Stability analysis for linear repetitive processes

Lecture Notes in Control and Information Sciences Editors: M. Thoma and W. Wyner

175

E. Rogers, D.H. Owens

Stability Amalysis for Linear Repetitive Processes

Spfinger-Verlag Berlin Heidelberg NewYork London Paris Tokyo HongKong Barcelona Budapest

Advisory Board L.D. Davisson • A.G.J. MacFarlane" H.Kwakernaak J.L. Massey .Ya Z. Tsypkin •A.J. Viterbi

Authors Eric Rogers Advanced Systems Research Group Dept. o f Aeronautics and Astronautics University o f Southampton Southampton, SO9 5NH United Kingdom David H. Owens Centre for Systems and Control Eng. School of Engineering University o f Exeter Exeter, EX4 4QF United Kingdom

ISBN 3-540-55264-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-55264-2 Springer-Verlag New York Berlin Heidelberg

This Work is subject to copyright. All fights 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 o fthis publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its current version and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by authors Offsetprinting: Mercedes-Druck, Berlin; Bookbinding: B. Helm, Berlin 60/3020 5 4 3 2 1 0 Printed on acid-free paper

PREFACE

Repetitive, or multipass, processes are characterised by a series of sweeps~ or passes, through a set of dynamics which in the simplest case is both linear and known. On each pass an output, or pass p r o f i l e , is produced which acts as a forcing function on, and hence contributes to, the next pass p r o f i l e . This so-called unit memory property is a special case of the more general s i t u a t i o n where i t is the previous M passes which contribute to the current one. The integer M is termed the memory length and such processes are simply termed non-unit memory. Industrial examples include tong-wall coat cutting and c e r t a i n m e t a l r o l l i n g operations. This interaction between successive pass p r o f i l e s is the basic source of the unique control problem f o r these processes. In p a r t i c u l a r , it is possible to generate o s c i l l a t i o n s which increase in amplitude from pass to pass. Such behaviour is c l e a r l y t o t a l l y unacceptable and hence appropriate control action is required. The concept of a multipass process was f i r s t introduced in the early 1970's as a result of work at the University of Sheffield on the modelling and control of long-wall coal cutting operations. This, in turn, led to systematic attempts at controller design for these and several other industrial examples based, e s s e n t i a l l y , on appropriately modifying existing standard linear systems techniques such as Nyquist diagrams. As the number of examples increased, however, i t gradually became clear that t h i s general approach was, at best, valid only under quite r e s t r i c t i v e conditions. ~ence the need for a rigorous control theory, where s t a b i l i t y is an obvious essential item of any such theory. Using previously published work as a basis, t h i s monograph presents a rigorous control theory, and associated t e s t s , for r e p e t i t i v e processes with linear dynamics and a constant pass length. This is based on an abstract representation formulated in functional analysis terms by, in e f f e c t , regarding the pass p r o f i l e as a point in a Banach space. All linear dynamics constant pass length examples are special cases of this abstract representation but t h i s work concentrates on so-called d i f f e r e n t i a l and discrete non-unit memory linear r e p e t i t i v e processes which are of d i r e c t industrial relevance. Three computationally f e a s i b l e sets of s t a b i l i t y t e s t s are developed together with some associated properties. These then lead to some preliminary r e s u l t s on feedback control which are included with the general aim of stimulating further research. A central theme in the work reported here is the use of s t r u c t u r a l links with other classes of l i n e a r dynamic systems. The work reported in t h i s monograph was undertaken during periods when one or both of the authors ~ere on the s t a f f of The University of Sheffield, The queen's University of Belfast and The University of Strathclyde. I t follows on from the original work of John Edwards at Sheffield to whom we owe a great debt of gratitude

VI as the pioneer of t h i s area. A number of former colleagues have also made very useful suggestions, p a r t i c u l a r l y Derek Collins and Ian ~il]son in the early days at Sheffield. Finally, we must thank Miss Yvonne Fleming ~or typing the f i n a l manuscript.

CON']~tTS

CHAPTER CHAPTER

CHAPTER

CHAPTER

I.

1

2.

5

INTRODUCTION PRELIMINARIES 2.1 ~IqL~FEATURES AND CONTROL PROBLEMS 2.2 CLASSICALSTABILITY ANALYSIS - A BRIEF CRITICAL OVERVIEW 2.3 A GENERAL ABSTRACT REPRESENTATION 2.4 STRUCTURALLINKS ~ITH OTHER DYNAMICSYSTEMS 2.5 TRANSFER-FLWCTION MATRIX DESCRIPTION

9 11 19 21

3. 3.1

STABILITYTHEORY ASYMPTOTICSTABILITY

32 32

3.2

BBUNBED-INPL~/BOUNOED-OUTPUTSTABILITY

41

3.3

STABILITY ALONG THE PASS

42

3.4

A 2D SYSTEMS APPROACH

58

4. 4.1

GRAPHICALAND ALGEBRAICSTABILITY TESTS ASYMPTOTICSTABILITY

64

4.2

65

4.4 4.5

STABILITY ALONg TIlE PASS - THE DIFFERENTIAL CASE STABILITY ALONG THE PASS - THE DISCRETE CASE APPLICATIONOF 2D SYSTEMS STABILITY TESTS APPLICATIONOF DELAY DIFFERENTIAL STABILITY TESTS

5. 5.1

SIMULATION-BASEDSTABILITY TESTS MATHEMATICALBACKGROUND

116

5.2

STABILITY TESTS

121

5.3

PERFOPJIANCEBOUNDS

140

5.4

INTERPASS SMOOTII[Ng

152

4.3

CHAPTER

CHAPTER

CHAPTER

5

64 76 84 I01

116

6.

CONTROLLERDESIGN - SOME INITIAL RESULTS

156

6.1 6.2

CONTROLPOLICIES ANO FEEDBACKCONTROL SCBEMES STABILITY ALONG THE PASS

156 177

6.3

TBE LIMIT PROFILE DESIGN PROBLEM

181

6.4

TIlE REPETITIVE SYSTEMS DISTURBANCE DECOOPLING WITH STABILITY PROBLEM

185

CONCLUSIONSAND FURTHER WORK

188

REFERENCES

194

7.

CHAPTER 1 INTRODUCTION The essential unique feature of a r e p e t i t i v e , or multipass, process can be illustrated by considering machining operations where the workpiece is processed by a series of sweeps, or passes, of a processing device. On each pass an output, or pass p r o f i l e , is produced and in a r e p e t i t i v e process t h i s acts as a forcing function on, and hence contributes to, the next p r o f i l e . In the simplest case, therefore, the output at any point on a p a r t i c u l a r pass is a function of the independent inputs/disturbances at that point and the pass p r o f i l e at the same point on the previous pass - the so-called unit memory property. Industrial examples include long-wall coal cutting and certain metal r o i l i n g operations. Repetitive processes also exist where, in e f f e c t , the current pass p r o f i l e is a /unction of the independent inputs/disturbances to that pass and a f i n i t e number, M, of previous pass p r o f i l e s . The integer M is termed the memory length and examples in this case are simply termed 'non-unit memory'. Such examples reduce to the case described above i f M is unity and hence in t h i s sense they can be regarded as the natural generalisations of t h e i r unit memory counterparts. One major example in this case is bench mining systems. In addition to that arising from the independent inputs and the memory property, some examples exhibit dynamic behaviour, termed interpass smoothing, between the production of successive pass p r o f i l e s . One such case is long-wall coal cutting where, as a r e s u l t of the basic process geometry, the machine's weight (up to 5 tonnes) causes considerable d i s t o r t i o n to the previous pass p r o f i l e as i t passes over. Hence, i f a physically r e a l i s t i c analysis is to be undertaken, a means of e x p l i c i t l y including t h i s feature is clearly required. The basic unique control problem f o r a r e p e t i t i v e process is the possible presence in the output sequence of o s c i l l a t i o n s which increase in amplitude from pass to pass. This behaviour is easily generated in simulation studies and in experimental studies on scaled models of industrial examples such as the long-wall coal cutter. Further, acceptable control of a given example c l e a r l y requires a suitable s t a b i l i t y and control theory. This monograph describes the development of a rigorous s t a b i l i t y theory, and associated s t a b i l i t y t e s t s , with p a r t i c u l a r emphasis on so-called d i f f e r e n t i a l and discrete non-unit memory linear r e p e t i t i v e processes which are of direct industrial relevance. Some highly promising results on using the developed theory and t e s t s for feedback control of these processes is also included with the general aim of stimulating further research. At the most general level, a r e p e t i t i v e process has nonlinear dynamics and a pass length which, by definition, is f i n i t e but varies from pass to pass. Hence the analysis of such a case would, at best, be a very formidable task. The special case of linear dynamics and a constant pass length is, however, t r a c t a b l e from an

analysis standpoint and includes the vast majority of currently known examples of p r a c t i c a l i n t e r e s t . Consequently t h i s monograph will follow a l l published work to date and r e s t r i c t attention to t h i s special case, with the observation that progres here may also suggest approaches to other cases such as nonlinear dynamics and a constant pass length. Early work considered only single-input/single-output systems for which an obvious i n t u i t i v e approach to s t a b i l i t y analysis and c o n t r o l l e r design is to attempt to make use of existing techniques. This b a s i c a l l y used a single v a r i a b l e , termed the t o t a l distance traversed, to convert the system into an i n f i n i t e length single pass process and requires following assumptions. (i) The pass length is 'long' and hence the e f f e c t s of the i n i t i a l conditions on each pass can be ignored. (if) The e f f e c t s of the previous pass dynamics can be represented by a 'long delay' term. tither work, however, has shown that the ' r e s e t t i n g ' action of the i n i t i a l conditions on each pass can act as a form of s t a b i l i s i n g action and hence prevent the growth of disturbances. Hence t h i s approach is, at best, only valid ' f a r enough' away from e i t h e r end of the pass. Further, no attempt has been made to use t h i s approach as a basis for developing a general control policy. Instead, only the problems a r i s i n g in a few well known industrial examples have been considered. As an a l t e r n a t i v e to the, e s s e n t i a l l y c l a s s i c a l l y based, approach described b r i e f l y above, suppose that rigorous s t a b i l i t y and control theories are developed from a general base, or abstract representation, with the following core features. (i) The e f f e c t s of the i n i t i a l conditions on each pass are e x p l i c i t l y retained. (if) Includes the previously studied examples as special cases but allows for others with a more complex, possibly multivariable, structure. Then, in principle, the limitations of the c l a s s i c a l l y based approach will have been removed. To provide a suitable basis, i t is obvious that any candidate abstract representation must e x p l i c i t l y include a l l of the c h a r a c t e r i s t i c s which define these processes. This p a r t i c u l a r problem has been considered in other work which has led to the development of a suitable representation f o r the most general nonlinear dynamics variable pass length case. Basically, t h i s regards the output on any pass as a point in a suitably chosen function space. Further, the sub-class of processes with linear dynamics and a constant pass length is a special case. Using t h i s a b s t r a c t representation as a basis, a rigorous s t a b i l i t y theory for the linear dynamics constant pass length sub-class has been developed. This consists of two d i s t i n c t concepts, termed asymptotic s t a b i l i t y and s t a b i l i t y along the pass respectively. Further, the former is a necessary condition for the l a t t e r which is known to be required in a l l p r a c t i c a l applications. The necessary and

sufficient conditions for these properties are expressed in terms of the spectral radius and resolvent of the linear operator associated with the a b s t r a c t representation. In terms of applications, a number of publications have reported the r e s u l t s of interpreting t h i s a b s t r a c t theory for certain p a r t i c u l a r cases. For example, the results for d i f f e r e n t i a l and discrete non-unit memory linear r e p e t i t i v e processes have been documented. These, however, are not computationally f e a s i b l e , where this is a common feature of the r e s u l t s to date for a number of other cases. Given the pivotal role of s t a b i l i t y , the development of computationally f e a s i b l e s t a b i l i t y tests is an obvious s t a r t i n g point for any further control r e l a t e d analysis of a given case. Consequently the development of computationally f e a s i b l e s t a b i l i t y tests for d i f f e r e n t i a l and discrete non-unit memory l i n e a r r e p e t i t i v e processes forms a substantial part of the work reported in t h i s monograph. One approach to the analysis of r e p e t i t i v e systems is to exploit, where possible, s t r u c t u r a l links which may exist with other well researched classes of dynamic systems. In t h i s work, previously documented links between d i f f e r e n t i a l and discrete non-unit memory linear r e p e t i t i v e processes and the following two classes of linear dynamic systems will form a central underlying theme. (i) Standard linear systems described by the well known state-space model or t r a n s f e r - f u n c t i o n matrix. ( i i ) 2D linear (image or signal processing) systems described by the so-called Roesser state-space model. Further, considerable use will be made of r e s u l t s from the s t a b i l i t y theory of certain classes of delay d i f f e r e n t i a l systems. One subsidiary benefit of t h i s general approach will be to strengthen the already known links between these areas. The material presented in t h i s monograph is organised into six chapters where the f i r s t of these introduces the basic features and unique control problems by reference to an industrial example. Some background r e s u l t s central to the analysis which follows are also developed, including a transfer- function matrix description for both the d i f f e r e n t i a l and discrete cases. Chapter 3 then d e t a i l s the s t a b i l i t y theory based on the abstract representation and applies i t to the d i f f e r e n t i a l and discrete processes to produce conditions suitable f o r the development of computationally f e a s i b l e s t a b i l i t y t e s t s . F i n a l l y , t h i s chapter considers the use of results from the s t a b i l i t y analysis of systems described by the Roesser model in the same context. Given t h i s basis, chapter 4 develops computationally f e a s i b l e s t a b i l i t y t e s t s for both d i f f e r e n t i a l and discrete processes which, in e f f e c t , use only standard linear systems t e s t s . The end product is two systematic t e s t procedures in each case, which are also compared from an applications standpoint with p a r t i c u l a r emphasis on Computer bided Design aspects. This chapter also considers the use of results from the s t a b i l i t y theory of systems described by the Roesser model and delay d i f f e r e n t i a l systems from the same standpoint.

Chapter 5 continues with the s t a b i l i t y theme by developing simulation-based t e s t s based on suitably well behaved plant step response data which is assumed to be available, or can be obtained by simulation studies. Further, i t is shown that these t e s t s produce, at no extra cost, computable information concerning the following features which are of significant importance in terms of the control of these processes. (i) The rate of approach of the output sequence to the so-called steady, or limit, p r o f i l e which is a consequence of s t a b i l i t y . (ii) Bounds on the performance along any pass. This information is unique to these t e s t s for which some i n i t i a l results on extending them to processes with interpass smoothing effects are also included. These results are the f i r s t reported output on the analysis of such cases. Following on from the previous three, chapter 6 presents the results of some i n i t i a l work on controller design. In p a r t i c u l a r , three control policies are formulated from practical considerations and feedback control schemes which use either state or output information are developed. Further, some candidate design algorithms are presented together with some relevant systems theoretic properties. Finally, chapter 7 summarises progress to date and b r i e f l y outlines some possible future research topics.

CIIAPTER 2

PRELIMINARIES This chapter introduces the unique features and control problems of r e p e t i t i v e processes by reference to an industrial example. Some background r e s u l t s central to the analysis of subsequent chapters are also developed. 2.1

Unique Features and Control Problems The essential unique feature of a r e p e t i t i v e , or multipass, process is the presence of a recursive action with interaction between successive outputs or pass p r o f i l e s . To formalise t h i s , f i r s t suppose, for simplicity, that the necessarily f i n i t e pass length a is constant and denote the pass p r o f i l e generated over a on pass k ~ 0 by Yk(t), 0 ~ t ~ a. Then a r e p e t i t i v e process is one where Yk(t) acts as a forcing function on, and hence contributes to, the next pass p r o f i l e Yk+l(t), 0 < t < a, k > O. Industrial examples include long-wall coal cutting and a b r i e f study of t h i s case is now given to introduce the basic unique control problem for these processes. In Great Britain, the most s a t i s f a c t o r y , and commonly used, method of mining coal is by a process known as advanced long-wall coal cutting. Figures 2.1a and 2.1b i l l u s t r a t e the basic operation of the long-wall system of working in which the coal cutting machine is hauled along the entire length of the face riding on the semi-flexible structure of the armoured face conveyor, or A.F.C., which transports away the coal cut by the rotating drum. These machines generally cut in one direction only, l e f t to right in Figures 2.1a and 2.1b, and are hanled hack in reverse at high speed for the s t a r t of the new sweep, or pass, of the coal face. Between passes, the conveyor is snaked forward hydraulically, as i l l u s t r a t e d in Figure 2.2, so that i t now r e s t s on the f l o o r p r o f i l e produced during the previous pass. During the cutting operation, the machines drum may be raised or lowered with respect to the A.F.C. by hydraulically t i l t i n g the body about a datum line on the drum or face side. The objective of t h i s is the v e r t i c a l steering of the entire long-wall i n s t a l l a t i o n (machine, conveyor and roof support units) to maintain it within the undulating confines of. the coal seam. h nucleonic coal sensor, situated some distance behind the drum, provides the primary control signal by measuring either the f l o o r or ceiling thickness l e f t by the machine. In order to obtain a simplified mathematical description of t h i s process, consider the idealised side elevation and plan shown in Figures 2.3a and 2.3b respectively. ~ere the constants F,R and W represent the f e e t spacing, drum o f f s e t , and the width of the machine (and drum) respectively, the variable Jk+l(t) represents the controlled drum deflection, and ek+l(t ) denotes the height of the A.F.C. on which the machine rides.

Suppose also that a l l angular deflections are

NUCLEONIC COAL SENSOR ~

MACHINE BODY ~

~

STONE× ALONG FACE DIRECTION

CUTTING

JACK

DRUM

FIGURE 2. l(a)

NEW COAL FACE

.,'c~::::31 DRUM I ~

OLD COAL FACE

FIGURE 2.1(b)

NEW COAL FACE ~ , / / / / / , / / / / / , / / / / J / ~

I

~

E

D

PUSHINGRAMS FIGURE 2.2

A.F.C. PANS

t

CUT ROOF STONE/COALINTERFACE STONE ~ ~ - ' / ~

/ /

F

DRUM ~

~

"~Bk+ I (t)

~

,{::::::= / ////III///III//~

CUT FLOOR .

.

.

.

.

.

k+l( .

.

.

.

t-X

I Z k+l (t) I t

I ek+ 1(t=R) ,t ..................... t+R

e k+I(I+R+F) I ALo.o FAC. ~.REC..ON t+R+F

(a)

NEW FACE!

I"

R ....

~ / / . / / / / / / / / . ~ .....

[ ~ l

I i

I

}'

t

1

CUTTINGDRUM I~ SKIDA

"CONVEYOR' '!

I =-

X

.

.

/ DIRECTION

W

MACHINEBODY

C .....

| FACE ADVANCE

B

OLD FACE

~'-JI

'

D . (b)

FIGURE 2.3

. ALONG FACE DIRECTION

]1

small. Then elementary geometrical considerations immediately y i e l d the following d e s c r i p t i o n of the coal c u t t e r dynamics Yk+l(t) + Zk+l(t ) = ek+l(t + R) + WTk+l(t + R) + R~k+l(t + R) + J k + l ( t )

o
(2.1)

where 7,~ denote the t r a n s v e r s e and longitudinal t i l t s of the machine r e s p e c t i v e l y and Zk+l(t ) denotes the height of the c o a l / s t o n e i n t e r f a c e above a fixed datum plane. The t r a n s v e r s e and longitudinal t i l t s of the machine are also those of the supporting conveyor s t r u c t u r e and are given by 7k+l(t) = (ek+l(t) - Ck(t))/W

(2.2)

Zk+l(t) = (ek+l(t) - ek+l(t + F))/F

(2.3)

and r e s p e c t i v e l y . F i n a l l y , suppose t h a t the A.F.C. moulds i t s e l f e x a c t l y onto the cut f l o o r upon which i t r e s t s - the s o - c a l l e d trubber conveyor ~ assumption. Then ek+l(t ) = k2(Yk(t ) + Zk(t))

(2.4)

where k 2 is a p o s i t i v e r e a l constant, and (2.1)-(2.4) form a complete d e s c r i p t i o n of the open-loop system. Conventionally, t h i s system is c o n t r o l l e d by manipulation of the v a r i a b l e J k + l ( t ) from a delayed measurement of the f l o o r coal thickness Yk+l(t - X), where X is the t r a n s p o r t delay, or lag, equal to the distance by which the coal sensor lags behind the c u t t i n g drum. More commonly, however, the roof coal thickness is used since i t can be r e l a t e d to Yk+l(t - X) on the assumption t h a t the seam thickness is constant. Suppose also t h a t the sensor and a c t u a t o r dynamics can be neglected and a s o - c a l l e d fixed drum shearer is under consideration, i . e . R = O. Then the control law in t h i s case takes the form J k + l ( t ) = kl(Rk+l(t ) - Yk+l(t - X)) - W7k+l(t )

(2.5)

where k 1 is a p o s i t i v e r e a l constant and Rk+l(t ) is a new external reference v a r i a b l e taken to represent the desired coal thickness on pass k + 1, k ~ O. Suppose now, f o r s i m p l i c i t y , that the v a r i a b l e Zk(t ) is set equal to zero. Then combining the above equations y i e l d s the closed-loop d e s c r i p t i o n

Yk+l(t) = -klYk+l(t - X) + k2Yk(t ) + klRk+l(t ) X > O, 0 < t < a, with assumed i n i t i a l conditions

k > 0

Yk+l(t) = O, -X < t < O,

k > 0

(2.6)

(2.7)

Figure 2.4 shows the response of this closed-loop system in the special case when k 1 = 0.8, k 2 = 1, X = 1.25, a = 10, to a downward unit step in Rk+l(t ) on each pass, i.e. Rk+l(t ) = -1, 0 < t < 10, k ~ O.

Note that the o s c i l l a t i o n s grow, or increase

in amplitude, severely from pass to pass. Hence the deterioration in system performance a f t e r the f i r s t pass must be due to the fact that the cut f l o o r p r o f i l e , or dynamics, on any pass acts as a disturbance on, and hence contributes to, the dynamics of the next pass. This interaction between successive pass dynamics is the essential unique c h a r a c t e r i s t i c of a l l r e p e t i t i v e processes and in cases such as that of Figure 2.4 strong control action is clearly required. Acceptable control of a r e p e t i t i v e process in a given case clearly requires a suitable s t a b i l i t y and control (feedback or otherwise) theory. This monograph describes the development of a rigorous s t a b i l i t y theory, and associated s t a b i l i t y t e s t s , for a special case described by a set of d i f f e r e n t i a l or discrete linear equations. These equations can be used to describe a number of industrial examples and in the penultimate chapter some i n i t i a l results on using the developed theory and t e s t s for feedback control of such examples will be presented. 2.2

~]assical S t a b i l i t y Analysis - A Brief C r i t i c a l Overview

I f the example under consideration is single-input/single-output (SISO), an obvious intuitive approach to s t a b i l i t y analysis and controller design is to attempt to make use of existing techniques in the form, for example, of the inverse Nyquist diagram. The essence of such an approach is to use the single variable Y = ka + t to convert the system into an i n f i n i t e length single pass process in which the relationships between variables are expressed only in terms of V. In particular, a variable, say, Yk+l(t), k > O,is identified as a function of Y(V) defined for 0 < V < + ~, where Y is termed the t o t a l distance traversed. Applying this approach to (2.6)-(2.7) yields Y(V) : - klY(V-X) + k2Y(V-a) + klR(V)

(2.8)

and this r e p e t i t i v e process is said to be stable if, and only i f , the system of (2.8) is stable in the standard sense. The r e p e t i t i v e process is now amenable to analysis by any of the well known classical techniques. Hence, for example, taking the Laplace transform with respect to Y and making use of the inverse Nyquist diagram leads to the result that the closed-loop system is stable in the standard sense i f , and only i f , k 1 < 1 - k2

(2.9)

The above analysis can, at best, only produce useful i n i t i a l guidelines since i t completely ignores the considerable d i s t o r t i o n caused to the previous pass p r o f i l e by the weight (up to 5 tonnes) of the machine as i t passes over. This problem is a common feature of a number of known examples of r e p e t i t i v e processes in that dynamic interaction, termed interpass smoothing, between passes causes

10

/

k

4

r

4

5

-1,0 FIGURE 2.4

6

7

8

11 distortion of the previous pass p r o f i l e . I t is clear, therefore, t h a t i f a physically r e a l i s t i c analysis of such examples is to be undertaken then a mathematical means of including t h i s interpass smoothing is required. In order to apply c l a s s i c a l s t a b i l i t y analysis and t e s t s to r e p e t i t i v e processes, i t is necessary to make the following assumptions. (i) The pass length a is 'long' and hence the e f f e c t s of the i n i t i a l conditions on each pass can be ignored. ( i i ) The e f f e c t s of the previous pass dynamics can be represented by a 'long delay t term. I n t u i t i v e l y , however, the ' r e s e t t i n g ' action of the i n i t i a l conditions on each pass could act as a form of s t a b i l i s i n g action and hence prevent the growth of disturbances. In p a r t i c u l a r , i t is easily shown, using a d i s c r e t i s e d form of (2.6)-(2.7) with ¥k+l(t), - X < t < O, appropriately chosen, that the i n i t i a l conditions on each pass can have a crucial e f f e c t on the performance of the simplified long-wall coat cutter dynamics. Suggesting that f o r systems with a lag, X, on the current pass the analysis based on the concept of the t o t a l distance traversed is valid only in the range ka + X << V << (k + 1)a, k > 0 (2.10) and for delay-free systems only in the range ks << V << (k + 1)a , k >0 (2.tl) Note also that no attempt has been made to use t h i s approach in formulating a general control policy. Instead, attention has been r e s t r i c t e d to the problems occuring in a few well documented industrial examples. Summarising, therefore, the c l a s s i c a l l y based approach to s t a b i l i t y analysis and c o n t r o l l e r design discussed b r i e f l y in t h i s section is limited by the following major factors. (i) I t completely neglects the e f f e c t s of the i n i t i a l conditions on each pass which are known to have a crucial e f f e c t on system s t a b i l i t y and performance in certain cases of p r a c t i c a l (and t h e o r e t i c a l ) i n t e r e s t . (ii) No attempt has been made to develop rigorous s t a b i l i t y and control theories for the wide range of known r e p e t i t i v e processes. Further, i t is by no means clear that such a development is possible even f o r suitably well defined sub-classes. 2.3

A General Abstract Representation As an a l t e r n a t i v e to the approach reviewed b r i e f l y in the previous section, suppose that rigorous s t a b i l i t y and control theories are developed from a general abstract representation with the following e s s e n t i a l features. (i) Explicit retention of the e f f e c t s of the i n i t i a l conditions on each pass. (ii) Treats the examples studied using the approach of section 2.2 as special cases and includes provision for others with a more complex, possibly multivariable, structure.

12 Then, in p r i n c i p l e , the limitations of the c l a s s i c a l l y based approach will have been removed. To provide a suitable basis, i t is obvious that any abstract representation must e x p l i c i t l y include the essential unique features. In the most general case of a variable pass length, these can be summarised as follows and are also i l l u s t r a t e d in Figure 2.5. (a) A number of passes through a known set of dynamics. (b) Each pass is characterised by a pass length , ak, which may vary from pass to pass and a pass p r o f i l e Yk(t) defined on 0 < t ~ ak. (c)

Note that the pass

p r o f i l e need not be a s c a l a r quantity. An i n i t i a l pass p r o f i l e Yo(t) defined on 0 ~ t ~ no, where a o is the i n i t i a l pass length.

The function Yo plays the role of an i n i t i a l condition f o r the

process. Each pass u i l l be subject to i t s own boundary conditions, disturbances and control inputs. (e) The process is unit memory, i . e . the dynamics on pass k + 1 depend only on the independent inputs to t h a t pass and the pass p r o f i l e on the previous pass k. 6iven ( a ) - ( e ) , suppose t h a t Yk(t ) is regarded as point in a suitably chosen function (d)

space.

In p a r t i c u l a r , suppose that Yk e Eak ,

k > _ 0

(2.12)

where Eak denotes a Banach space.

Then a general abstract model of these processes

can be formulated as a recursion r e l a t i o n of the form Yk+l = fk+l (Yk) '

k > 0

(2.13)

(where fk+l is an abstract mapping of Eak into Eak+l ) together with a rule for updating the pass length ak of the form ak+ 1 = gk+l(ak,¥k,Yk+l),

k >0

(2.14)

Repetitive processes also exist, f o r example so-called bench mining systems, uhere the current pass p r o f i l e is a function of the independent inputs to that pass and a f i n i t e number, M > 1, of previous pass p r o f i l e s . The integer ~ is termed the memory length and such processes are designated as 'non-unit memory of length M' or, more simply, 'non-unit memory'. Such processes are easily accommodated within the general structure of (2.13)-(2.14). Formally, a l l that is required is to replace these equations by Yk+l = f k + l ( Y k ' Y k - l ' ' " ' Y k + l - M) ' k > 0

(2.15)

and ak+ 1 =

gk+l(ak,ak_l,...,ak+l_M,¥k+l,...,¥k+l.~),

k > 0

(2.16)

13

I Y1 ! I I 1

I I I I

GO

o, 1

FIGURE

2.5

¥ ~2 I G2

14

respectively. This formulation is unnecessary, however, i f the ordered set (Yk,Yk_I,...,Yk+I_M) is regarded as a 'pass p r o f i l e ' in the product space Eak x Eak_l x . . . x Eak+l_~ , i . e . (Yk,Yk_l . . . . ,Yk+I_M) E Eak

Eak_l

.

Eak+l.N

In which case (2.15) and (2.16) become

(Yk+I,Yk,...,Yk+2_M) = (fk+l(Vk,...,Yk+l_M),Yk,...,Yk+2_M)

(2.1S)

ak+l : gk+l(ak'" "" 'ak+l- M'¥k+l ' ' ' " '¥k+1- ~)

(2.19)

and respectively which have an identical structure to (2.15) and (2.16). M points Yo,Y_I,...,YI_M are required to define the i n i t i a l p r o f i l e .

Now, however,

Any analysis of the abstract model defined above would d e a r l y be a formidable task. h d i f f i c u l t y which can be avoided by noting that the vast majority of processes of p r a c t i c a l interest are linear and of constant pass length. Hence from t h i s point onwards attention will be r e s t r i c t e d to linear processes with a k = a,

k >0

(2.20)

The following general definition characterises the unit memory version in t h i s case. Definition 2.3.1: h linear r e p e t i t i v e process of constant pass length

S(Ea,Wa,La)

a > 0 consists of a Banach space Ea, a linear subspace Wa of Ea, and a bounded linear operator La of Ea into i t s e l f .

The system dynamics are described by linear

recursion r e l a t i o n s of the form Yk+l = LaYk + bk+l '

k > 0

(2.21)

where Yk e Ea is the pass p r o f i l e on pass k and bk+ 1 e Wa, k > O.

Here the term

LaYk represents the contribution from pass k to pass k + 1 and bk+ 1 represents i n i t i a l conditions, disturbances and control input e f f e c t s . • In the non-unit memory case, l e t L~, 1 < j < i , be bounded linear operators mapping Ea into i t s e l f .

Then the most general representation of a constant pass

length non-unit memory linear r e p e t i t i v e process with memory length M takes the form

Yk+l = L1dk ÷ L Yk- 1 ÷ . . . + L % + , , a- , + bk÷l whereYk e Ea, k > 1 - M, bk+ 1 c Wa C Ea.

(2.22)

Note also that (2.22) reduces to (2.21)

with La s L1a i f M = t and hence i t can be regarded as the natural non-unit memory generalisation. Further, using (2.18), i t can be regarded as a process of the form (2.21) in the product space E~ = Ea× Ea x . . . x Ea (M times) by writing i t in the 'companion form'

15 .

I

Yk+2- ltI

-0

I

0

i'o

"Yk+l- M1

m

i

!

Yk+l --

0

0

LM a LM-I a

L2 a

+ La1

Yk J

I0

, k>O

bk+l

I

(2.23) and using the notation

,.:°o

I

M-1 La

o 2

La

(2.24) L

Hence r e s u l t s derived f o r the unit memory case can immediately be applied to the non-unit memory g e n e r a l i s a t i o n . To i l l u s t r a t e the g e n e r a l i t y of (2.23)-(2.24) (and (2.21)), the following examples are now considered. Example 2.3.1 - A delay - algebraic system - The s c a l a r equation Yk+l(t) = - koYk+l(t - X) + klYk(t ) + koRk+l(t ) O
k~O X< t ~ 0

(2.25)

where k o and k 1 are constants, has been shown to represent physical examples of r e p e t i t i v e processes such as long-wall coal c n t t i n g and metal r o l l i n g .

This

equation has the s t r u c t u r e of a unit memory l i n e a r r e p e t i t i v e process of pass length a, with Ea = Wa the vector space of continuous functions on [O,a] s a t i s f y i n g the i n i t i a l condition ¥(0) = 0 and norm

IIYII = max IY(t)l

O
(2.20)

0 < t < a

- X~ t ~ 0

(2.27)

Example 2.3.2 - Matrix recursion r e l a t i o n s - The d i s c r e t e s t a t e vector model Xk+1 = hXk + BUk, Xk e Ra, Uk E R~,

k > 0

can be regarded as a unit memory l i n e a r r e p e t i t i v e process with Ea = Rn' ~a = range of H and bk+ 1 = BUk, k ~ O. Example 2.3.3 - h d i f f e r e n t i a l non-unit memory l i n e a r r e p e t i t i v e process - The state-space model in t h i s case has the form

(2.28)

16 M

ik+l(t) = AXk+1(t) + BUk+1(t) + j~l BJ-IYk+1-j(t) M

Yk+l(t) = CXk+l(t) + D°Uk+1(t) + j~=i DjYk+1-j(t) Xk+l(t ) e Rn, Yk+1(t) e Rm, Uk+1(t) e R i 0 < t < a , Xk+l(O ) = dk+ I, k > 0

(2.29)

To write (2.29) in the form S(Ea,Wa,La) , as defined by (2.23)-(2.24), first note that Yk+l(t) = CI~eA(t-') (j~1 M Bj. iYk+l- j (r) + BUk+ 1(r)}dr M

+ C ehtdk+l + DoU,.+l(t)~ -- + E D-Yk+l_:(t) , J J- j=l

0

( t _( a,

k > 0

(2.30) Further, consider the problem in the context of the Banach space Ea = Cm(O,a) of bounded continuous mappings of the i n t e r v a l 0 < t < a into the vector space of real m-vectors Rm with norm }[Y{[ = sup {[Y(t){]m O
(2.31) {}P{Im = max ]Pi]" l
Then L],

1 < j < M, is defined by the r e l a t i o n • (L~Y)(t) = C~:eA(t - r)Bj_ 1Y(r)dv + DjY(t), 0 < t < a

(2.32)

and bk+ 1 by bk+ 1 = CI:eA(t-r)BUk+I ( r ) d r + DoUk+l(t ) + C ehtdk+l , 0 < t ~ a F i n a l l y , i f the system i n i t i a l

(2.33)

conditions dk+l, k > O, of i n t e r e s t l i e in a

subspace, W, of Rn and the control inputs Uk+l(t), k ~ O, are assumed to be piecewise continuous, Wa c Ea can be obtained by evaluating (2.33) f o r a l l such dk+ 1 and Uk+I. Example 2.3.4 - A d i f f e r e n t i a l unit memory l i n e a r r e p e t i t i v e process - Set M = 1 in the a n a l y s i s of example 2.3.3 to obtain t h i s special case. Example 2.3.5 - A d i f f e r e n t i a l unit memory l i n e a r r e p e t i t i v e process with interpass smoothing - Consider, f o r s i m p l i c i t y , the unit memory case and hence M = 1 in (2.29) of example 2.3.3. Then one p o s s i b l e method of modelling the e f f e c t s of interpass smoothing on the process dynamics is to assume t h a t the pass p r o f i l e at any point t on pass k + 1 is a function of the s t a t e and inputs at t h i s point on pass k + 1 and of the complete pass p r o f i l e on pass k. For example, a candidate r e p r e s e n t a t i o n is

17

ik+l(t) = AXk+l(t) + U +l(t) + Do I K(t,T)Yk(r)d Yk+l(t) = CXk+l(t) O ~ t ~ a,

Xk+l(O ) = dk+ 1 ,

where the i n t e r p a s s

k > 0

i n t e r a c t i o n term Bo

(2.34)

~K(t,r)Yk(r)dr

r e p r e s e n t s a 'smoothing

out' o~ the previous pass p r o f i l e in a manner governed by the p r o p e r t i e s of the kernel K ( t , r ) . Note t h a t the p a r t i c u l a r choice of K(t,r) = ~(t-

r)I m

(2.35)

where 6 denotes the Dirac d e l t a function reduces (2.34) to the case of example 2.3.4. I t is now e a s i l y v e r i f i e d t h a t (2.34) is a l i n e a r r e p e t i t i v e process in Ea = Cm(O,a ) with (LaV)(t) = CI r e A(t- r) Bo ~ : K ( r , t ' ) Y ( t ' ) d r ' d r , 0 and bk+ 1 = C f : e h ( t - t ' ) B U k + l ( t ' ) d t ' + C ehtdk+t ,

0 < t < a

(2.36)

0 ~ t < a

(2.37)

This approach can also be used to study the e f f e c t s of interpass smoothing on the dynamic behaviour of processes described by the equation of example 2.3.1. The next example provides a link between the unit memory version of example 2.3.3 and standard l i n e a r systems with a delay in the s t a t e . Example 2.3.6 - h d i f f e r e n t i a l unit memory l i n e a r r e p e t i t i v e process with i n t e r a c t i o n between pass p r o f i l e s and pass boundary conditions - Set M = 1, n = m, Do = O, D1 = O, O = I n in (2.29) and consider the case when ik+l(O ) = dk+l, k > O, is replaced by pass dependent i n i t i a l q KjXk(tJ) + ~;K(t)Xk (t)d t Xk+l(O) = dk+l + K°Xk(O) + j~l

conditions of the form (2.38)

where Ko,K1,...,K q are constant n × n matrices, K(t) is a piecewise continuous n x n matrix function of t on 0 < t < a and 0 ~ t I < t 2 < . . . ~ tq < a are q sample points.

Then t h i s process is a Linear r e p e t i t i v e process in Ea = Cn(O,a ) since i t

is e a s i l y v e r i f i e d t h a t the unit memoryversion of the construction given in example 2.3.3 s t i l l holds in t h i s case with La defined by (LaY)(t) = [ t e A ( t - t ' ) B o Y ( t ' ) d t ' ~0

+

eats,

0 < t < a

(2.39)

where Y = K°Y(O) ÷ j=l KjY(tj) ÷

C

K(t)Y(t)dt

(2.40)

18

A class of delay d i f f e r e n t i a l systems in Rn can be modelled by the state-space equations

X(t) = AX(t) + B 0 X(t-a) + BU(t) , t ~ 0 X(t-a):=

Xo(t ) ,

0
(2.41)

where A, Bo, B are constant n × n, n x n and n × / matrices r e s p e c t i v e l y .

I f the

delay a is interpreted as a pass length then i t is obvious t h a t these systems have c e r t a i n s t r u c t u r a l s i m i l a r i t i e s to l i n e a r r e p e t i t i v e processes described by a set of recursive d i f f e r e n t i a l equations. In p a r t i c u l a r , introduce the change of variables Uk+l(t ) = U(ka + t)

(2.42)

Xk(t ) = X((k-1)~ + t ) ,

0 < t < a,

and define the pass p r o f i l e s as Yk = Xk' k ~ O.

k > 0

(2.43)

Then (2.41) can be written as a

r e p e t i t i v e process of the form defined by example 2.3.6 with boundary conditions Xk+l(O ) = Xk(a), k > 0

(2.44)

i . e . a special case of (2.38) with q = 1, K1 = In, t 1 = a, dk+ 1 = O, Ko = 0 and

K(t) ~ 0. The next two examples are the natural d i s c r e t e analogues of the processes defined in examples 2.3.3 and 2.3.4. Example 2.3.7 - A d i s c r e t e non-unit memory l i n e a r r e p e t i t i v e process - This is the natural d i s c r e t e analogue of the process of example 2.3.3 and has s t a t e - s p a c e model i

Xk+l(P + 1) = ~Xk+l(P ) + AUk+I(P ) + E Aj_IYk+I_j(P ) j=l

Yk+1(V) = CXk+I(P) + D° Uk+I(P)

+

j~1DjYk+I-J(P)

Xk+l(P) e Rn, Yk+I(P) e Rm, gk+l(P) e Re 0 < P < a, Xk+l(O ) = dk+l,

k > 0

(2.45)

Further, define the pass profile on pass k to be the ordered set Yk = { Y k ( O ) ' Y k ( 1 ) ' " " Y k ( a ) }

(2.46)

and regard it as a point in the product space E a = Rm × R m x...x Rm with norm IIYktl :

max IIYk(P)llm O~PSa where, as in example 2.3.3, is any convenient norm in Rm.

II'llm

shown t h a t t h i s process can be written in the form

(2.24),

with

~y (L)(P)

S(Ea,Wa,La), as

(2.47) Then i t is e a s i l y defined by (2.23)

L~, 1 ~ j ~ M, defined by P-I c~p_1_rAj_ = E 1Y(r) + DjY(P), 0 < F < a r=O

(2.48)

19

and the disturbance bk+ 1 by P-1 E c~P'l-rAUk+l(r) + Do Uk+I(P) + c~Pdk+l , 0 <_ P _< a bk+l r=O

(2.49)

Example 2.3.8 - A discrete unit memory linear r e p e t i t i v e process - Set M = 1 in the analysis of example 2.3.7 to obtain t h i s special case. The situations covered by examples 2.3.5 and 2.3.6 also extend in a natural manner to the discrete case and hence the d e t a i l s are omitted. Further, examples 2.3.1, 2.3.3, 2.3.4, 2.3.7 and 2.3.8 have direct industrial relevance in the modelling f o r i n i t i a l simulation and control studies of industrial examples such as long-wall coal cutting, metal r o l l i n g and bench mining systems. Hence the examples used in the remainder of t h i s work will be exclusively drawn from these and/or extensions to include, f o r example, interpass smoothing e f f e c t s . Structural Links with other Dynamic Systems One approach to the analysis of r e p e t i t i v e systems is to exploit, where possible, s t r u c t u r a l links which may exist with other well researched classes of dynamic systems. In t h i s work such links between the processes of examples 2.3.3 2.3.4 and 2.3.7 - 2.3.8 and two other classes of linear dynamic systems will be extensively used. To introduce the f i r s t of these, consider the d i f f e r e n t i a l non-unit memory linear r e p e t i t i v e process of example 2.3.3 and suppose that the following operations are applied to i t s state-space model: (i) The previous pass terms are deleted or, equivalently, Bj_I = O, 2.4

Dj = 0 , 1 < j

<M.

( i i ) The subscript k + 1 is dropped. (ill)The concept of a pass length is i r r e l e v a n t . Then (2.29) reduces to X(t) = AX(t) + BU(t) Y(t) = CX(t) + DoU(t)

x(o)

=

d

(2.50)

which is just the well known state-space model from standard, or conventional, linear systems theory. Within the r e p e t i t i v e systems framework, (2.50) is termed the derived conventional linear system and, for notational s i m p l i c i t y , will be denoted by LD(A,B,C,Do) from t h i s point onwards. Use will also be made of i t s t r a n s f e r - f u n c t i o n matrix description in the case of d = 0 Y(s) = Co(S)U(s )

(2.51)

with Go(S) = C(sI n - A)-IB + DO

(2.52)

In addition to LD(A,B,C,Do) , use will also be made of the so-called associated conventional linear systems of (2.29) defined as

20 X(t) = hX(t) + Bj_IYI-J(t ) wJ(t) = CX(t) + DjyI-J(t) X(O) = O, 1 < j < ~ Suppose also t h a t dk+ 1 = O, k > O.

(2.53) In which case the ith element of (2.53) has, in

e f f e c t , been obtained from (2.29) by s e t t i n g B = O, Do = O, Bj_ 1 = O, Dj = O, 1 < j # i < ~, ignoring the pass length a, and dropping the pass subscript. Equivalently, (2.53) can be regarded as describing the contribution of pass p r o f i l e k + 1 - j to the current one. To see t h i s , r e s t r i c t t to [O,a] and set y l ' J ( t ) equal to pass p r o f i l e k + 1 - j. For notational convenience, the j t h , 1 < j < ~, element of (2.53) will be denoted by L~(A,Bj_I,C,Dj] from t h i s point onwards and use will also be made of the corresponding t r a n s f e r - f u n c t i o n matrix description wJ (s) = Gj (s)¥ 1- j (s) (2.54) with Gj(s) = C(sI n - A)-IBj_I + Dj

(2.55)

Finally, in the compact notation, the derived and associated conventional l i n e a r systems for the d i s c r e t e non-unit memory l i n e a r r e p e t i t i v e process of example 2.3.7 are defined by LD(~,A,C,Do) and L~(~,Aj_I,C,Dj] , I<j<M, r e s p e c t i v e l y with corresponding t r a n s f e r - f u n c t i o n matrices Go(Zl) = C(ZlI n - ~)-IA + DO

(2.56)

and Gj(Zl) = C(ZlI n - ~)-IAj_ 1 + Dj

(2.57)

The second area from which s t r u c t u r a l links are exploited is that of 2D linear systems described by the so-called Roesser state-space model. This has the following form f o r systems recursive in the positive quadrant Xh(i + i , j ) = A1Xh(i,j ) + A2Xv(i,j ) + B1U(i,j) Xv(i,j + 1) = A3Xh(i,j ) + h4Xv(i,j ) + B2U(i,j ) Y ( i , j ) = ClXh(i,j)

(2.58) n1 Here i , j are p o s i t i v e integer valued horizontal and v e r t i c a l coordinates, Xh e R , +

C2Xv(i,j)

+ DU(i,j)

n2 Xv e R are vectors which propagate information in the horizontal and v e r t i c a l d i r e c t i o n s r e s p e c t i v e l y , U e Rt and Y e Rm are vector inputs and outputs r e s p e c t i v e l y and A1,A2,A3,A4,BI,B2,C1,C 2 and D are real constant matrices of appropriate dimensions. Systems described by t h i s state-space model have been extensively studied in recent years using both state-space and t r a n s f e r - f u n c t i o n matrix techniques where the l a t t e r is two variable.

21

Comparing (2.58) with the r e p e t i t i v e processes described in section 2.3, and in p a r t i c u l a r the discrete process of example 2.3.8, indicates t h a t , despite notational differences, these r e p e t i t i v e processes have clear s t r u c t u r a l s i m i l a r i t i e s with 2D systems described by the Roesser model. In p a r t i c u l a r , the model of example 2.3.8 is a Roesser model where (i) X, the current pass s t a t e vector, plays the role of horizontally transmitted information; (ii) Y, the current pass output vector, plays the role of v e r t i c a l l y transmitted information; and ( i l l ) t h e f i n a l equation in (2.58) is redundant but could, if required, be used to represent other algebraic measurement equations associated with the application under consideration. This s i m i l a r i t y can be further highlighted by considering Figure 2.6 where the v e r t i c a l axis is taken to represent the evolution of a discrete process from pass to pass and the horizontal axis is taken to represent the evolution of the process state along the pass. In p a r t i c u l a r , note that each point in the (k,P) plane of this figure is associated with a s t a t e Xk(P), an output Yk(P) and a control input Uk(P).

Hence Figure 2.6 i l l u s t r a t e s the evolution in both k and P of the s t a t e

variables in terms of t h e i r values, the values of the outputs at points (j,P), j = k - 1, and the control input. This evolution of the 2D/repetitive process is uniquely specified by the boundary conditions along the axes {(k,O): k > 1} and {(O,P): 0 < P < a}, i . e . the s t a t e i n i t i a l conditions on each pass and the i n i t i a l pass p r o f i l e . 2.5

Transfer-Function Matrix Description The corresponding t r a n s f e r - f u n c t i o n matrices play a central role in the analysis and control of conventional, or standard, linear systems and 2D linear systems described by the Roesser state-space model. Hence, given the discussion of section 2.4, i t is to be expected that a similar role exists for appropriately defined t r a n s f e r - f u n c t i o n matrices in the analysis and control of the r e p e t i t i v e processes of examples 2.3.3 and 2.3.7. Before proceeding to consider t h i s matter further, a number of important preliminary r e s u l t s and observations axe required, the f i r s t of which is the fact that these processes are 'well posed' in the sense that they map sequences of inputs into sequences of outputs and each has a solution which is unique. Secondly, they exhibit multipass causality. In p a r t i c u l a r , noting that the following extends in a natural manner to example 2.3.7, consider the d i f f e r e n t i a l process of example 2.3.3. Then in t h i s case multipass causality means that the output, Yk(t), at any time t on pass k does not depend on information from the following s e t s , see also Figure 2.7,

22

PASS INDEX k+l k k-1

5-

I(k,t) I I I I Initial data specified on these boundaries

432}i=

0 FIGURE 2 . 6

23

N-1

k+l PASS l INDEX /

k

t k-l I I I I I I I 3

Non-causal information Causal information FIGURE 2.7

Yk (t)

24 X = {Xk(r):

D = {dg:

t < r < a}U{Xg(t):

0 < t < a,

g > k}

g > k}

O={Uk(r):

t
ok}

Y = {Yk(r):

t < r < a}U{Yi(t):

0 < t < a, g > k}

(2.59)

Continuing with the d i f f e r e n t i a l case, note that two parameters are required to specify a variable in (2.29), i . e . the distance, or time, t , along a p a r t i c u l a r pass and the pass number k. Hence any t r a n s f e r - f u n c t i o n matrix d e s c r i p t i o n must be two variable, or 2D, in s t r u c t u r e . Consider also the interpass dependence and regard the pass length a as a 'sample period' or i n t e r v a l . Then the following d e f i n i t i o n can be regarded as the natural r e p e t i t i v e process g e n e r a l i s a t i o n of the well known z-transform from d i s c r e t e conventional l i n e a r systems theory. Definition 2.5.1: The ' z - t r a n s f o r m s ' of the sequences Uk+l(t), Xk+l(t ) and Yk+l(t),

0 < t ~ a, k > 0 are defined by

u(t,z) = ul(t ) + z-lu2(t) + z-2Ua(t) + ...

(2.60)

X(t,z) = Xl(t ) + z-lX2(t) + z-2X3(t) + . . .

(2.61)

Y(t,z) = Yl(t) + z-iY2(t) + z-2Ya(t) + ...

(2.62)

and

respectively. Results on the convergence and existence p r o p e r t i e s of (2.60)-(2.62) are contained in the following r e s u l t . Lemma 2.5.1: Suppose t h a t the terms in (2.60)-(2.62) are bounded in the sense that there e x i s t s r e a l numbers Mi > O, 4 i > O, 1 < i < 3, such t h a t

~ MI 4~-1, k ~ 1

(2.6a)

IlXk(')l 5 ~2 4~-1, k ~ 1

(2.64)

IlYk(')l

(2.65)

II~(.)l and

5~3a~-1,

k~ 1

where [ I . I [ is chosen as any s u i t a b l e norm in g a.

Then (2.60)-(2.62) converge

absolutely in the regions Izl > A1, Izl > 42 and Izl > 43 r e s p e c t i v e l y . Proof:

Consider (2.60) and take the norm to yield

IN(.,z)ll ~ [Iol(.)ll + Iz-ll llu2(.)ll + ...

41

41 2

M1(1 + T~y + ( N I )

+ ...)

and hence absolute convergence since M1 liO("z)ll ~ T < + ~

(1 - T ~ )

(2.66)

(2.67)

2B provided [z[ > ~1"

The proofs for (2.61) and (2.62) follow in a similar manner and

are hence omitted.

• 0

At t h i s stage, d e f i n e ~ X ( t , z ) ~

X(t,z) = ~ X l ( t ) + z-

as

_2 3 ~ X2(t ) + z $~-X3(t ) + . . .

1 8

(2.08)

and consider, without loss of generality, the special case of zero i n i t i a l pass p r o f i l e s and zero s t a t e i n i t i a l conditions on each pass, i . e . Vl_j(t) = O, 0 < t $ a, 1 < j < M (2.69) dk+1 = O, k > 0

(2.70)

Hence X(O,z) = 0 and the 'z-transform' of (2.29) in t h i s case is easily shown to be 0 X(t,z) = (A + B(z)(I m - D(z))-lc) X(t,z) + {B +

B(z)(I m - D(z))-IDo } U(t,z) Y(t,z) = (I m- D(z))-I {CX(t,z) + DoU(t,z)}

(2.71) (2.72)

where M

M

$ Djz" j j:l j:l and the term (I m - D(z)) is always i n v e r t i b l e since limit B(z)

Bj_lz-J ,

D(z)=

(I m - D(z)) = I m which is obviously nonsingular.

(2.73)

Note also t h a t t h i s r e s u l t is

obvious i f z"1 is regarded as a backward s h i f t operator. Given (2.71)-(2.73), consider the problem of using the 'z-transform' to solve for a sequence of pass p r o f i l e s in the presence of a known input sequence Uk+l(t), 0 < t < a, k > O. In which case i t follows immediately that t h i s can be achieved by solving (2.71) for X(t,z), substituting the result in (2.72) to obtain Y(t,z) and then expanding the r e s u l t as a power series to obtain the pass p r o f i l e s in the order

{YI'Y2'Y3.... }" One method of solving for X(t,z), and hence Y(t,z), in (2.71)-(2.73) would be to employ the Laplace transform. Note, however, that the variables Uj(t), Xj(t) and Yj(t), j > 1, of the series U(t,z), X(t,z) and Y(t,z) r e s p e c t i v e l y are only defined on the f i n i t e interval [O,a] but use of the Laplace transform would require that they be defined on [0,+®). ~ence it would appear that the Laplace transform cannot be used in t h i s p a r t i c u l a r situation. The f a c t that t h i s is not the case is a direct r e s u l t of the multipass causality of (2.29) as defined by (2.59). In p a r t i c u l a r , by multipass causality, the r e s u l t will be unaffected i f the Laplace transform is applied to a r b i t r a r y extensions of the variables l i s t e d above from '

26 [O,a] to [0,+~), provided, of course, that these extensions satisfy the necessary existence conditions. Suppose, therefore, that the variables Uj(t), Xj(t) and Yj(t), j _> 1, have been suitably extended from [O,a] to [O,+m) and let the same symbols denote these extensions. Then the Laplace transforms, o r ' s transforms', are defined as follows. Definition 2.5.2: T h e ' s transforms' of the series U(t,z), X(t,z) and Y(t,z) are defined by U(s,z) = .2qJ(t,z) = ~ l ( t ) + z-1 .2qJ2(t ) + z-2 .2~3(t ) + ... (2.74) X(s,z) = .~X(t,z) : . ~ l ( t )

+ z-1 .~PX2(t) + z-2 .~C(3(t) + ...

(2.75)

and y(s,z) = ~ ( t , z )

+ -1

(2.76)

= ~l(t)

~ 2 ( t ) + z-2 ~ 3 ( t ) + ...

respectively where ~ denotes the Laplace transform with respect to the along the pass variable t. • Results on the convergence and existence properties of (2.74)-(2.76) are contained in the following result. Lemma 2.5.2: Suppose that there exists real numbers Mi > O, ~i > 0 and 2i > O, 1 < i < 3, such that ][Uj(t) [p < M1 e/?lt~i-1

(2.77)

][Xj(t) ]p < M2 e~2t~i-1

(2.78)

IIYj(t) Ip % M3

(2.79)

and e~3t~-1;

respectively, j > 1, Vt ~ O, where ]].]]p denotes any suitable vector norm. Then the series o~ (2.74)-(2.76) converge absolutely in the regions {[z I > ~1' Re{s} > Zl} , {[z[ > ]2' Re{s} > 82 ) and {Iz[ > ~3' Re{s} > ~3) respectively. Proof:

Consider U(s,z) and note that

IIUj(s) llp = ]lFe'Stjo Uj(t)dt[IP '

j ~ 1

_<M1 ~oe(~l"S)t2i -1 dt

M1 ~i -1 Rc((s_~l)} ,

(2.8o)

Re{s} > n 1

i.e. Uj(s) is well defined for Re(s} > ~1' j ~ 1.

Now write

27 U(s,z)

j~l_ z l - j Uj(s), then taking norms and using (2.77) yields M1

~I

j-1

[[U(s,z)[lp 5 Rei(s_~l)} j~l ( ~ ) M1

=

~1 Re{(s-fll)}(1 - ] ~ )

,

Iz[ > ~1

(2.81)

and hence U(s,z) is well defined, or converges absolutely, in the region {[z[ > AI' Re{s} > B1}. The proofs for X(s,z) and Y(s,z) follow in a similar manner and are

hence omitted. Applying the's transform' to (2.71)-(2.73) now yields Y(s,z) = G(s,z)U(s,z) where G(s,z) is the m x g 2D transfer-function matrix defined by

• (2.82)

G(s,z) = (Im- D(z))-IC(SIn- A- B(z)(Im- D(z))-Ic)-I{B + B ( z ) ( I m - D(z))-lDo } + (I m - D(z))-lDo

(2.83)

Hence application of the 'transforms' of definitions 2.5.1 and 2.5.2 respectively has resulted in an input/output description for (2.29) in the form of a 2D transfer-function matrix. Note also that (2.82)-(2.83) is invariant under the order of 'transform' application. In particular, it is easily shown that this description also results from f i r s t applying the 's-transform' of definition 2.5.2 to (2.29) and then applying the 'z-transform' of definition 2.5.1 to the result. Finally, this transfer-function matrix description extends in a natural manner to the case when (2.69) and/or (2.70) have non-zero entries and hence the details are omitted. By appropriate rearrangement, it is possible to use G(s,z) to obtain a clearer insight into the physical structure of (2.29). In particular, rewrite (2.82)-(2.83) in the form M Gj (s)z_ jy(s,z) Y(s,z) : Go(S)U(s,z) + j~1

(2.84)

where

Go(S) = C(sln - A)-IB + DO

(2.85)

Gj(s) = C(sln - A)-IBj_I + I)j, 1 < j < II

(2.86)

and Then M

G(s,z) = (Im- j~l GJ(S)z-J)-IG°(s)

(2.87)

and the block diagram structure of (2.84) is shown in Figure 2.8 or, equivalently, Figure 2.9. Either of these diagrams indicating that, in 2D transfer-function matrix terms, the dynamics of (2.29) are represented by a dynamic pre-compensator

28

U(s,z)

Y(s,z)

+

z-2G2(S)t~-~--

~ ' 1 z-MGM(S)I= FIGURE2.8

PRE-COMPENSATOR I

. U(s,z)

i•

Y(s,z)

I j~"lGj(s)z-J

IC.

REPETITIVE INTERACTION FIGURE2.9

29 followed by a positive feedback loop with unity gain in the forward path and dynamic elements in the feedback loop. These feedback elements are the repetitive interaction. At this stage, suppose that Bj_ 1 = O, Dj = O, 1 < j < M, the pass subscript k + 1 is dropped, and the concept of a pass length is irrelevant. Then i t follows immediately that G(s,z) reduces to ~o(S) of (2.85) which is just, see (2.51)-(2.52), the transfer-function matrix of the derived conventional linear system. Hence in this sense G(s,z) can be regarded as the natural repetitive process generalisation of i t s well known and extensively used conventional linear systems counterpart. In the unit memory case Gl(S ) of fl(s,z) is, see (2.54)-(2.55) with M = 1, the transfer-function matrix of the associated conventional linear system and hence, in effect, describes the contribution of Yk to Yk+l' k > O. Consequently i t is termed the interpass transfer-function matrix. For the more general non-unit memory case, element Gj(s), 1 < j < M, of (2.86) is, see (2.54)-(2.55), the transfer-function matrix of the jth associated conventional linear system and hence, in effect, describes the contribution of Yk+l-j to Yk+l' k > O. Further, let Y(s) denote the combined effects of the previous M passes. M

Then

M

Y(s) = j~l WJ(s) = j=lZ Gj(s)yl-J(s)

(2.88)

and this expression can be interpreted in unit memory form/by writing it as

(2.89) Y(s)

J

14(s)

62(s )

Gl(S

LY°(s) J

Hence the non-unit memory version of the interpass transfer-function matrix is taken to be the following N x N, N = n~[, block companion matrix

G(s) =

0 0

0

01

G2(s)

Ol(S)

Im

GM(S)

Im

(2.90)

The interpass transfer-function matrix will play a central role in subsequent chapters where use will also be made of the N x N constant coefficient block companion matrix defined as D = limit G(s) (2.91)

tsl~+®

and hence

30 [~

Im

D= DM

0

.]

0

Im

D2

D1

(2.92)

For the discrete process of example 2.3.7, the 'z-transforms' are defined as follows. Definition 2.5.3: The z-transforms of the sequences Uk+I(P ), Xk+I(P ) and Yk+I(P), 0 ~ P <

k > 0 are defined by

a,

U(P) U I ( P ) + z -1U2(P ) + z-2 U3(P) + ...

(2.93)

X(P) = Xl(P ) + z -1X2(P ) + z -2 X3(P) + ...

(2.94)

Y(P) = YI(P) + z-1 Y2(P) + z -2 Y3(P) + ...

(2.95)

and respectively. • A natural extension of the analysis of lemma 2.5.1 gives results on the convergence and existence properties of (2.93)-(2.95) and hence the details are omitted. Further, these series can be used in conjunction with the standard z-transform from conventional linear systems theory, termed the 'z 1-transform' in this context, to develop a 2D transfer-function matrix description of the state-space model (2.45). This invokes analysis which is just the natural extension of that used in the differential case and hence the details are omitted. The final result is Y(Zl,Z ) = G(Zl,Z ) U(zl,z ) (2.96) where G(Zl,Z ) = (I m - D(z))-lC(zlI n - ~ - a ( z ) ( I m - D ( ~ ) ) - l C ) - I { a A(z)(I m- D(z))-lDo} + (I m- D(z))-lDo

+ (2.97)

with

A(z)=

M

M

D(z)jz__ = Djz-j

(2.98)

g(zl'z) = (Im- j=~l GJ(Zl)Z-J)-lG°(Zl)

(2.99)

Go(Zl) = C(ZlIn _ ~ ) - 1 A + DO

(2.~0o)

1z-j

,

or

M

where and Gj(Zl) = C(Zlin _ ~)-1Aj_I + Dj, 1 < j < M

(2.102)

31 This 2D t r a n s f e r - f u n c t i o n matrix description has an identical block diagram interpretation to that of Figures 2.8 or 2.9 for i t s d i f f e r e n t i a l counterpart. Further, the t r a n s f e r - f u n c t i o n matrices Go(Zl) and Gj(Zl) , 1 < j < M, also have

)I

identical i n t e r p r e t a t i o n s to t h e i r d i f f e r e n t i a l counterparts and the non-unit memory interpass t r a n s f e r - f u n c t i o n matrix is the following N x N block companion matrix 0

G(Zl) =

0

OM(zI)

Im

0

0

O2(zI)

Im

(2.102)

Oi(z I

Use will also be made in the analysis of subsequent chapters of D = limit G(Zl)

(2.1o3)

which is just (2.92). Notes and References The industrial examples, unique control problem, and the abstract representation have evolved from the original work of Edwards (1974) and Owens (1977). Comprehensive d e t a i l s of the state-space model of (2.58) in section 2.4 can be found in Roesser (1975). For a comprehensive treatment of the transfer-function matrix descriptions see Rogers and Owens (1989a, 1990a). Edwards and Owens (1982) gives an extensive treatment of the c l a s s i c a l s t a b i l i t y analysis.

CHAPTER 3 STABILITY TEEORY

A rigorous s t a b i l i t y theory for the abstract representation of the l i n e a r constant pass length case given in d e f i n i t i o n 2.3.1 is developed. This is then applied to the special cases of examples 2.3.1, 2.3.3 and 2.3.7 to produce conditions suitable for the development of computationally f e a s i b l e s t a b i l i t y t e s t s in chapter 4. In the same context, i t is shown that an equivalence exists between s t a b i l i t y of example 2.3.8 (the discrete unit memory case) and the well established area of s t a b i l i t y analysis f o r 2D linear systems described by the Roesser model. 3.1

Asymptotic S t a b i l i t y An i l l u s t r a t e d by the simulation r e s u l t s of Figure 2.4, the e s s e n t i a l unique undesirable feature of a r e p e t i t i v e process is the possible presence in the output sequence of o s c i l l a t i o n s which grow in amplitude from pass to pass. Fence the natural i n t u i t i v e d e f i n i t i o n of asymptotic s t a b i l i t y is to demand t h a t , given any i n i t i a l p r o f i l e Yo and any disturbance sequence {bk}k> 1 which ' s e t t l e s down' to a steady disturbance b as k ~ + ~, the sequence {Yk}k>1 generated by

S(Ea,Va,La)

' s e t t l e s downI to a steady p r o f i l e Y as k ~ + ®. This idea is i l l u s t r a t e d in Figure 3.1 and i t s major drawback is that i t does not e x p l i c i t l y include the i n t u i t i v e idea that asymptotic s t a b i l i t y should be retained i f the model is perturbed s l i g h t l y due to modelling errors or simulation approximations. Consequently the following d e f i n i t i o n is preferred since i t ensures t h a t the ' s e t of stable systems' is open (in a uell defined sense) in the class of a l l linear r e p e t i t i v e processes. Definition3.I.l:

A l~near r e p e t i t i v e process

S(Ea,Na,La) of

constant pass length

a > 0 is said to be asymptotically stable i f there e x i s t s a r e a l scalar 6 > 0 such t h a t , given any i n i t i a l p r o f i l e Yo and any strongly convergent disturbance sequence {bk}k> 1 c Wa, the sequence {Yk}k>1 generated by the perturbed process Yk+l = (La + 7)Yk + bk+l' k > 0 converges strongly to a limit p r o f i l e Y e Ea whenever 117[I ~ 6. Note:

(3.1) •

Y does, of course, depend on 7, Yo and {bk}k> 1.

The use of the term 'asymptotic s t a b i l i t y ' in the above d e f i n i t i o n can be j u s t i f i e d by considering the case when bk+ 1 = O, k > O, which is strongly convergent to zero.

Hence, from the d e f i n i t i o n , asymptotic s t a b i l i t y requires that the

33

o,p

~'FIGURE 3.1

34 solution sequence {Yk}k>1 of (3.1) converges strongly to a p r o f i l e Y for each Yo and model perturbation 7. In this p a r t i c u l a r case, the solution of (3.1) has the form Yk = (La + 7)kYo' k > 0 (3.2) L and set 7 = ~

to yield 117I] = ~.

Then i t is immediately clear that the

sequence (La + 7)kYo, k > O, is strongly convergent and hence bounded for each Yo e Ea.

Application of the Banach-Steinhaus theorem now says that there exists a real

number Ma > 0 such that

l l(L a + 7)kll -< Ma , k > 0

(3.3)

or, equivalently,

+

)kl ILkll

_<

k _> 0

(3.4)

Given (3.4), consider now the case of 7 = 0 (the ' r e a l ' system) and define 5 Aa = (1 + ~ _ [ ) - 1 < 1 (3.5) In which case (3.4) takes the form

I lLk.II _<

k _> 0

(3.6)

and using (3.2) i t follows that k [IYkl[ = I igkYo[ [ _< [ILkaII [[Yol [ < M A~I IYoil

(3.7)

Hence, in the absence of disturbances, the output sequence {Yk}k>l converges strongly to zero for all i n i t i a l p r o f i l e s . Physically, t h i s requires that the effects of the i n i t i a l p r o f i l e are rapidly attenuated a f t e r a 'large number of passes' at a geometric rate. The key to constructing necessary and sufficient conditions for asymptotic s t a b i l i t y of 8(Ea,Wa,La) is to note i t s formal similarity with the matrix recursion relation which forms the basis of example 2.3.2. In particular, s t a b i l i t y of the l a t t e r , viewed as a conventional linear system, is governed by the relationship between i t s eigenvalues/poles and the unit c i r c l e in the complex plane. I n t u i t i v e l y , therefore, i t can be anticipated that asymptotic s t a b i l i t y of S(Ea,~la,La) will be related to the nature of the eigenvalues or, more generally, t h e spectral values of La which are defined as follows. Definition 3.1.2:

A complex number A is said not to be a spectral value of La i f ,

and only i f , the bounded linear operator AI - La, where I is the identity operator

35 in Ea, has range dense in Ea and a bounded inverse (hi - La)-I.

The set, ~(La) , of

a l l spectral values of La is called the spectrum of La and i t s spectral radius is defined to be the f i n i t e p o s i t i v e number r(La):= sup I~1

le~(L)

(3.8)

or, equivalently,

r(La) = limitIIL~[lI/k

(3.9)

k~+® • Theorem 3.1.1 below gives a necessary and s u f f i c i e n t condition f o r asymptotic s t a b i l i t y of S(Ea,~a,La) in terms of r(La) and i t s proof requires the following r e s u l t s . These are stated here without proof since they follow on straightforward use of standard r e s u l t s and concepts from functional analysis. Lemma 3.1.1: Let ¢ be a linear operator mapping Ea into i t s e l f with norm I1¢11 < 1. Then the operator I - ~ has a bounded inverse ~hich can be expressed in terms of the absolutely convergent power series.

(I-

¢)'1 = I + ¢ + ¢2 + ¢3 + . . .

(3.10)

and

II(I Lemma 3.1.2:

¢)-1II g i - 1 I1¢11 I f r(La) < 1 and r(La) < 2 < 1, then

fl(]):= izl> sup AI I ( z I Lemma 3.1.3:

(3i~11

La)-11 I

<+~

I f r(La) < 1 then there exists real scalars 5 > 0 and

r(La) < ~ < 1 such that r(L a + 7) < ~ whenever [i711 g 5. Lemma 3.1.4: ItTII ~ 5.

•

~ith the notation of lemma 3.1.3, suppose that r(La) < 1 and Then there exists a real number Ma(7) > 0 such that

II(L~ + ,)kli ~ M~(~)~k, k > 0 Lemma 3.1.5:

(3.12)•

(3i~3)

~ith the above notation, suppose that r(La) < 1 and l[71I g 5 and that

the sequence {bk}k> 1 converges strongly to b E Ea. Then k limit ~ (La + 7)J-l(bk+l_j - b ) = 0 (3.14) k~+® j=l • (the limit being interpreted in the sense of the norm) Lemma 3.1.6: With the above notation, suppose that r(La) < 1 and 1[7][ 5 5. Then the power series k (La + 7 ) J - l b

j=l

is absolutely convergent as k ~ + ~.

(3.15)

•

36 @iven the above results, the following theorem can now be proved which gives a necessary and sufficient condition for asymptotic s t a b i l i t y . Theorem 3.1.1:

The linear repetitive process

S(Ea,Wa,La) of

constant f i n i t e pass

length a > 0 is asymptotically stable i f , and only i f , r(La) < 1 Proof:

Suppose that

S(Ea,Wa,La) is

(3.1a)

asymptotically stable.

Then use of (3.5) and

(3.6) in (3.9) yields (3.17) r(La) = limitllL~ll 1/k < ha limit M1/k a = ha < 1 k~+oo k~+oo Conversely, suppose that r(La) < 1 and write the solution of (3.1) in the form k Yk : (La + 7)kYo + j~l (La + 7)J-lbk+l'J k : (La + 7)ky° + j~=l (La + 7)J-l(bk+l-J

b)

+ j~l (La + 7)j-lb~

(3.18)

Then, since [I(L a + 7)kYo[[ < ~a(7)hk[[Vo[[ by lemma 3.1.4, i t is clear that ( L + 7)kYo ~ 0 as k ~ + ~.

Further, i t is easily verified using lemmas 3.1.5 and

3.1.6 that the sequence {Yk}k>1 converges strongly to Y e Ea where limit Yk = Y:= E (La + 7 ) J - l b (3.19) k~+® j=l and the proof is complete. • The result (3.16) provides a necessary and sufficient condition for asymptotic s t a b i l i t y but l i t t l e or no information concerning transient behaviour. The following definition and results are a physically r e a l i s t i c approach to characterising system behaviour after a 'large number of passes'. Definition 3.1.3:

Suppose that the linear repetitive process

S(Ea,Wa,La) of

constant pass length a > 0 is asymptotically stable and let {bk}k>1 be a disturbance sequence which converges strongly to a disturbance b . Y:= limit Yk

Then the strong limit (3.20)

k~+~

is termed the limit profile corresponding to {bk}k>1.

•

37 Theorem 3.1.2:

Suppose that the linear r e p e t i t i v e process

S(Ea,Wa,La)of constant

:

pass length a > 0 is asymptotically stable and let {bk}k>1 be a disturbance sequence which converges strongly to a disturbance b®.

Then the limit p r o f i l e corresponding

to t h i s disturbance sequence is the unique solution of the linear equation. Y = LaY® + b® Proof: Use of (3.19) with 7 = 0 yields a power series representation of Y satisfies (3.21). The uniqueness of ~ form (I - La) Y (I -

(3.21) which

follows immediately on writing (3.21) in the

= b® and noting, by asymptotic stability, that r(La) < 1 and hence

La) has a bounded inverse in E a.

Equivalently, (3.21) has a unique solution

which can be written in the form V = (I - L a ) - l b ®

Corollary 3.1.2:

(3.22) •

Y is independent of the i n i t i a l pass p r o f i l e Yo and of the

direction of approach to b .

•

Note: Formally, (3.21) can be obtained from (3.1) with 7 = 0 by replacing all variables by t h e i r strong limits. To be of use in a p a r t i c u l a r application, the abstract r e s u l t s of theorems 3.1.1 and 3.1.2 must be converted (if possible) into a suitable computational form. No general rules exist for this procedure, other than the obvious necessity to compute the spectral values of La and hence t h e i r moduli. Further, severe d i f f i c u l t i e s can arise if the space Ea and the operator La have a complex structure. For the special cases of examples 2 . 3 . 1 , 2 . 3 . 3 , 2 . 3 . 4 , 2 . 3 . 7 and 2.3.8, however, the following analysis is possible. The approach to the spectral calculations used here is to consider the equation (zI - La)Y : ~ (3.23) and construct necessary and sufficient conditions on the complex scalar z to ensure that a solution exists for all ~ e Ea, and that this solution is bounded in the sense that [[Y[[ < Ko [[y[[ for some real scalar Ko > 0 and for a l l ~ e Ea.

This

yields the following results. Theorem 3.1.3: A delay - algebraic system - The linear r e p e t i t i v e process of example 2.3.1 is asymptotically stable i f , and only i f , (3.24)

Ikll < 1

Proof:

In t h i s p a r t i c u l a r case (3.23) can be written in the form zY(t) -

=

W(t) = -koW(t - X) + k 1 Y ( t ) , 0 < t < a W(t) = O, -X < t < 0

(3.25)

38

Further, by assumption, X > 0 and hence it is possible to choose an integer n > 1 such that (n- I) X < a < n X and, after a little manipulation, (3.25) can be written in the matrix form

| Y(t + x)

v(t + x)

L y(t +(n-1)X)

W(t +(u-1)X)

=

~(t + x)

(3.26)

~(t + (n-1)x)

E l i m i n a t i n g t h e W v a r i a b l e s now g i v e s

•[Y(t) (zI n -

]

K) / Y ( t + X)

~(t) =

y ( t + X)

L Y ( t + (n-1)x)

, 0 S t < X

(3.27)

~(t + (n-1)x)

where t h e nxn m a t r i x K i s lower t r i a n g u l a r with Kii = k l , 1 < i < n, K i + l , j + 1 = K i , j , 1 5 i , j < n -

1

(3.28)

Choosing z # k l , i t i s c l e a r t h a t (3.27) has a s o l u t i o n a t each p o i n t t E[O,X] and i t i s e a s i l y shown t h a t Y(O) = 0 and t h a t Y(t) i s continuous on 0 < t < a. F u r t h e r , c o n s i d e r , without l o s s of g e n e r a l i t y , t h e norm

]lxll'

= max ]x. I in Rn and apply this to (3.27) to yield l<j
IIYII' = I I ( z I n -

K)-lll'.ll~ll

'

(3.29)

Hence t h e only c a n d i d a t e f o r a s p e c t r a l value of La i s z = k 1.

In t h i s case

(zI - K) i s s i n g u l a r and i t i s a simple m a t t e r t o c o n s t r u c t ~ such t h a t (3.27) has

no solution.

Hence #(La) = {kl} , r(La) = [kll, and the proof is complete.

•

Suppose now that theorem 3.].3 holds and a strongly convergent sequence (Rk}k> 1 is applied. Then a representation of the corresponding limit profile Y(t) can be obtained by, see theorem 3.1.2, replacing all variables in the defining equation (2.25) by their limiting values.

This yields

Y ( t ) = - k 0 Y®(t - X) + k 1 Y ( t ) + k 0 R®(t), 0 < t < a o r , a f t e r a l i t t l e manipulation. ko -

~(t) : ~

~(t- x) + ~

ko

R(t)

(3.30)

(3.31)

Theorem 3 . 1 . 4 : The d i f f e r e n t i a l non-unit memory case - The d i f f e r e n t i a l non-unit memory l i n e a r r e p e t i t i v e process of example 2 . 3 . 3 i s a s y m p t o t i c a l l y s t a b l e i f , and only i f , a l l non-zero s o l u t i o n s of t h e r e l a t i o n IzI m - D1 - z-lD 2 - . . . have modulus s t r i c t l y

zl-MDMI = 0

l e s s than u n i t y .

(3.32)

39 Proof:

The abstract equation (zI - La) [Y1

:

E EM ff

(3.33)

YM ~M in this particular case, where La has the block companion structure of (2.24) as defined by (2.32), can be written out as

zYi(t)=Yi+l(t)+~i(t),

l_<j_<~- i

(3.34)

M

i ( t ) : AX(t) + j~l Bj'IYM+I'J(t)' X(O) : 0 M

z YM(t) = CX(t) ÷ j~l l)Jgl~l+l-J(t) + ~M(t) O
i ( t ) = AX(t)

+

.

j~l B j - l z l - J ~ ( t )

+

M

M-1

2

~

j=2 i=M+l-j

M

z YM(t) = CX(t) + j~l Dj zl'JYM(t ) +

M

Bj-lz~J-iyi(t)

M-i

P, DjzM-j'i~i(t ) j=2 i=M+l-j

+

~M(t)

x(o) = 0 , o < t ~ a Now define the polynomial matrix P(z) = zI m- D1 - z'lD2 - . . . - zl-MDM and suppose that P(Z)Mis nonsingular.

(3.37) (3.38)

Then (3.37) takes the form

X(t) = (l + j~l Bj-lzl-JP(z)-I C)X(t) M

+ {j~l BJ-]zI-J}P(z)-]{~M(t) M-I +

M-1

M

÷ ~ )'] D'zM-i-J~'(t)} i=l j=M+I-i J 1

M

~ B.~z~i-J~i(t ) i=l j=M+l-i j-i

YM(t) = P(z)-l{cx(t) + ~M(t) + ~ ~ Dj z ~ i - J ~ i ( t ) } (3.39) i=l j=M+l-i ~ d it is clear in this case that zI-L a has a bounded inverse and that the spectrum of La is hence a subset of the roots of IP(z)[ = O. Hence the proof will be

40

complete i f i t can be shown that the spectrum of La is equal to the set of solutions of ]P(z)[ = O.

To achieve t h i s set t = 0 in (3.37) and ~l(t) . . . . . .

~M.l(t) ~ O.

In which case P(z)YM(O) = ~M(O)

(3.40)

which has no solution i f y~(O) is not in the range of P(z), i . e . zI - La cannot be surjective if [P(z) l = O.

•

Corollary 3.1.4: The d i f f e r e n t i a l unit memory case-Setting M = 1 in theorem 3.1.4 gives the r e s u l t that the d i f f e r e n t i a l unit memory l i n e a r r e p e t i t i v e process of example 2.3.4 is asymptotically stable i f , and only i f , a l l eigenvalues of the mxm matrix D1 l i e in the open unit c i r c l e in the complex plane. • Note: I t is rather surprising to find that asymptotic s t a b i l i t y of the process of example 2.3.3 (and hence example 2.3.4) is independent of the system matrices and, in p a r t i c u l a r , independent of the eigenvalues of A. This is dfie e n t i r e l y to the f a c t that a is f i n i t e and will change d r a s t i c a l l y when the case of a ~ + ® is considered in section 3.3. Suppose now that theorem 3.1.4 holds and a strongly convergent sequence (Uk}k>1 is applied.

Then a representation of the corresponding limit p r o f i l e Y ( t ) can, as

for the case of theorem 3.1.3, be obtained by replacing a l l variables in the defining state-space model (2.29) by t h e i r strong limits. This yields i (t) = AX (t) + BU(t) + BY (t) Y ( t ) : CX (t) 0 < t < a,

+

Do~(t )

+

DY (t)

X (0) = d

(3.41)

where

j iBj

S Vj

(3.42)

or, since asymptotic s t a b i l i t y ensures that I m - D is nonsingular,

i®(t) = ( A , fi(I m - ~ ) - l C l X ( t ) Y®(t) : (I m - D ) - l c x ( t ) ,

+ (B + ~(I m - ~ ) - l D o ) \ ( t )

(I m - ~ ) - l D o ~ ( t )

(3.43)

Setting M = 1 in (3.41) - (3.43) gives the corresponding r e s u l t s for the unit memory case.

Consider now asymptotic s t a b i l i t y and the corresponding limit p r o f i l e s for the d i s c r e t e processes of examples 2.3.7 and 2.3.8. Then the following r e s u l t s are stated without proof since these follow from identical arguments to those used in establishing t h e i r d i f f e r e n t i a l counterparts.

41

Theorem 3.1.5: The discrete non-unit memory case - The discrete non-unit memory linear r e p e t i t i v e process of example 2.3.7 is asymptotically stable i f , and only i f , all non-zero solutions of (3.32) have modulus s t r i c t l y less than unity. • Corollary 3.1.5: The discrete unit memory case - The discrete unit memory linear r e p e t i t i v e process of example 2.3.8 is asymptotically stable i f , and only i f , a l l eigenvalues of the mxm matrix D1 l i e in the open unit c i r c l e in the complex plane. I f theorem 3.1.5 holds, the corresponding limit p r o f i l e state-space model is given by X®(P + 1) -- (,~ + X(I m - D)-lClX®(P) + (a o + ,~(I m - ]~)-IDolU (P) Y(P)

--

(I m - l))-1CXoo(P) +

0 < P _< a ,

(I m -

I'))-lOoUo(P)

X (0) = d

(3.44)

where M = j~l h j ' l '

M ~ = j=lI] Dj

(3.45)

Setting ~ = 1 in (3.44) - (3.45) gives the corresponding unit memory result. Equations (3.43) and (3.44) are just the standard state-space models from d i f f e r e n t i a l and discrete conventional linear systems theory respectively. Hence if examples 2.3.3 and 2.3.7 (examples 2.3.4 and 2.3.8 in the unit memory case) are asymptotically stable then, in e f f e c t , t h e i r r e p e t i t i v e dynamics can, a f t e r a ~sufficiently large I number of passes, be described by a conventional linear systems state-space model. The implications of this result from a feedback control standpoint are discussed in chapter 6 which follows on from the s t a b i l i t y related analysis of section 5.3. 3.2

Bounded-Input/Bounded-Output Stability An alternative approach to s t a b i l i t y analysis for

S(Ea,Wa,La) is

to demand that

bounded disturbance sequences generate bounded sequences of pass p r o f i l e s . Suppose also that this property is to be retained in the presence of small modelling errors. Then the following definition, compare with definition 3.1.1, is relevant. Definition 3.2.1:

A linear r e p e t i t i v e process

S(Ea,Va,La) of

constant pass length

a > 0 is said to be bounded-input/bounded-output (BIBO) stable i f there exists a real scalar $ > 0 such that, given any Yo and {bk}k>1C ~a bounded in norm ( i . e . llbkl l < c 1 for some constant c 1 ~ 0 and for a l l k > 1), the output sequence (Yk}k>1 generated by the perturbed process (3.1) is bounded in norm whenever 117]] ~ $. • The following theorem establishes the equivalence of the, apparently d i f f e r e n t , definitions of 3.1.1 and 3.2.1.

42

Theorem 3.2.1:

The linear r e p e t i t i v e process

S(Ea,Wa,La) of

constant pass length

a > 0 is BIBB stable i f , and only i f , i t is asymptotically stable. Proof:

Suppose that

S(Ea,Wa,La) is

BIBU stable.

Then consideration of the bounded

~La sequence bk = O, k > 1, and 7 = ~

leads to the boundedness of (3.2) and hence

to (3.6) where Aa is defined by (3.5).

I t now follows from (3.17) and theorem 3.1.1

that the system is asymptotically stable. Conversely, suppose that S(Ea,Wa,La) is asymptotically stable (and hence r(La) < 1) and that the disturbance sequence is bounded, i . e . k > 1.

[[bk[ I < cl,

Further, write the solution of (3.1) in the form k Yk = (La + 7)k Yo + j~l (La + 7)J'lbk+l-J

(3.46)

Then taking the norm of (3.46) and making use of the r e s u l t of lemma 3.1.4 yields k

]{Ykll ~ I{(La+ 7)kll IIYol] + jEI]I(La+ 7)J-lll IIbk+l_jll c1

Ma(~)Ak(ll¢oll + I----:--X) c1 M~(7)(IIYoll + 1 - A) whenever ][vll g 5. 3.3

(3.47)

I t now follows immediately that S(Ea,Ye,La) is BIBO stable.

S t a b i l i t y Along the Pass Under well defined conditions, asymptotic s t a b i l i t y of

S(Ea,Wa,La) in

the form

of theorem 3.1.1 guarantees the existence of a limit p r o f i l e . I t does not, however, also guarantee that this limit p r o f i l e has acceptable dynamic c h a r a c t e r i s t i c s . To i l l u s t r a t e this point, consider the following SISB d i f f e r e n t i a l unit memory process where ~ is a real scalar i k + l ( t ) : -Xk+l(t ) + Uk+l(t ) + (1 + Z)Yk(t ) Yk+l(t) = Xk+l(t) 0 < t S a,

Xk+l(O) = O, k > 0

(3.48)

Then, since B1 = O, use of corollary 3.1.4 immediately yields asymptotic s t a b i l i t y and substitution in (3.43) with M = 1 then gives the corresponding limit p r o f i l e description as

43 :

p

+

0_< t_< a,

~(0) : 0

(3.49)

Further, i t is e a s i l y shown that i f Uk+1 (t) e 1, Yo(t) = O, 0 < t _< a, k > O, then Yl(t) = 1 - e- t ,

0
(3.50)

and 1 = ~(e ~ t -

~(t)

I),

0 < t _< a

(3.51)

Hence, despite the f a c t that the f i r s t pass p r o f i l e Yl(t) is a quite acceptable ~classical ~ response to the unit step demand Ul(t ) -= 1, the limit p r o f i l e can have t o t a l l y unacceptable dynamic c h a r a c t e r i s t i c s . For example, i f ~? > 0 then the limit profile gro~s exponentially and can be said to be Wunstable along the pass' in an obvious i n t u i t i v e sense. The natural d e f i n i t i o n of s t a b i l i t y along the pass for cases such as that highlighted by the above analysis is to demand that the limit p r o f i l e is stable in the standard, or conventional, sense as a ~ + ~ ( i . e . B < 0 in (3.48)). Unfortunately, however, t h i s i n t u i t i v e l y appealing idea does not apply in a simple manner to processes such as those described by example 2.3.5. Consequently the following analysis develops the concept of s t a b i l i t y along the pass by considering the r a t e of approach to the limit p r o f i l e as a ~ + ~. Suppose that that can be modelled over the range of pass lengths

S(Ea,Wa,La)

a > ao, where a o is some nominal value of i n t e r e s t , and introduce the following terminology. Definition 3.3.1:

A collection of models of

S(Ea,Wa,La)with

pass lengths in the

range a ~ a o is termed an extended linear r e p e t i t i v e process and is denoted

S(Ea,Wa,La)a>ao. Further, suppose that a p a r t i c u l a r element of

S(Ea,~a,La)a>aoiSasymptotically

stable. Then the following r e s u l t shows that t h i s element can be p a r t i a l l y characterised by real scalars Ma > 0 and 0 < ~a < 1 describing the r a t e of approach to the l i m i t p r o f i l e . Theorem 3.3.1:

Suppose that the linear r e p e t i t i v e process

S(Ea,~a,La)of

constant

pass length a > 0 is asymptotically stable. Further, l e t t h i s process be subjected to a constant disturbance sequence bk+1 = b , k > O, which generates the limit profile Y .

Then there exists real scalars Ma > 0 and 0 < 2a < 1 such that

44

lib®l[ I tYk- LII -< M~k~{llVol[ +

, _ ~

}, k >_ 0

(3.S2)

a

Proof:

Since bk+1 = b®, k _> O, the solution of (3.1) with 7 = 0 can be written as k . Yk = LkYo+ j~1 L3a-lb® (3.53)

and, using (3.,9) with 7 = O, the limit p r o f i l e can be expressed as

L = ~ L~-I L

(3.54)

j=l Hence the ' e r r o r ' Yk - Y can be written in the form k Yk- Y~ = LaYo"

Oo

~]

L]-lb , ,

k >

j =k+l

0

(3.55)

or, taking the norm to obtain a (numerical) estimate of convergence, IIYk - Y I I -< []Lkall IIYol[ +

~ IIL~-I]I lib oil (3.56) j =k+l • The proof is now completed by using (3.6). Note: In e f f e c t , t h i s result s t a t e s that the output sequence {Yk}k>l approaches the limit p r o f i l e at a geometric r a t e governed by 2a"

This r e s u l t , together with

d e f i n i t i o n 3.3.2 below, plays a s i g n i f i c a n t part in the analysis of chapter 5. Civen theorem 3.3.1, the following d e f i n i t i o n of s t a b i l i t y along the pass is expressed in terms of the existence of f i n i t e bounds on the scalars He and l a as a ~ + ~. I t s e f f e c t i v e action is to demand that the r a t e of approach of the output sequence to the limit p r o f i l e has a guaranteed geometric upper bound independent of pass length for ~ > a o. Definition 3.3.2:

The. extended linear r e p e t i t i v e process

S(Ea,Wa,La)a~aois said to

be stable along the pass i f there exists f i n i t e real scalars M > 0 and 0 < ~ < 1 such t h a t , f o r each a > a o and for each constant disturbance sequence bk+ 1 = b®, k > O, the output sequence from the model S(Ea,W~,La) s a t i s f i e s the inequality

lib®il iiYk- YJI ~ M®~{llYoll + ~---:--X-}, k ~ 0 `

(3.57)

o

Despite i t s well defined physical meaning, t h i s d e f i n i t i o n is not in appropriate form f o r the derivation of s t a b i l i t y c r i t e r i a . A more useful d e f i n i t i o n is implied by lemma 3.3.1 below which leads to the central result o~ t h i s section in the form of theorem 3.3.2.

45 Lena 3.3.1:

The extended linear repetitive process

S(Ea,Wa,La)a>ao

is stable along

the pass i f , and only i f , there exists f i n i t e real numbers N > 0 and 0 < I < 1 such that (3.58) for all a ~ ao. Proof:

Suppose that

S(ga,Wa,La)a> a -

is stable along the pass and set b = 0 in Oo

0

(3.57) where Yoo = 0 is the corresponding limit profile.

Then t h i s inequality

reduces to I IYkfl -< ~ ~k® t tYoll for all ~ _> % and, since Yo is arbitrary and k Yk = La Yo' (3.58) follows t r i v i a l l y . Conversely, suppose that (3.58) holds. Then (3.57) follows in a similar manner to the proof of theorem 3.3.1, i.e. the extended process is stable along the pass. Theorem 3.3.2:

The extended linear repetitive process

S(Ea,~ta,La)a>_a°

along the pass i f , and only i f , (a) r := sup r(La) < 1 ®

is stable

(3.59)

a>__ao

and Mo:= sup sup I I(zI - L)" a II < + ® =_>% Izl>a

(b)

(3.60)

for some real number I ~ (r ,1). Proof[ To prove necessity, r(La) = limit[[Lk[I 1]k k~+~ a for a l l a ~ % and hence ~ auger in the range t

note from lemma 3.3.1 that < limit ~ M 1/k = I < 1 - k~+~ ~ ~ < t® < 1. Suppose also that I is any

< I < 1.

Then i t is clear that ( z I - La)-I can be expressed

as an absolutely convergent power series in z-1 for Iz I ~ I and I1( z I - La)-l[[ = Iz'll IIj~ 1 ~-~ L~II 1®1

-<~ j~O ~ IILjll I ® -<

~®j

~ j__zo (r)

(3.61)

46

M = i(i -

(3.62)

~/~®)

for a l l a > ao. For a proof of sufficiency, suppose that r

< 1 and consider the contour M in

the complex plane defined by the relation = {z:

lzl

= ~}

(3.63)

Further~ exploit standard results from operational calculus in Banach spaces to write LaK as the contour integral Lk a = ~ 1 I,, zk(zI - La)-ldz

(3.64)

where z = ~ eiOand i denotes the ~square root of minus one *. using (3.60) yields. 1 [2rkeikO(~ei 0 - La)-l~ieiOdO[[ IlL, I[ < ~--~11 JO

~'ol[2rk,

lll(~eie_

Then taking norms and

La)_llld e

(MoA)Ak

(3.65)

This v e r i f i e s (3.58) with M = Mo~ and ~ = ~ and the proof is complete. Note 1: I t can be shown that (3.60) can be relaxed s l i g h t l y to Mo:= ~% sup sup [[(zI - La)- 1 II < + ® Izl=~ Note 2:

(3.59) is equivalent to asymptotic s t a b i l i t y for a l l a > a o.

(3.66) Hence the

reason for retaining the separate i d e n t i t i e s of (a) and (b) in theorem 3.3.2, despite the fact that (b) does imply (a). The %oundedness' condition (b) of theorem 3.3.2 is equivalent to the existence of a ~ E ( r , 1 ) such that (3.23), i.e. (zI - La)¥ = y

(3.67)

has a uniformly bounded, with respect to a, solution Y e Ea for a l l choices of e Ea satisfying supll~ll < + ® and for a l l Izl > ~.

Further, i t is clear that, in

general, t h i s condition could prove very d i f f i c u l t to i n t e r p r e t . For the special cases of examples 2 . 3 . 1 , 2 . 3 . 3 , , 2 . 3 . 4 , 2 . 3 . 7 and 2.3.8, however, the following results are obtained.

47 Theorem 3.3.3:

S(Ba,~a,La)a>ao

A delay-algebraic system - The extended l i n e a r r e p e t i t i v e process generated by (2.25) of example 2.3.1 v i t h a ~ a o is stable along the

pass i f , and only i f ,

(a)

lkll < 1

(3.68)

(h)

Ikol < 1 - Ikll

(3.69)

and Proof: In t h i s p a r t i c u l a r case the extended process simply consists of the family of models (2.25) with a > ao f o r some a o of i n t e r e s t . Further, i t follows immediately from theorem 3.1.3 that r(La) = [kll f o r a l l a > 0 and hence (a) of theorem 3.3.2 in t h i s p a r t i c u l a r case reduces to (3.68). Considering now (b) of theorem 3.3.2, then, in t h i s p a r t i c u l a r case, (3.67) can be written as -z k o

(ko~(t - X) + ~(t))

Y(t) : ~ _ - : - ~ l Y ( t - x) +

z-

ks

(3.70)

Hence i t follows immediately that (3.60) holds i f , and only i f , -z ko

for some ~ E ( I k l l , 1 ) .

Elementary graphical considerations now reduce (3.71) to

(3.69) and the proof is complete. In the above theorem, (3.68) is a consequence of (3.69). Hence t e s t i n g example 2.3.1 f o r s t a b i l i t y along the pass reduces to the very simple task of checking (3.69). Consequently no further consideration of t h i s example is undertaken here except to note the analysis below which generalises theorem 3.3.3 to non-unit memory processes with a current pass delay such as bench mining systems. Here the dynamics are described by M

Yk+l(t) = -koYk+l(t - X) + ~ kjYk+l_j(t ) + koRk+l(t ) j=l

0
Yk+l(t) = O, -X < t ~ 0

k >0 (3.72)

~here k o and k j , 1 < j < M, are real constants of a r b i t r a r y sign.

Further, (3.72)

reduces to the equation of example 2.3.1 i f M = 1 and hence in t h i s sense i t can be regarded as the natural generalisation of t h i s unit memory process. Suppose also that Ea = Wa is chosen as the vector space of continuous functions on [O,a] with i n i t i a l condition Y(O) = 0 and norm written in the abstract form

IlVll

S(Ea,Wa,La) ,

= max IY(t)l. Then (3.72) can be O~t~a where La has the Icompanion form' of

(2.24) and L~, 1 < j < M, is defined by considering

4B

Yl(t) = -koYl(t - X) + kjYl. j ( t ) , 0 ~ t 5 a Yl(t) = O, -X 5 t 5 0

(3.73)

The following r e s u l t now provides necessary and s u f f i c i e n t conditions for s t a b i l i t y along the pass of (3.72) a~d generalises theorem 3.3.3 to t h i s case. The extended l i n e a r r e p e t i t i v e process S(Ea,Wa,La)a>_a° generated by

Theorem 3.3.4:

(3.72) with a _> oo is stable along the pass i f , and only i f ,

(a) and (b)

~up{lzt:

z~ -

M E kj z M-j = O) < 1 j=l

max If(z) I < 1 It f=l

(3.74)

(3.75)

where f(z) =

Proof:

ko M kjz-J

(3.76)

The f i r s t step in proving (a) is to compute the spectral values of La which,

since a > 0 is f i n i t e , are simply the eigenvalues,

llence the computation of r(La)

reduces to finding the complex scalar of largest modulus which s a t i s f i e s the eigenvalue problem La ~? = z~ E EMa (3.77) Writing out (3.77) yields zWi(t ) = Wi+l(t), I < i < M- i

(3.78)

M

zWM(t) : - k o z WM(t - X) + j~l kJ$/~t+l-j(t)

(3.79)

O
-z M koWl(t - X) wt(t) = M

(~.so)

zM -

~ kjz M-j j=l on using (3.78) to write

Wi(t) =

Zi-lWl(t),

2
(3.81)

and appropriate substitution in (3.79). fence, noting the initial conditions associated with (3.73), it follows immediately that the nomtrivial solutions of (3.77) are the roots of zM _

M

E kj z~ j = 0

j=1 and consequently

(3.82)

49

r(L a) -- sup{{zl:

M

kjzM-J = o} z (3.83) j=l The r e s u l t nou follous immediately on noting t h a t r(La) is independent of a.

zM

-

To prove (b), f i r s t note the i n t e r p r e t a t i o n of (b) of the general r e s u l t of theorem 3.3.2 in terms of (3.67) and u r i t e out the abstract equation (zI - La)

=

YM

~ EMa

(3.84)

1 < i _< M- 1

(3.85)

~M

for t h i s case over 0 < t < a as zYi(t)

- Yi+t(t) = ~ i ( t ) ,

z YM(t) - W(t) = ,M(t)

(3.86) M

~(t) = -k o W(t - X)

v(t) = o ,

-x
+

j_E_I kjYM+l_j(t )

<_0

(3.87)

(3.88)

or

z Yi(t) - Yi+l(t) : , i ( t )

,

1 < i < M- 1

(3.89)

M

z Y~(t) =-k o z YM(t- X) + j)1 kj YM+I_j(t) + ko ~M(t - X) + ~M(t) Using induction, i t now follows from (3.89) t h a t i-1 ¥ i ( t ) = z i ' l Yl(t) - j~l z i - j - l ~ j ( t ) , 2 -< i -< M

(3.90)

(3.91)

and use of t h i s in (3.90) yields 1 M-1 Yl(t) = f(z)Y1(t - X) + P-~[k° p=l P" z-PWP(t " X) -

M M-j r. P~ z ' P ' J y p ( t ) + ko~M(t - X) + ~M(t)] j=l p=l

(3.92)

where

p(z) = I - Z kj z-j j=l

(3.93)

A simple argument now indicates that the existence of a uniform bound f o r Yl(t) over the required ranges of a and z is equivalent to the existence of a uniform bound for the solutions of (3.84) over these same ranges f o r a l l permissible choices of Yi' 1 < i < M. Consequently i t remains to prove t h a t the existence of such a bound is equivalent to the v a l i d i t y of (3.75) with ]z I = 1 replaced by lzl _> ~. The f a c t that t h i s condition is s u f f i c i e n t follows immediately on noting that the terms in (3.92) a r i s i n g from yp(t - X) and yp(t) are uniformly bounded V [z I > e(r ,1).

To show necessity, f i r s t note that f ( z ) is a n a l y t i c V Izl >_ A and a l l

50 e(r ,1).

Further, (a) is equivalent to p(z) ~ O, V Izl ~ 1, and i t follows

immediately that the supremum of If(z) l occurs at a f i n i t e value, say, z = z . Now set z = z in (3,92) and note that i t is always possible to choose ~ i ' 1 < i < M, such that a l l terms in the resulting version of t h i s equation which do not involve Y1 are nonzero and constant. In which case a contradiction occurs i f If(z*)l ~ 1 since (3.92) is then unstable and hence Yl(t) is not uniformly bounded on [0,+®]. At t h i s stage, i t has been shown that s t a b i l i t y along the pass is equivalent to the existence of a ~ E (r ,1) such that

sup If(=)l < 1 Izlz~

(3.94)

Further, the proof to date has shown that the supremum of f ( z ) occurs at a f i n i t e value of z and hence, by the maximum modulus theorem, at the boundary, i . e . tzl > can be replaced by the compact set Izl = A. The proof is now completed by showing that (3.94) is a necessary and s u f f i c i e n t condition for (3.75) where the l a t t e r is obvious. To establish the former, suppose that sup I f ( z ) l < 1 (3.95)

Izl=l

Then continuity implies that for each Iz] = 1, there exists r(z) such that

If(zlit

(3.96)

< 1

for [z 1 - z[ < r ( z ) . A simple argument based on the compactness of ]z} = 1 no, yields the existence of a ~ such that (3.94) holds. • The following c o r o l l a r i e s yield useful information concerning the t e s t i n g of (a) and/or (b) of theorem 3.3.4 f o r a given example. Corollary 3.3.4: A necessary condition for (a) of theorem 3.3.4 is that p(z) of (3.93) s a t i s f i e s p(1) > 0 and hence M

1 - j~l kj > 0 Proof:

(3.97)

F i r s t note again that (3.74) is equivalent to

p(z) # O, V

Izl Z 1

(3.98)

Now consider p o s i t i v e real values of z and suppose that M

1 - j~l kj ~ 0

(3.99)

Then t h i s contradicts (3.98) since i t is e a s i l y seen that there e x i s t s a positive r e a l number z > 1 such that p(z ) = O. • Corollary 3.3.5:

Suppose (as in bench mining systems) that kj > O, 1 < j < M. Then

p(1) > 0 is a necessary and s u f f i c i e n t condition for (a) of theorem 3.3.4. Proof: I f (a) holds then p(1) > 0 is immediate from corollary 3.3.4. For the converse, i t is required to prove that i f (3.97) holds then p(z) ¢ O, V Izl > 1.

51

This follows immediately on noting that M

(3.1oo)

Ij~l kJZ-J[ ~ j=I ~ kj < 1, V tz[ _> 1 and hence M kjz_j[ > 1 - j~l M kj > O, V ]z I >- 1 IP(z)l = I1" j~l Corollary 3.3.6: 1 <

j <

M.

(3.101) •

Suppose (as in bench mining systems) that ko > 0 and kj > O,

Then

M

kj (3.102) j=l is a necessary and sufficient condition for (a) and (b) of theorem 3.3.4. Proof: Here it is required to prove that (3.102) is equivalent to both of these conditions. To prove the f i r s t , simply note that for (3.102) to hold ko < I -

M

1 - j~l kj > O, which is simply the result of corollary 3.3.4.

For the second, a

simple argument yields ko M $ k. j=l 3 A sufficient condition for (b) of theorem 3.3.4 is that

max lf(z)l [zl=l 1Corollary 3.3.7:

(3.103) •

M

[koJ < 1 - j=~llkjl Proof:

(3.104)

First note from (3.104) that M

M

_~l]kj[ < 1 and hence

j-

M

[j~ kjz-Jl _< ~ Ik~] < 1, V Iz] = 1 -1 j=l J Using (3.105) now yields M

(3.105)

M

[1 - j~l kjz-j[ -> 1 - j~l[kjl, V Iz[ = 1

(3.106)

and the result follows immediately since

Iko[ max

[z[=l

If(z)l _<

M < 1 1 - j~l[kjl

(3.107)

Testing (a) of theorem 3.3.4 in the general case is easily converted into the standard stability problem from discrete conventional linear systems theory. In the case of (b), let t denote the closed contour generated by p(z) as z traverses the unit circle in the complex plane either clockwise or anti-clockwise. Then this condition holds if, and only if, t lies completely outside the circle in the complex plane of radius [ko[ and centre the origin. This again is easily tested using elements of conventional linear systems stability analysis.

52 Considering now the processes of examples 2 . 3 . 3 , 2 . 3 . 4 , 2 . 3 . 7 and 2.3.8 yields the following r e s u l t s . Further, those f o r the discrete processes of examples 2.3.7 and 2.3.8 are stated without proofs since these follow from identical arguments to those used in establishing t h e i r d i f f e r e n t i a l counterparts. Theorem 3.3.5: The d i f f e r e n t i a l non-unit memory case - Suppose that (i) the p a i r {C,h}Mis observable,

(ii)

the p a i r {h, j~l Bj'lTJ-1) is controllable at a l l but a f i n i t e number of points 7 1 , 7 2 , . . . , 7 q in the complex plane,

and (iii)

[si n - A- j~l Bj_17~-1P(711)'lc[- has no roots on the imaginary axis

of the complex plane, 1 < i < q, where P(7) is defined by (3.38) with z replaced by 7Then the extended linear r e p e t i t i v e process generated by

S(Ea,Wa,La)a>ao

d i f f e r e n t i a l models of the form (2.29) in example 2.3.3 with a ~ a o is stable along the pass i f , and only i f , (a) r® = supflzl: P(z) = O) < 1

(3.1o8)

and

(b)

there exists real numbers e > 0 and r

< ~ < 1 such that

M

[si n - h - j~l Bj-lzl-JP(z)-IC[ # 0

(3.109)

for a l l complex numbers z,s s a t i s f y i n g [z[ > A and Re{s} > - e. Proof:

To prove (a), note from theorem 3.1.4 that r(La) is independent of a and

hence the r e s u l t . In the case of (b), f i r s t note again that (b) of the general r e s u l t , theorem 3.3.2, is equivalent to the existence of the uniform bound defined in terms of (3.67) which, in t h i s p a r t i c u l a r case, is equivalent to the existence of such a bound on the solutions of (3.34) - (3.35) as a - + ~. Given (3.35), t h i s is equivalent to the existence of such a bound on the solutions of (3.39) as a ~ + ®. Further, t h i s l a s t requirement is equivalent to (b) above since the f a c t that a l l of the c o e f f i c i e n t s of ~ l , Y 2 , . . . , y M in (3.39) are hounded in any region Izl > A with > r

reduces the boundedness condition to a s t a b i l i t y condition on the matrix ~(z) = h + j=lZ Bj_I zl-jP(z)-lC:

(3.110)

for Izl > ~. The f a c t that t h i s must be a strong s t a b i l i t y condition follous from the following argument based on the c o n t r o l l a b i l i t y and observability assumptions.

53

Suppose that it is only possible to choose e = 0 in (3.109). Then it is clear that a sequence (zj}j>1, with Izj[ ~ ], j ~ I, can be chosen such that one of the roots of the polynomial M Isln - A - j~l Bj'IZl'JP(zi)-Icl' i > I approaches a finite point s = i~o on the imaginary axis of the complex plane as i

+ ®.

Further, suppose, without loss of generality, that the sequence {zj 1}

converges to a point z = zo where the assumption (iii) guarantees that zo ¢ 7i, 1 i < q, and there are now two possibilities (a) z o = 0 and (b) z o ¢ O.

If z o ¢ O,

then the controllability and observability conditions imply that ll(Zo I I - La)'l[l +~ which violates the requirement of stability along the pass. Alternatively, if zo = 0 then A must have at least one pair of purely imaginary eigenvalues and hence, using the controllability and observability assumptions, [[La[ [ * +~ as a ~ +~ which

violates (3.60) of theorem 3.3.2 for s t a b i l i t y along the pass. Setting M = 1 in the above theorem now gives the following r e s u l t for the process of example 2.3.4.

•

Corollary 3.3.8: The d i f f e r e n t i a l unit memory case - Suppose that the pair {C,A} is observable and the pair {A,Bo} is controllable. Then the extended linear r e p e t i t i v e process

S(Ea,~a,La)a>aogenerated by the model of example 2.3.4

with

a ~ a o is stable along the pass i f , and only i f , (a)

a l l eigenvalues of the mxm matrix D1 l i e in the open unit c i r c l e

(b)

in the complex plane; and there exists realnumbers e > 0 and r

< ~ < 1 such that a l l

eigenvalues of the nxn matrix A + Bo(ZIm - D1)-Ic l i e to the l e f t of the line Re{s} = -e for a l l choices of [z t ~ ~. The corresponding r e s u l t s for the discrete processes of examples 2.3.7 and 2.3.8 take the following forms Theorem 3.3.6: The discrete non-unit memory case - Suppose that (i) the pair {C,~ is observable, (ii) the pair {~,jE1AJ. } m17J_I

is controllable at a l l but a f i n i t e

number of points 71,72,...,7q in the complex plane, and (iii) IZlIn - ~ - .~Aj_17 j-1 i P(7-I i )-1 c ] has no roots on the unit circle in j=l

54 the complex plane, 1 S i < q, where P(7) is defined by (3.38)with z replaced by 7. Then the extended l i n e a r r e p e t i t i v e process generated by d i s c r e t e models of the form of (2.45) in

S(Ea,Ya,La)a>ao example 2.3.7 with a ~ a o is stable

along the pass

i f , and only i f , (a) r = sup{Izl: P(z) = O} < I,

(3.111)

and (b)

there exists real numbers e > 0 and r

< A < 1 such that

M [zlI n - ~ - j ~ I A J _ l z l ' J P ( z ) ' I c [ ~ 0

(3.112)

for a l l complex numbers Zl, z s a t i s f y i n g ]Zlt > 1 - e and [z[ > 4. Corollary 3.3.9: The discrete unit memory case - Suppose that the p a i r {C,~} is observable and the p a i r {~,Ao} is controllable. Then the extended linear r e p e t i t i v e process

S(ga,~a,La)a>aogenerated

by the model of example 2.3.8 with a > ao is

stable along the pass i f , and only i f , (a) a l l eigenvalues of the m×m matrix D1 l i e in the open unit c i r c l e (b)

in the complex plane; and there exists real numbers e > 0 and r

< ~ < 1 such that a l l

eigenvalues of the nxn matrix ~ + Ao(ZIm - D1)" 1C have modulus s t r i c t l y less than 1 - e f o r a l l choices of ]z] > 4. Note: I t is assumed in t h i s work that the c o n t r o l l a b i l i t y and o b s e r v a b i l i t y assumptions of theorems 3.3.5 and 3.3.6 always bold. Consider now the problem of t e s t i n g the conditions of (a) and (b) of theorem 3.3.5 or 3.3.6 for a given example. Then i t is immediately c l e a r that t e s t i n g (b) in e i t h e r case is not a computationally f e a s i b l e proposition. Further, as a step towards the development of equivalent r e s u l t s which are computationally f e a s i b l e to t e s t , i t is convenient to introduce the following d e f i n i t i o n s at t h i s stage. Definition 3.3.3:

The asymptotic s t a b i l i t y polynomial, Pa(Z), f o r the process of

example 2.3.3 or 2.3.7 is defined by Pa(z) = IQ(z)[

(3.113)

where

Q(z) = I m- z'lD1 - . . . - z'MDM and is to be regarded as a polynomial in z -1

(3.114) m

66

It is now a simple matter to show that (3.32)(theorem 3.1.4) for asymptotic stability in either case can be replaced by IQ(z)l ~ O, Vlz I ~ I (3.1i5) Definition 3.3.4:

The s t a b i l i t y along the pass polynomial, Ap(s,z), f o r the process

of example 2.3.3 is defined byM Ap(S,Z) = Isln- A-

=~IBj-iz-JQ(z)'lcl

(3.116)

J and is to be regarded as a polynomial in s with c o e f f i c i e n t s which are r a t i o n a l functions in z " l . Definition 3.3.5:

The s t a b i l i t y along the pass polynomial, Ap(Zl,Z), f o r the

process of example 2.3.7 is defined by M

Ap(Zl,Z) = IhIn - t - j~I AJ-lZ-Jq(z)-lcf

(3.11~)

and is to be regarded as a polynomial in z 1 with c o e f f i c i e n t s which are r a t i o n a l functions in z-1.

•

A simple argument now shows that (b) of theorem 3.3.5 is equivalent to the existence of real numbers c > 0 and r < ~ < 1 such that

Ap(S,Z) ~ 0

(3.118)

for a l l complex numbers z , s s a t i s f y i n g Iz] > ~ and Re{s} ~ -~. Similarly, i t is easily shown that (b) of theorem 3.3.6 is equivalent to the existence of r e a l numbers e > 0 and r < 2 < 1 such that Ap(Zl,Z ) ¢ 0

(3.119)

for a l l complex numbers Zl,Z s a t i s f y i n g lZll > 1 - e and ]z I ~ ~. The following r e s u l t s now provide a l t e r n a t i v e sets of conditions for s t a b i l i t y along the pass of the processes of examples 2.3.3, 2.3.4, 2.3.7 and 2.3.8 which are computationally f e a s i b l e , see chapter 4, to t e s t . These express s t a b i l i t y along the pass in terms of the derived conventional linear systems of section 2.4 and the 2D transfer-function matrix descriptions of section 2.5. Their e f f e c t i v e action is to replace (b) in each case by two equivalent conditions which are compntationally feasible to t e s t . Further, the r e s u l t s for the d i s c r e t e processes of examples 2.3.7 and 2.3.8 are again stated without proof since they follow from identical arguments to those used in establishing t h e i r d i f f e r e n t i a l counterparts. Theorem 3.3.7: The d i f f e r e n t i a l non-unit memory case - Yith the assumptions of theorem 3.3.5, the extended l i n e a r r e p e t i t i v e process generated by

S(Ea,Wa,La)a>ao

d i f f e r e n t i a l models of the form of (2.29) in example 2.3.3 with a > ao is stable along the pass i f , and only i f ,

56 (a)

(b)

a l l eigenvalues of the NuN block companion matrix D, constructed from the 2D transfer-function matrix G(s,z) using (2.91) - (2.92), have modulus s t r i c t l y less than unity; a l l eigenvalues of the matrix k have s t r i c t l y negative real parts or, equivalently, the derived conventional linear system Ln(A,B,C,Do) of (2.50) is stable; and

(c)

a l l eigenvalues of the NxN block companion frequency response matrix obtained by setting s = i¢ in the interpass transfer-function matrix G(s), constructed from G(s,z) using (2.90), have modulus s t r i c t l y less unity for all real frequencies w > O. Proof: This consists of showing that (a) above is equivalent to (a) of theorem 3.3.5 and that (b) and (c) are, together, equivalent to (b) of this same result. Consider f i r s t , therefore, (a) and note that (3.108) is equivalent to

an(z) = zNIQ(z)l = [zMIm - zM-1D1 - zM-2D2

-...-

DMI # O,

V

Iz] > 1

(3.120) Further, use of induction on M and Schur's formula, yields pD(z ) = IzI N - D[

(3.121)

i . e . the characteristic polynomial of D. To generate (b), let I z l ~ + ~ in (3.118) &lad hence a l l eigenvalues of A must have s t r i c t l y negative real parts. Equivalently, LD(A,B,C,Do) must be stable. Given (a) and (b), note that si n At Ap(S,Z) = pD(z) jzI N - G(s)i -

and hence (b) of theorem 3.3.5 reduces to lzI N- ~(s)] # O, V ]z] ~ ~, Re{s} > - e Setting s = iw, the necessity that a l l eigenvalues of G(iw) Conversely, suppose that contour in the complex plane.

(3.122)

(3.123)

of (c) above follows immediately since (3.123) implies have modulus s t r i c t l y less than 1. (a) - (c) above hold and consider the usual Nyquist Further, let zj(s), 1 < j < N, denote the j t h

eigenvalue of @(s) and l j , 1 < j < N, the jth eigenvalue of D where [lj] < 1 by (a). In which case (2.91) indicates that the choice of limit z;(s) = Aj, 1 < j < N

Isl~+® J

(3.124)

incurs no loss of generality. Hence, using (a) and (c), i t is possible to choose a real scalar ~ in the non-empty range sup max Iz.(iw)[ < t < 1 (3.125) w>O I<j
57 Now suppose that s traverses the Nyquist contour in a clockwise manner. Then it follows immediately that the locus generated by the right-hand side of (3.122) does not intersect or encircle the origin of the complex plane for any choice of tz I >_ ]. M

Equivalently, all eigenvalues of the matrix A + j~l BJ-lZ-Jq(z)- 1C have s t r i c t l y negative real parts for a l l choices of Iz] _> ], i . e . Ap(S,Z) ¢ 0 for a l l complex numbers z,s satisfying Iz] >_ A and Re(s} >_ O. The f i n a l step in this proof is to demonstrate the existence of a suitable > O. To accomplish t h i s , f i r s t note again that the eigenvalues of M

/t + E Bj. lZ-Jq(z)-lc approach the eigenvalues of A as tzl ~ + ~. Hence (b) of j=l theorem 3.3.5 can be replaced by the requirement that a l l eigenvalues of M

A + j~l Bj. lz-JQ(z)-lC have s t r i c t l y negative real parts for a l l z lying in some compact set ~ _< [z I _< R with l~ ' l a r g e ' . The existence of a suitable e > 0 now follows by continuity and the consequent existence of a f i n i t e covering of this set by open balls. • Setting M = 1 in the above theorem nou gives the follouing r e s u l t for the process of example 2.3.4. Corollary 3.3.10: The d i f f e r e n t i a l unit memory case - With the assumptions of corollary 3.3.8, the extended linear r e p e t i t i v e process generated by

S(Ea,~a,L~)a>ao

the model of example 2.3.4 with a > a o is stable along the pass i f , and only i f , (a)

all eigenvalues of the mxm matrix D1 l i e in the open unit c i r c l e

(b)

in the complex plane; a l l eigenvalues of the matrix A have s t r i c t l y negative real parts or, equivalently, LD(A,B,C,Do) is stable; and

(c)

a l l eigenvalues of the mxm interpass transfer-function matrix G(s) ~ Gl(S ) of (2.90) with s = iv have modulus s t r i c t l y less than unity for a l l real frequencies v > O.

At this stage, consider the special case when the example under consideration is SISO. Then here (c) of corollary 3.3.10 reduces to the modulus condition I~l(iw)[ < 1, V real ~ ~ 0 (3.126) Equivalently, the frequency response plot generated by the interpass transfer-function Gl(S), s = i~, V real ~ > O, must l i e e n t i r e l y within the unit circle in the complex plane. Suppose also that zero state i n i t i a l conditions and control inputs are applied, i . e . dk+1 = O, Uk+l(t ) = O, 0 < t < a, k ~ O. Then,

58

a f t e r a l i t t l e manipulation, the process description in t h i s special case can be expressed as Yk(i#) : G~(i~)Yo(i~), k ~ 0 (3.127) Hence (3.126) requires that each frequency component of the i n i t i a l p r o f i l e is attenuated from pass to pass. A conclusion which provides a physical interpretation of (c) of theorem 3.3.7 in this special case. The corresponding results for the discrete processes of examples 2.3.7 and 2.3.8 take the following forms. Theorem 3.3.8:

The discrete non-unit memory case - With the assumptions of theorem

3.3.6, the extended linear repetitive process

S(Ea,Wa,La)a>ao generated by discrete

models of the form of (2.45) in example 2.3.7 with a > ao is stable along the pass

i f , and only i f , (a) a l l eigenvalues of the NxN block companion matrix D, constructed from the 2D transfer-function matrix ¢(Zl,Z ) using (2.103), have (b)

(c)

modulus s t r i c t l y less than unity; a l l eigenvalues of the matrix ~ have modulus s t r i c t l y less than unity or, equivalently, the derived conventional linear system LD(¢,A,C,Do) is stable; and a l l eigenvalues of the NxN interpass transfer-function matrix G(Zl), constructed from G(Zl,Z ) using (2.102), have modulus s t r i c t l y less than unity for a l l frequencies z 1 s a t i s f y i n g {Zll = 1.

Corollary 3.3.11: The discrete unit memory case - With the assumptions of corollary 3.3.9, the extended linear repetitive process generated by the model

S(Ea,Wa,La)a>ao

of example 2.3.8 with a ~ ao is stable along the pass i f , and only i f , (a)

a l l eigenvalues of the mxm matrix D1 l i e in the open unit circle

(b)

in the complex plane; a l l eigenvahes of the matrix ~ have modulus s t r i c t l y less than unity or, equivalently, LD(~,A,C,Do) is stable; and

(c)

all eigenvalues of the mxm interpass transfer-function matrix G(Zl) ~ Gl(Zl) of (2.102) have modulus s t r i c t l y less than unity for a l l frequencies z 1 s a t i s f y i n G {zll = 1.

3.4

A 2D Systems Approach This section considers the link between repetitive process s t a b i l i t y and BIHO s t a b i l i t y of 2D linear systems described by the Roesser s t a t ~ s p a c e model of (2.58). In particular, the link between s t a b i l i t y along the pass of example 2.3.8 and BIBU

59

stability of (2.58) is considered. The following results summarise the essential elements of the well established s t a b i l i t y theory for 2D linear systems (not necessarily described by the Roesser model). In general, a 2D linear shift invariant system can be described by a convolution of the input U(p,q) and the impulse response function h(p,q)Xere, however, i t is only necessary to consider i n i t i a l l y the special case of SISO systems described by the input]output map K L I J Y(p,q) =

Z

Z

a(k,g)U(p-k,q-g) -

r

r

b(i,j)Y(p-i,q-j)

(3.128)

k=O t=O i=O j=O Further, (3.128) is said to be spatially causal over the quadrant (p,q) > 0 since Y(p,q) depends only on input and output variables at points ( i , j ) < (p,q). Applying the 2D z transform to (3.128) yields a 2D transfer-function relating Y to U of the form A(Zl,Z 2) H(Zl,Z2) = ~ (3.129) where K

A(Zl'Z2)

L r a(k, t)ZlZ k 2t

(3.130)

k=O t=O

and I J B(Zl,Z2) = E Z b(i,j)z~zJ2 i=o j=o and, for simplicity, i t has been assumed that b(O,O) = 1. power series yields O0

(3.131) Further, expanding H as a

Oo

r h p,q)z z tt(zl,z 2) = z p=O q=O and (3.128) is said to be BIBO stable i f , and only i f , Oo

CO

z

z lh(p,q)I < +

(3.132)

(3.133)

p=O q=O The following standard result, stated without proof, now gives a condition for (3.133).

Theorem 3.4.1: Suppose that the two variable polynomials A and B are mutually prime and H(Zl,Z2) has no nonessential singularities of the second kind (i.e. there exists no (Zl,Z2) such that A(Zl,Z2) = B(Zl,Z2) = 0).

Then BIBO s t a b i l i t y of (3.128) is

equivalent to B(Zl,Z2) # O, V IZll < 1, lz2l < 1

(3.134)

Testing theorem 3.4.1 for a given example uould clearly be a formidable task. This i problem can, houever, be simplified by use of the following equivalent standard result which is again stated uithout proof.

60 Theorem 3.4.2: reqfiires that

With assumptions of theorem 3.4.1, BIBO s t a b i l i t y of (3.128)

(a) B(Zl,O) ~ o,

v

(3.135)

{Zl{ ~1

and

(b) B(z~,z2) # o, V $zll = 1, {z2{ 5 1

(3.136)

Note also that (3.135) and (3.136) are interchangeable in terms of z 1 and z 2. Further, additional simplifications of them have been derived and used, for example, to develop a 'Nyquist-like' s t a b i l i t y t e s t for 2D linear systems. Note: The 2D transform, and resulting 2D transfer-function matrix, used in this section is d i s t i n c t from O(Zl,Z ) of section 2.5. Consider now the Roesser s t a r , space model of (2.58). Then application of the 2D z-transform yields the following 2D transfer-function matrix

H(zl,z 2) = [c 1 c2]

(3.137) +1) Further, application of the above results to each element in turn of H(Zl,Z2) immediately yields that BIBO s t a b i l i t y , as expressed by theorem 3.4.1 or 3.4.2, is dependent on the roots of the characteristic polynomial In 1

ZlA1

ZlA2

(3.138)

p(zl,z2) = - z2A3

In2 - z2h4

Use of Schur's formula now yields p(zl,z2) = [Inl - ZlAlllln2 - z2A4 - ZlZ2A3(Inl - ZlA1)-IA2 }

(3.139)

and leads to the following r e s u l t . Theorem 3.4.3: The conditions of theorem 3.4.1 or 3.4.2 are equivalent to the following: (a) a l l eigenvalues of the matrix A1 have modulus s t r i c t l y less than (b)

unity; a l l eigenvalues of the matrix A4 have modulus s t r i c t l y less than unity; and

61

(c)

all eigenvalues of the transfer-function matrix

(3.~4o)

Q(z~ I):= A4 + A3(ZlIInl - AI)-IA2 with Iz1[ = I lie in the open unit circle in the complex plane. Proof:

Applying (a) of theorem 3.4.2 to p(zl,z2) requires that

p(zl,O ) = IInl - Zlkll ~ 0 for IZll < 1 and, by interchanging the roles of z 1 and z2, p(O,z2) = IIn2 - z2h41 ~ 0 for [z2[ < 1.

Hence (a) and (b) above follow

immediately and, using (3.139), (b) of theorem 3.4.2 reduces to

T(zl,z2)::

1In2 - z 2 q(z~l)l ~ O, V Iz1[ = 1, Iz21 ~ 1

(3.141)

i.e. a l l eigenvalues of Q(Zl 1) with IZl] = 1 l i e in the open unit c i r c l e in the complex plane. Conversely, suppose that (a) - (c) above hold. Then i t follows immediately that p(Zl,O ) ~ 0 for all [zll < 1 and hence (3.135) is valid. Further~ p(zl,z2) ~ p(Zl,O)T(Zl,Z2) ~ 0 for IZll = 1, lz21 < 1 as the eigenvalues of In2 - z 2 q(z~ 1) are non-zero in this domain and the proof is complete.

Theorem 3.4.3 is in i t s own right an alternative to theorems 3.4.1 or 3.4.2 for BIBO s t a b i l i t y of the Roesser model. In this context, (a) and (b) are equivalent necessary conditions and hence one of them could be dispensed with. They are retained here, however, since the primary purpose is to establish an equivalence with s t a b i l i t y along the pass of the discrete unity memory linear r e p e t i t i v e process of example 2.3.8. This is contained in the following r e s u l t . Theorem 3.4.4: Regard the model of example 2.3.8 as a 2D system described by the Roesser model and suppose that the corresponding 2I) transfer-function matrix of (3.137) has no nonessential singularities of the second kind. Then the extended linear r e p e t i t i v e process S(Ea,Wa,La)a>_a ° generated by t h i s model with a _> ao is stable along the pass i f , and only if, i t is ]~IBO stable in the sense of theorem 3.4.3.

Proof: This, in effect, consists of showing that the conditions of theorem 3.4.3 and corollary 3.3.11 are equivalent. Suppose first, therefore, that theorem 3.4.2 is applied. Then this requires the following conditions for BIBO stability which are precisely those of corollary 3.3.11 for stability along the pass.

62

(a) (b) (c)

a l l eigenvalues of the matrix ~ have modulus s t r i c t l y less than unity; a l l eigenvalues of the matrix D1 have modulus s t r i c t l y less than unity; and a l l eigenvalues of Gl(Zll):= C(z~1 I n - ~)-Ih o + D1

(3.142t

with [Zl[ = 1 l i e in the open unit circle in the complex plane. Conversely, suppose that (a) - (c) above hold. Then the proof that these imply s t a b i l i t y along the pass is identical to that of corollary 3.3.11 and is hence omitted. • The major conclusion to be drawn from this result is that any one of numerous tests for BIBO stability of 2D lineal" systems described by the Roesser model can be applied to the linear repetitive process of example 2.3.8. In particular, regard the model of this example as a 2D linear system described by the Roesser model with 2D transfer-function matrix (3.137) which is assumed to have no nonessential singularities of the second kind. Further, define the following two variable polynomial in terms of this transfer-function matrix I n -z 1 ~

-ZlAo

p(zl,z 2) =

(3.143) -z2C

I m -z2D 1

Then, for example, the following 2D s t a b i l i t y tests are applicable to example 2.3.8. Corollary3.4.4:

The extended linear repetitive process

S(Ea,Wa,La)a>aogenerated

by the model of example 2.3.8 with a > ao is stable along the pass i f , and only i f , p(zl,z2) # O, ¥ [ZlJ ~ 1, /z21 < 1 CpEollary 3.4.5:

The extended linear repetitive process

(3.144)

S(Ea,Wa,La)a>aogenerated

by the model of example 2.3.8 with a > ao is stable along the pass i f , and only i f , p(Zl,O ) # O, ¥ [Zl[ 5 1

(3.145)

p(zl,z2) # O, V IZll = 1, Iz2l 5 1

(3.146)

and Corollaw3.4.6:

The extended linear repetitive process

S(Ea,Wa,La)a>aogenerated

by the model of example 2.3.8 with a ~ a o is stable along the pass i f , and only i f there exists a,b with [al < 1, [b[ = 1, such that

63 (a)

p(a,z2) ¢ O, V [z2] ~ 1

(3.147)

(b)

p(Zl,b ) ¢ O, V [Zll ~ 1

(3.148)

(c)

p(Zl,Z2) ~ O, V IZll = 1, lz2[ : 1

(3.149)

and These and other conditions arising from theorem 3.4.4 are considered again in chapter 4 where the general subject is the development of computationally feasible stability tests. Notes and References The results up to and iacluding theorem 3.3.5 are based on the original work of Owens (1977) which was extended by Rogers (1987). For the necessary functional analysis see, for example, Taylor (1958). Theorems 3.3.7 and 3.3.8, which express s t a b i l i t y along the pass in terms of the 2D transfer-function matrix, are due to Rogers and Dwens (1989a,1990a) Section 3.4 has evolved from the york of Boland and Owens (1980), where the 2D z-transform and theorem 3.4.1 are due to Shanks, T r e i t e l and Justice (1972), theorem 3.4.2 is due to Huang (1972), and corollary 3.4.6 is from S t r i n t z i s (1977).

CHAPTER4 GRAPHICALAND ALGEBRAICSTABILITYTEST,$ This chapter develops computationally f e a s i b l e t e s t s for s t a b i l i t y along the pass of the d i f f e r e n t i a l and d i s c r e t e processes of examples 2.3.3 and 2.3.7 from theorems 3.3.7 and 3.3.8 respectively. The end product is two systematic t e s t procedures in each case. These procedures are also compared from an applications standpoint with p a r t i c u l a r emphasis on CAD (Computer Aided Design) aspects. The equivalence developed in section 3.4 between BIBO s t a b i l i t y of 2D linear systems described by the Roesser model and s t a b i l i t y along the pass of example 2.3.8 (the d i s c r e t e unit memory case) is considered in depth from a s t a b i l i t y a n a l y s i s / c o n t r o l l e r design standpoint. Finally, the application of r e s u l t s from the s t a b i l i t y theory of delay d i f f e r e n t i a l systems to example 2.3.4 (the d i f f e r e n t i a l unit memory case) is analysed from the same standpoint. 4.1

Asymptotic S t a b i l i t y Consider the extended linear r e p e t i t i v e process

S(Ea,Va,La)a>aogenerated by the

model of example 2.3.3 or 2.3.7. Then, by (a) of theorem 3.3.7 or 3.3.8 as appropriate, t h i s process is asymptotically stable f o r a l l a ~ a o i f , and only i f , a l l eigenvalues of the N×N block companion matrix

0 O=

Im

0]

0

0

Im

DM

D2

D1

(4.i)

have modulus s t r i c t l y less than unity. Note also that t h i s condition is necessary for s t a b i l i t y along the pass and hence no f u r t h e r t e s t s are required i f i t does not hold. Further, suppose that the elements of Dj, 1 < j < M, are known numerically. Then the obvious CAD orientated t e s t is to simply compute the eigenvalues of D and display them r e l a t i v e to the unit c i r c l e in the complex plane. Alternatively, write pD(Z) = t z % =

DI

aNzN+ a N_izN-1 + . . . + alz + a o

where the c o e f f i c i e n t s are r e a l scalars with a N = 1. 3.3.8 is equivalent to

(4.2) Then (a) of theorem 3.3.7 or

65

pD(Z) ~ O, V Izl > 1

(4.3)

i . e . a l l roots of pD(z) must l i e in the open unit c i r c l e in the complex plane.

This

is just the standard s t a b i l i t y r e s u l t from d i s c r e t e conventional linear systems theory and hence i t can be tested by applying any one of numerous well established t e s t s which avoid the need to compute the roots of pD(Z). At t h i s stage, however, only the Schnr-Cohn matrix t e s t is considered since, as shown in l a t e r sections, i t can also be used to develop t e s t s for other conditions of theorems 3.3.7 and 3.3.8. An a l t e r n a t i v e , the so-called Jury/Marden table t e s t , will be introduced in section 4.3 where i t will play a p a r t i c u l a r role in the development of one possible t e s t for (c) of theorem 3.3.8. The Schur-Cohu matrix t e s t , see the cited reference f o r the relevant background, converts the problem of determining whether or not a l l roots of pD(z) l i e inside the unit c i r c l e of the complex plane to one of determining whether or not a real symmetric matrix constructed from i t s c o e f f i c i e n t s is p o s i t i v e d e f i n i t e . In p a r t i c u l a r , suppose that the NxN symmetric matrix H = {hii ) where i hij = k~l(aN_i+kaN_j+k ai_kaj_k ), i ~ j (4.4) is constructed from the c o e f f i c i e n t s of pD(z).

Then i t can be shown that (a) of

theorem 3.3.7 or 3.3.8 holds i f , and only i f , H of (4.4) is p o s i t i v e d e f i n i t e . Further, t h i s new condition can be tested by any one of numerous equivalent c r i t e r i a . Hence, f o r example, the following r e s u l t c o n s t i t u t e s a t e s t f o r (a) of theorem 3.3.7 or 3.3.8. Lemma 4.1.1:

Consider the extended linear r e p e t i t i v e process

generated by the model of example 2.3.3 or 2.3.7 with a > a o.

S(Ea,Wa,La)a>ao_ Then (a) of theorem

3.3.7 or 3.3.8, as appropriate, for s t a b i l i t y along the pass holds i f , and only i f , a l l principal minors of the Schur-Cohn matrix (4.4) are p o s i t i v e . • Application of the t e s t of lemma 4.1.1, or any other one based on pD(Z) written in the form (4.2), is straightforward, given the c o e f f i c i e n t s . Clearly, however, obtaining these is not a p a r t i c u l a r l y f e a s i b l e proposition from a CAD standpoint. Further, these t e s t s do not provide easily used measures of r e l a t i v e s t a b i l i t y and/or performance indicators. Hence the major remit of such t e s t s is c l e a r l y in low order synthesis type problems where some, or a l l , of the elements of the matrices Dj, 1 < j < M, are design parameters. 4.2

S t a b i l i t y Along The Pass - The D i f f e r e n t i a l Case Suppose that condition (a) of theorem 3.3.7 f o r s t a b i l i t y along the pass of the extended linear r e p e t i t i v e process generated by the model of example 2.3.3 with

66 a > ao holds and consider the condition l i s t e d under (b).

Note also that t h i s

condition is necessary for s t a b i l i t y along the pass and hence no further t e s t s are required i f it does not hold. Further, suppose that the elements of A are known numerically. Then the obvious CAD orientated t e s t is, as in the case of (a), to compute the eigenvalues of A and display them r e l a t i v e to the open l e f t - h a l f of the complex plane. Alternatively, write PA(S) = IsI n - A[ = an sn + an. lsn-1 + ... + als + a O

(4.5)

where the coefficients are real scalars with a n = 1. equivalent to pA(s) ~ O,

Then (b) of theorem 3.3.7 is

V Re{s} > 0

(4.6)

i . e . all roots of PA(S) l i e in the open l e f t - h a l f of the complex pIane.

Further,

(4.6) can be tested without computing the roots of pA(s) by employing the classical Routh array. Application of the Routh array t e s t to (4.6) is straightforward, given the coefficients in (4.5). Clearly, however, obtaining these c o e f f i c i e n t s is not a p a r t i c u l a r l y feasible proposition from a CAD standpoint. Further, t h i s t e s t does not provide easily used measures of r e l a t i v e s t a b i l i t y and/or performance indicators. Fence, as with the t e s t s based on pD(Z) written in the form of (4.2) for (a) of theorem 3.3.7, the major remit of this t e s t is clearly in lo~ order synthesis type problems ~here some, or a l l , of the elements of A are design parameters. Suppose no~ that (a) and (b) of theorem 3.3.7 hold and then the p a r t i c u l a r example under consideration is stable along the pass if, and only i f , condition (c) holds. Further, consider again the interpass transfer-function matrix G(s) of (2.90), i . e .

0 G(s) =

0 ~(s)

Im

0 0 G2(s)

Im

] (4.7)

~l(s)

where

Gj(s) = C(sI n - A) -1 Bj_ 1 + Dj, 1 < j _<M

(4.8)

and set s = i~. Then i t follows immediately that this condition is equivalent to the requirement that the continuous curves, or c h a r a c t e r i s t i c loci, generated by the eigenvalues z j ( i ~ ) , 1 _< j < N, of (4.7)-(4.8) l i e e n t i r e l y within the unit c i r c l e in

67 the complex plane for a l l real z > O. Consequently t e s t i n g of this condition reduces to the evaluation and representation of these loci relative to the unit circle in the complex plane, h task which, with the additional simple operation of superimposing the unit circle onto the resulting plots, can be undertaken using standard CAD software for the derived system LD(A,B,C,Do). Note:

The zj(iv), 1 < j < N, are termed the repetitive system characteristic loci

to distinguish them from those associated with the transfer-function matrix, Go(S), of LD(A,B,C,Do). To develop an alternative t e s t for (c) of theorem 3.3.7 to that given above, f i r s t write p(z,s) = 0 where p(z,s):= IzI N- G(s) I (4.9) in the form z N + CN_l(S)ZN-1 + . . .

+ Cl(S)Z + Co(S) = 0

(4.10)

with coefficients Co(S), Cl(S), . . . . , CN.l(S ) which are rational functions in s. Further, l e t aN(s ) denote the least commondenominator of the c j ( s ) , j = O,I,...,N-1. Then (4.10) can be ~ritten as aN(s)z N + aN_l(S)ZN-1 + ... + al(s)z + ao(S ) = 0

(4.11)

where the coefficients no(S), al(s), . . . , aN(s ) are real polynomials in s.

In ~hich

case it follows immediately that (c) of theorem 3.3.7 holds i f , and only i f , the, assumed irreducible, polynomial p(z) = aN(s)zN + aN_l(s)zN-1 + ... + al(S)Z + ao(S ) (4.12) satisfies p(z) # o, v s: Re{s} = O, lzl ~ 1 (4.13) In order to develop a t e s t for (4.13), f i r s t suppose that the NxN Schur-Cohn matrix H = {hij } is constructed from i t s coefficients where i hij = k~l (aN-i+kaN-j+k- a i ' k a j ' k ) ' i < j

hij = hji , i > j

(4.14)

Then each element in H is a polynomial in s and/or i t s complex conjugate ~ and it can be shown that (4.13) holds i f , and only i f , the Hermitian polynomial matrix n(s) ~ H is positive definite for a l l s:Re{s} = O. Equivalently, for a l l constant complex vectors U # 0 of a unitary N-dimensional vector space, U* H(s)U > O, Vs:Re{s} = 0 (4.15) where * denotes the complex conjugate transpose.

68

A Hermitian polynomial matrix H(s) satisfying (4.15) is said to be axis positive and the following r e s u l t gives necessary and s u f f i c i e n t conditions f o r the existence of t h i s property. Theorem 4.2.1: only i f ,

The NxN Hermitian polynomial matrix H(s) is axis positive i f , and

(a)

H(O) > 0

(4.16)

(b)

]H(s)] > O, Vs:Re{s) : 0

(4.17)

and

Proof: I f H(s) > O, i . e . (4.15) holds, then clearly (4.16) and (4.17) hold. Conversely, suppose that (4.16) and (4.17) hold. Then, using (4.17), ]H(i~)l # O, V ~, i . e . the eigenvalues ~k(~), 1 < k < N, of H(i~) are non-zero for a l l ~.

Further, the ]k(~) are real continuous functions of a which are positive at

= 0 by (4.16) and hence positive V~. • The f i r s t condition of theorem 4.2.1 requires that the real symmetric matrix H(O) is positive d e f i n i t e . Hence i t can be tested by applying any one of numerous equivalent t e s t s . For example, this condition holds i f , and only i f , a l l principal minors of H(O) are positive. To develop a t e s t for (b) of theorem 4.2.1, set s = i~ and note that the determinant of a Hermitian matrix is real. Consequently q ( 2 ) : = [H(i~)[ must be a real polynomial of the form r

q(~2) = k~O q2kW2k

(4.18)

and i t follows immediately that (4.17) holds i f , and only i f , q(~2) has the so-called positive realness property q ( 2 ) > O, V w > 0 (4.19) Further, (4.19) can be expressed in terms of the roots of q ( 2 ) by means of the following easily proven result. Lemma 4.2.1: The polynomial q ( 2 ) s a t i s f i e s (4.19) i f , and only i f , i t has no positive real roots and q(w2) > 0 for some ~ > O. Note 1:

In the t r i v i a l case of r = O, lemma 4.2.1 reduces to q ( 2 ) ~ qo > O.

Note 2:

q ( 2 ) > 0 for some # > 0 can be replaced by e i t h e r qo > 0 or q2r > 0 [i.e.

q(+®) > O] which is easy to t e s t .

In particular, (4.19) is violated i f qo < 0 and

no f u r t h e r t e s t s are required. At t h i s stage, the classical Descartes rule of sign yields the following preliminary results concerning the positive realness property (4.19).

69

Lemma 4.2.2:

With qo > O, a necessary condition f o r q(w2) to have no positive real

roots is that there is an even number of changes of sign in the c o e f f i c i e n t s q2k when arranged in decending order.

•

I f the number of sign variations is zero, i . e . a l l c o e f f i c i e n t s are positive, the above r e s u l t is s u f f i c i e n t and is a special case of the following lemma. Lemma 4.2.3: A sufficient condition for q(~2) to have no positive real roots is that the coefficients q2k s a t i s f y qo > O, q2k ~ O, 1 < k ~ r

(4.20)

To t e s t lemma 4.2.1 in the general case, requires a means of determining the location of the real roots (if any) of the real polynomial q ( 2 ) . This is a well researched problem and numerous solutions exist one of which, for example, uses the concept of a matrix inner. Here, however, only the t e s t detailed below is used since it is known to be computationally less expensive to implement for numerical examples. First note that q(2) of (4.18) is a real even order polynomial and therefore has 2r roots symmetrically distributed with respect to both the real and imaginary axes of the complex plane. Suppose also that the polynomial r (_l)kq2k~2 k q(i~) = E (4.21) k=O is constructed from (4.18). Then, since (4.21) represents an anti-clockwise rotation of 900 , the symmetry discussed above is preserved but real roots, i f any, of q ( 2 ) become purely imaginary roots of q(i~). Consequently i f q(i~) has r roots with positive real parts then q(w2) has no roots with positive real parts and hence lemma 4.2.1 holds under the assumption that qo > O. To t e s t this new condition, replace ~2 b y ~ and form the so-called, see the cited reference for the necessary background, modified Routh array

ROW

2

2r + 1

2r

(_l)rq2 r

(" 1)r" tq2r- 2

. . . .

-q2

2r-1

(_l)rrq2r

(" 1)r" l(r" 1)q2r- 2 . . . .

"q2

N°

qo

qo

(4.22)

70 f o r (4.21) where (i) the e n t r i e s in row 2 are given by the c o e f f i c i e n t s of the d e r i v a t i v e of q(i~); and ( i i ) the e n t r i e s in row j , 3 _< j _< r + 1 are constructed as f o r the standard Routh array. Now l e t Var[(-1)rq2r, (- 1)rrq2r , . . . . ,qo] denote the number of changes of sign in the sequence [ ( - 1 ) r q 2 r , ( - 1 ) r - l r q 2 r . . . . . 'qo]"

Then RouthWs r e s u l t can be invoked to show

t h a t q(ix) must have Q = Yar[(-1)rq2r,

(-1)rrq2r,...,qo ]

(4.23)

roots with p o s i t i v e real p a r t s in t h i s case. Rence i t follows immediately t h a t q ( 2 ) has no p o s i t i v e r e a l roots i f , and only i f , t h e r e are r changes of sign in the f i r s t column of the a s s o c i a t e d modified Routh array (4.22), i . e . r = Y a r [ ( - 1 ) r q 2 r , ( - 1 ) r r q 2 r , . . . . qo] As an example of the use of ( 4 . 2 2 ) - ( 4 . 2 4 ) ,

(4.24) consider the case when

q(~2) = us_ 3~6 + 2~4 + ~2 + I

(4.25)

q(i~) = ~8 + 3~6 + 2~4_ 2 + 1

(4.26)

Then and the array of (4.22) in t h i s case is ROI~

s J

3 2

o

1

4 0.75 3.66 -0.64 11.12 0.51 -22.4 1

2 4 9 -0.75 1 -6.33 8 0.545 1 0 -0.61 1 0 3

-1 -1 1 0 0

(4.27)

Xere Var[1,4,0.75,3.66,-0.64,11.12,0.51,-22.4,1] = 4 (4.28) and hence, since r = 4, q ( 2 ) of (4.25) has no p o s i t i v e r e a l roots. F i n a l l y , note t h a t these t e s t s are not s u i t a b l e f o r CAD implementation. ~ence t h e i r major remit is c l e a r l y in low order synthesis problems where some, or a l l , of the elements in Oj(s), 1 < j < M, contain design parameters.

71 At t h i s stage, two a l t e r n a t i v e t e s t s have conditions of theorem 3.3.7. Further, i t is the order of (a) followed by (b) followed by tested does not hold. Suppose also that the the following steps represent an eigenvalue, procedure for t e s t i n g theorem 3.3.7.

been developed f o r each of the three clear that these should be t e s t e d in (e) ~ith termination i f the one j u s t f i r s t t e s t is used in each case. Then or graphically based, systematic

STEP 1: Test the necessary condition of (a) by computing the eigenvalues of the matrix D of (4.1) and displaying them r e l a t i v e to the unit c i r c l e in the complex plane. Stop if t h i s condition does not hold. STEP 2: Test (b) by computing the eigenvalues of the matrix h and displaying them r e l a t i v e to the open l e f t - h a l f of the complex plane. Stop if t h i s necessary condition does not hold. STEP 3: Compute the r e p e t i t i v e system c h a r a c t e r i s t i c loci generated by the eigenvalues of the interpass transfer-function matrix G(s) of (4.7), s = ix, for a l l real ~ > 0 and display them r e l a t i v e to the unit c i r c l e in the complex plane. The s t a b i l i t y along the pass c h a r a c t e r i s t i c s of tile p a r t i c u l a r example under consideration now follow immediately on visual inspection of the r e s u l t i n g plots. Using the above systematic procedure, ttlerefore, (a) - (c) of theorem 3.3.7 can be t e s t e d using t e s t s developed from the 2~ t r a n s f e ~ f u n c t i o n matrix G(s,z). These t e s t s are suitable for inclusion in a CAn package and are, in e f f e c t , well known t e s t s from conventional linear systems theory. To i l l u s t r a t e the use of t h i s procedure, consider the unit memory process described, under suitable choice of current pass s t a t e variables, by

ik+l(t)

=

Yk+l(t) =

0 ~ t < a ,

STEP 1:

0

1

0

0

0

1

- 24

- 26

-9

1

0

0

0

1

0

0

0

1

1

0

0

0

1

0

0

0

1

Xk+l(O ) : O,

Xk+l(t) +

2

0

0

0

3

o

0

0

4

Uk+l(t)

Yk(t)

Xk+1 (t)

k > 0

This step is redundant here sittce the matrix D1 = O.

(4.29)

72 STEP 2: h simple calculation yields the eigenvalues of the matrix A as A1 = -2, A2 = -3, A3 = -4. Hence (b) of theorem 3.3.7 holds. STEP 3:

A simple calculation yields that the eigenvalnes of G(s) ~ Gl(S ) are given

by 1

Zl(S) - s + 2 '

1

z2(s) - s + 3 '

1

z3(s) - s + 4

(4.30)

The r e p e t i t i v e system c h a r a c t e r i s t i c loci generated by the elements of (4.30), s = ix, V real ~ > O, are shown relative to the unit c i r c l e in the plots of Figure 4.1. Hence (4.29) is stable along the pass. Suppose now that each condition in theorem 3.3.7 is tested using the second of the t e s t s developed for it e a r l i e r in this section. Then the following steps represent an algebraic, or root clustering, based systematic procedure for testing this result. This serves as an alternative to the eigenvalue based procedure detailed above. STEP 1: Test the necessary condition of (a) by applying an appropriate t e s t from the s t a b i l i t y theory of discrete conventional linear systems. For example, the Schur-Cohn matrix t e s t , as defined by (4.4), could be used. Stop if this condition does not hold. STEP 2: Test (b) by applying the Routh array to the c h a r a c t e r i s t i c polynomial, ph(s) of (4.6), of the derived system LD[A,B,C,Do]. Stop if this necessary condition does not hold. STEP 3: Construct p(z) of (4.12) and hence the Schur-Cohn matrix H(s) of (4.14). Test if ~(0) is positive definite, and stop if this is not the case. STEP 4: Construct q(w2) = [I[(iw) l , and stop if lemma 4.2.2 does not hold for this polynomial. Then apply lemma 4.2.3 and stop if this condition holds, since the example under consideration is stable along the pass. I f , however, it does not hold, the final step is to test the modified Routh array condition of (4.24). As noted when each of them was considered separately, the t e s t s employed in the above procedure are not suitable for CAD implementation. Hence the major remit of this procedure is clearly in low order synthesis problems where some, or a l l , of the matrices of the example under consideration contain design parameters. To i l l u s t r a t e i t s application in such a case, consider the following SISO unit memory example where a 1 and a 2 are positive real scalars:

73

Im

1.Re

Im

Re

Im

Re

FIGURE 4.1

74

ol i]

Xk÷i(t) =

0

0

1

-3

Xk+l(t) +

-

!]

t~+ 1(t)

÷

Yk(t)

Yk+l(t) = [a I a 2 0 ] X k + l ( t ) 0 < t < a ,

Xk+l(O ) = O,

k > 0

(4.31)

STEP 1:

This s t e p i s redundant here s i n c e D1 = O.

STEP 2:

The c h a r a c t e r i s t i c polynomial of LD(A,B,C,Do) i s given by ph(s) = s 3 + 3s 2 + 3s + 1

(4.32)

A p p l i c a t i o n of t h e Routh a r r a y now shows t h a t LD(A,B,C,~o) i s s t a b l e and hence (b) of theorem 3 . 3 . 7 h o l d s . STEP 3: p(z) = (s 3 + 3s 2 + 3s + 1 ) z -

(a2s + a l )

(4.33)

H(s) = (s 3 + 3s 2 + 3s + 1)(§ 3 + 3§ 2 + 3 s + 1) -

(a2s + a l ) ( a 2 s + a l )

(4.34)

n(o) : 1 - a~

(4.35)

and ~(0) > 0 i f , and only i f , aI < I

STEP 4:

q(2)

(4.36) = x6 + 3 4 + ( 3 -

a~)w2 + (1 - a~)

(4.37)

F u r t h e r lemma 4 . 2 . 2 holds f o r any choice of a 2 and a 1 < 1.

Using lemma 4 . 2 . 3 , the

f o l l o w i n g c o n s t r a i n t s on a 1 and a 2 are a s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y

along

the pass. a 2 < 43 ,

a1 < 1

(4.38)

I f the c o n s t r a i n t on a 2 in (4.38) i s v i o l a t e d , then (4.24) must be t e s t e d and two options exist.

The f i r s t

of t h e s e i s to c o n s t r u c t the a r r a y with a t and a 2

a r b i t r a r y and then attempt t o o b t a i n c o n d i t i o n s on a 1 and a 2 uhich g i v e t h r e e changes of sign in t h e f i r s t

column and hence s t a b i l i t y

along t h e pass.

A l t e r n a t i v e l y , i f p a r t i c u l a r values of a 1 and a 2 a r e given, c o n s t r u c t t h e a r r a y and count the number of changes of sign in t h e f i r s t

column.

As an example of t h i s 1 second o p t i o n , (4.39) below shows t h e a r r a y f o r t h e choice of a 1 = - - and a 2 = 243.

42

75

~5 ~3 ~2

wO

-1 -3 1 24 5.56 8.34 0.5

3 6 6 10.5 0.5

9 9 0.5

0.5

(4.39)

Here there is only one change of sign in the f i r s t column and hence this particular case is unstable along the pass. The f i r s t of the two systematic t e s t procedures developed in this section uses, in e f f e c t , 'Nyquist like' t e s t s from the s t a b i l i t y analysis of LD(A,B,C,Do). In this l a t t e r context~ a major advantage of such t e s t s is that they provide easily used r e l a t i v e s t a b i l i t y and/or performance indicators such as gain and phase margins. These are extensively used in c o n t r o l l e r design (particularly in the SISO case) and i t would obviously be desirable to have similar measures available in the r e p e t i t i v e systems case. Extensive studies on a number of industrial examples have concluded that appropriately defined r e l a t i v e s t a b i l i t y and/or performance indicators in the s p i r i t of gain and phase margins should have a constructive role to play in c o n t r o l l e r design for r e p e t i t i v e processes. These studies have also shown, however, that computable information concerning the rate of approach of the output sequence to the limit p r o f i l e (see definition 3.1.3 and theorem 3.1.2) is at least of equal importance. Further, i t is clear that such information is not available from these 'Nyquist like' t e s t s . The only option being to inspect the r e s u l t of a closed-loop simulation study with the consequent prospect of a heavy computational load. The problems discussed above are considered again in section 4.5 and in the next chapter where alternative simulation-based t e s t s f o r s t a b i l i t y along the pass are developed from suitably ~ell behaved plant step response data which is assumed to be available. This leads to s u f f i c i e n t , but not necessary, s t a b i l i t y t e s t s which produce, at no extra cost, computable information concerning the rate of approach to the limit p r o f i l e in one special case of major p r a c t i c a l i n t e r e s t . Finally, the use of t h i s information in the formulation of c o n t r o l l e r design algorithms is considered in chapter 6.

76 4.3

Stability Along the Pass - The Oiscrete Case This section considers the testing of theorem 3.3.8 for s t a b i l i t y along the pass of the extended linear repetitive process generated by the model of

S(Ea,Wa,La)a>ao

example 2.3.7. Here, as in the case of theorem 3.3.7 for the d i f f e r e n t i a l process, the f i r s t step is to t e s t the condition listed under (a). This can be undertaken using either of the tests developed in section 4.1. As another alternative to these, the so-called Jury/~arden table t e s t is considered below since, see later in t h i s section, i t plays a particular role in the development of one possible test for the condition listed under (c). The Jury/Marden table t e s t , see again the cited reference for the relevant background theory, determines the location of the roots of pD(z) of (4.2) relative to the unit circle using a 'Routh like' array expressed in terms of the determinants of 2x2 matrices and proceeds as follows. First construct the so-called conjugate polynomial of pD(z) as ,

N

PD(z) = zNpD(z-1) : k~o akzN-k

(4.40)

and then generate the sequence of polynomials gj(z) : N~j a~J)zk k=o where go(Z) = pB(z)

(4.41)

(4.42)

and gj+l (z) = a~J)gj (z) - a(J)g*(z)N-j j j = 0 , 1 , . . . , N- 1 (4.43) This yields the follouing recursive relationship for the coefficients of gj+l' j = 0,1 . . . . ,N - 1

4 Further, denote the constant term, a~j),'" of gj+l(Z) by ~j+l' i.e.

and define the scalars Pk' 1 < k < N, as Pk = ~1 ~2 "'" ~k Suppose also that none of the Pk are zero and let V of them be negative.

(4.46) Then it

can be shown that the following result holds which serves as an alternative to lemma 4.1.1 for (a) of theorem 3.3.8 (or (a) of theorem 3.3.7)).

77 Lemma 4 . 3 . 1 :

Consider t h e extended l i n e a r r e p e t i t i v e

g e n e r a t e d by t h e model of example 2 . 3 . 7 u i t h a > a o. stability

process

S(Ea,Wa,La)a>ao

Then (a) of theorem 3 . 3 . 8 f o r

a l o n g t h e pass holds i f , and o n l y i f , V = N.

Given a p a r t i c u l a r

•

pV(Z), t h e f o l l o u i n g s t e p s r e p r e s e n t a s y s t e m a t i c procedure

f o r computing V. STEP 1:

C o n s t r u c t t h e f o l l o w i n g s o - c a l l e d Jury/Marden t a b l e from t h e c o e f f i c i e n t s

a o , a l , . . - , a N.

RO~

2N-1 2N 2N+l

z°

zI

z2

.

.

.

. z k.

.

zN-I

zN

ao

a1

a2

...

aN_ k

.,.

aN-1

aN

aN

aN_ 1

aN- 2

...

ak

...

a1

ao

~1 = bo

bl

b2

. . . . . . . .

bN- 1

bN_ 1

bN_ 2

bN_ 3

. . . . . . . .

bo

~2 = Co

Cl

c2

. . . . .

CN-2

CN_2

CN_3

CN.4

. . . . .

co

~N- I

=

ro

rI

rl ro (4.47)

¢~N = t o

Here t h e e n t r i e s i n row 2k+2 c o n s i s t of t h e e n t r i e s i n row 2k+1 w r i t t e n i n r e v e r s e order k = 0,1,2,... and ao

aN_ k

bk =

bo

bN. 1. k

bN- 1

bk

Ck= aN

ro

ak r1

(4.48)

t O -r1

ro

78 STEP 2:

The numbers ~l,~2,...,~N are now given as the f i r s t entries in rows 3 , 5 . . . ,

2N+1 of (4.47).

I t now follows immediately that a l l of the Pk of (4.46) are

negative, and hence lemma 4.3.1 holds, i f , and only i f , b o < O, c o > 0 , . . . ,

r o > O, t o > 0

(4.49)

The following special cases may occur in applying the above t a b l e t e s t to a given PD(z) (a)

Pk # O, but gj+l(Z) e 0 for some k < N

(h)

o, b.t 5k+1 = a ~ k+l) = (a~k)) 2 -

(a~k~) 2 = 0

(4.50)

In either case, the modification detailed in the cited reference should be used to continue the t e s t . Note also that i t may be necessary to use t h i s modification more than once f o r a p a r t i c u l a r pD(z) Suppose now that (a) of theorem 3.3.8 holds and consider the condition l i s t e d under (b). Note also that t h i s condition is necessary for s t a b i l i t y along the pass and hence no further t e s t s are required i f it does not hold. Further, suppose that the elements of ~ are known numerically. Then the obvious CAD orientated test is to compute the eigenvalues of ~ and display them relative to the unit circle in the complex plane. Alternatively, write P@(z I) = IZlIn - ~I n-1 : bnzln + bn_lZ 1 + . . . + blZ 1 + b o

where the c o e f f i c i e n t s are real scalars with bn = 1.

(4.51) Then (b) of theorem 3.3.8 is

equivalent to pa(zl) ¢ O, V JzlJ ~ i

(4.52)

i . e . a l l roots of p~(zl) l i e in the open unit c i r c l e in the complex plane.

Further,

(4.52) can be tested without computing the roots of p~(zl) by employing any one of numerous t e s t s such as the Jury/Marden t a b l e t e s t outlined above. Application of, for example, the Jury/Marden table t e s t to (4.52) is straightforward, from a CAD standpoint, given the c o e f f i c i e n t s . Clearly, however, obtaining these c o e f f i c i e n t s is not a p a r t i c u l a r l y f e a s i b l e proposition from the same standpoint. Further, t h i s t e s t does not provide easily used measures of r e l a t i v e s t a b i l i t y and/or performance indicators. Hence, as with the t e s t s based on pD(z) written in the form of (4.2) f o r (a) of theorem 3.3.7 or 3.3.8, the major remit of t h i s t e s t is clearly in low order synthesis type problems where some, or a l l , of the elements of ~ are design parameters.

79 Suppose now that (a) and (b) of theorem 3.3.8 hold and then the p a r t i c u l a r example under consideration is stable along the pass if, and only i f , condition (c) holds. Further, consider the interpass transfer-function matrix G(Zl) of (2.102), i.e. 0

G(zl) =

Im

0

0

0

Im

(4.5a)

where Gj(z 1) : C(ZlI n - ~)-lhj_ 1 + Dj, 1 < j < M and consider the case of [Zl[ = 1.

(4.54)

Then i t follows immediately that this condition

is equivalent to the requirement that the continuous curves, or c h a r a c t e r i s t i c loci, generated by the eigenvalues z j ( z l ) , 1 < j < M, of (4.53) - (4.54) l i e e n t i r e l y within the unit c i r c l e in the complex plane for all [Zl[ = I.

Consequently the

testing of this condition reduces to the evaluation and representation of these so-called r e p e t i t i v e system characteristic loci r e l a t i v e to the unit c i r c l e in the complex plane. A task which, with the simple additional operation of superimposing the unit c i r c l e onto the resulting plots, can be undertaken using standard CAD software for the derived system LD(~,A,C,Do). To develop an alternative t e s t for (c) of theorem 3.3.8 to that given above, f i r s t write p(z,zl) = 0 with

p(z,zl):=

IzlN - G(Zl) I

in the form z N + CN_l(Zl)ZN-1 +...+ c l ( z l ) z + Co(Zl) : 0

(4.55) (4.56)

where the c o e f f i c i e n t s Co(Z1) , e l ( z 1 ) , . . . , CN_l(Zl) are rational functions in z 1. Further, let aN(zl) denote the least common denominator of the c j ( z l ) , j = 0 , 1 , . . . , N - 1.

Then ~4.56) can be written as

aN(Z1) z N + aN_l(Zl)ZN-1 + . . . . + al(Zl)Z + ao(Zl) = 0 where the coefficients ao(Zl) , al(Zl) , . . . . aN(z1) are real polynomials in z 1.

(4.57) In

which case i t follows immediately that (c) of theorem 3.3.8 holds i f , and only i f , the, assumed irreducible, polynomial p(z) = aN(zl)zN + aN_l(Zl)Z N'I + . . . + al(Zl)Z + ao(Zl) satisfies p(z) # O, VZl: tzll : I, lzl > 1

(4.58) (4.59)

80 To develop a t e s t f o r (4.59), follow the analysis associated with (4.13) in the d i f f e r e n t i a l case and suppose that the NxN Schur-Cohn matrix H = {hij } of (4.14) is constructed from i t s c o e f f i c i e n t s . Then in this case each element of ~ is a polynomial in z 1 and/or i t s complex conjugate ~1" Further, i t can be shown that (4.59) holds i f , and only i f , the Hermitian polynomial matrix H(Zl) ~ H is positive d e f i n i t e V Zl: lZlt = t.

Equivalently, f o r a l l constant complex vectors U ¢ 0 of a

unitar~ N-dimensional vector space, U H(zl) V > O, V zl: Izlt = 1

(4.60)

where * again denotes the complex conjugate transpose. k Hermitian polynomial matrix H(Zl) satisfying (4.60) is said to be c i r c l e positive where this term can be regarded as the dual of the concept of axis p o s i t i v i t y introduced in section 4.2. Further, the following r e s u l t , which is proved using the same arguments as those used in establishing theorem 4.2.1, gives necessary and sufficient conditions for the existence of this property. Theorem 4.3.1: only i f , (a)

The Hermitian polynomial matrix H(Zl) is c i r c l e positive i f , and

t(1) > 0

(4.61)

IH(=I) I > o, V zl: I=11 = 1

(4im62)

and

(b)

The f i r s t condition of theorem 4.3.1 requires that the real symmetric matrix H(1) is positive d e f i n i t e . Hence, for example, i t holds i f , and only i f , a l l principal minors of tI(1) are positive. Alternatively, any one of numerous equivalent t e s t s could be employed. As a f i r s t step in developing a t e s t for (4.62), note that ~1 = Zll on the unit c i r c l e and that the determinant of an Hermitian matrix is r e a l . f ( z l ) : = I]l(zl) 1 must have the form q

•

Hence

•

f(Zl) :

V. cj(z~ + z13 ) j=O where the coefficients Co,C1,... , Cq are real scalars. i f , and only i f , f ( z l ) > O, V IZl[ = 1

(4.63) Consequently (4.62) holds (4.64)

and the following result now expresses this ne~ condition in terms of the roots of

f(zl). Lemma 4.3.2:

The polynomial f(Zl) s a t i s f i e s (4.64) if, and only i f , i t has no roots

on the unit c i r c l e and f ( z l ) > 0 for some [Zl[ = 1.

81

Proof:

The absence of roots on the unit c i r c l e ensures t h a t f(Zl) is nonzero on

this contour and f(Zl) > 0 f o r some lZl[ = 1 ensures that i t also p o s i t i v e . Note also that t h i s lemma reduces to f ( z l ) =- c o > 0 in the t r i v i a l case of q = 0 and no further analysis is required. II The location of the roots of f ( z l ) with respect to the unit c i r c l e can now be determined as follows.

F i r s t construct the polynomial W(Zl) as

W(Zl) = z q f ( Z l ) =

q cj(zq+J + z? - j ) (4.65) j=O and note that W(Zl) has the same number of roots on the unit c i r c l e as f ( z l ) and of the same m u l t i p l i c i t i e s . 4.3.2. Lemma 4.3.3:

Hence the following serves as an a l t e r n a t i v e to lemma

The polynomial f ( z l ) s a t i s f i e s (4.64) i f , and only i f , W(Zl) has no

roots on the unit c i r c l e and W(Zl) > 0 f o r some tZl] = 1. I

Note also that W(Zl) > 0 for some [Zl[ = 1 can be replaced by either W(1) > 0 or W(-1) > O, i . e . q ~] cj > 0 j--O

(4.66)

or O

(-1) q j~O(-1)J cj > 0

(4.67)

In p a r t i c u l a r , (4.66) and (4.67) are simple necessary conditions which should be tested before proceeding further. h polynomial of the form ~(zl) is one example of a so-called s e l f - i n v e r s i v e , or reciprocal, polynomial where t h i s term denotes the f a c t that the reciprocals of roots inside the unit c i r c l e are also roots. Suppose also that V(W) denotes the number of roots of ~(Zl) inside the unit c i r c l e . Then by the s e l f - i n v e r s i v e property ~(Zl) also

has V(W) roots outside the unit c i r c l e and, since i t s degree is 2q, p(W) = 2 ( q - V(~)) (4.68) roots on the unit c i r c l e . Hence no roots of W(Zl) on the unit c i r c l e requires that V(V) = q (4.69) Farther, V(W) can be computed by constructing the Jury/garden table of (4.47) for ~(Zl) and counting the number of positive products of the form (4.46). Note also that since V(Zl) is s e l f - i n v e r s i v e , the polynomial corresponding to gl(z ) in the sequence (4.43) will be identically zero. Consequently in t h i s p a r t i c u l a r application the f i r s t two rows in the table (4.47) must be constructed as per the

82 modification noted previously for the special cases l i s t e d under (a) and (b) immediately a f t e r (4.49). To i l l u s t r a t e the use of c i r c l e p o s i t i v i t y , as expressed by theorem 4.3.1, in the testing of (c) of theorem 3.3.8 consider the case when ( a f t e r clearing fractions) p(z) = (2 z~ + 10 z 1 + 12)z + (z~ + 5 z 1 + 6) (4.70) Then H(Zl) = 18 z~ + 105 z 1 + 186 + 105 z l l + 18 z12

(4.71)

for IZll = 1 and (4.61) holds since H(1) > O. Further, ~(Zl)

=

18 z~ + 105 z? + 186 z~ + 105 z I + 18

(4.72)

and (4.66) - (4.67) hold. The Jury/Marden table for (4.72) is shown below uhere the necessary modification has been used to construct the f i r s t two rows.

RfiW

0

1 2 3

72 105 51 = -5841

315 373 -16380

373 315 -6291

4 5

-6291 52 = -5.46x106

-16380 -7.37x106

-5841

6 7

-7.37×106 53 = 24.5~106

-5.46×106

mere 51 and $2 are negative and 53 is positive.

105 72

(4.73) Hence V(W) = 2 and therefore (c) of

theorem 3.3.8 holds in this particular case. At this stage, two alternative t e s t s have been developed for each of the three conditions of theorem 3.3.8. Further, as per theorem 3.3.7 in the d i f f e r e n t i a l case of section 4.2, these should be tested in the order of (a) followed by (b) followed by (c) with termination if the one just tested does not hold. Suppose also that the f i r s t t e s t is used in each case. Then the following steps represent an eigenvalue, or graphically, based systematic procedure for testing theorem 3.3.8. STEP 1: Test the necessary condition of (a) by computing the eigenvalues of the matrix D of (4.1) and displaying them r e l a t i v e to the unit c i r c l e in the complex plane. Stop i f this condition does not hold. STEP 2: Test (b) by computing the eigenvalues of the matrix ~ and displaying them r e l a t i v e to the unit c i r c l e in the complex plane. Stop i f this necessary condition does not hold.

83 STEP 3: Compute the r e p e t i t i v e system c h a r a c t e r i s t i c loci generated by the eigenvalues of the interpass transfer-function matrix G(zl) of (4.53) for a l l IZll = 1 and display them r e l a t i v e to the unit c i r c l e in the complex plane.

The

s t a b i l i t y along the pass characteristics of the p a r t i c u l a r example under consideration now follow immediately on visual inspection of the resulting plots. Using the above systematic procedure, therefore, (a) - (c) of theorem 3.3.8 can be tested using t e s t s expressed in terms of the 2D transfer-function matrix G(Zl,Z ). (Compare with the f i r s t procedure developed for theorem 3.3.7). These t e s t s are suitable for inclusion in a CAD package and are, in e f f e c t , well known t e s t s from discrete conventional linear systems theory. To i l l u s t r a t e i t s use, consider the unit memory process where Yk+I(P) = Xk+I(P), 0 ~ P < a, k > O, and, under suitable choice of current pass state variables, the dynamics are described by the single equation Xk+I(P + 1)

=

(I m - holhl)Xk+l(e ) + AolUk+I(P) + 7 ImYk(P)

0 ~ P ~ a,

Xk+l(O) = O, k > 0

(4.74)

where ho and h 1 are real constant mxm matrices and 7 is a positive real scalar. Further

~

let A~IA1 have eigenvector matrix T and eigenvalues 1 - 71, 1 - ~ 2 " ' "

1

~m satisfying T-1A~IA1w

diag{1

.

(4.75)

STEP 1:

This step is redundant here since the matrix D1 = O.

STEP 2:

~ = I m- Aolhl in this case and has eigenvalues 7j , 1 < j < m. Hence (b)

of theorem 3.3.8 holds i f , and only i f , max IYjt < 1 (4.76) l~j<m STEP 3: A simple calculation yields that the eigenvalues of G(Zl) ~ Gl(Zl) are given by 7 z j ( z l ) - Z l - YJ ,

1 <j <m

(4.77)

Hence, under (4.76), the process of (4.74) is stable along the pass i f , and only if, the r e p e t i t i v e system characteristic loci generated by (4.77) l i e e n t i r e l y within the unit c i r c l e in the complex plane for all IZll = 1. Equivalently, the unit c i r c l e with centre ~j must l i e e n t i r e l y outside the c i r c l e of radius 7 and centre the origin, 1 < j < m. Suppose now that each condition in theorem 3.3.8 is tested using the second of the t e s t s developed for i t e a r l i e r in this section. Then the following steps represent a root clustering, or algebraically based, systematic procedure for

84

t e s t i n g t h i s r e s u l t . This serves as an a l t e r n a t i v e to the eigenvalue based systematic procedure detailed above. STEP 1: Test the necessary condition of (a) by applying any one of numerous standard t e s t s from d i s c r e t e conventional linear systems theory to #D(z) of (4.2). Stop of t h i s condition does not hQld. Test (b) by applying any one of numerous standard t e s t s from discrete conventional linear systems theory to p~(Zl) of (4.51). Stop i f t h i s necessary

STEP 2:

condition does not hold. Construct p(z) of (4.58) and hence the Schur-Cohn matrix H(Zl) of (4.14).

STEP 3:

Test i f H(1) is p o s i t i v e d e f i n i t e and stop i f t h i s is not the case. STEP 4: Construct f(zl) = Ill(z1) 1 and hence W(Zl) = z~f(zl) of (4.65).

Stop if

(4.66) and (4.67) do not hold f o r t h i s polynomial. STEP 5: Test (4.69) by constructing the Jury/Marden t a b l e f o r W(Zl). As noted when each of them was considered separately, the t e s t s employed in the above procedure are not suitable for CAD implementation. Hence the major remit of t h i s procedure is c l e a r l y in low order synthesis problems where some, or a l l , of the elements of the matrices of the example under consideration are design parameters. The f i r s t of the two systematic t e s t procedures developed in t h i s section uses, in e f f e c t , ~Nyquist l i k e ' t e s t s from the s t a b i l i t y analysis of LD(~,A,C,Do). Hence t h i s procedure mirrors i t s d i f f e r e n t i a l counterpart of the previous section in terms of the production of easily used r e l a t i v e s t a b i l i t y and/or performance indicators. This problem is discussed again in section 4.4 and in the next chapter where a l t e r n a t i v e simulation-based t e s t s for s t a b i l i t y along the pass are developed from suitably well behaved plant step response data which is assumed to be available. Further, use of these, s u f f i c i e n t but not necessary, t e s t s produces, at no extra cost, computable information concerning the rate of approach to the limit p r o f i l e in one special case of major p r a c t i c a l i n t e r e s t . Finally, chapter 6 considers the use of t h i s information in the formulation of c o n t r o l l e r design algorithms. 4.4

Application of 2D Systems S t a b i l i t ~ T e s t s The central result of section 3.4, theorem 3.4.4, s t a t e s that BIBO s t a b i l i t y of 20 linear systems described by the Roesser model of (2.58) is equivalent to s t a b i l i t y along the pass for the discrete unit memory process of example 2.3.8. In t h i s section, the use of t h i s result in the development of s t a b i l i t y t e s t s , and r e l a t e d topics, for such r e p e t i t i v e processes is considered. Return, therefore, to the model of example 2.3.8 and regard i t as a 2D linear system described by the Roesser model with 2D t r a n s f e r - f u n c t i o n a a t r i x of the form (3.137) which is assumed to have no nonessential s i n g u l a r i t i e s of the second kind. Further, see also (3.143), define the following two variable polynomial in terms of t h i s 2D t r a n s f e r - f u n c t i o n matrix

85 In - Zl~

-ZlA o

p(zl,z2) =

(4.78) -z2C

I m - z2D1

Then corollary 3.4.4 is the basic s t a b i l i t y r e s u l t in t h i s context and i t s t e s t i n g in a l l but very simple cases is clearly not a f e a s i b l e proposition. This f a c t has led to the development of a large number of equivalent versions of t h i s r e s u l t , of which c o r o l l a r i e s 3.4.5 and 3.4.6 are examples, with varying claims of increased computational efficiency. Generally, a l l of the r e s u l t i n g t e s t s have been applied to low order SISO problems and the vast majority are root clustering, or algebraically, based, l[ence they can be regarded as a l t e r n a t i v e bases f o r the unit memory version of the second systematic t e s t procedure developed in section 4.3. At t h i s stage, define the so-called augmented plant matrix f o r example 2.3.8 as Ao A=

(4.79) C

DI

Then the following result gives three necessary conditions for stability along the pass expressed as stability in the conventional sense of DI, ~ and A, i.e. all eigenvalues lie in the open unit circle in the complex plane. Lemma4.4.1: Consider the extended linear repetitive process generated by the model of example 2.3.8 with a > ao.

$(Ea~ ,aL ,a)a>ao

Then the following are

necessary conditions for stability along the pass. (i)

The matrix D 1 is stable.

(ii) The matrix ~ is stable. ( i i i ) The matrix A is stable. Proof: Condition ( i i ) follows from direct application of (3.145) and (i) follows by reversing the roles of z 1 and z 2 in t h i s r e s u l t . To e s t a b l i s h ( i i i ) , consider the particular case of z 1 = z 2 = 1 in (3.144).

•

All of these conditions are easily tested using standard t e s t s from the s t a b i l i t y theory of LD(~,A,C,Do) and should be established before proceeding f u r t h e r . Note also that (i) is asymptotic s t a b i l i t y . In the development of a systems theory f o r the derived system, LD(~,A,C,Do) , the concept of a Lyapunov function has played a very important role. This is one method of s t a b i l i t y analysis for LD(~,A,C,Do) in the state-space domain and is based on the resulting matrix Lyapunov equation. The maturity and widespread use of t h i s approach in s t a b i l i t y analysis and c o n t r o l l e r design for LD(~,A,C,Do) has naturally

86 resulted in work on extending it to similar problems f o r 2D linear systems described by the Roesser model and hence example 2.3.8. A study of the published work shows t h a t t h i s general problem has been approached in two e s s e n t i a l l y d i f f e r e n t ways. The f i r s t of these consists of developing a 2D Lyapunov equation with constant c o e f f i c i e n t s and the second is based on a 1D Lyapunov equation with c o e f f i c i e n t s which are functions of a complex variable. In e i t h e r case, the basic objective is a suitable extension of the Lyapunov theory for LD(~,A,C,Do) which gives necessary and s u f f i c i e n t conditions f o r 2 0 / r e p e t i t i v e system s t a b i l i t y . The concept of an nD Lyapunov equation was f i r s t introduced in the development of a s t a b i l i t y theory for an nD continuous, or d i f f e r e n t i a l , system, where the problem under study was the development of conditions under which the r e s u l t i n g c h a r a c t e r i s t i c polynomial (a function of n variables) is s t r i c t l y Hurwitz and hence has no zeros (roots) in the region Re{si} > O, i = 1 , . . . , n . This was then extended to the Roesser model using the double b i l i n e a r transform to yield a 2D Lyapunov condition f o r s t a b i l i t y . In p a r t i c u l a r , consider again the augmented plant matrix h of (4.79). Then t h i s condition s t a t e s that there exists p o s i t i v e d e f i n i t e symmetric matrices Q,W1,W2 of dimensions (n + m) × (n + m), (n x n) and (m × m) respectively and W = WIG W2 such that the 2D/repetitive process Lyapunov equation ATwA - W = -q holds whereOdenotes the direct sum of W1 and W2, i . e .

[

W1

w =

0

(4.80)

0

(4.8~)

W2

Equation'(4.80) has constant c o e f f i c i e n t s but, unlike i t s conventional linear systems counterpart, is, in general, only a s u f f i c i e n t condition for 2D/repetitive system s t a b i l i t y . This is a well established r e s u l t (see, for example, the cited reference) and hence the proof is omitted here except to note that i t is based on the concept of a s t r i c t l y bounded real matrix from c i r c u i t theory. There are, however, a number of special cases when i t is necessary and s u f f i c i e n t and the following analysis considers the two of these which are most in keeping with the general aim of t h i s chapter. Suppose that the example under consideration is SISO. Then the following lemm&, which follows immediately as a special case of the general r e s u l t and is hence stated without proof, shows that (4.80) is necessary and s u f f i c i e n t in t h i s case. Lemma 4.4.2: Consider the SISO version of the model of example 2.3.8 and suppose

that the pairs (~,C) and (¢,A~) are completely reachable and observable respectively. Then the extended linear repetitive process

S(Ea,Wa,La)a>~o_

generated

87

by this model with a ~ a o is stable along the pass i f , and only i f , there exists symmetric positive d e f i n i t e matrices Wl,~2,~ = ~IQW2 and q such that (4.80) holds. The second special case of interest here is when A of (4.79) is normal and hence ATA = A AT (4.82) Further, note that (4.80) is equivalent to ATwA W < 0 (4.83) where the inequality denotes negative definiteness. This is structurally similar to the Lyapunov equation for LD(@,A,C,Do). Hence it follows that stability along the -

pass under (4.83) implies that A is a stability matrix in the conventional sense. This is now strengthened to a necessary and sufficient condition for stability along the pass by the following result. Equivalently, (iii) of lemma 4.4.1 is necessary and sufficient in this special case. Theorem 4.4.1: Suppose that the matrix A of (4.79) for the discrete unit memory linear repetitive process of example 2.3.8 is normal. Then the extended linear repetitive process generated by the model of this example with

S(Ea,Wa,La)a>~o_

a ~ ao is stable along the pass if, and only if, there exists symmetric positive definite matrices YI' N2' N = W I O W 2

such that (4.83) holds.

Proof: Necessity is immediate from the above discussion. To prove sufficiency, f i r s t note that if A is stable in the conventional sense then i t s a t i s f i e s the conventional linear systems Lyapunov equation, i . e . ATwA - W < 0 (4.84) Further, denote the eigenvalues of A by 2i, 1 < i < n + m, and the corresponding eigenvector matrix ~y R. Then, since A is normal, A =R h R (4.85) where * again denotes the complex conjugate transpose and h = diag{A.) 1 l
-

W = R({A{ 2- In+m)R*

(4.86)

Nhere tAl2 = diag{12il 2}

, and this matrix is negative definite since l
sense. Equivalently, (4.84) (the conventional linear systems case) holds under the choice of N = In+m and, since In+m is block diagonal under any partition, h also satisfies the 2D/repetitive process Lyapunov equation (4.83) and hence stability along the pass. • Suppose, therefore, that A of (4.79) is normal for the p a r t i c u l a r example under consideration. Then, in e f f e c t , theorem 4.4.1 states that s t a b i l i t y along the pass can be determined by testing A for s t a b i l i t y in the conventional sense. Further,

88

t h i s r e s u l t can be extended to a number of other sub-classes by use of appropriate s i m i l a r i t y transformations. The d e t a i l s of these can be found in the cited reference. Return now to the general case where (4.80) is a s u f f i c i e n t , but not necessary, condition f o r s t a b i l i t y along the pass. Then t h i s f a c t c l e a r l y reduces i t s effectiveness in t h i s context given the necessary and s u f f i c i e n t t e s t s of section 4.3. In e i t h e r of the special cases detailed here, however, i t could be used as an a l t e r n a t i v e route to t e s t i n g f o r s t a b i l i t y along the pass. This is p a r t i c u l a r l y relevant f o r the t e s t of theorem 4.4.1 which is c l e a r l y computationally more e f f i c i e n t than e i t h e r of the systematic t e s t procedures developed in section 4.3. Further, note t h a t numerous well defined algorithms e x i s t f o r computing p o s i t i v e d e f i n i t e solutions of t h i s 2D Lyapunov equation. Such algorithms have arisen due to t h e i r importance in other problem areas such as the design of low-noise and/or limit cycle f r e e 2D f i l t e r r e a l i s a t i o n s , the computation of grammians f o r 2D model reduction and (possibly) s t a b i l i t y t e s t i n g for nonlinear 20 systems. (See the cited references f o r complete d e t a i l s . ) One use of t h i s constant coefficient Lyapunov equation in the 2D systems context has been in the development of s t a b i l i t y margins with a view to use in control systems design. This is an active research area and numerous publications have appeared on various aspects. The extension of t h i s concept to example 2.3.8 is examined l a t e r in t h i s section a f t e r t h e second version of the Lyapunov equation has been considered. The second version of a Lyapunov equation for a 2D/repetitive process is, in e f f e c t , a 1D, or conventional linear systems, Lyapunov equation with c o e f f i c i e n t s which are functions of a complex variable. I t is, in e f f e c t , based on the following set of necessary and s u f f i c i e n t conditions for s t a b i l i t y along the pass (given (i) of lemma 4.4.1 by assumption): (a) the matrix ~ is stable in the conventional sense; and (b) the t r a n s f e r - f u n c t i o n matrix

Gl(z]l ) : C(z lIn_

+ D1

(4.87)

with [Zl-11 = 1 is stable in the conventional sense. Consider now the unit c i r c l e in the z l l plane C1 = {Zll:

lzl 1} = 1}.

each fixed z~ 1 e C1, Gl(Zl I) is, in general, a complex matrix.

Then for

Further, the

following r e s u l t gives a condition for an a r b i t r a r y matrix with a complex parameter, say F(z), to be stable for each Izl = 1 in the conventional sense. Theorem 4.4.2: Consider a matrix F(z) where the complex parameter z s a t i s f i e s tzt = 1. Then t h i s matrix is stable in the conventional sense i f , and only i f , for any given p o s i t i v e d e f i n i t e Hermitian, denoted P.D.H., matrix W(z) with Izl = 1 there e x i s t s a unique P.D.H. matrix H(z) such that

H-FtF=W

(4.88)

89

where * again denotes the complex conjugate transpose. Proof: Consider f i r s t s u f f i c i e n c y and take a fixed [z[ = 1. be any eigenvalue and eigenvector of F r e s p e c t i v e l y , i . e . Fv = ~v and

Further, l e t A and v (4.89)

u F = ~v* Pre-mu~tiplyi~g,(4.88) ~y v and post-multiplying i t by v now yields v Hu- v F ~Fv = v ]Iv(l - ]~I 2) = u*Wv Hence

(4.90) (4.91)

1~12 = v~, Wv ..... > 0

(4.92) v llv and t h e r e f o r e F is stable. To prove necessity, suppose that F is stable f o r any fixed tz] = 1 and let a P.D.H. matrix W be given. In which case consider 1 -

H = ~ (F*)qWF q (4.93) q=O and note t h a t t h i s H is well defined since F is stable and is c l e a r l y a P.D.H. matrix.

Further,

II . F'It F.

~ .(F*)qWF.q q~o(F*)q+lWFq+l W q=O = and suppose that ~1 is another solution of (4.88). Then H =

(4.94)

E (F*)qWFq = !o(F*lq(H1 - F *HIF)Fq. q=O q

oo ~ ( F* ) qlilFq - ~oo ( F* ) q+llllFq+I = t[ 1 (4.95) q:O q=O I Using t h i s r e s u l t now*gives the following set of conditions f o r s t a b i l i t y along the pass of example 2.3.8 which are equivalent to c o r o l l a r y 3.3.11. Theorem 4.4.3: The extended linear r e p e t i t i v e process S(Ea,Wa,La)a> a generated by =

-

0

the model of example 2.3.8 with a > a o is stable along the pass i f , and only i f , (a)

the matrix D1 is stable in the conventional sense;

(b) (c)

the matrix ~ is stable in the conventional sense; and the matrix e,quation H(z11) - Gl(Z11)lt(z11)G1(Zl 1) = W(Zl 1)

(4.96)

has a P.D.It. solution lt(Zl 1) f o r any given P.D.It. matrix W(zll ) and any z l l e C I. m

Recall now the unit memory versions of the systematic test procedures developed in section 4.3. Then, in effect, theorem 4.4.3 serves as an alternative to either

90 of these procedures. Further, these new conditions should obviously be tested in the order of (a) followed by (b) followed by (c) with termination if the one just tested does not hold. An additional step before proceeding to (c) being to examine the necessary condition l i s t e d under (c) in lemma 4.4.1, i . e . s t a b i l i t y in the conventional sense of the augmented plant matrix A. The t e s t i n g of (a) and (b) of theorem 4.4.3 is just the standard s t a b i l i t y problem for LD(O,&,C,Do). To develop a t e s t for (c), suppose that W(z~1) is given where ~(z] 1) = I m is an obvious choice.

In which case (4.96) can be solved to yield

a rational matrix solution |[(z~ 1) which is obviously ~ermitian.

Further, u r i t e z~ 1

in polar form as Zl 1 = e iO, 0 ~ O < 2~, and denote the j - t h order principal minor of H(Zl 1) by h j ( 8 ) , 1 < j < m. Then the hj(8) are functions of the single real variable 6 over the closed interval [0,2r] and the following corollary is e f f e c t i v e l y a systematic t e s t procedure for theorem 4.4.3. Corollary 4.4.3: The conditions of theorem 4.4.3 are equivalent to the following: (a) the matrix D1 is stable in the conventional sense; (b) (c)

the matrix ~ is stable in the conventional sense; and hj(8) > O, 0 < O ~ 2;, 1 < j < m

(4.97)

To i l l u s t r a t e the direct use of this corollary in a synthesis problem, consider the SISO case where D1 (a scalar in this case) s a t i s f i e s ]DI] < 1 and hence asymptotic s t a b i l i t y . Suppose also that ~ has eigenvalue-eigenvector decomposition T21O T2 " (4.98) = dlag{lj}l<j< n Then (b) holds i f , and only i f , l l j l < 1, 1 < j 5 n. Suppose nou that (b) holds and consider the case uhen ~j, 1 < j 5 n, is real. Further, define W as W = T 2 Q 1 and transform the augmented plant matrix to

i2

^

~ T-IAT :

. A3

(4.99) D1

where A2 : T21Ao ~ (dl . . . . 'dn )T

(4.1oo)

A3 = CT2 ~ ( f l " ' " f n )

(4.101)

~i = d i f i '

(4.102)

and Now define t < i ~n

and consider the special case when a l l of these numbers have the same sign. as shown below, (c) holds if, and only i f ,

Then,

91

max{ID1 + e(I n - ~)-lAol,1o 1 - c(I n ÷ ~)-lAol } < 1

(4.103)

To prove (4.103), take W(ei0) e 1 in (4.96) to y i e l d H(e i0) ~ hl(O ) with Zl 1 = e iO from h1(0)(1 - ID1 + 0 (z~lI n - ~ ) ' l a o l 2 ) = 1

(4.104)

Hence hl(O ) > O, 0 5 0 < 2~, i f , and only i f , [D1 + C(zllI n - ~)-IAo[ < 1, Zl 1 = e iO and necessity is immediate. I~(Zll)l ::

(4.105)

For sufficiency, note that

ID1 + C(~llIn - ~)'1~ol

= ID1 + i a ( ~ l I n - A)-1~21 n 5i = ID1 + S _ - : - - ~ 1 i=l Zl 1 2 i n

: ID1 + sgn(51) i~l ' i ( z ] l ) ]

(4.106)

where _ , ~i(z111 = Zl 1 " ~i

1 < i
(4.107)

and each of these functions maps the unit c i r c l e in the zl 1 plane onto a circle centred on the real line.

Itence the maximum value can only occur when Zl 1 = 1 or -1

and (4.103) follows immediately. Note: A similar analysis can be used to provide a s u f f i c i e n t condition for the case when the $i of (4.102) have d i f f e r e n t signs. Return now to the problem of t e s t i n g (c) in the general case. Then i t is clear that obtaining the hj(O) and t e s t i n g each of them for positive definiteness could lead to very severe computationai expense. The following analysis uses the Kronecker product, d e n o t e d @ , to obtain an equivalent r e s u l t to theorem 4.4.3 whose conditions are expressed in terms of the eigenvalues of constant matrices. This is a two stage operation as detailed below. Suppose, therefore, that a P.D.~. matrix W(ei0) is given. Then i t is required to show the existence of a P.D.H. matrix ~(e i0) which solves the following equivalent version of (4.96) V 0 E[0,2~]. lt(e i0) _ Of(e- iO)H(eiO)Ol(eig) = l{(e i0) Further, use of the Kronecker product enables (4.108) to be rewritten as (im2 _ Of(e- iO) ® GT(eiO))S[ll(eiO)] = S[W(eiO)]

(4.10s) (4.109)

92

where S[.] denotes the stacking operator. This now yields the following equivalen~ r e s u l t to theorem 4.4.3 which then leads to theorem 4.4.5 below whose conditions are expressed in terms of the eigenvalues of constant matrices. Theorem 4.4.4: The conditions of theorem 4.4.3 are equivalent to the following: (a) the matrix D1 is stable in the conventional sense; (b) (c)

the matrix ~ is stable in the conventional sense; i8 B ~ B(e o), the solution of T -iOo iOo)

B - Gl(e

)l! G l ( e

: W

(4.110)

is positive definite for any given positive d e f i n i t e matrix ~ and an arbitrary e o e [o, 2~] ; and

(d)

lira2- 6~(e-iO)®G~(eiO)l ¢ O, V 0 e[O,2,]

(4.111)

Proof: I t is clearly required to show that (c) and (d) above are, together, equivalent to (c) of theorem 4.4.3. First note, therefore, that (4.111) guarantees the existence of a unique solution, II(eiS), of (4.109). Further, H(e i8) is P.D.H. i f , and only i f , i t s eigenvalues are positive V 0 e[O,2r]. These are continuous i0 o functions of $ and will always be positive i f H(e ) is positive d e f i n i t e for an arbitrary 00 and (4.111) holds. Hence (4.110) and (4.111) are equivalent to (e) of theorem 4.4.3 and proof is complete. Consider now the problem of testing theorem 4.4.4 for a given example.

• Then, in

e f f e c t , t h i s consists of testing three constant matrices, DI,~ and H(eiO°), for s t a b i l i t y in the conventional sense and testing (4.111) V 0 e[0,2~]. Hence i t is clear that using this result as a basis for a systematic t e s t procedure will not lead to increased computational efficiency. The following theorem now expresses (4.111) in terms of the eigenvalues of constant matrices. Theorem 4.4.5: The conditions of theorem 4.4.4 are equivalent to the following: (a) the matrix D1 is stable in the conventional sense; (b)

the matrix ~ is stable in the conventional sense;

(c)

the matrix B(e iO°) is stable in the conventional sense for an arbitrary 0o

(d)

e[0,2~]; and [A2X1 + AX2 + X31 ¢ O,

V I2l : 1

(4.112)

where Xl

=

0 0

0 0

0 o

0 0

0

0

0

0

0

0 - ~nl ~v I

0

[

(4.113)

93

0 i

AoT®,T

o

_~T ® im

0

X2 =

% T ® AT 0

0

Imn

0

0

0

(4.114)

in 2 + ~T ® ~T

Im2 0

X3 =

]

CT ® I m DT ® CT

Imn 0 0 _ Im ® ~T

CT @ CT

0

0

:0 ] (4. i~s) - I n® ~T

ieo

Proof: In the case of (a) - (c), it is clearly only necessary to note that tt(e ) is the solution of the conventional linear systems Lyapunov equation and hence all its eigenvalues have modulus strictly less than unity. Consequently it remains to prove the equivalence of (d) in each case. Suppose, therefore, that f(eiO) = I im2 _ GT(e- iO) ® GTl(eiO) l (4.116) Then f(e iO) = II 2 - D1T®DT- e-iO'AT®DT)((I n - eiO~T) ®im )-I(CT(gI m) t0 m

_ (i m ®

AT)(i m ® ( i n e i O _ ~T))-

- eiO(ATo® ATolg(eiO)- I(cT ®

I(DIT® CT)

cTll

(4.117 /

where the matrix g(e i$) whose inverse appears in the last term is given by g(eiS) = - ( o T ® I n ) e i 2 0 + (in2 + ~T®¢T)eiS_ in OoT

(4.1181

flence f(e i#) = lIm2- O~®D~- uTv-IwI

(4.1191

where UT = [eiO(A~®D~), (I

QA~),

eiS"'T®A~)],n o

[iin ei'T°m ° V- 1

=

(4.1201

:J

imneiO - im ® @T 0 g(e i8

(4.121)

94

and

cT®Im W = DTQC T [cT®c T Further, (b) implies that

(4.122)

[Imn- eiO~T®Im I]Imn eiO- Im®~T[[g(eiO)] # O, V 0 e[0,2~]

(4.123)

and suppose that f(e i8) is pre-multiplied by the l e f t hand side of (4.123). In which case it follows immediately that f(e i0) # 0, V 0 6[0,2~] (4.124) is equivalent to (4.125) IXl ~ O, V 0 e[0,2~]

where X=

Im2- DT ®D T

e i O..T @ D1T)

CT ® I m

Imn- eiO~T ® I m

DT1® CT

0

CT ® CT

0

T im Q Ao

eiO(aoT ® aoT)

0

imneiO _ im ® oT 0

0

(4.126)

0 g(e iO)

Finally, it is clear that (4.126) and (4.112) are identical and the proof is complete.

II

To examine theorem 4.4.5 for a given example, it is necessary to test three constant matrices for s t a b i l i t y in the conventional sense and the second order matrix polynomial condition of (4.112). Note also that the matrix X1 of (4.113) is singular and hence further development is required for the two separate cases when is singular and non-singular. In particular, extensive, but routine, algebraic manipulations must be performed in both cases to reformulate (4.112) as a condition involving a f i r s t order matrix polynomial which can be easily tested with existing software for computing generalised eigenvalues. The details can be found in the cited reference. Consider now the problem of testing example 2.3.8 for s t a b i l i t y along the pass. Then it is clear that the obvious systematic procedure for testing theorem 4.4.5 serves as an alternative to either of the systematic procedures of section 4.3. Detailed comparative studies would, however, require the results from application of all of these procedures to suitably defined benchmark problems and this topic is not considered further here. To date, no consideration has been given in this work to the development of any form, CAD orientated or otherwise, of s t a b i l i t y margins. The following analysis represents a f i r s t attempt at this wide ranging problem for processes described by example 2.3.8. In particular, it extends some work from 2D linear systems described

95 by the Roesser model to t h i s case and b r i e f l y discusses the potential f o r further developments in t h i s area. Return, therefore, to the case of a constant c o e f f i c i e n t Lyapunov equation for example 2.3.8. In p a r t i c u l a r , suppose that there exists suitably dimensioned symmetric p o s i t i v e d e f i n i t e matrices q,W1,W2 and W = WI(~)W2 such that the 2D/repetitive process Lyapunov equation of (4.80), i . e . W - ATwA = Q (4.127) holds where the augmented plant matrix A is defined by (4.79). Then t h i s is, in general, a s u f f i c i e n t condition for s t a b i l i t y along the pass and hence p(zl,z2) of (4.78) s a t i s f i e s p(zl,z2) ¢ 0 ,

ViZl[ < 1, [z2[ 5 1

(4.128)

Suppose now that (4.128) holds and consider the 2D linear systems case. Then here the s t a b i l i t y margin has been introduced as a c r i t e r i o n f o r characterising the spatial domain performance of such systems. I t is defined using the largest bidisc where p(zl,z2) has no roots, i . e . p(zl,z2) ¢ 0 in U2al = {(Zl,Z2): [Zl[ < 1 + al, [z2] < 1}

(4.129)

p(zl,z2) ¢ 0 in U2¢2 = {(Zl,Z2): ]Zal < 1, [z21 < 1 + ~2}

(4.130)

p(zl,z2) ¢ 0 in U2~ = {(Zl,Z2): IZll< 1 + ~, Iz21 < 1 + ~}

(4.131)

Considerable e f f o r t has also been directed towards the development of algorithms for computing q1,¢2 and ¢. This has yielded numerous algorithms based on d i f f e r e n t approaches. For example, one set is based on minimising the distance between the roots of p(zl,z2) and the boundary of the unit bidisc W2 = {(Zl,Z2): [Zll = 1, Iz21 = 1}.

Alternatively algorithms based on the so-called resultant matrix could

he used. Further, i t has been shown that the s t a b i l i t y margin is related (in a well defined sense) to the minimal norm of the augmented plant matrix. In the 2D case, i t is not always necessary to know the exact value of the s t a b i l i t y margin. Instead, i t suffices to know that i t is greater than certain lower limits where one such limit, or bound, can be obtained as a function of the positive d e f i n i t e solution to the Lyapunov equation (4.127). The analysis which follows extends t h i s method to example 2.3.8 and discusses the outcome in terms of systems analysis. Consider, therefore, the case when suitably dimensioned symmetric p o s i t i v e definite solutions W and q exist for the Lyapunov equation (4.127) where W = WI(~)W2 and

Q1

Q2

q =

(4.132)

q~

Q3

96 Then this is, in general, a sufficient condition for stability along the pass and hence (4.128) holds. Further, the following analysis yields a lower bound for the stability margin as a function of the matrices Wl,W2,QI,~2 and Q3" First pre and post-multiply (4.127) by the matrix flllnQfl2Im where fll and f12 are positive real scalars and add W to both sides of the result. manipulations,

[:, oI_ "2

L#, °

I

fl2DlJ

w1 ?

This yields, after some algebraic

o ] [~1 ¢

w2

k#l c

#2%

=~ (4.133)

#2D~

where []~{]1 + ( i - fl~)W1 q =

#1f12Q2

~1/72~T

~113 + ( 1 - ] ~ ) W 2

(4.134)

Suppose also that Q is positive definite and hence, since W is assumed positive definite, a sufficient condition holds for

(4.135)

p(Zl,Z 2) ~ o , v Izll ~ 1, Iz21 5 1 where -Zlfl2Ao

I n - Zlfll~ ^

(4.136)

p(zl,z2) =

Im-Z2~2D1 -z2fllC The follouing result now establishes the relationship between the roots of p(zl,z2) ^

and p(Zl,Z2). ^

Lemma 4.4.1: respectively. ^

^

Let (Zl,Z2) and (Zl,Z2) denote the roots of p(zl,z2) and p(Zl,Z2) ^

Then

(Zl,Z 2) = (~1zl,~lz2) Proof:

(4.137)

By definition I n - ZlfllO

-Zl~2Ao

-Z2~lC

Im-Z2fl2D1

^

p(Zl,Z2) =

n

~ 1 I n - zlO

-ZlAo

- z2C

~211m - z2DI

m

= ~1~2

97 In

Zlfll~

-

-z2fl2C

"ZlfllA o Im

- z2fl2O 1

: p(~lZl,~2z 2)

(4.138)

and (4.137) follows immediately. • Using this result, it is possible to characterise the locations of the roots of p(zl,z2) as functions of fll and f12" In particular, if ~1 = /?2 = 1 then i t is ^

obvious that p(zl,z2) and p(zl,z2) are identical and s a t i s f y (4.128).

I f , however,

fli < 1, i = 1,2, the roots of p(Zl,Z2) move from (Zl,Z2) towards i n f i n i t y and for ~i > 1 they move towards the boundary, T2, of the unit bidisc and, eventually, some of them will be inside this bidisc. Further, note again that if ~ of (4.134) is positive definite for a given pair (~1,fl2) then this is a sufficient condition for the corresponding p(zl,z2) to s a t i s f y (4.135).

Consequently the range of (fll,fl2)

for which Q remains positive definite is closely related to the distance between the roots of p(Zl,Z2) and T2. Further, given that this is in general a sufficient, but not necessary, condition, it follows immediately that the range of (~1,~2) for which is positive definite can only give lower bounds, not actual values, for the s t a b i l i t y margin. The analysis below obtains lower bounds for ~t,a2 and q of (4.129)-(4.131) respectively in terms of the range of (~1,~2) for which q is positive definite. Considering f i r s t al, it is clear that a lower bound in this case can be obtained from the range of fll for which ~ is positive definite with f12 = 1.

This is

equivalent to Q3 > 0

(4.139)

z ql + (1 - z )w 1 - fllq2q3 2 -1 q2T > 0

(4.140)

and Further, (4.139) holds by assumption and hence it remains to consider (4.140) which is equivalent to (w1 - Q, + Q2U~IQ~) - Z~w 1 < o (4.141) and the values of fll for which this new condition holds can be obtained from the established theory of the extremal properties of pencils of quadratic forms with the structure F - 2B. In particular, i t is known that for B > 0 ×W Fx train[B-IF] 5 x-T-~B x 5 2max[~-lF], × ¢ 0 (4.142)

98

and F - ~max[B-1FJB < 0

(4.143)

where ~min[B'lF] and 2max[B'lF] denote the minimum and maximum eigenvalues of B-1F respectively (and < in (4.143) denotes the fact that the matrix on the left-hand side is negative semi-definite). Suppose also that B and F are defined by (4.144) B = W1 and F = W1 - q l +

q2qalq~

where ~1 > 0 by assumption.

(4.145)

Then the following lemma shows that t h i s particular

choice of F is positive semi-definite (written F > 0). Lemma 4.4.2: Suppose that ~l,~2,Ql,Q2 and Q3 are the (appropriately) dimensioned matrices of a positive definite solution to the Lyapunov equation (4.127). of (4.145) is positive semi-definite Proof: Rewrite (4.127) as

Then F

(4.146)

hT$1h

w2 Q3 where W > 0 and hence the right-hand side is positive semi-definite. that ~1 " Q1 ~ 0

This implies (4.147)

and Q3 > 0 since Q > O. Hence ~2~31Q~ > 0

(4.148)

and F > 0 follows immediately from (4.147) and (4.148). • Given t h i s result, (4.142) and (4.143) can now be used to obtain the values of fil ~or which (4.141) holds. This yields Zl 1 < 4~max[C1]

(4.149)

where C1 = B'IF = (I n - WllQ1

+

WllQ2Q31Q~)

(4.150)

and a l l eigenvalues of this matrix are real and non-negative since F > 0 and B > O. Further, the maximum eigenvalue of C1 is positive for an augmented plant matrix A ¢ 0 and the lower bound for ~1 is now given by

~1 ~ (4~max[cl])-1- 1

(4.151)

In the case of a lower bound for ¢2 of (4.130), a completely analogous analysis to that above yields ¢2 ~ (4~max[C2])- 1 _ 1

(4.1~2)

99 where C2 = I m - W21Q3 + w21Q~Q;1Q2

(4.153)

Similarly, i t is clear t h ~ a lower bound for ¢ of (4.131) c ~ be obtained by s ~ t i n g B1 = B2 = fl and determining the values of this parameter for which q of (4.134) is positive definite. This implies that : W- fl2(~q) > 0 or

(4.154)

(w- q)- ~2w < 0

(4.155)

The m~rix ~ is positive definite and (W- Q) is positive semi-definite, which follows immedi~ely from writing (4.127) as ~Wh = ~ - Q (4.156) Hence the range of ~ for which (4.155) holds can be determined in a similar manner to, for example, 81 of (4.149). ~ i s yields yl

< ~m~[C3 ]

(4.157)

where C3 = In+ m - W-1Q

(4.158)

and hence the lower bound for a as a > (~m~[C3]) -1 1

(4.159)

To i l l u s t r a t e these bounds, consider the special case when -015

-0.395 ]

A=

(4.160)

-0.01 Then the Lyapunov equation (4.127) has the following solution W=

,

q =

0 0.395 and hence (4.150) and (4.151) yield ~1 ~ 0.116 Further, (4.152) and (4.153) yield ~2 ~ 0.127

(4.161) -0.194

0.239 (4.162) (4.163)

and from (4.154) i t follows t h ~ W- ~2(W- ~ ) > 0 (4.164) for ~ < 1.083 and hence > 0.083 (4.165) As a comparison, the exact values of al,¢ 2 and ~, obtained using algorithms detailed in the cited references, are ql = 0.12, e2 = 0.282, ~ = 0.108

(4.166)

100

The lower bounds f o r the s t a b i l i t y margins developed above depend on the matrices {W,q}. In p a r t i c u l a r , d i f f e r e n t pairs yield d i f f e r e n t lower bounds and i t is known, see the cited reference, that the bounds which are closest to the actual value of the s t a b i l i t y margin are obtained from a pair {W,q} corresponding to the minimum norm of the augmented plant matrix A. Suppose, therefore, that s t a b i l i t y along the pass holds and p o s i t i v e d e f i n i t e solutions {~,Q} of (4.127) exist. Then the minimum spectral norm, ~, of the corresponding state-space model, or r e a l i s a t i o n , is defined as

= m~n[[T AT-l}}

(4.167)

where }}.[I is any suitable norm, and T = T I O T 2 where T1 and T2 axe real constant n×n and mxm matrices respectively. Summarising, therefore, t h i s section has considered in depth the development of s t a b i l i t y t e s t s f o r example 2.3.8 based on theorem 3.4.4 which shows the equivalence of s t a b i l i t y along the pass in t~is p a r t i c u l a r case and BIB~ s t a b i l i t y of 2D linear systems described by the Roesser model. P a r t i c u l a r a t t e n t i o n has been directed towards a Lyapunov approach and t h i s has yielded two e s s e n t i a l l y d i f f e r e n t systematic t e s t procedures and associated t e s t s . The f i r s t of these is based in a 2D Lyapunov equation with constant c o e f f i c i e n t s and the second is based on a 1D Lyapunov equation with c o e f f i c i e n t s which are functions of a complex variable. Further, the f i r s t approach is, in general, s u f f i c i e n t but not necessary but the second is both necessary and s u f f i c i e n t . Detailed comparative studies of these procedures with those of section 4.3 would, however, require the r e s u l t s from applying a l l of these procedures to suitably defined benchmark problems. Here this wide ranging area has been l e f t for future research with the note that i t s s u f f i c i e n t , ' b u t not necessary, basis will c l e a r l y reduce the general usefulness of the f i r s t Lyapunov based approach of t h i s section in terms of s t a b i l i t y testing. The application of the constant coefficient Lyapunov equation approach to the problem of developing physically meaningful s t a b i l i t y margins for example 2.3.8 has been considered. In paxticu]ar, some work from the area of 2D linear systems described by the Roesser model has been extended to t h i s case. Further, there are two ( i n t e r r e l a t e d ) areas to which future research e f f o r t could p r o f i t a b l y be directed. These are further development of the basic computational algorithm for increased e f f i c i e n c y , which may necessitate some reformulation of the existing r e s u l t s , and in depth work to establish the correlation ( i f any) with system performance. Consider now the f i r s t of these two areas. Then progress here will serve to strengthen the already documented links between example 2.3.8 and the Roesser model. In the case of the second area, the f i n a l objective here would c l e a r l y be to produce easy to use, ideally within a CAD environment, s t a b i l i t y and/or performance indicators. One obvious aspect to investigate in t h i s p a r t i c u l a r case is the links ( i f any) with the recently introduced concept of a pole for example 2.3.8, defined

101

in terms of the solutions of the two variable polynomial p(Zl,Z2) of (4.78), which is the most i n t u i t i v e l y appealing definition of a ' c h a r a c t e r i s t i c polynomial ~ for this case. Note, however, that the ~pole concept' for example 2.3.8 (and other cases) is s t i l l very much in the development stage and a review of progress to date can be found in the cited reference. The problem of developing s t a b i l i t y and/or performance indicators is considered again ia the next chapter where a l t e r n a t i v e simulation-based t e s t s are developed. These then lead to the production, at no extra cost, of computable information concerning the r a t e of approach to the limit p r o f i I e in one special case of major p r a c t i c a i i n t e r e s t . Finally, chapter 6 considers the use of t h i s information in the formulation of c o n t r o l l e r design algorithms. To conclude t h i s section, return to the more general non-unit memory case of example 2.3.7. Then an obvious question to ask is whether or not the analysis of this section generalises in a natural manner and to date no real e f f o r t has been directed towards t h i s area. This general question is not considered further here except to note that substantial progress in certain p a r t i c u l a r aspects should be achieved with r e l a t i v e l y l i t t l e e f f o r t . For example, it appears that the second Lyapunov approach should generalise in a straightforward manner and i f t h i s is the case then the next stage ~ould be to follow up on the s t a b i l i t y margin r e s u l t s . 4.5

tDnlication of D~!~yDjfferential S t a b i l i t y Tests A special case of example 2.3.6 has shown a s t r u c t u r a l link between d i f f e r e n t i a l unit memory linear r e p e t i t i v e processes and a p a r t i c u l a r sub-class of delay d i f f e r e n t i a l systems. The purpose of t h i s section, therefore, is to consider the application of r e s u l t s from t h i s well researched area to example 2.3.4 and, in p a r t i c u l a r , to the development of s t a b i l i t y t e s t s . As a primer to the analysis presented here, the following is a b r i e f summary of the relevant background material. Complete d e t a i l s can, for example, be found in the cited references. Consider the functional d i f f e r e n t i a l equation dn2 ~¥(t) dtn2

.

n2 n I di ~ ~ cij - - g ( t dt i i=O j=O

- jh) = 0

where the c o e f f i c i e n t s c i j are real scalars.

(4.168)

Then t h i s equation describes a delay

d i f f e r e n t i a l system with commensurate delays. Further, the following r e s u l t expresses one form of s t a b i l i t y for (4.168) in terms of i t s so-called c h a r a c t e r i s t i c function n2 n 1 . . C ( s ' e ' j h s ) : = sn2 + i=O ~ j~O "= c i j s l d j h s (4.169) Theorem 4.5.1: The delay d i f f e r e n t i a l system (4.168) is asymptotically stable independent of delay (I.O.D) i f , and only i f C(s,e -Jhs)" ~ O, V Re{s} > O, h > 0 (4.170)

102

Suppose now that the variable z:= e-hs, i.e. a l e f t shift operator of duration h, is introduced into (4.170). Then this yields the two-variable polynomial n2 n2 _nl C(s,z) = s + £ _~oCijsiz j (4.171) i=O jwhich can also be obtained directly from (4.168) by applying the joint (s,z) transform. Given C(s,z), it is always possible to realise (4.168) by the autonomous (no inputs 2D state-space model

i2(t)

A3

A4

Xl(t)] V(t) [e, 02] x2(t)J

x2(t) (4.172)

=

where X1 and X2 are terme, the delayed and continuous state vectors respectively. The characteristic polynomial of this model is =llnl - zAI C(s,z)

I [

-zh 2 (4.173)

-h3

SIn2-h 4

which can be written in the form (4.171). Note also that (4.172) realises both neutral (Cn2,J # 0 for some j e [1,nl] ) and retarded (Cn2,J = O, V j E[1,nl] ) systems. Ience s t a b i l i t y conclusions based on this model apply to both cases. Given (4.172), it is possible to derive sufficient conditions for asymptotic s t a b i l i t y of delay differentia] systems in terms of frequency dependent 1D Lyapunov equations. In particular, define the sets D and U by : {s: Re{s} > O} (4.174) and = {z: Iz[ < 1} (4.175) respectively and form the Cartesian product D ~ ~. Further, consider the following condition for so-called pointwise asymptotic sVability O(s,z) # 0 in D x U (4.176) i.e. the characteristic polynomial is void of zeros in the non-compact biplane composed of the closed right-half plane and the closed unit disc (or circle). Further, (4.176) is a stronger condition than (4.170) since it can be shown that D ~ U has more points than D x e (-D). Hence pointwise asymptotic s t a b i l i t y is more conservative, or stronger, than asymptotic s t a b i l i t y (I.O.D). The following result, for which a proof is given since it plays a central role in the analysis which follows, gives necessary and sufficient conditions for (4.176).

103

L e n a 4.5.1: The c h a r a c t e r i s t i c polynomial of the delay d i f f e r e n t i a l system described by the 2D model (4.172) s a t i s f i e s (4.176) i f , and only i f , the following conditions hold (a) a l l eigenvatues of the matrix h4 have s t r i c t l y negative real parts; and (b)

a l l eigenvalues of

Z(s)::

(c)

h2(sIn2 - h4)-lh 3 + h I

(4.177)

with s : iw have modulus s t r i c t l y less than unity for a l l real frequencies w > 0 or, equivalently, all eigenvalues of the matrix h 1 have modulus s t r i c t l y less than unity;

and (d)

Proof:

all eigenvalues of S(z):= h3(z-lI

n1

- h l ) - l h 2 + A4

(4.178)

have real parts s t r i c t l y less than zero for a l l Izl = 1. By Schur's formula

C(s,z) = Ilnl - ZAlllSIn2 - S(z) l = tInl - zZ(s){ISIn2 - k41

(4.179)

Using (4.179), the rest of the proof is a straightforward application of the Maximum modulus theorem. • Return now to the state-space model of example 2.3.4 and delete the current pass input terms to yield (with Yk+l(O) = Yk(a)) the autonomous version

Xk+l(t) = hXk+l(t) + BoYk(t) Yk+l(t) = CXk+l(t) + DiYk(t) 0 < t < a, k > 0 (4.180) Then ) t follows immediately that (4.180) can be modelled by the following special case of the 2D state-space model of (4.172)

x(t)1 LY(t+~)

DI

Y(t) ]

(4.1sl)

where here X(t) denotes the current pass state vector Xk+l(t ) and Y(t) denotes the previous pass p r o f i l e Yk(t). Further, the following r e s u l t now shows that an equivalence exists between s t a b i ] i t y along the pass of example 2.3.4 and pointwise asymptotic s t a b i l i t y of its delay d i f f e r e n t i a l interpretation.

104

Theorem 4.5.2: Regard the model of example 2.3.4 (in i t s autonomous form) as a delay d i f f e r e n t i a l system described by the 2D state-space model of (4.181). Then the extended linear r e p e t i t i v e process generated by t h i s model with

S(Ea,Wa,La)a>ao

a > a o is stable along the pass i f , and only i f , i t is poiatwise asymptotically stable in the sense of (4.176). Proof: This, in e f f e c t , consists of showing that the conditions of lemma 4.5.1 (in p a r t i c u l a r , the set consisting of (a) and (b)) and corollary 3.3.10 are equivalent. Oonsider f i r s t , therefore, the delay d i f f e r e n t i a l i n t e r p r e t a t i o n of (4.181). Then (a) and (b) of lemma 4.5.1 t r a n s l a t e to the following conditions for pointwise asymptotic s t a b i l i t y : (a) a l l eigenvalues of the matrix A have s t r i c t l y negative real parts; and (b) a l l eigenvalues of Gl(S ) = C(sI n -

A)-IBo + D1

(4.182)

with s = i~ have modulus s t r i c t l y less than unity for a l l real frequencies > O.

Further, i t follows immediately that a l l eigenvalues of O1 must have modulus s t r i c t l y less than unity ( i . e . asymptotic s t a b i l i t y ) in order for (b) to hold. Hence these two conditions are equivalent to (a) - (c) of corollary 3.3.10 for s t a b i l i t y along the pass. Conversely, suppose that (a) and (b) above hold. Then the proof that these imply s t a b i l i t y along the pass is identical to that of corollary 3.3.10 and is hence omitted. • Theorem 4.5.2 can be regarded as the analogous r e s u l t to theorem 3.4.4 which established the equivalence between s t a b i l i t y along the pass of the discrete unit memory process of example 2.3.8 and BIBO s t a b i l i t y of 2D linear systems described by the Roesser model of (2.58). The following analysis mirrors section 4.4 for the discrete case in considering the use of theorem 4.5.2 as the basis f o r the development of s t a b i l i t y t e s t s to serve as a l t e r n a t i v e s , and/or supplements to, those of section 4.2. A study of the published l i t e r a t u r e on the development of s t a b i l i t y t e s t s for delay d i f f e r e n t i a l systems shows that this problem has been studied from a variety of s t a r t i n g points, fine major approach has centred on applying a root clustering based argument to C(s,z) of (4.176) and uses techniques such as the Schur-Cohn matrix and modified Routh array, which were also used in section 4.2 to develop t e s t s f o r s t a b i l i t y along the pass of the d i f f e r e n t i a l non-unit memory process of example 2.3.3. Consequently the analysis below concentrates on a Lyapunov approach and complete d e t a i l s of the various other approaches can be found in the cited references. In a similar manner to that for 2D linear systems described by the Roesser model, the general problem of developing a Lyapunov approach to the s t a b i l i t y analysis of

105

delay d i f f e r e n t i M systems, and hence example 2.3.4 by theorem 4.5.2, has been studied in two ( e s s e n t i a l l y d i f f e r e n t ) ways. One of these consists of developing a 1D Lyapunov equation with c o e f f i c i e n t s which are functions of a complex parameter and the other is based on a 2D Lyapunov equation with constant c o e f f i c i e n t s . In e i t h e r case, the basic objective here is a suitable extension of the Lyapunov theory for the derived system LD(A,B,C,Do) which gives necessary and s u f f i c i e n t conditions f o r s t a b i l i t y along the pass of example 2.3.4. Considering f i r s t the 1D Lyapunov equation approach yields the following dual approach to that of theorem 4.4.2, and hence theorem 4.4.3, for the discrete unit memory process of example 2.3.8. Hence the r e s u l t is stated without proof. Theorem 4.5.3:

The extended linear r e p e t i t i v e process

S(Ea,Wa,La)a>Oogenerated by

the model of example 2.3.4 with a > ao is stable along the pass if, and only i f , (a)

(b)

a l l eigenvalues of the matrix A have s t r i c t l y negative real parts or, equivalently, the derived conventional linear system LD(A,B,C,Do) is stable; and the matrix Ly~punov equation H(s) - GI(S)H(s)GI(S) : W(s)

(4.183)

has a unique P.D.|t. solution It(s) for any P.D.H. matrix W(s), s = i~ for any > O. • Note: As in section 4.4, P.D.I1. denotes positive d e f i n i t e Hermitian and * denotes the complex conjugate transpose, i . e . l! (i~) = HT(-iv). Recall now the unit memory versions of the systematic t e s t procedures developed in section 4.2. Then, in e f f e c t , theorem 4.5.3 serves as the basis for an a l t e r n a t i v e to either of these. Further, these new conditions should obviously be tested in the order of (a) followed by (b) with termination if the former does not hold. Theorem 4.5.3 is the d i f f e r e n t i a l equivalent of theorem 4.4.3 for the discrete process of example 2.3.8. Further, the t e s t i n g of (a) in t h i s case is just the standard s t a b i l i t y problem for LD(A,B,C,Do). Tests for (b) can also be developed by following analogous steps to those used in section 4.4 for (c) of theorem 4.4.3. Consequently these are not detailed here and complete d e t a i l s can be found in the cited references. Instead, the following analysis gives an introduction as to how the Lyapunov equation (4.183) can be used to provide a physically based interpretation of s t a b i l i t y along the pass in t h i s p a r t i c u l a r case. Complete d e t a i l s , including the corresponding analysis for example 2.3.8, can again be found in the cited reference. Noting again the causality definition of (2.59) and Figure 2.7, apply the Laplace transform to the autonomous model of (4.181) to yield

lo6

Yk+l(S) = Gl(S)Yk(S ) ,

k >0

(4.184)

Further, suppose that the output Yk+l(S) is passed through a f i l t e r with ^

transfer-function matrix R(s) and denote the result of t h i s operation by Yk+l(S), k_>O, i.e. Yk+l(S) = R(S)Yk+l(S ) = R(S)Gl(S)Yk(S), k > 0

(4.185)

Then, by Parseval's theorem,

;5

+l(t

Yk+l(i~)Yk+l(iW)d~

(4.186)

= ~ Jo vk(io;)Gl(i~)R (i~)R(i~)Gl(i~)¥k(i~)d~

(4.187)

1

k+l(t)dt = ~-~

too,

.

,

Suppose also that If(s) = R (s)R(s) s a t i s f i e s the Lyapunov equation (4.183). which case

1;o,,

k(i~)(lt(i~)

÷l(t)Yk+l(t)dt = ~

In

- W(i~))Vk(iW)dw

l~y, k(iO))Yk(iW)dx ~ = ~-~ - ~l~ov, k(i~')~/(i~)Yk(iOJ)d~

(4.188)

and hence ~oY~+I(t)Yk+l(t)dt < ~oYT(t)Yk(t)dt, k _> 0

(4.189)

Equivalently, the f i l t e r e d output, Y, decreases in amplitude from pass to pass in an L2 sense. in the case of delay d i f f e r e n t i a l systems, the so-called continuous bounded real and discrete positive real lemmas from circuit theory have been used to develop 29 Lyapunov equations for the s t a b i l i t y conditions of lemma 4.5.1 and, in particular, (4.177) and (4.178) respectively. As shown below, the repetitive systems case requires only the former concept strengthened to so-called s t r i c t l y continuous bounded real which is denoted by S.C.B.R. Further, let W = W1 QW2 and q be real symmetric positive definite matrices. example 2.3.4 is

Then the proposed 21) Lyapunov equation for

pTwI'O + wI'Op + pTwO'Ip - W0'1 = - Q

(4.190)

where wI'O:= W1QOm, ~0,1:= OnQ~ 2 and P is the so-called augmented plant matrix defined as

P=

f:

(4.191) D1

The f i r s t step in relating (4.191) to s t a b i l i t y along the pass is the following formal definition of the term S.C.B.R.

107

Definition 4.5.1: Consider a square matrix Z(s) over R(s), the ring of polynomials in s over the real line. Then Z(s) is termed S.C.B.R. if the follouing conditions hold: (i)

Z(s) is analytic in D; and *

(ii)

I - Z (s)Z(s) > O, s = i~,V real a > 0

(4.192)

Further, interpreting this definition in terms of the interpass transfer-function matrix Gl(S ) and comparing the resulting conditions with those of theorem 4.5.3 immediately shows * that (ii) here is equivalent to (4.183) admitting the constant solution H = WT over the real line. Hence it follows immediately that S.C.B.R. implies s t a b i l i t y along the pass. In general, however, the converse is not true, a result which parallels that of section 4.4 for the 2D Lyapunov equations introduced there for the discrete process of example 2.3.8. This result is proved by straightforward modifications to the steps used in establishing i t s discrete counterpart and hence the details are omitted. Despite i t s sufficient but not necessary basis, which clearly reduces i t s usefulness in terms of s t a b i l i t y t e s t s given the necessary and sufficient alternatives of section 4.2, the 2D Lyapunov equation s t i l l has a (potentially) significant role to play in certain aspects of the analysis of example 2.3.4. One of these is to develop a f i r s t attempt at the wide ranging problem of constructing useful, and 'easy to use', s t a b i l i t y margins. This topic is returned to later in this section after s t a b i l i t y along the pass has been formally expressed in terms of (4.190). Suppose, therefore, that the quadruple {F,G,jT,K} is a minimal realisation of Z(s), i.e. Z(s) = jT(sI - F)-IG + K (4.193) Then the following lemma can be introduced and leads directly to the required result in the form of theorem 4.5.4 below. This so-called S.C.B.R. lemma is a well known result in circuit theory and its proof can, for example, be found in the cited reference. Lemma 4.5.2: Suppose that the quadruple {F,G,jT,K} is a minimal realisation of Z(s) of (4.193). Then Z(s) is $.C.B.R. i f , and only i f , there exists a positive definite symmetric matrix P1 such that QI:= I[FwP1 + p1F + jjT

(PIG +IJK)1 <0

L(P1G + JK)T

KTK _

i.e. ql is negative definite symmetric.

(4.194)

108

Theorem 4.5.4: Consider the d i f f e r e n t i a l unit memory linear repetitive process of example 2.3.4 under the controllability and observability assumptions of corollary 3.3.8, i.e. the pair {k,Bo} is controllable and the pair {C,A} is observable. Suppose also that there exists a matrix T, non-singular over the real line, such that ~l(S) = T[C(sI n A)-IB° + D1]T-1 (4.195) -

is S.C.B.R.

Then there exists positive definite symmetric matrices Wa,~2 and q such

that W = W I ( ~ 2 and q satisfy the 2D Lyapunov equation (4.190).

Conversely,

suppose that (4.190) holds for positive definite symmetric matrices W = WlE)W2 and q, then there exists a matrix T, non-singular over the real line, such that Gl(S) is S.C.B.R.

Proof:

Consider first sufficiency and note that G1(s ) has the minimal realisation

(A,BoT-I,TC,TDI T-I) and is S.C.B.R. Hence by lemma 4.5.2 there exists a positive definite symmetric matrix PI such that

I

ATpI + P1A + cTTTTC

Q2:=

PiBo T-I + cTTTTDI T-I <0

L(PIBoT-I +

cTTTTDIT-I) T

(T-I)TD~TTTDI T-I - I m

(4.196)

and this yields, after pre-multiplication by (InQTT) and post-multiplication by ( I n O T ) , the 2D iyapunov equation (4.190) with q = - (InOTT)q2(In®T) > 0

(4.197)

and W = (PI®TTT) > 0

(4.198) i

To prove necessity, suppose that (4.190) holds and set T = W~, F = A, G = BoT-1, jT = TC and g = TDIT-1. Then pre and post-multiply (4.190) by (InQ(T-1)T) and (In O T -1) respectively to yield (4.194) with P1 = ~1 and ql = - (In Q(T-1)T) Q(InQT-1).

Ilence Z(s) is S.C.B.R. as required.

•

As noted previously, S.C.B.R. is a special case of the 2D Lyapunov equation (4.183) and hence the following corollary can be stated. Corollary 4.5.4: Consider the extended linear repetitive process

S(Ea,~a,La)a>ao_

generated by the model of exmaple 2.3.4 with a > ao. Then theorem 4.5.3 for stability along the pass holds if there exists positive definite symmetric matrices = W I O N 2 and Q such that the 2D Lyapunov equation (4.190) holds.

109

In common with the 2D Lyapunov equation used in section 4.4 f o r the discrete unit memory process, a number of special cases exist where (4.190) yields necessary and s u f f i c i e n t conditions. 8he of these is the SIS8 case where i t is easily shown that corollary 3.3.10 holds i f , and only if, the interpass transfer-function Gl(S ) is S.O.B.R. This r e s u l t is stated formally in the following corollary. Corollary 4.5.5: The extended linear r e p e t i t i v e process

S(Ea,Wa,La)a>~ogenerated

by the model of example 2.3.4 with m = 1 and a > ao is stable along the pass i f , and only i f , the interpass transfer-function Gl(S ) is S.C.B.R. Vork is proceeding on the development of e f f i c i e n t algorithms for computing the matrix P1 ( i f i t exists). Further, other conditions for S.O.B.R. have been developed using, for example, algebraic Riccati equations. Oomplete details of progress to date can be found in the cited references. The remainder of t h i s section uses the 2D Lyapunov equation (4.190) as a basis for the f i r s t attempt at developing useful, and 'easy to use', s t a b i l i t y margins for example 2.3.4. In particular, i t follows the approach of section 4.4 and extends some work from delay d i f f e r e n t i a l systems to this case. Further, the potential for other developments in this area is b r i e f l y discussed. Suppose, therefore, that there exists positive d e f i n i t e symmetric matrices ~,W1,W2 and W = WI(~)W2 such that (4.190) holds. Then this is, in general, a s u f f i c i e n t condition for s t a b i l i t y along the pass and hence [si n - A " B° [ p(s,z) =[_zO[

I m - zD1

(4.199)

I

satisfies p(s,z) # 0 in D × (4.200) i . e . the s t a b i l i t y region excludes the non-compact biplane composed of the closed r i g h t - h a l f plane and the d o s e d unit disc (or c i r c l e ) . Further, as for the discrete process of example 2.3.8 considered in section 4.4, the s t a b i l i t y margins are defined (in common with the delay d i f f e r e n t i a l case) as the shortest distances between the roots of p(s,z) and the boundaries of D x ~. In p a r t i c u l a r , given s t a b i l i t y along the pass, these margins are defined as the largest values of the scalars 6 and • for ~hich 0 in

x

(4.201)

and p(s,z) # 0 in De × U

(4.202)

respectively where U6 = {z: [z I 5 1 + 6}

(4.203)

110

and D = {s:

Re{s} ~ - ¢}

(4.204)

Due to the non-compactness of D x U, a combined s t a b i l i t y margin (corresponding to ¢ of (4.131) in section 4.4 for example 2.3.8) is not defined in t h i s case since it has no c l e a r l y defined meaning. Consider now the case when p o s i t i v e d e f i n i t e solutions 14 and Q e x i s t f o r the Lyapunov equation (4.190) where 14 = 141@142 and

Q=

(4.205)

q2

QaJ

Then t h i s is, in general, a s u f f i c i e n t condition f o r s t a b i l i t y along the pass and hence (4.200) holds. Further, the following analysis yields lower bounds f o r 6 and ¢ in terms of 141,W2,ql,q2 and Q3" F i r s t pre and post-multiply (4.190) by (In OflIm) , where fl is a real scalar, to obtain

+

C

flDiJ

+

C

0

flD1 J

= C

flDlJ

I42

flD1J

(4.206) fl[}2

f12(q3- $/2)

Subtracting (27141 @W2) , where 7 is also a real scalar, from both sides of (4.206) now yields the following equation a f t e r some algebraic manipulations

~T141,0 + 141,0~ + ~TwO,I~ _ wO,1 = - {~

(4.207)

where

h- 7I n

fibo

(4.2os) and

flq2

J

(4.209)

52(~3 - W2) + 142

Suppose also t h a t q is positive d e f i n i t e and hence, since W is assumed positive d e f i n i t e , a s u f f i c i e n t condition holds f o r

p(s,z) # 0 in D × ~

(4.210)

111

where si n -

(A- 7In)

-fibo

^

p(s,z) =

(4.211) -zC

I m - zflD1

The following r e s u l t , whose proof follows similar steps to that of l e n a 4.4.1 for the discrete process of example 2.3.8 and is hence omitted, now establishes the relationship between p(s,z) and p ( s , z ) . Lemma 4.5.3: Consider p(s,z) and p(s,z) defined by (4.199) and (4.211) respectively. Then

p(s,z) = p(s - ~, ~ l z )

(4.212)

Using t h i s r e s u l t (compare with lemma 4.4.1 for the discrete case), i t is • possible to characterise the roots of p(s,z) as functions of 7 and^ft. In p a r t i c u l a r , i f 7 = 0 and fl = 1 then i t is obvious that p(s,z) and p(s,z) are identical and s a t i s f y (4.200). I f , however, fl > 1 then, for 7 > O, the roots of p(s,z) move from those of p(s,z) towards the boundary of D ~ U and ultimately cross i t . Further, note again that q > 0 is s u f f i c i e n t for p(s,z) to s a t i s f y (4.210) and hence the range of a and fl f o r which t h i s matrix remains p o s i t i v e d e f i n i t e can provide limits for the s t a b i l i t y margins defined by (4.201) - (4.202). Since, however, t h i s i s , in general, a s u f f i c i e n t , but not necessary condition i t follows immediately that the range of 7 and fl for which q is p o s i t i v e d e f i n i t e can only give lower bounds, not actual values, f o r these margins. The analysis below obtains such bounds. Considering f i r s t *, it is clear that a lower bound in t h i s case can be obtained from the range of fl > 1 for which

> 0

:

flQ2

(4.213)

Z2(Q3-W2) + W2

This is equivalent to Q1 > 0

(4.214)

w2 - Z2(w2 " ~3 + %q~1~) > o

(4.215)

and Further, (4.214) holds by assumption and hence i t remains to consider (4.215) which is equivalent to (W2 - q3 + Q2QllQ~) - ff2W2 < 0

(4.216)

where t h i s new condition is expressed in terms of a matrix pencil of the form F AB, with A = ~-2 B = W2

(4.217)

112

and F :

V2 - q3 + q2qllq~

(4.218)

Note also that this pencil is regular since W2 > 0 and F can be shown to be positive semi-definite on observing from (4.190) that ~2 - q3 = D~W2D1 ~ 0

(4.219)

The extremal properties of the characteristic values (eigenvalues) of the pencil F - ~B can now be examined using well established theory. In p a r t i c u l a r , the result l i s t e d under (4.143) in section 4.4 is relevant, i . e . (4.220) F ~max[B-1F]B < 0 -

where 2max[B-IF] again denotes the maximum eigenvalue of E l F .

Noting again that

B = ~2 is positive definite by assumption and F of (4.218) is positive semi-definite, (4.220) now yields the values of fl for which (4.215) holds as > ~max[B-1F] (4.221) or

< (¢2max(Im- ~/-21Q3 + ~lQ2Q~lQ~))-I

(4.222)

Further, i t is easily shown that all eigenvalues of B-1F are real and non-negatlve. Hence a lower bound for 5 can be written as 5 > (~max(Im- Y21Q3 + W21Q2~110~))-1

1

(4.223)

In a similar manner co the case of 5 above, a lower bound for ~ can be obtained from the range of 7 > 0 for which Q1 + 27~1

Qi] >o

(4.224)

q2

~3J Further, since q3 > 0 by assumption, t h i s is equivalent to Q1 - {]2T(]31112+ 27I/1 > 0 This is again a regular matrix pencil of the form F - 2B, with 2 = - 7 B = 2I~1 > 0

(4.225) (4.226)

and

r = Q1 " QTQ31~2 Using (4.190), i t is easily shown that F > 0 and F - ~min[B-1F]B > 0

(4.227) (4.228)

is a well known result from established theory where 2min[B-1F] denotes the minimum eigenvalue of II-1F.

These facts now yield a lower bound for • as

- ~,IQTQ3'q2] a _> ~1 ~min[Wllql

(4.229)

113

The s t a b i l i t y margin bounds 5 and z developed above have been derived assuming the a v a i l a b i l i t y of positive d e f i n i t e solutions for the 2D Lyapunov equation (4.190). As for the discrete case of section 4.4, such solutions may not exist but, i f they do, recently reported work, see the cited references for complete d e t a i l s , strongly suggests that they can be computed using similar algorithms to those for the d i s c r e t e case. This should then lead d i r e c t l y to e f f i c i e n t algorithms f o r computing the s t a b i l i t y margin bounds derived here. Summarising, therefore, t h i s section has considered in depth the application of r e s u l t s from the s t a b i l i t y theory of delay d i f f e r e n t i a l systems to example 2.3.4. The central r e s u l t is theorem 4.5.2 which shows that s t a b i l i t y along the pass is equivalent to pointwise asymptotic s t a b i l i t y when t h i s example is interpreted as a delay d i f f e r e n t i a l system. This r e s u l t has then been used to develop a Lyapnnov approach to s t a b i l i t y t e s t i n g which has yielded two e s s e n t i a l l y d i f f e r e n t approaches. In p a r t i c u l a r , t e s t s based on a 1D Lyapunov equation with c o e f f i c i e n t s which are functions of a complex variable and a 2D Lyapunov equation with constant c o e f f i c i e n t s have been developed. Further, as in the d i s c r e t e case of section 4.4, the f i r s t approach is necessary and s u f f i c i e n t but the second i s , in general, s u f f i c i e n t only. Detailed comparative studies with the systematic t e s t procedures of section 4.2 would, however, require the r e s u l t s from application of a l l of these t e s t s to suitably defined benchmark problems, ttere t h i s wide ranging area has been l e f t for future research with the note that i t s s u f f i c i e n t , but not necessary, basis will c l e a r l y reduce the general usefulness of the 2D Lyapunov equation approach in t h i s context. The application of the 2D Lyapunov equation approach to the problem of developing physically meaningful s t a b i l i t y margins for example 2.3.4 has been considered. In p a r t i c u l a r , some work from the delay d i f f e r e n t i a l systems area has been extended to t h i s case. Further, there are two (interrelated) areas to which future research e f f o r t could p r o f i t a b l y be directed. These are the devleopment of e f f i c i e n t computational algorithms and in depth work to e s t a b l i s h the c o r r e l a t i o n ( i f any) with system performance. Progress on the f i r s t of these areas will serve to further strengthen the already documented links between example 2.3.4 and certain classes of delay d i f f e r e n t i a l systems. In the case of the second, the f i n a l objective (as in the corresponding case f o r the d i s c r e t e process of example 2.3.8) would c l e a r l y be to produce 'easy to u s e ' , ideally within a CAD environment, s t a b i l i t y and/or performance indicators. One obvious aspect to investigate in t h i s p a r t i c u l a r case is the links ( i f any) with the recently introduced concept of a pole for example 2.3.4, defined in terms of solutions of the two variable polynomial p(s,z) of (4.199), which is the most i n t u i t i v e l y appealing d e f i n i t i o n of a c h a r a c t e r i s t i c polynomial f o r t h i s case. (See also section 6.2). (Again compare with the discrete case of section 4.4). Note also that the problem of developing s t a b i l i t y and/or performance indicators is considered again in the next chapter using the a l t e r n a t i v e simulation-based t e s t s

114

developed there. These t e s t s lead (as in the d i s c r e t e case) to the production, at no extra cost, of computable information concerning the r a t e of approach to the limit p r o f i l e in one special case of major p r a c t i c a l i n t e r e s t . Finally, chapter 6 considers the use of t h i s information in the formulation of c o n t r o l l e r design algorithms. To conclude t h i s section, return to the more general non-unit memory case of example 2.3.3. Then an obvious question to ask is whether or not the analysis of t h i s section generalises in a natural manner. Some promising preliminary r e s u l t s in t h i s area can be found in the cited reference. Notes and References The s t a b i l i t y t e s t s of sections 4.2 and 4.3 are due to Rogers and Owens (1989a,b,c, 1990a) and make use of the theory of axis and c i r c l e p o s i t i v i t y due to Siljak (1971,1973,1975). Complete d e t a i l s of both the conventional linear systems t e s t s used here and the modified Routh array can be found in Jury (1974), Gantmacher (1959) and Siljak (1971). Background on the c h a r a c t e r i s t i c locus can be found in Postlethwaite and MacFarlane (1979). Section 4.4 has evolved from the work of Boland and Owens (1980) as detailed in section 3.4. The background on the standard Lyapunov theory can, for example, be found in ~illems (1970) and that on the nD version is from Piekarski (1977). Anderson, Agathoklis, Jury and Mansour (1986) is the basis f o r the 2D Lyapunov equation, the special case of ]emma 4.4.2 is from the same source and that of theorem 4.4.1 is based on Fadali and Gnanasekaran (1989). Algorithms for computing solutions to the 2D Lyapunov equation can be found in Agathoklis, Jury and Mansour (1989). Theorems 4.4.2 and 4.4.3, the essence of the 1D Lyapunov equation approach are based on Lu and Lee (1985) as is the special case of corollary 4.4.3. Theorems 4.4.4 and 4.4.5 are based on Agathoklis, Jury and Mansonr (1990) and the background Kronecker product r e s u l t s can, for example, be found in Lancaster and Tismenetsky (1985). Agathoklis (1988) is the basis for the d e f i n i t i o n s of, and lower bounds f o r , the s t a b i l i t y margins. Rogers and Owens (1990b) contains preliminary r e s u l t s on poles for t h i s case, the role of the s t a b i l i t y margins in c l a s s i f y i n g system performance, and the extensions of these concepts to the non-unit memory case. Complete d e t a i l s of the delay d i f f e r e n t i a l s t a b i l i t y r e s u l t s which form the basis of section 4.5 can be found in llale (1977), Kamen (1982) and Agathoklis and Foda (1989a,b). The S.C.B.R. lemma is based on the work in Anderson and Vongpanitlerd (1973) and work on the use of algebraic Riccati equations in t h i s context can be found in Gu and Lee (1989). Agathoklis and Foda (1989b) forms the basis for the definitions of, and lower bounds for, the s t a b i l i t y margins. Rogers and Owens

115

(1990c) contains the d e t a i l s of the use of the 1D Lyapunov equation to provide a physically based i n t e r p r e t a t i o n of s t a b i l i t y along the pass. Details of the work to date on poles for t h i s case, the role of the s t a b i l i t y margins in c l a s s i f y i n g system performance, and the extensions of these concepts to the non-unit memory case can be found in Rogers and Owens (1990b).

CHAPTER 5

SlmtlLATIBN:BASEB STABILITYTESTS In t h i s chapter simulation-based t e s t s f o r s t a b i l i t y along the pass of the d i f f e r e n t i a l and discrete processes of examples 2.3.3 and 2.3.7 are developed. The s t a r t i n g point for these t e s t s is the assumption that suitably well behaved plant step response data is available or can be obtained by simulation studies. Further, i t is shown that these t e s t s produce, at no extra cost, computable information concerning the r a t e of approach to the limit p r o f i l e , together with bounds on the performance along any pass, in one special case of major p r a c t i c a l i n t e r e s t . This information is unique to these t e s t s and i t s use in the formulation of c o n t r o l l e r design algorithms is considered in the next chapter. Finally, some i n i t i a l r e s u l t s on extending these t e s t s to processes with interpass smoothing (see example 2.3.5) are presented. 5.1

Mathematical Background This section reviews the mathematical background necessary to develop the basic r e s u l t s of the next two sections. Complete d e t a i l s in a l l cases can be found in the cited references and the content of t h i s section begins with the following r e s u l t . Lemma 5.1.1: Suppose t h a t g e LI(O,T), d is a r e a l scalar and f(t):=

d + ~og(t')dt'

(5.1)

is bounded and continuous on the i n f i n i t e open interval 0 < t < + ~ with local maxima and minima at times t 1 < t 2 < . . . s a t i s f y i n g sup t j = + ~ in the extended h a l f - l i n e t > O.

Then, with t o = O, T NT(f) = Id] + ~ I g ( t ) i d t 0

(5.2)

where

NT(f):= If(O+)

k + k ~ l l f ( t k ) - f ( t k _ l ) ] + If(T) - f ( t k , ) j

(~.3)

and

(5.4)

Nc~(f) :=

where k

sup NT(f ) T>_O is the largest integer k such that t k < T.

Proof: Local maxima and minima of f correspond to points where g changes sign. r e s u l t now follows immediately on writing T k* tk T 0

k=l

tk_ 1

t , k

The

117 *

t

ldl + iIJ t -

g(t)dtl + I

k-1

,g(t)dtl

(5.5)

k

and noting that f(0+) = d and O

Jfl(t)dt = g(S) - g(a)

(5.6)

f o r any b > a > 0. • The quantity NT(f ) is simply the norm of f regarded as a function of bounded variation on the half-open interval 0 < t < T. Hence i t is termed the t o t a l v a r i a t i o n of f. For each f , NT(f ) is monotonically increasing and continuous as a function of T and therefore N®(f) can be obtained as N (f) = limit NT(f )

(5.7)

T~+~

Further, NT(f ) is easily computed from simple graphical operations on f ( t ) as i l l u s t r a t e d in Figure 5.1. These operations are easily included in a CAD environment and the cited reference gives f a r t h e r d e t a i l s on t h i s aspect. Note also that limitlN (f) - NT(f)[ = 0 (5.8) T~+oo

and consequently N ( f ) can be accurately estimated using data on a 'long enough' time interval 0 < t < T.

On such an interval, the continuity of NT(f ) as a function of

the stationary points t l , t 2 , . . , estimation.

implies that i t is insensitive to errors in t h e i r

I f f ( t ) is contaminated by noise n ( t ) , NT(f ) must be evaluated by

inspection o{ f ( t ) + n(t) where, under the assumption of a ' s u f f i c i e n t l y large' signal to noise r a t i o , the stationary points of f ( t ) can be estimated to 'reasonable' accuracy by visual smoothing of the recorded response f ( t ) + n ( t ) . This, together with (5.8), leads to the conclusion that the estimation of ~ ( f ) is a 'robust operation' in many p r a c t i c a l s i t u a t i o n s . In the d i s c r e t e case, the following is the equivalent r e s u l t to lemma 5.1.1. t t is stated here without proof since, in e f f e c t , t h i s follows identicai steps to that of ]emma 5.1.1 but without the complications introduced by continuity. Lemma 5.I.2: Suppose that the sequence defined by f(0) = d and r

f(r) = d + S g(j), r ~ I (5.9) j=l is bounded with local maxima or minima at sample instants 1 < r 1 < r 2 < . . . in the extended p o s i t i v e integers.

Then, with r o = 0,

r

Nr(f) = {d{ + j~lig(j){=

(5.10)

118

lI" 0

t~

tz T N r (f) = a + b + c + d

FIGURE 5.1

t3

119

where

r (r) Nr(f) = lf(O)[ +,j~1 lf(rj) - f ( r j _ l ) [ (5.11)

+ If(r) - f ( r (r))[

(5.12)

and N (f):=

sup Nr(f ) r~O where r (r) is the largest integer r l , r 2 , . . ,

satisfying r j < r.

•

The cited reference again gives further information on the evaluation of Nr(f ) for a given sequence f ( r ) . An essential underlying element of the analysis presented in the next two sections will be the following definitions and results from the theory of non-negative matrices. The proofs of the results l i s t e d are well known in the study of such matrices and are hence omitted. Definition 5.1.1: The p a r t i a l ordering < on n 1 x n 2 matrices is defined by the relation A
(5.13)

(5.14)

Aij < Bij , Vi,j Further, the 'absolute value' of an n 1

n 2 matrix A is defined to be the n 1 × n 2

real, or so-called non-negative, matrix

:

[ [AI~[

[A1n2l (5.15)

llAllp

L

JAn111

Lemma 5.1.3:

IAnln21

The absolute value, [[t[[p, of an n I × n 2 matrix A has the following

'norm like' properties

(i) [[AI[p > 0 (ii) (iii)

(iv)

[[Th[[p = [7[[[h[[p, for a l l complex scalars 7

(5.16) (5.I7)

If B is another n 1 x n 2 matrix then [[A + Blip < ]JAIl p + [[Blip

(5.18)

If B is another matrix compatible for pre-multiplication by h then ]IABJIp ~ ]]AJJpJlBIJp

(5.19)

(v) If t and B are square matrices then 0 < [[hl[p < B ~ r(A) < r(llA[Ip) < r(B) where r ( . ) again denotes the spectral radius.

(5.20)

120

Lemma 5.1.4:

I f A is an n 1 × n 1 matrix then (Inl -

Ilhllp) "l

exists and is

non-negative if, and only i f , r(llhllp) < 1

(5.21)

The following definitions and results summarise the essential required background theory from functional analysis. Again the proofs required are standard and hence omitted. Definition 5.1.2: Let X be a Banach space (subsequently specialised to X = ~ ( 0 , + ~ ) ) and Xd i t s dth Cartesian product regarded as the linear vector space of columns X = (Xl,X2,...,Xd)T of elements of X, Then the absolute value of x e Xd is defined by

]lxllp

= ( l l X l [ I , [ I x 2 1 l , . . . , l l X d t [ ) T e Rd

(5.22)

where I1.11 denotes the norm in X. Further, the norm in Rq is defined by max Ixil Ilxllq = l~i~q

(5.23)

where x e Rq is regarded as the column x = (Xl,X2,...,xq)T , and the norm in Xd is defined by

I1×11 = max llxil I

(5.24)

1Si~d

•

d2 d 1 Let B(K , i ) denote the space of bounded linear operators

Definition 5.1.3:

mapping Xd2 into Xdl.

Further, represent L e B(xd2,xdl) as

Y = Lx

(5.25)

or

Yi =

Lijx j

(5.26)

where the Lij are bounded linear operators in X.

Then the absolute value of L is

defined to be ][Ll1[[

IILld2[]

(5.27)

IILIlp = ][Ldll[[

][Ldld211

where [I-1[ is also used to denote the operator norm induced by the vector norm in X. Extensive use will be made of the following definitions and r e s u l t s for the • special case of X = L(O,+®). Definition 5.1.4:

The extended space of Xd = L~(O,+m) is denoted by Xd

e"

Further,

the natural projection of L ~ Xed into Xd(O,T) = L (O,T), regarded as a subspace of Xd, is denoted by PTL.

121

Lemma 5.1.5:

d2 d 1 Consider L e B(X ,X ) of (5.25) and suppose that its elements Lij of

(5.26) have the convolution form t (Lijxj)(t) : d i j x j ( t ) + ~oHij(t - t ' ) x j ( t ' ) d t '

(5.28)

Then PT Lij has induced norm r T [ [PTLij[ [ = Idij[ + J o ] H i j ( t ' ) [ d t '

(5.29)

iu

•

d2 d 1 Lemma 5.1.6: Suppose that L ~ B(X ,X ) has elements of the form (5.28) and denote the step response matrix of L by q(t) with elements {]ij(t). Then [[PTLij[[ = NW(qij), 1 _< i _< dl, 1 < j _< d2, and hence

[

NT(Qn)

IIPTLI[ p =

NT(qdll)

YT > 0

(5.30)

NW(qld2)] , VT > 0 NT(qdld 2)

(5.31) •

d2 d 1 Theorem 5.1.1: Suppose that the elements of L e B(X ,X ) have the structure of (5.28). Then, V T > O, [IPTLI] = ]](IIPTL]]p)]] d2 = max ~INT(Qij) 15i5d I j tlL}[ = I I ( I I P L l l p ) l l :

5.2

d2 max £1N®(qij) l
(5.32)

Stability Tests This section developes simulation-based s t a b i l i t y t e s t s for the differential and discrete non-unit memory linear repetitive processes of examples 2.3.3 and 2.3.7 respectively and their unit memory special cases. These t e s t s are based on suitably well behaved plant step response data which is assumed to be available or can be obtained by appropriate simulation studies. In the unit memory case, however, this data could (in principle) be obtained by appropriate experiments on the actual plant or process.

122

Consider f i r s t , therefore, the d i f f e r e n t i a l non-unit memory case with state- space model M

Xk+l(t) = AXk+l (t) + BUk+1 (t) + j~IBj.= 1Yk+l- j (t) M

Yk+l(t) = CXk+l(t ) + DoUk+l(t) + jEll}jYk+l_j(t)= Xk+l(t ) e Rn, Yk+l(t) e Rm, Uk+l(t ) e Rg 0 < t < a, Xk+l(O ) = O, k > 0

(5.33)

or, on solving the s t a t e equation, =

{j~IBj_ 1Yk+l- j (r) + BUk+1 (r) }dr M

+ DoUk+l(t) + ~iDjYk+l_j(t), 0 < t < a, k > 0

(5.34)

j-

Further, consider the problem in the context of the Banach space E a = Cm(O,a ) of bounded continuous mappings of the interval 0 _< t _< a into the vector space of real m-vectors R m with norm

I JYII = sup IJY(t) llm

O_
l
(5.35) Then, as shown

1

in the original analysis of example 2.3.3, (5.33) is a special case of S(Ea,Wa,La) in the product space E~a = Ea x Ea x . . . x Ea (M times) with dynamics described by the companion form recursion r e l a t i o n s of (2.23)-(2.24). 0 I 0

In p a r t i c u l a r ,

La =

(5.36) 0 LMa La M-1

0 L2a

I L1a

where (see (2.32) and (2.33) respectively) LJa, 1 _< j _< M, is defined by 4_

(LJY)(t)

CfoeA(t-r)Bj. iY(T)dr + DjY(t), 0 _< t _< a

(5.37)

and the disturbance bk+ 1 by

bk+ 1 = Cfoeh(t-r)BUk+l(r)dT + DoUk+l(t), 0 _< t <_ a

(5.38)

The analysis which follows uses as a s t a r t i n g point the so-called associated conventional l i n e a r systems of (5.33) defined by (2.53) as i ( t ) = AX(t) + Bj_IYI-J(t) $/J (t)

CX(t) + DjyI-j(t)

X(O) = O, 1 < j < M

(5.39)

123

and i t is assumed here that each member of this set is controllable and observable. (Note also again the interpretation of these systems given immediately a f t e r (2.53)). Further, the following assumptions concerning the step response matrix of each of these systems are invoked. hssu_mption 5.2.1: Write the jth element of (5.39) in the convolution form Wj = LJYl ' j where t • 1 " (LJyI-J)(t) = IoHJ(t')Y - J ( t - t ' ) d t ' + DjyI-J(t)

(5.40)

and H(t) is the m×m impulse response matrix Hi(t) = C eltBj_l

(5.41)

Then it is assumed that the step response matrix +

wJ(t) :

+ Dj, t

0

(5.42)

of t h i s element is available and i t is convenient to write t h i s matrix in the form

w1(t) wJ(t) =

.

W~l(t)

W m(t) (5.43)

.

W~m(t)

Here W~v(t) denotes the response of the pth output channel to a unit step in the vth input channel. kssumption 5.2.2: required that

wJ(t) is assumed to be a stable response.

Formally, it is

IlwJ(t)[lm ~ ~o]lHJ(t')]]mdt' + ]lDjllm < + ®

(5.44)

uhere I I . l l m = m.ax~](.)ijl is the matrix norm induced by the vector norm lJ ][.ll m = max (.)i [ in R m. 1

Note that for (5.44) to hold all eigenvalues of the matrix h must have strictly

negative real parts. This is also, see (b) of theorem 3.3.7, a necessary condition for s t a b i l i t y along the pass. Further, i t is assumed here that wJ(t) is available from appropriate simulation studies on the jth, 1 < j < M, element of (5.39). In the unit memory case, however, the corresponding matrix could (in principle) be obtained by appropriate experiments on the actual plant or process. Suppose now that Ea = Cm(O,a) is replaced by Ea = L:(O,+~) in the abstract model

S(Ea,Wa,La) where

La has the block companion structure of (5.36)-(5.37).

define L 6 B(xN,xN), XN = L~(O,+~),- N = mM, as

Further,

124

0 I L

:

0]

0

0

Z

LM

L2

L1

(5.45)

In which case i t follows immediately that the natural projection, see d e f i n i t i o n 5.1.4, of L E XNe into XN(o,a) = LN(0,a) is just L of (5.36), i . e . (5.46)

PaL = L a , 0 < a < +~ and

Poo L=L (5.47) Now apply the r e s u l t of lemma 5.1.1 to each element in turn of wJ(t), 1 < j < M, to construct the matrix I]P®L311p of (5.31) and hence the N×N block companion matrix

I ILl Ip =

Ii 01

(5.48)

0 Im ILMIIp t lL211p I lLll]p

Then it follows immediately that the following application of the partial ordering of definition 5.1.1 holds ItL~ltp~ fiLlip, 0 < ~ < + ~ (5.49) The following is the central result of this section and expresses stability along the pass of example 2.3.3 in terms of the matrix l]Ll]p of (5.48). Theorem 5.2.1: Suppose that the matrix JILl]p of (5.48) has been constructed for the differential non-unit memory linear repetitive process of example 2.3.3. Then the extended linear repetitive process generated by the model of this

S(Ea,Wa,La)a>ao

example with a ~ ao is stable along the pass if

r(llLllp)

< I

(5.50)

Proof: In effect, this consists of showing that (5.50) is a sufficient condition for the general stability along the pass result of theorem 3.3.2 to hold, i.e. (a) r = sup r(La) < I (5.51) ~O and (b) Mo:= sup sup ]J(zI- La)-1]] < + ~ (5.52) ~ a o Izt~ for some real number I e(r,l).

Further, in the case of (5.51) note again the

partial ordering of (5.49) and apply (v) of lemma 5.1.3 to yield r(La) < r(llLalip) < r(llLllp) < I, 0 < a < + ~ Hence (5.50) is clearly a sufficient condition for (5.51) to hold.

(5.53)

125

To prove (b), f i r s t note that if r(La) <

Izl

1 La then ~(I - ~_)-1 can be represented

by the absolutely convergent power series 1 La 1 LaLa

~(I -

z.

)-1

(5.54)

=~(i+~-+()2+...) Z

Further, for 0 < a < + ~,

I II(zI - L~)-IIIp < ~ -

IIL~IIp j~O

]~T j~O_

(5.55)

zl j

I zlj

1 LII~ _ E [j (5.56) < ~ j>O z on application of (i)-(iv) of lemma 5.1.3 to (5.54) and use of the partial ordering (5.49). Now consider the term = 1~

s

(5.57)

j~O Izl j of (5.56) and note that (5.50) implies the existence of a real number ~ in the range r® 5 r(llLllp) < A < 1 and hence

1

LII~

~ ~ j~O

Aj

= (AIN-

[LIIp)-l:= M < + ~

Combining (5.58) and (5.56) yields I I ( z I - L~)-IIIp~M, O< a< +~,

(5.58)

V Izl k ~

(5.59)

Taking the norm of (5.59) now leads immediately to the conclusion that sup sup [ l ( ] l ( z I - L~)-lIIp)ll = sup sup II(zI - La)-III

IIMII < +~

(5.60)

and the proof is complete. • At this stage, note that the i n i t i a l entries in wJ(t), 1 < j < ~, of (5.42) or (5.43) are simply the elements of the matrix Dj and hence the entries in [IDI[p = [[limit(PTL) IIp T~O+

(5.61)

O] 126

are given by 0

Im

llDllp = 0

l]VMllp

0

l lD211p

(5.82)

Im )[Dt))p

Further, note again from (a) of theorem 3.3.7 that (5.33) is asymptotically stable, and hence a necessary condition for s t a b i l i t y along the pass holds, i f , and only if, the spectral radius of the matrix D is s t r i c t l y less than unity. Application of the spectral radius inequality r(D) < r(I]Ol]p) (use (v) of lemma 5.1.3) now leads immediately to the following result which, given (5.49) and (v) of lemma 5.1.3, is, in effect, a simple preliminary t e s t for the applicability of theorem 5.2.1 to a given example. Lemma 5.2.1: The d i f f e r e n t i a l non-unit memory linear repetitive process of example 2.3.3 is asymptotically stable if r(lIDtlp) < 1 (5.83) where the matrix []DI] p is defined by (5.62).

•

Given theorem 5.2.1 and lemma 5.2.1 the following steps now, in effect, represent a systematic s t a b i l i t y t e s t procedure for example 2.3.3 in the general case.

STEP 1: Obtain the wJ(t), 1 < j < M, of (5.42) or (5.43) by appropriate simulation studies on the associated conventional linear systems of (5.39). Experience has shown that the 4th order Runge Kutta method for the numerical integration gives sufficient accuracy in most cases. For comprehensive details of this aspect, see the cited reference which reports progress to date on the development of a comprehensive computer aided analysis/design package for examples 2.3.3 and 2.3.7. STEP 2: Compute [[DI[ p of (5.62) either directly (since the entries in the matrices Dj, 1 < j < M, are assumed known here) or by use of (5.61) and hence t e s t lemma 5.2.1. Stop if this lemma does not hold since it is, in effect, a preliminary test for the applicability of theorem 5.2.1. STEP 3: Compute the matrix [[P LJ[[p ~ llLJ[[p of (5.31) and hence N × N block companion matrix ]]L[[p of (5.48), by applying the result of lemma 5.1.1 to each element in turn of wJ(t). STEP 4: Compute r(]]L]lp) and the example under consideration is stable along the pass by theorem 5.2.1 if r(tlL]]p) < 1.

127

Suppose, therefore, that wJ(t), 1 < j < Y, is available and step 2 yields a positive answer. Then, in effect, the above procedure reduces to the computation of lIP LJ[Ip, followed by the construction of IILllp and the evaluation of its spectral radius. Hence the computationally feasible s t a b i l i t y t e s t arising from theorem 5.2.1 is suitable for software implementation and therefore for inclusion within a CAD package. (See also the cited reference). Note also that a number of special cases exist where it is possible to obtain an explicit formula for lIP LJ[I p with the consequent possibility of obtaining 'synthesis type' results for use in design studies. The following examples i l l u s t r a t e this and other features of the application of theorem 5.2.1 using the above systematic procedure. Example 5.2.1 - The Monotonic Sign Definite Case - Suppose that entry (p,q), 1 < p,q < m, in wJ(t), 1 < j < M, is monotonic and sign definite. Then i t follows immediately that only its steady state value is required to compute the corresponding entry in []PLJ]Ip. Further, if a l l entries in wJ(t) are monotonic and sign definite then [[LJ[Ip = I[wJ(+~)llp

(5.64)

which is particularly easy to compute. Alternatively, since the entries in the matrices of i t s state-space model are assumed known here, write the j t h element of (5.39) in transfer-function matrix terms and apply the final-value theorem to yield IlwJ(+®)llp = IIGj(o)llp = tl-CA-1Bj_I + Dj][p (5.65) A number of practically relevant special cases exist where (5.64) and (5.65) can be used to great effect. For example, consider the SISO unit memory sub-class where the transfer-function of associated conventional linear system has the form n-1 (s - zi) i=l Gl(S) = ~ n (5.66)

H (s- ~i)

i=l where B is a positive real scalar.

Suppose also that the zi, 1 < i ~ n - 1 and the

2i' 1 < i ~ n, are real, distinct and negative and s a t i s f y the interlacing condition 21 < Zl < ~2 < "'" < Zn-1 < An

(5.67)

Then it follows immediately on considering the coefficients in the partial fraction

¢1(s)

expansion of

~

that wl(t) is monotonic and sign definite in this case.

(since IIL]]p ~ IILlllp in the unit memory ease)

Hence

128

n-1 zi

II

i=l

I ILl Ip : Pl--~--t

(5.68)

II hi i=l and use of theorem 5.2.1 yields s t a b i l i t y along the pass i f n-1 n

~iln1.= zii

<

lin 1 '~il

(5.69)

Example 5.2.2 - The SISO Underdamped Second Order Lag Case - Let the process under consideration be SISO and, for simplicity, unit memory. Suppose also that the associated conventional linear system transfer-function is an underdamped second order lag, i . e .

BJn Gl(S)

s 2 + 27~ns + ~2n

(5.70)

where 7 e(O,1) is the damping r a t i o , wn is the undamped natural frequency, and p is a positive real scalar.

Then in this case I ILl Ip can be computed from the closed form

,ILl,p= ;11 + e-a)e_ a

(5.71)

where 7~

a=41_ 2

(5.72)

To prove (5.71), f i r s t note that the unit step response in this case is easily shorn to be given by w1(t) = ~(I + - -

(~-3)

+

(3-~))

(5.73)

where

= "7 ~n- i ~ 41 - 72

(5.74)

and ~ is the complex conjugate. Further, solving the equation ~1(t) = 0 yields the time sequence k~ t k = Vn~l-72 ,

k = 0,1,2,...

(5.75)

and hence, using (5.73) and (5.75), ~l(tk) = Z(1 + (-1)k2A e-7~tkcos #)

(5.76)

where A and ~ are deduced from the polar decomposition _ ~ = A e i#

(5.77)

129

Hence wl(to) = 0 and wl(tk ) - wl(tk_l ) = ~((-1)k2A e ' ( k ' l ) a ( e ' a + l ) c o s ~), k ~ 1

(5.78)

which immediately yields IILtlp = p((1 + e-a)]2A cos ~lk~le'(k'l)a )

(5.79)

The closed form of (5.71) now follows immediately on summing the i n f i n i t e geometric series in (5.79). Use of theorem 5.2.1 now yields s t a b i l i t y along the pass in this case if fl(1 + e"a) < 1 - e-a (5.80) Finally, note that if 7 ~ 1 (the c r i t i c a l l y damped and overdamped cases respectively) then example 5.2.1 applies and I[LIIp = ~. Example 5.2.3 - A Numerical Study - To i l l u s t r a t e one element of the computer aided analysis/design package referred to e a r l i e r in this section, consider the unit memory process described by ik+l(t) :

Yk+l(t) =

Xk+l(t ) +

[::]

Uk+l(t) +

[0:0] 0.66

Yk(t)

1 Xk+l(t)

0 < t < a, Xk+l(O) = O, k > 0

(5.81)

The associated conventional linear system in this case has transfer-function matrix 1 Gl(S)

= s 2 + 1.6s

"0.33s + 0.33 + 4

0.33

and the elements of wl(t) are shown ia Figure 5.2. each of these in turn yields

I tLI

=[0.278

0.903]

tp [ °'139

o.636]

0.66s - 1.584]

0,66s + 1056jl

(5.82)

Application of lemma 5.1.1 to

(~.S3)

Further, r(]]L]]p) = 0.854

(5.84)

and this process is stable along the pass by theorem 5.2.1. At this stage, r e s t r i c t attention tO the unit memory case. Then for such processes an alternative version of theorem 5.2.1 exists which will find particular use in the next section. To derive this result, f i r s t suppose that IILltp z []L1]lp has been constructed and apply theorem 5.1.1 to compute the scalar IILII as

130

0.09 -

0.20

-0.03 -0.14 -

W],(tl

W ' l l z l t | O,ZO.

0.10

o0,37 °0.49 0.00

~

~

~

~

~

~

~

-0.60

,~ 1'o

TIME

0.50 ]

0.20

040 0.30"

w~,(t)

w ~(t~

0.10'

0.20. o.10

0.00

0

. 1

.

2

.

3

.

. 4

.

5

.

6

.

. 7

. 8

o.oo 9

10

o

i

~

~

~

~ TIME

TIME

FIGURE 5.2

~

~

~

~

1'o

131 m

IILII = II(llLIIp)ll = max ='SlN(W~J(t)) 1Siam j Further, combining the spectral radius inequality r(llLllp) < II(llLllp)ll = IILI] with (v) of lemma 5.1.3 yields r(L a) ~ r(llL~llp) S r(llLllp)

(5.85)

(5.86)

~ IILII, 0 < a <

+

~

(5.87)

and it follows immediately that ]]L][ < 1 is a sufficient condition for s t a b i l i t y along the pass in this case. h result which is stated formally as follows. Theorem 5.2.2: Suppose that IILII of (5.85) has been computed for the d i f f e r e n t i a l unit memory linear repetitive process of example 2.3.4. Then the extended linear repetitive process generated by the model of t h i s example with

S(Ea,Wa,La)a>ao

a > s o is stable along the pass if IILII < 1 (5.88) The result of theorem 5.2.2 in its current form cannot be applied to non-unit memory processes since the block companion structure of (5.48) in this case means that IILII is always at least equal to unity. Further, in the unit memory case it follows immediately from (5.87) that theorem 5.2.1 is less coavervative than theorem 5.2.2 in the sense that it can be applied to a wider class of examples. In particular, it can be applied to processes where r(]]L]]p) < 1 but ]]L]] ~ 1, a

situation which is highlighted by the following example where fl is a real scalar.

ik+l(t )

=

I::]

Xk+l(t )

Uk+l(t)

+

-

Yk+l(t) =

0

1

Yk(t)

+

0

1

0 < t < a, Xk+l(O) = O, k ? 0

(5.89)

I t is easily shown using, £or example, the f i r s t (eigenvalue based) systematic t e s t procedure developed in section 4.2 that (5.89) is stable along the pass for all possible choices of ft. Alternatively, suppose that theorem 5.2.1 or 5.2.2 is to be used. Then the associated conventional linear system of (5.89) has transfer-function matrix

[1/s+3 GI(S) = fl/(s+3)(s+2)

0 ] l/s+2

(5.90)

132

Further, inspection of the non-zero elements in Gl(S ) immediately indicates that they a l l have monotonic sign definite step responses and hence, by (5.64) and (5.65) Of examplelli[5'2'1'=[1/3

,,L,,p

L BI6

0 ]

(5.91)

1/2

I

Hence theorem 5.2.1 holds since r(llL]]p) = ~ and theorem 5.2.2 holds if 1 I I L I I = I I ( l l L l l p )11 = i + Z/6 < 1

(5.92)

In t h i s case, therefore, theorem 5.2.1 holds for a l l possible choices but theorem 5.2.2 produces an inconclusive r e s u l t for a l l fl > 3. Note: The constraint on applying theorem 5.2.2 to non-unit memory processes can (in principle) be removed by use of appropriate s i m i l a r i t y transformations. This p a r t i c u l a r aspect requires much further development ~ork, however, and is l e f t here as a topic for possible future research. The t e s t s based on theorem 5.2.1 or 5.2.2 are s u f f i c i e n t , but not necessary, and examples are easily generated where they produce an inconclusive r e s u l t . Further, the s t a b i l i t y along the pass c h a r a c t e r i s t i c s of such examples can only be determined by using necessary and s u f f i c i e n t t e s t s such as those developed in chapter 4. Consequently these t e s t s are not as generally applicable as t h e i r counterparts of chapter 4. Suppose, however, that theorem 5.2.1, or theorem 5.2.2 in the unit memory case, holds f o r the p a r t i c u l a r example under consideration. Then i t will be shown in the next section that these t e s t s produce, at no extra cost, computable information concerning the rate of approach to the limit p r o f i l e , together with bounds on the performance along any pass, in one case of major practical i n t e r e s t . This information is unique to these t e s t s and the next chapter considers i t s use in the formulation of c o n t r o l l e r design algorithms. To conclude t h i s section, the analysis below develops a refinement of the above t e s t s based, e s s e n t i a l l y , on f i l t e r i n g the elements in the step response matrices of the associated conventional linear systems. Following t h i s , the extension of a l l the results developed in t h i s section to the discrete non-unit memory process of-example 2.3.7, or i t s unit memory version of example 2.3.8, is noted. In the case of the former topic, i t is shown how the t e s t s developed to date in t h i s section can be refined by f i l t e r i n g ~ J ( t ) , 1 < j < M, of (5.39) with a f i l t e r defined in frequency domain, or transfer-function, terms. Its e f f e c t i v e basis is the following result. Lemma 5.2.2: Suppose that L is a bounded linear convolution operator mapping ~(0,+®) into i t s e l f with transfer-function L(s). Then

IL(s)l

(YO, v Re{s}

0

where YL(t) denotes the unit step response of L.

(5.93)

133 Proof:

Write Y = LU as t Y(t) = ~ H ( t ' ) U ( t - t ' ) d t ' 0 and note that

(5.94)

Y (t) : I:.(t')dt'

(5.95)

L(s) = ~0e-stH(t)dt

(5.96)

Further,

and hence IL(s)l < ~ole-StllH(t)Idt <~oIH(t)ldt,

VRe{s} >_ 0

(5.97)

The result now follows immediately on using the t o t a l variation result of lemma 5.1.1. • Consider now the SIS{} unit memory case and suppose that the step response of the associated conventional linear system s a t i s f i e s assumption 5.2.2 or, equivalently, a l l eigenvalues of the matrix A have s t r i c t l y negative real parts. In which case it follows immediately from lemma 5.2.2 that IGl(i~)l _< ]lGll I _< N (W1), V real u (5.98) where

I IGlll

= sapIGl(iu) l (5.99) w Further, it follows immediately from corollary 3.3.10 that this special case is stable along the pass i f , and only if,

I lCll I

< I

(5.100)

Suppose also that wl(t) is monotonic and sign definite. Then the following is a useful preliminary result which strengthens theorem 5.2.1 (or theorem 5.2.2) to a necessary and sufficient condition in this special case. Lemma 5.2.3: Consider the case when the unit memory process of example 2.3.4 is SISD. Suppose also that the step response of the associated conventional linear system is monotonic and sign definite and Nm(wl) = [wl(+m)[ has been computed. Then the extended linear repetitive process

S(Ea,Wa,La)a>aogenerated by the model in this

case with a > ao is stable along the pass i f , and only i f ,

~(W 1) < 1

(5.101)

Proof: Sufficiency follows immediately from (5.98)-(5.100). To show necessity, it is required to prove that IJ~llJ = ~(W 1) = IVl(+~)l (5.102)

134

where N (W1) = IWI(+®)I follows from example 5.2.1.

Further, t[Gll I < N (W1) by

(5.98) and (5.102) follows immediately since GI(O) = WI(+~) by definition. Note: Using lemma 5.2.3, (5.69) is necessary and sufficient for s t a b i l i t y along the pass of the sub-class of example 2.3.4 whose associated conventional linear system is defined by (5.66). Continuing with the SISfl unit memory case, return to (5.98)-(5.100) and discard the monotonic sign definite assumption. Then N(W 1) < 1 is an upper bound on the frequency dependent necessary and sufficient condition IIGll I < 1 for s t a b i l i t y along the pass. Further, it is clearly the best frequency independent upper bound on this condition. Hence it is to be expected that there is an infinite number of frequency dependent upper bounds on IIGlll which can be used to refine the sufficient condition N(W 1) < 1.

The following result is used below to characterise a class of bounds

which can be obtained by f i l t e r i n g operations on wl(t). Note that these bounds do not require detailed knowledge of GI(S ). (Recall from the discussion immediately after assumption 5.2.2 that wl(t) can (in principle) be obtained by appropriate experiments on the actual plant or process.) Lemma 5.2.4: Let L be a bounded linear convolution operator mapping L(O,+~) into i t s e l f with transfer-function L(s).

Suppose also that YL(t) denotes the unit step

response of L and let F# be a f i l t e r with the properties that:

(a) Y~:= F3YL e L (0,+~); and (b)

F~I(s) is bounded and analytic in the open right-half plane.

Then Ii(s) l < Aft(s):= tF~I(s)IN(Yg), V Re{s} > 0 Proof:

(5.103)

Write L = F~I(F~L) and apply lemma 5.2.2 to F~L.

Applying lemma 5.2.4 to the SISO version of example 2.3.4 now yields s t a b i l i t y along the pass if ll~lll < A < I (5.104) where A:= s~plF~l(iw) lN (W~) and N(W~)-- is the total variation of F~W1.

(5.105) Further, the choice of F~ = 1 reduces

this new condition to that of theorem 5.2.1 (or theorem 5.2.2), i.e. IlClt I < N(W 1) < 1

(5.106)

In general, however, F~ yields a frequency dependent hound capable of producing more refined results which approach the necessary and sufficient condition of (5.100) for

135 this case.

For example, i~ F#(s) = G~l(s) then W~(t) m 1 and hence (5.105) reduces

to (5.106) in this p a r t i c u l a r case.

In practice, however, t h i s choice of F~(s) is

not available but i t is i n t u i t i v e l y obvious that other choices, with a simpler structure, cam be used to produce easily computed intermediate estimates for the upper bound on IIGlll imposed by N (W1). The parameter h in (5.104) can be replaced by one obtained from the use of a collection, or set, of f i l t e r s . In particular, suppose that {Fi}l
IIGlll

< AA < 1

(5.107)

inf Ai

(5.108)

where AA:=

1~i~

and h i is computed as per A of (5.105) with F~ replaced by Fi and N®(W~) by N(FiW1 ). Finally, combining (5.106) and (5.107) leads to the following result which summarises the potential refinement of theorem 5.2.1 (or theorem 5.2.2) possible for the SISO unit memory case by appropriate use of the f i l t e r i n g operations detailed here. Its effective operating range is for processes where a A < 1 but N®(W1) > 1. Lemma 5.2.5:

The extended linear r e p e t i t i v e process

S(Ea,Wa,La)a>aogenerated by the

SISO version of the model of example 2.3.4 with a > a o is stable along the pass i f min[AA,N (W1)] < 1

(5.109)

Profitable use of these f i l t e r i n g operations requires, of course, the choice of an appropriate f i l t e r set {Fi}l
(5 ,,0)

whose unit stepresponse wl(t) is omitted here. Application of available software yields ~ ( v l ) = 1.64 and hence theorem 5.2.1 (or theorem 5.2.2) produces an inconclusive result. Consider, therefore, the use of the f i l t e r 0.1(1 + 4.5s) 2

F~sj''=

s(l+~s)

where 7 is a positive real scalar.

(5.111) Figure 5.3 then shows the graphs of &A(S) for the

particular choices of 7 = 0.1 and 1.0 which have been chosen here for

136

Noo(Wl )

.,..

o

.~=1

' ~:.17~ 0

Frequency FIGURE 5.3

..............

2.0

137

i l l u s t r a t i v e purposes only.

I t now follows immediately that AA < 1 for both of these

cases and hence s t a b i l i t y along the pass by lemma 5.2.5. To generalise t h i s f i l t e r i n g analysis, consider f i r s t the multivariable version of the unit memory process of example 2.3.4. Suppose also that the step response matrix of the associated conventional linear system s a t i s f i e s assumption 5.2.2 or, equivalently, a l l eigenvalues of the matrix A have s t r i c t l y negative real parts. In which case applying lemma 5.2.2 to each element in turn of Wl(t) yields ]lGl(S) llp A IlLllp , WRe{s} ~ 0 (5.112) Use of (v) of l e n a 5.1.3 now immediately yields r(Gl(i~)) < ~ < r ( l l L l l p ) , V r e a l w

(5.113)

where = sup r ( l l G l ( i u ) l l p )

(5.t14)

Further, i t is easily concluded from corollary 3.3.10 that (note again the assumption on the eigenvalues of the matrix A) any real number in the range ~ < x < r(llLIIp) satisfying x < 1 is a s u f f i c i e n t condition for s t a b i l i t y along the pass. I t is easily shown that the result of theorem 5.2.1, i . e . r(]lL]]p) < 1 is the best frequency independent upper bound on the necessary and s u f f i c i e n t conditions of corollary (3.3.10) for s t a b i l i t y along the pass. Further, i t is to be expected that there is an i n f i n i t e number of frequency dependent upper bounds on these conditions which can be used to refine the result of theorem 5.2.1. The following analysis characterises a class of bounds which can be obtained by f i l t e r i n g operations on wl(t) which are, in e f f e c t , just element by element applications of t h e i r SISD counterparts derived e a r l i e r . Note also that, as in the SISO case, these bounds do not require detailed knowledge of the elements of Gl(S ). Consider, therefore, element ( r , q ) , t ~ r < m, 1 < q < m, of ~ l ( t ) , written Wr,q(t), and l e t IF t ri ' q lJl
Suppose also that the members of this set s a t i s f y the conditions of lemma 5.2.4 in the sense that (a) ~1 := F~'qw I e ~ ( 0 , +~); and r,q 1 r,q (b) (F~,q(s))- 1 is bounded and analytic in the open r i g h t - h a l f plane. Further, construct the mxm matrix Fillip with element ( r , q ) , written ~r,q' given by ~r,q:

i~5i

(5.115)

~here ai

= supl(F~'q(i$))'l]N(g~,q)__-

(5.116)

138

Then i t follows immediately that example 2.3.4 is stable along the pass i f r(l[~llp) < 1

(5.117)

Finally, combining (5.117) with the r e s u l t of theorem 5.2.1 leads to the following theorem which summarises the potential refinement possible by appropriate use of the f i l t e r i n g operations detailed above. Theorem 5.2.3: The extended linear r e p e t i t i v e process generated by

S(Ea,Wa,La)a>ao

the model of example 2.3.4 with a > a o is stable along the pass i f min[r(llNllp) , r ( l l L l l p ) ] < 1

(~118)

Alternatively, compute [[~[[ = [ [ ( [ [ 5 [ [ p ) [ [ .

In which case, the following result

summarises the potential refinement possible by use of the (less generally applicable) result of theorem 5.2.2. Theorem 5.2.4: The extended linear r e p e t i t i v e p r o c e s s generated by

S(Ea,~a,La)a>ao

the model of example 2.3.4 with a ~ a o is stable along the pass i f

minEIl~ll, I lnll] < 1 (~110) Finally~ note that the effective operating range of theorem 5.2.3 is for processes where r(ll~lip) < 1 but r ( l i L l l p ) ~ I and that of theorem 5.2.4 is for processes where I1~]] < 1 but ILL[[ > 1. Further, note also that p r o f i t a b l e use of these f i l t e r i n g operations requires the development of rules for choosing appropriate f i l t e r sets. ~ere, however, the development of such rules is, as in the SISO case, l e f t as a future research area. The f i l t e r i n g operations detailed above extend in a straightforward manner to the non-unit memory process of example 2.3.3. In particular, suppose that a f i l t e r set, specified as in the unit memory case, is applied to each element of the step response matrix wJ(t), 1 < j < M, of the jth associated conventional linear system and denote the resulting matrix by ][~jllp. Further, construct the following N×N block companion matrix from the [[~j[[p, I < j < M 0

0

Im

II~llp = 0

ll~MIIp

0

Im

1

(5.120)

|

J

]]~2]]p ]]~lllp Then the following result is the formal statement of theorem 5.2.3 for the non-unit memory case. Theorem 5.2.5: The extended linear r e p e t i t i v e process generated by

S(Ea,Ya,La)a>ao

the model of example 2.3.3 with a > a o is stable along the pass i f

139

min[r(ll~llp) , r(I]LIIp) ] < 1

(5.121)

where ]l~[]p is defined by (5.120) and IILItp by (5.48). I

As the final item in this section, consider the discrete non-unit memory linear repetitive process of example 2.3.7 with state-space model M

Xk+l(P+l ) = ~ Xk+l(P) + A Uk+I(P) + jE1Aj_IYk+I_j(P)= M

Yk+l(P) = C Xk+I(P ) + DoUk+I(P) + jE1DjYk+I_j(P ) = Xk+l(F ) e Rn, Yk+I(P) e Rm, Uk+I(P) e R£ 0 ~ P ~ a, Xk+l(O) = O, k > 0

(5.122)

Then, as shown in the analysis o~ example 2.3,7, (5.122) is a special case of S(Ea,Wa,La) in the product space EM a = Ea x Ea × ...x Ea(~ times) with dynamics described by the companion form based recursion relations of (2.23)-(2:24). In particular, Ea is defined by (2.46)-(2.47), La by (2.24)(or (5.36)), L~, 1 < j < M, by (L~Y)(P) = P~lc~P-I'raj. IY(r) + DjY(P), r=O and the disturbance bk+ 1 by

0 < P _< a

(5.123)

bk+ 1 = P~Ic~P'I'rAUk+I(r) + DoUk+I(P), 0 < P < a (5.124) r=O Further, the associated conventional linear systems of (5.122) are defined by X(P+I) = ~ X(P) + Aj_IYI-J(P ) vJ(P) = C X(P) + DjyI-j(P) X(O) : O, 1 < j < M (5.125) Suppose now that each member of the set (5.125) is controllable and observable. (Note again the interpretation of these systems given in section 2.4). Further, introduce the assumptions detailed below concerning the step response matrices of the associated conventional linear systems. Then it follows immediately that all of the results developed in this section extend in a natural manner to (5.122)~ or its unit memory special case, with all required total variation computations undertaken using lemma 5.1.2. Hence the details are omitted. Assumption 5.2.3: Write the jth element of (5.125) in the convolution form ~J = LJY1-j where P (LJyI- j) (P) = E HJ(r)yI-j(P-r) + DjY(P) r=1 and HJ(r) is defined by HJ(r) = c~r'IAj_I

(5.126)

(5.127)

140

Then i t is assumed that the step response matrix wj(p ) = P ~ H J ( r ) + Dj, P > 0 (5.128) r=l of t h i s element is available and i t is convenient to write t h i s matrix in the form

W l(P)

w m(P)

wJ(P) =

(5.129)

Here W~q(P) denotes the response of the rth output channel to a unit step in the qth input channel. Assumption 5.2.4: required that

~J(P) is assumed to be a stable response.

]]wJ(P)[Im < r~llIHJ(r) l[m + llDjIIm < + ~

Formally, i t is

(5.130)

where Jl.l] m is defined as in assumption 5.2.2. 5.3

Performance Bounds The basic underlying theme of this section is the use of the simulation-based t e s t s of the previous section as a basis for the development of physically meaningful computable performance bounds for examples 2.3.3. and 2.3.7, or t h e i r unit memory special cases. In particular, i t is shown that these t e s t s produce, at no extra cost, computable information concerning the rate of approach to the limit p r o f i l e , together with bounds on the performance along any pass, in one special case of major practical i n t e r e s t . This information is unique to these t e s t s and the next chapter considers i t s use in the formulation of c o n t r o l l e r design algorithms. To motivate the results developed in this section, consider the d i f f e r e n t i a l process of example 2.3.3 under asymptotic s t a b i l i t y and hence, see theorem 3.1.4 and the discussion immediately following, the corresponding limit p r o f i l e is described by the state-space model ijt)=

(A +

m-

+ (B +

m

Y ( t ) = (I m - D)'Ic X ( t ) + (I m - D)'tDoU(t ) where

M

(5.131)

M

= ~IBJ_I , = ~ B. (5.t32) j j=l J Hence, in e f f e c t , the r e p e t i t i v e dynamics of the process of example 2.3.3 under asymptotic s t a b i l i t y can, a f t e r a ' s u f f i c i e n t l y large' number of passes, be described by a conventional linear systems state-space model. Further, other work, see the cited reference and the next chapter for f u l l details, has considered how this fact can be exploited in terms of the development of physically meaningful control policies and attendant controller design algorithms. In p a r t i c u l a r , consider the

141

often encountered physical situation where the control sequence applied is constant from pass to pass, i . e . Uk+1 = U , k > O, and hence bk+1 = b in (5.38) of

S(Ea,~a,La).

Then a number of control policies f o r this case have been formulated

where computable information concerning the following aspects is an essential item: (i) the rate of approach of the output sequence {Yk}k>1 to the limit p r o f i l e Y ; and ( i i ) bounds for the error Yk- Y~ on any pass k ~ O. fine common choice for U is U(s) = ~

(5.133)

where at least one element of the column vector ~ e Rg is unity and the r e s t are zero. This corresponds to the case where a unit step is applied in one or more input channels at t = 0 on each pass. Further, the following theorem is the basic underlying r e s u l t of t h i s section for the case of example 2.3.3 with a control sequence which is constant from pass to pass. Theorem 5.3.1:

Suppose that the extended linear r e p e t i t i v e process

S(Ea,~a,La)a>ao

generated by the model of example 2.3.3 with a > ao is stable along the pass and r([[Lllp) < 1

(5.134)

where [[L[]p is defined by (5.48).

Further, t e t the control sequence applied be

constant from pass to pass, i . e . Uk+1 = U , k > O, and hence bk+1 = b

in (5.38).

Then, for a e (0,+~), there exists an NxN non-negative matrix N and a real scalar 7 e ( r ( l l L I I p ) , l ) such that the error Yk - Y ' k > O, s a t i s f i e s [{Yk- ~ l [ p 5 W 7k{[[Yol[p + (I N- [ t L [ [ p ) - l [ I b lip} Proof: of

Since bk+1 = b ,

S(Ea,~a,La) ,

as

(5.~35)

k > O, the solution of the equation describing the dynamics

interpreted in the 'companion form' of (2.23)-(2.24), can be written

ky

k j.

Yk = La o + j~lLa lb~

(5.136)

Further, i t is easily shown that the corresponding limit p r o f i l e can be expressed as ~=

j!lL~-lb

(5.137)

Hence the error Yk - Y® can be expressed as

Yk Y° : L Yo

L 'b

j=k+l

(5,3S)

142

and therefore IIYk - Y®llp S IILkllp{llYollp +

~ IILIl~-l-kllb®llp} (5.139) j=k+l on application of ( i i i ) - ( i v ) of lemma 5.1.3 and use of the p a r t i a l ordering relation (5.49). To proceed, f i r s t note that since (5.134) holds then (I N - IILIIp) -1 exists, is non-negative by lemma 5.1.4, and i t is easily shown that (IN - I ILl Ip) -1 =

~ l iLl j - l - k (5.140) j=k+l P Hence i t remains to prove that there exists a non-negative matrix W > 0 and a real scalar 7 e ( r ( l l L [ I p ) , l ) such that IILk]]p < ~ 7k,

k >0

(5.141)

This follows on noting that r(L) < r(llLIIp)

< 1 by (v) of lemma 5.1.3 and therefore

i t is possible to choose real numbers ~ > 0 and 7 e ( r ( l [ L l [ p ) , l ) such that

IILkll ~ ~ 7k, k > 0 (5.142) Further, i t is clear that the p a r t i a l ordering IILkllp < Q holds where q is the NxN matrix with each element equal to IILkll. The result of (5.135) now follows immediately on using (5.140) and defining W as the NxN non-negative matrix with each element equal to ~. • Suppose, therefore, that the real scalar 7 is chosen as any number in the known range r ( l l L I I p ) < 7 < 1. Consider also the output sequence {Yk}k>l in terms of its convergence to the limit p r o f i l e . Then the computable information available from theorem 5.3.1 is the fact that this sequence approaches Y at a geometric rate governed by 7. Further, a number of refinements of this result can (in principle) be obtained by use of appropriate f i l t e r i n g operations similar to those detailed in the previous section. This particular aspect requires much further developmeut work and is l e f t here as a possible future research area. Consider now the unit memory case and suppose that theorem 5.2.2 holds. Then the following result provides an alternative to theorem 5.3.1 for this case. Theorem 5.3.2: Suppose that the extended linear r e p e t i t i v e process

S(Ea,Wa,La)a>ao_

generated by the model of example 2.3.4 with a > a o is stable along the pass and

(5.143)

IILII = II(llLIIp) ll < 1

Further, let the control sequence applied be constant from pass to pass, i . e . Uk+1 = U , k > O, and hencehk+ 1 = b in (5.38). Then, for a 6 (0, + ~), the error Yk - Y ' k > O, s a t i s f i e s

IIb®ll IIYk - •11

~ IILIIk{llYoll

+ i

- IILII }

(5.144)

143

Proof: Follows immediately on taking the norm of (5.139). • Suppose, therefore, that theorem 5.3.2 holds and consider the output sequence {Yk}k>_1 in terms of its convergence to the limit profile. Then the computable information available from theorem 5.3.2 is the fact that this sequence approaches YC~ at a geometric rate governed by fILl]. Further, in common with theorem 5.3.1, a number of refinements of this result can (in principle) be obtained by the use of appropriate filtering operations similar to those detailed in the previous section. This particular aspect requires much further development work and is again left as a possible future research area. Several potential refinements of theorem 5.3.2 exist which do not require filtering operations. Here, however, only the one based on the fact that [[Lkll _< IILl Ik is considered and the others are detailed in the cited reference. In this case, i f I ILk[[ < ][L]I k < 1 then ][Lkl[ provides an improved estimate of the r a t e of convergence of {Vk}k>l to Y . Further, [ILk[] is e a s i l y computed using the following systematic procedure which is s t a t e d here f o r the SISO case with an obvious g e n e r a l i s a t i o n to the multivariable version. STEP 1" Perform k simulations on yi = LUi where U1 is a unit step applied at t = 0 and U1 = y i - 1 2 < i < k. STEP 2:

Apply lemma 5.1.1 to compute I ILkl[ as I ILkll = N (Yk).

To i l l u s t r a t e the p o t e n t i a l f o r using theorem 5.3.2 in c o n t r o l l e r design, with an obvious extension to theorem 5.3.1, r e t u r n t o SISO case defined by (5.66) where, using (5.68), n-1 II z i i=l l lLll -- Z]--~----i (5.145) i=I Suppose also t h a t a s o - c a l l e d current pass s t a t e feedback control law has been designed, see the c i t e d reference f o r the necessary t h e o r e t i c a l background, which leaves the z i , 1 < i < n - 1, invariant and moves the 2i, 1 < i _< n, t o locations 7i which are r e a l , d i s t i n c t and negative and s a t i s f y the i n t e r l a c i n g condition 71 < Zl < 72 < . . . . .

< Zn-1 < 7n

(5.146)

Further, write the closed-loop associated conventional l i n e a r system in convolution form as W1 = LcY°. Then n-1 n zi i=l [ILcll -- ~ [ - V - - - [ (5.147) M 7i

i=l

and note t h a t IILcl I ~ 0 as 71 ~ - ~.

Equivalently, s t a b i l i t y along the pass coupled

144

with an arbitrary f a s t rate of approach to the limit profile results from an appropriate choice of 7i, 1 < i < n, satisfying (5.146) ( i . e . by placing 71 ' f a r enough' to the l e f t of the origin on the real line). In more general terms, suppose that the number 0 < b < 1 is available as a measure of the required rate of convergence of {Yk)k>1 to Y . Then clearly this requirement can be built into the design procedure as the constraint that IILcIl < b. This general area is considered again in the next chapter where a number of control policies and attendant controller algorithms are developed. Consider now the problem of obtaining bounds for the error Yk " Y ' k > O, from theorem 5.3.2.

In particular, suppose, for simplicity, that the i n i t i a l profile, ¥o'

is zero and note again the definition of ]l-I1.

IILjlk[[b~ll

lYe(t) - Y~(t) l < I]Yk- Y®I] ~

1-

I[L[[

Then it follows immediately that ' t >0

(5.148)

where the notation Y~(t) and Y~(t) denotes the ith, 1 < i < m, output channel of Yk(t) and ~ ( t ) respectively. Suppose, therefore, that [ I b [ [ is obtained as detailed below.

Then by (5.148)

Y~(t), t > O, lies in the 'band' defined by Y~(t)- 7k < Y~(t) < Y~(t) + 7k

(5.149)

where

7k:=

II~llkllb®ll 1 : ilLII

(5.150)

This band has the graphical interpretation shown in Figure 5.4 and it is obviously suitable for inclusion in a CAD environment. Further, its width decreases from pass to pass at a geometric rate governed by [[LI[. To compute lib II, f i r s t note that b® is the response of the derived conventional linear system LD(A,B,C,Do) to ~ ,

i.e. in state-space terms

~(t) : Ax(t) + B ~ ( t ) b (t) = CX(t) + DoU(t ) x(o) = o

(5.151)

Further, (5.15t) is stable since a l l eigenvalues of the matrix A have s t r i c t l y negative real parts. Suppose also that bm(t ) is available from simulation studies on (5.151) with U ( t ) .

Then it follows immediately that

145

#'J J

,I/I"

FIGURE 5.4

146

Ilb®{l = ma~ suptb~(t)l

(5.152)

lO where b~(t) denotes the ith element of b ( t ) .

In common with the conventional linear systems case, i t is to be expected that the systems response will be judged in terms of 'benchmark' choices of U ( t ) . Consider, therefore, the ith, i < i < m, channel of Yk(t).

Then one obvious choice,

again by analogy with the conventional linear systems case, is ei

v(s)

: -~

(5.153)

where e i is the ix1 column vector consisting of zeros everywhere except the ith position which consists of a unit element. Equivalently, a unit step is applied at t = 0 on each pass in t h i s channel with a l l others identically zero. An a l t e r n a t i v e to (5.152) exists in terms of an upper bound on l i b { I which, although more conservative in general, may prove computationally more a t t r a c t i v e in certain cases. To develop t h i s , f i r s t write (5.151) in the convolution form b = LbU , where Lb is defined by (5.38) with Uk+1 replaced by ~ .

Suppose also that the

step response matrix, denoted Wb(t), is available from appropriate simulation studies on (5.151) and note that this matrix s a t i s f i e s assumption 5.2.2.

IIence IILbl t can be

computed from Wb(t) as per the computation of IILII from wl(t) in section 5.2. Further,

lib®l{ ~ {ILb{{ IIUll

(5.154)

which gives a computable upper bound for lib J {

IIull

with

= ma~ suplui(t){

lO ~ where U~(t) denotes the ith input channel.

(5.155) Substituting in (5.150) now immediately

yields the following, generally more conservative, a l t e r n a t i v e 'band' to that of (5.152) where 7k :=

IILllkllLbllllUoolI i = {{L{{

As one example, suppose that ~ is specified by (5.153).

(5.157) Then 1]UJ{ = 1 and

this is one p r a c t i c a l l y relevant case where (5.156) may prove computationally more a t t r a c t i v e than (5.149). Finally, note that further consideration of the use of this computational information, together with that arising below from theorem 5.3.1, is postponed u n t i l the next chapter.

147

Note: The cited reference again details a number of potential refinements of the above analysis which do not require f i l t e r i n g operations. Consider now the problem of obtaining bounds for the error Yk " Y ' k > O, from theorem 5.3.1, where i t is instructive to consider the unit memory case f i r s t . In which case, i f ¥o is again assumed to be zero for simplicity, theorem 5.3.1 states

that IlYk - ~ l l p ~ IIMkllpllb®llp

(5.1581

IIMkllp := (I m- I[Ll[p)-lllL[Ipk

(5.159)

where and t h i s matrix has been introduced for notational convenience. Suppose also that b®(t) is again available from simulation studies on (5.151) with a given U ( t ) .

Then it follows immediately that

bmJT

(5.160)

1 ~ i <m

(5.161)

I Ib®l Ip = [bl,b2. . . . . where

hi:= supIb~(t) l,

t~O Suppose, therefore, that lib=lip has been computed.

Then it follows immediately

from (5.158) that

IIYk(t) - Y~(t)llp ~ IIYk - Y=llp ~ llMkllpllb=f[p, t ~ 0

(5.162)

Equivalently, Y~(t), t > O, lies in the 'band' defined by Y~(t)- m ~ Y~(t) < Y~(t)+ m~

(5.163)

where mki denotes the ith entry in the m×l column vector formed as the product of

l lMklIp and I1~1 lpThis band has the graphical interpretation shown in Figure 5.5 and is again obviously suitable for inclusion in a CAB environment. Further, i t s width from pass to pass is, in effect, governed by the relationship IIMk+allp = IIMkllpllLII p , k > 0 (5.164) In common with theorem 5.3.2, an alternative to (5.163) exists in terms of an upper bound on lib®lip which, although more conservative in general, may prove computationally more attractive in certain cases. To develop t h i s , f i r s t suppose again that the step response matrix, Wb(t), of b = LbU is available. Then IILbllp can be computed from Wb(t) as per the computation of ]lLItp from ~l(t) in section 5.2.

Further, lib=lip < I t L b l t p t l ~ l [ p

on use of (iv) of lemma 5.1.1 where ItVJtp = [U1,U2,... ,Vg]w

(5.165) (5.166)

148

.t~,¢I ffS" J ,,,,.,,11~ f J ..,pf~'~"*" J f jr /

FIGURE 5.5

149

and Ui:= suplUi(t) l t_>O

(5.167)

Hence

II~]lpltb®tlp ~

ll~]lpltLbllptl~llp:=

~k

(5.1~8)

and the following is a, generally more conserv~ive, alternative ~bandt to that of (5.163 /

where ~ denotes the ith entry in ~k" Finally, if ~ is specified by (5.153) then I,

i=j

=

(5.~o)

O, i ¢ j and this is one practically relevant case where (5.169) may prove computationally more ~ t r a c t i v e than (5.163). The generalisation of the analysis just completed to the ~plication of theorem 5.3.1 in the non-unit memorycase is a straightforward exercise. In particular, f i r s t note again that S(Ea,Wa,La) is defined by the Icomp~ion form I based structure of (2.23)-(2.24) i n t h i s case.

Further, I]L]lp i s g i v e n b y (5.48) andhence (5.159)

transl~es to

II~llp = (I N - IILIIp)-lflLII~

(5.171)

Suppose also t h ~ ltMkltp is written in partitioned form as [,,M~,,p

II~kllp

,,M~,,p]

=

(5.172)

IIM~llp IIM~llp where II~llp_ is of dimension m~m. Then (5.162) translates to

t~O IIYk(t)- ~ ( t ) l l p ~ l l Y k - Y ® l l p ~ l l ~ l l p l l b ® l l p , where lib®lip is again given by (5.160)-(5.161). Based on this

(5.173) result, the analysis

given ~ove for the unit memorycase generalises in a natural manner ~ d hence the details are omitted. Note: T h e r o l e s o f ~ ( t ) a n d ~ ( t ) in (5.149),(5.156),(5.163) and (5.169) c ~ , of course, be reversed. To conclude this section, the analysis below develops a refinement of the results given above on the error Yk-~' k ~ O, based, essentially, on filtering the elements in the step response matrices of the associated conventional linear systems. Following this, the extension of all results developed in this section to the discrete non-unit memoryprocess of ex~ple 2.3.7, or its unit memoryversion of example 2.3.8, is noted. In the former area, only the SISO unit memorycase is

150

considered since a l l others follow immediately as natural generalisations and the details can be found in the cited reference. Consider f i r s t XA(tA):= {f:eAtf e L (O,tA)} (5.174) and take the norm of f as the norm of eAtf in L®(O,tA).

Then the following is the

effective basis of the analysis given below for ~he SISO case and i t s natural generalisations. Lemma 5.3.1: Suppose that L is a linear convolution operator mapping L(O,+~) into i t s e l f with impulse response satisfying IhL(t) l < hoe-Tt, t ~ 0 for some 7 > 0 and let YL denote i t s unit step response.

Then for t2 < +~ the

induced operator norm of L restricted to X2(t]) s a t i s f i e s I{Lll ~ Nt (YL;~) , V~

(5.175)

where NtA(YL;A):=

IYL(O+)I +

N el3k(A) E (Ntk(YL)Ntk- 1 (YL)) k=l

(5.176)

with

~t k ,

~>0 (5.177)

Zk(~) -tk_l,

~ <0

and 0 = t o < t 1 <...< t N = t~ is any partition of [O,t2] with the property that 0 < h 1 < t k - tk_ 1 _< h2 < + ~ , k > 1.

I f t2 = + ~ , L maps X](t~) into i t s e l f for

< 7 with I ILll-< N~(YL; A) < + ~ Proof:

(5.178)

The induced norm of L in XA(tA) is tA I ILII = IYL(O+)I +-le~tlhL(t) ldt 0

N eflk(a ) i t k _< IYL(O+)I + Z IhL(t) ldt k=l tk- 1 which is just (5.176) and is clearly f i n i t e if t~ < + ~. and if ~ _< O, e~k(~) < 1 and therefore N (YL;a) _< N (YL) < + m

(5.179) If t A = + ®

then

N = + ®

(5.180)

151

Finally, i f 0 < 2 < 7 then tk ~k (~) r ]tke Jtk_llhL(t)ldt < ho(t k - tk_l)e

7tk-1

tk_l)e(~-7)tk + 7(t k - tk.1)

ho(t k

hoh2e(~-7)khle 7h2

(5.181)

and the (infinite) series in (5.176) converges. • I t is easily verified that Nt(YL;A ) is continuous and monotonically increasing in both ~ and t~ with No(YL;A) = 0 and Nt(YL;O) = Nt(YL) , i.e. the case of lemma 5.1.1 is recovered if ~ = O. Further, use of this lemma requires the availability of 7 > 0 or, at least, the availability of appropriate information to determine the admissible range, llere it will be assumed, for simplicity, that t h i s range is available in advance, with the cited reference giving f u l l details of how to obtain i t for a given example. Return now to the SISO version of example 2.3.4 and suppose that the step response, wl(t), of the associated conventional linear system is available and s a t i s f i e s assumption 5.2.2. Then this ensures that lemma 5.3.1 is applicable and use of (5.178) yields IfLtl < N®(W1;2) (5.182) where N (W1;A) is computed using (5.176).

Further,

N (W1;2) < I

(5.183)

is a sufficient condition for s t a b i l i t y along the pass and let I[.[[ 2 denote the norm in X] of (5.174), i.e. [[fll~ = sup e ~ t l f ( t ) l t~O In which case it is easily shown by mirroring the proof of (5.144), i . e .

(5.I84)

IIb®ll

[IVk - Y~II ~ IILIIk{llYoll + 1 - IILII }' k > 0

(5.185)

IlYk - Y®IIA ~ ]]LAII~{IIYolIA + 1 - IILAIIA }, k ~ 0

(5.186)

that

where IIL~II~ = N(W1;A).

~ence, with Yo again assumed to be zero for simplicity,

eAtlYk(t) - Y~(t) l ~ [[Yk - Y~IIA ~ ~k'

t >0

(5.187)

where

7k :=

1-

IILA[I A

(5.188)

152

Suppose, therefore, that [ ] b J ] ~ is available, where i t s computation is an obvious extension of the analysis detailed e a r l i e r in t h i s section for I t b ] ] hence the d e t a i l s are omitted.

and

Then by (5.187), Yk(t), t > O, l i e s in the 'band'

defined by IYk(t ) - ¥®(t) l < 7ke-At

(5.189)

Y=(t) - 7ke-At < ¥k(t) < Y=(t) + 7k e-At

(5.190)

or

This band has an obvious graphical interpretation and can easily be included within a CAD environment. I f A = O, t h i s band reduces to that of (5.149)-(5.150). Note also that the essential action of the exponential weighting, e At, is to introduce a refinement which can, in principle, be used to tighten t h i s band. In practice, it is envisaged that t h i s band will be computed for a variety of choices of A in a selection set. The cited reference contains complete d e t a i l s of t h i s p a r t i c u l a r aspect together with the r e s u l t s of some very promising i n i t i a l numerical studies. Note: The roles of Y ( t ) and Yk(t) in (5.190) can, of course, be reversed. As a final item in this section, return to the discrete non-unit memory linear repetitive process of example 2.3.7 or its unit memory version of example 2.3.8. Then it follows immediately that all of the results developed in this section extend in a natural manner to these processes where L~, 1 < j < M, is defined by (5.123) and b= by (5.124) with Uk+ 1 = U , k > 0.

IIence the details are omitted except to note

that all required total variation calculations are undertaken using lemma 5.1.2 or the discrete equivalent of lemma 5.3.1 as appropriate.

5.4

InterDass Smoothing To date, no consideration has been given in t h i s work to t e s t i n g for s t a b i l i t y in the presence of interpass smoothing e f f e c t s . This is a common feature of a number of known industrial examples, such as long-wall coal cutting, and is, in e f f e c t , the name given to dynamic interaction which occurs between passes and d i s t o r t s the previous pass p r o f i l e ( s ) . For example, the source of t h i s interpass smoothing in the long-wall coal cutting case is the machine's weight (up to 5 tonnes) as i t passes over. Consider again the simulation-based s t a b i l i t y t e s t s of section 5.2. Then the purpose of t h i s section is to consider the extension of these t e s t s to one possible method of modelling interpass smoothing e f f e c t s . This, in e f f e c t , assumes that the output at any point on the current pass is a function of the s t a t e s and inputs at t h i s point and of the complete p r o f i l e on the previous pass. In p a r t i c u l a r , the case studied is those d i f f e r e n t i a l unit memory processes which can be described by the state-space model of example 2.3.5, i . e .

153

Xk+l(t ) = AXk+l(t ) + BUk+l(t) + Bo Yk+l(t) = CXk+l(t) 0 ~ t ~ a,

Xk+l(O) = O, k > 0

(5.191)

Here the interpass interaction term Bof:g(t,r)Yk(r)dr represents a 'smoothing out' of the previous pass p r o f i l e in a manner governed by the properties of the kernel Note again that the p a r t i c u l a r choice of K(t,r) = 5 ( t - r ) I m (5.192)

K(t,r).

where 6 denotes the Dirac delta function reduces (5.191) to the case of example 2.3.4. Further, the analysis presented below extends in a natural manner to the non-unit memory version of (5.19t) (defined in an obvious manner) and to the corresponding discrete cases. Hence the details are omitted and can be found in the cited reference. The e f f e c t i v e basis for the analysis given below is the following result. Lemma 5.4.1: Suppose that L is a bounded linear operator mapping L(O,+~) into i t s e l f of the form (LV)(t) = ~:K(t,r)Yk(r)dr

(5.193)

IILII g sup ~ J K ( t , r ) Jdr t~O ~0 and equality holds in (5.194) if K > O, Vt, r. Proof: By definition

(5.194)

Then

ItLII =

sup t~O sup t~O sup su ~oIK(t,r)ldrlJY[p t~O }tY~I=I

(~.195)

sup ~OIK(t,~)ld~

t~O Further, equality obviously holds in (5.195) One choice of K(t,r) iSr[ K(t,r) = Ko e"flit-

if K > O, Vt,r.

where Ko and fl are positive real scalars and K(t,r) > O, Vt, r.

i

(5.196) This represents

so-called 'double sided exponential smoothing' and has been used, see the cited reference for f u l l details, to develop a f i r s t p r a c t i c a l l y r e a l i s t i c treatment of

154

interpass smoothing effects in the long-wall coal cutting example. easily shown that 2KO

Further, it is

(5.197)

IIsll = 7 in this case. Return now to the state-space model of (5.1911 and re-write it as Xk+l(t ) = AXk+l(t ) + BUk+l(t ) + BoVk(t) Yk+l(t) = CXk+l(t )

(5.1981

Vk(t ) = f~K(t,r)Yk(T)dT

(5.199)

where

Then, proceeding formally, the Laplace transform description of the associated conventional linear system is given by

wl(s) = ~l(s)v(s)

(5.200)

where Gl(S ) is (as before) the transfer-function matrix for the case of no interpass smoothing, and V(s) is the Laplace transform of V(t) = ~oK(t,T)Y(r)dv:= (KV)(t) where K is the integral operator with kernel K(t,T).

(5.201) It now follows that the L

induced norm of G1K is bounded above by

IILsll p = IILIIpllKIIp

(5.202/

where K:= sup r K ( t , r ) d r (5.203) tt0 J0 (The supremum being interpreted with respect to the partial ordering). The following result now provides a (computable) sufficient condition for s t a b i l i t y along the pass of (5.1911. Further, the proof of this result follows identical steps to that of theorem 5.2.1 and hence the details are omitted. Theorem 5.4.1:

Suppose that the matrix IILsllp of (5.202) has been constructed for

the d i f f e r e n t i a l unit memory linear repetitive process of (5.1911. Then the extended linear repetitive process generated by this example with a > ao is

S(Ea,Wa,La)a>ao

stable along the pass if

r(llLsllp ) < 1 Note:

As per section 5.2, r(]lLsl]p ) < 1 can be replaced by ]lLsll =

I}(JlLsllp)l}

< 1.

(5.204)

155

The cited reference contains the r e s u l t s of in depth t h e o r e t i c a l and numerical studies which serve to confirm the potential of t h i s means of representing interpass smoothing f o r c e r t a i n cases of p r a c t i c a l i n t e r e s t . As a sample of these, consider the SISO case where the unit step response of Gl(S ) is monotonic and sign d e f i n i t e and hence I[LIIp = IGI(O)I.

Suppose also that the interpass kernel is given by

(5.196) normalised such that oK(t,r)dr = 1 and hence ~ = Ko.

(5.205)

Consequently

IILstl p = IILIlp

(5.2o6)

in t h i s p a r t i c u l a r case, i . e . t h i s (normalised) interpass smoothing has no e f f e c t on the s t a b i l i t y condition of theorem 5.2.1 (or theorem 5.2.2). As a f i n a l point, note that no r e s u l t s are yet a v a i l a b l e on the extension of the s t a b i l i t y t e s t s of chapter 4 to processes with interpass smoothing. Further, i t is not immediately obvious how ( i f at a l l ) t h i s can be achieved. In p a r t i c u l a r , i t may well be t h a t the simulation-based approach of t h i s chapter is the only f e a s i b l e means of s t a b i l i t y t e s t i n g in t h i s case. h d e f i n i t e conclusion to t h i s conjecture, however, must await the outcome of much further research e f f o r t f o r which the r e s u l t s already available serve as an appropriate s t a r t i n g point. Notes and References The background material of section 5.1 is drawn from Ouens and Chotai (1983) and the references therein. Smyth (1991) discusses the associated numerical computations. Theorem 5.2.1 of section 5.2 is from Rogers and Owens (1990 d) and theorem 5.2.2 is from Rogers and flwens (1990 e). For a comprehensive treatment of the f i l t e r i n g r e s u l t s of section 5.2 see Rogers and Owens (1990 f) and Rogers and Ouens (1990 g) for the discrete versions of a l l of the r e s u l t s presented in section 5.2. Theorem 5.3.1 and 5.3.2 of section 5.3 are from Rogers and nuens (1990 d) and Rogers and Ouens (1990 e) respectively. The f i l t e r i n g r e s u l t s of t h i s section are from Rogers and Owens (1990 f , g ) . Finally, the r e s u l t s of section 5.4 are from Rogers and Owens (1990 h) which also contains d e t a i l s of the extensions referred to in the t e x t .

CHAPTER 6 CONTRflLLER DESIGN :SQMBINITIAL RESULTS This chapter presents some i n i t i a l work on c o n t r o l l e r design for the d i f f e r e n t i a l and discrete processes of examples 2.3.3 and 2.3.7. In p a r t i c u l a r , three control p o l i c i e s f o r these processes are formulated from p r a c t i c a l considerations and feedback control schemes which use e i t h e r s t a t e or output information are developed. Finally some candidate design algorithms are presented together with some systems theoretic properties, such as a return-difference matrix for the output feedback based schemes. 6.1

Control Policies and Feedback Control Sfhem~s By analogy with the conventional linear systems approach, consider the case of example 2.3.3 when there is 'no direct feedthrough' between input and output on any pass and hence the s t a t ~ s p a c e model Mp

i +l(t) = APx + (t) + BP k+ (t) + j l j_lYk+ _j(t) M

Yk+l(t) = cPx~+I(t) + j~ID~Yk+I - = j(t)

X~+i(t ) e R

nI

, Yk+l(t)

e Rm, Uk+l(t ) e Rg

ogt ~a, k>O (6.1) Then a study of industrial examples, such as bench mining systems, leads to the following three basic control policies. Note also that these p o l i c i e s extend in a natural manner to the corresponding discrete case of example 2.3.7. Hence the d e t a i l s , together with those corresponding to a l l other r e s u l t s presented in t h i s chapter, are omitted. S t a b i l i t y along the Pass - This is an obvious necessary item of any p r a c t i c a l l y f e a s i b l e control policy. The Limi~ P r o f i l e Design Problem - Suppose that the p a r t i c u l a r example of (6.1) under consideration is asymptotically stable, i . e . theorem 3.1.4 holds. Then the origin of t h i s control policy l i e s in the f a c t that the corresponding limit p r o f i l e is described by

i~(t)

= (AP + fiP(I m - DP)-IcP)x~(t)

Y ( t ) = (I m - DP)-IcPx~(t)

+ BPu(t) (6.2)

where M

BP

=

~

M

1, DP = z

(6.3)

j=l 3" j=l 3 Xence, in effect, the repetitive dynamics in this case can, after a 'sufficiently large' number of passes, be described by a conventional linear systems state-space model. Further, stability along the pass implies that (6.2) is stable, i.e. all

157

eigenvalues of the matrix AP + BP(I m - DP)'IcP have s t r i c t l y negative real p a r t s . To see t h i s , set z = 1 in the s t a b i l i t y along the pass polynomial Ap(S,Z) of (3.116) and use (3.118). Given s t a b i l i t y along the pass and (6.2)-(6.3) i t follows immediately that the r e p e t i t i v e systems behaviour a f t e r a ' s u f f i c i e n t l y large' number of passes, formally the limit p r o f i l e , can be c l a s s i f i e d in terms of well known conventional linear systems c r i t e r i a . Further, in depth studies on a number of i n d u s t r i a l examples has led to the following set of performance specifications which, in e f f e c t , c o n s t i t u t e the limit p r o f i l e design problem and which, f o r notational simplicity, is denoted by LPDP from t h i s point onwards. The use of quotation marks indicates t h a t the precise meaning of the terms within are a matter for judgement based on detailed knowledge of the p a r t i c u l a r application under consideration. (i) The process must be stable along the pass and hence the existence of a stable limit p r o f i l e described by (6.2)-(6.3) is guaranteed. Further, the limit p r o f i l e dynamics should s a t i s f y such other additional conventional linear systems performance c r i t e r i a as deemed appropriate. For example, the interaction e f f e c t s in reponse to unit step demands should be within 'acceptable l i m i t s ' . (ii) The output sequence {Yk}k>l must be within a ' s p e c i f i e d bound', or band, of of Y a f t e r a fixed number of passes, say k , and remain within i t for a l l $

successive passes k > k . (iii)

The error Yk - Y~' 0 < k < k , must be 'acceptable' Several v a r i a t i o n s of (i) - ( i i i ) above exist and are detailed in the cited reference. Further, much work remains to be done on developing rules for refining the terms in quotation marks into design c r i t e r i a which, where appropriate, should (ideally) display similar c h a r a c t e r i s t i c s to existing well used conventional linear systems ones. Recall also that the simulation-based s t a b i l i t y t e s t s of chapter 5 yield, at no extra cost, computable information concerning ( i i ) and ( i i i ) above in one special case of major p r a c t i c a l i n t e r e s t . This p a r t i c u l a r aspect is considered again in section 6.3 which presents some i n i t i a l r e s u l t s on the development of algorithms for designing the memoryless output feedback based schemes introduced below to solve t h i s problem. The R~petitive Systems DisturbanceDecouDling with S t a b i l i t y Problem - This control policy again has i t s origins in industrial examples and i s , in e f f e c t , based on regarding the previous pass p r o f i l e s as disturbances. I t s requirements can be summarised as follows and note that these obviously imply s t a b i l i t y along the pass. (i) The pass p r o f i l e Yk(t), 0 < t ~ a, should be independent of the pass p r o f i l e s Yk_j(t), 0 < t < a, 1 < j < M, for a l l passes k > k > 1 with an . optimum choice of k = 1.

158

(ii)

Suppose that (i) above holds. Then for a l l passes k > k > 1 the systems dynamics are, in e f f e c t , described by the derived conventional linear system LD(AP,~P,cP).

(Simply delete the previous pass terms.)

In e f f e c t ,

the limit p r o f i l e is reached exactly on pass k with dynamics parameterised by the t r i p l e (AP,BP,cP). Further, the dynamics of t h i s limit p r o f i l e should s a t i s f y such conventional linear systems performance c r i t e r i a as deemed appropriate. The minimum requirement here is s t a b i l i t y and hence the choice of name for t h i s control policy. This problem will, again for notational simplicity, be denoted by RSDDSP from t h i s point onwards. Further, i t has well defined s t r u c t u r a l links with both the LPDP introduced above and i t s well known, and extensively researched, conventional linear systems counterpart. The d e t a i l s of t h i s aspect can again he found in the cited reference. Section 6.4 presents some i n i t i a l r e s u l t s of the development of algorithms for designing the feedback control schemes detailed below to solve t h i s problem, fine aspect of t h i s is some i n i t i a l r e s u l t s on the extension of the well knoun geometric based theory for the conventional linear systems case, with the eventual aim of providing synthesis type conditions for the existence of a solution. I t is important to note that the control p o l i c i e s defined above are by no means exhaustive. For example, a linear quadratic optimal control problem for (6.1) can be formulated which is the natural generalisation of i t s well known, and extensively researched, conventional linear systems counterpart. This, and other p o l i c i e s , are detailed in the cited references. Consider now the problem of s a t i s f y i n g the various requirements of a given control policy f o r (6.1). Then, using the derived conventional linear systems case as motivation, one i n t u i t i v e l y obvious approach is to use an appropriately defined feedback control scheme. This is the underlying theme of the remainder of t h i s chapter f o r which the material below is essential background. In p a r t i c u l a r , some candidate schemes are defined and some relevant systems t h e o r e t i c properties of these are developed. The schemes presented below can be c l a s s i f i e d under the following two general headings. (i) Those which only e x p l i c i t l y use current pass information. These are termed current pass or memoryless. (ii) Those which e x p l i c i t l y use information from both the current and previous M pass p r o f i l e s .

159

Further, a l l of these schemes are the natural generalisations of a corresponding scheme f o r the derived conventional linear system LD(AP,BP,cP). In p a r t i c u l a r , they reduce to t h i s scheme under the following conditions (i) Any previous pass terms are deleted. (if) The pass subscript k + 1 is dropped. (iii) The concept of a pass length is ignored. Note: In the remainder of t h i s chapter, use of the term 'natural g e n e r a l i s a t i o n ' should be interpreted as the r e s u l t of applying ( i ) - ( i i i ) above. To introduce the f i r s t of these schemes, or laws, f i r s t note that a linear s t a t e feedback law f o r LD(AP,BP,cP) has the structure g ( t ) : FxP(t) + OR(t)

(6.4)

Here F and 0 are constant g × n 1 and g × m matrices r e s p e c t i v e l y to be selected and R(t) E Rm is a new external reference vector taken to represent desired behaviour. Further, (6.4) is a powerful and extensively studied control law where, for example, the cited text and the references therein give a comprehensive treatment of the known r e s u l t s . The natural generalisation of (6.4) for (6.1) is the so-called current pass, or memoryless, linear s t a t e feedback law Uk+l(t ) = FX~+l(t ) + GRk+l(t), 0 < t < a,

k >0

(6.5)

Again F and 0 are £ × n 1 and £ x m matrices respectively to be selected and Rk+1(t ) is a new external reference variable taken to represent desired behaviour on pass k + 1, k ~ O. Figure 6.1 shows a schematic diagram of t h i s control law. Substituting (6.5) into (6.1) yields the closed-loop state-space model

XPk+1(t) = (AP + BPF)x~+I(t) + BPORk+I(t) + Yk+l(t)--crx

,l(t)

BP iYk+l_j(t)

j=l 3-

+ j~IDJYk+I, p j (t)

OO (6.6) This system is said to be closed since i t has an identical structure to (6.1) and therefore necessary and s u f f i c i e n t conditions for closed-loop s t a b i l i t y along the pass immediately r e s u l t on interpreting theorem 3.3.7. Use of these, unlike the a l t e r n a t i v e set based on theorem 3.3.5, enable the computationally f e a s i b l e t e s t s of chapter 4 to be applied for a given F and 6. Two systems theoretic properties of (6.5) now immediately a r i s e which are of fundamental underlying importance in terms of potential applications. The f i r s t of these is the f a c t that the matrices D~, 1 < j < M, of (6.1) and hence, by theorem 3.1.4, asymptotic s t a b i l i t y is invariant under t h i s control law. Further, i t is clear that t h i s is also true for a l l multipass causal, see (2.59) and Figure 2.7, feedback control schemes. This follows immediately from the following facts. J

160

Yk+I.M (t)

Yk+2.M(t)

Yk.1(t)

+

t

I

+1

FIGURE 6.1

Yk(t)

161 (i) (ii)

P This system property only depends on the matrices Dj, 1 ~ j < ~. The output Yk+l(t) does not explicitly depend on the input Uk+l(t), 0 < t a, k > O.

Suppose, however, that there is ' d i r e c t feedthrough I between input and output on any pass, i.e. the output equation of (6.1) reverts to that of example 2.3.3. Then in this case asymptotic s t a b i l i t y is no longer invariant under a suitable choice of feedback control scheme. This subject is not considered here, however, and can be found in the cited reference. Return now to the large sub-class of processes described by (6.1). Then the invariance property of asymptotic s t a b i l i t y under multipass causal feedback control schemes has the following major implications (i) Any asymptotically unstable example cannot be stabilised by a multipass causal feedback control scheme. (ii) In systems theoretic terms, i t is by no means clear at t h i s stage how (if at all) this problem can be overcome. One approach may be to develop a more general representation which includes sub-classes such as examples 2.3.3 and 2.3.7 as special cases. No work has yet been undertaken in this general area, however, and it is l e f t here as an open research problem. In practical cases, however, it can be argued that asymptotic s t a b i l i t y is always present due to the s t a b i l i s i n g influence of resetting the i n i t i a l conditions on each pass. To discuss this point further, consider, for simplicity, the unit memory version of (6.1) with Uk+l(t ) = O, 0 < t < a, k > O, and zero state i n i t i a l conditions on each pass. Then the i n i t i a l ouptut on each pass is given by Yk(O) = D~Yo(O), k > 0 (6.7) and note again the condition of corollary 3.1.4 for asymptotic s t a b i l i t y . Further, in physical terms, asymptotic s t a b i l i t y requires that the i n i t i a l output on each pass does not become unbounded as k ~ + ®. This will most certainly be the case in industrially orientated examples where the i n i t i a l conditions on each pass are always f i n i t e . For example, in bench mining systems the cutting machine begins each pass from a fixed datum, or reference, level above the stone/coal interface. Using the above results and observations, it can be concluded that the d e - s t a b i l i s i n g influences (if any) in industrially orientated cases are induced by the along the pass dynamics. Consequently the design studies in the remaining sections of this chapter will assume asymptotic s t a b i l i t y . Note also that asymptotic s t a b i l i t y always holds if DjP = O, 1 < j < M, i.e. no ~direct feedthrough T from previous pass profiles to the current one. tience no loss of generality occurs if the terms arising from these matrices are deleted from the output equation of (6.1). This fact will also be exploited, where appropriate, in these design studies.

162

The second systems t h e o r e t i c property of i n t e r e s t at t h i s stage arises from the i n t e r p r e t a t i o n of (6.5) as the natural generalisation of (6.4). In p a r t i c u l a r , the closed-loop derived conventional linear system LD(AP + BPF, BPG,cP) is just the r e s u l t of applying (6.4) to LD(AP,BP,cP). Further, the standard design problem in t h i s case is the choice of F for closed-loop s t a b i l i t y , i . e . a l l eigenvalues of the matrix AP + BPF have s t r i c t l y negative real parts. This is the well known pole allocation, or assignment, problem and i t s basic form has a solution i f , and only if, the p a i r {AP,BP} is controllable.

Given a desired set of locations in the open

l e f t - h a l f of the complex plane for the eigenvalues of AP + BPF, the closed-loop poles, numerous algorithms exist for computing the corresponding F. One such set, for example, is based on the so-called controllable canonical form. Suppose now that (6.1) is asymptotically stable. Then intepreting theorem 3.3.7 in terms of (6.6) immediately yields that s t a b i l i t y of LD(AP + BPF, BPG, CP) is a necessary condition for closed-loop s t a b i l i t y along the pass. Equivalently, the existence of a solution to the corresponding conventional linear systems problem is a necessary condition for closed-loop s t a b i l i t y along the pass. As shown below, t h i s result also holds for a l l other feedback control schemes introduced here, and is used in the next section to develop one candidate systematic procedure for designing any one of these schemes for closed-loop s t a b i l i t y along the pass. Further, it provides a p a r t i a l answer to the basic underlying synthesis problem of determining under what conditions (6.5) can be designed for closed-loop s t a b i l i t y along the pass which is, as yet, unresolved in the general case. Note, however, that section 6.2 will also provide a complete answer to t h i s question for certain sub-classes of (6.1). Implementation of (6.5) requires measurement of a l l elements in the current pass s t a t e vector X~+l(t ). By analogy with the conventional linear systems case, this may not be physically possible or p r a c t i c a l l y f e a s i b l e on, f o r example, f i n a n c i a l grounds. In such cases, again by analogy with the conventional linear systems case, one possible option is to use a suitably designed observer or s t a t e estimation device. To date, however, no work has been undertaken on the development of an observer theory for (6.1) and t h i s topic is l e f t here as an open research problem. As an a l t e r n a t i v e , note that the output vector of (6.1) is available by assumption on each pass k > 1. Hence the material below follows the conventional linear systems case and introduces output feedback control schemes whose controllers e x p l i c i t l y use current, or a combination of current and previous, pass output information. Consider, therefore, the output of (6.1) at time, or point, t on pass k, k > 1. Then the information in the following set is multipass causal (see also (2.59) and Figure 2.7) and can therefore be used for feedback control purposes. Y::

{Yk(T): 0 _< W _< t}U{Vp(t): 0 _< t _< a, 1-M < P < k-l}

(6.8)

163

Clearly~ however, the most appealing from an implementation standpoint will be those control schemes which e x p l i c i t l y use only information at point t on pass k since they will obviously have a simpler structure. This approach is followed below to produce one sub-class of so-called current pass, or memoryless, feedback control schemes. In p a r t i c u l a r , a so-called memoryless dynamic unity-negative feedback control scheme is developed which is the natural generalisation of i t s well known, and extensively used, conventional linear systems counterpart. Suppose, therefore, that Rk+l(t ) e l~m again denotes a new external reference vector taken to represent desired behaviour on pass k+l, k >_ O. Further, define the so-called current pass-error vector as ek+l(t ) = Rk+l(t ) - Yk+l(t), 0 < t _< a, k > 0 (6.9) Then a memoryless dynamic unity-negative feedback contro]ler for (6.1) constructs the input Uk+l(t), k >_ O, as the output from c -- Ac Xk+l(t) +

eek+l(t)

Uk+l ( t ) = ~CxC . k+l(t) + DCek+l ( t ) 0 0

(6.10)

where X~+l(t ) e Rn2 denotes the internal s t a t e of (6.10). ]'he r e s u l t i n g control scheme is shoun in Figure 6.2 and i t is clear that (6.9) and (6.10) describe a memoryless dynamic unity-negative feedback control scheme for (6.1). This is the natural generalisation of i t s conventional linear systems counterpart and (6.10) is again termed the forward-path controller. In e f f e c t , (6.9)-(6.10) is just the conventional linear systems scheme applied on pass k + 1, k > O. Specific choices of the matrices in (6.10) can now be made to generate a wide range of special cases which are the natural generalisations of t h e i r extensively used conventional linear systems counterparts.

As one example, set kc = O~ Bc = O,

Cc = 0 to yield ,, . Uk+l(t) = DCek+l(t~

0 < t < a,

k >0

(6.11)

In which case (6.9) and (6.11) describe a memoryless constant, or proportional, unity-negative feedback control scheme for (6.1). To obtain the closed:loop s t a t e s p a c e model, f i r s t define

P

T X +I(t)T]T e Rn, n = n 1 + n 2

Xk+l(t ) = [Xk+l(t ) ,

(6.12)

Then combining (6.1) and (6.10) yields the following composite s t a t e - s p a c e model describing the forward-path system

m

c

i M

J~

165 M

Xk+l(t) = AXk+l ( t ) + Bek+1 (t) + jE1Bj_ 1Yk+l_ j (t) gk+l(t) = CXk+l(t) + jB=lI}jgk+l_j(t) O_O ~,here

,c 1

,

,c 1

B =

,c j c = [c P

0],

,

(6.13)

Bj_I

=

[B]I

, 1 < j < M

Be j

Dj = D~, 1 < j _< M

(6.14)

Further, combining (6.9) and (6.13)-(6.14) yields the closed-loop state-space model M

Xk+l(t) = (A-BC)Xk+l(t) + BRk+l(t} + j=~I(Bj- 1 - BDj)Yk+I- j ( t ) M

Yk+l(t) = CXk+l(t ) + jE1DjYk+I_j(t) OO

(6.15)

where

I

AP-BPDCcP

A- BC =

L BecP

BPcc 1 (6.16)

Ae J

Both (6.13) and (6.15) are closed in the sense that they have an identical structure to (6.1). Hence necessary and sufficient conditions for s t a b i l i t y along the pass in both cases, which are computationally feasible to t e s t , immediately result on appropriately interpreting theorem 3.3.7. Again, the matrices D~, 1 < j M, are invariant under this scheme and therefore, noting the conclusions based on (6.7), the analysis of later sections based on this scheme will assume open-loop asymptotic s t a b i l i t y . Further, since this scheme is the natural generalisation of its conventional linear systems counterpart, it follows immediately that s t a b i l i t y of the derived conventional linear system closed-loop is a necessary condition for s t a b i l i t y along the pass of (6.15). Equivalently, the existence of a solution to the corresponding conventional linear systems design problem is a necessary condition for closed-loop s t a b i l i t y along the pass. As in the current pass state feedback case, this result is used in the next section to develop one candidate systematic procedure for designing this scheme to give closed-loop s t a b i l i t y along the pass. Note also that it provides a partial answer to the synthesis problem of determining the conditions under which this scheme can be designed for closed-loop s t a b i l i t y along the pass. The transfer-function matrix description plays a central role in the design of the conventional linear systems dynamic unity-negative feedback control scheme. In

166

particular, a number of currently available design techniques are based on the so-called return-difference matrix defined in terms of the transfer-function matrix of the forward-path system. The following analysis introduces, and develops a major systems t h e o r e t i c property of, the natural generalisation of the return-difference matrix for memoryless dynamic unity-negative feedback control of (6.1). First r e c a l l from section 2.5 that the 20 transfer-function matrix description of (6.1) is Y(s,z) = 6P(s,z)U(s,z)

(6.17)

where the m x l 20 transfer-function matrix GP(s,z) is given by M

GP(s,z) = (I m- jXlG~(s)z-3)-IG~(s)

(6.18)

G~(s) = cP(Slnl - AP)-IBP

(6.19)

with

and P + DPj, G (s) : cP(Slnl - AP) - 1 Bj_I

1 <j <M

(6.20)

Further, consider the forward-path system (6.13)-(6.14) under the assumption of zero state i n i t i a l conditions on each pass and zero i n i t i a l pass p r o f i l e s . Then i t is easily shown that the 20 transfer-function matrix description of t h i s system is given by ¥(s,z) = Q(s,z)e(s,z) = GP(s,z)K(s,z)e(s,z)

(6.21)

where GP(s,z) is defined by (6.18)-(6.20) and K(s,z), the 2D t r a n s f e r - f u n c t i o n m a t r l x of the forward-path controller, is given by K(s,z) e G~(s) = CC(sIn2 - Ac)-IB c + Dc

(6.22)

This result is the natural generalisation of its conventional linear systems counterpart and states that ~(s,z), the forward-path 2D transfer-function matrix, is the product of the corresponding matrices for the plant and the controller. Finally, substituting e ( s , ~ ) = R(s,z) - Y(s,z)

(6.23)

into (6.21) and rearranging yields Y(s,z) = H(s,z)R(s,z)

(6.24)

where the m x m 2D closed-loop transfer-function matrix H(s,z) is defined by H(s,z) = (I m + ~(s,z))-lQ(s,z) (6.25) The block diagram interpretation of (6.24) is shown in Figure 6.3 and this is the natural generalisation of its conventional linear systems counterpart. Further, i t is a simple exercise to show that the block diagram algebra for the conventional

167

.,sz+,;

Y(s,z) II=

FIGURE 6.3

168

linear systems case also generalises iN a natural manner. Hence the details are omitted. In the case of the closed-loop derived conventional linear system, the return-difference matrix is defined by P c

T(s)

= Im

+ Co(S)Go(S)

(6.26)

Further, let Po(S) and Pc(S) denote the open-loop forward-path and closed-loop characteristic polynomials respectively. Then the following result expresses closed-loop s t a b i l i t y in terms of T(s)

pc(S) = IT(s) I

(6.27)

This return-difference relationship acts as the basis of a number of currently available design techniques (see, for example, the cited text and the references therein for comprehensive details of a representative cross-section), and the following analysis now develops the natural generalisation of this result for Figure 6.3. The natural generalisation of (6.26) is T(s,z) = I m + GP(s,z)K(s,z) (6.28) and in order to link this return-difference matrix to closed-loop s t a b i l i t y along the pass i t is f i r s t necessary to introduce a characteristic polynomial. This, in common with its conventional systems counterpart, should contain a l l the information necessary to determine s t a b i l i t y along the pass. Consequently an obvious candidate for this open-loop is pP(s,z) = Pa(Z)AF(S,Z) P P (6.29) U where, from section 3.3, Pa(z) and hp(S,Z) are the asymptotic and s t a b i l i t y along the pass polynomials respectively. In particular, from definitions 3.3.3 and 3.3.4 respectively,

F[(z)

=

Iqe(z)l

= [I m - z-lD~-...-z-MD~I

(6.30)

AP - BP(z)QP(z)-IcP 1

(6.31)

and A~(s,z) = l S I n l where M p

(6.32)

BP(z) = j=~IBj_1z- J

P Note also that Po(S,Z) is the natural generalisation of i t s conventional linear

systems counterpart p~(s) = IsInl - f f l .

Further, by Schur's formula

169

Slnl - AP

p~(s,z)

-uP(z) (6.33)

=

QP(z)

_CP and define the sets D and ~ by = {s: Re{s} > O} and

(6.34)

(6.35) = {=: Izl z 1} respectively. Then the following result characterises s t a b i l i t y along the pass of (6.1) in terms of i t s characteristic polynomial. Theorem 6.1.1: With the assumptions of theorem 3.3.5 (interpreted in terms of (6.1)) the extended linear repetitive process generated by (6.1) with a ao

S(Ea,Wa,La)a>ao

is stable along the pass if, and only if, pP(s,z) ¢ 0 in D × U (6.36) Proof: This, in effect, consists of showing that (6.36) is equivalent to the conditions of theorem 3.3.7 which are, in turn, equivalent to those of theorem 3.3.5. Suppose f i r s t , therefore, that (6.36) holds and hence by Schur's formula p~(s,z) = ] s I n l - APllqP(z) - cP(Slnl - AP)-IBB(z)I

[SInl - API IzI N - cP(s) l =

N

¢ 0 in D x U

(6.37)

Z

where GP(s) is the interpass transfer-function matrix of (2.90) for (6.1). Equivalently, lSInl - AP[ ¢ 0 in D

(6.38)

and IzI N- GP(s)I ¢ 0 in D x U (6.39) Condition (b) of theorem 3.3.7 is now immediate from (6.38) and setting s = iv in (6.39) generates (a) and (c) of this same result. Conversely, suppose that (a)-(c) of theorem 3.3.7 hold. Then following the second part of the proof of this result immediately yields p~(s,z) = Pa(z)Ap(S,Z) P P ¢ 0 in D x U (6.40) and the proof is complete. • One immediate use of this characteristic polynomial is to introduce the concept of a pole in terms of the solutions of p~(s,z) = O. This is again the natural generalisation of its conventional linear systems counterpart and has a well defined physical interpretation which can be used to provide a physically based explanation of the differences between asymptotic s t a b i l i t y and s t a b i l i t y along the pass. Computation of the poles of (6.1) is not a feasible proposition, however, and hence

170

they are not considered further in t h i s work. The d e t a i l s f o r the unit memory case can be found in the cited reference and i t should also be noted that the extension of some of the r e s u l t s given there to non-unit memory processes is s t i l l an open research question. Return now to T(s,z) of (6.28) and l e t Pc(S,Z) and Po(S,Z) denote the closed-loop and open-loop c h a r a c t e r i s t i c polynomials, i . e . si n - A+BC

ac(S,~)

-(B(z) - BD(z))

(6.4~)

=

-c

q(z)

and

Sln-A

-B(z)

-c

q(z)

%(s,z) =

(6.42)

respectively. Then the following result is the natural generalisation of i t s conventional linear systems counterpart and links t h i s return-difference matrix to closed-loop s t a b i l i t y along the pass. Theorem 6.1.2: The return-difference matrix for the d i f f e r e n t i a l non-unit memory linear r e p e t i t i v e process (6.1) under memoryless dynamic unity-negative feedback control s a t i s f i e s

Pc(S,Z) = lT(s,z)]

(6.43)

Proof: This follows immediately on use of Schur's formula and appropriate r e s u l t s from the theory of determinants, tlence the d e t a i l s are omitted. • Given that i t is the natural generalisation, i t can be conjectured that T(s,z) should play a similar role to i t s conventional linear systems counterpart in terms of the design of t h i s scheme f o r , say, closed-loop s t a b i l i t y along the pass. No work has yet been undertaken in t h i s area, however, and i t is l e f t here as an open research problem. Instead, the design studies of the next section, in p a r t i c u l a r , will develop a systematic c o n t r o l l e r design procedure based on the f a c t that a solution to the derived conventional linear systems problem is a necessary condition for closed-loop s t a b i l i t y along the pass. Further, t h i s procedure can be implemented using only standard conventional linear systems techniques such as the c h a r a c t e r i s t i c locus. The concept of a memoryless dynamic unity-negative feedback control scheme for (6.1) is easily extended to include a memoryless dynamic ( i . e . non-unity) feedback loop and/or memoryless minor loop compensation. The most general case is shown in Figure 6.4 where the notation L.R.P denotes (6.1) and M.L.R.P. denotes a process of the form (6.10). Further, it is easily shown that a l l relevant elements of the analysis given to date in t h i s section extend in a natural manner to these cases. Hence the d e t a i l s are omitted and can be found in the cited reference. Note also

171

R(s,z)= ,~

L.R.P. I

M.LR.P.

FIGURE 6.4

Y(s,z)

172

that these schemes by no means exhaust the p o s s i b i l i t i e s based on memoryless information and the cited reference again contains detailed information on this point. Suppose now that a case exists where the objectives cannot be met by a permissible memoryless scheme. Then one obvious option in such cases is to consider the use of controllers with memory, i . e . schemes where some, or a l l , of the controllers e x p l i c i t l y use information from some, or a l l , of the previous pass components of the causal set (6.8). Of all the various p o s s i b i l i t i e s , the analysis of section 6.4 makes p a r t i c u l a r use of s w c a l l e d proportional r e p e t i t i v e minor loop compensation schemes. Consequently only these schemes are detailed here with the cited reference giving a comprehensive treatment of a l l other p o s s i b i l i t i e s . Consider again, therefore, the process of (6.1). Then a current pass, or memorytess, linear s t a t e feedback la~ for t h i s case with proportional r e p e t i t i v e minor loop compensation has the structure.

Uk+l(t) = FX~+1(t ) + 6Rk+1(t ) - ~ KjYk+ 1 j(t), 0 < t ~ a, k > 0 (6.44) j=1 Here F~G and Kj, 1 < j < M, are g × nl, I x m and ~ x m matrices respectively to be selected and Rk+l(t ) is again a neu external reference vector taken to represent desired behaviour on pass k+1, k > O. Figure 6.5 shows a schematic diagram of this control law. Note also that it reduces to (6.5) if Kj = 0, I < j < M, i.e. the previous pass terms are deleted. Substituting (6.44) into (6.1) yields the closed-loop state-space model 'P

p

H

P

p

Xk+1(t ) = (AP + B F)Xk+l(t ) + BPGRk+I(t) + j~l (Bj_ I - IIPKj)Yk+l- j (t) H

Yk+1(t) = CPXkP+I(t) + zl)P.Yk+1 -(t) j=l J -1 OO (6.45) Further, (6.45) is closed in the sense that i t has an identical structure to (6.1). Hence necessary and s u f f i c i e n t conditions for s t a b i l i t y along the pass closed-loop, ~hich are computationally feasible to t e s t , immediately r e s u l t on interpreting theorem 3.3.7. Again, the matrices D~, 1 < j < M, are invariant and hence, using the J --conclusions of the analysis based on (6.6)-(6.7), i t is necessary to assume open-loop asymptotic s t a b i l i t y . The extra design freedom in t h i s control law is clearly the matrices Kj, 1 < j < M. Note also that these matrices only a f f e c t the previous pass driving terms in the state equation and hence only the interpretation of condition (c) of theorem 3.3.7 for closed-loop s t a b i l i t y along the pass. In particular, s t a b i l i t y of the derived system LD(AP + BPF,BPG,cP) is again a necessary condition for closed-loop s t a b i l i t y along the pass. Further, in common with the case of (6.5), this fact can, see the cited reference for complete d e t a i l s , be used to develop a candidate systematic procedure for design to ensure, say, closed-loop s t a b i l i t y along the pass.

173

Yk+t.M(t) Yk+2.M(t)

Yk.1(t)

Yk(t)

Yk+l.M(t) Yk+Z.M(t)

FIGURE 6.5

Yk_t(t) Yk(t)

174

This procedure requires rules for choosing the Kj, 1 ~ j < M, i.e. the repetitive minor loop compensation, and in a l l but a number of special cases, such as the one detailed in section 6.4 for the scheme introduced below, this is s t i l l very much an open research problem. Returning now to the case of output feedback based schemes, suppose that (6.9) again defines the current pass error vector. Then a memoryless dynamic unity-negative feedback controller with proportional repetitive minor loop compensation for (6.1) constructs the input Uk+l(t), k > O, as M

Uk+l(t ) = Y~+l(t) - j~lKJYk+l_j(t), O < t < a, k > 0

(6.46)

where the gj, 1 < j < M, are e × m matrices and Y~+l(t) is the output from

X~+l(t) = AcX~+l(t) + BCek+1(t) =

0 < t < a,

cex k+l(t) c + Dcek+l(t )

k >0

(6.47)

where X~+l(t ) E Rn2 denotes the internal state of (6.47). Note also that t h i s scheme reduces to that described by (6.9) - (6.10) i f Kj = O, 1 5 j < M, i . e . the previous pass terms are deleted. To obtain the closed-loop state-space model, f i r s t define Xk+l(t ) • Rn as in (6.12). Then combining (6.1) and (6.46) - (6.47) yields the following composite state-space model describing the foruard-path system M

Xk+l(t) = hXk+l(t ) + Bek+l(t) + j~IBJ_IYk+I_j(t) M Yk+l(t) = CXk+l(t ) + j~lDJYk+l_j(t) 0 ~ t ~ a, k > 0 where A,B,C and Dj, 1 < j < M, are again given by (6.14) but here Bj_I =

,

1 ~j ~M

(6.48)

(6.49)

0 Further, combining (6.9) and (6.48) - (6.49) yields the closed-loop state-space model M

Xk+l(t ) = (A - BC)Xk+I(t) + BRk+l(t ) + j~Bj_ 1 - BDj)Yk+I_j(t ) Yk+l(t) = CXk+l(t ) + jE1DjYk+I_j(t)= o ~ t ~ a, k~o (6.50) where A-nC is again given by (6.16). Both (6.48) and (6.50) are closed in the sense that they have an identical structure to (6.1). Hence necessary and sufficient conditions for s t a b i l i t y along the pass in both cases, which are computationally feasible to t e s t , immediately

175

r e s u l t on appropriately interpreting theorem 3.3.7. Again, the matrices DPj, 1 < j < M, are invariant and hence, see (6.6) - (6.7), i t is necessary to assume open-loop asymptotic s t a b i l i t y . As with i t s s t a t e feedback based counterpart of (6.44), the extra design freedom in (6.46) - (6.47) is clearly the matrices Kj, 1 ~ j < M, and these matrices again only a f f e c t the previous pass driving terms in the s t a t e equation. Hence they only influence the i n t e r p r e t a t i o n of condition (c) of theorem 3.3.7 closed-loop and s t a b i l i t y of the derived system LD(A-BC,D,C) is a necessary condition for closed-loop s t a b i l i t y along the pass. The cited reference shows how t h i s f a c t can be exploited to develop one candidate systematic procedure for design to ensure, say, closed-loop s t a b i l i t y along the pass. Further, section 6.4 uses t h i s scheme to solve the RSDDSP for one sub-class of (6.1) which is of industrial i n t e r e s t . This analysis also yields some i n i t i a l r e s u l t s on the development (see also the discussion a f t e r (6.45)) of rules for e f f e c t i v e l y selecting the r e p e t i t i v e minor loop compensation. As an a l t e r n a t i v e to the state-space approach detailed above, the scheme defined by (6.46) - (6.47) can be described in 2D t r a n s f e r - f u n c t i o n matrix terms. Further, a l l r e s u l t s for t h i s description follow immediately as straightforward extensions of t h e i r counterparts for the scheme of Figure 6.3. IIence only the f i n a l forms of the basic underlying r e s u l t s are stated here. The f i r s t of these is the closed-loop 20 t r a n s f e r - f u n c t i o n matrix description whose block diagram i n t e r p r e t a t i o n is shown in Figure 6.6

~(s,z) : H(s,z)R(s,z)

(6.5~)

where the m x m 2D closed-loop transfer-function matrix H(s,z) is defined by

~(s,z) : (I m + q(s,z))-lq(s,z)

(6.52)

Q(s,z) = (I m + GP(s,Z)KL(S,Z))-IGP(s,z)K(s,z)

(6.53)

with

Here GP(s,z) is the 2D transfer-function matrix of (6.1) defined by (6.18) - (6.20), K(s,z) is defined by (6.22), and M

~L(S,Z) ~ KL(Z) = ]~ 1Kjz-J

(6.54)

Suppose also that the return-difference matrix for t h i s case is defined as T(s,z) = I m + Q(s,z) (6.55) Then the r e s u l t of theorem 6.1.2 s t i l l holds with Po(S,Z) and Pc(S,Z) defined in terms of (6.48) and (6.50) respectively. As a f i n a l point in t h i s section, note that the schemes detailed or referred to here by no means exhaust the p o s s i b i l i t i e s for controlling (6.1), or i t s discrete counterpart, based on causal memoryless and/or non-memoryless information. This point is considered in detail in the cited reference which, for example, introduces schemes which include ~feedforward' elements. In terms of t h i s work, however, the schemes detailed in this section are s u f f i c i e n t to demonstrate the potential power of

176

R(s,z)

_~

GP(s,z)

FIGURE6,6

y(s,z),,

177

appropriately specified feedback control schemes in regulating the behaviour of d i f f e r e n t i a l (and discrete) non-unit memory linear r e p e t i t i v e processes. An a t t r a c t i v e feature of these schemes from an applications standpoint is t h e i r r e l a t i v e simplicity and hence the p o s s i b i l i t y of implementation without recourse to special purpose hardware/software. This is p a r t i c u l a r l y true f o r the memoryless schemes which, see the cited reference for a complete treatment, have the simplest structure in terms of the the information to be logged and/or stored in order to actuate the controller. Further, t h i s point strongly suggests that the potential of these schemes should be f u l l y evaluated before recourse to other schemes with a more complex structure. 6.2

S t a b i l i t y Along the Pass This section presents some i n i t i a l r e s u l t s on the design of the schemes introduced in the previous section for closed-loop s t a b i l i t y along the pass. In e f f e c t , the r e s u l t s presented consist of one candidate design algorithm plus an in-depth treatment of i t s application to certain sub-classes and d e t a i l s of some possible future research topics. These r e s u l t s represent the f i r s t output on this property, which is an essential necessary item of any p r a c t i c a l l y f e a s i b l e control policy and only apply (directly) to the memoryless schemes. (Consider again the discussion at the end of the l a s t section ou the 'complexity' of the schemes detailed there.) An important r e s u l t from section 6.1 is t h a t asymptotic s t a b i l i t y of (6.1) is invariant under a l l multipass causal feedback control schemes. Further, i t was argued there that t h i s property is always present in p r a c t i c a l cases due to the s t a b i l i s i n g influence of r e s e t t i n g the i n i t i a l conditions on each pass. Consequently the analysis of t h i s section assumes that (6.1) is asymptotically stable. Note again that asymptotic s t a b i l i t y always holds if D~ = O, 1 < j < M, and hence no loss of generality r e s u l t s from deleting the previous pass terms from the output equation of (6.1). Suppose, therefore that (6.1) is asymptotically stable and consider an a r b i t r a r y memoryless scheme from those introduced in section 6.1. Then interpreting theorem 3.3.7 and, in p a r t i c u l a r , conditions (b) and (c) of t h i s r e s u l t now gives necessary and s u f f i c i e n t conditions for closed-loop s t a b i l i t y along the pass. Further, this scheme is the natural generalisation of i t s conventional linear systems counterpart. Hence, as also discussed b r i e f l y in section 6.1, the existence of a solution to the corresponding conventional linear systems problem is a necessary condition for closed-loop s t a b i l i t y along the pass. This f a c t leads immediately to the following candidate systematic procedure, or design algorithm, for solving the r e p e t i t i v e systems problem ~hich assumes that the derived conventional linear systems problem has a solution. For example, in the case of linear s t a t e feedback t h i s assumption, J

-

-

in e f f e c t , requires that the pair {AP,BP} is controllable.

178

STEP 1: Solve the derived conventional linear systems problem using any appropriate design algorithm. STEP 2: I n t e r p r e t condition (c) of theorem 3.3.7 in terms of the closed-loop system r e s u l t i n g from application of the controller designed at step 1. Then proceed to t e s t t h i s condition by applying the appropriate step in either of the systematic procedures developed in section 4.2. On completion, the following two options exist. STEP 3: I f step 2 holds the c l o s e , loop system is stable along the pass. In which case proceed ( i f required) to examine other performance s p e c i f i c a t i o n s such as the r a t e of approach to the limit p r o f i l e . STEP 4: The closed-loop system is unstable along the pass i f step 2 does not hold. In which case return to step 1 and re-design ( i f possible). In e f f e c t , a l l steps in the above procedure can be completed using standard conventional linear systems techniques. Suppose also that step 1 is completed using a technique, f o r example the c h a r a c t e r i s t i c locus or inverse Nyquist array in the case of the memoryless dynamic unity-negative feedback control scheme, f o r which well tested software is already available. Further, complete step 2 using the f i r s t systematic procedure of section 4.2. Then the above procedure is clearly suitable for a CAD environment. This is the subject of on-going research, see the cited reference for a complete update, which is developing the necessary infrastructure for inclusion in a user friendly i n t e r a c t i v e package. Coupled with t h i s , research is proceeding in the following two general areas which, in e f f e c t , determine the applicable range of these memoryless schemes and t h e i r efficiency in terms of, f o r example, computational cost of the above design procedure. (i) The c l a s s i f i c a t i o n of the sub-classes ( i f any) of (6.1) for which t h i s problem can be solved by a p a r t i c u l a r scheme. Ideally what is required here are 'synthesis type' r e s u l t s similar to those for certain of the derived conventional linear systems problems, such as l i n e a r s t a t e feedback. This is termed the existence problem. (ii) The development of 'easy to use' design rules such as r e l a t i v e s t a b i l i t y and/or performance indicators. These can then be incorporated into the i n t e r a c t i v e package to give increased computational efficiency. This is an important aspect since, on the assumption that a solution e x i s t s , i t may require more than one i t e r a t i o n of the above procedure (or any a l t e r n a t i v e ) to yield a successful controller. All c o n t r o l l e r parameters appear in the closed-loop derived system of the memoryless schemes under consideration here. Further, s t a b i l i t y of t h i s derived system is a necessary condition for closed-loop s t a b i l i t y along the pass. Equivalently, a solution to the derived problem is also necessary for a solution to the r e p e t i t i v e systems problem. Hence a p a r t i a l answer to the existence problem already exists in a l l these cases.

179

The development of a complete solution to t h i s problem f o r a number of the p o t e n t i a l l y more common schemes, such as s t a t e feedback and unity-negative output feedback control, is the subject of an on-going research programme. This has already yielded solutions for a number of sub-classes characterised by certain s t r u c t u r a l properties and, see the cited references for complete d e t a i l s , provided useful pointers f o r more general cases. To i l l u s t r a t e the currently available r e s u l t s , and demonstrate the application of the above procedure, the following two eases are now detailed. Consider f i r s t , therefore, the SISO unit memory case where the derived and associated conventional linear systems are given by LD(AP,BP,cP) and L~(AP,koBP,cP ) respectively, where k o ¢ 0 is a positive real scalar. In which case it is easily shown that these systems have the same zeros which are assumed to be real, negative

and d i s t i n c t ,

i . e . a special case of the minimum phase property.

Suppose also that

cPBP ¢ 0 or, equivalently, these systems have n poles and q = n - 1 zeros. Then application of the above procedure to design a current pass s t a t e feedback law under these assumptions proceeds as follows. STEP 1: The derived conventional linear systems problem has a solution under the assumption that the pair {AP,BP} is controllable. In which case any one of numerous well documented algorithms can be used to compute the row vector F which assigns the eigenvalues of AP + RPF to a specified set of locations ] i ' 1 < i < n, in the open l e f t - h a l f of the complex plane. STEP 2: The closed-loop interpass t r a n s f e r - f u n c t i o n , denoted Gc(S), is given by q ko i~l ( s - zi) Gc(S) =

n •

(6.56) (s-

hi)

i=l where the z i are the zeros of LD(AP,BP,cP) which are r e a l , d i s t i n c t and negative. Hence s t a b i l i t y along the pass i f , and only i f , the frequency response plot generated by Gc(i~), V real ~ > O, l i e s e n t i r e l y within the unit c i r c l e in the complex plane. Suppose also that the h i are assigned to locations ?i which are r e a l , d i s t i n c t and negative and s a t i s f y 71 < z 1 < 72 < . . . < Zn_ 1 < 7n (6.57) Then, using the analysis of lemma 5.2.3 and example 5.2.1, t h i s special case is stable along the pass i f , and only i f , q n Ikol]i~lZi I < li~l~i ] (6.58) or, equivalently, the maximum value of the frequency response occurs at w = O. Note 1: The condition of (6.58) can always be s a t i s f i e d by placing 71 ' f a r enough' to the l e f t of the origin on the real line.

180

Note 2: The r e s u l t s presented here form part of the analysis for a sub-class of (6.1) with so-called 'fading memory' which can be found in the cited reference. The second case detailed here is where the derived system has the structure of a multivariable f i r s t order lag. In p a r t i c u l a r , suppose that m = g = n I and consider, for simplicity, the unit memory case. Then, a f t e r use of a s t a t e transformation, if necessary, the state-space model can be written as

ik+1(t) =

-

X;IAiXk+1(t) + A;IUk+1(t) + BoYk(t)

Yk+l(t) = ImXk+l ( t ) 0 _< t < a , k > 0 (6.59) where Ao and A1 are real constant matrices. Further, application of the above procedure to design a memoryless proportional unity-negative feedback control scheme for t h i s case proceeds as follows. STEP 1: Select the forward-path c o n t r o l l e r as ]{ = p Ao - A1 (6.60) where p is a p o s i t i v e real scalar. Then using the theory of a multivariable f i r s t order lag (see the cited reference for complete details) i t is easily shown that the closed-loop derived system is stable Vp STEP 2: The closed-loop interpass transfer-function matrix is given by

%(s) = (Im + Co (S)K) 1

= S+---pBe

(6.61)

Further, i t follows immediately on examining (6.61) that condition (c) of theorem 3.3.7 holds closed-loop i f , and only if,

; > r(Bo)

(6.62)

where r ( . ) again denotes the spectral radius. Finally, combining (6.62) with the result of step 1 yields closed-loop s t a b i l i t y along the pass V p > r(Bo) which can always be s a t i s f i e d by choosing a 'high enough' value of p. Note: The cited reference gives the generalisation of this analysis to the non-unit memory case, which is a straightforward exercise. Suppose now that a solution exists for the p a r t i c u l a r scheme under consideration. Then, as noted under ( i i ) above, i t may require more than one i t e r a t i o n of the above procedure (or any a l t e r n a t i v e ) to yield a successful c o n t r o l l e r with the consequent prospect of a heavy computational load. Hence the development of 'easy to use' design rules such as r e l a t i v e s t a b i l i t y and/or performance indicators is one obvious route to obtaining maximum computational efficiency. This general area is the subject of an on-going research programme which is being undertaken in p a r a l l e l with the development of the i n t e r a c t i v e package. The

181

d e t a i l s of t h i s are not discussed here and a comprehensive treatment of progress to-date can again be found in the cited reference. In summary, therefore, t h i s section has considered the design of an a r b i t r a r y memoryless scheme from those introduced in the previous section for closed-loop s t a b i l i t y along the pass. The end product has been a systematic design procedure for solving t h i s problem which, in e f f e c t , can be completed using standard conventional linear systems schemes and is suitable f o r a ChD environment. Further, an in-depth treatment of i t s application to two sub-classes of (6.1) has been presented. This has given a complete solution to the fundamental underlying existence problem for these cases and has also highlighted other possible future research topics. At t h i s stage, i t is not possible to f u l l y assess the potential of t h i s procedure, or the memoryless schemes of section ( 6 . i ) , in terms of solving t h i s most basic of control problems. In p a r t i c u l a r , much work remains to be done in the areas outlined here, and others detailed in the cited reference, before t h i s question can be r e a l i s t i c a l l y considered. The progress to date, coupled with the f a c t (refer again to the discussion at the end of the previous section) that these memoryless schemes have the simplest structure in implementation terms, strongly suggests that t h i s procedure should be f u l l y evaluated before considering other p o s s i b i l i t i e s . These p o s s i b i l i t i e s include the use of the s u f f i c i e n t , but not necessary, simulation-based s t a b i l i t y t e s t s of chapter 5 as a basis and the cited reference gives f u r t h e r information on t h i s p a r t i c u l a r point. 6.3

The Limit P r o f i l e Design Problem This section presents some i n i t i a l r e s u l t s on the design of the schemes introduced in section 6.1 to solve the LPDP. The format is similar to section 6.2 in that only the memoryless schemes are considered to produce the f i r s t output on t h i s problem. In e f f e c t , the r e s u l t s presented consist of one candidate design procedure plus an in-depth treatment of i t s application to one sub-class and some possible future research topics. Further, these r e s u l t s are based on use of the simulation-based s t a b i l i t y t e s t s of chapter 5 since these yield, at no extra cost, unique computable information concerning two components of t h i s problem in one special case of major p r a c t i c a l i n t e r e s t . Consider, therefore, (6.1) with D~ = O, 1 < j < M, for simplicity, and hence the corresponding limit p r o f i l e is described by the state-space model of (6.2)-(6.3) with DP = 0 or, in transfer-function matrix terms, by =

(6.63)

where

G~(S) = CP(SInl - hP - BPcP)-I~ P

(6.64)

182

Further, as a representative choice, consider the memoryless dynamic unity-negatlve feedback control scheme introduced in section 6.1. Then here the control action on the limit p r o f i l e is described in state-space terms (replace a l l variables in (6.9) and (6.10) by t h e i r strong limits) by

= ACx (t) ÷ U ( t ) = cCx~(t) + DCe (t) 0
(6.65)

e®(t) = R®(t) - Y ( t )

(6.66)

where

or, in transfer-function matrix terms, by U ( s ) = G~(s)e (s) where G~(s) is given by (6.22).

(6.67) lIence the closed-loop limit p r o f i l e can be regarded

as the conventional linear systems unity-negative feedback control scheme for G~(s) with forward-path c o n t r o l l e r dynamics defined by G~(s). Suppose also that the closed-loop specifications for the limit p r o f i l e can be achieved under the action of this scheme. In which case i t follows immediately that any appropriate technique from conventional linear systems theory can be used to design G~(s) and, simu]taneously, yield a candidate solution to the LPDP. S t a b i l i t y (in the conventional sense) of the limit p r o f i l e is a necessary condition for s t a b i l i t y along the pass, for which interpreting theorem 3.3.7 closed-loop gives a set of necessary and s u f f i c i e n t conditions. Further, these conditions can be tested by applying either of the systematic procedures developed in section 4.2. Suppose also that theorem 3.3.7 holds closed-loop and consider the remaining specifications of the LPDP, i.e. those relating to the rate of convergence of {Yk}k>1 to Y and the error Yk - Y ,k>O, respectively. Then the only effective option at this stage is to undertake detailed simulation studies with the consequent prospect of a heavy computational load. Note: In e f f e c t , the above approach to controller design examines two necessary conditions for s t a b i l i t y along the pass, i . e . the limit p r o f i l e and the derived conventional linear system. This is unavoidable, however, since s t a b i l i t y testing based on the s t a b i l i t y along the pass polynomial, Ap(S,Z), open or closed-loop is not a computationally feasible proposition. As an alternative to using theorem 3.3.7, suppose that theorem 5.2.1, or theorem 5.2.2 in the unit memory case, holds closed-loop. Then use of these simulation-based t e s t s produces, at no extra cost, unique computable information concerning the convergence rate of the {Yk}k>] and the error Yk - Y ' k > O. This information

183

r e s u l t s from interpreting theorem 5.3.1, or theorem 5.3.2 in the unit memory case, closed-loop under the assumption that the reference signal sequence is constant from pass to pass. Further, these f a c t s lead immediately to the following candidate design procedure f o r solving the LPDP using memoryless dynamic unity-negative feedback control. STEP 1: Use an appropriate technique from conventional l i n e a r systems theory to yield a candidate forward-path c o n t r o l l e r which s a t i s f i e s the limit p r o f i l e s p e c i f i c a t i o n s . This step assumes that these s p e c i f i c a t i o n s can, at l e a s t , be achieved to within 'acceptable bounds'. Further, the minimum s p e c i f i c a t i o n in a l l cases is obviously s t a b i l i t y in the conventional sense: STEP 2: I n t e r p r e t theorem 5.2.1, or theorem 5.2.2 in the unit memory case, in terms of the closed-loop system r e s u l t i n g from application of the c o n t r o l l e r designed at step 1. Then proceed to t e s t t h i s s u f f i c i e n t condition by applying the systematic procedure developed in section 5.2. On completion, the following two options exist. STEP 3: I f theorem 5.2.1, or theorem 5.2.2 in the unit memory case, holds the closed-loop system is stable along the pass. Then interpret theorem 5.3.1, or theorem 5.3.2 in the unit memory case, and proceed to step 5. STEP 4: I f theorem 5.2.1, or theorem 5.2.2 in the unit memory case does not hold, no defiuite conclusions can be drawn.

(i)

The feasible options then are to

t e s t the necessary and s u f f i c i e n t conditions of theorem 3.3.7 interpreted closed-loop and, if they hold, proceed to simulation studies to assess the convergence r a t e and the error Yk - Y~' k > O; or (ii) return to step 1 and re-design ( i f possible); or (iii) terminate STEP 5: Decide i f t h i s design s a t i s f i e s the specifications on the convergence rate and the error Yk " Y®' k > O. I f yes then stop, otherwise return to step 1 and re-design ( i f possible). All steps in the above procedure are suitable for a CAD environment on the assumption that step 1 is completed using a compatible technique such as the c h a r a c t e r i s t i c locus or inverse Nyquist array. This is the subject of on-going research, see the cited reference for a complete update, which is developing the necessary i n f r a s t r u c t u r e for inclusion in a user friendly i n t e r a c t i v e package, Coupled with t h i s , research is proceeding in a number of general areas which, in e f f e c t , determine the e f f e c t i v e operating range and efficiency in terms of, f o r example, computational cost of the above design procedure. These areas are not considered here and the cited reference again contains comprehensive d e t a i l s of progress to-date plus an update on some on-going current work. Instead, the following example is presented to i l l u s t r a t e the application, and p o t e n t i a l , of this procedure. Consider again the unit memory sub-class of (6.1) defined by the state-space model of (6.59). Then application of the above procedure to solve the LPDP for t h i s case proceeds as follows.

184 STEP 1: Consider again the use of the proportional forward-path c o n t r o l l e r defined by (6.60). Then, in t r a n s f e r - f u n c t i o n matrix terms, i t is easily shown t h a t the closed-loop limit p r o f i l e dynamics can be written as Bo (slm + p(I m - ~-))Y®(s) = p(l m

AolAI p )R®(s)

(6.68)

uhich is s t a b l e in the conventional sense i f , and only i f , p > max Re(Ji) where Ai l~i<m is an eigenvalue of Bo. Consider also the case of p ~ + ® ('high g a i n ' ) . In which case Y ( t ) is ' a r b i t r a r i l y close' to the inverse Laplace transform of P

~(s) = ~-~ ImR(S)

(6.69)

This is a t o t a l l y non-interacting conventional linear system with zero s t e a d y - s t a t e error to a unit step applied at t = 0 in any channel. STEP 3: In t h i s case i t is easily shown that the matrix ]lLIIp of theorem 5.2.1 or 5.2.2 closed-loop, denoted IILcllp , is given by 1 IILctl p : ~ llBoll p (6.70) This follows immediately since a l l entries in the step response matrix of the closed-loop associated conventional linear system are monotonic and sign d e f i n i t e , i . e . a special case of example 5.2.1. tlence theorem 5.2.1 holds V p > r(l]Boi]p ) and theorem 5.2.2 holds V p > ]l([]Bol]_)l ]. I n t e r p r e t i n g theorem 5.3.1 now yields that the closed-loop output sequence (¥k~k> 1 approaches Y at a geometric r a t e governed by

r(lIBofl p) 7 e ( p ,1), where t h i s set is non-empty for any choice of p > r (]lBo]]p), and interpreting (5.158) - (5.164) gives the error 'hand' for each element of Yk - Y®' k ~ O. Similarly, interpreting theorem 5.3.2 now yields that the closed-loop output sequence (¥k)k>l approaches Y at a geometric r a t e governed by

ll(tl~ollp)[I < I f o r suitable choice of p. Further, i n t e r p r e t i n g (5.148) - (5.150) P gives the single e r r o r 'band' for Yk - Y~' k 2 0 . STEP 5: Consider again the case of p ~ + ® and, for example, the use of theorems 1 5.2.2 and 5.3.2. Then I[Lcl[= ~[[(l[Bol[p) l I ~ O, i . e . the limit p r o f i l e dynamics of (6.68) are reached to within a r b i t r a r y accuracy on the f i r s t pass in t h i s case. Note: The refinements of sections 5.2 and 5.3 in the form of, f o r example, theorem 5.2.5 and the r e s u l t s derived from lemma 5.3.1 are e a s i l y included in t h i s design procedure. In summary, therefore, t h i s section has presented some i n i t i a l r e s u l t s on solving the LPDP by memoryless output feedback control. The end product has been a systematic design procedure for solving this problem which is suitable f o r a CAD environment. Further, an in-depth treatment of i t s application to one sub-class has

185

been given to highlight i t s potential. At t h i s stage, i t is not possible to f u l l y assess the p o t e n t i a l of t h i s procedure, or the underlying control scheme, in terms of solving the LPDP. In p a r t i c u l a r , much work remains to be done in a number of areas for which the cited reference gives a comprehensive overview. One such area in the unit memory ease is the development, and comparative studies of, an a l t e r n a t i v e procedure based, see also the discussion r e l a t i n g to (5.145) - (5.147), on combining a l l but the limit p r o f i l e specifications into the single constraint ttLcl I < b, where 0 < b < 1. Finally, note again that the memoryless schemes of section 6.1 have the simplest structure in implementation terms, tlence, as for s t a b i l i t y along the pass, i t is clear that the potential of these schemes should be f u l l y evaluated before considering other p o s s i b i l i t i e s . 6.4

The Repetitiv~System~Disturbance Decoupling with S t a b i l i t y Problem This section presents some i n i t i a l r e s u l t s on the design of the schemes introduced in section 6.1 to solve the RSDDSP. In p a r t i c u l a r , the p o s s i b i l i t y of obtaining a 'geometric s t y l e ' solution in the s p i r i t of that for the well known conventional linear systems problem is b r i e f l y explored. Further, the memoryless dynamic unity-negative feedback c o n t r o l l e r with proportional r e p e t i t i v e minor loop compensation introduced in section 6.1 is used to solve t h i s problem in one special case. Finally, some possible future research topics are noted. As a preliminary to the discussion which immediately follows, i t is instructive to b r i e f l y review the well-known disturbance decoupling with s t a b i l i t y problem for conventional linear systems. Consider, therefore, the system of (6.71) below where q(t) represents a disturbance which is assumed not to be d i r e c t l y measurable by the controller X(t) = AX(t) + BV(t) + Dq(t) y(t) = cx(t)

X(t) e Rn, Y(t) e Rm, U(t) e Rg, q(t) e Rv

(6.71)

Further, suppose that the linear s t a t e feedback law O(t) = FX(t) is applied to (6.71). Then the disturbance decoupling problem for the resulting closed-loop system is to find a suitable F such that q(t) has no influence on the controlled output ¥ ( t ) . Equivalently, t h i s closed-loop system is said to be disturbance decoupled r e l a t i v e to the pair Y(t), q(t) i f , for each i n i t i a l condition X(O) e Rn, the output Y(t), t ~ O, is the same for a l l q(t) e Rv. The above problem has been the subject of much research e f f o r t , fine element of which has been to use such geometric concepts as (A,B)-invariant subspaces to develop necessary and s u f f i c i e n t conditions for the existence of a solution which are, for example, given in the cited reference. Note, however, that these conditions do not guarantee closed-loop s t a b i l i t y , i . e . that a l l eigenvalues of A+BF have s t r i c t l y negative real parts, which is obviously essential for applications. This has led to

186

the so-catted disturbance decoupling with s t a b i l i t y problem for which the cited reference also gives necessary and s u f f i c i e n t conditions for the existence of a solution. Return now to the r e p e t i t i v e systems case and, since the following discussion generalises in a natural manner, consider the special case of a unit memory process. Further, interpret Yk(t), 0 < t g a, k > O, as a disturbance which is not d i r e c t l y measured by the c o n t r o l l e r on pass k + 1. Suppose also that the current pass linear state feedback law uk+l(t) = F o t k o (6.72) is applied. Then a clear structural similarity exists with the conventional linear systems problem for (6.71). To see t h i s , consider again the f i r s t requirement of the RSDDSP and interpret i t in terms of the closed-loop system. In which case i t follows immediately that t h i s requirement holds r e l a t i v e to the pair Yk-1( t ) ' Yk( t ) ' 0 < t < a, k > k

*

~ nl > 1, i f , for each i n i t i a l condition X (0) = dk ¢ R , the output Yk(t) is

the same for a l l Yk_l(t) E Rm. llence r e p e t i t i v e systems disturbance decoupling simply means that the contribution of the previous pass p r o f i l e to the current one is zero, 0 < t < a, k > k

> 1.

The second requirement of the RSDDSP for this case

requires, as a basic minimum, that F be selected such that a l l eigenvalues of AP+BPF have s t r i c t l y negative real parts. These facts now lead immediately to the conclusion that the RSDDSP in this case is structuraly similar to its well researched conventional linear systems counterpart. Further, i t can be conjectured that a solution to t h i s RSDDSP can be developed using geometric concepts such as (h,B)-invariant subspaces with the consequent p o s s i b i l i t y of a natural generalisation to the non-unit memory case. This general area is the subject of on-going research for which the cited reference gives a comprehensive treatment of the considerable progress to-date. On the assumption that a solution exists, implementation of a current pass state feedback solution to the RSDDSP would encounter the same potential d i f f i c u l t i e s as other uses of this law. In which case one option is to use output feedback based schemes. The analysis below uses the memoryless dynamic unity-negative feedback controller with proportional r e p e t i t i v e minor loop compensation introduced in section 6.1 to solve the RSDDSP in one special case. This analysis also represents the f i r s t output on the design of these minor loop schemes or, more generally, controllers with memory. Return, t h e r e f o r e , t o the closed-loop state-space model of (6.50) which results from application of the control law of (6.46)-(6.47) to (6.1). Consider also the special case when DP = O, 1 < j < M, n = m = g and IBP[ ~ O; a not uncommon 3 1 situation in industrial examples such as bench mining systems. Further, select the c o n t r o l l e r matrices K, as 3

187

Ki = (BP)_IB~_I,D 1 ~ j ~ M (6.73) Then Bj_ 1 = O, 1 < j < M, in (6.49) and hence in (6.50) the pass p r o f i l e Yk(t), 0 < t a, is independent of the pass profiles Yk_j(t), 1 < j < M, for a l l passes k > 1. Equivalently, r e p e t i t i v e systems disturbance decoupling is achieved in this case with o

the optimum choice of k = 1. Suppose, therefore, that (6.73) holds. Then i t is easily shown that the closed-loop limit p r o f i l e is described in transfer-function matrix terms by

Yo(s) o (Im+

(6.74)

where G~(s) and G~(s) are defined by (6.19) and (6.22) respectively. Equivalently, the limit p r o f i l e is described by the derived conventional linear system. Hence the design exercise can be completed by using an appropriate technique to design G~(s) to meet the required specifications. In summary, therefore, this section has presented some preliminary work on solving the RSDHSP by use of the schemes introduced in section 6.1. At best, these demonstrate the potential of these schemes in terms of this problem and i t is not possible at this stage to f u l l y assess them in this context. This can only take place a f t e r much further work has been undertaken in a number of areas which are detailed in the cited reference. Notes and References The three control policies of section 6.1 are from Rogers and Owens (1990 i , j , k ) respectively. The optimal control problem referred to is from Willson, Collins and Owens (1982) and Rogers and Owens (1990i) also details other candidate control policies. All control schemes detailed or referred to in this section are from Rogers and Owens (1990~). Owens (1978) and Wonham (1974), together with the relevant references therein, are two of numerous possible sources for the cited results from conventional linear systems theory. As noted previously, Rogers and Owens (1990b) details the work to date on poles and Smyth, Rogers and Owens (1990a) discusses some implementation aspects of the control schemes introduced to - date. Rogers and Owens (1990m) summarises the corresponding analysis of this section for the discrete case. Section 6.2 is based on Rogers and Owens (1990i) and Smyth, Rogers and Owens (1990b,c). Use has also been made of results from Rogers and Owens (1988a, 1989d) and Smyth (1991). Section 6.3 is based on Rogers and Owens (1990i,j) and Smyth (1991). Finally, section 6.4 is from Rogers and Owens (1990k) and has also made use of results from Rogers and Owens (1988b).

CHAPTER 7 CONCLUSIONS AND F1.~,TIIE1{, WORK Using previous work as a basis, a rigorous s t a b i l i t y theory f o r r e p e t i t i v e processes with linear dynamics and a constant pass length has been presented. This has been formulated in terms of a general abstract representation which, in e f f e c t , regards the output on any pass as a point in a Banach space. Further, t h i s model includes as special cases a l l unit and non-unit memory r e p e t i t i v e processes with linear dynamics and a constant pass length. Hence an obvious way to develop a rigorous s t a b i l i t y theory is to formulate t h i s in terms of the abstract model and then interpret the r e s u l t i n g conditions in terms of the p a r t i c u l a r example under consideration. The r e s u l t i n g s t a b i l i t y theory consists of two d i s t i n c t concepts termed asymptotic s t a b i l i t y and s t a b i l i t y along the pass respectively. Further, asymptotic s t a b i l i t y is a necessary condition for s t a b i l i t y along the pass which is required in a l l p r a c t i c a l applications. To provide a basic explanation of t h i s f a c t , r e c a l l that the e s s e n t i a l unique control problem for these processes is the possible presence in the output sequence of o s c i l l a t i o n s which increase in amplitude from pass to pass. Then, in e f f e c t , asymptotic s t a b i l i t y guarantees the existence of the limit p r o f i l e as a function of the ( f i n i t e ) pass length and s t a b i l i t y along the pass is independent of t h i s parameter. This, in turn means that asymptotic s t a b i l i t y alone would permit exponential growth terms in the dynamics along a pass an obviously t o t a l l y undesirable feature. Necessary and s u f f i c i e n t conditions for s t a b i l i t y are expressed in terms of the spectral radius and resolvent of the linear operator associated with the abstract representation. Hence application of t h i s theory to a p a r t i c u l a r example requires the i n t e r p r e t a t i o n of these r e s u l t s in terms of the parameters of i t s representation or model. No general rules exist for t h i s task and severe d i f f i c u l t i e s could arise i f the underlying Banach space or the linear operator have a complex structure. A s i g n i f i c a n t number of i n d u s t r i a l l y relevant special cases can, however, be dealt with in a r e l a t i v e l y simple manner. This has been demonstrated here by a detailed consideration of the long-wall coal cutting example and d i f f e r e n t i a l and discrete non-unit memory linear r e p e t i t i v e processes. In these l a t t e r two eases, the r e s u l t i n g conditions are expressed in terms of the matrices of the corresponding state-space descriptions. The basic d i f f i c u l t y with these conditions is that t e s t i n g one of them is not computationally f e a s i b l e , where t h i s is a common feature of the r e s u l t s to-date for a number of other cases. Further, given the pivotal role of s t a b i l i t y , the development of computationally f e a s i b l e s t a b i l i t y t e s t s is an obvious s t a r t i n g point for any f u r t h e r control related analysis of a given case. Consequently a s u b s t a n t i a l part of the work reported in t h i s monograph has been the development of

189

computationally f e a s i b l e s t a b i l i t y t e s t s f o r d i f f e r e n t i a l and d i s c r e t e non-unit memory linear r e p e t i t i v e processes. Previously reported work has established strong s t r u c t u r a l links between a number of sub-classes and other well researched classes of dynamic systems. In the case of d i f f e r e n t i a l and discrete processes, such links have been established with the following two classes of linear dynamic systems. (i) Standard or, within the r e p e t i t i v e systems framework, conventional linear systems described by the well known state-space model or t r a n s f e r - f u n c t i o n matrix. Basically, the r e p e t i t i v e systems s t a t e - s p a c e model reduces to i t s conventional linear systems counterpart under well defined conditions. ( i i ) 2D linear systems described by the Roesser s t a r , space model. Basically, s t a b i l i t y along the pass in the discrete unit memory case is equivalent to BIBO s t a b i l i t y of the Roesser model. All of the new r e s u l t s presented in t h i s monograph have, in e f f e c t , been developed by appropriately exploiting these links or, in the d i f f e r e n t i a l case, r e s u l t s from the s t a b i l i t y theory of certain classes of delay d i f f e r e n t i a l systems. As a r e s u l t of t h i s approach, three d i s t i n c t types of computationally f e a s i b l e s t a b i l i t y t e s t s have been developed which can be c l a s s i f i e d under the following general headings. (i) Graphical, or eigenvalue, based t e s t s . (ii) Algebraic, or root clustering, based t e s t s . ( i i i ) Simulation-based t e s t s . Further, (i) and ( i i ) share a common basis in that they are both based on a reformulation of the original state-space conditions in terms of the appropriate 2D t r a n s f e ~ f u n c t i o n matrix. These new conditions have then been used to develop two systematic t e s t procedures for each case which t e s t them in a p a r t i c u l a r order with termination i f the one just tested does not hold. Note also that in a l l cases the computationally most intensive condition is the l a s t to be tested ( i f required). The systematic procedure for each case based on (i) above uses, in e f f e c t , 'Nyquist l i k e ' t e s t s from the s t a b i l i t y theory of d i f f e r e n t i a l and d i s c r e t e conventional l i n e a r systems as appropriate. IIence these procedures are suitable for inclusion in a CAD package. As an a l t e r n a t i v e , the systematic procedures based on ( i i ) above make appropriate use of well known conventional linear systems root clustering based s t a b i l i t y t e s t s . For example, the procedure f o r the d i s c r e t e case uses the Jury/Marden table and the one for the d i f f e r e n t i a l case uses the Routh array and i t s modified version for real even order polynomials. These procedures are not suitable for inclusion in a CAD package and have t h e i r major remit in low order synthesis problems where some, or a l l , of the elements of the matrices of the example under consideration are design parameters. Previous work has shown that BInO s t a b i l i t y of 2D linear systems described by the Roesser model is equivalent to s t a b i l i t y along the pass of d i s c r e t e unit memory linear r e p e t i t i v e processes. Consequently a l l of the known s t a b i l i t y t e s t s for

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these systems can be applied to the r e p e t i t i v e systems problem. In this work, however, p a r t i c u l a r attention has been directed towards the use of Lyapunov equations. This has shown that, unlike the conventional linear systems case, two essentially different approaches are possible. One of these is based on a 2D equation with constant coefficients and the other uses a 1D equation with coefficients which are functions of a complex variable. Either of these Lyapunov equations could be used to form the basis of a systematic t e s t procedure to serve as an alternative to those based on (i) and (if) above. Note, however, that the constant coefficient version is, in general, s u f f i c i e n t but not necessary and this fact clearly reduces i t s effectiveness given the alternatives which t e s t necessary and s u f f i c i e n t conditions. Detailed in depth comparative studies of all of these procedures for the discrete unit memory case has, however, not been considered in the absence to date of results from applying them to suitably defined benchmark problems. The use of the constant coefficient Lyapunov equation in developing physically meaningful s t a b i l i t y margins for the discrete unit memory case has been considered. This i n i t i a l work has, in effect, been based on extending some results from 2D linear systems. Further, there are two (interrelated) areas to which future research e f f o r t could profitably be directed. These are further development of the basic computational algorithm for increased efficiency and in depth work to establish the correlation with system performance. Progress in both of these areas will also obviously serve to strengthen the already known links between these two, apparently d i s t i n c t , areas. Note also that it is by no means clear how, if at all, the procedures based on (i) and (if) above can be exploited in terms of s t a b i l i t y margins, except as a result of extensive simulation studies following on from the basic testing. In the d i f f e r e n t i a l unit memory case, i t has been shown that elements of the s t a b i l i t y theory of delay d i f f e r e n t i a l systems can be used to produce systematic t e s t procedures as alternatives to those based on (i) and (if) above. This arises from a proof of a previously conjectured result that s t a b i l i t y along the pass is equivalent to pointwise asymptotic s t a b i l i t y when this example is interpreted as a delay d i f f e r e n t i a l system, llence all of the known s t a b i l i t y t e s t s for delay d i f f e r e n t i a l systems can be applied to this example but here, as in the discrete case, p a r t i c u l a r attention has been directed towards the use of Lyapunov equations. The resulting analysis again yielding two essentially different approaches, based on a 2D equation with constant coefficients and a 1D equation with coefficients which are functions of a complex variable respectively. As in the discrete case, either of these Lyapunov equations could be used to {orm the basis of an alternative systematic test procedure to those based on (i) and (if) above. Again, however, the constant coefficient version is, in general, sufficient but not necessary and this fact clearly reduces its effectiveness given the alternatives which t e s t necessary and sufficient conditions. Detailed

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comparisons of a l l the available procedures for this case would, hoverer, require r e s u l t s from applying them to suitably defined benchmark problems. Hence this topic has not been considered further in this work. Using an analogous approach to the discrete case, the constant c o e f f i c i e n t version has been used to develop some i n i t i a l r e s u l t s on physically meaningful s t a b i l i t y margins. The results presented are, in e f f e c t , appropriate extensions of some of those currently available for delay d i f f e r e n t i a l systems. Further, tvo (interrelated) areas have been identified to which future research e f f o r t could profitably be directed. These are the development of a computational algorithm (which does not yet exist for delay d i f f e r e n t i a l systems) and in depth work to establish the correlation with systems performance. Progress in these two areas will again also serve to strengthen the links between these two, apparently d i s t i n c t , areas. Finally, note that, as in the discrete case, it is by no means clear how, i f at a l l , the procedures based on (i) and (if) above can be exploited in terms of s t a b i l i t y margins, except as a result of extensive simulation studies following on from the basic testing. In terms of controlling these processes, it is known that computable information concerning the following factors is of at least equal importance to stability margins. (i) The rate of approach of the output sequence to the limit profile. (ii) Bounds on the performance along any pass. Suppose also that chapter are ~sed. information on (i) the only possible

the simulation-based t e s t s listed under ( i i i ) e a r l i e r in this Then these t e s t s produce, at no extra cost, computable and (if) above. This information is unique to these t e s t s , with alternative being to inspect the results of simulation studies and

hence the prospect of a heavy computational toad. Further, this feature serves to o f f s e t the fact that these t e s t s are s u f f i c i e n t but not necessary. These simulation-based t e s t s have been developed from suitably well behaved plant step response data which, in this work, is assumed to be available or can be obtained by simulation studies. In particular, it has been assumed in this i n i t i a l study that the parameters in the sub-processes vhich define this step response data (the derived and conventional linear systems) are known exactly. By analogy with the use of such information in conventional linear systems (one form of robust control analysis) it ca~l be conjectured that this assumption can be relaxed in a r e l a t i v e l y straightforward manner. Further, some i n i t i a l highly promising results on extending these t e s t s to processes with interpass smoothing have been presented. These are the f i r s t reported output on the analysis of such cases and it is by no means clear at this stage how, if at a l l , t e s t s based on other approaches can be e f f e c t i v e l y used in this context. IIence it is clear that the simulation-based approach to the control related analysis of certain sub-classes of linear repetitive

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processes should, a f t e r appropriate development, be a powerful and f l e x i b l e technique. Return now to the basic problem of t e s t i n g a d i f f e r e n t i a l or discrete non unit memory linear r e p e t i t i v e process for s t a b i l i t y . Then the work reported here has produced a range of computationally f e a s i b l e s t a b i l i t y t e s t s grouped under three general headings. Note, however, that much work remains to be done which can be broadly c l a s s i f i e d as follows. (a) Further development of the specific areas detailed as appropriate in the main t e x t plus work on extending t h e i r e f f e c t i v e operating range. For example, further work on applying the simulation-based t e s t s to processes with interpass smoothing e f f e c t s should yield rapid progress. (b) Further work aimed at achieving maximum efficiency coupled, in the case of the eigenvalue and simulation-based t e s t s , with the development of a software i n f r a s t r u c t u r e to form the basis of a comprehensive computer aided analysis/design package. This should use currently available software as a basis and include the development of a suitable user interface f a c i l i t y . (c) The development of 'easy to use' s t a b i l i t y margins and/or performance indicators to a s s i s t in the formulation and solution of p r a c t i c a l l y relevant control p o l i c i e s . This area should i n i t i a l l y proceed from the highly promising work reported in chapters 4 and 5 for t h i s general area. (d) The development of a comprehensive 'systems t h e o r e t i c ' i n t e r p r e t a t i o n of s t a b i l i t y and related matters. One possible use of such a theory would be to provide indicators of control d i f f i c u l t i e s in the s p i r i t of Wonham's c o n t r o l l a b i l i t y result for pole assignment in conventional linear systems. Specific areas for i n i t i a l work could include (i) the precise implications of the c o n t r o l l a b i l i t y and observabitity conditions of the necessary and s u f f i c i e n t s t a b i l i t y r e s u l t s of chapter 3, ( i i ) the precise roles ( i f any) of appropriately defined poles and zeros where one candidate d e f i n i t i o n of the former in the unit memory cases has already been proposed, and (iv) the derivation of s t a b i l i t y conditions expressed in terms of the 2D t r a n s f e r - f u n c t i o n matrices as an entity instead of as in t h i s work where only i t s constituent elements have been used (in p a r t i c u l a r , the t r a n s f e r - f u n c t i o n matrices of the derived and associated conventional linear systems). The success in developing basic computationally f e a s i b l e s t a b i l i t y t e s t s has led to some i n i t i a l work on controller design. In p a r t i c u l a r , three control p o l i c i e s have been formulated from practical considerations and feedback control schemes which use either s t a t e or output information have been developed. Further, some candidate design algorithms have been developed together with some relevant systems theoretic properties. At a general level, the schemes developed in t h i s work have demonstrated the p o t e n t i a l power of appropriately specified feedback control schemes in regulating

193

the behaviour of d i f f e r e n t i a l and discrete non-unit memory linear r e p e t i t i v e processes. Further, an a t t r a c t i v e feature of these schemes from an applications standpoint is t h e i r r e l a t i v e simplicity and hence the p o s s i b i l i t y of implementation without recourse to special purpose hardware/software. This is p a r t i c u l a r l y true for the memoryless cases since they have the simplest structure in terms of the information to be logged and/or stored in order to actuate the c o n t r o l l e r . Hence i t is strongly recommended that the potential of these schemes should be f u l l y evaluated before recourse to others with a more complex structure. As a s t a r t i n g point, the s p e c i f i c areas detailed as appropriate in chapter 6 should be addressed. In conclusion, therefore, substantial progress towards the development of rigorous s t a b i l i t y and control theories for d i f f e r e n t i a l and d i s c r e t e non-unit memory linear r e p e t i t i v e processes has been made based, e s s e n t i a l l y , on an abstract representation of the general linear dynamics constant pass length case. This strongly suggests that a similar approach to other general cases should prove equally successful, p a r t i c u l a r l y if experience gained in the course of the work reported here can be exploited. Dne obvious area which should benefit considerably from such an approach is that of a constant pass length and certain classes of nonlinear dynamics.

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(Ed.)

Vol. 163: V. L. Mehrmann The Autonomous Linear Quadratic Control Problem Theory and Numerical Solution VI, 177 pages, 1991 Vol. 164: I. Lasiecka, R. Triggiani Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory XI, 160 pages, 1991

Vol. 172: H. Tolle, E. Ersii Neurocontrol Learning Control Systems Inspired by Neuronal Architectures and Human Problem Solving Strategies X, 211 pages. 1992 Vol. 173: W. Krabs On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes VII, 174 pages. 1992

Matrices

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