Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner
99 S. P.Bhattacharyya
Robust Stabilization Against Structured Perturbations
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
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 Author Shankar P. Bhattacharyya Department of Electrical Engineering Texas A & M University College Station Texas 7 7 8 4 3 USA
I S B N 3 - 5 4 0 - 1 8 0 5 6 - 7 S p r i n g e r - V e r l a g Berlin H e i d e l b e r g N e w Y o r k I S B N 0 - 3 8 7 - 1 8 0 5 6 - 7 S p r i n g e r - V e r l a g N e w Y o r k Berlin H e i d e l b e r g Library of Congress Cataloging in Publication Data Battacharyya, S. R (Shankar R), Robust stabilization against structured perturbations. (Lecture notes in control and information sciences; 99) Bibliography: p. 1. Perturbation (Mathematics) 2. Control theory. 3. System design. I. Title. I1. Series. OA871,B47 1986 515.3'53 87-16515 ISBN 0-387-18056-7 (U,S.) 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 version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-verlag Berlin, Heidelberg 1987 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 216113020-543210
This monograph is dedicated to my parents and to my brother Tarak P. Bhattacharyya
PREFACE
This book deals with the analysis and design of control systems for plants which contain physical parameters subject to perturbation. The physical parameters could consist of masses, incrtias, spring constants, aerodynamic coeflicicnts etc., that are required in the mathematical description of the dynamics of the plant. In engineering, one frequently encounters situations where the structure of the plant and the nominal values of these parameters arc known, but the parameters undergo large perturbations as the operating conditions of the control system change. This problem cannot bc treated within the framework of the familiar theory of robust control where transfer function norms are used to describe the plant perturbation class. The latter class of perturbations is unstructured as opposed to the highly structured class of perturbations that is relevant here. W e consider linear time invariant systems and focus on the problem of closed loop stabilily under perturbations of a real parameter vector representing the physical parameters. Our objectives are a) to analyze the stabilityof the closed loop system for prescribed ranges of perturbation b) to estimate the size of the stability region in this parameter space as a function of controller parameters and c) to thereby design controllers that provide adequate stability margins. Solutions to the above problems are developed in the transfer function and state space domains at.both the theoretical and the algorithmic, computational levels. Some auxiliary, related problems, dealing with feedback stabilization with controllers of low dynamic order are also considered. The results described in the book were mostly obtained by the author and his coworkers i~l the last two .years. I would llke to thank Leehyun Keel for several ideas that appear in Chapters 4-7, and for doing most of the computational work. I am grateful to Radek Biernacki for collaborating with me on work leading to the results of Chapters 2 and 3. Humor S. Hwang did the examples in Chapter 3. I thank Bob Barmish and R.K.Yedavalli for several useful discussions on structured perturbations, and John Fleming for suggesting
VI many improvements to the initial draft, of the manuscript. It is a pleasure to acknowledge tile support and encouragement of my longstanding friend and colleague, Jo Itowze. As for Boyd Pearson, it is impossible to thank him for everything he has taught me. I acknowledge the National Science Foundation's finandal support of this research. I am grateful to Didi who urged me to write this book. She and Supriya provided solid support at crucial periods over the last two years for which I am thankful. Finally I am very grateful to Mary D. Sehlhoff for her expert typing of the manuscript.
March 23,1987 College Station, Texas
S.P. Bhattacharyya
TABLE
CHAPTER
1
BACKGROUND I.I
OF CONTENTS
AND
PRELIMINARIES
Introduction ................................................................
1
1.2 Structured and unstructured perturbations ..................................
2
1.3 IIurwitz regions in coefficient space ..........................................
5
1.4 State space perturbations ...................................................
8
1.5 Discussion of contents ......................................................
10
1.6 Proof of Kharitonov's T h e o r e m .............................................
14
CHAPTER THE
2 STABILITY
HYPERSPHERE
IN P A R A M E T E R
SPACE
2.1 Introduction ...............................................................
23
2.2 Problem formulation .......................................................
24
2.3 T h e stability liypersphere: Linear case ......................................
26
2.4 T h e stability hypcrsphere: Affinc case ......................................
36
2.5 Proof of the main rcsnlt ....................................................
42
2.6 Solution of the robust stability problem ....................................
48
CHAPTER
3
STABILITY
ELLIPSOIDS
AND PERTURBATION
POLYTOPES
3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.2 T h e s t a b i l i t y ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.3 P o l y t o p e s o f p e r t u r b a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.4 C o n t r o l l e r d e s i g n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
: ...........................
3.5 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
VIII CHAPTER
4
ROBUST STABILIZATION: THE GENERAL CASE 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.2 Characteristic p o l y n o m i a l calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.3 Stability m a r g i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.4 Robustification procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.5 E x a m p l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
CHAPTER
5
STRUCTURED
PERTURBATIONS
IN S T A T E
SPACE
MODELS
5.1 Introduction .............................................................. 100 5.2 Stability margin and robustification ....................................... 100 5.3 Example .................................................................. 108 5.4 Appendix ................................................................ 111 CHAPTER
6
STABILIZATION
WITH
FIXED
ORDER
CONTROLLERS
6.1 Introduction .............................................................. 116 6.2 Necessary conditions using linear programming ............................ 117 6.3 Sufficient conditious using the stability hypersphere . . . . . . . . . . . . . . . . . . . . . . . .
124
6.4 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
CHAPTER
7
STATE SPACE DESIGN OF LOW ORDER REGULATORS 7.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
: .....................
133
7.2 T h e Sylvestcr e q u a t i o n f o r m u l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133
7.3 O u t p u t feedback controllcrs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
7.4 Stabilization a]gorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
7.5 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
IX 7.6 A p p e n d i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER SUMMARY
157
8 AND
FUTURE
RESEARCH
8.1 S u m m a r y ................................................................. 162 8.2 Research Directions ....................................................... 163 REFERENCES
.................................................................. 165
CHAPTER BACKGROUND
1
AND PRELIMINARIES
1. I N T R O D U C T I O N
The design of a control system is invariably based on an assumed nominal model of the plant to be controlled. After the usual shnplifications, such as linearization about an operating point, lumped parameter approximation etc., one ends up most often with a linear time invariant system described by a prescribed set of differential equations for the nominal model of the plant. At this point, various established design strategies such as the frequency domain methods of classical control, or the state space methods of optimal control theory can be applied to the nominal model to produce a feedback controller that yields closed loop slability and au acceptable output time response. An important and fimdamental practical requirement to be satisfied by the controller is the invarlance of the property of closed loop stability, under perturbations, from a suitable class, of the nominal plant modeh A controller satisfying this requirement is said to be robust with respect to the prescribed class of perturbations. The theory of analysis and design of such controllers is currently an active area of research in control theory (see for example, [1]-[43] and references cited therein.) The specific description of the class of perturbations against which robustness is required depends, of course, on the physics and engineering of the particular plant in question. The general theory, however, distinguishes broadly between two types of perturbation classes-structured and nnstructured.
2 2. S T R U C T U R E D
AND UNSTRUCTURED
PERTURBATIONS
In the traditional unstructured approach [1]-[4] to dealing with perturbaiions, the nonfinal plant is represented by the transfer function matrix Go(s) and the perturbed plant by the transfer function matrix G(s) = Go(s) + AG(a). The class of perturbations to be handled is described by assuming that, for a given stable proper rational function
R(~) [IAG(j,,)[I < IR(i,,)h
V~eR
(2.1)
where
IIAG(j~)II := a[AG(j~0)]
(2.2)
and ~[.] denotes the maxinmm singular value. Suppose now that C(s) is the transfer function of a feedback controller that stabilizes the nominal plant Go(S). The main result of the theory of unstructured robust stability is a necessary and sufficient condition for C(s) to stabilize the entire family of perturbed 1)]ants described by (2.1). This result states that, under some mild teclmical assumptions, C(s) stabilizes all perturbed plants determined by (2.1) iff
HC(I + GoC)-I(jw)H • [R(j,~)1_< I VwcR.
(2.3)
Tile above result, which was proved in [1], is useflll for checking the robustness of a given controller. The condition (2.3) has also been used in [2] to develop a synthesis procedure in tile case of single input single output systems. A similar condition has been stated in [1] for norm bounded nmltiplicative perturbations. The class of perturbations described by (2.1) is unstruclured in the sense that the norm bound (2.1) allows perturbations A G ( s ) to occur in '%11 directions" in the appro-
3 priate space of transfer functions. II is our view that many engineering problems cannot he dealt with adequately using this approach. This is due to the fact that the dynamical equations of most engineering plants, such as aircraft, robots, and chemical processes are usually known. Thus, good mathematical models are available and the system structure is well known qualitatively but there exists uncertainty regarding the numerical values of various physical parameters in the model. Spring constants, masses and inertias, reaction rates, and aerodynamic coemcients are conamon examples of such parameters. The uncertainly in turn may be due to the inability to measure various physical quantities, actual variations of parameters due to aging or to changes in the operating conditions of the system. There also exist uncertainties or errors ill the modelling process which take the form of changes in transfer function coemcients or perturbation of the state space matrices. Such perturbations are only remotely related to any transfer function norm. In fact the class of unstructured perturbations determined by a transfer fimction norm bound generates a very rich class of systems. Design based on protection against such a large class of perturbations may result in very conservative systems when only physical parameters are subject to perturbation.
For these reasons, there is a growing interest (see (5]-[43])
in the structured perturbation robust control problem. The distinct approaches that are develol,lng may be classified into the polynomial approach [5]-[23], the Lyapunov or state
space based approache~, [22]-[36], the tt synthesis approach [37]-[40], and the multi model simultaneous stabilization approach [41],[42].
In this monograph, we present some new results on this problem using the transfer function (Iiurwitz) and tile state space (Lyapunov) approaches. It. will be assumed that the
4 plant transfer function matrix G(s) or the plant state space model is dependent on a real I~arameter vector p representing physical parameters with nominal value p0 and A p := p _ p0 represents a perturbation. Clearly, larger (smaller) values of
IIApll~
correspond to
larger (smaller) perturbations. For a given stabilizing controller there exists a largest value p(p0) of JlAp IJ2 for which closed loop stability is preserved. This value therefore serves as a measure of stability margin. Based on these considerations we formulate several problems to be solved in the next few chapters. Problem A Determining the Largest Stability Hypersphere For a given stabilizing controller C(s) deternfine the radius p(p0) of the stability hypersphere centered at p0 defined by the condition that whenever [[Apl[2 < p(p0) tile closed loop system with plant parameter p0 + A p is stable and there exists at least one perturbation A---~with Ilt,---~[12 = p(p0) such that the closed loop system with the parameter p0 + ~
is not. stable.
Problem B Robust Controller Design Let. C(s) denote a stabilizing controller with an adjustable l~arameter vector x E R'. Determine a procedure to choose x so that the ra.dius p(p0, x) of the stability Ilypersphere is increased as a function of x until it contains a given class of perturbations {,tip}. Solutions to tile above two problems will be developed in Chapters 2-5. These solutions will also allow us to treat the following special types of perturbation classes : (i) Api, the
i th
component of Ap , is bounded by -71 < APl < e~
for given positive numbers ~ , el, i = 1, . . . ,k.
(2.7)
5 (it) The perturbation bounds are given by
-'u,i~ < Apt < wit , i = 1 . . . .
,k
(2.8)
where wi are weights and ~ is a positive constant. A natural solution to the problem with the above perturbation classes will be obtained within the framework of our approach by determining the largest stability ellipsoid in parameter space. The justification for using p ( p ° , x ) as a stability margin is that if xl and x2 are two ~'ontrollers with p ( p ° , x j ) > p(p°,x2)
(2.9)
then clearly x, is "more robust" than x2 because the correspo,ding stability i~ypersphere denoted by S 0 ( p ° , x l ) is larger than So(p°,x2) and in fact
So(p°,x2) C S p ( p ° , x , )
so that the family of perturbed plants guaranteed to be stabilized by xl contains tile family thai is guaranteed to be stabilized by x2. Generally speaking, the problem described above deals with the preservation of the Hurwitz property (i.e. roots in the open left half plane) of a set of polynomials generated by perturbing the coefficieuts of a nominal polynomial. We proceed to describe, in the next section, two important recent results on this problem. 3. H U R W I T Z
REGIONS
IN COEFFICIENT
SPACE
Let +
+ ...+
(3.1)
6
denote a Hurwitz polynomial that represents for example, the characteristic polynonfial of a closed loop control system for nominal values of tlle plant and controller parameters. Under plant parameter l)erturbations the coefficients of this polynomial change and i1 is of interest to determine a) whether all the polynomials obtained by perturbing each coefficient within a prescribed interval are Hurwitz and b) the size of the largest stability hypersphere in this coefficient space centered at the nominal coefficient vector. The first of these problems was solved in [5] and has come to be known as Kharitonov's theorem. The second problem was solved in [6]. We state both results below. Consider the family of polynomials {5(s)} given by
~(s) = '~o + tijs + .-- + ,S,s"
(3.2o)
with each coefficient varying continuously within a prescribed interval:
xi <_ bi <_"Yi, i = 0, l , . . . n .
(3.2b)
T h e o r e m 3.1 ( K h a r i t o n o v ' s T h e o r e m ) .The family of polynomials (3.2) are all Hurwitz if and only if the four polynomials
~1(~),~2(~),t;3(~) and t;4(~) given below are Hurwitz: ~1(S) = Yo + ZlS + Z232-{'-y3~ 3 + y484 +xss s +x6s 6+...
(3.3a)
~2(S)----yo + y l S + Z 2 a 2 + x 3 s 3 + y 4 s4 +yss 5 +z6s 6 +...
(3.3b)
~3($) = Zo + Zl s + y252 @ y3S3 + X4S4 +xss s+y6s 6...
(3.3c)
7 (~4($) = ;E° _{_ ~ I S "~- ~2S 2 "3" ~3 ~3 + X4 s4
+yss s + y e s e + ...
(3.3d)
This powerful theorem was first proved in [5]. An alternative system theoretic proof was given in [7]. It is emphasized that the theorem reduces file formidable task of verifying the Hurwitz property of the infinite family of polynomials (3.2) to the simple one of checking the stability of the four prescribed "extreme" polynomials (3.3). An independent, simple proof of this theorem, due to Herv~ Chapellat [69], is given in Section 6. This proof is similar to Kharitonov's but illustrates more clearly tile intuitive content of the theorem. The above theorem is useful for analysis because it solves the problem of determining the stability of the vectors 6° + A~ as A~; varies within a prescribed hypercube. For synthesis and design problems it is of importance to find out the largest such stability hypercube but this is a dimcult unsolved problem. On the other hand, the problem of determining the largest stability hypersphere in the space 6 has been recently solved in [6]. To state this important result let
:,o := {~16eR " + ' , 6o = 0}
(3.4a)
: ' , := { ~ I ~ R " ~ ' , ~ . = 0}
(3.4b)
A(w) := {~[6eR"+',g(s) = (s 2 + co2)e(s), t(s) arbitrary}.
(3.4c)
and for any real
Note that Ao, A , and A(w) are subspaces of R "+l and correspond respectively, to polynomials g(s) with roots at s = 0, s = eo, and s = :l:j¢o. Let do,d, and d(¢o) denote the Euclidean distances between tile nominal vector go and the subspaces Ao,A,, and A(w)
respectively. Let
d:=
inf
0<,,.,_<~o
d(w).
(3.5)
T h e o r e m 3.2 _T.he radius of the largest stability hypersphere in the coefficient space ~_ centered at b ° is given by
p(6 °) = min{do, d,, d}.
(3.6)
This theorem was proved in [6] where formulas for the calculation of do, d , and d(to} were also given. The result is nice because it gives a simple calculation to estimate the size of the Hurwitz region in the coefficient space 6 by fitting the largest stable hypersphere centered at ~o into this region. In Chapter 2, we solve a more useful and general version of this problem applicable to the plant parameter space. 4. S T A T E S P A C E P E R T U R B A T I O N S The state space description of dynamic systems are often based on tile choice of physical quantities as state variables. In this description tile matrices representing the model contain various physical parameters as entries. The robust stability and stabilization problems can be formulated more meaningfully in tile state space, in such cases. Let
~:(t) = A ° x(O
(4.1)
represent the dynamics of a stable system. Under parameter perturbations A ° changes to A ° + AA. The problem of determining bounds on the elements of AA that guaxantee stability of the matrices A ° + AA has been studied in the literature using Gershgorin's
9 theorem [36] and also Lyapunov methods [241-[28]. Patel and Toda [28] have given the following result: Theorem
4.1 [28]
If A ° is stable A ° + AA remains stable if the elements AAij of the n x n matrix AA satisfy IAAi;I <
1
namax(P)
(4.2)
where P is tile unique positive definite solution of the Lyapunov equation
ATp + PA + 2In =0
(4.3)
A structured version of this bound has been given by Yedavalli in [24], where the right hand side of (4.2) was replaced by 1
~'max(tPW.),
(4.4)
where U, is an n x n matrix wlmse entries are 0 or 1 corresponding to fixed or variable entries respectively, in A, IPI denotes the matrix P with each entry replaced by its absolute value, and (-)s denotes the symmetric part of the matrix in brackets. In the control problem the matrix A will represent the closed loop system containing plant and controller. Moreover, instead of arbitrary perturbations, the structure of the plant must be considered. In fact, the results of [43] have shown that any conclusion drawn by allowing arbitrary perturbations of state space models can be very misleading. Thus, certain elements of A must remain fixed and perturbation free either by definition of the variables or because they originate from controller parameters that are fixed. To synthesize tile controller the perturbation bounds should be determined as a function of controller
10 parameters so that tl,ey may be enlarged by appropriate choice of these parameters. This problem is treated in Chapter 5, using the Lyapunov based approach. 5. D I S C U S S I O N O F C O N T E N T S
The results described in sections 3 and 4 above are elegant, mathen~atically, but are not adequate to deal with the control problem, as they ignore all structural information about the plant and controller. This, in turn, is due to the assumption in both Theorems 3.1 and 3.2 that the characteristic polynomial coefficients perturb independently. Likewise, in Theorem 4.1 all elements of A are subject to perturbation. For the control problem, the closed loop characteristic polynomial coefficients are functions of the controller and plant parameter vectors and only the latter is subject to perturbations.
Kharitonov's
theorem, unfortunately, does not generalize in an obvious manner to the space of plant parameters. Despite this drawback, the same type of problem has been treated in several recent papers, on the stability of sets of polynomials, (see [5]-[16]). Of these papers [8]-[14] deal with independent perturbations of the coefficients of the closed loop characteristic polynomial and various sufficient conditions for stability are given. In [13] and [14] the geometry of Hurwitz polynomials for discrete systems systems has been studied, again in the space of coefficients of the closed loop characteristic polynomial.
The maximization of tile general "box-type" kind of perturbations described in (2.7) has been dealt with in [16]. A special case of (2.8) was treated in [151 where p was considered to be the closed loop characteristic polynomial coefficient vector and Kharitonov's theorem was used to determine the maximum value of t. However, these results are necessarily conservative because the coefficients are, once again, assumed to perturb independently.
11 These considerations motivate us to formulate the problem of calculating the stability margins and stability regions directly in terms of the parameter p. In Chapter 2, We calculate the largest stability hypersphere in tile parameter space p under the assumption that p enters linearly or affinely into the plant characteristic polynomial coefficients. In Ct,apter 3, under tile same assumption, we determine conditions for stability under the polytope of perturbations (2.7). From the point of view of analysis (i.e. controller is given) these results will constitute a generalization of Theorems 3.1 and 3.2 of this chapter. Tile synthesis problem will be treated by us throughout, under tile assumption that a real vector x of fixed dimension completely parametrizes the controller.This corresponds to fixing the controller order. Tile design problem in robust stabilization is to choose x so that stability is guaranteed for prescribed ranges of parameter perturbations. The solution of the design problem given in Chapters 2 and 3 is facilitated by tile fact that the stability regions are calculated explicitly in terms of controller parameters. The results of Chapters 2 and 3 are based on the report [20]. In Chapter 4, the general case ofnonlinear dependence of the characteristic polynomial coefficients on tim parameter
is considered. The results of Chapter 2 are applied to
this problem after a preliminary calculation of the characteristic polynomial in a hnear separated form. In this case, tile stability margin calculated may be conservative since it does not correspond to the largest stability domains. These results extend the earlier results of [18] and f191. Chapter 5 presents a state space treatlnent of the structured robust stability and stabilization problems. Tile Lyapunov based approach of [24] and [28] is extended to take
12 the structure of tlle plant perturbations into account and to define a stability margin as a function of the controller parameters. The design or robustifieation algorithm gives a nmnerical procedure to increase this margin. The results of Chapter 4 and 5 were originally reported in [23].
Tile procedures developed in Chapters 2-5 are applicable once a stabilizing controller has been found. An important practical consideration in controller design is that the design parameter vector that yields robust stability should not be of unduly high dimension as this bogs down the subsequent design process where adjustments to the controller must be made to attain other objectives. This motivates us to develop some results on stabilization with fixed or low order controllers. In Chapter 6, we use the characteristic polynomial fornmlation and results from linear programming and tile stability hypersphere calculation of Theorem 3.2 to give new bounds on the order of stabilizing controllers. In Chapter 7, an algorithm is developed for stabilization with a controller of low dynamic order, using the Sylvester equation formulation of [44][45] and [46] extended to the ease of dynamic compensation. These results were first reported in [47]. The contents of Chapters 4-7 were developed as part of the Ph.D. thesis of L.H. Keel [22]. Chapter 8 summarises tile results and discusses future research directions.
Ill Chapters 3-7, we have included some numerical examples to illustrate the design procedures. Tile examples in Chapter 3 were solved by Humor S. Hwang [17]. Tile algorithms and example problems in Chapters 4-7 were programmed by L.H.Keel [22]. Many of these involve tile optimization of a function (for example, stability margin) with respect to several variables (say, controller parameters). These have all been executed using the
13 standard optimization routines available in tile Harwell library [48]. Whenever possible the gradients of the objective functions have been explicitly calculated in the text. Ill each of the cases considered, however, nothing is known about the geometry of the objective function and MI that can be said is that the algorithms provide descent or ascent procedures to find local optima. This is not usually a serious limitation because these algorithms are to be viewed merely as devices to automate computational procedures to sequentiaUy improve a given design until a satisfactory controller is obtained. We emphasize that the results described here are necessarily preliminary because the problems treated here do not deal with performance measures related to system response such as tracking [49], disturbance rejection and transient response [50],[51]. Their aim, however is to provide effective and simple solutions to the problem of obtaining robust stability against real plant parameter variations occuring in prescribed ranges, with controllers of low dynamic order. Although this is an essential first step in many design procedures, it has received very little attention in the control theory literature. We expect that the results given here will provide some helpful tools to the practicing control engineer and will also spur the development of a more complete and effective theory of design. .Conventlon for r e f e r e n c e s :
In the rest of tile book we have adopted the following
convention to refer t.o equations, theorems and lemmas: A reference to equation (4.3) in a given chapter denotes equation (4.3) in Section 4 of the same chapter, while Theorem 6.5.4 denotes Theorem 5.4 in Chapter 6, Section 5.
14 6. P R O O F
OF KHARITONOV'S
THEOREM
The proof depends on the well known interlacing property of the roots of the odd and even parts of stable (i.e. strictly Hurwitz) polynomials, given, for instance, in [70, pp. 271]. Let p(s) be an arbitrary polynomial and write
p(s) =
p¢(s)
+
even degree terms
po(s)
(6.1)
odd degree terms
T h e o r e m 6.1170] The polynomial p(s) is stable if and only if the.leadin~ coe~cients of v~{s) and p,(s) are..of the same sign and the roots of pr(s) and pn(s) all lie on the imaginary axis, are nonrepeated~ and interlace:
"'"
< --Wol < --Wel < 0 < u)e I <
w o l < we~ ,
...
where :t:]wei are the roots of pels) --- 0 and O, ±~w,i are the roots of pa(s) : O. This theorem is illustrated in Figure flA. We shall say loosely, that p~(s) and po(S) interlace if they satisfy the conditions of this theorem. We will also need the following lemmas. L e m m a 6.2 Let
p,(s)=p~(s) +po,(S)
(6.2a)
p2(,)=p~(,) +po2(,)
(6.2b)
denote two stable polynomials of the same degree with the same even part p~(s) and differing odd Darts v.l(s) and w2(s) satis[vin~:
pol(jw)
<
Po~(j¢,')
V ~ e [0, ¢¢]
(6.3)
-
~ /Pe
(j~o)
Figure 6.1
I l l u s t r a t i o n of T h e o r e m 6.1
Po "xCj~)
16 Then
p(s)=po(,)+po(S) is stable for every po(s) satisfying P°'(J'~) < P°(J') j,, j,,,
v ,, • I0,oo].
< P°~(J~') j~,
(6.4)
Proof Since p~(s) and p~(s) are stable, po~(s) and also poz(S) interlaces with pc(s). It is easy to show that po(S) has the same degree as Pol (s) and po~(s), and that their leading coefficients all have the same sign.
Then the condition (6.4) forces po(S) to interlace with p¢(s).
Therefore, pc(s) + po(s) = p(s) is stable. This lemma is illustrated in Figure 6.2. The dual of this result is stated next, without proof. L e m m a 6.3 Let p~(s) = p~,(s) +po(S)
(6.5a)
p~(s) = p~2(s) + po(S)
(6.5b)
denote two stable polynomials of the same degree with the same odd part p,~(s) and differing even parts p~_l{s) and v~2(s) satisfying
po,(j~) < po2(j~)
v
~ • [0, ¢¢].
Then
p(s) = po(s) + po(S)
(6.6)
L__//
f
Figure 6.2
Illustration of L e m m a 6.2
Pe ( J ~ )
j~
po,(j (o)
Po (J ~ ) (j~)
j~
P02(] (~)
OJ
18 is stable for every pc(s) satisfying
p¢~(joJ) < p ~ ( j w ) < p~2(jw)
V
(6.7)
to e [0, oo1.
We are now ready to prove K h a r i t o n o v ' s theorem. Let us introduce the box B of coefficients of the p e r t u r b e d polynomials:
(6.8)
B : = {glt; ~ R"+~,z~ < & < y~,i = 0, 1, . - . , n ) .
T h e Kharitonov polynomials (3.3) repeated below, for convenience are:
bl(s)=yo+zls+z2s
2 + y3s 3 + y 4 s 4 + z s s S + z ¢ s 6 + . . .
~2(s)=yo+yls+z~s2+zss3+y4s4+yss
5+x~ss+
~3(s)=zo+Zls+y2s2+y3sa+z4s4+zssS+yssS
~4(S) = Xo "4"ylS + y2S 2 "4- Z3S 3 +
Z 4 s 4 "4" ySS5 "~- ySS 6
...
(6.9b)
+ ...
(6.9c)
~- " ' "
(6.9d)
These polynomials are built from two different even parts pemax(s) and p~i"(s) and two different odd parts pm"X(s) and poilU(s) defined below:
pemax(s} : = y o - 4 - ~ 2 S 2 -I-ylS 4 q - X s S 6 "4-ySS 8 " t - " "
(6.1o )
p~i"(s) :=Xo+y2s2+z4s4+yss6+zssS
(6.10b)
+...
and pmax($) . _ YI$ "4- ~353 q- ySS 5 -4- Z757 -4- ygS 9 "q- " ' "
(6.10c)
porain (s) : =
(6.10d)
z l s + y 3 s ~ + z s s s + yTs 7 + z g s 9 + . . .
19 The motivation for the superscripts max and min is as follows. Let /f(s) be an arbitrary polynomial with its coemcients lying in the box B and let/f~(s) be its even part. Then
ze,, ~ + yso., s
(6.11a)
a ¢ ( j w ) = ~fo - ~2w ~ + $4w 4 - g~w 6 + 6sw s . ' '
(6.115)
v~°"(.iw)
= yo -
z2w 2 + y4w 4 -
p e m i " ( j w ) = zo - Y 2 J
+ ~:4w 4 - Y6w ~ + Z s w s ' ' "
(6.11c)
so that
pm.X(j~) -- ~e(i~) = (y0 - ~0) + (s~ - ~:)~2 + (y4 - 6,)w ~ + (~. - ~ , ) ~ ' + . . .
(6.12~)
and
g~(j¢o) - p ~ i n ( j w )
= (,5o - z o ) + (Yz - ,Sz) to~ + (~4 - z 4 ) w 4 + (y6 - g a ) w e + ' "
(6.12b)
Therefore,
pT'n(J,,)
< s°(J,,) < p T = ( J ~)
v
~ e [0,oo].
(6.13)
Similarly, if ~ ° ( s ) denotes the odd part of/f(s), it can be verified that
poi",,w,<m(41 $°(jw___~)< p ~ " X ( j w ) jw
-
j~
-
V w
E[O,oo].
(6.14)
jw
To proceed, note that tile Kharitonov polynomials (6.9) can be rewritten as:
(6.15o.)
(6.15b) (6.15c)
2O
64(s)
~- pTin(s) -J- poaX(s)
(6.15d)
If all the polynomials with coefficients in the box B are stable, it is clear that the Kharitonov polynomials (6.9) nmst also be stable since their coefficients lie in B. For the converse, assume that the Kharitonov polynomials (6.9) are stable, and let S(s) = 6e(s) + S°(s) be all arbitrary polynomial with coefficients in the box B with even part 6e(S) and odd part
~°(s ). Since £ ( s ) and ~2(s) are stable and (6.14) holds, we conclude, from Lemma 6.2 applied to ~ ( s ) and S2(s) in (6.15a) and (6.15b), that
p~ax(s)+~°(s) is stable.
(6.16)
Similarly, from Lemma 6.2 applied to ~S(s) and 64(s) in (6.15c) and (6.15d) we conclude that pemln(S) + 6°(s) is stable.
(6.17)
Now, since (6.13) holds, applying Lemma 6.3 to the stable polynomials pemaX(s) + ~°(S) and Pemln(s) + 6°(s), we conclude that
6~(s) + S°(s) = 6(s) is stable.
This concludes the proof.
O.
Tile proof of Kharitonov's theorem is illustrated in Figure 6.3.
B1
0
rain
p
(J ~)
e
/
J
f
B4
B5~
F i g u r e 6.3
min
P ~ (J ~)
ma× (j ~)
Pe
I l l u s t r a t i o n of K h a r i t o n o v ' s T h e o r e m
."
B2
B3
~8.(j~)
j,,.,
8(j~)
o
(jco) j L,,,~
too×
Po
B6
22 The illustration shows how all polynomials with even parts bounded by pmaX(s) and pemin(s) and odd parts bounded by pToz(~l and P~""(~) on the imaginary axis satisfy tile interlacing property when the Kharitonov polynomials are stable. Figure 6.3 also shows that the interlacing property for the odd and even parts of a single stable polynomial, corresponding to a point 6 in coe~cient space generalizes to the box B of stable polynomials as the requirement of "interlacing" of the odd and even "tubes." This leads to the following • EtI&I(+ j,a¢~ alternative version of Kharitonov's theorem. Let + 3o;~ • m i n ) denote the roots of " max (0,+ 3"wrai~o, p~max (s)(p~mln ( $) ) and let 0,+3~o, , denote the roots ofpomax (s)(pomin (s)).
Theorem
6.4
The box B corresponds to stable polynomials if and only if
max < "" " 0 < ~ominel < ~Omaxet < Wolmin < Wmaxol < wmlne2 <~ ~oe2max < ~.~mlno2<( ~Ooa
This theorem states that stability of the polynomials in the box B corresponds to non overlapping of the frequency bands B1, B~, etc. shown in Figure 6.3. The implication of this fact on robust stability is that the perturbation box B can be enlarged until overlap occurs for some pair of adjacent bands.
CHAPTER THE STABILITY
HYPERSPHERE
2 IN PARAMETER
SPACE
1. I N T R O D U C T I O N In the standard feedback system of Figure 2,1, suppose that the plant transfer function contains the parameter vector p and the controller is characterized by the real vector x, i.e.
G(,) = G(,,p)
C(s) = c ( , , x ) and fixing the parameter vector p fixes the plant and the choice of the vector x is equivalent to picking the controUcr C(s). The characteristic polynomial of the closed loop system is then given by
=
i.
(1.1)
i=0
In this chapter, we consider tile case in which the 61(x, p) are linear or afflne in p and for a fixed x calculate the radius of the largest stability hypersphere centered at p0. Although this is mathematically, a special case, it will always hold in single input (multioutput) or single output (multiinput) systems if the p a r a m e t e r vector p is taken to be the list of plant transfer function coefficients. This problem is formulated in Section 2 and solved in Section 3. We then extend our approach, in Section 4, to treat the case where the plant transfer coefficients are linearly or affinely dependent on some "primary parameters" and calculate the largest stability hypersphere in this space. This allows us to treat the practically important case where the transfer function coefficients are interdependent. It
24 is to be noted that although linearity is essential for tile method proposed, fimctions of physical parameters can always be redefined as the primary parameters, in order to satisfy this assumption. For example, if J (inertia) and k (spring constant) are physical parameters and ~ is a transfer function coefficient, we label p := -~ as the new primary parameter of interest. Thus a broad class of practical problems can be effectively treated by this approach. In particular, tile radius of the largest stability hypersphere, for a given controller, measures the stability margin provided by the controller, and can be used to compare the robustness of alternative designs. 2. P R O B L E M F O R M U L A T I O N
Consider the feedback system of Fig. 2.1 where G(s) is the plant transfer fimction and C(s) is the controller transfer function.
=
(~)
J
I J I ~
1/
c (~)
Figure 2.1. Feedback System. We shall develop our results specifically for single input (multioutput) or single output (multiinput) plants.
Since tile theory for the two cases are analogous we restrict
considerations to the single input, case.
our
25 Let
G(s) =
. \
= ~
:=
nCs)d-1(s)
(2.:)
\..,-¢~)/
d.(s)
where d(s) is the least c o m m o n denominator of all elements of G(s) and write
d(s)=dqsq+ . . . . .
+do,
n(s) = nqs q + . . . . .
+ no •
(2.2)
where di are real scalars, nl are m × 1 constant real vectors and n and d are right coprime. T h e coefficients di and ni of a physical system are generally subject to perturbation and therefore we introduce the nominal values denoted by dO, n o and tile perturbations denoted by Adi, A n i ,
i = O,1,...,q. To rule out trivial cases, we will assume t h a t the
coprimeness assumed above continues to hold under the p e r t u r b a t i o n s considered. Let p : = [ n T,d0, . . . , n q , T d¢]r
(2.3)
denote tile plant p a r a m e t e r vector in R t', k = (1 + m ) ( I + q),
or
o
oT d~]r
p0 := [n o ,do, . . . ,nq
(2.4)
its uominal value and
Ap
:= [AnoT , A d o , . . . , A n o r , A d q ] w
(2.5)
its perturbation, so that
p
=
pO + A p .
(2.6)
26 The size of the real perturbation vector Ap is measured by its Euclidean length, denoted by II~PlI= and is given by
IIAplI~ = IIAn011~ + I I ~ m l l ~ + • • + II~nqll~ + ( A d o ) ~ + • • + ( A d q ) i
We shall assume that the controller C(s) stabihzes the nominal closed loop system when p = p0. The problem to be solved in tile next section is: Determine, for the given stabilizing controller C(s), the largest stability hypersphere Sp(p°), centered at p0, with radius p(p0) so that the closed loop system is stable for all peSp(p°), i.e. V l l ~ p l l 2 < p ( p 0 ) , and is unstable for at least one Ap with I I ~ p l l z = p ( p ° ) . 3. T H E S T A B I L I T Y H Y P E R S P H E R E :
LINEAR CASE
Let 1
c(.) = d--27/..o,(s). . . .
,....(s)): = a~'(.).~(s)
(3.1)
and write + dco
de(s) = dcp(s)s ~ + . . .
ny(s) = ny/+
..
+
(3.2)
T
where dci are real scalars, ncl are 1 × rn constant real vectors, and de, n Tc are left coprime. The characteristic polynomial of the closed loop system of Fig. 2.1 is given from (2.1) and (3.1) by the p + q degree polynomial
6(s) = dc(s)d(s) + ncT(s)n(s) .
(3.3)
Write g(s):=g0+gls+
...
+6,s",
n:=p+q
(3.4)
27 and define the closed loop characteristic vector by
<~ : = [ ~ n , ~ n - l ,
....
,~0] T E R n + l •
(3.~)
D e f i n i t i o n Tile characteristic vector $ E R "+1 is said to be Hurwitz if and only if the corresponding polynomial 6(s) in (3.4) is strictly Hurwitz (i.e., has roots in the open left. half plane).
Tile set of Hurwitz vectors in R "+1 corresponding to n th degree Hurwtz
polynomials is denoted by Hn C R "+1. Tile controller C(s) may be viewed as all operator that maps the plant parameter vector p into the closed loop characteristic vector ~. Let 6o denote the image under this mapping of tile nomitml plant parameter p0, as defined in (2.4). If A := {Ap} denotes a given class of perturbation of p0 against which stabilization is required, the conlroller C(s) must m a p p0 + A p into $ E Hn for every A p E A. These ideas can be sharpened by the following Lelnma which makes the above mentioned mapping specific. L e m m a 3.1 The closed loop characteristic vector 5 satisfies
Xp = ~
(3.6)
Px = ~
(3.7)
as well as
where X E R ( p + q + l ) × [ ( l ' l - m ) ( l + q ) ]
P E R (p'lq+l)×[(l+m)(p+l)]
X E R (l'trn){p+l)
28
and p E R (l+m}(l+q)
T
Il c p
neTv
T ncp
dcp
dcp
X =
BoO dco "T rico
nTo
deo
nT
dq
nqT
dco
/
dq
p
nqT
dq
nT
do
do
nT
x = ( nov
dcp
(3.8)
(3.9)
do
. . . . .
n¢o
d~0 )'1" .
(3.10)
The proof of this i e m m a is a simple calculation obtained by equating coefficients in (3.3) and is omitted. The l e m m a shows that tile compensator m a p s tim parameter p into t h e characteristic vector/i via the linear transfornlation X in (3.8). Analogously the plant
m a p s tile compensator p a r a m e t e r vec|.or x into b via the linear transformation P in (3.9). To avoid trivial cases, we will assume throughout that X has full rank. Now, define the following sets in tile space of tile characteristic vector 6:
Ao := {$[6 e R "+l , 60 = 0}
(3.11)
A . : = {,~16 e R - + ~ , ~,, = 0}
(3.12)
A(w) := {6}~ C R "+] , 6(s) = (s 2 + w ~ ) l ( s ) , l(s) a r b i t r a r y } •
(3.13)
and for ally real w,
29 Tile inverse image of each of these sets with respect to tile compensator map X in the parameter space of p is defined next:
Ao}
(3.14)
II,, := X - ' ( A , ) := {PIP • R ~ , Xp • A . }
(3.15)
II(~,) := X-'(A(oa)) := {PIP • Rk , Xp • A(oa)}.
(3.16)
Ilo := X-1(Ao) := {PIP • R ~ , Xp 6
Note that IIo, 1-I, and II(w) are subspaces of R k and A0, A,~ and A(w) are subspaces of Rn+l. Let to, r,, and r(oa) denote the Euclidean distances between p0 and II0, II,, and II(w) respectively. Then
ro := lip ° -t~l12
(3.17)
where t~ • Iio and
IItT~ p°lls _~ Ilto -
p°lls, v to • IIo.
-
(3.18)
Similarly
r. := lip ° - t : l 1 2
(3.19)
where t~, • H , and
IItZ
-
p°ll= _ lit.
-
p°lls, v t . • n .
(3.20)
and
~(w)
:=
lip °
- t ' ( ~ ) l l s
(3.21)
where t*(w) • II(w) and IIt*(w) - p°ll2 < lit(w) - p°[Is , V t(w) E lI(w) .
(3.22)
30 Let
r:=
inf
(3.23)
r(~o).
We are now ready to state one of the main results of this chapter. Theorem
3.1
Let C(s) be a fixed stabilizing controller as in (3.1) and (3.2). Then the radius of the largest stability.hypersp!lere in the space of p, centered at pO is given by,
e(p°) = mi,,(,-o, T, ,,-}
.
(3.24)
T h e proof of this theorenl is given in Section 5. Figure 3.1 illustrates this theorem. The intuitive content of the theorem is t h a t a perturbation of A ( p ) of p0 cannot destabilize the closed loop system nnless A p intersects rl0, I t , or II(w) for some we[0, oo], say w °. Intersection with rl0 results ill a root of ~(s) at t.tle origin, intersection with l'I, results in a root at s = c¢, and intersection with II(w) results in complex conjugate roots at s = + j w , all of which are destabilizing. An obvious extension of T h e o r e m 3.1 will allow us to solve the robust stabihty problem for a general perturbation region. This is stated in Section 6. Next we give formulas for the calculation of the distances r0, r,, and r. Let /f E A0, t E II0 and let w l = [0,0 . . . .
,0,1] T E R "+1. Then from (3.6)
6 = Xt
(3.25)
wT~ = ~o = 0 = w T X t .
(3.26)
and from (3.11),
31 7T-n 7ro
Sp(po) p(pO)
- -
r
~-(~) po
~ _______--------
The Stability Hypersphere Sp (pO) Figure 3.1
11"((323 = (= I)
32 Let, X! denote the last row of X. Then (3.26) can be rewritten
X,t = 0
(3.27)
which shows that tile shortest distance from pO to Iio must lie on the normal X T to II0. Therefore the shortest vector is given by
p0 _ t = a X T .
(3.28)
To determine a, premultiply (3.28) by Xt and use (3.27)
Xtp ° - Xtt =
ctXtXT
X~p ° = a x ~ x T
(3.29)
(3.30)
so that Xtp ° - XtXT.
(3.31)
Therefore the distance ro from p0 to tile hyperplane rl0 is given by 1
r02 -- x I x T { p ° T X T X I p ° ] .
(3.32)
By exactly analogous arguments the distance r,, is calculated as a r.
-
1 XIX~
lp°VXiX ' "
p°l,'
(3.33)
where X ! denotes the first row of X, Tile calculation of tile distance r(w) is now given. A representative vector ~ E A(~a) is given by = O(w) l
(3.34)
33 where
~(~)
=
w2
0 w2
1
E R I'+l)xt'-J)
(3.35)
0 td 2
and 1 E R "-1 is a r b i t r a r y . If t(w) e II(w) then Xt(w) = ~ ( w ) l .
(3.36)
The matrix X is of size (p + q + 1) x [(~ + 1)(1 + q)l- Consider first the case where X has at least as m a n y columns as rows. W h e n the plant order q is greater than or equal to the controller order p, X will have full row rank. CaseI : 7n+rnq._>p We partition X and t(w) as follows
x = [x~, X~l
t(w) = [ t I ( w ) l
[tj{~)J
where XI is square and nonsingular.
Such a partition can be m a d e without loss of gen-
erality. This is proved in Section 5 where a specific way of constructing the nonunique matrices X~ and X j is also given. Now,
Xltl(w)
so that
= ¢(w)l - Xjtj(w)
(3.37)
34 Then a representative vector t(w) E II(w) is given by,
t(~))-----(Kill (w) -
-X~IXs) (tsl ) I
p?~)
(3.3S)
I,
: = P(w) It
(3.39)
where P(w) is a fixed real matrix for each to and It is an arbitrary real vector. Note that tile dependence of t s on to can be dropped since t.~ can be any vector. By letting It sweep over all real vectors in (3.39) we generate all solutions of (3.36). Now
(3.40)
t(w) - pO = P(to)lt - pO
and
lit(to)
-
p°ll ~ _- p ° T p
° -- 2 1 T p T ( t o ) p ° + I T p T ( t o ) P ( c a ) l t
.
(3.41)
Tile vector It = l F which mininlizes the distance (3.41) for a fixed w is given by setting tile corresponding gradient to zero. This gives
It =
(vT(to)P(w))-'PT(to)pU
(3.42)
• ~nd
r2(to) = poT (I -- P ( w ) ( P T ( w ) P ( o v ) ) - I p T ( w ) ) p O .
(3.43-)
Q(,,) Case II
: p > m + m q
In this case the equation (3.36) m a y or m a y not have a solution for a fixed to and 1. If a solution exists, it is unique and is given by
t(w) = ( X T X ) - 1 X T ¢ ( w } I P(~)
(3.44)
35 where 1 satisfies [x(xa'x)-'x
T - I1¢(,.,)1 = [ x P ( ~ , )
- ~(~,)]l = o.
(3.45)
Let
l= N(~)l,
(3.46)
denote tile solutions of (3.45) with ], a r b i t r a r y so that
t(o~) ----P(o~)N(w)l, := P ( ~ ) l t .
(3.47)
P(,~)
Now (3.47) gives tile solution set of (3.36) as It ranges over all real vectors. Since (3.47) and (3.39) are of the s a m e form we can apply tile preceding development (3.39)-(3.42) to obtain the expression (3.43) for r2(co). Now, in b o t h cases we have r 2 = inf r2(~) = inf p ° T Q ( ~ ) p °.
(3.48)
Because ¢(o~) and hence P ( ~ ) and Q(w) are known, tile fimction r 2 ( - ,) can be determined and the minimization in (3.48) can be carried out numerically (for instance by graphing r2(~)) and tile global m i n i n m n l in (3.48) can be found over 0 _< w < co. Since r2(w) is a continuous function of w2 tile lninimuln of r2(~) occurs either for a finite value of w or at = c¢. T h e latter case corresponds to b~ = 0, 6n-l = 0 (i.e. a double pole at infinity) and since A(oo) C An, r(oo) >_ rn. Silnilarly, since A(0) C A0, if the m i n i m u m of r ( ~ ) occurs at. ~ --- 0, we ]lave r(0) _> r0. Therefore, the global m i n i m u l n of r(w) need be found only in the interior of tile interval 0 < ~o < oo. Before concludingthis section, we point out the interesting fact that the stability hypersphere determined here can be translated linearly to determine a larger conical stability
36 region. Let So(p °) be the stability hypersphere determined by Theorem 3.1, i.e.
(3.49)
Sv(P °) := { p i p e r ~, p = po + Ap, HAplI2 < p(po)}
and introduce the cone
Cp(p °) = {tlteR k, t = ¢xp, peSp(p°),0 < a < c~}.
.C.o.r o l l a r y
3.2
Under the conditions of Theorem 3.1, tile conical region Cp(p °) is a stability region. The proof of this result simply depends on the fact that if p is "stable" so is otp for 0 < ~ < oo because of the linear equation (3.6). Therefore, XSp(p °) C Hn implies that X C o ( p °) C /-/,. This corollary is illustrated in Figure 3.2. The above cMculations silow that Theorem 3.1 provides a constructive procedure for calculating the stability hypersphere. In the next section we show how this calculation can be extended to handle situations where the transfer function coefficients are interdependent. 4. T H E S T A B I L I T Y
HYPERSPHERE:
AFFINE
CASE
Let
(4.1) denote the vector of primary parameters ill R t,
a ° :--[aL
its nominal value and
4,
....
,
T
(4.2)
37
The Stability Hypercone Cp(p °) Figure 3.2
38 A a : = [Aal, Aaz . . . . . .
Aat] T
(4.3)
its perturbation, so t h a t a = a ° + Aa.
(4.4)
Let us assume t h a t the vector p in (2.3) of the plant transfer function coefficients depends affindy on a as p = Aa + b,
(4.5)
where A E R kxt and b E R k. W i t h o u t loss of generality, we can assume that A is of full column rank and therefore l _< k (otherwise the parameters a could be redefined). According to (3.6) and (4.5) tile closed loop characteristic polynomial vector ~ is now expressed as XAa + Xb = b
(4.6)
which shows the arlene transformation m a p p i n g the p a r a m e t e r vector a into the characteristic vector/g. As before, let us consider the sets (3.11) - (3.13) and denote the inverse images of A0, An and
A(w) (analogous
to (3.14) - (3.16)) in the space of a as I/0, I / , and
I](w). It is to be noted, however, that now some of these sets m a y be empty. Therefore, definitions (3.17), (3.19) and (3.21) with a substituted for p have to be augmented by
ro=OO
if r I o = O
(4.7)
r,~ = co if 1-I. = 0
(4.8)
r(w) = c~ if II(~) = 0 .
(4.9)
39 Defining r as in (3.23) we can generalize Theorem 3.1 as follows: T h e o r e m 4.1 Let C(s) be a fixed stabilizing controller as in (3.1) and (3.2). Then the radius of the largest stability hypersphere in the space of primary parameters centered at a °, is given by p(a °) = min{ro, r . , r} .
(4.10)
Tile proof of this theorem is similar to the proof of Theorem 3.1 and is omitted. We now give formulas for the calculation of tile distances ro, r . and r in the space of a. We note that t E I10 if and only if
XlAt + Xtb = 0 .
(4.11)
Tile above equation fails to hold if and only if the vector XzA = O, X l b # 0 and then II0 is empty (r0 = oo). Otherwise, if XtA # O, the distance r0 of the point a ° from the hyperplane (4.11) is given by the fornmla
ro-
1 [XtAa0 + XIB[ []XaAI[2
(4.12)
or
1
r2 -- XtAATX T
[a°TATXIAa ° + 2 a ° T A T x T x ~ b + b T x T x I b ] .
4.13
Tile distance r~ is calculated similarly as (¢~ r, =
~ 2,
[XlAa ° + X / b [ ,
if X j A = 0 and X l b # O; if X I A :fi 0
(4.14)
It should be mentioned that it is not possible that XtA = 0 and Xtb = 0 (or X f A = 0 and X l b = O) simultaneously, if the nominal point a ° is stabilized by C(s).
40 The calculation of the distance r(~) is now given. After exactly analogous derivations as in (3.34) - (3.38) the formula (3.38) can be written as
t(w) = At,,(.,) + b = P(w)lt
(4.15)
where now ta(~) E II(w) in the space of a. Since A is of full column rank it can be, after some possible row interchanges, partitioned as
A= where AI is a square nonsingular matrix,
(A1) A2
(4.16)
A1 G R t×t, A2 G R ct'-O×t. Equation (4.15), after
the same row interchanges, can be expressed as
(,1A 2 )
t'(~)=
(Pl(~)~ hi \P2(w)] l'-(b2)
"
(4.17)
From the first part of (4.17) we get
t.(~) = *71Pj(~)l, - h~-lbl
(4.18)
which substituted into the second part of (4.17) gives
!A~,,-'P,(y) - P,(~)~l, ~AI-Iyl - b~ =
(4.~o)
B(w) or
B(o~)It =
c .
(4,20)
Equation (4.20) is of primary importance in del.ermining whether 1-I(w) is empty or not. Let f~ : - {w[ rank [B(~)] = rank [B(~,)Ie], 0 < ~ < oo}.
(4.21)
41 Then II(~) ~ 0 if and only if w E fl ~ 0 and therefore we take r(~) = ~
for ~ ¢ ft.
(4.22)
For w G fl (if fl ~ 0) tile general solution of (4.20) is of the form It = D(~)it + e(~)
(4.23)
t , ( w ) = P(w)it + r(w)
(4.24)
and therefore
is the general solution of (4.15) with It an arbitrary real vector. Now, in a manner analogous to (3.40) - (3.43), we obtain r2(w) -----(a ° - r(w))TQ(w)(a ° - r(w))
(4.25)
and finally r2=
inf r2(w) .
(4.26)
Tile above derivation allows us to effectively deal with a more general class of perturbations than that covered in Section 3, i.e., some plant transfer fimction coefficient can be interdependent and some can be fixed. It is worth mentioning that the way we have dealt with equations (4.6) and {3.34) in this section gives a good theoretical insight into the problem. However, in a specific example it may be simpler to solve (4.6) directly with the right hand side set equal to (3.34). As a final remark, we note that Theorem 1.3.2 of Chapter 1 is a special case of Theorem 2.3.1 of the last section obtained by setting X = I,,+1, p = b.
42 5.
PROOF
OF THE MAIN
RESULT
This section contains the proof of T h e o r e m 3.1, and the demonstration of nonsingularity of Xx in (3.36). Proof of Theorem
3.1
Let C = C + U C - denote the complex plane with
C + = {siRe s > 0 } ,
C - = {siRe s < 0}
and let C! E C + denote the imaginary axis. Let Z(~) denote tile zeros of tile polynomial $(s) = g0 + gls + . . . .
~i.s n and introduce the function
t~(s) ~ [t~.,6._l, . . . . .
~0]T := ¢~6 R "+1 •
As in Section 3 let
~o := {~10 e z(~)}
(.5.1)
~x. := {~1~. = o}
(5.2)
A(.,) : = {gl~;(s) = (s 2 + w2)l(s), ICs) arbitrary}.
(5.3)
a, := {*Iz(*) n c, # ¢}
(5.4)
z~- := {~Iz(~) c c - }
{5.5)
a + := {~[z(~) n c + # o}
¢5.6)
and f o r 0 < w < o o
Define
43
and let H . denote the set of n th degree polynomials with zeroes in C - :
~.
:= {~1~c R "+*, ~. # o, z(~) • c - }
(5.7)
or
H.
=
A- \ A,.
(5.8)
We also note that
a~
=
U
(a(,,,)) u Ao.
(5.9)
0<w
Now consider the closed loop system of Fig. 2.1 and the equation
Xp=6 for the characteristic vector. With the compensator, i.e. X, fixed and the plant parameter p = p0 + Ap we have ~ = 6(x,p ° + Ap) and closed loop stability is equivalent to
6(x,p ° + Ap) 6 H , .
(5.10)
p(p0) = rain{r0, r,, r}
(5.11 )
Let
as ill (3.24) and let Sp(p °) denote the interior of the hypersphere of radius p(p0) centered at p0 in parameter space:
Sp(P °) := {PIP 6 R A', p = pO + Ap, flApt[2 < p(p0)}.
(5.12)
Let S ~ ( p °) denote the boundary of this hypersphere: SB(p °) := {pip 6 R k, p -----pO + Ap, llApll2 _-- p(pO)}.
(5.13)
44 The proof of the theorem now consists of showing that
$(x,p) E H a
V p E S . ( p °)
(5.14)
and ~(x,p*) ¢ H . for some p" e S f ( p °)
(5.15)
which together show that Sp(p °) is the largest stability hypersphere. Note that Sp(p °) cannot intersect II,, as otherwise ~(s) has a root at s = oc, which causes instability of the closed loop system. For clarity of presentation let us assume that
p = p(p°) = to.
(5.16)
Then there exists t~ E II0 (see 3.14) such that
ro = IItg -
p°II
_< Ilto -
p°[12 Vto ~ Iio.
(5.17)
Then B 0 to• e sro(p )
(5.18)
an d
xt;
:= ~; e a o
(5.10)
and therefore
x t ; ~ H,,.
(5.20)
With p" := t~ and/~(x,p*) = 65 this proves (5.15), which shows that at least one point on the boundary of S~o(p °) corresponds to all unstable system.
45 To prove (5.14) we note that
HP - P°II2 < ro _< min{r., r} Vp E S.°(p°).
(5.21)
Define II :=
U II(w). 0_<w
(5.22)
sro(p °) N IIo = 0
(5.23)
Sro(p °) n n . = 0
(5.24)
Sro(p °) n n = O.
(5.25)
x s . o ( p °) := {~l~ = Xp, p ~ S,o(p°)}.
(5.26)
Now (5.21) implies that
and
Let
Then (5.23) - (5.25) and the definition (3.14) - (3.16) of Ilo, 1-1, and II(~) imply that
XSro(p °) M Ao = 0
(5.27)
xsro(p °) n A . = 0
(5.28)
XSro(p °) f3 A 1 = 0.
(5.29)
and
Now, since the function Z(6) is continuous, tile following well known result will hold. Fact 1 If A C R T M is a simply counected region titan A C H , if and only if there exists ~" E A such that 5" E H,, and A A Hff = 0, where H ~ is the boundary of H , .
46 From the definition of H . H ,B C A I U A . = A o U A , , U A 1
(s.3o)
and (5.27) - (5.30) imply that
(5.31)
XS~o(p°) n H ." = ~.
Since C(s) stabilizes, by assumption, the nominal system we have
(5.32)
Xp ° E Hn.
We also note that XS,o(p0) is a simply connected region in R n+l since S~.(p °) is simply counected and /~ is a continuous function of p and the number of roots of 6(s) does not change for pe S,.o(p°). Therefore, the conditions (5.31), (5.32) and Fact, 1 imply that XS,o(p °) C H , , .
(5.33)
This shows that all parameter points p inside the open hypersphere S,.o(p °) of radius r0 centered at p0 result in stable closed loop systems and completes the proof for the case p = r0. When p = rn or p = r exactly analogous arguments apply. Tllese details are omitted.
<)
Construction
o f X/
a n d P r o o f o f its I n v e r t i b i l i t y
In this section we consider the equations (3.6) and (3.36) and show how to construct the nonsingular matrix X! in (3.36). In (3.36) ncp
dcp
nco
d~o
Sop ~¢p ncp
X=
nrp
dcp
dcp rico
nco k n¢0
de0
dco
dco
47 and
pT = ( n o t
,
do
nqT
. . . .
,
dq )
where X E R (q'+p+l)×[(l+m)(q+l)] and p E R [(l+m)tq+z)]. Consider first• the case m = l . In this case we m a y define
dcp d¢v r~.cp dcp
dco
ncp dcp
Xl =
ncp
E R (q+p+l)×(q+p+z)
dco
dcp rico
rico \ rico
dco
dco
dcl)
It is easily shown using the eliminant matrix t h a i Xz is nonsingular if no(s),
de(s) are
coprime as assumed. For the general case we let ncj(s) be eoprime with
de(s) for some 1 < j <_ m.. Then
the matrix
n~p
dcp
,~,~p dcp
n~p d~p E R ~p × 2p
Xj =
J
~cO 3 rico
i neo
d~o
d¢o
dco
48 is nonsingular as before. Now let ¢~cp"
dcp
Xj
XI =
d~o
E R (g+P+l) ×(q+P+l)
dco
Xj Xj Clearly X1 is nonsingular because Xj is nonsingular and dcv # 0. Now from the form of X we can write X p = XIp~ + X 3 p s by permuting the components of p to form Pl and pa. 6.
SOLUTION
OF THE ROBUST
0
STABILITY
PROBLEM
Tile theory developed in sections 3 and 4 was concerned with determining tile largest hypersphere centered at the nominal parameter that would retain stability. Theorems 3.1 and 4.1 show that this can be done by fitting the largest such hypersphere that just touches the closest, of the regions 110, Fin or II. From the proof of Theorem 3.1 it is clear that this result in fact hohls for any simply connected perturbation region containing the nominal parameter. This observation allows us to give a general solution to the robust stability problem. To state this, let p ° e R k in Theorem 3.1 or a % R t in Theorem 4.1 be the nominM parameter, let X correspond as in (3.6) to a nominally stabilizing controller, and let tile class of perturbations to be handled, A t. := {Ap}, or A__t := {Aa}, be given with A k C R k, A t C R e being arbitrary but simply connected. Let
•p := pO + Ak := {pip = pO + Ap, ApeA L'}
(6.1)
49
and
~t := . ° + A t := {al- = .0 + ~ a , Aa~A t}
(8.2)
and let II0, I1,, and II(w) be defined as in Section 3 and Section 4. Theorem
6.1
Let C(s) be a fixed stabilizing controller as in Theorem 3.1 or Theorem 4.1 of this chapter, which stabilizes tile closed loop system for the nominal p a r a m e t e r p = p0 or a = a °. Then closed loop stability holds for all parameters contained ill the simply connected region 7~ if and only if
"PNIlo ~'
Nn .
=
(6.3a)
= ~
(6.3b)
and
Similarly, closed loop stability holds for all parameters contained in the simply connected
region A , if and only if
~Nno=
and
This theorem is illustrated in Figure 6.1.
~
(8.4°)
50
Simple Connected SLabilit.y Region
7rn
/ 7/'0
"rr (~')
pO
/
Illustration
of T h e o r e m
Figure 6.1
2.6.1
51 The theorem given above is a complete solution to the robust stability problem in the linear or affine cases treated in this chapter. When "P or .4 is a regular geometric object such as a polytope or hypersphere or hyperellipsoid, the verification of the conditions (6.3) or (6.4) can be computationally relatively simple. We expect this result to play a useful role in synthesis and robustification procedures as well.
CHAPTER STABILITY
ELLIPSOIDS I.
3
AND PERTURBATION
POLYTOPES
INTRODUCTION
The spherical stability region calculated in the last chapter implicitly assumes that the worst case pertt, rbation in each direction is of the same magnitude. In practice, this may not be the case, for example, when the perturbations are prescribed to lie within a rectangular polytope. This motivates us to consider weighted perturbations in the next two sections and to solve the problem of determining the largest stability ellipsoid. In Section 4, we use the stability margins calculated so far to describe a numerical procedure to design a controller for robust stability. These ideas are illustrated by some examples in section 5 which concludes with a brief discussion. 2. T H E S T A B I L I T Y
ELLIPSOID
This section extends the method presented in the previous chapter to provide the solution to a more general problem, namely, that of finding the largest stability hyperellipsoid in the p a r a m e t e r space of a centered at the nominal point a °. As will be seen later, this is useful in applications where weighted perturbations occur. By "largest" in the above, we mean that the shape of the ellipsoid is fixed by specifying the ratios of the principal axes as
al
: az :-..
: c~t
(2.1)
where
ai>O, i=1,2, ....,l
(2.2)
53 and such an ellipsoid is enlarged to the m a x i n m m possible extent. Obviously, the largest stability hypersphere is a particular case of the largest stability hyperellipsoid if all a l in (2.1) are equal. Let E ~ ( a ° , a ) denote an ellipsoid centered at a ° with principal axes parallel to the coordinate axes and of lengths ca1, . . . .
, cal. Consider the family of ellipsoids
E(,,°.,~) := {E,(,,°,,~)IO _< ~ < oo}.
(2.3)
Let
-
( 2.4)
-
and define 5. by ,, = QS.
( 2.5)
so that 5. = Q - l a
.
(2.6)
Clearly, tile linear transformalion (2.4) m a p s tim set of all lJyperspheres ill the fi, space, centered at 5.0, onto £ ( a °, a ) and the m a p p i u g is one-to-one. Therefore, tile largest stability hyperellipsoid in the a space can be found by determining the largesl stability hypersphere ill the subsidiary space of 5.. This can be carried out by the m e t h o d described in Section 4 of C h a p t e r 2 with the m a t r i x .~ = A Q substituted for A, a replaced by 5., and a ° by 5.0.
(2.8)
54 Let =/~(~0) = rain{f0, ~n, 4}
(2.9)
denote the radius of the largest stability hypersphere in the fi, space. The above considerations lead to the following theorem. T h e o r e m 2.1 Let C(s) be a given stabilizing controller as in (2.3.1) and(2.3.2). Then the largest stability l~yp~rellipsoid E , . ( a ° , a } in the class £(a°,a} is given by
~'=p.
(2.10)
8. P O L Y T O P E S O F P E R T U R B A T I O N S In some applications, the plant parameters are known to lie within given bounds
0
a i -Ti
0
i
+¢i,
i=1,2,
...,l
(3.1)
or
-7;
i=1,2,
...1
(3.2)
and closed loop stability is required for all such values of the parameter vector. Equation (3.1) determines a rectangular polytope in the a space. It should be noted that whenever the parameters are perturbed independently, the stability polytope, rather than tlle stability hypersphere (or hyperellipsoid)~ is of primary interest. A procedure for treating tile above problem within the framework of this chapter is to find tile stability hypersphere (or hyperellipsoid) and ensure that it inscribes the polytope (3.1).
55 Since it is desirable to center the s t a b i l i t y hypersl)here (ellipsoid) at t h e center of t h e p o l y t o p e we redefine the n o m i n a l point and the tolerances as follows. Let
, := [,,,,~ .....
,~,l r , 7 := 17,,~,
....
,7,iT
(3.3)
and i n t r o d u c e the new n o m i n a l p a r a m e t e r vector 1
a ° = a° + ~ ( ~ - 7 )
(3.4)
and new tolerances 1
= ~(~ + ~).
(3.5)
Then (3.1) a n d (3.2) a r e equivalent to
-0
-0
-
ai -- ~i < a i < a i + ei,
i = l, ...
1
(3.6)
and -~i<
A~i<~i,
i=l,
...1.
(3.7)
This shows t h a t , for tile fixed p o l y t o p e p r o b l e m (3.1), the p e r t u r b a t i o n classes in (1.2.7) and (1.2.8) of Section 2 C h a p t e r 1 can b o t h be t r e a t e d within the s a m e m a t h e m a t i c a l framework. Therefore, w i t h o u t loss of generality, we will consider the class {ii), i.e., we aSSlllne
a oi - - w i 6
< a i < a oi ~ w i e
(3.8)
where e is a positive c o n s t a n t and wi, i = 1,2, . . . 1 are given positive weights defined in tile vector form as: w := [ w l , w ~ . . . . .
,u,t] T .
(3.9)
56
For a fixed controller C(s) that stabilizes the nominal plant with paraaneter a ° let ~(~0) denote the radius of the largest stability hypersphere in the space h calculated according to (2.9) and let Eh(a °, a) denote the largest hyperellipsoid determined according to Theorem 2.1. Let us also define the vector
W I
t/)]
10 2 ,
:=
tO! ,
...,
t:~.Ju)
where cti, i = 1, 2, • . .,I are the parameters characterizing the ellipsoid Eh(a°,a) as in Section 2. T h e o r e m 3.1 Let C(s) be a controller that stabilizes the plant with nominal parameter a ° and let E~(a °, c~) be the largest stability hyperellipsoid. T h e n t h e controller C(s) stabilizes tile closed loop system for all parameters lying in the polytope (3.8) if
~]}w'l[2 _< ~(~t °)
(3.11)
where w' is given by (3.10). The proof of this theorem is obtained by applying transformation (2.6) to the polytope (3.8) and then inscribing the corresponding polytope into the largest stability hypersphere in the space of fi centered at h0. The result of Theorem 3.1 can be strengthened by taking advantage of the fact. that If0 and II,, if nonempty, are (/-1)-dimensional hyperplanes in the a space. Thus, finding the conditions ensuring that the polytope does not intersect tile liyperplanes is a relatively easy task.
57 Theorem 3.2
Let C(s) be a controller that stabilizes the plant with nominal parameter a ° and let E~(a °, a) be the largesl stability hyperellipsoid. Then the controller C(s) stabilizes the closed loop system for all parameters lying in the polytope (3.8) if
e < miu
{ I X t A a ° + X ~ b l IXIAa ° + X s b l ~..,t i X l A i l w i ' El=1 t IXjAil~ ~i=1
~
],
IIw'l12 /
(3.12)
where Xi and Xi, are tile first and tile last rows of the matrix in (2.3.8), r e s p e c t tively Ai denotes the i 'h colunm of the matrix A ill (2.4.5) and ~ is the same as in
(2.9). Tile proof of this theoren, is omitted. The formula (3.12) is an improvement over (3.11) because the bound given by (3.11) can be rewritten as °<,,,in
-
I
ix,
+ x,bl
IX,A ° + x,bl
[ IlX~Aqll211w'll=' IIXiAqll211w'll~'
IIw'll2 J
(3.13)
and it call be shown, using tile Schwartz inequality, that the first two terms of (3.12) are greater than or equM to the corresponding terms ill (3.13).
4. C O N T R O L L E R
DESIGN
The stability margins p and/5 calculated in the previous sections are useful for comparing the robustness against variations of the parameter a of two given controllers. When the controller is not specfied but is to be synthesized by choosing a vector x of free parameters, we regard these margins as functions of x, i.e. p = p(x), ~ = ~5(x). Robustification of a controller then consists of iteratively choosing successive values of x to increase these margins until they are maximized, or the stability hypersphere or ellipsoid obtained contains the given range of parameter variations that are to be tolerated. The controller order
58 is successively increased until the above criteria are met. This problem is addressed in this section. For the sake of simplicity we will describe the algorithm in the plant transfer function coefficient space p. The same approach, with nfinor modifications of the formulas involved, is applicable to the a space and to the f~ space. The transfer function C(s) of the controller given by (2.3.1) and (2.3.2) specifies completely the controller p a r a m e t e r vector x in (2.3.10) or the matrix X in (2.3.8) and, conversely, the vector x or structured matrix X specfies the controller transfer function completely. Therefore, we refer to x or X as the controller and note that the distances v0, r , and r are functions of x and
p(x) = rain{to(x), r . ( x ) , r ( x ) ) := J ( x ) .
(4.1)
We define theoptinfization problem
max~ J ( x ) = m ax[min{7"o(x), r . ( x ) , r(x)))
(4.2)
and seek to maximize p(x) over all stabilizing x. Since a purely numerical gradient method will be used, clearly only local m a x i m a starting from a nominal stabilizing controller will be found. Also, the min operation in the calculation of p(x) means that a straightforward gradient method may not always work.
It is relatively easy to sidestep this problem
numerically, however, because r0(x), r , ( x ) and r(x) are continuous functions of x. We recall from the results of Section 3 of Chapter 2 that equations (2.3.32), (2.3.33) and (2.3.43) display the explicit dependence of r0, rn and r on x:
(4.3a)
59 r~.(x) =
X/-~ [p°TX~X/p °]
(4.3b)
and r2(x) = min p°TQx(w)p°
(4.3c)
qx(~) = I - V.(w)(PT(w)Px(w)) -~ PT(w)
(4.3d)
where
and P ~ ( w ) : = ( X } - I : (~)
-X}'IX3)I
(4.3e)
The quantities Xt, X t, XI, Xj are specified once the controller parameter x is chosen. A gradient based algorithm can be designed to update the controller by choosing the design vector xA-+l at the ktn iteration so that
p(x~.+~) > p(x~).
(4.4)
Tile update can be chosen as
O~A.
(4.s)
Axh
where the quantity Optx~) is determined by numerical means. Tile step size A call be chosen to maxinfize p(xL.+1) but since the closed loop system with the nominal plant and the controller x~.÷l must be stable it is required that P°xt.+j = 6t,+l G H .
(4.6)
where p0 is the matrix in (2.3.9) with nominal values for the plant parameters d,°., n °, i = 0,1,...,q and 6~+1 denotes the (k + 1) ~l iterate of the vector 6. Write P°(x~. + Axe,) = P°x~ + P°Ax~. = ~Sk + A~il, .
(4.7)
60
Let p(bl~) denote as in Chapter I the radius of the largest stability hypersphere in the characteristic vector space 15. Then the controller xL-+l preserves closed loop stability if
IIP°Axkll2 < p(,h.).
(4.8)
Now (4.8) can be satisfied by choosing Axk so that
p(s~) li~xkll2 < ~
HP°IiF
f4.0)
where [[.]lv is the Frobenius norm. There[ore the step size ~, in (4.5) can be chosen so that
A<
p(15~,)
Op( x~, ) IIP°IIF[I--~-~ 112
(4.10)
and this guarantees closed loop stability. For independent perturbations of plant parameters one should choose x such that the allowable "box-type" perturbations are maximized (variable polytope problem). In the framework developed here this problem is best solved by choosing x such that the hyperellipsoid with parameters c~; ± wi is maximized (the shape of the ellipsoid is best fitted to the shape of the polytope). The above robustification algorithm has been implemented and applied to various examples. Three such examples are given in the next section. 5. E X A M P L E S
Example 1 As all illustrative example a satellite control problen~ ]52, page 21] is considered. A sketch of the satellite is shown in Fig. 5.1.
61 For the salellite model we assume two masses connected by a spring with torque constant k and viscous damping constant d. The equations of motion from Fig. 5.1 are
J2~i2 + d(hz - h~ ) + k(02 - 01 ) = 0 where Tc is the control torque and Jl and J2 are inertias. If we choose, as the state vector,
T
(02 02 01 01)
tile state equations with Tc=u and 01=y are
(0
~, =
0
-
k s-7
- ~
J-7
d J~
k J,
o
y=(0
(0)0 x, +
o
o
J, 1
u
X
0)x..
Thus, the corresponding transfer function with J l - - 1 and J 2 = l is s 2 + ds + k G(s) = s~(s: + 2ds + 2k) "
(5.1)
Physical analysis of the boom leads to the conclusion that the k and d parameters vary within bounds given by 0.09 < k < 0.4
(5.2)
O.04~lk--O < d < 0 . 2 ~ 1~ .
(5.3)
62
81
el/
~2
Model
k
G
f d
F i g u r e 5.1 A sketch
of t h e
satellite
63 As a result, the vehicle resonance w,, can vary between 1 and 2 rad/s, and the damping ratio ( varies between 0.02 and 0.1. Our problem is to design a controller which stabilizes the closed loop system for all perturbations given by (5.2) and (5.3). We select k°=0.245 and d°=0.0218973 which are the middle points of the variation ranges of (5.2) and (5.3) as nominal values. Therefore, the nominal vector a ° of physical plant parmneters which are subject to perturbation is
Our objective is to obtain a low order controller x which will generate a stability hyperellipsoid inscribing the perturbation bound given by (5.2) and (5.3). We start with the stability hypersphere of radius p(a °) in the space of physical plant parameters a which is centered at a ° and with a 0 *h order controller. The controller is
C(s)-
ffcO dco
and the closed loop characteristic vector is
d3 =~
7/'2 d2 / n.l
0
no
0
000co0/() P
or
x rico •
0 0 nco 0
0 dco 0 0
d~o 0 0 0
0 0 0 0
0 rico 0 0
nl d2 d3 d4
, \ n
2 P
=/f.
64
XI = \ n~o
0
0
d~0
0
0
d~0
0
rico 0
0 0
0 0 0
0
, Xj =
00 0
From (5.1) there is the following linear relation between the transfer function coefficients of plant ni and di, and physical p a r a m e t e r s , d and k.
d3 d4
=
+
p
~,
"
b
From (2.4.12) and (2.4.14) we obtain
TO ~
k 0
~ Tn
~-
O0
~inc¢ X,A = l"c0, 0], X , b = 0 and X , A = [0, 0]. Now, following (2.4.15)-(2.4.20), we get tile solution (2.4.17)
ta(o~)= ( t#(vJ) ) = (I°~2/no°
and then we formulate equation (2.4.20) whose solution is
12 -- de0, Ii = 0, l0 = ne0 if o~ ~ 7X/~.co/2d~o. For w = ~
dco~ 2 - 7~co 2dc0W 2 -- rico
, t j ~- 1
equation (2.4.20) is inconsistent and therefore
= [(0,oo)\{~ ) } . For o~ E fl, t ( ~ ) E II(,J) takes the form (equation (2.4.24))
2d¢ow ~ -rico
\ ta(w)
=
0
65 Note that P(~0) and |t of (2.4.24) do not appear here so finally
r~(~') =
/¢o
~ 2 d,o.," . . . .
2a,o~2_,,o
-
It can be easily shown that r z =
2
+(d°) 2 otherwise.
rain r 2 ( ~ ) = ( d ° ) 2 , s o r = d
o and, since d o < k °,
p(~o) = do.
It is interesting that the above result does not depend on the controller x, i.e., it holds for any stabilizing controller C(s) of order zero. From Fig. 5.2 we see that the stability hypersphere So, with radius p(a°), does not inscribe the given perturbation bound. Because of the oblong range of the perturbation region, we now consider an ellipsoid E , ( a ° , a ) in the space of k and d, centered at (k°,d°), with a = [1,adj. The polytope which is to be inscribed into the ellipse is given by
-e
< Ak < e
-0.1168e < Ad < 0.1168e where • = 0.155 and w = [1,0.1168]. Now, from Theorem 3.2 we have the condition
e_<min
k°, a : ~ , /
where it can be shown that ~ = ~ = d ° / a d .
The above condition can be satisfied by
any ad < 0.079. For example, for a d = 0.07 we obtain the ellipse E with the semiaxes 0.3128186 and 0.0218973 containing the polytope given, as shown in Fig. 5.2.
66
.2 -
~
"-a
$1 by 1st Order Controller ~
/ "~..
/
F "NX/ /
X
Variation bound of of k and d /--E bY OthOrder
-.I rm
-.2
-.3 .05
J
w
I
!l
I
I
I
W
.1
.15
.2
.25
.3
.35
.4
.45
Constant,
k
Torque
Figure 5.2 Perturbation bounds and stability r e g i o n s in k and d s p a c e .
67 Note that the above solution is independent of tile controller as long as C(s) is of 0th order and stabilizes the plant. Therefore every 0 th order stabilizing controller will be robust for the perturbations given and the value nc0/dc0 can be used to satisfy other design req~lirements. In fact, it can be shown for the plant considered that for any 0 th order stabilizing controller, i.e., such that
nco/dco > 0, /he parameters /¢ and d can be
perturbed anywhere in the first quadrant. We have also calculated the largest stability hypersphere obtained for a "maximaUy" robust first order controller using the algorithm of Section 4 and this is shown in Fig. 5.2. as St. Example
2
As another illustrative example a control problem of a digital tape transport system [52, page 504) is considered. The model of a tape drive is shown in Fig. 5.3.
I XI-
i x3D
./
wheel wheel
dt 0 ~ ~ 1
eneoderand/A ~"
tach ~j
FI Vacuum chamber
R
r
O+
Figure 5.3 The model of a tape drive
68 The system is in static equilibrium when To = Fj and Kmio = r]To. We define the variables as deviations from this equilibrimn. The equations of motion of the system are given by tile laws of mechanics: J d.o~ 1~
+ 3 1 o,1 - r] T = K , , , i
;~1 = r l ~ l
Ldl
dt + Ri + K , ~ 1 = e ~2 = r2~2
dw2
J2 ~
+ 132toz = - r z T
T = K,(z3 - z]) + D](k3 '- k])
T =/GCz2
ZI = r l O l
- x 3 ) + D 2 ( ~ , 2 - ~'3)
~.~2 ~ /'202
~X3 ~-
x l -4- x 2 2
Assuming that IC]=K2:=I(, D] = D z : = D and substi|,uting the numerical values given in [52], we obtain the following transfer function
C(,)
=
X2(s) -
-
El(s) where, nl = 0.03D n0 = 0.3 x 1 0 - 4 K ds=l d4 = 0.025D + 2.25
-
,~1 s + no
dss 5 + d a s 4 + d z s a + d 2 s 2 + d l s + d a
69 d3 = 0.035D + 0.25 x 1 0 - ' K + 1.5225 dz = 0.01045D + 0.35 x 1 0 - 4 K + 0.2725 dl = 0.1045 x 1 0 - 4 K do = 0 . We assume that K and D are subject to perturbatlon.With nominal values D °=20 and K ° = 4 x 10 4. 0.6s + 1.2 G(s) = s5 + 2.75s4 + 3.2225s3 + 1.8815s2 + 0.418s Our aim is to find stabilizing controllers which maximize the radius of tile stability hypersphere in the space of plant, primary parameters K and D. We start with a 0 ta order controller
CCs) =
-~__o de 0 *
Tile closed loop characteristic vector is d5
a~t kaoo/
=/5
J
or
0 \ /'no i
n~o
0 0 0 0 n¢o 0
0 0 0 dco 0 0
0 0 d¢o 0 0 0
0 d~o 0 0 0 0
dco 0 0 0 0 0
0
2
P
=if
70
i o o o o o) 0 0 0
X1 \ n~o We have
/
rico 0
0 0 dco 0 dco 0 d¢o 0 0 0 0 0 0
T~o
0
nl d~ d3 d4 d5 dl
0.03 0.01045 0.035 0.025 0
=
0 0.3
0 0 0 0
0
, X./=
0
x010-4 )
0.35 x 10 -4 0.25 x 10 -4
+
0 0
0
0) /i 1.5225 [ 2.25 |
.
0.1045 x 10 -4
b
P
Now, following (2.4.15)-(2.4.24) we o b t a i n
t(w) = (tD(°:) )
\tK(~)
1
=ag-~e
(de-bfe)
a/
where a = 0.03 ~
- 0.03.Sdco
b = 0.1045 × 10-4 d--~ -- 0.25 × 10-4dc0 c = -0.01045dco + 0.025dco~ 2 d = 0.3 × 1 0 - 4 ~
- 0.35 x 10-4dco
c = 1.5225d¢0 - ~2dco f = 0.2725dco - 2.25~2dco Now, we get
,.2(~)
=
(D o
_
tv(~,))2 + (Ko
_
t~.(~,))~ .
A p p l y i n g the o p t i m i z a t i o n p r o c e d u r e of Section 4 we o b t a i n e d t h e following:
C(8) = 0.1/4.2214 = 0.02368
71 and r 2 = min r2(w)
= 2875.9946
Od
Therefore p = min{K °, co, r} = 53.618 is the stability margin obtained. Next, we wish to find a low order controller wMch will generate the largest stability ellipsoid. For this ellipsoid two weighting constants are chosen as follows based on the relative magnitudes of D o and K°: = 1, aK = 2000
ao
Similarly, we get
~(~')=
~
,~r, /
We obtained the followingrobnst controller: C(s) = 0.1/12.078 = 0.00827 and ~2 = min ~2(~,) = 99.801 tt$
Therefore, fi=min
{ 4 x 1 0 4 ,~,oo } = 9.99 Otk
is the stability margin. This gives about 35% allowable relative perturbations. Now we increase the order of the controller from 0 to 1. The controller transfer function is C(s)
-
n ~ l s + n~0
d~l s + d~o
72 Now P, X, p , x a n d X1 are given b y
0 0 0 0
ds d4 da
0 0 0
0 d5 d4
d2
0
d3
0
dl 0 0
0 nl no
d2 dl 0
0 0 0 no1 n~o 0
0 0 d~l d¢o 0 0
P=
no
oooo
X=Xz=
0 0 0 0 ncl n~o
0 d~l d~o 0 0 0
D'cl / '//'cO d¢o
o clo //o/
de1 dco 0 0 0 0
dco 0 0 0 0 0
0 0 0 d¢i d~o 0
, p =
rtl d~ d~
d4 ds dl
a n d tile relation p = A a + b is t h e s a m e as before. We get t.(w) = (tD(w) )
1
f de - b.f
where a = 0.025d¢lw 2 + 0.03~J ~ - 0.035d¢o - 0.01045dcl b = 0.3 x 10 .4 ~
+ 0.1045
× 10 -4 ~Y d
-- 0.25
x lO-4dco -
0.35 x l O - 4 d c l
e = 0.025dcow ~ + 0.035d~lw 2 - 0.03n¢1 - 0.01045d¢o d = 0.3 x 10-4"-~ z + 0.25 x lO-4w2dcl - 0.35 × 10-4d¢o - 0.1045 x 10-4dcl e = 1.5225d¢o + 0.2725d¢1 - w~d¢o - 2.25w2d¢l f = 0.2725dco - 2.25w2d¢o - 1.5225w~d~l + w~d~l and ~(~,)
= (D O _ to(.,))
~ + (K ° _ t~.(~))
~ .
Again, using t h e a l g o r i t h m of Section 4, we o b t a i n e d t h e following r o b u s t controller O.ls + 0.12 C ( s ) = 2.65s + 0.901
73 a.d r 2 = rain r 2 (u,) = 3061.7409 Therefore p = rain {k°,oo, r} = 55.333 is tile stability margin. With tile s a m e weighting coefficients as in tile 0 th order controller, we obtained the robust controller which generates the largest stability ellipsoid O.ls + 0.1 C(s) = 4.204s + 0.9296 and F2 =
rain "r2t~oj=158.0484. =' 0<~
Therefore t3= rain { 4 × 104 ,oo,~: / --- 12.57173 ~K is the stability margin which gives m o r e than a b o u t 44% allowable relative perturbations. Example 3 As another illustrative e x a m p l e a control problem of a V T O L helicopter,[53],is considered. T h e linearizefl model of the vehicle ill the vertical plane is described by the state equations ~=Ax+Bu,
9=Cx.
The state vector x E R 4 and the c o m p o n e n t s of x are: zl : horizontal velocity (knots), z2 : vertical velocity (knots),
74 x3 : pitch rate (degrees/s), :ca : pitch angle (degrees). The control vector u=[ul,u2] T, where =~ : collective pitch control, u2 : longitudinal cyclic pitch control. Following 153], the matrices A, B and C for typical loading and flight conditions at the airspeed of 135 knots are: A=
-0.0366 0.0482 0.1002 0
B =
0.0271 -1.01 a.3~ 0 0.4422 bzj
0.0188 0.0024 -0.707 1
-0.4555\ -4.0208/ a34 ] 0 /
0.1761 \ -7.5922 |
'o ° J c=(oloo). As the airspeed changes, all the elenleuts of t.he first, three rows of both matrices also change. The most significant changes take place in the elements a32, a34 and b21. Therefore, in the following, all the other dements are assumed to be constants. The nominal values of 032,
034 and 321 are 0.3681, 1.42 and 3.5446, respectively. Tile bounds on the variation
of the parameters depend on the desired equilibrium state. The following bounds on the parameters are given in [53} for linear controls: JAo32[ < 0.05, IAa34[ < 0.01, IAb211 < 0.04.
(5.4)
Given these bounds on the parameters , we use our approach to find a controller which stabilizes the family of plants given by (5.4). The transfer function of the system is
d(,) ,(~),n~(s)] G(s) = ~--L-ln
75 with d(s)--d4s4+d3s3+d~s2+dls+do
nl(s)=n.~l~ ~+n21s2+nlls+nol n2(s) = nn2s 3 + n22s 2 + n12s + no2 where
d4=1 d3=1.7536 d~=-as,t-O.O024 aa2+0.7737
dl =-1.0466 334 +4.0198 a3~ +0.0689 do=-0.0356 a34+0.1691 a32+0.0570 n31 =b2l n21 =0.7436 b21+0.0080 nil---b21 aa4 +0.0239 b21 +22.2045 no1 =-0.0366 b2j a34-0.0213 034 +0.04.56 b21 +0.7,553 n32=-7.5922 n22=-5.6262 n12=7.5922 a3~-18.2250 no2=0.2693 a34-1.1767 .
Now we select the vector a in (4.1) to "linearize" the problem:
t )
m_ |
a
|
a34
b21 \ b21a~4
76 T h e n A a n d b of (2.4.5) are given by
A =
0 0 0.1691 0 0 4.0198 0 0 -0.0024
-0.213 0.2693 -0.0356 0 7.5922 -1.0466 0 0 -1
0.0456 0 0 0.0239 0 0 0.7436 0 0
-0.0366' 0 0 -1 0 0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 0 0 0
Tile n o m i n a l value of a is
,
b=
0.7553 -1.1767 0.0570 22.2045 -18.2250 0.0689 0.0080 -5.626 0.7737 0 -7.5922 1.7536 1
/ 0.3681 \ .o= [ 1.42 / / 3.5446/ \ 5.0333 /
a n d t h e p e r t u r b a t i o n b o u n d s of a are derived from (5.4) as
o~° -
0.05
0_0.01
0 2
< al
< ,~o + 0 . 0 5
< _ a2
< o 20 + 0 . 0 1 _
(5.5)
a3° - O.O4 <_ a~ _< a ° + O.O4
a ° - 0.0918 < a4 < a ° + 0 . 0 9 2 6 .
Using tlle p r o c e d u r e of Section 4 we get t h e following 0 th o r d e r controller and stability margin
c(~) = ~ ( 0.213 0.104 -0.331 ; Po = 1.276 . 67
77 Assuming u,~ -- 1, i = 1,..,4 in Theorem 3.1 we get e _< -~.~ve T - = 0.638 which shows that the stability hypersphere contains the polytope of primary parameter perturbations (5.5). Therefore closed loop stability is guaranteed with this controller for all perturbations of the original parameters given by (5.4).
Discussion of the examples Ill example 1, the largest stability hypersphere for a 0 *h order controller is found to be independent of the controller parameters. Sinee this hypersphere did not contain the perturbation region, we attempted to solve the problem with a 1 ~* order controller. This was successful because the corresponding stability hypersphere did contain the perturbation region (Fig.5.2). However, it was interesting to observe, that by properly shaping an ellipsoid adjusted to the perturbation region, robust stabilization could be achieved with a 0 th order controUer also. Example 2 shows that tolerance of larger perturbation ranges ran be obtained by increasing the controller order, because the corresponding stability hyperspheres and ellipsoids enlarge significantly. Example 3 shows how nonlinear combinations of parameters can be handled, by introducing primary parameters to "linearize" the problem, transforming the original perturbation polytope to a corresponding one in the space of the primary parameters and inscribing the latter into a stability hypersphere or hyperellipsoid constructed, in this space, by choice of a controller.
CHAPTER ROBUST
4
STABILIZATION:THE
GENERAL
CASE
1. I N T R O D U C T I O N In this chapter, we continue our treatment of the structured robust stability and robust stabilization problem via analysis of the closed loop characteristic polynomial. Here, we drop the assumption made in Chapters 2 and 3 that tile parameter vector enters tile plant transfer function coefficients linearly or affinely, and let this functional dependence be arbitrary. In the next section, we give formulas for the characteristic polynomial in a particular "linearized" form that are applicable to MIMO systems. Based on this, a stability margin that measures the robustness of a given controller is defined in Section 3. A robustification procedure that attempts to increase this margin by redesigning the controller parameter vector is described in Section 4. The definition of the stability margin is such that if the margin exceeds a certain prescribed number, the corresponding controller guarantees closed loop stability for tile class of perturbations given. The results are illustrated by an example in Section 5.
2. C H A R A C T E R I S T I C
POLYNOMIAL
CALCULATION
Consider the standard multivariable feedback system ill Fig.2.1a with the plant transfer matrix G(s) and the feedback controller C(s). Let p be a vector of real physical plant parameters that is subject to uncertainty. ]n general the coefficients of the entries of the transfer matrix G(s) will be nonlinear functions of the parameter p. In this section we display a special form of the closed loop characteristic polynomial on which the transfer
79 function design procedure will be based. Let n be the order of the plant (Mcmillan degree of G(s)), t tim order of the feedback controller C(s) and let 5 ( s ) = 6 , + , s " + ' + . . . . . . + ~ls + ~5o
(2.1)
denote the closed loop characteristic polynomial. We refer as before to
6 := ( 6.+,
6.+,-i
""
6l
15o ) r
(2.2)
as the characteristic vector and for convenience say t h a t / f is Hurwitz if and only if 6(s) is Hurwitz. In the following, we show how to define the controller parameter vector x so that the closed loop characteristic vector 6 satisfies the equation
M(p)c(x) = 6
(2.3)
where M ( p ) is a matrix containi,g only plant parameters and c(x) is a vector containing only the controller p a r a m e t e r vector x. Tile vector x is a quantity tha~ completely defines tile t th order controller and conversely any t th order controller determines x. 2.1 Single I n p u t Single O u t p u t ( S I S O )
systems
Let p denote a vector of parameters that enters the plant transfer function coefficients. Then n n ( p ) s " + n . - l ( p ) s "-1 + . . . . . . + n l ( p ) s + no(p) G ( s ) q d . ( p ) s " + d . _ ] ( p ) s "-1 + --. + d , ( p ) s + do(p)
(2.4)
C(s) = fits' + fl,_,s'-' + ...... +fl, s +flo
(2.5)
~,s, + ,~,_,s,-, ~-::::.:: ~ - ~ 5 ¥ ~
'
80 With
c(x):=(,~,
'~,-1
"'"
00
t~,
...
B,-~
Z 0 ) r := x
(2.6)
we get tile closed loop characteristic vector 6 given by (dn(p)
0
......
n,,(p)
0
......
dn-a(p)
d,.,(p)
"" . . . .
nn-l(p)
n.(p)
"" . . . .
:
dn_~(p)
"" . . . .
:
nn-l(p)
:
•
...
"..
:
d0(p)
"
"'.
"'.
O
do(p)
"'.
•
-
-.
Or--1
"',
"'"
:
"..
"..
n0(p)
:
"'-
"'.
"'.
0
no(p)
".
:
•
"'.
"'.
•
".
~to
3t
..
e(x)
,Sn+t
¢'
~.+t-I
(2.7)
2.2 S i n g l e I n p u t M u l t i o u t p u t
Systems
Let
G(~)
=
(a,(~)
c(~) = ( c1(0
a~(~)
......
¢m(~))v
(2.8)
c2(+)
......
c.,(~) )
(2.0)
where Gi(s) = .i.(p)s"
+ n i n _ j ( p ) s ~ - 1 + . . . . . . + n i l ( p ) s + nlo(p) d , ~ ( p ) s n + d . - a ( p ) s "-1 + . . . . . . + d l ( p ) s + do(p)
c,(,) =
flits t + flit-is t-I + ...... ats t + at-is
t-I + ......
+ f l i j s + fliO + ~ l s + ~o
(2.1o)
(2.11)
81 Then With
c(x):=(at
"'"
ao
,fl, t
"'"
f l , o,
"",
flint
"'"
fl.=o)T:=x
as the controller parameter vector we get for the closed loop c h a r a c t e r i s t i c 4,(p) 0r._,(p)
o
...
"- . . . .
nj,,(p)
o
n~._l(p)
......
"" . . . . . . .
(2.12)
vector
n,,,(p)
o
-.,.-,(p)
"'.
:
"'.
0
"
"'.
0
...
:
"'.
:
"'-
,/.(p)
:
"'.
-1.(p)
-.-
:
"'.
do(p)
"'.
d,,_l(p)
n~o(p)
"'.
O
"'.
i
0
"-.
:
"-.
,Zo(p)
:
"'.
(Qt
n,,_~(p)
...
n,,o(p)
:
...
0
,~lo(p)
-.-
:
~
~n+g
n'O
tilt :
6.+t-i
=
~1o
(2.13)
c{x)
The multiinput single output case is tile dt, al of the single input nmltioutput c a s e with G(s) in (2.8) replaced by G(s) r and C(s) ill (2.9) replaced by C(s) T. The resulting equation for the characteristic vector is identical to (2.13). 2.3 M u l t i i n p u t - M u l t i o u t p u t ( M I M O )
systems
Let G(s) denote the plant transfer function matrix and (A, B, C) a minimM realization. Let ( D ( s ) , N ( , ) ) denote a left coprime factorizafion of G(s) so tllat G(s) = C ( s I -
A)-' B = D(s)-' N(s).
(2.14)
82
Now let C(s) be a tth order proper feedback controller with minimal realization (A¢, Be, C~, De). Let.
De
K, :=
o)
Bc
(
0
Bt :=
B 0
C, )
(2.15)
A~ 0 It
)
(co)
C, :=
0
Gt(s) := C,(sI - A t ) - ' Bt
It
(2.16) (2.17)
and note that the feedback system of Figure 2.1a has the same characteristic polynomial as that of the feedback system of Figure 2.lb. Now
G~(s) : =
C(sl- A)-' B
0
)
0 }I, (D(s) O)-'(.,'V(~) 0)
=
o
,;,
o
o.5,-,
(2.18)
xp
~,5,)
is a left coprime factorization of Gt(s) and
Kt := Kt
I -I
(2.19)
N¢(s)D~ls) -a
is a right coprime factorization of Kt. Then /~(s) = Det[D,.(s)D¢(s) + N.,(s)Ne(s)]
or
~(s) = Det
(a.(s)
N.(s) ) (No(,)
] j"
(2.20)
83
c
(~)
+
c
(~)
r
Feedback System
Figure 2.1a
I
+
Kt
Equivalent Feedback System Figure 2.1b
84 and from (2.18) and (2.19)
o
s.rt
i'
o
It
~
p~,)
O
:= Det[P(s)Q].
(2.21)
Let the vector x equal an ordered list of entries of /ft.
From the Binet-Cauchy
formula [54], Det[PQ] can be expanded as a sum of products of appropriate determinants of submatrices of P and Q. Let
i2
•
•
ip
•
denote tile determinant of the submatrix of P formed by the rows i l , i 2 , ' " , i v cohmms J ~ , j 2 , ' " , j p .
and the
Similar definitions apply to Q. Now
Det[P(s)Q]
i_<5,<...<~.___,,+~0
J~
j2
"'"
J,
2
--. (2.23)
= ~-~pi(s)qi , i=1
Therefore,
i=l
Note that tile polyxlomials pi(s) are functions only of tile polynomials derived from the plant and tile qi are likewise funtions only of x derived from the controller. Therefore, if p denotes a vector of plant parameters, the coefficients of the pi(s) are functions only of p. Similarly, the constants ql are functions only of x. It follows that the matrix M ( p )
85 consists of the coefficients of
pi(s) and c(x) consists of the qi respectively. Therefore we
have M ( p ) c ( x ) = 6.
(2.25)
3. S T A B I L I T Y M A R G I N Let p0 denote the nominal value of the plant parameter vector and p0 + A p a perturbation. Then
M ( p ) := M ( p ° + Ap) = M ( p °) + A M ( p ° , A p )
(3.1)
defines the perturbation of M(p°). Let x denote the controller parameter vector. Then the nominal characteristic vector is M ( p ° ) c ( x ) : = go. Under perturbations of p0 the characteristic vector S° suffers perturbations given by
M ( p ° + Ap)c(x) := S := 6 ° + AS.
(3.2)
In this setting, the problem of robust stabilization is to ensure that the characteristic vector g remains strictly Hurwitz (i.e. the corresponding 6(s) has roots in the open left half plane) for the given class of perturbations denoted by {Ap} := A. If, for example, the class of perturbations consists of perturbed ranges of parameter excursion,then
A : = { A p l - 7 1 < Apl <
¢i,i = 1 , . . . , k } .
(3.3)
Let p(S °) denote, as in Chapter 1, tile radius of the stability hypersphere centered at 6° in the space g, i.e.g ° + Ag is strictly Hurwitz for all t]AS]I2 < p(gO) and there exists with IIt,---~l12= p(s 0) such that 6o + ~
is not strictly nurwitz. An algorithm for calculating
86 p(ti °) is given later in this section. Prom (3.1) and (3.2) it follows that robust stability is achieved if VAp E A .
IIAM(p °, Ap)c(x)ll < p(6 °) Let [l"
(3.4}
I{r denote the Frobenius norm. The inequality (3.4) can be satisfied if IIAM(p°,Ap)Jlvllc(x)ll2 < p(,5°)
VAp E A
(3.5)
or equivalently
p(~o) - -
Ilc(x)ll2
> IlAM(p°,AP)llv
VAp 6 A_.
(3.6)
Let sup [IAM(p o, AP)IIF :=/3. APE#_
(3.7)
We have now proved the following result. T h e o r e m 3.1 .L.et x be .a stabilizing controller when p = p0. Then x stabilizes the closed loop system for all perturbations A p E A_ if
p(M(p° )c(x))
IlcCx)ll~
>/3
(3.8)
The formula (3.8) indicates that tile controller x must not only enlarge tile stability hypersphere,but do so with small "gain" IIc(x)[12. The quantity
t4~) :=
P(~°___L) _ P(M(p°)e(~))
I(c(x)H2
Ilc(x)lt~
(3.9)
87 is now proposed as a stability margin for the system with tile given controller x. The justification for using this quantity as a stability margin is apparent from (3.8) which shows that this margin serves as an upper bound on the level 3 of perturbations that can be tolerated in p0 with guaranteed stability. Note that the Frobenius norm was used in (3.5) - (3.7) instead of the sharper estimate ][AM(p °, Ap)l[2 because the supremum of tile latter quantity for A p E A is much more difficult to evaluate. Tile quantity 3 is independent of ttle controller and can often be easily calculated as the example in Section 15 shows. The above results hold in general. In the case of single input or single output plants a sharper result can be obtained using the results of Chapter 2 where the radius of the stability hypersphere was calculated in the space of transfer function coefficients of the plant. Single I n p u t or Single O u t p u t C a s e Consider a single input or sillgle output plant with parameter p entering the transfer function coefficients nonlinearly. In this case the characteristic vector is
Xrn(p0) -- ~0
(3.11)
where X is tile matrix of controller transfer function coefficients and m(p °) is tile vector of plant trausfer function coefficients, as in equation (2.3.8) in Chapter 2. Let re(p) := m(p °) + An,(p °, Ap). 78
(3.12)
88 Write re(p0) = y0
(3.13)
re(p) = yO + A y
(3.14)
and with x fixed, consider the largest stahility hypersphere in the space y centered at yO
with radius p~(y0) = p~(m(p0)). Then closed loop stability is preserved if
[IAm(p°,Ap)llz < p~(m(p°))
yAp E A
(3.15)
Write
sup IIAm(p°,AP)ll2
:-= ~.
"pea
(3.16)
T h e o r e m 3.2 Let x stabilize the nominal system. Then tile closed loop system remaius stable for all .per.turbations Ap E A if c, < p~(m(p°)) .
(3.17)
This theorem is an improvement over the previous one because of the following argument. Using the previous notation
X(m(p °) + Arn.(p°, Ap)) = 6 ° + Ab
(3.18)
aud X A m ( p ° , A p ) = A6 .
(3.19)
HXAm(p °, Ap)[l~ < p(6 °)
(3.20)
Now if
89 stability holds. A sufficient condition for this is: I[Am(p°'Ap)I[~ < P(8°) I[X[[---~"
(3.21)
Therefore if
sup IIAm(p°,AP)ll2 /,pet,
=,~
(3.22)
the condition given by Theorem 3.1 can be written as
<
p(6 ° ) IlXll-----~ "
(3.23)
The condition (3.17) given by Theorem 3.2 is better because
P(8°---~))< p~(m(p°)) IlXl12
(3.24)
and therefore the bound on allowable perturbations given by this theorem is bigger. These arguments also lead to the following useful result.
Corollary
3.3
Under the conditions of Theorem 3.2 the controller stabilizes the system for all pert.urbationsAp C A if
p(6 ° ) < IlXll---~ "
(3.23)
MIMO Case
In this case the nominal characteristic equation can be written as C ( x ) m ( p °) = 6o
(3.25)
and C ( x ) m ( p °) + C ( x ) A m ( p °, A p ) = S° + A6 .
(3.26)
90
Now pz(m(p°)) may be calculated as before with X in Theorem 2.3.1 replaced by C(x) and we can state the corresponding condition a < p:(m(p°))
(3.27)
for robust stability. Calculation of
p(if)
The main calculation in evaluating the stability margin p(x) in Theorem 3.1 is the determinition of p(~), the radius of the stability hypersphere. The quantity p , ( m ( p ° ) ) in Theorem 3.2 was calculated in Chapter 2 and although it is true that p(~) can be calculated similarly, or by using the formulas given in 16], we give here an alternative calculation of p(6) that may be simpler for computation. Write ~(s) = ~f0 + ~ls + . . . + &,s" =
~¢(,)
+
even degree terms
~o(~)
.
odd deKree t e r m s
From Theorem 3.2 of Chapter 1, p(~) = rain{d0, d,,, d}
(3.28)
where d :=
inf
O<w
d(w)
(3.29)
and do, d , and d(~a) are tile Euclidean distances respectively between t~ and tile subspaces Ao, An and A(w) defined in equation (1.3.4) in Chapter 1 and corresponding to polynomials with roots at 0, c~ and -I-jt~. We can easily see that
do = I~01, d .
= I~.l.
91 The determination of d(o~) follows. Proposition
3.3
T h e distance d(¢o) b e t w e e n / ; and A(¢0) is given by
i) .n ~..2p
[6o(~,o)12 [S~(jw)] ~ , ~ • dZ(~a) = 1 + ~ 2 + . . . +~4p + 1 + ~ 4 + . . . +~4(p-l)"
(3.30)
ii) n = 2p + 1
[ 6oC/,,,,,) ]~: d~(~) = [5.(j~)]2 1 + c, 4 + +. ,. . +J,~,,4p.
(3.31)
Proof Let "P. equal the set of polynomials of degree < n and let 7~(w) denote the subset of 7~. consisting of multiples of (8 2 +w2). Note t h a t 7~n is a vector space of dimension n + 1,'P(to) is a vector space of dimension n - 1, and
T = {s: + w 2, ~ + w 2 s , s ~ + w 2 s 2 , . . . , s n + ~ 2 s " - : }
is a basis for 7~(w). Define the inner p r o d u c t in 7~n :
< P, and let
P~
q
>= ~ Pi qi i=o
denote the orthogonal projection on "P(w). T h e n , it is clear t h a t
d~(,.,,) = 11,5-
PY(,5)II~.
Write
~(~)
=
(2 + ~)q(~)
+ ~ +
(3.32)
92
so t h a t ~f(jw) = $¢(jw) + ,~o(jw) = a j w + b
an d a -
- -
(3.33)
b = ~(j~,).
(j.,)
Now
P7[6(,)] = P.~l(, ~ + ~ ) q ( , ) + ~(,)] = (,~ + J ) q ( , ) + p T [ , ( , ) ]
so t h a t (3.34)
i) n = 2p: In this case
pj(s)
= 1 + ~a2s 2 + 0)484 "1-"'" 3t- ( - - 1 ) pu)2p-q2p
p z ( s ) = S -- ¢o2sa + w4s 5 + " " (--I){P-I)w2(P-I),~2P-1
(3.35) (3.36)
form a basis for 7~'L(w). Titus
lit
~ 2 --Pr(r)ll~--
pa>2 IIPIIP
b2
a2
liP1 II2
liP211~
p2>2 IIP~ll ~
and so the formula (3.30) follows from (3.33). (ii) n = 2p + 1: 11, this case p1(,9) in (3.35) is u n c h a n g e d a~,d
p2(s) = s - Js
2 + - - . + ( - 1 ) P - , 2 % 2p+1
and (3.31) once again follows from (3.37) and (3.33). This completes the proof. 0
(3.37)
93 The formulas (3.30) aald (3.31) aid in the calcualtion of d in (3.29). Write
(3.38)
t = I/~ and observe that d~
rain d(w) = min(d~, d2) o<_~<~o dl = rain
d(~)
(3.39a)
d(t).
(3.39b)
0<~<1
d2 = min o5t51
Thus the minimization over the infinite range in the determination of d can be replaced by the two minimizations over the finite ranges in (3.39). Proposition 3.3 given above and the formula (3.39) are reported in [69] and are due to Herv6. Chapellat. 4. R O B U S T I F I C A T I O N
PROCEDURE
Using the stability margins defined ill the previous section an algorithnl for controller design can be developed to iteratively upgrade the vector x of adjustable controller parameters to increase the stability margin. If xl, denotes the choice of controller at the k t~ iteration let the corresponding stability margin
V(x~.) :=
p(M(p °)c(xL, )) [l*(x~)lt2
(4.1)
Our objective is to choose xk+l so that
v(xk+1) > ~*(xk)
N.2)
and x~+l is stabilizing. From the equations
M ( p °)c(xt,+3 ) = ~k÷ I
(4.3)
94
and M ( p °)c(xk) = 6t.
(4.4)
we see t h a t if Hc(xk-4-1) - c(xk){l
xk+l is guaranteed to be stabilizing.
<
p(M(p°)c(xk)) {lM(pO)l{2
(4.5)
The correction Ax~ = x~.+, - xt. is chose,, via a
gradient m e t h o d based on numerical evaluation of the gradient of/z(x}. In the special case of single input or single output systems, the condition (3.34) given by Corollary 3.3 suggests that the stability margin can be taken to be the quantity p(Xm(p°)) IlXll
"
W h e n the perturbation ranges are given,/3 or a should first be determined (or estim a t e d ) for the prescribed controller order. Then the iterative improvement process can stop when p ( x ) > /3 or tt(x) > o' is attained because then stability with respect to tile perturbation range give,, is guaranteed. Note t h a t since the convexity of the function p(x} is not established there is no guarantee t h a t a global m i n i n m m will be found.
A connnon procedure in such cases is to
choose several initial guesses and to select the best answer. We next give an example of the robustification procedure.
95 5. E X A M P L E The calculations of this chapter will be illust,rtLted by considering the following multivariable system [23]:
, I
I I
,
!
Figure 2.2 Two mass - two spring multivariable System
'file trttnsfer function of this marginally stable system is
C;(s) =
(-,) Alsl
AlS)
where A ( s ) = rn]m.:~s 4 q- (k~m2 + kl;n.~ + k 2 m l } s :~ + klk2
96 We regard
p:--(rnl
ms
k~ k2)T
as tile physical parameter vector subject to perturbation. Now consider the 0 th order controller c(s)
{Ill ~12 ) ~21 0t22
=
with tile corresponding parameter vector
x=(3'
an
a12 a~1 a22)r.
The characteristic polynomial of the closed loop system becomes ~(s) = A(s)det{I + M ( s ) C ( s ) } = "rmlmzs 4 + (a12 + a2~)s s + (7k2rn~ + 7klm2 + 7k2m~ + aax + a21 )s ~
+ (2k2012 + 2k2a:v, + kla22)s + (Tklk~ + 2k~an + 2k2a2j + kla21) .
If we write the coefficients of this polynomial in vector form by separating tile plant parameters and tile controller parameters, we have mlm~
0
0
0
0
0
1
1
0
0
k~m2 + klm2 + k2ml 0 kak2
0 2k2
0 2k2 + kl
0
0
M(p) a12
~s
Or21
~0
c(x)
6
1 1 0 0 2k2 2k2 + / q
)
97 Now let m l = 1,m2 = 2, kj = 1,k2 = 2 with each element perturbing as follows
p°:=(1
A :=
2
1
2) "r
{Am1,Amz, Ak1,Ak2[ IAm,I ___O.Ol,lAmzl IAk, I ___O.Ol, IAk~l
_< 0.01,
_ 0.01}.
Then
M(p °) =
2 0
0 0 O\
0 8 0 2
1 0 5 0
1 0 4 0
0 1 0 4
0 1 0 5
)
and fl ~
sup ApC~.~
IIAM(p °, Ap)IIF = 0 0
0 0
0 0
0
0
0
0
2Ak2 0
2Ak~ + Akl 0
0 2Ak2
0 2Ak~+Akl
0 0
| Ak2m2+ L",tX l + ~ ' " '
k~Am2 + Ak2Arn 2
+ k,a,,.~ + Ak,~,,,.,
- [ +Ak..., + k~.,., + A~,,A,,,, k
A k l k : + k3Ak2 + AklAk2
=
'0.0301 0 0.0903 0 ,0.0301
0 0 0 0.02 0
° /ll
0 0 0 0.03 0
0
0
= 0.112099.
0
0.02
0.03/
p
Choosing the initialstabilizingcontroller as
XO ~--
2.2649934 13.898785 | -11.01622] 7.4233423 ] -6.212874!
we get the roots of the resulting closed loop system to be -6.5543880x10 4 ~ j0.4126079~ -0.317509796 ~ jl.99883004 ]
) F
98 The corresponding radius of the stability hypersphere and stability margin are
Since p ( x 0 ) < ,B the stability margin is inadequate and the initial choice of the stabilizing cotitroller needs t o be robustified. After 19 iterations of the robustification procedure of Section 4 we have a new controller
The cllaracteristic roots of the closed loop system are
The corresponding stabi1it.y radius and stability margin are
Since p ( x * )> /3, Theorem 3.1 shows tliat this controller gnarant.ees stability for the given range of perturbations.
Remarks Alt.lioug1i the formulatjon present.ed in i.l~ischapter is completely general the results obt.ained are coi~servativein relat.ion 1.0 the previous chapt.ers. This coilservatism stems
99
from tile use of the radius of the stability hyphersphere in coefficient space~ p(~), in calculating the stability margin. These results could be sharpened if larger stabilty regions could be determined in the parameter space. This is a difficult open problem in the general case which deserves much further study. On the positive side tile stability margins and robustification methods established here are computationally simple and provide some insight into the multivariable robust stabilization problem.
CHAPTER STRUCTURED
PERTURBATIONS
5 IN STATE SPACE MODELS
1. I N T R O D U C T I O N In the last three chapters~ we concentrated on the transfer function description of the plant and derived conditions for robust stabilization based on analysis of the closed loop characteristic polynomial. in this chapter~ we consider situations where the plant description is given in the state space format.
In such cases, the matrices that make up the state space model
contain various physical parameters subject, to perturbation and the robust stability and stabilization problems are solved most naturally in this setting. This is done in the present chapter. The problem formulation and the main results are given in the next section. This is followed by an example. The Appendix contains derivations of the gradient evaluations. 2. S T A B I L I T Y
MARGIN
AND ROBUSTIFICATION
2.1 P r o b l e m F o r m u l a t i o n Assume now that the plant equations are derived from physical considerations in the state space form
(2.1) y=Cx
and let the controller of order t be described by
~ = A~xc + Bey
(2.2) u = Ccx¢ + Dey.
101 Tbe closed loop system equations are
=
{/A 0
0t
At
+
0)/oc co)it
0
It
B,
~,
A,
E
0
It
z,
'
8,
Now (2.2) is a stabilizing controller if and only if At + BtKtCt is stable. Since we will consider the compensator order to be fixed at each stage of the design process we drop the subscript t henceforth and consider tile problem of robustifi'cation of A + B K O by choice of K when the plant matrices are subject to perturbation. Let p = (pl
P2
"'"
p . ) denote a p a r a m e t e r vector consisting of physical parame-
ters that enter the state space description linearly. This situation occurs frequently since the state equations are often written based on physical considerations. 111 any case, combinations of primary parameters can always be defined so that the resulting dependence of
A, B, C on p is linear. We also assume that the nominal model (2.1) has been determined with the nominal value p0 of p. This allows us to treat p purely as a perturbation with nominal value p0 = 0. Finally, since the perturbation enters at different locations we consider that A + B K C perturbs to A + B K C + ~,~'=i piEi for given matrices Ei which prescribe the structure of the perturbation. For fixed K our problem is to determine the allowable perturbation in pi that preserve stability. 2.1 S t a b i l i t y M a r g i n We now state a result that calculates tile radius of a spherical stability region in the parameter space p E R" when the controller is given. This result will also be a useful step in the robustification procedure to be developed.
102 Lel the nominal asymptotically stable system be
~(t) = M x ( t ) = (A + B K C ) x ( t )
(2.4)
and the perturbed equation be P
~(t) = ( M + ~-~piSl)z(t)
(2.5)
i=l
where the pi, i = 1 , . - . , r are perturbations of parameters of interest and the Ei, i = 1 , - . - , r are matrices determined by the structure of the p a r a m e t e r perturbations. Let Q > 0 be a positive definite symmetric matrix and let P denote the unique positive definite symmetric solution of
MTP + PM + Q = 0.
(2.6)
T h e o r e m 2.1
The system (2.5) is stable for all Pl satisfying
Ipd ~ < a'~""( Q2) E L , ~;
,=1
(2.7)
where
t,~ : =
lIETP + PE'II2
•
Proof Under the assumption that M is asymptotically stable with the stabilizing controller K , choose as tile Lyapunov function
V(z) =
zTpz
(2.8)
103 where P is the symmetric positive definite solution of (2.6). Since M is all asymptotically stable matrix, the existence of such a P is guaranteed by Lyapunov~s theorem. Note that V(~) > 0 for all z # 0 and V(z) ----, oo as [[zl[
, oo. We require ~'(x) < 0 for the
stability of (2.5). Differentiating (2.8) with respect to x along solutions of (2.5) yields
(2.9)
= zT(MTp + PM)x + xT(Zp, EyP + ~_.p,PEi)z. Substituting (2.6) into (2.9) we have r
P
~(:~) = _~TQ~ + xT(~_, p,E~e + ~ p, PE~)~. i=l
(2.10)
i=1
The stability requirement V(z) _< 0 is equivalent to r
p
zT(~p,ETp + ~'~p,PE,)~ < ~TQz. i=l
(2.11)
i=i
Using tile Rayleigh principle [54],
~.,.(O)
< ~rO---5~ < ~ . . . . C0) --
we
xT~
--
w ~ 0
(2.12)
have
ermi.(Q)xTx <
xTQx.
(2.13)
Tiros equation (2.11} is satisfied if r
•.T(ZpiE~P i=l
piPEi)z <_a.,I.(Q)zTz.
+ i=l
Since r
I:T(Zp, TP + i=1
i=1
(2.14)
104 r
r
~< II~TII~II(~p,ETp
+
i=1
~P~PE,)II211xlI~ i=l
t"
-< II~ll~()--~=Ipd[IET P + P Edlz )
(2.15)
i=l
(2.14} is satisfied if r
~-:~([pi[IIE[P+ PElf[2) <_om,.(Q).
(2.16)
i=1
Let #i :=
IIETp + PEII[2 = crm,,:(Ei'~P+ PEi).
Then (2.16) can be rewritten as
r
~(Ip~IIIETP + PZdl2) i=1
/t2
= ! Ip, I Iv21
......
tp:l)
-< ~.-.(Q)
(2.17)
Y
n_ g__ which is satisfied if
(2.18)
2
Using the fact that
(2.19)
Ilpll~ = ~ Ip~l2 i=l r
(2.20) i=1
we obtain r
(2.21)
ip, i~ < : l i . ( Q ) This t h e o r e m determines for the given stabilizing controller K , the quantity
p(K, Q) :=
~mi,(Q) ~/~---~.V .2.~=I..-5#i -
~
cretin(Q) + PE'lll
~--,7-=t]lETp
(2.221
105 which determines the range of perturbations for which stability is guaranteed. This therefore is tile radius of a stability hypersphere in p a r a m e t e r space. ~.3 R o b u s t i f i c a t i o n
Procedure
Using the index obtained in tile previous subsection, we now give an iterative design procedure to obtain the optimal controller K " so that (2.22) is as large as possible. For a given K the largest stability hypersphere we can obtain is
m a x p ~ ( K , Q) =
ai'"(Q)
(2.23)
Therefore the problem of designing a robust controller with respect to structured parameter perturbations can be formulated as: Find K to maximize (2.23), i.e.
0"2rain(Q) }
m.ax{maxp2(K, Q)} = m.a.x{n\a.x
(2.24)
subject to
~(A + B K C ) c C-. Equivalently 2
max K,Q
p2(K, Q)
= m a x '~n,,n(Q) v, r 2
(2.25)
K,Q ,f...~i=l ~ i
subject to
~(A + B K C ) C C-. Tile following constraJnted optimization problem is therefore constructed:
Given
(A, B, C) find a stabilizing controller K and a real matrix L to minimize J given below m i n J := rain "V~'''=IHETp + PEill~ K,L K,L crli,(LrL )
(2.26)
106
subject to
(2.27.)
(A + B K C ) T P + P ( A + B K C ) = - L T L and Jc :=
max
AEoiA+BKC)
Real(A) < 0.
(2.27b)
Note that the positive definite matrix Q has been replaced without loss of generality by LTL.
For any square full rank matrix L, LTL is positive definite symmetric. This
replacement also reduces computational complexity. In order to implement a gradient based descent procedure we derive the gradient of (2.27) with respect to K and. L. Before we state this result consider a slightly more general class of perturbations, i.e. r
A = Ao +
piAi,
B = Bo + E p i B i
i=1
(2.28)
i=l
Then we get
M = Ao + BoKC Theorem Let J
El = Ai + BIKC.
and
(2.29)
2.2 be defined as in (2.26) and let ( 2 . 2 6 ) - ( 2 . 2 9 )
hold. Then
(a) P
@J _ ali.(LTL)L{~min(LTL)vT2 OL
--E ~2 (EiTp+ pEi)(u.,vT + V..U.T)} i= 1
(2.3o) where V satisfies
(Ao + B o K C ) V + V(Ao + BoKC) T =
107 T T T - ~ a..,,f(ETp + PEi){Ei(u.iv~ + t,.u,,i) + (U.i,, T. + v.,u.,)E, }
(2.31)
i=l
where vai and uai are left and right singluar vectors corresponding to
~_m..(E~P + PE~L and v., and u., are left and right sin~:luar vectors corresoondimz to a,.~.!LTL).
(b) OJ 2 OK -- o'~i.(LrL ) r
{~-~ a...f(ETp + PEI)BT p(v.,uTi+ =.,v~) + B T p T v T } c T
(2.32)
i=1
(c) vT w
}
(2.33)
where v and w are the left and right eigenveetgrs of (A. + BoKC) corresponding to A..... the eigenvalue with max{Real(A)}. The proof of this theorem is given ill the Appendix. An algorithm for enlarging the radius of tile stability hypersphere p( K, Q) by iterating on (K, Q) call be devised using these gradients. Such an algorithm has been implemented using the Harwell optimization package [48]. The iterations stop when
p(K,,i,) >
ma llApll
is attained. Since little is known about tile geometry of the function p(K, Q) this procedure does not guarantee that a minimumim is attained but is nevertheless useful.
108
3. E X A M P L E As an example we again consider the VTOL helicopter [53] considered in Example 3,
()
Chapter 3. The linearized dynamic equation of the VTOL helicopter is shown below: d dt
~2 z3
z4
(_o.o3,6o.o271oo18 0.0482 0.1002
-1.010 pl
0.0024 -0.707
-4.0208 p~
z2 z3
0
0
1
0
z4
0.4422
0.1761 -7.5922[
-5052
y=(0
)
ul u2
1 0 0)x=z2.
Tile most significant changes take place in the elements Pl, P2 and P3. The following bounds on the parameters are given in [53]:
pj = 0.3681 + &pj
}Apl} < 0.05
AP2
lAp2[ _< 0.01
Pa = 3.5446 + Apa
lap31 < 0.04
P2 = 1.4200 +
Now we compute max
II~xpll2 = 0 . 0 6 4 8 .
Tile eigenvalues of tile open loop unstable plant are [ 0.27579 4- j0.25758 '~ -0.2325 ) . \ -2.072667
A(A) = |
A nominal stabilizing controller is given by -1.63522 /£o = \ 1.582236 J "
(3.1).
109 Ill Chapter 7 we develop an algorithm to obtain such a stabilizing controller for a given plant. Starting with this nominal stabilizing controller we performed the robustification procedure of Section 2. For this step we took the initial value
Lo=
1
0
-0.5
0.06
0.5
1
-0.03
0
-0.1
0.4
1
0.14
0.2
0.6
-0.13
1.5
/ "
The nominal values gave the stability margin
po = 0.02712 < 0.0648 =
IIApll2
which does not satisfy the requirement (2.7). After 26 iterations of tile robustification procedure we have p* = 0.12947 > 0.0648 = IlApll~
(3.2)
which does satisfy the requirement of Theorem 2.1. The robust stabilizing 0 th order controller computed is K-=(-0'99633989) 1.801833665 and tile corresponding optimal L*, P* and tile closed loop eigenvalues are
L* =
0.51243 -0.0004 0.12938 -0.0715
0.02871 0.39582 0.08042 0.34789
-0.1326 -0.0721 0.51089 -0.0253
0.05889 -0.3o04| -0.0145] 0.39751 /
p* =
2.00394 -0.3894 -0.5001 -0.4922
-0,3894 0.36491 0.46352 0.19652
-0.5001 0.46352 0.61151 0.29841
-0.4922\ 0.19652 | 0.29841 / 0.98734 /
-18.396295 ) A(A+ B K * C ) = [ - 0 . 2 4 7 5 9 2 ~ j l . 2 5 0 1 3 7 5 . \ -0.0736273
110 The robust controller guarantees stability for the class of perturbations given. It can be seen from (3.2) that the radius of the stability sphere is big enough that the feedback system with this controller remains stable even when the "worst" perturbation from the class (3.1) amplified by a factor of two is injected into the closed loop.
111
APPENDIX Proof
of
Theorem
3.2
Let J:=
ELl
2 am,,~:(E iT P + PEI) qii.(LTL)
(A.1)
El=, Trace{alo.( E T P + VEi)} Wrace{o'2mi.(LTL)} Then
1
A J_
4
O-min(LT L) P
{Z 2ama'(E/TP + PE')alm(LTL)Trace(Aa""f(ETp
+ PEI))
i=l
(A.2)
P
- Z 2a2ma~(ETp + PEi)a"In(LTL)Trace(Aami"(LTL))} " i=1
Here we note that
Aa,..~(ETP + PEi)
= vaiUT ai/k(Ei TP +
PEi)
(A.3) T T = vaiUai(E i AP w h e r e Vai
+ APEi)
a n d uai are right and left singular vectors corresponding to ¢rmaz(ETp + PEi),
respectively and
Aa,.i.(LT L) = v,.u~A(LT L) (A.4)
= vmuT(ALTL + LTAL) where vm and
~/'m
are right and left singular vectors corresponding
to
o'min(LTL), respec-
tively. (a) Calculation of @L P__4 Define M := A + B K C
(A.5)
Perturbing the Lyapunov equation (2.26) with respect to L and rejecting second order ternls~ w e h a v e
A/TAP + A P M
= --(ALTL + LTAL) .
(A.6)
112
From [61], the solution of the equation (A.6) is of the form rt
Ap=
~ ' ~ Z T j k ( M T ) J - ' ( A L T L ÷ LTAL)M ~-1. j=l
(A.7)
k=l
Substituting (A.7) into (A.3) leads to Trace(Aam~(ETP + PEi)) =
Trace(v~iu~i(ETAp + APEi)) = Trace{ E E 7j~ v,luTiE~( M~r )J(ALTL + LT AL)M k j k
+ E E 7J~valuT(MT)i(ALTL ~ LTAL)MkEI} = j A. T r a c e { E E 71kMJ(E'(u'iv£ + vaiuai) "1" + (uaiv5 + vaiuT)ET)(MT)kLTAL}
j and
(A.8)
k Trace(Aamin(LTL)) = Trace(v,.u~ALrL + vmu T.,Lr AL) ,T T AL + vmuTLTAL) = Trace(u.,~,.L
(A.9)
Substituting (A.8) and (A.9) into (A.2), we have
AJ _ a~,i,,~LTL ) (i~=1 Trace(2a . . . . (ET p + PEi )a~,,. (LT L) E E 7jkMJ(Ei(u"ivT + ?)aiitTi) -'}-(Uai't'T -'F valuT)ET)(MT)tLTAL} r
- E Trace{2a~.~(ETp + Pfi)~,,,i,~(L T L)(u.,v.,r + i=1
Now we have 2
AJ = a~i.(LTL ) {Wrace{ami.(LT L)
(A.IO)
113
E 7jkMJ ~ a=~=(ETP + PEi)(Ei(u~iv~ + v.o,~i) + (u=iv~ + v.iu~i)ET,)(Mr) ~ j,k
i:l
i,
r
- T r a c e { E crL.:( E~P + PEi)(u~v~ + v.,u~)} } LT AL i=1
2
- ~,~.j rsr 0
Trace{ami.lLTL)V_ ~ amaz(E 2 iT P + PEi)(umt ,T m + vmu~)}LT AL .
(A.11)
i=l
Therefore, OJ OL
2 L{ami.(LTL)V T aami"(LTL)
-
/--.~-"a .2. . . i=1
(E~T P +
PEi)(u~,'~
+ v . u . , )T}
(A.12)
where V satisfies
(A + B K C ) V + V(A + BKC) T = r
-
T T} ~ ~....(ETP + PE,){E,(..,v r + ~.,~o~) + ( ~ . , C , + v.,,,.,)S,
(A.13)
i=1
as claimed. (b) Calculation of os Consider perturbations in (A + BKC) and let
A :-- Ao + E p i A i
(A.14)
i=I
r
B := Bo + E p I B ~
(A.15)
i=]
where Ai and Bi are given matrices. Therefore
piEi := pi(Ai + BiKC)
(A.16)
114
and Ei is a function of K. Thus
AE~ = B i A K C .
(A.17)
M := Ao + BoKC.
(A.18)
Now let
Perturbing the Lyapunov equation with respect to K and rejecting second order terms, we ll a v e
MT A P + A P M = --(CT AKTBTo p + P B o A K C )
(A.19)
Ap = E E'/fl'(MT)i(CTAIfTBT°P + PB°AKC)Mk"
(A.20)
j
k
Now, differentiating (A.1) with respect to K, r
AJ=
1 T r a c e { E 2qm.z(ETp + PEI)Aer,n=z(ETp + PEi)} a~I~(LTL) i=1
2 ama=(ETp + PEI)Trace{Aa,,,::(ETp + PEi)} . a~I.(LTL) i=1
(A.21)
Hence
Trace{Aa,,,a=(EiTP + PEI)} = Trace{vaiu~iA(EiTP + PEi)} = Trace{v.iuT(AETp + E T A p + A P E i + PAE,)}
(A.22)
w ,7' + E i v . i u T ) A p } . = Trace{(uaiv£ + v . i u S ) V A E i + (v.lu.iE Substituting (A.17) and (A.20) into (A.22) we have
T r a c e { ~ ( E T P + PV,)} = Trace{C(u.,,,~ + . . , C , ) P B , AIC + C T T + Ei(ualv.iT + v.iuTi))(MT)~PBAK). E 7JA'MY((u°iv~ + v'iuol)Ei j,k
(A.:3)
115 Substituting (A.23) into (A.21) leads to
2
AJ = 2 7",T-, CTrace{Ea"*'( ETP + PEi)(u°ivT" + v"iuT')pB~+ O'min[d.J 1.~)
i=1
P
,T E Mi E (rma'(EiT P + P E i ) ( ( u a i v 'Ti + t'aiatai)E T iT + Ei(lt.ailal + VaiUai))(MT) k
j,k
i=l J
PB}a1¢ -
~
2
(A.24)
T " CTrace
amin(L L)
r
{ E a .... (ETp + PEi)(u~iv~. + v~,uTi)PBi + VPB}AA" .
(A.25)
i=l Therefore,
OJ OK = r
2
{~-~1ama~(ETp ÷ PEi )BiT P(t,ai~tal V + UaiVai T ) + BT pvT}cT
(A.26)
=
where V satisfies (A.13) and Ei = Ai + BiKC for i --- 1 , . . . ,r. (¢) T h e gradients of the constraint function Real {Am~z(A + BKC)} < 0 with respect to L and K are: Jc : = Real{Amaz(Ao + BoKC)}
(A.27)
0--L" = 0
(A.28)
OKii = Real{
vT w
}
(A.29)
with v and w are the corresponding left and right eigenvectors of A,,az(Ao + BoKC).
CHAPTER STABILIZATION
WITH
1.
FIXED
6 ORDER
CONTROLLERS
INTRODUCTION
Tile theory of the previous chapters assumes, as a starting point, that a nominal stabilizing controller has been found. This chapter and the next deal with tile problem of finding such a stabilizing compensator. When the dynamic order of tile controller is fixed a priori this is an unsolved problem. Existing solutions to the regulator problem can only generate controllers that are of high ez~ough order that arbitrary pole placement becomes possible. This includes the LQG theory [55], observed state feedback [56] and arbitrary pole placement approaches [57},[58]. Controllers that are robust with respect to unstructured perturbations evidently suffer from the same difficulty of high order (see the examples given in [2]). We also mention that adaptive control theory is notorious for producing high order solutions. It is certainly essential in practice, to have low order solutions to the stabilization problem. This requirement arises because tile controller must eventually carry out several functions such as tracking, disturbance rejection, desensitization against parameter varlations, provide good transient response, small steady state error, prevent various signals from saturating etc., in addition to the basic task of stabilization. Many of these requirements are in conflict with each other in ways that cannot be handled analytically, and the only recourse left to the designer is to iteratively redesign the controller using adhoc methods and graphical displays until a satisfactory solution is obtained. This redesign must be carried out in the parameter space of the stabilizing controller. If the basic stabilizing
117 controller order is high, so is the dimension of this parameter space and the subsequent design procedure can become unwieldy. From this perspective, the high order of controllers produced by "modern" control theory is one of the severest limitations of this theory. We attempt to alleviate this problem by presenting, in this chapter, some methods in the transfer function domain, for designing low order stabilizing controllers. These resuits were originally reported in [47]. The formulation presented in this chapter deals with stabilization of a system with a controller of prescribed order. The results obtained can be used to find low order solutions by successively updating the prescribed order until a solution is found. This formulation allows us to obtain a lower bound on the order of a stabilizing controller by means of a classical result of linear programming, known as Gotdan's Theorem [59]. Each stabilizing controller is found to correspond to a Hurwitz vector that lies in a linear subspace determined exclusively by the plant transfer function parameters, and the controller order. An algorithm for selecting Hurwitz vectors is described, that iteratively increases the radius of the largest stability hypersphere and reduces the distance to this linear subspaee. A stabilizing controller of the prescribed order ~ is found when this largest stability hypersphere intersects the subspace.
2. N E C E S S A R Y
CONDITIONS
USING LINEAR PROGRAMMING
Consider the problem of stabilizing the fixed
qth
order plant with transfer function
G(s) with a controller C(s)of dynamic order t. The class of all controllers of order t can be parameterized by a real vector x of fixed dimension as in Chapters 2 and 4. For single input or single output plants x can consist of the list of transfer function coefficients of C(s) as in equation (4.2.13). In multiinput multioutput systems x can consist of the list
118 of entries of the state space representation of a ttn order dynamic system with input y and output u as in equation (4.2.25). Let n := t + q denote the dynamic order of the closed loop system and
~(~) = ~ 6;(x)s;
(2.1)
i----0
the characteristic polynomial of the closed loop system. As before let
~(x) = [~0(x),~(x), . . . . . . ,~.(x) ~'1
(2.2)
denote the characteristic vector, and let H , :=
[61~eR"+~,,~(s)is of
degree n and Hurwitz] .
(2.3)
As shown in Chapter 2 and Chapter 4 we have
Mt x = g
(2.4)
for single input or single output systems, and
M,c(x) = ~
(2.5)
for nmlilinput and multioutput (MIMO) systems. The matrix M t is completely determined by the plant par:~melers, which we consider fixed here, and by |he order t of the controller. The reader is referred to equations (4.2.13) and (4.2.25) for the exact form of Mr. Let
R(Mt)
denote the image of the m a p Mr.
T h e o r e m 2.1 Let the plant be single input (multioutput) or single output (multiinput) and let Mt be defined as above. There exists a t th order stabilizing controller if and only if
R(Mt) f3 H .
# ¢
(2.6)
119
For the MIMO case equation (2.5) shows that (2.6) is also a necessary condition for stabilizability with a t 'h order controller. As t ranges over 0 , 1 , 2 , - - - , tile size of tile matrix M t and the numerical values of its entries change but the form of the equation remains unchanged. If t is high enough that R ( M t ) = R " + ' , ( 2 . 6 ) is trivially satisfied and (2.4) has a solution for every 6. This corresponds to the arbitrary pole assignment ease and will
occur
for t >_ q - 1 in SISO
systems, and generically for t > 9 Err - " in single input m output systems and t > ~r -~
in
single output v input systems. For low values of t the problem of checking (2.6) cannot be completely solved without grappling with the nonlinear Hurwitz conditions. Fortunately however, some useful necessary conditions for (2.6) can be obtained using a theorem of linear programnfing and this is described below. Let y > 0 denote that every component Yi is strictly positive and let
all
e, +, := {~ ~ R"+' I~ > o}
(2.7)
e~- := {6 e R "+' I~ < o}
(2.8)
d
denote all polynomials of degree n with strictly positive and strictly negative coemeients, respectively. Clearly,
n. c p2 o P~.
(2.9)
120
Lemma
2.2
If there exists
a t th
order stabilizing controller then there exists x so t h a t
Mtx>
O.
(2.10)
Proof Note t h a t there exists x so t h a t M t x >
0 if and only if there exists x such t h a t M t x < O.
Therefore if (2.10) fails R ( M t ) 13 P,+ = ¢ and R ( M t ) 13 P~" = ¢ and so R ( M t ) 13 Hn = ¢. T h e condition (2.10) can be checked by the following theorem [59]. Theorem
2.3(Gordan's
Theorem
of the Alternative)
For each given matrix A, Either ]. Ax > 0 ha.~ a solution x or
II.
ATy =
0,y _> 0 has a solution y
but never both. (Here y > 0 denotes t h a t at least one c o m p o n e n t of y is positive, and no component is negative). Geometrically we m a y interpret GordaJl's theorem as follows. Either there exists a vector x which makes a strictly acute angle (< -~) with all the row vectors of A, Figure 2 2.1a, or the origin can be expressed as a nontrivial, nonnegative linear combination of the
121
rows of A, Figure 2.lb.
A-
f
Figure 2.1a
Figure 2.1b
Gordan's theorem leads to the following useful result on low order stabilization. The result, is applicable to tile general multiinput multioutput case. T h e o r e m 2.4 If
M ~ y = 0,
y _> 0
has a solution y. then R ( M t ) N H,, = ¢~
and no ira order compensator can stabilize tile given plant. Tile c~mdition given by ('ordan's theorem can be checked by solving phase I (i.e. llnding a feasible solution) of the linear programming problem:
M T y = 0, The condition M t x >
Z yj = 1 Vyi >_ 0 {2.11} J 0 call also be directly checked by a slight modification of the
general linear progranuning prol)lem, [60], which is defined as follows:
122
(2.12)
Minimize (or Maximize j=l subject to n
i = 1,2,...,m, j=l
j = 1,2,--.,n.
z j > 0, Therefore we modify (2.10) to
,
Mr
> ¢
{2.13)
Xn
and set up the problem as mind:(::,,) = E
z,,j
(2.14)
subject to
EJ m , U j ) ~ , j _> ~, z,,j > O,
i = 1,2,.
,
j = 1,2, . . . . . .
where n~,¢ij} denotes the ( i , j ) th element of M , .
From the solution x , of this problem,
a solution x satisfying tlle inequality condition (2.10) is obtained with x = y - z. It is possible to avoid the strict inequality in (2.10) by introducing a positive slack variable ~. This slack variable may be chosen arbitrarily to be any positive vector without affecting the solvability of (2.10) as proved below.
123 L e m m a 2.5 Let ~ > 0 be an arbitrary positive vector. Then
Mix
> 0
has a solution
(2.15)
Mtx
_> ¢
has a solution .
(2.16)
if and only if
Proof The proof is by contradiction. Let us suppose that t[ > 0 and ~* > 0 are fixed vectors such that
6 M,x > ( =
has a solution
(2.17)
has no solution.
(2.18)
but
Mty > ~"
G =
\~,* Then let
max(~i,...,C,) r a i n ( f , , . - . , ~,)
(2.19)
and consider the vector a.~*. Clearly
min(a~l , - - - , a~,,) = m a x ( ~ ; , - . - , ~, ).
(2.20)
Now a~ > ~" so that Mt(trx) > cr~ > which contradicts (2.18).
~"
(2.21)
124 3.SUFFICIENT
CONDITIONS
THE STABILITY
USING
HYPERSPHERE
The results of the last section show bow linear programming can be used to obtain necessary conditions for stabilizability by a t th order controller. If these necessary conditions are satisfied there exists x such that M i x E P + .
In low order problems it will
frequently be true that this solution also satisfies M t x E H,, or M t e ( x ) E H,, and then x represents a stabilizing controller. In general, however, this will not happen, and this motivates us to develop a sufficient condition for (2.6) to hold. To state this result let 6 R "+1 be Hurwitz and let p(t~) denote the Euclidean radius of the largest stability hypersphere centered at 6 as in Theorem 1.3.2. Let d(6, M r ) denote the Euclidean distance between ~5 and the subspace R ( M t ) . Theorem
3.1
Let ~ denote any Hurwitz vector. If
d (~, Mr)
< 1
(3.1)
then the orthogonal projection t~/tt of t5 oi, t.o R ( M t ) satisfies
~M ~ R ( M t ) N H . .
(3.2)
Proof The proof is obvious from the geometrical construction shown below in Figure 3.1. T h e condition (3.1) guarantees that the stability hyperspbere centered at/~ intersects the subspace
R(Mt).
<>
125
p(s)
d (6, Mp) .R (Mp)
5~
5M
Illustration of T h e o r e m
Figure 3.1
6.3.1
126
We assume t h a t
M, has full rank. This entails no loss of generality as x can be redefined,
if necessary to m a k e it so. T h e equation for d(6, M r ) then, is
d(6, M r ) = I I M t ( M t T M t ) - ' M t T 6 - 6[[2.
(3.3)
T h e above t h e o r e m suggests the following minimization problem:
rain ~n,
d(~, M r )
(3.4)
p(~)
Once we find a g which satisfies (3.1) we call project g orthogonally onto }t(Mt) to obtain the vector/~,~I which is both Hurwitz and in t h e s u b s p a c e R ( M t ) . Then
~M = M , ( M ~ M t ) - I M ~ 6
(3.5)
is tile closed loop characteristic vector and
x = (M~Mt)-~M~
(3.6)
is the vector of transfer function coefficients of tile stabilizing controller. For tile case Mtc(x) = 6
(3.7)
where c(x) is a nonlinear hmction of the controller p a r a m e t e r x we m a y use the following strategy: Write M t y = if
(3.8)
y = c(x)
(3.9)
127 Let ¢S be a nominal choice satisfying (3.1). Then let
yO = (Mt~ M , ) - I M T ~
: - Nt~
(3.10)
and let p(y0) denote the radius of the largest stability hypersphere in the y space centered at y0. T h e calculation of p(y0) was given in T h e o r e m 2.3.1 of C h a p t e r 2. Now if x is such th at ilY0 _ c(x)ll 2 < p(y0)
(3.11)
it is clear t h a t M t c ( x ) is Hurwitz and a stabilizing controller has been found. Theorem
3.2
Let g be such t h a t d(~,Mt) - < 1 p(6)
(3.12)
IIN,g - c(x)ll~ < 1 p(Nt(~)
(3.13)
M,c(x) e ~'.
(3.14)
Then if x satisfies
it follows t h a t
and x corresponds to a t th order stabilizing controller. T h e above t h e o r e m suggests the following algorithnl:
rain IIN,~ - c(x)ll2 : = 3
, u' ~•t H
(3.15)
p(Nt~)
Tile mininlization of J can begin once a g satisfying (3.12) has been found since otherwise
p(Ntb) is
not defined. If J < 1 is attained a stabilizing controller has been found.
128 4.
EXAMPLES
E x a m p l e ..I Let
G(s)
-a s - 1 0 ~ 4 - 5 s a - 6 s 2 - 8 s + 4 2s s + s 4 - 20s s - 80s 2 - 80s
=
- - s s - - 10..¢ 4 - - 5 a a ' -
For
a 0 th
6s 2 -- 8s
order controller we have
( -: ) -
i
-2
-20 1 14 -14
0 --
-80 -80 -14
Using linear programming we obtained M o x > 0 with the controller parameter -0.001) x=
-1
.
-1 It turns out that M 0 x is Hurwitz.
We verify that the roots of the closed loop system
corresponding to this x are -0.480749 ± j0,674802~ -1.07459 | -39.1735 j -968.790 / ~tnd therefore C(s)
=
( -o.ool -'
-' -o.ool)
is a stabilizing controller. A pole placement controller would be of second order, with the
corresponding dimension of x being 9.
129 Example
2
Consider the following plant:
- 6 . 0 9 8 1 4 + 85.33813 - 415.3381~ + 1094.92758 l j G(8) =
815 _ 22.08814 + 29.47812 _ 71.95812 _ 5025811
-1-11051.568 ~0 - 19289 - 156,2588 - 1 1 7 1 2 . 9 4 8 1 ° + 36.1889 + 22.7488
- 4 1 9 . 6 8 8 T + 1069.5688 - 1298.428 s + 375.122584 - 1 9 1 . 6 4 8 7 - 907.886 + 1105.6185
- 1 3 6 5 . 3 5 8 3 + 2 5 9 . 2 5 s 2 - 62.228 + 198.096
÷1322.4583 -- 259.78~ - 3.348 - 208.55
For a 0 th o r d e r c o n t , r o l l e r
M0
1 0 -22.08 -6.09 29.47 85.33 -71.95 -415.33 -5025 1094.9275 -11712.94 11051.56 36.18 -192 22.74 -156.25 -191.64 -419.68 -907.8 1069.56 1105.61 -1298.42 0 375.1225 1322.45 -1365.35 -259.7 259.25 -3.34 -62.22 -208.55 198.096
130
y > 0 with
Using linear programming we }lave M ~ y = 0, 0 0 0 0 0 0.07915 0 0 0
Y=
0 0 0.21974 0.70109 0 0 0 as a solution 1.o the conditions required by G o r d a n ' s Theorem. It follows from Gordan's Theorem and Theorem 2.4 that there is no 0 *h order stabilizing controller. We increase the order of the controller to 1. This gives
M 1 =
1 -22.08 29.47 -71.95 -5025 -11712.94 36.18 22.74 -191.64 -907.8 1105.61 0 1322.45 -259.7 -3.34 -208.55 0
0 1 -22.08 29.47 -71.95 -5025 -11712.94 36.18 22.74 -191.64 -907.8 1105.61
0 1322.45 -259.7 -3.34 -208.55
0 -6.09 85.33 -415.33 1094.9275 11051.56 -192 -156.25 -419.68 1069.56 -1298.42 375.1225 -1365.35 259.25 -62.22 198.096 0
Using linear programlmng we found t h a t M i x > 0 for
o.ool \ -0.030557| x =
-0.008807/" -0.031460/
0 0 -6.09 85.33 -415.33 1049.9275 11051.56 -192 -156.25 -419.68 1069.56 -1298.42
375.1225 -1365.35 259.25 -62.22 198.096
131 llowever M i x is not Hurwitz. Therefore we adopt the minimization procedure (3.4). With an initial choice of g0 E R 17 corresponding to the polynomial g°(s) = (s + 1) 1¢ we find that p(g0) = 1 and d(g °, M1) = 3568.24, which does not satisfy the condition of Theorem 3.1. After "minimizing" (3.4) numerically we get a new value of g which gives
d*(g, M l ) = 1.0 x 10 - s
p*(g) = 1.0 X 10 -2 . Since d'{p,Mz} p'(M -- 10 -6 < 1~ we get from T h e o r e m 3.1 the stabilizing controller parameter vector 0.00249987 -0.0949889 | x =
-0.0300101 J"
-0.9999985 / The roots o f t h e closed loop system are -0.0054836 -0.2471693 -0.2189355 ± j0.7040592 -0.5174567 ~ j0.8936120 -1.0393600 i j0.2263056 -0.9189383 ± j0.6516824 -0.2942842 ~ jl.3348327 -0.5081362 ± jl.6949029 -2.8581280 ± j2.8312120J and the corresponding
1 st
order stabilizing controller is
C(~) =
-0.0300101s - 0.9999985 0.00249987s -- 0.0949889
A pole placement or observer based solution would be of 14 th order with tile corresponding dimension of the controller p a r a m e t e r vector x being 30.
132
Remarks
We have given some results and computational procedures that can aid the designer in generating low order solutions to the problem of feedback stabilization. Since no necessary and sufficient conditions for stabilizability with a fixed order controller are available as yet, these results are not final and it. is our hope that they will stimulate further work on this problem. It is clear that progress oll this problem can result if a better understanding of the geometry of the Hurwitz region and efficient ways of dealing with the Hurwitz conditions can be developed. This would sharpen the algorithm given here.
CHAPTER
7
STATE SPACE DESIGN OF LOW ORDER REGULATORS 1. I N T R O D U C T I O N
In this chapter, we continue our treatment of the low order feedback stabilization problem by developing a state space based algorithm. This algorithm first attempts to stabilize the closed loop system with a fixed order controller. This corresponds to an extended output feedback stabilization problem. We attempt to solve this iteratively. At each iteration a state feedback matrix assigning a prescribed set of eigenvalues is found and this matrix is approximated by output feedback. This is done successively by readjusting the desired closed loop pole locations in the left half of the complex plane to minimize a performance index thai. measures the deviation of the actual eigenvalues from the desired ones. A low order solution is found by sequentially increasing the controller order until stabilization is achieved.
The algorithm that is given depends on the parametrization of the state feedback pole assignment problem derived in [44]. This is briefly described in the next section. Ill Section 3, the fixed order output feedback stabilization problem is formulated as an optimization problem and Section 4 describes how the performance index can be decreased by increasing the controller order. Examples are given in Section 5 and some of the gradient evaluations of Section 4 are derived in the Appendix. 2. T H E S Y L V E S T E R E Q U A T I O N
FORMULATION
An algorithm was introduced in [44] for solving the pole assignment problem using
134 state feedback. This algorithm consists of solving for X and then for F
.4x - x A = - B c
Fx
(2.1)
= c
(2.2)
for given (A,B,.4) with an arbitrary choice of G. In (2.1) and (2.2) A, X and .A are n x n matrices. From a result in [61] the solution X of (2.1) generically has full rank if (A, B) is controllable and (G,J~) is observable. Let
Xi(T) denote
the i *h eigenvalue of T and )~(T)
the spectrum or eigenvalue set of T. It follows that if X has full rank the solution F has the property:
~(A
+
BF)
=
~(~).
(2.3)
The advantages of this algorithm are: a) The algebraic variety F(A) of matrices P which assign
a
prescribed set of elgen-
values A can be obtained by setting A = ~(,4) for a fixed .4, and letting the free parameter G run through the set of all possible real values. b) E~cient numerical procedures [62] are available for the solution of Sylvester's equation (2.1). Based on this parameterization of F(A) algorithms were given in [19] and ['15] for optimizing the conditioning of the closed loop eigenveetors and in [461 for minimizing the norm of the state feedback matrix F. Here, we extend these results by considering measurement rather than state feedback and by treating the problem of stabilization rather than arbitrary pole placement.
135
3. O U T P U T
FEEDBACK
CONTROLLERS
pth order
Consider the linear time invariant plant S cascaded with the
feedback corn-
pensatorC.
S:~=Az+Bu (3.1)
Ym C
:
k~
= Cx. +
B~y,,
= C~x, +
Dcy,-.
=
Acz~
(3.2) u The closed loop system is
or
•~ ~r
=
0
0p Ar
I,, Bp
B~
A~
0
Kt,
lp C~,
z~
(3.4)
zp
and the transfer function of the pt~ order compensator is
C(s) := Cc(sI - A¢)-IBe + Dc •
¢3.5)
The formula (3.4) shows that ally fixed order compensator design problem is equivalent to a static outpul feedback problem, tn particular tlle problem of stabilization with a fixed order controller p is equivalent to that of stabilizing Ap + BpI(pCp by choice of Kp.
The general solution of this problem is unknown.
The best available results on
the output feedack problem are those of Brasch and Pearson [57] and Kimura {58] which deal respectively with arbitrary eigenvalue assigmnent and "ahnost" arbitrary eigenvalue assignment.
136 Let A denote a symmetric set of n + p complex numbers (i.e. complex numbers
occur
in complex conjugate pairs) and let Kp(A) :--- {Kp]Kp E R ("+~)*(r+'),)~(A,, + Bph'pCp) = A}
(3.6)
where Ap E R (n+p)x(n+p), Bp E R (n+p)x(m+p), and Up E R ('+p)x(u+p) are as in (3.4). The result of Brasch and Pearson [57] states that if ( A , B , C ) is controllable and observable with controllability index v¢ and observability index vo, and p >_ min{vc,Vo} then Kp(A) ~ 0 for every choice of A. The result of Kimura [58] states that if p _> n - ra - r + 1 then )~(Ap + BpKpC~,) can be made arbitrarily close to any set A of n + p symmetric complex numbers. The upper bound on the order of a stabilizing controller established by the above results is in general too conservative. This stems from the fact that both results essentially require arbitrary pole placement. In fact for specific choices of the n + p complex numbers A, Kp(A) will "almost always" be empty unless p the compensator order is high. To lower the compensator order we therefore relax the specification of A in (3.6) to a simply connected region fl C C - , and consider the family
It is reasonable to expect that Kp(N) will in general be nonempty for values of p much less than the lower bounds given by the results of Brasch and Pearson or Kimura and numerical examples support this. The effective characterization of the family Kp(f~) is an unsolved open problem. Our approach to this problem will be to consider the state feedback family
F_p(n) = {FpIFP E R ('+p)x("+p), A(Ap + BpFp) C f / C C - }
(3.8)
137
~ . d de~ern, in~ ~ , G ~ -G(~) and then find X~'p such that, liG - G G I I
is ~matl in the
hope that such a Kv E Kp(f~). The advantage of this approach is that the family __p(fl) can be characterized conveniently as shown later. For the remainder of this section we drop the subscript p for convenience. In general, even if IIF - KCII is small it is not true that A(A + B F ) and ~(A + B K C ) are close. The ]atter can be achieved by making the eigenstructure of A + B F as orthonormal as possible. Let ~r,,,~(T) and ~'mi,(T) denote the largest and smallest singular values of T. It is web known [62],[63} that the perturbation of the eigenvalues of the matrix (A + B F ) for changes in the entries is small if the condition number k ( X ) :=
/I-¥ll2llX-' li2 : = °'max(-¥)/Crmin(-'¥) of the eigenvector matrix X is smafl. Let F - K C
:= T
so that A + B K C = A + B F - B T . Then using the formula in 163] we have I,~i(A
+ HA'C)
-
.~(A
+
B F ) I < IIBTII~k(X) _< IIBII211TII~k(X)
(3.9)
<_ IIBII=IIF - K 6 ' l l v k ( X )
which shows that control over the eigenvalue locations of A + B K C can be obtained only if both [IF - KC[{ and k ( X ) are kept small. One way of doing this is 1.o minimize
s = ~ , k ( x ) + ~ l l F - h'Cll~
a.,~.(.¥) = '~] , , , . , . ( X )
+ c~Trace{.(F - K c ) T ( F - I f C)}
(3.10)
by letting )~(A + B E ) range over the region f/ C C - . Similarly, by letting A + FDC = A + B K D C a dual problem can be formulated as
J n = fl, ¢( r. ...,. i . (XD) ( X n ) + fl2Trace{(Fo - B K o ) T ( F n
-
BKD)}.
(3.11)
138 The idea of simultaneously improving the conditioning of the eigenstructure and of minimizing the norm of 1;'- KG was first introduced in Keel and Bhattacharyya [64],[65]. Here an improved version of this algorithm is presented. In particular we convert the constrained optimization problem to an unconstrained problem and extend the class of regions f / C G - to more general and useful regions. These details are given next. 4. STABILIZATION A L G O R I T H M
In the Sylvester equation approach described in Section 2,
ax
- xA
= -Ba
Fx = ¢ aud let A(A) C 12 C C - .
(4.1)
(4.2)
Under the assumption A(A) N A(A) = 0 and ( A , B ) control-
lable, (G, f i) observable, tile unique solution X will 'almost surely' be non-singular by deSouza and Bhattacharyya [61] and then A(A+BF) = A(.~) with F = GX -1. By letting ~(.4) range over 12 and G tile free parameter run through all possible values this formula generates tile family F__(12)defined in (3.8). If .4 is a complex diagonal matrix in (4.1), it is clear that X in (4.1) is tile corresponding complex eigenvector matrix. However it is desirable that these matrices be real for computational convenience. The following Lemma 4.5 shows that A can be taken as a real matrix without loss of generality. Before we state Lemma 4.3 it is necessary to introduce some facts. D e f i n i t i o n 4.1 A real square matrix A is called a pseudo diagonal matrix if it is of the form
139
Q1
~2
as
(4.3)
A =
03
w~th ai, ~i real. D e f i n i t i o n 4.2 A complex square matrix is called normal if A * A = A A * . L e m m a 4.3 [54] A complex square matrix is unitary similar to a diagonal complex matrix if and only
if it is normal. L e m m a 4.4 Any real pseudo diagonal matrix is normal. Proof Taking the i ~h block from (4.3) such as
.4i =
(°,
-/31
we have
d4~A~
{ ~ + ~2 0
0
\
~, +Z~ }/
= A~.Ai.
(4.5)
An)
(4.6)
Thus, each block is normal. Now let
A=Diag(Al
A2
......
140
AA* = Diag( AIA~*
A
2A 2*
• ..
A.i;)
(4.~)
A * A = Diag(A~Aj
A~A2
• ..
A;A.).
(4.8)
Since AA* = A ' A , the statement is true.
0
L e m m a 4.5 Let (A + B F ) X = Xf4 and (A + B F ) Y = Y.4 where 1. A, B, A, X and F are .real matrices wit h appropriate dimensions, 2 . . 4 is real pseudo diagonal, .4 is complex diagonal, and 3. X and Y are nonsingular. Then, k(X)
= k(Y)
.
(4.9)
Proof From Lemma 4.3 and 4.4, A is known to he normal and unitary similar to the complex diagonal matrix A. Thus A = UAU*.
(4.10)
Write (A + S F ) X = X A = X U A U *
(4.11)
(A + s F ) x v = x u $
(4.12)
so that
and X U = Y.
(4.13)
141 Now,
k(X) = k(XU) = k(Y).
0
(4.14)
From this Lemma, minimizing ama:~(X )/o'min(X) ill (4.10) is equivalent to minimizing
amaz(Y)/trmin(Y). Therefore.
we can henceforth take .4 as a real pseudo diagonal matrix
without loss of generality. In order to use a gradient based descent algorithm the closed form expression of the gradient of the performance index (3.10) with respect to the variables G, K and the variable elements of A denoted hi is evaluated. The derivation is given in the Appendix. T h e o r e m 4.6 Given the performance index J in (3.10}, and constraints (4.1) and (4.2), the gradients of J with respect to the independent variables G, K, and A are as follows:
(a) O_.JJ = 2{a~(F - K C ) X -7" + B T u T} OG
(4.15)
where U satisfies
.~.U - U A
-
~r; . ( X 2 - )
- 2 & 2 X - I ( F T -- ( K c ) T ) F
(4.I6)
where va and ua are right and left singular vectors corresponding to a,~,,,(.X') and v~ and ui correspond to a.,~,{x)~ respectively.
(b) Let ai denote a variable element of .4.: c..gh--~.= - Trace
UX ~-ai
(4.17)
142
where U satisfies (4.16)
(c) °! -
OK
2 ~ 2 ( F - ~'C)C r
(4.18)
Equations (4.15) - (4.18) are used to devise a gradient algorithm that iterates on the free parameters G, K and the entries of A to reduce J . At each iteration of the algorithm we get Ai, Fi and Ki. Since ~(Ai) C f~ we have )~(A + BFi) C ~ for each i. However
~(A + BKiU) may or may not be in fl for each i, and the algorithm is designed to update Ki to drive A(A + BKiC) closer to A(.4i) = A(A + BF~) after some iterations. The following structure of tile closed loop eigenvalue matrix .4 ensures stability without constraints during the iterations: -
&~
-~2
&~ -~
-al ~i=
-a4
a, -a~ gt 5
Note that fii in the matrix .4 are the only nonzero parameters and furthermore the stability requirement,A(A) C G - , is automatically satisfied without constraints, for all real values of ~;. We can also parameterize ft, in such a way that the desired closed loop eigenvalue locations are automatically confined to some useful region fl as in Figures 4.1 and 4.2. In choosing A, a maximal number of 2x2 blocks are included in the initial choice• As the algorithm evolves some of the off diagonal terms may become very small. At that point
143
we start to vary the corresponding diagonal terms independently. In tile damping ratio
region described in Figure 4.2, 0 is also a free parameter. Marginal
Stability
Region
For this case we can simply modify tile matrix A to
-a2
-(a~ + 7) -(a~ + v) -a,
3,=
a, -(a~ + 7)
with fii as the real variable parameters and 7 is fixed. Tile eigenvalues of A are all to the left. of the line R e ( s ) = - 7 .
Im~ f
J f f J J
/
J J I f J J f
-"t
0
f I
Figure 4.1 Marginal Stability Region
Rc~
144
Damping Ratio Region (5~ + 7)tan¢sia0 -(a~ + ~) tw
=
O
U 0 0
t
~ Q
I rnag
--'7
~pl
jjP
Figure 4.2 Damping Ratio Region
Real
145 Now we discuss what happens when the proposed algorithn~ fails to find a stabilizing controller of order i. In this case~ we increase the controller order to i + 1. It is then necessary to have a way to select the initial values of Go, -40 and h'0 for the controller of order i + 1 to ensure that the performance index d keeps decreasing. The following theorem shows the way to select initial variables so that J always decreases with increasing controller order. T h e o r e m 4.7 Let J* be the optimal performance index with optimal variables G ' , 2i* and K* where .... (x')
d" = a l , a . . . ( X * )
+ a2llF* - K*CII~"
(4.19)
and X*.,,.and F* satisfy
AX'-XA'=-BG* F" = G ' ( X ' )
-I
Then for the extended system
0
O~
0
I~
0
I~
'
(4.20)
tile initial value of the corresponding performace index de is equal to J* if the initial choice of the free variables are as follows:
,. __
0) 0
X3AiX~ "~
where -4i is an arbitrary pseudo diagonal submatrix of the extended matrix
0
¢]~
(4.21)
146
and 0"1 0"2 X 3 --_
0.1
/
for arbitrary real ,lumbers 0./ satisfying 0.1 > 0.2 > "'" > ai > 0 with ai > 0.,,,~.(X*) and 0., < 0 . , , , , , . ( X * L
Proof Let the optimal values of J* be obtained by G* and K*, then the extended system becomes
0
~,
L
x,
)(x.
£ = -
-.¥,~
~,)
x, x , ) ( ~ L
~,
(o~ o)(~. o,) I~
G2
-x,.~,
Gs
) =_(Ba G,
G~ ]"
(4.22)
If we pick G1 = 0 and G2 = 0, t h e n Xa = 0 a n d X2 = 0 a n d X s f t i = Gs. Here we choose
~r2 X3 ~
°.
(4.23)
for a r b i t r a r y o1 > 0.2 > "'" > 0.i > 0 with at > a,..i,~(X*) and aa <_ a m a r ( X * ) . W i t h this choice of X3, we have
o , . . , , ( X ' ) = 0...,,(Xc)
(4.24)
g m o , ( X * ) = 0..,a~(Xc)
(4.25)
147
Therefore,
Now consider t h e t e r m I W -
(4.26)
ffmin(Xe)
~min (-~)
KCII~. Since
o)
X, =
Xa
'
we have (4.27)
where *ffl
X~ -1
=
.
..
1__
o)(x, o)
Now
F~ = G~X~-~ =
X~Ai
=
0
0
Xf 1
X3AIXf I
(4.28)
•
Let
IG =
K*
KI )
K2
Ks
(4.29)
"
Then
F,
-
K~C~
-K2C
XsAiX~
-
Ka
"
(4.30)
Here we choose K1 = 0 a n d A~ = 0. Also we can choose
(4.31)
1(3 = X s f i i X ; "
because Xs a n d .Ai are well defined. W i t h such a K we have
F , - K¢C, = ( G X - 1 0 - K C
o)
0
(4.32)
148
Thus,
IIF -/CCIl~
=
liFe -/~'¢c~112
(4.33)
Therefore, we conclude that ,,,,,a.(x') ,,mo~(x~) a, a , = i . ( x * ) + a2 ]IF* - K*CII2F = a~ a,.,,,(Xe) + a211Fe - KeC*H2F
(4.34)
with the choices of
Ge = (G* 0
0
)
X3.41
and
K~ =
with Xa as in (4.23). This concludes the proof.
( K* 0
0
X3fliX~'
)
(4.35)
<>
This theorem is useful for finding a low order stabilizing controller because it shows how by sequentially increasing tile order of tile controller, J can be guaranteed to decrease. Since a small enough value of each term of J confines the spectrum of A + B K C to f~ ( in accordance with (3.9))the algorithm eventually stabilizes the system. 5. E X A M P L E S
The algorithm developed in the last section is applied to several examples here. The gradient calculations of Theorem 3.4 are used Mong with the Harwelt subro.tine package for optimization. Example
1
The first example is a simplified model of the NASA F-8 Digital Fly-By-Wire(DFBW) airplane [66].]ts dynamic equations are as follows:
d =
-0.075 -0.27 4.4 0.078 -0.99-0.23 1.0 0.078
0 0.052 0
+
0.82 -3.2 00.046 0
~a br
149
(r) ¢
=
( 0 1 0 O) ( i ) oool
The given design specifications [66] are that the closed loop poles must be to the left of the line s = -0.2 i.e. 3' = 0.2, and the damping factor is > 0.7,i.e. ¢ = ~ in Figure 4.2. For the optimization problem the initial values are chosen to be
,4o=
(
-2
-3
1
Ks =
-5
3
-3
-5
-0.25
)
0.5
(0 0) 0 0 "
After 41 gradient iterations minimizing J in (3.10) the following 0 th order stablillzing compensator is obtained. •.. :
(3.24.51997 k 2.8359286
-0.379821 -0.055259j
The corresponding data is shown in Tables 1.1, 1.2 and Figure 5.1. For comparison, the same problem was run without including the condition number term in J (i.e. al = 0 in (3.10)). It is seen from the corresponding data, shown in Table 1.3, 1.4 and Figure 5.2 that the condition number increazes ~ignificantly, and although stabilization is achieved the closed loop eigenvalues fall to be in 12.
150 T A B L E 1.1
Eigenvalues for Example 1 al = l , a ~ = 1 , 5 = T ((->0"7),7=0"2 A(A)
A(A + BKoC)
A(A + BF'}
A(A + B K ' C )
- 2 . 3 9 =k j0.O0 +0.00 + j0.00 - 0 . 3 4 + j2.62
- 2 . 3 9 =i: j0.00 +0.00 + j0.00 - 0 . 3 4 =[=j 2 . 6 2
- 5 . 6 3 4- j l . 2 1 - 1.74 :i: j0.S9
- 2 . 7 8 + j2.49 - 3 . 0 1 2= j0.00 - 0 . 9 4 5= jO.O0
T A B L E 1.2 Performance Indices.
IIF- KClI~
~(X)
Initial
61.3301
94.572
Optimal
19.8917
38.759
P
I HAG
*3
• •
EIGENVALUg~ OF' A
0
E|GENVhLUES
•
P.~GENVALtlES OF A ~' B
OF
@ *2
& * HF m
0
*1
.'~
I
°b
_',
.~ : _'
-~/~°
' REAL
+1
0
V
°
-3
Figure 5.1
Eigenvalue locations corresponding to Table 1.1
151 TABLE 1.3 Eigenvalues for Example I (a, = 0, c~z = I, ~b = ~ (< > 0.7), 7 = 0.2) A(A)
BKoC)
A(A +
-2.39 + j0.00 +0.00 + j0.00 -0.34 4- j2.62
-2.39 4- i0.00 +0.00 + i0.00 -0.34 4- j2.62
A(A + BF') -2.39 + j0.01 -2.42 4- j0.32
A(A +
BK:C)
-1.44 -4-j2.54 -3.44 4- ]0.00 - 1 . 1 5 + ]0.00
TABLE 1.4 Performance Indices. IIF - K C I I } , hfitial
61.3301
94.572
Optimal
0.01530
233089
"
"-6
k(X)
x
•
EIGENVALUESOP A
•
EIGENVALUES OF A t BF* E I G E N V A L U E S OF A * BK*C
-'s
£4
~
"q
~2
-3
N
'~
-2
~-1
/I"
"i Figure b.2 Eigenvalue locations corresponding to Table 1.3
~I REAL
152
Example
2
Consider the symmetric vibration model of the standard D r a p e r / R P L satellite shown in Figure 5.3. The dynamic equations, taken from [67] are:
~// ql
(~,):c
q2 where
/i o o lOi) o
A=
o
o1~
0 O 0 0 14.8732 32.8086 0 0 -146.702 -7476.64 0 0 -41.8468 -2699.36 0 0
B =
C=
0 0 0 -0.04168 10.38611 3.725120
0 0 0 0.23623 -25.647 -9.1629
(0~00~) 0
0
1
0
0
where ql(q2)
is the vibration amplitude at z =
_L(~ =L).
153
tip
S
/,
Draper/RPL symmetric vibrational model. Figure 5.3
154
From the design specifications in [67], it follows that the closed loop system must have poles to the left of s = -0.5. For the minimization of J the initial values are chosen to be -0.7 -2 ,',~o =
10 -1
-1
-10
-0.5 -1
-
1/
2 -0.7
2.5
1.6
4
0.5
-0.5
-1
0) 0
After 67 iterations the following 0 th order stabilizing controller is obtained: K* = ( - 5 7 . 5 9 5 2 1 \ -20.50957
-482.41154~ -195.78886]"
Tables 2.1, 2.2 and Figure 5.4 display the performance indices and the corresponding eigenvalue locations. For the purpose of comparison, the problem was also run with the condition number term left out of tile performance index (i.e. cq = 0). In this case the algorithm failed to stabilize the system as shown in Tables 2.2, 2.4 and Figure 5.5. This example illustrates that both terms of the performance index need to be considered in the stabilization procedure.
155 TABLE 2.1 Eigenvalues for Example 2 al = 1 , a 2 = 1 , 7 = 0 . 5
A(A)
A(A +
BKoC)
+0.00 + j53.1 +0.00 + j5.43 +0.00 ± jO.O0
+0.00 ± j53.1 +0.00 + j5.43 +0.00 ± jO.O0
+o.oo ± jo.oo
+o.oo ± jo.oo
A(A +
BF')
A(A +
-3.13 + j41.7 -1.04 + j1.16 -0.68 ± j5.79
BK'C)
-0.86 + j46.0 -0.94 + j5.48 - 1 . 0 2 ± j1.16
TABLE 2.2 Performance Indices.
JJF - gcIl~-
k(x)
1196.5506 60.20818
409.2925 306.1105
hfitial Optimal
*$].1
t.46,0
o
~*4L*?
o EIGENVA/.UE$ Or A O
EIGEHVALURS OP A * B r *
•
~IGRNVALUE~ OF A * BR*~
.S
*I
P0 -S
-4
-3
-2
-I
*t
-1
~F
m.$
o ~"41. ? ~-46.0
) -$ I • L
Figure 5.4 Eigenvalue locations corresponding to Table 2.1
J" m r A r .J
156 TABLE 2.3 Eigenvalues for Example 2 a]=0, a2=l,'/=0.5
~(A)
~(A + BKoC)
~(A + B F ' )
~(A + BK'C)
+0.00 + j53.1
+0.00 + j53.1
-0.63 + jo.05
+178. + jo.oo
÷0.00 4- j5.43 +0.00 4- j0.00 +0.00 + jO.O0
+0.00 4- j5.43 +0.00 + j0.00 +0.00 4- j0.00
-0.66 4- j3.41 -0.59 + j7.86
-2.57 4- j6.19 +1.61 + j0.00 +0.83 4- jl.07
TABLE 2.4 Performance Indices. IIF - K C I I ~
k(X)
11965506 490.5633
409.292.5 383369.1
Initial Optimal
I~IAGt O O BIG[NVALUg~ Or A
~ * "L ll6
0 _~r~].41
O ~IGW.NVA~U[S Or & * RF* •
*S]. I.
E[GENVALUESOF & * BK*C
'"
• 00 I o°L
:l.'
.~
-~ . ~ : ~
*1
0
: -3.41 i -§.4]
Ir 0
I-?.t4t
"$$.L
Figure 5.5 Eigenvalue locations corresponding to Table 2.3
157 APPENDIX
Proof
of
Theorem
3.4
(a)
J = a a - -
o-,.~.(x)
+ a2Trace{(F - Kc)T(F
- KC)}.
(A.1)
Let J l :~--
~,.i.(X)
(A.2)
= Trace{ am.z(X)
~,.i.(X) }
so thai
A J, = T r a c e { - -
(a,.~,~(X)Aa.,..(X)) - a . . . . A ~ m i . ( X ) } .
(A.3)
Note t h a t
A a ~ . . ( X ) = v.ur. A X
(A.4)
Ao...(X)
(A.S)
= vi~T A x
where vi a n d ui are left a n d right singular vectors c o r r e s p o n d i n g to amin a n d va and us c o r r e s p o n d to er,,,~.. Thus,
AJ 1 -
1
-
T
T r a c e { a m i n ( X )v.uo - a , . . . ( X ) v i u T } A x
(A.6)
Now ]~: = Wrace{(F - A'C)r(F
= Trace{FTF
-
- KC)}
(Kc)TF
- -
F T ( K C ) + (KC)T(KC)}
= Trace(FTF) - 2Trace{(KC)TF} + Wrace{(gc)~'(KC)}
(A.7)
158
and AJ~ = 2Trace(FT AF) - 2Trace{(KC)T AF}
(A.8) = 2Trace{iF T - ( K C ) T I A F } Now we have
~L~.(x) Tr~ce{.~.(X),.,y
-
~.~(x),~,y}Ax
(A.9)
+ 2 a 2 T r a c e { ( F T -- ( K C ) T ) A F } .
From F = G X -1 , the gradient of F with respect to G is given directly as A F = A G X -~ + G A ( X -~) = A G X -1 _ G X - 1 A X X
-1
(A.10) = A G X -a _ F A X X
= (AG-
FAX)X
-I
-~.
Substituting (A.10) into (A.9) we have AJ = 2a~Trace{(F ~'- (IfC)T)AGX + Trace{ a~ c q
..~.(x)
(a..,~(X)%uy
_ 2a2X -I(F T - (KC)T)F}AX
-1 } - a m . ~ ( X ) v i u T)
(A.n)
.
Since (A.12)
AAX-AXA=-BAG
we have
L AX =
n ~70Ai-IBAGjP
i=1 j=l
-~.
(A.13)
159
Substituting (A.13) into the second term of (A.11) we have A J : = 2 a 2 T r a c e { X -a ( F T n
- (KC)T)AG}
n
+ T r a c e { Z Z 7iJ'4J-' i=1 j=l Otl .( _"2. . . ~- ~- ' , , .
)(~.(X
,-
T )v,,u.
-
," T o'ma~(X)viui ) - 2a2X-1(F T
-
(Kc)T)F)
(A.14)
AI-aBAG} =
2a2Trace{X-~(F w -
+ Trace{ Z "~,
(Kc)T)AG}
70.,aLi-l.a/~,.i-1 BAG}
i=1 j=l
,d
5 = Trace{2ct2X-'(F T - (KC) T) + B U } A G . F r o m (A.12) and (A.13) it follows that U is the unique solution of
AU
- UA
(A.15)
= X I.
Therefore 0__]_J= 2{a2(F OG
KC)X
-T
+ B T u T}
(A.16)
where U satisfies iU - UA =
a,
2 " O'mln (X)
{o,,,,,(X)v,,u T _ a,,,,,(X)viuT} _ 2a2X_,(FT _ ( K C ) T ) F .
(A.17)
(b) Now we evaluate the gradients of (3,10) with respect, to tile variable elements of i . Recall the equation (A.9)
160 A J-
2 a, " Trace{ermin(X)vauT _ o. ,,**(X)vluTi }A X
O'min(.X)
{A.lS)
+ 2a2Trace{(F T - ( K C ) T ) A F } . From F = G X -1, we compute( G is fixed) A F = - G X - I A X X -1 (Aag)
= -FAXX
-1.
Substituting A F into (A.18) A J = Trace
{_~ ~¢~1 ,., ( ~ . . , . ( x ) ~ . ~ °T • Omin[A
- 2~2x-1(F - (KC)r)F} ~X
~,....(x)~,,,r,)
]
j
(A.20)
= Trace{X1AX } . Since AAX
- AXA
n
= XA2
(A.21)
n
A X = E E "YiJAi-I(-XA'~)~4J-I"
(A.22)
i=l j = l
Substituting (A.22) into (A.20) n
n
AJ = Trace{E E 70XIA'-I (-XA~)AJ-' } i=l j = l n
i=1
(A.23)
n
= -Trace{E E
7 ' J A J - 1 X I A i - ' XAA} .
j=l
5
It, is clear that U is the unique solution of
.4U -
UA = Xy
as in (A.14). Now,
~ j = -Trace{VX~X~}.
(A.24)
161 Therefore,
OJ
- Trace{ U X ~
oai
}
(A.25)
As an e x a m p l e the following calculation is considered. Let.
\ u21
Then
U22
°~
051 = 2Trace
.T21
{(
ujl u21
ua2 u22
)(
~'22
~2
zH ~21
x12 ~22
~1 ~2~
z12 ~22
)(~o o)} 0
(A.26)
(A.27)
= 451(ullxll + u12z21)
o~
052 = 2Trace
{(
u11 u21
~(
ua2 u22]
)(0 o)} 0
255
(a.2s)
= 4a2(u21x12 + u22z22) or
3-~
\ a2(u21z12 + ~22~,22)
(A.29)
(c) Finally the g r a d i e n t of J with respect, to K is easily derived.
A J = - 2 a 2 T r a c e { C F T A K - C(KC):r A K } (A.30) = -2a2Trace{(CF T - C(KC)T)AK} Thus,
OJ = - 2 a 2 [ F - K C ] C :r OK
(A.a~)
CHAPTER 8 SUMMARY AND FUTURE RESEARCH 1. S U M M A R Y This monograph has dealt with some problems related to the robust stability and robust stabilization of systems containing a real parameter vector subject to perturbation. Specifically, the following results have been given: A. For systems where the closed loop characteristic polynomial coefficients are linear or afflne in the parameter, we have i) calculated the largest stability hypersphere (Theorems 2.3.1 and 2.4.1) and the largest stability hyperellipsoid (Theorem 3.2.1) for the case of weighted perturbations, and ii) given constructive conditions for determining if a given perturbation region in parameter space is a stability region (Theorems 2.6.1, 3.3.1 and 3.3.2). B. For the general case where the characteristic polynomial is a nonlinear function of the parameters, we have i) defined a stability margin (Section 3, Chapter 4) ii) given constructive sufficient conditions for determining if a given perturbation class is stabilized (Theorems 4.3.1 aud 4.3.2, and Corollary 4.3.3), and iii) established a robustification procedure to design controllers that enlarge these stability regions (Section 4, Chapter 4). C. For state space systems subject to structural perturbations, a stability hypersphere in parameter space has been determined using Lyapunov theory (Theorem 5.2.1) and
163 a robustification algorithm based on this calculation has been developed (Section 2.3, Chapter5). D. For the problem of stabilization with a low order controller, we have given a new lower bound on the order of a stabilizing controller using Gordan's theorem from llnear programming (Theorem 6.2.4). Transfer function and state space based algorithms for low order stabilization have also been developed, respectively, in Chapters fi and 7. 2. R E S E A R C H
DIRECTIONS
The results described here are initial attempts and have many limitations. There exist many interesting open problems that need to be worked on. We single out some of them below: 1. The extension of Kharitonov's result (Theorem 1.3.1) to the parameter space. 2. The development of necessary and sufficient conditions for robust stability and stabilizability, in the general case, extending the results given here for the linear and a/fine cases.
3. The development of computational methods to check the robust stability conditions given, for instance, in Theorem 2.6.1, in geometric terms. 4. Effective ways of designing robust controllers, directly and nonconservatively, as opposed to the iterative methods given in Chapter 4 and 5. This is required because lhe solutions produced by the iterative methods are strongly dependent on the initial choice of the controller. The solution of this problem will require a much deeper understanding of the geometry of the Hurwitz region. 5. The direct inclusion of response specifications, as in the recent paper [68], into the
164 design of robust controllers. 6. The development of necessary and sufiicient conditions for stabilizability with a fixed order controller, extending tile results given here in Chapters 6 and 7. This could require an appropriate definition of the largest instability hypersphere in the controller parameter space, in a manner dual to the largest stability hypersphere defined, in Chapter 2, in the plant parameter space. Some of these, in particular, items I and 2, are currently under study [69].
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