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
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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.