Lecture Notes in Control and Information Science-frontmatter

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner 92 Lj. T. Gruji6, A. A. Martynyuk, M. Ribbens-Pavella Large Scale Systems Stability under Structural and Singular Perturbations Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Series Editors M. Thoma • A. Wyner Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • J. Stoer • Ya Z. Tsypkin • A. J. Viterbi Authors Ljubomir T. Gruji6 Faculty of Mechanical Engineering P.O. Box 174 27 Marta 80 11001 Belgrade Yugoslavia A. A. Martynyuk Institute of Mathematics Ukrainian Academy of Sciences Repin Str. 3 252004 Kiew USSR M. Ribbens-Pavella Unversite De Liege Institute D'Electricit6 Montefiore Circuits Electriques Sart Tilman, B28 4000 Liege Belgique ISBN 3-540-18300-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-18300-0 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations f~.ll under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1987 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 2161/3020-543210 To Alek~and~ Miko~J~ovieh L~apunov ( 1857- 1918 } PREFACE This book const i tu tes an up to date presentat ion and development of s tab i l i ty theory in the Liapunov sense with various extens ions and app l i ca t ions . Prec ise de f in i t ions of wel l known and new s tab i l i ty p roper t ies are given by the authors who present general resu l t s on the Liapunov sta- b i l i ty p roper t ies of non-s ta t ionary systems which are out of the c lass - i ca l s tab i l i ty theory framework. The study invo lves the use of t ime varying sets and is broadened to t ime varying Lur 'e -Postn ikov systems and s ingu lar ly perturbed systems. A remarkable cont r ibut ion is proposed by the authors who es tab l i sh necessary and su f f i c ient cond i t ions , s imi la r to Liapunov's one, for uniform abso lute s tab i l i ty of t ime varying Lur 'e -Postn ikov systems. Comparison systems and comparison pr inc ip le are s tud ied , in general and par t i cu la r forms, and appl ied to large scal~ systems. In that sense various forms of la rge-sca le systems aggregat ion are ~tudied and various s tab i l i ty c r i te r ia are es tab l i shed under d i f fe rent hypotheses : with invar iant s t ruc ture , with Lur 'e -Postn ikov form and with s ingu lar ly perturbed proper t ies . Proposed resu l t s are broadened to s t ruc tura l s tab i l i ty ana lys is aimed at studying s tab i l i ty p roper t ies under unknown and unpred ic tab le s t ruc tura l var ia t ions . The c r i te r ia are developed both in a lgebra ic and frequency domains. They essent ia l l y reduce the order and complexi ty of s tab i l i ty problems. A number of various aggregat ion-decompos i t ion forms are also considered for power systems from the large sca le systems stand point . Prec ise de f in i t ions are in t roduced by the authors for various s tab i l i ty domains with app l i ca t ion to la rge-sca le systems in general and more spec i f i ca l - ly to power systems. S tab i l i ty p roper t ies and domains of d i s turbed power systems are es tab l i shed . vi Preface A number of examples and app l i ca t ions presented throughout th is book i l l us t ra te the various resu l t s . According to the amount and importance of de f in i t ions and s tab i l i ty c r i te r ia presented I cons ider that th i s book in i t ia l l y publ ished in Russian, represents the most complete one on s tab i l i ty theory proposed at th i s date. I t in te res ts a l l people concerned with s tab i l i ty problems in the la rgest sense and with secur i ty , re l iab i l i ty and robustness . Professor P ierre BORNE L i l l e , France FOREWORD Poincare's daring idea to obtain qua l i ta t ive in format ion on motion d i rec t ly from the d i f fe rent ia l equation descr ib ing i t , i . e . wi thout in tegrat ion , was rea l i zed by Liapunov 118921. With his abso lute com- p le teness and i r reproachable s t r i c tness , Liapunov la id the foundations of a conceptua l ly new approach to the qua l i ta t ive methods of the theory of d i f fe rent ia l equat ions. Nowadays, Liapunov's methods are recognized to be among the most powerful means of s tab i l i ty ana lys is in exact sc iences . These, along with the many extens ions fur ther developed, contr ibuted to broaden substant ia l l y the c lasses of problems able of being e f fec t ive ly analyzed by the d i rec t method. The present book contains an essay of development of the general theory of s tab i l i ty in the sense of Liapunov, elements of the s tab i l i ty theory of comparison systems (systems of ordinary d i f fe rent ia l equations with monotonous r ight -hand par ts ) , p resentat ion of the general methods for the ana lys is of s t ruc tura l s tab i l i ty of la rge-sca le systems, inc lud ing systems with s ingu lar per turbat ions . The Liapunov funct ions (sca lar , vector and matrix) and his d i rec t method for the s tab i l i ty ana lys is of the unperturbed motion are used throughout the book~ Some of the ob- ta ined s tab i l i ty resu l t s are appl ied to the ana lys is of la rge-sca le e lec t r i c power systems. The s tab i l i ty of these systems is a very impor- tant par t i cu la r case for which the d i rec t c r i te r ia show extremely use- fu l . The Russian vers ion of th i s monograph was completed in 1982, the 125th anniversary of Liapunov's b i r thday. S ince, new resu l t s of the authors have been added and inc luded in the present vers ion. More spec i f i ca l l y Chapter V has been thoroughly rev ised and completed. Overal l , th i s English vers ion is more than a mere t rans la t ion of the Russian one~ v i i i Foreword Our permanent concern has been to wr i te up in a c lear , easy to compre- hend, way, readable for both eng ineers who need conven ient mathemat ica l machinery for la rge-sca le system s tab i l i ty ana lys i s , and mathematic ians who are in teres ted in new problems of the qua l i ta t ive theory of d i f fe r - ent ia l equat ions . We have t r ied to do jus t i ce to sc ient i s ts who the f i r s t obtained re - su l t s in var ious areas of the la rge-sca le systems s tab i l i ty theory , and to re fer to the i r o r ig ina l papers. I t i s re f lec ted in the B ig l iograph ies which inc lude more than 400 re ferences . Cer ta in ly , even such a l i s t i s s t i l l i ncomplete . This can be par t ly exp la ined by £he in tens ive research e f fo r ts and developments in the area, and by the ext remely wide domains of i t s app l i ca t ion , beginning with techno logy and f in i sh ing with the problems of popu la t iona l dynamics. We apo log ize to a l l those whose work was not c i ted or p roper ly descr ibed . ACKNOWLEDGMENTS Academicians Yu.A. M i t ropo lsky and Ye.F. Mishchenko, Assoc ia te Member of Academy of Sc iences of the USSR, V . I . Zubov and Pro fessor Yu.A. Ryabov have got acquainted with the Russian manuscr ipt of the book. The i r deta i led remarks were ext remely va luab le . Many conversat ions of A.A. Martynyuk wi th P ro fessor A.B. Zhishchenko great ly in f luenced the presentat ion of problems connected with the a lgebra ic type of the ob- ta ined resu l ts . Co l laborators of the Processes S tab i l i ty Department of the Ins t i tu te of Mechanics of the Ukrainian Academy of Sc iences , I .Yu. Lazareva, Ye.P. Shat i lova have cont r ibuted much in the course of the techn ica l work on the manuscr ipt . Mrs. M.B. Counet-Lecomte did an outs tand ing job in typ ing the f ina l Engl ish vers ion . The qua l i ty of th i s camera- ready presentat ion owes enormously to her exper t i se . Th~ authors are cord ia l l y thankfu l to a l l of them. L j .T .G. A.A.M. M.R.P. Belgrade Kiev LiEge September 1987. CONTENTS List of basic symbols Chapter I OUTLINE OF THE LIAPUNOV STABILITY THEORY IN GENERAL I.I. Introductory comments 1.2 On definition of stability properties in Liapunov's sense 1.2.1 Liapunov's original definition 1.2.2 Comments on Liapunov's original definition 1.2.3 Relationship between the reference motion and the zero solution 1.2.4 Accepted definitions of stability properties in Liapunov's sense 1.2.5 Equilibrium states 1.3 On the Liapunov stability conditions 1.3.1 Brief outline of Liapunov's original results 1.3.2 Brief outline of the classical and novel developments of the Liapunov second method 1.4 On absolute stability 1.4.1 Introductory comments 1.4.2 Description of Lur'e-Postnikov systems 1.4.3 Definition of absolute stability 1.4.4 Liapunov'like conditions for uniform absolute stability 1.4.5 Criteria for absolute stability of time-varying systems 1.4.6 Criteria for absolute stability of time-invariant systems xiii 1 1 2 2 5 6 7 13 14 14 2o 42 42 43 44 45 46 51 X Contents 1.5 On stability properties of singularly perturbed systems 1.5.1 Introductory comments 1.5.2 System description 1.5.3 Liapunov-like conditions for asymptotic stability 1.5.4 Singularly perturbed Lur'e-Postnikov systems Comments on references References Chapter I I THE STABILITY THEORY OF COMPARISON SYSTEMS 11.1 Introductory notes 11.1.1 Original concepts of the comparison method 11.1.2 The Liapunov functions and comparison equations generated by them 11.1.3 Vector-functions and comparison systems 11.1.4 Matrix-functions 11.2 The Liapunov functions and comparison equations 11.2.1 On monotonicity and solutions estimations 11.2.2 Special cases of the general comparison equations 11.2.3 General stability theorems on the basis of scalar comparison equations 11.2.4 The generalized comparison equation 11.2.5 The scalar comparison equation construction 11.2.6 A refined method of comparison equations construction 11.2.7 Several applications of scalar comparison equations 11.3 Stability of the comparison systems solutions 11.3.1 The non-degeneracy of monotonicity. Definition 11.3.2 The basic statements of the comparison principle 11.3.3 Definitions of the comparison system stability 11.3.4 Linear comparison systems 11.3.5 Nonlinear systems with an isolated equilibrium state 11.3.6 The theorem of Zaidenberg-Tarsky and algebraic solvability of the stability problem 11.3.7 Nonlinear autonomous comparison systems with a non-isolated singular point 11.3.8 Several applications of nonlinear comparison systems 11.3.9 Reducible comparison systems 52 52 53 54 58 62 65 73 73 73 76 77 8O 83 83 9O 97 101 104 I08 111 117 117 117 119 121 124 126 128 129 135 Contents xi 11.4 Matrix-functions application to the stability analysis 11.4.1 Main properties of matrix-functions 11.4.2 Theorems of direct method based on matrix- functions 11.4.3 The scalar Liapunov function construction on the basis of matrix-functions Comments on references References Chapter I I I LARGE-SCALE SYSTEMS IN GENERAL III.i Introduction 111.2 Description and decomposition of large-scale systems 111.3 Structural stability properties of large-scale systems 111.4 Aggregation forms of large-scale systems and conditions of structural stability 111.4.1 Aggregation forms and solutions for the Problem A 111.4.2 Aggregation forms and solutions for the Problem B 111.4.3 The structural stability analysis of a large- scale system with non-asymptotically stable subsystems Comments on references References Chapter IV SINGULARLY PERTURBED LARGE-SCALE SYSTEMS IV.1 Introduction IV.2 Description and decomposition of singularly perturbed large-scale systems IV.3 Aggregation and stability criteria for singularly per- turbed large-scale systems IV.3.1 Introduction IV.3.2 Non-uniform time scaling IV.3.3 Uniform time-scaling IV.4 Comments References 137 137 138 143 149 151 155 155 157 160 163 163 1T6 214 221 223 231 231 231 233 233 234 243 260 261 xi i Contents Chapter V LARGE-SCALE POWER SYSTEMS STABILITY Notation 263 V.I Introduction 265 V.2 The physical problem and its mathematical modelling 267 V.2.1 Problem definition 267 V.2.2 Conventional problem formulation 271 V.2.3 Definitions of stability domains and their estimates 272 V.2.4 Liapunov's method applied to conventional transient stability analysis 273 V.2.5 System modelling 275 V.2.6 Mathematical formulation 277 V.3 Scalar Liapunov approach 279 V.3.1 Preliminaries 279 V.3.2 The "energy type" Liapunov function 280 V.3.3 Family of the "energy type" V functions 287 V.3.4 The Zubov method 290 V.3.5 Numerical simulations 291 V.4 Vector Liapunov approach 303 V.4.1 Introduction 303 V.4.2 Stationary large-scale systems decompositions and aggregations in general 305 V.4.3 General stability analysis of stationary large-scale systems 312 V.4.4 Power systems modelling 321 V.4.5 Power systems decompositions and aggregations 324 V.5 Conclusion 353 References 354 Postface 361 References 365 LIST OF BASIC SYMBOLS All symbols are ful ly def ined at the place where they are f irst intro- duced. As a convenience to the reader we have co l lected some of the more f requent ly used symbols in several places. The largest co l lect ion is the one given below. Addi t iona l list for later use can be found in the in t roduct ion to Chapter V. A,B,C,... A~ B AUB , AnB A,B,C,... a,b,c,... a,b,e,... Ba(t o) = {x:lJxJl]-v(t'x)- 0 :@÷0+} D*v(t ,×) d(X,A) = inf[llx-yll.:yeA] d(A,B) =max {sup [d(x,A):xeB], sup [d(X,B):xEA]} f: RxRn+R n I k He(-) i,j, k,...,N j:v~Y K[o,~ ] N N(t) N r = {( t ,x ) : t~Tr ,xO,VXoeBA, 3 T(to, Xo,P)e ] O, +~[, ~X (t ; to, Xo )eBp, VteTT ] ~M(to,E ) : Max {~:6:$( to ,e)~×oeBs( to ,e) x(t ; to,Xo)eSe ,VteY o} ~S ¢ 7re(to, Xo,P) = Min {T:~:r( to,Xo, p) 9X(t ; to ,Xo)eBp,VteY~ ) Xi(" AM<- ~m( • X(t; to,× o ) V 9 E II II [ ] ] [ ( ) the maximal A obeying the def ini - t ion of at t ract iv i ty the maximal ~ obeying the def in i - t ion of stabi l i ty the boundary of a set S the closure of a set S empty set the minimal r sat is fy ing the def i - n i t ion of at t ract iv i ty the i - th e igenvalue of a matr ix (') the maximal e igenvalue of a matr ix (.) the minimal e igenvalue of a matr ix (-) a mot ion of a system at tER iff ×(to)--X o , x(to;to,×o)--×o " impl ies" "iff" ("if and only if") "for every" "there exist(s)" "there does (do) not exist" "such that" "belongs to" the Euc l idean norm denotes a closed interval denotes an open interval a general interval which can be semi-open, open, or closed.