Lecture Notes in Control and Information Sciences Edited by A.V Balakrishnan and M, Thoma 29 M Vidyasagar Input-Output Analysis of Large-Scale Interconnected Systems Decomposition, WelI-Posedness and Stability Springer-Verlag Berlin Heidelberg New York1981 Series Editors A V Balakrishnan - M Thoma Advisory Board L D Davisson • A G J MacFarlane • H Kwakernaak J L Massey • Ya Z Tsypkin • A J Viterbi Author Prof M Vidyasagar Dept of Electrical Engineering University of Waterloo Waterloo, Ontario Canada ISBN 3-540-10501-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10501-8 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to 'Verwertungsgesellschaft Wort', Munich © Springer-Vedag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr 2061/3020-543210 This book is intended to be a fairly comprehensive treatment of large-scale interconnected systems from an input- output viewpoint Prior to treating the question of stability (and instability), we study both the decomposition posedness of such systems It is not necessary and the well- for the reader to have studied feedback stability before tackling this book, as we develop results concerning feedback systems as special cases of more general results pertaining to large-scale systems However, the reader should know some elementary analysis (e.g Lebesgue spaces, and have some general knowledge (e.g Perron-frobenius The first chapter is introductory, background material; after that, functional contraction mapping theorem), and chapters theorem) and contain the remaining chapters are essentially independent and can be read in any order I thank Peter Moylan for his careful reading of the manuscript and for several constructive ShakUnthala for her support suggestions, and my wife Virtually all of my research reported in this book was carried out, and most of the book was written, while I was employed by Concordia University, Montreal I would like to acknowledge research support from the Natural Sciences and Engineering Research Council of Canada, lesser extent from the U.S Department of Energy and to a Finally, thanks to Monica Etwaroo and Jane Skinner for typing the manuscript Waterloo September 29, 1980 M Vidyasagar my TABLE OF CONTENTS PAGE PREFACE v CHAPTER i: CHAPTER 2: INTRODUCTION ~THEMATICAL PRELIMINARIES 2.1 2.2 CHAPTER 3: 3.2 3.3 4.2 4.3 5.2 5.3 5.4 2~ 26 42 46 Some Results From the Theory of Directed Graphs Decomposition into Strongly Connected Components Results on Well-Posedness and Stability Weakly Lipschitz, Smoothing and Strictly Causal Operators Single-Loop Systems Continuous-Time Systems Discrete-Time Systems s7 57 73 81 88 88 94 95 103 Single-Loop Systems Criteria Based on a Test Matrix C r i t e r i a B a s e d o n an E s s e n t i a l S e t Decomposition 105 107 126 DISSIPATIVITY-TYPE CRITERIA FOR L2-STABILITY 133 7.1 7.2 7.3 CHAPTER 8: 12 SMALL-GAINTYPE CRITERIA FOR Lp-STABILITY • lO5 6.1 6.2 6.3 CHAPTER 7: Gain, Gain with Zero Bias, and Incremental Gain Dissipativity and Passivity Conditional Gain and Conditional Dissipativity WELL-POSEDNESS OF LARGE-SCALE I~TERCO~NECTED SYSTEMS 5.1 CHAPTER 6: Truncations, Extended Spaces, Causality Definitions of Well-Posedness and Stability DECOMPOSITION OF LARGE-SCALE INTERCONNECTED SYSTEMS 4.1 CHAPTER5: GAIN AND DISSIPATIVITY 3.1 CHAPTER Single-Loop Systems 134 General Dissipativity-Type C r i t e r i a 139 Special Cases: Small-Gain and Passivity-Type Criteria 144 L2-1NSTABILITY CRITERIA 164 8.1 8.2 8.3 164 168 175 Single-Loop Systems Criteria of the Small-Gain Type Dissipativity-Type Criteria Vl TABLE OF CONTENTS CONT'D CHAPTER 9: L~-STABILITY AND L~-INSTABILITY USING EXPONENTIAL WEIGHTING 189 9.1 9.2 9.3 190 198 205 General Special General Stability Result Cases Instability Result REFERENCES 213 INDEX 218 CIIAPTER i: INTRODUCTION D u r i n g the p a s t decade or so, there has b e e n a great deal of i n t e r e s t in the study of l a r g e - s c a l e systems as a s e p a r a t e d i s c i p l i n e in itself many factors, p h y s i c a l systems circuits, This i n t e r e s t is t r a c e a b l e to i n c l u d i n g the g r o w i n g r e a l i z a t i o n that many etc.) (e.g power networks, several s i m p l e r subsystems, and "structure" large-scale integrated can in fact be v i e w e d as i n t e r c o n n e c t i o n s of and that m u c h v a l u a b l e information is lost if the m e t h o d of a n a l y s i s does not take into a c c o u n t the i n t e r c o n n e c t e d nature of the s y s t e m at hand Moreover, several s u b j e c t s d e a l i n g w i t h reached m a t u r i t y , "small" systems have so that in order to expand the h o r i z o n s of k n o w l e d g e by t a c k l i n g new and c h a l l e n g i n g p r o b l e m areas, re- searchers have set their sights on l a r g e - s c a l e Some systems prime e x a m p l e s of this are o p t i m a l c o n t r o l theory, s t a b i l i t y t h e o r y of s i n g l e - l o o p and the f e e d b a c k systems It is as yet too soon to c l a i m that there e x i s t s a comprehensive theory of l a r g e - s c a l e systems stability theory of l a r g e - s c a l e Nevertheless, systems is a w e l l - d e v e l o p e d in w h i c h a large v a r i e t y of results is available effect two m e t h o d o l o g i e s in s t a b i l i t y theory, methods and i n p u t - o u t p u t methods While the area T h e r e are in namely Lyapunov there are some con- n e c t i o n s b e t w e e n L y a p u n o v s t a b i l i t y and i n p u t - o u t p u t stability, the actual t e c h n i q u e s used to e s t a b l i s h the two types of s t a b i l i t y are r a t h e r different; of l a r g e - s c a l e systems Lyapunov systems are w e l l - d o c u m e n t e d Miller [Mic this is e s p e c i a l l y methods so in the case for l a r g e - s c a l e in the r e c e n t books by M i c h e l and i] and S i l j a k [Sil i] contains come i n p u t - o u t p u t results, However, though [Mic i] there is not at p r e s e n t a c o m p r e h e n s i v e book on the i n p u t - o u t p u t a n a l y s i s of l a r g e - s c a l e systems In the same vein, Desoer and V i d y a s a g a r [Des the books by W i l l e m s [Wil 2] and 2] cover f e e d b a c k systems quite t h o r o u g h l y from an i n p u t - o u t p u t viewpoint, and it is n a t u r a l to attempt a s i m i l a r t r e a t m e n t of l a r g e - s c a l e systems This b o o k is i n t e n d e d to be a h i g h - l e v e l r e s e a r c h m o n o g r a p h t h a t sets forth m o s t of the a v a i l a b l e results on the decomposition, well-posedness, s t a b i l i t y and i n s t a b i l i t y of large- scale systems, that can be o b t a i n e d by i n p u t - o u t p u t methods Since m a n y r e s u l t s for f e e d b a c k systems can be o b t a i n e d as special cases of those given here for l a r g e - s c a l e systems, not n e c e s s a r y to have read [Wil book 2] or [Des 2| it is to follow this T h o u g h the e m p h a s i s h e r e is on i n p u t - o u t p u t stability, we note that i n p u t - o u t p u t m e t h o d s can be u s e d to e s t a b l i s h L y a p u n o v s t a b i l i t y as well is i n p u t - o u t p u t stable, In particular, also g l o b a l l y a s y m p t o t i c a l l y (see [Wil 3] , [Moy if a n o n l i n e a r system r e a c h a b l e and detectable, then it is stable in the sense of L y a p u n o v 4] ) T h r o u g h o u t this book, the e m p h a s i s is on t r e a t i n g the l a r g e - s c a l e s y s t e m at h a n d as an i n t e r c o n n e c t e d system, sisting of several s u b s y s t e m s c o n n e c t i o n operators 2.2) con- i n t e r a c t i n g through various inter- (For a p r e c i s e d e s c r i p t i o n , It is of course p o s s i b l e to "aggregate" s y s t e m o p e r a t o r s and the v a r i o u s see S e c t i o n the v a r i o u s sub- i n t e r c o n n e c t i o n operators, so that the l a r g e - s c a l e s y s t e m at h a n d is r e c a s t in the f o r m of a "single-loop" f e e d b a c k system W i t h this r e f o r m u l a t i o n , the s t a n d a r d s i n g l e - l o o p f e e d b a c k s t a b i l i t y results, those in [Des 2] and [Wil 2] b e c o m e applicable w h e t h e r a given s y s t e m is a "single-loop" connected" all of such as Therefore, s y s t e m or an "inter- s y s t e m depends on the m e t h o d of a n a l y s i s u s e d to tackle it However, it can be e a s i l y shown that c o n v e r t i n g the s y s t e m into a "single-loop" conservative f o r m u l a t i o n gives u n n e c e s s a r i l y s t a b i l i t y c r i t e r i a and w e l l ' p o s e d n e s s Therefore, criteria in this b o o k we only p r e s e n t results that p e r t a i n to i n t e r c o n n e c t e d systems, w h e r e b y the a n a l y s i s is c a r r i e d out in terms of the s u b s y s t e m o p e r a t o r s and the interc o n n e c t i o n operators; we avoid t r e a t i n g the s y s t e m as a w h o l e For this reason, we e x c l u d e linear t i m e - i n v a r i a n t systems f r o m our study The r e a s o n is that, and s u f f i c i e n t c o n d i t i o n s interconnected conditions though one can derive n e c e s s a r y for the s t a b i l i t y and w e l l - p o s e d n e s s linear t i m e - i n v a r i a n t systems, (of necessity) of the n e c e s s a r y involve t a c k l i n g the s y s t e m as a whole A s u b s y s t e m level a n a l y s i s can p r o d u c e s u f f i c i e n t c o n d i t i o n s s t a b i l i t y and s u f f i c i e n t c o n d i t i o n s n e c e s s a r y and s u f f i c i e n t conditions for instability, b u t not for The book is organized as follows: In Chapter 2, we introduce the concepts of truncations and extended spaces, which provide the mathematical setting for input-output analysis, we then give precise definitions of well-posedness and stability In Chapter 3, we introduce the concepts of gain and dissipativity, which play an important role in the various criteria for stability and instability, and give explicit methods for com- puting gains and testing dissipativity In Chapter 4, we present a few graph-theoretic niques for the efficient decomposition of large-scale connected systems Specifically, tech- inter- we show that by identifying the so-called strongly connected components (SCC's) of a given system, we can determine the well-posedness and stability of the original system by studying only the SCC's present some sufficient conditions system These criteria are graph-theoretic given a very nice physical In Chapter 5, we for the well-posedness interpretation In Chapter 6, we give some generalizations single-loop of a in nature and can be of the "small gain" theorem to arbitrary interconnected systems, while in Chapter generalizations 7, we state and prove several of the single-loop "passivity" Chapter 8, we derive several L2-instability scale systems Finally, theorem In criteria for large- in Chapter 9, we show how the technique of exponential weighting can be used to study L -stability and L -instability using the results of Chapters to CHAPTER 2: MATHEMATICAL PRELIMINARIES 2.1 TRUNCATIONS, In this notation section, and terminology particular notation Let functions X R+ = here and As introduce the m a t h e m a t i c a l is f r o m this book [Vid 4] and the set of all r e a l - v a l u e d into [0,~), measure we briefly employed R+ SPACES r CAUSALITY t h a t is u s e d t h r o u g h o u t denote mapping numbers, Lebesgue X EXTENDED R, w h e r e R denotes the m e a s u r a b i l i t y is c u s t o m a r y , The [Des measurable the s e t of r e a l is w i t h r e s p e c t we define 2] various to the subsets of as f o l l o w s : Definition For p [i,~), the s e t L P notes the s e t of all functions tion t + is i n t e g r a b l e f(.) E L [If(t) I]P for a f i x e d P p e f(.) [i,~) in over X such [0,~) if a n d o n l y = L [0,~) deP t h a t the f u n c - In o t h e r w o r d s , if If(t) Ip dt < Similarly, in X [0,-) L = L such that • If p [0,~) f(.) [i,~) denotes the set of all is e s s e n t i a l l y we d e f i n e , bounded the f u n c t i o n functions over I' f(.) the i n t e r v a l Ilp : Lp ÷ R+ by I tfI1p = [ If(t) lp dt] 1/p , vf e Lp If p = - , we define II-I I~ : L I IfEl = e s s ° t = inf where p e ~[.] [1,~], space denotes sup ÷ R+ by IfCt) j [0,~) {r : ~ [ t : I f ( t ) I > r] = 0} the Lebesgue measure ~f L of a set I t is w e l l - k n o w n [Dun i, p 146] the o r d e r e d (Lp , I.I I , Ip) pair , t h a t for e a c h constitutes a Banach In o r d e r can study to h a v e "unstable" the c o n c e p t of as w e l l truncated Definition is d e f i n e d a mathematical as "stable" functions and T < ~ ; then Let For b r e v i t y , refer the we use the XT(.) as to interval sense that introduce spaces the o p e r a t o r PT : X + X Vx• Note that that PT denotes fT( ) e L p belong to of the space X t > T notation the xT to d e n o t e truncation the of a g i v e n function function PT x, x(.) [0,T] the PT For the YT Lp) operator " PT = Definition L pe [0,~) we we t • [0,T] to the systems, extended whereby by setting (PTx)(t) = { x(t) and framework a fixed s e t of all < ~ The PT p • Lpe [i,~], functions (though space is a p r o j e c t i o n on X in symbol L = " f(.) the f(.) itself is r e f e r r e d in may pe such X or m a y to as the not extension L P Example e_~d s p a c e s Lpe The for the u n e x t e n d e d spaces tan t does C X Moreover, all finite T is t h e Then for p • Lp not belong It is c l e a r Lle function all for that, Lp , it is c l e a r Definition every set that p E , the The unextended fixed Lpe [i,-] c Lle to u s e be truncated to the e x t e n d - not belong to an[ function spaces f2(t) L of Vp • [i,~] in this fixed, norm L c L p pe [0,T] for L1 and Thus book let IIfl ITp T < is d e f i n e d IIf11 p = IIfTlIp= llpTfllp Let p = 2, a n d truncated inner let T < ~ product Then