystem is minimal while the characteristic numbers remain small and the control law rather simple. The example of a ten-dimensional state model missile will be considered in this respect. Section II fixes notations and introduces key results about noninteracting control theory via static state feedback 9 Section III is dedicated to practical applications to missile piloting. It also presents the simplification realized when separating the system into slow/fast subsystems, compared to a global solution9
il
Note on n o n i n t e r a c t i n g c o n t r o l via static state f e e d b a c k :
Notations : Consider a state-space representation of the foma : m
( O.L. )
J[ = A (X) + ~ u.1 B i=1
Y:
i
(X)
c (x)
where : 9 the state X evolves on a n-dimensional analytic manifold :E 9 for i E {1 ..... m}, the input ul is a real-valued piecewise continuous time function9 We shall use the n o t a t i o n U = [ U l , . . . , U m ] T . A is an analytic vector field on 3E for i E { 1 ..... m}, B i is an analytic vector field on :s 9 C is a IKp vector-valued function defined on :~ : for each X E X , C (X) = [C1 ( X ) , . . . , C p (X)] T. Consider a static state feedback of the form : (1)
U (t) = Y ( X ) + G ( X ) V (t)
where 9 V is a new input, a m x 1 matrix whose elements are real-valued time functions. 9 F is an analytic ]12,m vector-valued function defined on :~, and G is a m x m matrix whose elements are analytic real-valued functions :
F=
L&
j. j
Only feedback with G nonsingular for all X will be considered. Definition of the p r o b l e m : Given a system for which m=p, i.e. for which there are as many inputs as outputs, the purpose is to determine a static state feedback, as defined in (1), so that the closed loop system verifies that : 1. for i E {1 ..... m}, each new input v i affects only the output Yi 2. the input-output relation is linear.
29 Such a system is linear and decoupled.
Deeoupling and linearization :l" Let us bear in mind the following definitions : Consider j E {1 ..... p}. The characteristic n u m b e r pj associated with the output Yj is so defined : pj = i n f {i E lN/3k E { 1 , . . . , m } : LBkLkCj ~ 0} pj+l is the minimum number of integrations necessary to achieve a direct action of the input on the output Yj. 9 If all the characteristic numbers of the system are finite, the decoupling matrix A(X) of the system ( O.L. ) is so defined : A!=LBiLP~ci,
foriE{1,...,p}
, forjE{1,...,m}
1
We still suppose that m=p. Theorem : it is possible to find a solution to the noninteracting control problem for the ( O.L. ) system via regular state static feedback, i.e. with G nonsingular, if, and only if, the decoupling matrix A of the system is nonsingular. Hence, to determine a system for which the input-output relation is linear and decoupied, it is enough to : 1. consider F = A - 1 M and G = A - I N , M and N being so defined : f o r j E {1 ..... p} : PJ
MJ = E a~L~ Cj- L~+Icj i=o N~=0
if i7~ j and N J = k j
where ai,for i E (1 ..... pj}, and kJ are arbitrary constants. 2. make the coordinate change :
Vj E {1,...,p} : Vi E { 0 , . . . , p j }
2 i. = L~Cj J
It is to be noted that the linearized and decoupled system thus obtained is p nL-dimensional, with nL = ~ pj + p. If n g < n, part of the initial system is now unobj=l
servable. And these unobservable states might be unstable! However, it has been proved that a "wise choice" of the outputs, that may not be possible for practical applications, makes it possible to minimize the dimension of the unobservable submanifold. For k E IN, the qk distributions are so defined : Q~ = Span ( B I , . . . , B m )
and Qk = Span (adkAQ0, I~k-1 )
(~k being the involutive closure of Qk, i.e. the smallest involutive distribution holding Qk. t
For a closer study of these notions, see Isidori [10]. Other interesting references are mentioned,
30 Marino [14] proved that the minimum dimension of the nonlinearizable and thus n--1 unobservable state submanifold is ~ (dim (Qk) _ dim (Qk)). k=0 Let us now consider the application of these results to missile piloting9
III Application to missile piloting and simplification of control laws : 1 Modeling : The transverse acceleration of a nonrolling missile is to be controlled whiIe keeping its roll angle constant. The model is obtained through the flight mechanics equations. These equations have been simplified with respect to the following hypothesis : the attack angle c~ and the sideslip angle/3 are small the total velocity V is constant 9 aerodynamic coefficients Cyr et Czq are ignored 9 for reasons of symmetry, the Y and Z body axis moment of inertia, Iy and Iz, are equal, and products of inertia vanish.
Force equations 9 1 [rz (4,/3,,7) + mg cos 0 cos 9] & - q + p/3 = ~--~ 3 + r-
=
1 [py (ce,/3, () + mg cos 0 sin 99]
(The movement along the Gx axis is not considered). Moment equations : Ixl5 = L (o~,/3,() + Lpp IyCl-t- (Ix -- Iz) pr = M (a,/3, r/) + Mqq Izi: + (Iy - Ix) pq = N (oL,/3, ~) + Nrr
Kinematic equations : = p § t a n 0 (qsin c2 + r cos 9~) = qcosc2 - rsinc2 (These equations must be considered, as it is the roll angle c2 that is to be controlled). Fy,Fz,L+Lpp,M+Mqq,N+Nrr are aerodynamic forces and moments; ~,r/,( are equivalent roll, pitch and yaw control surface deflections, calculated from real deflections. Let us bear in mind that LpMo,N ( are never zero. Practically speaking, aerodynamic forces and moments are determined through wind tunnel testing; thus, it is discrete data that has been interpolated using spline functions. It is then wiser to use only the first derivative of these functions in the control laws. For the piloting of the missile, outputs to be controlled are : Y1 = c2 : roll angle 1 Y2 = I'~y = m [Fy (or,/3, ( ) "-t- mg cos 0 sin ~o] : acceleration along Gy axis Y3 = Pz = --1 [Fz (a,/3, 7/) + m g cos 0 cos c2] : acceleration along Gz axis m
31 Let us note that, as transverse accelerations are to be controlled, particularly along the vertical of the earth, the pitch attitude 0, may be a divergent state. This instability is of no consequence for the piloting of the missile, as 0 variations are taken into account in the guidance laws. Control surface jack dynamics will not be taken into account, and the following inputs will be considered, in order to get a state-space representation linear with respect to the control : Ul = ~ ,
u2=T},
U3=~
However, it is possible to include jack dynamics in the modeling and also in the decoupling and linearization. Moreover, if we supposed that the jacks have linear dynamics, e.g. a second order, the original model would then be linear with respect to the control. The system to be studied is the ( O.L. ) system previously introduced. Notations are the same as in part II.
2 Global approach : With respect to the chosen input, ~, r/et ~" are states of the system and the state vector [p, q, r, c~,/3, ~o, 0, ~, 77,~-]T is ten-dimensional. Let us first consider the decoupling and linearization of the global system as it has been modeled. The characteristic numbers associated with outputs Y1,Y2,Y3 are respectively Pl = 2, P2 = 0, P3 = 0. Hence, the unobservable submanifold is five-dimensional. Thus, even if theoretically the problem L~Fz,TFy could be solved, as the determinant of the decoupling matrix det. A _ ~ is never zero, these five unobservable states oppose this approach. This is all the more true that a more complete study proved that this unobservable submanifold corresponds to states other than 0 that may be unstable. As a matter of fact, in the case where a zero transverse acceleration is controlled, there is an initial state corresponding to a divergent trajectory in any neighbourhood of every equilibrium point of the asymptotic unobservable part. The approach does not give satisfaction. However, it is to be noted that, practically speaking, Fy and Fz depend mainly on and/3, 7/and ~ having a small influence. If they are neglected - - as has been proposed in [8] the control of slightly non-minimum phase systems - - the characteristic numbers are all two-valued, and the unobservable submanifold is one-dimensional, corresponding to 0. This could be expected, as the pitch attitude is naturally divergent when dealing with piloting. The state feedback law is then very complicated, as the characteristic numbers are high - - i t would be necessary to take into account the third derivative of the outputs. Such a state feedback does not seem realistic. Another approach of the problem could be a linearization of the dynamics of the system. According to Marino's proof [14], the linearized system is at best nine-dimensional. As the nonlinear model has ten states, again, one of the state variables is unobservable. The characteristic numbers are then all two. Again, the state feedback law is very complicated. Moreover, the outputs considered are not even the outputs to be controlled! It is then clear that, when keeping a global approach, any attempt to decrease the dimension of the unobservable manifold means generally an increase of the characteristic numbers and thus makes the state feedback more complicated. To stay out of this dilemma, let us separate the system into two subsystems, via the notion of multiple time scales. If a minimal dimensional unobservable submanifold is needed, as the value of the characteristic numbers is related to the dimension of the system, it is necessary - - except in certain cases - - to start from a small dimensional state manifold in order to get a simple control law. For such a case as the one considered here, decreasing arbitrarily the dimension of the system is out of question. It is, however, obvious from the study of the
32 system that some states are slow while others are fast. The notion of multiple time scales makes it easy to separate the system into slow/fast subsystems while keeping its whole dimension. The fast state variables are then used to control the slow ones. This approach has already been used in aeronautics by Menon et al. [15], Lane and Stengel [13], but that was for certain guidance and tracking problems applied to aircraft. Of course, taking into account these simplifications, the decoupling and linearization will not be exact but, practically speaking, is it really a problem?
3 Study of two cascaded subsystems : From now on, the state variables a,/3, 99, 0 will be considered slow, while the state variables p , q , r , ( , r / , ( will be considered fast. This hypothesis makes it possible to separate the state equations into two cascaded subsystems. Due to this dynamics separation, the slow variables are considered constant from the fast subsystem, and the fast variables are inputs or parameters of the slow subsystem. As before, Fy and Fz are supposed to be independent of ( and 7/respectively, so that the transverse accelerations to be controlled are outputs of the slow subsystem. The slow subsystem is denoted as (S) and the fast one is denoted as (F). The following schema illustrates the approach used here.
[~ u2
z
r_ ........
J
.
.
.
.
.
.
.
.
.
.
iT .... ii:i ..................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
?;-i .... i
I13
Iit
,'
Slow-time scale subsystem : The slow subsystem is modeled by the following state equations : &-q+pZ (S)
= ~--~ 1 [F,. (~, ~, 7/) + m g cos 0 cos 99]
/~ -t- r -- pc~ ----=~--~ 1 [Fy (cr,/3, 4) + m g cos 0 sin 99] ~ = p + tan 0 (q sin 99 + r cos 99) = qcos 99 -- r sin 99
The inputs to the system are the fast states p,q,r, and its outputs are those of the global system 99, Fy, Pz-
33 All characteristic numbers are zero and thus the decoupling matrix is : 1
t~n o ~in ~
t~n 0 cos ~
A= ~(-flFv,+aFye+mgcosOcos~o ) • v,, -~(-Fya+mgsinO [-~(-flFz,,+o~Fz,-mgcosOsin~o) m( 1 Fz~,-m9 sinO) - mI Fz e
,j Ir~ot,
Can this matrix be singular ? As [ ~ o l and IFz~I are large compared to IFy~l and if also it is supposed that Fyz ~ Fz~ -- f~, i.e. that Fy5 and Fz~ are almost constant, then ,.., If~ KgsinO'~ detA _ (m 2 m J (1 -t-/3sinc2tan0 + a c o s c 2 t a n 0 ). If also K
> igsin01, then detA = 0 ~ 1 +/3sine2 t a n 0 + a c o s g t a n 0
= 0
It is to be noted that the same result would have been achieved if the weight had been neglected from the start. Consider a c a n d / 3 c corresponding to the controlled transverse accelerations Pyr and Pzr and C2c the controlled roll angle. It is necessary to avoid the constant pitch attitude 0 such as 1 +/3c sin c2c tan 0 + O~ccos c2c tan 0 = 0, so that the equilibrium point is not a singularity. However, singularities correspond to large pitch attitudes as c~ and/3 remain small. However, practically speaking, it is perhaps not necessary any more to keep the roll angle constant. If the output is the integral of p, then there is no singularity any more resulting from the noninteracting control. Let us now determine a state feedback such that it is possible to decouple and linearize the slow subsystem. ""[@,~y,~zJ T is the closed loop input of the system. N o t e : (2)
rq
=F+G
Py
fz
As the characteristic numbers are all zero, it is sufficient to have : F = --A -1
(3)
(Iil)
+ LAC
A I! ~176
et G
~
0
o
where LAC
=
r• --
L ~w
[(Fz +
0
rag cos 0 cos g~) Fy~ + (Fy + mg cos 0 sin ~) Fy~]
[(F~ + ~
]
~os 0 ~o~ ~) F~ + (F~ + ~g ~os 0 sin ~) &~]
and TI,T2,T3 are any time constants, in order to get three linear decoupled first order systems, with a unit gain. A state-space representation of the subsystem is : 0 1
0 0
~
+
0
ry P~
34 Before decoupling and linearization, the state manifold was four-dimensional; now the state space is three-dimensional. A state variable, O, has become unobservable. Its variations are governed by the equation : 0 = q c o s ~ - r s i n c 2. If = 93 = Cte, r y = ~y = Cte, Pz = Pz = Cte, then0 = - [0 cos93 -- sin 93] A-1LA C. This differential equation shows the variations of the unobservable variable 0, when the observable state variables T, Py, Fz have reached their control value. Thus, it defines the asymptotic unobservable part. Let us note that it is not possible to find a more general decoupling law, such as those characterized in [7], that allows the stabilization of the unobservable part, as the pitch attitude can be naturally divergent depending on the controlled transverse accelerations when dealing with piloting. Fast-time scale subsystem : Let us now consider the fast part of the system, as defined by the state equations :
p=K1 [L (a, r , ~) + Lpp] ~l= ~1 [M (06/3, r/) + M~q -t- (Iz - Ix) pr] F)
1 [N (,~,/3, ~) + N ~ + (Ix - Iy) pq] /]~U 2
~----U3 A shorter notation will be : N: = A (X) + ~ bib i (X). i=i
The input to the fast subsystem is the input to the global system. The output is the input to the closed loop slow subsystem, i.e. 1-93, Fy, I'z]- T. G being nonsingular, relation (2) implies that the output of the fast subsystem is : L
.J
Let us bear in mind that the state variables of the slow subsystem are supposed to be constant during the evolution of the fast subsystem. Ttms, matrices F and G are constant also. For this subsystem, all characteristic numbers are one and the decoupling matrix is :
A = G -I
[!00 ~
0
As L(, M,7, Nr are non zero, det zX = 0 if, and only if, det G -1 = 0. (3) implies that
=
T2 0
A T3
35 Thus, if the fast subsystem (F) has singularities, they are those of the slow subsystem (S). As for the state feedback law, consider [~c,Fyr162 T the input to the closed loop system. Note :
u3
LPz~
As all characteristic numbers are one, it is sufficient to have :
~. = _ s
/~lr~ + bog LctPz + co~
+ L}~ r~ f'z
et ~ = s
~o 0 co
where
~y
L
= G -1
~ [M + Mqq + (Iz - Ix)p,]
L~[N+N,'+(Ix-I,)Pq]
and where a0, b0, co are non zero, to get three linear decoupled second order systems, with a unit gain. The study of this part of the system could be reviewed taking into account the dynamics of the actuators. The study of the slow subsystem would be left unchanged.
IV
Conclusion :
At this stage of the study, it is now clear that a global approach does not bring any realistic solution, either because there are too many potentially unstable unobservable states or because the controI law is too complicated for a practical realization. The separation of the system into slow/fast subsystems is a solution to this dilemma, as we tried to prove from a principle point of view. Many questions remain to be asked about the price paid by the decoupling as a consequence of this hypothesis, the general quality of the control law or its practical realization. Digital simulation of this method from realistic data is under study at the moment. As could be foreseen, direct action of control surfaces on aerodynamic forces having been neglected, transverse accelerations show a non minimal phase displacement in closed loop. It has also been noticed that, from a practical point of view, it would be interesting to impose on the slow subsystem a dynamics close to its own and to accelerate the fast subsystem within reasonable bounds. Thus, the dynamics separation between the subsystems is increased, even if it means the necessity to include the dynamics of the actuators in the model. The realization of this inner loop should make the synthesis of the external control law easier while ensuring the performance.
Acknowledgments The author would like to thank Professor A.J. Fossard for his helpful discussions during the development of this work.
36 References : [1] B. d' Andr6a-Novel. Commande non lindaire des robots. Herm6s, Paris, 1988. [2] D. Claude. Lin6arisation par diff6omorphisme et immersion des syst~mes. In A.V. Balakristman and M. Thoma, editors, Lecture Notes in Control and Information Sciences, volume 63, pages 339-351. Springer-Verlag, 1984. [3] D. Claude. D6couplage des syst~mes non lin6aires, s6ries g6n6ratrices non commutatives et alg~bres de lie. Siam J. Control and Optimisation, 24(3):562-578, May 1986. [4] D. Claude. Ddcouplage et lindarisation des syst~mes non lindaires par bouclages statiques. Th~se d'6tat, Universit6 de Paris-Sud - Centre d'Orsay, 17 juin 1986. [5] E. Freund. The structure of decoupled non-linear systems. Int. J. Control, 21(3):443450, March 1975. [6] J.W. Grizzle. Local input-output decoupling of discrete-time non-linear systems. Int. J. Control, 43(5):1517-1530, May 1986. [7] I.J. Ha and E.G. Gilbert. A complete characterization of decoupling control laws for a general class of nonlinear systems. IEEE transactions on automatic control, AC31(9):823-830, September 1986. [8] J. Hauser, S. Sastry, and G. Meyer. Nonlinear controller design for flight control systems. In A. Isidori, editor, Preprints of the IFAC Symposium,Capri,Italy, pages 136-141. Pergamon Press, June 14-16 1989. [9] L.R. Hunt, R. Su, and G. Meyer. Design for multi-input nonlinear systems. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory, pages 268-298. Birkh~user, 1983. [10]A. Isidori. Nonlinear control systems: An intoduction. In Lecture Notes in Control and Information Sciences, volume 72. Springer-Verlag, 1985. [11]A. Isidori and J.W. Grizzle. Fixed modes and nonlinear noninteracting control with stability. IEEE transactions on automatic control, AC-33(10):907-914, October 1988. [12]A. Isidori, A,J. Krener, C. Gori-Giorgi, and S. Monaco. Nonlinear decoupling via feedback: a differential geometric approach. IEEE transactions on automatic control, AC'26(2):331-345, April 1981. [13]S.H. Lane and R.F. Stengel. Flight control design using non-linear inverse dynamics. Automatica, 24(4):471-483, July 1988. [14JR. Marino. On the largest feedback linearizable subsystem. Systems and Control Letters, 6(5):345-351, January 1986. [15]P.K.A. Menon, M.E. Badgett, R.A. Walker, and E.L. Duke. Nonlinear flight test trajectory controllers for aircraft. J.Guidance, 10(1):67-72, January-February 1987. [16]S. Monaco and D. Normand-Cyrot. Nonlinear systems in discrete time. In M. Fliess and M. Hazewinkel, editors, Algebraic and Geometric Methods in Nonlinear Control Theory, pages 411--430. D.Reidel Publishing Company, 1986. [17]W.A. Porter. Diagonalization and inverses for non-linear systems. Int. J. Contol, 11(1):67-76, 1970. [18]S.N. Singh and W.J. Rugh. Decoupling in a class of nonlinear systems by state variable feedback. Trans. ASME J. Dyn. Syst. Meas Contr., 94:323-329, 1972.
NON LINEAR CONTROL CRYSTALLIZATION USING
OF A BATCH EVAPORATIVE AN ALGORITHM OF "L/A" TYPE
FEVOTTE G. (i), CANPONT D. (i), LAKRORI M. (2) (1) I.C.P.I. Laboratoire de contrtle des proctdts 31, Place Bellecour 69002 Lyon (2) L.A.G. (CNRS, UA 228), ENSIEG. B.P. 46 38402 St Martin d'H%res. Fraace
ABSTRACT : In spite of obvious industrial needs in crystallization little has been done till now to improve and optimize the control of the equipments. The field of crystallization process technology is very complex and the knowledge based industrial applications still remain remarkable. Moreover, it is generally difficult to propose some simplified approaches leading to effective improvements . We present here a simple application of the non linear "L/A" controller to a batch crystallizer and encouraging simulation results are shown . Firstly, we settle the difficulties arising from the technological choice of the evaporative crystrallisers : non linearities and specific industrial requirements in closed loop. After a brief recall of the principles involved by the L/A controller, a comparative study is proposed between the use of PlY) controllers and the L/A algorithm when some realistic industrial goals are settled. A trial based on an adaptive approach : the generalized predictive control (GPC) is also reported in this paper but it is till now impossible to conclude since some specific estimation problems have not yet been solved. The L/A algorithm which is very easy to implement has been found to be easier to tune and it leads to a more satisfactory behaviour in closed loop than the PID. KEY'WORDS : batch processes, crystallization, nonlinear control, adaptive control, identification.
Introduction
:
The batch crystallization is an industrial chemical process which consists in cooling a liquid phase, melt or solution till a crystalline solid appears : the nuclei. This nucleation is followed by a phase of crystal growth. An appropriate temperature trajectory is generally required in order to ensure a satisfactory production from an industrial point of view : quality reproductibility, minimization of the number of fine particles, ability to the filtration ...etc. From the process engineer's point of view, the plants are driven rather by an industrial know-how than by using recent theoretical advances. In fact, modelling the fundamental phenomena involved (nucleation, growth, crystal size dispersion...) is extremely difficult (.!3 and almost all reported results are laboratory ones. From the automatician's point of view, only few references are available, they are mainly concerned by the problem of the online estimation of variables linked to the quality of crystals. The goal being to find software solutions which would be less expensive than the use of particle analysers. See e.g. 2~, (5. So, it is hard to find some works trying to propose new control systems for crystallizers. Nevertheless, the industrial specialists have pointed out the fact that a better productivity could follow the implementation of such control systems (.43. The batch evaoorative crystallization The process involved here is a batch operation lead in order to get a temperature trajectory, as shown on figure 1. During the first period denoted i, the goal is to go quickly in the neighbourhood of the nucleation point (denoted* on figure 1). The second period consists then in tracking a slight downward slope setpoint trajectory favorable to a steady crystal growth. When a sufficient crystal population is ensured, the last phase (denoted 3) follows to achieve the growth but now it is possible to go faster : the temperature slope is increased up to 10-100 ~ / hour.
38 e(t),--
80
e*(t)
* = nucleation r
60
Input P(L)
800 600
.
400 ~
40
200 0
20
iO0
0
0
Time
Time
Figure 1 : PID control of a batch e v a p o r a t i v e crystallizer. P u r e soivant.
0 = temperature.
0* = setpoint.
P(t) = pressure
Despite the simplicity of the basic principle involved here : a strongly temperature-dependent solubiblity of the substance, many problems have to be taken into account. Notably, several crystallizers types in technological use can be differentiated according to the mode of inducing nuclei in the solution. This paper is concerned by cooling through vacuum : If a lower pressure than the vapour pressure is imposed by a vacuum pump (see fig. 2), the boiling point is reached. The solvant steam escapes from the stirred medium and is condensed at the top of the plant. As fax as heat transfer is concerned this is very efficient, more than the classical use of cold fluids. In fact, the evaporative crystallizers axe in many ways interesting from an industrial point of view (.~. However it is necessary to cope with some control problems arising from this technological choice.
OACCUNPU~
(/~
CO~[HS[I1 t
8(t),P(t)
J.~(/~'Po(t)
c) Figure 2 : A batch e v a p o r a t i v e crystallizer
Control
difficulties. ~oals
(1) Despite a possible flexible use of the crystallizer (i.e. various solvants and products) our aim is to ensure a satisfactory tracking of the setpoint temperatures. This fu'st goal is really very stringent.
39 In particular, the motrice force of the steam production is given by the difference : zkP = P(t) - Po (t,0) where P(t) Po (t,0) O(t)
(1)
is the vacuum imposed by the pump is the vapour pressure of the solvant is the temperature of the solution
From a batch to another if the solvant is recycled, the vapour pressure curve might be drastically modified by the increasing concentration of impurities in the medium. A real industrial example is shown on figure 3. A strongly nonlinear behaviour fotlows : one may compare for instance the slopes dP / dO between the curves (3-a) and (3-b) in the neighbourhood of the temperature denoted 01. 800
i
I
o
700 k
600
~o c~
000
400 no:
300
.g.
zoo _ _ i " = Samk
i
tOO
~
solvant
attar
"l r,e "~
0 0
10
20
30
TEMPERATURE
Figure 3 : Experimental solvant (.)
40
in Celsius
~0
60
70
deg.
c u r v e s P o ( 0 ) for a p u r e s o l v a n t (o) a n d a r e c y c l e d
(2) The manipulated input P(t) must be as smooth as possible otherwise the "flash" of the solvant will carry drops of mother liquor to the top of the plant. An undesirable crystaUization may take place on the condenser. It will then be necessary to wash it with hot solvant or, in difficult cases to break the crystal layer with hammer and cold chisel ! These operations are obviously always very time consuming. (3) The goal of flexible plant might be reached without needing to tune the controllers at each batch or each new user's requirements. The control system will have to be adaptive or, at least, somewhat insensitive to tuning parameters. This is very difficult to obtain since the operating industrial conditions are likely to be very various.
From the industriar s point of view, several profits are at stake: - a better control allows to shorten the batch-time : this is a consequence of a highest degree of reproductibility during the different periods - as direct measurements are almost impossible, the advanced cona'ol of the cryst~lizer is certainly the cheapest way to ensure in an optimized way the quality requirements - if the nucleation happens when the rate of temperature is too high the product will be fine grained with the attendant problems :
40 -> downstream filtering, centrifuging and washing will be difficult -> the crystals are likely to retain some impure mother liquor, they will dry badly and cake in storage : all these difficulties are also very time consuming -> lastly, we mentioned that it might be possible to avoid the appearance of crystal layers at the cooling surfaces. PID Control of the crystallizer Some industrial experiments have shown that it is impossible to get the 3 proceeding goals with PID controllers. In order to verify this point and to study the possibility of new solutions we designed a simplified simulation based on the following hypothesis. The thermal balance of the system leads to a state equation of the form : dO P(t) - P0 (t, 0) ~" = 7 1 + f(t, 0)
(2)
y, assumed to be constant is a ccefficient of mass transfer;, f(t,0) is an analytical function taking into account the heat of crystallization. We supposed that crystallization itself could be related by a classical equilibrium law of the form : x(t)
=
A e "B / (0(t) + 273)
where x(t) is the cocentration of dissolved solid (% mass) at the temperature 0. Po(t,0) is evaluated for the two extreme solvants (pure and recycled) from the experimental data given on figure 3. The temperature measurement is regarded as slightly corrupted by a white noise (Variance = 0,7 ~ Using a PID algorithm for P(t) in (2), it is difficult to get a satisfactory tracking and we are obliged to set different tuning parameters for the phases 1-3 and 2 (of. fig. 1). Some results have been shown on figure I in the case of pure solvant with optimized tunings for the 3 phases. Keeping the same parameters, the results obtained with the recycled solvant (figure 4) are far from the goals settled previously. For example, the oscillations of P(t) acceptable on fig. 1-b do not satisfy now the goal (3). Moreover, after the nucleation peak, in order to keep the solvant boiling, it is impossible to increase the pressure P(t) this is why the system remains in open-loop till the temperature crosses the serpoint curve again. This period denoted (OL) on the figure lasts too long.
Error O* - 0
800
* = nucleation
P o ( t ) ( - - ) ,Input P(t)
800 0 r
o E--,
--1
400 200
-2 0
500
(a)
500
0
Time
Figure 4 : PID control of the crystallizer - Recycled solvant
(b)
T~e
41 T r a c k i n ~ throueh GPC The adaptive control of the crystallizer might be able to bring solutions to the different difficulties encountered. For instance, the general aims proposed by Landau in (_6.)are in agreement with ours. So, we tried to apply the generalized predictive control (GPC) approach. The process identification has been lead using an extended least squares algorithm (ELS) with varying forgetting factor and constant trace ~ , 7_) : ^
Model
^
: 0(t+l)
=
T
O(t+l) . do(t)
(3)
with : doT(t)= [-0(t),-0(t-l), P(t),P(t-1), e(t), e(t-1)]
= observed data
e(t) is regarded as a disturbance. Algorithm :
with
O(t+l) = O(t) + F ( t + l )
F(t+l) =
e(t+l)
=
1 kl(t)
{P(t)
d0(t) E ( t + l )
(4-1)
F(t) ~ ( t ) d ~ T ( t ) F ( t ) - } kl(t)/'k2(t) + @T(t) F(t) d~(t)
AT O(t+l) - 0 (t). q)(t)
= "a posteriori" prediction error
1 +~T(t) F(t) O(t)
parameters :
(4-2)
(4-3)
Constant trace = 1 Initial forgetting factors = 0.95
Based on the model (3), a control law minimizing the following criterion is applied (see e.g. (.fi)). hp
=z
j=t
+ c<,-q-'>, u<,-,+j>j,}
where O*(t) = setpoint at time t ; hp = prediction horizon
(5)
; j>l => Au(t+j) = 0
This approach was found to be unsufficient as one can see on figure 5. This is not due to the control law itself but to the divergence of the ELS. Indeed, as far as the solution is boi!ing (i.e. AP < 0 in eq. (!)); the inputs P(t) are linked to the temperature through eq. (2). On the other hand, if the control leads to P(t) > P0(t, 0) the boiling stops. The inputs have then no more effect on the outputs and the ELS "explode". Several modifications could probably improve the robustness of the ELS, especially an information measure followed by an eventual parameter freezing in case of ebullition breakdown. Such solutions were proposed by M'saad (19 but the context here is different : the problem is not to freeze the parameters when a too poor information about the process is available in steady state conditions but to freeze when P(t) > P0(t) with obviously P0(t) unknown. Before developping on this approach we chose to try a more simple one which wouldn't be so depending on the process accidents : the L / A control.
42 80
~
e(t),-9=
60
e-(t)
Input
000
P(t)
600
nucleation
400
~9
40 200 20
0
5(10
0
500
0
Time
Time
Figure 5 : ELS divergence in case of boiling b r e a k d o w n : effect on the G P C T r a c k i n e throuvh L/A controller The principle of L/A control Consider a nonlinear continuous SISO (Single Input, Single Outpu0 process described by : Y(0 = f { y(t), u(t)} {u(t), y(t)} R. + x R. +
(6.1) (6.2)
where : R. + is the set of strictly positive real numbers; y(t) is the process output (the controlled variable) and u(t) is the control input; "f" is a nonlinear analytical function. Now consider an algorithm giving the conlrol signal: Uc(t) = c {y(t), y*(t) }
(7)
where: y*(t) is the reference signal (set point) and"c" is a nonlinear analytical function. To build up in a simple way a control law which respects the constraint (6.2) by generating only uc(t) > 0 we assume the following approach (fig. 6):
(1)- Consider two simple nonlinear transformations : y(t) = eh(0 , u(t) = eV(t) Then:
V{h(t),v(t)}E R •
(8) ==>
{y(t),u(t)}~ R [ x R +
(9)
it means that in the (h,v) domain, the condition (6.1) is automatically relaxed and the following process description is obtained : h(t) = g { h(t), v(t) }
(10)
Expression (10) defines a so called "nonlinear fictive process" (NFP - fig. 6)
43
Space(V~,,Y+,) U
Fictivespace(v, h) y
vI L
/
A
l~
~
p
I Y*
F
C
I h*
Figure 6 - Principal diagram of L/A control design
(2)- Define the reference signal: h*(t) = In y*(t)
(I 1)
and try to apply a linear or nonlinear controller over NFP; of course, from a simplicity point of view, it is more interesting to use a linear law. This implies the linearization of NFP around an equilibrium point; "a linear fictive process" is obtained (LFP). Then by any classical procedure we may design a "fictive linear controller" (LFC) which operates on the difference (h* - h) : v(t) = cf { h*(t) - h(t) }
(12)
where "cf" is a linear function.
(3)- Being designed the LFC, we come back from the fictive space (h,v) to the original one (u,y) by inverse transformations : h(t) = In y(t), v(t) = In u(0, h*(t) = In y*(t)
(13)
and obtain the corresponding nonlinear controller, denoted "L/A" : UL/A(t) = CL/A { y*(t) / y(t) }
(14)
Consider for example a numerical LFC controller of PI type: Vk = Vk-1 + KI (hk-1 - hi0 + K.2 (h*k - hi0
(15)
Then the corresponding real controller (denoted by ~A-2) is : Uk =Uk-1 + (Yk-I/YI0 K1 + (Y'k/y0K2
(16)
Other simple structures are available (2, T u n i n g the L/A controller In fact, from a pratical point of view it is sometimes difficult to find some good tunings for the parameters K1 and K2 of the control law (16).
44 Lakrori ~ I_!) proposed a simple way to introduce two new tuning paramete;-s : ct and A, which are more physically significant for the user and thus, easier to a priori locate. Moreover, these new parameters will allow an effective taking into account of the actuator's saturation (denoted Umax).We summarize this approach as follow : Let us consider a simple first order model of the process : 1: dy / dt + y = k. u with (u, y) > 0 where "gis the time constant and k is the static gain.
(17)
Suppose we have a system equipped with a zero holder (Bo) with period Tc and a sampler 0E) with period Te = Tc (fig. 7); the process is controlled by (16)- that is, in the ficfive space, a LFC like (15). Let be "Vmax" the upper limit corresponding to the maximal value Umax of the real action (Vmax = ha Umax).
(Y'k)
PROCESS ~
"--'~1 L/A- 2
~
(Yk)
I. F i g u r e 7- Control loop composed of a n u m e r i c a l L/A c o n t r o l l e r and a process with
bounded actuator. Consider the evolution in the (h, v) space (fig.8) and suppose that :
hk-I = h*, Vk.l = v*
(18)
Suppose now that a disturbance is applied upon the process between steps "k- 1" and "k" in such a way that (fig.8) : hk -
Vk -
(20)
In such conditions it is not difficult to show that if one introduces ~="~/Te and A = width band around the set-point up to which'the maximal realizable control action is demanded:
A= y* - ehA then we obtain (.9__,1_.0_,1.!.!): K1 = ct K2
(21)
In ( U m a x / U*) K1 + K 2 - In (y* / (y* - A))
(22)
45 h, h *~ h* h=h*
I
I
I
I
I
I
I
I
I
k-1
k
k+l
I
Vmsx
v
t,k
v(t)
V:*
k-1
k
k+l
t, k
F i g u r e 8 - Possible evolution in (h, v) u n d e r a fictive P I c o n t r o l l e r considering a c t u a t o r ' s saturation. Results and discus;igll ; The I J A algorithm has been applied in the conditions described previously for the PID. The figures 9 and 10 show the closed loop behaviour with pure and recycled solvant. In both cases, we needed only one set of tuning parameters : o~= 1 ; A = I~ So, our third aim is satisfied. Moreover, it is important to notice that we found the IJA controller easier to tune than the PID. It turns out that from the flexibility point of view, we obtained interesting performances. With respect to the industrial constraints ,especially the limitation of the input variations and of the duration of the openloop phase (OL) an effective improvement is obtained. One can compare the figures 1-b to 9-b and 4-b to 10-b. E r r o r 9" - e * = I
800
Po(t)(---)
,Input P(t)
nucleation
600
............................................................
400
0
F200
--1
0
-2 0
500
(a)
500
0
Time
F i g u r e 9 : L/A control of the crystallizer - P u r e soivant
(b)
Time
46 E r r o r O* - O * = nucleation
Po(t)(--) ,Input P(t)
800 600
I.I
0
400
-1
200
OL
0
-2 0
500
(a)
Time
0
500
(b)
Time
Figure 10 : L/A control of the crystallizer - Recycled solvant CONCLUSION : The encouraging results reported about the L/A control of industrial processes known for some real difficulties involved : biotechnology (.~, distillationI(.L0.), thermal processes with important time delay 1(.!2)-...are confirmed here. With respect to the basic idea of taking into account the actuator's saturation, the performances are beyond all what we could reasonably expect. The reason is probably a linearizing effect of the transformations applied. We are working now on this point 1(.!.~.). Despite what we showed about GPC, we expect that some further developments will lead to better performances. Notably, the use of reference models could enable the user to specify easily the actuator's behaviour and the adaptive control law could cope with the difficulty of varying parameters. REFERENCES: (1) Hulburt, H.M. (1983). Perspectives on crystallization in chemical process technology Aiche. Syrup.Series, Vol. 79, 235, pp 77-89. (2) Kotzev, A. ; Resnick, W. and Lavie,R. (1986).Evaluation of discrete approximations to continuous size distributions for control of crystallization processes.Chem. Eng. Science. Vol. 41, 12, pp. 3045-305 I. (3) Rohani, S. (1986). Dynamic Study of crystal size distribution (CSD) in a KC1 crystallizer. Canadian J. of Chem. Eng. Vol. 64, pp. 112-116. (4) Bennett, R.C. (1988). Matching Crystallizer to material .Chemical Engineering. Vol. 95, 8, pp. 118-127. (5) Klein, J.P. (1989). Procddds industriels de par dvaporation. Journdes SFT: Changements de phases darts les milieux dispersals. Paris. (6) Dugard, L. and Landau, I.D. (1988). Commande Adaptative.Mdthodologie et applications . Ed. Hermes. Paris. (7) Landau, I.D. (1988). Identification et Commande des syst~mes. Ed. Hermes. Paris. (8) M'saad, M; (1987). L'applicabilitd de la commande adaptafive. Th~se d'dtat. INPG Grenoble. (9) Lakrori, M. (1989). Control of a continuous bioprocess by simple algorithms of "P" and "L/A" type. IFAC Syrup. : Non-linear control systems. Capri-Italy. (10) Lakrori,M. ; Rey, D. (1989). On the use of "L/A" algorithms to control some chemical processes .IFAC Symposium on Dynamics and Control of chemical reactors, distillation columns and batch processes. Dycord' 89. The Netherlands. (11) Lakrori, M. (1989). Rdglage d'un L/A-2 numdrique en considdrant la saturation de ractionneur. Note interne LAG. Grenoble. (12) Fevotte,G. ; Lakrori, M. (1990). Application of the non-linear L/A algorithm to the adaptative control of a steam flow. CHISA' 90. PRAGUE. Aofit 1990. (13) Gibon, A.M. ; Fevotte G. (1989). Etude de l'effet lindarisant de la commande L/A sur certaines classes de processus. Note interne I.C.P.I. Lyon.
GENETIC BREEDING OF NON-LINEAR OPTIMAL CONTROL STRATEGIES FOR BROOM BALANCING
John R. Koza
Computer Science Department Stanford University Stanford, California 94305 USA E-MAIL: Koza@ Sunburn.S tanford_Edu TELEPHONE:415-941-0336 Martin A. Keane
Third Millennium Venture Capital 5733 West Grover Chicago, Illinois 60630 USA TELEPHONE: 312-777-1524
Abstract: This paper describes a search for the time-optimal "bang bang" control strategy for the three dimensional broom balancing (inverted pendulum) problem by genetically breeding populations of control strategies using a recently developed new "genetic computing" paradigm. The new paradigm produces results in the form of a control strategy consisting of a composition of functions, including arithmetic operations, conditional logical operations, and mathematical functions. This control strategy takes the problem's state variables as its input and generates the direction from which to apply the "bang bang" force as its output.
1.0 INTRODUCTION This paper describes a search for the time-optimal "bang bang" control strategy for the three dimensional broom balancing (inverted pendulum) problem by genetically breeding populations of control strategies using a recently developed new genetic algorithm paradigm. The problems of centering a cart and balancing a broom by applying a "bang bang" force are well-known problems in control theory. These problems have been previously studied by researchers in artificial intelligence, genetic algorithms, and neural networks. In particular, the cart centering problem has been previously studied in the genetic algorithm field in connection with Holland classifier systems (Goldberg 1983). The broom balancing problem has been
48 studied extensively in connection with neural networks (Widrow 1963, Barto et. al. 1983, Widrow 1987, Anderson 1989). We (Koza and Keane 1990) have previously successfully used the "genetic computing" paradigm described in this paper to rediscover the well-known analytic solution to the simple two dimensional time-optimal cart centering problem and to discover a non-linear control strategy for the two dimensional time-optimal broom balancing problem. In this paper, we search for a solution to the three dimensional time-optimal broom balancing problem.
2.0 BACKGROUND ON GENETIC ALGORITHMS
Genetic algorithms are mathematical algorithms that transform populations of individual mathematical objects (typically fixed-length binary character strings) into new populations using operations patterned after natural genetic operations such as sexual recombination (crossover) and fitness proportionate reproduction (Darwinian survival of the fittest). Genetic algorithms begin with an initial population of individuals (typically randomly generated) and then iteratively (1) evaluate the individuals in the population for fitness with respect to the problem environment and (2) perform genetic operations on individuals in the population selected proportionate to fitness to produce a new population. Professor John Holland of the University of Michigan presented the pioneering formulation of genetic algorithms for fixedlength character strings in Adaptation in Natural and Artificial Systems (Holland 1975). Holland established, among other things, that genetic algorithms operating on strings are a mathematically near optimal approach to adaptation for maximizing expected average payoff when the adaptive process is viewed as a multi-armed slot machine problem requiring an optimal allocation of future trials given currently available information. Recent work on genetic algorithms can be surveyed in Goldberg (1989), Davis (1987), Schaffer (1989), and Booker, Goldberg, and Holland (1989). 3.0 BACKGROUND ON HIERARCHICAL GENETIC ALGORITHMS
Populations of computer programs can be genetically bred to solve problems in several different
49 areas of artificial intelligence and symbolic processing (Koza 1989, Koza 1990a, and Koza 1990b). In this recently developed new "genetic computing" paradigm, the individuals in the genetic population are compositions of functions (i.e. LISP symbolic expressions), rather than f'txed-length character strings. In particular, the individuals in the population are LISP Sexpressions consisting of hierarchical compositions of functions and atoms appropriate to the particular problem domain. The set of functions used typically includes arithmetic operations, conditional logical operations, mathematical functions, and functions appropriate to the problem domain at hand. The set of atoms used typically includes various constants and particular inputs (state variables) appropriate to the problem domain. The search space is the hyper-space of all possible LISP S-expressions that can be recursively composed of the available functions and atoms. The results of this process are inherently hierarchical. The two basic genetic operations in the "genetic computing paradigm" using hierarchical genetic algorithms are fitness proportionate reproduction and crossover (recombination). Fitness proportionate reproduction for hierarchical genetic algorithms is the basic engine of Darwinian reproduction and survival of the fittest. This operation copies individuals selected proportionate to fitness in the same way as it does for conventional genetic algorithms operating on simple character strings. The crossover operation for hierarchical genetic algorithms is a sexual operation that operates on two parental LISP S-expression. In particular, the crossover operation creates new offspring S-expressions by exchanging sub-trees (i.e. sub-lists) between the two parents selected proportionate to fitness. It always produces two syntactically valid offspring S-expressions. This new genetic algorithm paradigm has been successfully applied to example problems in several different areas, including (1) sequence induction (e.g. inducing a recursive computational procedure for the Fibonacci and Hofstadter sequences), (2) automatic programming (e.g. discovering a computational procedure for solving pairs of linear equations, solving quadratic equations for complex roots, and discovering trigonometric identities), (3) machine learning of functions (e.g. learning the exclusive-or function, the parity function, and the Boolean 11-multiplexer function), (4) planning (e.g. developing a robotic action sequence to stack blocks in a specified order and programming an "artificial ant" to traverse an irregular trail), (5) pattern recognition (e.g. translation-invariant recognition of a simple one-dimensional shape in a linear retina), (6) symbolic function identification (e.g. symbolic regression,
50 symbolic integration, symbolic differentiation, and symbolic solution of differential equations), (7) empirical discovery (e.g. rediscovering Kepler's Third Law and rediscovering the multiplicative econometric "exchange equation" PQ = MV from time series data for prices, gross national product, money supply, and velocity of money of an economy), and (8) concept formation (generating a decision tree for inductively classifying data). The author's computer program used here (Koza 1990b), consisting of 11562 lines of Common Lisp code, was run on a Texas Instruments Explorer II+TM computer and a microExplorerTM computer (an Apple Macintosh IITM computer with a Texas Instruments LISP expansion board).
4.0 THE THREE DIMENSIONAL BROOM BALANCING PROBLEM
The three dimensional broom balancing problem involves a push cart with mass m c moving on a one dimensional frictionless track and a broom (inverted pendulum) of mass m b pivoting on the top of the cart. The broom has an angle 0 (measured from the vertical) and an angular velocity (o. The distance from the center of mass of the broom to the pivot point on the cart is ~.. There are four state variables of this system, namely, position x, velocity v, angle 0 , and angular velocity co. A "bang bang" force of fixed magnitude F is applied to the center of mass of the cart from either the left or fight direction at each time step x. The constants here are mc=0.9 kilogram, mb=0.1 kilogram, gravity g=1.0 meters/sec2, time step x x=0.02 seconds, length ~ =0.8106 meters, and force F--4.0 Newtons. The state transitions of this cart and broom system are expressed by non-linear differential equations. At each discrete time step "c, the state of the system at the next time step is computed from the current state of the system and the force being applied at that time step. In particular, the angular acceleration (I) (t) at time t is given by (Anderson 1988) as
~(t) =
gSin0 +Cos0 (-F - mp ~. (o 02 Sin0) rth +mp ~ [ 3 - m P C~ 0 me+mp J
51 The angular velocity co (t+l) of the broom at time t+l is therefore co (t+l) = co (t) + "~9 (t) and the angle 0 (t+l) at time t+l is 0 (t+l) = 0 (t) + x co (t), using Euler approximate integration. The acceleration a of the cart is given by
a(t)=
F + m p~. [0 ~ Sin0 - co co Cos01 me+mp
The velocity at time t+l is therefore v(t+l) = v(t) + x a(t) and the position x(t+l) at time t+l is x(t+l) = x(t) + '~ v(t). The problem is to find a control strategy (i.e. LISP S-expression) that specifies how to apply the "bang bang" force so that, after starting in any random initial state in a specified region, the broom becomes balanced with the cart at rest in minimal average time. The set of functions used for this problem consisted of addition (+), subtraction (-), multiplication (*), the sign function (SIG), the absolute value function (ABS), the square root of absolute value function (SRT), the square function (SQ), the cube function (CUB), and the greater-than function (GT). The greater-than function GT is a real-valued logic function of two arguments that returns +1 if the first argument is greater than (>) the second argument and returns -1 otherwise. The use of real-valued logic allows arbitrary composition of functions in the function set. The set of atoms for this problem consisted of velocity v, angle 0 , and angular velocity co. In addition, the set of atoms contained a smattering of random constants between -1.0 and +1.0. When a particular control slrategy (i.e. LISP S-expression) evaluates to any positive value for particular values of state variables v, 0 , and co at a particular time step, the force F is applied from the positive direction. Otherwise, the force is applied from the negative direction. Note that the control strategy can be viewed as defining a surface that separates the state space into parts. The environment consists of 10 randomly generated initial starting condition cases. Position x is chosen between -0.5 and +0.5 meters. Velocity v is chosen between -0.5 and +0.5 meters/ second. Angle 0 is chosen between -0.5 radians and +0.5 radians. Angular velocity co is chosen between -0.5 and +0.5 radians/second. We do not know of an analytic solution to this three dimensional broom balancing problem. However, an approximate optimal control strategy for a conventional linear approximation to
52 the solution has been computed by Keane. This analytic solution to the linear approximation calls for application of the positive force when co >if2, where A B (Sinh K(t + z ) - 2Sinh Kt)
I2=
K
and, otherwise, a negative force. Here t=z+
vA c
and
where the quantity A is computed as follows:
If
( v / C ) 2 ' then A = Sign O
>--- T --
Otherwise A : Sign (v/C). D, B, C, K are computed as follows: D= ~[4 B=
mc me +mp -F
D[m c + m,] C=
F [mo+mp]
K:~/D Since this solution to the linear approximation to the problem involved hyperbolic functions, we replaced the hyperbolic sine function by terms (up to the cubic level) from the Taylor series expansion to obtain an approximation to this approximation. The resulting "pseudo optimal strategy" serves as a benchmark for the genetically produced results below. Note that this
B3 "pseudo optimal strategy" is not used by the genetic algorithm in any way. Note also that the state transitions of the systems are controlled by Anderson's equations for 9 (t) and a(t) above. Time was discretized into 400 time steps of .02 seconds. The total time available before the system "times out" for a given control strategy is thus 8 seconds. If the square root of the sum of the squares of the velocity v, angle 0 , and angular velocity to (i.e. the "norm") is less than 0.07 (the "criterion"), the system is considered to have arrived at its target state. If a particular control strategy brings the system to the target state for a particular initial starting condition case in the environment, its fitness for that environmental case is the time required (in seconds). If a control strategy fails to bring the system to the target state before it "times out", its fitness for that environmental case is set to 8 seconds. The "fitness" of a control strategy is the average time for the strategy over all 10 initial starting condition cases in the environment. The process starts with the random generation of 300 random control strategies recursively composed from the available functions and atoms above. Examples of such random control strategies are linear control strategies such as v + 0 , co 0",/'~.57,and 0 > co and non-linear control strategies such as 02 + to. and 0 to31vl. As can be seen, the initial population of random control strategies in generation 0 includes many highly unfit control strategies, including strategies that totally ignore the state variables, strategies that repetitively apply the force in only one direction, strategies that are correct only for a particular few specific starting condition cases in the environment, strategies that are totally counter-productive, and strategies that cause wild oscillations and meaningless gyrations. In one run, the average time consumed by the initial random strategies in the initial random population averaged 7.79 seconds. In fact, many of these 300 random individuals "timed out" at 8 seconds for all 10 of the environmental cases (and very likely would have timed out even if more time had been available). However, even in this highly unfit initial random population, some control strategies are somewhat better than others. The best single control strategy for the initial random generation was the simple non-linear control strategy v 0 , which averaged 6.55 seconds. This control strategy correctly handled two (of the simpler cases) environmental cases and timed out for 8 of the environmental cases. Note that this control strategy is partially blind in that it does not even consider the state variable to in specifying how to apply the "bang bang" force. Nonetheless, in the land of the blind, the one-eyed man is king.
54 The genetic crossover operation is then applied to parents from the current generation selected with probabilities proportionate to fitness to breed a new population of offspring control strategies. Although the vast majority of the new offspring control strategies are again highly unfit, some of them tend to be somewhat more fit than others. Moreover, some of them are slightly better than those that came before. For example, the average population fitness improved from 7.79 to 7.78, 7.74, 7.73, 7.70, 7.69, 7.66, 7.63, 7.62 seconds per environmental case in generations 1 through 8, respectively, This generally improving (but not necessarily monotonic) trend in average fitness is typical of genetic algorithms. In generation 6, the best single individual was the non-linear control strategy t9 > v 3 I 0 + v I. It required an average of 6.45 seconds. In generation 7, the best single individual in the population correctly handled (i.e. did not time out) 4 of the 10 environmental cases. It required an average of 5.72 seconds. In generation 8, the best single individual was the non-linear control strategy [0 + m + 2.762v + 03 ]3. It required an average of only 2.56 seconds. This individual is not partially blind and considers all three state variables. It correctly handles 10 of the 10 environmental cases. Note that the number of correctly handled environmental cases ("hits") is a statistic that we find useful for monitoring runs, but this statistic is not used by the genetic algorithm. In generation 14, the average time required by the best single individual in the population dropped below 2 seconds for the f'trst time. In particular, the non-linear control strategy 0 + co + v + v 2 [ co - 0.734] 2 > v 0)2 required an average of only 1.74 seconds. It, too, correctly handled all 10 of the environmental cases. Two histograms provide a graphical picture of the learning of the population from generation to generation. The "hits histogram" shows the number of individuals in the population that correctly handle a particular number of environmental cases. The "fitness histogram" shows the number of individuals in the population with whose fitness value lies in a particular docile of
55 the fitness values. Both of these histograms have an undulating left-to-right "slinky-like" motion as the population progressively learns. In generation 33, the best single individual in the population was the non-linear control strategy 3 v + 3 0 + 3o~+v 0 2 > v r After its discovery, this single best control strategy was retested on 1000 additional random environmental starting condition points. It performed in an average of 1.76 seconds. In another test, it averaged 3.20 seconds on the 8 comers of the cube, but could not handle 2 of the 8 comers of the cube. The benchmark pseudo optimum strategy averaged 1.85 seconds over the 1000 random environmental starting condition cases in the retest. It averaged 2.96 seconds for the 8 comers of the cube. It was unable to handle the 2 worst comers of the cube. These results (in seconds) are summarized in the table below: P E R F O R M A N C E FOR THREE DIMENSIONAL BROOM BALANCING PROBLEM CONTROL STRATEGY
WORST 2 CORNERS
1000 POINTS
Benchmark Pseudo Optimum
1.85
2.96
Infinite
3v+3 0+3r
1.76
3.20
Infinite
5.0 CONCLUSION In summary, we used the new "computing paradigm" to discover a non-linear control strategy for the three dimensional broom balancing (inverted pendulum) problem which slightly outperformed our benchmark "pseudo optimum" solution for the problem. We believe our nonlinear control strategy is a very close to the time-optimal solution to the problem.
6.0 REFERENCES Anderson, Charles W. Learning to control and inverted pendulum using neural networks. IEEE
Control Systems Magazine. 9(3). Pages 31-37. April 1989.
56 Barto, A. G., Anandan, P., and Anderson, C. W. Cooperativity in networks of pattern recognizing stochastic learning automata.In Narendra,K.S. Adaptive and Learning Systems. New York: Plenum 1985. Booker, Lashon, Goldberg, David E., and Holland, John I-I.Classifier systems and genetic algorithms. Artificial Intelligence 40 (1989) 235-282. Davis, Lawrence (editor) Genetic Algorithms and Simulated Annealing London: Pittman 1987. Goldberg, David Eo Computer-Aided Gas Pipeline Operation Using Genetic Algorithms and
Rule Learning. PhD dissertation. Ann Arbor: University of Michigan. 1983. Goldberg, David E. Genetic Algorithms in Search, Optimization, and Machine Learning. Reading, MA: Addison-Wesley1989. Holland, John H. Adaptation in Natural and Artificial Systems, Ann Arbor, MI: University of Michigan Press 1975. Koza, John R. Hierarchical genetic algorithms operating on populations of computer programs.
Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI). San Mateo, CA: Morgan Kaufman 1989. Koza, John R. Econometric modeling by genetic breeding of mathematical functions. Proceedings of International Symposiumon Economic Modeling. Urbino, Italy: 1990a. In Press. Koza, John R. Genetic computing: a paradigm for genetically breeding populations of computer program to solve problems. Machine Learning. In press 1990b. Koza, John R. and Keane, Martin A. Cart centering and broom balancing by genetically breeding populations of control strategy programs. Proceedings of International Joint
Conference on Neural Networks, Washington, January 1990. Volume I. Schaffer, J. D. (editor) Proceedings of the Third International Conference on Genetic
Algorithms (ICGA). San Mateo, CA: Morgan Kaufmann Publishers Inc. 1989. Widrow, Bernard. Pattern recognizing control systems. Computer and Information Sciences
(COINS) Symposium Proceedings. Washington, DC" Spartan Books, 1963. Widrow, Bernard. The original adaptive neural net broom balancer. 1987IEEE International
Symposium on Circuits and Systems. Volume2.
P R O C E S S C O N T R O L USING N O N L I N E A R P R O G R A M M I N G T E C H N I Q U E S
B. Wayne Bequette Department of Chemical Engineering, Rensselaer Polytechnic Institute Troy, NY 12180-3590, USA
ABSTRACT Some of the most difficult problems associated with process control are due to process nonlinearity, manipulated variable constraints, uncertain parameters and unmeasured variables. In this paper a nonlinear programming approach is developed to estimate process parameters, unmeasured state variables and process disturbances.
A constrained optimization-based
procedure is also used to maintain a desired output variable trajectory, similar to techniques that have proven successful for linear systems. The process model, characterized by a set of nonlinear differential equations, is transformed into algebraic equations using orthogonal collocation on finite elements. A system with inverse response characteristics and a bioreactor model are used as examples. INTRODUCTION Within the past 10 years there has been an increasing use of advanced multivariable control system techniques which are known as model predictive control methods. A survey of the field has been performed by Garcia e t al. (1989). One of the most common predictive control methods is Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980). DMC is based on selecting a set of L future manipulated variable moves (control horizon), to minimize an objective function based on a least squares minimization of the differences between model predicted outputs and a desired output variable trajectory over a prediction horizon, R, as shown in Figure 1. Although the DMC optimization is performed for a sequence of future control moves, only the next control move is implemented. Model uncertainty and process disturbances are handled by calculating an additive disturbance as the difference between the process measurement and the model prediction at the current time step. It is assumed that the future disturbances are equal to the current disturbance, and a new trajectory is calculated. Notice that predictive control is really an open-loop optimal control strategy with feedback provided by the re-optimizing procedure. Predictive control strategies have been well-received by industry because they are intuitive and explicitly handle constraints. One limitation to the existing methods is that they are based on linear systems theory and may not perform well on highly nonlinear systems.
58 desired trajectory (future)
Y a
c I
t I
1
u 1
I
a 1
I
~ I
I
I
1
I
1
I
I
R t k current time
Prediction Horizon
maximum I
H
. . . . . .
I
,
I .............. I ["-1 1.... 1........................................ minimum
I
1
1
I
I
1
[
1
1
~I~,~-----
t
I
past control moves
1
1
I
1
I
Control Horizon Figure 1. Predictive Control Approach For these types of processes we feel that it is important to use nonlinear predictive control techniques with nonlinear dynamic models that have a physical basis. In this paper we first discuss dynamic process models in general. Next, we present a strategy for the regulation of the system, using a predictive control approach that accounts for constraints on manipulated and state variables. It is important to have a model that is representative of the current state of the process, so we then present a method to perform parameter estimation of the nonlinear dynamic system. Nonlinear Process Models This paper is based on the control of process systems that are described by lumped parameter models, that is, systems that can be modeled by sets of first order ordinary differential equations. In addition, we include deadtime on process inputs or output variables. Deadtime can be due to transport (flow through pipes) or measurement delays (analytical instrumentation, etc.). Mathematically, the representation is
59 dx
dt - f(x,u(t-0),p,l)
dynamic modeling equations
(1)
gl(x,u,p) = 0
algebraic equilibrium relationships, etc.
y = g2(x(t-qS))
state-output relationships
(2) (3)
where state variables
X
tl
=
manipulated variables
p
=
parameters
1
=
load disturbances (measured or unmeasured)
y
=
0
=
output variables deadtime between manipulated and state variables
00
=
deadtime between state and measured variables
This type of formulation may also be used for distributed parameter systems that have been "lumped" by discretizing the partial differential equations and forming ordinary differential equations. In order to si,alplify the solution of this problem, we use orthogonal collocation on finite elements to reduce the differenti',d equations to a set of algebraic equations. The dynamic model can then be stated as Ax - f(x,u,p,l,0) = 0
(4)
where A is a matrix consisting of collocation weights (Finlayson, 1980) and x is now a vector of state variable values at each collocation point. NONLINEAR PREDICTIVE
CONTROL
(NLPC)
The linear predictive control techniques can be extended to handle nonlinear systems by formulating a nonlinear programming problem (Eaton et al., 1989).
A single input-single
output system is shown as an illustration.
rain
tk+Tp k+RN c q~ (u,x) = fe 2 dt = Y. Y. Wj(Ysp(i,j)-Ypred(i,j)) 2 tk i=k j=l
u(k),...,u(k+L- I)
subject to Ax - f (x,u,p,l,0) = 0 U mni _
(5)
60
The objective function is least squares relationship between the desired output (Ysp(i,j)) and the model predicted value (Ypred(i,j)) at collocation point j on finite element i. There are N c collocation points on each of R finite elements. The current time step is i=k and Wj is the quadrature weight for collocation point j. In practice, if a prediction horizon of roughly R = 5 steps or larger is used, then similar results will be obtained with an objective function based only on the outputs at the end of each finite element. The approach is shown in Figure 1, where a sequence of L future control moves for each manipulated variable are used as decision variables in the optinfization procedure to minimize the objective function over the prediction horizon. Inequality constraints include absolute minimum and maximum values, as well as rate of change limits on the manipulated variables. This eliminates the excessive control variable action that can occur when too much emphasis is placed on the output variables. A more general formulation can include state variable constraints, however, great care must be taken to avoid infeasible solutions. Cuthrell
and Biegler (1987)
developed an efficient method
to optimize
algebraic/differential equation systems and used an open-loop optimal control example where the manipulated variable trajectory was determined. The optimal trajectory was calculated only once, then implemented, so there was no feedback control to account for model uncertainty and process disturbances. Since process models have some degree of uncertainty, we incorporate feedback by estimating loads and model error at each time step (as shown in the next section of this paper) and by calculating a new future trajectory. Rawlings and coworkers (Eaton et al., 1989) incorporate feedback through an additive disturbance term, similar to the DMC approach. Prediction using the additive output approach assumes that the future process disturbances are equal to the additive disturbance (including model error) at the current time step (k) d(k) = y(k) - Ym(k) Ypred (i) = Ym(i) + d(k)
for i=k+l to k+L
(6) (7)
ESTIMATION Many physical parameters for a chemical process are not known during the modeling phase, or change as a function of time or operating condition. Also some state variables are unmeasured and disturbances may effect the process. For tight control it is important to estimate parameters, unmeasured state variables and load disturbances. Our approach is to fomaulate estimation as a nonlinear programming problem (again, we use a single output system without deadtime as an illustration)
rain
k FI (Xo,P,1) = i=k~Y(i ) _ Ym(i))2
61 subject to Ax - f (x,u,p,1) = 0 g(x(k-E)) - y(k-E) = 0
(8)
Since the model predicted output is not exactly equal to the process output, an additive output disturbance is included, after the parameter estimation. d(k) = y(k) - g(x(k))
(9)
The objective function is based on the weighted (for the multivariable case) least squares differences between the measured variables (y) and the values predicted by the process model (Ym) at each measurement time step (ti), where the current time step is represented by tk. Notice that the decision variables for this problem are the process parameters (p), initial unmeasured state variable values (x(k-E)) and process load disturbances (1). The initial condition for the state variable vector (x(k-E)) is based on to = tk - T e. Jang et al. (1987) have used a similar approach for parameter estimation by integrating the ODE's and sensitivity equations, during each iteration of the optimization. Bequette (1990) presents examples where multiple measurements available at different sample intervals are used for effective disturbance estimation and predictive control of a chemical reactor. Bequette and Sistu (1990) demonstrate effective estimation and control of processes with significant time-delays and measurement noise. The importance of load disturbance estimation is shown in the following example.
E x a m p l e 1.
I n v e r s e Response System
Consider the Laplace domain model used by Maurath et al. (1988) to illustrate their linear predictive controller design technique. This model has a right-half-plane zero, yielding the characteristic inverse response to a step input change. (1 - 9s) (1 - 9s) y(s) = (15s+l)(3s+l) u ( s ) + (15s+l)(3s+l)
l(s)
(10)
For a control sample time of 1.5, Maurath et al. showed that a prediction horizon of 20 and a control horizon of 1 yielded good responses for setpoint changes. We will illustrate the advantage of load estimation for regulatory control of this process. The system is initially at steady-state, and a load disturbance of 1 = 1.0 occurs at t = 0.0. For the following simulation results, we use an estimation horizon of 10, a measurement sample time of 0.3, a prediction horizon of 20 and a control horizon of 1. Figure 2 shows that the response with load estimation is much better than assuming that an additive disturbance remains constant throughout the future horizon. These results are similar to those that would be obtained using the Generalized Analytical Predictor (GAP) of Wellons and Edgar (1987). The GAP was developed for linear systems, however, while our method is applicable to any dynamic process model.
62 0.8
0.6"
0.4
0.2
0 0.0
w/load estimation
-0.2
-0.4
0
1
I
I
I
I
30
45
60
75
90
Time Figure 2. Load Disturbance, Inverse Response Example Example 2.
Dynamic Nonlinear Bioreactor Model (Turbidostat)
In order to illustrate the advantages of the nonlinear model predictive technique, we consider a bioreactor model based on substrate inhibition (Agrawal and Lim, 1984) dx
dt = (g - D) x
(11)
ds gx ~-=(sf-s) D - Y I-t
(12)
~max s
(13)
k m + s + k l s2
where S
Sf
=
substrate concentration substrate feed concentration
X
----=
biomass concentration (measured)
D
=
dilution rate (manipulated)
g
=
specific growth rate
y
=
P'max ,km,kl
=
yield of cell mass parameters for the specific growth model
63 the parameter and variable values are shown in Table 1 for two operating conditions. The manipulated variable is the dilution rate and the measured variable is the biomass concentration. This system is characterized by input multiplicities since, at steady-state, any dilution rate value is associated with two possible biomass concentrations (one open-loop stable and the other open-loop unstable), in addition to the "wash-out" steady-state (x = 0). Table 1. Values of Parameters and Variables for Example 2. Variable
Steady-state I
Steady-state 2
x
1.500
0.995
s sf
0.250 4.0
1.512 4.0
D I.tmax
0.3325 0.53
0.30 0.53
km
0.12
0.12
k1
0.4545
0.4545
y
0.4
0.4
open-loop stable
open-loop unstable
This example presents a challenging control problem because of the possibility of operating at either an open-loop stable or unstable operating point. Our desire is to compare the nonlinear predictive control technique with that of a linear, proportional-integral (PI) controller (equations 14 and 15, Xsp is the setpoint (desired biomass concentration)). t D(t)
=
e(t)
=
DO + kc[e(t ) + 1 Ie(~)d~) ] xI 0 Xsp(t) x(t)
(14) (15)
The controller parameters used are k c = -1.0 and x I = 5.0, which yield a stable feedback control system at both the open-loop stable and unstable operating points. Servo Control The superiority of the NLPC scheme over the PI controller is shown in Figure 3, where the biomass concentration setpoint is changed from the open-loop stable to open-loop unstable operating point. The constraint handling capability of NLPC is shown in Figure 4, where the maximum dilution rate is 1.0 and the velocity constraint is lD(k) - D(k-l)l < 0.10 for the velocity constrained case. For the NLPC strategy the prediction horizon is 3, the control horizon is 1, the sample time is 0.2, and 4 intemal collocation points are used.
64
Regulatory Control The ability of the NLPC strategy to handle a substmte feed composition disturbance (sf) from 4.0 to 3.6 at t = 1.0 hour is shown in Figure 5. Biomass concentration is measured while the substrate and substrate feed compositions are estimated. An estimation horizon of 20 measurement samples is used, the measurement sample time is 0.04, the control sample time is 0.2, the prediction horizon is 3 and the control horizon is 1. The PI controller is moving to the washout condition of x = 0.0. NPSOL, a sequential quadratic programming code (Gill et al., 1986) was used for both the estimation and predictive control phases of this work.
SUMMARY A technique for feedback control that is based on a nonlinear dynamic representation of the process has been presented. The advantage of estimating load disturbances is clearly shown by example 1, for a linear system model. Effective estimation and predictive control of a nonlinear bioreactor system are illustrated in example 2. The NLPC strategy can operate over a wider range of operating conditions than can be handled by a typical linear control strategy.
ACKNOWLEDGEMENT This research was partially supported by a research initiation award from the National Science Foundation (NSF CBT-8910362).
REFERENCES Agrawal, P. and H.C. Lim, "Analyses of Various Control Schemes for Continuous Bioreactors," Advances in Biochemical Engineering/Biotechnology, Vol 31, 61, Springer-Verlag (1984). Bequette, B.W. "Nonlinear Predictive Control of a CSTR using Multi-rate Sampling Disturbance Rejection," submitted to Can. J. Chem. Eng. (1990). Bequette, B.W. and P.B. Sistu "Disturbance and Parameter Estimation for Nonlinear Dynamic Systems Using Nonlinear Programming Techniques", Proceedings of the 1990 American Control Conference (1990). Cuthrell, J.E. and L.T. Biegler, "On the Optimization of Differential-Algebraic Process Systems", AIChE J., 33 (8), 1257 (1987). Cutler, C.R. and B.L. Ramaker, "Dynamic Matrix Control - A Computer Control Algorithm," Proceedings of the 1980 American Control Conference, paper WP5-B (1980). Eaton, J.W., J.B. Rawlings and T.F. Edgar, "Model-Predictive Control and Sensitivity Analysis for Constrained Nonlinear Processes", Proceedings of the IFAC Workshop on Model Based Process Control, p 129, Pergamon Press (1989).
65 Finlayson, B.A. Nonlinear Analysis in Chemical Engineering, McGraw-Hill, New York (1980). Garcia, C.E., D.M. Prett, and M. Morari, "Model Predictive Control: Theory and Practice - a Survey", Automatica, 25 (3), 335 (1989). Gill, P.E., W. Murray, M.A. Sauders and M.H. Wright, "User's Guide for NPSOL (Version 4.0): A Fortran Program for Nonlinear Programming," Technical Report SOL 86-2, Department of Operations Research, Stanford University (1986). Jang, S., B. Joseph, and H. Mukai, "On-line Optimization of Constrained Multivariable Chemical Processes", AIChE J., 33 (1), 26 (1987). Maurath, P.R., D.A. Mellichamp, and D.E. Seborg, "Predictive Controller Design for SingleInput/Single-Output (SISO) Systems," Ind. Eng. Chem. Res., 27 (6), 956-963 (1988). Wellons, M.C. and T.F. Edgar, "The Generalized Analytical Predictor," Ind. Eng. Chem. Res., 26 (8), 1523 (1987).
1.4
\
]1" ::'. NLPC w/velocity II; ",~--------~ontrmnt
]\; 1"0 t 0.9
NLPC
\. ' ~ : ..... .... "
0
2
PI,,~,__
........................
"-"
4
6
8
1.0
Time (hr)
Figure 3. Response to a Setpoint Change for the BioReactor Example, Open-loop Stable to Open-loop Unstable Operating Point - Biomass Concentration
66
.ofl o.a
,
17
"l ,-" L t\L'.." ] 0.6 -[ i~t " ; . . ~ = "~
' 0.4
NLPC w/v.elocity constraint
',,'~,..,
0.2
0.0 0
2
4
6
8
10
Time (hr) Figure 4. Response to a Setpoint Change for the BioReactor Example, Open-loop Stable to Open-loop Unstable Operating Point - Dilution Rate
I.(150 ]
NLPC w/o estimation
e~
r
o
1.ooo "1
E
0.975
o
,isturbance
~/estimation
~
0.950
0.925
PI , 5
, 10
15
2
Time (hr) Figure 5. Comparison of Response to Feed Substrate Concentration Disturbance for the BioReactor Example, Open-loop Unstable Operating Point
C o n s t r u c t i o n of a u t o n o m o u s b o u n d a r y - v a l u e linear s y s t e m s from acausal i n p u t - o u t p u t functions Ran~ne Nikoukhah INRIA, Centre de Rocquencourt 78153 Le Chesnay Cedex, France Abstract
A frequency-domain realization theory is developed for tl~e class of autonomous, but not necessarily stationary, boundary-value linear systems. It is shown that this realization problem, which consists of constructing autonomous boundary-value linear systems fl-om prescribed input-output functions (weighting patterns), reduces to the factorization of several rational matrices in two variables having separable denominators. Tlfis factorizatioa problem is examined ~nd a method is given for constructing minimal factorizations for such rational matrices.
1
Introduction
In recent years, there has been a large effort to develop deterministic and stochastic system theories for the class of linear boundary value systems. In the deterministic case, basic notions of classical linear system theory such as controllability, observability, minimMity and irreducibility have been extended to the case of boundary value systems [1]-[5]. A class of boundary-value systems which is of particular interest is the class of autonomous (constant-coefficient, time-invariant) systems:
~(t) = Ax(t) + n u ( t ) ,
a < t < b,
v ~ ( a ) + vj~(~) = o,
v(t)
:
c~(t),
(1) (2)
a
<
t
< ~,
(a)
where A, C and B are n x n, m x n and n x p constant matrices respectively, and Vi and Vf are n x n bomldary matrices. Results o n autonomous boundary-value systems, and their discrete time counterparts, the two-point boundary-value descriptor systems [6], can be found in [2], [4],
[81-[lal. Our goal in this paper is to develop a method for realizing autonomous boundary value systems from their weighting pattern.~ (input-output maps). This problem has been considered in [12] for stationary systems (system having slfift-invariant weighting patterns [2], [4], [9]) which constitute a sub-class of autonomous systems. The outline of this paper is as follows. In Section 2, we present some basic results on tile wellposedness of boundary-value systems and establish the notation which will be used throughout the paper. We characterize the class of minimal boundary-value linear systems in Section 3. In Section 4, we present some results concerning factorization of rational matrices in two variables having separable denominators. We use these results in Section 5 to obtain necessary and sufficient conditions for realizability of candidate weighting patterns. We use the results of Section 4 also
68 in Section 6 to compute the dimension of minimal realizations of a realizable weighting pattern and to construct its minimal realizations. Due to space limitation, we present the results without proof; the proofs will be presented in a full length paper in preparation. Also, for simplicity, we only consider continuous-time systems.
2
Preliminaries
In this paper, we are only concerned with well-posed systems, i.e. systems where the state x(t) and consequently the output y(t) are uniquely determined in terms of the input function u(t) supposed to be square integrable. The boundary-value system (1-3) is well-posed if and only if
[21 det(Vi exp(aA) + V l exp(bA)) ~ 0.
(4)
Assuming (4) is satisfied, System (1-3) defines a well-defined input-output map
y(t) = f~ ~(t, 7")~(7")d7"
(5)
where w(t, "r) denotes the weighting pattern of the system given by w(t,T)={
Cexp((t-a)A)(l-P)exp(-(r-a)A)B, - C exp((t - a ) A ) P e x p ( - ( r - a ) A ) B ,
a < 7" < t < b, a < t < r < b,
(6)
arid where P is the canonic~I boundary-value operator [3] given by P = exp(aA){(Vi exp(aA) + Vf e x p ( b A ) ) - ' V I exp(bA)} e x p ( - a A ) .
(7)
Notice that the weighting pattern is completely defined in terms of the matrices C, P, A, and B. We call (C, P, A, B) a realization of the weighting pattern w; a boundary-value system having w for weighting pattern can easily be constructed from (C, P, A, B). 1 The problem considered in this paper then consists of constructing minimal realizations 2 given w. The causal and the anti-cansal parts of w are analytic functions and thus we can consider their analytical extensions to arbitrary intervals. Consider the extensions of the two parts of w to the interval [a, co), and denote the causal and the anti-causal parts respectively by w~ and w~:
wr 7") = w~(t, 7") =
C exp((t - a ) A ) ( I - P ) exp(-(7" - a ) A ) B , a < 7", t, - C exp((t - a ) X ) P e x p ( - ( v - a ) A ) B , a < v,t.
(s) (0)
We define the transforms of wc and w, as follows
Wc(s,a)
=
//// w~(t, r)e-~(t-=)-*(*-~)dtdr C(sI- d)-l(IA)-'B, ////
(10)
=
-C(sI-
(11)
=
P)(af+
A)-'P(aIA- d)-lS.
W, and W, are essentially the 2D Laplace transforms of wc and wa. Constructing Wc and W~ form w is straightforward and thus we shall assume that they are available. T h e realization 1Boundary matrices V/and 111 can be obtained as follows: V! = DPexp((a --b)A) and Vi = D(I - P) where D is any invertibte matrix. u(C, P, A, B) is a minimal realization of w if A has smallest dimension among all realizations of w.
69 problem considered in this paper then becomes the following. Given rational matrices W,(s, cr) and W=(s, o.) find (C, P, A, B) such that (10) and (11) are satisfied and A has minimal dimension. Of course not all rational matrix pairs {W,, W=} are realizable. In Section 5, we characterize the class of realizable rational matrix pairs and in Section 6, we present a method for constructing minimal realizations via factorization of rational matrices in two variables. But before we start, we need to characterize the class of minimal systems.
3
Minimal systems
In [4], neccssary and sufficient conditions for minimality of autonomous boundary-value linear systems have been derived.
Theorem 3.1 ([4]) System(1-3) is minimal if and only if '~-'
CA"P ] AJ CA'(1- P) = O,
(12)
V ImAi[ PAJB (I-P)AJB ] =IR ~,
(13)
N Ker
i,j=O
[
n-1
i,j=O n--I
n--I
N Ker CA'
C VIm AiB,
i=0
(14)
i=0
where P is the canonical boundary operator defined in (7). i~eachability and observability are not necessary for minimality, that is why minimal realizations are not necessarily similar; there are some additional degrees of freedom. We characterize exactly how minimal realizations are related in the next theorem and corollary. This result generalizes similar result obtained for stationary systems in [2]. 3.2 Suppose ( e l , Pi, A1, B,) and (C2, P2, A2, B2) are minimal realizations of a weighting pattern w, then there exists an invertible matrix T such that
Theorem
B2 = TBt, C2
01(A,-T-tA2T) (A, - T-~A2T)R,
=
Cl T-1 ,
(15) (16)
=
0,
(17)
=
0,
(18)
and Ol ( Pt - T-1P2T)I~I = O,
(19)
where 121 and 01 denote respectively the reachability and observability matrices 3 of System I. C o r o l l a r y Suppose (C1, P~, A1, B~) is a minimal realization of a weighting pattern w and T any invertible matrix. Then any (C:, P:, A:, B2) satisfying (15-i9) is also a minimal realization ofw. Theorem 3.1, Theorem 3.2 and its corollary completely characterize the class of minimal realizations. Note that for the special case of causal systems, from Theorem 3.1, we obtain the aThe reachability mat,rix of (Ai, Bi) is/~ = [ Bi AiBi ATo[ (AT-')rO[ ], i = 1,2.
is or~ = [ o r
..
A~-IBI
] and I.he observability matrix of (Ci, Ai)
70 classical conditions for minimMity, i.e. reachabiliLy and observability, and from Theorem 3.2, we get that all minimal systems are similar, which of course is a well-known result. So far, our analysis has been restricted to the time-domain. From this point on however we shall, be more concerned with the transform-domain (frequency-donmin). The transform domain analysis in our case involves a special class of rational matrices in two variables, in particular rational matrices in two variables which have separable denominators (in the causal ease, only rational matrices in one variable are involved). In the next section, we present some results concerning factorization of rational matrices in two variables having separable denominators. These results are essential for the development of our realization theory.
4
Transform
domain
The goal of this paper is to develop a transform d o m a i n realization theory for boundary-value systems. The transformation which we use for tlfis theory is the 2D Laplace transform introduced in Section 2. It is straightforward to see from (10) and (11) that we are dealing with rational matrices in two variables which are strictly proper and have separable denominators. A rational matrix H(s, or) is called strictly proper if lim tI(s, a) = lim H(s, ~7) = 0;
(20)
it has separable denominator if it can be expressed as ]V(a, or)
(21)
H(s, ~) = p(s)q(o)' where N is a polynomial matrix in two variable, a n d p and q are saclar potynon~als.
T h e o r e m 4.1 Let H ( s , a ) be a strictly proper rational matrix having separable denominator. Then there ezia~ strictly proper rational matrices L(s) and R(a) such that n ( s , a) = L(s)R(a).
(22)
This result is proved using the following lemma: L e m m a 4.1 Let N ( s , a) be a polynomial mairiz in s and cr. Then there exist polynomial matrices F(a) and a ( a ) such tha*
N(s,.) = F(s)G(a). In particular, in
(23,
(23)
we can choose F and G as follows F ( s ) = S(s)M,
where I
3
G(o') = E(r
(24)
.. ,at
S(8)=
/ 1
,
..
J~
C2~)
l s 1
1 ~r
z(~) =
1
a#r
..
,o,
71 and where M is a constant matrix, or the other way around:
.F'(s) = S(s),
a(g) = .~i~](a).
(27)
We could also factor M as M = U V in which case we can have F(s) = S(s)U,
C(g) = VZ(g).
(28)
T h e decomposition (22) is of course not unique.Two decompositions of particular interest in our case
are:
H(s,g)
L(a) q-~))
(29)
If(a,g)
~ n(g)
(30)
where S(s) and E(a) are as in (25) and (26). We say that the decomposition (29) is minimal if q(g) is a monte polynomial of least degree. If we require ~ be such that fli's are as small as possible, then the minimM decomposition (29) becomes unique. We shall only consider this latter minimal decomposition which can be constructed as follows. First express H as
N(a, g) H ( s , g) = v(s)q(g)
(31)
wherc the polynomial matrix N(s, g) and q(a) are coprime, i.e. there is no go such that N(s, g0) = 0 and q(go) = 0. Then decompose N(s, g) as in (24) and let L(s) = F(a)/p(s). Note that q(g) is the monte polynomial which is the least c o m m o n multiple of the denominators of all the entries of H in g. A n d so if ^
L(a)~ is any other decomposition of H, then q(g) divides c~(g). Similarly, we can define and construct the minimal decomposition (30). The construction is as follows. Express If as in (31) w i t h / V ( a , g ) and p(a) eoprime in a. Then, decompose N ( s , g ) as in (27) and let R ( a ) -- G(a)/q(g). D e f i n i t i o n 4.1 (C, A, l"(a)) is an s-~tate-space representation of a strictly proper rationM matrix having separable denominator If(s, g) if
If(s, g) = c(az - d)-~r(g),
(32)
where C and A are constant matrices and F(g) a rational matrix in a; we say that (32) is an s-factorization of H. Similarly, ( n ( s ) , E , B ) is a g-state-space representation of a strictly proper rational matriz having separable denominator H(s, g) if If(s, g) = ~2(s)(gI - E ) - I B
(33)
for some con3tant matrices E and B, and a rational matrix ~(a); we say that (33) i~ a gfactorization of H. Thanks to Theorem 4.1 any strictly proper rational matrix having separable denominator has both an s- and a g-state-space representation. For example an s-state-space representation of a strictly proper rational matrix having separable denominator H ( s , g) exists because thanks to Theorem 4.1, we can decompose H as in (22) and factor L (which is strictly proper) as follows
L(~) = c ( s I -
A)-'B
which m e a n that (C, A, BR(g)) is an s-state-space representation of H.
(Z4)
72 D e f i n i t i o n 4.2 The pair (A,P(a)) is called reachable if for all constant left eigenvector~ of A, v,
,r(~) r 0. :"he pair (a(~), E) is eaUed observable iy for all constant right ~igen~eetors oI A, ~, a(~)w # 0. Definition 4.2 generalizes the classical notions of reachability and observability in the 1D case. T h e o r e m 4.2 (C, A, P(a)) is a minimal s-~tate-apace representation of H(s, a) if and only if
1- (C,A) is observable,
e- (A, r(a)) is reachable. (i2(s), E, B) is a minimal ~-~tate-space representation of H(s, ~) if and only if 1- ( E , B ) is reachable,
~- (a(~),Z) is obser,able. A useful result which follows easily is the following: C o r o l l a r y Let (C,, A1, I"1(o")) and (C2, A2, F2(a)) be two minimal s-state-space representation3 of H(s, a). Then there ezists an invertible matrix T such thai r~(a) = ~ , ( a ) c~ = C,T -1 A2 = TA1T -1.
(35)
(36) (37)
let (a,(s), E,, B,) and (a~(s), E~, n:) be two minimal ~-s~ate-spaee representations o/a~(s, ~). Then ~here exists an invertible matrix U such that
a~(s) = a,(~)cr-'
(3S) (39)
E2 = UExU -1.
(40)
B~
=
UB~
For a rational matrix in one variable, the dimension of its minimal factorizations equals its McMillan degree, i.e. the degree of the least common multiple of the denominators of all of its minors. Similar rcsults can be obtained for rational matrices in two variables having separable denominators. D e f i n i t i o n 4.3 The s-degree of a strictly proper rational matriz with separable denominator H( s, ~r), denoted by #,( H (s, a ) ), is the minimal dimension of its s-state-space representations, i.e. /f (C, A, V(a)) is a minimal s-state-space representation of H( s, a ), then #~( H ( s, a)) = dim A. The a-degree of H(s, a), denoted by #~ (H(s, a)), is the minimal of i~s a-state-space representations, i.e. ff (a(~), E, Z~) is a minimal ~-state-space representation o/H(~, ~), then,,(H(s, a)) = dimE. Note that in general, the s-degree is not equal to the McMillan degree in s and that the a-degree is not equal to the McMillan degree in or. The McMillan degree in s for example corresponds to the dimension of minimal factorizations of ~he following type: H(s, ~ = C(a)(sZ - d ( a ) ) - ' Z ~ ( a )
(41)
which is a more general factorization than (32) where we require that C and A be constant. The additional degrees of freedom in (41) imply that the s-degree is always larger than or equal to the McMillan degree in s. Similarly, the a-degree is always larger than or equal to the McMillan degree in ~.
73 Theorem
4.3 Cor~ider the minimal decomposition
n(s, a) = L ( ~ ) ~ .
(42)
Then the s-degree of H, #~(H(s, cr)), equal8 the McMilIan degree of L(s). Consider the minimal decomposition S(s)_,
Then the cr-degree of H,
,
(43)
#~(tt(s, ~)), equal~ the McMillan degree of R(a).
An important consequence of Theorem 4.3 is that to construct minimal s- and a-factorizations, we simply have to decompose H as in (42) and (43), and then factor L and R using standard algorithms.
5
Realizability conditions
In Section 2, we defined the transfmrns of the causal and anti-causal parts of the weighting pattern of an autonomous, boundary-value linear system. We noted the transform of the causal part by Wc(s , a) and the anti-causal part by W~,(s, (7). We have already seen that, to be realizable, We and W= must be rational and strictly proper. However, these conditions are not sufficient for realizability. The following theorem characterizes completely the class of realizable matrix pairs. Theorem
5.1 The pair 4 rational matrices {Wo(s, ~), W~(s, a)} is realizable if and only ff
1- the two matrices are strictly proper and have separable denominators, 2- there exists a strictly proper rational matrix H(z) such that (s + a)(Wr
or) - W=(s, or)) = H ( a ) - H ( - a ) .
(44)
Condition 2 simply reflects the fact that wr r) - w,(t, r) depends only on the difference t - r. T h e o r e m 5.1 provides us with a simple realizability test. Given a candidate weighting pattern w ( t , r ) , compute its transforms ~ in (10) and (11), and see if they satisfy the two conditions of Theorem 5.1. Since w is made of two analytic functions, we can consider reMizing w on different intervals. The following result shows that the choice of the interval over which w is to be realized does not m a t t e r as far as realizability is concerned. Theorem
6
5.2 If w is realizable on any interval, it i~ realizable on all intervals.
Realization procedure
In this section, we derive an expression for the dimension of minimal realizations. We then use this result to solve tile realization problem via methods of factorization of rational matrices introduced in Section 4. Theorem
6.1 The
dimension
n
of
minimal
realizations
of
a
realizable
pair
{wo(~, ,,), w~(~,,,)) is given by the :oZlo~inj equation
~=..([ wo(s,~) Wo(s,~)])+u.(wo(s,~) )-..(Wo(s,~)-w~(s,~)) .,here re(X) a,~d z~(X) denote r~spectivelv the ~- and ~-degrees 4 X.
(45)
74 In addition to the dimension of minimal realizations, (45) allows us to test the reachability and observability of minimal realizations. In particular, if #4([ Wr W~(~,o') ]) which equals the rank of the observability matrix is equal to n, we can conclude that minimal realizations
are observable. On the other hand, if #~([ ~V:I;:;I ])equals n, then minimal realizations are reachable. If minimal realizations are reachable or observable, the realization problem is simple; it only involves one factorization. The situation is more complicated when minimal realizations are not reachable or observable. Let us start with the case where minimal realizations are observable. Observable case
Suppose that minimal realizations are observable, i.e. , , ( [ Wo(s,a)
Wo(~,a) ]) = n.
(46)
Thcn, a minimal realization can be constructed as follows: 1- perform a minimal s-factorization
2- compute B = li2~s a(A,(cr) - A2(cr)).
(48)
3- construct the minimal decomposition
(~Z+A)-~B
=
M
q(~)
and find a P that satisfies
(50)
P M = -IF. T h e n (C, P, A, B) is a minimal realization of {W~, W,}. Reachable case
Suppose thai; minimal systems are reachable, i.e.
#/ w~
)='~'
(51)
then a minimal factorization can be constructed as follows 1- perform a minimal r
[ wo(~,o) ] Wo(,,~) ] = [ ,I,=(,) '~'(') j (,,z + A)-IB.
(52)
2- compute
c = jim,(~,,(,) - r
(53)
3- construct the minimal decomposition
s(,)
[ c(~z- A)-'
(54)
and find a P satisfying M P = _It. T h e n ( C , P , A , . B ) is a minimal realization of {l~V~,Wa}.
(55)
75 U n r e a c h a b l e / u n o b s e r v a b l e case If minimal realizations are not reachable or observable, the situation is more complicated because the matrix A cannot be constructed by just one factorization as in the previous cases. In this case, we have to construct it from two different pieces which are obtained from two different factorizations. The following procedure can be used for constructing a minimal realization: 1- construct minimal s- and cr-factorizations
[ W~(s,~)
W~(s,~)]
=
[
~(~)
~(s) ] (al+d)-~/~
(57)
and let = lirns(~(s) - cZ~(s)) a = )imo(&(~)-&(~)),
(58)
(59)
2- decompose (d', fi~,A) in reachable/um-cachable parts and (~9, A, D) in observable/unobservable parts:
c= Iv, o5 ],
0
-A4
, A=
0
(60)
'
and
<.],
0
,44
'
1)5
(61)
'
3- construct system matrices as follows
A=
0
A1
0
0
-42 X,~
,
c=[0
cl
(6:)
B_0
where * indicates an arbitrary block entry and the matrix V is given by
v =/t_~(_~.~r) -~,
(G3)
where/~ a n d / ~ denote respectively the teachability matrices of (fi.~,/3~) and (-'i4,/~2)4- construct the minimal decompositions s(~)
(64) ~(o)
(65)
and find a P that satisfies
MIPM~
Then
(C, P, A, 13) is a minimal realization
= -Y.
of {W~, W~}.
(66)
76
7
conclusion
In this paper we have developed a realization theory for autonomous boundary-value linear systems. In particular, we have shown that by taking 2D Laplace transforms of the causal and anticausal parts of the candidate weighting pattern, we can examine its realizabiligy and construct minimal realizations by factoring rational matrices in two variables having separabie denominators. We have studied this factorization problem and have proposed methods foi" consLructing such factorizations.
References [1] A. J. Krener, Boundary value linear systems, Ast~rique, 75-76 (1980), pp. 149-165. [2] A. J. Krener, Acausal realization theory, Part I: Linear deterministic systems, SIAM J. Control and Optimization, 25 (1987), pp. 499-525. [3] I. Gohberg and M. A. Kaashoek, On minimalily and stable minimality of time-varying linear systems with well-posed boundary conditions, Int. J. Control, 43 (1986), pp. 1401-1411. [4] I. Gohberg, M. A. Kaashoek and L. Lerer, Afinimality and irreducibility of time-invariant linear boundary-value systems, Int. J. Control, 44 (1986), pp. 363-379. [5] I. Gohberg and M. A. Kaashoek, Minimal representations of semi-separable kernels and systems with separable boundary conditions, J. Math. Anal. Appl., 124, (1987), pp. 436-458. [6] R. Nikoukhah, System theory for two-point boundary value descriptor systems, M.S. thesis, Dept. of Elec. Eng. and Comp. Science, and Report L!DS-TII-1559, Lab. for hlformation and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 1986. [7] K. Nikoukhah, A deterministic and stochastic theory for two-point boundary-value descriptor systems, Ph.D. thesis, Dept. of Elec. Eng. and Comp. Science, Massachusetts Institute of Technology, Cambridge, MA, 1988. [8] It. Nikoukhah, A. S. WiUsky and B. C. Levy, Boundary-value descriptor systems: well-posedness, teachability and observability, h~.t.J. Co1~trol, 46 (1987), pp. 1715-1737. [9] R.. Nikoukhah, A. S. Willsky and B. C. Levy, Reachability, observability and minimatity for shiftinvariant two-point boundary-value descriptor systems, Circuits, Systems, and Signal Processing, 8, (1989), pp. 313-340. [10] 1~. Nikoukhah, M. B. Adams, A. S. Willsky and B. C. Levy, Estimation for boundary-value descriptor systems, Circuits, Systems, and Signal Processing, 8 (1989) pp. 25-48. [11] K. Nikoukhah, B.C. Levy and A.S. Willsky, Stability, stochastic stationarity, and generalized Lyapunov equations for two-point boundary-value descriptor systems, IEEE Trans. on Automat. Control, 34 (1989), pp. 1141-1152. [12] It. Nikoukhah, A. S. Willsky and B. C. Levy, A realization theory for two-point boundary-value descriptor systems, to appear in SIAM J. Control and Optimization. [13] It. NikoukhMl, A. S. Willsky and B. C. Levy, Reachability, obscrvability, minimality and extendibility for two-point boundary-value descriptor systems, Itapport INI~IA, no. 920, 1988. [14] B. L. IIo and K. E. Kallnan, Effective construction of linear state-variable models from input-output functions, Proc. Third Allcrton Conference, 1966, pp. 449-459. [15] D. C. Youla, The synthesis of linear dynamical systems fi'om prescribed weighting patterns, SIAM J. Applied Math. 14 (1966), pp. 527-549.
PC-TACT : Personal Computer Tool for Advanced Control Techniques
R.M.C.
De Keyser,
MSc.,
PhD.
Automatic Control Laboratory, University of Ghent Grotesteenweg Noord 2, B-9710 GENT/Belgium Telo : 32/91/225755 Fax : 32/91/228591
ABSTRACT Successful application of modern control methods to industrial processes assumes some expertise with the algorithms. This is necessary in order to gain some feeling for t u n i n g the design parameters~ Computerbased instruction, learning and t r a i n i n g of the user is of great value. An interactive personal c o m p u t e r t r a i n i n g and design software system is presented. It was m a i n l y d e v e l o p e d for 'in-house' training of p e o p l e who have to deal with dynamical processes and their automatic control systems. It c o n t a i n s process simulation modules and s t a n d a r d c o n t r o l l e r modules, as well as more a d v a n c e d a u t o - t u n i n g and a d a p t a t i o n modules besides m o d e l l i n g and i d e n t i f i c a t i o n modules. The use of a graphical a p p l i c a t i o n - b u i l d e r and a n i m a t i o n facilities makes the system attractive and easy to use for the n o n - e x p e r i e n c e d user.
KEYWORDS C o m p u t e r aided design; computer aided instruction; PID control; identification; software tools.
adaptive
control;
INTRODUCTION PC-TACT (Personal Computer Tool for A d v a n c e d Control Techniques) is an efficient software package for t r a i n i n g i n s t r u m e n t a t i o n and control engineers. It offers the user the o p p o r t u n i t y to become more skilled in topics such as control structure design, controller tuning, autotuning and adaptive control, m o d e l l i n g and process identification. It was developed p r i m a r i l y for control engineers working in everyday industrial practice. Indeed P C - T A C T never assumes that the user has a thorough knowledge of the control theory. It is rather the objective that the user compensates his lack of theoretical insight by gathering the necessary e x p e r t i s e at his PC-screen. This is done by experimenting, in a user friendly and interactive way, on real-time c o m p u t e r - s i m u l a t e d processes. The simulated process is d e f i n e d in a straightforward and pragmatic way by the user himself such that the training is effective : the experience is gained on a simulated system w~th a similar structure and similar dynamics as the user's real-life process. Many control principles are only s u f f i c i e n t l y well u n d e r s t o o d limited group of research people which are experts in the
by a field.
78 However the average control e n g i n e e r in industry is still rather unfamiliar with the methods. He is lacking the required feeling in order to answer the fundamental q u e s t i o n : "Is this type of control strategy p o s s i b l y the solution to my specific control p r o b l e m on my specific process?" Packages like PC-TACT might be a pillar to bridge this gap b e t w e e n theory and practice. There have been a lot of papers p u b l i s h e d on CAD of control systems (Astr~m, 1983), some of them c o n t a i n i n g a d v a n c e d topics (Chen and Fang, 1985; Clarke and Turfs, 1985; Schmid and Unbehauen, 1979; Duff, Dwolatzky, Sanoff and Wellstead, 1982; De Keyser, 1985) and powerful commercial packages are available. The approach of PC-TACT is somewhat different. Most CAD packages are built along the classical design techniques of linear (eventually nonlinear) systems and are operated in batch mode. Contrary to this, PC-TACT operates rather as a real-time simulator where the process and its controller actually are running c o n t i n u o u s l y in the computer. The user then changes some design parameter and immediately observes its effect on the process output, on the control variable, etc. . Based on these o b s e r v a t i o n s he proposes a new value for the design parameter, until a satisfying b e h a v i o u r is observed. The o p t i m i z a t i o n process is thus done in an iterative way with 'the user in the loop'. The advantage is that the user does not need to be skilled in control theory in order to use PC-TACT. PC-TACT has been written for nonexperts in order to familiarize themselves with the power of advanced control algorithms without being o b l i g e d to go into the mathematical details.
PC-TACT
HIGHLIGHTS
A 'real-time' user interface The PC-TACT package was developed to experiment with continuous-time dynamical (control) systems in real-time. O p e r a t i n g in 'real-time' is an attractive feature when c o m p a r e d to the numerous existing CACSD packages which operate in a less interactive 'batch' mode, according to a 3-step schedule : input of the system structure and parameters; computations; p r e s e n t a t i o n of the results. Contrary to this, in PC-TACT the user really looks at the 'movie' on his PC-screen, occasionally e n t e r i n g a new value for a design parameter and immediately observing its effect on the control loop as a function of time, e.g. on a trend screen. This is realized by structuring the software in a high-priority background task and a low-priority foreground task. The background task does the sampling and all r e l a t e d computations p e r i o d i c a l l y under control of the computer's real-time clock. The foreground task displays c o n t i n u o u s l y the new results while the simulation goes on and accepts c o n t i n u o u s l y eventual new p a r a m e t e r s from the user. S t e p - r e s p o n s e process models A~nother important difference with most existing software p a c k a g e s for simulation of dynamical systems is the way of d e s c r i b i n g the process dynamics. Traditionally, physical processes involving pressure, flow, temperature, level, speed, force, etc.., are s i m u l a t e d by c o n n e c t i n g basic s i m u l a t i o n blocks (e.g. integrators, Ist order blocks) or by s p e c i f y i n g the system's state equations.
79 In our a p p r o a c h the user sits down at his PC and just draws the step response of the process by means of a 'mouse' device. This is a very user-friendly method well a p p r e c i a t e d by users in industrial process control. These people frequently do not have a numerical mathematical model of their process dynamics, but they do have the necessary expertise to sketch the response of the process output to a fictitious step input. It is well known from the c o n v o l u t i o n theorem that, given the s y s t e m step response, the system output can be computed for any other input signal. Apart from this graphical input facility, P C - T A C T has also the option to define a process by entering the p a r a m e t e r s in a transfer function, i.e. time constants, static gain, time delay, n o n - m i n i m u m phase zeros, damping factor, natural frequency. This way of defining a process is mainly a p p r e c i a t e d by users dealing with the design and analysis of servosystems. A q r a p h i c a l application builder Building a PC-TACT application is very simple and attractive using a graphical a p p l i c a t i o n - b u i l d e r and an o b j e c t - o r i e n t e d environment. PCTACT is built along high-level modules (objects), such as process modules, regulator modules, auto-tuners, . . . . Using a mouse-device the user selects the desired modules from a menu, links them together by some clicks on the mouse and the system starts running. In fact the user simply draws the familiar block scheme of a control system, which then a u t o m a t i c a l l y becomes alive in the simulator! The procedure is i l l u s t r a t e d in fig. i. The f o l l o w i n g symbols are used : P = Process, R = Regulator, G = signal Generator. Ichooee
81ockllOelete
gleckll
Ornw
Connectlnf
Line
]oulTII
GI
PC-TACT
Fig.
1 : Screen dump of graphical
IN A
NUTSHELL
p R O I N T SUM G~EN 0 A C REG TU H IIOD s P C
application
builder.
Some basic applications only need 2 or 3 P C - T A C T modules, e.g. a process module, a controller module and a signal generator module. Other more advanced applications may contain an auto-tuner module or a process m o d e l l i n g module. A l t h o u g h the type of problems that can be tackled with PC-TACT is quite extensive, mainly dependent on the creativity of the user, a survey of the most obvious applications could look as follows. F a m i l i a r i z i n q with the process dynamics A basic u n d e r s t a n d i n g of the dynamical b e h a v i o u r of the process is a fundamental topic for any control engineer in industrial practice. E s p e c i a l l y on multi-input multi-output p r o c e s s e s with strong interaction some insight into the effect of specific process inputs on the primary (controlled) and secondary process outputs might be very helpful in order to design a good control structure. An example is given in fig. 2 where 4 process modules, each c h a r a c t e r i z e d by its own step response model, are used to investigate a 2 input/2 output plant.
80 D i s t u r b a n c e s i g n a l s z can be i n j e c t e d at s e v e r a l p o i n t s a n d the e f f e c t on t h e p r o c e s s o u t p u t s y a n d the i n t e r m e d i a t e o u t p u t s x can r e a d i l y be visualized. The u s e r can try to r e g u l a t e t h e p r o c e s s h i m s e l f by m e a n s of t h e i n p u t s u, w h i c h is a l w a y s an i n t e r e s t i n g a n d r e v e a l i n g e x e r c i s e b e f o r e d e s i g n i n g an a u t o m a t i c c o n t r o l s y s t e m .
Ul ' 'I~
Fig.
~
2 : Experimenting
'Yl
with
a MIMO-system.
Remember that all this is d o n e w i t h o u t a n y computational analysis, just by a p p l y i n g the c o n s i d e r e d p r o c e s s i n p u t v a r i a t i o n s (by m e a n s of the u p / d o w n - a r r o w keys of the PC) a n d t h e n w a t c h i n g the p r o c e s s o u t p u t v a r y i n g in r e a l - t i m e on a t r e n d s c r e e n . B e c o m i n q an e x p e r t in c o n t r o l l e r d e s i q n Designing a c o n t r o l s y s t e m in p r a c t i c e m e a n s d e c i d i n g u p o n the best control structure and then t u n i n g the regulators. Especially on processes with multiple inputs and/or outputs, many different control structures are p o s s i b l e s u c h as a b a n k of single-loop controllers, (partial) d e c o u p l i n g , feedforward/feedback c o n t r o l (fig. 3), c a s c a d e c o n t r o l (fig. 4), i n f e r e n t i a l c o n t r o l , etc . . . . ........................
Jl i leIr control
I
,
FF
Fig.
3 : Feedforward/feedback
control
structure.
z
i
Fig.
4 : Cascade
(Master-Slave)
-
control
!
structure.
Y
81 A l t h o u g h some general theoretical p r i n c i p l e s exist, it is difficult to state in advance which of the s t r u c t u r e s will be the best in a real situation. In PC-TACT, just by c o m b i n i n g some controller modules with some process modules, entering the (approximate) step response models in the process blocks, and d o i n g some e x p e r i m e n t s with setpoint or disturbance signals, the relative m e r i t s of a specific structure is found. Also the selection of a specific r e g u l a t o r a l g o r i t h m and the tuning of its parameters by trial and error can be t r a i n e d in PC-TACT and leads to a better skilled user. Several types of industrial control algorithms can be specified such as P, PI, PID, relay control, bang-bang or on-off control, etc. i n c l u d i n g p r a c t i c a l extensions such as antireset windup and hard limits on the c o n t r o l l e r output. The user can manipulate parameters such as p r o p o r t i o n a l gain, reset time and d e r i v a t i v e time independently, observe immediately (in real-time) the effect on the closed loop b e h a v i o u r and evaluate setpoint following and d i s t u r b a n c e rejection properties. If the process step response models c o r r e s p o n d to real-life plant units, all these features result in a powerful though e a s y - t o - u s e tool for optimization of control loops without interfering with normal p r o d u c t i o n operation. It is very s t r a i g h t f o r w a r d to i n v e s t i g a t e by means of PC-TACT "whatif" problems, such as : how much better is the c l o s e d - l o o p p e r f o r m a n c e when using a full PID-controller instead of a PI version? As will be done later in an example, it is quite easy to run in parallel, i.e. at the same time, several control loops, having identical process step response models, identical setpoints and disturbances, the only difference being the controller a l g o r i t h m or c o n t r o l l e r tuning. This is illustrated in fig. 5.
P,D
[-
ooo,,o,
j i
I'
,,
f/_ l
.
I
F-
~'---~z2
w +.,(~___~ ADAPTIVE control
Y
i
I
w +_K~ PREDICTIVE control
Fig.
5 : Comparing
L_Tzl
'6 u ,
different
....7 z2
~
control
,(~ [
strategles
in one
Y.
run.
82 E v a l u a t i n q pros and cons of a u t o - t u n e r s A new t r e n d in industrial control i n s t r u m e n t a t i o n is the use of autot u n i n g facilities. These algorithms are u s e d to compute a u t o n o m o u s l y a set of good parameters for the controller, based on the measured process input and output signals (self-tuning). Eventually these parameters can be adapted c o n t i n u o u s l y to v a r i a t i o n s in the process dynamics (self-adaptive). In PC-TACT, the a u t o - t u n i n g module computes the p a r a m e t e r s for a PID control structure (De Keyser, 1988). Tradit i o n a l l y in industrial practice, the t u n i n g of the control parameters in a r e g u l a t o r such as p r o p o r t i o n a l gain and reset time, has mostly been a c h i e v e d using trial and error methods, which are t i m e - c o n s u m i n g and disrupt process flow. It also requires the n e c e s s a r y expertise and in the p r e v i o u s section it was i n d i c a t e d how PC-TACT can be used to skill the user in this field. W i t h the auto-tuning facility, this tuning step is done by the c o m p u t e r itself. N e v e r t h e l e s s there remain some design parameters, specific for a u t o - t u n i n g strategies, that have to be chosen by the user. These are n o r m a l l y more convenient to handle, c o m p a r e d to the selection of P, I and D parameters, because they have a physical meaning d i r e c t l y r e l a t e d to the performance of the c l o s e d loop. As an example, the main design p a r a m e t e r of the autotuner implemented in PC-TACT, is the d e s i r e d closed loop settling time. The objective of the a u t o - t u n i n g modules in PC-TACT is to give the user the facility to skill himself in the use of these strategies. The design p a r a m e t e r s that are important for the user to f a m i l i a r i z e with, can be r e c o g n i z e d in any s e l f - t u n i n g algorithm. Therefore the PC-TACT auto-tuner can be well used to evaluate the pros and cons of commercial a u t o - t u n e r s in gener~l. Again, some related and p r a c t i c a l l y important "what if?" p r o b l e m s can be tackled, such as : "What will be the average return in control performance for my specific control a p p l i c a t i o n if I add an autotuning option to a standard PID c o n t r o l l e r ? " With PC-TACT a quick judgement can be gained, for example by experimenting with both a l t e r n a t i v e c o n t r o l l e r s in p a r a l l e l on identical process step response models (fig. 5). Process m o d e l l i n q and identification Another important topic in modern control engineering is process identification, i.e. experimental e s t i m a t i o n of a dynamic process model from m e a s u r e d input/output records. Once a numerical model of a real-life process is available, p o w e r f u l c o m p u t e r aided design packages can be used to analyse and s y n t h e s i z e control systems and highperformance control strategies such as MBPC (Model Based Predictive Control) can be implemented. Because of the importance to introduce process identification to control engineers in industrial p r a c t i c e and to train them in the selection of the related design parameters, PC-TACT also c o n t a i n s a m o d e l l i n g module. It can be used to estimate the "unknown" d y n a m i c s of PC-TACT p r o c e s s e s (in which case the process is in fact only assumed to be unknown) in order to f a m i l i a r i z e the user with the specific features of identification strategies. It could also be used to estimate the (really) unknown dynamics of real-life p r o c e s s e s in order to obtain a numerical process model. Model B a s e d P r e d i c t i v e Control MBPC seems to be a promising strategy for dynamic control and optimization of real-life processes. Many applications have been reported (Morari, 1988; De Keyser, 1990). The idea of MBPC is to compute the control action b a s e d on a long-range p r e d i c t i o n of the
83 process output. The forecast is done by means of an internal computer model of the process dynamics. The t e c h n i q u e has many advantages, e s p e c i a l l y on difficult processes. A typical MBPC strategy is available as a P C - T A C T module. The user can thus easily evaluate the pro's and con's of suchlike methods or quickly make a comparison with the performance of classical PID regulators. I n t e r f a c i n q to real-life processes PC-TACT includes A-to-D and D - t o - A m o d u l e s that can be used in c o m b i n a t i o n with an analog i n p u t - o u t p u t b o a r d in the PC. This allows the user to apply all previously d e s c r i b e d P C - T A C T modules to a reallife process. This leads to i n t e r e s t i n g new a l t e r n a t i v e s as illustrated in fig. 6, e.g. : - using the PC-TACT modelling module to i d e n t i f y a dynamical model of a r e a l - l i f e process - testing external commercial c o n t r o l l e r s on a benchmark process i m p l e m e n t e d in PC-TACT - experimenting on a real-life process with a PC-TACT controller module, eventually in combination with an a u t o - t u n i n g module.
ire............... ~ _
I'via
I ADC I
~'-~!
SIMULATED ~ proces
~m~ct
- - ~ REAL-LIFE
controller Fig.
PC-TACT
y Y.
REAL-LIFE I via I proces I
+
w
6 : Operational
[~6............... j !PC
modes
y via l ~-6,-t,, /
controller , PCI
w
of PC-TACT.
EXAMPLES
The user interface This is the most important part of the t r a i n i n g system. It allows the user to change in real-time (i.e. during a run and without interrupting the normal 'simulation and control'-cycle) any p a r a m e t e r of a PCTACT module and visualize the effect on several types of screen output. Changing the design parameters during o p e r a t i o n is done by means of the V(alue) command. For a controller module e.g. the design parameters are t h e s a m p l i n g period, the p r o p o r t i o n a l gain, the m a x i m u m limit on the c o n t r o l l e r output, etc. These can be v a r i e d by the user d u r i n g the run in order to familiarize with their effects. PC-TACT has currently 10 d i f f e r e n t types of modules. H o w e v e r several modules of the same type can be used in a PC-TACT application, e.g. 3 process m o d u l e s and 2 c o n t r o l l e r modules. Commands like V R2Kp=0.5; V R2Ti=I0; V R2Td=0 WQL~Id change the tuning parameters (Kp, T• Td) of the 2nd regulator. By the same token, the user can m a n i p u l a t e the p a r a m e t e r s of the autotuner, load a new process step response model for simulation, etc. He can readily observe in real-time the effect of his interactions by means of numerical output (numbers), t h e r m o m e t e r scale output (bar graphs) or historical trends (time histories). This is i l l u s t r a t e d in figs 7, 8, 9. The pictures on these screens are varying in real-time during the operation i.e. the user observes all signal v a r i a t i o n s on
84 animated screen output 9 He interactions with the control ters. GIo~t[--~--]=20.000
can learn system and
C2out[--M--]=O.0080
Piout[--~--]=74.631 XIo~t[--• P2o~ts $2out[--~--]=48.217 RITs [tunit]=l.0000 RIMin[--M----]=-O.100 RIKp [ ..... ]=999.00 R l m a x [ - - • RiT1 [tuni%]=999.OO
from the effects optimize the design
of his parame-
Xlout P l o t t t Glout [--W.--] [--Z--] I--X--] 1OO" 100j 100]
P 2 o t l t X2oLtt E--W.--] I--X--] 100.
80.0
80.0 ~
130.04
[email protected]
8
R2Ts [tuni%]=3.0000 TiTs [tunit]=5.0000 R2Kp E ..... ]=O.67OO TIKp [..... ]:O.7402 R2TI [tunlt]=34.940 TITx s R2Td [tun it]=-2.410 TIT~ [tu. it]=-2.116
60.O
60.0-
60.0-
60.0
6
l~'tun[O/I/2]=0.0000
40.0
4 8 90"
48.0-
40.0
4
20.0
20.0
:'0.0
[email protected]
2
@. 00 p c - t a c t
0 . 0 0 ~ c o ~ a n ~ :
@.
@.
~.
RIT~
[ t u n i t ] = 0 . 0 0 0 0
Tunit[55 p c - t a c t
Tlael[tunlt]=O.0800
Tl~esEtunit]=50.000 TIpeP[t~nit]=250.00
CountETunit]=1137.O
MS3=3.0000 c o ~ a n d :
Fig.
7
. . . . . . . . . . . . . . . . . . . . . .
: Numeric
screen.
Fig.
Plout [--z--] 100i
l
8
0 0
00
......................
: Bargraph
~
screen.
i Gloat [--~-- ]
. . . . . . . . . . . . . . . . . . . .
80.01
i 100
. . . . . . . j8,.O
o, o i / 40 @9 ~--__~ '20. @ 1 0.001
m
-
00 9 OO 9
-
P2outt--~--] 180: . . . . .
oo.o.t o.o. t
1
.
.
.
,
1 Glout[--Z--]
.
.
.
.
.
pc--tact
. 100
f-... ..
0
/
20 9 0 . 0 0 1
1 0 . 0 ~
0 -
,
.
,
co~Man~:
Fig.
9
.
.
,
-
-
,
.
-
9
1 0 . 0 0
...................... : Trend
screen.
The s y s t e m is t h u s e x t r e m e l y f l e x i b l e to i n v e s t i g a t e the sensitivity of t h e c o n t r o l l e r to a s p e c i f i c p r o c e s s p a r a m e t e r , to compare simultaneously the performance of t w o d i f f e r e n t controllers (e.g. adaptive a n d PID) on t w o i d e n t i c a l c o p i e s of t h e s a m e p r o c e s s , o r to g e t some feeling for a design parameter of a m o d e l l i n g module by "playing" with it and readily observing t h e e f f e c t s on o n e of the screens. Notice that t h e u s e r c a n p l a c e a n y of t h e v a r i a b l e s in a n y p o s i t i o n on any screen (in run-time) 9 For doing this he has 3 more commands : N(umeric), B(argraph), T(rend). For example the command 'T T I K p = 4 ' w o u l d p l a c e t h e t i m e h i s t o r y of t h e p r o p o r t i o n a l gain Kp estimated by the auto-tuner T1 in t h e 4 t h p o s i t i o n on t h e t r e n d s c r e e n . Concluding commands, This makes
we can state that the user can run PC-TACT with only a few all having the same structure : 'Command Variable=Number'. PC-TACT very user-friendly.
Example : Adaptive Control The b l o c k s c h e m e is g i v e n in fig. I0. Two parallel-running control loops are configured : - loop 1 is a r e l a y c o n t r o l l e r K1 w i t h i n t e g r a t o r Ii operating on process P1 - l o o p 2 is a P I D - c o n t r o l l e r R2 o p e r a t i n g on p r o c e s s P2 P r o c e s s P1 a n d p r o c e s s P2 a r e i d e n t i c a l , i.e. they are described the same step response model. The setpoints for regulators R1 a n d both come from the same signal generator G1 ( w h i c h c a n b e o p e r a t e d
by R2 by
85 the up/down arrow keys of the PC). S i m i l a r l y signal generator G2 used to introduce identical d i s t u r b a n c e s at the process inputs.
is
The only difference between the 2 loops is thus the controller algorithm. The configuration can be used to compare the control p e r f o r m a n c e of both strategies. O p t i o n a l l y the P I D - r e g u l a t o r R2 can be tuned by the auto-tuner TI. Notice that the a u t o - t u n i n g is based on the process input- and output signals of the first loop although it estimates the parameters for a r e g u l a t o r in the second loop. This illustrates that the auto-tuner, a l t h o u g h it is a direct self-tuner, can operate without the tuned c o n t r o l l e r b e i n g in the loop. This p r o p e r t y has important practical advantages.
set C-J-f----q
Fig.
~
C-J-/----U
I0 : PC-TACT block scheme for adaptive control.
CONCLUSIONS
A r e a l - t i m e software package called PC-TACT, was described. It was mainly d e v e l o p e d for design and for 'in-house' t r a i n i n g of people who have to deal with dynamical p r o c e s s e s and their automatic control systems. On the one hand the package can be u s e d to train people with a rather poor b a c k g r o u n d in the theory of control engineering. By e x p e r i m e n t i n g at the PC they get a feeling for process dynamics and the necessary e x p e r t i s e to operate and automate the u n d e r l y i n g real-life process. On the other hand the package also contains some currently 'hot' control strategies such as a u t o - t u n i n g and self-adaptive control, modelling and identification algorithms, Model Based Predictive Control. The interested user in industry can first experiment with these t e c h n i q u e s on the PC. A f t e r w a r d s he will be better armed to take the right decision whether or not to apply these advanced control strategies to his real-life process. Pc-TAcT is not complex and is intended to run on a p e r s o n a l computer. Thanks to its real-time and u s e r - f r i e n d l y operator interface, it is a useful instrument for practitioners and for instruction and training courses.
86 REFERENCES
A s t r 6 m K.J. (1983). Computer A i d e d Modeling, A~nalysis and Design of Control Systems - A Perspective. IEEE-Control Systems Magazine (2), 4-16. Chert Zhen-Yu, Fang Li-Zhang (1985). S e l f - t u n i n g cad-package and its application. 3rd IFAC/IFIP S y m p o s i u m CADCE '85, Copenhagen, Denmark. Clarke D.W., P.S. Tuffs (1985). A general p a c k a g e for implementing self-tuners. IEE Conference Control '85, C a m b r i d g e U.K. De Keyser R. (1985). Training software for adaptive control. In : H. U n b e h a u e n (Ed.), "Adaptive control of chemical processes", Pergamon Press, Oxford, 84-89. Also : Journal A 2 7 (3), 155-161. De Keyser R. (1988). A finite impulse response direct adaptive controller. In : S. Shah, G. Dumont (Eds), "Adaptive strategies for industrial use", Alberta, Canada. Duff J., B. Dwolatzky, S.P. Sanoff, P.E. W e l l s t e a d (1982) o Hardware and Software for Self-Tuning Control. IEEE Conference on Applications of A d a p t i v e and M u l t i v a r i a b l e Control, Hull, U.K., 85-89. Morari M., C.E. Garcia & D.M. Prett (1988). Model Predictive Control : Theory and Practice. IFAC W o r k s h o p "Model Based Process Control", Atlanta. Schmid Chr., H. Unbehauen (1979). I d e n t i f i c a t i o n and CAD of Adaptive Systems using the KEDDC Package. In : R. Isermann (Ed.), "Identification and System P a r a m e t e r Estimation", Pergamon Press, Oxford, 1087-]094. De Keyser R. (1990). Model Based P r e d i c t i v e Control Toolbox. ESPRIT workshop on Computer Integrated Design of C o n t r o l l e d Industrial Systems, E u r o p e a n Community, Brussels.
SOLVING SINGULAR REGULAR SYSTEMS A2X'" +AIX'+AoX=F(t) WITHOUT INCREASING THE DIMENSION OF THE PROBLEM. L.J6dar and E~ Navarro Departamento
de Matem~tica Aplicada, Univ. Polit~cnica
Apdo. 22.012
Summary.
Valencia
46071
Spain.
In this paper we study existence
conditions
lutions of Cauchy problems and boundary value problems
and closed form so-
related to the regular equa-
tion A2X'~+AIX'+AoX=F(t), in terms of a solution of the algebraic matrix equation A2X2+AIX+Ao=O
and without increasing the dimension of the problem.
lo INTRODUCTION Throughout
denotes the set of all nxn square matrices with comnxn plex entries. If S is a matrix in ~ , we denote by S + its Moore-Penrose pseudoinnxn verse and by S D we denote the Drazin inverse of S. An account of uses,properties and procedures
this paper, ~
for computing
these generalized
Second order matrix differential
inverses may be found in [ 3 , 16 , 17 ].
equations
of the type
A2X"+AIX'+AoX=F(t) where A i
(i.i)
appear in vibrational
for i=o,l,2,X(t)
and F(t) are matrices in g and F is differentiable nxn systems [14]~ in thermal and electrical problems [5 ] and from
finite approximations
to distributed parameter
'
rential equations
systems
described by partial diffe-
[I ].
For the case A2=I , Cauchy problems and boundary
value problems related to equation
(i.I) have been recently studied in [ 8 ] and [12] , under the hypothesis tence of a complete
set of solutions of the algebraic
The aim of this paper is to find existence
a solution of the associated
algebraic
equation X2+AIX+Ao=O.
conditions
of Cauchy problems and boundary value problems
of the exis-
and closed form solutions
related to equation
(i.i) in terms of
equation
A2X2+AIX+Ao=O and under the regularity the
(1.2)
condition of the existence
P( ~)=A2X~+AlXo+Ao It
of a complex number
~
o
such that
matrix
is
clear
that
if
the
matrix
A2 i s
singular,
to a monic equation as the one considered
is invertible then
equation
in [ 12 ] 9
(1.3) (l.1)
is
not
reducible
88 Note that if A2=I , then the condition (1.3) is satisfied by any complex number ~ that is not an eigenvalue of the companion matrix
C= -A
o
-A I
see [ 8 ,p.13] for details. Thus the condition (i.3) contains the ease A2=I as a particular case. Also results of [ii], obtained for the case A2=I are obtained as particular cases. The paper is organized as follows. Section 2 deals with the solution of Cauchy problems related to equation (I.i), and section 3 is concerned with the boundary value problem A2X''+AIX'+AoX=F(t),
O=
(1.4)
glX(O)+F1X(b)=O; E2X"(0) +F2X"(b)=O 2. THE CAUCHY PROBLEM FOR THE EQUATION A2X''+AIX'+AoX=F(t) We recall that as it has been studied by Campbell in [2 ], the Cauchy problem A2X"+AIX'+AoX:F(t) ; X(O)=Co, X'(O)=C I
(2.1)
is equivalent to the first order extended problem
(2.2)
AY'+BY=G(t) ; where YI=X, Y2=X ", and
[o i I
B= Ao
A1
,
[o]
G(t)= F(t)
(2,3)
Also, the regularity condition (1.3) is equivalent to say that the determinant of P(s)=A2s2+AlS+Ao, is not identically zero, and it means that the extended system (2.2)-(2.3)
is also regular,[ 7 , 9 ]. Hence, from theorem 3.1.3 of [2 ],if pro-
blem (2.1) is solvable, then the solution is unique. In this section we present an existence condition for the Cauchy probiem (2.1) and a closed form solution of the problem without increasing the dimension of the data. Let us assume the existence of a solution X
of equation (1.2), then an easy o computation yields that Z =X -~ I, is an invertible solution of the algebraic o o o equation
A2Z2+(2XoA2+A1)Z+P(Xo)=O for each complex number Xo
satisfying (1.3).
(2.4)
Premultiplying (2.4) by [P(Xo)]-i ,
it follows that Wo=Z:I, satisfies the monic equation W2+[P(Xo)]-I(eXoA2+AI)W+[P(Xo)]-IA2=O
(2.5)
Conversely, if W is an invertible solution of (2.5), then X=l I+W -I is a s o l u t i o n o f o
89 equation (1.2). Thus,under the condition (1.3), for obtaining a solution of equation (l.2),it is sufficient to find an invertible solution of the monic equation (2.5). Many different techniques for solving monic equations of the type (2.5) may be found in[ 4 , 6 ,13 ,15 ]. If Xo is a solution of equation (1.2), then Ul(t)=exp(tXo)
satisfies A2U ~" +AIUi+
+AoUI=O. Now, we are looking for a solution U2(t) of equatioD (i.i) of the form U2(t)=exp(tXo)U(t) where U(t) i s a new unknown. Taking d e r i v a t i v e s i t
U2(t)=exp(tXo)(XoU+U')
;
(2.6) f o l l o w s that
U~'(t)=exp(tXo)(X2U+2Xo oU'+U'')
If we impose that U2(t) be a solution of equation (l.1), then the matrix function U must satisfy A2exp(tXo)U''+(2A2Xo+Al)exp(tXo)U'=F(t)
(2.7)
Now, let us consider the change V(t)=exp(t(X -I I))U'(t) o o
(2.8)
Equation (2.7) may be expressed in terms of the variable V in the form A2V'+(A2Xo+A2lo+AI)V=G(t);
G(t)=exp(- lot)F(t)
(2.9)
Note that the algebraic matrix equation (1.2) may be written in the form (A2Xo+~oA2+AI)(Xo-~oI)+P(
~)=0
(2.10)
From the invertibility of P(~ ) and from (2.10) one gets that A2Xo+~oA2+A I is invero tible. If we denote by A the matrix A= (A2Xo+A 2 Xo+A I )-IA2=-Z ~ [ P (X o )] -IA2
(2. I i )
then from theorem 3.1.3 of [2 ] ,if F(t) is a k-times continuously differentiable matrix function where k is the index of the matrix A given by (2.11), it follows that for any matrix QC~nxn, the following expression defines the general solution of (2.9) t ^D^ V(t)=exp(-ADt)AADQ + Ioexp((s-t)AD)A G(s)ds + Vl(t)
(2.12)
where G(t)=-Zo[P(~o)]-iG(t)
; VI(t)=(I-AA
Dk-I ) Z (-1)i(A)iG(i)(t) i=o
From (2.8), the following expression defines solutions of equation (2.7) t U(t)= I exp(-s(X o-~o I))exp(-sAD)ds AADQ + o
(2.13)
90
+
itI s
k-I exp(-S(Xo- koI) )exp ((u-s)AD)AG(u) duds + ~ )OJ0 i=o
(s)ds
From (2.6),(2.12) and (2.14),it follows that U2(t)=exp(tXo)U(t),
(2.14)
is a solution of
(i.i) and U~(t):Xoexp(tXo)U(t)+exp(t ~I)V(t):XoU2(t)+exp(t ~I)V(t)
(2.15)
As Ul(t)=exp(tX o) is a solution of the homogeneous equation (i.i) with F=O,it follows that for any matrices P,Q in Cnxn' the expression X(t) given by X(t)=exp(tXo)P+( I~exp(Xo(t_s))ex ~ ( ~ I-AD)s)ds)AADQ+w(t) ^^ o
(2.16)
where ^ ^ W(t)=exp(tX o)it is (-s(X - koI))exp(AD(u-s))ADG(u)duds + 0 0 exp o k-i
(2.17)
t
+ i:oZ exp(tXo)
]oeXp(-s(Xo~o~))(z-XXD)(X)iS(i)(s)ds
Thus the following result has been proved: THEOREM i.
Let us assume that there exists a solution X
of equation (l.2),and o ^ that for some complex number l the condition (1.3) is satisfied. Let A be defined o ^ by (2.11) where Zo=Xo- ~I, and let k=Ind(A). If F(t) is k-times continuously differentiable matrix function and P,Q are arbitrary matrices in Cnxn,then the function X(t) given by (2.16)-(2.17) is a solution of equation (l.1). Now we are interested in the Cauchy problem (2.1). Under the conditions of the above theorem, the problem (2.1) is solvable, if there exist matrices P,Q, such that X(t),given by (2.16)-(2.17)
satisfies X(O)=C
o
and X'(O)=C 1
"
Note that from (2.15)-
(2.17),it follows that
X'(t)=Xoexp(tXo)P+XoU2(t)+exp(t XoI)V(t)=XoX(t)+exp(tkoI)V(t)
(2.18)
where V(t) is given by (2.12). In order to satisfy the Cauchy conditions of (2.1), the matrices P,Q, appearing in the expression of X(t), given by (2.16) must satisfy X(O)=P=C ~
and X'(O)=XoX(O)+V(O)=XoP+AADQ+R=CI
where
Hence, P=C
o
^^D k-I ..... R=(I-AA ) Z (-l)l(A)iG(1)(O) i=o
(2.19)
AADQ=c1-XoCo-R
(2.20)
and
^^ From theorem 2.3.2 of [ 17 ] ,and the fact that S=AA D is a g-inverse of itself, one gets that (2.20) is compatible if and only if, AAD(cI-XoCo-R)=CI-XoCo-R
(2.21)
91 and
in this case, a solution of (2.20) is given by Q:CI-XoCo-R, where R is defined by
(2.19). Taking into account that under the regularity (2.1) is solvable, then the solution is unique,
hypothesis
(1.3), if problem
see theorem 3.1.3 of [ 2 ] .The fo-
llowing result has been proved: COROLLARY i.
Let us consider the hypotheses
and the notation of theorem 1 and let R
be the matrix defined by (2.19). then the Cauchy problem condition
(2.21) is satisfied.
In this case the problem
(2.1) is solvable if the (2.1) has only one solution
given by t X(t)=exp(tXo)Co+( IOexp(Xo(t-s))exp((xoI-AD)s)ds)(CI-XoCo)+
W(t)
(2.22)
where W(t) is given by (2.17). REMARK 1.
^ Note that for the monic case A2=I, we have that the matrix A defined by
(2.11) is nonsingular and AAD=I. If ~ is any complex number that is not an eigeno value of the companion matrix C , then XoI-AD= XoI+P( Xo)Zol=-(Al+(Zo + XoI))=-(AI+X 0). Also, if F(t)=O, then the matrix R given by (2.19) is the null matrix and W(t)=O. Thus the corollary i generalizes
the lemma i of [ii] for the monic finite-dimen-
sional case. As we quoted above, thesolution
X of equation (1.2) may be obtained o in terms of an invertible solution of the monic equation (2.5). An effective method to obtain an exact solution of a monic equation of the type
X2+BIX+Bo=O
(2.23)
where B. ~ for i=1,2, is stated as follows: l nxn ' THEOREM 2. ( [ 14 ,p.49] , I 4
,p.836 ]).
If the matrix polynomial L(z)=z21+BlZ+Bo, Vectors, bll.... . ,bn, corresponding then X=QDQ
has n linearly independent right latent
to the latent roots
Zl,Z2,..,Zn,
is a solution of (2.23), where Q=[bl,...,bn]
i.e., L(z )b.=Ol 1 '
and D=diag(zl,...,Zn).
Other interesting iterative method for obtaining solutions of (2.23) is proposed in [ 6 ] . Further generalizations
of the above theorem 2 may be found in [13 ] .
3. BOUNDARY VALUE PROBLEMS Let us consider the boundary value problem
(1.4) and the hypotheses and the no-
tation of theorem I. The problem (1.4) is solvable if there exist matrices P,Q in gnxn' such that X(t) given by (2.16)-(2.17), satisfies of (1.4). These boundary value conditions
take the form
b EIP+FI[exp(bXo)P+([oeXp(Xo (b-s)lexp((XoI'AD)s)ds)AADQ+ ^^
the boundary value conditions
W(b)]=O
b
E2(XoP+AADQ+R)+F2(X ~ [ exp(bXo )P + !oeXp(Xo (b-s)lexp((koI-ADlldsXXDQI]
+F2(XoW(b)+exp(b Xo)V(b))=O Let W
i
be the matrix
(3.1)
92 EXAMPLE I.
Let us consider the Cauchy problem in g2x2
X"+
1
X" +
0
X=
exp(-t)
t J
[~0~]
x(~
equation
(1.2)
[o ~]
x 0
'
(3.9)
i
o
o n e ge~':.~
0
-2 ]
P(-1):A2-AI+Ao =
invertible -i
-2
Computing the matrices and functions appearing in theorem l, it follows that
k=l,
A=
:
{_~ o] ~!_~ oI ~
0
'
2
0
'
[lexp
._ {C
AAD= -
o] 0
~ R:<:_;,;,D>~:ro
~
o
]
[1 0
In this case we have
c xov~=[o 0] and thus the condition (2.21) is satisfied and in accordance with corollary i and the expression (2.22), the unique solution of problem (3.9) is given by
X(t)=exp(tXo)Co+W(t)=
where K(t)=
kll(t)
k12 (t) ]
k21(t)
k22(t)
k11(t):5/2-t-t2/2-(3+t)exp(-t)/2 ;
k21:-I/6-exp(-t)/2§
kl2(t)=(t3+2+6t2+6t)/3+(l+t/2)exp(-t)-3texp(t)/2-5/3 k22(t)=l+(exp(t)-exp(3t))/2+(2/9)(exp(t)+exp(4t)(3t/4-1))+t2/2 Analogous examples considering boundary value problems of the type (1.4) may oe given without difficulties.
93
I
bexp((s-t)AD)ADG(s)ds
+ (I-AA D)
wl-- o
k-i E (-A)iG(i)(b) i=o
(3.2)
then it follows that
V(b):WI+exp(-ADb)AADg
(3.3)
Let T=(T..) ij be the block partitioned matrix defined for l~i,j~2, TII=EI+Flexp(bXo ) ;
T21=E2Xo+F2Xoexp(bXo);
by the matrices
Tl2=FiI~exp(~(b-s))exp((kol~)s) ~ D
(3.4) T22:tF2•
s))exp(
;O)b +E2 I ;;0
;O)s)ds+F2explI
and let R 1 and R 2 be the matrices defined by the expressions
RI=-F1W(b) ; where R is defined
R2=-E2R-F2XoW(b)-F2exp(bko)W 1
by ( 2 . 1 9 ) , W 1 by ( 3 . 3 )
and W(b)by
With the above notation, the algebraic system
From t h e o r e m 2 . 3 . 2 titioned
matrix
of [17],
system (3.6)
T satisfies
case, the general
where I2n denotes
i f and o n l y i f , t h e b l o c k p a r -
R1
solution
= g
the form
the condition RI
and in this
(2.17).
(3.1) t ~ e s
is solvable,
(3.5)
§
of
(3.6)
-T+T)Y ;
takes
t h e form
Ye~2nxn
(3.8)
R2 the identity
matrix
in ~2nx2n and Y is ~
arbitrary
2nxn m a t r i x .
Thus the following result has been established: THEOREM 3.
Let us consider the hypotheses
and the notation of theorem 1 and let T=
(Tij) defined by (3.4), R defined by (2.19) and R i, for i=1,2, defined by (3.5). If the condition
(3.7) is satisfied,
then problem
taking matrices P,Q given by (3.8),
(1.4) is solvable and in this case,
the matrix function X(t) defined by (2.16),(2.17)
is a solution of problem (1.4). Now we will show with an example related to the Cauchy problem (2ol) that the compatibility condition of problem
(2.21) may be checked in an easy way and that the solution X(t)
(2.1) given by corollary 1 may be efficiently computed.
94 ACKNOWLEDGEMENTS.
The work in this paper has been partially [~upported by a grant
from the Direcci6n General de Investigaci6n Cientlfica y T~cnica, project PS87-0064.
REFERENCES I. M.J. BALAS, Trends in Large Space Structures: Fondest Hopes, Wildest Dreams, IEEE Trans. Aut. Control Voi. AC-27(1962),522-535. 2. S.L. CAMPBELL, Singular Systems of Differential Equations,
Pitman Pub. Co.,1980.
3. S.L. CAMPBELL and C.D. MEYER, JR., Generalized Inverses of Linear Transformations, Pitman Pub. Co.,1979. 4. J.E. DENNIS, J.F. TRAUB and R.P. WEBER, The Algebraic Theory of Matrix Polynomials, SIAM J. Numer. Anal., 13(1976),831-845. 5. R.C. DOHF, Modern Control Systems, Second. Ed., Addison-Wesley,
Reading 1974.
6. J.E. EISENFELD, Operator Equations and Nonlinear Eigenparameter Problems, J. Funct. Anal., 12(1973),475-490. 7. F.R. GANTMACHER,
The Theory of Matrices, Vol. II, Chelsea Pub. Co.,1960.
8. I.C. GOHBERG, P. LANCASTER and L. RODMAN, Matrix Polynomials,
Academic Press, 1982.
9. J.W. HOOKER and C.E. LANGENHOP, On Regular Systems of Linear Differential Equations with Constant Coefficients, Rocky Mount. J. Maths., Vol.12 No.4(1982),591-614. i0. L. JODAR, Explicit Expressions for Sturm-Liouville Operator Problems, Proc. Edinburg Math. Soe.,3C(1987),301-309. ii. L. JODAR, Boundary Value Problems for Second Order Operator Differential Equations, Linear Algebra Appl., 83(1986),29-38. 12. L. JODAR, Explicit Solutions for Second Order Operator Differential Equations with Two Boundary Value Conditions, Linear Algebra Appl., 103(1988),73-86. 13. L. JODAR and E. NAVARRC, On Complete Set of Solutions for Polynomial Matrix Equations, Appl. Maths. Letters, i(1990),i-9. 14. P. LANCASTER,
Lambda Matrices and Vibrating Systems, Pergamon, Oxford,
1966.
15. F.A. POTRA, On a Class of Iterative Procedures for Solving Nonlinear Equations in Banach Spaces, Banach Center Pub., Vol.13, PWN-Polish Pub., Warsaw, 1984. 16. C.B. MO[.ER, MATLAB User's Guide, Technical Report CS81-I(1980), Department of Computer Science, University of New Mexico, Alburquerque, New Mexico 87131. 17. C.R. RAO and S.K. MITRA, Generalized Inverse of Matrices and its Applications, John Wiley, 1971.
MATRIX SPECTRUM DICHOTOMY AND GENERALIZED LYAPUNOV MATRIX EQUATION A i d e r Ya. BULGAKOV I n s t i t u t e of Mathematics, S i b e r i a n Division o f t h e USSR Academy of Sciences, U n i v e r s i t e t s k y pr. 4, N o v o s i b i r s k - 9 0 , USSR
The
Problem
computation
of
of
computation
of
maximal
respective eigenvalues of
invariant
subspaces
non-self-adjoint
matrices
omitting
A is an
the
urgent
p r o b l e m of c o m p u t i n g m a t h e m a t i c s .
In t h e r e p o r t , t h e r e is given a s y s t e m o f m a t r i x
equations
including
matrix
subspaces
as
a
symmetrical
unknown.
In p a r t i c u l a r
H and
case,
the
when
we
s y s t e m is e q u i v a l e n t to a known Lyapunov m a t r i x p r o p o s e d s y s t e m is closed, and t h a t of
projectors
computation,
the
projectors deal
with
equation.
onto
a
sought
Hurwitz
It t u r n e d out
for
A, t h i s that
the
it can be u s e d in o r d e r to e s t i m a t e the e r r o r
algorithm
of
its
solution
including
the
parallel
c o m p u t a t i o n o f the c o n d i t i o n n u m b e r o f t h i s s y s t e m is given.
The
appearance
problems, seems
of
computers
not
only
enabled
to
but also c o n t r i b u t e d to c h a n g i n g o f s t a t e m e n t s
from
the
first
sight,cases.
So,
if e a r l i e r
the
solve
many
difficult
even in e l e m e n t a r y , as it
solution of
the
linear
system
Ax = f s e e m e d to c o n s i s t only in t h e d e t e r m i n a t i o n o f the c o m p o n e n t s x(i) of t h e v e c t o r x, a f t e r were
t h e w o r k s by J.von Neuman and H.H.Goldstine (1) and A.M.Turing (2)
written,
it
t~(A)=llA-lll'llAtl.
became
By t h e
necessary
parameter
accurately
s e t t i n g A and f) one can
which
be
can
problems
obtained
being
on
a
to
determine
/~(A) and
by
estimate
computer
or
ill-conditioned (tl(A)>-tt ~
a
else /z ~ -
also
the
the
accuracy
condition of
input
possible accuracy give is
up
of
some t h r e s h o l d
condition
number,
i.e.
number the
very
error
important
of closure of
is
the
the
value
IIAx - fll
approximated
due
the
connected
together with
connected
solution x
(by
solution to
value
w i t h t h e n u m b e r o f d i g i t s in a ceil o f t h e c o m p u t e r used). T h e r e a t , a
the
computation
number data
(or
with
this
simply,
the
e r r o r o f c l o s u r e ) , It is t h i s p a i r w h i c h g u a r a n t e e s t h e e s t i m a t i o n : fix - xll/lIxll ~ /a(A)IIAx - fll/llfll. Not too simple is the question: and as t h e e r r o r try
to
what
must
be c o n s i d e r e d as c o n d i t i o n n u m b e r
o f c l o s u r e in o t h e r c o m p u t a t i o n p r o b l e m s .
i n d i c a t e such notions f o r t h e c l a s s i c a l p r o b l e m -
In t h i s r e p o r t
we will
the p r o b l e m on d i c h o t o m y
of a m a t r i x s p e c t r u m by a i m a g i n a r y a x i s , i m a g i n a r y in its ample m e a n i n g , when it
96 is n e c e s s a r y to m a k e s u r e t h a t the m a t r i x axis
(such
maximal
matrices
invariant
are
called
subspaces
A h a s no e i g e n v a l u e s on t h e a n i m a g i n a r y
exponent~ally
L § (A)
and
dichotomic),
L (A),
but
corresponding
also
to
the
to
indicate
parts
of
the
s p e c t r u m lying in t h e l e f t and r i g h t h a l f - p l a n e , r e s p e c t i v e l y .
Thus,
for
separation
of
example, a
in
maximal
the
theory
invariant
of
automatical
regulation,
L § (A) p l a y s
a
subspace
very
theproblem
impotent
of
role.
If
a r i s e s w h e n c o n s t r u c t i n g a n o p t i m a l c o n t r o l u(t), w h i c h m i n i m i z e s t h e v a l u e CO
S o
( llCx(t)ll2+
in t h e s o l u t i o n s of t h e s y s t e m d x / d t
llBu(t)ll z) d t
: Ax. It is w e l l - k n o w n
that
to c o n s t r u c t
such
u(t), a s u b s p a c e L,(W) f o r t h e m a t r i x
[ ;.]
W=
A
-C'C
is n e c e s s a r y to be found.
It would s e e m q u i t e n a t u r a l
to solve it by f i r s t
computing the eigenvalues
W
lying in t h e l e f t h a l f - p l a n e a n d t h e n by c o m p u t i n g e i g e n - a n d p r i n c i p a l v e c t o r s .
It c a n be e a s i l y s h o w n t h a t such a s c h e m e if r e a l i z e d on a c o m p u t e r c a n p r o v e non s o l v a b l e . eigenvalues dealing
The
is
with
thing
not
the
is t h a t one
a computer
the
which
problem
can
we m u s t
matrix
Ao,
where of
close
to
it.
We know
computation
be c o n s i d e r e d
consider
owing to w h i c h we c a n only g u a r a n t e e ,
of
its
the
computer.
such
Thereby,
about
instead
det(A-Al)=O we deal
for
which
there
with exist
of
this
co-called
matrix
eigenvalues
a spot a
correctly
A general
stated.
"indefiniteness
In f a c t , principle",
t h a t i n s t e a d o f a m a t r i x A we h a v e a c e r t a i n
c is a s m a l l p o s i t i v e n u m b e r c h a r a c t e r i z i n g
equation
as
of m a t r i x '
of
that
IIA - Aoll -< cllAoll,
the number of
its
matrix
only
the
of d i g i t s
matrix
"c-spectrum", B(A),
so
in a cell
A satisfying i.e.
that
with
the
a set
of
IIB(A)II < r
d e t ( A + B(X) - M)=O.
It spots means
of
is
well
known:
"e-spectrum" that
the
conditioned.
This
if
self-adjoint
possesses o f
problem is
A is
not
of the
matrix
a diameter
eigenvalues case w i t h
non
of
which the
of
(A=A'),
does n o t
each
of
e x c e e d cllAll,
self-adjoint
self-adjoint
then
matrices.
matrices There
its
which
is
well
exist
the
97 e x a m p l e s w h e n t h e s p o t s of t h e " c - s p e c t r u m "
can have rather
c a n lead to t h e v a g u e n e s s or t h e d i m e n s i o n s of s u b s p a c e s typical
example
can
be
taken.A
two-parameter
family
large diameters which
L § (A) a n d L (A). Here, of
matrices
A (t)
of
a the
twenty first order: -20
-20.5L
0
A
b)
-19
-19.5L
(t)= 0
-2
0
-2.5L
0
0
-i
0
0
0
0
i
has a characteristic equation 20
20
I IC-j-k) -
(l-~)(
19
2
5 ~t
I--I (-j-o.s}) = o. J:t
J=l Let E)
=
1.5/-19
2tz(A~(l))=0.5
Ptt(A~(t))=l, dimension
( a t l=4,
of
L (A-(L)) --
l e t ~ = 3 2 -39 = 3 . 6 c - I 0 , are
is not
eigenvalues less
than
of 2,
at
l=10 l e t E) = 1.5e-19) t h e n
the
matrix
A-(1),~
whereas
the
dimension
consequently of
'.~
-
e q u a l to 1. It is to be n o t e d t h a t
the
L (A (l))
is
0
A-(l)o
do
of
the
determine
the
A, F a d d e e v D.K.
and
t h e e l e m e n t s of t h e m a t r i c e s Ao(l) a n d
n o t d i f f e r m o r e t h a n by t h e value ~ ( a t t = i0, ~ = 1.5e-19).
The
existence
eigenvalues quality
of
problem"
such for
examples
arbitrary
of s t a b i l i t y of a s e p a r a t e
is
connected
with
non-self-adjoint
matrices.
e i g e n v a l u e 2, of t h e m a t r i x
F a d d e e v a V.N. (4) p r o p o s e d to use one n u m e r i c a l p a r a m e t e r b e t w e e n t h e e i g e n v e c t o r s of m a t r i c e s this
parameter
is too
large,
"the
instability To
connected with the angle
A a n d A" c o r r e s p o n d i n g to }t. If t h e v a l u e o f
the problem
of
the
determination
of t h i s
eigenvalues
is unstable. Unfortunately, this parameter is not used as a measure of stability of eigenvalues
in the concrete corriputations.
Probably,
it
is connected with the
fact, that no efficient algorithm to compute this parameter was advanced.
There
exist
ill-conditioned, conditioned.
the
some
matrices
subspaces
L + (A)
A
whose and
separate
L (A)
eigenvalues
corresponding
to
though them
are
being well
98 The p r e s e n c e of t h e like e x a m p l e s compute
the
bases
preliminarily
computed
(Kublanovskaya adjusted,
V.N.)
and
Balzer)
of
a
was
sign-functions
methods
subspaces.
all
necessary
in
of
certain
and
the
special
worked
In
invariant
led t o t h e d e v e l o p m e n t of a l g o r i t h m s
subspaces eigenvalues.
It
are
applications.
be
destined
In
method
must
likelihood,
For
it
is
this
the
in
compute
the
first
the
A.A,
QR-algorithm were
Kublanovskaya
V.N.,
that
on
having
V.V.)
theoretically
maximal
and
without
(Voevodin
mind
information
case,
purpose,
(Abramov
to
L (A)
method
born
only
and
this
orthogonal-power
sign-functions
out.
L + (A)
which
these
invariant subspaces
foremost
matrix that
problem
the
is
becomes
their stable computation.
Before
separating
non-self-adjoint
matrices
the
projectors
onto
let
us c o n s i d e r
maximal
a more
invariant
particular
subspaces
and important
of
Hurwitz
p r o b l e m , h e r e a s p e c i a l a t t e n t i o n m u s t be p a i d to t h e L y a p u n o v m a t r i x e q u a t i o n .
As a c o n d i t i o n n u m b e r of t h e H u r w i t z p r o b l e m i t w a s s u g g e s t e d in w o r k (5) t o use
a
stability
quality
parameter
~:(A).
K(A)=2-11AII-IIHII, if t h e Lyapunov m a t r i x matrix
of
dimensions
determined
H,
NxN),
otherwise
has
~(A)=o,.
s o l u t i o n s of t h e s y s t e m d x / d t
Its
algebraic
expression
looks
as:
e q u a t i o n A*H + HA + I = 0, (I b e i n g s i n g l e
only
one
This
solution
parameter
is
in
the
directly
class
of
positively
connected
with
the
= Ax by t h e e q u a l i t y (6): T
It is obvious, dx/dt
that
tlx(t)ll 2 dt ))
K(A) = 211All (Sup ( m a x T->O IIx ( 0 ) I1=1
J" o
t h e less is ~(A) t h e m o r e
rough
is t h e
stability of the
system
= Ax w i t h r e s p e c t t o t h e d i s t u r b a n c e s o f t h e e l e m e n t s in t h e m a t r i x A.
As
an
error
of
closure
in
the
numerical
solution
of
the
Hurwitz
closure
of
the
Lyapunov
problem
o
IIA X + XA + Ill
is
used
equation.
is
the
True
which
is
theorem
the on
error the
of
error
of
closure
(7):
if
matrix
X=X" > O,
= IIA'X + XA + Ill < 1 t h e n t h e H u r w i t z ' A a n d
IIH-XJI/IIHII ~ A,
It Hurwitz
is
the
problem
theorem
which
Im(A)-2-11AIJ.IIXIIJ/~(A) "< /~.
allowed
(6) up to t h e a l g o r i t h m
to
close
the
algorithm
with a guaranteed
a n d d e s c r i b e d in i t s f i n i t e f o r m in w o r k (7).
of
solution
accuracy,
of
the
substantiated
99 Here
we
errors
in t h e
played
a
like
to
note
that
computation algorithm of
very
computation equation
would
important
algorithm
role
of
in
as
detailed
analysis
of
the
matrix
function
exp(tA) a t
estimating
positively
A ' H + HA + I = 0,
a
it
the
determined was
shown
influence
solutions in
(8).
the
of
of
computational
these
the
Thereat
Hurwitz A
errors
Lyapunov
in
a
matrix
estimation
of
the
m a t r i x e x p o n e n t i a l exp(tA) w a s amply m a d e use of:
Ilexp(tA)l] -< v/K(A)'.exp(-tllAII/~(A)),
t>--0.
Now, let us p a s s over to t h e s e p a r a t i o n o f m a x i m a l
invariant subspaces of a
non-self-adjoint matrix.
If
A is
the
exponentially dichotomic,
then
for
the
sought
for
projector
G
o v e r L + (A) a w e l l - k n o w n i n t e g r a l p r e s e n t a t i o n 00
1 G = ~-~ S (izI - A ) - l d z -CO
is valid. The value of the n o r m IIH;I o f t h e s y m m e t r i c a l m a t r i x 1 co H = ~-~ J" (A ~ + i z I ) - ! ( A - i z I ) - i d z
(1)
-00
can
be
there If
used
exists
as
a
only
dichotomy one
numerical
eigenvalue,
IIHII is n o t too large,
characteristic.
then
IIHll = ~,
If
since
the
i t can be e a s i l y o b t a i n e d t h a t
and H c h a n g e v e r y l i t t l e a t
small v a r i a t i o n s
of
A for
on
the
imaginary
integral
(1)
no
axis sense.
estimation showing that
G
w h i c h t h e p r o j e c t o r G and
t h e m a t r i x H have been built. This c o n t i n u i t y is c o n n e c t e d : i) w i t h t h e g u a r a n t e e of c o r r e c t n e s s o f t h e c o m p u t i n g p r o c e s s ; 2) w i t h t h e n e c e s s i t y t h a t t h e s o l u t i o n a d m i t o f a p r a c t i c a l u s a g e in p r o b l e m s .
It
is
interesting
to
note
that
if
A is
exponentially dichotomic,
then
for
H
f r o m (1) t h e p r e s e n t a t i o n C0
H = J" G ' ( t ) G(t) d t -00
is
valid.
Here
G(t)
is
d / d t ( G ( t ) ) = AG(t) + 6(t) determined
the
the
parameter
~(A) = 2.1IAII.IIHII.
Green
bounded of
matrix at
the
all
determined t.
Using
exponential
as
this
the
solution
equation,
dichotomy
of
of
equation
S.K.Godunov
(9)
the
A:
matrix
100 So, S.K.Oodunov proposed to
simultaneously compute the matrix
H and the
p r o j e c t o r O on L § (A) while s t u d y i n g t h e e x p o n e n t i a l d i c h o t o m y of A. In (10) i t
is s h o w n t h a t
to e s t i m a t e
matrices
G and
Ft, a s y s t e m
of
matrix
e q u a t i o n s c a n be used Gz-
G = O, GA - AG = 0, ~
HA
+
A
(GH) ~
~
H
+
GH = 0, (2)
9
G CG
w h i c h g e n e r a l i z e s t h e Lyapunov m a t r i x
(I-G)
-
C(I-G)
= 0
e q u a t i o n f o r t h e c a s e of t h e n o n - H u r w i t z A.
We h a v e p r o v e d t h e f o l l o w i n g t h e o r e m .
THEOREM: a) i f f o r some C=C'> 0 t h e r e e x i s t m a t r i c e s G a n d H(C)=(H(C)) ~ > 0 satisfying
system
if a m a t r i x
(2),
A has
then matrix
A h a s no e i g e n v a l u e s
no e i g e n v a l u e s
on t h e
imaginary
on t h e i m a g i n a r y
axis,
then
system
~
axis;
(2)
at
b) any
9
C=C > 0 h a s t h e only one s o l u t i o n w i t h r e s p e c t to G a n d H(C)=(H(C)) > 0 t h e r e a t 0O
G = l / ( 2 n i ) s (zI-A)- ldz;
where
~- is
eigenvalues
a A
closed
contour
having
located
strictly
in
theorem
presents
a
equation.
In t h e
same
error
of c l o s u r e
the algorithm control
and
the
computations
determination
of
of
either
correct
of
left
a
half-plane.
known
problem,
theorem
is o b t a i n e d ,
occurring
projectors
error
of c l o s u r e
In (11) t h e r e errors
over
the
was
exercise
assertion
in t h e
carried
clear
is the
to
important theorem
L (A)
and
based
(9). T h e
that
role
of
of
work
~:(A)>~:~
i.e.
that
matrix on t h e
in t h e in for
projectors
the
over
analysis
destined variant
of
for
the
of
the
in t h i s a l g o r i t h m
H a n d t h e v a l u e ~(A) w i t h t h e that
this
computations
a complete
on a
the
that
is u s e d
results
out
all
Lyapunov
in one a l g o r i t h m
by S.K.Godunov
G, m a t r i x
It
in t h e c o m p u t a t i o n s
L § (A) a n d
method suggested
in t h e p r o j e c t o r
contains
analogous
of
the
which
on
which plays an v a l u e ~(A). T h i s
subspaces.
or
the
self-crossings
G, H a n d t h e
computational
signs,
no
(lO) a t h e o r e m
of t h e e r r o r
the
orthogonal-power results
work
of matrices
invariant
influence
generalization
in t h e H u r w i t z
and play the role maximal
H = l / ( 2 ; r ) J ' ( A ' + i z l ) - l C ( A - i z I ) - 1dz
there
indication exist
no
" p r a c t i c a l " d i c h o t o m y of t h e s p e c t r u m A by t h e i m a g i n a r y a x i s .
L e t us p r e s e n t
one of t h e p o s s i b l e a l g o r i t h m s
of G a n d H s u g g e s t e d by A.N.Malyshev (12).
of a simultaaleous d e t e r m i n a t i o n
101 i) W i t h t h e h e l p o f t h e T a y l o r s e r i e s
A = exp(xA'),
B = -I,
0
and using
the
Choletsky
LL 9 =~. T h e r e a f t e r ,
2). AdBI;
We
Az, B2; ...;
a
we find
recurrent
by m e a n s
I
Aj_IO
0
X - -
m
A
)-ill = ~
approximations inequalities
) C,
m
invertability m
m
- B
= r to
O
~.
L by d e c o m p o s i n g
procedure
of
which
the
is r e d u c e d
transformation
0
A
matrix
pairs
at
every
P, s u c h a s
B
J
J
m --~
B*(A* - B') -t
m
determination
]I
" 9• !
Bj_I
It turns out that at
II(A - B
the
•
]=
to
Thereat
of
triangular
B o= L-XBo = L -I
to t h e c h o i c e o f t h e o r t h o g o n a l
P[Bj-tAj-IO
m
a lower
A 0 = L-IAo ,
successive
Are,Bin; ...
dt
0
decomposition
out
step of the recursion
e) = 211All f e x p ( t A * ) . C . e x p ( t A )
0
we s u p p o s e t h a t
carry
a t 9 = 1/(211AII)
G,
n~
(A - B )-ICA* - B* )-I m
at
m
rather
IIAII.IIHII a n d
"f2-'-IIAII.H
is
rn
large the
estimated
) 2UAII.H.
m
m
follows
velocity
of
beginning
from the
the
fact
convergency
from
some
m
that of
the
by
the
of the form:
II El" (A" - B * )-i in m tn
II (A - B rn
rn
G II :i C o n s t V/KCA)' exp(-2m/~c(A)),
)(A -B ) - 2 - t l A I I . H Ill
EP.
II -<- C o n s t KZ(A) e x p ( - 2 m / K ( A ) ) .
The now described algorithm presents an improved well-known method of the matrix-sign-function, and allowing to obtain the same accurate approximations to the projectors G, it also makes possible to find with minimal additional computing efforts
the approximations to H, and consequently, the parameter characterizing
the quality of the dichotomy ~:(A) = 2.11AII-IIHI]. Utilizing the dichotomy quality parameter and the error of closure of the Lyapunov equations generalized system, the said algorithm allows to write down the algorithm of a guaranteed accuracy in order to solve the problems of computation of maximal invariant subspaces. The result of this algorithm wold be either inequality ~(A)>-~:" providing a "actual"
102 absence
of the dichotomy of the m a t r i x
computation
of
the
matrices
G,
spectrum
H and
the
by the
value
imaginary
~:(A) with
axis,
indicated
or
the
the true
signs.
BIBLIOGRAPHY:
1. T u r i n g A.M. Rounding-off e r r o r s in m a t r i x processes. Quart.
J. Mech., 1, 1948,
287-308. 2. J.von Neuman and H.H.Goldstine. Numerical inverting of matrices of high order. - B u l l . Amer. Math. Soc., 1947, v.S3, no.ll, 1021-1099. 3.
Guaranteed
accuracy
Godunov S.K.,
for
the solving of
Antonov A.G.,
linear
K i r i l j u k O.P.
systems
and
in Euclidean
Kostin V.I.
-
spaces./
Novosibirsk:
"Nauka", 1988. (Russian). 4.
F ad d eev
D.K.
and
Faddeeva
V.N. Computational
methods
of
linear
algebra.-
Gos.lzdat.Fiz.Mat.Lit., Moscow, 1963. (Russian). 5. Bulgakov A.Ya. - Sibirsk.
Math. Zh.21(1980),
No 3, (32-41); English Transl.
in
Siberian Math. J. 21 (1980). 6.
S.K.Godounov
(Godunov)
calculatives dans calculatif (Proc.
du
Fifth
A.J.Boulgakov
(A.Ya.Bulgakov),
le probleme de Hurwitz et methodes a l e s
probleme Intcrnat.
Information
and
Sci.,
de
Hurwitz),
Conf.
vol.
Analysis
(Versailles,
44,
1982)),
and
surmonter
Optimization
Lecture
Springer-Verlag,
Difficultes
Notes
1982,pp.
(aspect
of
Systems
in Control
846-851.
and
(English
ab s t ract , p.845.) 7.
A.Ya.Bulgakov and Lyapunov's
S.K.Godunov,
equation,
Calculation
Computational
of
Methods
positive definite of
linear
solutions
algebra,
of
"Nauka",
Novosibirsk, 1985,pp. 17-38. (Russian). 8. Bulgakov A.Ya. Computation of the stable
matrix,
Computational
exponential
Methods of
linear
function of an asymptotically algebra,
"Nauka",
Novosibirsk,
1985, pp.4-17. (Russian). 9. Godunov S.K. - Sibirsk.Math. Zh. 27 (1986), no.5, pp.24-37. 10. Bulgakov A.Ya. - Sibirsk.Math. Zh. 30 (1989), no. 4, pp.30-39 11.
Bulgakov
A.Ya.
non-self-adjoint
Guaranteed
accuracy
matrix,Numerical
of
calculating
analysis,"Nauka",
or
invariaat
subspaces
Novosibirsk,
1989,pp.
12-93, (Russian). 12.
Malyshev
A.N.
Preprint
Novosibirsk, 1988 (Russian).
no.6,
Inst.Math.,
Siberian
Branch
Acad.
Sci.
USSR,
Discrete approximation of the minimal time function for systems with regular optimal trajectories
M. Bardi
M. Falcone
and
Dipartimento di Matematica P. e A. Universi~ di Padova via Belzoni 7 35131 Padova - Italy
Dipartimento di Matematica Unlversi~ di Roma "La Sapienza" P. Aldo Moro 2 00185 Roma - Italy
Abstract. In this paper we prove an estimate of the rate of convergence of the approximation scheme for the nonlinear minimum time problem presented in [2]. The estimate holds provided the system have time-optimal controls with bounded variation. This estimate is of order
v with respect to the
discretization step in time, if the minimal time function is H61der continuous of exponent v. The proof combines the convergence result obtained in [2] by PDE methods, with direct control-theoretic arguments.
1. I n t r o d u c t i o n . In this paper we continue the study of an approximation scheme for the classical minimum time problem for nonlinear systems which we began in [2]. We consider the continuous-time controlled dynamical system in IRN (1.1)
~ y' = b(y, (x) t y(0) = x
and the corresponding discrete-time system, with time step h > 0,
(I .2)
xj+ X0
= xj + h b(xj,aj) =
X
where the controls are taken in a given set A~IR M. For a given compact target set T we are interested in the minimum times T(x) and hNh(x) taken respectively by systems (1.1) and (1.2) to reach T , where Nh(x ) indicates the minimum number of discrete steps. Note that T is finite only on the set R of points controllable to the target T in finite time, that it tends to + ~ near D R and that R is not known a priori. We recall that the Dynamic Programming method provides time-optimal controIs in feedback form for the discrete-time problem, once the discrete Bellman equation for N h is solved. In [2] we considered the new unknown functions
104
(1.3)
v(x) :=
1 - e -T(x), if T(x)<+--,
L1 ,
if T(x)=+**,
Vh(X) : = f
1 -, e -hNh(x), if Nh(X)<+**,.. 1
if Nh ( x ' ) = + o * ,
and, under general controllability assumptions on both (1.1) and (1.2) around T , we proved that v h converge to v as h tends to 0, uniformly on compact subset of ]RN, which implies the convergence of hN h to T on compact subsets of R . This result was obtained by applying to the Bellman equations for the discrete-time and the continuous-time control problems a technique for passing to the limit in first order nonlinear equations first used by Barles-Perthame [5]. A crucial role in the proof is played by a comparison theorem between semicontinuous viscosity sub and supersolutions proved in [ 1]. In this paper we give an estimate of the rate of convergence of hN h to T using direct, controltheoretic methods. Roughly speaking, the main result says that if T is HOlder continuous of exponent v , for a compact subsets K of ~ there is a constant C such that (1.4)
Ih N h ( X ) - T ( x )
forany x ~ K ,
provided the system satisfies in K the property of the "bound on the variation of the optimal controls": (BV) there exists a constant C O such that, for all xe K , there is a time-optimal control with total variation less then C O , bringing system (1.1) to the target T. This property is well known for linear systems (see e.g. Hajek [16]), because the optimal control is bang-bang with a bound on the number of switchings. Many papers have been devoted recently to prove similar regularity properties for smooth systems, nonlinear in the state x and affine in the scalar control (x, in the case T = {0}. Sussman [31] has described a class of such systems which behave as linear systems. In general, however, the optimal control is known not to be bang-bang, and many authors have addressed the problem of proving that optimal trajectories are finite concatenations of bang and singular arcs, with a bound on the number of switchings. From these results one expects that (BV) hold, at least generically, if the state-space dimension N is small: see Sussman [32,33] for N=2, Bressan [6] and Sch~ittler [25,26] for N=3, Krener-Sch~ittler [19] and SchSttler [27] for some results in dimension 4. We recall also the negative results of Kupka [20] and Kawski [18] in higher dimensions. As regards the HOlder continuity of the minimal time function T , we refer to Liverovskii [21], Stefani [30] and the references therein for T = {0}, and to Bardi-Soravia [4] and Soravia [28] for more general targets. In a forthcoming paper [3] we treat the general case where no regularity of the optimal controls is known, by using a completely different method based on the theory of viscosity solutions and introduced by Souganidis [29] and Capuzzo Dolcetta-Ishii [10] for other control problems. This method leads to estimates worse than (1.4), namely of order v/2 with respect to h . In [I0] an estimate of order v is also proved, under additional assumptions different from ours, and by a different method. The reader will find in [ 1,2] a basic bibliography on the minimum time problem and on the theory of viscosity solutions for Hamilton-Jacobi equations. Our approach to numerical solutions of control
105
problems follows Capuzzo Dolcetta [7], Capuzzo Dolcetta-Ishii [9], Souganidis [29], Falcone [13,14], see also the survey paper by Capuzzo Dolcetm-Falcone [8] which traces its origin back to Bellman's time. The numerical solution of the minimum time problem has been studied by many authors by completely different methods and mostly in the linear case. See e.g. Falb-De Jong [12], 'Canon-CullumPolak [6], Neustadt [26], Eaton [10], Fujisawa-Yasuda [16], Hajek-Krabs [17], Rabinovich [27] and the references therein.
2. Definitions and preliminary results.
We assume that the set of adrm'ssible controls A is a subset of ~ M and we define the set A of admissible control functions appearing in (1.1) to be A := {to(.): [0,+,,0[ --~ A , measurable}. We will use a/so the following subset of contro/functions Ph := {cc(-)eA constant on [ih,(i+l)h[, V i e 1~}, for h>0. The general structural assumptions on b and T we need are the following b:]RNxA---~IR N is c o n t i n u o u s , l b ( x , a ) - b ( y , a ) ] < Llx-y I
(At)
and lb(y,a)l < K(l+lyl), V x , y e IR N and "v' ae A ; T i s compact ;
(A2)
there exists 7, 0<7~I, such that ]b(x,a)
- b(x,a')[ _
Let us denote y(x,t,ct) = y(t,o~) the solution of (1.1) and define ~h(X,t,cz) = ~h(t,cc):= x[t/hi for any oc(.)~ Ph where {xj} is the solution of (1.2) with aj = cc(jh), and [.] denotes the integer part. Using Gronwall Lemma it is easy to check that if y(t,cc)
_
V t ~ [0,s] , where 0c(.)a Pfi ,
then < K ( I + Y ) e Lth , V t ~ [ 0 , s ] .
We define the set R : = {x~ ~ N : there exists cc(.)~ A and t > 0 such that y(x,t,ct) ~ T}.
106 The time necessary to drive system (1.1) to the target T by applying the control function o~(t) is tx(C~) := inf {t : y(x,t,c~)~ T } < +,,0, where tx=+~, if y(x,-,(~) never reaches the target. The minimum time function is given by T(x):--
Notice that T : 5~---)[0,+~[. Obviously
inf tx(Ot). (x~ A
T~R
and T(x)=0 for any x~
A discrete version of the minimum time problem can be obtained replacing (1.1) with the Euler scheme (1.2) with step h, h > 0. The state xj of the discretized problem is then observed only at times tj = jh and aj ~ A for all j~ I~ (just to simplify notations xj and aj will sometimes denote also the whole sequences {xj} and {aj} ). We can define the analogous of R , tx(O0 and T(x) for this new problem, P~h := {x~ ]RN: there exists {aj} andj~ l~I such that xj ~ T}
nh(aj ,x):= min{j~ N :xj ~ T } < + ~ , where nh=+** if xj never reaches the target, Nh(X) := inf nh(aj ,x). {aj} Besides the previous assumptions, we need the small time local controllability of both the continuous-time and discrete-time systems, in a slightly stronger form than in [2]. Set d(x) := dist (x, 3 T ) ,
X 5 := {x : dist(x, ~X) < 5}.
We will assume for some 5 > 0 , 0 < 13, 4, 11 < I , (A3)
T(x) < C 1 d ~ (x) for all xE c ~ ,
(A4)
h Nh(X) < C 2 d~(x) + C 3 h1] for all x~ T5 .
In [2] we proved that a sufficient condition for both (A3) and (A4) to hold with J] = ~ = rl = C 3 = 1 is the following
107
T = { x : g i ( x ) _ < 0 ' v ' i = l .... M} where g i e C 2 ( l R N) and (A5)
IVgi(x)l > 0 for all x such that g i ( x ) = 0 ; V xe 3 T 3 a~ A such that gi(x)=0 i m p l i e s b(x,a).Vgi(x) < 0.
Hypothesis (A5) means that ~ T is piecewise C 2 and at each point of the boundary the controller can choose a vector field pointing inward T. A simple special case of (A5) is T is the closure of an open set with C 2 b o u n d a r y ; inf b(x, a)-n(x) < 0 for all xe ~ T ,
I
a~ A
where n(x) represents the outward normal to T a t x.
(A3) is essentially equivalent to the HSlder continuity with exponent 13 of the minimum time function T. Necessary and sufficient conditions for it have been studied e.g. by Petrov [23], Liverovskii [21], Stefani [30] for the case T={0} (see also the references~therein), and by the authors [2], Bardi-Soravia [4] and Soravia [28] for more general targets and l] = 1 or 1/2. Due to the discretization of the dynamics, a crucial role will be played by control functions in Ph' so it is interesting to approximate controls in A with controls belonging to T h. For a function o~ : m
I---fiRM we denote the total variation by V(a,I):--
~ V(c~i,I) , i=l,..,M where V(cLi,I) is the usual total variation of the i-th component of c~in the bounded interval I.
Lemma 2.1 I f ~ e A has 6ounded variation in [O,R], then for any h > 0 there e~-tsts an a h ~ Ph such that R
(2.2)
~
c~(s) - (Xh(S) ds < h V(a,[O,R])
q
0
Proof. Set Ik:=[kh, (k+l)h[ for k=0,1 ..... n , n such that (n+l)h < R < (n+2)h and In+l:=[(n+l)h,R]. We can define c~h in Ik as any value taken up by (x in the same interval, or for instance
ah(S):= lim a(t) forse Ikt--.->kh+ Then f Ik
~(s) _ O~h(S) ds < h V(a,I k) ,
and (2.2) follows easily. In the previous paper [2] we have proved the following result (the definition of v and v h is (1.3)):
108
Theorem 2,2 Assume (A1), (A3), (A4). Then v h converge to v uniformly on compact subsets of IRN. Corollary 2.3 Under the assumptions of Theorem 2~2 for any given compact subset K of R there exist -h and T, such that K~P,h
a n d hNh(x) <'~, V h < h , ' ~ ' x ~
K.
Proof. Since v h converges uniformly to v on K and v < 1 - • on K for some positive E, we have s V h < l - ~ , on K , for h < h, which gives the conclusion.
#
3. Estimates.
Lemma 3.1 Assume (AI), (A3), (A4) and let K be a compact subset of R . Then there exist two positive constants -h and C such that T(x) - hNh(x) < Ch 13 for any x ~ K
v h < h.
Proof. For x fixed in K we choose {aj} such that nh(a j, x) = Nh(X) and define a(s) = a[s/h ]. Let us choose "F as in Corollary 2.3. Then (AI) and Gronwall's Lemma imply (see Lemma 1 in [1]) the existence of a constant Y, depending only on K , such that y(x,t,(x) _.
K, t
Then (2.1) gives y(hNh(X , a ) - Yh(hNh(X), (x) ~ K ( I + Y ) e L T h =: C4h, 8 and we have, for h < - C 4'
y(hNh(X),
%.
Thus by (A3) T(x)
hl3 ,
#
Theorem 3.2 Assume (AI), (A2), (A4) and let xe R. Suppose there is a sequence a n in A with
109
tx(C~n) = T(x) such that
tim n---~+~
V(C~n,[0,T(x)+l]) < C O forall n.
Then there exist two positive constants -h and C, depending only on Ixl , an upper bound on T(x), and the constants appearing in the hypotheses, such that hNh(x) - T(x) < Ch v f o r off h < h ,
where v = min {7~, 11} . Proof. Let T be an upper bound on T(x). Since (A1) implies I y(x,t,o0-x I < (1+ l x I) (e K t l ) , we can choose a constant Y, depending only on txl, K and T such that ly(t,ct)l < Y, for all t < T(x)+l. Fix h > 0 and let e tne A be such that (3.1)
tx(C~n) - h < T(x).
By Lemma 2.1 there exist ~h = (an)he Ph such that T+I J
C~n(S)-~h(S)]ds
'v'n.
We have the following estimate t
y(t,Ct n) - y(t,~h) [ <: J
b(y(s,O~n(S)),CXn(S)) - b(y(s,ah(S)),~h(S)) ds _< t
ILly(S,an(S))-y(s,c~h(S))
ds ,
which implies ly(t,a n) - y(t,~h) I < L y CYo hYTI-1/Te Lt , V t < T + 1.
Combining this with (2.1) we get
(3.2)
y(t,Otn) - Yh(t,CCh) < C5h7, 'v' t < T + 1,
for a positive constant C 5 depending only on Ixl, T , C o, and the constants in (A1)(A2). Next (3.1), h
and obtain from (A4)
110 hNh(X) < T(x) + h + C 2 C ~ h ~'~ + C3 hrl ,
which concludes the proof.
#
The following result gives conditions under which the convergence of the discrete minimum time function to T(x) is of order 1 with respect to h.
Corollary 3.3 Assume (A1), (A2) with 7 = 1 and(A5). Suppose K is a compact subset of R where (BV) holds. Then there exist two positive constants "h and C such that I T(x) - hNh(X) I < Ch for arty x ~ K
V h <_-h.
Proof. By Lemma 5.1 in [2], (A5) implies (A3) with 13= 1 and the continuity o f T on the points of 0T. Then v is continuous in ]R N by the argument at the end of the proof of Thm. 3.3 of [2]. Since R={ x: v(x) <1 }, R is open. T h e n , T(x) _
V xe K.
By Lemma 4.1 in [2], (A5) implies also (A4) with ~ = 11 = C 3 = 1. Then the conclusion follows from (BV), Lemma 3.1 and Theorem 3.2.
#
Corollary 3.4 Under the hypothesis of Corollary 3.3, there exist two positive constants h and C such that ]v(x)-vh(x)]
Vx~
9( and h
Proof. We prove the estimate for V-Vh, the other being completely analogous. Without any loss of generality we consider the case T(x) > hNh(X ). We have
v(x) - Vh(X) = - e -0 (hNh(X) - T(x)) < e -T Ch where in the equality we used the mean value theorem, for some -3 ~ [hNh(X), T(x)].
#
111
References. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [ 17] [I 8] [19] [20] [21] [22] [23] [24]
M. Bardi, A boundary value problem for the minimum time function, SIAM I. Control Optim. 27, 1989, 776-785. M. Ba~'di, M. Falcone, An approximation scheme for the minimum time function, SIAM J. Control Optim. to appear. M. Bardi, M. Falcone, Numerical solution of the minimum time problem, forthcoming. M. Bardi, P. Soravia, On the minimum time problem for smooth targets, preprint Universita" di Padova, 1990. G. Barles, B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Mod61. Math. Anal. Num6r. 21, 1987, 557-579. A. Bressan, The generic local time-optimal stabilizing controls in dimension 3, SIAM J. Control Optim. 24, 1986, 177-190. M.D. Canon, C.D. Cullum, E. Polak, Theory of optimal control and mathematical programming, McGraw Hill, New York, 1970 I. Capuzzo Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl.Math. Optim. 10, 1983, 367-377. I. Capuzzo Dolcetta, M. Falcone, Viscosity solutions and discrete dynamic programming, Ann. Inst. H. Poincar6 Anal. non Lin. 6 (Supplement), 1989, 161-183. I. Capuzzo Dolcetta, H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim. 11, 1984, 161-181. J.H. Eaton, An iterative solution to time-optimal control, J. Math. Anal. Appl. 5, 1962, 329-344 P.L. Falb, J.L. de Jong, Some successive approximation methods in control and oscillation theory, Academic Press, New York, 1969 M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim. 15, 1987, 1-13. M. Falcone, Numerical solution of deterministic continuous control problems, Proceedings International Symposium on Numerical Analysis, Madrid, 1985 T. Fujisawa, Y. Yasuda, An iterative procedure for solving the time-optimal regulator problem, SIAM I. Control 5, 1967, 501-512. O. Hajek, Terminal manifolds and switching locus, Math. System Theory 6, 1973, 289-301. O. Hajek, W. Krabs, On a general method for solving time-optimal linear control problems, in Lecture Notes in Control and Information Science 38, Springer-Verlag 1982. M. Kawski, Control variations with an increasing number of switchings, Bull. Amer. Math. Soc. 18, 1988, 149-152. A.J. Krener, H. Sch~ittler, The structure of small-time reachable sets in low dimensions, SIAM J. Control Optim. 27, 1989, 120-147. I.A.K. Kupka, The ubiquity of the Fuller phenomenon, Trans. Amer. Math. Soc. to appear. A.A. Liverovskii, H61der's property of Bellman functions of plane control systems, Diff. Eqs. 17, 1981, 408-415. L. Neustadt, Synthesizing the Time-Optimal Control Systems, J. Math Anal. Appl. 1, 1960, 484493. N.N. Petrov, On the Bellman function for the time-optimal process problem, J. Appl. Math. Mech. 34, 1970, 785-791. A.B. Rabinovich, On a class of methods for the iterational solution of time-optimal problems, USSR Comput. Math. Math. Ph. 6, 1966, 30-46.
112
[25] [26] [27]
[28] [29] [30] [31] [32] [33]
H. Sch~ittler, The local structure of time-optimal trajectories in dimension three under generic conditions, SIAM J. Control Optim. 26, 1988, 899-918. H. Sch~ittler, Regularity properties of time-optimal trajectories of an analytic single-input controllinear system in dimension three, J. Optim. Th. Appl. 59, 1988, 135-146. H. Sch~ittler, On the structure of small-time reachable sets for multi-input nonlinear systems in low dimensions, Proceedings IFAC Symposium "Nonlinear control systems design", Capri, 1989. P. Soravia, Htlder continuity of the minimum time function with Cl-manifold targets, preprint Universita" di Padova, 1990. P.E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations 57,1985, 1-43. G. Stefani, Local controllability of order p, Proceedings Workshop on Differential Equations and Control Theory, Bucharest, 1983. H.J. Sussmann, A bang-bang theorem with bounds on the number of switchings, SIAM J. Control Optim. 17, 1979, 629-651. H.J. Sussman, The structure of time-optimal trjectories for single-input systems in the plane: the C~ nonsingular case, SIAM J. Control Optim. 25, 1987, 433-465. H.J. Sussman, The structure of time-optimal trjectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optim. 25, 1987, 1145-1162.
Analyse de l'algorithme multigrille FMGH de r~solution d%quations d'Hamilton-Jacobi-Bellman Marianne AKIAN INRIA Domaine de Valuceau-Rocquencourt B.P.105. 78153 LE CHESNAY CEDEX. Abstract In this paper, the resolution of Hamilton-Jacobi-Bellman equations by multigrid methods is studied. The Howard-multigrid algorithm FMGH is presented and, under some regularity assumptions, a convergence result is established. In addition, it is shown that the complexity of this algolithm is in the order of the nmnber of discretisation points. Some numerical examples are reported.
Introduction L'algorithme FMGH (introduit dans [1]) est un Mgorithme de r~solution num&rique d'&quations d'Hamilton-Jacobi-Bellman stationnMres utilisant les re&rhodes multigrilles. Nous en pr~sentons une nouvelle version, pour laquelle nons pouvons prouver un rdsultat de convergence; on pent, ~. partir d'un hombre fixe d'it&rations de l'algorithme de Howard approch& h chaque niveau de grille, obtenir une fonction valeur dont l'erreur est de l'ordre de l'erreur de discr&tisation. On en d~duit que la complexit~ de r~solution est dans la plupart des cas de l'ordre du nombre de points de dicr&tisation. L'algorithme a 5t& implement& sous forme de g&n&rateur de programmes [1,2], ce qui nous penner de r~soudre un large &ventail d'dquatiol~s d'Hamilton-Jacobi-Bellman stationnaires. Nous avons done test~ l'algorithme sur de nombreux probl~mes en dimension 1, 2 et 3. Les r~sultats num&riques obtenus confu'ment Ies rdsultats du thdorSme. De plus on peut constater que le nombre &it&rations n&cessaires k chaque xfiveau de grille est petit; la complexit~ de rdsolution des probl~mes test&s est done environ &gale h CN avec C petit (N &rant le nombre de points de discr&fisation).
1
P r o b l ~ m e s r~solus
On s'int6resse k la r6solution de probl~mes de contrSle stochastique; soit minimiser une fonction coflt J sur l'ensemble des contrSles admissibles Uad C )T/k, l'6tat &rant un processus stochastique )/-'t r6gi par une 6quation diff6rentielle stochastique dans ~ . Donnons l'exemple d'un problbme horizon infini dont l'6tat est arrSt& stir la fronti~re de l'ouvert Yl C JT~d, il s'~crit
dXt = g(Un X,) dt + a(X,) dWt x(o)
=
I'Vt processus de Wiener
zo
J(U(.),Xo) = E (fo*e-~tc(Ut, X,)dt + e-~*f(XT))
114 T 6tant le premier temps de sortie de Xt de l'ouvert fh La fonction coflt optimM
v(x) = min J(u(.), x)
~(.)
est alors solution (sous certaines conditions de r~gularit~) de l'~quation d'Hamilton-aacobi-Bellman[g]
mki {A(,,) v + ~(~) } = 0 dans a ~ v~
(1)
= f sur O ~
A(u) ~tant l'op~rateur aux d~riv~cs parfielles O~v A(u) v = -&v + ~ij a l j ( x ) ~ - . [ -
Ov ~j gj(u,X)~xj
1 a = ~o'o"
(2)
On obtient des conditions aux limites diffSrentes en modifiant un peu le probl~me (processus r~fldehis par exemple). Par la suite on s'intSressera k la r~solution numSriqucs d%quations d'Harnilton-Jaeobi-Bellman stationnaires avec des conditions aux limites quelconques. 2 2.1
Algorithmes Mdthodes
num6riques envisagdes
On a d'abord choisi pour des raisons de simplicit~ de discrStiser les 6quafions d'Hamilton-JacobiBellman par des differences finies et on se bornera au cas oh le domaine ~2 est le cube unit6 de/F~ a. On obtient Mors pour tout pas de discrdtisation h = 1/(n + 1) une &quation de la forme min
ueU~d
{A~,(~)Vh + Oh(")} = 0
V~ e Vh = ~-(nh, ~ )
(3)
oh ~h est la grille de points (ou nceuds) de ~, de pas h. I1 existe des discr~tisations diffSrences finics [14] et ~lSments finis [17] telles que le probl~me discret s'interpr~te en termes de probl~me de contrSle de chaincs de Markov. II vSrifie alors le principe du m a x i m u m discrct (PMD) i.e
Ah(U)Vh ~ 0 =~ Vh >_ O. C'est lecas par cxemple de la discr~tisation differences finies d'un problSme de Dirichlet sur l'intcrvalle ]0, 1[ d6finie par
dh(u)vh + Ch(U) = a,,(X)O~'V + g,(x, u)ohv + C(U, X) Ohv ~tant la discr~tisation de la dSriv~e de v (Ohv = O~_v = (v(x + h) - v(z))/h ou O~v = ( . ( ~ ) - . ( ~ - h ) ) / h salon le signe de j , da,~ le c ~ d~g~n~r~ (g,h > 2~,~) ou 0 h = (O_~ + a~)/2 d~rivSe sym6trique dad, s l e c a s non ddgSn~r~) et Oh = (c9h - cgJ:)/h ~tant la discr~tisation de la dSrivEe seconde de v. Lcs m~thodes habituelles de rSsolution du probl~me discret sont alors la m~thode d'itErations sur les valeurs qui converge avec la vitesse d'une i~Sration de Jacobi (facteur en 1 - O(h2)), ct l'Mgorithmc de Howard (iterations sur les politiques) : 9 u~ +1 E Argmin (dh(u) v'h''+ ch(u) ) 9 .~;~ est solution d e :
A,,(~,F'). + ~ t , ( ~ + ' )
= 0
115 qui n6cessite une inversion de matrice h chaque it6ration mais converge en g6n6ral avec la vitesse de la m6thode de Newton [13,5]. Les m6~hodes multigrilles par coni;re allient un taux de convergence ind&pendant du pas de discr6~isation et une complexi~5 de O(N). P o u r la r6solution d'6quations aux d6rivdes partielles non lin6alres il existe deux m6thodes multigrilles; la premiere est la m6thode FAS [11,10], qui a ~t~ 6tudi6e dans le cas d'~quations d'Hamilton-Jacobi-Bellman par IL Hoppe [12]. La deuxi~me est la m~thode Newton-multigrille [4]. Le fiaux de convergence par cycle multigrille est en g6n6ral semblable pour les deux m6thodes (cf. K.Stiiben et U.Trottenberg dans [11]). Pour une 6quation d'Hamilton-Jacobi-Bellman, l'it6ration sur les polifiques est idenfique h l'it6ration de Newton si les contr61es opfimaux sont r6guliers par rapport 5, la fonction valeur v [5]. Dans ce cas la m6thode Newton-multigrille n6cessite un hombre de calculs de contr61es optimaux moindre que la m6thode FAS. Nons avons done d6velopp6 un algorifihme de type Howard-multigrille pour les 6qua~ions d'Hamilton-Jacobi-Bellman g~n6rales. De plus les propri&~6s pa.rticuli~res de l'&quation d'Hamilton-Jacobi-Bellman nous ont pernfis de montrer un r6sultat de convergence de l'algorithme plus fort que dans le cas d'6quations non lin6aires quelconques. 2.2
Mdthodes
multigrilles llndalres
Avant d'aborder la description de l'algorithme nous allons rappeler la structure des m~thodes multigritles pour une ~quafion linSaire. Soit donc une ~quation aux d6riv~es partielles elliptique lin6aire dans l'ouvert born& I2 d e / R a q,m l'on ~crira sous forme variationnelle
Av+c=O
yEW
(4)
oh ]2 d6signe un espacc de Hilbert du type H'(12), H1(12), muni de la norme II-]]v et c d6signe unc foncfion r~guli~re de ~ d a n s / R (dans L2(fl) par exemple). On consid~re alors, comme dans le paragraphe pr~cddcnt, les grilles de points 12t, C ~ et les discr~tisations consistantes et stables de l'~quation (4)
Ahvl, + ch = 0
v/, ~ Yh
(5)
On suppose que la discr~tisation est d'ordre 8 pour la norme ]l-I]v, c'est; & dire qu'il existe un op6rateur d'interpolation ph : ~J, ---* Y v6rifiant IlPh~h -- ~IIv --< c~ J?
siv
cst solution de (4) et vh est solution de (5)
Dans ces conditions on cherche h r~soudre l'~quation (5) pour tin pas h assez petit. 2.2.1
Cycles multigrilles
Les m6thodes it6ratives cl~siques telles que les m6thodes de Jacobi ou Gauss-Seidel convergent avec un taux d&pendant du pas de dicr~tisation h, mais elles fournisscnt en g6n&ral un bon op6rateur de lissage (contraction des "hautes fr~qucnces" avec un taux ind~pendant de h). En revanche si I'erreur initiale est lisse, une bom~e approximation de cette erreur est fournie par la r~solu~ion d'unc ~quation dans une grille plus grossi&re. La mSthode multigrille consiste 5, combiner ces deux operations alCin d'obtenir une contraction globMe de l'erreur ind6pendante de h. Pour constl"uire un cycle multigrille on consid~re la suite de pas de discr~tisation h0 > -'" > ht > .-- tels que les grilles soient imbriqudes (ex : hk+l = hk/2 si i2 est le cube unit~). On notera
116 par la suite f/k = f/hk, ])k = ])hk,Ak = Ahk et plus g6n6ralement on remplacera hk par k lorsqu'il sera en indice. On introduit alors les op6rateurs &interpolation I~ +1 : l;~ ----* ])k+t et de restriction [~+1 : Vk+l ~ Vk. On se donne de plus sur chaque grille, u n op6rateur de relaxation (lissage) associ6 h l'6quation Akv + c = 0 : Tr : V~ ~ l/k, TC~(c)v = R~v + S~e. On peut alors d6finir r6cursivement le cycle multigrille associ6 k la r6solution de l'6quation A~1/+ c = 0 d'op6rateur
Adk(c) par [11 t (cf. figure 1) Z40(c) 1/ =
J~k(c) 1/ =
-AoXc (rfisolution exacte) si k = 0
V1
=
77.~)(c) v (lissage avant correction)
7"k
:
"7 k--1 0 Akv 1 + c, c~_ i = 0, ek-i = -/Mk_l(I~ rk)ek_l
I/2
:-
1/1 + I~_lek-1 (correction dans la grille grossi6re)
1/3
=
~;2(c)v2 (lissage apr~s correction)
A c e niveau on peut introduire la m6thode multigrille complete (FMG), dont l'algorithme FMGH est l'une des g6n6ralisations possibles. ul
u2
ul
u2
r4solution exacte relaxations
\ / V - c y c l e (3, = 1)
restriction du r6sidu interpolation L6gende
W - c y c l e (7 = 2) Figure 1: Exemples dc cycles multigrilles
2.2.2
C o n v e r g e n c e de la m g t h o d e m u l t i g r i l l e
Nous rappelons m a i n t e n a n t les r~sultats cla.4siques. On supposera par la suite que les pas de discr6tisation v6rifient hk+l = 6hk. On note -~'/k l'op4ratcur lin6aire de contraction associ6 s .Mk(c), i.e. tel que .Mk(c)v = Mkv + Nkc. Et on mtmi 1;~ d ' u n e norme hen-nitienne II.llv, telle que ~[Ivkllvk _<
Ilpk,~llv < cl[1/~ll,,
v1/~ e v~.
T h 4 o r ~ m e 1 : C o n v e r g e n c e des cycles m u l t i g r i l l e s On ~uppo~e A/or~ement ellip~ique alor~
llZ~Z~lln _< p
<
1, Vk > 0
~ou~ l'une des hypo~hd3e3 ~uppl~mcntairc~ suivantea 1. A e~t ~ymd~rique [6,7,10,16] 2. A quelconquc, g asscz rdguIi~re (Ca), ct ho aa~ez pc~it [10,15,8] P r o p o s i t i o n 1 : C o m p l e x i t 6 des m 6 t h o d e s m u l t i g r i l l e s (cf. K.Stgbcn et U. Trottenberg [11]) Si 764 < 1 le co4t d'un cycle muItigrilIe c~ gym h C(ul + u2) + C' .Nh CMa(h) ~1 - 76 d
117 2.3
Algorithme
FMGH
On cherche maintenant & r~soudre l'6quation d'Hamilton-Jacobi-Bellman discr6tis6e 3 pour un pas h assez petit. P o u r cela on se donne comme dans le paragraphe pr6cfident les grilles imbriqu6es f~. M6thode
multigrille :
Pour chaque 6quation lin6aire de la forme Ak(uk)v + Ck(U~,) = 0 oh,
uk E L/k = ~(f~k,U~d), on peut construire un cycle multigrille associ6. On notera A,lk(uk) et Mk(u~,) les op6rateurs affines et lin6aires du cycle multigrille associ6 & cette 6quation. On introduit de plus lcs op6rateurs d'interpolation (des fonctions et des contrSles)
•kT M :rbl~+1 : U~ ---, Mgthode de minimisation :
IA+I
Si l'op6rateur A(u) est donn6 par la forrnule 2, minimiser Ah(u)v+
Ch(U) revient 5. mininfiser l'hamiltonien H ( x , u, c_ghv) oh Ohv est la dicr6tisation du gradient de v et H est d6fini par H(x, u, p) = ~,i g~(x, u)pi + c(x, u). On consid~re alors une m6thode it6rative de minimisation de l'hamiltonien convergent vers le contrSle optimal uh(v), pour tout x E f2h et v E ];h, dont l'op6rateur sera not6 Afh(v). On utiliscra par exemple la m6thode du gradient ~ pas constant
( 2r
= u - po(O,,H(u, Ohv)) ) ou de Newton ( A/'h(v)u = u -- (O~2H(u, Ohv))-l(O,,H(u, cOhv) ). niveau #1 = 2
#2 = 1
minimisations
minimisations
17?,1 ~
de grille pas
k
hk
m2=2 cycles
1
cycles
i
t
c .
-
-
-
I
.
.
.
.
.
.
Jk_ :-J
Figure 2: Exemple d'algorithme F M G H avec fi = 2 []
mininfisation
O
r6solution exacte de l'&luation lin6aire ,
relaxations restriction du r6sidu ,
interpolation I interpolations de la fonction Z e~ des contr61es ZU Figure 3: L~gende
1/s 1/4
I
L_ _]
9
2
o
1/2
118 Algorithme FMGH
:
Oil se donne v0~ 6 ]20 et uo~ 6 14o et oll d6finit les suites v~" 6 ]}k et u~ 6 L/'k
pour 0 < n < h e t k 6 JN (cf. figure 2) par Pour0_
ck(u)). 9 v~+1 = A,tr"+l(u~+l)v~.', solution approchfe de l'6quation Ak(u2+l)v + c~(u~ +1) =0.
On obticnt done v;:~ valeur approchfie de vk et u~. vMeur approchSe de uk(vk), alors
9 ~h, : z~§ 4
3
Convergence de Pa]gorithme F M G H
Nous donnons maintenant le r6sulgat de convergence de l'algorithme F M G H dans lequel les contr61es optimaux sont calculds de faqon exacte. Pour cela nous raisons les hypotheses suivantes ( H 1) Ah(u) vdrifie le PMD, pour tout u 6 >'(f2h, U~d) ct tout h <_ ho.
lution unique, et f@,x) E C#(1Rd) pour 0 < t9 <_ 1, c'e~t i dire [~(p', z)
1.1~ et l.la
-
~(p, z)]k _< C~[p' - p[~
Vp, p' e JRd
dt~,,t des normes quelconques de ]~k et IT{d.
Cette hypoth~se est v~rifi~e avec fl = I si g est aft-me par rapport 5. u et c quadratique par rapport h u, ou si H est strictement convexe en u uniform6ment par rapport h x et p. ( I t 3) On ,uppo~e l'opgrateur Ah(u) coercif uniformdment en u et h < ho ( -Adu)v,
~, ) > c, Ilvll~.
L'op6rateur An v6rifiant l'infgalit6 de Gs
vh < ho, Vu ~ Uh , Vv e Vh (inddpendante de h) : (--Ah(u)v, v ) >__C,
Ibll~,
-
C=ll~ll~ avec c= = 0 si ~ >__ao >__0, cette hypothSse est vfrififie si A _> 10. ( H 4) L'opdrateur de la mdthode multigriIle vdrifie
tlMk(u)ltv,, _< p < Z
Vu ~ U.~ e~V,~ k o
La borne p d~pend 5. priori du contr61e, et its d6monstrations de convergence des cycles multigrilles fon~ in~;ervenir les bornes des d6riv6es des fonctions gi(u). N6anmoins en pratique cette hypoth~se est toujours v6rifide.
(H 5) ti:Z-~+'ilv,.+l_,,-,,_
V~ >_ O.
119
( H 6) La di3cr6tisation eat d'ordre 0 > 0 : Ilph.,, - vllv < Coh ~ Vh < ho. T h d o r ~ n a e 2 Soua lea hypoth~sea H 1, H 2, H 3, H 4, H 5, H 6 Ia suite v~ de l'algorithme F M G H avcc calcul exact de~ contrSle~ optimaux v6rifie pour tout 77 > 0 il exiate rn > 1 tel que ~i mk >_ m ~ partir d'un certain rang alor~ pour tout Vo ~ E ~(Q0,//2) il exi~te n tel quc 3i h >_ n alors
live- vdlv <_ ,7cd4 IIv~,r vll~, < (c,7 + 1)c,h~ La d6monstration de ce r~sultat se divise en quatre parties [2]. On montre d'abord la convergence d'ordre Q-(fl' + 1) de l'algoritlm~e de Howard (au moyen d'in6galit&s de Sobolev discr~tes), ce qui est similMre s la convergence quadratique de la m6thode de Newton. On en dSduit le r6sultat du thfiorSme pour un pa.s h0 petit eg une fonction initiale dans un certain voisinage de la solution. C'est un cas particulier du rfisultat de R.E. Bank et D.J. Rose [4]. Enfin on montre un r~sultat de contraction de la m~thode de Howard au d~part d'une sursolution (min,,(Ah(u)v + ch(u)). <_ 0), ce qui permet de prouver la convergence de l'Mgorithme Howard approch6 g niveau fixe pour tout niveau de grille et route valeur initiale. On en dfiduit clots le th~or~me dans le cas g6n~ral. Si les contrSles sont calculus par une mfithode de minimisation d'op6rateur 2fh on obtient un r6sultat semblable moyemmnt quelques hypotheses sur la convergence de A/h et sur les opfirateurs d'interpolation Z/A. Proposition
2 Si C~. d~igne le nombre de calcul~ d'une it6ration de minimisation en un point,
Ia compIexit6 de l'algorithme F M G H e~t
(m, + . . . + m,,)CMc(h) + (~C:r + C)N~ CFMGH(h) ~"
1
-- 5~
Donc sous les hypothb.ses du thfor~mc et si la complexit6 du cycle multigrille utilis6 est de l'ordre de Nh (cf. proposition 1) la complexit6 dc la r6solution des 6quations d'Hamilton-Jacobi-BeUman est cue aussi de l'ordre de Nh.
4
lZdsultats num6riques
On a contruit un g6n~rateur de programmes Fortran adapt6s au probl~me de contrgle stochastique r6soudre c t atLx m6thodes num~riques ddsir6es [1,2]. O n pcut clots r6soudre de nombremx probl~mes, en particulier nous avons test5 l'Mgorithme sur les ~quations d'Hamilton-Jacobi-Bellman suivantes Av+
rain ( u . V v + u . u ) + c = 0 u e ~d v= 0 sur O~t --4vWAv+
rain ( u . V v + u . u ) + c = O u
Ov
= 0
dans~2=]0,1[ a,d=l,2'3 (6)
dansgZ=]0,1[ d,d=l,2,3
E g~
sur On
La fonction c 6rant calcul~e en fonction de la solution exacte.
(7)
120 D a n s l ' a l g o r i t h m e F M G H on a utilisfi la m d t h o d e de Gauss-Seidel lexicographique p o u r l'op d r a t e u r de relaxation, l'interpolation linfiaire sur chaque coordonnfie p o u r les op~rateurs I~.+1, 2- et 2-/.,/et la restriction pleine (adjointe de l ' i n t e r p o l a t i o n ) p o u r Its opfirateurs I~+~. Les pas de discrfitisation vdrifient hk = 1/2 k, k &tant le n i v e a u de la grille. Les param~tres de la mfithode multigrille sont ui = v= = 2 et 7 = 1, pour les problSmes de Dirichlet et vl = v= = 1 et 7 = 2 en d i m e n s i o n supfirieure s 2 et 7 altern~ 1-2 en d i m e n s i o n 1, p o u r Its probl~mes de Neumarm (la eomplexit5 es~ alors encore en (9(N)). Ces pare.m~tres ~taie~t p e r f o r m a n t s pour les ~quations lin~aires. Les p a r a m ~ t r e s de I'algorithme F M G H utilis6s sont f~ = I ou 2 et r~h = m~ = 1. A u cours de l ' a l g o r i t h m e F M G H , on a calcul~ 5. c h a q u e n i v e a u de grille et aprb.s chaque minim i s a t i o n ou cycle multigrille, la norme L 2 et la s e m i - n o n n e H ~ discrete de l'erreur e~ = vk -- r~v" et la n o r m e L ~ discrete d u rfisidu f~ = A~(u~)v~ + c~. O n a de plus calcul~ les normes des rfisidus et erreurs des fonctions obtenues aprbs plusieurs irritations s u p p l 6 m e n t a i r e s a u niveau final, ee qui p e r m e t d ' e s t i m e r l ' e r r e u r de discrfitisation. Nous d o n n o n s en a n n e x e les t a b l e a u x de ces valeurs pour deux probl~mes de N e u m a n n en dimension 2, ainsi que les courbes de variation des normes ~ 2 d ' e r r e u r normalisfies Ile~llL=/h~ et le~-IH,/h~ en fonction d u n i v e a u de grille k, pour deux probl~mes
de Dirichlet en d i m e n s i o n 2 (on trace aussi Ies valeurs o b t e n u e s apr~s c h a q u e i t e r a t i o n de Howard suppl~mentaire a u niveau final). Lcs rfisultats sont s e n s i b l e m e n t les m~mes p o u r des probl~mes ~quivMents en d i m e n s i o n 1 et 3. Conclusion
des rdsultats num~rlques
Les r~suttats o b t e n u s sur des probl~mes 5quivalents sont e n g~n6ral meilleu_rs p o u r des conditions de Dirichlet que p o u r des conditions de Neumam~. Ceci p e u t s'expliquer p a r une meilleure efflcacit~ de la r e l a x a t i o n a u moins dans Its cas d'op~ratcurs lin~aires d~gdn~rds, ce qui e n t r a i n e des r~sultats plus p e r f o r m a n t s des cycles multigriIles. La contraction de
III~IIL=
5, ~ v ~ u
flxe est satisfaisante. P o u r des conditions de t y p e Dirichlet
elle est d ' e n v i r o n 1/100 dans les meilleurs cas (solution lisse) et d ' e n v i r o n 1 / 4 darts les moins bons. P o u r des conditions de type N e u m a n n elle varie e n t r e 1/10 et 1/2 environ. Ce qui explique qu'en g~nfral deux itdrations p a r niveau soient n6cessaires d a n s ces derniers cas. O n p e u t c o n s t a t e r a u v u des valeurs avant et apr~s m i n i m i s a t i o n a u dernier n i v e a u de grilIe q u ' h o r m i s les cas o h la contraction de ll/dlc2 est faible les m i n i m i s a t i o n s suppl~mentaires ne c h a n g e n t pas les valeurs des erreurs. Ceci s'explique p a r la convergence superlin6aire (d'ordre Q - ( ~ + 1)) de l ' a l g o r i t h m e de Howard qui l ' e m p o r t e sur la convergence lin6aire des cycles multigrilles, s u r t o u t si l ' e r r e u r initiale est de l'ordre de h ~ avcc hk petit.
Les valeu~ de te~.lm/hk et de ll~-ItL=/hk calcul~es 5 la fin de c h a q u e n i v e a u de grille se maintientent 5, une valeur b. peu pros constante et mfime dficroissante. O n a done bien u n e confirmation d u r6sultat du~th~or~me avec les valeurs m~ = f~ = 1; ce qui s'explique p a r le b o n t a u x de contraction des cycles nmltigrilles em-:-m~mes et de la m ~ t h o d e de m i n i m i s a t i o n (triviale dans les tests effectufis). La valeur d e II~llL,/h~ se m a i n t i e n t 5, une valeur 5, peu pros c o n s t a n t e pour les probl~mes h solution lisse. P o u r les probl~mes 5, solution plus oscillante, eUe p r e n d des valeurs i m p o r t a n t e s a u x n i v e a u x inffirieurs et d~croit ensuite vers une valeur voisine de la valeur limite. O n pouvait le pr~voir, puisque dans ce cas la vaIeur du g r a d i e n t de la fonction et done de g(u) est grande,
121
l'~quation est done d~g~n~r~e, ct la discr~tisation utilis~e est alors seulement d'ordre 1 en norme L 2 aux niveaux infSrieurs. Aces niveaux, on ne dolt done eonstater que la stabilit6 de la vateur de I[e~NL2/hk. Dans tes exemples h solution non lisse pr~ent6s la discr6tisation est d'ordre 1 jusqu'au niveau 4, et est d'ordre 2 ensuite. Pour des solutions de gradient plus important les rfisultats sont encore satisfaisants si les conditions aux limites sont de Dirichlet (stabillt~ de He~]]~.,/h~) mats pas si elles sont de Neumann. Ceci pcut s'expliquer par le manque d'efficacit~ (mauvais coefficient de lissage) de la relaxation utilis~e (Gauss-Seidel lexicographique) par rapport ~ la correction utiliste. I1 faut dans ces cas augmenter soit le sombre de relaxations ul + u2 soit 7, pour obtenir des r6sultats satisfaisants. Malgr6 ces derniers cas qni n'entrent pas dans le cadre des hypotheses (non coercivit~ de l'op6rateur), les r6sultats num6riques obtenus confirment le r6sultat du th6or~me 2; le comportement des cycles multigrilles et de l'algorithme FMGH ~tant en g6n6ral meilleur que celui esp6r6. R~f~rences [1] M. Akian, 126solution num~riquc d'dquations d'HamiIton-Jacobi-BelIman au moyen d'algorithroes multigrilles ct d'itdrations sur les politiques dans Analysis and optimisation of systems. Lecture notes in control and information sciences, n~ S. V., 1988. [2] M. Akian, Mdthodes multigriUes en contr~le stochastique. Th~se de doctorat, Universit~ ParisDauphine, 1990, 5. paraitre. [3] R.E. Bank, A comparaison of two multilevel iterative methods for nonsymetric and indefinite elliptic finite element equations. Siam J. Numer. Anal., Vol 18, n~ August 1981. [4] R.E. Bank, D.J. Rose, Analysis of a multilevel iterative method for nonlinear finite element equations. Math. Comp., 39, 453-465 (1982). [5] I't.. Bellman, S.E. Dreyfus, Applied dynamic programming. Princeton University Press 1962. [6] D. Braess, W. Hackbush, A new convergence proof for the Multigrid method including the V-cycle, SIAM J. Numer. Anal, 20, 11~ 967-975, (1983). [7] S. F. McCormick, Multigrid methods for variational probleroz: general theory for the V-cycle. SIAM J. Numer. Anal., 22, n~ 634-643, 1985. [8] S. F. McCormick, ed. Multigrid methods. SIAM frontiers in applied mathematics, vol 5, Dec. 1987. [9] W.H. Fleming, R. Rishel, Dcterminisc and stochastic optimal control. Springer Verlag. N.Y. 1975. [10] W. Hackbusch, Multigrid reecho& and applications. Springer Verlag Berlin, Heidelberg 1985. [11] W. Hackbusch, U. Trottenberg, Mt,ltigrid methods. Lecture notes in mathematics, n~ Springer Verlag, 1981. [12] II.H.W. Hoppe, MuItigrid methods for Hamilton-Jacobi-BeUman equations. Numer. Math., 49,
239-254 (1986). [13] R.A. Howard, Dynamic programming and markov process. MIT 1960. [!4] H. Kushner, Probability mc~hods ir~ stochastic control and for elliptic equations. Academic Press 1977. [15] J. Mandel, Multigrid convergence for nonsymmetric, indefinite variational problem3 and one smoothing step. Applied Mathematics and Computation, 19, 201-216, 1986. [16] S.V. Patter, Note on convergence of the Multigrid V-cycle. Applied Mathematics and Computation, 17, 1985. [17] J.P. Quadrat, Sur l'identifica~ion et Ie contr6le des syst~mes dynamiques stochastiques. Th~se de doctorat d'~tat (Univ. Paris Dauphine) 1981.
122
A
Annexe
I1" i 1,
,;00o .
0.50
1.36
120.2518.05
5.35
""
,00 1.16
1
0.50
r] 2 2
0.25 0.25
II H.,,,I,,.. I I,.l,,../,,: .
2
0.25
0.55
1.17
3.78 0.31
3
9.13
0..37
1.07
9.20
4
6.2510 -2
0.21
0.81
0.12
31~19-~,
698,
~85
3~0
3.13 l0 -2
0.15
0.71
0.08
5
3.13 1O- 2
6.98 0.10 0.10 9.29 10 - s 9.29 l0 - a 9.52 10 -4 9.52 10-'~
3.32 0.09 0.69 0.68 0.68 0.68 068
3.49 0.05 0.05 0.02 0.02 0.02 0.02
6 6 B 6 6 6 '6
1.56 1.56 1.56 1.56 1.56 1.56 1.56
1.56 1.56 1.56 1.56 1.56 1.5B 1.5B
10 -2 10 -2 10 -~ I0 - 2 10 -2 10 - 2 10 -~
[
744.00 87.20
0.13
4
i
[
86.40
6.2510 -2
5~1310-~ L0-~ 10 - 2 l0 -~ 10 - 2 10 -2 10 -2 10 -~
,o,,I,,,/,,. U
62.40
176.23
125.40 64.7l
249.55 128.98
508.47
104.98
35.40
1716.14
.
242.90
0864~7
127.54
25.10
1123.35
18.47
209.00 I0.10 10.10 0.71 0.71 9.90 10 - 2 9.90 10 -~
4493.08 868.81 868.81 476.98 476.98 456.97 456.97
108.77 7.21 7.21 8.31 8.31 8.55 8.55
T a b l e 1: I t d s o l u t i o n d e I ' 6 q u a t i o n d e N e u m a i m en d i m e n s i o n 2 d o n t la s o l u t i o n e.st v ~ =
T a b l e 2: R ~ o l u t i o n d e l ' d q u a t i o n d e N e u m a n n en d i m e n s i o n 2 d o n t la s o l u t i o n est v'(x,y) =
r
0.30 r
~os(~,j).
r
+r
~os(~u).
0.93
0.44
0.4|
0.03
J
2
3
4
5
$
G
i
0,40
(*
, 2
, ~
, 4
,
G
~
G
Figure 4: Courbes des valeurs Iek111~/th~ (s gauche) et IIekl]L~/h~. (k droite) pour la rdsolution de l'dquation de Dirlchlet en dimension 2 dont la solutiou est v'(x,y) = sin (rx)sin (,'ry).
7~.42
118.9
i
0.93
0,4~
[
!
2
3
4
~
6
6
$
F i g u r e 5: C o u r b e s d e s v a l e u r s [ek[H,/hk (~, g a u c h e ) et [le~lJL2/h~ (5. d r o i t e ) p o u r la r ~ o l u t i o n d e l ' 6 q u a t i o n de D i r i c h l e t en d i n a e n s i o n 2 d o n t la s o l u t i o n est v'(x, y) = 0.30 sin ( 1 0 ~ x ) sin (10,vy) + sin ( ~ x ) sin ( ~ y ) .
A DYNAMIC PROGRAMMING BASED GAS PIPELINE OPTIMIZER Hemant
S.
Lall
Perot Systems 1595 Spring Hill Road, S u i t e 777 Vienna, V i r g i n i a 22182 U.S.A. Peter
B.
Percell
Scientific S o f t w a r e - Intercomp, Inc. 10333 R i c h m o n d Avenue, S u i t e i000 Houston, T e x a s 77042 U.S.A. ABSTRACT
A dynamic programming based Gas Pipeline Optimizer (GPO) has been developed at S c i e n t i f i c S o f t w a r e - I n t e r c o m p for the HBJ gas t r a n s m i s s i o n p i p e l i n e s y s t e m in India. Used as an o p e r a t i n g and planning tool, the GPO will d e t e r m i n e the discharge p r e s s u r e s at the c o m p r e s s o r stations and the n u m b e r of c o m p r e s s o r trains to o p e r a t e at each c o m p r e s s o r station so that fuel consumption and s t a r t - u p / s h u t - d o w n costs for the entire HBJ system are minimized under steady state conditions. Further, the optimization will satisfy specified delivery flow rate and minimum d e l i v e r y p r e s s u r e r e q u i r e m e n t s at the r e c e i v i n g t e r m i n a l s and ensure that minimum line pack (inventory) r e q u i r e m e n t s are met for each section of the pipeline. Excessive s t a r t i n g and s t o p p i n g of c o m p r e s s o r trains will also be avoided. D y n a m i c p r o g r a m m i n g was selected as the o p t i m i z a t i o n technique for the GPO because it is robust and p r o v i d e s a g l o b a l l y optimal solution. The GPO will have the ability to h a n d l e pipeline networks with diverging branches. To the best of our knowledge, the dynamic programming a l g o r i t h m has been implemented in o t h e r e x i s t i n g c o m m e r c i a l software to m i n i m i z e fuel cost for straight line n e t w o r k s only. In this paper, two main c o m p o n e n t s of the GPO are discussed: the dynamic programmlng algorithm and the m a t h e m a t i c a l m o d e l s used to represent the p i p e l i n e and c o m p r e s s o r stations. A l s o i n c l u d e d is a d e s c r i p t i o n of some of the p r o b l e m s encountered d u r i n g i m p l e m e n t a t i o n and the t e c h n i q u e s used for t h e i r resolution. PROBLEM
DESCRIPTION
A f t e r completion, the HBJ gas t r a n s m i s s i o n p i p e l i n e system will t r a n s p o r t gas from a commercial gas field on the w e s t coast of India to six f e r t i l i z e r plants, three power p l a n t s and two LPG plants located between Bombay and Delhi. The p r o p o s e d p i p e l i n e e x t e n d s . a p p r o x i m a t e l y 1700 km and s u p p l i e s up to 20.5 MMSCMD of gas to the recelvlng terminals. In order to meet m i n i m u m delivery p r e s s u r e r e q u i r e m e n t s , the p r e s s u r e of the gas will be b o o s t e d at five compressor stations installed along the pipeline. At each c o m p r e s s o r station, b e t w e e n three and ten c o m p r e s s o r trains (consisting of a c e n t r i f u g a l c o m p r e s s o r and a gas t u r b i n e driver) will be i n s t a l l e d in parallel. The sole p u r p o s e of the GPO is to m i n i m i z e the fuel cost of o p e r a t i n g the pipeline. The o p t i m i z a t i o n d e t e r m i n e s a f e a s i b l e steady s t a t e strategy for o p e r a t i n g the compressor stations such that the overall fuel consumption cost plus the compressor train start-up/shut-down cost !herein a f t e r referred to as the O p e r a t i n g Cost) for the entire network is m i n i m i z e d . The formulation of the HBJ gas p i p e l i n e o p t i m i z a t i o n p r o b l e m is g i v e n below. D e t e r m i n e the d i s c h a r g e pressure setpoints and number of compressor trains to o p e r a t e at each c o m p r e s s o r s t a t i o n w h i c h m i n i m i z e the O p e r a t i n g Cost s u b j e c t to the following c o n s t r a i n t s : I. 2. 3.
m i n i m u m and m a x i m u m a l l o w a b l e o p e r a t i n g p r e s s u r e c o n s t r a i n t s , m i n i m u m and m a x i m u m line p a c k c o n s t r a i n t s , m i n i m u m and m a x i m u m s p e e d c o n s t r a i n t s on c o m p r e s s o r trains,
124 4. 5. 6. 7. 8.
surge and stonewall limits for compressor trains, available power limitations for the gas turbines, supply pressure setpoint, mlnlmum delivery pressure constraints, and specified delivery flow requirements.
The constraints outlined above will be discussed Mathematical Model section. DYNAMIC
in greater detail
in the
PROGRAMMING
in solving pipeline optimization problems, the selection of the technique to be used is a function of the complexity of the pipeline network, the objective function, and the type of constralnts. Dynamic programming was chosen for the HBJ network because it offers several major advantages over other mathematical optimization techniques. One of the benefits of dynamic programming is that it can handle nonlinearities easily. Utilization of a variant of the linear programming algorithm, such as sequential mixed-integer linear programming [i], is more difficult and less accurate because it requires linearizatlon of the equations for flow, horsepower, etc. Another advantage of dynamic programming is that, unlike other algorithms for nonlinear constrained optimization, a global optimal solution is guaranteed. Convergence problems are also eliminated due to the recursive, rather than iterative, nature of the algorithm. Thus the technique is highly robust. Furthermore, dynamic programming is effective in handling discrete decisions such as whether or not to bypass a compressor station. Other nonlinear programming algorithms do not possess these properties. Dynamic programming becomes more efficient as the number of constraints in the system increases. Since an increase in the number of constraints shrinks the feasible solution region, the total number of cases considered by the algorithm is reduced. This property is unique to dynamic programming. The principal disadvantage of dynamic programming is that its application is practically limited to simple structures. Otherwise, the computation time becomes too high for the solution to be of practical value. This is commonly referred to in the literature as "the curse of dimensionality". Fortunately, the HBJ pipeline network has only five compressor stations, all but one of which are in series and, therefore, computation time remains manageable. pa~t ApplicatiQns Dynamic programming was developed in the early 1950's by Richard Bellman for optimizing complex decision problems [2]. It was first used to optimize the operation and planning of gas pipeline systems by Wong and Larson in 1968 [3]. They successfully solved optimization problems occurring in both short-term terminal distribution and long-term transmission of natural gas. Since that time, dynamic programming has also been applied in the gas pipeline industry for the optimal design of pipelines [4]. A survey of the literature indicates several applications of dynamic programming in the operation of gas pipelines. These applications have resulted in reported fuel savings of as much as 8 to 12 percent [5]. With fuel costs relatively high (of the order of tens of millions of dollars annually for large systems), and computers becoming ever more powerful and affordable, the interest in pipeline optimization software has grown steadily over the past five years. Alqorithm Dynamic programming is a mathematical technique which decomposes the original optimization problem into a series of less complex subproblems which are computationally easier to solve. Each subproblem Is usually
125 referred to as a "stage". For a gas p i p e l i n e network, each compressor s t a t i o n c o r r e s p o n d s to a stage. T h e e s s e n t i a l c h a r a c t e r i s t i c of dynamic programming is the multistage nature of the o p t i m i z a t i o n procedure. Since d e c i s i o n s in dynamic p r o g r a m m i n g are optimized in stages rather than simultaneously, the computational effort required increases a p p r o x i m a t e l y linearly with the number of stages. If an exhaustive enumeration approach were used instead, the c o m p u t a t i o n a l effort would increase exponentially with the number of stages. Thus, dynamic pro~rgmm, ing can p o t e n t i a l l y reduce the c o m p u t a t i o n s p e r f o r m e d in m i n l m l z l n g the total cost for the HBJ s y s t e m by a factor of over 1012 . Each s t a g e (e.g., stage n) in d y n a m i c terms of four factors: PS n PD n CO n -
TR n -
programming
is
characterized
S u c t i o n pressure at the station. T h i s v a r i a b l e is referred to as the state variable. Its v a l u e d e p e n d s on the d e c i s i o n made at the preceding stage and on the n e t w o r k b o u n d a r y conditions. D i s c h a r g e pressure at the station. This v a r i a b l e is referred to as the decision variable. o p e r a t i n g cost of the station. Its v a l u e d e p e n d s on the suction and discharge pressures, and it is d e t e r m i n e d by s e l e c t i n g the number of trains to run w h i c h m i n i m i z e the o p e r a t i n g cost, and w h i c h do not v i o l a t e any t r a i n or s t a t i o n constraints. T r a n s i t i o n function: used to d e t e r m i n e the s u c t i o n pressure at the d o w n s t r e a m s t a t i o n for a g i v e n d i s c h a r g e pressure.
Cost
Cost
c o cP~.tP%.)
Suctl~
Station
con cPS P\
Transition
Suctlon~
Discharge Pressuro
Station n
Transition
TRn (PDr-)n
Discharge Pressure
P%.I
P%
The d y n a m i c p r o g r a m m i n g a l g o r i t h m for an N stage serial p r o c e s s on the a p p l i c a t i o n of the following r e c u r s i v e r e l a t i o n s h i p : FUn+I(PDn+I) where
= MINIMUM PD n
FUn+I(PDn+I) PS n PS N
in
{COn(PSn,PDn)
+ FUn(PDn) }
is
based
for n = l , . . . , N
~ m i n i m u m o p e r a t i n g cost of s t a t i o n s 1 t h r o u g h n, when the d i s c h a r g e p r e s s u r e at s t a t i o n n+l is PDn+I, = TRn+l(PDn+l) , = PDN+ 1 = s u p p l y p r e s s u r e setpoint,
and, for each v a l u e of PSn, the m i n i m i z a t i o n v a l u e s of PD n.
is o v e r all p o s s i b l e
The application of the recursive relationship is conceptually straightforward. A discharge p r e s s u r e grid is formed at each station by c h o o s l n g a range of values within w h i c h the d i s c h a r g e p r e s s u r e must lie, and selecting the number of grid p o i n t s in that range. The use of a d i s c h a r g e p r e s s u r e grid allows the r e c u r s i v e r e l a t i o n s h i p to be applied to a finlte number of d i s c h a r g e pressures. The optimal discharge p r e s s u r e at each s t a t i o n is s e l e c t e d from among the g r i d points by applying the recursive r e l a t i o n s h i p backwards, i.e., by first a n a l y z i n g the last s t a t i o n in the system, and then optimizing, by m o v i n g back one c o m p r e s s o r s t a t i o n at a time, until all c o m p r e s s o r stations are included.
126 At each compressor station, the recursive relationship is used to determine the d i s c h a r g e p r e s s u r e that y i e l d s the m i n i m u m o p e r a t i n g cost of all s t a t i o n s up to, and including, that station, for a g l v e n suction pressure. The optimal plan is obtained when the b a c k w a r d recursive r e l a t i o n s h i p is a p p l i e d to the first station, and it gives the set of discharge pressures at which to o p e r a t e the c o m p r e s s o r stations. The number of trains to operate at each s t a t i o n is o b t a i n e d as a b y p r o d u c t of the cost c a l c u l a t i o n s . Special c o n s i d e r a t i o n is given to the b r a n c h line in the HBJ system. The branch line d e s t r o y s the serial s t r u c t u r e of the network, and p r e v e n t s a straightforward implementation of the recursive relationship. This problem is o v e r c o m e by d e c o m p o s i n g the branched s t r u c t u r e into two inter-connected serial structures, and solving each structure individually by applying the serial o p t i m i z a t i o n technique, as before. The b r a n c h line and the main line are then r e c o m b i n e d by using a modified r e c u r s i v e relationship, which has an a d d i t i o n a l term for the branch line. This r e s t o r e s the serial structure of the n e t w o r k in such a way that the r e c u r s i v e r e l a t i o n s h i p for a serial p r o c e s s can a g a i n be used. MATHEMATICAL
MODEL
This s e c t i o n d e s c r i b e s the m a t h e m a t i c a l models used to represent the pipeline and c o m p r e s s o r stations. T h e s e models are used to c o m p u t e the o p e r a t i n g cost of each c o m p r e s s o r s t a t i o n at every discharge pressure grid point. Integration of t h e s e m o d e l s into the d y n a m i c p r o g r a m m i n g a l g o r i t h m will be d e s c r i b e d in the s e c t i o n on I m p l e m e n t a t i o n . In m o d e l i n g gas p i p e l i n e network systems, it is c o n v e n t i o n a l to b r e a k the network into components [6]. The components used here are legs, c o m p r e s s o r stations, external regulators, and nodes. In a pipeline network model, legs c o r r e s p o n d to s e c t i o n s of pipe w h o s e p r o p e r t i e s such as inside diameter, absolute roughness, heat transfer coefficient, and slope are constant. C o m p r e s s o r s t a t i o n s are g r o u p s of c o m p r e s s o r trains at a s i n g l e l o c a t i o n w h i c h are o p e r a t e d to achieve a given discharge pressure setpoint. External r e g u l a t o r s are model d e v i c e s in the network for e x t e r n a l flow into or out of the p i p e l i n e system. N e t w o r k nodes are just connection points where a combination of the p r e v i o u s network e l e m e n t s m e e t together. Pipe ~eq ~odel The pipe leg model c o n s i s t s of s t e a d y state e q u a t i o n s for the pressures, flow, and temperatures along the leg. The s t e a d y state leg equation r e l a t i n g flow and pressures is: (fL) WIWI/(2DA2)
= ~(Pu-PD)
+ p2g(hu-hD),
w h e r e W is mass flow rate, PU and PD are p r e s s u r e s at the u p s t r e a m and d o w n s t r e a m leq ends, respectively, f is the friction factor, L is leg length, D is inside diameter, A is c r o s s - s e c t i o n a l area, p is a v e r a g e d e n s i t Z, g is the g r a v i t a t i o n a l constant, and h U and h D are leg end elevatlons. The steady state leg t e m p e r a t u r e e q u a t i o n is: T D = e - a L T U + (b/a) (l-e-aL), where T U and T D are t e m p e r a t u r e s at the u p s t r e a m and d o w n s t r e a m leg ends, r e s p e c t i v e l y , and a and b d e p e n d on the heat c a p a c i t y of the gas, the flow rate, the pressure drop, the leg dimensions, the leg heat t r a n s f e r properties, the ambient t e m p e r a t u r e , etc. IDdividual
C o m p r e s s o r Mode~
In o r d e r to d e f i n e the c o m p r e s s o r s t a t i o n model, an individual c o m p r e s s o r train is first modeled. A natural way to c h a r a c t e r i z e the p e r f o r ~ a n c e of a c o m p r e s s o r is in terms of p o l y t r o p i c head, H, and p o l y t r o p i c efficiency, Ep,
127 H = ZsRTs((PD/Ps) ( ( n - l ) / n ) - l ) / ( ( n - l ) / n ) , where PD and P$ are the d i s c h a r g e and s u c t i o n pressures, respectively, T S is the s u c t l o n temperature, Z s is the a v e r a g e s u p e r compressibility, R is the u n i v e r s a l gas c o n s t a n t d i v i d e by m o l e c u l a r w e i g h t of gas, and n is the polytropic exponent. P o l y t r o p l c e f f i c i e n c y is a m e a s u r e of the p e r f o r m a n c e of a real c o m p r e s s o r r e l a t i v e to an ideal compressor, in which all energy used by the compressor goes into the gas being compressed. Polytropic head and e f f i c i e n c y d e p e n d in a ~ c o m p l e x way on the compressor speed, S, and inlet volumetric flow, Q. These r e l a t i o n s h i p s are usually r e p r e s e n t e d by p e r f o r m a n c e maps w h i c h are plots of polytropic head and efficiency as f u n c t i o n s of Q at a variety of speeds. For the m a t h e m a t i c a l model, t h e s e p e r f o r m a n c e maps are curve fit by cubic p o l y n o m i a l s as follows: Polytropic
Head:
H/S 2 = h 0 + h I ( Q / S ) + h 2 ( Q / S ) 2
Polytropic
Eff.:
Ep = e 0 + el(Q/S)
+ h3(Q/S)3 ,
+ e2(Q/S) 2 + e3(Q/S)3.
The p o w e r c o n s u m e d by a c o m p r e s s o r t r a i n
is given by
PWR = (WH/(EpEM)+AL)/ET, where E M and E T are mechanical e f f i c i e n c y and the thermal e f f i c i e n c y of the gas turbine, respectively, and A L is the a u x i l i a r y load. The thermal e f f i c i e n c y depends on the r a t i o s of (i) shaft power to rated power a d j u s t e d for site c o n d i t i o n s and (2) t u r b i n e speed to rated speed. The m a t h e m a t i c a l model used to r e p r e s e n t t h e r m a l e f f i c i e n c y is a cubic polynomial in these two v a r i a b l e s w h o s e c o e f f i c i e n t s are o b t a i n e d by a curve fit. The fuel use rate,
FU, for a c o m p r e s s o r
train
is g i v e n by
FU = PWR/HV, w h e r e HV is the Compressor
(lower)
h e a t i n g v a l u e of the fuel gas.
s t a t ~ o n Mode~
The p u r p o s e of the c o m p r e s s o r s t a t i o n m o d e l in the GPO is to determine the n u m b e r of c o m p r e s s o r trains to o p e r a t e and to c a l c u l a t e the fuel use rate for each train. O t h e r p a r a m e t e r s that are c a l c u l a t e d by the model and subsequently used to c h e c k constraints on the c o m p r e s s o r trains include speed, c o m p r e s s o r p o l y t r o p i c efficiency, required shaft power, available shaft power and turbine thermal efficiency. The station e v a l u a t i o n is done at given inlet c o n d i t i o n s of pressure, t e m p e r a t u r e and gas c o m p o s i t i o n and at given o u t l e t c o n d i t i o n s of pressure and mass flow rate. A c h e c k is done for feasibility with respect to m i n i m u m and maximum speed limits, stonewall and surge limits and a v a i l a b l e shaft power. The c o m p o n e n t s of the c o m p r e s s o r station model are several identical compressor trains which are c o n f i g u r e d in parallel, s t a t i o n inlet and o u t l e t r e s i s t a n c e elements (to a c c o u n t for p r e s s u r e losses due to station piping, filters and coolers) and an o u t l e t c o o l e r (which is m o d e l e d by a d i s c h a r g e t e m p e r a t u r e setpoint). Compressor fuel is taken from the p i p e l i n e at the suction slde of the station. Constraints Most of the c o n s t r a i n t s are a s s o c i a t e d w i t h the compressors, are additional important pressure constraints associated p i p e l i n e and the external regulators. i. 2.
but there with the
For each pipe leg, the p i p e l i n e p r e s s u r e must be b e t w e e n the least allowable o p e r a t i n g pressure, LAOP, and the m a x i m u m a l l o w a b l e operating pressure, MAOP, for the leg. For d e l i v e r y external regulators, there is a m i n i m u m p r e s s u r e constraint.
128 3.
For an active c o m p r e s s o r station, the c o n s t r a i n t s are: o s p e e d is between upper and lower bounds, o the ratio of flow to s p e e d is b e t w e e n the station surge and stonewall limits, and o the power consumed is less than the total p o w e r a v a i l a b l e from the active gas t u r b i n e drivers.
N e t w o r ~ S t a t e Equations The d y n a m i c p r o g r a m m i n g a l g o r i t h m uses the network state equations to determine the suction pressure and temperature at the downstream c o m p r e s s o r station(s), g i v e n the d i s c h a r g e p r e s s u r e and temperature at the upstream station. The r e s u l t s o b t a i n e d by s o l v i n g these equations are used by the c o m p r e s s o r s t a t i o n model to d e t e r m i n e the number of trains to run w h i c h will m i n i m i z e the o p e r a t i n g cost of the d o w n s t r e a m station(s). The primary v a r i a b l e s w h i c h d e s c r i b e a n e t w o r k state are: I. 2. 3. 4. 5.
the flows through external regulators, the flows through pipe legs, the flows through c o m p r e s s o r stations, the p r e s s u r e s at nodes, the t e m p e r a t u r e s at leg ends and at the s u c t i o n and d i s c h a r g e sides of c o m p r e s s o r stations.
T h e s e are c o n s i d e r e d to be the p r i m a r y state v a r i a b l e s b e c a u s e once they are known all other v a r i a b l e s of interest, such as c o m p r e s s o r station head, speed, p o w e r and fuel use rate, can be e v a l u a t e d using the formulas given above. A system of e q u a t i o n s w h i c h when p r o p e r l y f o r m u l a t e d is s u f f i c i e n t to u n i q u e l y determine the values of the primary network v a r i a b l e s c o n s i s t s of the following: i. 2. 3. 4. 5. 6.
This
For each external regulator, p r e s s u r e and mass flow rate s e t p o i n t equations. For each leg, the steady s t a t e m o d e l i n g e q u a t i o n as g i v e n above. For each c o m p r e s s o r station, d i s c h a r g e p r e s s u r e and t e m p e r a t u r e setpoint equations. For each node, a mass b a l a n c e e q u a t i o n w h i c h requires that the net flow at a node is zero. For each supply external regulator, a t e m p e r a t u r e s e t p o i n t for the supply flow. For each node, a t e m p e r a t u r e m i x i n g e q u a t i o n w h i c h r e q u i r e s that the t e m p e r a t u r e of each flow leaving the node be equal to the w e i g h t e d average of the flows t o w a r d s the node.
is the s y s t e m of n e t w o r k state equations.
The s y s t e m of n e t w o r k state equations is nonlinear because the leg equatlons are nonlinear. The s y s t e m is solved by an i t e r a t i v e scheme w h i c h l e a p - f r o g s between the pressure/flow and temperature equations. Each i t e r a t i o n is as follows: o W i t h t e m p e r a t u r e held fixed, the e q u a t i o n s (i), (2) and (3) are l i n e a r i z e d about the c u r r e n t v a l u e s of p r e s s u r e s and flows and the l i n e a r system is solved for the c h a n g e s in p r e s s u r e s and flows. The changes in p r e s s u r e s and flows are then added to the c u r r e n t values. This is just one step of Newton's method. o W i t h p r e s s u r e s and flows held fixed, the t e m p e r a t u r e s are c a l c u l a t e d using a G a u s s - S e i d e l s u b - i t e r a t i o n on e q u a t i o n s (5) and (6). The iteration is d e c l a r e d to have converged when pressures, flows, and t e m p e r a t u r e s are b e l o w a s p e c i f i e d
the changes tolerance.
in
It is o b s e r v e d from the system of n e t w o r k state e q u a t i o n s that the only controls that a pipeline operator has over the p i p e l i n e s t a t e are r e p r e s e n t e d by the setpoints in t h o s e equations and the number of c o m p r e s s o r trains to operate at each station. Normally, however, all the external regulator setpoints are considered to be fixed boundary conditions. The available c o n t r o l s are t h e r e f o r e reduced to d i s c h a r g e
129 pressure setpoint for each compressor station c o m p r e s s o r trains to operate at each station.
and
the
number
of
IMPLEMENTATION
T h e e x i s t e n c e of a diverging b r a n c h in the HBJ n e t w o r k necessitates the use of a backward recursive relationship in w h i c h the total m i n i m u m o p e r a t i n g cost is computed from the d o w n s t r e a m end of the p i p e l i n e to the u p s t r e a m end. This r e l a t i o n s h i p r e q u i r e s c o m p u t i n g the m i n i m u m operating cost of all d o w n s t r e a m c o m p r e s s o r s t a t i o n s for each station and every d i s c h a r g e p r e s s u r e grid polnt. T h e first step in the solution p r o c e s s i n v o l v e s s o l v i n g the n e t w o r k state e q u a t i o n s for a district, w h e r e a d i s t r i c t is the s u b n e t w o r k b e t w e e n each c o m p r e s s o r s t a t i o n and the first d o w n s t r e a m c o m p r e s s o r station on each branch. The compressor stations themselves are not included in the districts. If no d o w n s t r e a m c o m p r e s s o r s t a t i o n exists on a branch, then the last d e l i v e r y point on that b r a n c h is s e l e c t e d as the b o u n d a r y of the district. The equations are s o l v e d using the discharge pressure and t e m p e r a t u r e as the boundary c o n d i t i o n s at the u p s t r e a m c o m p r e s s o r station and the mass flow rate as the boundary condition at the downstream station(s). The value of d i s c h a r g e p r e s s u r e is o b t a i n e d from the grid, d i s c h a r g e t e m p e r a t u r e is a fixed setpoint, and the flow is the sum of d o w n s t r e a m d e l i v e r i e s and e s t i m a t e d d o w n s t r e a m fuel offtake. The a l g o r i t h m used to d e t e r m i n e the o p t i m a l d i s c h a r g e p r e s s u r e setpoints at the c o m p r e s s o r stations, and the n u m b e r of t r a l n s to o p e r a t e at each c o m p r e s s o r station, can now be s t a t e d as follows: 1.
2.
3.
4.
5.
6.
7. 8.
S e l e c t a district, in the order described in the Diverging Branches section, to w h i c h the b a c k w a r d r e c u r s i v e r e l a t i o n s h i p should be applied. The o r d e r i n g p r o c e d u r e s e l e c t s the districts s t a r t i n g at the d o w n s t r e a m end(s) and w o r k i n g upstream. For every discharge p r e s s u r e g r i d p o i n t at the u p s t r e a m station in the district, use the n e t w o r k model to d e t e r m i n e the suction p r e s s u r e and t e m p e r a t u r e at the downstream station on each b r a n c h w i t h i n the district. At the d o w n s t r e a m s t a t i o n on a b r a n c h w i t h i n the district, use the compressor s t a t i o n model to d e t e r m i n e the n u m b e r of trains to o p e r a t e which will minimize the operating cost of that station for a s e l e c t e d discharge pressure grid point. The p r i m a r y variables used in this c o m p u t a t i o n are: o S u c t i o n pressure and t e m p e r a t u r e , o D i s c h a r g e pressure, and o Mass flow rate t h r o u g h the station. A d d the minimum cost of operating the downstream station, obtained in step (3), to the minimum cost of o p e r a t i n g all s t a t i o n s d o w n s t r e a m from it (which has a l r e a d y been c o m p u t e d and s t o r e d by the dynamic p r o g r a m m i n g algorithm). R e p e a t steps (3) and (4) for e v e r y d i s c h a r g e p r e s s u r e grid point at the d o w n s t r e a m s t a t i o n and s t o r e the m i n l m u m of all the costs o b t a i n e d in step (4). The s t o r e d r e s u l t is the m i n i m u m cost of operating the d o w n s t r e a m station on that branch, and all s t a t i o n s d o w n s t r e a m from it (in the entire network), for the discharge pressure grid p o i n t s e l e c t e d at the u p s t r e a m station of the district. R e p e a t step (5) for each b r a n c h w i t h i n the district and store the resulting costs. This step p r o v i d e s the m i n i m u m cost of o p e r a t i n g all the d o w n s t r e a m s t a t i o n s in the district and the network, for the d i s c h a r g e p r e s s u r e grid p o i n t s e l e c t e d at the u p s t r e a m station. R e p e a t step (6) for e v e r y d i s c h a r g e p r e s s u r e grid point at the u p s t r e a m station of the district. R e p e a t step (7) for each d i s t r i c t in the network. This step provides the m i n i m u m cost of o p e r a t i n g all the s t a t i o n s in the network.
130 Bvmassinq
r
Stations
The d y n a m i c p r o g r a m m i n g a l g o r i t h m i m p l e m e n t e d for the HBJ s y s t e m provides a special capability to a l l o w b y p a s s i n g c o m p r e s s o r stations. When a s t a t i o n is bypassed, the d i s c h a r g e p r e s s u r e at that s t a t i o n is equal to the s u c t i o n pressure. It is not p o s s i b l e to s a t i s f y this c o n d i t i o n with a s t r a i g h t f o r w a r d i m p l e m e n t a t i o n of d y n a m i c p r o g r a m m i n g b e c a u s e discharge pressures are forced to be at g r i d p o i n t s w h e r e a s s u c t i o n pressures are c a l c u l a t e d from u p s t r e a m d i s c h a r g e p r e s s u r e s and, in general, will not be at g r i d points. One w a y to s a t i s f y this c o n d i t i o n is to add new grid points at each station and then set the d i s c h a r g e p r e s s u r e at the grid points equal to the s u c t i o n p r e s s u r e at that station. The d r a w b a c k to this approach is that the slze of the p r o b l e m c o u l d i n c r e a s e by a factor of almost N/2, w h e r e N is the number of s t a t i o n s in the system. Consequently, a h e u r i s t i c t e c h n i q u e has been d e v e l o p e d to reduce the number of extra grid points c r e a t e d when a station is bypassed. This technique estimates a lower b o u n d for the operating cost of all d o w n s t r e a m c o m p r e s s o r stations when the c u r r e n t c o m p r e s s o r s t a t i o n is bypassed. If this lower bound is greater than the m i n i m u m o p e r a t i n g cost of the c u r r e n t s t a t i o n and all d o w n s t r e a m c o m p r e s s o r stations, then the c u r r e n t compressor station is not a c a n d i d a t e for bypassing. Diverqing
B~anches
The presence of diverging branches in the network creates two complications in the a p p l i c a t i o n of the d y n a m i c p r o g r a m m i n g algorithm: First, the r e c u r s i v e r e l a t i o n s h i p for serial networks must be modified. However, as shown below, the m o d i f i c a t i o n is a s t r a i g h t f o r w a r d extension of the r e c u r s i v e r e l a t i o n s h i p for an N s t a g e serial process: FUn+I(PDn+I) where
=ZMINIMUM m PD m
m TRn+I, m FUn+I(PDn+I) PS m PS N
(COm(PSm,PDm)
+ FUm(PDm) )
for n = l , . . . , N
= index for the d o w n s t r e a m stations in the d i s t r i c t whose u p s t r e a m s t a t i o n is n+l, = transition function: used to d e t e r m i n e the suction p r e s s u r e at the d o w n s t r e a m station m for a g i v e n d i s c h a r g e p r e s s u r e at s t a t i o n n+l. = minimum o p e r a t i n g cost of stations 1 t h r o u g h n, when the d i s c h a r g e p r e s s u r e at s t a t i o n n+l is PDn+l, = TRn+I,m(PDn+I) , PDN+ 1 = s u p p l y p r e s s u r e setpoint,
and, for each v a l u e of PSm, the m i n i m i z a t i o n v a l u e s of PD m.
is over all possible
Second, the ordering of the compressor stations for the dynamic programming algorithm must be determined. When backward recursive r e l a t i o n s h i p is a p p l i e d to a serial network, the stations are simply o r d e r e d from 1 to n, starting at the d o w n s t r e a m end and w o r k i n g upstream. For n e t w o r k s w i t h d i v e r g i n g branches, however, d e t e r m i n i n g the ordering of the stations becomes a more i n v o l v e d i m p l e m e n t a t i o n problem, and t h e r e f o r e new t e r m i n o l o g y must first be introduced. Let S d e n o t e the set of stations and E denote the set of pipeline sections connecting the stations. (Each branch w i t h i n a d i s t r i c t is a p i p e l i n e section.) A "simple path" in [S,E] is a s e q u e n c e of a l t e r n a t i n g
131 s t a t i o n s and p i p e l i n e sections in w h i c h no s t a t i o n or p i p e l i n e section is repeated. A "tree" is a c o l l e c t i o n of stations and pipeline sections which have only one simple path b e t w e e n each p a i r of stations [7]. The labels or sets of pointers w h i c h will be u s e d to d e s c r i b e a tree can now be defined. Assume that station r is a root s t a t i o n (a station c o n n e c t e d to a supply point). T h e " d i s t a n c e " of s t a t i o n i, denoted Di, is the n u m b e r of plpeline s e c t i o n s in the s i m p l e path from station i to the root station. Dr is d e f i n e d to be zero. The "predecessor" of station i, d e n o t e d Pi, is the s t a t i o n on the s i m p l e path from station i to the root station whose d i s t a n c e is Di-l. Pr is d e f i n e d to be zero. A s t a t i o n j is called a "successor" of s t a t i o n i if i lies on a simple path from j to the root, and if Dj = Di+l. The "order" of the stations determines the sequence in w h i c h the b a c k w a r d r e c u r s i v e r e l a t i o n s h i p is a p p l i e d to the stations. The b a c k w a r d r e c u r s i v e r e l a t i o n s h i p is applied to a s t a t i o n with order (k+l) a f t e r it has been a p p l i e d to a station with order k. The ordering of the s t a t i o n s is o b t a i n e d by simply choosing the s t a t i o n (from among those not a l r e a d y selected) w i t h the largest distance label. T i e s are broken arbitrarily. The HBJ p i p e l i n e system currently has a tree s t r u c t u r e with one supply point, many delivery points and one d i v e r g i n g branch. To allow for future expansion, it is assumed that multiple dlverging branches are permissible.
SUMMARY
A d y n a m i c p r o g r a m m i n g based o p t i m i z e r p r o v i d e s an effective method for optimizing pipeline networks with tree structures. The a l g o r i t h m is robust and the solution o b t a i n e d is g l o b a l l y optimal. The o p t i m i z e r can solve a large class of pipeline o p t i m i z a t i o n problems.
132 REFERENCES 1. 2. 3. 4.
5.
6. 7.
J.G. Wilson, J.R. Mallinson, J.E. Cheney, "Simulation and Optimization of Gas Transmission Systems", 1986 International Gas Research Conference R.E. Bellman, "Dynamic Programming", Princeton University Press New York, 1957 P.J. Wong, R.E. Larson, "Optimization of Natural-Gas Pipeline System via Dynamic Programming", IEEE Transaction of Automatic Control - 13, October, 1968, p. 475-481 R. Tsal, E. Gordon, K. Simpson, R. Olson, "Optimal Gas Pipeline Design via Dynamic Programming with Variable Stages", Presented at the PSIG Annual Meeting, New Orleans, Louisiana, October 30-31, 1986 J. Gilmour, "The Development of an Optimization Program: New operating Strategies Through Application of the Optimization Concept", Presented at the PSIG Annual Meeting, Albuquerque, New Mexico, October 24-25, 1985 P.B. Percell, M.J. Ryan, "Steady-State Optimization of Gas Pipeline Network Operation", Presented at the PSIG Annual Meeting, Tulsa, Oklahoma, October 22-23, 1987 A. Ali, R. Helgason, J. Kennington, H. Lall, "Primal Simplex Network Codes: State-of-the-Art Implementation Technology," NETWORKS, Vol. 8, (1978) 315-339.
TWO EFFECTIVE APPROACHES FOR, HYDROELECTRIC GENERATION SCHEDULING jShi-Chung Chang, Wei-Lin Yang, I-Kong Fong, Chun-Hung Chen Dept. of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10764, R.O.C. ABSTRACT This paper presents and compares two systematic and effective approaches for the hydroelectric generation scheduling problem (HSP). The objective of the HSP is to determine the best substitution of hydro energy for thermal energy in electric generation so that the fuel cost is minimized while meeting the system load and constraints. In the first approach, HSP is formulated as an optimal control problem, for which multiplier method-based differential dynamic programming algorithm developed by Chen et. al. ([CCF88 D is adopted as the solution algorithm. The second approach exploits the network structure of the water balance relation and takes linear approximation of the first formulation to formulate HSP as a minimum cost, linear network flow problem. Bertsekas and Tsengs' RELAX code is applied to solve such a large---scale network optimization problem. Numerical results indicate that both approaches generate results close to TPC's empirical schedules, are efficient in computation and suggest predictive operation strategies. We address the differences between the two approaches in solution optimality and computational efficiency and provide necessary information for tradcoffs in selecting a suitable approach. I. INTRODUCTION tIydroelectric plants account for 16% of Taiwan Power Company (TPC)'s total generation capacity. Although it is far from dominating the supply of electricity, hydroelectric generation plays a crucial role to the reliability and economy of system operation because of TPC's system characteristics: 1) fast growth of electric energy demand; 2) no interconnection with other power systems; 3) large fluctuation of daily load pattern (6,800 MW - 11,000 MW); 4) lack of abundant and long-term dependable source of hydraulic energy. Currently TPC has a barely enough generation capacity and must judiciously operate its various generation facilities to meet the demand in a reliable and economic way. Among various kinds of generation units, hydro units are well known for their quick ~-
This work was. supported in part by the National Science Council of the Republic of China under Grants NSC-77--0404-E002--03 and NSC-78-0404-E002-17.
134
response, high reliability and almost zero operation cost. The major objective of scheduling the use of limited hydro resource is therefore to minimize the thermal fuel cost while meeting the system Ioad and maintaining certain spinning reserve level. TPC now has ten hydroelectric plants, some of which are hydraulically coupled. There exists a big-head-small-tail structure among hydraulically coupled plants, i.e., upstream reservoirs/plants have higher capacities than those of succeeding reservoirs/plants. How to effectively schedule the hydroelectric generation of such a complex system is a practical and challenging problem. There are many methods for solving HSP. One of the popular approaches is to exploit the network structure of the reservoir system and formulate HSP as a large-scale mathematical programming problem - the minimum cost network flow problem ([WMM82],[LSHS1]). When linear model is further considered, there exist many efficient linear network flow algorithms to solve it. In spite of its advantage for computational efficiency, a linear model is usually unsatisfactory when the system involves significant nonlinearities and a better solution is required. The solution obtained from such a static formulation is open-loop in nature. Another popular approach, which may generate a solution with feedback nature, is dynamic programming (DP) since HSP can generally be formulated as an optimal control problem ([Yak82],[Yeh85]). However, computationally DP has the famous problem of "curse of dimcnsionality" and can hardIy handle a problem involving more than four reservoirs. Instead of direct application of DP, the so--called differential dynamic programming technique turns out to be a promising DP-based solution method for a large-scale HSP according to our assessment ([CCF90]). In this paper, we study both the minimum cost linear network flow and the optimal control formulations for TPC's ttSP. A multiplier method-based differential dynamic programming (MMBDDP) algorithm developed by Chert et. al. ([CCF88]) is used as the solution method for the latter while the RELAX algorithm developed by Bertsckas and Tseng ([BeT88]) is adopted for the former. Both algorithms are considered quite efficient for solving the corresponding problems. Our objectives are not only to examine the computational features of the two approaches but also to investigate their operational implications to TPC's HSP. TPC's data of typical summer and winter weeks is used for scheduling. Numerical results demonstrate that both approaches are indeed quite efficient. MMBDDP algorithm generates better schedules than RELAX algorithm but i t usually takes a longer computation time to reach a solution. The solution quality of RELAX can be made close to that of MMBDDP by taking a fine linear approximation at the price of a longer computation time. Resultant schedules of both approaches are generally consistent with those obtained by TPC's experienced engineers. Specifically, we observe that our schedules suggest saving more hydro generation on Sunday for higher hydro generation during weekdays. Our study also addresses tradeoffs among generality of formulation, solution optimality and computational efficiency in selecting between the two approaches for real applications.
135
The remainder of this paper is organized as follows. An optimal control problem formulation of tISP and its MMBDDP solution algorithm are first introduced in Section II. Section III then describes the linear approximation of the optimal control problem and formulates a minimum cost linear network flow problem. Numerical results, comparisons and operational implications of applying these two approaches to TPC's HSP axe given in Section IV. Section V discusses tradcoffs between the two approaches and concludes our study. II. AN OPTIMAL CONTROL APPROACH Hydro generation scheduling is well known to be coupled with its thermal counter-part. Under present TPC system operation conditions, short-term commitment changes are not allowed for most of the base- and medium-load thermal units. The scheduling of hourly thermal generation over a one-week horizon is thus reduced to an economic dispatch problem. To meet the spinning reserve requirement, TPC keeps a certain percentage of the available thermal capacities (committed units and all the peaking units) as part of the spinning reserve. For a given load level, the remaining spinning reserve capacity that should be contributed from the hydro units can then be determined. According to the aforementioned operation of the TPC system, the coupling between hydro and thermal generation schedulings is only through the constraint that total generation should meet the system load. We decouple the hydro scheduling from the thermal part by first establishing the generation cost function of meeting the system load through purely thermal generation. For each given load level, the simple A-iteration method is performed to solve the economic dispatch over the set of available units ([WoW84]) and to evaluate the pure thermal generation cost of meeting the load. In other words, we aggregate all the available thermal units into one equivalent unit and construct its generation cost function. To mathematically formulate the hydro scheduling problem, let us define a few notations: Lk: system load at the kth hour; Jk(') : generation cost function at hour k; Xik: the water volume of reservoir i at the beginning of hour k; Uik: the volume of water released from reservoir i for generation during hour k; Sik: the spillage from reservoir i during hour k; Pi(.): the water-to---energy function of power plant i; Rik: the volume of natural inflow to reservoir i during hour k; Ni: the set of the immediate upstream reservoirs of reservoir i; SRk : the hydro spinning reserve requirement at hour k.
136
In TPC's hydro system, there is no significant flow delay relative to the one-hour time increment. The hydro scheduling problem over a one-week horizon (168 hours) can then be formulated as
(HSP)
168 Min [ J = P, J k ( L k - ~ Pi(Uik))] {Uik } k=l i
subject to (1) the water balance equation P, U j k - Uik + 1ENi S s lk - S i k + R i k , Xik+l = Xik + J~Ni (2) bounds on water releases Uik -
(4) the spinning reserve requirement r, [p.(Eik ) - p . ( u i k ) ] > s a k. i
1
1
Note that if we view the water balance equation as the dynamics of the reservoir system, the above (HSP) formulation turns out to be a discrete time, deterministic optimal control problem with constraints on both control (u and s) and state (x) variables. An effective, multiplier method-based differential dynamic programming algorithm (MMBDDP) is adopted to solve (HSP). It basic idea is to relax all constraints but the system dynamics by using the multiplier method. For a given set of multipliers, ~here is an unconstrained dynamic optimization problem to which the differential dynamic programming ([YaR84]) applies effectively. The optimal solution is obtained by iteratively updating multipliers and solving the corresponding dynamic optimization problems. Details of the MMBDDP algorithm can be found in [CCF88]. lII. A LINEAR NETWORK FLOW APPROACII It has been recognized in the literature that constraints of (HSP) have a network structure due to the nature of the water balance equation ([LSHS1],[WMM82]). To convey the basic idea, let us consider a cascaded, two-reservoir, two-plant system (Figure 3.1) over two periods of time. Figure 3.2 illustrates the network representation of the flow balance equation of this exemplary system: a node represents a reservoir at a time period; the vertical incoming (outgoing) arcs represent natural inflows, water releases and spillages immediately from the upstream (to the down stream) reservoirs; the horizontal incoming (outgoing) arcs represent the water volume at the beginning of the present (next) period of time. The flow conservation relationship at each node of the network thus corresponds exactly to the water balance of each reservoir. Bounds on water releases and storage volumes are modeled as bounds on arc capacities. Such a network
137
representation can easily be constructed for a general multi-reservoir system.
Q
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,,
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,.
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[ I
NATURAL INFLOW
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INFLOW RATE
Fig. 3.1 h Cascaded, Two-Reservoir, T~o-1)lant Hydro System.
Fig. 3.2 Network Representat iol~ over Two Stages.
Note that when tile water--t(>--energy conversion function P(.) is linear, the spinning reserve constraint can be easily incorporated into the network representation except that it creates arcs with gains, i.e., the network is no longer ordinary but generalized. In the following discussions, the spinning reserve constraint is not considered explicitly. We further approximate the cost function of (HSP) by a piecewise linear cost function so that the approximate (lISP) fits into the framework of a linear, minimum cost network flow problem. The key step of the approximation is to calculate the marginal generation cost of each hydro power plant. This is done by first equally dividing the range of water release of each plant into M intervals defined by grid points Uikl = l*(Uik -U-ik)/M' l=0,...,M. Note that the cost function in (HSP) is not separable with respect to water release variable Uik'S of individual power plants. The marginal generation cost of a plant therefore depends not only on the generation level of the plant but also on the total hydro generation level and the load. Motivated by the operational consideration that the hydro spinning reserve capacity is kept as high as possible, we assume that hydro units are loaded proportionally according to their generation capacities. Under such an assumption, the marginal cost Cikn for Uik e [Kik(n-1)' Uikn] is approximated by dJ I dTk Cikn = ~fr~k ILk- ~i Pi(Uik(n-1))] " u~-ik ' where T k = Lk - Ei Pi(Uik) is the thermal generation level. The derivative ~dJ
is the marginal
138 dT k
cost with respect to the thermal generation level while ~-ikl~ik n is the thermal power substitution factor of using water from reservoir i at time k.
The approximate cost function
^
Jik(Uik) of unit i is then defined as Jik(Uik) = Jik(~ik(n_l)) + Cikn.(Uik - Uik(n_l)), Uik e [Uik(n_l),Uikn], n = 1,...,M. The approximate total cost function is now separable and piecewise linear with respect to the decision variables. A final step of modeling is to make the transformation that M Uik = l~lUikl, where Uik1 E [0, ~ i k l - Uik(1-1) ]' I=I,...,M, i.e., replacing one arc variable by multiple arc variables.
Under the assumption that Jk(. ) is
convex, problem (HSP) can then be approximated by a linear minimum cost flow formulation
(MCF)
168 M Min E P. Y:. CiklUikl, {Uikl} k=l i l = l
subject to bounds on UikI and conservation of flow. We adopt the KELAX code devcloped by Bertsekas and Tscng to solve (MCF). It relaxes the flow conservation constraint to form a dual problem, iteratively improves the dual cost, and operates in a manner that is reminiscent of coordinate ascent and Gauss-Seidcl relaxation method. Rcaders may refer to [BET88] for the details. IV. EXEMPLARY SCtIEDULING RESULTS In this section, we apply the above two approaches to TPC's hydroelectric generation scheduling. The spinning reserve constraint is not considered in either approach for fairness of comparison. We examine the hydro scheduling of one summer week and one winter week, which are representative of the wet and dry seasons respectively. Ten grid points are taken for each decision variable in formulating (MCF). Various test cases are designed to investigate the influences of the number of plants and scheduling horizon on the two approaches, where the 10-reservoir cases simulate TPC's full scale hydro system. Each ease involves six runs with load variations of 0%, 3%, 7%, 10%, -5% and -10% respectively. Please refer to [CCF90] for the detailed data that we used. Both RELAX and MMBDDP codes were written in FORTRAN and all our numerical computations are performed on a VAX/780 minicomputer. The CPU time statistics of the two codes are listed in Table 4.1 and scheduling costs are in Table 4.2. We observe from Table 4.1 that the RELAX code on (MCF) takes 3-19 times shorter CPU time to convergence than the MMBDDP code on (HSP) does. Although this is a rough comparison since the convergence
139
criteria of the two are not the same, the result meets our expectation because (MCF) approximates (HSP) and RELAX codes exploits the network structure. If we take the schedules obtained from applying the RELAX code and substitute them into the nonlinear cost function of (HSP), we find that these costs differ from costs of MMBDDP schedules by less than 0.0570 for all our testing cases (Table 4.2). This indicates that the (MCF) using 10 grid points per decision variable provides a close approximation of (HSP). E~bAX ~BDDP Scheduling NO. of Horizon Plants Average CP~ Standard Average CPU Standard Time (sec.) Deviation Time (see.) Deviation 24 hrs.
3
19.56
4.75
5.94
0.35
24 hrs.
7
I00 ,!2
42.38
14.54
0.71
24 hrs.
i0
309. O3
64.44
15.97
0.52
72 hrs.
tO
I08~.00
173.00
102.39
8.21
10(WkS) 3317.66
305.68
511.17
49.68
168 hrs.
Table 4.1 CPU Time S t a t i s t i c s . Scheduling Nf. of Horizon Plants
Standard Average (R) - (M) Deviation
Average Standard (~)-(~)/(~) ~viation
24 hrs.
3
4.442e4
1.222e4
0.0345 Z
0.0055
Z
24 hrs.
7
2.fi55e4
1.624e3
0.0223 %
0.0027
Z
24 hrs.
lO
3.275e4
2.405e3
0.0260 Z
0.0017
Z
72 hrs.
lO
8.688e4
3.608e3
0.0233 %
0.0018
Z
168 hrs.
10(~tS)
3.464e5
1.309e4
0.0497 Z
0.0080
Z
(R): Scheduling Cost by ItELAX (M): Scheduling Cost by MMBDDP Table 4.2 Scheduling Cost Statistics. We notice that computation time of the MMBDDP code is approximately linearly proportional to the scheduling horizon and to the cube of number of hydro plants.
The time
complexity of the RELAX code depends on many factors such as number of nodes, number of arcs, range of cost coefficients, and range of flows ([BET88]). In the (MCF) problem, the number of nodes is determined by the number of reservoirs and the time horizon while the number of arcs depends on the topology of the hydraulic system, the number of grid points and the time horizon. From Table 4.1, the computation time of RELAX seems to grow quadratically with respect to the scheduling horizon. Schedules generated by our two algorithms are further compared with TPC's schedules. Fig.4.1 contrasts the three in terms of total hydro generation profiles for the winter week. All of them allocate hydro energy to reduce the need for thermal generation at peak hours. The diffcrenee between them is that schedules from our two algorithms suggest more savings of hydro energy ou Sunday and yield higher hydro generation during the peak hours of weekdays than
140
TPC's schedule. Although this comparison may not be fair since our algorithms utilized data that might not have been available when TPC's schedules were done, our approaches do provide systematic and efficient ways to generate a predictive schedule. i9 tlirllll-
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1200 i i 1000'
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,
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9
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160
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Figure 4.1 Contrast of Three Hydro Generation Schedules. We now take a closer look at the detailed schedules of individual units.
Figure 4.2
dcmonstrate the I ~ L A X and MMBDDP schedules for Ku-Kuan and Tien-Lun over a Sunday of the Summer week. The schedules are quite similar except that the MMBDDP schedule is less "blocky" than the RELAX schedules because of the linearizagion and approximation involved in (MCF).
The schedule out of RELAX can be made closer to MMBDDP by increasing the number ~llUlll"
;}linlhltJ ltlMII[15P|
~,,IH~l
~,liiiJl,i,J (ll[!ll'l~)
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'7 7 I
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hlUl'
Figure 4.2 Linear Approximation Effect.
-. ~l-I~ll,lll
ll: lil:ll-l.lill
hour
l~ll
141
of grid points of controls at the price of a higher problem dimension and longer computation time. This observation holds for all of our test cases. We further confirm the predictive nature of the two approaches by examining the Monday and Tuesday schedules (Figure 4.3) of the same week as that in Figure 4.2. The water storage and release capacities of the Tien-Lun plant are smaller than the K u - K u a n plant. Our results suggest an operation strategy for such a big-head-small-tall strategy that the storage level of Tien-Lun reservoir is lowered in the evenings of weekdays so that the Ku-Kuan plant can run in full generation capacity the next day during peak hours without causing forced spillage at Tien-Lun. This strategy applies to other pairs of big-head-small-tail plants. ~,~*cr RQrL and Tur (I'IPISDDP)
5~mmmm~r l~atl, 4rid Tuc. (RCLI%:D
k 140
t601-
140
L40 l-
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60
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io
15
2L)
-, Ku-K~an o, Ticn-L,~n
Z5
50
30
h,lgr
Figure 4.3 Optimal Schedule for a Big-Itead-Small-Tail Pair of Reservoir. V.CONCLUSIONS We have presented in this paper two approaches for the hydroelectric generation scheduling problem (HSP): the optimal control formulation with MMBDDP solution algorithm and the minimum cost, linear network flow formulation with the RELAX algorithm. The optimal control approach used a more accurate model of HSP than the linear network flow approach and was expected to result in a better schedule at the price of longer computation time. Application of the two approaches to TPC's weekly HSP justified such a prediction. Our numerical study was performed on TPC's system data of two typical weeks. Through reasonably extensive test runs, the MMBDDP algorithm generated all its schedules within one hour while the RELAX needed no more than ten minutes for scheduling a 10-reservoir system over 168 hours on a VAX/780 minicomputer. Considering the fact that TPC uses a mainframe computer which is at least an order of magnitude faster than the mini, we believe that both
142
algorithms are efficient for real applications. Furthermore, the resultant schedules not only are basically consistent with the schedules from TPC's expericpredictive operation strategies for TPC's system. In our study, the network flow approach generated schedules not so inferior to those from the optimal control approach while generally requiring much less computation time. Rescheduling of a whole week can be done very quickly by using the RELAX algorithm once the system states, such as load profile and amount of natural inflow, deviate from what are predicted. Although the MMBDDP algorithm schedules with certain feedback nature, the possibility of frequent rescheduling by using RELAX seems to make the linear network flow approach a better approach of the two. IIowever, the linear network flow approach easily falls short when any non-network type of constraints come into the problem formulation. Even when the linear spinning reserve constraint is added to consideration, a new solution algorithm for the resultant generalized network problem has to be developed. The MMBDDP algorithm, on the contrary, is applicable to a wide range of HSPs without requiring any extensions. REFERENCES [Ber82] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 1982. [BET88] D. P. Bertsekas and P. Tseng, "Relaxation methods for minimum cost ordinary and generalized network flow problem," Operations Research, Vol. 36, No.l, pp. 93-144, 1988. [CCF88] C. H. Chen, S. C. Chang, I. K. Fong, "A multiplier method-based differential dynamic programming algorithm," Proceedings of the 12th National Symposium on Automatic Control, Hsin-Chu, Talwan, Dec. 1988, pp. 481-491. [CCF90] S. C. Chang, C. I-I. Chen, I. K. Fong, P. B. Luh, "Hydroelectric generation scheduling with an effcctive differential dynamic programming algorithm," To Appear in IEEE Trans. on Power System Engineering, paper # 90 WM 014-1 PWRS. [LSH8I] D. Loucks, J. Stedinger, D. Halth, Water resource systems planning and analysis, Prentice---tIall, Englewood Cliffs, NJ, 1981. [WMM82] F. Wakamori, S. Masui, K. Morita, T. Sugiyama, "Layered network model approach to optimal daily hydro scheduling," IEEE Trans. on PAS, Vol. PAS-101, No. 9, 1982. IWoW84] A. J. Wood and B. F. Wollenberg, Power Generalion Operalion & Control, New York: Wiley, 1984. [Yak82] S. Yakowitz, "Dynamic programming applications in water resources," Water Resource Research, Vol. 18, No. 4, pp. 673---696, 1982. [YaR84] S. Yakowitz and B. Rutherford, "Computational aspects of discrete--time optimal control," Appl. Math. Comput. Vol. 15, No.l, pp. 29---45, 1984. [Yeh85] W. W--G Yeh, "Reservoir management and operations models: state-of-the-art review," Water Resources Research, Vol. 21, No. 12, pp. 1797-1818, 1985.
ITERATIV'E SOLUTION OF A FREE-BOUNDARY ARISING IN MICROSCOPIC
PARTICLE MANIPULATION
PROBLEM
INSIDE A LIQUID LAYER
P.K.C.Wang I and E.Behaegel 2 Department of Electrical Engineering University of California Los Angeles, California 90024, U.S.A. A.K.gunchal Analytic and Computational Research Inc. 3106 Inglewood Blvd. Los Angeles, California 90066, U.S.A. Abstract A free-boundary problem arising from the manipulation of microscopic particles immersed in a liquid film by means of a gas jet is considered. The objective is to determine the total force acting on the particle due to the flow inside the liquid induced by the gas jet. An iterative algorithm for the numerical solution of this problem is presented. The basic approach is to solve the flows in the liquid and gas separately with an assumed free-boundary shape, and then make corrections for the free-boundary shape using the stress balance conditions at the liquid gas interface. Some typical numerical results are presented.
1. I N T R O D U C T I O N In biomedical research and biosubstance processing (eg. separation of macro molecules or cells), it is of interest to select and manipulate particles or living organisms which are microscopic in size ranging from a few millimeters to several microns in diameter. In the case of living cells or organisms, it is essential that they are kept alive by immersing them in a liquid layer at all times. Due to strong surface tension of the liquid, it is undesirable to rupture the surface during manipulation. Recently, it was proposed that the particle manipulation be achieved by means of liquid surface deformations induced by controlled gas microjets [1]. This approach leads to a free-boundary problem involving the interaction of a gas jet with a liquid layer. In tlfis paper, an iterative algorithm for the numerical solution of this problem is presented. Attention is focused on the steady-state case. In the development of this paper, we begin with a brief description of the physical problem. It is followed by the mathematical formulation of the free-boundary problem. Then an iterative algorithm for the numerical solution of this problcm is presented. The paper concludes with some typical numerical results.
2. D E S C R I P T I O N
OF PHYSICAL PROBLEM
We consider the manipulation of microscopic particles immersed in a thin stationary liquid layer bounded below by a rigid plane and above by an interface separating the liquid from the gas. We as1Work supported by National Science Foundation Grant ECS 87-18473. 2Visiting Scholar supported by SAGEM-Centcr D'Eragny, Ccrgy-Poatoise cedex, France.
146 sume that all the particles are rigid and essentially spherical in shape and have approximately the same diameter. Moreover, the density of the particles is assumed to be approximately equal to that of the liquid so that the particles are completely submerged inside the liquid layer. We further assume that the tkickness of the liquid layer is slightly greater than the mean diameter of the particles so that only a single layer of particles can exist inside the liquid layer. The objective is to steer a specified particle to a specified target position along a given path. This is accomplished by introducing a moving probe consisting of a cylindrical gas microjet with an interior suction port near the liquid surface as shown in Fig.1. The basic idea is to introduce an appropriate amount of suction and outflow of gas in the probe such that the induced surface deformation of the liquid traps the particle inside a region directly below the probe. Figure 1 shows a sketch of the streamlines of the gas and liquid flows. The two minimum points of the interface deformation h correspond to the high-pressure stagnation points of the gas jet flows. The maximum point of h corresponds to a minimum pressure point of the suction flow. Due to viscous coupling between the gas and the liquid at their interface, circulating flow is induced inside the liquid layer. When the particle center is exactly on the suction-port axis, the circulatory flows are symmetric with respect to this axis. Consequently, the fluid pressure on the particle surface is also symmetric about the suction-port axis. Thus, there is no net force on the particle induced by the fluid motion. Now, we introduce a small horizontal displacement of the probe while keeping the particle fixed at the original position. In this case, the circulatory flows of the liquid about the particle are no longer symmetric. If the displacement is sufficiently small, there exists a net horizontal force which tends to restore the particle center to its equilibrium position along the probe axis. Thus, we expect that if the probe moves at a sufficiently low velocity, the particle will follow the probe movement. In this paper, attention is focused on the computation of the restoring force ou the particie for various values of jet and suction flow parameters, in particular, the gas velocities at the jet entrance and suction port, and the probe geometric parameters. Since the restoring force induces a potential-well centered about the probe axis, it is of interest to determine the shape and depth of this well as functions of the probe parameters. The foregoing problem requires the numerical solution of a steady-flow problem involving a free gas-liquid interface.
3. MATHEMATICAL
FORMULATION
OF
FREE-BOUNDARY
PROBLEM
To derive a mathematical model for the particle manipulation system, we introduce the following assumptions: (A1) Both the liquid and the gas are Newtonian viscous fluids having kinematic viscosities vI and vg; and constant mass densities P1 and pa respectively. (A2) The steady-state motions of the liquid layer and the gas are describable by the Navier-Stokes equations. (A3) The intermolecular or van der Wa,xls forces are negligible. (A4) The liquid layer is stable with respect to surface perturbations so that surface rupture does not OCCUr.
147 Let x =
('~1, Z3,553) denote the representation of a point in the 3-dimensional Euclidean space R 3 with
respect to the orthonormal basis (el,e2,e3} such that e3 is aligned with the probe axis as shown in Fig.1. Let ho denote the thickness of the unperturbed liquid layer. We normalize the length by ho. Let the rigid bottom plane be specified by SB = {x E R 3 : x3 = 0 , ( z l , z 2 ) e s
where FB is the rectangular
domain { ( z l , x 2 ) e R2 : xi,~i,, < zl < zl . . . . i = 1,2} containing the origin (zl,z2) = (0,0) as an interior point. The liquid-gas interface is denoted by SF = {x E R3 : x3 = h(xt,z2),(Zl,XZ) E FB}, where h is a C 1 nonnegative function (normalized with respect to ho) defined on s
h(xl,z2),(z2,x~) E F s } aald f / - = {x e /~3 : x3 < h(zx,z2),(Zl, X2) E s
Let fl+ = {x E R3 : x3 > The spatial domain occupied
by a spherical particle with normalized radius rp centered at xp is specified by B(xp) = {x 9 R 3 : [lx-xp[[ _< rp}, where [1. [[ denotes the Euclidean norm. Let S = {x 9 R 3 : 0 < x3 < ~3,(Xl,X2) 9 FB}, where ~3 is a specified positive number. For the single particle case, the spatial domains occupied by the liquid layer and the gas are given by s
= (f/-NS)\B(xv)
and ~2g = f~+ f3 S respectively. In f~g, the steady-state motion of the gas is described by the following dimensionless Navier-Stokes equations: 3
~--~uiDiu+ V p g - v a V 2 u = O ,
divu=O,
(1)
i=1.
where u = (ul,u~, u3) denotes the gas velocity (normalized by pgv~/h~); and D~ = O/Ozi. IIere, we have neglected the gravitational effects. Similarly, in fl/, the steady-state motion of the liquid is described by 3
C viDiv + V p / -- V / V 2 V = --gVX3,
divv = O,
(2)
i=1
where v = (vl, v2, v3) denotes the liquid velocity normalized by p/v//ho, p~ the liquid pressure (normalized by plv~/h~); and g the acceleration due to gravity (normalized by vl/ho).2 3 Assuming no slip at the rigid bottom plane $ 8 and at the particle surface, we have the following boundary conditions: v
where O ~ ( x . ) = { x ~ ~
= o
on
SsuOt~(xp),
(3)
: IIx - x~ll = ~ } .
When the liquid is confined to a container, we may also impose no slip conditions at the lateral surface SLy" = { x 9 /~3 : 0 < ~3 < h ( ~ l , ~ 2 ) , ( ~ , ~ )
9 ors}, V = 0
i.e. On SLF.
(4)
At the top boundary surface ST = {x E R 3 : x3 = k3,(xl,x2) 9 FB}, the gas velocities at the jet entrance and suction port are specified, i.e. u = (O,O,~,s)
on Ss,
u = (O,O,-~,s)
o,, Ss,
(5)
148 where u j and us are given positive constants; S j and S s are subsets of ST representing respectively the jet and suction port surfaces given by
s, = {x
-<4 + 4 -< r3o},
Ss = {x
ST: 4 + 4
-<
(6)
where r s < r j z < r j o are given normalized radii. Since tile liquid is exposed to the surrounding air, we impose the following Neumann condition on the lateral surface SLg = {x E /7a : h(zl,a:2) _< xa _< ~ 3 , ( z l , x 2 ) E FB} and the remaining portion of ST, i.e.
O(u ^ ~ ) / o n = o
on s,, %r { & ,
( & o Ss) u &~},
(r)
where n denotes the outward unit normal associated with Sg. At the gas-liquid interface SF. we have the kinematic boundary conditions: u3 = u l D l h + u2D2h
on SF,
v a = v l D i h + v2D2h
on E l .
(8)
The energy transferred from the gas to the liquid layer takes on three different forms: (i) displacement of the liquid (gravitational potential energy), (ii) increase of the surface area of the liquid (surface energy), and (iii) induced motion in the liquid due to viscous coupling. The viscous coupling is modelled by U ~. O~V
On
SF,
(9)
where a is a given positive coupling coellldent. The remaining boundary conditions are obtained from the equation for stress balance across the interface SF: q d-4fsf + sg + sst = 0,
(10)
where s.r and sg denote the stress on the interlace exerted by the fluid and the gas respectively, and s~ is the surface tension which is proportional to the mean curvature of the interface. The vector q can be resolved along the outward normal and two suitable local coordinates in the tangent plane at a point on the interface to yield three independent equations. The foregoing problem is a free-boundary problem in which the free surface SF and the gas-liquld velocity fields u = u(x) and v = v(x) are unknown quantities to be determined. The ultimate objective is to compute the hydrodynamic force f acting on the particle given by
f =
~=, fJo~(x,l"r(~)(r'(x) 9~)d(OZ3(~,))~,
(Zl)
where n ( x ) denotes the outward unit normal at the point x e 06(xp). In particular, it is of interest to determine the variables of the horizontal component of f as a function of the horizontal displacement of the particle from the probe axis.
149 4. I T E R A T I V E A L G O R I T H M Iterative solution of free-boundary problems in fluid mechanics involving gas and liquid interfaces was proposed by Ryskin and Leal [2]. They gave an iterative numerical algorithm for determining the buoyancydriven motion of a gas bubble through a quiescent liquid, and bubble deformation in an axisymmetric straining flow. Free-boundaxy problems associated with the motion of free-surface fluid flows were studied also using the marker and cell method by Harlow and Welch [3]. The free-boundary problem under consideration here is complicated by the presence of two fluids whose motions are coupled to each other at their interface. In what follows, we give an iterative numerical algorithm for the solution of this problem. The basic idea is to start with an initiaI guess for the interface SF, i.e. choose an interface shape function x3 = h ~
x2) defined on a specified domain FB along with initial guesses for the gas velocity components
ul = u~
x2) and u2 = u ~ ( x l , z 2 ) at the interface SF. From the kinematic boundary condition (7), we
obtain a guess for the gas velocity component u3 corresponding to u~ and u~. Using the interface velocity coupling condition (9) and the kinematic boundary condition (8), we obtain the corresponding guesses for the liquid velocity component vi = v ~
z2), i=1,2,3. Now, the gas (resp. liquid) velocity at the domain
boundary 0~ a (resp. Off/) is completely specified. Thus, the corresponding velocities u = u ~ v = v~
E ~2g; and
x E ~ f are uniquely determined. Consequently, the corresponding gas and liquid pressures
pg = p~(x) and p / = p~(x) are also determined. Now, we check whether the foregoing velocities and pressures satisfy the stress balance condition (10). If not, corrections for the interface shape h and the gas velocity components ua and u2 at the interface in the form:
h~ = h~ + 7 ~h~
ul = ~'~ + fl~6u~,
i = 1, 2,
are introduced, where 7 and fll axe given positive constants. The correction 6e = (Sh ~
(12) T are
determined by ,~e = - [ Oq/ Oe]lo~ q( c~
(13)
where q = (qx,q2,q3) T defined in (10); [Oq/Oe]]o is the Jacobian matrix associated with q, and c ~ = (h
o
,
o T ul,o ,,~) 9
The foregoing process is repeated. We observe that for any iterative step, the boundary-value problem for v corresponding to given interface shape h and boundary data with sufficiently small L3(~2I) norm has a unique solution in the Sobolev space H~(~f) [4]. But the boundary-value problem for u involves mixed Dirichlet and Neumann boundary conditions. The establishment of the convergence of the proposed iterative algorithm to a solution of the free-boundary problem is nontrivial. It will be discussed elsewhere.
5. N U M E R I C A L R E S U L T S Numerical solutions to the free-boundary problem are obtained for the case of a coa.xial cylindrical inner suction port and outer jet with radii r s = 0.225mm. r j r = 0.36mm, rao = 0 . S m m . and rpB = 2.1ram. (the probe outside radius). The numerical solutions to the boundary-vMue problems for the gas and liquid velocities u = u(x) and v = v(x) corresponding to specified interface shape and boundary data are
150 obtained by means of the "ANSWER," hydrodynamic software provided by the Analytic and Computational i~csearch Inc. Figure 2 shows the velocity vector field for the case where the particle center is on the probe axis. We observe the presence of vortex flows which are symmetric about the probe axis. In this case, the net force on the particle is zero. Figure 3 shows the corresponding results for the case where the particle is displaced from the probe axis. In this case, there exists a nonzero horizontal component of the hydrodynamic force exerted on the particle. The results indicate that when the displacement is sufficiently small, the hydrodynamic force tends to restore the particle to the equilibrium position where the particle center is along the probe axis. A more detailed description of the computational results will be presented in a forthcoming report.
References [1] P.K.C.Wang,'Manipulation and control of microscopic objects immersed in a fluid medium," Proc.IEEE
Micro Robots and Telcoperalors Workshop, IIyaanis, Mass. Nov. 1987, Paper No.18. [2] G.l~yskin and L.G.Leal,'Numerical solution of free-boundary problems in fluid mechanics," Part 1: The finite difference technique; Part 2: Buoyancy-driven motion of a gas bubble through a quiescent liquid; Part 3: Bubble deformation in an axisymmetric straining flow," J.Fluid Mech. Vo1.148, pp.l-17; pp.19-35 and pp.37-43, 1984. [3] F.II.Harlow and J.E.Welch,"Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface," Physics o/Fluids, Vol.8, No.12, Dec.1965, pp.2182-2189. [4] IL Tcmam,Navier Stokes Equations, Theory and Numerical Analysis, North Holland, Amsterdam, 1984, p.178.
151
GAS JET
J
SUCT] ON PORT
/
t
, ~
i
ir
INTERFACE
\
\
_~c--.'~--~-
i i ill
BOTTOM PLANE
~___ , /I
'--.'>
, , , i i , , , r l l , i i
U
7 "C
"'bt?" ((t \ !if ,J ii
GAS JET
? i
/ i
/ i
i
i
i
i
i
i
LIQUID LAYER
k "" i
I
PART lCLE
"ql"]
Xi
J I m
P
Fig.la Sketch of streamlines for the gas and liquid flows induced by the gas jet and the suction port.
SUCTION PORT
GAS JET
GAS JET
U INTERFACE ,. - 7 ~ r ~ ' = - - - . ~
,
~
,,'/ / .................
BOTTOM PLANE
.
- - - - -4- - - L
LAYER
......
I
Q
.
-- _
~4~S~-i.
,t~,~
../4,t
...
---
~,,-"
'~
;T ......
PARTICLE 0
' ....
J ..,'T
t
.............
X! ~
Fig.lb Sketch of the streamlines for the gas and liquid flows induced by the displaced gas jet and suction port.
1,52
GA.S Js
"''Li/I
~V/II Ill/ l/
ID./#1
~9.. 49
/,[/tl
1,~',';_~.,./III/
t'/ ~t i :
, l
i\ ~''--
/11/#
~ +l~'''" ......
~ I t ~ # ', *
l l ~ . -
-#
lit
I
........
-~-------,,',,\\\~..lz,[ . x \\
\x-'--..--
t
~\\\\'~'-'~,
'
/
-%%\~'~'--#a
i 39, 8E
39.165
3~
1 39.99
t
I i
I (JO. 1S
I qO. 32
]
I
gO. ~i9
VEUTORSI~II,.IEI 5.00E-Ol UNITS i
i
I
2?~''
-7, ; : . . . .
tt
"
I
~..-//t
. . . . . . .
"~
"r
.~,,
,
-411,.._" .
O
GAS ,/ET
~'IOR SOtL.~i ~'.OOE+O0UNITS SUC'CZON PORT
,~ , ''"
I - - 1
.,~\X..,
. . . . . .
'
i"
' ~"\
i "
"
,
"
Fig.2 Velocity vector field of liquid and gas with particle along the suction port axis.
153
G~
JET
~'TOR
S(~LEI
2.0(~+00 UNITS SUCTIOI~ POI~T
V i i i . / i / I
t/,7, /1
l
J""
GAS .TET
9 , , ~ "~ ", \
x/l/I/.,
,k~--.~
,,\\\\~
I i I ~ --
l,'l,
. .
.
.
•i ~ 3 . q9
t~//
%
.
.
.
.
t 39.8~
39165
~C'f~
~1.s
'I
.
.
.
I 39.99
S.OOE-OI
i
~.%
"/
#
/
I
.
.
I hlO. I S
I ~ 0 . 3 :>
I
i
YO.~9
IJIT$
I
;,,,-\\~
--////. - A " ' , , ,, , - ' M , --
_277~ .
9
~ ,
,
,
,
. . . . . .
'
9
/
l't
I
i
'
"
! i. , , , .... . . . . . .
!
::;iX'ix.
pARTICLE
.
,
i!!!i!!i ..:~
.....
'~'9. q9
I 39.65
i
i
,
~ ~ "
i'''l'','''i''','''I'' 39.82
39.99
; IJO. 15
'
i
iiO. 3e
liO. li9
Fig.3 Velocity vector field of gas and liquid with displaced particle.
OPTIMALITY CONDITIONS FOR ELLIPTIC VARIATIONAL INEQUALITIES
Liu Wenbin And J.E. Rubio School of Mathematics, University of Leeds Leeds LS2 9JT, U.K.
Suwa~ary. By
the
penalty method,
Ekeland's
Variational Principle
and
necessary conditions
of
lower-semicontinuity of some set-valued mappings, optimal
control for some abstract elliptic
obtained
in
the
conditions.
The
cases idea
where is
to
the find
convex
variational sets
optimality
inequalities are
satisfy
conditions
some
smoothness
first
for
some
penalized problems by Ekeland's Variational Principle then to pass limits to obtaln the optimality conditions. some
known
optimality
It is shown that these conditions lead to
conditions
in
necessary conditions for some problems.
many
cases.
They
also
yield
new
Our results give uniform forms for
several known optimality conditions for elliptic variational inequalities.
1.Introduction.
Optimal control problems for elliptic variational inequalities have been much
studied.
conditions,
One
of
the
main
difficulties
even though optimality
important cases (see for example,
conditions
is
to
obtain
optimality
have been obtained in some
[3],[4],[6],[7],[I0],[12],[13])o Recently,
Shi Shuzhong got some optimality conditions for abstract variational inequalities [14] [IS], which covers very wide range of problems. seem very complicated to apply.
But his results
In this paper we will, by using the
penalty
method, Ekeland's Variational Principle and lower semicontinuity of some set-valued
mappings,
problems. Then
establish
we show that
general
optimality
in many cases,
conditions
the conditions
for
this
lead to
known
results. We also show that they give new necessary conditions for a problem posed in [7] and [13]. It seems that our optimality conditions tend to unify known results. The methods used in this paper can be applied to obtain some necessary conditions for abstract strongly monotone variational inequalities
155 (see [14],[15]) without essential changes and we shall discuss this
problem
in some later papers.
2. Some studies about the lower limits of set sequences.
m
let V be a Banach space and V
its dual space.
As in iS], we give the
following definition: DEFINITION 2.1. Let A cV for n=l,2,..-. The lower limit of {A } is the set, n n denoted as lim A n-~
n=l,2,-- and
n'
given by lim A ={ y~V: there is {yn } such that yn ~ A n for n n-~
yn~y strongly in V}.
A well-known result is that lim A is always a closed set iS]. Let K be n n-~m a closed nonempty convex set in V; we will see later that lim(K-Yn)~(Yn-K), n-~ where Yn~K for n=l,2,-- and llyn-YoiIv~O,plays an important role. We note that li__mm(K-Yn)n(Yn-K] is not empty. The problem whether llm(K-Yn)n(Yn-K)=(K-Yo)n n-~ n-J<: (Yo-K) is apparently unresolved; we prove below this equality in two special cases to be used later. The equality may fail if K is not convex. Example 2.1.
Let V=H~(~), K= { y~V:yz~ a.e in ~, where ~HI(~),
~la~O}
and let yn4YO strongly in space V, where Yn~K (n=l,2,-..). We prove now that lim (K-Yn)a(Yn-K)=(K-Yo)n(Yo-K)n-~m To this end, note that (K-yn)~(Yn-K)={sgV:Isl~Yn-~} (Yn-K)
and Sn-~S, then s~(K-Yo)n(Yo-K).
Thus
and
we only need
if Sn~(K-Yn)n to prove
(K-Yo)n(Yo-K)clim (K-Yn)n(Yn-K). To this end, for any s ~(K-Yo)n(Yo-K], let Sn=max(-(yn-~),min(S,Yn-~))~
(K-yn)n(Yn-K).
that we
We know that Sn-~S strongly
in V from [2]. We can also prove lim (K-Yn)n(Yn-K)= (K-Yo)n(Yo-K), where K={y~V:IyI~} n-~ or K={y~Hl(~):ylO~=w, yzO in ~} or K={ y~Hl(~) :Y Ia~-~ ~la~} Example 2.2.
Let V=HI((o,I)), K={y~V:IVyl~l}
and yn~Yo , where Yn~K for
n=l,2,... We prove that lim (K-Yn)n(Yn-K)=(K-Yo)n(Yo-K)n-~: Similsrly, we just prove that (K-Yo)~(Yo-K)c li___mm(K-Yn)n(Yn-K). Note n-~ that (K-y)~(y-K)={ s~V: ms_x(-l-Vy,-l+Vy)~ Vss min(l+Vy, l-Vy)}. For any se (K-yO)~ (Yo-K),
let Sn= me~x(ms~x(-l-VYn,-l+VYn),
min (Vs, min(l+VYn, l-Vyn))-
156
Then Sn~ Vs in L2((O,I))
t
Let p = ~^s dx +s(O); we see that p n u n 2 l t 2 n By Schwartz's inequality, we see tlPn-SlIL2=~O(~O[Sn-VS]dx) dt -~
(K-Yn)•(Yn-K).
strongly.
611Sn-VSl122. That is llPn-Sllv-~3. Finally,
we note that if lim (K-Yn)n(Yn-K) = (K-Yo)n(Yo-K)
and Yn~KnBM+n
n-~
with llYn-Yollv~O, then,
lim (KnBM+n-Yn)n(Yn-KnBM+n)= lim (K-Yn)n(Yn-K), where n-~ --n~ IIyIIv-<M__+n,_M>-O}. Further, the lower-semicontinuity of the mapping
B_M+n={Y~V:
y-~ (K-y)n(y-K)
(see [i]) is equivalent to the fact that
lim (K-Yn)~(Yn-K) = n-~ for any {yn } such that Yn~K (n=l,2, ..) a~d llYn-YoHv-g3. Similar
(K-Yo)n(Yo-K) conclusion where
3.
holds
for
the mapping
(y,u)-~ Z[y,u)={s~K-y:v'Xv
=0},
FeC(K>
Optimality
conditions for
elliptic variational
Let V be a real reflexive Banach space,
V~
inequalities.
its dual space and U another o
reflexive Banach space.
Suppose that H is a Hilbert space such that VcH=H c
m
V , V is dense in H and the imbedding operator from V to H is continuous and compact.
Consider the following optimal control problems:
(ocv)
min g(y)+h(u), u~Uad
(3. i)
b(y,z-y)-(f+Bu, z-y)zO,
y~K, for all z~K,
where
(3.2)
(I)
K is a nonempty closed convex set in V,
Uad another nonempty
closed convex set in U. (2) b(-,.)
is a continuous and coercive bilinear form on V•
H, BgZ(U,H) and (f,w)m(f,w)H,
[4], f~
the inner product in H.
(3) g:H~R +, is locally Lipschitz for the H-norm and h, Uad~R +, U-norm continuous and convex on Uad, satisfying
the following condition:for
any Q>
O, there is P>O such that IIUlIu~P if (y,u)gKXUad and g(y)+h(u)~Q. TI~OREM 3.1.
[4] (OCV) has at least one solution.
[]
We consider now the optimality conditions for (OCV). hong
In [14], Shi Shuz-
gave some optimality conditions for it but unfortunately,
conditions complicated
one of his
is wrong as shown by V. Barbu and his corrected results seems very to use
[IS]. Nevertheless,
the
idea in [14]
seems applicable.
157 By using a different treatment,
we give the following conditions for (0CV).
THEOREM 3.2. Let (Y0,U0) be a solution of (OCV).If the mapping y~(K-y)n(y-K) is l.s.c at Y0'
then there are P0~V and ~e 0g(y0)cH
(ag is the generalized
gradient in [8]) such that for any (z,v)~K>4Jad,
(3.3)
b(Yo,Z-yO)-(f+Buo,Z-yo)ZO,
(3.4)
b(s,PO)=(~,s),
(3.s)
-(B(V-Uo), p o ) - h (Uo, V-Uo),
(3.6)
b(Yo,Po)-(Po,f+BUo)=O,
(3.7)
b(Po,Po)~(~,po),
f o r any se U k(K-Yo)n(Yo-K)H,
<
e
where ~H means the c l o s u r e of S in H norm and h'
is its
directional
deriva-
tive.
The left side of (3.4) must be understood as the H-norm continuous H extension of b(-,po) on U l(K-Yo)n(Yo-K) (see below). We call (3.4)-(3.7) l>O the weaker statement of Theorem 3.2. If further the mapping (y,u)~Z(y,u)= (K-y)n{s:b(y,s)-(f+Bu, s)=O} is l.s.c and g~CI(H),
(3.8)
b(s,PO)~(~,s),
and (3.8) is
then besides (3.S)-(3.7),
f o r any seU kZ(Yo,uo)V, k>O
stronger than (3.4).
We shall call (3.H)-(3.8)
the stronger
statement of Theorem 3.2. Proof.
+h(u)~inf
Let a(y,p,u)mb(y,p)-(f+Bu, p). {g(y(u))+h(u)}+l,
UeUad from ( 3 . 2 ) ( 3 ) , t h e r e hand t h e r e
where
u~y(u)
Let
(y,u)eKx4Jad
is the
such that O~g(y)+
solution
of
i s N>O such t h a t IlUIIu~N f o r any such ( y , u ) .
i s M>O such t h a t Ily(u)llv~M f o r u~y(u),
the solution
(3.1).
Then
On t h e o t h e r o f ( 3 . 1 ) such §
that
IlUlIu~N. Let (Yo,Uo) be a s o l u t i o n
d e f i n e d by Gn(Y,u)= g(y)+h(u)+llU-Uoli U 2+
o f (OCV) and
Gn;OnmKnBM+n>dJad~R u{+~}
NnSn(Y,U), where
BM+nm{YeV: Ilyllv~M+n}
and Sn(Y,U)=SU p - a ( y , p - y , u ) , which i s 1 . s . c . by B e r g e ' s r e s u l t [ 1 ] , [ 1 4 ] . PeK~BM+n Note S n ( Y ( u ) , u ) : O i f IlUllu~N. We c o n s i d e r t h e m i n i m i z a t i o n o f Gn on On . From Ekeland's Variational
Principle
(see [ 9 ] , [ 1 4 ] ) , f o r
any en>O t h e r e i s (Yn, Un)
158
~K~BM+n>
O~Gn(Yn, U )~inf Gn(Y,U) +e ~inf Gn(Y(U),U)+e n, n (y,u)~O n u~ BU N n
> 2 2 1/2 (3.10) Gn(Y,u)_Gn(Yn, Un)-Cn(lly-ynllv+llU-Un;lU ) , (y,u)eOn, where BUN ={u~Uad: ;lUlIu~N } and u~ y(u) (u~BUN)is the s o l u t i o n o f ( 3 . 1 ) .
Let
c n
and
en~O and Nn~+~. We can see t h a t
(Yn, Un)EKnBM+n>
(Yo,Uo)E K~BM>
t h e n Xmy(~) is
strongly
in K~BM as
in V and U. To do t h i s ,
sup - a ( y n , p-yn,Un)m6n~O. P~KnBM+n
liuliu~N.
that
we n o t e t h a t from t h e
Therefore,
t h a t cliYn-Xll~-(B(Un-N),yn-X)+~n.
We prove
yn~X and Un~~
inequality
(3.9),
a(yn, Yn-X, Un)~$n; t h e r e is c>O such
These i n e q u a l i t i e s
imply t h a t {;lYn-Xli V} is
bounded and ;lYn-XIIv~O as H is compactly imbedded in V~. Next, we prove t h a t ~=Uo, X=Yo and Un4U0 s t r o n g l y . (3. II) Thus,
Indeed, from ( 3 . 9 ) ,
g(yn) +h(Un) +IfUn-Uoll2+NnSn(Yn, Un )_
2< lim IlUn-UoIIu_g(yo)+h(Uo)-lim ( g ( y n ) + h ( U n ) ) _ ~ g ( y o ) + h ( U o ) - g ( y ) - h ( u ) .
R e c a l l t h a t ( X , ! ) i s a s o l u t i o n o f (3.1) and thus we i n f e r t h a t Un~U0 s t r o n g l y .
and (Yo,Uo) i s a s o l u t i o n o f (OCV)
Then X=yo and u=u O.
Now, d e f i n e Mn(Y,u)cKnBM+n, by Mn(Y,U)={peKnBM+n: - a ( y , p - y , u ) = S n ( Y , U ) } , which i s a weakly compact convex s e t in KnBM+n. From the r e s u l t about the continuity
o f marginal s e t s in [1],
( y , u ) ~ Mn(Y,U) i s u p p e r - s e m i c o n t i n u o u s
f o r the norm on K>
2.4).
in KnBM+n f o r f i x e d n ( a l s o see
Let seK-Yn, W~Uad-Un, and tk~O w i t h tk>O f o r k = l , 2 , . . .
Then, because (Yn, Un)~(Yo,Uo)~KnBM>
(Yn+tkS, un+tkw)e
On when k,n a r e l a r g e enough. Let pk(yn, S,Un, W)~Mn(Yn+tkS,Un+tkw).
By ( 3 . 9 ) ,
k g(Yn+tkS)+h(un+tkw)+llun+tkW-Uoll~ -Nna(Yn+tk s, P (Yn, S, Un, W)-Yn-tkS,un+tkw) ~
g(Un)+h(Un)+llUn_UoIi2_Nna(yn, pk(yn, S,Un, W)_Yn, Un)_Cntk(lls]l~+ll wliu )2 1/2. We here
differ
with the treatment of [14] in that in our definition of
Sn(.,-), the set KnBM+ n is independent of (y,u) for fixed n.
Let g~
be
159
the
Clarke's
directional derivative of g at y and divide the both sides of
the inequality above by tk. Suppose now that pk(yn, S,Un, W)9 p(s,w) weakly in V (possible a subsequence).
By the mean-value formula and continuity of the
derivatives,
o( g Yn, S)+h~(Un, W)+l~(Un, W)+Nna(Yn, S,Un ) -Nn[b(s , p(s,w)-Yn)-(p(s,w)-Yn, Bw)]m -en(llsll~+llwlI2"i/2 Uj , where l(u)=llU-Uoll~ and p(s,w)~ Mn(Yn, Un)
by the upper-semlcontlnuity of the
mapping (y,u)+Mn(Y,U) [I]. As in [14], by the Lopsided Minimax Theorem there is qn~Mn(Yn 'un)c K such that for any (z,v)~K>
(3. 12)
~nllZ-YnIIV +g (Yn'Z-Yn)+Nna(Yn' Z-Yn' Un)>-b(z-Yn' Nn(qn-Yn) )'
(3.13)
CnilV-Unllu+h~(Un,V-Un)+l~(Un, V-Un)a-(Nn(qn-Yn),B(V-Un)).
Let Nn(qn-Yn)=Pn and Z=qn in (3.12) and note that a(Yn, qn-Yn, Un)-~O. Thus we get enliPnllv +g~ Pn)a b(Pn,Pn), which implies that {Hpnllv} is bounded. Suppose that by extracting a subsequence, Pn -~ PO weakly in V. Let n-~ in (3.13) (see also [14] ) and note N 6 -~0. We get (3.5)- (3.6). n n We must treat (3.12) in a different way to prove (3.4). First of all, we introduce a set W= {s~V: there is {Sn},{k n} such that knSn-~S strongly in V, where SngK-Yn, kn>-O, and
lim Nna(Yn,knSn, Un)-~O}. Then, it follows that for n-~o0 any seW, from u.s.c of go(.,.) (see[8]), (3.14)
b(s,Po)~g~
If (K-Yo)n(Yo-K):lim (K-Yn)n(Yn-K),
then, for any s~(K-Yo)~(Yo-K), there is
{sn} such that Sn~ (K~M+n-Yn)n(Yn-K~M+ n) and Sn~ s in V strongly. other hand,
from (3.11), NnSn(Yn, Un)~O.
Thus, we see that
On the
Nna(Yn, Sn, Un)~
-a(Yn, Pn, Un)= NnSn(Yn, Un)~O, for any Sn~ Yn -KnBM+n' from the definition of Sn(Y,U). Consequently, for any s~(K-Yo)n{Yo-K), we have (3.14). If (y,u)~ Z(y,u) is l.s.c, then, we get (3.14) on Z(Yo,Uo). On the other hand, we know that Nn(qn-Yn)=Pn ~ PO weakly in V so, strongly in H (possible a subsequence) - -
and that Pn/NnEK-Yn and lim a(Yn, Pn, Un)~O. Thus, we get n-~r by taking Z-yn=Pn in (3.12), and then taking limits.
o
b(Po,Po)~g (yO,Po)
160 Now let us note that -(K-y)n(y-K)=(K-y)n(y-K)--S.
Thus,
V(S), where V(S) is the linear space generated by S. <
(3.14) holds on
We extend b(.,p0)IV(S) O
to H and denote the extension by ~, such that ~(y)-g (y0,y) on H (see [16]); we can see
that -g~
~(y)-
ag(y0); that is (3.4). b(P0, P0)-< ~(p0 ),
y) on H, which implies ~
H
and
~e
Further, we see from [16] that ~ can be made so that
if we can prove that b(P0+s,P0 )-< g~
for any se
U A(K-Y0)~(Yo-K)=V(S). To this end, let Sne (K-Yn)n(Yn-K) such that k0Sn~S k>0 strongly in V and let Wn= k0s n +pn ~ Nn(k0+l)(K-Yn), which converge to P0+S weakly in V.
Then let z-y n be w n in (3.12). Note that
lim Nna(Yn, Wn, Un)-~ 0
and by taking limits we have proved the inequality, so, (3.7). If ggCl(H) and Z(-,.) is l.s.c, then, (3.8) comes from (3.14).
(3.8) is stronger than (3.5)
as (K-Y0)n(Y0-K)cZ(Y0,U0). [] Remark 3.1. Even if g~ CI(H), we still can prove (3.8) by the separation theorems but we omit the proof here.
Later, we shall see that in many cases
the weaker statement is the same as the stronger one.
4.
Examples
of
some
applications.
In this Section, of simplicity,
we give some applications of Theorem 3.2. For the sake
let V=H~(fl) (or Hl(fi)),U=H=L2(n),b(y,p)=~n[(Vy, VP)Rn+aOpy]dx,
with the constant ao>O, (f'P)= ~n fpdx,
in (OCV)
(to be called (OCV)HIL2),
where n is an bounded open set in R n with a regular boundary. THEOREM 4.1. Let V=H~(n)- (or V=HI(n)) and K={yeV, yae a.e in n
with
eeH2(n)
v
and elan ~o}. Suppose that (Yo,Uo)e K~H2(n)XUad is a s o l u t i o n of (OCV)HIL2. Then, there are PoeV and ~eOg(yO)SUCh that
(4. i)
b(Yo,Z-yO)-(f+BUo, Z-yo)ZO, Yo~K, for any zgK,
(4.2)
b(s,P0)=((,s), for any seU A(K-Yo)n(Yo-K) H, k>O
(4.3)
(-AYo+aoYo-f-Buo)Po=O
(4.5)
-B p0eah(u0),
a.e in n,
m
where h(u)=~(u)ueUad , -~ ueUkUad
161
bCPo,Po)~(~,po) ,
(4.6) where U
k(K-Yo)n(Yo-K)={s~V,
there
IXsl~Yo-~}. If
is k>O such that
1~n~3,
k>O where n is the dimension of R n, then (4.2) implies: (4.7)
-APo+aoPo=~
a.e in ~+={xeR n, yo-~>O},
(4.8)
b(x(y0-~),p0)=(~,X(y0-~)) , where x~CI(~).
Then (4.2)-(4.6) can imply Theorem 3.2 in [4]. Proof. Example 2.1 implies that Theorem 3.2 can be used.
show (4.3) since (3.6) is a bit weaker than (4.3). -a(yn, XPn, Un)~-a(Yn, Pn, Un)-g3
We only need to
To do this, we note that
as qn E Mn(Yn, U n) and this leads to the inequa-
llty:b(Yo,~P0)m(f+Bu0,xP0),where
x~CI(~) and 0 ~ I .
With (3.6) we get (4.3).
If l~n~3,
then, ~+ is an open set and D(~+)cU A(K-Y0)~(Yo-K), so we get ~>0 (4.7). Take s=x(y0- W) for a xeCI(~), which is in the set U A(K-Y0)n(Y0-K)H, k>O in (4.2) we get (4.8).~ Remark 4.1.
We note that here,
{ s~H:
U A(K-Y0)n(yo-K)H= k>0
and U NZ(Yo,uo)V=[u A(K-Yo)]n{s~V:a(Yo,S,Uo)=O} V. k>O k>O From Remark 4.1, we see that in the case of Theorem stronger
statement
conditions However,
of
in Theorem
Theorem 2.1
3.2
in [13],
is
similar
which
was
to
we shall see that in some "regular" cases,
Theorem 3.2,
the
proved
s=O in n \ n + }
2.1
in [13]
"good" only
=L2(~).
optimality
as Uad=L2(~).
the weaker statement
is also similar to the "good" optimality conditions
not need the assumption:Uad
On the othen hand,
the
of
and we do
our conditions
are
valid for very wide range of g and h. THEOREM 4.2.
Suppose that w=O in Theorem 4.1.
in [13] and (Yo,Uo) the same as in Theorem 4.1. then, there is PO and ~
Let Zy and Sy be the same as If AYo-aoYo+f+Buo~O
in Zy O,
ag(y O) such that
(4.9)
PogSYo .
(4.10)
b(s,PO)~(~,s),
(4.11)
-B poeah(Uo ) ,
for any sESy 0,
which are the same as Theorem 2.1 in [13] if Uad=L2(~) same as in Theorem 2.1 [13].
and h and g &re the
162 Proof.
We just note that if -AYo+aoYo-f-Buo~O
in Zyo, then the set SYoC
U A(K_Yo)n(Yo_KlH={sEL2(n ]: s=O in ~\~ + }. Then from Theorem 4.1, we get our k>O concluslon, o Remark 4.2. We know that in the case above, and
-AYo+aoYo-f-BuoZO.
If -AYO+aOYO-f-Buo~O
(-AYo+aoyo-f-Buo)Yo=O,
in Zy O,
YoaO,
which is true in many
practical problems, then from [ii], we see the free boundary of this problem has some regularities.
That is why it is called a "regular" case.
We can give many other examples such as K={y~V:ylaHaO} and prove known results in, E.g. posed in [13],[7], THEOREM 4.3.
and K={yEV:lyi~l}
[4]. But here we turn to another problem,
in which K={y~Hl(~):lVy]~1}.
Let V=HI((O,I)),
K={y~V:[Vylsl}
in (OCV)HIL2
(Yo,Uo)E H2((O,I)}XUad is a solution of (OCV)HIL2.
and assume that
Let Sy={sEV: there is k>O
such that ma-x(-1-Vy,~y-1)~lgs~min(1+~y, 1-Vy)}. Then,
there are po e V and ~e
ag(y O) such that (4.12)
b(s,p)=(~,s),for any sESy O,
(4.13)
b[Yo,Po)=(f+Buo,Po},
(4,14)
b(Po,Po)~(~,po) ,
(4.15)
-B poEah(Uo ),
and further (4.12) can be expressed as:for any xECI[o,I],
~zO,
(4.163
-APo+aoPo=~,
(4.17)
b(X~yl,Po)=(~,X'yl),
where ~ yl-~oXmin(l-Vy O, l+VYo)dX,
[4.18)
b(x~y2, Po)=(~,Xiy2),
where X~y2=~xmax(-l-VYo,-l+VYo)dX.
Proof.
in fl+={xE(O,l}:IVy01
_
From Example 2.2,
X
we know that we can use Theorem 3.2
and
Sy =
U A(K-y)~(y-K). Then, note that D(~+)cSy 0 and X~yI,~y2E SYo. D l>O We shall discuss the stronger statement of Theorem 3.2 in later papers.
163 REFERENCES
[I] J.P. AUBIN AND A. CELLINA, Differential Inclusions, Springer Verlag, 1984. [2] C.P. BAIOCCHI AND A. CAPELO, Variational and Quasi-variational Inequalities, John Wiley and sons, 1984, pp.99. [3] V. BARBU, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80(1981), pp.556-597. [4] V. BARBU, Optimal Control of Variational inequalities, Research Notes in Math I00, Pitman, London, 1984. iS] C. BERGE, Espaces Topologiques et Functions Multivoques, Dunod, Paris, 1959. [6] A. BERMUDEZ AND C. SACUEZ, Optimal control of a Signorini problem, SIAM. J. Control and Opt., 25(1987), pp. 576-580. [7] A. BERMUDEZ AND C. SAGUEZ, Optimality conditions for optimal control problems of variational inequalities, Lecture Notes in Control and Inform Sci 97, Springer, 1987, pp. 142-IS3. [8] F.H. CLARKE, Optimization and Nonsmooth Analysis, Wiley-lnterscience, New York, 1983. [9] I.EKELAND, On the variational principle, J. Math. Anal. Appl., 47(1974), pp.324-353. [I0] A. FRIEDMAN, Variational Principles and Free Boundary Problems, John Wiley, New York, 1982. [II] A. FRIEDMAN, Optimal control for variational inequalities, SIAM. J. Control and 0pt.,24(1986), pp.439-451. [12] F. MIGNOT, Controle dans les inequations variationnelles elliptiques, J. Functional Analysis, 22(1976), pp. 130-185. [13] F. MIGNOT AND J.P. PUEL, Optimal control in some variational inequalities, SIAM. J.Control and Opt., 22(1986), pp.439-451. [14] SHUZHONG SHI, Optimal control of strongly monotone variational inequalities, SIAM. J. Control and Opt., 26(IS88), pp.274-291. [15] SHUZHONG SHI, Erratum: Optimal control of strongly monotone variational inequalities, SiAM. J.Control and Opt.,to appear. [16] K. YOSIDA, Functional Analysis, Springer-Verlag, 1965.
MINIMAX
~HAFE
FOR
Ol
OFTIMIZATION
VON
KAI~MAN
PROBLEM
SYSTEM
A. M y s l i n s k i System~ Research Institute 4 4 7 W a r s a w , ul. N e w e l s k a 6,
-
Poland
Abstract. An optimal shape design problem of an elastic body described by system of t w o n o l i n e a r elliptic equations of the fourth order is considered. The problem is to find the boundary of the domain occupied by the body in such a way that the cost functional approximating the stiffness o f t h e s y s t e m i n t h e equi librium state ks minindzed. It ks assumed that the volume of the body is constant. Moreover the function describing the boundary of the domain and its gradient are bounded. Necessary optimality condition for this problem is formulated using material derivative method.
I.
Introduction
an
elastic
This
paper
described of
the
by
[ B, The
is
shape
These
and yon
state
coupled
equations
for
optimization
equlibrium
nonlinear,
solutions
optimization the
respect
[ a,
g
variable
problem
cost
to
are
Karman
problem
of
this
elliptic
called
convergence
The
domain the
cost
] such
yon
of
systems
in
this
approximating
occupied
boundary
by
of
functional
problems
subject
considered
functional
the
describing
bounded.
ture
of
two
approximation
function are
of
a
The
for
plate
is
equations
Karrn~n e q u a di
-
the
finite
-
was
investigated
l~- ].
minimizing with
with
plate.
order.
Existence
mensional
concerned
system
fourth
tions.
in
is
nonlinear
is
where
the
the
plate.
domain
consists
plate
domain
its
are
In
occupied
called
shape
in
stiffness
Moreover
and
nondifferentiable.
the
to optimization
paper
the
by
the
gradient literathe
body
optimization
pr obl eros. In
such
literature
[ a,
optimization
author s
only
have
Q,
14,
problems studied
18
for
]
most
linear
shape
authors elliptic
have
considered
problems.
o p t i mi z a t i o n
problems
A
few for
165 nonlinear
elliptic
variational The
goal
of
optimization determine this
equations
inequalitiez thl ~
problem the
shape
[ 3,
occupied
by [
the
optimality
In
to
order
to
yon
of
do
]
or
elliptic
the
this
Karman
cost
we
to
the
employ
shall
shape
system.
We
functional
respect it
We
this
16
i nvesti gate
with
method. for
15,
nonlinear
derivative
speed
condition
8, ].
is
problem
plate. ]
IT
strongly
directional
IQ
5,
paper
for
optimization
7.oi e s i o
[ i0,
of
domain
developed
formulate
nondi f ferenti able
by
necessary optimization
pr o b l era.
~.
Formulation
of
Consider domain
Q
O
an
= < x
v ~ where We
U
assume
y
H os
= r~(v) We
by
we
[ l
two
parts
=
H2
stress
spaces
<
$ >
>
C e.l)
C2. e ) C a D
that Re:
model
V
= Hi
xI
of
connected.
an
[
u r e,
elastic the
@,
rI
0
<
and
].
Let
and
r~ F e
=~
< i
>.
Let
us
x3
plate.
plate
ii
The
continuous
= vCxe),
of
function the
simply
i.i p s c h i t z
deflection
bending the
6
by d
f be
= a
plate.
problem H~CCI3,
let
us
HaCO),
introduce
the
HO~CCI3 a r e
defined
= 0
>
necessary as
fol
-
] :
HeCO~
= < z E
LeCCO
: z i,
z lj
HoCCO
= < z E
HeC~%D
: z
@z/~n
us
0 • I >r
: < ae
is
is
such
: zi
the
plane
vCxeD ~ 0 < x e
C/
= < ( x l , x 3)
the
<
a I _< v e x e d
LaCeD
=
the
satisfies
C/C v)
nonlinear
formulate
in
constants.
= < z ~
~z/~n
Let
<
domain
r I and
wCvCx)D
~obolev
: 0
the
Airy' s
< xI
vCxeD
positive
VeCv)
force
: 0
H~C~
zi
is
R3
domai n
consider w
the
notation. 1 ows
that the
perpendicular Before
~
given
occupying
:
function
of
where
shall
fC vC x3 9
= C x i ,x 3 )
, ae. a r e
shall
denote
pl a t e
by
= < z ~ C0'I(0,19
consists V2
given
continuous
aI
bounder
problem
el a s t i c
= QCv)
Llpschitz
the
az/ax i
,
z ij
denotes
the
unit
outward
introduce
=
LaCeD,
= az/ax iaxj
outward
the
~
normal forms
normal
on
L~CCO
>
= 0
H
on
, i,j
to
a(. ,.D
the
V
Ce. 33Cb9
>
= l,e
derivative
versor :
E
z
of
z on
boundary
: H ~s( ~ D
H,
n
=Cn i ,n~)
r.
x H ~%)( C O
, R
,
166 bC
9
9 ,.
well
HI CQb
:
as
I C. 3
: H C fI>
=
I Cuiz I + uaz~D QC v)
=
=
CO,I), is
Note,
[ II
the
bilinear as
trilinear
[
is
@,ii,1- ~ 9 15
= - bCw,w,~?]
is
small then
a
Find
~unction
JCv)
= max
Uad
where
<
= { v ~ U
positive Remark, plate
- F~a
~
constant
:
x I
CO,a~3,
~
defined
[ Ii,
by
in
C~.~D,
14
and
equilibrium
the ~fuen
~tue~
w
the
body
symmetric,
bounded
state
system
of
e[eff~ent
d
os
the
[ 6
].
the
yon
fol I owing
L2CQ"D
~
/6nd
a
V r ~ Ho~CCz)
C~.7~
set
I
cost
@
].
of
If
[ 15,
k
compresive
is
or
sufficiently
condition
problem
/uncttonaL
x ~ ~Cv9
CD~.I2)
]
CP)
:
:
>
Ca. 89
b3J : < c I a.e.
Uad
functional This
intensity
optimization
in
~Cv9
1 X vC• 0 C2..79, cI
satisfies
The
[
solution.
Following
Id v / d x 2 1
].
symmetric
condition
v ,~tntmtztn&
cost
ape
respectively
+ 1C@~
the
lwCx,v) l
:
is
lot
C~.4]
and
HIcc/)
the
a unique
the
constants 9 the
Ca. Sb
]:
on
has
function
that
and
indicating
to v ~ U a d
the
C~.39
suitable
Ca. 79
consider
stlbject
Ce. 5]
V rl ~ H ~ C CD
acting
satisfies
problem
shall
forms
+ kbCw,r
parameter
forces and
- UllZ~Dro.]dx
< Cx I , x~]
=
the
described
aCf,D] R
+ CUl~.Z I
~;s i s
H~CnD
form
= bCf,w,r
~
C2.33
dx
C~..'~)
~)a
formulation
aCw,r
tensile
+ ~Ul~Zlyb
dx
element,
well
pl a t e
equations
:
- u2.2zl]r I
} where
variational
~ar man
as
constant.
as
The
[~
~
Ca. 6D
a.~ > ~
that
].
In
: [ Hoc)CCZ) ]3
dx
a given
a given
continuous
+ u2,~z22
s [CUl~Z ~ QC v3
I dz QC vD
bC . . . . . b
~ R give n by
bCu,zD
x~
the
,
I CUllZll L%C v)
d 6 L2C{)a ) i s
We
~
=
ICz9
k
~
aCu,zD
bCu,z,r)
a~
x. H I CCD
is
assumed
Ca.89
functional
to
,
= c~ and be
cs
are
given
nonempty.
approximates is
>
stiffness
nondifferentiable.
of
167 Lemma
2.1
C DE. l E D
]
optimal Proof
For
X
does
solution
shal i
v
given
section
more
3
we
t.
ping
of
C)t
[
].
Let
us
shape
vector
field
E
~- C C [ O , ~ , )
:
TtCVD
*%
f)
for
there
order
C)t
}
=
an
an
condition to
do
it
we
depending image
DO .
f)t w e
To
of
a
descri
shall
use
problem
this
exists
(2".g).
In
as
variational
and
:
<
f) i n t o
condition
be
the
in
the
variational
on map
-
the speed
domain
problem
we
problem.
[
O,
cb
~
[ O,
x
R 2"
Re
enough
regular
:
V(.
xO
domai
a
VC. ,.3
VCt,.) 9
domains
15,
optimality
C2..g).
considered
optimization
by
Uad,
necessary
domain
C2.73
E
[
(2. 8 3 ,
<2..89,
be
formulate
v
problem
of
will
the
to
the
than
condition
on
formulate
family
of
We
denote
the
r efer ence
Tt
corresponding
fo*-mulate
].
QL
the
Ig
of
the
Domain
transformation method
Uad
shall
suitable depend
[ 16
problem
introduce
parameter
not
~
in
general
Tt
satisfying
which
: is
In for
:
C 2 C R 2",R 2"3
R 2"
X
~
depending
on
satisfies
the
V
t
V x
R2"D
,
,
x
parameter
t
following
~ E
~D,
~
>
0
C2.. C O D
I~ 3
[
O,
equation
denotes
the
~3.
function
Re
family
of
mapplngs
xC. , X D
=
x(.)
:
d
d--s x C s , X J
We t
= VCs,xCs,XJ)
xCO,X)
= X
denote
by
e
[
O,e),
C~0 Qt
X (
~
= C~ = TLCVD(DO
family
the defined
= <
the
equation shall
is
a
nant
by
transposition of
~emark V
denote
=
C~.. 1 3 9 mapping
[ O,c*)
C2..llJ
domains
X
the the
R286
xC.)
for
s
_Tacobian matrix
depending
on
parameter
:
~
function C2..llD
DT t of
os
follows
such
~
[ 0,
matrix
D T t,
that
x
= xCtJ
satisfies
Jt
=
C-~
t
] }
of
the
det
n~pping
DT t
is
a
T t,
DT t
determi
-
D T t2..1
where
as
~ ~- R ~" : 3
x
where
We
~
R 2"
~t )
e > O,
~
: Note,that
C V i, v
~ we Tt
V2.D U
,
and
obtain of
the
selecting
VI
= XlVfXoD
taking that
into
domain
reference
the
following
,
V~
account flCVD
domain
field
C2.. l ~ D
as
= 0
by
[g]
:
C~-.133
(~. I 0 9 ,
given
vector
C~..ID
~10 = C O , i )
is
an
x CO,l).
well
image
as of
a
168 We
shall
consider
F~nd
domain
= max
o7% t A e
set
= < ~t
where
following
~]t ~ f n f ~ z ~ n ~
/(~t D
0
the
<
l~x,t9
cost
/unct~.onr, l
[
I
x ~ ~t
problem
CPt)
= c3
:
>
Ca. 1 4 )
>
a given
number.
~]~e p a i r
Cwt,ft)
~
the
(a. l SD
The
H~C~t)
following
set
x
= btCft,wt,~t)
at(ft,Dt3
= -btCwt,wt,'Ot3
0 is
assumed
H~Cf]tD
system
atCwt,9~t)
where
to wt
be
nonempty.
= w C x , tD,
is
arC. ,. ) Ca. 5 9 , the
+ kbtCwt,~tD
domain
+ itCg~t D
V 95t E
H~)COt])
~]t i n s t e a d
, among
Necessary
Our
] the
at
a
the
Ca.T].
system
iC. 9
are
defined
where
the
integrals
by
The
forms
Ce. 39 ,
Ca. 4),
are
taken
over
C~v).
in
solution [ 13
to
problem
Ca. 1 4 ) , C 2 . . I S 9
was
].
Y condition
to determine
respect
on
Definition
by
others
to
condition
[ ig
depending
J
is
with
optimality
of
optimal
optimalit
goal
Ca.15)
from
of
169
E H~C~t9
to
b t C ..... ),
,respectively,
existence
studied
V Dt
corresponds
b t C. , . 9, C2.6D
The
3.
Ca. 169
system
a
:
Ca. ~qne
:
:
: I dx nt
of
optimization
tAe
c 3 is
solution
shape
derivative
parameter
for
notion
the
the
of
t.
problem
Euler
of
Next
we
the
cost
functional
formulate
necessary
C2.149,C2.159.
derivative
of
Let
us
recall
the
cost
functional
the
cost
functional
y
determined
donu~in ~] :
3.1
point
: Euler
ders
in
direction
~
the
d J C Q , VD of
of
vector
field
is
: dJCs In
order
Ca. 1 4 D solution recall
= lira s u p t -~ 0
we
to
have
calculate to
C w t , f t) from
[ JCf] t)
[ Ig
Euler
determine ~
Hot,t)
] :
- JCC/9
C3. iD
derivative
material x
]/t
HoCf]t 9
and of
of shape
the
the
cost
functional
derivatives
system
C2.16).
of Let
the us
169 Definition
3.2
function lira t -~ where
Z]-~e m a t e r i a l
derivative
a point
I [ (w t ~ t - WoD/t
X ~ ~] i s
- w
II HaCr
0
wO
H~CQLD
:
w t ~ Ho~(~t 3 a t
space
HoCfO,
~
9
H CQD
of
0
by
[I-II
the
:
= 0
= w E HooCOD, otw oT t ~ HOOCf~3 i s
il-~ t h e
w
determined
C3. a)
an
image
denotes
of
the
function
norm
in
wt -~obolev
MaC
HaCCZb [ i ].
space In
order
Lipschitz
Lemma
3. I
with
: Let
a solution
H 0 CCO, Then
w
I I wt
Next,
I
in
of
the
respect
to
t at
mapping
steps
form
the
a point
follows
[ H~Cf) t) Let
[
us
[
6
]
t = O.
].
where
[
From
f
that
15
function
mapping
]a,
[ 0 , ~ 9 , ~ > O, t = wtoT t
: w
= fo
~ H COD.
:
[ I o < e t H~C CO First
wt
o'9
Frechet
we
and
theorem O,
t 6
denote
~ HoCOO"
I I .ft _ f
,
~
we have to show a ]a [ H O C Q t) of the
t.
e > 0 such
implicit
of
~
Ca'q-.ir.~).
= ft ~ Tt
two
canonical
employing
C w t , f t)
parameter
C w t , f t)
system ft
derivative
solution
< e t HeC O D -
consists
material
to
constant
differentiability
this
pair
the
exists
- w
:
Ca. iK~)
o%" t h e
the
of
the
respect
= w 0 E H 0 Cfl),
there
Proof
calculate
continuity
Ca. 169
system
be
to
~-
write
ft we
t
C3.3D
are can
--~ w t
problem
uncoupled.
show ~
Frechet
H 02 C O b
with
differentiability
of
(3.3). D
Lemma
3. S
: The
functions (a.
16)
wt
are
aC~,r
material E
given
derivatives
HOOCh%t) by
+ a'Cw,cb)
and
ft
~
w
~
Ho2Cf~L D
HaCOO
and
~" ~
satisfying
HaCOD
the
~y~tem
:
= bCf,w,r
+ bC~',w,r
+ b'Cf,w,r
+ kbCw,r
0 a C f , O) where
Cd,r
+ a'Cf,~))
of
= - b C 'w,w, D) - b C w, w, o) - b 'C w, w, D)
= Idiv
[dVCO)]Cdx
, VCO)
= VCO,X),
C3.4) 2 V D ~ HoC~'I3 a'C. , . ) , b ' C .
,.),
Q
b'C ..... D bC . . . . . ) = O.
These
are
Frechet
derivatives
,respectively, forms
are
with
given
by
respect :
of
the to
forms
parameter
aC. ,.), t at
a
bC. ,.), point
t
170
a'Cw,~D
= J
[ div
VCO)
- ~C
DVCOD
+
DVCODD]AwA~
dx
+ C3.5)CaD
f Q b'Cw,~D
dim
Cdiv
= g [ dim f~
VCOD
- CDVCOD
VC03
+ ~DVCODD]CAw~r
- CDVCOD
+ ~DVCODD]
+ X/wAr
~w~
dx
C3. S D C b D
C3. 6)Cc9 ~ZC~DVCOD~Zf
CDVCO) denotes
Proof
: Subtracting
domain t,
the
the
~] [ 1 9
passing
account
VC~fD~DVCODI~x
+ ~DVCODD
DVCOD
transforming
-
jacobian
from
] ,dividing
to
the
]Vf
+ ATV~w
matrix
of
the
integrals
both
the
t
C2. i O D - C ~ - . i ~ D , C 3 . 2 D , C 3 .
the as
SD
we
div
VCOD,
the
f]t i n t o
0
[
>
A~
as
+
C2.7D over
equations taking
C3.4D
dx
integrals
resulting
-
= ~ii
equations
the
well
obtain
VCOD
+ ~DVCOD)V~
matrix
of
--+
C div
C2.16)
domain
sides
with
~
CDVCO)
equations
over
limit
-
[ i~
by
into
]. o
Next
we
H o C ~ t)
determine
• H C Q t)
detern~nes the
the
increment
Definition function Cw t)
where
~
= w
II
H 2 C R ~'9 i s
shape
the [
system
the
parameter
19
]
HoCQtD
:
is
function
t value
The
shape
determined
the
C~.7).
The
pair
I IHeCC e
prolongation
~-
Cw t,ftD
shape
derivative
with
respect
value
to
only. W
derlvative
by
E
H2Cf)O
of
the
:
t W + oCtD
+
oCt)
of
of
derivative
the
increment of
3.3 wt
a
~atisfying
C~.~)
/t
-~
of
the
0
for
t
--~ O,
function
wt
w0
=
w
~ H ~ C Q t)
~
H COD,
into
wt
the
space
H'~ ~ 2 D . Let the
us
recall
functlon
holds W
[ l~
wt
~
] that
Ho~CQt D
if
the
exists,
shape then
derivative the
W ~ H~CC/]
following
of
condition
: = w
- ~w
VCOD
C3. VD
9
w
~
and
H~CCO
ia
~w
a
is
Integrating
the
,nate~'ial
gradient by
of
pa~ts
derivative
function the
w
system
of with C3.4D
the
function
respect two
to
wt
~
H
C~tD
xo
times,
taking
into
171 account
C3. VD
obtain
the
CW,FD of
~
the
and
eliminating
system
H~CC/9
x
system
of HeC{iD
CS. VD
derivative~
equations of
the
of
determining solution
the
shape
Cwt,ftD
VCOD
matrix
the ~
we
derivative
HO2Cs
D
x
HO2Cs
D
:
aCW,~D
= bCF,w,~D
+ bCf,W,~)
aCF,))>
= -~bCW,w.'Q)
+ kbCW,,~D
+ IICW, w,~D
HO2C~D
V 9~ ~
C3.83 I
li ( w ' w ' ~ j Taking that
into W and
W
= F
OW/"On where ing
of
point
[W
~-~A~
account
=
the
5
W
as
+ AwVCODnDA~
well
following
as
] dx
C3.7D
boundary
it
is
easy
conditions
[-O~'w.'3n a
]VCODn
,
OCOw."gh'~D/On.
integral
f] i n
the
OF/'On From
i i in
CS. 89.
dJff~,%O
of
direction
V,
=
[-@~'f/On 8
C3.~),CS.
the
cost
defined
IO>
]VCODn
it
functional
by
= max
wCxDD[WCx]
C3.19,
+ ~Tw(x9 x ,~ C/
= <
x ~ Q
function
sgn
In state
Cx9
= ~ L to
(p,q)
~
afq,))D
6C.D is
I 0 -i
is
vanish-
C~ given
VCOD]
by
at
a
[
T,
I
Ap ~ @n
given
a by
:
W
in
defined
+ k~Cp,~D
by
[ x
~ Q
>>
+ I F
:
(3.113COD
C3.11DCaD by
we
introduce
an
- -~
.[i,
adjoint
: - $ n
6C•
dx
V )) a
ill,
+
C3. i ~ 9
A q ~h9 d y an
measure
llgCa)
C3.1iJCbD
V ,~ ~ HCfI)
d~
Dirac
[wC'~b [
CS.
>
x > 0 x = O x < 0.
eliminate [HoCCD]~-
max
defined
if if if
= bCp,w,)))
is
= <
is
= bCp,f,~bD
$ V
HCC/9
: ICC/9
sgnCwO
order
aCp,~)
where
show :
C3.10)
results
9~
The
F
:
dJCf],VD
Q
to
on
C3. gD
derivative
]
C
V D ~ HoC~
= O
the
Ig
-
C~.8)CbD
F satisfy
02w/'On 2" =
The
1~,
= $ V
+ liCF,f,TI9
x
~
[]
HCfl)
and
CS. I S )
the
space
172 HCCI) Taking
= { r E into
symmetric
account
as
dJCE],V)
HeCC/)
well
: max
: r
the
= 0
on
fact
F
that
>
C3.149
the
forms
Ca. 3 9 , ~ . 4 3 , C ~ . 5 9
C3.8),C3.119Ca9,C3.17-9,C3.139
as ( C
sgn
wCx))[-
ICAwAp
we
+ ZIfAq)VCO)n
whet-e The
~
Is
necessary
Lemma all
3.3
dx
+
-Tt
constant Proof
=
I and
fields
V
+
VCODn
E Q~
has
Lagrange
defined
x
AC.)
condition
exists
Remark
3. i that with
the
I H F
det
DT t ,
described
: is
obtain C2-.89
CJ. l l ) C b )
optimality
c3.15)
]
=
> a2C. ) / a n a.
the
standard
multiplier
by
C2. i 0 )
~
the
e
R
form
:
such
that
following
for
condition
:
dJCQ,V)
where
by
: There
vector
holds
has
given
VCO)
:
obtain
r ~'.~x.)
are
given
: Taking the respect
= max
Jt
given
dx
by
- c3 =
C3. 1 5 )
0
and
c3
is
the
].
into
to
I n
C2.. 159.
[ 4
form
O,
is
account
directional
following
dJCv,9~)
->
d J C Q , VD
in in
dx
v
E
U
Remark
2.. i a s
derivative at
a
point
of v
~
well
the U
in
as
cost
C3. 1 5 )
we
functional
direction
~
~
U
:
< C sgn
wCx))C-
I
WlCX)
XlVCXa)~b
]
CAwZkp
+ ZlfAq)XlVCX2D~n
[ x E
Q~
I
dx
+
>
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(2]
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OPTIMALITY
CONDITIONS
FOR
Adam Institute
of
AI.
Automatics, Mickiewicza
A PARABOLIC
TIME
DELAY
SYSTEM
Kowalewski Academy
30,
of
30-059
Mining
and
Cracow,
Metallurgy
Poland
S U[l]l]lar y VaFious optimization problems associated with the optimal control of d i st ri b u t e d - p a r a m e t e r s y s t e m s with time delays appearing in the boundary conditions have been studied recently in R e f s . [i] + [7] a n d [ii] , [12]. In t h i s p a p e r , w e c o n s i d e r an optimal boundary control p r o b l e m f o r a l i n e a r p a r a b o l i c s y s t e m in w h i c h c o n s t a n t time delays a p p e a r in t h e e q u a t i o n a n d in the b o u n d a r y condition simultaneously. Sufficient conditions for the existence of a unique solution of the parabolic delay equation with the Neumann boundary condition involvlng a time delay are proved. The performance functional has the quadratic form. The time horizon T is fixed. Finally, we impose some constraints on the control. Making use of the Lions scheme [8] , necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional and constrained control are derived. The research reported here was ca~-ried o u t w i t h i n t h e r e s e a r c h p r o g r a m m e RP.I.02/AS0-2.2.
I.
Existence
and
uniqueness
Consider
now
the
[el l o w i n g
parabolic
ay j--~ + A ( t ) y
y(x,t')
Oy A
solutions
distributed-parameter delay
+ b(x,t)
=
y
(x,t)
=
o
system
described
by
the
equatlon y(x,t-r)
=
u
x ~ ~]
~o(X,t')
=
y(x,o)
of
x e Q
(x) c(x,t)
,
t ~
(Q,T)
(i)
t'E
(-r,O)
(2)
x e [~ y(x,t-z)
+ v
(3)
x E r
t e
x ~ r
t'
(0,T)
(4)
0"0
y(x,t') where: C ~
-
side
[] c
manifold of
= Rn
~P ( x , t ' ) o is a b o u n d e d ,
of
open
dimension
(n-l).
y -= y ( x , t ; v )
u -
set
with
Locally,
boundary
~]
is
V
v -
(-r,0)
, which
totally
r u(x,t)
e
v(x,t)
on
(5) is
a one
175 Q = O x
C0,T)
Q. = ~
x
[0,T]
Z b
is
a given
Feal
c r
is
a given
real
is
a specified
Qo
= F
o
x
= Q x
C OO
function
defined
on
Q
Coo
function
defined
on
Z
positive
{]umber
is
an
initial
function
defined
on
~I' o The
is
an
initial
function
defined
on
has
fol^m
ACt)
the
,
Z
= F
x
[0,T)
, ,
representing
,I,~
operator
(-'r,0)
(-T,0)
a
time
delay.
Qo o
n
ACt)g
=
-
Ox
aij(x,t)
Oy(x,tl Oxj
i, j = l and
the
functions
a..(x,t) zj
satisfy
n
i, j = l
> 0
Q) .
The
equations of
V
(x,t)
~: Q
9
V
{~
i
~
R
(7)
i=i are
a .lJ .(x,t)
of
side
condition
n a
where:
the
(6)
(4]
(i)
+
real
(5)
C OO
functions
constitute
is w r i t t e n
in
the
aij(x't)
c~
defined
a Neumann
following
on
[closure
The
problem.
left-hand
form
n Oy
_
O))A
Oy(x,t)
-
Ox.
i,j= 1
q[x,t)
C8)
3
a where:
is
a normal
derivative
at
r
directed
towards
the
0)? A
exterior being
of
the q(x,t)
First
we
unique
, c o s ( n , x i)
=
the
shall
For
this
introduce
case
the
F
prove of
Sobolev =
a Hilbert
an
y(x,t-r)
mixed
any
direction
~]
cosine
of
n
n
-
and (9)
conditions
for
initial-boundary
the value
existence problem
of
(i)
+
a (5)
L2(Z).
pair
space
of
Hr'S(Q)
HOc0,T;Hr(~]))
space
i-th to
+ v(x,t]
sufficient
v 6
for
is
exterior
the
where
purpose
Hr'S(Q)
Js
at
c(x,t)
solution
for
which
Q
normal
normed
n
real
numbers
([I0}o
p.
HSc0,T;H0(~]))
by
6)
r,s defined
~
0
we
by (I0]
176
II y ( t )
I[ 2
dt
+ II y II 2
H r (~]) where:
the
Chapter
The
spaces
i
[9]
of
problem
the
(5)
the
we
H s ( 0 , T ; H ~ (~]))
are
for
initial-boundary
defined
in
on
cylindeer first,
on
Q2
. In t h i s
way
introduce
15.2
the
=
Qj
Theorem
the
the Q
mixed can
solving ' etc. the
be
proved
(i)
until
+
the
solution
using
(5)
on
procedure
in t h e
a the
covers
previous
step
one.
((j-l)r.jr)
the
solution
in t u r n Q
next
simplicity,
Using
and
i.e.,
and
cylinder
determines
Ej
+
method, Q1
whole
For
a unique
(i)
constructive subcylinder
Hr(~2)
respectively.
existence
value
(11)
H s ( 0 . T ; H ~ (~]))
following
Q x Ej
([I0],
p.
, Ej
811)
notations: = F
• Ej
can
we
for
prove
j =
the
1 ..... K
fol l o w i n g
result. Lelrm~a i Let u ~ H -I/2'-I/4
f]
(Q)
~ H -I/2,-I/4
v e
L 2 (E)
(12)
(Q])
(13)
where f.(x,t) J yj_l(', qj
u(x,t)
(j-l)~)
- b(x,t)yj_l(x,t-r)
~ H I/2
(~])
(14)
e L2(Zj)
(15)
where qj(x,t)
=
Then,
there
exists
mixed
initial-boundary
+ v(x,t)
c(x,t)yj_l(x,t-r
a unique
solution
value
problem
~ H3/2"3/4(Qj)
yj (i),
(4),
for
the
(14).
Proof We
observe
that
for
j = 1
,
yj_llQo
(x,t--r)
=
9
o
(x,t--T)
and
177
yj_iIE,
(x,t-r)
respectively
= 9o(X't-r)
Then
the
assumptions
O
(13) 9 O
(14)
,
H3/2
E
and
3/4(Qo)
assumptions solution
are
(15)
are
Yo
HI/2(Q)
Yl ~ H 3 / 2 " 3 / 4 ( Q I )
f2
H-I/2"-I/4(Q2)
,
easy
to n o t i c e
the
that
Yl 6 H3/2"3/4(QI) 3.1
([9],
the
p.19)
we
[0,r] Trace
that
~
exists the
a unique
Theorem Let
,
for
that
and
ylIXI
follows
2 1
p.
9)
Qj
E
2)
a
uniqe
to v e r i f y L2
(X 1 )
the
using
that
fact
that
Theorem
implies
that
continuous HI/2(~).
(Ql)
. We
implies
Then,
shall
from
mapping
L 2 ( X I)
j = 3 ..... K
from
Again
H 3/2"3/4
is
It
the
continuous
ylIXl
Y2 e H 3 / 2 " 3 / 4 ( Q
~ yl ~
linear,
Thus,
for a n y
from
is yl(-,T)
,
[I0] a
~-
These
of
have
Then
that
L 2 ( X o)
Yl ~ H S / 2 " S / 4 ( Q I )
hence .
~
assume
existence we
HI/2(~) (13)
we
j = 2
yl(-,t)
is
solution
result
now
of there
sun~narize
.
1 and
9o ~ H3/2"3/4(Qo) there
exists
a unique
initial-boundary
value .
j = 1 ..... K
2. P r o b l e m
shall
Neumann
formulation.
now
formulate
problem.
horizon
The
time
The
performance
Icy)
:
u
be
" 9o ~ L 2 ( X o)
for
We
prove
(Theorem
yo,9o,9o,V
Then.
can
, H I ' I / 2 ( X I)
foregoing
~o
the
u ~ H-I/2"-I/4(Q)
YlIZ 1
H3/2"3/4(Q I)
~-
c HI/2(Q)
Theorem
Yl
and
ensure
condition
t
~ H3/4(~)
if
" Next
yl(-,r)
and
mapping
the
to
sufficient
fulfilled
problem
the
(I) +
optimal
T
is f i x e d
functional
by
t,v)
and
u ~
(5).
Moreover,
~
HI/2(Q)
H-I/2"-I/4(Q). for
the
y(-,jr)
mixed
~ HI/2(~)
Theorems
boundary control problem U = L2(X)
in o u r
is g i v e n
Yo
y ~ H3/2"3/4(Q)
Optimization
us d e n o t e
with
, v ~ L2(E)
solution
Let
SQly(x
given
the
space
of
Iz
(Nv) vat dt
for
the
controls.
problem.
by
- Zdl 2 dxdt
+
(16)
178 where: N
k I k
is
0
and
a strictly
Finally,
we
kl+k 2
positive
assume
the
> 0)
;
linear
zd
is
a
operator
following
on
constraint
given element L 2 (E) into on
controls
in L~(Q); L 2 (Z) . v
e
Uad
where Uad
Let
is
a
closed,
y(x,t;v)
value v E
denote
problem
Uad
(i)
We
~
control
problem
I (v o)
I (v)
Then
from
a unique by
the
from
the
c is
V
the
(x,t)
the
-
(17)
of
the
mixed
1 that
initial-boundary
to
for
a
any
is
given v
e
v 9
simplify Uad
solution
, we of
8p(v) at
p(:<.T;v)
the
. The
solving
of
the
to
seeking
a
v~
since
formulated -
optimal such
Uad
that
1.3
([8], v
p.
i0)
exists;
o
it
follows
moreover,
that v
for
is
o
)v2
>
0
characterized
condition
of
the
Z
0
V
performance
v 6
Uad
(18)
functional
(16)
we
can
express
(18)
form
Zd) (y(v)
(19),
we
define the
- Y(Vo))
dxdt
+ k 2 LNVo(Vg
- v o)
introduce the
the
adjoint
adjoint
variable
equation p
= p(v)
dr
dt
Uad
and
h
0 (19)
for
= p(x,t;v)
every as
the
equation
+ A
(t)
=
ap (v) at
Uad
wel l-defined
V v 9 To
control
p
(v - v o)
form
U
(16)
control
following
k I ~Q(y(v ~
Uad
of
corresponding
Theorem
L2(Q)
Theorem
following
i17 t h e
solution
at
equivalent
v 9
optimal
I' (v o) Using
(5)
subset
functional
H3/2"S/4(Q)
~
the
+
note
performance y(v)
convex
p(v)
kl(Y(v)
+ b(x,t+T) -
p(x,t+r);v)
zd)
x
= ~
~
,
t
9
(0,T-T)
(20)
(T-T,T)
(21)
*
+ A
=
(t)
0
p(v)
=
kl(Y(v)
-
z d)
x 9 r~ , t e
x
,-s_- f )
(22)
179 ap(v)
(x,t)
=
c(x,t+'r)
(x,t)
=
0
p(x,t+T;V)
X ~ F
, t ~
(0,T-r)
(23)
x
,
(T-r,T)
(24)
A ap(v)
E
F
t
~
A where n
A
(t)p
ap(v)
00
__ ~ i,j=l
(x,t)
,
existence
=
apply
the
Lemma
1
solution
zd
(with
v
t = T
and
in
of
Q
change time,
(20)
on
(20)
+
first,
this
t'
on
be
to
solved
solving
(20)
until
purpose, to
the
is e a s y
can
etc.
of v a r i a b l e s )
i.e.,
(24)
It
(24)
QK-I
. For
+
method.
, i.e.,
turn
cylinder
sense
problem
. problem
an o b v i o u s
reversed
the
a constructive
from
QK
whole
for
using and
startffng
the
(x,t)
+ tile
we
may
problem
(20)
= T-t)
2 hypothesis
z d ~ L2(Q)
We
given
covers
(with
the
p(v)
c o s ( n , x i) --~--]
proved
subcylinder
Theorem
(24)
Let
for
be
in t i m e
on
Op (v)
aji(x,t)
a unique
can
that
procedure
+
of
Q
backwards (24)
(25)
i,j= 1
cylinder notlce
[ a i j ( x , t ) a--~i lap
n ]~
A
The
~a
and
of
any
~ HS/2"3/4(Q)
simplify
purpose (20),
(19)
setting (21)
and
Q
(21)
we
x
get
X1
f
for
using
(y(v)
(T-T,T)
(Y(Vo)
1
v ~ L2(Z)
v = v
by
Theorem
the
the
satisfied.
there
exists
problem
adjoint
in
o - Y(Vo))
respectively
- z d) (y(v)
be
(20)
+
and
- Y(Vo))
unique
+
(24).
multiplying
both
+
(20)
integrating
then
adding
dxdt
glven
solution
(24).
equation
then
for
a
(20)
(24),
Then,
over both
~ sides
For
this
sldes x
of
(0,T-T) of
(20),
=
Q aP(v o ) . ] 0t + A ( t ) p ( v o) (y(v)
[ =
IQ
Y(Vo))
dxdt
+
T--T + f
f o
b(x,t+~)
p(x,t+T;Vo)
(y(x,t;v)
- y(x,t;Vo))
dxdt
=
180 =
#
p ( v o) ~
(y(v)
-Y(Vo))
dxdt
+
- Y(Vo))
dxdt
+
q + ~
A*(t)
Q
P[Vo)(Y(v)
T--T + f
f o
Using
the
(26)
b(x,t+r)
p(x,t+T;v
) (y(x,t;v)
f) equation
can
be
(i),
rewritten
~ Q p ( v o) - ~
-y(x,t:Vo))
dxdt
(26)
o
(y(v)
the
first
integral
on
the
right-hand
side
of
- Y(Vo))dxdt
-
as - Y(Vo)]dxdt
=
-~QP(Vo)A(t)
(y(v)
T - I
# P ( x ' t ; v o) b(x,t) ( y ( x , t - T . v ) - y(x,t-r;Vo)) dxdt = o ~] T-r = - f P ( V o ) A ( t ) (u - Y(Vo))dxdt -I J" P(x'L'+~" ;Vo)b(x't'+r) Q --T ~ (y(x,t' ;v)
The
second
Green's
- y(x.t' ;Vo]]
integral
formula,
A
on the
right-hand
can be e x p r e s s e d
(t)p(Vo](Y(V)
(27)
dxdt
side
- u
=
lay (v)
oy (Vo) ]
o Using
the
boundary
right-hand
side
T
of
condition (28)
[~y (v)
<&
Vo)t.
in
(26),
.F P ( V o ) A ( t ) ( y ( v ) Q
Q
T
of
view
of
as
T
ap(v ~ )
o
A
(4),
the
can be w r i t t e n
second
- u
+
component
on
the
as
#y (Vo) ]
] d dto
T =
#
T o
P ( X ' t ; v O) C(X,t)
(y(x.t--r:v)
--y(X
r
t--T;V
)) d r d t
*
o
"
T + J
# o
p ( x . t ; v O) (v - v O)
dFdt
=
F T-T
=
T
-r
; p(x,t'+r;v )C(X,t'+T) r o
(y(x.t';v)
-- y(X
"
t';v
o
)) d F d t
T + f
o
~r
P(X't;v~
(v -- V
o
) d[-dt
(29)
181 The
last
component
Jn
(28)
T ap(v ) o J" f a~ I (y(v) o r *
can
be
- Y(Vo))
rewritten
dFdt
as
=
A
T-r
Op(Vo) ~
T
fF ~
= ~O
(y(v)
- Y(Vo)dFdt
aP(V
+ ~T-r
(y(v)
- Y(Vo)dFdt (30)
A
A
Substituting
o )
0~7 .
(29
(30)
into
(28)
and
then
(27),
(28)
(26)
into
we
obtain kI f
=
(y(v ~
Q ~
-
- Zd) (y(v)
P(V o) A ( t ) ( y ( v )
- Y(Vo))
dxdt
- Y(Vo))
dxdt
=
+
Q o - I
~ b(x,t+T) C)
-r
p(x,t+r;Vo)
(y(x,t;v)
- y(x,t;v
o
)] d x d t
+
T-T - f
~ b(x,t+r) Q
o
p ( x , t + r ; v O) (y(x
t;v)
- y(x,t;v
"
+ j" p(v o) A(t) Q
(y(v)
- Y(Vo))
dxdt
)) d x d t
+
0
+
o + f
--T
f
T-T + J" o
r
c(x,t+r)
p(x.t+'r;Vo)(Y(x,t;v)
f c(x,t+r) F
p(x,t+r;Vo)
- y(x,t;v
(y(x,t;v)
o
)) d F d t +
- y(x,t;Vo))
d['dt +
T + I
f o
P ( x ' t ; v O)
T-T + f
(v-v O) d g d t
-
F f
o
ap (Vo) 0D [~ *
(y(x,t;v)
- y(x,t;Vo))
d['dt -
A
T + ~T-r
<
Op (Vo) a~
(y(x,t;v)-
y(x.t;Vo))dFdt
+
A
T-T +
fo
=
~
Substituting
b(x,t+r)
P(v o)
(31)
p ( x . t + T : V o)
(19)
we
-- y ( x , t ; V o ) )
dxdt
(31)
(v - v o) d F d t
into
(y(x,t;v)
obtain
182 f
(p(v o)
Z
+ X 2 N v o)
Theorem
~
For
problem
the
zd e
L2(Q)
and
exists
a unique
We
also
can
(i)
peFformar~ce
+
k2
(v -
v o)
dFdt
(5) w i t h
the
performance
> 0
consider
xll
an
is g i v e n
~ HI'I/2(Z)
c L2(Z)
X 1 Jz(Y(Vo)lZ + k 2 ~. We
introduce
on
which
O
optimal
controls
satisfies
control
Thus
Theorem
~(v)
o
zZd) (y(v)
IX k
dFdt
the
ad]oint
following
y
p.
+ A
o
there
maximum
where
the
(33)
~
(t)
p ( v o)
for
H3/2'3/4(Q)
well-defined.
each with
Then,
the
by
u
dFdt
0
+
V v E Uad
34)
equation
+ b(x,t+T)
p ( x , t + T ; v o)
0
=
x E t]
,
t
~
x ~ Q
, t ~
(0,T=T)
(35)
(T-T,T)
(36)
,
at
p(x,T;v
(17),
9),
Op(v o)
aP(V o )
with
the
Problem
([i0] ,
is
is c h a r a c t e r i z e d
-
+ A
(16)
f(NV)Vdrdt
solution
(NVo) (v - v o)
at
(32)
z
Y]E
v
functional
ly(v)] - Zdl 2 drdt +X
v ~
control
V v ~ Uad
by
the Theorem 1 a n d the T r a c e 2 L (Z) , t h e r e e x i s t s a u n i q u e
optimal
0
constraints v
al]alozous
z From
with
control
functional
l(v)
and
optimal
k
)
=
(t)
p ( v o)
=
0
0
x
~
Q
(37)
OP(V o ) 00
C(x,t+T)
,
p(x,t+T;v
o)
+ XI(Y(Vo)
A
I~
x ~ F,
zZd) t ~
(0,T-T)
(38)
ap(v o ) aO
kl(Y(V~
,
II - zZd) Z
x ~ F
, t E
(T-r,T)
(39)
for
given
A Using
the
LelJ]raa
3
Let
the
zZd
~
Theorem
1 the
hypothesis
L 2 (X)
and
following
of any
Theorem v
O
~ L 2 (Z)
lemma
1
be there
can
be
proved.
satisfied. exists
Then, a
unique
solution
183 p ( v o) e H 3 / 2 " 3 / 4 ( Q )
In this (32).
case
The
the
3
For
the
problem
Zzd
e L2(Z)
and
a unique
condition
the
theorem
(i) +
can
is n o w
(5) w i t h
k2 > 0 optimal
problem (34)
condition
following
Theorem
exists
to
and
(35)
be
+
(39).
also
performance
with
constraints
v
in
the
form
(33)
wit]]
(17),
there
fulfilled.
the
control
rewritten
o
which
functional on
control
satisfies
the
maximum
(32).
Re#erences [I]
[2]
[3]
[4]
[5]
[6]
[7]
[8] [9]
[I0]
[ii]
[12]
Knowles. G . , "Time-Optimal Control of Parabolic Systems with Boundary Conditions Involving Time Delays". J. Optim. Theory AppI., Vol. 25. 1978, pp. 5 6 3 - 5 7 4 . I
A Note about Singular Perturbations in the Filtering of a Markov Chain
Charles M a r c h e t ~ i T h o m s o n S i n t r a A c t i v i t e s Sous M a r i n e s Cagnes sur Mer Abstract We s t u d y t h e filtering p r o b l e m of a h o m o g e n e o u s M a r k o v c h a i n in a s i n g u l a r case, acc o r d i n g to t h e t i m e the scale of t h e o b s e r v e d p r o c e s s . U n d e r a s s u m p t i o n s , we o b t a i n a new a s y m p t o t i c e x p a n s i o n of the u n n o r m a l i z e d condit i o n a l d i s t r i b u t i o n of t h e Z a k a i e q u a t i o n , by i n t r o d u c i n g t h e t i m e scale a n d b o u n d a r y layer t e r m s . T h e t e r m s of t h i s e x p a n s i o n are c a l c u l a t e d e a s i e r by d e c e n t r a l i z a t i o n a n d a g g r e g a t i o n
C [~'1 ).
1
Introduction
We c o n s i d e r t h e h o m o g e n e o u s M a r k o v c h a i n ( x ~ ; t ~ O) w i t h v a l u e s in a finite s t a t e s p a c e E = {xl . . . . . x . } . B Let ~-+ A b e its g e n e r a t o r , w h e r e B a n d A a r e t w o n • n m a t r i c e s w h i c h t h e m s e l v e s g e n e r a t e M a r k o v c h a i n s , a n d 0 < ~ < 1. The matrix
B is a s s u m e d b l o c k - d i a g o n a l ;
e v e r y block r e p r e s e n t s the s t r o n g i n t e r a c t i o n s
b e t w e e n s t a t e s of t h e s a m e g r o u p ; a n d t h e m a t r i x ~A r e p r e s e n t s the w e a k i n t e r a c t i o n s or the slow t r a n s i t i o n s b e t w e e n the s t a t e s b e l o n g i n g to different g r o u p s . We c o n s i d e r t h e filtering p r o b l e m w h o s e s o l u t i o n satisfies t h e Zakal e q u a t i o n . Let (Yt; $ ~ O) be t h e o b s e r v a t i o n p r o c e s s m o d e l l e d by
CI)
dy,
= hC,~)at+
Y0
=
0
dw~
188 then the Zakai equation is d p ,r
= p r9( ~B- +
A)d~+
p r9 H , d y r
(2) p~
= p.
where H is the diagonal observation matrix, w h o s e diagonal is (h(zl) .....
h(z,,)).
T h e vector p~ is the n-dimensional row vector of probabilities p~(1) to be in state i at time t, given ~(y,;0 < s > t). In section 3, w e study the limit equation (for 9 = 0). In sections 4 and 5, w e assume the invariance of k e r B
by the operator A, so w e show that
in m e a n s q u a r e pC(t) a d m i t s a u n i f o r m e x p a n s i o n on [0, T]
pcc~) = po(O + Po(~)C~" - P~)~"-~ + ... + ~"(p,,,(t) + P,~(~)(1 - PB)e '-~) + 0(~ ''+~)
F i n a l l y , in s e c t i o n 6, b y d e c e n t r a l i z a t i o n a n d a g g r e g a t i o n , we s h o w t h a t t h e d i m e n s i o n of t h e e q u a t i o n s f o r Pk a n d P k m a y be r e d u c e d .
2 2.1
Preliminaries
S p e c t r a l projection and potentla] m a t r i x
We consider an n •
m a t r i x B , w i t h e i g e n v a l u e s (Ak). A s s u m e t h a t A 0 = 0 is a n e l g e n v a l u e .
We define the operators RA = (AI-
B ) -I
for
A ~ A~.
RA is c a l l e d t h e r e s o l v e n t of B (see [4] a n d [6]). We define the spectral projection P~
and the operator
H~k, a s s o c i a t e d w i t h t h e e i g e n v a l u e
Ak, b y
Px. = 2~ifc na d~ 1 Hx~
=
21ri
/c k (A ~k)-lRadA
w h e r e Ck is a p o s i t i v e l y o r i e n t e d c o n t o u r e n c l o s i n g Ak, b u t no o t h e r e i g e n v a l u e of B .
189 have t h e following results for the eigenvalue Ao = 0
We
PAoB
P~o = PAo PAoH~o = H~oPAo = 0 In fact, PAo is the eigenprojection
= B PAo = 0
BHAo = H~oB = PAo--I.
onto ker(B) along
R(B).
HAo is the inverse of B in the
subspace R(PA0 -- I).
HAo is c a l l e d t h e p o t e n t i a l m a t r i x , a n d we h a v e
H~o = (P~o - B ) - t
_ P~o.
In the sequel, w e shall write H/~ and P n instead of H~o and P~o"
Canonical decomposition
2.2 We
consider a h o m o g e n e o u s
M a r k o v chain (zt;t > 0) with values in a finite state space E,
with generator B where 0 is an eigenvalue of B of multiplicity r. Here, r is the n u m b e r of
r e c u r r e n t classes (see [21 and [4]). Let E be the set of these r classes with elements denoted ~'o We
r~
denote by q~y) =
inf{t _> 0; ~,
the probability
of ending in the class ~ starting from state y.
Let
~ r} then
q~y) = PvCrz < oo). T h e r functions (column vectors) qz--'('),~'EE, are solutions of the equation
Bq~=
0
and form a basis of the r-dimensional subspace k e r B = R(P/~).
T o each ~" 6 E
is also associated the invariant measure
(row vector) rn~- of the recurrent
subchaln defined on the class ~, by
m~-a
=
o
and
~ , )
=
(,~r,l)=
z.
VE~
These r measures m~- form a basis of kerB*, dual to {q~-,~q E-}, such that
(raz~q~-) = 1
and
(m~-, qF) = 0
if
~ ' ~ ~'.
190 If Q is t h e n X r m a t r i x d e f i n e d b y Q(y,x--) = q~-(y), a n d M is t h e r X n m a t r i x
defined
by M ( 5 " , y ) = mz-(y), t h e n BQ
=
0
MB
= 0
and
MQ
= I.
Finally the operator PB, spectral projection on k e r B = R(PB) along R(B) = R(I-- PB), admits the canonical decomposition PB = QM. We can deduce PBQ = Q
M PB = M.
U s i n g t h e G e r s h g o r i n t h e o r e m 1 we c a n s h o w Proposition
2.1
T h e nonzero eigenvalues of B have strictly negative real parts. Proposition
2.2
T h e spectral projection PB associated with the eigenvalue 0 of the matrix B satisfies
tim If:'- e ll = 0 tT~ (for the proof, see [I]).
2.3
Aggregate chain
Let A and B be two generators of h o m o g e n e o u s M a r k o v chains with values in a finite state
space E Csee 141 and
IS]).
Let F A and P B be the two spectral projections associated with the eigenvalue 0 respectively for the matrices A and B. W e d e c o m p o s e the operator PB as PB = QAI. Theorem
2.3
T h e restriction of the operator P B A P B {q~WE
E ' } by t h e m a t r i x
to the subspace R(PB) is represented in the basis
191 w h i c h is t h e g e n e r a t o r of a M a r k o v c h a i n on E . (for the proof, see [4])
3
The
Limit
equation
We c o n s i d e r t h e e q u a t i o n (1) in the S t r a t o n o v i c h f o r m
~' p'(o)
=
p'(~+
=
p
A)-
1 c ~-p u
2
+ p'g 9
If we n o t e A-=
A -- 1 H 2
we can w r i t e
/~, p'(o) Setting r
=
O,
=
=
p,(B___+
A-) + p ' H "y
p
we get 0 ----poB.
Remark
The
initial condition p0(O) belongs to kerB; the initial condition pS(O) = p is arbitrary,
and w e m u s t introduce in the asymptotic
expansion
the correction terms called b o u n d a r y
layer terms at t = O.
4
Asymptotic expansion
We c o n s i d e r t h e e q u a t i o n (1) a n d we w r i t e p~ in the f o r m
p'Ct) = P0(~) + PoCt)CZ -- P n ) s ;'~ + ~ (p,C~) + elC~)(x In
t h e s e q u e l , we s h a l l n o t e
c: = (I-
P , ) s , LB ,
~)e"-~)+...
192 and to simplify the n o t a t i o n s , we shall omit t h e d e p e n d a n c e on t of t e r m s in the expansiozl. We get formally (e -x) (e~
PoC'B
= (Po + PoC'") B P ~ C ' B = (Po + P o C ' ) A ' + (P~ + P ~ C ' ) B
~a + P o C ' +
+ (Po + P o C ' ) H 9
a n d , for k __> I,
(ek)
P k + /JkC'+ Pk+IG'B = (Pk+ PkO')'A+ (Pk+i+ Pk+IC')B + (pk+ PkC')H~
I t e m ark
If A be the s u p r e m u m of the real p a r t s of the n o n z e r o eigenvalues (AI) of the m a t r i x B , then, for any t,
IlCz - P~)s'~ll
_< c, ~'
w h e r e G is a p o s i t i v e c o n s t a n t , and k < O. T h e v e c t o r P k ( f . ) ( I -- PB)r ~B belongs to R ( B ) , and is called t h e k-th o r d e r b o u n d a r y layer term.
T e n d i n g 9 to O, w e get
(S_~)
0 = po B po(o) + Po(o)
(so)
Po
PoG"
= p
= poA + p i B
+ p o H ~I
= PoC'-A+ P o C ' H 9
a n d , for k > 1, p~(o) + pk(o)
= o
Pk = P k ' A + Pk+1 B + P k H ~
(Sk)
/6k(7" = Pk6"'A-+ P ~ G ' H 9 Using the decomposition of ~ n
p = p+ p w e get finally
with
for every p, w e have
~'B = 0
and
PPB = PPB = "P,
193
(s_~)
0
po(O) + ?o(O) (So)
= /3oB =
p
/~o = poA + ,6tB + PoHi)
? oC" = ['oc'-E + PoC "H a n d , for k >_ 1,
p~(o) + ?k(o)
(sk)
Pk
= 0 =
pkA
+
Pk+lB
+
pkH~l
?~c" = ?kC"T+ i'kC'U~ in I t b - f o r m , we get
And
po(0) + 2;0(0) (So)
=
p
dpo
= poAdt + p t B d t + poHdy
dPo
=
P o C ' A C -tdt + P o C S l t C - ' d Y
and, for k >_ 1,
pk(o)
+ ?k(o)
dpk
(Sk)
= o = pkAdt + @k+iBdt + pkHdy
di'k = @kC'AC-'dt + ['kC'~C-'dy Remazks
9
C o n s i d e r t h e s y s t e m ( S _ I ) ; a n d , we h a v e , for e v e r y t
poCO = FoCt) and,
d'fo
= ~oAdt + ~lBdt +'PoHdY
a n d , if we a p p l y t h e o p e r a t o r Pn~ we o b t a i n
d~o = "~oAPBdt + poH PBdy Now,
by the unicity of the decomposition
of semimartingales, w e have
~'oH = FoH pn. This m e a n s
that w e m u s t verify the following equality
PDH
= PBHPB
or
MIT
= MHPB.
1 94 F o r i n s t a n c e , t h i s e q u a l i t y is v e r i f i e d if t h e m a t r i x H is d i a g o n a l c o n s t a n t on e v e r y b l o c k of the matrix
B:
A s PB a n d B h a v e t h e s a m e b l o c k - s t r u c t u r e ,
we o b s e r v e e v e r y b l o c k of B .
P~ and H commute.
In the sequel, we shall assume that. 9
We
need to a s s u m e the invarlance of k e r B by the operator A in order to justify the
existence of the equations in ]~k. 9
T h e c o m p o n e n t Pk is obtained by solving the Poisson-equation
d (pk-l-- fo'pk-l(s)Hdy)--pk-tA Let J~fn be the inverse of B in the subspace R ( P n -- I) :
HBB
= B I I B = P D -- I
and
HBPD
= P D I I B = O.
Then
~k=--~d( P k - 1 - - fo'Pk-l(S)Hdy)~ + pk-l.4~v. 9
T h e c o m p o n e n t P'k satisfies the following equation
d'~k = p k A P B d t + ~'ktt P B d y , obtained by applying the operator PD.
5
Asymptotic
Result
In this section, we give an approximation result for the (m + 1) th order. Towards this end, we consider the following system of equations
p0(0)+ P0(0) (So)
a n d , for 1 < k < m ,
=
p
dpo
ffi poAdt + p l B d g .4- p o H d y
dPo
= P o G * A G - S d t + PoG*HG-Sd~/
195
pk(o) + e~(o) ( s k)
=o
tips, = p k A d t + p k + t B d t + p k H d y = PkC 9149
dPk We choose
Pm+ls u c h
+ PkC*HC- 9
that
p.~+~(0) dpm+t Proposition
= pm+J.Adt + Pm+lHdy
G.I
F o r a n y t E [O, T1, w e h a v e
E~Clp'(~) - ~ "kCPkCt) + PkC~)c'C~))123 = oC.~I",+~)) k=O
Proof We d e f i n e , for a n y t E [0, T],
r~.C~) =
p'CO - ~ ~kCp~C~) + PkCOc'ctl)) - ~ ' + I p . , + I C O k=O
It can be s h o w
that r m9 is solution of the following equation B
dr" =
r'~(O) = Thus
, ~ C - + Ale~+ r~,Uey
0
we have
(t(r:(s)(B+ A), B u t , for every p, we have C P ( ~ + A L p )
fot
fot
< 0, so we obtain
~(Id.(~)l ~) -< ~'('+~)o~ + I1~ 11'
/: ~ClrkCs)l~d.
Using proposition 4.3, and then the G r o a w a l t - B e llman inequality, we obtain, for any t E I0, T]
~.(l~c,)l ~) _< ~(..+,o and we get the desired result.
196
Decentralization and aggregation
6
We consider the systems (SO) and (Sk)
vo(O)+
?o(O)
(So)
= p
dpo
= poAdt + P t B d t + p o H d y
d~'o
= fioC*AG-*dt + PoC*HC-*dy
and, for k > 1,
pk(0) + ?k(0) (Sk)
dpk
=0 = p~Adt + ~k+xBdt + p ~ H d y
dPk = P k C * A G - C d t + P k G * H C - * d y If we apply the o p e r a t o r PB to the different equations of the systems (So) and (Sk), we get (S'o)
P~176 d~o
=F = p o A P B d t + poHPl3dy
and, for k _> I,
(sk)
LCo)
=0
d~k = pkAPmdt + p k H P B d Y B y subtraction, we have Po(o)
--
d]30 = p o G C A G - ' d t + ~6oC~HC-Cdy and, for k > 1,
~(o) + ?k(0) -- 0 (Sk)
d~k
= ~ k A ( I - - P B ) d t + Pk+IB d t + p k I t ( I -
dP k
= PkC*AC-'dt + Pk_xC*HC-*dY
PD)dy
Thus, by the block-diagonal structure of B, /5k and /Sk m a y be calculated by solving all equations on every block of B. The dimension of these systems is reduced.
Now, consider the systems (~'o) and (S'k) ~o(O)
=
CSo) d'fo = "foAPBdt + "foH PBdy
197 a n d , for k > 1, ~k(o)
CTk)
= o
d'pk = p k A P f l d t + pkH PBdy or,
cT,~)
~k(O)
P u t p ~ = PkQ , P ~ =
PkQ a n d A ~ = M A Q .
d'~
= 0
= "fkAPBdt + ~ k A P n d t + "fkH PBdy + ~kH PBdy T h e n we o b t a i n t h e a g g r e g a t e s y s t e m s
(s~) dp a = paAadt + paHdy a n d , for k > 1, p~(0)
dp~
= 0
= p~Aadt + ~ k A Q d t + "pkHdy
Remark T h e s e d i f f e r e n t e q u a t i o n s a r e of d i m e n s i o n
r, w h e r e r is t h e n u m b e r of t h e e r g o d i c s e t s
of t h e m a t r i x B . References [1] C o d e r c h - W i l l s k y - S a s t r y - C a s t a n o n ,
Hierarchical Aggregation of Linear S y s t e m s with Mttl-
tiple T i m e Scales , I E E E T r a n s . on A u t . C o n t r o l , A C - 2 8 , 1 0 1 7 - 1 0 2 9 , 1983. [2] C o d e r c h - W i l l s k y - S a s t r y - C a s t a n o n ,
Hierarchical Aggregation o/ Singularly Perturbed Fi-
nite Stats Markoe P r o c e s s e s , S t o c h a s t i c s , 8, 2 5 9 - 2 8 9 , 1983. [3] C o u r t o i s , Decomposability, A C M M o n o g r a p h [4] D e l e b e e q u e ,
S e r v i c e s , 1977.
A Reduction Process for Perturbed Markoe Chains , J. S i a m A p p l .
Math,
43, 2, 325-350, 1983. [5] D e l e b e c q u e - Q u a d r a t ,
actions , A u t o m a t i o n ,
Optimal Control Markoe Chains A d m i t t i n g Strong and Weak Inter17, 2 8 1 - 2 9 5 , 1981.
[6] K a t o , Perturbation T h e o r y / o r Linear Operators, S p r i n g e r - V e r l a g , 1966. [7] M a r c h e t t i ,
Mgthodss de Perturbations Singuli~res en Filtrage non Lingaire , T h ~ s e , U n i -
v e r s i t 6 d e P r o v e n c e , 1986.
FILTRES APPROCHI S POUR UN PROBLI ME DE FILTRAGE NON LINI AIRE DISCRET AVEC PETIT BRUIT D'OBSERVATION P a u l a Milheiro de Oliveira * INRIA centre de Sophia Antipolis BP 109 06561 Valbonne Cedex (France)
Abstract W e study the ~ymptotic behaviour of a nonlinear one-dimensional filtering discrete time problem, as
some parameter r tends to O. We treate the case of a nonlinear discrete time problem coming from a continuous time one with small observation noise. Finite dimensionM aproximate filters are proposed and results concerning estimations of their performance are stated and proved. For the result concerning the error between the approximate filter and the optimal one, only a sketch of the proof will be given. It makes use of probability changes and differentiation with respect to the initial condition. FinMly, we present the results obtained when applying those filters to an example and we notice that the propositions previously stated are verified numerically.
1
Introduction
On consid~re le prohi~me de filtrage non lin6aire unidimensionnel, eu t e m p s discret, suivant: On a un signal {X~} v6rifiant
x~+, = x~ + b (x~) zx, + . v ~
~ + , , x0 = ~
(1)
et on dispose d ' u n e observation y donn6e par g
o~ {wk} et { ~ } sont des bruits blancs gaussiens s t a n d a r d s ind6pendants et ~ une v.a. ind6pendante de {w~} et {~h~}. Le param~tre e eat suppos6 petit et A t = ~ , cz > 0. Le probl~me de filtrage consiste b. calculer, pour toute fonction mesurable r l'esp6rance conditionnelle de r 6taut donn6e l'observation yt j u s q u ' k l'instant k E {0, 1 , . . . , K}. On remarque qu'on peut regarder ce probl~me comme 6tant la discr6tisation d'un probl~me en temps continu, n s'agit, k cheque instant t E [0, T], d'estimer le signal {X,} solution de l'6quation diff6rentieUe stochastique dXt = b ( X t ) d t + a d w : , X0 = ~ (3) au vu de l'observation {Yt} donn6e par
dY~ = h(Xt) dt + ~dw~ , Y0 = 0 "et Faculdade de Eugenlmria da Universidade do Porto, Run dos Bragas, 4099 Porto Codex (Portugal).
(4)
199
oh wl,w 2 sont des processus de Wiener standards ind6pendants et ~ est une variable al6atoire ind6pendante de w 1 et w 2. (cf. [Le Gland], [Picard]). I1 est naturel de penser que si a est grand (donc At petit) le comportement de (1)-(2) est proche de (3)-(4) alors que, si a est petit, il s'en 6carte. I1 se trouve en falt que le cas critique est a = 1. Plus pr6cis6ment on s'int6ressera au calcul de l'estimation du signal {Xe} sachant l'observation j usqu'~, l'instant k, )(k = E[Xe[Y0k], laquelle minimise l'erreur quadratique moyenne. Vu les difficult6s num6riques qu'un tel calcul pr4sente, on cherche des approximations de {)(a} (filtre optimal) donn4es par des sch6mas r6cursifs. Le comportement asymptotique de ces filtres lorsque r tend vers 0 fern objet de notre 6tude.
2
Construction
des filtres approch6s
Pour des raisons li6es g l'application d'une m6thode de changement de probabilit6s, pour obtenir les estimations de l'erreur quadratique moyenne, il sera plus commode de consid6rer notre observation sous la forme suivante:
~+~ = h(x~) + ~
~+,,
(5)
off {va} est un bruit blanc gaussien standard ind6pendant de {wk}. Formellement, l'observation ffk+~ contient la m6me information que l'observation yk. On suppose que ~'~ est la tribu des observations jusqu'~ l'instant k:
Alors 17d = y0~-I ce qui signifie que le probl~me de filtrage pour 17"~ correspond en fait ~t une pr6diction pour Y0k. On consid6re le processus Me+t=Me+b(Me)At+0(~e+l-h(Me)),
Mo=mo
(6)
et on veut 6tudier la fa~on dont il approche Xk = E[XklY~]. Des rdsultats qu'on obtiendra sur l'erreur de pr4diction Xk - Me on d6duira immediatement les r6sultats 6quivalents sur l'erreur de filtrage, puisque, voir le paragraphe 2.1, on a
I s - .eel < c a t , o~ Re = E[Xe_d~?]. Pour la construction du flltre approch6 on peut s'inspirer de ce qui se passe dans le cas lin6aire et ainsi, par g6n6ralisation, choisir une approximation du gain. L'estimation de Xe - Me sera obtenue b. partir de l'6tude d'une suite r$currente. Par contre, pour estimer )~k - Mk, on introduira d'abord deux changements de probabilit6s et on obtiendra une expression pour Xe - M~ qui fait intervenir les d4riv6es de Xk par rapport ~ la condition initiale. Sur cette expression, on conditionnera les diff6rents termes par rapport ~ la tribu ~k des observations et on proc~dera ~ des estimations asymptotiques. Selon la valeur de c~, les changements de probabilitds choisis seront diffdrents et on aboutira, naturellement, ~. des expressions asymptotiques diff~rentes pour Xe-Me. Darts une premiere section on 6tudie le cas a > 1 (section 2.1) et ensuite le cas a = 1 (section 2.2). Le cas cr < 1 offrant certaines difficult6s dans l'application de cette m.4thode de d6monstration, on se limitera k faire une remarque proposant un filtre approch6 "performant".
200 HypothSses:
On suppose que
(H1) ( est une variable aJdatoire de loi de probabilitd po telle que I
poet', fl ol'Po( )d
<
(pour tout r ) .
oo
(H2) h est une fonction C 3 k d~riv~es born~es et [[ah'[[ >_ ah'(z) >_ eh > O, Vz. (It3) best une fouction C 2 ~. d~riv&s born~es, [b'(z)[ _< lib'I[, Vz. (H4) A t = ~ , 2.1
Le cas
a>_ l e t t , . = k A t . a>
1
On considSre d'abord le cas oh At = ~', avec a > 1. On propose un filtre approch~ unidimensionnel et, pour ce filtre, on estime XL. - M~ et )(~ - M,.. Le filtre a p p r o c h d On prendra 8 = a At darts le sch+ma (6), i.e. M,.+~ = Mk + b(M~) A t + a A t (Ok+x
h(M,.))
Mo = mo
(7)
On obtient facilement le r6sultat suivant: P r o p o s i t i o n 2.1 Mk dtant donng par (7), on a l'estimas r
tk+l 1
E[(Zt.+t - M,+,)2I 5 c e x p t - c T ~
+ ce.
(8)
Preuve On peut utiliser les d~veloppements de Taylor des fonctions b e t h, b(X~) = b(M,.) + b'((X)(x~ - M~) et h(X~) = h(M~) + h'((~)(X~ - Mk) , pour obteuir l'~galitd Xk+l - M,.+, = [1 - ~r At h,({x) + b,(c.x) At] (Xk - M~) + a v / ~ { w k + , - v k + 0 .
(9)
E
On utilise les hypothb.ses pour dfimontrer que, pour v "assez petit" (e _< (ah)-~l{"-l)), B EI(X%+, - M~+,) 21 _ (1 - d)k+'E[(Xo - M0) 21 + ~ - , o~t
At A --a- 1 - ( 1 - c h - - + t l b ' l l A t ) 2
<1,
= B n2aZAt.
g
Or
At At c-- < A < C
donc
B
= O(e) et (10) devient:
E[(Xk+~ - Mk+,) 2] < c e x P t - C - - / - - I + c~.
Quant h l'dcart Xk - M~., on montre le thdor~me suivant:
(xo)
201 T h 6 o r ~ m e 2.2 Le schdma (7) vdriJie 2~, - M~, = 0 ( ~ '~-~1~ v
~)
a~z s e n 8 0 ~
E[[)(t
-
+ c(d '-1/:
Mtl] < c e x p ( - c ~ }
v r
Preuve Elle sera divis6e en plusieurs parties, utilisant des changements de probabilit6s, une version discrete du Th6or~me de Girsanov et la d6rivation par rapport k la condition initiale. C h a n g e m e n t de p r o b a b i l i t 6 s . On introduit deux changements de probabilit6s. A l'aide d'une version discr6te du Th4or6me de Girsanov, on d6finit les probabilit6s . /5, telle que wa et Yt sont des 5vk- bruits blancs gaussiens ind6pendants 9 /5, telle que wt et Yt sont des ~ t - bruits blancs gaussiens ind4pendants, o~ wt = wk
vG7 -
-
-
[h(Xk_,)
-
h(Mt_l)] 9
E
On note
dP Lt= ~-fi~
At = ~dR ~'k "
et
Le nouveau systhme est Xa+l = a a_~ [h(Xt) - h(Mk)] + X t + b(Xt) At + ~ v / ~ @ k + l
(11)
Yt est un bruit blanc gaussien ind4pendant de wt. La densit6 de P par rapport ~./5 est Lt At. On a la formule de Bayes: E[ r
17~]
E[r If'd] = k i l t &i?tl oh r est une variable al6atoire P-int6grable et 9Vk-mesurable. Le calcul de log (Lt At) nous donne: 1 t
log (Lt &) = -~7 ~_,[h(X,_J - h( M,_,)I[(X, - M,) - (Xi_, - M,-1)] At t + - - y ~ [ h ( X , _ l ) - h(M,_~)lIb(X,_J --
b(M,_~)]
0"~r i = l
k
At
t
+ At 7 . =~ h(M,_,) 9, - ~ = ~
h2(M,_,)
"
(12)
D d r i v a t i o n p a r r a p p o r t h la c o n d i t i o n i n i t i a l e . On r lea vaxiables qui interviennent daals nos calculs comme des fonctions de X0 (condition initiale) et des processus ff;~ et ya. Soit ~bt = ~b(Xo, ~ , 9~). D6finissons slots les processus
Zk a= OXk OXo ' Z . , a= ~Zk~ z.
Zo=-I
202 On d6rive (11) et on obtient la majoration Z,k < c e x p { - c t , - t~}, pour n >_ k. r
(13)
En d6rivant (12), on obtient une expression pour Xk -- Mk : Xk - M,
-
1 h'(X~_l)
~
cat
,=1
~
~_.(Xl-t - M i _ J [ h ' ( X i _ j Zk-l,i-t - - -
h ' ( X / - J 2 Zk-1 i - , - h'(Xi-2) Zk-Li-2] '
At k +--h'(Xk_j i=t~-'(Xl-x - Mi-t) b'(Cxl) h'(Xi-1) Zk-l,i-t
cAt
~2h,iX ~ ( X i - ~ - M i-~) ~ i-W~h,,itx ~{~_~)Z~-~,i-~ 2 e h'(X~_~) i=~ c~ 0 h ' ( X ~ _ J OXo log (L,A~) Z~-t,0.
(14)
En utilisant (13), on montre que les premiers termes k droite dans cette ~galit~ sont d'ordre O(r r Quant au dernier terme, l'application d'une formule d'int6gration par parties sous /5 permet de conclure que sa ( ~ , P ) esp~rance conditionnelle est d'ordre 0(r 9 Cette d6monstration suit pas ~ pus la d6monstration d'un th6or~me 6quivalent pour les syst~mes en temps coutinu (cf. [Picard]). R e m a r q u e 2.3 Quand h est lin6aire, les estimations sur (14) sont respectivement d'ordre O(e~-t/2V r et 0(s douc ~'~ - Mk = O ( e ~-1/2 V r (15) E s t i m a t i o n d e l ' e r r e u r de filtrage Enfin, pour obtenir une estimation de i'erreur commise duns une 6tape de filtrage pour Y0k, iI nous suffit de remarquer qu'une approximation de )(~ = E[Xk_llYo ~-t] est encore donn6e par le sch6ma (7), puisque :?~ - :?~ = At E[b(Z~_l)1~3]
i.e. 12~ - 2~I _
(16)
L' estimation du th6or~nle 2.2 reste donc valable pour )(k - Mk. 2.2
L e c a s cz = 1
On consid~re maintenant uu syst~me d~crit par les ~quations:
{ X~+, = X~+b(X~)r
Xo=(
(1;)
En plus des hypothfises (H1) ~. (H4) on suppose que:
(ns) IIch'lJ < 2oh. R e m a r q u e 2.4 Une telle restriction signifie que notre 6tude n'est rulable que si la fonction h s'approche "suffisamment" d'une fonction lin6aire.
203 Le filtre a p p r o c h ~ On aimerait pouvoir construire un filtre approch6 unidimensionnel pour ce probl~me. Cependant cette thche se r~vNe d~licate, ce qui nous ram~ne b. la recherche d'un filtre du genre du filtre de Kalman ~tendu. On eonsid~re le filtre bidimensionnel suivant: Mo = t o o ,
(18)
h'(Mk)[ 0k-1 + a z h'(M~_,)] 0o = 0o. tik= hrZ(M~)[Sk_, + ~r2 h'(Mk_x)] + h ' ( M k _ , ) '
(19)
Mt,+t = M r + b(Mk)E + Ok (flk+l -- h ( M k ) ) , oh.
Pour b e t h lin~aSres, ce filtre correspond & prendre le filtre de Kalman - B u c y et n~gliger les termes en b At darts l'expression du gain. Comme dans le cas precedent (voir la section 2.1), on cherche d'abord &estimer X~ - M k et ensuite X~ Mk. Les m~thodes utilis~es pour obtenir ces estimations sont essentiellement les m&mes. Le lien entre erreur de filtrage et erreur de prediction reste le m~me, i.e., avec les notations du paragraphe 2.1, on a que -
De plus, on &tablit la proposition suivante: P r o p o s i t i o n 2.5 On a l'estimation E [ ( X k - Mk) 21 _< c e x p { - c ~ } + ee.
Le th~or~me suivant nous donne une estimation sur l'erreur de prediction X~ - M~. T h ~ o r ~ m e 2.6 On a l'esdmation E[12~ -
Mk[] _< cexp{-e~-} +
cr
R e m a r q u e 2.7 Dans le cas oh At = e~, avec cz < 1, on fait le raisonnement suivant: D'apr~s l'~quation d'observation le processus E
est un bruit blanc gausssien de variance r
done si on prend (20)
Mk = h - l ( y k ) ,
i.e. notre filtre approch~ consiste tout simplement ~ prendre l'image inverse par h de l'observation, on a
s E[(Xk
-
M~) .1 _< ll(h-')'ll 2 h-7"
Le filtre d6fini par (20) est d6j& "performant", pour a petit. Quant au r~sultat concernant l'estimation de l'erreur par rapport au filtre optimal, il semble difficile de trouver le bon changement de probabilit~ conduisant &une expression pour X k - M k dont les termes soient facilement estim6s.
204
3
Application num6rique
On applique le scMma (7) au syst~me
{ dXL = -Xtdt+c~dwit a~
=
( x , ~ + x , ) dt + ~ d~?,
off w i e t w 2 sont des processus de Wiener ind~pendants et X0 est une variable M~atoire gaussienne de loi A/'(3~o,Po), avec 9
cr=l.
9
~ = 1,0.1,
0.01, 0.001,0.0001
9
.~o=2.
9
p 0 = 1.
Cet estimateur nous permettra d'obtenir une approximation du filtre optimal facile i calculer (bien que l'hypoth~se (H2) ne soit pas v6rifi6e dans ce ca.s). Dans la suite on se propose de v6rifier num6riquement la performance du filtre approch6 par rapport a l'estimation donn6e par l'6quation de Zakai. Le comportement du filtre par rapport i la trajectoire simul6e est aussi 6tudi6 (voir le paragraphe 2.1). Les pa~ de discr6tisation b. considerer varieront selon la valeur de ~ e t on regardera l'intervalle de temps [0, 1].
pente z gamma
i
o.ooo
0.493 = 0,283
1o
O.OOlO0
0.010o0
0,1oooo
o o
ep$ilon
Figure 1: l'erreur quadratique moyenne entre le filtre approcM et la trajectoire simul~e, en fonction de e. Les figures 1, 2 et 3 repr~sentent l'erreur quaxtratique moyenne (eqm), obtenue sur 100 simulations de trajectoires ind~pendantes. On voit imm~diatement que, quand r est petit (s = 0.1, 0.01,...), la courbe repr~sent~e en ~chelle logarithmique s'approche d'une droite, i.e. e~['e~['e~['e~['e~[e~ ~ 10 - g a m m a Spe~e 9
On peut donc constater que la proposition 2.1 et le tMor~me 2.2, ~nonc~s pr~c~demment, sont v~rifi~s num~riquement et que la valeur de la pente est proche de la valeur obtenue tMoriquement. Dales la figure 4 on repr~sente, pour e = 0.1, l'~volution de l'erreur quadratique moyenne au cours du temps. Le pas de discr~tisation utilis~ a ~t~ At = 0.01. Le trait au dessus correspond i
205 peatc = l.OOO gamma = 0.388 0.00001-
d 0.00010-
O.OOlO0-
0.01000"
/ 0.10000-
-
o
o
p
p
epsilon
Figure 2: l'erreur quadratique moyenne entre le filtre approch6 et l'estimation donn~e par l'~quation de Zakai, ell fonctioa de ~.
pentc = 0,494 gamma = 0.275 0.00001-
0.00010-
oO,O01UO-
0.01000-
/
0.160002
-
~ o
P -
8o
P 2
~
O o~
-o
P w -
epsilon
Figure 3: l'erreur quadratique moyenne entre l ' e s t i m a t i o n donnde par l'~quation de Zakai et la trajectoire simul6e, en fonction de r
206 I'erreur entre [es deux filtres et les deux autres traits aux erreurs de ces m~mes filtres par rapport la trajectoire simul~e. La ressemblance entre le comportement du filtre approch~ et du filtre optimal est ~vidente. 0.0001-
0.0010m-
0.0100-
0.1000-
temps
Figure 4: l'erreur quadratique moyenne entre la trajectoire simul4e et les estimations et celles-ci entre elles, au cours du temps.
3,0002.500w ta
2.0001.500. 1,0000,5000.000 -0.500-
o
p
p
.~
9
temps
Figure 5: l'~volution des filtres approch~ et optimal sur une trajectoire simul~e. Dans la derni~re figure (figure 5), on repr~sente une seule trajectoire et le tiltre approch~, ainsi que l'estimation obtenue par r~solution num~rique de l'~quation de Zakai. On voit facilement que les
207
filtres (en pointill~ sur la figure) ~voluent de mani~re semblable et ne peuvent en fait ~tre distingu~s sur la figure.
4
Conclusions
Des filtres de dimension finie ont 6t~ obtenus pour le problSme de filtrage non lin6aire en temps discret 6nonc6. Selon la valeur du param~tre a c e s filtres sont d~crits par de diff~rentes ~quations. Si a > 1 on propose un filtre de dimension 1. En revanche, si a = 1, le filtre propos6 est de dimension 2. S i a < 1 la seule inversion de la fonction d'observation sera suffisante. Les rdsultats obtenus sur la performance de ces filtres permettent de conclure qu'ils sont asymptotiquement "aussi performaats" que le filtre optimal et qu'on peut vdrifier un tel comportement num~riquement.
R6f6rences [Bensoussan]
Bensoussan, A. : On some approzimation techniques in non linear filtering, Proccedings of June 1986 Workshop, Institute for Math. and Its Applications, University of Minnesota.
[Gelb]
Gelb, A. :Applied Optimal Estimation, written by the Technical Staff, The Analytical Sciences Corporation, M.I.T. Press, 1974.
[Jazwinski]
Jazwinski, A. H. : Stochastic Processes and Filtering Theory, Academic Press, 1970.
[Kallianpur]
Kallianpur, G. : Stochastic Filtering Theory, Applications of Mathematics, Tome 13, Springer-Verlag 1975.
[KBS]
Katzur, R. Bobrovsky, B.Z. and Schuss, Z. : Asymptotic analysis of the optimal filtering problem for one-dimensional diffusions measured in a low noise channe~ SIAM J. Appl. Math. 44 (1984), Part I: 591-604, Part II: 1176-1191.
[Le Gland]
Le Gland, F. : Estimation de param~tres dans les processus stochastiques, en observation incomplete - Application ~ un probl~me de radio-astronomie, Th~se pr~sent~e k l'Universit~ Paris IX-Dauphine le 26 Novembre 1981 pour obtenir le titre de Docteur-Ingenieur.
[Milheiro]
Milheiro de Ofiveira, P. : Filtrage non lingaire avec petit bruit d'observation. Comparaison numgrique de 2 fittres, rapport de stage de D.E.A., Universit~ de Provence, 1987.
[Picard]
Picard, J. : Nonlinear filtering of one-dimensional diffusions in the case of a high signal-to-noise ratio. 1986 - SIAM J. Appl. Math. (46),6,1098-1125.
[Picard]
Picard, J. : Asymptotic study of estimation problems with small observation noise, dans Stochastic Modelling and Filtering (Rome 1984), Lect. Notes in Control and Inf. Sci. 91, Springer, 1987.
[Picard]
Picard, J. : Filtrage de diffusions vectorielles faiblement bruitges, Proc. 7th Int. Conf. on Analysis and Optimization of Systems (Antibes 1986), Lect. N. in Control and Information Sciences, 83, Springer 1986.
[Picard I
Picard, J. : Nonlinear filtering and smoothing with high signal-to-noise ratio, Stochastic Processes in Physics and Engeneering (Bielefeld 1986), Reidel 1988.
DECENTRALIZED FILTERING FOR INTERCONNECTED SYSTEMS
Ali F e l i a c h i
Department o f Electrical & Computer EnKineering West Virginia Universi~,y Morgantown, West, VirKinia 26506-6101
ABSTRACT
The
paper
addresses
interconnected is
achieved
systems through
independent Sufficient
design
which for
of
independent
additional
filters condition
system.
the
using
decentralized local
measurements
use
local
stability
on
the
information
is s t a t e d
filters
dynamical
interface
only
in terms
models.
are
for
large-scale
The
decoupling
variables.
unbiased
These
and
of the observabiltiy
stable.
of original
An e x a m p l e i s g i v e n .
I. I N T R O D U C T I O N
Most
large-scale
together.
systems
A centralized
manner
filter
that
the computational
one
single
central
constraint the is
cost
to
independent
subsystem
are
the
vector
and the
than
knowledge
that
processors
are
coordinator
to g e n e r a t e
at
local
interconnected
at one central
and
centralized are
location
to
the global estimator.
are availahle
dispersed
alternative the
systems
subsystem
which in
levels
this using
T h e local f i l t e r s
The decoupling
to t h e i n t e r f a c e
at this
systems presented
local information. filter.
using
is achieved
variables.
These
parallel processing.
of decentralized
estimation
stations
This filter assumes
communication
at
related
distributed,
burden.
The
locally
to d e a l w i t h d i s t r i b u t e d
used
either
geographically
sophisticated
the problem
by independent,
schemes
subsystems
f r o m all s u b s y s t e m s
reliability.
equations
the
Decentralized
hierarchical
be done
for
information
measurements
satisfactorily.
state
Obviously
suitable for implementation
To t h e a u t h o r ' s solved
can
several
locations.
measurements
dynamical
dimensions
additional
structures
the
of
from the computational
reduce
process
of smaller
through
suffers tools and
extensive and
vector
at several
location.
requires
increase
are
composed
E s t i m a t i o n of t h e s t a t e
o r in a d e c e n t r a l i z e d
paper
are
is defined processors. measurement
obtain
estimation here
Most known gathering
estimates
This includes
has not
been
as the estimation
that
methods
of use
stations.
Local
are
to
sent
t h e w o r k of S p e y e r
a [1],
2O9 Chong
[2], Willsky
processors order
use
the
model.
addressed achieved
True
by
full order
[4], etc. system
decentralized
Sanders
through
addresses
[3]) Alouani
[5]) and
least squares
the same
problem
In
model
schemes Feliachi
estimation
but
the
most
or
of
these
a dynamic
aggregate
for
interconnected
and
Sperry
systems
[6], where
of the interface
solution
schemes,
is obtained
the
local
of the full have
been
decoupling
is
variables.
This paper
by
appropriate
finding
local models. T h e paper is organized as follows. T h e problem models
used
to design
the local filters are
is formulated
derived
these filters are designed and stability conditions are derived. in section V to illustrate the proposed
II.
Consider
•
where
in section If.
in section III.
A n example is given
technique.
PROBLEM
FORMULATION
the following linear time-invariant
stochastic system:
= X x(t) + ~ w(t)
x is an n-dimensional
E{[--(o)} = Xo; E{w(t)}
: 0;
The
In section IV
{I)
state vector and w a zero-mean white process with:
Var{i(o)}
= Zo
E{w(t)wT(r)}
= Q 6(t-T);
E{w(t)xT(o)]
= 0
E {' } denotes expected value The
system
has p
stations, SI, S 2 ... Sp) w h e r e
station S i the measurements
measurements
are taken.
are given by:
Yli(t) = L i x(t) + ~i(t) where
At
(2)
L i is a constant mixn matrix of full rank m i . /~i:
zero mean, white process with
E{w ~ } The
objective
= 0;
E{~(t)
~i(~)} = R i 6(t-T);
is to find an
unbiased
local information to estimate the observed
Ri:
constant matrix
stable filter of order
m i that uses
variables which are given by:
zi(t) = U i x(t)
The
filter is designed
only
(3)
so
that
it
optimized, i.e., if the error is defined b y
is unbiased
and
an
error
criterion
Ji is
210
(4)
ei(t) = zi(t) - zi(t ) then:
E { ei(t) } = 0,
t m 0;
WE
Ji = E{I
T
ei,Tf,}
ei(v) U i ei(v) dr + ei(T f) V i
(5)
o
where
Ui, V i m 0
constant weighting matrices.
IlL
INFORMATION
INDUCED
MODEL
Model is d e r i v e d at e a c h s t a t i o n Si u s i n g t h e o b s e r v e d v a r i a b l e s z i. will be called h e r e an i n f o r m a t i o n i n d u c e d model. t y p e R e p r e s e n t a t i o n [7]. Proposition:
=
zi(t)
Xri(t) Let
F i r s t , t h e following model is n e e d e d .
The g i v e n s y s t e m can be d e s c r i b e d b y t h e following e q u i v a l e n t s y s t e m :
zi(t)
Proof:
This model
I t is d e r i v e d from a H e s s e n b e r g
Ai
Mi
A~
Mi Ni] zi~t)
Ari
Mri
xi(t) =
o ]
Bi
zi(t)
Nri
Xri(t)
state
vector
+
B~
(6)
w(t)
Bri
zi(t)
~i(tl Xri(t) xi(t)
is
related
to
E(t)
by
the
following
similarity
transformation:
(7)
xi(t) = T2i Tli x(t) = T i x(t) where:
o o] Mli
M2i
0
I2 i
I l i , I2i a n d Imi a r e i d e n t i t y m a t r i c e s of a p p r o p r i a t e d i m e n s i o n s . Li is p a r t i t i o n e d as:
Li = [Lli
L2i]
S i n c e Li h a s full r a n k , t h e n Lli is s e l e c t e d so t h a t i t s mi columns a r e l i n e a r l y i n d e p e n d e n t , i.e., Lli is a s q u a r e i n v e r t i b l e matrix. A p p l y i n g t h e similarity t r a n s f o r m a t i o n T l i to t h e o r i g i n a l s y s t e m y i e l d s :
211 Xli(t) = Tli ~(t) where
Xli(t) = [ ]xi(t) zi(t)
(8)
Xli(t) = Tli A T[} Xli(t) + Tli B w(t) In partition f o ~ equation (8) can ~
zi (t) = Ai zi(t) + Mi x[(t) + B i w(t)
(9)
i[(t)
(1o)
= r l i z i ( t ) + r2[ x l i t ) + r 3 i w(t)
The matrix ~ Let
written
m i -~ P i .
Define is complete.
is an mi x ~
Then
matrix, Pi = n - ~ .
I t s r a n k qi ~ rain (mi, Pi).
q i -~ mi,
~i = Mi xl. But
If qi = mi
in general
then let M i = Mi and the decomposition
qi z m i.
In
this case
extract the
q5 linearly
independent components of $i: H r
= H Mi x:x =
[] Mi
xi
0
where rank (Nli) = P(Mi) -- qi, Y~ i s a qi x P i m a t r i x
and
H is an
elementary
matrix. (11)
Define
zi = M i x 1
Then
Mi xi = H-I [ Mi x : 110
=
Iqi 0
_
Therefore, equation (9) can be written as: zi
-- A i z i + M i zI + B i w
(12)
Consider equations (Ii). Partition M[: M~ = [Mli Define
M2i]
[
T2i =
where M[~ Mli 0
M2i I2i
T~.i is a similarity transformation.
x
exists since M~ has full rank
] Is is applied to equation (10):
(13)
212 The dynamics of xi' are described with: (14)
X[ = T2i xi = T2i r{i z i + T2i r2i T2i -I xi + T 2 i r3i w Partition eq,mtion (14) as Z i : A:I z i + M:l Z i + Ni Xri + Bs w
(15)
Xri = Ari z i + Mri z'l + Nri Xri + Bri w
(16)
Equations (12), (15) and (16) are the decomposition sought,
T h e m a t r i x T2i i s
given by:
T2i = [ Imi0
TTi0 ] Q.E.D.
Note:
zi are the observation
called
here
residual
the
interface
An u n b i a s e d filter
at station
The variables Xri are called the
S i.
variables.
IV, local
is obtained
criterion
Their dynamics are driven by zi which are
variables.
variables
by
filter
DECENTRALIZED is
derived
to e s t i m a t e
minimizing the error
can be written
FILTERING
criterion
the
observed
Ji g i v e n
variables.
by equation
(5).
This This
as:
Tf
Ji = Trace {
I Ui P(T) dT + V i P(Tf) )
(17)
O
To o b t a i n needed.
These
decentralization additional
additional
measurements
measurements are given
on the interface
variables
are
by:
Y2i = Di z'i + vi where
E{wvI} = 0;
E[vi(t) vI(T)] = S i 6(t--T); E{~ i vI(t)} = 0
Theorem 1: If D i has full rank qi then a stable, unbiased, local filter is given by:
~i(t) = F i zi(t) + K i Yli(t) + O i Y2i(t) where
Fi = Ai - K i
(18)
213 K i, G i minimize the performance
index (17)
K i =PR~* and P satisfies:
=AiP+PAI-PmI I P+miQm[ +QisiGI P(o) = L i ~o Li = Po
Proof:
The error
dynamics are described
el(t) = Fiei(t) + (Ai-Ki-Fi)zi(t)
by:
+ (Mi-GiDi)zi(t)
+ Bi(t)-Ki~i(t)-Givi(t)
(19)
F o r t h e f i l t e r to be u n b i a s e d , one n e e d s Fi to be s t a b l e a n d : Fi = Ai - Ki
(ZO)
Gi Di = Mi
(21)
The
matrices Ki, and G i are c h o s e n
D i is an
tixqi
parameters
matrix
of rank
If
to minimize Ji. ~izqi
to use in the minimization of Ji.
partitioned as Dri = [Dill of G i is given by [Gil Hence
qi"
Dli2] w h e r e
then
G i is an mixl i matrix and,
G i = MiD~iI,
In general
Dil is invertible.
4i > qi.
G i has Then
no
free
D i can
be
A compatible partitioning
Gi2]0
G i D i -- Gil Dil + Gi2 Di2 -- M i
(22)
Since Dil is invertible: Gil = [Mi-GizDi2] Di}
(23)
Gil is used to satisfy GiD i = Mi, a n d Gi2 is obtained by optimizing Ji.
The
matrix G i can be written: G i -- Gil E l + Gi2 E 2
(Z4)
where E 1 and E 2 are selection matrices. Substitute
(23) into (24):
G i -- H i + Gi2 T i where
(25)
H i = M i Di~ 1 E l T i = (E 2 - Di2 Di~ 1 E 1 ) It remains
propagates
as:
to determine
K i and
Gi2 that minimize
Ji"
The
error
covariance
214
# = FiP + PF~ + BiQB I + KiRiK ~ + (Hi + Gi2T i) Si(Hi + Gi2Ti) T
(26)
P(o} = Po The Hamiltonian corresponding to the optimization problem defined by equations 17, 19, 26 is given by: B = trace (UiP + [FiP + PF[ + BiQB l + KiRiK ~ + (Hi+Gi2Ti)Si(Hi+GizTi)T]A T where
A, m a t r i x of L a g r a n g e
multipliers, is determined
(27)
by:
(28)
=- aB =- [Ui + F i' A + ,%Fi]
A(Tf) = V i
(Z9)
K i and Gi2 are obtained from the necessary optimality conditions (30)
al~/aK i = 0 a n d a~/aGi2 = 0
These conditions give: K i = p R~ ~
(31)
Gi2 =
(32)
- H i S i TiT(Ti S i T~)-*
The gain matrix, given by equation (31) is the Kalman gain associated with the following subsystem:
z i = A i zi + [Bi
Gi][ wvi ]
(33)
Yli = zi + ;Ji S i n c e (Ai, I i) i s o b s e r v a b l e
(34) f o r a n y A i t h e n F i : [ A i - K i] i s s t a b l e . Q.E.D.
S u f f i c i e n t c o n d i t i o n f o r t h e s t a b i l i t y o f t h e l o c a l f i l t e r c a n be s t a t e d in t e r m s o f the observabilty
~t
of the original system.
Ei = [ Li
Theorem 2:
0]
If (A, L i) is observable then the local filter is stable.
Proof: Consider equation (6). Define :
=
[ -Alp
-MN] ;
~i = [li
O]
215 where
(A, Li)
is observable if and only if
(F, C i) is observable since they are
related by a similarity transformation (T i in eq. 7).
p( [ ~ Write
-T-r F Ci
(~I )n-l~lr
...
Therefore:
) = n
[Ai o]+[o
F =
0
0
Then:
ci~k [li~ where
•k(1,2)
o]
+
[o
li.k(1,21]
is t h e b l o c k (1,2) o f n k c o m p a t i b l e
with the partition
used
here.
The observability matrix corresponding to the system (F,Ci) is:
Ci F
[ Ii
0
i Ai
0 :
:
i
Am-I
0
0
0
a1(1,2)
o
0
am_l(1,2)
o
LG
nn_l(1,2)
+
J: : ".11-I
~i ~ n-I
The Therefore
n
.
columns
Ai
of
the
above
observability
matrix
are
t h e m c o l u m n s of t h e f o l l o w i n g m a t r i x a r e l i n e a r l y
[ I i T AI
""
independent.
(A~I~-I]T
This corresponds local filter designed
linearly independent.
to t h e o b s e r v a b i l i t y using
equations
m a t r i x o f t h e local s u b s y s t e m .
(33) a n d
(34) is s t a b l e .
Q.E.D. V.
Consider
the 3rd order
EXAMPLE
system:
[120] ill
x=
2
4
5
1
I
3
x
+
2
1
w
Hence the
216 Let x I be the observed variable at Station variables at Station
(I), a n d x 2 and x 3 be the observed
(2), i.e.,
Zl = Xl
Therefore,
it is s o u g h t , to e s t i m a t e z i a t S t a t i o n S i.
C o n s i d e r S t a t i o n (1).
The measurement
is g i v e n
by:
Ytl = zl + Pl Let the required
additional
m e a s u r e m e n t on t h e i n t e r f a c e
variable
z I = x 2 be
given by:
Y12 = [1 Let:
2] r z{ +
v1
Q = R 1 = I; S 1 = Zx2 identity matrix Using the notations of the paper: A 1 = I,
M I = 2,
Then:
GIDI=M1
gives
Let
Dll = I
Then
G l l = (2 - 2 G12) G 1 = [GII
=
E 1 = 2 [1 1]
-.
G1 = [2
0]
+ G12 [ - 2
2 [1
0] 0]
=
0]
+
=
[2 [-2
G12 [0
I]
0] 1]
1]
[:][ [:]1
0l - [Z 01 -
G 1 = [.4
It remains
2]
GII [I
T 1 = [0
GI= [2
Finally:
= [I
GIIDII + GI2DI2 = GII + 2G12 = Z
GI2]
H1 = M1 D [ }
And :
DII r
[ - 2 1] -
[ - 2 11
.S]
to s o l v e f o r K 1. U s i n g e q u a t i o n
(31}, t h e
T h e n , t h e local f i l t e r a t s t a t i o n (1) is g i v e n by:
z 1 = - 1 . 6 7 z 1 + 2 . 6 7 Y l l + [ .4 S i m i l a r l y , a local f i l t e r is d e s i g n e d
. 8 ] Y12
for station
{2).
steady-state
g a i n is 2.67.
217 VI.
The p a p e r
a d d r e s s e s t h e d e s i g n o f d e c e n t r a l i z e d f i l t e r s for l a r g e - s c a l e i n t e r -
connected systems. models.
The
variables. unbiased
These filters are
decoupling
These and
CONCLUSION
is
independent
stable.
designed
achieved filters
use
measurements
local
Sufficient condition for
o b s e r v a b i l t i y of original s y s t e m .
u s i n g i n d e p e n d e n t local dynamical
through
of
the
information only
and
interface they
are
s t a b i l i t y is s t a t e d in t e r m s of t h e
An example is g i v e n .
ACKNOWLEDGEMENT
This research was sponsored in par$ by the National Science Foundation under grant
ECS-8707139,
Research
Center
in part
under
by the
grant
West
PEL-21-88,
virginia University Energy and
in part
by
ONR
and
under
;r
contract
NO0014-K-0651.
REFERENCES
[1]
J.L. S p e y e r , "Computation a n d T r a n s m i s s i o n R e q u i r e m e n t s f o r D e c e n t r a l i z e d LQG C o n t r o l P r o b l e m s , " IEEE T r a n s . A u t o . Control, Vol. AC-24, No. 2, p p . 266-269, 1979.
[2]
C.Y. Chong, " H i e r a r c h i c a l E s t i m a t i o n , " P r o c e e d i n g o f 2 n d I~IIT/ONR C3 W o r k s h o p , M o n t e r e y , CA, J u l y 1979.
[3]
A.N. Willsky, et.al., "Combining a n d U p d a t i n g of Local E s t i m a t e s a n d Regional Maps Along S e t s of O n e - D i m e n s i o n a l T r a c k s , " IEEE T r a n s . Aurora. Control, Vol. AC-27, No. 4, pp. 799-813, 1982.
[4]
A.T. Alouani, J.D. Birdwell, " D i s t r i b u t e d Estimation: C o n s t r a i n t s on t h e Choice of t h e Local Models," IEEE T r a n s . A u t o m a t i c Control, Vol. AC-33, No. 5, pp. 503-506, 1988.
[5]
C.W. S a n d e r s , E.C. T a c k e r , T.D. L i n t o n , "A New Class of D e c e n t r a l i z e d F i l t e r s f o r I n t e r c o n n e c t e d S y s t e m s " , IEEE T r a n s . A u t o m a t i c Control, Vol. AC-19, pp. 259-262, 1974.
[6]
A. Feliachi, R.K. S p e r r y , " D e c o u p l e d F i l t e r s f o r I n t e r c o n n e c t e d S y s t e m s , " P r o c e e d i n g s o f 27th IEEE-CDC, A u s t i n , TX, p p . 2367-2368, December 1988.
[7]
E.C.Y. Tse, J.V. Medanic, W.R. P e r k i n s , " G e n e r a l i z e d H e s s e n b e r g T r a n s f o r m a t i o n f o r R e d u c e d - O r d e r Modelling of L a r g e - S c a l e S y s t e m s " , I n t . J. C o n t r o l , Vol. 27, No. 4, pp. 493-512, 1978.
SPLIT FORMS OF z - D O M A I N A L G O R I T H M S FOR LINEAR PREDICTION A N D S T A B I L I T Y A N A L Y S I S
Antonio Lepschy, Gian Antonio Mian and Umberto Viaro Department of Electronics and Informatics, University of Padova via Gradenigo 61A - 35131 Padova, Italy
A b s t r a c t - Many algorithms in linear prediction and s t a b i l i t y analysis may be expressed by the same general t w o - t e r m recursion involving two polynomials of consecutive degrees and the reciprocated polynomial of one of them. The properties o f these algorithms are interpreted from a geometrical point of view that refers to the loci described by the zeros of the polynomials in the related sequences as a characteristic real p a r a m e t e r varies. An analysis of all possible threeterm (split) forms involving only the symmetric and/or the a n t i s y m m e t r i c parts of three consecutive polynomials generated by the same t w o - t e r m recursion, is carried out. 1. I N T R O D U C T I O N A number of algorithms in linear prediction and s t a b i l i t y theory can be expressed by a recurrence relation for generating a sequence of polynomials of either ascending or descending degree [ I , 2] . The Levinson algorithm is perhaps the most popular of these. As is known, it was conceived in the c o n t e x t oF mean square estimation to find in a c o m p u t a t i o n a l l y e f f i c i e n t way an all-pole predictor. However, it may be used for d i f f e r e n t purposes; for instance, a tight relation exists between this algorithm and the s t a b i l i t y c r i t e r i a of Lehmer, Schur, Cohn, Marden, and Jury [ 3 ] , as well as the Nevanlinna-Pick problem ELI) 9 Another f a m i l y of z-domain algorithms comprises the so-called Routh-type methods, obtainable through a suitable variable transformation from the original s-domain three-term Routh procedure or from its t w o - t e r m form [ 5 ] . In particular, if the usual bilinear transformation is adopted, the algorithms for discrete-time systems modelling presented b y B i s t r i t z in E6] are obtained, whereas if the transformation s=(z +z- )12 is used, the B i s t r i t z s t a b i l i t y - t e s t procedure ~ 7 , 8 ] is obtained. Other families of z-domain algorithms may be found by transforming the s-domain Euclid-type a l g o r i t h m presented in ~ 9 ] . Much interest has been paid in the recent l i t e r a t u r e to unifying interpretations of some of the mentioned procedures. This has been done, for instance, by Benidir and Picinbono [ 10], by Delsarte et al. E 4 ] , and by Vaidyanathan and M i t r a [ I I ] according to either a network-synthesis approach or to an interpolation approach. Another a t t e m p t has been made in [ 12] for s-domain algorithms by adopting a different point of view; specifically, it has been shown that many algorithms can be regarded as particular cases of a unique general recursive procedure from which new interesting algorithms may be generated as well. A similar approach can be applied to zdomain algorithms as will be briefly outlined in Section 2 . In this way, it is also possible to give a geometrical c h a r a c t e r i z a t i o n of the various procedures along lines analogous with those followed in E13-] for the s-domain. The considered algorithms may be given either of the following recursive forms: ( i ) the t w o - t e r m form by which every polynomial in the related sequence is expressed by a suitable combination o f the preceding polynomial and its r e c i p r o c a t e d one, and [ i i ] the t h r e e - t e r m form which relates the s y m m e t r i c or a n t i s y m m e t r i c part of three consecutive polynomials in the sequence. The latter have been called in the literature split forms E14, 15], or immitance-domain forms
219
[ 16J as opposed to s c a t t e r i n g - d o m a i n ones. Such f o r m s are a t t r a c t i v e because they e n t a i l c o m p u t a t i o n o f a s m a l l e r n u m b e r o f c o e f f i c i e n t s and a l l o w one t o i n t e r p r e t the a l g o r i t h m s terms of reactance functions.
the in
In S e c t i o n 3 we c l a s s i f y and analyse f r o m the p o i n t o f v i e w o f the c o m p u t a t i o n a l c o m p l e x i t y all possible split r e l a t i o n s a s s o c i a t e d w i t h the same g e n e r a l t w o - t e r m r e c u r s i o n d e r i v e d in Section 2.
2. G E N E R A L
TWO-TERM
FORM
L e t us d e n o t e the i - t h degree p o l y n o m i a l in a s e q u e n c e by i
P.(z] = ~, a.. z j i iI j=0 and its r e c i p r o c a t e d p o l y n o m i a l by i P.(z) = z i P.(z -1] = ~ a.. z i-j . I
I
(1]
(2]
U
j=O The Common f o r m o f the t w o - t e r m
s t e p d o w n ( b a c k w a r d ) r e c u r s i o n s m e n t i o n e d in S e c t i o n 1 is
Ri(z) Pi I (z) = F.(z) P . [ z } + G.{z) P.(z) -
I
I
I
(3}
1
w h e r e Ri(z), Fi(z), Gi(z) are p o l y n o m i a l s o f s u i t a b l e d e g r e e (which is, in g e n e r a l , d i f f e r e n t From i and m a y d i f f e r From step to step) such t h a t Pi(z) is a Schur p o l y n o m i a l (all its zeros are inside the unit c i r c l e ) i f and o n l y i f Pi_l(Z) is Schur and the p a r a m e t e r s belong to c e r t a i n d o m a i n s . In p a r t i c u l a r , i f R.(z) = r
(4) = - ai0/aii w h e r e r i m a y be v i e w e d as a scaling f a c t o r ( w h i c h is equal to I - k 2 in the s t a n d a r d f o r m ) , r e c u r sion (3) r e d u c e s to the ( b a c k w a r d ) L e v i n s o n r e l a t i o n : i
z
i
'
F.Cz} = 1 I
"
Gi(z) = k i
=
-
PiC0)/Pi(O)
r z P (z) = P.(z} + k . P . ( z ) (S) i i-1 I I ; in which k i is the s o - c a l l e d i - t h r e f l e c t i o n c o e f f i c i e n t . As is k n o w n Pi(z) is Schur i f and o n l y i f P i _ l ( z ] is Schur and I k i I < 1 . A g e o m e t r i c a l insight i n t o this a l g o r i t h m can be g a i n e d by c o n s i d e r i n g t h a t the zeros o f both sides o f iS) b e l o n g t o the r o o t locus f o r the e q u a t i o n : P.(z) + c P.(z) = 0 1
(6)
i
f o r c = k i . This locus is s y m m e t r i c w i t h r e s p e c t t o the unit c i r c u m f e r e n c e and its " i " b r a n c h e s [which include the real axis) i n t e r s e c t this c i r c l e f o r ] c I = I . C l e a r l y , f i x i n g a r o o t o f (6) o,1 this locus c o r r e s p o n d s to assigning the v a l u e o f c and, thus, l o c a t i n g all o t h e r r o o t s too. The l e f t - h a n d side o f (5) has a zero at the o r i g i n and, t h e r e f o r e , the z e r o s o f Pi_l(ZJ lie on the o t h e r " i - 1 " b r a n c h e s o f the locus f o r the same v a l u e o f p a r a m e t e r c, i.e., ki, as the o r i g i n . The a l g o r i t h m m a y be e x t e n d e d t o the m o r e g e n e r a l case [4"] o f R.(z) : r .
(z-q),
I
q real,
] q I < I .
(7)
I
C o n s e q u e n t l y , the zeros o f Pi_1(z) (which c o r r e s p o n d t o the same v a l u e o f c as q) lie inside the unit c i r c l e i f Pi(z) is a Schur p o l y n o m i a l . The s i t u a t i o n is d e p i c t e d in Fig. I f o r a Schur p o l y n o m ial Pi(z) w i t h i = L I . If, instead, R.(z) = r. ( z + l } 2 , I
where
i
F.(z) = l+h. + (1-h.) z , I
I
a g a i n r. is a s c a l i n g f a c t o r and !
I
G . ( z ) = ( - I ) i h. ( z - l ) I
I
(8)
220 Im z
/
/
\
!
t
\
i
)<'
S \
Fig.
/---
/
/z
Root locus for equation (6) w i t h i -- 4 . ( 9 represent the zeros o f Pi(z) ; x represent the zeros of P (z) ; n r e p r e s e n t the zeros of Pi_1(z) and I the a d d i t i o n a l zero at q ).
P.(-1)
1
h = i 2
1 -
...
W.I
I
(the p r i m e denotes d e r i v a t i v e )
P'.(-1) - (-1) i P.'(-1) i
i
recursion (3) reduces to the R o u t h - t y p e a l g o r i t h m o b t a i n a b l e f r o m the t w o - t e r m f o r m of the classical s-domain Routh a l g o r i t h m via the u s u a l b i l i n e a r t r a n s f o r m a t i o n . This a l g o r i t h m too can be e x t e n d e d to the case o f R.(z} = r. (z - q) (z + I) . I
(9)
I
In fact, under s u i t a b l e assumptions on the value o f q (i.e., on the l o c a t i o n o f the corresponding zero of Ri(z)), Pi(z) is Schur if and only if Pi_1(z) is Schur and h i belongs to a given domain. To show this, let us consider the a d d i t i v e d e c o m p o s i t i o n o f Pi(z) into its s y m m e t r i c and a n t i s y m m e t r i c components: P.(z)l = Q',,l'(z) + (z+l) Qi,i_l(Z)
(10)
where the i - t h degree p o l y n o m i a l : Q . . ( z ) = 1 [ p i ( z ) + (_1)i P.(z)] I,I 2 I is s y m m e t r i c for i even and a n t i s y m m e t r i c for i odd, and the (i-1)-th degree p o l y n o m i a l :
(11)
1
(-1) I P.(z)_] Qi,i-1 (z) = 2(z+I) F Pi(z) l
(12)
is s y m m e t r i c f o r i odd and a n t i s y m m e t r i c f o r i even. Using (8) - (12), the basic recursion can be w r i t t e n as
221 r. (z - q) (z + I) Pi I (z) = Pi (z) - 2 h. (z - I) Q . . I
-
I
I,I--I
,(z).
(13)
The zeros o f both sides of (I 3) belong to the r o o t locus o f P.(z)l + c (z - I) Qi i-1(z) : 0
(14)
for c = - 2h i . It is easy to prove t h a t this locus is c o n f i n e d to the i n t e r i o r o f the unit circle, e x cept for the real axis, if Pi(z) is Schur. As shown in Fig. 2, if q lies on the real axis outside the unit circle, the zeros of Pi_1(z) (which c o r r e s p o n d to the same value - 2 h i of c) necessarily lie inside J z I = I and the Schur p r o p e r t y is r e t a i n e d . It is also possible to derive a s u f f i c i e n t condition for Pi(z) to be a Schur p o l y n o m i a l .
Im z
\ /
I
/
I
§
i
o
. . . .
Fig. 2 - Root locus for equation (14) w i t h i = 4 . ( 9 represent the zeros o f Pi(z) ; x represent the zeros o f ( z - l ) Q i , i _ 1 ( z ) ; o represent the zeros o f Pi_1(z) and the additional zero at q ) . In conclusion, by setting: 2 Qi_1,i_2(-1) - Qi_l,i_1(-1)
(is)
q c : = 2 Qi_1,i_2(-1) + Qi_1,i_1(-1) the f o l l o w i n g t h e o r e m can be stated: P.(z)i is Schur and I q I > I if and only i f Pi-1(z) is Schur and -l>q>qc
if q c < 1
t (16a)
or
{1 <-q oo t u iq>qct and
if qc > 1
1 hi > ~ (l+q)(l+qc)/(q-qc) > 0
whose p r o o f is o m i t t e d for b r e v i t y .
(16b)
222 if we set R.{z) = r.z(z-1), I
F.{z) = (f.-1)z2+3z-(f.+1),
I
I
I
G.(z) = -(f.+1)z2+z+(f.-1)
I
I
I
(17a)
I
with P.(0)
+
P.(0)
I f.
=
I
P.(O) - ~ . ( o 1 I
a = -
i0
+a.
I -k.
ii
-
ai0-aii
I
i
(17b}
I +k.i
recursion (3} reduces to the t w o - t e r m f o r m o f the o r i g i n a l t h r e e - t e r m B i s t r i t z a l g o r i t h m described in ~83 : riz{z-1) Pi_1(z} : E(f.-1)z2+3z-(f.+1)] I
P.(z} + ~--(f.+1}z2+z+(f.-1)]
I
I
I
P.(z}
I
(18)
I
which is r e l a t e d ( r a t h e r d i r e c t l y ) to the s-domain R o u t h a l g o r i t h m via the mapping s=(z +z )12 ~17-] ( a s i m i l a r expression holds for the a l g o r i t h m , suggested in E 7 ] }. Observe that, in order for equation (18) to be satisfied, the c o e f f i c i e n t s o f z I+2 and o f z 0 at the r i g h t - h a n d side should be zero; in o t h e r words, (fi-1)aii + (fi+1)ai0 = 0 , which is p r e c i s e l y the meaning o f r e l a t i o n (17b}. P a r a m e t e r fi plays the role of p a r a m e t e r 6 i in the B i s t r i t z p r o c e d u r e r 8 ] . The b i l i n e a r r e l a t i o n (17b) b e t w e e n fi and the r e f l e c t i o n c o e f f i c i e n t k i points out t h a t a necessary c o n d i t i o n for Pi(z} to be Schur is t h a t fi > 0 . Finally, a s t a b i l i t y - t e s t a l g o r i t h m that is n e i t h e r a L e v i n s o n - t y p e nor a R o u t h - t y p e a l g o r i t h m is o b t a i n e d E18_] by s e t t i n g R.(z) = r.(z-q)(z+1) , I
I
F.(z} = [(1-t.) + ( 1 + t . } z ] , I
I
G.(z) = 2 ( - l ) i t . .
I
I
{19)
I
This a l g o r i t h m m a y be r e l a t e d to the locus described by the roots o f Pi(z)+cEQi, i(z)+(z-1}Qi, i_1(z)]= = 0 , which is n e i t h e r s y m m e t r i c w i t h respect to I z I = I nor c o n f i n e d to tile i n t e r i o r o f the unit circle, as shown in Fig. 3 . Nevertheless, in this case too it is possible to f o r m u l a t e a s t a b i l i t y c r i t e r i o n ; this is m o r e cumbersome than the usual tests f r o m the c o m p u t a t i o n a l point o f v i e w but m a y be used to some advantage for o t h e r purposes.
Im z
...... ~
;.___2 ....
I
,
I
\
'"
oI,
',
]
,:
,
Re
7 iP
t
i
Fig. 3 -
Root locus corresponding to a l g o r i t h m (19) w i t h i = 4 . ( 9 represent the zeros of P i ( z ) ; x represent the zeros of Q i , i ( z } + ( z - 1 } Q i , i _ 1 ( z ) ; a r e p r e s e n t the zeros o f Pi_1(z) and the a d d i t i o n a l zero at q ) .
223 3. SPLIT FORMS The problem considered in this section is that o f d e t e r m i n i n g a component part o f polynomial Pi_2(z) (either its s y m m e t r i c or its a n t i s y m m e t r i c part) from one part of polynomial Pi_1(z) and one part of polynomial Pi(z). According to the t e r m i n o l o g y adopted in the l i t e r a t u r e , we shall name all the resulting t h r e e - t e r m relations the split forms of the original t w o - t e r m recursion. Due to the s y m m e t r y of the coefficients of the considered components, the global number of coefficients to be determined is about one h a l f o f that for the non-split form. This is an advantage from the computational point of view if, as usual, the amount of computation required to find each e n t r y is not too large with respect to that required in the non-split case. In the following, we first derive the general f o r m of all possible split relations associated with recursion (3). Then, we analyse some p a r t i c u l a r cases that are c o m p u t a t i o n a l l y a t t r a c t i v e . It turns out that the most convenient split forms do not necessarily originate from the simplest non-split structures. To this purpose: ( i ) we consider (3) for t w o consecutive steps, ( ii ) we separate each polynomial involved into its s y m m e t r i c and a n t i s y m m e t r i c part, and ( i i i ) we equate the s y m m e t r i c and a n t i s y m m e t r i c parts at both sides o f the resulting equations. In this way, we obtain a set o f four homogeneous equations in the six variables corresponding to the s y m m e t r i c and a n t i s y m m e t r i c parts o f the three polynomials Pi(z), Pi_1(z) and Pi_2(z). Once two of these variables have been fixed, each o f the remaining four variables m a y be expressed in terms of them (excluding the degenerate cases), which explains why it is (almost) always possible to find a t h r e e - t e r m s p l i t form o f a t w o - t e r m relation. The s y m m e t r i c and a n t i s y m m e t r i c part of Pk(Z), k = i, i-I, i-2 , will be indicated, respectively, by
sk(z) =~I [ p k ( z ) Ak(Z) =~]
+ ~k(Z)]
(20)
[Pk(Z) - Pk(Z)]
(21)
It is convenient to assume, without loss o f g e n e r a l i t y , that deg [Fk(Z)]=deg [Gk(Z) ] and that d e g [ R k ( Z ) ] = d e g [Gk(Z) ] + I . With reference, e.g., to (19), t h i s s i m p [ y means that the p o l y nomial G i ( z ) = 2 (_])i ti is regarded as a degenerate p o l y n o m i a l of degree I , like Fi{z), so that its s y m m e t r i c part is (-I) i t. (1+z) and its a n t i s y m m e t r i c part is (-I) i t i ( l - z ) . Similar considerations apply to Ri(z) in (17ai, which is to be regarded as a t h i r d - d e g r e e polynomial. To s i m p l i f y notation, the s y m m e t r i c and a n t i s y m m e t r i c parts o f Rk(Z), Fk{Z) and Gk(Z) will be denoted by s u b s c r i p t s " s " and " a " , respectively, and the a r g u m e n t " z " will be o m i t t e d . In this way, the four homogeneous equations mentioned above take on the form: Ris Si_ I + R.la Ai-1 - (F.is + G.is} Sl - (F.la - G..a) A., = 0
(22a)
R.,a Si-1 + Ris Ai-1 - ( F . za + G . la } S ..- ( F . .s - G . is ) A , = 0
(22b)
Ri_1,sSi_ 2 + Ri_ 1,aAi_2 - (Fi_ 1's + Gi_1, s} Si_ I - (Fi_1, a- Gi_1, a) Ai_ ] = 0
(22c)
Ri_1,aSi_ 2 + Ri_1,sAi_ 2 -
(Fi_l,a + Gi_l,a) Si_ 1 - (Fi_l, s- Gi_l, s) Ai_ 1 = 0 .
(22d)
By l i m i t i n g a t t e n t i o n to the t h r e e - t e r m stel~lown procedures associated with the same t w o t e r m recursion (3), the objective is to d e t e r m i n e e i t h e r Si_ 2 or Ai_ 2 from a part o f Pi and a part of P i - l " Since there are four possible combinations o f the j u s t - m e n t i o n e d parts, one has the eight split relations illustrated in Tab. ] where the black circles indicate the "initial conditions" for the current step. A split a l g o r i t h m uses only one part of e v e r y p o l y n o m i a l Pk in the sequence, but this part need not be always of the same kind. For instance, one may a l t e r n a t e l y determine the s y m m e t r i c part o f a p o l y n o m i a l and the antisymmetric part o f the successive one; in this way, to f o r m the e n t i r e sequence, one must use the relations corresponding to columns 3 and 4, a l t e r n a t l e l y .
224 TAB.
1
Sk
2
Ak
Sk
3
Ak
Sk
I
zl
Ak
Sk
5
Sk
Ak
6
Ak
o
k=i-1
o
o
k = i-2 !
o
o
o
9
Sk
7
Ak
Sk
8
Ak
o\T Oi>o
9 o"~o
O
Sk
Ak
9
O
O
If, instead, only s y m m e t r i c or only a n t i s y m m e t r i c parts a r e o f interest, the same split relation (either that corresponding to column 1 or that corresponding to column 2, respectively) will be used in all steps ( s t r i c t l y recursive split procedures). On the other hand, it may be shown that the relations corresponding to columns 3 and 4 (as well as those corresponding to columns 1 and 2, to columns 5 and 6, and to columns 7 and 8) can be given a common structure so that one might consider as being s t r i c t l y recursive also the procedure involving s y m m e t r i c and a n t i s y m m e t r i c parts alternately. Observe that the split algorithms must be p r o p e r l y i n i t i a l i z e d by computing the s y m m e t r i c or a n t i s y m m e t r i c part of Pn-I from the given original polynomial Pn" To this purpose, use can be made o f equations (22a) and (22b). Also, if the complete polynomial Pr' r < n , is to be recovered, as is the case in model reduction, both Sr and A r are to be computed from the same pair of parts of Pr+l and Pr+2 " These considerations must be taken into account when evaluating the computational c o m p l e x i t y of the algorithm. For b r e v i t y , we shall limit attention to the split relations corresponding to columns 1 and3,respectively. They turn out to be of the following form: Li Si-2 + Mi Si-1 + N.J S.z = 0
(23)
L'., Si-2 + M'.n Ai-1 + N'.z Si = 0
(2q)
and
where tile combination polynomials L., M., N. and L'., M'., N~ are as specified in the Appendix. I
I
I
I
I
I
Clearly, the computational complexity, i.e., the number of a r i t h m e t i c a l operations (additions/ subtractions and multiplications/divisions) necessary to compute every coefficient of Si_2, d i r e c t ly corresponds to the number of addenda in the combination polynomials and to the number of their coefficients d i f f e r e n t from + 1, as will be shown with reference to some specific case. The simplest forms of equation (23) are therefore obtained (possibly, after the e l i m i n a t i o n of factors common to L i, M i and N i) when Ni(z) = n i = const., and Li(z) and Mi(z) are (symmetric) polynomials of degree not greater than 2 and 1, r e s p e c t i v e l y . In particular, L i may also be o f degree 1; in this case Li(z) = I i z (which may be regarded as a degenerate second-degree polynomial) and Mi(z) = m i ( z + l ) . Indeed, this is the case for the split Levinson algorithm and for the B i s t r i t z algorithm presented in ]'8~ . Specifically, by r e c a l l i n g (5), the split Levinson algorithm takes the form: r i ri_ 1 z Si_ 2 - r i (z+l) Si_ 1 + n.z S.s = 0
(25)
with n i = (l-ki_1)(1+k i} and, by recalling (17) and the fact that R i = Ria i f R i is considered as a (degenerate) third-degree polynomial , the B i s t r i t z a l g o r i t h m is:
225 1
1
r i r i _ I z Si_ 2 -
~ r i fi (z+1)Si_ 1 +S. = 0 . I
(26)
Of course, one can exploit the scaling factors to further s i m p l i f y the computations, e.g., by f o r c ing to + I two of the coefficents I i, m i and n i. In particular, it is possible to set I i = I and m i = -I in (25) (i.e., r k = I, V k ), thus obtaining the simplest version of the split Levinson algorithm } 15~; the simplest version of (26) is obtained by choosing I i = n i = I (i.e., r k = 2, V k ). The number of a r i t h m e t i c a l operations that are needed to compute a c o e f f i c i e n t of Si_ 2 from those of Si_ I and Si is the same in the two cases: 2 additions and I m u l t i p l i c a t i o n , in g e n e r a l . In fact, referring for instance to the simplest split version of (26) (Bistritz algorithm) and denoting by bh, k = bh,h_ k the c o e f f i c i e n t of z k and z h-k in Sh(Z), h = i-2, i - I , i , we have: bi_2, 0 = fi.(bi_1,0 + bi_1, I) bi_2, k = fi.(bi_1,k + bi_1,k+ I) - bi,k+ I ,
(27a) 0 < k < i-2 .
(27b)
Notice that, although the computational c o m p l e x i t y of the simplest versions of (25) and (26) is the same, they do correspond to d i f f e r e n t t w o - t e r m recursions and, thus, to different sequences of complete polynomials Pk ' k < n, starting from the same original polynomial Pn" In other words they d i f f e r in the i n i t i a l i z a t i o n rule. From the previous considerations, it is also apparent that the s i m p l i c i t y o f a specific split form is not s t r i c t l y related to the s i m p l i c i t y of the corresponding non-split algorithm. The main feature of the z-domain Routh-type a l g o r i t h m (8) is to retain, except for the factor (z+l), either the symmetric part or the a n t i s y m m e t r i c part, in the same way as in the s-domain version either the even or the odd part is retained. It follows that equation (23) and the corresponding equation relating the a n t i s y m m e t r i c parts only, are a restatement of this simple rule. Equation (24) becomes instead r i ri_ I (z+1) 2 Si_ 2 + r i { hi_ I ~I - (-I) i'] + h.i ~1 + (-1)i~}(z-1) Ai_ I - S.j = 0
(28)
I
in which L i is a non-degenerate polynomial o f degree 2 . To construct the entire sequence, one also need the companion relation corresponding to column 4 of Tab. I, which allows one to obtain Ai_ 2 from Si_ I and A i. Obviously, the simplest form of (29,1 and of its companion relation is obtained for r k = I, V k . As a consequence of the bilinear transformation, starting from a Schur polynomial Pn' the zeros of polynomials Sk and (z+1)Ak_ I , V k , a l t e r n a t e along the unit circle, like the zeros of Sk and A k. It could be shown that a similar property also holds for the s y m m e t r i c and a n t i s y m m e t r i c parts of t w o consecutive polynomials generated according to (2S) and (26). ~. CONCLUSIONS A common form has been considered for the t w o - t e r m recurrence relations corresponding to: ( i ) the Levinson (Nevantinna-Pick) algorithm, ( ii ~ the z-domain algorithm obtained from the classical Routh algorithm via the usual bilinear transformation, ( i i i ) t h e B i s t r i t z algorithm,and ( iv ) other Euclid-type s t a b i l i t y - t e s t algorithms, A geometrical interpretation of their properties has been given. An analysis o f the eight possib|e t h r e e - t e r m (spiitJ forms associated with the general t w o t e r m form, has been carried out, Particular a t t e n t i o n has been paid to the relation involving the s y m m e t r i c (antisymmetric) parts only and to the relations involving the s y m m e t r i c and a n t i s y m m e t r i c parts, alternately. The conditions leading to the c o m p u t a t i o n a l l y most e f f i c i e n t formulae have been p o i n t e d out and discussed,
226 APPENDIX With reference to the notation adopted in (22), the combination polynomials in (23) and (24) take the forms: Li = F(Fis- Gis) R i a - ( F i a - G i a )
Ris] Ri-I Ri-1
(At)
Mi = E(Fi-l,s - Gi-l,s) Ri-l,a - {Fi-l,a - Gi-l,a) R i - l , s ] (Fi ~'i - Gi (~i) Ni = ~(Fi-1 ,a + Gi-l,a) R i - l , a - ( F i - l , s
(A2)
+ Gi-l,s) Ri-l,s'][-(Fis-Gis) R i a - ( F . , a - G.la } R.zs] +
+ E(F.. . ) R l.-.l , a - { F .l -. l , a - G .l .- l , a ) R .l--l,S . ] ~Fia-Gia)Ria-(Fis-Gis)Ris] I--I,S - G .I--I,S
(A3)
and L'., = [(F.la - G.,a) R.la - {Fis - Gis) Ris] Ri_ I Ri_ I-
(A4}
M'i = [(Fi_1, s + Gi-l,s) Ri-l,s - (F i-l,a + Gi-1, a } R i - l , a ] (Fi F:i - G i (]i)
(AS}
N'i = [{F.,s - G.,s) R.,s - (F.,a-G ia)Ria] [{F i-I ,a- G i- I ,a) R i - l , s - {Fi-i ,s- G i-1 ,s) Ri-l,a -] + +[{Fi i -
,a
+G
i-l,a
)R
i-l,a
-(F
i-l,s
+G
i-I,s
)R
. ] [ ( F . - G )R. - { F . - G . ) R . ] . i-~,s is is la Ja la ts
(A6)
REFERENCES
['1 ] T. Kailath, "A view of three decades of linear filtering theory", IEEE Trans. Inf. Theory, vol. IT-20, No. 2, pp. lZ16-181, 1974. [-2] J. Makhoul, "Linear prediction: A tutorial review", Proc. IEEE, vol. 63, No. 4, pp. 561-580, 1975. ~3] E. Jury, Theory 1964.
and Application
of the z-Transform Method. J. Wiley & Sons., New York,
F4] P. Delsarte, Y. Genin and Y. Kamp, "On the role of the Nevanlinna-Pick problem in circuit and system theory", Int. J. Circuit Theory Appl., vol. 9, pp. 177-187, 1981. [5]
K. ~striSm, Introduction to Stochastic Control Theory. Academic Press, New York, 1970.
[6]
Y. Bistritz, "Direct bilinear Routh stability criteria for discrete systems", Systems Lett., vol. 4, No. 5, pp. 265-271, 1984.
Control
]-7] Y. Bistritz, "A new unit circle stability criterion", in Mathematical Theory of Networks and Systems ( Proc. MTNS-83 Int. Symp., Beer-Sheva, Israel, June 20-24, 1983 ), pp. 69-87. [8]
Y. Bistritz, "Zero location with respect to the unit circle of discrete-time linear system polynomials", Proc. IEEE, vol. 72, No. 9, pp. 1131-1142, 1984.
Eg] A. Lepschy, G.A. Mian and U. Viaro, "Euclid-type algorithm and its applications", Int. J. Systems Sci., vol. 20, No. 6, pp. 945-956, 1989. [-10] M. Benidir and B. Picinbono, "Comparison of some stability criteria of discrete-time filters", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-36, No. 7, pp. 993-1001, 1988. [11] P. VaidyanathanandS. M i t r a , " A unified structural interpretation of some well-known stability-test procedures for linear systems", Proc. IEEE, vol. 75, No. 4, pp. 478-496, 1987. ['12] W. Krajewski, A. Lepschy, G.A. Mian and U. Viaro,"A unifying frame for stability-test algorithms for continuous-time systems", IEEE Trans. Circuits Syst., vol. CAS-37, No. 2, 1990.
227 E13] A. Lepschy, G.A. Mian and U. V i a r o , " A s t a b i l i t y test for continuous systems", Systems Control Lett., vol. 10, No. 3, pp. 175-179, 1988. ~14] P. Delsarte and Y. Genin, "The split Levinson algorithm", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, No. 3, pp. 470-477, 1986, [15]
P. Delsarte and Yo Genin,"On the splitting of classical algorithms in linear prediction theory", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, No~ 5, pp. 645653, 1987.
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Y. Bistritz, H. Lev-Ari andT. K a i l a t h , " I m m i t a n c e - d o m a i n Levinson algorithms", Proc. IEEE ICASSP-86, Tokyo, pp. 253-256, 1986.
[17]
Y. B i s t r i t z , " C o m m e n t on 'Zero location with respect to the unit circle of discrete-time linear system polynomials' '% Proc. IEEE, vol. 74, No. 12, pp. 1802-1803, 1986.
[18]
A. Lepschy, G.A~ Mian and U. V i a r o , " A common setting for some classical z-domain algorithms in linear system theory", Int. J. Systems Sci., vol. 21, ~990.
TIM E - D I S C R E T I Z A T I O N OF THE ZAKAI EQUATION FOR DIFFUSION PROCESSES OBSERVED IN CORRELATED NOISE* Patrick FLORCHINGER Universitd de Metz D~parternent de Math~matiques Ile du Saulcy F-57045 METZ C5dex
Franqois LE GLAND INRIA Sophia-Antipolis Route des Lucioles F-06565 VALBONNE C~dex
Abstract A time dlscretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional law of a diffusion process observed in white-noise. The case where the observation noise and the state noise are correlated, is considered. The numerical scheme is based on a Trotter-like product formula, which exhibits prediction and correction steps, and for which an error estimate of order ~f is proved, where 8 is the time discretization step. The correction step is associated with a degenerate second-order stochastic PDE, for which a representation result is available in terms of stochastic characteristics [8,9,11]. A discretization scheme is then provided to approximate these stochastic characteristics. Under an additional assumption on the correlation coefficient, an error estimate of order v ~ is proved for the overall numerical scheme. This has been proved to be the best possible error estimate by ElllottGlowinski [5].
1
Introduction
T h e purpose of this paper is to present a computable time discretization scheme for the Zakai equation of nonlinear filtering with correlated noises, and to provide an estimate of the rate of convergence. In the case of independent noises, the problem has been studied by Newton [13], KorczllogluMazziotto [6], B e n n a t o n [1], D i M a s i - P r a t e l l i - R u n g g a l d i e r [4], e i c a r d [14], Bensoussan-GlowinskiRascanu [2] and LeGland [12]. Some of these authors have actually considered the associated Zakai equation. T i m e discretization schemes have been provided with a rate of convergence of order 6, where 6 is the time discretization step. In the case of correlated noises, the problem has been studied by Elliott-Glowinski [5]. T h e bcst approximation of the continuous filter based on the values of the observation process at a regular
9Rcseareh partially supported by USACCE under Contract DAJA45-87-M-0296.
229 partition (with mesh 5) hm been considered, and it has been proved that the rate of convergence is of order V"~. However, no algorithm is provided to actually compute this approximation. The paper is organized as follows. In Section 2, the nonlinear filtering problem is presented. Some results are recalled in Section 3, on the Zakai equation, and on a related degenerate secondorder stochastic PDE. A Trotter-like product formula is then considered, with aa error estimate of order 5. However, this numerical scheme is not computable. In Section 4, a representation result in terms of stochastic characteristics is presented for the degenerate second-order stochastic PDE. This part follows mainly the work of Krylov-Rozovskii [7,8] - see also Kunita [11]. Under an additional assumption on the correlation coefficient, a time discretization scheme is presented in Section 5 based on an approximation of the stochastic characteristics - with an error estimate of order v/~. In addition, this numerical scheme is actually computable, as far as time discretization is concerned.
2
The
filtering
problem
On the probability space (~, F , P), consider the stochastic differential system dXt = b( X,) dt + a( X,) dW, + p(Xt) dV, dY~ = h(X,) dt + dVt
where {IV,, t _> 0} and {Vt, t > 0} are independent Wiener processes, with covariance matrix I (identity) and r respectively. For the clarity of exposition, it is assumed throughout the paper that r=I. Throughout the paper, the coefficients will satisfy the following hypotheses (i) b is a measurable and bounded function from R"* to Iq.'~, (ii} cr is a continuous and bounded[unction on R"*, with a ~=a n ' , such that 9 a is a uniformly elliptic m x m matriz, i.e. a(x) > cd, Oa~,~ 9 ~ ~= j=a ~ is a measurable and bounded function on R"*, (ii O p is a continuous and bounded[unction from I t " to the space of m x d matrices, with A C = pp', such that
i, ~ Op'k
9 ak =
is a measurable and bounded [unction on R'*,
9 ~ =~j=l ~Od,i
is a measurable and bounded function on R'*,
(in) h is a measurable and bounded function from It. '~ to R d.
With the diffusion signal {Xt, t _ O} are associated the two partial differential operators L =
' g i,j=l
L o A=1 ~
i,j=l
+ c"q
02 OxiOxi
+
.. i=l
a '." ' J ~ + ~ b -i O. OxiOxj i=a Oxl
Ozi '
230 An other family of partial differential operators to be considered is m
9 0
- ' B~ =z~ h~ + ~ P'~-Oxl
l <-k < d-.
Introducing
it is standard that~ for all T > 0 the original probability measure P is equivalent on [0, T] to the reference probability measure p t with Radon-Nikodym derivative ZT, so that under p t
dX,
= [b(X,) -
p(X,)h(X,)] dt + ~(X,) dW~ + a(X,) dY~
(2.1)
where {Wt, t >__ 0} and {Yt, t _> 0} are independent Wiener processes, with covariance matrix I (identity). By the Bayes formula E t ( f ( X , ) Z, l Y,)
E(f(S,)[y~)
=
In addition
Et(Z, [Yd f
E t ( f ( X ~ ) Z , l Y,) = J f ( z ) p t ( x ) dx ,
where the unnormalized conditional density {pt, t > O} satisfies the Zakai equation
d p =$ L * p~dt
+B~p~dYt k
The purpose of the next section is to recall existence, uniqueness and regularity results for this equation and related stochastic partial differential equations.
3
T h e Zakai e q u a t i o n , a n d r e l a t e d s t o c h a s t i c P D E ' s
For all n_> 0, let H " denote the usual Sobolev space, with norm [[. [[,. The following shorthand notations will be used throughout the paper: I" 1 -~ t[" ][0 and ![" I[ ~ I]" Ill 9 Consider the Zakai equation dpt = L*p~ dt + B~p, dY~~ .
(3.1)
The following existence, uniqueness and regularity result is proved in Krylov-Rozovskii [7] and Pardoux [15]. T h e o r e m 3.1 Let n>_O be fixed. Assume that the coefficients satisfy 9 a, c and p have bounded derivatives up to order (n+l), 9 b and h have bounded derivatives up to order n, and that the initial condition satisfies po E H". Then the equation (3.1) has a unique solution p E M2(O, T ; H"+I). In addition
9 pC L2(~; C([O,T]; H")),
231 . the following estimate holds Et[ sup [Iptll~] -< cllpoll~ 9 o
. . 02 89~-~ c ' ~ ij=l OziOxj
On one hand, the partial differential operator Lo generates a semi-group {Pt, t > 0}. On the other hand, it is possible to associate a stochastic semi-group {Q~, 0 < s < t} with the following degenerate second-order stochastic PDE dqt = A*qt dt + B~qt dYt k ,
(3.2)
which will be studied below9 Therefore, it is worth studying the following Trotter-like product formulas for approximating the original Zakai equation
(3.3) where 51 zx ti+l -- tl. The main interest of such product formulas is that the original equation has been splitted into a second-order deterministic PDE (prediction step), and a degenerate second-order stochastic PDE (correction step). In the non-correlated noises case, this stochastic PDE would reduce to a zero-order equation, for which there would exist a straightforward explicit solution. In the correlated noises case, a representation result is available by the method of stochastic characteristics (i.e. involving the stochastic flow of diffeomorphisms associated with a SDE driven by the observation process), see Krylov-Rozovskii [8]. As a consequence of the above discussion, there will be two steps in designing the approximation to the original Zakai equation (3.1) 9 first use a Trotter-like product formula, 9 then approximate the solution of the degenerate second-order stochastic PDE, by approximating the stochastic flow of diffeomorphisms involved in the stochastic characteristics method of [8]. It will be proved that the first step can be achieved with a rate of convergence of order 0(5), whereas the rate of convergence for the second step (and a fortiori for the global approximation procedure) is of order O ( v ~ ) only, where 5 zx maxi>0 51. Consider now the degenerate second-order stochastic PDE (3.2). Although no coercivity hypothesis is satisfied, the following result is proved by Krylov-R.ozovskii [8]. T h e o r e m 3.2 Let n>_l be fizcd. Assume that the coefficients satisfy
232 9 c and p have bounded derivatives up to order ( n + l ) , 9 h has bounded derivatives up to order n, and that the initial condition satisfies Po E H". Then the equation (3.2) has a unique weak solution q 6 M2(0, T; H"). In addition
9 q e L2(f$; C([0,T]; H " - ' ) ) , 9 q e L~(~; C~,([O,T]; H-)) the following estimate holds
Et[ sup Ilq,ll2,] < Cllq0ll2, 9 o
* Q~+~ffi.
Assume that the coefficients satisfy 9 a, c and p have bounded derivatives up to order 3, 9 b and h have bounded derivatives up to order 2, and that the initial condition satisfies Po 6 H 2. Then Pi approximates the solution Pt~ of the original Zakai equation (3.1) with a rate of convergence of order 0(6). Indeed
{Etlpl
4
Representation
in
-
p,,
21y x/2 < c S llpoll2 .
terms of stochastic characteristics
Going back to stochastic differential systems, the following result holds T h e o r e m 4.1 Let ~,t(') be the stochastic flow associated with the forward SDE d~t = P(~t) [dYt - h(~t) dt] . Assume that the coefficients h and p have bounded derivatives up to order (n+l). C"-diffeomorphism in R "~.
(4.1) Then ~s.t(') is a
Under the assumption that the coefficient p has bounded derivatives up to order 2, the inverse map -1 ~,,, (.) is given ezpliciteIy as the (backward) stochastic flow r/t,,(') associated with the backward SDE drh = p(rh) $ [dYt - h(r
dr] - po(yt) dr,
with
p~ ~=
opt _j
k=l j = l ~ X J pk "
(4.2)
233
The regularity of ~,.~(.) was first proved by Blagoveschenskii-Freidlin [3], whereas the rest of the theorem is proved in Kunita [10]. P r o p o s i t i o n 4.2 The Jacobian J,,,(x) (i.e. the determinant of the Jacobian matriz) of the diffeo-
morphism ~o,,(. ) satisfies Jo,,(z) ~= exp
{/' ~'(~.,.(~)) [dY. -
h(~.,.(x)) dd (4.3)
- f'ho(,,,~.(x))d-c- ft~(,~,,(x))d.c} with ~=V" Opk
a
,
--
-~s
o.d
~ E E Op'~o;{
k=l i=l
k=l i,j=l 0Xj 0X i "
R e m a r k 4.3 Note that [J,.,(rh,,(x))] -~ is actually the Jacobian of the inverse diffeomorphism rh.,(. ).
Define
h* ,,~ x
-~,,t x = e x p
dYr-
h ,,~ x
and |
o-
= =,,,(x) [J,,,(x)] -1 = exp
{/'
h'(r
dY~ -
Ih(6,~(x))l 2 dr
Introduce the following definition
Q;r
~ r
@,,t(qt,,(x)) ,
(4.4)
or equivalently Q;r
= r
e,,,(~)
.
where the same notation has been used as in the previous section. This will be justified by the Theorem 4.7 to be proved below. R e m a r k 4.4 Under the additional assumption that the coefficient p has bounded derivatives up to order 2, the Lemma 6.2 of [10, Chapter II] gives the following explicit expressions in terms of backward Itb stochastic integrals
J,,,(~t,,(x)) = exp
I f s ~Or*(Th,,(x)) $ [dY~ -
h(rh,,(x)) dr]
234 where the coefficients h0 and ao have already been defined as ho zx =
Z--, k=l i=I
~
zx ao =
and
Xi
Ootk i -ff---P~ = k=l i=I
d
~m
o'~2P kJ
i
Pk 9
s
Therefore 9 ,,o(x) zx Oo,,(rh.s(x) ) = exp
{f
h'(y,,~(x))~dY.
-
Ih(rh,.(x))l~dT (4.5)
R e m a r k 4.5 If p =- 0, then ~s,t(z) = x so that
Q;r
= r
e~p {h'(~)(Y, - Y,) - llh(x)l ~ (t - 4 } ,
which is actually the explicit solution of the equation dqt = hkqt dY~k ,
with initial condition r at time s. In this case, (3.3) reduces to the discretization schemes considered in [21 and [121. First, the following stability result holds P r o p o s i t i o n 4.6 Let n>O be fixed. Assume that the coefficients satisfy 9 h and p have bounded derivatives up to order ( n + l ) . I r e E H '~, then Q~r is a random variable with values in H '~, and in addition
'/~ < c I1r
{EtlIQ~r
9
The following representation result of the solution of equation (3.2) in terms of the stochastic characteristics rh,~(.), is the stochastic counterpart of the usual method of characteristics for linear first-order PDE's. It has been proved by Kunita [11] and Krylov-Rozovskii [8]. T h e o r e m 4.7 Let {Q~, s <_ t} be defined by (4.4). Then, the unique solution of equation (3.2) satisfies qt(x) = Q~tq,(x) . (4.6)
5
Approximation
of the
stochastic
characteristics
It has been proved in Section 4 that the stochastic semi-group {Q~, 0 < s < t} associated with the degeaerate second-order stochastic PDE (3.2) satisfies
q;r
= r
os,~(~.(x)),
(5.1)
235
where 7h,,(-) is the inverse of the stochastic flow of diffeomorphisms ~,,t(.) associated with the SDE (4.1). The purpose of this section is to investigate approximations of (5.1)9 Considering rh,o(.) as the stochastic flow of diffeomorphisms associated with the backward SDE (4.2), and rewriting (5.1) as
q~r
= r
,~,,,(~),
where (I)t,o(x) has been defined in (4.5), it is natural to consider the following approximatiou q,r
A
= r
(5.2)
~,',,.(x) ,
where 7,,,(x) =" x
-
p(x) ['1, -
Y.
-
h(x)(t
-
~)] + no(X) (t
-
4,
and
~,,.(x) ~ exp {h'(x) (Y, - Y.) - ~lh(x)l 2 ( t
-
~) -
~'(x)[Y,
-
Y. -
h(z)(t
-
s)]
+ ~ ( x ) ( t - s) + so(x) ( t - ~)} , are computable approximations of ~t.,(x) and Ot,,(x) respectively, both depending only on the incremeats (Yt - Y~). R e m a r k 5.1 One possible approach would be to approximate rh,,(.) by the stochastic flow of diffeomorphisms associated with the ordinary differential equation obtained from (4.2) by replacing the observation sample--path {Y~, 0 < o' < T} with some regular approximation, such as the Euler stepwise approximation or the polygonal interpolation. The numerical analysis of such an approximation should not be very difficult. However, the resulting approximation would not be explicitely computable.
The remaining of this section is devoted to studying the rate of convergence of this approximation. First, a stability result similar to Proposition 4.6 is needed. However, such a result can not be proved in the same way here, because ~'t,o(') is not a diffeomorphism. Therefore, an additional assumption is introduced on the correlation coefficient p. A s s u m p t i o n (A)
The m x d matrix p(z) has constant rank 0 < r < rain(re, d), and the r-dimensional vector space range(p(x)) does not depend on x. Therefore, there exists a m X r matrix p independent of x, and a r X d matrix -fi(x) with full rank r, such that
p(x) = p.'fi(x). It is assumed that the matriz'~ ~ ~'fi* is uniformly elliptic, i.e. ~(x) > 7I.
P r o p o s i t i o n 5.2 Let n>O be fixed. Assume that the coe~cients satisfy 9 p has bounded derivatives up to order (n+2), 9 h has bounded derivatives up to order n. I f r E H ~, then under Assumption A, -Q~r is a random variable with values in H ~, and in addition
(EGI~]r
'/~ _< m I1r
9
236
R e m a r k 5.3 The proof of this proposition is given in the Appendix 9 Since ~'t,,(.) is not a diffeomorphism, one can not use a change of variable as in the proof of Proposition 49 Instead, one uses the fact that ~t,,(x) and ;~t,,(x) are very simple functions of the Gaussian random variable (Y~ - 1i,). R e m a r k 5.4 The approximations ~'~.,(.) and ~ . , ( . ) are based on the explicit expressions for rh.,(') and (I),.o(.), given in (4.2) and (4.5) respectively. This explains why the regularity assumptions on the coefficients p and h are different in Proposition 4.6 and Proposition 5.29 Here is the main result of the paper. T h e o r e m 5.5 Consider the following approximation scheme ~i+1 = Pi', Q,,+, ~i
Assume that the coefficients satisfy 9 a and c have bounded derivatives up to order 4, 9 b and p have bounded derivatives up to order 3, 9 h has bounded derivatives up to order 2, and that the initial condition satisfies Po E H 2. Then ~i approximates the solution p~ of the original equation (3.1) with a rate of convergence of order O( v ~ ) . Indeed {Etlffl 21a/2 - p,, y < C v ~ llpoll~ . R e m a r k 5.6 The purpose of the Assumption (A) is only to provide a sufficient condition for the stability estimate to hold.
A further step in the time--discretization would consist in approximating the Fokker-Planck semigroup {Pt", t > 0}, using some classical approximation scheme. For instance, using the backward Euler scheme would result in the following global approximation scheme --t i
with same error estimate.
6
Conclusion
A time-discretizatiou scheme of the Zakai equation for diffusion processes observed in correlated noise has been proposed, based on the stochastic characteristics introduced in [8] and [11]. Under an additional assumption on the correlation coefficient, it has been shown that the rate of convergence of this approximation is of order v~, where 6 is the time discretization step. The same rate of convergence has been obtained in EUiott-Glowinski [5] for a different approximation 9 on one hand, the approximation considered in [5] has a probabilistic interpretation, which is not the case for the time discretization scheme presented here, 9 on the other hand, the latter is actually computable, whereas no numerical algorithm is provide to compute the approximation considered in [5]. Another point of interest would be to study some particle approximation for the degenerate second-order stochastic PDE, adapting the results presented in Raviart [16] for deterministic firstorder PDE's.
237
References [1] J.F. BENNATON, Discrete time Galerkin approximations to the nonlinear filtering solution, J.Math.Anal.Appl. 110 (2) 364-383 (1985). [2] A. BENSOUSSAN, R. GLOWINSKI and A. RASCANU, Approximation of Zakai equation by the splitting-up method, in: Stochastic Systems and Optimization (Warsaw-1988), (ed. J.Zabczyk) 257-265, Springer-Verlag (LNCIS-136) (1989). [3] Yu.N. BLAGOVESCHENSKII and M.I. FREIDLIN, Certain properties of diffusion processes depending on a parameter, Soviet Math. 2 633-636 (1961). [4] G.B. DI MASI, M. PRATELLI and W.J. RUNGGALDIER, An approximation for the nonlinear filtering problem with error bound, Stochastics 14 (4) 247-271 (1985). [5] R.J. ELLIOTT and R. GLOWINSKI, Approximations to solutions of the Zak~i filtering equation, Stoch.Anal.Appt. 7 (2) 145-168 (1988). [6] H. KOREZLIOGLU and G. MAZZIOTTO, Approximations of the nonlinear filter by periodic sampling and quantization, in: Analysis and Optimization of Systems, Part 1 (Nice-198.~), (eds. A.Bensoussan and J.L.Lions) 553-567, Springer-Verlag (LNCIS-62) (1984). [7] N.V. KRYLOV and B.L. ROZOVSKII, On the Cauchy problem for linear stochastic partial differential equations, Math. USSR Izvestija 11 (6) 1267-1284 (1977). [8] N.V. KRYLOV and B.L. KOZOVSKII, Characteristics of degenerating second-order parabolic Ito equations, J.Soviet Math. 32 (4) 336-348 (1982). [9] H. KUNITA, Stochastic partial differential equations connected with nonlinear filtering, in: Nonlinear Filtering and Stochastic Control (Cortona-1981) (eds. S.K.Mitter and A.Moro) 100169, Springer-Verlag (LNM-972) (1982). [10] H. KUNITA, Stochastic differential equations and stochastic flows of diffeomorphisms, in: Ecole d'Ete de Probabilites de Saint-Flour XII (1982) (ed. P.L.Hennequin) 144-303, Springer-Verlag (LNM-1097) (1984). [11] H. KUNITA, First order partial differential equations, in: Stochastic Analysis (Katata and Kyoto-1982) (ed. K.Ito) 249-269, North-Holland (1~984). [12] F. LE GLAND, High order time discretization of nonlinear filtering equations, in: #Sth IEEE COG (Tampa-1989) 2601-2606 (1989). [13] N.J. NEWTON, Discrete approximations for Markov-chain filters, Ph.D Thesis, Imperial College (1983). [14] J. PICARD, Approximation of nonlinear filtering problems and order of convergence, in: Filtering and Control of Random Processes (ENST/CNET-1983), (eds. H.Korezlioglu, G.Mazziotto and J.Szpirglas) 219-236, Springer-Verlag (LNCIS-61) (1984). [15] E. PARDOUX, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (2) 127-167 (1979). [16] P.A. RAVIART, An analysis of particle methods, in: Numerical Methods in Fluid Dynamics (Como-1983) (ed. F.Brezzi) 243-324, Springer-Verlag (LNM-1127) (1985).
F I N I T E - D I M E N S I O N A L S T O C H A S T I C F I L T E R I N G IN D I S C R E T E T I M E : TIIE ROLE OF CONVOLUTION SEMIGROUPS Fabio Spizzichino Dept. of Mathematics, University " La Sapienza " Piazzale A. Moro, 5 - 00185, R o m e
Summary.
We consider a stochastic dynamic system in discrete time with an observable
component Yn C ~ C ~Rw (observation), and a non-observable component X n E ~ C ~}~v (signal). The signal is a Markov chain and, at any step, the conditional density of the observation given the signal is fixed in the exponential class. We face the problem of finding transition kernels for X n for which the system admits filters of dimension v; we reduce this problem
to an analytical procedure in which the first step can be solved in terms of
convolution semigroups of probability distributions.
1. Introduction W e consider a stationary discrete-time stochastic process ({Xn}, {Yn})n=l,2,...(Xn E :~ C ~v, Yn e qJ C ~w) whose dynamics is described by the following properties:
{Xn} is a Markov chain with given
transition densities pn(x [ x'), Yn is conditionally independent on X (n-l) = (Xl, X2, ... , Xn-1) and y(n-1) _~ ( y t , y 2 ..... Yn-I). given X n (n = 2 , 3, ...), the conditional densities fn(Y 1 x) ----fyn(y [ X (n-l) = x(n-1);
y(n-l)=
y(n-1); Xn = x)
are given, n = 1, 2 , . . . . Let Pl be the probability density function of X1, pn(x [ y(n-1)) be the conditional density of X n given y(n-z) = Y (n-l) conditional density of X n given y(n) = y(n)
(prediction density),
(filtering density).
7rn(x ] y(n)) be the
The sequences of filtering
densities and prediction densities obey the "Dynamical Bayes Formula": 7rl(x ]Yl) oc Pl(X) 9 f l ( y l [ x),
...,
zrn( x [ y(n))
c< pn(X [ y(n-1)) . fn(Yn I x )
(i.i) pn(X [ y (n-l)) = J
Zr(n_l)(x' [ y (n-l)) 9 pn_l(x] x') dx'
the symbol cx indicates that two functions differ for a multiplicative quantity which does not depend on tile a r g u m e n t x E ~'.
239 In this paper we consider the time-homogeneous case in which pn(x t x') and fn(Y }x) do not depend on n. One is usually interested in finding out conditions on the triple Pl, p(x [ x'), f(y [ x), under which the recursive formula (1.1) actually allows for feasible computation of rrn and Pn. For this problem the following definition is of interest. Let cO. be an index set and B (~ be a ~r-algebra of subsets of ql.
Definition A family ~2 ---- {~u(x); x E E) of probability density functions on :s indexed by uE~
filter-conjugate
is
if there exists a measurable function T : ~ x ctJ --* % such that
[[- ~u(X') 9 p(x [ x') d x ' ] - f(y [ x) c< ~o(u.y)(x)
V u E ~
V y E ctJ.
(1.2)
E As it can be immediately checked by a direct application of (1.1): ( p l ( x ) = f ~uo(X') . p ( x ] x ' ) dx' , E imply
~Uo E ~ ,
~ filter-conjugate)
~rn(x [ y(n)) = ~Un(Uo;y(n))(x ) e ~, '~ y(n) e cLjn where u n : ~ x ~n .., cU. are recursivelly defined by ut(uo; Y,) = ~~
The ~
Yl), u2(uo; y(2)) = ~(ul ( uo, y,), y~), ...
(1.3)
stochastic process ul(u0,Y1) , u2(u0,Y1,Y2), ... is the "filter-process" and it
results to be a Markov chain as can be easily checked. The definition of a filter-conjugate family is the suitable translation of the concept of "conjugate family" in bayesian Statistics (see e.g. [2]) to the present case, where we must replace the usual Bayes formula with the dynamic Bayes formuta (1.1); see also the discussion presented in [6]. Filtering theory motivates the search for a filter-conjugate family with the following properties:
a) dim ~ < co,
b) There exists an algorithm for obtaining ~ and (u from the
knowledge of the elements E, ~, f(ylx), p(x[x'). W e consider here an exponential family of "observation densities" {f(y [ X)}xE t
(see
Sect. 2). Our aim is to build "compatible" transition kernels p(x ] x') which allow for the existence of filter-conjugate families with dim ~ = dim E = v and, in particular to show the role that convolutions semigroups of probability distributions can have in the problem. The results we obtained can be extended to the case in which p(x [ x') and f ( y [ x) depend 011 n. It has been preferred to treat the time-homogeneous case so as not to obscure the substancial aspects of the problems.
240 2. Sufficient conditions for the existence of a v-dimensional filter-conjugate family. W e assume that f (y ] x) is of the form f(Ylx) = a(x) 9 b(y) 9 exp { < x , d ( y ) > } x 6 ~" c ~v, y E ~ c ~v
(2.1)
with suitable functions a : t ~ RA', b : ~ -~ N4-, d : ~ -~ Nv and < 9 , 9 >
denoting the scalar
product. As shown in [1], [4], [7], under different kinds of conditions, the exponentiality of the family {f(y [ x), x 6 ~} is a necessary condition for the existence of a filter-conjugate family with % contained in a Euclidean space. Consider a quadruple ( ~ , %c~V,K:~xt~
K, c, p) with
+ U {0},c:~
and satifying the following conditions: I
e x p { < u , x ' > } 9 K(x;x') dx' cr e x p { < c ( u ) , x > } c(u) + d ( y )
6 ~
e(x') = ~ 9
V u ~ ~,
V x 6 t , V u 6 91.
Vy 6 ~
(2.2) (2.3)
I p(x) ~ 9 K (x;x') dx
(2.4)
t
V x' 6 :E and for some constant 7 > 0. I
p(x') e x p { < u , x ' > } dx' < ~o
u u 6 %
(2.5)
The elements t , y, a, d are those appearing in the representation (2.1). Remark
If K(x;x') satisfies (2.2) and xl, x 2 6 X are such t h a t xl-Fx 2 q t , it follows
V u 6 ~
J'~ e x p { < u , x ' > } 9 K(x I -4-x2 ; x ' ) d x ' =
= r
9 [ ; ~ e x p { < u , x ' > } 9 K(xl;x' ) dx'] 9 [ ; ~ e x p { < u , x ' > } - K(x2';x' ) dx]
where r
is a positive quantity depending on u, ~, c but non depending on xl,x 2.
V x, x' 6 t ,
define
p(x Ix')
7.
-
)
X(x; x,)
By the condition (2.4), it results
(2.6)
f t p(x I x ' ) dx = 1
V x' 6 it~ and so {p(x I x')}xE % can
be seen as a stationary transition kernel for the signal process {Xn}. Define, moreover, p(x) 9 e x p { < u , x > } ~~215 -- I ~ p(0) 9 e x p { < u , 0 > }
de
V x e ~,
~ -- {~.(x); x e ~ } , , ~
(2.7)
241 The
following result
extends
and
summarizes
arguments
contained
in the
paper
by Bather [1] and it is the starting point for the subsequent discussion.
2.1 T h e o r e m ~P is filter-conjugate with respect to the pair {f(y ] x)}, {p(x I x')}. Proof Our task is simply to check the validity of Eq. (1.2). Fix u E qJ., y E o~. By taking into account (2.1), (2.6) and (2.7), we have [J'a~ ~u(x') 9 p(x]x') d x ' ] 9 f(y I X) r [.rE p ( x , ) , e x p { < u , x ' > } 9 a(•
g ( x ; x') dx'] a ( x ) . b ( y ) . e x p { < d ( y ) , x > } or
i r e e x p { < u , x ' > } - K(x; x')dx'] p(x) 9 e x p { < d ( y ) , x > } cr p(x). e x p { < c ( u ) + d ( y ) , x > } c< r
~o(u,y) = c(u) + d(y) E ~
by the condition (2.3) and then ~o(u,y)(X) E 9.
[] For
a given f(y[x)
of the
form
(2.1),
the
problem
of finding
a filter-conjugate
family with dim ~ = v can then be solved by the following procedure: -
find ~ c Nu, c : q.t --* N v, K : E x ~: -* R + U { 0 } , satisfying the conditions (2.2) and (2.3) find p : t --* R-{- satisfying the conditions (2.4) and (2.5)
-
-
obtain the transition kernel p(xlx') by means of (2.6) and the conjugate family 9 by means of (2.7). In the terminology introduced by Bather, p(x) is an
invariant function and c(u) is a
connecting function. From a fixed K(x ; x') satisfying (2.2) we can obtain respectively one or infinitely many transition kernels p(x I x') depending on the sum of multiplicities of positive eigenvalues with positive eigenfunctions for the integral operator in (2.4) being one or greater than one. The procedure described just above will be illustrated in detail in the example shown in the last Section. Tile first step of the procedure asks for finding qJ., c, K satisfying the conditions (2.2) and (2.3). The form of the solutions for the Eq. (2.2), was already studied in detail by Bather [1] in the case with v = l , c linear. In terms of the symbols introduced here, one of his results can be stated as follows: if K(x;x') solves Eq. (2.2) with ~, ~
R open intervals and with c(u) = r.u
+ k (r > 0, k E ~), then it must result K(x;x') = g(x'-r-x)-exp{k.x} for some function g: ~
-*
~+ u
{o}.
W e are rather interested in obtaining sufficient conditions for (~
to satisfy Eq. (2.2).
This will be made in general in the Sect. 3; now we concentrate attention on the particular case c ( u ) = r. n (r e ~ ) .
242
For functions g, g: Nv _, R + U { 0 } , we shall use the notation Zg _= {x 9 Rv[ g ( x ) > 0 } . If Z1, Z 2 are two subsets of Nv, the symbol Z I + Z 2 denotes the set {x 9 ~v[ x
=
z 1--I- z2, zl 9 Z 1,z 2 9 Z2}
r . Z - - {z' 9 ~ V l z ' = r . z , z 9
and, for r 9 ~ , x 9 ~v, Z C ~v,
ZWx-
{z' 9 ~Viz'=z-J-x,z
9 Z}
2.2 L e m m a Fixr > 0andg:
~v._,~+U
{0} such t h a t
Zg + r . ~ c
~ ' a n d put
c (u) = r.u, %tO {u 9 Nv I J'Nv g(z) 9 e x p { < u , z > l d z < + c~}, K(x;x') ~ g(x'- r.x) The triple (c,~
(2.8)
K) satisfies the condition (2.2).
Proof J ' t e x p { < u , x ' > } 9 K(x;x') dx' = J'~. g(x'- r.x) 9 e x p { < u , x ' > } dx'. For any fixed x 9 i , g(x'- r 9 x) is positive if and only if x' 9 Z g + r 9 x and we can write J'~ e x p { < u , x ' > } 9 K(x;x') dx' =
J'tn(zg
+ r.x)
exp{} 9 g(x'-
r.x) d x ' .
So, by considering the transformation z = x' - r-x, we obtain f ~ e x p { < u , x ' > } - K ( x ; x ' ) d x ' = J'~.n(zg + r.x) - r-x e x p { < u , z + r - x > } 9 g(z) dz (2.9) The hypotesis Z o + r-x c :F, V x 9 ~:, implies that the set {[IM(Zg + r.x)] - r-x} does coincide with Zg, V x 9 ~; and so (2.8) entails J'~ e •
9 K(x;x') dx' = e x p { < r - u , x > } . J ' Z 9 e x p { < u , z > } - g(z) d z .
This completes the proof since the integral in the right hand side does not dcpcn on • [] The arguments contained in this Section can be resumed by the following s t a t e m e n t s 2.3 T h e o r e m Let the family of observation densities f(y[x) be of the form (2.1) and g (g: ~v_.. ~ + U { 0 } ) be such that Z g + r .~" c ~ for some r > 0. Suppose that for some 7 > 0 we can find a non null
p(x)
solution p(x) _> 0 for the equation p(x') = 7 9 J'~" ~
9 g(x'- r.x) dx
Suppose moreover we can find a (non empty) set % c ~.v such that r {u 9 ~v I J ' t p(x') 9 e x p ( < u , x ' > } dx' < ~ , d ( ~ ) -I- r. ~ c ~
J'Zg g ( x ' ) . e x p { < u , x ' > } d x ' < co} (2.10)
p(x)
F o r p ( x Ix')----- 7 " atx~./x,~.g(x'-~ J ~ ) r.x), there exists a f i l t e r - c o n j u g a t e f a m i l y g i v e n by
243 9~ = {~u(X); x E ~'}uEOd
with
p(x)-exp(} ~Ep(9).exp{}d 0 , ~o(u,y) = r.u + d(y).
~u(X) = Corollary
Consider t h e a s s u m p t i o n s of T h m . 2.3; suppose m o r e o v e r t h a t t h e "initial" density of X 1 is given by p l ( x ) cx p(x) . e x p { < u D x > } ' r
I YI . . . . . Yn) or p(x) 9 e•
pn(X
l Yl . . . .
with
u1 e ~
x E E, t h e n
{
,Yn-1) oc ~ . e x p { < : U n , X > )
Un =
rn-Z" Ul +
n-i rn_h E1 h " d(Yh)
3. C o n v o l u t i o n semigroups and solutions of Eq. (2.2) In this Section we w a n t to o b t a i n triples ( c , % , K ) satisfying t h e condition (2.2) with c : ~v _, ~ v of more general form. First we shall briefly recall, some well-known definitions and results (see e.g. [3]) a b o u t infinitely divisible d i s t r i b u t i o n s whose s u p p o r t is c o n t a i n e d in [0,+oo). This t h e o r y has indeed a f u n d a m e n t a l role in o b t a i n i n g our results (as it m a y be already clear by t a k i n g into a c c o u n t the R e m a r k presented in t h e Sect.2). Let M be a probability d i s t r i b u t i o n on t h e real line; M is infinitely divisible if for all n E N t h e r e exists a probability d i s t r i b u t i o n M n such t h a t M = Ma *...* Mn = (Mn) n* where * d e n o t e s t h e convolution operation. A d i s t r i b u t i o n M r is infinitely divisible if and only if it is the
probability
distribution
of X s + r - X s
(s>0)
for a
time-homogeneous
i n d e p e n d e n t i n c r e m e n t s {Xt}. T h e family of p r o b a b i l i t y d i s t r i b u t i o n s { M r } r > 0
process
with
associated to a
fixed t i m e - h o m o g e n e o u s stochastic process with i n d e p e n d e n t i n c r e m e n t s forms a convolution semigroup: Mt~+t2 = Mr1. Mt2. For a p r o b a b i l i t y distribution M on [0,+oa) d e n o t e by ~M its Laplace t r a n f o r m :
A
~M(s) = I 0 real f u n c t i o n
exp{- s 9 z} dM(z) ~
on
[0,+oo)
is t h e
Laplace
transform
of a p r o b a b i l i t y
distribution.
M if a n d only if ~(0) = 1 and it is completely monotone, i.e. it possesses derivatives ~(n) of all orders a n d (-1) n. ~(n)(s) > 0, s > 0. M is infinitely divisible if a n d only if, for n = 2, 3, . . . , the positive n - t h root !0n=(~M) 1/a is the Laplace t r a n s f o r m of a p r o b a b i l i t y d i s t r i b u t i o n ; this happens r
if a n d only if ~M = e x p { - r
where r has a completely m o n o t o n e d e r i v a t i v e a n d
= 0. If { M r } is a convolution semigroup a n d ~M1(S) = e x p { - r ~ M r ( s ) = e• Given r
{- ~-. r
V ,- > 0
we shall henceforth d e n o t e by M r (r
then (3.1)
t h e d i s t r i b u t i o n with Laplace t r a n s f o r m
244 exp {- r. r
In order to follow the line of the present discussion, the following remark is
now of interest Remark Let M r (r
be an infinitely divisible distribution on [0,+co)
0. T h e Eq. (3.1) can be written I
exp{- s. x ' ) m r ( r
admitting a density m r ( r
v ;,
') dx' = exp{- r. r
0
i.e. the function K(x;x') = mx(x') satisfies (2.2) with t stress t h a t M r (r
-- [0,-t-co) and c(s) = - r
We
can be seen as the distribution, at time t = r, of a process with
independent increments {Xt (~)
We recall finally that an infinitely divisible M is (strictly) slable if, in particular, M ---(Mn)n*where, V n, Mn is of the same type of M: M(z) ---- Mn( c~ )
(3.2)
for constants c n > 0. The norming constants arc necessarily of the form c n ~- n 1 / a with 0 <: < 2; the constant a is the characteristic exponent of M. A stable distribution with a E I1,2] cannot have support [0,-Fco) only. All stable distributions are absolutely continous with a regular density, tf r
---- s a, 0 < c ( < 1 , then M r (r
exponent c~; in this case, M r (r
is a stable distribution with characteristic
will be denoted simply by the symbol M r (a). It results
M r ( a ) ( z ) ___ Ml(a)( r-~/a ); for the corresponding density function we have mr(a)(z)
_-- ~
(3.3)
m:(a)(rl--~)
Fix now a function r : [0, +co) -. [0, + c o ) , with r
= 0, r
completely monotone and a
positive measure Q, concentrated on [0, + c o ) with a density q and ~oQ(s) < + c o
Vs > 0
and define c(u) -- - r
-u ), % -
(-co, 0],
x' K(x ; x') -- J" q(x'- z) dM• 0
(3.4)
3.1 T h e o r e m T h e triple (c,~
K) defined in (3.4) satisfies the condition (2.2) with t = [0,+co).
Proof T h e proof is now straightforward. V x > 0, K(x;x') is the density of the convolution of Q and Mx (r then, by the convolution theorem for Laplace transforms~ for s > 0 +00 4-00 +oo ]" exp {-s 9 x'} 9 K(x ; x ' ) d x ' = J" exp{-s 9 x'} 9 q(x') d x ' - f exp{-s.x'} dMx(x') o
= ~Q(s)
0
9 exp{-•
- r
r
exp {-x 9 r
0
= exp {x 9 c(-s)}.
[]
245
We remark that if r e.g. [3] pg. 295):
= r.s
Mr(r
(r>O), then { M r (r
= l[r.r ' +oo)(Z)
r>0, r > 0
K(x;x') ----q(x'- r-x); so we reobtain, for the trivial case 9 -
is a "translation semlgroup" " (see and K ( x ; x ' )
in (3.4) reduces to
[0,+oo) Z9 c [0,+oo), the result of
Lemma 2. When Mx(~b)admits a density mx (r
K(x ; x') in (3.4) can be replaced by
Xf
K(x ; x ' ) ~ f
mx(r
'-z)
dQ(z)
(3.5)
0
which shows that, in this case, we can consider also non absolutely continous measures Q. If in particular r K(x;x') = ~
= - s ~ (0 < a < 1), then by (3.3) T h m . 3.1 applies with 9 J'X'mt(e) ( x'-____s ) dQ(z) 0
(3.6)
x 1/a
Remark. We see that when we build K ( x ; x ' ) elements: the convolution semigroup {M• (r {Mx (r
by means of (3.4), we must choose two
and the measure Q. The form of the semigroup
is directly determined by the choice of the "connecting function" c(u); the choice of Q
has consequences on the form as well on the existence of the " i n v a r i a n t function" p(x). The latter aspect is clarified by means of the example presented in the next Section.
Analogously to what we did in T h m . 2.3 we can now draw the following conclusions: Let f(y[ x) be of the form (2.1) with d(y) _< 0, t -= [ 0 , + o o ) and let M r (r_>0) be a convolution semigroup of probability distributions on [0,+oo). Suppose we can find a density q on [0,+oo) such that, for some 3' > 0, the equation -~oo
p(x') = 7 " ]'o
e(x)
X'
a(x) "[~0 q(x'- z)dMx(z)] dx
as a non null solution p(x)_>0. Suppose moreover we can find so_> 0 such that
;
--}-oo p(x') 9 e x p { < - s , x ' > } d x ' < oo, o
-[-oo J" q(x') . e x p { < - s , x ' > } d x ' < oo V s > so. o
p(x) x' With respect to the transition kernel p(x [ x') = 3' 9 a(xi - p ( x ' ) J'0 q(x'- z) dMx(z) there
exists a filter conjugate family given by 9 ~ {~u(X)}uEq.t with ~ = (-co, -So] and ~u(~x) cr p(x) 9 exp{u - x}, indeed ~ satisfies the Eq. (1.2) with the expression -{-r
~0(u,y) = log [ f
exp{u 9 x'} dMl(x')] + d(y). 0
246 If 2e = (.co, X] or :E = [~, +oo) for s o m e X e ~ we can apply Thin. 3.1 in order to obtain triples (c,qs
satisfying the condition (2.2), by merely considering a linear transfor-
m a t i o n of the signal process; when X -- {x 6 ~v[ x = (xl,...,Xv) , xj > ~j, j =
1,2,...,v}
for a
fixed ~ E ~v we m a y use the theory of processes with i n d e p e n d e n t i n c r e m e n t s with s t a t e space ~ + v . Indeed let M z (~j) (7->0, j = l , . . . , v ) be t h e (absolutely continous) probability distribution at time t = v of a process {Xt (r
with i n d e p e n d e n t i n c r e m e n t s with s t a t e space ~ + v and, for
x -- (xl,...,xv), d e n o t e by K(x;x') the j o i n t p r o b a b i l i t y density function of the distribution MxI(T1)* ... * Mxv(!PV)(x'). T h e n
~+v
e x p { - < s , x ' > } 9 K(x;x') dx' = l]~ S
~+v
1-Ij exp{- r In
analogy
exp{-<s,x'>}
dMxj(r
x/ ---- e x p { - < r with
(3.5)
further
solutions
of
the
Eq.
(2.2)
can
be
obtained
by
c o n v o l u t i o n with positive measures Q on ~ + v .
4. Discussion of a particular case T h e p r o c e d u r e described in Sect. 2. and the results of Sect. 3. are now illustrated
by
discussing a p a r t i c u l a r case. Let f(Y I x) = ~
9 x 6 9 y6-i . exp {-x 9 y}
(4.1)
+oo x > 0, y > 0, 6 > 0, r(6) = ;
s 6-z - e x p {-s} ds. 0
We have
q.J = [0,+co,
E c [0, + c o ) ,
a(x) = x 6, d(y) = -y. Since d(q.]) = (-oo, 0], we need
cO. = (-oo, u0] for s o m e u 0 in order to satisfy the condition d ( ~ ) + r - eLI. c ~ with r > 0. W e put E - [0, + c o ) and consider the c o n v o l u t i o n s e m i g r o u p associated a stable process z-3/2
{Xt}t_>0 with cr = 1/2 ; the density of X t is m(1/2)(z) = ~ +co
f
0
exp {-s 9 z} 9
By (3.2) we h a v e Q(0) = 0,
Z-3/2 ,/~rr
9
exp {- ~ z } dz =
9 exp {- ~ z } :
exp {- -/2 9 s 1/2}
7-2 m r ( 1 / 2 ) ( z ) -- J 2 -1 ~ " z 3~2 exp {- ~ }. If we put in (3.5) Q(z) = 1
V z > 0
(4.2)
we o b t a i n K ( x ; x') = m x ( l / 2 ) ( x ') = J2. 7r
,
X
x,a/---5
X . ~2
9 exp {- 2-x' }
(4.3)
247 E q u a t i o n (2.4) then becomes ? ~176
p(x')oc3'.
0
4~-~r'
x (1-~) x ' ' 3 / 2 "
X2
"
exp { - 2.x-------3} dx
this a d m i t s solutions of the form p(x) oc x -6-1 which do not satisfy the condition (2.5). W e t h e n consider the measure Q such t h a t d Q ( z ) = z -1/2 dz. W i t h such a choice, (3.4) yields K(x ; x ' )
=
x , t1/ 2
. exp {- 2.x' x----~-2}" F u n c t i o n s of the form p(x) or x "6 are positive
eigenfunctions of the integral o p e r a t o r in (2.4); p(x) 0r x "6 satisfies the m e n t i o n e d condition (2.5) when 6 < 1, so we find t h a t , when f(y [ x) is given by (4.1), (0 < 6 < 1) and the t r a n s i t i o n s kernel is
X2
p(x I x') (x p ( x 'p(x) ) 9 a(x) 9 K ( x ; x ' ) = x -2"5 9 x '6-1/2 9 exp { - 2.x' }
a filter-conjugate family of densities 9 ~o(x) = (su ) 1 4
" x -6 9 exp {u 9 x};
--- {~u(X)}uEOd. exists
x > 0 . It results ~,(u,x) = -
with qJ. =
(-co,0)
and
J 2 . lu] - x. T l f f s i s a s l i g h t
extension of t h e situation considered in t h e e x a m p l e iv) of [1], where 6 -- 1 Aknowledgments I like to t h a n k M a r t i n J a c o b s e n and W o l f g a n g J. R u n g g a l d i e r for valuable discussions.
References [1] B a t h e r J.A.
Invariant Conditional Distributions. A n n . M a t h . S t a t . 36 (1965), 829-846
[2] Diaconis P. and Ylvysaker D.
Conjugate priors f o r exponential f a m i l i e s A n n .
Star. 7"
(1979), 217-226. [3] Feller W. [4] F e r r a n t e
P r o b a b i l i t y T h e o r y and its Applications (Vol. 2). Wiley (1970) M. and t~unggaldier W.J. On necessary conditions f o r the existence o f f i n i t e dim-
ensional f i l t e r in discrete time. To a p p e a r on Systems and C o n t r o l Letters. [s] Levine J. a n d Pignie G.
Exact Finite D i m e n s i o n a l Filters f o r a class o f N o n l i n e a r Dis-
c r e t e - T i m e S y s t e m s . Stochastics 18 (1986) , 97-132 [6] l~unggaldier W . J . and Spizzichino F. F i n i l e - d i m e s i o n a l i t y in discrete time non linear filtering f r o m a Bayesian Statistics viewpoint. S t o c h a s t i c Modelling and Filtering, G e r m a n i A. (Ed.),
Lecture notes in Control and I n f o r m a t i o n Sciences. 91. (1987) Springer Verlag
[7] Sawitzki G. Finite D i m e n s i o n a l Filter S y s t e m s in Discrete Time. Stochastics 5_ (1981), 107114 [8] V a n S c h u p p e n J.H. Stochastic filtering theory: a discussion on concepts, methods and resuits. S t o c h a s t i c control theory a n d s t o c h a s t i c differential systems.
K o h l m a n n M. a n d Vogel
W. (Eds.). Lecture notes in C o n t r o l and I n f o r m a t i o n Sciences 1 6 (1979), Springer-Verlag.
ON THE DECOMPOSITIONS OF OBSERVATIONS WITH NON-GAusSIAN ADDITIVE NOISE AND THEIR INNOVATIONS PROCESSES* YOSHIKI TAKEUCHI A-2361
Summary.
IIASA Laxenburg,
Austria ~
This paper is concerned with information structure of the observation with
additive non-Gaussian noise under the assumption that the noise belongs to a class of continuous martingales.
It is known that such an observation is decomposed into
a process with additive Gaussian noise and the quadratic covariation process of the additive noise.
It is also known that the innovations process is decomposed into a
standard Brcwnian motion process and the quadratic covariation process.
In this
paper, a number of sufficient conditions are obtained for the observation to have the information structure such that the information in the quadratic covariation process is not contained in the additive Gaussian part of the observation and/or the Brownian motion part of the innovation process.
I. INTRODUCTION Up to the present, many researches were devoted to nonlinear filtering and its associated innovations problems If] - [ 1 7 ]
Mainly,
they are concerned with the case
of additive Gaussian noise Ill - [12], and relatively small number of researches have been reported on the case of additive non-Gaussian noise. For the class of observations in which the additive noise is a non-Gaussian continuous martingale, we reported results on nonlinear filtering problems [13] and innovations problems[14]f" - [17]
In [13], it was shown that if the additive noise
is a non-Gaussian continuous martingale,
the observation is decomposed into two com-
ponents: one is a process with additive Gaussian noise and the other is the quadratic covariation process of the additive noise.
Also, the author showed in [15] that
the innovations process of such an observation process is a non-Gaussian martingale with the same quadratic covariation process as the additive noise.
The innovations
also decomposed into two components: one is a standard Brownian motion process and the other is the quadratic covariation process, and the latter is common in both decompositions of the observation and innovations processes. In this paper, we will give a number of sufficient conditions for having the unusual information structure such that the information in the quadratic covariation process is not contained in the additive Gaussian part of the observation and/or the Brownaian motion part of the innovations.
~
The first condition is that the family of
On leave from Dept. of Industrial Engineering, Kanazawa Institute of Technology, 7-i Ohgigaoka, Nonoichi-machi, Ishikawa 921, JAPAN
249 ~-algebras t.
generated by the observation or innovations
The second one is the stochastic
process is discontinuous
independence between
and the quadratic covariation process of the additive noise. the existence of a process which is computable
in
the Brownian motion part The third condition is
from the observation and contains a
martingale part that does not belong to the space of martingales Brownian motion part of the observation noise.
generated by the
These conditions cover a large class
of observations with non-Gaussian additive noise, and we will see that usually, information structure of such observation is different with additive Gaussian noise. understanding
We will also give a number of examples for better
of the results reported in this paper.
In this paper, mathematical denotes
symbols are used in the following way.
tile transpose of a vector or a matrix.
a nonsingular
square matrix, A -I denotes
The prime
The Euclidean norm is I.lo
the inverse matrix of A.
(~,~, P) is a complete probability space where ~ i s a s a m p l e
If A is
The triplet
space with elementary
events ~, ~ is a ~-algebra of subsets of ~, and P is a probability measure. denotes
the
from that of the observation
the expectation and E{.I~} , ~ C ~
the conditional
expectation,
E{.}
given ~, with
respect to P. ~[.} is the minimal sub-~-algebra
of ~ with respect to which the fami-
ly of >-measurable
is measuraie.
sets or random variables
{~
If ~I and ~2 are sub-
~-algebras of ~, then ~ I V ~ 2 denotes the minimal ~-algebra which contains b o t h ~ 1 and ~2"
Also, for ~ C ~ a n d
AE~,
{~t; 0 E t S T }
be a non-decreasing
{xt; 0 K t K T}
is said to be
~AA
denotes
the family { B A A ,
family of ~-algebras.
adapted to
E or
F-a~pted
all t E [0,T].
It is assumed that all random variables
~-measurable.
Unless otherwise stated, stochastic
BE~}.
Let F E
A stocastic process x E
if xt(~) is ~t-measurable
for
and stochastic processes are
properties are that with respect
to P.
2. OBSERVATION AND INNOVATIONS PROCESSES AND THEIR DECOMPOSITIONS Let y E {Yt; 0 J t iT}
be an m-dimensional
yt = O + f~hs(~)ds + n t ,
observation
process given by
t E [0,T] ,
where h E [ht; 0 ~ t i T }
is an m-dimensional
notes an m-dimensional
noise process.
(i)
signal process and n E {nt; 0 E t i T }
Most discussions
de-
on the innovations processes
have been made for the case where the noise n is a Gaussian process,
especially a
Brownian motion process.
In this paper, we are concerned with a more general case
where n is a non-Gaussian
continuous martingale with an absolutely
continuous quad-
ratic covariation process. Let (~,~, P) be a complete probability decreasing and right-continuous
space and F E { ~ t ; 0 K t K T }
family of sub-~-algebras
of 5.
We will assume that
all the stochastic processes including y, h and n are adapted to F. we will consider the case where n belongs that n has the form:
be a non-
In this paper,
to a class of continuous martingales
such
250 nt = fO Rs (~ where w - {wt; 0< t
t E [0,T] ,
(2)
is a d-dimensional ~t-Brownian motion process [12] and R(oJ) E
is an mxd-dimensional stochastic process adapted to Fo
If
R0(t ) - Ro(t,~ ) __ARt(0~)R~(~ )
(3)
is non-singular for all tE [0,T], then (2) is converted into a more convenient form: nt= So Ro(s) d~s, where w - {wt; 0 < t_
(4)
is an m-dimensional ~t-Brownian motion process and is defined
by
In what follows, we will assume the following conditions. (C- i) 0 is an ~0-measurabie random vector with E{I012} <=. (C- 2) E{f~lht(~ )I2dt}<~. (C- 3) E{/~]R0(t,~)Idt}<=. (C- 4) E{S~]R089
<~.
(C- 5) R OE{R0(t,~); 0 S tiT}
has a right-continuous modification, and for all
tE [0,T], there exists a positive constant C such that detiR0(t,~) [ ~ C
P-a.s.
Let us define an m-dimensional process y E Yt ~ O + ~ ~ 89
{Yt;
0 5 t i T} by (6)
s.
Then, from (I) and (2), we have Yt = 0+f~R0 89
(7)
(~)ds +wt"
As we can see from (7), y is a part of information in y with additive Gaussian noise. The other part of information in y is the quadratic covariation process Z E {Zt; 0J t T} of the noise n given by Z t $ f~ R0(s,~)ds,
tE [0,r].
We can see that Z is adapted to Y E { ~ t ; 0 J t iT},
(8) ~t ~ { y s ; 0 js Jr} by
Z t = y t y ~ - y O y ~- (f~YsdY~) - (S~YsdY;) '
(9)
Hence, we have i'heorem I ([13; Theorem i]).
The couple (y,Z) given by (7) and (8) has the
same information as y, i.e.,
%t =~tv ~t,
te [O,T],
(i0)
where
~t-~O{ys; O_<s_
(ii)
251
~t ~ ~{Ys; 0 fis ! t}
(12)
~t ~~
(t3)
and
0is! t}.
For the observation given by (I), let us define the non-Gaussian innovations process v Z {vt; 0 J t i T } by vt=Yt-
f~h(s)ds,
(14)
where fi(t) ~ E{ht(~)[gt ].
(15)
Clearly, v is a Y-martingale with the same quadratic covariation process as n.
~t ~ 0+ [~ R0"2(s) -~" dvs , then ~ E {vt; 0 i t iT}
t E [O,r],
is a ~t-Brownian motion process[10]
Let (16)
For the innovations
process v, we have the following decomposition. f~eorem 2 ([15; Theorem I]).
The couple (~,Z) given by (16) and (8) has the
same information as v, i.e.,
~rt =~ft ~/Z t,
tE [0,T] ,
(17)
where
"~t__A~7{Vs; 0 _<s _
(is)
~7t ~ o{~s; 0<_s_< t}.
(19)
and
In what follows, we are concerned with the conditions under which the decompositions given by (10) and (17) are explicit, i.e., ~ t ~D~t
(20)
~t ~Zt.
(21)
and
Clearly,
(20) and (21) are respectively equivalent to (22)
and
vt + t
(23)
As it is known ([i0; Lemma 5.3]), for the case where R 0 is given by a non-anticiparive functional of y, we have ~t = ~ t
(24)
and
However, it seems that for a large class of stochastic processes R0, (20) and (21) are valid.
If (20) and (21) hold, then the fundamental information structure of the
observation is different from that of an observation with additive Gaussian noise. In the next section, we will give sufficient conditions for (20),(21) and/or (22),
252 (23).
For simplicity, we will write "(22)" instead of "(20) and (22)", and also
"(23)" instead of "(21) and (23)".
3. MAIN RESULTS Let us first show the following theorem.
Theorem 3.
If
(i) Y ~ { ~ t ; 0 ~ t ST}
is discontinuous in t;
or (~) V ~ {q/t; 0 ~ t~ T} is discontinuous in t; then, (Jii) Z ~ {~t; 0 ~ t ~T}
is discontinuous in t.
Furthermore, if (i) is valid, then we have (22), and if (~), then (23).
Proof: Note that Y E {~t; 0 ! t i T }
and V E { ~ t ; 0 ! t i T }
are continuous in t. Then,
it is clear from (10) and (17) that (i) implies (~ii), and also, (~) implies (JJi). If (24) is valid for all t E [0,T], then Y is continuous in t. valid, then we have (22).
Example I.
Similarly,
Hence, if (i) is
(~) implies (23).
Let us consider the case where
o{R0(0,~)}CO{0}.
(26)
Z 0 = {r
(27)
Note that fl},
Z 0 + = O{R0(0,~)}
(28)
17/-0= o{e}.
(29)
and
Then, it follows that ~0 =
O{0}
(30)
and %0+ = ~ 0 V Z 0 + = o{O} V ~{R0(0,~) }. Thus, by (26), we have 40 + # 4 0 ~0+~V0
(iii).
and Y is discontinuous at t= 0.
and V is discontinuous at t = 0.
Remark I.
(31) Similarly, we have
Hence, by Theorem 3, we have (22) and (23).
In Theorem 3, (i) and/or (i)
are sufficient but not necessary for
In fact, if ~{R0(0,~ )} = ~{0}, then Z is discontinuous at t = 0.
continuous at t= 0 since
But Y is
~0+C~.
Now, let us give another class of observations for which (22) and (23) are valid. Theorem 4.
If R 0 is a non-deterministic process which is independent of w,
then we have (22) and (23).
Proof: Let
253 =
- 89
~t(t0) gexp{_f Oh,(~)R0
t
t
(s,~)d~s+ 2/0 IRo89
(32)
and define a probability measure P on (~,~) by P(A) __A/A~T(~)dp(~) ' Then, by Girsanov's
A E ~.
(33)
theorem [i0; Lemma 3.1], (~,P) has the same distribution as
(w,P) and is a standard Brownian motion process independent of R 0.
Hence, it fol-
lows that E{R0(t,0J) l~t } =E{R0(t,~)} =Deterministic where F. denotes the expectation with respect to P.
fn of time,
(34)
If (24) holds for all t~ [0,T],
then we have E{R0(t,~ ) l~t } = E{R0(t,m) I~t } = R0(t,~) .
(35)
This is a contradiction since R 0 is not a deterministic process. (22).
Similarly,
Hence, we have
to show (23), let P be a probability measure defined on (~,~) by
P(A) =A fA~T(~)dP(60) ,
A E 5 ~,
(36)
where t
+ 2f 0
IRO-89(s,~)Es(~)12ds}
(37)
and
Gt(~ ) =aht(~ ) _ h ( t ) .
(38)
Then, (9,P) is a standard Brownian motion process independent of R 0.
Hence, we have
(24). R~r~zrk 2.
It should be noted that under the assumption of Theorem 4, h and R 0
are not necessarily independent, and hence, not independent in general.
the signal h and the noise m are also
The stochastic independence employed in Theorem 4 is
concerned with only ~ and is described by (i) ~ is independent of R0; and (ii) for all 0< t< s_< T, ~s-~t is independent of {0, hu(0J) ; 0 < u <
t};
where (~i) is given by the fact that w is an ~t'Brownian motion process. By Theorems 3 and 4, two conditions on the observation: which (22) and (23) are valid were obtained. {y; 0, h, R 0, w}
which satisfy (22) or (23).
{y; 8, h, R0, w}
under
Now, let us give other classes of Because of Theorem 3, we will consider
only the case where Y and V are continuous in t.
[email protected] 1. stochastic process,
Let x - {x t E II; 0 < t < T} be a square integrable and Y-adapted x is said to belong to space $2(F;Y) if x has a representation:
x t = x 0 +foas (0j)ds +mt(~ ) , where x 0 is a ~0-measurable random variable with E{x02} <=o; a-{at(e) ; 0 < t <_ T}
(39) is an
F-adapted process such that E[ "u ~Tat (~) 2dr} < ~;
(40)
254
and m E [mt(~); 0! tiT}
Definition 2. f E {ft(~); 0 ~ t ! T }
is a square integrable F-martingale with m0(~) ~0.
Let us denote by ~2(F) the set of ~m-valued F-adapted processes with the property:
E{f~ Ift(~)I2dt} <~.
(41)
l~2(w, F) ~ {f0 f~(~)d~s; fE~2(F)}.
(42)
Let
Then, ~2(~, F) is called the space of square integrable F-martingales generated by ~.
~eorem 5.
If there exists xE$2(F;Y)
whose F-martingale part m does not be-
long to the space of square integrable F-martingales generated by w, i.e., m ~ ~2(w,F),
then we have (22).
~eorem 6. xE$2(F;V)
Let us define $2(F;V) by the same way as $2(F;Y).
If there exists
whose F-martingale part m does not belong to ~2(~, F), then we have (22)
and (23). The proofs of Theorem 5 and 6 need some preliminaries, and will be given in the next section.
In the remainder of this section,
Example 2.
let us give an example.
Let us consider the one dimensional case where y is given by Yt = e+ Soths(~) as +f~ ~ (S,~s,Ys)dws"
In (43), we assume that ~
(~t; 0~ t~T}
(43)
is a one-dimensional process given by
~t = ~0 +f~as(~)ds +~t 9 where ~ E {~t(w); 0 ~ tiT}
(44)
is a square integrable F-adapted process and ~ E {vt; 0E t
T} is a standard grownian motion process independent of w. (C- 5) and (5), we have l~2(w, F) = ~ ( ~ ,
F).
Note that because of
Then, it is clear that ~ ( w ,
F). If
r0(t,~t,y t) ~ (r(t,(t,Yt)} 2 is continuously differentiable in the first argument and twice continuously differentiabie in the second and third arguments, then ItS's stochastic differential formula implies r 0E$2(F;Y).
(The fact that r 0 is Y-adapted is
implied by (8) and (9).) Also, the martingale part m is given by ~r0 - t ~r0 mt(~) =f~-~(S,~s,Ys)d~s+I~-~-y (S,~s,Ys){r0(S,~s,Ys)}=d~ s. ~lus, we have m~IM2(~, F) unless ~r0/~ E0.
Remark 3. independent.
(45)
Hence, by Theorem 5, we have (22).
In Example 2, it should be noted that ~ and w are not necessarily Hence, it seems that an essential condition for (22) is R 0 is depend-
ent on an F-martingale which is independent of w.
Remark 4.
Let m be a martingale part of xE$2(F;Y).
Also, let <m,~> denote
the quadratic covariation process of m and ~, i.e., <m,w> is an m-dimensional Fadapted process such that {mt~ t- <m,~>t; 0 i t iT} well-known that there exists f ~ 2 ( F ) dition m ~ 4 2 ( ~ , F )
such that
is an F-martingale [I0] <m,G> t= f~ fs(~)ds.
is equivalent to <m>t ~ 0, where m E {Nt(~); 0! tiT}
nent of m orthgnnal to ~2(w,F),
i.e.,
It is
Then, the conis the compo-
255 ~t(~) ~mt(~ ) -f~f~(~)dOs,
(46)
and <~> is the quadritic variation process of ~, i.e., <~> is an F-adapted process such that {~t 2- <m>t; 0 i t iT}
is an F-martingale [II]
4. PROOFS OF THEOREMS 5 AND 6 In this section, we will use the following notation for simplicity. and ~2(y)
Let IM2(F)
respectively denote the space of square integrable F- and Y-martingales.
Let ~2(y) denote the set of ~m-valued Y-adapted f E[ft(~); 0~ t i T} with the property (41).
We can define ~i2(v,Y), i.eo, the space of square integrable martingales
generated by ~ similarly as (42). Now, we have the following lemma. Lemma 7.
Let E~$2(F;Y) and x be defined by
x t ~ t - ~ 0 - f ~ E { a s ( ~ ) I ~ s } d s E f~[as(~ ) -E{as(~ ) I~s}]ds+mt(~ ). Then, x E ~ 2 ( Y )
and xE$2(F;Y).
If m ~ 2 ( F )
(47)
is its martingale part, then we have
<x>tI~t = <m>tI~ t
(48)
<~,~>t[%t = <m,~>tr> t.
(~9)
and
Proof: Noting (39), we see that It$'s stochastic differential formula implies xt2 = x02 + 2f~xsdxs +<m>tI~t. By (50), it is clear that <m>tI~ t is ~t-measurable~
(50) Since
x02 + 2f~xsdxs = xt2 - <m>tl~ t (51) and the left-hand side of (51) is a Y-martingale, we have (48) by the definition and the uniqueness of <.>.
Similarly, by applying It$'s stochastic differential formula
to the F-adapted process {xtvt; 0N t ST}, we have xt~t=Xo~O+f~sdxs+S~xsd~s+<m,~>tI~t . (52) Since the sum of the first three terms in the right-hand side of (52) is a Y-martingale by
v, xE~2(Y), we have (49). This completes the proof.
L e n a 2.
If ~ is an orthogonal base of ~2(y),
then w is an orthogonal base of
~2(F;Y) ~{mE~2(F); m is a martingale part of x ~ 2 ( F ; Y ) } .
(53)
]M2(~,Y)=~(Y)
(54)
Namely, > ~2(~,F) ~ 2 ( F ; Y ) ,
where ~(Y)~
Proof:
{mE~i2(Y); m0(~) =0}.
Let x~$2(F;Y), and assume that its martingale part mE~I2(F;Y)
fies <m>tl~t# 0 and <m,w>tI~t = 0, ~(Y).
(55)
By eemma i, we have <x>tI~t#0 and <x,v>tI~t=0.
consequently, ]M2(v,Y) ~ ( Y ) .
satis-
Then, the process x defined by (47) belongs to Hence,
x~2(~,Y)
and
This completes the proof.
The following lemma is easily shown by the well-known results for the observation with additive Gaussian noise.
256 Lemma 3.
If v is not an orthogonal base of ~2(y),
i.e., if ~ 4 2 ( v , Y ) ~ ( Y ) ,
then we have (22).
Proof:
If ~t=q]t for all tE [0,T], then v is the Gaussian innovations process
for the observation 9 with additive Gaussian noise. Theorem 3.1], we have ]M2(~,Y) = ~ ( Y ) .
Hence, by (C- 3) and [i0;
This completes
the proof.
Now, we will give the proof of Theorems 5 and 6.
Proof of Theorem 5:
By Lemma 2, we have
~2(w,F) ~ 2 ( F ; Y )
> ]M2(~,Y) ~ ( Y ) .
Hence, if there exists xE$2(F;Y) ~2(w,F),
whose martingale part m does not belong to
then v is not an orthogonal base of ~ ( Y ) .
Hence, by Lemma 3, we have
(22).
Proof of Theorem 6: $2(F;V),
First, let us show (22).
Since q/tC~t, tE [0,T], if x E
Hence, S2(F;V) CS2(F;Y)
and we have (22) by Theorem 5.
then xES2(F;Y).
In order to show (23), note that from (i), (4), (14) and (16), we have (56)
v t = 0 + S~(hs (w)- h(s)}ds + S~ R~(s,~)dw s and vt =
ft ~-89 O+~ K 0 ~s,~){hs(~)-
fi(s)}ds+wt.
(57)
Let us regard v as an observation process and ~ as its component with additive Gaussian noise.
Then, we see that the innovations process and its Brownian motion
component of "observation v" are respectively v and ~ since E{ht(~)- fi(t)I~/t} = E{E{ht(~)- fi(t) I%t } 12/t} = 0.
(58)
Hence, we can apply Theorem 5 by replacing y and y by v and v, respectively. if there exists xE$2(F;V) ~2(w,F),
whose martingale part m E ~ 2 ( F ; V )
then we have (23).
Remark 5.
Namely,
does not belong to
This completes the proof.
As we can see from the proof of Theorem 6, for the observation with
feedback: yt=8+S~{gs(~)-~(s)}ds+f~ where ~(t) ~E{gt(~)i~t},
we have v = y
R~(s,~)d~s,
and the innovations informational
(59) equivalence
to the observation always holds, provided that (59) has a unique strong solution.
5. CONCLUDING REMARKS We obtained sufficient conditions for (22) and (23) and it was shown that for a large class of non-Gaussian martingales n, the information in Z is not contained in y and v.
We were not concerned with the innovations informational equivalence
to the observation in this paper.
It should be noted that the condition of innova-
tions informational equivalence is mainly related with the stochastic properties of h and {Ro89
0 ~ t~T}
whereas the condition for (22) and (23) is described
by the properties of R 0 and ~ and is almost independent of h.
257 REFERENCES [i] T. Kailath, "An innovations approach to least-squares estimation--Part I: Linear filtering in additive white noise," IEEE Trans. on Automat. Contr., vol. AC-13, pp. 646-655, 1968. [2] P.A. Frost, Estimation in Continuous Time Nonlinear Systems, P h . D . Dissertation, Stanford Univ., Stanford, Calif., 1968. [3] P.A. Frost and T. Kailath, "An innovations approach to least-squares estimation --Part I~: Nonlinear estimation in white Gaussian noise," IEEE Trans. on Automat. Contr., vol. AC-16, pp. 217-226, 1971. [4] T. Kailath, "A note on least squares estimation by the innovations method," SIAM J. Contr., vol. I0, pp. 447-486, 1972. [5] J. M. C. Clark, "Conditions for the one-to-one correspondence between an observations process and its innovations," Center for Computing and Automation, Imperial College, London, U.K., Tech. Rep. i, 1969. [6] V. E. Benes, "Extension of Clark's innovations equivalence theorem to the case of signal Z independent of noise, with fZs2dS< ~ a.s.," Mathematical Programing Study, vol. 5, pp. 2-7, 1976. [7] D. S. Cirelson, "An example of a stochastic differential equation having no strong solution," Theory Probability Appl., vol. 20, pp.427-430, 1975. [8] V. E. Bene~, "Nonexistence of strong nonanticipating solution to stochastic DEs: Implications for functional DEs, filtering and control," Stochastic Processes and their Appl., vol. 5, pp. 243-263, 1977. [9] D. F. Allinger and S. K. Mitter, "New results on the innovations problem for nonlinear filtering," Stochastics, vol. 4, pp. 339-348, 1981. [I0] M. Fujisaki, G. Kallianpur and H. Kunita, "Stochastic differential equations for the nonlinear filtering problem," Osaka J. Math., vol. 9, pp. 19-40, 1972. [ii] H. Kunita, Estimation of Stochastic Processes, Sangyo-Tosyo, Tokyo, 1976. (in Japanese) [12] R. S. Riptser and A. N. Shiryayev, Statistics of Random Processes I: General Theory, Springer-Verlag, New York, 1977. [13] Y. Takeuchi and H. Akashi, "Least-squares state estimation of systems with statedependent observation moise," IFAC J. Automatica, vol. 21, pp. 303-313, 1985. [14] Y. Takeuchi, "Generalized innovations equivalence for observations with nonGaussian noise," Proc. International Sessions 26th SICE Annual Conference, Hiroshima, Japan, July 15-17, vol. 2, pp. 1069-1072, 1987. [15] Y. Takeuchi, "On non-Gaussian innovations processes for observations with nonGaussian noise," Proc. 25th IEEE Conference on Decision and Control, Athens, Greece, Dec. 10-12, pp. 1035-1036, 1986. [16] Y. Takeuchi, "Innovations informational equivalence for observations with nonGaussian noise," Proc 26th IEEE Conference on Decision and Control, Los Angeles, California, Dec. 9-11, pp. 1163-1168, 1987. [17] Y. Takeuchi, "Innovations informational equivalence for a class of observations with independent non-Gaussian noise," submitted to IEEE Trans. on Inform. Thoery.
NONLINEAR THE
FILTERING
CASE
OF
WITH
UNBOUNDED
Patrick UF~A C N P ~
No
paper
noises
concerns
and
a one
coefficients. respect define
to
prove
paths
of
the
a formal
the
Malliavin
density
for
filter
is
the
the
formula.
unique
problem
continuity
Then,
we
filter.
Metz
by
prove
Finally,
solution
of
of
filter means
of
the
filter
With
this
and
we
~iven
of
a
the
with
aim,
we
prove
a result
that
Zakai
unbounded
the
existence
prove
correlated
with
process.
the
we
with
process
unnormalized
calculus, the
de
filtering
observation
continuous
Kallianpur-Striebel
Universit~
observation
we
the
COEFFICIENTS.
LORRAINE
a nonlinear
dimensional
First,
NOISES
FLORCHINGE~
399,
INDIA
This
DEPENDENT
a by
regular
unnormalized
equation.
INTRODUCTION
The
aim
compute given It
the
is
of
the of
known
that
minimize
that
the
nonlinear
history
well
which
in
"'at b e s t "
the
of
compute
this
obtain,
for
the
an
of
the
best
with
{yt}
is
to
the
following
process.
a
functional
to
to
any
t.
{x L }
of
xt
expectation
field
time
:
process
conditional
for
conditional
the
this
sigma
up
expectation
the
on
the
process
is
stochastic
estimation
respect
conditional t,
theory unknown
risk
observation
each
an
observation
quadratic
functional
paths
filterin~
paths
~enerated Notice
functional
distribution
by
that
to
reduces
of
xt
to
given
{ys,OSsst}. Some be
theorical
solved
9 The
are
continuity
observation 9 The state
an The of
scalar aim,
we
discuss
questions following
of
the
in
nonlinear
filterin~
which
have
to
:
filter
with
respect
to
the
paths
of
the
process.
existence
9 Gompute
case
the
the
of
uniqueness purpose
of
a nonlinear observation present the
a regular
general
main
form result this
ideas
the
paper
process of
and
for
nonlinear
for
filtering
a survey
density
of
is
the
solutions to
solve
problem
with
with the
unbounded previous
results
of
filter.
filtering to
equations
these
those
questions
dependent
works
on
[i?]
topics
[i8].
the
and
With
these and
in
noises
coefficients.
[i63,
and
equations.
a
this and
259 I STATEMENT
OF THE
We consider process
P~OBLEM. on a probability
- observation
stochastic
(xt,yt)~nx~P
differential
system
(O,~,P)
solution
the of
pair the
system
followin~
:
t=l
~=1
dyt=h(xt)dt
+ dvt
x o donn~
loi
de
space
(i.l)
mo
Yo=O where
w and
~m and Notice the
v are
two
independent
~P respectively. that since the process
system
process
and
is c o r r e l a t e d . Furthermore, assume 9 X o is a vector
the
the
field
9 XI,..,Xm,XI,..,Xp 9 h is a function
Wiener
v t appears
observation
followin~ on
are
in c ( ~ n , ~ p)
with
in
definitions
both
process,
hypothesis
~n with
bounded
processes
less
vector
such
than
X"o +
system
on the
in
of Ci.l)
coefficients
linear
fields
that
the
values
:
~rowth.
on ~n
~ hs163 h a s
less
than
t=l
linear For
~rowth
any
fin
t~s
associated
with
~o(s
order
and
any
the
system
p)
2 CONTINUITY
In this
we
OF THE
obtain
are
a paths remains
process of
of the notion
we are
function
~,
as
the
of
define
functional
= Es
t)
interested
This means that on the measures by
paths valid
i s used.
interest
by Chaleyat-Maurel Next, we study the paths That
solutions
(l~ filter
the
:
/ F~]
a.s.
FILTER.
section,
formula
observation which
(l.i)
> ~t~(y)
o f t h e f i l t e r ~t" sense errors made filter
bounded
explosive
O(Ys/OSs~t).
F~ =
hand,
avoid
>
y where
to
The
in nonlinear
in s continuity
of
in
continuity
properties
the model drives in the of the observation. On the formula when
which
an
different filterin~ the
filter
implies of
the of the continuity
been
discussed
approximation notions have with
right other
that
respect
observation p r o c e s s f o r t h e B a n a c h norm. of continuity in nonlinear filterin~
has
Co
the been
260 introduced bounded
by Clark
[8]
coefficients.
in t h e
We aim
F mappin~
Co([O,T],~P)
in ~,
norm
Co(s
such
on
That
notion
of
continuity
allows to estimate observation. The
continuity
where been
the
of
the the
studied
first
of
uncorrelated
the
continuous
that
filter
= ~t~(y) by
paths
one
has
with
by S u s s m a n n
for
by Fleming-MiLLer
[153
last results have continuity of the
been improved by Sussmann filler with respect Lo the that
Vc>O B Kc>O /
Banach
formula only
"cubic
Lwo the
such
the
which
a
unique system~
coefficients
and
h is a C 2 f u n c t i o n
h is
the
to
uncorrelaLed
unbounded
[303
when
with
with
a funcLional
a.s.
a paths
associated
of
respect
whenever
is s c a l a r
systems
existence wiLh
F(y)
~ives signal
observation
case
to prove
sensor
a polynomial
has
problem"
function.
These
[31] who proved Banach norm when
:
[Lh[
§ ~
~ X t h l 2 < c h 2 § Kc
L=I
The
of
continuity
the
(i.e.3~,i~p,s.t.
filler
X ~0)
associated
has
been
with
handled
correlated
by Davis
[9]
systems
in
the
case
t
of
a scalar
observation
with
bounded
Kolhman C123 w h e n t h e v e c t o r f i e l d s notice that Davis-Spathopoulos [113 of correlated system with unbounded Here,
we aim
to study
the case
observation
is scalar
Assume
XI,... ,Xm,X I are
that
function Denote
with
b y ~t
less the
with
than
of
the
~t ( x ) ' x § Moreover,
assume
that
a correlated
by
E11iott-
system
where
the
coefficients.
C 3b v e c t o r
of
and
X~, l ~ p commute. Moreover, have studied a particular case coefficients.
unbounded
exponential
solution
coefficients
and
fields
that
h Is
a
equation
:
Cs
~rowth.
followin~
~(~
integral
)(x)ds.
condition Tn
(H)
Vr>O,Vc>O,3K~>O
/[Xh~+
sup
[G
Isl_
where
order
differential
G =~ Xo is f u l f i l l e d . Then, we can slate
TIIEOREM 2 . 1 for
any
lipschitz In L h e need
: If
function
~ in
followin~,
Lo define
the
.
g--J.
[Xt
[2_
Isl_
operator
defined
by
L.
:
/or
continuous
sup ,
any
k~,
exp(k[Xo[)
C~(~n,o~),
the
is
filter
mo ~t~
inte6rable admits
a
then, locally
version. we present
the
unnormalized
main filler.
ideas
of the
proof
that
we
261 S i n c e we a r e w o r k i n g w i t h a c o r r e l a t e d s y s t e m w h e r e t h e f u n c t i o n h is u n b o u n d e d , we cannot, a s u s u a l y in n o n l i n e a r f i l t e r i n g p r o b l e m s (c.f. s a p p l y G i r s a n o v ' s t h e o r e m and d e f i n e an f i l t e r by m e a n s of the r e f e r e n c e p r o b a b i l i t y measure
unnormalized method. To
carry out that difficulty we define a locally lipschitz continuous formal unnormalized filter and compute a Kallianpur-Striebel formula
wich
links
the filter
~t
and
To d e f i n e t h e u n n o r m a l i z e d filter c o r r e l a t i o n t e r m in t h e d e f i n i t i o n
the
unnormalized
we aim to of the system
filter
Pt"
get rid of the p r o c e s s An order
to be a b l e to f i x y. Hence, we t r y t o d e f i n e t h e process xt a n y c o n t i n u o u s p a t h of Yt" Thus, f o r a fixed y, introduce process
xt
as t h e s o l u t i o n
of the
stochastic
differential
for the
equation
42.1) XO~XO using and by
the following
notation
F a C * ( N n , ~ n) f u n c t i o n , :
: if X d e n o t e s define
a vector
(@~-Ix)F(x)
field
for any
on
<(DOt(x))-Ix(ot(x)),DF(x)>.
Then,
following
[2] and
EQ],
fop
any
t~[O,T]
we h a v e
:
Xt =@Yt(~t)~ A s u s u a l y in n o n l i n e a r f i l t e r i n g , Girsanov exponential Z t associated
42.2)
define with
t foh(Xs)dYs
Z t = exp(
f o r a n y t in [O,T], t h e s y s t e m (1.1) b y
- ~ ~h2(xs)ds).
On t h e o t h e r hand, a c c o r d i n g w i t h f o r m u l a an i n t e g r a t i o n by p a r t s in t h e s t o c h a s t i c the definition
(2.3)
42.2), o n e integral
can compute appearing in
(x~))+(~-lXo)H(y~,xs)]ds
JO
aS
~
~
ft(o*-lx ~=1 Jo Ys
H is t h e f u n c t i o n
defined for t
(2.r
~S
- ~ where
any
~
)H(Ys,~
s
)odw ~) s
( t , x ) ~ [ O , T ] x ~ n by
Hft,x)-~ohO@sfx) ds. Hence,
Then,
we h a v e
the
of Z t and obtain
Zt=exp(H(Yt,X,)-rts
that
~n
(L,x)~ARx~ n
a paths
the process
by p a t h s
Z t is in L P f W ~
we c a n d e f i n e
formula o) a l m o s t
the u n n o r m a l i z e d
(2.5)
which surely
filter
permits in y f o r
to any
in t h e f o l l o w i n g
prove p~ way
:
262 DEFINITION 2.2 : For any t in [O,T] and any bounded function ~, d e f i n e the u n n o r m a l i z e d f i l t e r a s s o c i a t e d ~ i t h the s y s t e m ( l . i ) , d e n o t e d pt~, b N : pt~fy)
~here
y is
Then,
one
fixed
for a
any
t
)Z t dWfw)dm
0
(2.6)
( x )0
determinist.
: If for a n y kr/R, e x p ( k [ X o [ )
2.3
function
locally
[ [ ~(x
prove
can
PROPOSITION
and
=
C~(~n,~),
~ in
the
continuous
ItpsehitZ
is m o i n t e 6 w a b t e
unnormalized
filter
then
pt ~ admits
uersion.
Furthermore, by means of limitin~ arguments, we prove a Kallianpur -- S t r i e b e l formula which completes easily the proof of theorem 2.1
3 EXISTENCE
OF
In t h i s as
A REGULAR
section,we
a measure
admits
DENSITY
aim
to
a regular
TO
THE
prove
FILTER.
that
density
the
with
filter
~t
respect
considered
to
Lebes~ue
measure. The
problem
may
be
of
solved
techniques (See
the by
existence means
applied
s
The
for
a regular
Halliavin
to stochastic
Krylov-Rozovskii
Michel
os
of
this
[203, second
existence
partial
Fardoux point
of
density
to
calculus
[2?3
the
or
by
differential and
Tilter analytic
equations.
Chaleyat
Haurel
-
view).
d e n s i t y f o r t h e T i l t e r in t h e c a s e with bounded coefficients has been proved by means of Halliavin calculus, under Hormander's hypoellipticity hypothesis, by Michel [253, B i s m u t - H i c h e l [33 a n d Kusuoka-Stroock [233. O n t h e other hand, by a generalisation of the method developped i n [25], F e r r e y r a [13] has shown under some growth conditions on the function h the existence of a regular density to the filter associated with an unc'orrelated system where the observation coefficients are unbounded. Mow, b y m e a n s o T M a l l i a v i n c a l c u l u s , we aim to prove the existence of correlated
of a regular system
with
density a scalar
For a presentation to s In t h e
of a regular
systems
following,
and
that
h
are
with
less
to the Tilter observation
of the
assume
Halliavin
that
is a C ~ function than
the
case
unbounded
calculus,
XI,...,Xm, X I are
such
exponential
in
with
that
growth.
its
of
a
correlated
coefficients. we refer
Cb
derivatives
the
reader
vector
fields
os
any
order
263 Denote
,X ) t h e i d e a l
by ~(X I '"
ALfX,
"
"
~enerated
by
Lie
the
al~ebra
rn
X ) in t h e
Lie
al~ebra
ALfXo,
,X m)
Then we can state, THEOREM 3 . 1
: If
unnormalized
~(Xx,...,Xm)=~
PL
~ilter
n then,
admits
for
C~
a
any
t
]O,T],
in
density
~ith
in
is
the
to
respect
Lebes6ue measure. Since
the
stochastic
process
xt
defined
42.1)
the
solution
of a stochastic differential equation with coefficients dependin~ on t h e t i m e in an i r r e g u l a r way, w e use, f o r a f i x e d p a t h of the p r o c e s s Yt' t h e r e s u l t s f r o m s on the Halliavin calculus. Introduce
the
derivative
the stochastic Ft Then,
~t"
the
process
of xt"
= + t *-I I [ D(~y Xo)(X JO s Halliavin
Mr,
denoted
the
denoted
FL"
by
associated
flow
by
matrix
with
:
)F d s + ~ D ( * ~ I x ~ ) ( x s ) F g o d w ~
covariance
is ~iven
stochastic
associated
(3.i)
with
the
process
:
t JO In o r d e r t o it s u f f i c e s i) To ii) For
prove :
prove
Js
that
compute
To
all
k=1
an
that
/s
the
unnormalized
integration for
any
by
filter
parts
L~30,T],
Pt
admits
a density
formula.
(det
M r ) -I
is
in
L P f w @ m o)
p in ~*.
The inLe~ration by p a r t s f o r m u l a f o r t h e u n n o r m a l i z e d f i l t e r Pt is obtained by standard results of Malliavin calculus l i k e in s or s O n t h e o t h e r hand, t o p r o v e ii), it is e n o u g h t h a t F o r a n y t in ]O,T], t h e p r o c e s s x t p o s s e s s a C ~ d e n s i t y w i t h r e s p e c t to the Lebes~ue
measure.
And
is p r o v e d
this
THEO~EME 3 . 2
s z~
~here
the
derivatives any compact element in bounded.
by means
: Let = Zo
uecfor
in z,
of
the
zt~It~ n b e
the
t + ~Yo(S,z.)ds
[ietds
Holder
O~L~m, continuous
Yc,
followin~
result
stochastic
:
process
defined
by
ty + ~ are
~(s,zs)~ as
welt
in t uni/ormty subset K in ~n C~~ b o u n d e d in z ~hen [O,T] and that all their derivatives in
~ as in s t is z are
all
their
for a fixed uniformly
264 If 3 ( Y , , . . . , Y ) ( O , Z o ) = ~ z t admits
process pointin~
out
To obtain
that
e C ~ density
for
any
~ith
respect
t~]O,T],
the
stochastic
to L e b e s G u e
measure.
:
regularity
f i l t e r Pt' formula.
n then,
results
for
o n e h a s to i t e r a t e
the
like
density
in
of
[3] t h e
the
unnormalized
integration
by p a r t s
4 THE ZAKAI EQUATION. In t h i s s e c t i o n , we a i m to p r o v e t h a t the u n n o r m a l i z e d f i l t e r associated with a correlated system where the observation is scalar with unbounded coefficients solves a parabolic stochastic partial differential equation, the Zakai equation, for which C a n n a r s a and V e s p r i have proved an uniqueness result to the solution. T h e z a k a i e q u a t i o n h a s b e e n o b t a i n e d in t h e c a s e of a nonlinear filterin~ problem with independent noises and bounded coefficients by Z a k a i [32] a n d in the c a s e o f c o r r e l a t e d s y s t e m s with bounded c o e f f i c i e n t s by D a v i s [9], D a v i s - M a r c u s [I03 a n d P a r d o u x [2Z]. The c a s e of u n c o r r e l a t e d s y s t e m s w i t h u n b o u n d e d c o e f f i c i e n t s h a s been h a n d l e d by Y a r d o u x s and Hopkins [I9].Uniqueness results of the solution to the Zakai equation associated with uncorrelated s y s t e m s a r e ~ i v e n in t h e c a s e o f b o u n d e d coefficients by Kunita [22] and, f o r a v a r i o u s t y p e o f u n b o u n d e d o b s e r v a t i o n c o e f f i c i e n t s by P a r d o u x [28], Flemin~-Mitter [I~], Baras-Blankenship-Hopkins s H o p k i n s [193, F e r r e y r a s and Bensoussan s The case of correlated systems with bounded coefficients has been handled, by F a r d o u x [2Z], K r y l o v - R o z o v s k i i F a r d o u x EZ]. F i n a l l y , n o t i c e t h a t an u n i q u e n e s s
result
the observation Under
the same
THEOP~EN 4.1 filter
Pt
for the solution,
is s c a l a r
with
assumptions
: For
defined
(2.7)
L is
the s e c o n d
order
Holder
previous
in
C ~ ( ~ n,[P.),
~
solves
spaces,
the Zakai
section, the
it s t a t e s
unnormalized
equation
+ Pt(h~X~)dYt
differential
t:l
t
when
coefficients.
in t h e
I dPt~ = Pt(L~)dt
~here
in s o m e
unbounded
than
any /unction by
[21] a n d C h a l e y a t M a u r e l - M i c h e l Cannarsa-Vespri [43 h a v e proved
operator
(4.1)
def~ned
by
:
265 In
order
(4.1),
to
let
For every Cb(~n)
that
an
Xe30,1s
which
For any denote
state
uniqueness
us i n t r o d u c e
denote
are H o l d e r
measurable
Moreover,
, ~
i/2~
assume
result
the
solution
with
Z mapping
u~u and Wu a r e
(1.1)
of all
exponent
~n
into
equation
in C (~n)
in
every
conditions
in
and A in ]O,i[
u
for
functions
~.
s
functions
the following
of
:
C~(fl~n) t h e s p a c e
continuous
of s y s t e m
to
notations
t h e s e t o f all
that
the c o e f f i c i e n t s
by
function
by C 2 b'X(~n,z)
D~Dju
some
C 2 ( ~ n) i,j,
are
such
ISi,jSn.
fulfilled
by
in (4.2)
is
:
(C1) T h e v e c t o r f i e l d s X o , . . . . , X m , X i a r e in C~(~n). (C2)
The second
elliptic
with
(C3)
Fop every
(C4)
There
order
differential
an e l l i p t i c i t y x e ~ n,
exist
IX(x)[
0~s163
L=l
S ~
and
operator
constant
c>O
L defined
denoted
a.
for
some
constant
such
that
:
~ in
[0,I[.
L,j=i
D.h (C~)
For every
i,j,iSi,jsn,
the f u n c t i o n s
D~Dih
~
I/2 and
(I + h x) are
i + hz
in C~(~n).
Thus, a c c o r d i n ~ w i t h t h e o r e m 5.2 from [4], we can f o l l o w i n g u n i q u e n e s s r e s u l t to t h e s o l u t i o n o f e q u a t i o n
THEOREM
equa*ioa
4.2
: Se~
(4.1)
~(x)
has a u n i q u e
l" b (~n))
in C ( s
: i + h 2zf x )
n
~or
solution
p
every
such
C ( [ O , T ] ; C 2,1" b (~n,Z))
x in ~n.
state (4,1)
the :
Suppose
that
~ha*
e x p ( M ' ~ M) p.
for
every
H'
is and
O<X'
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(to
appear).
[ 3 0 ] H.J. S u s s m a n n , R i g o r o u s r e s u l t s o n t h e c u b i c s e n s o r p r o b l e m , in : H a z e n w i n k e l and W i l l e m s , e d s . , S t o c h a s t i c s y s t e m s the mathem a t i c s of f i l t e r i n ~ and i d e n t i f i c a t i o n and applications, ~eidel D o r d r e c h t (198&). [31] H.J. S u s s m a n n , L e s E q u a t i o n s d i f f E r e n t i e l l e s du f i l t r a t e non l i n E a i r e in : O u t i l s eL m o d E l e s m a t h ~ m a t i q u e s p o u r l'automatique, l ' a n a l y s e d e s y s t ~ m e s eL le traitement du signal vol 3 eds. C.N.~.S. s Z.
41983)
639-648.
H. Zakai, On t h e o p t i m a l f i l t e r i n ~ of W a h r s c h . verw. G e b i e L e 11 ( 1 9 6 9 ) 230-243.
diffusion
processes,
GROUP THEORY AND CONTROLLABILITY OF PARTIAL DIFFERENTIAL CONTROL SYSTEMS J.F. POMMARET Centre de Math4matiques (CERMA) Ecole Nationale des Ponts et Chauss&s (ENPC) La Courtine,93167 Noisy-It Grand Cedex,France
S U M M A R Y In the case of a classical control system of the form ~ = a(x) + b(z)u it is known that one can introduce the weak control distribution A w e spanned by the control Lie algebra C generated by the vector fields a, b on one side and the strong control distribution A s c such that Awc = a + A s c . The control system is then said to be controllable i[ Awe has maximum rank. However, in the linear case, when a(x) = Ax, b(z) = ]5 , the well known criterion of controllability just amounts to check that Asc has maximum rank. The purpose of our communication is to prove that the confusion made between A w e and Asc can only be understood through group theoretical arguments and a new definition of controllability which does not depend on the state internal representation and is also valid in the framework of partial differential control theory, that is whenever inputs and outputs are related by a (linear or nonlinear) system of partial differential equations (PDE) with any number of independent variables and any order. In this framework, the control Lie algebra is just the centralizer of the Lie algebra of the biggest group of transformations of the outputs preserving the control system for all possible inputs. A link with differential algebra is given and many explicit examples illustrate the main results.
Classical control theory is the study of the relations existing between input and output signals defined by systems of (linear or nonlinear) ordinary differential equations (ODE). Recently, we have extended most of the concepts involved (controllability, observability, invertibility, state, minimal realization, feedback, model matching, ... ) to partial differential control theory, namely to the study of the relations existing between input and output signals defined by systems of partial differential equations PDE). A summary can be found in (6) while the complete exposition appeared in (7). In order to give an idea of the problems involved, let us quote two examples that will point out the importance of these new techniques for appiications.
272
E X A M P L E : Maxwell equations with source (8). r o t E + O B / O t = 0, d i v B = 0
r o t H - OD / Ot = j, d i v D = p
Independent variables are (x 1, x 2, x3,t) for space-time, inputs are (p,j) while outputs are (E, B, D, H) related by convenient constitutive relations. It follows that the input must satisfy Op/Ot + divj = 0 . E X A M P L E : B6nard cells (4). When a thin layer of liquid is conveniently heated, stationary convective hexagonal ceils may appear. The independent variables are (x 1, x 2, x 3) for space while the 5 unknowns are temperature 0, pressure ~" and speed v = (v 1, v 2, v 3) related by the 5 Boussinesq PDE : divv = 0
A v -- aOg -- b~77r = 0
AO - cg.v = 0 where a, b, c are experimental constants and g = (0, 0, 0, - g ) is the gravity. If one does want to control the system by considering 0 as input and (Tr,v) as outputs, it is not evident at all that 0 m u s t satisfy : AAAO
-
9~ac(O~ + 0~2)0 = 0
otherwise no convective cell will ever appear. Also, it is highly non evident at first sight that each unknown is also satisfying one decoupled PDE of order 6 at least. The techniques involved in solving these problems (jet theory, homological algebra, diagram chasing, ...) are quite too sophisticated (though necessary I) in order to be described here and we refer the reader to our previous books (5, 7). However, we shall try to show simply how group theory can interfere with this scheme, while clarifying a few confusing concepts introduced in classical control theory because of their functional analysis origin. In particular,we shall focus on controllability but other concepts may be treated as well. In any case,the reader m u s t pay a t t e n t i o n to the following computations not f o r themselves but rather as examples o f a general procedure that can be applied on any system of PDE according to (5).
273
The confusing situation is as follows. If one considers a linear control system of the form :
= Ax + Bu with inputs u and n outputs x , we successively notice : 1) The use of an underlying dynamical system seems essential for defining controllability through the possibility to fill in the whole output (state) space close to x by varying the input u into the vector field (Ax + Bu)O,: . 2) However, though this definition is of a functional analysis type, the test to check it is purely formal, namely the system is said to be controllable if and only if :
rank(B, A B , ..., A ~ - I B ) = n
Of course, whenever PDE are used instead of ODE, no dynamical system can be exhibited and the previous definition/test of controllability no longer exists (See the example of B6nard cells !).Also, it does not seem possible to define controllability by refering to the i n p u t / o u t p u t relation only, even in the case of observable systems (where the knowledge of output is equivalent to that of state), as the exhibition of a dynamical system just amounts to introduce the state explicitly. Hence, even in the linear situation, we may have a doubt about the classical definition of controllability (but no doubt for the test !) as, indeed, it must be changed if we want to generalize it, or looses its importance if it cannot be generalized. The purpose of this communication is to prove that a general concept of controllability does exist in the partial differential framework, though quite different from the classical one, even if the test involved in order to check it is the same in the linear case. A t that time, we shall see how group theory interfers with this more general definition and allows for a better understaalding of a few differential geometric results that can be found in the litterature (1, 2, 3). In order to have a feeling about the confusion existing in the definition/test of controllability, we must turn to nonlinear systems as everybody agrees with the definition/test for linear systems. Let us consider a nonlinear control system of the form :
= a(x)+b(x)u where a, b are vector fields on the output (state) space. The differential geometric appoach to such a system brings out the weak controllability distribution A w e generated by a, b and all their brackets, that is to say A w e is the smallest distribution spanned by the control Lie algebra C of vector fields generated by a, b over the constants. By analogy with the linear situation,authors use to say that the control system is (locally) controllable if
274
and only if : rankAwc
= n.
However,an easy computation shows that such a definition/test does not agree at all with that o f the linear situation.Indeed, using indices i , j , k, l for o u t p u t (state) and r, s for the input,we have at once: a ~ . A~x',b~ . . 8 ~ , ( .[ a , b ] ) ~
A i' Br,' a = A x , b = B , [a, b] = - A B ,
...
and we discover that the linear criterion does only correspond to the strong controllability distribution A s c obtained from A w e by taking out a in order to have A w e = a + A s c It must be noticed that A w e is introduced by pure convenience by all the authors (1, p. 732 ;2,p. 253). It is also not explained why the equihbium condition x = 0 should be taken into account, especially whenever one looks for a global (nonlocal) definition of controllability, that is one valid at any x in the output (state) space. In any case, again, no one of these two definitions/tests can be generalized for P D E ! As a byproduct, one of the results of this communication will be to explain this confusion by means of group theoretical arguments, while extending all these concepts to the partial differential framework. Let us first sketch our purpose with a simple illustrative linear example (also see 7, p. 525 for another linear example) proving that group theory has to do with controllability. EXAMPLE:n = 2, a = cst
As d e t ( B , A B ) = 1 - a, this system is thus controllable if and only if a # 1 . Let us now look for the biggest group of transformations y = f ( x ) letting the system invariant f o r any i n p u t . We must have :
,)' = (=~ + u)ay'/a~' + (,~=' + , , ) a r /ax ~ = ,2 + u
y2 = (=2 + u)ay2/a~, + ( ~ ,
+ u)ay~/a=2 = ,~r + u
As this must hold for any u, we finally get the 4 PDE: z 2 O y l / O x l + azlOy~/Ox2 = y2, O y , / O x I + Oyl/Ox2 = 1
x2(')y2/Oxl + a x l O y 2 / O x 2 = a y 1, Oy2/Oz I + Oy2/Ox 2 = 1
275
By pure chance it is possible to integrate this system for various values of a after straightforward but tedious computations. One gets: a~0,1=~yl=x
1,y2=x 2
a = 0 =:~ yl = xl + y, y3 = X2
where A, #, u are arbitrary constants. Hence the group of output invariance depends on a and among the critical values we find a = 1. R E M A R K : I n general, it is not possible to integrate explicitly the system defining the group of transformations (pseudogroup) and it is for such a purpose that the formal theory of systems of PDE must be used for obtaining formal tests. Most of the computations can be done on symbolic computers (MACSYMA, REDUCE,...)but the corresponding algorithms are not known by engineers or even mathematicians. The pioneer of the formal theory of PDE has been D.C. Spencer in USA between 1960 and 1975. R E M A R K : T h e groups of transformations found in the example are Lie groups with a finite number of parameters (0 when a r 0,1 ; 1 when c~ = 0;2 when a = 1). However,in general, a partial differential control system may not be invariant by Lie groups but rather by Lie pseudogroups (like the one of volume preserving transformations). The general complete study is done in (5) and leads to delicate problems in the Galois theory of PDE or differential Galois theory. We are now ready to produce our main announcements. Using the methods of (5) it is possible to determine the system of PDE defining the biggest Lie pseudogroup F = FOUT of output transformations preserving the control system independently of the inputs. Its Lie algebra @ = @OUT may be obtained easily by linearizing the PDE defining F (exactly as in the case of volume preserving transformations where the corresponding Lie algebra is made by divergence free vector fields). In general @ is infinite dimensional. Now, we may look for the centralizer C(O) of O , that is to say the biggest Lie algebra of vector fields commuting with all vector fields of @.When F is transitive, that is to say when any couple of points in the output space can be joined by a transformation of F,it may even be proved that C(O) is a finite dimensional Lie algebra with dimension at most the number of outputs (5). D E F I N I T I O N : C ( @ ) will be called the control Lie algebra of the partial differential control system and the distribution spanned by C(O) is integrab!e because of Frob6nius theorem.
276
R E M A R K : T h o u g h the preceding computations can be done even when no dynamical system appears, they cannot be done even in the simplest situations (see (7) for many explicit examples) without the formal theory of Lie pseudogroups. Now,for any classical control system of the form :~ = a(•) + b ( x ) u , through a computation similar to that of the preceding example,we discover that @ is defined by the conditions [| a] = 0, [@, b] = 0 Which do not determine, in general, a formally integrable system of P D E (5, 7). This means (the reader may check it on the example) that, by means of derivations and linear eliminations, it is possible to find new first order PDE: this fact explains why one must introduce the successive brackets. From the Jacobi identity, we infer that [0, C] = 0 and thus C = C ( 0 ) ,a fact leading to exhibit A w c without a single reference to any concept of controllability whatsoever. Moreover, we have: P R O P O S I T I O N : I n the case of a linear control system of the form ~: --= Ax + Bu, then 0 = 0 if and only if the system is controllable and detA 7~ O. More generally, according to (J.F. Pommaret,Systems of P D E and Lie pseudogroups,Go and Breach,New-York,1978,p. 233), we have: D E F I N I T I O N : A Lie algebra of vector fields on a manifold is said to be non-stationay if the Jacobian matrix of its stationary functions (the functions expressing the relations of linear dependence among the generators) has rank equal to the dimension of the manifold. P R O P O S I T I O N : I n the case of an afflne control system of the form 2 = a(x) +b(x)u, then 0 = 0 if and only if A w c has rank n and C is non-stationary. E X A M P L E : I n the preceding linear example,as detA = - a , we obtain O = 0 r 1 and c~ r 0 .
c~ #
E X A M P L E : x 2 ~ 1 = u with one input u and two outputs (x 1, x2),an easy computation shows that: F = {yl = f ( x l ) , y2 = x 2 / ( O f / O x l ) l f a r b i t r a r y , invertible} depends on an arbitrary function of one variable and O = {h(xl)01 - x2(cgh/Oxl)O2lharbitrary}
c(e)
= {A='0,IA = cst}
277
It thus remains to find a new concept of controllability for partial differential control systems, with a test agreeing with the one known in the linear case and allowing to exhibit
A s c directly, that is with no reference to A w e through its construction. Using new methods from differential algebraic geometry, we have proved in (6, 7) that controllability is in f~ct related to the lack of any (self)-constrained output, that is to say the control system must be said to be controllable whenever it is not possible to find any output verifying a P D E by itself, independently of the inputs. Such a point of view strenghtens and extends certain recent results obtained by J.C. Willems in the linear situation (8). In the case of a control system of the form ~ = a(x) + b(x)u one has (1. Haddak,Proceedings IFAC symposium "Nonlinear Control Systems Design",14-16 june 89,Capri,ItMy, to appear): T H E O R E M : T h e maximum number of functionaly independent constrained outputs is equal to the corank of A s c . In particular, if A s c has m a x i m u m rank, then no constrained output may exist and the system is controllable in the previous sense. It follows from this result that the really important general constructions are those of (9 and the constrained outputs, not those of A w e and Asc. In the case of systems of algebraic PDE, differential algebra confirms this formal approach by contrast to the classical one as the search for constrained outputs is nothing
else than the differential geometric counterpart of the search for a differential algebraic closure for certain differential extensions determined by the control system (6, 7). E X A M P L E : I n the Bfnard example, the vertical speed and the temperature are constrained by the same PDE of order 6, but it should be quite difficult to exhibit the control Lie algebra without a symbolic computer. More generally, one can prove that any output is constrained ! (4, 7). EXAMPLE:In
the last example before, there is no constrained output.
E X A M P L E : I n the linear example already used,when a = 1, the output y = x 1 - x 2 verifies ~ + y = 0 and is thus constrained with transformations: a#0,1=~y--+y,a=0=~y--*y+u,a=l=~y---~(l+2A)y
E X A M P L E : I f &l = x l , x ~ = .T2, both x 1 and x ~ are trivially constrained but the biggest group of invariance may exchange them, contrary to what one could believe from the preceding example. EXAMPLE:If
k = 1, then x is trivially constrained but rankAwc = i while
r a n k A s c -- 0 in agreement with the theorem.
278
We hope that these few comments on partial differential control theory will open a new area of research, even in classical control theory, by adapting to such a simpler situation results and concepts originating from the powerful but rather unknown formal theory of systems of PDE and Lie pseudogroups. BIBLIOGRAPHY 1) 12..HERMANN, A.J. KRENER:Nonlinear controllability and observability, IEEE transactions on automatic control, 22, 5, 1977, p 728-740. 2) A. ISIDORhSur la thorie structurelle et le probl~me de la r~jection des perturbations dans les systmes non-lin~aires, in "Outils et modules math~matiques pour l'automatique ...",Travaux de la RCP 567, 1981, fiditions du CNRS,Paris, p 245-294. 3) A. ISIDORI:Nonlinear control systems,an introduction,Lecture Notes in control and information science 72, Springer Verlag, 1985, 297 p. 4) L. LANDAU, E. LIFCHITZ:Hydrodynamics,MIR, Moscow. 8) J.F. POMMARET:Differential Galois theory, Gordon and Breach,New-York, 1983, 760 p. 6) J.F. POMMARET:Probl'emes formels en th6orie du contr61e aux d6riv~es partielles,C. R. Acad. Sc. Paris, 308, I, 1989, p 457-460. 7) J.F. POMMARET:Lie pseudogroups and mechanics, Gordon and Breach,New-York, 1988, 590 p. 8) J.C. WILLEMS:Models for dynamics,preprint University of Groningen, to appear in Dynamics reported.
ALGEBRAIC CHARACTERIZATION OF INVARIANT ZEROS AT I N F I N I T Y F O R G E N E R A L I Z E D S Y S T E M S BONILLA E. Mois~s (*), MALABRE Michel Laboratoixe d'Automatique de Nantes, URA CNRS 823, 1, Rue de la No~, 44072 NANTES CEDEX 03, FRANCE
Abstrar ," We present here a Refined Structure Algorithm for Generalized (or Singular, Implicit, Descriptor,...) Linear Systems which separates some structural information coming from the "purely algebraic" part of the system and that coming from the "purely dynamic" part ; more particularly, this algorithm allows us to provide new algebraic characterizations for the four different types of invariant zeros at infinity.
INTRODUCTION : The set of the zeros at infinity is a very important structure, the knowledge of which is crucial in the solution of many control problems (see for instance [CLM 89] in the proper case and [CP 84] in a more general setting). In the case of generalized systems, it is well known, since the work of Rosenbrock [R 74], that all the zeros at infinity do not have the same nature. We shall deal here with the four types of invariant zeros at infinity which have been introduced in [LL 89]. Note that, for the time being, the distinction between these four classes comes from an algebraic setting, in the context of a canonical forna. More precisely, the separation in [LL 89] is not given in terms of transmission and input / output decoupling properties. In fact, this is actually an interesting direction for future results : since transmission structures are of great importance in many control problems where input / output behavior is fundamental (like, for instance inversion [CP 841, [WS 85], [CPW 88] .... ), geometric characterizations of transmission zeros at infinity will be of great importance. But this will not be discussed here. The aim of this paper is to provide an algebraic characterization for these four types of zeros at infinity with the help of some new Structure Algorithm (Refined Structure Algorithm). This algorithm comes from the work of Silvernlan and Lewis (see [S 69] [LB 87]) and the proposed refinement amounts to separating the "purely algebraic" part of the system from the "purely dynamic" one and, at each step, when row compressions are performed, to keeping memory of the rank increase due to each part. This is equivalent to the simultaneous use of two Lewis' algorithms in parallel. Section I is devoted to the notation and the geometric preliminaries. Section 2 is the main section where the Refined Structure Algorithm is detailled and its use in the characterization of the zeros at infinity enhanced (Corollary 1). For sake of clarity, most of the technical developments are sent in Appendix.
1 PRELIMINARIES :
1.1 Notation : ~ , ~ . . . . . denote real vector spaces (with elements x,y,...). For a given map M : ~ -->
(*) M. BONILLA is sponsored by CONACYT-MEXICO
280 When x i are vectors, I. (x i} J denotes the vector space spanned by the xi's. card{.J stands Ior the number of elements in {.}. 1.2 Geometric Preliminaries : We shall deal with generalized (or singular, implicit, descriptor,...) systems described by :
(1)
/E~(t)
= Ax(t)+Bu(t)
,t_>0,
l y(t) = Cx(t) where x(t) ~ ~ --- IRn, u(t) e r given as follows :
(2)
= [Rm, y(t) E qJ = IRP, rank E _
"U" --0, . ~
with the usual assumptions : (3)
KerB=0andImC=qJ.
We shall see in the sequel that the following decompositions for ~(~ and . ~ will play a key role : (note t h a t . ~ , , ~ , ~ , 81 and 8 2 are not unique)
{
'.z~_.
(4)
= ( I m E + Im A + Im B) (9 . ~
Im A = Im A n ( I m E + Im B) (9 o~o
.Im
B = Im B n I m E
(9
which implies : ~_~.
(4') and :
f
=
ImE
(9 ~ ' ~ 0
N o
~ (9~-i
N__1
&
(gN_
~
(5)
=
8 1 (9 K e r A (9 8 2
8 1 (9 K e r A = A - l ( I m Ker A (9 8 2
with :
B + Im E)
= A-I.,z~:.I A
A
This decomposition (via ~ ) allows us to define a new system ~ , A, B, C) for which the state equation no longer has an algebraic part :
PoA:
(6)
~X~ ---~.~
P0 B : ~J' _.q, C=C where :
:
~
281
(7)
PO: - ~ ~ ~ = ~ / ~
is the canonical projection onto._~ / ~----0"
The following non increasing algorithms will be used : (8)
c l f 0 = S ~ , c t f i t + l = ~ e r ~ n A - l ( I m B + E q f It)
It>0,
where, throughout the paper, fig, is taken as : (9)
~ = Ker C
and : (10) ~ 0 =
.~K,, ~lJ'+l= if(, h A - 1 (IrnB + E r It )
It>0.
These algorithms obviously satisfy : qf0 D ~.d0 D q f l D ~ 1 ~ ... D Clf~tD r
D ...
Hence the respective limits, qf* and r are equal. This limit is well known as being the supremal (E,A,B) invariant subspace contained in ffrv IV 81], [O 85]. Both algorithms will be used hereafter in order to describe some precise structure of (1). A
A
A
A
When dealing with (E, A, B, C), the associated recursions (8) and (10), say e~eit and ~0it obviously satisfy (see for instance (12)): ~ 0 D q~St0= ~V'1 = ~ 1 = ~ 2 ~... A
A
A
A
Hence, for (E, A, B, C), we shall only deal with : (11) ~ , e 0 = ~ , ~ , e I t + l = 5 % c ' ~ - 1 ( h n B + E ~ f I t )
It>0.
We fwst have the following result : Lcmma 1 : The following algorithms are equivalent to (11) (see (4)-(5)) : (12) ~ , e 0 = ~ ,
c~eit+l=,~nA-l(imB+E~lfit + .~)
(13) ~ l f 0 = ~ , c ~ e I t + l = f f ( , n [ i - l ( I m B + E C ~ e I t ) r Proof :
32 ]
It20, It>0.
i) the equivalence between (11) and (12) is direct since Ker P0= . ~ . ii) for the equivalence between (12) and (13), we shall first prove that, for any c~c ~(~ :
(14) A - I [ ( I m B + E ~ ) ~ . C ~ I ] = A - I ( I m B + E ~ ) $
$2 "
For that, let x e A-l[(Im B + E~) 9 -~1 ] ' then, because of (5), there exist Xle g l @ Ker A, x2e 32, t ~ ~, b ~Im B and XlE_~ 1 such that : x = Xl~ x 2 and AXl+ Ax2 = (b+Et) $ x 1, that is : Xle A'l(Im B + Er
and x2e A'I(_.~_1 )=: Ker A 9 82 , which imply :
A-l[(Im B + ECg) ~ -~1 ] c A-l(Im B + EcC) + Ker A 9 ~2 = A-l(Im B + E~) $ 82 . On the other hand, A'l(Im B + E~g) 9 82 c A'l(Im B + Ec~) + A-I(_.~I )c A-l(Im B + ECg+_~l ), which ends the proof of (14) and therefore that of ii) * The relation between algorithms (8) and (13) is described in the following Lemma :
282 Lemma 2
:
(15) ~ 0 nr Proof :
~cr c!,eIt nqATix+l=,cf(, n A -1 [( Im B + ECV'it-1)n( Im B + E r
Ix > 1.
First note that q.e 0 n~121=: ~(~ nSes = ,5~. Next, from (8) and (13), it follows :
q e Ix n~.eIt+l=5% n [A -1 ( Im B + E q ? I x - 1 ) l n [ a -1 ( Im B + E ~?It) @ g2 ]" On the other hand, let x 9 [A -1 ( I m B + EqfIt-1)] n [A -1 ( Im B + E r s 11 e A - l ( I m B + E q
?It-1) C 81 ~ K e r A , Sl2 9 A " I ( I m B + E
$ 8 2 ], there then exist
~eIt) c 81 ~ K e r A a n d s 2 9
that : x = s 11 = s12 + s2, which implies : s2= s 11 - s12 9 (81 ~ Ker A ) n $2 =: 0 (remember (5)). Therefore, x e A -1 ( I m B + Eq2 I t - l ) n A -1 ( I m B + E ~ i t ) and (15) follows,
2 STRUCTURE ALGORITHM AND ZEROS AT INFINITY : In the proper case (rank E = n), the infinite zero structure only depends on the transfer matrix of the system (see for instance [CD 82]). In that case, this structure at infinity has been characterized algebraically with the help of the famous Silverman's Structure Algorithm [S 69] (see [SK 83]). In the case of generalized systems, there exist zeros and poles at infinity, and the infinite structure is not completely characterized by the transfer matrix ; more precisely, the invariant zeros at infinity are not only transmission zeros but also include (possibly) some input / output decoupling zeros at infinity [R 74]. The Structure Algorithm has been extended by Lewis to this class of generalized systems (see [LB 85]) and it is thus possible to define some kind of structure of zeros at infinity directly from this Structure Algorithm in a similar way to that of [SK 83]. A preliminary result in that direction has been given in [M 89]. Meanwhile, as a part of a new canonical form ILL 89], four different types of zeros at infinity have been geometrically characterized as follows. Their respective orders, say mi,ni,Pi and q i ' satisfy : I c a r d {m i = g + l } = d i m [ ( K e r E n r (16)
q c a r d {n i = I x + l } [card
It) / (Ker E n q,e Ix+l)[ VIx >_ 0
= dim[(ImB n E"~ It-1)/(imBn
{Pi = g+1}
[,.card {qi = It+l}
Eq, e It)]VIx_> 1
d i m [ ( I m B n Eq,~ It) / (Im B n E~fa9- Ix)[
Vit >_ 0
d i m [ ( K e r E n q,e Ix) / (Ker E n r
VIx _> 0
Ix)[
The number of n i equal to 1 is given by : card{n i = 1 } = dim [ Im B / ( Im B n Im E)] = dim ( ~ ). For more details, see [LL 89]. Recall however that, for the time being, there exists no dynamic interpretation of these lists mi,ni,Pi and qi in terms of transmission or input / output decoupling properties. The main objective of this paper is to provide an algebraic characterization of the four lists described in (16) with the help of the following Refined Structure Algorithm (see Corollary 1) : Refined Structure Algorithm : Initialization (k=O) :
283 (17.a) T 0 [ E A B ] = [ E1 A1B1]
with TO a maximal row compression on E
(18.a) S0[~O0 ~COc]=[ DI00~0'Cl]
w~ ~o, ma~m~ ~ow com,~,~o~ ~176 [~0 ]
(17.b) I[ E A ; ] : [ E'I AI BI] (18.b) I [ O C ] = [ O ^
^
A
C1]
^
A 0, 13(i, D 1, and C 1 are empty. Iteration k :
j
(19.a) 0 (20.a)
0
Sk [ D k
Ek+i Ak+l Bk+l] withTk amaximal row compression on
0
7 Dk+x ck+l ]
with Sk a maximal row compression on []/Dk[
LB--kj
Ak ^ A A ^ Bk E k + 1 Ak+ 1 Bk+ 1 with Tk a maximal row compression on (19.b) ,', TkL 0~ 0 = 0 Ak B__kJ 0
(20.b)
(19.c)
A [o,c,I D,,, ,+i
AAEA 1
Sk B-k Akj ^ Ek Ek+l
0
Ck+ I
0
0
[A1
^ . Dk with Sk a maximal row compression on ^
L--BkJ
] =Zk
(20'c) ]I ^= D Ak 1L D k + Note that this proposed algorithm is exactly Lewis' one, but applied simultaneously (in parallel) to both systems (2) and (6). Then, the following hold for k >_.0 and directly follows from [M 89] and [LB 87] : Fac~ l, : f Ker Ek+ I = Ker E ~ qf, k (21) Ker Dk+ l = B ' I ( E qfk) qy, kr KerCk+ I = q,ok+l
284
Ker Ek+l (22)
= K e r E n c~ek
Ker Dk+l
= B-I(E
~/'k~_0)
~l-ekn Ker C^ k + 1 = d~ek+l A
for (22), note that Ker E = Ker E and Ker P0 = -~---0" We are now in a position to establish our main result : Theorem 1 : For any k _> 1, the following hold : A ,'~.k- 1 (23) Ker E k n Ker E k + l = Ker E n (24) Ker D k n Ker [ ) k + l = B-I(E ~ k - 1 ) A k (25) qfk n Ker C k + l = The following Corollary is thus immediate and provides an algebraic characterization (through our Refined Structure Algorithm) for the different types of zeros at inf'mity of (I) recalled in (16) : Corollary_ 1 : card { m i = ~t+l } = rank (El.t+2) - rank ( Z ~ + l )
"v' ~ > 0
card { n i = ~+1 } = rank ( Dlx+l ) - rank ( Atx) card {n i = 1 } = rank B(_B0)
k/~t > 1
card { Pi = p.+l } = rank ( A~t+l ) - rank ( D g + l )
V l.t > 0
card { qi = p.+l } =rank (E~+I) - rank ( Ep.+l )
'v' ~ > 0
The proof of Theorem 1 deeply relies on the following Lemma which depicts some interesting distributive properties and the proof of which is given in Appendix : Lemma 3 : For any k > 0, the following hold : (27) Ker E n (qfk + ~ , k + l ) = Ker E n cv'k + Ker E n ~ f k+l (28) Im B n (EC~"k + E~ ek+l) = I m B n Eq7 k + I m B n E ~ k+l (29) , ~ k = clpkn~lek+l Proof of Theorem 1 : i) ii)
(23) follows from (21.a), (22.a) and (29) 0 From (21.b) and (22.b), it follows : A
Ker D k n Ker Dk+ 1
-----
1
A
B- [ Eclpk-ln (E q ek ~ .r A
A
A
On the other hand, for any x e E%ek-ln (EClf'k ~ ~,q3), there exist v ~ q ek-1, v E c f k and -~)~-~:0 A
A
such that x = Ev = Ev 9 -~0" Then : -~0 = Ev - Ev ~ Im E n ~
=: 0 (remember (4')).
Therefore : E q f k - l n (E~ ek ~ ~...~_0) = E q / ' k ' l n EC~ k , which implies : KerDk nKer Dk+l
= B - l ( E q f k - l n E~l'ek) = B - I [ E ( q f k - l n c~k)] =
B-1 E ~ k - 1
, because of (27) , because of (29),
285 which establishes (24) 0 iii)
From (22.c), it follows :
clTk nc~ekc~ Ker
Ck+l= q2k n c~k+1
^ 1= then : cv'k n Ker Ck+
~k
(since
~k
c
~k
for all k and because of (29)), which ends the proof of
Theorem 1 .
CONCLUSIQN : We have presented here a refined version of the Structure Algorithm introduced by Lewis (See [LB 85]) in the case of generalized systems. This refinement amounts to performing simultaneously Lewis'Algorithm onto the given system and on some projection of this system, where the algebraic part of the state equation is eliminated. In this way, we have been able to characterize algebraically all the four types of invariant zeros at infinity of the system, which are geometrically described in iLL 89]. Furthermore, we have extended the well-known algebraic characterization of q ek (see (21.c)) [LB 87] to ~fa9-k (see (25)) : each step of these algorithms (8) and (I0) can be computed as the Kernel of some map derived from this Refined Structure Algorithm. Connections between the geometry of the system and that of its projection have also been established (see for instance Lemma 1 and (29)). Our future objective will be to use these results in order to derive a left inverse of a given (left invertible) system with a minimal pole structure at infinity, that is with the least possible number of differentiators. This minimal pole structure at infinity is in correspondence with the tran~mis~i0n ?:eros at infinity of the system (recall [CP 84]), which have to be further characterized from a geometric point of view.
REFERENCES : [CD 82] C o m m a u l t C., Dion J.M. : "Structure at infinity of linear multivariable systems : a geometric approach", IEEE Trans. on Automat. Contr., AC-27, n~ pp. 693-696, 1982 [CLM 89] C o m m a u i t C., Lafay J.F., Malabre M. : "Structure of linear systems : geometric and transfer matrix approaches", IFAC Workshop "System Structure & Control : State Space & Polynomial Methods", September 25-27, 1989, Prague [CP 84] Conte G., Perdon A.M. : " Infinite zero module and infinite pole module", Proceedings of the 6th Int. Conference on Analysis and Optimisation of Systems, Lecture Notes in Control and Information Sciences, Springer Verlag, Vol. 62, pp. 302-315 [CPW 88] Conte G., P e r d o n A.M., W y m a n B.F. : " Fixed poles in transfer function equations", SIAM J. Control & Optimization, Vol. 26, n ~ 2, pp. 356-368, March 1988 [LB 87] Lewis F., Beauchamp G. : "Computation of Subspaces for Singular Systems", MTNS' 87, Phoenix, June 1987 iLL 89] Loiseau J.J., Lebret G. : "A new canonical form for singular systems with outputs", in this book of Proceedings. [ L O M K 89] L o i s e a u J.J., Oz~aldiran K., M a l a b r e M., K a r c a n i a s N. : "A feedback classification of singular systems",IFAC Workshop "System Structure & Control : State Space & Polynomial Methods", September 25-27, 1989, Prague [M 89] Malabre M.: "On infinite zeros for generalized systems", MTNS'89, Amsterdam, June 1989 [M 73] Morse A.S. : "Structural invariants of linear multivariable systems", SIAM J. Contr.& Opt., Vol. 11, n~ pp. 446-465, 1973 [O 85] Oz~;aldiran K.: "Control of descriptor systems", P h . D . Thesis, Georgia Institute of Technology, May 1985 [R 74] Rosenbrock H.tl. : "Structural properties of linear dynamical systems", INT J. of Control, vol. 20, n ~ 2. pp. 191-202, 1974 iS 69] Silverman L.M. : "Inversion of multivariable linear systems", IEEE Trans. on Automat. Contr., AC- 14, n~ pp. 270-276, 1969
286 [SI~ 83] ~iiverman L.M., Kitap~i A. : "System Structure at infinity", Systems & Control Letters, 3, pp. 123-131, 1983 [V 81] Verghese G.C.: "Further notes on singular descriptions", JACC, TA4, Charlottesville, 1981 [WS 85] W y m a n B.F., Sain M.K. : " On the design of pole modules for inverse systems", IEEE Trans. on Circuits and Systems, CAS-32, n ~ 10, pp. 977-988, October 1985
A P P E N D I X : P r o o f of L e m m a 3 : Consider the generalization of Morse's group [M 73] corresponding to all possible changes of bases in the domain ~ (noted V), in the codomain . ~ (W), in ~U' (G) and in ~ (H) and all possible proportional state feedbacks (F) and proportional output injections (R). With each (E,A,B,C) system is associated an orbit under this group action. All equivalent systems within the same orbit have some common structural properties which are more nicely described on the particular element of the orbit called the canonical fomL As any element of the orbit, this canonical form (which is fully described in [LL 89]) is derived from (E,A,B,C) through an element of the transformation group, say (V c, Wc,Gc,Hc,Fc,Rc), namely :
(A.1)
Fc = ~A c = |B c = [C c =
W c E Vc ; where W c and V c are isomorphisms W c (A+BFc+RcC) V c Wc B Gc 9 where G c is an i s o m o r p h i s m Hc E Vc " where H c is an i s o m o r p h i s m
We shall first prove this Lemma 3 for systems in the canonical form : L~mm~ A,1 : For any k > 0, the following hold : (A.2)
Ker E c n (flec k + ~,eck+l ) = Ker E c n fleck + Ker E c n ~ c k+l
(A.3)
Im B c n (Ecfleck + E c ~ c k+l) = Im B e n Ecflec k + Im B c n Ecq~eck+l
Proof : The proof is based on both following properties : i) First of all, it is very easy, working on the canonical form (Ec,Ac,Bc,C c) (see the description of the foml in ILL 90]), to show that all the subspaces which appear in (A.2) and (A.3) can be spanned exclusively by vectors of the canonical bases of ~(~ and . ~ . ii) Next, it is relatively easy to see that for any triple ( ~ , 3 , ~ ) of subspaces of a vector space ~ , a basis of which can be chosen as a subset of a given basis o f ~ , i.e. : ff~, = {{ei }},S = {{ej }}, and cC= {{e k }}, where {ei }, {ej }and {ek }are elements of the basis of ~ , then distributlvity always holds, that is :
5%n($+~)=SZng+~n~. Therefore (A.2) and (A.3) directly follow , Now, a series of three results, which make the connection between Lemrna A.I and Lemma 3, is established : Fact A.1 : (A.4) (A.5)
Im E c = W c Im E Im B c = W c Im B
(A.6)
KerCc=Vc-1KerC
(A.7)
Ker E c = V c- 1 Ker E
Proof :
Directly from ( A . 1 ) ,
(=Vc "1 5ec,,)
287
Fact A,2 : With ~ (a.8) then :
defined in (4), let :
" ~ 0 = Wc " ~ {~
= Im Ec * ~:..c 0
~<~ c 0
(A.9)
Bc
~ c @-~:-c 1 Im B c n Im E c ~ ~ c (Im B c + I m Ec) ~ ~
1
PrgQf : Directly from (4) and Fact A.1 , Fact A,3 : For all k > 0 : =
Vc-1 cv'k
(A.10)
qfc k
(A.11)
~ag-ck
=
Vc-1
(A.12)
C~fck
=
Vc-I c~fk
gO. k
Proof : Only (A. 12) will be proved here, since (A. 11) and (A.10) are similar. First note that: r 0 = Vc-1 ~ f 0 . Now, suppose that (A.12) is true for some k _> 0, then, from (A.1), Fact A.1 and Fact A.2, it follows : ~fck+ 1= Vc-1 if.c n Vc-1 (A+B Fc+RcC )- 1W c- i (WcEd~fk+Wci m B+Wc._..~) = Vc-lc~ok+ 1 , We are now able to end the proof of Lemma 3 : Proof Qf Lemm~ ~ : i) (27) follows from (A.2), (A.7) and Fact A.3 0 ii) (28) follows from (A.3), (A.5), (A. 1) and Fact A.3 0 iii)
First of all, note that (29) holds for k = 0 since r -0 = q f 0 n e~;,1 . Now, suppose that (29) holds for
some given k > 1, that is : r = qfk-I n d~ek, then, from Lemma 2, (28) and (27) : q f k n c~ek+l = , ~ n A-l[( im B + Eqfk-l) n ( i m B + E~k)] = fff, n A - l [ I m B + E % ~ k - l n ( i m B + E ~ k ) ] = ~ e t . d k
,
THE DYNAMIC BLOCK DECOUPLING PROBLEM: A MINIMAL SOLUTION BY PRECOMPENSATION J. DESCUSSE Laboratoire d'Automatique de Nantes, URA CNRS 823, ENSM ,1 Rue de la No~, 44072 Nantes C~dex 03, France
Abstract : The purpose of the present paper is to provide a new dynamic solution to the Block Decoupling Problem based upon a procedure which extends to the block case that one previously given in [14] for Morgan's Problem. This procedure is minimal in the sense that it leads to the least infinite zero structure for any decoupled system reachable from the original one. Kevwords: Block decoupling problem ; block essential orders ; infinite zero structure ; minimal dynamic compensator.
1) Introduction . During the last 25 years, a great deal of interest has been brought to the theory of input-output decoupling for linear time invariant systems. Since the early works of Morgan[12] and of Falb and Wolovich [9], many authors have tackled this famous problem. It is rather difficult to give an exhaustive bibliography on this topic, so many papers have been published ! (Many references can be found in [10], [13] as well as in [7] for very recent ones). The theory is still in progress. Till a recent past , the general block decoupling problem remained unsolved without particular assumptions made on the system or on the feedback laws (see for instance [1], [5], [11], [13]). A complete solution with static state feedback is given in [18] When it is not achievable one can use dynamic state feedback [16] or precompensation.(See [10] for details on static or dynamic state feedback as well as precompensators). The scope of the present paper is to provide a new dynamic procedure which generalizes that one of Wang [14] for Morgan's Problem. Two of its main interests can be noticed : it leads to the least infinite zero structure for the dynamically decoupled system. Secondly, it can be extended with some arrangements to the nonlinear case [8]. Preliminary results are also presented in [6]. The paper is organized as follows : next section is devoted to notation and basic concepts. In the third section, we introduce the block decoupling problem with static state feedback. In the last one, we provide a new algorithmic dynamic procedure which can be used when static state feedbacks are not available. At last , we recall the definition of the "block essential orders" [17] , [18]. We also recall that they are the least infinite zero orders reachable for decouplable systems. 2) Notation and basic concepts We shall consider, in the sequel, linear time invariant systems described by :
289 ,T-,0 I ~ = Ax + Bu
[ y = Cx where xe ,,~ = ~n, u 9 ~ = ~m, y 9 ~ = [p,p; [p, denotes the field of real numbers. The set of natural integers will be written as I]t. Associated with 7_.0we shall consider a k-partition of the output y into k nonempty subsets of components Yi, each of dimension Pi (0 < Pi, '~ie k Pi = P). This partition induces a corresponding partition of C, written as {Ci} k. We shall assume, without loss of generality, that B is monic. The same symbols will be used to denote a map or any of its matrix representations in particular bases. Let 3 the image of 1 3 , a n d K e r C t h e kernel ofC. Let B b e a m a t r i x , t h e n sp{B}will denote 3 . For any space ~.,~, dim ~,~ is its dimension. If ely c ff-.~, ff.~/cV will denote the factor space of ~ - byC~. A set of p elements will be written as {.}p and k will be used for the (not necessarily ordered) set of integers {1, 2 ..... k}. The number of elements of a given list {.} is noted as card {.}. The symbol U will denote the union of sets with repeated common elements. The rank of (C, A, B) defined as the rank of its transfer matrix will be written as p ; Pi will denote the rank of (C i, A, B). In what follows, we shall use cFi* the maximal (A,B) - invariant subspace contained in Ker C i , also written as sup I(A, B, ker Ci), fi~i~ that one contained in nj~ i Ker Cj, while $~'i" will denote the maximal controllability subspace contained in ~i*Recall that %Pi* and "-~i* are the respective limits of "I.S.A." and"C.S.A." defined by : I.S.A. C.S.A. flYi 0 = ~
..~i 0 = 0
cFiP- = Ker Cin A "1 ( s
"1 + 3)
SLip. = ~i" 'q (A ,..%ip.-1 + 3)
We shall also use "C.I.S.A." defined by: 80 = o 8p. =
3
+ A(Ker
C N ,~,p._t)
of which the limit is written 8 * = i n f 0-.t (C, A, 3 ) Now, we recall possible definitions of the infinite zero structure of (C,A,B) (See for instance[2], [7]). Define P~+I = d i m ( 3 nCVp./ 3 n q P * ) # > 0 (2.1) = dim(C~/ where
{~PZ}
and %P* are
C~p.) related
p. > 0
(2.2)
to sup I(A, B, kerC).
The infinite zero structure of (C,A,B) is the list of integers {pp. } , or equivalently {n'i} defined by n' i = card {pp. ~ i}. has Pl infinite zeros of which the orders are the n'i's. Recall that Pl = P.
290 3) The Static State Feedbaqk Block Dec0vDling Problem (SSFBDP) SSFBDP is defined as follows: "For a given k-partition of the output {Ci} k find necessary and sufficient conditions under which there exist static state feedbacks u = Fx + ~-i~k Gi vi such that v i controls Yi without affecting yj, j#i, i ek". This problem remains unsolved in its most general setting. Up to now, the only available results deal with special situation when the static state feedback is "regular", which means that G = [G 1 I...I Gk] has rank m, or when it is "restricted", in other words when u = GFx + Gv, [5],[11]. Theorem 3.1 1"13] : Let ,T-.0 and a k-partition of the output be given. Then the regular SSFBDP is solvable if and only if : = )lie k ~ c~ ~i
(3.1)
As an immediate consequence, we have Corollary :~,1 : Let T.0 and a k-partition of the output be given. Then the regular SSFBDP is solvable if and only if : (3.2) = Zi~k ~ n n j ~ i 97*J The proof is not difficult to perform and is left to the reader. Lemma ~).1 : Let E,0 and a k-partition of the output be given. Then, if P = T-ie k Pi we have : = ,T-,i~k ~ n ~i r
ely* = c~iek clYi
(3.3)
Proof : It is quite similar to that one given in [3], for right invertible systems, turning Pi into pi.E] Now suppose that (3.1) or (3.2) fails. One can search for dynamic solutions. For this purpose an auxiliary system is joined to ~0. It is defined as T.a ,
/ xa = D l x + D2x a + Ev 7.a
] [ u = F l x + F2x a + Gv
where x a e "~a = I~n~ and v e Ig m. We shall say that decoupling is achievable with dynamic compensation if it is possible to find ,T-,asuch that the composite system T.0 x ,T-,adefined on [pn+no can be decoupled using static state feedbacks. Note that this problem is always solvable, under non restrictive conditions[13] , [15].
291 Our purpose here is to provide a new solution to this problem, when p = ,T_,iek Pi. This new solution will be based upon an algorithmic procedure, which generalizes that one proposed in [14] for Morgan's Problem (k=p). From it, it will be possible to define the essential orders of the output related to the given k-partition {Ci} k. 4 I The Dvnamic Block Decoupling Problem Consider the following [14] available when k=p.
Dynamic Algorithm (D.A.), which generalizes that one of
Steo 1 " From the given data ({Ci} k, A, B) compute ~1 = -T-,iek$ n nj~ i c]yj . If ~1 = ~, Stop ! If ~1 c ~,write ~ =
~1 @ ~2- Define Gl and G2 such that ~ I = I m B G 1
and ~ 2 = I m B G 2 Steo 3 : Consider ,T-,0and write u = G1 Vl + G2 v2 = A x + BG 1 v 1 + BG2v 2 assume that dim v 2 = q, and put integrators in series with the q corresponding inputs of v 2. One obtains rewriting v 1 as w 1 :~
~x + BG 2 v 2 _--
"w1 +
0
y = C(,T_,1
=Cx
Steo 4 - Go to step 1 and resume the procedure with T.1 instead of ,T-,0 Theorem 4.1 9 Let .T_,0 and a k-partition of the output be given. Then, if p = ,T-,i~kPi, "D.A." "converges" after a finite number of "loops", say N, towards an extended ,T-,Nfor which SSFBDP is solvable. Proof - The central idea of the proof amounts to show that dim n i c k c~i* (,T_,l) < dim ni~kClYi*(,T_,l+l), where I denotes the I-th loop of "D.A.". Indeed, if this inequality is true, after a finite number of steps say N, we shall fulfil clP*(T.N) = ~i~ k_ ClPi (,T-,N), and lemma 3.1 does the rest. To show the above inequality it is sufficient to perform the ;roof for I = 0. This is done in Appendix 1. [3 We shall now pay attention to the structural properties of TN, and mainly to its infinite zero structure. For this purpose, consider the infinite zero structure of the
292 subsystem ( Ci(T.N), A(,T_.N) + B(T.N)Fi(T.N), B(T.N)Gi(.T-,N)), where Fi(T.N) is any friend of ..9~i*(.T_,N)and Gi(,T_,N) satisfies Im B(,T_,N)Gi(,T_,N)= ~(T.N)r~,-q:~,i*(T.N). We shall note it simply as ~ (Ci(,~,N), ..%i*(T.N)). It can be computed using either (2.1) or (2.2). We shall note it { 5i1~ (,T-,N)} or equivalently {13iv(,T_,N)} with 13iv(,T_,N)= card { 5il~(,T_,N) > v}. The infinite zero structure of ,T-,Nwill be noted 0 i ~k (Ci(7-N), "-'%i*(,T-,N)) and, with the above notation , is defined by {T.i~kSit~(T.N)} or equivalently , from the above counting procedure , by 0 i ~k{ 13iv(,T--N)}" Lemma 4.1: Assume that p = T.ie kp i. Let N denotes the final loop of "D.A.". Then the infinite zero structure of ,T-,Nis not different from ui ~k (Ci('7--'0), "9~i*('T--0))' Proof: since .T-,N is decouplable using regular static state feedback, it follows that [4] ,T_,~(7.N) = Oi ~k,T_,~ (Ci (T.N), ,-gLi*(,T_,N)) Assume that C i ,.9:~i*P.(,~,0)=C i (T.N) ..9~.i*#(~N)
(4.1)
From (4.1) one can deduce that SIP'+1 (,~-.0) = 5il~+1 (,T-,N)
p2.0
which states the result. The proof of (4.1) is given in Appendix 2.F] Definition 4.2117]_ . [18]: The infinite zero structure ~
(C i , --%i*(.T_.0)) is said to be the
essential structure of the output Yi, i~k. It is written as ,T_,e(Ci,A,B).
The essential
structure of (C, A , B), relatively to the given output k-partition, is defined as ui~ k (Te(Ci,A,B)). The terminology of "essential orders" comes from the following property. ProPosition 4.1 [17]. [18]: Let ~
and a k-partition of the output be given. Assume that
P = .T-,iekPi. Then if SSFBDP is solvable for some feedback u = Fx + ~iek Gi vi then .T_~o(Ci,A+BF,BGi) >_T.e(Ci,A,B )
REFeReNCES
[1] Commault C., Descusse J., Dion J.M., Lafay J.F., Malabre M. " Influence de la structure & I'infini des syst~mes lin~aires sur la solution de probl~mes de commande. APII, vol. 20, pp. 207-252, 1986
293 [ 2]
Commault C., Descusse J., Dion, J.M., Lafay J.F., Malabre M. : About new decoupling invariants : the essentiel orders. Int. Journ. of Contr., vol. 44, 3, pp. 689-700, 1986
[ 3]
Descusse J. : Sur la structure & rinfini des syst~mes lineaires d~couplables : le cas des syst~mes inversibles & droite. Outils et ModUles Math~matiques pour I'Automatique, I'Analyse des Syst~mes et le Traitement du Signal, vol. 3, Editor I.D. Landau. Ed CNRS, Paris 1983
[ 4] Descusse J., Lafay JoF., Malabre M. : On the structure at infinity of linear block decouplable systems: the general case. IEEE Trans. on Automat. Contr., AC-280 12, pp. 1115-1118, 1983 [ 5]
Descusse J., Lafay J.F., Kucera V. : Decoupling by restricted static state feedback: the general case. IEEE Trans. on Automat. Contr., AC-29, 1, pp. 79-81, 1984
[ 6]
Descusse J. : A new approach for solving the linear decoupling problems. Proc. of the 10th World Congress on Automatic Control, IFAC, July 27-31, 1987, Munich, FRG
[7]
Descusse J., Lafay J.F., Malabre M. : A survey on Morgan's Problem. Proc. of the 25th IEEE Conf. on Decision and Control, Dec 10-12, 1986, Athens, Greece ; Solution to Morgan's Problem. IEEE Trans. on Automat. Contr., AC 33, 8, Aug. , pp. 732-739, 1988
[ 8] Descusse J. : Towards a dynamic solution to nonlinear block decoupling . Preprints of the International Conference "Automatique Non Lin~aire" of C.N.R.S. , June 13-17 , Nantes France , 1988; "New Trends in Nonlinear Control Theory", LNCIS, vol. 122, Springer Verlag.
[ 9] Falb P.L., Wolovich W.A. : Decoupling in the design and synthesis of multivariable control systems. IEEE Trans. on Automat. Contr., AC-12, 6, pp.651-669, 1967 [10] Hautus M.L.J., Heymann M. : New results in linear decoupling. Proc. of the 4th INRIA Conf. on Analysis and Optimization of systems. Springer Verlag, vol 44, 1980 [11] Kamiyama S., Furuta K. : Decoupling by restricted state feedback. IEEE Trans. on Automat. Contr., AC 21, pp. 413-415, 1976 [12] Morgan B.S. :The Synthesis of linear multivariable systems by state feedback, JACC 64, pp. 468-472, 1964 [13] Morse A.S., Wonham W.M. : Status of non interacting Control. IEEE Trans. on Automat. Contr., AC-16,6, pp. 568-581, 1971 [14] Wang S.H. : Design of precompensator for decoupling problem. Electronics letters, 6, pp. 739-741, 1970 [15] Wonham W.M. : Linear multivariable control : a geometric approach, 2nd edition, Appl. of Mathematics, Springer Verlag, vol. 10, 1979
294 [16] Commault C., Dion J.M., Torres J.:lnvariant spaces of linear systems. Application to block decoupiing. Proceedings of the 27-th IEEE CDC , Austin , USA ,Dec 1988.
[17] Commault C., Descusse J., Dion J.M., Torres J.: Block decoupling invariants. Geometric and transfer matrix characterization.-IFAC Workshop on Systems Structure and Control; 25-27 september, Pague, Tch~coslovaquie.
[18] Descusse J.:Block noninteracting control with (non)regular static state feedback. A complete solution. -IFAC Workshop on Systems Structure and Control; 25-27 september, Pague, Tch~coslovaquie.
Vl>0"
"dim niek cF i (7-1+1)< dim nie k cF i (ZI) ,
It is enough to perform the proof for I = 0. To proceed we need several auxiliary lemmas. Let P 9 ,~, x ~'a ~ ~ x '~'a the projection into ~, along "~'a, where ~. x '~'a is related to T.1 , then t
*
P c[Pi (7-1) = cFi (,T-,0)
i ek
The proof of this lemma is not difficult to perform and we shall omit it. For 7.1 = (Ae, Be, Ce) above defined, we have 9 niek flPi*(Z1) = P n i e k c/Pi*(Z1) Proof 9 Start from Ae n i e k flPi*(7.1) c flYi*(,T_,l)+ ~e c nie k (Vi*(T.1) + ~e)
x[-4] :,,,n[ {[Xo] l+
,o,1,,Jll
where sp{(Xli/UlI)} =flYi*(Z1). From the respective definitions of B 1 ( $1 = 7.iek ~ *
n nj~ i clYj ) and q'Pi (7.0) = sp {(Xli)}, by lemma A1 , we have also, using a repeated action of the modular distributivity rule "
Ix4
A e n ~. (Y. 1) c n sp { iek
I
iek
} + Sa
(A. 1 )
295 Now assume that r~ie k fiYi (,7-,1) = P n i e k c~Yi*(-T-,1)-This means that there exist
onto~.generat~of niek_ctYi*(Y4)written as [~I" From (A.1) we then obtain, by projection B 2 U e hie k 1,f'i*(~O) + @1
c $ r~c~iekqYi*(.~.,0)+ $1 = $1 which is impossible by the respective definitions of ~1 and ~ 2 13
Lemma A.:~ " Assume that SSFBDP is not solvable for ,T_,0. Then, for ,7_,1 = (C e, A e, Be) above defined, we have 9 dim hie k cFi*(,T_.,1)< dim nie_k (s
PrQQf " From the preceeding lemma we know that the generators of hie k c~Yi*(,T_,l)are written asIXI 1, where 0 ~ X I e hie k cFi*(.T_.0).
LU j So
dim nie_k ~qYi*(T.1) < dim hie k cV'i*(,7_,0)
(A.2)
Assume that equality holds. Then, from (A.1), one obtains A X I + B 2 U 1e hie k cFi*(T.0) + $1 which leads to A hie k flYi*(T.0) c rhiek cFi*(T_.0) + 9 The latter implies that chic k clYi*(,T_.0) is (A,B) - invariant, and so decoupling would be achievable for TO, which is impossible by assumption. Thus, (A.2) holds as a strict inequality. [] APPENDIX 2 "PI+I "-q~'i# (~1+1) = "-%ip" (E,I)
~' I> 0,
I~> 0,
e k"
It is enough to show that this equality holds for I = 0. The rest will follow by transitivity.
i) "P .~i I~ (,7_.1)c ..,%i# (-~,0)" The proof is performed by induction. This inclusion is true for I~ = 0. Assume that it is also true for I~ - 1. Then, from CSA, "-q~'p" i (TI) = ~'i* (,7-,1) n (A e ..%i1~'1 (,7-,1) + ~e)
296 By projection on to ~. along '~'a, we have 9 P --,~ip" (7-,1) = P (~i* (7-,1) c~ (A e ,~q,iIx-1 (,~,1) + Se))
= P ~'i* (7-,1) c~ P (A e ..~i #-1 (7-,1) + ~e)
1"'3 since Ker P =/D- I [q~l
It follows, from Lemma A.1, that 9 P ~"iix (7-,1) = ~i (T0) c~ (Sp {A X I + B 2 UI} + 51)
= P ~i (7,1) c~ P (A e -~,iix -1 (7_,1 ) + ~e)
sinceKerP;t]
It follows, from Lemma A.1, that : P -..~iP" (.7_,1)= ~i* (T.0) n (Sp {A Xl + B2 UI} + 51) where
(A.3)
sp
fx,]
(A.3)
} ---b21~ ~'1 (21)
From (A.3) we have : P ..~i # (T.1) c ~'i* (-7--0) n (A -.~i p--1 (7_,0) + 5) = ..~i p- (7,0) ii) "..~i p- (,T_,0) c P ...~i I~ (7_,1 )" It is enough to show that V X I e '~iix (7,-,0), there exists U I such that
The proof is still perform by induction. The inclusion is true for Ix = 0 . A s s u m e that it is a l s o true for g-1. Then V R I a v e c t o r of ~ i I x -1 , t h e r e exist X 1 , U I, U 1 such that :
"%iIx ) RI = A X I + B2U I + B1U I e ~di* (7,0) Moreover, from lemma A.1, there always exists W I such that :
RI = Ae
+
I+
Wl
(~1)
By assumption
and so"
R--L] E ~i (~I) ('h (A e,~i -I (~'.I) +~e ) = '-'~Pi(7,1) Wi which ends the proof. 0
Minimal rational interpolation and Prony's method A. C. Antoulas Rice University & E.T.H. Ziifich and
J. C. Willems University of Groningen
Abstract. A new method is proposed for dealing with the rational interpolation problem. It is
based on the reachability of an appropriately defined pair of matrices. This method permits a complete clarification of several issues raised, but not answered, by the so-called Prony method of fitting a linear model to given data.
1. Introduction.
Given the array of scalar pairs points
(xi, Yi), i E N , xi~xj, i ~ j ,
(1)
with finite entries, we seek all rational interpolants, that is rational functions
y(x) = n(x) d(x)'
(2.1)
n, d: coprime,
(2.2)
=y,, i
(2.3)
where
which interpolate the array (1):
N_
Moreover, we wish to keep track of the complexity of interpolants, which is defined to be their McMillan degree:
deg y := max {deg n, deg d}. The following problems arise: (a) Find the admissible degrees of complexity, i.e. those positive integers ~ for which there exist interpolants y(x) with deg y = ~c (b) Given an admissible degree ~r construct all corresponding solutions.
298
The first complete answer to the above questions was provided by Antoulas and Anderson [1986] using the so-called L6wner matrix as the main tool. For further developments, see Antoulas and Anderson [1989], [1990] as well as Anderson and Antoulas [1990]. In the present paper we will discuss a novel approach to answering the above questions. This allows a complete analysis of Prony's method of modeling, which for example, although widely used in digital signal processing is not well understood. Suppose that there exists an interpolant y ( x ) = n(x)/d(x) having complexity ~; let n(x) := no + n l x + . . .
+ n ~ ~, d(x) : = d o + d l x + " ' " + d)cx ~,
with either n): or d K different from zero. The c o e f f i c i e n t v e c t o r s corresponding to n, d
are
defined as:
c~(n) := (n0
nl
...
n~),, c K ( d ) : = ( d o d I . . .
d,)'.
Conditions (2.3) imply that the coefficients ni, d i satisfy the following sets of linear equations: n(xi) - yid(xi) = O, i ~ N.
In order to write the above system of equations in matrix form, the N•
constant matrices I~:
and I ~ which depend just on the data, are defined; their i th row is [I~]i := ( 1 x i . . . x~ r ), [ l ~ j i : = - y i ( 1 x i . . . x ~ ). With the notation introduced, the above system of linear equations can now be expressed as follows:
f~,c,(n) + ft,c,(d) = O,
(3)
while the coprimeness condition n, d: coprime,
(2.2)
must be fulfilled. It readily follows that every interpolant of complexity 1< must satisfy (3), (2.2) and vice versa. The former being a linear constraint it is straightforward to deal with. The difficulty lies in dealing with constraint (2.2), since as already mentioned we are interested in keeping track of interpolant complexity at the same time. A very common approach to rational interpolation is sometimes referred to as Prony's method. It is based on the assumption that N = 2rn+l data pairs are always interpolated by a rational function of complexity m, since such a rational function contains exactly 2m + 1 free parameters. This assumption however is true for g e r e r i c , i.e. randomly generated, data only. In such a case, (I~ra l~m) has size Nx(N+I) with full row rank; therefore, up to a non-zero constant, there are unique c~:(n), c~:(d) which solve equation (3). Moreover, the coprimeness constraint (2.2) is generically always satisfied. Consequently in the generic case, one interpolant (of minimal complexity m) can be computed this way. For n o n - g e n e r i c data however, trying to fit 2m + 1 data pairs with a rational function of complexity m, may or may not give a solution. Even if a solution is obtained
299
this way, questions (a) and (b) formulated above, remain unanswered and no understanding of the coprimeness constraint is achieved. In the sequel we will expand on the approach which is based on equation (3) and constraint (2.2), and show how the system-theoretic concept of reachability allows us to take care of this coprimeness constraint. 2. The main results.
elii
Consider the pair of NxN, Nx2 matrices defined in terms of the data array (1):
X2
q
F :=
G : = ( g t g2): = x/v
Y2
(4)
N
A closer examination of the matrices 1~ and 1~: defined in the previous section, shows that they are partial reachability matrices of the above pair, containing ~<+ 1 colunms: l~=(gt
Fgt
...
F~:gl), l ~ = ( g 2 Fg2 . . .
F~g2).
Thus the reachability matrix R~:=(G
FG
...
F~G),
of the pair (F, G) is, up to column permutation, the same as the matrix
(1~
1~,0,
involved in the solution of (3). Having made this observation, the rest of the development follows by applying the powerful theory of reachable pairs, since indeed, due to x i ~ x j , i ~ j , the pair (F, G) is reachable.
(5)
The first fact we wiII make use of (see e.g. Antoulas [198t] or Kailath [1980, ch. 6]) is the existence of right coprime polynomial matrices W(x), O(x) of size Nx2, 2x2 respectively, such that (xl - F)W(x) = G|
det |
~ 0.
(6)
Because of the reachability of F, G and the coprimeness of W, O, the polynomial matrices M - F and O(x) have both one non-unity invariant factor which is the same, namely
~(x) := :l-I~x - xi). Therefore, after a possible normalization by a (nonzero) constant, det |
= g(x).
Let Oij denote the (i, j)th entry of O. The i th row of (6) yields the equation
(7)
300
(x - xi)[W(x)] i = [G]i
I0u(X) O12(x)l=[Oli(x)-YiO21(x )
] 021(x)
O12(x)--YiO22(x)3 , i E N .
(8)
022(X)J
Evaluating this expression at x = xi we obtain
el l(xi) = yie21(xi), 012(xi)=yiO22(xi), A fact to be used below is that 021, 022
are coprirne
(9)
i E N.
polynomials. Because of (7), if 021 and
022
were not coprime, their greatest common divisor would have to be a product of terms (x-xi), where the x i are given in (1). Therefore, by (9), all four entries of (9 would have the same common factor. This however is a contradiction to the fact that one of the two invariant factors of O is equal to 1. The desired coprimeness of the entries of the second row of (9 is thus established. Main Lemma. The rational function (2.1) satisfying (2.2), is an interpolant for the array (1), if, and only if, there exist coprime polynomials p(x), q(x) such that = O(x)
Ld(x) J
(10)
Lq(x) j
and a(xi) = 021(xi)p(xi) + e22(xi)q(xi) ~ o, i ~ N.
(11)
Proof. Suppose that the rational function y = n/d is given in terms of (10). Multiplying (8) on the right by (p q)', yields
- [G]i = n(x) -yid(x). Lq(x) J Ld(x) Evaluating this expression at x = x i, we obtain n(xi) - yid(xi) = 0. Moreover, due to the fact that
02i(x), 022(x ) are coprime, (11) can always be satisfied. Therefore y(x) is an interpolant. Conversely, let y = n/d be an interpolant. We will show that
IP(X)l:=(9_l(x)
t.q(x)
1
Ld(x) J = ~
lln(x)l,
-e21(x) e~(x)
La(x) j
is a polynomial vector. Because of (8), and since by assumption n(xi) =yfl(xi), it follows that
e22(xi)n(x i) - el2(Xi)d(xi) = 0 and - e21(xi)n(xi) + ell(Xi)d(xi) = O, i ~. N, which implies the desired result. Furthermore, the coprimeness of n, d implies the coprimeness of
p,q.
1"1 Remarks. (i) The above lemma provides a paramelrization of all interpolants. In particular,
equation (10) shows that every interpolant can be expressed in terms of the linear fractional representation formula
301
y(x) =
P(x)OI l(x) + q(x)O12(x) p(x)e2i(X) + q(x)O22(x ) '
which involves the two parameter polynomials p, q and the four entries of the polynomial matrix O defined by (6). (ii) The coprimeness constraint (2.2) on n, d has been expressed equivalently as a coprimeness constraint on the parameter polynomials p, q together with constraints (11). The former is a nonlinear constraint in the space of coefficients of p, q ; it is automatically satisfied in the case of minimal interpolants discussed below. Constraints (11) are linear in the coefficients of p, q. For an example, see section 5. The matrix |
[]
in equation (6) is unique, up to right multiplication with a unimodular matrix.
In order to bring the complexity of interpolants into the picture, we use this freedom to transform O into column reduced form (this means no zeros at infinity). For simplicity we will use the same symbol, namely O, to denote the column reduced version of this matrix. Let the corresponding column indices be
1(, :=deg
1(,+
The column indices of column reduced polynomial matrices have two important properties. (a) Although the column reduced version of any polynomial matrix is non-unique, the corresponding column indices are unique. (b) Because of the so-called predictable-degree property of column reduced polynomial matrices (see e.g. Kailath [1980, ch. 6]), the degree of e(x)w(x), with O column reduced and w some polynomial vector, can be either ~:1 or greater than or equal to 1(2Combining the main lemma with the above considerations we obtain the Main Theorem. Consider the pair (4) and the resulting equation (6), where e(x) is column reduced, with column indices as above. ( i ) l f 1(i
5(Y mi~) = 1(t,
(12)
is the unique minimal interpolant. Furthermore, there are no interpolants o f complexity between 1(1 and 1(2. (ii) Otherwise, there is a family o f interpolating functions o f minimal complexity which can be parametrized as follows: ymin(x) = OI2(x)+p(x)OII(X) 8(ymin) = I<2 = N - l ( 1 , 022(x) + p(x)O21(x)'
(13)
where the polynomial p(x) satisfies deg p = 1(2 - 1(1, 022(xi) + p(xi)O2t(xi) ~ O, i e N.
(14)
302 (iii) In both cases (i) and (ii), there are families o f interpolants y = n/d o f every degree ~r ~ 1r2, satisfying (10), where deg p = } c - 1(1 and deg q = ~c - }c2.
Questions (a) and (b) posed in the introduction have thus been answered. In the next section we will make contact with Prony's method by expressing the above formulae in terms of the reachability matrix of (F, G). 3. The main results and R K.
Let the entries of a column reduced O(x) as defined in the previous section be: Oij(x ) :=e~J+ O~Jx+ . . . + o ~ x K~, i , j = 1,2.
A standard result in the theory of canonical forms of pairs (F, G), see e.g. Antoulas [1981], implies the following relationships 01j(F)g t + 02j(F)g 2 = 0, j = 1, 2,
(15)
where given a scalar polynomial s(x) := s o + six + 9 9 9 + s~x ~:, s(F) stands for the (square) matrix polynomial S o + s l F +
. . . +s~F~. Furthermore, the column indices ~q and ~c9 of (9 are the
reachability indices of (F, G). The coefficient vectors corresponding to each one of the entries of
(9 are:
Relationship (15) implies
R~fKj(01i) + l~r
= 0, j = 1, 2.
(16)
Given two polynomials p, q, deg p = ~c- ~q and deg q = }c - }c2 respectively, let Sp, Sq denote two matrices having }c + i rows each, and ~q + i, ~c2 + I columns respectively. The first column
of each matrix contains, in sequence, the coefficients of each one of the polynomials, starting with the constant term. The second column is obtained from the first by shifting it down by one (thus the first entry of the second column is 0, the second is the constant term of the polynomial, etc.) Similarly, the third is obtained by shifting the second column down by one, and so on. We are now ready for a reformulation of the results of the previous section in terms of Se, Sq and appropriate coefficient vectors. Theorem. W i t h the notation introduced above, every coefficient vector corresponding to an interpolant y = rdd o f degree ~c = ~ct or ~c > 1c2, can be represented in the f o r m
where
p, q
c~:(n) = Spc~l(Oll ) + Sqc~2(012),
(17.1)
c~(a3 = Spc~1(021) + Sqc~(Oz2)
(i 7.2)
are coprime polynomials of degrees ~c-~ct, ~c-~c2 respectively, and satisfy the
cop rimeness constraint (11).
303
Remarks (i) In view of the above theorem, here is the solution procedure involving equation (3) which is the starting point for Prony's method. First, construct the full reachability matrix R N of (F, G); as already mentioned, this is the same as the matrix (I/N I1N) which enters in (3), up to column permutation (this column permutation is crucial in the sequel). Proceeding from left to right, determine the first column, say FKIg2 of R N which becomes linearly dependent on the pre-
vious ones (i.e. linearly depend
on
FJgi, j < 1'~1,i =
1, 2).
The corresponding vector, after permuta-
tion of its entries, will be [ c~'(011) ]. %(021) J Because of reachability of the pair (F, G) the next linear dependence which is needed is that of
F':2gt, where lc2 : = N - ~cl, on the previous columns of R w. The corresponding vector, after permutation of its entries, will be [%(012)
r 1" The linear dependencies of all intermediate or subsequent columns are of no interest. All solutions to the problem are now parametrized by means of (17.1,2). (ii) The kernel of (ll N I~N) has dimension N + 2. But out of all coefficient vectors in this kernel, only two are needed. Their choice is dictated by the teachability property of the associated pair (F, G). Moreover, these two coefficient vectors are the basis for parametrizing interpolants of complexity below, equal to, and above N. (iii) The matrix (Sp Sq) is very closely related to the Sylvester resultant of p, q. The difference between the two lies in the fact that the Sylvester resultant has more columns. Therefore, because of the coprimeness of p, q, (Sp Sq) in the theorem above, must have full rank. [] 5. An example. Consider the data array containing 4 pairs: (0, 0), (1, 3), (2, 4), (1/2, 2), According to (4):
i0 i [!0] 1
-3
, G:=
2
-4
1/2 It follows that the fuU teachability matrix of the above pair is
-2
304 iI R4 =
]
0 -3
0 0 1 -3
0 1
0 -3
0 1
0 -3
0 1
0 1 -3
-4
2
4
-16
8
-32
16
-64
-8
-2 112 -1 1/4 -1/2 1/8 -1/4 1/16 -1/8
Recall remark (i) of the previous section. The first linear dependence occuring in R 4 is that of the fourth column: Fg2+6Fg 1+ g2=0.
It follows that x:t = 1. Therefore according to the same remark, ~c2 = N - ~cI = 4 - 1 = 3. This means that the next linear dependence is that of the 7 th column on the previous ones. It turns out to be 2F3gl - 9F2gl + 4Fgl - 2Fgl + g2 = O. The corresponding coefficient vectors are c1(011) = (0 6)', cl(O21 ) = (1 1)',
c3(012 ) = ( 0
4 -9
2)',
c3(022 ) = (1 - 1)'.
One column reduced version of O is therefore
e(x)= [ 6x
x(x - 4)(2x - 1) 1.
x+l
-
(2.x-
1)
J
By (i) of the main theorem, since 6x and x + 1 are coprime polynomials, there is a unique minimal interpolant with McMillan degree 1, namely y~in(x) =
6x x+l "
By (i) of the same theorem, there are no interpolants of degree 2. The next family of interpolants has McMillan degree 3. It can be parametrized in terms of the second degree polynomial p
as
follows: y(x) =
x(x - 4)(2x - 1) + (.o2x2 + p l x + po)6X -
(2x - I) + (p2x2 + p ~ x +po)(X+ 1)
The coefficients of p must satisfy the constraints (11), which in this case turn out to be po + 1 s O , 2p2 + 2pi + 2 p o - 1 s O , 4P2 + 2Pl + Po - 1 s O , P2 + 2pl + 4Po ~ O.
By letting p 2 = p i = 0
and p 0 = 2
in the above family of interpolants, we obtain the Lagrange
interpolating polynomial for our data array l(x) = 3 ( 2 x 2 - 9 x + 16).
The next family o f interpolants has McMillan degree 4. It can be parametrized in terms of a third degree polynomial p and a first degree polynomial q , as follows:
305
y(x)=
(p3x3 +p2 x2 + p l x + p0)6x + (qt x + qo)x(x - 4)(2x - 1) (P3X3 +p2 x2 + pl x + po)(X + 1) - (qlx + qo)(2x - 1)
Firstly, p, q must be coprime, i.e. p3q 3 -p2q2oql + p l q o q 2 - p o q 3 ~ O.
Then, constaints (11) must also be satisfied: po + qo ~ O, 2p3 + 2p2 + 2pl + 2po - ql - qo ~ O, 8p3 + 4p2 + 2pl + po - 2ql - qo ~ O, p3 + 2p2 + 4pl + 8po ~ O.
Notice that the first constraint is a non-linear one while the latter four are linear ones, as mentioned in remark (ii) of section 2. 5. Generalizations and concluding remarks. The above results can be generalized considerably. Consider the array consisting of the data
xi, vl, ri, i ~ N_, of size l / l , rixp, rixm respectively, satisfying the constraints x i ~ x j , i~:j, and r a n k V i = r i < p.
The former constraint is the so-called distinct-point constraint. The more general multiple-point interpolation problem involves information about derivatives. As it turns out, the latter constraint can be made without loss of generality. The left tangential or left directional interpolation problem consists in finding all p x m rational matrices Y(x) satisfying viY(xi) = Yi, i ~ N_,
keeping track of their complexity at the same time. The right tangential or right directional interpolation problem, as well as the bi-tangential or bi-directional interpolation problem, can be defined similarly. The procedure presented in the preceeding sections for the scalar interpolation problem, can be used to answer questions (a) and (b) formulated in the introduction, for all of the interpolation problems mentioned above. The complete solution of these problems, using the idea of defining appropriate reachable and observable pairs of matrices, is given in Antoulas, Ball, Kang, and Willems [1990]. In conclusion, the observation that the matrices 1~K and 1~ can be regarded as partial reachability matrices of a reachable pair (F, G), allows us to obtain complete insight into the rational interpolation problem. As a consequence, it becomes clear in what respect Prony's method of data fitting is deficient, and how to modify it so that it yields the complete answer.
306
6. References. A. C. Antoulas [1981], "On canonical forms for linear constant systems", Int. J. Control, 33: 95-122. B. D. O. Anderson and A. C. Antoulas [1990], "Rational interpolation and state variable realization", Linear Algebra and Its Applications, Special Issue on Matrix Problems, in press. A. C. Antoulas and B. D. O. Anderson [1986], "On the scalar rational interpolation problem",
IMA J. of Mathematical Control and Information, 3: 61-88. A. C. Antoulas and B. D. O. Anderson [1989], "On the problem of stable rational interpolation", Linear Algebra and Its Applications, 122-124:301-329. A. C. Antoulas and B. D. O. Anderson [1990], "State-space and polynomial approaches to rational interpolation", Proc. MTNS '89, Birkh/iuser, Boston. A. C. Antoulas, J. A. Ball, L Kang, and J. C. Willems [1990], "On the solution of the minimal rational interpolation problem", Linear Algebra and Its Applications, Special Issue on Matrix Problems, in press. T. Kailath [1980], Linear systems, Prentice Hall.
Discrete Normalized Coprime Factorization Peter M.M. Bongers, Peter S.C. Heuberger Laboratory for Measurement and Control Department of Mechanical Engineering and Marine Technology Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands
ABSTP~ACT
In this paper a reliable algorithm is developed to perform a normalized coprime factorization of proper discrete time finite dimensional linear time invariant systems. Instead of using the bilinear transform the factorization is calculated directly. The system is allowed to have a singular state-space matrix. It is shown that a modified discrete time Riccati equation plays a crucial role to obtain a state-space realization for the factorization.
I. INTI~ODUCTION
IlL {,he theoretical work of Desoer et al. (1980), Vidyasagar et a1.(1982), Vidyasagar (1984) the benefits of using coprime representations in stability analysis of controlled systems are shown. In the continuous time case Nett et. al. (1984), Meyer and Franklin (1988) and Vidyasagar (1988) derived state--space representations for the normalized coprime factors. Glover and McFarlane (1988), (1989), McFarlane (1988) established the importance of normalized coprime factors in the Hoo controller design. They explicitly solved a continuous time four block Hr control problem by using a normalized coprime representation of the plant. In practical situations a continuous time plant is controlled by a discrete controller using sampling and zero order hold circuits. So in order to design accurate discrete time controllers the controller design procedure has to be done in discrete time. The first step in discrete Hoo control design with normalized coprimc factors is to establish whether or not in discrete time normalized coprime factors exists and can be represented in state--space forms. Chu (1988) gave state-space representations for discrete coprime factors with an inner numerator under the condition that the system has no poles in the origin. Poles in the origin are of major importance since in discrete time very often time--delays have to be incorporated in the system model (appendix 1). In this paper we will show the existence of a normalized coprime factorization of a discrete time plant with possibly poles in the origin.
3O8 II. PB.ELIMINAILIES
Stable multivariable linear systems can be studied by considering them as transfer function matrices having all entries belonging to the ring 7/. Moreover, in many cazes (e.g. convolution operators) the ring 7/is commutative and is an integral domain (i.e. 7/has no divisors of zero). The class of possibly unstable systems are elements of the quotient field F of 7/. Throughout this paper we let (Vidyasagar et.al. 1982, Desoer et.al. 1980): :7: := { a/b [ a e 7/, b E ~ 0 }, a quotient field of 7/ G := a (not necessarily commutative) ring with identity. 7/:= a subring of G which includes identity I := { h ~ 7/I hq e G }, the set of multiplicative units of G 3" := { h e 7/[ h-l e 7/}, the set of multiplicative units of 7/ Note that:
Jczc~c~c7
(1)
In the sequel of this paper we will study real rational finite dimensional discrete time invariant systems. The ring {~ is identified with [RL~ the space of proper real-rational functions with no poles on unit circle with norm H-f]| []J(z)Hoo= sup ~[j(eJ0)]. The subring 7/is identified with U~IIco 0<0<2~r the subs~)ace of ~Lr with no poles outside the open unit disk, and analogously 7/• is idcntified with ~l-I~. The following notation is used. We will denote transfer functions as G(z) or if there is no confusion G, with a slight abuse of notation a transfer function is given by: G(z) : = D +
C(zI-A)qB : = [ ~ - ~ I ~ ]
(2)
A t denotes the transpose of A and G*(z) denotes Gt(z-1). For minimal plants G(z) E 7/, the controllability and observability Grammians P respectively Q are positive definite symmetric solutions of the following Lyapunov equations: APAt + BBt = P (3a) AtQA + CtC = Q (3b) DEFINITION2.1 (Vidyazagar 1984, Huang and Liu 1987) A plant G E :7 has a right (left) fractional representation if there exist N,M (I:/,IVI) E 7l such that: G = NM-' (= 1VI-'l~I) (6) Furthermore we say that the pair M,N (lVI,I'~) is right (left) coprime (rcf or lcf) if there exists U,V (0,'~) E ~ such that: UN + VM = I (lxi0 + iVIV = I)
(7)
The pair M,N (iVI,N) is normalized right (left) coprime (nrcf or nlcf) if in addition to (6): M*M + N*N = I (l~I1Q*+ I~N* = I)
(8)
3O9 Meijer and Franklin (1988) gave an explicit method to calculate the normalized right coprime factorization in continuous time.
I'It0POSITION2.1 (Heuberger t990) A plant O(z) := D + C(zI-A)qB e 7" is called inner: Gt(zq)G(z) = I, if and only if: a) AtQA+CtC=Q, Q=Qt b) DiD + BtQB = I c) CtD + AtQB = 0
(9a) (9b) (9c) []
IIL MAIN RESULT
The following theorem gives sufficient conditions for the existence of a state-space representation of a normalized right coprime factorization of a discrete time plant. In the proof we will frequently use system equivalent operations, described by Rosenbrock (1970). TIIEOItEhl 1 Given a minimal realization: G(z):=D+C(zI-A)qB:=
[~-i~]
E7.
(10)
and define:
F; I11
[M] = L -C+DF l)IIJ
then [M] is a normalized right coprime factorization of G(z)iff: a) b) c) d)
Ft= (AtQB+CtD)(I+DDt+BtQB)q Itllt = (I+DtD+BtQB)q Q-AtQA-CtC+(AtQB+C~D)(mQB+DtD+I)-~(BtQA+DtC)
Q PIOOF
= o
= Qt > 0
W e follow definition 2.1 to prove the theorem. From (12a,c,d) it can be seen (Anderson and Moore 1979) that (A-BF) is stable and hence M, N E ~. definition 2.1 (6): G = NM -1 Split the realization of (11) to M and N:
,'
ll,1
r I-A+nr ""1
llJ' N = L ~+DF
DIIJ
(12a) (125) (12c) (12d)
310 The inverse of M is determined by: FzI-A+BF BIIlO] T M'I= | F 1[ II / -B. u o -~ toj
o_!]
-,~_e,, o_i. 1~ FLIffIL.
,,
r162
,
.1
[ FII- i - i
1
-11- 1 ._~
~---FU"1. Evaluating the equations for NM q gives:
o
[zI-A 0 !llll. ] - I.L. NM q = [-BIIII'IF zI-A+BF L-DIlII-tF --C+DF t ,- I.
zI-A+BF 0 -C+DF DJ
,L-corZi,zi, ,0i]
r162
-O+DF
F~]=o
dcfinition2.1(7):Coprimenessof[M],ortheex]stenceofanV,
Uel(suchthat[Vu][M]=I
Let L = H -I and K be such that p(A-KC) < 1, The minimality of G(z) assures the existence of such a K. Define:
~v ~_-[~-~_~V~ ~] then U, V e "It',because (A-KC) is stable. Writing down the Bezout identity yields to: zI-A+BF
ev~[~]
:
[ zI-A+BF
~I~F-Kc
0
e,, I BII ] - I .
zI-A,~CB,,K,t
9
-LF
~=~
LII--]
I.
definition2.1(8):[M]ismner proposition 2.1 part c} Ft = (AtQB + CtD)(I + DDt + BtQB)q
proposition 2.i part b) [I-It I:ItDt] [HH] + HtBtQBH = I H t I t = (I + DtD + BtQB)-I
olL_~:,,J
zIOA+KC
-L~
BII
e~
01L~,,J
zI-A+BF o IBll ] zI-A+KC 0 | = L I I = I 0 -LF LI]-J
311
proposition 2.1 part a) ( A - B F ) t Q ( A - B F ) + [ - F ' Ct-FtDt] [ c - F F ] = Q Q - AtQA - CtC + (AtQB + CtD)(BtQB + D t D + I)q(BtQa + DtC) = 0 Q=Qt>0
The procedure to obtain a normalized right coprime factorization for the plant G is to solve the fticcati equation (12 c,d) to obtain Q, calculate F and choose an It. The equivalent for the normalized left coprime factorization is a direct result from theorem 1 and is given in the following corollary. COItOLLAR.Y 1 Given a minimal realization (10): G(z):=D+C(zI-A)'IB:= [ ~ ]
E5r
and define:
then [1q IN] is a normalized left coprime factorization of G(z) iff: a)
K = (APCt+BDt)(I+CPCt+DDt)q
b) c)
RtR = (I+CPCt+DDt)q P - A P A L- BBt+(APCt+BDt)(I+CPCt+DDt)-I(CPAt+DBt) = 0
d) p = pt > 0 Plt00F Let G = i'r
with (IQ,I~) E 2' a nlcf of G, then G t = ~@[-t with (~lt,l'r
a nrcf of G t, so the
realization of [IQ N] follows from theorem 1. cl I~.EMMtK Note that we don't need the assumption that A is invertible, as is the case in Chu (1988). Using proposition 2.1 it is straight forward to show that this assumption is indeed superfluous. This is of major importance since in discrete time very often time-delays are incorporated in the system. In order to solve the normalized coprime factorization for the plant in discrete time by means of standard techniques, the equation (12@ Q - AtQA - CtC + (AtQB + CtD)(BtQB + DtD + I)-I(BtQA + DtC) = 0 can be written as a standard Riccati equation. Define:
A,: [A
E0IJ,
:[0
The standard Riccati equation with AI,Bt,CI,RI,Q1 is : Q I - A I t Q t A I - CttCl + AItQIBI(BltQLBI + Rl)-lGtQiA1 = 0
312 Substituting the definitions of Al etc. gives:
[77]-r-ooo1[o evaluating this equation gives (12c). Which means that using standard techniques the discrete time normalized coprime factorization problem can be solved. Note that the sufficient conditions for the existence of a positive solution of the Riccati equation are still valid.
IV. CONCLUSIONS
Theorem 1 and Corollary 1 show that with standard mathematical tools the normalized coprime factorization can be calculated, which is necessary to design discrete time controllers,that satisfy H~o robustness bounds. Since in practical applications one will in general be dealing with a discrete time problem, this is an important step towards the solution of the Hoe control problem in discrete time.
V. LITEKATUgs
Anderson, B.D.O., J.B. Moore (1979), Optimal Filtering, Prentice Hall, Inc., Englewood Cliffs, N.J. Chu, C.C. (1988), On discrete inner-outer and spectral factorizations,Proc. Amer. Control. Conf., Atlanta, Georgia Desoer, C.A., R-W. Liu, J. Murray, It. Seaks (1980), Feedback System Design: The Fractional Representation Approach to Analysis and Synthesis, IEEE Trans. Automat. Contr., AC---25, 399---412 Glover, K., D. McFarlane (1988), Robust stabilization of normalized coprime factors: an explicit Hoe solution, Prec. Amer. Control. Conf., Atlanta, Georgia Glover, K., D. McFarlane (1989), Robust stabilization of normalized coprime factor plant descriptions with Ho0---boundeduncertainty, IEEE Trans. Automat. Contr., AC-33, 821-830. Heuberger, P.S.C (1990), PhD Thesis (to appear), Delft University of Technology, The Netherlands. Huang, Q., R. Liu (1987), A necessary and sufficient condition for stability of a perturbed system, IEEE Trans. Automat. Contr., AC---32, 337-340. McFarlane, D. (1988), Robust controller design using normalized coprime factor plant descriptions, PhD. thesis, University of Cambridge. Meijer, D.G., G.F. Franklin (1988), A connection between normalized coprime factorization and linear quadratic optimal regulator theory, IEEE Trans. Automat. Contr., AC--32, 1041-1047
313
Nett, C.N., C.A. Jacobson, M.J. Balas (1984), A connection between state---space and doubly coprime fractional representations, IEEE Trans. Automat. Contr., AC--29, 831---832. Rosenbrock, H.H. (1970), State-space and mullivariable Theory, Thomas Nelson and sons Ltd. London, Great Brittain. Vidyasagar, M. (1984), The Graph Metric for Unstable Plants and Robustness Estimates for Feedback Stability, IEEE Trans. Automat. Contr., AC--29, 403--418. Vidyasagar, M. (1988), Normalized coprime factorizations for nonstrictly proper systems, IEEE Trans. Automat. Contr., AC,---33, 300-301. Vidyasagar, M., H.Schneider, B.A. Francis (1982), Algebraic and Topological Aspects oI" Feedback Stabilization, IEEE Trans. Automat. Contr., AC-27, 880---894.
APPERDIX 1 In this appendix we show the existence of a singular state--matrix due to time-delays. Let the discretized plant be given by: Xk§ ---- A x k 4- Bvk Yk = Cx k 4- Dvk
In general the discrete time controller will have one time---step delay between the measured outputs and computed plant inputs. Using discrete time control design algorithms the time-delay has to be incorporated in the plant description. Let the time-delay be described by: Zk§ = Uk Vk ---- Zk
Then the augmented plant can be written as:
[x< -Zk*lJ
[::] +
Yk = [C D] Zk Hence the augmented state matrix is singular.
A FORWARD FOR
ACCESSIBILITY
NONLINEAR
DISCRETE
ALGORITHM TIME
SYSTEMS
J.P.BARBOT LABORATOIREDES SIGNAUX ET SYSTEMES CNRS-ESE PLATEAU DE MOULON, 91190 GIF SUR YVETTE, FRANCE.
ABSTRACT In this paper we present a SAMDI algorithm (Symbolic Algebraic Manipulation for Discrete time system), which permits us to know whether a system is accessible for nonlinear discrete time. This algorithm uses the symbolic language REDUCE. In our algorithm we use a set of Gij vector field, which allows us to work with the same geometrical tools as for the continuous time system. Therefore, the SAMDI algorithm uses the same procedures of the SAM (Symbolic Algebraic Manipulation) programme, which was realized for continuous time systems.
INTRODUCTION The local controllability of a linear analytic system is based on the concept of infinitely small displacements in the neighbourhood of a state point. This displacement is generated during an infinitely short period o f time, which is impossible to realize for discrete time systems, because there the sampling period is fixed. In order to create this infinitely small displacement in the neighbourhood of a state point, M. Fliess and D. Normand-Cyrot [1] have chosen a very short input variation. This new concept allowed us to introduce the notion of weak accessibility in discrete time [2], [3], which is an equivalent relation on the state manifold and allows us to use of the orbit's theorem [4]. Thus we obtain a criterion of weak accessibility in terms of the tangent state space at a point of the Lie algebra. This is generated by a set of Gij vector field, introduced by S. Monaco and D. Normand-Cyrot [5]. We may consider this vector field as a derivation of the input of a causal flow (which is the difference equation of the discrete time system) and a non-causal flow (which is the inverse of the difference equation of the discrete time system), or vice versa. This
315
composition of functions draws all the field vectors back to the initial point. For reasons of computing and because of its more physical character [6], we use in this algorithm the concept of "forward accessibility" [7], which is a restriction to the accessibility in the causal, followed by the non-causal, displacements. From the above-mentioned works, we will introduce below a SAMDI (Symbolic Algebraic Manipulation for Discrete time system) algorithm of forward accessiblity for the discrete time system. This programme has been realized with the symbolic language REDUCE [8] and also using some procedures of SAM (Symbolic Algebraic Manipulation) [9] (for continuous time systems). In order to simplify our calculations the SAMDI algorithm has only been realized for discrete time systems with a single input. Following theory is only used to introduce the algorithms easily and the reader can turn to the references to get a more extensive account on them.
Theory
on the Gid vector
fields
Consider a linear analytic discrete time system Z: X(k+l) A__f(X(k)) +g(X(k)) U(k) =Ah(X(k),U(k)) where X e R n, U e R and f, g, h are the vectorial analytic functions of the appropriate dimension. Definition 1 is the set of inputs U, such that h(.,U) is inversive for all X. Remark In this paper we consider the set of ~ , which is not empty and where all the inputs U belong to f2. We have to remind the fact that, if in a point X, we apply a composed function with its inverse, we remain in that same point. For the analytic discrete time system S. Monaco and D. Normand-Cyrot [5] introduced the following set of vector fields:
G(U) I XO = ~ h'l("U)'h("U+e)oe I XO, e = 0
316
We can interprete these vector fields graphically in the following figure:
/
( "
7,0 /'
Figure 1
\ xs" "
Prooosition 1 The differential operator G(U) applied to function h is equal to the Lie derivative of this function h on the vector G(U).Id (see [5]). From the G(U) vector fields we deduce of new set of vector fields Gij (where i e N andj Z). Thus
Gi,01x 0 =
~G(U) ~ U i [U--0,x0
with G0,0Ix 0 = G(U)Iu=0, x 0 In the same way we find
G0,j [X 0 =
f-J h(.,u)-l.h(.,U+e).fJ [ U=0, ~0,X 0 ~ e
and Gij Ix 0 = ~i+l f-j h(.,U)-l.h(.,U+e).fj [O=0,e--0, X0 ~UiOe
Remark The fJ function is equal to the fJth function of f and f0 is equal to the identity function.
317
FORWARD ACCESSIBILITY As a result of the research on weak accessibility of discrete time systems [1], [2], and [3], and on the notion of action [11], B.Jakubczyk and E.D.Sontag [7] introduced the notion of forward accessibility. Below you will find the definition of a linear system with discrete time system 2; with a scalar input. We assume that the elements of input A are open in ~ and zero inputs in element A. We first define A+(Xo ) as a forward accessible set of X0. A+(X0)= { X 9 R n / 3 k 9 N , ~ / i 9 k , 3Ui+,Ui - 9 Asuchthat X = h(.,U0" )'l....h(.,Uk')-l.h(.,Uk+)....h(.,U0 +) Ix 0 } From A+(X0) it is possible to define the forward accessibility. Dgfinition 2 A system is said to be forward accessible if, and only if, there exists V(X0) in the neighbourhood o f X 0 such that A+(X0) n V(X0) = V(X0). From this definition and the field vectors Gij [5], B.Jakubczyk et E.D.Sontag formulated the following theorem: Theorem 1 [7] The discrete time system 2; is forward accessible in X0 if, and only if, the vector fields generated in X 0 by the smallest Lie algebra F+(~) contains all the elements of Gij where id 9 N, is the dimension of n. R~mark The forward accessibility in X0 (r+(X0)) is the dimension of the general field vector generated in X 0 by F+(Y.). We find the proof of this theorem in [7] or in [6] with the mathematical tools introduced in [2]. The set of vector fields used in the above theorem is the only difference with the one given by D. Normand-eyrot [2]. TheQr~m 2 [21 The discrete time system s is weakly accessible at X 0 if, and only if, the vector fields generated in X 0 by the smallest Lie algebra F(Y.), which contains all the elements of Gij with i e N andj 9 Z.
318
Remarks a. F+(Y.) is included in F(~) and therefore forward accessibility means weak accessiblity. b. In a recent paper of A. Mokkadem [11], he introduces the concept of quasiaccessibility (on a Zariski topology) and studies the links of these concepts of accessibility. Thus, for some systems the results of our algorithm can be applied to the quasi-accessibility. COMPUTATION OF THE Gkj.VECTOR FIELDS As shown in [5], the Gi,j, with i, j e N, are easy to compute. Thus, like before, the vector fields G0,0 can be defined as follows
G0,01 X0 =
0 FG(.,0)-I.FG(.,e) de I X0,~0
which is equal to GO,O IX0 = j(f-1) I f(X0) -g [ x 0 where j(f-l) is the Jacobian matrix of function f-1. As a result of the following equation 3f-l.f
I X0= j(f-1) I rcx0) J(0 t x0 = Id
where Id is the matrix identity, we get: IS0,01 x 0 = J(f)-I ] x0.g [ X0l where the vector fields Gi,0 are equal to:
Gi,01 x 0 =
~i[J(f)+J(~).U]-I 0U i I X0,U=0 .g Ix 0
The properties of the derivatives of the matrix give us the following equation: [Gi,0 ]X 0 = (-1) i i! ([J(f)'l.j(g)]iJ(f)-l.g) I X0]
In the same way, the vector fields Gk,1 are equal to:
Gk,l IX 0 =
0k+l f-1 FG(.,U)-I.FG(.,U+e).fl 0 Uk 0 e [U--O'~O'Xo
319 which gives Gk,1 IX 0 = [j(f)]-I
1x0.Gk,01 r(x0) ]
Finally, from their recurrent property, we obtain: [Gk,j+l ] X0 =
[J(f)]-I I x0.Gkd l r(x0) ]
INTRODUCHTON OF THE "SAMDI" ALGORITHM AND CRITERIA TO STOP As shown above, the vector fields Gi,j are products of the Jacobian matrix, which are easily computable in the symbolic language REDUCE [8]. As an example we given the following little programme: FOR I:=1 : n DO BEGIN FOR J:=l : n DO BEGIN JF(I,J):=DF(F(I),X (J)); END; END; where DF(F(I),X(J)) is the symbolic function of the derivation of the first component of the vectorial function F, compared to the variable X(j). Note that the inverse of the matrices and their product are basic functions of the REDUCE language. Moreover, the calculation of the Lie brackets and the dimension of the vectorial space generated by a certain number of vectors are given in [9]. Therefore we limit ourselves to the following introduction of some stopping criteria. In this algorithm the first step must be to calculate the dimension of the space generated by the vector fields Gi,0 and X0 with i E N, for a finite number of vector fields. Therefore, as a result of equation (2) and the Cayley-Hamilton theorem, we find the following stopping criterion: CRITERION 1 We stop the calculation of Gj,0 when rank [Gj,0....G0,0] I x 0 = rank [Gj-I.0..... G0,0] ix0. The second step consists of calculating the dimension of the vector fields generated in X0 by the smallest Lie algebra 1-'+(~), which contains all the elements of Gi,j. For this we use the property of involution, see [12] and [9].
320 CRITERION 2 Consider a set of 13vector fields. If ~ Lf a 13and "~ Lg ~ 13we have [Lf,Lg] Ix ~ EB(X) where EB(X) is the vectorial space generated by the vector fields 1] en X. From this we can conclude that the dimension of EB(X) is equal to the dimension of space vector X, by means of the smallest Lie algebra containing 13. The following step is to calculate the vectorial space of the Gk,l where k and 1 are the integers. In order to stop this calculation we use criterion 1 for exponent k and the Cayley-Hamilton theorem for exponent I. However, in equation (4), the matrices are function of the exponents. For this, we need to assume that X0 is not a unique point. Also, the rank of a Jacobian matrix is a function for which there exists an open set of 7 state points, such that x ~ 7, and that the rank is maximal. Therefore we can introduce the following criterion: CRITERION 3 We stop the calculation of Gj,p as soon as rank [Gj,p+l ...... Gj,0] Ix 0 = rank [Gj,p ...... Gj,0] I X0.
The last criterion is evident, but we should not omit to use it in a programme: CRITERIQN 4 Whatever the vectorial space Eli generated by vector fields of n dimension, the dimension of El3 is smaller or equal to n. With the help of these criteria we can realize a SAMDI algorithm, represented in Figure 2. EXAMPLES The following examples are arbitrarily chosen to illustrate some properties of forward accessibility. Examole 1 Consider the following discrete time linear system: X(k+l) = A X(k) + B U(k) where X e R n, U ~ R and A, B are the appropriate dimensions of the matrices.
321
The forward accessibility is different from the usual discrete time commandability [12], by the fact that matrix A must be regular (see [6] for a comment on this subject). If matrix A is regular, all the inversive inputs are equal to R~ The forward accessiblity rank r + is then r+= rank [B,AB ........An'IB]
Consider the following linear analytic system Y= X1 (k) 3+X2(k) X(k+l) = FG(X(k),U(k)) - ~.X ] ( k + l ) = - [X2(k+l) = X2(k)+U(k) This system is inversive on levels XI>0 and XI<0; indeed, det(j(FG)) is equal to (3X12) -1. Computation of the vector fields Gi,j give: G0,0 = (3X12) -1 (-1,3X12) T Gi,0 = (0,0) T and
• i>0
G0,t = (9 X12 (X13+X2)2) "l (-3(XI3+X2)2-1, 9 X12 (X13+X2)2) T
The system is forward accessible, but we find a set of singularities for (X13+X2) 2 equal to zero, which correspond to Xl(k+l) equal to zero. Note that all the singularities found for this system are a result of the too restrictive definition of forward accessibility. Indeed, whatever the values of X(0), with the help of U(0) = - (X13+X2)3-X2 and U(1) = (X13+X2) 3 we find that state X(2) is equal to (0, 0)T. In the same way, from (0, 0) it is possible to go to any point (XI, X2) with the help of control law U(0) = X1 and U(1) = X2-X1.
CONCLUSION The above presented algorithm of forward accessibility is a symbolic tool which is useful when we study a linear analytic discrete time system with scalar input. Indeed, if the system is not forward accessible in an open set which is not empty, it is of no use to look for control law. However, if the system is forward accessible, it is not always possible to control all the states. We also have to limit ourselves to inversive systems, but this restriction does not hold for sampling systems [ 14]. Finally also note that the extension of SAMDI algorithms to linear analytic discretetime systems with multi-input only give some computational problems (sufficient memory, time of calculation, increase of the number of vector fiels index in order to take into account eachinput .... ).
322
ACKNOWLEDGEMENT I especially wish to thank M rs Doroth~e Normand-Cyrot and M r Salvatore Monaco for their precious help and suggestions. Without their publications [1], [2], [3] and [5], this article could not have been written. The research has been made possible by a grant from the European M.R.E.S.
,
t
GT,0 C,)HPIJTI~T I OH TR I RI IGU LP,~ 1Z,~T 1 C+
X ) ' ~ . 'i.~-
,+,,'+'O_.~F:~TE~XO~
I ~ '..t.o
-.,,
,p-
[
~/~-.
vm
'v~
TIlE
,f.
-,II--C}-~;:R I T E R I
fK' ~RITEF.~ON~ #
-'-...5/
C~r
.o ~".... I L
I,
-
INIjOLUT i C,N R~-IC'l T ~ IRNGULRR Z Z ~ T I O N
"NO
~
TIIE SYSTEM IS F011NAKD ACCESSIBLE
SVSTEH ISII'T 1
FOnHanD A C C E S S [riLE
Figure 2
323
REFERENCES
(1)
M.Fliess, D.Normand.Cyrot (1981) " A Lie-theoretic approach to nonlinear discrete-time controllability via Ritt's formal differential groups" Systems and Control Letters, Vol 1, N~ pp 179-183
(2)
D.Normand.Cyrot (1983) " Throrie et pratique des syst~mes non linraires en temps discret" Thesis of Doctorat d'Etat, University of Paris Sud, Orsay
(3)
B.Jakubczyk, D.Normand.Cyrot (1983) " Orbites de pseudo groupes de difffomorphismes et commandabilit6 des syst~mes non linraires en temps discret" C.R. Acad. Sciences Pads
(4)
H.J.Sussmann (1973) " Orbits of families of vector fields and integrability of distributions "Transactions of the American Mathematical Society Vol 180 pp 172-188
(5)
S.Monaco, D.Normand.Cyrot (1984) "Developpements fonctionnels pour les syst~mes non lineaires en temps discret" Report of I.A.S.I, C.N.R Italien (in French) and "Functional expansions for nonlinear discrete time Systems" Math. System Theory 21 pp 235-254 (1989).
(6)
J.P.Barbot (1989) " MEthodes de calcul appliqudes aux syst~mes non linEaires sous 6chantillonnage" Thesis of Docteur en Science, University of Paris Sud, Orsay
(7)
B.Jakubczyk, E.D.Sontag (1988) " Controllability of nonlinear discete time systems: A Lie-algebraic approach" Proposed to S.I.A.M J. Control and Optimization
(8)
A.C.Hearn (1984) " R.E.D.U.C.E User's manual " The Rand Corporation, Santa Monica USA
(9)
R.Marino, G.Cesaro (1982) " Nonlinear control theory and Symbolic Algebraic Manipulation " Ed Springer-Verlag Proc. Beer-Sheva Israel
(10) E.D Sontag (1986) "Orbit theorems and sampling " M.Fliess and M. Hazewinkel Eds Algebraic and Geometric Methods in nonlinear control theory pp 441-483 D.Reidel Publishing Company (11) A.Mokkadem (1989) "Orbites de semi groupe de morphismes rrguliers et systrmes non linfaires en temps discret" to appear in Forum Mathrmafique. (12) A.Isidori (1989) " Nonlinear Control Systems : An introduction " Communications and control engineering series (2nd Edition) Springer-Verlag (13) R.E.Kalman (1960) " On the general theory of control systems" Proc. First International Congress on Automatic Control, Moscou. London, Butherworth 61, Vol 1 pp 481-492 (14) S.Monaco.D.Normand.Cyrot (1985) "On the sampling of a linear analytic control system" 24 th Conf. Decision and Control Ft Lauderdale pp 1457-1462.
S t a b i l i s a t i o n globale de s y s t ~ m e s n o n - l i n ~ a i r e s p a r un contr61e positif. Jean-Luc Gouz4 INRIA Sophia-Antipolis 06565 Valbonne Cedex, France A b s t r a c t : We give sufficient conditions for global stabilization in a fixed domain around an equilibrium of a class of controlled systems of non-linear ordinaa-y differential equations in n-space, that can be expressed as the difference of a quasi-monotone increasing and a monotone increasing system. These systems are usual in biological modelling. We consider the cases when the control is scalar, or scalar and positive. We apply the theory to LotkaVolterra systems. R 4 s u m 4 : Nous donnons des conditions suffisantes de stabilisation globale dans une r4gion fix4e par un contr61e autour de l'4quilibre pour une classe de syst~mes diff6rentiels nonlin6aires en dimension n qui sont diff6rence d'un syst~me quasi-monotone croissant et d'un syst~me monotone croissant, fr4quents dans la mod41isation biologique. On consid~re le cas d'un contr61e scMaire et d'un contr61e positif scMaire. On prend pour exempte les modules de Lotka-Volterra.
I
Introduction
Nous nous int4ressons dans cet article au probl~me de la stabilisation globale vers l'4quilibre, dans une r4gion fix~e born4e de l'espace, d'une classe de syst~mes diff~rentiels non-lin4aires, par un contrSle en boucle ferm4e. L'approche la plus usuelle, et sans doute la plus g4n4rale, clans ce type de probl~me, consiste k utiliser des fonctions de Liapunov (voir [8]); mais il est souvent difficile, quand le syst~me est sp4cifi4, d'exhiber une telle fonction. D'autres approches existent, qui utilisent par exempte la th6orie de la vari4t4 centrale ([1]), mais elles sont surtout locales. Nous utiliserons ici une autre approche globale, ne faisant pas appel aux fonctions de Liapunov, mais nous restreindrons la classe des syst~mes non-lin4aires 4tudi4e. Ces syst~mes soar issus, dans notre cas, de probl~mes de mod41isation biologique, par exemple en dynamique des populations. I1 est alors normal que les signes de certaines grandeurs jouent un rSle important, car ils peuvent s'interpr4ter biologiquement (voir par exemple [7] et [2]). Nous consid4rons donc la classe de systbmes non-lin4aires n-dimensionnels: ~ = f(x)
- g(x)
(1)
avec les conditions suivantes vMables pour x dans un ouvert convexe W C R~: ~Ofi , (1:) >_0 pouri # j e t
( x ) > _ O ( i , j = l . . . . ,n)
(2)
Les fonctions f e t g sont suppos4es C 1. On dit que f e s t quasi-monotone croissante et que g est monotone croissante si elles v4rifient (2)(volt [13]). S i g -- 0, o11 dit aussi clue le syst~me
325 (1) est coop~ratif (voir [4]). Des conditions de ce type i n t e r v i e n n e n t souvent dans les syst~mes difffirentiels modfilisant des ph~nombnes biologiques ([7]), car, par exemple, le signe de ~ ( x ) traduit si la variation de l'esp~ce j influence positivement ou nfigativement 13 variation de l'esp~ce i. Remarquons que cette classe de syst~mes est d~jk tr~s vaste ([3]). Un exemple i m p o r t a n t est c e h i des modNes de Lotka-Volterra (voir [12,11]) qui s'fierivent:
,i:i = xi(bi + f i cqx~)
i =
1 ....
, n
(3)
j=l
Les syst~mes coop~ratifs et comp~titifs (en inversant le temps) ont ~t~ beaucoup ~tudi~ m a t h ~ m a t i q u e m e n t (voir [5,9]). Nous avons utilis~ ces r~sultats pour obtenir u n critbre surfisant de convergence globale pour le syst~me (1) vers un ~quilibre suppos~ unique ([3]). Nous appliquons ici ce critSre k u n probl~me de stabilisation globale dans une r~gion fix~e de ce systSme par u n contr61e scalaire et par un contr61e scalaire positif (la positivit~ dn contrSle a yant souvent un sells dans les probl~mes biologiques). Plus pr~cis&nent, nous cherchons k stabiliser (en boucle ferrule ou par des contr61es constants) le syst~me : 2 = f ( x ) -- 9(x) - ud(x)
(4)
o~ u est un scalaire et d une application que nous prendrons d'abord trbs simple pour l'~tendre ensuite. Dans la suite, quand nous parlerons de stabilisation globale, cell voudr3 dire que nous savons exhiber une r~gion born4e conten~m dans W, c o n t e n a n t un seul point fixe a s y m p t o t i q u e m e n t stable dont le bassin d ' a t t r a c t i o n contient la r4gion. Nous l y o n s limit~ notre exposition au cas u scalaire, mais on peut facilement ~tendre l' dtude au cas multivariable (volt la premiere remarque de la section 6). Dans la section 2, nous citons le crit~re de convergence globale vers l'~quilibre, que nous appliquons au probl~me de stabilisation dans la section 3, avec u n contr61e positif en section 4. Nous terminons en section 5 par une application k la stabilisation du module de LotkaVolterra et quelques remarques et interpr4tations biologiques. Notations: Pour x E R ~, on ~crit x >__ 0 si xi >_ 0 i = 1 . . . . , n et x > 0 si xi > 0 i = 1 , . . . , n . On note R~_ = {x E R"; x >_ 0} (c'est l ' o r t h a n t non-n~gatif fermi) et, si z < y, [z,y] = {z ; x < z < y} (intervalle ferm~ m d i m e n s i o n n e l ) . De m~me, on notera par e ~ le vecteur ( e ~ ' , . . . , e~"), par In x le vecteur (In a'l . . . . . In x,~). Si W C R " est ouvert, h : W ~ R " est C 1, et x E W, on note par Dh(x) 13 matrice jacobienne Ohi/O.rj(x). Soil x : J --+ R ~, o~l J e s t u n intervalle rdel, on note if: la d~rivfie de x pat" rapport an temps t. Si W C R ~ est ouvert, h : IV ~ R " est C 1, pour l'~quation diff~rentielle .i: = h(x) et une condition initiale x(0) = xo dans W, nous noterons .v(t, xo) ou parfois x(t) la solution maximale issue de x0. Enfin, on note (ei), i = 1 . . . . . n la base canonique de R ". 2
Un
crit&re
de
convergence
globale
Nous citons ici le crit~re de convergence globale en renvoyant ~ [3] pour t'exposition. Soil donc le syst~me (1) v~rifia.nt d,~ns I.V ouvert COllvexe les conditions (2), avec f et g suppos~es
326
C 1. Nous construisons alors un syst~me dans R2~:
{v
= f(v)
-
g(~)
= f(~)
-
g(v)
(.~)
ddfini pour (y, z) E W x W. On a Mors le th6or~me suivant: T h 4 o r ~ m e 2.1 S'il existe (yb,zb) E IV x W tels que :
z b <_ yb
f(vb) <_ g(:~) f(~b) >_ a(vb) et si le systDne (5) a au plus un dquilibre dans la rggion rectangulaire (z b, z b) <_ (y, z) <_ (yb, yb), alors il existe un et un seul dquilibre x" de (I) dans [z b,yb] et toutes Ies solutions x(t, xo) de (i) teIles que z b < xo <_ yb convergent vers cet gquilibre. Donnons une esquisse de la prenve (voir [3]): le syst~me (5) est coopdratif (au changement de signe en z pros); on montre qu'une solution (y(t), z(t)) encadre une solution x(t) de (1); sous les hypotheses du th6or~me, on montre alors que la solution de (5) issue de (yb, z b) converge vers (x*, x*), en d6croissant en y e t en croissant en z. C o r o l l a i r e 1 La r@ion [z b, yb] est iuvariante et globalement aysymptotiquement stable. Comme les signes des grandeurs utilis4es ont souvent une interpr&ation (biologique), il nous paralt plus clair de ne p a s s e ramener par translation g x* = 0. 3
Stabilisation
par
un contrSle
scalaire
Nous chercherons ici g stabiliser autour d'un 6quilibre x* le syst~me :
,~ =
f(x)
- ~(x)
~,el
-
(6)
o{t u est un contrSle scalaire (on peut agir sur la premiere composante de ./,). Les fonctions .f et g vdrifient les conditions (2). L'id4e du th4or~me suivant est simple: nous nous plaqons dans un cas o{t toutes les in4galitds du thdor~me 2.1 sont v4rifi4es, sauf pour la premiere composante; nous cherchons alors g utiliser u pour v4rifier cette in4galitd. De plus, nous choisirons u tel que f - uel v6rifie aussi la premiere des conditions (2). I1 faut aussi pour pouvoir appliquer le th4or~me pr4cddent que le syst~me (boucl4):
{v
= f(v)
-
,,(v)~
-
g(~)
:. = f(~) - .~(~)e, - g(v)
(r)
n ' a d m e t t e qu'un seul dquilibre (x*, x*), c'est g dire que le contr61e n'introduise pas d'&luilibre suppl4mentaire dans la r6gion stable. Cette condition est g6n&alement assez facile g v4rifier pour un u donnd. Nous chercherons u fonction (C 1) de xl seulement, et comme on doit avoir u(x~) = 0 pour pr4server l'dquilibre x* de (6), il en r4sulte que u(x~) = v(azh)(ah -- x~), 06 v est une fonction continue.
327
T h d o r ~ m e 3.1 S'il existe (gb, z b) E W x 1u tels que :
z b <_ x* <_ yb fi(yb) < ji(_b) i = 2,...,t~ /,(:) > ~;(r i = 2,...,,~ alors on peut trouver deux hombres positifs v(z~) et v(y~) teIs que s _ u(y~) < g~(z b) et f l ( z b) - u(z~) >_ gl(yb). Si, de plus on peut trouver un, e fonction V(Xl) telle que le .syst~me (7) n'ait que I'dquilibre (x*, x*) dans la r@ion rectangulaire (z b, z ~) <_ (y, z) <_ (yb, y~), alors routes les solutions x(t, xo) du syst~me boucId (6) telles que z ~ < xo <_ y~ convergent vers x*. La r@ion [z b, y~] est donc globalement aymptotiquement stable pour x*. Ddmonstration. On vdrifie d'abord les hypoth&ses du th~or~me 2.1 sur les fonctions f - ue~ et g. On cherche ensuite deux nombres v(y}) et v(z}) v4rifiant les in~galit~s: fa(y~) -v(y~)(y} - x'~) <_ gt(z ~) et (gl(~]b) _~ f~(z b) - v(z})(z} - xl) C'est possible en choisissant v(y ~) et v(z ~) positifs et assez grands. Si de plus le syst~me boucl~ (7), pour une fouction v donn~e, n'admet qn'un seul &luilibre (c'est (x*, x ' ) ) dans la rdgion consid~rfie, alors toutes les hypotheses du th~or~me 2.1 sont remplies. Remarques: 9 On volt qu'il s'agit de contr61er le champ au bord seulement de In r~gion stable. I1 est anssi clair que le contrSle u(,rl) est de signe oppos~ en z b e t .yb. 9 Le choix le plus simple pour v e s t une foncfion constani;e positive assez grande; on perd alors de in finesse par rapport au choix possible d'une fonction v avec v(z~) 7~ v(y~). 9 On peut aussi choisir pour v une fonction continue nulle par'tout saul vers yb et zb; on a alors un contr61e quasi impulsionnel. 4
Stabilisation
par
un contr61e
scalaire
positif
II arrive souvent, par exemple dans des probl~mes biologiques, que l'on ne puisse effectuer clue des contr61es avec un signe fixs Reprenons le systbme p%c~dent (6), et cherchons un contr61e u > 0. I1 est edors facile d ' a d a p t e r le th~or~me pr6e~dent, en supposant qu'il y a une indgalit~ non v~rifi~e seulement en un bord, et pas au• deux. T h 6 o r ~ m e 4.1 S'il existe (yb, z b) E I,V x |.V tels que :
z b <_ x* ~ yb .fi(y b) <__gi(z b) i --- 2 , . . . , n
L(?) > g i ( r
2. . . .
,,~
fl(Z b) ~___(Jl(y b) alors on peut trouver un contrSle u(xt) nul sur z b < x < x* et positif ou nuI sur x* < x < yb tel que fl(y b) - u(y~) <_ gl(zb). Si, de plus on peut trouver une telIe fonction u(xt) tdle que le syst~me (7) n'ait que l'@uilibre (x', x') dans la ,'@ion rectangulaire (z~,z b) <_ (y, z) _< (yb, yb), alors toutes les solutions x(t, Xo) du syst~me boucld (6) telles que z b <_ xo <_ yb convergent vers x'. La r@ion [yb z b]est gIobalement asymptotiquement stable pour x'.
328
La d&nonstration est identique k celle du tMor~me (3.1), sauf que l'on utilise u(x~) pour v&ifier l'in~galit~ senlement en y~. Remarque: 9 Le signe du contr61e conditionne le choix du bord off l'on fera a.gir ce contrSle; dans notre cas, si u est positif, on pourra agir en y~, et en z ~ s'il est n4gatif. 5
Stabilisation
par
contr61e
scalaire
positifagissant
sur plusieurs
composantes Nous allons maintenant consid6rer un cas o{t u est encore scMaire, mais agit sur plusieurs composantes de In vitesse, par un terme de type bilin6aire autour de l'6quilibre. Nous consid6rerons d'abord le cas particulier de deux composantes. Soit le syst~me : ,~ = f ( x ) - g ( x ) - u d ( x )
(S)
avec d(x) = (xa - x ~ ) q +(x2-x~)e2, o~t u est encore un scalaire positif.On a alors le thdor~me suivant, qui utilise le syst~me associ~ boucld : {{I = f ( Y )
-
u(y)d(y)
= j:(~) - . ( ~ ) d ( ~ )
-
g(z)
- g(y)
(9)
T h 4 o r ~ m e 5.1 S'il exiate (yb, z b) E IV • I.V tels que :
z b ~ x" < yb fi(u b) <_ gi(z b) i = 3 , . . . , n f~(?) _> g~(u b) i = 3 . . . . . , ~ alora on peut trouver deux nombres positifs a(z b) et u(y b) tels que fi(y b)-u(yb)di(y b) <_ gi(z b) et fi(z b) - u ( z b ) d i ( z b) >_ gi(y b) pour i = 1,2. Sous la condition que le systdme (9) n'ait que l'dquilibre (x*,x*) dana la ,'dgion rectangulaire (zb, z b) <_ (y,z) <_ (yb,yb), alors on peut choisir: 9 soit u(x) constant positif assez grand 9 soit U(Xl,X~) fonction positive C ~ telle que ,t~]gxo,~ ,, . ~ - x~) <_ 0 et (~"o,2
Jt'q
-
x~) < o
et toates les solutions x(t, Xo) du syst~me boucld (8) telles que z b < Xo < y~ convergent vers x*. La rggion [z b, yb] eat globalement asymptotiquement stable pour x*. Ddmonstration. Le jacobien du syst~me boucl4 (8) s'6crit: J = D r ( x ) - u ( x ) I 2 - I=(x - x*)(Ou(x)/Ox) - Dg(x) off I2 est une matrice telle que dn = d:= = 1 et nuUe ailleurs. Pour conserver la coop4rativit4 du champ f - ud, il faut et il suffit que U(Xl,X2) vdrifie les conditions donn4es dans le thdor~me. La suite de la ddmonstration est identique k celle du th4or~me (4.1), les indgalit& &ant vectorielles au lieu d'etre scalaires. Rem~rques:
329 * On peut prendre, par exemple:
avec kl et k2 constantes positives choisies pour que u reste positif et v6rifie les in6galit6s
en yb et z b 9 Ce th6or~me s'6tend de fagon claire au cas off on contr61e par un terme du type u ( x ) ( x x ' ) u n hombre k de composantes de la vitesse; on supposera alors que les (n - k) in6galit6s restantes sont v6rifi6es, et (sous r6serve que l'on n ' a j o u t e pas de points fixes au syst6me (9)) on pourra trouver u(x) positif c o n s t a n t stabilisant globalement le syst6me. Par contre, il semble difficile de prendre u autre que constant si k > 2. Extension
et exemples
biologiques
On peut 6tendre les r6sultats des sections 3 et 4 au cas de plusieurs contr61es E uiei. Si l'on a k contr61es, on supposera que les (n - h) in6galit6s restantes des th6or6mes 3.1 et 4.1 sont v6rifi6es, et l'on choisira ui fonction de xi. On peut 6tendre les r6sultats de la section 5 au cas o{t d(x) est une application diagonale (Dd(z) est une matrice diagonale) n6gative pour x _< x* et positive pour a" >_ x*, et donc S~allnulant en x ' . On va prendre comme exemp!e les modNes de Lotka-Volterra (3). On suppose que le systhme a un 4quilibre unique x* > 0 (voir ([11]), et, apr~s le changement de variable = In x, on obtient:
= ~ + ce~
(10)
On pose C = F - G, off F est une matrice & 416ments en dehors de la diagonale positifs et G une matrice positive. On suppose que l'on peut exercer un contrSle sur la premi&re composante, le syst6me boucl6 s'6crit donc: = F(d - d') - a(e ~ - d') - u(~)q
(11)
et le syst6me associ4 (7) s'6crit:
~
u
,,(z)e~ )
(12)
avec A=
-G
F
et
V = e z - e ~"
(13)
Nous allons choisir u(~) = k(e ~* - e~;), k constante positive; de plus, nous supposerons d bijective (volt ([3])); le syst~me (12)s'6crit alors:
~"
U
off le d6ternfinant de Al(k) est u n polyn6me du deuxi~me degr6 en k; si les hypotheses sur les in6galit6s du th6or6me 3.1 sont remplies, on pourra donc de plus choisir k pour
330
que Al(k) soit bijectif, donc que le syst~me (14) ait (~*,~') comme unique @quilibre. On pourra donc appliquer le th@or~me. Donnons deux exemples num@riques, le premier avec un contr61e u de signe variable, le deuxi~me avec un contr61e de signe fix@ : - Le syst~me (10) avec C =
4 -5
et b =
1
peut @ire stabilisfi vers son @quilibre instable (0, 0) avec un contr61e sur la premiere composante u(4) = 3 ( d ' - d r ) dans l'intervalle (dans R 2) ] - oo, ln2]. - Le syst~me (10) avec
50 1 ) ( : : ) C =
0
-2
3
I
i
--3
et b =
(16)
admet un @quilibre instable en (0,0, 0); il peut ~tre stabilis~ globalement dans l'intervalle (darts R a) [-ln(0.9, 0.88, 0.2), ln(1 + 10 -a, 1 + 2.10 -a, 1 + 10-3)] avec un contr61e positif sur la premiere composante que l'on peut prendre nul si ~1 _< ~ et @gal ~ u(~l) = 6.104(e ~ - e ~) si ~1 _> ~[- Remarquons que ce contr61e n'est pas d@rivable en ~*, mais on v@rifie que le syst~me boucld reste C 1 partout. * Biologiquement, ~ repr~sente un taux de croissance; l'exemple des modules de LotkaVolterra nous dit que, dans certains cas, on pourra stabiliser un syst~me biologique si on salt agir sur le taux de croissance d'une esp~ce dans les deux sens; et, dans d'autres cas, si on ne salt agir que dans un sens (en la chassant ou en augmentant le taux de reproduction par exemple). R e m e r c i e m e n t s " L'auteur tient & remercier les rapporteurs pour leurs remarques judicieuses. lZdfdrences [1] D. Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control, Systems control Lett. 5 (198.5), pp. 2S9-294. [2] J.L. Gouz6, Structure des modbles matMmatiques en biologie, in: A.Bensoussan et J.L. Lions, eds., Analysis and optimization of systems, Lecture Notes in Control and Information Sciences No. 111, Springer-Verlag, Berlin (1988) [3] J.L. Gouz4, A criterion of global convergence to equilibrium for differential systems, Rapport Inria No. 894, (1988) [4] M.W. Hirsch, Systems of differential equations which are competitive or cooperative I: Limit sets, SIAM J. Math. Anal.,13 (1982),pp. 167-179.
331
[5] M.W. Hirsch, Systems of differential equations which are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal., I6 (1985), pp 432-439. [6] M.W. Hirsch, The dynamical approach to differential equations, Bull. Amer. Math. Soc., ll (1984), pp 1-64. [7] R.M. May, Stability and complexity in model ecosystems, Princeton University Press, Princeton (1974) [8] N.Rouche et J.Mahwin, Equations diff~rentielles ordinaires, tomes 1 et 2, Masson, Paris (1973) [9] It.L. Smith, Systems of ordinary differential equations which generates an order preserving flow. A survey of results, SIAM Review, 30 (1988), pp. 87-113 [10] H.L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math, 46(1986), pp 8.57-874. [11] Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 10 (1980), pp 119-136. [12] V. Volterra, Variations and fluctuations in the numbers of coexisting animal species, (1927) in: F.M. Scudo and J.R. Ziegter, eds., The golden age of theoritical ecology: 1923-1940, Lecture notes in biomathematics No. 22, Springer-Verlag, Berlin (1978)
[13]
w. Walter, Differential and integral inequalities, Springer-V'erlag, Berlin (1970)
CONTROLLABILITY OF DELAY-DIFFERENTIAL SYSTEMS Paula Rocha Technical University of Delft P.O. Box 356 2600 AJ Delft
Jan C. Willems University of Groningen P.O. Box 800 9700 AV Groningen
A b s t r a c t : The concept of controllability is introduced and investigated for the class of AR delay-differential systems with separable AR descriptions. For this class of systems, it is shown that a system Z described by the AR equation
R(al,o'2)w=o (with a I the
differentiation- and a 2 the delay-operator) is controllable if and only if rank R(A,e-~) is constant for all A9 s
1. INTRODUCTION The aim of this paper is to investigate the concept of external controllability for delay-differential systems. The contributions to the area of delay-differential systems are vast, and can be divided into two main streams. On the one hand, the infinite dimensional state space approach of, among others, Bhat and Koivo, and Manitius Trigianni, [BK], [MT], and on the other hand the algebraic approach developed, for instance, in [L]], [M] and IS], which views delay-differential systems as systems over rings. Both these approaches are concerned with internal system properties, such as, for instance, controllability and observability of the state space. Recently, an external characterization of approximate controllability of the standard observable realizations
of delay-differential
transfer
functions has been given by Yamamoto, [Y]. However, although externally characterized, this controllability
property is defined on an
internal level,
as it concerns state
space realizations. Our purpose is to investigate for the particular case of delay-differential systems the concept of controllability introduced by Willems [W2] for general dynamical systems. In the behavioral framework of [W2], a system is viewed essentially as a set of external trajectories,
and thus controllability is defined as a property of these trajectories.
333
Here, we will consider the class of the s o - c a l l e d AR d e l a y - d i f f e r e n t i a l systems, and derive a controllability test in terms of the system AR representations. 2. CONTROLLABILITY OF DYNAMICAL SYSTEMS
We start with the basic definition of a dynamical system introduced in [W1].
D e f i n i t i o n 1. A dynamical system Z is defined as a triple Z := (T,W,~), where T c_R is the time set, W is the signal set and ~9 _c Wr := {w:T-+W} is the system behavior. Thus, a system is essentially characterized by its behavior, i.e. by the family of all trajectories in the domain W"T which are compatible with the system laws. Note that in this definition no distinction is made between inputs and outputs among the system variables. In fact, our approach is more general than the input/output framework, as input/output systems can be viewed as dynamical systems in the sense of Definition 1, if instead of inputs u and outputs y we speak of system trajectories = col(u, y).
In the sequel we will consider excIusively continuous time systems, i.e., systems with time set T=R. Intuitively, we will say that a system is controllable if it has limited range memory, i.e., if the values of the system trajectories on two intervals ( - c o , t ] and [t+d, +co) are in some sense independent provided that d > 0 is sufficiently large. Given two trajectories wl,w2eW R we define the concatenation of tox with w~ at time t e R as the trajectory w e W R given by w[(_|
and wl[t+|
We will
denote w := w 1 A w 2. D e f i n i t i o n 2. A dynamical system E = ( R , W , ~ ) is said to be controllable if the following condition holds. {Wle~ , W2e~ , teR}.~{3d>0
3w*e~B s.t. WlAUr*t+AdW2EB }
3. AR AND MA DELAY-DIFFERENTIAL SYSTEMS
The dynamical system E=(R,Rq,B) will be called an AR delay-differential system if its behavior B can be described as the solution set of a finite number of linear equations on the derivatives and delays of the trajectories, as indicated below.
334
~'*~ ~'~
d*
k=kl ~ ~=el Rke
VteR
--
dt k w(t - ~ ) = 0
with k l < k 2 positive integers, ~l<~2eZ, and Rat a g•
(1),
real matrix (for some positive
integers g) for kl <_k
by a I and the delay operator by r
Thus o'z:(Rq)~-->(Rq)~ is defined by a z w ( t ) = w ( t - 1 ) ,
for all teR. Using this notation,
equation (1) can be written in a condensed form as: -1
R(~l,a2,a 2 ) w = o
(2),
where R(sl,s2,s~ t) is a polynomial matrix
in the
determinates
Sl,S2,S ~-I defined by
e(sl,s2,s~l)."= Fk2 F~2 k 1.~ k=kl I..i ~=~1 Rke 81 s2. An important question is the specification of the domain where equation (2) should be solved.
Here,
in order
to
avoid cumbersome
technicalities,
we will
consider
the
trajectory domain to consist of the class of C~ functions in (Rq)e. We will denote this class of functions by C~(R,Rq). Thus the behavior ~ of an AR delay-differential system ,U=(R,Rq,~)
is described as the kernel of a polynomial delay-differential
operator
for some y e N , i.e., ~ = ker R(~l,a2,a21).
R(ai,a2,~21):C~(R,Rq)-~C|
Apart from AR delay-differential systems, we will also consider MA delay-differential systems, whose behavior can be described as the image of a polynomial delay-differential operator. More concretely, we will say that Z=(R,Rq, B) is an MA delay-differential
system, if there exists a qx~ polynomial matrix M(Sl,S2,S~l), for some ~et4, such that is represented by the following equation.
w = ~(al,a2,a~l)a
with
a ~ C|
H(al, a2,a~ 1) : C|
Re)
an
auxiliarly
variable.
(3),
Thus
~8= i m M(gl,az,a~ 1)
with
C|
Discrete time AR and MA systems have been studied in detail in [W1] and [W2]. In particular, it was shown that the classes of controllable AR systems and MA systems coincide. As we will see in the sequel, also MA delay-differential systems are strictly connected with controllability.
335
Convention.
From
delay-differential
now
on
systems
we
which
will
only
consider
are
respectively
AR and
of
the
MA descriptions for
form
and
R(~l,r
w = M ( a l , a 2 ) a , with R(Sl,S2) and M(Sl,S2) polynomial matrices. This does not imply any
loss of generality. Indeed, any AR d e l a y - d i f f e r e n t i a l equation P,(al,a2,a21)tv=o can be transformed
into
an
equivalent
equation
by
R(az,a2)w=o
premultiplication by
a
sufficiently high power ~2k of a 2. Similarly, given an MA delay-differential description w =~l(al,~2,a21)gt of a behavior B, an equivalent description w =M(az,~2)a can be obtained -
-1
k
as follows. Define a new variable a:=cr-2~, with k such that M(sl,s~,s z )s~ =: l~l(s~,s2) , with M a polynomial matrix in s~ and s~. Then, w = Y l g ~ , w = M~2a - ~ -~ ~ ~ r w = Ha. 4. CONTROLLABILITY OF AR DELAY-DIFFERENTIAL SYSTEMS.
In
this
section
we
will
derive
a
test
for
external
delay-differential system T with behavior B = k e r R ( a z , a 2 )
controllability
of
an
AR
in terms of properties of the
representation matrix R(sl,s2). For
the sake of simplicity, here,
we will restrict ourselves to
the study of
systems E = ( R , Rq,~) with a two-dimensional signal space, i.e., q = 2 . Moreover, we will consider
the
particular
class
of
systems
with
what
we
will
call
separable
AR
representations. Our first result establishes the connection between MA delay-differential systems and controllability. P r o p o s i t i o n 3. Every HA delay-differential system T = (R,Rq, B) is controllable. P r o o f . Let ~ = (R,Rq, B) be an MA delay-differential system. This means that there is e e ~ and a q•
polynomial matrix M(sl,s2) such that ~ = i m M ( a l , ~ 2 ) , i.e., ~$ is described by
the equation w = M ( a l , a 2 ) a , with aeC=(R,Re). Consider now two a r b i t r a r y trajectories w l , w 2 e ~ , and let ax,a2eC=(R,R ~) be such that w i = M a i ( i = 1,2). We will show that w 1 and w 2
can be concatenated yielding a new t r a j e c t o r y w*eB. For this purpose we will construct from a 1 and a 2 an auxiliary trajectory a* such that w* =Ma*. It is not difficult to check that there exists deR+ such that the following condition is satisfied. If ~IeC| such
that
ai[(t-d,t+d) = ai[(t-d,t+d),
with
teR,
then
6Ji := Mg i
~) is satisfies
ffJi(t) = w i ( t ) , ( ~ = 1,2). Thus, given any t e R , construct an auxiliary variable a* as follows: a*](_=,t+~)=all(.~,t+d ) w*=Ma*
and
a*l[t+d,+|174
will satisfy w*[(~,t)=Wlt(_|
is controllable 9
Then,
the
corresponding
and w*[[t+2d,+=)=w21[t+2~,+|
trajectory
This proves that
336
As we will see, under certain conditions, an All delay-differential system is MA, and therefore controllable. The obtained sufficient conditions for controllability turn out also to be necessary. A s s u m p t i o n . We will exclusively consider AR d e l a y - d i f f e r e n t i a l systems E=(R,Rq,~8) such that: (A1) q = 2 and (A2) ~ is described as Yg=kerR(al,a2) , with R(sl, s2) a 1•
separable
matrix, i.e. R is such that there exists a permutation matrix U satisfying RU =: [Piq] with p(sl,s2)=pl(sl)pz(S2), where Pl and P2 are polynomials in one indeterminate. T h e o r e m 4. Let i2 = (R,R2,B) be an AR d e l a y - d i f f e r e n t i a l system described by R(al,a2)w = o, and
suppose
that
R
satisfies
condition
(A2). Then
the
following statements
are
equivalent. (i)
12 is controllable
(ii)
rank R(A,e-;~) = constant, for all A~C.
Proof. (ii) .0 (i) Without loss of generality, we can assume that R=[p]q] with p(sl,s2) and
q(sl,s2) polynomials, and p(sl,s2)=pl(sl)pz(s2),
as in assumption (A2). Now condition
(ii) is equivalent to require that p(s,e -s) and q(s,e -s) have no common zero. We will prove that the behavior B=kerR(al,a2) can be described by the MA delay-differential equation w = M a with M(Sl,S~.):= col(-q,p). Clearly imM c_ ~3. To see that for every w e b there exists an a such that w=Ma, consider w partitioned in its two components, i.e., w = : col(wl,w2). Then w = M a is equivalent to { w l = - q ( a l , a 2 ) a
p(sl,s2)
is nonzero,
it can
be shown
that
and w2=p(al,a2)a). As
p(al,a2):C|176
is surjective.
Therefore, for all w2~C~(R,R) there exists a2~C~(R,R) such that w2---p(al,az)a~.. Thus,
wl= _qa2+ e
(4a)
w 2 = pa z
(4b),
with e := wl+q(al,a2)a 2. Note that e e k e r p(gl,aZ). Indeed, col(Wl,WZ)e~ means that
pwl=-qw2,
which together with equations (4) implies pe=o.
-q(ai,a2):kerp(al,a2)-~kerp(al,a2)
is
surjective.
This
We will next show that
implies
that
there
exists
u* ~ ker p such that e = - qa*, and hence, defining a := a 2 + a* there will hold: {w 1 = - qa and w2=pa}, yielding the desired result. Claim: Let p(sl,s2) and q(sl,s2) be two polynomials such that p(s,e -s) and q(s,e -s) have no common zeros. Then -q(al,a2):kerp(al,g2)-~kerp(ql,a2)
is surjective.
337 P r o o f o f t h e claim. We will start b y characterizing kerp($l,a2) b y means of a state space representation. Suppose that pl(sl) = Plni sinl +'.. + plsi 1 + Pl, 0 and p2( s2) = p2n2s2n2 +... + P2S2+ 1 0 w .l t h p i"1 ~ 0 r p2n29 P2, Note
that,
n2
w.l.g.,
0
we can assume P2 = 1
and p2#O.
Then,
it is easily seen that
ker p(crl,a2) has the following representation.
[ where
A12 :=
aixl =Aixl+Ai2x 2
(Sa)
~2x2=A2x2
(5b)
v=Cx~
(5c)
I:oi o] 0
A 1 :=
"
--Pl
1 --Pl
ff?l
~i
0
.~
0
.~176
I! ~
Thus v e k e r p ( a i , r
1
1
-
0
0
0
-
1
is
invertible,
pnl-I
_
nI
, A2 :=
0
~I
0
i
-202
n1
-P2
is a ni• 2 matrix, and C := [1
,n2-1
-
-e2
0 -.- 0] is of size 1xn1.
if and only if there exists (xi,x2) such that (v,(xl,x2))
satisfies
(5). As initial conditions for equation (5) we take Xl(0) and x2[[o,i ). Now given v and v* in k c r p ( a i , a 2 ) , let x and x* be such that (v,x) and (v*,x*) satisfy equation (5). Moreover define P(si,s2):=col([si-Al[-A12], if
q(O'l,a2) :ker P(al,a2)-->ker P(al,a2)
surjective.
Note
that
any
element
is
surjective
~ekerP
is
of
[0[s2-A2] ). Clearly
q:ker p-->ker p
also the
form
~,(r)=eAlr~i(O)+(cAi(~-OAi2~2(t)dt , for all r e R , and ~2(t-r)=A2~2(t), reT/. This implies that equation x=q(c,,a2)x* , with x~kerP(a,,a2)
will
~=col(~i,~2)
be with
for r e [ 0 , 1 ) and
given, is solvable for
x*ekerP(~l,a2) if and only if eA'~xi(O) = q ( a i , c 2 ) c a ' ~ ( 0 )
for
all
reR
(6)
and r
r
A2x2(t)=q(al,a2)A2x~2(t )
for r e [ o , 1 ) and f e z
A11" , It is easily seen that q(o'i,o'2)e xi(0)--q(Al,e-A1)eAl~-:c~i(0) r
r
,
r
(~). =eAlrq(A1,cAi)x*i(O), and
that q(al,a2)A2x~(t ) =q(al,A2)A2x2(t ) =A2q(at,A2)x~2(t). Thus, taking into account that A2 is invertible, (6) and (7) are respectively equivalent to:
338
(
(8)
xt(O)=q(Al,e-Ai)x~(O)
x2(t) =q(at,Aa)x~2(t )
t~[0,1)
(9)
Hence q:kerP.~.kerP is surjective if q(A1,r -al) is a surjective real matrix and q(al,A2) a surjective differential operator,
i.e., if q(sl,A2)
is a full row rank
polynomial
matrix. Using the assumption that q(A,e -A) and p(A,r -x) have no common zeros it can be shown that this is indeed the case. So q:kerp-~kerp
is surjective. This completes the
proof of the claim [:]
(i) =, (ii) First, we will show that if rankR(~,r -~) is not maximal at A=0,
then s
controllable. It is not difficult to see that if R(A,e -'x) drops rank at A=0, R(sl,s2) = stR(sl,s2)+ (s 2-1)/~(sl,s2).
is not then
Therefore Rw = 0 ~ aiR(al,o'2)w = (1 -a2)R(o't,~2)w ~,o
{ai~=(1-o'2)~ and col(~,~)=col(R,R)w}. Suppose that F:=col(R,R) is a full row rank matrix. Then it can be shown that for all C| O and ~ there exists weC|
2) such that
col (~,~)=Fw. Choose (V1,~I)= (o,o) and ( ~ 2 , ~ ) = (o,c~), with c~eR\{0}, and let w~ be such that col(O~,~i)=Fw ~ (i=1,2). As a~O/=(1-a~)~/, w i ~ k e r R (i=1,2), i.e., w 1 and w2 axe two system trajectories. We will see that w 1 and w~ are not concatenable in ~3. Suppose that
there
exists
to*ekerR
and
t o
such
that
ttr*l(-~,t0]=wll(-=,t0]
and
w*l[tr+=)=w~l[tr+r ). Denote col (ff,~'*):= Fw*. Then a~O*=(1-(~)~'*, implying that: 1+1
;i§
alO*dt= / tO
(10)
(1-a2)~'*(t)dt to
Equation (10) is equivalent to:
s.,(tl+l)-~(to)=-j
~*(t)dt+J "~'*(t)dt tO-1
(11)
t0
Clearly, O*(tl+l)=O2(tl+l ) and O*(to)=Ol(to) , and as 02(tl+l)=O=Ol(t0) the left-hand ,.h ,.~i+i ~to ,.'l ,.h ~ti+I side rt0~ (11) rt~+*is nul. On thertl+l~ hand, -Jr0_, 0 0~*+J,0V'*+Jtl r
j
=-j
~,+j tO-1
g-*=0+j
--t I
r
Ff. Thus, (11) can be written as J
- tO+l ,.t 1+1
r.t 1§
equivalent to require c~=J,1 "2(t)dt--Jt,
Ff(t)dt=O, which is
--t 1
~*(t)=0. This contradicts our assumption that
a # 0, and shows that w t and w 2 axe not concatenable. Hence S is not controllable. --
m
Suppose now that F := col (R,R) does not have full row rank. Then it is easily shown that
F=F1F 2
with
F1
and
F2
respectively
of
sizes
2xl
and
lx2.
So,
339
R(sl,s2)=[slls2-1]FiF2=f(sl,s2)F2, with f(sl,s2):=[slls2-1]F1 a scalar polynomial. I.e., R(sl,s2 ) is not a left-coprime polynomial matrix. In this case, it is not difficult to check that kerR(al,a2) is not controllable. This finally proves that if R(A,e-A) drops rank for A=0, then Z is not controllable. Assume now that R(A,e-A) drops rank for A=A*, and consider the system Z* with behavior B*=expA,~3 , with exp;~, defined by exp:~.(t)=c A-t, for all teR. Then ~* is described by a polynomial matrix R*(sz,s2) such that R*(A,e-;~)=R(A+A*,e-(X+:~*)). As rank R(A,e-~) drops for A=A*, rankR*(A,e -A) drops for A=0. Thus, by the foregoing L~* is not controllable. This implies that also E is not controllable, and therefore the proof of Theorem 4 is complete 9 5. CONCLUSION.
In this paper we introduced the notion of controllability for delay-differential systems as a property of the (external) system trajectories. We considered in particular AR delay-differential
systems with separable representations
R(sl,s2) and showed that
controllability is equivalent to the constancy of rank of R(A,e-~). In the special case where R(sl,s2) is a pure differential operator, i.e. R(sl,s2)=R(sl),
our controllability
test gives a characterization of controllability of AR differential systems which is similar to the one derived in [W2] for discrete time AR systems. The obtained results are a preliminary step towards the solution of the more complex (and interesting) question of characterizing the controllability of arbitrary AR delay-differential systems in terms of their AiR descriptions. REFERENCES
[Bc]:
Bellman, R., Press, 1963.
[BK]:
Bhat, K.P.M., and H.N. Koivo, "Modal Characterizations of Controllability and Observability in Time Delay Systems", IEEE Trans. A.C., Vol. AC-21, No. 2, pp. 292-293, April 1976.
ILl:
L~vy, B.C., "2-D Polynomial and Rational Matrices, and their Applications for the Modelling of 2-D Dynamical Systems", Thesis Stanford University, June 1981.
[M]:
Morse, A.S., "Ring Models for Delay-Differential Systems", Automatica, Vol. 12, No. 5, pp. 529-531, September 1976.
[MT]:
Manitius, A., and R. Triggiani, "Function Space Controllability of Linear Retarded Systems: A Derivation from Abstract Operator Conditions", SIAM Control & Opt., Vol. 16, No. 4, pp. 599-645, July 1978.
and
K.L. Cooke,
"Differential-Difference
Equations",
Academic
340
[s]:
Sontag, E.D., "Linear Systems over Automatica, Vol. 7, pp. 1-34.
[W1]:
Willems, J.C., "From Time Series to Linear System - Part I", Automatica, Vol. 22, No. 5, pp. 561-580, 1986.
[w2]:
Willems, J.C., "Models for Dynamics", Dynamics Reported, Vol 2, pp. 171-269, 1988.
[Y]
Yamamoto, Y., "Reachability of a Class of Infinite-Dimensional Linear Systems: An External Approach with Applications to General Neutral Systems", SIAM J. Control & Opt., to appear.
Commutative
Rings: A Survey",
Ric. di
GEOMETRIC APPROACH TO PARAMETRIC SENSITIVITY AND CAIN SUPPRESSION
W.A. Berger
Department of Physics/Electronics Engineering University of Scranton, Scranton, PA 18510 R.J. Perry
Department of Electrical Engineering Villanova University, Villanova, PA 19085
ABSTRACT This paper contains new results in the evaluation of sensitivity of eigenvalues of a closed-loop system to the perturbations in the state feedback gain elements using a geometric approach. A ha.sis for the gain vcc-tor space is developed using large-scale perturbations of the eigenvalues such that movement of the gain vector along a direction in this space affects the position of only one real eigcnvalue, or of a complcx conjugate pair of eigenvalues. It is shown that this basis can 1)c used to directly compute the incremental sensitivities of the eigenvatues to the individual gain vector elements. In addition, in the case where the system is not completely controllable, the degrees of freedom in selection of the feedback gain vector can be used to obtain gain suppression, thereby eliminating the sensitivities to the null gain elements. The set of possible solutions for the fcedbaek gain vector is identified for this case using a geometric interpretation, and optimal solutions are defined in the sense of minimum 2-norm, minimum infinite-norm (minimum m a x i m u m gain), and maximum number of zero gain elements. Nulnerical examples are included to illustrate these theoretical concepts. INTRODUCTION Tile eigenvalue assignment problem involves finding a state feedback vector k such that the eigenvalucs of A - b k T are in the desired locations. In this context, the concept of sensitivity of systems to parameter variations has been useful in the analysis and synthesis of linear timeinvariant closed-loop control systems [1-8] and active RC networks [9,10]. The normalized sensitivily (relative) ]1,9] is defined as OA Sk a
O,~
k
Ok ~
Ok
(i)
k interpreted as the ratio of the fractional change in some function ,~ to the fractional change in the parameter k, provided that all changes are sufficiently small (theoretically approaching zero). The unnormalized senMtivity (semirelative or absolute) is defined simply
342
(3k k The seltsitivity of a system can be cxprcssed in scveral ways. If the systcm is rcprcscntcd in Transfer Function Matrix (TFM) form, then the sensitivity of the system T F M can be used to characterize the sensitivity of the system. This idea was originally proposed by Bode [10] and has been widely used by control system designers as a design criterion in synthesizing feedback control systems. On the other hand, if a system is represented in the state-space form, then the s(msitivity of the system trajectories is a way of expressing the sensitivity of the system. In the state-space design of filtcrs a scalar sensitivity measure is often used [11]. Another way of expressing system sensitivity is in tcrms of eigenvalue sensitivity [12]. This is particularly true in cases where control systems are being designed to achieve specified closed-loop eigenvalucs, for example in flight control problems. GEOMETRIC
APPROACH
This section presents new results for a theory of cigenvalue modificatiolx through feedback gain vector parameter variation. A basis for the gain vector space is dcvcloped using large-scale perturbations of thc eigenvalucs such that movement of the gain vector along a dircction in this space affects thc position of only one real cigenvalue, or of a complex conjugttte pair of ~,ig~,nvalucs. This basis is also used to directly compute the incremental sensitivities of the ~,igenv~ducs to the individual gain vector elements. It was shown [13] that certain linear combinations of gain vector elcments can be used to affect lhe position of a single eigenvalue, without affecting the positions of the othcr eigenvalucs. This result is bmsed on the fact that the coefficients of the characteristic polynomial are determined by linear combinations of the gain vector elements when linear state-variable feedback is used to assign eigenvalues. However, the development in [13] was bascd on computations involving possibly ill-conditioned transformation matrices, and was valid only if the system was completely controllable from a single input. Here, these results are extended to produce a complete characterization of a "zhK-space", describing directions of gain vector changes which affect only one real eigenvalue or a complex conjugate pair of eigenvalucs. The characterization is complete bccause it describes gain vector directions for all of thc eigenvalues [14]. Furthermore, the development here can be applied to independent pole control, and also to control of individual zeros, not considered in [13]. The ,3I( directions are coml)uted by perturbing the desired eigenvalues and recomputing k using any numerically robust eigenvalue assignment algorithm. For real eigenvalues, the perturbation is performed by doubling the eigenvalue if it is non-zero, which represents a relative change of 1, or by changing a 0 eigenvalue to 1, which represents an absolute change of 1. For a complex conjugate pair of eigenvalues, two perturbations are performed. First, the eigenvalues are changed to repeated real eigenvalues, keeping their product constant and moving them along a circle to the real axis. Second, the imaginary parts are doubled, thus keeping their sum constant, representing a relative change of 1 in the imaginary parts.
343 By keei)ing either the sum or product of the eigenvalues constant in the perturbation, these large-scale perturbations will affect only that one real eigenvalue or complex conjugate pair, without affecting the other eigenvalucs [13] perturbations are chosen to represent relative or absolute changes of • in the eigenvalues only for convenience in using the t.M{ vectors t,o compute sensitivities, but larger perturbations could be used if desired. These types of perturbations are practical numerically because they involve large-scale changes (doubling) of the eigenvalucs and the computed zXK directions will therefore not involve a catastrophic cancellation of significant digits due to the subtraction of nearly cqual quantities as would occur if incrementally small perturbations were used. An inverse method, perturbing the gain vector then recomputing the eigenvalues, would not be practical, because it wouhl generally be impossible to determine which perturbed eigenvalue corresponds to which original eigenvalue. Example I For illustration tmrposes let us consider the eigenvalue assignment problem for an RLC electric network [7,81 with system matrices A=
0 1
B= -
[!!l -
desired cigenvalucs arc ( -3.3247 • j 1.1246, -2 ) and the required feedback gain vector to the first input is k T = [ 0.6624, 4.7420, 3.0796 ]. The
The rows of the following AK matrix represent the basis for the zXK-space for this example AI{ =
-3.4172 -3.4172 0.0000 0.0000 0.9485 0.9485 0.5000 3.5796 2.3247
If, for example, the feedback gain vector k T is modified by adding to it (0.5000, 3.5790, 2.3247) from row 3 of the ~ matrix, only the cigenvalue at -2 will change, shifting to -4. If ~ times row 3 of ~K. is added to k T, the eigenvaluc at -2 would change proportionately, and the other eigenvalues would not change. If row 1 of &K is added to k T, only the complex conjugate eigenvalues will change, both moving along a circle to ~r z - 3.5098. If a times row 1 of ,5I{ is added to k w (0 < a < 1), the complex eigenvahms would move a proportional amount along the circle, and tile other eigenvahm would not change. If row 2 of ,Sl( is added to k T, only the complex conjugate eigenvalues will change, moving to -3.3247 • j 2.2492, i.e. their imaginary parts are doubled. If row 2 of AK is scaled by some factor and added to k T, the imaginary parts of the complex eigenvalues will change proportionately, and the other eigenvalue would not change.
A chock of the eigenva, hms in all these cases confirms the results.
344 INCREMENTAL
VARIATIONS
The incremental sensitivities of the real eigenvalues to the feedback gain vector elements are easily computed directly from the AK matrix using the formula S~ = ~-T
diag (k T)
(3)
where A is the vector of eigenvalues. Sk~ is a n , n - m a t r i x whose (i , j ) element represents the incremental relative sensitivity of Ai to kj for non-zero eigenvalues, or the incremental absolute (un-normalizcd) sensitivity of Ai to ki for cigenvalues equal to 0. This formula for sensitivity is derived by considering how the eigenvalues would change if one element of k w was changed by a unit amount, e.g. a change in k r of [ 1 0 ... 0 ]. The equation y T A K = [ 1 0 ... 0 ] would have to be solved for yW which would represent the coordinates of this unit change in k I in the basis formed by the row-vectors of AK. Yi would indi.cate how Ai changes due to this change in k 1. Since each row of AK was computed by doubling an eigenvalue, the relative change of the eigenvalue is l; the unit change in k I considered here means Akj is 1; so simply multiplying y by k~ will change it into sensitivity values. For eigenvalues equal to zero, an absolute change of 1 was used in computing AK, so the sensitivity h~rmula gives an absolute (un-normalized) sensitivity in that case. Solve for yT using yT = [ 1 0 9 9 - 0 ] ~:XK-1, i.e. y = ~ - T [ 1 0 - 9 9 0 ]T. Do this for each unit change in k i and represent the result by a matrix of these y vectors, Y = A K - T I . Each column of Y is then scaled by the corresponding value of k to turn it into sensitivity values, i.e. Sk ~ = LsK-Tdiag (k) as in (3). The incremental sensitivities of the real and complex parts of complex conjugate eigenvalues to the feedback gain vector elements involves a more complicated derivation. What is needed is two ,Sk directions: one in which only the real parts of the complex eigenvalues change, and the other in which only the imaginary parts change. A complex eigenvalue, r 4- j i, with real part r and imaginary part i, has a polynomial rcprcsentation s 2 + sa + b
(4)
where a = - 2 r and b = rZ+i z If the gain vector is perturbed by &k, the coefficients a and b become a = - 2 r + yWzSk
(5)
b -- r 2 + i2 + z T & k where vectors yT and z T represent linear combinations of the gain vector elements. Let Ak 1 represent a direction that affects a without affecting b (zSkI orthogonal to z), and let Ak 2 represent a direction that affects b without affecting a (Ak 2 orthogonal to y). In the first case, changing a without changing b means that the magnitude (r2+i z) of the eigenvalues is staying
345 t h e s a m e , b u t the real a n d i m a g i n a r y p a r t s are c h a n g i n g , i.e. they are m o v i n g on a circle. In t h e s e c o n d case, c h a n g i n g b w i t h o u t c h a n g i n g a m e a n s that, t h e m a g n i t u d e of the eigenvalues is c h a n g i n g , b u t the real p a r t is not, t h u s only t h e i m a g i n a r y p a r t s are c h a n g i n g . T h u s , we can directly find a gain v e c t o r p e r t u r b a t i o n d i r e c t i o n w h i c h affects only the imag i n a r y p a r t s of t h e eigenvahles, b u t we can n o t find a d i r e c t i o n (using large scale changes) w h i c h affects only t h e real parts. S u p p o s e the real p a r t s of the cigenvalues c h a n g e to same.
r+br
a n d the i m a g i n a r y p a r t s stay t h e
T h e c h a r a c t e r i s t i c polynoraial is t h e n s z + sa + b = s 2 + s
(-2r-2~r)
+ r ~ + i z + 2r~ir + ~r 2
(6)
If b r is small (as it will be for i n c r e m e n t a l s e n s i t i v i t y calculations) we can ignore the 5r 2 t e r m and derive a b k direction in t c r m s of z ~ 1 a n d z~k 2 discusscd above. A s s u m e z_~kI m o v e s thc eigenvalues along t h e u n i t circle to t h e real axis, m a k i n g t h e m double positive real. A s s u m c '.3k z doubles t h e i m a g i n a r y p a r t s o f t h c eigcnvatues. a = -2r b
W e t h e n have
+ yTLkkl = - 2 Y~rZ~i -2
= r z + i z + zTL)&2
(7)
= r z + 4i 2
Solve for y'r,Ak l a n d zTAkz in t e r m s of r a n d i = 2r-
(s)
2
zYL~kz = 3i z We w a n t b r = r, so t h a t the relative c h a n g e in r is 1. T h e c h a n g e in a m u s t t h e n be
-2r
i n s t e a d of 2r - 2 V ~ r Z + i z a s it would be if ,_%kI were used, so we m u s t scale z:fl~l to get t h e desired amount r
Z~kl
(9)
T h e c h a n g e in b m u s t be 2 r 6 r = 2r ~ i n s t e a d of 3i ~ as it w o u l d be if ,._~k2 were used, so we n i u s t scale z:2ak2
Using an incremental 6k in the direction t = t I + t 2 will then change only the real parts of the eigenvalucs without changing the imaginary parts. In the case of complex conjugate eigenvalues with real parts equal to zero, the change in r must be br = i to give an absolute change in r of I. Since r = 0, formula (12) for yTLkkl r e d u c e s to yT,.'3k~ = - 2 i T h e c h a n g e in a m u s t be -2 i n s t e a d o f - 2 i , so s i m p l y scaling z:~kI
(li)
346 gives the required direction vector. The direction of gain vector perturbation needed to incrementally change the imaginary parts of the cigenvalues without changing the real parts is given by Alcz, but the magnitude of this vector must be adjusted to apply to incremental changes. Suppose the imaginary parts of the eigenvalues change to i + 6i and the real parts stay the same. The characteristic polynomial is then s z + s ( - 2 r ) + r 2 + i 2 + 2it~i + 6i 2
(13)
For incremental changes, tSi2 approaches 0 and can be ignored. We want 6i = i so that the relative change in i is 1. The change produced in b by Ak 2 is 3i 2 as in (12) but we want a change of only 2i 2. Thus, multiplying &k 2 by (2/3) will produce the desired vector
t = (2/3)~k~
(14)
To summarize, equation (3) is valid only for real eigenvalues. For complex conjugate eigenvalues, the direction vectors specified in the AK matrix can not be used directly in the sensitivity formula, they must be modified as discussed above to produce directions in which only the real or imaginary parts of the eigenvalues change incrementally. We therefore modify (3) to the following form which will be valid for both real and complex conjugate eigenvalues Sk X =
6K-Wdit,9(k)
(15)
where 6 K is a matrix whose rows are the same as the corresponding rows in AK for the real eigenvalues, but whose rows corrcsponding to complex conjugate pairs of eigenvalues are modified using (9) and (10) for non-zero real pazts, or (121 if the real part is zero, and (14) for the imaginary part. For complex conjugate eigenvalues, S k l ( i , j ) represents the sensitivity of the real parts to ki, Sk'~(i + l , j ) the sensitivity of the imaginary parts. Example 2 The b K matrix derived from the ,.%K matrix in Example 1 is [I.66247.18925.52691
6K = ]0.0000 0.63zz 0.6323[ [0.5000 3.5706 2.3247J Note that row 3 of &K is the same as row 3 of AK from Example 2, corresponding to the gain vector direction which affects only the real eigenvalue. The first two rows of 6 K differ from AK, row 1 of 6 K indicating the direction in which incremental changes in k affect only the real parts of the complex conjugate eigenvalues, and row 2 the direction which affects only the imaginary parts. Using (15), the sensitivity matrix is
t=
i
(12)
347 0.6624 -1.8894 1.2271 Sk A = /--2.5636 --6.5802 9.1438 [--0.8774 6.2818 --4.0796 T h e first two rows of Sk "~ indicate respectively the sensitivities of the real and imaginary p a r t s of the complex eigenvalues to k, a n d the last row indicates the sensitivities of the real eigenvalue. EIGENVALUE
ASSIGNMENT
Assume an n th-order linear system, described by state-space p a r a m e t e r s ( A , b ) , is reduced by an o r t h o g o n a l t r a n s f o r m a t i o n U to an U p p e r Hcssenberg F o r m (F,g) = ( U T A U , U T b ) and is not completely controllable, i.e. nc < n where nc is the size of the controllable subspace [15,16]. T h e n F =
Fz2]
g = [ g l 0 ... 0 ]T
(16)
and the m a t r i x F is block diagonal, where F l l is a s u b - m a t r i x of size (nc ,no), and F22, corre.~ponding to the uncontrollable subspace, is of size (nuc ,nuc ). Let the feedback gain vector v in the transformed coordinate space be p a r t i t i o n e d v T = [ v l T ] vz w ] where v I is of size n.c aml vz is of size nuc and let g be p a r t i t i o n e d similarly, gT = [ gl T I gz T ], noting t h a t g2 = 0. T h e cigenvalue ~ s i g n m c n t equation is then
F-gvT
=
[
FII- girl T 0
Flz-glv2 T F22
(17)
Since w does not modify the elements of Ii'lj or :F22, the eigenvalues of F are not affected by v z. This creates additional degrees of freedom in the gain selection for the assignment of desired eigenvalues [14]. Let the t r a n s f o r m a t i o n m a t r i x U be p a r t i t i o n e d by columns U = [ U l I U2 ] where U l contain.~ the first nc columns of U and U 2 contains the r e m a i n i n g nuc columns. Transforming the vector of gains v back into the original set of coordinates k = Uv
k I = Ulv 1
= k I +k
2
(18)
k2 = U2v 2
It can be n o t e d from (17) and (18) t h a t the eigenvalues of A - b k T = U ( F - g v T ) U T are not affected by the lincar c o m b i n a t i o n of the last hue columns of U . The c o m b i n a t i o n of gains k 2 can be a d d e d to k I without affecting the eigenvalue assignment.
348 MINIMAL
NORM
GAIN
Using a geometric interpretation, k 2 lies in the column space of the m a t r i x U2, which is the o r t h o g o n a l c o m p l e m e n t of the column space of U l , i.e. t h e null space of U1T. The last nuc columns of U therefore form an orthogonal basis for the space of possible s t a t e feedback gain vector modifications, k~. It is w o r t h pointing out t h a t the feedback gain vector k I is unique, a n d since the final k is a linear c o m b i n a t i o n of the o r t h o - n o r m a l columns of U , it will have m i n i m a l 2-norm if we let k2 = 0. Adding on any additional columns of U 2 would only increase the 2-norm of the final k. However, it is possible to use these columns to minimize the infinite-norm of k, or to maximize the n u m b e r of zero elements in k. By choosing v 2 in (18) to minimize the infinite-norm of k, the m a g n i t u d e of the m a x i m u m gain element in k m a y be reduced. An iterative non-linear fimction minimization, or a direct m e t h o d such as [17], could be used to solve for t h e required v 2. B o u n d s on the relationships between the 2-norm and infinite-norm [18] imply t h a t the m a x i m u m gain element may be 1
reduced by at most a factor of v/~_ n_ in m a g n i t u d e .
GAIN
SUPPRESSION
Since only nc eigenvalues are being assigned by the feedback gain vector k, it should generally be possible to find a solution with o n l y nc non-zero elements in k. T h u s , nuc is the m a x i m u m n u m b e r of null gain elements and is the n u m b e r of p a r a m e t e r sensitivities which can be therefore be m a d e zero. thereby eliminating the sensitivities to the null gain elements. Let s = ( s l . . . . , s,,uc ) represent the set of indices for which null gain elements are desired. Let P s represent a m a t r i x of sizc (nuc , n u c ) o b t a i n e d by taking the rows of U 2 as specified by s and let r s r e p r e s e n t a vector containing the nuc elements of - k I as sl)ecified by s. By solving the e q u a t i o n P,v 2 = r,
('19)
for vz, the resulting gain vector k will have the m a x i m u m n u m b e r of null elements. Example
3
Consider a 4th order system with a controllable subspace o f s i z e 1 a n d k I = [ -0.62802
U=
-0.69625
-0.19885
- 0 . 2 8 5 1 0 ]T
- 0 . 6 2 8 0 2 --0.03333 0.46926 --0.61990 --0.69625 --0.28844 --0.60114 0.26582 -0.19885 0.94995 - 0 . 2 3 8 9 9 - 0 . 0 3 0 5 3 -0.28510
0.11525
0.60109
0.73765
349 To obtain a solution with the maximum number of zero gain elements, let s = ( 2, 3, 4 ), solve (19) for vz, and use (18) to obtain k
[ -
0
.
2
8
8
4
4
-
0
.
6
0
U
4
0
.
2
6
5
8
2
[0.69625]
~
0.94995 -0.23899 -0.03053 0.11525 0.60109 0.73765 v z = [0.0531
k = [-1.5923
-0.7472
v2
= 10.19885 / [0.28510]
0.9871]T
o.o o.o O.OlT
In this solution, only k I is non-zero. Similarly, solutions with only k 2 or k 3 or k 4 non-zero may be obtained by using s = ( 1, 3, 4 ) or s = ( 1, 2, 4 ) or s = ( 1, 2, 3 ) respectively k = [0.0
-1.4363
0.0 0.0IT
k = [ 0.0 0.0 -5.0288
O.0 ]T
k = [0.0 0.0 0.0 - 3 . 5 0 7 6 ] T
CONCLUSIONS In this paper a geometric approach to the analysis and computation of the sensitivity of the eigenvalues to the perturbations in the feedback gain elements is presented. Also, a new concept of complete characterization of a space describing directions of gain vector changes for all eigcnvalue.s is introduced. The set of possible solutions for tile state ferdback gain vector in an uncontrollable system is identified using a geometric interpretation. The additional degrees of freedom in the gain ~elcction can be used to achieve some optimal results, such ms maximizing the number of zero coefficients, thereby eliminating tlw sensitivities to the null gain elements. The approach used ensures that the results are numerically robust. presented to illustrate the theoretical concepts. REFEREN
Numerical examples are
CES
[11
L.O.Chua and P.M.Lin, Computer-Aided Analysis of Electronic Circuits. Cliffs: Prentice-Hall, 1975.
Pl
R.J.Perry. H.H.Sun and W.A.Berger, "Sensitivity of Pole-Zero Configurations in Transfer Function Matrix Realization", Proc. I E E E 1987 ISCAS, Philadelphia, May 1987.
[3]
J.B.Cruz, Jr., J.S.Freudenberg and D.P.Looze, "A relationship between sensitivity and stability of multivariable feedback systems," I E E E Trans. Automat. Contr., Vol.AC-26, no.l, Feb. 1981.
[4]
L.F.Godbout and D.Jordan, "Sensitivity Int.J.Control, Voi.34, no.3, pp.561-576, 1981.
functions
for
linear
dynamic
Englewood
systems,"
350
[5]
S.D.O'Young and B.A.Francis, "Sensitivity Tradeoffs for Multivariable Plants," IEEE Trans. Automat. Contr., Vol. AC-30, no.7, July 1985.
[6]
A.Deif, Sensitivity Analysis in Linear Systems. Springer-Verlag, 1986.
[7]
K.V.Raman and A.J Calise, "Modal Decoupling Insensitivity", IEEB Trans. Automat. Contr., VoI.AC-32, no.6, June 1987.
[8]
A.Sambadan and P.C.Chandrasekharan, "Design of output feedback controllers with eigenvalue and eigenvector insensitivity," Int. J. Contr., Voi.33, No.5, pp.935-943, 1981.
[9]
L.P.Huelsman and P.E.Allen, Introduction to the Theory and Design of Active Filters, McGraw-Hill, 1980.
[10[ H.W.Bode, Network Analysis and Feedback Amplifier Design. Van Nostrand-Reinhold, Princeton, N.J., 1945. [11[ W.J.Lutz and S.L.Hakimi, "Design of Multi-Input Multi-Output Systems with Minimum Sensitivity", IEEE Trans. Circuits Syst., Vol. CAS-35, no.9, September 1988. [12] J.H.Wilkinson, The Algebraic Eigenvalue Problem. London, England: Oxford University Press, 1965. [13] R.Perry, "Synthesis of Transfer Function Matrices using Pole-Zero Control," Ph.D. Thesis, Drexel Uaivcrsity, 1981. [14] W.A.Berger, "Complete Parameter Control in Multivariable Systems, a Numerically Robust Approach with Applications", Ph.D. Thesis, Drexel University, 1988. [15] G.S.Miminis and C.C.Paige, "An Algorithm for Pole Assignment of Time-invariant Linear Systems," Int. J. Contr., Vol. 35, pp. 341-354, 1982. [16] R.V.Patel, "Algorithms for Eigenvahm Assignment in Multivariable Systems," Advanccs in Computer-Aided Control System Engiimering, C.J.Herget and M.Jamshidi, Eds. New York:North-Holland, 1985. [17] R.H.Bartels, A.R.Conn, and C.Charalambous. "On Cline's Direct Method for Solving Overdetermined Linear Systems in the l oo Sense," SIAM J. NUMER. ANAL., Vol. 15, No. 2, pp. 255-270, April 1978. [18] G.H.Golub and C.F.VanLoan, "Matrix Computations," The Jolm Hopkins Univ. Press, Baltimore, MD, 1985.
A
KIND
OF
NONLINEAR
REDUCTION
AND
Weilin
SYSTEMS
STRUCTURAL
Yang
siying
AND ITS CONTROL*
Zhang
D e p a r t m e n t of A u t o m a t i c C o n t r o l N o r t h e a s t u n i v e r s i t y of T e c h n o l o g y Shenyang, P . R . o f C h i n a 1 1 0 0 0 6 A b s t r s c t ....... In p r a c t i c a l s i t u a t i o n s we o f t e n e n c o u n t e r such kind of s y s t e m s w h i c h p o s s e s s s i m i l a r s t r u c t u r e . In a c c o r d a n c e w i t h the special structure of such k i n d of systems, we use modern differential geometric a p p r o a c h in this p a p e r to analyze the properties p o s s e s s e d by s i m i l a r s t r u c t u r e . B a s e d on the special structural properties, a n e w w a y of s y s t e m r e d u c t i o n is g i v e n in this paper. In addition, a relative structural control is also proposed.
Kcywords: similar structure, nonlinear system reduction, structural control
I.
In
a previous
symmctry control the
studied
system
system.
could and
was
could In
systems
which
be p r e s e n t e d
static note
He
that
into
possess were
a
shows out
feedback
for most difficult
often
some
will
those
systems
systems
which
that
method,
dimensional
addressed.
have
similar
The [3].
may
In his
system
control
too
control for
the we
ideal,
systems
such
to
paper,
However,
in p r a c t i c a l structure,
about
results
bc
a
subsystems
condition
problem.
condition
Fortunately,
the
for n o n l i n e a r
necessary
control the
that
of d e c o m p o s i n g
in
in
information
lower
contained
and
possessed
symmetry
a
shown
problem be
systems
also
decomposition
non{ntcracting
to verify.
the
structural
of
paper,the
sufficient
control
shown
cascade
by
[2],
it was
a feedback the
in
to d e d u c e
similarity
motivated
especially meet
was
particularly,
gives
state
also
It
loop. In this
Ninmcijer
systems.
.
geometric
Instruction
[!],
bc u s e d
be d e c o m p o s e d
a feedback
Henk
paper
system~
as
we the
352 control
systcms
generator
sets,
some
districts
such
special
the f e e d b a c k In
of c l e c t r i f i c d the
production
in social structure
has
to begin
static
feedback
of the
form
In this
special
of the
network
several
and m a r k e t
paper,
properties
systems
of
we find out
that
and could b r i n g
about
system.
our analysis,
nonintcracting
possessing
organization
systems.
decomposition
order
wire
here
problem.
we
briefly
Considering
review
a control
the
system
.I
=
A(X)
+ ~ Bi(X)u i is! H. (X) i=l ..... m
Z. = l
".:here
coordinates
of a s m o o t h
B I ..... B m are smooth v e c t o r output
map
C N i. we assume The seek
(l.lb)
1
X is local
smooth
(!.la)
a control
on
from M to a s m o o t h
that
static
fields
the system
state
feedback
n-dimensional M,
and H. :
Pi-dimensional
(l.la)
is s t r o n g
noninteracting
manifold
M~
>N.
manifold
is
M, a
(pi~!)
acccssiblco
control
problem
is
to
law of the form
u = a(X)+b(X)v whcrc
a:
M"
>R m, b: M -
>Kmxm
arc s m o o t h
maps.
b(X)
is n o n s i n g u l a r
;n
for
all
v
in
M.
Let
~.s
,~,(X)=A(X)+ ~
m
B i(X)a i(X)
and
B. (X)=
~-I
.(X)bj. is| dynamics
(X),
then in
suitable
m X = A ( X ) + ~ B- (X}v;
i
i=l I
xo
should
"
Xm
Am (Xm)
I
z 1 = H I (x I) z 2 = H2(x 2) z m = H m (x m )
coordinates
modified
v~
"{o(xo). o +
the
read
2(x 2) =
z
local
O
~ vl
Bm(Xm)
m
(1.2)
353 .... X = ( x I .x2,
where
The
x m )T w i t h e a c h
solution
x.i and
of above p r o b l e m
is
possible
Zi
decided
being
by
the
a vector
"
following
theorem. m
Theorem
1
[3] ;
Under
the a s s u m p t i o n :
~
KcrH.
i=l
assumption feedback fashion
that
each R. has l
nonintcracting
control
dimension,
problem
and
the
the
static
is s o l v a b l e
in
state
a
local
if and only if R~
+
1
K.
1
m* = s u p r e m a l ~'i:
whcrc
fixed
: 0;
l
{] and
=
TM
regular
local
controllablity
distribution
in
KorM~ K. = KerH~ I I"
In this p a p e r we c o n s i d e r hope
that
rC!cvant that
the s t r u c t u r a l to
the
the d e c o m p o s i t i o n
into
2.
In
propcrtios
s y s t e m with s p e c i a l
dccomposcd
this
system with
the s y s t e m w i t h
similar
At first Definition
structure
structure
section,
study
structure.
we give 2.1:
we
of s u c h
to
has
systems
arc
Our o b s e r v a t i o n
the p o s s i b i l i t y
is
to
bc
(1.2).
and p r o b l e m
formulation
the d e c o m p o s i t i o n
Here
wo
structure.
kind
of the s y s t e m s .
thc form s i m i l a r
similar
special
we o n l y
the d e f i n i t i o n
For tho f o l l o w i n g
consider
of s i m i l a r
problem affine
of
the
system.
structure.
system
(2.1)
Yl = hl (X) Y2 = h2 (X) where
X=(XI,X2)T,xI,X2C
{l.l).If
the f o l l o w i n g
(I)
K 1 + K 2 = TM
K n. The
required
conditions
arc
the
same
two c o n d i t i o n s (2.2)
as
354 (2) are
there
exist
two m a t r i c e s
PI,P2
by w h i c h
the
following
forms
hold
xI lip 1 (X)
nxn
-r
f2(PI(X)) (2.3)
0
nxn
J
{2.4) h 2 (X)
nxn are
satisfied, Now
the
we
then
study
the
t~-o s u b s y s t e m s
to us
that
the
we m a y
system
have get
a suitable
system
subsystems
with
dimension
such
kind
~;~,,
which
the
system.
forms
(I)
the
K. 1
By
Whether
to
control
How
to o b t a i n
The
key
of
the
ideas
(~.I),
then from with
the p r o b l e m
system
with
An
into if
can
[3],
idea
find
we w i s h
the
that
and
two
structurc o definition,
easily
can
could
to deal
to find
a
can be
statc
be
with
feedback
be d e c o m p o s e d
static
to
noninteracting
a way
problem
the
appear
coordinate
the p r o p e r t i e s
similarity
Therefore,
problem the
2n
we
above
structure
In a d d i t i o n ,
or not we can p r o v e
{2)
in
n.
(1.2).
control
similar
From
structure.
dimension
system,
the
system
similar
noninteracting
the
with
to have
properties.
the s i m i l a r
the
to d e c o m p o s e
is d e f i n e d
structure
decompose
used
h o (P2 ( X ) )
into
stated
as
feedback
is s o l v a b l e ;
feedback is to
similarity.
control
find
the
law. relation
between
,,. 1 D *
and
355 3. The r e d u c t i o n In the b e g i n n i n g Lemma
3.1:
For
of this
(2.1),
K1 N Proof:
that
of s y s t e m s
K2
section
if
=
(2.4]
with
similarity
we give
holds,
the f o l l o w i n q
lemma.
then (3.1)
0
We k n o w K1
=
Kerh 2
K2
=
Kerh 1
is < dh I ,x > = 0,
from
(2.4}
< dh 2
>
,y
= 0
x< M
,
y C [-I
[0 nliI [ "I =
Inx n If wc a s s u m e PCM
which
0
/
h2
(3.2)
h2 (P2 (X))
(3.1) does not hold,
there
should
exists
a
point
satisfies hI
= ~
Wo
note
that
result
(3.3)
Based
(3.3)
h2 the m a t r i x
in
is c o n t r a d i c t i o n .
on Lemma
3.1,
(3.2)
is n o n s i n g u l a r ,
therefore
the
So the Icmma holds.
we can b e g i n
our a n a l y s i s
,
= 0
of the
reduction
problcm. Since also
K1 + "'2v = TM
from
(3.2) we can e a s i l y get
N o w wc c o n s i d e r distribution Lemma R1
K1 {~
in ~
j~i
3.2: ~]
9
R.: 1
a2
Proof:
This
definition
of R~
that d i m ( K l ) = d i m ( K 2) = n
the s u p r e m a l
Next
For s y s t e m
K2
Icmma
(2.1),
regular
gives if
local
controllability
the c h a r a c t e r a s t i c
(2.2), (2.3), (2.4)
hold,
of E[
z
then
= 0 is
a direct
consequence
of
Lemma
3.1
and
the
1
The
assumption
Lcmma
3.3:
(2.2), (2.3), (2.4)
Assume
would
(2.2),(2.3),(2.4)
lead
hold,
to f u r t h e r
then
result.
356 D ~
"'I + Proof:
Since
(2.2),(2.3)
R2 =
thc
result
obviously
accessible.
From above Theorem
formulation
accessible,
addition,from
and
Lcmma
from 3.2,
dim TM
contradict
Therefore,
the
assumption
the L e m m a
our m a i n
For
a
affine
described
as
nonlinear
that the s y s t e m
is
holds.
we can o b t a i n
theorem. control
(2.1),(2.2),(2.3),(2.4),
system
with
following
the
are s a t i s f i e d .
R~ + K. = TM 1
from
<
results,
3.1:
similarity
i=I,2
l
Proof:
strongly
in fact that m e a n s
d i m ( R 1 + R 2)
strongly
is
system
we can get d i m ( R 2 ) = d i m ( R ~ ) . I n
if R 1 + E 2 = TM,
this
TM
Since
the r e s u l t s
directely
get
Based
on
R1 C
K2
, E
of L c m m a
C
3.1,
K1
Lemma
3.2,
and L e m m a
3.3,
we
can
the a n s w e r
of
the
the c o n s e q u e n c e . the above result,
we h a v e
gotten
f i r s t p a r t of our r e d u c t i o n p r o b l e m . 3.2:
Theorem can
A nonlinear
affine
control
about a feedback decomposition
bring
system with
similar
to
similarity
the
form
of
(1.2).
Proof:
This
result
is i m m e d i a t e
from Theorem
I.I and
Theorem
3.1. Remark
I:
system withe it
can
According dimension
be d e c o m p o s e d
dimension Remark
to T h e o r e m
2n,
3.2,
a nonlinear
if it p o s s e s s e s
into two
similar
noninteraeting
n. 2: For more g e n e r a l
Xl = f l ( X l ' X 2 ..... Xn)
+
situation g l ( X l ' X 2 ..... X n ) U l
affine
control
strurture, subsystems
then with
357 Xn = f n ( X l ' X 2 ..... Xn)
+
g n ( X l ' X 2 ..... X n ) U n
Yl = hl (Xl'X2 ..... Xn)
Yn = hn (Xl'X2 ..... Xn) where
xi
and Yi p o s s i b l l y
subsystems
has
similarity
w i t h s i m i l a r o r d e r n.
similar papers
4.
result
similar
being
structure,
like s y s t e m
In
structural
this
reduction
because
section,
that
can
For such k i n d
systems,
we can get
T h e r e w i l l be d i s c u s s i o n
with
the s e c o n d
that a c c o r d i n g
full use is a l s o
of some k i n d
the system.
section,
in o t h e r
similarity
part
That
of
In
our
the r e d u c t i o n
affine
we p r o p o s e
=
the f e e d b a c k
control
a(X)
system
the p r o p e r t i e s problem.
system
can make
property,
law w h i c h
I: C a l c u l a t i n g
and the s u p r e m a l
regular
Ki
=
local
K e r ( h i) controllability
a we
can lead
to
+ b(X)v a(X),
In
what
law of the form
to o b t a i n
method
If we have
control
control
to use s u c h
control
the f o l l o w i n g p r o c e d u r e
Ker(h~)
of
of the system.
Considering u
we can use
structure
control
convenient.
is the s t r u c t u r a l
order
system
properties
relative
systems,
we k n o w a n o n l i n e a r
dccomposition.
to
over general
of the
do now is to find the f e e d b a c k
Step
of s y s t e m s
consider
to i m a g e
Its r e a l i z a t i o n
with
feedback
~
the
s y s t e m have
problem
we w i l l
make
the p r o p e r t i e s
to d e a l
j$i
of
the w h o l e
c o n t r o l mode has the a d v a n t a g e
structure.
in
pair
problem.
to d e s i g n
should
If each
then we call
(2.1).
control
It is not d i f f i c u l t
above
.
of thc authors.
The
known
a vector
b(X)
i,j = 1,2 distribution
R~ l
358 Step
2: C h o o s e T = R I.
Step
choose
3:
a set
Step
coordinates { Z~
=
I i=l, ~
sp[dz
j=l, .... n i } ,
make
i=I,2
Let Di
= sp{~|
Di
=
of b a s i s
4: C o m p o s i n g
(i) C h o o s e
I j=l ..... n i I , i = ! , 2
of
D. (i#j) 3 G/D l .
the
feedbacklaw
a nonsingular
following
B ! ,B 2 f r o m
j=l,..,n i } which
matrix
b
o
which
is C ~ , a n d
obtain
formulation
I • gb~
as f o l l o w s
=
X
0
B1 B2
then bi =
(sTBi )-I
i=I,2
a = -U( 5, 5z ) (DTD) -ID T ( ~, ~,
)
b = bo(Inx n + b I + b 2 ) where D =Blb
I + B2b 2
L = ET(EET) -I E = Because
of
definition
of
the the
(BTB)-IBTG limited notion.
space, The
we
wouldn't
interested
reader
giv~ can
the see
detail reference
[7]
5. E x p l a i n a t i o n
Making new
full
research
use
of s y s t e m
direction
and
conclusion
structure
to d e s i g n
of c o n t r o l
theory.
The
control
basis
mode
of this
is
a
method
359 is
that
we must have
full
structure.
Fortunately
of
~hich possess
systems
similar
structure.
method ~:ay
,
understand in p r a c t i c a l
special
structural
such
control
not l i m i t e d
we use
to the s y s t e m r e d u c t i o n , obscrvability
arc a lot of ~.:ork r e m a i n
such as
a lot
and o b t a i n
include
and
of g e o m e t r i c a
a
the s t r u c t u r a l
and s y s t e m
to do in this
exist
proposc
they also
system
symmctry
the a d v a n t a g e
We also
Obviously,
of
there
structure
systcms.
approach.
of c o n t r o l l a b i l i t y ,
situations
the s i m i l a r
kind of
properties
structure,
In this p a p e r
to s t u d y and a n a l y z e
to r e d u c e
the
new
relatcd
control
is
the a n a l y s i s
stabilizability.
There
field.
REFERENCES
[!]
W c i l i n Y a n g and S i z i n g Z h a n g
Control
stsrcms
27th CDC, [2]
Austin,
Systems
AC-30,
[3]
No.
and S.I.
possessing 4, M a r c h
Honk Nijmeijcr
Systems",
symmetry
structure",
in N o n l i n c a r
Proceeding
of
the
1988
J.W.Grizzle
Control Vol.
and R e l a t e d C o n t r o l
"Rotation
IEEE Tran.
Marcus
Symmetries",
1985,
Structure
of
Nonlinear
Auto.
Control,
of N o n l i n e a r
Control
IEEE Tran.
pp248-258
"Fccdback Auto.
"The
Decomposition
Control,
Vol AC-28,
No.
8, August,
of
Mechanics",
1983,
pp861-862 [4]
R.Abraham
Benjamin [5]
and Cummings,
A.Isidori
Springer-Verlag, [6]
No.2,
MA.
Control
Systems
:
An
Introduction",
1985 and oct.
Geometric
April,
"Foundations
2rid e d . , 1 9 7 8
"Nonlinear
A.Isidori
Differential 26,
and J . E . M a r s d e n
1981,
"Nonlinear
Approach", p331
Decoupling
IEEE Tran.
via
Feedback:
Auto.control,
A
Vo!.AC-
36O [7]
D.
chinese),
*
This
Foundation
Cheng
Science
work
"The Geometric Publishing House
was
of Pcople's
supported Republic
Approach
to Nonlinear
of P.R.C.,
by
the
of china
Systems"
(in
1988
National
Nature
Science
RE JET DE PERTURBATION
DANS LES SYSTEMES
STRUCTURES
C. C O M M A U L T , J.-M. D I O N , A. P E R E Z Laboratoire d'Automatique de Grenoble E.N.S.I.E.G. - I.N.P.G, B.P. 46 38402 SAINT MARTIN D'HERES, France
Abstract. In this paper we solve the disturbance rejection problem for structured systems. A graph is associated with such a structured system. It turns out that disturbance rejection is solvable by state feedback whenever "the control inputs reach ale outputs more quickly than the disturbances" in the associated graph. We characterize in terms of input-output paths in the associated graph the in_fmitestructure of a structured system. Then we solve as a consequence the disturbance rejection problem. Keywords : Structured systems, Graphs, Disturbance rejection, Infinite structure. I. I N T R O D U C T I O N La plupart des systemes physiques pr6sentent une certaine "structure" qui r6side dans des relations parfaitement d6termin6es entre certaines variables repr6sentatives. I1 est ~ rioter que le plus souvent cette structure provient de la certitude que certaines variables n'ont pus d'interaction directe. Les autres relations entre variables sont plus ou moins bien connues et pour un module donn6 seront repr6sent6es par des pamm~tres sujets b,erreur d'identification. Les syst~mes structur6s sont un module s6duisaiat pour repr6senter ce type de situation. Une repr6sentation d'~tat structur~e est une repr6sentation d'6tat oft les 6I~ments des matrices (A, B, C) sont soit : - des z~ros - des param~tres ind6pendants Pour de tels syst~mes on peut 6tablir des proprigtr
structurelles, c'est h dire des propri6t6s
associ6es ~tune structure et qui soient vraies pour "presque toute" valeur des pamm&res. L'6tude des propri&gs structurelles trouve un outil puissant dans te repr6sentation graphique du syst6me. Une importante litterature a 6t6 consacr6e h l'6tude de la commandabilitr structurelle [1] [2], et depuis quelques ann6es des r6sultats commencent h &re obtenus sur la commande des syst~mes structur6s [31 [41. Darts un ouvrage r6cent [5] Reinschke fait le point sur les rgsultats existants et propose un certain nombre d'idfes nouvelles et int6ressantes sur le sujet. Darts cet article, nous donnons une interpr6tation graphique de la structure/~ l'infini qui nous parait plus naturelle que celle proposfe dans Reinschke. Nous en d&iuisons une condition n6cessaire et suffisante de rejet de perturbation structurel qui est extr~mement simple et intuitive.
362 Le plan de l'article est le suivant : - le chapitre 2 prdsente des rappels sur les graphes associEs 5, une matrice rdelle et leur relation avec le calcul du determinant pour les matrices carrdes. - le chapitre 3 introduit les notions de matrice et de syst~me structure ainsi que les graphes associEs. - le chapitre 4 est consacrE ~t I'interprEtation graphique de la structure ~ l'infini. - d a n s le chapitre 5 on donne la condition nEcessaire et suffisante de rejet de perturbation en termes de longueurs de chemins entrEe-sortie.
II. M A T R I C E S , 1) Q u e l q u e s
GRAPHES
ET DETERMINANTS
r a p p e l s s u r les g r a p h e s [6]
U n graphe G = (W, U) est un couple constituE : - par un e n s e m b l e W = {w I ...... Wn} de s o m m e t s - par une famiile U = {u 1....... Urn] d'arcs ElEments du produit cartEsien W.W. Nous nous limiterons daus la suite, sans le prEciser, aux 1-graphes, c'est-a-dire aux graphes pour lesquels il existe au plus un arc pour chaque ElEment de W.W. Autrement dit un arc u sera ddfini par u = (w i, wj) oh w i sera appelE extrEmit6 initiale et wj e x t r E m i t 6 finale. U n e sequence d'arcs distincts {ut, u 2 ...... uk} sera appelEe c h e m i n si pour tout i = 1, k-1 ; on a : extrEmitE initiale de u i = extrEmitE finale de ui.1. U n chemin sera dit 616mentaire s'il ne rencontre pas deux lois le marne sommet. U n c h e m i n {u I ..... Uk] est un c i r c u i t si : extrEmitE initiale de u I = extrEmit6 finale de u k. U n circuit est 61~mentaire si le seul s o m m e t rencontr6 deux fois est l'extrEmitE initiale du premier arc.
2) Graphes associ~s ~ une matrice [5] : Soit M une matrice rEelle (m x n), on associe ~t M un graphe G h (m + n) s o m m e t s b i p a r t i , autrement dit l'ensemble des sommets se decompose en deux sous-ensembles X et Y e t tout arc a son extrEmitE initiale dans X et son extrEmitE finale dans Y. Lorsque mij ~ 0 il existe un arc (xj, Yi) de p o i d s mij. Nous conviendrons de dire que cette representation est un g r a p h e de type 1 de la matrice. Si M est une matrice carrde (n x n) on pout lui associer un graphe ~ n sommets et tel qu'il existe un arc (xj, xi) si mij # 0. Le poids mij sera associ6 ~ l'arc (xj, xi). N o u s appelerons g r a p h e de type 2 ce nouveau graphe. I1 est h noter que pour une matrice carrEe le graphe de type 2 se dEduit du graphe de type 1 en repliant l'un sur l'autre les deux ensembles de sommets.
363 3) D~terminant et graphes [5] Si A est une matrice carr6e (n x n), le d6terminant de A s'6crit : n
det (A) =
~ ~e
o(g) 1"-[ an(i), i S(n)
i=l
oO : S(n) est l'ensemble des permutations de { 1...... n}~ r~ est un 616ment de S(n) r
est le signe de la permutation.
Pour calculer det (A) il suffit donc de d6terminer les permutations de {1...... n} telles que tousles
an(i), i soient
non nuls.
Dans le graphe de type i c e s permutations correspondent h des ensembles de n arcs qui 6tablissent une bijection entre les deux sous-ensembles de sommets darts le graphe biparti. En termes de th6orie des graphes, il s'agit de couplages d'ordre n (et donc d'ordre maximum) entre les ensembles X et Y. On replie le graphe biparti pour obtenir un graphe de type 2 en identifiant x i ~ Yi. De par les propri6t6s h6rit6es du couplage, ~ chaque sommet arrive un arc et un seul et repart un arc et un seul. Nous avons donc d6fini ici une famille ~l~mentaire de circuits d'ordre n. Par famille 616mentaire de circuit nous entendons un ensemble de circuits 616mentaires sans sommet commun. Pour obtenir le signe de la permutation g associde h une famille 616mentaire de circuits on utilise la r~gle de Cauchy : le signe de la permutation est celui de (-1) n-a ofa d est le nombre de circuits de la famille.
III. M A T R I C E S ET S Y S T E M E S S T R U C T U R E S
1) Matrice structur6e : L'616ment (i, j) d'une matrice structur6e est : -soit un z6ro fixe, l'ensemble des positions des z6ros fixes donne la structure de la matrice -soit un param&re
~'ij"
Si k est le hombre de param~tres
~'ij' le vecteur param&re A appartient
Exemples :
M1 = I
%1
0
%2
0
0
~.3
o
o
x4
est structur6e, Ae R4
"2tun ouvert V de R k.
364
M2 =
tl
0
~.2
~.2
0
~3
0
0
kl +~.3
2) Syst~me structur~ [1]
] n'est pas structurde, A ~ R 3
:
Un syst~me donnd par sa reprdsenmtion d'dtat
i Ax+Bu =Cx est (lit structur6 si la matrice
En pratique un syst~me structur6 permet donc de reprdsenter un syst~me pour lequel les seules informations sfires sont des absences de relations entre certaines variables, les autres relations mettant en oeuvre des param~tres mal connus et sujets ~t erreur d'identification. I1 est b, noter ici que la contrainte d'inddpendance des parambtres est. tr~s restrictive et parfois difficile/~ imposer pour une repr6sentation d'6tat. Une propridtd sera dite propri~t~ structurelle pour un syst~me structur6 de param~tre A~VcR k si la propridtd est vraie au sens usuel pour tout A de V sauf 6ventuellement sur rintersection de V avec une varidt6 algdbrique propre de R k. Rappelons qu'une varidtd algdbrique propre de R k est ddfinie par I'ensemble des x = {x i ...... Xk} annulateurs d'un nombre fini de polyn6mes non triviaux en x i ~ coefficients r6els. Autrement dit une propridtd structurelle sera vraie pour "presque tout" syst~me ayant la structure requise, de plus si pour un systbme donnd la propridtd n'est pas vraie, elle le sera pour un syst~me "proche". L'dtude d'un certain hombre de propridtds structurelles des syst~mes a d6j~t dtd r6alisde. Les efforts les plus importants ont portd sur la commandabilitd structurelle, des caract6risations tr~s int6ressantes de cette propridt6 ont pu ~tre donndes [I], [2]. 3) G r a p h e
d'un syst~me structur~
:
Au syst~me structural Y. ddfini par ses matrices (A, B, C) on associe le graphe G(Y$ = {Z, W} oh l'ensemble des sommets est : Z=UuXuY oh card (U) =nombre d'entrdes,card (X) =nombre d'dtats,card (Y) = hombre de sorties. Aucun arc n'arrive sur U ni ne part de Y. De plus, il y a un arc (ui, Xj) si bji ~: 0, un arC (Xi, Xj) si aji ~ 0 et un arc (x i, yj) si cji # 0.
365 Exemple :
Xl A=
0
0 B=
~.2
3.3
0
~-4
0
X5
C = (~.8
0
(
k6
0
0
~,7
0
0
x~)
G(E):
u2
X7
~.~
IV. I N T E R P R E T A T I O N
GRAPHIQUE DE LA STRUCTURE A L'INFINI
1) Structure .h I'infini : Rappelons que nous appelons z6ro A l'infini d'une fonction de transfert propre t(s) = n(s)/d(s), la diff6rence deg (d(s)) - deg (n(s)). Soit T(s) une matrice (p x m) rationnelle propre, T(s) peut ~tre factoris6e en :
)[ s'nt'" T(s) = BI(s
".s'n, 0
B2(s) = Bl(s) A(s) Be(s) 0
B i(s) et B2(s ) sont des matrices bicausales, c'est4t-dire des matrices inversibles propres et d'inverses propres. Les matrices bicausales n'ont ni z6ro ni p61e A l'infini et se caractdrisent par Det ( lira
Bi(s)) = constante non nulle, i = 1, 2
S--.>oo
r e s t le rang de T(s) ; n I < n 2 ..... < nr Les n i sont les ordres des z6ros h rinfini de T(s). La matrice A(s) est appel6e, forme de Smith McMillan "~l'infini de T(s). La liste des n i constitue la structure ~ l'infini de T(s).
366 Les n i peuvent se calculer comme suit :
- n t = plus petit ordre des zdros h l'infini des 616ments de T(s) - n 2 = plus petit ordre des zdros ~ l'infini des mineurs d'ordre 2 de T(s) - n 1 k-1
nk = plus petit ordre des z6ros ~ rinfini des mineurs d'ordre k de T(s) - ~
nj
j=l
La structure ~ l'infini peut 6galement etre vue & travers la "matrice-syst~me" correspondant ~ une r6alisation (A, B, C) de T(s). Appelons P(s) la matrice syst~me :
P(s) = (sI - A C
B t 0
et intIoduisons la notation : M~}...... ly Jl,----, Jq
pour dgsigner une sous-matrice (p x q) extraite d'une matrice M e t correspondant aux lignes i 1...... ip et aux colonnes j~ ...... jq. Nous utiliserons un r6sultat dfi & Rosenbrock : det P(s)ll';~]. n. n+i, ..... n+,,/
(- 1)q det (sI-A) det P(s~ ......
Par ce rdsultat, ~ tout mineur d'ordre q de T(s), on peut associer un m i n e u r d'ordre (n + q) de P(s). I1 est h noter, par exemple que le zdro ,5 l'infini d'un m i n e u r de T(s) vaut : n - (degr6 du mineur correspondant de P(s)). 2) C a r a c t 6 r i s a t i o n
g r a p h i q u e [5] :
Th6or~me 1 : Soit ~ un syst~me structur6 et G(Y~) le graphe associ6. On a alors : i) rang structurel de ~ = n o m b r e de z6ros h l'infini = ordre m a x i m u m d'une famille 616mentaire de chemins entrde/sortie dans G(Y.). ii) les z6ros `5 l'infini sont caract6ris6s c o m m e suit : - n 1 = (longueur m i n i m u m d'un chemin 616mentaire entrde/sortie dans G ( S ) ) - 1
- n k = (longueur minimale totale d'une famitle 616mentaire de k chemins entrde/sortie
367 k-1
dans G(Y.)) - ~ nj - k j=l
Dgmonstration
:
i) Le rang du systtme est l'ordre du plus grand mineur non nul que l'on peut extraire de T(s). I1 est doric associ6 au plus grand mineur non nul que l'on peut extraire de P(s). Considtrons le graphe de type i associ6 ~t la matrice P(s), la variable s 6tant considtrte ici comme un param~tre. Ce graphe comprend au moins n circuits correspondant aux boucles des sommets 6tats. A une sous-matrice du type : pr
~1..... n, n+j~ ..... n+jq 'S-'l,..., n0 n+il ..... n+iq
on peut associer un graphe de type 2 qui revient ~t rtaliser les associations entrfe-sortie (Ujl, Yil) ..... (Ujq, Yiq). Le dtterminant de cette sous-matrice est difftrent de ztro seulement si dans le graphe correspondant il existe une famille 616mentaire de circuits touchant les n + q sommets. De la formule du dtterminant et de la propritt6 d'indfpendance des param~tres cette condition est suffisante structurellement. Les circuits contenant des sommets du type entrte-sortie se traduisent sur le graphe de type 1 de P(s) par des chemins entrte-sortie 616mentaires, les m~mes chemins entrte-sortie 616mentaires se retrouvent sur le graphe G(Y.). Pour rtcapituler, rechercher le rang de T(s) 6quivaut ?arechercher la plus grande sous-matrice de type p t , s,~l,..., n, n+j . . . . . . . +J,t k / 1 ...., n, n+il,..., n+iq
dtterminant non identiquement nul et ceci revient ~t chercher darts G(Z) la plus grande famille 616mentaire de chemins entrte-sortie. ii) La determination des ztros ~ l'infini va consister ?~rechercher les plus petits ztros ~ l'infini de determinants du type
ce qui revient ~trechercher le plus grand degr6 des det (,;::. P(s)
o+4.1 n, o-J, n+il..... ..... rt+lq/
Pour une matrice de ce type, de determinant non strictement nul, le degr6 correspond &la famille de circuits ~t (n + q) sommets comprenant le plus de sommets d'gtat (chaque sommet d'ttat apportant un facteur du type (s - aii)dans le produit). Les sommets d'ttat qui n'interviennent pas dans cette famille de circuits par une boucle du type (s - aii) sont simEs sur des circuits 616mentaires touchant les entrEes-sorties sur le graphe de type 2 de P(s). Ces families de circuits correspondent sur le graphe de type 1 de P(s) &des families 616mentaires de chemins entrge-sortie touchant un minimum de sommets d'gtat. Ces families de chemins correspondent 6galement aux plus courtes families 616mentaires de chemins entrte-sortie sur G(5-'.). Des remarques prtctdentes il vient :
368 n 1 = minimum du nombre de sommets d'dtat contenus darts un chemin dldmentaire entrde-sortie de G(Y.) = longueur du plus court chemin entrde-sortie de G(Y~) - 1. La diffdrence de 1 provient du fait qu'un chemin entree-sortie de longueur k contient k-I sommets d'dtat, De mame : n k = (minimum du nombre d'dtats contenus dans une famille dldmentaire de k chemins k-1
entrde-sortie de G(Y.)) - ~ nj = (longueur minimum d'une famille dldmentaire de k chemins entr~ej=l k-I
sortie de G(Y,)) - ~ nj- k j=l
Exemple : Considdrons un syst~me structure dont le graphe est le suivant : x1
ul
~
U2 ,
~ x4
X2
x5
x6
>
Yl
~"
Y2
x7
I1 existe une famille dldmentaire de deux chemins entr6e-sortie, le syst~me est donc de rang 2 n1=4-1=3 n2=(4+5)-2-3=4
V. R E J E T DE P E R T U R B A T I O N I) Presentation du probl/~me Considdrons le syst~me suivant
{
~=Ax+Bu+Eq y=Cx
Le probl~me du rejet de perturbation consiste/l chercher une commande du type : u=Fx+Gq qui permet de rendre la sortie insensible ~ la perturbation q, autrement dit : C(sI- A - B F) -1 ( E + B G ) = 0 Les conditions ndcessaires et suffisantes d;existence de solutions ~, ce probl~me sont bien connues par rapproche gdomdtrique [7]. Nous prdf6rons ici utiliser la condition suivante :
369 Le probl~me du rejet de perturbation a une solution si et seulement si les syst~mes (C, A, B) et (C, A, (B, E)) ont m6me structure h l'infini. Autrement dit le syst~me non perturb6 et le syst~me perturb6 doivent avoir m6me structure h l'infini [8]. Posons : G(s) = C(sI - A) -I B H(s) = C(sI - A) -1 E La formulation suivante de la condition de rejet sera plus utile. Lemme [9] : Le rejet de perturbation est possible si et seulement si :
-
G(s) et [G(s), H(s)] ont mEme rang
- G(s) et [G(s), H(s)] ont mSme somme des ordres des z6ms infinis. L'utilisation de ce r6sultat et de la caract6risation gmphique de la structure ~ rinfini pemaettent d'6noncer le r6sultat principal de cet article : Th6or~me 2 : Soient Y. et Y~p les systEmes avec et sans perturbation et G(T~), G(Y.p) les graphes correspondants. Le rejet de perturbation est possible si et seulement si : i) L'ordre maximum r, d'une famille 616mentaire de chemins entr6e-sortie est le m~me dans G(Y~) et G(Y.p). ii) La longueur minimum d'une famille 616mentaire de r chemins entr6e-sortie est la mSme dans G(Y.) et G(Ep).
La condition i) est la condition de conservation du rang, c'est-~t-dire le hombre de z6ros h rinfini. La condition ii) indique que la somme des ordres des z6ros h l'infini doit &re conserv6e. Exemples : Soient les syst~mes structur6s dont les graphes sont les suivants : Xl x2 u ~ ~ ' - " ~ " ' ~
q
y
perturbation rejetable
370 Xl
X2
perturbation non rejetable q
Xl
x2
perturbation non rejetable (crit~re du rang) x3 q ,"'/
x4
\. ~"~ Y2
Remarques : - Les conditions de rejet de perturbation peuvent facilement se traduire en un probl~me de riot maximal ~ cofit minimal dans un graphe. De nombreux algorithmes existent pour traiter ce type de probl~me de mani~re efficace. -
Les rEsultats obtenus dans cet article peuvent se transposer au cas oh la description initiale est en
termes de transferts structures comme darts [4]. -
Lorsque la perturbation n'est pas mesurable, s-IG(s) et [s-lG(s), H(s)] doivent avoir mame
somme des ordres des zeros infinis. I1 suffit doric de rajouter au graphe G(Y.p) un arc sur chaque entrde de commande du syst~me pour se retrouver dans les conditions d'application du thEor~me 2.
REFERENCES
[1] LIN C.T. : "Structural Controllability", IEEE Trans. on Auto. Contr., Vol. 19, pp. 201-208, 1974. [2] GLOVER K., SILVERMAN L.M. : "Characterization of Structural Controllability", IEEE Trans. on Auto. Contr., Vol. 21, pp. 534-537, 1976. [3] LINNEMANN A. : "Decoupling of Structural Systems", Syst. and Contr. Lett., Vol. 2, pp. 71-86, 1981. [4] COMMAULT C., DION J.M., BELMEDHI A. : "Structural Systems within the Transfer Matrix Approach", Syst. and Contr., Lett., Vol. 9, pp. 335-339, 1987. [5] REINSCHKE K.J. : "Multivariable control, A Graph-theoretic Approach", Lectures Notes in Control and Information Science, Springer Verlag, 1988. [6] BERGE C. : "Graphes", Gauthier-Villard, 3~me Edition, 1983. [7] W O N H A M W.M. : "Linear Multivariable Control : A Geometric Approach", Springer Verlag, 1985 [8] MALABRE M. : "Structure ~ l'infini des triplets invariants : Application ~ la poursuite de module", ConfErence INRIA Analyse et Optimisation des Syst~mes, 1982, Proceedings Springer Verlag, Vol. 44, pp. 43-53.
A N E W C A N O N I C A L F O R M FOR D E S C R I P T O R S Y S T E M S WITH O U T P U T S .
J.J. LOISEAU ECAN d'lndret 44620 La Montagne-FRANCE G. LEBRET LAN ENSM,1 rue de la No~, 44072 NANTES Cedex 03-FRANCE ( Research supported by the C.N.R.S. and the Region Pays de la Loire)
Abstract : We construct a new canonical form for descriptor systems (E,A,B,C) under proportional feedback, proportional output injection and change of bases of input space, output space and state space. This form is compared with that of Van Der Weiden and Bosgra (1980).
I INTRODUCTION : We consider the linear time invariant system :
(1.1) (1.2)
Ex=Ax+Bu y=Cx
where UEEZ~.,~~]~m, X~,,%~]R n, y ~ ~ A P . E can be non-invertible and even non-square; ,~-,~]R.q is the codomain of E and A. Such a system is called "descriptor system" or, in our text, the quadruple (E,A,B,C). There has been a lot of interest for some years in the study of these systems. A central question is to define canonical forms under some "control groups". In the strictly proper case (E=I), it is well-known than the Kronecker's theory of matrix pencils [2] provides the basic tool to construct a powerfull canonical form under state feedback, output injection and basis changes of ~ , 'lJ', ~ ( see Morse [3], Loiseau [8] ). In the case of descriptor systems, Van Der Weiden and Bosgra [1] decompose the state space into a dynamical part and a static one. This allowed them to use Morse's results [3]. We have to emphasize that the physical interpretation of the control group considered in [1] is not clear : the inputs (resp. outputs) of the canonical form are combinatons of the inputs (resp. outputs) and of the state of the system.
The aim of the present paper is to give a canonical form for descriptor systems (E,A,B,C) under the action of the proportional group ~ for which admissible transformations are state
372 feedback F, output injection R, changes of the basis of the domain V, of the codomain W, of the input space G, and of the output space S. In other words, each (W,V,G,S,F,R)e:E" where W,V,G and S are invertible, transform (E,A,B,C) in
(W-IEv,W-I(A+BF+RC)V,W'IBG,SCV).
We shall construct this proportional form In section II. This construction is based on the result of [4], in which such a form is introduced for the systems without output. Then canonicity will be tackled by characterizing the list of invariants which describes the orbit of (E,A,B,C) under the action of ,2". Before the conclusion (section IV), we shall discuss, in section I11,the difference between our work and that of Van Der Weiden and Bosgra [1].
I1 CANONICAL PROPORTIONAL FORM; I1.1 CONSTRUCTION OF THE FORM; To the quadruple (E,A,B,C) we can associate the restricted pencil N(sE-A)K (see Jaffe and Karcanias [5] for the strictly proper case.), where N is a matrix representation of the canonical projection ~ --->,9~./ImB and K is a matrix for the insertion map KerC -->..,~2,. Note SEk-Ak= P[N(sE-A)K]Q the Kronecker form of N(sE-A)K; P and Q correspond to changes of bases. The restricted pencil is a subsystem of the quadruple where one reduced the domain of sE-A to KerC and his codomain to ,~x~./ImB. Preserving SEk-Ak for the restricted part of sE-A, one can represent the quadruple by: (Wo,Vo,Go,So,Fo,Ro)o (E,A,B,C)=(Eo,Ao,Bo,Co) with:
qoj r Co=[
r ][p
J 0 ]
where s V i ~ { 1,2,3 } Liand Kiare real matrices ; fl'I is the (I.I) identity matrix. If the group E" had also allowed the use of derivative feedback (FD) and output injection (RD), one would be able to cancel each ~i(s), i ~ { 1,2,3 }. In other word, we can choose Fo, Fp, Rp, RD such that:
,
o
Wo (s(E+BFD+RDC)-(A+BFp+RpC))Vo=[ 0 SEkoAk
SoCVo--[
]
o ]
373 Clearly, this gives a canonical form under the action of proportional and derivative feedbacks, proportional and derivative output injections and changes of bases. Here, the use of feedback and output injection is restricted to a purely static action. If it is always possible to cancel Ki ,i E: { 1,2,3 }, this is no longer the case for Li, iE: {1,2,3}. However, one can take advantage of the well-known blocks of the Kronecker Normal Form. Using linear combinations of rows and columns of SEk-A k, (~I(S), (~2(S)and (~3(S), one can reduce the matrix Li, i es 1,2,3 }, without canceling them completely9 In fact, these were the basic ideas used in [4] to construct a canonical form for (E,A,B) triples In this spirit, one can give a constructive proof of the theorem 2.1 (see below) which describes our" proportional form" for (E,A,B,C) quadruples under the action of E'. For the sake of clarity, and because of the lack of space, this procedure is not exposed here9 We shall introduce our form with the help of the results of [4]. By duality, we shall extend it to a particular form for systems without input. Then, a combination of these two forms allows us to justify the form for the quadruple (E,A,B,C). The starting point in the obtention of the canonical form of the triples (E,A,B) is to take the system in the form:
L
r
J
oo47]
where (SEk-A k )TB is the Kronecker form of the restricted pencil N(sE-A). There are classically four types of diagonal blocks in the Kronecker normal form (see Gantmacher [2] ):
I s-(zi 1 9
,
Lkij =
I 1" 1 kij
$
Lv[ sl
s-co i
associated to the finite elementary divisors
sl ~i
-
1"
<--vi--->
associated to the infinite elementary divisors
1" .
.
Ei
sl
$
< - - - 8 i + 1 --> associated to the column minimal indices
Lqi.
9
"qi+l
L ;J * ~-- qi --> associated to the row minimal indices
374 The main result of [4] is the following one. The reduction of
r
by the use of
transformations of the subgroup E'F={(W,V,G,0,F,0)} of E', gives, after reorganization of [sEo-A o Bo] a split in two of the lists {ci} and {vi}. Any triple (E,A,B) can be given, through the action of J'F, the following equivalent form: [ sE-A ]=block-diag [ sE i -A i ] * [ sE 1 -A 1 ]= block-diag [Lkij] with * [ sE 2-A 2]= block-diag [Lqi]
B= [Bij]
Lkij
i=1 to 6 , j=l to 2
unchanged and BII =0 'v'j
with L'qi unchanged and B2j =0 ~/j
* Associated to the two types of {si} there are two types of blocks:
[ sE 3 -A 3 ]= block-diag [Ldi]
with
Lo'i=
" 9
1 B3j=0 Vj sl
<'--
s
-->
[sl. 1 [ sE 4-A 4 ]= block-diag [LTi] with
s
4.
4--- '~I --> * Associated to the two types of
[ sE 5-A 5 ]= block-diag [Lmi]
{Vi}
with
B42=0
there are two types of blocks:
Lmi =
1 s 9
1" mi sl
4-
[ sE 6 -A 6 ]= block-diag [Lni]
with
41:b~ }
Lni =
B5i=0 Vj
4,
mi-->
. .
ni
Bsz=block-diag
L 'sJ 4---- hi-1 -"-->
9
?J B61=0
Now, if one applies this result to systems without input but with outputs (E,A,0,C), one obtain the following result that is deduced by duality9 Split in two of the lists {qi} and {vi} under the action of ..q'R={(W,V,0,S,0,R)}:
375 Any triple (E,A,C) can be given, through the action of E'R, the following equivalent form: [ sE-A ]= block-diag [ sEi -Ai]
C = [Cij]
i=1 to 2, j=l to 6
* [ sE 1 -A 1 ]= block-diag [Lkij] with Lkii unchanged and Cil =0 'Vi * [ sE 2 -A 2 ]= block-diag [L~i] with
Ls unchanged and Ci2 =0 'v'i
* Associated to the two types of {qi} there are two types of blocks:
[ sE 3-A 3 ]= block-diag [L~i]
with L(i =
is 1 . -11, . 9 ~i
Ci3=0 Yi
L 1J* 4--- ~i-1 --~
[ sE 4-A 4]= block-diag [L~i]
with L~i =
I
s
1`
1
is $ 4--+
4--
.ol]}
014---- block-diag { [ 0
024=0
* Associated to the two types of {vi} there is two types of blocs:
[ sE 5-A 5]= block-diag [Lmi]
with
S . Lm i =
.
Ci5=0 Vi
| mi
sl -J $ is1 [sE6-A6]=block-diag[Lqi ]
.
11,
~
with Lq,= L
.
jqi-1
sl 4--
qi
C26= block-diag {[1 0 . .
4.
"-')'
0 ]}
C16=0
Let us now come back to general systems with input and output equations; The reduction procedure of ~l(s), ~2(s) and s lead to a split in two of the lists {~i} and {qi}, and a split in four of the list {vi}; the list of the finite elementary divisors is unchanged.
376 We are now able to state our main result : Theorem 2,1 9 Any (E,A,B,C) quadruple (1.1) can be given through the action of ,.q"the following equivalent block-diagonal form :
II
II
_~
j i
-,Z"
3 m
3
/
I / ~ Oj
II
I ,_S
377 One can remark the following important particular cases : oi =1 corresponds to a zero column in sEc- Ac and Cc [,i =1 corresponds to a zero row in sEc- Ac and Bc qi =1 corresponds to a zero column in sEc- Ac but a non zero column in Cc ni =1 corresponds to a zero row in sEc- Ac but a non zero row in Bc Note again than some zero columns (respectively zero rows) appear in Bc (resp. in Cc) when B is not monic (resp. if C is not epic).
11.2 INVARIANTS. CANONICAL FORM : Let us now introduce the following algorithms. (2.2.1)
q2"~
q/~+]=KerCc-~-](E~/~+lmB)
(2.2.2)
ff.,~0=,,~'C "t~#+l=KerCc'~A'i(E~'ro~-~'+lmB)
J
(2.2.3)
8~
8#+i=E'l(A(KerC~3~)+ImB)
l,U, oc$oc:,U,1 c ... ~j,~S~..."
(2.2.4)
flJ'~
"lJ'~'+l=E'l (A(Ke rCc~fl.Y~)+l m B)
J
(2.2.5)
-~'~
%~+l=E(KerCnA'l(~'-#+lmB)) l ,,~o ~q/,oD~l ... ~ p - _ ~ . . .
(2.2.6)
~o=,,~,
,,~+l=E(KerC,qA-1(,~#+lmB)) J
(2.2.7) _8~
l ~2"0Dff-~'0D"V1...Dq/'#D'V'~'...
-
-
~#+l=A(KerCc~E "18_~)+lmB
(2.2.1), (2.2.2) and (2.2.4) were introduced by 0zs [6]; (2.2.3), (2.2.5) and (2.2.7) were defined by Malabre [7]. The convergence of (2.2.6) is a consequence of the above mentioned inclusions. We shall now characterize the indices which appear in theorem 2.1 in terms of the dimensions of the steps of these algorithms. Prooosition 2.1 9 {(S-Or.i)kij} is the list of the invariant factors of (Eq/*/E ,,,,,,,,~*[IA+BF+RCII"V*/,.%*), the map induced in the, quot,ient subspa,ces,~"/,.,%* and Eq/' /F_.,~ where F and R are such (A+BF+RC)'],/' c:E'12" and ~ ='1,/" n3 9
-
.
[ W ' * n . ~ '-I ]
- card {oi > P.}=d'm tfl),,n flj,~._ll
V #>1
card {~>__p.}=dim Lfl), n8 ~-I
V p.>l
378
-
card {r~i> i~}=dim I --~+'~#-tl
9 rKe rE n ff.,.9.~-i] -card{mi=~}=omlKerEnfls I
- card {pi = #}=dim
~'lmBnE~"-'l
llmBnEff~_] j
Particular case:
v
1
V ~>1
V p.> 1
card {~i> I~}=dim I--~*+-~-' l
V I1> 1
card {hi = p.}=dim I ImBnE%~rL-2IV i imBnE1s j 1~>-2
card
= ,}=dim
IKerEn f17~-' 1
V
>_ 1
card {ni = ;}=dim ,J ImB "[ llmBnlmEJ
Proof: The characterization of the polynomials {(s-oq)kij} is due to Malabre [7]. The other relations can be obtained by direct counting of the number of vectors in bases of the considered subspaces (see for ideas Loiseau [8]). We can remark that the algorithms (2.2.1) to (2.2.6) are invariant under feedback, output injection and changes of basis. This shows us that the indices {cri}, {7i}, {~J, {~}, {mi}, {ni}, {Pi}, {qi}
and the polynomials {(s-ai)kij} constitute a complete set of invariants for quadruple (E,A,B,C) under the transformation group S. In other words, this proves the canonicity of the form described in Theorem 2.1 . It is now important to compare this form with other existing ones and particulary the canonical form of Van Der Weiden and Bosgra.
III COMPARISON WITH THE CANONICAL FORM OF VAN DER WEIDEN AND BOSGRA; Van der Weiden and Bosgra (VDW & B) introduced [1], for somes quadruples (E,A,B,C), a canonical form which looks like our form. We shall compare both canonical forms. Two facts must be pointed out : 1)In the VDW & B's process, only the dynamical part of Dg is considered as the "actual" state space. Thus the system is described by the so-called "strictly generalized state space form" where E has a particular description "
o] From our point of view, the system is in a general form (see (1.1)): the state space is .,%. This is not a restrictive constraint : one can also, in our case, consider systems in strictly generalized state space form ( this is just a matter of basis transformations.).
379 2)The authorized transformations, in both cases, are in close connextion with the underlying state space. - This is why VDW & B use the" strict pencil equivalence" : PVDW&a(S)= M P(s) N with
[ o u=~,o//-A=I
Mll M12RI]~-sI-A11"A12B1]rNll
-
0 y~2]
-A=2B2/|N21 N22
=Pvow~.a(s)
0 X2 S JL -01 -02 0 JL F1 0 R~ and F1 only act on the dynamical part of 3(5 X2 and Y2 allow combinations between the other part of ~ and the inputs or the ouputs. In our case, the authorized transformations can be formalized by :
wl, w12RllEs A1 AI , :tE v11v12i]
W021W22 R2 0 S
-A21 -A22 -C1 "C2
V21 V22 F1 F2
=
Po(S)
R21 and [ F1 F2] act on the full state space 3P.,. JR' Only internal transformations (changes of bases) are allowed on the inputs and outputs. This second point shows that the "strict pencil equivalence" and our "proportional equivalence" are really different. We shall now illustrate this difference with an example. Consider the following systems which are equivalent in the sense of VDW & B :
l-s lolo- I 9 [-sE1-A1B1
I0s
P2cs~;EsE2A2~I
111 1
~'IS~=L ci o]=Ioo ~IoI
02
L. q-5-5 o[6-_.1
Esl~ o silo
=
0 0 1 I1 100
Indeed, they have the same VDW & B canonical form:
,oo iooiI oiOOI
r,oOOoqro,_, 0
s
sll
010
o., .,Jl., 0 1 0 j l 0 0 1 ~oo ~ 1 7060 0 0 0 0
s01 = 1 0 0 1 0 =PlvDW&B(S) L1 0 0 0
loolioooj lOOI
rio oo O]ro s,o 1-1 0
L~176 I~ oo,
P1VDW&B(s)=P2vow&B(S)
loo roso, oll II oo,,1 -L~olo _-P2vow,o~s~ 00 0 / L 0 0 0 -1
0 00
Now look at their canonical form under the action of our group (W,V,G,S,F,R).
380 s01 ~1 ro '~176 =Plo(s)
LSOo'tJL? o Oo'jLOo o Oo;, ~J--I oo,o
Pl =2 "ml =1
L1 o o o
o1 Fo,oo sl0
?JL?'o'oJL~ oj=Looo,
= P2o(S)
ql =3 ;nl =1
1000
Since they do not have the same lists of invariants, these systems do not belong to the same orbit. This implies that both transformation groups are actually different: and the associated canonical form too.
IV CONCLUSION: We have introduced a new canonical form for descriptor systems with an output equation. In other words, we have characterized by nine lists of invariants the orbit of any (E,A,B,C) quadruple under the action of the proportional group. It generalizes the form given by Morse [3] in the strictly proper case. The considered systems have been taken in a general form without any restriction of regularity, regularizability, controllability. E and A are not even assumed to be square. More properties of this canonical form will be investigated in future work. AcknowledgemQn(; The authors gratefully thank Dr. M. Malabre for helpful conversations.
References: Van Der Weiden ,A.J.J. & Bosgra,O.H., "The determination of structural properties of a linear multivariable system by operation of system similarity.", Int.J.Control, 1980, Vo1.32, N~ pp 489-537. [2] Gantmacher.F.R (1959), "The theory of matrices, Vol.2, Chelsea, New York. [3] Morse,A.S. "Structural invariants of linear multivariable systems", SIAM.J.Control, Vol.11, N~ August 1973. [4] Loiseau,J.J.;C)zs & Karcanias.N, "A feedback classification of singular systems", IFAC WORKSHOP "System Structure & Control :State Space & Polynomial Methods" sept 25-27,1989 Prague. [5] Jaffe,S. & Karcanias,N. (1981), " Matrix pencil characterization of almost (A,B)-invariant s..ubspaces: A classification of geometric concepts", Int.J.ControI,Vol.33, n~ 1, pp.51-93. [6] Ozs of descriptor systems", PHD thesis Georgia institut of technology, May 1985. [7] Malabre,M., " A structural approach for linear singular systems", International Mini Symposium on Singular Systems, Atlanta, 4-6 December 1987. [8] Loiseau.J.J., "Some geometric coniderations about the Kronecker normal form", Int.J.Control, Vol42, n~ pp. 1411-1431, (1985). [1]
E v a l u a t i o n T r a n s f o r m a n d S y m b o l i c C a l c u l u s for N o n l i n e a r
Control Systems
V. Hoang Ngoc Minh ~ G. Jacob, L.I.F.L. - U.A. 369 C.N.I~.S. Universit6 Lille I, 59655 Villeneuve d'Ascq, France. A b s t r a c t : Given a nonlinear control system, one can view its output function as a signal, parametrized by the primitives of the input functions. This signal can be formally described by its Fliess' power series, that is a formal power series on noncommuting variables. The temporal behaviour of the system can be derived h'om this symbolic description by a transform, that we call "Evaluation transform" and that generalizes the inverse Laplace transform to the nonlinear area. We developpe here the basic tools of that symbolic calculus by introducing a "kernel" for our Evaluation transform. This kernel can be viewed as some "temporal memory" of the systena in the Volterra's meaning as well as in the programmation meaning.
1o I n t r o d u c t i o n
and notations
Following M. Sliess ([2]) the output Y(~) of any nonlinear control system can be symbolically described by its generating series S = ~ < SIw > w (also called Fliess' wEZ ~ power series) that is a formal power series on noncommuting variables belonging to an alphabet % -- {z0, Zl,...,zm}. This series is nothing else as the adequate generalization of the Heaviside calculus ([2], [3], [8], [9]) in the nonlinear area : the notions of transfer function and of impulsive response, coding signals produced by linear or multilinear control systems, axe generalized to generating series and Volterra series, coding signals produced by nonlinear control systems. The inverse Laplace transform allows to recover the signal symbolically described by some transfer function. In the same way, the Evaluation transform ([5], [6]) allows in return to easy derive the temporal behaviour of the system from its symbolic description by computing the Evaluation Eu(w)
of each word w, for the i n p u t a :
integral
( a z~
a z~
...
a ~m), as b e i n g tile i t e r a t e d
/0'
~uw. It follows from the Fliess' f u n d a m e n t a l f o r m u l a (also called Peano-
Balker formula) which presents the o u t p u t y(t) as the E v a l u a t i o n of the g e n e r a t i n g series S : y(t) =
~ < S l w > g~,(w)(~). So the o u t p u t y(t) can b e viewed as a signal wEZ9
d e p e n d i n g on the parameters ~(~) =
/0'
a~(~-)dT,
z E Z ([5], [6]).
We develop here, in c o n t i n u a t i o n of [5] a n d [6], the basic elements of this symbolic calculus using the notations of E v a l u a t i o n via kernel functions which plays the rule of t e m p o r a l m e m o r y of systems in the Volterra's m e a n i n g as well as in the p r o g r a m m a t i o n m e a n i n g . So the Evaluation transform becomes a f u n c t i o n a l d e p e n d i n g o n the kernel a n d on the inputs. This systematic t r e a t e m e n t has been used in [7] to write M A C S Y M A programs by use of its recursive definition a n d of the recursive i n t e r n a l r e p r e s e n t a t i o n by the b i n a r y tree of the linear c o m b i n a t i o n s of the n o n c o m m u t a t i v e r a t i o n a l fractions describ ed by (Coz j o ) * P ~ z q ( c l zj~ )*P~zi~ (c2 zj~ )'1'~... zi~_~ (ck-1 z j ~ _ ~ )*P~-~ z i k (cl, zj~ )*pk, where p o , 9 9 9 Pk a r e integers, Co, .. 9 ck are complex n u m b e r s , Zjo , . . . , z j k a n d Zio , 9 9 9 zi~ are letters in Z.
382
Recall that a f o r m a l p o w e r series o n t h e a s s o c i a t i v e v a r i a b l e s z E Z ( n o n c o m m u t i n g if card Z > 2) with coefficients in/it(([1]), is any m a p p i n g : S:Z*
~ ~( w l
~< S I w >,
and the set of ~11 formal power series over Z is denoted by Zr( << Z >>. A formal power series S will be written as a formal sum E < SIw > w, where w6Z" < SIw > is the coefficient of the word w in S. A formal power series S E ~r( << Z >> will be said q u a s i r e g u l a r if and only if its constant term < S]~ > is equal to O. The s u m of two formal power series S and T is the formal power series S + T in ~f << Z >> defined by : V w E Z*,
< S + T I w > = < S [ w > + < TIzu > .
The C a u c h y p r o d u c t , noted by ".", of two formal power series S mad T is the formal power series S.T in iK << Z >> defined by : V w E Z*,
< S.TIw > =
~ ltjvEZ*
< S[u > < T I v > . ~It l J : l o
The symbol "." will be omitted when there is no ambiguity. For aa~y quasiregular formal power series S in/T*2 << Z >>, S* represents classically the formal power series E S'~' In commutative variables, it coincides with the rational n>0
fractio~ 1/(~
-
s ) , ~nd in ~his case ,,~ have S "~ = (1/(~ - S)) ~. 2. S y m b o l i c c a l c u l u s
2.1. E v a l u a t i o n o f f o r m a l p o w e r s e r i e s Let Z = {zo, z l ; . . . , zm} be a finite alphabet. D e f i n i t i o n 2.1.1. : W e wii1 call i n p u t r e l a t i v e to Z t/~e given of a vector a = ( a z~ a z' ... a z~ ) of" p i e c e w i s e c o n t i n o u s rea~ vaJued f u n c t i o n s defined on a c o m p a c t intez~val [0, t] C ~ + . ConventionaJly t h e O_component o f a n y i n p u t is a z~ ~ I. F o l l o w i n g K . T . Chen ([2]), we will coil p a t h a s s o c i a t e d t o t h e i n p u t a --(a
z0
a"
...
a zo,
), the time depenae~t vector r = ' ( ~,o
~,,
...
~,m ) de~ned
as follows :
/"
/o"
In all the sequel, we suppose that the function f is Stieltjes integrable with respect to { ~ } ~ z defined over [0,t], (t _> 0), and f vanishes at zero. The unit step (vanishing at zero) is noted " u n " : v , E ]O,t],
~(~)
= 1,
~(0)
= O.
383
D e f i n i t i o n 2.1.2. : T h e E v a l u a t i o n o f t h e w o r d w in Z* w i t h r e s p e c t t o t h e k e r n e l f , for the i ~ p u t a = ( a ~~ a ~ . . . a ~" ) related to t h e finite a l p h a b e t Z , is defined by i n d u c t i o n on the l e n g t h o f w as follows :
&(f;w)(O
=
,
&(f;v)(r) d~,(r)
if
w = vz.
This definition is extended to g ( << Z >> in the following way : D e f n i t i o n 2.1.3. : W e will call E v a l u a t i o n o f t h e f o r m a l p o w e r series S in g~" << Z >> w i t h r e s p e c t to t h e k e r n e l f , for theinpu~ a = ( a ~~ a ~* . . . a ~=) related to the finite a l p h a b e t Z , w h e n it is defined, the functionaJ : Ca(f;S) =
E
< SIw > Sa(f;w).
wEZ ~
In particular, for f = u n , the E v a l u a t i o n o f t h e f o r m a l p o w e r series S in f f ( < < Z > > , f o r the i n p u t a = ( a ~~ a ~ . . . a == ) r e l a t e d to the finite a j P h a b e t Z , is the function ([5], [6]): & ( s ) = & ( ~ ; s).
According to the Fliess' fundamental formula, we see the Evaluation of a formal power series S can be viewed as a transform that associates to S the signal depending on the primitives {(~ },~z of the inputs functions, and the Evaluation transform is notlfing else as a generalization of inverse Laplace and Fourier transforms ([5], [6]). The two following lemmas can be obtained easily by induction : L e m m a 2.1.1. : L e t u, v be two w o r d s in Z * . T h e n : Ca(f; UV) -'- Ca(~Ta(f ; u); V).
L e m m a 2.1.2.
:
L e t z be a letter in Z . For any n ~_ 1, we h a v e :
C,(/; z'~)(t) =
fo' f ( r ) (r
7n_~ - r ~).v
d~,(r).
I~ particular, for z = z0, we h a v e : Ea(f;z'~)(t) =
~
t
(t --
f(r)
T) n-1
(n-1)!
dr.
From the lemma 2.1.2., we can see that tile Evaluation transform is an "extension on noncommutative variables" of the exponential transform of the "ordinary generating
series" S = E
c n z " , which associates to S the "exponential generating series" ga(S) =
n>0
c,, ~ n>O
well known in "combinatorics" ([4]):
384
C o r o l l a r y 2.1.1. : Let z be a Jetter in Z. T h e n for any n >_ O, S~(z'~)(t) = ~"n-~. " zn p ~ r t i c ~ , ~or z = zo, ~hen ~o~ ~ y ~ >__ o, s a ( ~ ) ( t ) = ~t,~.
T h e o r e m 2.1.1. : Let S and T be two formal p o w e r series in g~ << Z >>. Let r be a scalar. T h e n : Z~(f; S + r.T) = E~(f; S ) + r.E=(f; T).
P r o o f : The proof is immediate since the linearity of the ordinary integrals
9
T h e o r e m 2.1.2. : Let S and T be two #orrnaJ power series in ~ ( << Z >>. T h e n : E~(f; S . T ) = E~(E~(f; S); T).
P r o o f : Actually, by the definition 2.1.3., we have : &(E=(f;S);T)=
~
~
< S]u > < T l v > E ~ ( E ~ ( f ; u ) ; v )
uEZ* v6Z*
= ~
~
< Sl~ >< TI~ > eo(f;~)
(lemma 9LL)
uEZ* vEZ*
\ u E Z * vEZ*
= Eo(f; S . T )
9
The introduction of the kernel f for the Evaluation function G ([5], [6]) allows to give a notion of memory for the system. This kernel can be viewed as the temporal memory of the system in the Volterra's meaning as well as in the p r o g r a m m a t i o n meaning ([7]), that justifies our approach. So the Evaluation transform becomes a functional depending on the kernel f and on the inputs {aZ}zEz. For implement these functionals, we used the A-notation of MACSYMA and the recursive internal representation by binary trees of the linear combinations of the noncommutative rational fractions described by
(co Zio )*P~ z , (Cl z~, )*p, z,2 (c~ zj~ )*~.~.. z~ _, (c~_~ zi~ _, )*~-'z,~ (ok zj~ )*~, where p 0 , . . . , ;k are integers, Co, 9
ca are complex numbers, Z j o , . . . , zjk and Zio , . . . , zik are letters.
T h e o r e m 2.1.3. : (Convolution theorem) L e t z be a letter in Z. Let H be the formal p o w e r series on the only letter in z. T h e Evaluation with respect to the kernel f , for the input a = ( a z~ a zl ... a zm ) related to the finite aiphabet Z , of the formal p o w e r series H is : E~(f; H ) ( t ) =
/o'
h(~z(t) - ~z(~-))df(r),
385
where h(~z(t)) the EvaIuation of the formal power series H. In particular, if H is a formal power series in g( << zo >>, then :
g~(f; I t ) ( t ) =
/'
h(t - , ) d r ( r ) .
Proof. : (i) F i r s t ease : H = z", n _> 0 : For n = 0, the result is i m m e d i a t e . If n > 0 then we have : 8 ~ ( f ; zn)(t) =
f ( T ) ((~(t)--U ( n - -~-z(r))"-I 1)! d~z(r')
= /'f(~)d[ - (4z(t)-~-~('))"]~,
(leumaa 2.1.2.)
J
(integration by parts)
=
/'
h(~,(t) - ~ , ( r ) ) d f ( r )
(ii) G e n e r a l case : H = E
( / ( 0 ) = 0)).
H " z " ' we have :
n>__o
E=(f; H)(t) = ~
H , e ~ ( f ; zn)(t)
n>O
~(~))~
(~(t)
-~i
df(~) (case (~))
n>o
=
i'
h((,(t) - ~(r))df(r).
In p a r t i c u l a r , for z = zo, since ~ 0 ( t ) = t, t h e n we have result
9
2 . 1 . 2 . : Let G E f~2 << Z >> be a s series, and let H be a power series on the only letter in z. T h e Evaluation with respect to the kernel f , for the i n p u t a = ( a ~~ a ~ . . . a ~'~ ) r e l a t e d ~o t h e / ~ n i t e alphabet Z, of the formM power series C H is : Corollary
s
~ ( a H ) ( t ) = < al~ >
h(~(t)) +
/' h(~.(t) -
~(T))dEo(a)(,-),
where h((~(t)) the E v a / u a t i o n o f the f o r m a / p o w e r series H. /'ormai power series in g ( << z0 >>, then :
e ~ ( c ~ ) ( t ) = < cl~ >
h(t) +
In particular, i f 2I is a
/' h(t-,)aeo(C)(,).
386 P r o o f : Actually, since G H = < Gle > H + G I H , where G1 is the quasiregular formM power series Lr.2 << Z >>. Hence by the theorem 2.1.1, the Evaluation of the formal power series G H , for the input a = ( a ~~ a z~ ... a ~ ) related to the finite alphabet Z, is < G]e > h ( ~ ( t ) ) + g ~ ( G l t I ) ( t ) . Using the Convolution theorem in the particular case where the kernel f is the Evaluation of the quasiregular formal power series G1, we have the expected result * 3. C a l c u l u s e x a m p l e s Example
3.1.
In this example, we c o m p u t e the Evaluation of the series z *'~ for any letter z and for any integer n > 0. Lemma
3.1.1. : For any integer n > 1, we have g~((c~z)*~)(t) = exp(~(~(t))g~(c~(z(t)) ,
where the g,, are polynomials and verifie the following inductive equations :
{
g,,(a(~(t)) =
/o
g~-l(a(~(t)) + a
g~-l(a((~(t) - (~(r)))d~0-)
if
n > 1.
P r o o f : Since ga((az)*)(t) exp(c~(z(t)), we can write gl((~(z(t)) = 1. ~Ve suppose that the result is true for any u, 0 _< u < n - 1. For v = n, we have ( a z ) * ' * = ( a z ) * ( a z ) *~-~. By the induction hypothesis and by the corollary 2.1.2, we obtain : =
sa((~z)*")(t)
=
exp(~(z(t))j._l(~4z(t)) + ]o exp(~((~(t) - (~(~)))g.-l(~((~(t)
= exp(c~,(t)) g , - l ( a ~ ( t ) ) hence we have the expected result
Lemma
3.1.2.
:
9
+ a
- (~0-)))dexp(~(~0-))
g , , - l ( a ( ~ ( t ) - ~('r)))d~z(T) ,
_i()
The f~ilj, j.(~(,(t)) = j----0 ~ '~ -J i (~(,)J(~) j! , f o r n >_ I, is the
unique solution o f the inductive, equations :
~,,(~(t)) =
I
j,,-l(,~(t)) +,~
/
1 if ~ = 1 t
9,,-~(~(~(t)- ~O-)))d~(T)
if ,, > i.
387
Proof:
Given a I = 1 E •(
<
Z ~,
Ea(G1)(t) = 1.
w e h a v e c l e a r l y gl(OL~z(t)) =
Suppose that for any n _> 1, we have g , ( a ~ ( t ) ) = E~(Gn)(t), with G,, E f/( << Z >> . Thus we have (see the corollary 2.1.2.) E~(G,,) = $~(G,,-1)+S~(azG~-l). This equation is true if G~ = (1 + az)G~_l. In other words : n--1
~n : ('l'JvOlZ)n--i = ~ ( ~ j=O Since $~((az)J)(t) =
j!
1}(O~Z)J.
, j >_ 0, tl~en we have for any integer n >__1 : n--1
By the l e m m a 3.1.1. and the l e m m a 3.1.2., we have the following proposition : Propositon
: For any positive integer n, for any complex number a, we
3.1.1.
.,/]ave :
e~[(c~z)*~](t) =
exp[c~z(t)]~.; n J- i [c~G(t)]Jj! if
n > O.
In particular,/'or z = z0, we have : C
,.~[(aZo)
*n
](t) =
n--1
exp(at) ~
j=o
1 (at) j
n-
if
7z = 0
if
n>O.
J
By the theorem 2.1.2. and the proposition 3.1.1, we deduce the following theorem Theorem
3.1.1. : For any positive integer n, for any complex number a, we have
f(t)
[a(4~(t) j~ ~ ( r ) ) ] j df (7)
if
n=0
if n > O .
In particulaz, for z = zo, we have :
g~[f; (azo)*"](t) =
~ j=O
n J- 1
exp[a(t - ~-)] [a(t J[ - T)] j df(-r)
if
n=O
if
n>O.
388
E x a m p l e 3.2. In this example, we compute the Evaluation of the formal power series S that verifies the following polynomial equation :
S + #lSz + fl2Sz 2 + ... fl, Sz" = G, where z is a letter of Z, and G a formal power series of ~" << Z >>. We have S K = G, 71
where K is the formal power series of Zrx"<< Z >> defined by E
flkzk with/30 = 1.
k=0
Since the constant term < Kle > = fl0 = 1 does not vanish, then the formal power series/x~-1 exists and is a formal power series in the single commutative variable z. Suppose that K admits r complex distinguished roots # 1 , . . . , #r of respective multiplicity order m l , . . . , mr. One can express mfically K -1 under partial fraction decomposition form
~_~ Ern~ (_-"~-l)k ~l,k f~l,k,
where for any I e [1..r] and for any k e [1..rn,], A,,k e ~T, and
l=l k=l
f V' " Let hl,k({~(t)) be the Evaluation of Hi,k:
each Hl,k is \-fiT]
hl,k(~z(t))=exp(~l))~(k--1
So S = GK -1 can be expressed as ~ m ,E
1
(proposition 3.1.1.).
Al,k GHl,k. By the theorem 2.1.1. and by (_#l)k
l=l k=l
the corollary 2.1.2, we obtain the Evaluation of S :
Za(S)(t) =
(_#l)k < Gle > hl,k(~z(t)) -t-
hl,k(~z(t)
--
G(r))dg.(G)(r)
.
1=1 k = l
Let us indicate that in the particular case z = z0, the above calculation corresponds mutadis mutandis to the inverse Laplace transform in the study of linear control systems. As conclusion, we can say that if the Fliess' series is considered as a symbolic encoding of input/output behaviour of the nonlinear control systems, then the Evaluation transform allows in return to easy derive the temporal behaviour from this symbolic description, (see [6] for a simple obtaintion of the Taylor expansion of the Volterra kernel). Thus, we get a generalization of the notion of transfer function (generating series on one variable) and impulse response, encoding signals produced by linear or multilinear systems, and the Heaviside calculus (Fourier and Laplace transforms) as already pointed out by M. Fliess and al. ([2], [3], [8], [9]). In the symbolic calculus for linear control systems area, the integration operator is noted by ,,1,, ~ . Here, it coincides with the letter z0. And the letters z of the Fliess' encoding alphabet Z plays an analogous
389 part : they encode the "Stieltjes integration operators". And we have the following Evaluations of some usual formal power series : S
$.(f; S) f(t)
d Zn
~f
~ (n - 1) f t exp({z(t)- {~(r)) ({*(t) -{*(r))Jdf(r) ,=o J ,o J! f ' (r - ~(,-))~ an ni df(r)
z*",n > 1
E
e/o-)
E
CnZ n
n>O
Jo
n>O
then, in particular Z* ~ ~
Z ~"
' e~p(r
n_~O
/
n:>O
~--~(_1)% 2"
'(r
t
~z(,))df(~)
-
-
~(~))
e~p(r
-
r
cos(~.(t) - (~(r))df(r)
n~O
~--Lj(__ 1)- z2n+ i
t sin({,(t) -
~.(r))df(r)
n>O
In particular, for f = un, we have the following Evaluations ([5], [6], [7]):
s
&(s) 1
s
Zn ,n
z ,n>_l
exp(~z(t)) ~
C(t)
~ - ~ Cn Zn n>O
n>O
then, in particular Z* ~ ~
Zn
exp(~(t))
n>O
Z
nZ n
4~(t) exp(r
n>O
~-~j(-1)~z 2~
cos(r
n)O
sin(~(t)) n~O
j
j[
390
4. R e f e r e n c e s [1] [2] [3]
[4] [5]
[6] [7] [8] [9]
J. Berstel and C. R e u t e n a u e r , S6ries rationelles e~ leurs langages, Masson, Collection E.r~.I., 1984. M. Fliess, Fonctionnelles causales non lindaires et ind~termin~es non commutative, Bull. Soc. Math France, 109, 1981, pp. 3-40. M. Fliess, M. L a m n a b h i and F. Lamnabhi-Lagarrigue, An algebraic approad~ to nonlinear functional expzmsions, IEEE Trans. Circ. Syst., CAS-30, 1983, pp. 554-570. D. Foata, La s~rie g~n~ratrice exponentielle dans les probl~mes d'~num~ration, S4mlnaire de Math. Sup., Presses de L'Universi[5 de Montr4al, Montr6~tl, 1974. V. Hoang Ngoc Minh, ElEments d'un calcul symbolique pour les syst6mes dynamiques non lindaires, Journ6es-S6minaire "TraJtements Alg6briques ct Informatiques des S~ries Formelles Non Commutatives", Lille, December 1988. V. Hoang 1Ngoc Minh and G. Jacob, Symbolic calculus and Volterra series, IFAC Symposium "Non Linear Control Systems Design", Capri, Juin 1989. V. Hoang iNgoc Minh and G. Jacob, Transformation d'Ewlua~ion et Calcul Symbolique pour les Systbmes Non Lindaires, LIFL technical report, IT 168, 1989. M. Lamnabhi, A new symbolic calculus for the reponse of nonlinear systems, Systems ~z Control Letters, 1982, pp. 154-162. M. Lamnabhi, Functional za2aJysis of nonlineaa" circuits : a generating power series approach, IEE proceeding, Vol. 133, Pt H, N ~ 5, pp. 375-384.
IMMERSION IN INFINITE DIMENSION H. Hammouri & S. Othman Laboratoire d'Automatique et de G6nie des Proc~d~s Universit~ Claude Bernard Lyon I, 43 Bd de 11 Nov. 1918; 69622 Villeurbanne FRANCE.
Abstract
:
The input-output map is the intrinsic object of a dynamical system . The immersion problem consists of sending a dynamical system to another one via a transformation which preserves the input-output maps. Many authors have studied the immersion of nonlinear systems into a finite dimensional state affine systems ( up to output injection ). This result can be applied to the synthesis of observers of nonlinear systems. In this paper we prove that under some assumptions, any autonomous single output system can be immersed into an infinite dimensional linear system on some Banach space. Keyw0rds : Nonlinear systems, infinite w
dimensional systems, immersion.
Introduction.;
The immersion problem is introduced by Fliess and Kupka [3] . They proved that if the observation space of some nonlinear system is finite dimensional then we can immerse the nonlinear system into a state affine system. We recall the definition of the immersion. Let (.T_.I), (~2) be the two following systems : 1
(T.1) k= F(u,x) y
(T_.2) ~'=
x(t) ~ M 1 , y(t) ~ Bm , u(t) c Rp
Ht(x) F2(u'z)
z(t) r M2 , w(t) c Rrn , u(t) c B p
W = H2(Z)
Where M 1 and M2 are two manifolds. We say that (T-l) is immersible into (~,2) if there exists a continuous function : T" M 1 ~ M2 SuChthat: If x(u,xO,t) and z(u,T(xO),t) are the unique trajectories of (~1) and (]~2) which start at xOand :(x O) respectively at t=to, then y(u,xO,t)=Hl(x(u,xO,t)) coincides
392 with W(U,T(X0),t)=H2(z(u,T(X0),t)) for all x0 and all admissible controls u(.) along some interval of t. Note that to construct an observer system, many authors classify the systems which are immersible into state affine systems up to output injection. See [1], [4], [5], [6], [7], [8]. All these immersions are finite dimensional. Under some assumptions, the authors in [2] constructed an infinite dimensional unitary immersion. In this paper we give an immersion of some autonomous nonlinear system into two infinite dimensional linear systems. The generalization to the non autonomous case can be obtained in the same w a y . w 2 An infinite dimensional immersion : A)
Banach
immersion
:
Let (T_.) be the system : {~( = f(x) y = h(x)
x(t) c M
y(t) E R
Where M is an analytic manifold, f a vector field and h a real analytic function Such (T_.) is said to be analytic. Let O be the observation space ( i.e. the R-vector space generated by the family { Lkf(h), kr N }, where Lkf(h) is the kth Lie derivative and L0f(h) = h ). Remark that if this family is not free then O is finite dimensional and by Fliess-Kupka's theorem [3], we can immerse (~) into a finite dimensional linear system. Along this paper the family { Lkf(h) , kc N } is free and satisfies the following assumptions : (A1) I I Lkf(h) II oo = Pk < oo Where I I I loo is the norm of L~176 functions on M ). (A2)
sup~ o
~k+l
( the space of essentially bounded mesurable
= ;k < +oo
IJk Now set e k = Lkf(h) and consider the R-vector space : E=
ale i
s.t:
~lailPi<+oo i=0
Claim 1 : (i) E is a functional space, its elements are continuous functions . (ii) E with the norm :
I T_.,i.oe~a,I IE =~0=lal~ is a Banach space.
393 (iii) T h e topological dual space of E denoted by H is a Banach space which can be identified with :
ai
a
~ < + o o
s.t :
suPizo tJi
}
induced by the norm :
. =. lail II]-[z,i=0 I I"=supi~~ *=o
where
I - [ a i = ( a o , a 1 ..... a . . . . ) is an o r d e r e d i=o P r o o f of c l a i m 1 :
(i) c o m e s continuous (ii) since summable complete (
of r e a l s .
from the fact that a uniform limit of continuous fonctions is a function. the space II(N ) (the space of ordered sequences which are ) is complete, one can use the same argument to prove that E is in fact E is isomorphic to II(N) )
*~"
(iii) Let ]-~b i be such that : i=o §
sequence
supi ~
Ibil
. < +oo.
~i
oo
]-[b i i=0
gives rise to the continuous linear form :
E
,B
§162
§
Converslv 9 Let I: E ----, B be a continuous linear form and set Using the continuity of 1, we can verify that : a) suPi>o ~
li = I(ei) .
< +oo. ~ui
b)
III II = suPi~ o
where II i II is the norm of the continuous linear form .
In this way, we identify a continuous linear form on E with
. ..
an
ordered
lail
i['[0ai ( supi~o ._~ < +o= ) and we denote the norm by II II, 9 =
~.Li
N o w let L be the shift ooerator on E defined bv L(e=~ = e=..l,
for i r N
sequence
394 Remark 2 : 1) L is well defined and continuous on E §
§
§
Since L ( ~ ale i ) = % aiei+ 1 and II i=0
i=0
,.I- ee
+~
,Y_.,aiei+ 1 II = ,Y_.,la~l#~§ < ~,% la~l#~ i=0
i=0
i=0
~li+ 1
where
;~ =supiz o - -
which is finite by (A2) .
2) L is exactly the extension of the Lie derivative Lf to the functional this m e a n s that L restricted to O coincides with Lf.
space E,
NOW let A be the adjoint operator of L, A is a linear operator of H into itself defined by : = "=i--~oai+l
A(~C[ ) i = oai Let
T " M
T(x) : E - +~,o
--, H be the valuation function defined by : 9B +ee
for every x r M
,
i
where e i = Lf(h).
aie i ----, ~ aiei(x) I-0
i=0
By claim 1 part (i) T is well defined. In this paper we consider on H two topologies : (i) Its natural Banach topology as above. (ii) T h e *-weak topology. The second topology is defined by ' A s e q u e n c e {fn}ncN of H converges to fcH if and only if 9 V zcE ;
limn-.,o~ (fn-f)(z) = 0
Definition
3 :
We say that a function g : M ----. H is "-weakly continuous if and only if g is continuous with respect to the "- weak topology. Lemma 4 : (i) (ii)
The operator A is continuous. T h e function T is "-weakly continuous.
395 Proof (i) By a s s u m p t i o n
(A2) :
supi~0
~'li+ 1
= ;~ < +oo
Pi Let ~ E H ;
~= ~
,
i=0
II ~ IIH = sup=z 0 p. I
I ~:i+~l II A~; II H =suPiz0 ~
I ~i+1 I ~i+1 _< supiz0 _ _ 'Ui+l
~i
_ _ _<~II ~ II H ~li
Then : I I A I ! -< ~, where J JA I J is the norm of A. M o r e precisely it is easy to see that ] IA J I = ;~. (ii)
If {Xn} c o n v e r g e s
to
T(X).
to x in M w e have to prove that
{T(Xn)}
converges
*-weakly
§
§
i
'
I (~x n) -~x))(~ ae~)l = I ~ a~(Lf(h)(Xn) - (~f(h)(x)) I i=O
i=O
-< Z lalIIl(h)(Xn) - kf(h)(x) i I i=O §
_~ 2}-'. lal~ i i=O
Since ~ I a~l~i is summable then 9 i=0 §
V E_>0
;
3 No
,
V N>_N0
;
0 < ~ lanlPn < n=N - "~
A n d then : +~
oo
NO
I (~xo) - T(x))( T. ae i ) I i=0
-<
+~
~'. lal II~(h)(x ) - Lf(h)(x)l '
n=0
+ 2
~, n=N0+l
lal~o
Using the continuity of Lfi(h) , 0 _< i < N O , there exists rl > 0 such that if d(x n, x ) < q ( d ( . , . ) is a metric on M which is compatible with the topology of M ) then I Lfi(h)(xn) - Lfi(h)(x) I -< d K for 0 ~; i < N O , wh e re q depends on K and ~. +oo
Set K =
~ l a n l , w e can choose n=0 This m e a n s that :
Let ~ C H ;
~=
.=
~;~ ,
N 1 E N, such that
II ~ IJH = supi~0
Pi
Vn_>.N 1 ; d(Xn,X)_< q .
396 +
eo
I ("Kx.) - ~x) X)-'. aie i ) I ~ r
[]
i=O
Now consider t h e system :
(S)=
(t)=Cg t
~tcH
; w(t) c R
Where C is the continuous linear form : C : H - -
-'[a i i=O
.B ._, a o
Because A is a linear bounded operator, it becomes an infinitesimal generator of an analytic group e tA . And V ~0r H ; :3 ~(t) a unique analytic solution of ~t = A ~;t such that ~(0) = ~0 Theorem 5 ; Under the assumptions (A1), (A2), the system (~) is immersible into (s) for the "-weal~ topology on H . Proof of theorem 5 : Let xOr M and y(t) the output of the system (T_.) with respect to the initial state x O . Let w(t) be the output of (S) with respect to T(x0) . Let us prove that : y(.) - w(.) in some interval [0,T] . By analyticity of (T.) : +=o
13
+~
rl
t (n)/n ~ t Ln h o y(t) = ~: ~y ~v, = :~ ~ f()(x) n=o n=o Since
on some interval
[O,T1], TI>O .
w(t) is analytic, 13
+co
w(t) = :~ -~w(n)(o) n=O
on some interval
ill
More generally : Since C is a continuous linear form :
[O,T2], T2>O 9
397 n
d w = CAn~(t)
,
where A n is defined as follows :
dt n
for n>l for
An~ = A(Anq~)
V~r H
A ~ is the identity operator.
n=O
This gives " w(n)(0) = CAn~(0) = CAnT(x 0) + oo
= C [ ' ~ T i + n ( X 0) i=O + ~ i+n
=Cl-[Lf
0
(h)(x)
i=O
= Lnf(h)(x0) This implies that w(t) = y(t) on [0,T] where T = inf (T1,T2). [7 In order to construct an observer for the above infinite dimensional linear system, sometimes we minimize some quadratic convex criterion, for this we need a smooth norm. For this reason we have to embed H into a Hilbertian vector space FI for which the old immersion can be extended. B)
Hilbert
Now Let
l
immersion
H be the Hilbert space :
§ a i s.t 9 +~ ~ "=
:
i=O
Jaij2 < +co / 2
P i (i+1)
2
The scalar product and the norm of H are defined by : + ,=
+ .
+ ~
akb k
<,__l~oa'j~obj>=~ 2 )2 = "= k=0 ~Jk(k+ 1
II ~IIR = , / ~ : ,
~> ,
v ~~H
Remark that I:! ~ H Using (A2), the operator A can be extented to a continuous operator ,~ on I:1 defined by :
398 +o,~
§
p,(IIai ) = r[ai+, i-O
i=O
*~
Jai+ll 2
i=0
~i2 (i+1) 2
II ~,(lr[ai ) I1~ = ~ "
Note t h a t :
i=O
+= lai+ll
-<E i=0
2 2
2
~i+1(i+2)
2
2 2
Pi+l (i+2) ~i (i+1)
2
+oc
2
_<2x II ]-[ai I1~ i=0
where
;k
~'li+ 1 = suPi~0 - < +oo
( By
(A2))
.
Pi
the
As above, ,&, generates an analytic group e t'~, and the valuation function has natural extention
,--,R
~-M
+
c,o
i
x - -
*riLl(hi(x) i=O
l_emma 6 " -~ is continuous . Proof Let {Xn} be a sequence which converges to x in M, (that means d(xn,x)--. 0 for s o m e metric which is compatible with the topology of M ) . Since I Lfk(h)(xn) - Lfk(h)(x) I -< 2pk, we obtain : +o~ L~ k 12 I (h)(x n) - Lf(h)(x)
II~"-~x)ll2=",v n T_,
2
k=0
Pk
1
+~ _< 4T_.'
(k+l) 2
1
k=0 ( k + l ) 2
C h o o s e N 0, N 1 two integers such that : (1)
(2)
2
T_., 1 k=N0+ 1 (k+1)2
V n>N 1 ;
2
I Lfk(h)(xn) - Lfk(h)(x) J < d M
for 0< k
399
/ where
No
M =2//~
1
1
k=0p2k(k+ 1 )2 This implies :
II "KXn)-{(x) I IR <.~.v n > N 1 Denote by (,S)
[]
the following dynamical system evolving on the Hilbert space I:1 :
{~t = A~t
~t r ~ ; Yt c B
(S) Yt = C~t
-,B
C : H - §
'-[a i
i=0
.... a o
Theorem 7 9 (T.) is immersible into (,S) . the proof is the same as that of theorem 5 . Conclusion : Let (T.1) be the nonlinear system :
(T- 1)
k = f(x,x) y = h(~.,x)
where ,~s R m is an unknown system :
constant parameter.
This leads to the augmented
)~ = f(X,x)
t
(D ~, o l y =h(x) A s s u m e that (T.) satisfies the above assumptions (A1) , (A2) . To identify the parameters ,~ it is sufficient to construct an observer for its associated linear system (S) . [ REFERENCES ] [1] D. Bossane, D. Rakotopara, J.P. Gauthier : "Local and global immersion linear systems up to output injection" . Submitted to M.C.S.S.
into
400 [2] F. Celle, J.P. Gauthier, D. kasakos, Go Sallet : "Synthesis of nonlinear observers, a harmonic analysis approach", Math. systems theory, vol. 22, pp 291-322. [3] M. Fliess, I. Kupka : " A finiteness creterion for nonlinear input-output differtial systems", SIAM journal on control and opt. vol. 21 N~ 5, pp 721-728, Sep. 8 3 . [4] H. Hammouri, J.P. Gauthier : "Bilinearization up to output injection Systems and control letters 11,'1988, pp139-149 . [5] H. Hammouri, J.P. Gauthier : "The time varying linearization up to output injection", 28th CDC, Tampa, Florida, Dec. 13-15 1989 . [6] H. Hammouri, F. Celle, "Some results on nonlinear systems equivalence", Joint conf. on new trends in systems theory, Genova, Italy, July 9-11, 1990 . [7] A oJ. Krener, A. Isidori : "Linearization by output injection and nonliear observers", Systems and control letters 3, 1985, pp 47-52 . [8] J. Levine, R. Marino : "Nonlinear systems immersion, observer and finite dimensional filters", Systems and control letters 7, 1986, pp 133-142 o
On Observers for S m o o t h N o n l i n e a r Digital S y s t e m s J.W. Grizzle* Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109-2122/USA
P.E. Moraal'* Department of Applied Mathematics University of Twente PO Box 217, 7500 AE Enschedc Enschede, The Netherlands
A b s t r a c t : The problem of constructing an observer for a nonlinear discrete-time system is analyzed as one of left-inverting a certain map from the states to the measurements. Practical and theoretical consequences are derived.
1.
Introduction
The introduction of differential geometric and algebraic techniques has brought a phenomenal increase in our ability to modify the behavior of a nonlinear control system, as long as the full state of the system is available for feedback. In practice, however, economic and/or physical constraints usually dictate that only a small subset of the state can actually be measured. Hence, if performance considerations motivate the implementation of a full-state-feedback controller, then one must reconstruct the state fiom the measurements. A dynamical system which does this is called an observer [Lull. To be precise, consider a discrete-time system on R ~, xk+l
=
F(xk,uk)
=
h(xk)
(1.1)
where y E R ~' and F and h are smooth functions. A second system,
zk+l = G(zk,yk, uk) , xk = H(zk,yk, uk)
(1.2)
with zk e R t, some l >_ 0, is an asymptotic observer for (1.1) if it satisfies: (A) V x~, 3 zl such that & k = x k for a l l k > 2 , and (B) for allx~ E R " , z ~ E R ~, lim I I : ~ k - x k l l = 0 . If --
k.--*oo
the read-out map H in (1.2) is the identity, so that n = t, then (1.2) is called an identity observer [Lul]. For later use, the observer (1.2) will be said to be dead-beat of order d, if, upon writing GUk'"k(Zk) = G(zk, Yk, uk), and Iluk'~'k(zk) = H(zk, Yk, Uk), then H ~'''" o G u'-''''-'
o...o
GU'"(z~)
= x,~ ,
(1.3)
independently of the particular observer initial condition zl, where x~ is the state of (1.1) at time d and where " o " denotes composition. It should be remarked that dead-beat "Work supported by the National Science Foundation under contract NSF ECS-88-96136. ** Work performed while a visiting scholar at the Department of EIecgrica!Engineering and Computer Science, University of Michigan at Ann Arbor; work supported by grants from Netherlands Universities Foundation for International Cooperation and the University of Twente.
402
observers are of interest for stabilization problems, because, if uk = ce(zk) is a stabilizing feedba& for (1.1), then u~ = a(:~k) will always result in an internally stable closed-loop system whenever the observer (1.2) has the dead-beat property. This is one of the rare instances of a nonlinear separation principle. Designing a nonlinear observer which meets the convergence requirement B has proven to be very difficult. In the mid 70's there were attempts to use Lyapunov techniques in order to establish sufficient conditions for the asymptotic convergence to zero of the observer error, e = x - 3: [KET]; of course, the major problem is to produce a "natural" Lyapunov function candidate. In the mid 80's, considerable progress was made on using coordinate transformations in order to represent a given plant as the composition of a linear system and nonlinear output injection [KI] or as a bilinear system plus nonlinear output injection [HG]. The main shortcoming of this approach is that only a few systems admit such a representation (see [Kr, Ka]). This becomes even more clear when one looks at the class of sampled-data systems (that is, time-wise discretized continuous-time systcms), where the requirement that a plant can be represented as the composition of a linear model and nonlinear output injection, for an open set of sampling times, is so stringent as to reduce it, when n = 2, to the class of systems that can be exactly linearized by coordinate transformations alone [CG]. Let us close by noting that the work of [LM] enlarges the class of systems for which observers can be designed with "linear thinking" by considering immersions rather than diffeomorphisms of the state variables; [Kr, Ka] transform the system to an approximate normal form, with the goal of increasing the accuracy of a classical linearized observer. The papers [BZ, Ze] take a similar approach, transforming the system to an observer canonical form, where the transformation may depend on the input and its derivatives; unfortunately, even in canonical coordinates, choosing the observer "gains" so as to achicve an asymptotically stable error dynamics is not totally transparent. Many structural aspects of the general nonlinear asymptotic observation problem are analyzed in [vS], and a comparative study of several observation techniques has been carried out in [WCZ]. Finally, [FI] has approached the problem from the point of view of differential/difference algebra; the usual notion of observability, involving injectivity of the map fi'om states to measurements, has a natural analogue in terms of differential dependence of tile statc on the outputs and controls; this establishes a relation between observers and elimination theory. The remainder of this Note presents some preliminary results on thc construction of observers for discrete-time systems, obtained by viewing the problem as one of leftinversion of the state-to-measurements map. The basis for this perspective is prescntcd in Section 2. In Section 3, the connection between the observability rank condition and the existence of smooth observers is explored. Section 4 discusses a practical means of implementing a dead-beat observer via lookup tables; a simple example is considered. Section 5 shows that an old fricnd, Newton's Algorithm, can be used to construct a nonlinear observer under quite general conditions. Some concluding remarks are made in Section 6.
403
2.
Observation
=
Recursive
to-Measurements
Inversion
of
the
State-
Map
To simplify the presentation, it is hcreafter supposed that the system (1.1) is autonomous; that is, we are dealing with a system
~+, = F(~) y~ = h(xk)
(2.1)
which does not depend on any exogenous variables. The composition of F with itself i-timcs will bc denoted F(;), with the convention that F (~ = identity. If F is invcrtiblc, F ( - 0 thcn has the obvious meaning, ( F - l ) (0. It is emphasized that the restriction to autonomous systems is not essential and is only done to simplify the presentation of the basic ideas. As usual, the system (2.1) is said to be observable [NC, Ni, So] at a point 2 E R" if there exists an integer 1 < N < cr such that 2 is the unique solution of thc set of equations (Yi=hoF(O(x) , i = 0 , . . . , N - 1 (2.2) where 9i = hoF(O(i). It is locally obsewable at ~ if there exists an open set O about i and an integer 1 < N < cr such that i is the unique solution of (2.2) over all x E O (instcad of R"). Finally, the system is uniformly observable if there exists an integer 1 < N < cr such that the mapping
ru:=
[h ] !
:R"~(R") N
(2.3)
h o F (~v-1) is injective. The concept of local uniform observability should bc clear. Whenever (2.1) is uniformly observable, the system of equations,
!
=
i
Yk
(2.4)
h 0 F(N-1)(Zk_N+I)
can be uniqucly solved for Xk-N+l, and the current state xk obtained by
xk = F (u-')(xk_~+l)
9
(2.5)
Of coursc, if F is a diffeomorphism, as is the case for sampled-data representations of continuous-time systems, the two steps (2.4) and (2.5) can be combined to yield
!
=
i
(2.6)
which can be solved directly for xk. A left-inverse of (2.6) can then be used to construct a dcad-bcat observer for (2.1) [G1]. Following Glad, let L be such an inverse, that is
L(h o F ( - ~ + l ) ( x ) , . . . ,
h(~)) = ~ ;
(2.7)
404
then letting z~ = Yk-N+i, one has that zk+ 1 =
z
i=l,...,N-2
=
(2.8)
=
is a dead-beat observer of order N. Conversely, suppose t h a t
zk+l = ~k =
G(zk,yk) H(zk,yk)
(2.9)
is a dead-beat observer of order N. Then H ~ o G~-' o...o
G~'(zl) = x ~
(2.10)
for all Zl E R t ; thus the left-hand side of (2.10) does not depend on zl, and
L ( y l , . . . ,YN) := f I T M o G ~ - ' o . . . o G y~
(2.11)
is a left-inverse of (2.6). In the same vein, an a s y m p t o t i c observer constructs a left-inverse to the system of equations (2.6) as N ~ oo. Hence, there is an essential equivalence between constructing an observer and leftinverting the system of equations (2.6), either exactly or asymptotically (see also [HS]). Clearly, for nonlinear systems, insisting that this can be done explicitly, in closed-form, is very restrictive. It is therefore proposed ~o formulate an extended concept of an observer as an implicitly defined dynamical system, involving successive approximation routines, logical variables a n d / o r lookup tables in order to dynamically "estimate" the state of a deterministic nonlinear system. This is further pcrsued in Sections 4 and 5. First, we make a slight detour and discuss the smoothness of observers.
.
The
Observability
ness
of an
Rank
Condition
and
Smooth-
Observer
T h e system (2.1) is said to satisfy tile observability rank condition at 2 if there exists N < oo such t h a t the function FN in (2.3) has rank n at ~ [So]. T h e observability rank condition is usually emphasized as being i m p o r t a n t because it assures the local injectivity of (2.4). However, it also plays an important role in the existence of smooth observers. To see this explicitly, consider first the simple s y s t e m on R,
Zk+l = zk
Yk ---- (xk) 3
(3.1)
It is observable at 5: = 0; indeed, the m a p p i n g x --+ (x) s is a homeomorphism. But the state can not be recovered smoothly (differentiably) from the measurements because the rank of (x) 3 at the origin is zero. More generally, recall t h a t whenever an analytic m a p F : R" ~ R" is a bijection onto its image, then there exists a n analytic inverse r -1 : F ( R " ) ~ /~n only if, rank F(p) = n at every p E R", as the chain rule applied to the analytic m a p F -1 o F : R" ~ R ~ shows immediately. T h e consequence of this for observers is the following.
405
P r o p o s i t i o n 3.1: Consider the system (2.1), where F and h are analytic, and suppose that F : R" -+ ft ~ is a diffeomorphism. Under these conditions, there exists an analytic dead-beat obscrver for (2.1) only if, there exists N < c~ such that (i) the rank of Fg in (2.3) equals n at every point 2 E R ~ and (ii) FN is injective on R~; that is, P~v is an injective immersion. P r o o f : By hypothesis, L(y~,...,yN) defined in (2.11) is analytic and a left-inverse of (2.6). tIcncc the chain rule implies that the right hand side of equation (2.6) has rank n for all x E R ~. Since F is a diffcomorphism, this in turn implies that (2.3) also has rank n at all points of R ~. Of course, L being a left inverse of (2.6) and F a diffeomorphism imply that FN is injective, rn The hypothesis in Proposition 3.1 that F is a diffeomorphism can be weakened, but some restrictions on F are inevitable. Consider the system Xk+l
=
0
yk
=
(zk) 3
(3.2)
The system does not satisfy the observability rank condition at x = 0, but xk+l = 0
(3.3)
is ccrtainly an analytic dead-beat obscrvcr. For systems with inputs, some sort of strong acccssibility condition would probably bc sufficient for establishing the necessity of a "uniform obscrvability rank condition" in an extended vcrsion of Proposition 3.1.
4.
An up
Approximate
Dead-beat
Observer
via
Look-
Tables
The main motivation for using lookup tables in order to implement a nonlinear observer is to minimize the number of calculations that have to be done in real-time in order to produce an estimate of the state fl'om the available measurements. The approach we have taken is to discretize the image of the set of equations (2.6), by covering it with a grid, and then solving for the state xk at the grid points, using a Newton algorithm. All of this was done, of course, off-line. The observer was constructed by producing an approximate solution to (2.6), on-line, through an interpolation/extrapolation sdleme based upon tile solutions at the grid points. In the case that F in (2.1) is not invertible, then one would solve the set of equations (2.4) for Xk-N+a and obtain zk via (2.5), once again, all done numerically and off-line. As an example, consider the following set of equations describing the growth of the methanol utilizing organism Methylomonas in a continuous stirred tank biological reactor (CSTBR). The model is taken from [HK]; the relevant equations are:
{ ~1= #(x2)zl
-
Oxt
= f(x) ~ = -o(x:)x, + D(xzz - x:) y= ~ = h(z)
(4.1)
406 where
0.504x2(1.0 - 0.204x2) #(x2) = 0.000849 + x 2 + 0.406x~
' x2 _< 4.89
~ ( ~ ) = =:(1.32 + 3.86=: - 0.661~) 0.000849 + x: + 0.406x~ xl x2 t #(.) a(.) D x:,f
: : : :
, x2 _< 4.89
cell density of Methylomonas mcthanol concentration time in hours specific growth rate
: specific substrate consumption rate : dilution rate = 0.4 : concentration of substrate in feed = 1.8
Potentially, D and x~,I are control variables, though we have assumed them to bc constants. The system (4.1) has three equilibrium points, namely: x e'l = (0, 1.8), x e'2 = (0.24,0.39), x ~'3 = (0.68, 0.0033); the first and third are asymptotically stable. We constructed an obscrvcr around the unstable equilibrium point, x ~'2 in order to emphasizc the diffcrences with the classical linearization approach. The first stcp was to time-wise discretizc the model, or more precisely, obtain thc sct of equations (2.6) for the discretized system. Since it is known that [Va]
y(t)
=
~
L ~i h ( = o ) ~ t,
,
(4.2)
i=O
one easily obtains that
,43, yk
=
y(kT)
h(~k)
where xk = ~ ( k T ) , y, = y(kT), L 1 h ( ~ ) = ~0h( = ) f ( )=, and L~§ = Lf(Lifh)(x). In the actual numerical implementation, wc only kept terms up to order T 2 in (4.3), and calculated the resulting finite sum as a function of xk with REDUCE. The image of (4.3) was discretizcd with a 5 x 10 grid, concentrated along the diagonal of R • R, which represcnts the product of the measurement space. The truncated cquations (4.3) were then solved for the corresponding valucs of the state at the grid points with a NcwtonRaphson algorithm, oncc again, implemented on REDUCE. The resulting tablc was then passed to M A T R I X x , where the observer was implemented with the linear interpolation function. The readcr desiring more details should see the report by [Mo]. Figurc I at the end of the paper shows the error dynamics of this obscrvcr in comparison with a classical linear observcr (with eigcnvalues placed at .4 and .5), for two initial conditions of (0.25, 0.3) and (0.25, 1.6). Both observers use only linear calculations online. Thc tablc look-up approach, howevcr, is accurate over a much larger range of initial conditions. The table look-up scheme scems to bc very cgicient when the number of states is small. For larger systcms, thc size of the table necdcd to achieve sutIlcicnt accuracy can bc too
407
large for practical implementation. In this case, a method that we are currently exploring is to obtain an initially "crude" estimate of the state xk via a lookup table, and then to refine the estimate, on-line, via Newton iteration. In the next Section, it established that Newton's algorithm can be interpreted as an asymptotic nonlinear observer.
5.
Newton's
Algorithm
as an
Observer
The goal of this Section is to interpret Newton's algorithm as an asymptotic observer. For simplicity, we will focus on systems for which the set of equations (2.4) is square, injective, and satisfies the observability rank condition. Let Yk = column(yk_N+~,...,yk) be the last N + 1 measurements and let I f ( z ) = column ( h ( x ) , . . . , h o F(N-~)(z). Then the Ncwton-Raphson algorithm for y~ - a ( x ~ _ N + ~ ) = 0
is (i+, = (i +
((i)
]'
(5.1)
(Y~ - t I ( ( i ) )
.
(5.2)
The standard convergence theorem for this algorithm can be found in fLu2], for instance. Let I" [ a norm on R ~, let II" II the induced norm on/2(/~"), the set of linear operators from 12" to a " , and let ill" III be the induced norm on E(s T h e o r e m 5.1 f l u 2 ] : Suppose that H is twice differentiable and that IIIH"(~)III ~ 1( for all z E R"; suppose there is a point (~ E R" such that Po := H'((~ is invertible with IIPo--lll __ 0 and converges to a solution of (5.1). If instead III/V'(x)tll _< s( only in a neighborhood B of (o with radius
">- E
1-
~o
(5.a)
then the successive approximations generated by Newton's algorithm remain within this neighborhood and converge to a solution of (5.1). [-1 The most interesting point of the Theorem is that it gives an estimate of how good the initial estimate of zk should be before a few iterations of (5.2) will generate better estimates. In particular, the quantity [][II"(x)lll, which measures the degree of nonlinearity of the equation (5.1), is seen to be of central importance in determining how coarse the grid discussed in Section 4 can be taken. For a linear system, IIIg"(x)lll ----0, and the grid can be infinitely coarse, and when lllH"(x)lll is large, the grid needs to be finer. Let
/~ oY'(r = ( + L0z and as before, let (Ork) (d) represent |
l
(Y~- H(())
(5.4.)
composed with itself d-times.
T h e o r e m 5.2: Suppose there exists an open bounded set O C R ~ that is forward invariant for the system (2.1); that is F ( O ) C O. Suppose further that F and h in (2.1)
408
are at least three times differentiable, and that there exists an integer 1 _< N _< n such that the set of equations (2.4) is square, injective on O, and satisfies the obscrvability rank condition at every point of O. Under these conditions, for every e > 0 there exists a 8 > 0 and an integer 0 < d < oo such that, for k > N + 1, =
&k = F(N)(Zk)
(5.5)
is a local asymptotic observer for (2.1) is the sense that: (A) if 371 E (~ and z~v+l = x~, then ~:k = x~ for all k _> N + 1 and (B), if xl E O, IZN+~ -- x~] < ~ and for all k >_ 1, distance (xk,,-~ O) > e, then Iim [&,~- xkl = O. (,-~ 0 means the complement of 0 ) . -k.--*oo P r o o f : The complete proof is too long for this short communication, but the basic idea is to view the observer problem as one of solving a sequence of nonlinear inversion problems, each described by (2.4). Since the set O is relatively compact and invariant, the NewtonRaphson algorithm can be shown to have a uniform rate of convergence over the entire sequence of problems. The idea then is to iterate long enough on each problem (the parametcr d) so that F applied to the solution of the k-th problem is a very good initial guess for the (k + 1)-st problem. For all of this, the convergence analysis of [Lu2J is very useful. El R e m a r k s : The open set O is assumed to be bounded, but not necessarily small. If F ( O ) ~ O, then only finite time estimates are possible; the same is true of the linearization approach of [KI] (for discrete-time system, see [CG]). Theorem 5.2 states, essentially, that if one uses the lookup table approach with a sufficiently small grid for initializing (5.5), then an asymptotic observer can be found without explicitly knowing a left-inverse to (2.4). The observer (5.5) is coordinate dependent. It is interesting to note that the coordinate transformation approaches, in general, would only favor convergence of (5.5) if they reduce IIIH"(~)III- In particular, eliminating low-order polynomial terms in favor of high-order terms will not always accomplish this task. Finally, if the system to be observed is not autonomous, the (5.1) is parameterized by the inputs. As long as the inputs take values in some bounded domain, the same estimates needed for the construction of the observer (5.5) can still be performed.
6.
Conclusions
In this paper, building upon an idea of Glad, we have shown that viewing the observer problem as one of inverting a certain map from states to outputs leads to asymptotic observers for general nonlinear digital systems. In particular, the Newton-[~phson algorithm was interpreted as a means of constructing such an observer. In its practical implementation, in order to guarantee convergence within only a few iterations, it would be necessary to have a good initial guess for &k. This could be obtained by the table lookup scheme of Section 4. Other practical considerations involve work on efficiently representing and manipulating nonlinearities in a real time programming language, numerical studies aimed at developing rules of thumb for choosing the sampling time of a digital representation (because longer times between samples means that more computation time is available for increasing the accuracy of the estimate of the state, but
409
also possible degradation of the accuracy of control), studying algorithms other than Newton's, incorporating noise effects, etc. Some results on globally convergent forms of Newton's alogrithm are available [NW], and could be investigated in this context. Finally, if one calculates the Jacobian linearization of the observer (5.5), one will find that it is nilpotent. For robustness reasons, one should investigate modifications of Newton's algorithm that have better linearized error dynamics.
References
[BZ] [CC]
D. Bestle and M. Zeitz (1983), "Canonical form observer design for non-linear time-variable systems", Int. J. Control, Vol. 38, No. 2, pp. 419-431. S.T. Chung and J.W. Grizzle (1989), L'Observer error linearization for sampleddata systems",Proceedings of the 28th IEEE Conf. on Decision and Control, Tamr~a, December, pp. 90-95. ,, [Fl] M. lVliess (1987), 'r remarques sur les observateurs non lindaires , Onzi~me Colloque GRETS[, Nice, France, June. [Cl] S.T. Glad (1983), "Observability and nonlinear deadbeat observers", Proc. IEEE Conference on Decision and Control, San Antonio, December, pp. 800-802. [HG] H. tIammouri and J.P. Gauthier (1988), "Bilinearization up to output injection", Sys/. ~ Control Letters, Vol. 11, pp,,. 139-150. [HIq 9 . Hoo and J.C. Kantor (1986), Linear feedback equivalence and control of an unstable biological reactor", Chem. Eng. Commun., Vol. 46, pp. 385-399. . [HSI M.L.J. Hautus and E.D. Sontag (1980~, " An approach to detectability and ob.servers", in Lectures in Applied Mathematics, Algebraic and Geometric Methods in Linear Systems Theory, Vol. 18, pp. 99-135. [Ka] S. Karahan (1989), Higher Order L{near Approximations to Nonlinear Systems, Ph.D. Dissertation, Mechanical Engineering, University of California, Davis. [KET] S.R. Kou, D.L. Elliott and T.J. Tarfi (1975), "Exponential Observers for nonlinear dynamic" systems", Information. and,.C~176 Vol. 29, pp. 204-216. [KI] A.J. Krener and A. Isidori (1983), Lmearization by output injection and nonlinear obscrvers", Syst. ~ Control Lellers, Vol. 3, pp. 47-529 [Kr] A.J. Krener (1988), "Nonlinear controller design via approximate normal forms", reprint. .G. Luenberger (1966), "Observers for multivariable systems", IEEE Transac[Lull lions on Auto. Control, AC-11, pp. 190-197. [Lu2] D.G. Luenberger (1969), Optimization by Vector Space Methods, John Wiley & Sons, Inc., NY. [LM] J. L~vine and R. Marino (1986), "Nonlinear system immersion, observers and finite dimensional filters", Syst. ~J Control Letters, Vol. 7, pp. 133-142. [Mo] P.E. Moraal (1989), "Multi-rate Sampled-Data Observers for Nonlinear Systems", Tectmical Report, May. [NC] D. Normand-Cyrot (1983), Thdorie et Pratique des Syst~mes Non Lingaires en Temps Discret, Th~se de Docteur d'Etat, Univ. Paris-Sud. [Nil H. Nijmeijer (1982), "Observability of autonomous discrete-time nonlinear systems: a geometric approach", Int. J. Control, Vol. 36,pp. 867-874. [NW] N. Navid and A.N. Wilson, Jr.(1979), "A theory a n d a n algorithm for Analog Circuit Fault Diagnosis", IEEE Trans. on Circuits and Systems, Vol. 26, No. 7, Ju~, pp. 440-4579 [So] 9 . bontag (1984), "A concept of local observability", Syst. ~ Control Letters, Vol. 5, pp. 41-47. [va] V.S. Varadarajan (1984), Lie Groups Lie Algebras and Their Representations, Springer Verlag, NY. [vs] A.J. van der Schaft (1985), "On nonlinear observers", IEEE Trans. Auto9 Control, Vol. 30, pp. 1254-1256. [wcz] B.L. Walcott, M.J. Corless and,, S.H. Zak, (1987) "Comparative" study of non-linear state-observation techniques , Int. J. Control, Vol9 45, No. 6, pp. 2109-2132.
410
M. Zeitz (1987), The extended Luenberger observer for nonlinear systems", Syst. g_4 Control Letters, Vol. 9, pp. 149-156.
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F i g u r e I: Comparison of a classical linear observer (dotted line) and the table lookup scheme (solid line) as described in Section 4. Upper graph: initial condition is (0.25, 0.3). Lower graph: initial condition is (0.25, 1.6).
An Interior-Point Algorithm for Large-Scale Quadratic Problems with Box Constraints Panos M. PardaIos Computer Science Departmcnt The Pennsylvania State University, University Park, PA 16802 Yinyu Ye Department of Management Science The University of Iowa, Iowa City, Iowa 52242 Chi-Geun Han Computer Science Department The Pennsylvania State University, University Park, PA 16802
Abstract We present computational experience with an interior-point algorithm for large-scale quadratic programining problems with box constraints. The algorithm requires a total of O(x/~'L) number of iterations, where L is the size of the input data of the problem, and O(n 3) arithmetic operations per iteration. The algorithm has been implemented using vectorization and tested on an IBM 3000-6008 computer with vector facilities. The computational results suggest that the efficiency of the algoritlim depends on an appropriate choice of paralneters. Computational results with various large-scale problems, including exainpies of obstacle problems, are presented.
1.
Introduction
Quadratic progranlming problems with box constraints have the following form: m i n f ( z ) = l x T Q x + cTz
(1.1)
s.t. l < x < u, where l , u , c , x E R ~ and Q is an n x n symmetric matrix. Problems of the form (1.1) can be found in many applications, such as journal-bearing problems [3], elastic-plastic torsion problems [4], and contact and friction problems in rigid body mechanics. It is also clear that linear least squares problenm with upper and lower bounds rain It .4x + b I1=
(1.2)
414
s.t. l < x < u, form a special case of (1.1). In addition, linear complementarity problems with upper bounds on the primary variables [5] can be transformed to (1.1). The general problem (1.1) is well known to be NP-hard. In this paper, we are concerned with the case where Q is a symmetric positive semidefinite matrix. Many algorithms have been proposcd for the solution of (1.1). Dembo and Tulowitzki [4] proposed an algorithm that uses conjugate gradients and a projected gradient method. More recently, a direct method for sparse least squares problems with upper and lower bounds was considered by Bj5rck [2]. Mor5 and Toraldo [10] introduced an algorithm that combined standard active set strategies with the gradient projection method. Problem (1.1) can be transformed to the following form by a simple linear transformation: m i n / ( x ) = 2xrQx + cTx (1.3) s.t. 0 < x
<e,
where e E R" is a vector of l's. In this paper, we propose and implement an interior-point algorithm for solving (1.3) and present preliminary computational results on a variety of test problems.
.
Interior-Point Algorithm with Box Constraints
for Quadratic Problems
Convex quadratic programming has been shown to be polynomially solvable by using the ellipsoid algorithm [8]. After Karmarkar's new polynomial time algorithm [6] for solving linear programming, many researchers have developed different interior-point algorithms for convex quadratic programming problems. Ye and Tse [14] proposed a projective algorithm for convex quadratic problems. This algorithm is based on the iterated application of the projective transformation technique followed by optimization over an interior ellipsoid centered at the current approximation to create a sequence of points that converge to the solution. This algorithm requires O(n4L) arithmetic operations, where L is the size of the input data of the problem. Monteiro and Adler [9] applied their interior path following primal-dual algorithm to these problems. Recently, a potential reduction O(n3"SL) algorithm has been proposed and implemented for a class of quadratic problems by Pardalos et aI. [13]. In this section, we outline the interior-point algorithm for solving the convex quadratic problem with box constraints (1.3). This is a special form of convex quadratic programming. Due to its simple structure, we show how the recent potential reduction algorithm solves it efficiently. Problem (1.3) can be tralasformed to the standard form m i n f ( x ) = lxTQx + cTx
(2.1)
415
s.t.x+z=e
and x,z>_O,
where z E R '~. T h e dual of (2.1) can be written as
max eTy -- l x T Q x s.t.s~=Qx+c-y>0
(2.2)
and sz=-y_>0,
where y C R '~. Using the Kuhn-Tucker conditions, ~:* is an optimal feasible solution if and only if 1. x* is a feasible solution of (2.1). 2. T h e r e exists a y* such that x* and y* are feasible for (2.2). 3. ( x * ) ~ ( Q x * + c - y - ) - ( z ' ) % "
= (x.)T
. + (z.)~
:
= 0.
Also note t h a t the duality gap of (2.1) and (2.2) is given by
A = l x T Q x + cTx -- (eTy -- l x T Q x ) = xT,sa~ "-~ ZTSz > 0. 2 2 We use the primal-dual potential 5 m c t i o n to characterize the interior points x and y. r~
r
z, s~, s~) = p ln(xT s~ + zTs~) -- ~ ln(xi(s~),) -- ~ ln(zi(s~)i), i=1
(2.3)
i=1
where p >_ 2n + V~-nn. We can i n d u c e the following system of bilinear equations from (2.3).
XS.e-
A - - e = O, P
(2.4)
A
Z S , e - - - e = 0,
(2.5)
P ~-
(Qx + c - y) = 0, x + z-
(2.6)
e = 0,
(2.7)
s~ + y = o,
(2.8)
where X = diag(x), Z = diag(z), S~ = diag(s~), and S= = diag(sz). We use a modified Newton's m e t h o d for solving (2.4) - (2.8). Given xk,where 0 < xl k < I for i = 1 , . . . , n , z k, w h e r e 0 < z i k < l f o r i = l , . . . , n a n d z
k+z k=e,
_ s k = yk, where (a~k)i > 0 for i = 1 . . . . ,n,
416
and s~ j' = Qz k + c - yk, where (s~k)i > 0 for i = 1 , . . . , n , the modified Newton's method is formulated b y
(yk)T)T, 0
where v k = ((xk) T, (szk) T, (Zk) T, (Szk) T, denotes a step length, yk is the Jacobian matrix of (2.4) - (2.8) at v k, and g(v J') denotes the left hand side of (2.4) (2.8) at v k. Note that
~(~) :
((X'S2e
e)Z, -
zx'
P
(Z'S)e
Ak 9T
-
--e) p
T , o, o, o)
,
where
p,~ (~')~2+(z~)~2. =
j k is given by
jk =
S. k X k 0 0 0 0 0 S, k Z k 0 -Q I o o I I 0
0 0
I 0
0 I
(2.9)
0 I
If --(Jk)-~e(vk ) = Av, then
~j(v ') = -:'zxv.
(2.10)
From (2.9) and (2.10), we have the following system of linear equations. We generate the moving direction (Ax, Az, Ay) by solving the following system of linear equations. Ak X ~ A s ~ + S~k~_.x = - - e -- XkS~ke, P
(2.11)
Ak Z k A s ~ + S, k A z = - - e - Z k S , ke, P As~ = QAx Ay,
(2.13)
Ax = -Az,
(2.14)
A~
(2.15)
(2.12)
- -
Equations
= -Av.
(2.1 I) - (2.15) can be formulated
o I
s?(z~) -I - z I
0
as follows;
~z Ay
=
~(Z~)-le-~? 0
.
(2.16)
Since Az = --Ax, the system (2.16) can be anaplified as Ak (Q + S~a(Xk) -1 + S,k(Zk)-i)Ax = --((Xk)-'e - (Zk)-le) -- (s~a - s, k) P
(2.17)
417
and ~?k?j = S z k ( Z k ) - I
~L ( - - A X , ) -- ( - ~ ( Z k ) - l e
_ ,$zk).
(2.18)
(Note that because z k = e - x k and s , k = Qx ~, + c + s~ k, the system of (2.17) and (2.18) can be further simplified.) Then, we can choose 0 either performing a line-search to reduce the potential function (2.3), or according to heuristics based on the problem, =flmax{0
: x k +O/kx ~_ O,e-- (x k T O A x ) ~_ O,--yk--O/ky ~_0, and s.~ '~ + O(QAx - - Ay) > O} for some 0
(2.19)
Next we summarize the steps of the potential reduction algorithm. Potential Reduction Algorithm
Choose ~o = .5~ and ~ = m~x{1, 3 II Q~O + c - (~_(Q~O + c))~ II _ ( ~ ( Q ~ O + c))}; Setz ~ ~176 ~ ~176 ~176 Let a ~ = (xO)rs o + (zO)r 2 and k = 0; w h i l e A k _> 2 -L do begin Compute Ax and Ay from (2.17) and (2.18); Let 0 be given by (2.19); x k+l = x k -[- O/ks; Z k+l ~ e -
X k+l "
yk+l __=yk + 0Ay; s~ k+~ : Qx k+~ + c - yk+l; 3zk+l :
__yk+l;
/xk+, = (xk+~)rs ~+~ + (Zk+~)Ts)+~; k=k+l; end. The following results can be developed for the algorithm.
T h e o r e m 1 The algorithm terminate~ in O((p - 2n)L) iteration~ and each iteration usea O(n 3) arithmetic operations. ProoL Note that
z~ o = =
(xo)r~. o + ( z ~
~
(xo)r(Q~O + c + ~ ) + ~(~ - ~ o ) ~ = (1/2)~r(Q~/2 + c) + ~
_< m a x { ( 1 / 2 ) ~ r ( Q ~ / 2 + c) + ,~,3,~ II Q~/2 + c H - ( 1 / 2 ) ~ r ( Q ~ / 2
+ ~)} < 2 L.
On the other hand, it can be verified that 7~
2n ln(A ~ - ~ ln(x,~176 i=1
- ~_, ln(zi~176 i=1
< 2n ln(2n) + O(1).
418
Therefore, ~(zO, zO, sO, sO) _< O ( ( p - 2~)L). From the potential reduction theorem [7] for p _> 2n + V"2"n, we have that
~(~+,,~+~ ~+,,~k+1) _
~ , ~ ) _ 3/16.
Thus, in O ( ( p - 2n)L) iterations,
~(~k, zk,~k, ~k) < _ ( p _ 2~)Z, and hence, (p -- 2n)ln((zk)T.s= k + ( z k ) T s z k) < --(p -- 2n)L -- 2n ln(2n), i.e.,
~k = (zk)rs k + (zk)rs k < 2-C.
3.
Computational
0
Results
The interior-point algorithm for convex quadratic programming problems with box constraints has been implemented using vectorization on an IBM 3090-600S with Vector Facilities utilizing the VS Fortran compiler. T h e ESSL (Engineering and Scientific Subroutine Library) is used for vector and matrix operations [1]. Vectorization is an architectural method used to pipeline the execution of implicit or explicit vector operations, hnplicit vector operations are in the form of do-loops in a language without explicit vector constructs like FORTRAN. A vectorizing compiler analyzes do-loops to determine if a vector construct implicitly exists. If one does exist, then the compiler generates machine vector operations that are significantly faster than the equivalent scalar operations needed to perform the do-loops. The IBM 3000-600S Vector Facility has a set of 16 vector registers of single precision or 8 registers of double precision. Each vector register may contain as many elements as the section size of the system, which is 256. Each vector is partitioned into a sector in order to execute on tile vector hardware. Compound vector operations such as multiplyand-add, nmltiply-and-subtract, and multiply-and-accumulate can be pipelined by the vector facility. For these 3 instructions, two F L O P s are executed per cycle after pipeline startup. The VS Fortran compiler checks 4 stages of qualification before a D O loop can be compiled into vector instructious: Analysis Eligibility Stage, Recurrence Detection Stage, Operations Support Stage, and Economic Analysis Stage. ESSL is a collection of high-performance mathematical subroutines coded in VS Fortran and Assembly language; ESSL has a good performance for matrix-vector linear algebra and matrix multiplication. It uses state-of-the-art algorithms tailored to the IBM 3090 Vector Facility. ESSL's peak performance is about 75 MFLOPs. T h e practical vector program performance goal of the system is about forty to fifty MFLOPs.
419
For our computational experiments, we use two classes, dense and sparse matrices , of test problems. The method used to generate the first class of the test problems is well explained in [51. The o~her class consists of obstacle problems (see [11]). We solve the test problems using our potential reduction algorithm introduced in the previous section. /k k > 2 -L is used for checking optimality during the algorithm. We use a fixed small positive number 10 -5 as 2 - c , and l ( f(xk-1) - f(zk)) / f(zk) I < 10-s is used for the stopping criteria additionally. All computations are performed with double precision. In the potential reduction algorithm, a linear system (Q + Dk)x ~ = bk is solved at each step k, where D e = diag(dlk, d 2 k , . . . , d , k) and di k > 0, i = 1 , . . . , n . Since Q is symmetric positive definite, Q + D k is also symmetric positive definite. The linear system solver is a m a j o r time consuming step of the interior-point methods and many researchers have focused on reducing CPU time for this step. There are two categories of linear system solvers, itcrative methods and direct methods. A direct method is used for the first class of the test problems and an iterative method for the second class.
3.1.
C o m p u t a t i o n a l Results w i t h D e n s e M a t r i c e s
We use the D G E I C D subroutine of the ESSL for solving linear systems during the potential reduction algorithm. Two parameters, fl and p, are used in our algorithm. In order to find parameter values with which the algorithm converges to the optimum objective function value quickly, we perform experimental computations by fixing one p a r a m e t e r value and changing another p a r a m e t e r value using the first class of the test problems. We find that as/~ approaches 1, our algorithm converges quickly. On the average, the n u m b e r of iterations is very large for small values of p, decreases rapidly as p increases up to the value of n l's, and then slightly increases. When p is about r~x's, we obtain the m i n i m u m average number of itcrations. From the experimental computations, we conclude that the good combination of p a r a m e t e r values is given by fl = 0.99 and p = n ''5 (for details see [5]). Next, we are going to investigate the relationship between different parameters of the test problems ha(x*), cond, dc9, and average number of iterations, Itr., required to solve the test problems using out" algorithm ([5], [10]). T h e na(x') is a p a r a m e t e r that specifies the n u m b e r of active constraints at x*. As ha(X*) increases, the n u m b e r of active constraints at x* increases. The cond determines the condition number of Q so that log(condition nmnbcr (Q)) = cond. The deg decides the amount of degeneracy of the problems. Extensive computational results with various test problems suggest that I t r . increases as the cond increases. In our computational results, we can't find a significant dependency between de9 and [tr. But, we have a very small variation of I t r . for different deg compared with the results in [10]. This means that our algorithm will terminate within the expected number of steps, independent of the degeneracy of the problems. It is interesting to consider the difficulty of the problems as a function of the number
420
of active constraints at z*. Our results show that as x* increases, the computational difficulty grows. If there are m a n y active constraints at x ' , our algorithm needs more steps to converge to the optimal solution. Table 1 shows the average number of iterations and average CPU times (in seconds) needed to find the optimal solution for various problems of dimension 100 to 1000. Ten test problems are solved for each n, 100 < n < 500 and one test problem for n >__600.
n 100 200 300 400 500 600 700 800 900 1000
Itr. 14.90 15.10 15.40 15.70 16.10 16.00 16.00 16.00 16.00 16.00
Time 0.57 3.27 10.98 25.02 47.72 83.99 129.29 197.37 274.35 372.31
Table 1: fl=0.99, p=n Ls, cond=6, deg=6, na(x')=n/2.
3.2.
Computational l~esults with Sparse Matrices
Next, we use our algorithm for solving large-scale practical engineering problems. The obstacle problems, elastic-plastic torsion problems, and journal-bearing problems can be formulated as (1.1). Using instances of tile obstacle problem, we are going to test our algorithm. T h e obstacle problems are generated in the manner explained in [11]. For these problems, Q is a special block tridiagonal matrix. Superdiagonal and subdiagonal blocks of (2 are - I E/~,~x,~, where I is an identity matrix, and diagonal blocks are also tridiagonal matrices whose diagonal elements are 4 and their superdiagonal and subdiagonal elements are -1. All blocks of Q have a same size rn by m, where rn 2 = n. Note that Q is a very sparse syrmnetric positive definite matrix. Let c=-h
2 where h =
1 m + I"
Upper bounds and lower bounds of variables are given by (sin(9.2al) sin(O.3a2)) 3 < x, __<(sin(9.2a,)sln(9.3e=)) 2 + 0.02, for I < i < n, wherec~l=(i-LT
i-1
] xm) xhandc~2=
[
~_
] xh.
421
We use the DSDCG subroutine of the ESSL that uses the conjugate gradient method for solving linear system. This method takes advantage of the sparsity and the special nonzero pattern of the matrix Q. The quantity II (Q + z)~) ~ - b~ II / II ~k i1_< ~ is used as a stopping criterion for the linear system solver. Our computational results are obtained by setting e = 10-4 and e = 10 -s. Table 2 shows the computational results. CPU times are given in seconds. From Table 2, it is clear that the total CPU times depend on the tolerance e in the
10000 40000 00000 115600
=Itr. 9 11 10 10
10 -4 Time 4.0 30.7 72.3 98.5
(5 = Itr. 13 12 14 13
10 -s Time 8.8 48.1 180.8 213.8
Table 2: 8=0.999, p = n l"s.
stopping criterion of the linear system solver.
4.
Concluding Remarks
In this paper, we consider an interior-poin~ algorithm for quadratic programming problems with box constraints and present preliminary computational results. Also, it is shown that the algoritlma is robust for the general problems and, in particular, for degenerate problems. It is shown that our potential algoritlma can be a good solver for practical engineering problems that are very large and sparse. Vectorization techniques can be used for solving problems of higher dimensions efficiently. Furthermore, by using a cyclic reduction method and the special stlnaeture of the obstacle problems, an improvement in parallelism can be achieved. T h e proposed algorithm can be used for the solution of the general, nonconvex, global optimization problems. M a n y algoritlmas for such problems (see [12]) solve a convex subproblem at each iteration. These subproblems can be solved efficiently using the interior-point algorithm described in this paper.
422
References [1] Engineering and Scientific Subroutine Library Guide and Reference. edition, Sept. 1987.
IBM, 3rd
[2] A. Bjhrck. A direct method for sparse least squares problems with lower and upper bounds. Numerische Mathematik, (54):19-32, 1988. [3] C.W. Cryer. The method of cristophcrson for solving free boundary problems for infinite journal bearings by means of finite differences. Mathematics of Computation, 25(115):435-443, July 1971. [4] R.S. Dembo and U. Tulowitzki. On the minimization of quadratic functions subject to boz constraints. Technical Report YALEU/DCS/TR-302, Department of Computer Science, Yale University, Feb. 1984. [5] C.-G. Han, P.M. Pardalos, and Y. Ye. Computational aspect of an interior point algorithm for quadratic programming problems wi~h box constraints. Technical Report CS-89-35, Computer Science Department, The Pennsylvania State University, Oct. 1989. [6] N. Karmarkar. A new polynomial time algorithm for linear programming. Combinatorica, (4):373-395, 1984. [7] M. Kojima, S. Mizuno, and A. Yoshise. An O(x/'~L) iteration potential reduction algorithm for linear complementarity problems. Technical Report, Department of Information Science, Tokyo Institute of Technology, Tokyo, Japan, 1988. [8] M.K. Koslov, S.P. Tarasov, and L.G. Khachian. Polynomial solvability of convex quadratic programming. Soviet Math. Dokl., 20:1108-1111, 1979. [9] R.D.C. Monteiro and I. Adler. Interior path following primal-dual algorithms part II: convex quadratic programming. Mathematical Programming, (44):43-66, 1989. [10] J. Mor6 and G. Toraldo. Algorithm for bound constrained quadratic programming problems. Numerische Mathematik, (55):377-400, 1989. [11] J. Mot6 and C. Toraldo. On the solution of large quadratic programming problems with bound constraints. Teclmical Report MCS-P77-0589, Mathematics and Computer Division, Argonne National Laboratory, 9700 South Cass Avenue Argonne,Illinois 60439, May 1989. [12] P.M. Pardalos and J.B. Rosen. Constrained global optimization: algorithms and applications. Zecture Notes in Computer Science 268, Springer-Verlag, 1987. [13] P.M. Pardalos, Y. Ye, and C.G. Han. Algorithms for the solution of quadratic knapsack problems. Teclmical Report CS-89-10, Computer Science Department, The Pennylvania State University, April 1989. [14] Y. Ye and E. Tse. An extension of Karmarkar's projective algoritlma for convex quadratic programming. Mathematical Programming, (44):157-179, 1989.
O P T I M I S A T I O N DES R E S E A U X E L E C T R I Q U E S D E G R A N D E TAILLE* G. Blanchon - J.C. Dodu EDF, Direction des Etudes et Recherches, BP 408 92141 Clamart - France
J.F. Bonnans INRIA, Domaine de Voluceau, BP 105 78153 Rocquencourt - France
II6sum6 Cet art.icle d6crit une m6thode de programmation quadratique successive permettant de r6soudre le probl~me de la r6partition optimale des puissances dans un r6seau 6lectrique. Cette m6thode converge globalement et superlin6airement. Des techniques onL 6t6 raises en oeuvre pour traiter des r6seaux de grande taille. Le programme quadratique de chaque it6ration est r6solu en utilisant un algorithme de gradient r6duit avec factorisation LU des matrices de base et conjugaison des directions r6duites. La m~thode a 6t6 appliqu6e au probl~me particulier de l'optimisation du plan de tension ~ puissances actives fix6es. Le r6seau 6tudi6 poss~de 838 noeuds et 1330 lignes ou transformateurs. 1
Introduction
Le probl~me de la r6partition optimale des puissances est l'un des principaux probl~mes rencontres par les ingdnieurs d'6tudes de r6seaux ~lectriques. Depuis le d6but des ann6es 60, la plupart des mdthodes d'optimisation non tin~aire ong 6t6 expdrimentdes pour le rdsoudre. Parmi les m6thodes utilisdes en premier, on peut citer Ies m f t h o d e s du gradient rdduit gdn4ralis@ [8]. A partir des ann4cs 80, ont ensuite 4t6 appliqudes des m~thodes de prograrranation lindaire successive [3], de lagrangiens augmentds, puis de programmation quadragique successive. Parmi ces derni~res, on distingue deux classes : 1. Les m4thodes de Newton, dans lesquelles la matrice des d~riv4es secondes du crit~re quadratique est le hessien du lagrangien du probl~me [17,13]. 2. Les m4thodes de quasi-Newton, dans lesquelles la matrice des ddriv4es secondes du crit~re quadratique est une approximaticn du hessien du lagrangien. Cette approximation est raise h jour b. chaque itdration s l'aide d'une formule telle que la formule D F P ou la formule BFGS [11,101.
5es m~thodes de p r o g r a ~ a t i o n quadratique successive sont consid6r~es aujourd'hui comane Ics plus efficaces par de nombrcux auteurs. On sait qu'elles poss~dent une vitesse de convergence superlin5alre (et m~me quadratique dans le can des m~thodcs de Newton), ce qui, de ce point de vue, les rend sup6rieures aux autres m6thodes. Dans une m6thode de quasi-Newton, le hessien du crit~re quadratique est une matrice pleine, ce qui limite la taille des probl~mcs dtudi6s. La dimension du hessicn du crit~rc quadratique ne semble pas pouvoir d~passcr quelqucs cengaines 1. Dans une m6thode de Newton au contraire, le hessien du crit~re quadratique (c'est-h-dire le hessien du lagrangien du problSme non lindaire) est une matrice creuse qui poss~de le mSme creux que le jacobien des 6quations que mod~lisent un r6seau 61ectrique. De plus, le calcul des ddriv~es secondes est simple (au moins dans son principe !), car les fonctions non lin6aires song analytiques. "Pr6sent6 b. la 9~m~ Conference Internationale Analyse et Optimisation des Syst&mes, 12-15 Juin, Antibes, 1990 ; Lectures Notes in Control and Information Sciences, Springer-Verlag, Berlin, s paraitre 111 ne semble pas, ~. cc jour, quc les formules de qua.si-Newton qui permettent de pr6server ]e creux du hessien du lagrangien du problSme non-lin6aire aient 6t~ utilis6es.
424 On peut alors envisager des tailles de hessien pouvant atteindre 4000 x 4000, ce qui correspond h l'6tudc dc r~seaux pouvan~ eomporter jusqu's 2000 sommets [7]. Dans cet article, nous proposons de r~soudre le probl~me de la r~partition optimale des puissances s l'Mde d'une m~thode de Newton. Dans les autres ~tudes utilisant la m6me approche, on n'effectue pas de recherche lin~aire apr~s r~solution du progroanme quadratique. A la difference de ces ~tudes, nous allons munir l'algorithrne d'outils issus d'une r~fiexion th~orique qui permettent d'assurer la convergence globale tout; en gardant le pas ~gal s 1 (done la convergence quadratique) pros de l'optimum. Ces outils sont l'id~e de convexification du probl~me quadratique s'il ne donne pas une direction de dcscente d'un critbre de recherche lin~aire, ainsi qu'un nouveau crit~re de recherche lin~aire. De plus, l'impl~mentation du probl~me quadratique est adaptSe s la grande taille : r~solution par gradient rdduit avec factorisation LU creuse ct conjugaison des directions r~duites lorsquc cela a un scns. I1 est int~ressant de noter que si, s la suite des travaux de Han [12], les m~thodes de programmation quadratique successive ont ~t~ largement utilis~es, les codes publids dans la litt~rature ont dtd implant,s essenticllement pour des probl~mes de petite taille. Au contraire, les implSmentations concernant les probl~mes de grande faille ont port~ sur des m~thodes de type Robinson (cf. [14]) ou des m~thodes de gradient r~duit conjugud pour des systbmes s contraintes lin~aires (cf. [16]). Notre impl~mentation, s laquclle nous avons donn~ le nora OPSYC (optimisation de syst~mes creux), est done une des premieres implementations de la mSthode de Newton pour les problbmcs de grande faille, qui reprenne les outils de globalisation de l'algorithme. 2
F o r m u l a t i o n d ' u n e classe de p r o b l ~ m e s d ' o p t i m i s a t i o n des p u i s s a n c e s d a n s u n r~seau ~lectrique
On entend ici par "rdseau ~lectrique" un syst~me constitu~ de moyens de production alimentant des consommateurs par l'interm~diaire d'un rSseau de transport maill~ ; lui-m~me compos6 de lignes et de transformateurs. Ce syst~me peut 6tre sch~matisd par un graphe comprenant N nocuds et L arcs. Un noeud reprdsente un site de production et/ou de consommation. Un arc reprdsente unc ligne ou un transformateur. Les lignes et les transformatcurs sont paxcourus par du courant alternatif. L'~tat d'un r6seau 61ectrique en rdgime stationnaire est compl~tement ddfini lorsque l'on connait le module V/et la phase 0, de la tension en chaque noeud i. La puiss,'mce active P, injectde au noeud i et la puissance r6active Qi inject~e au noeud i s'expriment en fonction des modules et des phases des tensions par des relations de la forme : P, = ~, v? - ~
,s~ v~ vk cos(O, - ok + (,k )
(2.I)
,J~kV~Vksin(0~ -- Ok + (~k)
(2.2)
kEAi
Q~ = 6,v~ ~ - ~ kEAi
off al, b,, Y~k, (ik sont des constax~tes et os Ai repr~sente le sous-ensemble des noeuds reli6s au noeud i par ~ arc. P, (resp. Q~) est le bila~ alg6brlque des productions et des consommations actives (resp. r6actives) au noeud i. Dans ce qui suit, les 6quations (2.1) et (2.2) seront respectivement 6crites sous la forme P = ~o(0,V) et Q = (r off P,Q,0 et V sont des veeteurs de dimension N ayant respectivement pour composantes Pi, Qi, Oi et V/. Dans les dtudes de planification, de gestion prfivisionneUeou d'exploitation des r6seaux ~lec~riques, on est amend ~ r~soudre des problSmes d'optimisation consistmat s minimiscr une fonction des
425
variables P, Q,O et V, tout en respectant les 6quations alg6briques du r&eau (2.1)(2.2) et des contraintes d'in~galitd qui traduisent les limites de fonctionnement des composants du %seau (centrales ghermiques et hydrauliques, lignes, transformateurs, ...). Un tel probl6me se pose sous la forme d ' u n programme math~matique non lindaire qui s'&rit :
Minimizer f(P, Q, O, V) sova les conditions : P
=
g(P,Q,O,V) <_ o;
~o(O,V); Q = r
p'* <_ P < P M ; Q m < Q < _ Q M ;
Vm
Dans cet article, nous &udierons plus sp&ialement le probl~me de l'optimisation du plan de tension. Dans cc probl6mc, on souhaitc que la puissance activc injectde au noeud i soit fix& ct 6galc s une valeur donnde po. Ceci n'est p ~ possible a priori car on dolt r6aliser l'6quilibre : N
E P, = ~
(2.3)
i=1
off 7r repr&en~e les pertes actives du r~seau (dissip&s par effet Joule dans les r6sis~ances dcs fignes et des transformateurs), et ces pertes ne song pas connues ~ l'avance. En pratique, l'dquation (2.3) est satisfaite grs ~ l'action des r6gulateurs automatiques de vitesse des alternateurs. Nous supposerons que la log de r~glage automatique conduit ~. des relations de la forme : p~=po+A~p,
rinG,
off G C { 1 , . . . ,N} repr&ente le sous-ensemble des noeuds qui parficipent au rdglage, et off les cocfficients Ai song domlds et gels que : A~ > 0; lEG Oil
aura
:
p , = po,
N
vies;
p=~-EP ~ {=1
La puissance r6active inject6e an noeud i sera de la forme :
Qi = Qi~ + Ci - Si - Qi~, o• Qia est la somme algdbrique des puissances r~actives fournies ou absorbSes par les alternateurs, lcs condensateurs et les selfs. Qia satisfait des indgallt& du type : m <- Qia g Q M, Q,a
i E R,
(2.4)
oh R C { i , . . . , N} est le sous-ensemble des noeuds off il y a au moins un alternateur, u n conden-
sateur ou u n self. Dans (2.4), Q~] peut &re n~gatif. Ciet S; song des variable suppl~mentaires, que nous d&igncrons par vaxiables arfificielles, et qui sont telles que :
C~>O,
S~>_O,
i=l,...,N;
Qir cst une d o n n a qui rep%sente la puissance rdactive consommde au noeud i.
(2.5)
426 Le probl~me de l'optimisation du plan de tension consiste s minimiser une fonction de la forme : N
N
F = o~p+,8~_,Ci + 7 ~ _ , S i , i=1
ou bien de la forme :
(2.6)
i=l
N
N
N
F = c~--~(V~M - Vi) + f l ~ C i i=1
+TESI
i=l
(2.7)
i=1
oh. ~ est un coefficient positif et o~ fl > 0 et 3' > 0 song choisis trSs grands par rapport s c, La forme (2.6) correspond ~ la minimisation des pertes actives, alors que la forme (2.7) correspond h la recherche du plan de tension le plus ~lev~ possible. 0 u t r e les infigalit~s (2.4)(2.5) et les in~galitds V " s'5crivent : Pi~ + s = vi(O, V), Pi~ = VI(0, V),
V i e G,
Vi r G,
Q,g + Ci - S, - QI~ = r Ci - Si - Qir = r
<_ V < V M, les contraintes s respecter
V), Y),
(2.8) (2.9)
V i e R,
Vi r
(2.10) (2.11)
Les variables "axtificielles" Ci et Si song introduites darts la formulation du probl~me pour qne celui-ci air toujours une solution r~alisable. St, 5. Poptimum, ces variables song positives, elles pcuvcnt 6tre interprdtdes comme le volume global de condensateurs et de sells suppldmentaires qu'il faudrait installer au noeud i pour avoir un fonctionnement du rdseau dans les limites de tension admissible. Le probl~me de l'optimisation du plan de tension entre dans le cadre de la formulation suivante : I Minimiser f(=) + ~_, ~j(ej);
e = g(x);
xm < x < x M
(2.12)
I'=1
o~t f et gi,J = 1 b. p song des fonctions de R" darts R, et oh ffj : R --~ /~ :=] - c% +c,~] est une fonction convexe "~tendue" (c'est-s pouvant prendre la valeur +c~). Rappelons que le "domaine effectif" d'une foaction convexe (&endue) h :/~'* ~ _~ est, pax d~finition, l'ensemble :
dora h := {= e R" 11,(=) < + ~ } . On a donc h(x) = + ~ s i x f[. dora h. Une fonction convexe (~tendue) h : R n --~ /~ est dire "propre" si dora h n'est pas vide . Les fonctions ql correspondant au probl~me de l'optimisation du plan de tension song des fonctions convexcs propres, b. valeurs positives ou nulles. EUes song de plus lin6aires par morceanx dans leurs domaines effectifs. Pax exemple, en posant x = (0, V,p) E R 2N+1, a la contrainte (2.10) on peut faire correspondre la contrainte el = gi(x) avec :
gi(x) = r
V) + QIr
la fonction ~i ~tant ddfinie pax :
9 ~(el)
min{/~Ci + 7S~; O~ + Ci
Sl = e,, O,~ < Oi9 < QiM, el > o, s~ > 0}.
Puisque ~i(el) est la valeur optimale d'un programme lin~aire ayant pour second membre ei, ~i est une fonction convcxc de ei, lin~aire pax morceaux dons son domaine cffectif. Cette fonction cst rcpr6sentSe sur la figure 1.
427 ~i(ci) pente - 7
/
pellte
QM
" ei
De m~me, & la contrainte (2.8) correspond la contrainte el = gi(x) avec :
g,(:~) = ~,(o, v ) - :,,p - eo, la fonction r
dtant ddfinie par : dam~,={0},
3 3.1
~,(0)=0.
U n a l g o r i t h m e d e t y p e N e w t o n a d a p t d & la f o r m u l a t i o n (2.12) Fornmlation du probl~me quadratique.
Analyse locale.
Le fait que les fonctions ffj de la formulation (2.12) sont lindaires par morceaux permct de formuler un probl~me de programmation non lint~alre ~quivalcnt, h donn~es rdguli~res, dans lequel certaincs variables interviennent lin6airement. Pour ceci on note que ~j 6tant le marx d'un nombe fini de fonctions lin6aires s'6crit commc 'l~j(t) = =
max{ajk 4- fljkt;k = 1 . . . s j , t e dora (ffj)}, rain{7 6 R;c~j~ + f l j k t , - 7 <- O,k = 1 . . . s j } .
Le probl~me (2.12) revient done & rdsoudre P
minf(x)+ vie
dom(~j);
~-'~vj;g(x)=e; j=, ajk+fljkcj--vj
x m <x <xM; k=l...sj,
(3.1) j=l...p.
On peut alors vSrifier que la formulation du probl&me quadratique associd & la mdthode de Newton pour (3.I) peut alors se ramener & la rdsolution du probl~me suivant (of. ddtails dans [4]) qui est l'analogue du probl5me quadratique associ6 & un probl&me de programmation non lindaire standard :
Q(z,H) minVf(z)'d+ (d,~)
d'tId+ ~Oj(ej); j=l
g(z)+g'(z)d=
e; x '~ < z + d
--
<
--
z M.
Ici la matrice n x n H est le hessien du lagrangien L(x, A) := f ( x ) + Atg(x) par rapport h x (ou une approximation si une m~tbode de quasi-Newton est utilis5e). La valeur initiale de e n'intervient pas (ceci est dfi au fair que e intcrvient d'une fa~on "lindaire" dans (2.12). Une m5thode de type Newton sans recherche lindaire se formule done de Ia mani&re suivante :
Algorithme
1
428 0) Choisirx ~ n e t A~ ~;k#--0. 1) Choisir H ~, approximation de V2L(x ~', A~) et calculer d ~, solution locale de multiplicateur associ6 ATM. 2) x ~+~ ~ x ~ + d ~. k ~ k + l . Allerenl.
Q(x ~, H ~) et
le
Dans le cas oh H ~ = V~L(xk, A~), et pour le probl~me de programmation non lin6aire dquivalent on obtient une convergence quadratique si les gradients des contraintes actives sore linfairement inddpendants, la condition du deuxi~me ordre 6rant v6rifi~e ; la complfimentarit6 stricte n'est pas n~cessa]re (cf. [51). 3.2
G l o b a l i s a t i o n de l ' a l g o r i t h m e
L'extension des id6es de Han [12] sur la globalisation de la mdthode de Newton par une recherche lin6alre dans la direction d ~' fondde sur la d6croissm~ce d ' u n e fonction de pdnalisation exacte nous amine 5. consid6rer la fonction p
P,(x,e) = f(x)
+ ~-~ 4}i(el) + dl~(~) - ell,, i=1
oh I1.111est la norme s On peut alors remplacer la phase 2 de l'algorithme 1 par une recherche lin6aire dans la direction d ~. Une condition suffisante pour que d ~ soit une direction de descente de P~ est que r > ]IA~[[ooet que H ~ soit d6finie positive. Cette derniSre condition peut dventuellement ~ r e relax~e s i x k est proche de ~: et H k = (voir [5]). L'acceptation du pa~ pk = 1 ne peut ~tre gaxantie m~me si (x k, H k) est proche de (:~, V2L(~, A)) ; c'est l'effet Maratos. La m6thode de [6] qui dvitc l'effct Maratos peut ~tre ~tendue au probl~me (2.12) en consid~rant la fonction
V2L(x k, Ak)
P
P~.~(x, e) = f(=) + ~ r
+ ~*(g(x) - e) + Hit(=) -- ell,.
/=1
Ici # est une approximation de A mise s jour dans l'algorithme suivant les iddes de [6] et on base la recherche lindaire sur la d6croissance de _/5 ,,. Comme le probl~me de minimisation de P par rapport 5. c (soit /s,.,(x) la fonction ainsi obtenue) se ddcompose, composante par composante, en la minimisation d'une fonction lin6aire par morceaux d'une variable scalaire, on pcut le rdsoudre explicitement 5. un cofit mod6r& On se f a m i n e ainsi 5. la minimisation d'une fonction de p6nalisation exacte ne d~pendant que de x. 3.3
A l g o r i t h m e de r 6 s o l u t l o n d u p r o g r a m m e q u a d r a t l q u e
En raison de la grande taille du problbme nous avons choisi tm algorithme de type gradient r6duit conjugu6 avec recherche lin6aire exacte. Le calcul d'un point admissible est effectu6 en minimisant une fonction p6nalisant les contraintes (analogue b. /5 ,,). L'efficacit6 pratique du code d6pend de fa~on critique de la strat6gie de lib6ration des variables. Une variable n'est lib6r6e que si son gradient r6duit n'est pas trop petit ; de plus le nombre de variables !ib6r6es simultan6ment est limitd. La matrice de base est factoris6e et raise 5. jour en utilisant le code de factorisation LU creuse dfi 5. J.K. Reid [15]. 4
R 6 s u l t a t s et a p p l i c a t i o n s n u m 6 r i q u e s
L'algorithme a 6td mis an point et test6 tout d'abord sur des r6seaux d'6tude de 22 sommets, puis 193 sommets, enfin sur u n r6seau de grande taille (838 sommets). Nous allons d6crire et examiner
429 les rdsultats obtenus sur ce dernier r~seau. C'est un r~seau 5. tr~s haute tension (400-225 kV) comprenant 838 sommets, 1171 lignes et 159 transformateurs. La consommation est de 65 00O MW. La production cst assurde par 168 groupes thermiques et des groupes hydraulique raccordfis cn 67 sommets du r~seau. On dispose, en outre, de 443 sources de r~actif se ddcomposant en : 168 groupcs thermiques, 67 sommets de production hydraulique, 42 compensateurs synchroncs ct 166 sommets de raccordement de condensateurs ou sells. Le probl~me tralt6 sera l'optimisation du plan de tension dans deux configurations particuli~res. Le r6seau considdrfi cst, en fair, un r~seau de planification ~tudi~ h l'ann6e (N + 5), ce qui signifie que les charges rfiactives sont tr~s insuffisamment compens~es et un programme de load-flow classique (calcul des transits connaissant les puissances) ne peut fournir de solution et divcrge, paxce que, e n c e r t a i n s sommets, la limite de puissance transmissible est dfipass~e. Nous allons donc d'abord effectuer une fitude, classique en planification, qui consistc 5. ddterminer les moyens de compensation suppldmentaires h installer sur le r~seau pour assurer un fonctionnement dans les limites de tension admissibles. Ce sera notre premiere ~tude sur le r~seau dficrit ci-dessus que nous appellerons "r~sean non compensr Dans un deuxi~me temps, nous supposerons que les charges rdactives de ce rdseau ont dt~ pr6alablement compens~es et nous effecterons sur ce rfiseau, que nous appeUerons "rdseau compens6", une 6tude de type exploitation : h partir d'un plan de tension "plat" (mfime tension initiale en tousles noeuds) nous d~terminerons la meilleure rfipartition des puissances r6actives. Dans les deux problSmes, le crit~re d'optimisation correspond 5. la minimisation des pertes actives. Les variables sont : - les les les - les -
-
tensions en phase et module : 0, V en chaque sommet, soit 1676 variables, pertes actives sur le r~seau : 1 variable, productions r~actives des sources de r6actif, soit 443 variables, variables artificielles C et S e n chaque sommet, soit 1676 variables.
I1 y a donc au total 3796 variables. Chaque probl~me comporte : - 1676 contraintes d'6galit6, - des contraintes de bornes inf6rieure et supdrieure sur les modules de tension et les productions r6actives des sources de r~actif, - des contraintes de positivit~ sur les variables artificiclles. 4.1
l:t~sultats
Les principaux r~sultats figurent darts les tableaux 1 et 2 darts lesquels : PQD num6ro du programme quadratique, NIT hombre d'it~rations de chaque programme quadratique, FACT nombre de changements de base effectu6s dans chaque programme quadratique, C O N J hombre de conjugaisons des directions dans l'algorithme du gradient conjugu&, CYCL nombre de cycles. Nous avons adopt~ une strat~gie de libdration des variables hors base qui consiste s ne rendre superbasique qu'unc pattie des variables dont le gradient projet~ est non nul, les antres variables restant bloqu6es. Un cycle est la rdsolution du prograrmne quadratique darts lequel on ne remet pas en cause cette partition,
430 PAS pas de la recherche lin~aire d'Armijo, C R I T E R E valeur, apr~s chaque programme quadratique, de la fonctiotmelle/5~,,r(x, e), RESIDU Mesure en nomre gx des conditions d'optimalit@. grandeur de la distance & l'optimum.
Cette mesure donne l'ordre de
Les prineipaux points & noter sont : - la tr~s rapide croissante du crit~re, - la qualit6 de la solution obtenue (valeur de R E S I D U ) , - l'efficacit6 de la stratdgie de libdration des variables : en effet, Ie programme quadratique tend & ddbloquer trop de variables, ce qui g@ne l'identification des contraintes actives, - la mise en oeuvre effective de la conjugaison des directions, de fa~on plus nette 6videment pour le r6seau non compens6 qui comporte 42 variables superbasiques & l'optimum, alors que le r~seau compensd n'en comporte que 4. Tous ces rdsultats ont pfi ~tre compares avantageusement k ceux obtenus par un algorithme de programmation lindaire successive disponible b, E D F [17] ; en particulier, les temps de ealcul sont divis~s par 2. PQD 1 2 3 4 5
NIT
C R I T E R E RESIDU 1117 617.5 13.6 97 489.8 384.8 0 18 0.50 78 310.7 910.6 49 6 1.00 77 795.5 178.7 67 3 1.00 77 766.6 4.2 3 0 1.00 77 766.6 0.4 0 0 1.00 calcul sur IBM 3090 = 19.392 secondes
FACT
88 68 69 4 79 3 6 0 1 0 Temps de
CONJ
CYCL
PAS
Table 1: l:h!seau 838 sommets non compens6
PQD 1 2 3 4 5
NIT
FACT
90 56 65 27 24 3 1 0 1 0 Temps de
CONJ
CYCL
PAS
C R I T E R E I RESIDU 25 095.3 15.8 19 480.9 72.2 0 10 0.50 13 656.7 143.4 0 13 1.00 13 555.3 10.2 9 3 1.00 13 549.8 0.2 0 0 1.00 13 549.8 0.2 0 0 1.00 calcul sur IBM 3090 = 16.308 secondes
Table 2: R6sean 838 sommets compens6
References [1] K. Aoki, M. Kaneshashi, A modified New,on method for optimal power flow u~ing quadratic approzimated power flow, I E E E Trans. on PAS Vol. PAS-104, n ~ 8 (1985), 2119-2125. [2] D. Bertsekas, Constrained optimization and Lagrande multiplier rnethod~, Academic Press, New York, 1982.
431
[3] G. Blanchon, J.W. uoclu, A. Media, Developing a new tool for real-time control in order to coordinate the regulation of reactivepower and the voltage schedule in large-scale EHV power systems, CIGRE,-IFAC Inetrnational Symposium on Control Applications for Power Systems, Florence, 1983. [4] G. Bla.nchon, J.F. Bonnans, J.C. Dodu, Application d'une mgthode de programmation quadratique successive ~ l'optimisatlon des puissances dons les r~seaux 6Iectriques de grande faille, Bull. Dir. Etudes et Recherches, ser.C. EDF, h paraitre. [5] J.F. Bonnans, Local study of Newton type algorithms for constrained problems, in Lecture Notcs in Math. 1405, p. 13-24, S. Dolecki ed., Springer Verlag, 1989. [6] J.F. Bonnans, Asymptotic admissibility of the unit stepsize in exact penalty methods, SIAM J. Control Optim. 27, p. 631-641, 1989. [7] R.C. Burchett, H.H. Happ, D.R. Vierath, Quadratically convergent optimal power flow, IEEE PES Winter Meeting, Paper 84 WM 045-1, Dallas, Texas, 1984. [8] J. Carpentier, Differential injections method : a general method for secure and optimal load flow, Proe. of the IEEE PICA Conference, Minneapolis (1973), 255-262. [9] R. FleLcher, Practical methods of optimization (second edition), Wiley, Chichester, 1987. [10] L. Franchi, M. Innorta, P. Marannino, The Han-PowelI algorithm applied to the optimization of the reactive generation in a large-scale electric power system, Proc. of the IFAC Symposium Large Scale Systems, Warsaw, 1983. [11] T.C. Giras, S.N. Talukdar, Quasi-Newton method for optimal power flows, Int. J. of Electrical Power and Energy Systems, Vol. 3, n ~ 2 (1981), 59-64. [12] S.P. Han, A globally convergent method for nonlinear programming, J. Optimiz. Theory Appl. 22, p. 297-309, 1977. [13] G.A. Maria, J.A. Findlay, A Newton optimal power flow program for Ontario Hydro EMS, IEEE Trans. on Power Systems, Vol. PWRS-2, n ~ 3 (1987), 576-584. [14] B.A. Murthagh, M.A. Sounders, A projected lagrangian algorithm and its implementation for sparse nonlinear programming bases, Math. Programming Study 16, p. 84-117, 1982. [15] J.K. Reid, A spar~ity exploiting variant of the Bartels-Golub decomposition for linear programming bases, Math. Prog. 24, p. 55-69, 1982. [16] D.F. Shanno, R.E. Maxsten, Conjugate gradient methods for linearly constrained nonlinear programming, Math. Programming Study 16, p. 149-161, 1982. [17] D.I. Sun, B. Ashley, B. Brewer, A. Hughes, W.F. Tinney, Optimal power flow by Newton approach, IEEE PES Winter Meeting, Paper 84 WM 044-4, Dallas, Texas, 1984.
A d a p t i v e Variable M e t r i c M e t h o d s for Nondifferentiable O p t i m i z a t i o n P r o b l e m s S.P. U RYAS'EV
Research Scholar International Institute for Applied Systems Analysis A-2361 Laxenburg, Austria
Summary.
This paper deals with new variable metric algorithms for nonsmooth optimization
problems, so-called "adaptive algorithms." The essence of these are as follows: there are two simultaneously working gradient algorithms, the first in the main space, the second with respect to the matrices that modify the space variables. The convergence theorems for these Mgorithms are given for different cases.
1
Introduction
Many efficient algorithms have been developed for nonsmooth optimization problems (see, for example, Refs. 1-8). Most of these algorithms require the solution of some complicated linear or quadratic programming subproblem at each algorithm iteration. The first subgradient algorithm proposed by N.Z. Shor (Ref. 9) requires only subgradient calculations and projection operations at each iteration. It appears that tiffs a/gorithm has a low convergence rate for ill-conditioned problems. Variable metric algorithms lie somewhere in between: they require not too many calculations at each iteration, but nevertheless have a good practical rate of convergence for ill-conditioned functions. Variable metric algorithms are widely used for smooth optimization problems (see, for example, Ref. 10). As a rule, these algorithms cannot be generMized to nonsmooth optimization problems. The difficulties are connected with the fact that even if the first and second derivatives exist at some point, they do not give the full local description of the function. Because the function is nonsmooth, a point of nonsmoothness can be arbitrarily close to the point where derivatives exist. These difficulties have lead to the appearance of new ideas in the construction of variable metric algorithms. In the works of N. Shor and his coworkers (see, for example, R.eL 11), so-called "space-dilation algorithms" were developed. Such an approach offers the opportunity to construct practieM and effective algorithms, but the most effective algorithm from this family, the r-algorithm, is not sufficiently understood from the theoretical point of view. l~efcrences to this and related topics can be found in IteL 12. This author proposes an alternative "adaptive" approach that is applicable to optimization and game theoretic problems. The first variable metric algorithms in the framework of this approach are discussed in R.eL 13 for stochastic quasigradient algorithms. (Different optimization algorithms for stochastic optimization problems can be found in 1Lef. 14 and others.) The results of this paper are based upon the working paper (R,ef. 15).
433
2
E s s e n c e of t h e A p p r o a c h
Let us consider a convex optimization problem
(i)
f(z) ~ min, xER
n
where the function f ( z ) is convex on the Euclidean space R'*. We use the following recurrent algorithm to solve this problem: x "+1 = z ' -
p,,H'g',
s = 0,1,...
(2)
where s is the iteration number; p, > 0 is a step size (scalar value); H" is an n x n matrix; and g* is a subgradient from the subdifferential Of(x) of the function f ( x ) at the point z*, i.e. g* E Of(z*). We recall that the subdifferential of the function f ( x ) at the point y E -R~ is given by the formula (see, for example, l~eC 16)
Of(y)
= {g ~ R " : f ( x )
- f ( y ) >_ ( g , x - y)
Vx E 22~}.
At the s th iteration, the natural criterion defining the best choice of the m a t r i x 11 s is via the function
~ , ( H ) = f(xs
_ p,_Ug').
T h e best m a t r i x is a solution of the problem
Ts(ll)'-~
min HER.
.
(3)
x-
It is easy to see that problem (3) is a reformulation of the source problem (1), since if H* is a solution of (3) then the point x s - p,H*g" is a solution of (1). Moreover, problem (3) is more complex t h a n (1) because the dimension of problem (3) is n times higher than t h a t of (1). llowever, at the s th iteration of algorithm (2) we do not need the optimal matrix, it is enough to correct (update) the m a t r i x H*. If we already have some m a t r i x H~, then the direction of adaptation can be defined by differentiating, in the general sense, the function ~os(II) at the point t[~. If the function f ( z ) is a convex function, then the function ~os(H) is also convex. We can use the following formula (ReL 17) for t h e differentiation of the complex function ~o, : 0~,(1:;)
= -p, {~/:9 ~ o : ( ~ " - pj:;,:)}
;
here and below, the superscript T means transposition. If g~ 6 Of(z" -pstI~)g'), then _p,g~g,r E O~(H~)). 9
.
T
W i t h respect to the m a t r i x / / , in the &rectmn g~g" , one can do a step of the generalized gradient method: s
s
H~=H~+Aogog
s T
s
,
Ao > 0 .
It is possible either to take I P = H~ or to continue the iterations of the generalized gradient algorithm with respect t o / / : s
a
3
$
Hi+ 1 = H i + Aigig
s T
,
a
Ai > 0 ,
(4)
434 where g: E Of(x"
p j I : g ' ) and _p,g~g,T E O 1, assume H" = I/'~, ()" At the next
-
_
iteration, II~ +1 = H'. The number i(s) can be taken independently of s, for example, i(s) = 1 for all s. GenerMly speaking, algorithm (2) is not monotone with respect to the objective function f(x). However, one can chose i(s) such that
f ( : . p.lt,'{o)g')
< f(:
-
p.n~g')
and, with each iteration, the objective function decreases. Note that matrix updating requires additional calculations of objective function subgradients. This can be avoided by taking g,+X = 98, i(s) = 1 and using the matrix H~ at iteration ( s + 1). Therefore, we propose the following formula for matrix updating: H s+l = H a+)%gs+lg "T,
As > 0.
(5)
In formula (5), additional subgradient calculations are not required.
3
C o n v e r g e n c e for S m o o t h F u n c t i o n s
First let us give the convergence theorem of algorithms (2) and (5) for the case of a differentiable function f(x). Denote g" = V f ( x ' ) . For algorithm (2), as the direction of motion we choose the normalized
{:
subgradient
~c, =
~
, for Ilg'll #
o;
0
, for
Ila'll =
0.
The algorithm can be rewritten as follows:
x s+l =
x'-
p,H'~"
]
H" + P'g'+I~jT I '
H "+1 H~
(6)
I
where I is a unit matrix. (Thus, for the parameter A, in formula (5), we choose the value p,/llg'll.) Denote $
@4 = ~"~ptgt+l(tr,
D(K) = max IIx - yll, y,xEK
I=0
f"
=
minf(x), xER ~
.~=
min f(xt), O<_l<s
i.e., D(K) is the diameter of the set K, and f~ is a record of the lowest value of the function f during t h e previous s iterations. We also denote by T r ( Q ) the trace of matrix Q. Let us now formulate a theorem about the convergence of algorithm (6). T h e o r e m 3.1 Let f: R '~ --* R be a convex smooth Lipsehitz function
f(~) - f(y) ~ Lxll~ - yll for ~,y e le*
(T)
with Lipschitz gradient llVf(~) - vf(y)ll ~ L2llx - YI[ for ~, y ~ R ~,
(S)
435 and let there exist a compact set K such that
(9)
IWf(~)ll _> ~g > 0 for ~ ~ K.
If the step sizes p,, s = O, 1 , . . . , in algorithm (6) are nonnegative and if for all s greater than some S we /save
6K >
p~ L~2 -1 +L:2
n+2
f(x O)-f*
+L 1
Pl
+ f ( x O)
f"
,
(10)
then:
~) for s > S: L-y"
p? z~2 -~ + L2 n + ~. / ( : )
<
- 1" + z~ ~ p~
+
t=O
+ f ( z ~ - f* - 2 - 1 T r ( O ' O : )
D(K)
p,
,
e) for Pl = (s + 1) -1/2, l = 0 , . . . ,s and s > $ (where c = const):
:.-:.<_
+:(:>-r- 2 -1 r r ( O ' O : ) ] V ( K ) ( , + 1)-1/2
<
c(s + 1) - V 2 '
s) for E % p . ~ = or
-,/2
p-~l < 2L1L2D(K); if p, = s-l~ 2 then:
li_m (L - F ) (hO)) -1/: s ' n <_ r
a~oo
If the number of iterations of the algorithm is chosen before the start, then Statement 2 of the theorem implies that the convergence rate of algorithm (6) is not worse than the generalized gradient algorithm with matrix H s = I , s = 0, 1 , . . . . For H ' = [, s -- 0, 1 , . . . the following estimate (see, for example, Itef. 18) is known:
L - / " <_(s + 1)-'/2~(::~ where ~(x,X') = ~ix.. tlx -YlI,
X" = { : : f ( : )
=/'}.
(11)
Note that in the estimate in Statement 2 of the theorem, there exists an additional term Tr(OSO *r) that increases the convergence rate. Because of this term, the algorithm does not hold when the objective function is ill-conditioned.
436 4
Convergence
for
Nonsmooth
Functions
Let us consider algorithm (2). We assume t h a t at the s th iteration of the main algorithm for u p d a t i n g m a t r i x II~), formula (4) is used i(~) times. At iteration 3 + 1 we take tI~ +1 = II:(~). At iteration zero, fi~ = I, where [ is a unit matrix. Let us fix some ~ > 0. We choose i(s) to be the minimal n u m b e r such that :
(
-
p.~i,(.), ) <_:(x') -,.
It is convenient to normalize the test vector g:; therefore denote 0
ifg~' = 0,
gi ,Igi I]
otherwise.
(12)
For each s = O, 1 , . . . let the sequence { ~ i } , i = O, 1 , . . . of positive values be given. Wc express the algorithm in more detail. A l g o r i t h m 4.1 Step I
Initialization
s = O, x ~ = xi,m, gO E O f ( z ~
i = - 1 , H o 1 = I.
S t e p II
1.
l I ~ = II~+~,i=O.
2.
x~ = x" - p,H~g s.
3.
C o m p u t e g~ E Of(x~), if g~ = 0 then STOP, otherwise ~ = gr
4.
//Z+I
5.
If f(x~) < f ( x ' ) -
6.
i = i + 1, return to point 2.
=
//:+;~
~' :
si'~i g
-x.
9
e, then i(s) = i and go to step III.
S t e p III
x 3+I = x~, g,+1 = g~.
S t e p IV
s = a + 1 and return to step II.
W e n o w formulate a theorem about the convergence of Algorithm 4.1.
Theorem
4.1 Let f: Rn --~ t2 be a convex function, lhe set of minimum points X* of the function f ( x ) be
non-empty and bounded, {p,} be a sequence of positive numbers, and (As/}, ~ = 0, 1, . . . , i = 0, 1 , . . . be a given sequence of positive numbers satisfying oo
oo
~,, = oo, ix0
~
g , < oo,
for ~ = O, 1 . . . . .
(13)
ix0
Then there exists 5 such that f ( x ~) <_ e + f " and d(z~, X ~ minimum value of f on R".
-~
O, f ( x f )
-~
f " for i ~ ~ , where f* equals the
437 Algorithm 4.1 has a substantial deficiency connected with the fact t h a t the step size p~ does not change in the internal iterations i = 1 , . . . , i(s). Let us consider an algorithm with a steepest descent control of p, at each iteration i = 1 , . . . , i(s). Such a modification considerably improves the algorithm. Let v > 0 be a given number and {)~J}o be a sequence of positive numbers. A l g o r i t h m 4.2 Step I
Initialization
s = O, IIo 1 = I, j = - 1 , i = - 1 , x ~ = xi~it, go E Of(x~ S t e p II
1.
II~ = II~+:, i = O.
2.
P~i = argminp>o f ( x ~ - piing" ).
3.
j = j + l , j ( s , i ) = j.
4.
x~ = x ~
5.
C o m p u t e gl E Of(z~) such that (g~',lIly") <_ O.
6.
H:+ 1
7.
If IIx," - x q l >__ - tl, en i ( s ) = i a n d go t o S t e p III.
8.
i = i + 1, return to point 2.
-
psiII~g s.
II: + A * ' c :
S t e p III
x ' + l = z~,g "+t = 9~.
S t e p IV
s = s + 1, return to Step II.
Let us introduce some additionM designations. Let
T ( x ) = {y E an: f ( y ) <_ f ( z ) } and let L be a Lipschitz constant of the function f on the set T(z~
Recall that the function f is considered
strictly convex on the set T ( z ~ if
alf(x) + a2f(y) > f(alX + c~2y) for all cq, a2, x, y such that al+a2=
1,
al >0,
a2>O,
xET(x~
yET(x~
We next formulate a theorem about the convergence of Algorithm 4.2. Theorem
4.2 Lct a function f: R '~ ---* 12 be strictly convex (possibly nonsmooth) on the set T(x~
be given, and let the sequence {Aj}, j = 0, 1 , . . . of positive numbers satisfy the conditions
i=0
i=0
let v > 0
438 Then there ezists 5 such that
f(x ~) - f* <_ 2vL. Remark
In practical situations, the steepest descent is impossible (Step II-2 of Algorithm 4.2) and one
is forced to use approximations. Some heuristic formulae for stepsize control were used for the practical algorithm. This stepsize control is analogous to the control used for space delatation algorithm (llef. 11). l l a t h e r rough approximations are used for stepsize control.
5
Algorithm with Symmetric Matrices
T h e algorithms discussed above have the following deficiency: one m u s t store an n • n m a t r i x where n is the dimension of the source problem. We next propose an algorithm with s y m m c t r i c matrices; to store such matrices requires only (n 2 + n)/2 numbers. T h e function ~ , ( / / ) = f ( x ~ - p~Hg ~) characterizes the choice of a m a t r i x H. Denote by G the set of s y m m e t r i c n x n matrices. T h e set G is a linear space. For the a d a p t a t i o n of the m a t r i x we can consider the problem ~ , ( H ) ~ mln . II EG Analogous to (4) one can use the gradient algorithm with projection onto the set G : . . . . r~) , I17+1 = I-In ( H"i + A.iyis
A ,i > O,
where g~' E a f ( z ~ - p, tt~g ~) and l I a is the projection operation onto the set G. Lemma
5.1 If H~ E G, then
T h u s to update the m a t r i x 1I~ one can use tile algorithm
9 = H'~ + AI, ( ,gig : Hi+l
+gsg~r ) .
(14)
It is convenient to normalize tile vector g~; therefore we rewrite formula (14) as
z;+, = H," + a," ( ~ S
+
r
(15)
Tile s y m m e t r i c formula for m a t r i x modification can be combined with Algorithms 4.1 and 4.2.
6
Algorithm with Positive Matrices
Note t h a t in the algorithms described above, the m a t r i x / / "
can be, generally speaking, nonpositive. If a
function f ( x ) is convex and g" E O f ( z ' ) , then the m i n i m u m point of problem (1) belongs to the subspace
A, = {zERn:(x-x',g"
) < 0 } . It is possible that the point x "+l = x ' - p ,
Hg" does not belong to the
subspace A, if H* is nonpositive. To guarantee positiveness of tlle matrix H*, let us consider the case where m a t r i x H" is represented as follows:
439 H s = B S B "r, where B s is an n x n matrix. In this case the iteration of the algorithm is given by the formula x s+l
=
x s
_
psBSB'Tg s,
where g~ E O f ( z ' ) .
Tile function ~o,(B) = f (x" - p , B B T g ") defines tile choice of a m a t r i x B. If the
function f ( z ) is convex, then it can be proved t h a t the function ~o,(B) is weakly convex. Next we will use the family of weakly convex functions that were investigated in 1Lef. 19. (Other analogous families of functions can also be used; see, for example, I~ef. 20.) Let X be a convex subset of R ~ (possibly X = R~). A continuous function f on X is called weakly
convez on the set X if, for all z E X, the set O f ( z ) consisting of vectors g such that f(y)-
f ( x ) >_ ( g , Y - z) + ( ( z , y )
for all y E X
is nonempty, where ((x, y) is uniformly small with respect to [Ix - yl] on each compact subset I( C X, i.e. for each e > 0 there exists 6 > 0 such that I1~;(=, y)ll/ll= - yll < for =, y ~ so, 11= - yll < ~.
L e m m a 6.1 Let the function f : R n --~ 1~ be convez on 1~", and the set 13 E R nx,~ be convez.
Then the
function (p(B) = f ( x - pBBT~) is weakly convez on 13 and O~( B ) = { - p ( ~ ( ~ + r
e Of(= - pnnrr
L e m m a 6.1 gives a formula for the subdifferential of tile function qo,(B). For the a d a p t a t i o n of matrix B ' , the following gradient method can be used: B~+, = B~ + %1 (~k'9 "T + 9 " ~ r ) B~,i = 0, 1 . . . . .
(16)
where ~' denotes the normalized vector g~ E Of(x~) (see (12)). Analogous to Algorithm 4.2, we write an algorithm with t h e m a t r i x modification formula (16). Algorithm Step I
6.1 Ini tialization s = 0, Bo 1 = f , i = - 1 , j = - 1 , z ~ = xinlt,g ~ E Of(z~
S t e p II
1.
B8 =
2.
i=0.
S, B.%l,
if IIB$+~g'll = o otherwise
440 3.
P,i = argmino> o f (x" - pB~B~rg').
4.
j=j+l,j(s,i)=j.
5.
x~ = x" --psiB~B~Tg ,.
6.
Compute g~ e Of(x~) such that (g~, 23~B~'g ") < O.
7.
Bt+l = B Z + ) U ( ~ g ~ r
8.
If ]]x~ - x'll :> v, then i(s) = i and go to Step III.
9.
i = i + 1 and return to point 3 of Step II.
~_
g s ~is T )B~"
Step III
x~+x = -,,~z'~ a~+l = g~.
Step IV
s = s + 1 and return to Step II.
We formulate a theorem about the convergence of Algorithm 6.1 for smooth objective functions. T h e o r e m 6.1 Let the function f: 12'~ ~ R be strictly convex and smooth, Ll be a Lipschitz constant of the function f on the set T(x~
and L2 be a Lipschitz constant for gradient V f ( x ) on the set
T~(xo) d=ef[~'x:~eT(~0)min[Ix--y]l<_
v}.
Let there be given a value v > 0 and a sequence of positive numbers {A./} satisfying co
./=o
j=o
Then for Algorithm 6.1 there exists ~ such that
IIg~ll ~ 2~L=. R e m a r k As was mentioned before the exact steepest descent (see Step II-2 of Algorithm 6.1) is not implementable. Rough approximations are used for practical purposes.
7 1. 2. 3. 4.
5. 6. 7. 8.
References EAVES, B.C., ZANGWJLL, Generalized Cutting Plane Algorithms, SIAM Journal on Control, 9, pp. 529-542, 1971. BERTSEKAS, D.P., At~o ~r S.K., A Descent Numerical Method for Optimization Problems with Non-Differentiable Cost Functionals, SIAM Journal on Control, 11, pp. 637-652, 1973. WOLFS, P., A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions, Mathematical Programming Study 3, pp. 145-173, 1975. LSMAltECIIAL,C., STrtODIAT, J.J., AND BHIAIN, A., On a Bundle Algorithm for Nonsmooth Optimization, Nonlinear Programming 4, Edited by O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, Academic Press, New York, I981. MI~F'LIN, R., A Modification and an Extension of Lemareehal's Algorithm for Nonsmooth Minimization, Mathematical Programming Study 17, pp. 77-90, 1982. KIWlEL, K.C., Methods of Descent for Nondifferentiable Optimization, Springer-Verlag, Berlin, 1985. DI~M'JANOV, V.F., AND VASlL'I~V,L.V., Nondifferentiable Optimization, Springer, New York, 1985. ROBINSON, S.M., Newton's Method for a Class of Nonsmooth Functions, SIAM Journal on Numerical Analysis, to appear.
441
9. Sltolt, N.Z., On a Structure of the Algorithms for Numerical Solution of the Optimal Planning and Designing, Kiev, PhD Thesis, 1964 (in Russian). 10. DENNIS,J.N., AND Moltl~, J.J., Quasi-Newton Methods, Motivation and Theory, SIAM Review, 19, pp. 46-89, 1977. 11. SltoR, N.Z., Minimization Methods for Non-Differcntiable Functions, Springer-Verlag, 1985. 12. POLJAR,B.T., Subgradient Methods: A Survey of Soviet Research, Proceedings of a IIASA Workshop "Nonsmooth Optimization," Edited by C. Lemarechal and R. Mifflin, 1977. 13. UR.YAS'EV,S.P., Stochastic Quasigradient Algorithms with Adaptively Controlled Parameters, IIASA, Laxenburg, Austria, WP-86-32, 1986. 14. EJtMOLIEV, Yu., AND WETS, R.J.-B., EBS. Numerical Teehiques for Stochastic Optimization, Springer-Verlag, 1988. 15. U~tYAS'SV,S.P., Adaptive Variable Metric Algorithms for Nonsmooth Optimization Problems, IIASA, Laxenburg, Austria, WP-88-60, 1988. 16. ROCKA~'ELLAR,R.T., Convex Analysis, Princeton Mathematics, Vol. 28, Princeton Univ. Press, 1970. 17. PSIIENYClINYI,B.N., Necessary Conditions for an Extremum. Dekker, New York, 1971. 18. NESTEROV,YU.E., Minimization Methods for Nonsmooth Convex and Quasiconvex Functions, Economika i mat. metodi, USSR, XX, pp. 519-531, 1984 (in Russian). 19. NIJRMINSK1, E., Numerical Methods for Solving Deterministic and Stochastic Minimax Problems, Naukova Dumka, Kiev, 1979 (in Russian). 20. HOVFMAN,A., Weak Convex Functions, Mullifunclions and Optimization, 27. IWK d. TH Ilmenau, tieft 5, pp. 33-36, 1982.
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OPTIMIZATION:
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Alexander Ioffe of M a t h e m a t i c s , T h e T e c h n i o n H a i f a 32000, I s r a e l
Department
ABSTRACT.
ORDER
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problems
programming,
optimization
in o n e or a n o t h e r
sense,
let us c o n s i d e r
determined
-
with
relaxed see
reduced
the p r o b l e m
[Ill,
by
the
constraints optimal
that
many
(mathematical
control
[I2],[RI],
to a p r o b l e m
fact
problems,
[DZ])
can
of s u c h sort.
be
443
(p)
fo: X ~ R,
where
fo(X)
minimize
G(x) E K,
subject to
Set
G: X ~ U, E: X ~ Y
Y = RxUxV,
K = {u:O(u)
~ O}
and
take a continuous
and
dist(K,u)
E(x) =0, K
is a closed convex cone in
sublinear function
s ce+(u)
for some
function of the intersection of the polar of K
~(u)
U.
such that
c > 0 (say,
the support
w i t h the unit ball in
U~ )
and consider the functions
g1(y) = g1(=,u,v)
= max {~,e(u),llvll},
L
g2(y) = max {~,O(u)} + kllv]l; + Ilvll)
g3(y) = ~ + k(~+(u) +
(where
~
:
max { ~ , 0 } ) .
Finally,
we fix an
x ~ X
and set
F(x) = (fo(X) - fo(~),G(x),E(x)), fi(x) = Ei(F(x)),
i =1,2,3.
Then the following "reduction principles" I. If
x
is a strict local minimizer
hold:
to eiteher of
f.
and an admissible
1
element for II. local
(P), then
is an isolated local solution to the problem;
If the equality constraint
in
(P)
solution
f
attains
sufficiently III.
x
to
large
(P),
then
is an
in
(P)
are
f
Q(x)
r
=
{x:
G(x)
~ G}.
If
(cf.
[Ii]),
C
if, say, x
x ~ x
s u f f i c i e n t l y close to
=
x
(i.e.
x
x
x,
x
close to
{0},
then
at
is a x
for
the cost then
x
function is
is
a local
x.
means
that
x
we have
x ~ G and
a
fIT]
sufficient is that
condition
Q'(x)
is an isolated local solution of
we have (other than
for
is onto.
[P].)
satisfying
G(x)
f (x) > f (x). 0
of a neigborhood of
x
~ r.dist(O,Q(x)),
(For a survey of regularity results see Indeed
and
minimum
(P),
at
regularity given by the theorem of Ljusternik
for any admissible
at
in
~ C
such that for any dist(A,x)
A
local
x
attains a local minimum at
3
(Regularity of the constraint there is a constant
regular
admissible element
solution to (P) if and only if
where
a
at
k;
If the constraints
Lipschitz and
2
is regular
~ K, E(x)
(P), then
= O) which
is
This means that for any
x
0
x
itself)
either the inequality above
444 holds,
IIECx)II > 0
or
or
@(GCx)) > 0 which means that
f (x) > 0 , whereas 1
f (x) = O. On the other hand, if the latter is true and I than fo(X) > fo(X) which means that x is an isolated (P). This proves Principle Let
us
minimizer
prove for
f
valid and let
x
admissible other hand,
also
I for
Principle
III.
It
is
such
that
fo
since
~ fo(U)
of
local
to
that
CP).
solution
x
Suppose
is the
x. By regularity,
(r+l)(dist(K,G(x))
Ux - ull ~
is Lipschitz,
~(x) = s
obvious
solution
be any point close enough
u
is admissible, of
g1"
if it is a local
3
x
a
local
latter
is
there is an
+ llE(x)ll). On the
we have
~ fo(X)
+ lllx - ull
fo(X) + 1(r+l)(dist(K,G(x)) + llE(x)U) ~ f3(x). Using principle (P)
from
I, we can obtain sufficient
sufficient
conditions
for
either
and Ill, we can do the same for necessary f3
([BP],
[Ii],
[12],
reduction
to
procedures
for problems
[R2]).
composite
Thus, of
composite
unconstrained
optimization
advantages; of certain this
theory.
it also
connection
we
This
offers
difficulties
a good
number
of questions
(e.g.
condition
of
of [R2],
why
by
epi-derivative
of
instrument
technical
many
point"
them
problems
describing
the standard
sometimes
that
contain
a
composite
to many and
problems,
in specific formula
function
aesthetical the nature
better a pair
branches
seen.
In
of
"twin"
that
cover
[I4] and actually clarify a
Hessian
must be solved
a
4 and 5 stated
from which
are
[K2],
numerical
optimization.
and the other sufficient, [KI],
complicated
pair of second order conditions begin
has
second
order
conditions
or why the necessary
[KI] cannot have a sufficient
what kind of technical
We
approach
3
develop
[W]. Theorems
a unified
2 and
one necessary
cannot work in sufficiently order
of
to
via composite
"observation out
Theorems
the most recent developments
[FS],
analysis
approach
and ways
mention
second order conditions,
optimization
offers
for
f.; using principles II 1 associated with f and 2 makes it possible to use
III
with constraints
optimization
of
(of any order)
conditions
Principle
in section 4 show a way to sensitivity
conditions
"twin"
in order
ete.).They to obtain
second
also
show
a workable
cases. for
which
in proofs of all the above mentioned
the is
the
results.
lower
second
principal
order
technical
445 2.
Let
LOWER SECOND ORDER E P I - D E R I V A T I V E OF A COMPOSITE FUNCTION
f
be a function
order e p i - d e r i v a t i v e
X
which
is finite
at
x
along
h E X
f
at
with respect
The
is
limit as
x" = O, we write basic
if
t 9 0
0
for
[RI]
of
the
lower
all
second
nonzero
h,
then
A = F'(~),
= {y* 9 a g ( y ) :
K(x') = {h: g ' ( y ; A h ) (we write simply
~
and
K
if
f
w(h)
A'y"
c)
is
C 2 near
then
S
be
where
if
L
at
containing
defined by
x
a
local
x.
(I) . We set
= F"(~)(h,h);
s <x',h>}.
x;
is the linear subspace of and
7 + L
the collection
converging to zero. For any #s(h,w)
attains
x" = O) and assume that
L + Im A = Y
continuos with respect to Let
that
Y.
(regularity condition)
dom g - Y,
epi-derivative
= x'},
a) g is finite and lower semicintinuos at b) F
f
linear manifold
In what follows we consider functions 7 = F(~),
order
is the following:
m i n i m u m on e v e r y f i n i t e d i m e n s i o n a l
~(x*)
).
f"(x;h).
property
>
- t<x~,h'>)
of the epigraphs of the functions (see
its role in o p t i m i z a t i o n
f'~(x;h)
to
second
this means that the e p i g r a p h of the function h ~ f " ( x , x ~ ; h )
h ~ t-a(f(x + th) - f(x) - t<x",h>)
determines
lower
is
the upper Kuratowski
If
The
[RI]:
f'i(x,x';h)- = lim inf t-2(f(x + th') - f(x) h'~ h t 9+0 Geometrically,
x.
x" 6 X ~
defined as follows
of
on
at
there
z ~ X
such
spanned b y that
g
is
y + Az
of sequences
s E S
is a
Y
and any
= tim sup inf t 2(g( 7 + tn Ah" ng~ h'gh
s =
{t } of p o s i t i v e n h e X, w E Y, we set + t n% )
- (7)
- tn < x ' , h ' ~ ) ,
numbers
446 lim sup inf = sup lim sup n 9 m h'9 h e > 0 n ~ ~ i. The following
THEOREM
- either or
-
f"_(x,x~;h)
f2(x') ~ ~
= - ~
alternative for some
In
the
last
holds:
h;
and
f"(x,x';h)
+~
case
min ~s(h,w)). sES w -> ~s(h,w) is either =
the f u n c t i o n
or it is convex
continuous
with
in case w h e n g(-)
is a continuous
of the first
convex
is continuous,
f"_(x,x';h)
part
function.
g ( F ( x + th)) = g ( y +tAh w(.)
identically
equal
to
d o m ~;(h, ") = ~(x').
We shall give a sketch of the proof
and since
inf IIh'-hll<e
In this case
+ t2w(h))
it follows
of the theorem
+ o(t 2)
that
= lim inf t-a(g(y +tAh + t2w(h')) h'-) h
-g(y)
- t<x',h'>)
t ->+0 min
=
s ~S
lim sup tn~(g(y
+tnAh"
+ t2w(h))n - g(Y)
- tn<x%h'>)
h'-~h n-~oo
=
If
min #s(h, w(h) ). s E S
f[(~,x';h)
> - m
for any
lim inf t-1(E(~
h,
then
+tAh + t2w(h))
- g(y)
- t<x~,h>)
z 0
t 4+0 and,
consequently,
~(x ~) ~ ~.
g'(y;Ah)
In a similar
z <x',h>
for any
way we conclude
that
h
which
implies
h ~ K(x ~)
if
that
f[(~,x';h)
is
finite.
3. SECOND O R D E R
It is not a difficult second
order necessary
THEOREM
is that
2.
~ ~ ~
to d e r i v e
from Theorem
1 the f o l l o w i n g
condition.
A necessary and
matter
CONDITIONS
condition
#s(h,w(h))
~ 0
for
x
for all
to be a local h ~ K,
s E S
minimum or,
of
f(.)
equivalent-
447 ly, that for any
h E K, s ~ S
there is a
A
y* E ~
- ~[(h,y*)
such
that
~0.
For the "twin" sufficient condition we need some attraction property for
the cone
K
of critical directions.
(AP) ~ ~ z
and any bounded
This is a convenient one:
{h n}
sequence
a subsequence converging to a certain We observe
THEOREM 3
(AP)
for
and all nonzero h E K
0
for all
condition).
If
provided
that
f(.),
h E K
there is a
a
h
if
has
~ ~ z.
dim X < m.
holds,
(AP)
~s(h,w(h))
or, equivalently,
y* E ~
g'(y;Ah n) ~ 0
that
h E K.
is trivially satisfied if
(sufficient
minimizer
local
g'(y;Ahn)
that
The property
such
then
x
is a strict
> 0
for
a11
s E S
that for any s E S and any nonzero
such that
- #~(h,y*)
> 0
(2)
It follows from Theorems 2 and 3 that in every specific s i t u a t i o n we must be
able
to
calculate
@s
second order conditions. case
when
the
programming any
original
say,
problem
Since
with the Lagrangian, has
If,
#g
to
non-positive
(P)
order
to
be for
obtain
is a polyhedral is a
standard then
is p r e c i s e l y
a workable
function problem
~(h,y*)
the H e s s i a n
observed
in
this
connection
> O,
V
h E K,
condition but a weaker
natural necessary counterpart
that
calculated. terms.
in finite dimensional so called piecewise of them the
g
function
on
~
for
associated
is
always
(3)
In fact,
in
[RI]
(that is both
linear quadratic
g(').
is actually
(3)
>
by
ill]).
in some specific
Rockafellar
spaces
it does not have a
(by which we mean that replacing in
is either affine or quadratic.) f(-)
= 0
h ~ 0
the f u n c t i o n
situations
considered
dim X < ~
composite and
#s can
this has been functions
dim Y < m) with
(The latter means that the domain
can be broken into finite number of polyhedral g(.)
is the
mathematical
#~(h,y*)
It is possible to mention a few more cases in w h i c h
in different
of
function
one because
does not make the condition necessary - see
done
(which
of
y* ~ ~. Therefore the standard Lagrangian c o n d i t i o n
is still a sufficient
be e x p l i c i t l y
pair
we obtain the standard second order conditions.
a
g
in
with finitely many constraints),
h E K,
It
or
pieces and on each of
It turns out
epi-differentiable
and
that
in this case
a formula
for
the
448 epi-derivative class
was
obtained.
of functions
(see
This
[C2],
hold already for a function on quadratic functions.
result
[14]) R2
hidden
in
function of tiability
[K2]
extended
too more
to a more
general:
is the m a x i m u m
general
it may
of a linear
not
and a
A formula for the e p i - d e r i v a t i v e of the m a x - f u n c t i o n
in
t (and
of
not
which
f(x) =
is
can be
but
case
x
when
f(t,x)
max
f
has
a
polynomlal-type
is finite dimensional).
general
max-functions
(involving
behavior
a
A c r i t e r i o n for epi-differenmaximization
over
arbitrary
compact spaces and also defined on infinite dimensional Banach spaces) ven in [14]. We also mention the p a p e r b y Darkowskii and Levitin similar work was done in case when m a x i m i z a t i o n
as
is gi-
[DL] where a
is a p r o b l e m of concave prog-
ramming. In all
these
cases
epi-differentiable. involved
infinite
~s
does not
actually
This, unfortunately, number
possible
condition
sequences
not
taking
s
and
into
there
account
the
on
s
and
f(.)
is
is not a typical case when problems
of constraints
is considered,
semi-lnfinite p r o g r a m m i n g or optimal control. all
depend
is
such
as
problems
of
In general we have to scan over no
variety
hope of
that,
say,
second
a
order
necessary
directional
behaviors of a convex function can be s u f f i c i e n t l y powerful.
4. SENSITIVITY ANALYSIS
We now c o n s i d e r what happens Namely,
if we begin to slightly perturb
p(u) = inf g(F(x) x (For
simplicity
infimum above fixed small We
the map
F.
we consider the "value" function
shall
we
consider
is calculated
the
case
subject
of
a
+ u). "local"
to the condition
value
function
when
llx - xll < ~
for
the some
e > O. This does not cause any real loss of generality.) formulate
second order sufficient order directional
two
theorems
with
estimates
condition is satisfied,
derivatives of
p
at
and,
in
case
exact formulas for
when
the
the first
zero. (It should be remarked at this
449 point
that
order
derivatives
classical
we
typically of
problems
need
value
with
second
order
functions
equality
in
information
extremal
constraints
standard problems of mathematical
to
calculate
problems
and
-
see
first
lIT]
[ B ] , [ G ] , [GJ],
for
iS], for
programming.)
We set
p [ ( O , u ) = lim inf t-*(p(tu) - p ( O ) ) t ~ +0
and f o r
any
s
e S,
t~1(p(tn) -
p~+C0;u) : lim sup n9 THEOREM 4. For any
s ~ S
p(0))
~
we have
p~+(O;u) ~ inf ~s(h,w(h) + u) h If the sufficient condition of Theorem 3 is satisfied, p~(O;u) = Therefore,
if
Cs
differentiable
at
does 0
depend
along every
p'(u) = To formulate
Cs(h,w(h) + u)
min min s~Sh~K
not
on
u
~s(h ) = {ym ~ ~:
s,
then
p
is
directionally
and
r
min h~K
the second theorem,
then
+ u).
we set
for
h ~ K
- r
z 0}.
THEOREM 5. We always have
p~(O;u) ~
This
becomes
Moreover,
if
an
equality
Cs
.
min
min
max
s ~ S
h ~ K
y~ ~ ~ s ( h )
if
the
sufficient
does not depend on
condition
s, then
p(.)
of
Theorem
3 holds.
is directionally diffe-
rentiable and p'(O;u) =
The
last
theorem
analysis
developed
obtained
in [G],
is normal,
in
[GJ],
embraces [G],
III).
the
essence
[GJ],
[B].
[B] to infinite
reinterpretation
(using Principle
max . y~ ~ ~(h)
min h ~ K
of Theorem
Otherwise,
of It
the
approach
allows
dimensional 5 in terms
to
to
extend
sensitivity the
problems
like
of
is not
it is a more difficult
(P)
(P).
results If
(P)
difficult
work because
in this
450 case the relation between the value function of perturbation of constraints and
p(.)
(P)
connected with additive
is not so obvious.
As far as Theorem 4 is concerned,
it seems to have no predecessors.
We
observe finally that the fact that we consider only additive perturbations of F
is not a cause of any loss of generality.
form
F(x,u)
A general perturbation of the
can be eassily reformulated in terms of additive porturbation
of a certain other map.
REFERENCES
[B]
J.-F. Bonnans,
Directional
derivatives
of
optimal
solutions
in
smooth
non-finite
convex
non-linear programming, to appear. [BP]
J. Burke
and
R. Poliquin,
Optimality
conditions
composite functions, Math. Programming,
for
to appear.
[Cl] R.W. Chaney, Second order directional derivatives for nonsmooth functions, J. Math. Anal. Appl. [C2]
128 (1987), 495-511.
, Optimality conditions for piecewise ming, J. Optimization Theory Appl. 61 (1989),
C 2 nonlinear program-
179-203
[DL] B. Darkowskii and E. Levitin, Quadratic optimality conditions for problems of semi-infinite programming, Trans. Moscow Math. Society 48 (1985). [DZ] M.C. Delfour
and
J.P.Zolezzio,
Shape
sensitivity
analysis
via
min-max
differentiability, SIAM 3. Control Optimization 26 (1988), 834-862. [FS]
R. Fletcher
and
E. Sainz
de
Maza.
Nonlinear
programming
and
nonsmooth
optimization by successive linear programming, Math. Programming 43 (1989), 257-276. [G] B. Gollan, On the marginal function in nonlinear programming,
Math.
Operation Research 9 (1984), 208-221. [GJ] J.
Gauvin and R. Janin,
non-llnear
mathematical
Directional programming,
behaviour Math.
of optimal
Operation
solutions
Research
in
13
(1988), 629-649. ill] A.D. Ioffe,
Necessary and sufficient conditions for a local minimum 1-3,
SIAM J. Control Optimization 17 (1979), 245-288.
451 [12]
, Second order conditions semi-infinite programming,
in nonlinear nonsmooth problems of
in "Semi-lnfinite Programming and Applicati-
tions " Lecture Notes in Economics and Math~ Sci. 215, Springer 1983, pp. 262-280. [13]
, On some recent developments optimality conditions,
in the theory of second order
in "Optimization", S. Dolezki
(Ed.), Lecture Notes
in Math. 1405, Springer 1989, pp. 55-68. [14]
, Variational analysis of a composite function: the
[IT]
lower second order epi-derivative,
A.D. leffe
and
V.M.Tihomirov,
a formula for
to appear.
"Theory
of
Extremal
Problems",
North
Holland 1979. [KI] H. Kawasaki,
An envelop-like
second
necessary
order
effect
of
conditions
infinitely many for
constraints
minimization
on
problems,
Math. Programming 41 (1988), 73-96. [K2]
, The upper and lower second order directional derivatives for a sup-type function, Math. Programming 41 (1988), 327-339.
[P]
J.-P. Penot,
Metric regularity,
multifunctions,
Nonlinear
openness and Lipschitzian behaviour of
Analysis~
Theory,
Methods
&
Appl.
13
(1989),629-643. [RI]
R.T. Rockafellar,
First
and
second
order
epi-dlfferentiability
in
nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), 75-108. JR2]
,
Second
order
optimality
conditions
in
nonlinear
programming obtained by way of epi-derivatives, Math. Operation Research
iS] A. Shapiro, Sensitivity analysis of nonlinear program and differentiability properties of metric projections, SIAM J. Control Optimization 26 (1988), 628-645. [W]
S.Wright, On exact algorithm for composite non-differential optimization, Math. Programming 44 (1989), 221-234.
A N E W SUFFICIENT CONDITION FOR. T H E W E L L - P O S E D N E S S O F NON-LINEAR LEAST S Q U A R E P R O B L E M S ARISING IN IDENTIFICATION A N D C O N T R O L Guy Chavent C E R E M A D E , 75775 Paris C4dex 16 - INRIA, BP 105, 78153 Le Chesnay C4dex
Abstract We show how simple I-D geometrical calculations (but along all maximal segments o[ the parameter or control set!) can be used to establish the wMlposedness of a non-linear leastsquare (NLLS) problem and the absence of local minima in the corresponding error function. These sufficient conditions, which are shown to be sharp by elementary examples, are based on the use of the recently developped "size x curvature" conditions for proving that the output set is strictly quasiconvex. The use of this geometrical theory as a numerical or theoretical tool is discussed. Finally, application to regularized NLLS problem is shown to give new information on the choice of the regularizing parameter. Key words Non-linear least squares, parameter estimation, identification, inverse problems.
1
Introduction
Parameter estimation problems are often set as output-least square problems and solved numerically as such. But usually many theoretical and practical problems stay open, as existence and uniqueness of the global minimum, possible existence of local m i n i m a (which are highly undesirable fl'om a computational point of view), stability of the global minimum with respect to tile data ... The reason is that the parameter --* output mapping usually exhibits no interesting mathematical properties except regularity. Hence we have tried to develop, in a seriesof papers [1] [2] [3] [4] [5], some quantitative sufficient conditions for the well-posedness of the non linear least squares problem, following the intuition that this well-posedness should hold as soon as the size of the output set (on which one projects the data) is "not too large" with respect to its curvature. We illustrate in this paper how the most precise "size x curvature" conditions developped in [5] can be used to obtain insight into well-posedness, stability and local minima of non-linear least square problems. We refer to the introduction of [3] for a more detailed discussion of the motivations to this approach, and to reference [6] for an exaanple of application of these technique to a plane wave inversion problem.
2
T h e n o n - l i n e a r least s q u a r e p r o b l e m
We consider a "parameter space" E, a "set of admissible parameters" C, a " d a t a space" F and a "parameter ---* output mapping" r satisfying E C F c2
= = = :
Banach space, with norm ][[[E Closed convex set of E Hilbert space, with scalar product < , > F C ~ F a C2-mapping, with ~ , ~ ' and ~" bounded over C
(2.1)
453
The inversion of a given "data" z e F
(2.2)
find ~ e C such that J ( z ) = [[~(x) - zl[~ = min over C
(2.3)
in the least-squares sense reads then :
which is the non-linear least-square problem we want to investigate. Of course, well-posedncss of problem (2.3) does not hold under the sole hypothesis (2.1), and a classical way to enhance the behaviour of (2.3), in the absence of other information on ~, is to replace (2.3) by its Tychonov regulaxization
(2.3)~
find ~
E
CI7 $such that
4(=)
=
ll~(z)-zll~+PIl=-=oll~
=minover C1-1s
where Xo
C E E$ > 0
is an Hilbert subspace of E, with scalar product <, >~ is a given " a priori guess" for the unknown parameter is the regularization parameter
(2.4)
Usually, s is chosen to imbed compactly in E, which automatically ensures existence of a solution to (2.3)~, but does not give any hint on uniqueness, stability of ~ and local minima of J~. On the contrary, the Hilbert-type techniques used in this paper will allow to skip the compactness hypothesis (for example chosing s = E - provided E itself is an Hilbert) and yet prove the well-posedness of (2.3)~ for large enough ~. Of course, (2.3)~ can bc written in the same form as (2.3) by a proper choice of E, F, ~ and z, namely : replaced C replaced F replaced replaced Z replaced
byE,= s by C , = C Q s by F , = F x s ~ = (~,r by z , = ( z , ~ x o )
the scalar product < X , Y
>F+<
x , y >~
(2.5)
Hence we shall first state our results for problem (2.3), and then specialize them to the case (2.5) of the regularized problem.
3
P a t h e s and R e l a t e d g e o m e t r i c a l a t t r i b u t e s
The main step to obtain the well-posedness of (2.3) will be to prove that ~(C) is a strictly quasiconvex sct, as this will imply the existence of a neighborhood of ~(C) on which the "projection on ~(C)" is well behaved. Hence we need to equip ~(C) with a collection 7) of pathes, which we simply choose to be the image by ~ of all segments of C : 7) = { ~ ( [ x , y ] ) , x , y e C}
(3.1)
for which a subfamily of maximal pathes (cf [5]) is obviously given by
7)~ = {~([=,y]),=,y ~
ac]}
(3.2)
where c9C is the relative boundary of C. A path P = ~([x, y]) of 7) is naturally parametrized by
e [o, l]: P(t) = ~((1 - t)z + ty)
(3.3)
454
and we can define the velocity V(t) and acceleration A(t) associated to this parametrization by V(t)
~oi(z,)(y-x) ~o'(z,)(V - z, y - x)
=
A(t)
(3.4)
where ~(z,) [resp. ~"(x,)] is a linear [resp bilinear] mapping from E to F , and 9 , = (t - t)~ + tv.
(3.5)
The theory requires that pathes can be reparametrized as C2-function of the arc-length v, which satisfies : du = IIV(*)lldt (3.6) Hence at points t of a path where V(t) ~ O, the velocity v({) and acceleration a(t) with respect to are length v are given by
v(t)=
V(t)
a(*)=
Ilv(t)ll
,
A(Q
A(t)
ilv(t)ll-------7 - ,, < ,,, [tv(oll------7 >
(3.7)
and their norms by : lt("(t)ll = ~ ,
lb(O}l = IlA{t)l} ( 1 - < . , iiy(oil~.
A(t)
>=)~
(a.s)
At points to of a pa.th where V(to) --- 0, one may still define v(t) and a(t) by continuity provided that { T h e fight-hand sides in (3.7) are defined (3.9) V(to) = 0 =, and have a limit when t --+ to, t 7~ to Notice that (3.9) holds automatically as soon as ~o'(x) is injective for any z in C! Now that 90(C) is equipped with C2-pathes with respect to the arc-length, we define the r a d i u s o f c u r v a t u r e p(t) of a path P = ~o([z,y]) at parameter t by : p ( 0 = lla(t)lV ~ =
Itv(t)ll=cx- < v, A(t) ilA(t)ll" ~
>=)-~, > [IV(~)ll 2 -IIA(t)II -
(3.x0)
-
Following [51, we introduce also the g l o b a l r a d i u s o f c u r v a t u r e pa(t,t') of P at ~ seen from ~', which is given by the formula 0
pa(t,t') =
Sgn(f - t ) < P ' - P , v ' > ( 1 - < v,v' >=)89 Sgn(t'-Q
ifSgn(t'-t)
<0
ifSgn(t'-t) and < v,v' > 2 0
>0
ifSgn(t'-*)
>0
and <_O
(3.11)
where P, P', v, v'... stand for P(t), P(t'), v(t), v ( t ' ) . . , with the convention that Pc(t, ~') = + c a if Sgn(r < P ' - P,v > > 0 and < v,v' > = 1. Geometrically, pa(l, *') is the distance of P ( t ) to the intersection of the two haK spaces normal to each end of the subpath of P located between P(t) and P(t'). It is related to the usual radius of cm'vature p(t) by limp(t,t') = p(t) (3.12) t~t I
455
We denote also by 5(t,t') the arc l e n g t h between the two points P(t) and P(t') of the path P = ~([~, v]) :
a(t,t') = / " IlV(r)lldr
(3.13)
and by O(t, t') the d e f l e c t i o n between the path directions at the two saane points : @(t,t') = c o s - l ( < v(t),v(t') >)(in radian),
(3.14)
which statisfles (cf [5]) :
O(t,t')<_/ e dr(t)
=
p(t)
[,'
IlA(r)ll(l_<, .', IIV(r)ll"
' " IlA(")lld~" > 2 )rdr
A(r)
' ~
(3.15)
These geometrical quantities are used to associate to the non-linear least square problem itself, ie to C, ~ and 7~, the following numbers, which shall be relevant for the study of its wellposedness : (3.16)
R(C, bg, T')
=
(3.17)
Ra(C, % 7~) =
(3.18)
A(C, 90,'P)
=
inf
inf p(t)
(smallest radius of curvature)
mf "
mf "
sup
sup 5(t,t')
x,yEO t,t'E[O,1]p
c(t,t) ' "
(smallest global radius of curvature) "
(largest path length)
z,~EC t , t ' E [ O , l ]
(3.19)
|
=
sup
sup @(t,t')
(largest path-deflection)
x,yEC t,t'E[O,1]
Of course, from (3.12) we see that
~ ( c , ~, 7:,) < re(c, ~,'p).
(3.20)
In practice, our sufficient conditions for wellposedness will require only to know lower and upper bounds to the above quantities, which we shall denote by the same letter, but without arguments : _R _< R(C,~o,P), A > A(C,%P) (3.21)
< e,~(c,~,p), e >_ e(c,~,~)
In view of (3.20) it is reasonable to suppose that R a n d / ~ has been chosen such that as to satisfy
<_ R
(3.22)
For example,it is easy to check that the following choices for R, A m~d O satisfy (3.21) : R
=
,', = e
.
i~ i~ Ils~ x)H2 ,,,,~oo,~Io,,] II~"(=,)(Y - =, v - :':)11
sup /" II,e'(=,)(y - =)lldt
x,yEOC JO
. sup. [ '. I J. ' ( x ,. ) ( v - = , v - =)ll dt ~,~oc ~o II~'(~,)(v - =)ll
(3.23)
(3.24) (3.25)
where xt = (1 - t)x + ty. We recall also theorem 6.7 of [5] which allows to obtain a lower bound R.c to RG(C, c2, P ) in terms of R, A and 0 : T h e o r e m 3.1 Let 12, A, 6) satisfy (3.2i). Then :
R -%=
~i.r
(s.2i) ~,,,z (s.2#.
Rsine+(~-Re)cosO
if 0 < e < rr/2 ifrr/2<@<_~r
(3.26)
456
4
Wellposedness
of the non-linear
least square
problem
We consider now the non-linear least square problem (2.3), which throughout all this paragraph is supposed to satisfy hypothesis (2.1) and (3.9) at least. When it comes to actual numcrical resolution of (2.3) by an optimization algorithm, it is of utmost importance that the objective function J(x) has no local minima. Hence we shall incorporate this property into our definition of wellpose&~ess : D e f i n i t i o n 4.1 (Q-well posedness) The non-linear least square problem (2.3) is said to be Q. wellposed on ,ome open neighborhood 0 of qo(C) for some pseudo-disgance 6(x,y) on C if and only if :
(i) for any z 9 o,
there e=i~U a
.,~ique ,olution ~ of ( e . s )
(ii) the z-,-+ & mapping is Lipschitz continuous from (0, ItlIF) to (C,~(=,y)) (iii) for any z 9 O, d is ~trictly quasiconvez, (ie ha, no local minima distinct from &), and any minimizing sequence converges to & for 6(x, y). If we take 6(x,y) = [Ix - YIIE and drop condition iii), the above definition reduces to Output Least Square Identifiability (OLSI, cf [2]) introduced in the context of parameter estimation. In the gcncral case, Q-wcllposcdness is both weaker (because stability is required only for ~(x, y) instead of Ilx - Y[[E) and strongcr (because it requires strict quasiconvexity of J) than OLSI. The pseudo-distance ~(x, y) on C which appears quite natural for the NLLS problem (2.3) is the arc length, in data space F , of the path ~2([x, y]), i.e., with notations (3.5) : "1
5(~'Y) = J0 iiC(x,)(y - z)llrdt
(4.1)
Of course, if one makes the additional hypothesis that beside being bounded over C as stated in (2.1), ~p'(x) admits a pseudoinverse which is also bounded over C, or in other terms that { 3a'M > a,, > 0
Ot,nllY]]E <__]I~9'(X).yllr <__OtMllY[IE
such that VZ 9 C, Vy 9 Z
(4.2)
W,y 9 c
(4.3)
then 5(x, y) bccomcs equivalent to the usual norm in E :
~-,11=-ullz < 6(=,y) ___~MII~ - u t l E
It is now very easy to use the results of [5] to establish well-posedaaess results for (2.3). We begin with the most precise condition : T h e o r e m 4.1 (P~-su~cicnt condition) Zet hypothesis (2.1) and (3.9) hold, and R >_ I ~ be lower bounds to R ( C , ~ , P ) and Rv(C,w, 79) as defined in (3.15) (3.17) and (3.21).
Y: P~ > o
(4.4)
~(C)closed in F
(4.5)
Then : i) The NLLS problem (2.3) i~ Q-wellposed on 0 = {z 9 FId(z,~(C)) < / ~ }
(4.6)
for ~he p.~eudo-dis~ance 6(x,y) on C defined in (~.1). ii) More precisely, if zj 9 O,j = O, 1 satisfy
Iiz0 - z, lir + m ~ d(zj,~(C)) < d < ~ i=0,1
(4.7)
457
for some d, then the corresponding solution~ &j,j = O, 1 of (2.3) ~atisfy the s t a b i l i t y e s t i m a t e :
6(:~0,:~1) _< (1
d/R)-'ltzo - =,ll~
-
(4.8)
P r o o f : Hypothesis (2.1) (3.9) and (4.4) imply, using theorem 5.12 of [5], that r is strictly quasiconvex with a neighborhood v~ given by (4.6), which, together with the fact that ~(C) is closed, implies existence, uniqueness and Lipschitz stability of the projection on r all over ,9, and absence of local minima by theorems 3.6, 3.9 and 3.5 of [5]. [] Of course, if the strong property (4.2) on the derivative c2'(x)-the so-called "sensitivity Matrix" in the finite dimensional case-holds, then one obtains stability for the ]Ix - Y[IE distance on C, and c2(C ) is necessarily closed : C o r o l l a r y 4.1 Let hypothesis (2.1) and (.{.2) hold, and R >_ Ra be lower bounds ~o R(C,~,79) and RG(C,~,79) a~ defined in (3.16) (3.17) and (3.21).
If: RG > 0
(4.9)
Then T(C)
is closed in F
(4.10)
and conclusion~ i) and ii) of theorem (4.2) hold, with (4.8) replaced by o',,,ll:~o - :~,IIE
< ( I --
dlR)-'llzo
-
~,11,~
(<1.11)
The above theorem can be seen as a generalization of the local inversion theorem or implicit function theorem to the case where the range of ~'(x) can be strictly included in F, so that tlie equation ~(u) = z can be solved only in the least-square sense. We come now to a slightly less precise condition, based on the lower bound to Ra(C, 9%79) given in therorem 3.1 : T h e o r e m 4.2 (O-sufficient condition)
Let hypotheses (2.1) and (3.9) hold, and R , A , O be given satisfying (3.21) (for ezample u~ing formula (3.23 thru 25)), and suppose that ~o(C)
is closed.
Y: 0 <_ (9 <_ 7r/2
and R > O
(4.12) (4,13)
OT
7r/2<_ e < ~ r
andA/R
(4.14)
Ra>O
(4.15)
Then Ra defined by (3.26) satisfies and eonelusion~ i) and it) of theorem 4.1 hold for thi~ value of Ra. P r o o f : It results immediatly from theorems 3.1 and 4.1. [] Notice (cf the properties (3.15) of the deflection 0 ) that the conditions O _< 7r/2 or r, are actually integral size (du)x curvature (1/p(u)) conditions for the output set ~o(C), and that condition A / R < ( 9 - tan O is also a size ( A ) x curvature ( l / R ) condition, but which involves the largest size and the smallest curvature ! We now specialize theorem 4.2 to the case where ~o'(x) and its pseudoinversc are uniformly bounded ovcr C, i.e. when (4.2) holds. Using the least hypothesis of (2.1), we know that :
3# > 0, Ik,"(=)(y, y)ll~ < #llyll :v= e C, Vy e c Then we have the
(4.1~)
458 C o r o l l a r y 4.2 Let hypoihe~i~ (2.1) and (4.2) hold, and define : r: =
,~1~
,
e
=
(~l,~..)diam c
(4.17)
and : P~ =
R 0 < O <7r/2 R(sin | + (C~M/e,~ -- 1)(9 cos @) 7r/2 < {9 _< ~r
(4.1s)
If: 0 _< 19 < 7r/2
(4.10)
or
7r/2%@
andP~ >0
(4.20)
Then fl, e NLLS problem (2.3) is q-wellpo~ed on the cylindrical neighborhood (~. 6) of ~( C') for fl,.e Ilx ~ yllz distance on C, and the slabiIity estimate (4.11) holds a8 soon as fl~e data zo,zl satisfy
(4.7). Notice that, under the hypothesis (2.1) and (4.2) of the above corollary, the Q-wellposedness of the NLLS problem will be ensured as soon as the dianaeter of C is small enough. Notice also that the conditions (4.19) or (4.20) on the admissible size for C are sharp, as it can bc verified by considering the two following simple examples :
ezample 1 : given X > 0, choose, E = R , C = [0, X] and (cos x, sin ~)
(4.21)
~(x) =
{ ezampIie:givenY
choose, E = R , C = [ Y , X ] a n d
for Y < x < 0
(cosx, sin~) for
(4.22)
0 < 9 < X
(The r mapping of example 2 is not C2 at x = 0, but the whole theory still holds for path is which are only C' and picccwise C~). We conclude this paragraph with a (tentatively critical ...) discussion of the applicability of the above results to parameter estimation problems. The fundamental prerequisite for the use of this theory is that ~(C) is a "variety" which carries pathes 79 having radii of curvature larger than some R > 0. (This of course requires that the "variety" r itself has a "bounded curvature"). The condition Re: > 0 then can ahvays be satisfied by reducing the size of C, and is hence less critical. Let us first consider the case where E is finite dimensional, for example in parameter estimation in O.D.E.s, or in P.D.E.s after discretization has been performed. This case is very important practically, as it is the only one which one can actually attempt to solve on a computer ! One may then try to use the above geometrical theory to determine if the setting of the parameter estimation problem is satisfying, ie if the knowledge of C and ~ allows for a unique, stable determination of the parameter, and if so, which accuracy is required on the data. This requires the estimation of lower bounds R a n d / ~ to the smallest radius of curvature (3.16) and global radius of curvature (3.17), which for sure is not an easy task. When the dimension of E is not too large, one can try a numerical determination of R using (3.16) (3.10) and Ro using (3.17) (3.11), and use theorem 4.1 which given the most precise sufficient condition. This includes intensive computation (namely, along all segments [x, y] with extremities x and y located on the (relative) boundary OC of C[), which may quickly become unaffordable when the number of parameters is larger than a few units... But the reward for this computational effort is a treasurable ixfformation on the wellposcdncss of the NLLS problem and the absence of local minima in the objective function for any z in O, which is practically very useful both for the engineer who has set the
459
parameter estimation problem ("do I have enough information for recovering my parameter in a unique and stable way ?") and the numerical analyst in charge of the computations ("is my optimization routine going to be stuck in local m i n i m a 7"). W h e n this numerical approach is impossible, one may think of calculating analytically (i.e. with paper and pencil) a lower bound R using (3.16) (3.10), upper bounds @ using (3.25) and A using (3.24) (all these quantities are expressed by simple formula involving only r - x ) and ~"(z)(y - z , y - z), and then use theorem 4.2 to get information on the wellposedness of the NLLS problem. There are yet no example where this approach has been used, but the corresponding theory is just being released now in [5] and in this paper, so we hope that some application along this line will show up in the future. Let us turn now to the case where E is infinite dimensional, as for example in parameter estimation in P.D.E.s. We expect here that the generic (in an imprecise sense ...) situation is R(C,~o,'P) = 0, so that the above geometrical theory does not apply. As a support for this assertion, we refer to [2] where it was shown, for the model problem of the estimation of a diffusion coefficient function in a 1-D elliptic equation, that one can find, when the discretization is refined, a sequence of pathes on which the smallest radii of curvature tends to zero (these small radii of curvature are obtained for perturbations of the diffusion coefficient having smaller and smaller support containing one stationary point of the solution to the elliptic PDE). It would however be maybe possible to prove some wellposedness results in the somewhat academic case where the solution to the elliptic equation does not possess any stationary point. A bright singular point in this dark picture of the situation for the ilffinite dimensional case is given in [6], where tl~e estimation of the shape (i.e. a function) and the phase (a number) of a plane wane is discussed, and analysed using the above geometrical theory. To conclude on ilffinite dimensional E, let us mention that the geometric theory may reveal as a useful tool for analysing how the well-poscdness of the NLLS problem deteriorates when E is approximated by larger and larger finite dimensional spaces - multiscale aa~alysis of functions should play a crucial role here (see [2] and [71 for vet3 preliminmT results).
5
Well-posedness of the regularized N L L S problem
We investigate in this paragraph the wellposedness of the regularized problem (2.3) under the minimum set of hypotheses (2.1) (2.4). Of course, as we have not required any compact injection from s into E, this minimum set of hypothesis does not ensure in general even the existence of a solution .~, in opposition to the linear case (i.e. ~2 E s F)) where the same hypotheses ensure the existence of a unique f&. We shall be able in this paragraph to quantify the natural intuition that "a m i n i m m n amount" of regularization should be added in order to compensate for the non-linearity of ~o, mad restore a situation similar to that of the linear case. As mentioned at the end of paragraph 2, the study of the wellposedness of the regularized problem (2.3) can be made very simply by applying all results of paragraph 4 to the NLLS problem (2.3) with a proper choice for E, C, F, ~ m~d z as exptaned in (2.5). For sake of simplicity, we shall explicit this approach only for the case of corollary 4.2. Using hypothesis (2.1) and (2.4), we know that :
3&M > 0, II~,'(~).yllr -< aMII'JII~,W E C n C,V,j e C
(5.1)
3B > 0, IW'(z).(y, Y)IIF -< Bllyllg,w E c n E, vu E x
(5.2)
(notice that if E itself happens to be aa~ Hilbert space and the choice e = t3 is made, then &M coincides with aM defined in (4.2) and r coincidcs with/3 defined in (4.16)).
460 As in corollary 4.2, we shall need the size of C in the parameter space g :
m~(C,g)
= sup II= - Y l l ~
(5.3)
aa,yEC
but the position of the a-priori guess x0 with respect to C with also play a role through the "radius of C seen from x0 in g" : rad(C, xo, $) = sup
a:EC
11~ -
~01l~
(5.4)
In order to express the results in a simple form, we introduce the dimensionless quantities : (5.5) (5.6) (5.7) (5.8)
q g d(z,z') r
= = = =
rad(C, z0, s e/(/~diam(C, $))
S)
d(z,z')/(fldiam(C,s &M/(3diaan(C,$))
(position index of x0 w.r.t. C) (adimensional regularization pm-ameter) (adimensional distance in data space) (adimensionat upper bound to sensitivity)
Notice that 1/2_<,1_<1
as soon a s z 0 C C ,
(5.5)
which is the only (sensible) case which we shall consider, and that
(
71 q
close to 1/2 close to 1
r x0 is "close to center" of C 4:~ z0 is "far from center" of C
(5.G)
Notice also that the unit fl diam (C, s used in (5.7) to define the adimensional distance in data space is a measure of the maximum deviation across C of ~o from its linear approxinaation. We caaa now state the rcsult. Theorem
5.1 (Wellpo,,edneaa of regularized problem)
Let E,C,~o,e,xo s,ai4V (~.i) an,t (e.4) o,~Z~, ,1,r
be defined by (5.5) (.5.8) (a.a), a,,.,l
g, ui,,, a ..... by : 9
for 2/rr
< 't < 1 ("Xo
far from center of C"), and ~ >_0
:
~.~ = ~
(5.7)
am.~ = r{z ~ - ,#}'/=
(5.s)
9 [or 1/2 _< q <_ 2/~r ("zo close to center of C"), and r >_ 0 : "~mi,, =
71m~x =
largeat root, in [q,2/rr] of the g ~ r s i n 1/g + ((1 + r '/= - 1) cos 1/g - 7? /unction.
{
(5.9)
~{~ -,7~}'/~ for 2/~ <_ g{(gsin I / g + ( ( I + r
_
I) cos 1/Z') 2 -- r/2} 1/2
(5.10)
for'g~, < g < 2 / r
y: g > groin
(5.11)
Then the regularized problem (2.3), ia Q-wellposed on the cylindrical neighborhood of ~(C) of adimenaional size (defined in (5.7)) ~ for the ]Ix - vll~ di~*anee on c ,
461
P r o o f : Rewriting problem (2.3), in terms of E~, C,, F~, c2~ and z, as indicated in (2.5), one cheeks easily that : I .7: )Yllf. < ( d + a~)'/~llylle ~llyll~ --- II~,,(
w e c n ~',v~ ~ E
II~"(~)(y,y)ll~, - BII~II,~ w e c n E , vy e c
(5.12) (5.13)
Following now corollary 4.2, we define : @: = (/~/r /~,~ =
C = 1/g
(5.14)
{ [2/3(diamC)2 for 0 _< 1/g < 7r/2 g/~(diamC)2{gsin 1/g + ((1 + ~2/e2)1/2 - - 1) cos l/Y} for rr/2 < 1/g < rr
(5.15)
Corollaz'y 4.2 implies now that, if 2/~r_
or
(1/Tr < r < 2 / T r
and/~., >0)
(5.16)
then problem (2.3)~ is Q-wellposed on the neighborhood ~ of ~,(C~) in F~ of size Rc,~. But, for any z~ = (z, sx0) one has :
4 ( - ' - ~,dc,)) ~ = ~ J ~ { l l ~ ( = )
- ~11} + dll~ - ~ollg}
i.e.
d,(,-,, ~,o(C,)) ~ _< d(z,~,(C))~
+ d,'ad(C,~o,~)
(5.~)
Hence z~ will belong surely to O, independently of the choice made for x0 as soon as z belongs to the neighborhood 0 of ~(C) of size
d~,x
{
d,-ad(C,~o,~)~} 'n,
~,~
(~.lS)
provided of course tha~
~,~ > ~
(5.zo)
,-ad(C, zo,~).
Using the adimensional variables (5.5 thru 8), we rewrite (5.23) (which implies/Q;.~ > 0!) as : g > 77
rsin 1/z + ((1 + ~ / ~ ) ~ / :
- 1) cos 1/g > r~
if 2/rr _< g if 1/~r < g < 2 / ~
(5.20)
and (5.22) as : ~
= ~{(~sln 1/~ + ((1 + r
- 1) cos 1/~) ~ - ,7;}'/~
(~.21)
if l/;r < ~ < 2/7r Hence we see that if g satisfies (5.24), then the regularized problem (2.3)~ is Q-wellposed on the neighborhood O of c2(C ) of adimenslonal size ~n.x given by (5.25), which is the announced result. [] \u have illustrated on figure 1 the q -+ g.fin function for various values of (, which makes clearly visible that the choice > m ~ { 2 / , ~ , ~} (5.22) ensures the Q-wellposedness of the regularized problem independently of the sensitivity index (. Figures 2 and 3 illustrate how the size dm~ of the neighborhood of ~p(C) depends on ~ (ta.kcn larger than g.~. of course !), ~7 and (. These curves can be used for example to determine, given an estimation of an upper bound dm~ of the measurement and model error, the smallest mnount of rcgularization g to be used in order to restore wellposedness of the NLLS problem and suppress local minima on a neighborhood of ~(C) large enough to contain the expected data,.
462
.p._t,o,' _~, .
I
1.0 from top to bottom : zeta = 10 zeta = 5 zeta= 1
O.B
/ /
"
/
0.7 2/.rr.
0.5 '
'
+
'
0.5
I
'
'
'
0.7
'
I
'
'
O.B
'
'
1.0
Figure 1: T h e n f i n l m u m value ~ m i n of t h e r e g u l a r i z a t i o n p a r a m e t e r as f u n c t i o n of the positio1~ index q of t h e a-priori guess and t h e sensitivity i n d e x ~ of t h e ~ m a p p i n g
I 25.0
2.2
/
20. o
/ /
from left to fight : eta = 1/'2, zeta = 0 eta = 0.57, zeta = 0 eta = 2/pi ~ eta=0"76~an zeta eta=0.88/ Y eta=l.OI
.ts. 0
10.0
-
/ r
/ / / /
/
from ]eft to fight : eta = 1/2, zeta = 0 eta = 0.57, zeta = 0 eta = 2/pi / eta = 0.76~ an zeta eta=0.88/ Y = .
1. 5
r r
ops_bor
'
/
#/// //7/// ////// ////// //////
0. 7
5.0
0.0
''l'l''J'l''''l''''l~l''
0.0
1.0
2.0
0.0 3.0
4.0
5.0
0.0
0.5
1.0
1.
.p~_bor Figure 2: T h e d.,~x f u n c t i o n giving Lhe size of t h e c y l i n d r i c a l n e i g h b o r h o o d for t h e regularized p r o b l e m , as f u n c t i o n of 7, for five values of t h e p o s i t i o n i n d e x 77 of t h e initial guess z0, a n d a zero value of t h e s e n s i t i v i t y p a r a m e t e r s ~ for t h e two curves 77 = .5 a n d q = .57 c o r r e s p o n d i n g to ,7 < 2 / 7 r . Left : general
overview
; Right
: close
up on the
[0,1.5]
interval.
463
0.40
,-~~
I
0. 3 0 eta = 1/2
0. :30
0. 2 0
eta
from left to right :
0, 2 3
zeta = 0 z e t a = 1/2 zeta = 1 zeta = 5
zeta = 10
~
0. 15
0. 1 0
0. 0 0
/
0.57
from left to right : zeta = 0 z e t a = 1l'2 zeta = 1 zeta = 5 zeta= 10
0. 0 8
I I ~ 0. 5 0
0. 55
I I 0. 60
0. 0 0
1 I 1 I I I 0. G5
[
.p,_b~
0. 50
O. 7 0
0. 55
]
0. 60 I
0. G5 0. 70 ,P,-~o' ]
Figure 3: Influence of the sensitivity parameter ( on ~/-~x for values of the position index 71 of the in]tiM guess smaller than 2/~.
References [1] C. CHAVENT,"Local Stability of the Output Least-Square Parameter Estimation Technique'", Mat. Apllc. Comp., Vol 2 no1, pp 3-22, (1983) [2] G. CHAVENT, "New Trends in Identification of Distributed Parameter Systems", Proceedings of the IFAC World Congress, Mnchen, Pergamon Press, (1988) [3] G. CHAVENT, "On the Uniqueness of Local Minima for General Abstract Non-linear LeastSquare Problems", Inverse Problems 4, pp 417-433, (1988) [4] G. CHAVENT, "Quasiconvex Sets and Size • curvature condition ; Application to non linear inversion", INR.IA Report 1017, April 89 (Submitted to JAMO) [5] G. CHAVENT, "New size x curvature conditions for strict quasiconvexity of sets", INRIA Report to appear, 1989 [6] W. W. SYMES, "The Plane Wave Detection Problem", to appear [7] G. CHAVENT, J. LIU, "Multiscale parametrization for the estimation of a diffusion coefficient in elliptic and parabolic problems", Preprint IFAC Symposium on Control of Distributed Parameter Systems, Perpignan, France, June 26-29, 1989
AN IDENTIFICATION T E C H N I Q U E FOR A D A P T I V E S Y S T E M S IN THE CASE OF POOR EXCITATION
Hong Yao Xu and Co Robert Baird Department of Electrical Engineering Technical University of Nova Scotia Halifax, Nova Scotia, Canada B3J 2X4
Abstract. An Adaptive Identification Technique (AIT) is presented primarily for adaptive systems with insufficient persistent excitation. In this technique, the P(t-1) matrix is reset reasonably back to P*(t-1), and a new forgetting factor ~.(t) is designed to track the parameters of slowly timevarying processes. In addition, a weighted filtering algorithm is proposed such that a modified least squares algorithm is presented in terms of the procedures mentioned above. Theoretical analyses and simulation results have demonstrated that the proposed AIT methodology can overcome the bursting problems of adaptive systems in the case of poor excitation. Key words:
Identification; adaptive systems; least squares algorithm; bursting; excitation;
simulation.
1
Introduction
Many identification techniques have been proposed for adaptive systems in recent years. However, the estimated parameters may burst from instability in adaptive systems, if the preconditions of these identification techniques cannot be satisfied. It is well-known that when persistent excitation is gone in adaptive systems, most existing identification algorithms will not only fail to identify the parameters correctly, but also exhibit a bursting phenomenon [1], or so-called "estimator windup" [2]. A variable forgetting factor was presented [3] to solve this problem, which can be only used in deterministic systems, so the identification problem is still unsolvable. Isermann has proposed a simple remedy so-called "switch off" when identification conditions are violated [4]. However, the method [3] is not suitable for stochastic systems, and to "switch off" the identification is not effective [5]. If there is poor persistent excitation and when the adaptive control systems run for a certain time, the bursting phenomena may occur sometimes in practical applications. The proposed AIT has a strong
468 robustness to avoid this problem and has a good convergence property. Simulation examples are given to illustrate its results.
2
Bursting
Phenomena
In
Adaptive
Systems
Consider a linear system described as follows: A(q "1 )y(t)=B(q" 1)u[t-d(t)]+C(q "1 )0~(t)
(2.1)
where, A(q'l)= a o + alq "1 + . . .
+ anq-n
B(q'l)= b o + blq-t + . . . + bnq -n , b o ~ 0 C(q-1)= 1 + c l q -1 + . . .
+cnq -n
and y(t), u(t) and ~(t) represent the output, input and noise sequence, respectively, t is the sampling instant and d(t) denotes variable time delay of the system. The noise sequence {~(t)} is a Martingale difference sequence defined on a probability space (~, E, P) and adapted to (E t, t e h0, where Et is generated by the observations up to t and Eo includes initial condition information. Considering the system (2.1), we propose: Assumption set A : (i)
(ii) (iii)
E{('~
Et-t} = 0
a.s.
(2.2)
E{ o)(t) 2 [ F.t-1} =0"2< oo
a.s.
(2.3)
N lim 1 N._~=,sup~ ~ (0(02<00 t=1
a.s.
(2.4)
It is supposed that the coefficients in A(q'l), B(q "1) and C(q "1) are unknown, but input {u(t)} and output {y(t)} are available here. Assumption set B: (1) An upper bound for n in (2.1) is known; (2) B(q -1) and C(q -I) are stabie polynomials. Assumption set C: I y*(t)I < Ym <=~, g t > 0
(2.5)
Consider that the system model (2.1) is time-invariant and can be written in the standard form as y(t)= ~(t-d)T0(t)+~(t),
d = do(t)= d(t)
A basic property of the recursive least-squares (RLS) algorithm is denoted as follows:
(2.6)
469 V(t) A__-~(t)Tp(t.1).l-0(t)
(2.7)
~(t) A=~(t)-e
(2.8)
Also, we know that P(t-1)'I=p(-1)-I+
t-1 ~,~(i-d)~(i-d) T i=0
(2.9)
The fact that "0(t) will converge to zero provides that lim t.-) oo 1-)minP(t'l) "1=~176
(2.10)
lira t-1 t~ool.)min ,~, ~(i-d)(~(i-d)T=oo i=0
(2.11 )
and
where l.)mi n denotes the minimum eigenvalue. Equations (2.10) and (2.11) are the conditions for parameter convergence using the RLS algorithm. Here, (2.11) implies that if the det[~(i-d)~)(i-d) T] ^
= 0, 0(t) will not be able to approach the true value e. Theoretically, if the conditions (2.10) and (2.11) are violated, the RLS algorithm cannot converge and bursting phenomena will occur in an adaptive system. In practice, the bursting problems will likely occur: 1. When little dynamic information can be obtained from the measured input and output signals in an adaptive system. Especially when a fixed forgetting factor ~. is used in a time-variant system. 2. If there is no enough persistent exciting signal into the adaptive control system, the covariance matrix P(t) may grow very quickly. 3. If the process is not properly excited, some rows in the matrix H(t)=~[t-d(t)]~[t-d(t)] T will be linear dependent such that det[H(t)]=0.
3
Adaptive Identification Method
In order to solve bursting phenomena, based on the Recursive Least Squares (RLS) algorithm we add several modifications into it and present a Modified RLS (MRLS) algorithm, respectively. 1 ) to use the prediction time delay do(t ) to substitute for all known and constant delays d or d=l. do(t ) is estimated to equal the variable process time delay d(t) using a time-delay tracking technique[9] 2) to present a forgetting factor ~.(t) ~ { ~'min, ~'mid, ~.rnax }. 3)
to reset matrix P(t-1) as P*(t-1) on-line by a diagnosing method.
4 ) to employ some remembering units to supply important data for resetting and filtering. Hence, the Modified RLS algorithm is described as follows: 0(t) = O ( t - 1 ) + K ( t - l ) { y ( t ) - $ [ t - d o ( t ) ] T e ( t - 1 ) }
(3.1)
470 P(t-2)e[t-do (t)] K(t-1) = X ( t . t )+~,[t_do ( t ) ] T p ( t _ 2 ) ~ It_do (t) ] ~[t-do(t)]
= [y(t-do(t)) . . . . .
y(t-do(t)-n),
(3.2)
u(t-do(t)) . . . . .
A
u(t-do(t)-n) . . . . . A
A
- y ( t - d o (t)) . . . . .
- y ( t - d o (t)-n)] T
(3.3)
A
y(t) = ~ [ t - d o ( t ) ] T ~)(t); y(m) = 0 , ' ~ < 0 Pl(t'l)
I X(~'.l)
-
P(-1)=Ro[,
[
(3.4)
P(t-2)-K(t-1)~[t-do(t)]mP(t'2)
]
Ro > 0
(3.5)
where Equations (3.3)_(3.5) are described similarly to Sin and Goodwin [10] in which do(t)=1 and X(t)=l. r(t-1)=r(t-2)[;L(t-1)+~[t-d
o(t)] TP(t-2)~[t-d
o(t)],
=:(-1)>0
(3.5)
If [r(t-1)13minP1(t-1)<
K],
0 < K < co
(3.7)
then P(t-1)
= Pl(t-1)
(3.8)
else K
P(t-1) =
Pl(t-1)
(3.9)
"r(t-! )13 min P ! ( t - ! ) On the other hand, assume the P(t-1) matrix is reset at {ti} = { t 1, t 2, t3 . . . .
}, when t ~ {ti}, we
adopt the Equation (3.8). Otherwise, for t e {ti}, P(ti-1 ) is reset by using the diagnosing method expressed below: First check if II K(t-1) }]2 < p = 0
V t>0
(3.10)
then let P(t-1)=
P*(t-1) = diag [ P'11 ' P ' 2 2 . . . . .
(3.11)
P*nn ]
t
with
0 < Rmi n < p ii < Rmax < ~ '
for i=1,2 . . . . .
(3.12)
n
where p ii is stored in the remembering units. Otherwise, no reset is given to P(t-1). The forgetting factor ~.(t) is selected and determined by the following: rain ~-min > O, ;~(t)=
Xmax Sl, Xmid =
i
{ II
~(i)
- e*t(t-M)ll 2 } > 8max
max ^ i { {{ e(i) - e*t(t-M)ll 2 } < 8mi n ( ~-min" '~maxj'
otherwise
.....
t ]
(3.13)
V i e [t-M+1 . . . . .
t j
(3.14)
~'ie
[t-M+1
(3.15)
471 w i t h 0 < 6 m i n < 6max,6min and 6maxe ~[. where e*(t-M) is the estimated parameter at the sampling instant (t-M). ~.min=0.855 and ~Lmax = 0.995 as usual. Lemma 1. The above algorithms (3.1) to (3.15) are subject to the system (2.1), it results in that ^
^
(i) Ile(t)-ell < l l e ( t - l ) - e l l < Ile(o)-ell;
t > 1
(3.16)
N (ii)
lim
_
~
N~
Hmin~ t=l
e(t) 2 ~.(t)+Rmax#[t.do(t)]T~[t.do(t) ]
(3.17)
where (3.18)
e(t) = y(t) - +[t - do(t)] T ~)(t-1) This implies lim (a) t ~
e(t) {~.(t)+Rmax~[t.do(l)]T~[t.do(t)]}l/2
lira ^ ^ (b) t_., ~, lie(t)-e(t-k)ll = 0
= 0
(3.19)
(3.20)
for any finite k
Proof. A straightforward extension of Lemma 3.3.8 of Goodwin and Sin (1984) is made in terms of do(t), X(t), P(t-l) and the filtered parameters.
Through both experimental and theoretical analyses, we know that the bursting phenomena are unavoidable in some adaptive systems. In order to establish the AIT technique for solving this problem, we propose the new procedures summarized as follows: Step 1. do(t } is adjusted on-line to keep do(t)=d(t), see [9]. Step 2. We define the function ~.(t)e {~'min , ;Lmid , ~'rnax} as above (3.13)~(3.15). The point that ;L(t) is designed here is to revitaiize the algorithms (3.1)~(3.15) according to the different working conditions. Step 3. In order to avoid the growth of the P(t-1) matrix, P(t-1) is reset as P*(t-1) reasonably instead of P(-1) periodically. Step 4. Remembering units periodically store the updated parameters e*t(t ) and the matrix P*(t1). We believe that historical records are of useful and valuable references in a period of time. Use e*t(t ) and P*(t-1) to substitute respectively ~)(t) and P(t-1) which are going to deteriorate currently. ^
Compare e(t-M+l) . . . . .
A
9(t-1),
A
e(t), then
(i) If there exists max { i then let
II ~(i)-e*t(t-M)ll2
} < P.
Vie
[ t-M+1 . . . .
t]
(3.21)
472 ^
e*t(t)= e(t)
(3.22)
and P*(t-1)= diag { P'11' P'22 . . . . . (ii)
P*nn }
(3.23)
Otherwise, if there is min. ^ i { II e(i) -e*t(t-M ) II2 } > p., V i e [ t-M+1 . . . . .
t ]
(3.24)
check the diagnosing condition (3.10) and the filtering condition (3.25) so as to decide if some proper actions need to be taken correspondingly, and keep e*t(t ) and P*(t-1) constant until (3.21) is satisfied. Step 5. The weighted filtering procedures of e(t), y(t) and u(t) are implemented through a detecting criterion such that the system can be protected to avoid such a bursting phenomenon.
(1) If the condition A
Q(t) = inf II e(k)-e*t(t-M) {I2 > E max > 0
for E: max chosen
(3.25)
ke [t-M,t] is fulfilled , we obtain the filtered parameters: k e ( t ) = ~ ) f ( k ) = e * t ( k - M ) + 1--~ ,~, {Ae(i)-e*t(t-M)} M. i=k-M+l
y(t)=yf(k)={
k ~,e'~ i=k-M+l
u(t)=uf(k)={
k ~L, e "~i u(i)}/{ i=k-M+l
y(i)}/{
k ~/_, e'F,i}, i=k-M+l
V k ~ [ t-M+1 . . . . .
> 0
k ~/_~e-~i}, i=k-M+l
> 0
ibid
ibid
t ]
(3.26)
(3.27)
(3.28)
(2) If we have A
R(t) = sup II e(k) - e*t(t-M){{ 2 < s rain
forE; min chosen
(3.29)
k e It-M+1, t ] where 0 < Emi n < E.max, k = t-M, t-M+1 . . . . .
t. Hence, no filtering procedure is needed.
A
(3) Otherwise, we only require e(t) to be filtered using Equation (3.26). Equations (3.28) and (3.29) show the weighted filtering, where e-~ i represents a weighting factor, its decreasing rate is depended on ~. M denotes the length of measurements, M=10~20, ~=0.01~0.1 are recommended here. In (3.27) and (3.28), we use a weighting factor e-~ i to emphasize old data which indicate that the system operated better in the past. Simulation results have shown that the variations of e(t), y(t) and u(t) are reduced obviously by (3.26)~(3.28).
473 It must be pointed out that in Equations (3.13), (3.14), (3.21), (3.24), (3.25) and (3.29), ^
since the same term {{0(i)-e*t(t-M){{ 2 {s calculated only once every M sampling intervals, a number of computational efforts can be saved. The MRLS algorithm has good convergence properties. Its theoretical analyses will be derived from the following lemma 2 and lemma 3. Lemma 2. For the system (2.1), the MRLS algorithm (3.1)_(3.15) is used in order to achieve:
II~(t)-ell 2
(1)
_< K l l l ~ ( 0 ) - e l l
2,
t > 1
(3.30)
where K 1 = condition number of [P(-1) " l ] - - A l . ) m a x P ( - 1 ) ' l / ' O m i n P ( - 1 ) ' l = ' o m a x and.Orni n denote the maximum and minimum eigenvalue, respectively.
lira " ~ (2) N--,o~ t=l
e(t) 2 ;L (t) +,~ [ t . d o (t)]T p (t. 2),~ [t_do ( t ) ]
(3,31)
and implies that lim (a) t-~oo e(t){ ; L ( t ) + K 2 ~ [ t - d o ( t ) ] T ~ [ t - d o ( t ) ]
}-1/2 = 0
(3.32)
K 2 = l.)ma x P(-1) N lim ~ ^ ^ II e ( t ) - e ( t - 1 ) t l 2 < oo 2.., (b) N ~ =, t=l N lira ~ l l ~ ( t ) . ~ ( t . k ) l l 2 < N ..~ oo t=k
(c)
lira ^ ^ (d) t - - , ~ I l e ( t ) ' e ( t k ) l l = 0
oo
(3.33)
for any finite k
v ke ~
(3.34)
(3.35)
where e represents a true parameter vector, ~(t) indicates a vector of measurement {y(t-1) . . . . . y(t-r), u(t-1) . . . . . Proof.
u(t-s)}, and e(t) the prediction error shown in (3.18).
It is similar to [6] in which do(t)=1 and ~.(t)=l were given, but here parameters do(t), X(t),
P(t-1) and the filtered parameters are different. Lemma 3. If assumption set A and set B are satisfied and the system has the transfer function 1 1 C'i'q) " 2"
(3.36)
which is both input and output strictly passive. Then the algorithms (3.1)_(3.15) ensure that with probability one
474 (i)
(a~
lira
V(t)
t-~
sup r ( t - 1
< oo
(3.37)
where V(t) is given in (2.10) and r(t-1)= r(t-2) + e [ t - d o ( t ) ] T ~ [ t - d o ( t ) ] , N lim (b) N-->,*, Z c ~ ( t ' l ) ~ [ t ' d o t=l (ii)
I
-
r(-1) - zr(-1)
(t)]TP(t'2)~[t'do(t)]'
(3.38)
v(t)
< oo
r(t-1)
(3.39)
same as (3.34) N
(iii)
lira ~ N~,~
[e(t~-0)(t)] 2 r ( t- 1 )
< oc
(3.40)
t=l Proof. This is a similar lemma to lemma 3.3 in Sin and Goodwin [10], but do(t), ;L(t), P(t-1) and the filtered parameters are different.
4
Simulation
Results
As we know, in many implicit and explicit adaptive techniques, for instance Self-Tuning Regulators (STR), only if there is enough persistent excitation in the RLS algorithm, can it converge [1-2] in a closed-loop control system. Otherwise, the RLS algorithm cannot converge to the true value. In Example !, for instance, if the persistent excitation is poor, RLS algorithm cannot converge well. However, the proposed AIT using MRLS algorithm can guarantee a good property of parameter convergence, even if the persistent excitation is poor. Examples 2 will illustrate this argument. Simulation experiments have been done by VAX-11/780. Example 1. Assume a system described by y(t) - 0.6y(t-1) + 0.7y(t-2) = u(t-2) + 0.5u(t-3) + ~(t)
(4.1)
When 0 < t < 40, the while noise re(t) with 02=0.4, At t = 40, the RLS algorithm basically converged to the true value 6=[ -0.43, -0.34, 1.10, 0.30 ]. But when t > 40, ~(t) was very largely reduced to 0"2=0.005 which implied that the persistent excitation became poor. As a result, all the estimated A
parameters e(t) shifted to the wrong area. ~2, one of the estimated parameters, is shown in Figure 1. ^
During 500 < t < 1000, several bursting phenomena occurred and then e(t) all became false value: e(2500) = [ 0.41, -0.302, 0.005, -1.01 ]. However, in the same case, the proposed AIT can demonstrate its strong robustness, one of its results is shown in Example 2. Example 2. Consider that the system and simulation conditions are the same as those in Example I. Assumed that at t > 40 the persistent excitation o)(t) was numerously decreased to be insufficient. In this case, we selected M=15, s s 6ma• 6min =0.015 and ,u.=0.0015. The A
estimated parameters 0(2000)=[-0.428, -0.344, 1.096, 0.303] almost exactly converged to the true values 0=[-0.43, -0.34, 1.10, 0.30] for the minimum variance control of STR. This indicates
475 that the AIT can still identify the real parameters even if the input exciting suddenty becomes insufficient. One of the estimated parameters, 92, is shown in Figure 2.
5
Conclusion
Adaptive identification becomes a very difficult task of an adaptive system in the case of poor excitation. The proposed AIT can fulfill it and overcome the bursting problem. Theoretical analyses and simulation studies have demonstrated the AIT methodology that is basically established by four proposals: 1 ) The P(I-1) matrix is reset as P*(t-1) based on Equations (3.5)~(3..12); 2) Weighted filtering is given to 8(t), y(t) and u(t) by (3.25)~(3.29); 3) Remembering units are employed to store some historical data through (3.21)~(3.24) for filtering and resetting; 4 ) A forgetting factor function ~.(t) is designed according to (3.13) ~ (3.15). However, it is recognized that if system excitation is too insufficient, the AIM may not work well. Therefore, further research work needs to be done to find out the minimum persistent excitation that the AIM can tolerate.
6
References
I . Anderson, B. D. O. (1985). "Adaptive Systems, Lack of Persistency of Excitation and Bursting Phenomena". Automatica, Vol. 21, No. 3, pp.247-258. 2 . Astrom, K. J. (1983). "Theory and Applications of Adaptive Control--A Survey". Automatica, Vol.19, No.5, pp.471-486. 3. Fortescue,T. R., L. S. Kershenbaum and B. E. Ydstie (1981)."Implementation of Selfquning Regulators with Variable Forgetting Factors". Automatica, Vol. 17, No. 6, pp. 831- 835. 4 . lsermann, R. and K.H. Lachmann (1985). "Parameter-adaptive Control with Configuration Aids an~ Supervision Functions". Automatica, Vol. 21, No. 6, pp.625-638. 5. Foley, M., K. Walgama, Y. H. Jin and G. Fisher (1988). "Adaptive Control Based on a Modified Kalman Filter Predictor". Preprints of the 8th IFAC Identification and System Parameter Estimation. Beijing, PRC. Vol.1, pp.342-347. 6. Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering, Prediction and Control Prentice Hall, New Jersey. 7. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1980). "Discrete-time Multivariable Adaptive Control". IEEE Trans. Automatica Control, AC-25, pp.449-456. 8. Anderson, B. D. O. and C. R. Johnson, Jr. (1982). Exponential Convergence of Adaptive Identification and Control Algorithms ". Automatica, Vo~.t8, pp. I.
476 9. Xu, H. Y. (1988). "Robust Protection-Time Delay Tracing Adaptive Control and Its Application". Preprints of the 8th IFAC Identification and System Parameter Estimation9 Beijing, PRC. Vol. 2. pp.1064-1069. 10. Sin, K. S. and G. C. Goodwin (1982). "Stochastic Adaptive Control Using a Modified Least Squares Algorithm". A u t o m a t i c a , Vo1.18, pp. 315-321.
1
t32
9 .
1.00,50.0-0.5 -1.0 -1.5
'
I
100
'
I
'
200
I
'
300
I
'
400
I
9
500
I
'
600
I
9
700
|
'
800
I
"
/
K
900 1000
Fig. 1. The bursting parameter [32 of the RLS in Example 1
0.6
132
0.4 0.20.0-0.2 -0.4
9
0
I
|
50
100
9
I
150
'
I
200
9
I
250
9
I
300
9
I
350
9
I
400
9
I
450
9
I
500
Fig. 2. The convergent parameter ~2 of the AIT in Example 2
Asymptotic
Properties
in Rational
/2-approximation
L. Baratchart, l~f. Olivi, F. |Vielonsky Institut National de Recherche en Informatique et Automatique Avenue E. Hughes, Sophia-Antipolis 06560 Valbonne (France)
Abstract: This paper is concerned with the problem of best rational approximation of given order n in the Hardy space/:/2. We show that, generically, all critical points converge to the fnnction in tI~ as n increases to infinity. This property shows in turn that local maxima can appear only for a finite range of orders. This has consequences on an algorithm to find local minima previously described by some of the authors [3].
I
Introduction
Let us recall briefly the/2-approximation problem as described, for instance, in [2]. Consider the real Hardy space H~,I~ of functions f analytic in the complement of the closed unit disk 0 , vanishing at infinity, that can be written f ( z ) = F_.k~__l.f~z -k with .fk E R. and such that the norm Ilfll 2 = g:k .t2 is finite. The assertions that the coefficients fk are real and square integrable are respectively equivalent to the facts that the function f maps the real line into itself and satisfies a growth condition at the frontier T of U: sup- 1 r>l
27~
fo ~,"]f(rei~
< c~.
As in [2], let P~ be the set of real polynomials of degree at most n, and T'~ the subset of monic polvnomials, of degree n whose roots are in U. Moreover, let ",x'l C H~,I~ consist of all rational fractions h = p/q, where p E P,,-1 and q E T'~. The problem is to minimize, for arbitrary n, the criterion F)(h) = I[/- hll ~ where h is in x-1 The relevance to system theory of this question arises from the need of describing the input-output behaviour of a given system (i.e. its transfer function) by a finite dimensional model, whose transfer function is therefore rational. For more details, and a stochastic interpretation of the 12 norm, we refer the reader to [1]. Of course, in this context, our problem can only appear as the prototype of an integral criterion, which can admit many variations and generalizations, putting for instance weight functions, aclditional constraints, a.nd going over to the multi-input multi-output case. In the sequel, however, we shall restrict ourselves to the simple formulation given above. Apart from a classical formulation using the so called normal equations [4], this problem has been mostly tackled using differentiation, namely a gradient algorithm [6]. In [:3], an algorithm is described, which converges to a local minima by integrating numerically a differential equation. In all three cases, critical points (i.e. points where the derivative is zero) are on the saddle, because they are the ones that can be computed, and the trouble comes of course from the possibility of having several local minima. 9a . ~ r L
9
478
Let us take a closet" look at ti~e criterion F}(p/q) that we want to minimize. As q is settled, the polynomial p is uniquely determined as the orthogonal projection of f onto the n-dimensionM linear subspace 1.~ of H~,it defined by lq = P~-l/q. We shall denote by L'~(q) this projection and replace tile former criterion by the following one:
~L,~(q) = f
L}(q) ~" q
At each order, we get a set of critical points for this criterion and the goal of this paper is to describe their asymptotic behavior. These critical points may be of two different kinds: either L~(q) and q are coprime, in which case the point q is sa.id to be irreducible, either they share a common factor and we get a reducible point. When q is reducible, the variational argument (due to Ruckebush) used in [1] to show that it cannot be a minimum applies as wel! to show it cannot be a maximum either unless .f is already rational. In the sequel, we shall suppose that the element f of H~.it to be approximated is not already a rational function a.nd we shMl make the mild assumption that it is analytic on a domain strictly containing the complement of the open disk U. Then we show that, generically, critical points converge in /2-norn- to the function f as the order (i.e. tile degree of the polynomials) growths to infinity. Using this fact, we deduce that criticM points which are local maxima can only appear for a finite range of orders.
2
P r o p e r t i e s o f t h e c r i t e r i o n d,i a n d its c r i t i c a l p o i n t s 9
,;'7
As in [2], we introduce a.n other Ilardy space H+lZ, which contains analytic functions g in the unit disk U, naapping tile real axis into itself and such that
sup
Ig(rJe)!=g0 < ~ .
r
The operator f ---+ f~ given by F ( - ) = if(z) where f(z) = f ( ~ ) , deanes an application from t i l t ~ onto H+rL and conversely. Moreover the orthogonal sum s = If/.a 9 is a Hilbert space with the following scalar product i
r
dz
~Jrf(z)g(z)T.
< f,g > =
As 5 = z -~ on the unit circle T and the coefficients in the power series expansion of 5 are real, we also have I ,;" ; , I dz < f ' g > = 2 i T r J ~ .f~-:JJ(-)--. z z This scalar product verifies the two following obvious properties which will be used in the sequel: 1) for all/," E Z, the multiplication by z k is an isometry of L~.~(T), i.e. for IT' ), all .f, 5' E L2.r~t< zkf, z~g > = < f, 9 >; (1) -
-
2) for al! f , g, h in L2,rc(T) such that f9 and f h are in L=,Iz(T).
< fg, h > = < s, .fh > .
(:2)
479
Let f be the function of H~,rt to be approximated and let g = f~, its image in the space FI2,1L" + W i t h the assumption made on f, there exists a real ,\ > 1 such that g is analytic in the open disk Ua centered at 0 of radius ,\. Let q in 79~ and (7 defined as in [2] in the following way: ]
~(:) = ~-q(:). Z
T h e Weierstrass division theorem (cf [5]) or more precisely its one dimensional version (also known as the Hadamard representation, cf [7]), applied to the function cq~ of/:f~rt shows ~hat there exists a unique function v(g , q) analytic in Ua and a unique polynomial w(g, q) of degree n - t such that: 9~ = o(g, q)q + w(9, q). It follows from [2] that if you seek in the space of rational fractions Vq for the minimum Lj(q)/q of ~by, you get Lf(q) = w(g, q). The quotient v(g,q) of the former division which we shall simply denote by v when no confusion can arise, possesses the helpful property to give the value of the criterion 02 at ;he corresponding point q: p r o p o s i t i o n 1 Let q be eLpoint of 79~ and v the correspondin 9 quotient, II~ll ~ = eTfq).
Proof: Using (1) and (2), the value of the criterion at q is:
Z(q) ' ' _ , g - L(q) _ >.
L ( q ) , f _ L(q) > - - < g -
62(q)=
q
q
q
q
Then
the last equality coming out directly from the definition of the scalar product in L2.Fc(T). [] \\% come now to a divisibility property at critical poilits wkictl ensures the existence of many zeros in the unit disk for the associated quotients v when the points are irreducible: p r o p o s i t i o n 2 Let q, a point of'P], the lwo following assertions are equivalent:
(i) q is a critical point of 6~. (ii) q dicides vL(q). Then, if q is an irreducible critical point, q divides v. Proof: Let q = z" + %_lz ~-I + ... + qo E 72~, q is a critical point iff: or
,
V I E { 0 .... , n - l } , - ~ q i ( q ) = 0 . These partial derivatives are equal to Od~ n
~(q)
= -2< f
L(q), O ( L ( q ) ) > q Oqi q
=
- qL(q)
_ . _? < f
,
- -q ~({)}~,,,,.Lq.
L(q) ziL(q) > + 2 < f
- q- , - -q2
>.
480
As L(q)/q is the orthogonal projection of f on the space Vq, the first term of the right-hand side is zero, and
Og,} 0~1~(q) = 2 < f
L(q) z'L(q) q , q------y-> .
This set of derivatives vanishes iff for all polynomials P in C ~-1 [z]
< f -L(q) q
'
pL(q) >=0. q2
From the definition of the scalar product, we get
or
q
q
If a is a root of order m of the polynomial q, this implies f ,_, L ( q ) , , , dz Vl ~ {L...,m},jT v t , ) - - ~ - - t ~ j i : _ a)~ - 0. Then, hy the residue formula, the following derivatives should vanish:
W 6 {O,...,n~- i}, rvZ(~)] ~''
L--~J
B y induction, o' is a zero of order m
of
vL(q).
(~) = 0.
This proves the equivalence of the two
assertions and proposition 2.
3
A s y m p t o t i c behaviour of critical points
We denote by Cn the subset of T'~ containing the critical points at order n, and we put C = U,C,. By choosing a point in C, for each n, we construct a sequence of quotients (v,~). In order to prove that the family of functions (v,,) is normal, we use the integral representation:
1 f~ g(r162
v ( z ) = 2i---~..,
q(()
de
(-
z
where it is any real such that 1 < ;L < .\. Using this expression we prove the l e m m a 1 Let /
a real number such that 1 < #' < ~, there exists on the open set Uu, an uniform bound for the set of functions (v~) which depends only on the function g.
Proof: On the unit circle T, the quotient i[/q is of modulus 1. Then by using the maximum principle over the complement of the unit disk U, we get
v~ ~ c-u, lq-~I ~(() _
481
This inequality is true on the circle T~ so that:
w ~ u.,, I,v(~)l _< ~ ,,J~L=.
~l
But for z E U,,, 1~ - z[ is greater than # - ~t' and we get the bound ill question. [] We shall first consider the case when our sequence consists of irreducible points only. Note that this is the case when it is obtained by repeatedly using a gradient algorithm for increasing values of n as described in [3]. From such a sequence, take a subsequence (wp) which converges to a limit Wlim uniformly over all compact subsets of Uu. Let td such that 1 < #' < # and suppose that the analytic function wti,~ has no zeros on the circle Tu,. Then
3N, V,~ > N , W e :r.,, Iw.(:) - w.~(~)l < Iw.m(z)l. By Rouch~'s theorem, w, and wum will have the same number of zeros in the open set U,,, but using proposition 2, a quotient corresponding to an irreducible critical point of order n has at least n zeros in U. As the order of points in the subsequence (wp) tends to infinity, wti,,, must be equal to zero. This is a contradiction with the assumption made on the circle T~,,. By letting IF vary continuously, we get a compact circular annulus containing infinitely many zeros for wti~ and thns this limit n-lust vanish on the open disk Uu. We just showed that every convergent snbsequence of (v,a) converges to zero uniformly on every compact set of U,,. Since it is normal, this is true for the sequence (t,,) itself. By using tim proposition 1, we get the F-convergence of any sequence of irreducible critical points to tile function to be approximated as the order of points tends to infinity. In order to generalize this fact to sequences containing also reducible points, we shall have to restrict ourselves to functions with tile property that Cn is finite for each n. This can be proved to be generic in various contexts. For instance, it is shown in [1] that such functions form a set of first category in the disc algebra of U u where # > 1. We first prove the p r o p o s i t i o n 3 Let p E 79~, a critical point such that the fraction Ly(p)/p is irreducible. Let r ET)~ a n d q = p r . Then (i) LI(cl) = rLl(p) ~ff r divides v(g,p). (ii) if (i) is verified, we have the following equivalence: q is a critical point iff p is a critical point and r divides the quotient v(g,p)/p.
Proof:
Apply the above mentioned division theorem to gc7 and gl'5:
J~7 = v(j, q)q + rz(q ) and ~ = ~,(g, p)p + LAp). Multiply the second equation by F:
g~7 = v(g,p)~p +/:LAp). Let us denote v(g,p) by vp and divide vpF by r:
vpF = v(vp, r)r + L~g(r). Plugging this expression in (3), we get
s~ = v(,,,, r)~ + (Lo#(r)p + ~L:(?))
(3)
482
and the second term on d~e right-hand side is of degree strictly lower than that of q. Thus we have t,~F = v(9 , q)r + Lff.(r)
(4) In order to prove (i), suppose first that L/(q) = rL/(p) holds. Applying to the second equation of (4) the assumption that p divides Lj(q), and hence that + divides Lf(q) we get (nsing the nsual notation for division) dpL~(~). As roots of/5 lie in the complement of the unit disk, r divides L~g(r). But L~-(r) is the remainder of a division by r and then it must be zero. The previous pair of equations beconles
I zj(q) = ,'ZAp),
(5) t,pr
= v(g,q)r.
The second equation of (5) shows that r divides vp. Conversely, suppose that @% The first equation of (4) implies that rIL~7,(r ). As the degree of L~7,(r) is strictly lower than that of r, it must be zero and (4) reduces as in the former case to (5). Suppose now that the assertions of (i) are verified and let prove (ii). By proposition 2, the fact that q is a critical point means that qlu(9, q)Ll(q) i.e.
p,-1~,(7)~G(p) or
pq~,~&(,,)
which yiekls that p]vpL1(p) i.e. p is critical. This yields also that r ] ( ~ ) L f ( p ) . Moreover as (1) is verified, r ] ( ~ ) p . As LZ(P) and p are relatively prime, we deduce that r]-~. Conversely, this last relation implies that q[vp. By the second equation of (5), we get q]u(g, q)r~. As roots of ? lie in the complement of the unit disk, q divides v(9, q)r and in particular v(9, q)rLl(p). Then using the first equation of (.5), q divides v(fl, q)Lf(q) and q is critical. O If critical points are irreducible, there exists an order over which the corresponding quotients v have more than any preassigned number of zeros. To get our generalization, we prove that such an order exists even in the case of reducible points. Following proposition 3, these points are generated by adjoining to irreducible critical points q of lower order, zeros from v/q. \u show that for a fixed order, the number of such zeros is bounded from above. Let I,~ be the subset of C, containing irreducible critical points of order n and let q in I , . We denote by Z(v/q), the number or zeros of the quotient u/q in the disk U. Then Z(v/q) is finite. Indeed, with the assumption made on f , the quotient v is a.nalytic on the open set U.\ which conta.ins the compact disk C. If Z(v/q) is not finite, v vanishes in U which means that the function f to a.pproximate is already a rational fraction, but we discarded this case ill the introduction. Let us set one more notation:
R. = max{Z(,,/q),q ~ z.}, then R , is finite. It is obvious when I , itself is finite. Otherwise, let suppose that R , is not finite, then we can select a sequence of critical points (qt) in In whose corresponding
483 quotients (vl) have a n u m b e r of zeros growing to infinity. From this sequence, we can extract as before a subsequence which tends to zero. But this means that there is a. sequence of critical points of order n which converges to the function f and then f is again rational. Indeed, we have
gqt = vzql + Lf(ql).
(6)
T h e functions vt converge uniformly to zero on ~" and the polynomial.~s ql and (7~ are also b o u n d e d on/_7 as their degree and their coefficients are. T h e n by (6), Lf(q~) is bounded. Vve can successively extract two subsequences such that Li(qt ) and ql will converge respectively to some polynomials p and q, uniformly on U. By taking the limit, the equation (6) becomes gq = p on U and thus f is equal to/3/(~. As a conclusion, at order n + R~, quotients v corresponding to irreducible critical points as well as reducible ones which come fl'om irreducible points of order n have all a.t least n zeros. At order maxp<,{p + Rp} + 1, no critical point comes from an irreducible one of order less t h a n or equal to n. Thus all the corresponding quotients have more t h a n n zeros. This is the result we needed and finally, we proved the 1 Let f , be a function of the Hardy space H~.R, distinct from a rational fraction, analytic on a open domain containin.g the complement of the unit disk U. Let (u~) be a scquence of quotients corresponding to irreducible critical points q. whose orders tel~d to infinity. Then the sequence (vn) converges u,tiformly to zero on every compact subset of an open set containing the closed unit disk (7.Consequently, the sequence of critical points (Li(q~)/%) tends to the function f , accordingly to the l'2-norm. Generically, the assumption7 that q, is il-redtzcible can be dropped. theorem
4
Finiteness
of the
number
of orders
where
local
maxima
appear
Let us restate theorem 1 in the following manner: if f is non rational, then for any e > O, there exists no such that any irreducible critical point q of order n > no satisfies t9~(%) < e. Now, let q be a critical point of order n. We shall show that it cannot be a local m a x i n m l n if n is large enough. First., we can assume (1 is irreduciMe. The partial derivatives of the criterion < f - ~ q ~ .~" > at q vanish ie: l - L-kOq V i E { O .... , n - 1 } , < . f
q 'Oqi
>=0
or
gi E {0 .... , n -
1},< f
L(q__i),.o~7(L(q)) . > . < f _. L(q), ziL(q) > = 0 . q q q q=
As L_~ is the orthogonal projection of f on tile space l,~, we know that q Vk E { O , . . . , n - 1 } , < f
L(q) z k --, q
q
>=0,
(7)
so the last equality reduces to Vi E { O , . . . , n - 1 } , < f
L(q) -iL(q) - -q, - q2 > = O.
(s)
484
Combining (7) and (8), we get
q2
q
where 7"1 and r2 are any polynomials in P , - I . But q and L(q) are relatively prime and the last equality is equivalent to < f-
z(J__2 q , q_P2 > = ~
(9)
P being any l)olynomial of P2n-1. Now, we come again to equalities (7). Taking partial derivatives leads us to V i e { O , . . . , n _ 1}, _ < ~ q /
, I .q_ > _
q , q2 > = 0 .
Using (9), the second term in the left-hand side is zero and we get the orthogonality relations: ,~q > = O.
VkE { O , . . . , n - 1},< ~
(10)
Next, we shall use the following l e n a m a 2 Let h be a function of H~rt orthogonal to the linear space I~, where p is any polynomial of T)~. Then D divides h. Proof: Let (i, be the complex roots of the polynomial p ie:
p(~) = H ( ~ - ~,). For each root (i, the function h will be orthogonal to the quotient 1/(z - (i). Using the definition of the scalar product, we get
z-~i
= 0.
From Cauchy's formula applied to the function h ( 1 / z ) / z , wc deduce that vi ~ {0 .... , , , ) , h ( ~ ) = o which means that (~ divides h. [] Now, apply this lemma to the functions 5~q,~~ (L(_l_q.l~q J which are orthogonM to the space Vq (cf (10)). There exist polynomials ul of J')n-t 1 such that: V i e { 0 ..... n - 1 } , ~
--
=
~,~.
Moreover, the polynomials ul are linearly independent. Indeed, let (Ai), a family of real nmnbers such that ~ )~illi .-~ O. Then ~ ,\i(lU; = 0 or
485
This yields
q E,,,
OL(q)
L(q)
= o
and the polynomial q must divide tile sum ~ Aiz i which is of degree n - 1. As q is of degre~ we get that
n,
~_, Aiz i = 0 and each real number Ai is zero which proves the independency of the polynomials ul. Now, we can evaluate the variation of the criterion at the critical point q, using the hessian matrix H whose entries are by definition:
05
L(q),
L(q) > q
- - < f - - - f OqiOqj q or
_90
Z(q)
"Oql < f =-2
- <~
'~ =2
=2
0 Z(q) > q ' Oqj ~ -
\ q ]
q 'OqjOqj
<--,-->-
q q
<--'--> q-
uj
q ' Oqi
q 'q2Oqi > - < f -
q
>
>
J~q/
~
>
and using (9), we gct
2 < 7' q
q
Oqi
> "
But
=
[Cl2Z'~-i _ 2(lqz i] =
2q~13,
so by (9) again, we obtain 05
OqiOqj < f -
_L _( q ) , f _ q
L(q) > = 2 [ < - -v,, - -vj> + < f q q q
L(q),9(tzivj > J . q ~ q3
Now, the variation of the criterion in a neighbourhood of the critical point q following a direction given by tile real vector (.\1, ..., An) in the space 7~ is
;~ As the family of polynomials (ui) is independent, we choose the numbers ,\i such that
486
The value of (})Aq(,\I,..., ,\,) becomes
[ ui u,_>+
~
L(q) L(q) q q On the other hand
(OL(q)~ oiL(q) + gtul. \ oq~ ] q = Using this equality together with (9) gives the following expression for (89 + < f =
L(q) _9q" ~ klui > (1 - qa
_ 9 < f
"
L(q)
71
7S L(q) >
As the order of q growths, f _ L_~ tends to zero, then the variation following the chosen II q Ii . . . . direction becomes positive which means that the critlcat pomt q ma.y not be a maxnnum. t h e o r e m 2 Let f , be a function as in theorem 1. maxima can only appear for a finite range of orders.
Then, critical points which are local
References [1]
L. Baratchart, Sur l'approximation rationnelle l a pour les syst~rnes dynamiques lin4aires, Th~se de doctorat d'dtat, Universitd de Nice, September 19S7.
[2]
L. Baratchart and M. Olivi, Index of critical points in /=-approximation, Systems & Control Letters 10 (19SS), 167-174.
[3]
L. Baratchart and M. O!ivi, New tools in rational L 2 approximation, proc. of tile 8th IFAC Syn~positun on Identification and System Parameter Estimation, Vol. 2, Beijing 198S.
[4] J. Delia Dora, Contribution ~ l'approximation de fonctions de la variable complexe au seas de Hermite Paddet de Hardy, th~se d'dta.t Univ. Sc. et Mdd. de Grenoble, 1980. [5]
R.C. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, Englewood Cliffs, N J, 1965.
[6]
B. Wahlberg, On system Identification and Model Reduction, rep. LiTHISY-I-0847, LinkSping University, 1987.
[7]
J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, AMS colloq. Pub. XX, 1969.
A M i n - M a x T e s t i n g A p p r o a c h to Failure D e t e c t i o n and I d e n t i f i c a t i o n Elias Wahnon IRISA/Centre IINRIA de Rennes 35042 Rennes Cedex, France
Abstract The problem of failure detection and identification in stochastic linear timeinvariant systems is approached as a Min-Max Hypothesis Testing problem. A statistical test is proposed and a Min-Max testing technique is followed, leading to a linear transformation of the residual vector, which results in a new statistical decoupling of failure influences. The resulting problem is to detect a change of the mean in a gaussian vector sequence and to estimate the random time of failure occurrence.
1
Introduction
Detection and identification of actuator and sensor failures is an important problem in engineering systems which rely on sophisticated control systems to achieve high Performances. The methods to deal with the Failure Detection and Identification (FDI) problem are generally classified in two main categories : 1. Redundancy Methods : the simplest way is the hardware redundancy, i.e., several identical measurement and/or control units, such as the quadriplexed digital flight control system, where the failure decision is generally made by a majority voting scheme. However, this passive m e t h o d increases the system cost and complexity. An active way is to exploit the analytical redundancy or parity checks, i.e., temporal relationships between the actuator inputs and sensor outputs which are, for perfectly known deterministic systems, zero when the system is operating normally and deviate from zero when a failure occurs, (see [7]). 2. Filtering Methods : known also as Detection Filters and originally proposed by Beard[2], this methodology is based on the design of a full-order observer where the gMn is chosen so that the direction of the observer output error vector in the output space, allows to identify the failed component. White and Speyer[ll] reformulated this theory as an eigensystem assignment problem, developing the required linear implicit equations for the observer gain and the solvability conditions. Massoumnia
488
et al. [8], relying on geometric concepts and exploiting the dual relationship between the FDI and the Control Decoupling problems, presented a method for designing a dccoupling filter. Viswanadham et ai.[10] proposed a frequency d~,main approach to develop generalized parity space techniques based on the theory of stable factorizations and showed the equivalence of this generalized parity vector approach with the concept of detection filters. Essentially, the so far cited references concern the problem of FDI for deterministic linear-time-invariant (LTI) systems. However, as was recently pointed out by Kerr[5], the original motivation was the problem of failure detection in navigation applications which are described by stochastic LTI models. In this paper, the FDI problem for stochastic LTI systems is approached as a Min-Max Hypothesis Testing problem. A 'detection filter', i.e., a linear filter, based on the no-failure hypothesis H0, where the gain matrix is chosen so that, upon the failed system hypothesis Hi, the mean of the filter residual vector propagates along distinct but fixed directions for each failure, is implemented. A statistical test is proposed and a Min-Max hypothesis testing technique is followed (see [9], [6], where this technique was proposed for the detection of changes in the AR part of a multivariable ARMA process). This technique leads to a linear transformation of the original residual vector which results in a statistical decoupling of failure influences. The problem, finally being a problem of detecting a change of the mean in a non-white gaussian random vector sequence and the estimation of the random time of occurrence, simple well-known detectors can be used. Two methods are also discussed for the special case where the number of failures is higher than the dimension of the measurement vector. In Section 2, the FDI problem for stochastic LTI systems is formulated. The Min-Max hypothesis testing approach is developed in Section 3. Extension to the special case of more failures than measurements is presented in Section 4. Simulation results of a working example are discussed in Section 5. 2
Problem
Formulation
Consider the normally operating system described by the following state space model :
x(k+l)
= Ax(k)+Bu(k)+w(k) y(k) = Cx(k) + v(k)
where x E R" and y E R "~.
w(k)~N(O,Q) v(k) ~ N(0, R)
(1)
489 The failed system is modeled as :
x(k + 1) = Ax(k) + Bu(k) + w(k) + ~ I{k>_n}fimi(k)
i=l
(2)
y(k) = Cx(k) + v(k) where fi 6 R '~, assumed known, is the failure direction associated with the i th actuator failure, rni(k) is a general time-varying unknown scalar which may be a function of x(k) or u(k), 7/ represents the random time of the i-th failure occurrence and IA is the indicator function of the event A. The FDI problem can be stated as the problem of detecting on-line a failure and to isolate, statistically, the influence of the i-th failure from the j-th, j # i, so that an identification of the failed component should be possible. For simplicity of notation, take 7i = T, i = 1 , . . . , r. Assume now that a linear filter, based on the nominal model, Eq. (1), has been implemented where the gain matrix, K, should be later specified :
u(k) = ,.,(k)-C~o(k) ko(h + 1) = A~:o(k) + Bu(k) + IC,(k)
(3)
Let H0 be the hypothesis which states that the system is described by the nominal model, Eq. (1), and H1 the hypothesis that the system is described by the Eq. (2). Under Ho, u(k) is a zero mean non-white gaussian sequence with variance matrix, V(k), given by :
v(k) = cP(k)c T + R P(k + 1) = (A - I(C)P(k)(A - KC) T
(4)
+ KRK T + Q and assume that the steady state value of V(k) has been attained, denoted here by V. Under HI, decompose u(k) into : ~(k)
=
y(k) - c ~ 0 ( k )
=
v(k) - c ~ ( k )
+ c [ ~ , ( k ) - ~o(k)V{k>,}
~ ( k ) + ~(k)hk_>, } where ux(k) ,,, N(O, V) and/~(k) is given
by:
o~(k -I- 1) -- (A - KC)o~(k) -F f, ml(k) ,(k) = Ca(k) ; ~(k
where: o/(k) ~ :~l(]g) -:~o(k).
-F'.'-F f~rn~(k)
(5)
490
Remark: The case of sensor failures, represented by the following model for the i-th sensor failure : x(k+l)
,j(k)
= dx(k)+Bu(k)+w(k)
c~(k)
=
+
v(k)
+
z{~>_~,)e~,(k)
where ei is the i-th m 9 1 unit vector, leads to the following equation for ~he mean variation :
o,(I~ + 1) = (A - Z(C)~(k) - K~i6,(k) The presence of the i(ei as a failure direction is a problem since K is not known. However, as pointed out by White and Speyer[11], taking gi being any direction such that e{ : Cg~, this equation can be written as Eq. (5) and #(k) can be made to lie in the plane generated by CKel and ei. Therefore, we use the actuator failure case, Eq. (1) as the general problem. The F D I problem can be formulated as the following hypothesis testing problem :
U,~de,'Ho: ~,(k) ~ iV(O, V) UnderHl : 3% k < T : V ( k ) ~ N ( O , V ) k >_ ~: . ( k ) ~ N ( , ( ~ ) , V) 3
The
Min-NIax
Testing
Approach
The statistic :
(G)
to 2 .r(k)v-l.(~) allows to test beLween H0 and Hi. Furthermore, Eo(to) =
m
El(to) = I m
m +
,~(k)v-b(k)
kT
where Ei(') is the expectation under Hypothesis Hi. Now, assume that there exists a gain matrix K such that the failures associated with the fl, i = 1 , . . . , r, in the system described by the mean Eq. (5), are dctcctablcs according to the definition given by Beard [2], i.e., #(k) maintain a fixed direction in the output space due to fi and all eigenvalues of A - K C can be arbitrarily specified, except for the constraint on the conjugate symmetry. This gain matrix ensures the decoupling of the failure set, { f l , . . - , fr}, influences in tlle output space and can bc computed, for cxamplc, by the method presented in
491
[11]. Therefore, assuming that a such gain matrix has been found, we have that for each fi,i = 1 , . . . , r :
rank[Cfi, C(A - [(C)fi, C(A - KC)2fi,...] = 1 and
assume
(7)
also :
rank[CfI,Cf2,...,Cf~] = r, r <_ m Define the following r * r nonsingular symmetric matrix :
(8)
a= [Cfl,..., Cf~]TV-l[Cfl,..., Cf,]
(9)
The test to, Eq. (6), is a X2 distributed random_ variable with non-centrality parameter, 7, under HI (for k >_ r) , given by :
Under the condition in Eq. (7), #(k) maintain a fixed direction, Cf~, in the output space due to {mi(1);l = k - 1, h - 2,...,7"},i = 1 , . . . , r . Therefore,
(adl,. .,Cf,mx){[k 1,] + +
i A21rnl(k - 3)
;~2~mr(k 3)
i
+ ....
2,1
(Cft,Cf2,...
Cf~)
I
3.42
and 7, Eq. (10), can be formally writen as : =
....
We look now to isolate, statistmally, the different failure influences by building, for each failure, a test which should be insensitive to the other failures. Take, for example, the first failure fl and partition the matrix )c =zx [ 3S cTl i - T t2] ,whereSll F22 is a scalar and ~c22 is a ( r - 1) * ( r - 1) matrix. We shall follow a Robust Min-Max hypothesis testing technique (see [9], [61), by considering for the failure f;, the least favorable case for the remaining failures ( f 2 , - . . , f.)- For a fixed test leve! c~, given by : PHo(to > A) < a', where, A is the threshold fixed by c~, the power of the X2-test to, Ptil(to > A) is an increasing function of the non-centrality parameter 7, Eq. (10). By the Min-Max approach we are looldng to find :
min
(~,-..,~,)
7 =
min
(~,...,~r)
[.All, (.s
, .s
[-7-11 F12 ] ] .T-r .T.22
3,41 A'42 :
3~I,
492
The minimum, obtained for : (.M2,...,.M~) r = --~-~19rT.A41, is given b y : min 3' = 3Al(Scm - ~129r~)'Su~)-A41 (~4~,...,M,)
-~175axl ] = 3/li{(1,-.7-i2.7-~i)(cfl,..., C f ~ ) r v - i } v {y-l(c
f l , . . .,
cA)(1,-f~27~)r}ax~
Tlicrefore, considering the following transformation of the original residual vector sequence, u( k ), : ~,,(k) = ( 1 , - 7 1 2 : r ~ ) ( c f ~ , . . . ,
cA)rv-l,(k)
(11)
we have that ul(k) is a non-white gaussian scalar sequence, zero mean under H0 and witll a non-zero mean under Hi (for k > r) given by :
E(vt(k))
= (1,-.7"i2.Y~i)(cfi,...,Cf,.)Tv-1/.z(k)
= (l'-212"~)(CI~'""C/~)rW-~(C/~'""C/~) =
(~,-j:,~:#)j:
J(4.
~
and, V a r ( vl ( k ) ) = .T m - .T12.T~I.T'f~2. i.e., ul(k) is only sensitive to the failure fl and is not affected by the remaining
failures (f2,.-., A)Finally, returning to the original problem stated in Eq. (2), given the failure set {f~,.-., fr} with the unknown random times of occurrence {T1,..., Tr}, we build for eacli failure, f/, the required transformation, ui(k), of the residual sequence u(k) which is essentially a statistical decoupling of failure influences. The problem being now of detecting a change of the mean and the estimation of the r a n d o m time of occurrence, simple detectors can be proposed. In particular, the well-k/iown Hinldey detector can be useful, implementing for each ui(k) two sucti detectors, to detect an increase or a decrease in the mean of ui(k). The Hinldey detector, introduced in [4] and analysed in [1] is given (for example, to detect an increase
493
in the mean of/]i(k)) by the following recursive procedure : TO
~
0.
~ffk
:
T k _ 1 "-~ Y i ( ] r
Mk
=
Alarm
5
- - ry
rain Tj
(12)
l<j
: whenTk-M~
>
where A is the threshold and 5, fixed apriori, is the minimum jump to be detected. Remarks: 1.) The Min-Max technique leads to a linear transformation of the residual vector v(k) which hides, for each failure, the subspace spanned by the remaining failures in the output space. Consequently, the existence of a gain matrix satisfying the condition in Eq. (7) is required. Otherwise, different failures m a y span the same subspaces and no hiding should be possible. 2.) Even though the Hinkley's detector was developed to detect a change of the mean in a white gaussian sequence, the experience shows that this detector is robust with respect to the whiteness assumption. 4
More
Failures
than
Measurements
In the special case, r > m, i.e., more failures than measurements, the assumption, Eq. (8), cannot be satisfied and some modification are required for the identification of the failed components. Two methods are here discussed : 1. Partial Filters : Take, for simphcity, rn = 2 and r = 3, i.e., (fl, f2, f~). Then, if we assume that the probability of simultaneous failures is very low (i.e., zero), then by building two filters, according to Section 3, the first for f; and f2, and the sccond for fa and, for example, ft, we obtain, by adding an heuristic reasonning, a complete failure identification system. However, if simultaneous failures are assumed, we are not generally able to identify the failed components, except for the case where the mean jump size m a y be used to make the failure decision. 2. Clustering : Define the following distance measure between fi and fj (see an analogous definition in [3]) :
(Cfl)T(cfJ) d(fi, fj) = l - pii = l - { Cfi ll C f j l
(13)
and make the required clustering, i.e., replace by a single failure vector fi the failure vectors fj which are close in the sense of the distance measure, Eq. (13), such that we stay with the required number of failure vectors. Note that pij is nothing but the cosine angle between the images of fi and fj in the measurement space. Let us r c m ~ k that with the clustering method, some failed components m a y not be
494
well identified since the failed component direction in the output space has a main projection in the clustered direction but also a smalt component in other failure directions.
5
Simulation Results
Given an stochastic LTI system described by the Eq. (1), where : A=
034 123
-3
; /3 = [fl, f2] ; fl ----
0 2 5
1 0
=
11 ] -i/2 i/2
1
C=
[0101 0 0 !
; Q = G G T;
G=
1/2
t/3
0
.05
the condition in Eq. (8), rank [Cfi, C f2] = 2 = dim y, is satisfied and the gain matrix K, calculated by the method presented in [11], which ensures the condition 25/8 91/24 in Eq. (7) is given by : f ( = 5/4 53/12 . Furthermore, the closed-loop matrix 2 29/6 A - I ( C is stable and its eigenvalues are freely chosen ( here, Ii = 1/2, A2 = 1/4, and Aa = 1/6). The steady-state variance matrix, V, has been computed according to Eq. (4) and the required transformation of the residual vector is given, according to Eq. (11), b y :
where z,r(k) isolates the influence of {ml(j))~=r and u.~(k) the influence of {m:(j))~: T. Since the system matrix A is not stable a feedback control has been used : r .4 l.S9 4 . a 7 ] u(k) : - D ( x ( k ) - Xre/), where D = ] j and ~.T : (1, 1, 1). 1.27 8.52 17.33 "~ef L Fig. 1 presents the simulation results for the case where at the instant 300 the second actuator faiied (a constant bias failure of 10). For the nominal residual vector, both elements of this vector jump at the failure instant while for the transformed residuals, only u~(k) jumps while u~(k) rests insensitive to the failure f2Therefore, the transformed residual sequence performs a statistical decoupling of failure influences, where u~(k) is only sensitive to the failure fl and u~(k) to the failure f2-
495
6
Conclusion
A Min-Max hypothesis testing approach for failure detection and identification in stochastic LTI systems has bcen presented. A statistical test has been proposed and a Min-Mmn approach has been developed. This approach leads to a linear transformation of the original residual vector which results in a statistical decoupling of failure influences. Simulation results of a working example have also been presented. The problem of failure isolation by the Min-Max approach, when the optimal gain matrix is used in the filter which generates the innovation sequence, is actually investigated.
Acknowledgment I would like to thank Dr. A. Bcnveniste and Dr. M. Basseville from IRISA/INRIA, Rennes, France, for the fruitful discussions.
References [1] M. Basseville, 'Edge detecticn using sequential methods for change in level Part II : Sequential detection of change in the mean', IEEE Trans. Acoust. Speech Sig. Process., vol ASSP-29, 1981, pp. 32-50. [2] R.V. Beard, 'Failure accommodation in linear systems through self reorganization', Mass. Inst. Technol., Cambridge MA, Rep. MVT-7I-1, Feb. 1971. [3] A. Benveniste, M. Basseville and G.V. Moustalddes, 'The asymptotic local approach to change detection and model validation', IEEE Trans. Automat. Control, vol AC-32, 1987, pp. 583-592. [4] D.V, Hinkley, 'Inference about the change-point from cummulative sum tests', Biometrika, voI 58, 1971, pp. 509-523. [5] T.H. Kerr, 'A critique of several failure detection approaches for navigation systems', IEEE Trans. Automat. Control, vol AC-34, July 1989, pp. 791-792. [6] G. Le Vey, 'Analyse modale et surveillance vibratoire des machines tournantes', Thhse de l'Univ. Rennes 1, 1988. [7] X.C. Lou, A.S. Willsky and G.C. Verghesse, 'Optimally robust redundancy relations for failure detection in uncertain systems', Automatica, vol 22, 1986, pp. 333-344.
496
[8] M.A. Massoumnia, G.C. Verghesse and A.S. Willsky, 'Failure detection and identification', IEEE Trans. Automat. Control, vol AC-34, March 1989, pp. 316-321. [9] A. Roug6e, M. Basseville, A. Benveniste and G.V. Moustakides, 'Optimum robust detection of changes in the AR part of a multivariable ARMA process', IEEE Trans. Automat. Control, vol AC-32, 1987, pp. 1116-1120. [10] N. Viswanadham, J.H. Taylor and E.C. Luce, 'A frequency-domain approach to failure detection and isolation with application to GE-21 turbine engine control systems', Control Theory and Advanced Technology, vol 3, MITA Press, March 1987, pp. 45-72. [11] J.E. White and J.L. Speyer, 'Detection filter design : Spectral theory and algorithms', IEEE Trans. Automat. Control, vol AC-32, July 1987, pp. 593603.
1
0.5 0 -5 -0.5 -I0
0
200
400
i
i
15
600
-1
0
*,,(k) : N.mi.al II~idual
200
400
600
*.~(k) : 'l'r,.~h.rm.M Ih:~id..I
10 5 0
0
-5 -10
9
0
a
t
200
400
_~
600
0
l
a
200
400
600
Fig 1 Time histories of Nominal and Transformed Residuals for the case of a Constant Bias Failure of the Second Actuator.
I
ON-LINE DETECTION OF MINIMAL ORDER FOR LINEAR PIECEWISE I STATIONARY SYSTEMS
t
M. GUGLIELMI Laboratoire d'Automatique de NANTES U.R.A. 823 Ecole Nationale Sup6rieure de M6canique 1 rue de la No~ 44072 NANTES Cedex 03 Membre du GRECO Traitement du Signal et des Images Abstract: This work deals with the detection of minimal order for linear systems with piecewise constant coefficients. In this case, it is impossible to use off-line algorithms. We propose, here, a real-time method able to detect the minimal order of a linear system. The basic mathematic tool is given by a simple divisibility condition of the charateristic polynomial of the differential equation for stationary systems. This necessary and sufficient condition is used to define an algorithm able to give the reduced equation. Then, we show that this result can be directly extended to linear piecewise stationary systems for which we search a change if the order. The real-time method that we obtain, is checked in simulation on deterministic systems and on systems with measured noisy informations. I- INTRODUCTION: The knowledge of the minimal order for dynamical systems is a very important information. In control systems (where supplementary modes may be a drawback) as in the pattern recognition problem in the aim of classification (diagnostic...), it is fundamental to describe the linear observed system by a lowest order differential equation. For stationary system, it exists well-known algorithms able to get the exact minimal order. In the determistic case, the principle of these algorithms, like the HO's one, are based on the calculation of the rank of the HANKEL matrix [1] build with the input/output of the system. In the stochastic environment, the order is also obtained by using algorithms which compute the rank of matrix build with the variance/covariance informations [2]. The mimal dimension of the system can also be directly obtained by using the AKAIKE's criterion [3]. Finally, some parametric identification methods can be implemented with a on-line test to check the process order (BURG-LEVINSON [3]). The most important drawback of these methods is that they need a large sample of signal, so they are unable to work on real time. In many cases, it is very interesting to have a mean able to get the order information of a dynamical system from real time input/output information, specially if we turn our attention on systems for which the
498
order can decrease. The reducibility of a differential equation can be studied by the means of the one-parameter groups [4]. But this approach is not quite easy to use in an algorithm based on input/output informations. Our paper deals with the definition of areai time algorithm able to detect a change in the order for piecewise stationary linear systems. Its organization is: after a simple example, in the paragraph 1I, we give the necessary and sufficient condition which must be satisfied by a nth linear stationary system to be represented by a n-1 th differential equation. This condition is a relation concerning the initial values, which uses the quotient of the division of the characteristic polynomial of the differential equation by its factors. From this condition, we obviously get a generalization (paragraph III) for a preducibility greatest than one. In the IV paragraph, we propose a algorithm able to detect on-line a change in the order when the linear system has piecewise stationary coefficients. Two examples are given in paragraph V. I[: N E C E S S A R Y and S U F F I C I E N T R E D U C I B I L I T y C O N D I T I O N : II-1: A simole example: Consider the following second order linear stationary system: y(2)(t) + aly(1)(t ) + a 0 y(0)(t) = 0 diy(t) . dt I Among all the trajectories obtained in the phase plane @~ ) when the initial conditions y(0)(0) and y(1)(0) are varying, two of them are straight lines. In this case, the trajectories can be represented by a first order differential equation. For example, the figure I shows that for the system described by: y(2)(t) + 3 y(1)(t) + 2 y(0)(t) = 0 In the paper, we note: y(i)(L) =
.............................. p 09, -2.00
-1.50
-1.00
Notice that the two straight line trajectories are the solutions of the 2 nd order differential equation: y(2)(t) + a!y(D(t ) + a 0 y(0)(t) = 0
-0.50
- 1 . 0 0 -"
(only if the initial conditions verify y(1)(0) + 2 y(0)(0) = 0 case [~ or y(1)(0) + y(0)(0) = 0 case [~]). Here, we
- 2 . 0 0 "3
~3.00 -~ [rojectolce de :~ + AY + BY = 0 (A=3, B=2) en fonctlon des conditions inltio}es
FIGURE 1
propose a method which detects directly, in the general n th order case, the presence and which deletes the supplementary useless dynamics in the differential equation.
499
II-2: Reducibility of 1:
Let's have the linear homogeneous system (Y) described by the linear differential equation with constant parameters: y(n)(t)+ aly(n-1)(t)+ a2y(n'2)(t) + ......... + any(0)(t) = 0
(eq. tI-1)
With the n initial conditions y(n-1)(0), y(n-2)(0) ..... y(0)(0) We suppose that y(t) belongs to C ~176 diy(t) We note: y(i)(t) - dt i -T :~ (t) = [ xl(t ), x2(t) ......... Xn_l(t)] T = (t) = [ Xl(t), x2(t) ......... Xn_!(t), Xn(t) ] P()~) is the characteristic polynomial of the differential equation II-1 When it will be necessary and when no ambiguity will be possible, we shall use equally, in the paper,P@) like a differential operator or like a polynomial in of X.When P(X) is the fifferential operator, we note: P(X) o x(t)= x(n)(t)+ alx(n-1)(t)+ a2x(n-2)(t) + ......... + anX(0)(t) II-2-1:Theorem: The trajectory described by the equation (II-1) can be obtained by a n-1 th order linear differential equation if and only if it exists a set of n-1 coefficients {a i ) such as if the polynomial: Q(X) = ) n - l + o~ ~n-2+ c~2 )~n-3 § ....... + O~n.1 1 we have: y(n'l)(0)
+ Cr
divides
) + cc2y(n'3)(0) + ....... + c~n_ l y(0)(0) = 0
In other words:
y(n'l)(0) = -
where
y T ( 0 ) = [y(n'2)(0), y(n-3)(0)...y(0)(0)]
P()~)
(eq ii-2)
~Ty(0)
Demonstration: 1 ~ Necessary_ condition: Suppose that y(t) can be represented by the n-lth order differential equation: y(n-1)(t)+ C~l y(n'2)(t)+ (z2y(n-3)(t) + ........ Then, the characteristic polynomial:
+ Ctn_ly(0)(t) = 0
(eq. II-3)
5OO
Q(X) = xn-l+ oc1 ~n-2+ a2 )~n-3 + ....... + an_ 1 divides P(X). From the division of P(~,) by Q(X) which gives: P(X) = (7~ - ~0) Q 0 0 + K
where K is a constant.
we have: P()0 o y(t)= [(X- X0 ) Q(L) + K] o y(t) Thanks to the properties of the differential operator, we get: 0 = [ (X- %0) Q(X) ] o y(t) + K o y(t) 0 = (7~ - 7~0) o[Q(X) o y(t) ] + K o y(t) Thus:
0= 0 + K o y(t) = K y(t) Finally K=0 so
Q(X) divise P(X)
In the other hand, the condition II-2 is obviously satisfied: Q.E.D. 2 ~) Sufficient condition: Let's have the differential equation with the characteristic polynomial Q(X). By hypothesis, we suppose that: Q(~) = ~n-l+ o~1 ~n-2+ cc2 ~n-3 + ....... + ~n-1 divides P(X) If w(t) is a trajectory defined by: w(n-1)(t)* a 1 w(n-2)(t) + ........ +con_ lW(0)(t) = 0
(eq II-5)
with: w(i)(0)= y(i)(0) for i varying from 0 to n-2. We define e(t) by:
e(t) = y(t) - w(t).
From the expression: e(n)(t)+ a l e ( n - 1 ) ( t ) § a 2 e ( n - 2 ) ( t ) § .........
+ ane(O)(t)
We can easily show (Annex 1) that e(t) satisfies: i=n i=n-1 e(n)(t) + ~ ai e(n-i)(t) = - anW(0)(t ) + ~ (~ - ai)w(n-i)(t ) i=1 i=1 The second member of this equation is equal to 0 due to the fact that the characteristic polynomial of this part is:(7~ Q(X) - P(X)). Using the hypothesis, we have:
P()0 = (7~ - X0)Q(X).
Thus: (Q(X) - P(X)) ow(t) = {9~ - (9~ - 9~0) 1o [Q(k) o w(t)] = ~0 [Q(X) o w(t)] = 0 So e(t) -- 0
t , if and only if all the initial conditions are equal to zero. For
i from 1 to n-2, it is satisfied by definition of e(t).
501
To get e(n-l)(0) =0 we have: e(n-1)(0) = y(n-1)(0) - w(n-1)(0) y(n-1)(0) = w(n-1)(0 ) = . {o~1 w(n-2)(0) + ........ +t:Zn_lW(0)(0)} = - {(z1 y(n-2)(0)+ ........ +an_ly(0)(0)} = -
-T ~ Y(0)
So, the trajectory w(t) is the same that y(t) if the above condition is satisfied and we can say that: y(t) has a model described by a n-1 o r d e r d i f f e r e n t i a l e q u a t i o n Q.E.D.
II-3: Reducibility of order >1; The above result can be enlarged: 11-3-1 :Theorem: The following linear differential homogeneous system: y(n)(t)+ aly(n-1)(t)+ a2y(n-2)(t ) + ......... + any(0)(t) = 0 can be represented by a linear differential homogeneous equation of order 1 (l
+ (x1y(0)(0) = 0
This result is easily obtained by generalization of the demonstration of II-2.
II-4: E x a m p l e ; Let's call (E) the following 2 nd order system: y(2)(t) + 3 y (1) (t) + 2 y (0) (t)= 0 with the two initial conditions: y(1)(0), y (0)(0) Its characteristic polynomial is P(~) = X2 + 3 ). + 2 = (X+I)(X+2) If we use Q(3.) = (3.+2) y(t) can be described by a 1st order equation if we have also: y(1)(0) + 2 y (0)(0) = 0.
If use call Q(~,) = (3.+1) y(t) can be described by a 1st order equation if we have also: y(1)(0) + y (0)(0) = 0.
502 This is, in the phase plane, the two straight line trajectories presented in figure I.
III: A L G O R I T H M ; When we turn our attention on y(t) defined by a linear differential equation with all the initial conditions, we can find a reduce equation by the means of the following algorithm: Factorization of the characteristic polynomial P(~) j=p
P(~') =r-[ (z. _ ~j)rj j=l J:=0 repeat j:=j+l i=p Compute Q(X) := U (~" - ~i )ri icj i=l Compute R := Q(~.) o y(0) until R=0 or j=p+l If R=0 then reduce equation is Q ( k ) ~ y(t)
IV: O R D E R R E D U C T I O N for P I E C E W I S E S T A T I O N A R Y
SYSTEMS:
IV-l: Al~,orithm: Now, we look after linear differential homogeneous system (I2) descibed by a differential equation E1 for t [tO , tl] and E2 for [t 1 , tf] with an order smaller than El. We suppose that E1 is known. The figure II shows, in the phase plane, a particular trajectory for a linear system (E) descibed by a second order equation E1 for t varying beetwen 0 and 80 mS and a first order equation E2 for t>80 ms. The figure III shows the time trajectory:
503
1.00
2.00
D~de
1.50 o.ao ~
DETECTION de RUPTURE
O.bO
RUPTUR
IO0
" =
0.50
0.40 0.00
0.20
0.00
-0.50 1.00 2.00 ,3.00 4.00 5.00 Tcmos
6.00
0.00
0.20
FIGURE II
0.40
FIGURE III
The problem is to detect the time of the order change. We can use the earlier algorithm like that:
Factorization of the characteristic polynomial P(%) j=p P(%) = I - I (% - ~j)rj j=l T:=0 repeat t;=t+l J:=0 repeat j:=j+l
i=p Compute Q(%) := I ~ (% " ~'i)ri i~j i=l Compute R := Q(%) o y(t) until R=0 or j=p+l until R=0 or t= tf If R=O then reduce equation is Q(X)e y(t)
0.60
0.80
1.00
504
IV-2; Examole: Let's have (E) described by the equation El: y(2)(t) + 3 y (1) (t) + 2y (0) (t) = 0 for t 0 and 80 ms (y (0) (0) = 0.5, y (1) (0) =2) and E2 : y (1) (t) + 2y (0) (t) = 0 for t>80 ms. At every time, we compute the expressions: Y(I)(t)+2y(0)(t)
and
y (1) (t)+y (0) (t) and we compare the
results to zero. The change in the expression y (1) (t) +y (0) (t) for t= 80 ms gives the change order time.
DETECTION de RUPTURE
(T=80)
2.00
1.00
...... ~176176
l'.&
........ 2.oo'....... ~.'& . . . . . . . . 4.oo'. . . . . . . .~'&. . . . . . . ~ i & o Temps
FIGURE 4 Without doing any analysis of the properties of this detector when we use noisy signals, we have done a simulation of the previous system with noise on the two infomaations y(0)(t) and y(1)(t). It seems that the detector has a robusteness when we introduce noise in the signals.
505 2.00 DETECTION de RUPTURE (7=80) 1.00
0.00
- I .00
-2.OO.o O,1 1 1 1 1 1 1 { 1 1 1 1 1.00
ItlllJlllll
2.00
Iltl
IIIIIIllll
3.00
[ llllllitl
4.00
lllillll
5.00
11 j
6.00
FIGURE 5 VI: CONCLUSIQN; The method we have presented here, is able to pointout the decrease of the order for differential equation from the study of one trajectory. In the stationary case, a necessary and sufficient condition must be satisfied by the initial conditions. For dynamical systems with order variation, from this condition, it is possible to implemente an algorithm able to detect the order decrease from the only signal (and its derivates). With the mathematical tools used here (factorization of characteristic polynomial), we theink that it is obvious that this algorithm can be used for discrete-time systems. One interesting point of this detection is in the fact that this algorithm can be jointly used with parametric identification methods (like Extended KALMAN Filtering) which need the order as an a priori information. The real-time order estimation increases the performance of the identification methods. Notice that we have got an algoritm which checks the order decrease. This is a more difficult problem that the increase's one. In fact, when the order grows up, the identification is not good and it is possible to detect that by the means of other information (like innovation ...). Our work can be carried on. The first direction may be the study of the performances of this algorithm in a stochastic known environment (Gaussian...). The second interesting study is to try to enlarge this result to non-stationary systems (i.e: with time varying parameters). Here, the difficulty is that the differential operateur becomes non-commutative and, by the way, the factorization of the characteristic polynomial is more difficult [5].
506
Vll: BIBLIOGRAPHY: 1- R.E. K A L M A N - P.L. F A L B - H.A. A R B I B : Topics in mathematical systems theory Mac Graw 1970 2 - G . A L E N G R I N - G. F A V I E R : New stochastic realisation algorithms for identification of ARMA models IEEE ASSP conf. TUSLA 1978 3 - L. M A R P L E : Digital spectral analysis with applications Prentice Hall 1986 4 - J.M. H I L L : Solution of differential equations by means of one parameter group Pitman Research notes in Math. 1982 5 - E. K A M E N : The poles and zeros of a linear time-invariant system Linear algebra and its application 1988 - 98 PP 263-289 ANNEX 1: Calculation of the differential equation satisfied by e(t): By definition, we have : e(t) = y(t) - w(t). From the expression:: e(n)(t)+ al(t)e(n-1)(t)+ a2(t)e(n-2)(t ) + ......... + an(t)e(0)(t) = y(n)(t)+ al(t)y(n'l)(t)+ a2(t)y(n-2)(t ) + ......... + an(t)y(0)(t) -{ w(n)(t)+ al(t)w(n'l)(t)+ a2(t)w(n'2)(t) + ......... + an(t)w(0)(t) } By hypothesis, the first part is equal to 0 ( about y(t)). We can compute: : d w(n)(t) = ~t (w(n" I)(0 =- ~" ((C~1 w(n'2)(t) + ....... +0%lw(0)(t)) = _ (oqw(n-l)(t) + ........ +O~n_lW(1)(t) ) Thus: e(n)(t)+ al e(n-1)(t)+ a2 e(n'2)(t) + "'" +ane(O)(t) =
i--n-1 E (cq - ai)w(n-i)(t ) - a n w(0)(t) i=2
A
SIHPLE
HETIIOD FOR IDENTIFICATION OF
LINEAR
DYNAHICS NITH llYSTERESIS NONLINEAR INPUT
Lang Zi-qiang Department of Automatic Control, Northeast University of Technology, Shenyang 110006, Liaoning, P.R.China Summary In with
this paper,
a simple parametric identification method for linear dynamics
hysteresis nonlinear input is
presented.
The
parametric identification
of
the linear dynamics with deadzone nonlinear input, as a particular situation of the considered problem,
also can be realized by this method.
The effectiveness of the
method is illustrated by simulation studies. Remarks on tile theoretical analysis of this method and its application in engineering are made at the last.
I. Introduction Generally
speaking,
discontinuous complicated
the
nonlinearities, model
identification
such
as
of
the
multivalued
establishing problem,
dynamic
and/or
systems
memory,
is
since the applications of the
with a
mere
parameter
estimation procedures having been well developed for linear dynamic systems to such ones
have
been
difficulties
very limited due to tile formidable nature
beth
special cases,
at
algorithmic and numerical levels
of
the
problem
[1]. llowever,
with
in
some
by taking advantage of the special features of the nonlinearity
in
the considered dynamic system, simple identification strategy could be presented to cope
with
example, preload
its the
[2]
for
parametric identification of the linear dynamics with dead zone
and
nonlinear
identification some
parametric identification problem.
input
was
carried
out
only
In our previous work
by
doing
Least
Squares
calculations in twice and combining the two calculation results
(LS) in
way.
In this paper, hysteresis problem
parametric identification problem of the linear dynamics
nonlinear
is presented.
input is considered,
a simple method for solution
The basic feature of the method is that two
to
with this
identification
experiments are carried out for parameter estimation computations , with two Pseudo Random Ternary Signals (PRTSs) of different amplitudes being used as input sequence
508 in
the
dynamic
two
experiments respectively.
The parametric
system with dead zone nonlinear
input,
identification
of
linear
being as the particular situation
of tile considered problem, also can be done by using this method. The effectiveness of tlle method is illustrated by simulation studies. Remarks on the analysis of this method and its application in engineering are made at the last.
~I{ { ( t ) )
A(-')
=
l
-
.
A(q-' )
Fig.l. Discrete Time Linear Dynamics with Hysteresis Nonlinear I n p u t
I I.
The D e s c r i p t i o n esis
The
block
nonlinear
input
of the Linear
Nonlinear
diagram
of
Dynamics with Hyster-
Input under the Excitation
tile discrete
i s a s shown i n F i g . 1 .
Its
time
linear
difference
o f PRTS
dynamics
equation
with
model c a n be g i v e n a s
A(q -~ ) y ( t ) = q - " B(q - ' ) ~ [ u ( t ) ] + ~ ( t ) llere,
{u(t)}
and {y(t)}
hysteresis
(2.1)
are the system input and output
sequence respectively
A(q- ' )=1+at q- ' + . . . + a , q-"
B(q-' )=b. +b, q-' +...+b,. q-"
b, ~0
are polynomials with respect to the backward shift operator q-' , d is time delay of the system,
{~(t)) represents disturbance
noise with zero mean,
sequence,
which is assumed to be white
~[u(t)] denotes the hysteresis nonlinearity,
the analytical
input and output character of which is as follows
I
k[u(t)-b]
for u(t)-u(t-1)>O and #[u(t-1)]-k[u(t)-b]=O
~[u(t)] =]k[u(t)+b] # ~r
for u(t)-u(t-1)
where k and b are the parameters of the nonlinear element.
(2.2)
509
For
system
(A.1)
(2.1),
following assumptions are made in
Tile structure
parameters of the system,
n.
this paper m and
d,
are
known
a
priori. (A.2)
Polynomials
A(q-' )
and
B(q-' )
are
and
coprime,
is
A(q-')
a s y m p t o t i c a l ly s t a b l e . Apply a PRTS as input s e q u e n c e { u ( t ) } different
levels
of which are -D,
O,
to excite the
system
(2.1), the
and D
and D respectively,
three
satisfies
the
condition
(2.3)
D>b~O Fig.2
illustrates
the
input
and
output
relation
of
the
hysteresis
nonlinearity under the excitation of the PRTS.
r
V[u(t)] k(D-b)
-k(D-b) -D
0
B
I
~(t)
t Fig.2
From expressed
Fig.2,
it
The Input and Output Relation of Hysteresis Nonlinearity under the Excitation of PRTS
is
obvious
that in this particular
case
~[u(t)]
be
as
(2.4)
~[u(t)]=K(D)u(t)+L(D)s[u(t)] where,
can
K(D)
and
L(D),
as
marked in
Fig.2,
are the parameters
of
a
preload
nonlinear characteristic (slope and preload parameter respectively) which is of the same input and output relation as the hysteresis nonlinearity under the of the given input sequence -- PRTS with the levels -D, O, and D ,
excitation
510
I!
s[u(t)]=
u(t)~O+ u(t)=O
here the implications u(t)=O"
(2.5)
u(t)~O"
-
of expression
u ( t ) = O + , u ( t ) = O and u ( t ) = O - a r e a s f o l l o w i n g
i f u ( t ) = O and t h e r e e x i s t s and u ( t - # + l ) = O 1=1,2 . . . . .
u(t)=O-
u(t)=O
i f u ( t ) = O and
reference
a positive
1=1,2 . . . . .
p a l making
u(t-p)>O,
integral
p~l making
u(t-p)
p.
f o r any p o s i t i v e
with the expression
integral
p .
i f u ( t ) = O and t h e r e e x i s t s and u ( t - p + l ) = O ,
In
a positive
integral
(2.4),
it
#~1, u ( t - p ) = O
should
be
pointed
out
that
if
( D - b ) k ~ h k , t h e n L(D)=bk, o t h e r w i s e L ( D ) = ( D - b ) k i s a f u n c t i o n o f D , t h e a m p l i t u d e
of
t h e PRTS, F i g . 2 o n l y s h o w e s t h e c a s e when ( D - b ) k > b k .
Substituting
(2.4)
into
(2.1) gives
A(q")y(t)fq-'B(q-')(K(D)u(t)+L(D)s[u(t)]}+~(t) =q-'K(D)B(q-')u(t)+q-'L(D)B(q-')s[u(t)] (2.6)
will
be
the basis
method f o r s y s t e ~
for us to derive
the
+~(t)
simple
(2.6)
parametric
identification
(2.1).
III.
Identification
Assume t h e l e a d i n g t e r m o f B ( q " )
Algorithm
being
u n i t y and
that
(u(t)}
is
t h e PRTS
mentioned in the above section.
Express ( 2 . 6 ) componentwise as Y(t)=x(t)To+~(t)
(3.1)
where x(t)r=(-y(t-1) Or=[a, Since carry
a,',
..... -y(t-n),u(t-d)
a,,K(D) . . . . . K(D)I~,L(D) . . . . .
.....
~(t)
n,
u(t-d-m),s[u(t-d)]
LS method i s d i r e c t l y
applied
c o m p u t a t i o n f o r t h e p a r a m e t e r s e from ( 3 . 1 ) .
K(D)" and L ( D ) ' ,
the estimates
and L(D) c a n be o b t a i n e d i m m e d i a t e l y . to hold approximately
..... s[u(t-d-m)])
L(D)b,]
i s a s s u m e d t o be w h i t e n o i s e ,
out parameter estimation i=l .....
.....
for parameters a,,
Since the following relations
to
Thus,
i = l . . . . . n , K(D) can be t h o u g h t
511 [L(D)b~I*
= [L(D)]'bj
[K(D)b l" = [K(D)]'b~ ~le determination
j=l ..... m, the estimates of parameters
of bj',
in B(q-' ), can be
done as below b,'=([K(D)b+]'K(D)'+[L(D)b~]'L(D) ") / [ K ( D ) " + L ( D ) " ] where
[K(D)bj]"
and [L(D)b~]',
5=I ..... m
(3.2)
3=1 ..... m are the estimates of K(D)bj and
L(D)bj,
j=l ..... m, being obtained by LS parameter estimation computation from (3.1).
Carry coming
out
tile a b o v e i d e n t i f i c a t i o n
f r o m two d i f f e r e n t
-D, ,O,D,
and
-Dz ,O,D~
identification (D~)D~ ) b )
experiments respectively. the
linear
being
the
c o m p u t a t i o n s i n t w i c e by u s i n g experiments, taken
One can t h e n o b t a i n
as
with input
PRTSs o f sequence
data
the levels in
the
two g r o u p s o f p a r a m e t e r e s t i m a t e s
two of
dynamics as i=l . . . . . n,
a , " ",
j = l . . . . . m,
h~",
h=l,2
and K(D,)',
L(D~)',
, the estimates Further, parameters
h=l,2
o f K(D, ) , L(D, ) and K(D~ ) , L(D~ ) r e s p e c t i v e l y . the final
estimation
results
f o r a, ,
i = l . . . . . n,
b e , j = l . . . . . m, t h e
i n t h e l i n e a r d y n a m i c s , c a n be d e t e r m i n e d a s
aa, "~ +Ba, "2
i=l,...
a b e " +Bb~" ~ ,
j=l ..... m
,n
and
w h e r e a~O, BzO, a+B=l.
From Fig.2 , parameters PRTS,
it can be easily seen that t h e relation
between L(D), K(D), the
of the preload nonlinearity dependent on the amplitude D of the applied
and k, b, the parameters of the hysteresis nonlinearity
system,
is
as
in
the
follows
k(D-b)=DK(D)+L(D) Furthermore,
consider
respectively,
one h a s
~
identified
(3.3) the
two
different
cases
that
D
equals
to
D+
and
k(D,-b)=D,K(D,)+L(D,) k(D2-b)=D~K(D2)+L(Dg)
(3.4)
D~
512
By
solution
to
the
coupled equations (3.4),
k and b can be
expressed
in
the
following way k=([D1K(D~ )+L(D~ )]-[D~K(D~ )+L(D~ )]) / (D,-D2)
(3.5)
b=(D, Dz [K(D, )-K(D, ) ]+DI L(D~ )-D2 L(D, )) / ([D~ K(D~ )-D, K(D, )]+L(D~ )-L(D~ )) Therefore,
(3.6)
k" and h*, the estimates of k and b, can be determined as
k'=([D~K(D,)'+L(D~)']-[D~K(D~)'+L(D~)'])
/ (D~-Dr)
(3.7)
b" =(D, D, [K(D, )" -K(D, )" ]+D, L(D~ )'-O2 L(D~ )" } / ([DI K(D, )'-D, K(D, )* ]+L(D, )'-L(DI )')
(3.8)
Thus, the parametric identification procedures for system (2.1) are completed.
The
parametric identification problem of the linear dynamics with
dead
zone
nonlinear input also can be dealt with by the above procedures. Obviously, if the nonlinear element in system (2.1) is dead zone nonlinearity, t h e n in t h e o r y one has L(D,)=L(D2)=O
(3.9)
Therefore. providing the parameter estimation results L(D,)" and L(D~)" obtained in the above procedures satisfy the conditions L(D,)'zO,.
L(D~)':O
(3.10)
and I L(Oh)" << b', L(D~)" (( min [k'b', (Dh-b')k*] the
structure
of
the
h=1,2
(3.11)
nonlinear element in the identified
system
should
be
determined concretely to be of the dead zone nonlinear form rather than its general representation -- hysteresis. This fact indicates the versatility of the identification algorithm to account for related models which belong to a subclass of the general representation.
IV. S i m u l a t i o n S t u d i e s Example 1: The l i n e a r dynamics o f t h e f i r s t
example system i s
y(t)-l.Sy(t-l)+O.7y(t-2)f#[u(t-l)]+O.5~[u(t-2)]+~(t) where, t h e s t a n d a r d d e r i v a t i o n o f z e r o mean w h i t e n o i s e ~ ( t ) i s 0 . 1 , r
is
513 the h y s t e r e s i s
nonlinear
0.07 r e s p e c t i v e l y
The
character
p a r a m e t e r s k and b b e i n g 0 . 8
and
section 3 i s used to do parametric identification of
the
.
The algorithm system.
input with its
in
applied
PRTS i s of order number 4,
equation D**=Dx'O, Dx '
different levels
its
signal length 80
are - 1 ,
used
for
parameter
estimation in
each
of
character
I ( D , = l ) and -1.5, O, 1.5
O,
(D,=I.5) in the two identification experiments respectively. T, data
and
the
two
the number of the LS
identification
computations, i s 168.
Table I and 2 p r e s e n t the parameter estimation r e s u l t s for the system by using the
data coming frem experiment I and 2 respectively.
system parameters
in
beth the linear dynamics and the
computed by using the r e s u l t s in Table I and 2
Table. parameters
a,
true values
- 1.5
es t i mates
- i . 5048
parameters
0.7 0.7039
a, - 1.5
estimates
-1.5031
0.7 0.7025
Table. parameters values
estimates
Example first
one.
2
:
Tile
a,
-1.5040
with ,,=~=172, are shown in Table 3.
b,
K(D, )
L(D, )
0.5
0.688
0.056
0.4853
0.6605
O. 0761
b,
K(D, )
L(D2 )
0.5
0.7253
0.056
0.4916
0.7070
0.0762
b,
k
b
0.5
0.8
0.07
3
a,
- 1.5
h y s t e r e s i s nonlinearity,
2 (D,=I.5)
as
true values
the
1 (D,=I)
as
Table.
true
The final estimates of
0.7 0.7032
0.4885
0.8001
0.0792
The l i n e a r
d y n a m i c s o f t h e s e c o n d e x a m p l e s y s t e m i s same a s t h e
nonlinear
element
#[u(t)]
here
is
a
deadzone
nonlinear
514
characteristic,
its
dead zone parameter b and gain
parameter k
are
0.I and 0.8
respectively.
The are
same input sequences and identification algorithm as in the first example
used to excite the
T = 100.
system and to do parameter estimation
The p a r a m e t e r e s t i m a t i o n
identification
experiments
and
d y n a m i c s and t h e n o n l i n e a r i t y
results
the
obtained
final
are presented
by u s i n g t h e d a t a from t h e two
parameter in F i g . 4 ,
computations. Ilere,
estimates
for
the
linear
5 and 6 s e p a r a t e l y .
Table. 4 (D,=I) parameters
a,
a2
true values
- 1.5
0.7
estimates
-1.4980 0.6938
b,
K(D~)
L(D,)
0.5
0.72
0
0 . 5 5 8 2 0 . 6 8 9 4 0.01088
Table. 5 (D~=I.5) parameters
a~
true values
- 1.5
estimates
-1.4975
a=
0.7 0.6949
b,
K(D~)
L(D,)
0.5
0.7467
0
0.5366
0.7265
0.01052
Table. 6
Since (3.11).
the
parameters
a,
a~
true values
- 1.5
0.7
estimates
-1.4978
L(D,)"
b,
0.6944
k
b
0.5
0.8
0.1
0.5474
0.8000
0.1247
and L(D=)" in Table 4 and 5 satisfy the conditions
structure
of
the nonlinearity ~[u(t)]
here
should
be
(3.10)
and
determined
concretely to be deadzone rather than hysteresis. Therefore, the versatility of the identification algorithm is verified by this example.
V.
Remarks on the Theoretical Analysis of the Identification Method and Its Application in Engineering
(i) It can be proved theoretically that under the assumptions (A.1) and (A.2), the
parameter estimates
of system (2.1)
o b t a i n e d by t h e i d e n t i f i c a t i o n
algorithm
515
in section3 are all of consistentpropertyprovidedthatN, appliedPRTS is largert h a n m+n.
u(t)
[
m(s)
O,(t) ~
O,(t)].
the length
6(s)
of t i l e
y(t)
(a) 0
I
u(t)
e, (t)
0
I
J I
a c t u a t o r motor
y(t)
actuator
controlled ] plant
L (b)
Fig.3
(ii)
A Kind of Dynamic System w i t h llysteresis Nonlinearity Existing Widely in Engineering
The system with hysteresis nonlinearity in Fig.3 (a) exists
engineering.
Its
practical
widely in
s t r u c t u r e b l o c k diagram in e n g i n e e r i n g i s as shown
in
Fig.3 (b).
Since m(s), the transfer function of the actuator motor can be known a
priori
practice,
in
the
estimation
of
b,
the
parameter of
the
hysteresis
nonlinearity, can be realized by the identification method proposed in this paper. Thus,
by compensating the influence of the nonlinearity in some way
obtained hysteresis parameter estimate nonlinearity compensation signal
of
b* being as
the
the system, beth
basis
the
of
w i t h the determining
identification and
controller design problems of the system in Fig.3 can be well transformed to problems for
a
linear system. T h i s is a practical example for
the
the
proposed
identification method to bring into play i t s action in engineering.
References
[1].
R. V. J a t e g a o n k a r , P a r a m e t r i c I d e n t i f i c a t i o n of Discontinuous Nonlinearities, Preprints of 7th IFACSymposium on Identification and System Parameter Estimation, York, UK, pp 167-172 (1985).
[2]
Gu Xing-yuan, Dao Yu-an and Lang Z i - q i a n g , P a r a m e t e r I d e n t i f i c a t i o n o f a C l a s s o f D i s c r e t e Time N o n l i n e a r S y s t e m s , P r e p r i n t s o f 12th IMACS World C o n g r e s s on S c i e n t i f i c Computation, P a r i s , F r a n c e , pp 627-629 (1988).
A p r i o r i e s t i m a t e s for o p e r a t i o n a l d i f f e r e n t i a l i n c l u s i o n s a n d n e c e s s a r y c o n d i t i o n s for o p t i m a l i t y H a l i n a Frankowska Centre N a t i o n a l d e la l : t e c h e r c h e S c i e n t i f i q u e , F R . A N C E C E R E M A D E , U n i v e r s i t d P a r i s - D a u p h i n e , 75775 P A K I S a n d I n t e r n a t i o n a l I n s t i t u t e for A p p l i e d S y s t e m s A n a l y s i s , A U S T I ~ I A
Abstract Derivation of necessary conditions for optimality in optimal control theory relies on variational calculus of control systems. We provide here such calculus for the semilinear differential inclusion x' e A x -t- _F(t,x), x(O) = Xo
(1)
where A is the infinitesimal generator of a Co-semigroup o n a separable Banach space X and F : [0, T] x X -,~ X is a set-valued map. Such inclusion encompass a number of control systems, including those with state dependent controls. We prove a relaxation theorem and investigate infirdtesimal generators of reachable sets and variationM inclusions. R.esuits are applied to derive necessary conditions for optimality for some sen'dlinear optimal control problems.
Introduction We are concerned with the multivalued operationM equation (differential inclusion) (1), where A is the infinitesimal generator of a C0-semigroup (G(t)}t>0 on a separable Banach space X and F : [0, T] • ,\" .-.z X is a set-valued map. Such inclusion is a convenient tool to investigate for instance the semilinear control system =' e A= + / ( t , x , u ) , u e U(t,x), x(0) = z0 (2) where U : [0, T] x X -,-* X is a set-valued map of controls (depending on the time and on the state). Setting
F(t, x) = f(t, x, U(t, z)) we reduce the above control system to the inclusion (17. Differential inclusion
='(t) e F(t,=(t))
(3)
in fiMte dimensional context was extensively studied since the 30'ies. Filippov and Wa~ewski [19] have shown that under very mild assumptions the control system
x' = f(t,x,u(t)),
u(t) e U is measurable
(,17
may be reduced to the differential inclusion (3). This helped to study the closure of trajectories to (4) and led to the celebrated Filippov-Wa~ewski relaxation theorem. Kecently it was generalized in [20] to unbounded differential inclusions in nonseparable Banach spaces. Control system (2) with state-independent control subset U can be considered as a family of differential equations: with every control u(.) (measurable selection of U) one can associate the semilinear diffcrentiM equation
z' = A= + ~(t,=) where ~ , ( t , x ) = f ( t , x , u ( t ) ) . expression.
Differential inclusion (17 unifies all solutions of such family in one single
520 It is natural to wonder why to perform such changing. We provide here conditions under which solutions of (1) and (2) do coincide, lIence, by so doing, we do not get additional solutions. We do lose the inputoutput representation of our system. So once results about differential inclusion (1) are derived, they should be translated using the initial representation involving dynamics f and the map of controls U(., .). W h a t are the advantages? On one hand differential inclusions admit the calculus very similar to the one of differential equations, which can be exploited in the framework of control systems. On the other hand this allows a unified approach to various systems. Indeed differcntial inclusion also encompass much more sophisticated control systems: 1. dosed loop control systems
x'(t) = Ax(t) + f ( t , x ( t ) , u ( t ) ) ,
u(t) E U(t,x(t))
(5)
= O, u(t) E U(t,x(t))
(6)
2. implicit control systems
f(t,x(t),x'(t)-Ax(t),u(t)) 3. systems with uncertainties
x'(t) E Ax(t) + f ( t , x ( t ) , u ( t ) )
4- E(t,x)B,
u(t) e U(t,x(t))
(7)
where E(t, x) is a function incorporating errors of the model. Setting F(t, x) = f(t, x, U(t, x)) in the first case, F(t, x) = {v + Ax I O E f(t, x, v, U(t, x))) in the second one and F(t, x) -- f(t, x, U(t, x)) + v(t, x)B in the third one, we replace the control systems (5)-(7) by the dilTerential inclusion (1). We refer to [1], [7], [4], [8]-[13] for many results on differential inclusions and their applications to control theory for finite dlmensior.al control systems. In this paper we study (1) and its mild trajectories. Wc start Section 1 by a theorem analogous to the Filippov theorem [7], a kind of set-valued Gronwall's lemma. This allows to prove a relaxation thcorem for (1). Namely, under some technical assumptions, the mild trajectories of (1) are dense in the mild trajectories of the convex]lied inclusion:
x'
G
Ax + "~ F ( t ,x),
x(O) = xo
In Section 2 we investigate infinitesimal generators of the reachable map a~sociated to (1) and the variationai inclusion for (1) and provide a comparison of solutions of differential inclusions a~ld control systems. We prove in Section 3 necessary conditions for optimality for some infinite dimensional optimal control problems. 1lesults in this direction were already obtained in [15]. We relax here some assumptions and provide further applications.
1
Quasitrajectories and convexified dynamics
Consider a separable Banach space X and a strongly continuous semigroup G(t) E s t _> 0 of bounded linear operators from X to X having the infinitesimal gcnerator A. Let 0 _< to < T be given and F be a set-valued m a p from [to, T] • X into closed nonelapty subsets of X. We associate with it the differential inclusion x'(t) e Ax(t) + F ( t , z ( t ) ) (8) Denote by C(to,T;X) the Banach space of continuous maps from [t0,T] to X with the norm [Ixilr = sup~e[~o,T] Iix(t)ll and by Z~l(t0, T; X ) the Banach space of Bochner integrable maps (see for instance [16, i).78]) from [to, T] to X with the norm []xlil = ftTo ]lx(t)lidt. Set Ll(to, T) --- Z:l(to, T; ]EL+). A continuous map x G C(to,T;X) is called a mild trajectory of (8) from the initial condition x0 E X if there exists a Bochner integrable map f E s X ) such that
f(t) E F(t,x(t)) a.e. in [t0,T]
(9)
521 and Y t e [to, T], x(t) = a ( t - to)Xo + i.e. f is a Bochner
integrable selection of the set-valued map
]~ia(t -
.~)f(s)ds
(I0)
t ~,z F(t, z(t)) and x is the mild solution of
the initial value problem
x'(t) = A:~(t)+ f ( t ) ,
t ~ [to,T],
x(to) = ~o
(11)
We shall call (x, f ) E C(to, T ; X ) x s (to,T; X ) a trajectory-selection pair of (8) if f verifies (9) and x is a mild solution of (11). Recall that a set-valued map ~ : X -,~ X is called L-Lipschitz on K C X if for all x E K, O(x) it 0 and for all x , y E K , ~(x) C O(y) + L[Ix - y[IB, where B denotes the dosed unit ball in X. Throughout tile whole paper we shall consider the Lebesgue measure on 1~. T h e o r e m 1.1 ([3]) Let U : [t0,T] "-* X be a set-valued map with closed nonempty images. Then the following statements are equivalent i) - - U is measurable: for every open set (9 C X the set {t E [to,T] I U(t) M (9 ~ ~} is measurable. it) - - There exist measurable selections un(t) e U(t) such that for every t E [to,T], U(t) = U~>>.lun(t) . Moreover if U is single-valued, then the above statements are equivalent to iii) There exist a negligible (i.e. of measure zero) set .M C fro, T] and measurable single-valued maps u,, : fro,T] ~-* X assuming only finite number of values such that u,~ converge pointwise to U on [10, T]\Af. Consider the solution set of (8) from the point x0 E X on fro,T]: ,S[to,T](xo ) = {x I x is a mild trajectory of (8) on [to,T], x(to) = xo} Let Yo E X , g E s
and y E C(to, T ; X ) be a mild solution of the Cauchy problem y'(t) = Ay(t) + g(t),
y(to) = yo
We estimate the distance from y to the set ,S[to,~(xo) under several assumptions on F:
/
H~) Vx E X the set-valued map F ( . , x ) is measurable I12) 3 fl > O, k E f_fl(to,T) such that for almost all t E [to,T] the map F(t,.) is k(t) - Lipschitzian on y(t) + f i b Ha) The function t~-.~ dist(g(t),F(t,y(t))) belongs to s )
From [2, Theorem 8.2.8 and Corollary 8.2.13] we deduce that the function t ~ dist(g(t), F(t, y(t))) is measurable whenever assumptions H1), II~) are satisfied. Theorem
=
1.2 ([15]) Let 6 > 0, M = sup~e[o,T_to ]HG(t)[I. Assume that l i t ) - II3) hold true and set
re(t):
(. Z
,(t)= re(t)(, + Z (s)ds)
If , ( T ) < fl, then for all xo E X with I[yo-xo[I <_ 5 and all r > O, there exist x e S[to.T](X0) and f Es T; X ) satisfying (g), (10) such that for all t E [to, T] fix(t)
-
y(t)l I <
71(t ) + s(t - to)re(t)
and for almost every t E [to,T], l[f(t) - g(t)[[ _< k(t)(~(t) + r
- to)re(t)) + 7(t) + e.
Remark -W h e n X is a finite dimensional space, the above estimation holds true with r = 0. This follows from the Filippov theorem [7]. [] Consider the following norm on C(to, T; X ) • s
v ( ~ , f ) ~ C(to, T ; X ) x L~(to,T;X),
T; X):
II(~:,f)llc•
= IIxltc + tlfll~
522 C o r o l l a r y 1.3 ( L i p s c h i t z d e p e n d e n c e o n t h e i n i t i a l c o n d i t i o n ) Let (y,g) be a trajectory-seleclion pair of (8) on [to,T] and assume that F, y satisfy H1) - 113). Then there exists L > 0 such that for all 71 near y(to) we have
d i s t c x r ((y,g), { ( x , f ) is a trajectory-selection pair o[(8) on [to,T] ,x(to) = ~}) _< LILT]- Y(to)]l We compare next trajectories of (8) and of the convexifted (relaxed) differential inclusion:
z'(t) e Ax(t) + ~'~ F ( t , z ( t ) )
(12)
We say that a set-valued map U : [to,T] "-~ X has an integrable selection if there e ~ s t s an integrable single-valued map [to,T] ~ t ~ u(t) E U(t). T h e o r e m 1.4 Let (y,g) be a trajectory-selection pair of the relaxed inclusion (12) on [to,T]. Assume that F and y satisfy all the assumptions of Theorem 1.2 and that F(.,y(.)) has an integrable selection. Let ~](.) be defined as in Theorem 1.2. If rl(T ) < fl then for every s > 0 there exists a mild trajectory x of (8) on [to,T] satisfying x(t0) = y(to) and Itx - Y][c <- r To prove the above we need T h e o r e m 1.5 Let U : [to,T] "-* X be a measurable set-valued map with closed images having at least one integrable selection. Then for every r > 0 and intcgrable selection ~(t) E ~ U(t) there exists an integrablc selection u(t) E U(t) such that sup tE to,T]
[tG(t-s)u(s)dsJt~to
ftG(t-s)'ff(s)ds
< e
(13)
Jr0
Proof-Fix ~ > 0, an integrable selection ~(~) E ~ U(t) and let no(.) be an integrable selection of U(.). Define measurable set-valued maps U,~ : [to, T] ~ X with closed nonempty images by V t E [t0, T], U,~(t) = uo(t) U ( U ( t ) N n B ) aad set r = dist(-ff(t), -d6 U,~(t)). Then the sequence {E,(-)}=>I is integrably bounded. Furthernmre lim,~_oo r = 0 almost everywhere in [t0,T]. Using [2, Corollary 8.2.13] we deduce that for every n > 1 tl, ere e . ~ t s a measurable ~ele~t~on ~ ( t ) C ~ U~(t) ~uch that for almost every t e [to,T], llu~(t) - ~ ( t ) I I < r + 88 Therefore u,~ converge to ~ in El(to,T; X ) . Set M = supte[o,T_to] [[G'(t)i [ a~d observe that for all large n
sup
a(t-s)u~(s)ds-
aCt -
f
_< M sup ~ I I ~ ( s ) - ~ ( s ) l l d s _< ~
t elto,T]
~e[t0,T] J to
It remMns to show that for every n > 1 there exists an integrable selection u(t) E U,,(t) C U(t) such that sup
tE[to,T] Jto Fix n >_ 1 aa~d observe that the set-valued map U,~ is integrably bounded. Hence to deduce this last inequality we may apply the R.ela~xation Theorem 2.1 from [15] to end our proof. [] Proof of Theorem > 0 and let
1.4 - -
0 < a < rain
It is not restrictive to assume that to = 0 and that f T k ( t ) d t > 0. Fix
2 m ( T ) f o r k ( t ) d t ' 2m(T)(fs
' 2'
From Theorem 1.5 we deduce that there exists a measurable selection f ( t ) E F(t, y(t)) such that sup ~ot G(t - s ) f ( s ) d s - /o t G ( t te[0,T]
s)g(s)ds
~ a
523 Set xo(t) = G(t)y(O) + fo G(t - r ) f ( r ) d r . Then for every t e [0,T], Hxo(t) - y(t)H _~ a. Observe that for all t E [0, T], F ( t , . ) i s k(t)-Lipschitz on x o ( t ) + ~ B . tIence d i s t ( f ( t ) , F ( t , xo(t))) <__k(t)a. Furthermore, by the choice of a, M e x p ( M foT k(s)ds) foT k(t)adt < ill2. By Theorem 1.2 applied with s = a there exists a trajectory x of (8) ~atisfying x(0) = Xo(0) = y(0) and
I . this way we obtain Ilx - YI[c -< [Ix ' xollc + Ilxo - ylk < c/9- + c/2 = e.
[]
The above implies the following generalization of a result from [15]. Let S~~ trajectories of (12) defined on the time interval [to, T] and satisfying x(to) = xo.
denote the set of mild
T h e o r e m 1.6 (1R.elaxation t h e o r e m ) Let F : [to,T] x X ...* X be a measurable with respect to the first variable set-valued map with closed nonempty images and let xo E X . Assume that the function t ~, diat(O,F(t,O)) is integrable and that there exists k E L t ( t o , T ) such that for almost every t E [to,T], F(I, .) is k( t)-Lipschitz. Then the mild trajectories S[to,T](X0) of (8) starting at Xo are dense in the mild trajectories starting at xo, S[~~ of the relaxed inclusion (12) in the metric of uniform convergence. 2
Generator
of reachable
map
and
variational
inclusions
Consider a set-valued map F : [to,T] x X ---* X with closed images, where 0 < to < T. For all t E [to,T[ and t < t ' < T , x o E X set a ( t ' , t ) x o : { x(t') I x e SEt,t,j(xo ) } (14) This is the so-called reachable set of (8) from (t, xo) at time t'. T h e o r e m 2.1 Assume that F : [to,T] x X ",* X is continuous, has closed nonemply images and for some > O, k E s and for almost every t E [to,T], F(t,.) is k(t)-nipschitz on xo + 5B. Then for every u e eaF(to,xo) dist ( a ( h > o + hu, ~(to + h, to)xo) = o(h) whcre limh~o+ o(h)/h = O. Fix u e ~ F(to, x0) and set y(to + h) = C(h)zo + f,~+t, a ( t o + h Set r = sup,e[O,h] dist(u, ~6F(to + s, y(to + s))). Then, by continuity of F , Theorem 1.2 yield that for some C > 0 and every small h > 0, there exists ]ly(to + h) - xh(to + hll < Cr Applying Theorem 1.4 we end the proof.
Proof--
s)uds = a ( h ) x o + l~u + o(h). limb-o+ ~(h) = 0. This and co X xh E $[to,to+h](o) such that El
C o r o l l a r y 2.2 Under all assumptions of Theorem 2.1 ~6 -/;'(to, Xo) C lim inf /~(to + h, to)xo - G(h)xo h~o+ h
Consequently, if xo E Dora A, then Axo + e-O F(to,xo) C liminf R(to + h, t o ) x o - xo h~o+ h We introduce next an analog of the variational equation for differentia] inclusions. For this we need to extend the notion of derivative to set-valued maps. D e f i n i t i o n 2.3 Let ~ be a set-valued map from a Banach space X to another Y and lel y E .T(x). d e r i v a t i v e d ~ ( x , y) is the set-valued map from X to Y defined by
v e dT(x,y)(u)
~=.
hn~+_ d ( "~,g ( x + h u h )h- - Y )
= 0 for some uh -+ U
When .T is locally Lipsehitz at x then the above definition may be rewritten as v e d.T(x,y)(u)
~
lim dist h~O+
v,
= 0
The
524 We refer to [2] for properties and to [8] - [12] for the applications of set-valued derivatives in the finite dimensional context. Below dF(t; x,y) denotes the derivative of the set-valued map F(t,., .), i.e. its partial derivative with respect to the state variable. Let (y,g) be a trajectory-selection pair of the differential inclusion (8) defined on the time interval [t0,T]. We "linearize" (8) along (y,g) replacing it by the " v a r i a t i o n a l i n c l u s i o n " :
w'(t) E Aw(t) + dF(t;y(t),g(t))(w(t)), w(to) = u Consider the solution map Sc,z from X to the space C(to,T;X) • s
(15)
T ; X ) defined by
5cx(~) = { ( z , / ) i s a trajectory-selection pair of (8) on [t0,T], z(t0) = ~ } The following result was proved in [15]. T h e o r e m 2.4 ( V a r i a t i o n a l i n c l u s i o n ) If F and y verify assumptions HI) - It3), then every trajectoryselection pair (w,~r) of the linearized inclusion (15) on [t0,T] satisfies (w, ~) E dSc,L(y(to),(y, g))(u). A stronger result may be proved when the map t ~ F(t,y(t)) contains an integrable selection. Considcr the "convex" linearization of (8) along (y,g):
w'(t) E Aw(t) + d~-6 F(t;y(t),g(t))(w(t)), w(to) = u
(16)
In the theorem stated below we consider the solution map Sc(() = 3[t0.Tl(~) as the set-valued map frmn X to the space C(to, T; X). T h e o r e m 2.5 Under all assumptions of Theorem 1.2 assume that F(.,y(.)) has an integrable selection. Then every mild trajectory w of the lincarized inclusion (16) defined on [to, T] satisfies w E d Sc(Y( to), y)(u). Proof--
By Theorem 1.4 we may replace F by ~ F . Then the result follows from Theorem 2.4.
[]
The derivative of the set-valued map ~ F(t,x) has the following useful property: If F(t,.) is locally Lipsctfitz on a neighborhood of x, then for every y E F(t,x)
dF(t;x,y) + T~F(t,~)(Y) C d-~'hF(t;x,y)
(17)
where T-~f(t,~)(y ) denotes the tangent cone of convex analysis to "~F(t, x) at y. See [15] or [2] for the proof. We provide next two comparison theorems for trajectories of the control system (2) and the differential inclusion (1). Let Z be a complete separable metric space, X be a separable Baaach space and f : 1Z+ x X • Z ~ X be such that for all (x, u) E X • Z the map f(-, x, u) is measurable and for every t E 1%+, f(t,., .) is continuous. Consider a set-valued map U : It+ x X .-~ Z with closed nonempty images and define the set-valued nlap
F : IZ+ • X...* X by V ( t , x ) ~ It+
x
X, F(t,z) = I ( t , x , U ( t , z ) )
T h e o r e m 2.6 If U is measurable with respect to t and continuous with respect to x, then solutions of (1)
and (2) do coincide. Proof-It is enough to show that for every trajectory-selection pair (x,v) of the differential inclusion (1) defined on [t0,T] there exists a measurable selection u(t) E U(t,x(t)) such that for almost every t E [t0,T], v(t) = f(t,x(t),u(t)). By [2, Theorem 8.2.8] the map t ",-* U(t,x(t)) is measurable and for every u E Z the single-valued map f(.,x(.),u) is measurable. Applying [2, Theorem 8.2.9] we end the proof. [] We observe that in general the images of the map F defined above are not closed, while the calculus of differential inclusions developed in previous sections deals only with closed-valued maps. For this reason we provide one more comparison theorem. Denote by s the set of maps f : It+ ~ X such that for
all 0 <_to <_T, f E s
T;X).
525 T h e o r e m 2.7 Assume that U does not depend on x and is measurable with respect to t. Further assume
that there exists k E ~.~o~(IL+) such that for every (t, u) E 1Z+ x Z, f ( t , . , u) is k(t)-Lipschitz and that the set-valued map t".., f(t, O, g(t)) has a selection v E s X ) . Set F ( t , x ) = f(t, x, V(t)). Then for every solution x of (1) defined on the time interval [to,T] and e > 0 there ezists a solution y of (2) such that 9 (to) = y(to) and for every t C [to, T], II=(t) - y(t)ll < E. Proof-Fix a trajectory-selection pair (x,g) of (1) defined on [to,T]. By [2, Theorem 8.2.9] and our assumptions there exists a measurable selection uo(t) E U(t) such that v(t) = f(t, O, uo(t)) almost everywhere in [t0,T]. By the Lipschitz continuity, f(.,x(.),uo(.)) e s From Theorem 1.1 ii) there exist measurable selections u,~(t) e U(t) such that for every t e [to,T], U(t) = lJ,,>l u,~(t). Define measurable set-valued maps G,, : [t0,T] • X --* X with closed nonempty images by G-~(t) = U~=of(t,x(t),ui(t)) and set en(t) = di~t(g(t),G,,(t)). Then e,,(-) are measurable, integrably bounded and for almost every t E [to,T], limn~c~e,(t) = 0. Let g~(t) E G~(t) be such that dist(g(t),G,~(t)) = lig~(t)- g(t)l I. By [2, T h e o r e m 8.2.9] there exist measurable selections ~n(t) E U(t) such that g,~(t) = f(t,x(t),~r~(t)) almost everywhere in [to,T]. For every n _> 1 consider the semilinear system
y'(t) = Ay(t) + f(t,y(t),~,~(t)),
y(to) = z(to)
and let y~(-) denote the corresponding solution. By Theorem 1.2 there exists C > 0 such that for all n > 1
sup
tfi[to,T]
II~(t) - y:(t)ll
Thus y,, converge to x.
_< c
/,;"
Ila(s) - f(s,~(s),~,,(s))ll
ds = C
j:
~:(s)ds
[]
Remark -Naturally a similar result can be proved when U depends on z as well. But in this case we have to assume Lipschitz continuity of f with respect to u and of U with respect to x. [] 3
Application: semilinear opt i m a l control problems
Let Z be a complete separable metric space, X be a separable Banach space and f : IL+ • X x Z ~ X be such that for all (x, u) E X • Z the map f ( . , x , u) is measurable, for every t E I~+, f ( t , . , . ) is continuous and for every (t, u) E 1Z+ • Z the map f ( t , . , u) is differentiable. Consider a measurable set-valued map U : R + --* Z with closed nonempty images. We assume: a) There exists k E L~oc(R.+) such tha~ for almost every t E tZ+ and for all u E U(t), k(t)-Lipschitz, i.e., v z', ~" e x , V u e U ( t ) ,
b) There exists v E s
IIS(t,x',u)
-
f(t,z",u)ll
f ( t , . , u ) is
< k(t)I1~' - ~"ll
such that for edmost every t E tL+, v(t) E f(t,O,U(t)).
Let K C X . Recall that the contingent cone and the Dubovitskij-Miljutin tangent cone to K at x E K are defined by
TK(x) = llmsup h~O+
K-x ~z
, DK(x) = { w G X l 3 e > O
suchthat Vhe[O,e], z+hB~(w) CK}
respectively. See [2, Chapter 4] We apply results obtained in the previous sections to derive necessary conditions for optimality for three optimal control problems. Consider a continuous, Gs differentiable function ~p : X • X ~ R. aald closed subsets K0, K1 C X. For every T > 0 set UT = {u : [0,T] ~ Z I u(t) e U(t) is measurable}.
526 3.1
Problem
with fixed end time and free end point
Consider T > 0 and the optimal control problem minimize qo(x(0), x(T))
(18)
over mild solutions of the semilinear control system
z'(t) = Am(t) + f(t,x(t),u(t)), u E LtT, z(O) E I(o
(19)
Let (z,~) be a trajectory-control pair of (19). We associate with it the linear equation
z'(t) = AZ(t) + ~(t,z(O,~(t))z(t)
(20)
and denote by S~(t; s) the solution operator of (20). That is the only strongly continuous solution of the operator equation
VO<s
S~(t;a)~ = S ( t - s ) ~
+
S(t-a)
(~,z(a),g(~))Sg(cr;s){da
T h e o r e m 3.1 Let (z,g) be an optimal lrajectory-control pair of the conlrol problem (18)-(19). Then lhe
map
p(t)
=
-S~(T; t)* ~-~ (z(0), ~(T))
(21)
satisfies the maximum principle: < p(t), f(t,z(t),-~(t)) > = max < p(t), f(t,z(t),u) > a.e. in [0,T] ,~eU(t)
(22)
and the transversality condition: p(O) e ff'~xl(Z(O),z(T)) + TKo(z(O))-
(23)
where Tlco(z(O))- denotes the negative polar cone to the contingent cone Ti(o(z(0)). ProofDefine the set-valued map F : [0,T] x X -,~ X by F(t,x) = f(t, x, U(t)). From [2, Theorem 8.2.8] follows that F is measurable with respect to t. By the assumption a) it is also k(t)-Lipschitz with respect to x. Consider the differential inclusion
x'(t) E Ax(t) + F(t,x(t)), z(O) E Ko
(24)
By Theorem 2.7 and continuity of ~v, the trajectory-selection pair (z(.),f(.,z(.),~(.))) is optimal for the problem (18), (24). Thus we may replace the control system (19) by the differential inclusion (24). Consider the linear control system w'(t)
=
Aw(t)+ ~
y(t)
e
Tazi(t,~(t),cr(,))(f(t,z(t),~(t)))
w(O) E TKo(z(0)) (25)
The reachable set/~L(~) of (25) by the mild trajectories from ~ at time T is given by RL(() =
{
S~(T;0)~+
/;
S~(T;s)y(s)ds l Y E s
e T~/(,M,),u(,))(f(s,z(s),g(s))) a.e.
}
From Theorem 2.5 we deduce that every mild trajectory w E C(0, T; X) of the "linear" differential inclusion
w'(t) E Aw(t) + d'd-S F(t;z(t),f(t,z(t),g(t)))(w(t))
527
on [O,T] verifies w E dSc(z(O),z)(w(O)). From the definition of the derivative for almost every t E [0,T]
V w E X , ~x(t,z(t),-ff(t))w E dF(t;z(t),f(t,z(t),~(t)))(w) I[ence, using (17) we deduce that every solution w E C(O,T;X) of the linear control system (25) verifies w E dSc(z(0), z)(w(0)). Since z is optimal, using Corollary 1.3 we deduce that for every trajectory w of (25) we have (U(p(z(0),z(T)),(w(0), w(T)) I >_ O. The last inequality yields that for every integrable selection y(t) E "d'5f(t,z(t), U(t)) and every ~ E T1r we have
Define p(.) by (21). Setting y(t) = f ( t , z ( t ) , g ( t ) ) i n the above we get p ( 0 ) - -~7(z(O),z(T)) E TK0(z(0))-. llenee the transversality condition (23) holds true. On the other hand applying (26) with ~ = 0 we obtain: for every integrable selection y(t) E ~ f(t, z(t), U(t))
fo T (3~(T; t)*p(T), y(t)) dt <_ ~0T (S~(T; t)*p(T), f(t, z(t),~(t))) dt llence sup~,eu(t) < p(t), f(t, z(t), u) > = < p(t), f(t, z(t),g(t)) > almost everywhere in [0, T]. This yields the ma.ximum principle (22). t::l 3.2
Problem
w i t h free e n d t i m e a n d f r e e e n d p o i n t
Consider the optimal control problem minimize { (p(x(O),x(T))iT > 0 }
(27)
over mild solutions of the semilinear control system (19). T h e o r e m 3.2 Let (z,~) be an optimal trajectory-control pair of the above problem and T denote the cor-
responding optimal time. Then the same conclusions as in Theorem 3.1 are valid. Furthermore if f(.,x, u) and U are continuous at T and z(T) E Dora A, then sup (p(T),f(T,z(T),u))
~ (p(T),-Az(T))
~eu(o P r o o f - - By the proof of Theorem 3.1, z is optimal for the problem (27), (24) and it is enough to prove the last statement. By Corollary 2.2 and optimality of z, for every u E ~5 F(T, z(T)), 0~2 (z(0), z(T))(Az(T)+ u) _> 0. tfence the result. [] 3.3
Problem
w i t h fixed e n d t i m e a n d e n d p o i n t c o n s t r a i n t s
Consider T > 0 and the optimal control problem minimize { V(x(O),x(T)) [ x(T) E Kx }
(28)
over mild solutions of tile semilinear control system (19). T h e o r e m 3.3 Let (z, ~) be an optimal trajectory-control pair and let Q C DKr(z(T)) be a convex cone with nonempty interior and P C T1~'o(z(0)) be a convex cone. Further assume that for every (t,x) E [0, T] x X
the set f ( t , x , U ( t ) ) is closed. Then there exist A > O, ~o ~ P - , ~T E Q- not vanishing simultaneously such that the map pit)=
S~T;t)* (-)~ ff--~x (z(O),z(T)) - ~T)
(29)
satisfies the maximum principle (22} and the transversality condition (p(0), -p(T)) The proof can be found in [15].
=
~V~(z(0),z(T)) + (r
~r)
(30)
528 R,eferences [1] AUBIN J.-P. ~: CELLINA A. (1984) DIFFERENTIALINCLUSIONS.Springer-Verlag, Grundlehren der Math. Wissenschaften, Vol.264 [2] AUBIN J.-P. & FI~ANKOWSKA H. (1990) SET-VALUED ANALYSIS,Birkh~,user, Systems and Control: Foundations and Applications [3] CASTAING C, & VALADIEI~ M. (1977) CONVEX ANALYSIS AND MEASUR.ABLEMULTIFUNCTIONS. Lecture Notes in Mathematics, n ~ 580, Springer Verlag, Berlin [4] CLAR,KE F. (1983) OPTIMIZATIONAND NONSMOOTII ANALYSIS.Wiley Interseience [5] DUNFOR, D N. & SCIIWAI~TZ J.T. (1967) LINEAIL OPERATORSPart I: General theory. Interscience Publishers, Inc., New York [6] FATTORINI II. ~z FR,ANKOWSKA H. (to appear) Necessary conditions for infinite dimensional control problems. Mathematics of Control, Signals, Systems [7] FILIPPOV A.F. (1967) Classical sobttions of differential equations with multivalued right hand side. SIAM J. Control & Optimization, 5, 609-621 [8] FR,ANKOWSKA H. (1987) The maximum principle for an optimal solution to a differential inclusion with end point constraints. SIAM J. Control ~z Optimization, 25, 145-157 [9] FR,ANKOWSKA II. (1987) Local controllability and infinilesimal generators of semi-groups of set-valued maps. SIAM J. Control ~: Optimization, 25,412-432 [10] FI~ANKOWSKA ]I. (1989) Set-valued analysis and some control problems. Proceedings of the International Conference 30 years of Modern Control Theory, Kingston, June 3-6, 1988, E.R,oxln Editor, Marcel Dekker [11] FI~,ANKOWSKA H. (1989) Local controllability of control systems with feedbacks, Journal of Optinfization Theory and Applications, 60, 277-296 [12] FI~ANKOWSKA tI, (1989) Contingent cones to reachable sets of control systems. SIAM J. Control &: Opt. 27, pp. 170-198 [13] FR.ANKOWSKA lI. (1989) Optimal trajcctories associated to a solution of contingent lIamillonJacobi equations. Applied Math. & Optim., 19, pp. 291-311 [14] FII.ANKOWSKA II. (to appear) Some inverse mapping theorems, Ann. Inst. IIenri Poincar~, Analyse Non Lin~aire, [15] FR,ANKOWSKA II. (to appear) A priori estimates for operational differential inclusions, J. Diff. Eqs. [16] HILLE E. & PIIILLIPS R,.S. (1957) FUNCTIONALANALYSISAND SEMI-GKOUPS.American Mathematical Society, Providence, l~hode Island [17] PAZY A. (1978) SEMI-GR,ouPs OF LINEAI~.OPER.ATOItSAND APPLICATIONSTO PARTIALDIFFEItENTIAL EQUATIONS.Springer, Berlin [18] TOLSTONOGOV A.A. (1986) DIFFERENTIALINCLUSIONSIN BANACfl SPACES. Nauka, (in Kussiaa) [19] WAZEWSKI T. (1963) On an optimal control problem. In: Differential Equations and Appfications, Proc. Conf. Prague, 1962 [20] QIJI ZIIU (to appear) On the Solution Sets of Differential Inclusions in Banach Space, J. Diff. Eqs.
Relaxed Controls for Time Delay Systems by l's
B. Vinter
(Department of Electrical Engineering, Imperial College, Exhibition Road, London SW7 2BT, England)
Despite the attention timt optimal control problems involving time delays have received over the last two decades, little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers. Recently Warga has proposed a relaxation procedure for fully nonlinear problems with delays in the dependent variables and in the controls, and showed that the resulting relaxed problem has a solution. We show, through an example that this relaxation procedure can fail to give a p r o p e r e z t e n s i o n , i.e. the effect of relaxation can be to reduce the infimum cost. Sufficient conditions are given for a proper extension. A new relaxation prot:edure is provided for which the extension is proper in certain situations where the former extension is not,. Abstract:
1. Introduction. There is a substantial literature treating optimal control problems with time delays. In a number of respects our understanding of problems with time delays matches that of delay free problems. Wc note in particular that necessary conditions of optimality, akin to the Pontryagin Maximum Principle for delay free prob[enls, have also reached a high level of refinement [4] for problems with time delays. Itowcver one important strand in the delay free theory is the study of relaxation procedures and of conditions for c•
of minimizers. This is an area which, in the context of time delay problems,
has received very little attention. Our objcct here is to describe some of the difficulties involved, and to indicate how, in some respects, they can be overcome. The optimal control problem we address is Minimize g(x(1)) subject to
(e)
x(t) = f(t,x(t), u(t), u(t-01), u(t-02)),
a.e. [0,11,
x(0) = •
u(t) ~ ~,
a.e. [--02,11.
The data associated with this problem are: real numbers 0 < 0i.< 0 2 < 1, a point x0E Rn, a set f~ C Rm and functions g: Ru
R f R 1+n Win3
Ru
We have chosen this problem for simplicity of exposition, and to keep notationM complexity to a minimum.
No difficulties arise in extending the theory to follow to apply to control problems where
delay terms are present also in t.he state variable. We could also have looked at problems involving more than two delays. Our reason for clmosing precisely two delays is that cert,ain difficulties encount.ered with two delays typify those for several delays, and are not shared by problems wit.h one delav. Notice lhat we have adopted dynanfical equations where there isageneral .onlinear dependence
530 on u(t), u ( t - - 0 t )
and
u(t--02). (In the special case that the dependence is 'separable',i.e. f is
expressible as f(t,x(t), u(t), u ( t - 0 1 ) , u ( t - 0 2 )
= fl(t,x(t), u(t)) + f2(t,x(t), u(t--01) ) + f2(t,x(t), u(t--O2)
the task of finding an appropriate notion of relaxation is a simple adaptat, ion of standard delay free theory). T h e hypotheses we impose are f(-, x, u0, Ul, u2) is measurable and f(t, x,.,.,.) is continuous, f2 is compact, g is continuous, 3 a constant c and a function k(-) E L 1 such that Ill(t, x, u o, u 1,
u2)ll _< c(
t + Ilxll), for all (t, x, (u0, ux, u2) ) E [0,1] • R n x f23 , and
Hf(t, x, u 0, Ul, u2) - f ( t , y, u0, Ul, u2)l[ < k(t) l l x - y l l for all (t, (u0, ul, u2) ) E [0,1] x f23 and x , y E N n. W a r g a [5] was the first, to call attention to the speciat difficulties encountered in addressing existence questions for problems with non-separable time delays in the control. relaxation procedure ( 'weak' rela.xation, as we shall call it,).
Warga also proposed a
The contributions of this article are,
firstly to illustrate through an example that weak relaxat.ion can have the undesirable effect of reducing the m i n i m u m cost (in other words weak relaxation m a y fail to be proper), to give a simple testable criterion for properness of weak relaxation and, finally, to supply a more refined notion of rela~,:ation which is proper in certain situations where this criterion fails.
2. Delay Free Problems. For purposes of comparison we briefly review a standard procedure for relaxing delay free problems. Throughout this section then we look at Minimize g(x(1)) subject to
.~(t) = f(t,x(t), u ( t ) ) ,
a.e. [0,q,
x(0) = x0, u(t) e f~,
a.e. [0,1].
The hypotheses on the data are those of the previous section, when tile delay free problem is regarded as a special ease of (P). A measurable function u: [0,1] ~
R m satisfying u(t) E f2 a.e. is called an ordinary control. A
pair (x(.), u(-)), comprising an ordinary control u(.) and an absolutely continuous function x(-) which satisfies the differential equation is called an ordinary process. called admissible.
If x(0) = x 0 the ordinary process is
The optimization problem, posed over admissible ordinary processes, is called the
original problem (Poriginal)"
Two questions now arise. When does rite original problem have a
minimizer and. in the event that there is no minimizer, how can we supplement the class of admissible ordinary processes to ensure existence of a minimizer? 13orb have very satisfactory answers. As for the first question, a simple geometric condiLion "f(q x, .Q) is convex for all (t, x) E [0,1] x ~n,, is I~nowll to assure existence of an optimal admissible ordinary process (See, e.g. [3]). An existence
531 theory, applicable for nonconvex 'velocity sets' f(t, x, f~), involves introduction of the notion of relazed process.
A relazed control is a measurable essentially bounded function, often written ~ or t---~ ~t' taking vMues in the space of regular probability measures on ft. ('Measurable' here is understood in the sense that t - - < g , # t >
is measurable for each g E C(fl).)
We denote the set of relaxed controls by Jtl,. The
set .46 can be regarded as a subspace of the topological dual space (L[0,1; C(fi)])*; it acts on elements r in the primal space according to 1
r ~f
J" O(t,u) d#t(u ) dt. 0
We equip it with the relative weak* topology. A relazed process is a pair (x, #) comprising a relaxed control and an absolutely continous function x which is a solution to the differential equation, in the sense that :~(t) = f f(t, x(t), u) d , t ( u ). It is admissible if x(0)=x 0. (Under the hypotheses, given /~, there is a unique x such that (x,#) is an admissible relaxed process.) An ordinary control u(.) can be regarded as an element in .21,; we identify it with the point t ~ 6 ( t ) , where 6~ denotes the unit measure concentrated at the point a.We find that .At, is a non-empty, compact set. .At, = closure{ordinary controls} The mapping
u - - x(-) : .Ag ~
C([0,1];~I n) is continuous, where (x,,u) is the admissible
relaxed process associated with /~. The problem posed over admissible relaxed processes is called the relazed problem
(Prelaxed).
We
deduce immediately from the preceeding assertions:
Theorem 2.1 The relaxed problem has minimizer, and tile infimum cost for the original problem coincides with the minimum cost for the relaxed problem. In symbols Inf{ Poriginat } = Min{Prelaxed}. The fact that the infimum costs coincide is summarized by the statement "tile relaxed problem is a
proper extension of the original problem".
Properness of the extension is desirable for a number of
reasons. It means that there is a close connection between the problem of primary interest (the 'original' problem) and the problem which replaces it.
It also suggests a methodology for finding
ordinary admissible processes which come close to achieving the infimum cost in situations where existence of a minimizing ordinary admissible process is not assured: we should solve the relaxed problem and then approximate the 'relaxed' minimizer by an ordinary one.
A full account, of these
ideas is to be found in Warga's book [4].
3. Relaxation Via a Reduced Problem. We return now to problem (P). In relation to this problem we term ordiT~ary controls measurable fuuctions u(.): [ - 0 2 , 1 ] ~ n such that u(.) E f~ a.e.
Ordin~Lr~
532 processes and admissible ordinary processes have the obvious meanings (c.f. Section 2). Problem (P), posed over admissible ordinary processes, is the original problem (Poriginal). A well-known technique (see e.g. [4]) reduces a control system with commensurate time delays to a delay free problem. ('Commensurate' means that 0t/0 2 is a rational number.)
To avoid some minor
technical difficulties, in describing this technique we assume a little bit more than commensurability, namely 9
01 and 0 2 are rational.
This means that there exist integers 0 < N 1 < N 2 < N such that 01 = N1/N and 0 2 = N2/N.
We
write A = 1/N. The basic idea is to section ordinary controls and the corresponding trajectories into segments of length A, and to stack these segments to form higher dimensional
vector valued functions
on the interval [0,A]. The resulting functions satisfy a delay free differential equation. The reduced problem is expressed in terms of a partitioned state vector Y~ = col [x0,.., XN_l] and control vector fi = col [U.N 2 .... u 0 .... UN_l]. It is
Minimize ~(2(A)) subject to subject to ~,(s) = f(s,~(s), fi(s)) , (R,P)
a.e. [0,~],
fi(t) E f2, a.e. [0,A], together with the mixed endpoint conditions x0(0 ) = x 0 and xi(0 ) = Xi.l(1), for i = 1,2,..,N-1 .
The d a t a for this problem is constructed from that for (P): = ~ (N2+N), ~(.~) = g(xN_i) and the components {fi}iN-~ of f are fi(s,~.fi) = f(iA+s, xi, ui, Ui_N1 , Ui_No).. It is a straightforward matter to show that (RP) and (P) (posed over ordinary processes
are
equivalent, in the sense that there is a one to one mapping from ordinary admissible processes for (P) and ordinary processes for (RP) which satisfy the mixed boundary conditions. This mapping carries an ordinary process (x,u) for (P) into (2,fi) where ~(s) = col[x(s), x ( A + s ) , x(2/',+s),..,x([N-1]A+s] and fi(s) = col[u(-N2A+s ) .... x([N-IlA+s]. This reduction procedure, together with the theory of the preceeding section, suggests a relaxation procedure for the time delay problem.
We can relax (R.P), which has no delays, along the lines of
Section 2. It is not difficult to show that the relaxed extension of (RP) has a solution and is a proper extension (the a r g u m e n t s involved in establishing riffs latter property are a little bit different to those underlying Section 2, since we must ensure that. given an 'admissible' relaxed process for (RP), we can find a suitable ordillary process approximating it and which satisfies the mixed botmdary comlitions). The procedure of replacing (P) by the reduced problem (liP) and relaxing (R.P) we shall refer to ;us 'relaxation
via the reduced problem'. This procedure in certain respects achieve~, the desired
533 objectives. The effect of relaxtion via the reduced problem is to guarantee existence of minimizers, and the relaxed problem is a proper extension of (P). IIowever it is unsatisfactory ill certain respects. Notice first of all that the dimension of the state and control spaces in (P~P) can be very large ((Nxn) and ([N2+N]xm respectively). The fact that we have posed (P) on the time interval [0,1] is merely a normalization procedure, and the above technique for eliminating delays works on an arbitrary finite time interval; however we note that, for fixed time delays 01 and 02, t.he dimension of the spaces involved increases rapidly with the length of the underlying time interval. Apart from this, in passing to the reduced problem on the time interval [0,A] the connections with the original problem are somewhat obscured.
Finally, relaxation via the relaxed problem is only possible for commensurate
delays.
4. Weak Relaxation. We now describe another relaxation procedure for the time delay problem, due to Warga [5]. Take an ordinary process (x(-), u(.)), and define u0(t ) = u(t), ul(t ) = u ( t - 0 1 ) and u2(t ) = u ( t - 0 2 ) , for t e [0,1].
Then (x(-), (u0(-), u l ( ) , u2(.)) ) satisfies C .~(t) = r(t,x(t), u0(t), Ul(t), u2(t)),
a.e.
[0,1]
(4.1) /(u0(t.), ul(t), u2(t)) ~ f~3,
-
x(0) = x 0
together with Ul(t ) = u 0 ( t - 0 t )
,01
and u2(t ) = u l ( t - - [ 0 2 - 0 1 ] ), [ 0 2 - 0 i ] < t < l .
In this way we separate the facts that three functions on [0,1] are inserted as the last. three arguments of f in the dynamics, and that these functions are delayed versions of each other. standard
theory
(Section 2) to define
relaxed controls, etc., for the system
We can use the (4.1),
treating
Ul(t ), u2(t ) and u3(t ) as independent functions; our starting point then is to take a relaxed control to be a measurable function with values regular probability measures on the Borels set o f ( f l x ~ x f l ) .
We
then impose some condition on # which corresponds to property (4.2).
Definition 4.1
We say that a measurable function
# from
[0,1] to the linear space of regular
probability measures on the Borel sets of ~x.O.x.Q is a weakly relaxed control for (P) if it satisfies tile compatibility conditions
(4.3)
~0 ~t-0i(u0' ul' u2) = ~1 t~t(u0 ' ul' u2)' a.e. [01, I] and
91 ~t_[o.,.0~](u0, u r u 2) = 9 2 ~,t(u0, u 1, u~), ~.e. [(o2-0l), Ile,'c 9 0 , 9 1 . 9 2 :
C * ( f ~ x f ~ x f 2 ) - - - C (9.) denote projections onto tile u0, uland
respectively. An alternative, equivalent, statement of tile compatibility conditions in Definition 4.1 is
11. u,) coordinates
534
1-01
l
; ]'r 0
and
t, u0) d,,t(u0, u p u2)dt
:
; Ir 01
u 1) d~t(u 0, u t, u2)dt
t-(~176 J"
J'r , u 1) dpt(u 0, u 1, u2)dt for everyOr e LI([0,1]; C(f2)).
1 J" Ir
=
u 2) d/~t(u 0, u I, u2)dt
02-01
These conditions are at first sight complicated, but they just express the fact, in linear functional terms, that the second and third components of #t are associated with delayed versions of the first and second components respectively. We now associate with a weakly relaxed control weakly relaxed processes and an admissible
weakly relaxed process, in the obvious way.
Problem (P), regarded as a minimization problem over
weakly relaxed processes, will be call the weakly relaxed problem (written (Pw-relaxed) ' If a weakly relaxed control arises from some measurable function (u0,ul,u2)(.): [ 0 , 1 ] ~ f ~ x ~ x ~ then the compatibility
conditions (4.2) and
(4.3) are equivalent.
It wilt be clear then that
(Pw- relaxed ) is an extension of (Poriginal). The following result is due to Warga [5].
Tlmorem 4.1 (Pw_relaxed) has a minimizer.
Proof.
T h e topology we take on the set of weakly relaxed controls N is that induced by the weak*
topology on (L1([-02, 1]; C(f2xf2xf2)))*.
With this topology N is a nonempty compact set (to straw
this, we note in particular that the compatibility conditions defining membership of 3{ are expressible in terms of a set of continuous linear functional constraints).
Also the mapping # ~ x, where (x,/~) is
tile unique admissible weakly relaxed process corresponding to /z, is a continuous mapping from Jf to C([0,1];R n) . Since g is continuous then, (Prelaxed) has a solution. Adopting this notion of relaxation has numerous advantages.
Unlike relaxation via a reduced
problem, the range space C*(f2xf2xfZ) does not depend on any relationships interval [0,1].
between 01, 02 and the
It can be applied even when the delays are noncommensurate, and the weakly relaxed
problem has a solution. Whether or not it provides a proper extension will be discussed shortly. 5. Strong Relaxation.
For r e ~ o n s that will become a p p a r e n t presently, we find it convenient now to
introduce another, slightly more elaborate, notion of relaxation.
This relates to the special case of
evenly spaced delays. Throughout this section we assume * q
0<0_<1 such that 01= 0 and
09 =
20.
Suppose that u(.) is an ordinary control for (P). Once again set u0(t ) : u(t), Ul(t ) : u ( t - O ) and u2(t ) = u ( t - 2 a ) , for tE [0, i]. \Ve can organize these relatiotlships as (a.l)
(u0, u i ) ( t - 0 ) =
(u[, u2)(t )
for
tE [0,1].
This suggests a new compat.ibillty condition for defining relaxed controls which iuvolves projections
535 ~201,912: C*(f2xf2x~)---, C'~(f2xf2) onto the coordinate pairs (Uo,Ul), (Ul,U2) respectively.
Definition 5.1
We say that a measurable function # from [0,1] to the linear space of regular
probability measures on the Borel sets of f2•
is a weakly rela::ed control for (P) if it satisfies the
following compatibility condition: (5.2) 901 ~'t-0(u0 ' ul' u2) = ~12 Ut(u0' UP u2)' a.e. [O,a 1. The compatibility condition here too can be equivalently expressed in terms of continuous linear functional constraints: 1-0 J" fr
1 UO,Ul) d/~t(uo, Ul, u2)dt = J" J'r
0
for every r e LI([0,1];
Ul,U2) d#t(Uo, Ul, u2)dt
0
C(f2•
By considering special choices of test functions r we deduce that the 'strong' compatibility condition of this section is more stringent than the 'weak' compatibility condition of the last section, tlowever, if a strongly
(or
[0,1]~f2•
weakly)
relaxed
control
arises
from
some
measurable
function
(u0,ul,u2)(.):
then all the compatibility conditions we have considered ((4.2), (4.3), (5.1) and (5.2))
are equivalent. Strongly relaxed processes, etc., are now defined by analogy with earlier definitions. problem
(P)
posed over admissible strongly
We call
relaxed processes the strongly relaxed problem
(Ps-relaxed)" (Ps-relaxed) is another, tighter, extension of (Poriginal). Essentially the same argument.s as those employed in proving Theorem 4.1 yield:
Tlieorem 5.1 (Ps-relaxed) has a minimizer. 6.The Main Results: Conditions for Proper Extensions. So far we have not addressed tile question of whether the 'strong' and 'weak' relaxation procedures we have described give proper extensions of (Poriginal)" The fact that, for strongly (or weakly) relaxed controls # arising from some measurable function (u0,ul,u2)(.): [0,1]---,f2xflxf2, the compatibility conditions ((4.2), (4.3), (5.1) mad (5.2)) are equivalent, encourages belief that, the extensions are proper. (Pw.relaxed) can fail to be a proper extension.
It conies as a surprise then that
This is established by the example we shall study in
Section 7. '].'lle question of when, precisely, (Pw_relaxed) is a proper extension of (Poriginal) remains open, but we are at least able to give sufficient conditions for this to be the case.
Theorem 6.1. Suppose that 01 and 02 are rational ,mmbcrsand 0.27~201. "l'hen min(Pw_rela•
= inf (Poriginal).
536 The final result tells us that strong relaxation gives a proper extension when the condition ' 0 2 # 2 0 I' is dropped.
Theorem 6.2. Suppose that 01 and 02 are rational numbers and 02=201 . Then min(Ps_relaxed) = inf (Poriginal).
T h e proofs of these two theorems are quite elaborate; full details will be given in [1]. However a brief outline of the techniques involved are appropriate here. Given a weakly or strongly relaxed control
~*: [0,1] - - C*(f~ 3) we appeal to known conditions, concerning existence of probability distributions whose
marginal
distributions
construct a relaxed control [*: [0,A] ~ rational. (}tere A = I / N
satisfy
certain
relationships
(see
[29,
to
C*(Q N2+N1) for the reduced problem when 01and 02 are
where NI, N 2 and N are positive integers such that 01= N t / N and
02=N2/N. )
This can ahvays be done if either 01=202 and /~ is a strongly relaxed control, or if 0 2 # 2 0 1 a n d # is a weakly relaxed control.
We then construct an ordinary control fi(.) for the reduced problem using the
theory of Section 3, whose cost is arbitrarily close to that of/~. An ordinary control for (P) whose cost is arbitrarily close to that of ~ is then obtained via die inverse of the one to one mapping from ordinary controls for (P) to ordinary controls for the reduced problem introduced in Section 3.
7. An Example.
We now give an example to illust.rate that, in the absence of additional hypotheses,
(Pw_relaxed) may fail to be a proper extension of (Pordinary). Minimize x2(1 ) subject to i2(t) =
•
(Xl--0.5t)2+
d(u(t), u ( t - - A ) , u ( t - - 2 A ) ,
a.e. [0,i],
= u(t),
u(t) 6 [0,i]and Xl(O)= x2(O)= O.
Here A is a rational ,lumber, 0< A<_ 0.5, and d: p : 3
iR is any continuous function such that
i~ (ul,u2,u a) e{1,t,0}u{0,0,1} d(u l'U2'U3)
0
ot, herwise.
Consider tim fuuction &, _= v where v = 0.6
6{t,~,o]+ 0.5 6[o,o.U.
\,Ve see that this function is a weakly relaxed control. (Notice that the relevant compatibilit, y condition is satisfied sitlce~0// = ~1// = x.~(t)) = (0..St. 0) . O_
~,y
= 0.5
b[o]+ 0.5 6{t]. )
and the correspo,ding
The associated state trajectory is (Xl(t.),
cost is zero. Since the cost ca:mot be negative, we
537 Min (Pw-relaxed) = 0. Now suppose ((Xl(-),x2(.)), #t) is any admissible strongly relaxed process with cost zero. This is only possible if x l ( t ) = 0.5t, tE [0,1] and s u p p i p [ } C {1,1,0}U{0,0,1},
a.e. [0,1].
This last condition means that there exists a measurable function ~(-): [0,1.]---*[0,1] such that
y[ = c~(t)6 {1,1,o} + (1-c~(t))8 {o,o,H ,
a.e. [0,1].
However we deduce from the condition ' x l ( t ) = 0.5t' and the dynamical equation that 0.5 = a(t)x1. + ( 1 - - a ( t ) ) x 0 , i.e. a ( t ) #[
(=
a.e. [0,1],
--': 0.5. We have shown that y[ must satisfy
0.5~{Ll,o} + 0.56{0,o,l}) = u,
a.e. [0,1.].
0.56{L U + 0.5~{o,o}:/:
But #[ = u does not satisfy the 'strong' compatibility condition. (~20,1u =
0.56{1,o } + 0.56{o,1 } = 91,2).
Since (Ps_rela.xed) has a solution, and admissible strongly relaxed
processes define admissible weakly relaxed processes, it follows that Min (Ps_relaxed)
> Min (Pw_relaxed).
(strict inequality). But problem (Ps_relaxed) is an extension of problem (Poriginal) so Inf (Poriginal) _> Min (Ps_relaxed)
> Min (Pw_relaxed).
We summarize our findings:
Proposition 7.1 The data for problem (P) can be chosen so that 0 i and 02 are rational, 02= 201_ , and (Pw-relaxed) is not a proper extension of (Poriginal). Observe that Proposition 6.1 predicts that, in this example, (Ps.relaxed)
is a proper extension of
(Poriginal)" There is one further respect in which this example is revealing. Define
11(01,02) : = {the infimum cost of the original problem for delay parameters O1 and 02}. It is of interest to know if this function is continuous, for if it is not the well-posednness of problem (17) is called into question in circumstances where the delay parameters have uncertain values,
l:;xamining
can be discoa~inuous. Let 0 be a fixed rational number, 20 and {0(I)},{0( ')} be sequeuces of rational points such that 0(1)~ 0 and 0(.2)
variants on the above example tells us that r/ 0 < 0 < 0.5.
Let
for which 0f'e)~90(. I) for i = 1,2,.. l
---
1
For each i sufficiently large, the function #t = u is a weakly
relaxed control in tile above example when we adopt
0 (.I) and 0(e) as delay parameters, and the cost is 1 1
zero. Since the cost cannot be negative, anud in view of Theorem 6.1, we must have
,1(0[12,0(e)) = 0, llowever ~/(fl, 20) > 0
as
for all i sufficiently large.
we have shown. Tiros ~l is discontinuous at (0,20).
538 REFERENCES
[1]
T. Andrews, J. Rosenblueth and R. Vinter, Relazation Procedures for Time Delay Systems, in preparation.
[2] [3]
C. Dellacherie and P. A. Meyer, Probabilit~s et Potentiel, Iterman, St,rasbourg, 1975 W. It. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975.
[4]
J. Warga, OpIimal Control of Differential and Functional Equations, Academic Press, New York,
[5]
J. Warga, Nonaddilively Coupled Delayed Controls, preprinL
!966.
MAXIMUM
PRINCIPLE DIFFERENCE
FOR
NONCONVEX
CONTROL
FINITE
SYSTEMS
Boris S. Mordukhovich Department
of Mathematics,
Wayne
State University
Detroit, M i c h i g a n 48202, U S A Optimal control problems are considered for a family of finite difference systems with constraints on the control and phase variables. Such systems arise in computer calculations of optimal control systems with continuous time and aiso in the simulation of discrete large-scale optimization problems. We regard discrete approximation systems as a process with decreasing time quantization period. New necessary optimality conditions for such systems are obtained in the form of ~he approximate maximum principle which is fulfilled the more precise the less is the time quantization period. We consider arbitrary nonlinear control systems without any asssumptions about convexity. Constructive methods are proved for approximating the state constraints so thai, the stability of the Pontryagin maximum principle is ensured in computer calculations of nonconvex control systems with coutinuous {~ilne. The results obtained are used for the analysis o[ qualitative aspects and the development of numerical algorithms of opt,treat control. 1. I n t r o d u c t i o n The paper deals with optimal control problems for finite difference systems of the form
z(t + h) = z(t) + hf(x(t),u(t),t), v(t) E U(t),t E T/, = {to, to + h , . . . , t l - h},
(t.l)
(1.2)
with various constraints on the state variables. Such problems arise in computer work when the time derivatives are replaced by finite differences : ~:(t) ~ [x(t + h) - x(t)]h -~, et,c. It is then natural to consider the problems ~ a constructive interpretation of the corresponding problems for continuous dynamic systems without using the abstraction of continuous time. We note also that finile difference dynamic systems can be used for modeling a variety of static problems of discrete optimization of large dimensiouality. Optimization problems for systems (1.1), (1.2), regarded as a process as h ; 0, occupy an intermediate position between optimal control problems for systems with continuous and discrete time (with fixed h) and lead to the appearance of essentially new effects some of which are analyzcd below. I~ is well-known that the Pontryagin maximum principle [P 61] gives some necessary conditions for an extremum in general optimal control problems for continuous-time systems described by ordinary differential equations. But, for its discrete analogue to hold as necessary optimality conditions, certain assumptions about convexity [B 73, CCP 70] are required, since wil,hout ~hem the
540 discrete maximum principle is invalid even in the simplest problems with a fl'ee right-hand end. The connection between convexity and maximum type conditions in the theory of extrernal problems in reflected in the classical Kuhn-Tucker theorem [R 70] and follows naturally from the general results of nonsmooth analysis [C 83, Mot 88] (the d~aracteristic form of the normal cone to the convex set). On the other hand, the unconditional satisfaction of the maximum principle in systems with continuous time is linked with the "hidden convexity" property inherent to such systems fIT 74], which can be traced back to Lyapunov's theorem on convexity of the values of a nonatomic vector measure [L 40]. It is important to emphasize that, in order to utilize the Pontryagin maximum principle correctly in computations of actual control systems with continuous time on computers operating discretely, we have to prove the stability of necessary optimality conditions supplied by the maximum principle in difference approximations of nonconvex systems with constraints on control and state variables. Our main result is to prove the necessary optimality conditions, spccifically for finite difference control systems with constraints on the trajectory, in the form of the approximate maximum principle which is valid when the perturbation of constraints is constructively matched with the step of a difference mesh without any assumptions about convexity. In control problenls where the right-hand end of trajectory is free and the left-hand end is fixed, our result is in accord wit,h the quasimaximum principle of [CK 71]. During the proof of the approximate maximum principle, an approximate analogue of the hidden convexity property is found to be inherent to finite difference systems so that we can solve the problem posed in [Mo 71] of finding the reasons why finite difference systems with a small quantization period behave like any discrete systems with a priori convexity conditions. We shall use the following standard notations: R" is the n-dimensionalEuclidean space of column vectors (they are written in rows) with the norm II "II and the scalar product < -,. >; coX, d X are the convex hull and the closure of the set X; E is tile identity matrix; o(h) and O(h) a.re vector quantities such that
o(h)/h
--, 0, IlO(~dll _< K~ as z, s 0. In addition to the usual notation for a
matrix product, we put
.l:=i
( AiAi-1 ... Aj for i > j,
fiat= J
J E 0
for i = j - 1 , for i < j - 1
2. P r o b l e m F o r m u l a t i o n and Definitions. Wc take the control problem
5c = f(x,u,t), u(t) E U(t), t E T = [to, t1],
(2.1)
~ ( z ) < O,i= 1 , . . . , m ; ~ ( z ) = O,i= m + 1 , . . . , r e + p ,
(~.2)
I = ~Oo(Z)~ min,
(2.3)
541 where z = (xo, xx) = (x(to), x(t~)) E R 2~ and T is a fixed intervM. For each positive integer we consider the quantization step hN = (q - to)/N and construct, a sequence of finite difference problems of minimizing the functional (2.3) with z = ZN = (XN(t0), zN(t~)) and the constraints
xN(t + hN) = XN(t) + hNf(xN(t),uN(t),t),
(2.4)
UN(t) ~ U(t),t ~ TN = {to, t0 + hN,... ,t, -- hU),
(2.5)
(PI(ZN) <_ "/iN, i = 1 , . . . , m ,
(2.6)
--5iN <_ qoi(ZN) <_ 61N,i = rn + 1 , . . . , r n + p .
(2.7)
We assume henceforth that U(t) is Hausdorff-continuous with compact values in a metric space, that the set of trajectories optimal in (2.3) - (2.7) is uniformly bounded as N ~
eo, that the
functions f,~oi, i ~ 0, 1,... ,Trt + p, are continuous at the point considered with respect, to their variables and are continuously differentiable with respect to x wigh derivatives that are uniformly bounded in the compacta. W'e inl:roduce the functions
H(x, ~b, u, t) = (~,, f ( x , u, t)), ~(x,~b,t) = m a x { l t ( x , ~ b , u , t ) : u C U(/)},
which are called respectively the Hamilton-Pontr~jagin function and the Hamiltonian of (2.1), and wc form the adjoint system to (2.3)-(2.7):
,/,u(~) = r
+ hN) + h~~ .
O~Oo.
(x,,,(t), ,~u(t + h,,,),,~N(t), t), t e T~, ,
_
O~o,.+p.
~bN(tl) = -- aON-~zl t ZN ) - - . . . - - Z m + v , N ~ t
zN )
(2.S)
(2.9)
D e f i n i t i o n 2.1. We say that the approximate maximum princole holds for the optimal processes {.z'~v(.),u~
in (2.3) - (2.7)if there exist sets of multiplier {AiN,i = 0,1 . . . . . m + 1,) and
corresponding trajectories ~/,~,(.) of the system (2.8), (2.9), such that
s_r(:~(t), CX,(~ + 1~,,,), ~,~(1), t) = '~(~,~(t), ~/,~(t + hN), t) + ~(t, h,-,,), t e T~,, 0~o
o
O~o,.+p
o
4,X,(lo) - [Ao,,,37s (z~,) + . . . + .~,,,+,,.N----g~o (z,)] = Co(/,N),
(2.10) (2.11)
542 A,N(~o,(~) - "I~N) = e,(h~,), i = 1 , . . ,
m,
(2.12)
)~;N --> 0,i = 0,1,...,m,()~oN) 2 + . . . + (.~+p,N) 2 = 1,
(2.~3)
where z} = (x~(to),x~(t,)),ei(h) --+ 0, e(t,h) + 0, as h .L 0 uniformly with respect to t C TN, N = 1,2,.... It follows from (2.10), (2.12) and (2.13) that e(t, hN) <_ 0, ei(hjv) _< O,i = 1,... ,rn. Ife(t, hN) --
O, ei(hN) -- 0, then the relations (2.8) (2.13) define the (ezact) discrete maximum principle [B 73] wlfich is not necessary for the optimality of {x~v(-), u}(-))}, N ---+0% in even the simplest systems with the nonconvex vectogram f ( z , U(t), t). We prove hdow that the relations of the approximate maximum principle are necessary for the optimality in (2.3) - (2.7) under the accurate coordination of state constraint perturbations with the step of a difference mesh. D e f i n i t i o n 2.2. Let to(N) = to + O(N)h~v, 0 <_ O(N) _< N - 1, with integers O(N), be points of the mesh TN. The sequence of discrete controls {uN(/)},N = 1,2 . . . . . is called regvlar if for any increasing subsequence of integers {N} and any sequence of points rO(N)(E TN) ---+t E T are given, we have, with any integer q = coast,
fl(tt(TO(N),ttN(TO(N)+q)) --+ O, N --+ o~, t e [to, tl), p(~t( rO(N), tt( To(l,')_q) ) "--+O, N -+ oo, t = t, , where p(., .) is the metric in U(t). This concept of a regular sequence of discrete controls postulates a natural property which corresponds to tile property of regular points (points of approximate continuity) of a measurable control in continuous-time systems [P 61]. D e f i n i t i o n 2.3. Let r be an integer, 1 < r < N - 1. We call s
an r needle-shaped variation
of the control uu(.) if ~ j ( N ) , t = ,-0,(N),
fiN(t)=
u,v(t),t E TN,t # r o , ( N ) , j = l , 2 , . . . , r ,
whcrc the parameters {Oj(N),uj(N)}~= 1 satisfy the conditions uj(N) E U(ro,(N)),O < Oj(N) <
N - 1,0j(N) # Ok(N)j • k ; j , k E {1,2 . . . . ,r}. If r = 1 we call (2.1,1) an one-needle-shaped (or simple needle-shaped) variation of the control ujv(.). D e f i n i t i o n 2.4. Let iN(') be the trajectory corresponding to (2.14) of the systeln (2.4) with the initial condition iN(to) = xN(t0). The increment i N ( ' ) -- aN(') of the trajectory zN(.) is denoted by A~,ZN(.) with r r 1 and by Ao~zu(.) with r = 1 and is called respectively a r-needle-shaped and
one-needle-shaped increment of the trajectory. At the terminal point t = tl we say of a teT"minal increment of XN(').
543 3.
T h e A n a l o g u e of t h e H i d d e n C o n v e x i t y P r o p e r t y for F i n i t e D i f f e r e n c e C o n t r o l
Systems We consider here an important structural property of terminal increments of a trajectory ia finite difference systems (1.1), (1.2) as h I 0, whereby we can approximate up to terms of order
o(h) an integral combination of trajectory increments in an one-needle-shaped type. This property is in essence similar to the hidden convexity in continuous type control systems and plays a key role when proving in Section 4 the approximate maximum principle in nonconvex finite difference systems with constraints ca trajectories. Let {UN(t)}, t E TN, ]V = 1 , 2 , . . . , be a sequence of controls admissible in (2.5), {Oj(N), uj(N)}~=l, At = p, p + 1 , . . . , be a sequence of parameters of their needle-shaped variations; mj, j = 1 , 2 , . . . , p, be a set of nonnegative integers (p, mj are fixed and independent of N). We consider terminal increments of the trajectory AoivxN(tl) corresponding to one-needle-shaped variations of the control with parameters {Oj(N),vj(N)) and denote them, for simplicity, by Ao~jxN(tt).
We form the integral
combination P
aN(p,-0 = ~-~sa~J~N(e,)
(3.1)
j=l
T h e o r e m 3.1. Let, a control sequence {utv(t)},t E TN, be regul~'. Then with any p, rnj there
/kr&,xN(tt),
exist an intcgcr r independent of N, and terminal trajectory variations
corresponding
to r-needle-shaped variations of the control u~v(t), such that we have the representation for (3.1):
AN(p, mj) = A~,x,v(t,) + o(hN),N --+ ~ .
(3.2)
P r o o L It can be assumed without loss of generality that the numbers r6(N ) = to + Oj(N)hN, N co, convcrgc to ej C T , j = 1 , 2 , . . . ,p. We first assume that
(3.3)
c i ~ ck,j # k;ej # ti,Vk, j E { 1 , 2 , . . . , p }
From the parameters (3.1) we construct the number r = mi + . . . + m p and the r- needle-shaped trajectory increment A~ZN(h), generated by the variation
(iN(t) =
{ u./q(N), t = reAN)+q, q = 0, 1 , . . . , rnj -- 1, UN(t),t E TN, t # rOa(N)+q,j = 1 , 2 , . . . ,p,
where uj~(N) is the element of the set U(roj(N)+q)) ncarest to ,j(N).
(3.4)
The controls (3.,1) can be
constructcd in view of (3.3) and are admissible in (2.5). Using (2.4), (3.4), we can prove that p
mj k=N-1
fqs
~,,:~,,,(t,) = h,,,{E ~C FI rE + hN~.(:~,,,(T~), UN0-~),'~)] X j = l q=l
~+~/
(3.5)
544
where Wiq = f(xN(ro,+q), vjq, Te,+q) - f(XN(TO~+q),UN(VO,+q),ro,+q). Since {UN(.) },N = 1,2, . . . , is regular and f, U are continuous, we obtain wjq - wj0 ~ 0 as N ~ ee, whence (with wj = wjo)
p
k=N-1
_~
ArOuXN(tl) -~- hN{EITZJ I-I [E-Ij=l
(;EN(Tk),UN(Tk),Tk)]UJj} ~- O(hN)
04+1
and the relations (3.2), in the light of (3.1), (3.5). If assumption (3.3) is violated, the construction of vairation (3.4) has to be corrected. simplicity, we shall consider two extreme cases a) el . . . . .
% r tl, b) el . . . .
For
% = tl, which, in
conjuction with (3.3), cover the general case of the disposition of points ej in T. Since the proof of (3.2) in cases a) and b) is similar to the case (3.3) considered already, it suffices to give the method of constructing the r-needle-shaped variations of uu('). Consider the case a). We renumber the parameters {0j(N), u./(N)},j = 1,2 . . . . ,p, in such a way that 01 < ... < 0F, and we identify the subscripts {Oj}, with the corresponding points {roj }. At the points 0~,0~ + 1,... ,01 + ml - 1, we vary the control uu(.) in accordance with (3.4). We assume that the control variations corresponding to parameters {0k, vk}, I < k _< p -
1, have been
constructed, and we construct them for {Ok+x,vk+~}. Let 0* be the greatest (extreme right-hand) point 0, in which the control variation has been made. If 0* < 0k+1, we vary the control u~v(-) at the points Ok+l, Ok+l + 1,..., 0k+l + mk+l - 1 according to
(3.4).
If 0* > Ok+l, the variation is performed
at the points 0* + 1,0* + 1 , . . . , 0* + mk+~. 1,1 case b) we renumber the parameters {Oj(N), uj(N)} in such a way that 01 > ... > 0F, and we construct the variation of uu(') in the order opposite to that in case a). O Remark
3.2.
It follows from the proof of Theorem 3.1 that its result is due to the specific
nature of discrete approximation systems with an infinitesimal time quantization period. It is wellknown [P 61] that, for systems with continuous time, the set of terminal increments of trajectories corresponding to all possible collections of needle-shaped control variations is a convex cone. This is connected with the fact that the control can be varied transversely in intervals of the infinitesimal length e and appears in the fact that any convex combination of needle-shaped increments can be approximated up to o(s) by an admissible increment which corresponds to a special set (packet) of needle-shaped variations.
Results of this kind do not hold for arbitrary systems without a
priori convexity assumptions. In discrete approximation systems the role of a small time interval is pla.yed by the discretization step (parameter) h J. 0, so that, in the limit, we can organize transverse variations over the time axis which are asymptotically small in h. We prove in this way the assertion of Theorem 3.1 which is the finite difference analogue of the hidden convexity property. 4. T h e A p p r o x i m a t e Constraints
Maximum
P r i n c i p l e in C o n t r o l S y s t e m s w i t h I n e q u a l i t y - T y p e
Wc prove here {,he approximate maximum principle stated in Section 2 for problcms with incquality-type constraints.
In this Section and in the next Section 5 we assumc that the as-
545
sumption of Section 2 holds and that the sequence of optimal controls in (2.3) - (2.7) is regular in the sense of Definition 2.2. D e f i n i t i o n 4.I. An inequality-type constraint in (2.6) with fixed i E { 1 , 2 , . . . , m } is called
essential for the sequence of admissible trajectories {xN(.)} along the sequence of integers A = {N} if (c21(z/v) - ")'i/v) = O(hN) as h/v J. O, i.e., for s o m e / ( i > 0 we have
-K,h/v < ~,~(z/v(to),~/v(q))
- " r . v < 0 , N --+ oo, N c A.
(4.1)
The constraint is called inessential for {XN(')} alone A if for any given K > 0 there exist No such that
~oi(x/v(to),ZN(tx)) --7i/v <-- - K h N , N
>_ No, N e A.
(4.2)
We note that the concept of an essential constraint in a sequence of finite difference optimization problems corresponds to the concept of an active inequality-type constraiat in usual problems of mathematical programming and optimal control.
It is easy to see that, if the constraint is
not essential along the sequence A = {N}, then it is inessential along the sequence A, C A. IIence it can be assumed without loss of generality that, for the sequence of optimM trajectories {x~v(')} in problems (2.3) - (2.6), the first g constraints in (2.6) are essential and the remaining
m -g.,O < g < m, are inessential along all N = 1 , 2 , . . . . From the optimal trajectory increment A z } = (Az~v(10) , Az~v(q)) we construct the (g + 1) -dimensional column vector
e/v(Az~) .
((-~e~ . .
and wc consider the vectors b/v = Ax~
.
,(Tgz ~~176
(~.3)
g/v(b/v, O, u) = g/v(b/v, A0,.x}(t,)), and the set
C-'/V = {aN(b,,O,u) : bN e B/V,O < 0 < N -- 1,u e U(ro)},
(4.4)
where B/V = {x E /{", [[xl[ _< h/v}. We form the convex hull COGN of the set (4.4). The following lemma shows that the set coG/v can be shifted by an amount of order o(h/v) as hN ~. 0 in such a way that it does not intersect the negative orthand in Re+x:
Y = {y = ( v ~
e R TM : v' < 0,i = 0 , 1 , . . . , e } .
(4.5)
L e m m a 4.2. There exists a sequence of (g + 1)-dimensional quantities of order o(h/v), h/v J. O, such that
( c o a ~ + o(h/v)) a z = r N > No
(4.6)
Using this result, we can prove the approximate maximum principle in problems (2.3) - (2.6) in the case of any perturbations ")'iN of inequality-typed constraints.
546 T h e o r e m 4.3.
Let {x~
be optimal processes in problems (2.3)- (2.6) where the
inequalties in (2.6) arc essential along the sequence {x~
N = 1 , 2 , . . . , for i = 1 , 2 , . . . , g and are
inessential for i = g + 1 , . . . , m. Then the relations of the approximate maximum principle (2.8) (2.13) hold for these processes, and
e,(hN) = O(h,v),i = 1,2,...,g;AiN = 0, i = g + 1 , . . . , m , N
>_ No.
(4.7)
P r o o f . By Lemma 4.2, the separability theorem for convex sets and the structure of sets (4.4), (4.5), there exist vectors #N = ( # ~ . . . #~) E R ~, (#~ i rtOtPi/ o',
O~al
+ . . . + (#~)2 = 1, N > No, such that
o
ttNK~xo~ZN), b~) + (3-~-(z,-,,, ao,,x~(t,))] + o(1,,) ___0
(4.S)
for any bN e BN, IIbNII _< hN, and parameters {0(N), u(N)} of one-needle-shaped variation, where It'y >_ O,i = 1,...,g. We put Auv = #'N,i = O, 1,...,g;.~iN ----0, i = e + 1 , . . . , m , and obtain sets of Lagrange mutipliers which satisfy (2.12), (2.13), (4.7), while el(h) = O(h) follows from (4.1). Using thc standard transformation of the method of increments [GK 71] we obtain from (4.S) the relation
O~~ o O~t o (AoN-~x (zN) + . . . + ,~eN-~x (zg) -- ~b~
bN) -- hN[H(x~
(4.9)
v
r b-0+,), ,,,-0) - ~r(~o (To), r
,6(,-0), ~-0)] + o(I,.) >__o
along the trajectories (2.8), (2.9) corresponding to {x~
With bN = 0 we get (2.10) from
(4.9), and with u = u~(ro) we obtain (2.11) by reduction ad absurdum. [] 5. T h e A p p r o x i m a t e M a x i m u m P r i n c i p l e in G e n e r a l C o n t r o l S y s t e m s w i t h E q u a l i t y Type Constraints Let us study the discrete approximations (2.3)- (2.7) of problem (2.1)- (2.3) with cquality type constraints. Our example below shows that if 51N = 0 or 5iN J. 0 too rapidly, the approximate maximum principle may not be satisfied in ( 2 3 ) - (2.7). E x a m p l e 5.1 We consider the two-dimensional control problem =
v,y(0) = 0,.~ = w,s(0) = 0, x = (y,s) E R 2,
=
(,,, ~) e
(5.1)
m , t e [o, 11,
I = ~o(x(1))
= s(1)
--~ min
(5.2)
with fixed left-haad end and the constraint at the right-hand end of the trajectory ~,,(x(1)) = y(1) = 0
(5.3)
with the quantized control domain in (2.1):
V = {(0, 0), (0,-1), (1, -2), (vq, -3))}
(5.4)
547
It is easy to show that, in the corresponding finite-difference approximation (2.3) - (2.5), (2.7) of the problem (5.1) - (5.4), there exists the unique sequence of solutions u~v(t) = ( 0 , - 1 ) , x ~ ( t ) = (0,--t),t e T N , N = 1 , 2 , . . . , where r
(5.5)
- (--~lN,--MN) for the corresponding trajectories of the adjoint system (2.8), (2.9)
with Lagrange multipliers ~dmissible in (2.13). The Hamilton-Pontryagin function along {X~v('), r h ~ the form [IN(U, t) = - - & N V - - ~ o u W , and for the optimal control (5.5) we obtain H N ( u ~ ) = ~o~v. It can be shown directly that max{I-IN(u) : u e U} - HN(U~ in
(2.13),
>__ 1 for any {-'~0N,/~IN} a,dmissible
i.e. condition (2.10) does not hold.
We introduce a coordination condition between perturbations 51u and the step hN: . 5~1v lm - - = oo, i = m + l , . . . , m + p ,
(5.6)
n~oo hN
and we show that in the caze of (5.6) the relations (2.8)- (2.13) are necessary for optimality in (2.3)
-(2.7). T h e o r e m 5.2.
Let the processes {x~(-),u~(.)} be optimal in problems (2.3) - (2.7) with
any perturbations 7iN and the coordination condition (5.6) for &N- Then approximate maximum principle (2.8)- (2.13) holds in (2.3)- (2.7); moreover ei(hN) = O ( h s ) , i =
1,... ,m.
P r o o f . We can assume for simplicity that ~ol -= 0 in (2.6), since these constraints were considered in Theorem 4.3. We put ~a+N(z) = ~a~(z) - 5m, ~a~-u= -~o~(z) - 5,v, i = m + 1 . . . . , m + p, and write constraints (2.7) as
~+,u(ZN) <_ O,~07~v(Zu) <_ O,i = m + 1,...,,~ +1"
(5.7)
Applying Theorem 4.3 to (2.3) - (2.5), (5.7), we find the sets of multipliers {Aog, A~, A,x, i = m + 1 , . . . , m + p} along which the corresponding relations (2.8) - (2.13) and (4.8) hold. Let the constraint qO+u(ZN) < O be essential for {x~(.)) along the scquencc A = {N}.
Then, by tile
coordination condtion (5.6), we have A+NA(U = O,i = ra + 1 , . . . , m + p , N >__No,
whicl, proves (2.8) - (2.13) for •iN = )~+N -- ) ~ , i
= m + 1 , . . . , m + p []
R e m a r k 5.5. The approximate maximum principle has a variety of applications in the qualitaLive theory of optimal processes. These include, in particular, finding thc convergence conditions of discrete approximations with rcspect to a functional, a Hamiltonian, controls, trajectories, velocitics, or obtaining existence theorems for optimal controls, optimality and suboptimality conditions ill systems with continuous time, etc. [Mor 88].
548 REFERENCES
[B 73] V.G. Boltyanskii. Optimal Control of Discrete Systems. Nauka, Moscow, 1973 (in Russian). English Translatiou in Wiley, New York, 1979. [C 83] F.tI. Clarke, Optimizalion and Nonsmooth Analysis. Wiley, New York, 1983. [CCP 70] M. Cannon, C. Cullum, and E. Polak. Theory of Optimal Control and MalhemaLical Programming. McGraw-Hill, New York, 1970. [GK 71] R. Gabasov and F.M. I(irillova. Qualitalive Theory of Optimal Processes. Nauka, Moscow, 1974 (in Russian). English Translation in Marcel Dekker, New York, 1976. lIT 74] A.D. Ioffe and V.M. Tikhomirov. Theory of Eztremal Problems. Nauka, Moscow, 1974 (in Russian). English Translation in North-Holland, Amsterdam, 1979. [L 40] A.A. Lyapunov. On completely additive vector functions. Izv. Akad. Nauk SSSR. Set. ll,[alem., 3, 6,465-478, 1940 (in Russian). [Mo 71] N. N. Moiseev. Numerical Methods in Theory of Optimal Systems. Nauka, Moscow, 1971 (in Russian). [Mor 88] B.S. Mordukhovich. Approzimation Methods in Problems of Optimization and Control. Nauka, Moscow, 1988 (in Russian). [P 61] L.S. Pontryagia et al. Mathemalical Theory of Optimal Processes. Fismatgiz, Moscow, 1961 (in Russian). English Translation in Wiley, New York, 1962. [R 70] R.T. Rockafellar. Convez Analysis. Princeton Univ. Press, Princeton, N.J., 1970.
Functional Viability Constraints l ~ e m i D u l u c 8,= C h r i s t i n e V i g n e r o n CErs
UNIVErtSIT]~ DE PARIS-DAUPtIINE F-75775, Paris cx(16) France
Abstract We shall consider systems whose evolution is described by a differential inclusion with memory; this mezms that the velocity ~(t) depends in a set-valued map way not only on the state x ( t ) but on the whole history of the state up to the time t. Moreover, the system is subjected to functional viability constraints, that ~s constraints that involve at each time t, the history of the state up to the time L Haddad's general functional viability theorem gives us a necessary and sufficient condition for the existence of viable solutions, that is solutions to the differential inclusion that satisfy the viability constraints. Yet, this condition needs to involve the terms in which the viability constraints are defined in more explicit instances. In tlfis paper, we will study two different categories of problems where the condition caal be expressed with relative simplicity: Viability problems with delays azld Volterra viability problems. In tlfis respect, we propose an explicit condition for the existence of viable solutions.
Introduction Let F be the set-valued map describing the dynamics of the problem and I f the viability set describing the constraints imposed on that system. The usual viability theorem (see Aubin [3]) puts that, as long as at each point of K , the interscction of the contingent cone 1 with the value of F is non empty 2, there exists viable solutions to the problem. Haddad ([5],[6]) showed that in a more general context of fmlctional differential inclusions (also called differential inclusions with memory), similar conditions can be used to prove the existence of viable solutions. But the "tangent" set to the functional viability constraints set is no longer the usual contingent cone, which has been already the object of a rich calculus (see Aubin [3], Aubin & Frankowska [2], Aubin & Cellina [4]). In this paper, we shall prove that, for two cases, wlfich are the problem of constraints depending on delays and the Volterra viability problem (see Hale [7]), this set can be ILet us recall the contingent cone at z to a subset K:
7~z~(=) := (v I lim infd(z + h~O+
hv, K ) / h = O}
2We can state this theorem as follows: Tlmorem 0.1 Let us assume thai i) it) iii)
F : X ".-* X is upper semiconliuuous the images o f F are convex and compact K is locally compac~
Then K enjoys the local viability properly if and only if
(1)
VzEK, F(x) n T g ( z ) # O
550 characterized in terms of the contingent derivative of some maps. On the technical side, a recent version of the Constrained Inverse Function Theorem is very useful for our demonstration. Before going any further, we need to set up some notations. X is a finite dimensional vector space. C is the space C(] - co, 0], X ) supplied with the topology of uniform convergence on every compact. C~ is the subspace of ),-lipschitzean functions of C. If ~ E C, we denote by: 9 %%0: The function of C := C(] -- o o , + o o [ , X ) defined by: { % V ( s ) = ~ o ( s ) if s _ < 0 ~q,~(,s) ~(0) + ,sv if ,s >_ 0 9 T(t)~,: The function of C(] - oo, 01, X ) defined by: (T(t)~)(s)=~(t+s)
for all.~
andt <0
I f / 4 is a subset of C(] - 0% 0], X ) and ~o belong to 35, we denote by: 9 Z)~c(c;): The closed subset of X defined by:
9hi ---4 0 + , pl E C(] - 0% h,], X ) such that
i) T(0)~,~ = ii) T ( h 0 ~ i E K iii) (~i(hl) - ~o(O))/hl := vi ---~ v The functional viability theorem deals with systems of functional differential inclusions such &3:
for almost all t E [0, + o ~ [ , x'(t) E F(T(t)x) (2)
T(O)x =
and wlfich ar'e subjected to functional viability cons[ralnts:
(3)
V/: E [0, +oo[, T(t)x E 35
where F : C ( ] - c ~ , 0], X ) ~ X is an upper-semicontinuous set-valued map with non empty compact and convex values, ]C is a closed subset of C~ and ~ 6 35. Haddad's functional viability theorem can be stated as follows: Theorem
0.2 35 cnjoy~ the fanctional viability property, that is, V~ E 35, ~x(.) E C-~- satisfying (2)
and (3)
if and only if K is a functional viability domain, that is
v , ~ 35, z~,=(~,) n F ( ~ ) r Given a closed subset K of C~, we would like to know whether there exists a solution to differential inclusion (2) starting at r E 35 and viable on 35 (condition (3)). We therefore need to characterize, for a given ~p in 35, the set 7)~c(~y) in order to check whett~er 35 is a functional viability domain. For this reason, our work here will exaanine two cases - the viability problems with delays and tim Volterra viability problems - where we give to the condition v E 73~(r a more appealing aspect, considering the definition we started from.
551 The viability problems with delays are the case when the actual state of the system is constrained by a fiifite number of previous states. T h a t is when: V t e [0,+oo[,
x(t) E M(z(t - O~),.. .,z(t - Op))
where ( 0 1 , . . . , 0p) E ( R : ) P are the delays and M : X p ~ X is a set-valued map. Therefore the constraints set is: :={~eCa I ~(0) eM(~0(0,),...,qo(0,))} In the Volterra viability problems, the state of the system is constrained by the cumulated consequences A(t - ~-)z(r) of past evolutions x(T), that is:
v , e [0,+oo[,
z(t) e M (f'__~oA(t - ~-)=(t)dr )
where, if C~ denote the set of function with compact support, A 6 C~(R.,s and M : X ~ X is a set-valued map. Therefore the constraints set is:
K.= {~p E Ca I ~o(O)E M ( L
X), Supp(A) C R*+
A(-~)~(s)d~)}
These kinds of problems are motivated by biological and economical problems. For instance, if we consider the dynamical version of the Walrasian allocations of ressources model developed by Aubin in ([1]), we can say that consumers alloeaf~ions are constrained by the production of firms. The planning of such a production can depend on the zmticipations of the producer which are made after having observed the past demand of the consumers. Therefore the allocations can be constrained by Y(z(t - 0)) if the producers observe the demand with a delay O or by Y(f'-oo Aczp(A(o - t))z(O)dO) if the producers observe an average of the past demand (A being a discount factor).
1
Viability P r o b l e m with delays:
Let us set up the general model. We have got a finite number of delays, ( O h ' ' ' , 0p) E (l:t:)P where p C N* and a set-valued map M : X p ~ X. Therefore the viability constraints set will be: K = {~ e C ( ] - o % 0 l , X ) [ ~p(0) e
M(~o(Ot),... ,cp(0,))}
One of the assumptions the Functional Viability Theorem requires that the viability domain nmst be included in the set of the A-lipschitzean functions for some )~ > 0. Thence, we will study the set s = h?. M Ca In this section we will first prove results about the simplest case which happens when the dynamics are just constrained by one delay. T h e n we will see that the other case can be generalize without great efforts. 1.1
The
one
delay
case:
We suppose that p = 1 and that:
;c -- {~ e c(] - oo, 0], x )
I ~o(0) E M ( ~ ( O ) ) } , 0 < 0 is given
No other assumptions need to be made about the set valued-map M in the first result we shail obtain. But the generality of the proposition does not provide us a useful way to compute iPA:(c;). T h a t is why we wiI1 need further results.
552 Proposition
1.1 Dpc(%0) = [D(M o %0)(0,%0(0))](1)
Proof-The next proposition involves the contingent derivative of the set-valued map M o %0.3 Let v E D~c(%0). By definition, we know that it is equivalent to: 3hl ~
0+ and %01E C ( ] - o o , h i ] , X )
such that:
{ 0
{
0 + and vl - - ~ v
such that:
0
%0(0) + h,vi e (M o ~)(0 +
hl)
We can write it too in terms of contingent cone (1,v) G TGraph(Mo~,)(O,%0(O)) and thus in tcrm of contingent derivative: v 6 [D(M o %0)(0, %0(0)](1) Conversely, Iet us consider v such tha~ (1, v) E TGraph(Mo~0)(0, ~(0)). Then, there exists vl ---+ v and hi ~ 0, which verify %0(0) + hlvl 6 ( M o ~)(0 + hi) If we construct %oi E C(] - 0% h;],X) by qol := 7r~,%0, then %0; satisfies the requirements of the definition of 'Djc(%0) and so v belongs to D~c(%0). [] We can show now a more precise result. Proposition ~zd if." (4)
1.2 If %oE Ca, M, ~ are cloaed and aleek4 respectively ~t (%0(0),%0(0)) and (0,%0(0)) ImDqo(0, ~,(0)) - DomDM(%0(O), %0(0)) = X
then V~c(%0) = [DM(%0(O), %0(0)) o D%0(0,
[D(A~I o %0)(0,%0(0))](1) C D-~I(c2(O), ~(0))(D%0(0)(1)) 3Let F : X -,-* Y be a set-valued map from a normed space X to another normed space Y and y 6 F(x). The contingent derivative DF(z, Y) is the set-valued map from X to Y defined by Graph(Of(z,y)) := TGraph(F)(x,y) 4We shall say that a set-valued map F is sleek at ( z , y ) E Graph(F) if and only if (z', y') ---* Graph(F) is lower semicontinuous at (x, y) and it is sleek if it is sleek at every point of its graph.
553
Indeed, let v E [D(M o ~o)(0,~(0))1(1). T h e n there exists hi ~ 0 and v, - - + v such that 90(0) + hlvl E M(~o(O + h~)). So ul := (~o(0 + hi) - ~o(O))/hl is a sequence of A B x , which is compact. We caa~ therefore extract from ul a subsequenee again denoted by ui such that ul ---+ u. Therefore u E [D90(0, ~o(0))](1) and
90(0) + hlvi E M(~(O) + hlu,) ~
v E [DM(90(O), q~(0))l(u)
so that
v E [DM(90(O), ~(0)) o D~(O, ~o(O))](1) [] 1.2
The
multidelay
case:
We now examine the case with p delays: ~C={~,EC(I-~,0I,X)
I ~,(0) E M ( ~ ( 0 i ) , . . . , V ( 0 p ) ) }
We also need some new notations. Let be ~5 the function defined by: ~(~) = (~,(01 + t ) , " - , ~,(0, + t)) Then we can write the functional viability constraints set in a new way: a: = {~ e C ( ] - oo, 0],X) [ ~,(0) E lVqr The next two propositions are the analogues of Propositions 1.1 and 1.2, and the proofs are enough similar to enable us not to exhibit them here. P r o p o s i t i o n 1.3 59~(~o) = [D(M o @)(0,~o(0))](1) P r o p o s i t i o n 1.4 As in ~he one delay case, if ~o E Cb,~, M,q~ are closed and sleek respectively in ('~(0),90(0)) a,~d (0,~(0)) and (5)
ImD~(0, @(0)) - DomDM(@(0), ~(0)) = X p
fl~en ~ : ( ~ ) = [DM(q(0), ~(0)) o D@(0, ~5(0))](1) We have characterizcd 9 and would like now to determine 79~(~o) where ,~ > 0 because, in fact, it is the set that will be useful in the application of the theorem (0.2). A new proposition will help us to do so. P r o p o s i t i o n 1.5 ~D~(~o)fl I n t B x ( A ) C 2?~c,(~) C g)~(~o) Cl Bx(A) Bcforc thc proof, let's state a preliminary result.
l ~ m m ~ 1.6 ~o~(90) = Bx(A) Proof-Let v E 7?jc(~o) Cl I n t B x ( A ) , then 3hl ~
0 , vl ~
Vk=l,...,p,
v such that:
0
II~dl < ,x ~o(0) + hlvl = 90i(hi) 6 M(@(hl)) where ~ol E C(] - oo, hi], X ) is defined by
{ r(0)~,~ = ~, ~ ( s ) = ~o(0) + sv~ , V~ e [0, hd ~, E C(] - oo, 0], X) and ~i(hi) E M('I'i(h,)) ==* T(h,)~i E I(. ]lvil] < A and T(0)qol = ~o ==> T(hl)~o, E C~ Therefore T(hi)~ol E KT~ and so v E D~:~(~o). []
554
2
Volterra viability problems:
Instead of involving only a finite number of delays, the Volterra viability constraints take into account the whole history of the state. We recall that the constraints set in this case is *~ :=
{9 e c~
I 9(0) e M
(i o~ a ( - e ) 9 ( O ) d e )
= ~V_r((A 9 ~ 0 9 ) ( 0 ) , 9 ( 0 ) )
}
In this section as in the first one, we prove that, for all 9 in ,U~, the "tangent" set to ~A at 9 can be computed in terms of contingent derivatives. But before we proceed in establishing the results, we need to make the following assumptions. * d E L t ( R , L : ( X , X ) ) ) . d(t)is bounded around 0 by [JAil. We denote by Supp(A) the closure of {t E EL / A(t) r 0}, and we assume that Supp(A) C 1%+\{0}. 9 M
:
X ~ X is a set-vMued map. We assume that 9(0) e M(f%o A(-a)~o(s)ds) = M ( ( A * rrog)(O)) ii) M is sleek at ( ( a * rro9)(0),9(0)) iii) D o m D M ( ( A * rrog)(O),~o(O)) = X i)
(6)
Notice that the one delay problem ea~d the Volterra problem can be seen ~ two instances of the s,'une b ~ i c problem: weighting the past states of the system with a measure on R. In the one delay problem, the measure is the Dirae measure 8o, and in the Volterra problem, the measure has the "density" A(.) with respect to the Lebesgues measure. Therefore, we can approximate a one delay problem by a Volterra problem by taking A(a) = .:\(s)frt. where A: 1% ~ R+,]IA]JL~ = 1 and A is "concentrated" around O. Before sketching the main results, we establish some preliminary lemma. L e n n n a 2.1 9 Proof--
C [DM((A
*
rro9)(O), 9(O))]((A
*
rrog')(O)) n B,\
Let v E T~jca(9). Then there exist
h.i ---+ 0+ ii) 9, ~ c ( ] i)
~,
h,],x)
bounded
a n d A - lipschitzean
such that
T(O)91 : 9 ii) 9,(h,) e }L((A 9 ~o9,)(<)) iii) ( 9 , ( h 0 - ~ o ( 0 ) ) / h { = .{ - - +
v
Let ul be the sequence of vectors defined by (A * ,'~og~)(h,) - (A * %9)(0) hi (A
Since ~r0~o is .k-lipschitzean, it is almost everywhere differentiable and thus ul converges to ~
*
~ [(A. ~o9,)( < )
= -
hi
hi
a(-s)(9,(h,
-
(A .
%~,)(h,)]
+ ~) - 9 ( 0 ) ) d ~
555
= T, .1_~, A ( - , ) ( ~ , i ( < + ~) - ~oi(O))d~ We know tha~ ~ol is ~-Iipschitzean and that for a large enough i, we have:
w e [ - h , , 0 ] , IIA(~)II _< IIAII Tlfis implies that:
ii,v. ff~, o A(-s)G~
+ s) - ~o,(0)&ll
<:
1
0
_ x;llAII f-L,, .x(< + ~)& •h, II [I 2 -- 89 =
Therefore:
(7)
--7[(A * 7ro~Oi)(hi) - (A * 7ro~o)(hl)] ---+ 0 Moreover ZoC2being A-lipschitzean, it is almost everywhere differentiable. So that: Vi
l[7r~176 '
-'~~176 ll < hi
-
and
~o~(h, + ,) - ~o~(,) ---. ~o~'(~) for ~nost all s e R
hi Since A E L~(Y{, Z:(X, X)), Lcbesguc's convergence thcorem implies that:
hi
co
Therefore :
(8)
~ [ ( A * ~o~)(h,) - (A * ~o~)(0)] ---* (A * ~or Properties (7) and (8) imply that ul converges to (A*~roT')(0). We can then write: ~o(0) + hlvi = T i ( h 0 E M ( ( A * ~ro~,)(0) + hiu 0
where hl ---+ 0 +, ul ---+ (A * rro~O')(0) and vi - - + v. Tlfis implies that: v E [ D M ( ( A * rro~O)(0), ~o(0))]((A * rro~O')(0)) [] P r o p o s i t i o n 2.2 Z)~,(~o) C [D(M o A*~o~o)(0,~o(0))](1) FI B~ Proof -M is pseudo-lipschitzeml around ((A * ~'oqO)(0), ~o(0)) because X is finite dimensional, M -1 is a
closea set-valued map sleek at (~(0), (A* ~oV)(0)) =d (cf set-valued Graves' theorem(tel)) ImDM-l(~o(0), (A * rro~O)(0)) - D o m D M ( ( A * 7ro~o)(0), ~o(0)) = X Let v E :D,c~(~o). according to lemma 2.1 , there exist two sequences hi ----+ O, vl ------+ v such that ~(0) + h,~ e M ( ( A * ~o~)(<) + o(hJ) Siuce M is pseudo-lipschitze= ~ o = d ( ( A * ~o~O)(0), ~(0)), the~e exists Z > 0 = d a sequence v ~ such that ~o(0 ) + hlv ~ E M ( ( A * 7ro~o)(/z0)
al~a h, ll~? - viii _< Zo(<).
556
Tlfis implies that v ~ ----* v so that r
+ hlv ~ 9 M ( ( A * ro~)(hl)) ~
v 9 [D(M o A * 7r0~)(0, T(0))](1)
[] We need the following lem_ma. L e m m a 2.3 T being a function of C and v an element of X , ~hen (A * %c2)(h) - (A * 7r,~)(h) = h ~ h ) v where ~ h ) 9 s
and [l~h)l[ _< ]ld[Ih for any ~mall enough h > O.
Proof-(A*%~,)(h)
( A * % ~ ) ( h ) + f ~oo, , A ( - s ) ( s + h)vds = ( A * %r + h[~ fo A ( - s ) ( s + h)ds]v =
and s
We cml then establish the main proposition. Proposition
2.4 [D( M o d * %~,)(0, io(0))](1) n [nt(13A) C 79~ (~)
Proof-Let v e [D(M o A * %~p)(0, ~(0))](1)Cl Int(B~) , then there exist hi ~ (see proof of prop 2.2) ~o(0) + hiv~ 9 M ( ( A * %~')(hl))
0 +, v ~ ----* v such that
Let us divide the proof in two steps. Step t: We define qa! 9 C(] - oo, h d , X ) by
then (cf lemma 2.3) ~,~(hl) 9 M ( ( A * rr0qo~)(hl) - hie-'(hl)v ~ Since M is pseudo-llpschitzean around ( ( d * rr0~)(0), qo(0)), there exists l > 0 and a sequence v~ such that
~(o) + h,.~ 9 M((A . %~)(h,) a,nd
h&4 -v~ll _< lllh,~h,),?ll Thence, we can write,
{ ll(A*~ro~)(h,)
(A* ~ro~)(O)l[ ~ Ahd[A[IL, Ilv~ -v~ ~ lllAllh, ll~ll ]I(A 7ro~o~)(hl) (A 7r0cy)(hl)ll _< )IAllh~]Iv~ -
*
-
*
and v~ ~
v and ~(0) + hiv~ e M ( ( A * ~roqo~)(h0
557
stcp 2: We define ~ E C(]
-
0% hi],X) by
Then there exists a sequence v~ - - ~ v satisfying
~o(0) + hiv~ E M((A * ~ro~o~)(hl) and such that
{ I M - ~?11 < ZllAIIh, llvt - v~ II(A * ,.0~(h~) - ( A . ~0~)(h,)ll _< IIAllh~llv~ - v~ By repeating these steps we construct a family (i,k) E N x N*
{
W E c(] - oo. h,]. x ) r(0)~, ~ = ~,~(.) = ~ ( 0 ) + ~.~-~ for ~'~+t(hi) E M ( ( A * r
{~/k}(i,k)ENxN. of functions satisfying for all
~u s e [0.
hd
I1.~*' - ~,~11 < llldllhill,r - .~-111
II(A * ~ro~*G(hl) - (A* 7r0~p~)(hi)ll - IIAIIh~llv~ - .~-~11
We then have for all (i, k) E N
x
N*
Ilv? - v~ll --- IM - "?II(E~=~(IIIAIIA')"-~) ___(llv~ll IIAIlhl)(1 - tllZllhl)
(9) which implies (10)
II(A * #o~O~+X)( hl) - ( A . ~o~)(hi)ll < E~=, k IIA[I I M ' - vlp - 1 [lfq2 + IIA[I IIv~ < (llv~ IIAIIhY)/(1 - ZlIAIIhY)
_
Therefore, if i is sufficiently large, then we actually stay, while constructing the sequences ~ and v~, in the neighborhood of ((A * 7roqa)(0),~p(0)) where .~I is pseudo-lipsclfitzean. Moreover, since llvll < ;~, we have for i sufficiently large:
vk e N ,
I1~r <
Let us fix i "sufficiently large". Then vlk is a convergent sequence whose limit we denote by ~. Let us define the function j l 6 C(] - co, hi], X ) by
{ T(O)%i= ~,(a)
=
~,
T(0) + s ~ f o r all s E [0, hl]
Since I]~7]1< A, ~6, is A-Iipsclfitzean. r is also bounded. Then we can write Vk ((A * Tro~pki)(hi),qa(O) + hive) 6 GraphM ( A . ~o~t )( hi ) ---* ( A . ~o~i )( h, ) when k ----* +oo ~(0) + h i ~ ---* ~(0) + hlW, w h e n k ~ +~ Moreover GraphM is closed, so that r
E M((A * 7rog~i)(hi))
558 Consequently, we have
r ~ c(] - oo, h,], X)
T(hl)r e ~C~ (r
- ~(0))/(h,)
= <
The relation (9) implies that lllAllh, [ 1 ~ - v~ ~ 1 = ~ h ,
o
I1~'II
Therefore v; ----* v and thus v E DK~(~). []
References [1] AUBIN J.-P. (1981) A dynamical, pure czchangc economy with feedback pricing, J. Economic Behavior and Organizations, 2, 95-127 [2] AUBIN J.-P. &: FRANKOWSKA H. (1990) SET-VALUED ANALYSISBirkhs Control: Foundations and Applications
System~ and
[3] AUBIN J.-P. (to appear) VIABILITY THEORY [4] AUBIN J.-P. & CELLINA A. (1984) Grundlchren der math. Wiss. # 264
IFF ERENTIAL INCLUSIONS, Springer-Vcrlag,
[5] HADDAD G. (1981) Monotone trajectories of differential incIuaiona with memory , Isr. J. Math., 39, 83-100 [6] HADDAD G. (1981) Monotone viable trajcctorie~ for functional differential inclusions, J. DifF. Eq., 42, 1-24 [7] HALE J. (1971) FUNCTIONALDIFFERENTIALEQUATIONS, New-York, Springer-Verlag
Maximum PrincipleOf Optimal ControlsFor A Nonsmooth SemilinearEvolution System* Jiongmin Yong Department of Mathematics, Fudan University, Shanghai 200433, China
Su rrnrn~,-'y For finite dimensional case, Clarke proved a Pontryagin type maximum principle for controlled system (1)
~Ct) =
f(t,z(t),uCt)),
with f being Lipschitz continuous (not necessarily C 1) in x. However, to our knowledge, for semilinear distributed parameter system (2)
~(~) = AxCt)+ f(t, ~(~), u(t)),
similar result was only proved for the case that f is C l in x. The purpose of this paper is to relax this condition for a class of systems of form (2) which contains systems governed by semilinear parabolic partial differential equations and ordinary differential delay equations. The idea is to approximate the original nonsmooth problem by a sequence of smooth problems. For these smooth problems, we use the Ekeland's variational principle to obtain a kind of approximating maximum principle. Then, we manage to take the limits to obtain the maximum principle for the original non~mooth problem. The point is that we only take the limits for the maximum conditions and the adjoint systems and it is no need to discuss the limit of the associated variation systems! We know that since our original system is not necessarily smooth, the associated variation system is not easy to treat. The avoidness of such a system allows us to present the Pontryagin type maximum principle for our optimal controls successfully. By assuming the target set Q to be of finite codimension in the state space, the costate is ensured to be nonzero.
* This work was par~ially supported by the Chinese NSF under Grants 0188416 and 1880414.
560 w
Introduction. For finite dimensional case, Clarke proved a Pontryagin type maximum principle [5] for controlled system
(1.1)
:~(t) =
fCt, xCt),u(t)),
with f being Lipschitz continuous (not necessarily C I) in z. However, to our knowledge, for semilinear distributed parameter system (1.2)
:~(t) =
AxCt) + fCt, xCt), uCt)),
similar result was only proved for the case that f is C 1 in z ([7-9] and see also [1,10]). The purpose of this paper is to relax this condition for a class of systems of form (1.2) which contains systems governed by semilinear parabolic partial differential equations and ordinary differential delay equations. The idea is to approximate the original nonsmooth problem by a sequence of smooth problems. For these smooth problems, we use the Ekeland's variational principle ([6]) to obtain a kind of approximating maximum principle. Then, we manage to take the limits to obtain the maximum principle for the original nonsmooth problem. The point is that we only take the limits for the maximum conditions and the adjoint systems and it is no need to discuss the limit of the associated variation systems! We know that since our original system is not necessarily smooth, the associated variation system is not easy to treat. The avoidness of such a system allows us to present the Pontryagin type maximum principle for our optimal controls successfully. By assuming the target set Q to be of finite codimension in the state space, the costate is ensured to be nonzero. w
Optimal Control Problem, M a i n Result. Let us start with basic assumptions. (H0) X is a reflexive Banach space with strictly convex dual X*, U is a metric space with metric p and T > 0 is a positive number. By [2], we know that when X is reflexive, one can always replace the norm by an equivalent one so that its dual is strictly convex. (H1) A : P(A) C X --, X generates a compact Co-semigroup eAt on X and there exists a sequence of finite dimensional projections P , : X ---, X,, such that
(2.1)
P . e At
=eAtp,,
lim
sup
.-.oo .eU,[xl
Vt > O, n >
1;
lira
[eAt(I-P,)f(r,x,u)]=O,
I P . z - zl = O,
Vz ~ X,
VtE(O,T],rE[O,T],g>o.
(H2) The maps f : [0, r] • X • V --, x and fo : [0, r] x X x U ~IR are strongly measurable in t, Lipschitz continuous in x and continuous in u. Moreover, there exists a constant L > 0, such that for all t 6 [0,T], z,~ 6 X and u E U,
(2.2)
if(t, = , - ) - f(t,~,u)l, If~
(2.3)
If(t, x, u)l , If~
f~ z, u)] < L(1 + Ixl).
< LI=- ~1,
561
We define U = {u(.) : [0, T] --* U, u(-) is measurable }. As usual, a function ~(.) 9 C([0, T];X) is called a mild solution of (1.2) with initial state x0 G X, if it satisfies the following integral equation:
(2.4)
t.t
x( )=eA%+Jo
teI0,rt.
I
u()X(:)
It is clear that under (H2), for any zo 9 X and u(.) E U, there exists a unique solution of (2.4). We define such an z(.)to be the response of thesystem under the control with the initial 8tare xo. Next, associated with the system, we have a cost functional of following Lagrange form: (2.5)
J(u(.)) =
//
f~
Concerning the terminal constraint, we a..~sume the following: (H3) The set Q c X is convex and closed. Moreover, it is of finite codimension in X, in the sense that there exists a q e Q, such that span{Q-q} - span{z-q [ x 9 Q} is a finite codimensional subspace of X, in which Q - q has a nonempty (relative) interior part. Then, the terminal constraint of our problem will be (2.6)
z(T) E Q.
Now, we can state our optimal control problem. Problem C. Minimize functional (2.5) subject to state equation (2.4) and the terminal constraint (2.6). Any pair of state and control (.~(-), ~(.)) satisfying (2.4) and (2.6) and minimizing (2.5) is called an optimal pair. Remark 2.1. Actually, the constraint (2.6) gives a restriction on the control u(.). Usually, we call a control u(.) E U to be admissible, if the corresponding state ~(.) satisfies (2.6). If we denote all such controls by Uad, then Problem C is equivalent to minimizing (2.5) over U~,~. Remark 2.2. It should be pointed out that only the Lipschitz continuity of f and f0 in x have been assumed. Thus, the system as well as the cost functional are not necessarily smooth. This is the main point of this paper. Remark 2.3. Hypothesis (HI) will be used for approximation. It is not hard to see that if X is a Hilbert space, A is a self-adjoint operator with a countable set of spectrum
and (2.3) holds, then, (2.1) hold. This actually contains the case of semifinear parabolic partial differential systems. (It4) For any sequence (Urn(')} 9 U with property
(2.7)
Q) = o, rn--+ oo
562
one has lim J(um(.)) > inf J(u(.)). m--.oo - t~(-)ett
(2.8)
We note that for the case Q = X, i.e., the terminal state is free, the above is automatically true. Also for the case that
/(t,~., u) = ACt, z ) + B u , f~ z, u) is convex in u and U is a reflexive Banach space (the situation similar to that given in [4]), the above (H4) holds. Next, we introduce the usual Hamiltonian (2.9)
H(t'x'u'r176162 =r176176
+ (r
v(t, x, u, r 1 7 6 1 6e2 [O,T] x X x u x B x x*,
where (., .) is the duality between X and X*. Our main result is the following Thereom 2.4. (Maximum Principle) Let (HO)-(H4) hold. Let (~(.), f~(.)) be an optimal pair for the Problem C. Then, there exists a pair (r r e/RxC([5, T]; X*), such that (2.10)
(r
(2.11) (2.12)
r r
r
=eA'(r-t)r
+
fT
r 0, < 0,
eA'('-t)f(r)dr,
~ E [0, T]
r176162 (2.13)
H(t, ~(t), ~(t), r
(2.14)
(r
a.e. r 6 [0,T], r
= men~ H(t, ~(t), u, r ~ ( T ) - Yl) < 0,
r
a.e. t C [0, T],
Vyt E Q,
where OxH(r, z, u, r
r in (2.12) is the Clarke's generalized gradient ([5]). We see that the statement of the maximum principle is almost the same as that for smooth problems except for the adjoint system (2.12) which, roughly speaking, says that the function r satisfies the following evolution equation: (2.1/;)
(b(t) 6-A*r162176162
t 6 [0,T].
w Approximation of Control Problem. In this section, we are going to construct a sequence of smooth problems to approximate the original nonsmooth one. The idea is adopted from [41. Without loss of generality) we ~sume the following:
563
IleatllL(X) < Me w',
(3.1)
Vt >_O,
for some constants M > 1 and w 6IR. Also, we may let dimX,~ = N,, < oo,Vn _> 1. Then, we can choose a basis {e~', 1 <_ i <_ N,~} for X , and for the later purpose, we assume that ~ 1 [e['[2 = 1, n > 1 (In the case that X is a Hilbert space, we may let (el', 1 < i < N,} be an orthonormal basis for X,~ instead). Then, one has a~' 6 X*,I < i < N,, such that 2/.
P,x = E(a'~,z)e'~,
(3.2)
Vz e X.
i=1
From (2.1) and the uniform boundedness theorem, we know that there exists a constant Lo > 0, such that
(3.3)
liP,IlL(x) _< Lo,
v,~ _> 1.
Next we define A, :JRto. ~ X , as follows
(3.4)
A, Ce) = ~ e : ~ ' ,
w = (e,,
,e,,.) e ~t"~
i=1
Then, we define V(t,z,u) e [0,T l x X x U,
(3.G)
{/-(t,~,~) = :(t,e:.-~.~,,o,,4e.(o)do, fa"~
:o,,,(t, ~,,,) = f~..~ /o(t, p: - 1A.o,, ,,):.(o)~o, where : . ( ) ~ C~(a#-) is a usual mo~U~er We have the following result which is important but not surprising. Lemsna 3.1. The functions f " and/o,, are continuously Frechet differeatiable ha z ~nd far aN t E [0, T], x, "~C X u C U and n >_ 1,
{ [/'(,,=,u) - f'(t,~,u)l , l/~ (3.6)
~) -/~
~,~)l< Lo/;l~- ~I,
if.(t,z,u)_f(t,z,u)l, lfo,.(t,z,u)_fo(t,z.,u)l
Next, for any u(.) ~//, we consider the following (smooth) evolution system
/: We will show that the above (smooth) system approximates system (2.4) in a satisfactory sense. Before that, let us first give the following which reveals the role of (2.1) in our approach. L e m m a 3.2. Le$ (I:tl) and (1:I2) hold. Then,
564
(3.8)
lira I(I- P,,)z(t)[ = o,
Vt e [0,T],
uniformly in u(.) E U. The proof is straightforward. By the above lemma, we can show the following Theorem 3.3. Let (ttl) and (112) hold. Let Ua('), u(.) E U satisfy the foflowing lira u,(t) = u(t),
(3.9)
a.e. t E [0, T].
I'I,--I.QO
Let Xa(') and =(.) be the solutions of (3.7) and (2.4) corresponding to Ua(') and u(.),
respectively. Then,
(3.10) (3.11)
I=.(t)- =(t)l--o, ~m
f~
uniformly in t E [0, T].
zn(t),u,(t))dt =
f~
Now we would like to construct a sequence of smooth control problems. To this end, we let (~(.), e(.)) be an optimal pair of the Problem C (which is assumed to exist). Let :~a(') be the solution of (3.7) corresponding to ~(.). Then, by (3.10) we see that (3.12)
nlim
I~a(t)-- ~(t)l =0,
uniformly in t E [0, T].
Thus, by the terminal state constraint .~(T) E Q, we may find a sequence of convex and closed subsets Qa of X, such that
(3.13)
Q c Qa,
n >_ 1;
~,a(T) E Qa,
lira PH(Qa, Q) : o,
a --,,.~ o o
where PH(', ") is the Hausdorff metric. Then, we introduce the following approximating control problem: P r o b l e m Ca. Minimizing the following functional
JaCu(.)) =
(3.14)
//
f~
z,~(t),uCt))dt,
subject to evolution system (3.7) with u(.) E ~/and
xa(T) E Qa.
(3.15)
It is seay to see that for some constant C > O,
,(.)eu
//
/~
> -C, -
> 1. -
Now, we let tl~ be the set of all controls u(.) E U, such that the corresponding solution Za(') of (3.7) satisfies constraint (3.15). Then, by the construction of Q,, we see that
(3.17)
~(.) E Ua~d, Vn >_ 1.
565 Next, let us denote (3.18)
J,,=
inf
,,(.)eu2~
J,,(u(.)).
Then, we have the following result.
Theorem 3.4. Let (H1),(H2) a~d (H3) hold. The~, (3.19)
nm J: = J(~(.)) = J. f/,----+ OO
Pro@ First of all, by (3.17), we have J,, < Jn(fi(')). Thus, it follows from Theorem 3.3 that
~l~ J,, < J".
n-..-+ (:~
Now, suppose (3.19) is not the case. Then, one can find a constant 6 > 0 and a sequence
{urn(.)} c u~, such that Jm(um(')) <
(3.20)
f _ 6.
From Theorem 3.3, we see that (3.21)
lim [Jm(um('))- JCum(.))] =0.
m , - - ~ OO
Thus, for rn large enough, we have (3.22)
g(um(')) <
-~.
On the other hand, form (3.13), we can prove that (3.23)
lim yEQ inf Iz(T;um(-))- vl =0. m---*oo
Then, we see that (3.22) contradicts (H4). Hence, (3.19) holds. [] We should note that even though we assume the original nonsmooth problem having an optimal pair (~(.), a(.)), the Problem C,~ need not have an optimal pair. w
P r o o f of the M a y l m u m Principle. This section is devoted to prove the maximum principle stated in Section 2. Let (~(.), fi(.))be an optimal pair of Problem C. We adopt all notations introduced in Section 3. Let d(u(.),v(.)) = meas{~ e [0, T] I u(t) r ~(~)}, for all u(.),v(.) e Zl. Then, (/~, d') is a complete metric space ([r,gJ). For any n >__1, we define (4.1)
a ... rz ~,*,rT ,,~2x~/2 r,,(u(.)) = {dq.(z,~(T)) 2 +-(-oo,J.-,/,,lt , , Vu(.) E/./,
where x,,(.) is the solution of (3.7) corresponding to u(.) and
566
f' J0 dQ. ( z ) =
inf{ly- zlly 9 O,,},
d(_~,~.o_,/.j(=o)
=
(~o _ (2. - ~1)+,
Vx 9 Vz ~ 9 IR.
It is clear that F,(.) is continuous. Also we see that
(4.3)
F,(~(.)) > 0,
v~(.) 9 U.
Now, we let 2,,(.) be as in Section 3 and 2 % ) be defined as in (4.2) replacing (x,(.), u(.)) by (2,~(.), a(-)). Then by Theorem 3.4, we know that as n - , oo, (4.4)
0 < a(n) = Fn(~(.)) = d(_oo,j'_,/a](J,(~(')) <_ IJr,(~(')) -- 2,1 + 1 __, 0.
Thus, by nkeland's variational principle ([6]), we have a u'~(.) 9 U, such that
~ ~(~"()'~()) <- 4~, r,(~-(.)) ~ r,(~(.)), [ F(~(.)) >_F.(~"('))- v~-~'(."C'),4")), w(.)eU.
(4.~)
Next, we let z"(-) be the solution of (3.7) corresponding to u"(.) and
(4.6)
z~
2' I~
Vt9
r].
Now, we fix any u(.) 9 U. As in [8,91, for any P 9 (0,11, there exists a measurable set E o C [0, T], with measEr = pT, such that if we let u~ (.) = u" (')X[o,T]\s, (') + u(')XE, (') and let x~(.) be the solution of (3.7) corresponding to up(-) and
/:
(4.7) then, one has
{ ~(t) ~~
(4.s)
:
:
~-(t) + p~.(~) + o(p), ~~ + pC~ + o(p),
uniformly in t E [0, T], where rt
(4.9)
,,(t) =
'
Jo
~('-~lf2(~,~(,),~,(~))~,(~)d~
+ fo' ~*('-~) [F(~, ~"(,), ~(,)) - f"(~, ~"(,), ~"(~))]d,,
t 9 [0, T],
567
~oct) =
(4.1o)
/:
yo,,,C~,~.Cr),,,.C~))~.C~)d~
+ fo~ [fo,n(~, ~.(~), ~(~1) _ fo..(~, ~.(~1, ~.(~1)]~, t 9 [o, T]. Then, proceeding as in [9], we have
(4.11)
0 ~o -V'Y(~T < (-~.,.',,(T))+ 4...(T),
where
(4.12) 4, = dQ.(z"(T)) Fr,(ur,(.) ) a,,, a,, 60dc2.(z"(T));
.~o = d(_~,:~
(z~
F.(,,"(-))
'
and
(4.13) By (no), ~ in [0], we k.ow that O~Q~ = {a.} ~s ~ si.glton. Thus, ('~,,, ~o) is independent of (~,,(.), r A~, by the con,.exi~y of Q,, a.d from (4.5) and (4.~1), one has
(4.14)
(,~.,y, - .~(T)) _< I="(f) - .~(T)[ = 5, ~ 0,
(n --~ oo),
Vy~ E Q,~.
Since Q is of finite codimensional in X, thus, by a similar argument as in [9] (see [7] also) together with (3.13), we can obtain a subsequence (still denoted by (~r,, 40)), such that
(4.15)
(~-,4 ~ -~ (4, 4 ~ e (X* • ~) \ {0}.
From (4.12), we see that
r
(4.16)
_- _,~o < O.
On the other hand, we let
(4.17) and let r (4.18)
Cr,(T) = -~,~,
r = -r -o
Vn >_ 1,
be the unique solution of the following system: ~r~(t) = eA'(T-t)42,,(T)+ fTeA'(r-t)Hn(r, xr*(r),u'*(r),42~
e [o, T], where
(4.m)
It'*(t, x, u, 42~ 42) =--42~176(t, x, u) 4- (42,f'*( t, x, u)), ~r ,o , ~,,) E [ 'O , T ] • 2 1 5
"*,n>_I.
568
Substitute (4.1S) in to (4.11), we have (see [9]) v~T (4.20)
_> (r
~,,(T)) + r
o 0
= /oT[H,~(t, xn(t),u(t),r162
_ Hr,(t,z,~(t),un(t),r162
This can be regarded as an approximating maximum principle. Then, it is natural to expect that we can take the limits to get the final result. The rest of the proof is to achieve this. We note that (by (4.5)) (4.21)
{ Iz•(t)- ~(t)] ~ 0 ,
ir
/
_
r o 0,
uniformly in t E [0,T],
r
• r
On the other hand, from the equation (4.18) and the Gronwall's inequali~, we see that (4.22)
Ir
_< c,
Vn >_ 1,t 6 [0, T].
Thus, by the reflexivity of X, we may assume that
{
/~(.,x"(.),u"(.))*r
(4.23)
~ r
in L2(GT;X),
fo,,(.,x"(.),u"(.)) L A(.), in L2(O,T;X), r ~ r in L2(O,T;X),
for some functions r A(.) and 0(.). Next, we will derive the equation which the function r satisfies. To this end, we need the following result. Lemma 4.1. Let eAt he a compact semigroup on reflexive Banach space X. Let gu(') 6 L~176 with
(4.24)
Ig,(.)l~(o,r~x) <-- C,
W >_ 1.
Then, there exists a subsequence of ga(') (stiU denoted it by itself) and a g(.) E L ~176 (0, T; X), such that for all t 6 [0, T],
~m I
(4.25)
/:
eX(~-r)Cg,~(r) - g(r))&" l = 0.
By noting the fact that the semigroup eAt is continuous in the uniform operator topology for t > 0 ([11]), one can easily prove Lemma 4.1. Now, by the above lemm% we can assume the following (see (4.18), (4.22) and
(4.23)): (4.26)
r
o r
strongly in X,
Yt 6 [0, T].
569 Then, for t < T, we have (4.27) ir ) - ea'(r-t)~b(T) _< I~P.Ct) - r
/"
ea'(,-t)--,, ~,~ ~-, ~"(~-), ~(~), r176r
+ leA'(r--t)CCr, CT) -- r
+ ML
+ Omens(u" # a) + 01r ~ - r176 I -~ o,
.
-*
f
e~Cf-t)lr
) - ~p(T)ldr
oo.
While, by [4] (Proposition 1.12 and Lemma 5.4), we know that if (4.28)
H~"(',z"('),~('),r176
~s'('),
inL~176
then, (4.29)
;(r) eO~H(r,~(r),~(r),r176162
a.e. r E [0,T].
Thus we obtain (2.12). While, the rest conclusions follow easily.
[]
References
[1] N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, North Holland, New York, 1981. [2] E. hsplund, Averaged norms, Israel J. Math., 5 (1967), 227-233. [3] A.V. Balakrighnan, Applied Functional Analysls, Springer-Verlag, New York, 1976. [4] V. Barbu, Optimal Control of Variational Inequalities, Pitman, Boston, 1984. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [6] I. Ekeland, Nonconvez minimization problems, Bull Amer. Math. Soc. (New Series), 1 (1979), 443--474. [7] H. O. Fattorini, A unified theory of necessary conditionz for nonlinear nonconvez. control systems, Appl. Math & Optim., 15 (1987), 141-185. [8] X. Li and Y. Ya~, Maximum principle of distributed parameter systems with time lags, Distributed Parameter Systems, Lecture Notes in Control and Information Sciences, Springer-Verlag, New York Voi.75,410--427, 1985. [9] X. Li and J. Yong, Necessary conditions of optimal control for distributed parameter syatems, submitted [10] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. [11] A. Pazy, Semigroups od Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [12] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York, 1962.
NE~ ALGORITHMS O~ SOLVING EXTREN~L PROBLEMS Rafail Gabasov Byelorussian State University,
220080,f/insk,USSR
Faina M.Kirillova Institute of )~thematics,
22060~,
rvinsk, USSR
Algorithms os solving mathematical programming and optimal control problems using a new approach to the solution of linear programming problems are discussed. This approach has been developing in Einsk (USSR) from the begimling of the 70th. Numerous experiments by computer carried out for linear programming problems (in three computer centers independently) have demonstrated advantages of the adaptive method (R.Gabasov, F.N.Kirillova,
O.I.Kostuykova,
1980) and
another methods regarding to the simplex method and other known methods. 2rinciples of the adaptive method were used while algorithms of solving mathematical programming and optimal control problems were constructed. The paper gives a brief account of recent results obtained by participants of the Y~nsk Seminar on constructive problems.
theory of extremal
They deals with exact finite algorithms for solving optimal
control and nonlinear programming problems tion of nonlinear functions), ciples.
(using network interpreta-
support maximum and
6 -maximum prin-
Consider the classical optimal control problem
1 .
J(u) i
(1) - ~ Cx, u , ~ : ) ,
~(~(~),-~)
~0
h (.:~ C~'))
-
~(~.)
=:r.
,
(2) (3) (4)
,-t. e T , o,
(5) , Piecewise-continuous
R"
Rt
vector-functions
R"
~(s
k
s E T , which take on the
values from the set (5) generate continuous trajectories ~(%), s ~ T, of the system (2) satisfying the phase
(3) and the terminal constraints
571 (3),(4).
They are usually called admissible.
is called optimal if the cost function Complexity
I) infinite number of nonlinear dimensionality
are chosen;
3) nonlinear nature
Lhe problem
(I)-(5)
constraints
in finding
of transformations.
can be considered
efficient
methods
to the problem
of formation
which allow to solve
and numerous
lems are known.
methods
operation gradient
~rom the general point of
of passing feature
to finite-dimensional
the limit
(or its analogue) processing
dimensional
simplicity
of the basic method
consists
in calculating
of current
information
On the other hand,
finite-dimensional methods as well.
we introduce
problems However,
are always finite and determined
are used in the mathein our methods
interval
They accumulate
of the problem
and linear-quadratic
only by optimal control differ fundamentally
optimal
(I)-(5).
optimal
intervals
2. ;~&ile solving the problem
,
d ' 6 ~ ~,
Various methods known among
6.
,
(I).-(5) in
shortening
the time
The methods are finite for linear
control problems,
(I)-(5)
i.e. their realization
6"e A"
,
linear programming
A ~ R""
of solution are known for the problem
them is the simplex method.
of primal and
the principal part is ful-
filled by a linear model which is an interval problem:
d,
They
structures~
from the tradi-
by computer needs a finite number of full integrations conjugate systems and a finite storage.
c,x,
do
to solve the extremal problem.
As a result we obtain a method of solution of the problem approximations
a finite-
of solution.
unlike with mathemati-
the sizes of auxiliary problems
not depend on the accuracy needed
tional ones.
the
and a step along it. Instead for a com-
matical programming
which successive
is
of these principles.
linear problem with an efficient algorithm
cal programming
prob-
of the suggested approach
which in many known approaches
prehensive
(I)-(5) with a given
to the object being opti-
method of realization
On the one hand we abandon
consists
is not new: both various linearization
Characteristic
defined by a particular
(I)-(6)
of this comple-
of linear finite-dimensio-
the problem
mized and a finite storage of computer. view the idea of the approach
By virtue
as one of the ultimate
accuracy using finite number of references
methods
on output .signals ( ~ ) ,
ofthe space from which elements ~ s
The main idea of our approach
nal problems
control
the optimal value.
of the extremal problem is defined by three factors:
2) infinite
xity.
An admissible
(I) reaches
(6)
(6). The most
This is a cause for many
572 scientists
to confine
their investigations
reduces to the classical ting effective constrained
algorithms~
The same conclusions
optimal control problems
zation problems when methods
reduce
(the penalty function
of solution are grounded
However,
of solving complex optimization
For example
(mathematical
and finally
there exists a nt~ber of optimal control
package
LP ASY realizing
problems
such peculiar programming
solution
the known methods
software
of the phenomenon
me-
as methods
The fact is that an increase
which caWt be solved using the most powerful
source
to finite-dimenprogramming
to a more complicated
problems
programming
minimi-
The same is said
on a reduction
problems.
of solution leads
auxiliary problems
become helpless.
are often used when
in reality such a way cannot be considered
of a given accuracy of "simplified"
"the problem
to unconstrained
methods).
sional nonlinear programming problems thods).
to the phrase
L2 problem which can be solved by the exis-
and perfect
the simplex method
[d]
is clear enough for specialists.
generated from optimal
The Linear
control problems,
have
features which are seldom occured in general.linear
problems
and they are not considered
by the simplex
me-
thod created in the middle of the 40th. In this connection the problem method
a new method was created in ~ n s k
(6). Its characteristic
called a support preserves ons: The support unlike
are as follows:
the feasibility
is similar to the basis
of vectors
result
of decreasing
all the variables varies;
iteration
at iterati-
solution and
element;
a suboptimality
which do not satisfy
rion vary at the iteration. variable
~
of the simplex method but
in the method as an independent
is based on a principle
3) an iteration estimate.
the optimality
solution is improved at an
but the support is improved
5) the solution can be stopped using the suboptimality s -optimal
As a
crite-
As for the simplex method there only one
4) not only a feasible
(as in the simplex method),
on fo~ any
I) the
~ ; 2) a special construction
the latter itTis separated from the feasible
participates
well;
features
starts with any initial vector
to solve
feasible
solution with a preassigned
9 The method called "adaptive"
as
criteriaccuracy
~ 0
experiments
programmed
both on general problems
random-number
generators
mal control problems. published [ L , 3 ] . 3. The continual tion elements
~ny
and
and on special problems results
by different
obtained from opti-
of the experiments
constraint-inequality
of extremal problems.
~ested in numerous
(6) constructed
have been
is one of the complica-
Such a constraint
can be taken
573 into account
using the semi-infinite
;
d. -<~:~; d"
k C+-) * c , d . , d . ,
x
linear programming
problem:
~ . ; ~ , (-L) , ~ "Cr ) ~
~
The optimal control problem
](u)
--c'x(~')--ma.z
, A: : , 4 ( + . ) z
+5(~-)u
, x(t)
--x.
,
with phase constraints
in
the
class
of
functions k I
where
the base functions
set of parameters
reduces plicated
~j
~j ( ~ ) ,
i " ~
, j - ~ ,k
d, i to the problem
,
, 4; & T ,
satisfies
, (7). The problem
than the problem
, (7) is essentially
(6). It is investigated
Slater condition is fulfilled.
a n d the
the constraints
more com-
as a rule when
7~aen the constructions
of adaptive
method are generalized to investigate the problem (7) new essential features arise, So, the support ~:~p of the problem (7) is now a
IJ,
rl->-lI,[
While passing
P J .pl
from the problem
menon is observed: problem
(6)
there
are alwavs
basic
is no longer
appears
sible solutions
to be true:
among optimal
The theoretical Dr. O.i.Kostyukova computer programs zations.
(6) to the problem
if among the optimal
then this statement the statement
grounds
feasible
solutions
P. V. Caishun.
4. The second complexity
(7). Instead
there are always superbasic
for the problems
and Dr.
of the
solutions(:zi.
feasible
feasible
for solution
(7), a new phenosolutions
true for the problem
element
fea-
of the problem
worked out
(7) and its generali-
(infinite
dimensionality
was investigated
of a for
the optimal control problem: :
cz(~:')----
m~
,
& - Ax
(7).
(7) were developed by
V.C.~edvedev
of the problem
space on which the optimal problem is given)
]'(~)
/(~,,e
Ik
Ck r T , k e K,~p , t h e set of induces , k6 , possessing the following properties:
totality of moments and the set ]~up ~
.~,
(s)
574 This problem
reduces
~c(t)~(~)o~
-ma..z
to the semi-infinite
,
SM~)~(~)d~
T
linear programming
problem
= ~ , lu(~)l , ~ , ~ ~ 7
(9)
T
Its complexity
is defined by infinite
from which oNtimizing
elemengs
dimensionality
~(@)
It is known that the use of L.S.Pontryagin's to solve
the problem
(8) leads to a nonsmooth
lem which cannot be efficiently cannot also manage function L-problem optimal cases
to overcome
being nonsmooth.
maximum principle
boundary value prob-
solved yet. Dynamic programming difficulties
because
of the Bellman
Long ago hopes were set on the method of
of moments worked out by N.N.Krasovsky
control problems
of a space
are chosen.
to dual problems
(8) the dual problems
which reduced
of finite
are finite-dimensional
dimension. and convex.
the In Then
it seemed that using the "well"-developed convex programming methods for their solution one can effectively construct optimal controls for the problem
(8). Practical
showed that for optimal
a ~ 3
controls
experience
of numerical
very often it is impossible
by this method because
solution
to obtain
of strong influence
off errors on an acuracy of calculations the dual stage.
of switching points
A new approach following elements.
(8) is based on the
to solve the problem
of roundduring
Generalizing -the adaptive method constructions. Along with the admissible control m = (a(~), ~ ~ 7 ) we introduce the support
, ke K~.p),
T,.p={~ assigned
~a
as
sets of integers
a totality of moments ~ such that ~
P = m , p - ( kj ( ~ k ) , j e J , . ~ e K.. r ) ,
Q
The vector $ = Q'Cs.p . C,r e : ( C ( l ~ k ) , j 6 is called a vector of potentials.
The trajectory the trajectory @
:
- A'~
are brought
,
Support a
= to
control
= ~, , a 6 Ksu P ) ,
of the primal
of the conjugate c
-
H' 0
the support
and
system
(I) and
system
,
control
~,
~e~
-
to prove
(t) , ~ ~ T ,
~(~). ~(%).~ an H ( ~ , ~ , ~ )
T ,
Y~ximum Principle.
and sufficient support
•
~(s
to correspond
This allows control
~(r
~ (t) , t ~ T ,
~k ~ ~ =m
] ~
For the optimality
in the normal problem
that such a support {U, T ~ p ~
~s~p
exists
and the corresponding
T, of the primal : ~'(A~ * 6~)
of an admissible
(8) it is necessary that along the trajectories
and the conjugate systems Hamiltonireaches the maximum value
575
H(~({),~({),~({)) Let
Support optimal Tsup
6 it
is
S
lu I _ ~ .
along {~,-[,.~,x(s
that
control
there
~(%), h ~ - [ ,
to be 6
exists
-
a support
the condition
holds
.<
{u,T~.rJbea
Let
,
F o r an a ~ a i s s i b l e
n e c e s s a r y and s u f f i c i e n t
such that
{~T
number.
-rv~ximum P r i n c i p l e .
T
not •
if
= ma~ H(z({),~({),u),
E ~ 0 be an a r b i t r a r y
support
control
for a givenE~,O.
the corresponding
satisfies
on w h i c h
trajectory
the terminal
the m e n t i o n e d pT.inciple is
Construct
~({)
constraint
the function
9 s e -[ ,
then
cU({)
of
the system
, ~ 6T
,
(8)
is an optimal
c ontro i. iVhen l~Ha(z')-gll-~ i (
d
is a p a r a m e t e r Consider
I{'-{~l
the case ~I H~c(s
constraints is equal
of the function
s & ~
~(s
the optimal
and localizes
to ~ ,
the switching p o i n t s
by the Newton method.
the limited on-line
storage c a p a c i t y
on the dynamic
s~stem order.
successive
Such a p p r o x i m a t i o n s
with m u l t i - d i m e n s i o n a l
r lset
conditions
from items control
C
on trajectories
characteristics).
is required. a finite
sizes
The
the finishing proce-
system of n o n l i n e a r Regardless
time which
The last p r o p e r t y
and time
characteristics)
this m e t h o d by computer of a
depends
is a result
only
of the
in the course of which o p t i m a l of the p r o b l e m
can be called l o n g i t u d i n a l
The constructions
poly
a finite
approximations
are a c c u m u l a t e d
and
of the support p r o b l e m
(its discrete
To realize
the solving occupies
type
(6), ~. - ~~
is the number of zeros
(continuous
given a c c u r a c y
segments
of the form
is o b t a i n e d using
The latter consists::in s o l v i n g
special
with the help of functions
~,
Solution
control structure
final form of optimal control dure.
,
I 9 Taking into account
in this p r o b l e m is equal to rn
the n u m b e r of variables
equations
is a r e c o m m e n d e d value).
s 6 T , we form a support p r o b l e m
The number of basic
defines
to the finishing procedure
- ~I17 o. 0~ ~t'-s
=T'(s
zeros of the function A(s c(~), I(s
we pass
of the method, ~ = 0 0 ~
(8) shorten.
ones.
3,4 can be g e n e r a l i z e d
which takes on the values
: can be given in the form:
for p r o b l e m s from the
hebo
dary
576
systems are changed for non-stationary
Stationary
any difficulties. to consider
Instead of the fixed interval
the intervals
[ i, , i" ]
systems without ~
it is possible
with non-fixed
allows one to solve time optimal control problems More complicated
and concrete
results
including
are given in the papers by Drs. Glushenkov Workshop
on Control Applications
5. If in the problem by phase constraints
~. ~ F I ~ ( ~ ) ~ ' ,
Az + 6 u ,
I~c~l~,
on of the method we described,
programming
constraints
are replaced problem
(lO)
(10) represents
a generalizati-
In line with new type of constraints
using the structure
of phase
Now the support is
constraints
Here we have no possibility
and becomes
to present all the
results in detail. We should note that they are complete as ones for the problem (8). At present
in 1988 [ 4].
~CO)=Xo ,
a basic notion of the method is again a support, more complicated.
experiments
~T=[o,~']
A method of solving the problem
constructed
for linear systems.
numerical
we obtain a new very complicated
= c'~(C)--,m.a.z, i .
~(u)
This
and S.V. Gnevko at the
of Nonlinear
(8) the terminal
endpoints.
in nature
a computer p~ogram have been composed for the problem
which can be considered as a particular which is of a considerable lem of optimization controls
practical
of dynamic
case of problem
interest
(10) but
and is called a prob-
systems in the class of inertial
6. Let us discuss situations when there is a third complexity element in the problems, i.e. non-linearity. Among them a problem of optimization of linear dynamic considered first:
J(u)
- c'=(~')~ma~
object with nonlinear
, ~
= A~
input is to be
6(u),
§
(11)
~(o)-
~.
,
Dr. 0.!.Kosty~cova
Hz(~')
-~
u(~)eU,s
and Yu,L.Kruohenok
for two classes of admissible finite
,
set of values
U
investigated
controls:
=(U,,~,
=[0,~'] the problem
(11)
I) for relay controls with
..., U k ~ ;
2) for piecewise-
577 continuous
functions bounded to the absolute
It appeared
that the technique
used for the problem
described
value= G - ~U: lul# { ~
in the items 3,4 could be
(11).
The second class of simple nonlinear is formed by problems
of optimization
optimal
control problems
with respect
to cost quadratic
functions
J, ( u )
- ~(~;d~
,
(12)
Y
] , (u)
. S~'({)s
(13)
J~ ( ~ )
= i'(t~(~),
]. (u)
= ff,(U)
(14) * ]~(u)*
]~(u)
(15)
The replacement of the cost f u n c t i o n by (12) r e q u i r e s no i n t r o d u c t i on of new constructions. Consider the problem (8) in which instead of a linear cost function we take one of cost functions To take into account a new element a new construction, introduced because
i.e. a support
The support
of cost function accumulates
of constraints
for linear-quadratic
those described
of nonlinear
problems
for linear problems.
is the construction
optimal
into account.
equality-conditions
of linear problems
are characteristic
The support we
of terminal constraints
to take these constraints
which being not characteristic obtained
of cost function.
before can be called a support
it was introduced
with finite number
problems.
8. Return to the problem ness of numerical
(I)-(5).
across
in the nonlinear
~. (~) Therefore
--ma~
,
the problem
It is well-known account
analogue
~(~)
~0
(I)-(5)
depends
of these functions
of the problem
,
d.
~
(6)
~d'.
(1)-(5)
it
(16) is naturally
to
(16).
from linear programming
of the condition matrix
construct
character
lie
Effective-
But the problem in a "pure" form comes
before studing the problem
investigate
~(~,{),
of the problem
on to a great extent how fully specific is taken into consideration.
A.S.Chernu-
Its individual properties
functions ~(~),~.(x~u,s163 methods of solution
result here
for another class of extremal
problems. The results were obtained by Drs. V.M.Raketsky, shevich and A.V.Luboc~cin, A.I.Lutov [ # ] 9
in the nonlinear
The results
are of the same nature as
The most important
of finite methods
(13)-(15).
of complexity we introduce
that an adequate
~
of the problem
(6) allows one to
methods whose efficiency
is essentially
higher than
578 general methods. information
Here the network
is given by network
Developing
network methods
transportation
problems
are a classical
for nonlinear problems
we have con-
cluded that the adequate method for taking into account complexity
element of extremal problems
of nonlinear continuous
functions.
It appears
in which
example. the third
is a network representation
that values
of any nonlinear
function can be obtained as an outputs
of networks
const-
ructed in a special way. This fact is based on A.N.Kolmogorov's theorem
(1957) about a superposition
network model represents
on. Using it it is not difficult linear function. nonlinear
complexity
to calculate
It is particularly
functions
of continuous
graphically
suitable
because it allows
ftmctions.
The
of a concrete
derivatives
functi-
of a non-
at approximation
of
one to detect non-linearity
sources. All these questions and results are considered in detail in [ 6 ] .
of numerical
experiments
9.Now all complexity elements of the problem (I)-(5) have been investigated. To unite them in a method of solution of (I)-(5) consider the problem
L(~(s
=~
luc~)l
,
~{
, ~E T - l o , ~ ' ]
The basic result of the qualitative is Pontryagin
maximum principle:
theory of optimal processes
for a normal optimal control U(s
s & T, there exist such Lagrange multipliers the trajectories = -
A'({)~
,
~ ..... ~
,
that along
~(t), ~(%), % e ] , of primal and conjugate ~(s
=c
- H'P,
A({)
systems
=D}(m(a)>/mz,
(18)
the maximum condition
~'(r
=
m~
~'ce) 6 u
,teT
,
(19)
is true. The prototype Lagrange
of such optimality
multipliers
of variational
calculus
carried over to nonlinear programming method and the adaptive methods ther form.
conditions
which has been later
problems.
the optimality
They come not from the multipliers
is the rule of In the simplex
conditions
have ano-
but from the basis and
the support. The Lagrange multipliers (called simplex multipliers, potentials) are obtained from the basis and the support. The support
of the problem
(17) is closely connected with
579 locally controllability of the system. The support $,up = ~ s ~ p , ~ p ~ r e p r e s e n t s such a set of moments from the i n t e r v a l T and a set of natural numbers assigned to each moment that the system (17) is locally controllable. The Support ~aximum Principle: for each normal optimal control there exists such a support that along the accompanying trajectory of the conjugate system (1~)the maximum condition (19) is true. The statement extends the Pontryagin maximum principle: in the Support ~iaximum principle only special multipliers generated by the support are used. The situation is the same as in classical result of the simplex method, i.e. among optimal feasible solutions there are always basic feasible solutions. By virtue of this fact we can speak of the Lagrange basic vectors. Algorit]~ms of solving nonlinear optimal control problems are given in [ 5].
~EFEREUC ]~S [I] Gabasov R., liirillova F . M . 1984). Consideration of Optimal Control Problem Specifity of Generalizing 1~lathematical Programming.- In: Preprints of IFAC 9th Triennial World Congress, v.5, Budapest, 264-269. 2 ] Oabasov R. Xirillova F.~I. (19~I). Constructive ~iethods of Parametric and Functional Optimization.- In: Preprints of IFAC 8th Triennial World Congress, v.4, Kyoto, 111.-116. [ 31 Gabasov I{., Kirillova F. ~,{., Kostyukova 0.I., Pokatayev A.V. (1987). Optimal Program Controls and Flexible Feedback.- In: Preprints of IPAC Xth Trieilnial World Congress, v.8, Munich, 119-124. 4] Abstracts of International Soviet-Polish Seminar on ~hthematical J r/lethods of Optimal Control and their Applications (1989)~ Institute of ~athematics, f~insk, USSR. [ 5 ] Oabasov R., Kirillova F.~L (co-authors Thatyusl~in A.I., Kostyukova 0.I., Raketski V.M., Pokataev A.V.)(1984.-1990). Constructive ~ethods of Optimization, parts I-5. Linear Problems. Control Problems. Network Problems. Convex Problems. Nonlinear Problems. University Press, ~Insk,USSR. [6] Gabasov i~., Kirillova F.M., Pokataev A.V., Kalinin A.I. (1990). General Approach to Construction of Algorithms of Optimization of Nonlinear Control Systems with Nonsmooth Characteristics. In: Preprints of IFAC Xlth Triennial V~orld Congress, Tallinn, USSR (to appear).
CONSTRAINED
CONTROLS
IN L I N E A R
OSCILLATING
SYSTEMS
F.L.Chernousko Institute
f o r P r o b l e m s i n M e c h a n i c s , USSR Academy o f p r . V e r n a d s k o g o l O 1 , M o s c o w , 1 1 7 5 2 6 , USSR
~_U_~.@_EX.- We c o n s i d e r linear state
controlled by
The
means
of
Nell-known
control
as
a
linear
obtained is
Some
to
a
open-loop
the
system,
example
are
control
obtained;
also
of
arbitrary by
on
as
a
well
depend
on
initial
in
of
the under
which
constraints. number
a
absence
of
This linear
scalar
constrained
as
time
the
of
eigenfrequencies
and
which
obtaining
in
given
of
law
considered
are
imposed
a
constraints.
motions
applied
of
to a t e r m i n a l
for
natural is
the
they
and
applied
conditions
system
transfer
state
geometrical
is
controlled
on c o n s t r a i n t s
is
to
technique
satisfies
control.
An
initial
subject
sufficient
(pendulums)
process
point-to-point
combination
control applied
a
technique
this
oscillators
control
of
a given
control
system;
of c o n s t r a i n t s .
approach
a
from
Kalman's
non-controlled
the
a problem
system
Sciences
terminal
the
states.
system
has
the of An
zero
eigenvalues. INTRODUCTION Property systems paper
of
controllability
without [i]
proposed
Kalman to
consider
on
control.
satisfied. number control.
of
a
We
use
conditions These linear
with
Kalman's under
results
are
oscillators
on
as
solutions
systems
usually
considered
control.In
conditions
control
(natural)
linear
is
imposed
obtained
find
non-controlled we
sufficient
constraints
of linear
of
the
method
for the
applied controlled
combination
system.
control
a by
In
this
constraints
imposed to
linear
well-known
controllability
a
geometrical
which
for
his
and
scalar
give
of
of paper
imposed
constraints
system a
and
some are
arbitrary
constrained
581 GENERAL RESULT Ne c o n s i d e r = A(t)x lu(t)l Here
x
The
functions
problem
(i)
x(t
(2)
to all
satisfies
0
x~
fixed
or
x~
x(T)
are
control (1) (2)
n
vector, x m
u
is
matrices
a control, which
are
A and
find t
a
control
u(t)
satisfying
c
[t o , T]
and
such
that
and
terminal
conditions
the
= X1
given
the
solution
(3)
vectors,
t
is
fixed,
T may
be
either
o
free
(T
> t
). Me
denote
by
~t)
the
n~
matrix
defined
o
by
= A(t)m, where
E
~t 0) is
a
= E
(4)
unity
n
x
n
matrix.
The
solution
of
(i)
n
satisfying
x(t)
the
initial
= r176
condition
(3)
is
(5)
+ ~ ~-l(s)S(s)u(s)ds~ t o
Substituting
(5)
into
the
terminal
condition
(3)
we
obtain
T
~-1(t)B(t)u(t)dt
(6)
= X*
t o
where X)K :
~-I(T)x%
The c o n t r o l ( see
obtain
Xo
(7)
must satisfy
(6)
and (2).
Ne s e e k
it
as
[1] )
u(t) where
u(t)
-
= OT(t)c, c
O(t)
is a c o n s t a n t
B
piecewise
t
initial
) = x O,
Here
n,
of
is
a constrained
u c Rm),
a state
x
for
with
> 0
is n
system
c Rn ,
a = const
is time,
constraint of
(x
respectively
continuous
dynamic
+ B(t)u,
~ a,
t
are
a linear
(8)
= ~-1(t)B(t) n-vector.
Substituting
(8)
into
(6)
we
582
(9)
R( T )c = x * where
R(T)
is
a
symmetric
n
x n
matrix
T
R(T)
(lO)
Q( t )Or( t ) d t
= t o
Consider
a quadratic
f o r m (v
is a c o n s t a n t
n-vector)
T
(R(T)v,v)
= ~
IQT(t)vI z dt
(11)
> 0
t o
It f o l l o w s
from
the
(i)
system
v = O, the
see
R(T)
is c o n t r o l l a b l e
[2],
linear
and
R(T)
is a n o n - n e g a t i v e then
matrix.
is p o s i t i v e
definite.
In
If
for
all
this
case
solution
x
%.bAgr__e_m_. Let (i)
(ii)
is p o s i t i v e
s y s t e m (9) has a u n i q u e
c = R 1 (T)
system
(II) that
(i2)
the is
matrix
R(T)
be
controllable)
and
positive for
definite
(the
n-vector
the
any
inequalities T
IQ
(t)
K(T)
IR(T) K(T) hold
for
vl
vl
g ~(T)Ivl,
V t,
t
e [t
o
, T]
(13) (14)
Z %(T)Ivl
some
T > t
~T)
and
. Here
K(T)
is
n x n
matrix
(det
K
o
0), arbitrary
Ix*l then
~-I(T)
control
u(t)
The c o n t r o l
According lu(t)l =
to =
lOT(t)
(8),
are
positive
scalars,
v
is
an
If
< a %(T)
the
Ey_o.~.
n-vector.
%(T)
(8),
(15)
given
by ( 8 ) ,
(12)
satisfies
(12)
IQT(t)R-1(T) K(T) K-I(T)
x*l
=
R-I(T)
x*l.
(12) (2)
satisfies by i t s
(2),(3). definition.
583 Using
(13)
lu(t)l
we o b t a i n
~
/ZT)IK-I(T)
Now we s u b s t i t u t e lu(t)l
R-~(T)
x
= R(T)
~ ~T)~-I(T)I
The p r o o f
is
over.
R#_m#.!K.
The
matrix
matrix:
K(T)
= E
K(T)
R(T)
v
K(T)
K(T) In
x*l.
in
this
and use
vl
= ~T)
(13),
case
(14)
and (15)
~1 ( r ) l x
(14) MT)
is
can
be
a
lower
I*~ a
equal
to
bound
unity
for
the
n
minimal
eigenvalue
of
the
matrix
SYSTEM Ne
consider
a system
+ z
= u
I
Here
i
~i
are
describe
by
means
constraint
(2)
oscillators
..,
is
of
pendulums
of
springs.
a scalar
control
(16)
constants
a control
boundary
with
n
,
u
and
= ~l'o
~l ( 0 )
'"
coot dinates
a system
body
OSCILLATORS
linear
i = 1 '
eigenfrequencies,
a
of
OF
R(T).
or
e
subject a
to
system
> 0 (2).
of
u(t)
Control
are Equations
masses must
(16)
attached satisfy
to the
conditions
%i(o ) = ~l' o
i
= 1, ....
n, (17)
1
~(T)
. ~,,
We a s s u m e
&l(T)
that
.
all
n~,~
i
are
positive
= 1 .....
n.
and
different.
Denote
I
e
=
0
o
and
Q =
min
(~
1§
- ~ ) > 0 1
(18) 0 = ~
The of
0
system change
< e
1
<...<
(16) of
e
n
is c o n t r o l l a b l e
variables
if
and
only
if
9 > 0
By
means
584 "~F
=
we
,
y
1
reduce y
The s t a t e
g'IZ
+ u,
1
Z
of
oly
i
(20)
1
, i
includes
system (20)
o-t) ~
diag
(-sin
diag
(sin
o,~)
diag
(cos
-1 =
(20)
The f u n d a m e n t a l
z
matrix
o.t) 1
w,~)
(21)
(t)
~
(a%) d e n o t e s
elements
equal
are
(8),
= I,
1 ,. 9 - , n
T (t)
(see
(19)
is
(cos
where diag
n
y,
I diag t) =
B&
1 .....
i'
to
vector
the
i =
|
(16)
1
for
= l
-0 z
m~
1
(4)
~
l
to
a.
(21))
n
a diagonal
.Matrices
x n
and
B
matrix
Q
for
with
the
diagonal
system
(20)
2n-columns
B.+z. = O,
Q~(t)
= cos
e.t%
(22) Q
, = -sin
It f o l l o w s
QQT
w t,
i = 1 ..... n
(22),
from
(i0)
=
R(T) QZ
o
=
'
RZ
R~
I
(23)
T R k =
Here
Qk,
~ Qk o
dt,
k =
O,
Rk
are
n x n
Q 1
= COS
(~ t C O S
~ t,
QZ
= sin
e t sin
e t,
ij
13
Qo
ij
i
i
= -cos
i,
2
matrices
with
elements
j
J
o t sin i
o t; j
i,
j =
i ..... n;
585 RI,Z
= T _ _ + sin 2
II
4
sin(~
R1, 2 . ij
2( cos
R0 il
(24)
l
)T
~
2(
-
l
e )T j
2( r i -
1
a)j)
+
l
(o)T J
2 ( to I +
(18)
~lzQ ,
I(o i
toil z Q ,
i,
1 ..... n
-
cos(ca
+
ej )
from
(25)
+
el
)T
'~
1
follows
Using
+
~
i
cos(a)
j =
sin(
+
6)J )
l
Ij
It
-
6)1
e~T
2~ T -
4e
.
6) -
~
=
Ro
2
-
1
, i#
j
e j)
that
we
obtain
the
T/21
g (4D)
- 1
+
ei
a) j
Z 3Q,
following
i
#
estimates
j
(25)
,
for
the
elements
(24) IR[l
+ 0.5(e
IR
i +
~
ii
We p u t
from
IQT(t)vl Therefore
in
we c a n
(13).
From
its
2n
in
IR ~
'
i
(14)
ejl -1
+
(26)
I < (2Q) -1
li
2;
-
#
j
and
p,
find
k
.
Using
(22)
= nl/21vl
(27)
(14)
we o b t a i n
-
a symmetric
elements
.
take
~ (T/2)IvI
is
(13),
IQ(t)l-lvl
= n 1/z
M
E
k = 1,
ei
< 0.51
(13)
~
IR(T)vI
for
=
/J ( T )
Here
< 2(3{)) -1
~ 4(3Q) -1,
K(T)
obtain
~j)-~
IR [ j I
stem
IMvI,
2n from
M = R(T)
x 2n (23),
-
matrix. (26)
(i,
(T/2)E2n
The j=l
(28)
following .....
n;
estimates i
~
j)
we
586 IM IM
lJ
I ~ 2(30)
"1,
IM
I g 4(3#)'1,
l,n+J
n+i,n+J
IM
II
I ~ 2(3~)
Ig ( 4 0 )
"1 ,
-1,
IM
IM
i,n+i
n+l,n+t
I ~ ( 2 Q ) "~
I~ ( 4 0 ) "1
it follows from these e s t i m a t e s zr, Z ~. M < i , J = l lJ
IIMII z =
2n - (40~
2( n --2n)'4 +
+
(30~
2n - (20~
+
+ 2 ( n Z - n ) "16 _ ( ki { } - 1 ) 2 (30) 2 where
k
=
n
Using
[5n(64n
these
55 ) / 7 2 ]
estimates
IR(T)vl The
-
(14)
T z 2k
n z 1
we o b t a i n
z (T/2)lvl-
condition
l/z
from
IIMII-Ivl is
0 -1 , ~ ( T )
z
satisfie&
= T/2
-
(29) (28)
[(T/2)
-
k
0-1],1vl
if k
0 -1 > 0
(30)
n
The
inequality I x*l
~
(15) an -~/z
can
be written
[(T/2)
-
as
(see
(27),
(30))
k.0-1]
It f o l l o w s from this inequality
T > 2n 1/z
a'llx*l
+ 2k
(31)
0 "1 n
Here
x*
X"
is
given
= (r]~,
by (7)
a)l~[}T
,
while s = O,
x~
1 are
1
obtained
from
(17),
(19) (32)
The control law follows from ( 8 ) , (22) u(t)
= ~ iml
(CiCOS
a~lt -
c.+1
sin
a)lt)
(33)
587 According (with
to T h e o r e m , c
satisfies the
the
initial
increases
T
if
determined
(12)
constraint
to
the
satisfies (31)
by
(2)
and
terminal
when a and
0
and
then
R(T)
transfers
state
decrease
(17). and
the control ( 3 3 )
given
the
Note
when
by
(23),
system that
(24)
(16)
the
from
time
Ix*l increases,
T see
(31).
SYSTEM N I T H
ZERO EIGENVALUES
Now we consider a system with two degrees one
eigenfrequency
(t All
+
~
of
~o
= u,
notations
change
the
2
~t
equal
in
(34)
variables
= u-2aY'l system
(34)
=
to z e r o
u,
lul
are
and
of freedom and with
the
~ a same
(34) as
before.
By
means
of
the
parameters
~o = u~ 2 a z '
t
= e~t'~,
is reduced
to
the
u = au
form c o n t a i n i n g
(3s) no
parameters
('is o m i t t e d )
y + y = u,
Ne
seek
a
z = u,
control
lul
u(t)
~ 1
(36)
transferring
the
system
(36)
from
the
given initial c o n d i t i o n y(o)
= yO
y(o)
= v o,
z(O)
= z o,
z(O)
= u~
(37)
to the given terminal c o n d i t i o n y(T)
= yl,
y (T)
The terminal time
= v 1,
T>O
Z(T)
= z "l',
z(T)
= w1
(38)
is not fixed.
Using the approach similar to the p r e s e n t e d above, we found an e x p l i c i t solution of the problem (36) - (38). We give here only the final
result;
the complete proof see in
[3]
588
Denote +3 AO
_
/(2_
3(1I + 2) ly't_
6
yOl+
_i_ I v 1 - v ~ /(
+
2(~-
6)
Iw 1 + w ~
+
(39) 1 + ~
The
lwl -w
terminal
T = 2~k,
Here
k
time
or,
time
T
any
we
3(1I + 2) 2~ 2_6)
Iz 1 _ z O l
is g i v e n
[Ao+
k ~ kX =
is
T
Bo
_
integer
should
by
( A oz + 4B o ) ~ / 2 ] / 2
satisfying
choose
k
(40)
(40).
=
entk
In
x.
The
order
to
control
decrease
the
is
by
given
formula
the
u(t)
= -c sin t + c cos I
where c
1
c
2
constants = 2[TZ(y1
_ yO)
= 2(v 1 -
3
c 1, c2,
v ~
- T(W 1 + w e ) ]
c3,
c 4,
4
are
+ 12(z 1 _ zo)
T -1, S -1,
c c
4
(41)
t - c t + c
2
3
-
= 614(Yl
= 216T(y1
equal
to
6T(w o + wl)]
S-1
_ yO)
+ 2(z 1 -
z o) _
_ yo)
+ 3 T ( z I _ z O) _
- (T z + 12) W1 - 2(T z- 6 ) w~S - I S = T(T z -
Formulas
24)
(39)
satisfying
(42)
-(42)
all
define
explicitly
the
control
u(t)
conditions (36)-(38).
CONCLUSION
It for
is
linear
case o f are
shown t h a t
laws proposed by
systems w i t h o u t c o n s t r a i n t s can be used a l s o
g e o m e t r i c c o n s t r a i n t s imposed on c o n t r o l .
applied
to c h o o s e for
continuous c o n t r o l
to
the
arbitrary
systems
total
of
time
boundary
linear T
oscillators.
. Explicit
conditions.
formulae
The for
Kalman in
the
These c o n t r o l s main T
point are
is
given
589 Time-optimal specific
control
cases,
see
the minimal time
shown
in
[3].
can o b t a i n
for
To
systems
are
[5].Though
times
Therefore,
quite
such
[4],
T
and
using
satisfactory
known only
T
is
T ~ are comparable
explicit
in some
greater
continuous
than
as it was
controls
we
results.
REFERENCES
1. K a l m a n , R . E . On the general t h e o r y of control systems.Proc.
First P. 2. 3.
IFAC
Congress,
v.
Butterworth,
London,
1960,
481-500.
Krasovsky,N.N.
Theory
1968,
Russian).
476p.
(in
Chernousko,F.L. oscillating v.
1.
52,
N 4,
On
systems. 1988,
of
Control
of
construction Applied p.
Motions. of
bounded
Mathematics
549-558
(in
Nauka,
Moscow,
control
in
and M e c h a n i c s (PMM),
Russian;
translated
into
English). 4.
Pontryagin,L.S.
et
al.
Processes. Nauka, Moscow, 5.
Chernousko,F.L.,
Mathematical 1976, 392p.
L.D.Akulenko,
O s c i l l a t i o n s . Nauka, Moscow,
and
Theory
of
Optimal
(in Russian). B.N.Sokolov.
Control
1980, 384p. (in Russian).
of
Fuel Savings by Optimal Aircraft Cruise with Singular and Chattering Control Gottfried Sachs Klaus Lesch Institute of Flight Mechanics and Flight Control Technische Universits M/inchen Arcisstral3e 21, Mfinchen, Germany
Abstract Fuel minimization in aircraft cruise is considered as a~l optimal periodic control problem. Optimality conditions for flight trajectories with singular arcs and state veriable constraints are derived. For the numerical investigation, an optimization program based on the method of multiple shooting is applied. It is shown that singular arcs in optimal periodic flight paths for minimizing fuel consumption per range exist. Equivalently, chattering arcs may be considered as an alternative. However, results are presented for a modeling more realistic where chattering arcs are the only solution. Furthermore, the improvements due to optimal periodic cruise are considered. It is shown which level of fuel consumption reduction may be achieved when compared with the best steady-state cruise.
Nomenclature
CD CL D g H h J If L
drag coefficient lift coefficient drag acceleration due to gravity Hamiltonian altitude performance criterion lift dependent drag factor ( C D ~--"C D 0 -{- I~ C2L ) lift
rn
nlass
P
propulsive power ( P = reference area switching function
S
TV)
591
T V x y 7 5 A ~0
thrust speed horizontal coordinate state variable flight path angle throttle setting Lagrange multiplier propulsive power specific fuel consumption athmospheric density thrust specific fuel consumption
Introduction Periodic cruise of aircraft has recently received much attention because it offers a possibility for reducing fuel consumption (Refs. 1 - 22). This type of flight no longer consists of a rectilinear trajectory like the classical steady-state cruise but shows a periodic behavior each period of which may be described as a climbing flight at a high thrust setting followed by a sinking flight at a low or zero thrust setting. Accordingly, the control and state variables show a periodic behavior. In the results presented up to now, the controls behavior is of a nonsingular type. This particularly concerns thrust control because the Hamiltonian is usually considered to be a linear function of thrust setting. The nonsingular thrust behavior means that it is a bang-bang type control where it switches between its boundary values and does not tal<e on values interior to its admissible set. A singular arc would mean that thrust setting is at interior values for a finite interval of the optimal flight path. However, results for such a thrust setting behavior have not been presented up to now. It is the purpose of this paper to show that singular arcs exist in optimal periodic cruise problems. Based on the equivalence between singular and chattering arcs, it will be shown that chattering control solutions exist in cases where the fuel consumption characteristics are changed to a modeling more realistic.
Problem Formulation The optimal control problem consists of minimizing the fuel consumption for a given range of the aircraft. This can be formulated by introducing the performance criterion
]_
m (xc o)
(i)
Xcyc where m / i s the fuel consumed and xcyc is the (horizontal) length of one period of the trajectory. The whole trajectory is composed of the individual periods.
592
The performance criterion is subject to the equations of motion which may be written as
dV T - D - mgsin 7 d-'~" = mVcos7 '
d7 _ L - mgcos 7 dx rnV2cos'~ '
dh dx = tan-),,
drn I _ dnI dx Vcos7
(2)
The aerodynamic model for the lift and drag forces is L = CL(~/2)V2S,
D = C0(~/2)V2S
(3)
where
Co = CD0 + KC2L
(4)
The thrust model accounts for the effect of speed, altitude and thrust setting. It may be expressed as T(V, h, 5) = 6 Tma=(V, h) (5) Fuel consumption characteristics are described by the following rclation ,~i = ~,(V, 6) T(V, h, 6)
(6)
whcrc a is the thrust specific fuel consumption. The mass of the aircraft can be considered constant for one period since the fuel consumed during such a time interval is small as compared with the total of the mass, i.e.
- u ( x c . c ) - m i ( 0 ) << m
(7)
Periodicity of the flight path implies the following boundary conditions V(xcyc) = V(O),
7(xcyc) -- 7(0),
h(xcyc) = h(0)
(8a)
The initial condition for the fuel mass can be written as mI(0) = 0
(8b)
The control variables are the lift coefficient CL and the throttle setting 6 which are subject to the following inequality constraints < CL < CL . . . .
0 _< 5 _< i
(9)
The atmospheric model which is used for describing air density and thrust dependence on altitude corresponds to the ICAO Standard Atmosphere (Ref. 23). The periodic control problem is to find the control histories CL and 6, the initial states (V(0), V(0), h(0)) and the periodic cycle length xc~c which minimize the performance criterion J = mf(xeyr162 subject to the dynamic system described by Eq. (2), the boundary conditions given by Eqs. (8a, b) and the inequality constraints for the control variables, Eq. (9).
593
Optimality
Conditions
Necessary conditions for optimality can be determined by applying the minimum principle. For this purpose, the Hamiltonian is defined as H = Av T - D - r n g s i n 7 mVcos7
L-mgc~ +AT r n V 2 c o s 7
rhf +)~htanT+/kfv--~os7
(10)
where the Lagrange multipliers .k = (Av, A.~, Ah, A f ) T have been adjoined to the dynamic system of Eq. (2). The Lagrange multipliers are determined by *
day. = A v T - D - r n g s i n T + V ( D v - T v ) 2(L - rng cos 7) - V L 7 + A-r dx rn V ~ cos 7 rn V z cos 7 6zy - V m l v + Af V 2 cos 7 dATdx = Av ( D - T) sin 7 + rng _ A7
m v cos -
L sin 7
Ah
mV cos 7
cos 7
AI rhy sin 7
(11)
dab _ Av Dh -- Th _ A7 Lh . __ /~f lTtfh dx rnV cos 7 rnV 2 cos 7 V cos 7 dAf _ 0 dx with the following boundary conditions Av(xovc) = Av(0),
A,(xcvc) = ~ ( 0 ) ,
Ah(Xc,c) = Ah(0), Ai(xovc) = 1/x~v~ (12)
The optimal controls CL and 8 are such that H is minimized. For this reason, CL is determined either by
OH/OCL = 0
(13)
or by the constraining bounds of Eq. (9). In regard to the other control, H is linear in 6. Therefore, a bang-bang type of control a n d / o r singular arcs can exist. This case is treated according to the procedure described in the following. The throttle is bang-bang 6 --- 0
if
S ( y , A , CL) > 0
5 = 1
if
S ( y , A , CL) < 0
* Partial derivatives of D, L, T, r h / a r e denoted by subscripts, e.g. D v = cgD/OV.
(14)
594 where -
0
s(y, ~, eL) = N H(y, ~, eL, 6)
(15)
is the switching function and y = (V, 7, h, ms) T A singular arc exists if S(y, A, CL) = 0 for a finite interval of x. The order of the singular arc is p = 1. This is because
z~---m I) 05
= o
(16)
~ ~<~) r o
(17)
06
where di
~(1)(y, A, CL) := dx--"7"S(y, A, CL),
i = 1, 2
(18)
The relation ~(2)(y, A, CL, 8) = 0 can be uscd together with Eq. (17) to give the singular control 6 = 6,i.g(y, A, Co) (19) The switching structure of the optimal control is considered to be known. It may be expressed as ,~ = h k ( y , . ~ ) ,
z k < z <_ z k + l ,
k = O,...,s
(20)
where 0 = x0 < xl < ... < xs < x~+l = xcyr
(21)
6k E {0, 1, 6,i,,g}
(22)
Referring to Eqs. (16) - (22) and the above assumptions, the switching conditions described in the following exist. Denoting by xj a switching point between subarcs where the throttle is on its boundary, the switching condition is
~(y(x~), ~(x~)) = 0
(23)
For a singular subaxc xj ~ x ~ xj+l, the following switching conditions exist
~(k)(~(~j), ~(~j)) = 0,
k = 0, 1
(24)
The system described by Eq. (2) is autonomous so that the Hamiltonian H is constant. Since furthermore the cycle length xc~c is considered free, the following relation holds
H-
mAxc'c) Z2
(25)
eye
Additional conditions exist for constraint arcs where the altitude constraint
C(h) = - h _<_ o becomes active. These conditions are formulated according to Ref. 24.
(26)
595
In the numerical investigation, an optimization program based on the method of multiple shooting was applied (Refs. 25, 26). In these references, the numerical procedure is described in detail.
Singular Control From a physical standpoint, the occurrence of singular arcs with optimal periodic trajectories is primarily due to fuel consumption characteristics. This holds when thrust specific fuel consumption (rhl = ~rT) shows a linear dependence on speed, i.e., a = vV
(27)
where ~, is a c o n s t a n t Such a type of specific fuel consumption may be considered as idealized characteristics of propeller driven aircraft, The model according to Eq. (27) is used as a reference case for the following considerations. An optimal period of a flight path for minimizing fuel consumption per range is shown in Figs. 1 and 2. The aerodynamic data applied may be considered as representative of propeller driven aircraft. The control variables history presented in Fig. 1 shows the singular behavior of thrust setting and how large the length of the singular arcs are in relation to the length of the arcs where fllrust is at its boundary values. T h e singular arcs may be understood as a smooth transition from one boundary value to the other, thus avoiding abrupt changes. Fig. 2 shows the history of state variables. According to altitude behavior, the optimal periodic trajectory consists of a climbing flight phase at maximum thrust setting followed by a sinking phase at zero thrust. The singular thrust arcs may be considered as elements connecting these phases. It may be of interest to note that a state variable constraint (h >_ hrai,, = 0) becomes active. Considering the reduction in fuel consumption due to optimal periodic control, an evaluation is presented in Fig. 3. This Fig. shows how optimal periodic control compares with the best steady-state cruise. The advantage provided by periodic control is comparatively small. This holds for a wide range of maximum thrust level considered.
Chattering Control Optimal periodic cruise may show a chattering thrust behavior when a more realistic type of fuel consumption model according to Fig, 4 is considered. It accounts for a non-zero fuel consumption at idle power setting, For the periodic cruise investigation, it was presupposed that the engine is switched off for phases where ~ = 0 is required by the optimality conditions. Practically speaking, this means that the propeller is in a feathered pitch position for a gliding flight condition.
596 The optimal trajectory is then the same as considered in the preceding chapter with the singular control case. However, the controls show a difference in as much as the singular thrust arc no longer exists but is replaced by a chattering arc. The reason for this replacement is due to the equivalence existing between a singular and a chattering arc for the idealized fuel consumption characteristics considered in the preceeding chapter.In case of the idealized fuel consumption characteristics, a chattering arc is equivalcnt to a singular arc because the chattering behavior representing a switching between minimum and maximum thrust at an infinite frequency may be understood as producing a kind of average value which is equal to the value of the singular arc at the same time. When changing from the idealized fuel consumption characteristics to the model used in the present chapter, both control histories could be applied and the same periodic trajectory would result. However, the fuel consumption would differ so that the two control techniques are no longer equivalent. This is illustrated in Fig. 5 which shows a momentary value of fuel consumption at a certain time. For the chattering control, the fuel consumption remains the same for b o t h fuel consumption models considered. This is indicated as fuel consumption A in Fig. 5. An application of the singular control of the preceeding chapter to the case considered in this chapter would result in an excess fuel consumption indicated as B in Fig. 5. As a consequence, the periodic trajectory with chattering control now represents the only solution for optimal cruise. Considering again optimal periodic cruise in relation to the best stcady-state flight, greatcr advantages in favor of periodic cruise now result. This particularly holds for aircraft where the maximum altitude admissible is restricted such as for aircraft with non-pressurized cabins (Ref. 27). An evaluation is presented in Fig. 6 which shows the reduction in fuel consumption per range due to optimal periodic cruise. It may be of interest to note that the advantage of optimal periodic cruise now reaches a level which can be considered as of practical significance.
Conclusions It is shown that periodic trajectory solutions with singular arcs exist in optimal control problems for minimizing fuel consumption. This holds for fuel consumption characteristics which show a linear dependence of thrust specific fuel consumption on speed. Equivalent to singular arcs, a chattering behavior may be considered as an alternative. However, solutions are presented for a fuel consumption modeling more realistic where chattering arcs are the only solution. Furthermore, the improvements due to periodic control are also considered and it is shown for a wide range of maximum thrust levels available which are the reductions in fuel consumption as compared with the best steady-state cruise. For singular control solutions, the improvement appears to be comparatively small. However, the fuel consumption reduction due to optimal periodic cruise reaches a level of practical significance for the more realistic modeling yielding a chattering type of control.
597
References [1] Speyer, J.L., "On the Fuel Optimality of Cruise," Journal of Aircraft, Vol. 10, 1973, pp. 763-765. [2] Speyer, J.L., "Nonoptirnality of the Steady-State Cruise of Aircraft," AIAA Journal, Vol. 14, 1976, pp. 1604-1610. [3] Speyer, J.L., Dannemiller, D., Walker, D.,"Periodic Optimal Cruise of a Hypersonic Vehicle," AIAA Paper No. 80-1777, 1980. [4] Speyer, J.L., Dannemiller, D., Walker, D.,"Periodic Control of an Atmospheric Vehicle," Collection of Papers of the 25th Israel Annum Conference on Aviation and Astroautics, 1983, pp. 245-255. [5] Speyer, J.L., Dannemiller, D., Walker, D.,"Periodic Optimal Cruise of an Atmospheric Vehicle," Journal of Guidance, Control, and Dynamics, Vol. 8, 1985, pp. 31-38. [6] Gilbert, E.G., Parson, M.G., "Periodic Control and the Optimality of Aircraft Cruise," Journal of Aircraft, Vol. 13, 1976, pp. 828-830. [7] Gilbert, E.G., "Vehicle Cruise: Improved Fuel Economy by Periodic Control," Automatica, Vol. 12, 1976, pp. 159-166. [8] Lyons, D.T., "Improved Aircraft Cruise by Periodic Control," Ph.D. Dissertation, The University of Michigan, 1980. [9] Gilbert, E.G., Lyons, D.T., "Improved Aircraft Cruise by Periodic Control: The Computation of Optimal Specific Range Trajectories," Proc. of 1980 Conf. Sci. Syst., Princeton Unversity, 1980. [10] Breakwell, J.V., Shoee, H., "Minimum Fuel Flight Paths for Given Range," AIAA Paper, 1980, No. 80-1660. [11] Breakwell, J.V., "Oscillatory Cruise - A Perspective," in: Lecture Notes in Control and Information Sciences, Vol. 95, Optimal Control, 1987, pp. 157-168. [12] Houlihan, S.C., Cliff, E.M., Kelley, H.J., "Study of Chattering Cruise," Journal of Aircraft, Vol. 19, 1982, pp. 119-124. [13] Bilimoria, K.D., Cliff, E.M., Kelley, H.J., "Classical and Neo-Classical Cruise-Dash Optimization," Journal of Aircraft, Vol. 22, 1985, pp. 555-560. [14] Bhardwaj, P., Kelley, H.J., Cliff, E.M., "Aircraft Cruise Performmlce Optimization Using Chattering Controls," Proceeding of the 10th IFAC Congress, Munich, 1987, pp. 200-205. [15] Grimm, W., Well, K.-H., Oberle, H.J., "Periodic Control for Minimum Fuel Aircraft Trajectories", Journal of Guidance, Control, and Dynamics, Vol. 9, 1986, pp. 169-174. [16] Menon, P.K.A., "A Study of Aircraft Cruise, Proceedings of the AIAA Atmospheric Flight Mechanics Conference, Williamsburg, Aug. 10-20, 1986, pp. 518-532. [17] Sachs, G., "Verringerung des Treibstoffverbranchs dutch periodische Optimalflugbahnen," DGLtt Jahrbuch, 1984, 090-1 - 09-17. [18] Sachs, G., "Flugzeitsteigerung dutch zyklisch gesteuerten dynamischen Danerflug," Zeitschrift fiir Flugwissenschaften und Weltranmforschung 9, 1985, pp. 42-52. [19] Sachs, G., Christodoulou, T., "Endurance Increase by Cyclic Control," Journal of Guidance, Control, and Dynamics, Vol. 9, 1986, pp. 58-63 [20] Sachs, G., Christodoulou, T., "Reducing Fuel Consumption by Cyclic Control," ICAS Proceedings, 1986. pp. 153-160. [21] Sachs, G., Christodoulou, T., "Reducing Fuet Consumption of Subsonic Aircraft by Optimal Cyclic Cruise," Journal of Aircraft, Vol. 24, 1987, pp. 616-622. [22] Sachs, G., Christodoulou, T., "Effects of Fuel Consumption Characteristics on Optimal Cyclic Endurance Flight. Technika Chronika, Scientific Journal of Greece-Section B, 1988, pp. 23-39. [23] ICAO Standard Atmosphere, International Civil Aviation Organization, Montreal, 1984.
598
[24] Bock, H.G., "Numerische Behandlung von zustandsbeschrgxtkten und Chebyehef-Steuerungsproblemcn," Course R1.06 of the Carl-Cranz-Gcsellschaft, Oberpfaffcnhofcn, Germany, 1983. [25] Oberle, H.J. "Numerische Berechnung optimaler Steuerungen yon Heizung und Kiihlung fi]r ein reatistisches SonnenhausmodelI," Institut f/it Mathematik dcr Technischen Universitg.t M/inchen, TUM - M8310, 1983. [26] Bulirsch, R., "Die Mehrzielmethode zur numerischen LSsung yon nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung," Report of the CaxlCranz-Gesellschaft, Oberpfaffenhofen, Germany, 1971. [27] ICAO Standard and Recommended Practices, Operation of Aircraft, Annex 6, 1986.
h
3.0
[kmJ
"iV CLICL''09,0 1 SINGULAR ARC
0
0.0
0.2
0.4
0.6
0.0
0'2
'
0 2
0:4
0'.4
0 6
0 6
0:8 1 X/Xcyc
V
018 l.O [m/s] X/Xc•
l0~
0.0
0.2
0.4-
0'.6
0'8
r .0
X/Xcyc
,oo.j 0.0
Fig. 1 Optimal periodic trajectory with singular arcs, control variables ( L / D ) m a = = 15 P , , , ~ f ( m g V o * ) = 0.17 xcyr = 82.6 km
1.0
X/Xcyc
O.
0.0
0.8
I 0.2
0.4
0.6
0.8 1 .0 x/Xcyr
Fig. 2 Optimal periodic trajectory with singular arc, state variables P . , ~ / ( m g V o ) = 0.17 xcyc = 82.6km
599 1.0
.002 Amf [mr]sLeadyl
(
~? I mo x
M O D ~ / / 0.5
.001
{m?] IdLe /
(rhr ]mox
/ / /
.0
O.
0
2
0.0
3
0
O.
o:s
,
P~o• [mgV#]
Fig. 4 Fuel consumption model with non-zero fuel consumption at idle thrust setting
Fig. 3 Fuel savings Amy due to optimal periodic cruise with singular control
0.2
&r 1 .0 { rn:-r'}~~•
o 6
&mr MOMENTARY
FuEL CON-
J.,"
I
(mr)sLead
~, I
O.1 {mr} IdLe
~176
0.0
.1
0.2
0.3 Pmo• [mgVo]
Fig. 5 Momentary fuel consumption for both control techniques considered
Fig. 6 Fuel savings A m I due to optimal periodic cruise with chattering control, (ml)~=dy: fuel consumption of the best steady-state cruise with a 3.5 km altitude limit, (~hd~d(ms)m,,
= 0.1
Optimal Path Planning of Manipulators with Singular Configurations, Workspace and Collision-free Constraints
D. WANG, D. GEOI~GES, Y. HAMAM D~partement Automatique, E.S.I.E.E. BP 99, 93162 Noisy Le Grand Cedex, France
Abstract This paper presents an optimal control approach for the off-line path planning of manipulators with a large number of non-convex constraints relating the kinematic performances and the environmental limits. The main contribution is in the analysis of the singular configurations and workspace of manipulators in dynamical path planning problem, the developpement of a new method for three-dimensional collision avoidance of revolute manipulators in an environment clustered with obstacles defined by convex surfaces, and the use of the augmented Lagrangian for the numerical solution of this highly nonlinear and nonconvex path planning problem. This approach is based on the formulation of all these constraints as special state space constraints, in the form of inequalities, and the path planning problem as an optimal control problem. The solution is then obtained by UZAWA's algorithm. Some simulation results are presented for the 3-dof robot PAMII~ built at ESIEE by P. Durand and F. Viette for the purpose of research and education [1].
1
Introduction
Optimal off-line path planning of manipulators is an important research subject in robotics since it is related to a wide variety of problems in its general form. In particular, the dynamic behaviour of a revolute manipulator is highly non linear and coupled and the robot ldnematics and environment limits may lead to some very complicated constraints for the formulation of the problem and its numerical solution. Previous work in this area has been usually concerned with either dynamical control or geometrical path planning of robots [1],[2],[3],[4], and no e~sting method seems to be powerful enough to solve the problem by a unified approach in taidng into account both the dynamical and kinematical constraints.
2
Analysis of singular configurations and workspace
The singular configurations and workspace of manipulators have been studied in terms of robot kinematical performances [5],[6]. We consider them below from the viewpoint of robot dynamical control, and complete some of their analysis.
601
The workspace is the reachable work area of the end-effector position and orientation in the operational space. It is a volume M transformed from the joint coordinates domain N by the geometrical model f of the robot. For a n-link manipulator, N may be represented as a generalized torus in the joint space. Defining X as the vector of position and orientation of the end-effector, and q the vector of joint coordinates, the geometrical model of a revolute manipulator is a nonlinear function
X = f(q). The singular configurations of a manipulator represent these configurations where the manipulator loses some degrees of freedom, and becomes uncontrollable. They are a set of hypersurfaces in the joint space, determined by vanishing the determinant of the Jacobian matrix of the geometrical model f . The off-line path planning problem consists of generating trajectories between initial and final configurations, which may be then used as reference trajectories to be tracked in real-time control. Therefore, singular configurations must be avoided in order to assure the feasibility of the generated trajectories. The hypersurfaces representing the singular configurations and the joint motion constraints may decompose the domain N into a set of n-dimensional subspaces Ui with N = {Ui,i E I}. Suppose that the transformation f maps each subspace Ui to a volume Mi in the operational space. Since it is a diffeomorphical mapping, we have the following propositions :
Proposition 1 (1) Every singular configuration on the boundary of Ui is mapped by f to a point on the boundary of M~; (2) The pre-image f - l ( X ) of every point X in the workspace M may be at most one point q in each subspace Ui E N. It is thus necessary to determine only all the volumes Mi by the transformation f . The workspace M is the intersection of all the volumes 33"/. A point X in M may be reached by k configurations if its pre-image f - l ( X ) is contained in k different volumes Ui. The calculation of Mi from Ui and f may be carried out with an algebraic projection method [7],[8]. Conceptually, the interior of U~ may be considered as an n-dimensional manifold, and the following projection procedure may be used repeatedly for k = n, n - 1,. --, 2 to project Ui into an infinite set of 1-dimensional manifolds. P r o j e c t i o n P r o c e d u r e This consists of projecting a k-dimensional manifold into an infinite set of (k - 1)-dimensional manifolds with constant value of the variable q,~-k+l , and finding the critical points on the axis q,~-k+l where the topological property of the (k - 1)-dimensional manifolds is changed. Since the boundary of Ui is the intersection of a finite set of n - 1-dimensional hypersurfaces, the above procedure will find only a finite number of critical points for k = n, and it is the same case for other values of k. So Ui may be constructed from the critical points and the 1-dimensional manifolds, and Mi may be obtained by sweeping successively in the operational space the images of the k-dimensional manifolds from k = 1 to n - 1. In general, the volumes M~ may be overlapping. Hence,
602
one position and orientation of the end-effector may be reached by several configurations in different subspaces U~. On the other hand, some inequality constraints corresponding to the surfaces of the subspaces Ui may be written explicitly in order to specify the subspace Ui. They also ensure that the positions and orientations of the end-effector are in the workspace, if they are reachable. From the above analysis it the following proposition is concluded Proposition
2
(1) The singular configurations may be avoided if and only if the path planning is carried out in a single subspace Ui of the joint space; (2) If the initial and final positions and orientations of a path to be generated may be reached in k subspace Ui, the path may be searched in each of the k subspace Ui.
3
W o r k s p a c e constraints
The workspaee determined above is generally defined by a However, it is a natural assumption in spatial planning for the workspace of a robot is constrained in the operational or walls. Assume S = (z, y, z) T to be an arbitrary point in constraint means that :
S E W = { (z,y,z)
[ a~x + biy + ciz ~ di,
set of complex surfaces. a robotic workshop that space by a set of planes the robot segments, tlfis
i=l,2,.--,n}
(1)
where W is the constrained workspace defined by n planes. In order to simplify the problem, a finite set of significant points on the robot segments may be used to represent the constraint. It may be noted that a convex potyhedra is collision-free with a wall if all of its vertices are in the free half-space which is the complement of the wall. Therefore, if the robot segments are described by a set of convex polyhedra with a total number of m vertices, it is sufficient to write at most mn inequalities as follows : aixj+blyj+cizj
4
Collision-free constraints
The collision-free path planning for revolute manipulators has been for a long time an open problem, especially on the practical level, because of its computational complexity. We present below a new formulation based on the transformation described in [9] to represent each convex object by a penalty function. Namely, suppose that an objet contained in the above bounded region W of the cartesian space is the intersection of k > 1 regions Ri, where each/~; is defined by an inequality gi(x, y, z) ~ 0
603 and the function gi is convex, differentiable and normalized so that [ grad gd >- 1 on the surface g; = 0. The penalty function G is defined for the object in terms of the functions gi by = (d
+
+ g,,
a=
v
=
v,
(y- V)+c
where 0 < t << 1 and c = - k t with k > 0. The function vi is always positive and it is an increasing function of gi, and has the following property :
vl =
{
2g~+ O(t2),
g~>0,
t,
gi = O,
O(t2),
gi < O.
The function V contains one term vi for each region R~. The value of this function is small compared with t at those points in the intersection I of the regions _R~ that lie sufficiently far from the boundary B of I; and it is greater or equal to t outside I, where at least one of the variables g; is greater than or equal to 0. The function G is also convex. The surface G = 0 lies within a distance O(t) of the boundary B of I. Except in the vicinity of B, the function G is always positive outside I and negative within. Hence, the surface G = 0 may be considered as the boundary B of I. Although this is inherently an approximation, the degree of this approximation may be made arbitrarily small to lie within the actual physical tolerances of B. Suppose that a convex object of the robot segments is defined by a set of functions
gli(X, q) with i = 1,--., r, and that a convex object of the obstacles is defined by a set of functions g2j(X) with j = 1 , . - . , s. Introducing the following penalty function : a
=
q)..., g,(x, q), g:x(X),
9. . , g
s(x))
and G(q) = min G(q, X)
(3)
X6R 3
their collision-freeconstr~nt may be denoted by G(q) > 0
(4)
However, the calculation of the function G(q) would lead to some very time-consuming optimization subproblem. In order to avoid this drawback, an approximate method is developped. Namely, we consider a mesh on the surface of each convex object of the robot segments, where neighbouring nodes have a fixed equal distance d. The obstacles are expanded with the distance d, and a penalty function is defined for each convex object of
604
the expanded obstacles in the Cartesian space. The collision-free constraints are then specified by restricting the displacements of the finite set of node points on the surfaces of the robot segments to be outside the obstacles. This approximate method allows the cartesian variable X to be eliminated by substitution for a finite set of points on the robot segments, whose coordinates may be determined as functions of the joint variable q.
5
Solution of the path planning problem
From the above discussion, the singular configurations, workspace and collisionfree constraints may be formulated as a set of inegalities S(q) <_ O. It is known that the dynamical model of a manipulator may be written in a compact form by M(q)~ + N(q, q) = U, where M is an n x n, symmetric, positive definite matrix, N an n vector of all the resistant torques, and U the n control vector. In the state space equation form, it may be denoted by 2 = F ( X , U), where X = (q, (l)T -= (X1, X2) T. Other path planning constraints are the joint positions and velocities limits : Xmi,~ <_ X < X r ~ , the control vector constraint used to eliminate the saturation of the actuators : U,,~i,~ _< U _< Um~, and the geometrical path tracking constraint E(X1) = O. If a performance index relating the energy consumed by the actuators is used, as proposed in [11], the path planning problem may be formulated as an optimal control problem : this consists of finding an optimal solution X(~), U(t) for t E [0, T] with the above constraints so that the robot may move from a fixed initial state X(0) = X0 to a known final state in the operational space f(X(T)) = YF : min J(U) =
kuTRudt
X,U
(5)
subject to
2 = F(X, U), Z/(Xl)
X(O) = )20
= 0, i : 1 , h e ,
S/(Xl)
and
f ( X ( T ) ) = YF
_~ 0, i = 1,77,s
where T is the robot travelling time, R a n • n , symmetric, positive definite matrix, and k > 0 a constant. The above problem includes a large number of nonconvex kinematical constraints, and the robot dynamics are highly non-linear. An efficient numerical approach is thus necessary to ensure the convergence of the optimization problem. The chosen method is based on a discretization of the problem and on a dual formulation using an augmented Lagrangian (see [10],[11]). The discrete augmented Lagrangian
605 is defined as follows : N-1 i=0
+~o(~, u,~i~ - ud + e~(r
r/~ - u ~ ) }
(6)
N-1
+ ~
h~{~c(~i,s(x~)) + ~o(~;,E(x~0)}
i=0
where
~=(p,O) = 9 c(p,0)
{11 Max(O,p+
= (;+
cO)112-
I1p 112}/2c
~c0)r0
and c > 0 denotes the coefficient of the augmented Lagrangian, (t3i, 7i E 1~2'~, 6j, Cj E R '~, q = (711,...,q,~,) E /~'~',# = (#~, ...,#,~,) E /~'~) are the Lagrange multipliers. The problem is to find a saddle point of the following dual problem :
Max
{ M i n Lc}
(#~, 7. 5~, r ~j, m)
( x . ud
(7)
subject to :
Xi+l = F(Xi, Ui),
f ( X N ) = YF
(8)
where
(#i, 7. 6. r
vj) > 0, U.~i~< U~ < Um~ and j = 1 , 2 , . . . , N - 1
i=0,1,.--,N-1
Assuming that the second-order Kuhn-Tucker and constraints - qualification conditions hold for the constrained optimization, a local saddle point of the Lagrangian may be found if ~ large positive value of c is chosen.
6
Algorithm
and
Simulation
The above method was applied for the optimal path planning of the first three finks of robot PAMIIL In the joint space defined by Q3 = {-~r <__qi ~ ~r t i = 1,2,3}, the singular configurations are determined by the following equation : sin q3(I~ sin q~ + sin(q~ + q3)) = 0
(9)
These are a set of hypersurfaces decomposing Q3 into four subspaces (see Fig.l) : Q3 = {D1, D2 = {D21, D22}, D3, D4 = {D41,D42} }
(10)
The workspace generated by each subspace is delimited by two spheres and one cylinder. The singular configurations constraints in the subspaces D1 may be written as follows :
606
q~,e
.,
to
J.o @
-::
Figure i: (a) Decomposition of joint space into four subspaces by singular configurations; (b) Surface approximated by a penalty function of a cylindrical obstacle. q3 - r e - arcsin(~ -,3 sin q2) + q 2 _< 0 -q3 - arcsin(~ sin q2) - q2 _< 0 -q3 < 0 q3-rr_< 0
(n)
The workspace is constrained in the operational space by six planes or wMls paral1.el to the a~,:es of the reference coordinates frame of the mamipulator. With some simplification of the geometrical modelling, 60 inegalities were used to specify the constraint. Fig.1 shows the approximate surface of a cylindrical obstacle defined by a penalty function. The approximation of surfaces of convex polyhedra presents also a good precision. 65 node points were used to define a mesh on the surface of the PAMIIL arm. Therefore, if there are m obstacles in the environment, 65m constraints must be introduced. UZAWA's algorithm has been used in order to solve the path planning problem with the following considerations according to [11],[12]. The final constraint f ( X N ) = YF is considered as a primal constraint defined in the operational space without using the inverse geometrical model. This is realized using a projected gradient approach defined by the orthogonal projection matrLx on the tangent space of the final constraint :
P = S~ - D(S)RD(S)O(S)T)-ID(S) Algorithm 1. initialization (k=0) (a) choose the sampling number N and the robot travelling time T, and let h = T / N ;
(12)
607
(u) choose ;0 =
o
o o r
0)r,
(c) choose the initial and final position of the end-effector verifying the geometrical constraints (note that there are four possible configurations for one position in the workspace, and the initial and final configurations must be within a same subspace Di ); (d) determine an initial trajectory/~,., X ~ verifying the final constraint YF =
H(XN); 2. At iteration k : calculate M i n L c ( X 1 , . . . , X N - 1 , U o , . . . , U N _ I ) local minimum in U = (U0, ..., UN-1) of the Lagrangian
by searching a
N-1 i=0 N-1
+ F_, i=0
subject to Xi+l = F ( X i , Ui), i = O , . . . , N - 1. (a) compute the gradient VuL~(U k) ; (b) compute the descending direction d k = - P V u L c ( U ~) ; (c) update U k : U k ~-- U k + td k, t > O. (d) if [[ V u L r k) [[< eu, then do U k+l = U k, X TM = X~, i = 0, ...,N and go to 3(a), else return to 2(a) . 3. Dual iteration : (a) evaluate the constraints and update the Lagrange multipliers as follows : i. for the equality constraint E ( X 1 ) = 0 : #k+l = #k + p E ( X k l + l ) ii. for the inequality constraints 0 < 0, let p = (fl,7, 5,(,rl) :r : p k + l = p ~ ( 1 - p/c) + ( p / c ) M a x ( O , p k + cO(X TM,Uk+l)) with0 < p < 2c (b) if all the constraints are satisfied, then finish, else do k ~- k + 1 and go to 2(@. This algorithm has been implemented in Fortran on a VAX-II/GPX, and a graphical simulator has been developped in DAL-Fortran of the CAD software CALMA. We show in Fig.2 and Fig.3 one example of a generated trajectory. The simulation of a pick-and-place operation of the robot PAMIi% in a constrained environment is presented. The following numerical values have been used : the initial position
608 1.1
~
I.
.$
6
e
9
o
-g
-.6
..IL ..11
-~ ,...!
.... .3
f ....
t
1.....f
....
1.3
2
(,~,,
L, , ~ )
f ....
2~
f ....
.1
.%5
-I-2
. . . .
"!
. . . .
f . . . .
! . . . .
(~.,
|
. . . .
! . . . .
! . . . .
~, , - ~
Figure 2: The driving currents and the joint velocities of the generated trajectory.
Figure 3: Some intermediate configurations of the generated trajectory (the first is the initial configuration and the last is the final configuration).
609 (0.3, 0.3, -0.2) and finn position (-0.25, 0.3, 0.35) (in meters); the sampling number N = 41; and the constraints: - 3 < Ui _< 3 (in amp.); - 3 _< qi _< 3 (in rad.) for the subspace D1 ; - 1 < 0i _< 1 (in rad./sec.) ; and W = {-0.6 < x < 0.6, -0.1 < y < 1.0, - 0 . 6 < z < 0.6} (in meters) for i = (1,2,3).
7
Conclusion
We have developped a method of optimal path planning for robot manipulators with a large number of nonconvex geometrical constraints. The proposed method used for the analysis of the workspace knd singular configurations of robot manipulators is simple and easy to implement. The consideration of these kinematic performances in off-line path planning is important to ensure and exploit the capabilities of manipulators to perform tasks in a complex environment. The new collision-free constraints specification method presents not only a good approximation of this difficult prob.lem, but alsa improves the numerical feasibility of the solution of the path planning problem with an optimization approach. The augmented Lagrangian method is useful for the solution of problems with nonconvex constraints. Trajectory planning algorithms implemented within CAD software such as ILobot-Sim of CALMA do not take into account robot dynamics. The method proposed in this paper has not this drawba& and might be an important complement to such a CAD software.
References 1. M. Jamshidi, J.Y.S. Luh and M. Shahinpoor, Recent Trends in Robotics, Modeling, Control, and Education, North-Holland, 1986. 2. J. T. Schwartz, M. Sharir and J. Itopcroft, Planning, Geometry and Complexity of Robot Motion, Ablex Publishing, 1987. 3. J. Canny, The Complexity of Robot Motion Planning, ACM Doctoral Dissertation Award 1987, The MIT Press. 4. J.T. Schwartz and C. YAP, Advances in Robotics : Algorithmic and Geometric Aspects of Robotics, Volume 1, LEA Publishers, 1987. 5. I~. G. Selfridge, The Reachable Workarea of a Manipulator, Mechanism and Machine Theory, Vol. !8, No. 2, 1983. 6. J. I~astegar and P. Deravi, Methods to determine workspace, its subspaces with different numbers of configurations and all the possible configurations of a manipulator, Mech. Much. Theory Voh 22, No. 4, 1987. 7. M. Ben-Or, D. Kozen, and J.Reif, The Complexity of Elementary Algebra and Geometry, J. Comp. and Sys. Sciences, Vol. 32, 1986. 8. Kozen and Yap, Algebraic Cell Decomposition in NC, IEEE symp. FOCS, 1985.
610
9. Paul G. COMBA, A Procedure for Detecting Intersections of Three-dimensional Objects, Journal of ACM, Vol. 15, No. 3, July 1968. 10. G. Cohen and Dao Li Zhu, Decomposition - Coordination Methods in Large Scale Optimization Problems : the Non differential Case and the Use of Augmented Lagrangians, Advances in Large Scale Systems, Vol. 1, JAI Press Inc., 1984. 11. D. Georges, Planification Optimale de Trajectoire et Commande des r~obots Manipulateurs, Th~se de Docteur-Ing6nieur, ENSMP, 1987. 12. D. Georges, Y. Hamam : "Planification Optimale de Trajectoires pour Robots Manipulateurs", APII - P~AIK0 1987, Vol. 21 - 2, pp 129 - 150.
ON S I N G U L A R O U T P U T T R A C K I N G IN M U L T I V A R I A B L E N O N L I N E A R ' S Y S T E M S
M.A. FAYAZ and F. LAMNABHI-LAGARRIGUE Laboratoire des Signaux et Syst~mes, SUPELEC 91192 Gif-sur-Yvette CEDEX, FRANCE
Abstract: In this paper we generalize some results from Crouch, Ighneiwa and Lamnabhi-Lagarrigue [1], to multi-input, multi-output nonlinear systems. The notion of rank of singularity is also introduced to quantify the regularity of the solutions to the singular tracking problem. Necessary and sufficient conditions for the trajectory tracking are then derived. A two degrees of freedom manipulator is chosen to illustrate the result.
1. I N T R O D U C T I O N When studying control systems we often need to determine whether we can make the output of the system follow a desired trajectory for a specified time interval. We here use the term trajectory tracking to denote the ability to realize a set of trajectories as outputs of the system by using a given set of inputs and initial conditions. While the time interval is [0, ts[ such that the internal trajectory Xd(t) corresponding to the external one yd(t) does not cross the singular manifold, we are in the regular case. Here we are concerned with trajectories Xd(t) crossing singular manifold. In other words, we assume that the strong relative degree is not well defined. This study was started by R. Hirschorn and J. Davis[3], in the case of single-input,single-output nonlinear control systems. Crouch and al. [ I] continued this study, introducing further structures, which are important in quantifying the multiplicity and smoothness of the solutions of the problem. In this paper we generalize some of the previous results in multi-input multioutput cases. Recently some other approaches to this problem have been studied, see Grizzle and al. [2] and Fliess and al. [4]. After giving the definition of singularities for the multivariable systems, we generalize the notion of the rank of singularities (see [I] for the SISO case) also denoted by r(x0, Ytl). This rank and the introduction of multivariable singular differential systems of order k, 0 < k _
612 2. R E G U L A R T R A C K I N G Let us first recall some facts of the regular tracking. In order to avoid unnecessary complications, we will restrict our analysis to systems having the same number m of inputs and outputs. A multivariable nonlinear system of the form :i = f(x) + ~ g(x)u i i=1 y=h(x), x(O)=x O,xe M,
(*) ye ~l
has a relative degree o~ = {al, r . . . . . am} at a point xo, if (i)
Lgj Llkhi(x) = 0
for all 1 _<j < m, 1 < i _<m, k < c q - 1 and for all x in the neighbourhood o f x 0, (ii)
the matrix Da ( x ) = I Lg:Ltat-lhl(x)"'" LgmLfal-lhl(X) 1 [_LglLtam-lhm(x)
LgmLlarn-lhm(x).]
is not singular at x0. The matrix Da(x) is sometimes called the decoupling matrix. We will see in the example, that the singularities introduce an intrinsic coupling into joint variables of the robot manipulator. Let us consider a C** vector function (Ydi) on (--,~, 4-oo), 1 < i < m. Assuming the relative degree o~ < ,~ (i.e. cq < ** for all 1 _
f
Sa
x9
m
f(x) + i~ml g , ( x ) u , .
.
t y(a) = Ca(x) + Da(x)u
with
Y(a)(t) =
1
! Lym(am)(t)J
,
Ca(x)=
ILI
9 LLtam-lhm(x)j
and
We know tlmt if
and
i)
Ydi(ki)(O)= (Lf(ki-l)hi)(xo)
ii)
Da(xo) is regular,
1 <_k i _<(zi-1, 1 _
Da(x)
defined as above.
613 then Yd can be tracked on a maximal time interval, for which Dcx(xd(t)) is not singular, t e [0, ts). Moreover the solutions of system (*) xd, subject to the feedback control ua(x, yd(Ct)(t), t) = [ Dct(x) ]-1 (yd(Ct)(t) _ Cot(x) ) with y d (a) = Ydi(Cq), 1 --
3. S I N G U L A R Definition
TRACKING
1
xs is a singular point for output tracking if rank Dct(Xs) < m. The set of all such singular points Xs is denoted by S. S = {xs ~ M, rank Dct(Xs) < m}. Note that in the M I M O case the singularities are of different types. W e can classify them as follows (see Tchon [6]): m
Sk=
{ x s e M,
rankDct(xs)=m-k}
and
S =
t..)Sk. k=l
Definition
2
W e shall say that Yd can be tracked through a singular point xs e S, if there exists an admissible pair (rid, id) defined on some interval [0, ts+ e) such that rid(t) = Ud(t) and id(t) = xd(t) for t e [0, ts) and Xd(ts) = xs- As in [1] we begin our approach by defining the rank of singularity, which gives us an information about the existence and m a x i m u m s m o o t h n e s s of an admissible solution to the singular tracking problem. To simplify the problem, we assume that the state trajectory crosses the singular manifold S at time ts = 0 and x0 e S. Let us consider again the system Set and a singular point xo e S with rank (Da(xo)) < m. Differentiating with respect to time, k times the second equation, we obtain y(C~+ k) = C~ + k(X, u, fi. . . . . u(k + l)) + Dt~(x)u(k),
where y(Ct + k) = (yi(ai + k)) , 1 < i < m.
Definition
3
G i v e n a singular point x0 and a function Yd. Let r ; r(x0, Yd) be the largest integer for which there exists (, r= (v0, Vl vr) E ] ~ r + l , such that the equations . . . . .
y ( = Ck-~(xo, C t v0, + Vk-z) k +) Dct(x0)vk d
are satisfied.
"'"
with
0 < k < r and v i = u(i) (0)
614 If these equations are satisfied for all 1 2 0 then r = ~ . If these equations are satisfied for 0 < 1 < or+r0 then r = r0. If these equations are not satisfied for all 0 < i < ~x then
r is not defined.
W e also define Ek(x0, Yd) = {~k E ~xmx(k+l), Y~I - (k) (0) = Ck(x0, ~' Z-I) + D~(x0) Vk}
Note that if r is defined, then Ek(x0, Yd) # (~.
Definition 4 The singular system of order k, 0 _< k -< r(x0, Yd), determined by the singular point x0 and the function y a ,
Z k is given by the equation:
= f(x) (~0
.o
+ )". .o
= (u 1, u 2 . . . . .
it
=
gi(x)ui,
,o
um ) = u
t
u 2 (1)
~1k-2 = u k - 1
y = h(x) with the initial conditions x(0) = x0, ui(0) = v i , i = 0 .... k-1
and
.~k ~ Ek(x0, ya).
Prooosition 1 : Consider a singular point xo ~ S for the system (*). A necessary condition for tracking Yd through xo is that the rank r(x0, ya) is defined. If r(x 0, Yd) > I and r > k > 1, then there exists a C 1 soludon
(x, u
(-e, E) with
0 ....
, U m,
1
1
U 1 .....
x(0) = xo, (u~
tl m ,
2
2
k-1
k-t
u I . . . . , u m, ..., u 1 . . . . . u m ) o f a singular system o f order k on
. . . . . u~-t(0)) ~ Ek(x0, Ya) if and only if, Y,l can be tracked through x o on
(-~, ~) with solution (ua, xa) satisfying u d e C g and x d a C k+l. Sketch o f the proof: T h e necessity o f r(x o, Yd) being defined in order to track Yd is clear. If YO can be tracked through a singular point x0 ~ IR n with xd E C k+l, Ud E C k, then a solution to the singular system o f order k exists. Conversely, if there exists such a solution, we show that the output y o f the singular s y s t e m coincides with Yd- Differentiating y(t) = (hi(x) . . . . . hm(x)) T along a solution of the singular system, we obtain
y(J)(t) = Cj(xo, u0(0) ..... uj-=-l(0)) + Da(xo)uj-ct, with
615
Uj_a_ 1 =
:
0 -- ji < Oq + k - 1.
,
~Umjrn-Ctm-IJ Differentiating one more time, we obtain
y(~+k)(t) = C~t+k(X, uo ..... Uk-l) + Dcx(X)Uk-I
which is equal to yd (~+k)from (i). Hence y(J)(t)= yd(J)(t)for t ~ (-E, +g) and for 0 <j < o~ + k - I. Clearly uo = Ud ~ C k provides the desired control components for a solution to the singular tracking
problem. W e shall show that the m a x i m u m possible smoothness of x d is C T M and of u d is C a. in another publication. In order to illustrate our results, we consider the following example. 4. E X A M P L E Let us consider the robot manipulator with two degrees of freedom of Fig.l. T h e dynamic equation of the manipulator is given by :
D(0)i~ = F(0, ~9) + G ( 0 ) G + U
where 0 = (01, 0z) r , u = (ul,u2) v, D(0) is the initial matrix, F(0, 0) represents the centrifugal and Coriolis
forces, G(0) represents the gravitational torques and U the external applied torque on the
manipulator. The kinematic equations are given by yl(t) = hl(0(t)) = It cos (0t(t)) + 12 cos (0t(t) + 02(0) y2(t) = h2(0(t)) = 12 sin (0l(t)) + 12 sin (01(0 + 02(0). The state form equations which are obtained after a usual change of coordinates and the use of a static feedback are[5] :
/~ = =
~
o(0) = o0
v
~(0) =
to
y = h(0) Differentiating the ouput function y(t) with respect to time gives = J(0) ~ Therefore we obtain
and
~ = ~:(0, ~) + J(0) v
C~ (0, ~) = K:(0, ~), D~ (0, ~) = J(0), where J(0) represents the Jacobian matrix
of the output function with respect to 0:
616
/
/
L301
bO2J
and det(J(0)) = 1112sin 02. The singular state set is then given by S={(0,~)e
/RZxlR z ,
02=kg
w i t h k e Z}
If the tracking time interval [0, T] is such that the state trajectory (0(t), ~(t)) does not cross the singular manifold for t e [0, T] we are in the regular case and the solution of the tracking problem is well known. In order to illustrate the tracking problem through the singular set and to determine the rank of singularities, we shall consider the following parabolic trajectory,
Yd
= [Ylo(t) = [sin(c~t)]2/2p (Y2d(t) = s i n a t
(2)
the reason of this choice will be explained later. In order to determine the rank of singularity at t = 0 we calculate the derivatives of the desired trajectory and the system output at the singular point. In the following for the sake of simplicity we assume 1t = 12=1: yi(0) = 0 and Yid(0) = 0 Differentiating one more time gives ~r
= 0,
for i = 1, 2
~,2(0) = 1620 ,
yld(0) = 0
and
y2d(0) = ct
If 520 ~: call, then r is not def'med and this trajectory cannot be tracked through the singular point (01, 02) = (0, n ) .
If 0zo = ~ 1 , we differentiate again,
y t ( 0 ) = 1620(620
+ 2010)=a(cr
+ 2010)
and
y2(0) = -1 v20 This gives 610 = a( P-1)/21p
91d(0) = =Z/p '#2d(O) = 0.
and
v20 =0
Then r is defined (r > 0). Similarly differentiating one more time, we get
yt(3)(0)=31(020 v20 + 610 v20 +
620 r i o )
and
Yld(3)=0
617 y2(3)(0) = 1(020 3 + 3()10 Z 1320+ 3()10 ()20 2 _ v20)
Y2d(3) = . ~ 3 .
Therefore VlO = 0 and v2o = a2(3cd2 + 4cclp2 - 3~p2+41p 2) / 412p2 At this stage we can say that the rank of singularity r is greater or equal to 1. Now from proposition 1, if a C 1 solution of the singular system ~ i exits then we shall know that xais at least C 2 and ua at least C 1. We can continue this procedure for finding more information about the regularity of the solutions at the singular point and for finding the initial conditions for the singular differential system ~ k with k >
2.
Fig.2
and Fig.3 show the controls (vial(t), v2a(t)) and the velocities (01a(t), 02d(t))
corresponding to the output trajectory ya(t) for two different values of the parameter or. We observe that these curves are regular at the singular point. Moreover the constraints imposed on 0o(t) and Vd(t) at a singular point are all satisfied. Denoting sin0i , cos0i , sin(0i
+ 0j)
and cos(0i + 0j) respectively by
si
, Ci , Sij
i,j = , 2, the singular systems Z 0 and 2] 1 associated to the manipulator may be written
= V
0
20 t_v2~ =
=(
[.~2~t.]kllSl+12st2
lie=/+/
12st2Jk
/ et
2.1 C212s121
Yl = hi(O) = l l c t + 12c12 Y2 = h 2 ( 0 )
= llSl + 12s12
with the initial conditions 0(0) = (0, ~) and e(0) = ((:z/l, offp-l)/21p) and
"o= ~=o vO=
V1
=D(
Lv2U
-
~3
Ly2d(B)J L-l c -l ct -l ct
-3
.,=o,=
and cij,,
618 Z 1
l=c,=lF lvo ] Llls,+12sl=
l=c,= Llzs,z
l=s,=JL=v1~
Yl = hi(O) =11cl +12c12 Y2 h2(0) = 11sl + 12slz
with moreover v(0) = (vt ~ V20) found previoulsly
F-ltSl -12s12 -11s121-I and where D =J(0(t))-l=/ L llCl-12c12 12c12 J
Using symbolic computations (REDUCE), we were able to characterize the trajectories for which the right hand side of Z 0 and E 1 contain the determinant of the matrix J(0(t)) in their numerator. For doing this we first substituted the values of si, ci, slj and cij in terms of Yid and )'id, i = I, 2 in these expressions. For parabolic trajectories the only allowed parametrization is given by (2). REFERENCES [1]
P.E. CROUCH, I. IGHNEIWA, F. LAMNABHI-LAGARRIGUE, "On the singular tracking problem", to appear in "Mathematics for Control, Systems and Signals".
[2]
J.W. GRIZZLE, M.D. DI BENEDETrO, J. HAUSER and S. SASTRY, "Approximation by regular input-output maps, preprint 90.
[3]
R.M. HIRSCHORN and J. DAVIS, Global output tracking for nonlinear systems, SIAM J. Contr. Optim., 26, pp 1321-1330, 1988.
[4]
M. FLIESS, P. CHANTRE, S. ABU EL ATA, A. COIC, "Discontinuous predictive control, inversion and singularities: Application to a heat exchanger, preprint 90.
[5]
K. TCHON, External trajectory reproduction problem for manipulators with degenerate kinematics, preprint 1989.
619 Y2,
Yl
Fig. 1
Y|CI) Y]Tl(I) &t(J(I)) ( ~ m ~ 0 - ~ p--I L,~3) 0-4
0.3 0.2 .
i
i
i.......... t
/
! ,,,wn~
!,,
t
i
~
.1
!
!
~ /L..-I
a4
..~
0
T 'r \ - ~~ . '
',
,
?--!--"
i
i
,~/~
', ~
L _ _ _ . "~
,--"
/~ \~/ !',', "
"
.,L__L/
L_',,. !
!/
!
!
!
! ..,
~-':~'-d~,,I
! /
I
----i----';
Y ' ~ /
-0.1
.4).'1
'v'70) vrr~*) ,~t(J(t)) (~io, ~ o A p - I L.-3) I '
ol
,>{
&l
i
~[~ I
x: ! ] !-'?
i
~v-40 "--r - - ~ - -
I
E - " ~ ')
t
t "J
'.J./
~f
I,/ I
Fig.2
V2(t) VIT2~t) det(ICt)) (wim .~"l p - I L-3)
vKt) vrrzo) ~t(J(t)) (.,~m ~ z p - z 1..-.3) IL
__._1 ~ l ; ~ _ ~ : , a _ i
[
'
r
.
.
.
. . . . . .
,
q w
o
~'/ ( ~I '~•163 " _,__ ',
-o.4
-o.~
..__~.
',.
I : .--~,-l\ '; ._~_2,__ \ ",,~ ~__....i. __LL2z
:
.
...
'l i -6
Fig.3
..4
-2
0
FORMAL CONTROLLABILITY AND PHYSICAL CONTROLLABILITY OF LINEAR SYSTEMS Claude Lobry and Pierre Delfini Drpartement de mathrmatiques, Universit6 de Nice. Parc Valrose, 06034 Nice, France.
Introduction In a recent book [10] E. Nelson shows how a very small amount of nonstandard analysis (N.S.A.) is able to provide an elementary theory of Wiener process avoiding all technicality related to mesure theory. The present paper is an attempt, in the same direction, to provide a theory for the control of distributed parameters systems. Let's consider the rather classical following linear control system :
~2N
X'(t)
=
F X(t)
Y(t)
= C X(t)
+ G u(t)
where the vector X describes the state and assume that N is an infinitely large fixed integer and define the infinitesimal dx by the relation Ndx = 1 (see w 1). The system 3SN is not a standard system because N is not standard. Nevertheless any classical theorem relative to this system is a theorem. For instance Kalmann rank criterium is valid. Let's consider the case where F is the matrix
I
-2
F=
0
where
A =
dx 2
1
1
the matrix B is the column
I~ 1
and the matrix C is
c--(I,o)
1 . -2
624 A discrete modelisation of the dynamics of a string fixed at its left extremity and controlled by is right one can be recognized in Z N. The state vector X(t) is :
X(t) =
x(t)
where
x(t) =
f
x1(t) 1 et
f X'l(t) "~ x'(t)= ~ x ' z ( t ) /
x'(t) \XN-l(t)
k,.X'N_l(t) J
The observation of the string is its position. Thus every state is reachable in an arbitrariUy short time by an appropriate control. ( The vector x(t) lies on ~N-1 because for all values of t > 0, x0(t) = 0 ) On the other hand, another idealization of such a string is given by the following P.D.E.
{~ 2w
(P.D.E)
~2W (x,t) = ~ (x,t)
w(0,t) = 0
w(1,t) = u(t)
whitch is known to be exactly controlable (opposed to approximatly controllable) in a time greater than 2. There is a discrepancy between the predictions of the two models. The objective of this paper is to show how an inacurate definition of "controllability" is responsible for this descrepansy, and to give reasonnable definitions, and to compare the two approaches.
1
S o m e concepts of n o n s t a n d a r d analysis :
We will use the following notations ( see [10] ) : Ifx and y are two real numbers we say that, x = y in case x-y is infinitesimal x < < y in case x < y and x e y x << ~ in case x is not infinitely large The few basic concepts of N.S.A. required for a good understanding of this article can be found in the following books ; [4], [5], [7], etc... Nonetheless we will develop the non standard concept of shadow. Given N infinitely large such that Ndx = 1, we call "doted-line U" the following subset of [R •
:
U = {(ndx,Un) ; n =O,...,N }
625
The s h a d o w o f U is the standard set including all the standard points o f ~
which are infinitely close to
x l~
some points o f U. The following examples illustrate this concept. If U n = (-1) n the shadow o f U is the set [0,1] x {-1,1 } If U n = sin(3-fndx) the shadow o f U is the graph o f the function x --->sin(r~x) The first example is clear. The second one c o m e s from the fact that pdx infinitely close to qdx induces Up = sin('ffpdx) infinitely close to Uq = sin('ffqdx) because the function x ---) sin(~:x) is standard continuous. The illustration o f U n = sin(kTrndx) where k is not standard
(
k cannot be considered as small towards
N ) shows the shape o f the shadow and s h o w s that it is not necessarily a graph. A theory o f these drawnings can be found in [1]. :. .... :,
'/'?'
?/U,:
Ti7:!7!{!77:7!;{:+77:;!;':7:i: ~ : 177'i.
,;;;;:7~7 ;7{;:;7,; ',7 :;17 L
! )!7!:i!:i!-i!:~!:!: ;?~!:-!::f.:;!::!~!? ;! :?:! :! !:;!:$!:;! ::!~!~:-!!:!!:!!:::!:!:;:}!:::7::!!?i_:: i!~7::!L!!:!7:!!:!!:!!2:!i:!!!~ !
!.i~':.)
i
i
.
~ il
~ 71 .
- -{-:-":-;':-: - f-:{- I- (-X- :-"
,!JilLL}i;s ; ! , ~ ; J : : : i ; J L ; i . L:L:~.;!:::,!2.i::L;IM]~2~II;;~L
Iq:130
~.:3z6
h : 2~30 k: 500
it=- 4930
: '1000
I!~:~:!i7~!~7:7i':!!!i~!7i;!~/~s!~'~::!!7::;:Lii !!-~Z
/~ A A r ii
if
'
,,
~'
:
A r, A A I
;
I
,
I
,
i
+
7i 7 ;i 7t 7 7,7 f
!iiii!iil-Ti- iiiiiiii;'!iii-:i:!':iiiiiii v
t
l". f
t
t
,
,
7 ', 7t
;:
! V V t,~ V V V \i V
For different values o f k, the doted-line U : U n = sin(k3-fndx) N = 1/dx = 5.103
The state vector x in
~N-I mentioned in the introduction is indifferently interpreted as a vector or as a
doted-line. Definition :
A vector x = (xl,x2,...,XN) o f 112N is S-continuous if : (p-q)dx - 0 ~
Xp-Xq "- 0
T h e o r e m : (of continuous shadow) The shadow o f a S-continuous vector is the graph o f a continuous standard function.
626
2
Formal
controllability
versus
physical
controllability
:
The system Z N is not standard. The infinitesimal dx describes an infinitesimal interval of •. Within the context of our modelisation process, any event which size is smaller than dx is considered as definitely inaccessible. Conversely some magnitudes are infinite, i.e. not physically achievable. These considerations (and others which we cannot develop in this short note) suggest the following definitions : Definition
1 :
The system 51.N is formally controllable if and only if it is completely controllable in the usual sense. Nil : If X is a vector of ~ n we note II X II = M a x ~ 1 IXkl
Definition
2 :
The system ZN is physically controllable if for any given expected state Z e such that II Ze II < < ~o and any given observation Y(t), t E I-h,0] ( h >> 0, 11Y(t) II << ~ ) there exists a control t ~ u(t) t E [0,T]
( lu(t)l << ~ , T << oo )
such that for every solution X0(t) of ZN with u = 0 which satisfies : C X0(t) - Y(t)
t C [-h,0]
the corresponding response satisfies : II X(t, X0(), u()) II << ' ~
t ~ [0,T]
X(t, X0(), u()) = Z e Comment : this definition is by no means the only possible definition of the "physical controllability". The choice of a definition is rather a matter for modelisation than a problem of mathematics. The definition we choosed is an adequate one among others. The main idea here is that N.S.A. provides a mathematical formalism to handle the physical concept of "order of magnitude". Proposition
1 :
Given the system
51Nwith N not infinitely large, the formal controllability is equivalent
to physical controllability. Proof : it is a very simple exercise of N.S.A. using the continuous dependence of solutions of O.D.E. with respect to the initial conditions and the fact that Z N is standard. Thus physical controllability has to be explored only for N infinitely large.
627
3
C o u n t e r e x a m p l e to p h y s i c a l c o n t r o l l a b i l i t y
:
In this paragraph we show that the system YN is not physically controllable. For this purpose we use a slight modification of a result by Callot-Messirdi [9]. Theoi~m :
Consider the differential equation (N = 2p) x'(t)
=
Fx(t)
Y(t) = C X(t) There exist a solution X(0 with the following properties : :=> IlY(t)ll"0
-l
Xp(lO) = 1 The details of the proof are given in [6]. The numbers 1 and 10 are given just for convenience and could be replaced by any other standard integers. A numerical evidence of this solution is given below for 1'4 =1000, where X(t) is given by the explicit formula : N-1 Xn(t) =
Z
u k cos(COkt ) + ~ s i n ( ~ k t ) _
k=l
sin(k'ffndx)
j
ca k
where uk and v k are the coefficients of x(0) and x'(0) in the basis of the heingen vectors of the matrix A. The n th component of the vector V k is sin(lcrt'ndx). We have : N-1 uk = 2 &x ~ Xn(0) sin(k'ffndx) n=l
N-1 v k = 2 dx ~ X'n(0 ) sin(k"ffndx) n=l
and 2
. (k.rr,:tx'~
a 6QQO
0.2000 D,JJ" -02000
-0 ~000f
-t.O000L"
WiU.~J~II gtJ,J,',.u JU,IJ~,, .~u..,,v "~..vu.
J,.~.
W~'~ --
Yr..
...:.~
.,
627
3
C o u n t e r e x a m p l e to p h y s i c a l c o n t r o l l a b i l i t y
:
In this paragraph we show that the system YN is not physically controllable. For this purpose we use a slight modification of a result by Callot-Messirdi [9]. Theoi~m :
Consider the differential equation (N = 2p) x'(t)
=
Fx(t)
Y(t) = C X(t) There exist a solution X(0 with the following properties : :=> IlY(t)ll"0
-l
Xp(lO) = 1 The details of the proof are given in [6]. The numbers 1 and 10 are given just for convenience and could be replaced by any other standard integers. A numerical evidence of this solution is given below for 1'4 =1000, where X(t) is given by the explicit formula : N-1 Xn(t) =
Z
u k cos(COkt ) + ~ s i n ( ~ k t ) _
k=l
sin(k'ffndx)
j
ca k
where uk and v k are the coefficients of x(0) and x'(0) in the basis of the heingen vectors of the matrix A. The n th component of the vector V k is sin(lcrt'ndx). We have : N-1 uk = 2 &x ~ Xn(0) sin(k'ffndx) n=l
N-1 v k = 2 dx ~ X'n(0 ) sin(k"ffndx) n=l
and 2
. (k.rr,:tx'~
a 6QQO
0.2000 D,JJ" -02000
-0 ~000f
-t.O000L"
WiU.~J~II gtJ,J,',.u JU,IJ~,, .~u..,,v "~..vu.
J,.~.
W~'~ --
Yr..
...:.~
.,
629
~000
0.2000
O0 -0.200
-0.6000
-I.0000
&~O00
0.2000
-0.200(
-06000
Now let's suppose that our observation of Y on [-1,0] is 0 and that our expected state is 0 at time 10. The rest solution 0 and the solution of the above theorem are both initial states compatible with this observation. The control 0 is the good one for the rest initial state, but not for the "Callot-Messirdi" solution. Comment : this example shows the existence of states w h i c h , in some sense, are not "physically observable" for this system, although being "formally observable". The question whether such state are exceptional or not is an another matter of reflexion.
4
Systems with limited energy :
The mechanical energy of the string idealized by the system ~:N is given by the formula :
E(t) = ~-
[
dx
dx + ~" =
630 The mechanical energy is constant. It is not absurd to suppose that under certain circumstances the assumption that the energy of the initial state is Iimited (not infinitely large) is reasonnable. In this case we have the Theolt-~m Within the class of states with limited energy the system is physically controllable, in a time greater than 2. Proof : it is proved in [6] that: Theorem : If E(0) is limited, then U(t) the solution of Z N (with u = 0) is S-continuous. From this explicit solution we conclude with the same arguments than those used for modelisation by P.D.E. (an explicit solution for u is given if we imagine that the string is not controlled on its right end but is prolongated rightwards up to infinity. The value at 1 for the solution of the semi-infinite string is the right control).
Conclusion
Firstly, this note is a cautionary note regarding P.D.E. models for the control of distributed parameter systems. The choice of a P.D.E. model is justified on the basis of assumptions which are not explicit. It is not difficult to imagine (at least academic) mechanical system (say I0 4 masses connected by springs) which is better modeled by Z N with N infinitely large than by a P.D.E. For this system, accident like those described here are theoretically possible. We are not technicians, and therefore have not a clue whether or not such things are feasible in the actuM mechanical systems. However, why not ! The second point, in fact our main point, is that the system NN with N infinitely large is a better mathematical model for the control of distributed parameter systems for the following reasons : _
because of their great conceptual simplicity (the state is a vector in ~N-I and O.D.E. have always
solutions under simple assumptions) they are very close to the physical system and are more flexible for physical discussion about legitimacy of certain assumptions (cf in w and w
is limited energy an
acceptable assumption?). _ when suitable assumptions are physically acceptable it is possible to prove the equivalence with some P.D.E. models (cf w and then recover all already known results. Control of large flexible structures is a popular problem. Many important methods and results are already available (see [8] [I 1] for survey papers, [2] [3] for recent contributions). Unfortunately these results are usually set in very abstract spaces and physical interpretation is not obvious. Radically elementary nonstandard analysis is probably the language in which implicit assumptions underlying modelling by continuous systems are the most transparent and thus provide a bridge beetwen actual problems and sophisticated mathematical results.
631 R~f6renccs :
[11
BOUDAOUD A. : Moddlisation de phdnomdnes discrets et approxhnations diophantiennes
infinitdsimales. Th6se, Mulhouse, 1988. [21
CHEN G., HUANG F.L. and LIU K.S. : Exponential stability of a long chain of coupled vibrating strings with dissipative linkage. S I A M J. Appl. Math., vol 49,No 6, pp 1694-1707, 1989.
[3]
CHEN G., KRANTZ S.G., RUSSEL D.L., WAYNE C.E., WEST H.H. and COLEMAN M.P. : Analysis,
designs, and behavior of dissipative joints for coupled beams. S I A M J. Appl. Math., vol 49,No 6, pp 1665-1693, 1989. [4]
DELEDICQ A. et DIENER M. : Lemon de calcul infinitdsimal . A r m a n d Colin, 1989.
[5]
DIENER F. et REEB G. : Analyse Non Standard. Hermann, 1989.
[6]
DELFINI P. : Mdthodes finitaires appliqudes d la rdsolution de certains probldmes d'dvolution. Th~se. Universit6 de Nice, 1990.
[7]
GOZE M. et LLrlZ R. : Non standard analysis : a practical guide whith applications. Lecture Notes in Math .881, Springer-Verlag, Berlin et New-York, 1981.
[8]
LIONS J.L. : Exact controllability, stabilization and perturbations for distributed systems. S I A M Rev., 30 (pp.l-68), 1988.
[9]
MESSIRDI
M. :
Discr~tisation spatiale d'une ~quation de propagation . Th~se de Magist~re,
Universit6 d'Oran, 1987. [10]
NELSON E. : Radically elementary probability theory. Annals of Mathematics Studies. Princetown University Press, 1987.
[11]
RUSSEL D. L. : Controllability and stabilizability theory for linear partial differential equations :
recent progress and open questions. S I A M Rev., 20, (pp.639-739), 1978.
Conditions Yielding Weak ControUability for a Class of Linear Hereditary Systems
J. A. Reneke* and A. K. Bose Mathematical Sciences, Clemson University Clemson, SC 29634-1907
Abstract. Examples of hereditary systems are provided by integral equations, functional differential equations, and systems with delays in the control action. Most discussions in in the literature of controllability for such systems are based on reformulations of the short memory systems as infinite dimensional state space systems. The obvious extension for the concept of controllability for finite dimensional systems has proven to be too strong, i..e., intended applications are excluded. Alternatives can be thought of better in terms of the original hereditary system formulations as terminal interval conditions, i.e., the system should be approximately controllable to a target function defined on a terminal interval. This is the notion of weak controllability. The problem is to find criteria for the original system description which imply weak controllability. 1. Introduction. Extending the concept of controllability to hereditary systems, delay differential equation systems or systems with dynamics given in terms of integral equations, has proven to be difficult. The extensive literature, summarized in [5], is largely devoted to semigroups and evolution equation methods, i.e., reformulations of the original dynamic equations as infinite dimensional state space systems [3]. The obvious extension of the concept of controllability from finite dimensional systems has proven to be too strong, i.e., intended applications are excluded [2]. We present here an alternative which can be thought of in terms ot the original hereditary system formulation as a terminal interval condition, the system should be approximately controllable to a target function defined on the terminal interval. After the appropriate function spaces are introduced this notion ot weak controllability will be made precise. The problem is to find criteria for the original system description which imply weak controllability. Indirect conditions on the system parameters, coefficients oi coefficient functions for example, which imply weak controllability can be developed using reproducing kernel Hilbert space representations of the input/output operator. The requisite calculations and the relationship between controllability for state space systems and weak controllability for hereditary systems will be illustrated with examples. *Partially supported by N00014-86-K-0693.
633
2. Preliminaries. We begin by presenting our abstract setting. One should refer to the text [6] by Willems for a basic discussion of the operator equation approach to systems. For completeness, we present only necessary background information. Let r be a nonnegative number, S the number interval I-r, oo), d a positive integer, X = R a, <-, .> the usual inner product on X, and I 9 I the corresponding inner product norm. Let G be the class of functions from S into X to which f belongs only in case t is continuous on [0, ~,) and if r > 0 then f is continuous on [-r, 0) and frO-) exists. Let {Nx, x in S } be the family of pseudonorms defined on G by Nx(f) = lub [ I f(t) I,-r < t < x}, f i n G a n d x in S. Let B denote the class of linear transformations of G to which B belongs if and only if 1) [Bf](x) = 0 for each f in G and x in I-r, 0] and 2) for each compact subinterval [u, v] of [0, ~ ) there is a number b such that I [Bf](t) - [Bt'](s) I < b
fs Nx(f) dx, for each f in G and subinterval [s, t] of [u, v].
The nondecreasing function bI is called a variation function for B on [u, v]. We will reserve C to denote the element of B defined by 0t
[Cf](t) =
- r < t_<0
f 0 f(s) ds
0_< t
for f in Gand t in S. The simple state space system given formally by h(t) = f(t) + t
c~f o h(x) dx where o~ is a d x d matrix and f is in G, would be written as h = f + c~Ch. The general class of hereditary systems to be considered are of the form h = f + Bh with B in B. One should realize that our abstract setting is based upon an integral equation description of the system dynamics. We can obtain an equivalent class of models as follows. Let A denote the class of linear transformations of G to which A belongs if and only if case A - I is in B where I is the identity on G. If B is in B then I - B is a reversible map from G onto G and (I - B)-I is in A . Similarly, an element A of A is a reversible map from G onto G and I - A-1 is in B. Thus we have equivalent system descriptions in terms of B = I - A -1 or A = (I - B) -1 [4]. For the system as diagrammed in Figure 1, if B = I - A -1 and D is in B then h = A(f + Dh) = (I - B)-l(f + Dh). Thus h = f + Bh + Dh or h = (I B - D)q f. Here(I - B - D) q exists since B + D belongs to B. D
f --
v
v
v
A Figure 1. Feedback Control System
v
634
Our analysis takes place in a reproducing kernel Hilbert space associated with the operators in A and B. Let k denote the increasing function defined on S by
+2r+t if0
~lf(Sp)-f(Sp.1) 12/ (k(Sp.1) = ~ ldf 12/dk _<M p=l s I1
for each partition {Sp}0 of [a, b]. The least such number M is denoted by f : I df 12/dk Note that elements of GH are absolutely continuous on compact subintervals of [0, oo] and [-r, 0) if r > 0 [3]. Furthermore, if each of f and g is in GH and [a, b] is a b compact subinterval of S then the integral f a < df, d g > / d k exist as a limit through refinement of partitions of [a, b] of approximating sums of the form ~s /dk. Let Goo denote the subspace of GH to which f belongs only in case .~ I df 12/dk < oo -r r
and Qoo the inner product for Goo given by Qo,,(f, g)= + f < df, dg>/dk. -r
The space {GooQoo} is a complete inner product space with norm Noo and has a reproducing kernel K given by K(s, t)x = k(min(s, t))x for (s, t) in S x S and x in X. Thus K ( , t)x belongs to Go,, for each t in S and x in X and = Q~(f, K(,t)x) for each f in Go., t in S and x in X. Let {Px, x in S} denote the family of projections on G given by If(t) -r < t < x [Pxf](t) = ~l.f(x)
x
for each f in G and (x, t) in S • S. It is important to note that operators B in B are causal, i.e., PTB = PxBPT for each T > 0. It may be shown that the norm N~
T restricted to PTGH, T > 0, is equivalent to the Sobolev norm ( f {I F(t) 12+
0
I f' (t) 12} dt) 1/'z. However, use of the norm Noo or equivalently the kernel K results in simplified representations and approximations of continuous linear operators. One can show [4] that an element B of :B maps G into Gn and hence an element A of ,,71.maps GH onto Ga. We will be concerned with the restrictions of elements of ,21. and :B to GH but will not introduce any special notation for these restrictions.
635
Suppose that p, the input operator, is a function from (I - P 0 ) G H into G H with the following properties: 1) p is linear and causal. 2) For each t in S, PtP is a continuous function from (Pt" P0)GH into PtGH, i.e., continuous with respect to N,~. Note that we would expect p to have a nondegenerate null space. If A is in A then, for each t in S, PtAp is a continuous linear operator from (Pt - P0)GH into PtGH. Since each of {(Pt- P0)GH, Q~} and {PtGH, Qoo} is an R K H space, PtAp(I-P0) has a matrix representation [1], i.e., there is a function L t from S x S into the continuous linear operators on X such that 1) Lt(, s)x is in (Pt - P0)GH for each s in S and x in X and 2) <[PtApf](s), x> = Q,,. (f, Lt(, s)x) for each s in S, f in (Pt- P0)GH, and x in X. The matrix representation L t of PtA9 may be computed as follows. For x and y in X, = Qoo(K(, sl)y, Lt(sl, s2)x) = <[PtAp(t-P0)K(,sl)y](sa), x> for each Sl and s2 in S. Note that if sl -< s2 < tl < t2 then Lt(1) (Sl, s2) -- Lt(2)(Sl, sz). Furthermore, if sl < s2 < t then Lt(s 2, Sl) = Lt (sl, Sl). Hence, we will speak of L as the matrix representation of A 9 without reference to t. 3. W e a k Controllability of (A, 9)- We will assume in this section that A is a fixed element of,21., p is a fixed input operator, L is the matrix representation of A9, and "lis a fixed positive number. The system (A, p) is said to be weakly controllable on I-r, T] provided 1) [A9(I - P0)GH](T - r) = X and 2) (I - PT.r)GH is contained in cl (I - Px-r) A9( I - P0)GH 9 W h e n r = 0, as would be the typical case for state space systems, the second condition is trivially satisfied and weak controllability becomes equivalent to the usual notion of controllability. Theorem 1. These are equivalent: 1) [A9(I - Po)GH](T - r) = X. 2) If L ( , T - r)x = 0 then x = 0. 3) rank [Ap(I-Po)p*A*K(, T - r)](T - r) - d.
f~r(dL(,
4) rank T-r))* dL(, T-r) / d I = d. Theorem 2. Given that 9 is a function from [0, T] x [0, T] into the continuous t
linear transformations of X such that [Af](t) = q)(t, 0) f(0) + s 9 (t, "~) df('0 for 0 < t < T and f in G H, and [phi(t) = 13h(t) where 13 is a d x d matrix, these are equivalent: 1) [Ap(I - P0)GH](T - r) = X. T-r
2) rank f0
q)(T- r, "~)13(I-Po)g*O*(T - r, x) d'c = d.
636
Theorem 3. If L is suitably differentiable then these are equivalent: 1) [Ap(I - Po)GH](T - r) = X. 2) There is an s" in [0, T - r] and a positive integer q such that dL(s,T-r)
I s = s'
2
L(s, T - r) [ s = s"
rank
d
q
d_d__.L(s, T - r) [ s = s' --
ds q
Corollary. If 13 is a d x d matrix, [pf](t) = [3f(t) for f in (I - Po)GH and t in S, g is a function from S into the d x d matrices with all but finitely many values in the se! {d x d matrices c~ I rank [13, cc~ ..... cxd-I[3] = d}, and A = (I - B) -1 where [Bf] (t)= { ; t 0 g(s)f(s)ds
-r
for f in G and -r < t < T, then [Ap(I - P0)GH](T - r) = X . For each positive number b and f in G let -r<_t
=
{o
f(t - b)
b
and 0
-r
f(t + b)
0_
[S.bf](t) = The operator Sb "shifts" the graph of a function in (I - Po)G to the right and S.~ "shifts" the graph of a function in (I - Pb)G to the left. Consequently, we say the operator D from G into G is time invariant provided S.bD Sbf = D f for each f in (I P0)G and b > 0. Theorem 4. If Ap is time invariant and b > 0 then these are equivalent: 1) [Ap(I - Po)GH](t) = X for b < t. 2) If L ( , b)x = 0 then x = 0. Any function f on S of the form f(t) = ~]p_--o.nK(t, sp)xp where s = {sp}p=O.n is an increasing sequence in S and {xp}p=o,n is a sequence of vectors in X is called a Kp o l y g o n a l f u n c t i o n . Clearly, any K-polygonal function is in G,~ since each component is a linear combination of functions in G,~. Also the subspace of all K-polygonal functions determined by a fixed partition s of [-r, T] is a closed linear subspace of G~,. Let 1-Is denote the orthogonal projection of G ~ onto this subspace. The union of the finite dimensional subspaces I-IsG~. is dense in PTGH with respect to the Hellinger norm N H. To see why this holds, suppose f is in PTGH and is orthogonal to K ( , t)x for each t in I-r, T] and x in X. Then = QH(f, K(, t)x) = 0 and f is the zero function. The orthogonal complement of the closure ot
637
the union of the subspaces IIsG~, contains only the zero function, i.e., this union is dense in PTG For each part {tp}~ of [T - r, T] and g in (I-PT.r)GH, let gt denote the block d(n+l)-vector with the (p + 1)th block defined by gt(P + 1) = g(tp), for p = 0, 1, ,., n. Let Kt(p, q) = K(tp, tq). Also let H t denote the block d(n + 1) x d(n + 1) matrix with the (n + 1)2 blocks defined by 1 t:) ; Ht( p, p + 1) = Ht( p + 1, p) Ht(1, 1) = dk(to,
=
-1 dk(tp-1'
tp)
I , l < p < n-
Ht(P + I ' P + 1)= { dk(tp_1 :, tp) + dk(tp,tp+,)l } I , I < p _ < n 1
Ht(n + 1, n + 1) = dk(tn. 1' tn)
I
where I is the d x d identity matrix. Theorem 5. These are equivalent: 1) cl.R.((I - PT_r)Ap(I - P0)) (I - PT-r)GH =
2) N ( ( A p ( I - Po))*(I- PT-r)) = {01 3) The only function g in (I - PT-r)GH with the property that for each ~ > 0 there is a partition s of [T - r,T] such that if {tp } refines s then NH( 2 n
p=0
L(, tp)[Htgt](p)) <
where L is the matrix representation of Ap(I - P0), is the constant function g = 0. Lemma. For each partition t of [T-r, T] and g in (I - PT_r)GH, K t H t g t = gt" n
For each partition {tp} o of IT - r, T] let U t denote the d(n +1) x d(n + 1) block matrix with the (n + 1)2 blocks defmed as lbllows: Ut(1, 1) = 0 ; Ut(p, q) = 0 ifp > q Ut(p+ 1, p + 1)= U t ( p , p + 1) =
1
x/dk(sp- 1' So) -1
~dk(Sp. r Sp)
I,l
I,l
Ut(p, q) = 0 if p < q + 1 where 0 and I are the d x d zero and identity matrices, respectively. Let L t denote the d(n + 1) x d(n + 1) matrix with the (n + 1) 2 blocks defined as follows: Lt(P, p) =
1 I, l_
L t ( n + 1, n + 1)= Lt( p + 1, p) =
1
,/dk(s _ 1' s.) -1
,/dk(sp_ :, sp)
I
I, 1 _
638 Lt( p, q) = 0 if p < q or p > q + 1 II.
L e m m a If {to} o is a partition of [T - r, T] then H i = UtLt. Let M t denote the d(n + 1) x d(n + 1) block matrix with (p, q) the block defined T
by Mr(p, q)= .s (dL(, tq))*dL(, tp) dk,, where L is the matrix representation of Ap(I - P0)" Assume the d(n + 1) eigenvalues of (Ut)*MtU t, counting multiplicities, are labeled and ordered )~l((Ut)*MtUt) _< )v2((Ut)*MtUt) _< ... _< ~d(n+l)((Ut)*MtUt). Theorem 6. These are equivalent: 1) clR((I - PT_r)Ap(I - P0)) = (I - PT.r)GH 2) "tmf ~'d+l ((Ut)*Mtut) > 0, where the infunum is over all partitions t of [T - r,T]. 4. Examples. Our first example is a time invariant finite dimensional state space system. We want to illustrate the computations for A and L. Also we show that the first condition for weak controllability can hold and the second condition not hold. C o n s i d e r the s y s t e m h(t) = h(0) + cc[Ch](t) + 13u(t), w h e r e c~ =
[.10 1],13=[00 0], and S - [-r, T]. t ~cos(t - s) [Apu](t) = "[oksin(t - s)
The input/output operator can be represented as sin(t
-
s) [
cos(t_ s)j[ d [ o0
OJ u(s),
[o]
for each u in (I - Po)Gw We might as well think of u of the form u(t) = u(t) , t > 0. Thus I fro sin(t - s) an(s)] [Apu](t) =
Lftocos(t
s) du(s)]"
dt
Notice that ~--{f o sin(t - s) du(s) = f 0 cos(t - s) du(s), i.e., if f =
L;J
is in Ag(I - P0)
G H then fz = fl". The projection 1-I of (I - Po)GH onto clAg(I - P0)GH can be represented as follows:
,,-, where ~ =
,tl..,-:,:,_ -1 cosh T
,..,.:
de
639
If f =
is not Ap(I - Po)GH, i.e., f2 :~ f ; then f - FI f is orthogonal to Ap(I - Po)GH.
Clearly, for any 0 < r < T, (I - PT-r)GH is not contained in clAp(I - Po)GH. On the other hand, the first condition for weak controllability does hold. Since rank [13, oc 13] = rank [ 0
1] = 2 the first condition must hold for 0 < r < T.
Still computing L and checking the column independence condition for L ( , T - r) is instructive. If 0 < s < t _ = QH ((I - Po)K(, s)y, L ( , t)x) = <[Ap(I - P0)K(, s)y](t), x > , i.e., L(s, t)* = [Ap(I - Po) K ( , s)](t), and 0
IsO sin(t- 't:) d'c
[Ap(I - Po)K(, s)](t) =
r
_0
I o cos(t - ~) dE_]
-0
cos(t - s) - cost] / -sin(t - s) + sintA
_0 If x is in X then
0 L(s, T - r)x = [ (cos(T- r - s ) - c o s ( T - r)) XI] 0 +[(-sin(T- r - s) + sin(T - r)) x2]
.1
Clearly, if L ( , T - r)x = 0 then x = 0. For our second example consider the variable coefficient finite dimensional state space system h ( t ) = h ( 0 ) + [C cch](t)+ ]3u(t), where cz = [ 1
~1' 13=I00 0], and
S = [0, T]. Since r = 0, the second condition is trivially satisfied. The corollary implies that the first condition also holds, but again we will compute the matrix representation L and see directly that the linear independence condition holds. The input/output operator can be represented as I1/2t2et
Hence for s < t
Ifoe
[ 1/2t2et][0 L(s, t)* = Le t
0J
0
duU)
]
eS-1 (-1/2s 2 + s - 1)e s + 1
Clearly, since the rows of the last factor are linearly independent, if L ( , T - r)x = 0 then x = 0.
640
Finally, for the last example consider the hereditary system f(t) -1 _
h(t)
I
t
(frO) + J'0 (t- s) h(s) ds + [9ul(t)
0 _< t
Note that S = [-1, ~,), d = I, and X = R. Recall that the control u is in (I - PO)GH. The input/output operator m a y be represented as JO
-1
[Aou](t) cosh(t - s) d[pu](s)
0
Suppose that [pu](t) = u(t - 1) for u in (I - Po)GH and t in S. If g is in (I - PT-1)GH and 0 l
t
_t+l
g(t + 1) - J'T-I (t + 1 - s)g(s) ds
t> T -2
then A p u = g. That is (I - PT-1)GH is contained in (I - PT_I)A9(I - Po)GH. Therefore the second condition of weak controllability holds for T > 2. If T = 2, then [A9(I - P0)GH](T - 1) = [Ap(I - Po)GH](1) = {0} ~ X. That is the first condition of weak controllability does not hold for T = 2. For the
0 case T > 2, let c be in X = R and u(t) =
-1 _ < t < 0 C
Fmh(-T-
2)(t+l)
t>0
Then [Apu](T - 1) = j.~-l- cosh(T - 1 - s) du(s - 1) = J ' lT-1 c o s h ( T - l - s )
C
ds = C 2) Hence [A9(I - P0)GH](T - 1) = X. That is the system is weakly controllable. Still sinh(T
computing L and checking column independence condition for L ( , T - 1) is instructive. Here -1 < s < 0 , 0 < s + 1< 1 0 sinh(t - 1) - sinh(t - s - 1)
0<s,0<s+l_
0
0
sinh(t- 1)
0
L(s, t) = <
Hence L ( , T - 1)x = 0 implies x = 0.
Bibliography 1. N. Aronszajn, T h e o r y of reproducing kernels, Amer. Math. Soc. Trans., 68(1950), 337-404.
641
2. J. C. Louis and D. Wexler, On exact controllability in Hilbert spaces, J. Diff. Equations. 49(1983), 258-269. 3. A. Manitius and R. Triggiaui, Function space controllability of linear retarded systems: A derivation from abstract operator conditions, SIAM J. Cont. and Opt., 16 (1978), 599 - 645. 4. J. A. Reneke and R. E. Fennell, An indentification problem for hereditary systems, J. Applicable Anal., 11(1981), 167-183. 5. D. Salamon, Control and Observation of Neutral Systems, Pitman Advanced Publishing Program, Boston, 1984. 6. J. C. Willems, The Analysis of Feedback Systems, The MIT Press, Cambridge, Mass. 1971.
NONLINEAR
EXPONENTIAL
STABILIZATION
BOUSSINESQ
Summary.
For
the
one-dimensional
utt in a b o u n d e d that
global
I.
with
homogeneous
feedback
exponential
Boussinesq
of
+ ~(u 2) Xx
boundary
solutions
= f
conditions,
+ cu 3t + du 5)
f = -(aut+bu
decay
of M i n n e s o t a USA
equation
+ DUxx + ~ u x u x x
+ Uxxxx
domain
a nonlinear
EQUATIONS
Yuncheng You of M a t h e m a t i c s , University Minneapolis, MN 55455,
School
OF
in
the e n e r g y
of
the
it
is p r o v e d
achieves
the
space.
Introduction General
Boussinesq
(i)
utt
The
Boussinesq
waves
systems
temperature
typical
are
form
+ ah2u
+ ~hu
+ g(Vu)hu
+ ~ A ( u 2k)
equations
arise
from
physics
in s h a l l o w
spring
equations
water [7],
near
[6.11].
and
of
Hamiltonian
thermomechanical
constant
version
some
[10],
etc.
problems
mechanics
such
[8].
phase
transitions
these
mathematical
In
the o n e - d i m e n s i o n a l
= f .
Boussinesq
equation
as
long
nonlinear with models
has
a
the
form, (2)
utt Assume
Boussinesq
+ aUxxxx
that
a
> 0
+ ~Uxx
.
In
+
~u X u XX
[12],
we
have
+ aA2u
+ ~Au
=
f
u
(3)
u(O,x)
,
a
linear
(~)
can
u2
=
)~x
shown
in
tt
F
~C
f
that
. the
linear
equation: u
where
+
fl C ~ n be
= Ou/Ou
= Uo(X )
. ut(O,x
is
and
open
exponentially
) =
bounded
stabilized
~+x
~
, F
0
on
~+x
Vo(X )
in
D .
=
with
a
uniform
in
the
space
Lipschitz
boundary
H = H2(~)
x L2(~)
o
feedback f(t,x)
= -(au
t
+ bu)
t
,
2 0
.
by
643 We
have
also
proved utt
in
[12] +
+ Uxxxx u(t,O)
that
Uxx(t,1
)
u(O,x) with
~ =
= Uo(X )
(0,1)
,
can
)
=
in
fi(t) s
, ut(O,x ) = Vo(X)
be
achieved
by
fl(t)
exponential
for
t
~ 0
.
for
t
2 0
,
for
t
2 0
in
boundary
= -Utx(t.l
decay
of
R+x
= 0
=
Uxxx(t,1)
global
= 0
~Uxx
= Ux(t,O
(s)
the
~
,
dissipative
feedback,
)
(6) f2(t) In
this
dimensional
paper
= -~Ux(t,l
the g l o b a l
nonlinear
) + ut(t,l )
exponential
Boussinesq
.
stabilization
equations
by
of
nonlinear
one
feedback
is
presented.
2.
Nonlinear
Exponential
Consider
utt
the
Stabilization
one-dimensional + ~Uxx
+ Uxxxx
nonlinear
+ ~u
u x xx
+ ~(u 2)
(7)
u u(0,x)
where
fl
is
consists f
is
of a
control
study
a
Uo(X )
bounded
= u
f
= 0
x
,
u t ( O , x ) = Vo(X)
interval
Let
=
xx
equation:
~,~.x
of be
any
R
and
its
in
~2x
on
~+x
in
~
real
.
,
boundary
constant
' P
F
numbers,
and
function.
by
the
existence
open
endpoints.
Denote
We
an
=
Boussinesq
problem'of
and
feedback
can
implement.
the
exponential
take
linear
in
terms
as
possible.
of
a
a
no
feeback. simple Below
global Hence
polynomial is
the
feedback
decay
be m a t h e m a t i c a l l y
However,
to
finding
main
of
conclusion
of
look u
result.
to e n s u r e
solutions
elaborate,
we
f
in
H
the .
Although
it h a s
to be
easy
is k n o w n
if we
are
for
some
and
ut
nonlinear with
the
global such
to confined
feedback order
as
f low
644 THEOREM. a nonlinear
For
the
f = -(au t + bu
with
a,b,c
(Uo. Vo)
and
e H
d
the
positive
+ cu~
constants,
solution
(u.ut)
(9) a uniform
exponential
Proof.
We w r i t e
+ du 5) such
exists
lim [[rU(t[ ") t-~+ m ~ u t ( .))ll
as
equation
(7),
exists
there
feedback
(8)
at
Boussinesq
nonlinear
.
that
uniquely
for
any
initial
in
C([0.~);H)
data . and
H = 0
rate. the
Boussinesq
equation
(7)
with
the
feedback
(8)
follows. utt
+ aut
+ flUxx + b u
+ Uxxxx
u = u
(I0) u(O,x) Then
(i0)
can
be
+ d u 5 + ~u x u x x
= 0
x
= Uo(X )
formulated
+ cu~
on
ut(O,x as
an
~+x
F
'
,
) = Vo(X )
abstract
+ ~(U2)x x = 0
, in
fl .
semilinear
evolution
equation,
d u ~-y(v)
u = A(v ) + r(u.v)
(11)
(u(O.:~ the
linear
operator
A [$]
A:~(A)
is
~ H
I
= [ -(A2+~A+bI)
and
the
nonlinear
part
is
,
u
0 (12)
2 0
) = (v ~o
v(O. where
t
defined
][:]
by
,
~(A)
=
E
-al
defined
by 0
FCu,v)
= [
] -CgCu)
(13) g(u) Me h a v e semigroup
on
concerned,
for
(u(t.x).ut(t.X))
= d u 59 + KUxUxx + shown H .
by
cuct.i!
(11) ))
~ ,[u~ x2x,
that and
( U o . Vo) of
the
[12]
By [ 5 ]
any
ut(t
Moreover.
in
A its
if
generates
there
h(v) an
exponentially
with
exists
= cv 3
the
a unique
local
stable existence
solution
that
s C([0.v];H).
regularity,
and
references,
s H ,
such
+ hCv))
with (Uo.Vo)
T = r(Uo. Vo) s E
,
then
the
. corresponding
645 solution
referred
Now t a k e (10)
to a s
the
v = ut
1
~-E[llutll2
satisfies
inner
product
.
CI([o.T];H)
n
of
both
sides
of
the
+ Ilu
xx
II 2 - /Ilia
u6dx
x
II 2 + b l l u l l 2 ]
r
+
~: j u x u x x u t
= 2I - -dd ~ [ l l u t l l 2 + IlUxx II2 -
~x +
2~
+ allutll2
+ cfu4tdx
I( U U x ) x U t d x
/]llu x I12 + bllull 2 + ~"
u6dx
Then an
take
the
uu
dx]
L2(D)
uncertain
constant
P{ d - - ~ [ I u t u d x +
Summing g1
(I0)
to b e
chosen,
we
obtain
+ d
5~
K
up
(ld)
and
~u0~x 5
-
dx
pu
= 0
.
. where
p
> 0
/3llu x 112 + bllull 2
Iu x u x x x ~x+~i~OUx,xUaX, Q
llut 112 + Ilu xx 112 + /]llu x 112+ bllull 2
-
(15),
+ 6
-
with
D
+ IlUxx II2 -
D
D
utu
Ilu tll 2 + IlUxx II2 -
-
3 + d iuO~x-~~ lu:d x lu
dd t r l l u t [ 1 2
(16)
+ ~
of
D
+ c
tdx
product
d [ I u t u d x + 2Uull 2 ] = P{--dT n
(15)
c
inner
+ 2a Ilull2]
s u t~u d x
c
+ allutll2 +
u dx
-
n 2~
equation
, we g e t
6 dt +~-•
(1~1)
solution
E C([O,r];E)
L2(~)
with 2
strong
(X(t.;! t( t ))
we h a v e
- 2~
the
u dx} lu:
following
= 0 identity,
J311ux II2 + bl/ull 2 + pallull 2 + 2 p f u t u d x
ix~d x ]
2~ u u
+ {(a-p)llutll2
+ PHUxx
,,~.
D
p/]llUxll2
+ obllull 2 + e
d x + "~
t u d x + pc u t u d x
. ;4uO~x~ i+x_ ~,++x, :o. n which
is
valid
Denote
C(t) (17)
for
t
in
the
n
existence
interval.
by
= G(t;p.a,b,c,d) + 2p
t udx D
= llutll2
+ IlUxxll 2 - / ] l l U x l l 2
+ E fl
D
D
+ bllull 2 + pallull 2
is
646 and N(t)
9a , b , c , d )
= N(t;p
<,8,
,
+o (16)
can
in
which
p > 0 those
~
written
2I
d dt
a.b,c is
an
d
1~
Ilull
We
Ilull
also
where
in
C(t)
we c a n and as
the
~@
s e~P
> 0
Holder +
can
estimate
inequality
c(e)@q
. !
~
P arbitrarily
be the
terms
2pi~utudx[
f/
~ J
in
g pllutll2
be
designed,
In
order
, we r e c a l l
~ m-1
and to
estimate
the
, p 2 1
m ~ k > 0 -
and i
to
,
=
the
I
q small G(t)
<
and
as
0 g O ~ 1
,
Young
I
p
where if
> 0 k
. n p
inequality:
< ~
C{e)
e
=
. ~.@
~
o
(p-1)l(pqe
.
II(p-1))
follows.
+ pllull 2
[2 (20)
I v~
~u3dx] ~
b
]~l~: IlUx 113~L,, ~ c ~
elllUxxll 2 + Ci(~:.e1)llul126 in
which
we u s e d
the
Cagliardo-Nirenberg
1 so
that
equivalent the
Young Next
1
0 > ~ to
, and Ilu
xx
inequality
II
the due with
51 < 0 ( 2
Poincare to eI
the
-
inequality
~1)
2 ~ with
(1-0)~ -
inequality
that
homogeneous
being
9
small
IlUllH2(f})
boundary as
we w a n t .
we h a v e
]2"r~uu2dxl
~
"
follows:
+ K Ilull ,0 e Lp
Wm , p
need
UUxaX
~
= 0
choose.
N(t)
< constllull 8 Ilulll-O Wk , P wm, q Lr n n oCm~) - (i-o) r .
e We
parameters
constant
~ ellull
+ N(t;p,a,b,c,d)
positive
inequalities
Wj , p 2~
are
terms
Cagliardo-Nirenberg
5:ax- o
as
intermediate
non-square
Q
G ( t ; p . a ' b , c ' d)
and
2
[2
be
(19)
- ~llu x [I2 + btlull 2 )
+ p(llUxxll2
.ru,-,<,x+,,oru.u<,x+,,<, t ~ O ~
12
Then
= (a-p)llutll2
12a'ij~iullUx[2dx f2
(by
Holder
inequality)
is
conditions,
and
.
647
(211
12~lHU,L6Clluxll2/Sdx)5/6rv
~
(by Holder
inequality)
f2 const(7)llull L
611u 1124 x L.~
(by
Cagliardo-Nirenberg/
6[lUxxllS/411ul]3/4 (by Young inequality) L 6211Uxxl12 + C2('f,6211lullS~/311ul13/4 (since I l u l l 4 / 3 < Cllull 4 / 3 ) Lb L6
< const(~)llu[l
e211Uxxll2 + C 2 ( - f , e 2 ) l l u l 1 4 ^ l l u l 1 2 / 3 Lb e2llUxxlI 2 + C 2 ( - r where
again Then
62 for
Lb
can be arbitrarily
the
terms
in
N(t)
(by
Young
+ C2('T 62)11uli 2
small
as
, we h a v e
we w a n t .
the
following
]'r I Ilu t IIL4C~[Ux 1 8 / 3 d x ) 3 / 4 f/ Ilu t i l L 4 c o n s t (~fl IlUxx 115/411ull 3 / 4
f~
6311Uxx 112 + C 3 ( ' r , 6 3 1 1 1 u
II4 . + C t ( ~ , a g ) l l u l l 6 t
63
can
We also
be a r b i t r a r i l y
L6
small.
3 I,ocJ'utudxl ~; pcC 73 pcllu C5
L5
have
(23)
where
estimates:
]~utu2dx[ ~
(221
where
inequality)
is
a constant.
fu4dx
+
1 ~u4dx )
II4 . + pcC Ilull 6 + pcC511ull2 t L5 5 L6 Similarly
(241
we h a v e xx
L6
and
(251
12p UUx2d• Based G(t)
on
the
pr62,,Uxx,2 + c2( above
= C(t:p.a,b,e,d) + (d/6)fu6dx fl -
(261
(CI(~:,al)
= (l-p)llutll
estimates,
we c a n
Lb deduce
c2( ,2),u,2] that
> l l u t l l 2 + llUxxll 2 - /~llUx[12 + ( b + p a ) l l u l l - pllutll2
- pllutl 2 -
(el+e2)llUxxll2
+ C2(~r.62))llul16^ - C2(~,62)11u112 Lb 2 + (l-el-e2)llUxxll2
. (h+p(a-1)-C2(~. e2) ) Hull2
- flllUxll2
'
648 d
_
+ (6
Cl(K'el)-C2(~'e2))~
u6dx
'
n
and N(t)
= N(t;p.a.b,c.d)
~ ( a - p ) l l u t l l 2 + plluxxll 2 - P/]IlUxII 2 + pbl lull2
+ c ~u'tt d x + pd ~ u 6 d x -
( e 3 + ~3 p e 1 + P e 2 ) l l U x x ,l 2
- C3(-T.e3)II u II4 . - C 4 ( - T , e 3 ) l l u l l 6 a - p c ] l u . l l 4 A - pcC Iluli 6 t L'i L~ L-~ 5 L6 3 ( ~ : , e )1l l u l l L 6 - PCsllul[2 - 2PC1 6 - PC2(~'e2)
+ (pb - p o %
tlull 2
3 - 2"Pel - Pe2 ) ilu x x iI2-p~llu x II2
= ( a - p ) l l u tl12 + ( P - e 3
(27)
L 6 -pC 2 (~ ' e 2 )
Ilull 6
- PC2(..~2)),u,
2 + (r162 fl
+(pd Note
that
- C4(~.e3)-pcC
3 5 - ~pC 1 ( K., e 1 )_PC2 (.T, e 2 ) ) J'u6dx fl
by t h e @ a g l i a r d o - N i r e n b e r g
I-P,Uxl121 < ellUxx ,2
(2s) where
c a n be a r b i t r a r i l y
e > 0
(1-p)liutII2+ (29)
small.
(1-e-el-e2)ilUxxll2
inequality +
1 ~ . we h a v e
K(/],e)llull 2 From ( 2 6 )
and
(2S)
we h a v e
+ (b+p(a-1)-K(~,e)-C2(~.e2))tlull2
+ (6 -CI(K'el)-C2(~'e2))
u6dx
~ G(t)
=
G(t;p,a,b,c.d)
9
(l+p)llutll2+(l+e+el+e2)llUxxll2+(b+p(a+l)+K(p,e)+C2(~,e2)) + ( ~d ' + C 1 ( K . e l )
Ilull2
+ C2(~,e2))lu6dx
fl From ( 2 7 ) N(t) (30)
and
(28),
it
s
that
= N(t;p,a,b.c,d)
~ (a-p
)llutll2
+ (p-pc
- ~3 p e l - P e 2 - e 3 ) l l U x x
+ (pb-pcCs-PC2(~,e2)-PK(~,e))llull2 + ( (l-p) c-C3(~, e3) )Zu~dx
9 + (pd - ~1 p3 C
Now we r e q u i r e
(K,el)-PC2(~,e2)-Cd(~,e3)-pcCS)~U
that 1 > p > 0 1
-
e
-
e I
a
-
p
> 0
-
,
e 2
>
0
6dx
-
II2
649 3 - ~el-e2)
p(1-e
-e 3 > 0
,
b+p(a-1)-K(p.e)-C2(~,e2)
(31)
b -
cC 5 - C 2 ( ~ . e 2 )
(l-p)c d
- Cl(~,el)
choose
parameters
requirements. e.el,e2.e and
e3
First
3
small 3 32
-
-
K(~,e)
in
take
C2(~.e2)
enough
' so
order 1
p
= ~
and
fix
-
-
> 0
,
> 0 ,
- ~3 p C 1 ( ~ , e l ) _ P C 2 ( ~ ,
pd
We c a n
- C3(~,e3)
-
> 0 ,
> 0
,
e2)-C4(~,e3)-pcC
5
as
follows
the
so
that
the
to 6
<
values,
meet p
say
<
> 0
1 a =
.
above Then
e2
choose 1 16'
-
el=
1 8'
that 1
e
e I
-
e 2
=
3/4
>
0
and 3 - ~
p(l-e Then
take
a
=
1
1 > p
= ~
eI -
.
Choose c =
Next
choose
b
> 0
such
b = max(K(~.e) with
c
chosen
and
e2)
-
e3
c
> 0
2C3(~,e3)
+
=
1/d such
i
> 0 that
.
that
+ C2(~,e2).cC5 fixed
as
above.
+ C2(~,e2) At
last
+ K(p,e)}
we c h o o s e
+ 1 d > 0
satisfying 3
d=max{6Cl(~,el)+6C2(~,e2),~Cl(~,el)+C2(~,e2)+cCs+2C4(~,e3)}+3 With obtain
such
from
a
(29)
(non-unique)
choice
and
following
(30)
gI { l l u t l l 2 (32)
the
of
+ IlUxx 112 + llu[I 2 +
~ max(3/2,2b,d/3){llutll2 ~ (2b+d){llutll2
all
the
involved
parameters,
relations,
fuedx) ~ G ( t ) + IlUxxll2 + Ilul[ 2 + f u 6 d x }
+ IlUxxll2 + Ilull 2 + f u 6 d x }
,
n and
(33)
1,ut,2 + 1,Uxx,2 + 89 ~1{ n u t l t 2
+
llUxx
]12 + llull2
i; q d x § 3;uedx + fu~dx + fuedx) +
[2
9
we
650 Substitute
the
relations
(32 / a n d
(33)
into
the
identity
(19).
we
obtain I
d
It follows
1
dt G(t)
2
+ 4(2b+d~
G(t) also
Define
d
~ 2
dt G(t)
shows
that
an equivalent
(34)
and
where
(32)
C(Uo,Vo)
u t = v~
, and
By
G(t)
is a L y a p u n o v - l i k e
norm of
yields
= G(O)
H
functional
of s o l u t i o n s .
by for
(~)
6 H
that
is d e f i n e d
the p a r a m e t e r s
the c o n t i n u a t i o n
of h y p e r b o l i c
as mild
solutions
remain
strong
these
solutions
solutions
of
by
(u,v)
(17)
as
chosen.
exist
[5],
uniquely
(Uo.Vo)
H
at a u n i f o r m
the [5, Thm.
(u,~/
1.1]
of
with
semilinear
since
in
and
of
by
equations
(11),
decay
fixed
theory
evolution
solutions
Finally.
^
.
t II(: )II~ ~ 2G(t I ~ 2G ( U o , V o l e x p (- 2 ( 2 g + d ) ) t o
(35)
and
= 0
t 2(2b+d))
~ G(O)exp(-
H(~)JI~ = ,,UxxH2 + ,,uH 2 + Hvl, 2 Then
+ N(t)
that
(34) (32)
1
G(t)
and
(11)
evolution
(35) for
6 E
t = O, u = u ~
.
implies
with
that
t 6 [0.~) Moreover.
exponential
its
equations
(u.ut)
and (35)
that
they
shows
that
rate.
references,
initial
[g]
for
two
states
(Uo.Vo)
and
^
(Uo,Vo)
respectively,
there
is a relation t
(361
H (vu((ttl)) -
for
t
in
continuity and
( ~^ I ) tH) H t)
( ^u~ v~
the
common i n t e r v a l
of
g : H~(O) ~ L 2 ( ~ )
the G r o n w a l l
there
~ I I (uv ~ o
exists
corresponding
a
inequality,
sequence
of
solutions sup
+ f "g(u(s))-g(:(s)) 0
of existence. , the
From t h e
denseness
it f o l l o w s
that
points
of
~)
local
in
(Uo.Vo)
s H
such
that
the
of
the
satisfy
If(u(t)v(t)) - (vn(t)un(t))llH ~ 0
as
n ~
te[u.T] for
each
compact
interval
[O,r]
in
the
existence
interval
ds
Lipschitz
E = ~(A)
for any I c E
ILL2(
H ,
I
651 solution
(u,v)
continuity
os
globally initial
starting Go(-,.
)
in
C([O,m);H)
data
(Uo,Vo)
from
(Uo,Vo)
implies
that
Besides in
H o
.
the
Thus
(35)
solution
(35)
holds
The p r o o f
for
is
together
(u,v) all
with
exists solutions
with
completed.
REFERENCES [13
O.P. Bhutani, P. Mital, and G. Chandrasekaran, On invariant solutions of the generalized Boussinesq equaitons, Intern. J. Engr. Sci., 26 (1988), pp. 307-310.
[2]
P. C o n s t a n t l n , C. F o i a s , B. N i c o l a e n k o , a n d R. Temam, Integral and Inertial Manifolds for Dissipative Partial Differential Equations, Sprlnger-Verlag, New Y o r k , 1 9 8 8 .
[3]
J.M. Chidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), pp. 273-319.
[4]
J. Hale, Asymptotic behavior of Dissipative systems, Math S u r v e y s & M o n o g r a p h s , V o l . 2 5 , AMS. P r o v i d e n c e , 1988.
[~]
A. H a r a u x a n d E. Z u a z u a , D e c a y e s t i m a t e s f o r some s e m i l i n e a r damped hyperbolic problems, ARch. R a t i o n a l Mech. Anal., 100 (2) (1988). pp. 191-206.
[G]
R. H i r o t a . Exact N-soliton solution long waves in shallow-water and in Math Phys., 17 ( 1 9 7 3 ) , p . 8 1 0 - 8 1 4 .
sol the nonlinear
wave equation lattices, J.
[7]
J.L. Lions, Control of Distributed Grauthier-Villars, Paris, 1985.
Singular
Systems,
[s]
H.P.
Appl.
McKean, Math.,
Boussinesq's equation on 34 ( 1 9 8 1 ) , p p . 5 9 9 - 6 9 1 .
the
circle,
Comm.
A. P a z y . S e m i g r o u p s partial differential
[10]
J. Sprekels. One-dimensional thermomechanical phase transitions with non-convex potentials of Ginzburg-Landau t y p e , IMA P r e p r i n t , ~505 (1989).
[11]
F. U r s e l l . The l o n g - w a v e p a r a d o x i n t h e waves, Proc. Cambridge Philos, Soc., 49
[12]
Y. You, T e c h n i c a l University. 1989.
of linear operators and applications equations, Springer-Verlag, 1983.
~102,
Center
of
Pure
[9]
Report
to
theory of gravity (1953), p. 685-694.
for
Appl.
the
Math.,
Purdue
DIRECT ADAPTIVE STATE-SPACE
CONTROL IN A SETTING*
Edward W. Kamen Department of Electrical Engineering University of Pittsburgh Pittsburgh, PA 15261 (USA)
SUMMARY Direct invarlant
adaptive
n-dimensional
2n-dimensional difference
adaptive
discrete-time
state representation
equation.
simple procedure
minimal
control of a single-input
The central
system is approached
constructed
result
The only assumption
is not required to be minimum phase.
(4n-dimensional) relationship
regression
estimation
this paper Elliott
schemes.
The estimation
exciting
system i3 exponentially desired poles
The approach
parameters
estimation
is that the
the given system
taken in the paper is based of an expanded
vector of the linear
parameters.
In this way the
is reduced to a
The particular
parametrization
of the parametrization
of controller
parameters
If the expanded
(as opposed to persistently
large values
constructed
in
derived by
i3 based on the standard
regression
exciting),
stable with the closed-loop
for sufficiently
of a
in
problem which can be solved using existing
version
least squares algorithm.
sufficiently
in particular,
vector with the coefficient
is a "state-space"
[I].
recursive
2.
on the system model
the desired controller
of the controller
linear-in-the-parameters parameter
parameters
of the system input as a linear function
containing
determination
in terms of a
from the input/output
the controller
order of the system is known a priori;
on the expression
directly
linear time-
of the paper is the derivation
for directly determining
pole placement.
single-output
poles
vector is
the closed-loop
located close to the
of the time index.
Preliminaries Consider
the single-input
single-output
time system defined by the input/output
y(k)
=
n Z aiY(k-i) i=l
In (i), k is the integer-valued output,
and u(k)
is
the
+
n Z b.u(k-i)~ i=0 discrete-time
real-valued
linear time-invariant
difference
control
equation
o index, input.
discrete-
(ARMA) model
(I) y(k)
is the real-valued
Note that
if
the
*This work was supported in part by Eglin Air Force Base under Contract F08635-90-K-0210.
No.
656 coefficient
b 0 is n o t
zero,
there
is a d i r e c t
feed between
the input
and
output. The coefficients assumptions
on the
is t h e m i n i m a l transfer
a. a n d b.
in
system model
order
of the
(i) a r e
unknown
(1) a r e t h a t
system.
the
fixed
order
Here minimal
constants.
n is k n o w n
order means
The only
and that
that
the
function n Z b,z -i 1 a(z)
n -i 7. a.z i=l
i-
does
not have Now
let #(k-l)
#T(k-l) where
any common
-
denote
[y{k-1)
"T" d e n o t e s
system
at t i m e
poles
the
a n d zeros.
the 2 n - e l e m e n t
y(k-2)
...
transpose
the
y(k-n)
vector
u{k-l)
operation.
u(k-2)
Defining
defined
by
... u(k-n)]
the
state
representation
(see
x(k)
of t h e
k to b e x(k)
we h a v e
regression
following
x(k+l)
= ~(k-l)
2n-dimensional
= Ax(k)
,
state
[2, p. 75])
+ bu(k) (2)
y(k)
~ cx(k)
+ b0u(k)
,
where
A=
b -
bI
b2
9 . .
0
0
0
0
~
0
0
...
1
0
0
0
.,.
0
0
0
...
0
0
0
0
9 . .
0
0
...
0
0
1
0
.,,
0
0 0
:
:
a2
...
1
0
...
0
0
0 0
an_ 1
b n 84
an
"aI
0
0
...
0
0
0
0
0
0
...
0
0
0
0
[b 0
0
...
0
1
0 th
...
0
bn_
...
1
0
0
1
0
th +(n+l) position
0] T
(n+l) position
c = Now
[a I
consider
a2
9 -.
an_ 1
the control
u(k)
an
bI
b2
generated
... b n _ 1 by
b n]
n
657 n
n
u(k) = - Z diu(k-i) i=l
-
Z ciY(k-i) i=l
,
(3)
where the c. and d. are the real-valued controller parameters. Note that in l l (3) we are not allowing the input u(k) at time k to depend on y(k) at time k since there may be a direct Let 8
c
feed from u(k)
denote the 2n-element
to y(k)
in the given system
column vector of controller
(i).
parameters
defined by 8~ = [c I
c2
... Cnd I
d2
Then the feedback
control
where x(k)=~(k-l)
is the state defined
u(k) = -e~x(k)
feedback
control
... d n]
(3) can be written
in the form
(4)
, in the previous
(4) into the state representation
state description
of the resulting
x(k+l)
closed-loop
section.
Inserting
the
(2), we have the following
system:
= (A-b0~)x(k)
(5) y(k) = cx(k)
+ b0u(k)
The 2n poles of the closed-loop of the closed-loop following
system are of course equal to the zeros
polynomial
l:
If n is the minimal
order of the system
all 2n poles of the closed-loop state feedback of the form
(ii)
det(zI-A+bsT). c
We then have the
result.
Proposition (i)
characteristic
the state representation teachability U = [b
system
(I), then
(5) are assignable
by using
(4);
(2) is reachable;
that is the 2nx2n
matrix Ab
...
A2n-lb]
is invertible. Proof:
From known results,
control
of the form
system
(I).
all 2n closed-loop
Thus if n is the minimal
assignable
by using state feedback
Since pole assignability
implies
order,
in [2, pp. 75-77];
manipulations
teachability,
however,
we also have
(2) is pole which proves (ii).
(i).
9
of the state representation
(2) is
the above proof avoids the matrix
{Pi:
i=I,2,...,2n}
within the unit disk of the complex plane, there i8 a unique controller det(zI-A+bST)c In addition,
by using a
given in [2].
Now given a desired set
reachable,
the state model
of the form u(k)=-8~x(k),
It should be noted that teachability proved
poles are assignable
(3) if and only if n i8 the minimal order of the given
=
(z-Pl) (z-P2)
using Ackermann's
formula
of closed-loop
if the state representation
parameter ... (see
poles
vector 8
c
located (2) is
satisfying
(z-P2n) [3]), we can express
(6) 8
c
in the
658 following
form.
~(z)
First,
=
writing
...
(z-pl) (z-p2)
(z-P2n)
2n-i Z ~i zi i=0
= z 2n +
define ~(A)
2n-1 ~ ~'iA i i=0
= A 2n +
w h e r e A is the s y s t e m m a t r i x of the s t a t e solution
8
satisfying
c
is the i n v e r s e
In the next be e 3 t i m a t e d
3.
Direct Again
0
...
0
(7)
i]
(8)
matrix
defined
we show that the c o n t r o l l e r
from measurements
Determ/nation consider
[0
of the t e a c h a b i l i t y
section
directly
Then the u n i q u e
(2).
8 T = ~U-I~(A) , c row v e c t o r g i v e n b y =
and U
representation
(6) is g i v e n b y
w h e r e ~ is the 2 n - e l e m e n t -i
,
of the
of the C o n t r o l l e r
the s y s t e m
system input
Parameter
(i) with t h e s t a t e
x(k+l)
= Ax(k)
+ bu(k)
y(k)
= cx(k)
+ b0u(k)
in P r o p o s i t i o n
parameter
vector 8
i. can
e
a n d output.
Vector
representation
(9) w h e r e A, b, c are as d e f i n e d controller Theorem
i:
parameter Suppose
in S e c t i o n
vector 8
c that n is the m i n i m a l
state m o d e l
(9) is r e a c h a b l e
closed-loop
characteristic
denote
Then the s y s t e m
u(k) where
8
= z
= x(k)
input u(k)
is the c o n t r o l l e r c d e f i n e d b y (8).
Proof:
With ~ defined by ~U-iAib-
the s t a t e
the
k e y result.
of the s y s t e m
vector
defined
2n-i + • ~ix(k+i-2n) i=0
Given
(I) so that the
a desired
parameter
vector
(9) y i e l d s
.
(10)
,
given by
(ii) (7) a n d ~ i3 the row
of U we have
i=0,1,2,...,2n-2
= 1 .
by
in the f o r m
- 8Tx(k)
(8), b y d e f i n i t i o n
equation
to d e t e r m i n i n g
2n-I i Z ~i z , i=0
can be e x p r e s s e d
0 ,
aU-iA2n-lb Solving
+
= ~u-l~(k+2n)
vector
order
(i.e. U i3 invertible) .
the 2'n-element column
~(k)
Our approach
polynomial 2n
~(z) let ~(k)
2.
is b a s e d o n t h e f o l l o w i n g
(12) (13)
659 .
x(k+i)
Multiplying
i-l.,
= Alx(k)
+
b o t h sides of
E A~-3-1bu(k+j) j=0 (14)
,
i=l,2,...
on the left b y ~U
-i
(14)
and using
(12) a n d
(13), we
obtain ~U-ix(k+i)
= ~U-IAix(k)
~U-ix(k+2n) Multiplying
,
= ~U-IA2nx(k)
i=0,i,2,...,2n-I
(16)
b o t h sides of the ith e q u a t i o n
in
~U -I ~i x (k+i) = ~U-iZiAix(k) Adding
(16)
and
Ix
~U -I
(k+2n)
+
(15)
+ u(k) (15) b y Zi yields
(17)
i=0,1,2, ...,2n-I
(17), we have
n
7. (k+i) i=0Zi x
]
~U-I
2nl
2n
=
]
7. ziA~ x(k)| i=0 ]
+
+ u(k)
.
Then s e t t i n g
~(k)
= x(k)
~(A)
and rearranging Throughout order
terms,
+
= A 2n +
2n-1 7. ~i Ai i=0
8T . ~U-I~(A) c we o b t a i n (ii).
the r e m a i n d e r
of this
so that the state m o d e l N o w let w(k)
2n-i 7. ~ i x (k+i-2n) i=0
denote
,
9 section
(9) is r e a c h a b l e
the e x p a n d e d
we a s s u m e
that n is the m i n i m a l
and the f o r m
4n-element
regression
(II) exists. vector defined by
~(k) ] w(k)
=
Lx(k-2n)] a n d let 8 d e n o t e
the 4 n - e l e m e n t
parameter
vector defined
by
= [~u-I _8T]
eT
c Then
(ii)
can be r e w r i t t e n
in the f o r m
u(k)
= 8Tw(k+2n)
,
or u(k - 2n) From
(18)
we see that 8 can be e s t i m a t e d
s y s t e m inputs direct
and outputs,
procedure
parametrization Elliott's
and since
for e s t i m a t i n g (ii) or
parametrization
The e s t i m a t i o n parameter
= 8Tw(k)
estimation
(18)
8
directly
from measurements
is c o n t a i n e d
o the c o n t r o l l e r
turns
(18) of the
in 8, we t h e r e f o r e
parameter
vector
out to be a " s t a t e - s p a c e "
8 . c version
have a
The of
[I].
of 8 b a s e d on schemes
(18)
c a n be c a r r i e d
such as g r a d i e n t - t y p e
out u s i n g e x i s t i n g
algorithms
or r e c u r s i v e
660 least squares
(RLS).
We shall use the RLS a l g o r i t h m
which is defined as
follows. Given a 4nx4n positive sequence
definite matrix Q, let P0,PI,P2,...
of 4nx4n symmetric matrices
denote the
defined by
P0 = Q
PkW(k+l)wT(k+l)Pk Pk+l " Pk -
,
k~0
(19)
.
1 + wT(k+l)PkW(k+l) Then the RLS estimate
~(k)
of 8 is given by
~(0) = e
o
= initial estimate
PkW(k+l) ~(k+l)
= ~(k)
+
[u(k-2n+l)
-wT(k+l)~(k)]
, k>0
. (20)
l+wT(k+l)PkW(k+l) Defining ~(k-2n) we can interpret time k-2n.
~(k-2n)
= wT(k)~(k-l)
as the estimate
,
at tim~ k of the system input u a t
We can then define the "input estimation e(k) = u(k-2n)
Rewriting
(20) in terms of e(k),
- ~(k-2n)
error"
.
we have
PkW (k+l) ~(k+l)
= ~(k)
+
e(k+l)
,
k>0
(21)
l+w T (k+l) Pk w (k+l) From known properties Proposition
~:
of the RLS ~igorithm,
The estimate
k*~, where 8* is an element
~(k) defined by of 4n-dimensional
By the above proposition, e*.
Unfortunately,
we have the following
(19) and
(20) converges
Euclidean
the RLS estimate
~(k)
result.
to 8* a3
space.
always
converges
to some
e* is not equal to 8 in general.
Thus, letting 8* denote c to 8 , we have that 8* is not equal to 8 in c c c if the expanded regression vector w(k) is sufficiently
the part of 8* corresponding general. exciting
However,
(SE), we can get 8* to be as close to 8 as desired. c c
First,
w(k)
SE if for some k~4n, k Z w(i)wT(i) i-0 Then applying
is positive
known results
(e.g. see
that w(k)
definite. [4]), we have the following
Theorem 2:
Suppose
is SE and that P0=Q=~I,
algorithm.
Then given any C>0, if ~ is sufficiently
lle*-ell< where
l J l J is the Euclidean
norm.
~>0, large,
result.
in the RLS
is
661 in T h e o r e m 2 i m p l i e s
The result
that we can get
as desired by taking the initialization that w(k)
is SE).
[4] to guarantee of ~(k)
corresponding
The Adaptive
to be
large
small
as
(assuming
One could also use the modified RLS algorithm developed that ~(k)~e as k~m and thus ~
in
(where ~ (k) is the part c c c that w(k) is SE. Thus it is possible to
to 8 ) assuming c parameter vector
identify the controller
4.
Ile~-el I
P0 to be sufficiently
8
c
(k)~8
directly
from input/output
data.
Controller
Again let ~
(k) denote the part of the ~(k) corresponding to the C controller parameter vector 8 . Then the adaptive controller is simply c realized by setting u(k) = _~T(k)x(k) . c closed-loop system is given by the state equation
The resulting
x(k+l) which can be rewritten x(k+l)
= Ax(k)
- b~T(k)x(k)
,
(22)
in the form = (A-becT)X(k)
Since the desired pole locations
+ b[sT-~T(k)c c ]x(k)
were chosen
to be within the unit disk,
the
system x(k+l) is exponentially
stable.
=
(A-bS~)x(k)
Then if
11%is sufficiently
small for k large,
II it follows
(22) is exponentially
from standard
closed-loop
system
closed-loop
poles will be "close to" the desired poles.
then have the following Theorem 3: (19),
Suppose
(20).
and in the limit as k ~ ,
the
Using Theorem 2, we
result.
that w(k)
is SE and P0=Q=~I,
Then if ~ is selected
(22) is exponentially
stable
results that the
sufficiently
~>0 in the RLS algorithm large,
stable with the closed-loop
the closed-loop
system
poles located close to the
desired poles in the limit. As noted in the previous algorithm
in
closed-loop
5.
section,
if w(k)
i3 SE and if the modified
RLS
[4] is used, ~ (k) will converge to e . In this case the c c poles will be assigned to the desired locations in the limit as
A n Example
Consider difference
the discrete-time
system given by the second-order
equation y(k) = y(k-l)
- y(k-2)
+ u(k-1)
+ 2u(k-2)
input/output
662
The poles of the system are located at 0.5+j.866 circle,
and thus the system is unstable.
which is outside the unit
The system is also not m i n i m u m phase
since there is a zero at z=-2. Letting x(k)
the coefficients
A =
denote the state vector d e f i n e d by T x (k) = [y(k-1) y(k-2) u(k-l) u(k-2)] of the state model
,
(2) are given by
1 0 0
c
-
-i
[i
1
2]
The teachability matrix is
U = [b
Ab
A2b
The objective
A3b] = 0
0
1
0
is to design the adaptive
closed-loop poles are at z=0,0. parameter vector 8
which accomplishes
c
8T = ~U-IA4
c
=
controller so that the
By Ackermann's
the controller
this is given by
[ 2
1
-7
-7
AS discussed in the previous section,
formula,
1
2
(23)
7]
the adaptive
controller
is realized by
setting u(k) = -~T(k) x(k) , (24) c where ~ (k) is the estimate of 8 generated from the RLS algorithm (19) and c c (20). Since w(k+l) depends on y(k-5) and u(k-5), to evaluate (19) and (20) for k>0 we need initial values of y(k)
for k=-5,-4 and initial values of u(k)
for k=-5,-4,-3,-2,-i.
(20) and the control law
Equations
(19),
(24) were
p r o g r a m m e d with the initial conditions y(-5) u(-5)
- 0, u(-4)
= S, y(-4)
= 0, u(-3) P0 = ~(0)
It should be n o t e d that u(-3) (Since ~(0)=0,
if u(-3)=0 by
The computer Figures
8 . c
- 0, u(-l)
= 0
(104) I =
o
was set equal to 1 to insure that ~(i)~0. (19) we see that ~(k)=0
implementation
1 and 2, respectively.
converge to zero.
= 0
- I, u(-2)
for all k>0.)
resulted in the control and output shown in Note that both the control and ~he output
Shown in Figure 3 is the estimate of the third element
This estimate does converge to the correct
value
(see
(23)).
The
of
663 e3timates
of t h e other e n t r i e s
Thus the c l o s e d - l o o p desired
poles
of e also c o n v e r g e to the correct c are p l a c e d (after an initial t r a n s i e n t
values. at the
locations. ACKNOWLEDGEMENT
The a u t h o r
wishes
tO the work p r e s e n t e d relevance
to t h a n k John C h i a s s o n in this paper,
of A c k e r m a n n ' s
for s e v e r a l
a n d in p a r t i c u l a r
discusslons
for p o i n t i n g
relating
out the
formula.
REFERENCES [i]
H. Elliott, "Direct A d a p t i v e P o l e P l a c e m e n t with A p p l i c a t i o n to N o n m i n i m u m Phase Systems," IEEE Trans. A u t o m a t i c Control, Vol. AC-27, 720-722, 1982.
[2]
G.C. G o o d w i n and K.S. Sin, A d a p t i v e Filtering, P r e n t i c e Hall, E n g l e w o o d Cliffs, NJ, 1984.
[3]
T.K.
[4]
E.W. Kamen, Convergence Processing,
Kailath,
Linear Systems,
Prentice-Hall,
Prediction
Englewood
and Control,
Cliffs,
NJ,
1980.
"A R e c u r s i v e P a r a m e t e r E s t i m a t o r Y i e l d i n g E x p o n e n t i a l u n d e r S u f f i c i e n t E x c i t a t i o n , " Circuits, Systems a n d S i g n a l Vol. 8, No. 2, pp. 207-228, 1989.
y (k)
8
u (k) '
1.2
6 9
~
4
0.4
2
. . . . . . . . . . . . e~--~ k 9 o9 -0.4 ) -0.0 "o -i. 2 9
Figure i. The control signal.
:=::: -2
......... 6 ......
9
Figure 2. The output response.
1.0~ 0.8 0.6 0.4 0.2'
.99 "::-.-e.
00,0,00
00000 4 ...................
Figure 3. Estimate of third entry of 8c.
pp.
k
Pole Placement via Generalized Predictive Control Youbin PENG"and Raymond HANUS Laboratoire d'Automatique, C.P.165,'Universit~ Libre de Bruxelles 50, Avenue F.D. Roosevelt, 1050-Brussels, Belgium Abstract: T h i s paper presents a tuning strategy for generalized predictive control(GPC). I t is shown that the GPCdesign can be made to be equivalent to the usual pole-placement technique by the introduction of suitable f i l t e r s and the selection of particular horizons. Hence, the desired closed-loop behaviour can be obtained and the s t a b i l i t y of the closed-loop system is guaranteed. The resulting GPC proves to be robust even i f overparameterization or near pole-zero cancellation is encountered. Two simulation examples i l l u s t r a t e the good performances of the proposed algorithm.
I. Introduction The basic idea behind long-range predictive controllers, is to build a predictive model of the plant over a range which is larger than the plant delay and to use this model in the control law design. Several long-range predictive control designs have been developed by many authors, for instance, Mosca et al.(1984),De Keyser et ai.(1985) and Clarke et ai.(1985). In particular, the generalized predictive control(GPC) approach derived in Clarke et ai.(1985) is able to handle control problems such as unknown and possibly variable time-delay, non-minimum phase behaviour and plant-model mismatch. GPC is essentially an algorithm in which the output y(t) is predicted over f i n i t e horizon and a sequence of control signals are calculated to minimize a quadratic cost function subject to an assumption made about the future control increments. I t offers several possible tuning knobs such as costing horizons and control horizon. However, there is no straightforward relationship between the cost-function used in GPC and the characteristic polynomial of the resulting closed-loop system. Usually, the desired dynamic behaviour can be approached by introducing suitable design f i l t e r s or polynomials. In particular, Gorez et ai.(1987) imposes the desired closed-loop poles by f i l t e r i n g the input, output and reference signals and using the f i l t e r e d signals in the cost function. The f i l t e r poles coincide with the desired closed-loop poles. However, there is s t i l l one part of the poles of the closed-loop system which depend on the horizons of GPC. These poles are not necessarily inside the unit circle, and the s t a b i l i t y of the closed-loop system is not always guaranteed. Here, we show that these poles can be assigned at the origin of the q-plane by the selection of particular Un leave from the Department of Automatic Control and Computer Engineering, Huazhong University of Sciences and Technology, Wuhan, China
665 horizons of GPC without influencing the poles assigned with the f i l t e r i n g process. In this case, the GPC design is equivalent to the usual pole placement technique. The major theoretical and algorithmic problem of the pole placement technique lies in the solution of the Diophantine equation. For instance, i f the system model equation is overparameterized, then a common factor between the numerator and denominator polynomials of the system transfer function w i l l certainly occur and the Sylvester resultant determinant w i l l be n u l l . In this case, the algorithm is ill-conditioned. When GPC is made to be equivalent to the pole placement, the same ill-conditioning problem w i l l occur i f overparametrization is encountered and there is no precaution. However, i f we use the recursive matrix inversion algorithm proposed by Favier(1987) in GPC, such drawback can be avoided by simply tuning the control horizon. Clarke et al.(1987a,b) and Mclntosh et ai.(1989) also derived conditions on the horizons of GPC so as to assigne the poles of closed-loop system at the origin of the q-plane. However, their conditions are less general. Furthermore, overparameterization can not be tolerated when a deadbeat control is achieved. This paper is organized as follows. Section I I defines the modelling assumptions and predictive model. Section I l l recalls the generalized predictive controller. Section IV investigates the closed-loop performances. Section V presents a new tuning strategy of GPC. Section VI shows the simulation results, and conclusions are given in section VII.
II. Modelling Assumptions and Predictive Model Consider a single-input single-output(SISO) discrete linear time-invariant system described by the following CARIMAmodel:
A(q~)y(t) =B(q") u (t)+C (q") ~ ( t ) / &
2. I
where { y ( t ) } and { u ( t ) } are sequences of outputs and inputs respectively. { ~ ( t ) } is a sequence of random variable with E{~(t)/F,.~)=O, E{~(t)2/F,.~):o ~ and F,., is the sigma algebra generated by {y(t-1) . . . . . y(O)} together with i n i t i a l conditions. A:l-q" is the differencing operator. A(q"), B(q") and C(q") are polynomials of degrees n,, nb and no in the backward s h i f t operator q": A(q"):l+a,q% ... +ao.q''~ B(q") = b,q"+ . . . +b,bq~b C(q'1)=1+c,q"+ . . . +c=q~ The optimal j-step-ahead predictor is given as follows. See Goodwin and Sin(1984). Theorem 1: For the system described by (2.1), given measured input and output data up to time t and any given u(t+j) for j>O, the optimal j-step-ahead predictor y ' ( t + j / t ) satisfies
C(q")y" ( t + j / t ) =Fj(q")y (t) +Ei(q")B(q") Au(t+j)
2.2
where
y" (t+j/t)=ECY(t+j)/F,}=y (t+j) - Ej{q") ~(t+j)
2.3
and
C(q") :Ej(q") A(q")+q JFj(q' )
2.4
666
where A(q"):AA(q"):I+A,q"+...+A~q ~, ni:n,+l, and Ej(q"):eo+e,q"+...+eF~qJ+', Fj(q")=~+~.,q"+...+~q', n,=n,, also,
I-~
E{ [y(t+j) -y* (t+j/t) ]2)=E{E[Ej(q")~ (t+j) ]~F,)= Z e,2o2
2.5
VVV
I-0
For the predictor (2.2) to be optimal for all t, i t is necessary that the initial conditions for y ~ be appropriately chosen. However, since C(q") is asymptotically stable, the effect of arbitrary i n i t i a l conditions will diminish exponentially. Thus we can simply define a prediction 9(t+j) with any i n i t i a l conditions as C(q-1))(t+j)=Fj(q~)y(t)+Ej(q1)B(q-')Au(t+j)
2.6
In the sequel, we will replace y * ( t + j / j ) by ) ( t + j ) . For j~k(k is the system time-delay), the prediction 9(t+j) depends entirely on available data, but for j>k, assumptions need to be made about the future control actions. It is generally assumed that the controls are to be performed in open-loop by ignoring the future noise sequence {~(t+j)}. III. G e n e r a l i z e d P r e d i c t i v e C o n t r o l
The generalized predictive controller(GPC) minimizes a multi-stage cost function of the form: N2 N2 J=E{Z [y(t+j)-w(t+j)]%>,Y-~u(t+j-l) 2} 3. I l-N1
1-1
where N, i s the minimum c o s t i n g horizon, N2 i s the maximum c o s t i n g horizon and ;~ i s a
nonnegative weighting factor, {w(t)} is the sequence of the reference signal. The expectation in (3.1) is conditioned on data up to time t assuming no future measurements are available. One can then ignore the future noise signals and use the prediction ~(t+j) to generate the control law. The prediction .~(t+j) can be separated in two parts as follows: Partioning Ej(q")B(q") according to Ej(q") B(q") =q"C(q") Gj(q-')+q+'Qj(q~)
3.2
where Gj(q")=go+g,q"+...+gj.,q-j§ Qj(q~)=qj.o+qj.lq"+...+qj.,qq"~, nq=nb-2 Substituting (3.2) in (2.6) leads to ~ (t+j)-Gj(q") Au(t+j- l)+~(t+j/t)
3.3
where ~(t+j/t) is the prediction of the output assuming there are no future changes in Au(t), i.e. C(q")) (t+j/t) :F~(q")y (t)+Qj(q")Au (t- I)
3.4
In the GPC, a control horizon Nu is introduced. That is, after an interval NuNu. Taking this hypothesis into account, the minimization of (3.1) with respect to the future auxiliary control sequence {Au(t) . . . . . Au(t+Nu-1)} yields
Or=[G~TG,+~I]"G,T(~-? (/t))
3.5
667 where
~r=[9(t+N,) . . . . . ~(t+N2) ] OT:[Au(t) . . . . . Au(t+Nu-I) ]
W%[w(t+N,). . . . . w(t+N2) ] ~(/t)r= [~(t+N,/t) . . . . . 9(t+NJt)]
3.6
and a (N2-N,+I)xN~ matrix
I gN,-, "'" gm-Nu gm 9gNI-Nu+I GI = 9 gN2-, " g:Nu
3.7
in which gj=O i f j
IV. Closed-loop Analysis GPC can be represented in a classical feedback structure by expressing (3.5) in polynomial form. Here we assume that w(t+j)=w(t) for regulation problem( We should use the estimated values of w(t+j) for tracking problem, but this does not change the closed-loop performances). Note that only the f i r s t element of 0 gives the current control value. Writing the f i r s t row of [G,rG~+>,I]"G,r as KT=[KN,. . . . . KN2] and using (3.3) and (3.4) results in R(q") u(t)=T(q" )w(t )-S (q')y (t) N~
4. I N2
where R(q")=[C(q")+~ KjqIQ~(q~)]D(q"), S(qI) = Z K~Fi(q"), T(q")= ~.=KjC(q") I'N1
J-N1
4.2
I-NI
Taking (2.4) and (3.2) into account, the closed-loop polynomial can be written as
A'(qi):C(q ') [A(q")+q" N~ Kjqj(B' (q") -A(q")Gj(q") ]
4.3
J-N|
Hence, the closed-loop system has the poles corresponding to the roots of C(q") and that of Am(q")=A(q")+q' ~2KJqj(B' (q")-A(q")Gj(q"))
4.4
l-N1
By modelling assumption, C(q") is asymptotically stable. However, Am(q") is not always stable and more analysis must be undertaken. We note that Am(q") is depending on the choice of N~, N2, Nu and ~. In fact, i t is not a simple task to adjust the weighting factor X in such a way that Am(q") is stable. Here, we only discuss the case when ~=0. I f ~ is increased, the controller is progressively detuned. However the
668 closed-loop behaviour caused by non null I ,
if
it
is good and known, can be
alternatively obtained by introducing suitable design f i l t e r . This w i l l be discussed at the end of this subsection. As the calculation of K involves the inverse of matrix G~TG~, the rank of G, should be equal to N,. We now show when this condition is satisfied. The proof can be found in Peng(1990). Theorem 2: Two polynomials ~(q") and B(q") of order na and nb are r e l a t i v e l y prime i f and only i f rank(G,)=N, with either of a)N,>%, N2>_n~+N,-I, N.:nl; b)N,=%, N2>Nu+nb-1, Nu>nA. VVV Now, l e t us show when A'(q ~) is reduced to C(q"). Theorem 3: For the GPC approach described above, i f A(q") and B(q") are r e l a t i v e l y prime and i f N,>nb, N2>_%+N,-I, No=%, >.=0, or N,=%, N2>_Nu+n:-l, N,>%, >.:0, the closedloop c h a r a c t e r i s t i c equation (4.3) is reduced to A'(q")=C(q"). Proof: From theorem 2, G,TG, is i n v e r s i b l e . Let
~ Kjq~B' (q"):B~
"~. KiqJA(q-')G1(q"):A.(q~)+a,(q)
|-NI
4.5
i-~1
where B.(q") and A.(q") are polynomials without positive powers of q, Bp(q) and Ap(q) are polynomials with positive powers of q only. From the fact that the f i r s t j terms of Gj(q~) are the f i r s t j terms of the impulse response of B'(q;')/A(q'~), we deduce that Ap(q)=Bp(q). Hence, (4.4) can be rewritten as Am(q')=A (q") - q"A. (q") +q~B. (q '' )
4.6
A.(q") can be derived from (4.5) as N2
A.(q")= Z KjqJA(q")G~(q")-A.(q) J-N1 N2
= ~. Kj[(gF,~,+...+g~.~)+.. "+ (gF,a.A-,+gF2aJ - q~.2+ gH~.Aq-.~+,]
4.7
I'NI
Noting that KTG,=[I N2
0.....
0], we have
N2
Kjgj.,=1,
~ Kjg~=O, i=2 . . . . ,N.
4.8
I f Nu>nl, we can substitute the f i r s t nA equations of (4.8) into (4.7). This yields A.(q')=q(A(q")-l). Moreover, i f N,_>%, we have B~
Hence, for the values of N,, N2
and N, mentioned in the theorem statement, A~(q")=1 and A'(q")=C(q~).
W/V
I f C(q") is taken as unit, which is often the case of adaptive control, the selection of particular horizons described in theorem 3 w i l l lead to a deadbeat control law(i.e. A'(q')=1). In this case, the system output has generally a large over-shoot to reference and disturbance changes. In order to penalize this over-shoot, a desired characteristic polynomial can be assigned. This is the case of the pole placement control I er. The GPC algorithm can be easily adapted to provide a standard pole placement controller by considering the augmented system: A(q")y,(t) =B(q")Au,(t) +P(q") C(q~)~(t)
4 .g
where y,(t)=P(q")y(t), u,(t}:P(q")u(t) and P(q~) is a suitable polynomial.
669
We can derive the control law by minimizing the cost function which uses the f i l t e r e d signals: N2
N2
J=E{~ (y,(t+j) -w,(t+j) )2+lT_~u,(t+j- I )2) J-N1
4.10
l-I
Since the role of P(q")C(q") in the augmented system is similar to that of C(q") in original system, all the subsequent results of the original GPC s t i l l hold only with C(q") replaced by P(q")C(q") and y ( t ) , u(t) and w(t) replaced by y,(t), u,(t) and w,(t). However, the control law is s t i l l valid for y ( t ) , u(t) and w(t) since the common factor P(q") may be dropped in both sides of (4.1). Hence, the closed-loop characteristic equation is A" (q")=P (q'~)C (q") Am(q ~)
4.11
where Am(q" ) is the same polynomial as t h a t defined by (4.4). We can choose P(q") so as to assign part of the closed-loop poles to a desired l o c a t i o n . The closed-loop system has also the poles of Am(q" ) which is independent of P(q')C(q"). However, i f we take N,>nb, N~>n~+N,-I, No:nA, I=0, or N~:%, N=>N,+%-I, N,>nA, X=O(i.e. Am(q':)=1), the closed-loop system has only the poles of P(q')C(q") and poles at the o r i g i n of the q-plane. In t h i s case, the GPC design is equivalent to a standard pole placement c o n t r o l l e r where C(q") is the optimal observer polynomial. I t is c l e a r t h a t more general alternatively
Am(q") caused by non n u l l
X, i f
it
is good and known, can be
obtained by introducing i t to P(q").
V. Tuning Strategy of GPC The condition required when solving the Diophantine equation for a pole placement controller, is that A(q') and B(q~) are relatively prime. Here, we see from theorem 2 that this condition is also necessary for assuring the i n v e r t i b i l i t y of G,TGI. However, we can avoid the singularity of GITG~by simply ruducing the control horizon N. until G,TG, is regular. To this end, we use the recursive algorithm for inverting G,TG~with respect to N~. This recursive algorithm was f i r s t proposed by Favier(1987) in order to decrease the computation time of GPC. Here, i t is also used for avoiding the singularity problem of the matrix inversion. We summarize the idea as follows: The matrix M=G,TG, has the following particular form: N2-1
N2-1
I-NI-I
i-NI-I
7. g Z
N2-1
Z g,.,g,
M =
T. g,g,.,
N2-1
...
N2-1
Z g,.2
X g~g~-.u+,
I-NI*I N2-1
...
~' g~-Ig~N.*,
I-NI-1
I-N1-1
I-N1-1
N2-1
N2-1
N2-1
Z g~N.+,g, i=Nl-I
or in a concise form
2 g~.u.,g~, --i-N1-1
~ g~,=.2 I-N1-1
5.1
670
i M, : 9 : M :
with
5.2
MN~ 1
.2-1 F MH M, =,-NI-17" g2, MN:M and M'=|'NL"L
N2-1
with
M2
.........
N2-I
N,=[ -~a igH+'g} . . . . . j i-
N2-1
~" gH+'gH+2 ] ' LI=N'T and ~,= T. gj_i+,2
I-NI-I
where Mr, is a ( i - 1 ) x ( i - 1 ) matrix and e, is a scalar.
J-NH
matrix,
L, is a ( i - 1 ) x l
column matrix, N, is a I x ( i - l )
By using the block matrix inversion f o r m u l a [ K a i l a t h , algorithm f o r i n v e r t i n g M.
row
1980], we get the recursive
N2-1
I . Calculate MI'= [ 7. g2 ].i; I-NI-I
2. For i : 2 1 . . . , N , ,
calculate M,I by using the formula M,.,"+M,.II L~N~M,.I-,/#' _M,.I-,LJ#,"
M,I :
5.3 -NjM,.,"/#,
1/#,
where #h=e,-N,M,.I~L~ is known as the Schur complement of Mrl. The above recursive algorithm is v a l i d i f and only i f M,I e x i s t s f o r i : l , . . . , N
u.
In f a c t , M,~ e x i s t s i f and only i f MHI e x i s t s and #,~0. Hence, we can f i n d the maximum achievable value of Nu by examining when #N.+, is n u l l . We now present a tuning strategy of GPC: I . Choose P(q17 as the desired c h a r a c t e r i s t i c polynomial and use the augmented system (4.97 to derive the control law by minimizing the cost f u n c t i o n (4.107. 2. Assign the f o l l o w i n g values to the design parameters: NI>%,
N2>%+N,-I,
3.N, is set as f o l l o w s : replaced by N,+I.
~=0
5.4
Select N, so that #N. is non n u l l but ceases to be i f N, is
For p r a c t i c a l implementation, we require R(q17, S(q I ) to have bounded c o e f f i c i e n t s and hence near s i n g u l a r i t y of M must also be avoided. This is necessary f o r numerical
stability. To avoid near singularity of M, we proposed to stop step 3 when #,.+, is smaller than a lower bound EMI, where ~ is a small positive number for numerical robustness. Remark: From theorem 2, we have some interesting results when N,>%, N2_>%+NI-I and ~,=0: i) I f A(q" 7 and B(q") are relatively prime, M,~I exists (i.e. GI has full rank) i f and only i f N.
671 polynomials Ao(q~) and Bo(q") of degrees n, and n~ with Bo(q")/Ao(q~):B(q")/A(qa). As the matrix G= is s t i l l valid for ~(q") and B:(q~), step 3 gives N~:nznl-m. The above remark shows that the
ill-conditioning
problem of pole-placement
technique does not exist in our tuning strategy. In fact, we can assign the closedloop poles to the desired location even i f A(q") and B(q") are not r e l a t i v e l y prime. We can also avoid the s i n g u l a r i t y of G,TGI by reducing the minimum costing horizon N,. But a tuning strategy for N~ seems more complicated than that for Nu. The presentation has so far contained discussions of what can be achieved when the system parameters are known. The case of unknown parameters w i l l be t r a i t e d by using the certainty equivalence principle, i . e . the control law is designed as i f the estimated parameters were the true system parameters. We can use the ELS(ExtendedLeast-Squares) method to estimate parameters. The U-D factorization with forgetting factor can be used in the i d e n t i f i c a t i o n schemes.
VI. Simulation Results The indirect adaptive GPC is applied to linear systems for tracking a square-wave reference trajectory. The f i r s t simulated system is described as follows: At the f i r s t 100 samples, the input-output model is
(l-0.Tq")y(t)=(5q"-8q2)u(t)+~(t)/(l-q ')
6.1
At the second 100 samples, the input-output model is (l-l.6q"+O.63q2)y(t)=(O.5q2-O.Sq3)u(t)+~(t)/(1-q -')
6.2
where ~(t) is taken as white noise with zero mean and variance o~:0.01. Such system is nonminimum-phase. In order to demonstrate the effect of overparameterization, a second order discrete model with 2A(qI) and 3B(q") parameters is estimated. The design parameters are N,=4, N2=8,~=0, ~:0.001 and P(q"):l-O.5q ~. Figures I and 2 show the output signals and the control signals respectively. The closed-loop performances are excellent. The control horizon N, reaches i t s upper bound(2 or 3) thanks to the proposed tuning strategy. The second simulated system is described by (l-2q~)(l'-0.71q")y(t)=(Sq-'-8q2)(1-0.lq-~)u(t)+~(t)/(l-q~)
6.3
where ~(t) is taken as the same noise as that of simulation I. Such system is
both nonminimum-phase and open-loop unstable.
In order to
demonstrate the effect of near pole-zero cancellation, a second order discrete model with 2A(q") and 3B(q") parameters is estimated. The design parameters are the same as that of simulation i . Figures 3 and 4 show the output signals and the control signals respectively. The closed-loop performances are also excellent. The proposed tuning strategy of GPC gives a good choice of control horizon (N~:2).
672
VII. Conclusions The GPC design can be made to be equivalent to the usual pole-placement technique by the introduction of suitable f i l t e r s and the selection of particular horizons. The closed-loop system obtained with the GPC design then shares the same dynamic response and stability performance as that obtained with the pole-placement controller. A tuning strategy is proposed in order to prevent the singularity problem of the matrix inversion. Hence, the ill-conditioning problem does not exist when using the algorithm. In this paper, we have only discussed regulation problem. For tracking we can simply incorporate the internal model principle into GPC or add a feedforward action to GPC. It is shown in Peng(1990) that the proposed tuning
proposed problem, suitable strategy
is applicable with these extensions.
Acknowledgements The authors would like to thank Dr. Kinnaert for his useful remarks.
Reference [I] Clarke,D.W., P.S.Tuffs and C.Mohtadi(1985). Self-tuning Control of a Difficult Process. In Barker, M.A. and P.C.Young(Eds), Identification and System Parameter Estimation, pp. 1005-1014. Pergamon Press, Oxford. [2] Clarke,D.W., C.Mohtadi and P.S.Tuffs(1987a). Generalized Predictive Control-I. The Basic Algorithm. If. Extensions and Interpretations. Automatica, 23, 137-160. [3] Clarke,D.W. and C.Mohtadi(1987b). Properties of Generalized Predictive Control, 1Oth World Congress on Automatic Control, Munich, F.R.G., Preprints, Vol.iO,pp.63-74. [4] De Keyser,R.M.C. and A.R.Van Gauwenberghe(1985). Extended Prediction Self-adaptive Control. In Barker, M.A. and P.C.Young(Eds), Identification and System Parameter Estimation, pp. 1255-1260. Pergamon Press, Oxford. [5] Favier,G.(1987). Self-tuning Long-range Predictive Controllers, 10th World Congress on Automatic Control, Munich, F.R.G., Preprints, Vol.10, pp.80-87. [6] Goodwin,G.C. and K.S.Sin(1984). Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, NJ. [7] Gorez,R., V.Wertz and K.Y.Zhu(Ig87). On a Generalized Predictive Control Algorithm. System and Control Letters. Vol.9, pp.369-377. [8] Kailath,T.(1980). Linear Systems. Prentice-Hall, Englewood Cliffs, NJ. [9] Mclntosh,A.R., S.L.Shah and D.G.Fisher(Ig8g). Selection of Tuning Parameters for Adaptive Generalized Predictive Control. 1989 American Control Conference, Vol.2, pp.1846-1851. [10] Mosca,E., G.Zappa and C.Manfredi(1984). Multistep Horizon Self-tuning Controllers: The MUSMARApproach. gth World Congress on Automatic Control, Budapest, Hungary, pp.935-939. [11] Peng,Y.B.(Iggo). On Adaptive Control: Pole-Zero Placement and Generalized Predictive Control. Ph.D. Thesis, Universit~ Libre de Bruxelles. (To appear)
673
I ,
:',
Zi' ,'
~' fl!,,,, ,- '
I\',
,
,/" ~il/
i-;~,,;~, ..... ~ ;.---.-...-+-C,,,. ~ ,,,,,-
V
- 2
i
0
40
20
60
i
L
i
i
l
i
80
100
120
140
t60
t80
200
Figure ]: Simulation I, output and reference signals versus tlme(In sampling periods)
I 0
V
-I i
i
i
t
i
i
i
i
I
20
40
60
80
tO0
120
140
160
t80
-2 o
Figure 2: Simulation I, control slgnal 4
200
versus tlme(in sampling periods)
....
A
2 r'"'~
o
..+
.
.
.
1
Ij
.
'
Z~
_+ . . . . . .
,+
., ....
it
0
l
l
-2 o
20
40
i
i
i
i
i
f
i
60
80
I00
120
140
t60
180
!00
Figure 3: Simulation 2, output and reference signals versus time(in sampling periods)
! 0
0
i
i
i
i
i
i
i
t
i
20
40
60
80
I00
120
140
160
180
Figure 4: Simulation 2, control signal
versus tlme(In sampllng periods}
200
SEPARATION PRINCIPL~ IN OPTIMIZING CONTROL OF STATE-CONSTRAINED DYNAMICAL SYSTEMS UNDER B o ~ D E D
UNCERTAINTY
M.A.Brdys School of Electronic & Electrical Engineering, University of Birmingham, Birmingham, U.K. B15 2TT B.Ulanicki Water Control Centre, School of Electronic & Electrical Engineering Leicester Polytechnic~ Leicester, U.K. LEt 9BH
ABSTRACT.
Conslderation is given to the problem of the design of a control law for a
class of state-constrained dynamical discrete-time systems under bounded uncertainty. A set of recursive equations of a dynamic programming type is developed to characterize a closed-loop minimax control law. It is shown that a separation principle holds for the class of systems under consideration.
The difference between stochastic and set-
bounded models of uncertainty is pointed out in the context of the applicability of a separation principle.
I.
Introduction The problem of finding a closed-loop minimax control law for state constralned
discrete-time dynamic system under uncertainty is considered. A prohabilistic description of the system uncertainty is not available; moreover~ due to a presence of state constraints,
the probabilistic model is not sufficient to design a control law which
is feasible when applied to the real system, that is feasible independently on realization of the uncertainty. The problem is performance index based. Similar state-constrained problems but without the performance index have been considered in e.g., [I], [3], [9], ~J~ [11]. The papers [7] and [11] deal with time-continuous systems. In the paper a minimax principle of optimality is derived in a general form. The principle is then applied to derive a recursive set of equations of a dynamic programming type which characterize the control law. The same problem has been considered in [2]. However~ the state constraints there have been incorporated directly into a performance index by using an indicator function which takes infinite values for points nor satisfying the state constraints. The approach is completely not practical because of numerical problems created by those sort of functions. On the other hand, however, a general concept of [2] of using dynamic programming is similar to our approach. In this paper, the state constraints are taken into account explicitly and the minimax control law is chosen from a set of feasible control laws. A safe tube is designed in state-time (x,t) space analogously as "stable bridge" in [11]. It is identified that the main result constitutes a separation principle which is well known in a stochastic control. Using the separation prineiple~ an important difference between stochastic and
675 set-bounded models of uncertainty is pointed out. A case of a perfect state information is treated separately. 2.
Problem Formulation Given the discrete-time dynamic system xk+ I : fk(Xk,Uk,Wk) , k c O,N-I
(I)
where x k c R n, is the state vector; u k c R m is the control vector; w k g R r is the unknown input vector (input disturbance). Available are measurements of the following structure
5< = h k ( ~ , v k) , k ~ I,N
(2)
where zk E R s, is the measurement vector and v k e R p is the corresponding error (noise). The system initial state x ~ is not known exactly;
it is known, however, that it
belongs to a bounded set X . The uncertain quantities can be represented jointly by 0
vector
A q : (Xo,Wo,W 1 .....
No statistical
WN_I,Vl,V 2 . . . . , v N)
(3)
information about w k, k = O, ..., UN_ 1 and Vk, k ~ I,N is available.
It
is known, however, that these uncertain quantities lie in the bounded sets Wk,k e 0,N-I, and Vk, k e 1,N, respectively. The above mathematical model of uncertainty is deterministic and we shall call it set-bounded
(or just bounded). The sets describing
possible variations of uncertain quantities can be written jointly as Q ~ [qeRn+Nr+(n-1)P:
XoEXo,WkCWk,ke
O,N-l,VkeVk,k e ~
(4)
There are instantaneous control and state constraints UkCU k , k e 0,N-1 where ~
(5)
and
Xkg~,
k ~ I,N
(6)
~ad Uk are known bounded sets. There is a fundamental difference between
control constraints and state constraints for control problems under uncertainty. Namely, the control constraints are not uncertainty dependent so that any control function which is feasible for the model with an arbitrary nominal value of uncertainty is also feasible for real realization of the uncertainty. Information available to the controller at time k
consists of the measurements
z I, ..., zk and the past control activities u o, u 1, ..., Uk_ I. We shall lump the information into a vector ~
=
~erefore, Uk(') with values
(z I . . . . .
%~ Rk(s+m)
%.,Uo,U 1 . . . . .
Uk_l), kc 1,N-1
(7)
the considerations are restricted to control laws of the form : R k(s+m) R m, k g O ~ N - 1 u k = Uk(~k)
(8)
676 We shall define the ~ystem control law u(-) on the time interval 0,N as composed of the control laws Uk(-) as follows :
u(.) = (Uo,U1(.) . . . . .
uN_1(-))
(9)
Clearly the control law at time zero u (-) is a constant vector function, that is o u (-) = u ~ R m o o A control design problem (CDP) is to find a control law (if it exists) ~(-) such that : i)
The constraints (5) and (6) are fulfilled for every q C Q and for every k g O,N.
Such a control law will be called feasible; we shall denote the class of all feasible control laws ii)
by~,f.~
The control law minimizes over the class of all feasible control laws the system
performance index (cost functional) defined as : max
q~Q
F(Zl,Z2,
.... zN,~,Uo,U 1 . . . . , UN_1)
(10)
where F(') is an additive function N-1
k=O In other words, the
~(-) solves the following closed-loop minimax (worst-case)
problem : rain
max
F(z I .... , ZN,XN,Uo,Ul(.) , ..., UN_1(-))
(12)
u(- )~If qeQ An open-loop minimax (worst-case) formulation of the design problem would consist in minimisation of a worst-case form, given by Eq.10, of the system performance function, with respect to all sequences of control variable values
~h ~ ' % ~k' k~o,N-1 such that the state constraints (6) are satisfied (feasible controls). Clearly, the closed-loop solution yields a superior value of the worst~case performance index than the open-loop one. ApaPt from the performance value, however, an important and favourable property of the closed-loop solution is that it exists even if the uncertainty bounds Xo,V k and Wk are too wide in relation to the control bounds Uk for an open loop feasible control to exist. We are not interested in this paper in the investigation of sufficient and necessary conditions for existence of the closed-loop solution. This will be reported elsewhere. We shall concentrate in the sequal on deriving equations and other quantitative relationships characterizing the solution.
677 3.
Minimax Principle of Optimality A necessary condition for control law to be a solution of the CDP will be derived.
The condition, called minimax principle of optimality, will have a form suitable for further development of dynamic programming (DP) type of recursive equations characterizing the solution. We shall begin with some definitions. Given the control law u(-) and the uncertainty realization q, the pair (u(-),q) applied to the state equations (see Eq.(1)) and to the measurement equations (see Eq.(2)) generate unique state and measurement trajectories. Thus the corresponding unique value of the function F(-) (see Eq.(11)) is determined. Let us denote this value by J (u(-), q). Clearly for every q e Q the following holds :
j(~(.),~) > J(~(.),q)
(13)
where ~(-) is the solution of CDP and the pair (~(-),~) solves the problem described by Eqi12). Given at a time stage k an information vector % . Def.1 (Consistency of q with {k ) An uncertainty vector q ~ Q is said to be consistent with %
if the kth subveetor
(Xo,Wo,Wl, ..., Wk_l,Vl, ..., vk) of the vector q satisfies together with the vector {k the state equations for i s O,k-1 and the measurement equations for i ~ 1~k . A set of all uncertainty vectors which are consistent with the observation ~
is denoted by
Q(%) Def.2 (Consistency of xk with {k ) A state vector X k e ~ k
> O, is said to be consistent with ~
vector q e Q which is consistent with ~
if there exists uncertainty
and the xk is generated by (Xo,Wo, ..., Wk_ 1,
Uo,Ul,Uk_ 1) according to the state equations (1) for 1 ~ O,k-1" where ui~ i c 0,k-1 are the control components of the vector % , x ~ and w i, i ~ O,k-1 are the state uncertainty components of the vector q. A set of all state vectors xk which are consistent with the observation %
is denoted by X({k).
Clearly the set X ( ~ )
constitutes the best estimation of the unknown real state
vector at the time k in the sense that it is the smallest possible set in the state space in which the real state is contained. On-line recursive algorithms for finding X(~)
or its outer approximations have been proposed and investigated in, e.g. [33,
[83,[9],[Io]. A pair(u(-), q) applied to the systems produces observation {k' control variable at the kth time stage equal to U k ( % ) and set X ( % ) . The control law restricted to the last N-k time stages is defined as follows : uN-k(-) ~ (Uk=Uk(%),Uk+1(.) Similary for XkBX (%),
Un_1(.))
(14)
qN-k8= (Xk,Wk,Wk+ I' ..., wN_1,Vk+1, ..., vN)
(15)
. . . . .
We shall use the definitions (14) and (q5) also in a more general sense, that is when:
678 uN-k(.) a__ (Uk,Uk+l(.) ..... UN_1(-)) , where uk eUk and when xk ~Xo,k~ k in Eqn.(15) where Xo, k is a bounded set in the state space. As previously, the pair (uN-k(-), qN-k) determines uniquely the state and measurement trajectories over the time period k,N and consequentiy, a unique value JN_k(uN-k('), qN-k) of the performance index corresponding to that time period is produced. Thus the function JN_k (- ,. ) is well-defined. Def.~
N-k (consistency of q with ~k ) N-k A vector q is said to be consistent with ~
if XkCX(~ k) and wiCWi, for
igk,N-1 and vigVi, for iek,---N . A set of all vectors qN-k consistent with the is denoted by ~-k(~k). Let us notice that the set QN-k(~) constitutes the best possible prediction of uncertainty realization over the time period k,N.
Let us consider the control design
problem (CDPk) at time stage k ~ 0 with initial information ~ tion of past controls and the corresponding measurements,
~k = (~1 . . . .
%'Uo'Ul(~l,Uo
) ....
in a form of collec-
that is
uk-1 = u k _ l ( Z l . . . .
%-1'Uo'Ul ....
~ - a ))
Thus the CDPk problem can be formulated as min uN-k(.) c~fN-k
max N-k ~N-k,~ , JN_k(uN-k(.), q e~ ~k J
qN-k)
(16)
Clearly the class ~ f N - k of feasible control laws on k,N depends on the uncertainty bounds given by the set
(~N-k(.),qN-k), value of performance Principle
QN-k(~).~et us
where ~N-k(.) by ~(%).
denote the solution by
= (~,~+I(.)
.....
UN_1(-)) , and the corresponding
of optimality
The pair (uAN-k(.),f -k) constitutes the solution of the CDP at time stage k with A ~k = ~k" The principle of optimality says that closed-loop minimax control strategy has the property that independently of the uncertaintyrealization over the first k time stages the control ~
A (A .
= uk _~) together with the rest N-k-1 control laws
~.I(-),
i e k+l,N constitute closed loop minimax control strategy based on information available after the k-th control stage. Proof of the minimax principle of optimality is given in [4]. 4.
Separation Principle We have already introduced the notation of state estimates as a point to set-
valued mapping which maps a current information vector %
into the smallest set X(~ k)
which contains the true state vector. We shall introduce some notions needed to handle
679 the feasibility problem (see for comparison, sequence of feasible controls uk gUk k g O,N-I
e.g. [2],[3],[5],[6],[11]).Given
the
. Let us define a sequence of sets in
a state space by the following recursions.
xDN:x ~
(17a)
XDk ( ~ ' ~ + I ..... uN_ ~) : ~ a
[~n:Q(~,%,wkh
m k + 1 ( ~ + ~ ..... uN_~)~
(~?b) where k ~ O,N-I ; % ( ~ , % , W k) is the image of the set Wk ~ d e r given ~ and uk by the mapping fk(~,%~,.) The sequence [rOk(%~ .... uN_1)], k C O,N-I constitutes a time-varying tube in a state-space of the property that at any time stage k the set XDk(Uk,
.., UN_ 1) is the largest feasible set of state vectors x k such that the con-
trol uk is feasible independently
on the realization of uncertain quantity Wk, if the
system starts from the Xk; moreover~ the resulting states Xk+ I have got the same property as the xk in connection with Uk+ I. Def.4 (Safe tube) Let us define at the kth time stage a family of sets as follows :
~X k ~ (xe2Rn: X = XDk(~ . . . .
uN_q), UkeUk . . . .
UN_leUN_1]
where 2 Rn is the power set of R n. A sequence [ ~ k ] , k g O,N-I
is called safe tube. A physical meaning of the safe tube
is clear. A true state vector at any time instant k is known only with accuracy to its set estimation X(~ ). Therefore, it is sufficient and necessary for feasible control strategy over k,--~ to exist, that is ~ f N - k ~ ,
that the set X ( % ) c ~ .
Let us notice
that it is not enough here, although this may seem to be true, that X(~k) is covered by a number of different sets XDk. The safe controls uk associated with the sets XD k ~ee (17b)) may be different~ Because the set X ( % )
constitutes the smallest approxi-
mation of the unknown state vector, we must have exactly one control which is feasible for all points of the estimate X ( ~ ) . The notion of the safe tube is similar to a notion of the stable bridge introduced in [11] for time-continuous systems. Def.> (Feasible control set) Given set XcR n and a time stage k, a set of feasible controls at the k-th time stage for given X is defined as
uok(x) ~ {~cuk: x~xok(,~,~+ 1 . . . .
,%_~), ~+~uk+ 1 . . . .
If a given control law u(.) is feasible, that is u ( - ) C ~ f UDk(~)
(~9)
u~_lcuN_ 1] , then U k ( % )
g
for every q s Q. We shall also introduce a set of estimation type of notions
we have introduced so far by defining prediction ~ + 1 ( % , U k )
of measurement ~ + I
the k+1-th time stage which is consistent with information available at the k-th stage in a form of (%,Uk).
at
680 Def.6
(Measurement prediction)
zk+l(%, ~ ) ~= (Zk+ 1 a s 9 ( satisfy
the state
qeQ) (Xo,Wo,W1 , . ,-
wk ,
Vl,V 2 . . . .
Vk+l,~,UkJ
(20)
a n d measurement e q u a t i o n s f o r i r O,k .
We a r e now i n a p o s i t i o n
to f o r m u l a t e t h e main r e s u l t .
Theorem 1 An o p t i m a l v a l u e F o f t h e s y s t e m p e r f o r m a n c e i n d e x (10) i s g i v e n by :
= min
max
Uo~Uo
[g ( z . , u )
ZIBZl(Uo).
o
I
The f u n c t i o n H l ( Z l , U o) i s g i v e n by t h e l a s t
~(~) subject
where
to
= rain
step of the recursive
max [gk(Zk+l,Uk) + ~ + l ( ~ k , Z k + l , U k ) ] ~+1
uk r
Un_ 1
max Zn,X n
, for k ~ 1,N-2
(22)
(23)
[gN(ZN,UN_l) + f(xN) ]
subject to UN_I~UDN_I(X(~N_I)) , ~eZN(~N_I,UN_I),
If
procedure
),
, and Z k + l r
~_1(~N_1) = rain
XNr
(21)
+ Hl(Zl,Uo)]
o
: XN = fN-1 'UN-1 'WN-1)C
and (24)
~-I~X(~N-1)'WN-lCWN-1 ] "
uA(") is the CDP solution, then the following hold uAo = Arg
for every
min[ UoeUo
max
(25)
[go(Zl,Uo) + Hl(Zl,Uo) ]] ;
ZleZl(Uo )
%, Uk(~k) = Arg
subject to (22), for kcl,N-2
minimax
J
(26)
+ f(~)] ]
(27)
[gk(Zk+l,Uk) + ~ + l ( ~ , Z k + l , U k ) ]
,
and for every ~N-I' ~-1(~N-I ) = Arg
rain [ max Un_ 1 ~ ' ~
[gN(~,~_l)
subject to (24) A proof of the theorem is given in [4 ] A structure of the set Q (see Eq.4), Def.2 and Def.6 imply that
~ + I ( ~ % ) = [zk+1~s: ( ~ x ( ~ )
A ( 3wk~Wk) ^
( ~ Vk+leVk)Zk+ 1 = ~ + l ( X k + l ,Vk+ 1 ), where Xk+ 1 = fk+l (Xk'Uk'~k)] Therefore, Eqn.(20) can be w r i t t e n in the f o l l o w i n g form :
(28)
681
z.,k+~(~k,,.,k) = Zk+l(X(~),~)
(29)
Let us notice now that due to (29) the equation and inclusions (21) .. (27) do not depend explicitly on ~k; the information,vector ~k affects them only through that state estimation set X ( ~ ) .
We shall formulate this fact in terms of a sufficient
information function [2]. Corollary I The mapping ~
~ X(~)
is the sufficiently informative function for the CDP.
The Theorem and Corollary I together constitute the separation principle. The principle says that at any time stage k the closed-loop minimax control signal can be generated in the two following steps : first, the state estimate X ( ~ )
is pro-
duced and then based on this estimate a deterministic minimax control problem is solved and the control signal ~ ( ~ )
is obtained.
It is not difficult to show that the separation principle doesn't hold if the k-th time stage cost function depends explicitly on xk. Therefore, it is not applicable if
N-I
F (z,~,u)
= F(x 1 . . . .
xN,u 1 . . . .
UN_1) = ~ g k ( x k , ~ ) k=0
+ f ( x N)
It is well known that for such strucfures of the performance indices the separation principle is applicable in the case of stochastic systems. This is an important difference between stochastic and set-bounded modelling of uncertainty in optimal control of dynamic uncertain systems. An applicability of a separation principle can be achieved by introducing a new state variable Xn+ I in a standard way [2] Xn+1,k+ 1 = Xn+l, k + gk(Xk,Uk), with initial condition Xn+l, ~ = 0 The performance index then becomes equal to Xn+1, N and the Corollary 1 and consequently the separation principle is applicable to the extended system. Another interesting problem arises where some of the components of the vector Wk, k ~ O,N-I are constant but unknown parameters. Let us denote the corresponding vector of parameters by' a. Clearly, there is now a time structure in uncertainty associated with the state equations ~id the previous model is no longer valid. Again, by introducing new state variables in a form of the parameters, the applicability of the theory presented in the paper is achieved where the additional state equations are as follows : ak+ I = ak with an initial condition a ~ = a.
The exact solution of
the problem in this case and some suboptimal schemes in a form of adaptive algorithms will be reported elsewhere. 5.
Perfect State Information. If a state vector is available we have a qualitatively simpler case. We do not
now need to define a safe tube through the family [ ~ k ]. As previously, we define
682 recursively the following sets: XD k and
= xz ~
XD N = ~
[~n
(3~Uk)fk(~,~,Wk)C_XDk+1 ] for k ~ 0,N-I
(3O)
.
Clearly, if the state vector belongs to the set XD k then there exists at the k-th time stage control
uk
which independently on uncertainty realization guarantees that
X k + 1 ~ k + I and the same applies for k+1~ ..., N-I. The feasible control set at the k-th stage has the form :
~k(x) : [ ~ U k : fk(x,~,Wk)c_~k+1~ and the set is non-empty if
xgXD k .
The formulae (21) .. (24) are now simplified to where the operation max is taken with respect to the variable wk subject to WkeW k and where the state estimates X({ k) do not occur. For example, ~ ( x k) :
subject to
min uk
Ukr
and t h e r e s u l t i n g 6.
max wk
the formulae (22) and (23) are now as follows :
[gk(xk+q,Uk) + Gk+1(fk(Xk,Uk,Wk)) ]
k) and Wkr k , A A o p t i m a l c o n t r o l law i s a f u n c t i o n o n l y o f Xk, t h a t i s uk = Uk(X~).
Conclusions A problem of design of closed-loop minimax control strategies for state-
constrained dynamical discrete-time systems, with uncertain initial conditions, parameters (possibly time-varying) and exogenous input signals, and with noisy measurements has been considered. The uncertainty has been assumed to be modelled through bounds which define fixed sets within which lie unknown constants (certain of the parameters and initial conditions) and tubes for time-varying quantities (other parameters, plant disturbances and measurement noise).
A DP type recursive procedure has
been derived to design the control rules. It has been shown that the separation principle holds for .the class of problems considered in the paper. Based on applicability conditions of the separation principle and the conditions for the separation principle derived for stochastic problems, an important difference between deterministic set-bounded and stochastic modelling of uncertainty has been pointed out. 7. [1]
References D.P.Bertsekas and l.B.Rhodes (1971). On the minimax reachability of target sets and target tubes. Automatica, Vol.7, pp 233-247.
[2]
D.P.Bertsekas and I.B.Rhodes (1973). Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems. AC-18, No.2, pp 117-124.
IP;~;~;Trans.on Aut.Gontr.,
683 ~]
M.A.Brdys and S.Nowacki(1982). Design of safe controls and feasible control rules for uncertain dynamical systems under state constraints with applications to water systems. Technical Report, Institute of Automatic Control, Warsaw University of Technology (in Polish).
[4]
M.A.Brdys and B.Ulanicki (1989). Separation principle in optimizing control of state-constrained dynamical systems under bounded uncertainty. Research Memorandum No.20, School of Electronic & Electrical Engineering, University of Birmingham.
[5]
J.D.Glover and F.C.Schweppe (1971). Control of linear dynamic systems with set constrained disturbances. IEEE Trans.on Aut.Contr., AC-16, No.5, pp 411-423.
[6]
W.Findeisen (1982). Some definitions and notions of feasible control. Unpublished notes. Institute of Automatic Control, Warsaw University of Technology.
[7]
A.B.Kurshanski and I.Valyi (1988). Set values solutions to control problems and their applications. In 'Lecture Notes in Control and Information Sciences INRIA'. A.Bensoussan and J.L.Lions (Editors). Analysis and Optimisation of Systems. Springer-Verlag, pp 987-988.
[8]
A.B.Kurzhanski (1988). Identification - A Theory of Guaranteed Estimates. Working Paper. International Institute for Applied System Analysis. Laxenburg, Austria.
[9]
J.P.Norton
(1987). Identification and application of bounded parameter models.
Automatica, 23, pp 497-507. [10]
F.C.Schweppe (1968). Recursive state estimation : Unknown but bounded
errors
and system inputs. I~;~;~iTrans.Automat.Contr., Vo.AC-13, pp 22-28. [11]
A.J-Subbotin and A.G.Chencow (1981)- Optimization of Guarantee in Control Problems. Nauka (in Russian).
UNI~?CATION OF
5ONE
HI~*IO ADAPTIVE
CONTROL
ALt~ORI~I~ISAND GLOBAL CONVERGENCE ANALYSIS
T i a n You C h a i Department of Automatic Control, Northeast University o f T e c h n o l o g y , S h e n y m l g 1 1 0 0 0 6 , L i a o n i n g , P. R. C h i n a
Su~taary This
paper
schastic some
presents
multivariable
HIMO a d a p t i v e
cases.
the general
control
structures
wide variety
of adaptive
viewpoints.
tile
adaptive
adaptive
control.
~daptive
For t h i s
scheme
approaches
arc
special
schcxae u s i n g a m o d i f i e d
control
theorist
which includes
(1980),
Allidina
and
has been significant Dugard
control,
and
arbitrary
peper,
other
progress
Dion
schemes and extensions
this
control
effort
least
adaptive
a
in extending
(1985) pointed open,
out
form
many
one of challenges
a unifying
view
of
has been devoted to determining
adaptive
schc~es and presenting
and self-tuning
SISO d e s i g n s that
in particular,
in
results
albeit
in
this
the fact
t o t h e Mlr~} c a s e .
multivariable
global
a
schemes in Egardit
Ilewcver,
systems have not been given yet,
some p r o b l e m s a r c s t i l l
In
out that
Hughes ( 1 9 8 3 ) a n d Chai ( i 9 f f 7 ) .
for multivarlable
addition,
have been developed
i s Cite p r o b l e m o f p r o v i d i n g
between the seemingly disparate
algorithm
direction
algorithms
Ljung and Anderson (1984) pointed
general
certain
by s p e c i f i c
.
linear
Introduction
control
F o r 5150 a d a p t i v e
tile relationship
tbcrc
matrix
control
algoriClu~s is also given.
different to
scheme which can
interactor
dcrived
convergence of this
1. A
adaptive
systems with arbitrary
A proof of global
squares
direct
stability
that In
adaptive analysis
of
o f known s c h e m e s c a n a n d m u s t b e c o n s i d e r e d .
general
adaptive
s c h c m e s f o r Nlr~0
systems
interactor matrix is presented and the relationship bctwocn
having
the
the
scheme
proposed and other scheraes developed by specific approches is discussed. The global convergenoa analysis for the sche~ao will be established.
685 2 , General Adaptive Scheme Consider the ~It~ system described by the linemr vector difference equation A(z-' )y(t)=B(z" ' )u(t)+B, (z")v(t)+C(z- ' )w(t) where
u(t),
(2.1)
y(t) and v(t) arc n-vectors defining the system input output
and measurable disturbance vector respectively,
and A,
matrices in unit delay operator z-' such that A(O)=I, (I
is the (nxn) identity matrix).
B, ,
B(O)=O,
C are
polynomial
B, (0)=0, and C(O)=I
The nxl disturbance sequence w(t) is assumed to
be a real stochastic p r o c e s s defined on a probability the
B,
vectors
D-- algebra generated by the observations
space (l),~
,
F). Ft denotes
up to and including time t.
It
is
a s s u m e d that E(w(t)/Ft - ,)=0
a.s.
E(w(t ) w ( t ) ' / F t - , ) = ~ , with trace ~<|
(2.2) a.s.
and
lira sup fIN ~ llw(t) If'<|
Assumption A:
(I) dot C ( z " )
It is well known that,
made t h a t the
time
(2.3)
asstociat~d
(2.4)
~ O, for Izl~l . for
tile time delay associated delay
a.s.
(2) B(z-')-'B2(z-')
the menovariable
case,
with the control
with the measurable
is proper.
the assumption
input
is less
disturbance
to
is usually
than or equal obtain
a
to
causal
control law. ~le assumption A(2) i s a natural extension of the above assumption
to
the multivariable case. Assumption B:
The
T(z-' )=A(z-' )-'B(z-'
system
(2.1)
), between u(t)
is
s t r i c t l y proper and
and y(t)
has full
the
transfer
function
rank a.e.z.
The l a t t e r ass.umption i s a necessary condition for a r b i t r a r y output tracking ( Elliot
and Nolovich 1982).
A consequence of t h i s assumption i s t h a t there
a interactor matrix ~(z) (Wolovich and Falb 1976 ) ,
exists
which i s a unique nonsingular
nxn lower left triangular matrix such t h a t
lira t ( z ) T ( z - '
)=g
g : nonsingular
(2.5)
The control objective i s to find a control law to minimize the variance of the fol lowing generalized tracking error vector function: e(t+k) =P(z" ' )~(z)y(t)-R(z" ' )u' (t)+S(z- ' )v(t)+Q(z- ' )u(t)
(2.5)
686 where P ( z - ' ) , such
that
R(z-'),
P(O)=I,
S(z-') and O(z-') are weighting polynomial matrices in
and
u'(t)
i s an n - v e c t o r d e f i n i n g t h e known
and k i s a maximum order o f any polynomial in E(z)~
siglml
reference
The cost
z" input
function
is
g i v e n by J=E(ile(t~k)
ll'/Ft)
(2.7)
TImorum 2.1
(I) The ~encralized tracking error vector function e(t+k) is decribcd by e( t+k)=~, (z-')-' (G(z-')y(t)+(ll(z-' )+~, (z-')Q(z-' ) )u(t)+(D(z- ' )+~o (z-')S(z- ' ) )v(t) -~~ (z-')R(z-' )u" (t))~^, (z-)-' F(z-' )C(z-' )w(t+k)
(2.8)
where ~, (z-'), G"(z-'), If(z-' ), D(z-' ), ~ ( z " ) are polynomial matrices z-' such that det A, (z")~O,
for Izl~l, and ll(O)=K.
(2) The causal optimal control law minimizinK (2.7) is g i v e n by ~(z-' )y(t)+(ll(z-' )+A• (z-')Q(z-' ))u(t)+(D(z-' )+A~(z-')S(z-' ))v(t) -~, ( z - ' ) R ( z - ' )u" (t)=O
(2.9)
(3) For the system (2.1), i f the control
law
(2.9)
is
used,
the
minimum
p o s i b l e v a l u e of quadratic c o s t function ( 2 . 7 ) for stochastic case i s J=E(Ile(t+k)
II' / F t ) = E ( fl F ( z - ' ) w ( t + k ) II ' l F t ) = 7 '
(2.10)
and tile minimum possible value for deterministic case is J=O
(2.11)
Proof : (1)
As shown i n Dugard e t .
a l (1984) and Dugard and Oion ( 1 9 ~ ) ,
using
thc
fol lowing relations
where
A, ( z - ' ) A ' ( z - ' )=A(z-' )~, ( z - ' )
(2.12)
P ( z - ' ) ~ ( z - ' )~, ( z - ' ) = F ( z - ' )A(z- ' )+z-" G ( z - ' )
(2.13)
A0 ( z " ) F ( z - '
) = F ( z " )A, ( z - ' )
(2.14)
A, ( z - ' ) G ( z - ' )=~(z- ' )~, ( z - ' )
(2.15)
~(z-')=z-'~(z)
and ~~ (O)=A, (O)=I, (2.15), ~
have
F(z-')=F,+F,z-'+ ..... +F,.,z ......
'~ detA~
A, (O)=A, (O).
and d e t ~ , ( z - ' ) = d e t A , ( z - ' ) ,
From (2.1),
(2.6), (2.12)-
687
e ( t + k ) - - ~ , - ' ( ~ y ( t ) + ( i l + ~ Q)a(t)+(D+A~', S ) v ( t ) ) - ~ , Ru ' ( t ) ) + ~ ~ '~Cw(t+k)
(2.16)
where I I ( z -~ ) = z " F ( z - ' !1, = 0 , D(z-'
(i=
)B(z" ' ):ll ......
-1 .....
....
§ .....
+11., z + l l . +tl, z - ' + . . . . .
+11.. z - "~ ,
-k+l)
)=z' F(z- ' )B, (z-')=D~
D, =0 ( i = - I
z' "
-..,,
z' - ' + .....
g - , z + g . +D, z - ' + . . . . .
+i). 9 z" ' ,
-k+l)
(2) and ( 3 ) . S u b s t i t u t i n ~
(2.16) into (2.7) results
in
J=E(fl h , - ' ( ~ y ( t ) + (ll+~~ Q)u(t)+(D+A, ~ S ) v ( t ) - h~" , Ru" ( t ) ) + ~ , " FCw(t+k) For s t o c h a s t i c
It"/F,
) ~ 7'
(2.17)
c a s e A, i s c h o s e n such t h a t A , ( z -1 )=C(z-' ) . From ( 2 . 1 4 ) and ( 2 . 1 7 )
J=E(U ~ ' ( ~ y ( t ) + ( l i + ~ ) n ( t ) + ( m ~ S ) v ( t ) - ~ ' C R u " ( t ) ) | l ' / F , )+E( [ZFw(t+k)t/'/F, ) 9E( ~le
first
[IF(z"
)w(t+k)g'/Ft
)=1'
term on t h e r i g h t hand s i d e o f tile above e q u a t i o n i s
e q u a l t o z e r o and i s b r o u g h t t o z e r o by t h e c o n t r o l For d e t e r m i n i s t i c
c a s e A~
w(t)=O,
greater
than
or
law ( 2 . 9 ) .
( 2 . 1 7 ) can be u r i t t e n
as
J= II ~. -' (~y(t)+(ll+~,Q)u(t)+(D+~S)v(t)-~,Ru" (t) If' 20 Obviously, control law (2.9) makes J equal to zero. From (2.12)-(2.15), we can obtain the another expression of optimal control law in the deterministic case.
G ( z - ' ) y ( t ) + ( l l ( z - ' ) + Q ( z " ) ) u ( t ) + ( g ( z -t ) + $ ( z - ' ) ) v ( t ) - [ { ( z - ' )u" (t)=O
(2.18)
F and G are g i v e n by P(z -I )~(z -I )=F(z "~ )A(z-' )+z-' G(z- ' )
(2.19)
where ll(z-' )=z'F(z" )B(z" ), D(z-' )=z'F(z -~ )B2 (z "I ), II,=D, =O(i=-I ..... -k+l), II,=g. If the
system parameters are unknown,
controller
we use the adaptive control algrithm where
parameters may be estimated from tlle estimation
equation
derived
from the error vector function equation (2.8) and the control law obtained from the control law equation generated from (2.9).
3. Examples of tile General Adaptive Scheme multivariable self-tuning regulators and controllers and model adaptive schemes will be shown to fit into the general description. Example 1 : Borrison's scheme
reference
688
It
is
possible to treat the scheme proposed by Borrison (1979) as a
special
case of the general algorithm. Choose tlle polynomial matrices as ~(z)=z" , & = C , P=I, R=Q=S=O and
B,=O
"Equations (2.8) and (2.9) then transform into
The
y(t+k)=~ ' (~y(t)+llu(t))+Fw(t+k)
(3. I )
[Iu(t)+~y(t)=O
(3.2)
model (3.1) is the basis for the multivariable self-twling regulator.
Since C
is unknown, the estimation c~luation is chosen as in Borrison (1979) with ~ rcplaced by
unit matrix.
The equation (3.2) is used to generate the control input
in
the
laultivariable self-tuning regulator. 'Example 2: Keviczky and colleagues' schc~e Keviczky ~lis
is
a
aald colleagues (1978) presented special
case of the problem
HIHO minimum
considered
with
variance [(z)=z ~
rcgulator.
A0=C,
P=R=I,
B~ =q=S=O. Example 9 3: Koivo's scheme. Koivo scalar
(1980)
derived a multivariable self-tuning
version of Clark and Gawthrop (1975).
scheme within
tile general
structures.
Example 4: Goodwin, Ramager The and
treat the
tile
Koivo's
[ ( z ) = z ~ , B~ =0, S=O a n d A,=C.
and Caines's scheme
(1980) has global convergence properties.
to
extending
It is possible to treat the
model reference adaptive control algorithm proposed by
Caines
l~0ssible
If cl~se
controller
Goodwin,
Ramager
It can be shown that it
above scheme as a special case of Lhe general scheme
is
with
[(z)=diag(z-''), A(z-')=diag(A,(z-')), i=1 ..... n, A,=I, P=I, Q=S=O, v(t)=O. Example 9 5: Hul~ivariable adaptive algoritlua of Goodwin and Lon~ The algorithm (Goodwin et al. Goodwin
and Long(1980).
general
class
generated
from
of the
1980) was extended to a wide class of system by
The global convergence of this algoritlua applied
stably invertible system was established. general structure
if
d~oose
A,=I,
This scheme
Q=S=O,
P=I,
to
the
can
be
R(z)=~(z)
v(t)=w(t)=O.
Example 9 6:
General multivariable minimum covariance controller
Dugard
al.
et
(1984) presented a minimum covarianco
controller
using
the
689 genera l
interactor
stochastic
matrix.
It
can be shown t h a t i t i s p o s s i b l e
to
controller as special case of the general algorothm with
treat A =0,
this P=I,
Q=5=O, B,=O. Inserting this into (2.8) results in
~ ( z " ) y ( t ) + l l ( z - ' ) u ( t ) = ~ ( z - ' )R(z- ' )u" ( t ) lqlis
equation
denotes
(3.3)
t h e c a u s a l minimum c o v a r i a n c e c o n t r o l l e r
(l}ugard
et
al.
1984). Example 7: F I n l t l v a r i a b l e s e l f - t u n i n ~ c o n t r o l l e r . Recently,
the
global
convergence
of
self-tuning
m u l t i v a r i a b l e systems was e s t a b l i s h e d by Chai (1988). and
usiltr ( 2 . 6 ) ,
(2.8),
(2.9) r e s u l t s
controller
Choosing
for
A.=C,
i n t h e model and c o n t r o l law
general
B2=O.
S=O,
used
by
Chai
(1988). Example 8: S e l f - t u n i n g c o n t r o l b a s e d on t h e m o d e l - f o l l o w i n g approach Bittanti based
on
~md
~uttolin
(1985) proposed a multivariable
model-followir~ approach so as to cope with
Choose A0(z-')=C(z-'),
P(z-')=A"(z-'),
self-tuning
nonminimum
R(z")=l~(z"),
phase
scheme systems.
S=O and B,=O. Thus, (2.8)
can be transformed into the estimate equation as follows. ~ (t)=G~(z-' )y( t-k)+ll(z- ' )u(t-k)+( l-~(z- ' ) ) f (t/t-k)+F(z" ' )w(t)
(3.4)
(2.9) can be writtcn as
(ii(z- ' )+~(z- ' )Q(z-' ) ) u ( t ) : ~ ( z " Thus,
polynomial
m a t r i c e s ~,
)[~ ( z - ' ) u ' ( t ) - ~ ( z " ' ) y ( t )
(3.5)
II and ( I - ~ ) can be e s t i m a t e d on Lhe b a s i s oE
(3.4)
with an EIS-typc procedure and thc control input may be generated from (3.5).
4. Direct Adaptive Control Algorithm Choosing A0 (z- ' )=C(z-' ) and using the identity
l=F(~-' )~(~-' )+z-' ~(z-' ) where F,
G are polynomial matrices of dcgrces (k-l),
(4.1) n4, respectively and F(O)=I~
we have
# ( t ) = a ( z - ' ) y ( t ) + ~ ( z - ' ) u ( t ) + ~ , ( z - ' ) v ( t ) ~ G ( z - ' )~" ( t - k / t - 2 k ) + F ( z - ' ) w ( t ) where
a=~-'G *'*, I~="~! and
(4.2)
~,=FI). The d e g r e e s o f polynomial m a t r i c e s a, ~, and i~ a r e n l ,
n2 and n3. The optimal control law (2.9) becomes
a ( z - ' ) y ( t ) + ~ ( z " ' )u(t)+B, ( z - ' ) v ( t ) + G ( z - ' )d" ( t / t - k ) = Y " (t+k) where
(4.3)
690 y' (t+k)=R(z -' )u' (t)-S(z-')vCt)-Q(z-' )u(t)
The
above
consideration
motivates the following
direct
adaptive
control
algorithm.
O(t)=O(t-k)+a(t-k)(#(t)-O(t-k)X(t-k))X(t-k) P ( t - k )
(4.4)
O(t)X(t)=y" (t+k)
(4.5)
where
~(t)
initial
is
a recursively computed parameter e s t i m t e
matrix
with
specified
v a l u e ~ ( O ) , . . . , O ( k - 1 ) ^ f o r 0 g i v e n by
O=[a, , a , . . . .
;B, ,B, . . . .
;G0 ,G, . . . . ; ] '
and X(t)=[y(t)" , y(t-l)' .... ;u(t)' ,u(t-l) T .... ;v(t) T ,v(t-l)' .... ; ~ ( t ) " , ~ ( t - 1 ) ' , . . . ; ]' and where y(t) is the a posteriori prediction which replaces 4" (t/t-k) and given by A
~(t)=O(t)X(t-k)
(d .6)
w i t h ~(z)=O f o r ~Sk-1 P(t-k)=P(t--2k) - [P(t-2k)X(t-k)X(t-k)' P(t-k)]
/ [l+X(t-k)' P(t-2k)X(t-k)] (4.7)
w i t h PC-I) = , . . . . = P ( - k ) = c l , r a(t-k)=l
(4.8)
Condition h r ( t - k ) t r P ( t - k ) ~gl <|
(4.9)
r ( t - k ) =r ( t - k - I ) + X ( t - k ) ' X(t-k)
(4. I0)
where
with r (-k) =, .... r(-I )=n(nl+n2+n3+n4) Condition B
X ( t - k ) ' P ( t - 2 k ) X ( t - k ) ~g, <~
(4.11)
I f e i t h e r c o n d i t i o n h or B i s not s a t i s f i e d , then P ( t - k ) =[ r ( t - 2 k ) / r ( t - k ) ]P(t-2k)
(4.12)
a ( t - k ) = l l [ l+X(t-k)" P ( t - k ) X ( t - k ) ]
(4.13)
5. S t a b i l i t y The
global
convergence
and Convergence A n a l y s i s
proof for the d i r e c t adaptive control
algorithm
b a s e d on t h e f o l l o w i n g a s s u m p t i o n C b e s i d e s a s s u m p t i o n s g i v c n i n s e c t i o n 2. Assumption C:
is
691 (I) [(z) is known. (2)
Upper bounds for the degrees (nl,n2,
n3,
n4) of polynomial matrices
a(z"),
B(z -(), ~1(z "t) and G(z-') are ~ . (3) [~(z-')-'-ll2 l] is strictly positive real. ~(z-') is defined in (4.1) as
~(z-' )=~(z' )~(z' )=l-z-'~(~ ' )
(5.1)
(4) The off-line choices of P and Q are such that det[P(z")[(z-')B(z")+z-*Q(z-')~(z-')]~0 where
d e t ~ ( z - ' )=detB(z-' )
and
for :z:~l
B(z" )A(z-' )-'
is
the
(5.2) right
matrix
fraction
rcpresentmtion of the systom transfer matrix T(z- ' )=A(z" ' )- ' B(z- ~ ). Theorem
5.1:
Subject to Assumptions
(2.2)-(2.4),
A,
control algoritl~m (4.4)-(4.13) is used to system (2.1),
B and C, then
if
the
adaptive
it follows that with
probability one (1) lim sup fIN ~- ', y(t) ;' (~
(5.3)
(2) lira sup tIN Z -' u(t) ;' <~
(5.4)
N~
t-o
(3) lira l/r,/ Z E(If
e(t+k)
II " / F , ) = f
'
(5.5)
where u" ( t ) and v ( t ) a r e assumed t o be bounded. P r o o f : Wc b e g i n by d e f i n i n g t h e f o l l o w i n g g u a d r a t i c form i n ~ ( t ) V ( t ) = [ t r O ( t ) P ( t - k ) - ~O"(t) T] / r ( t - k ) where 0 ( t ) = 8 ( t ) - 8 .
Using a s s u m p t i o n C ( 3 ) , a s shown in Chai (1988), r e s u l t s
lim [ N / r ( N ) ] | l / N ] ~ ~ b ( t ) N4o0
where
(5.6)
|!2 =0
in (5.7)
a.s.
t-o
r(N)/N~[C~/N] Z |! z ( t ) O{~+C2~[C,/N] Z |! b ( t ) II~+C, ~" ~ 9 ~
(5.8)
b(t)=-~(t)X(tLk)
(5.9)
C(z-')z(t)=b(t)
(5.10)
From ( 5 . 7 ) and ( 5 . 8 ) we have lim
[I/N] Z
I I z ( t ) I1~=0
a.s.
(5.11)
From ( 5 . 8 ) lim sup [r(N)IN] Equation
(5.3)
and
< | (5.4)
a.s. now f o l l o w from ( 5 . 1 2 ) .
(5.12) Along t h e
line
depicted
by
692 Chai(1988), we have e
lira tIN
Zlle(t~k)-F(z-')w(t+k)[l'
N ~
=0
t-o
llence lim
1IN
~ E(lle(t+k) t|'l Ft)=lim 1/N ~ [ lle(t+k)-F(z-')w(t+k)[['+ t-e
~4~
t*e
E(I|F(z- ' ) w ( t + k ) | l 9
]=f
a.s.
T h i s c o m p l e t e s t h e p r o o f o f t h e theorem.
6. Conclusions In linear
this
paper,
a general direct adaptive control s c h e ~
for
multivariable
stochastic systems having the general interactor matrix and corrupted by
colour noise and measurable disturbance has bccn presented.
l~sed
on
error
disturbance
addition,
f u n c t i o n which i n c o r p o r a t e s t h e s y s t e m o u t p u t ,
input
and
systems
The control
and
set-point variations
is
appl i c a b l e
is able to cancel the effect o f mcasurable
appropriate
definitions
o .v
the polynomial
algorithm
control
to
input
,
nonminimum-phase
disturbances exactly.
matrix
a
A0
enable
In
both
deterministic and stochastic systems to he dealt with the same basic algorithm. It
was
also shown how the general adaptive scheme encompasses
other
scheme
developed by specific approaches. The
global convergence analysis of this general adaptive algorithm have
been
~iven in t h i s p a p e r .
Referenccs Alliiaina, Bittanti,
A. Y., and llu~has, F . ~ . , S . , and ~ a t t c l i n i ,
1983, I n t . J . C o n t r o l , 38, 1!16.
R., 1985, I n t . J . C o n t r o l , 27, 1 ~ 7 .
B o r i s s o n , U., 1979, Automatica, 15, 209. Chai0
T.
u
1988,
8th
IFAC Symposium on System I d e n t i f i c a t i o n
and
Parameter
' E s t i m a t i o n , B e i j i n g , PRC.;1987, C o n t r o l and D e c i s i o n , 3, 1. Dugard, L . , and Dion, J . H . ,
1985, I n t . J . C o n t r o l , 42, 1251
l)ugard, L . , Goodwin, G. C., and Xianya, X . , 1984, Automatica, 20, 701. E g a r d t , B., 1980, I . E . E . E . T r a n s . ^atom. C o n t r o l , 25, 693. Elliott,
II., and •olovich,
~.A, 1982, I . E . E . E . T r a n s . Aurora. C o n t r o l , 27, 340
Goodwill, C . G . , and Long, R. S . ,
1980,
I . E . E . E . T r a n s . Aurora. C o n t r o l , 25, 1241.
693 Goodwin,
C.
G.,
Ramadge,
P.J.,
and C a l n e s ,
P.
E . , 1980, IEEE, T r a n s . Automa.
Control, 25, 449. Keviczky,
L.,
llctthcssy, J., llilger, H., and Kolostori, j., 1978, Automatica,
526. l~oivo~ll.N., 1900, Automatic.a, 16, 351. Ldung, L . , and Andcrson, B.D.O.,
19~d, Autor
20, 499.
Wolovich, W.A., and F a l b , P. L . , 1976, SIAM d. C o n t r o l O p t i m i z . ,
14, 996.
14,
BOUNDARY
CONTROL
OF
WITH
A STOCHASTIC
NONBNOOTH
PARABOLIC
FINAL
EQUATION
COST
F~ancl] F l a ~ d o l [ Dipartimento
di
Matemat[ca
Via P~incipe
Abs~ao~.
A Riccati
stochastic
data
obtained
system
pro~rammin~
of the
in t h e
is s t u d i e d
by
associated
alon~
L-Q-R
Italy
boundary
a direct
to t h e p r e v i o u s
method,
di T o t [ n o ,
TORINO,
with
control
of a
approach.
The
works. certain
problem
final
The solution a priori
is s o l v e d
is
bounds.
by a d y n a m i c
INTRODUCTION The
boundary
control
in t h e
last
of parabolic two
associated
Riccati
[LT],
a common
[F1],
[F23)
(backward) final
[DI]),
applications The
treated
of
from
in [ L T ] ,
Recently,
i.e.
at
case
where
least P(T)
the author
has
of determlnisticparabolic The
aim
considered
here
type,
bounded
with s
the
are
and Neumann
abstract
unbounded
a direct data,
conditions.
operator,
in
precludes controls of
it)
are is
is
for a class
of
in t h e b o u n d a r y the results
is to o b t a i n The state
stochastic
as coefficients control
but
the
problem.
extendln~
paper
of the
slightly
is a s s u m e d
solution
arlsin~
P(T)
equation
L-Q-R
[DI],
wei~htin~
boundary
Ricoati
Of the
systems.
bilinear
value
(or a multiple
of the
[B3,
assumption,
o f view,
Dirlchlet
systems,
parabolic
operators
boundary
final
of the present
case o f s t o c h a s t i o
in t h e
when
proposed
nonsmooth
final
one
of
from the work
[AFT],
operator
regularity
point
solution
control
[FS].
with
the
aspects
as the
A part (of.
(the weaker
= identity
a solution
variational
on
linear
This
the mathematical
equations
See
analyzed.
on t h e
extensively
various
a s well
approaches
works
Riccati
[LT].
been
functional.
interest,
via the
problem,
of regularity
mentioned
"non-smooth"
only
constructed
cost
in t h e a b o v e
is n a t u r a l
used.
equation,
in t h e
(L-Q-R)
have
has been
In p a r t i c u l a r ,
of the various
is a n a s s u m p t i o n Ricoati
state
different
equation
feature
systems
decades.
the L i n e a r - Q u a d r a t i c - R e g u l a t o r
and
I0123
argument.
investigated
EI],
Unive~sita'
8
arising
in c o n t r a s t
by a monotonicity
The synthesis
1.
equation
parabolic
is n o n s m o o t h ,
Amedeo
similar
of results
equations
e q u a t i o n s of p a r a b o l i c
of the noise covering
terms
both
the
(as
in
Dirlchlet
695 Notation~
1.1.
Let
X and
Y be
@eneralIy operator will
from
be
linear
two
by
X to
denoted
selfadjoint
bounded
of
w(t),
such
for
M~(O,T;X),--
while
its
continuous
function
Hypo%helem
in
in R,
X,
set
of
(when of
all
nonne~ative
will
be
denoted
of
all
defined) bounded
definite
b Z L(X,Y)
and
by
C$(I;L(X,Y))
processes is
processes
be
denoted
of
f(t),
finite,
of
P0rmuli%ion
[5rownian
the
t ~ [O T3,
will
such by
motion
be
w(t),
adapted
denote
and
P
H and
be
U be
linear
two
separable
operators
satisfying
t h a t E l f ( t ) ] ;~ is a
L-Q-R
spaces
the
to
Cw(O,T;X).
S%oohai%Io
Hilbert
the
by
Problem J
Let
)-§
continuous
strongly denote
be
linear
).
dt
t will
space
space
be
will
is a
operator
all
will
norms
If T
The
m-dimensional
subspace
and
adjoint
P(.):I~L(X,Y)
If(t)
of
products
its
the
the
Stochastic
~0
and
and
and
Cs(I;~-*(X))
M-valued E
1.2.
Y,
a standard
that
inner
I . I , respectively.
T:~, r e s p e c t i v e l y .
operators
~iven
all
space
X to
functions
meanin~
Finally,
and
interval
operator-valued
spaces, > and
i~s domain
D(T)
from
I is a n
(similar
<..
Y,
by
operators
if
Hilbert
denoted
and
following
let
A,
G,
D
, M,
assumptions:
T (At)
-A:D(A)
C
S(t)
(A2)
G 6 L(U,D(A~))
(A3)
D
I
Dm
,,..,
~enerator
of
an
analytic
in H; for
6
infinitesimal
is t h e
H ~ H
semigroup
some
~ s 40,1];
L(H);
+
(A4)
n ~ Z
(AS)
P
(S);
E ~+(H), T
and
there
exists
a sequence
P
s [
(H)~
L(H,D(A))
such
that
P
T,n and
operators
~ P T,n
< P x,x'> = T
Under quadratic
of
+
these
llm n~ ~
assumptions
regulator
x x>
minimize
J(u)
T
V x ~ H.
T,n
we
consider
the
followin~
linear-
problem:
~
2 41,1)
= E I {<My(t),y(t)>
*
lu(t)!
}dr
*
2 over
all
Vn { N ,
~ P T,n*1
u s N
(O,T;U), w
subject
to
the
state
equation
(T),y(T)>
696 (1.2)
Here
y(t)
w =
=
I
(w
probabillty
+
wm)
... space
expectation. ~s H and
S(t)~
(~,~,p),
Equation
(1.3)
E
clear
(see
iF2] o r so that
:=
Indeed that on
from
the
Note
right
that
Remark
the assumption
y(T),y(T)> of
this
I.
limit
Under
of a class
control
u actlng
conditions Remark
(see
2.
interest
Assumption
term
in
of assumption
(A5)
w e set:
~
(H)~ iF2]
L(H,D(A)) or
monoconocity
[F33),
through
Is f o l l o w s and
assumption
for certain
controls
equation
stochastic
boundary
(AS)
is ~ a t i s f i e d
applications.
PT ~ ~H)"
Various
satlsfyln~
(AS)
exist
operators
determlnlstlc with
(
(AI)-(A3),
of bilinear
Isomorphlsms
in t h e
belon~
last
(see
infinite,
of operators
problem
P~R
by the
on the
of the
for every
not
y(T),y(T)>.
exists be
y does
T,n
defined
can
However,
help
mathematical
y s M~(O,T;H)
41.2)
parabollc Dirichlet
is
tl~e l i m i t in
(A5).
u. an abstract
equations or Neumann
with boundary
[F2,3]).
for the
slmple
the
a complete
correspondln~
meanin~
E
tha~
assumptions
version
the
is w e l l
(1.3)
AS(t-s)Gu(s)ds.
[O
on
solution
[F33).
Mith
lim n~
T
t
§
motion
the
has a unique
a priori.
y(T),y(T)>
Brownian
and E denotes
(2.2)
u E H~(O,T;U)
is n o t
S(t_s)DJy(s)dw](s)
0
~s a s t a n d a r d
t o C w ( O , T ; H ) ' in ~ e n e r a l , (1.1)
jt
s j=1
it
PT
classes,
dlscussed which
do
is p o s s i b l e
that
by several
Instance,
other
are
PT~ Z*(H)
case
nonsmooth
For
cannot
in not
to
opera~ors
it h o l d s and
concrete
[F5].
~Ive
an
(AS),
example
be satisfactory
of
examples,
However,
satisfy
PT
f o r all
there
and
even
of L-Q-R
solved
(see
iF6]).
2.
SOLUTION
2.1-
S~atement The
Riccatl
(2.1)
OF
RICCATI
o~ the
Rain
sysnthesls equation
EQUATION
Result
of problem
(t ~ t O , T ) ,
(1.1)-41.2)
is b a s e d
on the
followin~
x,y & D(A)):
d
-dt
m ~J=l
+
(2.Z)
THE
P(T)
= P T
-
J
J x,D y>
-
§
<MX,y>
-
§ =
0
697 In
order-
r.o o b t a i n
a solution
approximacin~
[nte~ral
(2.3)
: S(T-t>~:P ~T,
P
(t) n
of
Riccati
S(T-t)
has
a
~ ~ [O,I) to
be
unique
(cf.
t").
interpreted
monoConicity
the
L-Q-R
problems
42.4)
P
of
(t)
<
P
~
P
PT, n
and
for
from
42.3)
S(T-t):~P
to
that
P~(t),
the
~
D.
(H)~L(H.D(A~)))
quadratic
term
monotonici~y
we
in
(2.3)
has by
Moreover,
arguments
for
the
have
V t 9 CO,T],
we
V n e N.
have
S(T-t)
(alon K with (t)l
I"
+
~
P(t)
c
E
+
s
c>O{
(s)DJ3s(s-t)ds
D: j P
j=l
n
~ O)
+
constant
S(s-t):t[M
t
c
+
(s)13ds
IP
t
some
(s)D
(s)]S(s-~)ds
standard
(t)
n
such
j =I
( Cs([O.T];
T,n
IP
j
D~: P
C A , I-~ p ~ ( s ) ] , : A ~ G E A ~ G ] * ; A ~ 4 - ~ p ~ ( s ) ) .
P
n when~
j
,,-
n*1
addition, (t)
~_ S(s-t)m:EM
s
recall
associated
n In
(we
as
ti~,~
n
solution
iF2])
jT
~
n whicl1
Lntroduce
equation
(s)AGGXA~:P
P
we
(2.1)-(2.2),
n from
thus,
the
Gronwall
lemma
there
exists
c>O
that
42.5)
IP
(t)l
~
c
s [O,T],
V t
V n s
N.
n
By
a
well
known
selfadJoint
there
42.6)
convergence
operators,
exists
strongly
and
of
next
course
(2.4)
P(.):ZO,T]
as
P(T)
result
from
n
~ ~
= PT"
for
monotone
and
+
(2.5)
Z+(H)
sequences
it
follows
such
that
of
that
P~(t)
~ P(t)
V t ( [O,T],
The
main
properties
of
PC.)
are
~iven
in
the
theorem.
THEOREN
1+
Assume
(AI)-(AS),
and
let
PC.)
be
~iven
by
(2.~).
Then
+ i)
V~E
[O,1),
exists (2.7) ii)
p(.)
a
P(,)~
constant
{A:~FP(t)} satisfies
strongly
C&(CO,T);
as
t
c(M) ~
(H))~
> O such
c(~)
condition ~ T;
~
/
C&(CO,T);L(H,D(A*~))),
that
(T-t) K (2.2),
and
0 in
the
! t sense
< T; that
P(t)
~ PT
there
698 jT i[i)
'2
Vx 6 H,
IG*A~-P(t)S(t-s)xl
ds
is finir.e
0 iv)
PC.)
satisfies
(2.8)
P(t)
<
the
integral
Riccati
S(T-t)~*P(T)S(T-t)
(and
t E CO,T]
)
+
SCs-t):~[M
+
j=1
D. jp(~)D
j
P(s)AGG*AXP(s)3S(S-t)d~
-
for
2
inequality
t
accordin~
<_ clxl
(the
integral
to the
of
property
the
last
in p a r t
term
is u n d e r s t o o d
ill));
-I v)
if
in a d d i t i o n
and
also
2.2,
Proof
LEMMA
2.
the
A
is c o m p a c t ,
differential
(2.9)
IA*u
(t)l
there
Proof.
the
In
sequel,
Let
evolution
(2,10)
r,
Ua(s,r),
operator
exists
~(~)
~
n
r ~ s ! t.
=
42.11)
s and
(r)
by t h e
numbers
A
rewrite
(as
(t)U
(r)
in
It0
4.
(v)AGGXA*P
equation
A*~P
~
t.
n
(u)U
(u,r)du. n
(2.3)
(u,r)
From
(v,r)*[M
(v)]O
in t h e
P
(s)D
D*JP
(s)D
J
n
(v,r)dv n
it
S(v-r)*Etl
~"
j
(v,r)dv. n
with
=
]U
the evolution
A*~S(t-u)*P
property
(t)U n
t A.ZS(v_u).EM U
(2.11),
+ j=1
*
n
i < u
(t,r)
n
n
(along
(u)U n
r
that
continuous
EF2,Z]):
equation
n
j
j=l
for
such
r
(t)U
D*
42.13)
E0,T3
strongly
(cf.
[F2,33)
n
= S(t-r)~:P
m
last
n6N.
n
(t,r)
n
n
The
42.8),
that
S(S-u)A~GG*A*P
n P
in
the
be
equation
I
P
(2.12)
in
forms = U (t,r)*P n
n
true
is s a t i s f i e d .
, O ~ t < T,
t are
S(s-r)
we c a n
two
P
holds
(2.1)
r
U~(s,r)
followln~
identity
> 0 such
O ~ r ! s ! T,
n
Usin~
c(~)
/ (T-t) ~
defined
U (s,r)
the
equation
1
of T h e o r e m V ~ s EO,1)
then
Riccati
for
every
of U ( t , s ) ) n
(t,r) n
m
~-J=l
J
D* P
x e H we
(s)D n
have
J
]U
(v,r)dv n
yields
699 1/2 ~P
(~)
2 U
1/2
(~,r)~l
=
(~)U
n Thus t ha t
from
(2.5)
there
(~,r)x,(J
n
n
exists
a
(~,r)x>
IP
(t)U
(t,r)l
n
This H
is
into
the
basic
D ( A ~)
for
(2.15) The
same
(2.5)
IP
estimate every
result into
constant)
42.167
IU
(t)
proof.
and
there
holds
for
t#
/
~e
< --
n
S(t)*
and
A *p
and
(2.15),
~
(u)U n
Let,
s
(t,z-)
1 --9{ ]IA*
I -e(
<
maps
such
that
(c
P
l)~(v,r)l
; r
(2.14),(2.15)
denotes
a
6eneric
(u,r)ldu
(u)U n 1 -o(
j tu
c/(t-u)
{l(t-v)
such
S(t)
c(~)
bounds
n
n
= sup
the have
yr
(u,r)l
r,
c-
that
constant
we
1 -d,
P
t and
~
recall
a
9 gsin~
[c/(s-u)
1 -~
IA:~
(2.17)
n,
t E CO,T3.
(2.10)
*
c
now
exists
s (s,r)l
[c/(v-u)
~c]IU
(v,r)
Idv-
n
<
v <
*
c
t } . Then
(2.17)
yields 1 -e( I A :~
1 -~. P
(u)U
(u,r)l
n
bound
[nco
c
/
1-2~
(t-u)
f
n
(t,r)
/
(2. 16)
1-,'~
1-"( ( ~.- v )
we
have
*
c(s-r)
1-2o( ]dv
(t-or
s
<
<
).
c/(~.-u)
I]sin~
(s,r)l
~
c
~
/
this
t) 1 -~
IU
(t-u)
n
J t ,, [ 1 1 ( v - u )
( b~cause
<
~
(t-s)
* c
f
n
(t,r)(s-r)
1-2~
/
(t-s)
n 2~
whence exists
ff~(t,r)
~
h ~(O,T),
c
such
~
c(t-r)
that
for
~(t,r).
every
r
~
Therefore,
s
< t
with
there t-r
~
h
it
holds
1-~ (2.
18)
IU
(s,r)t.
!
c
/
(t-s)
n Thus,
6iven
9 E (0,1),
from
(2.12),
(2.14)
and
(2.15),
t 42.19)
IA:~fP
n
(r)l
c/(t-r) Ta~Ing in
t
CO,T-hi
= T,
this
(2.9)
~
~
bound follows
o/(t-r)
+
~
*
J
cl(t-r) yields from
2 ~1.
n of
U (t,r)l n
n
the
equations
positive
lip
of
c(~)
(r)
1/2
(t)
~ > O,
~
=IP
n independent
n
IA~Stt)I
and
~
n
(r)x,x>
n constant 1/2
(2.14)
<~ < P
(2.9) (2.19)
we
have
1-~ c/s
r
W
r
in
the
with
t
]dv
<
t
with
interval = r
+ h/2.
t-r [T-h,T],
hwhile
700 COROLLARY
3.
A*u
~
For
in addition
Proof.
we
Iemma
4-
Part
The
weak
t E EO.T)
(weakly)
assume
From
COROLLARY Proof.
every
AxrP(t)
that
A -!
2 and
iii)
the
and
. Thus
is c o m p a c t ,
strong
P(t.) s L ( H , D ( A : ' V ) ) (2.7)
1
result
holds
of
holds
(2-6),
IGXA*P
every
x 6 H and
for
corollary
a~
in
[PS].
3 yields 2
(s)S(s-~)x{
>
IG~A*P(s)S(s-~)xi
t ! s < T.
Thus
from
Fa~ou
2
v
lemma
~v
2 ds
IG~A*P(s)S(s-~ ) • I
lira i n f n~
t
every
~ A*~P(t)
n
r
for
and
If
true.
2 itm inf n~
true.
A*YP~(t)
then
convergence
of theorem
convergence
~ & [0,I),
n ~ ~
as
t
n
< v
< T-
t
But,
analogously
to
42.3),
P (t) n
satisfies
the
equation v
P
(t)
= S(V-t)~P
(V)S(v-t)
n
+
n P
(s)AGG*A*P
that,
with
taMin~
inner
the
J
~-- D *
j
P
j=1
(s)D
-
n
n
lim
inf of
with
products
~
(s)]S(s-t)ds
n so
m
( S(s-t)*EH ~t
both
elements
sides
xEH,
(after
i.e
rewritln~
(t)x,x>
=
this
...
) we
equation obtain
n (2-20)
P(t)
~
S(v-t):~P(v)S(v-t)
jv t S(s-t):r
~
s
D,J
~ j--l=
J P(s)D
P(s)AGG*A*P(s)]S(s-t)ds. Part
iii)
follows
PROPOSITION conwer~es Proof.
5to
S~ep
Indeed,
from
P(t)
P(.)x I-
For
~ PT
every
x,x>
(t)
inequality
strongly
uniformly
=
~ P(t)
over
x s H,
), w h e n c e
as
as
[O,T],
lira < P ( t ) x , x > t~T n
T n (since
thls
the
~
thesis
v
9
T.
t ~ T. for <_
Moreover,
every
P~ ( . ) x
xtH.
Iimt~infT < P t t ) x , x > -
lim inf t * T by
(AS).
n S~ep
2.
For
Indeed, P
it) n
whence
from ~
every (Z.3)
x E H, we
S(T-t)~P
limt ~SUPT < P ( t ) x , x > -
have S(T-t)
T,n
_>
*
I T S ( s - ~ ) : ~ [ M ~ j=l ~ D:, j P ( ~ ) D J ] S ( ~ - t ) d ~ t
n
701 P(t)
<
S(T-t):r
S(T-t)
-
The
T
claim
Step
3.
lim t+T
From
it
readily
is t h e
(.)x,x>.
I an~
Thus,
in t s EO,T].
2,
this
for
the
of
x s H we
xEH,
the
limit
Dini,
=
implies
(t)3fP(t)-P
(t)l
<[P(t)-P
Pz'oof. with iv)
is c o n t i n u o u s
functions
the
limit
x,CP(t)-P
(t)3
is u n i f o r m
(t)]
1/2 x>
n
(t)]x,x> conver~es
uniformly
continuous
and
v)
convergence
of
of
of
1 hold
the
part
to P(-)x.
This
also
t=T.
Theorem
result
property
at
true.
previous
ill),
we
proposition,
alon~
readily
obtain
part
convergence
stated
in
(2.20).
Nhen Corollary of
(2.20),
iv)
Integrabillty
from
proof
the
P~(.)x
is s t r o n g l y
Parts
Usin~
the
that
P(.)
6.
=
n
follows
COROLLARY
that
n
n
that
of the continuous
this
n
finally
]S(s-t.)ds-
1/2
(t)]xl IP(t)-P
implies
J
P(s)D
inequality
n
if
have
function
2
IEP(t)-P
J
inequality.
every
fixed
D:t
j=1
monotone
a theorem From
from
for
pointwise
By
s
S(~-t):=[M
t
follows
steps
.
and
jT
*
A -t
is c o m p a c t ,
3.
Usin~
Corollary as
this 4 to
claimed
is ~ t a n d a r d
3.
SOLUTION OF THE L - Q - R the
obtain
hypotheses
the
convergence
in p a r t
to
have
obtain
It
Nith
we
v)
the
an of
stron~ one
can
identity the
(more
Theorem.
differential
easily)
in p l a c e
of
Finally,
RIccatl
repeat
the
from this
equation
the
inequality
(2.1).
PROBLEN and
notations
introduced
in s e c t i o n
1.2,
we
have: THEOREN J(u)
7.
over
(3.1) where
Given
u*(t) P
=
Is ~ i v e n
correspondln~ (3.2)
to
~tep
d
(u) n
1.
= E
and
there u*
u~.
Let
exists
a unique
is c h a r a c t e r i z e d ,
-G:=A~P(t)y*(t)
by Theorem
P~oo~.
x E H,
N~(O,T;U).,
1, a n d
control
by
the
u :t m i n i m i z i n g
feedback
formula
t ~ [O,T),
y:r i s t h e
solutlon
of
(I.2)
Moreover, =
J(u*),
J~(u)
be
the
<My(t),y(t)>
cost
*
functional
lu(t)l
)dr
defined
+
E
a~
y(T),y(T)>. T n
in
identity
702 Equat, i o n of
(2-3)
is t h ~
min[m[z[n~
gqua~lon Since
Jn(U)
(1.2).
J(u)
Riccati
over
Thus
equation
all
< J(u),
we
correspondinK
u 6 ~(O,T;U), <
conclude
to
subject
J~(u)
for
every
t.h~ p r o b l e m
to
the
state
xeH
and
,IEMZ(0,T;U).
that 2
(3.3)
<
d(u)
Vx6
H,
u
(O,T;U)w
Step
2.
y*(t)
Con~ider
= S(t)x
*
obtained
from
From
estimate
the
equation
has
Moreover,
closed
s j=1
~t S ( t - s ) D j y * ( s ) d w j (s) ;0
the
state of
T'
< T,
of
minimizing
and
(u)
under
the
usual
required
in
J
whence
(u*),
uS
to
for
to
the
of
is easy
;H),
belongs for
choice
it
y* ~ Cw(O,T'
control
~t AS(t-s)GG~A*P(s)y:~(s)ds "0
-
the
1,
u*
to
T'
C w ( O , T ' ;U) over
in
43.1).
that
this
as
see
every
problem,
T'
< T.
for
every
[O,T'],
2 {<My(t),y(t)>
§
)dr
lu(t)l
+
E
O
state
iF2]
by
theorem
= -G*A*Py*
optimal
I
= E
T'
equation
(1.2)
i) o f
solution
control
is t h e
loop
equation
part
a unique
the
J
the
equation
solve
(note
the
L-Q-R
in p a r t i c u l a r
it
that
P(T')
problem).
follows
has
the
Thus
that
regularity
u:~ s M L ( O , T ; U )
= (T'
bein~
T' arbitrary).
Definin~
J
(u)
similarly
to
J
n,T' particular,
with
PI%(T' ) a s
> J
final
f o r every
(u:r
(u)
but
over
[O,T']
(in
n weight T'
< T-
oparator),
we
thus
have
But
n,T' (3.4)
lim T'*T
(that d
we
prove
(uS).
( T ' ) y ( T ' ) , y ( T ' )>
holds
below)
Thus
Alon~
so
that
wlth
true.
consequence (3.4).
The of
Recall
operator,
it
Fix
such
the
Riccati
strongly complete.
a
yCT),y(T)> T,n
n,T' J(u*).
=
n
(3.37
uniqueness
that, follows
~ > 1/2 equation
continuous
by
proves
tame
of
the
the
that
that
(2.3)
it
t=T.
of
A ~ ~ L(H)
A-@y(.)
control Ne
to
(3.4)
T'
A*P%~ for
see
and
have
in > -
that
(3.2)
easy to
prove
is a b o u n d e d
every
that
holds
§ T
(P(0)x,x>
is a n
only
is c o n t i n u o u s
is easy Thus
as
is o p t i m a l
J.
that
limit
finally,
u*
optimal
assumption A*~P%~
the
whence,
coercivity
- ~ . Then
at
can
> J (u*), n
this
the.strict
we
~ <
at
I/2
t=T,
A*~P(.)A ~
true
and
the
([F.~]).
and
from
is proof
is
7O3 REFERENCES
tAFT] A c q u i s t a p a c e P . F l a n d o l i F. , T e r r e n i B. problem and optimal control for non-autonomous J. C o n t r o l O p t i m i z . to a p p e a ~ .
,
"Inltlal-boundary value parabolic system~", SIAM
[hi B a l a M r i s h n a n A.V. , " B o u n d a r y c o n t r o l of p a r a b o l i c e q u a t i o n s : theory'* in T h e o r y of N o n l i n e a r O p e r a t o r s , Proc. F i f t h Intern. S c h o o l , B e r l i n , 1977. [DI] Da P r a t o coefficients"
G., I c h i k a w a Ann. Nat.
L-Q-R Summer
A., " R i c c a t i e q u a t i o n s w i t h u n b o u n d e d P u r a Appl. 140 (1985), 2 0 9 - 2 2 1 .
[DS] D e l f o u r M . C . S o r i n e M., " T h e l i n e a r q u a d r a t i c o p t i m a l c o n t r o l problem for parabolic systems with boundary control through a Diricblet c o n d i t i o n " , in: C o n t r o l of D i s t r i b u t e d P a r a m e t e r S y s t e m s , Proc. 3rd IFAC S y m p o s i u m , T o u l o u s e 1982, J.P. B a b a r y & L. Le L e t t y e d i t o r s , P e r g a m o n P r e s s , O x f o r d 1983, 87-90. [FI] with 86.
F l a n d o l i F., " R i c c a t i e q u a t i o n a r i s i n G in a b o u n d a r y c o n t r o l p r o b l e m d i s t r i b u t e d p a r a m e t e r s " , S I A M J. Control Optimlz. 22 (1984), 76-
[F2] F l a n d o l l F., " R i c c a t l e q u a t i o n a r i s i n g in a s t o c h a s t i c o p t i m a l control problem with boundary control" Boil. U . N . I . , Anal. Funz. Appl. Serle VI Vol.l -C, N.I ( 1 9 8 2 ) , 3 7 7 - 3 9 3 . [F3] F l a n d o l i F . " A l g e b r a i c R l c c a t i e q u a t i o n a r i s i n ~ in b o u n d a r y problems" S I A M J. C o n t r o l O p t i m i z . 25 (1987), & 1 2 - 6 3 6 . [F4]
Flandoli
boundary [F5]
FIandoIi
boundary
F, , "A new
control F,
control
theory",
p r o o f o f an a p r i o r i e s t i m a t e a r i s i n g Appl. Math. L e t t e r s , to a p p e a r .
"On t h e d i r e c t s o l u t i o n theory", preprint.
iF6] F l a n d o l i F , , "A c o u n t e r e x a m p l e in t h e s y s t e m s )', AppI. Math. L e t t e r s , to a p p e a r . [I] I c h l k a w a A , equations", SIAM
of R i c c a t i
boundary
equations
control
" D y n a m i c p r o g r a m m i n ~ a p p r o a c h to s t o c h a s t i c J. C o n t r o l O p t l m i z . IF (1979), 152-173.
In
a r i s i n G in
of p a r a b o l i c
evolution
[LT] L a s i e o k a I. T r i g g i a n i R., " D l r i c h l e t b o u n d a r y c o n t r o l p r o b l e m p a r a b o l i c e q u a t i o n s w i t h q u a d r a t i c cost: a n a l y t i c i t y a n d R i c c a t i ' s feedback synthesis" S I A M J. C o n t r o l O p t i m i z . 21 (1983), 4 1 - 6 7 . [L] L i o n s J.L., O p t i m a l Differential Equations,
C o n t r o l of S y s t e m s G o v e r n e d by P a r t i a l Sprlnger-Verla~, N e w YorM, 1971.
[P] P a z y A. ~ S e m i G r o u p s Differential Equations,
of L i n e a r O p e r a t o r s a n d A p p l i c a t i o n Springer-Verla~, N e w York, 1983
s Sorlne M. " S u r le s e m i g r o u p e n o n RiccatI" R a p p o r t C R M A n. 1055, Univ.
lineaire associe' M o n t r e a l , 1981.
a
control
for
to P a r t i a l
l'equation
de
APPROXIMATION OF LYAPUNOV EXPONENTS OF STOCHASTIC DIFFERENTIAL SYSTEMS ON COMPACT MANIFOLDS Axel GRORUD
Denis TALAY
U.F.R. de Math~matiques Universit5 de Provence 3, Place Victor Hugo 13331 MARSEILLE FRANCE
INRIA 2004, Route des Lucioles Sophia-Antipolis 06560 VALBONNE FRANCE
Abstract We consider a stochastic diffcrential system on a compact manifold ; under our hypotheses, its Lyapunov exponents are deterministic. We propose an algorithm of numerical computation of these exponents, and we give a theoretical estimate for the approximation error. The method is based upon the discretization of the linearized stochastic flow of diffeomorphisms generated by the differential system.
1
Introduction
T h e L y a p u n o v e x p o n e n t s of a stochastic d y n a m i c a l s y s t e m enable to s t u d y its stability. A survey of this i m p o r t a n t theory, for linear a n d non linear systems, may be found in [3], and in Arnold[l] (we will use the n o t a t i o n s of this last reference). F r o m an applied point of view, it is necessary to c o m p u t e the numerical value of the Lyapunov exponent, s. For the linear case, an a l g o r i t h m has been proposed to c o m p u t e tile u p p e r one, and its t,heoretical convergence r a t e given, in Talay [11] (where an a p p l i c a t i o n to an industrial p r o b l e m is also described). T h e aim of ihis p a p e r is to treat the non linear case, and to show t h a t one m a y also c o m p u t e all the exponents (not only the top one). Let us consider a d-dimensional C~ c o m p a c t manifold .A4, and A, Bj (j = 1,... ,r) C~176 vector fields on ,~'f. W e also consider a r - d i m e n s i o n a l s t a n d a r d W i e n e r process (W~). W e will deal with the stochastic differential s y s t e m in the Stratonovich sense on .M :
dxt = A(.v,)dl + ~ Bj(x,) o dW[ j=l
(1)
705
This system defines a stochastic flow of diffeomorphisms (x,(z)) (cf. Ikeda & Watanabe [8] e.g.) ; if Tx,(~) : T.Ad ~ T~,(=)Ad is the linear p a r t of zt at z, and if the vector fields TA, TI?j are the linearizat.ions of A, B i, then the m a p p i n g Tzt from TAd to T.A-t defined by ( x , v ) --. (x,(x),Tz~(::)v)is a flow on on the tangent bundle T.M, generated by the system r
dT , = ra(r
,)Jt + E
o
dw/
(2)
j=l
Now, for x E Ad, let P~:.M be the projective fibre over z, and PAd - U=e~4{x } x P=lt4 bc the projective bundle over Ad. Let Pxt be the equivalence class of (x(, i"-~,l) in PAd ; this proc,'ss on PAd solves a stochasl, ic differential system (cf Arnold & San Martin [2] or Carverhill [7]) :
dPx, = PA(Px,)dt + ~ PBj(Px~) o dW j
(3)
j=l
(We will not give here the explicit expression of the vector fields PA, PBj). We suppose that the following hypothesis holds : ( H 1 ) (x~) is ergodie, and tim differentia1 operator P A + ~ E]=~ ( P B j ) 2 is strongly elliptic. Under (H1) (cf. Ikeda & W a t a n a b e [8] o.k.), the above differential operator induces on PAd a Riemannia.n m e ~ u r e dr, and the process (Px,) has a unique invariant probability law u, and u has a srnoot.h and strictly positive density q(-) with respect to dr. Moreover (//'1) (and even weaker assumptions, cf. Arnold & San Martin [2])implies: there exists a real number A such that, for any (x,v) in TAd : ~=
lim
!loglTx,(::)vl t
and there exists a smooth function ~ on P A d such that :
x=
r
(4)
T h e n u m b e r ,~ is called the top Lyapunov exponent of the system (2). In Section 2, we will propose an. algorithm which permits the numerical computation of .~ (at least if tile geometrical structure is not too complex) : we will construct a Markov chain (which can bc simulated on a computer) whose top L y a p u n o v exponent approximate )~ ; in Section 3, we will give a theoretical estimate of the approximation error. In the last Section, we will generalize to the approximation of the other Lyapunov exponents of the system. We will use the results due to Arnold & San Martin, Baxendale and Caverhill for the continuous time processes (cf. tile papers in [3]), and the results due to Bougerol ([5] and [6]) for the discrete time approximating process. Another paper, generalizing these results to the case of non compact manifolds, and presenting examples, is in progress.
706
2
Discretization
scheme
We discretize thc system (2). Let us begin by an elementary remark.
2.1
Preliminary
Proposition
2.1 We may choose an atlas A on Ad such that :
(i) A has a finite number of charts (r Dora(e)) ,'for each r Val(r unit ball of lR a ;
is included in the
5i) let (r Dorn(r an arbitrary cliart, and ce (resp. ~j) the expression in local coordinates oJ' the vector field ,4 (resp. B j ) ; then, for any multi-index I, and any l < i < el, 1 < j < r, the derivative Otc~~ (resp. Otfl]) is bounded on Val(r ;
(~ii) ~
e Ia; such that : Vx ~ a ~ , 3(r Oo~(r C A : B(r a chart will be called in the sequel a large chart around x).
R) C Vat(e) (such
P r o o f Let us suppose : for any atlas satisfying (i) and (ii) (easy to obtain), the following is true : V(R,, n 6 IN) such that R , J. 0, there exists x , E 3.4, such that, if (r Dora(C)) is any chart around x~, B(r R~) is not included in Val(r This is impossible : as Ad is compact, a subsequence (x~,) converges to a limit x ; ( r 1 6 2 being a chart around x, I.he continuigy of r implies: for k large enough, l ? ( r C Val(r []
For any vector-valued function 7, let 07 denote the matrix [C?k~ti]i.k. T h e previous Proposition implies : C o r o l l a r y 2.2 For any random variable U with a compact supported law Pu, there exists ho(Pv) > 0 satisfying,/or any h <_ h0(Pu) : Vx e ~ , there exists a chart (r Do,~(r in .,4 such that, if a, flj denote the expression of the vector fields A, Bj in this chart, and if 1
r
a := ~ + [ X](o3j)sj j--1
the'n ;
r
r162162
e
Val(r
, a.s.
j=l a l/. d a.s.
j=l
707
2.2
Algorithm
The initial value is any deterministic pair (xo, Vo) in TAd ; let us describe the passage from (xv, %) to (zv+,, vp+~) : let (r D ~ 1 6 2 Zp ~ O'p (rosp. be a large chart around -~ be thc represcntation of the vector ficld A (resp. Bj) in local coordinates ; we set :
ap :=
1
+7
r
(ois.)p.
j=l
Cp induces a basis % in Tx)M given by : eJf = Oj(f o r162 function f on M and 1 _< j _< d.
for any smooth
Let, V ) denote the vector of IRa whose coordinates in the canonical basis are (,hose of -l~ 'Up in %, and
--h Xp
=
=hp ) . Cp(X
Thcn ~h+l is computcd according to the formula : r j=l where
:
9 the random variables Up+ J 1 will be supposed to satisfy the following rcquirement : ( H 2 ) the (U~+T)'s are i.i.d., and the following conditions on the moments are [ulJ j ]2 = 1 ,9 the common law Pu of the filled : E[U~+,] = E [ C fJ p + d 3 = 0, ErU t p+lJ
zJ 's has a continuous density w.r.t, the Lcbesgue measm'e, whose support (Lp+l) contains an open interval including 0 and is compact, ; 9 the step-size h is less than ho(Pj) (in the scnse of Corollary (2.2)). Next, --hvv+l will be the vector of T-~)+1,,t4 whose coordinates in the basis (-p+l are the coordinates in tile canonical basis of 1Rd of --h
.'-:'=h
='~h
Yv+ 1 = M p + l V p
W]lcrc
--h
--h
r j=l
708
2.3
Remark
The condition (H2) does not allow the simulation of a gaussian la.w, wherea.q one could expect that u J + , v ~ would be an approximation of W~p+~)h - Wda. The restriction to compact-supported laws ensures that for h small enough r
j=l
belongs to Val(r , so that gh+l may be defined without any change of chart (in particular, for technical reasons appearing in the proof of the convergence, it is very important that it is obtained with the chart (r D~162 which is measurable w.r.t, the or-field generated by (U~, , 0 < k < p , 1 _<j _
3
A p p r o x i m a t i o n of t h e u p p e r L y a p u n o v e x p o n e n t
h mh ' Obviously, the process (2~, Mp) is a multiplicative Markov process in the sense of Bougerol [5] : if/~h'p is its transition operator, for any Borel sets E~ in ,M and E2 in GI(IR a) :
/~h((x0, M ) ; Z, x E 2 M ) = / ) h ( ( z 0 , [ d ) ;
Ea x E2)
, Vp e IN , Zo E M
, M e GI(IR d)
where: E2M = { N M E GI(IR a) ; N e E2}. Besides : L e r n m a 3.1 Under the hypotheses (H1), (H2), for any h small enough :
(i) the process (a:v) --h is ergodic ; let -fia be its unique invariant probability law ; then sup E,~a(LogllMpt ~ h I + LoglI(--M~)-IlI) < eo
(ii) The system (2},'-M~) is strongly .irreducible, in the sense that there does not ezist a finite union F of proper sub-spaces of IRa such that, for any integer p, -MhF = F. P r o o f First, we remark that our construction ensures that (~-h) is a Feller process on a compact manifold, so that it has at least one invariant measure ; tile hypotheses ( H I ) --i--zh and (H2) imply that, from any starting point X 0 = z, the probability to reach any set of stl~ctly positive dr-measure in a finite time is strictly positive (at each step, the
709
process may go in any direction), therefore (~)) is an ergodic process. The last part of the statement (i) comes from the Corollary (2.2). Next, we will use the following consequence of a result due to Bougerol [5] (Proposition 5.2) : if ~ multiplicative Markov process satisfying (i) were not strongly irreducible, then there would exist a continuous finite family ( E l ( x ) , . . . , E,(x)} of proper sub-spaces of 112.J, with same dimension, such that, for all x in .M : --h
--h
{MtEl(x),..., M, E,,(~)) = {E~(:,:),..., E,,(~)} ,a.~. Let us suppose that this property holds. Then we would have : r
(Id + O~o(~o)h + ~ OA,o(~o)Ui,ATO(u;'=lEd = u,% E,
(s)
j=t
Ta,king the expectation of each side of the previous equality, one would get : O d 0 ( U ~ , Ei) = Ui%lEi
Multiplying by Upj + t before taking the expectation, one would also get : Un n O?3j,o(~=lEi) u~=~& =
rt
Thus, using (5), F = Ui=lEi would be a proper sub-space of IR~ such that 3o0(F) = O~j,o(F) = F , Vj = 1 , . . . , d But this would contradict the hypothesis (H1). [] A Corollary of this Lemma (for the proof, cf. Bougerol [6], Section 2) is : T h e o r e m 3.2 Under the hypothescs (H1), (H2), for any h small enough, there exists -~l, E IR such that, for any xo in ./td, for any vo in T,o.A4 : lim 1
,",'-oo N log [M~cvol =
-~h
,
P, oa.S. --h - - h
The number ~h is called the top Lyapunov exponent of the Markov process (xv, My). Numerically, we will compute, for N large enough, the quantity :
l log I ~ , o i N
The theoretical approximation error is given by the
(6)
710
T h e o r e m 3.3 Under the hypotheses (H1), (H2) : [A - ~ h [ = O(h). P r o o f The proof will be a succession of Lemmas. Let Pg) be the equivalence class of (~'vh, ~ ) i n
P.M. We begin by proving:
L e m m a 3.4
E log 1~72§
= Zlog I~v01 + Er
+ g)h 2
(7)
with : r defined in (4), ]g)l <- C (for some constant C independent from p, h). P r o o f For any Stochastic Differential Equation in ]1:(d whose coefficients are smooth and bounded, as well as their derivatives :
dy, = .(y,)d~ + ~ b~(~,) o dw: j=l
and any smooth bounded function f with bounded derivatives, the Euler scheme 1~
_,,
--h
--,,
j=l
~
-Z,
j
j=l
satisfies : Ef(yh+,) = Ef(y~) + E L f ( ~ ) h
+-~+,h 2
whcrc L is the infinitesimal generator of (Yt), and ]$h+~I --< C (C independent from p, h). Bcsidcs, we know (Baxendalc [4]) that, for any initial condition (x0,v0) of the system (2), if (Px,) is the solution of (3) with the corresponding initial condition Pxo, then : d z E log ITz,(Xo)V01 = E r In local coordinates, choosing f ( x , v) := log ITxdx)v[, we get (7). Q
~terating (7), on~ gets : Slog I ~ o l
= log Iv0l + E,~=5' E ~ ( P ~ ) h + ~V"N-1 = ~ ~hh2 ~
Lct us dividc cach term of the previous equality by N and make N tend to infinity. Wit,h arguments already used, we can easily check t,hat the process ( P ~ ) is ergodic ; let ph be its unique invariant probability law ; then the limit N
lim
1 ~Er
N~+oo "N p=l
o•
and is equal to
f.~ r
Therefore, the conclusion of our Theorem will be a corollary of the following
711 L e m m a 3.5 For any smooth function f on PAd :
lip, I(0).(d0)- f,.S(O)-~h(dO)l = O(h) P r o o f In Talay [11] (Lemmas 4.3 and 4.4), a similar result has been obtained for processes on the sphere S d-l ; the construction of the approximating process was different, but the proof can be adapted without difficulty for the present situation. Let us just give some indications. Let s be the infinitesimal generator of the process (Pxt) solution of (3) on PAd, and, for a given smooth function f on PAd, let u(t, O) := Eef(Pz,). It is well known that
~a4 t , o) = ~(t,o) u(O,O) = f(O)
(8)
Wc may construct an atlas on PAd sat,isfying the requirements of the Proposition (2,1). Proceeding as in Lemma 4.3 of [11], one can prove : L e m m a 3.6 fender (H1) :
(i) Th.ere exist strictly positive cons'~ants F and 7 such that : VO E Pj'td
:
lu(t,O)
-/p 2,4 f(s)d~,(s)]_
(9)
(ii) for any multi-index I, there ezist strictly positive constants FI and 7x such that, for any o in P.,t4, a . y ,j in Vat(Co) (where (r Dora(Co)) is a large ch~rt around 0 (in the sense of Proposition (2.1))), the spatial derivative Oru(t, r l(y)) satiges : lOin(t, Col(y))] ~ FI exp(--Tlt)
(10)
Then tedious computations (where the remark in Section (2.3) concerning the mensurability of the charts around 2"~, and the property (2.1)(i) of the chosen atlas play a role) show that'our construction implies :
vk , vp , F~u(kh, P-e~+~) = E~,(~h, P-e~) + ZC~,(kh, ~ , ) h
h h2 + -r~,~+~
where Ighk,p+~l <- C~ e-c~p1' (for some strictly positive constants C~, C2 independent from p, h). Then one may proceed as in the proof of Lemma 4.4 of [11]. []
712
4
Approximation
of the spectrum
The approximation of all the Lyapunov exponents may be performed in the following way. For x in M , and any integer k (1 < k < d), let Gk.(Tx.M) be the Grassmann manifold of all k-dimensional subspaces of the tangent space T,.M, and Gk(T.M) be the Grassmann bundle tJ~e~ Gk(T,:M). For any I ( in
Gk(T~:M), let Kt = Txt(x)I( C T~:,(,)A.4, and Jr(I() = [det(Tz,(x)]K)].
Let us suppose that the above hypothesis (H1) holds and that the infinitesimal generators of the processes (Kt) are strongly elliptic. In this context (and under even weaker hypothescs : Baxendale [4]), for any 1 < k < d, there exist real numbers A1,... ,)~k and a smooth function Ck on Gk(T.s such that, for any x in .44 and K in Gk(T,l~.t), if u now denotes the unique invariant probability law of (/it)
: ~..~.
lira -1t log
Jr(t,') = A, + 9 .. + Ak = ] a
t--+co
k(TM)
Ck(O)~(dO)
To approximate the exponents, we first choose k = 1 and approximate Aa = A as described in the previous Scctions ; then we choose an orthonormal basis (v~,v02) of a sub-space A" of T~.oAd tot initial value of F), and we compute --
--.h2 1 logldet(v- - h~l ,v74 )[
(ii)
N
Going on up to d, we succcssively get approximations of all the sums .~ + . . . + A~. This provides the approximate values of the cxponents. With the same technique as above, we may prove that the approximation error on each of these sums is of order h, and therefore the error on each Ak is also of order h.
5
Conclusion
We have proposed an algorithm of approximation of the Lyapunov exponents of stochastic differential systems on a compact manifold, and given a theoretical estimate of its convcrgcnce rate. From a numerical point of view, as in the linear case, the pertinent choice of the number N of steps in the formula (11) may present important difficulties. Further studies in that direction arc necessary.
References [1] L.ARNOLD : Lyapunov exponents of Nonlinear Stochastic systems , Nonlinear Stochastic Dynamic Engineering Systems , G.I. Schneller & F. Ziegler (Eds.), Proceedings of the IUTAM Symposium, Innsbruck, 1987, Springer-Verlag.
713
[2] L.ARNOLD & L.SAN MARTIN : A control problem related to the Lyapunov spectrurn of stochastic flows , Matematica Aplicada e computational 5 (1), 31-64, 1986. [3] L.ARNOLD & V.WIHSTUTZ (Eds.) : Lyapunov Ezponents, Lecture Notes in Mathematics II86, Springer, I986. [4] P.BAXENDALE : The Lyapunov spectrum of a stochastic flow of diffeomorphisms , Lyapunov Ezponenls , L.Arnold & V.Wihstutz Ed. , Lecture Notes in Mathematics 1186, Springer, 1986. [5] P.BOUGEROL : Comparaison des Exposants de Lyapounov des processus Markoviens multiplicatifs, Annales Inst. H.Poincar6, 24(4), 439-489, 1988. [6] P.BOUGEROL : Thdor~mes limites pour les syst~mes linfaires 5. coemcients Markoviens, Prob. Theory and Related Fields, 78, 193-221, 1988. [7] A.CAVERHILL : A formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation problem, Stochastics 14,209-226, 1985. [8] N.IKEDA & S.WATANABE : Stochastic Differential Equations and Diffusion Processes, North-tIolland, 1981. [9] E.PARDOUX & D.TALAY : Discretization and simulation of S.D.E. , Acta Applicandae Mathematicae 3,23, 1985. [10] D.TALAY : Efficient numerical schemes for the approximation of expectations of functionals of S.D.E. , Filtering and Control of Random Processes , H.Korezlioglu & G.Mazziotto & J.Szpirglas (Eds.), Proceedings of the ENST-CNET Colloquium, Paris, 1983, Lecture Notes in Control and Information Sciences 61, Springer-Verlag, 1984. or
D.TALAY : Discrdtisation d'une E.D.S. et calcul approch6 d' esp6rances de fonctionnelles de la solution , Mathematical Modelling and Numerical Analysis 20(1), 141, 1986. [11] D.TALAY : Approximation of upper Lyapunov exponents of bilinear stochastic differential systems, (submitted for publication), INRIA Report 965, 1989.
3 Y N T H E ~ I 3 OF CLOSED-LOOP ~ r E H
Wladyslaw Instytut
Elektrotechniki ul.
ABSTRACT. that
discontinuous oF
The motion
system
are assumed
object
has
been
approach The paper
in c l o s e d - l o o p
the use of some
in t h e w o r k
can
structures,
which
the time
close
application
oF the
which
elements
A creation of
structure
Some
of
the
has
system structure
is s u g g e s t e d
The examples adaptive
system
suggestions
creation
The
control
structure
oF
has
oF the
the controlled
closed-loop
theory.
design
(Filippov)
dynamics
equations.
of this
be a b l e t o t r a n s f e r one.
For
and
discontinuity
on t h e
suboptimal
decision
in p r a c t i c a l
minimum
in a d a p t i v e
oF
system
by p l a n a r
~-solutions
dynamics
used
the concept
closed-loop
of this
impinge
Thus,
Krakowskiej
POLAND
is d e f i n e d
differential
been therefore
system.
would
Because
in p a r t i c u l a r
by r a n d o m
be u s e f u l
to the
oF which
equations.
shows
Politechniki
a creation
dynamics
solutions
has
KULCZYCKI
Krakow,
of a random type.
stochastic
Piotr
31-155
resistances
described
synthesis.
with
24,
with
the
differential
generalized
been used.
acting
deals
an object
and
i Elektroniki
Warszawska
The paper
controlls
theory
HEJMO
CONTROLLING A RANDOM OBJECT
comprised
real
to t h e t a r g e t
in
as to practical
been
given.
I~ A certain
class
two
control
oF position
dynamics F
other
oF
between
oF these
+ Ku,
y(O)
of mechanism
industrial
technological changing
= Yo'
]ul ~
represents
the
control,
is a p o s i t i v e
ensured
K iF:
resistances
control
can
constant.
problem (CLS)
theory there
which
results
should
u
any
continuous
y e
. In t h e
makes
the solution
oF the
initial
Following state
z~ =
problem. = Xo;
The y =
and a velocity
function
that
Function
of
object
paper
consideration
[6]
both the and
time--optimal
closed-loop ( X o , Y o)
solved
oF
equations of this
is
a problem
has been
because
impossible
differential
position
x(O)
of this
inequalities
makes
of
From
the concept
transfer
= y,
is a m e a s u r a b l e
under
f which theory
oF optimization.
economical
y are a position
of diFFerential
classical
their
the minimum-time
as:
Controllability
oF an object
oF the ~unction
standard
x,
0 < ~ < K,
discontinuity
of the
important,
of motion,
of theory
so that
be d e s c r i b e d
i, w h e r e
on the ground
application
need to change
F is a p i e c e w i s e
If(y) ] < K - ~,
of time-optimal
is v e r y
mechanisms
respectively,
devices
operations,
~ ~z
system to the
715 target
z
=
(0,0)
=
y,
in
minimum
a
time:
i
x(O)
=
xo,
y =
9(y)
where
T-
=
(-~,0)},
{(x,y)
R-
=
(-~,+~)5, The curve
by:
the
dh"(y)/dy -
K),
and
T
T
:
•
U
= T-
x
=
f(~:),
x
=
+ K),
y/(~(y) (-~,0]
becomes
closed--loop
the
=
(x,y>
~
y ~ y
C(~ i ) class
a.e. the
CLS
acting
resistances
can
appear
because
the
motion
resistances
can
differ
considerable
resistances. become We curve
non can
Consequently~ time-optimal
distinguish
inaccuracy:
-~
b)
the
the
=
describes
[6].
the
main
cases
oE
curve
T
the more
on
CO,m)
--~ <
h" (y)
<
-- K)
< 0
a.e.
on
(-~,0],
is
y/(E(y)
curve
under
f(y)
f§
and
E-
T
are
less
steep
the
time
h'(y)
< 0
a.e.
on
[0,~)
- K)
<
h' (y)
<
a.e.
on
(-~,0],
to
the
0
passive
impinge
consideration.
motion
Thus,
f+ (y)
if
0
if y
=
O,
f- (y)
if y
<
O,
piecewise
y
this
J
and
on
function
the
right-hand
side
of
one
CLS
T*,
i.e.
~
(4)
J the we
because
dynamics define
of
by:
> O, (5)
continuous.
the
i.e.
(3)
resistances,
essentially
switching
T*,
2. P R E L I M I N A R I E S . As
can
I
optimal
<
resistances
=
than
and
K)
subjected
mechanism
controller
above
a.e.
motion
the
creation
motion
the
0
of
one
Thus, T
real
than
<
paper
h" (y) y/(9(y)
real
[5].
curve
mentioned
+ K)
< y/(f(y).+
= =
in m o t i o n
steep
< y/(f(y)
switching
object
[I].
In t h e
the
by
(i), (2)
is
y
switching
time-optimal
switching
created
h(y>,
the
dh(y>/dy
reasons
which
<
dh*(y)/dy
the
T~
for
the
(~i)
variations
technical
T
~
h" (y)
taken
curve
as
(a.e.):
h" (y)
kind
where
that,
switching
--~ < y / ( f ( y ) This
many
•
one. two
a)
of
:
y
= O.
role
E 6
curve
h(y),
controlling
becomes
the
{(•
<
-~
this
from
(1),(2)
(1),(2)
function
same
and
=
: x =
h(O)
system where
switching
upon
, R§
everywhere [0,~),
{(x,y)
and
the
--* ~ i
on
time-optimal
systems
of
plays
almost
=
(-~,+~)}
feedback
: [0,~)
(2)
T+
is o f
T+
(i)
U R+,
h(y),
U
= Yo'
(0,0) T§
(0,+~)}, E
y(O)
R-,
>
true
then
U
T-
h
{(0,0)}
are
on
(x, y)
1,
time-optimal
following
a.e.
O,
h(y),
function
a classical
described If
: x =
{(•
the
curve of
=
Ku(x,y),
(x, y) e
I -1, u(x,y)
+
subjected
to
=
716 discontinuities,
the
standard
( C a r a t h e o d o r y o r ~)
inappropriate
[2].
discontinuous
differential
equation
(Filippov
solutions)
DEFINITION 2 1 ~
with
The most p e r f e c t
non-empty interior)
compact s u b i n t e r v a l differential a.e.
on I ,
B
is t h e
M
denotes
Then x
open
unit
closure
E
= ~ ~>o
REMARK the from
2.2.
z
In this
q
:
THEOREM 2 . 3 . starts
will
[0,~)
of
If
the
be
(4)
occurs
infinite
of
is
The p r o o f THEOREM
2.4.
~-solution
Let
given (3)
curve
point
z
which
tends
T
occurs.
and
~ T\{O}
there
q
the ~ ~[g](t,x)
play
the
solutions
solutions
q§
: [0,~)
with
jr-solution, performing
in
in
~2\{0}
and a f t e r it
a ~inite
of
starting
where
( X o , Y o)
=
cvx
c o n s i d e r e d CLS.
and u n i q u e n e s s i s
equal
target
s e t Z~
same p h y s i c a l the
This
of the
9
respectively,
reaches
a)
the z"
starts
•
in
on each
[6]).
9
there the an
time.
9
to
point
[4].
will
-I.
existence
Then:
equal
in t h e
role u ~
identically
[7].
measure
has t h e
t h e n from each z ~
in
an i n t e r v a l
of
iff
[3].
formula:
( 1 ) , (2)
or
of
encirclements
identically
switching
q+
is
function
an j r - s o l u t i o n
t h e movement o f
w h i c h goes r o u n d t h e
number o f
(I
x ~n --* ~ n
o9 M c ~n
essential
(proof
~
can become of
(~-solution)
continuous
is Lebesgue
hull
noted
solution
g ( t , (x + c B ) \ Z ) ,
u ~ +I
the unique ~-solution
trajectory
: I
by t h e
t h e CLS
when
~z.
~
of
x
: I
given
convex
paper
object,
that
called
in ~ n , H ( Z )
ball of the
e ~Z\{O}
~z , o
is
because t h e y c h a r a c t e r i z e
considered
Filippov
g
~ cvx ~(z)=o
B o t h ~ and j r - s o l u t i o n meaning,
is
Let
is
x = g(t,x),
where o p e r a t o r
~[g](t,x))
of
be an a b s o l u t e l y
of I.
equation:
solutions
definition
From
~ T\{O}
the
each
Jr--solution
unique
2 ~ ~Z\T there starts o which reaches the
in a f i n i t e jr-solution
time. the
b)
From
trajectory
a
each
o~
o
TX{O}.
to
If t h e
reach
following
-~
y
<
,
-~21Yl where
ol
> O,
jr-solution
the
~2
target.
None
o~
the E-solution
h" (y)
_
y/(f(y)
h'(y)
> 0,. (9i
starting
from
z~
hold: +
K)
on
}
[0,r
-
/92 ~
starts
from
< y/(f(y)
> O, z
/92
> O,
~ T\{O}
- K) then
reaches
on the the
161
(-~,0] trajectory target
of the
in a 9 i n i t e
time.
o
The
trajectory The p r o o f
of
this
is
given
We c a n i n t e r p r e t
jr-solution in
[7].
lies
totally
on
the
switching
curve
T.
9
the jr-solution
from the
physical
point
of
view.
The
717 CLS
(i), (2)
type of
is
only.
the
R§
(3)
Besides,
after
to
the
re-penetrate
oscillate
around
depending
on
process).
The
process delay
the
by
tends
to
NOTATIONS:
(Q~S,P)
elementary
events
is
an
interval
S
Given ~ I
in
(~S~P)
and I .
DEFINITION
on
in
(~,S,P)
the
random
and
the
X, 2.7.
iii>
i.
The
the
is
the
limit
from
of
T+
~ ~ ~,
amplitude (sliding
oscillatory when
process
non-empty
the
appears.
set
of
measure>.
I
9
The f u n c t i o n
process
and
a probabilistic
interior.
that
starts
operation
same
a
q
immediately
(I),(2>,
the
(~ is
is
of
trajectory
the
upon
into
trajectory
frequency
acting
P
it
switching
space
and I
the
(or
X : ~ x
I ~
over
such (~,S,P)
random f u n c t i o n >
the function
X(~,.)
a function
: ~
n-dimensional I,
P)
if a f a m i l y
be
function
given: set
of
of
i)
L
a mini-max
G
will
called
~
will
defined
be c a l l e d
a
all
in
by in
: D x
d
loss
1
defined
all
o~
(towards
process
X are
a fixed
elementary
and
(N,S,P)
can
~
(or
oE
I. 9
space
(which
N -~
X
I derivatives
(~,S,P)
a probabilistic
I --* ~
(or ~ ) - s o l u t i o n s
stochastic
almost
x
process
if a l m o s t
equation,
decisions
x ~n
stochastic
G(~,X,t)
process
D o~
function of
be
differential
is a s t o c h a s t i c
loss
X =
realizations
above
Let
will
equation:
non-empty
the
I and
An
interval
moreover,
ii)
it);
the
starting
(~,S,P),
factor.
measure
and
DEFINITION
where
trajectory
none
the
n - d i m e n s i o n a l random v a r i a b l e
For a f i x e d
Given
in
oF
realizations
a)
is
differential
probabilistic
states;
(~,S,P)
R+
"bang-bang" the
penetrate
that
starts
certain
is a ~ - a l g e b r a ,
it
random
~)-solutions ~
a
to
implies,
practice,
in
non-empty
tends
there
the
starts
9 2.6.
dependent
event
with
n-dimensional stochastic
realization.
In
structure
When
with
that
be c a l l e d
.
of
there
that
into
therefore
real
q
T-
is a p r o b a b i l i s t i c ~,
in ~ i
each t
T
T
T
(i), (2)
inherent
is
zero.
DEFINITION 2 . 5 . for
curve
the
of curve
curve
time
q-solution
generated time
the
delay
to
penetrates
the
process
curve
equal
the
T-
control
the
side
from
curve
the
from
right-hand
Thereby,
leaving
forced
generate then
identically
the
in R +.
to
hold
~-solution
exists
to
able
I~
be
of
selected
nature from
. : D -~
~
U
{+~}
and
m
b>
mini-max a)
I
strategy (d>
=
sup n
2.
(decision)
e
L(d,n>;
loss
16
:
D --, ~ i
U {• by:
i
(d
D
are
) =
that
and
for
give
inf d
towards the probabilistic
given respectively
~
N
We assume a d d i t i o n a l l y
L(d,.)
b)
do
E
i
respectively (d).
(7)
D
each d E D t h e r e e x i s t s
measure P.
by:
an i n t e g r a l
a> The f u n c t i o n
b> B a r e s s t r a t e g y
(decision)
of
of
Bayes
do e D a r e
718
a)
ib(d)
= a ~ L(d,n)
dP(n);
b)
ib(d
o
) =
inf d
N
~
i D
b
(d) 9
(8)
3.PROBLEM STATEMENT. Identification brought to
of
effect
the functions
because o f
approach t o t h e problem s o l u t i o n of
motion resistances will f(~,y)
whereas t h e
real
= Y,
X --*
= -b(~)
X0
x,
w h e r e x ~ and y~ a r e r e a l c o n c e n t r a t e d i n Zo =
dimensional stochastic The g o a l (10)
of
this
i n w h i c h t h e CLS possible
to
we g e t
to
(10)
is
Such a c o n c e p t o f
in
real,
The c o n t r o l l a b i l i t y P(~
:
b(~)
fixed (2),
~,
~ [O,K -
For base
any
on
=
of
~
decision
going
expectation would
be
The
(10)
of the
considered
(10), (2) the CLS
to the
as close
CLS
because the
of
and
whereas
theory. h(Y)
= -
select time, as
So,
B such
approximation
of
(12)
system
object,
by fixed by
that
will
(I0). in
the time
is
will
useful
measure
~ollows that
(10)
will
~or a
be d e f i n e d by
by t h e f o r m u l a :
+
I)]}
values
I)]
and
sgn
in the
this
the
value
be
it
sgn[Y]
(II)
control b(~)
selected
with
structure
will the
be
use
will
replaced of
by
some
[Y]
CLS
system
(12)
(10),(2)
is b r o u g h t
and to
(12)
the
the
origin,
minimum.
presented
the
(10),(2)
and
+
to the the
of
[-I,I]
we get:
in w h i c h
upon
random
be
[YZ/2(B
possible
a probabilistic
sub--optimal
unknown
will
two
w o u l d be as c l o s e as
Function creation
c u r v e ~or
of
2.6..
a suboptimal structure
[7]
be g i v e n
concept
distribution~
Definition
the expectation of
ensured i f
{YZ/[2(b(~)
I
of which
acting
described
closed-loop
the
(II),
value
X = are
h(Y)
and
B the
elements
X =
(10)
structure.
From t h e w o r k
h will
equation,
~ u n c t i o n u : ~z --*
the origin,
the
(i>:
YO = Y ' '
unipoint
such t h a t
switching
into
into
u n d e r s t o o d as a s o l u t i o n
control
(10> i s
= 1.
function
non-Eixed
(1),(2)
parameter
We
of s])
the time-optimal
whereas t h e
is
a control
industrial
(9)
+ u(X,Y),
The above i m p l y t h a t
be c o n s t r u c t e d .
O,
a random f a c t o r
with
define the
brought t o
=
after- setting
[Y]
a random
consideration:
[0]
In accordance w i t h
random f a c t o r )
a minimum.
synthesis of
sgn
process IX,Y]
Thus,
a random d i f f e r e n t i a l
random v a r i a b l e
paper i s
(independent o f
sgn
reason~
-b(~)
(Xo~Yo).
be p r a c t i c a l l y
The F o l l o w i n g model
into
b introduces
By t h i s
Y =
taken
sgn [ y ] ,
Y --* y ,
= x';
cannot
reasons [5].
appears a v a i l a b l e .
random v a r i a b l e
--* f ( y ) ,
and f -
be t h e r e f o r e
system under c o n s i d e r a t i o n . f(~,y)
f+
many t e c h n i c a l
concept
function
concept B of (12)
denoted
of
(5) the
a random will 3CL3.
will by
be
(9)
sub-optimal
and
of a use,
deterministic
variable.
be called
one for
CLS
In w h a t
sub-optimal
follows
719 4.SWITCHING The
CURVE
set
where
o~
bmt n
<
decisions:
(B,b) can
the
object
are
=
-B
follows
target
sgn
L
4.1.
i)
If
SCLS
has
and
B < the
B E
above
with
(3)
motion F(.)
model
~
takes and
=
a
- ~]
switching
the
properties
I~ B =
b,
then
h ~
h ~,
properties
8=b
SCLS
and
becomes
(6)
shown
hold. in
i.e.
we
oF b)
[O,K
- s]
the
switching defined
From the
[7]
to
the
curve shown the
following
(12)
remark.
satisfies
in t h e o r e m
switching
(4)
2.4
curve
one
too
the
solution
(see
fig.3).
(see
9
e>b
x B
F;g.1 Some
examples
will
define
different
ways
o~
choice
oF
and
(see
is
time-optimal
2.3
by: it
loss
Consequently,
theorem
SCLS
controlled
resistances (12).
pair
the
the
K-E],
o~
understand
a Fixed
in w h i c h
[.]~
[0,
set
value.
leads
the
the
for
by
x
finite
has
(3)
sgn
described
(4)
X ] c the
decision
a time
-b
> b then
and
a
motion
SCLS
one
oE
resistances
oE
[O~K
[bmtn,b
ThereFore,
the
b then
N =
Consequently,
express
is t h e r e f o r e (B~b)
N.
D.
will
the
the
by:
~
undertaking
formula: on
it
as
an
L
given
b(~)
d =
a)
pair"
(12)
IF B
oF
ii)
time-optimal iii)
the
each
o~
As
loss
based
determined
solution I).
by
[.],
Comparison REMARK
the
be
n =
i.e.
when:
is
for
will
].
B,
oF
defined
that
function
Fig.
value
definition
f(.>
the
[bmtn~bmax
=
the
the
states
Obviously,
Function
reach
curve
bmax.
D
a selection
SELECTION.
nature
B value.
Fig. oF
2).
the
720 The p r e s e n t e d
instances
consecutively
chosen
will
steps
be u s e d
for
in a d a p t i v e
an o p e r a t i o n
o~
system
this
synthesis
as t h e
system.
ill
T
X
Fill. 2 l EXAMPLE 4 . 2 .
Assumptions:
w h i c h has a d e n s i t y ~orm N = [ b n , b 9
the set
mQx
b is
a random v a r i a b l e
E u n c t i o n ~; ]
C [O,K -
the support of
~],
"
I
> O,
d > O.
- c - (B -
simulations
[8]
I
and i s
(B)
L follows
+ b m~x
E
if
{-c.(B
(14)
o~ t h e
>: ~ i n t ( N ) ;
)/2.
b d O,
i~ B -
b ~ O,
from p r a c t i c a l Taking
-
into
bm ~ x ) , d . ( B
together
be g i v e n
1 [bmtn'bm~x
c
B -
in
L(B,b) of
point
(B)
with
(7b)
b y B = bm~ n , = I
in
~J
(b
mt
--
n
description.
numerical of
view.
account the form o f
the mini-max loss Punction w i l l
inf B
if
f r o m some r e s u l t s
strate~.
= sup
c < d t h e n Prom
Similarly~
b)
recommended
function
mini-max d e c i s i o n w i l l
mLn
> 0 if
(13>
nO Appl~cat~o~ o / M[ns density
b)
Any o t h e r f o r m u l a can be used
The a b o v e s i m p l e f o r m o f
(b
m~x
~(x)
oG
= d-(B -
If
mtn
;
Eunction is
we d e ~ i n e by:
L i B , b)
support
< b
distribution
o~ d e c i s i o n s D = [bmkn ~bm~x ] "
The l o s s f u n c t i o n
where c
b
the
density
bm~n)}. it
(14)
follows
that
the
because (15)
).
]
> d t h e n B = bm ~ x .
Instead,
the
be g i v e n by:
by c = d,
it
~ollows B =
721 bD A p p L ~ c u ~ o ~ o~ Ba~es struLe~y. T h e by
(Sa)
will
take
here
the
function
o~ the
Bayes
loss
given
~orm: b
B
ib(B)
=
m~•
I
d- ( B - b ) ~
from
(16)
b)db
- I m
b After
calculating
i
(B)
b
=
0
c- ( B - b ) ~ ( b I d b .
we
(16)
get:
B
Using value
(171
o~
B.
I
~(b)db
mL
n
we c a n
= c/(c
realize
Equation
(17)
+
the can
d).
(171
Bayes
be
strategy,
replaced
by
selecting
the
respective
equivalent
~ormula:
b ~ x
I
B
~(b)db /
I
B
B(b
~
~(b)db = dlc.
9
(18)
b
B:b
F;~.3 EXAMPLE
4.3.
Assumptions:
Let
the
distribution
uniGorm i n N = [bm~n'bmax] ~ [O,K. be g i v e n t h e r e f o r e x(b)
=
the
bmtn<
I
1/(b
max
-b
m~n
)
if
b ~ [b
mtn
the
random
variable
be
density ~unction will
T h i s case i s a p a r t i c u l a r
,b
m~x
],
cases,
l o s s E u n c t i o n L i s t h e same as p r e v i o u s l y ~ (13).
of
bmax;
by
0 in the other
by
~3,
i.e.
is given
one shown i n Example 4 . 2 .
(19)
722 o9 ,4ppLs163
o/ M s 1 6 3
previously, of
because
random
ss
the
variable
The
mini-max
distribution
(with
t2D A p p l ~ c a s 1 6 3 o/ Ba3les sCz'as B
=
b
+
ms n
[c/(c
+
The v a l u e o6 B c a l c u l a t e d
in
EXAMPLE 4 . 4 .
the
Assumptions:
p a r a m e t e r s p > 1,
r
results
strategy
does
the
m~x
-
b
same
as
the
detailed
e}:ception
of
the
support).
(19)
(17)
into
we
~=orm
get:
).
n~s n
(20)
such a way s a t i s f i e s random v a r i a b l e
> 1~ o v e r t h e s e t
the
use
Setting d)](b
are not
t h e Bayes s t r a t e g y .
has b e t a d i s t r i b u t i o n
N = [bms
c
[O,K -
9
with
~],
bmLn<
b ; t h e s e t o~ d e c i s i o n D and t h e l o s s 9 u n c t i o n L a r e t h e same as rn~x p r e v i o u s l y . T h i s case i s a l s o a p a r t i c u l a r one shown i n Example 4 . 2 . So,
the
density I
(b)
k-(b-
k
b
=
is
a
) p - i (b ms
0 where
9 u n c t i o n can be g i v e n b y :
in
real
the
other
o9
monotonic
b.~ MppLs163 [bmtn,b
] by
-
n =
broth),
strcxte6~). T h e (see
standard
one
-bms
zlr-ldz
=
mB
(21)
are
the
same
as
in
the
4.3). to
replace
the
beta
distribution,
+
z =
n~
interval we
mb
+
n,
z P - i (I
-
zlr-ldz.
take
where
m
into
=
l/(b m~x
We g e t :
).
[c/(c + d ) ]
o
Using
], bm~x
I.
results
order
9or
Z =
b
=
Example
In
transgormation:
z p - * (I -
[b ms
[~(b)db -00
distribution
[0, i], a
b ~
cases,
o/ B a w e s ss
consideration
i9
such t h a t
constant
A p p s 1 6 3 1 6 3 1 6 3o/ M~.ns case
-b) r-i m=x
(221
o
the
e>:pression
backward we
get
transformation a
value
oi= B.
we [8]
can
convert
shows
the
Z
way
into
B
and
9tom
(22) c r e a t i o n .
o~
this 9
5.SUGSESTIONS FOR ADAPTIVE SYSTEM SYNTHESIS Themes o f
this
p a p e r has r e s p e c t t o
s a d d l e s o~= machine t o o l s , certain
k i n d o9 r o b o t s ,
o9 them r e q u i r e around i t .
to
In t h i s
variation
support
such
machines,
or the position
be b r o u g h t t o
industrial industrial
is
able to
o9 m o t i o n r e s i s t a n c e s .
without
the 9irst
data concerning the motion r e s i s t a n c e s . [0,o],
resulting
0 < o < K,
where ~ i s
from experiences in
technological
cycle
is
obligatory
by c << d.
p r o c e s s ~ o r each
s t e p s t h e s y s t e m has none Thus,
it
should take the
a su~:~iciently great
device exploitation.
the adaptive level
Some
any e n c i r c l e m e n t s
ensure the sliding
In
d e v i c e s as t h e m a n i p u l a t o r s and
mechanisms and a u t o m a t a .
a target
case t h e mini-max s t r a t e g y
T h i s way o~ a t r e a t m e n t
o~ t h e
tracer
heuristic
o9 t h e s y s t e m c o m p l e t e s t h e
o9 t h e v a l u e s o9 t h e m o t i o n r e s i s t a n c e s .
Having got
value
A ~ t e r each
su~gicient
data
quantity
723 of
those,
it r e p l a c e s
the
interval
[0~]
by
[b
,b mtn
determined
border
fills
up t h e
terns
to account
the
next
held
The other
the
too.
is a c a s e t h e p r o c e s s
o~ the
value
the
and using
cycle.
or
methods, completed
or
weighers.
of c and
d in t h e
adaptive
level
parameters
of
Bayes
In t h e
adaptive of from
B calculation.
of
should
intervals
loss f u n c t i o n
simulating
of
the SCLS
be u s e d
in
the (13).
but
its t e c h n i c a l its r e a l i z a t i o n
level
Next,
effectiveness with
the
and
is u n d e r t a k i n g mentioned
the measurements
the use of assigned each control
can change
the
concept
values of an
is r e c o m m e n d e d
on practical
use of
strategy
data are
the other one
act
the uniform~
the above the
to
will
of the
with
there
D E B.
of the way
which
mini-max
after
level
Obviously,
based
in w h i c h
from
with
the presented
calculations,
the value
creation
among
more
determines
the statistic
i.e.
o~ the
is m u c h
i.e.
strategy
be e x e c u t e d adaptive
case,
type
the quantity
resulting
treatment
on any
IG t h i s
decides
curve
of one
new values
c a n be c r e a t e d ,
showed
possibilities
time
and
b e used,
] or
step
This operation
In c e r t a i n
some
The next
may
a target
calculates
makes
dependence
,b
be u s e d f o r
the
Function
method [b
system
switching
on
a choice
and corrected. for
will
of B adaptation
strategy
the
the
distribution.
will
be a c c e p t e d
because
decision
target
in t h e p r e v i o u s
mini-max
Obviously~
concerning
which
as
resistances
[0,~]
beta
can
steps,
value o~ B for
data any
a decision
starts
the ~irst
of the motion
next
devices
The system [0~]
empirically the system
o@ B which
that
to
triangle
them
creation
cycles
cycle. be brought
the support
cycle.
curve
new value
the
received
can
new switching
the
] with max
control
around
calculate
with
consecutive
and calculates
considered
executing
in t h e
In t h e
may spiral
support
values
for
which
complicated.
A~ter
data
technological
trajectory, device
values.
values
of
a large scale
of
typical
technical
means.
REFERENCES. [i] A t h a n s M.. F a l b P-L. O p t i m a l C o n t r o l . M c G r a w - H i l l , Inc. (1964) [2] B r u n o v s k y P. T h e C l o s e d - L o o p T i m e - - O p t i m a l C o n t r o l . I: O p t i m a l i t y . S I A M J. C o n t r o l . V o l . 1 2 , N o 4, 1974, p p . 6 2 4 - 6 3 4 . [3] F i l i p p o v A.F. D i f f e r e n t i a l E q u a t i o n s w i t h D i s c o n t i n u o u s R i g h t - H a n d S i d e . T r a n s . Amer. M a t h . S o c . , S e t . 2 , V o i . 4 2 , 1964, pp. 1 9 9 - 2 3 1 . [4] H a j e k 0. D i s c o n t i n u o u s D i f f e r e n t i a l E q u a t i o n s . J. Diff. Eq. N o 32~ 1979, pp. 1 4 9 - 1 8 5 . [5] H e j m o W. S t a b i l i t y o f a T i m e - O p t i m a l C l o s e d - L o o p S y s t e m w i t h P a r a m e t e r C h a n g e s . Int. J. C o n t r o l , Vol. 45~ N o 4, 1987, pp. 1161--1178. [6] H e j m o W., K l o c h J. O n t h e T i m e - O p t i m a l P r o b l e m o f P o s i t i o n a l C o n t r o l with Discontinuous Resistances of Motion. RAIRO - System analysis and C o n t r o l . Vol. 18, N o 3~ 19847 p p . 3 2 9 - 3 4 1 . [7] H e j m o W., K l o c h J. O n t h e P e r t u r b a t i o n s in a T i m e - O p t i m a l C l o s e d - L o o p S y s t e m . Ann. P o l o n . M a t h . , Vol. 50~ N o 1, J989. [8] K u l c z y c k i P. T i m e - O p t i m a l S t o c h a s t i c C o n t r o l o f t h e D i s c o n t i n u o u s D y n a m i c O b j e c t (to be p u b l i s h e d in 1990).
M a x i m u m Principlefor Stochastic Optimal Control with N o n Convex Control Domain by Shige Peng Department of Mathematics, Shandong University Jinan, 250100, China Abstract. Our objective is to derive the stochastic maximum principle for the following optimal control problem: minimizing a cost function J(v(.)) = Eh(z(T)) +
EI[ LCy(0,
subject to 9 Co)
where the adapted process v(.) iS valued in a control domain U C ~ , and is called admissible control. A well knowldifIicult situation is when the control enters the diffusion term : ~z = a(z, v), and when U is non convex. In this paper, we use a new method to overcome this problem. The main idea is to introduce a "random valued function" W(z, t, w) (see section 3.1). The adjoint processes and the the variational inequality can be derived in a natural way via this function. w
Intrductlon
The maximum principle for stochastic control systems is studied by many authors. Among those works one can see [1], [2], [3], [4], [9]. Recently, the author [10] introduced a new adjoint equation, called the second order adjoint equation and obtained the maximum principle for general case: the diffusion coefficient can contain control variable, and the control domain can be non convex. The result shows that the maximum principle in general case is novel. And it recovers the preceeding results. We also derived a new adjoint equation, called the second order adjoint equation. In this paper, we treat the above problem by a new appraoch: for a given optimal control, we can introduce a random %-alue function" W(z, t, oo) (see 3.1 for detail). The first order adjoint process (the classical one) and the second order adjoint proccss can be regarded as the first and the second derivatives of W respectively with respect to z, at the point of the corresponding optimal trajectory y(t). This gives an intrinsic relation between the above two processes. Further essential relation (see (20)) can be also easily derived. Thus we think that this approach is more natural and more intuitive. This paper was organized as follows, In section 1, we introduced some basic notations and stated problem disposition. In section 3, we introduced the function W(x, s) which plays a cruciM role in this paper. The ~ j o i n t processes and the corresponding adjoint equation were derived in this section. The maximum principle was obtained in section 4.
725
w Statement of Problem Let (fi, jr, P) be a probability space equipped with filtration 7t. L~t B(.) be a /~-valucd standard Wiener process (our method is valid for general iRd-valued Wiener process). We assume = ~{B(~); o < s < t}. We denote by ~q(H), q _> 1 the set of progressively measurable random processes valued in an Euclidean space H such that
E
/;
Iz(t)lqdt <
oo,
vz() e ~(H),
Let U be a non empty subset of R~ (called control domain). The set of admissible controls is defined by U = (v(.) e .Mq(Rk); v(t) E U, a.s., a.e. ,Vq = 1, 2, ...} For any given v(.) E U, consider the following stochastic control system
dz(t) = g(zCt), vCt))dt + a(x(t), v(t))dB(t), (1)
z(0)
zo
where g(z, v), a(x, v) are ~"-valued mappings defined on ~ problem is to minimize the following cos~ function
• ~k. The optimal control
J(v(.)) = Eh(z(T))
(2)
over the set of admissible controls, where h(x) maps R r~ to/R. Our method is also valid for a general cost function J(v(.)) = EifoT L(xCt),v(t))dt + hCx(T))]. Our main assumpions are g, or, h are twice continuously differenfiable with respect to x. They and all their derivatives 9x,ax, h~, gzz~axz,hxx are continuous in (x,v). 9z, a,, gzz, axx, hxx are bounded, g,a are bounded by C(1 + ix] + iv]). h is bounded by C(1 + ix]2)
w Adjoint Processes and A4joint E q u a t i o n s
3.1. First Order and Second Order Adjoint Processes Let u(.) be an optimal control and let
y(.)
be the corresponding trajectory. We
can define a family of adapted proce~es parameterized by (x,t) e ~"~[0, T]: ~(s) = y(s; x,t); t < s < T by solving the following SDE
(3)
~ d~(s) = g(y(s), u(s))ds + ~(y(s), u(s))dB(s)
(
yCt)=~
With this processes, we can introduce a notion which plays important role in this paper
W(z,t)= E~V'h('~(T;z,t))
726
In general W(z, .) is a stochastic ~-adapted process. According to Krylov [6], y(s; x, t) is twice continuously differenfiable with respect to z. We denote by Yx (resp. Yxx ) the first (resp. the second) derivative of ff with respect to z. They are solutions of
{ dy~ = g~Cy(~),u(s))Y.Cs)ds + ~(y(s), u(s))y~(~)dB(s) y~(t) X, t < s < T ,
(4) and
I d'yzz = g=(~(s),u(s))~z.(s)ds + a=(~(s), u(s))~.z(s)dB(s)
(5)
+ g=Cy(s), u(~))y~(s)~( s)ds + ~=(y( s), ,,(s) )y=(~)~( s)dB( ~) y:=(t)=O, t < s < T , N
N
N"
2
~"
N
~"
9
where fzxyxy= stands for 3": : y~_*o~__y~, and fxYz= stands for y'].~~y~. WIx, t) is also twice continuously differentiable. Iia fact, thrir derivatives have the following form z.....d~2
.i.,
UZIOZi
.
.
.
.
.
.
i
Wx(z, t) = E ~'[h=(~'(T; z, t))Yx(T; z, t)] W**(z;t) = E;"[h::(y(T))~=:(T) + h~=(y(T))y~(T)y::(T)]
(6)
It is also easy to check that
E~'ly(t;z,s)l ~ < C(1 + Izl ~) E~',]y~(t; x, s)[ 2 + E~', ]y=(t; z, s)[ u _< C
IW(~, t)l + IW.(x, t)l _< c(x + Ixl ~) We take the following notation
p(t) = Wx(y(t), t), Q(t) = Wzz(y(t),t). p(.) and Q(.) are respectively called the first and the second order adjoint processes. It is not hard to check that
p(t) = ZS'[hz(y(T))Y(T,t)]
(7) (8)
Q(t)=
where Y and
Zi
E ~' Eh=i(y(T))Zi(T,t)+ ET'hz=(y(T))Y(T,t)Y(T,s)
J are solutions of
(10)
+ ftg~z(y(r),u(r))Y(r,s)Y(r,s)dr + fta~x(y(r),u(r))YCr, slY(r,s)dB(r)
727
3.2. First order and Seconder Order Adjoint Equationz We first consider equation for the first order process p(.). Multiplying any p(-) C ~ ( / R ~') on both side of (7), and then taking integral and expectation, we have (11)
E foT(P(t), p(t))dt = E foT(p(t),h~(y(T))Y(T,t))dt = E(hx(y(T)),z(T))
with z(t)= f~ r(T,s)p(s)ds. One can check that z(.) solves (12) (with r
(13)
{ dz(t)=(g:(y(t)u,(t)z(t)+p(t)0dt+(o:'(y(t)u,(t)z(t)+r = 0). According A.Bensoussan [1], there exists k(.) 9 N~(/R '~) such that
E(h~:(y(T)),z(T))= E
r149
/o[(p(t),~o(t))+(k(t),r
with r = O, in case (11), and such that (p(.),k(.)) is the unique solution of the following backward SDE
-dp(t) = (g~(y(t), u(t))p(t) + c~;(y(t), u(t))k(t))dt - k(t)dB(t) p(T) = hx(y(T)) Or
(14)
-alp(t) = H::(y(t), u(t),p(t), k(t))dt - k(t)dB(t) p(T) = h::(y(T))
where we denote the Hamiltonian by H(=, o,p, k) = (p, g(x, v)) + (k,~(x, v)). The above equation is called the first order adjoint equation. It is a classical equation. We now consider the equation for the second order adjoint process Q(.). From (11), we can write the first term of the right hand of (8) as
Ea{ +
I f
Y*Cs, t)(k(s),ax::Cy(t),u(t))Y(s,t)ds
728
It
follos
that Q(.) can be interpreted as a quadratic form of Y(.).
QCt) = E7'{
f"
r*(s,t)H~xCYCt),uCt),pCt),qCt))YCs, t)ds
+ Y*(T,t)h:~:dy(T))Y(T,t) Thus Q(.) has quardatic form with respect to Y. Where Y is the solution of linear equation (9). But following idea of [1], it can be expressed as a linear form of certain linear equation. Indeed let S n be the space of all symmetric n x n matrices with scalar product
(,~, ~).
=
t,-[,~d,
v~-, ~ e s".
For any f(.) in .M#(S"), we have
E
(QCs),f(s)),ds = E
/o f 9 tr[Y'(r,s)Hx~Cr)Y(r,s)fCs)]drds
+E
tr[Y*CT, s)H==CT)r(T,s)f(s)lds
+ Etr[Hx==(T) -L r Y(T,s)~(s)Y*(T,s)ds]
= E[frcH~Cr),eCr)).dr-jo
+ (h~(y(T)),~(T)),
Thus
(is)
z
/o"(O(s),f(s)).ds= E[/o (~=(r),~(r)).~r+(h=(y(T)),~(T)).
where we denote
H=(,.) = H::~(y(,'l, ,,(~), p(,'), k(~) ) y(r) =
~0r Y(r,s);(s)Y*(r,s)ds
Since for any ~" in S '~, one has
Y*(t,s)fY(t,8)=f +
[Y(r,s)fr*(r,s)g;(r)+gx(r)Y(r,s)fY*(r,s)
+ ~x(r)y(r,s)fy(r, s)4(r)ldr
with
729
Thus y(.) is the solution of the following linear SDE
{ d Ct) =
+
+ (t)lat
+
+ [a::(t)~(t) + ~(t)~r*(t) + ~(t)]dB(t)
(16)
7(0) = o
with ~(.) - 0 in this case. Notice that S '~ is also an Euclidean space. So, for (15), (16), we have the same situation that we treated in (11), (12). Thus, following the same routine, one can show that, there exists M(.)) in .M~(Sr'), such that (Q(-), M(.)) solves uniquely the following backward matrix-valued SDE
-dQ(t ) = [Q(t)g::(t) + g; ( t)Q( t) + a;( t)Q( t)a~:(t) + a;(t)M(t) + M(t)r - M(t)dB(t) Q(T) = h~(y(T))
(17)
The above equation characterizes uniquely the second order adjoint process Q(.). Tha~ equation is then called the second order adjoint equation. w
B a c k w a r d Stochastic P D E and M a x i m u m Principle
Recently Peng [11] formulated a kind of stochastic Hamilton Jacobi BeUman equation arising in the situation where the coefficients are random. It can be formulated as a backward semilinear stochastic partial differential equation. We can apply this result to treat the random function W(z, t, w). Indeed, since this function is defined via the ~state equation" (3.1), where,for the given optimal control u(.), thus, in general, the coefficients g(x, u(t)) and a(x, u(t)) are random. So we have a situation of [11] (and this is the simplest situation.). It follows that W(z, t) can be formulated as a solution of the following linear stochastic backward PDE
(18)
W ( z , t ) = h(z) +
l
l rn(z,s)aB(s)
+
The problem is to look for aa adapted pair (W, m) which solves the above equation. Where/~ and ~ are differential operators with random coefficients 0~ S,J
S
!
with a(x, v) = 89 v). We can derive easily the maximum principle by (18). Indeed, for any given s E [0, T), and T - s > e > 0 we can construct a classical ~spike variation ~ as follows u~(t ) = / v ' s
730 Where v(w) is a ~-measurable bounded random variable with values in the control domain U. Obvioursly, u ~ is an admissible control. Let ye be the corresponding trajectory. One has y~(s) = y(s)
Since u(.) is optimal, then
JCu~(.)) - J(u(.)) = EE(h(y~(T))]~+~) - EE(h(y(T))l~+c ) >__0 From the definition of W(x, t) have
E(h(y(r))ls = W(y(s + e),s + c) E( hCy~(T) )l ~+~) = W(Y~( ~ + ~), s + ~) Using generalized Ito's formula (see Kunita [7]), we can deduce
E[W(yCs + ~),s + ~) - wCy(s), s)]= 0 and
E[W(y~(s + e),s + e) - W(y(s),s)] = E
[-~(yr
+ ~(y~Ct),t).,Cy~(t),t)J~t
i
+Z
s+e
2
[ZaiiCyE(t),,) 1,3
0"~ W(yeCt), t) OxiOxy
$
I
It follows that
s
t,3
|
!
!
$
Let ~ - ~ 0, fro,- the de~nition of p(-) and q(.), we obtain 1
.
~,~,-I~-~-(yCs),v)QCs)] + (aCy(s), v),.,.CYCs), s)) + (g(yCs), v)),p(s)) (19)
.~r[~'CyCs),,(s)lOCs)] + (~(yCs),,(s)),,,,CyCs), s)) + (g(y(~l,,(s))), p(s)) vv E u, a.e., a.s.
731
This is the maximum principle. Moreover, we can derive the following relation
m~(y(s), s) = g(s) - O(s)(r(y(s), u(s))
(20)
Indeed, in the case where g, ~r, h are sufficiently smooth wi~h respect to x, one can check easily that W(z, s) is also sufficiently smooth in z. It follows from (18) that w~(~,t) = h~(~) +
//
O~[.e.(~,s)W(~,s) +
9(~,s)m(~,s)]as-
m~(~,s)dB(s)
Again using Ito-Kunita's formula, we see that the martingale part of the process
w~(y(~),s) is f T[m~(y(W=(y(~),s)~(y(s),~(~))]aB(s) 8), S) =
[m~(y(s),s)- O(s)~(y(s),~(s))laB(s)
But on the other hand, since W..(yls), s) = p(s), from the first order adjoint equation (14) we know this martingale is - ft k(s)dB(s). Thus we obtain (20). For general case, we can obtain (20) by approximating g, (r, h with their mollified function~. It follows that the maximum principle can be written as
ltr[~* (y(s), ,~)Q(s)I + H(y(s), ~, p(~), ~(~) - Q(s),~(~,(s), ~,(s))) (2o)
1 , > it~[~ (y(s),~C~))Q(~)] + nCy(~),~,Cs),p(s),k(s)- QCs),~(~(~),,,(s)))
Vv E U
a.s., a.e.
Remark. In the case where (21)
a{u(s); s > t} is independend of ~,
we can prove that (22)
K(t) = Q(t)a(y(t),u(t))
Remark. The maximum principle can be also derived by viscosity solution approach, see recent work of Zhou [12]. But an additional assumption similar to (22) is still needed. Referent e s [1] A.Bensoussan, Lecture on Stochastic Control, Lecture Notes in Math. No. 972, Nonlinear Filtering and Stochastic Control, Proceedings, Cottons, 1981 [2] J.M.Bismut, An Introductory approach to duality in Optimal Stochastic Control, SIAM,Rev. Vol 20,No. 1, June, 1978 [3] U.G.Haussmann, General Necessary Conditions for Optimal Con- trol of Stochastic system, Math.Programming study 6(1976)
732
[4] Y.Hu, Mazimum Principle of Optimal Control for Markov Processes, Proceedings of Symposium on Control Theory and Applications, PRC, Sept., 1987 Submitted to Acts Mathematics Sinica [5] N.IKeda, S.Watanabe, Stochastic Differeutial Equations and Diffusion Processes, North-ttolland Publishing Company, 1981 [6] N.V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York, 1980. [7] tt.Kunita Ecole d'Ete de Probabilite de Saint-Flour XII, 1982 [81 tt.Kunita and S.Watanabe, On Square integrable MartingIrs, Nagaga Math. Journal, Vol 30, August 1967 I91 H.J.Kushner, Necessary Conditions for Continona Parameter Stochastic Optimazation Problems, SIAM, J. Contr. Vol. 10, pp550-565, 1972 [101 S. Peng, A General Stochastic Mazimum Principle For Optimal Control Problem, SIAM J. Control and Optim.. Vol.28, No.4, 1990. [111 S. Peng, Stochastic ttamilton-Jacobi-Bellman Equationz, Submitted. [12] X. Zhou, Ph.D. Thesis of Fundan University.
NASH VS STACKELBERG STRATEGIES IN A CAPITAL ACCUMULATION GAME Jacques Th~pot Universit~ Louis Pasteur 4 rue Blaise Pascal - 67070 STRASBOURGC~dex (France).
This vestment (1983)
paper is
problem as
aimed at giving new i n s i g h t s on the s t r a t e g i c in-
formulated by
Spence (1979)
and Fudenberg
& Tirole
[F-T hereafter]. These authors analyse the investment intertemporal
strategies
simultaneously used
by a firm established in a market and by a
new entrant when investment is assumed to be i r r e v e r s i b l e and when there is no capital depreciation through time. The s t r a t e g i c investment level of the established firm is then defined as the minimum i n i t i a l prevents
the new entrant to
get nonnegative
capital stock which
p r o f i t s in
the future. The
s t r a t e g i c investment depends on the strategies used in the post-entry game. The
analysis of
the post-entry game w i l l be made in terms of D i f f e r e n t i a l
Games. Two dynamic
e q u i l i b r i u m concepts
w i l l be
discussed : the perfect
Nash e q u i l i b r i u m (closed loop) and the Stackelberg e q u i l i b r i u m . T e c h n i c a l i ties
as well
Section and 3.
as comparisons
with F-T r e s u l t s are given in Th~pot (1989).
I is devoted to the d e f i n i t i o n of the D i f f e r e n t i a l Game i the Nash
the Stackelberg
strategies are r e s p e c t i v e l y analyzed in Section 2 and
1. THE MODEL
tic
We consider two firms, f i r m I and firm 2, involved in a duopoliscompetition on a market over an i n f i n i t e horizon [0,+~[. Let K i ( t )
( i = i , 2 ) be the capital stock associated with firm i at time t . A time O, f i r m 2 has j u s t entered the market and has no capital K2(O) = O, while cap i t a l stock KI(O) = K~ of firm I is determined by preemptive investment. In the f o l l o w i n g , firm i w i l l also be termed the "established" f i r m and firm 2 the "entrant" one. Let Hi(KI,K z) be the instantaneous net revenue of firm i which is assumed to depend on the capital stocks of the two competitors. As i n F-T we assume that function Hi is d i f f e r e n t i a b l e , concave in Ki and that a2Hi/aKiaK j and aHi/aK j are negative (i ~ j, i = 1,2). Let l i ( t ) be the firm
i ' s investment
rate at
time t, the price of which is supposed to be
736 constant
and
equal
to
one.
Thus,
each
firm's
instantaneous p r o f i t
is
( H i ( K I , K 2) - l i ) . T h e r e is no depreciation so t h a t K i ( t ) = I i . Investment is assumed to be i r r e v e r s i b l e (I i ~ O) and upper bounded by a constant level Ti . Each f i r m wishes to maximize i t s net present value of p r o f i t s Vi . Accordingly, the problem can be stated as the f o l l o w i n g d i f f e r e n t i a l game : V1 = ~0 [ ~ l ( K l ( t ) ' K 2 ( t ) ) - l l ( t ) ] e ' r l t d t ' (1) V2 = ~0 [ E 2 ( K I ( t ) ' K 2 ( t ) ) - 12(t)] e - r 2 t d t ' KI ( t ) = l l ( t ) ,
(2)
K2 ( t ) = 12 ( t ) ,
(3)
0 < 11(t) 4 y 1 0 ~ 12(t) < y z (4) Kl(O) : K~, K2(O) = O. (5) where r i denotes f i r m i ' s discount r a t e . KI and K2 are the state v a r i a b l e s of the game while 11 and 12 are the c o n t r o l v a r i a b l e s of f i r m I and f i r m 2 respectively. 2. THE PERFECTNASHEQUILIBRIUM In t h i s section we are going to c h a r a c t e r i z e strategies
by using
the Nash e q u i l i b r i u m
the standard techniques u s u a l l y c a l l e d as the "method
of c h a r a c t e r i s t i c s " [ c f . Case (1959)]. Let Hi be the current value Hamiltonian of f i r m i defined by : HI = ( H i ( K I , K 2) - I i ) + x i l i + ~ i l J , where x i and
~i are the costate variables r e s p e c t i v e l y
associated with d i f f e r e n t i a l
equations Ki = i i and KJ = l J . Under the assumptions made, Hi is a s t r i c t l y concave f u n c t i o n of Ki and l i n e a r in I i ; in a d d i t i o n , we w i l l assume t h a t the dynamic system is stable,
as indicated by the f o l l o w i n g
[. 02~1
02~2
02~I
02~2
[(aK1)Z
(aKz )z
aKlaK 2
aKZaKz
O.
condition
: (6)
Applying the Maximum Principle with I i and IJ functions of K1 and K2 results in the following conditions : 9 ii = rik i ST[i ali 81J + (I - x i ) - (p' , aK i OKi aKt ~oi : ricoi
8/I i +
(i
- Xi)
aKJ
OII . _ q~i c~lJ aKa OKJ '
Xi < i ~ I i : O, X i = 1 ~ I i undetermined, Xi > 1 ~ I i = T i . lim x i ( t ) e" r i t t-~oo
=
lim t-~
q>i ( t ) e" r i t
(7)
(8)
(g) (i0) (11) = O.
(12)
737
9 The
three policies
liable to
be used along the equilibrium path
are determined by relations (9)-(11) : ( i ) Policy 1 = no investment policy : (I i = O) : firms i does not invest ~ i t s capital stock is kept constant since no capital depreciation is allowed ; this policy has to be implemented along the equilibrium path when x i < 1. ( i i ) Policy2 : steady state policy : (x i = 1) : The capital stock is determined by the condition that its marginal value (measured by adjoint variable x i) equals the inves-
Policy3 : maximum investment policy : firm i invests at
tment cost. ( i i i )
the maximum level ~i ; capital stock increases ; i t has to be implemented along the equilibrium path only when xi > 1. A sequence of policies over the horizon [0,+~] w i l l be called a strategy 9 A duopoly regime is defined as a combination of policies ; regime ( k - l ) stands for firm i using policy k and firm 2 policy I. Clearly all the regimes encompassing policy
1 or 3 are determined by the open loop condi-
tions (obtained by dropping partial derivatives of investment rates in relations (7) and ( 8 ) ) , since the upper and lower bounds o f 11 and 12 are independent on Kz and Kz. Regime (2-2) d e f i n e s the steady s t a t e o f the game which n e c e s s a r i l y holds w i t h I I = 12 = O.Consequently, open loop and closed
loop
s o l u t i o n s coincide.
Accordingly,
the c h a r a c t e r i s t i c s
of the p o l i c i e s
are summarized in t a b l e i :
Policy
ii
I
0
feasibility conditions
equilibrium conditions
none
xi < I
arl i _
r i
Xi = I
aK i
3
Ti
none
xi > I
Table 1 : characteristics of the policies. - Regime (2-2) defines the Nash long run equilibrium point (KI(N),K2(N)) of the duopoly where KI(N) and K2 (N) are solutions of the system :
anl - r I ' aR2 - r 2 " ~K1
aK2
(13)
738
These relations define the classical reaction functions K2 = RI(K1) and K2 = R2(KI) associated with each firm. A perfect equilibrium path is determined by a sequence of duopoly regimes along which equilibrium conditions given in table I, transversality conditions (12) are satisfied. I t is quite clear that adjoint variables m~and~ have to be continuous along the equilibrium because of the regularity conditions of the problem and the absence of the
constraints on the state.
All these conditions ensure the unicity and
s t a b i l i t y of the equilibrium path as i t will be seen now. Let C be the
upper envelope of RI and R2. I t is quite clear that any equilibrium path starting above C will necessarily remain at the i n i t i a l point. Since there is no depreciation, steady state (KI(N),K2(N))is only on the equilibrium path when the i n i t i a l capital stocks are on line ZIN. I t remains then to characterize the equilibrium path for other i n i t i a l conditions. This will be done in determining the equilibrium path ending at some point (R1,~2) of C. The path is univocally defined by Proposition i and 2.
Proposition 1 : Let capital stocks ~l and kz such that ~(i >I KI(N) and 7{2 = R2(TCI). (i) There exists an unique perfect Nash equilibrium path of the game ending at point (~1,?
Proposition 2 : Let ~'1{7(i) the capital stock of firm 2 held at the {eventual) switching point of firm 1 on the equilibrium path ending at (~KI,R2CKI)). Let I be the set of values of ~(i such that the switching point exists. (i) v I E I ~'I{KI ) < RI{KI ) ; ( i i ) ~" is a continuous and decreasing function such that ~'I{KI{N))=I(Z{N). ( i i i ) I is an interval of the form
[KI (N),x I 1.
P r o o f : see Th~pot (1989).
739 Thanks to these propositions, i t is easy to construct the e q u i l i brium path starting from a point {K~,O} (or more generally {K~,K~}): ( i ) i f K~ > x I , firm 1 does not invest while firm 2 does as quickly as possible u n t i l capital stock K2 equals R2(K~). ( i i ) i f K~ < X1, both firms invest as quickly as possible until time t~ when KZ(tl) = ~1(K1(t1)) ; then firm 1 stops i t s investment while firm 2 goes on investing u n t i l time t~ when i t s capital stock equals R2(KZ(tl)) (curve Au on figure I ) . Because of the Jrr e v e r s i b i l i t y of investment, the equilibrium strategies of the firm do not lead in general to the Nash steady state of the game {KI(N),K2(N)}, although there exists a feasible path leading to i t when K~ < KI(N). Accordingly, in the long run, capital stock of the established firm w i l l be fixed at a higher level and the one of the entrant at a lower level than those prevailing at the steady state of the game ( ~ i ~ RI(N), R2 ~ K2(N)). Figure I depicts situation.
the various
equilibrium paths according to
the i n i t i a l
K~
P2
Z'I2
I
ZI
A
B
M1
\ Z2
Figure I : Nash equilibrium.
PI
~i
740
3. TIIE DYNAMICSTACKELBERGEQUILIBRIUM
Stackelberg equilibrium is
a standard concept in static duopoly
theory. The dynamics of this concept, which derive from the extensive form formulation [cf. Basar & Haurie (1982)], rely upon asymmetries of information between the players. At any point in time the follower is assumed to observe the decision of the leader. In the following, firm I is the leader and firm 2 the follower. We assume that the discount rates of the firms are identical : r i = r z = r. The follower, firm 2, is assumed to observe, at any time, t, the rate of investment I i chosen by the leader. Firm 1 faces an optimal control problem, which determines its optimal response to any value of I i . As in the Nash equilibrium context, we have
an2
i2:r~2
an2
~
:
r92
_
aK z ' lim t
e -rt
.k2(t)
t
12 = h ( X 2)
-
The
= 0
x2 ~ i ,
if
leader, f i r m 1, then
relations
(16),
q)2(t)
T2
if
determines his
(17) and
into f i r m l ' s c o n t r o l problem [ c f . Stackelberg Hamiltonian Hi as :
HI = (Hi-I i ) + X i l i
e -rt
+ ~lh(x2)+
(18) :
_
s t r a t e g y by
~i(rx2-~]
aK 1aKz
a~ s
= r91
- ~
+ a i I r ~ 2 - OE2).aKI)
a2n2 (aK 2 )2 '
aZll2 + EI ~
aKz
taking
into
Wishart & Olsder (1979)] by d e f i n i n g a
+ El
aKI
(18)
these elements are incorporated
"costate" variables associated with. We have : an1 a2n2 ~ + .ql = rxl
of costate va-
x 2 > 1.
When x 2 and 92 are considered as additional state variables,
~i
(17)
= 0
-~ + oo
12 is determined as a f u n c t i o n
Where the rate of investment, r i a b l e x2 :
account
lim
=
-~ + oo
(i6)
__, OKi
a2112 + Ii I
(aK z )2
aK z aKz '
~I
(19)
and ai the
(2o)
(21)
E1 : _ # h , ( x 2),
(22)
A i : O, Ii = 0 if x I g i , Ti i f x i > i , lim e- r t x l ( t ) : l i m e - r t ~ ( t ) = O.
(23)
t-*oo
(24) (25)
741 Variables dingly initial
~2 and ~
have
only terminal conditions (17) ; accor-
the costate variables associated with have to s a t i s f y the following conditions : ~i ( 0 )
= ~1(0)
= O.
From relations (23) and (26), we deduce o l ( t )
(26)
= 0 Vt, so that t h i s variable
can be dropped out the formulas. Function h is not d i f f e r e n t i a b l e at point ~2=i in the usual sense. I t is a 6-function [ c f . Wishart & Olsder (1979)] and h ' ( x 2) stands for the d i s t r i b u t i o n derivative of 8-function h. Let t be any date at which investment rate 12 switches from ~2 to 0 ; then relation (22) implies : ~1 (t+O) = ~1 (~) + ~2~i (~). (27) Clearly,
~l as to be considered as a state variable of the Stackelberg ga-
me, with an i n i t i a l condition which has to be consistent with the const r a i n t s of the problem. Variable Ez can be interpreted as the evaluation of
the
marginal
profit
of
the
follower,
as made by the leader. Relations
{ ( 1 6 ) - ( 1 7 ) - ( 1 8 ) - ( 2 0 ) - ( 2 1 ) - ( 2 3 ) - ( 2 4 ) - ( 2 5 ) - ( 2 6 ) - ( 2 7 ) } define
the
necessary
conditions determining the Stackelberg equilibrium path. Each firm is l i k e ly to use the three policies presented in section 3.2. ; they are characterized
in the
same way as in
the Nash case (cf. table 1), except for the
steady state policy of the leader, which is determined by the r e l a t i o n : a~Z a2ii2
aKz
-
r + ~z ~
aKzaKz '
(28)
where ~I depends on the path. Stackelberg equilibrium, as before, is determined by assuming that costate variables ~ and ~ are continuous along the equilibrium path. Because of relation (27), state variable ~z is not continuous at points where ~2=i ; t h i s makes variables x z and @z not d i f f e r e n tiable
at such switching points. Accordingly s i g n i f i c a t i v e changes in the
strategies occur as compared with the Nash case. Regime (2-2) definies the steady state,. (KZ(T),K2(T)) of the Stackelberg dynamic duopoly. KZ(T) and K2(T) are solutions of the system (cf. Th~pot (1989)) :
~aKI
= r + 112, = r. LaK2
LaK2
~ / aKlaK 2
( r / l 2) + (aK2)z
'
(29) (30)
742
Relation (29) defines the generalized reaction function, K2 = G (KI) of the a2H2 leader. In t h i s paper we only consider the case when : ( r / l z) + ~ 0.
(aN 2 )
This means t h a t the upperbound of investment T2, which determines the speed of adjustment of f i r m 2, is s u f f i c i e n t l y high ; Note t h a t t h i s steady state coincide with the classical Stackelberg long-run {KI(S),K2(S)) for r=O or ~z = +~. With condition (30), we have KI(T) ~ KI(S) and KZ(T) ~ K2(S). The
Stackelberg equilibrium has to be determined for any i n i t i a l values of the capital stocks together with the additional condition that ~Z(O)=O.As i t will be seen later, this is not possible everywhere. Two situations have to be considered according to the number of switching points from investing to not investing of the follower : (i) Stackelberg strategies for no switching point of the follower In this case, variable ~i has to remain equal to zero along the entire path. The Stackelbergpath coincide with the Nash path for trajectories where firm 2 is never investing. On figure i, this corresponds to area IV2, namely for K~ ~ KI(S) and ~2(K~) ~ K~ ~ RI(K~). ( i i ) Stackelberg strategies for one switching point of the follower The following proposition gives the ending point characterization of the Stackelberg path in this case.
.Proposition 3 : Let ~i be any value of firm l ' s capital stock : (a) For ~l <_ KI {T), there exists an unique Stackelberg equilibrium path ending at point {Td,Rz{Td)), with at most one switching point for the leader, located on a switching curve such that cl (Kz(T)) = KZ{T). The switching point of the follower arises when curve Rz is reached. (b) Thereexists a lower bound Li < KI (T) such that an unique Stackelberg equilibrium path exists with an ending value of 7(l belonging to the interval [LI,KZ(7)[. The ending point CKI,T(z) is located on a stopping curve of the leader r I such that rI(KI(T)) = KZ(T) and Rl(K) < ~- I(KI ) (c~ (K1) V K1. The switching curve of the follower, cz has the following properties = ( i ) oz(KI(T)) = KZ(T) ; ( i i ) c2(Kz) <~'z(K1), V KI . The lower bound, LI is defined by ~2{0) = TI(LI ] . Proof : see Th@ot (1989).
743 The
various
Stackelberg
equilibrium
paths
are
depicted
on
Figure 2. KL
C
(Zl,)
Ml , WI,
Qm.
K"
Figure 2 :
Stackelberg equilibrium.
Point T is defined as the i n t e r s e c t i o n of curve o of equation K2 = | (K 1 ) and curve R2 . The IGP going through p o i n t T crosses axis OKI at p o i n t Xz of coordinate (xz,O), w i t h x I ~ O. Unlike the Nash case, a subgame p e r f e c t e q u i l i b r i u m is not found when the switching point of the f o l l o w e r takes place before the terminal p o i n t . The state-space, {KI,K 2} can be d i vided
i n t o several
regions : ( i ) Region I is defined as in the Nash case.
( i i ) Region 111 ( Q I T P I ) and region 112 (W2NP2) : The e q u i l i b r i u m path is (1-3) ~ ( I - 2 ) and (3-1) --~ (3-2) r e s p e c t i v e l y . Coincidence with the Nash case arises in region I I 2. (iii) Region I I I ( l i n e XIT ) : The e q u i l i b r i u m path is (3-3) --~ (2-2). ( i v ) Region IV 1 and IV 2 (OXITQ2 and XzTQI ) : The e q u i l i b r i u m path is (3-3) ~ ( 3 - I ) --~ ( 2 - i ) or (3-3) ~ (1-3) --~ (1-2).
(v) Region V (Q2WNT). There is no e q u i l i b r i u m path c o n s i s t e n t with the initial c o n d i t i o n , El = O. I f such a path e x i s t e d , i t would have no s w i t ching p o i n t f o r firm 2 and accordingly be of Nash type with f i r m 2 not i n vesting over the e n t i r e h o r i z o n and Ez=O at every time. But a Nash path in t h i s region n e c e s s a r i l y has to s t a r t with an investment period of f i r m 2.
744
Hence the contradiction. Region V w i l l be termed the time inconsistency region. Clearly, the subgame perfectness property in space {KI,K 2} is violated
around the
dubious region
:
let
us
consider,
for instance, the
Stackelberg path starting from point A on figure 4. At point A', the subgame perfectness property starting
there
starting at
;
nor
this point
does not hold since there is no equilibrium point it
holds
consists of
at point A" where the
equilibrium path
sticking to i t . However, i t turns out
that the subgame perfectness property is valid everywhere in space {KI,K2,~I}. This exemplifies the crucial role of variable ~i, which induces a strong path-dependency in the firms' strategies. State variable ~z summarizes the evaluation of the follower's marginal p r o f i t made by the leader : it has here
is a
specific ingredient of the dynamic Stackelberg equilibrium, which
no counterpart
in the pure Nash context. The perfectness
issue raised
in a particular way. The Stackelberg e q u i l i b r i u m is perfect except in
the time inconsistency region.
REFERENCES
.BASAR T., Time consistency and Robustness of Equilibria in non cooperative dynamic games in DynamicPolicy games in Economics, F. van der Ploeg & A.J. de Zeeuw eds., North-Holland (1989), 9-54. .BASAR T. & HAURIE A., Feedback-Stackelberg solution in continuous ti-time multi-level games, in : proceedings of the 21st IEEE Conference on Decision & Control (Orlando, FL) (1982), 664-668. .CASE J.H., Toward a Theory of ManyPlayer Differential Games, SIAMJ. Control, 7 (1969), 179-197. .FUDENSERGD. & TIROLEJ., Capital as a Commitment : Strategic Investment to Deter Mobility, Journal of Economic Theory31 (1983), 227-250. .LEVIRE J. & THEPOTJ., OpenLoop and Closed Loop Equilibria in a Dynamic Duopoly, in, Optimal Control and Economic Analysis, G. Feichtinger ed. (1982), North-Holland, 145-156. 9SPENCEM., Investment Strategy and Growth in a New Market, Bell Journal of Optimizatio~ Theory and Applications 10 (1979), 1-19. .STARRA. & HOT., Non zero-sum Differential Games, Journal of Optimization Theory and Applications 3 (1969), 184-21g. .THEPOT J., On the Multiplicity of Perfect Equilibria in a Dynamic Oligopoly, EconomicsLetters 24 (1987), 127-131. .THEPOTJ., The Strategic Investment Revisited, mimedBETA, 1989. .TIROLE J., The Theory of Industrial Organization (1988), MIT-Press. .WISHART D. & OLSDERG., Discontinuous Stackelberg solution, International Journal of Systems Science (1979), 10, 1359-1368.
A two player dynamical game with imperfect information Odile P O U R T A L L I E R Institut National de Recherches en Informatique et en Automatique Route des lucioles, 06560 Valbonne
France abstract We intend to study a dynamical game with imperfect information : "the game of hunter and rabbit". This game was already studied in [1] to [5], but either with different sets of states, or iv~ cases which make it equivalent to a very short game. In this article we use theoretical results stated in [2], in order to give sufficient optimality conditions for a couple of two players' strategies. These conditions will enable us to compute without any difficulty, the hunter's strategy. The imposed condition on rabbit's strategy doesn't imply existence, but nevertheless proves that the hunter's computed strategies, in the worse case, give us a lower bound for the payoff of the game. This work was part of a doctoral work performed at INI~IA, supervised by P.BER.NHAI1.D.
1 1.1
D e s c r i p t i o n of the game, o p t i m a l i t y conditions Description
of the
game.
We intend to s t u d y a zero sum game with imperfect information. It is a two player game : 7-/, the hunter, and "R., the rabbit. T h e r a b b i t can move, jumping right or left at each step of time, along an infinite discrete line Z. T h e m a x i m u m size of his jump is r. A t every instant, the hunter standing at a distance, shoots a bullet aiming at a position of the line. T h e bullet reaches the line two time steps later. Of course, 7~ doesn't know where ~ has aimed at. We will consider two different versions of this game : First version : T h e fact to be hit by a bullet doesn't avoid the game to continue, until a fixed final time T. T h e r a b b i t wants to minimize the n u m b e r of bullets, t h a t hit him, while the hunter wants to maximize this same number. Second versioll ; T h e game is finished as soon as the r a b b i t is hit by a bullet, or at the latest at a fixed m a x i m u m final time T. T h e rabbit ~vants to minimize his expectation to be killed, while the h u n t e r wants to maximize it. 1.2
Dynamics
and
Notations.
We denote : 9 y~ E Z, 7Us position at time t, 9 ut E U = { - r , - r + 1 , . . . , + r } , 7~'s control at time t, 9 Yt = ( y l , y 2 , . . . , y t ) the sequence of t R ' s successive positions until time t ; S t the set of such sequences. 9 IfT~ is at y at time t, it will be on one of the 4 r + 1 possible positions : y - 2 r , y - 2 r + 1 , . . . , y + 2 r at time l + 2. For every y E Z, we denote V ( y ) = { y - 2r, y - 2r + 1 , . . . , y + 2r} ; with this definition,
746 if ~ is on y at t, vt E V(y) is the hunter's control. 9 Let y~ and y be two r a b b i t ' s successive positions, we define the set : V'(yo,,j) = {yo - 2 r , . . . , y - r - 1 , 7 . / + r + 1 , . . . , y o + 2r}. 9 At time t we denote z , the position the flying bullet is aiming at. T h e s t a t e of the game at time t is hence defined by the couple xt = (Y, z,), and its dynamics by the equations :
/
Yl
=
Y~+x = zl Zt+ 1
0
yt .-1-ut irrelevant
~
Y~ .
Furthermore we introduce the notations : 9 9 E" : the simplex of ~ ' , that is, the set of vectors ( v a , . . . , v , ) , such t h a t vi > 0 and ~-~vi = 1. i
9 We define the operator ~i, such t h a t for Y E S t, ~i(Y) is the element of the sequence Y whose rank is t - i. Particularly ~o(Y) and ~ ( Y ) stands for ~ ' s positions at time t and t - 1, if Y~ = Y. 9 for Y E S t-x and y ~ Z, Y.y is the sequence of S ~ made by concatenation of Y and y. 9 (It), and (| are the vector of lenght s : (1, 1 , . . . , 1)', and (0,0 . . . . ,0)' 9 Let M be a matrix, we use the notation M (~} for the matrix, with the same elements as M except the row n u m b e r u + r + 11, which is made up of ls. 9 Let A be a set, I](A) is the set of probability laws over A. 1.3
Observations,
controls,
and
strategies.
9 At every instant, the rabbit only knows the sequence of his own past positions, while the hunter knows the complete state, and particularly, the place where the flying bullet is aiming at. For a given time t and a s t a t e (Y,z), Tg's control will depend only on Y, while 7-{'s control may depend on Y a n d o n Z. In the first version of the game, it is i m p o r t a a t e to remark tllat r a b b i t doesn't know, whether or not he has been reached by a bullet. 9 We call "rabbit's mixed control" for a given time t and s t a t e Y E S t, a probability law p over U. Likewise we call "hunter's mixed control" a probability q over V(5o(Y)). 9 We call '*l~'s behavioral strategy" any sequence of maps, ~p~, ~ : 1 , . . . , T - 1 : = (~1 . . . . ,~T-1) with ~, : S ' - - - , 1 ] ( U ) : Z ~ - * v , ( Y ) 9 Likewise, we call "H's behavioral strategy" any sequence, ~b, of maps z,bt, t = 1 , . . . , T = (~bx,.'..,r
r
2 :
X t - - - * 1--~(V) : ( Y , z ) ~ - - ~ r
We will use the notations : r
= (~,(Y)(-r),...,~,(Y)(r)
)' E ~
9 ,(Y,z) = ( r - 2 r ) , . . . ,r + 2r) )' E 2 "~ ~ t ( Y ) = ( q 2 t ( Y , ~ l ( Y ) - 2 r ) , . . . , '.~t(Y, 5,(Y) + 2r) )' E (E'~) "~ 9 1.4
Payoff
and
optimal
strategy.
In tills part we transliterate, without any proof, the results proved in [2]. We note Vt(Y, z) a n d ~ ( Y ) the 7 / a n d 7~'s respective value fonctions for a given time t a n d state (]~ Z), y = ~0(] F) a n d Y a ~" ~1(]I) 9
747
9 Let (% r
be a couple of 7~ and ?-/'s optimal behavioral strategies, we have : V((Y, z)
and
~bt(Y)is an
=
max ~,~v y~ r qEFI~v(~)) ~ev(~)
argument of the above maximum, the map V~,=(Y,z') being defined by :
First version : v,,,(y,
z,)
=
Second version :
{
v,(Y, z,) if z # ~o(Y) 1 + V~(Y, z') else
v,,(Y, z,)
=
{ V((Y,z')if z # 5o(Y) 1 else.
'
9 On the other hand : ~(Y) =
rain ~ peI-I(U),ev(y.)
Rt[Y](z) ~, ~ p(u)V~+,.,(Y.y+ u,v)r ~eu ~ev(~)
~'t(Y) is an argument of the above minimum, where RdY](z ) is the a posteriori probability to have z~ = z, if we know the ?-/)s strategy. First version : TL~+I[Y.y+ u](z) = F.jev(y.)R,[Y](j)r Rdz](~) = r Second version: ( y =
5o(Y) and
y. =
5,(Y)) R,[Y](j)r
R,+,[Y.y + u](~)
= J~"r176 J+~+"
~_,
R,[YI(j)
jev(u~), d#~+.
adYl(=)
= ~,(o)(~)
Looking for the two players' optimal strategies is equivalent to compute in E" x (E'~) "~, the saddle points of bilinear maps whose matrices are defined by : Mt,z = ( ~I,.y,y.-2, I/~/,Y.uo-2,+' t " " I G.V.~o+:~ ) where
~?,,v,~ = R,[Y](z)M,,v,~ and (M,,r,A~,~ = V,+,,~(Y.v + ~ , , ) . In general we will note M instead of l~t,Y, and _M~ instead of Mt,y,~. A fixed point notion arises here : the values of R, that occur in matrix M, must be exactly those computed with the T/'s strategy that has been given by the saddle point computed with matrix/~I.
2 2.1
Main
results.
Assumption.s.
A s s u m p t i o n (A1) Let ~ = (~1 . . . . . ~T-~) be a Td's behavioral strategy. Let Y E S t-1 be a sequence of t - 1 Ti's positions, and let y = 5o(Y) be the last one, we denote for i = - r , . . . , r the row vector of lenght m : (0._=j0 ~ , ( Y . y + i ) ( - r ) . . . ~ , ( V . y + 0 ( r ) 9_~_q), i+r
2r+t
r--i
and NL[Y] the n x rn matrix made up of the row L{[Y]: ( L[~[Y] ) N,[Y] =
L[[Y]
L~[Y]
748 T h e n we s u p p o s e t h a t there exists a strategy, ~, such t h a t , for every t i m e t < T - 1, and every sequence Y E S t of r a b b i t ' s positions, we can write t h e e q u a l i t y i n / R '~ :
Assumption (A2j) r = (r CT-~) is a ~ ' s s t r a t e g y such t h a t : -for t r 1, a n d Y E S t a 7Z's sequence of positions, we h a v e :
~r -for t = i :
~I(0)'=( i
i )EZ'-
Assumption (A2a) r = (~bl,...,r is a 7-/'s s t r a t e g y such t h a t : - T h e m i x e d controls at t i m e t # 1 are :
~',(V, z)' = ( ~ ,
for z e V'($I(Y),6o(Y))
1 - 4ru,~)
2r
2r
2r+l
2r--1
v t ( Y , 6o(y) - ~ + i)' = ( ~ , '
1
i--1
q2t(Y, 6o(Y) + r ) ' = ~ , 1
for i = 1 , . . . , 2 r -
1
2r--l--i
2rq-1
- (2r - 1 ) # , ~ )
2r--1
, 2r+l
with 1
u=4-; - T h e c o n t r o l at t i m e t =
and u-
fit-A-,,
4~#, ' #,=
R,[Y](z).
is : %(0)=(--,
2,2
~
zEV'(u~,u)
1 rn
,..,
1 --)en'. m
Conclusions.
V e r s i o n 1 : U n d e r (A1 a n d (A2.1) w e h a v e : C1.1 : t h e a posteriori p r o b a b i l i t y m e a s u r e to have zt = z, given ~bt_l, is 1/m for every Y, z a n d t :
Rt[Y](z) = ~. C2a : t h e pair of TCs a n d 7-/'s strategies, (~, r w h e n it exists, is optimal. C3.1 : t h e value of Tr et 7~'s criterions are, for e v e r y t i m e : Vt(Y, z) = { :~(T~(T- it_- 1)1)+ ~ , ( Y ) ( z - y) ~(Y) = ~(Tt h e payoff is : I ( T
otherwise.if z - y E U, y = 6o(Y),
t).
- 2) .
rn
V e r s i o n 2 : U n d e r (An) a n d (A~a) w e h a v e : C1.~ - W h a t e v e r t h e sequence Y E S t o[ T/'s positions, Q ( Y ) = y, t h e a posteriori p r o b a b i l i t y measure to h a v e zt = z, given ~bt-l, satisfies : n , [ Y ] ( y - ~)
. . . . .
n , [ Y ] ( v + ~) = ~, ,
749
3.2 3.2.1
Conclusions Time
t = T-
(C2.2) a n d
(C~a).
1
For each 7r sequence of successive positions Y E S T-a, y = 5o(Y), a n d for each z E { y - r , r }, by a s s u m p t i o n we have : RT_,[Y](z) = a T - a ,
... , y +
Therefore every r a b b i t ' s control choosen at t i m e T - 1 is o p t i m a l ( t h e h u n t e r d o e s n ' t shoot any more at T - 1 ), a n d so ~ T - l ( Y ) is optimal. T h e n we h a v e : { ~ T _ , ( Y ) ( z -- y) i f z - y C { - r , . . . , r } V:r_a (Y, z) = 0 otherwise, and ff'T-,(Y) = aT-1 ~
VT_,(Y,z) = aT-1 9
zEy+U
3.2.2
Time
T-
(~ + 1).
P r o p o s i t i o n h a v e becn proved a t time T - 1, let's suppose it t r u e up to time t = T verify it a t t - l = T - ( s + l ) .
s, and let's
Let Y E S t-~ b e a Tg's positions sequence, y = 6o(Y) a n d y~ = 5~(Y) his positions at times t - 1 and t-2, w=y-y~ his control at t - 2 . Especially we suppose, t h a t 7~'s criterion can be w r i t t e n : j" p, + ~ . ~ , ( z ) ( ~ p~
v,(Y, Z)
- ,j)
if z - y e u otherwise.
Game matrix : Using t h e special f o r m of ~ ' s criterion, we h a v e : v,(Y.y - ~ , y - 2~) V~(Y.y-r+l,y-2r)
v , ( y . y - ~ , ~ - 2~ + 1) V~(Y.y-.r+l,y-2r+l)...
:
:
Vt(Y.y + r,y - 2r)
Vt(Y.y + r,y - 2r + 1)
...
v , ( y . y - ~,,j + 2~) V~(Y.y-r+l,y+2r) :
= p.(Jt)~•
...
Ve(Y.y + r,y + 2r)
+ ~.N,[Y],
a n d the g a m e m a t r i x M at time t - 1 and s t a t e Y is w r i t t e n : M = (fll,,M~,-2,I...Ifl,.~,,M~.-lic~M~-,[...Ic~My+~lfl,.+,,,+L,My+,.~t]...lfl~,,tM~,+~,)
,
where M. = ( p.(1).~
+ ~.N,[Y]
)(~) for each ~
= y + ~,
~ e U,
(This n o t a t i o n is defined in section 1.2). M , = p,(lt)~•
+ c~,N,[t;] for each z E V'(y~, y ) .
Optimality of r Let's prove t h a t ~ t - l ( Y ) is an a r g u m e n t ok" : maxQr r M Q, or similarly t h a t ~P,-I(Y, z) is a n a r g u m e n t of : maxQe~.= r M, Q , for all z E Y(y~).
750
Whateverz=y+u,
(#t-~(Y)'M:,
=
uEU: + cr, N~[Y] )(~)
~ , - a ( Y ) ' ( p,(1t),•
(| (|
--= 62t_1(Y)'(pn(TJ.)nxm Jr a,Nt[Y] +
(]L)',, - (p~(]l)~ + a,L~[Y])
(| (|
o'..._
= p,[~)'~'-"-+ - : ( I I )~, + ~o,_,(Y)(u)(1, - p , , . . . , 1 - p,,, 1 - p, - a _ , , . . . , 1 - p, - a~!. - p . . . . . ,1 - p.,)" m
~ r+u
r--u
W h e r e a, = cr,~,(Y.y + u)(i) is a positive integer. H e n c e ~ , ( Y , y + u) only has to weight the 2r elements of rank 1 to r + u and 3r + u + 2 to 4r + 1. ~Pt(Y; y + u)' m u s t look like : (X,
. . . x ,x 0 , ., . . 0 ,,. x , 2r+l
r+u
...
,
),
r--u
which is satisfied by formulas given by (A~.2). For each z e V'(y~, y) :
r
= +,_,(Z)'(p~(~).• + ~.N,[Y]) p , ( ~ ) k + --~(~)",
which d o e s n ' t i m p o s e any condition on ~e-x(Y, Z), Z E V'(y=, y). T h e s e c o m p u t a t i o n s p o i n t out the o p t i m a l i t y of r O p t i m a l i t y o f ~P~-1 Let's check t h a t 'I~t_x(Y) is an a r g u m e n t of : m i n , o e s . P'A/ffQ_a(Y). c o m p u t e Mqlt_I [Y]. We call c, t h e (u - r + 1) 'h c o o r d i n a t e of this p r o d u c t .
In order to do t h a t , let's
c. = ( R,_,[Y](yo - e~)A I --. I n , _ , [ Y ] ( y + ~ - 1)A i n , _ , ( y + , ) ( ~ ) " I
Rt_I[Y](y + u + 1)A I . - . ] Rt_t[Y](y~,.+ 2 r ) A ) @,_~(~") where A is a row vector :
A = ( Vt(Y.y + u , y Cu
= R~_l[Y](y+u)+
2r) . . . . . V,(Y.y + u,y + 2r) ), ~
= n,_,[Y](,j + ~,) + ~
V,(Y.y+u,v)
~
with y = 6o(Y). Rt_~[Y](z)r
v~(z.~ + ~ , , ) ~ , [ Y . y + q ( v ) (1 - n,_,[Y](y + ~))
and using t h e C1.2 result H- ( 1 -
a,-i)Z(Y.y + u)
=
oq_,
=
~,_, + (i - ~, _ , ) (p.+, +
=
P ~ + 2 "~- O Q - 1 0 " s - b 2 9
~,+,~,)
Since t h e e l e m e n t s of Mql~(Y) are all equal, any P minimizes P'MqI~(Y), ~ ( Y ) -- ~tc%+i + Ps+i -
and we chech t h a t :
751 where (a,,),, is a sequence defined by : 1 ai = 0, and a~+l = m(1 - as) " We notice that these values depend only on time. C~a -The couple of 77. and ?{'s strategies, (~, ~/,), when it exists, is optimal. Ca.~ -Let the numerical sequences {p~}~ and {(r,,}~ be defined by : 1 Pn+l
=
O'n+ 1
=
- - O ' n -[- fin 777, 1 -- Pn
Pl
=
0
O"1
=
1
,
the values of ~ and ~ ' s criterions, for each time, are given by : f PT-,+~T-*~o~(Y)(z--Y)
vdy, z )
ifz--yE
U
otherwise,
PT-t
~ ( Y ) = PT-'+I + O'T-t+iaz , the payoff is : PT-1
,
Additional notations: When any ambiguity on Y is possible, we will use the notations : at
=
R ~ [ Y ] ( y + u ) for each u E U
~,,, = _~,[r](~) i = i , . . . , 2 ~ fl,
=
i
-
ha,
,
m being the i *h element of the ordered (2r + 1)tuple : (yo - 2 r . . . . . y - r - 1, y + r + 1 . . . . . y~ + 2r),
yo = 5x(z). 2.3
Note
:
Assumption (Ax) is an equirepartition condition on rabbit's positions for each time. As a matter of fact, if ~. uses a strategy satisfying (At), if his position at time t is Y, he will be on one of the m positions {y - 2 r , . . . , y + 2r} two steps of time later, with an equal probability measure 1 / m . The bunter that shoots a bullet at time t, that will reach the line at time t + 2, doesn't think that rabbit will be on one of the m possible positions with a highest probability measure. He doesn't have any further information to help him shoot. The rabbit's choice of such a strategy may be interpreted as a will to reduce as much as possible the hunter's a priori information on his coming moves. It is natural to think, in case of zero sum games and particularly aon cooperative games, that the attempt to reduce the information available to his opponent, when it is possible, is a good tactics. Assumptions (A2.x) and (A2a) may be interpreted in the same way.
3
Proof
(Version
2).
We will only give the proof concerning the second version of the game, since the others rely upon the same ideas and computations is easier due to the simplest expression of R~[Y]. Let a It's strategy : T = (qOl. . . . ,~T-1) , satisfying (A,), and a ~ ' s strategy : ~ = ( ~ ' l , - . . , ~'T-2) , satisfying (A~a). First of all, we will prove ( C l a ) , then with this result, we will prove, using dynamic programmation, conclusions (C~.2) and (C3a).
752
3.1
Conclusion
Time
t = 2
(Ca.2).
R2[YI(z) = r
= __1 for all z E V(0) m
so (CL~) is t r u e a t t i m e 2, we suppose it t r u e for t > 2, a n d we w a n t to prove it at t i m e t + 1. Time t > 2 w h e n e v e r Y E S t, y = 6o(Y), we s u p p o s e t h a t R , [ Y ] ( y + u) = a, for each u E U. Using t h e filter expression t h a t c o n n e c t s p r o b a b i l i t y R at a t i m e t to n e x t t i m e t + 1, we can s t a t e : = (A"+"'+#~")~'+a'( 89
nt+~[Y.y+~,l(y+i)
1
,
for/=-2,',...,-2,2,...,2,"
at
-
ify+iEy+u+U
= (fl,,, + . . . R,+,[Y.W + u l ( y
-
1)
l~t+l[Y.y -~- u ] ( y
-~ 1)
=
+ #2r.,)u + a t ( 1 -- ( 2 r -- 1)#) , 1 - a,
(fll,t d~....~ ~2r,t)!2 .~_ a t ( 1 1 --
r~+,[Y.y + u](y)
= (A,, +...
+ &r,,)(1
-- (2". -- 1 ) # ) ,
ify-lCy+u+U
ify+lEy+u+U
o~ t
4"..)
-
i -- a t
E a c h of t h e s e values have to b e equal to
, so # a n d u m u s t satisfy the following linear
1 m(1-~,)
system :
a,( 89 + ~) + # , , = -
1 1
a,(1
-
A(1
-
(2". -
4".~) =
1)~)
1 -
-
m
+ Au
=
-m
.
an easy c o m p u t a t i o n makes t h e proof. Nevertheless, it r e m a i n s to check w h e t h e r , at each i n s t a n t , t h e control belongs to (Era) " , t h a t is to say t h a t # a n d u m u s t respectively stay in t h e intervals [0, 2~-a] a n d [0, ~ ] . T h e only difficulty is to verify t h a t u is positive, i.e. to prove t h a t fit > ~ , hence, a , < A.~. In order to d e m o n s t r a t e this last inequality, we first notice t h a t (at)terv is an increasing, recurrent m - ~ - 4m sequence, t h a t converges toward therefore 2m m -- , / m 2 -- 4rn 2m a n d on t h e o t h e r h a n d :. -- 4 m 4r <-2m -- n m n2(rn 2 -- 4 m ) > (St -- 4 r n -- n) 2 m
r
-- ~
r162 16r 3 - 1 4 r
2-4r-
1:> 0 .
T h i s last e q u a l i t y is t r u e for r > 1, (one needs only to consider successive derivatives), t h a t ends the proof of t h e first p a r t of the proposition. In t h e case r = 1, a detailed ( a n d c u m b e r s o m e ) analysis shows t h a t t h e r e d o e s n ' t exist a 7t% strategy, satisfying a s s u m p t i o n (A2.2), for each time t _< T - 2, as soon as T > 6.
753 We notice that ~,(Y) is equalizing : against it, any 7Us strategy gives the same value of ~ ( Y ) . More generaly this computation proves that all strategies satisfaying Ci.~ are equalizing. Computation
of V~_1(Y, z)
V~_t(Y, y - r) = r
M~,_,. ~,_~(Y, y - r) 0
( LIZ[)[y ] = r
0
P'(~):-
+
a,
i -- (2r -- i)# #
4:),it]
=
1 P~ i
1 ,r,-- + r m
=
P, (1G, + p,(1-~,_l(Y)(-r))
=
(--or, + p, + ~,_i(Y)(-r)(l -- p,)
=
p,+, + ~ , + ~ , _ ~ ( Y ) ( - ~ ) ,
+ ~,_l(Y)(-r)
m
and similarly :
Vt-l(Y,y+u) Vt_t(Y,y+i)
= p,+, +cr,+iqot-,(Y)(u) =
p,+,
for u E U fori=r+l
. . . . . 3r.
The proposition is therefore proved until time l = 1, that remains to deal with now. 3.2.3
timet=l=T-(T-1)
Let's notice that there is not flying bulle*, at this time. Game m a t r i x : M = pT_2(ll)~xm + O'T-2N2[O]. O p t i m a l i t y of Pl 1 M , l ( 0 ) = (pr-2 + --~T-2)(1)n ~Z hence (Ih(O) is an argument of :
mineez- P ' M ~ i ( O ) .
O p t i m a l i t y of r r
=
r
p~--~(~).x,,. + ~.r-~ N~[o] ) i
=
pr-,(~)",
m
hence qi(O) is an argument of :
maxoez~ ~i(O)'MQ
eayo~
and furthermore :
= ~(O)'M,~(o) =
PT-1
9
754
4
Conclusions
1- F e a s i b i l i t y of (A2.1) : The following simple strategy satisfies (A2.~) :
r
= (1/m,...,1/,,), rn
r
1 0
ifi=j otherwise.
foreachi,jE
{-2r,...,+2r}.
Observe that we have pure controls as soon as t _> 2. The fact that 7-{ used mixed control at t = 1 is enough to avoid giving more information to ~ . 2- F e a s i b i l i t y of (A2.2) : we have exhibited a ~ ' s strategy that satisfies (A2.2) for each r > 1, and whatever the duration of the game is. In the case r = 1, such a strategy exists only for a game duration, T, not superior to 5. 3- F e a s i b i l i t y of (A~) : We cannot raise a general method to study the existence of strategy satisfying (At). We have therefore showed that in case where r = 1, it is impossible to compute such a strategy as soon as T > 5 4- L o w e r b o u n d o f g a m e payoff: The impossibility to find (or nevertheIess to prove existence of), ~ ' s strategy satisfying (A1), doesn't make the above developpement useless. As a matter of fact, at each step of time, the ~ ' s chosen control being equMizing, the found value of the payoff exhibited is a lower bound when A1 cannot be satisfied. Futhermore we can see that this lower bound is swiftly increasing toward 1, and so the 7~'s strategy as not so bad.
Bibliography - [1]- T.Ba~ar, "On a class of zero-sum discrete games with delayed information", Lecture notes in control and information sciences 119. pp9-34, Springer, Berlin, t989. - [2]- P.Bernhard, "information and strategies in dynamic games 982, 1989.
", rapport
de recherche I.N.R.I.A.
- [3]- P.Bernhard et A.L.Colomb, "Saddle point conditions for class of stochastic dynamical games with imperfect information", IEEE trans, on Automatic control, AC 10-23, pp. 98-101, i98. - [4]- P.Bernhard, A.L.Colomb et G.Papavassilopoulos, "Rabbit and Hunter game : two discrete stochastic formulation", Comput. Math. Applic. 13, No 1-3, pp. 205-225, 1987. - [5]- A.L.Colomb, "l~tude de jeux k deux joueurs en information incomplete ", thb.se, Universi~5 de Provence, Marseille, France, 1986.
AN INTERACTIVE MULTIPLE CRITERIA DECISION SUPPORTING TOOL WITH APPLICATION TO A SIMPLIFIED REGIONAL DEVELOPMENT PROBLEM CAO DONG and M. INSTALLE Laboratoire d'Automatique, de Dynamique et d'Analyse des Systemes Bg.timent Maxwell, Place du Levant, 3 B-1348 Louvain-la-Neuve - BELGIUM
ABSTRACT In this paper, an interactive multiple criteria decision supporting algorithm is proposed which combines a Pareto solution determination method based on the reference point technique of Wierzbicki (1982) with the use of an interactively updated explicit utility function. This function which is recursively updated by the decision makers of the problem is used to represent their preferences among the various objectives of the problem to be solved. The maximization of the non linear utility function is done through an original procedure which works in a progressively enlarged optimization space of small dimension. This procedure makes the algorithm efficient even for large scale probiems. Finally, an application of the algorithm to a simplified regional development problem is presented.
INTRODUCTION The solution of regional developement problems asks usually for the consideration of many objectives among which some may be in conflict. Those objectives represent the interests of the actors or decision makers involved in the development of the region. In this paper, a decision supporting tool is proposed that implies - through an interactive procedure - those actors in a multiple criteria decision making (MCDM) procedure. In the past, interactive MCDM methods have been proposed which allow for the intervention of the DM in the course of the planning procedure in order to progressively identify their preference structure and find eventually a satisfycing compromise solution. Among those algorithms , the one developped by Rosenthal (1985) combines a Pareto solution determination method with the use of an explicit function which was developped by Keeney and Raiffa (1976) from global utility theory. In this algorithm, the explicit utility function is interactively and iteratively updated by the DM on the basis of the sequentially generated Pareto solutions and the associated trade-off informations. Hence, the practical problem of the "a priori" determination of the utility function representing the preference structure of the DM is avoided.
756 However, this algorithm suffers from an important drawback : since each time that the utility function has been updated, the corresponding Pareto set of decisions optimizing this utility function has to be determined, one has to solve a great number of non-linear constrained optimization problems. Hence, for practical dynamical problems like regional development ones with a constraints set of great order, the search for a satisficing compromise solution may be very time-consuming. In the second part of this paper, a new interactive algorithm is described which can be used for large scale MCDM problems with linear constraints and linear objective functions. At each iteration of this algorithm, an explicit utility function is first interactively updated through information given by the DM. This information consists in a minimum acceptable value and in a full satisficing value for each objective of the problem. Then, the maximization of this updated utility function provides a feasible solution vector in the criterion space. Finally, this solution vector is projected on the efficient surface through the reference point method developped by Wierzbicki (1982). The maximization of the utility function is done through an original optimization procedure. In the third part of this paper, the algorithm is applied to a simplified regional development planning problem. DESCRIPTION OF THE ALGORITHM A linear MCDM problem is represented as : max Z(x) = [ z 1(x), z2(x) ..... Zp(x)]T = [C1x,...CpX]T s.t x~X = { x / A x < b , x > _ o } [P.1] where x is an n-dimensional vector of decision variables, zi(x) is the ith objective function, A is a given m x n matrix and X is the feasible set of the constrained decisions. Generally, for this vectorial optimization problem, one cannot obtain a solution which maximizes simultaneously the p objectives because of the presence of conflicts. However, a often used solution to this problem is a Pareto or efficient solution. In the next sections, it will be shown how : 1) the DM interact with the optimization procedure through the recursive updating of a utility function representing their preference structure; 2) a reference solution Z is determined by maximizing the corresponding updated utility function; 3) a Pareto solution Z corresponding to the reference solution Z may be found; 4) the complete algorithm is organized.
757 Interactive updating of the utility function A well-known additive utility function is chosen in order to represent the preference structure of the DM "
u(z, ..zp)=
1-e I]~(zr-11)
=
u,
where ~i is the minimum acceptable value of the ith objective, 5 i is the full satisficing value of the ith objective and ~i < 0 is the preference coefficient parameter. ui is the "attribute" of the ith objective and takes values in [0, 1] for ~'i < zi -< 8i. This utility function is relevant with the general decision making principle: The more, the better for each objective and the choice of the exponential says that the smaller the objective value, the greater the increasing tendency of U(zl ..... Zp). For the interactive determination of the parameters describing the preference structure U(zl .... Zp), the following information was used : at the current Pareto solution, the DM are asked to give their "margin of allowance", that is, the ~ and 8i for each objective. The J3i are then automatically updated such that for the middle value of the allowance interval, ui has a value equal to (~ where 0.5 < ~ < 0.9.
Maximization of the utilitv function As stated before, the reference solution is obtained by maximizing the function U(Zl...Zp). To overcome the computation inefficiency caused by a constraint set (Ax ~ b) of great order with a given non-linear preference structure U(zl...Zp), a reference solution is first determined in the approximated criterion subspace formed by the convex combination of the sequentially generated Pareto solutions [z i, i = 1 ,...k and the extreme points (zJ, j = 1 ...p) obtained from the individual optimization of each objective. Hence, the problem is : max U(z 1...zp) X,z
s.t. Z= ~, ~7'+ i=l...k
and
~
~.i =1
[Umaxl] ~_, ~
:~i
j=l...p
X.~>0
i=l...k+p
where k is the iteration number. We call the solution of [Umaxl] : [z~. If this solution is identical to one of the former Pareto solutions, the dual information is then used in order to extend the criterion subspace through the cutting planes defined below. Hence, the new problem is 9
758
max U(z, ..... zp) z
p
p
i=1
i=1
[Umax2]
where PJi, ZJi are respectively the dual information and the Pareto solution obtained at the jth iteration (j = 1 ..... k). W e call the solution of [Umax2] : ~2.
Generation of the Pareto solutions AS written before, the reference point method is used to obtain the Pareto optimal solution corresponding to the solution
Z maximizing U(Zl,..., Zp).
This method is based on the
minimization of a generalized Tchebychev distance from a given reference point Z. : min Jr max 1:i(zi-z,) ] + (g,...,g)(z- z ) } z ~.Li=l...p s.t. Z--- [ClX ..... CpX]T xeX
[P.2]
or its equivalent problem " min {(z + (8,...,E) (Z - Z)} z.cx s.t. cc>'~i(Zi-Z i) (i=l,...p), z i - c i x = 0
[P.3] 0 = 1 .... p), A x ~ b
where cc is any real number, the weighting coefficient "q > o (i = 1,...P) and E is a very small positive value. The supplementary term (E.... ~) (~. [z) guarantees the efficiency of the solution found by [P.3]. Two theorems based on the duality theory are easily derived " Theorem l "The optimization gf [p.3J gives a Pareto solution Theorem 2 "The ootimal solution of [P,3] ootimizes at the same time the followina dual oroblem 9 P max - ,~.pizi s.t. z i = cix and Ax < b i-1
and where the variables #i, i = 1..... p are the dual variables corresponding to the objectives zi = cix, i = 1 ..... p. The geometrical interpretation of [P.3] is i~ustrated at the next figure for a problem with 2 objectives -
759 Z2 Cutting plane
Criterion set
Zl p
p
The equation - ~_,lAz~ = - ~ . ~ z i i-1 ~1 generates a cutting plane of the efficient subspace which pass through the efficient point [z.
The comolete algorithm The organization of the algorithm is as follows : steel : For i = 1 ..... p, each objective is optimized independently and a pay-off table with ~i is obtained. steo 2 : Given this pay-off information, the DM are asked to specify their margin of allowance "yi and 8i for each objective, k is set equal to zero. steo 3 : The problem [Umaxl] is solved and Zl is obtained.
This point is projected on the
efficient region by [P.3] in order to have an initial Pareto solution [z. steo 4 : k = k+l. The DM decide whether the last Pareto solution ~,k is satisficing. If yes, the problem is terminated. If no, the step 5 is executed. step 5 : The DM modify their "margin of allowance" % 8i in order to have a newly updated U (zl ..... Zp).
steo 6 : The problem.[Umaxl] is solved. If ~1 is different of ~k, Z.1 is projected on the efficient region by [P.3] in order to have a new Pareto solution ~k and the algorithm goes back to step 4. If ~lis equal to ~Tk,step 7 is executed. steo 7 : The problem [Umax2] is solved. If ~2 is equal to ~1, the algorithm goes back to step 5. IfZ 2 is different of Z1, the step 8 is executed. steo 8 : ~2 is projected on the feasible region through [P.3]. The efficient solution ~k is used to extend the criterion subspace. The algorithm goes back to step 6. The next figure illustrates the various steps of the algorithm 9
760 Z2
~
Z~
The convergence of the algorithm towards a solution maximizing a given U(zl ..... Zp) is guaranteed by the fact that the combination of [Umax2] with [P.31 is essentially a cutting plane method applied to a non linear optimization problem where the following requirements are satisfied : 1) the real polyhedral objective region {Z(x) s.t.xe X} is bounded; 2) the function U(zl,..., Zp) is quasiconcave (Luenbergher, 1984).
ASIMPLIFIED REGIONAL DEVELOPMENT PROBLEM A simpliefied regional development problem within a chinese context with a planning horizon of 6 years is considered where the production system is composed of 2 sectors : the agricultural sector and the rural industry sector which receives the raw materials from the agricultural sector and does a value-adding transformation. In the agricultural sector, two types of land are considered : type 1, irrigated flatland and type 2, dry flatland. By building the appropriate hydraulic installations, lands of type 2 can be transformed into lands of type 1. Five production activities can be performed as follows : Activity/land :
type 1
type 2
Soy beans
X
X
Rice
X
Fruits
X
X
where "X" indicates that on the land of type i the production activity j is possible. Finally it is supposed that the rural industry sector of the studied region absorbs all the harvested soybeans in order to transform them into a value-added, commercialized product.
Descriotion of the model Let Xij(k) be a state variable which represents the area of land of type i devoted to the jth activity during year k. Dynamic equations relating the state variable Xij(k) to the land allocation decisions are the following ones :
761 Xij(k+l) = Xij(k) + U~ij(k) - U~ij(k)
(E.1)
where the decision variables U+ij(k), U-ij(k) are respectively the area of land of type i added, substracted, at the kth year for the activity j. The transformation of the land of type 2 into that of type 1 is described by the following equations : X l ( k + l ) = X](k) + TR(k) X2(k+l) = X2(k) - TR(k)
(E.2)
where Xl(k), X2(k) are respectively the total area of land of type 1 and 2 and TR(k) is the area of land of type 2 transformed into land of type 1 at year k. The static constraints associated with the land allocations are the following ones :
EJ Xij(k) < Xi(k)
(i = 1.2)
(CA)
For the industrial sector, the assets of the rural industry are imbedded into a state variable Xr(k) with which the production capacity is associated. The dynamic equation describing the evolution of those assets is as follows : X,(k+l) = o~X,(k) + U,(k)
(E.3)
where Ur(k) is the investement level in the industry at year k and CCris a redemption factor. Directly associated with this equation is the limit of fabrication capacity of the industry : ~11X11(k) + J321X21(k) -< PrXr(k)
(C.2)
where ~r is the fabrication yield (Ton/kS) and ~}11, ~21 are agriculture yields(Ton/Ha). Finally, the dynamic equation for the regional cash-flow is the following : Xa(k+l ) = asXB(k) + ,7_.~oijXij(k)- 7~rTR(k) - Ur(k) - Up(k)
(E.4)
where Xa(k) is the available cash-flow at year k, ZPijXij(k) is the net profit obtained from the production activities Xij(k), ~rTR(k) is the cost for the land improvement, Ur(k) is the investment in the industry, Up(k) is the budget for public welfare (the income budget for the workers is included in this term) and cq3 is a money erosion factor. Other static constraints are the following ones : non-negativity of the state and decision variables. - minimum quantity of produced rice : -
~12X12(k) > 4Q(k)
(C.3) (C.4)
762 where Q(k) is the quantity of rice necessary for the population at year k (prediction). - necessity of preinvestment in production activities : ~,'~jXij(k) + 7trTR(k) + Ur(k) < XB(k) (C.5) where "~1jXij(k)are costs of production. limit on available man-power for production : 7_,~ijXij(k) + ~rTR(k) ~ MP(k) (C.6) where =r,/qj " man-hours/Ha and MP(k) is the predicted available man-power on year k. limit on available water for irrigated crops : -
-
3
(C.7)
~__,~Xij(k) -<Wa(k) j=l
where % is the water demand coefficient for culture j (M3/Ha) and Wa(k) is the prediction of available water resource at year k. constraint that guarantees the continuity of the production at the end of the planning period of 6 years: XB(6) -> XB(5) (C.8) -
Definition of the ob!ective~ In this regional development problem, the planning horizon is divided into 2 development stages of 3 years each. At each development stage, 3 objectives are considered which represent the different interests of involved parties in this production system : objective 1 maximize the total profit of production; objective 2 - maximize the budget for public welfare and objective 3 - minimize the erosion caused by the production activities. At each development stage, these objectives are measured by their average over the years covered by the stage. Hence, the problem contains 6 objectives given by the following expressions : 3
6
objl" max k__~l{ ~,RjXij(k)t/3 ij ~
obj4 9max ~ , t ~__,PijXij(k)l/3 k=4~ ij J
2
5
obj2" max~__,{Up(k)}/3 k=0
obj5 : max ~_.~{Up(k)}/3
3
6
obj3 " min ~__~f~/___,eijXii(k)l/3 k=lL
ij
J
obj6 " min ~/_.,I ~_~eijXfi(k)t/3 I:~--4L ij
J
In these expressions, the eij, i = 1,2; j = 1, 2, 3, are erosion coefficients which may be given, for example, in M 3 of soil removed per hectarea of land of type i with an activity j.
763 Solution of the oroblern The above problem was resolved on a PC/AT computer with 640 KB memory. First, a pay-off table was computed :
Z1
Z2
Z3
Z4
Z5
Z6
max Zl
18700
8580
-5550
5910
9110
-1770
max Z2
12800
14800
-3830
5910
2840
-1770
max.Z3
5380
7430
-1610
5930
2880
-1780
max.Z4
18700
5400
-5540
26400
8970
-7470
max.Z5
17400
5400
-5220
16600
24000
-5010
max.Z6
16300
5400
-4910
5750
16100
-1730
The diagonal elements of the above table are the maximum values of each objective and were chosen as the initial full satisficing values 5i for each objective. Furthermore, the initial minimum acceptable value of each objective were chosen as follows : Y1 = 10000; )'2=7000; )'3 = - 5500; )'4= 11500; )'5 = 10000; )'6= -6000. Then, three iterations were made interactively in order to obtain a psychologically satisficing Pareto solution. The table at the next page gives, for each iteration, the margins of allowance that have been modified by the DM as well as the reference and Pareto solutions. The underlying course of actions giving rise to the last, satisficing solution is the following one: at the first stage, because of the budget limitations, the main production activity consists in fruit production which doesn't need great investment. Gradually, the investment is put in the transformation of land of type 2 into that of type 1. On the transformed land, the most profitable production activity, that is, soybeans is taken.
CONCLUSION In this paper, an interactive multiple criteria decision supporting algorithm and its application to a simplified regional development problem have been presented. The reduced dimension of the non-linear problems [Umaxl] and [Umax2] makes it possible to work in an interactive mode even for a large scale linear problem. This interactive mode was implemented through the use of an adjustable utility function which represents progressively an approximation of the preference structure of the decision makers.
764 obj.1
obj.2
obj.3
obj.4
obj.5
obj.6
51
18700
24000
-1730
10000 16400 16800
-1610 -5500
26400
Y1 z1 zl
14800 7000
11500
10000
-6000
8070 832O
-4910 -4810
16300 16800
13800 16200
-4770 -4620
i
52
t e r 2
72 z2 z2
id. id. 17200 17200
id. id. 7710 7720
id. id. -4960 -4960
id. 18000 18800 18900
20000 id 14700 16300
id. id. -5210 -5210
83 "t'3 z3 z3
id. id. 17500 17500
id. id.
id. id.
id. 20000
id. id.
id. id.
7210 7210
-5080 -508O
20500 20500
14700 16400
-5700 -5700
~t e r 1
t e Ir 3
REFERENCES Dyer,J.S. (1973) A time-sharing computer program for the solution of multiple criteria problems. Management Sci., 19, 1369-82. Keeney, R.L. and H. Raiffa (1976) Decision with multiple objective : preference and value trade-off. John Wiley, New York. Luenbergher, D.G., (1'984) Linear and non linear programming, Addison-Wesley, Reading, M.A. Rosenthal (1985) Concepts, theory and techniques : principles of multiobjective optimization. Decision Sciences 16, 133-152. Wierzbicki, A.Po (1982) A mathematical basis for satisficing decision. Mathematical Modeling, 3, 391-405.
CRONE CONTROL : PRINCIPLE, SYNTtlESIS, PERFORMANCES WITIt NONLINEARITIES AND ROBUSTNESS-INPUT IMMUNITY DILEMMA A. OUSTALOUP Equipe Systrmes et Commande d'Ordre Non Entier L.A.R.F.R.A. - ENSERB - Universitd de Bordeaux I 351, cours de la Librration - 33405 Talence Cedex Abstract The area of this paper concerns the robustness of stability degree, and more particularly the robustness of the damping of the control versus the parmneters of the plant. The approach of the CRONE Control is presented as resulting from the non integer order differential equation which represents the dynamic model directing a natural robust relaxation. A frequency illustration of robustness is given in the Nickol-Black plane through an open loop frequency template. Such a template is synthesized by means of a CRONE variable phase regulator from the phase diagram of an agricultural mobile robot using a highly non linear error detector. The corresponding robustness performances are presented. At last, a dilemma is established by considering two templates of different lenghts. The imput ot the plant is given for step responses to the reference input. It shows that the greater the robustness, file lower the input immunity. 1 - Introduction
Robustness is a very wide concept, even in a same domain such as the automatic control one. In fact, robustness is a notion which always translates the same idea, namely insensitivity. In automatic control, it is frequent to consider the robustness of stability (case of Hod approach). In the CRONE Control, a French abbrevation of "Commande Robuste d'Ordre Non Entier", namely "Non Integer Order Robust Control", the robustness considered is much stricter, that is to say the robustness of stability degree. More precisely, the robustness which is at stake translates the insensitivity of the damping factor or the stability degree of the control to the parameters of the plant. That amounts to saying that the robustness considered is the robustness of the control damping versus the plant parameters. A frequency illustration of damping robustness is deduced from the non integer order differential equation which represents the dynamic model governing a natural robust relaxation, that of water on a porous dyke. Particularly, in the Nichol-Black's plane, robustness is illustrated by an open loop frequency response locus which is reduced to a vertical straight line segment of abscissa between - x / 2 and - x around the axis OdB. This segment, defined for the nominal parametric state of the plant, is called open loop frequency template (or more simply template ). In this paper, the template is synthesized by means of a CRONE variable phase regulator from the phase diagram of a mobile robot using a highly non linear error detector. The performances obtained through a numerical simulation show that the control is robust, not only versus the open loop gain, but also versus the non-linearity of the error detector, whether the non-linearity is intrepreted as an increasing gain or a decreasing gain. In order to study the price of robustness, the last part of the paper deals with the robustness-input immunity dilemma. For two different lengths of the template, the noted performances reveal that the input of the plant is higher in the case when the template is longer and consequently when the robustness is higher. 2- From strategy
t h e r o b u s t n e s s o f s t a b i l i t y d e g r e e in n a t u r e : The non integer approach of the C R O N E Control.
2.1- O b s e r v a t i o n
to a n e w r o b u s t
control
of a natural robust relaxation
Our approach, the aim of which is the conception and the application of a new robust control strategy, makes use of the observation of a natural relaxation, that of water on a porous dyke. Already in the 17th century, the constructors of dykes had noted the damping properties of the very disturbed dykes and particularly those forming air pockets which can be compressed by the advance of water. Otherwise, an attentive observation of the relaxation of water on fluvial or coastal dykes, reveals that in the case of very damping (or absorbing) dykes through a porous volurnic structure and a rough surfacic structure : - the natural frequency of the relaxation is different whether the dyke isfluvial or coo.~tal ; - the damping of the relaxation seems to be independent of the dyke, whether it isfluvial or coastal.
768 Given that the fluvial and coastal tests can be distinguished by very different carried water masses, the observation seems to show that the relaxation is characterized by a natural frequency which depends on the motion water mass and by a damping which is independent of it. Although it should be paradoxical in the integer approach of mechanics where any relaxation presents a damping linked to the carried mass, this result reveals the insensitivity of the damping factor to a parameter, in this case the motion water mass. So, it translates the robustness of the relaxation phenomenon as for stability degree. 2.2- Non integer o r d e r differential equation as a dynamic model governing the relaxation After trying to determine the mathematical origin of the robustness of such a phenomenon, it appears that it resides in non integer derivation. Indeed, by taking into account the fractality of porosity and the corresponding recursivity, we show [5] that the relaxation satisfies a linear differential equation of non integer order n' between 1 and 2, namely : zn'(d~tt)n'p(t) + P(t) = 0 ;
(1)
P(t) designates the dynamic pressure at the water-dyke interface ; x is a transitional time constant which is a function of the water mass. 2.3- F r e q u e n c y t e m p l a t e The differential equation (1) leads to a symbolic equation of the form : ('r
+ P(S) = O,
(2)
from which one draws :
(3) This operational equation is translated by the fonctional diagram shown in figure 1. Because of a unit feedback, the direct chain determines an open loop transmittance of the form : ~(s) = ( 1 ) n' ,
(4)
which is the transmittance of a non integer integrator and which defines an open loop frequency response of the form :
in which mu = 1/'c designates the unit gain frequency. Given that arg ~3(jm) = -n' ~/2 with 1< n'< 2, the black locus of [3(jr~) is a vertical straight line of abscissa between - ~/2 and -~.
Figure 1 - Functional diagram defining an open loop transfer Physically, a water-dyke interface cannot be characterized by a non integer derivation in all frequency domain [5]. In fact such a derivation is limited to a range of medium frequencies. So, the vertical straight line so defined is reduced to a vertical straight line segment (fig. 2). This segment is called open loop frequency template (or more simply template). When the water mass M changes, the frequency cou is modified in conformity with the relation C0u = (K / M) 1/n', (6)
769
where K is a constant depending on the water-dyke interface. That amounts to saying that the template so defined slides on itself at the time of a variation of the water mass. Such a vertical displacement of the template insures the constance of the phase margin and, consequently, the invariance of the corresponding damping factor in time domain, so translating the robustness of the damping.The longer the template, the greater the robustness.
A
-7~
I I
I ~(j~)l dB
I I
0~u I
0 dB
-n'rd2 -r~ / 2
0
arg ~(jf.0)
I
-%-
t I
Figure 2 - Illustration of robustness in the Black plane: theform aM the vertical sliding of the template AB insure the robustness of tt,e damping through a constant phase margin ~)rn 2.4- The template in automatic control In automatic control, the aim is to obtain such a behavior, namely : - an open loop frequency response Black locus which forms the template so defined for the nominal parametric state of the plant ; - a vertical sliding of the template at the ume of a reparametration of the plant. Trying to find the synthesis of such a template defines the approach (said non integer) that the CRONE control uses.
3- Synthesis
of
the template
3.1- I n t r o d u c t i o n Although several synthesis methods of the template exist, here, we simply present the method which corresponds to the application considered in this paper, namely the wire guidance of a mobile robot. Let 13(jr and G(j~) be the open loop frequency response of the control and the frequency response of the plant. We assume that the phase diagram of G(jo) (fig. 3) is invariant, which is the case when only the variations of gain are considered (case of our application).
0i
argGfjr
argl3(j(o)
I .n t~/"2 -
I .
/D
.
~
Figure 3 - Synthesis of the template from the phase diagram of the plant.
770 To pass from the argument of G(jc0) to the argument of 13(jco) for coA < co < C~ the observation of figure 3 shows that the regulator CN(Jco) placed in cascade with the plant must provide both a phase delay and a phase advance which should be functions of frequency. 3.2- Idea of the synthesis : smoothing of crenels with variable cyclical ratio One considers that the phase diagram of the regulator results from a smoothing of crenels constituting a phase asymptotic diagram. So, one must introduce the notion of phase smoothing curve which replaces thephase smoothing straight line one usually used in the non integer approach. In order to obtain a smoothing whose value should be a function of frequency, the idea is to vary locally the cyclical ratio of the crenels. This is translated by the consideration of crenels whose cyclical ratio is a function of their ranks (fig. 4). The distribution of the zeros and the poles is defined by the relations:
ai = coi/co'i and with
rli = co'i+I/coi,
cq rli = 0~'i+l/co'i = cte > I
(7)
(8)
k/i,
where a i andrli are called recurrent factors. phase smoothing curve
OJl
r
0,~
0)2
0)2
0)'i
0)i
0)5+1
_L__2_L2__0)N 0)'N-t
CON-I CON fOB
Figure 4 - Obtaining of a phase smoothing curve through a modulation of the cyclical ratio of the crenels of a phase asymptotic diagram. 3.3
-
Synthesis
process
The synthesis process successively consists : 1 - in fixing a recursive distribution.of the zeros co'i, namely co'i+l/co'i = oti Tli = cte V i ;
(9)
2 - in considering as data, the values of the phase that the regulator must provide at the frequencies co'i, namely r = arg CN(.jco'i)= (Pd - arg G(jr
(10)
where (Pd = -n' n/2 is the open loop desired phase that the phase locking defines around cou ; 3 - in determining the distribution of the poles, that is to say that of the recurrent factors a i given that ai = coi/co'i. 3.4 - Exact algebraic method The algorithm which results from this exact method is that used by the CRONE software. A regulator frequency response in conformity with the phase asymptotic diagram shown in figure 4, is defined by an expression of the form : (1 + jco / co'l).-.(1 + jco / co'i)...(1 + jco / co'N)
CNOco) = Co
,
(I I)
(1 +ji0/col)...(1 + jco/coi)...(1 +JCO/CON) the corresponding argument satisfying the relation : N
N
arg CN(jco) = ]L tan -1 co/ co'i - ~] tan'l co/ coi. i=l i=l
(12)
771 For the frequency ~ = t0'i, which suggests to replace the index i by the new index j, the phase of the regulator becomes : N N arg CN(Jm'i)= ]~ tan "1 o ' i / 0 ; j - ~ tan "I to'i/o~j, j=l j=l
(13)
from where one draws, taking into account (10) : N
t a n ' l to'i/c0j = A(o'i),
(14)
j=l in putting : N
(15)
A(to'i) = ~ tan -1 ~o'i/o'j - q~d + arg G(j0~'i). j=l
So, one obtains the system of equations : tan -1 o' / 0h + tan "l o' 1 / co2 + ... + tan "I to'l / ~ tan -I 0;2/ro I + tan-1
o'2/a,, 2 + ... +
= A(to'l)
tan-I t 0 ' 2 / o N
= A(to'2)
tan- 1 ~'N / Ol + tan" 1 o, N / c02 + "-- + tan" 1 to,N / m N = A(to'N),
(16)
which constitutes a non linear system of N equations, the N unknowns of which are the toi. One puts ct0ai = C and xi=l/c q. By taking the tangente of each m e m b e r of each equation, the line of rank i admits the reduced writing : N
tan (]~ tan -I aij) = tan A(m'i) = A i with aij = Ci-Jxj,
(17)
j=l
of which one tries to determine the expression of the first member, namely: N PN q~ tan ( • tan -1 aij) = Z (-1)P s2p+l(i) / Z (-1)q S2q(i), j=l p=0 q=o
(i8)
expression in which the different magnitudes are defined as follows : PN = IN - 1 / 2], integer part of N - 1 / 2 ; qN = [N / 2], integer part of N / 2 ; S0(i) = 1 ; Sl(i) = ~ aijl;
Jt I~jl-
(19)
772 T h e sums of orders 0, 1, 2 ..... 1 corresponding ot the line i, namely s0(i), sl(i), s2 (i) ..... sl(i), are the "elementary symmetrical functions of the roots" introduced by Viete; the sum of order 1, sl(i), is the Viete function of rank 1. From the calculations already published [9], relation (18) can be written: n
(-I)P C(2P+I) (i-l)s29+i(1) _ Ai ~ (-l)q cT"q (i-l)S2q(1) = Ai p=o q=l
(20)
which can bc broken down by developing term by term, line by line : Sl(l) + A 1 s2(1) - s3(I) - A 1 s4(I) + s5(I) + A 1 s6(1) -.
.....+ (-I)PN S2p+1(1) - A I (-l)qN S2qN (I) = A I
C Sl(l) + A z C 2 s2 (1) - C 3 s3 (1) - A 2 C 4 $4(I) + C 5 $5(I) + A 2 C 6 $6(1) - . ..... + (-1)P N C2pN+ 1 S-2pN+I(1) - A 2 (-1)q N SZqN(O= A 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
=
. . . . . .
CN-1 S1(1) + A N C2(N-t) szO) - C3(N-1) $3(1) - A N C4(N-l) $4(1) + C5(N-1) S5(1) + AN C6(N-1) S6(1) _ . ..... + (_l)PN C(2pN+I) (N-l) S2pN+I(I) _ AN (.1)qN C2qN (N-l) S~N(1) = AN . Finally, given that if 2pN+l = N - 1, 2qN = N and if 2qN = N - 1, 2pn+l = N,
(21)
(22)
one obtains a linear system of N equations, the N u n k n o w n s of which are the sums of orders 1 to N corresponding to the first line of (16), namely the Viete functions slO), sz(1) ..... SN0) ; in fact, these ones constitute the new u n k n o w n s in relation to which the non linear system (16) has been linearized. The solving o f this system providing the values of these unknowns, it only remains from these ones to express the initial u n k n o w n s xl, x 2 ..... x N, in order to then go back to the genuine u n k n o w n s cq, ctz ..... a N, given that aj = 1/xj. st(l), s2 (I) ..... SN(I) being the elementary symmetrical functions o f the roots Xl~ x2]C, .... XN/C N ' I , let us p u t Yl = Xl, Y2 = x2/C ..... YN = XN/CN'I, in which case YI, )'2 ..... YN are the solutions of the polynomial equation of degree N : N
yN + ~ (_l)i si(l) yN-i = 0, i=1
(23)
Si(l') = Z Yjl Yj2 "'" Yji" Jl <J2 <''" <Jl l<jk
(24)
in which
The solving of this equation making it possible to have the yj defined by yj = xj]C i-l,
(25)
one easily deduces the xj, namely
xj = cJ -I yj,
(26)
then finally the otj: aj = l/xj = 1/cJ -1 yj.
(27)
773
4 - Application to the wire guidance of a mobile robot 4.1 - Plant to be controlled
The study plant to be controlled is an agricultural mobile robot which collects saffron and is directed by a real time local area network [8]. It is made up of a trolley which spans the rows of saffron flowers, and is fitted with two motor wheels behind and two multidirectional free wheels in front. It is automatically guided by a conducting wire which is buried in the middle of each row by the farmer. An electromagnetic signal is transmitted at the back by a transmitter. The signal is received by the conducting wire which acts as an antenna. Two receiving coils placed in front of the robot receive the signal and generate a differential speed reference input which makes it possible for the robot to return to the centre of the row. This wire guidance device with passive support continues to work even if the buried wire is interrupted in several places. We have carried out a frequency indentification of the robot through an harmonic analysis between the differential voltage applied to the motors and the position of the robot in relation to the wire. Such an identification has permitted us to obtain the phase diagram shown in figure 9. 4.2 - N o n - l i n e a r i t y of the error detector m o t i v a t i n g the use of a C R O N E r e g u l a t o r
In a hostile environment, the detector of the error signal e delivers a signal V, the variation of which is a non linear function ofe which is not widely known. A characteristic example is that of the wire guidance of an agricultural robot : indeed, the wire, resting on the soil, follows more or less the irregularities of the soil, so that the distance between the wire and the coils of the electromagnetic captor varies at random, thus inducing a random variation of the sentitivity of the error detector.There are two phenomena which simultaneously affect the transfer of the error detector : the first one is the non linear distorsion noted for the high values of e (curve in S) and due to a large asymmetry of the wire-coil configuration ; the second one is the random variation of sensitivity which is translated by a random variation of the slope at the origin, thus leading to a graphic representation of V(e) by a family of curves in S, of which it is impossible to specify which one should be taken into account at a given time. It is well known that the nature of these two phenomena allows us to interprete the error detector as a linear highly non stationary device, the instantaneous gain (A(t) = V(t)/E(t)) of which varies as fast as the variables. Given the mbusmess performances of the CRONE control, even if the parameters of the plant vary as fast as the variables, the consideration of such a non-stationarity is sufficient for justifying the synthesis of a template in open loop. The corresponding functional diagram of the control is shown in figure 5. The equivalent amplifier A(t) is highly non stationary. The features of the regulator CN(jO) are given in paragraph 4.3. EQUWALENT AMPLIFIER
I
CRONE REGULATOR
PLANT
-I
-I
--
Figure 5- Block-diagram of the control : e(O is the reference input equal to zero ; s(t) is the position of the robot in relation to the conducting wire. 4.3- S y n t h e s i s o f the r e g u l a t o r
In order to synthesize two templates of different lengths, we have used the CRONE software to synthesize two regulators covering two different frequency ranges. These are described by :
r1+JL)(1 +6)(1 1,256
Cm(s) = 50 ~
+__s_
+s 2 s(1 + 0 . 6 s 6.25) (1 + ' i - ~ )
6.28)
(28)
774 + __s__ 1 +s~_) andCNz,(s)=170(l+0.--~281(l+l.--~57)( 1 2.5)( +4.-~)(I+~)(I+TSo0)( 1 20 (29) s
(1+0.95
s
+ 1s2 + 0 "/(3 1 . 5 8 /
s
+~)/1+1-~2)(119.36 '
+
s 800)
In each transmittance, the highest pole is introduced to limit the gain at high frequencies. The static gains of the regulators insure an open loop unit gain frequency equal to 0.7 Hz. 4.4- Performances obtained through a n u m e r i c a l simulation A simulation has been carried out from the regulator of transmittance CN2(S). As tar as the non linearity described in paragraph 4.2, passing from a curve in S to another one can be modelized by a generalized Duffing's function defined as follows. For the linear area, namely I E I g b with o < b -< 1 : V( e ) -- a ~ with a > o. For the non linear area, namely b < I ~ I < 1 : V( e ) = ( (x + ye2) ~ with V(b) = ab and V(1) = 1 then V(-b) = -ab and V(-1), which entails the relations : (x= a - b2 a n d y = l - a 1 - b2
1 - b2
(30)
Figure 7 presents the step responses to the reference input for a = 0.2 and b = 0.6 (figure 6.a) and a = 5 and b = 0.6 (figure 6.b). The robustness versus such non-linearities is remarkable since the first overshoot remains practically constant despite the importance of the non-linearities.
/
/
0.5
(a)
/ 0
(b)
-2
-t -t
-0.5
0.5
\,
/
\ /
0 -!
-0.5
/
/
-3
/
, ',\
-i
/
-0.5
0.5
t
Figure 6 - Non-linearities used in the simulation i.Si
1
0.5
M Figure 7 - Unit step responses o f the CRONE Control : non-linearity (a) ; - -
- with the non-linearity (b)
without non-linearity ; - -
-- with the
775
5-Robustness-input
immunity
dilemma
5.1- I n p u t s e n s i t i v i t y in t r a c k i n g a n d in regulation
Let us consider the funtional diagram shown in figure 5, in wich we intoduce a disturbance P(t) and a noise b(t) at the output of the plant. Moreover, we consider that A(t) = 1 for separating the problems and so simplifying the study. One can then express the following frequency responses (given that ~ (jco) = CNfjco)G(jco)) :
u(jco)]
= CN(jco)
E(JCO)J[~.~))--=~
1 + [30(o)
-u(jco) --] = - CN(jm) LP(Jco)+ BGCO)JE~)= o 1 + ~Gco)
and[
(31)
The input sensitivity in tracking and in regulation can be defined as the modulus of these frequency responses, namely :
S ( c o ) U(jco) ~(j)=
=
UGco)
= ~ ;
P m o P(jco)+ B(jco) L(jco) = 0 EG~)= 0 1~ (co) defines the input immunity. For agcou << 1, I 13(jco) ] being then great in comparison with unit, S(co) is reduced to : S (co)= ~
=IGOr],
(32)
(33)
namely in dB (fig. 8 ) : S (co) dB = - [ G(jco~ dB. For co/cou>> 1, I [3(.]o)) I being then small in comparison with unit, S (co) is reduced to :
(34)
namely in dB (fig. 8 ) :
(36)
S (co)dB--I G(jco) I dB = -I GGco)I dg + I ~Gco) [dB.
The intersection of the gain diagrams defined by (34) and (36) is obtained at the frequency cou, since i CNGcou) I I G(,jcou) t = 1 from the definition of cow In view of figure 8, it appears that the curve of S(CO) is successively imposed by : - the plant at the frequencies inferior to o.~ ; - the regulator at the frequencies superior to cou. Moreover, this figure shows that the input sensitivity is higher and consequently lower the immunity in the case when the template is longer. This phenomenon clearly reveals the robustness-input immunity dilemma.
776
sg0)
i. o/%,>>I /
o~/o~u<
[ CN~m)~dB ~ N
[
I
: ~
(a)
I
'( a _ _ _ ) + ( b )
'- / ~ ~ 1 " i
I (a)+(c)
0dB
,
Figure 8 - Frequency variation of the input sensitivity in tractdng and in regulation: A iB l : template (1); A2B2 (A2B2>AIB1): template (2); along AIB 1 or A2B2 , [ flO'co)[ dB= (cou/co)n' dB; along C1DI Icu6oo)l aB +lcoo )l dB. or C2D2, I oco)l
5.2. Performances obtained through a numerical simulation A simulation has been carried out with the regulators defined by relations (28) and (29), without considering the non-linearities of the error detector. Figure 9 gives the open loop Bode diagrams corresponding to the two regulators. It shows that the template obtained with the regulator CN2(S) is longer than that obtained with the regulator CNI(S). Figure 10 gives the variations of the input of the plant corresponding to the step responses of the control successively obtained with the regulators C N 1(s) and CN2(S). It appears that the input is greater for the regulator CN2(S). I1
lit
1era
et
Ii
I1
d II
,Q--~
-1~ d B
-ZT~
-2~
x \,b
i
111-'t
-1111
i
Illlll
I
1~ - Z
I
tlllll
I
llI -I
I
t[llll
I
I0 R
I
--368
Iltlltl
1D I
1B Z
11t]
Frcquencg [112:]
Figure 9 - Bode diagrams of the plant and of the open loop control ." gain; ..... phase ;(a) plant ; (b) plant corrected by the regulator CNI(S) ; (c) plant corrected by the regulator CN2(S)
777
0 r.s
0 1.Sf
t ,tSB
1.21
m ~8~
.91
t
.6~ .31
1~
81 -li]~
,
1
-..J~
I
.1 Ti~e(~)
II
.81
.82
Tiae(~)
Figure 10 - Variations of the input of the plant corresponding to the unit step responses of the control obtained with the regulator CN1(s)(a) and the regulator CN2(s)(b ) : the initial value is much greater with CN2(S)
6. C o n c l u s i o n The approach of the CRONE control arises from the non integer order differential equation which represents the dynamic model directing a natural robust relaxation, that of water on a porous dyke. A frequency illustration of robustness is deduced from such an equation. Particularly, in the Nichol-Black's plane, robustness is illustrated by an opeu loop frequency response locus which is reduced to a vertical straight line segment of abscissa between - re/2 and - ~ around the axis OdB. The template that this segment defines, slides on itself when the parametric state of the plant changes. In this paper, the template is synthesized by means of a CRONE variable phase regulator from the phase diagram of a mobile robot using a highly non linear error detector. The performances obtained through a numerical simulation show that the control is robust, not only versus the open loop gain, but also versus the non-linearities. A new dilemma, the robustness-input immunity dilemma is established by taking into account two templates of different lengths. The input of the plant is given for step responses to the reference input. It indeed reveals that the greater the robustness, the lower the input immunity, since the initial value of the input varies by an important factor when the template lengtht increases towards the high frequencies. References [I] - I. Horowitz and M. Sidi - Synthesis of feedback systems with large plant ignorance for prescribed time domain tolerances. Int. I. Control, vol 16, n~ 287-309, 1972 [2] - A. Oustaloup - Linear feedback control systems of fractional order between 1 and 2.IEEE Int. Symposium on Circuits and Systems, Chicago (USA), April 27-29, 1981 [3] - A. Oustaloup - Syst~mes asservis linraires d'ordre fractionnaire : Th6orie et Pratique - Ed. Masson, Paris, 1983 [ 4 ] - A. O u s t a l o u p and B. B e r g e o n - Frequency space synthesis of a robust dynamic command.IFAC'87, 10th World Congress on Automatic Control, Munich (FRG), July 27-31, 1987 [5] - A. O u s t a l o u p - From fractality to non integer derivation: a fundamental idea for a new process control strategy.8th Int. Conf. "Analysis and optimization of systems".INRIA, Antibes (FRANCE), June 810, 1988 [6] - A. Oustaloup - From fractality to non integer derivation through recursivity, a property common to these two concepts.Survey, session "Fractality and non integer derivation", 12th IMACS World Congress on Scientific Computation, Paris, July 18-22, 1988 [7] - A. Oustaloup - From the robustness of stability degree in nature to the control of highly non linear manipulators.Proceedings edited by Springer Verlag, Colloque Int. CNRS "Automatique Non Linraire", Nantes (France), 13-17 Juin, 1988 [8] - P. Baylou - L'automatisation de la cueillette du safran.Production des Plantes Aromatiques et Mrdicinales. Nions (France),Octobre 1988, 1988 [9] - A. Oustaloup, A. El Yagoubi, P. Baylou, J.F. Samson and P. Melchior - An algorithm of the CRONE software: Application to the guidance of a mobile robot. Eighth IFAC Workshop "Control Applications of Non linear Programming and Optimization", Paris, France, June 7-9, 1989. Proceedings Pergamon Press
V e r s u n e s t a b i l i s a t i o n non l i n 6 a i r e d i s c o n t i n u e Michel FLIESS I, Franqois M E S S A G E R
1z 3
Laboratoire des Signaux & Syst~mes. CNRS-ESE, Plateau de Moulon 91192 Gif-sur-Yvette Cedex (France). TEl : (1) 69.41.80.40 - Fax : (1) 69.41.30.60 2
Applications MathEmatiques et Logiciel. 6, rue AmEdde BollEe 92500 Rueil-Malmaison (France). TEl : (1) 47.49.14.00 - Fax : (1) 47.51.10.89
ABSTRACT : Since Sussmann [18], it is now well-known that, in opposition to the timeinvariant linear case, controllability of a nonlinear system does not necessarily imply the possibility of stabilizing it by a smooth feedback. This paper approaches the stabilization from another point of view, the discontinuous control, and therefore abandons the constraint of smooth feedbacks. It is interesting to see that this technique has already existed for a long time for relay systems (Cypkin [6]), sliding modes (Utidn [19]) and, of course, optimal control. Our method is based on a generalized controller canonicaI form [9, I0] which has recently been derived by using the analogue of the theorem of the primitive element in differential algebra (cf. Kolchin [14]). This canonical form is used to obtain an equation which defines a linearizing state feedback. By choosing the right parameters we are able to express a discontinuous stabilizing state feedback. The introduction of switching curves permits to take the structure of the system into consideration. We illustrate this technique by applying it to an example taken from Aeyels [I], which was shown not to be smoothly stabilizable.
1. I N T R O D U C T I O N I1 est aujourd'hui connu (cf. Sussmann [18]) que, contrairement au cas linEaire stationnaire, la commandabilit6 d'un syst~me non linEaire n'implique pas nEcessairement la possibilit6 de le stabiliser par un bouclage d'Etat rEgulier ou lisse, c'est9h-dire indEfiniment diffErentiabte. De nombreux travaux, souvent illustrEs par des exemples fort instructifs, ont depuis pennis de mieux cerner ce probl~me ( Brockett [3], Aeyels [1], Kawsky [13], Sontag [17], Dayawansa et Martin [7] .... ). Parall~lement, Byrnes et Isidori [4,5] ont entrepris une recherche de grande ampleur pour, gr~ce ~t la dynamique des zEros,.obtenir des conditions de stabilisation par bouclage rEgulier. Cette communication aborde la stabilisation par un autre point de vue, h savoir la commande discontinue et, donc, abandonne l'exigence de boucle rdguli~re. Quoique Sussmann [18] ait ddjh dcrit, il y a plus de dix ans, que "the introduction of disconfinous vector fields is unavoidable", en dehors, bien entendu, de la commande optimale bangbang, une tetle approche n'a re~u chez les thEoriciens qu'une attention fort limitde. Elle est cependant fort ancienne comme en t6moigne le livre de Cypkin [6], et se perpdtue, aujourd'hui, surtout par Ies modes glissants (cf. Utkin [19]), bien que certains travaux de 3
Travail de th5sc effectualdans lc cadre d'unc convcntion CIFRE entre lc CNRS ct AML.
779
nature plus th6orique paraissent de loin en loin (cf. Arstein [2], Hermes [12], Sontag [ 16], et aussi [ 11 ]). Rappelons encore que, sous certaines conditions de commandabilit6, Sussmann [18] a d6montr6 que la stabilisation est possible avec une commande analytique par morceaux. Nous proposons une approche de la stabilisation par bouclage discontinu en utilisant une forme canonique de commande [9,10], g6n6ralisant celle bien connue pour les syst6mes linEaires stationnaires monovariables, obtenue par des m6thodes d'alg~bre diffErentielle que l'un des auteurs a dEveloppEes [8]. Nous illustrons notre approche par un exemple qui, d'apr~s Aeyels [1], ne peut 6tre stabilis6 par bouclage r6gulier. II. R A P P E L S D ' A L G E B R E D I F F E R E N T I E L L E
(voir Ritt [15] et Kolchin [14])
On suppose le lecteur familier avec les propri6tds de base des corps commutatifs. Pour simplifier, les corps consid6r6s sont de caract6ristique nulle. Undcorps diff6rentiel est un corps commutatif K muni d'une d6rivation unique notee ~ - = , qm sausfait les propnetes usue les : d V a , b e K, ~ - ( a + b ) = a + b ?
IIIM
.
.
.
.
I
p
1
d (ab) = ~ + ab dt Une constante clans K est un 616ment c e K tel que c = 0. L'enscmb~e dc~; constantes de K est un sous-corps de K appeltd te corps de constantes. Une extension diffdrentielle L / K est donn6e par deux corps diff&entiels K, L avec K C L. Un 616ment de L e s t dit diffdrentiellement algdbrique sur K si, et seulement si, il satisfait une 6quation diff6rentielle algdbrique ~ coefficients dans K. L'extension L / K est dite diff&eutiellement alg6brique si, et seulement si, tousles 616ments de L sout diff~rentiellement alg6briques sur K. Dans le cas o~ L / K est liniment engendr~e, on a l e r~sultat fondamental suivant: T H E O R E M E : Pour une extension L / K finiment engendrde, les deux conditions suivantes sont dquival~ntes : (i) L/K est diffdrentiellement algdbrique; (ii) le degrd de transcendance de L/K est fini. Rappelons que ce degr6 de transcendance peut s'interpr6ter comme le nombre minimal de conditions initiales n6cessaires pour intdgrer le syst~me diff6rentiel repr6sent6 par L/K. I I I . I N T R O D U C T I O N DE L A F O R M E
CANONIQUE GE.NERALISEE
Soit k un corps diff6rentiel de base. On note k < u > k le corps diff6rentiel engendr6 par k et les composantes de l'entrEe u = (u 1. . . . . urn). Une dynamique est une extension diff6rentiellement alg6brique K/k finiment engendrEe. Soit x = (x 1. . . . . x n) une base de transcendance de K / k < u > , qui, comme on l'a vu, est de degr6 de transcendance t2ni. Les d6rivdes x sont k-alg6briquement d6pendantes de x. I1 en ddcoule (D)
Ai ~x ," i . -~,,u . . . . . . . . . . .,,(m)~=O J
( i = l ......n),
780
oh les A i sont des polyn6mes ~tcoefficients dans k. En invoquant des spEcialisations dans les reels ou les complexes, on voit que l'dcriture explicite (D')
x i = ai(x,u,~l ..... u (m)) = 0
(i = 1..... n),
a seulement une valeur "locale". (D) et (D') redonnent une image plus famiti~re de Ia notion de dynamique. Une autre base de transcendance x = (xx ..... xn) de K / k < u > conduit, 6videmment, ~ des expressions semblables. On sait que les composantes de (resp. x) sont k-algdbdquement dEpendantes de celles de x (resp. x). Ce sont les formules de passage d'un 6tat ~ l'autre qui font intervenir l'entrde et ses dErivEes. Pour des extensions algEbriques non diffErentielles usueUes, qui sont finiment engendrEes, la notion d'gldment primitif est bien connue. I1 en existe un analogue diffErentiel (cf. Kolchin [14]) qui affirme l'existence d'un ElEment ~ 9 K, dit dldment primitifdiffdrentiel, tel que K = k. C'est dire que K est diffErentiellement engendrd par k et ~. Le degrE de transcendance n de K/k est le olus petit entier n tel que ~(n) soit k-algEbriquement dependant de ~,~ .... ~(n-1). Posons ql = ~ ..... :'(n-l) ' . . q~ = ~ . Alors, q = (ql ..... qn) est une base de transcendance de K / k < u > , qm fouralt une gdadralisation implicite de la forint canonique de conmamlde :
I]
ql = q2
qn-1
=
qn
k.C(Ctn,q,u,u ..... u(n)) = 0 06 C est un polynEme ~ coefficients dans k. On en ddduit donc une forme explicite "locale":
!l = q2
I
qn-1
=
qn
ctn = c(q,u,u ..... u(n))
IV.
METHODE
DE
STABILISATION
PAR
RETOUR
D'ETAT
DISCONTINU Considdrons la dynamique, de dimension n : (Z)
{~~ = f l ( ~ ' u )
fn(~,u) 06 ~ = (~I ..... ~n) et u = (u 1..... urn). On suppose, pour simplifier, que fl ..... fn sont des fonctions polynomiales des arguments h coefficients reels. I1 est facile de ramener l'dtude de (E) ",i celle des corps diffdrcntie!s ~ S c e ~ !'idda! diffdrentiel premier [14], qui lui
781
correspond. L'analyse du paragraphe prdcddent montre l'existence d'un dl6ment primitif x I qui conduit &la forme canonique suivante : Ix~ = x2
(Y'g)
/Xn_ 1 = Xn k.G(xn,X,U,U ..... u(m)) = 0
o ~ a x = ( x 1..... xn). Parun argument du type "thdor~me des fonctions implicites", il vient :
I
xI = x2
v
(Eg)
IX--1 =
Xn
1
k.x.n = g ( x , u , u ..... u(m))
C'est la forme canonique explicite, qui, en gdndral, n'est valable que "[ocalement". Dbs lors, il est possible d'expfimer le bouclagc lindarisant, cn 6cfivmlt : (z)
n
p
g(x,u,u ..... u(m)) = i~t aixi + i~l bivi
off les a i e t b i sont des constantes rdelles et v = (v t ..... vp) est la nouvelle entrde. I1 s'agit alors de rdsoudre rdquation (E) en prenant l'entr6e u c o m m e inconnue. On obtient la stabilisation en injectant cette commande dans le syst6me (Zg') durant un petit intervalle de temps At et on rdsoud & nouveau (E) & l'instant t + At. V. E X E M P L E
DE S T A B I L I S A T I O N
DE SYSTEME
Le systSme ?~ entr6e monodimensionnelle v : x=x+ (2)
3t
y3
f(x,y) + v
off f(0,0) = 0, n'est pas, d'apr~s Aeyels [1], stabilisable par retour d'dtat r6gulier. V.1. E t u d e p r d l i m i n a i r e (Y.) peut, par un bouclage r6gulier de la forme u = fix,y} + v, 6tre mis sous la forme :
(]~')
{i = xy+3 u
11 faut distinguer quatre cas selon le signe de x et de x, donc selon la position du couple (x,y) par rapport ~ la courbe x + y3 = O.
782
Y
B' A
x A A'
<
3 x+y=0
Le plan est partitionn6 en quatre r6gions : A = B = A'= B'=
{(x,y) ~ ~ 2 tel que x > 0 et x + y3 > 0} {(x,y) ~ I R 2 t e l q u e x > 0 e t x + y 3 < 0} {(x,y) ~ ] R 2 t e l q u e x < 0 e t x + y 3 < 0} {(x,y) e ] R 2 t e l q u e x < 0 e t x + y 3 > 0}
Remarque : Les fl~ches repr6sentent le sens de d6placemcnt du couple (x,y) dans le plan avec une entrde u nulle. Si (x,y) e B U B', x et x ont des signes opposds. Aeyels [1] d6montre l'existence d'une commande u guidant l'6tat (x,y) de (T,') vers l'origine. Si (x,y) e A U A', x et x ont marne signe; le syst~me ne peut 6tre stabilis6 en restant darts A U A'. Par un argument analogue ~ celui du paragraphe pr6c6dent, il existe une commande qui permettra au couple (x,y) de passer de la r6gion A (resp. A') ~t la rdgion B (resp. B'). V.2. Calcul et choix des c o m m a n d e s
a) R6gions A et B a.1) (x,y) E A On peut v6rifier tr~s simplement que (x - u) est un 61dment primitif diff6rentiel, ce qui permet de mettre ( Z ' ) sous la forme ddcrite prdc6demment, en effectuant le changement de variable suivant : (F~)
XI=
X - U
x2
x1
x + y3 _ u
d' off
(Zgl)
r~xl = x2
Lx2
-u + (x2+u) §
(xx+u))2/3
Exprimons ensuite le bouclage lin6~arisant en dcrivant : (El)
-'6 + (x2+h) + 3u(x2+h - (xl+u))2/3 = axx I + a2x 2 + bw
off a l, a 2 et b sont des constantes rdeUes et w une nouvelle entr6e, qui sera supposde nul!e. Pour des raisons ~videntes de stabilisation, a 1 e t a 2 seront pris r6els et n6gatifs.
783
On consid~re (El) comme une Equation en u, u et "u que l'on rEsoud aux instants d'dchantillonnage : U=-X tl=-
1 X2
"11= - a l x 1 - a 2 x 2
Connaissant les conditions initiales de u ~ l'instant d'Echantillonnage t=nAt, nous calculons les param~tres A, B e t C de la fonction u(t)=At2+Bt+C, pour nAt
;
x=-x<=>x+y3
=.x<=>2x+y3
=0
On vient de faire apparaitre une courbe de commutation. Quand l'on se situe au-dessus de cette courbe on conserye la commande du paragraphe a. 1) qui aura tendance ~ dEplacer le couple (x,y) vers cette courbe donc dans une region intErieure ~t B off x sere proche de -x. B t
B'
Y A
9
.<..
x+y :0 J
2x+ y3= 0
A' B
784
D~s que (x,y) traverse cette courbe, on change de type de commande de sorte que (x,y) reste dans un voisinage de la courbe de commutation en favorisant la stabilisation de y. L'616ment primitif choisi est x + u, d'oh le changement de variable : Yl = x + u (F2) y2 = y t = x + y 3 + u qui conduit ~ :
it = Y2
(7"g2)
2 = "u + (Y2- in) + 3 u ( y 2 - u - (Yl-U)) 2/3 On exprime ensuite le bouclage lindarisant en 6crivant : (E2)
"u + (y2-tl) + 3u(y2-u - (yl-u)) 2/3 = aly 1 + a2y 2
Une solution de (E2) est : u = (Yl - Y2) 1/3 = Y2 - Yl + (Yl- Y2) 1/3 "u= axy 1 + a2y 2 - Yl + (Yl - Y2) I/3 Cette commande conduit (x,y) au-dessus de la courbe de commutation en stabilisant y. Kawski [13] a utilis6 une commande continue du m6me type, c'est-~-dire avec une puissance non enti6re de l'6tat, mais pr6sentant l'inconv6nient de poss6der une dddv6e tendant vers l'infini ~t rorigine. C o m m e dans le paragraphe pr6c6dent, on rencontre encore des probl~mes de discontinuit6 de la commande lors de l'utilisation des formules de passage (F2) qui seront r6solus par une r6-initialisation de la commande sur l'origine. L'enchainement de ces deux types de commandes a.1 et a.2, de part et d'autre de la courbe de commutation, emm~nem l'6tat sur l'origine (0,0) ou dans un voisinage par un d6placement en "esFalier" (cf. figures 1 et 3). Lorsque tdut en restant darts B, l'6tat "s'approche tr~s pr6s" de l'origine, le procdd6 de c o m m a n d e discontinue mis en oeuvre risque, si l'on prend un pas d'6.chantiUonnage trop grand, de le faire sortir de cette r6gion, c o m m e le montre le dessin SUlvant :
Y
A )
B ~ B A 2
x
x + Y 3---0 x +y3_-0
C'est pourquoi le systSme n'atteindra pas, dans certains cas, une stabilit6 sur l'origine asymptotique usuelle mais plut6t dans un "petit voisinage" de l'ori~ne. I1 est clair que d'un point de vue pratique cela ne porte pas ~ cons6quence.
785
b) Rfgions A' et B' Les entr6es u calculdes dans le paragraphe a et le syst~me ( ~ ' ) sont sym6triques par rapport ~t l'origine. On conserve donc dans A' (resp. B') les commandes de la rdgion A (resp. B). b.1) (x,y) E A' La commande utilis6e est celle calcul6e en a. 1, mais ici dans le but de se diriger vers la zone B'. b.2) (x,y) ~ B' Quand le couple (x,y) se situe au-dessous de la courbe de commutation 2x + y3 = 0, la commande du paragraphe b.1 est conservEe, pour les mSmes raisons qu'en a.2. Puis quand (x,y) se situe au-dessus de cette courbe de commutation, la commande utilisfe est celle calcul6e au paragraphe a.2. V.3. M 6 t h o d e de calcul L'enchainement des calculs ne d6pend pas de la r6gion o~ est l'6tat (x,y). Seules changent les 6quations permettant de calculer la commande. L'algorithrne de calcul est le suivant : 1~ Echantillonnage de (Z') ~ l'instant t = nAt : saisie de l'6tat (x,y). 2 ~ Calcul de l'6tat du syst~me virtuel (~gi) par l'utilisation des formules de passage (Fi), avec r6-initialisation de la commande. 3 ~ Identification des param~tres A, B et C de la nouvelle commande u(t) par r6solution de (Ei). 4 ~ Envoi de cette commande pendant une petite p6riode de temps At dans (Z'). On recommence le traitement ~t la 1~ 6tape un At plus tard soit h l'instant t = (n+l)At. On prend i = 1 ou 2 selon le signe de x et de la position de 1'6tat (x,y) par rapport ~ la courbe de pommutation. VI. R E S U L T A T S
DE S I M U L A T I O N
Les figures 1 "~4 pr6sentent deux cas de stabilisation sur ordinateur, avec les param~tres a I = -10, a 2 = -10 et At = 0.02 s. Les points initiaux de l'6tat de ces deux cas sont, pour les figures 1 et 2 : (x,y) = (5,10) et pour les figures 3 et 4 : (x,y) = (0,-10).
786
100~
I0 8 6
60
4 40
2 0
20
90
-2
~ y S---V
-4
-20
-6
Figure 2
Figure 1
20
8
y
0 4
1'5
-20 0
-2'o
-120
-4
~ v
X
10
-40 -60 -80 -100
9- 1 2
Figure 3
-120
Figure 4
V. B I B L I O G R A P H I E
[1] D. AEYELS.Stabilization of a class of nonlinear systems by a smooth feedback control. Systems & Control Letters, 5 (1985) 289-294. [2] Z. ARSTEIN Stabilization with relaxed controls, Nonlinear Analysis, Theory, Method & Applicadon, 7 (1983) 1 I63-1173. [ 3 ] R.W. BROCKETT Asymptotic stability and feedback stabilization, in "Differential Geometric Control Theory", R.W. Brockett, R.S. Millman and H.J. Sussmann eds, Birkhtiuser, Boston, 1983, p. 181-191. [4] C.I. BYRNES and A.ISIDORI New results and counter examples feedback stabilization. Systems & Control Letters, 12 (1989) 437-442.
in
nonlinear
787 [5] C.I. BYRNES et A. ISIDORI Rdgulation asymptotique des syst~mes non lindaires. C. R. Acad. Sc. Pads, 1-309 (1989) 527-530. [ 6] Ja.Z. CYPKIN Th~ode des asservisscments par ptus-ou-moins (traduit du russe), Dunod, Pads, 1962. [7] W.P. DAYAMANSA and C.F. M A R T I N Two examples of stabilizable second order systems, in "Computation and Control", K. Bowers and J. Laud exis, Birkh~iuser, Boston, 1989. [8 ]
M. FLIESS Automatique ct corps difftrentiels. Forum Math., 1 (1989) 227-238.
[ 9 ] M. FLIESS Gdndralisation non lindairc de la forme canoniquc de commandc et lindarisation par bouclagc.C.R. Acad. Sc. Pads, 1-308 (1989) 377-379. [10] M. FLIESS Generalized controller canonical forms for linear and nonlinear dynamics, IEEE Trans. Automat. Control, 5. para~tre. [11] M. FLIESS, P. CHANTRE, S. ABU EL ATA and A. COICDiscontinuous prcdictivc cona-ol, inversion and singularities. Application to a heat exchanger. These proceedings. [ 12] H. HERMES On the synthesis of a stabilizing feedback control via algebraic methods, SIAM J. control and Optimization, 18 (1980) 352-361. [13] M. KAWSKI Stabilization of nonlinear systems in the plane. Systems & Control L~tters, 12 (1989) 169-175. [14]
E.R. KOLCHIN Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
[15] J.F. RITT Differential Algebra, Amer. Math. Soc., New York, 1950. [ 16 ] E.D. SONTAG Nonlinear regulation : The piecewise linear approach, IEEE Transactions on Automatic Control, 26 (1981) 346-357. [17] E.D. SONTAG Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34 (1989) 435-443. [ 18] H.J. SUSSMANN Subanalytic sets and feedback control. Iournal of Differential Equations, 31 (1979) 31-52. [19] V.I. UTKIN Sliding Modes and their Applications in Variable Structure Systems (translated from the Russian), Mir, Moscow, 1978.
A C O M B I N A T O R I A L A P P R O A C H OF T H E N O N L I N E A R S A M P L I N G P R O B L E M
&Monaco* and D. Normand-Cyrot** * Dipartimento di Informatica e Sistemistica, Universi~ di Roma, "La Sapienza",Via Eudossiana 18, 00184 Rome, Italy. ** Laboratoire des Signaux & Syst~mes, CNRS-ESE, Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France.
ABSTRACT In this paper, results of combinatorial interest taken from the well-known Baker-Campbell-Hausdorff formula are proposed. They are used to analyze algebraic relations between nonlinear continuous time systems and their discretized analogues.
1 - INTRODUCTION Given two elements X and Y belonging to an associative algebra, we note DX = adX the formal derivative operator: DX(Y) = XY - YX If t belongs to R, the following equalities of formal series are satisfied [3]: d
d-'i"/t = 0 (e-X e X + tY) = E l f x ) ( y )
(1-1)
andreciprocally: d eX Y =di"/t = 0 eX+tB I (X)(Y)
(1-2)
with:
E l ( X ) ( y ) _ 1 -Dxe-DX (Y) = ~ (l~'~(-1)kT)!D x k ( y ) k>0
(1-3)
and
BI(x)(Y) = ~ DX ( Y ) = ~
(1-4)
1 bk D X k ( y ) ~.~ (bk, Bernoulli numbers )
E 1 and B 1 are Lie series with only one occurrence of the variable Y. In (1-3) the fraction means the simplification of the series numerator by the denominator, this represents a compact expression which will be extensively used lateron. In this paper, the formal equality (1-3) is first extended to higher order derivatives with respect to t and then to the exponential term:
789
X
+Z
tk ~T".rYk
(1-5)
k>l
instead o f X + tY, where the Yk, k > 1 denotes a family of formal variables. For, we define repeatedly a family of Lie series denoted: Ei(x,Yjl ..... YJi-1)(YJi), Ji > 1 and we show that each derivative (1-5) with respect to t, of order p, satisfies a homogeneous, of degree p, polynomial in terms of these elements E i. These combinatorial equalities which are variant and extensions of the well-known BakerCampbell-Hausdorff formula are directly applied in this paper to the sampling problem and to explain relations between nonlinear differential equations and their discretized analogues. The present work goes further than the results proposed in [7] and leads to compact solutions which are much more appropriate for digital computing making use of formal languages. Moreover, we can show that the same algebraic manipulations can be used to analyze functional expansions associated to nonlinear discrete time equations, this gives strong analogies between nonlinear continuous, discrete and discretized systems which are extensively studied in [10, 11] The use of formal algebraic methods in nonlinear control theory were initialized in the continuous time case [4] and further developed in both mathematics and control domains [2, 5]. The paper is organized as follows: some definitions and notations are recalled in section 2, section 3 presents a combinatorial study of parametrized exponential series. The results are applied in section 4 to the sampling problem of a nonlinear differential equation.
2 - P R E L I M I N A R I E S AND F O R M A L N O T A T I O N S Consider a family (or an alphabet) denoted {Xl .... Xk} of elements of an associative algebra. Some definitions and notations from the theory of formal languages are recalled. We denote: Xil...Xip; ij E {1 . . . . . k}, p > 1 any word of length p in terms of the indeterminates {X 1,...,Xk }; 1 is the word of length zero or neutral element and: Xik = Xi... Xi (k times) denotes the word composed by the concatenation k times of the same letter Xi. According to these notations the following exponential type and logarithmic type series are easily defined. We note by: e Xi = I + Xi + (1/2!)XiXi + ... + (1/k!)Xi k + ...
(2-1)
the exponential series and in the same way: Log (I + Xi) = Xi - (1/2[)XiXi+ ... +((-1)k/k!)Xi k + ...
(2-2)
the logarithmic series. Stating by convention that any element Xi is said of degree i, any homogeneous of degree p polynomial with real coefficients is written as:
P=~ m>l
Z
ail"'aimXil'"Xim
(2-3)
il,...,im>l m
with: all.. aim • R and ~
ij = p.
j=1
We note DXi = adXi the formal derivative operator which acts on any word Xi 1..... Xij according to the rule: DXi(Xil ..... Xij) = XiXil ..... Xij - Xil ..... XijXi (2-4)
790 The shuffle product [12] is defined in a recursive way as follows: lmXi = Xi ml = Xi XiO)Xj = Xj(.oXi = XiXj + XjXi
(2-5)
Xjl .... Xjpf-OXil .... Xij = Xjl(Xj2 .... Xjpr-OXit .... Xij) + Xit(Xjl .... Xjp(OXi2 .... Xij)
3 - SOME COMBINATORIAL FORMULAE Hereafter, we will successively denote as {X,Y} and {X,Y i, i > 1} formal variables and t any element of R. The aim of this section is to generalize the formal equalities (1-1) et (I-3) to higher order derivatives with respect to t and then to the exponential term referred as the multivariable case:
tk X + kaY-'~1 ' " Yk The monovariable case: Considering equality (1-3), or equivalently at t r 0 the equality:
EI(x+tY) =
1 - e-D(X+tY) (- 1)k n r X + D(X + tY) =k>~_o(k+l)! ~" ty)k
(3-1)
we define: (3-2)
E2(X,Y) = (d/d0/t = 0 EI(x+tY)' and iteratively for i > 3: Ei(x,Y,...,Y) = (d/d0/t = 0 Ei-I(X + tY, Y,...,Y)
(3-3)
In this way, we define a family of Lie series denoted by Ei(x,Y,...,Y)(Y). Since each series Ei(X,Y,...,Y)(Y) contains i occuring of the variable Y, it will be said of "degree i". For the First elements, we obtain the compact expressions generalizing (1-3):
el(x) =
l-e -DX
D------g---= ~
(- 1)k ~ v k
~"'~
k~O
E (X,Y) =
,~
E3(X,Y,Y) =
r~-z2
20
(~ ~) = 2 , ( ~
DXk(mDY)
(3 -4)
I-DX+(DX2/2!) -e'DX . . . . . 2 x-', (-1) k+2 " ~ tc0t)x) = L ~ DXk(mDY)2 k_>0
and finally for i > 1: Ei(x,Y,...,Y) =
1-DX+(DX2/2!)+...+((-1)i'I/(i'I)!)DXi'I-e'DX.(mDY)i-I DX i
(3-5)
With these notations, it becomes possible to extend (I-1) and to express any derivative as a homogeneous polynomial in terms of these Lie series. We state:
791 Theorem 1: The following formed equality of series is verified:
e - x e X + tY = 1 + ~
( t k / k ! ) a k ( X , Y ..... Y)(Y)
(3-6)
k_>l
where:
ak(X,Y,...,Y)(Y) = (d/dtk)/t = 0 e -X e X + tY
can be decomposed as a homogeneous polynomial of degree k in the formal indeterminates of "degree i": E i ( x , Y ..... Y)(Y). Several techniques to prove this theorem are possible. W e will choose the more intuitive one which consists in the successive derivation with respect to t, expressed at t = 0, of the equality (1-1). In fact, according to (1-3) and (3-I), we can rewrite equality (1-1) at t ;~ 0 as: (d/dt)/t * 0 (e X + tY) = e X + t Y E l ( X + t y ) ( y ) = e X + t Y a l ( X + t Y ) ( y )
(3-7)
Taking into account the non-commutativity of the formal variables X and Y and applying the usual rules of derivation, with respect to t, of products of formal objects, we alternatively compute the polynomials ai (X,Y,...,Y)(Y) as follows: By definition and because of (3-7): (d/dt2)/t = 0 eX + tY = e X a 2 ( X , y ) ( y ) = (d/d01 t = 0 ( e x + tYEI(X+:Y)(Y)) = (d/dt)/t = 0 (e X + tY)EI(X)(y) + e X ( d / d 0 / t = 0 (EI(X+tY)(Y)) = e X E I ( X ) ( y ) E I ( X ) ( Y ) + eXE2(X,Y)(Y) in other words: a2(X,Y)(Y) = {E 1(X)(Y) }2 + E 2 ( X , y ) ( y )
(3-8)
from the definition: E2(X'Y) (Y) = (d/dt)/t = 0 E l (X+tY)(Y) Proceeding this way we can d e c o m p o s e successively the elements ak(X,Y ..... Y)(Y) as a homogeneous polynomial. For k = 3, we obtain: (d/dt3)/t = 0 eX + tY= e X a 3 ( X , y , y ) ( y ) = ( d / d 0 / t = 0 (eX + tYa2(X+tY'Y)(Y)) = (d/dt)/t = 0 (eX + t Y ) ( E I ( X ) ( y ) E t ( X ) ( Y ) + E2(X'Y)(Y)) + eX(d/dt)/t = 0 (EI(X+tY)CY))(EI(X)(Y)) + eXEX(X)(Y)(~Vdt)/t = 0 (EI(X+tY)(Y)) +
e X(a/dt)/t = 0 (E2(X+tY'Y)(Y)) which is: a3(X,Y,Y)(Y) = {EI(X)(Y)} 3 + 2 E I ( X ) ( y ) E 2 ( X , Y ) ( Y ) + E2(X)(y)EI(X,Y)(Y) + E3(X,Y,Y)(Y) (3-9) under the definition: E3(X,Y,Y)(Y) = (d/d0 / t = 0 E2(X+tY'Y)(Y)
792
For the higher order derivatives, we obtain the recurrent formula: (d/dtk)/t = 0 eX + tY = eXak(X,y,...,y)(y) = (d/dt)/t = 0 (eX+tYak'l(X+tY'Y"'"Y)(Y)) =
eXEI(X)(Y)ak_I(X,Y,...,Y)(Y) + eX(d/dt)/t= 0 ak.l(X ?- tY,Y,...,Y)(Y)
(3-10)
Denoting with L the Lie algebra generated with X and Y, L0 the Lie ideal of L generated by Y and successively with Lop , for p > 1, the decreasing series of ideals defined as: Lo 1= Lo, Lop= [Lo,L0P-1], p > 2. We verify: Corollary_ 1: Each element Ei(x,Y,...,Y)(Y), i _> 1 is a Lie element of degree i, belonging to L0 i, and including i occuring in the variable Y. Theorem 2: Using the compact notation: Ei(x,Y,...,Y)(Y) = Ei(DX), we verify the following equality of series: e-XeX+tY=l+ ~ (tk/k!) ~ k>l
~
c01 ..... im)Eil(DX)-..Eim(DX)
(3-11)
k>l il,...,im21
= 1 + T(Eil ..... Eim) = S(Eil ..... Eim), with ~. ij = k. Moreover, the coefficients c(il,...,im) satisfy the recurrent formulae of a shuffle product, j=l
which is f o r k > 1: C(il)C(i2) = c(il,i2) + c(i2,il) = C(ilf.Oi2) C(il)C(i2,i3) = C(il,i2,i3) + c(i2,il,i3) + c(i2,i3,il) = C(ilO~i2i3)
The following corollaries are easily deduced from Friedrichs theorem [12] or related ones. Corollary 2: The series Log( 1 + T(Eil ..... Eim)) is a Lie element in the variables Eil(DX) ..... Eim(DX) if, and only if, the coefficients c(i 1..... im) satisfy the recurrent formulae of a shuffle product. This Lie element corresponds to the Baker-CampbeU-Hausdorff exponent of the exponential product: e-XeX + tY. Log (1 + T(Eil ..... Elm) )= BCH(-X, X + tY)
(3-12)
1 + T = e BCH(-X, X + tY)
(3-13)
that is:
Corollary 3: Denoting with E i et F i two families of indeterminates, we verify: T(Eil+Fil,..,Eim+Fim) = T(Eil ..... Eim)+T(Fil,..,Fim)+T(Eil,..,Eim)coT(Fil,..,Fim) (3-14) and in a similar way: S(Eil+Fil ..... Eim+Fim) = S(Eil ..... Eim)(oS(Fil ..... Fim).
(3-15)
793
The multivariabl~ case: Let us now generalize these results to the exponentia! expansion associated with the multivariable series which is nonlinear in t : X + ~ (tk/k!)Yk. k>l
We will exactly follow the same procedures as in the previous case, in terms of a new family of formal Lie series, defined with degree i and denoted in a compact form by EI(DX) for i > 1. Because of the non linearity with respect to t, each EJ(DX), forj > 1, is now expressed as a J linear combination of Lie series of degree i = ~ ip, denoted by EJ (X,Yi 1..... Yij. 1)(Y ij), which are p=l
straightforward extensions of the elements defined in (3-5). In fact, for ij > 1, we easily define the elements: EJ(x,Yil ..... Yij.1)(Yij) (of degree ~ij), by the compact expression: 1 - DX + (DX2/2!) +...+ ((-1)j'I/(j-1)I)DXJ -1 - e -DX DX i coDYi1...mDYij_1(Yij)
(3-16)
In the first terms, we compute : El(X)(Yi) =
1 - e -DX DX (Yi) said of degree i
E2(X,Yi)(Yj) =
1 - DX-e -DX DX 2 (mDYi)(Yj) said of degree i + j
(3-17)
Besides these elements, we define by successive derivatives with respect to t the series Ek(DX) of degree k. According to (3-1), we can write at t = 0:
E1(DX) = EI(X)(Y1) =
1 - e -DX
DX
d X + ~ (ti/iI)Yi) (Y1) =di"/t = 0 (e-x e i_>1
and equivalently at t # 0, 1 - e-D(X+~
E t ( x + ~ (ti/i!)Yi)( ~ (ti/i!)Yi+l) = i>1
(ti/i!)Yi)
i_>l D ( X + E (ti/i!)Yi) i_>l
i~0
( ~ (ti/i!)Yi+l) (3-18) leO
For higher order terms, proceeding as previously, we compute the successive derivatives with respect to t and we define for example for k = 2: E2(DX)=(d/dt)/t = 0 E I ( X + ~ k_>l
(ti/iI)Yi)(~
(ti/i!)Yi+1)
i>0
= E2(X,Y1)(Y1) + EI(X)(Y2)
(3-I9)
ProPosition: Any element Ek(DX), k _> 1, defined of degree k, is a linear combination of elements of degree k of the form EJ(x,Yil ..... Yij_l)(Yij). We obtain: k
Ek(DX) = ~ j=l
~
c~(il...ij) EJ(x,Yil ..... Yij_l)(Yij),
il,...,ij> l
(3-20)
794
with ct(il...ij ) e R, Y~ij = k, EI(DX) = EI(X)(YI) = al(X)(Y1) and according to the recurrent law : k
Ek+I(DX) = EJ(x,Yil+I ..... Yij-1)(Yij) +"'+ Z
Z
j=l
a(il...ij)
i 1 ..... ij_>l
k
EJ(x,Yil ..... Yij_l)(Yij+l) + ~ j=l
~
a(il...ij) Ej+I( X, YI, Yil ..... Yij-1)(Yij)
(3-21)
il,...,ij_> 1
For the t-u-stterms, we compute: E2(DX) = E2(X,Y1)(Y1) + EI(X)(Y2) (3-23) E3(DX) = E34X,Y1,Y1)(Y1) + 2 E2(X,Y1)(Y2) + E2(X,Y2)(Y1) + EI(X)(Y3), E4(DX) = 2E3(X,Y2,Y1)(Y1)+2E3(X,Y1,Y2)(YI)+3E3(X,Y1,Y1)(Y2)+3E2(X,Y2)(Y2) +2E2(X,Y 1)(Y3)+E2(X,Y3)(Y1)+E1(X)(Y4)+E4(X,Y 1,Y 1,Y1)(Y1)+E2(X,Y1)(Y3). (3-24) Remark: When only one indeterminate is involved Y1, we recover the equality: Ek(DX) = Ek(x,Y1 ..... Y1)(Y1). According to these notations, the multivariable analogues of results stated previously can be written with respect to the generalized elements Ek(DX) for k > 1: Theqrem 3 : The following equality of series is verified: e-X
e (X + ~" :"""2.ttVx!)Yi)= 1 + ~
(tk/k[)Ak(X,Y1 ..... Yk)
(3-25)
k>l
with : Ak(X,Y1 ..... Yk)= (dk/dtkt/t = 0 e-X e(X +E (ti/i!)Yi) which can be rewritten as an homogeneous polynomial of degree k in the formal indeterminates: Ei(DX), i>l. Generalizing (3-8) and (3-9) according to the notations (3-23) and (3-24), we directly obtain: A 1(X,Y1)=EI (DX)=E1 (X)(Y 1) A2(X,Y 1,Y2)= {E 1(DX) }2 + E2(DX) A3(X,YI,Y2,Y3)= {EI (DX) }3+2E I (-DX)E2(DX)+E2(DX)E I(DX)+E3(DX)
(3-26) (3-27) (3-28)
Denoting with L the Lie algebra generated by X and Yi. i >1, with iLo the Lie ideal of L generated by Yi and successively denoting by Lo p, p>l, the decreasing series of ideals of "degree p" defined as: Lo 1= 1Lo, Lo 2= [1Lo, lLo]+2Lo, Lo 3= [[1Lo,[1Lo,ILo]]+[2Lo, ILo]+3Lo =[Lo2,1Lo]+3Lo, (3-29) Lo 4= [1Lo,[ iLo,[ ILo,1Lo]]]+[1Lo, {2Lo, ILo]]+[ 1Lo,3Lo]+[2Lo,2Lo]+4Lo = [Lo3,1Lo]+[Lo2,2Lo]+4Lo. We obtain: Corollary 5: Each element E'(DX), for i >_1 is a Lie element of "degree i" that is belonging to Loi.
795 Theorem 4: The following equality of formal series is verified: e-XeX+Y'(ti/it)Yi= 1+ ~
(tk/k!) ~
k_>l
m>l
~
c01 ..... im)Eil(DX) ... Eim(DX)
(3-30)
il,...,im>_l
= 1 + T(Ei! ..... Eim) = S(Eil ..... Eim), with ij > 1, ]~ ij = k. Moreover, the coefficients c(i 1..... ira) satisfy the recurrent formulae of the shuffle product.
4 - NONLINEAR SYSTEMS UNDER SAMPLING When we consider a nonlinear differential equation driven by controls, which are constant over time intervals of amplitude 5, the question of finding a nonlinear difference equation, so that its state evolution corresponds, at sampling times, to the solution of the nonlinear differential equation, when the same initial states are assumed, is referred to as the sampling problem. With respect to usual numerical integration methods, the solution of this problem leads to an implicit sampled representation described by a nonlinear difference equation. On the basis of these equations we can design various digital control schemes and study the preservation of special properties under sampling [8, 9]. This problem, solved in the linear analytical case (equation 1-1) [7] is posed here in a strictly formal way. This approach enables the extension to differential equations including non-linearities with respect to the controls; the solution appears as a direct consequence combinatorial formulae proposed in section 3. It has to be noted that the recursive properties of the solutions are very adequate to the use of symbolic programming language as REDUCE language for example already tested in [1]. Consider a nonlinear differential equation:
x(t) = f(x(t) + ~
(ui(k)/i!)gi (x(t)
(4-1)
i>1
defined on R n by the vector fields f and gi : Rn .... R n, assuming the controls constant over time intervals of amplitude 5 : u(t)=u(k), pour k5 < t <(k+l)& k > 0, the question which is posed is to find a nonlinear difference equation of the form:
x(k+l) = F(x(k)) + ~
(ui(k)/i!)Gi(x(k)
(4-2)
i~l
such that the state evolutions of (4-I) and (4-2) coincide, at sampling times, when x(t -- 0) = x(0); that is x(t) = x(k), for t = k& k >- I. Denoting by Lf and Lgi, the formal directional derivatives associated respectively to the vector fields f and gi, we define an alphabet of formal variables on the basis of which computing rules established in section 2 and 3 can be applied. More precisely, elements Lf and Lg i are called of degree 1 and i respectively; the operation of concatenation of two letters corresponds to the composition of differential first-order operators and consequently exponential and logarithmic expansions, Lie and shuffle products are defined in a very natural way with respect to this composition.
796 Integrating during time interval [kS,(k+l)5[, the solution, at time t = (k+l)8, of the equation (4-1) can be expressed as the flow associated to the constant vector field : f(x(k)) + ~ (ui(k)/il)g i (x(k)). i>l We obtain the formal solution : x(k+l) = eS(L~Y, (ui(k)/i!)Lgi)(Id)/x(k) i_>l
(4-3)
The results proposed in section 3, in particular th e application of theorem 4 to (4-3), directly lead to a compact solution of the sampling problem. Comparing it with (4-2), we can write: F(x) = eSLf(Id)/x Gk(X) = (d k/duk)/u= O jS(Lf+X(ui(k)/i!)Lgi )(Id)/x, k > 1 = e~SLf ~ ~ c(il ..... im)Eil(DSLf) ... Eim(D~SLf)(Id)/x, m_>l il,....im>l with ij >- 1 and Y. ij = k
(4-4)
For the first terms, we compute: G 1(x) = erLf E 1(D~SLf)(Id)/x G2(x) = eSLf{ {El (DSLf) }2 + E2(DrLf) ] (id)/x G3(x) = eSLf{ {E l(DSLf) }3 + 2E 1(D~SLf)E2(DSI_,f) + E2(D~iLf)E I (DSLf) + E3(DSLf) }(Id)/x
(4-5) (4-6)
with: E 1(D~Lf) = E 1(SLf)(SLg I) E2(DfiLf) = E2(SLf, SLg 1) (SLg 1) + E 1(SLf) (SLg 2) E3(DSLf) = E3(SLf, SLgl,SLgl)(SLgl) + 2E2(SLf, SLgl)(SLg2) + + E2(SLf, SLg2)(SLgl)+ El(SLf)(SLg3),
(4-7) (4-8) (4-9)
The elements Ei(SLf, SLgil ..... 5Lgij.1)(fLgij ) are defined by the expressions (3-4) and (3-16) : For the first terms, we compute : El(SLf)(SLgi) = l'e'DSLf = ~ ~(-1) D S kL t k ( S L g l ) , DSLf ~o
k _>0
(4-10)
-- 5Lg i - (52/2!)adLf(Lgi) +(~3/3 !)ad2Lf (Lgi) + 0(53) E2(SLf, SLgi)(SLgj) =
1-DfLf-e-DSLf DSLf 2 (c~
1)k+ 1 DSLfk(c0DSLgi)(SLgj) =~z~ (-(k+2)! k>O = _ (52/2[)adLgi(Lgj) + (53/3 0adLfadLgi(Lg j) + (53/3 !)adLgiadLf(Lgj) + 0(53)
(4-11)
797
E3 (SLf, SLgi,SLgj)(SLgk) =,
I - D 8 L f - + (DSLf2/2!) - e - D S L f (coD8Lgi)(coDSLgj)(8Lgk) DSLf3
= ~ ~ ( - l ) k + 2 DSLfk(o~DSLgi)(coDSLgj)(8Lgk) k>0 = (5313!)adLgiadLgj(Lgk) + O(83)
0-12)
Remarks; l~i)Defining, according to (3-29), the successive Lie ideals associated to the elements 8Lf and 8Lg i ,for i > 1, it follows that Ei(D8Lf) belongs to L01. (it) Considering now the reverse problem which consists of associating to any difference equation of the form (4-2), a differential equation of the form (4-1), in such a way that the solutions coincide at sampling time, some remarks can be deduced from the previous analysis. Let us first assume the sampling period equal to 1 and the existence of a vector field f, such that eLf (Id), is equal to F. According to the definition (1-4) of the reverse series, we see that the vector fields gi can be successively computed as follows: gl(x) = B l(Lf)(G1)(Id)/x g2(x) = B 1(Lf)(G2_G1G I_E2(Lf,Lgl)(Lg 1))(Id)/x g3(x) = BI(Lf)(G3+2G1GIGt-2G1G2-GIG2-2E2(Lf,Lgl)(Lg2)-E2(Lf,Lg2)(Lgl))(Id)/x
(iii) For practical purpose, approximated solutions only are computed, this corresponds to the truncation of the series solutions, with respect to the sampling period, at a fixed order which define the order of approximation of the sampled system. REFERENCES
[1] J.P. BARBOT, A computer aided design for sampling a nonlinear system in "Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Lect. Notes in Comp. Sc. 357, T. Moraed. pp. 74-88, 1989. [2] P. CROUCH, F. LAMNABHI-LAGARRIGUE, Algebraic and multiple integral identities, Acta Applicanda Mathematicae ~ pp. 235-274, 1989. [3] R.GOODMAN, Lifting vector fields to nilpotent Lie groups, J. Math. Pures & Appli., 57, pp.7786, (1978). [4] M. FLIESS, Fonctionnelles causales non lindaires et ind6terminres non commutatives, Bull. Soc. Math. France, 109., pp. 3-40, 1981. [5] M. FLIESS and D. NORMAND-CYROT, Algrbres de Lie nilpotentes, formule de Baker-CampbellHaussdorff et int6grales itErEes de K.T. Chert, s~minaire de probabilit6s, 1980-1981, 16, Lect. Notes in Math., J'. Azema and M. Yor eds., pp. 257-267, Springer-Verlag, 1982. [6] P. LEROUX and X.G. VIENNOT, A combinatorial approach to nonlinear functional expansions: an introduction with an example, Proc 27-th IEEE-CDC, Austin Texas,USA, 1988. [7] S. MONACO and D. NORMAND-CYROT, On the sampling of a linear control system, Rap. 6-84, DIS, Rome, Proc. 24th IEEE CDC, Fort Lauderdale, USA, pp. 90-95, 1985. [8] S. MONACO and D. NORMAND-CYROT, Invariant distributions under sampling, Theory and Applications of Nonlinear Control Systems, ( C.I. Byrnes and A. Linquist eds.), North Holland, pp. 215-221, 1986. [9] S. MONACO and D. NORMAND-CYROT, Zero dynamics of sampled nonlinear systems, Syst. and Contr. Letters, 1._LL1pp. 229-234, 1989. [10] S. MONACO and D. NORMAND-CYROT, Functional expansions for nonlinear discrete time systems, Mathematical Systems Theory, 21, pp. 235-254 1989. [11] S. MONACO and D. NORMAND-CYROT, Combinatorial relations between discrete and continuous time nonlinear systems, submitted for publication, 1990. [12] R.REE, Lie elements and an algebra of shuffles, Ann. of Math., 68, pp.210-220, 1958.
N O N L I N E A R MRAS IN ROBOTS M O T I O N C O N T R O L Bernard BROGLIATO Laboratoire d'Automatique de Grenoble - ENSIEG-INPG-CNRS BP 46 38402 Saint-Martin d'H6res, France
abstract : Inttds paper, we provide a brief overview o f the principal adaptive control laws that have been developped in the field of rigid manipulators during the last ten years, that can be turned into a common MRAS framework and analysed using passivity arguments.
1) I N T R O D U C T I O N Robot manipulators control has been a major topic for an increasing number of researchers during the last decade [1]-[15].Lndeed, besides the fact that new schemes had to be developped in order to improve the performances of new manipulators (particularly those employing direct drive arms), making it mandatory for designers to take the inertial effects into account in their totality (such as Coriotis and centrifugal torques), it appeared that those systems belong to a very particular class of non-linear systems, and even of finite-dimensionnal physical systems (ie systems described by a finite set of generalized coordinates ).Many control schemes have emerged, a large proportion of which are explicitly or implicitly inspired from the Model Reference Adaptive Systems (MRAS) approach studied in the linear case (see eg [16]).Two main phases can be distinguished concerning robots control through a model-reference approach : i) the first one consists in adaptive control laws designed as parallel Adaptive -Model -Following -Control (AMFC) schemes.(figure 1) ii)the second one consists in adaptive control laws derived from the computed torque method, based on the linearization- wit- the- unknown- parameters property of manipulators, and using some physical properties of robots.They can be interpreted as series-parallel AMFC .(figure 2) The general problem of controlling rigid frictionless manipulators can be stated as follows : Find a control law that asymptotically (in the adaptive case) linearizes that non-linear plant, ie makes it asymptotically follow a prespecified linear decoupled reference model.It is in fact essentially this model which will make laws of type i) differ from laws of type li) above. The paper will be organised as follows: in section 2, we provide a brief recall of linear AMFC.In section 3, we present an overview of the "first generation" schemes studied in [1]-[7].In section 4
799
we present the second type of adaptive laws studied in [8]-[15].Section 5 is devoted to unify stability analysis of those schemes. Conclusions are given in section 6.
~_~
reference ] ym model u
1u
riga : parallel MRAS (trajectorytracking : r ~ 0 )
fig. 2 : series-parallelMRAS (state regulation : r = 0 )
2) L I N E A R A M F C 21) Generalities Model reference adaptive systems have been studied extensively in the literature .As noted in [17], designing MRAS requires, from a general point of view, two steps: i) an algebraic part : One has to find a class of reference models such that a bounded control signal u and a controiler C(0*) exist such that lim t~+,,o ]] Xp - Xm[I =0 ,where x m and Xp are the model and plant state (or output) vectors respectively, O* being the true unknown parameters of the plant. ii) an analytic part : it consists in determining the adaptive law for adjusting the parameters estimates 0, such that the overall system will be stable. Point i) obviously requires prior knowledge about the plant that has to be controlled. It is strongly related to minimum-phase properties of the plant.It is called the perfect model matching problem .It can be considered either as a transfert matrix matching, or by establishing algebraic conditions on the error equation.It leads in that case to the well-known Erzberger's conditions on the plant and the model [16] .Note that these are sufficient only and may or may not be verified[27].AMFC problem can then be seen as an MR.AS problem for which a solution of asymptotical model matching exists. 22) A s s u m p t i o n s on the model and the plant W e give here a list of the properties that are generally required in the parallel AMFC problem for SISO systems [17][18][19] and that allow one to solve points i) and ii) above.The plant and the
800
model are described as follows: (P) (RM)
:~p=Apxp+Bpu xm=Amxm+Bm r
; yp=Cpxp ; ym=Cmxm
where r is a bounded piecewise continuous signal. i) the plant is minimum-phase and stabilizable ii)an upper-bound on the plant order is known iii)the relative degree p of the plant is known, and p < Pm (Pm is the relative degree of the model) iv)the sign of the high-frequency gain (transfert function approach) is known. Note that i) is necessary for (P + controller) to be detectable[17,p.375], as the eventuals poles-zeros cancellation that may appear between the plant and the controller (and lead to unobservable modes) are stable.It is shown in [18] that iii) is the only necessary assumption.If the plant zeros are exactly known (which may not be the case in practical applications), then i) is not necessary, but the model has to contain the plant unstable zeros, in order to avoid unstable poles-zeros cancellations.One could ask here whether this still represents a reference-model problem, or simply pole-placement.ii) can be rendered unnecessary by relaxing the objectives on the error [18]. iv) is not a necessary condition : controllers can be found that achieve the MRAC problem although it is not verified.The most general approach for which this condition is unnecessary can be found in [28]. Several remarks arise : i) All that has been done is concerning SISO systems.It can be translated to the MIMO case [17] [19], taking into account the particular structure of multivariable systems, especially for those systems that admit a Hermite normal form .But this is far from trivial, particularly in the adaptive case, and is still being under investigations. ii) The case usually studied in the literature is the parallel MRAC problem, with an input / output (transfert function) approach.The approach which seems to be the most widely used [20]- [23] is based on an augmented state vector xT= [ XpT, wiT, w2T], where w I and w 2 are n-dimensionnal vectors obtained by filtering the input and the output of the plant.It is shown [20] that a non-minimal representation of the reference model can be constructed in terms of the plant true parameters, that allows to derive an appropriate error equation, and then conclude on the convergence of the state error vector. It presents the great advantage of using the plant output instead of the state vector in the control law. Schemes designed from a state space approach have been seldom studied.An approach proposed in [16, p. 218] is based on hyperstability's arguments for the stability analysis.The control law
801
consists in output and state feedbacks, the gains of which are adaptively adjusted by the so-called integral+proportionnal adaptation mechanism .The main drawbacks of this approach are that the input matrix Bp has to be known (or at least its structure) for stability conditions,and that the plant state vector has to be measurable.Though these assumptions could appear at first sight quite stringent, there exist particular cases (see next section) for which they are verified. However, it presents the advantage of being available for MIMO systems, as no assumption is done on the output and input dimensions. remarks a) In this approach, the relative degree problem related to the SPR condition is avoided, as one can consider the output to be such that Cp Bp # 0, ie the relative degree of that "equivalent" system is one. b) MR.AS are often presented as an example of direct control.As noted in [17], it is not a necessary condition, and MRAS can also be designed through an indirect approach .In the litterature, MRAS are almost always considered through a direct approach, as it can be shown that for minimum-phase systems, a reparametrization of the process is always possible so that the estimated parameters turn out to be those of the controller .The most commonly used algorithms are gradient type, as the proportionnal + integral adaptation law [16].This is due to some nice properties of such algofithrr~ such as passivity, which is not met in least-squares algofithrr~ for instance. 23) Robustness issues Adaptive laws lead in the "ideal" case to boundedness of all the signals in the closed-loop system.Actual implementations of theses schemes make it necessary to take the "non-ideal" (ie perturbed) case into account.Several types of unmodeUed dynamics can be considered : i) internal and external bounded disturbances on the state or the output of the plant. ii)Plant parameters variations in those schemes where the process to be controlled is tinae-varying. iii)Plant dynamic uncertainties. A general way to design adaptive schemes with unmodeIled dynamics is to relax performances objectives [ 17][24], ie such algorithms should at least insure BIBO stability, which constitutes in fact the main theoritical problem.A fast approach is to take "small" disturbances only into account [t7][40,p.16] : ~is is based on the total stability concept and is related to the uniform asymptotic stability of the unperturbed system.Considering general unmodelled dynamics, two approaches are available: either modifying the adaptive law (e 1 modification[22],cr modification[21], bounds on
110"11124],dead-zone[25]), or increasing the "richness" of the reference input [26].Note also
that when no disturbance is present, those algorithrr~should at least lead to finite parameters and output errors, and verify exact or asymptotical model matching conditions if possible.Application
802
of the first type of modifications to the series-parallel case should not cause any difficulty.However, persistent excitation arguments on the reference model output are to be reconsidered, as Ym depends on yp and therefore cannot be assumed bounded (which is a necessary condition of excitation). 3) NONLINEAR AMFC 31)Generalities The problem of compensating a nonlinear system in order to make it match a linear reference model has been invastigated (see eg [29],[19, chp 7] for references, [27]).Techniques described therein may not always be very useful for the manipulator's case which represents, as noted in the introduction, a narrow class of nonlinear plants, for which global results can be derived. (Indeed, the general techniques applied to smooth affine in the input nonlinear plants of the form ~ = f(x) + g(x) u often lead to local results only).[30] provides a comparison work on several dynamic linearizing feedbacks studied for rigid robots, in the non-adaptive case.Note that all those linearizing tools present the major drawback that if there is any uncertainty on f and g, then the linearizing properties vanish. Adaptive control has recently been studied.J31] solves the nonlinear AMFC problem for a particular class of time-varying perturbed plants, employing alternatively a discontinuous VSS approach, and a continuous control law derived from a work presented in [16] for the linear case. A more general approach can be found in [19], for continuous SISO and MIMO square systems, linearizable and decouplable by static state feedback, which can be seen as an extension of the work done in [32] for robots controLIt represents an attractive pioneer work, but its major drawback is that estimated terms appear on the denominator of the control law, and must therefore absolutly be kept away from zero.This problem is far from obvious, and has been recently solved for the linear case in [28]. Due to the difficulties encountered, the nonlinear AMFC problem has been much less studied than its linear counterpart.However, it seems that similar problems arise for the design of suitable schemes, especially concerning assumptions i) and iii) in paragraph 22) (SISO plants).The main distinction lies in the definitions of those properties, particularly for the MIMO case, where several definitions of the zero-dynamics can be given [ 19, p.305].Fortunatly, for systems decouplable by static state feedback (as rigid robots) they are equivalent.
32) manipulators dynamics Let q, ~t, and ~ denote the robot's joints position, velocity and acceleration n-vectors.Rigid frictionless manipulators dynamics are then given by the set of nonlinear differential equations : M(q) ~t'+ C(q, ~) ~ = u- g(q)
(1)
803
where :
M(q) is the nxn inertia matrix, C(q, ~t) (} contains the centripets and Coriolis torques,
g(q) stands for gravity torques, u is the input vector torque. The main properties of equation (I) are : i) M(q) is a symmetric strictly positive definite matrix, bounded in q, composed of smooth functions ofq. ii) M(q) - 2 C(q, ~t) is a skewsymmetric matrix. iii) M(q), C(q, ~1) and g(q) can be expressed as Y0, where Y is composed of known functions of q and q, while 0 stands for unknown physical parameters. iv) (1) is static state feedback linearizable and decouplable by the so-called "computed-torque" control law :
u = C(q, ~t) q + g(q) + M(q) a
, a being any output signal.
v) q and q are both physically accessible for measure. i) and ii) are fundamental properties used to derive stability proofs of many schemes .i) insures among other things that global solutions of (1) exist [17, p.55], using the smoothness property of M(q).ii) shows that any system with input v and ouput M(q)~ + C(q, ~t)v, v a n-dimensional vector, represents a passive system [33]. iii) and iv) have been extensively used in the literature, at least for a certain class of schemes [8]-[15]. v) means that the state vector xT = (qT, ~1T) is entirely measurable.This is worthnoting, as minimum-phase properties of systems (linear or not) are obviously related to their output.In the finear case, transfert zeros'dynamics are given by forced (u ;e 0) state trajectories that nullify the output. In the nonlinear case, techniques have been developped to caracterize t h e so-called zero-dynamics [19, chp 7].They are also related to the definition of the output y = h(x).So if the state is known, the designer can choose among several possible definitions of h.In the case of robots, it has been shown that choosing y = q leads to a minimum-phase plant [ 19, p.322].This fact has certainly played a key role in the design of robots AMFC in joint space coordinates.Note that as long as the jacobian matrix J(q) remains nonsingular, one can easily translate joint space design into task space design [10]. The next paragraphs are devoted to a tentative classification of several schemes studied in the field of MRAS applied to robots.We will not consider in the following pioneer works (see eg [34]) that were based on "large" approximations of equation (1), due to their lack of global stability. 33) Parallel A M F C 331)Schemes derived from the linear case Chosing xT = (qT, ~tT) as a state vector, (1) can be rewritten :
~=
dl - M-l(q) [C(q, c].) ~ - g(q)]
+
u
M-l(q)
(2)
804 or :
= A(x) + B(x) u
(3)
It is then attractive to consider the manipulator as a 2n-dimensional plant, and try to make it match a linear decoupled model : im = Am Xm + Bm r
Ama 2n x 2n
Hurwitz matrix
Such is the purpose of papers [ 1]-[5].These control laws were inspired by the work done in [16,p.218].The main distinction lies in the adaptation laws, that include specific terms to compensate for the nonlinearity of (3). Several solutions are proposed, some of which are based on discontinuous (or "unit vector") control laws [1][2][5 part 1], or on continuous control laws [3][4][5 part 2].Properties i) and v) have been used.Note that property iii) has been used in [3] to rewrite the error equation derived from (3) (see [3] for details). As noted in [1], these algorithms require less computations than computed torque techniques. However, they also lead to less precise results. A major drawback of the discontinuous (sliding modes) approach is the so-called chattering phenomenon. No robustness analysis is available in the above references.It has been shown that including a dead- zone in the update law can reduce these negative effects [35].
332) Adaptive computed torque A second type of parallel schemes has been developped in [6]-[7].They appear as the first application of computed torque techniques in an adaptive context. Unfortunatly, on the contrary of schemes [8]-[ 15], no use is done of properties ii) and iii). Unknown time-varying terms are then updated using adaptive algorithms similar to those proposed in [16], and the closed-loop stability is insured by adding a large enough feedback PD term in the control law (see references for details). The reference model is then given as in the exact linearization case by a n th order decoupled double integrator that the plant + controller will asymptotically track. The main drawback of this approach is that the stability proof (as in [4]) hinges on the assumption that M(q) and several other time-varying terms in (1) remain constant during the adaptation process. 4) SERIES-PARALLEL AMFC Another way of ~onsidering equation (1) is to simply write it as : ~1"= M-l(q) [ C(q, q) q + g(q) ] + M'l(q) u
(4)
Chosing the "state" vector as x = q, (4) reads as : x = f(x) + g(x) u
f and g being this time n-dimensional vectors.
(5)
8O5 A whole class of adaptive schemes [8]-[15] can be interpreted as laws achieving asymptotical tracking between (5) and a series-parallel reference model given by : xr = - A x + r , where r = A Cla +/:ia
(6)
One key point in the stability analysis is that the error vector is given by : v = x - x r = H(s)- 1r q , with H(s) = ( sI + A) -1 and qtw = q - qd (see [36] for details) These algorithms can be considered as the most accomplished ones to date. Properties i), through v) are used. Various control laws have been proposed, from the pioneer works on the subject, derived almost simultaneously in [8] and [9], which can be seen as a continuation of schemes developped in [6]-[7], to more elaborate schemes [10]-[15]. The basic idea remains the same from one to another. Modifications essentially concerns update laws, with the so-called composite [11] or repetitive [14] controllers, or update laws using only desired trajectories (instead of measured signals) in the regressor matrix [12]. Exponentially stable schemes have also been studied [13], [15]. 5) S T A B I L I T Y ANALYSIS An interesting point is that all those schemes [1]-[15] can be interpreted as a feedback connection of a strictly-passive block with a passive one (figure 3). Some of them have explicitly been designed using that tool [I]-[3],[5],[6],[7]. The others [4], [8]-[15] have been studied employing a Lyapunov approach. It can be shown [37] that they can be turned into such a passive interpretation.
~ L~
str'~y-passive
passive
1
figure 3
All those convergence proofs have been lead in the ideal case.Robustness analysis similar to those already done in the linear case have been rarely published.Pioneer works on that subject can be found in [38] and [39].[38] is devoted to the robustness analysis of the scheme developped in [32].The main drawback of the approach is that the size of the instability domain depends on M-l(q), where M(q) is the estimate of the inertia matrix. Therefore, M(q) must be kept nonsingular.In [39], the authors have shown that the ~-modification can be applied to the laws
806 [8] and [9], considering almost all the possible unmodeUed dynamics i) to iii) (section 23)).At this stage several renmrks arise : i) Parasitic noises on the output (especially on the velocity q measures) have not been considered. ii) It would be interesting to see if the other techniques developped in the linear case can also be applied to the robots case. iii) Those robustness tools have been developped for parallel schemes. Results obtained in [39 ] confirm the remark done at the end of section 2). 6) C O N C L U S I O N In this paper, we have provided a tentative summary of the most significant adaptive control laws for manipulators studied in the last ten years. The most important fact is that all those controlled systems can be interpreted through a common framework (ie MRAS ), and analysed using passivity tools, either they were explicitly designed like that, or through a Lyapunov approach. references [ 1] Nicosia S.,Tomei P.:"MRAC for industrial robots";Automatica,vol.20,n~ 1984 [2] Balestrino A.,DeMaria G.,Sciavicco L.:"An AMFC for robotic manipulators";ASME JDSM&C,vol. 105; 1983 [3] Nicolo F.,Katende J.:"A robust MRAC for industrial robots";2nd IASTED Symposium on Robotics and Automation, Lugano; 1983 [4] Lim YK.,Eslami M.:"Adaptive controllers designs for robots manipulators systems yielding reduced cartesian errors";IEEE TAC, vol.AC-32,n"2; 1987 [5] Balestrino A.,DeMaria G.,Sciavicco L.:"Hyperstable AMFC of nonlinear plants";Systems & Control Letters,vol.l,n~ 1982 [6] Tomizuka M.,Horowitz R.,Landau ID.:"On the use of MRAC techniques for mechanical manipulators";2nd IASTED Symposium Modelling Identification Control and Robotics,Davos,March;1982 [7] Horowitz R.,Tomizuka M.:'An adaptive control scheme for mechanical manipulators;compensation of nonlinearity and decoupling contror';ASME JDSM&C,vol. 108,n~ [8] Sadegh N.,Horowitz R.:"Stability analysis of an adaptive controller for robotic manipulators";IEEE International Conference on Robotics and Automation, Raleigh; 1987 [9] Slotine JJE.,Li W.:"Adaptive manipulator controha case study";IEEE TAC,vol.33,n~ 1; 1988 [10] Kelly R.,Carelli R.:"Unified approach to adaptive control of robotic manipulators";27th IEEE Conference on Decision and Control,Austin,Texas; 1988 [11] Slotine JJE.,Li W.:"Composite adaptive control of robot manipulator";Automatica,vol.25,n~ [ 13] Sadegh N.,Horowitz R.:"An exponentially stable adaptive control law for robot manipulators";submitted for publication [ 12] Sadegh N.,Horowitz R.:"Stability and robustness analysis of a class of adaptive controllers for robotis manipulators";To be published in the International Journal of Robotics research,1987 [14] Sadegh N.,Horowitz R.,Kao WW.,Tomizuka M.:"A unified approach to design of adaptive and repetitive controllers for robotic manipulators";presented at the USA-Japan symposium on flexible automation, Minneapolis ; 1988 [15] Bayard DS.,Wcn JT.:"New class of control laws for robotic manipulators: part 2: adaptive cam"; International Journal of Control,vol.47,n~ 1988 [16] Landau ID.:"Adaptive control: the model reference approach";Dekker,New-York;1979 [17] Narendra SK.,Annaswamy MA.:"Stable adaptive systems";Prentice Ha11;1989 [18] Miller DE.,Davison EJ.:"On necessary assumptions in continuous time MRAC";28th IEEE Conference on Decision and Control,Tampa,Florida,December;I989 [ 19] Sastry S.,Bodson M.:"Adaptive control: stability, convergence and robustness";Prentice Hall; 1989
807 [20] Narendra KS.,Valavani LS.:"Stable adaptive controller design-Direct control";IEEE TAC,vol.AC-23,n~ [21] loannou PA.,Tsakalis K.:"A robust direct adaptive controUer";IEEE TAC,vol AC-3 l,pp.1133-1143;1986 [22] Narendra SK.,Annaswamy MA.:"A new adaptive law for robust adaptation without persistent excitation";IEEE TAC,voI.AC-32,n~ 1987 [23] Sastry S.:'MRAC :S tability,parameter convergence and robustness";IMA Journal of Mathematical Control & In formation,vol. 1,pp.27-66;1984 [24] Kreisselmeir G.,Narendra KS.:"Stable adaptive MRAC in the presence of bounded disturbances";IEEE TAC,voI.AC-27,n~ 1982 [25] Peterson BB.,Narendra KS.:"Bounded error adaptive control";IEEE TAC,voI.AC-27,n~ [26] Narendm SK.,Armaswamy MA.:"Robust adaptive control in the presence of bounded disturbances";IEEE TAC,vol. AC-31,n~ 1986 [27] Marino R.,Nicosia S.:"Linear model following control and feedback equivalence to linear controllable systems";International Journal of Control,vol.39,n~ 1984 [28] Lozano-Leal R.,Collado S.,Mondie S.:'Model reference robust adaptive control without a-priori knowledge of the high frequency gain~;IEEE TAC, to appear;1990 [29] Di Benedetto M.:"A condition for the solvability of the model matching problem";Conferenee on nonlinear systems analysis,Nantes, France; 1988 [30] Kreutz K.:"On manipulator control by exact linearization";IEEE TAC,vol.34,n~ [31] Balestrino A.,DeMaria G.,Zinober ASl.:"Nonlinear AMFC";Automatica,vol.20,n~ [32] Craig JL:"Adaptive control of mechanical manipulators";Addison Wesley Publishing Company;1988 [33] Landau ID.,Horowitz R.:"Synthesis of adaptive controllers for robots manipulators usinf a passive feedback systems approach";International Journal of adaptive Control and Signal Precessing,vol.3,n~ [34] Dubowsky S.,DesForges DT.:"The application of MRAC to robotic manipulators";ASME JDS M&C,vol. 10 l,September; 1979 [35] Slofine JJE.:"On modelling and adaptation in robot control";IEEE Conference on Robotics and Automation,San Fransisco,CA; 1986 [36] Ortega R.,Spong MW.:"Adaptive motion control of rigid robots: a tutorial";27th IEEE Conference on Decision and Control, Austin,Texas;1988 [37] Brogliato B.,Landau ID.,Lozano-Leal R.:"Adaptive motion control of robot manipulators:a unified approach based on passivity"; American Control Conference, San Diego,CA; 1990 [38] Kang H.,Dawson D.,Lewis FL.:"A robust adaptive controller for rigid robots";28th IEEE Conference on Decision and Control,Tampa,Florida;1989 [39] Reed JS.,Ioannou PA.:"Instability analysis and robust adaptive control of robotic manipulators"; 27th IEEE Conference on Decision and Control,Austin, Texas; 1988 [40] Anderson BDO.,Bitmead RR.,Johnson CR.,Kokotovic PV.,Kosut RL.,Mareels IMY.,Praly L.,Riedle BD.:"Stability of adaptive systems : passivity and averaging analysis";MIT Press; 1986
Adaptive Control of Feedback Equivalent Systems J.-B. Pomet*
A b s t r a c t We address the problem of stabilizing a nonlinear system depending on some unknown parameters in suck a way that M1 the systems obtained by varying these parameters are equivalent to one system, supposed to be state-feedback stabilizable. The description of the adaptation iaws make use of passivity. We consider both the "general" case, and the case where some "matching assumptions" hold.
and
L. P r a l y t
on the p a r a m e t e r p; g is the matrix field defined by (3). One particular p* in Ill will be called the "true value of the p a r a m e t e r p", and our problem is to stabilize the system gp,, p* being unknown. This will be done by means of a dynamic controller, i.e. of a system with a certain state, to be determined, input x, and output u : State: x U ~
1
~.~p.
X
Introduction
We consider a family of nonlinear affine-inthe-control systems, indexed by a parameter vector p: P =
(Pl
...
Pl) T
e
RI
.
(1)
State: to determine
u
Adaptive Controlier
Ix
Fig. 1 : Closed-loop system
T h e state of the overall closed-loop system The system Sp corresponding to a given is composed of x and the dynamic variables value of p is described by : (or the state) of the controller. By "stabilize Sv." we then mean that both x and Sp : ~ = f ( p , x ) + g(p,x) u (2) the dynamic variables of the adaptive conm troller must be bounded, and x must tend f(p, x) + x (3) k=l to a certain point 0 of M " , for all the solutions of the closed loop system (global where the state x lives in an n-dimensioned properties), or only for some (local propC ~ manifold M " and is completely meaerties). By "p* being unknown", we m e a n sured, that the adaptive controller must not deu = ( u , , . . . , u,~) (4) pend on, or use the value of, p*. is in tZ"~, and f and the gk's are k n o w n This general problem, as well as the smooth vector fields smoothly depending distinction between global and local results, is extensively discussed in [10], [8J, *Univ. of Toronto, Department of Mathematics, or [11], where a complete bibliography m a y Toronto, Ontario, M5S IA1 Canada tEcole des Mines, CAI Section d'Automatique, be found. T h e present paper is specifically 35 rue St Honor4, 77305 Fontainebleau, France devoted to the case when all the systems
809
@ are equivalent to one another by state feedabck and diffeomorphism. In section 2, we state precisely our assumptions about the systems Sp. In section 3, we describe some general adaptation schemes. Section 4 presents some adaptive controllers designed for the general case where no more assumption is satisfied. Section 5 is devoted to the case where some "matching assumption" is satisfied and presents some modified controllers. Finally, in section 6 we give some geometric conditions allowing to construct a diffeomorphism meeting this "matching assumption".
is such that
rankg(p, x)Z(p, x) = r = k g ( p , x) (7) 2. There is a system
= i(=) +
independant of p such that~ for any p,
u
=
(9)
=
c~(p,x) q- fl(p,x) w (10)
transforms (2) into =
2
(s)
f(~) + O(~)w
(11)
Assumptions
Linear Parametrisation (LP) assumption: The fields f and g in (2) depend linearly in the parameter p : I
f(p,x)
-= a~
+ ~--~piai(x)
(5)
i=l 1
k(p, x) =
+
b (x) , (6)
Local Stabilizability of the Transformed System (STS(~2) ) assumpt i o n : There ezist k n o w n fltnctions Vnom and U, of class (at least) C I and C 2 respectively, from a neighborhood ~ of 0 in M ~ to R "~ and to l:t respectively, such that: 1. U(~) is n o n n v a t i v e , and zero if and only if ~ is zero, and for any I ( > O,
i=1
{ ~ / Y(~) < I ; } where the ai's and the b~ 's are smooth vector fields 5ndependant of p). Numerous practical examples, as position control of a DC-motor [6] or control of biochemical processes [4] satisfy this linear dependance on the parameters. All the existing schemes for adaptive nonlinear control require linear parametrization. Feedback and Diffeomorphism Equiv a l e n c e ( F D E ) a s s u m p t i o n : There exists three smooth maps, o~ from R z x M ~ to I t ' , t3 f r o m R ' • M ~ to Mm• and ~o from R l x M ~ to M ~ such that
(12)
is a bounded subset of M ' L 2. For all ~ in M ~, we have:
U~=~(() : L~U(() <_ -cU(()
(13)
where c is a strictly positive constant and ~ denotes the "nominal transformed closed loop field":
~(~) ~- f(~) q- f(~)Vnom(~)
(14)
Global Stabilizability of the Trans1. For each p, ~o(p,.) is a diffeomorphism f o r m e d S y s t e m (global S T S ) a s s u m p on M '~, and the "matrix field" 13(p, .) Lion : The same as STS(~2), with ~2 = M ~.
810
For the sake of simplicity, we suppose that ~, a and fl are defined globally in assumption FDE. If they were defined only for p in a certain domain, but for any x, we could modify our algorithms, adding a projection to maintain the estimates of the p a r a m e t e r in this domain (see [10] or [11]). If they were defined locally with respect to x too, only local results could be obtained. The most touched-on situation, in the present litterarure is concerned with fully state-feedback linearisable systems. Assumption F D E includes it as a particular case and is clearly more general. For linearisable systems, STS is satisfied too, V~om being a stabilizing linear feedback. Notice that with assumptions F D E and STS, defining U,om by :
Setting u
=
ul + u2 ,
(17)
the family (@) is imbedded in the family (S~,q) in an obvious w a y :
(is)
sp - s L
We m a y consider that the system @, to be controlled is just $~,.q,, with q* = p*. In the following, it will often be convenient to write p* and q*, keeping in mind that q* : p*.
The controls ul and u2 will be computed thanks to some "estimates" of the parameters : Ul will depend on/5, and u2 on/5 and ~, some "estimates" of p* and q*. The adaptation is the part of the controller which gives these estimates t5 and ~. Unom(P, X) : O~(p, X)+]~(p, X)Vnom(qO(p, X)), T h e following remarks are some guides to (:5) design the adaptation. for any p the feedback u = U,om(P, X) staConsidering W, c~ and/3 given by assumpbilises Sp. In particular, u = U.om(P*, X) tion FDE, if we define ~ and wl by stabilises Sp,. This is not however a solution to our problem, because p* is un= (i9) known, and the controller must not depend U 1 = Or(/5, Z) -1- fl(/5, X) W 1 , (20) on its value. we have, from assumptions F D E and LP,
3
Adaptation
laws
=
From the f and g in (2) and (3), we define a new family (S'p.q), in which, for any p and q, the system S'p,q is : with
S;.q : ic = I(P, z) + g(p, Z)Ul + g(q, z)u2
/ ( ( ) + 9 ( ( ) W l + A(/5,x,wl)(p* -/5)
+3-;(/5,
A(p,x,u)
+
=
(16) Remark 1 : In general, q = (qa,... ,qt) will have the same dimension as p, but we m a y omit the qi's such that the bki's (see (6)) are identically zero, i.e. such that g(q,x) does not depend on ql In particular, if g does not depend at all on the parameters, q may be dropped in the parametrization (16). This is also the case if for some reason w2 is identically zero. II
...,
x) a'(x) +
kbi(x) k----1
, (22)
where the u~'s are the functions of/5, x and wl given by (20). T h e adaptation will be based on linear estimation, and must rely on an "observation equation", i.e. an equation, satisfied
811
by the signals present in the system, which is linear with respect to (p*, q*). If h is any function of 4, we may choose as an observation equation the one describing the evolution of h(~(t)), n a m e l y :
p* z [(~) - (~.)]
strictly passive filter
C
Fig. 2 : The filtered equation error such that e may be computed from available signals (e is defined by fig. 2 but cannot be obtained this way, since p* and q* Oh , c &o , ^, + g(~ ~G~P ~ ) g ( q * , ~ ) ~ are not available). The ideas of hyperstability and passsivOh. 0~o + ~ ( ~ ) ~ p (/~, x)D. ( 2 3 ) ity were introduced in adaptive control in [7], and one may find in [8] what we precisely mean here by "stricly passive". To stress the fact that this equation is linA possible choice of the filter in fig. 2 is : ear with respect to p* and q*, we may rewrite it as : Oh --}- ~-(~)A(~, x, Wl) (p* - ~)
+re
9
z(h,~,~,~,~,~) = z ( ~ , ~ , { , ~ , u ) where
Z
Oh
=
Oh + -~(~)A(p, x, w~)~
(,.)r
=
Z[
q. ]
(27)
where v is a positive function of the avail able signals. Using (25). e may then be (24) obtained by : ~ =
-re+(J~-z)+Z(/3)
(28)
e =
q - h(~) .
(29)
(28)-(29) only makes use of available signals (from (25), h - z only depends on Oh O~v controls and state variables), and gives k=l an r which, from (24) and (25), satisfies Oh. Oto x " ~-(~)~p(16, )/~, (25) (27). This adaptation scheme is the one described in [11] (with S~,q here instead of G there). We then have the following propand the matrix field Z is naturally defined erty : from (23), (24), and (25). Then, a general, and implementable, L e m m a 1 ( [ l i D No matter what the control laws U 1 and u2, or the smooth function gradient-type adaptation algorithm is h are, for any x(t) solution of the controlled system (ul,u2)-$~,,~,, and any so(D<) ) = -ZTe (26) lution of (26)-(27), defined on [0,T) (0 < T < +oo), we h~ve, for all t in [0, T), where the filtered equation error e is of the , , ~ + rll~N ~ ) _ (,.) + ~lbft)ll same dimension as h, and is given as the $ 0 1 2 output of a "strictly passive" filter : -< ~ (,lol)- (~:)~ + ~]I4o)II -
/0
(30)
812
The various adaptation schemes we will use in the two following sections a r e obtained by precising the choice of both the function h and the filter in figure 2. This filter may be different from (27).
4
Adaptive the
General
Control
in
close enough to p* is bounded and such that x(t) goes to zero. As a global result, we have (from [8]),
If FDE and LP are satisfied, and S T S is satisfied globaIIyaIly, a n d if in a d d i t i o n all the vector fields are globally Lipschitz, then any solution of the closed-loop system is bounded and such that x(t) goes to zero.
Case
2. Another possible choice for h is the real positive function U given by assumption Here, we make no more structural assumpSTS. The filter of figure 2 is again of the tion besides LP, FDE and STS, and do not type (27), and we altogether obtain the foluse the control u2; ul is given by (32) : lowing controller, which is the one we preu2 -~ 0 , (31) sented in [11] or [10], particularized to the = u~ = Uno~(p,z) . (32) case of feedback equivalent systems : A d a p t i v e C o n t r o l l e r ACI(U) : To describe completely the adaptive controller, we have to define the adaptation, U ---- Unom(P, 2:) (33) i.e. how/3 in (32) is obtained. The adap" e Z(~5, x, Unom(/~,x)) T (34) tation scheme (section 3) will contain no OU = - r e + 5-;(~(~, x)) ~(~(p, x)) (the term depending on q in (16) being now identically zero, we just omit q, see remark + w ( ~ , ~).~ (35) 1). We propose two choices for the function h, with corresponding choices of the filter where of figure 2, both of the type (27). e = q - U(cp(5, x)) (36) 1. A first possible choice is to take for h r - 1 + IIZIlllWII (37) some coordinates on M ~ and to define the OU filter (fig. 2) by (27), with no ~ and q*, and w(p,x)~ ( ~ ( p , ~ ) ) ~0~o ~p, xi~38 ~ ) r a constant real positive number. We only Z(p,x,u) = (zl(p,x,u),... mention the results obtained when the systems are fully feedback linearizable, and we ..., z'(p, x, u)) (39) choose for h some coordinates in which (11) z~(p,z,u) = OU O~ and $ are linear (~ = $(~) implies ]~ = Ah 57(~(p, ~ ) ) ~ (>, x) with A a Hurwitz matrix). Then, the adaptive controller (26).(60)-(28)-(29) for feed[a'(x) + ~__lukb~(x)t40) back linearisable systems is similar to those described in [3], though the point of view Notice that q is the state of the small sysis different there. We have the following tem necessary to realize (27) using only results for these controllers (see [3] or [8]): available signals.
If FDE and LP are satisfied, and S T S is satisfied locally, i.e. STS(~) is satisfied for T h e o r e m 1 ([11]) some ~, then any solution of the closed- If assumptions LP and FDE hold, and asloop system (figure I) such that 4 0 ) and sumption STS holds locally, there exists an Ko) are closed enough to zero and p(O) is open neighborhood of (p*,O,O) such that
813 any solution (~,x,~) with initial condition (i5(0), x(0), 7/(0)) in this neighborhood, exists on [0, oo), remains in a compact set and its (x,~)-component tends to zero. In addition, the point (p*,O,O) is a (non asymptotically) stable equilibrium point. T h e o r e m 2 ( [ i i ] ) I[ assumptions LP and FDE hold, and assumption S T S holds globally, and if in addition there exists a C o function d on II such that for all (p, x) in II x M , with Z defined in (39),
IIZ(p,=)ll ~ ( p , x ) < d(p) (1 + V(p, x) ~) ,
meeting, Ior any (p, q,b, ~), x)v2(p,q,[o,x)+-fi-j(p,x)[o = 0
(43) We will use here/~ to compute ul with (see (15)) : Ul = Unom(P, X) ,
(44)
and the estimate ~ of q* to compute u2 by :
u~ = v2(p,~,/~,~) .
(45)
(41) We then have =
,~(~) -]- A ( p , x, u . . . . (]), x)) (p* -- p)
then all the solutions are defined on [0, (x)), + A2(15, x, v2(i5, ~,p, x)) (p* - ~X46) remain in a compact set and their (x,q)with A2 the row vector defined by component goes to zero. This algorithm therefore either gives only local results or requires the bound (41) on the growth of the different vector fields. This was also the case for the first algorithm (h = some coordinates), but, as stressed in [11] or [10], if the systems are such that $ is not Lipschitz in the coordinates h, we cannot guarantee global stability in the previous algorithm whereas we can here.
~ ( p , x) [g(ql, x) - g(q~, x)] ~ = A~(p, ~, u~) (q~ - q~) (47) We will again use the a d a p t a t i o n schemes proposed in section 3 to obtain (#, ~). We propose three different choices :
1. First, as in section 4, we m a y chose for h some coordinates on M " , using the filter (27) with r some positive constant. AgMn, we only mention the results obtalned if the systems are fully feedback linearizable and the coordinates h are these 5 Adaptive control with in which ~ is linear. Thee adaptive controller is given by (17)-(44)-(45)-(26)-(28)matching assumptions (29). The same controllers are described in We call matching assumptions the fact that [3], but without the reparametrisation (and F D E holds with the following restriction on consequently with # in place of ~), and unthe dependance on p of ~ : der an assumption different from FDEM2, see remark 2 below9 We have the following A s s u m p t i o n F D E M 2 : Assumption F D E result : is satisfied, and ~o has the property that If FDEM2 and LP are satisfied, and S T S there exists a smooth map v2 : is satisfied globally (resp. STS(~2) is satisfied for some ~), then any solution of (p,q,~,x) ~ v2(p,q,p,x) E R ~ (42) the closed-loop system (resp. any such that
814
z(0) a~d 7(0) a~e closed enoueh to ~e~o and /5(0) is close enough to p*) is bounded and such that x(t) goes to zero. Notice that, unlike in the situation of the preceeding section, we need no additionnal assumption like global Lipschitzness to get global stability. This would not be the case if the vector field h was not linear in the coordinates h (see [8]). This is one of the reasons to prefer the following choice of h.
(p*,p*, O, O) is a (non assymptoticaIly) stable equilibrium point of this dynamical system.
This vields the following controller : A d a p t i v e C o n t r o l l e r .AC~(U) :
We then define the following modified algorithm which is defined only for U(~) smaller than [To : A d a p t i v e C o n t r o l l e r .AC~'(U, [70) : The same as .AC'I(U), but U is replaced by
Here, we get a global result as soon as assumption STS is global, without a restriction like (41) (but under assumption FDEM2, which is stronger than FDE). Yet, STS being globally satisfied is very restrictive. If it is satisfied only locally, AC'I(U ) gives a good beheviour for only some so2. Another choice for h is h = U, the filter lutions : f has to start close to 0, which being given by (27) with r equal to 1. The is natural, but (/5, q) also has to start close expression of Z is then to (p*, q*), which is more unconvenient. We will modify ACI(U ) to get a better local reZ(/5, q, b, x) = ( Z I Y ) , (48) sult. Suppose that STS(f~) is satisfied for a certain f/, and Uo is a positive number with, A2 being defined by (47), such that Y(/5, x, v2(/5,p, q, x)) (49) 0 ~ , ^, u ( ~ ) < Uo ~ ~ ~ a . (55) = -~ztp x)A2(/5, z,v2@,/5,~,z)).
p
=
U.om(/5,~) + ,~(/5,/5,0,~)(50)
=
-- (~ Z(/5, X, Unoitl(/5 , X))
(51)
- eY(/5,~,,:(/5,b,O,~))"
(52)
OU~ ).~( s~ ) + -~;(
(53)
q = =
-~
uq
(s6)
in (53) and in the definition of Z and Y. We have the following result :
where e = U - U(~),
UoU uo - u
(54)
and Z and Y are given by (39)-(40) and (49)-(47). and the following result :
T h e o r e m 4 If FDEM2, LP and STS(~) hold, and Uo is chosen according to (55), the solutions @(t), ~(t),~(t), =(t)) of the
closed.loop
system S,.-.aCi(U, Uo) such
T h e o r e m 3 If FDEM2, LP and S T S that U(~(0)) < Uo are defined on [0, oz), (resp. STS(f~) for a certain neighborhood remain in a compact 3et and their (x,7])ft of(O, O) in R zx M '~) hold, then all the so- component goes to zero, U(~) remaining lutions (/5(t), ~(t), ~(t), x(t)) of the closed. smaller than Uo. In addition, (p*,p*, O, O) lOOp system Sp.-~Ctl(U ) (resp. the solu- is a (non assymptotically) stable equilibtions with initial conditions in a certain rium point of this dynamical system. neighborhood o f ( / , / , O, 0)) are defined on [0, oe), remain in a compact set and their S k e t c h o f p r o o f : We only state the proof (x, U)-component goes to zero9 In addition, of theorem 3; the proof of theorem 4 goes
815 the same way, replacing U by (56) and considering that (56) is infinite when U is Uo. Consider a solution of the closed-loop system. Lemma 1 (30) implies that i5 and e = 9 - U are bounded on [0, T), the right maximal interval of definition of the solution, and that the time-ruction e(t) is in L2([0, T)). In addition, from (53), (13) and (54), we have :
This controller is the same as these described in [5], except that the double parametrization is not used there (replace by/3 and (62)-(63) by/~ = - Z r - y r . The approach to the synthesi presented here is however different : the authors design the adaptation, and the control u2, to make the positive function - Oil + 1-Iq* =
y(~) + 89
i1 < - c q
+ (l+e) e ,
(64)
(57) decrease
ong t h e s o l u t i o n s
(in fact, U(4)
is a quadratic function of { and the third term is absent there). The authors encounter the problem about implicit definition of i3 which we mention in remark 2. We can, as for AC'I, modify it AC~ into iA d a p t i v e C o n t r o l l e r AC,,i.rr : ~ , Uo). : t The same as .AC:(U), but U is replaced 3. A third choice for h in the algorithm of by section 3 is UoU which, together with e being L 2, implies that q is bounded too. This implies that the solution itself is bounded on [0, T) and therefore that T = +oo. Then (57) and (30) applied for T = +oo give e and 7, and therefore U, going to zero. []
i
-
<
:
.
-
u
(65)
(58)
in the definition of Z and Y. In this case, we chose as a passive filter (fig. T h e o r e m 5 The adaptive con2) the following, with state X on M = : trollers .aC'=(U) and .4Ci(U , Uo) have the ,~aTrl,e pTOperl~ie8 (z8 ,~Ctx(~) r) agl,d ~ C l ( U , Uo) = a(X) _ Z [(~) _ ~* o u , ,T (59) given by theorema 3 and 4, replacing (x, 7) e = --s 9 by x. From assumption STS, this filter is (see [8]) "strictly passive with respect to U". Now, See [8] for a proof using hyperstability. A from (46), a particular solution (for X) of direct Lyapunov argument, using the functhe first equation in (59) is precisely ~ (no- tion (64) is also described there. It is simtice tllat this would be faulse without as- ilar to a proof given in [5]. Our al.gorithms are sumption FDEM2, (42) and (45)). This R e m a r k 2 : exlicitely defined, i.e. give u,/3, q (and 7}) as means that we may compute e by explicit functions of the state variables x, OU ~ -~ (60) 15, ~ (and 7). Let us check this for .AC'I(U ) e = -of ( ) . (or .AC'{(U)) : since ~ stands for qa(/5,x), and considering (54),/3 and r~ are given by Doing so~ we obtain the following (51) and (53) as functions of/~, 7, and x; A d a p t i v e C o n t r o l l e r AC'=(U) : then (52) gives ~ as a function of x, /3, 0 and 7. For .AC'2(U ) (or AC~'(U)), the same U = Unom(~,X ) -J- I]2(~,~,~,X)(61) holds, without 7. v = (62) This would not have been the case if we 1, q = - Y ( ! 5 , x, v2(~,},~,z)) r (63) had not used the two estimates/5 and 4 for
816 p*. In fact, using only /3, we would have Let, for any p, ~p be the distribution defined u2, instead of (45), by spanned by the control vector fields of Sp : u2 =
v2(/~,#,p,x).
(66)
~p(x) = rangeg(p, x ) ,
(69)
Using then the same adaptation law as ~p be the distribution made of the vector above, but without overpaxametrization fields in ~p whose Lie bracket with any vecwould give, for .AC':(U), instead of (61)- tor field in ~p is in ~v, and Z;p be the dis(52), tribution made of the vector fields in ~v whose Lie bracke~ with the drift vector field u = U.om(16,x) + v2(15,/3,15,x) (67) f(p, .) of the system Sp are in ~v. p --- - Z ( p , x , u ) T (68) which no longer gives explicitly u and 15, because ~ depends on Z which depends on u which depends o n ). Indeed, (67)-(68) g.ive an algebraic equation in (for example) /3, which may fail to have a solution. This problem of implicit definition had been first mentionned, in the context of feedback linearisation, in [5]9 The overpaxametrisation (16) is the only way we presently know to solve this problem. Notice however that it may just not occui, for instance if g does not depend on p at all, because then Z does not depend on u (this was pointed out in [5]), but we saw (remark 1) that in this case, we may drop all q in our repaxametrization, which is therefore ineffective; indeed, v2 in (43) does not depend on q in this case, and is not needed in our algorithms (if we keep
~:p =
{ x e G~/[x,G~] c G~}
s
{ X E ]Cp/[X, f(p,.)] E gp}(71)
=
(70)
P r o p o s i t i o n 1 Suppose that the dimen-
zion of (around Then, some 9, only if
Gp(x) and Z.p(4
are constant
(p, 2)) and that ~p(x) is involutive.
there exists locally, around (fi,'2), a and/3 satisfying FDEM2 if and
1. Gv(x) doe~ not depend on p, 2. for i =
1,...,l,
we have,
locally
around (p, ~ ), ai 9
~, + [f(P,'), ~,l
(72)
P r o p o s i t i o n 2 Suppose tha~ the dimen-
sion of ~,(z) and Z . & ) are constant (around (p, e)) and that for any p, the distribution Gp(x) is involutive. it, we jnst get ~ = 0). Also, in [3], an asThen, there exists locally, around (fi,~), sumption ("strong feedback lineaxisation") some ~o, a and fl satisfying F D E M 2 if and is made that allows to avoid this problem. only if, for i = 1 , . . . , l , we have, locally In these cases, since (43) is only used for around (p, ~), p = q, FDEM2 can be replaced by the apa i 9 Gp + If(p,.), /Cp] (73) paxently weaker assumption FDEM1 (see section 6). II The proof of proposition 1 may be found in [8], and the proof of proposition 2 in the forthcoming [9]9 Some more general prop6 Geometric Conditions erties, including global results, as well as We give here some geometric conditions for a more extensive discussion may also be assumption FDEM2. allowing to build the found in these references. Proposition 1 diffeomorphism ~ locally around a certain and 2 axe only concerned with local exis(p,~). tence of the diffeomorphism ~o, whereas our
817
assumption FDE needs a global definition. In fact, if the conditions of our propositions are satisfied everywhere, building ~o locally may (or may not !) result in a global definition of qo. See also the comment after assumptions FDE and STS for the case where (and a and fl) are defined only for p in a certain domain, but for all x. In remark 2, we saw that using a double adaptation prevents the problem of implicit definition of the controller from occuring, but requires assumption FDEM2 instead of the weaker FDEM1 : A s s u m p t i o n F D E M I : FDE is ~ati~fied, and ~ has the property that there exists a smooth map ~ meeting, for any (p,p,x), Ocp~ ,
0~o. ,
_-)g(p, x)0 (p,p, x) + N ( p x)p = o
(74) This is the assumption required in [5] ("extended matching assumption") for linearisable systems (though written a bit differently). Clearly, if FDEM1 is satisfied and the control distribution ~p does not depend on p, FDEM2 is satisfied too. But (see [8]), if Up is involutive, FDEM1 implies that ~p does not depend on p. Therefore, we could substitute FDEM1 to FDEM2 in our proposition 1. Indeed, the conditions given in [5] for the "extended matching condition" (equivalent to FDEM1) are exactly conditions 1 and 2 of proposition 1.
References
"
[1] d'Andrda-Novel B., Pomet J.-B., Praly L.: Adaptive stabilization for nonlinear svstern.~ in the. nlnne ] l th
Internal Report AP89.09, Louvain, Belgique 1989. [3] Bastin G., Campion G.: Indirect adaptive state feedback control of linearly parametrized nonlinear systems 9 Internal Report AP89.09, Louvain, Belgique 1989. [4] Bastin G., Campion G.: Indirect adaptive nonlinear control control of Compartiment Systems. in New trends in Nonlinear Control Theory (Nantes, 1988), Springer-Verlag, 1989. [5] Kanellakopoulos I., Kokotovic P.V., Marino R.: Robustness of Adaptive Nonlinear Control under an Extended Matching Condition. IFAC symp. on Nonlinear Syst., Capri, 1989. [6] Kanellakopoulos I., Kokotovic P.V., Marino R.: An Extended Direct Scheme for Robust Adaptive Nonlinear Control. Urbana IL, 1990. [7] Landau I.D. : Adaptive Control : the model-reference approach Dekker, 1979. [8] Pomet J.-B.: Sur la commande adaptative des syst~mes non lindaires. Th~se de l'Ecole des Mines, Paris, 1989. [9] Pomet J.-B., Kupka I.: Feedback equivalence of a family of control systems. Forthcoming, Toronto, 1990. [10] Pomet J.-B., Praly L.: Adaptive nonlinear regulation: equation error from the Lyapunov equation9 28th IEEE CDC, Tampa, 1989.
Viability Kernels, Controlled Invariance and Zero Dynamics for Nonlinear Systems
Jean Pierre Aubin
Christopher I. Byrnes
CEREMADE
Dept. of Systems Science and Mathematics Washington University
Universite de Paris - Dauphine
St. Louis, MO 63130
F-75775 Paris cx(16), France
USA
Alberto Isidori Dipartimento di Informatica e Sistemistica Universita di Roma Via Eudosianna 18 00184
I.
Roma, Italy
Introduction
In recent years, a combination of techniques drawn from geometric nonlinear control theory and nonlinear dynamics have made possible the formulation of a nonlinear enhancement of some of the basic concepts and constructions of the frequency domain approach to classical automatic control.
One such tool, the concept and methodology of
zero dynamics, provides a nonlinear counterpart to the transmission zeroes of a linear system.
Remarkably, zero dynamics have turned out to play a fundamental role in important
design problems such as feedback stabilization, asymptotic tracking and disturbance rejection (see [ ]-[ ]) in very much the same way as transmission zeroes are used in the linear theory.
There is however a limiting factor in the current formulation of zero
dynamics, which requires the development of ad hoc extensions in several interesting applications, viz. the essential use of various regularity hypothesis.
In this paper we
show how the zero dynamics of a nonlinear system have a very natural interpretation in terms of nonsmooth analysis, especially Viability Theory.
Moreover, the interpretation of
zero dynamics in terms.of viability theory and set-valued analysis provide an even wider range of potential applications, such as the recent characterization by Cannarsa and Frankowska of optimal trajectories of Meyer problems as viable trajectories.
In section 2 we illustrate the concept and use of zero dynamics for the problems of nonlinear feedback stabilization and output regulation.
We begin section 3 with the
calculation of Cannarsa and Frankowska, motivating the need for a more general definition of zero dynamics.
Results from Viability Theory are then applied to formulate a general
existence theorem for nonlinear zero dynamics, in the absence of the standard regularity conditions.
This existence result is then used in Section 4 to derive a necessary
condition for local asymptotic stabilization b y smoeth feedback.
There must exist a
continuous output function such that the resulting input-output system is minimum phase. Research supported in part by grants from AFOSR and NSF.
822 2,
Preliminary Remarks on Zero Dynamics
We begin by considering a nonlinear input-output system which is affine with respect to the control.
x - f(x) +g(x)u
(2.1a)
y - h(x)
(2.1b)
We suppose
xEIR n,
uEIR m
and
yEIR P
continuous and that h is continuous.
and for now we shall assume f, g are Lipschitz The concept and construction of zero dynamics for
(2.1) has recently found several applications in nonlinear feedback control in problems such as stabilization, tracking and disturbance attenuation [1]-[4]. On the one hand, typically such applications presume the existence of zero dynamics, either of the system to be controlled or an auxiliary system, taking advantage of the dynamical properties of zero dynamics in much the same way as classical automatic control uses properties of transmission zeroes of a frequency response function.
On the other
hand, however, until now general existence results for zero dynamics have not been discussed.
In this section, we motivate the development of a rather general concept of
zero dynamics by indicating its use in problems of feedback stabilization and output regulation.
We begin section 3 with the recent characterization hy Cannarsa and
Frankowska [5] of optimal trajectories for Meyer problems, via viability theory, as trajectories of the zero dynamics of an augmented system.
Example 2.1
We first consider the problem of feedback stabilization.
(2.1) with m-p-i and for which 0 is an equilibrium,
i.e. f(0)-0.
Consider a system
We also assume the
generic condition
- D h(0)~O g
(2.2)
is satisfied where, for a function F and a vector field derivative of F in the direction V(x) by DvF(X).
V, we denote the directional
In the light of (2.2), treating y-h(x)
as a coordinate function we may choose complementary coordinates zi,
i-l,...,n-I
satisfying
- 0
(2.3)
so that in the coordinates (z,y), z T- (z ..... Zn.l) the system (2.1) locally takes the form
z - fo(Z,y)
y - Dfh(z,y) + Dgh(z,y)u
(2.4)
823 Before deriving an explicit feedback law achieving local asymptotic stahillzatien, we illustrate the construction of the zero dynamics of (2.4). condition for which Yo - h(Xo)-0
we seek a control law u(x)
Thus, imposing the constraint Yt- 0 control law u(t). V
Setting
If x ~ - (Zo,0) is an initial such that Yt- 0
for ta0.
imposes a constraint on the state vector x and on the
y(t) - 0 in (2.4) yields in fact the constraint set
- {(z,y): y=0)
(2.6)
and a control law u(t) in a coordinate free feedback form (2.7)
u.(x) - Dgli(x) -I (-Dfh(x)) which maintains the constraint (2.5) for any initial condition XoEV .
The zero dynamics
of (2.1), expressed in the local coordinates (2.4) is the induced dynamical system
z - fo(Z,
0)
(2.8)
evolving on the submanifold V*, which is invariant for the closed-loop system (2.1)-(2.7). If the origin is asymptotically stable for (2.8), then a center manifold argument shows [6] that the feedback law u(x)-u.(x)-Ky locally asymptotically stabilize (2.4), for any k>O.
(2.9) We remark that (2.9) can be regarded
as a proportional error feedback law for a "minimum phase" nonlinear system with a correction term u.(x) which, however, cannot in general be suppressed.
Remark 2,2(i)
A similar result holds for systems having higher relative degree p,
which a generalization of the normal form (2.4) exists (see. e.g. [4]-[6]).
for
In this case,
by tuning k in (2.9) asymptotic stabilization can be achieved on arbitrary compact sets, provided an auxiliary "non peaking" condition is satisfied ([6]).
Moreover,
the case p-i
and in the case p-n, this "non peaking" condition is automatically satisfied.
(ii)
When the system does not have a relative degree p, the zero dynamics -- as
computed by the zero dynamics algorithm -- may still exist.
And, in this case, it is
still true that minimum,phase nonlinear systems are locally asymptotically stabilizable
[6]. (iii)
These results extend, mutatis mutandis, to the multivariable case. as the next
example illustrates ([4], [6]).
E•
2,3
As an explicit example consider the angular momentum equations of a rigid
spacecraft, controlled by two pairs of gas jets
824 wI - uI
w2 - u2
(2.10)
w3 - wI w2 As is well k n o w n e q u i l i b r i u m 0o
(see e.g.
[7]),
(2.10) can be
locally asymptotically
s t a b i l i z e d about the
C h o o s i n g the outputs
Yl " Wl'W3
w2+w~
Y2 -
it is easy to see that the c o r r e s p o n d i n g zero d y n a m i c s manifold,
rendered
evolves
on a o n e - d i m e n s i o n a l
i n v a r i a n t by the f e e d b a c k law
U.l - w I w 2 u.2 - 2w I w 2 w 3 and that the zero d y n a m i c s
takes the form
2 w 3 - -w 3 so that the s y s t e m is " m i n i m u m phase".
Accordingly,
for k>0
the f e e d b a c k law (which
c o i n c i d e s w i t h the law d e r i v e d in [7] w h e n k-l) u I - w I w 2- k ( W l - W 3) u 2 - -2w I w 2 w 3 - k ( w 2 + w 3 2) locally asymptotically
Example
2,4
stabilizes
Zero d y n a m i c s
and d i s t u r b a n c e
(2.10).
also p l a y a f u n d a m e n t a l
rejection.
- s(w)
the s y s t e m
role in p r o b l e m s
C o n s i d e r an e x o g e n e o u s
of a s y m p t o t i c
tracking
system
w E I R 2, s(0) - 0
producing a reference
trajectory
(2.11a)
to be a s y m p t o t i c a l l y
tracked
YR - r(w)
(2.11b)
and a d i s t u r b a n c e
signal d(w)
x - f(x) + g ( x ) u + p(x)
to be a s y m p t o t i c a l l y
d(w)
(2.11c)
Two m a i n p r o b l e m s are to d e s i g n a f e e d b a c k scheme, only error m e a s u r e m e n t , achieving asymptotic e(t) - h(x(t))
w h i c h stabilizes
tracking;
Under mild assumptions,
x
-
~(w)
u
w h i c h are s o l u t i o n s f(~(w))
-
(2.11a)
u s i n g e i t h e r full state m e a s u r e m e n t
- (2.11c) w h i l e at the same
as t ~
(2.12)
n e c e s s a r y and s u f f i c i e n t
in [2]-[3]
conditions
in terms of the e x i s t e n c e
for the s o l u t i o n of these
of f u n c t i o n s
c(w)
to the p a r t i a l d i f f e r e n t i a l
+ g(=(w))c(w)
or
time
i.e.
- r(w(t))~ 0
problems were derived
rejected
+ p(=(w))d(w)
- a~s(w) @w
equation (2.13a)
825 There is a good geometric existence c(w).
Moreover,
(2.13b)
theory for ~(w) in (2.13a) given any choice of
under certain regularity hypotheses
is automatically
satisfied.
Indeed,
c(w) may be chosen a priori so that
viewing e, as defined
in (2.12),
as the
output of the augmented system (2.11co)-(2.11a)
one sees that if w o- 0 then, for any Xo,
to say y(t) - 0 is achieved with a control u(t)
is to say e(t) - O.
dynamics
That is, if the zero
of both the augmented system and the system to be controlled
the zero d y n a m i c s of the a u g m e n t e d s y s t e m c o n t a i n s
sets V*a and V ~ are smooth,
in case both zero constraint
system.
Moreover,
(2.13)
is e q u i v a l e n t
(2.11c)
to the e x i s t e n c e of a smooch,
exist,
then
the zero d y n a m i c s of the control s o l v a b i l i t y of
to V ~ in V a~ (see
invariant complement
[2]-[3]).
In the discussion of feedback stabilization or smoothness
and output regulation,
certain regularity
e.g. we asked that relative degree * such as (2.2) were satisfied or that V itself be a smooth submanifold.
conditions
surprisingly, and Marino
conditions were assumed to hold,
such conditions
are not necessary
as the following example,
[8], shows.
Example 2,5
[8]
Consider the system
-x 2 (-x+2y 3)
x -
(2.14a)
y - u
(2.14b)
The feedback law hand,
u-x-y 3 renders the origin locally asymptotically
if a smooth real-valued
function h(x,y)
is any feedback law maintaining constraint theorem,
Not
due to Boothby
set h(x,y)-0
stable.
On the other
is such that (2.2) is satisfied and u(x,y)
the constraint h(x,y) - 0
which may be computed
then the flow induced on the
from (2.14a) and the implicit function
is unstable.
Remark 2,6
In particular,
for the system (2.14) one cannot find an output function
for which both the regularity condition
(2.2) and the asymptotic
resulting zero dynamics hold.
one does not need relative degree hypotheses for w to a smooth V , but in this case we can show that
stability of the
Of course,
the zero dynamics
algorithm to converge
the zero dynamics
algorithm cannot converge while yielding asymptotically
stable dynamics.
826 Section 3
V~abi~tu
Ke~els
add Existence Theorems for Zero Dynamics
There are two regularity issues in the construction of the zero dynamics:
the
existence of a smooth V* and the existence of extra piecewise smooth feedback laws making V
invariant and inducing a system evolving on V .
The next calculation, due to Cannarsa
and Frankowska [5], shows that both may fail in an interesting class of problems for which, nonetheless, computing both V* and the associated feedback laws is extremely important.
We begin with some preliminary general remarks about V
in the context of a
general approach to geometric nonlinear control.
Fix a time-interval (0, T] with F C IR n
T S ~
and a constraint set
is controlled invariant for (2.1) provided for each
U C IR m.
XoE F
admissible piecewise continuous control u(t), such that the solution for tE [O, T] and is contained in F.
Given the closed subset
A subset
there exists an x t of (2.1) exists
K - h'l(0) for a continuous
function h, the maximal closed controlled invariant subset of K, provided it exists, is denoted by V (K).
Example 3.1
[5]
Consider the optimal control problem for (2.1) in Meyer form over a
finite time interval [O,T), with T fixed.
That is, for some constraint set U c IR m we
seek to minimize the end point functional
g(x(T)) over all solutions to (2.1) with
-
fo
and
u(t)
a piecewise continuous admissible control.
x(o)
It is well-known that the
value function
V(to,X o) - inf (g(x(t)): x(to) - x, to ~ T)
is nondecreasing along trajectories of (2.1) and is constant along optimal trajectories. Therefore, solving the optimal control problem is equivalent to computing V*(K)
control laws renderinE V*(K)
x -
f(x)
and the
invariant for the following system and choice of K-h'l(0):
+ g(x)u
w - 0
x(0)
-
fo
w(0)
- v(0,f)
r - i h(x,w,r)
- w-v(r,x)
In general, the existence of V*(K) and a description of those controls which render V (K) invariant can be obtained using methods from set-valued analysis. we express (2.1) as a differential inclusion
More explicitly,
827 x E F(x),
F(x) - (f(x)+g(x)u: u E U)
(3.1a)
and, without loss of generality, we set
K - h'l(0)
(3.1b)
By a solution of (3.1a) with initial condition x ~ we mean some absolutely continuous trajectory x(t), tz0 satisfying (3.1a) and x(0)-x o.
In the language of nonsmooth
analysis, a closed subset K has the property local viability if for any x ~ E K, there exists a T>o
such that some solution x(t) to (3.12) exists and belongs to K for tE[0,T].
If, moreover,
T-~, then K enjoys the global viability property.
controlled invariant subset (for infinite time).
Denoting by
In particular, K is a
TK(X )
the contingent cone
of K at x (see e.g. [9], [I0]), Haddad has shown that K has the local viability property if and only if K is a viability domain, in the sense that
Vxek
F(x)
Tk(X)~ ~
(3.2)
Furthermore, if F is bounded on K then, if K is a viability domain, K enjoys the global viability property.
In this language, then, if it exists,
V*(K) is the maximal closed viability domain
contained in K, referred to as the viability kernel of K (see [9], [i0]).
Moreover, it is
known that V*(K) exists (although perhaps empty) provided that F is a Peano map; i.e.
(i)
Graph (F) is closed;
(ii)
F(x)~@ and convex;
(iii)
F(x) - c (i + II x If)B, for some c>0 and some ball B.
From this "viability kernel" theorem, we conclude an existence result for zero dynamics. For (2.1) condition (ii) is always satisfied if U is convex.
Theorem 3,2
Suppose (2.1)-(3.1) satisfies the Peano conditions (i)-(iii).
Then
(2.1) has a zero dynamics evolving on the maximal closed, controlled invariant subset Z -V (K).
Moreover the zero dynamics is the system defined by the differential inclusion
zeZ (z), where
zeZ*
Z(z) - F(z)
T ,(z) '
Furthermore, if 0E U
and f(x) has a nontrivial
Z invariant set in K, e.g. an equilibrium,
Remark 3,3
then Z ~@.
The system (3.3) is a classical autonomous dynamical system only if Z(z)
is a singleton, for each zEZ ~
This situation occurred in the earliest works
feedback stabilization of invertible, nonlinear minimum phase systems.
[4] on
Of course this is
828 not always the case as a differential controls,
evolving on Z .
minimum phase
z(t)EW
W
of
0
if f(0)-0,
then we say that (2.1)
and any control u(t) containing
z~
suc~ that for
on the
(or(3.1))
is
t~0, z(t)EZ*,
there
such
(3.4a) as t ~ .
Remark 3,4 autonomous.
(3.4b)
Even when Z(z)
is single-valued,
tl.e zero dynamics
(3.3) may not be
This will be the case only if a selection for the set-valued map
exists.
are, of course, (see e.g.
For example,
constraint
Z
z
Z*
so that the zero dynamics should in general be thought of
or a control system with invariance
if for every zEZ*
exists a neighborhood
on
[ii],[12],
inclusion,
[13]).
For general Z(z), some general
it is also interesting
to find selections
theorems concerning Lhe existence of continuous
Z
defined
and there selections
829 4.
Necessary Conditions for Local Feedback Sta~ilization
Consider again the control system (2.1a) ',~here we assume f, g are C I and f(0) - O. The fundamental problem of (local) smooth feedoack stabilization discussed in Examples 2.1, 2.3 and 2.5 is the existence of a C I fee,Lback law
u - =(x)
(4.1)
so that the closed-loop system (2.1a)-(4.1)
(4.2)
x - ~(x) - f(x)+g(x)=(x)
has 0 as a locally asymptotically stable equilibrium.
Theorem 4,1
Suppose (2.1a) satisfies the Peano conditions (i)-(iii).
A necessary
condition for (2.1a) to locally asymptoticllly stabilizable by smooth feedback is the existence of a continuous output function, That is, there exists a neighborhood
W
y-h(x),
of 0
such that (2.1) is minimum phase.
so that for each compact subset
KCW and
each compact, convex constraint set UcIR m, the zero dynamics of (2.1) exists and satisfied (3.4).
Example 2,5 (his) about
0.
The system (2.14) is locally asymptotically stable on some ball
graph so that the Peano condition (i) is satisfied. the controls, condition (iii) is satisfied on
W.
For any compact constraint set And, with the output function
h(x,y) - (y3-x)
(4.3)
the system (2.14)-(4.3) is minimum phase.
Indeed
u(x,y) - -y 7/3
(4.4)
renders the constraint set
(4.5)
Z* - ((x,y)e h(x,y)-O}
invariant.
Z
- Z
Proof. strategy
W
Because it is in "normal form," as a differential inclusion it has a closed
In particular if
{y:
WEIR m
is a closed ball of radius
IJYlJ ~ (3 c)i/7}.
Choose a C I stabilizing law (4.1) on a ball about 0.
c, then
(4.6)
Using the feedback
on
830 (4.7)
u - =(x) + v
we obtain the open-loop system
x - ~(x) + g ( x ) v
(4.S)
We vote that (4.8) expressed as a d i f f e r e n t i a l (i)-(iii)
if (2.1a) does.
In particular,
inclusion satisfies
if uEU
the Peano conditions
, a compact c o n v e x constraint set then
vEV(x) - =(x) + U
(4.9)
where V(x) is compact and convex valued.
In p a r t i c u l a r
(4.1), c o m p u t e d relative to the constraint (4.8) relative to the constraint
44.9).
uEU
In this sense,
as (2.1a), c o m p u t e d relative to input constraints, particular,
the zero dynamics on B for (2.1a)-
corresponds
to the zero dynamics for
the zero dynamics of a system such
is invariant under feedback.
In
it suffices to prove the T h e o r e m for the C I s y s t e m 44.8).
Choose a C I Lyapunov function for ~(x)
d e f i n e d on
B
and define the continuous
output function
y-
h(x) - DgV(X)
(4.10)
Along trajectories of (4.8) we have
a.e.
V(x t) -< u(t)y(x t) + D V(x t)
For any Zo~Z
, if
(4.11)
u(t) is chosen so that z(t)~Z, we must have h(z(t)) - 0.
Therefore for
any trajectory of the zero dynamics we have
V(z ) _< D V(z ) ~ 0 t ~ t
a.e.
(4.12a)
where
D V(Zto ) - 0
In particular,
<-> Zto - 0
(4.12b)
for z ~ near 0, z t remains in a n e i g h b o r h o o c of 0.
negative d e f l n i t e n e s a
V(zt) ~ - X(V(zt) )
of D V
and the positive definitenes~
a.e.
Moreover
of V imply
44.12),
the
831 for X a continuous monotone increasing function defined on IR +. continuous in t, for
V(z t) - V(Xs)
Since V(xt) is absolutely
OSSst
t S - Is X(V(zz))dT
from which follows
V(zt) S ql(V(Zo )) Pl (t)
where ql is monotone increasing and Pl is strictly decreasing with Pl(t) ~ 0 ~ t ~ ~ (see [14] p. 103).
II AtU
Since V(x) is positive definite it then fellows that
-< q2(]]Zo]]) " P2 (t)
where again P2 is strictly decreasing and P2(t) ~ 0
as
t ~ ==.
832 5.
[i]
References
C.I. Byrnes and A. Isidori,
"A Frequency Domain Philosophy for Nonlinear Systems
with Applications to Stabilization and Adaptive Control," Proc. of 23rd IEEE Conf. on Dec, and Control, Las Vegas, NV, 1984.
[2]
C.I. Byrnes and A. Isidori,
Regulation asymptotique des systemes nonlineares, C.R.
Acad. Sci. Paris, 309 (1989), pp. 527-530.
[3]
A. Isidori and C.I. Byrnes, Output Regulation of Nonlinear Systems, iEEE Trans. Aut. Contr., AC-35,
[4]
(1990), 131-140.
C.I. Byrnes and A. Isidori, Asymptotic Stabilization of Minimum Phase Nonlinear Systems, submitted to IEEE Trans. Aut. Control.
[5]
P. Cannarsa and H. Frankowska, "Some Characterizations of Optimal Trajectories in Control Theory," preprint, 1989.
[6]
C.I. Byrnes and A. Isidori, Local Stabilization of Minimum Phase Nonlinear Systems, Sys. Contr. Left., 10 (1988), pp. 9-17.
[7]
R.W. Brockett, Asymptotic Stability and Feedback Stabilization, D.i.fferential Geometric Control Theory (1983), pp. 181-191.
[8]
W.M. Boothby and R. Marino, "Feedback Stabilization of Planar Nonlinear Systems," Sys, and Contr, Lett, 12 (1989) 89-92.
[9]
J.P. Aubin, Viability Theory, preprint 1989.
[i0]
J.P. Aubin and H. Frankowska, Set Valued Analysis, Birkhauser-Boston,
in]
A.J. van der Schaft, "Clamped Dynamics for Non-lnvertible Systems," Nonlinear
1990.
Control and Robotics, North Holland, 1988.
[12]
X.M. Hu, Robust Stabilization of Nonlinear Control Systems, Ph.D. dissertation, Arizona State University.
[13]
J.P. Aubin and A. Cellina, Differential inclusions, Springer-Verlag, Berlin 1984.
[14]
W. Hahn, Stability of Motion, Springer-Verlag, Berlin 1967.
O u t p u t R e g u l a t i o n of a Flexible R o b o t A r m
A. De Luca, L. Lanari, G. Ulivi Dipartimento di Informatica e Sistemistica Universits degli Studi di Roma "La Sapienza" Via Eudossiana 18, 00184 Roma, Italy
ABSTRACT The problem of controlling the motion of a one-link flexible robot arm is considered. A nonlinear dynamic model is derived assuming that the flexibility is represented by elastic springs along the link. Taldng as output the end-effector angular position, the resulting zero-dynamics is unstable. Therefore, inversion-based controllers cannot be used for tracking of output trajectories. Instead, it is shown here that a nonlinear regulator approach solves the problem, allowing asymptotic trajectory tracking with internal stability. Simulation results show the good performance of the overall controller and in particular the benefits achieved by a nonlinear design of the regulator. INTRODUCTION Trajectory tracking in nonlinear systems has motivated interesting studies [1-3] which have found application in relevant areas like robot motion [4], satellite attitude [5] and induction motor control [6]. Although different terminologies are used for denoting the same control law, exact reproduction of output trajectories is essentially based on feedback controllers which invert the input-output behavior of the plant. The feasibility of an inversion-based approach relies on the properties of the socalled zero-dynamics of the system [7]. In particular, in case of nonlinear systems with no zero-dynamics output tracking is achieved quite easily. Plants that are fully linearizable and decoupable via static or dynmnic nolflinear state feedback belong to this class. More in general, the problem of output trajectory tracking can be solved assuming asymptotic stability of the zero-dynamics. Under this hypothesis, the closedloop system consists of an input-output lineaz" and controllable part, and possibly of an unobservable stable dynamics. From the point of view of system inversion, these nonlinear systems behave like linear ones with no right-half plane zeros, and therefore are usually denoted as minlmmn phase systems. On the other hand, direct inversion of non-minimmn phase nonlinear systems leads to unstable closed-loop behavior. In fact, reproduction of a desired output evolution is attempted in a way equivalent to cancellation of unstable transmission zeros in linear systems. For these cases, control by inversion is not a feasible solution and the tracking problem becomes much more complex. Robot arms with flexible links under end-effector position control typically display non-minimum phase features [8]. This property is inherited by any realistic finitedimensional dynamic model of the flexible arm, both in the linear and nonlinear case.
834
Limitations arising for end-effector tracking in one-link flexible arms have been analyzed in [9,10], together with possible solutions. It is easy to show that assigned joint trajectories can be exactly reproduced in a stable fashion. Extrapolating on this argument, an alternate output point can be defined along the link, and used in place of end-effector position for designing a stable inversion control law. Although satisfactory performance was obtained in terms of tip motion, it is hard to generalize this idea. Moreover, a small but persistent tracking error is present at the end-effector level. A basically different approach to trajectory tracking has been recently proposed in [11], where the output regulation problem is solved for general nonlinear systems. The underlying idea is that tracking of a prescribed output trajectory can be achieved witAout the need of forcing a linear dynamics in the input-output behavior. Under mild assumptions, a bounded evolution of the state can be associated to a given reference trajectory producing the desired profile at the system output. The regulator is designed so to drive the actual state towards this paa'ticular steady-state solution, asymptotically obtaining stable output reproduction. As a result, this approach is well suited for facing the output tracking problem in nonlinear non-minimum phase systems. In this paper, the problem of end-effector motion control for a one-link planar flexible arm is approached in a direct way and solved by means of nonlinear regulation theory. A great effort is expected in order to bring the regulator into action for more complex flexible robotic systems, but the problem considered here, and the particular form of solution proposed, contains most of the aspects which are relevant in the more general case. In paxticular, in the robotic application it is useful to combine the regulator approach with an inversion-based control law. The controller design proceeds in two phases: first, a nonlinear inversion feedback is applied to the original plant bringing it to a more convenient canonical form; then, for the compensated system the regulator is set-up and takes care of the possible induced instabilities. A similar control structure has been proposed in [12]. In the following, a nonlinear dynamic model of the flexible arm is introduced. The steps involved in the inversion and in the regulation phases are presented in detail. Numerical simulations are reported illustrating the good performance of a fully nonhnear regulator control design.
MODELING
OF A FLEXIBLE
ARM
Consider a one-link planar robot with pure bending flexibility. Different finitedimensional models of flexible arms can be derived but when the analysis is limited to small deflections a linear dynamics is obtained. However, in presence of heavy carried loads a n d / o r in case of fast motion, nonlinear effects arise due to the larger deformations coming into play. For general multi-link flexible arms, the system is governed by equations with even harder nonlinearities, including those due to rigid body motion and couplings of rigid with flexible dynamics. A modeling technique which simply complies with nonlinearities considers the flexible link subdivided into a chain of rigid segments and assumes the deformation concentrated in elastic springs. Multi-link flexible robot arms can be treated in the same way. This representation is similar to ~he finite-element me~hod, but the model needs not to be limited to small deflections.
835
With reference to Fig. 1, the Lagrangian dynamics is obtained in the standard form as
B(q)cl + c(q, cl) + e(q) + I ( q + D(t = Gu
(1)
with q E A N. Explicit expressions parametrized in tile number of segments can be given to the terms in (1), so that the model order can be varied easily to achieve the desired accuracy. y
L
s' J
Fig.1 - Discrete modeling of a one-link flexible arm Taking N equal segments of uniform mass, and denoting by L and M the total link length and mass, the elements of the inertia matrix B ( q ) are
ML 2 . 1 bii=~(-~+N-i),
i=l,...,N,
ML 2 .1 bij = ~ ( ~ + N - max{i,j})cos(ql - qj),
i,j = 1 , . . . , N .
The components of the centrifugal force c(q, cl) are
ci = ~
M L3 (12 + N - m a x { i , j } ) s i n ( q i - q j )
qj,.2
j=l
while those of the gravitational term e(q) are
el=
MZg(1 N2 2 + N - i ) c ~
c~
where 7 is the angle between the plane of motion and the vertical plane and g is the gravity constant. The linear elasticity matrix K takes on the form
K =
",~
",.
",
--ki--1
~i--1 "~-t~i 9 .|
--]Ci ".
".
0
--ION--1
]ON-1
...
,
where ki is the elasticity constant of t h e / - t h spring i = 1 , . . . , N-1. The viscous damping matrix D has the same structure of K , with the spring damping coefficients di's in place of the ki's. A term do should be added in the (1, 1)-element of D, representing the viscous friction at the joint axis. FinaLly, vector G = (1 0 ... 0) T selects the only equation where
836 the scalar torque u performs work. The choice of absolute generalized coordinates q results in a more compact model since no Coriolis terms are present. However, it may be convenient to rewrite the system dynamics in terms of relative angles 8i between segments so to display directly the deflection variables. The transformation between absolute and relative coordinates is a linear one, with 81 = ql and 8i = ql - qi-1, i = 2 , . . . , N. Note that using variables 8, matrices K and D become diagonal. When the end-effector angular position is chosen as output, a non-minimum phase system is obtained from (1). Moreover, this output has always relative degree 2, independently of the number N of assumed segments f For the sake of clarity, and without loss of generality, the following treatments will be limited to the N = 2 case, on a horizontal plane (7 = 90~ 9 The dynamic equations can be explicitly written as (2) where b11(o2) = ~ + 2ccosO2,
b12(82) = b + ccosO2,
cl (82,01, b2) = - c (0~ + 26102) sin 82,
and a = 5ML2/24, b = ML2/24, c = ML2/16.
b22 = b,
c2 (82, bl) = c b 2 sin 82, The linearized expression of the
end-effector angular position
v = 8 ~ + ' ~ 1 82
(3)
will be taken as controlled output for system (2). INVERSION-BASED P R E C O M P E N S A T I O N The design of an inversion-based controller for output tracking is accomplished deriving twice (3) and setting ~ = v. Solving for the input torque u yields [9] 1 ( o 2 ) - 2612(o2) ( c 2( 0 2,0,)+klO2+d102)-~ u = c1(02,01,02)+d001-~ ~ 12-~22"--'-bx2"-2~2)
2detB(02) 2b22 b12(82) -
-
v,
(4) which is in the form of a nonlinear state feedback applied to the flexible arm. After replacing 81 with y - 82/2, the closed-loop equations become
02 = 2(c2(82, 02, ~1) + k182 + d~2) + b12(82) - 2622
2b12(82) b~2(82) - 2622
= ~(82, ~2, #) + Z(o2)~.
(5)
To track a desired output trajectory yd(t), the linear input-output behavior in (5) is stabilized by =/~d + ko(~d - b) + kp(Vd -- V), (6) ? This is true in a generic sense, i.e. except for a unique combination of a r m mechanical p a r a m e t e r s [9].
837
but the critical issue here is the instability of the unobservable closed-loop dynamics. As a m a t t e r of fact, the associated zero-dynamics obtained with y -- 0 //5 =
(c/2)@ sin 02 + 2(k10~ + d10~) b - c cos 05
(7)
turns out to be unstable. Therefore, the inversion-based controller (4)-(6) cannot be used as such for output reproduction, and this situation calls for a nonlinear regulator approach. However, there is still a benefit in using (4) as a precompensation on the original plant. The resulting internaediate system (5) is in fact already in a nice canonical form, with the main nonlinearities removed by feedback from the input-output path. The design of a regulator will take advantage of this internal structure.
N O N L I N E A R R E G U L A T O R DESIGN The instability of the unobservable states in (5) requires counteractions to be taken with the new control input v, other than just stabilization of the input-output behavior. Instead, v will be designed so to attract the state trajectory to a particular steadystate manifold, which depends on the reference trajectory. State evolution on this surface, which will be invariant for the closed-loop system, produces as output the desired trajectory. Thus, if the initial state lies on the manifold output tracking will be exact. Otherwise, the output will not follow the desired one during a transient phase until the state has (exponentially) converged to the manifold, so that asymptotic tracking is obtained. The necessary and sufficient conditions found in [11] for solving the regulation problem in nonlinear systems are recalled. Consider a system given by ~: = f ( x ) q- a ( x ) u ,
y = h(x),
(8)
and assume that its linear approximation at x = 0 is stabilizable by means of a linear feedback u = F x . A reference trajectory yR(t) is supposed to be generated by an autonomous dynamic system (i.e. the exosystem) = s(w),
YR = q(w),
(9)
for which every point in a open neighborhood of the stable equilibrium w = 0 is Poisson stable (see [7], p.352). Vector functions f and h are assumed to be zero at x = 0, as well as s and. q at w = 0. Define two mappings u = c(w) and x = 7r(w) being at least C 2, with c(0) = 0 mad a'(0) = 0. A state feedback of the form u = c(w) + F(x -
(10)
solves the Regulator Problem [11] if and only if c(w) and 7r(w) satisfy the following equations 0rr = q(w) =
+
(n)
838 The resulting nonlinear regulator (10) is made of a feedforward term c(w), providing the desired steady-state response, plus a feedback designed around the surface x = ~r(w), which is a manifold in the extended state space. This control approach will be applied next to the flexible arm, assuming the inversion feedback law .(4) has already been used for precompensation. The system state is x = (y, t), 82,82). Stabilizability of the linear approximation of (5) (or equivalently of (2)) is easily verified. Consider as reference trajectory a sinusoidal function yR(t) = A sin(c0t + r which can be generated by a properly initialized second-order exosystem ~v=
w2
-WWl
Equations (11) become in tlds case
O~rl Owl~w~ 07r 3
07rl
07r2
- _~---~wl = ~(w),
07r2 ~
- x--~l
UtD2
COWl
07r 3
07r 4
97[ 4
Owl
c,w2
Ow2
UW 2
= c(w),
~,~ = ~ ( ~ ) , (13)
with a and fl defined in (5). Exploiting the particular form of these partial differential equations, it is straightforward to see that 'ffl(W) = Wl,
7r2(w) ~-- 02w2, c(w) = -o32Wl.
(14)
Note that the feedforward term c(w) is just the desired output acceleration. By using (14), system (13) is reduced to the last two P.D.E. which are the first-order counterpart of the unobservable dynamics in (5). Therefore, once ~ra(w) is determined analytically, ~'4(w) can be obtained by explicit time derivation. Since a closed-form solution is hard to find, standard methods of approximation can be used, e.g. by polynomial expansions. In particular, assuming for ~ra a complete polynomial of third degree ~a .~ a~ ~,~ + a~,~ + a~, w~ + a ~ , ~ w: + a ~ w i + a ~ ~,~ + , ~ : w ~ w~ + a~:w~ w I + a~:w],
(15) a solution has been computed substituting (15) into (13), expanding in Mac Laurin series the nonlinear functions a ( w ) and fl(w), and retaining terms up to the third order. Simple calculations show that the coefficients a11, a12, and a22 are zero in the solution. The regulation control law is then fully determined as 4
v = c(wl) + ~
F, (x, - ~ri(wl, w2)).
(16)
i=1
It should be stressed that the actual input torque u to the arm is the composition of (16) and (4). A block diagram of the overall controller is reported in Fig. 2. As an useful comparison, a llnearized version of the regulator (16) can be obtained within the present framework, by simply limiting the above expansions to first order
839
terms. Equivalently, a standard regulator may be designed on the linear approximation of system (5). Note however that this does not yield a fully linear regulator for the original model (2), because the inversion-based nonlinear precompensation is still present.
SIMULATION RESULTS The nonlinear regulator for the one-linl~ flexible arm has been tested on two types of end-effeetor output trajectories, a continuous sinusoidal motion and a point-to-point task with a half-wave sinusoidal velocity profile. The mechanical parameters of the flexible arm are L = 1 m, M = 0.2 kg, kl = 5 Nm/rad, do = dl = 0.01 Nm sec/rad. The simulations were run using a third order Runge-Kutta method with adaptive integration step. For the sinusoidal motion, an amplitude A = 0.05 tad, an angular frequency w = 20~r rad/sec and a phase-shift r = rr were selected as exosystem parameters. Note that the chosen signal frequency is well beyond the natural resonance of the arm. The arm starts from the undeformed rest position with an initial tip error of 0.1 tad. The linear feedback matrix F is selected so to place the poles of the linear part of the closedloop system at [-20,-20,-30,-30]. The obtained tip angular motion is reported in Fig. 3, together with the desired one which is perfectly reproduced after about four periods. Fig. 4 shows that the input torque applied to the arm has a very smooth behavior even during the transient phase. Moreover, the system nonlinearities are clearly displayed by the non-sinusoidal behavior at steady-state. The deflection amplitude 82, representing the unobservable part of the closed-loop system, and the joint position 81 are reported in Fig. 5; as expected, 82 remains limited. Note also that tip as well as joint motions are in opposite phase with the flexible variable. For comparison purposes, Figs. 6 and 7 report the output error, i.e. the difference between actual and desired output, obtained with nonlinear regulation or, respectively, with the linear approximation discussed in the previous section. Although the two waveforms look similar, the second one shows a residual error even at steady-state, because of neglected nonlinearities. As a typical robotic trajectory task, a point-to-point tip motion of 90 ~ in one second has been considered next and the results with nonlinear regulation are shown in Figs. 8-11. The trajectory tracking performance is very good. The regulation torque is compared in Fig. 9 with the one computed for an equivalent rigid arm (kl = oo), giving evidence of the reduced effort needed during the travel in the flexible case. Joint and tip velocity, reported in Fig. 10, differ only during the transients due to unmatched initial conditi6ns. In fact, even if this significant reference trajectory can be generated by a sinusoidal exosystem, the starting alld stopping processes act as abrupt changes in the angular frequency w or as reset operations of the exosystem internal state. Some transien~ error will then arise even if the states on the input-output path axe matched with the desired trajectory and the corresponding derivatives. This condition is not sufficient to keep the output error at zero. Such a behavior is illustrated in Fig. 11.
CONCLUSIONS The problem of tracking a desired end-effector trajectory for a flexible robot arm is conveniently solved in two steps. First an inversion-based feedback control is used as a
840
precompensation on the original system, then the design of the controller is completed by means of the nonlinear regulator approach. Of course, one may use the nonlinear regulator approach directly on the original plant. Changes result in the transient behavior but not at steady-state. The overall feedforward/feedback approach is a viable design tool which avoids hand-crafted solutions, but still retains space for intuition. For robotic applications, the need of an exosystem for defining reference trajectories may be somewhat limiting. For example a slowly-varying sinusoidal profile and a ramp (unbounded) reference can be asymptotically tracked but are generated by exosystems which are not in the theoretical frame worked out until now. REFERENCES [1] R.M. Hirschorn, "Invertibility for multivariable nonlinear control systems", I E E E Trans. Automatic Control, vol. AC-24, no. 6, pp. 855-865, 1979. [2] ll.M. Hirschorn, "Output tracldng in multivariable nonlinear systems", I E E E Trans. Automatic Control, vol. AC-26, no. 2, pp. 593-595, 1981. [3] S.N. Singh, "Generalized functional reproducibility condition for nonlinear systems", I E E E Trans. Autoraatic Control) vol. AC-27, no. 4, pp. 958-960, 1982. [4] T.J. Tam, A.K. Bejczy, A. Isidori, and Y. Chen, "Nonlinear feedback in robot arm control", in Proc. ~3rd Conf. on Decision and Control (Las Vegas, NV, Dec. 12-14, 1984), pp. 736-751. [5] T.A.W. Dwyer III, "Exact nonlinear control of large angle rotational maneuvers", I E E E Trans. Automatic Control, vol. AC-29, no. 9, pp. 769-774, 1984. [6] A. De Luca and G. Ulivi, "Design of an exact nonlinear controller for induction motors", I E E E Trans. Automatic Control, vol. AC-34, no. 12, 1989. [7] A. Isidori, Nonlinear Control Systems, 2nd Edition, Springer Verlag, Berlin, 1989. [8] R.H. Cannon, Jr. and E. Sclunitz, "Initial experiments on the end-point control of a flexible one-link robot", Int. d. Robotics Res., vol. 3, no. 3, pp. 62-75, 1984. [9] A. De Luca, P. Lucibello, and G. Ulivi, "Inversion techniques for trajectory control of flexible robot arms", ]. Robotic Syst., vol. 6, no. 4, pp. 325-344, 1989. [10] A. De Luca and B. Siciliano, "Trajectory control of a nonlinear one-link flexible arm", Int. d. Contr., vol. 50, no. 5, pp. 1699-1715, 1989. [11] A. Isidori and C. Byrnes, "Output regulation of nonlinear systems", I E E E Trans. Automatic Control, vol. AC-35, no. 2, 1990 (To appear). [12] P. Lucibello, "Nonlinear regulation, with internal stability, of a two link flexible robot arm", in Proc. 28th Conf. on Decision and Control (Tampa, FL, Dec. 13-15, 1989), pp. 1645-1650.
This paper is based on work supported by the Ministero deH'UniversitA e della Ricerca Scientifica e Tecnologica under 40 % and 60 % funds.
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A n Analysis of R e g u l a r i t y C o n d i t i o n s in N o n l i n e a r Synthesis M.D. Di Benedetto* Istituto Universitario Navale Istituto di Matematica Via Acton, 38 80133 Napoli, Italia
J.W. Grizzle'* Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109-2122/USA
A b s t r a c t : Many results are now available for output nulling, system inversion and dynamic input-output decoupling, each with its own set of regularity conditions. In this paper, we present a framework, based upon the canonical map from initial conditions and inputs to outputs, in which these regularity conditions can be reformulated and compared. As a byproduct, it is seen how to enlarge the class of systems for which an inverse systcm can be constructed. K e y w o r d s : Dynamic Decoupling, Inversion, Output NuIling, Regularity, Zero Dynamics
1.
I n t r o d u c t i o n and P r e l i m i n a r i e s
During the last years, conditions have been found which guarantee that a nonlinear system possesses such properties as output nulling [1,2,3,4,11,12,14], left-invertibility [7,9,15], dynamic input output decouplability [5,6,7,10,13,16]. Given a method from one of these references for verifying one of these properties, certain constant rank conditions arc needed to guarantee smoothness of various functions computed at each step and/or finiteness of the number of the computations. We propose an analysis of the regularity conditions in terms of the canonical map from initial conditions and inputs to outputs, for a general nonlinear nonsquare system. By reformulating the existing rcgularity conditions associated with steps of particular algorithms into a common framework, it is possible to understand their relations. In Section 2, we analyze the regularity conditions for the zerodynamics algorithm [3,4,12]. Sections 3 and 4 address respectively the inversion [15] and the dynamic decoupling algorithms [5,6,7,16]. It will be seen that the regularity conditions associated with the inversion algorithm are stronger than those associated with the zero-dynamics and dynamic decoupling algorithms; the conditions for the latter two are not comparable in the sense that neither one implies the other. It will also be seen that the procedure used by [6] for dynamic decoupling can be profitably incorporated into the inversion algorithm, in order to extend its applicability. "Also at: Dipartimento di Informatica e Sistemistica, Universits di Roma, ((La Sapienza)); work performed while visiting the Department of Electrical Engineering and Computer Science, University of Michigan at Ann Arbor. **Work supported by the National Science Foundation under Contract, NSF ECS 8896136.
844
We now introduce the framework in which the comparisons will be carried out. Consider an affine nonlinear control system E
:
& = f(x) +g(x)u
(1.1)
y = h(x)
where, x(t) e X, a simply connected open subset of R", u(t) e U = R "~, y(t) E Y = R ~', and f(.), the columns of g(.) = [g,(.),... ,g,,(.)] and the rows of h(.) = col(hi(.),..., h~(.)) are analytic functions of x. Define ~(x) = s p a n { g l ( x ) , . . . , g , , ( x ) } and suppose that dimG(x) = m for every x e Z . We let y(~ = y(x) = h(x) and define, by differentiating along trajectories of (1.1), y(k+l) = y ( k + l ) ( x , u , . . . ,u(k)) = ~ [ / ( z ) + g(x)u] + ~ -i=00u(O 1 0__~v_~(k),(i+1) We may view y(~) as an analytic function y(k) : X • Tk-xU ~ R~ (where T k - I U is the (k - 1)st order tangent bundle of U), or as a rational function of the components of u , o . . , u (k-l) with coefficients analytic in x. Adopting this latter point of view for the moment, define, following [7], Ifj to be the field of rational functions of (the components of) u , . . . , u (j-l) with memormorphic coefficients in x and set K := Kn. Let t; denote the vector space over K spanned by {dx, d u , . . . , du (,`-1)} where the differential of a vector valued quantity means the differential of each of its components. Define the nested sequence of subspaces of E by ~'o = s p a n { d y , . . . , dy(k)}, k = 0 , . . . , n. In [7] it is shown that dimS,, - dimE,`_1 = dim.T,` - dim.T,`_1 =: p* ~
where p*, the rank of the system (2.1) [9], is a limiting value in the sense that, if one extends $ and K in the obvious way, p* = dim$,`+k - dim$,`+k-1 = dimhr,,+k -- dim.T,`+k_l for
k>_0.
The above framework was very convenient for proving the conceptual equivalence of several algorithms, which compute system inverses, decoupling compensators and ranks associated with left- and right-invertibility, without specifically addressing the issue of singular points; indeed, they were effectively incorporated into the field K. tIowever, when actually constructing a compensator or an inverse, one must work over the field of reals and thus one requires the existence of an open set in which the necessary operations are well-defined. The following functions will be useful for stating the required regularity conditions [8]. Let Fo denote the map Fo : X -~ Y by x ~ y = h(x). For all k > 1, let F~ denote the map Fk : X x T k-~ U --~ T k Y by (x, u , . . . , u (k-~)) , (y, y 0 ) , . . . , y(k)) and let Ek denote the map Ek : X x T k - I U ~ X x T k Y by ( x , u , . . . , u (k-~)) ----* ( x , y , . . . , y ( k ) ) . In the obvious way, for 0 < k < n, Fk and Ek can be viewed as functions on X x T ,`-1U. Let F and E denote F,` and E,` respectively, and let ~" : X x T " - x U ~ X be the canonical projection.
2.
R e g u l a r i t y and o u t p u t nulling
The concept of the zero-dynamics, introduced by Byrnes and Isidori [1], and analyzed in a sequence of papers [2,3,4,11,14], has proven to be important through its applications to local stabilization problems, model matching, output reproducibility, exact linearization via dynamic feedback and asymptotic disturbance rejection (see [12] for a complete set of references and further results). This Section develops an alternate characterization of the
845
regularity conditions given in [4,12] for the purpose of comparing it to those associated with other algorithms important for nonlinear synthesis. For example, it will be clear that the regularity conditions for output nulling are weaker than the notion o[ strong regularity studied in [8] which arises from the inversion algorithm [15]. To do this, we rewrite the zero-dynamics algorithm using the formalism presented in Section 1. Let xo be a given point of X such that f(xo) = 0 and h(xo) = 0. Set Mo = {x E XIh(x ) = 0}. Assume that there exists an open neighborhood U~ of xo such that oh has constant rank on Mo N U~. Let M~ be the connected component of Mo gl U~ which contains the point xo. S t e p 1: Calculate y(1) = Cl(X) _{_d,(z)u and define DI(z) := Hi(z). Suppose there exists an open neighborhood U0 of Xo, contained in Ug, such that Dl(x) has constant rank, denoted rl, on M~ gl U0. Hence, there exists a permutation of the outputs, and a partition, such that upon writing y = col(if1, yl), where yl has rt components and yl has # - rl components, then
=
al(x) + 21(z)
/
satisfies rankRdl(x0) = rl. Since rankRDl(xo ) = rl, there exists an analytic matrix RI(x), defined on MGn Uo, such that d~l(X) = Rl(x)d:(x) on M~N Uo. Therefore, ?j~l) can be expressed as
~1)
----. ~I(X ) _[_ n1(X)(9~1) __ ~I(X)) =:
;
that is ~)~)(x,u)= 01(x,[/[l)(x, u)). It follows that
y~2)(X, U, U) = u,,(1): ~(1} -(2) "~ 1 (x, yt ,Yl ,u) 001 .:. = LfO 1 + ~ - [l) y~ ) +L~Olu .
(2.1)
Note that as M(~ VI Uo is an embedded submanifold of X, after possibly shrinking U0,/~l can be extended to an analytic function on all of Uo. Define U 1 {x E M~ 71 UolOt(x,O) = 0}. Assume that there exists an open neigh=
borhood U~ of Xo such that
a~
has constant rank on M1 A U~; let M~ be the
0z
connected component of M1 fl U~ containing Xo. S t e p 2: Differentiate !)[1). From (2.1), this can be written as
' Yl -C1),Yl ) + = e2(x, Define D 2 ( x , ~ D) col(dl(:~), d2(x,~l))). Suppose there exists an open neighborhood /_:1 of x0, contained in U~, such that D2(x, 0) has constant rank, denoted r=, on ~I~ VI UI. ----
For b e F/-I(0) n
n
U1),
rank R E2(b) = rank R
[.x] d~
d#
Jb
846
=
rank R
I
0
0
*
d 1(x0)
0
*
aYl(Xo)
*
* =
d2(xo, O)
. + rankn
0
+ rankR
=
+
+
Therefore there exists a permutation of the components of ~)x, and a subsequent partition, such that ~)1 = co1(92, ~)2), where ~)2 has r2 - rl components, and, upon writing
c2+d2u=[ ~~+J2u] + 2,~u
'
then rank R col(Jx(Zo), d~(zo, 0)) = r2. Therefore, there exists an analytic matrix R2(z, ~)~)), defined on ( M f n U1) x V~, where 1/1 is a sufficiently small open neighborhood of the origin in /:/r~, such that d2(x,~ 1)) = R2(x,~I)). col(dl(x), a~(x,~l))). It follows that 7)~~) can be expressed as
_.
--"
0 2 ( X , ~(1) -(2) -(2)-~ Yl
,Yl
,Y2
/
9
Etc. The regularity conditions of [4] for the zero-dyanmics algorithm can therefore be restated as, for 1 < k < n, a) rank R Dk(x,O,...,O) constant on M~_ 1 N Uk-1 (2.2)
Oh T(x)
b) rank R
-~-x (x,0)
constant on M~ N Uk
(2.3)
00k c) rank R Dn(xo, 0 , . . . , 0) = m (2.4) for a sequence of nested open neighborhoods Uo D .-. D Un of Xo; the superscript c denotes the connected component containing Xo. By construction M~ n Crk = ~r(F~-' (0)) c n U~ (2.5) We now show that (2.2), (2.3) and (2.4) are equivalent to: a') rankREk is constant on F/-21(0) n 7r-l(Uk_l) (2:6) b') r a n k R F k is constant on F~-~(0) N rr-l(Uk) (2.7) c') rankRE,,(b ) - rankRE~_l(b ) = m, b 6 Fs ) N 7r-l(U~_a) (2.8) To start, condition (2.2) is equivalent to rank R Ek is constant on F[J I(O)N~"-I (lr (F[_.x~(0))CA Uk-1) which, after possibly shrinking Uk-1, is the same as (2.6). By construction of the
847 algorithm, span{dy, dy(1),..., dy (k)} =
span{dy, d~!jl, 1 < i < k, i _< j _< k. dr163 1 _< m _< k} = Oh s p a n { ~ d z , d,)!J), 1 < i < k, i _< j _< k ,
00~, ~ ,a, ~ 00.. d_(j ) 0~ + Z . , 2 _ . ~ y~ , 1 <m_<]r i=1 j=i
(2.9)
91
Oh
9
span{d~! J),l < i < k, i < j _< k} @span{~xxdz ,
OOmdx 0--2
' l <-m<-k}
Furthermore, d~!j), 1 < i < k, i <_ j <__ k, are linearly independent over R for every b e F/_~I(0) n rr-l(Uk_l) and hence also for b e FIX(0) fl ~r-t(Uk); we have used condition (2.2). Condition (2.3) gives that span{~dx, o0i, o~:ax, 1 < j < k} has constant dimension over R for every b e Fs Cl ~r-~(U~), after possibly shrinking O'k. Therefore, (2.2) and (2.3) together imply (2.6) and (2.7). The converse is similar. Finally, at each step of the algorithm, rk = rankRDk(xo, O,...,O) = rankREk(b ) -rankREk-~(b ) for b E F/_~I(0) fl ~r-l(Uk_l). Therefore, condition (2.4)is equivalent to (2.8). Whenever the system is square, it is possible to show that (2.2) and (2.4) together imply
-~(~o) b")
has rank over R equal
the matrix
~ (x0, 0,..., 0) to the number of its rows. Indeed, (2.8) implies that the Jacobian linearization of (1.1) is right-invertible and therefore dim spanR{dy(xo),... , dy(")(Zo, 0 , . . . , 0)} = (n + 1). # (2.10) In view of (2.9), b") follows. In conclusion, for square systems, (2.6) and (2.8) are equivalent to the regularity condition of Proposition 1.7, p. 302 of [12].
3.
Regularity
and
system
inversion
In [11] it is shown that, in general, the notion of the zeros of a linear system may be extended to nonlinear systems in three distinct ways: (1) the dynamics associated with y -- 0 (i.e., the zero or output hulling dynamics), (2) the dynamics of a left-inverse system, and (3) the dynamics associated to a maximal loss of observability. Under the hypotheses that the system is square and the inversion algorithm [15] can be carried out, the first two notions always coinicide [11]. In [8], it is shown that the required regularity conditions
848
for the inversion algorithm are
rankREk(b ) = dimK Ek ,
(3.1)
1 < k < n, for b E F-~(0) N 7r-l(O), (L) an open neighborhood of z0. If, in addition, one wishes a reduced-order left-inverse, then the condition rankRFk(b ) = dimg 5vk , 0 < k < n, for b E F-~(0) A ~--~(O) is also required. These two conditions constitute local strong regularity as defined in [8].
4.
Regularity and dynamic decoupling
Starting with [5], and continuing with [6,7,9,13,16], several solutions have been developed for the dynamic input-output decoupling problem. Even though all the algorithms are variations on the theme of adding integrators, which goes back to [17], the required regularity conditions turn out to be quite different. In this Section, we express the regularity conditions, of several of the algorithms from the above papers, using the formalism presented in the Introduction. The dynamic extension algorithm of [7] constructs at each step a static state variable feedback and a dynamic extension in order to maximally decouple the outputs. In [8] it is shown that the required regularity condition is
rankREk(b ) = dimK gk , 1 < k < n
(4.1)
where b e F-i(0)f'l~r-l(O) and O is an open neighborhood of xo. In [16] it is shown how to modify the rule used in [7] for adding integrators in order to achieve a minimal order compensator. The required regularity conditions are unchanged. Until now, all of the regularity conditions presented have involved evaluating, in a sequential manner, conditions on the differentials of y, ~ , . . . ,y(k), where each output yl was always differentiated up to the same order. This simple pattern no longer holds for the algorithm of [6] where, when evaluating the constant rank conditions, (i) each component of the ouput may be differentiated a diffcrent number of times and (ii) certain lower order derivatives may be excluded. To make this precise, the algorithm of [6] is rewritten in an abstract form. We initialize the algorithm by setting Po = spang{dx) and I0 = r Assume that Pk (~,~) has been defined. Let n/k be the smallest integer such that ay i ~ Pk; these integers correspond to the relative degrees of the extended system at step k of the algorithm of [6]. Choose Ik C { 1 , . . . , # } such that .
(a) s ~ l~_,, k
k
(b) Pk + span~-{dy~ =') 1 < i <_ #} = Pk 9 span1r EIk}. Choosing Ik corresponds to selecting independent rows of the decoupling matrix at step k. Define Jk = {i e Ik]3j r Ik s.t. dim1r "~) , r e (/k\{i}) U {j}] = card Ik}. The index set Jk specifies the inputs on which an integrator is added because, after performing a nonsingular static feedback, there are at least two nonzero elements in the corresponding columns of the decoupling matrix at step k. Finally, define Pk+t = Pk @ spanK{dY["~)]i E Jk} ; Pk+, specifies the extended state space. In [6], it is shown that under right-invertibility [9], the algorithm converges in a finite
849
number of steps k*, when Jk" = r In order that the feedback computed at each step of the algorithm [6] be smooth, the corresponding decoupling matrix is assumed to have constant dimension at the "point of interest". For simplicity, we suppose that (Xo, u0 = 0) is an equilibrium point of (1.1) and that the "point of interest" is (x0, 0 , . . . , 0), an equilibrium point of the extended system. Hence, the regularity condition required is that there exists a choice for the index sets Io C I1 C ... C Ik-, such that, for 0 < k < k* dimg spang{dx , dy} "i) , dy! "k') , i E Ji, r EIk, 0 <_j <_ k - 1} =
dimRspanR{dz, dy["~)(zo, O,...,O), dy(~"~l(zo, O,...,0), i E Jj, r EIk,
(4.2)
0_<j~k-1} These conditions are obviously weaker than (4.1). To make this even clearer, we specialize them to two-input two-output (TITO) systems. Let nl and n~ be the relative degrees of the outputs and let n~ and n~ be the essential orders, as defined by [10]. The regularity conditions can be shown to be equivalent to either:
(a) dimnspann{
,
=
+
and (b) dim/:/spanR{dx , dy~"~)(Xo, 0,..., 0 ) , . . . , dy~'q)(Xo, 0,..., 0), dy~'~)(Xo, 0,..., 0)} =n+n~ -nl +2, or, the same conditions with yt and y~ interchanged.
5.
Concluding
remarks
In conclusion, we give two examples to highlight the comparisons of the regularity conditions studied in the paper. Consider the systems
{ :~1 "~"XlUl + X3 k2 = z2u~ + x4
22 :
[ :~I = X3Ul + X'4 :~2 X3 9
/
with Yl = xl and y~ -- x2 in each case. It is easily checked that the regularity conditions (3.1) for the inversion algorithm [7,15] and the dynamic extension a]gorithms [7,16] do not hold. However, for El, the regularity conditions (4.2) for the dynamic decoupling algorithm [6] are not met whereas the conditions (2.6), (2.7) and (2.8) for output nulling [12, Proposition 1.7 p. 302] are met. The contrary holds for E2. Nevertheless, for E2 a zero-dynamics manifold can still be computed using the conceptual version of the zero-dynamics algorithm [12, p. 290] which, apparently, requires the weakest possible conditions. The analysis carried out in Section 4 suggests an improvement of the inversion algorithm [7,15]. Instead of calculating the derivatives of the output step by step, one should adopt the strategy proposed in [6]. For example, on E2, one would proceed as follows:
Yl = X3Ul -I- X4
850
Write Yl = x3ff2 -~ Y4 and differentiate again to obtain ffl = "T3Y~3) "]- (if2) 2 "~ /12- Finally,
9= Acknowledgement: The authors thank J. Hauser and A. Isidori for many inspiring conversations on the subject of this paper. References 1. C.I. Byrnes and A. tsidori, "A frequency domain philosophy for nonlinear systems, with applications to stabilization and adaptive control", Proc. 23rd IEEE Conference on Decision and Control, Las Vegas, NV, Dec. 1984, pp. 1569-1573. 2. C.I. Byrnes and A. Isidori, "Local stabilization of minimum-phase nonlinear systems", System ~J Control Letters, 11, 1988, pp. 9-17. 3. C.I. Byrnes and A. Isidori, "Asymptotic properties of nonlinear minimum phase systems", Colloque International Automatique non Lindaire, Nantes, France, June 1988. 4. C.I. Byrnes and A. Isidori, "Feedback design from the zero dynamics point of view", Workshop on Computation and Control, Bozeman, Montana, Aug. 1988. 5. J. Descusse and C.H. Moog, "Decoupling with dynamic compensation for strong invertible a/fine nonlinear systems", Int. J. Control, 42, 1985, pp. 1387-1398. 6. J. Descusse and C.H. Moog, "Dynamic decoupling for right-invcrtible nonlinear systems", Syst. Contr. Lett., 8, 3, 1987, pp. 345-349. 7. M.D. Di Benedetto, J.W. Grizzle and C.H. Moog, "Rank invariants of nontincar systems", SIAM J. Control, 72, 3, 1989, pp. 658-672. 8. M.D. Di Benedetto and J.W. Grizzle, "Intrinsic notions of regularity for local system inversion, output nulling, and dynamic extension of nonsquare systems", submitted to C-TAT, April 1989, revised November 1, 1989. 9. M. Fliess, "A note on the invertibility on nonlinear input-output differential systems", Syst. Contr. Lett., 8, 2, 1986, pp. 147-151. 10. A. Glumineau and C.H. Moog, "Essential orders and the non-linear decoupling problem", Int. J. Control, Vol. 50, NO. 5, !989, pp. 1825-1834. 11. A. Isidori and C.H. Moog, "On the nonlinear equivalent of the notion of transmission zeros", in Modelling and Adaptive Control, C.I. Byrnes and A. Kurszanski, Eds., Lccture Notes in Control & Information Sciences, Springer-Verlag, Vol. 105, 1988. 12. A. Isidori, "Nonlinear control systems", 2nd Ed., Springer Verlag, Berlin, Communications and Control Engineering Series, 1989. 13. H. Nijmeijer and W. Respondek, "Decoupling via dynamic compensation for nonlinear control systcms', Proc. 25th CDC, Athens, 1986, pp. 192-197. 14. A.J. van der Schaft, "On clamped dynamics of nonlinear systems", Proc. M.T.N.S., Phoenix, AZ, Analysis and Control of Nonlinear Systems, C.I. Byrnes, C.F. Martin, R.E. Saeks, Eds. North-Holland, 1988. 15. S.N. Singh, "A modified algorithm for invertibility in nonlinear systems", IEEE Trans. Automatic Control, AC-26 (1981), pp. 595-598. 16. X.-H. Xia, "The essential structure of nonlinear systems", preprint, 1989. 17. S.H. Wang, "Design of precompensator for decoupling problem", Electronics Letters, 6, 1970, pp. 739-741.
DISCONTINUOUS PREDICTIVE CONTROL, I N V E R S I O N AND S I N G U L A R I T I E S . APPLICATION TO A HEAT EXCHANGER.
Michel Fliess 1. Salwa Abu el Am 3* Anne Coi'c 3*
(1) (2) (3)
Pierre Chantre 2
Laberatoire des Signaux et Syst6mes, CNRS - ESE, Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France. Laboratoire d'Automatique du CNAM, 21 rue Pinel, 75013 Pads, France. ADERSA, 7 Boulevard du Mar6chal Juin, B.P. 52, 91371 Verri6res-le-Buisson Cedex, France.
A B S T R A C T : Predictive control by inversion is applied to nonlinear systems. Problems related to singularities and non minimum-phase, which often arise in this context, are most simply overcome by introducing a discontinuous control approach. Such a solution is illustrated on a heat exchanger example.
1. I N T R O D U C T I O N Model Based Predictive Control (MBPC) is now well established and, from its successful application in various fields, one can evaluate its performances, disadvantages and advantages (see for example Clarke et al. [6], Richalet et al. [14]). A nonlinear control strategy based on MBPC principles has recently been proposed (Abu el Ata et al. [1, 2]) by employing techniques stemming from the input-output inversion of nonlinear systems (Fliess [9]). The main novelty of our communication, with respect to the above references, concerns
* Work partially supported by a DRET Contract n~ 89-1159.
852
singularities and non minimum-phase problems which almost always occur for nonlinear systems. We propose in this paper, to consider these problems through a discontinuous control approach instead of using smooth control as in Hirschorn et al. [ 12] and LamnabhiLagarrigue et al. [13]. Discontinuous control allows, in fact, to circumvent the dangerous zone where the singularities and the unstabilities are living. This avoids the necessity of a full mathematical treatment of those problems which are well-known to be most difficult. Notice that the idea of using discontinuous control is not new (see, e.g., Cypkin [7] and Utkin [ 16]). The discontinuities are here related to an algebraic understanding of nonlinear systems structure. This has also been proved to be fruitful for nonlinear stabilization (Fliess and Messager [10]). The validity of the proposed discontinuous control approach is demonstrated with two academic examples, and with the case study of a heat exchanger. This real process was previously studied and controlled with nonlinear predictive control approach by Chantre [4, 5]. This process has the advantage to possess a simple nonlinear modelization which exhibits singularities and non minimum-phase problems. The simulations have been done with the help of the MATLAB CAD software package. 2. P R E D I C T I V E C O N T R O L BY I N V E R S I O N A brief review of the principles of predictive control by inversion is given here for the SISO case. For more details, see [I, 2]. In this technique, the control variable is calculated according to a receding horizon strategy. That is, at each sampling instant (see Fig. 1): - The process output has to rally a set point trajectory in the future. - A reference trajectory initialized on the actual process output defines the way to rally the set point on a prediction horizon. The characteristics of this trajectory are chosen in relation to the desired closed-loop dynamics. This trajectory must also satisfy some compatibility conditions with respect to the initial conditions, according to the inversion theory. - The future control values are provided by the left inversion of the process model, but only the first determined control value will be applied, the whole procedure being repeated at the next sample instant. Illustrative examples of application of this technique are given in [ 1, 2].
853 3. W H A T D I S C O N T I N U O U S
CONTROL
IS G O O D F O R
As often encountered in nonlinear systems, singularities might occur and have to be taken into account in the control strategy. The approach adopted in this paper consists of treating singularities through discontinuous control which permits us to avoid the dangerous neighbourhood around the singular points. We first give a brief mathematical definition of singularities and, for the sake of simplicity, we restrict ourselves to the SISO case. Let us assume that the system is given by an input-output differential equation:
F(u, u . . . . . u (a), y, ~ . . . . . y(~)) = 0
(1)
where u and y are respectively the input and the output variables. Notice that if we start with the usual state variable representation, this setting implies eliminating those variables (see Diop [8]). The equation (1) also defines the inverse system, which permits us to calculate u, when we know y. In order to express the highest derivative u (~) as an explicit function u (cO = f(u, u . . . . . u (cc-I), y, y . . . . . y(~)), the implicit function theorem has to be employed. A singularity occurs therefore when the OF Jacobian 0 7 vanishes. A possible way for keeping this coefficient far enough away from zero consists of introducing discontinuities in u and/or in some of its derivatives, while the output variable remains smooth. Another disturbing phenomenon appears with unstabilities, i.e., when the inverse system is unstable. This is, for example, the case when the linear tangent system: FuAu + F,',Au + ... + Fu(a)Au (a) + FyAy + FyAy + ... + Fy(~)Ay(~) = 0
(2)
is non minimum-phase around a constant operating point, that is where u (v) = y(V) = 0 for v >_ 1 (see B y m e s and Isidori [3] for a more general definition of the nonlinear minimumphase concept).
854 Without entering into too many details, we can see that the fact of being minimum-phase or non minimum-phase for (2) defines a partition o f the space (u, y). Once again, taking discontinuous values for u and/or some of its derivatives, while y remains smooth, might lead our system from a non minimum-phase domain to a minimum-phase one.
4. A P P L I C A T I O N S In all the SISO applications which we present here: - The control law is obtained by inversion. -
The set point trajectory c(t) is a step function joining two constant operating points of the system 9
For the two academic examples, the reference trajectory yr(t) is defined via a linear fast order differential equation
~+'~=0 where "~is the time constant and e = Yr c. For the last heat exchanger example, the reference -
trajectory is defined via a linear third order differential equation
E + 3"~ + 3~2e + 3"~3"e" = 0. 4.1
Academic examples 9
,
S I N G U L A R I T Y : Take the input-output system y + "g'y = u + # ' u e u , where 'g' and z" are positive constants. There is an obvious singularity at u = 0. We want to rally the previously defined set point trajectory from a constant operating point where u is negative, to another one where u is positive. The usual continuous control leads to the singularity at u = 0, where a conventional computer integration stops (see Fig. 2a). If we impose a discontinuity on u to avoid this particular zero value of u, the integration works and the possibly difficult problem of analyzing the singularity disappears (see Fig. 2b).
855 UNSTABILITY: Now consider the following system: y + "~'~,= u + "c"u u, where "c' and z" are positive constants. The tangent linearized system around the constant operating point (u, Y) Ay + x'A~' = Au + z"uAu is non minimum-phase for u < 0. So, if we want to rally the same set point trajectory, the usual continuous control is divergent (see Fig. 3a). On the contrary, the introduction of a discontinuity, which yields u positive, brings back to stability (see Fig. 3b). Notice that we have also avoided the singularity at u = 0. 4.2
Heat exchanger
A brief physical description of this process is given here. For more details about this description, for the complete modelization and for nonlinear continuous control applications of this process, see [4, 5]. This system, used for energy recovery in the petrol industry, is composed o f two countercurrent liquid-liquid parallel heat exchangers (see Fig. 4). The objective is to control the output temperature y, by means of the flow repartition rate u. The other variables are here supposed to be constant. The input-output behaviour is very nonhnear in the operating domain: - The static characteristic (Fig. 5a) presents a hill top, i.e., the sign of the static gain changes.
- The dynamics (Fig. 5b) changes from a well damped behaviour to an oscillatory one, with minimum-phase or not. The non minimum-phase domain of the linearized model can be located on the static characteristic. Physical laws give a simplified state variable representation of dimension three:
Xl = a l X l + b l u + c l
x2 = a2x2 + b2u + c2 = a3y + uxl + (1 - u)x2 where the state is composed of the intermediate temperatures xl, x2 and the output temperature y. The coefficients al, a2, a3, bb b2, Cl, c2 are constants.
856
It is possible to eliminate state variables xl, x2 for such a bilinear system [8, 11, 15]. It yields the following input-output equation: [ ( a 2 - al)(U - 1)u + u ] [ - " y + a3"y+ ( a l b l - a2b2)u 2 + 3(bl - b2)ux + + (alcl - a2c2 + a2b2)u + (2(cl - c2) + b2)u + a2c2] + + [al2u + 2alu + u][(1 - u ) y - ((a2 + a3)(1 - u) - L1)~r + + a3(a2(1 - u) - u)y - (1 - u)(bl
-
b2)u 2 + (b2+ C l - c2)u + c2)] +
+ [aE2(U - 1) + 2a2u + l J ] [ u y - ((al+ a3)u + u)y + + a3(alu + h)y -- u((bl - b2)u 2 + (b2+ Cl - c2)u + c2)] = 0 Singularities arise when the coefficient of u vanishes:
)' + ((al - a2)u - a2 - a3)y + a3((al - a2)u + a2y) - ((bl - b2)u 2 + (b2 + cl - c2)u + c2) = 0 W i t h o u t m a k i n g a complete analysis o f this last equation, we notice that a particular singularity is obtained from its static solution (), = y = 0).This static singularity is located in the non minimum-phase domain (see Fig. 5a). W e want to rally the previously defined set point trajectory, from a constant operating point which lives outside the non m i n i m u m - p h a s e domain, through this domain. The usual continuous control stops near a singularity which is located in the non m i n i m u m - p h a s e domain (see Fig. 6a). The introduction o f a discontinuity on u leads the system outside the non minimum-phase domain, and therefore stabilizes it (see Fig. 6b). In the same time, the problem o f the singularities is avoided. 5. C O N C L U S I O N W e d e m o n s t r a t e d that the introduction o f control variables discontinuities p e r m i t s a straightforward solution o f classic control problems, which seem to be very difficult in the setting o f smooth controls. Note that our analysis does not apply if the Jacobian ~
3F
related to (1) does not contain u
nor any o f its derivatives. This is also the case when the inverse system has no dynamics, that is ~x = 0 in equation (1):
857 ~
F(u, y, y ..... y(B)) = 0. In order to bypass these important particular cases, which are often encountered in practice, we can possibly reverse the preceding analysis by now introducing other types of discontinuities. The problem will be dealt with in future publications. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13]
[14] [15] [16]
S. Abu el Ata and M. Fliess: Nonlinear predictive control by inversion, Proc. IFAC Symposium on Nonlinear Control Systems Design, A. Isidori ed., Capri, Italy, June 1989. S. Abu el Ata, A. Co'fc and S. Chavy: Commande pr6dictive non lin6aire par inversion. Application hun probl~me de guidage d'avion. Actes Colloque SMAI "L'Automatique pour l'A6ronautique et l'Espace", Paris, Mars 1989. C.I. Byrnes and A. Isidori: Local stabilization of minimum-phase nonlinear systems, Systems Control Lett., 11, pp. 9-17, 1988. P. Chantre: Commande et optimisation de processus industriels non lin6aires. Mise en oeuvre sur un train d'6changeurs thermiques en parall61e, Th6se Doct. Ing., Universit6 Paris VI, Paris 1983. P. Chantre: Commande pr6dictive non lin6aire h l'aide d'un mod61e de Volterra, IASTED International Symposium, Grindelwald, Switzerland, February 1989. D.W. Clarke, C. Mohtadi and P.S. Tufts: Generalized Predictive Control - Parts I and II, Automatica, 23, pp. 137-148 and pp. 149-160, 1987. J.Z. Cypkin: Thforie des asservissements par plus-ou-moins (translated from Russian), Dunod, Paris, 1962. S. Diop: Elimination in control theory, Math. Control Signals Systems, to appear. M. Fliess: Automatique et corps diff6rentiels, Forum Math., 1_, pp. 227-238, 1989. M. Fliess and F. Messager: Vers une stabilisation non lin6aire discontinue, These Proceedings. M. Fliess and C. Reutenauer: Une application de l'alg6bre diffdrentielle aux syst6mes r6guliers (ou bilin6aires). In "Analysis and Optimization of Systems", A. Bensoussan and J.L. Lions eds., Lect. Notes Control Inform. Sci 44, pp. 99-107, Springer, Berlin, 1982. R.M. Hirschorn and J. Davis: Output tracking f0r nonlinear systems with singular points, SIAM J. Control Optimiz., 25, pp. 547-557, 1987. F. Lamnabhi-Lagarrigue, P.E. Crouch and I. Ighneiwa: Tracking through singularities. In "New Trends in Nonlinear Control Theory", J. Descusse, M. Fliess, A. Isidori and D. Leborgne eds., Lecture Notes Control Inform. Sci. 122, pp. 44-53, Springer, Berlin, 1989.' J. Richalet, S. Abu el Ata, C. Arber, H.B. Kuntze, A. Jacubash and W. Schill: Predictive Functional Control, Application to fas~ and accurate robots. Proc. 10th IFAC Congress, Munich, 1987. E.D. Sontag: Bilinear realizability is equivalent to existence of a singular affine differential i/o equation, System Control Lett., 11, pp. 181-187, 1988. V.I. Utkin: Sliding Modes and their Application in Variable Structure Systems (translated from Russian), Mir, Moscow, 1977.
858
Prt~s5
Pa~
out:put:
<---
I
Present
.."-
--->
Fig. 1 Predictive
Future
control
0.:1
~149
y
Or!
O
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2+6 t.
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9
. . . . .
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z
2.:1
6
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4
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U
)
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0.5
O
-0.S ......... " ' ' ' " I
1.6
2
~6
2
6.6
4
-I
+I.5
0
t.
ao Continuous u
I
.
0.5
I
t,.
b. Discontinuous
control singularity
example
( r =0.7, ~=i,
control ~=20
)
. 0.0 01!
y
9 .
-I
-i
-1
. .
-0.~. ......... "
-0.6 ......... '
U
principle
." e+"'" ' '
...
+ .
o.o!_i 2 -O.6 ......... "' O'll
-0.5 ......... ""
I
1,6
2
2,6
3
:1.6
4
, 4.0
-I 0'6
I
i.5
, 2
i L2 t
, 6
, :1.6
, 4
, 4.6
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6
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i
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l
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2
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4
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t a. Continuous Fig.3
'o
:! i -++ ...... : 0
I
|.5
2
t
control Unstability
b. Discontinuous example
( r=0.7,
~=i,
control ~'=i
)
859 secondary
u
T
(x-
t
>~h
J x2
2
ex_~.anger "
U Q, QI, Q2 T, TI, T2 y, xl, x2
~ ~ %
secondary fluid 2 Q2 T2
: f l o w r e p a r t i t i o n r a t e of the v a l v e (0
Fig.4
"C
Parallel
heat
exchangers
I
static optimum
80
/o ~2
70
/ o
o/ A
o"
9
Fig.5b
60
........~o:
, \~
i
50-
/0 Fig.5a
,
0.5
Static
~
I
1
characteristic
Dynamics
860 611
61
I
('Clm[ y
61 1
,
,
.
66
.. 9.-
...........
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+'
t.................. 9............. t ;o
o
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8o
. . 12o. . 14o.
1o0
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minlmum-phaae
'
1~ non
domain
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0.2
0
"~"
F.O
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8O
t
IoO
120
140
160
180
20o
0
i.~...................
a. C o n t i n u o u s
,
P.O
4o
80
oo
t
(second)
control
domaln
'~ " " "
0
0
ulnlmum-phase
,
,
*
,
100
lzo
;4o
100
(second)
b. D i s c o n t i n u o u s c o n t r o l
Fig. 6 C o n t r o l of t h e h e a t
exchanger
( r=20s.
)
,
laO
20O
THE ANALYSIS OF SINGULARLY PERTURBED ZERO DYNAMICS A.Isidori Universith di Roma, Rome, Italy, and Washington University, U.S.A. S.S.Sastry University of California Berkeley, California, U.SA.
1.
P.V.Kokotovic University of Illinois Urbana-Champaign, Illinois, U.S.A. C.I.Bymes Washington University St.Louis, Missouri. U.SA.
Introduction
In the last few years, there has been a great deal of interest in the design of o f f e e d b a c k laws for nonlinear s y s t e m s . A particularly a p p e a l i n g design technique, substantially proposed as early as in the work [1] of Porter on nonlinear noninterating control, is the one based on the use of state feedback in order to achieve a linear input-output behavior. This technique, which relies upon exact cancellation of nonlinear terms via feedback and is, to some extent, a nonlinear equivalent of pole-zero cancellation achieved by incorporating into the control law an inverse of the controlled plant, may produce undesirable closed loop b e h a v i o r if the system zero d y n a m i c s are unstable. The system zero dynamics represent the nonlinear analogue of the zeros of the transfer funcion of a linear system and are defined as the internal dynamics arising in a system when input and initial conditions are chosen as to constrain the output to remain zero for a some interval of time (see [2], [3], [4]). Nonlinear systems whose zero dynamics are asymptotically stable are called minimum phase systems. In the recent paper [5], motivated by the study [6] of the linearization by nonlinear state feedback of a class o f slightly n o n - m i n i m u m phase systems encountered in the flight control of V T O L aircraft, it was pointed out that regular perturbations of the parameters of a nonlinear model may result in a singular perturbation of its zero d y n a m i c s and, in particular, in a two time scale decompositon [7] of the zero dynamics of the perturbed system. In the present paper, which continues the analysis o f singularly perturbed zero dynamics initiated in [5], we introduce a new formalism which somehow simplifies the previous analysis and provides additional insight.
2.
Notations
and
basic
assumptions
Consider a family of SISO nonlinear systems depending parameter e, described by equations of the form: = f(x,e) + g(x,e)u y = h(x,e)
on a real-valued
(2.1)
862
where
f(x,e),
g(x,e), h(x,E)
are a s s u m e d
to be smooth
functions
d e f i n e d in a
n e i g h b o r h o o d of (x~ in R n x R . The system corresponding to the value e = 0 of the p a r a m e t e r is c a l l e d the u n p e r t u r b e d s y s t e m , while any s y s t e m o f the family c o r r e s p o n d i n g to a nonzero value of e is called a perturbed system. The state x = x ~ is an e q u i l i b r i u m point of the unperturbed system, i.e. f(x~ = 0 and, without loss of generality, it is assumed that h(x~ = 0. In what follows, we assume that each system o f the f a m i l y (2.1) has relative degree, noted r(e), at the point x ~ Thus, by definition, r(e) satisfies: Lgh(x,e) . . . . .
LgLfr(e)'2h(x,e) = 0
LgLfr(E)-lh(x~
# O.
for all x near x ~
Clearly, either r(e) is constant in a neighborhood o f e = 0, or it abruptly changes at e = 0. In the latter case, which is of m a j o r interest in the analysis of the zero dynamics, there exist two positive integers, noted r and d with d # 0, such that: r(e) = r r(0) = r + d.
for all e > 0 in a neighborhood of e = 0
S y s t e m s in which this situation occurs are said to have a singularly perturbed zero dynamics. The integer r is the r e l a t i v e degree of the perturbed systems, and the integer r + d is the relative degree of the unperturbed system. By definition, the change from r(e) = r to r(0) = r + d implies that: Lgh(x,e) . . . . . LgLfrqh(x~
LgLfr'Zh(x,e) = 0 # 0
LgLfrqh(x,O) . . . . . LgLfr+d'lh(x~
for all x near x ~ and all e
for all e > 0 LgLfr+d-2h(x,O) = 0
(2.2a) (2.2b)
for all x near x ~
# O.
(2.2c) (2.2d)
The f o l l o w i n g d e v e l o p m e n t s are based on an assumption about the e - f a m i l y of systems (2.1) which, as we shall see later on, allows the singularly perturbed zero d y n a m i c s to possess only two time scales. First of all we assume, in view of (2.2b) and the first on o f (2.2c), that: LgLfr'lh(x,e) = ema0(x,a) where a 0 ( x , e ) is a smooth function, nonzero at (x~
(2.3) and m is an integer. Without
loss o f generality, m can be set equal to d. Otherwise e can be replaced by e d/re. Then, our two-time-scale assumption is that, for all 0 < k < d,
863
LgLfr'l+kh(x,e) = ed'kak(X,e)
(2.4)
where each ak(x,e ) is a smooth function in a neighborhood of (x~ Note that for k = d in (2.4), the condition (2.2d) implies ad(X~ r 0. In particular, this assumption is e.g. satisfied when g(x,E) has the form: g(x,e) = go(x) + egl(x) and f(x,e), h(x,e) are independent of e (see [5]). 3.
The
analysis
of a
singularly
perturbed
zero
dynamics.
We proceed now to identify the zero dynamics of the peturbed system and to show how they are related to the zero dynamics of the unperturbed system. In doing so, we take the advantage of the fact that the zero dynamics of a system are not altered by addition of integrators to the input channel (see [8, page 389]). Because of this property, we can identify the zero dynamics of any system of the family (2.1) with those of the following family of e x t e n d e d systems: = f(x,e) + g(x,e)z 1 EZ 1 = Z 2
ez 2 = z 3
~Zd =
(3.1)
v
y = h(x,e). In order to identify the zero dynamics of (3.1), we set to zero the output y and its derivatives, until we arrive at a relation that can be solved for some v = v ( x , z , e ) . Then, we set v = v(x,z,e) on the system (3.1) and we consider the restriction of the systems thus obtained on the submanifold (of the state space) on which the output and its derivatives are vanishing. We first obtain: y = h(x,E) y(1) = Lfh(x,e) . . ,
y(r-1) = Lfr-lh(x,e) y(r) = Lrrh(x,e) + a0(x,e)eazl Because of the special structure of (3.1), the (r+l)-th time derivative of y has the form: y(r+l) = Lfr+lh(x,E) + al(X,E)Ed-lz1 + (~o(X,E)ed-lz2 +
864
+ (Lfao(X,s
+ (LgaO(X,s
2.
In a s i m i l a r way, we obtain: y(r+k) = Lfr+kh(x,E) + t~k(X,s
+ ... + Cto(X,e)Ed-kzk+1 +
+ ed'k+lpk(X,Z 1. . . . . Zk,s where Pk(X,Zl . . . . . Zk,e) is a p o l y n o m i a l in z 1. . . . . f u n c t i o n s of x and e. T h e last two iterations give:
z k, whose coefficients are smooth
y(r+d-1) = Lfr+d-lh(x,e) + ad.1(x,e)eZl + ... + ao(X,e)eZd + + e2pd.l(x,z 1. . . . .
Zd.l,s
y(r+d) = Lfr+dh(x,e) + CZd(X,e)Zl + ... + Oh(x,e)Zd+ C~0(x,e)v +
+ ePa(x,zt . . . . . Zd,e). R e c a l l that, by a s s u m p t i o n , o:0(x~
~ 0. Thus, in a n e i g h b o r h o o d of x = x ~ z 1. . . . .
z d = 0, e = 0, the e q u a t i o n y(r+d) = 0 can be solved for v, as a s m o o t h x,z 1. . . . . Zk,e. This function is a feedback law of the form: d v(x,z,e) = ~ ai(x,~)z i + b(x, e) + eQd(X,Z,e) i=l
f u n c t i o n of
(3.2)
where z = (z I . . . . . zk) and al(x~ = (cq(x~176 # 0. E q u a t i n g to zero y a n d its first r + d - i d e r i v a t i v e s p r o v i d e s a set o f r+d c o n s t r a i n t s d e l i n e a t i n g the s u b s e t of the state space on w h i c h the zero d y n a m i c s of (3.1) are d e f i n e d . These c o n s t r a i n t s are c o n v e n i e n t l y e x p r e s s e d in the form: 0 = H(x,e) + eR(x,z,e)
(3.3)
where: H(x,e) = col(h(x,e), Lfh(x,e) . . . . .
Lfr+d'lh(x,e))
(3.4)
a n d R(x,z,e) is an ( r + d ) - d i m e n s i o n a l v e c t o r of s m o o t h f u n c t i o n s satisfying R(x,0,e) = 0. O b s e r v e n o w that, at e = 0, the r i g h t - h a n d - s i d e of (3.3) reduces to H(x,0) and that, at x = x ~ the d i f f e r e n t i a l s of the r + d e n t r i e s o f H ( x , 0 ) are l i n e a r l y i n d e p e n d e n t b e c a u s e the s y s t e m (2.1) has r e l a t i v e degree e x a c t l y equal to r + d at
865 (x,a) = (x~ (see [8, page 148]). As a consequence, at (x,z,a) = (x~ matrix of (3.3):
the jacobian
[ ~-~(H(x,e) + eR(x,z,e)) e~ R(x,z,e)] has rank r+d. It follows that - for each e near 0 - the set of points (x,z) which s a t i s f y (3.3) n e a r (x~ is a s m o o t h ( r + d ) - d i m e n s i o n a l s u b m a n i f o l d . This s u b m a n i f o l d , noted Z*, is rendered i n v a r i a n t by the f e e d b a c k law (3.2) and the restriction o f the flow of the c l o s e d - l o o p system (3.1)-(3.2) to Z* describes the zero dynamics of (3.1), i.e. that of the family (2.1). In order to represent these zero d y n a m i c s e x p l i c i t l y , we note that since the differentials of the entries of (3.4) are linearly i n d e p e n d e n t at x ~ for each small e, the f u n c t i o n s : ~i(X,E) = L f i ' l h ( x , e )
1 _< i < r + d
can be used as a (partial) set of new local c o o r d i n a t e s for (2.1). The partial t r a n s f o r m a t i o n thus defined is smooth in x and e. If r+d is strictly less than n, then the c h o i c e of new coordinates m u s t be c o m p l e t e d by a set of additional smooth functions rli(x,e), 1 _< i ~ n-r-d. Since, for all (x,e) near (x~ we have: (span {g(x,e)})lq (span {dh(x,e) . . . . .
dLfr+d-lh(x,e) }).1. = {0}
we can always choose rl(x,e) (see [8, page 150]) in such a way that Lgrl(x,e) = 0 for all (x,e) near (x~ Without loss of generality, it is possible to choose rl(X,e ) so that rl(X~ = 0. In the new coordinates: = (~1 . . . . .
~r+d)
= (rll . . . . . "qn-r-d) the system (2.1) assumes the form:
= c(~,n,e) + d(~,n,e)u
=q(~,q,e). Recall that, by definition, the system: = q(0,n,0) r e p r e s e n t s the zero dynamics of the unperturbed system (2.1). In the new c o o r d i n a t e s , the e x t e n d e d s y s t e m (3.1) with f e e d b a c k (3.2) assumes the form:
control
866 = c(%, 11, e) + d(~, rl, e)Zl fi = q(~, 11, e) E7.1 = Z 2
(3.5)
ET.2 = Z 3
d I~Zd = E ~i(~,rl,E)zi + ~(~,TI,E) + E(~d(~'iq'Z'E)" i=l Note that, in the new coordinates (~, rl), the constraint (3.3) identifying the zero dynamics manifold becomes:
0 = ~ + el~(~,n,z,E). Solving this relation for
~
yields:
= eF(rl, z, e) where F0q, z, e) is a smooth function. We see, then, that in the (~,rl,z) the zero dynamics manifold Z* of the perturbed system (3.1) can be the graph of a smooth mapping; in particular, on this manifold the ~ are of order e. In order to render the zero d y n a m i c s of (3.1) explicit, is substitute (3.6) into (3.5) to obtain:
(3.6) coordinates, expressed as coordinates suffices
to
= q(eF(rl, z, e), rl, e) F~ 1 = Z 2 s
(3.7)
= Z3
d ezd = E ~i(eF(q, z, e),rl,e)z i + ~(FA::(rl, z, e),rl,e) + i=l + e(~d(eF(rl, z, e),rl, z, e) T h e s e equations describe the zero dynamics of the perturbed systems for each small nonzero e. They appear in a singularly perturbed form:
ri = f(n, z, e) e~. = g(B, z, e) with jacobian
matrix:
(3.8)
867
0
1
0
...
0
0
0
1
...
0
az(n,z,O) ) =
o.,
0z
0
0
0
(3.9)
...
1
...
~d(0,1],0)
which is n o n s i n g u l a r for all aq near 0, because
~1(0,0,0)
~ 1(0,1],0)
is nonzero
(recall that
A
~(x~ = 0, ~q(x~ = 0, and that ai({,rl,a) derives from ai(x,e) with x expressed as a function of (~,rl,e)). Hence, (3.8) is in the standard two-time-scale form [7]. F r o m the above analysis we see that, under the assumptions (2.2), (2.3) and (2.4) the zero dynamics of the family of perturbed systems (3.1) exhibit the twot i m e - s c a l e b e h a v i o r of a s i n g u l a r l y p e r t u r b e d s y s t e m . Its " r e d u c e d system", defind by setting e = 0 and eliminating the variable z, is exactly the system: fi
=
q(O,~,O)
(3.10)
which d e s c r i b e s the zero d y n a m i c s o f the u n p e r t u r b e d system. Its " b o u n d a r y l a y e r system", obtained by rescaling time as x = t/e and then setting e = 0, is that of a family of linear systems p a r a m e t r i z e d by a constant vector rl, n a m e l y : 0
1
0
...
0
0
0
1
...
0
0
0
0
...
1
dz
Z+
d~
~(O,n,o)
4.
... ~d(O,n,O)
0
(3.11)
(ore,o)
Remarks
Remark
1. There are cases p o l y n o m i a l o f (3.11) vanish, instance - when the system is d i s c u s s e d in [5]. In each of system are the roots of: xa _ ~(0,n,0)
in which all the c o e f f i c i e n t s of the c h a r a c t e r i s t i c e x c e p t for ~ l ( 0 , r l , 0 ) . This situation occurs for linear and g(x,e) = b0 + b l e or h(x,e) = co + cle, as these cases the e i g e n v a l u e s of the b o u n d a r y layer
= 0
and therefore the boundary layer system is unstable if d > 2.
868 R e m a r k 2. A more detailed inspection of the constraint (3.3) which identifies the
zero dynamics manifold Z* of the unperturbed R(x,z,e) can be rewritten as:
system shows
that the vector
R(x,z,e) = [ 0 .1 [S(x,z,e)] with: S(x,z,e) = A(x,e)z + eQ(x,z,r where: 9
a0(x,e)e
d-1
A(x,z) =
0 ao(X,e.)e,d'2
9 ..
9~
0
0
~
...
ao(X,e)
and Q(x,z,E) is a function which vanishes at z = 0 together with its first order derivatives with respect to z. Therefore, choosing ~' and ~" to denote:
~" = colG+x . . . . . ~r+d) the constraint (3.3), expressed in the (~,~,z) coordinates, has actually the form:
0=~' (4.1)
0 = ~" + ~(~',~",n,E)z + E2(~(~',~",n,z,~) A
where A(0,0,0,e) is nonsingular for small nonzero e. In the previous analysis, we have solved (4.1) for ~' and ~" in order to express Z* as a graph of a mapping. In A
view of the nonsingularity of A(0,0,0,e), one could have solved these equations for ~' and z, thus obtaining Z* in the form of a graph of a mapping:
~'=0 z = G(~",~,e) and express the zero dynamics obtained is not defined at e = 0.
accordingly.
However,
the transformation
thus
869 Remark 3. If the basic assumption (2.4) fails to hold, i.e. if any of the functions ai(x,e), 1 < i < d-1 has a singularity at e = 0, most of the previous analysis is still valid, and in particular the expression (3.7) of the zero dynamics of the perturbed system. However, the equations in question are not anymore in the standard twotime-scale form, and the singularly perturbed zero dynamics may exhibit more than two time scales.
Remark 4. If a system is modeled by an input-output differential relation of the form: F(y,y 0) .....y(n),u,uO) .....U (n-r)) = 0
then its zero dynamics, those induced by the constraint y(t) = 0, can be clearly identified as those of the autonomous system: F(O,O ..... O,u,u (1)..... u (nr)) = 0 This point of view suggests a simple interpretation of the calculations illustrated in the previous section and, in particular, of the e-scaling introduced in the extended system (3.1). To this end, note that the constraint y(t) = 0 necessarily induces, on the input u to the system (2.1), a constraint of the form: Lrrh(x,e) + a0(x,e)edu = 0 Differentiating this d times with respect to time, after simple manipulations one obtains a differential equation of the form: CCo(X,E)Edu(d) + ... + OCd_I(X,E)EU(1) + OCd(X,E)U+ eP(x,u,eu (1)..... 8d'lu(d'l),s + +Lfr+dh(x,e) = 0 where P(x,v 1. . . . . v d ,e) is a polynomial in v 1 ,Vd, with coefficients which are smooth functions of x,e. Replacing x by its expression as a function of the (~,~q) coordinates, adding the differential equation . . . .
= q(~, T1, e) and finally eliminating ~ (which can be expressed as a smooth function of rl,u,eu (D ..... ed'lu(d'D,e) yields an alternative description of the zero dynamics of the system (2.1). Clearly, the equations thus obtained can be given the form (3.8) derived in the previous section, after having set: z i = Ci'lu (i'D
1 ~ i ~ d.
870 5.
References
[1] W.A.Porter, Diagonalization and inverses for nonlinear system, Int. J. Control, 21 (1970). [2] A.J.Krener and A.Isidori, Nonlinear zero distributions, 19th IEEE Conf. Decision and Control, Albuquerque (1980).
[3] C.I.Byrnes and A.Isidori, A frequency domain philosophy for nonlinear systems with applications to stabilization and adaptive control, 23rd IEEE Conf. Decision and Control, Las Vegas (1984). [4] A.Isidori and C.H.Moog, On the nonlinear equivalent of the notion of transmission zeros, Modeling and Adaptive Control, Springer Verlag L.N. in Contr. and Info. Scie 105 (1988). [5] S.S.Sastry, J.Hauser and P.V.Kokotovic, Zero dynamics of regularly perturbed systems may be singularly perturbed, Syst. Contr. Lett., 13 (1989). [6] J. Hauser, S.S.Sastry and G.Meyer, Nonlinear controller design for flight control systems, IFAC Symposium on Nonlinear Control Systems Design, Capri (1989). [7] P.V.Kokotovic, H.K.Khalil, J.O'Reilly, Singular Perturbations in Control: Analysis and Design, Academic Press (1986).
[8] A.Isidori, Nonlinear Control Systems (2nd Ed.), Springer Verlag (1989).
glargissement des ohjectifs de robustesse des syst~mes de co,-n=nde
David Bensoussan D~partement de g~nie ~lectrique Ecole de technologie sup~rieure Universit~ du Quebec 4750. avenue Henri-Julien Montreal, Quebec H2T 2C8
Abstract
:
In this paper, we will review the theoretical background which led to the emergence of the Hoo theory. The advantages offered by the Hoo theory, particularly in the domain of robustness, are weighted against the Hoo theory, limitations which are the conservativeness of the results. It is proposed to use the Hoo theoretical framework, a) to reconsider the role of disturbances so as to use them beneficially through the feedforwarding of their estimates, b) to study variable structure systems, the variable structure being used as an extra degree of freedom. The combination of the variable structure with the feedforwarding of disturbance estimates improves performance objectives while aiming to achieve robustness objectives.
Introduction
Les deux classique bas4 descriptions par
Zames
traditions
connues
sur les
r4ponses en
de variable d'~tat. [i]
a
permis
de
de
la
th~orie
L'approche reformuler
gain en relation avec les non-lin~arit~s
pourraient Feintuch
illustrer
[6].
Feintuch [8]
un
niveau
sont
le contr61e
presenter le
sectorielles
plus
entree-sortie,
introduite
abstrait
th~or~me du petit
[2,3] puis ~ reformuler nombre d'alg~bre
d'op~rateurs
et que
[5], et de Saeks et
initi~es par Zames [7] et
resum~es dans
ont permis la d~finition plus compl~te du problAme de contr61e dans le
a ~t~
pour presenter d'importants
rer ces
servi ~
Les ~tudes de r~alisabilit4,
syst~mes entr~es-sorties
traduits dans
d~veloppements
de systemes
les travaux de Saeks [4], Tung et DeSantis
contexte d'une alg~bre d'op~rateurs
ont ~t4
contr61e
le probl~me du contr61e dans un contexte
d'alg~bre norm~e; cette approche a d'abord
probl~mes de contr61e ~
du
fr~quences et le contr61e moderne bas~ sur les
causaux ou
reprise par
r~sultats de stabilit~ et de sensibilit~
les espaces
de Hardy
dans le domaine du contr61e.
deux theories
et de reconsid~rer de
strictement causaux.
La th4orie des
Zames dans une publication c~l~bre [9] : ces r~sultats
[10], ouvrant la porte ~ des nouveaux Ii est bon ~ ce
stade-ci de reconsid4-
en mettant en valeur leurs points forts et leurs limitations nouvelles
questions
quant
aux
directions
d'une recherche
future dans le but d'~largir le domaine d'application de ces theories.
874 I. Commande moderne et commande H~
La th~orie
de contr61e
moderne bas4e sur les m4thodes d'analyse des variables
d'~tat a pris un essor extraordinaire
depuis les ann~es soixante.
Malgr~ les succ~s
ind~niables des theories de Wiener Hopf Kalman (WHI<), l'on recourt encore aux m4thodes classiques,
utilisant
retard) lorsqu'il
y a
s'agisse de r~duire s'agisse de
les
representations
lieu de
la
r~duire la
sensibilit~
aux
fr~quences
compensateurs
perturbations
sensibilit~ aux
pour de nombreux contr61eurs
en
construire des
incertitudes
(r~seaux d'avance-
de sensibilit4,
externes
qu'il
ou
encore qu'il
de mod~lisation.
C'est le cas
industriels.
Les m~thodes d'analyse classique et les m~thodes d'analyse de WHK s'attachent r~soudre des
probl~mes de
moderne pr~supposent precision,
diff~rente nature.
l'existence de
leurs variations param~triques de tels syst~mes est L'incertitude de
d~finis
approch~s sinon incertains,
sont partiellement
faisable
grace
aux
connues.
propri~t4s
avec grande
Dans la r4ali-
et leurs structures ou
Ii reste que la commande
de
la
boucle
de r~action.
mod~lisation est difficile a exprimer dans le contexte des espaces
d'~tat lin~aires quadratiques non param4trique
peut ~tre
part, les m~thodes d'analyse
structures,
et qui peuvent ~tre repr~sent4s par des ~quations d'~tat.
t~, les modules sont g4n4ralement
dans le
D'une
syst~mes bien
et
gaussiens.
De fait,
les probl~mes d'incertitude
semblent ind~pistables au moyen des techniques de WHK.
contexte de
la representation
exprim~e par
une bande
classique,
de tol~rance
une
Toutefois,
incertitude de mod~lisation
autour des
valeurs nominales;
par
exemple, un diagramme de Bode ou de Nyquist peut ~tre extrapol~ en bandes lesquelles d~limiteront
l'ensemble
des
variations
du
module original par rapport au module
nominal.
Quant au probl~me de rejection des est
une
m~thode
de
projection
perturbations
externes,
la m~thode
quadratique ~ un vecteur connu ou encore ~ un processus al~atoire dont t~s de covariance sont bien connues.
En pratique,
incertaine,
incluse
et le processus
dans un ensemble de covariances.
impliquant des spectres de puissance variables en fr~quence ne peuvent avec les m4thodes d'optimisation une fr~quence particuli~re, peuvent fort
bien se
des m~thodes de WHK. peut admettre
quadratique.
les performances
d4t~riorer.
al~atoire a Les probl~mes ~tre r~solus
En voulant optimiser les performances du syst~me
sur le
reste du spectre
C'est encore i~ un probl~me non r~solu au moyen
Dans les m~thodes
de r~ponses
l'existence de perturbations
des signaux d'~nergie finle.
les propri~-
le vecteur en question appartient
un certain ensemble ~tant donn~e qu'il est incertain, une covariance
de WHK
qui consiste ~ trouver la meilleure approximation
en fr~quence
dans l'espace
par contre,
l'on
L2(-~,~) , i.e. l'ensemble
875 Les descriptions syst~mes
comme
entrges-sorties
41~ments
d'une
syst~me par des r4ponses d'addition et
en
produit de
syst~mes alors
systgmes,
qui
norm~e,
fr~quences,
de multiplication
d'4tat ne le permettent pas.
de
alg~bre
de
permettent
de syst~mes,
de
alors que
La norme quadratique est
~
d4finir les
preserver
de
des op4rations
les m4thodes de variables
diffieile g
estimer pour le
que la norme de produits de syst~mes d'entr4e-sortie
plus facilement estimable du fait de la verification telle norme
consistent
m~me que les descriptions
l'in4quation de
est
Schwartz.
Une
permet de d~river des r~sultats de robustesse bas~es sur le th~or~me du
petit gain et sur la positivit~ des systAmes.
Elle permet ~galement d'utiliser l'interpolation
encore l'utilisation
pour l'approximation stable.
gamme d'outils
de Nevalinna Pick, la g~n~ralisation
espaces de Krein ou
de la
L'ensemble de
seulement du
une large
norme de
math~matiques
tels
du th~or~me de Beurling-Lax aux
des r~sultats
d'Adramjan, Arov
et Krein
Hankel d'un op4rateur instable ~ un op4rateur
ces m4thodes
permet d'analyser
point de rue de la stabilit~
une boucle
de commande non
ussi du point de vue de la robustesse en
rapport avec un ensemble de modules que l'on peut retrouver dans un domaine d'incertitude inh4rent
g la
description nominale
peuvent 4galement mettre en valeur
les
du syst~me
limitations
~ commander.
des
Ces m~thodes
performances
pour chaque
domaine d'incertitude.
La th~orie [10].
Ace
de commande
jour, il
optimaux pour
dire H~
est possible
les syst~mes
a fait d'4normes progrAs depuis sa conception de d~velopper
d4crits par
des algorithmes
des matrices de r~ponses en fr4quence.
reste toutefois qu'en d~pit de l'~l~ganee math~matique tats obtenus
sont assez oonservateurs
le supremum sur l'ensemble matrice de
r4ponses en
de la
valeur singuli~re
pour l'analyse
~ savoir
maximale de la
Cette norme limite grandement la performance du
syst~me compens~ ~tant donn~ qu'elle utilise le pire des comportements comme base
I1
de l'approche H~ , les r4sul-
et ee en vertu de la norme utilis~e,
des fr~quences
fr~quenee.
de compensation
du syst&me
Comment lors viser ~ une amelioration
compens~ sur des
tout le
performances
en fr~quence
spectre de fr~quence.
optimis~es
sur
une telle
base?
Quant ~
la commande adaptative,
tion param~trique,
que ce soit par
elle r~pond aux probl~mes des modules ~ varia-
l'identification
des
param~tres combin~e
~ une
compensation appropri~e ou encore par la modification de la compensation par rapport un module de r~f~rence ehoisi. xes
impliquant
~conomiques)
un
haut
degr~
Ii est fort peu probable que les d'incertitude
s'appr~tent ~ une identification
syst~mes comple-
(e.g. petrochimiques,
biologiques ou
aisle ou g une commande adaptative.
876 II. Au del~ de la robustesse
L'approche que norm~e et
nous proposons
tient ~
de l'optimisation H~; elle
conserver les
avantagaes de l'alg~bre
vise ~ initier des modifications
de structure
AP au syst~me P de faGon ~ introduire un nouveau degr~ de libert4 dans la conception des syst~mes
de commande.
Trois avenues principales
premiere consiste ~ utiliser mesurables,
de
faGon
b~n~fique
plut6t que de chercher ~ en minimiser
de recherche se dessinent~ les
perturbations
les effets.
La seconde consiste
mettre en place une structure pouvant absorber les effets de perturbations, toutefois ~tre
mesurables,
demeurent
caract~risables
La troisi~me consiste ~ introduire des structures non r~pondre ~
des probl~mes
de nature
particuli~re,
au
La
qui soient
qui sans
sein d'un ensemble donn~.
lin~aires lesquelles pourront
les structures flexibles consti-
tuant une solution de choix.
Ii est propos~ de reconsid~rer commande.
Le but
le
r61e des
g~n~ralement accept~
sur la sortie et l'entr4e du l'optimisation H~,
syst~me.
o~ l'on
C'est i~
propos4e nous diff~rencierons
l'effet du
par exemple
les boucles de
un des
additives d'~nergie
objectifs de
: celles qui ont des
et qui peuvent ~tre mesur~es comme c'est le
la trajectoire
d'un avion,
entre la
finie. Dans l'approche
entre deux types de perturbations
caract~ristiques
vent sur
dans
recherche ~ r~duire la norme des transmissions
sortie et l'entr~e et les perturbations
comportements
perturbations
est de minimiser l'effet des perturbations
et celles
cas pour
qui appartiennent
hun
ensemble bien d4fini, tels les cassis et crevasses sur une autoroute.
Pour le premier type tiendra
compte
de
la
de perturbation, variation
commandable et/ou observable, perturbations
mesurables.
un comportement
type et
les
de
voilier avanGant Les perturbations
AP
~ une
du
faGon
objectifs
brider l'~nergie de
commande
b~n~fique
contre le
commande qui va permettre de neutraliser
contenue dans
P,
qui
qu'elle soit
l'effet
de ces
vent illustre fort bien
caract~risables
d~sir~s
stabilisation
syst~me
n'engloberont pas les perturbations
L'id~e maltresse consiste ~ atteindre
nous viserons
structure
afin d'utiliser
Un
cette cat~gorie de probl~mes.
de
auront g~n~ralement
purement statistiques. les perturbations
pour
et au besoin, trouver la meilleure
les perturbations
purement contraires aux
objectifs de commande.
Le probl~me
reviendra donc
~ minimiser
les transmissions
la boucle de eommande que nous les localiserons du syst~me
Pet
le signal
sont les perturbations
d'erreur de
~ la
sortie du
entre les sorties de syst~ne P, l'entr~e
la boucle e, et les entr~es du syst~me qui
additives ~ la sortie du
d3, le bruit des senseurs d 2 et la r~f~rence u.
syst~me d I,
~ l'entr4e
du syst~me
877 Nous consid~rons
queen
plus de la compensation
parall@le
F, des compensations
les sont
activ4es
lorsque
C' = C + &C, F' = F AP.
L'incertitude
+
s@ries
la structure
&F offrent
plus
de mod~lisation
ques de robustesse
appliqu~es
l'optimisation
variable
entre
de latitude
6P pourra
au nouveau
H~ de ce nouveau syst~me r@sultera
s~rie C et de la compensation
AC et des compensations
en jeu.
AF lesquel-
Les compensations
pour l'4tude du syst~me
toujours
syst~me
parall~les
~tre ~tudi~e
nominal
en des r~sultats
P', C',
P' = P +
selon les techniF'. L'optimisation
beaucoup moins conservateurs
que
H~.
I dl
I
~ Y
9 figure
1
iPiFc PctF1 iPcF PcFip fluI
-
(I + F'P'C')"I
- F'(I + P'C'F')'[
F'(I + P'C'F')'I
- (I + F'P'C')"I F'P'
C'(I + F'P'C')"I
- C'F'(I + P'C'F')-1 - C'F'(I + ?'C'F')'L + (I + C'F'P')"~
dl
d2
d3
Une seconde possibilit@ fectuer
une optimisation
syst~me
~ structure
tions externes,
variable
afin de
reli@e aux cas
des perturbations
mesurables
de type H 2 de faGon ~ ce que la transmission vienne
maintenir
exemple,
on pourra chercher
suivante
:
annuler
celle qui
la boucle
la variation
due ~
de commande
de structure
l'effet
est d'ef-
g4n~r~e
par le
des perturba-
dens son etat nominal.
&P visant
~ l'optimisation
Par H2
rain II PC(I + FPG)-Ir
AP C', F'
stabilisants
Pour ensemble
le
- P'C'F'
- P'C'(I + F'P'C')-Ir
(I +
P'C'F')-Id2
second type de perturbation,
bien d~fini D, nous viserons
~
- (I + P'C'F')-Id 1
+ (I + P'C'F')'Id3112
celui des perturbations la caract~risation
appartenant
de l'ensemble
~
un
des varia-
878 tions
de
structure
maintiendront voiture qui
AP
qu'elles
la performance de r~pond entre
soient
la boucle
autres au
commandables de commande.
probl~me des
et/ou Le
cassis at
observables
qui
cas d'amortisseurs
de
crevasses de la route
illustre bien ce probl~me~
Finalement on pourra reconsid~rer est une
les trois conceptions pr~cddentes
lorsque AP
structure flexible non lin4aire et d~river des r~sultats de robustesse pour
chacune desdites conceptions.
Dans un certain sans, la recherche se propose de relier les r~sultats d'optimisation H~
qui constituent une th~orie math4matiquement
bas~s sur de la commande adapt~e aux perturbations
bien fond~e, ~ ceux qui sont
[ii] qui offrent des performances
4tonnantes sans avoir la rigueur ou l'~l~gance de l'optimisation H~.
L'une des
difficult~s majeures pr~visibles
Plusieurs paramdtrisations
sont possibles,
r4side en la caract~risation
nonobstant
la
caract~risation
de AP.
fr~quen-
tielle. L'effet transitionnel durant la mise en jeu de la structure variable constitue la difficult~ d~terminante. relatifs aux besoin.
modes dits
Ii y aura
glissants
[12]
lieu de et de
recenser les
d~velopper de
Ceci constitue un d~fi de taille en sus
travaux tels ceux
nouveaux crit~res au
de l'optimisation
d'op4rateurs
ou
de r~ponses en fr~quence propos~e dens les trois types de conception.
L'approche de robustesse.
recherche propos~e d~passe donc l'interpr~tation
La reconception du syst~me de commande incorporant une
structure devrait ticuli~rement
pallier ~
l'une des
limitations
classique de la modification
de
de l'optimisation H~, plus par-
calla du conservatisme des r~sultats.
Entre la conception visant ~ l'~tude
de stabilit~
tenant compte
des crit~res
des pires cas pour l'~tude de la performance en l'absence d'incertitude de mod~lisation, et pires cas
la conception d'incertitudes,
(ind~pendante) la
visant
~ l'~tude
recherche propos~e
de la
peut offrir
stabilit~ pour les un acc~s ~ une ~tude
combin~e de la performance am~lior~e et de la robustesse.
BiblioBraphie
[I]
G. Zames, "Functional Analysis Applied to Non Linear Feedback Systems", Transactions on Circuit Theory, vol. CT-10, pp. 392-404, Sept. 1963.
IEEE
[2]
G. Zames, "On the Input-Poutput Stability of Time Varying Non Linear Feedback Systems", part I : "Conditions using Concepts of Loop Gain, Conicity, and Positivity", IEEE Transactions on Automatic Control, vol. AC-II, no. 2, pp. 228-238, April 1966.
879 [3]
G.Zames, "On the Input-Output Stability of Time Varying Non-Liear Feedback Systems", Part II : "Conditions Involving Circles in the Frequency Plane and Sector Nonlinearities", IEEE Transactions on Automatic Control, vol. AC-II, no. 3, pp. 465-476, July 1966.
[4]
R.Saeks, "Resolution Space Operators and Systems", Lecture notes in Economics and Mathematical Systems, vol. 82. R. DeSantis, R.Saeks, L.J~ Tung, "Basic Optimal Estimation and Control Problems in Hilbert Space" Mathematical System Theory 12, pp. 175-203, 1978.
[5]
[6]
A. Feintuch, R.Saeks, "System Theory. A Hilbert Space Approach" Academic Press, 1982.
[7]
G~ Zames, "Realizability Conditions for Nonlinear Feedback Systems", IEEE Transactions on Circuit Theory, vol CT-II, pp.'186-194, June 1964.
[8]
A. Feintuch, "Strong Causality Conditions and Causal Invertibility", SIAM Journal of Control and Optimization, vol. 18, pp. 317-324, May 1980.
[9]
G.Zames, "Feedback and Optimal Sensitivity : Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses", IEEE Transactions on Automatic Control, vol. AC. 26, no. 2, pp. 301-320, April 1981.
[i0] Dorato P., "Robust Control", IEEE Press, 1987. [ii] C.D. Johnson, "Theory of Distrubance Accomodating Controllers", Advances in Control and Dynamic Systems, vol. 12 (C.T~ Leondes ed.) Academic Press,1976. [12] A.S.I. Zinober, series, 1989.
"Variable gtructure Control Systems", lEE Control Engineering
RIGHT HALF PLANE POLESAND ZEROSAND ROBUSTNESS LIMITATIONS IN FEEDBACKSYSTEMS Michel KINNAERT " Laboratoire d'Automatique, C.P. 165, Universit~ Libre de Bruxelles Avenue F.D. Roosevelt 50, B-1050 BRUXELLES, BELGIUM
Abstract Upper bounds on achievable robustness for the control of LTI systems with right half plane (RHP) poles and/or zeros are derived. Robustness limitations with respect to a class of multiplicative uncertainties are computed using a specific minimax weighted complementary s e n s i t i v i t y . The minimax problem amounts to an eigenvalue problem for single-input-single-output (SISO) sytems, which is the class of plant models we consider here. For systems with a single RHP pole, an e x p l i c i t formula is given for the minimax weighted complementary s e n s i t i v i t y . Application of the results to an inverted pendulum shows that ,thanks to the computation of the robustness limitations, tedious t r i a l and error procedures might be avoided. I. INTRODUCTION
Our aim is to quantify the limitations due to plant RHP poles and zeros on achievable robustness in LTI controller design. We would l i k e that the computed limitations be independent of any particular design method. They should thus depend exclusively on the plant model. Much work has been recently devoted to the study of the limitations due to plant RHP poles and zeros in LTI controller design. These limitations have been expressed in terms of constraints on the sensitivity function S and the complementary s e n s i t i v i t y function T. Integral relations which must be satisfied by IS (J~)I and
IT(J~)I were developed in
[ I ] . They are independent of the controller design method as long as the controller does not have any RHP pole or zero. However, i t is not always possible to stabilize a LTI system with a stable minimum phase controller (consider for instance a s t r i c t l y proper second order system with a nonminimum phase zero and an unstable pole to the right of that zero in the s-plane). Hence, this approach cannot be used in our work since we want to find indices that are independent of a prior controller design. In his recent papers [21, [3], Glover introduced an index which characterizes the achievable optimal s t a b i l i t y margin for a system described by a l e f t coprime factorization subject to H bounded perturbations on the numerator and the denominator. The index which is introduced here gives similar information for a class of multiplicative " Senior Research Assistant of the Belgian National Fund for Scientific Research
881 uncertainties.
The case of additive uncertainties has been considered in [4] but i t is
not reported here due to a lack of space. The definition of the index is a consequence of the results of [5-7]. Its computation is performed via an eigenvalue problem using a procedure based on [7]. An e x p l i c i t Formula is derived for systems with a single RHP pole. The paper is organized as Follows. The notations are described in section 2. The d e f i n i t i o n of the index is j u s t i f i e d in section 3, and i t s numerical computation is discussed in section 4. An e x p l i c i t Formula for systems with a single RHP pole is derived in section 5. Its application to an inverted pendulum, in section 6, i l l u s t r a t e s the usefulness of the index.
The proofs of the theorems are omitted here due to a lack of
space. They can be found in [4]. 2. NOTATIONSAND DEFINITIONS
We consider a standard feedback configuration made of a SISO plant H, and a Feedback controller G. We assume that the nominal mathematical models for H and G are Finite dimensional linear time invariant (FDLTI) systems described by proper transfer functions H(s) and G(s) respectively. For the closed-loop system to be well defined, we require lim
H(s)G(s)~-1.
S- >~
As in [6], we assume that the "true" plant lies within some domain of uncertainty containing H, and we l e t HT(S) denote the true plant transfer function. Moreover, we introduce a prespecified stable rational function F(s). Then, M(H,F) denotes the class of plants HT(S) such that HT(S) : (I+L(s)) H(s]
(I)
where IL(j~) I < IF(J~)l For all ~
(2)
and such that HT(S) and H(s) have the same number of RHP-poles. Thus, multiplicative uncertainties affect the plant model. ~(H) denotes the class of all proper controllers G(s) that stabilize H(s), and RH| denotes the set of stable proper rational functions with real coefficients. In the sequel, the plant transfer function H(s) is written in the form H(s) : n(s) / d(s)
(3)
where n(s) and d(s) are coprime real polynomials. The degrees of d(s) and n(s) are denoted dd and dn respectively. We introduce the following factorizations n(s) : n+(s) no(s) n.(s) ; where n.(s), no(s), n.(s), d§
d(s) : d+(s) do(S) d.(s) (4) do(s) d.(s) are real polynomials with degrees dn+, dno,
dn, dd§ ddo, dd. respectively such that n.(s) and d+(s) have all their roots in the open r i g h t - h a l f plane , no(s) and do(s) have all their roots on the imaginary axis, and n.(s) and d(s) have all their roots in the open l e f t - h a l f plane. The roots of n+(s) and d+(s) are denoted respectively b,, i : I, 2, . . . . dn+ and a,, i = I, 2. . . . .
dd.. Hence the plant
transfer function can also be Factorized as follows :
H(s) : H,(s) Bo'(s) B,(s)
(S)
882
where dd, Bp(s) : ~ i=l
acs ...... ~+s
dn§ b,-s ~(s) = 0 -- .... i = l b,+s
and
are the Blaschke products of open RHP poles and zeros r e s p e c t i v e l y , and H~(s) is proper and has no pole or zero in the open r i g h t h a l f plane. 3. DEFINITION OF THE INDEX
The definition of the index is based on the following theorem [5],[6]. Theorem 1 A c o n t r o l l e r G ~ ~(H) s t a b i l i z e s a l l plants in the class M(H,F) i f and only i f IIT
(6)
Eli. < i
Now, l e t us consider the following minimax weighted complementary sensitivity l, : inf G~Z(H)
JJ WT 16
II T ( j ,
=
(7)
where W is a rational weighting function whose choice w i l l be discussed below. Let GT, ~, and Tdenote the optimal controller, the optimal complementary sensitivity , and the optimal weigthed complementary sensitivity corresponding to ~T 9 Then, from theorem I, GT stabilizes a l l the plants of the class N (H,W/AT). To obtain an index characterizing robustness using (7), the weighting function W should be linked to a bounding function for multiplicative plant uncertainties. Moreover, the structure of W should be chosen in such a way that T corresponds to a meaningful design. This problem is considered next. 3.1. Structure of the weiqhtinq function W As discussed in [5], a usual form for the gain of the modeling error bound for multiplicative uncertainties,
JF(J~)I, is
as depicted in Fig. I. Thus, the curve of W(jw)
versus w should have a similar form. Additional constraints on the structure of Wmust be satisfied i f we want~to correspond to a meaningful design. Indeed, i f the plant has pole excess E > 0, ]/H(s) = 0(s') as Isl tends to i n f i n i t y . Therefore, in order to make sure that IT(jw) / H(j~)I (which is the gain of the transfer function from disturbance, measurement noise,
and command input
to
plant
input) remains bounded as ~ -->m,
one should choose W(s) = 0(s') as (s( --> ~ [7]. From these considerations, we deduce the following possible structure for W(s)
W(s)
:
(T.s+2)
'~w
(8)
where I/T, is the frequency from which the modeling uncertainty starts increasing significantly, and w is an integer ~ 0 which characterizes the rate of increase of the modeling uncertainty.
883 We can now define a robustness indicator with respect to m u l t i p l i c a t i v e plant uncertainties, IM, as the maximum static gain that can be tolerated For a bounding function F of the form K(T,s+I) (+" (where K is a constant), i.e. the maximum value of K such that all the plants of the class M(H,K(T,s+I) ~'') can be stabilized. Obviously, From (7) and theorem i , IM = I/~ T , where AT is computed with the weighting function (8). 4. SOLUTION OF THE MINIMAX PROBLEM
For the sake of generality, we shall consider a weighting function W(s) of the form
W(s) = n.(s) / dw(s) where nw(s) and dw(s) are real
(9) polynomials with degrees dnW and dd~ respectively.
Moreover, nw(s) and d~(s) are coprime, t h e i r roots are located in the open l e f t - h a l f plane, and dnW> dd~ + E where c is the pole excess of the plant. The computation of the minimax weighted complementary s e n s i t i v i t y Function is now studied. Plants with imaginary axis poles and/or zeros require a special treatment, and they w i l l be dealt with in a separate section. 4.1. Plant without pole or zero on the imaqinary axis (no(s)=do(s)=1) This is a particular case of the minimax mixed s e n s i t i v i t y problem presented in ~7]. The results are dual to those derived in section V of that paper. Theorem 2 ==========
Consider the factorization (4), define W(s)
(5) of the plant transfer function H(s) and l e t (9)
a) I f d+(s) = constant, AT • i n f (I W(j~) T(j~) If- = II ~ ( J ~ ) I f . = 0 GEZ(H)
b) I f d~(s) ~ constant, AT is the largest (in absolute value) real value of ~ such that the solution of the polynomial equation I - - - n~ n'+ q'T : d+ PT + dw n~ qT (I0) consists of two real polynomials, PT and qT, whose degrees are dn~ + dnW - I and dd§ -I respectively. Moreover the optimal value AT is reached with the proper c o n t r o l l e r d. d, q~ G~ . . . . . . . . . ~ - - - where qT and P~Tare the solution of (10) corresponding to ~T, n. PT and the resulting closed-loop system has a l l i t s poles in the open l e f t h a l f plane. Part (a) corresponds to the obvious fact that, i f the plant has no RHP pole, one can make IT(j~)I uniformly a r b i t r a r i l y close to zero by making the controller gain small enough. The solution AT, p~, ~T of (10) can be computed by solving a generalized eigenvalue problem, following the same p r i n c i p l e as [7]. 4.2. Plant with poles and/or zeros on the imaqinary axis For a plant with poles on the imaginary axis, the closed-loop system obtained with the c o n t r o l l e r GT computed using the theory of [7] is not asymptotically stable.
884 Moreover, i f the plant has zeros on the imaginary axis, these zeros should be introduced in dw(s) to assure s t a b i l i t y of the closed-loop system [7]. However, i f we perform this modification of dw(s), we shall loose the l i n k between W(s) and the uncertainty bound F(s). Thus another approach is needed for systems with poles and /or zeros on the imaginary axis. [g] gives such an alternative approach. The results presented here for the complementary sensitivity function are dual to those developed in [9] for the s e n s i t i v i t y function. Their proofs follow the same lines. Theorem 3 ==========
Consider the following factorization of H(s) H(s) = H2(s) Ho(s) (11) where Ho(s) has poles and zeros only on the imaginary axis, and H2(s) has poles and zeros only off i t . Let ~'denote the optimal weighted s e n s i t i v i t y function for the factor H2(s). Under the assumption that IW(J~)I ~ II Y" I I - at each imaginary axis pole of H,
(12)
i t is possible to build a sequence {Y'w}in RH~ such that lim k->|
IIY',II.
=
II Y" II.
(13)
and such that Y, satisfies the interpolation constraints needed to ensure s t a b i l i t y of the closed-loop system. Thus imaginary axis poles and zeros do not affect the optimal weighted complementary s e n s i t i v i t y subject to assumption (12). 5 . EXPLICIT FORMULA In this section, we derive an e x p l i c i t formula for the value of %Tcorresponding to a plant with a single RHP pole. Theorem 4 ========~=
We consider a proper plant H(s) with one RHP-pole, a, factorized according to (5). Moreover we assume that the weighting function W(s) is of the form (9). Then %T= i n f II T G(~(H)
Wil, =
IB,(a)" W(a)]
(14)
Using the weighting function (8), (14) can be written : ~T = I/IM =
]
(I + 7.a)"*
B,(a)" I
(15)
For a minimum phase system, B,(a) = I. In this particular case the study of the evolution of IM, the maximum admissible static gain uncertainty as a function of T,a yields the following conclusions.
If
the
gain of
the
modeling uncertainties
increases at
885 a rate of 40 dB/decade, or more at high frequencies (( + w Z 2), IM decreases at least quadratically when rwa increases. In this case, for a fixed value of a, a value of I/T, smaller than a yields an admissible static gain uncertainty smaller than 25 %. I f the modeling uncertainty only increases as j~ at high frequencies (E + w = 1), the previous number becomes 50 % (see table). Thus, the gain of the modeling uncertainties should not increase significantly in the frequency range [O,a] in order to be able to accommodate sufficient static gain uncertainties. This limitation is strengthened i f the plant has nonminimumphase zeros, because JB,(a)'l > I in this case. For a system with a single RHP pole and a single RHP zero, the situation where the RHP pole lies to the l e f t of the RHP zero in the s-plane is more favourable than the converse situation. This can be checked by applying (15) to two transfer functions with a single RHP pole and a single RHP zero, in which the location of the RHP pole and the RHP zero is interchanged. 6. NUMERICAL EXAMPLES i . H(s) : e (s-2) /
((s-5)
(s+6))
,
~ ( R
Fig. 2 gives the evolution of IM as a function of 7W. The type of results that can be derived from f i g . 2 is the following. I f the multiplicative plant uncertainties can be described by (I),
(2) with F(j~) = F,(j~) = 0.1 (0.1 s +I) there exists a linear
controller stabilizing the class of plant M(H,F,). However, i f the bounding function is F2(j~) = 0.3 (0.1 s+i), there is no linear controller stabilizing M(H,F2). Indeed, the largest static gain that can be accommodatedwithout loss of s t a b i l i t y for a bounding function with a zero at s = 10 is IM = 0.286. 2. Inverted pendulum We consider the inverted pendulummounted on a cart sketched in Fig.3. The dynamic equations of this system are derived in [11] and they are linearized around the state (Y,Y,8,8) :
(Y,O,O,O) f o r u : O. The f o l l o w i n g t r a n s f e r f u n c t i o n between the force u
a c t i n g on the c a r t and the p o s i t i o n o f the c a r t y w i t h respect to Y has been obtained
H(s) : K(s-b) (s+b) / (s2 (s-a) (s+a)) where K = the
I/M , b = / g / l
,
pendulum concentrated at
a: ]
(16)
(M+m)g / MI, w i t h M, mass o f the c a r t ,
the t i p ,
I,
length
of
the
pendulum,
m, mass o f
and g,
gravity
acceleration. For 1 = ] meter, M : m : 0.5 kg, g=10 ms2, we obtain a = vr-2-O-s' and b = ~ s " . In the sequel, those numerical values are always used in model (16). Since the system has poles a t the o r i g i n , we use the r e s u l t s o f section 4.2. where
H2(s) = K (s-b)(s+b)/((s-a) (s+a))
(17)
To be able to use theorem 3, we have to check condition (12).For the class of weighting functions (8), we obtain IW(o)J=1. Applying (15) to system (17) we obtain
886 ~T : II T II- ~ l(a+b)/(a-b)I which is larger than I. Hence a sequence such as defined in theorem 3 always exists in our case. In model (16), the transfer function of the device generating the force u is assumed to be unity.
Let us assume that this transfer function can actually be modelled by a
f i r s t order system whose time constant is approximately 10 times smaller than I/a. Thus a more accurate model for the linearized plant would be : HT : H(s) / (0.03 s + 1)
(18)
Since the nominal model H(s) has relative degree 2, and since the modelling uncertainty w i l l increase significantly for frequencies larger than 33.333 rad/s, we choose : W(s) : ( 0.03 s + I) 2 The corresponding value of IM, given by (15),
(19) is 0.133.
I t can be checked that
belongs to the class (I),(2) with F(j~) = 0.133 ( 0.03 j~ + I )2 .
HT
(20)
We can now analyse the achievable robustness with respect to parameter variations. Assume that the mass M varies in the range [0.4
0.6]. The transfer function which is
equivalent to (18), for M=O.4, is : H' T = 2.25 (s2 - 10) / [ s2 (s2 - 22.5) (0.03 s + I) ]
(21)
Fig. 4 gives the Bode plots of H(s), H'T(s) as well as the bounds corresponding to the class ( I ) , (2) for H(s) and F(s) given by (16) and (20) respectively.
Notice that the
factor I/s 2 has not been included in the Bode plots, in order to obtain a better i l l u s t r a t i o n of the differences between the curves. following conclusion.
From Fig. 4, we can draw the
There exists no controller based on the nominal model (16) and
on the unstructured uncertainty model ( I ) , ( 2 ) , with F(s) = K (0.03 s + I )2, that can stabilize the actual plant subject to a 20 % variation of the mass M. Besides, no satisfactory controller was obtained for system (16) in [11]. To obtain adequate performances for the control of the inverted pendulum, an additional measurement was performed and a cascade control structure was used. In this way, the transfer functions appearing in the modified control problem do not have any nonminimumphase zero [11]. However, no unstructured modeling uncertainties were introduced in this work. 7. CONCLUSIONS
We have presented an index which gives an upper bound on achievable robustness with respect to a class of multiplicative uncertainties, in LTI controller design, for systems with RHP poles. This index t e l l s the designer whether the required closed-loop system properties can possibly be reached so that tedious t r i a l and error procedures can be avoided. Its computation is particularly easy. I t amounts to an eigenvalue problem. Moreover, an e x p l i c i t formula exists for systems with a single RHP pole. A similar index has been derived to characterize robustness with respect to additive unstructured uncertainties in [4]. Similar indices can be defined for multivariable systems. However, their computation will
be more involved. I t does not amount to eigenvalue problems. An interesting
computational approach is given in {12].
887 Acknowledqement This research topic was suggested to the author by Professor GrahamC. Goodwin. Some helpful discussion with him and with Doctor Rick H. Middleton at an early stage of this project are gratefully acknowledged REFERENCES
[ I ] J. S. Freudenberg and D.P. Looze (1985). Right Half Plane Poles and Zeros and Design Tradeoffs in Feedback Systems. IEEE Trans. Automat. Control, AC-30,6, pp. 555-565. [2] K. Glover and D. McFarlane (1988). Robust Stabilization of Normalized Coprime Factors: an Explicit H. Solution. Proceedings of the 1988 American Control Conference, Atlanta, Georgia. [3] D. McFarlane and K. Glover (1988). An H. Design Procedure Using Robust Stabilization of Normalized Coprime Factors Proceedings of the 27th. IEEE Conference on Decision and Control , Austin, Texas, pp. 1343-1348. {4] M. Kinnaert (1989) Robustness LimitationsDue to Right Half Plane Poles and Zeros in LTI Controller Design. Internal Report. Laboratoire d'Automatique. Free University of Brussels. [5] J. C. Doyle and G. Stein (1981). Multivariable Feedback Design : Concepts for a Classical/Modern Synthesis. IEEE Trans. Automat. Control, AC-26, I, pp. 4-16. [6] M. Vidyasagar and H. Kimura (1986). Robust Controllers for Uncertain Linear Multivariable Systems. Automatica, 22, pp. 85-94. [7] H. Kwakernaak (1985). Minimax Frequency Domain Performance and Robustness Optimization of Linear Feedback Systems. IEEE Trans. Automat. Control, AC-30, 10, pp. 994-1004. [8] R.L. Kosut (1986). Adaptive Calibration : An Approach to Uncertainty Modeling and On-Line Robust Control Design. Proceedings of the 25th IEEE Conference on Decision and Control, Athens, pp. 455-461. [9] B.A. Francis and G. Zames (1984). On H" Optimal Sensitivity Theory for SISO Feedback Systems. IEEE Trans. Automat. Control, AC-29, i , pp. 9-16. [IO]H. Kwakernaak (1986). A Polynomial Approach to Minimax Frequency Domain Optimization of Multivariable Feedback Systems. Int. J. Control 44, pp.117-156. [11]R.H. Middleton and G.C. Goodwin. Digital Estimation and Control : A Unified Approach. Book to be released in 1990. [12]E.A. Jonckheere and J.C. Juang (1987). Fast Computation of Achievable Feedback Performance in Mixed Sensitivity H" Design, IEEE Trans. Automat. Control, AC-32, pp. 896-906.
888 TABLE T.a
IM
IM
E+w=l
E+w=2
0.5 0.4 0.33 0.286 0.25
I 1.5 2 2.6 3
0.25 0.16 0.11 0.081 0.063
Table 1 : Evolution of index IM as
IM (+w=3 0.125 0.064 0.037 0.023 0.016 function of r.a
FIGURES
Gain Nominal model t
Envelope
~ Locj ~,
IF(j~II
Log
Fig. 1 : Typical behavior of multiplicative
uncertainties
889 0.45
0.4 0.35 0.3 0.286 O,2fi 0.2 0.[5 0,1 0.05 ..................................
0 0.5
0 0 1
Fig. 2
:
l
1.5
I m h l mR
2.5
2
Plot of IM(rw) for H(s)=a(s-2)/((s-5)(s+6)) w=0 , --- w=l , ..~ w=2
,/?
m
Fig.3
: Inverted pendulum mounted on a cart
u (force)
5
///!
4.5 4 3.5 /; /,
3 // 2.5 _ =:~.v21
.....
2 1.5 1
0.5
-1
-0.5
0
{),5
1
1.5 log w
Fiq. 4 : Inverted
pendulum
__
IH(jm) l
....
IH'T(j~)I
....
IBoundaries
example
for the class of plants
)with H(s) and F'
(i),(2)
given by (16) and (20)
STABILIZING CONTROL OF A SINGULARLY P E R T U R B E D SYSTEM DRIVEN BY WIDE-BAND NOISES
Mohamed E1-Ansary Department of Mathematics California State University, Bakersfield 9001 Stockdale Highway, Bakersfield California 93311
ABSTRACT
A nonlinear singularly perturbed system is considered where the state equations as well as the output equations are corrupted by wide-band noises. The purpose of this paper is to spell out sufficient conditions under which an output feedback stabilizing control designed for a reduced-order model, via the use of stochastic observer, will stabilize the full-order system. Our results, which were established earlier on orderreduction and stability are applied to justify the use of the reduced-order models and the stability of the closed-loop systems which result from applying output feedback to the open-loop systems.
1. Introduction
Until recently, singular perturbation techniques [3] have primarily focused on state and output feedback design of linear systems. Advantages of these techniques, such as order reduction and seperation of time scales, are expected to have a more dramatic effect on feedback design of nonlinear systems. Stabilizing deterministic nonlinear singularly perturbed systems have been considered, for example, in [4-6]. Stabilizing nonlinear stochastic systems via the use of an asymptotically stable stochastic observer has been considered in [7], where the work has been a generalization of the Kalman filter structure [8]. In this paper, motivated by the work in [7], in which a stabilizing feedback control for a system represented by an It5 equation (a Markov model) has been designed using an observer, we designed an output feedback stabilizing control for a nonlinear full order singularly perturbed system driven by wide-band noise by designing a stabilizing control for the corresponding reduced-order open-loop Markov model. To assure the readability of this paper we will state in this section the basic ~ sumptions and the main theoren~ tha~ were developed in [11 and [2t. Section 2 consists of the main result of this paper, an output feedback control has been designed.
891 Reduced-Order Model and Convergence Theorem
In [1] and [2], we studied the nonlinear singularly perturbed system: = a l ( x ) + A z 2 ( x ) y + B l ( X ) V ' , x(O) = x o (1.1) /~) ----a21(x) § A2y § B2(x)v e, y(0) : Yo (1.2) where xeR", yeR m, v~eR r. v ~ is the wide-band noise and was defined as v~(t) = ~ v ( t / e ) . It is wide- band in the sense that thepower spectral density matrix S~(w) = S(w/e) will have a frequency band Wo/e when S(w), the spectral matrix of a stochastic process v, has a frequency band w0. # is a small positive parameter representing parasitic elements. The process v(t) satisfies: (A1) v(t) is a stationary, zero mean, right continuous, uniformly bounded on [0, co). The a-algebras induced by v(t) are assumed to have a mixing property with an exponential mixing rate [9], 81tpA,,t IP(A2/Az) - P(A2)I ~ g e - a t (1.3) for some a > 0, where Alea{v(a), a < t} and A2ecr{v(8), s > t + r}. The exponential mixing rate assumption was taken for convenience but can be replaced by a more general mixing rate as in [10-12]. The following conditions were also imposed: (A2) The coefficients az, a21, A12, B1 and B2 are continuous in x and have continuous partial derivatives up to the second order which are bounded uniformly in x. (A3) The constant matrix A2 is Hurwitz, i.e. ReA(A2) < O. (A4) The positive parameters e and/z satisfy e >/Z'~o where To > 0 is arbitrary but fixed. The reduced-order model can be represented by a diffusion process ~(t) and its operator L ~ , where ~(t) is the solution of the It8 equation: d~(t) = b(~c(t))dt § a(~(t))dw(t), ~.(0) = Xo (1.4) where b and a are given by (1.6) and (1.10) respectively, and the operator L ~, whose form was obtained from the proof of the convergence theorem, is given by:
L'~f(x) :
~bi(x)i:, (x)+ 2 i , i : , (1.5)
where
b(x) = ao(x) + hi(x) - A 1 2 ( x ) A 2 Z h 2 ( x ) + h3(x), A(x) = Bo(x)S(O)B~(x) = [aq(x)] 1, ao(x) = al(x) - A~2(x)A-~'a21(x), Bo(x) = B , ( x ) - A12(x)A-~lB2(x),
= S(to) is the spectral matrix of v,
1(') denotes transposition.
(1.6)
(1.7) (1.s) (1.o)
(1.1o)
892 ! hi, = tr[D~BoW' + D,Ax2A z- - 1 El, $ l ! h2i = tr[EiBoW + E i A 1 2 A 2 - 1 E ], h3i = tr[-F[BoW'Bt~(At2) -1 -- F[BoE'(At2) -1 + F[Ax2A;1P],
(i.Ii)
Di = [Vzr162162
(1.14)
Zi = [Vzr/ii!Vzr/i2i...iVzrlir]nxr; Fi = [V,-~iliVz~i2i...iVz~/m],,xm;
BI = [r
B2 = [rli.i]mxr, Ai2 = [~ii]r*xm,
(1.12) (1.13)
(1.15) (1.16)
oo
w = f R(~)a~,
(11z)
0
R is the correlation matrix of v, oo
r. -_ f
eA,'t,B2R'(r)dr, for some "7e["7o,oo), "70 > 0
(1.18)
0 oo
P = f eA2A(B2a' + aB~)ea'2~dA.
(1.19)
0
The following assumption was also required: (A5) b(x) and Bo(x) satisfy the growth and Lipschitz conditions:
[b(#l + IB00:)l __ K(1 + I~:l), Vx~n"
IbCx) - b(z)l + ] B o ( z ) - B o ( z ) l < K l z - z l , Vz, zeR" The probability space in which v(.) is defined and the notion of weak convergence were introduced in [1] and the following convergence theorem was proved: Theorem 1 Under the assumptions (A1)-(A5), the process x~m(.), defined by (1.1) and (1.2), converges weakly to the diffusion process ~(.), defined by (1.4)-(1.19), as e -+ 0, # --+ 0, and r --+ "7. The explicit form of the limiting diffusion model shows the interaction between two asymptotic phenomena through the dependence of the matix ~, defined by (1.18), on "7 = lira * (The implication of such dependence on the engineering practice of neglect9 ~/~--*0 ~ "
ing parasitic elements was discussed in details in [1]). The first phenomenon arises in singular pertubation analysis due to the existence of parasitic elements (/z --, 0) while the second phenomenon arises in asymptotic stochastic analysis of systems driven by wideband noise (r -+ 0). The Stability Result Most of the work that was done in the area of stochastic stability, was dealing with systems represented by R6 equations (Markov models). The stochastic Lyapunov method, that is analogous to the deterministic Lyapunov method, was effectively used [cf. 13, 14]. One of the first attempts to study stability properties of dynamical systems which are driven by wide-band noise, has been made in [12]. We utilized some of the techniques of [12] and extended them to the singularly perturbed system given by (1.1)
and (1.2).
893
In [2], we proved a Lyapunov like stability theorem about the slow variables which was based only on conditions uppon the approximating diffusion process. A Lyapunov function V(x), xeR ~ was considered with the following properties: (a) V(z) is real valued, positive definite, V(x) = 0 =:a x = O, V(=) -~ co as [x[--* co, and has continuous partial derivatives up to the third order. (b) For any vector or matrix valued function g(x,t) =- O(x) for te[O,T] and xeR '~, we have: ]V~(x)g(x,t)l < K V ( x ) , (1.20)
(v-c~)g(~,,)),~(~, t) l _< gv(~), ~OZIOZjOXk glgZ.g3 < KV(=~ 9
3
lc
--
x
(L21) (1.22)
~"
where i,j, k = 1, 2, , n and a.1. #2., and g,Z are components of vectors or matrices which are O(x). The constant K in (1.20)-(1.22) may not be the same and it is independent of T. Also the following assumption was made: (A6) The coefficients a~(z), a~2(z) and B~(z) vanish at x = 0 and for some M > 0 and 9 .,
,mr7
J
VxeR ~, [al(x) I + ]A12(=)] + ]BI(Z)] _~ M Ix[. Now we state the main theorem which is recalled from [2]. Theorem 2 Suppose that there exists a Lyapunov function V(x) defined on R ~ satisfying (a) and (b) and for some A > O,
L'W(x) <_ -AV(x)
VxeR"
Suppose that the assumptions (A1)-(A6) are satisfied. Then there exist r such that for a l l C a n d / ~ s a t i s f y i n g 0 < e _ < r
(1.23) /2o, and ~/o
[ ~* - - "~]
solution of (1.1) and (1.2), is uniformly stochastically asymptotically stable as t --* co, i.e. for any rh and r/z, there is a 6 > 0 such that if lxo[ < 6 then: (I) P{]x(t)l_ 0 } _ > 1 - W x for s o m e 0 > 0 (II) P { li,m I~(t)l = 0} = 1 In proving theorem 2, we proceeded in a way similar to [12] except that the averaging method of the Lyapunov function is almost the same as the method used in the
proof of theorem 1, which was introduced in [15] and further developed by [10], [16], and [17].
2. Output Feedback Stabilizing Control
We consider the system:
= ~(~) + A~C~)y + BI(~)~p + aI(~). ~# = a~lC=) + A~v + B~C=)v~' + a ~ ( = ) . * = *,(~) + *~v + B3(=)o~ 2
(2.1) (2.2) (2.3)
where u is a control vector in R p, z is the output vector in Rq (q < n) represents the observed variables, v~' eR r and v ~ e R a are defined in terms of the independent processes vl (t) and v2(t) which have the same properties as v ~, which is defined in section 1, where ex and r are different in general, so that, if the observation noise v 2,2 has spectrum
894
which is wider than that of the system noise v~ ~ then we expect s2 to be much smaller than ~1. Moreover, let Rl(r) = E(Vl(t)v~(t + r)), R=(r) = Z(v:(t)v~(t + r)), W1 = OO
OO
f R,(r)dr, W2 = f R2(r)dr and Sl(w) and S2(w) denote the spectrum matrices of o
o
vx and v2, respectively. The matrices G1,G2,el and Ba satisfy certain smoothness conditions which are specified later. The main assumption here is that the states of the system are not available for perfect measurement. Now we proceed with the procedure that leads to the design of the stabilizing control is, as follows: The open-loop reduced-order model of (2.1)-(2.2) takes the form (see [1]): d~ = fl(~)dt + F1 (~.)udt + al ("2)dwl (2.4) d2 = f2(~)dt + F=C~.)uat + a=C~.)dwl + aa(~)dw= (2.5) We consider a controller of the form u = g($) where ~ is the output of the observer: d~ = fl(~.)dt + Fl($)g($)dt + g [ d z - f=(~.)dt - F=(~)g(~.)dt] (2.6) for some constant gain matrix g . The vectors fl (x), f2(x) and the matrices F1, F2, al, a2 and aa are to be determined. The closed-loop reduced-order augmented model which follows from (2.4)-(2.6) with u g(~) is: d~ = f l (~-)dt + F1 (-~)g(~:)dt + o'1 (~)dwl (2.7) d~ = fl (~)dt + Fx ().)g(e)dt + K [ A (e) - A (e)]dt (2.8) +g[F=(~) - F=(~.)]g(~)dt + ga=(~.)dw~ + gaz(~)dw= To determine the exact form of fx, f2, F~, F2, a~, az, and a3 we propose an observer for the full system (2.1)- (2.3), to reconstruct the states x, in the form: x = f,(~) + F,(~.)u + g [ z - f~(~) - F=(~)u] (2.9) The gain matrix K is the same as the one appearing in (2.8). Now applying the same control law u = g(~) to (2.1), (2.2) and (2.9), as a function of the reconstructed states ~, then the closed-loop full- order augmented system takes the form: =
~: = f~(~,) + F~(~,)g(~.) + Kick(x) - f~(:~) ,9 =
+
+
--
F= (x:)g(x)] + gc=y + gBa(z)v~ =
+
which can be simplified to 2 = ~l(X) + A;=(X)y + & ( X ) v ~ ~1 = a'21(X) + A=y + B~(X)v ~ where X =
(2.10) (2.11)
,
pC =
,
+
Kl(X) = ( fx(ff:) +
FI(~)g(~) + g[c,Cx) - f2(~-) - F2C~)g(~)]
/
}
(2.12) (2.13) (2.14)
(2.15)
&(x) =
o).
(2.16)
895
Systems (2.10) and (2.11) are basically in the form of (1.1) and (1.2) and under additional assumptions, which will be stated later, it cart be shown by the the convergence result (Theorem 1) that X(t) of (2.10) and (2.11) converges weakly to a Markov process )( as r --~ 0, r -* 0, /z --* 0 and ~ --+ ~/. By tracing the steps of the convergence proof (see [1]), in the case of only two parameters r and/~, we will find that in the case of three parameters, the ratio is given by ~ and it does not depend on r and so we will require e~ _+ "7e['71,co). Moreover, all the upperbounds that we obtained in the steps of the proof which were O(# + ~) or O(v:fi + v/~) will depend here on r ~2, /z and on the fact that ~ > ~h for "h > 0 arbitrary but fixed. Now we derive the diffusion operator corresponding to X with the aid of (1.5)-(1.19), where the assumptions that will be listed later, will validate this derivation. As in (1.6), the drift coefficient is b(X) = rio(X) + t~l(X) - ,~2(X)A;XI~2(X) + I;3(X) (2.17) The diffusion coeffiecient is A(X) = 5(X)g'(X) (2.18) where =
( ~ll (~12)
, \521 b~2 rio(X) = ri, (X) - i , ~ (X)A~Xah (X) = a0C~) + a0(~)~(~)
(2.19)
/x(~) + [F,(e) - KF~(~) - K~A~Xa~(~)]~(e) + K[~(~) - f~(~) - ~ : A ; ~ ( ~ ) ]
)
(2.20)
where
(2.21)
Go(x) = Gl(x) - A12(x)ATtGz(x),
go(X)=( B0()
0 ),
-KczA21B2(~,)
(2.22)
KB3(~.) (2.23)
5=
o)
(2.24)
where
(2.25) 0
From(2.15) and similar to (1.14) we have: ( f o r X =
o x,(:
oX,)o
(:))
fori----1,2,...,n
(2.26)
for i = n + 1 , . . . , 2 n
(2.27)
896 where /S,(x) = [V,,ail(x)i ... iVzaio(z)],xo ; KBs(x) = [a~j(x)],xo for i = n + 1,...,2n From (2.14), (2.16) and similar to (1.16) and (1.15) we have
~,(x) = P,(x) =
(?')
0~,•
,
i=
1,2,...,~
(2.29)
i = ~+ 1,...,2~
(2.30)
E,(X) = ( ( E , ) , • :) i = 1,2,...,m 0 Then similar to (1.11) and from (2.22)-(2.24), (2.26) and (2.27) we have: hxi = tr[Dibol4;" +/9~AI2A~'IE] = tr[D~BoW + D~At2ATXE] = hli for i = 1 , 2 , . . . , n From (2.27) we get: hil = 0 for i = n + 1,...,2n
(h,:,)
Hence
(2.28)
(2.31)
(2.32) (2.33) (2.34)
Similar to (1.12), we have: = h2i for i = 1,2,...,m where h2i is defined as in (1.12), and similar to (1.13) we have:
hs, = t r [ - ~ " D o ~ ' g ~ A ; = hzi
for
1 - F'[3o~'(A;1) ' +
(2.35)
~'L2A;'~I
i = 1,2,..., n
(2.36)
i=n+l,...,2n
(2.37)
and hsl = 0 for Hence ~sCX) =
(2.38)
where h3i is defined as in (1.13). Notice that, it can be verified that/5 = p. Now, since ,iCX) = &CX)SC0)B~(X) where "~(0)= ( S l i 0 )
(2.39)
$20)(0)
Then from (2.18), (2.19), (2.22), and (2.39), we get:
a(X) = ( B~ \-gc2A;1B2(x)~
0 ] KBs(x)~-C~J
(2.40)
Then the reduced-order closed-loop model corresponding to (2.10) and (2.11) takes the form:
d.~ = [b(2) + Go(.~)gC~:)ld,t + B o C x ) v / ~ e w x
(2.41)
897
d~ = [fl (x,) + (F, (k,) - KF~ ($z) - Kc2 - Kc2A~IG2(~z))g(~) + g { o ( ~ ) - f2(~) - c2A;la21(~) - c2A;lh2(2,)}]dt - K c 2 A ; 1 B 2 (~) S~/~)dwl + KB3(~-) V / ~ ( ~ d w 2
(2.42) Now we impose a consistency condition which is stated as follows: the reduced-order closed-loop model (2.41) and (2.42) has to be the same as the closed loop reduced-order model which is given by (2.7) and (2.8). Hence by comparing the coefficients in the two systems and insisting that they must be equal for all K and for all functions g(x), we have: fl i x) -= b(x) (2.43) F1 (x) = Go(x) (2.44) 0.1 (x) = Bo(x) ~ (2.45) f2 (x) = el i x) - c2A 2' a21 (x) - e2 A 2' h2 (x) (2.46) F2 (x) = - e 2 A ; 1G~ (x) (2.47) Hence, the open-loop reduced-order model (2.4) and (2.5) can be written in the form: d.~ = [b(~) + Go(~)u]dt + 0.1(~,)dwl i2.48) d2 -- leo (x) -t- Fo (x) u]dt + 0.2 (x,)dwl + a3 (2,)dw2 (2.49) where b(x) and Go(z) are defined by (1.6) and (2.21) respectively, and
C0(X) ---~C1(x)
-
-
c2A21a21C x)
-
-
c2A21h2Cx)
(2.50)
Fo(x) = -c2A~lG2(x)
(2.51) (2.52) (2.53)
0"1 (.T) = B 0 (x) ~
0.2 (x) = -c2A~ 1B2 (x) ~ o3 = B3( )
(2.54)
and the proposed observer (2.6) takes the form: d~ -- [b(}) + Go(~,)g(~,) + g{c0C~) - c0(}) + (F0(~) - Fo(k,))g(~)}]dt (2.55) +K0.2 (~,)dwl + K0.3 (~)dw2 The design problem, is to choose a function g(x) which is smooth enough and a constant matrix K such that the state x and the error e = ~ - $ will be stochastically asymptotically stable. Let us write the the It6 equations (2.48) and (2.49) in the form
,using (:), (2.56)
d)( = b()Odt + 5(2)dw where
b(X) = (
bi~') + G~
)
(2.57)
b(~,) + Go(~,)g(~) + K{eoi~) - coi~) + [F0(~) - F0(~:)]g(~)} and
0 )
(2.58)
K0.2 (5;) K0.3 Suppose we have found u = g(z) and the gain matrix K which stabilize (2.56), then the next step is to apply the same control law to the open-loop full-order system (2.1)(2.3) where u = g(~) and ~ is the reconstructed states and satisfies the equation of the following observer: ~ = b(~:) + Go(~:)g(~) + g[cl(x) - co(:~) - Fo(~)g(~)] + gc2y + gB3(~)v~ 2
(2.59)
898 Now we are ready to spell out all the conditions, under which the stability of (2.56) will imply that of (2.1), (2.2), and (2.59), when u = g(~) is applied to (2.1) and (2.2). This will be done with the aid of the stability result (Theorem 2). We require the following assumptions: (A) al,b, Bo,B3,A12, and cl to vanish at their respective origins. (B) b and 5 are required to satisfy: i
~(x) -~(Y)[ +]6(X)-5(Y)[ _
VX, Y eR 2,*
b(X) ~ + lh(z) ls <_ c(1 + Ixl ~) v z e R ~" (c) The coefficients all(X), s 1~1(X), a-ul(X) and ]~2(X) are continuous and have continuous partial derivatives up to the second order which are uniformly bounded in XeR 2" in addittion to a'21(Z) and sB2(X).
(E) Ido(X) - a-o(Y)l+ l~o(X) - Bo(Y) <_K IX - Y] VX, Y e R ~" and for some i (F) v, (t) and v2 (t) satisfy the same type of conditions which are stated in (A1). (G) The constant matrix A~ satisfies (A3). Now we state the main theorem of this paper:
>
O.
Theorem 3 Suppose that there exists a Lyapunov function V(X) which is defined on R 2'* and satisfies the same type of assumptions as in (a) and (b) of section 1, but for X e R 2'*. Moreover, suppose that the assumptions (A)-(G) are satisfied and let L be the diffusion operator, whose form is the same as of L "1, given by (1.5), but its drift and diffusion coefficients are those of X, given by (2.56)-(2.58), and for some A > 0
Lvcx) < - ~ v c z ) Then there exists r
I~ - ff "
< -
vx~R ~"
(2.6o)
#o, and "70 such that for all 0 < e{ < r
X(t)
0 < ~ _< ~o, i = 1, 2, and
the solution of the closed-loop system resulting from \~(t)
(2.1), (2.2), and (2.59) after applying the feedback control u = g(~) is stochastically asymptotically stable as t ~ co. Proof: From the assumptions and (2.60), the solution of the closed-loop reduced-order model, represented by (2.56), is stochastically asymptotically stable then by imitating the steps of the proof of theorem 2, after the necessary modifications concerning the following estimates given in [2],the proof of the theorem will be completed. SN here is a subset of R 2~. Also the following estimates depend on the fact that ~ > "/I. Iv~,1(x,t)] <_ (glv~7 + K 2 v ~ + gsv~)V(X), IV~v,2(X,t)[<_ (K4~i + gs~ + Ks~,)V(X),
[~(x,0~-o(X)
< (gTv~ + g s v ~ + Kgv~)vCx),
~x~CX, t)r + %~--~CX,t)Cs + ~Cx)CvP,v~)') <_(K, o v ~ + K ~ i v ~ + K,2V~ + K,3~, + K,,~: + KI~)VCX). All the above inequalities are true for XeSN.
899 REFERENCES:
[1] M.G. E1-Ansary and H.K. Khalil, ~ On the interplay of Singular Perturbations and Wide-Band Stochastic Fluctuations," SIAM Journal on Control and Optimization, Voh24, No.l, January 1986. [2] M.G. E1-Ansary, " Stability of Nonlinear Singularly Perturbed Stochastic System," Dynamic Systems Modelling and Control, ASME, Vol.1, pp.115-119, November 1985. [3] P.V. Kokotovic and H.K Khalil, " Singular Perturbatins in Systems and Control," IEEE Press, 1986. [4] J.H. Chow and P.V. Kokotovie, " A Two-Stage Lyapunov-Bellman Feedback Design of a Class of a class of Nonlinear Systems," IEEE Trans. Automat. Contr., Vol. AC-26, pp. 656-663, June 1981. [5] J. Chow and P. Kokotovic, " Two-Time Scale Feedback Design of a Class of Nonlinear Systems," IEEE Trans. Automat. Contr., AC-23, pp.438-443, 1978. [6] A. Saberi and H. Khalil, " Stabilization and Regulation of Nonlinear Singularly Perturbed Systems- Composite Control," IEEE Trans. Automat. Contr., Vol. AC-30, pp. 739-747, Aug. 1985. [7] T. Tarn and Y. Rasis, "Observers for Nonlinear Stochastic Systems," IEEE Trans. Automat. Contr., AC-21, pp.441-448, 1976. [8] H. Kwakerna~k and R. Sivan, "Linear Optimal Control Systems," John Wiley and Sons Inc., 1972. [9] P. Billingsley, "Convergence of Probability Measures," John Wiley, New York, 1968. [10] H.J Knshner, ~Jump-Diffusion Approximation for Ordinary Differential Equations with Wide-Band Random Right Hand Side," SIAM J. Control and Optimization, Vol. 17, pp. 729-744, Nov. 1979. [11] G.C. Papanicolaou and W. Kohler, ~Asymptotic Theory of Mixing Stochastic Ordinary Differential Equations," Comm. Pure Appl. Math., Vol. 27, pp. 641-668, 1974. [12] G. Blankenship and G.C. Papanicolaou, ~Stability and Control of Stochastic Systems with Wide-Band Noise Disturbance," SIAM J. Appl. Math., Voh 34, pp. 437-476, May 1978. [13] F. Kozin, "A survey of Stability of Stochastic Systems," Automatica, J. IFAC, 5, 1969, pp.95-112. [14] H.J. Kushner, ~ Stochastic Stability and Control," Academic Press, New York, 1967. [15] T.G. Kurtz, "Semigroups of conditioned Shifts and Approximation of Markov Processes," Ann. Probability, Vol. 3, pp. 618-642, 1975. [16] H.J. Kushner and Y. Bar-Ness, "Analysis of Nonlinear Stochastic Systems with wide-Band Inputs," IEEE Trans. Automat. Contr., AC-25, pp. 1072-1078, 1980. [17] H.J. Knshner, "A Martingale Method for Convergence of a Sequence of Processes to a Jump-Diffusion Process," Z. Wahrscheinlichkeits theorie Verw. Gebiete, 53, 207-219, 1980.
CONTROL
Marcelo
OF
UNCERTAIN DYNAMICAL POSITIVE REAL
Car~alho
Minhoto
Abstract-
This
c o n d i ti o n s
to
outputs,
inputs,
m
measurements
into Two
forward of
feeds
the
made
where
outputs
Zak
In
using
only
Positive
Real
presented.
two
In
static
and
the
problem,
the
G are These
result
and
system
inputs
the
with
and
first,
compensators,
second
s u f f i ci e n t p
outputs
(~-q~}~9.
back
pr e s e n t s
vari able
this
then
F and
by
necessary ti m e - i n v a r i a n t
and
are
second
and
synthesis.
STRICTLY
the one
feeds
the
system o f which outputs
system.
The
_> m
a Strictly
the
nominal
Wal c o t t
i i near
results
is
compensation
presents
a p
USING
T e i x e i r a x'z
paper turn
SYSTEMS SYSTEMS
the
constant
results
str uctur e
if
compensated
are
c o n d i ti o n s
the
is
system given
compensation useful
in
the
o b s e r v e r - c o n t r el I e r
nominal
system
for
the
is
by
given
by
matrices. control
system
with
u n c e r t a i n t i es.
Complete Marcelo
and
Preferred
Carvalho
Mailing
Minhoto
Adress
Teixeira
FEI SwrUNE~-~ Depto.
de
Engenharia
Caixa
P o s t a l , 31
IS378
- Ilha
Telefone
El~trica
Solteira
Number:
-
(0187D
S.P, - BRASIL - @P--3482
Keywords: Strictly Positive Real Systems, Control OT Uncer tain Feedback S t a b i i i z a t i on.
Matrices, Dynamical
IThe author is with the Department of F E I S - U N E S 7 >, I l h a S o l t e i r a - S P , Brasil. zThis
work
was
supported
by
CNPq.
Variable Structure Systems, Output
Electrical
Engineering,
901 The
first
stabilization
of
the
and
Steinberg
gives
necessary
result
and
I
error
and
be
used
dynamical
Corless
problem
sufficient
observer-controller trial
can
uncertain
in
of
output for
1985.
conditions
synthesis;
the
system,
for
and
Walcott
The
feedback
example, second
result
Walcott
Zak
in
present
and
7~k
only
one
procedure.
- I NTRODUC-TI ON The
have
a
uncertainties on
the
[I]
SPR,
system
using
The systems
systems the
is
of
control
the of
Popov' s uncertain
to
c o n d i ti o n s
p
outputs, outputs
constant are
m
[I]
synihesis
of
to
inputs,
a
turn p
>
m, The
matrices.
only
systems
turn
measurements.
useful
the
wi t h
available
conditions
in
dynanucal
results
the
usiI~g
e x a m p l e,
matrices
s y ~ t eros
example,
and
paper
uncertainties,
.An
inputs
the
control
and
outputs
stabi I i zati on
of
in
the
variable
Walcott
and
~k
and
[2].
STATEMENT OF THE PROBLEMS Consider the
k
=
y
= Cx,
where Define
The
this
observer-controller
structure
y
with
with
of
for
presented
inputs
built
for
nonlinear/uncertain
-
are
results
with
measurements,
II
paper
only
compensation
For
useful.
transfer
important
S-PR m a t r i c e s ,
very
time-invariant a
the
CS~R)
contr ol
systems, [3].
using
this
the
because the
these
a ~-PR i s In
of
of
systems
into
linear
, [2] ,
Real
i n
hiperstability
dynamical
into
Positive
i mpor t a n c e
stability
asymptotic
system
Strictly
gr eat
~
+
x
Bu rank ~
~n
F E ~
,
[C]=p, u
~
system rank
D~m,
C2.1D C2.~-)
[CB]=m, y
G ~ ~n~
E
~P
constant
and
p_>m.
matrices,
and
= Fy .
C2.3)
conditions
PCA+C~2) BTp = FC
from
the
SPR
matrices
+ CA+C~2) IP : : - Q < 0 , w h e r e P = p T > 0.
for
{A+C~,B,FC}
are
[13: (2.4) C2.59
go2 Problem
8.1:
desired
to
For
the
find
existence
of
(2. 5 )
satisfied
are
Remark
matrices
2,1:
Problem
In
2,~-:
Remark
Is
the
problems
the
but
with
K
problem 2.1
0
is in
using
To
knowledge,
question.
In
matrices error
F
such III
cal i y
that -
Lemma x
and
is G
and [i]
B. 1:
Consider
u
e ~m,
there
and
a
outputs of
G. used
recipe
in
system
solution
for
the but
choose
are
considered,
non~nal
problem
step
1
for
and
the
in
[4]
the
of
problem dynamical
measurements.
the
then
is
observable;
the
in
a linear
and
outputs.
is
nonlinear/uncertain
2.2,
and
the
2.1
The
problem
the
C2. 43
matrix
the
the
Y e ~P
exists
that and
of
solution
(2.5)
is the
conditions
problem
in
and
that
p r o b l em
m=p:l.
control
stable ;
such
sufficient
the
choose
is
construction it
G
Ci f
2.1
is
a
for
a
open
of
trial
and be
to
(A+GC)
p o s s i bl e)
the
F
and
P
satisfied.
CONDITIONS TO TURN A SYSTEM Consider the lemma:
~ ~n
Then,
(2.4)
F
it for
[ 2].
for
given
C2.3),
a arbitrary
assumed
inputs
with
conditions
K feedback
this
controllable
the
procedure:
asymptoti
is
are
only
[E]
with
of
In
and
and
matrix
2.1
8.2.
the
s)~tems our
the
considered
useful
mxp
necessary
systen~s
2. I
is
=
K~
synthesis
and
C2.2)
sufficient
P.
some
result
presents
compensated
= BK,
problem
er
2.1
C2.1),
and
problem,
The
observer-controll paper
G for
this
2, 2:
This
system
necessary
INTO
C2.19
SPR
and
C2.2D,
where
p_>m. transformation
T:~n
~n
such
that
~i] = Tx,
C3.1)
C3. 2) A4
B
--[o Theorem
existence
B. 1 : of
C3. 3)
The F and
necessary K in
the
and problem
s u f f i ci e n t ~-.I,
is
the
c o n d i ti o n s exi s t e n c e
for of
903 P-'~3 ,e ~ m x p - m
for
the
system
T B
v=[o to
be
+
T A T - 1 I- I n _ p
z,
w=
stable,
P23
where:
Theorem
3.2.
existence
of
1
!A41
T is defined
(A. 1)-(A.
in
:
J
The
9)
and
necessary
F and
G in
the
CCBD
and
asymptotically
The
in
theorem
condition
theorical of
the
as
theorem
unstable CAT,
of
2.~
well
3.2
is
3.1 :
T
independently
of
(~. 4D
(~.~9
In
if
AT I and
the
(B. 19
( A . I ) - C A . g)
the
K,
been
points to
X
Ul
where
for
C3. 7D
~ ~p-m
the
treated of
view
(A1)c_~,
,,,','R., t h e
1 emma
theorems
of
only
condl ti on
is ~ ~n-p
some
has
equivalent
in
of
beginning and
3.1
numerical
of
ci ent
CB. 3 3 .
after
matrices
are
extensively [5].
The
where
from
condition
X+CA
)
is
the
s u b s p a c e of
controllability
[0].
AT[cB]I3
conditions
a
for
z
defined
3. 1.
theorem
subspace
Remark
stable
is
1
suffi
problem
AITz § AT[ CB] lUl , u I =Kz,
defined
C3.59
,
]
A3 i A 4 a
be
v
i 0
[
to
C 3 . 49
w
]
I p_ m
asyTnptotically
=
z
A3
a
are
and is
3. I
particular
necessity
are
presented
is
choice of
for
not
3. 7:
and
proof
satisfied
(B. 29
3. I
of
T;
theorem G
satisfied
=
BK
for
a systematic
unique, can
be
note
that
3.1, and
some way
we
but
verified from have
P, i f T = P N > O.
some
PN
to
the
choose
one
the that and
T.
I V - CONCLUSIONS This time-invariant
paper system
shows t h a t CLTISO
the
into
conditions SPR
using
to only
turn
a
linear
outputs
and
904 inputs
given
measurements
LTIS.
are
These
equivalent
results
nonl i near luncer t a i n
dynamical
variable
structure
APPENDIX
A
Define
B I : = [bi,i," " " b l , n - m ]
rank
-
[Bl]=n-m"
of L e m m a
the
useful
stabilization
in
s~tems,
and sliding
Proof
to
are
the
for
of
a
control
of
example
using
modes.
2. I ~nxn-m
B1TB=O and bTl,i
bl,o'=O'
a n d consider
for
i~j,
that
i and j
~
{i ..... n-m}. Then, C can be made equal to C = M IB T + M2BIT
CB=MIBTB;
and so,
1 M2
N o t e that
1
~pxn,
e
definition
where
from t h e c o n d i t i o n r a n k
then,
rank
CA. ID
,
=p.
there
Therefore,
exist
p-m
to m I ..... rap_m, s u c h
fwhere a := b denotes
N.l E
two c o l u m n s
~n-mxn-m
from
[CB]=m it f o l l o w s
(A. 29 and
columns
os
N i _ I,
M@,
equal
for
that
a=b by definition9
i = 1 ..... q a r e n o n s i n g u l a r
of H~N 0 ...
that
where
NO: = In_re.
and i n t e r c h a n g e
9O5 It i s
easy
to show
Hence
from
CA. 39
Now,
N -I BT 1 has
positions.
T BOI
where
the
Ol
follow~
same r o w s
~n-pxn
~
it
from
of
BTB=o
from
CA. 5]),
N-I=N q ...N1.
that
BT ,
but
and
b l,i T
with
some c h a n g e s o f
b l , j = O,
i~j,
define
BOI
B OIT BOI
=0.
C = M1BT
T + N3BoI
C A. 79
and
-T BOI
Therefore,
CA. 49 a n d
From
e CA.S)
Therefore,
BI =:
N i- / = N i and s o
that
+
~
~p-mxn
from
CA. 89
; then and
T BOI
CA. 79
B=O,
it
-T B=O BOI
and
follows
that
-T N4BoI.
C e ~pxn,
CA. 8D Now,
consider
the
matrix
CA. gD
Then, -I
T
-i
.
CA. i09
where
CA. 11D
M 3 + M I K 4 + M 4 K 5 = M 3 + M 4 B-oTI B-o I K ~
K3
= -BoI
C
that
ccT=o
such
that
CA. 123
ngul
ar .
and
CA. IPO
CA. 13)
C
Note
nonsi
+ M I B T B K 1 = O,
and is
from
CA. 8 ) ,
satisfied,
there because
exist -T BOI
K 1,
K 2,
K4
BOI
and
BTB
e
K5 are
906 Therefore, that
TB
from
=
=:
C2.1),
B N,
CA. g )
CP-.2),
and
ro ]
CT -I=
I
B
=:C N
CA. I O )
is
easy
to
,
show
CA. 14)
P
=:
.... :" ""
= : A N , Tx=:
1
,
CA. 159
[A3 i A4] where
A1
Hence,
~
the
lemma
is
B - Proof
APPENDIX Necessity: C2. 49
~n-pxn-p,
Note
and
A4
of
are
if
Hence,
K,
CB.I)
if
and (B. 2)
Consider Then,
T
from
in
AN=TAT-1
IRp x p - m
have F
A2
~
~n-pxp ]
,
BN
= "I'B a n d
FC N =
F C T =-1 9
then
T > O. = PN
F and p=pT > 0 s u c h t h a t
and o n l y
if
there
exists
C2. 4)
and
C2. 5)
PN=PN>O s u c h
that
are s a t i s f i e d . CA. 9)
and
AN , BN
and
CN
in
CA. 14)
and
CA. Ig)
Q
(B. 29,
e D~n-pxn-p'
but
P2
[CCB)I ] = p - m ,
rank
arbitrary
and
to
CB)
where PI
~pxn-p
3. I
equivalent
pT-i
there exist
~
I
Theorem
where, PN = CT T ) - I
are satisfied
A3
proved.
that
(2.59
~ D~pxp,
such
that
9
RPXP'
1E
P2
PI =PIT>o'
( C B ) Ts C B ) I =0
PN>O
An
]
(B. 39.
and
P2=pT>o, ~
9
Therefore,
CCB)I
9
~p-mxn-p
is
from
we
(B. 39
that C CB) T p 2 ' pp=pT>_ 0 Then,
transformation,
the has
.
lB. 49
system from
< A + B K , B, F C }
CA.g),
CA. 149,
after CA. 15)
the and
i i near
CB. 49
the
907 following
form:
+
where
Pp_
51)
B
T P2 > O,
=
CB.
U,
A4
P2 e ~ p x p
Now,
consider
the
linear
tr a n s f o r mmti on: "In-p
0
T: =
9
9
s
,Y
~m
CB. "7)
[CB]
. . . .0. . . . i[~. ~. . . ~ ~ ji where
[CB]I~pxp-m
. [CB] F[CB]=O
and r a n k [ [ C B ] l ] = P - m .
N o t e that
. cB. s~
i-I=
Then,
from
CB. 5)-CB. 89 it f o l l o w s
that
u = ~ + K~,
. . . . . . . . .
[CB]ITA 3
~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ECB]ITA4p21[CB]I
T ~-1 [CB] 1 1-1 rCB]IP
(B. g3
".,.
908
CB. i0)
BN=[O" Now,
0 " [ c B ] T p ~ [ C B ] I T and C N = [ O "
it is e a s y to s h o w
PN=~T>O
such
0"
Im]
(B. ii}
that t h e r e e x i s t s
that CB. l~-D
BNPN-T- = if
and
only
if
are satisfied.
Therefore,
there
exists
Note that from
from
CA.'149 ,
P N B N K C N T -1 has z e r o s
such
CB. II9
a n d CB. 139
and
elements
and CB. 149
a s y m p t o t i cali y
stable.
A 2 = A21 [CB] T+A22[ CB] T I'
that
CB. 149
C2.49
it
and
we h a v e
C2.5)
that
follows
for i , j e f l ..... n-nO.
that Hence,
is n e c e s s a r y
Now,
for A~_, A 4 a n d p~l
A4 = A41[cB]T+A4~[cB]TI'
CB. 13])
P=pT>o
CB. i19
Ci,j9
f r o m CB. I O D - - C B . I ~ 9
representation
~'N
consider
the
fol 1 owl ng
:
CB. i0}
and
CB. 17D
909
e~. 1 : ~cmP21 Ecm%rcB31Paaccs3~+ + [ C B ] P 2 3 [ CB]IT+[ CB]I P 2T3 [ CB] T Hence,
using
X I _--
CB. 1 6 ) - C B - 1 8 D
An
CB. 18D
CB-15D
it f o l l o w s
that
A1 A22 1 +
CB. IgD
Now,
define
in
CB. 19D,
CB. 20D where
P'-23 E
~mxp-m.
Then
I~ is
asymptotically
stable
for
some
P23
and
the
necessity
is
proved.
Sufficiency: > 0 such
Note
that
Consider is a s y m p t o t i c a l l y Then,
from
that
in
CB. 1 8 ) ,
PP~I = ~ I m and that
there
for
a given
> O, exists
for
P23'
any
P--~3 s u c h
there
exists
P~p_ > 0 g i v e n .
that
A1
in
CB. 219
stable.
CB. I S g , C B . QD-CB. II>
it
follows
that
fop
=EI m,
910
C B. 2~-D
--[o from
CB. IO9,
3
X4
where,
AN=
~
Define
and
Kl=-kI
=pT>o,
P a
CB. P-S)
P
a
m
, k>O e
A 4 = A4
C B. 243
+ K1
and
~n-mxn-m
such
that
a
pa XI + -T Pa =: -Qa <0 A1 Therefore,
Pt=
it
CB. 259
for
[~ a
[
0
[ CB] TP2 [ CB]
follows that for
] -I
1
T
=Pt e
~nxn,
C B.
289
g r e a t v a l u e s o f k > O,
CB. ?.79
[o and
so,
APPENDIX
CB. 289
the
sufficiency
C
Proof
-
N e c e s s s ty: CA. 14)
and
CB. 3),
CB.:9
CA. 15).
PN CAN + TC~: N)
proved.
of T h e o r e m
Consider
and
is
the Then,
so (2.43
+ CAN
3.2
i i near from is
I
tr a n s f or m a t i on
the
proof
equivalent
+ TC~ZNDTP N
=
of to
theorem
T
in 3.1
C A. g) , we
have
]
911
=
[ ]
-
Now,
%
%
0_~
Q4
from
+
-Q
Sufficiency:
and
=
-~aP~ any
<
P~
-
- aIp
ks
+
it
o
%1
0
CG
follows
"
that
(C.
-%
=
asymptotically is
-%
<
in
PIAI
+
stable
for
o
some
proved.
0
for
where
CA.I~)
asymptotically
is
T
CC. 1 9
from
1
<
A~CCB31K9
aIp] -T I A~,
and
=
o.
then
~erefore,
-A 4
TC-.~CN
CA. 1 5 )
necessity C A 1T
If
T~[ -A 4
CA. i 0 9 . CG
The
-- K--~,
~
=
where
C AT+ATc CB91KD
e ~p-mxn-p.
G
O,
+
so
so~e
<
CB. 3 )
+
and
[
_
=
so~e
P~
=
[T I :
-1
we and
have CB.~D
in
CCI~o
T~] that
it
stable
is
Choo=e gi y e n
B OTI G
follows
for
=
that
in
-A~, -%
O; t h e r e f o r e , -Q < 0 for great values of a > 0 and T = P ~ > 0 i n lB. 3). H e n c e , the sufficiency is proved.
= for
ACKNOWLEDGEMENT I advisor
of
would
like
Ph.D. thesis,
to
thank
Prof.
for
his
guidance
Pedro and
M.G.
Ferreira,
my
encouragement.
V -- REFEP.zENCES [1 ]
~--YEIN B E R G , A. stabi i i zati on ~r4xun~. ~ ~.
and M. CORLE~, "C~utput feedback of uncer tain dynand cal s y s t e m s , "" ~ ~ n 2 ~ . , Vol. A C - 3 0 , 1025-1027, C~zt. 1 9 8 5 .
[2]
W A L C O T T , B . L. a n d S . H . ZAK, "Combined observer-controller synthesis for uncer tain dynami c a l s~tems wi t h a ~ p l i c a t i o n s , "" ~ O~n~n/~. ~&. ~4aA% ~ e n 2 % . , V o l .18, n-- i, 8 8 - i 04, i g 8 8 .
[ 33
L A N D A U , I. D. , ~ d ~ e E~n/n~Z: &~A~ ~ a ~ , New York : Dekker , i gvg.
[4 ]
S-YEI N B E R G , A. , "'A s u f f i ci e n t c o n d i ti o n for output feedback stabilization of uncertain dynamical s~stems," ~ 7ax~n~. r ~. E~. , Voi.33, n ~ 7, 6 7 @ - 6 7 7 , 1988.
[ 5]
PATh, R.V. and P. MI S ~ A , eingvalue assignements by f e e d b a c k , "" ~ ~ . ~ 57g-BBS, Ig8g.
[ O]
WONHAM, W. }4. , .~o.4~z,L ~ u ~ ~ Z e ~p4%n~o~, Berlin: Springer Verlag,
~
"Numer i cal al g o r i t h m s for constant and dynamic output ~. E~. , Vol. 3 4 . n ~ @, ~n/n~Z: 1974.
~
~
~
P A R A M E T E R ROBUST C O N T R O L DESIGN BASED ON P A R A M E T R I C I D E N T I F I C A T I O N Q U A LI TY M.GAUVRIT and G.GOMES Centre d'Etudes et de Recherches de Toulouse (CERT) D~partement d'automatique (DERA) BP 4025 31055 - TOULOUSE Cedex (France)
ABSTRACT In this paper, we deal with the problem of designing a feedback controUer for a linear discrete time plant that is liable to perturbations of a physical parameter vector. The new robustness measure is based on the link between the poorest quality of the closed loop parameter identification and the performance and stability robustness for regulators. Reversely, ttds means that the better the robust control, the worse the closed loop identification quality. Our first objective is not to identify unknown parameters but to derive an analytical expression of the covariance matrix of a bayesian estimator. We obtain then the controller gain matrix which would minimize this quality if the identification algorittun was implemented. Important relationships between PRLQG method and this new technique called PRCBI are obtained and it is shown that the PRCBI is a generalization of the PRLQG method. Furthermore a flexible structure application shows that the best results are achieved using the PRCBI method. INTRODUCTION In this paper we deal with the problem of designing a feedback controller for a linear discrete time plant in which perturbations of a physical parameter vector occur. The LQG/LTR (loop transfer recovery) Ill techniques and Hm optimization theory are well suited for treating unmodeled dynamics and stochastic uncertainties but they are not very adequate to handle structured parameter variations. The first difficulty arises from the fact that parameter uncertainties are usually given in state space forms (physical model) while the conventional uncertainty models are based on transfer function description. Norm bounds of the frequential model error (whether additive or multiplicative) cannot be easily obtained from norms of the real parameters. Therefore, the use of conventional error models for Parameters variations leads to a loss of information about the magnitude of these variations. The second drawback of these classical methods is due to the fact that a small parameter variation can produce a very large gain variation or (and) phase variation [5]. In this case, a robustness measure based on gain margin or phase margin is inadequate to design a parameter robust control. Furthermore, the LQG/LTR technique can be used only under minimum phase assumptions on the system. This limitation becomes a major problem when the parameter variations transform minimum phase systems into nonminimum phase. For all these reasons, TAKK and SPEYER [31 proposed a parameter robust LQG (called PRLQG) based on the input-output decomposition. This asymptotic procedure generalizes the LQG/LTR design to a larger class of parameter variations.However this technique is very dependent upon the structure of the LQG method. We will focus this paper on a new parameter robust design grounded on a very different approach, the robusmess measure depending on the quality of bayesian parameter estimates. There are three main reasons to use such a criterion : a) - The adaptive control of a regulator system tries to achieve a trade off between a good quality of the parameter estimates and acceptable control performances. Unfortunately, this compromise is often
913 unrealistic because very good performances of the closed loop regulator lead to poor qualities of the parameter estimates. Inversely, a criterion using the worst quality of the closed loop parameter identification leads naturally to robust control. In this sense robust control is conflicting with adaptive control. b) - A very great error variance of a parameter estimate means that the input-output behavior has low sensitivity to the variation of the parameters. On the contrary, the quality of the closed loop identification would become excellent. A poor identification quality can therefore be associated to a performance robustness. c)
-
The classical robustness criteria using singular values, gain or phase margins, H ~ norm, radius of stability hypersphere and as forth, don't depend on the nature of disturbances which act on the plant. However, a closed loop system can be robust for a disturbance class and very sensitive to another. The robustness measure based on the covariance matrix error of the estimates will be a function of the statistics of the nature of disturbance.
To define a scalar measure it is important to obtain the exact covariance error matrix of a bayesian estimator because the CRAMER-RAO bound cannot be used. The first part of this paper will be devoted to a study of the asymptotic behavior of the bayesian parameter estimates in open and closed loop. The mathematical results for the multivariable case are presented here more extensively including convergence properties and asymptotic behavior. They extend work on SISO systems presented in [9]. The second part of the paper deals with parameter robust design, using this new robustness measure. This synthesis is called PRCBI (Parameter Robust Control by Bayesian Identification) and we will exhibit some very important links between the PRLQG and PRCBI methods. Numerical examples illustrate the performances achieved for various methods (LQG/LTR, PRLQG, PRCBI) in the case of the spring mass systems proposed by TAHK and SPEYER [3]. 1
-
A S Y M P T O T I C B E H A V I O R O F B A Y E S I A N E S T I M A T E S IN C L O S E D L O O P
The closed loop structure for this study must be adapted to parameter robust control which is the main objective of the research. To achieve this, we need the closed loop identification architecture for the nominal parameter set 0o though identification is not on main concern and though 0o is known in practice, The question is : if the parameter vector 0o of the plant was unknown what would be the quality of the estimate A
0o (in closed loop) ? The answer to this question will provide us with a robustness measure. The control loop is thus built of three blocks as shown of figure 1 (left) : 9 the plant denoted S(0o) 9 the Kalman filter operating with parameter vector 0o 9 the linear regulator with gain matrix Ge The closed loop bayesian identification of vector 0o will have here parallel structure shown fig. 1 (right)
914 ! Kalrnan Falter
I-'%-'t----1
f,lt~r I-" 1 ~
I
~
1
Ka~anm~r
~
Figure 1 : Closed loop bayesian identification of 0o ^
Referring to bayesian identification, the parameter estimate 0 o and the covariance error matrix P0o can be obtained with r closed loop linear control operating in a parallel structure.The discrete time linear equation of the system can be written under this form : S(Oo) I xk+l = Ao Xk + Bo Uk + D Wk /Zk = C Xk + Vk
(1) (2)
where Xk, Zk, Wk and Vk are respectively nxl, m x l , pxl and rex1 vectors.Without loss of generality, the parameter 0o belongs to the Ao and Bo matrices; and the sequences Wk and Vk are independant gaussian white noises with : =
=o;
TI=
=
Each Kalman filter operates according to :
^
^
X(k+Uk) = (Ao+AA) X(k/k) + (Bo+AB) Uk ^ A ^ X ( k + l / k + l ) = X ( k + l / k ) + (Ko+ AK) (Zk+ 1 - C X(k+l/k+l))
=
} (3)
When k becomes very large, only density values p(0 o + A0/Zk) with a very small norm I A0 I are not equal to zero. Therefore, we can write to the first order : ^
~
^
Uk = - Ge X(k/k)o = - Ge X(k/k) The a priori estimate error covariance matrix can be written : ^
Po + A p ' = E [ ( X k - X (k/k-l)) (Xk- X (k/k-l))] To obtain the variation AP' due to the variation AA, AB and AK the term in square brackets is written as a ^
function of Xk. 1 - X (k-I/k-2) and the stationary property of second order moment is applied. Introducing the following quantities : L = E [ X(k/k-X) A ^ (X k - X(k/k-1) "r] X(k/k-1) T] N = E[ X(k/k-1) ^ ^ we obtain the following matricial equations to evaluate AP'
AA*=AA-ABGe
AM = CAP' C T
(4)
AP' = Ao(I-KoC) AP'(I-KoC) T A T +(AoAK+ AA*Ko)Mo(AoAK+AA*Ko)T + AA* L AA *T -Ao(I-KoC) N AA *T - AA* N(I-KoC)T A T
(5)
However, the control vector modifies the quantities L and N as follows. By using Kalman relation (3) we can write : A
^
A
X(k/k-1) = (Ao+AA*-BoGe) X(k-1/k-2)+(Ao+AA*-BoGe)(Ko+AK)(Zk-I-CX(k-1/k-2)) ^
A
(6)
^
Xk-X(k/k- 1) = Ao(Xk_l-X(k+l/k-2))-AA*X(k- l/k-2) A
- (Ao+AA*)(Ko+AK)(Zk. 1-CX(k- 1/k-2))+DWk- 1 If we multiply (6) by (7) and neglecting high order terms, we obtain :
(7)
915
N = (Ao-BoGe) N(I-KoC)TAT-(Ao-BoGe) LAA*T-(Ao-BoGe) KoMo(AA* Ko+AoAK)T L = (Ao-BoGe) L (Ao-BoGe) T + (Ao-BoGe) KoMo KoT (Ao-BoGe) T
(8) (9)
Relation (9) gives L. Substituting L into equation (8) gives N which is then substituted into Eq(5) ; AP' is then given as a function of A0 which leads to : Tr(Mo 1 AM) = Tr(Mo 1 CAP' CT) = A0 Gel~ A0T
(10)
Referring to our previous works [9], the matrix error covariance for large k is then given by relation (11)
t,oo ~Oo
(1 1)
The control modes of the nominal closed loop system are given by the eigenvalues of the matrix Ao-BoGe. So, the quantity L (9) expresses the regulator sensitivity to the parameter uncertainties. In the same way the Kalman filter sensitivity is represented by N because the eigenvalues of the matrix Ao(I-KoC) are the estimation modes of the basic LQG design (8). Relations (4-11) give the analytical expressions needed to compute the asymptotic behavior of the bayes optimal estimates in closed loop. Therefore, we are able first to define a robustness measure and second to design a parameter robust controller. 2 - P A R A M E T E R ROBUST C O N T R O L We consider the stochastic linear discrete time system S(0) where the parameters 0 =(01, 02,..0r} can be viewed as the independent parameters associated with each parameter variation of the matrices A and B. The aim is to obtain tile optimal gain matrix Ge of the controller minimizing a robustness measure based on the quality in closed loop of the bayesian nominal parameter identification. For the deterministic case we must at the same time look for the optimal gain matrix Ge and the optimal direction of the fictitious process noise added directly to the control input of the plant. This input matrix leads to the optimal gain matrix of the Kalman Filter. 2.1 - Choice of the robustness measure For large k, the a posteriori conditional density of parameter 0o + A0 can be written [9] : p(0o + A0/Zk) = C' (1 + AO ~loAOT)k/2
(12)
To obtain a very bad quality of the closed loop bayesian identification the conditional density must be insensitive to the variation A0. Therefore, the values of the quadratic form AOGeloAOTmust be very small whatever the direction of vector A0 may be. We can therefore consider various robustness measures based on the matrices G~1or G.such as vO
~O
a ) - A optimality : rain [Trace (G0~ ]
Ge b) - D optimality : min G ~ Ge
,-t
and the corresponding criterious based on G0o.An important drawback of D-optimality is that the condition G o ~ 0 is an identifiability condition of the parameter vector(referring to theorem I). If IGor=0, the data set is not informative enough to distinguish between different parameter vectors and then the bad identification concept has no meaning. The robustness measures using the matrix P0o(Or Got) lead to the same
916 conclusion. The A-optimality represents therefore the best robustness measure which avoids a singular matrix G~1 Furthermore, this criterion leads to a quadratic optimization problem which is easy to solve. 2.2 - Robust controller design The objective is now the design of a feedback parameter robust controller. The robustness assessment is performed by a measure of the parameter identification quality. The algorithm that we describe will be called PRCBI : it computes the parameters of the regulator and Kalman gain matrix which minimize the A optimality. In the previous sections, we have obtained the asymptotic behavior of the covariance error matrix of the vector 0o. In the control design, the uncertainties parameters appear around the known nominal vector 0o- The design is based on this nominal value 0o of a mathematical model of the plant and the aim is to assess the system robustness to parameter uncertainties in order to define a robust controller. The closed loop structure is given on fig.2, where the Kalman filter operates with known nominai value 0 o ~wi
I
G~
vi
II ......
I
Fig. 2 : Closed loop structure of the robust control If the plant and disturbance statistic parameters are known the Kalman gain matrix is obtained by the classical filtering equations and the only parameters to be optimized belong to the regulator gain matrix : J = m i n [Trace (G0~ ] (13) Ge The matrix trace is computed by using the algebraic linear equations (4), (5), (8), (9) and (10). Referring to these relationships, the algorithm calculates the variations AMi(i = 1..... r) corresponding to the r vectors A0 i taken under the following form : (AOl...AOi...AOr)T = I where I is the unity matrix (rxr), and we have from (10) : Trace (Golo) = ~
Tr [lvI~1 A M i ]
(14)
Then, the optimal parameters of the regulator gain matrix are obtained by classical non linear programming methods. Note that since nothing is known about the convexity of the function Trace ( G ~ there is no guarantee that a global minimum will be found. A common procedure in such cases is to choose sereval initial guesses, attempt to find local optima and to then select the best workable answer. The different steps of this new design are shown on fig. 3. If the noise input matrix D and the noise statistic parameters are unknown these parameters become free and appear in the optimization procedure. We f'md here usual assumption of a "fictitious noise" to represent uncertainties in the system. The LQG/LTR and PRLQG methods use this concept. The PRCBI algorithm seems a priori very far from the LQG/LTR and PRLQG methods. However, we will show in the next section that the PRCBI algorithm is a generalization of the PRLQG method.
917 Oi (i=l,..or) • 0 0_LA 9
t Plant model [
~'~-q~
A A i .D Q
Ao~
;;
Relations (4),(5)(8),(9) [~ Mo ; ~,~(i=l,..,~)
IRetati~ (X4)[ Non linear
Trace(C-~lo )
Fig. 3 : Steps of the PRCBI algorithm 2.3 - Relations between P R L Q G and P R C B I methods In the first step, we use a decomposition method called the internal feedback loop (IFL). In IFL modelling a parameter variation AA* is split up into three parts : the input, output and feedback matrices [3].Suppose that AA and AB is parametrized as a function of independent variables A0 (A01,.., A0r} and given as : r'
r
~ A = Y. A i A 0 i ~ B = ]~ B i A 0 i i=l i=f+l We can write AA* = AA - AB Ge as : AA* = - S T (A0) V(Ge)
(15) (16)
Where S E R nxps, V E IRqvxn are matrices and T(A0) E 1::1psxqv is a matrix function of A0 = (A01,A02 .... A0r).To show the link beween PRLQG and PRCBI methods we choose, in accordance to PRLQG method, the noise input matrix D column similar to the matrix S and we investigate the effects of this condition on the PRCBI method. It results from the similarity condition that there exists a matrix P such that : S = DP
(17)
The discrete Kalman filter equations lead to the following relation : Pk+l/k+l = fl-KoC) Ao Pk/k A T (I-KoC) T + (I-KoC) DQDTfI-KoC) T + KoR K f
(18)
In particular let Q = ct2I.By analogy with the continuous time case [1], the steady state solution P/ct 2 of the Kalman filter approaches the zero matrix (as ct goes to infinity) if we have the following conditions: CD is an inversible square matrix and the open loop system C Ao,D possesses no zero outside the unit circle. Then, we obtain from (18) : Ko = D(CD) "1 as ct ~ +co
(19)
This equation shows thus that the nominal Kalman gain matrix does not depend upon the matrix Ao. The variation ~ due to the variation AA of the parameter incertainties is then given by AK = 0
(20)
So, we deduce from equation (16) and (19) : (I - KoC)S = 0
( I - KoC) A A * = 0
(21)
The relationships (15) to (21) have been obtained in accordance to same results of the PRLQG method (see Theorem IV in [3]). Now, we study the effects of these conditions on the PRCBI method.The solution of the matrix equation (equ.80) reads using (20) and (21) ; N = - (Ao-BoGe) (L + Ko Mo K T) ~ * T
(22)
918 Substituting this solution into (5) we obtain the variation AP' and the solution of the matrix equation (5) can be written under this form : &P' = AA* KoMo K T &A *T + &A* L &A*T
(23)
The last step of the PRCBI method before the criterion minimization,is now the derivation of an analytical expression of G~lobY (19). We have : Tr(Mo 1AM) = Tr (&A*T I~Io ~xA* E)
(24)
E = (A o - Bo Ge) E (A o - B o Ge) T + K o Mo KoT
(25)
Mo = CT Mo I C
(26)
with
Using the decomposition (15) the robustness measure can be written under this form : J = T r ( G ~ i o ) = T r [ ( Q c + G~ R c Ge) I7]
with
Qc = Z i=l
A T I~o Ai
(27)
Re=
BiT i~o Bi
(28)
i=r'+l
The minimization of criterion J where the symmetric matrix ~ is submitted to the constraint (25) gives the optimal gain matrix Ge of the controller. Finally, the optimal gain matrix satisfies system (29) :
Go = (Boa` r" Bo + Re) -1 Bov r" Ao r
= ATF
Ar+ Qc+O~RcGe
f
(29)
with Ar = Ao - Bo Ge. These relationships are identical to the LQ steady state control acting on the system (0o) and which minimizes the following linear quadratic criterion : N N-1 J= • X kT Qc x k + ]~ U kT Rc Uk
(30)
k=l k=0 where Qc and Re axe given by equation (28). Here, weightings Qc and Re uses in the general case, total information on the structure of parameter vairations (matrices Ai and Bi). By the choice of a matrix D column similar to S, our method leads to a PRCBI suboptimal robust technique if the following conditions and properties are satisfied : a) - the closed loop system is stable and C, Ao, D is detectable and stabilizable b) - C(zI-Ao)-IS has no zero outside the unit circle c) - matrix D is choosen such that S = DP and CD is an inversible matrix d) - Kalman weighting matrix is 132DDT with 13~co e) - Control weighting matrices Qe and Re are given by (28). If the conditions (a), (b), (c) hold the sub-optimal PRCBI method leads to the same properties that the PRLQG method. The Kalman filter becomes insensitive to the parameter variations. Furthermore, the PRCBI appears as a generalization of the PRLQG method because the constraint (17) limits the optimization set. The PRLQG solution can thus be used to choose the initial regulator gains and process noise matrices in the nonlinear programming part of the PRCBI algorithm.
919 3 - PARAMETER
ROBUST CONTROL OF A MASS SPRING DAMPER SYSTEM
We consider the example presented in [4].We have c h o s e n 2 input output configuration as shown in fig.4 :
D-%Tq
s,'temc (cokr.ated)
(non colocated) Figure 4 The linear state equations are given as : Z
= Ac X + Bc U~ CX J
(31)
where
A~=
0
1
0
0
0
0
0
0
-k ml 0
-c ml 0
k ml 0
c ml 1
0
0
0
0
0
0
0
0
_.k_ m2 0 -2_2K m3 0
~:_ m2 1 -2c m3 0
0
0
_k_ __c_ -_2k m2 m2 m2 0 0 0 0 0 __k_ m3 0 0 0 0
System N :
System C
0
0
-_2c, m2 0 __c_ rn3 0 0
_..k__ ._C_ -k rn4 rrg m4
BCN
=[0
CN
=[00000010]
BCC CC
=[0 0 1 0 0 0 0 0]
0
0
0 0 __k_ _c_ m3 m3 0 1
10m2
0
0
-c m4
0IT
= BCN
The parameter nominal values are choosen as follows ml0 = m20 = m30 = m40 = 1
ko = 4
Co = 0,02
It is first necessary to discretize equ (31), to use the robust design PRCBI : X k + l = Ad Xk + Bd Uk~ Zk = CXk I
(32)
Now suppose that each system is subject to a single parameter variation Aml. This uncertainty causes a variation ~Ac but we have : ABCN = A B c c = 0 For the discrete system, this effect causes a variation on AAd and ABd and we must take it into account in the parameter robust design. To compare the performances of LQG/LTR, PRLQG and PRCBI methods the regulator for the methods is designed with the cost : N Ji=100~
N-1 X kT C T C X k
k= 1
+ Z Ud k=0
with N - - ~
920
The LQG/LTR and PRLQG designs are asymptotic methods. The Kalman filter is selected to produce asymptotic insensitivity to parameter variations. The weighting 132BBTdue to a "fictitious" process noise is used for the LQG/LTR design and 132SST for the parameter robust LQG design with AAc = -STV. For the PRCBI method, we compute the optimal vector D, which appears in the weighting 132DDT, and which minimizes the following criterion : J2 = Trace [Gm~10]
(33)
The parameter 13 is equal to 100. The nominal and perturbed models are discretized. The Kalman filter, according to two discrete models, computes in steady state the nominal values Mo, Ko and the variations AK due to a scalar variation. The solution of the implicit equation (4) to (11) supplies the values Tr[Mo 1 AM] and Tr [ G m : l ] . A non linear programming method computes the optimal D such as: I_
ILl..]
-1 J2M=minTrace [ G mll0] D The following vectors D for the three methods are : Method LQG/LTR
D = BCN D = S = [0 1 0 0 0 0 0 0] T f o r s y s t e m C S [0 I 0 0 0 0 0 0] T f o r s y s t e m N
MethodPRLQG MethodFRCBI
(34)
[ Dopt = [0.31 -0.12 -0.018 0.12 -0.41 0.59 -0.004 Dopt = [1.17 0.70 -0.59 1.22-0.21-1.54 0.02
|
-0.58] -0.31]
for system C forsystemN
As shown in table 1, the LQG/LTR controller for the continuous and discrete models has poor robustness because the precise range of the tolerable Aml is very small. For the system N, a very small variation of ml produces an unstable mode. This stability robustness is increased when the PRLQG design is used. Furthermore as shown in table 1, the PRCBI design gives a very important improvement of the stability range. For this robustness measure, PRCBI method achieves the best results. Method LQG/LTR PRLQG
#
Method
Range m t 0.96 0.50
1.53 1.94
Range m 2
LQG/LTR
0.99
1.06
PRLQO
0.87C
1.253
0.6
4
LQ
0.4
2.1
LQ
LQG/LTR
0.98
1.37
LQG/LTR
0.99
1.04
0.87
1.22
0.35
2.67
PRLQG
0.56
1.93
PRLQG
PRCBI
0.10
2.7
PRCBI
System C
System N
Table 1 Figure 5.a shows the zeros and poles of the plant and the compensator for the LQG/LTR design. In this figure we see that all the plant zeros are cancelled by the compensator poles and this produce a weak robustness to parameter variations. The position of these zeros and poles appears on figure 5.b for the PRLQG design. We see that the f'trst and third zeros are cancelled while the second zero remains intouched. It can be said the parameter robust LQG avoids pole-zero cancellation which can be a source of poor robustness because the second zero moves very much when the parameter ml varies. Figure 5.d shows the
921
root locus for the PRCBI design of system C. There are no pair of quasi-cancellation of plant zeros by compensator poles. The third system zero (letter C) is not cancelled at is moves along the unit circle when ml varies : robustness is achieved by locating the compensation pole-zero inside the unit circle from the plant zero. The ffequential characteristics of this robust compensation are quite like notch f'fltefing. The time response characteristics are given in Figure 6 when the control is applied after 10 seconds of openloop. The sampling period is equal to 0.4 second. For the nominal system C (Figure 6.a) the LQG/LTR, PRLQG and PRCBI methods give good time response though ligh oscillations appear in the control by LQG/LTR and PRLQG methods. For these two techniques these oscillations become worse when the plant is perturbed by a parameter variation. LQG/LTR method leads to instability for ml = 1.5 because the plant zeros are almost cancelled by the compensator poles for the nominal value ml = 1, and the closed loop root goes outside the unit circle when ml varies. Futhermore, under these conditions PRLQG method though less sensitive to parameter variations that LQG/LTR method doesn't give satisfactory robustness : poorly damped oscillations. On the contrary a best performance and stability robustness is achieved by the PRCBI technique and we see the positive effect of the compensation for the stability robustness of system C. We may draw similar conclusions for the robust control of system N. CONCLUSIONS Tiffs paper describes a parameter robust control based on the worst quality of closed loop bayesian estimates. Theoritical results show the links between PRLQG and PRCBI methods; Furthermore a suboptimal technique gives a very simple control design. The new PRCBI techniques leads to a controller design which gives on the classical example of a mass-spring damper system, good performances and stability robustness. These numerical results show that this technique is potentially a practical tool for engineers. REFERENCES [1] J.C Doyle, G. Stein. Robustness with observers.IEEE Trans. Automatic Control, vol.AC24, Aug. 1979 [2] J.C Doyle, G. Stein. Multivariable Feedback Design concepts for a classical/modern synthesis.IEEE. Trans. Automatic, Control, Feb 1981 [3] M. Tahk, J.L Speyer. Modelling of parameter variations and asymptotic LQG synthesis. IEEE Trans Automatic, Control Vol AC. 32, sept 1987 [4] M. Tahk, J.L Speyer. A Parameter robust LQG Design Synthesis with Applications to control of flexible structures. American Control Conference - Minneapolis, 1987 [5] U.Shaked and E.Sikoka.On the stability of the continuous time LQG optimal control. IEEE Trans. Automatic Control, vol AC30, October 1985 [6] L.H.Keel, S.P.Bhattacharyya and J.W.Howze. Robust control with structured perturbations. IEEE Trans. Automatic Control, vol.33, jan. 1988 [7] C.B.Soh, C.S.Berger and K.P.Dabke. On the stability properties of polynomials with perturbed coefficients. IEEE Trans. Automatic Control, vol. AC30, Oct. 1985 [8] R.K Mehra. Optimal Input Signals for Parameter estimation in Dynamic Systems :Survey and New Results. IEEE Trans. Automatic. Control Vol AC-19, Dec 1974 [9] M. Gauvrit, C.Fargeib, P.Rodrigo. Synthesis and performance of bayesian estimators Control and Dynamics Systems, vo1.19, Academic Press, 1983 [10j J.L.Alspac, L.L.Sharf, A.Abiri.A Bayesian solution to the problem of state estimate in an unknown noise environment. InternationalJournal Control., vol. 19, n~ pp 265-287, 1974 [11] P.C.Muller, H.I.Weber.Analysis and optimization of certain qualities of controllability and observability for linear dynamic systems. Automatica, vol.8, pp 237-246, 1972 [12] H.Kwakernaak and R.Sivan. Linear optimal control systems. New York - Wiley, 1972
922
o.
o.
o
:o
,
~l,, .
.
.
.
.
.
.
r"
.......
---.::=
.
i-
,-i
7
I
Y
(.9 CI '-I I rf o,
(-, ,.-1 L~ I CI
(J o
'El
0 r-
T
O ,C
"r
o
.,r o
-o
~D
u u
tO
g
:o. :o
--o. f
L . ~ ~
...........i: o
y
T
i .........
--L~i i
..........
o ....... r
:
bJ~
:
LE
~o
~
o
923
i~
+*
*I*S
i
m , *, ,,+ i.i ~l~+,,, ,l+, ,i~.b, , , i ] ,+,+b++* J.i
i D
, + . , , + + , , i , , , * l , + , + l l l + , + , , , *1.+ +.I l.l
, l+l
' - - r - ' r22. 9
............. sy,to, c M.Ihoe
............
Figure 5.a
U~LT!
Figure 5.b
Sr.t.- r ~.t~d p~LOc;
,-t
i +
io
I . . . . . . .~. *~......... i ~ ................... J . . . . . . . . . 8 .... z .... +l,~ .*.J *,J, . . . . . . . . . ;~. ,-7a=;7,.,.;-~ - - = =
Figure 5.c
/ I.||
""1
-3.1
[
~-~-"~ ~; Loeu* st*t*- c x*Lho+~C|X
Figure 5.d
ROTATION SYSTEMS
SYMMETRIC STRUCTUKE IN CONTROL AND R E L A T E D S T A B I L I T Y A N A L Y S I S *
W e i l i n Yang
S.Y.
Zhang
D e p a r t m e n t of A u t o m a t i c C o n t r o l N o r t h e a s t U n i v e r s i t y of T e c h n o l o g y Shenyang, P . R . o f C h i n a 1 1 0 0 0 6
Abstract The concepts of rotation symmetric structure, s y m m e t r i c c a n o n i c a l form and s y m m e t r i c o r d e r arc d e f i n e d for linear c o n t r o l s y s t e m s . It is shown from the r e s u l t s of this p a p e r that the control systems w i t h r o t a t i o n s y m m e t r y can be t r a n s f o r m e d to the symmetric c a n o n i c a l form u s i n g an a p p r o p r i a t e t r a n s f o r m a t i o n . From this, a new m e t h o d for p r o v i n g the s t a b i l i t y of control systems with the s p e c i a l s t r u c t u r e os p r o p o s e d . The r e l a t e d s t a b i l i a b i ! i t y a n a l y s i s and c o n t r o l s t r u c t u r e are a l s o g i v c n in this paper. Keywords: Rotation symmetric structure, symmetric form, l i n e a r c o n t r o l systems, s t a b i l i t y a n a l y s i s
canonical
I. I n t r o d u c t i o n In the analysis of large-scale control systems severe difficulties arise from the h i g h d i m c n s i o n a l i t y of the system e q u a t i o n s . M e t h o d s for d e a l i n g w i t h such s y s t e m s m u s t take a d v a n t a g e of s t r u c t u r a l p r o p e r t i e s in o r d e r to a c h i e v e c o n c e p t u a l or n u m e r i c a l s i m p l i f i c a t i o n s of the overall p r o b l e m . In a d d i t i o n , for p u r p o s e s of a n a l y z i n g a c o n t r o l system, it is o f t e n u s e f u l to have a d e s c r i p t i o n of the s y s t e m in t e r m s of i n t e r c o n n e c t e d s u b s y s t e m s . O n c e one has such description, in pronciple it is p o s s i b l e to relate the stability, controllability etc., of the overall system to corresponding properties of the various subsystems and their interconnections. In the f i e l d of s t a b i l i t y a n a l y s i s , e s p e c i a l l y the r e s e a r c h on l i n e a r c o n t r o l systems, there are m a n y m e t h o d s and results. The m a i n d i s a d v a n t a g e of the m a j o r i t y of t h e s e s t a b i l i t y m e t h o d s is that t h e y arc not v e r y e f f e c t i v e for the a c t u a l high d i m e n s i o n a l systems, some restricted conditions are needed in many situations. The disadvantage provides the n o t i v a t i o n of a c o m p l e t e l y new way for tcstin~ the s t a b i l i t y of a c o n t r o l system possessing special structure with high dimension. We note that there exist many p r a c t i c a l s y s t e m s p o s s e s s i n g s y m m e t r i c s t r u c t u r e , such as m a n y p o w e r s y s t e m s and s o c i a l systems. If we can find a n e w m e t h o d to analyze the stability of the systems u s i n g the properties of symmetric structure, an e f f e c t i v e way is p r o v i d e d for the s t a b i l i t y a n a l y s i s of some k i n d s of a c t u a l systems. * T h i s w o r k ~as s u p p o r t e d by the N a t i o n a l cf P . R . o f C h i n a
Nature
science
Foundation
925 This p a p e r i n v e s t i g a t e s a p a r t i c u l a r i n s t a n c e in w h i c h a c o n t r o l s y s t e m p o s s e s s e s the r o t a t i o n s y m m e t r i c s t r u c t u r e . R e l a t e d work on system a n a l y s i s about s y m m e t r i c s t r u c t u r e can be found in [1]-[4]. In [I] Grizzle studies various technical eonditiond under which nonlinear control systems with symmetries admit local or global d e c o m p o s i t i o n s in terns of l o w e r d i m e n s i o n a l s u b s y s t e m s and f e e d b a c k loops, the s t r u c t u r e of the i n d i v i d u a l s u b s y s t e m is d e p e n d e n t on the s t r u c t u r e of the s y m m e t r y group. In [2], J . L e w i s s t u d i c s some l i n e a r s y s t e m s t r u c t u r e but more a b s t r a c t problem, i.e., g e n e r a l s t r u c t u r e of s y m m e t r i c systems, he s h o w s that e v e r y m o d u l e of e v e r y finitedimensional associative algebra is a s s o c i a t e d with the state space of some class of s y m m e t r i c s y s t e m s . However, the c o m m o n p r o p e r t y of these results is that they d e f i n e d on the abstract mathematical concepts and it is v e r y d i f f i c u l t to a p p l y to actual situations according to presently tcchniqucs. Borrowing some ideas from [2],[4], we d e f i n e r o t a t i o n s y m m e t r i c s t r u c t u r e of w h i c h the l i n e a r control systems can n e c e s s a r i l y be t r a n s f o r m e d to a symmetric c a n o n i c a l form. O b v i o u s l y the s u b s y s t e m s and their i n t e r c o n n c c t i o n s of s y m m e t r i c c a n o n i c a l form can p r o v i d e us with a more e f f e c t i v e way to s t u d y the ststems. By u s i n g the canonical form, we study the relationship between s t a b i l i t y of c o n t r o l systems and the lincar control structure with symmetry. It will also be shown later in this p a p e r that the r e l a t i o n s h i p can be a p p l i e d to the system s t a b i l i a b i ! i t y p r o b l e m and get c o n t r o l s t r u c t u r e . The o r g a n i z a t i o n of the p a p e r is as follows. S e c t i o n 2 gives the definitions of r e t a t i o n s y m m e t r i c s t r u c t u r e and o t h e r c o n c e p t s u s c d in the paper. S e c t i o n 3 i n v e s t i g a t e s the s t r u c t u r a l p r o p e r t i e s of the s y s t e m s with r o t a t i o n s y m m e t r y . It is shown, through a special transformation, that the p r e s e n c e of a r o t a t i o n symmetry always ~ i v c s a s y m m e t r i c c a n o n i c a l form. S e c t i o n 4 d i s c u s s e s the s t a b i l i t y of linear control systems with symmetric structure. Section 5 investigates the s t a b i l i a b i l i t y problem. Section 6 contains the c o n c l u s i o n s and c o m m e n t s .
2.
Definitions
Consider that a linear following dynamic equations: (2.1] where
control
and c o m m e n t s system
is
described
by
the
X = AX + Bu X C R kn,
A C R knxkn,
BC R knxm
uC R~
B e c a u s e the s t r u c t u r e of r o t a t i o n s y m m e t r y is d e p e n d e n t u p o n structure of r o t a t i o n group. So at the b e g i n n i n g we give r o t a t i o n group.
the the
Definition I: A g r o u p c o m p o s e d of all the nxn real orthogonal m a t r i c e s w i t h p o s i t i v e d e t e r m i n a n t is called r o t a t i o n group. If the d i m e n s i o n of the a l g b r a on this g r o u p is k, then the g r o u p k can bc d e n o t e d by SO(k). Using above d e f i n i t i o n , this paper. Definition
2:
Let
we may
system
introduce
the main
(2.1) be a class
of
concept
linear
of
finite-
926 dimensional systems of fixed d i m e n s i o n , its e l e m e n t s are ordered p a i r s of m a t r i c e s (A,B); if t h e r e e x i s t s a set of all p a i r s m a t r i c e s (G,T), and G C S O ( k ) , T C SO(I), s u c h that for (A,B) we have (2.2)
GA = AG
then the symmetric
,
GB
= BT
s y s t e m has r o t a t i o n symmetric structure. The o r d e r is d e f i n e d to be the d i m e n s i o n of r o t a t i o n
For edemtity
a s p e c i a l situation, m a t r i x ), then the symmetric s t r u c t u r e . concept
The following structure. Definition f o l l o w i n g form
if T ~ I s y s t e m is
reflects
3:
For
system
(2.1),
A1
A2
---
A2
A2
A1
---
A2
A2
--.
A!
A2
A2
---
A2
A1
rotation g r o u p G.
( I is an a p p r o p r i a t e sized said to have absolutely
the m a i n
if
the
property
system
of
symmetric
matrix
A has
the
A --
A1
,A 2
are
rotation
appropriately
symmetric
Considering below.
If
there
canonical
control
Definition
4: For
exists
F: X
the )u,
then the system called s p e c t r u m of the matrix.
3.
Transformation
then
the
system
has
form.
stability which
for
we
should
region: system
C-
give
the
; =[
s: Re(s) < 0 } .
(2.!)
concept
as
satisfies
Cto be
definitions, with rotation
In this paper, we are t r a n s f o r m the s y s t e m (2.1) We system
matrices,
structure,
~ (A + BF) ~
Having above s t u d y the s y s t e m s
sized
stabiiizable.
we m a y be symmetric
theorem
Here
convenient structure.
and
the
is
the
to a n a l y z e
and
stability
~
analysis
i n t e r s e t e d in the n e c e s s a r y c o n d i t i o n s to to the r o t a t i o n s y m m e t r i c c a n o n i c a l form.
begin by c o n s i d e r i n g the t w o - o r d e r can be w r i t t e n in the f o l l o w i n g form
symmetric
system.
The
927
31
IAIA]IxII[BII Xo
where nxm
T
~
A4
X o~
A 3,
A 4 arc
~3
XI,X 2 ( = . n . .AI. . A 2
Bo
nxn
(3.2)
G 9
definition
2,
there
=
"G
A4
~ ~3
Lcmma i: structure,then matrix
For the
,3.3
system (3.1), matrix G must
E:[ ~ the
system
can
will
if be
matrices
=
G
and
"T
B2
help
it h a s similar
us
to
rotation to the
get
the
symmetric following
oIxn] nxn
be
transformed X1
to
+
the
following
form:
u
iii [AIAi{i111
4)
v
A2
A1
Xo
Proof: As w e k n o w , if G a n d exists a matrix, which satisfies (3.5)
P G P -I
(3.6)
PG
Since
= E
similar
to
a rotation
or
F
group,
T
that PGG T
(3.6) PG
matrices,
or
G belong
GG T =
hence
E are
Bo
= EP
matrix
(3.7)
from
of
B2
lemma
nxn
(3.8)
a pair
, G"
A4
The following technical transformation theorem.
#urthermore,
exists
[AIA2]IA!A21 IBI]ill ~3
and
BI,B 2 are
matrices.
According to which satisfy
~~
matrices,
and
=
EPG T
(3.8)
= EP
= E EPG T
= EPG T
then
then
there
g28 since
E'E
= I2nx2 n
,therefore
PG : PG T It
is
matrix,
so
not
or
P(G-G T
difficult
r a n k ( G - G T) < thus, there must satisfies equation From
the GA
obviously,
for
that
(G-G T
is
)
a
skew-symmetric
exist (3.5).
a nonsingular
of
rotation
transformation
symmetric
structure,
P
which
wc
have
= AG P in
l~ ....I 0 ~. . . ~. G ~ D -I that
to sec
2n
definition
(3.9)
) = 0
(3.5}
= FAGP -I
is
~. . 11. ~ . . . P.G P .- I p ~ - I
PAP - I p G F- 1
=
therefore (3.12)
E P A P -I
=
PAp-IE
let
PAP
-I A3
then
(3.12)
A 4
becomes
[AIA21 [Al }E A4
~3
it
is
apparent
complete
the
=
A4
,
using
Lemma
A 2
=
A 3
proof.
The a b o v e l e m m a s h o w s that structure c a n be t r a n s f o r m e d to By
A 4
that A1
which
A 3
i,
we
get
the s y s t e m w i t h t w o - o r d e r symmetric canonical form.
transformation
theorem
as
Theorem I: For s y s t e m (2.1), if it p o s s e s s e s symmetry, a n d the r o t a t i o n s y m m e t r i c o r d e r is k, t h c n transformation by w h i c h the s y s t e m c a n be n e c e s s a r i l y the f o l l o w i n g canonical form:
symmetric
follows. the rotation there exists n transformed to
929
(3 .13)
where
A!
A2
---
A2
A2
A1
...
A2 v +
v =
Proof: a group
on
space
A2
...
Ai
A2
B -I
A2
.-.
A2
A1
B
According G CSO(k). Vn
different
9
U.
nxn ,~
A1, A2 ~
is
B1
B
,
basis
to the d e f i n i t i o n S u p p o s e ~ (G) is
the
/7(G~, w o u l d
of V n
,-
of the
be
rotation symmetry, there representation of g r o u p O
difference
therefore,
if
the
according
state
space
to of
the system
(2.1) is t r a n s f o r m e d , the c o r r e s p o n d i n g group will be changed either. From the properties of SO(k), the elements of SO(k) correspond the r o t a t i o n of c e r t a i n e o r r d i n a t e s s y s t e m ; so t h e r e m u s t e x i s t an a p p r o p r i a t e b a s i s on w h i c h the r e p r e s e n t a t i o n (G) has the f o l l o w i n g form I1 -..
9
Ii- 1 ~(O)
=
{ gi
E
: gi
0
, i=l,...,k
}
Ii+ I I k
where
E is
Now Lemma of
we
I,
we
From k
therefore, The omitted
the
same
consider it
which
t.•
the
is not
as
Lemma
the
I,
IiC
R nxn
tranformation.
difficult
to
get
is
the
identity
From
the
similar
matrix. procedure
to
P l ( t I ..... t k) ..... P k ( t l ..... t k)
have P l ( t l ..... t k ) G i P ! - l ( t l
..... tk)
= gl
P k ( t ! ..... t k ) G k P k - l ( t l
..... tk)
= gk
above analysis, which satisfy
there
P1
= P19
the
transformation
following here.
part
must
exists
an
appropriate
set
of
, i = ! ..... k
can
of p r o o f
is
be
available.
similar
Having the t r a n s f o r m a t i o n theorem, f o l l o w i n g k e y r e s u l t of t h i s p a p e r .
we
to
that
are
of L e m m ~ 1 , VTZA
in p o s i t i o n
it
is
to p r o v e
930 If the s y s t e m ( 2 . 1 ) has rotation symmetric structure, T h e o r e m 2: stable if the and the symmctric o r d e r is k, t h e t h e s y s t e m is following two subsystems (3,14)
X1
:
( A1 -
A2
)X
+ ~I
u
Xo. :
( A1 +
A o.
)X
+ ~ o. u
and
(2.15) are
stable.
and (3.16) where
Here
_ B1 = the
Bi
X!
,
X2
K-I
I
~
B 1 - -~
are
the
R n,
A I,
A o~ a r e -Bo
k
.~_ l:z
same
Bi
'
as T h e o r e m
the
(3.17)
Q :
....
|
(3.13)
- }< [ I
v
=
I
...
0 [
A
0
...
A
0
. . .
0
|
KI
o]o
I I }< 0
by a transformation P By means of another
I
yield A
(3.181
-
k-I
" I' "r .... equation
I,
either.
K-I T
~
as T h e o r e m
! k K ~ ' Bi [=l
-
Proof: According to T h e o r e m I, t h e r e is which system (2.1) b e c o m e to s y s t e m (3.13). transformation X = QX with
--~-I
same
.. S
.
X
+
~
u
n S
A
where
o As
: A 1 - A o~
AO
: A~
+ .A o
k
and
-~-Sl y
k '~z i
= l
~-l
!
l
-V ~, Bi + ~ ~ Bk-1 - -fBk I
k ,=.!
{3.18)
shows
the
conclusion
Bi of
the
theorem.
The system (2.1) is a d y n a m i c a l order of kn. However, its stability can be checked by considering the two n-th order subsystems only. This simplication of t h e s t a b i l i t y analysis is gained from the properties of r o t a t i o n symmetric structure. It is also shown that the rotation symmetric order indicates the simplification d e g r e e of s t a b i l i t y a n a l y s i s .
931 4.
Stabiliability
and
control
structure
Now attention will be t u r n e d to the relationship between s t a b i l i a b i l i t y p r o p e r t i e s and the r o t a t i o n s y m m e t r i c structure. For effective introduced.
analysis,
below
definition
is n e c e s s a r y
to
be
Definition 5: The mean of minus trace of system matrix A a c c o r d i n g to the number of s u b s y s t e m s is d e f i n e d to bc the p r i m i t i v e s t a b i l i t y of system (2.1). D e n o t e d by r , i.e. ~n
(4.1)
= - >- aii/k i=l where k is the number of subsystems. Obviously, the primitive s t a b i l i t y can express the p o t e n t i a l s t a b i l i t y of the system. Using
~(A)
above
Theorem
3:
concept,
we can get
Let a system
if the p r i m i t i v e (4.2)
= -Tr(A)/k
stability
be
the
of the
the f o l l o w i n g
theorem.
same as T h e o r e m
following
=.,~ and
~-,, ~. 2,
system
A X : Ao• + B-u
is less than zero, then the p r i m i t i v e s t a b i l i t y of s y s t e m l a r g e r than that of every s u b s y s t e m if the system p o s s e s s e s s y m m e t r i c structure.
(2.!) is rotation
Proof: Because the tr~ce of s y s t e m m a t r i x A is i n v a r i a n t t h r o u g h a nonsingular transformation, therefore Tr(A) = (k-!]Tr(A } + Tr(A ). In ~.Ad l ~-.lOn (~.3)
(k-l)Tr(A s) + Tr(A o)
:
( k - I ) T r ( A I - A 2) + T r ( A I + A 2)
: k T r ( A I) - (k-2)Tr(A2) thus ~(A)
= - T r ( A I) +
(k-2)Tr(A2)/k
if (A 2)
< 0 , and
k .~ 2
then (k-2)Tr (A2) > 0 so Above theorem shows that r o t a t i o n symmetric structure can i n c r e a s e the p o t e n t i a l s t a b i l i t y of a system. It is also s h o w n from the t h e o r e m that only p o s i t i v e i n t e r c o n n e e t i o n of s u b s y s t e m s yields the increase of p o t e n t i a l s t a b i l i t y .
932 Next
we give
Theorem subsystem
4:
(4.4)
v =
the s t a b i l i a b i l i t y Under
analysis
the same h y p o t h e s i s
of the
system.
as in T h e o r e m
3,
if
the
A1 x + B2 u
are stable, and the primitive s t a b i l i t y of system (4.2) is less than zero, then the system (2.1) is s t a b i l i a b l e through a local control s t r u c t u r e of d i m e n s i o n n. Furthermore, the local control s t r u c t u r e F should s a t i s f i e s (4.5) where
(AI+A2+Bk F) C
Bk
i
=
C-
k
~ I B.:
k
Proof: hypotheses
Erom the proving p r o c e d u r e of T h e o r e m of this theorem are satisfied, then the k-I
(4.6)
Xi :
(AI-A~
""
are stable. Therefore, following subsystem "'k
, i=l, ~ .... k-I
from T h e o r e m
___ (~i+~2)'" z ~ v
v
(4.7)
+ ~i u
3, if the subsystems
2,
we
merely
consider
the
-u ' BkU
so that, if there exists (4.5), then thc systcm (2.1)
a control s t r u c t u r e can 5c stabiliable.
F
which
satisfies ~Z~
Kemark: If thc system (2.1) is a b s o l u t e l y symmetry, that is T _=_ I , then a c c o r d i n g to the t r a n s f o r m a t i o n theorem, the system becomes I A Ao
n2
9. .
A1
...
A2 A2
B! 1 BI
~176 (4.8)
where
B1 C
Ao
9 -- A 1
A2
B I J "u
~2
A2
A1
B~
{"
K nxm
By using
0
(4.9)
v +
X =
i =
: 0 0
A0s
...
... ...
Theorem
2 we obtain
00 1 a~
A
0
v
+
I ~U
S
0
A 0
if the c o n d i t i o n s Therefore, thus the s y s t e m is u n c o n t r o l l a b l e . T h e o r e m 4 are not satisfied, the s y s t e m could'nt be stable.
in
It is not d i f f i c u l t to see from T h e o r e m 4 and Remark that the control structure is relevant to the group T. the transformation involved in T h e o r e m I; it is also r e l e v a n t to the c o n t r o l l a b i l i t y of
933 the system. The f o r t h c o m i n g paper~
detailed
5.
discussion
Conclusiona
will
be
described
in
and c o m m c n t s
In practical s i t u a t i o n s wc o f t e n e n c o u n t e r some systems with special structure, such as symmetry, similarity, etc. Using structure properties of c o n t r o l systems, we could find some more e f f e c t i v e ways to deal w i t h the systems. In this paper, a new m e t h o d for p r o v i n g the s t a b i l i t y of l i n e a r c o n t r o l systems has been d e r i v e d which takes a d v a n t a g e of the properties of rotation symmetric s t r u c t u r e . The r e l a t e d s t a b i l i a b i l i t y a n a l y s i s is also g i v e n in this paper. It is of i n t e r e s t to i n v e s t i g a t c other s t r u c t u r e p r o p e r t i e s in c o n t r o l systems, such as the s i m i l a r structure. In a d d i t i o n , some other d i r e c t i o n s arc p r e s e n t l y b e i n g p u r s u e d in o r d e r to increase the applicability of the s y s t e m s possessing rotation symmetric s t r u c t u r e . T h e s e r e s u l t s will be r e p o r t e d at a late date.
REFERENCES
[i] Jessy W . G r i z z l e and S . I . M a r c u s "The S t r u c t u r e of Nonlinear C o n t r o l Systems P o s s e s s i n g S y m m e t r i e s " , IEEE Trans. on Auto. Control, ,VoI.AC-30, M a r c h , 1 9 8 5 , p p 2 4 8 - 2 5 8 L~Jr~ J.Lewis, R.McCas!and S t r u c t u r e of the State Space", pp 59-64
and C.Martin Int.J.control,
"Symmetric Systems: Vol. 43, No.l, 1986,
[3] Jan Lunzc " S t a b i l i t y A n a l y s i s of L a r g e - S c a l e S y s t e m s of S t r o n g l y C o u p l e d Similar S u b s y s t e m s " , Proc. 10th W o r l d on A u t o m a t i c Control, IFAC, M u n i c h , 1987, p p 7 - 1 2 [4] W.L. Yang "Rotation R e l a t e d C o n t r o l Structure", and Control, Austin, 1988
S y m m e t r y in N o n l i n e a r C o n t r o l S y s t e m and Proc. 27th IEEE C o n f e r e n c e on D e c i s i o n
[5] W.Murray Wonham "Linear Multivariable A p p r o a c h " , S p r i n g e r - V e r l a g , 1979 [6] Jr.W. Miller A c a d e m i c Press, 1979
Composed Congress
"Symmetry
Groups
and
Control:
their
A
Geometric
~ppllcatlons~ " " ",
[7] Jan c . W i l l a m s " S y m m e t r i e s in D y n a m i c a l Systems", Proc. 25th C o n f e r e n c e o n D e c i s i o n and C o n t r o l , ~t.hen~, Jre~c,., l ~ 6 , p p 5,~0 5=~
A I I Y D I ~ O D Y N A M I C L I M I T FOR, A L A T T I C E CAI~ICATUI~E O F D Y N A M I C R O U T I N C IN C I I ~ C U I T S W I T C H E D N E T W O I ~ K S
V. A n a n t h a r a m * School of E l e c t r i c a l Engineering Cornell University, I t h a c a , NY 14853
ABSTR, A C T
In s i m u l a t i o n s circuit switched n e t w o r k s with d y n a m i c a l t e r n a t e r o u t i n g exhibit hysteresis p h e n o m e n a , which suggest t h a t u n d e r d y n a m i c r o u t i n g there m a y be more than one s t a b l e regime of o p e r a t i o n for the same offered traffic. This p o s s i b i l i t y also shows up in some a n a l y t i c a l models of d y n a m i c routing : one can write O D E limits for network o c c u p a n c y probabilities which a d m i t multiple equilibrium p o i n t s for ccrta.in ranges of p a r a m e t e r s . These ODE limits a v e r a g e out the s p a t i a l variation of the network. We a t t e m p t to preserve the s p a t i a l c h a r a c t e r i s t i c s by considering a l a t t i c e m o d e l of d y n a m i c routing. We derive a h y d r o d y n a m i c equation for this lattice m o d e l . This is an integro-differential equation which describes how the s p a t i a l o c c u p a n c y profile of the n e t w o r k evolves over time, and it a d m i t s m u l t i p l e s p a t i a l l y homogeneous equilibrium solutions for certain ranges of the p a r a m e t e r s . These solutions may be loosely t h o u g h t of as the different o p e r a t i n g regimes. Using this equation one can s t u d y questions like "for w h a t p a r a m e t e r values is a hot s p o t of h e a v y loading in the system likely to take over the whole network by knock-on effects ?"
* l~esearch s u p p o r t e d by N S F u n d e r NCI~ 8710840 and a P Y I award, by I B M u n d e r a F a c u l t y D e v e l o p m e n t A w a r d , and by BellCore Inc.
938 1. INTI~ODUCTION
Dynamic routing schemes in networks adaptively adjust traffic patterns in response to demand, so as to make better use of spare capacity, and to provide robustness to failures or overloads. Such schemes have been the topic of considerable recent interest, primarily because it has only recently been possible to implement them in practice, and because they offer improved performance over the traditional hierarchical routing schemes. A difficulty associated with dynamic routing schemes is the potential for metastable states. Several simulation based studies of such routing schemes have revealed the existence of hysteresis phenomena, suggesting that the network may have several qualitatively different regimes of operation for the same offered traffic, spending long periods of time in one or the other regime and sometimes moving from one to the other in response to fluctuations in the demand.
Intuitively a situation where most calls are
using alternate routes is likely to persist for a while because arriving calls will then find the network close to saturation and will be unable to make their direct connections. On the other hand, for the same offered traffic, it might also be the case that if most of the calls ia progress are using their direct route, arriving calls will be able to make their direct connection. Important performance characteristics of the network such as blocking probabilities typically differ considerably between regimes. All the same, the improvement in performance over hierarchical routing schemes is such that dynamic routing schemes are being implemented in real world networks, with control schemes such as trunk reservation for directly routed traffic. These, if suitably chosen, mitigate the effects of the potential multiplicity of operating regimes. The possibility of metastable regimes of operation is also predicted ia analytical models for dynamic routing such as the ones studied by Kelly, [7], Krupp, [8], Marbukh, [10], [11], and Gibbens, Hunt and Kelly, [6]. In [7] and [8], simple fixed point approximations for the blocking probability are written, and it is found they have multiple
939
solutions for certain ranges of the p a r a m e t e r s . T h e models in [6] and [10], [11], are more detailed. O D E limits are found for the fraction of n e t w o r k links t h a t are in a given s t a t e as the network size becomes large. In Section 2 we briefly discuss the m o d e l of G i b b e n s , H u n t and Kelly, which p r i m a r i l y m o t i v a t e d this work. In this p a p e r we r e p o r t on an a t t e m p t to u n d e r s t a n d the i n t e r a c t i o n between the o p e r a t i n g regimes using particle system techniques, [9], [12]. To describe the d y n a m i c exchange between different o p e r a t i n g regimes, one needs simple equations t h a t describe how the s p a t i a l l y d i s t r i b u t e d network s t a t e evolves over time.
M o t i v a t e d by this we
consider a l a t t i c e m o d e l in Section 3, which is analogous to the m o d e l of G i b b e n s , Hunt and Kelly, [6]. We find a h y d r o d y n a m i c e q u a t i o n for this lattice model.
This is an
i n t e g r o d i f f e r e n t i a l equation describing the time evolution of the s p a t i a l l y d i s t r i b u t e d network o c c u p a n c y profile. This e q u a t i o n also a d m i t s m u l t i p l e s p a t i a l l y h o m o g e n e o u s t i m e - i n v a r i a n t solutions for certain ranges of the p a r a m e t e r s , which may be loosely t h o u g h t of as the different o p e r a t i n g regimes. T h e main results are s t a t e d as T h e o r e m 1 and T h e o r e m 2 in Section 3. T h e key i d e a of the p r o o f is sketched in Section 4. Full details are available in [1].
2. ODE
LIMITS
In this section we review the O D E limit of G i b b e n s , H u n t and Kelly, [6], in order to m o t i v a t e the investigation in the following section. G i b b e n s , H u n t and Kelly, [6] consider a simplified m o d e l for d y n a m i c a l t e r n a t c r o u t i n g which bypasses the spatial features of the network.
Consider a collection of
N links, each of which consists of C circuits. A t each link, calls arrive according to a Poisson process of r a t e v. If its link is not s a t u r a t e d , the call occupies one circuit on the link. If its link is s a t u r a t e d , the call chooses two distinct links at r a n d o m from the remaining N-1
links, and if neither is s a t u r a t e d , the call occupies one circuit from each
of these two links. Otherwise the call is blocked and rejected from the s y s t e m . Each
940 o c c u p i e d circuit is held for an i n d e p e n d e n t e x p o n e n t i a l time of mean 1. (Note t h a t when a call occupies two circuits after m a k i n g a successful choice of a l t e r n a t e route, it is a s s u m e d t h a t these circuits are released i n d e p e n d e n t l y ) . Let 7 ~ ( t ) , 0 <_ k <_ C, denote the fraction of the N links t h a t have k occupied circuits at time t. d i m e n s i o n a l simplex.
(70~ , 7 ~ , . .
9, 7 Cg ) evolves as a Markov process on the C-
In [6], an O D E limit is found for the evolution as N --+ co.
Namely, if the initial conditions ( % ~ ( 0 ) , 7 ~ ( 0 ) , . . . , 7 c :r ( 0) )
converge weakly to a limit
( % ( 0 ) , 7 1 ( 0 ) , . . . , 7 c ( 0 ) ) , then the process converges to the d e t e r m i n i s t i c process given by the e q u a t i o n s ~/0 = 7, - (v + 2V7c (1 - - 7 c ) ) 7 0 ,
-(k+
~c = - c ' i c
t,+ 2~7c(i--7c))7k ,0 < k < C ,
+ (~ + 2~,-rc(1 - ~ o ) ) ~ c - ~
,
(2.z)
with the a p p r o p r i a t e initial conditions. W h e n one looks for equilibrium p o i n t s of eqns. (2.1), one finds the following
The
equilibrium p o i n t s are given by the solutions of 7; = ( , + 2~,7~(1 - 7 c ) ) ~ C ! 7 ; (, + 2,~(1 ~b))c~! ' "rc = E ( ~ + 2 ~ ( 1
- ~c),C) 9
(2.2)
W h e n C is large enough, one finds t h a t there is a range of ~, in which eqns. (2.2) a d m i t three solutions, two of which are stable. To the left and right of this range there is a unique s t a b l e solution. Notice t h a t the s p a t i a l characteristics of the network are averaged out in w r i t i n g the a b o v e differential equations.
941 3. L A T T I C E CAP~ICATUI~E
We now introduce a lattice caricature for d y n a m i c a l t e r n a t e r o u t i n g which has the virtue of preserving spatial features of the system. We will write a h y d r o d y n a m i c limit for this lattice model. This is an integrodifferential equation which describes how the spatially distributed network state evolves over time, see eqn. (3.1). T h e main results are T h e o r e m 1 and Theorem 2 below. Let Z ~ / M denote the lattice in R. d consisiting of points all of whose co-ordinates T h e points of Z d / M are called sites. Let
are r a t i o n a l with d e n o m i n a t o r dividing M . 1u denote { 0 , 1 , . . . , C } .
We consider a Markov process (77~',t _> 0) on W zd/M which
caricatures a circuit switched network with d y n a m i c r o u t i n g (the s t a t e m e n t s below are true for any d, but the situations d = 1, and d = 2 are likely to be of most interest). We use ~/ to denote a generic element of W zdlM, and call r/(z) the value at site x. Let M* denote ((2M+l)~-l) 2
The Markov process is described by the t r a n s i t i o n s
~(.)
, .(~)
-
1
at r a t e ~ ( ~ ) ,
T/(z) -----+r/(x)+ 1 at rate u if ~(x) # C ,
(~(z), ~(y), ,7(z))
, (~(x), q(y) + 1, ~(z) + 1) at rate . / M " if z , y , z are distinct sites with
~7(x) = C,~(y) < C,~(z) < C and y , z E x § [ - 1 , 1 ] ~ . There is no difficulty constructing such a Markov process even though the n u m b e r of sites is infinite. See Liggett, [9], C h a p t e r 1, Section 3, for details; T h e o r e m 3.9 of that section applies directly. We t h i n k of each site in the 'lattice as representing a link in our network, which consists of C circuits. We think of the value at a site as giving the n u m b e r of occupied circuits in the corresponding link. Occupied circuits become free at rate 1. At each link there is a Poisson process of calls with rate v. Each call occupies one circuit on its link if available; if the link is saturated the call r a n d o m l y picks two other links which are
942 in its [--1, 1]d n e i g h b o u r h o o d , and uses one circuit from each of these links if possible. Otherwise the call is blocked and rejected from the system. Note that because we have a compressed lattice, the i n t e r a c t i o n actually has range M on the scale of links. For x E Z a / M , let u M ( t , x , k ) denote P ( q M ( x ) = k), 0 <_ k <_ C.
We extend
the definition of u M ( t , ' , k ) to 1Ed by setting u M ( t , x , k ) = UM(t,[X]M,k) for x E l t a , where [X]M denotes the m i n i m u m e l e m e n t in Z d / M which dominates z in the usual partial order on I t d . Let u(O,x,k),O <_ k <_ C, be continuous functions with b o u n d e d derivative and with F_.C=ou(O,x,k) = 1. Let u ( t , x , k ) , 0 < k < C denote the solution of the integrodifferential equations
O,,(t,z,o) at
=~(t,~,l) 21-2tiff
-u(t+
q,r E [-1,1] d
u(t,x + q , C ) ( 1 - u ( t , x
+ q+ r,C))dqdr)u(t,x,O) :
Ou(t,x,k) - (k + i ) K t , = , k + 1)
Ot
+v(1 +
-(k+di+
2~ - ~ f f q , r E [--1,1] d
21- 2df f q , r E [--1,1] d
u ( t , x -F q,C)(l - u ( t , x
u(t,x + q , C ) ( 1 - u ( t , x +
Jr q-F r , C ) ) d q d r ) u ( t , x , k
-- I)
q+ r , C ) ) d q d r ) u ( t , x , k ) ,
forO
OKt, x,C) Ot
Cu(t,=,C) 2i-~, f f
+ v ( l + q, r E [ - 1 , 1] d u(t,x + q, C)(1 - u ( t , x + q + r,C))dqdr)u(t, x, C - 1) .
(3.1) T h e n we have the following : Theorem
1 : Fix T < oo. Suppose t h a t we start (7]M)t with initial configuration the
p r o d u c t measure having P(rIM(x) = k) = UM(O,x,k), 0 <_ k <_ C. If UM(O,x,k) ---+ u(0, x, k) u n f o r m l y on c o m p a c t sets, 0 _< k <_ C, then UM (t,x, k) ---+ u ( t , z , k) for all
t E [ O , T ] , x E l t d and 0 < k < C .
943 Theorem 1 is a statement about pointwise convergence of probabilities. There is a corresponding functional limit theorem. This functional limit theorem allows us to describe the limit behaviour of an arbitrary choice of spatial integrals r times t l , . . . , t ,
E [0,T] as long as the r
r
at
decrease sufficiently rapidly. This allows, for
example, to describe the evolution in time of spatial averages of the state over compact regions of the lattice (which caricatures compact regions of our network with dynamic routing). Let S(tL d) denote the Schwarz space of rapidly decreasing functions on 1Kd and S ' ( R d) the space of Schwarz distributions, which is its topological dual. We recall that ,6"(1:1.d) consists of precisely those infinitely differentiable functions f E C ~ (IKd) such that, with
II/l]~,~ = sup IzaD~/(z) t ~Ert ~ 0~ 1+...+~,l
where a = ( a , , . . . , a n ) , fl = ( f l , , . . . ,flu), x" = [ I ; = 1 x ; ~ and D P f - o~-~..... o'"'~'~, f ' we
have each Ilfll~,p < oo fox- all ~, 9. Further, S(R. d) is a locally convex topological linear space with topology given by the family of seminorms Ilfll~,p. Then we have :" Theorem
2 : Given Ck E S(R.d), 0 < k < C, let C
X M (r
=
1 M~
~
~ r
xEZ'I/M
View Z M as an element
(x)= k).
k=O
of O([O,T],(S'(R~))c+~).
Then Z M =:~ Z., where =~ denotes
weak convergence in D ([0, T], (S* (R. d ) ) c + ' ) , and C
x(r
P
= ~ J k=0
r
~.
~EIK~
When we look for spatially homogeneous solutions of eqns. (3.1) which are time invariant, we are led to the same equations (2.2) found by Gibbens, H u n t and Kelly. Thus we see t h a t for large enough C, there is a range of u over which eqns. (3.1) admit three spatially homogeneous solutions.
These may be loosely thought of as different
phases associated to the uetwork. The exchange between the phases can be studied by numerically integrating the eqns. from the appropriate initial conditions. Such work is
944 c u r r e n t l y in progress, [4]. Similar equations can also be w r i t t e n for more complicated d y n a m i c r o u t i n g schemes, including t r u n k reservation, using the techniques of the next section. The s t u d y of these equations may also be useful in m a k i n g comparisons between schemes.
4. S K E T C H O F T H E P ~ O O F S
T h e m a i n idea is to consider a b r a n c h i n g tree r u n n i n g backwards in time, which is in some sense a dual to the forward time process. We start from z E R d at time t, with a single "particle" alive at z. At time t - s (reversed time s), there is a certain set 7), of "particles" which are alive. A particle alive at reversed time s stays alive oll reversed time [s,t]. Each live particle p is at a p o i n t x(p) E R, a and does not move. F u r t h e r , at each reversed time s, we are given a m a p
F, :W_..+ 2 w'" If C E W I'" is such that C E F,(l), this m e a n s t h a t if the p o i n t x(p) has va|ue C(p) for each p E "P,, then tracing the forward time process from time t - s to t, the resulting value at x is I. Each particle p alive at reversed time s has associated with it several i n d e p e n d e n t Poisson processes. There are C Poisson processes D(p,k),l ~_ k ~ C of rate 1 : at a time of D(p,k), F,(l) is reevaluated for each l C W by t h i n k i n g of this p o i n t as requiring a call to leave in forward time if the n u m b e r of calls in progress is at least k. T h e r e is a Poisson process A(p) of rate ~, : at the times of A(p) F,(l) is reevaluated for each l E W by t h i n k i n g of such a point as an exogenous arrival in forward time. (This process does not p a r t i c i p a t e in a l t e r n a t e routing). Finlly, there is a Poisson process Q(p) of rate 2~, : at the times of Q(p) p generates z(1),z(2) E [ - 1 , 1] d i n d e p e n d e n t l y and uniformly and places one new particle at z(p) + z(1) and one new particle at x(p) -b z(1) q- z(2).
F,(l) is reevaluated for each l e W by t h i n k i n g o["
945
z(p) + z(1) as being a link which is p o t e n t i a l l y s a t u r a t e d and is p e r f o r m i n g a l t e r n a t e
routing to x(p) and x(p) + z(1) + z(2). F r o m the b r a n c h i n g tree on R. ~, we c o n s t r u c t a b r a n c h i n g tree on Z a / M by moving particles to the nearest lattice p o i n t t h a t d o m i n a t e s their assigned p o i n t . Now, more t h a n one p a r t i c l e may be alive at a site.
W e need to show t h a t this h a p p e n s with
vanishingly smaU p r o b a b i l i t y as M ---+ co. We also need to show t h a t if we c o n s t r u c t b r a n c h i n g trees s t a r t i n g from two d i s t i n c t sites x, y E Z d / M , the p r o b a b i l i t y t h a t they i n t e r s e c t , (i.e. t h a t some site is occupied by particles from each process) has vanishingly small p r o b a b i l i t y as M ---* oo. This ensures t h a t the joint p r o b a b i l i t y terms t h a t show up in the g e n e r a t o r equations for the finite M process a s y m p t o t i c a l l y become p r o d u c t s and give the desired h y d r o d y n a m i c e q u a t i o n s for the evolution of the o c c c u p a n c y profile.
ACKNOWLEDGEMENTS
I am greatly i n d e b t e d to l~ick D u r r e t t for his considerable i n t e r e s t in this work, and for guiding me to the relevant p a r t i c l e systems l i t e r a t u r e .
I would also like to
t h a n k l~oland D o b r u s h i n , G a s t o n G i r o u x , L a w r e n c e Gray, Vadim M a l y s h e v , l l o b e r t o S c h o n m a n n , and Glen Swindle for useful c o m m e n t s .
REFERENCES
[1] V. A n a n t h a r a m , "A H y d r o d y n a m i c Limit for a L a t t i c e C a r i c a t u r e of D y n a m i c R.outing in Circuit Switched N e t w o r k s " , P r e p r i n t , 1989. [2] J. M. A k i n p e l u , " T h e Overload P e r f o r m a n c e of Engineered Networks with Nonhierarchical and Hierarchical R o u t i n g " , A T 8~ T Bell Laboratories Technical Journal, Vol. 63, No. 7, pp. 1261-1281, 1984. [3] V. A n a n t h a r a m ,
" M e t a s t a b i l i t y and P h a s e T r a n s i t i o n s A s s o c i a t e d to D y n a m i c
l~outing ill Networks", Proceedings of the 28th IEEE CDC, T a m p a , F l o r i d a , 1989.
946
[4] M. Benchekroua, PhD. Dissertation, in preparation, Cornell University, 1990. [5] I'L. J. Gibbens, Some Aspects of Dynamic Routing in Circuit-Switched Telecommu-
nication Networks, l~ayleigh Prize Essay, University of Cambridge, 1986. [6] IL. J. Gibbens, P. J. Hunt and F. Kelly, "Bistability in Communication Networks",
Preprint, 1988. [7]" F. Kelly, "Blocking Probabilities in Large Circuit Switched Networks", Advances
in Applied Probability, Vol. 18, pp. 473 -505, 1986. [8] 1L. S. Krupp, "Stabilization of Alternate Routing Networks", International Com-
munications Conference, 31.2.1 -31.2.5, Philadelphia, 1982. [9] To Liggett, Interacting Particle Systems, Grund.
der Math.
Wiss., Vol.
276,
Springer Verlag, 1985. [10] V. V. Marbukh, "Asymptotic Investigation of a Complete Communications Network with a large number of points and bypass routes", Problems of InformatioT~
Transmission, pp. 212 -216, 1982. [11] V. V. Marbukh, "Fully Connected Message-Switching Network with a Large number of Nodes, Bypass routes and a limited number of Waiting places at the nodes",
Problems of Information Transmission, pp. 154 -161, 1985. [12] G. Swindle, "A Hydrodynamic Limit for the Contact process with Large range",
Ph.D. Dissertation, Center for Applied Mathematics, Cornell University, Ithaca, NY, 1988.
A n algebraic m e t h o d for optimizing resources in timed event graphs Stdphane Gaubert INt~IA, Domaine de VoIuceau-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France.
Abstract A min-linear system theory has been developed for timed event graphs using dioid algebraic structures. This in particular allows modeling and evaluating performance of flexible workshops and distributed processing systems. In this paper, we use this theory to study the problem of resource optimization, i.e., minimizing the cost of some resources (machines, pallets, processors) in order to achieve a given rate of production, or optimizing the rate of production itself. This problem reduces to an integer linear programming problem. We use rational symbolic computation iu the dioid algebra to simplify and generate the corresponding simplex.
1
Introduction
By Discrete Event Systems, one usually means systems involving logical or synchronization constraints, which differ in nature from the differential or recurrent systems we are used to working with. Discrete event models appear as the natural framework for dealing with manufacturing workshop or parallel processing problems. Among the various questions that arise in these fields, we shall consider two intimately related topics: performance evaluation and optimization This paper follows the conceptual and theoretical work that has been done by Cohen et al. [1],[2]. They have shown that a restricted class of Petri nets, namely timed event graphs, can be described with linear equations in the so-called MinMaz: or dioid algebra. A fundamental result on timed event graphs analysis (see Cohen et aI. [1], and Chretienne [3]) m a y be stated as follows: (i) there exists a periodic behaviour over a period 0 that the systems follows after a transient. The average throughput of the system is defined by: ~=
quantity produced from date t to date t + 8
At is constant after the transient. ultimate value )~.
We call periodic throughput of the system the
(ii) for each circuit c of the timed event graph, we define the weight w(c), as the total number of tokens of the initial marking in all the places along the circuit. The length of c, l(c) is the sum of the holding times of the same places. Then, A is equal to the least average weight of all the circuits of the graph, namely:
958
,k = min 9w(c) c
(1)
l(c)
We now assume that the quantities of some of the resources of the system, R1, . . . , Rk (machines, buffers, etc.) are unknown. Let qi be the quantity of resource Ri. The periodic throughput ,k obviously depends on q l , - . - , q~. The form of the function ,\(q) is given by the structure of the system. Our goal is to realize a given throughput ~. Knowing the price pi of resource Ri, the total cost of resources is given by: J(q) = Plql + 9 .. + Pkqk Then, our goal is to solve: min_S(q) (2) ~(q)>a From equation (1), we can split ,k(q) > ~ into a set of constraints: for all circuit c,
~(~) Z(c)
> -
The unknown quantities ql, 9 9 qk are clearly associated with the initial marking of the graph. It means that the weight of the circuit is an affine function of q: w(c) = a ~ q l + . . . + a~.qk + tic, where a~ = 1 if r e s o u r c e R i belongs to circuit e and 0 otherwise, tic denotes the total of other initial markings along circuit c (those that are not associated with resources). Therefore, problem (2) may be rewritten as an integer linear programming problem: min (Plql + . . . + P~qk) for all circuit c, a~q~ + . . . + a~qk + tic >_ l(c)-A
subject to the constraints (3)
A priori, the set of constraints is indexed by all the circuits of the graph. This leads
to practically intractable complexity. Hillion and Proth (see [7], [8]), who had already recognized the possibility of writing the resource optimization problem (2) as an integer linear programming problem, had proposed heuristics to solve it. Here, we work in a different direction. It is clear that all the constraints are not independent. We introduce an algebra of rational functions over a dioid that allows computing and simplifying the set of constraints. So, we get a simplex of smaller size. 2
Example
Consider the event graph of Figure (1,(a)) (cf. next section for more explanations about timed event graphs). It may be a simple model for a production line with two tasks (associated with circuits C1 and 6"2), with input arc u and output y. We assume there is no input constraint (an infinity of tokens are available from the input arc u). What is the periodic throughput? Here, it is not too difficult to see that we only have to consider the three circuits labeled Cl = (zx, x~), C2 = (x2, x~), and C3 = (xl, x~, z2, x~). A direct application of (1) gives: 9 , w(c1) w(c2) w(c3)) = min(2, 1 1 1 Some cycles of the graph have a natural interpretation as resource cycles. Because the total number of tokens in a cycle remains constant, tokens of the initial marking may be interpreted as "resource tokens" which indicate how many tasks may be performed
959
(a): initial event graph C1
c,2
c3 (b): event graph with unknown resources C1
u
uz 2
ux I
9~
y
c3
(c): unconstrained timed event graph ~-~~ ~,~ [~ ~
initial marking [ : of the place (tokens)
c4
~q
u
t C2
I
~*~
] holding time 11: of the place (in time units)
J ~
~f'~
~
Figure 1: Optimizin/~ resources in a simple timed event graph in parallel in the same cycle. For instance, cycle C1 corresponds to a job that may be performed by two units of production in parallel (two Lokens in the initial marking), and taking two units of time. Storage resources may also be modeled with cycles: let us denote by z21z~ the place between transitions x~ and z2. Then z2}z~ is a storage place between tasks (1) and (2). Let us consider the circuit 6'4 = (zl, z~,a:2) in Figure (1 , (b)). The number of tokens in place x21x[ is at most ql. ql has the meaning of storage size of z~Ix ~. In Figure (1, (b)), two kind of resources are unknown, ql, and q2 (i.e. the initial marking of place zl Ix2 is ql and the initial marking of place z~ lY is q2). Let A(ql, q~) be the periodic throughput. By putting an infinite amount of resources available (ql = q2 = +co), we achieve the "best" throughput, A(+c~). This corresponds to deleting the resource arcs. We thus get the unconstrained graph of Figure (1,(c)). We have A(+e~) = rain(w(6'1) w(c2)) = 1 z(cl) ' l(c2)" 2 What is the minimal amount of resources necessary to achieve A(+c~)? By enumerating the elementary cycles in figure (1, (b)), we get:
2 1 q2 ql
A(ql,q2)=min( 2' 2' 4 '
7)
which provides the solution qt = 1 and q2 -~-- 2. The purpose of this paper is to describe an algebraic way for doing this in the general case.
960
before
after
(a): firing a transition
(b): not allowed in event graphs
Figure 2: Petri nets and timed event graphs 3
Event
graph
modeling
We refer the reader to Chretienne [3] for an extensive study of timed Petri net. We just recall here that Petri nets are bi-partite graphs, with two kinds of nodes, places (denoted by circles) and transitions (rectangles). Tokens circulate along the arcs according to the following rule: transition i is firable iff there is at least a given number of tokens g - (i, P) in each upstream place P. When the transition is fired, x-(i, P ) tokens are consumed in each upstream place P and ~+(P~, i) are produced in each downstream place P~. We simply write the numbers x-(i, P) and x+(P', i) along the arcs (see Figure 2, (a)). Event graphs are a subclass of petri nets in which (i): there is a single transition upstream, and a single transition downstream every place, and (it): firing a transition consumes exactly one token in each upstream place and produces one token in each downstream place (i.e. x - ( i , P ) = g + ( P ' , i ) = 1). In Figure 2,(b), situations such as (1) (confluent place), (2) (or place), and (3) (multiplying tokens) are forbidden. These restrictions in particular imply that the behaviour of the graph is determined by the initial marking. For instance, allowing the or place ( case (2)) would mean "undeterminism", since the left downstream and the right downstream transition might be simultaneously firable. In limed event graphs, we associate with every place a holding time r(P). Tokens must stay at least for r(P) in the place before being available for firing downstream transitions. We now go back to the first example of Figure (1,(a)). The counting variable associated with transition zl, denoted as the map t ~ x~(t), is defined by: z~(t) =
number of the last firing of transition xx up to time t
When no confusion is possible, we denote by ilj the (unique) place from transition j to transition i. We have r(z~lzl) = 2 and r(z21z'l) = 0. The arc from z~ to z2 induces the following constraint: z~(t) < z~c(t) + r(x2lx~) = z~c(t) + O. Considering the others arcs, we get similar inequalities. Thus: z~(t) < min(x~C(t) + 2, yC(t) + 1, uC(t))
(4)
i.e., counting variables abide by "rain, + " inequalities. Let us make the following fundamental change of notaLions: min is denoted by @ (e.g. min(3, 2) = 3 9 2 = 2), and + is
961
denoted by | (0 | 2 = 0 + 2 = 2). As usual, we sometimes omit | of a | b. Then, (4) is equivalent to
and write ab instead
x~(t) <_ 2xlC(t) @ lu~(t) | ue(t)
(5)
We have similar inequalities for the others transitions. If every transition is fired as soon as possible, the counting variables x~ must be maximal. We shall now study the maximM solutions of systems of linear inequalities in the (@, | algebra.
4
S o m e e l e m e n t s of dioid t h e o r y
An extensive account of this algebraic structure can be found in Cohen et al. Gondran& Minoux [6], and Cuninghame-Green [4].
[2],
D e f i n i t i o n 1 We call dioid a set 79 together with two laws, 9 and | such that (79,@) is an idempotent commutative monoid with neutral element e, (D \ {e},| is a monoid with neutral e, product is distributive over addition, and e is absorbing for the product, i.e.: (aeb)(~e=ae(bec),
a~e=a,
(,~Ob) O ~ = ~ e ( b e r
~Ve=~e~,=e
a(~b=bea,
aea=a
For instance, (• U {+co}, min, +) is a dioid. ~ = + c o is the neutral element for min, and is absorbing for ordinary addition ( + c o + x = +co). e = 0 is neutral for +. D e f i n i t i o n 2 The canonical ordered structure associated with 79 is defined by: x ~ y iff x~y=y. x @ y is the -~-upper bound of z and y. In the case of (R u {-+-oo}, min, +), ~ is the dual order of the usual order g . 2 ~ 3 -- 2 ::~ 3 _~ 2. Owing to the distributivity axiom, multiplication is isotone: x ~ y :::>x z -'< yz. This allows us to solve some implicit equations by an iteration technique. Consider x solution ofx=ax(~b. Wehavex~-b,x=ax@b~ab(gb, x > . - a ( a b @ b ) @ b = a 2 b @ a b ( t ) b . By a straigthforward induction, we get that x dominates all the akb (k E N). We are thus led to introducing the fundamental concept: D e f i n i t i o n 3 ( C o m p l e t e dloid) 79 is a complete dioid if and only if (79,-~) is complete as a partially ordered set (each subset X of 79 as an upper bound, denoted as the possibly infinite sum ( ~ e x z), and the distributivily property eztends to infinite sums:
Y( @ ~ x ~) = @ ~ x u~ and ( @ ~ x ~)Y = @ ~ x ~u We can now state a simple (but useful) result: Theorem
1 Let 79 be a complete dioid. The least solution Z of z ~- a x @ b
is given by Z = a*b whilh a* = e @ a @ a 2 (9 a 3 (~ .... E x a m p l e : Let R" = R U {+0% - c ~ } . (~', min, + ) is a complete dioid. The absorption property implies e | x = + c o + x = + ~ , and in particular, + c o + ( - c o ) = + c o . From now on, we shall always use the usual order < in (R', min, +). Then, the lower bound of the set {a ~ ~,a ~,aa,...} = {0, a , 2 x a,3 x a . . . . } isa* = - o o . Thus, the maximal solution of x < min(a + x, b) is - c o if a < 0 and b otherwise.
962
5
State
space
representation
E x a m p l e : The event graph of Figure (1,(b)) is described by the following equations:
9 ~(t) _< .i(~) ~ i ~ ( 0 ,
.~(~) _< ~ ( ~ - 2)
(6)
Eliminating xl,x2 ' ' , and y is immediate. Thus:
In the general case, we describe the event graph by the column vector x ( o = [~(0 . . . . . . . (0]', the entries of which are the counting functions of all the transitions, cr(o = [~ (r . . . . . . AO]' denotes an input vector. Then, the counting function X'(t) corresponding to the earliest behaviour of the system is the greatest solution of a state equation of type: X(t) <<_AoX(t) ( 9 . . . (9 A T X ( t - T) (9 BoU(t) ( 9 . . . (9 Br, X ( t - T')
(8)
T h e o r e m (1), gives an explicit form: X(t) = aoA1X(t-1)@...@AoArX(t-T)(gAoBoU(t)(gAoBiU($-I)(9...(gAolir, X(t-T')
By introducing the state vector of the system X ( t ) U(t) = [U(t), U(t - 1 ) , . . . , X ( t - T')]' we obtain X(t) = A Y ( t - 1) ~ /3N(t). We shall restrain constraint /g. Then, given the initial condition recurrent equation: X(t) = AZ(t
- -
(9)
= [X(t), X ( t - 1 ) . . . . , X ( t - T + I ) ] ' and the standard state space formulation: to the case where there is no input X(0), the system is described by the
i)
(iO)
To get this standard form, we need computing A~. A0 must have no cycles with negative weights (this would mean a deadlock constraint). This implies that A~ is given by the finite sum: A 0 = I d @ . . . @ A o .
6
Asymptotic behaviour
We now study tile periodic properties of the min-linear time invariant dynamic s y s t e m (10). T h e y are closely related to eigenvalues of the matrix .4 . 6.1
S p e c t r a l t h e o r y of i r r e d u c i b l e m a t r i c e s
We associate a weighted graph G ( A ) with the matrLx A = (aid) in the following way: G ( A ) has n nodes 1 . . . . , n, and there is an arc (i, j) fi'om i to j if and only if aj,, r ~ . Then, the weight of ( i , j ) is ai,~. We define the weight w'(c) of a circuit c = ( i i , . . . , ik) of length l'(c) = k by w'(c) = aq,~ ~ | a~k,i~_ , @ . . . | ai~,h. We say that the m a t r i x A is irreducible if and only if the graph G(A) is strongly connected. From the results of Gondran &: Minoux [5] , and Cohen et al. [1], we have Tlmorem
2 A s s u m e .4 is irreducible, then there exist a unique eigenvalue o f .A, A, solution to the equation A z = Ax with x ~s r A is characterized by: A = max{u; (u-i.A) * > - o o }
(ii)
963
A is also equal to the minimal mean weight of all the circuit c of G(A): )~ = rain w'(c)
o t'(~)
/
Going back to the initial event graph, this gives a proof of formula/il). The following states the connection between eigenvalues and asymptotic throughput: T h e o r e m 3 ( C o h e n e~ al.[1]) Le~ 2d be the least solution of the autonomous recurrenl
syslem (10), and ~ be ~he eigenvalue of *he irreducible matrix .4. Then, there exists E N* (~he period), and tt C N (*he length of~he transienl} such ~hal:
x(t + e) = ~ex(t) for t > t~ If X(t) represents the quantities produced, then )~ has the obvious meaning of rate of production, or periodic throughpul. In the conventional algebra: X(t + 8) = 0)~ + X(t) i.e. the vector quantity OA is produced during ~ units of time. 6.2 G e n e r a l case If the matrix A is not irreducible, we partition the graph of G(A) into k strongly connected components. By renumbering the nodes of G(A), we may assume that A has the block-triangular form A1,1 A2,1
9 A2, 2
. .. ~
An,1
A~,2
...
~" t
A= Ak,k
where A t , t , . . . , Ak,k are strongly connected. Let AA~,~ be the eigenvalue of the component Ai,~. Then, the spectrum ~r(A) (the set of the eigenvalues of A) is a subset of {AA~,~. . . . . AAk,~}. AA~,~is an eigenvalue ofA iff the upstream eigenvalues AA.... . . - , AAk_,.k_~ are greater than AA~,,. We shall only consider the smallest eigenvalue ~- = rain ~(A) = min{AA, . . . . . . AAk,k} which corresponds to the slowest part of the system. Then, ~ is also characterized by (11).
7
Symbolic
computation
of the maximal
eigenvalue
When A is a constant matrix, a well known algorithm (Karp [9]) allows computing A. In our case, A (more precisely A(q)) is a matrix function of the undeterminates q l , - . . , qk associated with the resources (cf. equation (7)). The classical analogous for (#-lA(q))* would be ( I - #-tA(q))-l. A symbolic computation of this rational matrix function of q and # would obviously be out of range. In the dioid case, we shall show that absorption properties make this computation simpler. We now recall the Gauss-Jordan algorithm for computing the * of a given matrix. 7.1
The Gauss-Jordan algorithm
Let A be a n • n matrix with entries ai,j in a complete dioid, and b a vector. ~" = A*b is the least solution of x = Ax @ b: xi = a l , l Z l (D a l , 2 X 2 . . . ~ a i , n x a ~ bl x2
=
a 2 , 1 x l $ a 2 , : x 2 . . . $ a 2 , n X n (~ b2
(12)
964
We can eliminate xl by substituting zl = a * l , l ( a l , : x 2 . . . 9 al,,~z~ (9 bl) in the n - 1 last equations. This means that the usual gaussian elimination algorithm allows the computation of A*b. There is a simple way of writing the algorithm by considering the matrix equation X = A X ~ A. This is equivalent to solving n systems of type (12) where b is replaced by a column of A. The least solution of X = A X (~ A is A * A . We have A * A = A A * , and A* = AA* (9 I d (this simply is the definition of A*). So, we can recover A* from A* A. It should now be clear that A* A can be computed by the following Gauss-Jordan algorithm (cf. Gondran & Minoux[6]): start with the matrix A (~ -- A define by induction A ('n+l) = ra (m+l)~/ by: al,(re+t) (,~) 71"mare, (ra)~ ~. l,t t = al,(m) t ~ tgd al~rn with 7rm = (a('~,))* Then, we get A (n) = A * A . It is essential to notice that A* is finite iff A * A is finite which is equivalent to all the ~'m being finite. 7.2
Rational closure of the dioid of polynomials
We need to perform the Gauss-Jordan algorithm with the initial matrix A (~ = l ~ - l A ( q ) , the entries of which are monomials in # - 1 , ql . . . . . qk. Our goal is to compute the set of constraints in problem (3). The aftlne constraint ,-~-5ql + . . + ,-~q~ +
>__~"
can be written as a monomial with rational exponents using the dioid algebra:
Therefore, we introduce: D e f i n i t i o n 4 A generalized polynomial f u n c t i o n o f q t , . . . , q ~ and t~- t is a finite sum: f ( q l . . . . . q k , # - l ) = ( ~ a a q ~ X . . . q ~ k # - - ~ k + l , with (~ = (cq . . . . ,~k+l) E (Q+)k+t and a,~ E R.
We denote by R < q t , - - . , qk, ~ - l > the dioid of polynomial functions with indeterminates and /~-1, endowed with the natural sum and product. We keep the usual
ql,...,qk
11
1
algebra for the exponents (e.g. 2q 3 | Oq~ = 2q~-). For instance, 0q~ is a monomial, which is the dioid version of "0 + ~-" a in the conventional algebra. We now study the * properties of polynomials. In a commutative dioid, we have
(o): (aeb)* =a'b* Then, we only have to consider the * of a monomial. We have already pointed out that in the scalar case, a* is 0 or - c o . This leads to defining the indicator function X by: X,(t)
f 0 L -co
if t < s if $ > s
965 Let P E R- < q l , . . . , qk, #-1 ~>, and Q, R E R < q t , . . . , qk >. The main algebraic features of indicator functions are:
(0: 9 ..q~ ) = X ~
~(~)
ql p ...qk p
(.i): Xq(t) r X~(t) = Xqe~(t) (i~): (,): [)/~(~)]* = X Q ( 0 (~): [P X,~(,~) X~(o)]* = P* X,~(,~) XR(o) We denote by ~" tile set of all the functions f of the form:
f(q,l ~) = P(q,#) Xq(q)(#) XR(q)(0)
(13)
for some P E R < ql . . . . , q k , # - i > and Q , R E R ' < q t , . . . , q k >. Then: 1 The dioid F is stable under the * operation.
Proposition
.T" is the rational closure of R < q l , . . . , q k , # -1 >, i.e., the least extension of R- < q t , . . - , q k , # -1 > which is stable under @,| and *. Thus, the entries of the matrix A ('~) defined in the Gauss-Jordan algorithm will have the form (13). As the entries of the matrix A (~ only have powers of #, we never use the (i) property. Moreover, formulas (o), (ii) and (iv) show that the* of a polynomial is an indicator function. Then, we have the stronger result: P r o p o s i t i o n 2 At the end of the Gauss-Jordan algorithm, we get: ~r,,(q,#) = for some P E R < q t , . . . , q k >
Xp(,r
T h e eigenvalue A is characterized by (11), thus it is the maximal solution of X p ( # ) > - c ~ , which gives A = P. Let us explicitly write the polynomial P: P ~- ( ~ , a ~ q ~ . 99qkak 9 Going back to the conventional algebra, we get: A = P(q) = min,~(a~ + atq~ ... + akq~) This is the symbolic expression for A(q) which we need for solving problem (2). 7.3
Complexity
estimation
This algorithm is useful when (i) the size n of the system (of the matrix A) is large, and (ii) there is only a few indeterminates (k << n). After reordering the lines if necessary, we partition the matrix as follows: "~(q)=
~
E(q) F(q)
Using block inversion formula, we get that [#-lA(q)]* is finite iff the following k x k matrix [F'(q, #)]" = [ p - i F ( q ) 9 # - t E ( q ) ( # - I c ) * # - t D ] * (14) is finite. Reduction to matrix F'(q, #) has a O ( n 3) complexity. The application of the algorithm to F'(q, #) is ahnost equivalent to a symbolic inversion. So, k m u s t be small (a few units). Example:
We apply the algorithm to the case of the initial example (Figure (1),(b)).
]
]
966
which gives:
~~(~') = [~'-he~ ' - ~ X ~ , .
(~,)~,-~(q~eZ.q~)]" = X~,~ (/~) X ~ , ~
(#) X4~,~(~,),~, 0,)
With the usual notations, the conditions for 7r2(A+oo) = ;2(1) being finite are ~ > ~-, ~ > ~-, l+n > ~- We find the same optimal values ql = l a n d q2 = 2 as the ad hoe approach of section (2). Conclusion: Dioid theory allows adapting some algorithms from usual linear algebra to problems of performance evaluation and optimization of timed event graphs. Instead of enumerating all the circuits of the graph, the use of rational properties of path algebra gives us a sinapler integer linear programming problem after some symbolic computations. This allows us to distinguish an artificial difficulty coming from the complexity of the event graph from an effective complexity due to the number of unknows. A c k n o w l e d g e m e n t : The author is much indebted to the INRIA working group on Discrete Event Systems. Particularly, to Guy Cohen, Ramine Nikoukhah, Jean Pierre Quadrat and Michel Viol. References [1] G. Cohen, D. Dubois, J.P. Quadrat and M. Viol. "Analyse du comportement p@riodique des syst~mes de production par la th~orie des dioides", Rapport INRfA N. 191, Le Chesnay, France, 1983. [2] G. Cohen, P. Moller, J.P. Quadrat and M. Viot. "Algebraic tools for the performance evaluation of discrete event systems", IEEE Proceedings, Special issue on Discrete Event Systems, 77, No. i (Jan. 1989). [3] P. Chretienne. "s Rdseauz de Pelri Temporisgs", Th~se de l'Universit@ Pierre et Marie Curie (Paris VI), Paris 1983. [4] l't.A. Cuninghame-Green, "Minimax Algebra", Lectures notes in Economics and Mathematical Systems, N. 166, Springer, Berlin, 1979. [5] M. Gondran and M. Minoux, "Valeurs propres et vecteurs propres duns les dioi'des et leur interpretation en th@orie des graphes" EDF, Bulletin de Ia Direction des Eludes et Recherches, Serie C, Math~matiques Informatique, 2, pp. 25-41, 1977. [6] M. Gondran and M. Minoux, "Graphes el Algoriihmes", Eyrolles 1979. [7] H.P. Hillion and J.M. Proth "Performance evaluation of job-shop systems using timed event-graphs", IEEE Trans. on Automatic Control, 34, N.1, pp. 3-9, Jan 1989. [8] II.P. Hillion. "Mod~lisation el Analyse des Syst~mes de Production Discrets par les Rdseaux de Petr~", Th~se de l'Universit@ Pierre et Marie Curie (Paris VI), Paris, Janvier 1989. [9] 1-s Karp. "A characterization of the minimum mean-cycle in a digraph", Discrele Maths., 23, p. 309-311, North Holland 1978.
WIP and CSP-1 Quality Control in a Two-Stages Queue Like Production By Charles S. Tapiero and George Tsiotras * IAE, Universite Aix-Marseille and Hebrew University ** AT&T BeU Laboratories, Middletown N.J.
Summary The paper considers a two stages queue like production system with limited buffer when production is unreliable. Analyses performed on the model are then used to obtain some basic insights regarding the management of unreliable Just in Time manufacturing systems. In particular issues relating to quality improvement in JIT systems as well as substitution between quality control versus zero defects technologies are clarified. Introduction The current propensity to commonly recommended by Just
reduce inventories "at in Time (JIT) "gurus"
any cost" as is not always
desirable (Schonberger 1982, Hall 1983, Monden 1986). While, for the most part, inventory costs have been understated, primarily due to a managerial focus on direct rather than the more substantial indirect costs. There is today much more awareness of the role of inventory in manufacturing management and its direct and indirect costs. For example, the recent paper by Conway, Maxwell, McClain and Thomas (1988) has pointed out, on the basis of analytical and simulation studies, the conditions under which it might be beneficial to reduce work in process (WIP)
inventory.
This
paper
considers
the
mutual
effects
of
WIP
inventory,
manufacturing process reliability and quality control. We study the effects of JIT concepts, consisting in limiting the WIP in a queue like manufacturing system on processes that might be unreliable, combined with the basic requirement that outgoing output from such systems must be
of
"high
conformance
quality". and
Quality
inspect
the
control
is
used
manufacturing
to
assure
process
product
operational
performance in so far as product quality is concerned (see also, Hsu and Tapiero 1988, 1989, Tapiero and Hsu 1988, 1989, Tapiero 1987 and related studies). On the basis of the model studied we shall reach the following conclusions: (1) In JIT systems quality improvement processes are more critical provides control
to the proper functioning of a shop floor. incentive can
for
shop floor quality
compensate partially
only
As a result,
improvement.
costly
zero
defects
(2)
JIT
Quality
technologies.
968 Particularly,
we compute the
cost
minimizing
parameters
for
a
CSP-1
sampling (Dodge 1943, Duncan 1974) and compare the "willingness to pay" for quality improvement as a function of the sampling effort. In stations
addition,
it is shown that
dependencies
and
as
buffer
a result,
reduction
the
for
variability
WIP
of
increases
flows.
For
unreliable manufacturing systems, this problem becomes acute, which means that larger buffers may be required to reduce consecutive work stations's
dependencies.
Alternatively,
it
is
argued
that
automation
which reduced process time variability may be more economical in JIT than say in inventory intensive manufacturing systems. When comparing processes that rework defective units in station compared to scrapping of detected defective units, the conditions under which one approach or the
other
is
interpreted,
desired
provide
a
is
shown.
justification
Such for
the
observations, basic
appropriately
premises
of
JIT
manufacturing systems, which require a flawless, continuous flow process with zero defects. Quality Control of a Two Stations J I T System Consider for simplicity a two stations tandem queue like manufacturing system with a f'mite and intermediate buffer as shown in Figure 1. Jobs arrive at a first, ample waiting space station, at a Poisson rate 2. For simplicity, it is also assumed that the queue is saturated such that whenever a server is free, a job enters the process. The
amount
of
time
required
to
process
a job
is
assumed
to
be
exponentially distributed with parameter /z I while the process is assumed to be perfectly reliable. A job processed at the first station is routed instantly to the second station for further processing. We assume that this station has a limited buffer (waiting space) with R spaces, a single server with general service time whose probability distribution function (pdf) is G(.). In addition, it is assumed that the server can process jobs in a defective manner. That is, the process propensity to perform its function in a faulty manner is known and given by three parameters [o 1, o 2 , 0]. When the server is in state "1" it produces a fraction defectives o I which will be called the "standard" mode, while in state ~2", it produces a fraction defectives o2 , o2> o , which is called the faulty mode. The probability of switching from a standard to a faulty mode is known and given by 0.
For simplicity, this
paper will assume that o 1=0, corresponding to a perfectly reliable server and let the switching probability be constant, independent of the number of units produced. The switching process is therefore as that
969 shown in Figure 2. In such situations, unlike queueing models with breakdowns, the process is neither stopped, nor is it taking a vacation, but operates continuously, albeit producing, with some probability, a faulty output. In this sense, the manufacturing technology of the production process is represented by the parameters [o 1'02'0]" The smaller these parameters, the more effective (in a quality production sense) the manufacturing system. For quality conformance a CSP-1 Continuous Sampling method is applied (Dodge 1943, Duncan 1974). For such quality control, we can compute the various probabilities of producing defective units, detecting such units and of course the probability distributions of basic quality states of the queue-like production process. We summarise essential results. A production cycle is shown in Figure 3, during which N jobs are processed and the following is shown. Proposition 1: Let i be the number of defective units produced after the production system has switched to a defective mode and let k be the total number of units produced in the defective mode. Define by Pi(k) the probability that the ith defective unit is detected, then these probabilities are given by the following relationships, (1) Pi(k)= b(1-b)i'~NBD(i,k), k=i, i + l , i + 2 ..... i=1,2,3.. while the probability of producing i defectives in a production run is given by, oo
(2)
qi = ~ Pi (j) j=i
where NBD(i ,k) is given by the negative binomial distribution, (3) NBD(i,k)= ( k-l i-I(l'v ~ k-b '2i k = i , i + l , i + 2 , i + 3 . . . . . . Proof: (See Tapiero and Tsiotras 1990) The expected number of defectives production cycle is,
and
their
variance
in
a
oo
(4)
i = ~ iqt , i~i
i
i i-I
By the same token, the expected number of defective units which leaves A the production system undetected is y =E(i-1), while its variance is a 2y ' or ^
(5)
co
r
y = T. (i-1)qi; a 2= ~. (i-l-y)2qi i-I
Y i=l
If b/ is the total expected number of units produced in a cycle, then of course, the expected number of non-defective units produced in the cycle
970 ^
is
given by,
^
N-y.
In the values to these parameters.
following
proposition,
we
provide
explicit
Proposition 2: Let N be the number of units produced in a cycle. expected production in a cycle and its variance are given by, A (6) N = ( 1 - 0 ) ~ + 1/0 where Oo
=
P
~.
(8)
v a r ( N i)- 1 = k=i (1-0)~
X
kPi(k)=(l/ 02
b)o 2 A
+ P2 + 2nP]
+
(1-0)/02+(1-0)2nP 2
OO
GO
P2 =
the
O0
(7)
(9)
Then,
}:
~:
i--I
k~Pi(k)=(l+~176176176
k=i
Proof:
see (Tapiero and Tsiotras 1990)
Proposition 3: In the long run,^the probability of inspecting a unit is given by, (10) n = b + (1-b)nl/N while the probability of detecting
a
defective unit
and
shutting
down
the system is given by, A (11) J = I / N On
the
basis
of
these
probabilities,
the
production
system's
performance can be assessed. Consider first the amount of time required to perform a job. Let r be the amount of time required to produce a unit standartly, r I is the time required to inspect the unit and z2 the time to shut down the system and attend to repairs, then the production time of a job is given by the mixed distribution T below, r + z (12) The
T
w.p.
=
z + z I + z 2 w.p. r w.p. 1 - ~ 5 - ~ subsequent analysis will integrate
considerations
of
quality
and
quantity issues in the manufacturing process.
The Two Stations With Sampling And Limited Buffer
Here blocking of the first station occurs when upon completion of service in the first station attempts to move to station 2 when it is full are made.
In this case,
station 2 occurs. general and
mixture
repair.
blocking occurs until
a departure
from
In this system, the service time in station 2 is a distribution
Attendance
of
the
service
times,
can
be
time
computed
including without
inspection altering
the
971
basic structure of the queueing model.
Namely,
the mean service time
and its second~, moment are,, given by, (13) (14)
l//.t = z + (~r+~)z t + ~z"2 ET 2 = ( 1 - ~ - r 0 E r 2 + n E ( z + z l ) 2 + J E ( r
+rl+z2) 2
Since the manufacturing system has a limited buffer, under an assumption of heavy loading at the first station, the output rate of the first queue is /~ , (which equals the service rate of result, the second station behaves as if it were an Poisson server
arrival acts
as
rate
/t i
an
since
additional
during
space
the
that
this
station). As a MIGIIIR+2 queue with
blocking
belongs
to
period the
the
second
first queue.
Furthermore, due to the blocking period, the server in the first station does not process unit and remains therefore idle. That is, no more arrivals
occur
at
the
second
station.
Due
to
the
exponential
assumption, this is equivalent to stating that arrivals do occur at the second station at a rate of /~i but they are lost. In view of this observation, the second station becomes an MIGIIIR+2 queue. Such queues have been studied previously (e.g. see Makino 1964, Keilson 1966, Lavenberg 1975) and their operating characteristics are known. As result, using the adjusted service rate /t and E(T 2) we can obtain
a a
direct relationship between the buffer size R, the manufacturing process unreliabilities and the manufacturing system operating performance. These effects are two fold, first on the manufacturing capacity and second on the blocking probabilities for prior queues (causing thereby inventory accumulations and forced idleness). Let fl be the probability that the system is full, blocking thereby the first station. Then, for a buffer size R, (15)
fl = 1 - I / ~ + P
+2], with
p=l~lll.t,
R+I
PR+2= l / l ~
flj]
j=0
with (15)
1 ifk ~k
=0
all(I-at)
"~
if
{Efljak+l. j +
k--1
ak}l[l'a 1]
if k > 2
and (16) a k = f0{[1-FT(t)]e'/Zlt /Zk1 tk-l/(k_l)!} dt If we denote by F~(s) the Laplace Transform of Fr(t),
then using
the
well known transform relationship, (17) LT(tk" IFT(t)) = ('l)k" IF'T (k- 1) (S) where
F~(k'l)(S)
easily shown that,
is
the
(k-1)st
derivative
of
the
transform,
it
is
972
o~ = ~/(k-1)l][F(k)/,u~- (-1) k-I F TO ( k - 1 )(~ l ) / ~ l ] with F;(S) given by, (19) F ; ( s ) = F;(s){F;i(s)[~+SF;2(s)]+I-5-~} (18)
The expected time a job will spend in the second station is thus, R+I
(20)
W = (R+2)//,t - E JPj//t 1 j-I while the variance of the waiting time is, R+I
(21)
var(W) =(R+2)[ET2-1//~ 2] +
E
R'4-1
[j ( j + I ) P j ] / / ~
j=l
- [ T. jPj//~I] 2 j=1
( R + 2 ) (R+ 1)PR+2//~I/~ A Similarly, the expected number of jobs in the second station is L=/~IW, while the expected number in qeue, Em q is given by, L q = Emq= /Jl(1-fl) Wq - ft. Finally, the throughput of the system (at station 2) is d R, and given by, ~R = ]gl/[jO+PR+2] , p=/~t//~ and thcrcforc the loss of throughput ,4 due to limited buffcr in the station is given by, d = 5R(X+~).
These terms, combined with the production system's quality performance characteristics, can bc used to assess the overall performance of the production system. Throughout our applications, define the following cost parameters, C a = the cost per unit of blocking the first station, measuring the forced idle time imposed on the first queue. C = the inventory cost in the second station, per unit per unit time and expressing the cost of WIP in the second station Ct = the inspection cost per unit C d = the post process cost of a dcfcctuous unit, expressing the penalties associated to jobs processed in a faulty manner and that have left, undetected, the system. C = the cost due to the loss of output because of the inspection T
and defective units. C = the cost of adjusting the process, once a defective unit is F detected. These include f'Lxed stoppage, re-start costs as well as the repair or maintenance costs associated to the manufacturing system. Since our process is a renewal-reward process, the long run average cost is equal to the average cycle cost (Cinlar 1975) and therefore this cost, denote by 4 , is given by, (22) 9 = [ C
+Ci[bN+(1-b)nl]+
CdQ]/[~q/~]+
CILq+ CTZ1 + CBfl
973
where N//a is the expected cycle time, C fl is the cost of blocking per unit time of the second station, or the probability of the system being full. A A [bN+(1-b)nl] is the expected number of units inspected in a cycle which was computed earlier, CILq is the inventory cost per unit time since, L q is the average number of units in the queue, CTA is the average loss of output due to inspection and repairs/maintenance, stoppage and restart. Finally, CaQ is the expected cost of non-detected defective units which have exited the system. Since in a cycle, we produce qi defectives on the average, one of which is detected, the expected number of undetected faulty units in the cycle is given by, O0
(23)Q = ~ (i-1)qs with
qi
defined earlier
Numerical analyses were used and the following conclusion were reached. 1. When the buffer space is reduced, the production variability increases. Since such variability has negative effects on the smoothness of the production flow, it is possible to compensate such variability through an increase in automation (which reduces service time variability). We might conjecture therefore that JIT provides an incentive for automation. When we consider the quantity produced in a cycle, we note that while with increases in inspection the expected production cycle becomes smaller, its coefficient of variation increases. 2. When the buffer space R is reduced, the cost of process unreliability decreases. Further we note that A~/AO is largest at R = I , which means that it is most advantageous to invest in process quality when the buffer space equals one. When buffer space increases, the relative advantage in improving quality is reduced. In fact, when b=0.05, we see from that A ~ / A 8 is largest when R=0, at which time investment in process reliability improvement reduces costs the most. Close to full inspection (b=.8), an investment in process reliability improvement can at best be used to reduce inventory costs (explaining thereby the growth with respect to R). 3. In J r r systems, flexibility reduces the blocking probability. Thus the value of flexibility at a given station is given by the cost reduction effects such reduction in blocking has. Of course, the
974
probability of blocking can be reduced by increasing inventory, but this incurs at the same time several other costs. 4. The marginal cost of inspection
are
greater,
the
greater
R.
5. The probability of blocking decreases when the buffer space increases, but it declines faster when the buffer space is smaller. At the same time, while the loss of throughput is smaller when R increases, this loss occurs also at a declining rate. This ~improvement ~ in operating performance involves a cost reflected in the increased average buffer space (inventory cost). These costs increase when we increase inspection. Thus, if the process reliability is not increased, quality can be improved only with inspection which induce detrimental operation of the manufacturing process. For in- versus out of station repair we saw that the greater the cost of production due to process unreliability,
the
more
beneficial
it
will
be
to
inspect
in-station
rather than out. These conclusions, having a strategic implication to the design and the management of unreliable Just in Time Manufacturing Systems are in part
accepted
as
"common
knowledge".
This
paper
has
provided
a
justification based on the modeling of JIT as tandem queue like manufacturing systems, reinforcing both current experience and quantifying it to value JIT, quality control and the process of quality improvement REFERENCES Cinlar, E., Introduction to Stochastic Processes, Prentice Hail, Englewood Cliffs, N.J., 1975. Couway R, W.Maxwell, J.O. McClain and L.J. Thomas, The role of Work in Process Inventory in Serial Production Lines, Operations Research, 36, 1988, 229-241 Dodge H.F., A Sampling Inspection Plan for Continuous Production,
Annals of Mathematical Statistics, vol.14, 1943, 264-279 Duncan A.C., Irwin, Ill. 1974
Quality
Control and Industrial Statistics,
4th.ed.,
Hall R.W., Zero Inventories, Homewood, Ill. Dow Jones-Irwin, 1983 Hsu L.F. and C.S. Tapiero, A Bayes Approach to Quality Control of an M/GI1 Queue, Naval Research Logistics Quarterly, 35, 1988, 327-343 Hsu. L.F. and C. S. Tapiero, Quality Control of the M/G/1 Queue,
Euro. J. of Operations Research, Forth., 1989
975 Hsu L.F. and C. S. Tapiero, Quality Control of an Unreliable Flexible Manufacturing System: with Scrapping and Inf'mite Buffer Capacity, Int. J. of Flexible Manufacturing Systems, Forth. 1989 Keilson J., The er.godic queue length distribution for queueing systems with finite capacity, J. Royal Stat. Soc., Series B, 28, 1966, 190-201 Lavenberg S.S., The steady state queueing time distribution for the M/G/1 finite capacity queue, Management Science, 21, 1975, 501-506. Makino T., On the Mean Passage Time Concerning some Queueing Problems of the Tandem Type, Journal of the Oper. Res. Soc. Japan, 7, 1964, 17-47 McClain J.O. and L.J. Thomas, Operations Edition, Englewood Cliffs, N.J. Prentice Hall, 1985
Management,
Second
Monden Yashiro, (Ed.), Applying Just in Time, The American~Japanese Experience, Industrial Engineering and Management Press, Atlanta, 1986 Schonberger R.J., Japanese Manufacturing Techniques: Lessons in Simplicity, New York, Free Press, 1982 Tapiero C.S., Production Transactions, 19, 1987, 362-370
Learning
and
Quality
Nine Hidden Control,
liE
Tapiero C.S. and L.F. Hsu, Quality Control of an Unreliable Random FMS: with Bernoulli and CSP Sampling, Int. J. of Prod. Res., 26, 1988, 1125-1135 Tapiero C.S. and G. Tsiotras, JIT, and the Quality Improvement Process in Queue Like Production Systems, Working Paper, 1990
976 Figure 1: The Production Model
Limited Butler [][]
Saturated o ....
Re.~=esta.o.
R
M/G/1
~
~R~7;k
Unrell,lble Station
Figure 2: Production System Switching %Defectlvell Produced Failure Mode Probability of mwitch
:StandardMode # Produced
Figure 3: The Production Cycle Start I
b Probabllletlr Failure n
~Sample ~
D
~
Repair ProductionCycle
Re,tart
MODULOIDS AND PSEUDOMODULES 3. T h e L a t t i c e S t r u c t u r e P r o b l e m E. Wagneur* Abstract We determine here sufficient conditions for finite dimensioned modulo~'ds and pseudomodules to be lattices. In particular, we show that completeness of the dioM D of scalars is such a condition. The simplicity conditions for pseudomodules, which make the classification problem tractable, are also shown to be sufficient for the lattice structure. Since these conditions are clearly unrelated, both results show that neither one is necessary. A concluding example illustrates this remark.
1.
Introduction
In traditional dynamical systems discribed by differential or difference equations, the underlying time set is independent of the evolution of the system, and sudden changes in the state are associated to singularities or bifurcations. In contrast, the dynamics of deterministic discrete event systems (DEDS) is characterized by the occurrence of sudden changes in the state, appearing at instants which are determined by the dynamics of the system. For example the message flow in communication networks, or the material flow in automated production lines may be modelled as DEDS. The mathematical modelling of DEDS aims at : 1. Providing the analogue of the powerfull classical models for the analysis of the dynamics involved. 2. Giving a tool for the analysis of the relation design/performance of the underlying physical system. 3. Developing an instrument for observation and control of the system described. Although various models have been proposed in relation with one or more of these goals (e.g. 1, 3 in [7] and [11]), we will concentrate here on the model introduced in [3],[4], which has proven to be relatively efficient in connexion with objectives 1 and 2. In this approach, a DEDS is modelled as an event graph (a particular class of Petri nets), and its evolution discribed by a set of equations of the type : ;gi(~ ~- 1) = Max{ Max {aij + xj(k)}, bi + u(k)}, l<j_<, y(k) = Max {cj + xj(k)}.
1 < i < n,
l<j
In many applications, the underlying set of parameters is (]RU {-cr max, +), an idempotent semiring which have been widely used in connexion with various optimization problems. With this model, the set of equations above is linearlized and becomes : * GERAD, Centre conjoint Ecole des It.E.C., Ecole Polytechnique, McGill University. This research has been supported by the program M.E.S.S., Quebec, Actions Structurantes, NSER.C, and F.C.A.R. grants 86CE-130, 89EQ3528, and CrtSNG, grant OGP38915.
978 =,(k + l) =
+ b,u(k),
1 < i < n,
j=,
where + stands for the max operator and 9 for the sum (as usual, unless needed, the dot will be omitted). In vector form, we may write the system as :
x(k + 1) = Ax(k) + bu(k), y(k) = c'=(k), with c t the transpose of c. Clearly, x(k) belongs to some kind of vector space, or semiring module (cf [14]), called moduloi"d over (lit U { - ~ } , m a x , +). This structure exhibits interesting properties, which, as a rule, do not hold in more general semiring modules. Also, since "additive" inverses are not available, most of the results of classical linear algebra do not apply. Various efforts have been given in the recent years, to extend some of the concepts and theorems of linear algebra to moduloi'ds. In particular, Cunnighame-Green [5] defined the notion of permanent of a matrix, Gondran and Minoux [6], that of bideterminaat and versions of C r a t e r ' s rule together with Cayley-Hamilton's theorem have been proved ([8], [10]), and regularity properties of matrices have been studyied [2]. Assuming the dioi'd D of scalars is completely ordered, yields the essential result that, for finitely generated modulo~ds, we have a weak notion of independence, for which existence ([12]) and unicity ([9], [12]) of a basis is guaranteed. Restricting to the case where the dimension is finite, we proved in [13], that when D is also assumed to be a group with respect to its second composition law, then the classification problem (up to a pseudomodule isomorphism) has a nice solution for a relatively wide class of D-modulo'/ds (called simple pseudomodules). In particular, such a pseudomodule is then completely charecterized by the substructure generated by its basis over {0, 1}. Since this defines a finite semilattice with 0, it is also a lattice. Hence it is important to be able to know when the moduloid itself is also a lattice, for lattices exhibit interesting properties which have been widely investigated (cf [1]). We will always assume here that the set of scalars D is completely ordered (hence is a lattice). The aim of this paper is to investigate the lattice properties of moduloi'ds and pseudomodules. In particular, we show that a finite dimensional moduloi'd (or pseudomodule) is a lattice whenever either D is complete (Corollary 4.1 to Theorem 1), or M is simple (Theorem 2). In Section two below, we recall the basic definitions and essential results of [12] and [13]. In Section three, we introduce the concept of saturated linear combinations and then show that, in the finite dimensional case, such combinations always exist and axe unique when D is complete. In Section four, we prove that either the existence of saturated linear combinations, or the simplicity conditions imply the lattice structure. 2.
Basic d e f i n i t i o n s
Definition 2.1 A dio~d is a triple (D, +, .) where D is a completely ordered set whith a minimM element 0 ; + and 9 are two binary operations on D satisfying :
979 a + b _= max{a, b}, (D*,-) is a commutative, divisible monoi'd with neutral element 1 (where D* = D \ { 0}); 0.a=a.0=0
VaED;
9 is distributive with respect to + ; If (Z), .) is a group, then (D, +, .) is called a pseudoring. The dioYd will be restricted here to the case where (D*, .) is not a group, and, unless otherwise stated, all elements in D* are supposed to be _> 1. A subset A C D of a dioi'd (resp.. pseudoring) D which is also a dioi'd (pseudoring) is called a subdioYd (subpseudoring) of D. Note also that a pseudoring contains a unique maximal (w.r. to C) subdioYd, called the positive cone of D. The definition of a D-moduloYd M (or moduloYd over a dioi'd) and of a D-pseudomodule (or pseudomodule over a pseudoring) is defined in a similar way as module over a ring. Internal and external composition in M are also denoted by -t- and .. Remarks 2.2 R1. Idempoteney also holds in M , for monoi'd, M is (join) semilattice.
x+x
= (1 + 1)x = l x = x. Hence, as an idempotent
R2. M is partially ordered by x < y *~ x + y = y. R3. Although many moduloYds axe also lattices (e.g. E1 of Example 2.3 below), which need not be modular (E2), M is not a lattice in general (E6). R4. We will write x A y for over A.
rain{x, y},
whenever it is defined, and assume 9 is also distributive
The definition of morphisms (of moduloid or pseudomodule) is similar to that of morplfism of modules. Also the prefixes mono-, epi-, and iso- are used with their usual meaning. It should be noted that moduloi'd (pseudomodule) morphisms are isotone (z _< y :=> ~o(x) _< ~o(y)). Examples 2.3 E l . (IN U { - o o } , max, + ) , is a dioid. E2. (IR U { - o o } , max, + ) , is a pseudoring. E3. D ~ is a moduloi'd (resp. pseudomodule), whenever D is dioid (resp. pseudoring). Since D is completely ordered, D " is a distributive lattice. E4. For D an arbitrary dioi'd or pseudoring, let xl = (1, 0, 0), x2 = (1, 1, 0), xa = (0, 1, 1) E D a 9 3
The pseudomodule M = {y~ Aixi I Ai E {0,1}} = {0, z l , x2, z3, xl + za} generated by the
xi's
i=l
over the pseudoring {0, 1} is a nonmodular lattice.
Eb. For D arbitrary, let xl = (,~, 1), x2 = (1, # ) , with 1 < # < A. The set M = {~lxl+2~2x2 I )~1, A2 E D} is a proper submoduloYd (subpseudomodule) of D 2 .
980 Although Xl and x2 are not comparable, we have xl < ,~x2, as well as z2 < ~xl. Note also that the semilattice X + = {xl, x2, x l + x 2 } is isomorphic to the semilattice ((1, 0), (0, 1), (1, 1)}. E6. Let O = ( q u { - o o } , m ~ , + ) c I3 = ( r t u { - o o } , m ~ , + ) , a n d ~ = 0 , 1 , 0 ) , ~ = (0,1,1), ~ = (0,-v~,0) E D ~ 9 Then X = {~1,~,~} generates a Z ) - p , e u d o m o a u l e M , which is not a lattice, since the set (x E M [ z _< z i , x _< x2} has no 1.u.b. in M. Again, as a poser, X is isomorphic to Y = {(0,1,0), (1,1,0), (0,1,1)} C D 3 , and Z + and Y + are isomorphic semilattices.
Let xi E M , i E I , where I is an arbitrary index set. Definition 2.4 We say that X = (zi)iE I generates M iffVx E M may be written as a finite linear combination 9, = ~ ~ i X i , with xi E X, ,ki E D, Ai = 0 except for a finite number of values i E I. iE1 Let Xz stand for the span of X , and consider the following properties, where I1,12 C I. P1.
VI1,I2, XI, f] XI2 = XltnI2.
P2. VI1,/2, /'1 N/2 = O ~
XI1 N XI~ = O.
P3. V{ E I, VIx, { f! 11 => X{i} f] Xx, = 0. P4. Vi E I, VI1, ,i ~ Ix :=> xi ~ X q . Clearly these properties are listed here from the strongest to the weakest. They are equivalent in a vector space, but not in a moduloi'd as simple examples show. W h e n M is a pseudomodule, then P4 r P3. In.[12], we show that P4 is equivalent to the condition that X is the set of irreducible (x = y + z => x = y or x = z) elements of the semilattice M . Also the existence theorems for bases of [12] imply that if M is finitely generated then it has a unique system of generators satisfying P4. T h u s weak independence is defined by P4, and independence by P3. A weakly independent system of generators is then called a weak basis and its cardinality defines the weak dimension of M . In case M is a pseudomodule, a system of independent generators is called a basis. It is unique up to a rescating map of the form zi ~ ;bz~, (LJ e I). Examples 2.5 E l . Let xl,a:2 as in E5 above, then X = {2.1,22} is independent and generates a twodimensional moduloYd, which is not isomorphic to D 2 , since morphisms are isotone. E2. For z l , z2 as above, let Yl = A - l a x , Y2 = x2. T h e n Y = {Yl,y2} is independent. Now Yx < Y2, hence, even as a poser, Y is not isomorphic to X . However, as a submoduloid of D 2 (i.e. with the relation Y2 < A#yl), the moduloid generated by Y coincides with that generated by X . E1 above suggests that the order structure on an independent system of generators is not sutficient for the characterization of the moduloid it generates. Nor is it necessary, from E2. It follows that the classification problem in the general case is a m a t t e r of non trivial combinatorial complexity.
981 Given z , y E M let L ( x , y ) = {k E D I x _< A y } . Also, any basis X generates a semilattice X + over the pseudoring {0,1}. Whenever these operations are defined, we will write AM (resp. A+) for the meet in M (resp in X + ) . Definition 2.6 We say a pseudomodule M is simple iff it has a basis X = (xi)iex such that : $1.
Vwi, x j E X , O ~
XiAMX, j = x E M
s2. vx, y E x + , Z(x, y) r
r
xiA+xj
=xEX
+ ,
~ 1 e L(x, y).
Remark 2.7 If M has a finite basis X , then X + = X + U {0} is a finite semilattice with universal lower bound, hence is a lattice. Examples 2.8 E l . D'* is a simple pseudomodule, with basis X . i = 1,...,n,.
(el)'], . . where . .ei
.
(~,
~n),i
E2. Let xl = (A, 0 , 1 , 0 ) , x2 = ( 0 , # , 0 1 ) , xa = (0, 1, A, 0,), x4 = ( 1 , 0 , 0 , # ) , with 1 < A < # . T h e n X = (xi) 4 g e n e r a t e s a s u b m o d u l o ~ d o f D 4. We l e a v e i t to the reader to check that it is not simple. Note that not all bases of a simple pseudomodule satisfy the simplicity conditions $1, $2 above. Hence when it does, such a basis is called a canonical basis. In [131 the following results are proved. Proposition 2.9 If X -- ( x l ) , e i , and Y = (Yj)jEJ are two canonical bases for some simple finite dimensional pseudomodule M ~ then there is a semilattice isomorphism ~0 : X + ---* Y + . Proposition 2.10 T h e semilattice X + generated by a canonical basis X of a simple finite dimensional pseudomodule M characterizes M up to a pseudomodule isomorphism. 3. S a t u r a t e d
linear combinations
The following concept of saturated combination will be used. The definition just requires the existence of a set of generators. Let M be a pseudomodule (or a modulo~d) generated by a set X = (zdiel. Definition 3.1 We say that x = ~ Ajxj (where J C t ) is a saturated combination of x, if the following two
jEJ
conditions hold : 1: Vj E J VA > Aj, x < x+ Axj,
2. Vi q~ J , V A E D*, x < x + Axi.
982 Proposition 3.2 In a finite dimensional modulo'id (pseudomodule) M over a complete dioid (pseudoring) D , saturated combinations always exist and are unique. Proof Let x E M , V x k E X , l e t F-k = {A E D I Axk ~ x } . We h a v e 0 E --k, h e n c e E k # (3. Since Ek is bounded and D is complete, 3Ak = Sup {~k} E --k. Let I ( = {i E I [ ;~i # {0}}. Then x = y~ Akxk is a saturated combination for x. Uniqueness is straightforward. kEK
When I is finite (say I = { 1 , . . . n}), a saturated combination is easily completed to a maximal combination involving all the x i ' s i = 1, . . . , n by setting Ai = 0 for i 6 K . 4. ModuloYds and t h e lattice s t r u c t u r e
Let M be a finite dimensional D-moduloid (rasp. a D-pseudomodule). Theorem 1 If saturated linear combinations exist and are unique, then M is a lattice. Proof Let x, y E M .
Just complete the saturated combinations for x, y given by Proposition 3.2
above into maximal combinations x = ~ Aixi, y = ~ i=1
Then z = ~
YiZi
=
for if
X A y,
u < x,
#ixi,
and let
Vi ~--- /~i
A/-zi ,i = 1 , . . . , n .
i=l
and u _< y for some u E
M,
then the (completed)
i=1
saturated combination u = ~
T]ix i
necessarily verifies
qi <_ vi , i = 1 , . . . ,
n .
i=1
Theorem 1 and Proposition 3.2 yield the following corollary. Corollary 4.1 If D is complete, then M is a lattice. Remark 4.2 Neither completeness of D , nor finite dimensionality are necessary, as the following simple examples show. Examples 4.3 El. D is a moduloid (pseudomodule), which is a lattice. E2. Let X = ( x i ) i E I N , X + the free semilattice generated by X , and X + = X + U {0}. T h e n X + is a pseudomodule over {0,1}. It is a lattice since any element in X + is a finite sum of generators. E3. Let X as above, with xi < xi+x, V / E IN. T h e n X U {0} is a pseudomodule over {0,1}. Since it is a chain, it is a lattice.
983 T h e need to classify moduloYds a n d pseudomodules up to i s o m o r p h i s m led to the concept of simple pseudomodule. As recalled in P r o p o s i t i o n 2.10 above, they are characterized by their s t r u c t u r e semilattice. T h i s result m a y b e seen as n a t u r a l generalization of the classical t h e o r e m asserting t h a t a finite dimensional m o d u l e over a ring is completely chararecterized by its dimension. T h e o r e m 2 below says t h a t simplicity is also a sufficient condition for the lattice s t r u c t u r e in t h e finite dimensional case. Theorem 2 A simple p s e u d o m o d u l e with finite basis is a lattice. Proof Let X = (=i)'~=~ b e a canonical basis of M , a n d X + the semilattice g e n e r a t e d by X . As a finite semilattice, X + is a quotien~ of the free semilattice w i t h n generators, a n d X + a quotient of the boolean algebra 2 " , for some n. Let ~r s t a n d for t h e quotient m a p : 2 " ~ X +. T h e kernel of zr is a n ideal K of 2". Define D I ( = {~k I A E D, k E K ) . It is easy to see t h a t D K is an ideal in D ", which is t h e kernel of the m a p H : D " - - ~ M , defined by t h e linear extension of ~r. As a direct p r o d u c t of distributive lattices, D " is a distributive lattice, hence t h e relation defined by D K (u _-- v 4=:=~ ~,~k E D I ( - ) - u + A k = v + A k ) is a congruence relation with respect to the lattice s t r u c t u r e of D " . Therefore, the semilattice M of its equivalence classes is also a lattice. T h e o r e m 1 and T h e o r e m 2 give sufficient conditions for the lattice structure. Notice t h a t simplicity condition S1 in Definition 2.6 a b o v e falls short to require t h a t M b e a lattice. One would t h e n be t e m p t e d to conclude t h a t S1 is necessary for the lattice structure. E x a m p l e 4.4 below shows t h a t it is not t h e case. E x a m p l e 4.4 Let D b e a complete pseudoring,
a n d xl
=
(0,0, A t , 0 , 0 , 0 ) ,
x2
=
(0,0,0,#t,0,0),
=s = (0,~2, ~3,0,0,0), =, = (0,0,0,~2,~s,0), =5 = (~,, ~ , ~,AT, 0,0), =6 = ( 0 , 0 , ~ , , ~ 5 , ~ 6 , ~ T ) E D 6 , where t h e Ai, # j ' s are a r b i t r a r y in D*. T h e x i ' s are i n d e p e n d e n t . Hence X = {xl , . . . , x6} is the basis of a 6-dimensional s u b p s e u d o m o d u l e M of D 6 . By T h e o r e m 1, M is a lattice. Assume simplicity condition S1 holds for some basis Y = {Yl, - . . , Y6}, w i t h yl = 4 i x l , i = 1 , . . . , 6. Since M is a lattice, we have : yl ^us # 0, us ^y~ # 0, y3 ^y6 # 0, ~2 ^y4 ~ 0, y4 Ays # 0, Y4 Ay6 ~ 0. If $1 holds, t h e n we must have : Yl A Ys = Yl~ Y3 A Y5 = Y3, Ys A Y6 ----- Yl, Y2 A Y4 = Y2, Y4 A Y5 = Y2, Y4 A Y6 =- Y4, a n d Yl + Y2 = Ys A Y6 9 This leads to t h e following set of equations a n d inequalities (where we assume w i t h o u t loss of generalty t h a t ~1 = 1 ) :
These have t h e following sets of solutions :
As : Az = P 4 ~ 6 , #1~2 = P2~4 =-- # 5 ~ 6 . A4 : )q : / z 4 ~ 6 , ]-t 1~2 = "~7~5 9
984
Now it is easy to see that :
Ai ==~A2A6 _~4A7 _~ P5A5 .
The contradiction comes from the fact that the parameters Ai,/~1 (1 _< i,j <_ 7) can be chosen arbitrarily. REFERENCES [1] BIRKHOFF, G., Lattice Theory.
A.M.S. Coll. Pub. Vol. XXV.
Providence, R.I., 1967.
[2] BUTKOVI~, P. HEVERY, F., A Condition for the Strong Regularity of Matrices in the Minimax Algebra, Discr. Appl. Math., 11, pp 209-222, 1985. [3] COHEN, G., DUBOIS, D., QUADRAT, J.P., VIOT, M., A Linear System Theoretic View of Discrete-Event Processes and its use for Performance Evaluation in Manufacturing, IEEE Trans. on Automatic Control, AC-30, pp 210-220, 1985. [4] COHEN, G., MOLLER, P., QUADRAT, J.P., VIOT, M., Une th4orie lin4alre des syst~mes h ~v6nements discrets. Rapport de recherche INRIA # 362. Le Chesnay, 1985. [5] CUNINGHAME-GREEN, R.A., Minimax Algebra, Lecture Notes in Economics and Mathematieal Systems, vo1.166, Springer Verlag, 1979. [6] GONDRAN, M., MINOUX, M., Valeurs propres et vecteurs propres dans les dioYdes et leur interpr4tation en th4orie des graphes. E.D.F. Bulletin de la Direction des ls et Recherches. Sdrie C-Math. Info. #2 (1977), pp. 25-41. [7] INAN, K., VARAIYA, P., "Finitely Recursive Process Models for Discrete Event Systems", IEEE Trans. on Automatic Control, 33, 7 (1988) 626-639. [8] MOLLER, P., Th4or~me de Cayley-Hamilton darts les Dio~des et Applications b. l'I~tude des Syst~mes g t~.v4nements Discrets, Lect. N. in Contr. and Inf. Sc. 83, Springer Verlag, 1986. [9] MOLLER, Notions de rang dans les diogdes vectoriels, Comptes Rendus de la Conf&ence "Alg~bres Exotiques et syst~mes h 4v4nements discrets", C.N.E.T. 3-4 juin 1987. [10] OLSDER, G.J., ROOS, C., Cramer and Cayley-Hamilton in the Max-Algebra. Linear Algebra and its Applications, 101, (I988) 87-108. [11] RAMADGE, P.J., WONHAM, W.M., "Modular Supervisory Control of Discrete Event Systems. Proc. 7th Inter. Conf. Anal. et Opt. des Syst., Lect. Notes Control and Information Science, 83, Springer Verlag, 1986. [12] WAGNEUR, E., "ModuloYds and Pseudomodules. 1. Dimension Theory". Discrete Appl. Math., (to appear, 1990). [13] WAGNEUR, E., "ModuloYds and Pseudomodules. 2. Classification of Simple Pseudomodules. Cahier du GERAD G-88-38. [14] ZIMMERMANN, U., Linear and Combinatorial Optimization in Ordered Algebraic Strutures, An. of Discr. Math., 10, North-Holland, New-York, 1981.
MODELLING AND ANALYSING A CLASS FLEXIBLE MANUFACTURING SYSTEM
OF
Liu C h a n g y o u D e p a r t m e n t of A u t o m a t i c C o n t r o l N o r t h e a s t U n i v e r s i t y of T e c h n e l o g y Shenyang, ii0008, P. ~. of C h i n a
ABSTRACT In this paper, a special algebraic structure called path algebra or m a x - a l g e b r a is u s e d for m o d e l l i n g and analysing nxm f l e x i b l e m a n u f a c t u r i n g s y s t e m (FMS) in w h i c h m jobs are p r o c e s s e d on n m a c h i n e s . The p s e u d o - l i n e a r s y s t e m m o d e l with f i n i t e b u f f e r is established. The m o d e l l i n g a l l o w s for e v a l u a t i n g the d y n a m i c behavior of the s y s t e m u n d e r a d e t e r m i n i s t i c and c y c l i c production process. Given any f i x e d p r o c e s s i n g time and routing of processing activities, the s t e a d y - s t a t e r u n n i n g r h y t h m of the s y s t e m can be d e t e r m i n e d from e i g e n v a l u e of the system, i.e., the average w e i g h t of c r i t i c a l c i r c u i t of c l o s e d - l o o p n e t w o r k . Some r e s u l t s a b o u t the d y n a m i c b e h a v i o r of nxm f l o w - s h o p and job-shop FMS and i l l u s t r a t i v e e x a m p l e s are given.
I.
INTKODUCTION
Discrete event s y s t e m (DES) p r o p o s e d in 1980 [IJ is a new kind of s y s t e m s w h o s e b e h a v i o r can be d e s c r i b e d by m e a n s of a series of time-consuming activities, p e r f o r m e d a c c o r d i n g to a c e r t a i n order. The o c c u r r e n c e s of d i s c r e t e e v e n t s corresponding tc starting times or e n d i n g t i m e s of some a c t i v i t i e s can cause the s y s t e m to move from one s t a t e to a n o t h e r . As a t y p i c a l DES, f l e x i b l e m a n u f a c t u r i n g s y s t e m (FMS} h a v e b e e n paid m u c h a t t e n t i o n and s e v e r a l a p p r o a c h e s w e r e d e v o t e d to m o d e l l i n g and a n a l y s i s of FMS. The pseudo-linear system can be set up by using path algebra[Z-41 . In a FMS w i t h m jobs p r o c e s s i n g on n m a c h i n e s , called nxm FMS for short~ there are n x m k i n d s of different activities. S e t t i n g that x { [ - l ~ § (k) d e n o t e s the kth s t a r t i n g time of job W~ processing on m a c h i n e M, , t(i,j) d e n o t e s the t i m e - c o n s u m i n g of job W] p r o c e s s i n g on m a c h i n e M~ , u(k) and y(k) d e n o t e the kth input and o u t p u t time of r e s o u r c e s r e s p e c t i v e l y , then the a b o v e nxm FMS can be f o r m u l a t e d as the f o l l o w i n g p s e u d o - l i n e a r system: (I} X(k) = A X(k) e B U(k} (Z) Y{k} = C X{k) (n+m)xnm where A, B, C are m a t r i c e s w i t h nmxnm, n m x ( n + m ) and dimension respectively. I t e r a t i n g e q u a t i o n (I) r e p e a t l y , y i e l d s {3) X{k) = A"" X(k) ~ A* S U(k) where A* = I ~ A ~ A ~ @ . . - ~ A "'-I (4)
986 If the order-arrangement of m a n u f a c t u r i n g is reasonable, i.e., no v i c i o u s c i r c l e w h i c h r e s u l t s in t h e d e a d l o c k of m a n u f a c turing exists for resource requirmcnt, t h e n A is a nilpotcnt matrix and equation (i) h a s a n u n i q u e o p e n - l o o p solution A*BU(k). Thus, the state and output equations are changed into X(k) = A* B U(k) {5) Y(k} = C A* B U(k} (6} If inputting resources i n t o t h e s y s t e m a g a i n is r e l a t e d to r e l e a s e of r e s o u r c e s l a s t time, i.e., U(k) = G Y ( k - l ) , t h e n t h e following closed-loop s y s t e m is f o r m e d X(k) = A* B G C X { k - l ) (7} Y(k) = C A* B G Y { k - l } (8) where G is a d i a g n a l f e e d b a c k m a t r i x . It is u s u a l l y chosen as u n i t m a t r i x I. The shortcoming of the a b o v e m o d e l is that blocking, an important phenomenon in m a n u f a c t u r i n g s y s t e m is n o t considered. An i m p l i c i t a s s e n t of s y s t e m w i t h i n f i n i t e b u f f e r l e a d s to this inadequacy. In this paper, modelling and analysing nxm FMS with finite b u f f e r is d i s c u s s e d by using path algebra, and a new model taken blocking phenomenon i n t o a c c o u n t is s e t up. In s e c t i o n II, some basic concepts on path algebra and description of orderarrangement of manufacturing are introduced. In section III, modelling on n x m F M S in f l o w - s h o p and job-shop form with finite buffer are analysed. I n s e c t i o n IV, the steady-state running rhythm of s y s t e m is d e t e r m i n e d f r o m s o l u t i o n of t h e eigenvalue p r o b l e m of c l o s e d - l o o p system, and some illustrative examples are given.
II.
BASIC
CONCEPTS
AND
DEFINITIONS
Path algebra, o n e of d i o i d s , is a s p e c i a l a l g e b r a i c s t r u c in w h i c h t w o b a s i c o p e r a t i o n s ~ a n d Q a r e d e f i n e d as a ~ b ~ m a x |a, b} , a O b @ a + b , (9} u a, b ~ D = R U [ - ~ } It is obviously that operation ~ and O are commutative, associative and distributive in t h e s e t D. I n p a t h a l g e b r a , e = 0 is i d e n t i t y e l e m e n t a n d E = - ~ is n u l l c l e m e n t . In t h e c a s e of matrix operation, the m u l t i p l i c a t i o n a n d the a d d i t i o n of m a t r i x in p a t h a l g e b r a w i l l b e d e f i n e d as ture,
s
C = A O B
,
c~j
= 7-'@ ks1
ark
@
bkj
~ max
[aLk
+ bkj }
{I0)
k
C = A O B , cl j = at j 8 bl j = m a x {at ~ , bl j } (II} In path algebra, one can associate a square matrix with a graph. The order of m a t r i x A is t h e n u m b e r of n o d e s in graph G(A) . If al j --k= 6 , t h e n e x i s t s a n a r c f r o m n o d e j to n o d e i, a n d a[ j is called its weight. It is w o r t h p o i n t i n g out that Ak , r a i s e d m a t r i x A to t h e k t h p o w e r , h a s an o b v i o u s g r a p h i c m e a n i n g . (Ak)Lj d e n o t e s t h e m a x i m u m w e i g h t i n a l l p a t h s of l e n g t h k from node j to n o d e i. T h e p a t h in w h i c h i n i t i a l n o d e a n d terminal node coincide is c a l l e d c i r c u i t . The ratio between weight and length, for a n y c i r c u i t in G ( A ) , is c a l l e d i t s a v e r a g e w e i g h t . The circuit with the maximum average weight is called the critical circuit.
987 In a j o b - s h o p m a n u f a c t u r i n g system, the sequence of m a c h i n e s for each job visiting and the sequence of j o b s p r o c e s s i n g on each m a c h i n e m a y be d i f f r e n t each other, w h i c h is u n l i k e in t h e case of f l o w - s h o p manufacturing system. In o r d e r to d e s c r i b e the above sequences, some definitions w i l l be g i v e n as b e l o w . Definition I. aJ ...al) a r e n d i f f e r e n t discrete events, where n-m events not occur, v e c t o r v = lit ...ia] is called sequence vector, if (a). ik = -i, w h e n e v e n t ak n o t o c c u r s , (b}. the integer a r r a y i~ ...i,, is a p e r m u t a t i o n of integer numbers l,...,m, (c) . ik (I ~ k $ m) is t h e f i r i n g o r d e r of e v e n t ak. Definition 2. F o r a n y s e q u e n c e v e c t o r v = [il ...i~ ] w i t h n - m e n t r y -i, v e c t o r w = [jr ...j~] is c a l l e d p e r m u t a t i o n v e c t o r of v, if (at . jk = -I, k = m + l , m + 2 . . . n , (bt. j~ = h, w h e n iL = k. Sequence vector is c l a s s f i e d as r o w a n d column one. For instance, row sequence vector v = [-1,2,-1,1,3,-1] indicates that events fire according to t h e o r d e r of a4 , az , a~ , which is denoted by permutation vector w = [4,2,5,-1,-1,-1]. Definition 3~ A m a t r i x w i t h n x m d i m e n s i o n w h i c h is f o r m e d b y juxtapolation of m n-dimensional column sequence vectors is called column sequence matrix. Consequently, a matrix with nxm dimensien w h i c h is f o r m e d b y j u x t a p o l a t i o n of n m - d i m e n s i o n a l row sequence vectors is c a l l e d r o w s e q u e n c e matrix. Definition 4. For any sequence v e c t o r v = [ii . . . i , ] , v e c t o r u = [j~...j~] is c a l l e d d i r e c t i o n a l v e c t o r of v, if (at. j~ = h = a r g {iL = ik + I} , w h e n ik < n , (b). jk = 0 , w h e n ik = n . Definition 5. For a column (or row} sequence matrix P, matrix P with the same dimension is c a l l e d c o l u m n (or row) d i r e c tional matrix of F, if e a c h c o l u m n (or row) v e c t o r of ~ is column (or row) d i r e c t i o n a l vector corresponding to the same column (or row) s e q u e n c e v e c t o r of P. Based on the above preparations, medelling on n x m F M S c a n be done without a hitch.
III. I.
MODELLING
ON
nxm
FMS
WITH
FINITE
BUFFER
Modelling on nxm Flow-Shop FMS In a flow-shop FMS, jobs visit machines in seri~s and machines process j o b s in t h e s a m e s e q u e n c e . Whether or not a machine fulfils its service to a j o b a n d c o n v e y s it to next machine depends on t h r e e c o n d i t i o n s , i.e., (a}. m a c h i n e is r e a d y , (bt. j o b to be s e r v e d h a s a r r i v e d , (c). t h e r e e x i s t s a t l e a s t o n e i d l e p l a c e in b u t l e r of t h e n e x t m a c h i n e . Setting t h a t Z { L - I ) ~ + j (k) d e n o t e s t h e K t h o u t p u t t i m e of j o b W~ on m a c h i n e Mi , bl d e n o t e s the capacity of b u f f e r of machine Mt , the nxm flow-shop F M S c a n be f o r m u l a t e d as the following pseudo-linear system by using path algebra: Z(k) = A0 Z(k) ~ AL Z ( k - l ) O B U(kt (12) Y(k) = C Z(kt (13) U n d e r the f e e d b a c k input U(k) = G Y(k-l) = G C Z(k-l) , leads to t h e f o l l o w i n g closed-loop system model Z(k) = A~* (At 9 B G C) Z ( k - ! ) ~ M Z ( k - l ) (14) Y(k) = C Z(k} (15t
988 Modelling on n x m F M S is to a s s i g n t h e n o n z e r o matrices in the above expressions according to Considering the above statements about manufacturing the nonzero e l e m e n t s in m a t r i x A0 , AL , B a n d C c a n by the below rules: A0 (im+j, ( i - l ) m + j ) = t ( i + l , j) , Ao ( ( i - l ) m + j + l , ( i - l ) m + j } = t(i, j + l ) , Ao ( ( i - l ) m + j , im+j-b1+~-l) = 0 , Ai ( ( i - l ) m + j , (i+l)m+j-b,§ = 0 , B((i-l)m+l, m+i) = t(i, I) B(j, j) = t(l, j) , c(j, { n - l ) m + j ) = 0 , C ( m + i , im} = 0 ,
elements of some rules. activities, be assigned
i=l ..n-l, j=l...m i=l .n, j = l . . . m - I i=l .n-l, j > b [ § i=l .n-l, j < b i + , + l i=l .n j=l .m j=l. .m i : l . .n
16) 17) lS) 19) 20] ?i) 22) 23)
2.
Modeling on nxm Job-Shop FMS In a job-shop FMS, it is i m p o r t a n t to give a distinct description to t h e s e q u e n c e s of j o b s a n d machines. If column sequence m a t r i x P is u s e d f o r d e n o t i n g t h e s e q u e n c e of e a c h job visiting machines, row sequence matrix Q denoting t h e s e q u e n c e of each machine processing jobs, column directional matrix denoting t h e s e q u e n c e of m a c h i n e f o r e a c h j o b n e x t v i s i t i n g , row directional matrix Q denoting t h e s e q u e n c e of job for each machine next processing, matrix R denoting the distribution of state variables and permutation matrix S denoting t h e s e q u e n c e of j o b s q u e u i n g in b u f f e r of e a c h m a c h i n e , then the nonzero elements in m a t r i x Ao , A~ , B a n d C c a n b e a s s i g n e d b y t h e b e l o w r u l e s : A0 (r(~lj ,j), r ( i , j ) ) = t ( ~ , j , j) , ~j > 0 A0 ( r ( i , ~ i j ) , r ( i , j ) ) = t(i, ~ j ) , ~i~ > 0 A0(r(i,j}, r(pij, s(51j, q(p,j, j)-b(5,~}-l)}) = 0 , q(PtJ , J) > b(~lj )+I, DLJ > 0 Ai ( r ( i , j ) , r(p~j , s(pij , q(pij , j ) + m ( ~ i j ) - b ( p I j ) - l ) ) ) q ( ~ j , j) " b(~1 ~ )+I, ~,~ > 0 B(r(i,j), j} = t(i, j) pij = 1 B ( r ( i , j ) , m+i) = t(i, ji , qlj = 1 c(j, r ( i , j ) ) = 0 , ~ij = 0 C(m+i, r(i,j)) = 0 , ~ij : 0
IV.
DYNAMIC
BEHAVIOR
OF
CLOSED-LOOP
(24 (25
= 0
(26 , (27 (28 (29 (30} (31)
SYSTEM
F o r a n x m FMS, if t i m e - c o n s u m i n g m a t r i x T, column sequence m a t r i x P, r o w sequence matrix Q and state variables distribution matrix K are given, then column directional matrix ~, row directional matrix Q and row permutation m a t r i x S c a n be d e r i v e d . The psuedo-linear system model can be established by using rules (24)-(31}, and its dynamic behavior can be determined from the eigenvalue a n a l y s i s of c l o s e d - l o o p system. L i t e r a t u r e s [ 2-s] s h o w that the eigenvalue is j u s t t h e s t e a d y - s t a t e running rhythm of closed-loop system, so w e p u t e m p h a s i s u p o n e i g e n v a l u e analysis of s y s t e m . N o w s o m e r e l a t i v e t h e o r e m s a r e g i v e n . Definition 6 [z] F o r a s q u a r e m a t r i x A, if t h e r e e x i s t a r e a l n u m b e r A a n d a v e c t o r V w h i c h e n t r i e s a r e n o t a l l e q u a l to 8, s u c h t h a t A V = A V, t h e n A is c a l l e d e i g e n v a l u e a n d V is c a l l e d eigenvector.
989 Theorem 1 [z] The eigenvalue of a n i r r e d u c i b l e square matrix j u s t is t h e a v e r a g e w e i g h t of c r i t i c a l circuit in g r a p h G ( A ) . The eigcnvalue of a n x n s q u a r e m a t r i x A in m a x - a l g e b r a sense c a n be c a l c u l a t e d as f o l l o w ; A = v-~ {32)
A
i=i
where
~' = ~ ~ ~z ~ ... 8 ~" 9 Definition 7 [2] . A s y s t e m is s t a b l c i f f t h e r e number A , such that for all initial conditions and i, it h o l d s t h a t l i m [XI (k)] *''k = A 9
(33) exists a real all subscript
Theorem 2 [=] . A s y s t e m is s t a b l e i f f t h e r e e x i s t a real n u m b e r A, i n t e g e r d a n d k0, s u c h t h a t f o r a l l i n i t i a l c o n d i t i o n s , it h o l d s t h a t X ( k + d ) = A d X(k) , ~ k ) k0 . The above definitions and theorems suggest a relation between the eigenvalue and the stead-state running rhythm of system. N o w w e t a k e s o m e s i m p l e F M S as t h e i l l u s t r a t i o n s . Example i. In a f o l l o w - s h o p FMS with three jobs processing on two machines, its p a r a m e t e r matrixs a r e g i v e n as f o l l o w :
T=
P= 6
5
O=
4
2
2
2
1
Assuming that the capacity of b u f f in m a c h i n e t h e n t h e m a t r i x Ao , Al., B, C, Ao* a n d M in 12) determined as f c l l o w b y u s i n g (16)-(23}
A0
=
~ 3 . . 0 . . . . . . 9 5 . 5 . 4.4
. . . . .~ ] ~q B = . . 3 . _|
.
.
At
=
.
.
!!!iliI
.
.
A0*=
.
0 .... 0 C . . . . . . 0 0 . . . . . 0
M
2
3
Mz is b~ a n d (14}
=
0, are
..... 0
12 5 0 5 0 6 . 0 125 . 5 0 1 6 9 4 9 4 0
2 2 . 0 9 9 ~ 7 14 14 12 12
=
8
8
14 18
14 18
6 12 16
where
12 16
t h e d o t in m a t r i x d e n o t e s 6 . L i t e r a t u r e [~] s h o w t h a t t h e e i g c n v a l u e of c l o s e d - l o o p system with infinite b u f f e r is A = 15. In this example, the eigenvalue of c l o s e d - l o o p system (14) is A = 16, which shows that blocking phenomenon leads to the i n c r e a s e in the steady-state running rhythm of system and the system model with finite buffer can given more suitable description for maunfactuing system 9 Example 2. In t h i s e x a m p l e , we consider a c a s e of 3 o b s h o p manufacturing s y s t e m . Its p a r a m e t e r m a t r i x T, P, Q, E, ~, Q a n d S a r e g i v e n as f o l l o w ;
T
=
p = 6
5
4
Q = 2
2
!
3
2
1
990
[: .o.-3] [2 2 o] R=
B'= 4
5
[2 3 o]
[~. 2 ~]
~=
6
0
0
s =
1
0
1
2
3
2
!
Assuming that the capacities of b u f f e r in m a c h i n e M~ a n d M~ bl = bz = 2, t h e n m a t r i x Ao, Ai , B, C, Ao* a n d M in ( 1 2 ) - ( 1 4 ) determined as f o l l o w b y u s i n g r u l e s ( 2 4 ) - ( 3 1 ) .
arc are
~
~
~0
~
.
r~
~
AI
Ao*
=
|i0
3
1
5
[:!,:
0
-' . 0
5 0
0
B
=
.
.
.
2 2 9 9 7 12 12 I0 20 20 18 14 14 12 4 4
0
2..2 7
. . . . .
C =
.
9
0
.
M
0
=
0 0
]
.
The
cigcnvalue of c l o s e d - l o o p s y s t e m is A = 20. w h e n bt < 2 (i = 1,2), Ao is n o t a n i l p o t e n t r e s u l t s in d e a d l o c k p h e n o m e n o n 9 E x a m p l e 3. In this example, m a t r i x Q, ~ a n d e x a m p l e is c h a n g e d i n t o
Q =
~_ : 3
2
=
0
1
=
3 in m a c h i n e C, A0, a n d (24)-(31)
Ao
2
1
Mi a n d M~ arc M in ( 1 2 ) - ( 1 4 )
* =
35 o 3 6 11 o ; : : 5
C
=
.
.
o 5
.
.
. :2 9 12
o..] . . . .
The
above
the
01j iii?ii I 50
~: :::~| A,
6;.. 6; "o''"
2.~:~ :] =
S in
2
.
B
which
s =
1
Assuming t h a t t h e c a p a c i t i e s of b u f f e r bl = bz = I, t h e n t h e m a t r i x Ao , A~ , B, arc dctermincd as f o l l o w b y u s i n g r u l e s
A0
matrix,
0
.]
o
0
.
0
. 0
M
=
12 7 12 . 18 15 18 12 9 12 4 4
.
eigenvalue of c l o s e d - l o o p s y s t e m is ~ = 15. W h e n b~ < 1 (i = 1,2), Ao is n o t a n i l p o t c n t m a t r i x , which results in deadlock phenomenon. In job-shop manufacturing system, cloumn sequence matrix P usually is g i v e n a c c o r d i n g to t h e t e c h n o l o g i c a l requirements of jobs, but w e c a n c h o s e the s e q u e n c e of j o b s o n e a c h m a c h i n e to reduce the steady-state running period. The difference of d y n a m i c
991 behavior of closed-loop system between different routings has b e e n s h o w n in the l a s t two e x a m p l e s 9 E x a m p l e 4. In t h i s e x a m p l e , t h e m o r e g e n e r a l c a s e of j o b s h o p is c o n s i d e d . Its p a r a m e t e r m a t r i x T, P, Q, K, P, Q a n d S a r e g i v e n as f o l l o w :
T=
3 [
4
a =
P--
22 3-1
3-1
4 7 -
~ =
3 0
3 0 0 -I
o _ - - 1 2 3 1 2-1
~ :
2 2
3 0 0 -i
s :
i 1
Assuming t h a t the c a p a c i t i e s of b u f f e r in m a c h i n e a r e bz = b3 = 0, t h e n t h e m a t r i x A0 , AI , B, C, A0* a n d (14) a r e d e t e r m i n e d as f o l l o w b y u s i n g r u l e s ( 2 4 ) - ( 3 1 )
AG
=
9
,
5
. . 0
2
. 2
9
.
9
1
9
o
0
.
.
--
.
.
.
.
5
.
.
.
.
.
.
I
9
1
.
.
.
.
.
.
.
.
.
.
3
.
.
.
c
4
0
83030 4. . 0 .J
5
.~
o
.
]
Ao*=
At
3
.
.
~
.
Nz a n d M3 M in (12)-
00
]
~ . 0
3 B
.
. . .
"2 2 -
. .
.
.
.
.
.
0
.
.
.
9
0
.
.
.
M
.
0 .
.
.
0
=
6
.
.
5
3
-
3 3 0 q I0 i0 9 9 . . Ii II 7 7 i ; i [ I0 I0 .
0
.
.
.
.
. 0
.
.
7 3 . 3 0
.
.
.
The eigenvalue of the c l o s e d - l o o p system is A = Ii and the period-order is d = I. Now, we consider the influence of the c a p a c i t y of b u f f e r o n t h e d y n a m i c b e h a v i o r of c l o s e d - l o o p s y s t e m . In t h i s e x a m p l e , w h e n the c a p a c i t y of b u f f e r in m a c h i n e M2 is i n c r e a s e d to bz = I, the eigenvalue of c l o s e d - l o o p s y s t e m is A = I0. A n d w h e n t h e c a p a c i t y of b u f f e r in m a c h i n e M3 is a l s o i n c r e a s e d to b3 = bz = i, then the eigenvalue a n d the p e r i o d - o r d e r of c l o s e d - l o o p system are c h a n g e d i n t o A = 9.5, d = 2, w h i c h is s a m e to the c a s e of s y s t e m w i t h i n f i n i t e b u f f e r s h o w n in l i t e r a t u r e [7]. T h u s , d e s i g n of t h e capacities of b u f f e r in e a c h m a c h i n e is a p r o b l e m of c o n s i d e r a b l e i n t e r e s t in f l e x i b l e m a n u f a c t u r i n g system.
V.
CONCLUSIONS
In this paper, Modelling o n n x m F M S is a n a l y s e d by using p a t h a l g e b r a 9 B a s e d o n the p s u e d o - l i n e a r system model with finite buffer, the d y n a m i c b e h a v i o r of c l o s e d - l o o p s y s t e m is analysed from the eigcnvalue of s y s t e m 9 The illustrative examples show that this new model can give a more suitable description for manufacturing system 9
992 REFERENCES
[I]
Ho, Y.C. and Cassandras, D i s c r e t e - E v e n t Systems, on D e c i s i o n and Control,
[2]
Cohen, G., Dubois, D., Quadrat, J.P. and Viot, M., A LinearS y s t e m - T h e o r e t i c View of D i s c r e t e - E v e n t Processes, Proc. of 22nd IEEE C o n f e r e n c e on D e c i s i o n and Control, San Antonio, Texas, 1983, pp. I039-I044.
[3]
Cohen, G., Moller, P., Quadrat, J.P. and Viot, M., Linear System T h e o r y for Discrete Event Systems, Proceedings of 23rd IEEE Conference on D e c i s i o n and Control, Las Vegas, Nevada, 1984, pp.539-544.
[4]
Cohcn, G., Dubois, D., Quadrat, J.P. and Viot, M., L i n e a r System-Theoretic V i e w of D i s c r e t e - E v e n t P r o c e s s e s and Its Use for P e r f o r m a n c e E v a l u a t i o n in Manufacturing, IEEE Trans. on A u t o m a t i c Control, AC-30, 1985, p p . 2 1 0 - 2 2 0
[5]
Xu Xinhe, Decision,
[5]
Xu Xinhe, Modelling and D y n a m i c Analysis for A class Discrete-Event Systems, J. of Northeast University Technology, No.3, 1987.
[7]
Dubois, D. and Stecke, K., U s i n g Petri Nets to R e p r e s e n t P r o d u c t i o n processes, P r o c e e d i n g s of 22nd IEEE C o n f e r e n c e on Decision and Control, San Antonio, Texas, 1983, p p . i 0 6 2 - ! 0 6 7
C., C o m p u t i n g C a s t a t e Variables for P r o c e e d i n g s of 19th IEEE C o n f e r e n c e Albuquerque, 1980, pp.697-700.
Linear Discrete Event Systems, No.3, No.4, 1987.
J. of Control and
of of
