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 ( 1 8 5 7 - 1918 }
PREFACE
This book c o n s t i t u t e s an up to d a t e p r e s e n t a t i o n and d e v e l o p m e n t of s t a b i l i t y t h e o r y i n t h e Liapunov s e n s e w i t h v a r i o u s e x t e n s i o n s and applications. P r e c i s e d e f i n i t i o n s of w e l l known and new s t a b i l i t y p r o p e r t i e s are g i v e n by t h e a u t h o r s who p r e s e n t g e n e r a l r e s u l t s on t h e Liapunov s t a b i l i t y p r o p e r t i e s of n o n - s t a t i o n a r y s y s t e m s which are out of t h e c l a s s i c a l s t a b i l i t y t h e o r y framework. The s t u d y i n v o l v e s t h e use of t i m e v a r y i n g s e t s and i s broadened to t i m e v a r y i n g L u r ' e - P o s t n i k o v s y s t e m s and s i n g u l a r l y p e r t u r b e d s y s t e m s . A r e m a r k a b l e c o n t r i b u t i o n i s proposed by t h e a u t h o r s who e s t a b l i s h n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s , s i m i l a r to L i a p u n o v ' s one, f o r uniform a b s o l u t e s t a b i l i t y of t i m e v a r y i n g L u r ' e - P o s t n i k o v s y s t e m s . Comparison s y s t e m s and comparison p r i n c i p l e are s t u d i e d , i n g e n e r a l and p a r t i c u l a r forms, and a p p l i e d to l a r g e scal~ s y s t e m s . In t h a t s e n s e v a r i o u s forms of l a r g e - s c a l e s y s t e m s a g g r e g a t i o n are ~ t u d i e d and v a r i o u s s t a b i l i t y c r i t e r i a are e s t a b l i s h e d under d i f f e r e n t h y p o t h e s e s : w i t h i n v a r i a n t s t r u c t u r e , w i t h L u r ' e - P o s t n i k o v form and w i t h s i n g u l a r l y p e r t u r b e d p r o p e r t i e s . Proposed r e s u l t s are broadened to s t r u c t u r a l s t a b i l i t y a n a l y s i s aimed a t s t u d y i n g s t a b i l i t y p r o p e r t i e s under unknown and u n p r e d i c t a b l e s t r u c t u r a l v a r i a t i o n s . The c r i t e r i a are d e v e l o p e d both i n a l g e b r a i c and f r e q u e n c y domains. They e s s e n t i a l l y r e d u c e t h e o r d e r and c o m p l e x i t y of s t a b i l i t y
problems.
A number of v a r i o u s a g g r e g a t i o n - d e c o m p o s i t i o n forms are a l s o c o n s i d e r e d f o r power s y s t e m s from t h e l a r g e s c a l e s y s t e m s s t a n d p o i n t . P r e c i s e d e f i n i t i o n s are i n t r o d u c e d by t h e a u t h o r s for v a r i o u s s t a b i l i t y domains w i t h a p p l i c a t i o n to l a r g e - s c a l e s y s t e m s i n g e n e r a l and more s p e c i f i c a l l y to power s y s t e m s . S t a b i l i t y p r o p e r t i e s and domains of d i s t u r b e d power s y s t e m s are e s t a b l i s h e d .
vi
A number of examples and a p p l i c a t i o n s illustrate the various results.
Preface
presented throughout this
book
According to t h e amount and i m p o r t a n c e of d e f i n i t i o n s and s t a b i l i t y c r i t e r i a p r e s e n t e d I c o n s i d e r t h a t t h i s book i n i t i a l l y published in Russian, r e p r e s e n t s t h e most c o m p l e t e one on s t a b i l i t y t h e o r y proposed a t t h i s d a t e . I t i n t e r e s t s a l l p e o p l e concerned w i t h s t a b i l i t y problems i n t h e l a r g e s t s e n s e and w i t h s e c u r i t y , r e l i a b i l i t y and r o b u s t n e s s .
P r o f e s s o r P i e r r e BORNE L i l l e , France
FOREWORD
P o i n c a r e ' s daring i d e a to o b t a i n q u a l i t a t i v e i n f o r m a t i o n on motion d i r e c t l y from t h e d i f f e r e n t i a l equation describing it, i.e. without i n t e g r a t i o n , was r e a l i z e d by Liapunov 118921. With h i s a b s o l u t e comp l e t e n e s s and i r r e p r o a c h a b l e s t r i c t n e s s , Liapunov l a i d t h e f o u n d a t i o n s of a c o n c e p t u a l l y new approach to t h e q u a l i t a t i v e methods of t h e t h e o r y of d i f f e r e n t i a l e q u a t i o n s . Nowadays, L i a p u n o v ' s methods are r e c o g n i z e d to be among t h e most p o w e r f u l means of s t a b i l i t y a n a l y s i s i n e x a c t s c i e n c e s . These, along w i t h t h e many e x t e n s i o n s f u r t h e r d e v e l o p e d , c o n t r i b u t e d to broaden s u b s t a n t i a l l y t h e c l a s s e s of problems a b l e of being e f f e c t i v e l y a n a l y z e d by t h e d i r e c t method. The p r e s e n t book c o n t a i n s an e s s a y of d e v e l o p m e n t of t h e g e n e r a l t h e o r y of s t a b i l i t y i n t h e s e n s e of Liapunov, e l e m e n t s of t h e s t a b i l i t y t h e o r y of comparison s y s t e m s ( s y s t e m s of o r d i n a r y d i f f e r e n t i a l equations with monotonous r i g h t - h a n d p a r t s ) , p r e s e n t a t i o n of t h e g e n e r a l methods for t h e a n a l y s i s of s t r u c t u r a l s t a b i l i t y of l a r g e - s c a l e s y s t e m s , i n c l u d i n g s y s t e m s w i t h s i n g u l a r p e r t u r b a t i o n s . The Liapunov f u n c t i o n s ( s c a l a r , v e c t o r and m a t r i x ) and his d i r e c t method for t h e s t a b i l i t y a n a l y s i s of t h e u n p e r t u r b e d motion are used t h r o u g h o u t t h e book~ Some of t h e obt a i n e d s t a b i l i t y r e s u l t s are a p p l i e d to t h e a n a l y s i s of l a r g e - s c a l e e l e c t r i c power s y s t e m s . The s t a b i l i t y of t h e s e s y s t e m s i s a v e r y import a n t p a r t i c u l a r case for which t h e d i r e c t c r i t e r i a show e x t r e m e l y u s e ful. The R u s s i a n v e r s i o n of t h i s monograph was c o m p l e t e d i n 1982, t h e 125th a n n i v e r s a r y of L i a p u n o v ' s b i r t h d a y . S i n c e , new r e s u l t s of t h e a u t h o r s have been added and i n c l u d e d i n t h e p r e s e n t v e r s i o n . More s p e c i f i c a l l y Chapter V has been t h o r o u g h l y r e v i s e d and c o m p l e t e d . O v e r a l l , t h i s E n g l i s h v e r s i o n i s more than a mere t r a n s l a t i o n of t h e R u s s i a n one~
viii
Foreword
Our p e r m a n e n t c o n c e r n has been t o w r i t e hend,
way, r e a d a b l e
for
e a s y to compre-
both e n g i n e e r s who need c o n v e n i e n t m a t h e m a t i c a l
machinery for large-scale who a r e i n t e r e s t e d ential equations.
up i n a c l e a r ,
system stability
analysis,
i n new problems of t h e q u a l i t a t i v e
and m a t h e m a t i c i a n s t h e o r y of d i f f e r -
We have t r i e d t o do j u s t i c e t o s c i e n t i s t s who t h e f i r s t o b t a i n e d r e s u l t s i n v a r i o u s a r e a s of t h e l a r g e - s c a l e s y s t e m s s t a b i l i t y t h e o r y , and to refer to their original papers. It is reflected in the Bigliographies which i n c l u d e more than 400 r e f e r e n c e s . C e r t a i n l y , even such a l i s t i s still i n c o m p l e t e . T h i s can be p a r t l y e x p l a i n e d by £he i n t e n s i v e r e s e a r c h e f f o r t s and d e v e l o p m e n t s i n t h e a r e a , and by t h e e x t r e m e l y wide domains of i t s a p p l i c a t i o n , b e g i n n i n g w i t h t e c h n o l o g y and f i n i s h i n g w i t h t h e problems o f p o p u l a t i o n a l d y n a m i c s . We a p o l o g i z e to a l l t h o s e whose work was n o t c i t e d
or p r o p e r l y d e s c r i b e d .
ACKNOWLEDGMENTS
Academicians
Yu.A.
M i t r o p o l s k y and Ye.F.
of Academy of S c i e n c e s
of t h e USSR,
V.I.
Mishchenko,
A s s o c i a t e Member
Zubov and P r o f e s s o r Yu.A.
Ryabov have g o t a c q u a i n t e d w i t h t h e R u s s i a n m a n u s c r i p t of t h e book. T h e i r d e t a i l e d r e m a r k s were e x t r e m e l y v a l u a b l e . Many c o n v e r s a t i o n s of A.A. M a r t y n y u k w i t h P r o f e s s o r A.B. Z h i s h c h e n k o g r e a t l y i n f l u e n c e d t h e p r e s e n t a t i o n of p r o b l e m s c o n n e c t e d w i t h t h e a l g e b r a i c t y p e of t h e obtained results. Collaborators
of t h e P r o c e s s e s S t a b i l i t y
D e p a r t m e n t of t h e
Institute
of Mechanics of t h e U k r a i n i a n Academy of S c i e n c e s , I . Y u . L a z a r e v a , Y e . P . S h a t i l o v a have c o n t r i b u t e d much i n t h e c o u r s e of t h e t e c h n i c a l work on t h e m a n u s c r i p t . Mrs. M.B. C o u n e t - L e c o m t e did an o u t s t a n d i n g j o b i n t y p i n g t h e f i n a l E n g l i s h v e r s i o n . The q u a l i t y of t h i s camerar e a d y p r e s e n t a t i o n owes e n o r m o u s l y to her e x p e r t i s e . Th~ a u t h o r s
are c o r d i a l l y
thankful
to all
of t h e m .
Lj.T.G.
A.A.M.
M.R.P.
Belgrade
Kiev
LiEge
S e p t e m b e r 1987.
CONTENTS
List of basic symbols
xiii
Chapter I OUTLINE OF THE LIAPUNOV STABILITY THEORY IN GENERAL
1
I.I. Introductory comments
1
1.2
On definition of stability properties in Liapunov's sense
2
1.2.1
Liapunov's original definition
2
1.2.2
Comments on Liapunov's original definition
5
1.2.3
Relationship between the reference motion and the zero solution
6
1.2.4
Accepted definitions of stability properties in Liapunov's sense
1.2.5
Equilibrium states
1.3
1.4
7
On the Liapunov stability conditions
13 14
1.3.1
Brief outline of Liapunov's original results
14
1.3.2
Brief outline of the classical and novel developments of the Liapunov second method
2o
On absolute stability
42
1.4.1
Introductory comments
42
1.4.2
Description of Lur'e-Postnikov systems
43
1.4.3
Definition of absolute stability
44
1.4.4
Liapunov'like stability
45
1.4.5
Criteria for absolute stability of time-varying systems
46
Criteria for absolute stability of time-invariant systems
51
1.4.6
conditions for uniform absolute
X
1.5
Contents
On stability properties
of singularly perturbed
1.5.1
Introductory
1.5.2
System description
1.5.3
Liapunov-like
1.5.4
Singularly perturbed Lur'e-Postnikov
systems
comments
52 52 53
conditions
for asymptotic
stability
systems
54 58
Comments on references
62
References
65
Chapter I I THE STABILITY THEORY OF COMPARISON SYSTEMS
73
11.1
73
Introductory 11.1.1
Original
11.1.2
The Liapunov functions and comparison equations generated by them Vector-functions and comparison systems
11.1.3 11.2
concepts of the comparison method
11.1.4 Matrix-functions The Liapunov functions and comparison
equations
11.2.1
On monotonicity
estimations
11.2.2
Special cases of the general comparison
11.2.3
General stability theorems on the basis of scalar comparison equations
11.2.4
The generalized
11.2.5
The scalar comparison
11.2.6
A refined method of comparison construction
11.2.7 11.3
notes
and solutions
Stability of the comparison
equations
equation construction
11.3.4
Linear comparison
11.3.5
Nonlinear systems with an isolated equilibrium state The theorem of Zaidenberg-Tarsky and algebraic solvability of the stability problem Nonlinear autonomous comparison systems with a non-isolated singular point
of the comparison
system stability
systems
Several applications systems
of nonlinear comparison
Reducible
systems
comparison
104
111 systems solutions
Definitions
11.3.9
97 101
of scalar comparison
11.3.3
11.3.8
83 9O
I08
The non-degeneracy of monotonicity. Definition The basic statements of the comparison principle
11.3.7
77 8O
equations
11.3.1 11.3.2
11.3.6
76
83
comparison equation
Several applications equations
73
117 117 117 119 121 124 126 128 129 135
Contents
11.4
xi
Matrix-functions application to the stability analysis
137
11.4.1
Main properties of matrix-functions
137
11.4.2
Theorems of direct method based on matrixfunctions
138
11.4.3
The scalar Liapunov function construction on the basis of matrix-functions
143
Comments on references
149
References
151
Chapter I I I LARGE-SCALE SYSTEMS IN GENERAL
155
III.i
Introduction
155
111.2
Description and decomposition of large-scale systems
157
111.3
Structural stability properties of large-scale systems
160
111.4
Aggregation forms of large-scale systems and conditions of structural stability
163
111.4.1
Aggregation forms and solutions for the Problem A
163
111.4.2
Aggregation forms and solutions for the Problem B
1T6
111.4.3
The structural stability analysis of a largescale system with non-asymptotically stable subsystems
214
Comments on references
221
References
223
Chapter IV SINGULARLY PERTURBED LARGE-SCALE SYSTEMS
231
IV.1
Introduction
231
IV.2
Description and decomposition of singularly perturbed large-scale systems
231
IV.3
IV.4
Aggregation and stability criteria for singularly perturbed large-scale systems
233
IV.3.1
Introduction
233
IV.3.2
Non-uniform time scaling
234
IV.3.3
Uniform time-scaling
243
Comments
References
260 261
xii
Contents
Chapter V LARGE-SCALE POWER SYSTEMS STABILITY Notation
263
V.I
Introduction
265
V.2
The physical problem and its mathematical V.2.1 Problem definition
V.3
V.4
V.5
modelling
V.2.2
Conventional
V.2.3
Definitions
V.2.4
Liapunov's method applied to conventional stability analysis
267 267
problem formulation
271
of stability domains and their estimates
272
transient 273
V.2.5
System modelling
275
V.2.6
Mathematical
277
formulation
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"
287
V.3.4
The Zubov method
290
V.3.5
Numerical
291
V functions
simulations
Vector Liapunov approach
303
V.4.1 V.4.2
Introduction Stationary large-scale systems decompositions aggregations in general
303
V.4.3
General stability analysis of stationary systems
V.4.4
Power systems modelling
V.4.5
Power systems decompositions
Conclusion
and 305
large-scale 312 321
and aggregations
324 353
References
354
Postface
361
References
365
LIST OF BASIC SYMBOLS
All symbols duced. more
are
fully
defined
As a c o n v e n i e n c e
frequently
used
to the r e a d e r
symbols
is the one g i v e n
below.
the i n t r o d u c t i o n
to C h a p t e r
A,B,C,...
at the p l a c e
we have
in several
Additional
where
The
list
for later
upper
case
union, upper denote
of the collection
can be found
intersection case
in
boldface
case
lower
Ba(t o) = {x:lJxJl
the h y p e r b a l l
case
letters with
and r a d i u s
family
of all
differentiable
on
differentiable
on
the f a m i l y
the
Tr×N
A
and
Gothic
A
B
and
letters
constant~or
except
V
Gothic
letters
vectors
a,b,e,...
ous on
sets
with
lower
the
letters
of sets
boldface
matrices
entries
origin
Script
between
functional
denote
c(i'J)(TrxN,R k)
use
intro-
sets
AUB , A n B
C(TrxN)
some
largest
boldface
a difference
c(i'J)(TrxN)
first
V.
A~ B
a,b,c,...
are
collected
places.
denote
A,B,C,...
they
of all
denote the
scalars
center
equal
to
A(to)
functions TT
in the
and
i-times j-times
N functions
continu-
functions
mapping
TTxN
family
into c(i,J)(rrxN)
of all Rk
which
are
in
B
xiv
List of basic symbols
D+v(t,X) = lim sup { v t t + @ ' x < t + 8 ; t ' x ) ] - v < t , x ) O the of
:8+0+}
upper v
right-hand
along
X
at
Dini
derivative
(t~x)
D+v(t,X) : liminf{ y[t+o'x(t+o;t'x>]-v(t'x)0 :@÷0+} the of
lower v
denotes
D*v(t,×)
right-hand
along that
D+v(t,X)
d(X,A) = inf[llx-yll.:yeA] =max {sup [d(x,A):xeB], [d(X,B):xEA]}
d(A,B) sup
f: R x R n + R n
X
Dini
both
D+v(t,×)
X
to
A
a distance
between
A
and
function
B
mapping
RxR n
~n
Ik
the
He(-)
the H e r m i t i a n
i,j, k,...,N
integers
j:v~Y
the
imaginary
K[o,~ ]
the
class
on
[0,~ ]
kxk
identity part
the o r i g i n first
matrix of a m a t r i x
of c o m p a r i s o n
of N
(-
unit
a time-invariant
the
and
can be used from
into
derivative
(t,X)
a distance
a vector
N
at
functions
neighbourhood Rn ,
natural
or the
of
set of
numbers
:
N : {I,2,...,N}
N(t)
a neighbourhood
of the o r i g i n
at
of the o r i g i n
of
of the o r i g i n
of
tER Nr = { ( t , x ) : t ~ T r , x < N ( t
)}
a neighbourhood
over
RxR n
N= { ( t , x ) : t e R , x e N ( t ) }
Tr
a neighbourhood R×R n
0 : {x: x:O}
the
singleton
of
Rn
containing
the
R
the
set of all real n u m b e r s
R+ = [O,+~[CR
the
set of all n o n - n e g a t i v e
the
set of all p o s i t i v e
origmn
numbers
o
R+ = 10,+~[
real
numbers
L i s t of b a s i c
symbols
xv
Rk
k-th
RxR n
the
S
a time-invariant
S(t)
a time-varying
Ss
a structural
dimensional cartesian
ing all via 8T : { ( t , x ) : t e T m , x e S ( t ) }
product
vector of
subset subset
R of
of
matrices
of
TTxR n
of
RxR n
Rn
Rn Rn
structural
structural
space
and
set of a s y s t e m
system
a subset
real
defin-
variations $
associated
with
S(t) a subset
S = {(t,x):tcR,×eS(t)}
S(t)
sign ~ =~l~l -I iff ~#0 and
Sr=S
,
the
signum
the
largest
associated
iff
type
with
TT:R
nonlinearity
sign 0 = 0
T o = [to~+~] = { t : t o ~ t ~ + ~ }
with
to
T : [-~,+~] : { t : - ~ t ~ + ~ }
the
largest
T o = ]to,+~ ] = { t : t o < t ~ + ~ }
the
left
= {t:togt<+~)
the right
, meR
moments (or,
to
a time
toeR
an initial
V~(t)
is the
x(t)
time
initial
consideration to )
instant
×=0
connected associated
positive
definite
v(t,x)<~
, Vx~V~(t)
a state
unbounded
instant
largest
of
time
to
with
admissible
variable,
to
with
of all
under
of all
time
unbounded
semi-open
teR
hood
with
associated
interval
beginning
unbounded
semi-open
the right
a time
interval
associated
interval
Tier
time
associated
interval Tr = [f,+~[ = {t:r-
interval
semi-open
interval
To : [ t o , + ~ [
time
vector
v
neighbourwith
such
of a s y s t e m
a
that
at
teR
X:(Xl ,Xz ,...,Xn)T
~,#,7,...
Greek
letters
otherwise
denote
specified
scalars
unless
xvi
List of basic symbols
AM(t o ) : M~x { A : A : A ( t o ) , V p > O , V X o e B A ,
the maximal
A
obeying
the defini-
tion of a t t r a c t i v i t y
3 T(to, Xo,P)e ] O, +~[, ~X (t ; to, Xo )eBp, VteTT ]
~M(to,E ) :
the maximal
Max { ~ : 6 : $ ( t o , e ) ~ × o e B s ( t o , e )
tion of stability
~
obeying
the defini-
x(t;to,Xo)eSe ,VteY o} the boundary
~S
the closure
¢
empty
7re(to, Xo,P) = Min {T:~:r(to,Xo, p)
nition
9X(t;to,Xo)eBp,VteY~
of a set
S
of a set S
set
the minimal
r
satisfying
the defi-
of a t t r a c t i v i t y
)
Xi("
the
i-th
eigenvalue
AM<-
the maximal
of a matrix
(')
eigenvalue
of a matrix
eigenvalue
of a matrix
(.) ~m( •
the minimal
(-) X(t;to,× o )
a motion
of a system at
tER
iff
×(to)--X o , x(to;to,×o)--×o "implies" "iff" V
("if and only
if")
"for every" "there
exist(s)"
"there
does
(do) not exist"
9
"such that"
E
"belongs
II II
the E u c l i d e a n
[ ]
denotes
a closed
] [
denotes
an open
( )
a general
interval
semi-open,
open,
to" norm interval interval which
can be
or closed.
CHAPTER I
OUTLINE OF THE LIAPUNOVSTABILITY THEORY IN GENERAL
1.1.
INTRODUCTORY COMMENTS
In the late years of the XIXth century H. Poincar~
(1881-1882)
started
qualitative analysis of non-linear differential equations considering those of the second order. His ideas inspired A.M. Liapunov
(1892, p.8)
for more general qualitative analysis of the non-linear differential equations which resulted in a genious scientific contribution presented in his famous fundamental dissertation
(Liapunov,
1892). Among numerous
basic contributions the following are in particular important for what will be considered in this book. A.M. Liapunov initiated qualitative study of relationships between perturbed
(real) motions and an unperturbed
nal) motion. Moreover, along the motions.
(reference,
desirable,
nomi-
he considered behaviour of suitable functions
These functions can represent different types of
energy and/or material flow, which illustrates and emphasizes physical importance of the problem.
Relying on the concept of continuity of
functions, A.M. Liapunov naturally introduced a general definition of stability of the reference motion with respect to some functions requiring
e-8
closeness of the values of these functions taken along
perturbed motions an~ the reference motion.
Although A.M. Liapunov did
not explicitely define either attraction or asymptotic the reference motion his dissertation ,
(with or without respect to some functions)
it
(~946), J.L. Massera (1952), I.G. Malkin
stability of
has
been
often
accepted
in
[see N.G. Chetaev
(1949,1956), E.A. Barbashin and N.N. Krasovskii (1954), H.A. Antosiewvicz
(1966) and N. Rouche,
P. Habets and M. Laloy
(1958), T. Yoshizawa (1977)] to associate the
notion of asymptotic stability with Liapunov's d i s s e r t a t i o n .
The
reason for it is likely following Liapunov's remark on his theorem on
2
Ch. I
Outiline
stability of the reference motion
"Remark 2.
I f a function
v,
of the L i a p u n o v
(Liapunov,
stability
theory in general
1892, p. 61)
:
s a t i s f y i n g the conditions of the
theorem, admits simultaneously an i n f i n i t e l y small upper l i m i t , and i t s d e r i v a t i v e is a s i g n - d e f i n i t e function, then i t
can be
proved that every perturbed motion, s u f f i c i e n t l y close to the unperturbed one, w i l l Altogether,
approach i t asymptotically."
A.M. Liapunov founded a general concept of stability of
motions, which has been becoming more and more important for dynamic analysis in mathematics,
mechanics, automatic control,
systems theory,
ecology, biology, economics etc. Another remarkable Liapunov's contribution is his second method that is also called Liapunov's direct method
[see Lasalle and Lefschetz
(1961)]. The aim of this method is a stability problem solution via system state differential equation w i t h o u t
use
o f its
solutions.
Infor-
mation about a stability property is to be deduced directly from the differential equation despite the corresponding stability property definition is phraised in terms of system motions.
A.M. Liapunov estab-
lished the concept of semi-definite and definite functions and used them as a
keystone
of his second method. Then, A.M. Liapunov trans-
ferred stability problems to the problem of behaviour of a corresponding definite function along system motions.
In order to solve this pro-
blem A.M. Liapunov considered the sign of the total time derivative of the chosen function along system motions.
The sign has to be tested at
all points belonging to a neighbourhood of the reference motion. The precise requirements on the function and the sign of its total time derivative along system motions were established by A.M. Liapunov for (uniform)
stability and (uniform) asymptotic
stability of the zero
state representing the reference motion.
1.2.
ON DEFINITION OF STABILITY PROPERTIES IN LIAPUNOV's SENSE
1.2.1. Liapunov's original definition A.M. Liapunov started his investigations with the following punov
(1892), p. 12]
"1.
Let us c o n s i d e r Let
be
k
tion.
[A.M. Lia-
: any m a t e r i a l
system w i t h
k
degrees
of
freedom.
its
posi-
ql,q2,...,qk independent
variables,
which we use to d e t e r m i n e
I. 2
Stability properties
We s h a l l
3
assume t h a t
quantities
taking
real system p o s i t i o n s are taken f o r
real
values for
all
such v a r i a b | e s .
Considering the mentioned v a r i a b l e s as f u n c t i o n s in time we shall denote t h e i r f i r s t
t
time d e r i v a t i v e s by !
!
!
ql,q2~'",qk
In every dynamic problem, in which forces are p r e s p e c i f i e d in c e r t a i n way, such functions w i l l differential
s a t i s f y some k
second order
equations.
Let any p a r t i c u l a r s o l u t i o n f o r such equations be found ql=fl(t)
,q2=f2(t)
in which the q u a n t i t i e s t
, which at every
t
qj
,... , q k = f k C t )
,
are expressed as real f u n c t i o n s in
give only p o s s i b l e values te them. I
To t h a t p a r t i c u l a r s o l u t i o n w i l l of our system. Comparing i t
correspond a d e f i n i t e motion
in a known sense with o t h e r s , which
are p o s s i b l e under the same f o r c e s , we s h a l l
unperturbed, and a l l o t h e r s , w i t h which i t
call
t h a t motion
is compared, pertur-
bed. Understanding by
to
a given i n s t a n t , l e t us denote the
values corresponding to i t t i o n with
of q u a n t i t i e s
qj,q~
along any mo-
qjo,q~o .
Let qlO : fl(to)+cl !
v
qlO =fl[to
where
ej,e~_
)+c
: f2(to)+e2 v
t
v
,'", q k o
determined.
Let
QI,Q2,...,Q n
= i'k( o ) + e k
We s h a l l
By s p e a k i n g a b o u t p e r t u r b e d m o t i o n s , close comprehend m o t i o n s ,
b a t i o n s are n u m e r i c a l l y
t
, t
,
be c a l l e d p e r t u r b a -
which w i l l
them e v e r y number s u f f i c i e n t l y
t u r b e d o n e , we s h a l l
= fk(to)+ek
T
constants.
the c o n s t a n t s ,
a perturbed motion is
we may p r e s c r i b e
,"" , q k o t
' q20 : f2(to)+~2
are r e a l - v a l u e d
Prespecifying
tions,
, q20 ~
sufficiently
for
assume t h a t
small. to the u n p e r which the p e r t u r -
small.
be any given continuous r e a l - v a l u e d func-
t i o n s of q u a n t i t i e s I
!
ql,q2,--.,qk,ql,q2,--.,q
I
k
Along the unperturbed motion they become known f u n c t i o n s of t
, which w i l l
be denoted by
motion they w i l l
Fz,F2,...,Fn . Along a perturbed
be f u n c t i o n s of q u a n t i t i e s
i It can happen that the quantities qj by their choice do not take all real values but only those not greater than - and not less than certain bounds.
4
Ch. I
Outline of the Liapunov stability theory in general
v l 6~ t ' 6 1 ' e2 ' " " 6k ' 61' 62 ~'°" k v
When a l l
are equal
~j:6j
to
zero,
then
the
quantities
Q1-F1 , Q2-F 2 , ... , Qn-Fn will
be e q u a l
6:]
6j, tion
are rises
to
zero,
:is
it
bounds on t h e grater
than
turbed the
t o
choice
of
we a c c e p t Let
of
the
motion,
the
constants then
infinitely latter
the
a ques-
small
never
of
topic
the
functions choice
become
of
our
considered the
unper-
by d e t e r m i n i n g
the in
a feature that
and
latter,
characterize or
inves-
Qz,Q2,...,Qn of
will
stability,
instant
herein
positive numbers
in
question
some
of
the
contrary
those
following
cases
does n o t
t o , when p e r t u r b a t i o n s
the
LI,L2,...,L n
interested
considered
to
it,
in
which
d e p e n d on a
are acting.
Thus
definition.
be arbitrary given positive numbers.
L s , nevertheless
a~l
the
instability. of
the
is
a character
be c a l l e d
be e x c l u s i v e l y
solution
the
which
respectively,
unperturbed
if
small,
such
that
Under a s p e c i f i c
whichwill
We s h a l l
specify
question,
question,
be c a l l e d
infinitely
?
and a c h o i c e
the
sense the
the
to
d e p e n d s on b o t h
to
will
the
t . However,
are
Qs-Fs
values
of
motion
latter,
every all
quantities
instant
reply
for but
possible
their
A solution tigations,
zero
not
If for
how small they are, can be selected so that for a~l real
EI,E2,...,Ek,E~,E~,...,E ~
ej, 6~ ] , satisfying the conditions
f~jl ~ E j , 1 6 1 1 ~ E j and for all
t,
greater than
(j=~,2,...,k)
'
t o , the inequaZities
IQI-FzI < L l, IQ2-F21 < L
2,
,
I%-F~I < ~
,
are satisfied then the unperturbed motion is stable with respect to the quantities
QI,Q2,...,Qn ; otherwise
- unstable* with res-
pect to the same quantities*." l In general Ixl means the absolute value of a real-, or modulus of a complex quantity x.
The
Remark
1.1 has
2 o n p. 61
been
asymptotic
commonly
stability
in L i a p u n o v
(1892)
used
as a b a s i s
o f the
reference
reproduced
for notions motion.
on p . 2
of our
of attraction
§
and
I. 2
Stability properties
5
1.2.2. Comments on Liapunov's o r i g i n a l d e f i n i t i o n
Comment I. on
The
IQj-Fjl
the same yields tion.
inequalities
are
type
definitions
of the r e f e r e n c e
A.M.
to a r b i t r a r y
T h i s has b e e n
Qj
can r e p r e s e n t
Liapunev
introduced
and a c c e p t e d
the
IQj-Fjl
, which
original
defini-
proved. motion
was d e f i n e d
Qj
that
are
by A.M.
flow.
Lia-
continuous
and p h y s i c a l l y
or m a t e r i a l
new variables
following
by i m p o s i n g
and
very thoughtful
X i : Qi-Fi
avoided
to L i a p u n o v ' s
functions
energy
are w e a k a n d t h o s e
is u s u a l l y
l e j l , fell
Stability
qi , q~
because
on all
equivalent
c a n be e a s i l y
Comment 2.
fell
and
unsymmetry
equivalence
punov with respect all
lejl
on
This
of i n e q u a l i t i e s
stability This
strong.
in
important
In this
connection
xi ,
, i=i,2,...,n ,
(Liapunov,
1892,
"We shal] assume t h a t the number n
p. 15)
:
and the functions
such, t h a t the order of the system is
Qs
n and t h a t i t
, are
is reduc-
i b l e to the normal form dx I dx 2 dt : XI ' - ~ :
dx n X2 '"" ' dt : Xn
'
(I)
and everywhere in the sequel we s h a l l consider these l a s t equat i o n s , c a l l i n g them the d i f f e r e n t i a l equations of a perturbed motion. All
Xs
in the equations (1) are known f u n c t i o n s of quan-
tities
xl,x2,...,xn,t
vanishing f o r
Comment
5.
X I : X 2 :... : X n : 0 .
Stability
of the r e f e r e n c e
ness of the p e r t u r b e d
motions
sufficient
is a s s u r e d
closeness
Comment 4. motion i.e.
The
quired
t
in the either
not less t h a n tem motions Cammgnt
closeness
is to be r e a l i z e d
for all
glected
5.
cases w h e n
to
Liapunov
.
has b e e n
T o : ]to,+~] commonly
,
ne-
has b e e n c o m m o n l y
T O : [to,+~[ can be c r u c i a l
, i.e.
for aZZ
in c a s e s w h e n
re-
t sys-
t=t o
defined
it is not i n f l u e n c e d
point
to
close-
their
to the r e f e r e n c e
interval
the c l o s e n e s s
or on
at
instant
motions
time
arbitrary
motion provided
initial
. This
difference
are d i s c o n t i n u o u s A.M.
at the
Namely,
9 o : [to,+~]
t o . This
motion requires
o f the p e r t u r b e d
greater t h a n
"
to the r e f e r e n c e
over unbounded
literature. on
,
by
stability to
of the r e f e r e n c e
. However,
the i n i t i a l
motion moment
for
6
Ch. I
can e s s e n t i a l l y when
influence
system motions
tially
influence
even w h e n
all
Comman£ 6. Liapunov nates
stability
qj , q$
are
values
of the r e f e r e n c e
respect
motion
in g e n e r a l
in cases
to
of all
in
can e s s e n -
Ej
and
E~
t.
motion
perturbations
than with
theory
t . Besides,
continuous
to i n i t i a l
, rather
in
admissible
motions
stability
of the r e f e r e n c e
continuous
the m a x i m a l
with respect
of the L i a p u n o v
stability
are not
system
The
Outline
was d e f i n e d
of the
general
to p e r s i s t e n t
by A.M. coordi-
external
dis-
turbances. Commen£ Ej
7. Ej
of all they
The T
and
Lj
Let
useful
definition they
must
does
not
care
be positive.
admissible
Ej
for e n g i n e e r i n g
r
and
about
Hence,
Ej
the values
for large
values
can be so small
that
needs.
Relationship between the reference motion and the zero solution
2k be the order
state
that
, the m a x i m a l
are not
1.2.3.
stability
except
variable.
energy
conservation
for a large following
of the
Using
class
scalar
and the
of systems form
equivalent
and
Yi , i=1,2,...,2k ,be
physical
laws
(e.g.
law of the m a t e r i a l get
state
the
its
conservation)
differential
i-th
law of the we
equations
can
in the
:
dYi dt or in the
system
the basic
= Yi(t,Yl vector
"'"'
Y2k )
'
i=1,2,...,2k
(i)
'
form
dy = Y ( t , y ) dt where *
y = (yl,Y2,...,Y2k) T e R 2k
Y : TxR 2k ~ R 2k
. A motion
of
and
, and the r e f e r e n c e
physical
of view
the
system.
reference
is a s o l u t i o n
of
by
motion
the r e f e r e n c e
F r o m the m a t h e m a t i c a l
motion
Y = (YI,Y2,...,Y2k)T
(2) is d e n o t e d
~(to;to;Yo) ~ Y o point
(2)
~r(t;to;Yro)
motion
point
,
~(t;to;Yo)
should
of view
, . F r o m the
be r e a l i z a b l e
this
means
that
(2),
ddt ~ r ( t ; t o ; Y r o ) ~ y [ t , ~ r ( t ; t o ; Y r o ) ] Let
the L i a p u n o v
transformation
of c o o r d i n a t e s x=y-y
where
Yr(t) ~ r ( t ; t o ; Y r o ) f(t,x)
It is evident
Let
by the
r
3)
be used,
,
4
f : T x R 2k ~ R 2k
= Y[t,Yr(t)+x]-Y[t,Yr(t)]
be d e f i n e d .
by 5
that f(t,O)
• In Liapunov's notation
~0 r
!
6 l~T
y = (ql,q2,...,qk,ql,q2,...,qk)
.
1.2
Stability properties
7
Now (2)-(5) yield
dx d--@ : f(t,x)
In this
way,
the b e h a v i o u r
ence m o t i o n
(in total
the state
deviation
reference
motion
zero d e v i a t i o n mind,
the
of p e r t u r b e d
coordinates) x
with
in the x=0
result
punov's
second
method
for the
system
(7).
deviation
emphasizes
and r e s u l t s
Let
to the
coordinates
in state
following
motions
related
is r e p r e s e n t e d
respect
total
(7)
zero
state
Yi
complete
presented
behaviour
deviation.
is r e p r e s e n t e d
coordinates
Q :R2k~R n , n=2k
to the r e f e r -
by the
by the
x i . With
generality
in A.M.
this
of b o t h
Liapunov
is a d m i s s i b l e
of
The
in
Lia-
(1892)
but
not re-
quired.
Theorem I .
x:O
Stability of
Q:X
of the system (?) with respect to
is necessary and sufficient for stability of the reference motion of the system tinuous in P~OO{. in
y
(2) with respect to every vector function
y .
Necessity. of
Sufficiency.
Let
Q
in
y
to
with
, such
that
of
(7),
to
~i>0
Ixiol < 6i , i=i,2,...,2k , where
the p r o b l e m
reduces
(2) with
(7) w i t h r e s p e c t
respect to
to
Q
of
, implies
, implies (7)
(with re-
, ~ = (£1,£2,...,
Ixi(t;to;Xio)l
, X(to;to;×o) for every such
e ×o
stability
stability
< £i ,
, is the
Li>0 ,
that
lYjo-Yrjol
< L i , Vtmt o , i=l,2,...,n
of the
to the
Continuity
, 6 i=Si(~)
, ~3 : ~2 ~J ' j : l , 2 , _ , n ,
theorem
chosen.
×=0
• Finally,
-Qi[Nr(t;to;Yro)]l
is i m p l i e d
.
of
x(t;to;×o)
× = (XI,X2,...,X2k)T
x
< £i ' Vi:i,2,_.,2k
. Stability of
is c o n t i n u o u s
to
, £i = ~ i ( L , Y r ) • [ : (LI,L2,'",
lYi-Yril
IQi[N(t;to;Yo)]
tion of
Q(y) : y respect
Q(y) = y
~±>0
existence
i=1,2,...,n , there is ~>03 - ] ,j=l,2,...,n , implies
This
with
,be a r b i t r a r i l y
of
that
because
x:O
respect
< L i , i=i,2,...,n
Vt~t o , VtoeR , i=i,2,--,2k solution
of
existence
× ) guarantees
~2k )T , such
is true
Li>0 , i:l,2,...,n
implies
IQi(y)-Qi(yr)l
part
stability
~r
Ln )T , i:i,2,...,2k
spect
This
and e v i d e n t l y
by s t a b i l i t y
of
~r
Q that i8 con-
.
of the r e f e r e n c e
problem
Q~
×=0
• mo-
of
x ; it is stated and p r o v e d h e r e i n for the first time.
1.2.4. Accepted definitions of s t a b i l i t y properties in Liapunov's sense By the very whose toeR
definition,
motions
. However,
(time-varying)
stationary
are not e f f e c t e d such
property
systems.
by
(time-invariant) (the c h o i c e
is not
of)
characteristic
It is t h e r e f o r e
natural
systems the
are those
initial
instant
for n o n - s t a t i o n a r y
to c o n s i d e r
influence
8
Ch. I
of
to
on s t a b i l i t y
motivation Liapunov Massera Malkin R.E.
properties
for a c c e p t i n g
(1892),
K.P.
(i949,i956), (1954),
H.A.
V.I.
(1966), N.P. W. H a h n
Zubov
K.S. N a r e n d r a
(1964)
(1958)
and J.H. T a y l o r
Chetaev
Coppel
(1967)],
B.P.
and N. Rouche,
Definition
i.
The state
×=0
(1952),
I.G. (1959),
Stepanov
(1965), T. Y o s h i z a w a E.A.
(1967),
Barbashin
(i975,i977)
the h i s t o r i c a l
until
P. H a b e t s and M. Laloy
is
Krasovskii
and V.V.
Grujid
P. H a b e t s and M. Laloy ( i 9 7 7 ) ] - F o r of s t a b i l i t y d e f i n i t i o n s
which
Demidovich
(1969,1974),
(i973), Lj.T.
in g e n e r a l
(1946), J.L.
Krasovskii
Nemytskii
[see also W.A. Siljak
systems,
[see also N.N.
(1960), V.V.
theory
b a s e d on those by A.M.
(1933), N.G.
E.A. B a r b a s h i n and N.N.
[see also D.D.
the d e v e l o p m e n t (i966)
Persidskii
Bertram
stability
of n o n - s t a t i o n a r y
B h a t i a and G.P. Szeg~
(1967)
N. Rouche,
of the L i a p u n o v
the next d e f i n i t i o n s
Antosiewiez
K a l m a n and J.E.
(1960)I,
Outline
(1970)],
[see also
reviews
of
1975 see T. Y o s h i z a w a
(1977).
of the s y s t e m
(7) is
: o
(i) stable with respect there exists
Ti
to
iff for every
8(to,e) > 0 , such that IIx(t;to;×o)H
(ii) uniformly
stable with respect
<e
toeT i
and every
If×ell< 8 ( t o , e )
eeR+
implies
, vt~l o
Ti
to
iff both
(i) holds and for
o
every
eeR+
the c o r r e s p o n d i n g
maximal
i n f [ 6 M ( t , e ) :teTi] (iii) stable
in the whole with respect
8M(t,c) ~ + ~ (iv) un£formly and
(i) s a t i s f i e s
> 0 ;
to
e~+~
obeying
Ti
iff both
(i) holds and
, vteY i ;
in the whole with respect
Ti
to
iff both
(ii)
(iii) hold.
(v) unstable and
8table
as
8M
with respect
to
Ti
iff there are
T 6 T o , ~ > t o , such that for every
ll×oU <8
toET i , ee]0,+~[
8e]0,+~[
there
is
×o
, for w h i c h
HX(T ; t o , X o ) U ~ (vi) The e x p r e s s i o n iff These
li:R
stability
Example
1 (Lj.T.
.
"with respect
Ti "
to
to=l
(i)-(v)
properties Gruji6
hold as
1975) .
t~+ ~
Let
but not for
t:+~
.
x~ R
and
for
dtdX _ i-tl x • Then,
to#l
and
vt o~]-~,1[
,
t#l
.
the m o t i o n is not d e f i n e d and
Ix(t;to;X6)
Hence,
from
•
X ( t ; t o ; X o) = ( t - l ) - l ( t o - l ) x o For
is o m i t t e d
I ~
+-
as
t ~(1-0)
,
V(Xo~0)eR
.
,
I. 2
Stability properties
9
8M(t,e) However, The
6M(t,e ) :e
state
x=O
T i c ]-i,+~[ Example
: 0 , VeeR+,
is u n i f o r m l y
, but
it is not
2 (Lj.T.
is d e f i n e d
Gruji6
Vt • ]-~,1]
, Vt 6 ]1,+~[
stable
in the whole
with
respect
to every
stable.
1975).
The
first
order
non-stationary
system
by dx
(I + t s i n t + t 2 cos t) x • exp (-7 1
dt
Solutions
are f o u n d
in the
1 ~
x(t;to;Xo)
~ • exp (-t s i n t )
+ t • exp (-
~)
form
i + t o exp ( - ~
+ t o sin to)
:
~ ' to # - 7
Xo
~+
½
texp
(-
' t #-7
'
~ +t sint)
so that l×(t;to;Xo) I ~ + ~ This
result
as
t ~ ( - ~ - 0) , V t o e ] - ~ , - ~ [
and a n a l y s i s
BM(t,e ) :
of
X(t;to;Xo)
, V(Xo#0)eR
yield
0
t ~
]-~,-~]
e
t e
]-~,01
e ~ [ ~ + 2 t exp ( - ~ + t
s i n t ) ] -I
, t~
ff
[0,+~[ ff
The
state
x=O
uniformly
]-~,+~[
is stable
stable
examples,
Proposition I. X:0
of
with
with
respect
respect to every
the m o t i o n s
× ( t ; t o ; X o)
×
are not
i8 continuous
of the system
×=0
]-~,+~[
to
bounded
and
Ti c
stable. continuous
If there is a time-invariant
such that
stability Ti
in the whole
, but it is not
In these
in the whole
in
in all
teR
NcR n
neighbourhood
(t,to,Xo) E ToxRxN
(7) with respect
of then
to some non-empty
implies its stability.
This r e s u l t
can be e a s i l y
Proposition
2.
If
x=0
tively, it is uniformly bounded T i e r .
ExampTe 3 ( L j . T . tionary
proved
as well
as the
of (7) is stable stable(in
Gruji~ 1975).
following
(in the whole)
the whole) with respect
Solutions
o f the first
system dx _ dt
are g i v e n
by
~ + 27t x , a>O ~ +~t +Tt 2
, ~2 < 4 ~
, 7>0
:
then, respecto every
order
non-sta-
i0
Ch. I
Outline of the Liapunov stability theory in general
× ( t ; t o ; X o) = (~ + ~ t o + T t 2 ) ( a + ~ t + 7 t 2 ) -I x o . In this case 8M(t,e)
=
(4a7 - ~ 2 ) e 8 7 ( a + ~ t + 7 t 2)
[1 - s i g n (t +
)] + ~ [ l + s i g n
(t +
)]
Hence, inf
[6M(-t,e ) : tER] :0
, Ve6]O,+~[
,
and 8M(t,e) -*+~ The state
×=0
it is u n i f o r m l y
for any
~ E ]-~,+~[
(i
(ii
~-*+~ , V t e R
is stable in the whole but not uniformly.
However,
Definition
as
2.
The
stable
in the w h o l e w i t h r e s p e c t
state
x=0
of the s y s t e m
attraative with respeet to Z~(to) > 0
and for every
such that
II×oli< A ( t o )
true and for every
Ti
~>0
there exists
with
toET i
there exists
t o -uniformly there is
~>0
respect
there
exists
there
every
£xamp]e
4.
Yi
,VtE
iff
]to+
both
(i)
is
and for every such that
] = Tu[to,a(to),~ ] ;
to
Yi
iff
(Xo~ ~ ) e B~x]0,+~[
: to 6 T i]
(i) is true, there exists
: Tu(Ti,Xo,~ )
with respect (i) is true,
is
to
Yi
hold
"with respect
to
For the s y s t e m of E x a m p l e
AM(t)
=
Tm(t,x,~)
:
(ii) and
A>0
(iii)
and for every
such that = Tu(Ti,A,~)
"in the whole" iff
and every
;
iff both
there e x i s t s
T u ( T i , A , ~ ) e [0,+~[
(i)-(iv)
A(to) e ]0,+~[
(vi) The e x p r e s s i o n
~ ( t o ; X o , ~) e [0,+~[
A(to) > 0
sup [ T m ( t o , X o , ~ ) : (to,Xo) ~ T i × B A ] (v) The p r o p e r t i e s
there exists
such that
attractive
that is, that
~ e ]0,+~[
to~T~-
~u[to,Z~(to),~] e [0,+=[
and for every
sup [Tm(to;Xo;~)
hold,
to
attractive with respect
r u ( T i , X o , ~ ) e [0,+~[
(iv) uniformly
T i = [~,+=[
:
llx(t;to;Xo)II <~
sup [rm(to,Xo,~ ) : X o e B a ( t o ) (iii
(7) is
iff for every
implies
+ T(to,Xo,~),+~[ ; x o -uniformly attractive ~ ]0,+~[
to
.
.
(i) is true for
toel i Y i " is o m i t t e d
iff
Yi=R
i the f o l l o w i n g are found
0 , te]-~,l[ +~ , t e ]i,+~[ +~
-~A I=1 + l
, t e ]-~,1[ ,
t ~
]~,+oo[
. • :
I. 2
Stability properties
The state
x=O
a) a t t r a c t i v e
is
ii
:
in the w h o l e
b) t o - u n i f o r m l y T i C ]i,+~[
with
respect
to
T i = ]1,+~[
attractive
in the whole
with
attractive
with
to
respect
, to any b o u n d e d
,
c) x o - u n i f o r m l y d) u n i f o r m l y
attractive
with
respect
respect
T i = ]i,+~[
to any b o u n d e d
,
Tic
]i,+=[
,
e) not a t t r a c t i v e . The next
results
Proposition
3.
is continuous in
X(t;to;Xo)
attraction of T±
verified.
If there is a time-invariant neighbourhood
such that
X=0
can be e a s i l y
of the system
x=0
of
NCR n
then
(t,to,Xo) 6 T o × R × N
(7) with respect to some non-empty
implies its attraction.
Pr0p0siti0n
If
4.
of (7) is attractive
x=O
attractive with respect to every bounded Example once
5 (Lj.T.
again
and
Grujic
find
1975).
We c o n s i d e r
the
system
of E x a m p l e
3
: inf [AM(t ) : tER]
max[O rm(t,A,~)
then it is uniformly
Tia R .
=
= +~
,
, (27) - I { [ ~ 2 _ 4 ~ 7 + 47~ -I - a ( ~ + # t +7t2)] I/2 - ~ }
for
A a (4~7 _ ~ 2 ) ~ [47( a + ~ t + 7 t 2 ) ] -I
0 , for
A < (4a7 _ ~ 2 ) ~ [47( a + ~ t + 7 t 2 ) ] -I
,
Hence,
sup [rm(t,A,~) The
state
x:O
a) a t t r a c t i v e
: t~R] is
for
: +~
A >_ (4a7 _ @ 2 )
:
in the whole,
b) x o - u n i f o r m l y
attractive
in the whole,
c) t o - u n i f o r m l y
attractive
in the w h o l e
Ti c R not
attractive
uniformly
Definition
with
(iii)
respect
to any b o u n d e d
3.
in
the
whole
with
respect
The
state
respect
x=0
of the
system
~table with respect to
to
Ii
and a t t r a c t i v e
(7) is
Ti
any
bounded
Tier
with
respect
stable with respect to
stable
with
to
respect
to
respect
li
and
:
iff it is both
equi-asymptotically
Ii
x o -uniformly
to
li
stable
;
iff it is both attractive
with
Ti ;
quasi-uniformly asymptotically is b o t h
to
attractive.
(i) asymptotically
(ii)
with
,
d) u n i f o r m l y e)
~ [47(a + ~ t + 7 t 2 ) ] -I .
uniformly
stable
with
stable with respect to respect
to
Ti
and
Yi
iff it
t o -uniformly
,
12
Ch. I
attractive
Outline
with respect
of the
to
Liapunov
stability
stable w i t h r e s p e c t
tive with respect (v) the p r o p e r t i e s sponding X:O
to
to
Ti
li
iff it is
and u n i f o r m l y
attrac-
Yi ;
(i)-(iv)
stability
in g e n e r a l
Ti ;
(iv) uniformly a~ymptotieally stable with respeot to both u n i f o r m l y
theory
of
hold X:O
"in the whole" iff both the correand the c o r r e s p o n d i n g
attraction
of
hold in the whole; Yi
(vi) exponentially stable with respect to real n u m b e r s
~1
and
~>0
such that
llx(t;to;Xo)ll ~ ~llXoll exp [-~(t-to)]
"with respeot
Example 6 (Lj.T. Gruji6 1975). d_xx : dt
and its s o l u t i o n s
A(t)
X
to
A>O
and
implies
, VtET O , VtoET i
This holds in the whole iff it is true (vii) The e x p r e s s i o n
iff there are llXo~ < A
for
A = +~ .
Y i " is o m i t t e d
The second order
iff
Yi:R
s y s t e m is d e s c r i b e d by
,
l+t 2
are found in the f o r m
X(t;to;Xo)
1 +tot
i
:
Lto_t
, t -t o
X0
-
, =+ ot
Hence, e
~M(t'e) which
: 7 1 1 + (l+t 2)
-i
• ( ~ - s i g n t ) + sign t]
,
implies inf [SM(t,e): t e R ] : 0 , V c E ]0,+~[
,
and [(1+t2) I/2 ~ x l l ~ - l - 1 ] I/2 Tm(t,llxll,~)
, for
Ilxll ~ ( 1 + t 2 )
-I/2 ,
= 0
,
for
0 < Ilxll ~ ( 1 + t 2 )
-1/2
,
which yields sup [ T m ( t , a , ~) : t ~ R ] Therefore,
the state
a) a s y m p t o t i c a l l y c) u n i f o r m l y
x=0
stable
b) e q u i - a s y m p t o t i c a l l y
+~
:
is
bounded Notice
0 < ~ ~ a ( l + t 2 ) I/2
,
V A •
]0,+~[
.
:
in the whole,
stable,
asymptotically
d) not e q u i - a s y m p t o t i c a l l y e) not u n i f o r m l y
for
stable w i t h r e s p e c t
to any b o u n d e d
Tier
asymptotically
stable
in the w h o l e with r e s p e c t
to any
TicR.
that the s y s t e m is linear•
The next r e s u l t s
,
stable in the whole,
are s t r a i g h t f o r w a r d
corollaries
to P r o p o s i t i o n s
I-4.
1.2
Stability properties
Proposition
5.
If there is a time-invariant
euch that
X:0
13
stability of
non-empty
Ti
Proposition
is continuous
x(t~to~Xo)
asymptotic
If
in
(7) with respect
of then
to some
stability.
of (7) i8 asymptotically
x:O
uniformly asymptotically
NCR n
(t,to,Xo) eYoXRXN
of the system
x:O
implies its asymptotic
6,
neighbourhood
stable
then it is
stable with respect to every bounded
TicR•
1.2.5. Equilibrium states For
the
sake
of c l a r i t y
Definition 4. over
Ti
State
we
x*
state of the
x(t;to,X*) The
system
(7)
equilibrium
is its
state
iff
"over
expression
Proposition 7. (7) over
Ti
Yi "
, vteY ° , VtoeT i
is o m i t t e d
XwERn
For
: x*
iff
TC=R
to be an equilibrium
it i8 necessary and sufficient
( i ) for every
toEY ±
.
.
•
state of the system
that both
X(t;to;X* )
there is the unique solution
(7)j which i8 defined for alZ
tel
(8)
of
o
and (ii)
:0
f(t,x*)
Necessity.
Proof. rium
, vteY o , Vtoel i
state
of
(7)
Sufficiency.
If
Necessity is e v i d e n t l y
x~
satisfies
x ( t ; t o , X ~) = X ~ , V t e Y o d dt Hence~
×(t;to;X*)
(8)
man
in
the
(1964),
Kalman
for
Let
arbitrarily toeY i
8.
due
existence by E . A .
by
for
x~
to be a n e q u i l i b -
(8).
condition
= f[t,X(t)]
to t h e
(ii)
then
x(t) =
VteY o , VtoeT i
of
(7) at
condition
If
and uniqueness Coddington
(1966)
Bertram
x:O
( t o , X *)
for a l l
(i).
(7)
Then
xo=0
and
of
the
solutions
a n d N. L e v i n s o n
L.S.
Pontriagin
(1955),
(1970)
can P.
be Hart-
[see a l s o
R.
(1960)].
x=O
of
small.
because
implied the
(ii)
VtoeY i , obeys
of the system
then it is an equilibrium
P&0of.
and
is a s o l u t i o n
Halanay
a n d J.E.
Proposition Yi
(i)
•
books A.
= x~
is u n i q u e
holds.
The c o n d i t i o n s found
and
x(t) : O : f ( t , x * )
toeY i , which
Hence
of
.
and
(7) is stable with respect
state of the system over
be s t a b l e
with
UX(t;to;O)ll < e
respect
to
for all
llXoll = 0 < 6 M ( t o , e )
. Let
to
Yi .
Ti
and
e>0
teY o
and
every
XI
and
X2
be
be
14
Ch. I
two s o l u t i o n s
of
Outline of the Liapunov stability theory in general
(7) t h r o u g h
(to,0)
IIXl(t;to;0)-x2(t;to;0)ll for all
teY o
and e v e r y
. Then,
, toel i
< lIXl(t;to;0)ll +IIX2(t;to;0)ll
toeY i
(9)
< Cn
because 6n
IlXoU : 0 < 8 M ( t o , ~ ) Let
en + 0
as
×2(t;to;O)ll Hence,
n ~+~
is less
. It n o w f o l l o w s than
En
from
no m a t t e r
Xl(t;to;0)
(9) that
how large
llXl(t;to;0) -
integer
n
is taken.
~ X2(t;to;O)
and )IXi(t;to;0)ll < en for a r b i t r a r i l y the u n i q u e
large
solution
integer
of
X:0
is an e q u i l i b r i u m
Let
g :Rn~R n
define
n.
(7) on state
It f o l l o w s
Yo of
, i:i,2
for all
(7) o v e r
an a u t o n o m o u s
,
that
x(t;to;0)
~ 0
is
toET ± , which proves Ti .
that
•
system
dx
d--t-: g ( x ) Every
stability property
Proposition Corollary
of
x=0
(10)
of
(i0)
is u n i f o r m
in
toER
of the eystem
(10)
is its e q u i l i b r i u m
. Besides,
8 yields
I.
some non-empty
If
x=O
interval
TicR
then it is an equilibrium
state
state
over
of the
system.
1.3. ON THE LIAPUNOV STABILITY CONDITIONS 1.3.1. B r i e f o u t ] i n e of Liapunov's o r i g i n a l r e s u l t s A.M.
Liapunov
to s o l v i n g
(1892,
stability
p. 25)
defined
problems
two e s s e n t i a l l y
as f o l l o w s
different
approaches
:
" A l l ways, which we can present f o r s o l v i n g the question we are i n t e r e s t e d i n , we can d i v i d e in two c a t e g o r i e s . With one we associate a l l those, which lead to a d i r e c t i n v e s t i g a t i o n of a perturbed motion and in the basis of which there is a determination of general and p a r t i c u l a r s o l u t i o n s of the d i f f e r e n t i a l equations ( I ) . In general the s o l u t i o n s should be searched in the form of infinite
s e r i e s , the s i m p l i e s t type o f which can be considered
those from the preceding paragraph. They are series ordered in terms of i n t e g e r powers of f i x e d v a r i a b l e s . However we s h a l l meet in the sequel series of another character.
1.3
S t a b i fits-
conditions
15
The c o l l e c t i o n
of all
which a r e i n t h i s
ways f o r
category,
the s t a b i l i t y
we s h a l l
cal]
With a n o t h e r one we a s s o c i a t e a l l on p r i n c i p l e s
All
sequel,
of integrals
there will
variables
there
and an
o f the e q u a t i o n s ( I ) ,
and i n gen-
o f them, which we s h a l l
meet i n the
of functions
according to given c o n d i t i o n s ,
by t h e i r
under an a s s u m p t i o n t h a t satisfying
total
derivatives
xi,x2,...,x n
in
of
which
t , taken
are f u n c t i o n s
of
t
the e q u a t i o n s ( I ) .
The c o l l e c t i o n
duced
an i n v e s -
in the case t h a t
be always a d e t e r m i n a t i o n
xZ,x2,...,Xn,t
s h o u l d be s a t i s f i e d
In order
the w e l l - k n o w n way f o r stability
function.
i n the b a s i s o f a l l
call
o f any s o l u t i o n
t h e s e ways can be reduced t o a d e t e r m i n a t i o n
investigation eral
example, is
o f the e q u i l i b r i u m
is a force
which a r e based
equations of a p e r t u r b e d motion.
Such one, f o r tigation
the firot method.
those,
independent of a determination
o f the d i f f e r e n t i a l
investigation,
of all
ways o f
such a c a t e g o r y we s h a l l
the second method," to e f f e c t i v e l y
the
concept
tion of d e c r e s c e n t
"We s h a l l
develop
functions
T and
functions
p. 56)
:
of real (39)
(s=1,2,.,n)
(40)
o f the form
Jxsl ~H
,
H are c o n s t a n t s ,
t h e f o r m e r o f which * may be a r b i t r a r i l y
can be small
than z e r o ) *
Then we s h a l l
consider only functions
X I
Such p r o p e r t i e s it
introduce
will
: X 2
:... : X n
possess a l l
features;
several
for
if
small,
in them it
which
are c o n t i n u -
and v a n i s h a t
,
functions But,
c o n s i d e r e d by
besides that,
definition
they
o f which we
terms.
L e t be c o n s i d e r e d a f u n c t i o n the c o n d i t i o n s ,
(40)
: 0
were n o t m e n t i o n e d ) .
can possess s p e c i a l
zero,
1892,
intro-
and the
,
ous and o n e - o n e under t h e c o n d i t i o n s
sufficiently
Liapunov
l a r g e and the l a t t e r
(but different
shall
A.M.
functions
(Liapunov,
Xl,Xg,...,Xn,t
t~T
us (even i f
method
consider herein real-valued
obeying conditions
arbitrarily
second
and d e f i n i t e
as follows
variables
where
the
of s e m i - d e f i n i t e
T
V which is
is
can t a k e ,
o n l y v a l u e s o f one a r b i t r a r y
sufficiently
such t h a t large,
under and
H
a p a r t from t h o s e equal t o sign.
no-
16
Ch. I
O u t l i n e of the L i a p u n o v
or
sign,
then we s h a l l
negative funotion. In a d d i t i o n to t h a t ,
t
theory in general
signoonstant, When we wish say t h a t i t is positive
Such a f u n c t i o n we s h a l l c a l l to underline i t s
stability
, and the c o n s t a n t
that,
if
the f u n c t i o n
V d o e s not depend on
H can be chosen s u f f i c i e n t l y
under the c o n d i t i o n s (40) the e q u a t i o n
V=O
small
so
can hold
o n l y f o r one set o f values o f the v a r i a b l e s Xl : x2 :,--:Xn
then we s h a l l
call
: 0
,
V signdefinite one, and wishpositive-definite or negative-def-
the f u n c t i o n
ing to u n d e r l i n e i t s
sign -
inite. We s h a l l
use the l a s t n o t i o n s a l s o w i t h r e s p e c t to f u n c t i o n s
depending on is
t.
However, in such a case the f u n c t i o n
signdefinite o n l y under the c o n d i t i o n , i f
be c a l l e d
p o s s i b l e to f i n d such a
function
V will
for it
it
t-independent positive-definite
W, f o r which one of two expressions or
V-W
-V-W
would r e p r e s e n t a p o s i t i v e f u n c t i o n . Hence, each of f u n c t i o n s
x~ + x~ -2XlX 2 cost
, t(x[ + x~) - 2x Ix 2 cost
is s i g n c o n s t a n t . However, the former is o n l y s i g n c o n s t a n t , and the l a t t e r ,
if
n=2 , i s s i m u l t a n e o u s l y s i g n d e f i n i t e .
Every f u n c t i o n
V,
f o r which the c o n s t a n t
H can be chosen
so small t h a t f o r numerical values o f t h a t f u n c t i o n under the c o n d i t i o n s (40) t h e r e is
an
upper
bound, w i l l
be c a l l e d
bounded.
In view o f the p r o p e r t i e s which, under our assumption, possess all f u n c t i o n s every
function
considered
independent
A bounded function nevertheless which
for
how s m a l l ,
all
values t~T
will
can
,
will
be s u c h ,
for
example,
t .
be such
there
of
by u s ,
of
is
variables,
Ix~l ~
that such
for
every
non-zero
satisfying
positive
number
e ,
h , for
conditions
(s = 1 , 2 , . . . , n ) ,
hold the f o l l o w i n g :
This c o n d i t i o n w i l l independent o f
f o r example, e v e r y f u n c t i o n
t . However f u n c t i o n s depending on
bounded, can v i o l a t e i t . a function
satisfy,
t , even
Such a case r e p r e s e n t s , f o r example,
I. 3
Stability conditions
17
s i n [ ( x I +x 2 +...+xn) t ] When the f u n c t i o n we shall
speak
Such i s ,
that
V fulfils
the preceding r e q u i r e m e n t , then
it admits infinitely small upper bound.
f o r example, a f u n c t i o n (X 1 + X 2 +.,.+ X n )
Let
sin t.
V be a f u n c t i o n a d m i t t i n g i n f i n i t e l y
Then, i f
t ~T
where
small upper bound.
we know t h a t the v a r i a b l e s s a t i s f y a c o n d i t i o n
~
is
, IVl ~ ~ ,
number, hence we conclude t h a t there i s
a positive
another p o s i t i v e number ~ , less than which cannot be the g r e a t est q u a n t i t y among I X l ] , I x 2 1 , . . . , I X n l In order along
to e x a m i n e
behaviour
system motions
(1892,
without
p.58 ) p r o p o s e d
the
"
of the v a l u e s using
of a d e f i n i t e
the m o t i o n s
following
function
themselves
A.M.
V
Liapunov
:
"Simultaneously w i t h the f u n c t i o n
V we s h a l l o f t e n consider
an expression V'
~V X1 + BV aV X BV : 8X----~ ~--~2 x2 +'''+ 8X n n + 8--t '
representing i t s tion that
total
time d e r i v a t i v e , taken under an assump-
x i , x 2 , .... xn
are f u n c t i o n s o f
t,which satisfy dif-
f e r e n t i a l equations of a perturbed motion. In such cases we s h a l l always assume t h a t the f u n c t i o n is
such that
V'
V
as a f u n c t i o n of the v a r i a b l e s (39) ~ would
be continuous and one-one under the c o n d i t i o n s (40) * Speaking f u r t h e r about the d e r i v a t i v e of the f u n c t i o n we s h a l l mean t h a t i t These
concepts
have
for a s o l u t i o n
of
been
is the t o t a l
the k e y s t o n e
(uniform)
"Theorem I .
stability
of the of
V,
second
x:0
Liapunov
(Liapunov,
method
2892,
and
p. 59) :
If the differential equations of a perturbed
motion are such that it is possible function
V,
derivative."
the derivative
to find a signdefinite
V' of which in view of these
equations would be either a signconstant function with the opposite sign to that of
V,
then the unperturbed motion is In a d d i t i o n
to this
reproduced
on p.2
asymptotic
stability
ic s t a b i l i t y
of
result
of our
x=0.
A.M.
stable."
Liapunov
§ l.i that
concept
or identically equat to zero,
has
(1892) become
and for a s o l u t i o n
made
the
"R£mark 2"
the f o u n d a t i o n of
(uniform)
of the asymptot-
18
Ch. I
In o r d e r to i l l u s t r a t e
Outline of the Liapunov stability theory in general
deepness,
generality
and i m p o r t a n c e
of L i a p u n o v ' s
r e s u l t s once again, let f o l l o w i n g his r e s u l t s be cited (Liapunov, 1892, P-75):
"ThgO~£m
I.
When the roots
of the character-
KI,~2,..-,K n
istic equation are such that for a given natural number i8 impossible
any relationship
mlK I + m 2 K 2 +...+mnK n
in which all equal to
are non-negative
ms
m,
satisfying
:
0
V
of the power
m
giving their sum
to find just one whole of the quantities
for arbitrarily
@iuen whole homogenous
quantities
of the same power
xs
"Thzo~zm If. U
form
V
: U
function
signdefinite of the power
Gantmakher
results
form of any even power m
(1974b)
satisfying
the s t a b i l i t y 5.
Definition
(i) positive
xTHx
theory.
k-th
H = (hij)
e Rnxn
(negative)
(negative)
definite
(negative)
order p r i n c i p a l
iI
i2
--
iI
i2
... i k
then the i8 also sign
potential
( T h e o r e m 3', p.189) theorem
is
the f o l l o w i n g
iff its q u a d r a t i c
theorem
for
is needed.
respectively.
m i n o r of the m a t r i x
H
hili I
hili 2
-'-
hi2i 1
hi2i 2
...
ik] :
.
hlki 2
form
V(x) :
respectively;
iff its q u a d r a t i c
hiki I
w h i c h has
:
semi-definite,
definite,
of these
[see S. Bar-
t h e o r e m is a f u n d a m e n t a l
(negative) semi-definite
is p o s i t i v e
positive
This
are
U ."
the f u n d a m e n t a l
For its p r e s e n t a t i o n
A matrix
is p o s i t i v e Let a
(1970)].
m,
that equation
r e f e r r e d to as the L i a p u n o v m a t r i x
nett and C. S t o r e y
(ii)
recognized
Ks
(9) there is the function
and d e d u c e d from t h e m a t h e o r e m
been commonly
U of the
When the real parts of all roots
definite with the opposite sign to that of F.R.
(9)
m."
and when in the equation
being
Ks
the equation n ~V Z (PslXl + P s 2 X 2 +...+PsnXn ) ~ s:l
negative
it
,
integers,
then it is always possible
homogenous function
m
of the form
.
""
.
form
V(x) : xTHx
• he d e n o t e d by hili k hi • 2mk
.
. °
hiki k
where ij ~ {l,2,...,n} , ij < ij+ 1 , j = 1,2,...,k , k = 1,2,...,n .
I. 3
Stability
The
leading
conditions
principal
2
The
following
Theorem 2.
19
minor
o f the
order o f
k-th
hll
h12
... hlk
h21
h22
--- h 2 k
hkl
hk2.
... hkk
,
H
is
k : 1,2,...,n
.
..,
criteria
are
well
known
[see
Cantmacher
(1974a)]
Necessar U and sufficient for a symmetric
n×n
:
matrix
H
to be : (i) positive
semi-definite
is that all its principal
miners are non-
negative
H
~ 0 , l~il iI
(ii)
i2
negative
-.
ik
semi-definite
is that both all its even order principal
minors are non-negative
and all its odd order principal minors
are non-positive
iI (iii)
positive
definite
i2
... i k
_<0 , k : 1,3,...
is that all its leading principal
minors
are
positive
H
> 2
(iv)
negative
definite
0
negative
definite
iff
and a square i f f its is the
a square its
Hermitian transpose
matrix
symmetric
matrix
A
part
leading principal
leading principal
> 0 2
that
;
minors are alter-
and positive
(-I) k H
Notice
k : 1,2,...,n
is that both its first order
minor is negative and all its natively
,
,..
,
k = 1,2,...,n
.
...
A with part
a l l r e a l v a l u e d e l e m e n t s is ( s e m i - ) i A s = ~ (A + A T ) is ( s e m i - ) d e f i n i t e ,
with
c o m p l e x v a l u e d e l e m e n t s is ( s e m i - ) d e f i n i t e i A H = ~ ( A + A *) is ( s e m i - ) d e f i n i t e , w h e r e A~
conjugate
matrix
of the
matrix
A.
20
Ch. I
Now,
the
matrix I and
fundamental
theorem
Outline of the L i a p u n o v stability theory in general
theorem
o f the
- can be s t a t e d
stability
theory
- the L i a p u n o v
as a c o r o l l a r y
to the
preceding
Theorem 3.
In order that real parts of all eiyenvalue8
, be negative
A E R n×n
tive definite solution
H,
it i~ necessary
symmetric matrix
G,
H E R n×n , of the
which i8 also positive solving
Storey
of a matrix
=
-G
A,
that for any posi-
G E R n×n , there exists
+HA
definite
the L i a p u n o v
and sufficient
(Liapunov)
ATH
For
Theorems
!! by L i a p u n o v .
the unique
matrix equation ,
symmetric matrix.
matrix
equation,
see
for e x a m p l e
Barnett
and
(1970).
1.3.2. Brief outline of the classical
and novel developments
of the Liapunov second method Following method
Liapunov
consists
conditions
are
time
derivative
hood
of
×=O
Liapunov
1.3.2.1.
V
(iii)
of the
classic
in the
books
(iv)
time
the class
functions
stability
and
its t o t a l
neighbour-
development
by Y o s h i z a w a
o f the (1966)
the class ~ ~ +~
the agars
KR
the class
estimates
are u s u a l l y
investigation
is H a h n
(~967).
of the
denoted
funcby
of p r o p e r t i e s What
follows
~ , of
is
and results.
¢ , ¢ : R+~R+ , b e l o n g s t o
increasing
K iff
They
to the
K[O,~ [ , 0 < ~ ~ +~
strictly
or l o w e r
derivative.
contributor
A function
on
(i) h o l d s
iff b o t h
, iff b o t h [0,~[
for
and
~ = +~
it b e l o n g s
it is d e f i n e d ,
continuous
¢(0) = 0 ; , K = K[O,+~[
to the
class
K
; and
¢(~) ~ + ~
; L[O,~ [
ly d e c r e a s i n g (v)
V
(1977).
on his d e f i n i t i o n s
the class
as
second
providing
expositions
are u s e d as u p p e r
its t o t a l
. The m a i n
based
and (ii)
theorems
of his
scalar function
can be f o u n d
and Laloy
functions
D e f i n i t i o n 6. (i)
development
o v e r a time-invariant
system motions
use of the c o m p a r i s o n
mainly
of s t a b i l i t y
on a p p r o p r i a t e
method
Habets
and
: R+~R¢ and
along
classical
Comparison functions
Comparison tion
the
of a n u m b e r imposed
. Adequate
second
and Rouche,
(1892),
on
L iff
iff both [0,~[
it
and
(iv) h o l d s
is d e f i n e d ,
continuous
l i m [¢(~) : ~ + ~ ] for
a = +~
and
= 0 ;
, L = L[0,+~[
.
•
strict-
1.3
Stability conditions
Let
@I
21
d e n o t e the i n v e r s e
f u n c t i o n of
The next r e s u l t was e s t a b l i s h e d Proposition
(i) If (ii) If If
(iii)
by H a h n
@EK
and
@EK
then
¢(~) • K
;
@eK
and
eel
then
¢(0) E L
;
@ eK[O,e [
and
¢(~) =~
lim[¢(~)
Cz(~) <@I(~) Definition
7.
[0,#1
on
A function
(i) the class
and
(ii) the class
KK
(iii) the class
iff bOth
and KL
(iv) the class
1.3.2.2. Pzfinitz
(iii)
throughout
the book, w h i c h
(1892)
Definition
8.
a)
N
and H a h n Function
v
of
~ R+ , b e l o n g s
×:0
@(o,f) e K [ o , e [
v
is n o n - n e g a t i v e vanishes
¢(o,~) 6 K [ O , ~ [
all
w i l l be used
is
definition
by
Def. 24.3a). :
N : v(×) e C(N) on
neighbour-
,
N : v(x) a 0 , V x e N
at the o r i g i n
N:S
:
(-V)
,
v(0) : 0 ; S
x:O
of
iff
(i)
;
semi-definite
respectively).
It is to be n o t e d that is b o t h p o s i t i v e
(i) holds
(on a n e i g h b o u r h o o d
is p o s i t i v e
or in the whole,
xeR n
;
•
iff there is a t i m e - i n v a r i a n t
(iv) negative semi-definite
I.
for every
~ :~ = ÷~ .
(iii) positive semi-definite in the whole iff
Remark
;
~ 6 [0,~[
(ii) positive semi-definite on a n e i g h b o u r h o o d
S
for every
~ • [0,~[
~ = ~ = +~ ;
holds for
Rn~R
on
v
iff
:
, N ~ R n , such that
c)
whole)
to
is based on the c o r r e s p o n d i n g
b)
for
then
.
for every
(1967, p.98, v :
is c o n t i n u o u s
holds
[O,~[
on
the f o l l o w i n g d e f i n i t i o n
(i) positive semi-definite hood
is not defined on
f ~ n c t i o n s and comparison f u n c t i o n s
For the sake of p r e c i s e n e s s
Liapunov
both
] ;
¢I
:@(a)
for
¢(~,o) e /[0,~[ iff
then
for e v e r y
(i) holds
KL[o;~,~ [ i f f
6 [0,8[
#
where
¢(~,o) • K [ 0 , # [ iff
¢I6K[0,~
@(~) >@(~)
and
¢ , ¢ : R+xR+
KK[o;=,D [
e [0,8[
then
: ~-~+o~] : ~
¢ eK[O,~ ] , @EK[O,~ ]
If
(1967).
9.
(iv) If # • K and ]~,+~] ;
(v)
¢ , ¢I[¢(~)] ~ ~ .
of
for
N:R n ;
x:O
or in the
(on the n e i g h b o u r h o o d
•
function
and n e g a t i v e
can be a v o i d e d by i n t r o d u c i n g
S
v
defined
by
semi-definite.
the n o t i o n of s t r i c t l y
v(x) : 0
for
This a m b i g u i t y
positive
(negative)
22
Ch. I
semi-definite
positive
function.
and there
Then
is s t r i c t l y
Necessary
is
yen
such that
H is strictly positive
spectively. v(X) : xTHx
and
positive
sufficient
positive
semi-definite
and
there
that
Necessary that
is
and
j • {1,2,...,n}
sufficient
semi-definite there
is
odd
semi-definite,
for a s y m m e t r i c
is o b v i o u s l y
that
such
i 2
is o b v i o u s l y
Definition
9.
Function
(i) positive N ~R n N
and
both
...
(iv)
x=0
both
such
that
ii
i2
.. ij
v :Rn~R
is
:
ii)
>
such that
both
H
of T h e o r e m
and
neighbourhood
it is p o s i t i v e
semi-definite
N , on
;
S of
X=0
iff
(i) holds
;
negative definite
X=0
(1967)
2 holds
neg-
0
for
whole,
2 holds
to be s t r i c t l y
positive definite in the whore iff (i) holds (-v)
respective-
to be s t r i c t l y
(i) of T h e o r e m
matrix
that
v(x) > 0 , V ( x # 0 ) e N
N:S
iff
Hahn
H
ij
(ii) positive definite on a neiyhbourhood for
matrix
definite if there is a t i m e - i n v a r i a n t
, of
, re-
that
for a s y m m e t r i c
j • {i,2,...,n)
(negative)
(v(y) < 0)
(negative) semi-definite iff
(-i) j H
(iii)
to be strictly
is said
v(y) > 0
(negative)
i 1
ative
v : Rn-~R
A function
(negative) semi-definite iff b o t h it is p o s i t i v e
semi-definite
ly.
Outline of the Liapunov stability theory in general
(on a ne~ghbourhood
is p o s i t i v e
respectively).
definite
S of
N=R n ; or in the whole)
(on the n e i g h b o u r h o o d
S
or in the
•
proved
Necessary and sufficient for positive definiteness of v on a neighbourhood N of x:O is existence of comparison functions ¢i • KE0,e[ , i:i,2 , where ~ : sup {llxll :x•N} , such that both v(x) Proposition
C(N)
and
10.
¢l(llxll) ~ v ( x )
Definition
10.
(i) positive
Function
~¢2(IIxlI)
v : R×R n ~R
semi-definite on
time-invariant
, VX•N
connected
is
. :
T~ = [~,+~[ neighbourhood
, 7•R N
of
, iff there x=0
is a
, N C R n , such
that a)
v
is c o n t i n u o u s
in
(t,x) • T T × N
:
v(t,x) • C ( T T × N )
;
1.3
stability conditions
b)
23
v
is n o n - n e g a t i v e
on
N
:
v
vanishes
origin
v(t,x) >_0 , V ( t , x ) e Y ~ x N
;
and c)
at the
d) Iff the c o n d i t i o n s yen
such
that
dGfinit8 on
:
(a)-(c)
v(t,0) = 0
hold
v(t,y) > 0
and
, then
v
toy T
there
is strictly positive
is
semi-
l~ ;
(ii) positive
semi-definite
on
(iii) positive N=R n ;
semi-definite
in the whole on
(iv) negative
semi-definite
(in the whose) on
(-v)
, VteT T .
for every
is p o s i t i v e
lT xS
semi-definite
iff
(i) h o l d s
T~
for
iff
Yr
(in the whole)
N:S
;
(i) holds
for
(on
Yr× N )
iff
on
T~
Y~× N ),
(on
respectively. (v) The
expression
ments
Definition
hold
II.
Y T " is o m i t t e d wer
Function
(i) positiue
definite
connected positive inite
"on
for every
on
TT , reR
semi-definite
N
of
on
iff all
corresponding
is
:
,iff x=O
lrxN
there
is a t i m e - i n v a r i a n t
, N C_R n , such that
and there
on
N,
(ii) positive
definite
on
T~xS
(iii) positive
definite
in the whole on
negative
definite
(in the whole)
on
T~
(on
I~×N)
positive
definite
(in the whole)
on
T~
(on
T~×N)
(iv)
w : Rn~R , o b e y i n g
exists
w
elTxN
require-
•
: RxRn ~ R
v
neighbourhood
function
.
both
it is
a positive
w(X) ~ v ( t , x )
def-
, V(t,x)
; iff
(i) holds
TT
for
iff
N:S
;
(i) holds
for
N:R n ;
iff
(-v)
is
, respective-
ly. (v) The
expression
ments
hold
The f o l l o w i n g inition
result
(ii)
~ER
.
is o b t a i n e d
Necessary
ii.
to be positive
(i)
l T " is o m i t t e d
iff all
corresponding
require-
•
directly
from
Proposition
i0 and Def-
i0.
Proposition
hood of
"on
for every
TTxN
when
N
for a function
v :RxRn~R
i8 a time-invariant
v(t,x) e C ( T T x N ) =0
;
, vteT T
and (iii)
there is
CeK[o,~
¢(llxll) ~ v ( t , x )
[ , where
, V(t,x) e Y T x N
~ = sup{llxll : x e N }
,
n
neighbour-
is that
x:O
v(t,0)
on
definite
and sufficient
such that
24
Ch. I
Definition ×=0
at
so that R~mark
tance w
12. teR
Set which
X E V~(t)
of L i a p u n o v ' s
be c o n s i d e r e d ,
where
N
V~(t)
is the
concept
v,w : R × R n ~ R
v
is p o s i t i v e
definite
w
is p o s i t i v e
semi-definite
Let
V~(t)
tion
12. Then,
(a) there v~ •
and
is
.
on
Let
but it is not p o s i t i v e W~(t)
be a s s o c i a t e d
the f o l l o w i n g ~ • ]0,+~[
is true
such that
of
v , v : R×R n +R
deepness
functions
they of
Yr ×N on
neighbourhood
a function
and a p p r e c i a t e
neighbourhood
(i)
connected
theory in general
,
. m
of d e f i n i t e
(ii)
,
with
v(t,x) <~
to u n d e r s t a n d
is a c o n n e c t e d
E YrXN
largest
can be a s s o c i a t e d implies
In o r d e r
2.
Outline of the L i a p u n o v stability
obey X=0
let
the
and
impor-
functions
following
v
on
and Yr xN ,
:
; Yr xN
and
definite with
w(t,x) > 0 , V(t,x¢0) on
v
and
Yr ×N w
in sense of D e f i n i -
: (Fig. i)
V~(t) ~ N
, VteY r
,
]o,~ ]
Vf(r) For
OF (t)
example,
v(t,x) : (l+t2)llxll 2 , v(t,x) = (et+e-t)llxll 2
(b) for any
~ • ]0,+~[
such
(Fig. 2)
that
Figure i
for w h i c h W~(t)\ N # ¢
W~(r) ~ N
there
is
teY r
.
\
/
/,
etc; , t >r
,
I. 3
Stability conditions
For example, Definition (i)
13.
w(t,x)
Function
dear•scent
bourhood N
on
N
(ii)
decrescent
on
deorescent
in
The
Proposition
still
Proposition 12. TT×N
when
N
that
there
v(t,x)
iff
whole
T~
is a t i m e - i n v a r i a n t
for
every
z•R
and
function
.
to c l a r i f y
the
following
on
for
w
T r ×N
be
and
v
is p o s i t i v e
definite
and
deerescent
is p o s i t i v e
definite
but
not
(a)
Let
~ > 0
Then,
for
be the every
neighbourhood VteTr
v~
be
the
associated
following
a = sup {RXU
importance
number
~ • ]0,~]
there
on
Tr xN
with
v
on
on
existence
: וN}
,
such
of decrescent . Let
and
w
; Tr ×N
.
in t h e
sense
of
:
obeying
V~(t) c N
is a c o n n e c t e d
X:0 , N~ ~ N , s u c h t h a t
of
is
v,w : R×R n ~R
decrescent
holds
largest
N~
deorescent
×=0
:
w
W~(t)
of
considered,
(i)
Then,
condi-
.
meaning
and
to be
v
neighbourhood
(ii)
and
N=R n
for
•
¢ • K[o,a [ , where
they
12.
;
(i) h o l d s
sufficient
, V(t,×) •T~×N
v
V~(t)
N=S
on
;
13 i m p l y
a time-invariant
functions
Let
w
, V(t,x) •TT×~
for
iff
neigh-
function
TT " is o m i t t e d iff a l l c o r r e s p o n d i n g
let
Definition
= (l+t2)-iIlxll 2
definite
~w(x)
(i) h o l d s on
Definition
In o r d e r
is
a positive
that
functions obey
, iff
and
"on
v ( t , × ) ~¢(H×II)
Ramark 3.
:
Necessary is
of a comparison
v : RxR n ~R
TTxS
hold
1O a n d
w(t,x)
, TER
the
expression
tions
= e-tljxJl 2 , o r
×=0 such
(iii) (iv)
TT
of
w : Rn~R,
,
25
, Vt•T T
.
time-invariant V~(t) D N ~
,
(Fig. 3) •
(r)~~ ".,W
Nf
For
example,
v(t,x)
-
Figure 3.
2+t2 Ilxll2 , v ( t , x ) = (2 + sin t) Ilxll2 l+t+t 2
26
Ch. I
(b) let
~ > 0
for any
be the
largest
~ e ]0,~]
and
can be a r b i t r a r i l y there of
is
N6
Outline
number
for any
small,
obeying
stability
W~(t) c N
connected
e >0
tl(~,Ne. ) 6 ]T,+~]
, W~(tl) c N e
of the L i a p u n o v
the
W~(tl)
in general
, VtET 7
neighbourhood
, and d e n o t e s
such that
theory
. Then,
N e qN
diameter
is a p r o p e r
w(t,x) = (l+et)llxll 2 , w(t,x) = (l+t2)llxll 2
Barbashin
and K r a s o v k i i
unbounded
functions.
ity
subset
(1952,
They
1954)
showed
discovered
necessity
the
4
etc.
concept
of r a d i a l l y
of it for a s y m p t o t i c
stabil-
in the whole.
Definition
14.
Function
v : RxR n -~R
(i) radially unbounded on +~
(ii)
Ne )
(Fig. 4) .
re
For example,
(e
of
V~E R
.
, iff
Ilxll ~ + ~
implies
v(t,x)
the whole
Jlxll~ + ~
implies
v(t,x) ~ + ~
13.
(on
(1967)
the next
can be e a s i l y
,
verified.
Necessary and sufficient for a positive definite in T T ) function
v
to be radially unbounded i8 that there
@ e K R obeying~ respectively,
v(t,X)
-> @ ( l l x l l )
, VXER n
( Vt e Tr )
For examp]e, v(t,x) is not r a d i a l l y
unbounded, v(t,x)
= (2 + t 2 ) ( 4 + llxll) llxlJ 2. + IIxll2 but -- (2 + t 2 ) ( 4
+ IlXU) IIXII
2 + llxll
is r a d i a l l y
, vteT r
•
Hahn
Proposition
exists
, TEE
:
, VteTT ,
radially unbounded iff
Following
lT
is
unbounded.
In this
v(t,x)
case,
>-@(llxll)
,
@(~) = ~ e K R V(t,x)
ER×R n
obeys
, VtcR
I. 3
Stability conditions
1.3.2.5.
Dini derivative
In this
section
: R~R
(ii)
of upper and lower limit
[see M c S c h a n e
they can be e x p l a i n e d
tk
~i) and
and E ~ l z r i a n d z r i v a t i v 6
the notions
are n e e d e d
(1967)I Let
27
be a m e m b e r
for every
tk~r-
( t k ~ r +)
integer as
In brief
[see D e m i d o v i c h
as follows.
of a sequence
tk6R
Definition
(i944)].
of a f u n c t i o n
ST
(S:)
k , tk
k~+~
obeying
(tk>7)
.
15.
(i) N u m b e r
~eR
sequence
is the partial
S~
such that
(S:)
k>N
(ii) the symbol
limit of the function
iff for every
implies
a : +~ (~ =-~)
over the sequence
S~
e> 0
there
~ over the
is an integer
is the partial (S~)
limit of the function
iff for every
e 6 ]0,+~[
there
is an integer N such that, r e s p e c t i v e l y , k>N implies 1 I >~ (~(t k) < - ~ ) ; (iii) the greatest (smallest) p a r t i a l limit of the function @ all sequences
S~
respectively,
which
(lira inf [¢(t) : t ~ T (iv) right upper
is its
left upper
is d e n o t e d
by
(lower)
limit
at
@(tk) over
t=T
,
lim sup [@(t) : t ~ ~-]
]) .
(lower)
when e v e r y w h e r e
N
l@(tk)-a I < e ;
limit of
in (iii)
T-
~ at and
t= 7 $7
is a n a l o g o u s l y
defined
are r e s p e c t i v e l y
replaced
+
by
r+
Definiti0N
and
16.
Let
ST
.
•
V be a continuous
(either
scalar
or vector)
tion,
V : T~xR n ~ R s , v(t,x) 6 C ( T z x N ) , and let solutions
system
(7) exist and be d e f i n e d
(i)
D+V(t,x)
: lim sup
on
TrxN
{~ [t+@,x(t+@;t,x)]@
is the upper right Dini derivative (t,×) (ii)
is the
(iii)
-v(t,x)
(t,×) 6 T r x N : 8 ~ 0 +}
v along the motion
X
at
= lim inf
{v [t+O~X(t+@~t~x)]. _ @
lower right Dini derivative
of
-V(t~x)
×
at
: 8 ~ 0 +}
v along the motion
;
D-V(t,x)
= lim sup
is the upper (t,x)
of
for
func-
of the
;
D+V(t,X)
(t,x)
. Then,
X
;
{v [t+@~x(t+@;t;×)]e
left Dini derivative
of
-v(t,x)
: 0~0- }
V along the motion
X
at
28
Ch. I
(iv)
Outline of the Liapunov stability theory in general
D_V(t,x) = lim i n f {V [t+8;X(t+~;t,x)]@ - v ( t , x ) is the lower left Dini derivative of
(t ,x)
(t,x)
v
v has Eulerian derivative
along the m o t i o n
: D_v(t,x) If
V along the motion
X
at
.
(v) The f u n c t i o n at
: 8~0-}
: Dv(t,x)
is a scalar
X
iff
and then
function
av = ~+
v(t,X)
at
(grad v) T f(t,x)
v(t,x)
= D+V(t,x)
v(t,x) : D v ( t , x )
and d i f f e r e n t i a b l e
i892)
v , v(t,x) : ~
DtV(t,x)
,
= D-V(t,X)
•
(t,x)
then
(Liapunov
,
where av 8v 8v grad v = [Bx i , Bx 2 ,... , ~ n ] Effective method
application of
Theorem 4. and
D+v
in the f r a m e w o r k
without
will mean
: lim sup
Following
If
system
motions
which
enables
themselves.
D+v
(1892),
of
(7)
X +@f(t,x)]s - V ( t , X )
of the system
and
X over
in
is
D+v
Persidskii
: 8 + 0 +]
(t,x) E Y r X S
(7) at
.
can be used.
on t i m e - i n v a r i a n t
the next result x
•
{V [t+S
conditions
Liapunov
(1966),
Theorem 5.
X
that both
S£ability
Halanay
utilizing
second L i a p u n o v
(1966),
be an open set. Then,
holds a~ong solutions
1.3.2.4.
of the
by Y o s h i z a w a
V be continuous and locally Lipshitzian
Let S
D + V(t,X)
O*V
D+v
is based on the next result
calculation
TTxS
of
T
(1933),
sets
Yoshizawa
(1966)
and
is obtained.
c o n t i n u o u s on
ToX RxN
(on
To x TTxN )
then existence of both an open c o n n e c t e d time-invariant n e i g h b o u r h o o d S
x=O
of
T T xS)
and a decrescent positive
definite function
D+v(t,x)
such that, respectively,
v on
~ 0 , V(t,x) 6 RxS
e I xS ) i8 necessary and sufficient for uniform stability x
:
0
of the system
P~oof.
e >0
Necessity.
S
(on
(V(t,x) 6 (on
Y T ) of
(7).
Let
be a r b i t r a r i l y
x:O
chosen.
of (7) be u n i f o r m l y Let
B~ ~ N
stable
(on
, ~ = min [~,6M(e)}
YT ) and
, ¢ eK[o,e [
and v(t,x) Stability ¢ eK[O,e [
of
x=0
imply
= sup {¢ [llx(t+o;t,x)ll] :o • [0,+~[}
and c o n t i n u i t y continuity
of
v
of in
X
in all arguments
(t,X) E R × S
(YTxS)
.
(11) together
where
with
S=B~
I. 3 Stability conditions
Stability
of
x:0
implies
×(t;to,0) ¢ eK[0,e [ From
29
8)
: 0 , V(t,to) • T o X R
now proves
(11) it follows v(t,x)
(Proposition
:
(V(t,t o) • Y o X T r )
v(t,0) = 0 , v t e R
(VteTT)
.
:
z~(~xll)
(v(t,×) • T r × S )
, v(t,x) •R×S
.
Altogether, v is positive definite on S (on TTxS ). Uniform stability of ×:0 of (7) and continuity of X imply [Halanay (1966)] existence of
~•K
VteYo
such that , VtoeR
that
x~ =X(t+e;t,x)
is
v
implies
, re>0
~ ¢ [~(llxll)]
v(t,×)
which proves
IIxoll < $(e)
(VtoeYT)
, V(t,x)
decrescent
x=×(t;to,x
and
v(t+~,× ~ ) = sup
llx(t,to,xo)ll<
o)
eRxS
on
S
(V(t,x)
(on
eR×S
eTT×3)
,
Let
e>O
TTxS).
{~[llx(t+e+a;t+e,x~)ll]
< sup { ~ [ l l X ( t + a ; t , x ) U ]
v(t,x)
,
so t h a t
. Then,
= sup { ¢ [ i l x ( t + e + a ; t , x ) l l ] : v(t,x)
~(~Xoll)
. Hence,
:a • [0,+=[}
:oe :ee
[0,+=[}
[0,+=[}
,
(v(t,x)
eTr×S)
Hence,
D+v(t,x) Su~j'ia~2ncy.
, V(t,x) e R x S
Let the conditions
be arbitrarily and
<0
chosen
¢1,@2 E K[0,[ [
(V(t,X) e Y T x S )
of the t h e o r e m
and
~ be such that
be such
(Propositions
¢i([Ixli) -
be satisfied.
6~ C_N . Let 11-12)
, V(t,x) e R x 6 [
Let
e>0
~ =min{e,~}
that (V(t,x) e Y T x S [ )
(i2)
I Now, B is selected as &(¢) = ~ 2 [ ¢ i ( ~ ) ] , ~ :~(e) ~ Evidently, 8(e) > 0. Since D+v is non-positive and since F~(t) c_B~ , V t e R ( V t e Y r) , for all
~e]0,¢2(~)]
, then
U×oll <~(¢)
implies
< ¢2(6(e)) = ¢ 1 ( ~ ) , and therefore v[t,X(t;to,Xo)] t e Y o and all toeR ( t o e Y ~) , which together with ¢i(llx(t;to,Xo)ll ) _< v [t,x(t;to,Xo)]
(VtoeT~)
Altogether, since ~leK[0,[[ , llXoll < 5 ( e ) implies U×(t;to,Xo)~ < ~ , V t e Y ° , V t o e R Utilizing
CeKR
in (11),
following
I and by r e f e r r i n g
it is easy
to prove
~V(to,Xo) (12) yields
_
VteT o , VtoeR
Definition
v(to,x o) -<¢2(UXoll)
<¢I(e)
,
.
, ~(c)= (VtoeTT)
the proof
to Barbashin
for all
I
¢2 [~i (~(c))] > 0 , Ve>O
•
of the p r e c e d i n g
and Krasovskii
,
theorem,
(1952,1954),
30
Ch~ I
Theorem
If
6.
existence
x v
V(t,x) E R × R n
stability
(on
TT ) such that,
(V(t,x) E T
If
f
TTxN ) w h i c h
Rzmark 5.
The
Following
theorems
enables
effective
(1944)
Liapunov
Halanay
(1966)
Theorem
7.
(1892),
and Hahn
If
X
theorems
is continuous
i8 necessary
and sufficient
Proo~. (on
(7)
Necessity.
T T ). Let
AE]0,+~[
e >0
and
and
v(t,x)
The
function
the next
(on
on
loxRxN
Let
x=0
of
is d e c r e s c e n t
D+v
@[=(UXlI)]
5. Let
X ~ =x(t+8;t,x)
is d e t e r m i n e d
v(t+@,X*)
(on
v on
S
. The
is g u a r a n t e e d
(l+~a)(l+a) -I
(1966),
(on
S of
x=0
, a
and a posi-
TTxS)
respectively,
( v ( t , x ) ely×S)
,
(15)
stability
(7) be u n i f o r m l y chosen,
~
x=O
of
S:B~
. Let
asymptotically
be such ,
and p o s i t i v e
, VteT o
and u n i f o r m
, ~e
of
,
attraction
a*E
of
(on
, @(0)=0
, and
part
, where
of T h e o r e m
-I : a E [0,+~[}
-I : o E [0,+~[} ,
Vt~R obeying
the
~ E K[0,e [ , c o n t i n u i t y
X=0
.
Tr×S )
so that
[0~ru(Ajv)]
X,
]i,+~[
, VteR
(Vtoe T r) , v x e S
, 8 >0
of
S
,
~ CK[O,e[
of the n e c e s s i t y
, x = x ( t ; t o , X o)
by c o n t i n u i t y
on
stable
B~c_N
@~K[o,e [
definite
, VtoeR
in the p r o o f
existence
that
+ ~ s ) ( ] + ~ ) -I : o E [0,+~[}
~ [llx(t+0+a~;t,x)l[] ( l + ~ s * ) ( l + a ~ ) -I
v = min {i,~}
equation
by
ToXl~xN ) then exist-
: sup {~ [llX(t+@÷s;t,X)ll](l+~a)(l+a)
Let
4.
is o b t a i n e d .
= sup {~ [llx(t+8÷a;t+8,x~)ll](l+~a)(l+a)
:
v
(on
is r e p l a c e d
Yoshizawa
b e c a u s e X is c o n t i n u o u s in all its a r g u m e n t s , (i + o~e) ( l + a ) - 1 is also c o n t i n u o u s , X ( t ; t o , 0 ) - 0
[
RxN
via T h e o r e m
for uniform asymptotic
be a r b i t r a r i l y
=EK[0,e
on
T r ).
~ : min{6M(e),a,~}
@(UXlI)-
for uniform
TTxN ) t h e n
neighbourhood
, v(t,x)~R×S
: sup {~[llx(t+o;t,x)ll](1 v
D+v
result
S such that,
D*v(t,x)-<-@(x)
of the system
of
(1949,1956),
time-invariant
@ on
(on
-< 0 ,
(7).
Lipschitzian
also w h e n
definite function
tive definite function
RxN
in the
(1976)].
Massera
(1967),
ence of an open connected decrescent positive
hold
D+v(t,X)
and sufficient
on
locally
calculation
and L a s a l l e
definite
of the system
X=0
then
ToXTrxR n )
positive
respectively,
Lipschitzian
is also
preceding
[McShane
T~ ) of
(on
ToXRXR n
is necessary
xR n)
(on
is l o c a l l y
in the p r e c e d i n g
on
unbounded decrescent
in the whole
Remark 4.
D+v
(7) is continuous
of a radially
whole function
Outline of the Liapunov stability theory in general
. Let
o =8+o*
. Then
last of
I. 3
31
Stability conditions
(1966)],
[Halanay
1 +~a*
-
i+o*
1
+so
[1-
l+a
(~-1)
~
)1
> 0
so that v(t+O,x*)
=
¢ [~x(t+o;t,X)~] v(t,x) [1 -
1 +~a ~
(~-1)
( V t o e T T)
, Vt•T
0 +~a)
0
(1 + ~ * ) ( 1 + a o )
VtoeR
(~-i) (1 + o * ) ( 1
[1-
]
o ,
or v(t+0,X*) - v ( t ~ X ) 0 This i n e q u a l i t y
~ _
(s-l) v(t,x) (l+a*)(l+sa* +~0)
in limit as
D*v(t,x)
0+0 +
~ -~(x)
VtoeR
(VtoeTr)
'
, VteYo
takes the form
, V(t,x) • R × S
(V(t,x) • Y
xS)
where ~(x)
is
continuous
function,
ly attractive Hence,
@
(a-l) ¢ ( l l x ) l ) [i + T(A)][I +~T(A)]
:
T(A) e [ru(A,v),+~
[Halanay
'
[ , because
x:0
is u n i f o r m -
(1966)].
is positive
definite
on
S
(TTXS)
Sufficiency. U n d e r the c o n d i t i o n s of T h e o r e m 7 all c o n d i t i o n s of T h e o rem 5 are fulfilled. Hence,
x=0
Its u n i f o r m a t t r a c t i o n
lr ) is p r o v e d as follows.
Let
~
be such that
(on
8~c_S
of (7) is u n i f o r m l y
. Let
¢i and ¢2 ~ K [ 0 , ~ [
¢l([IXll) ~ v ( t , × ) _<¢2(IIXli) , V(t,X) E R x B ~ Let
I A : ¢2 [¢i (~)]
As shown in the p r o o f of the s u f f i c i e n c y tions
(13) and
(14) g u a r a n t e e
that
v [ t , x ( t ; t o , × o ) ] _< V(to,Xo) and that
v
is d e c r e a s i n g
in
t
Obviously
~ >0
. If
u >0
obey
(V(t,x) e T r x B ~ )
part of T h e o r e m
llXoU < A
Yr )"
(12) (14)
5, the c o n d i -
implies
, VteT o , VtoeR
: ~ , VtoER
X
(VtoeYr)
,
of (7). Let
( V t o ~ Y r)
, Ilxo~ < ~ .
(15)
then D*v [ t , x ( t ; t o , X o ) ]
where 7 :
(on
"
along motions
inf { v [ t , X ( t ; t o , X o ) ] : t e T o }
stable
inf{@(X)
< -7
xeS\Bp,p : ¢ I ( v ) }
:
Therefore, v [t,X(t;to,×o)]
<- V ( t o , X o ) - 7 ( t - t
o)
,
so that V(to,Xo)
v [t,x(t;to,Xo) ] < v which contradicts
(15). Hence,
for ~=0
t • ]
7
- v
+t° '+= [ '
w h i c h t o g e t h e r with (13),(15) and posi-
32
tive d e f i n i t e n e s s
of
Ch. I
Outline of the Liapunov stability theory in general
v
S
on
TrxS)
(on
prove that
lira [llx(t;to,Xo)il : t + + ~] = 0 , V t o E R i.e.
that
and/or
x=0
is a t t r a c t i v e .
Let
~ <+oo , and
p >0
(VtoeYr)
,
be a r b i t r a r y ,
:
Then,
7 (P ) D*v(t,X)
and for ,
<-7
, g(t,X) e R x ( S \ 8 $ )
t = ru(A,p ) + t o , ru(A,p) > 0 grey o , which happens
only
(g(t,x) 6 Y r x ( S \ B ~ )
, [ru(a,p) =0
for
) ,
implies
rlx(t;to;Xo)ll
A = ~ =+= ] ,
v [ t , x ( t ; t o , X o ) ] g v ( t o , X o) - 7 ( t - t o) _< @2(A) - ¢ 2 (A) + ¢ i ( ~ ) so that ¢l[llx(t;to,Xo)ll] -< el(P) , V t o e R ( V t o e Y r) , yields }JX(t,to,Xo)ll
implies
¢ 2 ( A ) _~i(~ )
ru ( a , O )
< #
llxoll< A
< @l(p)
,
t e]ru(ZX,p),+~[ v [t,X(t;to,Xo)]
< v [to+ru(A,p) ; X ( t o + r u ( A , p ) , t o , X o ) ]
K el(P)
SO that llx(t;to,Xo)ll < P which proves Following
,
Vt e ] t o + r U ( A , P ) , + ~ [
that a t t r a c t i o n
Barbashin
in w h i c h we choose
T h e o r e m 8. tence
If
X
of radially
function
v
such that,
(on
is
continuous
(on
, VXoeB A
Yr ) .
•
and the p r e c e d i n g p r o o f
YoXRXR n
(on
positive
YoXYrXR n )
definite
definite function
then
exis-
in the whole
~ in the whole
respectively, , g ( t , x ) ERxRn
x=O
of the system
Krasovskii
(1959)
If
x
and u t i l i z i n g
is continuous
real numbers
v(t,x)E C(ToXN )
r/.l_llx}l p - < v ( t , x )
,
stability
in the
(7).
9.
tence of a vime-invariant
( g ( t , X ) 6TrXR n)
for uniform asymptotic
Theorem
(i)
(1952,1954)
on
Tr ) and positive
7 it is easy to prove
tha~
is u n i f o r m
unbounded decrescent
Theorem
positive
(Vto~/r)
it is easy to p r o v e
is necessary and sufficient
Following
x=0
and K r a s o v s k i i ¢ eKR
D * v ( t , X ) <-@(X)
whole of
of
, VtoeR
on
ToXRXN
neighbourhood
~I,~2
and
and both,
-<~2nxllP
¢(~) : ~ P
~3
(on
S of
in the p r o o f of
ToXTrXN ) then exis-
x:O
, a function
and a positive
integer
respectively,
, V(t,x)
6RxS
(V(t,x)
ETrXS ) ,
v,
p such
I. 3
Stability conditions
33
and
(ii)
D * v ( t , X ) ~ -N3HxlIP , V ( t , x ) eRxS
is necessary and sufficient of the system
Theorem 10,
( V ( t , x ) eT~xS)
for exponentia~
stability
(on
T r ) of
x=O
(7). If
x is
c o n t i n u o u s on
To XR×Rn
ToXTrxR n )
(on
then
existence of a function v, positive real numbers ~1,~2 and ~ 3 and a positive integer p such that v(t,x) 6 C ( T o × R n) and both, respectively,
(i)
nlUxll p ~ v ( t , x ) ~ n211xllp , V ( t , x ) e RxRn
( V ( t , x ) e TrxR n) ,
and
(~i)
D * v ( t , x ) ~ -~3Ilxll p , V ( t , x ) •RxR n
is necessary TT )
Remark
and sufficient
x=O
of 6.
f
If
is d i f f e r e n t i a b l e
sets u s e d
In the case
principle
[see L a S a l l e
Stability
The n a t u r a l s h o u l d be Let
sets
on w h i c h
time-varying
dynamical
set of
S(t) ~ S
d /x,S(t)] Let the distance example
Lee
between
and M a r k u s
d [Sl(t),S2(t)]
(10)
properties
the m o s t
powerful invariance
of t i m e - v a r y i n g
. A mapping
teR
then
is
is a set-
S : R~P
S(t)
that
will
i~ is t i m e - i n v a r i a n t
x from
S(t)
be d e f i n e d
be c a l l e d
set. by
: inf [llx-Yll : y e S ( t ) ]
and
Sz(t)
(1967)]
S2(t)
be d e f i n e d
by
[see for
:
=
= max {sup (d [ Y , S 2 ( t ) ] : Y e S l ( t )
) , sup ( d [ y , S l ( t ) ]
:YeS2(t))}
Definition 17. (i) A set
i c Rn
is a limit set of
lira [S(t) : t ~ T ] (ii)
Set
valued
systems
sets.
Rn at
distance of
- the
system
on the
10.
on t i m e - v a r y i n 9 s e t s
Its set v a l u e
set. I f f
d/x,S(t)]
is d i f f e r e n t i a b l e
is a c h i e v e d by t h e L a S a l l e
are t i m e - v a r y i n g
P = 2 Rn be the p o w e r
Let
: L , iff
function
lira [S(t) : t ~ T ]
(on
(1976)].
conditions
studied
valued function.
method
v
5 -Theorem
of stationary
of the L i a p u n o v
1.3.2.5.
then
in T h e o r e m
extension
stability in the whole
(7).
of the system
corresponding
Remark 7.
for exponential
( V ( t , x ) elTxRn )
S
= S(T)
S , S : R~P
lira {d[i,S(t)]
is continuous
at
, at : t~T}
t: r
t :T , : 0
iff
34
Ch. I
(iii) Set valued function ous at every The set
teR
{x :x-0}
Definition
S
contracts
Te~
set
TT ) iff it is continu-
S(t) eC(R)
(S(t) eC(TT )) .m
¢ .
S(t)
to
(on
te/r ) ,
will be denoted by
(or asymptotically (i) there is
is continuous
(at every
Time-varying
18.
every
Outline of the Liapunev stability theory in general
is asymptotically
contractive
0 ) iff both
such that
S(t)
is a neighbourhood
of
x:0
for
t~Y T
and (ii) the closure
S(t)
This is asymptotic Def~nltlon
19.
for every
S(t)
contraction
A set
(i) positiveZy
of
S(t)
obeys
of
is
lim IS(t) :t-*+~] = 0 . •
S(t)
as
:
invariant with respect toeR
t~+ ~ .
(toeYT)
to function
there is
v
(on
~o: ~ ( t o ) e R
TT ) iff
such that both
V~o(to) c_S(to)
a) and b)
V~o(t ) c_s(t) , V t e Y °
(ii) invariant with respect that
V~(t) $S(t)
The notions
;
to function
, VtER
.
v
of asymptotical, ly contractive
ant sets with respect
iff there is
~6R
such
•
sets and
(positively)
to functions were discover@d by Gruji6
invari-
(1975).
Their importance was shown for stability analysis of non-stationary systems. then
N
Theorem and
Let
:
N : {(t;x) : t e R ,
is replaced by 11.
N(t)
If
X
×eN(t)}
is continuous (on
in
and
t6R
~K
(tEY~)
(i) (ii) (iii)
is replaced by
(t,to,X) e / o X N
T T ) neighbourhood
( t e Y T ) , which may be time-invariant, V
R
l~
NT
i8 continuous
either asymptotically
If
contractive
((t,to,X) e Y o X N T)
of
x:O
a~ every
teR
N(t) --N , or time-varying
or not,
then existence
, and continuous neighbourhood
S(t)
of
of functions
X:0
at every
such that, respectively,
v(t,x) eC(S) v(t,0) =0
(v(t,x) eC(ST))
, VteR
(VteT~)
,
,
v(t,x) ~¢(llxll) , V(t,x) e S
(V(t,x) e S z )
,
and (iv)
D+v(t,X) < 0
, V(t,x) e S
is necessary for ~tability
P~OOf.
Let
x=0
be arbitrarily
(on
(V(t,×) e S r ) Y~ ) of
of (7) be stable
chosen.
Let
V(to)
(on
X=0 T T ),
, of the system e>0
and
(7).
toUR
( t o 6 Y T)
be the largest positive number obey-
1.3
stability conditions
ing By(to) _cN(to) of x in all its teR
(teYT)
continuity in
teR
( t ~ Y 7)
~(t)
¢
by
in
(Ii) and , and
YT ) implies
e>0
teR
for every
for a r b i t r a r y
((to,X o) e S r ) (on
and ~(to,e) : min {V(to) , ~M(to,e)} Continuity a r g u m e n t s ~mplies c o n t i n u i t y of ~M(t,e) in
for every f i x e d of
(tET~)
for such
35
. Continuity
(teYT)
fixed
e >0
. Let
S(t) = B~(t,e)
fixed ~K
e>0
X(t;to,0) :0
N(t)
((t,x) e S r )
satisfied
due to
, 0 • ]0,+~[
v
be d e f i n e d
that is c o n t i n u o u s of
X
in
(i) of the Theorem. , VteY o , YtoER
(il).
is c o n t i n u o u s
and
Let
in
teR
(to,Xo) e $
Stability
of
x=0
( V t o e Y ~) . H e n c e ,
of the T h e o r e m h o l d s i n view of (11). The c o n d i t i o n
is o b v i o u s l y
guarantees
~(t,e)
CeK
. Continuity
imply
of
. Hence,
(ii)
(iii) of the T h e o r e m
X* =X(t+9;t,X)
, (t,x) e S
. Then,
v(t+0 ;X ~) = sup {¢ [IIx(t+e+e;t+@,x*)ll] : o e [0,+~[} = sup {~ [IIx(t+0+o;t,×)II] : o e [0,+~[} -< sup {~ [Nx(t+ol;t,x)II] : O l e
[0,+~[}
: v(t,x) Hence, v [t+0 , X ( t + 8 ; t , X ) ] - v ( t , X ) 0
_< 0
V(t,X) 6 S
Taking right u p p e r limit of this i n e q u a l i t y The c o n d i t i o n s
conditions
is a s y m p t o t i c a l l y
ally c o n t r a c t i v e s t a b i l i t y on such that
we get ( i v ) o f t h e T h e o r e m .
of this t h e o r e m are only n e c e s s a r y
The f o r m of s u f f i c i e n t S(t)
(V(t,x) 6 S ~ )
essentially
contractive,
either as
t ~-~
or not.
or
depends
, Vt e [-~,+~]
on that w h e t h e r
If it is not a s y m p t o t i c -
t ~+~
(only as
Y T ) then there is a t i m e - i n v a r i a n t
Sc_s(t)
( V t e [T,+~])
t ~+~
neighbourhood S of × = 0
that are also n e c e s s a r y w e r e e s t a b l i s h e d
Liapunov
In o r d e r to clarify
(1892).
P r o p o s i t i o n 14.
If
it is i n v a r i a n t definite
The same
is true
ally
for
as
tive d e f i n i t e
and
the cause of this p h e n o m e n o n
we
Gruji~
(1975).
continuous
at every and
tER
of
( t E T T) (as
x:O
of
respect
invariant x:O
set
and
t ~÷~
neighbourhood
For example, S2(t)
T~ ) with
time-varying
t ~+~
contractive
Sl(t)
on
nei~bourhood
then
to every
Ir ) , respectively.
nor p o s i t i v e l y
function.
Sl(t)
by
invariant (on
t ~-~
asymptot~cally
neither i n v a r i a n t bourhoods
every x:O
of
contractive
However,
function
suffi-
already
is a t i m e - i n v a r i a n t
S
(positively
positive
a neighbourhood
by f o l l o w i n g
for
. In such cases
cient c o n d i t i o n s
state the next result
•
and not s u f f i c i e n t .
,
: {x : IlxII < e x p ( - t ) }
that
is
i8 not a s y m p t o t i c -
) , respectively.
S(t)
with respect
asymptotically
S(t)
of
x:O
is
to every posi-
contractive
neigh-
36
ch. I
Outline of the Liapunov stability theory in general
and S2(t)
= {x : ~ aiixil < ( l + t 2 ) -I • a i @ i:l
are not p o s i t i v e l y tions
vI
and
invariant
v2 ,
v1(x) : xTHx
v2(t,X) : (2 + sint) xTHx Rgmark
8.
If
f
is c o n t i n u o u s
tively.
If
sufficient of
x=0
Example
stability
N(t)
(on
of
(7)
T~ ) of
x=0
v(t•x)
~(t,x)
(Gruji6•
The
S(t) set
S(t)
contractive.
not p o s i t i v e l y
of
(1977)]
to the f o l l o w i n g
(i)
is p o s i t i v e
If there
(t,X) • T r × N
of
¢l(nxll)
v(t,x)
m
~(t,x)
~-¢2(IlxU)
Tr )
definite
x=0
at every
teR
and
on
S(t)
• VteR
for all
teR
state
to the f u n c t i o n
for u n d e r s t a n d i n g
is a t i m e - i n v a r i a n t
for some
both
(on
• V(t,x) 6 R × R n
T eR
and
and p o s i t i v e x=0
S(t)
, i=1,2
is
v.
Habets
neighbourhood
¢ieK
of ~he sys-
the set
the c o u n t e r e x a m p l e
[see also p.32 in the b o o k by Rouche•
and (ii)
v
with respect
(19~9)
all
which
The cause of this is that
7 can be also h e l p f u l
I.
of d i f f e r e n t
stability
then
v , the e q u i l i b r i u m
Massera
Hypothesis
a number
(i+ e x p t ) -l +IIXI121 X
neighbourhood
Example
v(t•x) • C(I'I)(TrxN)
(7), r e s p e c -
Obviously•
of
unstable.
invariant
,
1975).
< 0 , V(X~0) •S(t)
this n e g a t i v e n e s s
t e m is e v i d e n t l y
N(t)
(teTr)
but not s u f f i c i e n t
for a s y m p t o t i c
• [(l+expt) -I -ilx~ 2] llxll2
is c o n t i n u o u s
d e f i n i t e n e s s in the w h o l e
• and
teT
= {x : llXl[2 < (i + e x p t ) -I}
~(t,x)
Despite
NT)
of the s y s t e m
is chosen•
in the whole•
= - [I + e x p ( - t ) ]
asymptotically
or
Let
= [I + exp(-t)]211xll 2
Let
func-
definite,
at every
contractive
d x = ! [ I + exp(-t)1 -I [2 e x p ( - t ) dt 2 If
((t,x) e
x=0
11 are also n e c e s s a r y
can be o b t a i n e d
(Gruji6, 1975).
and d i f f e r e n t i a b l e
definite
is p o s i t i v e
(t•x) e N
is a s y m p t o t i c a l l y
of the s y s t e m 7
in
of T h e o r e m
conditions
to p o s i t i v e
, H =H T
Tr ) neighbourhood
then the c o n d i t i o n s
• i:i,2•...•n}
• etc.
is c o n t i n u o u s
(on
for a s y m p t o t i c
with r e s p e c t
]0•+~[
by
and L a l o y
N
of
, such that
x=0 for
•
I. 3
Stability conditions
then
x:0
of the
The h y p o t h e s i s ~> 0
the
is
is not
(on
has
Y~ )
decrescent
such that
v
~S(t)
may
such
Tr
•
that
that
for any
should
be
is the boundary f u n c t i o n of a both
at every
:~
be
on
contractive
respectively,
w(t,x)
stable
function.
w : RxR n ~ R n
iff,
the b o u n d a r y
~6R
function
asymptotically
A function
S(t) a R n
(i) S(t)
the
is not
v
20.
(7) is a s y m p t o t i c a l l y
because
g~(t)
because
Definition
set
system
fails
set
required
37
teT
, VX • aS(t)
(teTr)
, VtER
, and
there
(Vt•TT)
,
and (ii)
for e v e r y
teR
(t6Sr)
Vx•N[~(t),aS(t)]\ bourhood
of
Def~nlti0n
21.
on
(on
aS(t)
~(t) > 0
~S(t) The
w[t,(I+XI)X] The next
boundary
that
result
~(t) > 0
is
where
such
N[~(t),aS(t)]
that is
w(t,x)
~(t)
~}
,
neigh-
•
aS(t) × T T )
such
, there
~S(t)
function
iff for
X e ~2(t)
is radially
S(t) t•R
(t•T~)
increasing
, there
is
implies
< ~ < w[t,(I+X2)X] generalizes
of
every
Lemma
, VIXII
, ~2•
]0,~(t)[
2.1 by H a r r i s o n
(1979)
, ~i < 0
•
to t i m e - v a r y -
ing sets.
1.
Lemma
If
loXRXR n
is continuous (on l~ ) and , then, respectively, both
S(t)
( T o × T T x R n)
the boundary function
(i)
~S(t)
(on
w
~S(t) x lr)
of
×
is continuous on
is radially
S(t)
increasing on
,
and (ii)
D+w(t,X)<0
, V×eSS(t)
S(t)
are ~ufficient for
, Vt•R
(vt•TT)
to be invariant
with respect to motions of the system Proo~.
Let all
(o,y) • R S(t), in
. Hence,
, where
teR
of the
~(e,y)
~ • ]o,T[
exists
(positively invariant on
hold
and =~
and
which
in limit
dicts
the
as
O ~ O+
condition
that of
let
such
there that
~[t,x(t;a,y)]
because
> 0
0
point
and
7>0
X
and
S(t)
exist X(t;s,y) >~
,
are
continu-
(teYT) . Hence,
~[_ ~ + O ; X ( o + O ; o , y ) ] - w ( ~ y )
Notice
Yr )
(7).
lemma
((a,y) • T T × aS(o))
Vt • ]a,7[
Vt • ] G , v [ ous
conditions
×aS(~)
,
~(t,z)>
S(t)
and
(ii)
implies of Lemma.
~(t,x) x
VOe]O,v-o[ J
D+~(o,y) The
is a d m i s s i b l e
is its b o u n d a r y
>0
. This
contradiction even
point.
z
result proves
is the
contrathe
Lemma.
interior
•
38
Ch. I
Theorem 12.
Let
×
be continuous
be continuous
(i) S(t)
Outline of the Liapunov stability theory in general
on
ToxRxR n
neighbourhood
of
(ToXTrxR n)
and
at every
tER
x=0
(tETT) ;
be positive definite on U [S(t) : t E R ] ( U [S(t) ."t a T T] ) ; the boundary function o~ of S(t) be radially increasing on
(ii) v (iii)
(on
aS(t)
, VxeS(t)
, VteR
Then,
, vxeaS(t) be asymptotically
respectively,
(a) the conditions
system
(i)-(v)
(7) (on
tion of
(i),
(iii),
(v) and
of the system
x:0
(c) the conditions
(i)-(vi)
of the system
x=O
are sufficient
for stability
of
of the
x=0
T T );
(b) the conditions
Proo
( v t e T r) ;
, v t c R (Vt~T T) ; contractive.
(v) D + ~ ( t , X ) < 0 (vi) S(t)
is admissible;
aS(t) x TT ) " w =v
(iv) O + v ( t , x ) ~ 0
(7)
(vi) are sufficient
(on
T r );
are sufficient
(F) (on
for attrac-
for asymptotic
stability
of
T~ ).
~.
A. The conditions tively
(i),
invariant
(iii)
(on
and
(v) prove
Yr ) with respect
X(to) ~ S ( t o) ~ x ( t ; t o , X o ) 6 S ( t ) due to L e m m a i. B. The c o n d i t i o n
(vi) means
C. A and B prove
(b).
D. Let
toeR
Let
( t o e Y 7) , and
~(to,e) e ]0,+~[
Existence (ii))
, vt 6 [to,+~[ lim
e>0
be a r b i t r a r i l y
number
~(to,e )
of
S(t)
of the
( v t o e Y T)
(7), ,
= 0 .
chosen.
obeying
definiteness
BE
is posi-
system
F~(to,e)(t)c_ B e , v t e
by both the p o s i t i v e
and t i m e - i n v a r i a n c e
set
, VtoeR
[S(t) :t-~+~]
be such that
of the p o s i t i v e
tion is a s s u r e d tion
that
that the to motions
[to;+~[
the last of
(Proposition
v
.
condi-
(the condi-
14).
Let ~(to,e) e ]0,+~[ be chosen so that B~(to,e ) c Fo~ ( t)o _e)(t The c o n d i t i o n (iv) and the p o s i t i v e i n v a r i a n c e of S(t) (A) now prove
that
Xo e B~(to ,e)
implies
X ( t ; t o , X o ) E V ~ ( t o , e)(t)c- B e which
proves
stability
E. The a s s e r t i o n proof. Example
(on
(c) is i m p l i e d
• Vte/o
T T ) of
x=0
, VtoeR
( V t o e Y ~)
. Hence,
(a) holds.
by (a) and
(b), which
n-th
system be d e f i n e d
,
completes
the
•
8 (Gruji6,
1975). dx _
Let the f u n c t i o n
Let
X + (4 + 2 t
dt v be a c c e p t e d
order
+t2)j[xjI2
in the form
X
by
.
v(X) = l[Xll2
so that
it is
1.3
Stability conditions
positive
definite
39
in the whole and ~(X)
This result
suggests S(t)
The b o u n d a r y aS(t) w
aS(t)
of
8S(t)
so that on
2]
should b e in the form
S(t)
S(t)
is now easily found,
-- {x : JJxlJ2:~(4 + 2 t + t 2 ) -I} function
,
of the set
(~(t,×) : (4 + 2 t +t2)ilXl[ 2
S(t)
. It is r a d i a l l y
increas-
. Furthermore, ~(t,X)
:
aS(t)
~(t,x) For
that
+t2)llxll
= (X : llxl12< a ( 4 + 2 t + t 2 ) -I } , ~ e ] O , l ]
is the b o u n d a r y
ing on
: -211X112[1 - ( 4 + 2 t
-211 xll 2 [ 3 + t + t 2 - (4 + 2 t + t 2 ) l l X l l 2 ]
,
,
= + 2 ~ ( 4 + 2 t + t 2 ) - i [(~-i) t 2 + (2~-i) t + (4~-3)]
~(t,x) <0
on
aS(t)
it is s u f f i c i e n t
that
~ : 0.70 , w h i c h
implies also ~(t,x)
<0
, V(Xf0)~S(t)
, VteR
,
and S(t)
Obviously,
the set
: { x : llxll 2 -< 0.70 (4 + 2 t + t 2 ) - I }
S(t)
is c o n t i n u o u s
and
lira [S(t) : t ~ + ~ ] All c o n d i t i o n s
of T h e o r e m
tem is a s y m p t o t i c a l l y Theorem 13. (i)
S(t)
(ii) v
(iv) S(t)
The
ToXRXRn X=0
be oontinuo~(s neighbourhood of
be positive
(iii) S(t)
12 are satisfied.
state
x=0
of the sys-
stable.
X be continuous on
Let
: 0 .
definite
be positively
on
U[S(t)
: t6R]
invariant with respect
be asymptotically
(ToXTTxRn)
and t6R
at every
( t E T T) ;
(U IS(t) : t E T T ] )
to
v
(on
;
T T );
contractive
and (v) O + v ( t , X ) - < 0
, VxeS(t)
, VteR
(Vtel~)
Then, respectively~ (a) the conditions X:0
(on TT (on TT
and [v) are sufficient
for stability
(i)-(v) are sufficient for asymptotic
stability of
) .
P~oo ~.
A. The c o n d i t i o n
of
) ;
(b) the conditions X:0
(i)-(iii)
(iii) i m p l i e s
{/~(to)(to ) c S ( t o )
implies
existence
of
~(to) > 0
such that
40
Ch. I
Outline of the Liapunov stability theory in general
V~(to)(t ) c_S(t) This r e s u l t t o g e t h e r w i t h
, VtcT ° , VtoeR (ii) and
(VtoETT)
(v) g u a r a n t e e s
that
x o • V~(to)
implies ×(t;to,Xo) •V~(t) B. Let
toeR
Let
, VteT o , VtoeR
( t o e T T) , and
~(to) e ]0,[(to) [
Such
~(to)
exists
definiteness Let
of
v
be a r b i t r a r i l y
be such that
(Proposition
, V~ e ]0,~(to)[
.
chosen.
V~(to)(t) c_B e , V t e T o
due to t i m e - i n v a r i a n c e
~ ( t o , e ) e ]0,+~[
x° e B B ( t o , e )
e >0
( V t o e T ~)
of
Be
and p o s i t i v e
14).
be such that
BT(to,e ) cV~(to)(to)
. Hence,
implies X ( t ; t o , X o) • B e , V t e T o •
due to the r e s u l t under A• w h i c h p r o v e s C. Let
A(to) e ] 0 , + ~ [
The c o n d i t i o n s lim
obey
(i)-(v)
guarantee
[×(t;to,X o) : t - ~ + ~ ]
that
Example
9
(a) p r o v e s
(Gruji6, 1975).
, ~e]0,~(to)[
x o e BA(to )
= 0 , VtoeH
due to the result under A. Hence, which together with
(a).
BA(to) c V~(to)
X=0
implies
( V t o e T T)
is a t t r a c t i v e
• (on
T~ ),
(b). •
A positive
definite
function
v
in the whole,
w h i c h is not d e c r e s c e n t , 2 (i + e x p t ) l l x a 9
v(t•x)
:
- 2 (l+expt)IIxll4 + ( ] + 2 n xll 2
is c h o s e n
, Irxll2 _< [2 (i + e x p t ) ]
,
~ xll 2
for s t a b i l i t y
dx = i dt - {~[1+2
-I
expt) IIxll2 • [ 2 ( i + e x p t ) ] -I _ <~xll 2 _< i
_> 1
analysis
of the
n-th
order system
exp(-t)] + 2 e x p ( - 2 t ) } × + [i+ 2 e x p ( - t ) ] 2 IIxll 2 x .
Hence • v(t•x)
= - 2 {l + q exp(-t) [i + e x p ( t ) ] [ l + 2 e x p ( - t ) ] [ l -
T h i s result so that
suggests
S(t)
is a s y m p t o t i c a l l y
lira [S(t) : t ~ + ~ ] The set because
S(t)
the choice of
contractive,
T ER
invariant
, the choice
~M(r ) _ 2 +2 expT 2 +exp T
0cS(t)
with respect
, VteR
~ •]0,~M(r)]
imply V[(t)
, and
to the f u n c t i o n
of and
.
, S(t) = {x : IIxll2 < (2 + expt) -I}
= 0 .
is p o s i t i v e l y
for every
S(t)
(2 + expt) llXll2]}
= {x : llxl12< ~[2 (1 +expt)]-l}_c S(t)
, VtETT
.
v
1.3
Stability conditions
All c o n d i t i o n s
of T h e o r e m
is a s y m p t o t i c a l l y £xamp]e
41
13 are
satisfied.
Hence,
x:O
of the
stable.
10 (8ruJi6,
1975).
The
n-th
order
linear
system
is d e s c r i b e d
by dx dt Let
_
2-1(~ + 2 7 t ) ( ~
function
Its total
v
time
n
Let
~ E ]0,+~[
V[(t)
and
with
S(t)
respect
S(t)
for any
~ • ]0,+~[
conditions
that
on
continuity,
of T h e o r e m
In the
motions
v
respect
the next preceding
that
v
v(t,X) = 0 ,
, vte
]-~,+~[
$ = +~
S(t)
The
contractive
, vteR
state
]0,~]
.
x=0
of the
sys-
follows
application
of H y p o t h e s i s
with
then
invariance
v.
Asymptotic
contract/on
respect
to
v
prove
asymptotically
S(t)
from T h e o r e m
of the of
the
of set
S(t) set
contractive.
S(t) - R n , but then set
that
weaker
because
(positive)
as a new
directly
13 under
is p o s s i b l e
and
function
, is also
of T h e o r e m
1, w h i c h
the
set
of T h e o r e m 15 by
V[(t) 15.
relying
,
In fact,
on the
explanation.
Corollary 2 (Barbashin and Krasovskii, 1952; Gruji6, 1975). exist a time i n v a r i a n t function
set
, V[e
and a s y m p t o t i c a l l y
v(t,x) : 0 , Vx e S ( t )
13 are f u l f i l l e d .
, can be a c c e p t e d result
continuous
invariance , $<+~
that
. The
because
contraction
to the
(positive)
case
obeys
= {X : UXU2< ~ (a + ~ t + ~ t 2 ) -I} to
10 i l l u s t r a t e s
than
, ~ e ]0,$]
[ e ]0,+~[
It is not d e c r e s c e n t .
stable.
asymptotic
with
and its V[(t)
in the whole.
system
Besides,
Example
condition
S(t)
along
is o b v i o u s l y
tem is a s y m p t o t i c a l l y Note
definite
= {x : llxll2< [(~ + # t + 7 t 2 ) -I} c_ S(t)
The set
All
. Evidently,
.
VXER
is i n v a r i a n t
; a,7 e R +
_> (4~7-t32)(4"'/)-111xll 2
ls p o s i t i v e derivative
, VteR
o
, 4a7 >~2
+~t +Tt2)-ix
v(t,×) : (a + ~ t + 7t2)~×II 2 v(t,x)
The
system
v
on
neighbourhood
N a n d a real n u m b e r
N of
X:O , a p o s i t i v e
~ E ]0,+~[
If there definite
~uch that both
J
and (ii)
D+v(t,x)
_<0 , V(t,x) e R x N
then the state
x=0
(V(t,x)eY~xN)
of the system
,
(Z) is a s y m p t o t i o a Z ~ i
stable
(on
Tf ) , r e s p e e t i v e Z y . If, in addition, of the s y s t e m
N=R n
(Z) ~
and
v
is r a d i a ~ y
a~ymptotioa~ly
stable
unbounded
(on
in the whole
Y~ ) then (on
Y~ ) .
x:0
42
Ch. I
Outline of the L i a p u n o v
Notice that in the case when orem 42.3 by Hahn ×:0
N:R n
stability
this corollary corresponds
(1967), which shows that asymptotic
is then equi-asymptotic
theory in general
to The-
stability of
stability of it. This assertion
can be
easily broadened to Corollary 2.
1.4. ON ABSOLUTE STABILITY 1.4.1. Introductory comments The absolute
stability problem has been one of the central problems
the stability theory Lur'e and Postnikov lic servo-systems system,
for its theoretical (1944)
and engineering importance.
showed that mathematical
had a particular
models of hydrau-
form referred to as Lur'e-Postnikov
or in short as Lur'e system.
has also become the classical
of
The absolute
stability problem
problem of the control
theory and con-
trol engineering. Since 1944 the problem has been attacked via various approaches. of them were aimed to establish
sufficient
conditions
stability,
which was f~rstly done by Lur'e himself
conditions
are purely algebraic
of solutions obstacle Elegant 1964,
of non-linear
equationB,
of existence
which has been the key
for their effective application. frequency
1973,
research
for absolute
(1951). The Lur'e
and require verification
algebraic
Most
criteria by Popov
(1959,
1960,
1974) have become most applicable
1961,
1962,
1963,
and fruitful for further
in various directions.
The ingenious approach by Popov inspired a great number of scientists for its extensions,
whose contributions
The crucial extensions Yakubovich (1962,
(1962,
1963,
i963,
and Rekasius and Lee
1964,
1964), Kalman
tant contributions
Sun (1971, Piatnitskii
1967,
(1963) and Sz6ge
(1966), Dumkov
(1969), Bertoni,
(1973). Excellent
together with significant Aizerman and Gantmacher
to in brief.
(1963,
i975), Tsypkin
1964).
(1964), Desoer
surveys
(1972),
(1968),
Par-
Garg and Rabins
(1972) and
of these and other contributions
original results (1968),
(1966), Meyer
(1970), Siljak and
can be found in the works by
(1963), Gantmacher and Yakubovich
Piatnitskii
Ibrahim
(1965), Jury
(1965), Dewey and Jury
Bonivento and Sarti
Other impor-
(1964),
(1967), Moore and Anderson
1972), Anderson and Moore
(1965),
1968, 1970,
are due to Gelig (1964), Halanay
(1965), Tokumaru and Saito
tovi and Nahi
schetz
1965,
(1964), Naumov and Tsypkin
(1965), Anderson
will be referred
of the Popov approach were at first achieved by
Siljak
(1969,
(1965), Lef-
1974) and Narendra and
1.4. Absolute stability
Taylor
43
(1973).
The problem of the necessary stability in particular (1969), Piatnitskii Serikbaev
and sufficient
conditions
cases was solved by Nelepin
(1970,
1971), Maigarin
for absolute
(1967),
Persidskii
(1970), Muhametz'anov
and
(1970).
The necessary and sufficient Liapunov-like stability were established show that a functional
by Gruji~
conditions
(1978a,b,
1980,
for absolute 1981b), which
family should be used rather than a single func-
tion whose form is not influenced by the form of system nonlinearities.
1.4.2. Description of Lur'e-Postnikov systems Non-stationary d_xx : dt
Lur'e-Postnikov A(t)
x+ B(t)
systems are described by
f(t,~)
Z :
C(t)
It is accepted that the matrix functions C e C(R,R mxn)
tion
, D • C(R,R
f E C(RxRm,R m)
However,
~xm
)
fined in the sequel.
Notice
of
, B • C ( R , R n×m) , The vector func-
form need not be known.
that and
N : RxR m ~ R mxm {¢l(t, Z) al
: diag
(16)
f are to be known. They will be de-
f : (~l,~2,...,@m) T
N(t,x;f)
A • C ( R , R nxn)
and that they are known.
and its real functional
certain properties
A matrix function
x+ D(t)f(t,E)
Z = (al,a2,...,om)
T
•
is associated with
¢2(t, N)
f by
Cm(t,2) }I
a2
,
Rm
aL-
,...,
:~(t,x,f)
In case
f(t,~) 6 C(I'J)(RxR m)
, j 6 {0,1}
Nt(t,x;f) and in case f~
f(t,Z) 6 c(J'I)(R×Rm)
elementwise
matrices ment of
N(t,x;f)
, j e {0,1}
is defined by
, then its Jacobian matrix
~a¢i(t,z)~ fz(t,~)
Let
= ~
, then
constant
:
matrices
L
]
J,L,M
N t , N and fE , respectively.
•
E Rmxm
be of
For example,
the
the
same t y p e
(i,j)-th
as
ele-
M is equal to zero iff a¢ i
aa. e 0 3
With the matrices respectively.
a
, L and
.
M will be associated
It is accepted that
(a=0) • ]
and
sets
J ,L
(M:0) • M
and
M ,
. These sets
44
O u t l i n e of the L i a p u n o v s t a b i l i t y t h e o r y in general
Ch. I
may be u n b o u n d e d . In case C(t)
Notice
that
f(t,Z) EC(I~I)(R×R m)
and
D(t)
x=0
h o l d s always.
then it is a c c e p t e d
are also d i f f e r e n t i a b l e
The v e c t o r n o n l i n e a r i t y (I)
MDL
f ENo(L )
is the unique
in
that
A(t) , B(t) ,
tER
iff
equilibrium
state of (16),
and (2)
N(t,x;f)
eL
Iff in a d d i t i o n
, V ( t , x ) eRxR n to
(i) and
(3)
f ( t , Z ) EC(I'I~R×R m) ,
(q)
Nz(t,~ ) eJ
X ~0
(2) the f u n c t i o n
f
obeys
, v ( t , Z ) ERxRm
and (5)
f z ( t , Z ) EM , V ( t , Z ) ERxRm
then of
f E N 2 ( J , L , ~ ) . Iff
j , JEJ
, and
are n e g a t i v e ) The sets
then
J ,L
J
M, MEM f
and
and
obeying
M
~
are such that
, can be a r b i t r a r i l y (i)-(5)
belongs
can be p r e s c r i b e d
some
large to
will be d e n o t e d
The m o t i o n of the s y s t e m
by
or to be d e t e r m i n e d .
the a r g u m e n t
f o r m of
f.
f
N2
that
is a s s u m e d
Usually
No(K) , NI(L)
and
respectively.
by
, X(to;to,×o;f)
emphasizes
Besides,x
and
(16) is d e n o t e d
x(t;to,×o;f) where
No , N I
elements
NI(L)
they are fixed and then for the sake of s i m p l i c i t y N2(J,L,M)
(or all)
(or small w h e n they
~ ×o
the form of
, X
depends
on the
c o n t i n u o u s i n all its a r g u m e n t s .
1.4.3. Definition of absolute s t a b i l i t y The concept
of a b s o l u t e
stability
was o r i g i n a l l y
related
ant L u r ' e - P o s t n i k o v
systems
and later on it was b r o a d e n e d
ing L u r ' e - P o s t n i k o v
systems
(16).
Deflnition 22. (i) absolutely stable
stable on
in the w h o l e
(ii) uniformly ically
The state
absolutely
stable
(iii) completely ly stable
X=0 Nj
~ j E {0,1,2}
for every
uniformly
(to,f) 6 R × N j
Nj
for every
absolutely
o
~ ER+
feNj
stable on
in the whole
and in a d d i t i o n
(a) for every
of the s y s t e m
there
(16)
to t i m e - i n v a r i to t i m e - v a r y -
is
, iff it is a s y m p t o t i c a l l y
; iff it is u n i f o r m l y feNj
stable on
aaymptot-
; Nj
iff it is a b s o l u t e -
iff b o t h is
, such that
6(e) > 0 , w h i c h H×oU < $ ( e )
is i n d e p e n d e n t
implies
of
1.4. Absolute stability
45
llx(t;to,Xo;f)ll
< e , VtET O , Vtoe R , VfeNj
,
and
(b) for e v e r y which
A E ]0,+~[
is i n d e p e n d e n t
JlXon < A
that c o m p l e t e
absolute
stability
nonlinearities
uniform absolute
,
, such that
stability
, Vtoe R
of
x:O
~ vfeNj
means
•
its
(to,Xo) e R × R n
in both i n i t i a l d a t a
both necessary
and s u f f i c i e n t
e v e n for a s y m p t o t i c
(7) in general, on
7(A,p) • [0,+~[
and
f e N . , j e {0,1,2} ]
been d i s c o v e r e d stability
is
< p , Vt • ] t o + f ( A , p ) , + ~ [
uniformly
Since L i a p u n o v - l i k e system
there
(to,Xo,f) • R x R n x N j
implies
ll×(t;to,Xo;f)ll
Notice
p > 0
and of
Nj
stability
then such c o n d i t i o n s
, j 6 {0,1,2}
, of
x=O
conditions
of
x=O
have not
of t i m e - v a r y i n g
are not k n o w n for a b s o l u t e
of the s y s t e m
(16).
1.4.4. Liapunov-like conditions for uniform absolute s t a b i l i t y v f : RxR n ~ R+
Let of
f.
be a f u n c t i o n w h o s e
functional
form
depends
on that
Then, Fj = {v f : v f ( t , x ) • c(i'i)(RxRn)} i:0 iff j=0
is a f u n c t i o n a l
family.
, i:l iff j • {1,2} Besides,
,
, j • {0,1,2}
the f o l l o w i n g
,
functional
f a m i l i e s will
be needed,
¢.]
:
{¢f : ¢f•KR
@j = {@f : 0 f E K
Theorem
14.
bility on tion
It is necessary Nj , j • {0,1,2)
~ •KR
(i) for every
v f • Fj
f •N i
(ii) for every
there is
Preof. Nj
Neoessity.
, j e {0,1,2}
6 ]1,+~[ fixed,
,
x
Let
. Let
×=0 feNj
arbitrary,
,
j •{0,I,2}
for uniform absolute
Fj
,
cf •~j
~ - $f(llx~)
~j
and
h
such that both
obeying , V(t,x)
v f E Fj
and
, V(t,x)
•RxR n ,
@f • %
satisfying
•RxR n
of (16) be uniformly absolutely be a r b i t r a r y xeR n ,
A
but fixed.
be a r b i t r a r y
in
s t a b l e on
@ e
Let
KR
: sup {@ [ l l X ( t + s ; t , x ; f ) l l ]
1l ++a~ o
;
]llxll,+~[ and
and
vf(t,x)
sta-
of (16) that there exist a funa-
x=O
~¢f(llxll)
there are
D*vf(t,X)
,
and sufficient families
~(llxn) ~vf(t,x)
and
j•{0,i,2}
,
, fENj}
, of
and functional
f•Nj)
'
. s e [o,rf(2a,
~1 a ) ] }
46
Ch. I
Outline of the Liapunov stability theory in general
Hence, ~ ¢(llXll)
vf(t,x) Following ately fied
now the n e c e s s i t y
follows for
that
the
~
is d e f i n e d
superscript to
f
in
Sufficiency. Let
conditions
=
proof
(i) and
and
of T h e o r e m
(ii)
~fe K
( ~ - 1 ) ¢(@xll) +a Tf(
and
~f
part
denotes
14 are
satis-
that
2A, ~1 a ) ]
of the p r o o f
that
7 it i m m e d i -
of T h e o r e m
such
[1 + rf(2a, ~1 a ) ] [ 1
in the n e c e s s i t y ~f
14, all
feNj
~f
and
of T h e o r e m ~f
7 and
are r e l a t e d
be a r b i t r a r y but f i x e d . Under the c o n d i t i o n s o f
conditions
is u n i f o r m l y
of T h e o r e m
asymptotically
j e {0,i,2}
•
The p r e c e d i n g
proof
sult
of the
feNj
Theorem (16)
part
@f(II×II) =¢[~f(IIxll)]
~f(~xtl) where
, V ( t , x ) ERxRn , v f E N j
and
(iii)
8 are
stable
fulfilled.
in the w h o l e
of D e f i n i t i o n
22 show
Hence,
X:0
for any
that
of
fENj
the next
,
re-
holds.
Theorem 15.
It is necessary
lute stability
on
there exist a functional i = 1,2
, of
family
Fj
for complete
uniform abso-
of the system
x=0
and functions
(16) that
¢i • ¢i E KR ,
@EK such that both
, and
Cz(~xlt) ~ v f ( t , x )
(i)
and sufficient
Nj , j 6 {0,1,2}
~ ¢2(ttx~)
, V ( t , x ) ERxRn , v v f e F j
•
and (ii)
D * v f ( t , X ) ~ -@(~xll) , V ( t , X ) E R x R n , v f e N j
Theorems
14,
studying
absolute
is not
15 show that
influenced
a functional
stability by the
rather
family
than
Fj
a single
is to be used function
form of the n o n l i n e a r i t y
v
for
whose
form
£6Nj
1.4.5. Criteria for absolute s t a b i l i t y of tlme-varying systems The
problem
of a l g o r i t h m i c
corresponding (16)
even when
Therefore, be found Theorem
the
under
requirements f
is fixed
form of which
%4 or T h e o r e m
of a f u n c t i o n
for a s t a b i l i t y and k n o w n
vf
it will
property
has not b e e n
can be a c c e p t e d
vf
obeying
of a g i v e n
solved
and t h e n
the
system
in general.
conditions
are to
satisfy
the
corresponding
conditions
of
family
Fj
is a Liapunov
functional
fam-
15.
Definiti0N
23.
ily on
, j ~ {0,1,2}
Nj
construction
A functional
, of the
system
(16)
iff it obeys
Theorem
14. •
1.4. Absolute stability
47
iY(t,x)
Let
vf(t,X)
= ( x T , f T) H ( t ) ( x T , f T ) T + 2 ~
f T ( t • Z ) ®dE
(17)
where H(t)
~ H(t) ~ C ( 1 ) ( R • R ( n ÷ m ) × ( n + m ) )
and 0 E R mxm
In order
to e s t a b l i s h
is a L i a p u n o v ing n o t a t i o n
family
be used
: [Im - D(t) L] -I C(t)
F(t,L)
= A(t) + B(t) LE(t,L)
=
[: ,o
n • ET(T,L) L
{01,02
under
which
N2
on
, ...,0 m }
F 2 = {v f : vf(17)
of the
system
(16)
, f ~ N 2}
the
follow-
:
E(t,L)
R(t,L)
@ : Niag
conditions
functional will
,
det [ I m- D(t) L] f 0 , v ( t , L ) ERxL ,
iff
,
I[i H(t)
n • 0 • LE(t,L)
7 I + £ T ( t , L) L O E ( t , L )
J
Q ( t , J , L , M ) = [ I m - D ( t ) M] - I { C ( t ) + C(t) A ( t ) + + [ D ( t ) + C(t) B(t) L + D(t) d] E ( t , L ) } iff
det [ I m - D ( t ) M] # 0 , V(t,M) ~RxM ,
T ( t , J , L , M ) = H(t) + F T ( t , L ) H(t) + N(t) F ( t , L ) + E T ( t , L ) L O Q ( t , J , L , M ) +QT(t,J,L,M) eLE(t,L) + £ ? ( t , L ) J ® E ( t , L ) Here•
Im
is
Theorem 16.
m×m
identity
L is open set then for functional family
If
F2 = { v f to be Liapunov functiona~ sufficient
:vf(17)
family on
• f e Ny(J•L•M)}
N2(J,L~ )
it is necessary
and
that
(i) the matrix function
(ii)
matrix.
R(t•L)
be positive
definite
on
R×L ,
sup { ~ R ( t , L ) l l : ( t , L ) ERxL} < +~ ,
and
( i i i ) the matrix f u n c t i o n Proof.
Necessity. Let
functional LeL
family
. Hence,
of T h e o r e m
of
to (16)
(ii)
o£ the
. Now,
system
f(t,E) = LZ
of D e f i n i t i o n
it follows
RxExLxM.
be negative d e f i n i t e on
F2 = {v f : v f ( 1 7 ) , f E N 2 ( / , L , M ) } N2(J,L,~)
feN2(],L,M)
16 due
Along motions
on
T(t•J,L,M)
for any
22 and
(16).
Let
implies
be a Liapunov f(t,~) = LE (i) and
(i) of T h e o r e m
f EN2(3•L•M ) •
(ii)
14.
•
48
Ch. I
Outline of the Liapunov stability theory in general
ddt v f ( t ' X ) Since that
L is open and for every
: vf(t'x'N'fz"
vf
is continuous in a l l arguments i t follows
(t,J,L,M) E R x ] x L x ~
d vf(t,x) dt This
result,
sity of
(ii)
(iii)
of T h e o r e m
of T h e o r e m
I ~TNtod~)
there
is
f E N2(],L,M )
x.
: xTT(t,J,L,M)
14 and
(ii)
such that
(18)
of D e f i n i t i o n
22 imply
neces-
16.
Sufficiency. The f o l l o w i n g are o b v i o u s l y true : (a) for every i = 1,2
(t,x,f) 6 R x R n x N 2 ( J , L , M )
there
are
Li = Li(t,x,f) e L ,
, such that
xTR(t,LI ) x ~ vf(t,x) and (b) f o r every
(t,x,f)
~ xTR(t,L2 ) x ,
ERxRnxN2(J,L,M)
there are
(J,L,M) eJxLxM
such
t h a t (18) holds. Now,
(i) and
(iii)
completes Ramark R×L
(ii)
of T h e o r e m
of T h e o r e m 16 t o g e t h e r
the proof.
9.
Notice
iff there
that
the m a t r i x
e>0
such that
is
definite
Rzmark
Conditions
I0.
cient
In case
for u n i f o r m
Example
for every
where
now
L
absolute 1981b).
different
~2 The v e c t o r
imply
(i) of T h e o r e m (ii)
of T h e o r e m
function
is p o s i t i v e
14,
and
]4, which
nonlinearity
the m a t r i x
R
definite
on
[RT(t,L) + R(t,k) - e l n]
(t,L) 6 R × L
of T h e o r e m
that
ii (Gruji6,
tem possess
(a)
(b) implies
•
is p o s i t i v e
conditions.
16 and with
16 are p u r e l y
is compact stability Let the
time-varying
LO.O] c o s t , f = (~1~2)
then
on
N2
second
terms
linear
these of
system
conditions ×:0
order
described
stability are
suffi-
.
Lur'e-Postnikov
sys-
by
O.lt T
is
time-dependent
and
f E NI(L )
,
1.4. Absolute stability
In t h i s
case
49
t :
{L : 0 ~ L ~ 912}
]
:
{J:J
M
=
{ M : M : d i a g { m l , m 2} , m i e R , i = 1,2}
D(t) ~ 0
F(t,L)
=
[-1
, L : diag {£1,£2}
: diag {jl,j2}
. Hence,
,Ji
L(0"5 + 0 . 0 1 £ 2 t) cos t
It c a n be e a s i l y negative
verified
definite
H(t) ~ [2
and
FT(t,L)
that
for every
0=0
. Hence,
+ F(t,L) 16 a r e
fulfilled.
absolutely
stable
on
FI is o n e m e m b e r
dv d~ ~ -~xTx Hence Theorem
x=0
t]
, -1 - (1 - 0 . 1 ~ 2 ) t 2
]
+ ~ I 2]
, c g ]0,1.8[
(t,L) E R x L
, which
, IIR(t,L)U e 1 , a n d
definite
The
state
Derivative " e • ]0,1.8[
on
suggests
Rx]xtxM
x=O
= NI(L)
= {v f : v f ( t , X )
is c o m p l e t e l y
, (0.5 + 0.01 £i t) sin
R ( t , h ) e [2
N2(],L,~)
family.
and
[FT(t,L) +F(t,L)
is n e g a t i v e
of Theorem
e ~ , i = 1,2}
E(t,L) =C(t)
- (1 - 0.1 ~1 ) t 2
,
. All
of the
. In t h i s
the
is u n i f o r m l y
example,
= xTx , Vf ~ N I ( t ) } of
v
obeys,
along
absolutely
motions,
, Vf 6 NI(/)
, V(t,x) e RxR n
uniformly
system
stable
on
NI(L)
_
d---~
-
i2o3ot2 2t ]FII 1+ t 2
The nonlinearity
x +
2t
f
D(t)=[-t2],
finally,
f(t,~)
,
30 + 2 0 t 2 l+t 2
2 ] x + ( - t 2)
scalar
then
d e t [II - D(t) L]
E(t,L) and
is
Lur'e-
'
= [1,t
Since
due
15.
dx
and
II -D(t) ¢ 0
on
= [I-D(t)
of
T(t,J,L,M)
conditions
system
Example 12 ( G r u j i 6 , 1978a}. The second o r d e r n o n - s t a t i o n a r y P o s t n i k o v system i s d e s c r i b e d by
tion
choice
is
R×L
f(t,Z)
f~No(L)
L =[i + L t 2 ] ,
, where LeL
L = [0,1]
, obeys
. Hence,
L]-Ic(t)
=
l + L1t 2
[1,t 2 ]
,
the
condi-
to
50
Ch. I
F(t,L)
:
Outline of the Liapunov stability theory in general
A(t) +B(t)
LE(t,L)
20 + 30t 2 i + t2
2t
- 2t +L____L_~ i +Lt 2
The m a t r i x (t,L) 6 RxL Hence,
suggests
IIR(t,L)]I ~ i
The matrix
function
V(X) : xT×
and
and T
is n e g a t i v e
H(t) ~ 12
T(t,L)
and
definite
O:0
definite
on
R×L
V ( t , X ) ~ RxR n , V f e N o ( L )
which
show that all c o n d i t i o n s
that
x:0
for every
so that
R(t,L) ~ 12
: FT(t,L) + F(t,L)
is n e g a t i v e
dv ~ _ v(X) dt
]
_ 30 + (29L + 20) t 2 + 19Lt 4 (I + t 2 ) ( l + Lt 2)
'
[FT(t,L) + F(t,L) + 12] , which
1
of T h e o r e m
of the s y s t e m is c o m p l e t e l y
. Therefore, ,
15 are satisfied.
uniformly
It follows
absolutely
stable
on
NO([0,1])
In special system
cases
D(t) ~ 0
dx
-dt
Following Gruji4
Theorem 17. X=Q
and
f is i n d e p e n d e n t
= A(t)
(1977a),
x+ B(t) f(Z)
t •R
. Then the
, Z =C(t)
For complete uniform absolute stability on
of the system
ei~envalues of
[A(t) +~In]
ative diagonal matrix (a) the matrix function R,
(19)
NO([O,K])
of
(19) it is sufficient t~at :
g>O
for which real parts of all
are negative for every
there exist a positive definite matrix
on
x
we state
(i) there exists a real number (ii)
of
(16) takes the f.ellowing form
0 6 R £x£
H = H T E R n×n
t E R j and non-neg-
obeying both
[2K -I - ~ C B - B T c T ~ ]
is positive definite
and
(b) the matrix function [AYH + HA + ( A T C Y ® + C T + HB)(2K -1 - e C B - BTcTo) -1 (ATCYO + C Y + HB) Y] is negative definite on Application ly p e r t u r b e d
R.
of this t h e o r e m w i l l be shown in the f r a m e w o r k large-scale
systems
stability
analysis.
of s i n g u l a r -
1.4. Absolute stability
51
1.4.6. Criteria for absolute s t a b i ] i t y of time-invariant systems In the
case
rather
than matrix
A ,B ,C
independent
then
and
the
In some
cases
are m a t r i c e s
the
and the
Lur'e-Postnikov
dx : dt
Let
D
functions
Ax + B f ( Z )
linear
part
such L u r ' e - P o s t n i k o v
real
valued
elements
nonlinearity
system
is t i m e - i n v a r i a n t ,
• Z :
of the
systems
with
vector
f
is time-
Cx + D f ( Z )
system
(20)
is not
be d e s c r i b e d
stable.
by
dt
where the
fEN0(L
system
lowing
Tsypkin
provided
that
: f(Z) - F Z The
)
accepted Notice
(1962b),
that
K = K-P
notation
A
means
set
Let
W(s)
of the put.
the
F <_ K
L : [F•K[
(20)
with
hermitian
and
, f(~) :
D =0 .
system
(21)
in g e n e r a l i t y
this
sector
into
the
can be closed,
is an
L = ]F•K[
infinite
, where
be the t r a n s f e r
part
of
form
of the
it may be
P
(F,K) : IF•K] ,
diagonal
entry
is a d i a g o n a l
as input
matrix
and with
of the
Yo = - Z
the pair
is completely controllable
rank
the matrix
A
(B,AB•...,An-IB)
is observable
(A,C)
part out-
: ~1 [W(j~) + w T ( - j ~ ) ]
of the system. (20) that (A,B)
linear as its
W(j~)
rank
(iii)
~ =~
• and
It i8 sufficient for absolute stability on
(i) the pair
(ii)
x:x
losing
Yi = f
HeW(j~)
x=O
that
by fol-
the
or
= C(A- Sln)-iB-D
18.
sector
. Then•
elementwise.
system
Theorem
is stable,
. Iff there
Then
is the
matrix
h EL
without
that
of
then
every
matrix.
(F•K) : [F,K[
K
a diagonal
and
• L = (0•K)
that
is stable
(F•K)
A
transforms
or s e m i - o p e n ,
matrix,
f = [Z
we d e f i n e
It is c l e a r
that
is such
for
is such that
. Hence
(20).
[ = (F•K)
is s t a b l e
F
preceding
system
and
(21)
=
n
,
:
(cT,ATc T , . . . • (AT)n-IcT)
= n
is stable : He hi(A)
:
< 0 , Vi : I•2,...,n ,
•
No[(O,K)]
of
52
Ch. I
Outline of the Liapunov stability theory in general
and that there exists positive
(iv)
matrix
diagonal matrix
and diagonal
T ~ R mxm
such that
e e R mxm
He { T [ C ( A - j w l n ) - I B- D+ K- l ] + j w ® C ( A - j w I n ) - l B} is positive
Proof
of this
(1973).
semi-definite
theorem
can be found
The requirements
(A,B)
and
Popov
(1960)
[see also T o k u m a r u
Theorem 19.
, respectively,
D=O
In the case
No((O,K) )
of
is stable matrix,
(i) A
~+~
(1965)].
.
and Taylor
and o b s e r v a b i l i t y
can be removed
and Saito
of the
by following
•
it is sufficient for absolute
of the system
x:O
and for
in the book by N a r e n d r a
for c o n t r o l l a b i l i t y
pair
on
(A,C)
w E [0,+~ [
for all
stability
(20) that both
Re ki(A) < 0 , vi : 1,2,...,n ,
and
(ii)
@ E R mxm
there exists diagonal matrix K -I + H e [(I m + j w e ) W(jw)]
and for If.
Remark
w~+~
stability
analysis on
Greater blem.
w
number
Greater
needed
order
n
systems
[see Siljak
definite
w~ 0
(Theorems
for all
of
18,19)
(semi)-definiteness This
of nonlinearities,
blem even for digital higher-order
criteria
to positive
for all m
is positive
w~
[0,+~[
.
The Popov-like
dependent
such that the matrix
is difficult more
test of
difficult
calculation.
new approaches
m×m
matrix problem.
computational computational
It is evident
to their
absolute
computational
A , again more difficult
computer
reduce
stability
that
propro-
for
analysis
are
(1971)].
1.5. ON STABILITY PROPERTIES OF SINGULARLY PERTURBED SYSTEMS 1.5.1.
Introductory comments
A phy s i c a l of their scale.
appears
sible nected
can be composed responses.
of subsystems
of each would be studied meaningful
additional
to infer the over-all subsystems.
disconnected
from Others
system property
This p h e n o m e n o n
subsystems
and
whether
subsystems
from that
of different
is m a t h e m a t i c a l l y
time scale.
and to explore
on interactions
time
expressed
speeds time
then a dynamical
in the c o r r e s p o n d i n g
to use such i n f o r m a t i o n conditions
with different
Each of them has its own natural
If all of them were d i s c o n n e c t e d
prope r t y
simple
system
transient
It under
it is pos-
of all disconscales related
by existence
of
to
1.5. Singularly perturbed systems stability
53
arbitrarily small positive parameters
~i
in differential equations. When all
~i
at the highest derivatives are formally set equal to zero,
then the number of differential equations is decreased.
This is singu-
lar case. The singular p e r t u r b a t i o n approach to systems analysis and design has been successfully used for model-order reduction and separation of time scales. A number of results obtained on various singular p e r t u r b a t i o n problems were excellently reviewed by K o k o t o v i 6 , . O ' M a l l e y Jr. and Sannuti (1975,1976). Liapunov stability properties of singularly perturbed systems were studied by Gradshtein (1962), Hoppensteadt
(1951), T i c h o n o v
(1952), Klimushev and Krasovskii
(1966,1967,1968,1974),
Desoer and Shensa
Shensa
(1971), Wilde and Kokotovi~
(1972), Siljak
Porter
(1974,1976,1977a,b),
(1974a,b)
(i975), Gruji6
Habets
(1972),
(1970),
Zien (1973),
Geraschenko and Geraschenko
(1976a,b , 1977c,d , 1978c , 1979a,b,c , 1981a), Suzuki and
Miura (1976), Young, Kokotovi6 and Utkin (1977), Javid (1978), Chow and Kokotovi6 Martynyuk and Gutowski
(1978), Khalil and Kokotovi4
(1978), Chow (1979), and
(1979).
1.5.2. System description A singularly perturbed system
S
to be considered is governed by two
non-linear vector differential equatiOns d_XX = f(t,x,y,~) dt
,
(22a)
dy ~ - ~ : g(t,x,y,~)
.
(22b)
The state vector of the whole system is y e R m , f(t,x,y,#) • The p a r a m e t e r ted that
~
C(RxRnxRmxR+,R n)
(xT,yT) T , where
and
g(t,x,y,#) ~
x ER n
and
C(RxRnxRmxR+,R m)
is positive and can be arbitrarily small.
It is accep-
~ 6 ]0,i]
Open connected neighbourhoods respectively,
N× c R n
Ny g R m
of
x=0
and
y=0
so that
(xT,yT) T = 0
unique e q u i l i b r i u m state of the system
(22) in
NxXNy - the cartesian
If
~
Nx
and
~y , for every
,
is the
~ • ]0,1]
is formally set equal to zero, then the system (22) degenerates
into the r e d u c e d - o r d e r due to
f and
and
g
product set of
are related to
(or, the degenerate)
system
SO
(subscript zero
~=0 ) d_xx = f(t,x,y,0) dt 0
= g(t,x,y,0)
,
(23a)
.
(23b)
.
54
Ch. I
It is a c c e p t e d x •N x
iff
steadt's tion
that
y:0
g(t,x,y,0)
usage
of c o o r d i n a t e s
system
becomes
. The j u s t i f i c a t i o n
effective
framework
Outline of the Liapunov stability theory in general
of s i n g u l a r l y
can be put
1892,
perturbed
in the
If
is s u i t a b l e
p>0
part
for the
is s m a l l
and a fast
obtained
from
= t-t° P
enough
part.
(22)
. Then
The
after
the
1966)
and
and
is b a s e d
of L i a p u n o v ' s
Comment
systems.
teR
2 herein)
Hence,
on H o p p e n transforma-
in the
the r e d u c e d - o r d e r
form dx dt
which
for e v e r y
for this d e m a n d
(Hoppensteadt,
(Liapunov,
zero
f(t
,X,
stability then fast
the
0,0)
analysis
system
(or,
changing
the
of the
(22)
system
equilibrium
is c o m p o s e d
boundary
the time
boundary-layer
(24)
,
layer)
scale
state.
of a slow
system
ST
by i n t r o d u c i n g
is o b t a i n e d
in the
form
d_/y : g ( ~ , b , y , 0 )
(25)
d~
Here,
~
and
parameters,
b , b = (~l,~2,...,~n)T respectively.
They
is
~,
, are real
valued
scalar of
and v e c t o r
are
introduced
instead
t
zero
for e v e r y
t • R , X ~ Nx
and
x •N x ,
respectively. It is a s s u m e d • [0,i]
iff
Separation owing tem
that
g
y:0
.
of time
becomes
scales
to the p o s s i b i l i t y S0(24)
of the
and the
stability
is a d v a n t a g e o u s of a s e p a r a t e
fast
problem.
system This
ST(25)
means
from
the
analysis , and
that
the
and
stability
viewpoint
of the d e g e n e r a t e for the o r d e r
following
sys-
reduction
problem
can be
solved. Under
what
can u n i f o r m inferred
qualitative asymptotic
from
A solution
for
properties stability
such a p r o p e r t y this
problem
of
will
of
f
and
g,
which
of the e q u i l i b r i u m (24)
and
(25)
be p r e s e n t e d
hold
state
on of
Nx×Ny (22)
be
? by f o l l o w i n g
Gruji6
(1981a).
1.5.3.
Liapunov-like conditions for asymptotic s t a b i l i t y
Let
Nxo = {x : x • N × , x#0} and f o r
,
Ny ° -- {y : y • N y
,y#0}
,
v(~,b,y) •C(I'I"I)(RxRnxRm,R) , 8V va =~-~
The p r o b l e m
solution
,
Vb
is b a s e d
= ( 0V
8V
a~ I , a~2 ,
On the f o l l o w i n g
8V )T , two a s s u m p t i o n s
that
,
I. 5. Singularly perturbed systems stability
relate
some p o s i t i v e
Assumpti0n
i.
and p o s i t i v e soon as
definite
There
exists
definite
on
functions 8 6
Nx
nite on
Nx
# ec(Rn,R+) and
Ny
~e c(Rm,R+)
and
, respectively•
(b)
8~(t~x)[f(t~x,y,~)-f(t,X•0,0)] V(t~X,y~)
asymptotic
of A s s u m p t i o n
stability
of
2.
is d e c r e s c e n t
uniformly
in
and r a d i a l l y
and
so that
~i < I
~2 defi-
and
,
(a) g u a r a n t e e
uniform
of (24), w h i c h is g l o b a l as soon as (b) is the r e q u i r e d
x EN x
such that
(a)
v~ g(~,b,y,O)
on
qualitative
property
as soon as
numbers
and r a d i a l l y
that
unbounded
Ny : R m , or
as soon as
~I < ~ ' ~2 < i
e C(I,I,I)(RXNxXNy,R+)
NxXNy
that is d e c r e s c e n t
and there exist n o n - n e g a t i v e ~ ~ i
v(t•X,y
definite
unbounded
ger
and p o s i t i v e
Ny = R m}
definite
on
,
~i , ~2 , ~3
and
~4
and an inte-
and
~ - @(y)
v ( ~ , b , y ) •RXNxXNy , or (b)
~i
that are p o s i t i v e
as
•
There exists either and p o s i t i v e
{v(t•y) 6 C ( I • I ~ R x N y o , R + ) Ny
numbers
unbounded
NxxNy .
Assumption y
(25).
that is d e c r e s c e n t
~ ~l@(X)+~2~(y)
.
N x = R n . The second c o n d i t i o n
on
(24) and
radially
i and the c o n d i t i o n
x=0
to
~ - @(x) • V ( t , x ) •RXN×o ,
6 RXNxoXNyx]0•l]
The h y p o t h e s e s
v
c(l,l)(RX~xo~R+)
Gt(t,x) + 8~(t,x) f(t,x,O,O)
in
and
and, in a d d i t i o n j
(a)
f
8
N x = R n , and there exist n o n - n e g a t i v e
and f u n c t i o n s
of
55
{ v ( ~ , b , y ) •RXNxXNy o} • r e s p e c t i v e l y ;
v~[g(~,b,y,p) -g(~,b,y,O)]
~ ~l#~@(b) +~2@(y) ,
v ( ~ , b , y • # ) •RxN×xNyx]O~l] , or
{ v ( ~ , b , y , p ) •RXNxXNyoX]O,l]} ,
respectively;
(c)
~3~(b) +~4¢(y) , •RxNxxNyX]0,~] or { V ( ~ , b , y , # )
v~+v~ f(~,b,y,#) ~ V(~,b,yo#)
,
The n u m b e r s ~I ' f2 ,~I , ~2 , ~3 be c h o s e n as small as possible. If
v
is i n d e p e n d e n t
positive dition
definite
eRXNxXNyoX]0,1]} ,
•
respectively.
on
(c) v a n i s h e s
of Ny
x,
and
which
only.
~4
of A s s u m p t i o n s
is a d m i s s i b l e ,
i and 2 should
then it should be
If it is also t i m e - i n v a r i a n t ,
in A s s u m p t i o n
Let
2. I-
_
2
~2 + ~4
"
then con-
56
Ch. I
It is a lower
estimate
Outline of the Liapllnov stability theory in general
of the u p p e r
bound
of a l l o w a b l e
#
as e x p r e s s e d
by Theorem
If Assumptions
20.
and for
P 6 ]0,~[
1 and 2 hold,
then the equilibrium state
of (22) is uniformly asymptotically
(xT,yT) T-- 0
If, in addition,
# -~ 0 as Soon as i > ~i + ~i ~ - I + ~3 N×XA/y = R m+n then the equilibrium ~tate is uniformly
asymptotically stable in the whole for every
P~oof.
Let
p
stable for every
be d e f i n e d
by
v = 8 +v
and for
/I 6 ]0,~[
~ ~0
.
. Then,
v (t,x,y) e c(l'l'l)(RXNxoXHyo ) and
it is d e c r e s c e n t
tions tion
i and of
0
and p o s i t i v e
2. Its e u l e r i a n
(22)
is found
definite
derivative
in the
following
{z(t) = 0 , t e [to,+= [ , means
therefore,
it will
due to
(22).
follows
The r i g h t - h a n d
due
form
for
to A s s u m p along
a mo-
z(t) = ( x T ( t ) , y T ( t ) ) T state
henceforth}
is r e a c h e d
and,
:
f+vt+v f+ i V yTg
side
of this
equation
is t r a n s f o r m e d
as
: T
T
= 8t + Sx f ( t , x , 0 , 0 ) 1
+ Sx [ f ( t , x , y , # )
T
1
+~vyg(t,x,y,0) The
Nx×Ny
the e q u i l i b r i u m
not be c o n s i d e r e d
; : 0 +0 x
on
~[t,x(t),y(t),M]
conditions
(a),
- f(t,x,0,0)]
T
+~Vy [g(t,x,y,~)
(b) of A s s u m p t i o n
+v t +v T f(t,x,y,~)
-g(t,x,y,0)]
i and
(a)-(c)
. of A s s u m p t i o n
2
yield
P(t,x,y,#)
-< - ( i
i
- ~i - ~i ~ - I
V#e]0,~[
- ~ 3 ) ¢(X) - ~ [ I
and # ~ 0
- ~ 2 - g ( ~ 2 + ~ 4 ) ] ~(Y)
, V ( t , x , y ) eRXNxoXNyo .
' (26)
Let Nox :
{z : x : 0 , y e N y o }
It is e v i d e n t
, Noy :
{Z : X E N x o , y = 0 }
that
NxXNy : NxoXNyo uN o u {z : z:0} Let
uM
be the m a x i m a l
neighbourhood
VpM(t)
positive of
Z=0
cause
p
of
N =NxXNy
is p o s i t i v e
number
on
hood
of
z=0
(Proposition
Let
ri
and
T i , t o -< T i < T i ~ +~ Z(t) e V P M ( t )
for w h i c h
, V(x,y) e V ~ M ( t )
for every
definite
. the l a r g e s t
connected
obeying
Y ( t , x , Y ) E [0,~M[ is a subset
, No =NoxUNoy .
N
t e R and
, vteR
. Such
N
a
,
v M >0
exists
is t i m e - i n v a r i a n t
14). , denote
\ N o , Vt e ] ~ i , ~ [
instants
such that
, ~i > t o
,
be-
neighbour-
1.5
Singularly perturbed system stability
and
57
z(t) e N ° , V t 6
If
Z(to) 6 V V s ( t o ) \ N O
first If
interval
to be
Z(to) eN O , then
to be In
considered,
what
and
follows,
then
i:0
considered
, T~ = t o
the
next
i ~ 0
, ro : to
and
i=~
the
, [TO•TO[ : [to,To[
next
and
one
is a n
[Ti_l,Ti]
one
is
[ r o , r I]
is
]~I,T~[
is t h e
[ r o • r I]
is the
first
interval
.
integer.
Let ~(t;to•Zo;#
) : (xT(t;to,Zo;#),~T(t;to•Zo;#))
be t h e m o t i o n
of the
system
(22)
through
T ,
zo
at
~(t;to,Zo;#) t = to
for
EZ o • # > 0 .
t Proposition
~ong
15.
motions
Proo~.
The function
i~ ~triotly decreasing in
of (22) for every
~(t;to,zo;p)
Part I. L e t
p
Bt 6 [T ~i _ l , r i [ v[t,x(t)•y(t)]
such
t~[Ti_l,T i ]
and for
# 6 ]0•#[
~ ~0 .
that
g w[t•x({)•y(t)]
w for s o m e ~i < ~2
t 6 ]Ti_l•fi_l[
• such
. If
t : Ti_ * 1
to the
plied
BTI•~2 e ]~i_l,Ti_l [ ,
that g V[~2,X(T2)•Y(;2)]
V[rl•X(~l),y(?l)] due
then
continuity
by c o n t i n u i t y
of of
~ f
and
and
BT 3 6 [ T I , ¥ 2 ]
However, t h i s
contradicts
~
in
t eT o
g • and
due
for w h i c h
, Vt o e R , w h i c h
to c o n t i n u i t y
~
z 0
of
im-
~ . Hence,
.
t=T 3 due t o p o s i t i v e definiteness
(26)
is
of
¢
and
and
(1-
[ i - ~ 1 # ~ - 1 - ~3) > 0
Hence • Let
t : T i-i ~
tions Let
be t h e
Y2 ~ [ 7 ~ - i , ~ i [
Since
z(t) = 0 are
be
is e x c l u d e d ,
continuous,
Yl = [ ~ i - l , T i [ 8[t,×(t)]
~[1-
is i m p o s s i b l e •
Yl ~ [ T i - I " T i [
and l e t
1
,
= 8(t,0)
then f o r the
~2 - # ( ~ 2 + ~ 4 ) ] > 0 , V# e ] 0 , ~ [
and
t 6 ]T:_I,Ti]
set
of all
the
set o f a l l
instants
Vt g [ t o , + ~ [ either sake
t
such
instants , and
the
or
of preciseness.
that
obeying
since
w Yl = [ T i - l , r i [
, Vt E T I , a n d
is to be
considered. ×(t) = 0 , y(t) = 0
system
.
mo-
Y2 = [ z i - l , r l [ •
Then,
p[t,×(t),y(t)]
= p[t,0,y(t)]
Further, d d-T v [ t , 0 , y ( t ) ]
d = ~-~ v [ t , 0 , y ( t ) ] t e l I , V# • ]0,~[
This
disproves
t •Yl
• Let
now
1 g -F(1 and
-~2-P~4) # ~0
T2 = [ T ~ - I ' T i [
@IT(t)]
.
" Then
, (27)
y(t) =0
, Vt • 1 2
•
58
Outline of the Liapunov stability theory in general
Ch. I
Hence,
p[t,x(t),y(t)]
and
d P[t,X(t),O] d~
which
disproves
defined
= v[t,X(t),0]
g -(i
- ~ i - ~3) ¢ [ X ( t ) ]
t • Y 2 . Altogether,
conditions niteness
2.
Results
(26),
i > fl +~i ~ - I
of
[ri_l,ri[
@
and
(27),
+~3
¢ , prove
some
3.
Let
Part
P
B~ 6 [r~_l,rl]
t • ]ri,ri[
. Hence
to c o n t i n u i t y ~3
Parts
of
(22),
of
@
~
and the p o s i t i v e decreasing
, the defi-
over
every
~ P[t,x({),y(t)]
¢ ,
, and
Proposition
than a negative
. This
result,
that
z:0
first
statement radially
stability
of the
unbounded,
statement
Application
, such that
~
in
t , and Part
2.
~ 0
t=f3
in Part
15, w h i c h
prove
for
due
i. together
(Hahn,
# ~0
1967)
. Further, > 0
and
to the d e f i n i t i o n
definite
function
definiteness
on
with positive
uniform
defi-
stability
the p o s i t i v e
definite-
(1 - ~i -~i ~ - I ) of
~,
NxoXNyo
prove , on
and d e c r e s c e n c y
> 0 ,
that
Nox
of
of
v
and on prove
a t t r a c t i v e w h i c h c o m p l e t e s the p r o o f of the t h e o r e m • In the case NxXNy = R m + n then ~ is which
of the
of this t h e o r e m
analysis
~
and
positive
is u n i f o r m l y
second
of
# ~0
and
I
(I - ~ i -~i #~-I - ~ 3 )
for
' [i < }2
that
as e x p l a i n e d
V# e ]0,~[
and
V# e ] 0 , ~ [
also
(26)
i-3 prove
z=0
the
of
such that
p[t,x(t),y(t)]
such
and d e c r e s c e n c y
ness
is less
as
~ v[r2,X(}2),Y(¥2)]
of b o t h
• {Tl,r2{
contradicts
niteness
Noy
t e [T~_l,ri[
definition
is s t r i c t l y
d p[t,x(t),y(t)] dt
Now,
I, the
3 T I ' T 2 • ]ri'Ti[
V[rl,X(~l),Y(rl)]
which
is not
, ~2 > 0 , ~3 > 0
that
v[t,x(t),y(t)]
Hence,
Vt 6 T 2 ,
, ri_ I ~ t o , V i a i
Part
due
there
,
above. Part
for
, Vt E T 2 ,
together
theorem. will
of s i n g u l a r l y
with
the p r e c e d i n g
result
proves
•
be shown
in the
perturbed
framework
Lur'e-Postnikov
of a b s o l u t e systems.
1.5.4. Singularly perturbed Lur'e-Postnikov systems Let
system
(22) be
of the L u r ' e
form
(Gruji6,
1981a)
:
= A l l X + ql@l(Ol) + Al2Y
,
al
#Y : #A21x + A22Y + q 2 ¢ 2 ( a 2)
,
o 2 : # c 2 1 x + c22Y .
:
T T ClIX + cI2Y T T
(28a)
(28b)
1.5. Singularly perturbed system stability
The m a t r i c e s ate order.
A(.)
and vectors
C(.)
The n o n - l i n e a r i t i e s
~i(O) : 0 , and in Lur'e
59
¢i
sectors,
and
¢i
and
y:O
are the unique
tem
~i
equilibrium
are c o n s i d e r e d
states
assumptions
We accept pair
that
dy d--~ : A22Y + q2@2(o o) 2 , 02 : CT2Y •
(30)
system,
are g u a r a n t e e d
respectively,
and that
(Barnett there
.
and Storey,
exist
numbers
1970), @i e [0,+~{
such that
-
_
J~In)-lql ~_ 0 , v ~ e
Then,
O(X) function
[O,k I] , where
Popov,
T -1 cii Aii qi > 0 , i:&,2
by
is a stable m a t r i x
kl I + Re (i + j ~ l ~) cITI(AII
is a L i a p u n o v
sys(29)
All
e I • ]0,+~[
for which
,
(All,q I) - c o n t r o l l a b l e ,
and
,
of the d e g e n e r a t e
dx
These
continuous,
[O,k i] , k i • ]0,+~[
-dt - : A l l X + qzCz(o ) , a~ : c i x
and of the b o u n d a r y - l a y e r
in
are of the a p p r o p r i -
, are one-one,
in
~i (ol) E [O,ki] , i:1,2 Vo i • I-~,+~[ oi In addition, only those n o n - l i n e a r i t i e s x=O
q(.)
, i=1,2
T
+ J ~ I n)
-I
(All - J ~ I n )
-i
ql
[0,+~]
i (xTHx+@I *l(o~) dUl)
I]
of the d e g e n e r a t e
H1
1973; N a r e n d r a
=
T
-elql(All
subsystem
is the solution and Taylor,
T T AIzHI + HIAII +glgz
(29) for every
(Yakubovich,
1973)
1962;
@i
Kalman,
1963;
of
:-eli1
,
hl+Hlql
:-~/~igl
(31)
for -
T
i
~i : kl I -~i Cll ql The a s s u m p t i o n s
8(X)
(32)
1,2 can be now verified.
Test o f Assumption i . Hence,
T
, hi : ~ (~i All ell + ell)
Let
@ ( x ) and
is d e c r e s c e n t ,
HI
be determined as above.
positive-definite
on
R n~ and r a d i a l l y
bounded. Let c o n d i t i o n (a) In this
(a) be t e s t e d
case
8t : 0
first.
and
0 xT(x) f ( x , O , O ) -< -~i eln21UxU , V(x~O) ER n where
and
7 2 = AI/2(H I + ~il k l A(.)
is the m a x i m a l
eigenvalue
Cll
cT1 )
of the matrix
(-)
. Hence,
un-
60
Ch. I
¢(x)
and
Outline
Nx =R n ,
= ~ 1
--
8(x)
- f(x,0,O)]
-I
in g e n e r a l
Rn
V(x~0)
'
theory
'
Nxo = {x : x#0 , x e R n}
(b) The accepted choice of eX [T f ( x , y , p )
stability
, ~3 = ~ c z n 2
< -¢(x)
X
which implies
i
= n3llxll
@ t + eT f ( x , 0 , 0 )
o f the L i a p u n o v
.
yields
:
x T ( 2 H i + @ i ~i(°~) o o1
~I ¢ ( x ) +~2@(Y)
Cli) {Al2Y + ql[@1_ (al) _@l(O~)]}
cii
' Vx•N×o
, Vy•R m
V# • ] 0 , 1 ]
for
@(y)
= P311yll ,
Cz = k z ( n z ~ 3 ) - l " n 2 l i q z U ' l i c z z
where
~2 = ( w z P 3 ) - 1
~ 2 ( k z l l c z 2 1 1 " l l q £ II + BAz211) ,
~i
'
k(-)
= ki/2(Hl)
denotes the minimal eigenvalue
The value of ~i
and
I[ ,
~2
P3 > 0
will be determined
and functions
Test of Assumption
2.
Let
8 ,¢
and
v(y) :llyg
of the matrix
in the sequel. •
The numbers
satisfy Assumption
be a tentative
v
in order to show another alternative
struction of a Liapunov
function.
definite
on
Rn
and radially
v
is decrescent,
system in the Rosenbrock dy _
d--t-
where
for con-
positive-
unbounded.
(a) In order to test this condition of Assumption boundary-layer
1_.
function.
This choice is accepted
Hence,
(.)
-
form
2, we present
the
:
u22k~2) Y 0 °
2
(1977e),
Referring to Gruji~ is negative definite
the matrix
for every
D22(~2)
= D~2(~2) + D22(e2)
(o,¢ 2 ) e R × N o ( [ 0 , k 2 ] )
iff both
D22(0) and D22(k2) are negative definite, which is accepted the analysis. Finally, ~(Y) = P3UYll and v~ g(~,b,y,0) ~ -@(y) V(y#0) e R m verify condition (a).
in
m
(b)
v~[g(cz,b,y,/~) - g(c~,b,y,O)]
: ~l_yT {gA21b + q 2 [ ~ 2 ( o 2 )
-~2(°~)]}
V ( y # 0 ) e Rm Let ~'l
= 2 c-lr/2
sup •
[0,k
]IA21 +cxq2cT1 II 2 ]
,
T ~2 = k211q2C2lll P3l
,
1.5. Singularly perturbed system stability
It
is a s s u m e d
that
61
~2 < 1 . T h e n
v~[g(~,b,y,~) - g ( ~ , b , y , O ) ] ~ ~l~¢(b) +~2@(y) , V(~,b,y,~) ~RxRnxRmx]o,+~[ , which (c)
implies
In this The
case
lower
= : I
in
v~ ~ 0
estimate
(b)
and of the
of A s s u m p t i o n
v b ~0
, which
upper
bound
F
-
2. yield
~3 : 0 , ~4 : 0
of a l l o w a b l e
p
.
is g i v e n
by
I - ~2 t2
Now, the
i > fl + ~ I system
(28)
implies
absolute
for every
~•]0,~[
Example 13 (Gruji6, 1979c).
All
:
on
of
z : (xT,yT) T = 0
No(1) , L=[0,K]
of
, K = diag{kl,k 2} .
Let
'
•
stability
ql
:
'
-2
CII
:
•
1/10
Al2:I el2
ki:2
and
A21 : 10 -3 12
, 621 :
A22 =
We
select
@i : i
1
and
'
e I : i/10
•
q2 :
so
k-~ + Re (i + j@io~) CTii(Aii- jo;12)-i
[7] '
k2 : I
C22 :
that
T
T
i
k-~-- ~ > o . Hencej
H1:[h11hi21 h12 from
)-i
ql-ezql(A11 +J~I2 i
is d e t e r m i n e d
•
• h22 j
(Ail-J~I2)
-I
qz
62
Ch. I
in the
Outline of the L i a p ~ o v
stability theory in general
form M1
Therefore,
W1 = 0.16
The m a t r i x
D22(~2)
and
W2 : 0.45
is o b t a i n e d
D22(~2)
D22(0)
Hence, Finally,
.
as
=
D22(1)
and
~i : 0.05
=
+ ~22 are both
, ~2 : 1.88
' negative-definite.
• ~i : 0.02
and
~2 : 0.002
. Hence,
: o.52
Since
~i + ~! = 0.55
of the s y s t e m # e ]0,~[
defined
, i.e.
order
reduction.
system,
1 > ~i +~i
example
separation Instead
two
, L : [0,K] scales
were
advantage way
Z = (xT,yT) T = O
stable
for e v e r y
, K = d i a g {2,1}
is the a c h i e v e d
straightforward
effective
that
is a b s o l u t e l y
No(L)
systems
Another
is the more
i , it follows
of time
of the
2nd-order
was tested.
ity a n a l y s i s
than
~ • ]0,0.52[ , on
The a d v a n t a g e of the problem
is less in this
analysis
considered of this
and
the
approach
for a s y s t e m
order
of the
4th-
condition
to the
Liapunov
stabil-
function
construction. However, (29)
if the d i m e n s i o n s
m
and of the b o u n d a r y - l a y e r
further
reduction
of the order
and
n
of the r e d u c e d - o r d e r
system
(25)•
problem
(30)
system
are high,
then
(24), a
is needed.
COMMENTS ON REFERENCES 1. In 1 8 8 1 - 1 8 8 6 , eral
title
which
an idea
equations simple
directly
had p u b l i s h e d
Curves
on n e c e s s i t y
functions,
In L i a p u n o v ' s 1892),
Poincar~
: "On the
Defined to study
and not by means was r e a l i z e d
work
his d o c t o r a l
: "General
his
four m e m o i r s
by D i f f e r e n t i a l functions• of their
under
Equations"•
defined
reduction
a genin
by d i f f e r e n t i a l to the more
successively. Problem
dissertation,
of M o t i o n
an e v o l u t i o n
Stability" of "the
(Liapunov,
fundamental
Comments on References
variant
63
of the stability theory"
and having arisen twenty-two
(Moiseyev, 1949) ascending t o A r i s t o t e l
centuries ago, is completed.
This is the
main merit of Liapunov in front of the science. 2. The precise definition of uniform stability was given by Persidskii (1933), and uniform asymptotic
stability by Malkin
of the corresponding
theorems
5. Definition
is based on those of Liapunov
1.a-e)
(i933), Chetayev C1952), Malkin
(1946), Massera
[see also Coppel Demidovich
(1965),
(1967), Hahn
Habets and Laloy
Yoshizawa (1967)
(1980)],
Anapolsky and Vasilyev stability definiitons
Levinson
(1975,1977)
(1967)], Barbashin
[see alsoRouch,
(1979), Matrosov, of
introduced till 1975 can be found in works by Habets and Laloy
: Nemytsk~and
Stepanov
(1970), Gutowski
(1980). can be found in
(1960), Codington and
(1971),
Pontryagin
known theorem,
of expansion of a system of differential
stability
(1961)
(1960)].
that Poincar~'s
(1892) general theorem,
of motion
(1964)
survey of development
in ser~es in powers of a small parameter,
of Liapunov's
Zubov
(1969,1974a),
Martynyuk and Gutowski
(1950) illustrates
states possibility
method
(1960)I,
(1966), Bhatia and Szeg~
(1980). A historical
(1955), Hartman
solutions
[see also Krasovskii(1959);
[see also Siljak
[see also the paper by Kalman and Bertram 5. Duboshin
(1892), Persidskii
for solutions existence and uniqueness
the following books
Statements
Barbashin and Krasovskii
Stepanov
(1973), Gruji~
(1966), and Rouch,
4. Conditions
(1958)
(1960); N e m y t s k ~ a n d
(1970)], Narendra and Taylor
Yoshizawa
(1949,1956),
(1954), Antosiewicz
Kalman and Bertram
(1954).
on stability also belong to them.
which
equations
is a special case
which was the basis of his first
analysis.
As it is known,
Poincar~ proved
his theorem for a case of a system of the order not greater than the third order,
having mentioned in Notes a possibility
tion to a system of an arbitrary order.
of its generaliza-
In 1894, Picard proved Poin-
car@'s theorem by another method and also for a special case. i.e. after 20 years of Liapunov's some years of its publication
book
in French,
theorem for a system of an arbitrary only
(in Liapunov's
(1892) publication
In 1911,
and after
Moulton proved Poincar@'s
order involving one parameter
theorem there can be several
(p)
parameters).
6. Survey of works on solution of the Liapunov matrix equation is given in the book by Barnett
and Storey
(1970). An effective method for its
solution is also given in the work by Karpinskiiand 7. Liapunov method by v .
(1892) proposed
to denote auxiliary
Larin
functions
(1976). of his second
V ; in the present book we denote scalar functions by
V
and
64
Ch. I
Outline of the Liapunov stability theory in general
8. The concept of asymptotically variant Gruji6
contractive
sets and positively
sets with respect to a function is introduced
(1975). The meaning of these concepts is illustrated
stationary
in-
in the paper by in the non-
systems analysis.
9. Popov's ingenious approach has inspired a great number of scientists, who contributed much to the development bility.
Yakubovich
(1962-1965,1967,1968,1970,1975)
1964), Kalman (1963) and Szeg6 Popov's method. Considerable
of the theory of absolute
(1963,1964)
(1962-
were the first to develop
results were also obtained by Gelig (1964), Halanay
Ibrahim and Rekasius
(1964), Naumov and Tsypkin
Jury and Lee (1965), Tokumaru and Saito and Jury
and Tsypkin
(1966), Anderson
(1968), Partovi and Nahi
(1964), Desoer
(1965), Meyer
(1966), Dymkov
sta-
(]964),
(1965),
(1965), Dewey and
(1967), Moore and Anderson
(1969), Bertoni,
Bonivento
and Sarti
(1970),
Siljak and Sun (1971,1972), Siljak (1972)~ Anderson and Moore (1972), Garg and Robbins (1972), Piatnitskii(1970). The sufficiently complete outlines
together with considerable
original results
the works by Aizerman and Gantmakher
(1963), Gantmakher
(1968), Lefschetz (1965), Piatnitskii Narendra and Taylor (1973). For some particular absolute
can be found in
(1968), Siljak
cases the necessary and sufficient
stability are stated by Nelepin
and Yakubovich
(1969,1974), conditions
(1967), Persidskii
of
(1969),
Piatnitskii (1970~1971), Mukhametzyanov and Serikbayev (1970). Necessary and sufficient Liapunov like conditions for absolute stability of and Aizerman conjecture Gruji~
i0. Numerous
results,
are well described Sannuti
for any Lur'e-Postnikov
system are due to
(1978-1981). obtained
for the singularly perturbed
in the following works
(1975,1976).
: Kokotovi6,
Stability of the singularly perturbed
according to Liapunov was studied by Gradshtein Flatto and Levinson
(1955), Pontryagin
Klimushev and Krasovskii
systems
(1951), Tikhonov
(1957), Mischenko
(1962), Razumikhin
systems,
O'Malley and (1952).
(1959),
(1963), Hoppensteadt
(1966,
1967,1968,1974), Desoer and Shensa (1970), Shensa (1971), Wilde and Kokotovi~ (1972), Siljak (1972), Vasiljeva and Butuzov (1973), Zion (1973), Porter Geraschenko
(1974,1976,1977a-b),
(1975), Mischenko
Habets
1978e , 1979a-b-c , 1981a), Suzuki and Miura Young, Kokotovi6 Kokotovi~ (1979).
(1974a-b),
and Rozov (1975), Gruji£
and Utkin (1977), Javid
(1978), Khalil and Kokotovi~
Geraschenko
and
(1976a-b , 1977c,
(1976), Kuzmina
(1977),
(1978), Chow (1978), Chow and
(1979), Martynyuk and Gutowski
References
65
Aizerman, M.A., and F.R. Gantmacher (1963), Absolute Stability of Control Systems. A.N. SSSR, Moscow (in Russian). Anderson, B.D.O. (1966), Stability of control systems with multiple nonlinearities. J. Franklin Institute, 282, No.5, 155-160. Anderson, B.D.O. and J.B. Moore (1972), Construction of Liapunov functions for non-stationary systems containing non-inertial nonlinearities. Avtom. i Telem, No.5, 14-21 (in Russian). Antosiewicz, H.A. (1958), A survey of Liapunov's second method. In Contributions to the Theory of Nonlinear Oscillations, edited by S. Lefschetz, Vol. IV, Princeton University Press, Princeton, 141-166. Barbashin, Ye.A. (1967), Introduction to the Theory of Stability. M. Nauka, 225 p. (in Russian). Barbashin, Ye.A. (1970), The Liapunou functions. Nauka, Moscow (in Russian). Barbashin, Ye.A.,and N.N. Krasovskii (1952), On the stability of motion in the large. Dokl. Akad. Nauk SSSR, 86, No.5, 455-456 (in Russian). Barbashin, Ye.A.,and N.N. Krasovskii (1954), On the existence of Liapunov functions in the case of asymptotic stability in the whole. Prikl. Mat. Meh., XVIII, 345-550. Barnett, S., and C. Storey (1970), Matrix Methods in Stability Theory. Nelson, London. Bertoni, G., C. Bonivento, and E. Sarti (1970), A graphical method for investigating the absolute stability of time-varying systems. Ricerche di Automatica, I, No.I, 102-111. Bhatia, N.P., and G.P. Szeg~ (1967), Dynamical Systems : Stability Theory and Applications. Springer-Verlag, Berlin. Chetaev, N.G. (1946), Stability of Motion. OGIZ, Moscow (in Russian). Chow, J.H. (1978), Asymptotic stability of a class of non-linear singularly perturbed systems. J. Franklin Inst., 305, 275-281. Chow, J.H., and P.V. Kokotovi~ (1978), Near-optimal feedback stabilization of a class of nonlinear singularly perturbed systems. SIAM J. Control and Optimization, 16, No.5, 756-770. Coddington, E.A., and N. Levinson (1955), Theory of Ordinary Differential Equations. McGraw Hill, New York. (Russian translation published in 1958 by Inostranoi Literaturi, Moscow.) Coppel, W.A. (1965), Stability and Asymptotic Be~auiour of Differential Equations. D.C. Heath and Co., Boston. Demidovich, B.P. (1967), Lectures on the Mathematical Theory of Stability. Nauka, Moscov (in Russian). Desoer, C.A. (1965), A generalization of the Popov criterion. IEEE Trans. A.C., AC-10, No.2, 182-185. Desoer, C.A., and M.J. Shensa (1970), Network with very small and very large parasitics : natural frequen~es and stability. Proc. IEEE, 58, 1933-1938. Dewey, A.G., and E.I. Jury (1966), A stability inequality for a class of non-linear feedback systems. IEEE Trans. A.C., AC-11, No.i, 54-62. Duboshin, G.N. (1950), On one Poincar@'s theorem. Yestn. Mosk. un-ta, Ser. Yestestv. nauk (Natural Sciences), No.12, 35-38 (in Russian).
66
Ch. I
Outline of the Liapunov stability theory in general
Dymko~, V.I. (1967), On absolute stability of frequency-modulated systems. Avtom. i Telem, 109-114 (in Russian). Flatto, L., and N. Levinson (1955), Periodic solutions of singularly perturbed systems. J. Rational Mech. Anal., 4, No.6, 943-950. Gaiduk, A.R. (1976), Absolute stability of control systems with multiple nonlinearities. Avtom. i Telemeh, No.6, 5-11 (in Russian). Gantmacher, F.R. (1974a), The Theory of Matrices. Vol. I. Chelsea Publ. Co., New York. Gantmacher, F.R. (1974b), The Theory of Matrices. Yol. 2 . Chelsea Publ. Co., New York. Gantmacher, F.R., and V.A. Yakubovieh (1965), Absolute stability of nonlinear control systems. In Analytical Mechanics~ Stability of Motions and Space Balistics. Nauka, Moscow, 30-63 (in Russian). Garg, D.P., and M.J. Rabins (19Y2), Stability bounds for nonlinear systems designed via frequency domain stability criteria. Trans. of the ASME J. of Dynamic Systems, Meas. and Control, 262-265. Gelig, A.H. (1964), Investigation of stability of non-linear discontinuous automatic regulating systems with non-unique equilibrium states. Avtom. i Telemeh, XXV, No.2, 153-160 (in Russian). Gelig, A.H., and Komarnitskaya (1966), Absolute stability of non-linear systems with non-unique equilibrium state in critical cases. Avtom. i Telem., No.8, 5-14 (in Russian). Gerasehenko, E.I. and S.M. Geraschenko (1975), Method of Motion Decomposition and Optimization of Non-Linear System~, Nauka, Moscow (in Russian). Gradshtein, U.S. (1951), Application of Liapunov's stability theory to the theory of differential equations with small multipliers in the derivative terms. Dokl. AN SSSR, 81, No.6, 985-986 (in Russian). Grujid, Lj.T. (1975), Novel development of Lyapunov stability of motion. Int. J. Control, 22~ No.4, 525-549. Grujid, Lj.T. (1976a), General stability analysis of large-scale systems. Proc. IFAC Symp. on Large-Scale Systems Theory a~d Applications, 203-213. Gruji£, Lj.T. (1976b), Vector Liapunov functions and singularly perturbed large-scale systems. Proc. 1976 JACC, 408-416. Gruji£j Lj.T. (1977a), Un lemme matriciel r6ciproque; application ~ la stabilit@ absolue. C.R. Acad. Sci., Paris, Ser.A, 384, 1409-1412. Gruji6, Lj.T. (1977b), Is the stability theory consistent and complete ? First World Conf. on Mathematics at the Service of Man, Barcelona, July 11-16, 20. Gruji~, Lj.T. (1977c), Stability theory of sets and singularly perturbed large-scale systems. Ibidem, 22. Gruji£, Lj.T. (1977d), Converse Lemma and singularly perturbed largescale systems. Proc. 1977 JACC, 1107-1112. Gruji6, Lj.T.
(1977e), Stability and instability of product sets. Sys3, No.l, 14-31.
tems Science,
Gruji~, Lj.T. (1978a), Absolute stability of non-stationary systems : resolutions and applications. Proc. 1978 JACC, Philadelphia, 327337. Gruji6, Lj.T. (1978b), Solutions for the Lur'e-Postnikov and Aizerman problems. Int. J. Systems Sci., 9, No.12, 1359-1372.
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68
Ch. I
Outline of the Liapunov stability theory in general
Kalman, R.E. (1963), Liapunov functions for the problem of Lur'e in automatic control. Proc. Nat. Acad. Sci. U.S.A., 49, No.2, 201-205. Kalman, R.E., and J.E. Bertram (1960), Control system analysis and design via the "second method" of Lyapunov, I. Trans. of ASME : J. Basic Eng., 82, 371-393. Karpinsky, F.G., and V.B. Larin (1967), On algebraic and differential Riccati's equations solutions. Mat. Fizika, 19, 36-41 (in Russian). Khalil, H.K., and P.V. Kokotovic (1979), D-stability and multi-parameter singular perturbation. SIAM J. Control Optim., 17, 56-65. Klimushev, A.I., and N.N. Krasovskii (1962), Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms. J. Appl. Math. Mech., 25, i011-i025. Kokotovid, P.V., R.E. O'Malley Jr., and P. Sannuti (1975), Singular perturbations and order reduction in control theory - An overview. Prepr. 6th IFAC World Congress, IC, 51.3. Kokotovi6, P.V., R.E. O'Malley Jr., and P. Sannuti (1976), Singular perturbations and order reduction in control theory - An overview. Automatica, 12, 123-132. Krasovskii, N.N. (1959), Certain Problems of the Theory of Stability of Motion. FIZMATG!Z, Moscow (in Russian). Kuzmina, L.K. (1977), On solutions stability of some systems of differential equations with a small parameter for derivatives. Prikl. Matem. i Mekhanika, No.5, 567-573 (in Russian). LaSalie, J.P. (1976), The Stability of Dynamical Systems. SIAM, Philadelphia. LaSalle, J., and S. Lefschetz (1961), Stability by LiapunoV's Direct Method. Academic Press, New York. Lee, E.B., and L. Markus (1967), Foundations of Optimal Control Theory. John Wiley and Sons Inc., New York. Lefschetz, S. (1965), Stability of Nonlinear Control Systems. Academic Press, New York. Liapunov, A.M. (1892), General P~oblem of Stability of Motion. Harkov Math. Soc. (Published in Collected Papers, 2, Ac. Sci. USSR, MoscowLeningrad, 1956, 5-263.) (in Russian). Lur'e, A.I. (1951), Certain Non-linear Problems of the Theory of Automatic Control. Gostehizdat, Moscow (in Russian). Lur'e, A.I., and V.N. Postnikov (1944), On the stability theory of control systems. Prikl. Mat. Meh., VIII, No.3, 246-248 (in Russian). Maigarin, B.Z. (1970), Certain criteria for absolute stability of control systems. Avtom. i Telemh., No.I, 188-191 (in Russian). Malkin, I.G. (1954), On the question of the reciprocal Liapunov's theorem on asymptotic stability of control systems. A v t o m . i T e l e m e h . , No.I, 188-191 (in Russian). Malkin, I.G. (1968), Motion Stability Theory. Nauka, Moscow (in Russian). Martynyuk, A.A., and R. Gutowski (1979), Integral Inequalities and Stability of Motion. Naukova dumka, Kiev (in Russian). of stability.
Ann. of
Massera, J.L. (1949), On Liapunov's Math., 50, 705-721.
conditions
Massera, J.L. (1956), Contributions 64, 182-206.
to stability theory. Ann. of Math.,
References
69
Matrosov, V.M. (1962), To the theory of stability of Motion. Prikl. Matem. i M e k h a n i k a , 25, No.5, 885-895 (in Russian). Matrosov, V.M., Yu.L. Anapolsky, and S.N. Vasilyev (1980), Comparison in Mathematical Theory of System~. Nauka, Novosibirsk, 479 p. (in Russian). McShane, E.J. (1944), Integration. Princeton University Press, Princeton. Meyer, K.R. (1965), Liapunov functions for the problem of Lur'e. Proc. Nat. Acad. Sci. U.S.A., 53, 501-503. Mishchenko, Ye.F. (1959), Asymptotic methods in the theory of relaxational oscillations. Uspekhi Matem. Nauk, 14, No.6, 229-236 (in Russian). Mishchenko, Ye.F., and N.Kh. Rozov (1975), Differential Equations with a small parameter and relaxation o~eillgtions. Nauka, Moscow (in Russian). Moiseyev, N.D. (1949), Essays on Oevelopment of the Theory of Stability. GITTL, Moscow-Leningrad (in Russian). Moore, J.B., and B.D.O. Anderson (1968), A generalization of the Popov criterion. J. Franklin Institute, 285, No.6, 488-492. Moulton, F. (1902), An introduction to Celestial Mechanics. MacMillan, New York. Mukhametzianov, I.A., and S.S. Serikbayev (1970), On necessary and sufficient conditions for absolute stability of certain nonlinear systems. Avtom. i Telemeh., 11, 11-18 (in Russian). Marendra, K.SY, and J.H. Taylor (~973), Frequenoy Domain Criteria for Absolute Stability. Academic Press, New York. Naumov, V.N., and Y.Z. Tsypkin (1964), Frequency criteria for absolute stability of processes in non-linear automatic control systems. Avtom. i Telem., XXV, No. 6, 852-867 (in Russian). Nelepin, R.A. (1967), On the problem of exact boundary of the region of absolute stability of control systems. Avtom. i Telemeh., No.4, 30-37 (in Russian). Nemytskii, V.V., and V.V. Stepanov (1960), Qualitative Theory of Differential Equations. Princeton University Press, Princeton. Partovi, S., and N.E. Nahi (1969), Absolute stability of dynamic systems containing non-linear functions of several state variables. Automatica, 5, No.4, 465-473. Persidskii, K.P. (1933), On stability of motion at first approximation. Mat. Sb., 40, 284-293 (in Russian). Persidskii, S.K. (1969), On the problem of absolute stability. Avtom. i Telemeh., No.~2, 5-11 (in Russian). Piatnitskii, E.S. (1968), New investigations on absolute stability of automatic control systems (Survey). Avtom. i Telemeh., No.6, 5-56 (in Russian). Piatnitskii, E.S. (1970), Absolute stability of nonstationary nonlinear systems. Avtom. i Telemeh., No.i, 5-15 (in Russian). Piatnitskii, E.S. (1973), On existence of absolutely stable systems which do not obey the criterion by Popov. Avtom. i Telemeh., No.l, 50-37 (in Russian). Poincar6, H. (1881-1882), Sur les courbes d6finies par une 6quation diff6rentielle. Journal de Mathdmatiques, s~rie 3, 7, 375-422; 8, 251-296.
70
Ch. I
Outline of the Liapunov stability theory in general
Pontryagin, L.S. (1957), Asymptotic behaviour of systems of differential equations solutions with a small parameter under the highest derivatives. Izv. AN SSSR, Ser. Matem., 21, No.5, 605-626 (in Russian). Pontryagin, L.S. (1958), Systems of ordinary differential equations with small parameters for high derivatives. In Proo. of the 3rd AllUnion Math. Congress, Moscow. Izd-vo AN SSSR, 3, 570-577 (in Russian). Pontryagin, L.S. (1970), Ordinary Differential Equations. Nauka, Moscow (in Russian). Popov, V.M. (1959), Criterii de stabilitate pentru sistemele neliniare de reglare automata, bazate pe utilizarea transformatei Laplace. Studii si cerretari de energetica, Acad. R.P.R. anul, IX, No.4, 6~7-680. Popov, V.M. (1960), Noi criterii de stabilitate pentru sistemele automate neliniare. Studii ~i cercetari de energetiaa, Acad. R.P.R., anul., X, No.5, 7~-88. Popov, V.M. (1961), On absolute stability of nonlinear automatic control systems. Avtom. i Telem., XXII, No.8, 961-979 (in Russian). Popov, V.M. (1962), On a critical case of the absolute Avtom. i Telemeh., XXIII, No.I, 5-24 (in Russian).
stability.
Popov, V.M. (1963), Solution of a new stability problem of control systems. Avtom. i Telem., XXIV, No.I, 7-28 (in Russian). Popov, V.M. (1964), On an absolute stability theory problem of control systems. Avtom. i Telem. j XXV, No.9, 1257-1262 (in Russian). Popov, V.M. (1973), Hyperstability New York.
Popov, V.M.
of Control Systems.
Springer Verlag,
(1974), Dichotomy and stability by frequency-domain
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of IEEE,
62, NO.5,
Porter, B. (1976), Design of stabilizing feedback controllers for a class of multivariable linear systems with slow and fast modes. Int. J. Control, 23, No.I, 49-54. Porter, B. (1977a), Singular perturbation methods in the design of stabilizing state-feedback controllers for multivariable linear systems. Int. J. Control, 26, 585-587. Porter, B. (1977b), Singular perturbation methods in the design of full-order observers for multivariable linear systems. Int. J. Control, 26, 589-594. Razumikhin, B.S. (1963), On stability of the systems of differential equations solutions with a small parameter by multipliers for the derivatives. Sib. Matem. Zhurnal, 4, No.l, 225-230 (in Russian). Rouche, N., P. Habets, and M. Laloy (1977), Stability Theory by LiapunoV's Direct Method. Springer-Verlag, New York. (Russian translation published by Miz.) Shensa, M.J. (1971), Parasitics and the stability of equilibrium points of non-linear networks. IEEE Trans. Circ. Theory, CT-18, 181-484. Suzuki, M., and M. Miura (1976), Stabilizing feedback controllers for singularly perturbed linear constant systems. IEEE Trans. A.C., AC-21, 123-124.
References
71
Szeg~, G.P. (1963), On the absolute stability of sampled-data control systems. Proc. Nat. Acad. Sci. U.S.A., 58, 558-560. Szeg~, G.P. (1964), On the absolute stability of sampled-data systems : the indirect control case. IEEE Trans. A.C., AC-9, 160-163. SzegS, G.P., and R. Kalman (1963), Sur la stabilit~ absolue d'un syst~me d'6quations auxdiff~rences finies. C.R. Acad. Sci. (Paris), 252, 388-390. Siljak, D. (1969), Nonlinear Systems. John Wiley and Sons, New York. Siljak, D.D. (1971), New algebraic criteria for positive realness. J. Franklin Institute, 291, No.2, 109-120. Siljak, D.D. (1972), Singular perturbation of absolute stability. IEEE Trans. AC, 720. Siljak, D. (1974), Stability of Control Systems. Faculty of Electrical Engineering, Belgrade (in Serbo-Croatian). Siljak, D.D., and K.C. Sun (1971), Exponential absolute stability of discrete systems. Z. Angew. Math. Mech., 271-275. Siljak, D.D., and K.C. Sun (1972), On exponential absolute stability. Int. J. Control, 16, 1003-1018. Tikhonov, A.N. (1952), Systems of differential equations containing small parameters in the derivatives. Math. Sbor., 31, ?3, No.3, 576-586 (in Russian). Tokumaru, H., and N. Saito (1965), On the absolute stability of automatic control systems with many nonlinear characteristics. Memoirs of the Faculty of Engineering, Kyoto University, 27, No.3, 347-379. Tsypkin, Y.Z. (1962a), On stability in the whole of non-linear and sampled-data automatic control systems. Dokl. AN SSSR, 145, No.l, 52-55 (in Russian). Tsypkin, Y.Z. (1962b), On certain properties of absolutely stable nonlinear sampled-data automatic systems. Avtom. i Telem.~ XXIII, No.12, 1565-1570 (in Russian). Tsypkin, Y.Z. (1963), Absolute stability of equilibrium states and processes in non-linear sampled-data automatic systems. Avtom. i Telem., XXIV, No.12, 1601-1615 (in Russian). Tsypkin, Y.Z. (1964), Frequency criteria for absolute stability of nonlinear sampled-data systems. Avtom. i Telem., XXV, No.3, 281-289 (in Russian). Vasilyeva, A.B., and V.F. Butuzov (1973), Asymptotic expansion of the singularly perturbed equations solutions. Nauka, Moscow (in Russian). Wilde, R.R., and P. Kokotovi6 (1972), Stability of singularly perturbed systems and networks with parasitics. IEEE Trans. A.C., AC-17, 2~5-246. Yakubovich, V.A. (1962), Resolution of certain matrix inequalities appearing in the theory of automatic control. Dokl. AN SSSR, 143, No.6, 1504-1307 (in Russian). Yakubovich, V.A. (1963a), Absolute stability of non-linear regulating systems in critical cases. I. Avtom. i Telem. o XXIV, No.3, 293-303 (in Russian). Yakubovich, V.A. (1963b), Ibidem. II. Avtom. i Telem., XXIV, No.6, 717-731 (in Russian).
72
Ch. I
Outline of the Liapunov stability theory in general
Yakubovich, V.A. (1963c), Frequency conditions for absolute stability of regulating systems with hysteresis nonlinearities. Pokl. AN SSSR, 149, No.2, 288-291 (in Russian). Yakubovich, V.A. (1964a), Absolute stability of non-linear regulating systems in critical cases. III. Avtom. i Telem., XXV, No.5, 601-612 (in Russian). Yakubovich, V.A. (1964b), Method of matrix inequalities in the stability theory of nonlinear regulating systems. I. Avtom. i Telem., XXV, No.7, 1017-1029 (in Russian). Yakubovich, V.A. (1965), Ibidem. 590 (in Russian).
llI. Avtom.
i Telem.,
XXVI, No.4, 577-
Yakubovich, V.A. (1967a), Frequency conditions for absolute stability of control systems with several non-linear or linear non-stationary blocks. Avtom. i Telem., XXVIII, No.6, 5-30 (in Russian). Yakubovich, V.A. (1967b), Absolute stability of sampled-data systems with several non-linear or linear non-stationary blocks. I. Avtom. i TeZem., XXVIII, No.9, 59-72 (in Russian). Yakubovich, V.A. (1968), Ibidem. II. Avtom. i Telem., XXFIX, 81-101 (in Russian). Yakubovich, V.A. (1970), Absolute instability of non-linear control systems. I : General frequency criteria. Avtom. i Telem., XXXI, No.12, 5-14 (in Russian). Yakubovich, V.A. (1975), Frequency conditions for oscillations in nonlinear regulating systems with single one-one or hysteresis nonlinearity. Avtom. i TeZem., XXXFI, 51-64 (in Russian). Yoshizawa, T. (1966), Stability Theory by Liapunov's Second Method. The Mathematical Society of Japan, Tokyo. Young, K.K.D., P.V. Kokotovi6, and V.I. Utkin (1977), A singular perturbation analysis of high-gain feedback systems. IEEE Trans. A.O., AC-22, 931-938. Zien, L. (1973), An upper bound for the singular parameter in a stable, singularly perturbed system. J. Franklin Inst., ~95, 373-381. Zubov, V.i. (1964), Methods of Liapunov and their AppZications. P. Noordhoff Ltd., Groningen. Zypkin, J.S. (1963), Die absolute Stabilitgt nichtlinearer systeme. Regelungstechnik, No.4, 145-148.
Impulsregel-
II
CHAPTER
THE STABILITY THEORYOF COMPARISONSYSTEMS
II.l.
INTRODUCTORY NOTES
11.1.1. Original concepts of the comparison method An idea
of a c o m p a r i s o n
We shall
deal w i t h
has
numerous
led to a n u m b e r problems
of f u n d a m e n t a l
concerning
the
topic
results. of this
chap-
ter. In L i a p u n o v ' s blem the
in the
memoir
first
comparison
(1892)
(Liapunov
approximation
was
tion
A in
terms system
is an
nxn
constant
× , components
not
smaller
than
of e q u a t i o n s
matrix,
of w h i c h that
following
of the
functions
(Liapunov,
"Theorem 1.
the
s t a b i l i t y prosolutions
of
first
1935,
X
second
is a h o l o m o r p h i c in their order,
vector-func-
decompositions with
solutions
the of a
approximation -
A×
X . Namely, pp.
vx~N
contain
of the
dx dt for the d e f i n e d
1935)
by c o m p a r i n g
system
dx : Ax + X(x) dt where
,
solved
Liapunov
had
stated
the
95-96).
When the defining equation corresponding to a system
of d i f f e r e n t i a l equations of the perturbed motion has only roots with the negative real
parts, the unperturbed motion is stable and
in such a way that any perturbed motion, for which perturbations are small enough, w i l l asymptotically approach the unperturbed one."
"Theorem 2.
When the defining equation has roots with the negative
real parts, then, nevertheless which would be i t s remaining roots, a
74
Ch. II
known c o n d i t i o n a l s t a b i l i t y
The stability theory of comparison systems
will
e x i s t f o r an unperturbed motion.
Namely, in case of existence of
k
such r o o t s , the m o t i o n w i l l be
s t a b l e f o r p e r t u r b a t i o n s subordinate to some equations o f the form F j ( a l , a 2 , . . . , an) = 0 , ( j = 1,2,...,n-k) in which as
Fj
are the holomorphic f u n c t i o n s of the i n i t i a l
of f u n c t i o n s
x s , t u r n i n g i n t o zero, when a l l
and which a l l o w us to represent a l l
as
values
vanish,
these values as holomorphic
f u n c t i o n s o f independent v a l u e s . " "Theorem I l l .
When among roots of the d e f i n i n g equations we f i n d
such ones t h a t t h e i r real parts are p o s i t i v e , then the unperturbed motion is u n s t a b l e . " V.V.
Nemytskiiand
V.V.
Stepanov
(Nemytskii,Stepanov,
1949, p.168) w r o t e
" S t a b i l i t y a n a l y s i s and also d e f i n i n g of the f a m i l i e s of
:
0 -curves
can be mainly done via the comparison method. E s s e n t i a l l y , t h i s method i s the f o l l o w i n g . Let there be given dxi dt = fi(t,xl,...,xn) which we c a l l
, i = 1,2,-.,n
,
(A)
the comparison system, and l e t the analyzed system
have the form : dx i d---~- = fi(t,xl,...,xn) + X i ( t ' x z ' " " x n ) where
XL
'
(B)
are " s m a l l " in d i f f e r e n t senses; then we t r y to make
conclusions on behaviour o f the i n t e g r a l curves of the system (B) by means o f t h a t o f i n t e g r a l curves o f the system (A). P a r t i c u l a r l y , much have been done in s u p p o s i t i o n t h a t the comparison system i s a l i n e a r system with the constant or v a r i a b l e c o e f f i cienzs." Concretizing P. 79)
systems
considered
(A) and
(B),
the Cauehy
d_yy = dt P(t,y) + F(t,y)
dx dt = P(t,X) The
system
Under
the
(A1)
Problem
construction properties
(At).
is r e f e r r e d
and
I, we
of s o l u t i o n s
Martynyuk
= A Y(t,y)
, y ( t o) =yo ,
mean
the
system
(1979,
(BI)
, X(to) =yo = Xo
(A1)
as a c o m p a r i s o n following
of s t a t e m e n t s ,
of the
and R. @ u t o w s k i
for e q u a t i o n s
to h e r e i n
shall
the p r o o f
A.A.
problem
(B1)
allowing
system.
: the a l g o r i t h m ' s us to d e t e r m i n e
on the basis
of the
the
system
Introductory notes
Let
75
yr:X(t;to,Yo)
y = r+x
be a s o l u t i o n
and transform
the
of the
system
(BI)
system
to the
dr : Y(t,r+×) -Y(t,x) dt Let
(AI).
form
We d e n o t e
by
:
: ¢(t,r)
us d e n o t e ~¢
: A(t)
ar
Further
, F(t,r)
: ¢(t,r) -A(t)r
on we s h a l l
consider
system
of e q u a t i o n s
d__rr : A ( t ) r + F(t,r) dt dS
_ A(t)s
dt
Similarly,
under
construction
and the
termine
the p r o p e r t i e s
, r(to) : r o
s ( t o ) : So : ro
ProbZ2m I I
the
rithms'
(A2)
.
r:o
we s h a l l
mean
•
(B2)
system
(A2)
the
p r o o f of s t a t e m e n t s
of the
(B2)
following which
solutions
:
allow
the a l g o us to de-
on the b a s i s
of the
system analysis.
As far as the that
they
comparison
are e i t h e r
systems
completely
(A)-(A2)
are
integrable
concerned,
or more
we
suppose
accessible
for a n a l -
ysis. Grebenikov
and R y a b o v
(1979, dz
dt where tem
zEG , OcR n , # e M
(B3)
they
:
is c a l l e d
In c o n c r e t e that
of first In this sults tems
interpretation
(AY),
dium,
t
o f the
qualitative
(B2),was
approach
on the
Liapunov's scalar
system
part
on
,
Alongside equations
w(O):z
o
for the
of the
(B3)
' the
sys-
,
(A3)
system
system
or so that the
with :
(A3)
(B3). can be c h o s e n
system
(A3) has
so
a number
comparison
behaviour
obtained
method
a great
of s o l u t i o n s ,
(Bellman,
1954;
number
particularly,
Demidovich,
1967,
of reof sysCesari,
et al.).
Another based
: zo
parameter.
W(t,w,~)
a system
integrals.
on the
1964,
the r i g h t
not d e p e n d
z(O)
"
of d i f f e r e n t i a l
the c o m p a r i s o n
cases
it s h o u l d
considered
Z(t,z,p)
is a small
used a system d_~w : dt
which
p.38)
in w h i c h
Liapunov
functions
equations
t i o n of the
scalar
of f u n c t i o n s .
- scalar,
usually
respectively.
the i d e a of a c o m p a r i s o n
method
set
vector
It is i m p o r t a n t or v e c t o r
or m a t r i x
in a v e c t o r
ones
- w e come
leads
of
to a
comparison
the f o l l o w i n g
function
is
on the n a t u r e
f o r m or m a t r i x
to note
Eiapunov
is u t i l i z e d ,
Depending
me-
: an a p p l i c a -
to a c o m p a r i s o n
76
Ch. II
equation
(system)
of the
but the d i m e n s i o n nyuk, tial
At the
tems
of e q u a t i o n s matrices.
11.1.2.
time
functions
classical
simpliest
of the
initial
matrix-functions
of the
state
space
system,
(Marty-
of the
matrix
mediums
solutions
of the
corresponding
ordered
by a cone
ini-
can be more
of n o n - n e g a t i v e
sysdefi-
and comparison equations
defined
development
differential
that
comparison
since
The Liapunov functions generated by them
The L i a p u n o v
that
the
are p a r t i a l l y
than
comparison
to e x c e e d
same
for the a n a l y s i s
nite
order
of the L i a p u n o v
1984b) is a d m i t t e d system.
appropriate
the
smaller
The stability theory of comparison systems
in 1 . 3 . 2 . 2
of the
second
are
applied
method,
in the
basing,
trends
in fact,
of
on the
inequalities
D*u(t,x) < 0 , D*u(t,x) -< - a ( u ( t , x ) ) These
inequalities
initial
ones
(Melnikov,
1956;
Corduneanu
(1960)
have
found
principle The
with
stability
Assumption
applied
a
general
a functional
Liapunov
Opial,
(1958)
being
theory
development
by m e a n s
in a n u m b e r
of p a p e r s system
d u : ~(u) dt t h e n and
I) a p o s i t i v e
only
then,
definite
on
when •
2) a c o n t i n u o u s ,
called
non-negative
(i0),
1964)
comparison
comparison
Ch.l,
equations
a comparison
equa-
ueR+ ' exist
:
(in the whole)
function
(u : Rn-~R)
function
~(u) e LiPu(N)
(1963, the
function.
of the ~4
'
there
u : N-~R
inequalities
1960).
of a L i a p u n o v - l i k e
For an a u t o n o m o u s
and they w e r e
complicated
, Lakshmikantham
principle
is shown I.
himself
of the more
1958;
, Antosiewicz
tion exists
by
a consideration
Antosiewicz,
quite
application
were
until
~
;
,
~(0) : 0 ,
(~(u) e L i P u ( R n ) )
such that D+u(X)
Assumption 2.
V(x~O) E N
,
For a n o n - s t a t i o n a r y
system
d u =~(t,u) dt
'
(V(xiO) e R n) . (7) of Ch.l,
a comparison
equation "
teT r
14 See i) of Comments on References to Ch. II.
ueR+
"
reR
(2)
Introductory notes
exists
77
t h e n and
2) a p o s i t i v e
only
then
definite
on
v : RxN~R
function
2) a c o n t i n u o u s ,
YrxR+ ~R+
when
there
Yr , ~ e R
,with
(v : RxR n ~ R )
non-negative
exist
:
respect
to
N
(in the whole)
;
in the d o m a i n
, w(t,O) : 0 , w(t,u) •LiPu(N)
~ ×R+
function
~:
, (w(t,u) • L i P u ( R n ) )
such
that D+v(t,×) ~ w ( t , v ( t , X ) ) Item
2 of the
conditions of the tems
link with
(7),(10)
II.I.3. Let
present
of the
fine
to the a n a l y s i s
of e q u a t i o n s
conditions
( V ( X # O ) ~ R n)
(2),(2)
for the
state
•
of s t a b i l i t y
and to the x:O
of the
study sys-
I.
Vector-functions and comparison systems
the
the
function
Definition
i.
i) positive point
totality
vi
b)
V
e)
vi
functions
Vector-function
v : Rn~Rm
N , vi(x) E C ( N )
is n o n - n e g a t i v e
on
N , V(X)~O
vanishes
4) negative
i) of the
semi-definite fulfilled
bourhood
i) positive
S 2.
3) positive
(on the neighbourhood is p o s i t i v e
x:O
exists
for
S
N:S
x:O
,
;
I) of the
of the point
Defini-
x:O
(on the n e i g n -
•
is to be c a l l e d
a time-invariant
on w h i c h
:
neighbo~rhood
it is p o s i t i v e
semi-definite
;
on the nei@hbourhood
i) of the D e f i n i t i o n
definite
point
for
semi-definite
V : Rn~Rm
fulfilled
of the
iffconditions
Vector-function
definite
S
;
N=R n ;
definite iff there
conditions
;
, Vie[1,m]
i is f u l f i l l e d
in the whoZe,
iff (-V)
, V(X~0)eN
N of the
;
, vi(O)=O
respectively.
V(x)>O
, vie[1,m]
on the neighbourhood
point
:
neighbourhood
, VxEN
or in the whole)
N , N c R n , of the
2) positive
x=O
Definition
for
semi-definite
or in the who~e),
Definition
at the point
semi-definite
I are
we de-
such that
on
iffcondition
, i:l,2,...,m ,
, for w h i c h
is to be c a l l e d
continuous
3) positive
v i : Rn~R
v = [Vl,V2,...,Vm IT
V(X) = max {vi(x) , i=l,2,...,m}
, NSR n, exists
are
2) positive
tion
of real
of a v e c t o r
semi-definite, iff a t i m e - i n v a r i a n t
X:O
a)
are
, V(X#O)~N
is d e v o t e d
u=O
stability
the c o m p o n e n t s
and
chapter
solution
of C h a p t e r
us c o n s i d e r
being
, VteT 7
in the whole, N:R n ;
2 are
S
of the point
fulfilled
iffconditions
for
N:S
1) of the
X:O
, iff
; Definition
2
78
Ch. II
4) negative
definite
the whole) iff or in the
(on the neighbourhood
(-V)
whole).
is p o s i t i v e
Prop0sition
I.
neighbourhood
sufficient a =
sup
and
In o r d e r
that
there
: ×6N}
{lIxH
N
exist
3.
For
son
(an a g g r e g a t e
the
point
for w h i c h
are
2)
a positive
with
system
,
definite
on
, vi : R n ~ R
respect
for
be p o s i t i v e
defiand
~ where
v i ( × ) 6 C(N)
, ViE[l,m]
,
(i0)
of C h a p t e r
I,
a compari-
UER m
(]) '
to
u
: (ml(U),W2(u),...,Wm(U)) 7
~(u)
~k(u)-<~(u')
V
, it is n e c e s s a r y
system)
v : [vl,v2,...,Vm ]T
a continuous
S
fulfilled.
dt
exists
tion
X=0
conditions
, VxEN,
an a u t o n o m o u s
iff
i] t h e r e
or in
x:O
neighbourhood
~i E K [ 0 , ~ [ ~ i = i ~ 2
d__uu : ~ ( u )
exists
of t h e p o i n t
(on the
vector-function
of the
functions
~2(rlxJl)
Assumption system
that
, such
~l(llxII) ~ V ( X )
S
definite
•
nite
on the
The stability theory of comparison systems
N
(in the
, viCC(N )
MCR m
in
vector-func,
vector-function
, ~ieLiPu(M)
u k--uk' . u,~u~
whole)
(v i e C ( R n ) )
, i@[1,m] , exists
. v~k
. (u.u'>e~
and
(u,u')eR m)
(or
V(~,k)e It,m] ; 3) a n
inequality
D+v(x) ~ ( v ( x ) ) is
fulfilled.
Together
with
function v i : ~xR
, v(x~O) cN
a system
of e q u a t i o n s
, Vie[1,m]
= max {vi(t,×)
,
•
(7)Ch.I,
V , V(t,x) : [vl(t,X),...,Vm(t,X)] T n ~
( v ( x i O ) eR n)
. A s it was
, vie[1,m]}
shown
(Matrosov,
1968)
we shall consider
the
components
above and
we
avector-
of which
determine
formulate
are
V(t,×)
a number
=
of
definitions. 0efinition
3.
1) positive
A
semi-definite
time-invariant a)
vi
are
Vi~[l,m] 8)
V(t,×)
c)
vi
3 are
continuous
Definition
Tr
with
is to be
, ~ E R ,iff t h e r e N
of t h e
respect
to
point
exists X=0
called
a connected
, N~R,
(t~x) ~ TTxN
:
such
that
, v i eC(T~xN)
,
;
is n o n - n e g a t i v e for
x=0
8emi-definite
fulfilled
5) positive
on
neighbourhood
vanishes
2) positive
V : RxR n ~R m
vector-function
for
semi-definite 3 are
on
N=S
on
Tz×S
, V(t,X) ~Tz×~
= 0 , Vt~T
iff c o n d i t i o n s
, Vi~[l~m] i) of the
; ; Definition
;
in the
fulfilled
~ : V(t,x)zO
, vi(t,×)
for
whole
on
N=R n ;
Tz
iff c o n d i t i o n s
i) o f the
Introductory notes
4) n e g a t i v e positive In the its
79
semi-definite
(in
whole)
on
T~
(on
TTxN )
semi-definite
(in the whole)
on
TT
(on
T~xN)
above the e x p r e s s i o n
"an
definition
conditions
are f u l f i l l e d
Definition 4. i) p o s i t i v e
on
N
semi-definite
function
w
on
2) p o s i t i v e are
definite
fulfilled
N ,
3) p o s i t i u e nition 4) n e g a t i v e tive
definite
4 are
T~xS
N:S in
definite
the
definition fulfilled
are
Proposition tive the
2.
definite point
ditions
×=0
T~×N
whole)
that
vi(t,0)
= 0 , vteT T
exists
Assumption tem
4.
For
then
i) there
function 2) there
n(t,u)
is
defi-
;
I) of the
Definition
iff c o n d i t i o n s
on
TT
TT
(on
(on
4
and
i) of the
T T x N ) iff
Defi-
(-V)
is p o s i -
TT×N ) . T T " can
"on
be o m i t t e d
iffl it s
. • V : RxR n ~ R m
is a t i m e - i n v a r i a n t sufficient
that
be posi-
neighbourhood the
of
following
con-
; ;
C E K [ 0 , ~ [ , where
~ = sup {If×If : xeN}
and
Ch.Z,
sys-
.
a non-stationary
_ ~(t,u)
system
(7)
a comparison
(t,u) e Y T x M
(TTxR m)
(4)
then w h e n
a positive
v , v(t,x)
exists
v
system)
and only
exists
N
, v(t,x) e T T x N
du dt exists
T~
TCR
, ViE[l,m]
a function
(an a g g r e g a t e
that
a positive
:
b)
there
time-in-
the c o n d i t i o n
the v e c t o r - f u n c t i o n
, where
v i ( t , X ) e C ( Y T x N ) , vie[l,m]
~(llXll) ~ 9 ( t , x )
:
;
the e x p r e s s i o n
a)
c)
on
, it is n e c e s s a r y
be f u l f i l l e d
,satisfying
on
N=R n
for all
In order on
, such
exists
, V(t,×) • Y r x N
whole
(in the whole)
In the p r e s e n t
a connected
x=O , NgR
, iff c o n d i t i o n s
for
(in the
conditions
iff
;
fulfilled
definite
w : Rn~R
.
is to be c a l l e d
exists
and there
~ V(t,x)
on
for
,iff there
TT×N
is
. •
of the point
on
w(×)
TER
iff (-v)
T~ " can be o m i t t e d
V : RxR n ~ R m
T T ,T@R
neighbourhood
positive
for all
A vector-function
definite
variant
nite
given
the
definite
on
TTxN
(in the whole)
:(Vl(t,X),V2(t,X),...,Vm(t,×))
a continuous
with
respect
to
wk(t,u) gWk(t,u') for Uk:U ~ V(U,U') e R m) , k , u e [ 1 , m ] ;
, upgup'
, k#v
T , v I :RxR n ~ R ;
(t,U) E TfxM
= (~1(t,u), ~2(t,U),...,~m(t,U)) T , W k ( t , 0 )
vector-
function
= 0 , ~k e L i P u ( M ) ,
V(u,u')eM
(or
80
Ch. II
The stability theory of comparison systems
3) an i n e q u a l i t y D+v(t,x) is fulfilled. These
systems
~ ~(t,v(t,x))
(Y~xRm)
, V(t,x) 6 Y r x N
•
are a n a l y z e d
somewhat
later
in S e c t i o n
II.3.
~s
11.1.4. M a t r i x - f u n c t i o n s The
concept
of the L i a p u n o v
a vector-function. Let
us c o n s i d e r
j:l,2,...,m
We
shall
Its c o n t e n t
a totality
denote
Let
this
determine
solutions
where
following
of f u n c t i o n s
uij
the concept
(Martynyuk,
: Rn~R
If e l e m e n t s
u12(x) U22(×)
... Ulm(×) ... U2m(×)
Uml(X)
Um2(X)
...
totality the
by
B(x)
of
1984a).
, i = 1,2,...,m .
: max {uij(x)
(i0)
Ch.I
call
given
, ie[l,m]
is an
mxm - m a t r i x ,
follows.
; je[l,m]} B
along
(B(×(e;x)) -B(x)U) Uij = I , V(i,j) e [l,m]
of the m a t r i x - f u n c t i o n
: I
the Liapunov
by f o r m u l a
: Lim inf ~i @~0 +
B(x)
it
in what
of the m a t r i x - f u n c t i o n
B(x)
uij(x)
Umm(×) and
conditions
a derivative
of s y s t e m
U : (Uij)
u11(×) U21(X)
under
B(X) shall
is the
generalizes
:
matrix-function ~
We
matrix-function
-"
L ~B(x)
B
are
iff
(i,j) ~ T ( X )
;
iff
(i,j) • Y ( x )
,
continuous,
then
where l(x) Definition
5.
i) positive point
The m a t r i x - f u n c t i o n
uij
b)
B(x)
c)
uij
:uij(×)
:B(x)}
B :Rn~R m×m
semi-definite iff a t i m e - i n v a r i a n t
x:0 , NCRn ,
a)
: {(i,j)e[1,m]
are
exists
continuous
such on
is n o n - n e g a t i v e vanishes
for
X=0
that
N ; on
N ;
.
is to be c a l l e d neighbourhood
N
: of the
:
uij(x) • C ( N ) B(X)~0
, V(i,j)e[l,m]
15 See 2) of Comments on References to Ch. II. 16 See 3) of Comments on References to Ch. II.
, V(i,j)e[l,m]
, VxeN ;
;
;
Introductory notes
2) positive
81
semi-definite
iffall c o n d i t i o n s
5) positive
1) of the D e f i n i t i o n
semi-definite
Definition 4) negative
semi-definite
Definiti0n
6.
i) positive
-B
is p o s i t i v e
The m a t r i x - f u n c t i o n
B(X) >0
definite
4) n~gativ~
S
6 are f u l f i l l e d
3.
x:0
or in the whole).
there exist
sup {U×II : xeN} , s u c h that ~B(X)
~2(llxll)
, Vx~N
.
x:0
iff con-
1) of the D e f i n i t i o n
S
of the point
functions
B
be p o s i t i v e
that in a n e i g h b o ~ r h o o d
N
, i:~,2 , w h e r e
, V(i,j)e[1,m]
of
~ :
, and ¢l(IIxll )
of f u n c t i o n s
uij(t,X)
: T T x R n ~ R , V(i,j)
and d e t e r m i n e
The d e r i v a t i v e of the s y s t e m
of the m a t r i x - f u n c t i o n
B: ~ × ~ n ~ R m x m
.
along solutions
(7) Ch.I we define by
°
B(t,x)
1
: lira inf ~ { B [ x ( t + 8 @~0
U =(Uij)
6
or in
•
¢iEK[o,~[
uij(x ) e C(N)
×:0
(on the n e i g h b o u r h o o d
B(t,x) = max {uij(t,x ) , ie[1,m] , j e l l , m ] }
where
N , and
•
Let us c o n s i d e r a t o t a l i t y ell,m]
neighbourhood semi-definite
N=S ;
for
In order that the m a t r i x - f u n c t i o n
it is r~ecessary and s u f f i c i e n t x=0
:
N=R n ;
definite (on the neighbourhood iff -8 is p o s i t i v e d e f i n i t e
Propositi0n the point
is to be c a l l e d
of the point
in the whole i f f c o n d i t i o n s
for
of the point
X=0
(on the
;
the whole)
definite
of the point
B is p o s i t i v e
on the neighbourhood
i) of the D e f i n i t i o n
are f u l f i l l e d
S
S
semi-definite
8 : Rn~R m×m
×=0 , on w h i c h
, V(X~0) ~ N
definite
3) positive
N=S ;
1) of the
N=R n J•
definite iff there exists a t i m e - i n v a r i a n t
2) positive ditions
for
for
×=0
or in the whole).
NcR n , o f the point and
5 are f u l f i l l e d
(on the neighbourhood
or in the who~e) i f f S
of the point
in the whole, iff all c o n d i t i o n s
5 are f u l f i l l e d
neighbourhood
S
on the neighbourhood
is an
If e l e m e n t s uii(t,x ) E Y~ xR n ) then 8(t,x)
+
x+8f(t '
mxm-matrix,
Uij = 1
are c o n t i n u o u s = I
x))] - B ( t , x ) U } ,
,
, V(i,j)~
for all
/i,m]
(t,X) 6 T T x N
-~
iff
(i,j) e Y(t,x)
;
l D+B(t,x)
iff
(i,j) e T(t,×)
,
where l(t,×) = max {(i,j)e/1,m]
:uij(t,x) =B(t,×)}
(V(t,×)
82
Ch° II
Definition called
7.
The
matrix-function
B : T f × R n ~ R mxm
is to be
:
i) p o s i t i v e
semi-definite
time-invariant a)
uij
on
Y~ , T E R ,
neighbourhood
are
continuous
N
with
b)
B(t,X)
c)
uij
2) p o s i t i v e 7 are
~emi-definite
fulfilled
3) p o s i t i v e
4) n e g a t i v e
7 are
Definition
8.
tive
N ,
x=0 , NSR n, such
(t,x) E T ~ × N
that
TrxS
= 0 , VtEY~
iff c o n d i t i o n s
;
, V(i,j)e[1,m]
;
i) of the D e f i n i t i o n
;
in the whole
on
YT
(in the whole)
T T (on
on
(in the whole)
for all
reR
iff c o n d i t i o n s
l) of the
N:R n ;
for
"on
7 the e x p r e s s i o n
on
Ir
YtxN ) (on
iff
(-B)
lrxN ) .
Y T " can be o m i t t e d
iff its
condi-
. • B :TTxR n ~R mXm
The m a t r i x - f u n c t i o n
definite
on
neighbourhood
N
semi-definite
function
w
on
Yz, T e R , iff there of the point
on
N ;
Y~xN
and
w : Rn~R,
is to be
2) p o s i t i v e fulfilled
5) p o s i t i v e nition
definite for
definite
8 are
on
N:S
TTxS
is a c o n n e c t e d
x:0 , NOR n, such there
exists
satisfying
w(x) ~ ( t , x )
time-invari-
that
B
a positive
is posi-
definite
the c o n d i t i o n
, V(t,x) e T r x N
iff c o n d i t i o n s
;
i) of the
Definition
8 are
i) of the
Defi-
;
in the whole
fulfilled
for
on
N:R n
YT
iff c o n d i t i o n s
•
4) n e g a t i v e
definite
(in the whole)
on
T r (on
T~xN )
positive
definite
(in the whole)
on
Y~
T~×N) , r e s p e c t i v e l y .
In the D e f i n i t i o n ditions
are
8 the e x p r e s s i o n
fulfilled
Proposition
4.
be p o s i t i v e
definite
hood
:
, uij(t,x )
B(t,x) ~ 0 , V(t,x) E T T x N
on
N=S
a connected
:
i) p o s i t i v e ant
on
fulfilled
fulfilled
to
, uij(t,0)
semi-definite
In the D e f i n i t i o n
of the point
X=0
semi-definite
is p o s i t i v e
called
for
semi-definite
Definition
are
for
exists
;
is n o n - n e g a t i v e
vanishes
iff there
respect
eC(YrxN ) , V(i,j)e[l,m]
tions
The stability theory of comparison systems
In order
of the point
ing c o n d i t i o n s
for all
be
on
×=0
reR
"on
Yr " can be o m i t t e d
that
the m a t r i x - f u n c t i o n
TT×N
, where
fulfilled
u~j(t,×)eC(TT×~)
b)
uij (t,0) : O , V t e / T
; ;
:
iff
is
(-8)
iff
its con-
. •
N
, it is n e c e s s a r y
a)
(on
B :~×~n ~ R m x m
is a t i m e - i n v a r i a n t
neighbour-
and
the
sufficient
that
follow-
Comparison equations
c) there such
83
exists
a function
¢(llXll) Remc~k I. (Martynyuk, appear order
in case than
consider
the
~B(t•x)
autowski,
there
is n e e d
first
one
• V(t,x)
~ : sup {llxll:xeN}
i979).
ETTxN
v(k)(t,×)
system
(7)
vector-Liapunov
functions.
= (vl(t,x),v2(t,X),Vm(t,X))
= ~vj/at + (Vvj) T f(t,x)
(~)vj (t,x)
: ovj
Further
.
we a s s u m e
right
Really•
we
r , the
full
deriva-
in virtue
of
by f o r m u l a s
v~l)(t,×)
Naturally,
of h i g h e r
V,
J
Ch.I
naturally
of d e r i v a t i v e s
vj(t•x) E C £ ' £ ( T T x N ) and s u c c e s s i v e l y c a l c u l a t e t,x • and its c o m p o n e n t s up to £-th - o r d e r
tires
smooth
,
. •
The m a t r i x - f u n c t i o n s
for a c a l c u l a t i o n
of the
a vector-function V(t,X)
where
~eK[0,e [ , where
that
part
(4-i)
here
that
together
on we can make
/at + (Vv
~-I))T
system
with
, je[1,m]
f(t,x)
(7) Ch.l
functions
has
v.] ,
je[l,m] a sufficiently
je[i,m]
.
a table
Vl(t,X)
v2(t,x)
...
Vm(t,x)
v~l)(t,x)
v$1)(t,x)
"'"
vm(I)(" X) ~D~
; "
,.,,....,.,,.b.,....,..,....,.,oo
V~~)(t'x) from which
the
corresponding
II.2. We
matrix-function
conditions
are
fulfilled
v(~) m (~,X)
. . .
can be
,
constructed
(Definition
21,
if the
Section
4.1).
THE LIAPUNOV FUNCTIONS AND COMPARISON EQUATIONS
shall
study
equation which (see
Liapunov
V~£)(t,x)
(7)
stability
Ch.l
the point
via the u=O
of the
state
analysis
is the
x=0
of the
equilibrium
of the
system
comparison state
(the
described
equation zero
by
(2) for
solution)
1.2.3).
Further ential
on we
shall
have
need
for
some
results
of the
theory
of d i f f e r -
equations.
11.2.1. On monotonicity and solutions estimations According
to the
general
concept
of the
comparison
method,
the
rela-
84
Ch. II
tionship and
between
the
of o b t a i n i n g (2). (2)
stability
solution
The
u:0
A necessity in this
to
of their
However,
study
state
for
5.
rectangle
such
properties
Let
functions
D : t o st Sto+a
satisfy
cases
of the
interval
u=0
is to
equation
~(t,u)
(2)
Ch.l
necessity
of the
integrate
system
equation
exceptions.
solutions
u(t) , ~(t)
[to, to+8]
the
initial
are
(0<6~a)
arises
P&OO~.
It
consider
and
~dz
, V(t,u) •D
of c o n t i n u i t y where
the
satisfy
exist
connection
a
U(to) = Uo
in a
,
value
z(t) > 0
.
d~fferentiable
on the
equations
, ~(to) = Uo for
~(t;to,Uo)
a function
z(t)
to
<
t
, then <
to+6
= ~(t) - u ( t )
. and
its E u l e r
that
of
z(t)
function
of c o n t i n u i t y
there
and
: d--~ d~ t:to _ _dull _ dt ~t=to
t - t_o
On a c c o u n t
[to,to+6]
<
is o b v i o u s
point
to
be d e f i n e d
; a,beR+
continuous and
conditions
We shall
Z(to) : 0
8(t,u)
dt
u(t~to,Uo)
derivative.
and
, ~(t,u) ~ C(I'I)(D)
dt
there
z
: ~(to,Uo)
exists
is p o s i t i v e .
and d i f f e r e n t i a b i l i t y t I (t o < t I < to+6) for
-~(to,Uo)
a neighbourhood Let
of the
at an i n t e r v a l
of f u n c t i o n s , where
t o < t < t I . Besides,
> 0
u
and
z(t I) : 0
In
at the
point
t : tI
is ~-~dzIt:tl:
~(t I , ~ ( t l ; t o , U o ) ) - ~ ( t ~(tl;to,Uo)
which
solution
(7)
to the
an i n e q u a l i t y
If f u n c t i o n s
there
system
lead
are, in reality,
, lU-Uol ~ b
~(t,u)
this
the
(2)
determination
~(t,u) eC(1'Z)(D)
with
of the
system
connection.
Proposition
and
x:0
solution
conditions
way
in q u a d r a t u r e s .
of the
of the
stability
simpliest
The stability theory of comparison systems
is i m p o s s i b l e
due to the
of the
Proposition the
5. It f o l l o w s
Proposition
:u(tl;to,Uo)
,
condition
~(tl,u)
hence,
I , u(tl;to,Uo) ) ~ 0
-~(tl,u) f r o m this
5 is proved.
•
> 0 contrast
that
Z(tl)> 0
and,
Comparison equations
Definition
9.
85
The
solution
an upper
is called
(~ower)
u(t;to,Uo) for any
< r+(t)
solution
u
r(t)
defined
( u ( t ; t o , U o) > r-(t))
of the
equation
(2) w i t h
(to,Uo)eD
.
Theorem i .
(The comparison p r i n c i p l e ) .
be c o n t i n u o u s
and
let
there
du dt Let
us
suppose
[to,to+6l[ such
that
_ w(t
'
Cauchy
u)
for
solution
d_~ ~dv
t=t o
the
initial
value
~:
uo ,
RxR+~R+
u ( t ; t o , U o)
pass-
problem =Uo
"
u+(t;to,Uo )
is a d i f f e r e n t i a b l e
V(to) g u o
, Vt ~ ]to,to+B[
solution
U(to)
'
[to,to+6]
(2) iff
Let a function
a unique
of the
an u p p e r
and there
that
exist
(to,Uo) ~ R×R+
ing t h r o u g h
on an i n t e r v a l
solution of the equation
is d e f i n e d
function
over
v : [to,to+~l[ ~ R
and
w(t,v)
, Vt • [to,to+61[
,
(5)
then v(t) ~ u + ( t ; t o , U o ) The
theoremremains
in i n e q u a l i t i e s in e s t i m a t i o n
correct
(5) and
The p r o o f
of the
theorem
and
(1980)
,7
Laloy
Proposition defined
6.
Let
on an open
if we
(6) and
(6) i n s t e a d
set
t • [to,to+61[ the
consider
of an u p p e r
functions domain
0(t,u)
D'c D
"~"
a lower
(6) of
u - ( t ; t o , U o)
"~" solution
one. by Rouche,
w(t,u)
satisfy
• instead
in the book
and
and
<w(t,u)
sign
u*(t;to,Uo)
is i l l u s t r a t e d
e(t,u) and
for
be
Habets
continuous
and
an i n e q u a l i t y
, V(t,u) • D '
,
also
/o(t,u)/ Suppose
that
u - ( t ; t o , U o)
~m
6 =min(a,b/m) upper
and
l~(t,u)l
,
and
lower
let
and by
~+(t;to,Uo)
, ~ - ( t ; t o , u o)
equation
by
u+(t;to,Uo )
and
of an e q u a t i o n U(to) = U o ,
upper
and
lower
solution
of an
d~ dt
passing
, m~+.
us d e n o t e
solution
du _ w ( t , u ) dt
~m
through
T h e n on a joint
the point interval
-
s(t,~)
,
(to,Uo) of e x i s t e n c e
of the
considered
solutions
obtain ~ - ( t ; t o , U o) ~ * ( t ; t o , U
o) < u - ( t ; t o , U o )
Jv See ~) of Comments on References to Ch. II.
~ u+(t;to,Uo ) .
•
we
86
Ch. II
The
proof
orem Let
of the P r o p o s i t i o n
The stability theory of comparison systems
6 follows
f r o m the
Proposition
l and The-
i. us c o n s i d e r
a question
nection
with
Propositions
Suppose
that
we w o u l d n ' t
son e q u a t i o n We shall satisfy
5,6 and be able
Luzin's
in the
bounds
such
of the
with
problem
solutions for
(Luzin,
a solution
1951).
of the
compari-
two
functions
g(t,u)
and
~(t,u)
which
domain
, V(t,u)6D
D , moreover,
,
the d i f f e r e n t i a l
equations
them,
be i n t e g r a t e d
Cauchy
results
in con-
relationship
d~ _ g(t,~) dt would
application
to o b t a i n
g(t,u) < ~ ( t , u )
connected
theorem
(2).
try to give the
of C h a p ] y g i n ' s
d~ , ~-~ : ~(t,~)
exactly.
connected
~(t;to,Uo)
Then
with
and
, ~(t o) : ~ ( t o ) : u
an , u n k n o w n
the e q u a t i o n
~(t;to,Uo)
o ,
to us,
solution
(2) can be found
passing
through
of the
between
the point
uo
t:t o , i.e. ~ ( t ; t o , U o) < u ( t ; t o , U o) < ~ ( t ; t o , U o)
on
a general
Thus,
interval
Propositions
by means
of e x i s t e n c e
5,6 j u s t i f y
of its substitutior~
of the
considered
thc c o m p a r i s o n by
a simpler
solutions.
equation
nne,
(2) a n a l y s i s
admitting
the
exact
integration. The
proof' of the P r o p o s i t i o n
orem Let
6 follows
from
the P r o p o s i t i o n
5 and The-
i. us c o n t i n u e
inequality fulfilled (HI).
of i ~ e q u a l i t y
with
it. Namely,
(5) t o g e t h e r let the
To
exist
an i n t e g r a l conditions
be
functions
w(t) , k(t)
continuous
and n o n - n e g a t i v e
;
(H2).
There
(HS).
There
exists
exists
tion
g(z)
(t,u) e D
a functiorl
and n o n - n e g a t i v e a continuous, for
z~0
c(t,u) for
, continuous u~0
, moreover,
is a d m i s s i b l e .
: w(t)u+c(t,u)
respect
and n o n - d e c r e a s i n g
g(z o) > 0
A representation ~(t,u)
with
to
;
non-negative
a @ R+ ; (H4).
with
following
:
There on
analysis
connected
, v(t,u)~D
for
z o ~ ]0,a]
func,
Comparison equations
Theorem
2.
Let
1) a s s u m p t i o n s 2)
87
(HI)-(H3)
be f u l f i l l e d ;
inequalities
a)
u(t)
~ a+
[w(s)u(s)
÷e(s,u(s))l
ds
,
o
b)
e [t,
zexp
w(s)ds)]
exp(-
w(s)ds)
to
~ k(t)
g(z)
,
t~T o
,
to
hold. Then
for all
t e [to,r [
u(t)
the
following
s G- 1 [ G ( a ) +
estimation
k(s) ds]
exp (
w(s) ds)
t o
where
G- 1
is
a function, :
,
(7)
t o
inverse
G(W)
is v a l i d
to
ds
that
~
of
0
~a~u
j
o
and
the
value
r
c a n be d e t e r m i n e d ds
from
= G(a) +
o
If to the 3)
there
enumerated exists
conditions
a constant
s
then
the
Proof.
inequality We a s s u m e
(5)
is
note c
(s) ds .
one
such
ds
:
that
fulfilled
for a l l
t6T o
and
Va 6 ]0,5[
that
that [ t, z e x p (
(s) ds )
y
w(s) ds) ] ~ k(t) e x p (
t o
c(t,u)
more
~
z : u e x p (and
If:
we a d d
~6R+
a relationship
-< u ( t ) e x p (
w(s) ds) g(z)
, Vter o
t o
w(s) ds)
g (uexp
[-
(s) d s ] )
Jt o We
u(t)
transform
-< a +
the
inequality
w(s) u(s) + k(s) to
2.a)
of the
exp (
Theorem
2 to the
w(s) ds) g(u(s) to
exp [-
form
(s) ds]
ds
88
Ch. II
The
stability
theory
of
comparison
systems
t
= R(t) exp (
i
w(s) ds)
to
It
is
not
difficult
d---R-Re x p ( dt
to
i
w(s)ds)
find
÷R(t)
out
that
w(t)exp
(
to
-< w ( t )
i
w(s)ds)
~
to
R(t) exp (
w(s) ds) +k(t)
exp (
(s) ds)g(R(t))
to Hence
As
far
as
a>0
in
virtue
dR dt
< k(t) g(R(t)) -
of
the
Condition
(H3)
of
the
Theorem
2,
we
obtain
IR(t)dr --a
Itk(s)
< -
g(r)
ds
to Further
o n we
shall
obtain
G(R(t))
< G(a
+
k(s)
ds
to taking
into
Let
pay
us
account our
R(t)
can
make
side.
Let
us
the
notation
attention
to
inadmissible
define
G(-)
The by
condition an
fact
dr
of unbounded
that
unbounded
which
:
in a g e n e r a l continuation
R(t) =~
: G(a) +
continuation
~ _
to
to
the
right
equation
(s) ds .
of
the
dr
:
inequality
will
dr
+
for
those
Uo
be
correct
function
R(t)
~t o a<~ values
i.e.
we
have
t e [to, ~[
~=~ an
for
dr
~-~7"
uo
k(t) dt _< g-CFY <
For
an
function
to k ( t ) dt
for
from
case
inequality
or
This
the an
~ , for
C(w)
ae
inequality
]0,~[
ae
]0,~I
dr
for
which
is g i v e n
Comparison
equations
89
R(t) and
-< G -I [G(a) +
(s) ds]
estimation u(t)
-< R ( t )
exp (
(s)
ds)
-< G - 1 [ G ( a )
+
k(s)
ds] exp
(
(s) ds)
o
will
be
Example filled
correct.
•
I.
w(t) =1
Let
conditions
and
o f the
u(t)
c(t,u)
Theorem
-< a +
=u 2
2 we
[u(s)
for all
obtain
+u2(s)l
t->t o
from
ds
the
, t->t o
Having
ful-
inequality
,
t o
an
estimation a exp (t-t o ) -< ~ + a [i - e x p ( t - t o)]
u(t) where
t e [to,r [ ,
l+a = to+ln
Really,
functions
Theorem
2 are
k
the
and
g
far
T
mentioned
in e s t i m a t i o n
as
l[
: e x p ( t - t o)
l dr
a(u)
:
1
~
, g(z) : z 2
.
u :G-I(k)
=
: ~
7V : -~+uo
the
-x+---1 ' Uo
k ( s ) ds
~d r+
=
to
Further
of
1
'
o
then
a ~ a(a) +
2 b)
following k(t)
AS
for
exp(s-to)
o
1 exp(t_to -- - g1 + u--g
ds
) _ 1_
to
on i G-l(p)
:
- [G(a) +
i
a
t
i +a[l
-exp(t-to)]
"
u(s) 0s] + !u o
to
Now, T
basing
can
on
inequality
be d e f i n e d
from
(7),
an dr
Theorem
3.
Let
1) c o n d i t i o n s function 2)
the
desired
estimation.
Value
equation dr
7~ :
~-~
o
o
e x p ( s - t o) ds
+
to
:
(H1)-(H3) w(t) = - ~ ( t )
inequalities
we o b t a i n
be
fulfilled, holds
moreover,
f o r all
t~t o
in c o n d i t i o n ;
(H1)
the
90
Ch. II
The
stability
theory
of
comparison systems
[t a)
u(t)
b)
c [t,
g a + i [- ~ ( s ) u(s) + c ( s , u ( s ) ) ] ] to
zexp (-
I)
Then
for
all
t~
u(t)
is
valid
and
S
the
[to,r[
an
(s) ds)
T
If
to
the
3)
there
enumerated exists
the
a constant
estimation
The
proof
Let
us
the
condition
of
note
the that
(8)
Theorem the
Case
defined
(s) ds)
(8)
from
s) ds
Theorem such
3 we
add
one
more
and
all
ae
:
that
o
holds
for
5 is
all
t->t o
similar
inequality
2 a)
to
that
must
be
of
the
]0,~[
Theorem
fulfilled
. •
2.
together
with
:
2,3
can
equations
i.
Let
all
b~
u +c(t,u)
, VteY o
utilized
for
obtaining
solutions.
Let
us
conditions
(t) dt
, U(to):a
of
< ~
the
various
consider Theorem
and
several 2 be
dr'
< ~
there
exists
a constarlt
~e
]0,+~[
(t) dt
=
such
estimations
, to¢I
that
of
com-
cases.
fulfilled.
uo
and
, t_>t o , z_>0
Special cases of the general comparison equations
Theorems parison
of
~cR,
d~__{ < - ~ ( t ) dt -
II.2.2.
e x p (-
: G(a) +
conditions
t
then
(s) ds]
carl be
dr
_< k ( t ) g ( z )
estimation
O -I [G(a) +
value
,
f;o
( s ) d s ) ] exp (
t
hold.
ds
,
Suppose
that
Comparison, equations
91
T h e n there exists
a constant
M(a,t o)
such that
u(t) s M ( a , t o) < +~ for all
t e [to,~ [
Example 2.
a 6 ]0,~[
Let,
w(t) : ~-~ 1 In case
and
and
c(t,u)
= -t21 - exp [_(i__~_i )to + u exp (~i
_~_l)to]
i from an i n e q u a l i t y
1 t~!u(s)+~_~
u(t) ~ a~
exp
__/_1)
[-(
exp
to +u(s)
i (~-
__i)]) d s
, t~t o ,
to
an e s t i m a t i o n I u(t)
(lo)
_< exp [ - ( i - t~)] in i + [exp(-a) _ _ 1 ] t to
is f u l f i l l e d Functions
for all
k
and
t E [to,+~[
g
and
of the T h e o r e m 1 k(t) : ~ -
a 6 ]0,~[
, where
~: into
.
2 have the form
, g ( z ) = exp(z)
as far as - to
,
(%- - to
~-~ exp -
Functions
G
- to
([ - to
1 t2 exp(z)
G -I
and
G(u)
=
exp(-r)
dr = -exp(-u)
+ e x p ( u o)
;
0
-u
= in
u = G-I(x) Further
[-X÷exp(-uo)l
= -In
[-X+exp(-uo)l
Orl,
I/ # :
tdr exp(-r) d r +
= - tn[-(G(a)
+
_
-
]tr2
0
G-I(#)
_
[
___1)
1
exp(-a) + e x p ( u o ) + ( - ~ + t o
1 1 = - in [%--~o +exp(-a)]
k(s) ds) + exp(-uo)l 0
Taking into account estimation (7). Value
~
can
be
all
defined
I;:
(t) dt
th~s,
we o b t a i n
the
from ~'ondition
(9)
=
]
<
~
,
I>
inequality
:
exp(-a')
(10)
<
~:
,
from
the
92
Ch. II
The stability theory of comparison systems
_~1 : e x p ( - ~ ) to
Case 2.
Let
all
conditions
o f the
[k(t) + w ( t ) ]
Then
there
exists
E : in t o
< ~
dt
a constant
Theorem
2 be f u l f i l l e d . <
,
M(a,to)
such
dr
Suppose
that
:
that
u ( t ) - < M ( a , t o) <~, for all
t e [to,~ [
E x a m p l e 3.
ae
]0,~[
, where
~
can
be a r b i t r a r i l y
large.
Let
1 w(t) =~-~ In c a s e
and
and
2 from
an
u(t)
ul/3
= ~1
e(t,u)
~t
exp(-
)
for
all
t->t o
inequality
-< a +
[
u(s) + ~
1 ul/3 (
s) e x p ( -
)] ds
t>_t o ,
to an e s t i m a t i o n
1 u(t)
and
are
2/3 2
_!_1) 3/2 ]
e x p (-i)
(12)
value
fulfilled
arbitrarily
for
M(a,to)
= [(aexp (
all
[to,~[
large.
c [t , 2 exp (
t~
Let
us p r o v e
I
w(s) ds)]
exp (_1)
1
Conditions
i [w(s)
k
(11)
+k(s)]
and
g
k(t)
: ~exp
are
ds
]0,~[
w(s) ds)
, where
fact
~
we d e t e r m i n e 1
fulfilled
:
exp
_1_1) :
1
:
on
be
2
(-3-~jo) as
far
2
1
expressions
, g(z)
=
2
~--~ zl/3exp (-3---~o) in the
r -I/3dr
:
A2/3 < ~
as
dt +
;
form
z z/3
i t2
~
dr
:
1
to
~
) + _ to _<~;
r -I/3 dr
=
r 2/3 :
to
dr
can
that
=
exp ( ~ - - t o
to
Further
ae
It is t h e
Jt 0
: c [ t , z exp ( - ( ~ - - t o ) 1 functions
~Jo 2/3 2 3/2 )) +3--~o]
and it.
Jt 0
For
1 -~(~-to
-< [(a exp (~o))
Comparison
equations
93
= Jut-i/3 dr :
G(U)
3 ( U 2 / 3 -Uo2/3") = k
;
"U 0
2 u = G - i ~ X j "" = ( ~ X + u
p
r -1/3dr +
=
exp (-
2/3.3/2 o ~
;
) dr = ~_(a2/3_u2/3) + (~__tol i)
exp (-3t~-)
;
o
G-I(#)
+ t k ( s ) ds) + u2/3] 3/2
i
_2 (O(a) = [7
[a 2/3
=
-
1 j_l) ~2 (~-to exp
3/2 2 (-3--~)]
at o
On the basis of e s t i m a t i o n Case
3.
(7) we o b t a i n
Let all c o n d i t i o n s
the i n e q u a l i t y
of the T h e o r e m
[w(t) + k ( t ) ] dt < ~
(12).
2 be fulfilled.
,
< [ g--~
Suppose
that
=
Jto
and there exists
a constant
~ 6 ]0,~[ (t) dt =
T h e n t h e r e exists
a constant
such that ~
M(a,to)
such that
u(t) < M ( a , t o ) for all Example
t e [to,~[ 4.
and
a E ]0,~[
, where
In Case
<~ M(a,to) ~ 0
for
a-*0 , t o > 0 .
Let
w(t) =~2 ' e ( t , u ) for all
.
=u2~
exp ( )
t>-to>O . 5 from the i n e q u a l i t y u(t)
-< a +
[-~-iu(s) +
u2(s) exp ( )1 ds
, t->t o
an e s t i m a t i o n a exp [ _ ( ~ _ _ i ) to u(t)
is f u l f i l l e d
for all
and the c o n s t a n t
-<
1 i) 1 + a (F- t o
t ~ [to,~)
M(a,to)
and
M(a,to)
exp (t-~)
a 6 ]0,~[
can be d e f i n e d a
]
, where
~ = t o exp ( - ~ )
by an e x p r e s s i o n
t o exp (t-~)
: t o - a exp (~-:-) ~o
94
Ch. II
TO c o n v i n c e contained
ourselves
in that, we must
in e s t i m a t i o n
c [t , z e x p ( - ( ~ -
t~
The stability theory of comparison syst~n~s
determine
functions,
w h i c h are
(7). F r o m the fact that
1 1)= ))] exp ( [ - to
1 (~) z 2 (_~+2) ~-~ exp exp to
exp
1 1__) ( ~ - to
= z 2 ~ 2 e x p ( ~ j o) we d e t e r m i n e
k(t) Functions
p
and
G -I
have
dr
p =
= ~-Texp(
) , g(z) = z 2
the form
exp(
) dr
1 +~
=
1 i) + (-~+to
exp(
) ;
0
a
e-~(p) : By c o m b i n i n g
these
functions
(13). An e x p r e s s i o n
~
1 ÷ a ( 1~ - 1__) to e x p
(t~)
into i n e q u a l i t y
(7), we o b t a i n
exp(
Case 4.
estimation
can be d e f i n e d f r o m an e x p r e s s i o n
Let all c o n d i t i o n s
) dt
:
of the T h e o r e m
[w(t) + k(t)] dt < ~
,
dr . ~-~
2 be f u l f i l l e d .
(')dr |g(--~-y =
dr
Suppose
that
=
to Then there
exists
a constant
M(a,to)
such that
u(t) g M ( a , t o for all
t e [to,~ [
Besides, £×amp]e
and
M(a,t o) ~ 0 5.
Let
a e ]0,~[
for
<
, where
~
can be a r b i t r a r i l y
large.
a~O , to>O
i 1 w(t) :~-~ , c(t,u) :%-~u
for all
t~to>0
. In Case
4
from the i n e q u a l i t y ,t u(t)
~ a + 2 1 s~ u(s)ds u(t)
is f u l f i l l e d
for all
large;
~ a exp [ 2 (_i_i i
to -E)]
t 6 [to,~ [
the c o n s t a n t
and
M(a,to)
a e ]0,~[
To o b t a i n e s t i m a t i o n We d e f i n e
(14) , where
~
can be arbi-
can be d e f i n e d by an e x p r e s s i o n
M ( a , t o) = a exp ( ~ )
one.
t~t o
~0
an e s t i m a t i o n
trarily
,
to>O
(14) we must act s i m i l a r l y
i
as w i t h the p r e v i o u s
Comparison
equations
95
I
w(t) : ~ and
further
, g(z) : Z
on [w(t)
+k(t)]
dt
=
~-
~to
dt
= ~
o
A
A --
:
~
i n s
:
0°
,
:
i n s
:
o Functions
p
and
G -I
have
A
the
form
a + i +~) P : in u ~ (-t Do On the b a s i s estimation to>0
.
Case
5.
of i n e q u a l i t y
(14).
Let all
be f u l f i l l e d .
(7) and the
It is o b v i o u s
conditions
Suppose
G-l(p)
"
that
there
exists
: aexp
(~ i t o -t)
calculated
lim M(a,to)
of the T h e o r e m
3
functions : 0
for
(w(t) = - ~ ( t )
we o b t a i n
a~0
and
, Vtm t o > 0 )
that dr
Besides,
< ~ ,
a constant
~e
<
]0,~[
, such
that
(t) d t :
Then
there
exists
a constant
u(t) for Case
all 6.
t e [to,~[ Let
be fulfilled.
all
and
a•
]O,g[
for
all
t e [to,~[
(s) ds)
3
(w(t) : - ~ ( t )
, Vt~ t o >0)
that
I. [
t 0
exists
that
.
of the T h e o r e m
I Then there
such
~ M ( a , t o) e x p ( -
conditions
Suppose
M(a,t o) < ~
•
a constant
M(a,t o) <~
such
that ds)
u(t)
~ M ( a , t o) e x p (-
~(s)
and
ae
~
10,~[
, where
can
be a r b i t r a r i l y
large.
96
C h . II
Case be
7.
Let
all
fulfilled.
conditions
Suppose
The
of the
stability
Theorem
3
< ~
dr
f
<
tO
there
of
comparison
(w(t) : - ~ ( t )
systems
, Vt~ to>0 )
that
k(t) dt
Besides,
theory
exists
a constant
~•
k(t) dt
]0,~[
=
~
~
:
such
that
.
to
Then
there
exists
a constant
M ( a , t o) < ~
such
u(t)
~ M(a,to)
exp (-
~(s)
and
a ~ ]0,~[
, where
that
ds)
t o
for
all
t o >0
t • [to,~[
Case 8.
Suppose
~0
for
there
for all
exists
M ( a , t o) ~ 0
Case
9.
c(t,u) : 0
Let
,
:
such
~ M(a,to)
exp (-
and
a e ]0,~[
, where
and
to>0
.
(H4)
there
for
a~0
in C o n d i t i o n to
dr
M(a.,to) < ~
u(t)
. According
of a comparison
< ~
a constant
t c [to ~ [
and
is
(Rouche,
that
(s) ds)
~
be the
Habets
c a n be a r b i t r a r i l y
function
and
Laloy~
w zp 1980)
stable
o)
p(s) de ~ a
;
o
stable
if
(Sa>O)(VtoeYo)(Vtat
o)
p(s) ds
g a
to c)
and a solution
(15)
if
is u n i f o r m l y
large;
equation
(Vtoero)(3a>O)(Vt~t b)
and
:
du d~ : p(t) u a)
a~O
that (t) dt
Then
M(a,to)
.
is e q u i - a s y m p t o t i c a l l y
(VtoE ~ )
stable
if
p(s) ds ~ - ~ o
for
t4+ ~
;
Comparison equations
i0.
Case
c(t,u)
Let
in
97
Condition
= a exp[-ku]
(H4)
+ ¢(t) - a
there
, where
be
the
¢ : R~R
function is
w(t)
: 0
a continuous
and
function,
a,keR t A solution
of
the dv -dt
has
the
is
= a exp[-kv]
defined
us
some a)
pay
for
our
all
~(t)
: 0
on
is
defined
if
¢(t)
c)
for
:
defined
if
to
for
defined
the
,
(16)
e x p [-k ~ ( s ) ] ds}
, where
, ~
¢(t)
, to= 0
$2
equation
function
, heR÷
(16)
and
.
consider
it
under
:
: - a t + k -I in { e x p ( k c ) tot
o
,
is a
c6]0,~[
small
(b-a) t + k -I in
v(t)
II.2.3.
~(s) ds-at
the
all
all
v(t o) = c = 0
is
v ( t o) = c
{ e x p (kc) + ak
s;
:
= b , heR+
v(t) is
,
, then v(t)
b)
- a
tot o , c E ]0,~[
attention
hypothesis
if
+¢(t)
: ~(t) + k -I in
~(t)
Let
equation
form v(t)
and
comparison
, ~
value,
exp(kc)
-
t~Y o , c6]0,~[
he~+
- i} ;
then a
, ~
[exp (-k(b-a)t) , heR+
, b la
-i] ;
, then
= O(t)
for
+ exp(kat)
all
+ k - 1 i n {1 + a k
teTo
exp (-kO(s))
ds}
,B.
Genera] s t a b i l i t y theorems on the basis of scalar comparison equations
I n t h e comparison method, a c o n n e c t i o n between s t a b i l i t y and
stability
which
follow
of
the
from
solution
the
of
comparison
equation
(2)
is b a s e d
principle
(Liapunov,
the
of
o f the s t a t e
on
estimations
1955;
Chaplygin,
1919). G.I. the
Melnikov's
(1956)
result
is
basis
sequel.
18 See 5) of Comments on References to Ch. II.
the
statement
given
in
98
Ch. II
Proposition 7. tinuous
on
a) there
If the v e c t o r - f u n c t i o n
T×N
(on
exists
we
This
is i n c l u d e d statement
equations
equation
ec(l'l{TrxS)]-
the m e t h o d
8.
the
of the
~rthe
by the
Ch. I, is con-
function
N
v
on
S (on
;
function on
v
(on
comparison
the A s s u m p t i o n
In order
that
the
(i) be a s y m p t o t i c a l l y
tion
in v i r t u e
Tr × N )
,
of sys-
then
equation
construction,
construction
of c o m p a r i s o n
i.
direct
state
stable
u =0
of the
(in the whole)
comparison it is n e c e s s a r y
:
function
~(u)
satisfies
the
condition
2) of the A s s u m p -
i;
b) that
~(u) < 0
(For the P r o o f Theorem
4.
ous
Tx N
on
i) there the
, [v(t,X)
definite
(7),
:
11.5).
sufficient
a) that
in s y s t e m
definite
derivative
into
f
besides
positive
can be p r o v e d
(see
Pr0posit~on
and
Dini
and,
Ch. I , is n e g a t i v e
can i l l u s t r a t e
which
)
EC(I'I)(s)
b) the r i g h t - h a n d (7),
Tr×N
a decrescent
TrxS ) , v(t,X)
tem
The stability theory of comparison systems
fulfilled
see the S e c t i o n
(on
u >0
Tr x N ) , then
an open
(u E R + )
•
3.5.) f
iff
connected
in s y s t e m
i)-4)
(7),
Ch. I, is c o n t i n u -
hold,
time-invariant
neighbourhood
S
of
x :0 ;
state
exists
a decrescent
TT × S ) and a f u n c t i o n
exists
a function
the P r o p o s i t i o n
positive
a,
a(~xll) ~ v(t,x) 5) there
for
If the v e c t o r - f u n c t i o n
exists
2) there
is
which
definite
belongs
which
class
v
on
S
(on
K , such that
(V(t,x) 6 Y r X S )
, V(t,x) 6TxS
~
function
to the
satisfies
conditions
a)-b)
of
7 a n d the e s t i m a t i o n
D*v(t,X)
~ ~(v(t,X))
, V(t,x) eYxS
(V(t,x) e T r x S )
is f u l f i l l e d ; 4) c o n d i t i o n s the sSate
x=O
Theorem
5.
ous
TxS
on
i) an o p e n
x: 0 ;
a)-b)
of the
of s y s t e m
Proposition (7),
Ch.I,
If the v e c t o r - f u n c t i o n (on
Y r x S ) and there
connected
time-invariant
8 are
fulfilled,
is a s y m p t o t i c a l l y f
in s y s t e m
exist
(7),
stable. Ch. I,
is c o n t i n u -
:
neighbourhood
S
of the state
Comparison equations
99
2) a d e c r e s c e n t
positive
function
belonging
a
definite
to tne class
a(llxll) ~ v ( t , x ) 3) a f u n c t i o n
~:
(2), p a s s i n g Vte T
$ (on
~×S
) and a
(V(t,x) e~×S) ,is c o n t i n u o u s
solution
the point
on
such that
, (t,u)~(t,u)
of the unique
through
K
v
, v ( t , x ) eYxS
RxR+ ~ R +
sures e x i s t e n c e
function
u ( t ; t o , u o)
and it en-
of the e q u a t i o n
(to,u o) ~ int (TxR+)
; ~(t,0) = 0 ,
and D+v(t~X)
~ ~(t,v(t,X))
(V(t,x) E ~ x S )
, V(t,X) E T x S
then a) s t a b i l i t y
of
of s y s t e m
u :0
of the e q u a t i o n
(2) implies
stability
of
x =0
(7), Ch. I ;
b) a s y m p t o t i c
stability
of
u=0
of the e q u a t i o n
asymptotic
stability
of
X:0
of the s y s t e m
If in a d d i t i o n
to 1)-2)
4) a f u n c t i o n
b , belonging v(t,X)
of T h e o r e m
5, one more
to the class
K
(2) implies
equi-
(7), Ch.l. condition
is f u l f i l l e d
:
obeys
~ b ( l l x l l ) , V ( t , x ) eTxS
(V(t,x) e~xg)
then c) u n i f o r m bility
stability of
x=0
of
u=0
of the e q u a t i o n
of the s y s t e m
(2) i m p l i e s
stability
of
u=0
of the e q u a t i o n
uniform asymptotic
stability
of
x:0
of the s y s t e m
For a p r o o f
see T h e o r e m s
5, 7, Ch.l,
(2) implies
(7), Ch.l.
and the book by Rouche,
Habets
and
(~980, p.66).
Example
6.
In the p r o b l e m
in the sense of L i a p u n o v Gutowski,
of s t a b i l i t y
(Aminov,
1979) we c o n s i d e r
pseudo-linear
form
dt For this
s y s t e m we shall
-
A(t,x)
consider
in v i r t u e
x
,
xTB(t)
motion
in a
form)
definite
function
v,
X
of e q u a t i o n s
D*v(t,x)
Martynyuk,
xeR n
a positive :
1962;
of the p e r t u r b e d
: the R o s e n b r o c k
v(t,x)
w i t h the d e r i v a t i v e
of a body of the v a r i a b l e
1959; M a t r o s o v ,
equations
(also c a l l e d dx
of the p e r t u r b e d
motion
: x T dB d-C x ,
where AT(t,x) B(t) + B(t) A(t,x) and
sta-
(7), Ch.l;
d) u n i f o r m a s y m p t o t i c
Laloy
uniform
0
is a zero-matrix.
= 0
for all
(t,x) e ~ × S
,
mass
100
Ch. II
According
to the T h e o r e m
function
v(t,×)
D*v(t,X)
are
not
coincide
variable parison
with
mass.
I, p. 85 , the p o s i t i v e
and a c o n s t a n t
sufficient
negativeness
conditions
the n e c e s s a r y
The
equations
The stability theory of comparison systems
situation
for
ones
of the
stability.
(Aminov,
can be s o m e w h a t
application.
For the
definiteness
derivative
These
1959)
conditions
for bodies
changed
expression
of the
by the
D*v(t,×)
do
of a
scalar
com-
we have
an
estimation
IId~ll p ( v ( t , × ) )
D*v(t,×) where
p(v) = (a-l(v)) 2 . Here
class
K,
is such
consider
a comparison
~ v(t,x)
du
:
dt
and
far as
the
a(-)
, which
belongs
to the
v(t,x)
-
U
is a p o s i t i v e
condition
for
, V(t,x)e~xS
.
equation
-
As
function
that
a(ltxD) We
the
,
stability
II p(u)
definite
of
the
form,
then
solution
u=O
is
Ila-6~ dt <
A)
According
to T h e o r e m
of the i n i t i a l
5 :
system.
As
1) we
conclude
far as
stability
(Matrosov,
1962)
of the
solution
for b o d i e s
x=O
of a vari-
able mass
I
I
~ dbij
--~--dtl
= Ibij(~)
-bij(O)l
< ~
o
holds,
then
the
V(i,j) • [l,m] ber
times.
condition
This
situation
blems
for b o d i e s
Thus,
on the basis
que
sufficient
A)
of a v a r i a b l e of a s c a l a r
condition
[0,~[
is quite
form
v(t,×)
condition
often
also
change
probable
if d e r i v a t i v e s the
sign the
in n u m e r o u s
dblj/dt
finite
practical
numpro-
mass. uomparison
of s t a b i l i t y
the q u a d r a t i c was
can be f u l f i l l e d ,
on a s e m i - a x i s
. As it was
necessary.
equation
we d e f i n e
the
: the p o s i t i v e - d e f i n i t e n e s s stated
by A m i n o v
(1959)
uniof
this
Comparison equations
10l
I I . 2 . 4 . The generalized comparison equation Alongside
w i t h the s y s t e m of e q u a t i o n s
(7), Ch.l,
we shall c o n s i d e r
a
scalar equation d u : ~(t,u,x) dt determined
on
YoXR+×N
(N ~ R n)
and n o n - d e c r e a s i n g
from the d o m a i n
Yo .
Let us n o t e that for a s o l u t i o n
9.
as g i v e n
with respect
of the p r o b l e m
for values
~(t,u,X) to
(u,x)
(17)
the f u n c t i o n
that the f u n c t i o n
~ : YoXR+xN ~R+
is c o n t i n u o u s
and n o n - d e c r e a s i n g
with respect
fixed
which
the u n i q u e
t,
(17), p a s s i n g By
[to,P[
solution
ensures
through
(~ > t o )
we d e n o t e
u ( t ; t o , U o)
ferentiable,
r(t o) ~ u o ;
2)
dr ~ ~ ( t , r , x ) ~-[
on
t
x(.)
the m a x i m a l
interval r(t)
, (t,u,x)
to
(u,x)
for
of the p r o b l e m for
x(t)eN
of e x i s t e n c e
be c o n t i n u o u s
of the
and dif-
such that
[to,~[
Then
r(t)
(For a p r o o f
see p a r a g r a p h
Theorem
u ( t ; t o , u o)
(to,Uo) e int (ToxR +)
. Let the f u n c t i o n
r : [to,~[ ~ R +
i)
solution
some p o i n t s
be de-
for fixed
teY o .
~(t,u,x)
Proposition
Suppose
(17)
. Let the f u n c t i o n
fined, c o n t i n u o u s
can be c o n s i d e r e d
, u(t o) : u o ,
~ u ( t ; t o , U o)
5 is g e n e r a l i z e d
, Vt E [to,~[
and
3 of the p r e s e n t
x(t) e N
.
•
Chapter).
in case of the c o m p a r i s o n
equation
(i7) as
follows.
Theorem 6. tinuous
on
f
If the v e c t o r - f u n c t i o n TxN
(on
i) an open c o n n e c t e d 2) a d e c r e s c e n t function
a,
T ~ x ~ ) and there time-invariant
positive
definite
which belongs
a(llxll) ~ v ( t , x )
of the s y s t e m exist
neighbourhood
function
to the class
, V ( t , x ) eTxS
(7), Ch.l,
is con-
:
v
on
K,
S S
of the (on
state
x:0 ;
Y~ xS ) and a
such that
( V ( t , x ) eTrxS ) ;
3) a f u n c t i o n : T×R+×S
continuous,
~R+
,
non-decreasing
t e n c e of the u n i q u e the s y s t e m
(t,u,×)
with respect
solution
(17) p r o v i d e d
D+v(t,X) ~
~ ~(t,u,×)
, ~(t,0,0)
to
u(t;to,Uo)
, Vt~T
,
(u,×) and e n s u r i n g
exis-
of the C a u c h y p r o b l e m
(to,Uo) e int (T×R+)
~(t,llXll,x) , V ( t , X ) eYxS
: 0
and
( V ( t , x ) ETTxS ) ;
for
102
Ch. II
4) a s o l u t i o n
of the s y s t e m d_xx = f(t,×) dt
for all a)
tET
(for all
u-stability
b) a s y m p t o t i c
du = ~(t,a-l(u),x) ' dt
t6T~
×=0
of the s y s t e m
of the system
u-stability
equi-asymptotic
(18)
) , then
of the z e r o - s o l u t i o n
ity of the state implies
The stability theory of comparison systems
of the z e r o - s o l u t i o n stability
(18) implies
stabil-
(7), Ch. I ; of the system
of the state
×=0
(18)
of the s y s t e m
(7), Oh. Z. If in a d d i t i o n
to i)-4)
5) for a f u n c t i o n
b,
v(t,x)
one more c o n d i t i o n
which belongs
~ b(llxll)
is f u l f i l l e d
to the class
, V(t:x)
eTxN
(V(t,x)
:
K,
eTTxS ) ,
then c) u n i f o r m uniform
u-stability stability
d) u n i f o r m a s y m p t o t i c (18) implies system Pr00~.
of the z e r o - s o l u t i o n
of the state u-stability
uniform asymptotic
x=0
of the s y s t e m
of the s y s t e m
of the zero s o l u t i o n stability
(18) i m p l i e s
(7), Ch. I ; of the s y s t e m
of the state
x=0
of the
(7), Ch. I .
F r o m the fact that the
function
,~(t,u,x)
is n o n - d e c r e s c e n t ,
we d e f i n e ~(t,][xl],x) ~ ~ ( t , a - l ( v ( t , x ) ) , x ) for
llxJl ~ a - l ( v ( t , x ) )
. Hence,
D+v(t,x) A comparison
equation
~ oJ(t,a-l(v(t,x)),x)
has the form
:
d u = ~(t,a-1(u),x) dt In view of T h e o r e m any
teT T
determine
i for any
, for w h i c h
, U(to ) = U o
(to,Xo) 6 Y l x S
x(t;to,Xo)
and
and
. (to,Uo) e T i x R +
u(t;to,Uo)
are defined,
an e s t i m a t i o n a(llxll) ~ v ( t , x ( t ; t o , X o ) ) ~ u ( t ; t o , v ( t o , X o ) )
On the b a s i s stability,
and we
of e s t i m a t i o n
stability
be d e t e r m i n e d
(19) and c o n d i t i o n s
of the state
similarly
x=0
as in the T h e o r e m
Example ? (Hatvani, 1975).
We c o n s i d e r
described
equation
by a d i f f e r e n t i a l
19 See 6) of COmments on References to Ch. II.
of the s y s t e m
of the s y s t e m 6
(19) (18)
u-
(7), Cho I,
can
~9. •
a mechanical
system,
w h i c h is
Comparison equations
103
_d2y _
~dy + b(t) g(y)
+ a(t)
: 0
dt 2 where
the
function
(a g e n e r a l i z e d function
g
a
(damping
rigidity)
coefficient)
is c o n t i n u o u s l y
is c o n t i n u o u s
on
]-~,+~[
is c o n t i n u o u s
differentiable
motions
sufficiently
small
substitution
are
considered
initial
dx I : X2
dt
We i n t r o d u c e
over
the
perturbations
y = xI , y = x2
dx 2 dt
-
b
]0,+~[
;
-b(t)
•
interval
y(O)
it is o b t a i n e d - -
'
and
on
and
Xl ~ 0 The p e r t u r b e d
t >0 -
'
, y(O)
[0,+~[
. By u s i n g
for the
that
g(xl)-
a(t)x
2
.
the n o t a t i o n s a
if
a _>0 ;
0
if
a <0
: {
[a]
O
if
a->O
-a
if
a <0
;
[a] + Let
us c o n s i d e r
a positive
definite
v(t,xl,x where that
q
is
q(t
for
Io r~(t)]
Lq(t)
System
(18)
2)
a continuously
->qo > 0
has
, function
= -~- + q ( t )
Ii
g(s)
differentiable
ds
,
function
t -~0 , qo = c ° n s t
dt < ~
v,
on
To
such
and
lq(t)-b(t)
I exp[2t]
dt
<
+
the
form
dXl dt : x2
'
dx2 dt
:
-b(t)
g(xl) - a(t) x 2 ,
du d-T : p(t) u + (q(t) - b(t)) g(x I) x 2 where p(t) Having
fulfilled
: rq(t)l
the r e q u i r e m e n t 2 lq(xl) x21
and t a k i n g stability y=y=O
into of the
+ 2[a(t)]
account
that
zero-solution
-< (I +u2)llxll 2 , u : c o n s t llx(t)il -< K H x ( O ) N and,
hence,
exp (t)
stability
, , we define of the m o t i o n
, if
dg(~) dy
-< u
; -~
;
(la(s)l + Ib(s)l) ds
<
u-
104
Ch. II
II.2.5. We
The scalar comparison equation construction
consider
a system
of equations d_~x : dt
in w h i c h nomials lutely
X E R n ,- P
is an
of smallness converging
contains
bounded
fs
are
of the p e r t u r b e d
Px +
q(x)
on
YoxN ,
N
q(x)
second
contains
order,
, which
satisfy
continuous
poly-
or series
of the s t a t e
VtET ° ,VxEN
some n o n - n e g a t i v e ,
(20)
matrix;
t h a n of the
in the n e i g h b o u r h o o d functions
motion
+ h(t,x)
nxn -constant
not m o r e
lhs(t,×)I ~ f s ( t ) where
The stability theory of comparison systems
x=O
the
;
abso-
h(t,x)
estimation
, Vse[1,n]
,
(21)
and bounded
functions
of
time. By applying the
a linear
non-singular
s y s t e m of e q u a t i o n s
form
transformation
(E0) we r e d u c e
the m a t r i x
P
(det A ~ 0)
to
to the d i a g o n a l
: d_yy = AF + w ( y ) + P ( t ) dt
where with
yER n ;
A
is a d i a g o n a l
the c o m p o n e n t s
vector
function,
which
Following
the r e s u l t s
for v a r i a b l e s Ys
Let
us n o t e
complex
that
values
of the
complex
moduli
will
take
connected
pressions
-
conjugate
be o n l y
nI
, Ys ~s
Ys
the
(22)
(1969)
we
q ;
p
is a
shall
consider
ex-
= rs e x p ( - i 0 s )
variables
coincide,
. For
function
vector
(21).
are complex.
variables (n I < n )
work
,
is a v e c t o r
estimations
values
variables
w
account
of T i k h o n o v ' s
for r e a l
Ys As
then
are real,
and
far as m o d u l i rs
for rs
of d i f f e r e n t
a function
nl V : ~ rs s:l
-
1 + ~
nI ~
" " (Ws e-18s + W s eles)
i + ~
s:l
It is o b v i o u s
that
An e s t i m a t i o n
of terms
first
into
= rs e x p ( i ~ s )
~s
dv nlZ : ~srs dt s:l
realized
matrix;
with
we define
The
y = A×
v
in the d o m a i n
nI Z
" i@s) (Ps e - 1 8 s + p s e
s:l
is a p o s i t i v e of the r i g h t
definite part
function.
of e x p r e s s i o n
N = {y : lYsl < h , h e ~ + ]
t e r m can be e s t i m a t e d
in the
(25)
following
(23) w i l l be
. way
n1 ~srs s:l
~ ~v
, ~ : max{a s , se[1,nl]
}
(24)
Comparison equations
105
For the second t e r m it will be
nl I~
n1
Z (Ws e-i0s + W s e i O S ) l s=l
N v2
~ ~s s:l
~v2
=
(25)
where nl Z ~s s=l
:
and
(ml---mn) I h k - 2 ~ las ml+...emn:k k:2,3,...
~s :
if mn a(ml...mn) Ylml Y2m2 ""Yn
WS = ml+m2+...+m n : k k=2,3,... are the c o n v e r g e n t T a k i n g into account follows
series in the d o m a i n estimation
N.
(21) the t h i r d t e r m can be e s t i m a t e d
as
: nI Z (Pse -I"e s + S:l
l g1 where
"
nl ~(t)
is a b o u n d e d
non-negative
From estimations
nI z Ips(t)l sml
~se~Os)l~
:
Z ~ [asj I Ps(t) s=l j:l
function.
(24)-(26)
we o b t a i n a d i f f e r e n t i a l
d--t < ~v + ~ v 2 + ¢ ( t ) for all
A comparison
t~Y o
equation, du dt
--
is an e q u a t i o n
and
v
, heR+
corresponding
inequality (27)
,
.
to the i n e q u a l i t y
(27)
+~u 2 :
(26)
n
dv
determined
~ ¢(t)
~U
of the R i k k a t i
+~(t)
,
type and,
U(to)
: vo
(28)
,
in general,
it is not i n t e g r a t -
ed in q u a d r a t u r e s .
Remark
I.
If ( V t o e T i) ( V t e T o)
vo +
ft
then an i n e q u a l i t y
v(t) ~ a +
~(s)ds
-< a ,
aER+
o
(~v(s) + # v ~ ( s ) ) ds Jt o
can be a n a l y z e d
Remark 2.
on the b a s i s of T h e o r e m s
Possibility
of the a p p r o x i m a t e
be c o n n e c t e d w i t h the e x p o n e n t i a l correlation
2,5. analysis
majorization
of e q u a t i o n
(28) can
of the right part of
106
Ch. II
The stability theory of comparison systems
~v + / 3 v 2 -< Z(e-kV-l) where
values
~
and
k
are
chosen
bo obey
max [ ( ~ v + # v 2) - ( £ e - k V - i ) ] v~h* The
obtained
comparison
can be c o m p l e t e l y
The
(1969),
constant
h # >0
condition
rain
for
h* < h
.
equation
analyzed
values
the
~
d u : £(e-kU_l) dt
paper
,
(see 2.2).
£
and
k
k
'
k:~
+~(t)
As it is shown
can be d e f i n e d
in T i k h o n o v ' s
by f o r m u l a s
+7
is d e t e r m i n e d h*
from
-<
the
condition
that
keR
,
3~
8~
Finally, h* : m i n { h
Example with
8 (Tikhon0v,
1974).
lengthwise
symmetry
the
is d e f i n e d of such
by the initial
a lengthwise
in the p r o c e s s metry
of motion.
sider
a non-controlled
such bounds we
With
the
motion
of motion.
"p(ml...m4) Psk , s
initiating fourth
in t h e i r
order,
s e [1,4]}
the
of the body parameters
simple;
decomposition
moreover,
atively
large"
Vedrov,
1938).
hl
(over t h e i r
initial
which
and
motion
moment
violate
the
of the p r o b l e m the p e r t u r b e d
variation.
(ml'"m4)
Ps
constants,
and sym-
we
con-
motion
variation,
Equations
Xs
with
ml...x~4 + X s
in
that
of the
the roots
and
:0 ~2
absolute
, S e [1,4]
Xl
are h o l o m o r p h i e
the
terms not lower
in the n e i g h b o u r h o o d
det (P - h l ) are
body
this
c o n d i t i o n Ibr r e a l i z a t i o n
of the m o t i o n
of the m e d i u m
rigid
plane;
form
and c o n v e r g i n g that
of the
(at the
of the p e r t u r b a t i o n s
motion
are
. We a s s u m e
The
is the a b s e n c e
dx s 4 - ~ PskXk + Z dt k:l mlim2+msim4:m m:2,3 where
a motion
in the v e r t i c a l
the a i m of s i m p l i f i c a t i o n
density
have
~ ~ <0
consider
plane
data
of the k i n e m a t i c
can n e g l e c t
perturbed
We
motion
of motion)
, - 3 ~}
N c
, P : [Psk have
]4
s,k=l
"small"
value)
and
negative
functions than
of the
{IXsl ~ h
of a c h a r a c t e r i s t i c
,
,
equation
• ~3 ,~4 real
parts
"compar(see
Comparison equations
By m e a n s tem
to
of
the
107
non-singular
a diagonal dys dt
Further
- XsYs +
on we
transformation
we
reduce
the
initial
sys-
form
~
~ P ml'"m4) yml ml*...*m4 : m m:S,q
introduce
ym~ + y s
' s e [1,4]
variables
: YS --
ZS
...
(mlm2)
Z
m!
Bs
m2
Yl
Y2
, s = 3,4
,
ml+m2: m m=2,3 so t h a t
ddt zs s : 3,4 tions that
they
could
- ksZ s +
b~
lower
third
the
function
) • b~
than
solutions
the
inequalities
4 [ 4 (i) b(ml...m4) mI m4q ~ E b s k Yi + sk Yl ""Yq J k=3 i=l ml+...+m4 : 2
, where not
satisfy
of
z3 , z4
order)
1"''m4) the
(if w e
quickly
the
neighbourhood
tend
It
the to
Zs
are
is n o t
terms
zero.
nonlinear
difficult
higher
Really,
than let
us
to
' func-
verify
those
of
consider
4 z I%12 s:3
:
Ny c R 2
constants, order.
neglect
enough
v
In
are
fourth
Zk + Zs
this
function
satisfies
a differential
inequality d v g 2 (-a + h'H) dt where
a : max (~3,~4)
M = Hence,
for
4
4
Z s=3
~ k=3
4
vo = v ( t o )
as
far
as
the
(i)
( ~ Ibsk i=l
I +
IZs[
quickly
gibly
small
. It
h*
(ml'"m4)
Z Ibsk ml+...+m4= 2
I)
we h a v e
value
then
,
;
IZs(t)l
v
a
is
[-(a
large
decreasmand,
means
that
it
- h~M) t ] , enough
beginning is
possible
s = 3,4
according with to
use
to
, the
t o >0
condition,
become
approximate
tions Ys
:
B(mlm2)
~
s
ml m2 Yl Y2
, S = 3,4
ml+m2: m
m:2,3 for
a transformation
dy~ dt
of
= ~sYs +
the
two
~
ml+m2= m m=2,3
remained Q ~mlm2)
equations
Ylml Y2m2 +
~.~
to ,
the
s =
3,4
form •
negli-
correla-
108
Ch. II
Here
~s
third The
are nonlinear
order,
obtained
functions
]Ys] -< h
' O ~s( m l m 2 )
equations
by means
Zs
The stability theory of comparison systems
: YS ÷
with are
power
higher
constants
than
which
those
c a n be
o f the
calculated.
of a substitution
~
C (sm l m 2 )
Ylm l Y2m 2
, S : 1,2
ml+m2= m
m=2,3 under
a suitable
choice
of
C (mlm2)
are
s
reduced
to the
form
d Z l : k l Z l + ~ ( 2 1 ) z~z. 2 d---~ ul ; dz2 dt where
and We
, C~ 12)
uI
Further
on w e
take
into
represent
are
calculated.
introduce
account
: h 2 z 2 + C ~ 12) z I z~
the
notations
that
variables
zI
~1,2
: - e _+ i~
and
z2 = ~i
z I = p exp (i0) It
is n o t
difficult
to d e t e r m i n e P
8
: pt
-
E1
that ¢~1 - 1/2 --2-)
in l e x p ( - 2 ~ t ) +
CI~ I + Co O~I
2~i
C I = (-~-I - ~ I ] e x p ( - 2 ~ t o) 2 ~ Pc the
solutions
zI
C 2 = 8o-~to
' and
z2
are
of the
solution
a)
~i < 0
(~ > O)
b)
~i > 0
(~ > O)
illustrates
that
p
tends
zl, 2
to
for
zero
~ In [ e x p ( - 2 ~ t o ) + 2~ I in t h e
.
:
the
domain
A refined method of comparison equations construction course
of construction
of
comparison
of modulus
of c o m p o n e n t s
of
derivative
of the
t h e total
initial
whole
II.2.6. the
small
over
perturbations
estimations
CI e ~i I .
form
[±iS(t;to,~o)l
either
too
~'
initial
In
or under
+
obtained
Zl, 2 = P ( t ; t o , P o) e x p Analysis
form
z 2 : p exp (-i8)
= (C I e x p C 2 ~ t ) +
where
Thus,
,
in the
Liapunov
of
the
equations right
function.
o f the
perturbations,
we
part All
often
use
in e x p r e s s i o n
this,
as a r u l e ,
Comparison equations
simplifies leads
construction
to m o r e r o u g h
of the
comparison
(Tikhonov,
1965,
We c o n s i d e r
dYs
dt
where
a special
p~
are
existence
case
By m e a n s
with
the
condition
tem
the
system
zero
only
In this for the
~ 0
that
system
precise
definition
Let
there
be given
equations
(yV)
sense (29)
(v • [2,m])
having
the
the
(29)
f o r m of poly-
of c o n s t a n t l y
satisfies
of the
effecting
condition
corresponding
b s(ml...mn ) , w h i c h
of i n v a r i a b i l i t y
of e q u a t i o n s
special
In the
, s • [l,k]
Cauchy
of pro-
b(ml...mn) Zm ~ s Zlml z~2 ""Znmn k:2 ml+...+m2: k
the
(29)
constants
values
considered
chosen part
in c o n f o r of the
sys-
is r e d u c e d to a n o n l i n e a r canons Pv can be d i f f e r e n t f r o m
of the
case
first
are
of a linear
indices
~,
of a s p e c i a l
equation
of the
i.e.
root, system
when
divisors
for
kl =~i ~ 0
(29)
takes
the
m
dYl dt In the r e m a i n e d
one
or h i g h e r
- X
equations
coefficient
tiplier
Let
it
transformation
form
the
time method
(29).
connection
~n 1 ~ ~ n l - i g'"g ~2 < 0
the
solution
coefficients
mity w i t h
form.
root.
terms
have
system
: Zs +
constant
(29),
same
in 11.2.5
has a m o r e
+ Ys(t,y)
nonlinear
continuous
Ys
vA - ~ s
s v
+ Pp Y
We a s s u m e
of the
ical
at the
1975).
of a zero
Ys(t,y)
of a n o n l i n e a r
the
but
illustrated
construction
Melnikov,
constant
connected
The
motion
functions
perturbations.
equations,
forms.
equation
: AsYs + # s - l Y s - 1
nomials;
blem,
of these
their
1969;
of the p e r t u r b e d
with
109
p~
of the
power.
us d e t e r m i n e
+
IYl
Z ~ +Yl(t,y) gk Y k:2
of the
different
system
from
"non-special" Assume
the
also
(29)
(30)
each
nonlinear
zero o b l i g a t o r y
variables
that
P~
V :
nl ~ rs
y2,y3,.-,y n
0...0 = 0
term with
contains
for
as a mul-
in the
first
s ~2
function
s:l
in the
form
v = Vl+V o , where nl Vo
corresponds ative
terms,
of
to the
non-special
function
corresponding
v
to
:
~ r s s=2
variables. in
s:1
virtue
of
, namely
In
expression
the
system
of (29)
the we
total
deriv-
isolate
some
,
llO
Ch. II
nI __dr : ~ ~srs dt s=l
nl ~ ;Is-l(Ys-i s:2
+ ~I
n] E
1
The
stability
e-lOs+
theory
Y s - I eiOs)
(p~ y e e -iO s + p-s y v e i0s)
comparison systems
of
+ gk Y t~"- l r l
+
n1 ~ (Ys e-i0~o + y
+ ~1
S:2
" eZ0s)
s:l
(31) We
estimate the
the
first
terms
term
right
nI
~srs
: al
rs
+
s=l
term
~
~
the
~
rs
~ alv
(31)
as
follows
+ (~2-~1)
Vo ;
nl-i g
s-i Ys-I e - l O s
Z
#r s ~ gv
, ~s-i ~#
, s e [2,n]
,
s:l
is a s u f f i c i e n t l y
third
expression
:
s:2
where
(as-~±)
s=2
n1 Re
of
nI
~
s=l
second
part
:
nl
the
of the
term
small
constant;
:
m m k-i * k V 1 -< ~ + ~ klgkl Z gky I k 2 gkv k=2 k:2 m
k-i VoV
where
* the i
fourth
term
.
[
gk
: ~
gk
for
k
odd
,
]gk]
for
k
even
;
:
nI Z (p~yVe-iOs
+P2 ~VeiOs)
s
s:2
n1 Z
Ipvr Sl
I~,I-i
Vo
:
m 52
Ck V k-I
Vo
•
k--2
s:2 nl
Ck : *
the
fifth
term
ship
taken (31)
into
takes
nl ~ (Ys e - i 0 s ) s=l
account
the
Taking absolute
is
such
into
v +
account
value
the
~
n1 N ]Ysl s-i
illustrated
~ ~(t)
.
estimations,
the
relation-
form m
d--tdv _< ( ~ l + g )
s:2
:
Re
Having
;
4p;l
lUt=k
*
k
~ gkv k:2 the
is not
v o
fact
small,
that
m
+
[~2 - e l that we
the
can
+
X k:2
root
show
(ck+k[gk]) v k-l]+ ~(t) ~2
that
is n e g a t i v e the
m
~2
:
~I + ~ (Ok +k]gk]) vk-1 k=2
domain
and
its
v ~h ~ , h*
:
Comparison
In this
111
equations
connection m
d___~v < ( ~ i + ~ ) v + dt in the d o m a i n A comparison
v-~ h ~
and
equation,
for all
E
gkvk + ~ ( t )
,
(32)
k=2
teT o
corresponding
to
the
inequality
(32)
m
~ k d__uu : +~) u + E gk u + ~ ( t ) dt (~1 k:2 has
the p e c u l i a r i t y
k,
can
Rzm~rk
be
that
5.
In case
of a pair
for odd values
of the
index
which
+
similarly
corresponding du --dt : ( ~ l + ~ ) u
Inequalities
(32),
roots
m g~ vk +@(t) E k:3
is c o n s t r u c t e d
equation
of special
~I ~ 0 , ~nl ~ n l _
I ~...
inequality
d~v ~ ( ~ i + ~ ) v dt holds,
gk
negative.
~2 < 0 , a d i f f e r e n t i a l
parison
coefficients
(53)
(34)
v ~ h*
for as the
, k -odd
inequality
,
(32).
(34)
A com-
to it is m * k E gku + ~ ( t ) k=3
+
can be a n a l y z e d
, k-odd
on the
basis
.
(35)
of T h e o r e m s
2, 3.
11.2.7. Several applications of scalar comparison equations A. S u p p o s e turbed
that
for a t i m e - i n v a r i a n t
following
i) m a t r i x 2) in
P
conditions
3) t h r o u g h bounded
are
is stable,
per-
Cr
each point for all the
is a n u m b e r from
]lq(x)]1 ~CrHX]J
N
there
passes
at least
t ~ T o , and b e l o n g i n g following
notations
of c o m p o n e n t s
; is fulfi]]ed,
(
to
(Nemytskii,
n 1/2 z Pik ) i,k:l
qi
one
solution
X(t;to,Xo)
N. 1955)
:
,
of a v e c t o r
(ql,q2,
,qn )T
which
are
zero; eM
By e x t e n d i n g
< -b 9 < 0 , Vi E [l,n]
is constant;
us a s s u m e
different
(36)
:
an e s t i m a t i o n
HPII : m
of the
Px + q ( x )
fulfilled
Re Xi(P)
N : {x : llxll ~ r }
where
Let
of e q u a t i o n s
motion dx : dt
the
system
the d o m a i n
-
n-1 1 n n~/'~-~
of d e t e r m i n a t i o n
b2 IIPII of the
system
(36)
right
part
112
Ch. II
on the w h o l e the
system
: k-i
Zk
complex
(36)
plane
linear
, 6 e ]0,1]
and
part,
The stability theory of comparison systems
fulfilling
after
, we o b t a i n
a unitary
a substitution
Here
the
functions
components
proaches Theorem tem
~l,.--,~n
X(t;to,×o) zero,
we
(36)
be u n i f o r m l y
2) w h i c h
to f u l f i l l
satisfies
+ ~i
,
+ ~
~n
into
account
the
transformation
of
ql(×),...,qn(X)
call
7 (Nemytsk~1, 1955).
sufficient
take
remaining
shall
Yk =
l = knZn
of n o n l i n e a r i t i e s
If m o t i o n
of v a r i a b l e s
of
a system
dZldt - klZl +6 b12 z2 + "'"+ e n - l b l n Z n
dZn at
transformation
in
N c R n , v t e T o ,for
it the
t-~+=
ap-
0 + -motion.
In o r d e r
that
the
state
X=0 n-i
of the sys-
attractive
from
NM = {× : Ilxll < e M
r}
conditions
i)-3)
with
in c o n d i t i o n
the
constant
it is
an i n e q u a l i t y c r < b2 n - 2 m - i / 2 E M - i
P~00~.
By m e a n s
of the
function
dv dE < - 2 E-b2 we c o n s t r u c t
a comparison
v-- izr 2
for an e s t i m a t i o n
IIAll e + ~ Cr
+ n~-~
n v~
v
equation
du _ 2 ~(6 ) dt ~ u
, u(0) : u o ,
(37)
where ~(e)
_ b 2 6n-1
_
n%/-m
nA/~l
IlPll e n
nv~
As far as
u ( t ; t o , U o) : u o exp ~2 e(e) and h a v i n g see that
fulfilled
the
B. We c o n s i d e r ly, we a s s u m e i') m a t r i x
P
conditions
statement system that
(t_to) ]
of the
of the t h e o r e m (36)
the
is stable,
Re Ai(P)
2') qi(x) = x i It(x) , w h e r e
H(x)
3') a s o l u t i o n
of the
x(t;to,Xo)
theorem,
type
eonditiens i < ~
<0
< 0 , we can
•
of n o n l i n e a r i t i e s ,
are f u l f i l l e d , Vie
is a q u a d r a t i c system
VteT °
~(e)/n%/m
is evident.
for a special
following
VtoeTi
(36)
[l,n] form
:
;
of v a r i a b l e s ;
is d e f i n e d
for
e Tix R n The next
theorem
is due
to M a r t y n y u k
and V e r b i t s k i i
name-
(1982).
(to,Xo)
Comparison equations
Theorem 8. formly
113
In order
attractive
that
the
state
N : Ix :v(x) it is s u f f i c i e n t
x:O
of the
system
(36) be uni-
from
to fulfill
<
~¢X
conditions
}
i')-3').
•
Here #i
: Pii
+
Pik
and
vi
1 are m a x i m a l
in m o d u l u s
eigenvalues av : ~
el(X) respectively; tion
~
is a m i n i m a l
v(x) = x T B x
Proof.
For
scalar
of the m a t r i c e s xi
and
eigenvalue
Hi(x)
of b o t h ,
of the m a t r i x
an e s t i m a t i o n
Liapunov
of the
function
v(-)
following
of the
func-
total d e r i v a t i v e
of the q u a d r a t i c
,
D v(x) the
B
.
~ ~v(x) + Z ¢ i ( x ) H i ( x ) i is valid :
estimation
O*v(x) ~ Kv(x)
,
v2(x)
+ Z ~il~il
(38)
as far as Xr2~v(×)
and
~Zi v(x) A comparison
> el(X)
v(x)
> ~i(x)
IViIH2>Hi(x)
~iR 2 >el(X)
R : dim N : INI
equation du dt
--
which
lull
corresponds
:
KU
2
+.~U
to the
' ~ < 0 , ~ >0
differential
, u(0) = u o
inequality
(38),
, has the
solution
-I u(t;uo) From
the c o n d i t i o n
value Theorem C. Let
uo
8 is proved. us e s t i m a t e
of b i a x i a l
According the
u(t;Uo)
is s u b o r d i n a t e
law on the d o m a i n rail
= ( -~ ~0
+ C I exp [-Kt])
, Vt~T °
to the
the
influence
of a t t r a c t i o n
vehicle
model
of
C l a~ > 0
. The
0 < u o < ~/~
initial
. Hence,
the
• of n o n l i n e a r i t y of an u n p e r t u r b e d
on the basis
to the w o r k by L a z a r y a n ,
investigated
we have
condition
of the
rail
of the p s e u d o - s l i d i n g motion
of the T h e o r e m
Korotenko, vehicle
Demin,
of the m o d e l
8.
Radchenko
is a m e c h a n i c a l
(1969)
system
which
114
Ch. II
consists
of three rigid bodies,
connected
(Figure 5). For a d e t e r m i n a t i o n and rails we a p p l y a n o n l i n e a r radii
of the s u r f a c e with respect
profile).
The m o d e l
equations
of the 14th order.
sponds
by v i s c o - e l a s t i c
of i n t e r a c t i o n
to the rail t h r e a d s
elements
forces b e t w e e n w h e e l s Variation
linearly
of
(a conic w h e e l s
by a s y s t e m of n o n l i n e a r
The zero s o l u t i o n
motion.
For some
m o t i o n of the v e h i c l e
the m o t i o n of a s e p a r a t e l y
systems
r o l l i n g d e p e n d i n g on t h e i r t r a n s v e r s e
is d e s c r i b e d
to an u n p e r t u r b e d
unperturbed
theory of comparison
t h e o r y of p s e u d o - s l i d i n g .
of w h e e l s
movements
The stability
values
of p a r a m e t e r s
is a s y m p t o t i c a l l y
taken w h e e l e d
differential
of the s y s t e m correstable,
an
though
couple w i t h conic w h e e l s
is
always unstable.
E i.
0
~o ~M
x
ky
Equations
Figure
2u
4
of a r e c t i l i n e a r
m o t i o n of the m o d e l have the f o r m
my + 2 k y ( 2 y - 2 H e Jx ~" + 4 k z b 2 0
+ 4~zb20
Jz ~ + 2 k y a ( 2 a @ - Y l
-Yl-Y2
- 2kyH(2y-2H0
+Y2)
-Yl-Y2)
+ 2kxb2 ( 2 @ - ~ i
Jzl~ j + 2kx b2 (@j -@) Here Qyj i
=
force at the point component
, Xji
of the
of the p s e u d o - s l i d i n g
i - t h wheel.
j-th
is a l e n g t h w i s e
of contact
The p s e u d o - s l i d i n g
= 0 ,
-@2 ) = 0 ,
= QYi
'
, j : 1,2
.
(39)
2 ~ Yji" i=l
is a n u m b e r of the w h e e l of the (Xjl - X j 2 ) d
: QCj
:
) = 0 ,
myj + 2 k y ( y j - y + ( - 1 ) J a @ + H $ )
Q@j:
5
wheeled
couple;
component
i-th
wheel;
force at the point force as a f u n c t i o n
of p s e u d o - s l i d i n g Yij
is a t r a n s v e r s e
of contact of r e l a t i v e
of the sliding
Comparison equations
1 15
can be d e t e r m i n e d
by formula kle V/1
where
kI
is a c o e f f i c i e n t
of Coulomb's sliding,
friction;
e --v~/v ;
is the v e l o c i t y
p
v
ek I 2 + (kfp)
of p s e u d o - s l i d i n g , is a loading
is the v e l o c i t y
of s l i d i n g
function
the first
two terms
e is a r e l a t i v e
of the wheel
motion;
contact.
v.
We de-
3 kl
1 k3 - 2 and define
is a c o e f f i c i e n t
of an u n p e r t u r b e d
at the point
note
kf
on the wheel;
(kfp) 2
of an e x p a n s i o n
in powers
of
E of the
F : V . x.
V~
3
F : - k l e + k3 e3 : - k I - ~ + k 3 ( 7 ) The l e n g t h w i s e determined
and t r a n s v e r s e
by e x p r e s s i o n s X : F V~x
components
of p s e u d o - s l i d i n g
3 (V~x) v - +
-
kl V~x + k3-
Vt
k3
in w h i c h sliding
=
F
V~y
= _kl
V.
the l e n g t h w i s e are d e t e r m i n e d
are
v~× v 2~ V 3
v
vty + k3 V
( v
and t r a n s v e r s e respectively
) +
k3
(V~x)jl
(i=2)
(V~x)J 2 : -d~j - ~ o T y j
V3
components
as follows
(i : i)
: d~j + ro V Tyj
;
'
2 v~y v.x
V~y3 Y
force
:
of the v e l o c i t y
of
:
(V~y)
Jl
: yj - v ~ j
,
V
Here r ° is a radius icity of the wheel• With this
of
the
middle
circle
Qyj : 2Yj into account
expressions
pj : u j - v ~ j we can w r i t e
down the system dy _ u d-T -
d~ dt
--
=
X
(V~y)j2 of
: yj -v*j
rolling;
~
is
a
con-
in view Xjl : - X j 2 =Xj
Taking tution
;
•
"
;
; Yjl : Y j 2 :Yj
;
(4o)
; QCj : 2Xjd of forces
; qj
: $wj-~o
(39)
m du ~-~ : - 4 k y y
(40) and a p p l y i n g yj
' j : 1,2
in new v a r i a b l e s +4kyH8
,
in the
+ 2ky ( y l + Y 2 )
Jx d__XX = 4kyHy - 4 (kz b2 + k y H 2) 8 + 4 ~ z b 2 X dt
a substi-
form of ,
2kyH(Yl+Y2)
;
116
Ch. II
: ~
dt
;
The stability theory of comparison systems
Jz ~d-~ -- -4 (kya 2 + k x b2) ~ + 2 ~ a ( Y l - y : pj
dt
+ v@j
•
j : 1,2
+ ( m I v 7 + 2 k y ) yj -
2k I
:
-
7m I
v
PJ
~
~
v~ I - $ r o YJ + ~ qJ
qJ +
(p~+q~) v3
;
2 (Jzl v 7 PJ + dr o
JzlV7 + ~
I Jzl dqjdt : 2 k x b 2 ~
;
2
m~ dP~dt= 2ky(y-~0-(-1)~a~) d#j dt
2) + 2k×b2 (@l+ @2 ) ;
2 kl~ 2 k3~qJ v qJ + v3
2k×b2) ~J
2 2 (Pj+qj)
5:1,2 (41)
For a linear quadratic Jzl = m l d2 tem
approximation
Liapunov
of system
function
for a full
(41) we construct
v(×) = xTAx
derivative
numerically
, X e R I~ . A s s u m i n g
of the
function
a
that
(in view of the sys-
(4i)) we have an e s t i m a t i o n 2k3 mlv3
2
2
8v
~v
2
2 ( av aP 2
av q (42)
2 2 Hj = pj + qj
Let us denote
A(IIj)
, A(¢j)
maximal
are maximal
eigenvalue
estimation
;
eigenvalues A.
to the
j :1,2
connection
;
A(A)
is the
vi : I . The
:
-< ~v + ~ (1-[1¢ 1 +112¢ 2 ) < ~v + mlv3
(41)
H~,¢j,
. "
G(/~A(A) 1 +/~ 2)
v2 '
(43)
2k 3
comparison
sponds
of
In this
form
where
The
8v 8v apj PJ + --Sqj qJ
Cj
of the m a t r i x
(42) is reduced D*V(X)
_
to state
equation
the ×=0
given
inequality can
N
(43).
be :
in
given
{
x
The by
:v(x)
"
B, which domain
an
is
easily
integrated,
of attraction
of
the
corresystem
expression A2(A)
<
}
c
RI4
~ ( # 1 +~2) and any m o t i o n
initiated
inside
this
domain,
of radius R : ~
~ A2(A) ~(#1+~2)
will not
leave
the sphere
Stability
II.3.
of
the
solutions
117
STABILITY OF THE COMPARISON SYSTEMS SOLUTIONS
II.3.1.
The non-degeneracy of monotonicity. Definition
We c o n s i d e r
a system
of d i f f e r e n t i a l
equations
d__UU = ~(t u) dt " where
u •S cR m
, ~ : To×S ~ R m
space,
an o r d e r
in w h i c h
, (t,u) ~ ~ ( t , u )
is given
K : {u E R m , u i a0 ,i : 1,2,..,m} set
{il,...,ip}
lel : p < m
;
{il,...,ip} with
ordered
& by
crossing
coordinates
by the
formula
by a cone
with
by means
is a n o t h e r
the
Yil,...,Yip
;
. Let
K
. By
of c o r r e l a t i o n s
K~
be a v e c t o r elements
we d e n o t e
from the
ij • [il,i p]
is a cone
~
a
i ~ i I < i 2 <... < i p g m
is o b t a i n e d
value
Rm
of n o n - n e g a t i v e
interior
set w h i c h
out
(44) '
in
;
set
Ye
Rm
is a v e c t o r
which
Ke : {y • R m ,ye ~ 0 , ys = 0 , I~I : p }
,
~ :
is d e f i n e d
; its i n t e r i o r
is
o
Ke
, ye > 0 ; a cone
We d e n o t e
by
segment
e 6R m
is
- {y : y E R m , a i g y i g b i
a vector
e :
l,l,...,i T
and
suppose
' i 6 [l,n]}.
that
S D
D {u : o ~ u ~ e } Definition IO. which
(Martynyuk,
I~ I ~ I~I
Obolenskii,
, it follows
from
~(-)
functions which
is a vector,
constructed
~l(t,U),...,~m(t,U)
is i n d u c e d
by s y s t e m
Corrolary i .
If in the d o m a i n the
~s(t,u')
following
,
by m e a n s
satisfies SxT o
~ ~s(t,u")
of the
the
are
u s. :.u .s
for
VtET o
P
, for
,
set
the v e c t o r
~
from
field,
W l~l - c o n d i t i o n .
for c o m p o n e n t s
inequalities
set
that
, then we say that
(44),
Lion
~(t,U)
u-v • K ~
If for any
e K~ \ 0 , V(U,V) e S
~(t,u)-~(t,v) where
1978).
of a v e c t o r
func-
fulfilled
., u~ ~u~'
(v~s)
(45) U : (ul,...,Um~eS cR m ,
then the
system
Inequalities
(45)
Definition 11. right
Farts
II.3.2. The
satisfies
are k n o w n
A system
of w h i c h
monotonicity
system.
(44)
the
s e It,m] •
condition
differential
condition
condition),
,
W°-condition.
as W a z e w s k i ' s
of o r d i n a r y
satisfy
(Wa~ewski's
the
teY o
(1949).
equations
of u n i f o r m
is to be c a l l e d
(44),
the
non-singular
Chaplygin-Wa~ewski's
•
The basic statements of the comparison principle
following
important
property
of the
system
(44)
can be d e r i v e d from
118
Ch. II
the c o m p a r i s o n
principle
t e m of d i f f e r e n t i a l and
~(t,u)
tion,
and
The stability theory of comparison systems
(see 11.2).
equations
; g : To×S~R m
Let for the right part
of the sys-
(44) the two v e c t o r - f u n c t i o n s
be given,
which
satisfy
, V(t,u)
ETo×S
;
g(t,u)
the W a { e w s k i
eondi-
such that
a)
g(t,U)
b)
systems
< ~(t,U)
of e q u a t i o n s dw = g ( t , w ) dt
,
(46)
d_~V : g ( t , v ) dt
are i n t e g r a b l e conditions
tions
all s o l u t i o n s
point
of the systems
on a g e n e r a l holds
(46)9(47)
interval
also
in
the f o l l o w i n g
a)
g(u)
b)
systems
for
t:t o
of the
the s y s t e m
w-(t;to,Uo)
i.e.
w(t) , V(t)
w i t h the initial
are d e f i n e d
system
(44),
on
starting
T# C T o . from the
(to,Uo) e T o x S " lie b e t w e e n the two s p e c i a l
and the m a x i m a l - o f
This
solutions
W(t o) : v(to) : u o
In a d d i t i o n , initial
and t h e i r
(47)
< n(t,u)
of e x i s t e n c e
case
one of the s y s t e m
< V+(t;to,Uo )
(48)
of the c o n s i d e r e d
functions
g and
g
solutions.
do not d e p e n d on
teT o ,
:
, v(t,u)
ETo×S
;
of e q u a t i o n s dw _ g(w)
dt
,
(49)
dV = g ( v ) dt
are i n t e g r a b l e
and t h e i r
solutions
above; hence the e s t i m a t i o n ma~ s o l u t i o n s Estimation state
(46)
i.e.
< U(t;to,Uo)
is v a l i d <~(u)
(the m i n i m a l (47)),
solu-
systems
us to a n a l y z e
of the n o n - i n t e g r a b l e
the two systems
(46),(47)
are d e t e r m i n e d
(48) is valid
of a u t o n o m o u s
(48) a l l o w s
u=0
(5O)
as it was g i v e n
for the m i n i m a l
and maxi-
(49)9(50).
stability
(at all)
(instability)
system
of the
(44) on the basis
(or (49),(50) 9 w h i c h are either
of
integrable
or a u t o n o m o u s ) . Let us r e c o l l e c t The s o l u t i o n ditions one
the known
concepts.
U + : [to,#[ ~ R m
(to,u o)
u : [to,~[ ~ R m
of the s y s t e m
is c a l l e d the
(right)
(44) w i t h the initial
maximal
w i t h the same initial
solution
conditions
if any o t h e r
satisfies
equality u + ( t ; t o , U o ) ~ U(t;to,Uo)
, Vt 6 [to,#[ n [to,~[
con-
.
an in-
Stability of the solutions
Similarly solution
we define which
the
lustrate
inequality
. In this
connection
same
of e s t i m a t i o n by u t i l i z i n g
Habets
monograph
10.
(48)
(1980)
ing an i n e q u a l i t y
~
function
Rm
(47)
be any
Proposition
through
ing an i n e q u a l i t y
~>
function
system
Rm
(46)
be any
g
are
con-
which
we
il-
of the b o o k by
5.29
of G u t o w s k i ' s
be c o n t i n u o u s ,
(t,u) 6 T o × 3
system
and let
(44)
g
(to,Uo) e T i x S passing
satisfy-
satisfy
be a m a x i m a l
. Then
~,g :To×S ~R m
for all
passing
solution
(to,Uo) E T i × S
Ch. IX
statement
the point
functions
W ° - condition, the
u
the
solution
of
, u : [to,~[
through
the point
u ( t ; t o , U o) < v + ( t , t o , U o ) , Vt e [to,~[ n [to,~[
Let
the
and
statements,
2,3,
v + : [to,~[ ~ R m
of the
. Then ii.
on two
U-
as a ,
(to,Uo)
Q , g : ToXS ~ R m
for all
passing
solution
(to,Uo) • T i x S
solutions
items
and the
functions
W ° -condition; the s y s t e m
u- : [to,~[ ~ R m
u - ( t ; t o , U o) -< U(t;to,U o)
condition
is b a s e d of the
solution
1976).
Let
the
initial
lemmas
and Laloy
(Gutowski,
Proposition
minimal
the
for one and the
The p r o o f
Rouch,
(right)
satisfies
V t e [to,~[ n [ t o , ~ [ sidered
119
(t,u) e T o × S
and
w- : [to,~[ ~ R m
through
of the
the point
system
u(t;to,U o)
be c o n t i n u o u s , let
g
> w-(t;to,Uo)
through
the
solution
(to,Uo) E T i x S
(44) p a s s i n g
satisfy-
satisfy
be a m i n i m a l
.
of
, U : [to,~[ ~ the point
, Vt E [to,~ [ N [to,~[
.
11.3.3. Definitions of the comparison system stability For the
system
(44)
Vt E T o . It a l l o w s following
exists
12. w~th
The
state
respect
6(to,6 )
uniformZy tion
12.a)
mal
6
denoted
Ti
stable tion
d
solutions
corresponding
that
with
for any
respect
fulfilled by
and
6M(t,~)
in the whole
12.a)
uniformly
are
stable
the D e f i n i t i o n s
with
fulfilled in
U(t;to,Uo)
concepts
in the
respect and
are
:
and any
ee]0,~[
there
an e s t i m a t i o n
teT o ;
YI
obeys
is
u o ,0 _
for all to
(44)
toeT i
iff c o n d i t i o n s
for every
the whole
12.b),c)
system
iff for any
inf IBM(t,6) c
non-negative the
of the
is f u l f i l l e d
stabZe are
u:0
to
such
u + ( t ; t o , U o ) < ee b
can c o n s i d e r
form.
Definition a) stable
we
us to f o r m u l a t e
ee]0,~[
of the D e f i n i -
the corresponding m a x i -
: : t e T i] > 0 to
I i iff c o n d i t i o n s
SM(t,e) ~ + ~ with
;
~ Ve-~+~
respect
fulfilled.
•
to
Ti
of the D e f i n i -
, VteT i ; iff c o n d i t i o n s
of
,
120
Let
Ch. II
us note
valid
that the e x p l a n a t i o n
for the D e f i n i t i o n
Def~n~t~0n
13.
The
>0
and
T(to,Xo,~)
state
fulfilled b)
Uo
for any
e[0,+~[
u=0
to
Ti
iff
and
that
and
and
T(to,Xo,~)
for any
: (to,U o)eTi}
d) uniformly attractive with respect
and
are
fulfilled
Tu(A,~ ) 6 [0,+~[
sup {rm(to,uo,~) Definitions whole"
13.a)-d)
Def{niti0n
and
14.
spect
to
Yi
to
such
formly respect
e) e x p o n e n t i a l l y real
values
: Tu(Ue,~)
A >0
to
Ti
iff c o n d i t i o n s
for any
~ 6 ]0,+~[
of the Defi-
there
: ru(a,~ ) .
•
"attractive in the
of are
u=0
of the
system
fulfilled
with
to
(44)
for any
to
li Yi
uo - u n i f o r m l y a t t r a c t i v e
with
respect
is
:
T i iff it is stable
respect
stable with respect to
and
Yi
and
re-
iff it is stable with
respect
stable with respect to to
with
;
to
Yi
t o -uniformly
with Yi
;
iff it is attractive
Ti ; stable
with respect
to
with Ti
respect
to
and u n i f o r m l y
T i iff it is uniattractive
with
; stable
~ _> I
U(t;to,U o) •
exist
that
with
and
respect
~ >0
to
Ti
such that
iff there for
exist
0 <_ u o ~ A e
A >0 the
ity is valid.
of a)
and
;
13.a)
asymptotically
Yi
exist
;
iff c o n d i t i o n s exist
those
state
stable
to
Ti
there
~table with respect to
Yi
stable
under
there
that
into
and a t t r a c t i v e
with respect d) u n i f o r m l y
~ e ]0,+~[
of the D e f i n i t i o n
c) quasi-uniformly asymptotically uniformly
iff c o n d i t i o n s
: (to,Uo) • lix [0 < u o ~ a e ] }
b) equi-asymptotica~Zy respect
is
is
teST i .
The
a) asymptotically
and
are m o d i f i e d
if c o n d i t i o n s
A(to) e ]0,+~[
{e
; li
such
~ e]0,+~[
exists there
such that
sup{Tm(to,Uo,~) 13.a)-c)
there
U+(t;to,Uo)<
and any
e[0,+~[
:
teST i
,+~[
toe/i
is
:0 <-U o <-A(to) e ] = T u [ t o , A ( t o ) , W ]
for any
ru(Uo,~) • [0,+~[
is also
: 0 -
attractive with respect to
are f u l f i l l e d
A>0
Uo
an e s t i m a t i o n
~u[to,A(to),W]
c) t o - u n i f o r m l y
I (1.2.4)
(44)
for any
and for
t Cite+
sup [Tm(t,Uo,~)
nition
system
attractive with respect to
a) are f u l f i l l e d A(t o) > 0
of the
~>0
such
for all
-uniformly
to the D e f i n i t i o n
12.
a) attractive with re~peet a(to)
The stability theory of comparison systems
-< ~u o exp [-~(t-to)]
,
Vt•Y o ,
Vto6T i ,
and
inequal-
Stability
of
the
Definitions
solutions
14.a)-d)
121
become
the
corresponding
~tability in the whole p r o v i d e d
both
ity in the w h o l e
in the whole
and
"with respect to
14, t h e e x p r e s s i o n
II.3.4.
attraction
the
li"
hold.
type
of stabil-
In D e f i n i t i o n s
can be o m i t t e d
iff
li:R
12.
Linear comparison systems
We c o n s i d e r
an a u t o n o m o u s
linear
system
du --: dt where
of asymptotic
definitions
corresponding
UER m ;
Definition Metzler's
P
is an
1959).
(an
(51) P : (Plj)
A real
M-matrix)
equations
Pu
toxin-matrix,
15 (Newman, matrix
of d i f f e r e n t i a l
if
mxm-matrix
(i,j)-th
W
element
is to be c a l l e d wij
of
W
satis-
fies wij Definition parison
16.
iff
12.
V(i,j) • [1,m]
i#j
An a u t o n o m o u s
system
Proposition
O,
Pij ~ 0
Let
linear
, i ~j
P be an
, i ~j
, V(i,j) c [ 1 , m ]
. If
system
(51)
is to be c a l l e d
, V(i,j) • [1,m]
(mxm)-matrix
w : Eto,~[ ~ R m
. •
such that satisfies
a com-
PCj ~ 0 ,
an i n e q u a l i t y
d_ww ~ Pw dt
and
V : [to,~[ ~ R m
then
an e s t i m a t i o n
is a s o l u t i o n dV :
of the C a u c h y
Pv
V ( t o) : vo
dt
The p r o o f
if only
w o ~ vo .
,
'
w(t;to,W o) ~ V(t;to,V o) is v a l i d
problem
, Vt 6 [to,~[ n [to,~[
,
•
of this p r o p o s i t i o n
can be found
in B a r b a s h i n ' s
work
(1970,
p.87).
Proposition lowing
a) m a t r i x equation b) the
13.
P
P in
Let
statements
are
is stable
(all
det (P - h l )
following
system
equivalent
(51) be an
Then
Re ~j(P) < - 8
, where
Xj
hold
Pll
PI2
""
P21
P22
"'" P2k
°..°.,.
Pk i
....
Pk 2
the
are roots
:0 ) ;
inequalities
(-i)
M-matrix.
fol-
:
: Plk
.°,°,
"'" Pk k
> 0 , k = 1,2,-.,m
;
of an
122
Ch. II
(this
condition
Hick's
is c a l l e d
The stability theory of comparison systems
the S e v y a s t y a n o v - K o t e l y a n s k i i c o n d i t i o n
condition);
c) for any C > 0 , c = (cl,c2,.-.,Cm)T T •..,bm) , obeying
, there
pTb
d) a d i a g o n a l matrix
matrix
pTB + BP
In s p e c i a l pT+ p
case,
ically
9.
The
stable
B with
when
B=I
state
Necessity. Then
acteristic
equation
Vj e [i,m]).
The
uo>0
the
Sufficiency. tion
v(u)
the
state P
ficult
that
Together
to d e t e r m i n e
with
This
proves
(51)
~'. •
D+v <0
Proposition tively
of n o n l i n e a r
:
m×m -matrix),
system
exists
the
the m a t r i x
(51)
is a s y m p t o t -
a positive
vector
(i.e.
satisfy
the
(51) be a s y m p t o t i c a l -
roots
the
equivalence
13 is that
Xj
of the
condition between
in this
case
the
there
char-
Re ~ j ( P ) < 0 , statements exists
a
(u o ~)
system
. After
.
of s y s t e m
(51) we s h a l l
some
consider
transformations
uT(pTB + B P ) obeying
u
the
func-
it is not
dif-
holds,
13 t o g e t h e r
with
conditions
autonomous
of the
the T h e o r e m
pTB + BP
state
13 due to the
is n e g a t i v e
u:0
definite
of the s y s t e m
9 can be d e f i n e d
by the
effec-
stability
o f the
state
x=0
of the p e r t u r b e d
motion
(10),
of a s y m p t o t i c
equations
Vu E R +
c) of P r o p o s i t i o n
the m a t r i x m V(uf0) GR+
stability
m .
,
9, t h e n
asymptotic
verified
Uo > 0
U:0
of the
M-matrix
of T h e o r e m
and h e n c e
such that
that
P is an
far as
condition
unit
is s t a b l e
P u° < 0
(u ~ 0 )
D+v As
,
det (P - k l ) : 0
consequence
: uT B u
exists
such that
the m a t r i x
such
, b : (bl,b2,
•
of the c o m p a r i s o n
a) a n d b) of the P r o p o s i t i o n vector
diagonal
only t h e n w h e n t h e r e
Let
ly stable.
b>O
;
definite.
PTu o < 0
P~oo~.
is a l w a y s
20
u:0
u o : (ulO,U20,...,Umo)T
-C
( I is
definite
t h e n and
:
the p o s i t i v e
is n e g a t i v e
is n e g a t i v e
Theorem
or
Example 9 (Martynyuk, O b 0 l e n s k i i , 1 9 8 0 ) . We c o n s i d e r portional-integral c o n t r o l , r e d u c e d to the f o r m
2o See 7) of Comments on References to Ch. II. 2J See @) of Comments on References to Ch. II.
a
Ch. I.
system of pro-
Stability
of
the
solutions
123
dxi dt
_ PiXi +a
da
n ~ a i x i - pO - f(o) i=l
=
--
dt where
Pi > 0
Let
illustrate
us
for w h i c h stable.
, p >0
the
, of(a)
state
shall
reduce dYi dt
for o f the
(x--0>, o = 0 )
_
the
system
: -Pixi
(52)
the
on asymptotic
Proposition
_<
13 a n d
stability
the
state the
of the
is a s y m p t o t i c a l l y
9 will
state
y:0
(y=O
, z=O)
-
i 6 [1,n] n
-
follow
,
fail
X i:l
(×:0
, z:0
a2 ) -~- - a f ( a )
Pi
,a:0)
from
according
conditions
to
for
of a system
1 +~i z ,
-- - P i Y i
Jail P l Y i
la2
(2p
-
state
n
~
i=l
(2p -
fail
~
) z - g(z)
i=l is a s y m p t o t i c a l l y
Pi stable
then
and
only
then
of inequalities o
Yl
(2p-
X 2
Theorem
o f the
dt
system
(52)
values
form
n
d-iz : The
the
, z =~
lailP i -~
Z i=l
stability
dYi dt
when
system
[1,n]
to
n
asymptotic
(52)
of the parameters
1 2 1 a2 -< ~ P i x i + 2 Pi '
+ xi°
Z aixi o -po 2 -of(o) i=l
Further
,
, f(0) --0 .
domain
o f the
i 2 = 2 xi ' i ~
n
dz dt
o ~0
,
of a substitution Yi
we
>0
estimations
By means
, i : 1,2,...,n
Z i=l
Z
o
> 7-2 " Yi > 0 Pi z° + g ( z °)
, i = 1,2,...,n
>
,
Z lailpiy i , i:l
wi
>0
is j o i n t . After
some
transformations
of t h i s n
E i:l which
determines
z = O)
is a s y m p t o t i c a l l y
Remark 4. and
For
Rutkovskaya
domain
restrictions
the
Jail
we
define
an
estimation
-< P
(53)
#i
on p a r a m e t e r s
for which
the
state
for
Piontkovskii
(y=O
stable.
system
(1967)
system
of
equations
determined
the
(52)
following
n:4
estimation
of the
of p a r a m e t e r s i
~4
( rain (Pl)) 2 i=l i
lal 12
<
(~)P2
(54)
,
124
It is o b v i o u s a larger
that
domain
Let
us give
the
system
the
form
by
both
for
compared
a criterion (5~)-
Q
that
present
R we
R,QR,Q2R,...,Qm-IR
. Let
of the m a t r i x
Ak
and
R~0
Theorem
iff I0.
Ak~O
rank
H =m
Paoo~.
definite, R >0,
Besides,
D+v <-$v
as
(P +P)
. According
to
an
m×m 2 -matrix
of w h i c h R>O
defined
are m a t r i c e s
iff all m a i n
diagonal
:
> 0 , k = 1,2,-.,m ,
state
u=0
of the
or
of the
Proposition
far as R~0
Pij a 0
comparison R>O
a n d the
13.d)
, i ~j rankH
system
or
(51)
is
R~O
a n d the
the m a t r i x
pTB + BP
exponential
, V(i,j) ell,m]
=m
the f u n c t i o n
, are n e c e s s a r y
v(u) : U T B U
stability
(Ua0)
o f the s t a t e
U:0
is
In this and
suffi-
for w h i c h ,
~ :
•
Nonlinear systems with an isolated equilibrium state an a u t o n o m o u s
nonlinear
system
duu : n ( u ) dt
Assumption 5.
Let
right part
tinuous cally
UEU
Theorem
ueR m
(55)
: of s y s t e m
and a solution
(55)
satisfy
o f the
Caucby
the
W ° -condition;
problem
for a n y
it
is con-
UoER m
is lo-
s u c h that
for all
unique;
2) t h e r e
exist , U~0 II.
t i o n 5 are the
a
neighbourhood
, ~(U) ~ 0 An i s o l a t e d
asymptotically
that
, R : -~
..- rkk
we can give
guarantees
consider
i) the
in
t h e n and o n l y t h e n w h e n
: km(-pTB -BP) AMI(B) II.3,5,
of
(51)
.
connection cient.
stable
As a c o n s e q u e n c e
negative
U=O
system
, k = 1,2,...,m .
An isolated
exponentially
state
of the
to w r i t e
rll ..- rlk ..........
of the
(54).
P
columns
are positive
(53) d e f i n e s
estimation
stability
construct
the
estimation
by the
the m a t r i x
(p_pT)
Irkl
the
ensured
Q :
us a g r e e R
:
n>4
shall
H = (R,QR,Q2R,...,Qm-IR)
minors
and
for e x p o n e n t i a l
We shall
and
n=4
with
Q-R : P , w h e r e
the m a t r i c e s
We
The stability theory of comparison systems
ch. II
stable
fulfilled
and
U
~(u) : 0
state
u=O
then and
only
and there
state
for of the
u:O
.
comparison
then when
exists
u=O
•
system
conditions
a positive
vector
s y s t e m of i n e q u a l i t i e s ~i(Ulo,...,Umo)
is joint.
of the
< 0 , vi 6 [1,m]
(55)
of the
is
Assump-
Uo E K n U
such
S t a b i l i t y of the s o l u t i o n s
The p r o o f is b a s e d
illustrated
the set
14.
of a d d i t i o n a l
If c o n d i t i o n s
function
to zero on
7(u)
and Obolenskii(1980)
statements.
o f the A s s u m p t i o n
A r = {u : u E B r & 7 e R ~ ~ ( U )
and the r e a l equal
in the p a p e r b y M a r t y n y u k
on a number
Proposition
125
= 7u}
fulfilled,
is c o m p a c t
= (~(u),u)/lul 2
and not
is c o n t i n u o u s
then
empty,
and not
: lu:l + lu21 +... + luml
lul
A r n U , where
5 are
and
m
Br
=
; Z
{U : u ~ K
ui:
, re
.
i=l
Proposition
15.
If c o n d i t i o n s 9(U)
then
in
K
there
: max
u ~x~
exists
ities
of the A s s u m p t i o n
(n(u),u) lul 2
a vector
~i(Ulo,...,Umo) < 0
fulfilled
and
5 • aB r ,
< 0
uo
5 are
such that
the
, vi ~ If,m]
system
of i n e q u a l -
,
is joint. Proposition
16.
Let
HT(u)
o f the c o n i c
segment.
and if t h e r e
exists
be a local
If conditions
a vector
semi-group
of transformations
of the A s s u m p t i o n
~e Br nU
s u c h that
the
5 are
fulfilled
system
of in-
equalities (Ei,...,~m)
n is j o i n t ,
then we
can
find
~0
such
equality
Proposition
if o n l y 17.
and if t h e r e
~ >0
for one
that
component
I f the c o n d i t i o n s
exists
a sequence
and for a n y v e c t o r
, for all
0
us
is f u l f i l l e d rs
for
Proposition exists
18.
a strict
such that
an in-
inequality 5 are
u s E K nU
holds.
fulfilled
, lim
us = 0
an i n e q u a l i t y , i =1,2,...,m
i = 1,2,...,m , t h e n
s u c h that
~i
of the A s s u m p t i o n
us
~i(Us) ~0
there
vi•[1,m]
HT(~) a ~ ,
is f u l f i l l e d ,
value
,
{u : u
=
for a g i v e n
u(t;0,U~)
If c o n d i t i o n s
a positive
vector
,
,Vt >rs}
s
of the P r o p o s i t i o n U° e K nU
we
can find
the
n 0 : 5 are
for w h i c h
fulfilled
a system
and
o f in-
equalities ~i(Ulo,...,Umo) < 0 , V i e is j o i n t , stable.
then
the
state
u=0
o f the
[l,m]
system
(55)
, is a s y m p t o t i c a l l y
•
Let us n o t e
that
Propositions
14-17
prove
the n e c e s s i t y
of c o n d i t i o n s
126
Ch. II
of the T h e o r e m
Ii, w h i l e
The stability
the P r o p o s i t i o n
theory of c o m p a r i s o n
i8 proves
sufficiency
systems
of its
conditions.
Theorem 12.
The
state
u:0
of the
if
:
a)
c o n d i t i o n s of the A s s u m p t i o n
b)
there tor
exists us
the
a sequence system
comparison
5 are
us •K
if only
(55)
is u n s t a b l e
fulfilled;
, lim
u s = O , such that
S~t~
for each
vec-
is s t r i c t l y
ful-
of i n e q u a l i t i e s
~i(]l,~2,...,~m) is joint,
system
for one
e0
, i • [l,m]
j e [l,m]
the
,
inequality
filled; c) t h e r e set The
exists
proof
II.3.6.
a neighbourhood
Kusn U
a vector
of the T h e o r e m
U
field
of the
~(u)
state
does
12 is s e t t l e d
not
u=O
such
vanish.
by P r o p o s i t i o n s
that
on a
•
16,
17.
The theorem of Zaidenberg-Tarsky and algebraic solvability of the s t a b i l i t y problem
In c o n t r a s t solvable
to the
one
general
Zaidenberg-Tarskii
Assumption 6.
A comparison
the
i) f u n c t i o n s parameters
is a l g e b r a i c a l l y o f ~he
theorem
by
1980).
system
conditions
~i(Ul,U2,.-.,Um,~)
, i : 1,2,...,m
domain
~i(0,0,...,0,~)
m z 2 , ~ = (~l,...,~p) E,
(56)
:
are p o l y n o m i a l s ,
are v a r i a b l e s ,
from the
there
on the b a s i s
: ~i(Ul,U2,...,um,~)
following
(~l,u2,...,Um)
problem
system
(Lopatinskii,
dui dt satisfies
stability
for a c o m p a r i s o n
p ~0
are
:0 ,
variable
;
2) p o l y n o m i a l s 8uj
(ul'uy'""Um'~)
are n o n - n e g a t i v e 3) a g e n e r a l comparison tions
for all
criterion system
of the
pEE
, i ~j
following
f(u1~..,,Um,~) : 0
,
;
for a s y m p t o t i c
(56)
, V(i,j) • [l,m}
is c o n t a i n e d
stability in the
(instability)
finite
system
of the of equa-
types , g(ul,...,um,~) > 0
, h(u1,-.,Um,~)
<0
, (SF)
k(Ul,...,Um,~) ~ 0 , ~(Ul,...,Um,~) ~ 0 ~ m(ul,...,Um,~) s 0 . Functions
f , g , k ,h , ~ , m
which
are
constructed
according
to the
Stability of the solutions
right part of s y s t e m
127
(56),
ensure
fulfillment
of a s y s t e m of i n e q u a l -
ities ~i(Ul,...,Um,~) < 0
, u i >0
;
~i(Ul,...,Um,~) > 0 , u i < 0 , and also an i s o l a t i o n tion of the values
of the e q u i l i b r i u m
of the p a r a m e t e r s
ities w o u l d e n s u r e a s y m p t o t i c u =0
of the
Theorem
13.
on s y s t e m
system
(56),
order that
and i n e q u a l i t i e s
solution
and s u f f i c i e n t
ZI,Z2,...,Z r •
Theorem
~i : ~ I , ....,~p :,~p
Ul : ~ i • u2 = ~ 2 ,"", Um : ~ m
values
~l,...,~p ( s u b s t i t u t e d of e q u a t i o n s
13 via a n u m b e r of e x a m p l e s
Let a linear c o m p a r i s o n
The c o l l e c t i o n
illustrated
•
.
°
°
o
Pkl
o
°
.
.
.
.
.
Pk2
.
°
°
""
o
the s y s t e m it is i n s t e a d of
and i n e q u a l i t i e s
and s u f f i c i e n t
(Martynyuk,
, V(i,j) e [l,m]
in T h e o r e m
conditions
I > 0
,
13 is w i d e l y
,
(58)
k n o w n and
inequalities
k : 1,2,...,m
(59)
•
Pkk I
As it is k n o w n these i n e q u a l i t i e s
(Proposition
for a s y m p t o t i c
13.b))
stability
are the n e c e s s a r y
of the state
u=0
(58).
ii.
Inequalities
(59) are n e c e s s a r y
s t a b i l i t y of the state dUidt :
where
: in
system
by S e v a s t y a n o v - K o t e l y a n s k i i ' s
(-l)k
[xamp]e
~ , but
1980).
it is r e p r e s e n t e d
totic
ZI,Z2,...,~ r
contain
the f o l l o w i n g p r o p e r t y
d--{du= Pu , P = [Pij ]in , Pij _>0 , i ~j
of s y s t e m
operations
•
Obolenskii,1978,
be given.
(57).
n u m b e r of systems
only one of the systems
Let us i l l u s t r a t e
10.
that
inequal-
of the state of
of the same form, w h i c h
for the real p a r a m e t e r v a l u e s
~l,-..,~p ) s a t i s f y
£xample
(instability)
Ul,...,um ,and w h i c h p o s s e s s
(57) have only one real
. The determina-
n u m b e r of a l g e b r a i c
(57) we can d e f i n e the finite
do not c o n t a i n
necessary
stability
u=0
for w h i c h these
is the p r o b l e m of a n a l y s i s
By means of the f i n i t e
of e q u a l i t i e s
state
~ • E
aij ~ 0
for
( Zm a i ]
j:l
i ~j
u=0
and s u f f i c i e n t
of the c o m p a r i s o n
u~ £ j + l ) 2 m i + l
, i,j
system
: 1,2,...,m
J
;
~i , mi
are n a t u r a l
numbers.
for a s y m p -
128
Ch. II
Example 12.
The stability theory of comparison systems
Let a s y s t e m dui dt
2£i,i :
-Pi
2£~.,i
2£i÷i
U.• ~
+ ui+ I
, i = 1,2,...,m-I
, (60)
du m m-i dt = X l a i l i--i be given, bers;
where
fall > 0
Pi >0
2£i+ I
ui
; £i,ViE
the
[l,m]
,are
the
arbitrary
natural
num-
. If
p~: then
2£m+ I -PmUm
"straight
line"
m-i
m-i
i=l
s=l
1)2£m+i
~ }ai}(n
given
Ps
by an e q u a t i o n
m-i
: ( H i)Um Ui s:l Ps is a s y m p t o t i c a l l y The
state
turbations
u=0
i : 1,2,...,m-i
'
,
stable. of the
system
of p a r a m e t e r s
Pk
(60)
is a s y m p t o t i c a l l y
and
fail
, Vie
stable
[i,m-1]
are
if the p e r subordinate
to the r e s t r i c t i o n m-i
m-i fail ( H __11)
Z
Let
in the
system
(60)
there
be
m=2
. Then,
du I 2£I+1 2£i+i dt - allUl + a12u2
(61) du2 dt where The
a12,a21 a0
necessary
state
Ul=0
;
and
£1,£2
are the n a t u r a l
sufficient
, u2=0
of the
all < 0
II.3.7.
2~2+i 2£2+i - a21ul +a22u2
conditions
system
,
(61)
2£i+ 1 all 2£1+ 1 a21
numbers.
for a s y m p t o t i c have
the
form
stability
of the
:
2£2+1 a12 I 2£2+ 1 > 0 a22
Nonlinear autonomous comparison systems with a n o n - i s o l a t e d s i n g u l a r point
We c o n s i d e r
a comparison
system
du i dt = ~i(Ul,U2j...,u m)
, i = 1,2,...,m .
(62)
Stability
of
the
solutions
Assumption 7. i) the right of the
129
The
system
(62)
parts
of the
system
corresponding
locally
satisfies
Cauchy
(62)
the
are
problem
conditions
continuous
for
field,
condition, 3) for every
generated
where
by the
21~ I + 4
point
U•M
system
(62),
is
exists
termined
a strictly
for
14.
fixed
totically
stable
The p r o o f
of the t h e o r e m
Obolenskii, 1980)
Proposition
19.
the i n t e r s e c t i o n
and
t >0) , w h i c h
~(u)
O}
the
W l~i -
'
continuous
, where of the
:
i
i=l
conditions
curve
6(U) > 0
system
(62)
(a set
in the p a p e r
on two a u x i l i a r y
C(S) 6 ~
, C(0) : 0
of the A s s u m p t i o n
is i l l u s t r a t e d
is b a s e d
, de-
.
~ ) are a s y m p -
7 are
fulfilled.
(Martynyuk
statements.
and
•
If c o n d i t i o n s
of the A s s u m p t i o n
7 are f u l f i l l e d ,
of the
with
cube
set
{u e
:
and there
relative
Z
monotonous
points
if all
Qh is compact
'
s e ]-6(u),6(U)[
Many
satisfies
, where m
ant
solution
~ m ;
: {u • R m
Theorem
and the
(Ul0,...,Umo) • S c R m
unique;
2) a v e c t o r
there
:
Rm ,
exists
to the
M
lull
< h,
N
of the
of t r a n s f o r m a t i o n s
by a vector
then
h ~ ~+ , i • [1,m]}
a neighbourhood
semi-group
is g e n e r a t e d
an a r b i t r a r y
field,
set
M , invari-
Ht(Uo )
corresponding
(u o e Qh to s y s t e m
(62).
Proposition
20.
If c o n d i t i o n s
for an a r b i t r a r y
point
solution
to
II.3.8.
belongs
an
~-limit
7 are
point
fulfilled,
of the
system
then (62)
M .
Several applications of nonlinear comparison systems
We c o n s i d e r
a large-scale dx i dt
where
of the A s s u m p t i o n
u° eQh
x i • R nl , A i
of i n t e r a c t i o n s
system
of e q u a t i o n s
Aixi + h i ( x l ' x 2 ' " " X m ) are
among
nlxn i -matrices;
the i n d e p e n d e n t dx i dt - Aixi
Let us suppose
that
the
roots
' i E [1,m] h i : R n ~ R nl
(free)
" ie
of the p e r t u r b e d ,
motion (63)
are
functions
subsystems
El,m]
.
of c h a r a c t e r i s t i c
det (Ai - k i l i) = 0 , i E [1,m]
(64) equations ,
(65)
130
Ch. II
where real
Ii
is an
parts.
We apply
illustrated formation
nixn i -unit
we r e d u c e
theory of c o m p a r i s o n
are simple
of the
and by means
x i : Biz i dPi dt
matrix,
the m e t h o d
in 11.2.5
The s t a b i l i t y
ones
comparison
with
equation
of the n o n - s i n g u l a r
the
system
(63)
to the
systems
the n e g a t i v e construction,
linear
trans-
form
i,~ -i~ i i8 i + ~ <~ie + Fie ) , i 6 [l,m]
- ~iPi
(66)
where
~.i With
the
= Re { d i a g ( ~ , . . . , X ~ i ) }
system
(63)
F i : B [ I Hi (BlZl,...,BmZm)
we a s s o c i a t e
a tentative @
nI
nm
: ( z Plj,,
V(PI~...,P m)
n i < n i , i • [1,m]
teristic
equations
Assumption
8.
1) the roots negative 2) there which
large-scale
the '
system
(63)
is such
equations
s)]
+Fie
u*eK
(~IUl
of the
(65)
that are
charac-
;
simple,
= (~l(Ul,...jum)
and
with
the
,--,~m(Ul,...,Um)) T
inequalities
S ~i(Ul,...,Um)
of the A s s u m p t i o n
such
that
the
system
+ ~-~I(Ul,...~Um)
< 0
~
mum + ~m(Ul,...,Um) then
~(u)
W °-condition
If c o n d i t i o n s
a vector
is solvable,
roots
"
1
j=l
IB.
of v a r i o u s
parts;
satisfies
Theorem
is a n u m b e r
a vector-function
]~ niz
exists
j:l
of the c h a r a c t e r i s t i c real
function
(65).
The
exists
Liapunov
s Pmj)T
j=l where
vector
@
the
state
, Vi • [1,m]
8 are
fulfilled
(U I > 0,...,u m > 0)
,
< 0 , (U I >O,...,U. m >0)
,
X=0
of s y s t e m
and there
of i n e q u a l i t i e s
(63)
is a s y m p t o t i c a l l y
stable. Pao0~.
Under
the c o n d i t i o n s
of the T h e o r e m
15 we obtain
a system
of
equations du
whe re
: ~ * u +n(u)
"
: dlag{~l,~2,"~,
which
is a c o m p a r i s o n
Theorem
11 this
m}
system
completes
of the
,
as
for the
the p r o o f
Example 13 ( M a r t y n y u k , N i k i t l n a , motion
, uER m
dt
self-oriented
wheel
j •
equations
(63).
of the T h e o r e m
1982),
'
Together
15.
with
the
•
We c o n s i d e r equations o f a
of chassis
with
a weightless
stand.
Stability of the solutions
131
We a s s u m e are
that
the
fulfilled,
theory
for w h i c h
of r o l l i n g
pneumatic,
being
(1945),
x,y,z
be the C a r t e s i a n of the
of the
largest
drawn
with
dinates
of a n e l a s t i c
and a l s o by d e f o r m a t i o n s fying p r o p o s i t i o n rotation
about
Equations velocity
: we
wheel
in the
by an a n g l e ~ ,{
shall
the axis
Ox
. Let
inertial
us i n t r o d u c e
consider
a wheel
o f the p e r t u r b e d
motion
of an u n p e r t u r b e d
motion
of the
is a v e l o c i t y
o f the
forces, cients H0
are
parameters,
stipulated o f the
transverse
linear
Cx
(67) , (68) u+~
are
by a d e f o r m a t i o n
friction;
tive to the a x i s
Equations
which
stand rigidity
is a c h a r a c t e r i s t i c
ticsof
the
following with
x,y
simpli-
respect
to the
,
:
: 0 ,
(67)
(68)
o f the w h e e l
in the
of p n e u m a t i c ;
:
to the damp; and
expressions
by , b 0
flexion
and
h0 , h u a moment
are
connections ~+b-(~.~
-~)
v:
o
of
are c o e f f i the t o r s i o n ; characteris-
of i n e r t i a
respectively.
, of n o n - h o l o n o m i c : o
at a c o n s t a n t form
(69)
are a m a s s
o f the w h e e l ,
the
= o ,
removal
contained
relative
M,J
-b~
wheel
,
of a nonlinear
+v(e+~)
of c o o r d i n a t e s
@ , coordinates
have
+ v ~ ) - b y y -a~
u = -y a , b , h~
elastic
v :const
J8 +hob +Hoblbl + h o e u
a weightless
6).
system
rigid
sys-
of m a s s e s
.
m~+huU+h~(u+ve
where
connected mechanical
with
(Figure
of r o t a t i o n
o f the
system of coor-
centre
of the w h e e l
of
its c e n t r e ,
of the road.
CXlYlZ I
into the
of
is
plane
the b e g i n n i n g
w i t h the m o v i n g
stand
line
which
through
non-stationary
tem
crossover
slope,
the p l a n e
We l o c a t e
is valid.
straight
in the m i d d l e
the w h e e l ,
w i l l be d e t e r m i n e d
by
Keldysh
the
Oxyz
developed
N.V.
coordinates
The p o s i t i o n
the
of an e l a s t i c
Let
point
Figure 6
conditions
rela-
132
Ch. II
and
an e x p r e s s i o n
of
y(t),@(t),~(t),¢(t)
In
variables
(69)
yl=8
differential
constitute
equations
93
a closed
, y3:~
of the
, :~2 -
= -~vY3
, y5=y
, y6=u
perturbed
motion
takes
bo
~e
+~vY4
Further
on
in s u b s y s t e m s
variables Let
us
nents,
zk
-Y2
, Y4
choose
the
Vk : P k several Vl
determination
we
the the
= -vY3 - v Y l
shall
pass
of
form
H0 Y3-~Y2]Y2[
j
system
•
-Y6
(7o)
,
(71)
a
+~
Y4
on to the
(72)
canonical
.
Liapunov
function
to s u b s y s t e m s
, Pk : Z 2 k - l e
with
(70)-(72)
-iO k
transformations
we
the
following
compo-
: i0 k
* @k : z2ke
, k : 1,2,3
.
obtain
+ a12v2 -Tv 2 I
: -allVl
the
h~ Y6 ---~- v(YI+Y3)
m
(70)-(72)
vector
corresponding
After
Y5
(k = 1,2,...,6)
+b
Y z - 3 - Y2
j
Y6 = ~m
•
for
, y4:~
(hu+h ~ )
Y5 : -Y6
system
, VteT o .
, y2=~
YI:Y2
The stability theory of comparison systems
V2 : a 2 1 v I - a 2 2 v 2 + a 2 3 v 3
,
(73) v3
= a31vl + a32v2 -a33v3
•
where all
hs/2J
= a21
a31
; a12
= - b S 2 sin @i / (VJkl)
= v (Q cos 01 - S 1 s i n 02 ) / k 2
= -vh~
sin 83 cos 81 / (mk3)
k I :~468J
-h~
/ (2J)
;
Q : ~2 s i n 0 2 - k 2 S2
fixed
pressions The
a32
= QS3/k 3
: h/(2m)
; a33
k 2 : v 4~V~-~ 2 /2
;
k 3
:~4bym-h
2 / (2m)
: ~i c°s 01 + k l
sin 01
;
= ~2 cos e2 + k 2 sin @ 2
; $3
: ~3 cos 83 + k 3 sin 83
;
(73)
are
; ~2 ~
: ~v/2
(k : 1,2,3)
made
so t h a t
of a t t r a c t i o n
of the
;
= sin 03(a cos 82 - h ~ S 2) / (mk 3)
; S1
values
domain
a23
sin 01 / (Jkl)
c°s 02
~i : h8/(20) For
;
;
; 7 = 2HoSIISll
; ~3
: h/(2m)
estimations
singular
derivatives
conditions
point
;
; h = h~ + h u
of the
the Wazewski
;
can
are
in ex-
fulfilled.
be d e t e r m i n e d
by
inequalities o D2a12 Yl > o
Y2
oD
> Yl
2/DI
-allDl '
7D 1 o
' Y3
(74
> E~D3/DI
'
(75
where D1 = a22a33 -a23a32
;
D2 :
~21a33 +a23a31
;
D3 = aola32
+a31a22
•
(76
Stability of the solutions
Here
the b a r o v e r
coefficient
after
133
letters
in the
formulas
the e s t i m a t i o n .
tion o f t h e n o n - z e r o
singular
(76)
Existence
point
means
the c o r r e s p o n d i n g
of the d o m a i n
of the c o m p a r i s o n
of a t t r a c -
system
in case
D2~I 2 > a l l D I
of indicates
boundedness
the p a r a m e t e r s
plane
of the huV
system
solution.
can be d e f i n e d
The
from
stability
the
domain
in
inequality
DI > 0 . We c o n s t r u c t e d plane
huV
curves
1-3),
in F i g u r e
for the
which were
stability values
a n d we c o n s t r u c t e d
obtained
v for the
7 the
following
following
according values
domains
of
a :
in F i g u r e
to
of the
in the p a r a m e t e r
104 , 2 . 1 0 4
, 5.i04
8 the a t t r a c t i o n
(74),(75),
depending
remaining
parameters
:
h u : 200
Y :
h~
: 100 k g . s e c / m
H~
: i000
a : 104 k g / m b : 0,5.i04
kg/m
: 5,3 m -2 h u = 4j5 m -I
kg.sec/m
kgm.sec 2
b0 = 2.106
kgm
b u = 2.106
kg/m
h~
domains
on the p a r a m e t e r
m : 105 k g . s e c 2 / m 90 k g . s e c 2
(the
= 200
kg.sec/m
V2 , VI , V3 , M 0.06 - O.O06 0.05 - 0.005
hu,K2M.C 4000
0.04 - D.O04 V1
3000 ~
0.03 - D.O03
3
.V2 2000
0.02 - 0.002
1000 '
2
I
20
30
Theorem
system
40
domain.
Let
80
90
0-
V,M/C
0 30
40
50
V,M/C
Figure 8
in the p r o b l e m U = u*
0.001
of e s t i m a t i o n
be an i s o l a t e d
of the a s y m p t o t i c
equilibrium
state
o f the
(55).
For a s y m p t o t i c sary
50 6'0 70 Figure 7
11 is u t i l i z e d
stability
0.01-
~'
and
stability
sufficient
that
in the w h o l e the
same
o f the
state is
state
both
u = u*
stable
it is n e c e s -
in the w h o l e
134
Ch. II
and a t t r a c t i v e Example
14.
in the w h o l e
Let us
the e q u i l i b r i u m sis.
and
of an i n e r t i a l
to the t h e o r y
a mechanical
connections~
Nikit ina,
(see the D e f i n i t i o n
5, Ch.l).
c o n s t r u c t an estimation of the
state
According
of such
The stability theory of comparison systems
of K e l d y s h
system,
have
the
part
together
form
of the
(1945)
with
(Keldysh,
attraction double
domain
wheeled
the e q u a t i o n s
of
chas-
of m o t i o n
the d e s c r i p t i o n of the d a m p e r
1945;
Goncharenko,
Lobas,
1981)
J~ + h@~ + [bll + 2(PN + aN r + Cb d2 ) ] ~ + 2 Iv~/r + (512 + 2IV/r) @ + bl3Z -2h~r~ J*$" + h s ~ + 5 2 2 8 -d21x
-2(at
+aN) ~ - h~xsin
-2IV/r
(1+ d2/r 2) ~ + (521 - 2 f C b d2) @ + b22 z -
- 2 5 ~ - hsx c o s x
mz + b 3 3 z
= 0 ,
+ (b31 + 2 a N )
~
~+
b328 - d 3 1 X - 2 h ~
~ig~l
: k(×-×
~'3 + ~ + r 6 + V O -~v~
+~
k(X-X Here
8 ,~ ,z
dinates.
and
1)
+v~
+five
-- e l l ~
are
inertial,
and
difficult
to find
reduce
coordinate.
a system dx i = dt
Let
introduce
,
of e q u a t i o n s
and
are
that
us i n t r o d u c e
•
(77)
inertia-free
in the
system
a notation
correlations
sign o + eia
(77)
coor-
(77)
there
a = X-Xl to the
,
form
, i = 1,2,...,8 ,
a
ba >0
(78)
functions
where
za
to the
canonical
are
of
,
v2
= Za eiSa
canonical
variables
form w i t h (79)
, V 3 : Ze signz e
respect
we c o m p o s e
after
of the
a comparison
system
+ B i %/~ + siu
n-N d_/u = Z kjuj - b a V ~ dt j=l
- ~ i = Re (Xi)
; Xi -
are
reduction
to a part
du i dt = - ~ i u i
Here
: o
~ ,~ ,X I ,X
8 Z ~ik Xk + b I V l a l k=l 8 d__~a = Z a k x k dt k=l
V I : Zae -iSa
On the b a s i s
o ,
:
-~v~
out
= 0
i) ,
-boVla I sign
We
-2a~
+el28 +el3 s -gllX
It is not
is an e x c e s s i v e
X = 0
roots
, W = a sign a of the
system
(79) (78)
variables.
' (80)
, i = 1,2,...,n-N
of the
.
characteristic
polynomial
Stability of the solutions
of the
first
complex
equations
conjugate
nary m e t h o d Let
n
ones;
in the
us s t i p u l a t e
o f the
Values
following
problem
of the
system
(78) a m o n g w h i c h
coefficients
of c o n s t r u c t i o n
functions
Cauchy
only one b r a n c h solutions
of the
all o t h e r
course
the
derivatives ofthe ness
135
: for the
(79) (80)
there
are o b t a i n e d
of the
comparison
zero v a l u e
are
system
of t h e i r
considered;
the
solution
corresponds
to a c h o i c e
which will
(80).
arguments
are not
solution,
N
by an o r d i -
local
be an e s t i m a t i o n
uniqueof
for o t h e r
from above.
of the p a r a m e t e r s
must
satisfy
the i n e q u a l i t y
b aala2...an_ N > B l k I a2a3.-. ~ n - N + B 2 k 2 ala3...~n_N +,..+Bn_ N kn-Nal~2... ~ n - N - i The
vector
of an i n v a r i a n t eTu
The a t t r a c t i o n
has a r e s t r i c t i o n
< A2/B 2 , e :
> (Bi~o
from above
1,1,...,1 T
u=0
+ SiUo)/~i
can be e s t i m a t e d by i n e q u a l i t i e s , i = 1,2,...,n-N
.
Reducible comparison systems
The necessary state
U
d o m a i n o f the s t a t e Uio
II.3.9.
cone
•
u=0
alization
and
sufficient
of the
one in L i a p u n o v ' s
i f f the
do not systems
systems
admit 22.
stability
the
direct
At the
c a n be a n a l y z e d ,
same
of the gener-
time
in p a r t i c u l a r ,
sense.
1975).
following
for a s y m p t o t i c
(55)
comparison
comparison
D e f i n i t i o n 17 ( S i l j a k ,
M -matrix,
conditions
(51) or
to n o n - s t a t i o n a r y
some n o n - s t a t i o n a r y reducible
systems
P(t)
Matrix conditions
are
is c a l l e d
a non-autonomous
fulfilled
<0
, Vt E T o , i =j
;
0
, Vt E T o , i # j
, V(i,j) • [1,m]
Pij( t )
Definition
18.
Non-autonomous
linear
system
d_uu : P(t)
dt is c a l l e d
reducible
M -matrix
and
duced
to the
comparison
if by m e a n s system
•
system if
of L i a p u n o v ' s
(81)
u P(t)
is the n o n - a u t o n o m o u s
transformation
it can be re-
dU
--:
dt
Bu
2~ See 9) of Comments on References to Ch. II.
(82)
136
Ch. II
with Let
the
constant
us r e c o l l e c t
defined
by
that
Definition
19.
fulfilled
:
a)
and
Matrix
[(t)
L(t)
•
L(t)
matrix
: VteTo}
,
properties
mxm-
with
for all ,
in general,
following
one
matrix. ele-
conditions
are
t e T o , i.e.
sup{~[(t)II
, VteY o can be
a linear
Liapunov
complex,
iff the
< +~
aER+
transformation
is an
k(t) e c l ( T o )
are b o u n d e d
Idet L(t){ ~ a > 0
The m a t r i x
where
the L i a p u n o v
sup { U L ( t ) I I b)
8.
we call L i a p u n o v ' s
u = [(t) v,
merits is c a l l e d
L(t)
M-matrix
The stability thoery of comparison systems
: VteTo}
< +~
,
•
found
in the book
by D e m i d o v i c h
(i967). Theorem
16
ducible
then
(Erugin,
1946).
and only
can be r e p r e s e n t e d
The
then
in the
when
Proof.
Let
us note
, i ~j
that
It follows exp [Bt]
system
fundamental
(81)
is re-
matrices
X(t)
(8b)
: L(t) e x p [ B t ]
B,
for a m a t r i x
, (i,j) c [i,m]
non-negative.
differential
of its
form X(t)
blj z O
linear any
, elements
from
the
Bt N = ( exp [-~-])
satisfying
fact
that
,
is a p o s i t i v e
N
the c o n d i t i o n
of the m a t r i x
exp [Bt]
are
value
and Bt t 2 t2 exp [-N--] = I+ B ~ + B 2 - ~ + '"
Theorem 17.
An i s o l a t e d
is a s y m p t o t i c a l l y a)
system
(81)
b) a p o s i t i v e U ° >O
.
tem
(81)
bii ~ O
U:O
of the n o n - a u t o n o m o u s
system
(81)
if
is r e d u c i b l e ; vector
Uo :
ul,'",Um°o T
exists
such that
BTu ° < 0
•
18.
Theorem
state
stable
•
The
isolated
is stable,
, i ~j
state
u=O
if the m a t r i x
, (i,j) 6 [ l , m ]
,and
B
of the n o n - a u t o n o m o u s of the r e d u c e d
the
Liapunov
llL(t)II ~ a ~ e x p [ B t ] ~ -I
VteT o
system
matrix
linear
sys-
is such
that
L(t)
satisfies
an e s t i m a t i o n
P~oo~.
Statement
the
system
(81)
the
fundamental
of the T h e o r e m
solution matrix
in the X(t)
.
18 follows form •
from
aER+ the p r e s e n t a t i o n
u(t) = X(t) u o
and
from
(83)
of for
Stability
of
the
solutions
Definition 20. elements
137
Non-autonomous
Plj(t)
M-matrix
P with
the c o n t i n u o u s
quas£-dom£nant d£agonal
has a
iff
constant
bounded
di
exist
such that m
djlpjj(t)l
19.
Theorem
The
is e x p o n e n t i a l l y elements P&oo~.
>
Z dilPlj(t)l i=l
isolated stable
state
if the
Siljak's
v t e r o , Vj e [ 1 , m ]
of the
u:O
M-matrix
and a quasi-dominant We u t i l i z e
,
•
non-autonomous
P has
system
continuous,
(81)
bounded
diagonal.
method
(Siljak,
1978).
For a f u n c t i o n
m
P(U)
it is not d i f f i c u l t holds for e v e r y
to
teTo
=
~ diu i i=l
find an e s t i m a t i o n
.Here
~ER+
and
tion
v(u) ~ - ~ p ( u )
it is d e t e r m i n e d
, which
from
the c o n d i -
m
Ipjj (t)I-d?
I ]
w h i c h is v a l i d dominant
because
diagonal.
of %he
Taking m
Z dilPij(t)
I ~ ~ ,
VtEr °
,
i:l i/j fact
that
into a c c o u n t
the the
2 1/2
IluU : ( Z u i )
M-matrix fact
that m
lul
and
has a q u a s i -
UuII s lul
, where
lull
= z
i:l
P
i=l
we d e f i n e
lu(t)i
~ ~lUol
exp[-=(t-to)].
Here : dMdm I
II.4.
we introduced
(i0),
(7),
Ch. I.
In this
section
B in
the
state
vector
condition
(t,x)
we c o n s i d e r
dx
the L i p s c h i t z
B in
the m a t r i x - f u n c t i o n
Ch.l, a n d the f u n c t i o n
d-{ = g(x) where
.
Main properties of matrix-functions
In 1 1 . 1 . 4
tem
; d m : m i n {d i : i e [l,m]}
MATRIX-FUNCTIONS APPLICATION TO THE STABILITY ANALYSIS
II.4.1.
system
; d M = m a x {d i : i e [l,m]}
X E R n , and
to
only
the a u t o n o m o u s
,
g(X)
with respect
x for a n a u t o n o m o u s
for a n o n - a u t o n o m o u s
(84) is c o n t i n u o u s
x
sys-
system
in
N ; the
and s a t i s f i e s
solution
X ( t ; x o) o
is u n i q u e
and it c o n t i n u o u s l y
depends
on
to
and
Xo , to ~0
, xo EN
.
138
Ch. II
Let C
UcN
and
B : R n ~ R mxm
The stability theory of comparison systems
be the m a t r i x
Liapunov
function
defined
on
as follows.
Definition 21. A m a t r i x f u n c t i o n B in x is to be c a l l e d the matrix Liapuno~ funation fo~ the system (84) on U iff the c o n d i t i o n s
I)
B(x) e Cl(U) ;
2)
B(x) ~ 0 , V(X # 0) E U
are
fulfilled.
Let
us
those
state which
mulate
the p r o p e r t i e s are
Let
b)
B
and
existence
of a n e i g h b o u r h o o d
for
there
yet
from
holds
I : i,j e [ 1 , m ] }
x*~N
if
B
the
N
fact
for
that
y
for-
Then
is c o n t i n u o u s
proof
functions
from
:
at
XeW
such
the c o n t i n u i t y
B(x) • C I ( U )
:
max
[
and
X(t;x)
that of
to
it follows Uij(X)
=
eT(x)
from the
continu-
x; that o
max
uij(X)
•
be the
largest
(~,~)~ T(x) and
let
M
in-
B_ = {x :B(x) < c 1 c
B
Let
for the
defined on
system
to that
for any given
0
by L a S a l l e
(84)
there
U and the s o l u -
o f the system (84) be precompact on ~ M n 8_ c
is s i m i l a r (LaSalle,
holds
close
E : {x : B(X) ~ O , x E U n N }
in
X(t) = × ( t ; X ) . Then
II.4.2.
let us
;
of the point
follows
sufficiently
e x i s t the Liapunov m a t r i x - f u n c t i o n
The
B(x)
: ( i , j ) max e T ( x ) (uij(Y) - u i j ( X ) ) on
us d e n o t e set
£(x) = L
Theorem 21 (The Invariance P r i n c i p l e ) .
t ~0
and
to
case.
be a m a t r i x - f u n c t i o n .
T(y) c T ( x )
(i,j)
tion
similar
N;
on
~(y) - ~ ( x )
variant
for the given
B
at the point
B
vector-functions
~ max{luij(Y)-uij(x)
B(X) Let
(84),
point;
ity of e)
to the
principle
x,y • N
is c o n t i n u o u s
B(x) d)
Invariance
21.
liB(y)-B(x)rl
c)
of
of the m a t r i x - f u n c t i o n
characteristic
a)
this
motions
•
LaSalle's
Proposition
, along
and for the
all
U for
t ÷+~
.
vector
• Liapunov
1976).
Theorems of direct method based on matrix-functions
In o r d e r
to f o r m u l a t e
we state
the
following
and p r o v e
theorems
on s t a b i l i t y
and i n s t a b i l i t y
Matrix
functions
Proposition terval if
application
ZZ.
]a,b[
. Then
$(t) s 0
2aoo6.
Let
139
,D(t)
be a c o n t i n u o u s
matrix-function
@(t)
is a d e c r e a s i n g
on
for all
@(t)
then
the
are
sO
]a~b[
l
.
the s y s t e m
cases
and p r o v e
(84)
the
utilizing
Theorem
22.
tion
8 ; N ~ R m×m
If there
that
functions
D+uij(t)
. In both
Let us state
to p r o v e
decreasing
derivative
D+uij(t)
matrix-function
t 6 ]a,b[
It is s u f f i c i e n t
tion
]a,b[
on an in-
all e l e m e n t s
on
theorems
uij(t)
if
(i,~) ~ T ( x )
(i,j) • T ( x )
will
on s t a b i l i t y
the L i a p u n o v
, then
be d e c r e a s i n g
and
on
instability
for
matrix-functions.
a positive
such that
. Really,
< 0 . In case w h e n
functions
exists
]a,b[
of the m a t r i x - f u n c -
definite
for any
x •N
Liapunov
matrix-func-
its d e r i v a t i v e
in virtue
o
of s y s t e m
Proof.
(84)
is
B(x) ~ 0
As far as
B
function
B
B(0) :0
Besides,
ous and
B(0) : 0
for all
x ° eS~
continuous
there
. Let
, we
on the b a s i s
(Rouche,
Habets,
22,
tion
x(t;Xo)
Theorem tion the
23.
1977).
system
(84)
asymptotically Paoo{. those
Since
is
exists
B(x) < 0 , then
conditions
the
we
is c o n t i n u -
B(Xo) < e l ( e )
< ~01(e)
definite
B(x(t;Xo) ) s 0 obtain
,
the
llx(t;xo)II < e
initial
for all
Liapunov
of the
condita0
. •
matrix-func-
its d e r i v a t i v e
solution
in
matrix-function.
shall
(I) w i t h
(x~0) e N
that
for
of the t h e o r e m
i, the
solution
xo E~
an e q u a l i t y lim
is valid,
where
X(t;Xo)
According
to the
condition
~k • K [ o , ~ [
that
derivative,
) <- B ( x o)
a positive for any
such that
B(x(t;xo) ) ~ 0 , then,
t~0
system
, the
in view
system
(84)
of
is
stable.
of the T h e o r e m
illustrate
of the
N
N ; moreover
B(x)
is a d e c r e a s i n g
for any
) -< B ( X ( t ; x o )
such that
such
on
on
¢i • K [ o , ~ [
on the Dini
Then
is stable.
definite
definite
far as
~I e K [ o , ~ [ , t h e n
If there
B : N + R mxm
. As
(84)
. As far as
B(x(t;Xo) )
is a s o l u t i o n
x o . As far as
e>O
6 : 6(e) > 0
: llXll < ~}
~l(llX(t;Xo)ll where
a function
of the t h e o r e m
Laloy~
and p o s i t i v e
be given
can find
: {x • N
of s y s t e m
and p o s i t i v e
exists
there
view of the P r o p o s i t i o n Hence,
x:0
is c o n t i n u o u s
is also
B(X) a~l(ilxl})
, then
, k = 1,2
such
are the
of s y s t e m
llx(t;xo)ll
is a n o n - t r i v i a l of the t h e o r e m that
intensification (84)
is stable.
of Let
us
= 0 , solution there
of the
exist
system
functions
(84).
140
Ch. II
~l(llxll) Since ~(t)
B(x(t,Xo) ) <0 = B(x(t;xo) )
from below,
~ B(×)
, then
it h a s
us i l l u s t r a t e
pose
that
~>0
the s o l u t i o n
~ ~2(llx~)
a finite
that
. Then,
.
and h e n c e ,
decreasing,
and
the
since
function
it is b o u n d e d
limit = i n f ~(t) t
the n u m b e r in v i e w
×(t;xo)
, VxEN
~(×(t;Xo) ) <0
is m o n o t o n o u s l y
lim ~(t) t++ ~ Let
The s t a b i l i t y theory of c o m p a r i s o n systems
~=0
= e z 0
. In o r d e r
o£ c o n t i n u i t y
satisfies
llX(t;×o)ll ~
(85) to show
of the
it~ w e
function
an inequality >0
for
0 ~ t <÷~
,
(86)
where # = e o n s t . But as far as $(t) < 0 ~ it f o l l o w s f r o m (86) a @(t) ~ - b , b = c o n s t . H e n c e , for all t > to z0 an i n e q u a l i t y
~(t)
= ~(t o) +
sup-
~(s) ds
that
~ ~(t o) - b ( t - t o )
to w i l l be
fulfilled.
equality ness
T h a t is i m p o s s i b l e , b e c a u s e
is n e g a t i v e
o f the m a t r i x
for
lira Hence,
in c o n s e q u e n c e
we o b t a i n Let
the
us s t a t e
Theorem function 1) the
domain
origin
B(×)
some
system
of having
fixed
sign of
B(×)
(84)
similar
there
to that
exists
of C h e t a y e v .
such a matrix
Liapunov
:
B(×) = 0
P = {X : B(X) > 0 , X e N }
domain
P
is f u l f i l l e d
on the d o m a i n
o
there
holds
of Hahn's
adjoins
the
Proof.
that
Let
us p r o v e (84) w i l l
In v i e w o f c o n d i t i o n B(x o) > 0
leave
, where
S~
set
~p n N
;
K,
~ @(B(×)) where
, where
0 < ~ < +~
@ eK[o,~ [
is
,
is u n s t a b l e .
during the
i) o f the
in the o r i g i n .
on the
P ;
B(×)
class
motion
the c e n t r e
B(x(t;Xo) ) = 0
on i n s t a b i l i t y
t h e n the u n p e r t u r b e d
that
the p o s i t i v e -
•
of its p o s i t i v e n e s s
function
the s y s t e m
= lira
(85).
theorem
is b o u n d e d
4) on the
contradicts
of the in-
of c o o r d i n a t e s ;
2) an i n e q u a l i t y 3)
~(t)
o f the p r o p e r t y
If for the
B that
, which
part
. Thus,
inequality the
24.
t ++~
B(×)
the r i g h t
the
evolution
domain
= IX : × • N
t
one
solution
of
N.
theorem,
We c o n s i d e r
of
we
can
, IIXII < ~} the
find
×o e P n S 6
is an o p e n
solution
X ( t ; × o)
such
sphere
with
. As far
Matrix functions application
as
~(x)
141
is b o u n d e d on the d o m a i n
is also b o u n d e d
on this d o m a i n s i.e.
of the c o n d i t i o n
: ~(t o) +
Example 15.
: B(X(t;xo) )
. Hence,
on the basis
~ @(to) + ¢ ( ~ ( t o ) ) ( t - t o )
. That
P w i t h the i n c r e a s i n g ' o f
~P n N
dzl -dt
$(s) ds
to X(t;xo) 6P
is f u l f i l l e d as long as the d o m a i n
$(t) ~ e < + ~
~(t)
4), the i n e q u a l i t y
~ @(t)
through
P, the f u n c t i o n
due to the c o n d i t i o n
We c o n s i d e r
t.
is why
But
X ( t ; x o)
X ( t ; X o)
2), h e n c e
a s y s t e m of e q u a t i o n s
must
cannot
it leaves
N.
leave
P
•
of the p e r t u r b e d
3 2 : - p l ( Z l + sin2(z2-i) -a) 3 + (z 2 + z 3 + c o s 2 ( z 2 - 1 )
leave
motion
-b) 3
- s i n 2 ( z 2 - 1 ) ¢ ( Z l , Z 2 , Z 3) dz2 dt dz3 dt
~ ( Z l , Z 2 , Z 3) P (Z 2 + Z 3 + C O S 2 ( Z 2 - 1 ) -b) 5 + (z I + z 3) + (sin2(z2-1) - i ) ¢ ( Z l , Z 2 , Z 3 ) -
+
(87)
p3(Zl + z3 )7
where ~ ( Z l , Z 2 , Z 3 ) : _ #2(z 5 2 + z3 + cos2(z2_l) _ b)5 + (z I + z3 ) 5 _
- p4(zl + s i n 2 ( z 2 - 1 ) -a) 7 - p5(z2 + Z 3 + COS2(Z2-1) - b ) 7 3 - # l ( Z l + sin2(z2-1) -a) 3 + (z 2 + z 3 + c o s 2 ( z 2 - 1 ) a : sin2(-i) We d e t e r m i n e
restrictions
z I = z 2 = z 3 =0 stable. We i n t r o d u c e
, b : cos2(-l) on p a r a m e t e r s
of the s y s t e m
B(z) uij(z)
: 0
Ull(Z)
= {
u22(Z)
:
u33(z)
#i
, i : 1,2,...,5 .
for w h i c h the state
(87) will be stable or a s y m p t o t i c a l l y
a matrix-function
where
• Pi > 0
-b) 3 ;
in the f o r m
: {uij(Z)}i,j:l,2,
for
i ~j
(Zl + s i n 2 ( z 2 - 1 )
(88)
3 ,
, i,j : 1,2,]
;
-a) 2 ;
( i ~ ; ~ 2 (z 2 +z3 + c o s 2 ( z 2 - 1 ) - b )
2
(l+e) 4 2 2 2 2 (Zl +z3)
PlP2
We c o n s i d e r sets n i : {Z ; Z E R 3 : (ull/21 >U22 %/2 A u $ $ ) A ( U l i _>u22 A u i l >_u33)}
142
Ch. II
: {z
z
R 3 :
The stability theory of comparison systems
A Ull ~ U 3 3 ) A
-~22
(Ull ~ U 2 2 A Ull ~ U 3 3 ) }
,
~3 : {Z ; ZER3 : (U2~22 Z UlI~I A U ~2 ) A (U22 ~ Ull A u22 ~ U 33 ) } 22 >U3~3 ~4
:
1~
<. ~2
A
{Z; Z R 3 : (u22 _ Ull
~2
~2
U22 ~ U33 ) A (u22 ~ Ull A u22 ~ u33) } ,
£5 :
z~ ->u~2 1~ >-U22 Z~) A (u33 _> U l l {Z ; Z ~3 : (U33 11 A U33
A U33 ~U22) } '
£6
{Z ; Z R 3 : (U ~2 < U~2 A V2 ~ u ~ 2 ) 33 - l l u33 22
A
= :
£8
:
A (u33 z
Ull
u33 z u 2 2 )
}
•
u ~2 .
,
(u~2
{Z" Z ~3
On the set
<
1~,}
: . Ii - u 2 2 )
~"1 U ~"2 '
B(Z) : U l l > 0
, and the total
derivative
of
B(z)
derivative
of
B(Z)
derivative
of
B(z)
is
fiCz) :
-~ --
On the set is
~3 u £4
,
--ee
B ( Z ) = u22 >0
~(z)
, and the total
:
0.22 --co
On the set
~S U £6 •
B ( Z ) = u33 > 0
, and the t o t a l
is
fi(z)
.
.
.
.
.
--co
It is not d i f f i c u l t the f o l l o w i n g
to n o t i c e
estimations
that
for d e r i v a t i v e s
[ii(Z)
, i : 1,2,3 ,
hold
QII (z) ~ (Zl + s i n 2 ( z 2 _ l ) _ a ) 4
3 (i - (i+c) 3) Pl (1+6)3
VZE~ '
; z
5
U22(Z) •
-<-< (Z 2 + Z 3 + C O S 2 ( S 2 - 1 ) - b )6
(1-(1+e)5)/32 2(1+E)3 PJ.
,
VZe ~3
;
Matrix functions application
U33 (z)
~ ( Z l + Z3 )
6zz(z)
-<
i Pi +¢
143
8 (l+e) q ( i + e ) lq (l+e) ? 2 2 ( P4 ? 7 + P 5 - - p ) P1 P2 Pi P2 p~ 3
(Z2 + z 3 + C O S 2 ( Z 2 - 1 )
d22(Z) < 3 - (i+6) 2 Pl P2
(Zl
-b)a(1 -(1+~)3)
)~(1-(I+e)5)
+z3
v z • ~2
'
vz6 ~4
'
7 7 #iP2 P 3 " ( 1 + e ) 1 4 ) (zl + s i n 2 ( z 2 -l) - a ) 8
,4
~33(Z)
VZ 6 ~5
'
< (l+c)"plP2 (P4 + P5 ( I + E ) ~
'
V z c $Z7n~6
p2]_P2 ( pP4q ( i + e
1~33(Z ) -< (1+6) 4
)6
;
P~ 2 8 a3-C~T~- @ ) ( z 2 +z a +cos (~2-i) -b) ,
P5 + 1 +6
Vz E ~ 8N
~6
(89)
Let us d e n o t e
P4
-
T a k i n g into a c c o u n t
that
7 7 PlP2
P5
+
--
P?2
.
(90)
6 U
~i
= R3
,
i=l it follows
from e s t i m a t i o n s
respond to the a s y m p t o t i c state
z=0
. Namely,
if
(89) that cases
stability P3 > ~
P3 > ~
and s t a b i l i t y
we can d e t e r m i n e
and
P3 = ~
cor-
of the e q u i l i b r i u m E>0
such that
o
B(z) > 0
and
we o b t a i n
B(z) < 0 ,
B(z) > 0
V z e3R_
and
. If
P3 = ~
, then,
assuming
that
~--0 ,
B(z) : 0 , V z e R 3
11.4.3. The scalar Liapunov function construction on the basis of matrlx-functions (84) there be c o n s t r u c t e d
Let for the s y s t e m B(×)
a matrix
Liapunov
: {uij (×)]ml,j :1 " Uij (0) : 0 • v(i•j) e [1,m]
function
,
(91)
and a)
B(x)
b) the d e r i v a t i v e
of
B(x)
where
U
is an
B :
: max uij(x) i,j
, V(i,j) e [1,m]
along solutions
of the s y s t e m
tlim + O + inf ~1 ( B ( × ( t ; x ) )
mxm -matrix
-B(X)
with the e l e m e n t s
U)
;
(92) (84) obeys
,
(93)
Uij : l , V(i,j) e [1,m] ;
c) the f u n c t i o n v(x) where
C :
Cl•...~cm
d) the d e r i v a t i v e
T 6
= cT B(x)c
Rm • c i f 0
of the f u n c t i o n n+v(×)
,
(94)
are c o n s t a n t
scalars•
i : 1,2,...,m ;
(84) has the form :
c T B(x)
c
,
(95)
144
Ch. II
where
B(x)
is d e t e r m i n e d
For a statement of the
system
possibility
tion
(91)
and
(94)
and
derivative
to be an i n i t i a l in terms
II.4.3.1.
Thaorams on s t a b i l i t y
The
criterion
theorem Basing case
of s t a b i l i t y ,
on the
the
Let
the
exists
an
function
v
4) the total
the
the
the
general
of c r i t e r i a
of
B .
the
first
is w i d e l y - k n o w n .
we state
for the g i v e n
D+v
(95) on
exists
is
on the
¢~0
definite
N;
: domain
N ;
(componentwise); on
N;
: or
to zero on
N,
(84).
equilibrium state
x=0
of the T h e o r e m
is r a d i a l l y
(84)
The
of the t h e o r e m
proof
func-
function
a matrix-func-
constitutes
method,
that
vector
is p o s i t i v e
equal
of s y s t e m
Hahn,
be such
B : N ~ R mxm
(94)
system
(94)
(84)
semi-definite
If all c o n d i t i o n s function
of the
i n t h e whoZe)
of w h i c h
direct
m-dimensional
b) i d e n t i c a l l y
Then
(~tability
the basis
system
derivative
a) n e g a t i v e
along
to p r e s e r v e
statement
the
of the
to c o n s i d e r
of the m a t r i x - f u n c t i o n
I) the m a t r i x - f u n c t i o n
3) the
application
x=0
we have
following
Theorem 25.
2) there
the
(91)
of the m a t r i x - f u n c t i o n
of L i a p u n o v ' s
concept
of the state
application
us, t r y i n g
to take
of e x i s t e n c e
on s t a b i l i t y
~mmediate
of s t a b i l i t y ,
i.e.
stability
stability
Secondly,
permits
criteria one,
the
(93).
(95)
on
(93).
of the m a t r i × - f u n c t i o n
: at first,
its d e r i v a t i v e
form of the w e l l - k n o w n tion
formula
theorems
on the basis
following
its
by the
of the basic (84)
The stability theory of comparison systems
is s t a b l e
of the
system
25 are
fulfilled
unbounded,
then
in the whole. is s i m i l a r
the
(84)
is stable.
for
N=R n
equilibrium
a n d the
state
×=0
•
to the k n o w n
ones
(Krasovskii,
1959;
1967).
Let there
exist
a domain
which
D+v(x) s O
which
D+v(X)
Theorem 26. i) c o n d i t i o n s 2) on the
set
D+v(x) s O
~
. We denote
= {X : X e N ,0 $ v ( X ) by
M
the
< a ,a E R + }
c N
invariant
set
largest
over in
:0 Let
the
i)-3) ~
system
(84)
of T h e o r e m
be such
2 5 are
the total d e r i v a t i v e
that
:
fulfilled; (95)
satisfies
the
condition
~
on
Matrix functions application
Then the motions
set
M
is a t t r a c t i v e
originating
The p r o o f taking
of the
into
II.4.3.2.
theorem
account
27.
following
Let
the
I) c o n d i t i o n s
the
~
is s i m i l a r fact
that
system
to the
tend
to that
domain
to the
set
M
of Theorem
the f u n c t i o n
(94)
~
, i.e.
for
26.1
all
t ~+~
.•
(Hahn, 1967)
is scalar.
stability
(84)
be such
that
along
its
solutions
the
:
i)-3)
of the T h e o r e m
2) the total d e r i v a t i v e Then
respect
domain
on a s y m p t o t i c
the
holds
with
from the
Theorems
Theorem
145
equilibrium
(95)
state
25 are
fulfilled;
is n e g a t i v e
definite.
x=0
of the
system
27 are
fulfilled
(84)
is a s y m p t o t i c a l l y
stable. If all
conditions
function of the
(94)
system
(84)
In view
Proof.
we c o n s t r u c t its total
25.2 27.
G
entire
be the
28.
i) c o n d i t i o n s 2) f u n c t i o n 3) a l o n g
Then
the
v
(95)
(Hahn,
largest
Let
of the
the
i)-3) (94)
{n~ae{ant
solutions
Theorems
(84)
on
N
and
conditions
the
statement
that
on
not
the point
contain
X=0
the
.
:
25 are f u l f i l l e d
for
N=R n
;
unbounded; system
outside on
×=0
(84)
the
total
derivative
(95)
is
G; G; of the
system
(84)
is a s y m p t o t i c a l l y
instability
theorems
on i n s t a b i l i t y there
does
•
we state
that
all
proves
R n , which
be such
of the
state
in
set
of T h e o r e m
At first
such
the x=0
system
definite
Hence,
which
(84)
is r a d i a l l y
equilibrium
w
definite.
except
system
in the whole.
that
state
27 for the
is p o s i t i v e
fulfilled,
and
N=R n
in the whole.
(84),
semi-definite
Suppose
which
for
equilibrium
system
b) n e g a t i v e
II.4.3.3.
x,
are
definite
the
stable
is n e g a t i v e
~967)
the
i) of T h e o r e m
in
a) n e g a t i v e
stable
then
•
trajectories
Theorem
condition
function
derivative
of T h e o r e m
unbounded,
is a s y m p t o t i c a l l y
of the
the
of T h e o r e m
Let
of the T h e o r e m
is r a d i a l l y
two t h e o r e m s
exist
on i n s t a b i l i t y ,
(Liapunov,
1935).
positive
constants
D+v(x)
generalizing
g' , g
: g' C T B(X) C
and
the
Liapunov's
function
(96)
146
Ch. II
or
D+v(x)
Functions
v,
the o p p o s i t e
= g cTB(x)
: cTB(x)C
v(x) sign
iff
The stability theory of comparison systems
, and
w
EN
the
V(X~0)
(v(x) ~o & w(x) ~o)
c+ w ( x )
(97)
are not
semi-definite
following
and of
two r e l a t i o n s h i p s
(v(x) ~o ~ w(x) ~o)
;
do not hold. Theorem 29.
Let
the
system
i
the
conditions
I)-2)
2
the
derivative
(95)
fixed
opposite the
Theorem
30.
a
definite,
the
The w e l l - k n o w n alization G*
ries
of the
Theorem 31.
(95)
of the
sign o p p o s i t e state
be such
25 are
system
in
(84)
has a
function
5) the
derivative G*
of the
sign
N,
identically
to
i)-2) v
which except
w(X)
on
Liapunov
does
not
point
D+v
(95)
equilibrium
state
outside
×=0
semi-defi-
is u n s t a b l e .
also
the
x:O
.
admits
•
a gener-
entire
trajecto-
•
:
fulfilled;
semi-definite
of f u n c t i o n
or it is semi-
is not
(84)
contain
25 are
is not n e g a t i v e
(97);
function.
be such that
of T h e o r e m
v
such that the f o r m
N.
system
the
w
has
vanishes
on i n s t a b i l i t y
(94)
and p o s i t i v e
v (94)
function
of the
(84)
•
:
and a f u n c t i o n
case the
X=0
system
that
is unstable.
fulfilled; g
of the m a t r i x
(84)
the
conditions
N
(84)
either
Krasovskii's t h e o r e m
system
system
function
w
Let
on
of the
latter
i) the
on
of the
constant
in the
2) the
v (94)
on
N;
is p o s i t i v e
semi-def-
G*
of the
system
(84)
is u n s t a b l e .
•
Applications
Let us c o n s i d e r which
D+v
be a set
II.4.3.4.
fulfilled;
semi-definite
(84)
of T h e o r e m
on the basis
Let
the
system
positive
the e q u i l i b r i u m
inite
25 are
solutions
x:O
function
and
of the
is not
state
i)-2)
derivative
nite
the
:
(95).
the
exists
moreover
Then
(94)
D+v
Let
i) c o n d i t i o n s
Then
v
to
equilibrium
2) there the
of T h e o r e m along
that
sign;
the f u n c t i o n
Then
(84) be such
leads
the
known
problem
to a c o n s i d e r a t i o n
of the p e r t u r b e d
motion
of s t a b i l i t y
of the
system
in
the p r o d u c t
of d i f f e r e n t i a l
space,
equations
Matrix functions application
147
a)
d y = y(y) + Y * ( y ; z )
b)
dZ d~
;
dt
(98)
Here
yER p
,
Z* : R P x R q ~ E
z~R q , Y : Rp~ R p q ; in a d d i t i o n ,
RP,R q ; R P x R q what
manner
(98)
on
Z(z) + Z * ( y ; z )
=
and
vanish
stability
RPxR q
, Y* : R P x R q ~ R p
functions
for
y = z : 0 . The
of the
equilibrium
is c o n n e c t e d
with
a)
d~
b)
d!
, Z : R q~ R q
Y,Z ; Y * , Z W
that
are
problem
state
,
continuous
on
is to d e t e r m i n e
x :y : 0
of nonlinear
of
the
in
system
approximation
y(y)
l
dt
(99)
Here
functions
duce
some
9.
N z ~R q
and the
Y,Z
are
the
same
as
in the
system
(98).
Let
us
intro-
assumptions.
Assumption and
: Z(z)
dt
Let
there
of the
exist
time-invariant
equilibrium
states
y=O
neighbourhoods and
z:O
Ny !
Rp
consequently,
matrix-function
B(y,z)
with
elements
istic
to t h e
Vii
vll(Y)
v12(Y,Z)]
v21(Y,Z)
V22(Z)
=
,
; i,j : 1,2
quadratic
, which
satisfy
estimations
character-
forms 2
v(y~O) eNy ;
v22(z) m c2211zll 2
v(z# O) e Nz ;
Vll(Y)
(100)
~
czlllYll
(i01)
Assumption such
that
vl2(Y,Z)
a cl211Yllllzll
v ( y # O , ziO) eNyXN z ;
v21(Y,Z)
:
V(y~O, z#O) GNyXNz
I0. on
Let Ny,N z
Vl2(Y,Z)
there
exist
or on
aVll
( - - ~ - , Y) ~ Pi1IlYn
NyXN z
2
av22 2 (---~--~--,Z) ~ P2111zll aV12 8v12
(--~--
' Y)
, Y*)
~ #I4HY~
2
constants
, i = 1,2
; j = 1,2,...,8
:
8Vll
;
( ~ y - - , Y*) ~ p1211y112 +plaI1y[lllz[l
;
BY22 (--~--,Z*)
+#lsHYllllzll
~ Ozsllyll 2 +
Pij
;
~ P2211zll2 +P2311yllnzll ; (Io~)
;
nz711yllllzll +#1811zll 2
;
148
Ch. II
2
aVl2
,Z) ~ p2411xll + P2511yllllzll
3v12
(-YF- ' z~) ~
Theorem 32.
The stability theory of comparison systems
#26 IIylI2+
Let for the system
;
#9711yll II zll + #2811 zll
(98)
:
1) all conditions of A s s u m p t i o n s 9, 10 be fulfilled; 2) the matrix fell 6
Ci21
:
,
C12
=
021
,
'
~12
:
°21
'
L°21 022J be positive definite; 3) the matrix S
=
I°ll L°21
a121 a22J
be negative definite, where
(Djordjevi@,
1983)
2 oil : ~1(#ii+#12) + 2 ~ i ~ 2 ( P i 4 + P i 6 + P ~ 6 ) ,
a22 : f/2(P21+P22 ) + 2 ~ l f 1 2 ( P l s + P 2 4 + P 2 8 $ 2 2 o12 : ~ (f/lPl3 +P23f/2 ) + f l l ' @ 2 ( , o 1 5 + P 2 5 + P 1 7 + P 2 7 ) Then the e q u i l i b r i u m state
y :z : O
of system
)
,
,
~1,~2 >0
(io3)
(98) is a s y m p t o t i c a l l y
stable. If the conditions of Assumptions N z =R q
9, 10 are fulfilled for
Ny =R p ,
and the conditions 2)-3) of the t h e o r e m hold, then the equi-
librium state the whole.
y :z :0
of the system (98) is a s y m p t o t i c a l l y
stable in
•
The proof is similar to that of (Djordjevi6, estimations
(101) it is not difficult
1983). On the basis of
to illustrate that the estima-
tion from below v ~ UT c T @
Cu ,
(i04)
where
UT
=
holds for the function
( IIY II, IIZ El )
v(y,z)
On the basis of estimations function
;
C =
I::J
= ~T B(y,z)~
, ~ : (~1 ~2 )T
(i01) for a derivative
v defined by the expression
v(y,z)
of the
Matrix functions application
it is not difficult
149
to find the estimation
~(y,z)
~ uTs U
(105)
In view of the conditions 2)-3) of T h e o r e m 32 and estimations
(104),
(105) all conditions of the Theorem 27 are fulfilled by the function v and its derivative. If in estimations
(101) we replace the sign of inequality by the oppo-
site one, then by using the method similar to the one given above, we can obtain an estimation
~(y,z) ~ uTS'u , which allows us to state conditions of instability of the equilibrium state
Y =Z =0
of the system
(84) on the basis of the T h e o r e m 29.
Theorem 32 illustrates that the asymptotic brium state
y =z =0
of the system
the equilibrium state
y=O
stability of the equili-
(98) can hold even in case when
of the system
(99)-a) and
z=0
of the
system (99)-b) has no property of asymptotic q u a s i - s t a b i l i t y of Theorem 6.
COMMENTS ON REFERENCES i. Fundamentals of the theory of differential and integral inequalitics are contained in the works by Chaplygin Wa~ewski Alekseyev
(1950), Azbelev (1965), Zubov
(1975), Antosiewicz (1967), et al. Walter
(1956), Azbelev, Tsalyuk (1957), Luzin
tham, Leela
(1919),
1962, 1964),
(1967), Melnikov
(1966), Conti
(1956), Hahn
Numerous results are summarized in monographs by Bellman
(1969), Mitrinovi~
Sharova
2. Definitions
(1958,
(1951), Matrosov
(1958,1965), Yoshizawa
(1964), Bekenbach,
Filatov,
(1919), Gronwall
(1965), Szarski
(1970), Beesack
(1976), Martynyuk,
Gutowski
:
(1967), Lakshmikan-
(1975), Rabczuk
(1976),
(1979).
in 1.3 are stated by taking into account
some results
of the works given in 3 of Notes and References to Chapter I. The vector L i a p u n o v functions were a topic of analysis in a number of works [Bellman (1962), M a t r o s o v 1963), LaSalle
(1962), Dec
(1971), L a k s h m i k a n t h a m
(1965,
(1975), et al.], and they found a wide application
while a n a l y z i n g stability of large-scale
systems
(see I of Notes and
References to Chapter IIl). 3. The Liapunov matrix functions originally have arisen in consideration of the derivatives higher than of the first order of the vector Liapunov functions.
As far as the authors know,
for the first time
150
Ch. II
they were described by Martynyuk (1979);
see also
4. The comparison
The stability theory of comparison systems
in the book
[Martynyuk~
Martynyuk
(1984a].
principle
in the motion stability theory is developed
on the basis of both scalar Liapunov functions Martynyuk
(1985)I and vector
Martynyuk
(1975), Michel,
polskii, Vasilyev Bitsoris, (1979),
(1976), Gentina,
cou~arison principle
Martynyuk,
[see monographs
(1977), Siljak
Gentina, Borne,
Borne,
Burgat,
initial
Burgat,
Bernussou, (1978),
inequalities,
(1979), Tikhonov
developed
(1963) while
in the present
(2975), Martynyuk
7. Statements14.a) Kotelyanskii(i952); Persidskii (1969); 8. The Theorem 0ho]ensky
considered
Ana(1976),
Grujid
in which the
in 2.2, were the
Radziszewski
(1970,1971),
of quasi-stability,
formulating his theorems book,
is based on the idea of an scalar or vector equation
(1975,1984b) ].
and b) are proved in papers
by Sevastyanov
i4.c) in the papers by Fiedler and Ptak 14.d)
which was
of stability
is not utilized here.
system extension by the auxiliary
[Hatvani
(1972),
(2965,1969).
on the basis of the comparison method, Approach,
by Gruji~
is also defined via vector norms.
6. In the Theorem 6 Persidskii's concept applied by Matrosov
by
(1978), Matrosov,
Bernussou
in the works by Gutowski~
Gutowski
[see An O u t l i n e
Let us also mention papers by Bitsoris
5. Integral and differential topic of analysis
functions
Miller
(1980)].
Burgat
Gruji~,
Gutowski
in that by Araki
(1962) and
(1975) and also LaSalle
11 is proved independently
(1978) and Burgat, Bernussou,
(1951), (1976).
in both papers by Martynyuk,
Grujid,
Gentina,
Borne
(1978). 9. Without the monotonicity stationary
property
of
system, criteria of autonomous
have not a direct
generalization
the
right
comparison
to non-autonomous
10. In 4 the results of papers by Martynyuk lllustrative
ly with Shegai
Shegai,
(see Martynyuk,
of the non-
systems
stability
ones.
(1984,i985)
in which ideas of the Liapunov matrix-functions stability theory is developed,
part
are presented,
application
to the
results are derived joint-
1986).
References
151
Alekseyev, V.M. (1965), Theorem on an integral inequality and severalofits applications. Mat. Sbornik, 68, No.2, 251-273 (in Russian). Aminov, M.Sh. (5959), Some problems of mo~ion and stability of a rigid body of a variable mass. ft. Kazan. aviats, in-ta. Set. Matematika i Mekhanika, X. Antosiewicz, H.A. (1958), A survey of Lyapunov's second method. In Contributions to the Theory of Nonlinear Oscillations. Princeton Univ. Press, 14i-166. Antosiewicz, H.A. (i965), Recent contributions to Lyapunov's second method. Colloq. Int. CNRS, 148, No.2, 29-37. Araki, M. (1975), Application of M-matrices to the stability problems of composite dynamical systems. J. Math. Anal. and Appl., 52, No.2, 309-321. Azbelev, N.V. (1956), On boundaries of Chaplygin's theorem on differential inequalities applicability. Mat. Sbornik, 38, No.2, 161-178 (in Russian). Azbelev, N.V., and Z.B. Tsalyuk (1958), On Chaplygin's problem. Ukr. Ma~. Zhurnal, I@, No.I, 3-12 (in Russian). Azbelev, N.V., and Z.B. Tsalyuk (1962), On integral inequalities, I. M ~ . Sbor~ik, 56, No.3, 325-342 (in Russia~). Azbelev, N.V., and Z.B. Tsalyuk (1964), On differential and integral inequalities. Prec. of the 4th Math. Congress, Moscow, 2, 384-391 (in Russian). Barbashin, Ye.A. (i970), Liapunov's Function. Nauka, Moscow (in Russian). Beckenbach, E.F., and R. Bellman (1965), inequalities, second revised printing, Springer Verlag, Berlin. Beesack, P.R. (1975), GronwhaZl Inequalities. Cavleton Math. Leczure No~es, No.11. Bellman, R. (1954), Theory of Stability of Ordinary Differential Equations Solutions. lzd-vo Inostr. Lit., Moscow (in Russian). Bellman, R. (1962), Vector Lyapunov functions. J. SIAM Control, Ser. A, 1, No.l, 32-34. Bitsoris, G. (1976), Sur la stabilit@ asymptotique des syst~mes interconnect@s. C.R. Acad. Sci. A, 282, 301. Bitsoris, G., and C. Burgat (1976), Stability conditions and estimates of stability region of complex sys<ems. Int. J. Systems Sc£., 7, No.8, 128-135. Burgat, C., J. Bernussou, Lj.T. Gruji6, P. Borne, and J.C. Gentina (1978), Les perturbations structurelles arbitraires et p6riodiques. RAIRO Automa~./Sys~. Anal. and Contr., 12, No.3, 245-267. Cesari, L. (1964), Asymptotic Behaviour and S~abi~ity of Ordinary Pifferentint Equations Solutions. Mir, Moscow (in Russian). Chaplygin, S.A. (19i9), New method of integration of a general differential equation of the train motion. Byul. Nauch.-Eksperim. in-va Putej Soobshchen, la, No.9, 308-334 (in Russian). Chaplygin, S.A. (1976), New method of appro×imate integration of differential equations. Selected Works, Nauka, Moscow, 307-360 (in Russian). Conti, R. (1956), Sulla prolungabilita della soluzioni di un sistema di equazioni differenziali ordinarie. Boil. Ynione Mat. Ital., Ser. II!, A~no XI, NO.4, 510-514. Corduneanu, K. (1960), On application of ~ifferential inequalities to zhe theory of stability. AnaLele Stint. ale Univ. "Al. i Cusa" din Tasi, see.l, 6, No.l, 47-58. Demidovi~, B.P. (1967), Lectures on Mathematical Theory of Stability. Nauka, Moscow (in Russian).. Dee, S.G. (1971), On vector Lyapunov functions. Prec. Amer. Math. See., 31, No.3, 575-580. Djordjevi~, M.Z. (1983), Stability Analysis of Large-Scale Systems whose S~bsystems may be Unstable. EHT, Zurich, Report No. 83-01. Erugin, N.P. (19~6), Reduced systems. Tr. Mat. in-ta AN SSSR, 13, No.3, 3-140 (in Russian). Fiedler, M., and V. Ptak (1962), On matrices with non-positive off-diagonal elements and positive principal minors. Czech. Nat. J., ~2, No.87, 382-400. Filatov, A.N., and L.V. Sharova (1976), Integral Inequalities and the Theory of Nonlinear Oscillations. Nauka, Moscow (in Russian). ~entina, J.C., P. Borne, C. Burgat, J. Bernussou, and Lj.T. Gruji~ (1979), Sur la stabilit@ des syst~mes de grange dimension, normes vectorie!les. RAIRO Automat./Syst. Anal. and Control, 13, No.I, 57-75-
152
Ch. II
The stability theory of comparison systems
Goncharenko, V.I., L.G. Lobas, and N.V. Nikitina (1981), On a single statement of the problem on shimmi of eastoring wheels. Prik~. Mekhanika~ 17, No.8, 82-88 (in Russian). Grebenikov, Ye.A., and Yu.A. Ryabov (1979), Constructive Methods of Nonlinear Systems Analysis. Nauka, Moscow (in Russian). Gronwall, T.H. (1919), A note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 20, No.2, 292-296. Gruji~, Lj.T. (1972), Large-Scale Systems Stability. Ph.D. dissertation, Faculty of Mechanical Engineering, Belgrade (in Serbo-Croatian). (Published in 1974) Gutowski, R. (1976), Rownania Rozniazkowe Zwycajne. Wyd-wo Naukowo-Techniczne, Warsawa. Gutowski, R., and B. Radziszewski (!970), Asymptotic behaviour and properties of solutions of a system of non-linear second order ordinary differential equations describing motion of mechanical systems. Arch. Mech. Stosow, 22, No.6, 675-634. Gutowski, R., and B. Radziszewski (1971), Asymptotic behaviour and properties of solutions of nonlinear ordinary differential equations ~f first order describing the motion of a mechanical system, ibid. j 2~, No.l, 17-25. Hahn, W. (1967), Stability of Motion. Springer Verlag, New York. Ha~vani, L. (1975), On application of differential inequali
References
15 3
Martynyuk, A.A. (1984b), Extension of the space state off dynamic systems and the problem of stability. Colloquium on Qualitative Theory of Differential Equations, Szeged, August 27-31, 58-59. Martynyuk, A.A. (1985), Scalar equations of comparison in the stability of mo~ibn theory. Prikl. Mekhanika, 21, No.8, 3-19 (in Russian). Martynyuk, A.A. (1986), On application of the Lyapunov matrix-functions in the theory of stability. J. Nonlinear Analysis : Theory~ Methods a~d App,., 9, No.12, 1495-1501. Martynyuk, A.A., and R~ Gutowski (1979), Integral Inequalities and Stabilivy of Motion. Nauk. Dumak, Kiev (in Russian). Mar~ynyuk, A.A., and N.V. Nikitina (1981), On nonlinear systems of ccmparison in'stability problems of large-scale systems. Prlkl. Mekhanika, 17, No.12, 97-102 (in Russian). Martynyuk, A.A., and N.V. Nikitina (1982), The comparison mezhcd in the problem of shimmi. Ibid, 18, No.11, %02-107 (in Russian). Martynyuk, A.A., and A.Yu. 0bolenskii (%978), Stability of Aut#~oma~s Systems of Comparison AnaSys~s. Izd-vo in-ta Matemag., Kiev, An Ukr. SSR, ?reprint 78.28 (in Russian). Martjnyuk, A.A., and A.Yu. Obolenskii (1980), On stability cf ~utznsma ~s .',a~ewskii's systems solutions. Pifferents-Uravn~nia, 16, No.8, 13921407 (in Russian). Martynyuk, A.A., and V.V. Segai (1986), To the theory cf stability of ~he autonomous systems. Prikl. Mekhanika~ 22, No.4, 97-102 (in Russian2. Martynyuk, A.A., and V.G. Verbitskii (i982), The domain of attraction estimation for nonlinear systems of the definite form. : r ~".~~ .]ekhanika, i8, No.10, 102-107 (in Russian). Matrosov, V.M. (1962), To the theory of stability of mo~ion. PrikZ. Mav~m. i Mekhanika, 26, No.6, 992-1002 (in Russian). Matrosov, V.M. (1963), To the theory of stability of Mo~ion. Ii. Tr. Kazan. aviats, in-ta. Set. Matematika i Mekhanika, No.80, 22-33 (in Russian). Matrosov, V.M. (1967a), On differential equations and inequalities with discontinuous right parts. I. Differents -Uravnenia, 3, NO.3, 395-409 (in Russian). Matroxov, V.M. (1967b), On differential equations and inequalities with discontinuous right parts. II. Differents -Ur~vnenia, 3, No.5, 839848 (in Russian). Matrosov, V.M., L.Yu. Anapolskii, and S.N. Vasilye@ (1980), Mevhod of Comparison in Mathematiaa~ Theory of Systems. Nauka, Novosibirsk (in Russian). Melnikov, U.I. (1956), Some problems of the direct Liapunov me~hod. Pok~. AJ SSSR, II0~ No.3, 326-329 (in Russian). Melnikov,~G.I. (1975), Dynamics of Non,{near M~ohanical and Electro-~ie~h~nical Systems. Mashinostrojenije, Leningrad (in Russian). Michel, A.N., and R.K. Miller (1977), Qualitative Ana~ys~s af Large-Scale Dynamical Systems. Academic Press, New York. Mitrinovi6, D.S. (1970), Analytic Inequalities. Acad. Press, ~erlin. Nemytskii, V.V. (1955), Estimation of the domain of nonlinear systems asymptotic stability. DokZ. AN SSSR, 101, No.5, 80}-80u (in Russian). Nemyzskii, V.V., and V.V. Stepanov (1949), Qualitative Theory 2f D{ff~rential Equations. G!TTL, Moscow-Leningrad (in Russian). Optial, Z. (1960), Sur l'allure asymptotique des solutions de aertaines @quazions diff6rentielles de la m6canique non lin6aire. Ann. pol. ma~h., 8, No.2, 105-124. Persidskii, K.P. (1951), Some critical cases of denumerable systems, izv. AN Kaz. SSR. Ser. Mat. i Mekh., No.5, 3-24 (in Russian). Persidskii, S.K. (1969), On some theorems of the second Liapunov methc~. Differents -Uravnenia, 5, NO.4, 678-687. Piontkovskii, A.A., and L.D. Rutkovskaya (1967), Investigations of certain problems of the stability theory via the vector Liapunov function method. Av~. i Tel~m., 28, No.10, 23-31 (in Russian). Rabczuk, R. (1976), Elemen~y Nierownosci Ro~niczkowych. Panstw-wyd-wc ~aukowe, Warsawa. Rouch, N., P. Habets, and M. Laloy (1980), Stabi~ivy Theory by Liap:~ncv's Direst method. MIR, Moscow (in Russian).
154
Ch. II
3evastyai.~v, B.A. Mal~m.
N~k,
The stability theory of comparison systems
(1951), Theory of bvaching stochastic process. Uspskhi
6, 47-49 (in R u s s i a n ) .
Siljak, D.D. (1978), Large-ScaZP Dynamic Systems : Stabitity and Stl,uctuft. Narth-I[okla~d~ New York. Szarski, J. (1967), Differentia[ Inequalities. Panstw. Wyd-Wo Naukowe, Wa r.q.'lw;]. Tikhonov, A.A. (1965), On stability of motion under the constantly effeeting perturb:~t[onu. Yevtn. geningr, un-Lan No.l, 95-101 (in Russian). Tikhonov, A.A. (1969), To the problem oi' stability of motion under the constantly effeeting perturbations. Yearn. Lenlngr. un-ta, No.19~ 116122 (in Russian). Tikhonov, A.A. (1974), O1~ Some NonLinear ProbLem~ of Dynamics. Dissertation, Lengosuniversitet (in Russian). Vedrov, V.S. (1938), Dynamic Stabitity of the Aeroplane. Oborongiz, Moscow (in Russian). Walter, W. (1964)~ Differentia~ and integral Inequalities. Sprinzer-Verlag, Berlin. Wazewski, T. (]950), Syst6mes des 6quations et des in@galit@s (Jiff6rentie] kes ord~maives aux deuxi~.mes membres monotones et leurs applications. Ann. Soc. Po~. Math., 23, 112-166. Yoshizawa, T. (1966), Stabitity Theory by LiapunoV's Second Method. Math. See. ,J;Lpall, Tokyo. Zubov, V.[. (1957), A.M. Liapunov'S Methods and Their Application. Izd-vo Lenil~gr. un-ta, Leningrad (in Russian).
Ill
CHAPTER
LARGE-SCALESYSTEMS IN GENERAL
111.1.
INTRODUCTION
A system can be considered complex
for the great number of its nonlinear-
ities and/or their form, for its nonstationarity and for its rich structure.
A system can be considered large-scale due to several in-
trinsic features among which we quote the following 1966; Martynyuk,
1975)
(see a l s o K u k h t e n k o ,
:
i) high-dimensiona]ity; 2) manifold of the system structure structures,
(networks,
trees, hierarchical
etc.);
3) multiple connections of the system elements nection in one level and between different 4) manifold of the elements nature
(machines,
(sub-systems interconlevels of hierarchy); automata,
robots, people-
operators); 5) recurrence of change of the system composition and state ity of the system's structure,
connections,
(variabil-
and composition);
6) multiple criteria of the system (difference between local criteria for sub-systems and global criterion for a system in the whole, their inconsistency). A direct analysis of a dynamical property ability,
observability,
optimality,
can be cumbersome or even yet impossible. bility analysis,
this means that the
general and powerful method
(such as stability,
robustness)
system
In the framework of the sta-
(second) Liapunov method,
for the stability analysis,
fectively directly applied to large-scale
controll-
of a large-scale
the most
cannot be ef-
systems. The cause is the
lack of an algorithm for constructing a system Liapunov function. A trial can be made to test indirectly a stability property of a large-
156
Ch. III
Large-scale systems in general
scale system by solving one of the following problems P~oblem A.
Decompose
the whole system into interconnected
and then find an aggregation under which its Liapunov its disconnected
:
stability
subsystems
subsystems
form of the system yielding conditions can be deduced
and from qualitative
from stability of properties
of their
interactions. ProbZam 8.
Decompose
the whole system into interconnected
and find an aggregation simultaneous
reduction
under which a Liapunov
form of a large-scale
of the number and complexity
Bailey
1968) and Aoki
vector Liapunov 1963,
functions
discovered by Bellman
1968), the concept
application
and in certain
and Wa~ewski
of
(1962) and Matrosov
of matrix Liapunov
functions by Martynyuk
(II).
analysis
(at least at the last stage of the anal-
cases on the comparison principle
(1965) and Kayande
tor Liapunov functions
Bailey
of
of vector norms discovered by Robert
(1950), which was further developed
Lakshmikantham invariant
aggregation
are the concept
of these concepts to the stability
relies on the Liapunov method ysis)
subsystems.
initiated effective
tools for solving these problems
(1964) and the concept However,
(1968)
subsystems without using
of disconnected
systems.
New mathematical (1962,
of interconnected
about stability properties
(1966,
large-scale
of the conditions
stability property of the whole system is in-
ferred from the properties information
subsystems
system which enables
and Lakshmikantham
to study conditional
by Kamke
(1932)
in the second Chapter. (1966) used vec-
stability
and conditionally
sets, respectively.
(1966~
1968) was the first to apply effectively
vector Liapunov
functions to the Liapunov
the concept
stability analysis
of
of large-
scale systems by solving the Problem A. This problem was later on studied
(in the framework of continuous-time
large-scale
- exponentially
stable subsystems
and Rutkovskaya (1970,1971), (1972,1973), Weissenberger
extensively
for
Gruji6
(1963,1972),
(1971-1973 , 1975 ,1977 ,1978),
(1978), Gruji£,
Gentina,
(1972),
(1973,1974),
Zemlyakov and Matrosov
(1974), Martynyuk
Saeki, Araki and Kondo
Thompson
(1972), Gruji~ and Siljak
(1972 , 1974b,c , 1976b), Michel
(1973), Vakhonina,
Piontkovskii
(1970), Porter and Michel
(1972), Thompson and Koenig
Siljak and VukceviE (1977), Araki
by Matrosov
(1967), Michel and Porter
Siljak
Araki and Kondo
(1978),
systems)
systems with
(1973),
(1975), Michel and Miller Borne, Burgat and Bernussou
(1980) and Ikeda and Siljak
(1980);
Description and decomposition
-
~57
asymptotically stable subsystems by Michel and Porter (1970), GrujiE and Siljak (1972, 1973), Siljak (1972a, 1978), Gruji~ (1972, 1974b,c, 1976b), Michel (1973, 1974), Araki (1975, 1978), Blight and McClamroch (1975), GrujiE, Gentina and Borne (1976), Bitsoris and Burgat (1976), Michel and Miller (1977), Araki (1978) and Sinha (1980);
- unstable subsystems by Thompson (1972), Gruji~ (1972, 1976b), Gruji~ and Siljak (1972, 1973), Michel (1973, 1974), Michel and Miller (1977), Morari, Stephanopoulos and Aris (1977), Araki (1978) and Siljak (1978); -
equi-ultimately bounded subsystems by Kloeden (1975);
- partially stable subsystems by Bondi, Fergola and Gambardella
(1979).
Matrosov (1962, 1968) was the first to treat the Problem B, which was further studied by Gentina, Borne and Laurent (1972, 1973), Grujic (1974a,c ,1977a), Gruji~, Gentina and Borne (1976), Borne and Benrejeb (1977), Borne, Benrejeb and Cocquerelle (1977), Gruji6, Gentina, Borne, Burgat and Bernussou (1978), Gentina, Borne, Burgat, Bernussou and Gruji~ (1979), GrujiE and Burgat (1979, 1980), Burgat and Gruji~ (1979) and Vidyasagar (1980) in the framework of time-continuous large-scale systems. Different stability domains of time-continuous large-scale systems were studied at first by Weissenberger (1973) and later on by Bitsoris and Burgat 1976)~ Morari~ Stephanopoulos and Aris (1977), Siljak (1978) and aruji6 and Ribbens-Pavella (1978, 1979). The surveys together with authors' original results were given by Siljak (1972a, 1978), Martynyuk (1975), Michel and Miller (1977), Araki (1978), Gruji~, Gentina, Borne, Burgat and Bernussou (1978), Burgat, Bernussou, Gruji~, Gentina and Borne (1978), Sandel, Varaiya, Athans and Safonov (1978) and Gentina, Borne, Burgat, Bernussou and Gruji6 (1979).
III.2.
DESCRIPTION AND DECOMPOSITION OF LARGE-SCALE SYSTEMS
A large-scale system
S of dimension dx dt
where
:
n is governed by
f(t,x,P)
f is a member of functional family
,
(1)
F,
feF ,
F = {fl,f2,...,fN} , fk e C(TxRn×Rsxq Rn) , Vk = 1,2~...,N ,
(2)
158
N
Ch. III
is a n a t u r a l
internal all
number.
(e.g.
The
parameter)
admissible
and/or
matrices
P
P :
The
matrices
PI
and
external
is d e n o t e d
are
singleton
{0}
. The
number
The
i-th
interconnected
S.
dxl dt
and The
n : n I + n 2 +.-.+n s number
N , the
fi e[i
tem
(possibly
(Siljak,
Gruji6,
turally
invariant
is
singleton,
structures
F
of the
The iff
function X=0
and
disconnected
tem
vector Si
,
N = {K}
. The
whole
defined
describes
the
subsystem
Si
(3)
the
dxi d~ :
and
,
(4)
GrujiC, The
o f the i972;
whole
k(t) Z K
is the
(possibly
the
set
,
variations
condition
whole
Burgat,
system , i.e.
is the
number
of all
fi(t,x,Pi)
= 0
sysBer-
strucset
N
possible
- gi(t'xi)
vector the
whole
c a n be n o w
gi(t'xi)
: fi(t,xi,O)
describe
the
all
teR
and i-th
(5),
= fi(t,x,Pi) o f the
for
x i : [0 T ,0 T ,...,xi,O T T ,...,0T] T
gi(t,xi)
state
that
N
and
ever
, vteR
1978).
notation
. Evidently
action
(3)
system.
Si
is the
hi(t,x,Pi)
.
k = k(t)
i978;
number
by
subsystem
xi
be
by
x R l X q , R ni ) , V k : 1,2,...,N
is c o n s t a n t ,
satisfies
X'eRniz
may
subsystems
x : [x~,x~,...,x~] T
, k(t) • N
1977,
dxi dt The
P
is d e s c r i b e d
F i (Vi = i,2,...,s)
and Borne, k(t)
Pi : 0 . T h e
gi : Y x R n i ~ Rni
as
set
o f the
Si
' vi = i,2,...,s
structural
1975,
iff
fi
and
number
superscript
arbitrary)
Gentina
of t h e
the
The
.
1971-1973,
nussou,
of
,
N : {i,2,...,N} describe
class
.
defined.
subsystem
fik c C ( T x R
families
variations
denotes
partitioned
= {f~ , f i2 , " , f lN } ,
arbitrary)
~ P2 , v t ~ T }
- fi(t'×'Pi)
The vectors x and f are f : [ f ~ , '~T ~ T I T 2,'",'sJ so that Fi
s
reflects
The
P,
completely
Si
of
perturbations.
by
{P : PI ~ P ( t )
P2
[ T T T]T Rs×q Pl,P2,'",P~ e
P :
matrix
Large-scale systems in general
of
"
the
i-th
function
hi
-gi(t,xi) system
S
described
+ hi(t'x'Pi)
(5)
'
disconnected defined
, Vi = 1,2,...,s on its
i-th
subsys-
by ,
(6)
interconnected
by
v i : 1,2,...,s
(7)
Description and decomposition
Let
Hi
denote Hi
the
159
set of all p o s s i b l e
hi ,
{h~,hi,... , 2 h~) , h i ( t , x , p i )
:
: fi(t,x,Pi)
-gi(t,xi)
Vj = 1,2,...,N , Vi = 1,2,...,s . Evidently, In order scale
f~ e F i
implies
to d e s c r i b e
system
S
more
let the
h~ • H i , and vice
concisely
following
al parameter
sij: T ~ { 0 , 1 }
sij : Y ~ [0,I]
, and r e p r e s e n t s
S I : R~R nixNni
matrix
i-th
that
plies
Slk(t) : 0
it may
The f u n c t i o n
be,
but
for all
need
tem
all p o s s i b l e
S i (7) that dxi dt
provided Let
all
- gi(t'×i)
and will
=
I
S(t)
be c a l l e d
all p o s s i b l e structura~
S(t)
structur-
t , or
of the structural subsystem
Si ,
that
Slj(t)
=
i
im-
, Vi = 1,2,...,s ,
of the
i-th
subsys-
, Z~ : 1,2,...,s ,
valued
(9)
functions.
hy
013
.-
Ois ]
S2
023
..,
02s
0s2
describes
interconnected
by
012
"I ,
0ij E Rnixnj
-ssj all
structural
the structural will
The
of
,
are b i n a r y
be d e f i n e d
021
required
+ Si(t) hi(t,x,Pi)
sij : Y ~ {0,i}
S
The m a t r i x
.
interactions
SI
element
large-
I i - diag {i,I,...,1} • R nixni
can be now d e s c r i b e d
Si : T ~ R n×Nn
function
interconnected
not be,
: R x R n x R l×q ~ R Nni
hi
variable
k~j
hi =
defines
,
Vi : 1,2,...,s .
be i n t r o d u c e d .
valued
(i,j)-th
of the
versa,
structurally
notation
is b i n a r y
S i : [SilIi,si21i,...,SiNIi] Notice
the
(8)
matrix
be d e n o t e d
set o f t~e system,
by
variations
of the
of the system
S.
Ss
and r e f e r r e d
system
The
S
set of
to as the
160
Ch. III
S :S :
systems
i11o12ols I 02 i
Ss :
Large-scale
$2
.- 02s
_
_
, S i = [SilIi,si2Ii,...,SiNIi]
, sij e {O,i}
}
"'" Ss
Os2
sl
in general
(10) In
(10)
there
rather If we
directly
system
that
:
g(t,x)
sequel, along
Many
is o b t a i n e d
then
the
over-all
by h(t,x,P)
of the
,
pep
, S(t)eS s , vteR
can be r a n d o m
.
representing
or can be i n t e n t i o n a l l y
control
if it is not
motions
(9)
T T T T h : [hl,h2,...,h s]
and
variations
variable
in cases
(5)-(8).
perturbations
in s t r u c t u r a l l y
III,3.
(1),
+ S(t)
structural
structural
taken
via
T
can be d e s c r i b e d dx d~
In the
is p e r m i s s i b l e
g = [g[,g~,...,gT]Ts-
S
Notice
than T
set
able
sij ~ [0,1]
(11)
unpredict-
designed
so as
systems.
otherwise system
defined,
D+v
and
v*
will
be
(11).
STRUCTURAL STABILITY PROPERTIES OF LARGE-SCALE SYSTEMS structural
Liapunov,
stability
but
only
I.
The
those
properties
can be d e f i n e d
of t h e m w h i c h
will
be u s e d
in the
sense
of
in the
sequel
are
defined. Definition
equilibrium
- structurally asymptotically is a s y m p t o t i c a l l y
stable
state
x=0
stable
of the
system
(in the whole)
(in the whole)
for e v e r y
(11)
over
is
PxS s
iff it
(P,S) e P×S s
, re-
spectively, -
structurally PxS s every
-
uniformly asymptotically
iff it is u n i f o r m l y
structurally exponentially stable spectively.
cant
structural connective
We d e n o t e
(in the whole)
stable
over
(in the whole)
for
(P,S) e P×S s , r e s p e c t i v e l y ,
is e x p o n e n t i a l l y
The
stable
asymptotically
stable
(in the whole) over
(in the whole)
for every
PxS s
iff it
(p,s) 6 FxS s , re-
• stability stability
a vector
concept
generalizes
concept
[xT,uT] T E R n+s dx d--t = g l ( t ' x )
by S i l j a k by +S(t)
z
and
involves
(1971-1973, and
h(t,x,P)
consider ,
the
signifi-
1978). a system
Structural stability properties
du
where
and,
161
= g2(t,u,x,P,S*)
g 2 : R x R S x R n x p x S s ~R s ;
besides,
ence of the unique
solution
[xT(t;to,Xo) through
each point
Definition
2.
of
The
smoothness,
(12)
[xT,uT] T = 0
e RxpXSs
ensuring
,
exist-
:
, uT(t;to,Uo)] T : zT(t;to,Zo)
RxNxR S . Let G r e e k
state
= 0 , V(t,P,S*)
possess
of e q u a t i o n
(12)
S*eS S ,
g2(t,O,O,P,S~)
g l , h , g2
functions
,
letters
of s y s t e m
,
denote (12)
numbers.
is
: o
u-staDle
a)
there ity
with respect to
exists
6(to,e) > 0
llU(t;to,Uo)ll < 5
b) uniformly inition value
u-stable
a) are ~M
T i , iff for every such
, vtel o
that
with respect to
fulfilled
satisfies
and
the
u-stable
a) are
d) uniformly
of the d e f i n i t i o n s
for every
ted,
iff
3.
The
u-attractive there
exist
eeR+
the
of the def-
consequent
Y i iff c o n d i t i o n s
~M(t,e) ~ + ~
for
e ~+=
in the whole with respect to b) and
c) are
fulfilled.
Ti "
maximal
of the and
teT i ;
Y i , iff c o n d i t i o n s
•
in d e f i n i t i o n s
[xT,uT] T : 0
with respect to and
an i n e q u a l i t y
Uo-Uniform~y
u-attractiVe
the D e f i n i t i o n 6 ]0,+~[
state
~(to) > 0
llZoU < A(t o) +~[ holds; b)
an i n e q u a l -
a)-d)
can be omit-
Yi : R .
Definition a)
eER+
: t e T i] > 0 ;
and
"with respect to
An e x p r e s s i o n
every
condition
fulfilled
u-stable
J~zoIJ< ~ ( t o , e )
Y i , iff c o n d i t i o n s
in the wh~Ze with respect to
definition
and
is f u l f i l l e d ;
inf [~M(t,e) c)
for
toEY i
3.a)
there
are
exist
of s y s t e m
Y i iff
(12)
for any
and
such
that
fulfilled
and and
~ e ]0,+~[ for
, Vt E ]t o + r(to,Zo,~)
with respect to
A(t o) > 0
:
toeY i
r ( t o , Z o , ~ ) E [0=+~[ IIu(t;to,Uo)ll <~
is
for any
Yi
iff c o n d i t i o n s
toeY i
and any
fu(to,A(to),~)
e [0,+~[
of
such
that sup [Tm(to=Zo,~) c}
to-uniformly
u-attractive
the D e f i n i t i o n exist
A>O
: zoeB~(to)
3.a)
and
d)
uniformly
with respect to
are f u l f i l l e d
Tu(Zo,~) • [0,+~]
sup [Tm(to,Zo,~)
u-attractive
] : ~u(to,A(to),~)
and
for any
;
T i , iff c o n d i t i o n s ~ e ]0,+~[
of
there
such that
: t o e T i] : ~u(Zo,~)
with respect to
;
Y i , iff c o n d i t i o n s
,
of the
162
Ch. IIl
Definition 3.a)
3.a),c)
hold and
[0,+~[
are f u l f i l l e d ,
for e v e r y
e) P r o p e r t i e s
3.a)-d)
Definition
3.a)
are f u l f i l l e d
are v a l i d
"with respect to Ti = R .
iff 4.
The
state
a) asymptotically with respect
and
Tu(A,~)
Ii
ble w i t h
respect
to
in d e f i n i t i o n s
3.a)-e)
of s y s t e m
u-attractive
(12)
respect
with respect to
Ii
Uo-Uniformly
and
is
of the
toe T i . • can be
:
T i ,iff it is
with
u-stable
u-stable
to
Ii ;
T i ,iff it is
u-attractive
u-sta-
with
re-
Ti ;
C) q~asi-~iformly asymptotically is u n i f o r m l y attractive
u-stable
uniformly
u-stable respect
e) exponentially constant
u-stable
with respect
with respect
uniformly asymptotically ive w i t h
"
and
with respect to
and
to
Ti
iff c o n d i t i o n s
A(t o) ~ ]0,+=[
[xT,uT] T : 0
u-stable
to
b) equi-asymptotieally
and
A >0
: ~u(A,~)
in t h e w h o l e ,
for any
omitted,
d)
of the D e f i n i t i o n
exist
: ( t o , Z o) e TixBA]
An expression
spect
conditions
there
s u c h that sup [ T m ( t o , Z o , { )
Definiti0n
i.e.
~ e ]0,+~[
Large-scale systems in general
to
to
Ii
and
T i , iff it
to-uniformly
u-
Ti ;
u-stable
with respect to
with respect
to
with respect to
to
Ti
I i , iff it is
and u n i f o r m l y
u-attract-
Ti ;
u-stable values
with respect to
~i
and
~>0
Ti
iff t h e r e
such that
for
exist
A>0
UZoU < A
an in-
equality
Ilu(t;to,Uo)ll
~ ~liZoll exp [ - ~ ( t - t o ) ]
,
VtmT o , toET i ,
is f u l f i l l e d ; f) exponentially tions g)
It p o s s e s s e s types
of
iff
properties
u-stability
In D e f i n i t i o n s omitted,
u-stable
of the D e f i n i t i o n
4.a)-g)
in the whole with respect to
Ti
4.e)
•
are
4.a)-d)
and
fulfilled
for
A : +~
iff c o n d i -
in the whole, iff the c o r r e s p o n d i n g
u-attraction
an e x p r e s s i o n
hold
in the whole.
"with respect to
T i : R ~s
2s See 2) of Comments on References to Ch. III.
Ti "
•
c a n be
Aggregation
III.4.
forms and conditions
of structural
stability
163
AGGRESATION FORMS OF LARGE-SCALE SYSTEMS AND CONDITIONS OF STRUCTURAL STABILITY
The a g g r e g a t i o n
form of a l a r g e - s c a l e
cance for o b t a i n i n g aggregation
effective
method
consists
scale system by a system At first
of smaller
n-th
Further
on we illustrate
closely
connected
tem
so that
could be found
such a s u b s t i t u t i o n
order has
for a definite
n
that correctness
with the c h a r a c t e r i s t i c
solutions
in o n e - t o - o n e motions,
of structural
each of
perturbations.
of the a g g r e g a t i o n features
of the
large-
is not correct,
characteristic
realization
signifi-
The essence
of the initial
dimensionality
systems
it may seem that
for a system of the which can exist
has an essential
conditions.
in a s u b s t i t u t i o n
of the initialone andof o b t a i n e d manner.
system
stability
form is
of the initial
sys-
S.
III.4.1.
Aggregation forms and solutions for the Problem A
III.4.1.1.
The f i r s t
For the subsequent notation
(Gruji£,
aggregation form and s t a b i l i t y
analysis 1972)
+ Dtvi(t,Xi)
it is useful
to introduce
the f o l l o w i n g
:
: lim sup { vi(t+@'xi) @
~ij : { Ol ,'i:ji~j , 6ij llj :
criteria
vi(t'xi)
is the K r o n e c k e r
: @~ 0 +}
symbol
,
,
diag {(i-61j),(~-62j),...,(i-~jj),O,...,O } E R nixni xi(t+@;t,×i)
+
: Xi +~x i ,
D x i j v i ( t , x i) = v i ( t + @ • Xi + li,j+iAxi) - v i ( t + @
, xi + Iij&xi)
: lim sup
8 --0 +
]
: ilaX ill ~ 0
~xij and
+ vi(t,Xi)) T D + vi(t,X i) = (D~ ivi( $ ,x i) Dxi2V i (t,xi) ,... , D + Xi i ' Xini It is easy to verify
that
and Bernussou,
:
1978)
(Grujid,
1972;
+ D+vi(t'Xi ) ~ Dtvi(t~xi) This i n e q u a l i t y Now, the first Assumption
1.
Gruji£,
form can be i n t r o d u c e d
There
an open connected
exist
and
Burgat
this chapter.
aggregation
v i : RxR ni-~R*
Borne,
+ [Dx+ i vi(t ' ×i)] T d×i dt
should be in mind throughout
x i = 0 , functions
Gentina,
by
neighbourhood
~i : R x R n l - ~ R +
Ni
of
and real numbers
164
Ch. III
aii
and
aij(S i ) and
that
systems
in general
:
i)
vi
2)
~ij($i)
3) 4)
a i i < 0 , Vi = 1,2,...,s , + + ]T Dtvi(t,xi) + [Dxivi(t,xi) g i ( t , × i) ~ a i i ~ i ( t , x i ) , V ( t , x i ) ~ R × N i Vi = 1,2,...,s ,
5)
+ ]T [ D x i v i ( t , × i) S i h i ( t , x , P i)
~0
V(P,S) where This
are positive
definite
on
, Vi,j = 1,2,...,s , i #j
N i , vi : 1,2,...,s
,
, VSegs,
,
s
~
~ ~ij(Si) j=l
~j(t,×j)
, V ( t , x ) e RxN
,
e PxS s , Vi : 1,2,...,s ,
N ~NlXN2x...xNs aggregation
(1972, Let
~i
such
Large-scale
,
N
is a C o n n e c t e d
form slightly
neighbourhood
generalizes
that
of
of G r u j i d
X=0
.
and S i l j a k
1973).
aij(Si)
be the
A(S) = ( a i j ( S i ) )
(i,j)-th
, aij(Si)
element
of the
= aii6ij + ~ij(Si)
aggregation
matrix
A(S) ,
~ Vi,j = 1,2,...,s , V S e S s .
(~) Evidently, tion The
A is
a Metzler
matrix
(Chapter
II.5.4)
due t o
2) o f Assump-
1. condition
connected
4 of A s s u m p t i o n
subsystem
is i m p o s e d
S i (5),
I is a r e q u i r e m e n t and
on the i n t e r a c t i o n
hi
the c o n d i t i o n in o r d e r
imposed 5 of the
on the dissame
to e s t a h l i s h
assumption
a solution
the P r o b l e m
A.
Theorem i.
(I) It is sufficient for structural asymptotic stability
of
x=O
of the system
(a) Assumption
PxS s
real number
~i
is asymptotically
Via(t)
that
I holds,
(b) there is positive every
(11) over
or
~± = +~
contractive
such that the set
for every
~ 6 ]O,~i[
and
i = 1,2,...,8 ,
(c) there is
8uoh that
A e R sxs
elementwise,
where
(13) obeys
A(S)
A = A(S ~)
for some
S~ESs
A(S) ~ A , V S e S s ,
is admissible,
and (d) the aggregation
(-i) i
matrix
A
is stable,
all
al2
... all I
a21
a22
-..
a2i I
lail
ai2
...
aiil
f
I
or equivalently,
> 0 , Vi : 1,2,...,s
for
Aggregation forms and conditions of structural stability
(II) bounded
and
Ifj
is s t r u c t u r a l l y Proof.
(I) Let
The v e c t o r (d)
[Fiedler
that
imply
(1962)].
and
bi
existence
The
be the
to 2) and
condition
is the
largest
V~(t)
i-th
and the
of a c o n n e c t e d
element
vector
.
3) of A s s u m p t i o n
, where
connected
1
that
for every V =
neighbourhood
. Hence
of
that can
b =-(AT)-Ic
(b) g u a r a n t e e s
where
un-
system
PxS s .
contractive
, VtcR
~ e ]0,~[
of the
over
from
~(t,x) = bTv(t,x)
for every
the h y p o t h e s i s
due
is r a d i a l l y
x=O
positive
is a s y m p t o t i c a l l y
v(t,x) < ~ , V x e
contractive
i = 1,2,-,s} Further,
positive
V~(t)
vi
in the whole
be d e t e r m i n e d
• Vi = 1,2,...,s . Let
such
N i : R ni ,
be e l e m e n t w i s e
×...× Vs~s(t)
[vl,v2,...,vs] T , and
totically
stable
bcR s
and Ptak
(I),
i = 1,2,...,s , then
ccR~
and
is e l e m e n t w i s e
Vl~l(t) x V 2 ~ ( t )
×=0
to
every
a vector chosen
b
~i ~ ]0'~i[
for
asymptotically
be a r b i t r a r i l y
and
in a d d i t i o n
~i : +~
165
V~(t)
of
is asymp-
~ = min {hi~ i :
b = [bl,b2,...,bs IT
conditions
4) and
neighbourhood
N
5) of A s s u m p t i o n
of
x =0
1
, N CNlXN2×...×N s
such that
D+v(t,x)
s bTA(s) W(t,X)
,
V(t,x,P,S)
ERxN×PxS s ,
(14)
where w
Positive
definiteness
N , positiveness
of
[nz,n2,
=
~i
of all
on
bi
,as] T
N i , vi = 1,2,...,s , the
of
b,
the
condition
definition
(c) of T h e o r e m
of
i and
(14) y i e l d D+v(t,X) The f u n c t i o n 1.3.2.5
v
(II) guarantee positive radial
N I = R ni
positive
Notice
vi
The s t r u c t u r a l If the
another
Theorem 2. stability
of of
(I)
(a) A s s u m p t i o n
with
All
PxS s
asymptotic
structural
of
v.
on
used
o f the
I holds;
of T h e o r e m x=0
of
(15)
13 of S e c t i o n
(ii)
is s t r u c t u r a l l y
the h y p o t h e s i s vi
and
~i
which
system
vi
implies
•
deorescent.
by T h e o r e m
stability
for s t r u c t u r a l (11)
over
l
2 of S e c t i o n
the proof.
I are not
guaranteed
asymptotic
of all
of C o r o l l a r y
completes
in T h e o r e m
Assumption
, Vi = 1,2,...,s , hence,
unboundedness
conditions
of the
i is not uni-
is d e s i r a b l e
be used.
It is s u f f i c i e n t
X=O
of
stability
uniform
should
6 R×NxpxS s .
PxS s .
v . Radial
functions
criterion
conditions
definiteness
satisfied
that
over
together
definiteness
are
all
, V(t,x,P,S)
(P,S) 6 PxS s . Hence,
stable
unboundedness
1.2.3.5
form.
satisfies
for every
asymptotically
~ -cTw(t,x)
P×S s
uniform that
asymptotic
then
166
Ch. III Large-scale systems in general
(b) all
vi
are decrescent functions;
(c) there is
such that
A E R sxs
elementwise,
where
(13) obeys
A(S)
A = A(S ~)
for some
A(S) ~ A , v S ~ S s ,
S*6S s
is admissible;
and {d) the aygreEation matrix (II) I ~
A
is stable.
in addition
u~bounded for every
Let
function under
v
and
tion
i, and
1.3.2.4
Theorem
v
due
the a s s e r t i o n
and the p r o o f
(ii)
Theorem
3.
under
(a) Assumption
of T h e o r e m
(11) over
P×S s
and
such
A e R s×s
for
aij(Si)
S~6Ss
of
8 of Sec-
stability and
of
×:0
of the
~i
exponentia~ stability
that
that
and
~i3
obeys
A(S) = (aij(Si))
= [~ijSij + a i j ( S i ) ]
suah that both
=~isIlXill
~i(t,xi)
nj3
,
held for all A(S) ~ A ,
where
A = A ( S ~)
is admissible;
is stable. If, in addition
(II)
then
Proo~.
b and
Let
condition
tive and
v
be d e t e r m i n e d
(b) and the
proof
numbers
~i , ~ 2
~3
D+v(t,x)
~ -WslIxl)
assertion
Example
for every
N i E R ni
under under
I.
subsystems
i = 1,2,...,s
stable in the
P×S s
the
that
to (I),
of the system is structurally exponentially
x=0
whole ouer
and
and
i : 1,2,...,s ;
for some
assertion
due to T h e o r e m
vi
~ii , ~ 2
(t,xi) e RxN i
VSES s ,
the
and the p r o o f
1 holds;
~ ~i21(Xil(
(c) there is
Now,
i. The
•
of all
~i111xill ~ vi(t,xi)
A
i.
exponential
features
(b) there exist positive numbers
(d)
l.b).
holds
PxS s
of T h e o r e m
1.3.2.4
(I) It is sufficient for structural
of the system
X=0
condition
(II)
is radially
of the system is structur-
x=0
7 of S e c t i o n
under
structural
special
v~
as in the p r o o f
to the
to T h e o r e m
2 guarantees
system
due
and
stable in the whole over
be d e t e r m i n e d
is d e c r e s c e n t
(I) is true
Theorem
of
b
N i = R nl
i = i~2,...,s , then
ally uniformly asymptotically Proo~.
to (I),
are
(I) is true (Ii)
Let be
and
the
is true fourth
considered,
as
in the p r o o f
of T h e o r e m
such valid
that
I imply
both
for all
of T h e o r e m existence
~. Then
of posi-
~IIIxU ~ v(t,×) ~ 2 1 1 x U
(t,×) e R × N
. Hence,
the
in v i e w
of T h e o r e m
9 of the S e c t i o n
in v i e w
of T h e o r e m
I0 of the
order
system
composed
of two
1.3.2.4
same
section.
second
order
•
Aggregation forms and conditions of structural stability
dx1
_
i
dt
-(1+2t) X 1 + s i i (t)
167
+ s12(t)
0
+ Sl3(t)
+
[°]t 0./x21
dR 2 dt
1
-2(i+t) x 2 + s21(t)
+ s22(t)
+
~-~
0.4X12
sat 0.2 x22
where
sat ~ : ~
example,
for
P = {0}
I~I ~ 1
sil(t Si(t)
and
sat~
and the structural
: sign ~
matrices
for
I~I ~ 1
Si(t)
have
0
si2(t )
O
si3(t )
0
sil(t )
0
si2(t)
0
si3(t)
In this the form
] vi:1,2
: 0
[s 0]
and
S(t)
The structural
Ss :
:
S2(t)
set of the system
S(t) : S(t)
is found
as
:
S2(t) Si(t)
: [Sil(t) 12 , s i 2 ( t ) 1 2
sij(t) ~ {0,I} , V t e R Notice
that
al set
Ss
the system
bility concept = 1,2 by
and
structural
are not admissible and that
j,n = 1,2,3
Sl(t) hl(t,×)
and
,
, Vi:1,2 , V j : l , 2 , 3 } .
variations
described
in the f r a m e w o r k
sij(t) : Smn(t)
, si3(t) 12]
is a d m i s s i b l e
. All p o s s i b l e
system
S2(t) h2(t,x)
where
by the structur-
of the c o n n e c t i v e for all
interactions
sta-
i,m :
are d e s c r i b e d
168
Ch. III
Large-scale systems in general
0.4 Xzl
s a t O.IXll
0
0
hl(t,x)
0
0.5 x22
=
,
h2(t,x)
=
0
0.4 x12
0
0
The
form of d i s c o n n e c t e d
subsystems
dXl dt suggests
the
form of
l+t l+t ~ xl
sets
Vi~(t)
gi~ ( t ) which
shows
e ]0,+~[ i=1,2 tions
{×i
they
: are
all = - 1
A(S)
2+t l+t 2 x2
+ Ixi21)
, i:1,2
, ~II(SI)
.
by
asymptotically obtained
' Vi=l,2
contractive
that
: 0.1sli
= 0.4 (s21 + s22 ) , ~22 = -2
of A s s u m p t i o n
-
IXil I + Ixi2 I < ~(l+t2)-l}
It is f u r t h e r
, and
~21($2)
= (l+t2)(IXill
are now d e t e r m i n e d
:
that
d×2 dt
'
vi ,
vi(t,Xi) The
0.2x22
sat
O.ix21
~i(t,×i) , ~12(SI)
, ~22($2)
,
for every
= IXill + Ixi21 = 0.5 s12 + 0.1s13
: 0.2 s23
obey
, ,
all
condi-
all
condi-
1. Hence, q , 0.5 s12 + 0. i s131
- 1 + 0.1Sli
:
0.4 (s21 + s 2 2 )
,
I
-2 +0.2s23
j
Evidently,
0.8 ,
which
] I , vSeSs - 1.8 ] 0.6
O.9
A(S)
,
yields -0.9 A
,
0.6]
.
=
0.8 , - 1.8
The m a t r i x tions of the Ss
•
of
A
(II)
system
obeys
the
o£ this
condition
theorem
is s t r u c t u r a l l y
are
(d) of T h e o r e m
i. Since
satisfied
it follows
asymptotically
then
stable
that
x=0
in the w h o l e
over
Aggregation
forms and conditions
of structural
stability
169
III.4.1.2.
The second aggregation form and ~ t a b i £ i t y c r i t e r i a
The s e c o n d
aggregation
Assumption 2.
There
xi= 0 , f u n c t i o n s
form
exist
vi
is d e f i n e d an open
and
£i
by
connected
and real
NI
neighbourhood
numbers
eli
and
of
~ij(S)
such that I)
vi
and
~i
are p o s i t i v e
definite
2)
aii < 0 , Vi = 1,2,...,s ,
3)
aij(Si)
4)
D t+v i ( t ' x i ) + [D~ivi(t'xi )]T gi(t'xi) ¥i : 1,2,...,s ,
z0
, vi,j = 1,2,...,s , i l j
on
,
N i , Vi : 1,2,...,s ,
vges s , _< ~ii £ 2i( t , x i )
, V(t,xl) E R x N . l ,
and 5)
+
V(P,S) e P x S
S i h i ( t , x , p i) ~ n i ( t , x i) j:l~iJ~ (S i) ~ j ( t , x j ) , V(t,x) E R × N
, Vi : 1,2,...,s .
form
generalizes
s
This a g g r e g a t i o n
Theorem 4.
s
]T
[ D x i v i ( t , x i)
slightly
(I) For structurdl asymptotic
system (11) over (a) Assumption
•
that
of M i c h e l
stability of
(1975,
1974).
of the
X:0
it is sufficient that
PxS s
2 holds,
(b) there i8 positive number
~i
or
such that the set
~i = + =
is asymptotically contractive for every
Vi~(t )
and every
~ 6 ]O,~i[
i : 1,2,...,s ,
(e) there is where
Ae R sxs
such that for some
A : A(S ~)
A(S)
S*6S s
(13) obeys
A(S) ~ A , V S E S s ,
is admissible,
and (d) the aggregation matrix (II)
bounded and
A
is stable.
If, in addition
~i = +~
to (I),
for every
is structurally asymptotically
Proof.
(I) The
(d) of T h e o r e m
conditions 4 guarantee
2) and
D = d i a g {dl,d2,...,d s } for
for the v e c t o r vector
U
the
such
function
v
d = [dl,d2,_.,ds] T
d , positive
and the p r o p e r t i e s
definiteness of all
Ni
N ~NI×N2×...×Ns , of
x=0
tion
existence
(b) g u a r a n t e e s
of the system
x=O
5) of the A s s u m p t i o n
(LaSalle,
diagonal
such
then
stable in the whole over
nite.
Let
is radially un-
N i = R ni , v i
i = 1,2,...,s
the
that
ATD + DA
of
v
vi
on
and
is n e g a t i v e v(t,×)
defi-
= dTv(t,x) of the
for all
i = i,2,-.,s
of a n e i g h b o u r h o o d
is p o s i t i v e
of a p o s i t i v e
by
(a),(c)
of a p o s i t i v e
positiveness
Ni
existence
2,
existence
be d e f i n e d
Elementwise
imply
on w h i c h
1976)
PxS s
number
definite.
The
~ , or
~ = +~
,
N ,
condi, such
170
Ch. III
that
the
largest
connected
neighbourhood
v(t,×) < ~ , V(t,×) E R x V ~ ( t ) ~ ]0,~[ (a) and
D+v(t,x)
V~(t)
of
is a s y m p t o t i c a l l y
. In v i e w of the c o n d i t i o n s (c) of T h e o r e m
Large-scale systems in general
x=0
obeying
contractive
for e v e r y
4) a n d 5) of the A s s u m p t i o n
2 and
4, it f o l l o w s
~ w T ( t , x ) ( A T D ÷ DA) w ( t , x )
, V ( t , X ) e RxN , V(P,S) ~
P×S s
(16) Positive
definiteness
of
~i
negative
definiteness
of
ATD + DA
right-hand
side o f
of T h e o r e m
13 of the S e c t i o n
6 PxS s over
Hence,
(16) on
x:0
also radially
imply negative
for e v e r y 1.3.2.5
(ii)
N i : R ni
unbounded
conditions
every The
N
N i , Vi : 1,2,...~s ~ N c N l × N 2 x
(P,S) c PxS s
are
satisfied
is s t r u c t u r a l l y
.xNs
definiteness . All
and
of the
conditions
for e v e r y
(P,S)
asymptotically
stable
PxS s (II) S i n c e
all
of
on
structural
, which
asymptotic
for w h i c h
Theorem 5.
(11)
(a) A s s u m p t i o n
is
where
completes stability
guaranteed
are
v
is
~i : +~
satisfied
for
• by T h e o r e m
4 is not
is e s t a b l i s h e d .
uniform asymptotic
P×S s
then
In v i e w of all
1.3.2.5
the p r o o f .
criterion
structural over
definite.
unbounded,
it is sufficient
stability
X:0
of
that
2 holds,
(b) the function (c) there
is r a d i a l l y
2 of the S e c t i o n
the next
(I) For
of the system
vi
and p o s i t i v e
of Corollary
(P,S) 6 P×S s
uniform,
and
is decrescent
vi
A 6 R Sxs
A : A(S ~)
such
that
for some
on
N i , Vi : 1,2,...,s ,
(13)
A(S)
S~Ss
obeys
A(S) ~ A , vSe S s ,
is admissible
and (d) the aggregation (If) If,
unbounded
matrix
for every
Proof.
D
and
~
be d e f i n e d
condition
(b),
neighbourhood
shown
v
all
is p o s i t i v e
conditions
proves
the
unbounded still
then
valid.
filled,
of T h e o r e m
assertion
which
v
All
(I),
asymptotically
appropriate that
to
N i : R nl
i : 1,2,..~s , then
uniformly
Let
is stable.
in addition
structurally
In v i e w of the
A
N
in the whole
s u c h as in the p r o o f
×:0
definite
v
(II)
of T h e o r e m
7, the S e c t i o n
1.3.2.3,
are
N I : R ni
and
is r a d i a l l y conditions
unbounded
of T h e o r e m
the proof.
•
and p o s i t i v e 8, the S e c t i o n
4 it was Hence,
satisfied vi
4.
on the
(16) holds.
all
PxS s .
of T h e o r e m
a n d that
(I). W h e n
{s
over
is d e c r e s c e n t
. In the p r o o f
on
is radially
N
under
completes
vi
of the system
X:O
stable
the f u n c t i o n of
and
which
are r a d i a l l y
definite. 1.3.2.3,
(16) are
is ful-
Aggregation forms and conditions of structural stability
Structural (11)
exponential
is g u a r a n t e e d
Theorem 6.
stability
of
x:0
of the
large-scale
system
by
Por struotural exponential
(I)
171
system (11) over
PxS s
it is sufficient
X:0
stability o f
of the
that
(a) Assumption 2 holds, (b) there exist positive numbers ~illlXill
(t,x I) • R x N i
all (c)
2 ~ v i ( t , x i)
there
is
A E Rsxs
V S • S s , for
(d)
and
i : 1,2,...,S ,
such
that
A(S)
713
such that both
= 7i311Xill
~i(t,xi)
obeys
: [aij(Si)]
hold for
A(S)
N i 3 N j 3 , where
: [eii6ij +~ij(Si)]
g A ,
A =A(S*)
i8 admissible,
A is stable. If, in addition
(II)
then
of the system
x:O
the whole over
Proof. and
u2
and
a i j ( S i)
S*•S s
for some
and
~il " ~i2
~ ~i2[Ixi
tive
~M
be
and
the
definiteness
conditions
(b)
and
~3
and
D+v(t,×)
proves When
(16)
completes
Example
2.
(I)
, the
i : 1,2,...,s
that
to T h e o r e m
same
section
satisfied
A simplified control
~i ' ~ 2
for
all
~ ~211xll 2 , which
Section
and all
1.3.2.3.
conditions
model
c a n be g i v e n
o f an o r b i t i n g
in the
dxi dt - = AIIX 1 + Ai2x 2 + Ai3x 3 +uBf(~)
Lur'e
astronomical
like
form
, ~ : Cx
,
Vi=1,2,3
nI : n2 : n3 : 3 ,
All
=
a2
,
-a 5 ,
-a 3
0
,
a4
,
AI2
=-AI3
, -a 5
Aij = 0
of
(P,S) • PxS s ,
1972),
where
The
•
mathematical
system
numbers
7111×II 2 ~ v ( t , x )
N:R n
4
of nega-
is n e g a t i v e .
9 o f the
then
preof.
~M
(t,x,P,S) •RxNxpxS s
i : 1,2,...,s
the
of Theorem
. In v i e w
of positive both
all
are
proof
A T D + DA
eigenvalue
f o r all
due
as in t h e
of
existence
, such
hold
under
for
10 of the
observatory
imply
73 = - X M
such
eigenvalue
A T D + DA
~ -7311xll 2
assertion
Theorem which
and
be defined
v
maximal
of
, in fact
N i : R nl
for every
N i : R ni
PxS s .
D ,d
Let
let
to (I),
(11) is structurally exponentially stable in
, i--2,3
, j--1,2,3
=
-a 6
,
0
,
0
-a 7
,
0
,
0
, j # i ;
(Gruji6,
172
Ch. IIl
0 A22
= A23
=
, -a I ,
2a 2 , -a 3 , -2a
,
0
0
0 •
aq
B
=
,
5ii
0
l 0 |
,
[0 ,i~j
' 6i3J , 0
, 5i2 , O
0
, -a 5
Large-scale systems in general
'
$ij = [ i , i=j
C =
,
rT
: [ 1 2 3 Pli~Pli~Pli]
li
f(E)
The
fT.
"
33
, rLJ
= [pj
1
e [0,1]
,
Vo-ieR
x:O
state
is the
of all
numbers.
The
;
for
unique
matrices
is
equilibrium
Aij , B
and
C
form
stable,
then
a choice
of A s s u m p t i o n
definite
Hi
is a l s o
of all 2
vi
vi=1,2,3
of the
as w e l l
vi:i,2~3
=
Am(G i )
F
is the
its m a x i m a l
symmetric positive
the
as
v
system. are
The
known
real
,
matrix
definite
following
G i e R 3×3 symmetric
matrix
A
. Since matrix.
is o b t a i n e d
, 211HI!I(UAL211 +P[Ir12U)
, -Xm(G2) + 2vllH2ll .Ur2211 ,
0
,
minimal
0
eigenvalue
0
, -Am(G 3) + 2vllH 3 U• II r3311
of the
matrix
8 i • and
eigenvalue. Am(Gi) ilr'i i II <
all
For
in v i e w
, 2(IHIII(UALall +vllrz311)
0
If
conditions
Nil =Am(Hi)
Aii
:
Am(G ) +2~IIHLII'UrLlll
then
state
,
form
arbitrarypositive
A
2,3
subsystems s u g g e s t s the c h o i c e of vi functions T v i ( X i) = x i H i x i , V i : 1 , 2 , 3 , where Hi is the m a t r i x s o l u T Liapunov matrix equation A i i H i + HiAii : - @ i , v i : i , 2 , 3 ,
of the
such
"
disconnected
in the tion
i : 3,2,3
,Pj3 ]
~i(ai) e C(R,R)
d× i dt - A i i x i ~ the
2
l.
elements
of
,pj
] T , E : (GI,02~0"3)T
= [ ~$(al),~2(G2),,3(03)
@i(~i ) G
o
of
(II)
, Ni2 = A M ( H i )
2z, llHill
,
of T h e o r e m
, Ni3 = i ,
Vi:i,2,3 6 are Vi:i,2,3
satisfied
for
kM(Gi)
is
Aggregation forms and conditions of structural stability
Hence,
x~O
III.4.1.3.
of the
system
is e x p o n e n t i a l l y
stable
in the whole.
forms,
we c o n s i d e r
The t h i r d a g g r z g a t i o n form
In c o n t r a s t
to the p r e v i o u s
form for w h i c h
an a g g r e g a t e
Assumption
There
of the
173
3.
states
Xi=0
exist
two a g g r e g a t i o n system open
(a c o m p a r i s o n
connected
, functions
system)
neighbourhoods
vi ,~i
and real
such
a
is n o n l i n e a r . Ni
numbers
, N i c R ni
~i ~ 0
,
Vi = 1,2,...,s • such that i)
vi
are p o s i t i v e
2)
~±
satisfy
of the
a
3)
there all
4)
problem
and they
exists U~U
,
N i , Vi = 1,2,...,s ; ensure
existence
for a c o r r e s p o n d i n g
are
of a local
system
solution
of d i f f e r e n t i a l
continuous;
a neighbourhood u~0
D t+v i ( t , x i )
on
W ° -condition,
Cauchy
equations,
definite
U
, ~(t,P,S,u)
of the point
f 0 ; ~(t,P,S,0)
+ [ D ~ . v i ( t , X i ) ] T gi(t,Xi) z
u=0 :0
such
that
for
;
g ~ivi(t,Xi)
, V(t,Xi) e R x N i ,
Vi : 1,2,...,s ; 5)
+ ]T [ D x i v i ( t , x i) S i h i ( t , x , p i) ~ ~i(t,P,S,vl,...,Vs) V(P,S) e P x S s , vi : 1,2,...,s
This
form of a g g r e g a t i o n
Theorem 7.
,
allows
us to state
the
following.
Let
(a) c o n d i t i o n s (b) there
, V(t,x) 6 R×N
•
o f the A s s u m p t i o n
exist positive
Vif(t)
are
values
asymptotically
3 be f u l f i l l e d ; ~i
or
~i = +~
contractive
such
for any
that
the
~ 6 ]0,~[
sets a n d aZl
i : 1,2,...,s ; (c) there
exist
5~6S I
such
~i(t,P,S,u)
estimation
that f o r f u n c t i o n s
~ ~i(t,P,S~,u)
~i
= ~ i (t,P,$,u)
, V(t,P,S,u)
eRxpXSsXU
an ,
Vi : 1,2,..,s , is f u l f i l l e d ; (d)
the
state
U : 0
o f the c o m p a r i s o n
system
du i dt : ~iui +~i(t'P's~'ul'''''us) be a s y m p t o t i c a l Z y Then
x=O
the s t a t e
stable
on
o f the s y s t e m
(a)-(d) radially
are
x=O
the w h o l e
(11)
is s t r u c t u r a l l y
asymptotically
for
N i : R ni , U : R ~
and functions
P×S s
vi
state
,
stable.
If conditions which
, vi : 1,2,...,s
of system on
P×S s .
are f u l f i l l e d unbounded [21)
for
~i = + ~ • i = 1,2,...,s , then
is s t r u c t u r a l l y
asymptotically
stable
the in
174
Cb. IIl
Proof.
Relying
3, we ically of
on the
conclude
that
contractive
conditions
along
the
conditions
the for
4),
set
(a)
~i ~ ]O,~i[
every
of the
of T h e o r e m
system
7 and
3 ensures
the
The
corresponding
, Vi
following
Uio E V i i i ( t o )
Assumption
3 and
Conditions ensures
Proposition
(a),
of
the
If
conditions
U=R+
S
whole
system
of
view
of c o n d i t i o n s
Ch. II, a l l o w s
U o , P , S ~)
Theorem
asymptotic
the
the
Theorem
radial
functions
rem
ensure
the
comparison
rem
7•
Theorem
of the
, Vi = 1,2~...,s
,
7 and
stability
the
2),
3) of the
us to o b t a i n
VteR
on
,
, i : i,2,...,s
obtained P×S s
estimations .
estimation
oF the
state
x:0
Ni=R
, ~i = +~
(11).
, then of
10,
s ui(t;to,
(d)
structural
.
system
, i : 1,2,...,s , i n
vi(t,xi)
estimation
,
1,2,...,s
:
du i dt - ~ i u i + ~ i ( t ' P ' S * ' U l ' " " U s ) for
is a s y m p t o t -
. Fulfillment
(ll)
~ RxNxPxS s
comparispn
in general
i) of A s s u m p t i o n
, Vi = 1,2,...,s
D + v i ( t , × i ) ~ a v i +~i(t,P,S,vi,...,Vs) V(t,x,P,S)
systems
V l ~ l ( t ) x V 2 ~ 2 ( t ) x...x V S ~ s ( t )
5) of A s s u m p t i o n
motions
Large-scale
vi
asymptotic
7 are
fulfilled
unboundedness together
stability
system.
This
and with
conditions
in the
proves
for positive
the
whole
(b)-(d)
of the
second
ni
definiteness of
zero
statement
in the
the
theo-
solution of the
,
of
Theo-
• 8.
Let
(a) conditions
of the Assumption
(b) al~ functions (c) there exist
be decrescent
vi S*ES s
estimation
3 be fulfilled; on
N i , Vi = 1,2,...,s ;
such that for functions
~i(t,P,S,u)
s ~i(t,P,S~,u)
~i(t,P,S,U)
, v(t,P,S,u)
an
eRxpxSsxU
holds; (d) the state
u = 0 of the comparison dui dt
be uniformly Then the state asdmpvotically If conditions
system
= a i u i + ~ i ( t , P , S ~ " U l ' " " U s ) ' i : 1,2,...,s
asymptotically ×=0
stable.
of the system
stabZe on
(11) is structurally
uniformly
PxS s
(a)-(d) are fulfilled for
N I = R ni
and functions
v±
are
Aggregation forms and conditions of structural
stability
radially unbounded for all
i : 1,2,...,s
, then
system (11) is structurally
uniformly asymptotically stable in the whole
on
175
the state
of the
×:0
PxS s .
Proof.
We
obtain
estimations
vi(t,xi) similarly state
as
U= 0
functions
~ ui(t;to,
in t h e is
Theorem
uniformly
vi
that
asymptotieally
7-
It
,
state
on
VtER
follows
asymptotically
the
stable
u o ,P,S ~)
x=0
of
, i = 1,2,...,s
from
the
stable
and
the
system
8 are
in t h e
i = 1,2,...,s
and
of f u n c t i o n s state
u=0
rem
•
8.
Theorem 9.
decrescent
vi
and
ensure
from (11)
that
the
decreasing
of
is
structurally
P×S s
If c o n d i t i o n s (a)-(d) of the Theorem s U=R+ , radially unbounded functions whole
condition
for
uniform
vi
all
asymptotic
fulfillment
of
the
fulfilled
, that
N i = R ni
positive
, then
stability second
for
are
in
the
estimations
the
statement
,
definite
of
whole
of
the
Theo-
the
Let
(a) conditions of the Assumption
3 with constants
ai
, vi = 1,2,...,s
be fulfilled; (b) there exist positive values
such t~at
~il " ~i2
n i l I I X i l ] ~ v i ( t , X i ) ~ Ni211xill for all
( t , x i ) ERxNi
(c) there exist
, i =1,2,...,s ; such that for functions
S*ES s
an esti-
~i(t,P,S,u)
mation ~i(t,P,S,u)
~ ~i(t,P,S*,u)
, i = 1,2,...,s
, v(t,P,S,u)
•RxpxSsxU
holds; (d) the state
u=O
dui dt
- ~iui
be exponentially Then the state stable on
x=O
PxS s
If conditions then the state
of the comparison system + ~i(t,P,S~,Ul,...,Us
Proof.
Due
to
i = i,2,...,s
stable. of the system (11) is structurally exponentially
.
(a)-(d) are fulfilled for x=O
of the system
stable in the whole on
mations
)
conditions
PxS
N i : R ni , U = R ~ ,
Vi•
[1,s]
,
(11) is structurally exponentially
s
(a)-(e)
of
the
Theorem
9 we
determine
esti-
176
Ch. I I l
vi(t,xi)
~ ui(t;to,
where
Ulo
As far as the state
u:0
stable,
we can find such
it follows
u o , P , S ~) , v t e R
~ ~i211Xioll
systems in general
Large-scale
, i = 1,2,...,s ,
, i : 1,2,...,s
of the c o m p a r i s o n
s y s t e m is e x p o n e n t i a l l y
A>0
values
from c o n d i t i o n
and p o s i t i v e
Ulo < A
7~1
and
~>0
that
that
s
S
ui(t;to,
uo,P,S* ) g 7
Z ulo e x p [ - P ( t - t o ) ] i:l
i:l From conditions
(a) of the t h e o r e m and 1) of the A s s u m p t i o n s
~lllXll s v(t,x)
=
3, we get
s
~ v i ( t , x i) s 7 Z Uio e x p [ - ~ ( t - t o ) ] i:l i:l
Hence
s
UX(t;to,Xo)II ~ ~
~
Uio exp [ - ~ ( t - t o ) ]
,
i:l
where
%*
=~i17 -
This p r o v e s
.
the first
If c o n d i t i o n s s
U=R+
(a)-(d)
, i 6 [1,s]
estimations the s y s t e m
and,
Solutions systems
of the T h e o r e m the value
hence,
an e x p o n e n t i a l
9 follows
disconnected
stability properties
ditions
interactions son f u n c t i o n s
(~i)
The n u m b e r of r e q u i r e d n u m b e r of s u b s y s t e m s
subsystems
(Si) (Si)
functions.
statement
test.
of the l a r g e - s c a l e
is s u i t a b l e
of all d i s c o n n e c t e d
exact
Furthermore,
tests
for e f f e c t i v e
information
subsystems
(Si)
is
the a g g r e g a t i o n
con-
1 and 2) i m p o s e d on the
in terms of the f u n c t i o n s
stability
of
so that the form of the
Then,
used as c o m p a r i -
of the L i a p u n o v
of the d i s c o n n e c t e d
(i.e.
of the s o l u t i o n
The second
decomposition
for Dini d e r i v a t i v e s
(v i) t a k e n a l o n g m o t i o n s
in
f o r th~ Problem A
5 of the A s s u m p t i o n s
are e x p r e s s e d
,
+~
•
subsystems
for the final s t a b i l i t y (the c o n d i t i o n s
estimation
in the whole.
for the P r o b l e m A r e q u i r e
of their L i a p u n o v
N i = R ni
can be c h o s e n equal to
from all this.
into i n t e r c o n n e c t e d
construction needed
A
on ~oZ~tion~
9.
9 are f u l f i l l e d for
,then
Conal~ion
corresponding about
of the T h e o r e m
(11) w i l l be f u l f i l l e d
of the T h e o r e m
III.4.1.4.
statement
subsystems
is for one g r e a t e r
functions (Si) • than the
s+l ).
111.4.2. Aggregation forms and solutions for the Problem B U n l i k e the f o r m e r a g g r e g a t i o n related
to i n t e r c o n n e c t e d
forms,
subsystems
the next a g g r e g a t i o n S i . Let
forms are
v i E C ( R × R n i , R)
and
Aggregation
forms
and conditions
I
vi
is the
function
III.4.2.1.
v,
) + Sihi(t,x,Pi)]
,
i : 1,2,...,s
,
i-th
component
.
of the
vector
The f o u r t h aggregation form and s t a b i l i t y
Assumption
4.
form
There
, functions
bij(Si)
177
and
is d e f i n e d
exist
vi
cij
(tentative
Lia-
v = [Vl,V2,...,Vs ]T
The f o u r t h a g g r e g a t i o n
xi:0
stability
D~vi(t,X i) + [D~ivi(t,Xi)]T[gi(t,Xi
[D + v i ( t , x )
function
punov)
structural
: 1 or
v~(t'x'Pi'Si)
The
of
such
1)
a i j ( Z i) ~ 0 , b i j ( S i )
2)
vi
5)
ti(O) = 0 , [i(pl)
4)
0 ~ fi(Pi) / pi < ~ i
is p o s i t i v e
connected
and
~i
criteria
by neighbourhoods
: RxR~R
and
real
N i , N i c R ni
numbers
, of
aij(Si)
,
that ~0
, cij a 0
definite
on
~ t i ( p ~) ,
, Vi,j
= 1,2,...,s
N i , vi : 1,2,...,s
, i #j
, vSeS s ,
,
, Vp[, Pi'. • R. ,. Pi. ~ Pi ' vi : 1,2,...,s
VPieR
, pi f 0
, 0 < Ki g + ~
,
, where
s
Pi
=
Z C i j V j , Vi = 1,2,...,S j:l
s
5)
vi(t,x,Pi,$i) Vi = 1,2,..,s
~
,
s
~
j=1
a i j ( S i) v j ( t , X j )
, V(t,×,P,S)
eR×N×P×S
+
~
j:1
bij($i)
~j(pj)
,
s ,
and
6)
there
are
some
S*ES s
aij 6 R are
Vi,j : 1,2,.-.,s This
aggregation
B : (bij) tor
4)-6)
, VSES s . form
, aij = a l j ( S * )
inequality
v*(t,x,P,S)
Lur'e
comparison
matrices
A = (aij)
and
4 imply
r : [pl,P2,...,ps IT the
comparison
. The
vector
, vec-
condidifferen-
type
~ A v ( t , x ) + Bz(r)
, r = Cv(t,x)
, v(t,x,P,S) e
RxNxpxS s
or s i m p l y
,
(17a) v*
Let
for
,
A,B,C 6 R sxs , a n d the c o m p a r i s o n
, where
Assumption
of the
bij = b i j ( S * )
, b i j ( S i) ~ b ± j
•
z : [~l,~2,...,~s] T of the
and
a i j ( S i) ~ a i j
introduces
C : (cij)
functions
tions tial
and
, bij • R admissible,
~ Av + B z ( r )
,
r :Cv
(l?b)
.
K = d i a g {KI,~2,...,K s } .
Theorem lO. system
(11)
( I ) For s t r u c t u r a l over
P×S s
asymptotic
it is s u f f i c i e n t
stability
that
of
x=O
of the
178
Ch. Ill
Large-scale systems in general
(a) Assumption 4 holds, (b) there is positive number
~i
or
~i = +~
such that the set
i8 asymptotically contractive fo~ ~very i
: 1,2,...,s
,
(c) either the matrix (d) the pair matrix
(A,B)
A
B = (blj)
or
i~ positive diagonal,
C = (cij)
is controllable,
(A,C)
is observabZe and the
is stable,
(e) there is a non-negative matrix
diagonal
matrix
s×s
®
such that the
Q ( w 2) ,
Q(w 2) = K - Z + H e ( I
+jw®)C(A-jwI~
is positive definite for every
we
1 B ,
(18)
[0,+~]
If, in addition to (I),
(If)
radially unbounded for every
Ni :Rnl
i : 1,2,...,s
then
ueR s
(I) Let
equation
be d e s c r i b e d
of the Lur'e
by the
The
conditions result,
Matrosov,
(c)=(e)
the c o n d i t i o n s vi
that
(a) and over all
see also i,3-6
(b) prove
Uniformity
of the
of the system
II)
that
differential
under
vi
(19)
is v a l i d
over
[O,K[
1950; due
definiteness
of
to
of all
conditions
x=0
of true
unbounded.
because
• gua-
For structural uniform asymptotic stability of
x=O
(11) over
P×S s
which
(ll)
is not
10, is e n s u r e d
stability,
on
PxS s
is e v i d e n t l y
are r a d i a l l y
asymptotic
u=0
Wa~evski,
4 and the
stability (II)
of
1932,
4, p o s i t i v e
asymptotic
and all
structural
by T h e o r e m
(I)
(11)
P×S s .
.
stability
(Kamke,
by 2 of the A s s u m p t i o n
the a s s e r t i o n
, ~i = +~
Theorem 11.
Chapter
p =Cu
absolute
principle
structural
P×S s . Then,
,
of the A s s u m p t i o n
is g u a r a n t e e d
N i = R ni
ranteed
guarantee
the c o m p a r i s o n
1968,
comparison
is
vi
of the system
type = Au + B z ( p )
This
and
, ~i : +~ ×:0
is structurally asymptotically stable in the whole ove~ Proof.
Vi~(t )
and every
~ E ]O,~i[
by
it is sufficient that
(a) Assumption 4 holds, (b) the function
vi
(c) either the matrix (d) the pair matrix
(A,B)
is decrescent on B : (bij)
or
C = (cij)
is controllable,
(A,C)
is positive dia@onal, is observable and the
A is stable,
(e) there is a non-negative diagonal matrix
N i , Vi : 1~2,...,s ,
Q ( w 2) (18) is positive (II)
If, in addition
unbounded for every
i = l~2,.--,s
s×s
matrix
0
definite for every
to (I), then
x=O
N i = R nl
and
such that the wE vi
of the ~ystem
[0,+~]
is radially (11) is
.
Aggregation forms and conditions of structural stability
179
structurally uniformly asymptotically stable in the whole over
PxS s .
P~OO{.
of
(I) T h e
u:O
o f the
conditions
comparison
parison principle also C h a p t e r
If)
that
PxS s
If,
then
Notice
of
uniform
(li)
over
that
~i ) 0
~
is
system
~i = +~
(19)
teed v i a Example
of
comparison The
sixth
and
of
vi
uniformly
comsee
conditions vi
that
conditions
stability
form
x:0
1, is
(a) and
of
(b)
(11) e v e r
are r a d i a l l y
unbounded
asymptotically
stable
is u s e f u l
• Otherwise
takes
v
the
or
of the
then absolute
stability
the 3.
case
is not p r o v e d
exponential
o f all
the
only
all
(i o f the A s s u m p t i o n
(17)
(A + B K C )
w h i c h is a s p e c i a l there
~i = +~
bij ~ 0
inequality V*
definiteness
the
1968;
in
•
fourth a g g r e g a t i o n
and
result,
to the
and
stability
Matrosov,
due
N i : R n~
is s t r u c t u r a l l y
This
1950;
P×S s
assumption
all
for w h i c h
differential
over
absolute
[O,K[
Wa~ewski,
asymptotic
PxS s .
the
i e {l,2,...,s} due to
same
guarantee
(19) on
4, p o s i t i v e
in a d d i t i o n ,
x=O
the w h o l e
1932;
is v a l i d
by 2 of the
prove s t r u c t u r a l
system
(Kamke,
3-6 of the A s s u m p t i o n guaranteed
(c)-(e)
Av
stability
system
order
of
(ii)
of
over
stability
system
4),
comparison
, A
the
: A + BKC
aggregation
to be e x p o n e n t i a l , x=O
is
and h e n c e ,
form
v~m
f~rst
if t h e r e
~i < +~
form.
u=O and
is d e s c r i b e d
In the
of the
cannot
case
comparison
therefore,
P×S s
in such
,
structural
be g u a r a n -
a case. by
(n i = 2 , i = 1 , 2 , 3
s : 3)
dXl dt
: xl +
[:]
¢l(Sl) +
]
6 + 12t 2 -
- F + - J -
Xli + (Ix211 - 0.3 Ix311 - 31xi~I - 21Xl~I) 3
sign
+
5 + 15t 2
dx2 dt
_
x12 + (0.7 I x 2 2 1 - 21×111- 51×~21) 3 s i g n xz2
Xll I
]
x I:l If: °If:°If:s°II: S2
I'12
I
i
I
i
~
(23
IA
-D
IA
g.,o
o
o
~o
o
A
I
~
iI
N.
2
~
2
N
+
~ N
N
~
I-,
X
fD
fD
I
I
v
!
I
m
X
+
I
+
v
I
~
A
+
I
!
I
I
II
+
CO
-
r~
r~ -
O
~
O
+
X
Ca
I
!
t,O 4 =
t,.)~
II
1-o
I
I
roco
il
I
I
I
N N) t~o
0q
~-'
T
~ hO
I "-~
. t~
Pd
?
+
ro
N3 r~
-I-
I
~
II
x
0q
~
tO tO
I k.N
-2 t-d
~
?
4-
X
rx3 ~-~
+
I
I
I
iI
k.s
t~bo
I
L
I
N N3
v
+ O
÷
~O ro
r.o
?
.4-
~.3 r.o ~J
.4-
I
Ii
--
0
-t-
r~ ho
I
?
+
F .~
I-o F~ v
-I-
I
1
fD
).-,
fD
~r
I
Aggregation
forms
and conditions
of s t r u c t u r a l
stability
181
{[ 12oo][oosl S = d i a g { $ 1 , $ 2 , S 3}
,
I2
Ss :
These
S :
disconnected dxi dt
0
0
0
12
-<
subsystems,
S2
0
0
S3
which
- ~iXi + bi@i(ai)
are of the L u r ' e - P o s t n i k o v
can be d e s c r i b e d
the
0
0
a i : c~x i
"
type,
-<
0 (10101010, ,01010101 )
first
third
2
by
i : 1,2,3
'
and
1
'
of w h i c h
are un-
stable. The
f o r m o f the
interactions
vi(x
i)
suggests
= ]nil
the
] + Ixi21
form of
,
vi ,
Vi : 1,2,3
.
It n o w r e s u l t s V~ ~ - 2 v + 2 z ( r )
,
r : Cv
together with A :-21
,
B = 2I
,
r = (pl,P2
~P3) T
,
z(r)
C =
o] 3
-2
1
0
-2
Hence, ,
~2(P2) =p~
, [~(P~) -<~2(p~ )
All c o n d i t i o n s
K -i + H e ( l
x=0
ly s t a b l e in the
In this
, h(P~) <~3(p~ ) 4 have
is o b s e ~ v a b l e
of
(II)
sector
example
comparison
~
of T h e o r e m
o f the w h o l e in the w h o l e
system
over
been
|
,
for
p;~ p.]
,
!
for
V!
p 3 -< P 3 .
verified.
The
and the m a t r i x
-1
4
-i
-1
-i
4
11 are
, i:I,2,3
Tl
Pl < Pl
> 0
satisfied.
is s t r u c t u r a l l y
P×S s
, ~i:~
for
[ ]
+j~J®) C(A-joJI) -1 B
All c o n d i t i o n s state
[i(P[) -< [i(Pi)
o f the A s s u m p t i o n
is c o n t r o l l a b l e ~ (A,C) 1 Lez O: [ I so that
order
31/3)T
vl
[i(Pl) = P~
[3(p3) =p~/3
@i
5
: (pl,P2,p
A
pair
(A,B)
is stable.
, VCO~ [0,+~1
The
equilibrium
uniformly
for all a d m i s s i b l e
asymptotical-
nonlinearities
[0,+~[
the
stability
system
test
despite
has b e e n
the w h o l e
carried
system
out
via the t h i r d
has b e e n of the
sixth
182
Ch. IIl
order.
The
vector
Liapunov
function
used
Large-scale
in t h e
systems
example
has
in general
had
the
form
Llxazl + Ixa2l III.4.2.2. Although special tion
The f i f t h aggregation form the
case
form,
aggregation
obtained
it
Assumpt~0n xi=O
fifth
for
B=O
is in g e n e r a l
5.
There
, functions
i)
aij(Si)
2)
vi
3)
v~(t,x,Pi,S$)
are
v±
~i
are
c a n be
(hence,
another
~±
C=0
)
from
and
real
positive
, VSCSs
the
as a
fourth a g g r @ g a -
defined
by
N i , N i ~ R ni
numbers
definite
considered
form
neighbourhoods
, Vi,j = 1,2,...,s both
sometimes
aggregation
connected
and
~ 0 , i ~j
and
form
a i j ( S i)
such
, of that
,
on
N±
, Vi = 1,2,...,s
,
s
~
~ aij(Si) j=l
eRxNxpxS s
V(t,x,P,S)
nj(t,×j)
, Vi = 1,2,...,s
,
,
and 4) t h e r e a r e r e a l numbers a i j , b i j f o r some S~eS s a r e a d m i s s i b l e , Vi,j This the
= &,2,...,s
form
, VSeS s .
introduces
comparison
, a i j = a i j ( S ~) and b i j = b i j ( S i ) such t h a t aij(S i) ~aij ,
only
vector
v~(t,x,P,S)
one
(sxs)
differential
~ A(S) w ( t , x )
aggregation
matrix
A = (aij)
and
inequality
g Aw(t,x)
, V(t,x,P,S)
eRxNxpxS
s
(20) Theorem 12. system
For structural asymptotic stability of
(I)
(11) over
it is sufficient
PxS s
x=O
of the
that
(a) Assumption 5 holds, (b) there is positive number is asymptotically
~i
or
~i = +~
contractive for every
such that the set ~ E ]O,~i[
Vi~(t)
and every
i : i,2,...,s ,
and (c) the matrix (II)
A
is stable. If, in addition
radially unbounded for every
to (I),
Ni = R n i
i = 1,2,...,s , then
(11) i8 structurally asymptotically
' ~i = +~ x=O
and
vi
of the system
stable in the whole over
PxS s .
is
Aggregation forms and conditions of structural stability
Proof.
(I) In view
of the A s s u m p t i o n matrix tence
A
1962)
condition
5, it follows
are n o n - n e g a t i v e .
of
vector
of the
s-vector c
b
elementwise
elementwise
positive
the
conditions
off d i a g o n a l
condition
elements
of the
(c) g u a r a n t e e s
such that
following
I) and 4)
holds
exis-
for a r b i t r a r y
(Fiedler
and
s-
Ptak,
:
For such a v e c t o r v(t,x) = bTv(t,x) the
5 and
b
conditions
(21)
a tentative
asymptotic all
Hence,
-c
(21)
•
Liapunov
function
v
is d e f i n e d
by
(a) of the
theorem,
i),3)
and
4) of the A s s u m p t i o n
yield
result,
obeys
=
.
v~(t,x,P,S) This
all
the
positive
ATb
Now,
(a) and the
that
Hence,
183
~ -cTw(t,x)
positive
definiteness
contraction conditions
×:0
of
(ii)
(II) Now, for every
of
for all
i : i,2,...,s , prove
that
the
~i
v
radial
obeys
structural
Ni
stable
conditions
completes
asymptotic
the
and that
over
v
P×S s .
over
unhoundedness
all
P×S s , w h i c h
on
1.3.2.5
asymptotically
and
that
over
and
13 of the S e c t i o n
, ~i : +~
1.3.2.5
If it is d e s i r a b l e
vI
is s t r u c t u r a l l y
Ni :Rni
~ RxNxpxS s
of both
Vi~(t)
of T h e o r e m
i = 1,2,...,s , show
2 of the S e c t i o n
, V(t,x,P,$)
P×S s .
of
vi
of C o r o l l a r y proof.
stability
•
be u n i f o r m
then Theorem
13.
(I)
of the system
For structural PxS s
(11) over
(a) Assumption
uniform asymptotic it is sufficient
stability
x=O
of
that
5 holds,
(b) the function
vi
is decrescent
on
~i
, Vi = i,2,...,s ,
and (c) tke matrix
A
is stable.
(II) If,
unbounded then asymptotically Pree~.
condition
s-vector
c>0 b
the
v~(t,x,P,S) result,
positive
to (I),
positive
definiteness
there
function
~ -cTw(t,X) definiteness of
~i
conditions
s-vector v = bTv
on
i2,
i) and
guarantee
b>0
uniformly
4) of the that
satisfying
for arbi-
(21).
For
obeys
, V(t,x,P,S)
e RxNxpxS s
and d e c r e s c e n c y Ni
is radial'ly
vi
PxS s
of T h e o r e m
is
and
N i = R ni
(11) is structurally
(a) and the
5, as in the p r o o f
such a v e c t o r
This
in addition
of the system
stable in the whole over
(I) The
Assumption trary
×=0
for every
of
vi
(22) on
Ni
i = i,2,...,s , prove
and
184
Ch. III
that
the f u n c t i o n
1.3.2.4
over
v
asymptotically
stable
(II) When, unbounded ditions pletes
obeys
over
7 of the S e c t i o n
is s t r u c t u r a l l y
uniformly
PxS s . to
(I),
8 of the S e c t i o n
structural
exponential
the
functions
vi
and
N i = R ni
and
vi
function
1.3.2.4
stability
~i
as
over
is r a d i a l l y
v
obeys
all
PxS s , w h i c h
con-
com-
requires
exponential
it is sufficient
PxS s
specific
properties
of
s h o w n by
For structural
(11) over
(a) Assumption
of T h e o r e m
(11)
•
The
(I)
of
i = 1,2,...,s , t h e n the
of T h e o r e m
Theorem 14.
conditions
x=0
in a d d i t i o n
for e v e r y
the proof.
system
all
P×S s . H e n c e ,
Large-scale systems in general
stability
x=O
of
of the
that
5 holds,
(b) there are positive
numbers
~ilIlXiR ~ v i ( t , X i ) ~,i2Hxill (t,xi) e RxN i
and
and
~ii , ~i2 , and
obeyin@
~13
~i(t,xi)
=~i3nxill
for every
i = 1,2,...,s
and
(c) the matrix
A is stable.
(II)
then
If, in addition
of the system
x=O
the who~e over
PxS s .
Proof.
the v e c t o r
(I) Let
in the p r o o f (b),
off T h e o r e m
it n o w f o l l o w s
and a connected
that
which
shows
that
satisfied
nentially
N i : R ni
(I) s h o w s satisfied
exponentially Example order
4.
that over stable
A sixth
interconnected
are p o s i t i v e of
x:0
~ v(t,x) ,
numbers
~ ~211xll
of
i=1,2,...,s stable in
be d e f i n e d
~i , ~ 2
so as
condition and
73
both
,
V(t,x,P,S)
x:0
v
In v i e w of the
such that
of T h e o r e m
P×S s . H e n c e ,
over
exponentially
(22) holds.
N
conditions
for every
N I = R ni
a n d the f u n c t i o n
~ -~311xn
all
over
stable
(If) with are
there
~lIIxll v~(t,x,P,S)
are
b>0
13. T h e n
neighbourhood
and
to (I),
(11) is structurally
~RxNxPxS s
,
9 of the S e c t i o n (ii)
1.3.2.4
is s t r u c t u r a l l y
expo-
PxS s .
, Vi : 1,2,...,s , i m p l i e s all
conditions
N:R n , which
of T h e o r e m
P×S s . H e n c e ,
then
in the w h o l e
over
order Lur'e
system
subsystems
(Si)
x:0
together
~0 of the S e c t i o n of
P×S s .
(Ii)
is s t r u c t u r a l l y
•
is c o m p o s e d described
1.5.2.4
by
of t h r e e
second
Aggregation forms and conditions of structural stability
Xi
: dt
ai
:
-i0
2Xil
+xi2
+
185
[01 [2 i21 +
¢i(~i)
Sil
, ¢i(ai)
oi I • [0,2]
¢i(oi)
• C(R,R)
,
,
~i(~i)
Vi
=
1,2,3
,
25 si2 ]
@i(oi)
$~1 e [ 1 , 2 ]
, ~ i ( a i ) e C(R,R)
,
Vi = 1,2,3
,
,
and
The s t r u c t u r a l
Si :
°l
: -2(Xll
+ x12) + 0.005 (x21 + x22)
,
~2
: - 2 ( x 2 1 + x22) + 0.005 (x31 + x32)
,
a3
:
0 . 0 0 5 (X21 + x 2 2 )
matrices
Si
and
[0si2] for
Sil
S
- 2(x31 + x32)
have
the
form
hi :
,
si2
.
Vi : 1,2,3
,
25@ i
and S
:
diag {SI,S2,S 3 }
is a d m i s s i b l e for where sij(t) : S i k ( t ) s t r u c t u r a l set S s is d e t e r m i n e d n o w by
,
j ~k
, Vi,j,k : 1,2,3
. The
:],[::]}} Let v i ( x i) : IXill + Ixi2 I Vi : 1 , 2 , 3 ,
,
.
Vi : 1 , 2 , 3
Then,
~i(xi)
: vi(xi)
,
N i :R 2 ,
a l l ( S I) : -6 + 4 S l l
,
a l 2 ( S I) = 0 . 5 s 1 2
,
a l 3 ( S I) : 0
a2l(S2)
= 0
,
a 2 2 ( S 2)
: -6 + 4s21
,
a23($2)
= 0.5s22
a 3 1 ( S 3)
= 0.5s32
,
a32($3)
:
,
a33($3)
= -6 + 4s31
and S* : d i a g
{[oi][oi][0I]} ,
1 o b e y all
conditions
0
,
1
of A s s u m p t i o n
0
1
5. In this
~ o.~ o] A
=
0.5 obeys
A zA(S)
, VSESs , because
-2
0.5
0
-2
1 case
1
,
186
Ch. III
vi
of T h e o r e m
0.5 s22
.5 s32
0
-6 + 4s31
for every
fifth
stable
aggregation
first a g g r e g a t i o n
III.4.2.3. The
-6 + 4s21
13 are satisfied.
ly a s y m p t o t i c a l l y The
0
=
is p o s i t i v e
radially unbounded
o1
o,5 12
A(S)
The f u n c t i o n
Large-scale systems in general
definite
in the whole,
i = 1,2,3
. Hence,
The e q u i l i b r i u m
in the w h o l e
over
decrescent
all c o n d i t i o n s
state
×=0
and of
(II)
is structural-
S~
form can be c o n s i d e r e d
as an a d a p t a t i o n
of the
form For the P r o b l e m B.
Thz ~ixth aggregatian farm
sixth a g g r e g a t i o n
Assumption
6.
There
functions
vi
and
1
aij(S i) ~ 0
2
vi
and
form is d e f i n e d by e×ist a c o n n e c t e d
~i
neighbourhood
and real n u m b e r s
a i j ( S i)
Ni
of
xi=0
,
such that
, i ~ j , Vi,j : 1,2,.-,s ,
~i
are both p o s i t i v e
definite
on
N i , Vi = 1,2,...,s ,
s
3
v i ( t , × , p i , $ i) ~ ~ i ( t , × i ) V ( t , x , P , S ) ~R×N×PxSs
~ aij($ i) ~ j ( t , × j ) j=l
, vi = 1,2,..o,S ,
,
and
4)
there
is
S~eSs
such that for
aij = a i j ( S ~)
,
aij(S i) ~ a i j
,
vi,j = 1,2,...,s , V S E S s Let
A = (aij)
Theorem 15. system
(11)
be d e t e r m i n e d (I)
For structural
over
(a) A s s u m p t i o n
by the A s s u m p t i o n
PxS s
asymptotic
it is sufficient
6.
stability
of
x=O
of the
that
6 holds,
(b) there is positive is asymptotically
n~mber
~i
or
contractive
~i = +~
for every
such
that
~ E ]O,~i[
Via(t)
the set and every
i = 1,2,.,.,s , and (c) the aggregation
matr{x
A
is stable.
(II) If, in addition radially totically
unbounded stable
This t h e o r e m
then
x=O
in the whole
is p r o v e d
to
(I),
Ni =~n1
of the system over
" ~i = +~
and
(11) is structurally
PxS s .
in the same w a y as T h e o r e m
4.
•
vi
is
asymp-
Aggregation forms and conditions of structural stability
187
Uniformity of the structural asymptotic stability is guaranteed by Theorem 16. of the system
x=O
(I) For structural uniform asymptotic stability of (11) over PxS s it is sufficient that
(a) Assumption
6 holds,
(b) the function
vi
is decrescent
N i , Vi = 1,2,...,s ,
on
and (c) the aggregation
matrix
A is stable,
(If) If, in addition to (I), Ni=R ni and v i is radially i = 1,2,. ,s then ×=0 of the system (11) is structurally uniformly asymptotically stable in the whole over PxS s unbounded for every
Proof of this theorem is essentially the same as that of Theorem 5. " Refined properties of the functions v i and ~i lead to the exponential character of the structural asymptotic stability, which is precisely expressed by Theorem 17. system
(I) For structural exponential stability of P×S s it is sufficient that
x=O
of the
(11) over
(a) Assumption
6 holds,
numbers 751" ~i2 and ~i3 such that both ~illlXill2 ~vi(t,xi)~i211Xill 2 and ~i(t,xi) =~i311XilI hold for every (t,xi) ERxN i and i = i,2,...,s ,
(b) there exist positive
and (c) the aggregation matrix
A
is stable.
(II) If, in addition to (I), then
x=O
of the system
the whole over
Ni=R ni
(11) is structurally
i=l,2,..,s
for every
exponentially
stable
PxS s
This theorem is essentially proved in the same way as Theorem 6. • £xamp]e 5. A sixth order Lur'e system is composed of three second order interconnected subsystems ($i) with unstable linear parts,
dXi I d -t [ ~
~] Xi + [ iO -
Sl2
Sll] [~il(Sil)] + [ s13] ~i( )s io
[
2
°il :-Xil ' 0i2 :-Xi2 ' a : 0,5 (Xll +x21 +X31 ) , ~ij(aij) a -I ~[1,2] 13 ~ij(aij) 6C(R,R)
~
~i(e) o -I E [0,i]
, ~i(a) EC(R,R)
~
Vi : 1,2,3
,
Vj =1,2
, Vi = 1,2,3 , Vj =1,2
,
in
188
Ch. III
The
structural
matrices
and
Si
s
S
have
0 s l0 si2
for
structural
Ss =
0
set
Ss
are
bounded.
si4 T
,
Vi
= 1,2,3
,
is d e t e r m i n e d
by
{ {[:000:][:0oo:][:000:]} S : diag
functions
They
1
{ S I , S 2 , S 3}
= diag
,
,
001
The
form
s31
hi = (10~ii,¢ii,~i2,10¢~2,~i)
$ The
the
Large-scale systems in general
vi
are
positive
accepted
definite
_< S _<
001
in
in t h e
the
0 0 1
form
whole,
vi(x i)
decrescent
= llxill 2 ,
and
Vi=1,2,3
radially
.
un-
Now, ~i(xi)
= llxill ,
all(S I) =-9.5+sll+s12+0.5(s13+s14)
N i = R 2 , Vi = 1 , 2 , 3
, al2(S I) =0.5(s13+s14)
,
, al3(S I) =0.5(s13+s14)
,
a21(S 2) :0.5(s23+s24 ) , a22(S2) =-9.5+s21+s22+O.5(s23+s24 ) , a23(S 2) =0.5(s23+s24 ) , a31($3)
=0.5(s33+s34
) , a32(S3)
and
obey
all
structurally
It
is
a21 : 1
, a22 : - 6 . 5
, a23 : 1
a31 = 1
, a32 = 1
, a33 = - 6 . 5
of Assumption
to n o t e
of the
Assumption
7.
There
functions
vi
and
i)
v i is p o s i t i v e there
is
tractive and
6 and
stable
that
the
comparison are
~i
such
every
on
(II)
of T h e o r e m
the w h o l e
over
aggregation
17. H e n c e ,
) ,
x=0
Ss
form
admits
sign
functions.
real
that
in
sixth
a connected and
definite
~i > 0 for
=-9.5+s31+s32+0.5(s33+s34
, a13 : 1
indefiniteness
2)
a33(S3)
, al2 : 1
exponentially
interesting
,
all : - 6 . 5
conditions
is
=0.5(s33+s34)
neighbourhood
numbers
N i , Vi = i,2,...,s the
~6 ]0,~i[
set
Ni
aij(Pi,Si)
Vi~(t)
, Vi = 1,2,...,s
of such
Xi:0
,
that
, is a s y m p t o t i c a l l y ,
con-
Aggregation forms and conditions of structural stability
189
s
3)
vi(t,x,Pi,Si) v(t,x,P,S)
Now,
s ni(t,xi)
= [aij(Pi,Si)]
system
, Vi : 1,2,...,s ,
matrix
A(P,S)
depends
on
(P,S)
, A(P,S) =
• (I) For s t r u c t u r a l
18.
(11)
ni(t,xi)
6RxNxPxS s . •
the a g g r e g a t i o n
Theorem
~ aij (pi,Si) j=L
over
(a) A s s u m p t i o n
PxS s
asymptotic
it is s u f f i c i e n t
stability
that
of
o f the
x:O
both
7 holds,
and (b) the a g g r e g a t i o n
(II)
If,
unbounded
in a d d i t i o n
for every
(11) is s t r u c t u r a l l y
(I) Let
Proof.
tion
(a) and
. The
of
on w h i c h
This r e s u l t
: vl(t,Xl)
such
for
3) of A s s u m p t i o n
~ w T ( t , x ) A(P,S) w ( t , x )
the c o n d i t i o n
All c o n d i t i o n s
of T h e o r e m
PxS s . Hence,
x=0
of
of
neighbourhood
is a s y m p t o t i c a l l y along
motions
, V(t,x,P,S)
of
P×S S
The
condi-
of
v
v
on
guaran-
V~(t)
of
contractive. (ll)
~ RxNxpxS S .
(b) p r o v e
~ 0
, V(t,X,P,S)
13 of the S e c t i o n
(ll)
definition
largest
7 impli@s
definiteness
is
system
over
+... + v s ( t , x s )
vi
and
o f the
in the w h o l e
positive
tER
• ~i = +~
×=0
then
7 and the
the
for all
D+v(t,x,P,$)
When,
semi-definite
Ni = R n i
+ v2(t,x2)
7 prove
that
v(t,x) < ~
and
(I),
stable
2) of A s s u m p t i o n ~ >0
D+v(t,x,P,S)
to
asymptotically
v(t,x)
condition
The c o n d i t i o n
PxSs
is n e g a t i v e
i = 1,2,...,s
i) of A s s u m p t i o n
tee e x i s t e n c e x:0
A(P,S)
( P , S ) E PxS s .
every
radially
matrix
is s t r u c t u r a l l y
eRxNxpxS s . 1.3.2.5
are
satisfied
asymptotically
stable
over
over
.
in a d d i t i o n
bounded,
then
Vf(t)
and
All c o n d i t i o n s PxS s , w h i c h
(I),
N:R n , v
ly u n b o u n d e d , 6 ]0,+=[
to
is p o s i t i v e
D+v(t,x,P,S) of C o r o l l a r y
Example
6.
systems
is d e s c r i b e d
, ~C : + ~
fourth by
~ 0
2 of S e c t i o n
order
system
vi
is r a d i a l l y
in the w h o l e
contractive
, V(t,x,P,S)
the proof.
and
definite
is a s y m p t o t i c a l l y
completes The
N c : R ni
are
and r a d i a l -
for e v e r y
6RxRnxPxS
1.3.2.5
un-
.
satisfied
over
• composed
of two
second
order
sub-
190
Ch. IIl
Large-scale systems in general
9 + t + 5t 2 i + t2 Xll + 2 P l l S l l X 2 1
] + PllSllX22
I + t--2 X 12 + ~ P l l s llX21 + P l I S I I X 2 2
4 +t +4t 2 6P21S21Xll
X2 :
+ 2P21S21X12
1+t
X2+ t i + t 2 X22
~-2P21s21Xll + P 2 1 S 2 1 X 1 2
The
system
S
=
structural
matrix
S
is found
as
ii[1 ,sll 1o i o o , oil Lo,u ,sll,
, ,
0 0
o
o
,S21 , 0 ,
0
[
{,
0
S21
, S21
O,
and
S s : {S : O - < S - < I 4} The
parameter
perturbation
matrix
P
has
the
form
Pll
P and the
set
P of all a d m i s s i b l e
P : Let
P :
v i : ( l + t 2) II x i l l 2 , i : 1,2
~l(t,x,pl,S1)
Lp~ j P
l
, be accepted.
and
v2(t,X,p2,S
2) <
V(t,×,P,S) ~Rxl~xPxSs
, which
is
0 . 2 5 J -< P -< 0 25
_<
yields
x1]
Then,
S2
'
Aggregation forms and conditions of structural stability
a z ( t , × i) = i v ~ + t 2 (3xlz + x z 2 ) The f u n c t i o n s
~i
and
x2 , respectively.
The
~2
• e2(t,x2)
are
sign
aggregation
191
=Iv/T~
indefinite
matrix
(2x2z + x 2 2 )
and
A(P,S)
linear
in
xI
now r e s u l t s
and
in the
form -i
A(P,S)
, ~ PllSll
= 2P21S21
It is n e g a t i v e The set
definite
Via(t)
is a s y m p t o t i c a l l y . All
librium
state
stable
PxS s .
=
X i : Xil + x i 2
contractive
conditions x=0
over
functions
time-invariant,
of
of the
in the whole
When the
-i
,
Vi~(t)
Vi--1,2
on
,
gi
then
<
for all (II)
, i = 1,2 ~ e ]0,+~[
of T h e o r e m
system
and
the
hi
in
system
v i , hence
18 cannot
~i = +~
18 are verified.
The
, equi-
asymptotically
PxS s (9),
(ii)
be a p p l i e d
v,
g
and
h
in
(11),
be
cases,
are
by
h(x,P)
should
in such
hence
is d e s c r i b e d
dx
Theorem
. Hence,
is s t r u c t u r a l l y
d~ : g ( x ) + S ( t ) and the f u n c t i o n s
,
(23)
chosen
independent
for w h i c h
the
of
t .
following
can be useful. Assumption
8.
There
functions
vi
and
1)
vi
~i
is p o s i t i v e
2) a l o n g m o t i o n s
are a c o n n e c t e d and real
definite
of the
v.*(x,Pi,Si) i Vi =1,2,...,s
on
system
neighbourhood
numbers Ni
and
Ni
aij(Pi,Si)
of such
time-invariant,
xi:0
,
that Vi:l,2,...,s
,
(23)
_< n i ( x i )
, V(x,P,S)eN×P×S
s _Z a i j ( P i , S i ) j l
•j(xj)
,
s ,
and 3) the z,
singleton z
:
Theorem 19. system
{0}
is the
largest
invariant
:
:
(I)
(23) over
(a) Assumption
For structural P×S s
8 holds
asymptotic
it is sufficient
set of
o}
stability
that both
(23)
in the
set
•
of
x=O
of the
192
Ch. III
Large-scale systems in general
and (b)
the
aggregation
(P,S)
e pxS s
(II) unbounded
for
Proof.
(I)
condition nite
on
Let
addition
i = 1,2,...,s
v(×)
(a) and N.
in
asymptotically
The
XM(P,S)IIw(x)II
The
If,
every
structurally
A(P,S)
matrix
condition
is n e g a t i v e
(b)
to
stable
= vl(xl)
N i = R ni
(I),
then
×=0
in
the
+v2(x2)
1) of A s s u m p t i o n
condition 2 ,
This
result
ness
of s o l u t i o n s
tion
8 and
of the
ensures
• NxpxS s ~M(P,S)
,
<0
<0
,
S0
, VxEZ
to
Rn
(23)
Fied
then
For
on
relative
(P,S) • PxS s
For every
8 now yields
of
v
N.
on
The
asymptotically
stable
Examp]e
7.
order
fourth
the
defi-
T
s
•P×S
stable
N
i976)
the
(23)
stationary
3) of A s s u m p -
and
vi
is r a d i a l l y
over
system
relatively
principle
x=0
of PxS s
x=0
(P,S) • PxS s .
are p r e c o m p a c t
invariance
in the w h o l e
precompaet-
p r o v e now that
for e v e r y
N i = R ni of
ensure
condition
(LaSalle,
(I),
and
of
v*(x,P,S)
w : (nl,n2,...,ns)
(P,S) • PxS s . Hence,
turally
The
to
view
is p o s i t i v e
,
to
solutions
v
,
is a s y m p t o t i c a l l y in a d d i t i o n
is
, V(P,S) • PxS s . Hence,
Vx•N/Z
principle
N = R n , all
every
Rn
(23)
(23)
P×S s .
. In
function
for
definiteness
invariance
(II) When, unbounded
every
is r a d i a l l y
system
over
v(P,S)
and p o s i t i v e
system
for
vi
and
the
+...+Vs(Xs)
8 the
2) of A s s u m p t i o n
V(x,P,S)
of
of whole
v*(x,P,S)
the
definite
.
(li) .
is satis-
is struc-
•
is d e s c r i b e d
by
l dX I = [-6Xll + 2x12 + sat o i + 4 sin o"2/
L-sXl2+ 2Xll
dt
d× 2 _ [sat
Lsat
dt The
structural
SI
:
+ sat a 1 + /4 s i n
u I -2X21
J
0.i = Xil + Xi2
+ S 2 i X 2 2 - 3 sat 0.2 - 2
sign
, Vi : i,2
e21 o
U I
-
5X22 + S22X21
matrices
Sl , S2
3 sat a 2 and
S
[: i] [s:11o ,
2 sign e 2 are g i v e n
S2 :
0
for
a2
s22
i]
J by
, S = diag{S z $2}
,
Aggregation
forms and conditions
of structural
stability
I - 6 X l l + 2x12 + sat a I + 4 sin a21
:
: hl
193
, h2
2 3 4 ]T [h~,h2,h2,h 2 ,
L-SXl2 + 2Xll + sat a 1 + 4 sin a 2
where h~(x) = x22
, h~(x)
= sat a I - 2 x 2 1
-0.1sat
a 2 - 2 sign a 2 ,
h~(x) = x21
, h~(x)
= sat a I - 3 x 2 2 - 0 . 1 s a t
a 2 - 2 sign a 2 .
The s t r u c t u r a l
Ss :
set
Ss
takes
5 : diag
the
form
,
of
~ $ ~ diag
, 01
0 0
Let
vi(xi)
definite, obey
the
for
:ISxil] 2 , Vi : 1,2,3
decrescent condition
Z
Hence, there
xeZ
=
{x
iff both
• However,
,
a21(S 2) : 0
, a22(S2)
,
= IXll + x 1 2 I
and
case
Xll = - x 1 2 set for
in
Z
×eZ
that
{0}
the
Xil i xi2 largest
. Since
and of the
x21 = - x 2 2 system
and
from
is n e g a t i v e
for e v e r y
the m a t r i x
definite
x=0 over Notice
of the
system
A(S)
(If)
, which
if
show that
Xll = - 5 1 2
the
state
and
equations
x2: [(s21-1)x21 ],
i : 1,2
set of the
, as soon as system
in
Xll ~ 0 , Vi : 1,2 . Z
is the
singleton
,
(because of
.
( s 2 2 + 3 ) x21
invariant
then all c o n d i t i o n s
= Isat a21
by
L-10x111 i.e.
:-3
~2(×2)
it r e s u l t s
[ 8xi] Hence,
parameters
8, , al2(S I) : 4
in this
is p o s i t i v e
following
(Xll + x 1 2 )2 + [ sat (x21 + x 2 2 ) ] 2 = 0}
:
is an i n v a r i a n t
x21 = - X 2 2
vi
The
all(S 1) : - 2
is d e t e r m i n e d Z
every
unbounded.
2) of A s s u m p t i o n
N 1 = N 2 = R , ~l(Xl)
The set
. Evidently,
and r a d i a l l y
its
symmetric
of T h e o r e m
is s t r u c t u r a l l y
part
19 are
is n e g a t i v e
satisfied.
asymptotically
stable
in the w h o l e
Ss . that
~i
and
~2
are not p o s i t i v e
definite
definite),
Therefore,
functions.
194
Ch. III
III.4.2.4. The
s~vgnth aggregation form and s t a b i £ i t y ~ r i £ e r i a
The
seventh
aggregation
Assumption
9.
functions
vi
1)
vi
2)
~i
3)
~i(O) : 0
Large-scale system in general
There , ~i
form
are
a connected
and
{i
is d i f f e r e n t i a b l e
N i , Vi : 1,2,...,s is p o s i t i v e
is d e f i n e d
and
in
vz --~i(~) d~
neighbourhood
real
numbers
'(t,xi) E R × N i
NI
aij and
of
and
xi:0
bij
positive
such
that
definite
on
, definite
on
R+
, 0 <~i(vi)/~i(vi)
I
by
, vi : i,2,...,s
,
< ~i _<+~ , Vvi ~ R +
is d e f i n e d
and
continuous
+~
only
in
, Vi : 1,2,...,s
vi
for
,
v i e R+
all
0
and
5)
tends
there
to
is
tractive
[i > 0 for
if a n d such
all
that
if
the
~ • ]0,~i[
v i ~+~
set
,
Vi{(t)
, vi = 1,2,...,s
is a s y m p t o t i c a l l y
con-
,
and s
6)
~i(t,x,Pi,Si) vi : 1,2,...,s
This and
, V(t,x,P,S)
aggregation matrix
K,
non-linearity Theorem 20.
system
A
form
eR×N×PxS
determines
A : (aij)
Z {aijf~j[vj(t,×j) j:l
two
, B : (bij)
s
] +bij~j[vj(t,×j)]}
,
"
aggregation
matrices
A
and
, K : d i a g {Kl,gr,...,~s} , and
B,
vector
z : [~l,~2,...,~s IT (I)
For structural asymptotic stability of
(11) over
(a) Assumption (b)
_< ~ i [ v i ( t , × i ) ]
it is sufficient
Px2 s
x:O
of the
that
9 holds,
is stable,
(c) there is a non-negative diagonal 1) 2K - I - O B - B T ®
matrix
s×s
0
such that both
and
2)
-[AT +A+ (ATo+ I +B)(2K-L-OB-BTo)-L(ATO+I
+B) T]
are positive definite. (If)
If, in addition to (I),
radially unbounded then
×=0
totically stable in the whole over
Proof.
(I)
Let
v
be
defined
Ni : R n i
of the system
for
and
- ~i : ÷~
vi
is
(11) is structurally asymp-
P×S s . (9 : diag{@l,Sr,...,Ss)
by
~vi(t,xi)
(2a)
]o
Aggregation forms and conditions of structural stability
The c o n d i t i o n s
(a) of the
imply p o s i t i v e
definiteness
assumption
guarantees
asymptotically
theorem of
H : 711
u
existence
contractive
and on
of
i), N.
3) and The
~ >0
for every
, D = ~~( A T ® + I )
195
4) of A s s u m p t i o n
condition
such
~ E ]0,~[
that
9
5) of the
the
set
same
V~(t)
is
. Let
, r = K-l_ 71( O B
+BT@)
Then
AT+A+(AT@+I+B)(2K - I - OB-BTO)-I(AT®+I+B) T : ATH+ HA+(D+HB) r-I(D+HB) T. The r e q u i r e m e n t s Gruji6, Q
1977)
i) and
existence
2) of the
condition
of
and p o s i t i v e
X, Y
such that
and the
definite
(due to s×s
matrix
yTy : F
(25a)
ATH + HA + XX T = - Q
(25b)
XY+HB This result
(c) g u a r a n t e e
condition
:-D
(25c)
6) of A s s u m p t i o n
9 yield
_<
~)(t,x, P,S)
-< wT(t,x) Aw(t,x) + wT(t,x) Bz(t,x) + zT(t,x) ®Aw(t,x) + zT(t,x) e)Bz(t,x) -- wT(t,x)(ATH + HA + XXT) w ( t , x ) - [xTw(t,x) + Y z ( t , x ) ] T [ x T w ( t , X ) + Y z ( t , x ) ] - [wT(t,x) - z T ( t , x ) K-z ] z ( t , x ) S -wT(t,x) QwT(t,X) Hence,
all
conditions
fied over
P×S s
and
(II) When, radially
in a d d i t i o n
contractive
for
the S e c t i o n
1.3.2.5
to
under
(I),
(I)
unbounded,
and the
~ E ]0,+~[
are
. All
satisfied
set
, ~i : +~
and
is p o s i t i v e V~(t)
conditions
over
are
satis-
is true.
N i : R ni v
1.3.2.5
vi
is
definite
in the
is a s y m p t o t i c a l l y of C o r o l l a r y
P×S s , w h i c h
completes
2 of the
•
in the case tion form
the
5 of A s s u m p t i o n
system
(ii)
can be r e l i e d
Assumption
10.
functions
vi , ~ i
vi
13 of the S e c t i o n
Vi = 1,2,...,s , then
every
When the c o n d i t i o n
i)
of T h e o r e m
ERxN×PxS s
the a s s e r t i o n
unbounded,
whole and r a d i a l l y
proof.
V(t,x,P,S)
,
There
are a c o n n e c t e d
and
is p o s i t i v e
on
satisfied, then
which
the
happens
sixth
aggrega-
on
~i
is d i f f e r e n t i a b l e
and d e c r e s c e n t
9 is not
is t i m e - i n v a r i a n t ,
and real in
neighbourhood
numbers
aij
Ni and
(t,xi) E R x M i , p o s i t i v e
of bij
xi:0 such
definite
, that
on
N i , Vi : i~2,...,s ,
2)
~i
definite
on
3)
~i(O) : 0 , 0 g ~ i ( v i ) / ~ i ( V i )
R+
, Vi = 1,2,...,s ,
< g i ~ +~
' Vvi e ~+
, Vi : 1,2,...,S ,
~i
196
4)
Ch. 111
Ivi{i(~)
d~
is d e f i n e d and c o n t i n u o u s
0 ~
if and only
if
vi
Large-scale
v i 6R+
in
t e n d s to
+~
systems
in general
and tends
to
+~
,
and s
5)
vi(t,x,Pi,Si)
E {aij~j[vj(t,xj)] j:1 v i : i,2,...,s , V ( t , x , P , S ) ERxNxpxSs •
Theorem 21.
(I)
of the system
A
+bij~j[vj(t,xj)]
}
x:O
For structural uniform asymptotic stability of
(11) over
(a) A s s u m p t i o n (b)
~ ~i[vi(t,xi)]
£t is sufficient that
PxS s
10 holds,
is stable,
and (c) there is a non-negative diagonal I) 2K -I -OB -BTo and 2) - [AT S A + ( A T e + I + B ) ( 2 K - I are positive
s×s
-OB
(II) If,
- BTO)-I(ATe+
in addition
to (1),
i = £,2,...,s
then
(I) Let the f u n c t i o n
and the c o n d i t i o n s definite
the fact that hypothesis
to
v
on
vi
on
N.
such that both
[ + B) T]
v
be d e f i n e d by
3) and 4) of A s s u m p t i o n N.
and
vi
of the system
is radially (11) is
stable in the whole over (24).
on
PxS s .
The c o n d i t i o n
10 p r o v e
that
The well d e f i n e d form of v f o r a l l
is d e c r e s c e n t
v
~i
(a)
is
and
~i,
N i , w h i c h are i m p l i e d by the
of the t h e o r e m and c o n d i t i o n s
and the c o n d i t i o n crescent
i),
Ni= R mi
x =0
structurally uniformly asymptotically
positive
~
definite.
unbounded for every
Pros~.
matrix
i) and 3) of A s s u m p t i o n
4) of the same a s s u m p t i o n
F o l l o w i n g n o w the part
guarantee
that
of p r o o f of T h e o r e m
v
10,
is de-
20, related
so that
~(t,x,P,S)
~ - w T ( t , x ) Qw(t,x)
in v i e w of the c o n d i t i o n s
(c-i)
ized by a n e g a t i v e
definite
positive
Hence,
1.3.2.4
definite.
are s a t i s f i e d (II) When,
on
for every
and obeys
all c o n d i t i o n s
completes
the proof.
(c-2),
function
to
(I),
of T h e o r e m
on
~ i : R ni v
that
v
N because
of T h e o r e m
PxS s , w h i c h p r o v e s
i : 1,2,...,s , then •
it f o l l o w s
(-wTQw)
all c o n d i t i o n s
in a d d i t i o n
unbounded
and
, v ( t , x , P , S ) eRxN×P×S s ,
vi
is
under
(I).
is r a d i a l l y
is also r a d i a l l y
8 of the S e c t i o n
is majorQ
7 of the S e c t i o n
the a s s e r t i o n and
(26)
unbounded
1.3.2.4,
which
Aggregation forms a n d c o n d i t i o n s of s t r u c t u r a l s t a b i l i t y
The r e q u i r e m e n t
that all
~i
be p o s i t i v e
the s y s t e m
(23) as s h o w n in what
Assumptlon
11.
functions
vl , ~ i
bers
and
1)
aij vi
There
~i
definite
can be o m i t t e d
for
follows.
are a c o n n e c t e d
and
bij
197
neighbourhood
all i n d e p e n d e n t
Ni
of time
of
xi=0
,
t , and real num-
such that
is d i f f e r e n t i a b l e
in
xI eN i
and p o s i t i v e
definite
on
Ni ,
Vi = 1,2,...,s , 2)
~i(O) : 0 , 0 ~ ~ i ( V i ) / ~ i ( V i ) < z i S +~
3)
I vi
~i(~) a~(~----7d~
is d e f i n e d
0 4)
+~
' Vvi 6 R+
and c o n t i n u o u s
if and only if
vi
of the s y s t e m
(23)
a l o n g the m o t i o n s
in
, Vi : 1,2,...,S ,
v i •~+
tends
to
and tends
+~
to
,
S
vi(×,pi,$i)
s ni[vi(xi) ]
Z {aijnj[vj(xj) ] +bljfj[vj(xj)]} j=l
Vi : 1,2,...,s , V(X,P,S) e N x p x S
,
,
and 5)
the s i n g l e t o n
invariant
set of the s y s t e m
s +a~(x 2) +...+a2(x
) : o}
(a) Assumption
PxS s
it i8 sufficient
(23)
• x:O
(I) For struatura7 a s y m p t o t i c stability of
system (23) over
(b)
is the largest
Z : {x : ~i2(x 1)
in the set
Theorem 22.
{0}
of the
that
11 holds,
A is stable,
and (c) there is a non-negative diagonal 1) 2K - 1 - OB -BTo
sxs
matrix
®
such that both
and
2)
- [ A T + A + ( A T e + I + B ) ( 2 K - I - ® B - B T ® ) - I ( A T ® + I + B ) T]
are positive
definite.
(If) I t
in addition to (I),
unbounded for every
i : 1,2,...,S
then
N i = R nl
X:0
and
of the system
(I) Let the f u n c t i o n
proofs of T h e o r e m definite
on
N,
(26) is v a l i d Hence,
for
Am(Q)
be d e f i n e d
20 and T h e o r e m
and a l o n g
for
v
Q
(24).
21 it f o l l o w s
solutions
symmetric
by
positive
b e i n g the m i n i m a l
that
of the s y s t e m of
(23) is
Following v
the
is p o s i t i v e
(23) the i n e q u a l i t y
definite matrix
eigenvalue
is radially
PxSs
structurally a ~ y m p t o t i a a l l y stable in the whole over Proof.
vi
obeying
(25).
Q , Am(Q) > 0 ,
198
Ch. III
5(x,P,S) Since
in the set
largest and
since
set
V~
all
all in
of the
of
x=0
solutions of
of
x=0
over
PxS s
•
E x a m p l e 8.
The
(25)
order
I111~i+t + 2 t 2)(xll2 2
(23)
is the
on
(27)
to the
~
and
1976)
holds, largest
(27), are
is s t r u c t u r a l l y
to
(I),
~ i = R ni
N=R n
~
~
and
is also
relative
to
composed
is r a d i a l l y unbounded
R n , which
asymptotically
system
vi
radially
stable
of two
prove
in the whole
second
2 )(xll+
2
i~
+ x 1 2 ) - 2 kXll
2 2 xl2) - 2(Llxll
21,5 +
x Ii
bXl2)
order
]
2 1/5 +x12) xj_ 2
+ S12
Sll
+
0
o
-0.1 (1 + sin t) x~ixl2
1 SZ
o
•
Sll
, s~
,
,
•
i
O
2
o / S12
2
0.05 (1.5 + cos t) xllx21
13x21 - 0.5 (x21 + 4x22) X21
d× 2 dt 10x22
2 2 2(x21 + x22) x22
+ S21 0
[ 0 ÷
S22 0.10 x 1 2 x 2 2
$2 =
i '
o
, s21
sub-
13 x22Xll
[ sic ~ ,
then
satis-
by
dX 1
12(l+t
u
(LaSalle,
system
are p r e c o m p a c t
fourth
{O} 11,
relative
of
principle
is s t r u c t u r a l l y
is d e s c r i b e d
=
are p r e c o m p a c t definiteness
of the
in a d d i t i o n
(23)
singleton
P×S s .
Vi = 1,2,...,s , then
that
dt
(23)
over
the
(27)
due to 5) of A s s u m p t i o n
invariance
stable
(II) When, unbounded,
systems
(23)
P×S s . Hence,
asymptotically
and
of
due to p o s i t i v e
conditions
fied over
set
solutions N
~ -Xm(Q)llw(x)ll 2
Z = {x : ilw(x)}l 2 : O}
invariant
Large-scale systems in general
0]
" s22
'
,
, s22
0
]
]
Aggregation forms and conditions of structural stability
199
The form of nonlinearities suggests vi(xi) : l l x i l l 2 , which is positive definite, decrescent and radially unbounded. Now, along motions of the system, vl(t,x,S1
g Vl(Xl)[-2OVl(×l)
v2(t,X,S 2
<
V~/2(X2
-Vl(Xl)
-v~/5(Xl
[0-5Vl(X l) + 0-5VI(X i) + 0.Sv#/5(Xl
) +0.5v$/2(x2
)] ,
) - 2 0 V 2 ( X 2) - v ~ / 2 ( × 2
)] ,
V(t,x,S) eRxRnxSs where Is I
]
i
0I = SS
=
$2
0 Sll
SI2 0
J
0 s12
]
o
0
g
s21
0
s22
0
0
S2l
0
s22
[:oo °1j 101
_<
101 0 1 0
0
It follows that
N:R
4
,
n~(v~) =v~ , a2(v 2) :v~/2
, ;l(v~)
= v~+v~/~
, ~2(v2) :v~/~
,
and Vl -< ~ 1 ( - 2 0 ~ 1 - ~1 +0"5 [ 2 )
, v2 -< ~ 2 ( 0 " 5 ~ 1 - 2 0 ~ 2 + 0 " 5 ~ 1 - ~2)
so that ~l(vl)
• [o,+~[
~2(v2____~)• [o,+~[
~l(Vl )
Hence,
K-I : 0 and -20 , 0 ] A
=
,
B
0,5 , -20
2K -1
=
-
"
•
[_i,0.:]ivii] o.5
The matrix A is stable, Let
and
vvz,v 2 •~,
' n2(v2 )
,
-
e = 20-112
(@B + BTe)
= ~
,
z(v)
which implies
-0.5
-°:1
L ~2(v2 that
both
,
200
Ch. III
Large
scale
systems
in general
Fo
l
L 4 are p o s i t i v e
definite.
fied.
the
Hence,
uniformly
All
conditions
equilibrium
asymptotically
state
stable
The zigh£h a g g r e g a t i o n
The
aggregation
Assumption
12.
functions
vi , ~i
~)
vi
2)
there
and
There
~i
is
is the most
~ij
over
general.
21 are
satis-
is s t r u c t u r a l l y
Ss
It is d e f i n e d
neighbourhood
definite
such that
the
on
set
f 6 ]0,~i[
S
system
30 J
Ni
of
by
xi:0
and
such that
are p o s i t i v e
for every
of T h e o r e m
,
form and ~ t a b i £ i £ y a r i t z r i a
are a c o n n e c t e d
and
~i > 0
tractive
form
(II)
of the
in the w h o l e
III.4.2,5, eighth
of
x=0
75
•
N i , Vi : 1,2,...,s ,
Via(t)
and
every
is a s y m p t o t i c a l l y
con-
i = 1,2,...,s ,
S
3)
v i ( t , x , P i , S i) ~
Z ~ij(t,x,P,S) i,j=l
i=l V(t,x,P,S)
•RxNxpxS
there
~ij e ]-~,+~[
s
~i(t,xi)
~j(t,xj)
,
,
and
4)
are
and
{-
~ij e ]-~,+~]
~ij ~ #ij (t,x,p,5) Vi,j This
:
i,2,...,s
aggregation
N(t,×,P,$)
, V(t,x,P,S) form
The
set
all
its v e r t i c e s
L
is
Lv :
(bounded
M(L,e)
all
#ij < +=
lity
condition
, where
" The m a t r i c e s
(I)
(11) over
test
£ij
if
N,
matrix
• and
~ i j : + ~ , Vi,j : 1,2,...,s}
hyperparallelepiped.
+ #ij
e >0 M
• #ij : +~
iff there
and test
L
The
set of
are
is
, vi,j = 1,2,...,s
#ij : +~
introduced
of a f u n c t i o n a l
of the m a t r i x
For structural P×S s
a functional
Lv ,
from a s t a b i l i t y
to a s t a b i l i t y
Theorem 23. system
: L + el
,~ij~ij<~ij
by
or
, ~ij < + ~
L E R s×s , L = (£ij)
or u n b o u n d e d )
is d e n o t e d
-
~ij
in g e n e r a l
Let
~ij<+~
L : £ij : [ij
Let
R×N×P×S s
introduces
if
, such that
eRxNxPxS
= [~ij(t,×,P,S)]
L : {L : ~ i j ~ i j ~ i j
, ~ij ~ i j
asymptotic
it is sufficient
M(L,e) stability
that both
and
e=0
iff
to r e d u c e
stabi-
matrix
on
N
for every of
x=O
keL v . of the
Aggregation forms and conditions of structural stability
(a) A s s u m p t i o n
201
12 holds
and (b) the m a t r i x
M(L,e)
(II) If,
radially
unbounded
structurally
is n e g a t i v e
in a d d i t i o n
for every
asymptotically
Proof.
(I) In v i e w of the
follows
that
the set
to
same
V~(t)
The c o n d i t i o n
stable
The c o n d i t i o n
Ni :Rn~
assumption
(a) and
is a s y m p t o t i c a l l y
4) o f A s s u m p t i o n
for e v e r y
N(t,x,P,S)
= L . Hence,
M(L,e)
condition
1974c-1976b) for e v e r y
then
which,
~ >0
it condi-
such
that
~ ~ ]0,~[
.
, V(t,x,P,S) e RxNxpxS s .
there
of
is
L ,M
LEL
such
and
e
that
L,
= w T ( t , X ) Lw(t,x)
definite
for e v e r y
it is n e g a t i v e together
i2,
N . The
for e v e r y
12 and the d e f i n i t i o n
for such
is n e g a t i v e
(b),
on
of
: wT(t,x) M(L,e)w(t,x)Since
is
is
PxS s
definite
contractive
vi
system
12 y i e l d s
(t,x,P,S) e R x N x p x S s
w T ( t , x ) N ( t , x , P , S ) w(t,x)
over
existence
~ w T ( t , x ) N ( t , x , P , S ) w(t,x)
prove that
and
of the
i) of A s s u m p t i o n
is p o s i t i v e guarantees
LEL v .
" ~i = +~
×=0
in the whole
condition
3) o f A s s u m p t i o n
v*(t,x,P,S)
(1),
for every
i = i,2,...,s , then
v = v I + v 2 +...+v s
tion 2) of the
definite
definite
with positive
i = 1,2,...,s , p r o v e s
negative
ellW(t,x)ll 2
LEt v
for
according
every
LEL
definiteness
definiteness
of
to the
(Gruji@, ~i
on
Ni
of
wT(t,x) N(t,x,P,S)w(t,x) on
N
over
PxS s . All
have b e e n v e r i f i e d asymptotically (II)
If,
ally u n b o u n d e d conditions
in a d d i t i o n then
of
(11)
13.
functions
vi , ~ i
to
(I),
N ~ : R ni
~i
is s t r u c t u r a l l y
and
v
2 of the S e c t i o n the proof.
structural
, ~i : +~
and
is r a d i a l l y 1.3.2.5
are
vi
is r a d i -
unbounded. satisfied
All
over
• asymptotic
stability
There
is p o s i t i v e
and
are a c o n n e c t e d ~ij
neighbourhood
the n e x t
is p o s i t i v e
Ni
of
s u c h that
definite
on
Ni
and decrescent
definite
on
N i , vi : 1,2,.-.,s ,
Vi : 1,2,...,s , 2)
1.3.2.5
PxS s
N = R n , ~ = +~
o f the
x=0
13 of the S e c t i o n
assump-
be a c c e p t e d .
Assumption vi
of T h e o r e m
PxS s .Hence,
over
completes
For u n i f o r m i t y
i)
over
stable
of C o r o l l a r y
PxS s , w h i c h
tion w i l l
conditions
on
Ni ,
xi:0
and
202
Ch. IIl
3)
E vi(t,x,pj,$i) i:l
~
Pij(t,x,P,S)
Large-scale systems in general
~i(t,xi)
,~j(t,×j)
,
i,j:l
V(t,x,P,S)
eR×NxpxS s
there
~ij 6 ]-~,+=[
,
and
4)
are
~ij
and
{-
~ij e ]-~,+~]
~ ~ij (t'x,p's)
T h e o r e m 24.
(I)
of the system
•
' Mij = +=
~RxNxpxS s
•
x=O
For structural uniform asymptotic stability of
(11) over
(a) Assumption
, such that
-
~l~ij_ ' gij_ < +~ <~ij
Vi,j : 1,2,...,s , V ( t , x , P , S )
, ~ij s ~ i j
it is sufficient
PxS s
that
13 holds
and (b) the matrix
is negative definite for every
M(L,e)
If, in addition
(II)
radially unbounded for every
to (I),
~ ~i = +~
Ni =Rai
then
i = i,2,...,s
LEL v
and
vi
is
of the system
x=O
(11) is structuratly uniformly asymptotically stable in the who~e over PxS s
Proof.
(I) The
definiteness crescent
on
definition
condition
of N. of
The e
conditions
guarantee
e RxNxpx$ The
such that s c o n d i t i o n (b) and
ments
of T h e o r e m
Hence, over
x=O
unbounded,
over
1) of A s s u m p t i o n
of
3) and
existence
0*(t,x)
N
as well
13 prove as that
4) of A s s u m p t i o n of
LeL
13 show now
1.3.2.4
is s t r u c t u r a l l y
are
(t,x,P,S)
-ellw(t,x)U 2 that
satisfied
uniformly
positive
it is de-
13 and the
for every
~ wT(t,x)M(L,c)w(t,x)
2) of A s s u m p t i o n
7 of the S e c t i o n (11)
on
all r e q u i r e over
asymptotically
PxS s stable
PxS s
(II)
ed.
(a) and
v = v I + v 2 +...÷v s
All
When,
in a d d i t i o n
Vi = 2,2,-.,s
conditions
PxS s , w h i c h
Exponential special
Assumption
14.
functions
v±
completes of the
of all There
and
, then
of T h e o r e m
character
features
to
vI
(I), N=R n
and
8 of the the
structural and
v
and
vi
is also
Section
proof.
is r a d i a l l y
radially
1.3.2.4
are
unbound-
satisfied
• asymptotic
stability
requires
£i
are a c o n n e c t e d
~ij
N i = R ni
and p o s i t i v e
neighbourhood numbers
~il
N± and
of ~i2
xi=0 such
that i)
~il[IXi ~2 ~ vi(t,xi)
~ ~i2~xi B2
, V(t,xi) C RxN i , Vi : i,2,...,s ,
Aggregation
forms and conditions
S
2)
of structural
stability
203
S
Z v~(t,×,Pi,Si)~ i:l
N
P i j ( t ,X, P,S) I l x i l l • I l x j U , Z i;j:l
V(t,x,P,S)
•RxNxpxS s ,
there
~ij e ] - ~ , + ~ [
and
3)
are
~ij Vi,j : 1,2,..-,s
Theorem 25.
system
and
~ ~ij ( t ' × ' P ' $ )
, V(t,x,P,S)
(a) Assumption
PxS s
<~ij
•RxNxpxS
(I) For structural
(11) over
#ij • ] - ~ ' + ~ ]
, ~ij sl]ij
' ~ij = + ~
s
, such
that
'
•
exponential
it is sufficient
stability
x=O
of
of the
that
14 holds,
and (b) the matrix
If,
(II)
system
is negative
M(L,c)
in addition
(11) is structurally
definite for every
to (I),
then
N i = R ni
exponentially
stable
L•L v
of the
x=O
in the whole over
PxS s . Proof.
(I) T h e
tence
of positive
vl + v 2 +...+v s The
~ wT(t,x)
together
with
tion
14 g u a r a n t e e
L6 L
such
that
v*(t,x,P,S)
then
N(t,x,P,S)
the
definition
that
for
N(t,×,P,S)
Since
both
it is n e g a t i v e
there
is p o s i t i v e
L
of
every = L
conditions
exis-
of Theorem
over
PxS s . H e n c e ,
over
PxS s
N:R n
and
and
number
x=0
, V(t,x,P,S)
e
and
the
e RxNxpxS s ,
condition
(t,x,P,S) ERxN×P×S
s
3) o f A s s u m p -
there
- ~ f l w ( t , x ) l l 2 , V ( t , x , P,S)
M(L,e) for ~3
are
every such
9 of the
exists
is
to
(I),
<+~
LcL v
•R×N×PxS
then
~=0
definite,
(Gruji~,
s
and
VLeL
,
1974c-1976b),
that
Section
(11)
#ij
negative
~ -~311xll 2 , V ( t , x , P , S )
of
v =
and
definite
v*(t,x,P,S) All
w(t,x)
w = (UXlll,Ilx211,...,IIXsll)T . When a l l
= L
14 g u a r a n t e e
14 i m p l i e s
~ wT(t,X) M(L,c) w(t,x)
where now
because
i) of A s s u m p t i o n
7 1 and 72 s u c h t h a t the f u n c t i o n 2 2 71ilxll ~ v ( t , X ) ~211xll , V(t,X) 6R×N .
2) o f A s s u m p t i o n
v*(t,x,P,S)
M(L,0)
(a) a n d
numbers
obeys
condition
which
condition
•~xNxpxSs
1.3.2.4
structurally
have
been
verified
exponentially
stable
.
(II)
When,
all
in a d d i t i o n
conditions
of Theorem
N i : R ni
i0 o f
the
, Vi : 1,2,...,s Section
1.3.2.4
, then are
sa-
,
204
Ch. III
tisfied In the
PxS s , w h i c h
over case
functions
of the ~i
completes
system
(23)
in A s s u m p t i o n
the
Large scale systems in general
the proof.
condition
•
on d e f i n i t e n e s s
13 can be a v o i d e d
as shown
of the
in what
fol-
lows.
Assumption
15.
functions
vi ,~i
1
vi
2
along
There
is time
Pij
invariant
motions
s Z v[(×,Pi,Si)
of the
such
neighbourhood
Ni
of
xi=0
and
that
and p o s i t i v e system
s ~
~
i:l 3
are a c o n n e c t e d
and
definite
on
N i , Vi : 1,2,...,s ,
(23)
#ij(x,P,S)
~i(xi)
~j(xj)
, V(x,P,S) e N x p x s s ,
i,j:l
the
singleton
in the
set
is the
{0}
largest
invariant
2(x1)+n~(x2)
Z : {x
set of the
system
(23)
+ "'"+~2(x s s) : o} •
: ~I
and 4)
there
~ij e ]-~,+~[
are
~i~
and
~ ~j(x,P,S)
#ije
]-~,+~]
{ ~ij
Vi,j : 1,2,...,s , V(x,P,S) E N x p x S s
Theorem
system
(I) For structural
26.
(23) over
(a) Assumption
PxS S
, ~j
~ij
, such that
' #ij < +=
•
asymptotic stability of
it is sufficient
of the
x=O
that both
15 holds
and (b) the matrix
M(L,e)
(If)
is negative definite for every
If, in addition to (I), the set
radially unbounded for every i8 structurally asymptotically
Proof.
(I) The
1976b)
negative
(a)
and
2) and
condition
of
4) of A s s u m p t i o n < 0
,
theorem
M(L,E)
Vx E N / Z
,
V(P,S)
is
(23)
Px$ s . (Grujid,
Let
along
vi
of the system
guarantees
for every
15 g u a r a n t e e
and
~ i : R ni
x=O
stable in the whole over
(b) of the
definiteness
D+v(x,P,S)
then
i = 1,2,...,s
LEL v
. The
motions
of
1974c-
condition (23)
e PxS s
and D+v(x,P,S) ~0
where These
, VX EZnN
, V(P,S) e PxS s
v : V 1 + V 2 + .-.+ V s . results
and p o s i t i v e
definiteness
i) of A s s u m p t i o n
15 g u a r a n t e e
tive
PxS s . All
to
satisfied
N
over
(LaSalle~
1976)
~
precompactness
conditions
over
of
of the
PxS s . Hence,
on
N
that
of s o l u t i o n s invariance ×=0
of
is i m p l i e d of
(23)
principle (23)
by
relaare
is s t r u c -
Aggregation forms and conditions of structural stability
turally
asymptotically (II) When,
unbounded
then
Example 9. systems
and
(I),
N i : R ni
and
is also r a d i a l l y
asymptotically
stable
vi
is r a d i a l l y un-
unbounded.
in the w h o l e
The f o u r t h o r d e r s y s t e m c o m p o s e d
is d e s c r i b e d
dt
u
PxS s
over
in a d d i t i o n to
N=R n
is s t r u c t u r a l l y
stable
205
over
Hence,
x=O
PxS s
•
of two s e c o n d o r d e r
sub-
by
: - ~ X i + Sil
0.5
¢(o)
, o : [-16,-I,-8,-2] x
, i =1,2
,
together with ¢(~) e o
]0,+~[
,
vaeR
a#O
, ~ e ]0,+~[
, ~ = const.
,
and
0
Sll
0
0
S9
0
s21
0
0
0
S21
S : S :
Ss :
Let Vl(Xl) now d e r i v e d
• sil : s 2 i e { O , i }
2 = 2x i + x12
, v 2 ( x2)
: x $ i + 2x$ 2
and
u : vi + v2
It is
where ~i Hence, and
Ell M(L,e)
Since nite
~
: ~
and
=#11 =-2m , ~12 =#12 =0 =-2~I , L = {L : L =-2~I}
is p o s i t i v e
VLeL
2 2x211 + x12
. The
vI
and
whole and r a d i a l l y
unbounded,
positive
All c o n d i t i o n s
Hence,
x=0
of the s y s t e m
stable in the w h o l e £xample
I0.
over
then v2
=0
M(L,e)
of
~l
(II) of T h e o r e m
is s t r u c t u r a l l y
uniformly
system is d e s c r i b e d
[~ ~Xil + ~Sil sat xj2] ~xi2 + ~si2 sat xjl j
; ~,~ e ]0,+~[
=-2o~
is n e g a t i v e
are p o s i t i v e
N L = R 2 , N 2 = R 2 , and
•
, ~-22 = # 2 2
Ss
The f o u r t h - o r d e r
d×idt : (-i) i xj +
2 : J x $ 1 + 2x22
, ~21 =#21 , e=O
valued parameter
functions
definite.
~2
by
,
definite and
~2
,
defiin the are
24 are satisfied. asymptotically
206
Ch. IIl
In this
case
hi(x)
S i l f si2
(-1) I xj
=
is a d m i s s i b l e
and
2 h i ( x ) = ~.
at
~x i
]2
'
-
hi
Large-scale systems in general
h3(x) i
'
= [h~,h~,h~]
T , Vi,j
= 1,2
, i Rj
:
, S
[ ] 0
= ~.
,
sat xj 1
,
so that 1
0
Sil
0
0
0
0
1
0
0
0
si2
Vi -i
The
structural
set
Ss
is d e f i n e d
1,2
:
diag {SI,S 2}
by
_< S <- d i a g 1000
Let of
vi(x i) : [Ixiu2, V i : 1,2 (If)
P l l : -~ ~22 =-~
of T h e o r e m
' ,
Assumption
~12 : Ni :R2
26,
~(Sll and
. The
Vi:l,2
function
. Now,
+ s12) E [0,2~] ~i(Xi) :IIxcII
15. E v i d e n t l y
e:0
M(L,O)
in
1000
vi
all c o n d i t i o n s verified
that
, ~21 = ~ ( S 2 1 + S22) e [ 0 , 2 ~ 1 , , Vi = 1,2 , obey all conditions M(k,e)
, where
]tll ~12l
:
obeys
it can be e a s i l y
:
L
of
now
,
L =l L
:
L
:
=< L
=<
and
Hence,
[>2[
L ~ L v . All fore,
x:0
whole
over
Notice
that
on s e p a r a t e
guarantees conditions o f the
negative
definiteness
of
M(L) : L
of
(Ii) o f T h e o r e m
26 are
satisfied
system
is s t r u c t u r a l l y
asymptotically
for all and,
there-
stable
in the
Ss application majorization
o f the s i × < h of each
vi
aggregation , yields
form,
which
is b a s e d
Aggregation
forms and conditions
~11 :-~ #21
of s t r u c t u r a l
, P12
stability
= 2 +~'(s11+s12
= 2 + ~'(s21+s22)
L
:
) e [2,2+2~]
• [2, 2+2~']
[::,] ~
L:
-< L -<
207
[
,
, ,u22 : - ~
-,~
2+2~
2+2t
-~
,
-
and Lv =
,
~
•
2 Hence,
in
is m o r e
this
III.4.2.6. The
next
of
Problem
Xi:0 vi
i : 1,2,...,s 2) a l o n g
stability
condition
a development
A more
exist
the
-~
2+2~"
~ > 2~
property,
which
.
complete
result
systems
open
of
with
connected vi
is
third
one
obtained
precise
~i
such
positive
for
here
a for
a
estimations.
neighbourhoods
and
time-invariant,
the
that
definite
Ni
, N i ~ R ui
: on
Ni
,
, of
the
system
(23)
the
following
estimations
are
ful-
:
v~(×,Pi,S where
i ) ~ ~i(P,S,vl,-,v
N = NI×_.×N s
functions ensure
~i
are
existence
dui dt
all
is
-~
form
, functions
are
motions
filled
4) t h e r e
former
large-scale
There
states
i) f u n c t i o n s
3)
B.
of autonomous 16.
form
2+2~"
guarantees
the
The nin£h a g g r e g a t i o n
Assu~ptl0n of t h e
~ >2+2~ then
aggregation
solution class
case
restrictive
,
-~
of
'
, = 0
satisfy
solution
Vi : 1,2,-.,s U
of
~i(P,S,Ul,--.,Us) , Vi = 1,2,-.,s
, V(×,P,S)
neighbourhood
they
a continuous
a neighbourhood
, u#O
~i(P,S,O,...,O)
Theorem 2 7 .
a connected
continuous;
: ~i(P,g,ul,...,Us)
exists ueU
is
s ) , vi : 1,2,...,s
the # 0
.
'
of
x=O
the
,
,
W ° -condition
V(P,S,u)
state and
of
the
• N×P×S s
and
system • PxS ×U ~
u=O
such
V(P,S)
6 PxS s
'
that
for
,
•
Let
(a) conditions of the Assumption (b) there exist estimation
S*ES s
16 be fulfilled;
such that for functions
~i(P,S,Ul,...,Us)
V(P,S,u) e PXSs×U is valid;
~ ~i(P,S*,Ul,...,Us)
an
~i(P,S,Ul,...,Us) , Vi : 1,2,...,s
,
208
ch. III
(c) there exist
Then
on
x=O
are radially
PxS s
unbounded
(23)
solution
vi[×i(t)]
conditions
for the
S
N i = R ni
U=R+
and functions
i = 1,2,...,s , then the state
asympto~icaIZy
stable
x=O
in the whole
ui(t;to,
. For
s ui(t;to,uo,P,S*)
of the
system
Ch. II ). C o n d i t i o n s positive
(23)
~i 6 K [ 0 , ~ [
of this
definiteness
vi
will
the
comparison
and
i) of the
function
Ch. I , there
~ = sup {~xll : × 6 N }
.
(28)
be c o n s i d e r e d
for
along
principle Assumption
the (see 16
v,
= v l ( x 1) + v 2 ( x 2) + . . . + V s ( X s) 10,
(a),
system
condition
, is f u l f i l i e d
of the
ofthe
to the A s s u m p t i o n
' where
system
, Vi E If,s] in v i e w
16 and
a comparison
, V(P,S~,u) epxgsxU
functions
(a) of the t h e o r e m
v(x) according
(23) we get
, Vie [1,s]
u o , P , S ~)
, Vi 6 [i,s]
5) of the A s s u m p t i o n
2),
system
= ~ i ( P , S* , U l , . . . , u s )
uio ~ v i ( × i o )
and,
for
for every
is structurally
theorem,
dduti
ensure
asymptotically
.
(b) of the
motions
of inequalities
is valid.
(23) is structurally
are f u l f i l l e d
In view of the
Proof.
The
of the system
(a)-(c)
of the system on
that the system
PxS s .
If conditions vi
such
0
,Ul,...,us) < 0 , Vi = 1,2,...,s
the state
stable
U°~U
a vector
0
~I(P,S
Large-scale systems in general
such
,
exists
a function
that
s
~l(ll×(t;to,×o)ll)
~ v(x(t;to,Xo))
=
~ vi(xi(t;to,×io)) i=l
s
Z ui(t;to, i=l Condition
(c) of the
stability
of the
for any
theorem
state
c>0 , t o E R
and T h e o r e m
u=0
we can
Uo)
of the find
, 11,
system
N~>O
VtET o , V×oEN
.
Ch. II
asymptotic
(28).
and
ensure
In this
u(E) > 0
connection
such
that
S
ui(t;to,
Uo)
< ~l(e)
, Vt ~ t o + o
,
i=l follows
from the
Further
on,
follows
from the
inequality
Uio < N ~
due to c o n t i n u i t y condition
we can ll×ioll < ~
, vi 6 It,s] find
~>0
such
that
vi(xio)
. Hence, s
llX(t;to~Xo)ll ~ ~ l ( v ( X ( t ; t o , × o ) ) )
~ll(~l(e))
~ ~i ~ ( Z u i ( t ; t o , i=l
= e , Vt e t o + S
,
uo ))
Aggregation
and the
forms and conditions
state
×=0
When c o n d i t i o n s S
(a)-(c)
and radially
function for
×:0
(28)
, but
conditions
that
(23)
Assumpti0n
(b) are
×=0
17.
stable
and c o n d i t i o n
fulfilled
There
1) for f u n c t i o n s
e NxpxSsxU
vi
exist
open
the
stable
in the
27 e s t i m a t i o n s
sign of e q u a l i t y ,
27 are
connected vl
the p o s i t i v e estimates
the n e c e s s a r y
in
respec-
and
(in the w h o l e )
, ~i
neighbourhoods and
numbers
constants
~il ' ~i2
~illlxil I ~ v i ( x i )
(23)
£i(P,S,Vl,...,Vs) of a c o n t i n u o u s
the next
satisfy solution
suffi-
on
P×S s
N i , N i c R ni ~i < 0 ,
s u c h that
~i2Hxill
:
, Vx i e N i ,
estimates
are
fulfilled
:
, Vi : 1,2,...,S , V ( x , P , S , U) 6
corresponding
exists
ueU , u~O
£i(P,S,u)
set
(a) conditions b) there ~xist
initial
a neighbourhood
,
the
W ° -condition
of the C a u c h y
and e n s u r e
problem
for a sys-
equations
du i dt : ~ui +~i(p'S'ul'''''us)
estimation
state
(23).
o f the s y s t e m
t e m of d i f f e r e n t i a l
28.
of the the
;
3) f u n c t i o n s
Theorem
asymptotically
stability
v~( x , P i , S i) ~ ~ i v i +~i(P,S,Vl,...,Vs)
4) t h e r e
,
, are valid;
2) a l o n g m o t i o n s
for the
N I : R ni
U:0
Hence,
16 and T h e o r e m
with
asymptotic
of s y s t e m
Vi : 1,2,...,s , a n d also
existence
for
(c) is f u l f i l l e d
state
in the whole.
o f the T h e o r e m
×i : 0 , f u n c t i o n s
[l,s]
fulfilled
v i , Vi : 1,2,...,s , t h e n the
is s t r u c t u r a l l y
for s t r u c t u r a l
states
27 are
, Vi : 1,2,...,s . The
if in the A s s u m p t i o n
2) and
of the s t a t e
Vie
asymptotically
•
then conditions
cient o n e s
unbounded
Uio ~ + ~
system
PxS s .
Let us n o t e
of the
is s t r u c t u r a l l y
functions
is a s y m p t o t i c a l l y
of the
whole on
tively,
(23)
o f the T h e o r e m
unbounded
v in X is r a d i a l l y
uio ~ + ~
system
system
209
stability
PxSs .
stable on
U=R+
of the
of structural
#0
,
, Vi : 1,2,...,s , V ( P , S , u ) E P x S s X U
,
conditions; U
of the p o i n t
£i(P,S,O)
=0
,
u:0
s u c h that
V(P,S) ePxS s
.
for •
=
of the Assumptign
S~ESs
17 be fulfiZled;
such that for functions
~i(P,S,ul,...,Us)
an
~i(P,S,Ul,...,u s) ~ ~i(P,S~,Ul,...,Us ) , vi = 1,2,...,s ,
epxSsxU ,is fulfiZZed; o) the ~tate u:O of the system v(P,S,u)
du i dt : a i u i +~i(P,S~,ul,...,Us)
, Vi = 1,2,...,s ,
210
be exponentially Then the state stable on
state
×=0
of the system
×=0
proof
the
In the
framework concept
AssumptiOn
There ×i=0
functions
vi
to
similar
Problem
exist
to t h a t
B,
the
and
definite
V(t,×,P,S,u)
9.
•
form
on
is b a s e d
27
neighbourhoods ri
such
that
N i , N i ~ R nl
:
N i , Vi = 1,2,...,s
, satisfy
the
6RxN×P×Ss×U
;
W ° -condition
, ri(t,0,P,S,0)
with = 0 ,
; constants
~i>O
contractive
the motions
for
of t h e
such
every system
that
(11)
~ r i ( t , × , P , S , v ] , - . , v s) are
exists
set
the
Vi~(t)
next
, vie
is a s y m p -
, Vi = 1,2,..,s estimates
[1,s]
; :
, V(t,×,P,S,u)
e
fulfilled;
a continuous
solution
d---tdu : r ( t , x , P , S , u ) aggregation
the
~ e ]0,~i[
v ~ ( t , x , P i , S i)
This
aggregation system
connected vi
, Vi = 1,2,...,s
and
of the Theorem
next
comparison
e R×N×PXSs×U there
~ then the
form
open,
are positive
u
exist
totically along
is
, functions
r i =ri(t,×,P,S,u)
3) t h e r e
5)
o f the
18.
V i e [1,s]
4)
theorem
of a generalized
states
respect
S
N I = R ni , U = R +
(23) is structurally exponentially stable in
The t e n t h a g g r e g a t i o n
on the
2)
(23) is structurally exponentially
P×Ss .
of
III.4.2.7.
of the
of the system
(a)-(c) are fulfilled for
the whole on The
stable.
PxS s .
If conditions
1)
Large-scale systems in general
Ch. III
form
leads dx dt
_
o f the
, V(t,x,P,S,u) to
the
system
e RxNxpxSsxU
consideration
g(t,x) + S ( t ) h(t,x,P)
•
of an extended
system
(29)
S*eS s
d UU : r ( t , x , P , S * , u ) dt
Theorem 29.
Let
:
(a) conditions of the Assumption (b) there exist
S*ES s
18 be fulfilled;
such that for functions
ri(t,×~P,S,U)
an
estimation
ri(t,x,P,S,u)
~ ri(t,x,P,S*,u)
, V C t , x , P , S , u ) e RxNxPxSsxU
is fulfilled; 27 See 4) of Comments on References to Ch. III.
,
Aggregation forms and conditions of structural
(o) the state
stability
of the system
(xT,uT) T = 0
(29)
211
be a a y m p t o t i c a l l y
u-
stable. Then the state 8table on
conditions
If
of the system
×=0
(a)-(c)
dially u n b o u n d e d
Proof.
It
follows
18 t h a t
are f u l f i l l e d
functions
(11) is s t r u c t u r a l l y
tion
the
from
of
the
that
V{(t)
is
the
set
conditions
obtain
an
(b)
aggregated
du i dt = r i ( t , × , P , S which
for
conditions v
the
from asymptotic
s
. The
from
the
U:R$
for
tion
is
v
It f o l l o w s whole
of the
stable
in
In o r d e r
6,
(a)-(c)
, ~i : + ~
bounded
of
the
theorem is
4)
on
and
existence
of
ra-
of
Assump-
definite of
for
the
,
PxS s . I)
positive
contractive and
= +~
of the system
x:O
in the whole
ensures
, i : 1,2,_.,s
on
{ >0
every
such
{ e ]0,~[
Assumption
, V(t,x,P,Z,u)
has
a solution
first
statement
.
18 w e
of
the
ERxN×PxSsxU
ui(t;to,uo;pi,S of
state
the
i)
Theorem
(xT,uT) T : 0
29 of
,
for
all
follows the
system
the
and
for
all
and
the to
x:0
of
whole
obtain the
on
aggregation 19,
There
.
are
, Vi : 1,2,...,s
2) r i : r i ( t , x , P , S , u
of
fulfilled vi
system is
for
being , ~:+~
asymptotic
the (11)
the
the
form,
we
exist
must
open
N i : R ni
radially and
and
, un-
the
u-stability
(29)
structurally
uniform
system
func-
in
the
estimates
(30)
asymptotically
structural
(11)
formulate
connected
functions
positive
,
• for
x:0
and
from of
system
P×S s
state
xi:0 vi
the
conditions
the g i v e n
states
also
(xT~uT) T : 0
are
N:R n , U:R s
from
that
29
functions
, then
unbounded.
state
, Vi ~ [l,s]
(30)
Theorem
radially
state
i) f u n c t i o n s
of
, vtET o
II.
also
the
Assumption
Chapter
i = 1,2,...,s
of
of the
) ~ ui(t;to,Uo;Pi,Si)
, i 6 [1,s]
all
stability
Ni
18
theorem
u-stability
Theorem
If c o n d i t i o n s
that
(a)
U = R +s , ~i
estimates
vi(t,Xi(t;to,Xio;Pi,Si) and
asymptotically
system
,Ul,...,u s)
ER×N×PxS
the state
stable
asymptotically
of
•
N i = R nl
: v I + v 2 +...+ v s
Assumption
u i o ~ v i ( t o ~ x o)
(t,x,P,$)
(29),
v i , then
function
5)
to
for
asymptoticaZly
N . Condition
Due
(11) is structurally
PxS s
vi
definite
in
the
asymptotic framework
a number
of
neigbbourhoods
and on
ri Ni
such and
that
of
conditions. N i , N i c R ni , :
decrescent
on
; ) , Vi : i,2,...,s
, satisfy
the
W ° - condition
with
212
Ch. III
respect to
u , and
[l,s]
vie
5) a l o n g
v ( t , x , P , S , u ) eRxNxpxSsxU
motions
of the
system
(11)
the
estimates
s ri(t,x,P,S,vl,...,v s)
V ( t , x , P , S , U ) ERxNxPxSsxU
,
[1,s]
Vi~
,
,
fulfilled,
4) there du i dt
ri(t,O,P,S,O ) =0 ,
,
;
v ~ ( t , x , P i , S i) are
Large-scale systems in general
exists
a continuous
solution
of the
ri(t,x,P,S,Ul,...,u s ) , i = 1,2,_.,s
=
Theorem 30.
Let
(a) conditions (b) there
system
, V(t,x,P,S,u)
•
RxNxPxSsXU..
:
of the A s s u m p t i o n
exist
S~ESs
~uch
19 be fulfilled;
that for a function
ri(t,x,P,S,u)
an
estimation
ri(t,x,P,S,u) ~ ri(t,x,P,S*,U) , v i e [ 1 , s ] , V ( t , x , P , S , u ) ERxNxpxSsxU , (c) the state ally Then
(xT,uT) T = 0
the state
asymptotically If conditions vi
X=O
(a)-(c)
(11)
the whole
P×S s
The
show t h a t
on
on
(a) of the
N.
Using
the
30~
and
the T h e o r e m
27 we c l a i m
structurally
uniformly
and
radially of the
asymptotic-
that
the
2)-4)
state
principle
the
asymptotically stability
i) of the A s s u m p t i o n
19
on
state
system
for
the
N = R n . As x=0
estimates
(11)
of
is
N i = R ni
function
a result
of the
system
in the w h o l e the
19,
to the p r o o f
fulfilled
stable impose
we find
and
P×S s
v i , then on
definite
of the A s s u m p t i o n
of the
30 are
and d e c r e s c e n t
in
similarly
stable
x=O
stable
principle
x=0
functions
, and functions the state
is p o s i t i v e
Hence,
of the T h e o r e m
for e x p o n e n t i a l vi
uniformly
asymptotically
and
comparison
asymptotically
unbounded
uniformly
on f u n c t i o n s
the
unbounded
comparison
structurally Conditions
(a)-(a)
radially
theorem
conditions
~ ui(t;to,Uo)
and
uniformly
v = v I + v 2 +... + v s
(b) of the T h e o r e m
If c o n d i t i o n s
N i = R ni , U = R +
i = 1,2,...,s , then
is s t r u c t u r a l l y
function
s
for
for al~
vi(t;xi(t;to,×io))
also
be uniformly
(11) is structurally
are f u l f i l l e d
conditions
the
decrescent
U=R s
(29)
P×S s .
unbounded
of the system on
of the system
stable
are r a d i a l l y
Proof.
of the system
u -stable.
special
on
, v
is
of this (11)
is
PxS s . •
requirements
Aggregation
forms a n d conditions
Assumption
20.
There
of the states
xi=0
exist
of structural
stability
open c o n n e c t e d
, functions
vi , ri
213
neighbourhoods
and p o s i t i v e
N i , N i ~ R ni
numbers
Nil ,Ni2
such that
I)
~il~IXill ~ v i ( t , X i) ~i2Jlxill
2)
r i = r i ( t , x , P , S , u ) , Vi 6 [1,s] respect
to
U , and
, V ( t , x i) ~ R X N i , Vi E [1,s] ,
satisfies
the
V(t,x,P,S,u) 6 RxNxpXSsXU
;
W ° - condition
with
, ri(t,0,P,S,O ) = O ,
Vi : i,2,...,s ; 5)
along motions filled
of the s y s t e m
(11)
the f o l l o w i n g
estimates
: v ~ ( t , x , P i , S i) ~ ri(t,x,P,S,Vl,...,v s) v i e [1,s] , V(t,X,P,S,u) e R x N x p x S s x U
4)
there
are ful-
exists a c o n t i n u o u s
solution
du i dt - ri(t,x,P,S,Ul,_.,Us)
Theorem 31.
, ;
of the s y s t e m
, i : i,2,...,s , V ( t , x , P , S , u ) E R × N × P × S s × U
.m
Let
(a) conditions
of the A s s u m p t i o n
(b) there exist
S~6S s
such
20 be fulfilled;
ri(t,x,P,S,u)
that for functions
the
estimates
ri(t,x,P,S,u) ~ ri(t,x,P,S*,u) , V ( t , x , P , S , u ) ERxNxpxSsxU , are fulfilled;
(xT, uT) T = 0
(c) the state
of the system
(29) be exponentially
u-
stable.
x=O
Then the state stable on
If conditions state
×=0
the whole Proof.
(a)-(c)
are f u l f i l l e d
of the system on
(71)
is s t r u c t u r a l l y
exponentially
(11)
N i = R ni , U = R $
for
is structurally
, then
exponentially
the stable
in
P×S s .
Conditions
sure e x i s t e n c e function
of the system
PxSs
(a) of the t h e o r e m and
of the p o s i t i v e
v : v I + v 2 +...+v s
numbers
71
i) of the A s s u m p t i o n and
72
satisfies
an e s t i m a t i o n
principle
conditions
20 en-
such that the ~111xlI ~ v ( t , × )
~211xll , V ( t , x ) eRxN Together w i t h the c o m p a r i s o n and 2)-4)
of the A s s u m p t i o n
(b) of the t h e o r e m
20 lead to e s t i m a t i o n s
v i ( t , x i ( t ; t o , X o ; P i , S i ) ) ~ u i ( t ; t o , U o ; P i , S i)
, Vi E [1,s]
,
s
v(t,X(t;to,xo;P,S))
~
Z
ui(t;to,Uo;P,S)
,
i=l
respectively.
Hence,
in view of the c o n d i t i o n
(c) of the t h e o r e m
it is
214
Large-scale
Ch. I I I
e a s y to f i n d (11) the
out
that
an e x p o n e n t i a l first
statement
If c o n d i t i o n s U : R +s
for a s o l u t i o n
estimation
(a)-(c)
state
x:0
PxS s
and
of the T h e o r e m N:R
estimates
o f the
on
of the
system
P×S s , w h i c h
proves
o f the t h e o r e m .
, Vi : 1,2,...,s , then
son p r i n c i p l e
x(t;to,xo;P,S)
is f u l f i l l e d
systems in general
system
o f the
(11)
31 are
fulfilled
for
and as a c o n s e q u e n c e
n
funciton
N i : R ni
of the
v we d e t e r m i n e
is e x p o n e n t i ~ l l y
stable
,
compari-
that
the
in the w h o l e
on
•
III.4.2.8.
C o n c l u s i o n on s o l ~ t i o n s
Solutions
for the p r o b l e m
tion
into
interconnected
lity
or s t a b i l i t y
enable
one
shot
aggregation lishing enable
forms
have
test
been
of the
on the
subsystems.
of' the w h o l e
developed.
Seven
They have been
stability
either
However,
system.
properties.
form of i n t e r a c t i o n s
Liapunov
system decomposi-
do not r e q u i r e
on t h e
stabi-
they
different
used
for estab-
Besides, choice
they
of the
form
function.
The structural s t a b i l i t y analysis of a large-scale system with non-asymptotically stable subsystems
In e x a m p l e s pedent
They
of disconnected
for d i f f e r e n t
influence
III.4.3.
B have been based subsystems.
stability
criteria
of a t e n t a t i v e
test
f o r t h e Problem B
3, 5 of the p r e s e n t
subsystems
xi:0.
H e r e we
scale
systems,
stable"). possess,
The
chapter
of a l a r g e - s c a l e
shall when
analyze the
obtained
as a rule,
a case
equiiibria comparison
a linear
we
system
consider have
of the
a case,
the u n s t a b l e
stability
of s u b s y s t e m s
theory
are
which makes
inde-
states o f large-
s t a b l e ("neutrally
systems(aggregated
part,
when
their
systems)
do not
analysis
more
difficult.
III.4.3.1. The
Eleventh aggregation
present
aggregation
t i o n o f the P r o b l e m
Assumption of the
21.
states
f o r m is r e a l i z e d
A and
There ×i=0
is c o n n e c t e d
exist and
form
open
connected
functions
~) f u n c t i o n s
vi
are p o s i t i v e
2) f u n c t i o n s
ql
are
locally
in the
with
vi
definite
and on
Lipschitzian
the
of the
system
(II)
of a solu-
form.
neighbourhoods such
Ni,
Vi = 1,2,...,s ;
and they
an e s t i m a t e
that
N i ~ N i ~ R hi,
qi
condition; 3) a l o n g m o t i o n s
framework third
satisfy
:
the
W° -
Aggregation
forms and conditions
of structural
stability
+
T
DSvi(t,Xi) + [Dxivi(t,xi)]
215
9i(t,xi)
~ O ,
, V ( t , x i ) • RxN i
vi : 1,2,...,s is f u l f i l l e d ;
~)
[D~vi(t,xi)] V(t,X,P,S)
T Sihi(t
•RxNxpxS
x,Pi')
s , and
< qi(t
P,S,Vl,...,Vs)
qi(t,P,S,O,...,O)
:0
Vi • [i s]
, Vi • [l,s]
,
Vt• R . •
Theorem 32.
Let
(a) conditions of the Assumption
21 be fulfilled;
{b) there exist positive numbers such that the sets and every
• ]O,~i[
(c) there exist
S~•Ss
i8 f ~ f i l l e d
~ qi(t,P,S*,u) •R×P×Ss×U
- qi(t,P,S
be asymptotically
stable on
~i = +~ )
, i ~ [1,s]
;
of the system
dui dt
Then the state
that
contractive for any
i : 1,2,...,s ;
V(t,P,S,u)
u=O
(it is possible
such that an estimation
qi(t,P,S,u)
(d) the state
~i
are asymptotically
Vi~(t)
x=O
,Ul,...,u s)
, i = i,2,...,s
stable. of the system
(11) is structurally asymptotically
P×S s
If conditions
(a)-(d) are fulfilled for
are radially unbounded and
N i = R ni , U = R $
~i = +~ " Vi • [l,s]
bility in the whore of the state
u=O
of the
chapter.
•
Theorem 33.
Let
theorem
is s i m i l a r
(b) functions
vi ,
(c) there exist
of the system (11).
of the Theorem
S~CSs
such that estimation
V(t,P,S,u)
is fulfilled u=O
dul dt
21 be fulfilled;
i = 1,2,...,s , be decrescent;
qi(t,P,S,u)
(d) the state
to t h a t
x=O
:
(a) conditions of the Assumption
_< q i ( t , P , S * , u )
ERxpxSsXU
, Vie [l,s]
,
;
of the system - qi(t,P,
S~
vi
sta-
of the system ensures structural
asymptotic stability in the whole of the state The p r o o f
, functions
, then asymptotic
,Ul,...,Us)
, i : 1,2,...,s ,
? of this
216
Ch. III
be uniformly asymptotically Then the state
asymptotically stable on If conditions
stable.
of the system
x=O
(11) is structurally uniformly
P×S s .
(a)-(d) are fulfilled for
are radially unbounded
Large-scale systems in general
N i = R ni
~ functions
Vi 6 [1,s]
schitz conditions and that of Wa~ewski for
J
U = R +s , functions
UER$
vi
satisfy the Lip-
qi
, then the state
x=O
of the system
(11) is structurally uniformly asymptotically stable in
the whole on
PxS s
The
proof
Theorem
is
similar
to t h a t
of t h e T h e o r e m
8.
•
Let :
34.
(a) conditions of the Assumption
21 be fulfilled;
(b) there exist positive numbers ~iilIxil I ~ v i ( t , x i )
(c) there exist qi(t,P,S,u)
S~ESs
~CI"
such that
~C2
~ ni211xil i , V ( t , X c ) e R x N i , i = 1,2,. ,s ;
such that the estimate
~ q i ( t , P , S ~ , U ) , Vi 6 [ 1 , s ]
, V ( t , P , S , u ) eRxpxSsxU
,
is fulfilled; (d) the state
of the system
u=O
du i dt - q i ( t , P , S * , u )
, i = 1,2,...,s
,
be exponentially stable. Then the state stable on
If conditions qi
of the system
x=O
(a)-(d) are f u l f i l l e d for of the system
x=O
stable in the whole on proof
is s i m i l a r
III.4.3.2. We
structurally exponentially
N i = R ni , U = R $
, and functions
satisfy the Lipschitz condition and that of Wazewski for
then the state
The
(11)is
PxS s .
P×S s .
to t h a t
of the Theorem
9-
•
in t h e
the
system
(11)
in the presence
interconnections
of
of a s m a l l
subsystems
positive
the
assumptions
invariant on the
structure, independent
i.e.
S(t) ~ I
subsystems
28 See 5) of Comments on References to Ch. III.
para-
28
dx _ g ( t , x ) +# S ( t ) h ( t , x ~ P ) dt under
,
The £welfth aggregation form
consider
meter
uER$
(11) is structurally exponentially
. We m a k e
(32),
(31) the
following
Aggregation
forms and conditions
dxi dt Vector-functions
gi
of structural
= gi(t'xi)
r :
(t,x) : t e [ o , + ~ [
The e q u i l i b r i u m Functions
state
h(t,x,P)
unique s o l u t i o n
,
xi:0
sequence •..,s)
xi
xi
there
is
the
, kGR+
. (33)
to -uniformly
stable.
to be such that there exists for the s y s t e m
in a s e q u e n c e
~i = 1
(t'xI'""Xs'P)
~i1(t,x~i,...,Xs,P)
, i:i,2,...,s
, Vi : 1,2,...,s , is
is equal to the p r e v i o u s ¢i
and they s a t i s f y
the
(31), and
.
be o r d e r e d
{~i }
(32)
in the d o m a i n
z Ixjl < k j:l
of the C a u c h y p r o b l e m
h(t,0,P) = D , v(t,P) 6 ~ + × P Let v e c t o r s
continuous
to
are s u p p o s e d
217
' i = 1,2,...,s ,
are defined,
Lipschitz c o n d i t i o n w i t h r e s p e c t
stability
~i
, i = 1,2,o..,s . In the
and each s e q u e n c e one or to index
element
k.
~k
(k = 2,
Let us denote
= [Dxivi(t'Xi )]T hi(t'x'P)
'
: [ D x i v i ( t , X i ) ] T hi(t,0,...,0,x~.,...,Xs,P) i
,
- ~i ~i (to'X=io'""Xso'P) = ~t {¢i~i (t,xei(t),...,Xs(t),p)} ,
where
~t
is an a v e r a g i n g o p e r a t o r
tained time P (33))
with respect
t , Xk(t) : x k ( t ; t o , × k o )
(for
We d e t e r m i n e
a vector-function
are c o n t i n u o u s
We denote
p (X i , E ( Q i : 0 ) )
: inf
V~i,~ i (0 < ~ i < ei
~i
Assumption
~ lJXioJl ~ ei
2Z.
of the states
say that
T h e r e exist xi=O
vi
and d e c r e s c e n t 2) f u n c t i o n s
vi
than zero in the set
~i(t o ,X~io'""Xso, < ri
.
p) <-$.1
, Vt ° e [0,+~[
open c o n n e c t e d n e i g h b o u r h o o d s vi
and
vi
such that the
•
Hi , N i c R ni following
:
are p o s i t i v e on
is s m a l l e r
1
• E (vi:0))
, functions
c o n d i t i o n s are f u l f i l l e d i) f u n c t i o n s
~i
& 6(Ni,ei) > 0 ' P(Xio
,
t
We shall
of
Pi = {xi : llxill < k } .
[llxi-x~l] , xie E ( ~ i : 0 ) ]
£ (vi:0)
3ri(Ni,ei) >0
in the d o m a i n
__A {x i E F i : ~i(xi) : 0 }
Definition 5. iff
(32).
v : (Vl(Xl),...,Vs(Xs)) T , c o m p o n e n t s
and d e f i n e d
E (vi:0)
con-
from the d o m a i n
, k : 1,2,...,s , is a s o l u t i o n of the s y s t e m
which-v i
for
to an e x p l i c i t l y
to,×ko
definite,
N i , Vi E [1,s]
are c o n t i n u o u s
continuous,
differentiable
;
and n o n - p o s i t i v e
on
N i , V i e It,s]
;
218
Ch. IlI
3) a l o n g
motions
of s u b s y s t e m s
Dtvi(t,xi)
(32)
:
+ [Dxivi(t,xi)] T gi(t,xi) V(t,xi) 6RxN i
4)
functions with
5)
~i
the
, Vie
constant
functions
hi
Theorem 35.
(a) conditions
If,s]
, are
of the Assumption
the
;
Lipschitz
condition
on
N
bounded
on
N.
•
22 be fulfilled;
~i
, uniformly
, Vi 6 [~,s]
with respect
;
to,X~io,...,Xso
to
, satisfy
[l,s]
~ 0 ,
:
(b) there exist sequence of (c) ~ i
, Vie
~ vi(xi)
L;
, vi = 1,2,...,s
Let
Large scale systems in general
be defined in the set
E (vi:O)
, smaller
, i 6 [1,s]
than
zero.
Then the state Proof. will of
Let
there
be p r o v e d
the
of the system
x=O
be g i v e n
e e ]0,k[
if it a p p e a r s
system
(31)
(31) is
that
and
for
where
7(e) >0
and
filled
for all
t >t o
Let
us a s s u m e
that
the
Assumption
22
for
>0
will
c a n be ~s-I can
= cs
~ [IXio B <~ i=l
for
the
the
that
any
satisfy
function
this
~s-i
• Thus,
for
the
that
for
t zO
< e
for
t ~ 0
Estimates
all
illustrated
be
ful-
,
ilXioll < ~ i above
and
can
are
of
v i ( t , × i)
, ~i
and
general
surface .
a number
will
satisfies
constants
Cs_ 1 ,
es-i ,~s-i
6i(~i,ei)
he
of t h e T h e o r e m
, i : i,2,..,s
and
we
, i = s-2,...,1
x i , satisfying
Xi(t)
i) of
c s >0
exists
choose
by m e a n s
for
condition
es_ I : m i n { ~ s / 6 L ( ~ s - 1 ) , C s }
we
~i ~ }IX~l} ~ ei/2
, Vi : 1,2,...~s and
,
n s ~ ll×s)l s e s / 2
there
and
, e~+l}
conditions
of the
o f the m o v i n ~
theorem
points
v i ( t , x ~) = e i , an i n e q u a l i t y t ~ 0 . Fulfilling
<#o
t o , will
constants
xs
process,
Vs_l(t,xs_l)
value
any
on
0 <~
an i n e q u a l i t y
~s ~ < -~s(~s,es)
Continuing
In a d d i t i o n ,
some
point
(c) o f t h e
e~ = m i n { B i + i / 6 L ( a i + l - 1 )
defined.
vs
will
condition
the
depend
. As a c o n s e q u e n c e
function
such t ~0
for which
defined.
not
,
.
6 s = 6/s for
for
to t h e
determine
tants
Mo(e)
c a n be g i v e n
8s(~s,es ) >O
theorem
x i ( t ) : X i ( t ; t o , x o)
s
< e
i=l
According
. The
solution
an e s t i m a t e
[Ixi(t;to,Xo)ll
Vs(t,Xs)
t o 6 ]0,+~[
every
S
~s > 0
stable.
t o -uniformly
the
can
consbe
condition
fulfilled 35 we c a n
an estimate
for verify llxi(t)~
.
for all
i e [l,s]
. We
consider
Aggregation forms and conditions of structural stability
a solution
xi(t) ' t = tio
a moment
l e a v i n g the d o m a i n intersecting
219
determined
llxill<~i
by
the h y p e r s u r f a c e
and at
v i ( t , x i) : c i
at the
!
point
Xio
. For
be e s t i m a t e d
vi(t,xi(t)
t >t[zo
as f o l l o w s
) ~ ci +
Estimating
behaviour
of the f u n c t i o n
and
two p o s s i b l e
Ii
d r + g Itt ¢ia i ( r , x Z(r) ,'"' Xs(r) ) dr .
~i(xi(r)) lo
o
t ~t o
solutions
(34)
(34) we a s s u m e
xj(t)
remain
that
in
rj
p (Xlo
l) Let there be
, E (vi:0))
and the c o n d i t i o n
l¢~l(t,xl,...,Xs,P)l ~ u i
for
Pi = lxi
~ ri(~i,ei)
. We d e t e r m i n e
4) of the A s s u m p t i o n
(t,x) e R + x N
<~' < 7 i / 2 u i , (T i = inf l~i(xi)l)
, u i •R+
are
> ri/2
, ~i ~IIxill ~ el}
v i ( t , x i ( t ) ) _< c i - 7i(t-tio)/2'
for w h i c h
• P(xi(t) , E ( v i = 0 ) )
IIxi(t)II < N i
xi(t)
and will not i n t e r s e c t
far as the e s t i m a t e 2) Let there be
P(xi(t)
. Chosing
,
: P(Xi , E ( v i = O ) }
to obtain that the c o m p o n e n t
from
22 that
, i • [1,s]
we o b t a i n an e s t i m a t e
estimate
for all . There
cases.
hi(t,0,...,0,P) = 0
llxiU < e i
.can
:
the t h i r d term in f o r m u l a
j e [i+l,s]
vi(t,xi(t))
> ri/2
for all
. Hence,
it is easy
will not leave the d o m a i n
the h y p e r s u r f a c e
, E (~i:0)) >ri/2
t >t;o given by
vi(t,xi(t) ) = c i
as
is correct.
P ( x ~ o ,E ( v i : 0 ) )
> ri(~i,ei)
. The third term in the
(34) will be r e p r e s e n t e d
in the form
:
t
¢~l(r'Xl(r)'""Xs(r))
dT
:
io ~ i i (~,xl( ~),...,x~ (~) ) - ~ i ( ~ , R ~ . (~) ,...,R~(~))] d~ +
--
(35) +
~i(r,~
(r)
,t~o where
i
Xs(r)) dr '""
"
Xk (t) :xk(t;tko,' Xko) ' • Xko ' : { k (tko,tko,Xko) ' " ' , k :~i,...,s • is a (52). A c c o r d i n g to the c o n d i t i o n 4) of the
s o l u t i o n of the s y s t e m Assumption
22 ~i -I
r~(t,x~,...,Xs)-~(t,~i,...,K)l
~ L(
n:l In the c o u r s e of this c o n s i d e r a t i o n we d e t e r m i n e
II×kl < en
of
(n = i,...,~i-i )
×i(t) for
s
Z IJXnn+
lwxk-~ku)
z
k:a i (index
i
is i n c r e a s i n g )
~
into
220
Ch. III
account
the c h o i c e
of
cn
, n : 1,2,...,~i-i
Large-scale systems in general
for
I~ < ~ . _ i
we o b t a i n
1
~i -I Z n=l
L Using
the
conditions
at an i n t e r v a l
[t~
large,
term
the
iO
last
(b) and ,t[
iO
will
t
the
condition
6
where
i
(55)
t > r i o + £ . As a c o n s e q u e n c e and
<"
(c) of the T h e o r e m
+2£.]
in
6i
IlXnll
£i
be s m a l l e r
than
of the L i p s c h i t z
5) of the A s s u m p t i o n
55, we d e t e r m i n e
can be c h o s e n
22,
that
sufficiently
5~i(t-t~o)/6
condition
by means
on
of the
for
gi(t,x) choice
of
T!
~i
, we
will
show
not
that
leave
rJxi(t) -xi(t)ll < ei/2
the domain
I(xiU < 6
at this
s Z
, i.e.
JRxk -~kJl
the
solution
interval
xi(t)
and
~. _~l . < 6L
k:~ i Hence,
for
t e ]t[o+ii,t~.o+2£i[
I~
and
~ < min { , l , , ~ i _ l }
6i ) dz < -~-(t-tio)
~I(T,XI(Z),...,Xs(r)
an i n e q u a l i t y
,
o
is valid. negative
It means for
that
t >-tio+£i
ing the h y p e r s u r f a c e v i ( t , x i) -
the
and,
. That
respect
xi(t)
to
moreover,
will
not
for
for
one of the two leave
If a f t e r
vl(t,xl(t))
terms
i • [2,s]
of
in the
is why the
some
vi(t,xl)
t'io ' Xio '
interval
term
solution
after
defined
: min {~,~i,~" ~i_l } given
all
will
is valid.
for any
t_>t o
vi(t,xi(t) ) to
by
come
at every sub-
above
it is not d i f f i c u l t
leav-
of e s t i m a t e s
cases
we e s t i m a t e
be
xi(t)
# <~i
IfxiH < e
will
domain
: c i . From uniformity
the d o m a i n
, then
in the
(34)
t i E ]t~o,tio' +£i [
$
Hence,
in s e q u e n c e
find out
in
that
s
7~ IIxi(t)I q < 6 i:l Here
expression
v i ( t , X i) = c i , r e m a i n s
the h y p e r s u r f a c e
sequent
last
~o = m i n { ~ i }
t o • Y o . The Example
dx 1 dt
t >_t o _>0
, ~(e)
theorem
and
~ ]]Xio]] < ~ i:l
= min{~i } , ie
is proved,
11 ( K o s o l a p o v ,
the p e r t u r b e d
dYl dt
for
1979).
[l,s]
, do not
for
~ <~o
depend
'
on
m We
consider
a system
of e q u a t i o n s
of
motion
2 -Xl + x 2 + ~ x 2 Y l
'
-Yl + Y2 +~x22
dY2 J "dr
_
dx2 dt
_ - ~ ( x 2 Y 2 - X l X 2 + x l z +x12 s i n t )
'
2 2 2 : ~(-x2YQZ - Y l Z 2 + (YI+Y2) c o s t )
, (36)
Comments on references
221
dz _ z3 2 z2 d--~ - ~ (+ YlY2 - X I Z + COS t) For
~:0
we get
dXl dt
_
Solutions
three
independent
subsystems
2 dx 2 dYl - X l + x2 " - ~ - : 0 ; dt
of these
subsystems
(36)
_ -Yl + y 2
have
the
dY2 - 0 dt
•
; dz ~-~:0
form
X 1 = X~o + (Xlo-X~o) exp [-(t-to)]
, x 2 : X2o
,
Yl
• Y2 = Y2o
,
: Ylo + (Ylo-Y2o)
Z
=
(37)
exp [-(t-to)]
Z O
Referring vatives,
to s u b s y s t e m s and
all
~i
(37),
the L i a p u n o v
can be d e f i n e d
functions,
their
full
deri-
as f o l l o w s
1
V l ( Y ) : y~ + (yl-y2)2
dvl '
--2(Yl-Y2
V2(Z ) : Z 2
2
v3(x)
All
conditions
2
2)
2
:-2X4o
of the
2
2
2
Y2oX2oZo
- ~2(Z
'
;
) E 0
dt
for
dv 3
Yl : 0
35 are system
Yl : 0
~ Y2:0
2(xl-x~) 2
_
dt
' for
of the T h e o r e m
Xl = x2 = Y l = Y2 = z = 0
2
Y2o-
2 4 : - 2 z o4 - 2 Z o X 2 o
: x~+(xl-x
-3(x o) ¢3
4
dv2 '
(Zo,Xo)
< 0 -
-i ~l(Yo'Zo'Xo ) :-
~
)2 : ~l(y)
dt
, Y2:0
fulfilled. (36)
is
: ~3
; ~ 0
,
, z :0 Hence
the
state
t o - uniformly
stable.
COMMENTS ON REFERENCES Brief
review
of c o n t r i b u t i o n s
will
be g i v e n
only
for
1. B a i l e y
the
concept system
and Aoki
of all
1950),
a development
(1968)
of a v e c t o r
three
functions•
comparison
(until
system.
stability
is the
properties
systems
of a l a r g e - s c a l e
part
tor L i a p u n o v
to l a r g e - s c a l e
stability
time
(1966,1968)
large-scale essential
for
continuous
aggregation to apply
only
principle
proposed Bailey
by m e a n s
function
is g i v e n
theory
sense
and
1919,
the
Kamke,
first of the
A solution.
(the m e t h o d
norms,
in C h a p t e r
of e f f e c t i v e
was
to a n a l y s i s
approaches
of v e c t o r
(Chaplygin•
of w h i c h
the m e t h o d
(1966,1968)
of the P r o b l e m
illustrated
stability
1982).
Liapunov
the m e t h o d
systems
in L i a p u n o v ' s
the m i n i m a x 1932,
An
of vecmethod)
Wa~ewski,
!I of the p r e s e n t
222
Ch. III
Large-scale systems in general
book. Lakshmikantham (1965) and Kayande and Lakshmikantham (1966) applied vector Liapunov functions to analysis of stability conditions and conditionally-invariant sets, respectively. Considerable development received ideas of Bailey (1966,1968) while investigating the large-scale systems in the framework of the Problem A with the following subsystems : * exponentially stable [Matrosov (1963,1972); Piontkovskii and Rutkovskaya (1967); Zadorozhny (1969); Michel, Porter (1970); Porter, Michel (1970,1971); Siljak (1971-1973 , 1975,1977,1978); Zemlyakov (1972); Thompson (1972); Araki, Kondo (1972); Thompson, Koenig (1972); Gruji~, siijak (1972,1973); Gruji~ (1972 ,1974b,c ,1976b); Michel (1973,1974); Weissenberger (1973); Vakhonina, Zemlyakov, Matrosov (1973); Siljak, Vuk~evi~ (1974); Martynyuk (1975); Michel, Miller (1977); ~raki (1978); Gruji~, Gentina, Borne, Burgat, Bernussou (1978); Saeki, Araki, Kondo (1980); Ikeda, Siljak (1980)]; * asymptotically stable [Michel, Porter (1970); Gruji6, Siljak (1972, 1973); Siljak (1972a,1978); Gruji~ (1972 , 1974b,c , 1976b); Michel (1973,1974); Araki (1975,1978); Blight, McClamrock (1975); Gruji~, Gentina, Borne (1976); Bitsoris, Burgat (1976); Michel, Miller (1977); Araki (1978); Sinha (1980)]; * unstable
[Thompson (1972); Grujig, Siljak (1972,1973); Gruji~ (1972,
1976b) ; Martynyuk (1972,1973,1975); Michel (1973,1974); Goisa, Martynyuk (1974); Michel, Miller (1977); Morari, Stephanopouios, Aris (1977); Araki (1978); Siljak (1978)]; equi-absolutely bounded [Kloeden (1975)]; * partially stable [Bondi, Pergola, Oambardella (1979)]. In the framework of the Problem B solution continuous large-scale systems were studied by Matrosov (1962,1968); Gentina, Borne, Laurent (1972,1973), Oruji~ (1974a,c , 1977a); Gruji~, Gentina, Borne (1976); Borne, Benrejeb (1977); Gruji~, Gentina, Borne, Burgat, Bernussou (1979); 0runiC, Burgat (1979,1980); Burgat, Gruji~ (1979); Vidyasagar
(1980). Domains of attraction of large-scale systems were analyzed by Weissenberger (1973), and somewhat later, by Bitsoris, Burgat (1976); Morari, Stephanopoulos, Aris (1977); Siljak (1978). Gruji~ and Ribbens-Pavella (1978,1979) studied the problem of asymptotic stability domain estimation in the framework of large-scale systems. 2. The concept of stability of motion with respect to a part of variables belongs to Liapunov (1893). The first theorems on partial stability on the basis of Liapunov's functions were obtained by Rumyantsev (1957)° Let us mention here the works by Peiffer, Rouehe (1969) and by
Reference s
223
Rouche and Peiffer
(1967) which greatly influenced the development of
the partial stability theory. An outline of this concept development together with the original results is contained in the paper by Oziraner and Rumyantsev
(1972) and in the book by Rouche, Habets and Laloy
(1980). 5. The word "sat" is an abbreviation of the expression
"saturation
nonlinearity". 4. An idea of the analyzed system extension by means of a generalized comparison system [Matrosov (1963)] is Hatvani (1975) and by Martynyuk
utilized
(1975) and it
the monograph by Martynyuk and Gutowski
in the works by
is widely applied in
(1979). Application of this
idea to analysis of the large-scale system structural stability is realized here for the first time. 5. Large-scale systems with weakly interconnected subsystems
[Martynyuk
(1972a,1975)] admit an effective application of the method of averaging, developed in nonlinear mechanics together with the method of Liapunov's
[Bogolyubov, Mitropolsky functions
[Liapunov
(1974)],
(1892)].
The given form of aggregation was developed by Kosolapov under the guidance of Martynyuk
[Martynyuk, Kosolapov
(1978); Kosolapov
(1979)].
6. Siljak pointed out to the authors that Persidskii used the notion of structural stability in a sense different than he defined connective stability and than structural stability defined in this book. The notion "structural stability" was used also by other authors, even they studied only influence of system parameters tural variations on stability.
instead of system struc-
There is not any ambiguity herein be-
cause structural stability is used only in the sense defined in Chapter III.
Anashkin, O.V. (1978), On investigations of stability under persistently acting disturbances in the neutral case. Different Uravn, 14, No.6, 1124-1127 (in Russian). Aoki, M.
(1968), Control of large-scale dynamic systems by aggregation. on Aut. Control, AC-13, No.3, 246-253.
IEEE Trans.
Araki, M. (1975), Application of M-matrices to the stability problems of composite dynamical systems. J. Math. Analysis and Applications, 52, 309-321. Araki, M. (1978), Stability of large-scale non linear systems-quadraticorder theory of composite-system method using M-matrices. IEEE Trans. on Aut. Control, AC-23, No.2, i29-142. Araki, M., and B. Kondo (1972), Stability and transient behaviour of composite systems. IEEE Trans. on Aut. C~ntroZ, AC-17, No.4, 537-
541.
224
Ch. Ill
Large-scale systems in general
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Zadorozhny, V.F. (1969), Asymptotic stability analysis of many-dimensional systems by means of the method of vector Liapunov functions. Kibernetika i Yychislitel'naya Tekhnika, No.l, 92-97 (in Russian). Zadorozhny, V.F., and A.A. Martynyuk (1972), Estimation of influence of connections between sub-systems on stability of a linear non-stationary system. Prikl. Mekhan£ka, 18, NO.9, 65-71 (in Russian). Zadorozhny, V.F., and A.A. Martynyuk (1973a), On a general problem of aggregation solution as the problem of moments. Matematieheskaya Fizika, No.14, 49-54 (in Russian). Zadorozhny, V.F., and A.A. Martynyuk (1973b), On solution of the general aggregation problem as the moments problem. Mat. Physics, 14, 4955. Zadorozhny, V.F., and A.A. Martynyuk (1975), The and L-problem of moments in the problems on mensional systems. In Problems of Analytical of Stability and Control, Nauka, 151-154 (in
direct Liapunov method stability of many-diMechanics, Theories Russian).
Zemlyakov, A.S. (1972), On a problem of the comparison truction. Tr. Kazan. Aviats. in-ta, No. 144, 46-54
system cons(in Russian).
CHAPTER IV
SINGULARLYPERTURBEDLARGE-SCALESYSTEMS
IV.I.
INTRODUCTION
Singularly existence
perturbed of several
parameters on them. bounds
often
parameters
as
construct
~i
point
and whose
order of the system. values
of view, matrix
Also,
of small parameters.
Stability
the usual
by
of these
the upper
assumption
problems
1976-1979,
it is important
it is important Pi
matrix
allowable
was
that
Such
to find a way to
lower
to estimate
function
independence in what
construction
of singularly Liapunov
implies
of small than the
upper
• Solving these problems,
or up to a certain degree)
as explained
studied via vector
Grujid,
to estimate
that will be independent
system Liapunov
(complete,
of subsystems,
. Existence
of the system a g g r e g a t i o n
order will be essentially
of small parameters
blem of the overall blem of the
characterized
~i+0
the a g g r e g a t i o n
parameters
Besides,
are usually
~i , ~i >0
it was difficult
of small parameters.
From the e n g i n e e r i n g
IV.2.
small
systems
led to the dependence
In such cases
~i+i/~i + 0
been
large-scale
mutual
different
allowable
we meet
a pro-
and the proindependence time
scales
follows.
perturbed function
large-scale method
since
systems 1976
have
(Lj.T.
1981).
DESCRIPTION AND DECOMPOSITION OF SINGULARLY PERTURBED LARGE-SCALE SYSTEMS
A singularly
perturbed
structurally
variable
large-scale
system
is de-
scribed by dxi - fi(t,x,y,Pi,Si) dt
i:i,2,...,q
(la)
232
Ch. IV
~i where
Xi e R n i
appropriate small ble
denoted
: gi(t,x,y,M,Pq+i,Sq+i)
, Yi e R m i
order,
and,
value
hence, of
by
, fi
~i
and
#ie]0,1]
~i
gi
, i:l,2,...,r
are
is a p o s i t i v e
continuous
parameter
by
~im
• The
set
,
(ib)
functions
that
, M =diag{~l,~2,...,~ r}
is d e n o t e d
can
. The
of all
of the
be a r b i t r a r i l y upper
admissi-
allowable
M
is
M , M:
and
Singularly perturbed large-scale systems
{M:O~M~I}
,
then, ~m : {M:O < ~i < # i m • Vi:l,2,...,r}
~ere, When
q+r = s the
system
.
small
has
parameters
#i
essentially
r
t-t o #i
ti In
such
cases
scales 1981)
ti
ri
be m u t u a l l y
then
0 < [i ~ ~i < +=
(Lj.T.
(2)
Gruji5
, i:i,2,...,r
• Vi:l,2,...•r
#1 ~i : ~ i
1977,
1978,
[ I = T I = ~ I =i
(3)
, i:i,2•.-,r
(4)
. In the
cases
when
subsystem
' i:i,2,...,r
be r e f e r r e d of the
(5)
(2)
and
and
.
described
by
(6), '
(6a)
gi(t,x,y,M, pq+i,Sq+i)
to as
the
whole
system
gidd~
i-th
interconnected
(i),
(6b) s£ngularly
perturbed
and
(0T ...,0T,yi,0T,...,0T)TT 6 R m
In the
: @i(t,xi,yi~M,pq+i,Sq+
T ~oT)T E ~n X i : (0T~...,0T~xi,0T~...
the case
(7a) i) ,
(7b)
• n : nl+n2+...+n , X i E ~ n i
, m = ml+m2+...+mr
, y i 6 R m1" , w i l l
it
can
be u s e f u l
to c o n s i d e r
• yi :
be r e f e r r e d
i-th disoonnected singularly perturbed subsystem. q=r
it r e (5)
dxi Xi • dt - fi(t' ,Yz,Pi,S i ) ,
to as
(4) h o l d
(3)
.
~idd~:
where
time
1979a,b,
,
d× i dt - f i ( t ' × ' Y ' P i ' S i )
subsystem
whole
bounds•
Evidently,
will
the
.
tim~-seal~ng of the system is uniform a n d f r o m
i-th
then
ti
,
sults
The
unrelated scales
, i=l,2,...,r
[i 6 [[i,~i] where
time
related
ti tl -~i in c e r t a i n
mutually
time-saa~ing of the system is non-uniform. H o w e v e r ,
can
via
are
unrelated
Aggregation and stability criteria
233
dXi : f i ( t , x i , y i , p i , S i ) dt
which w i l l be c a l l e d w h o l e system,
(8a)
0 :
gi(t,xi,yi,0,pq+i,Si)
the
i-th disconnected
,
(8b)
degenerate
subsystem of the
and dYi dt : gi(~'bi' yi 'O'pq+i' S i )
that will be c a l l e d the
i-th
system of the whole system. bT,0 T,...,0T) T , bier ni
disconnected
In
(9),
. When all
~R
#wi
fast and
(9 )
(boundary-layer)
sub-
bier n , b i = (0T,...,0T,
are
(formally)
'
i:i,2 ...,q ' '
set equal
to zero
in (I) then dXi : f i ( t , x , y , p i , S i ) dt O: gi(t,×,y,O,Pq+i,Sq+i)
the interconnected
w h i c h will be c a l l e d
system of the w h o l e system,
the interconnected
will be c a l l e d
degenerate
fast
(reduced order)
sub-
, i=1,2,...,r
,
(boundary-layer)
(ii)
subsystem of
that
0 : gi(t,x,y,O,pq+i,Sq+i)
is true for every 0 :
(P,S)EPxS
iff y:0
, V ( t , x , y ) eRxNxXNy , , and that
gi(t,xi,yi,0,pq+i,Sq+i).,
is true for any
(P,S)EpxS s iff y i = 0
V ( t , x i , y i) e R X N x X N y
. Hence,
dXi : f i ( t , x i , 0 , P i S i) dt ' (10) is e q u i v a l e n t
,
(8) is now e q u i v a l e n t
to
(12)
'
to
dXidt - f i ( t ' x ' O ' P i ' S i )
IV.3.
(iOb)
system.
I t is a c c e p t e d
and
,
and
= T i g i ( ~ , b , y , 0 , p q + i , S i)
dt I
the w h o l e
, i=l,2,...,r
(10a)
, i:l,2,...,q
•
(15)
AGGREGATION AND STABILITY CRITERIA FOR SINGULARLY PERTURBED LARGE-SCALE SYSTEMS
IV.3.1.
Introduction
The p r e c e d i n g
analysis
forms of s i n g u l a r l y bined with d i f f e r e n t
showed existence
perturbed
large-scale
aggregation
of d i f f e r e n t systems.
decomposition
W h e n they are com-
forms of l a r g e - s c a l e
systems
in gen-
234
Ch. IV
eral,
a great v a r i e t y
turbed large-scale ferent
of a g g r e g a t i o n
systems.
time scales
Singularly perturbed large-scale systems
ti
forms r e s u l t s
However,
the c r u c i a l
are m u t u a l l y
for s i n g u l a r l y
per-
is w h e t h e r
dif-
s y s t e m is d e c o m p o s e d
into
dependent,
point or not.
IV.3.2. Non-uniform time scaling IV. 3.2.1. In this q
General
section
q=r
interconnected
following
holds.
singularly
assumptions
aggregation
i.
There
NiyCR mi
numbers
perturbed
form of the w h o l e
Assumption and
The l a r g e - s c a l e
of
~il
exist
06R mi
and
~i2
system
connected
Pi(t•xi) e
C(I'I)(R×Nixo,R+) ~i>0
neighbourhoods vi , ~ i
, so that
~6]0,~i[
is p o s i t i v e
definite
on
Nix
,
5)
~i
is p o s i t i v e
definite
on
Niy
•
4
~it(L,Xi) + V TiX i ( t , x i ) f i ( t , x i • 0 • 0 • 0 ) Pit(t,0) ~ 0 , VteR
6
vTiXi (t ' X i ) [ f i ( t
~i2~i(t,y xi
Assumption
=
(0 T
i=1,2,...•r
definite
on
: Nix
and
is a s y m p t o t i c a l l y
con-
N-~i(t~xi)
• V(t,x)ERxN x ,
2.
x i • y i , p i , S i ) - f i ( t , x i , 0 • 0 , 0 ) ] < filni(t,Xi ) +
is
tractive
functions
and
,
EL = (0 T •'",0 T ~Yi, , T 0T ,'"•0 T ~) T e R m
v i • non-negative
.m
numbers
~i1
~il
, ~i2
C(1'1'1)(R×Nixo×Niyo,R+) and v i is p o s i t i v e : {vi(t~Yi) e c(l'l)(R×Niyo,R+) and vi
is p o s i -
~i4 and a p o s i t i v e i=l,2,...,r :
~ixXNiy
R×Nx×NyXPXS s
, V(t,X,y,P,S)•
T h e r e exist
vi(t~xi•Yi) •
there
, and
,'"~0 T ,XiT,0T•...,0T)T@~n
integer
p
such that
, or
tive d e f i n i t e 2)
,
i ) , xi~O
~i2 , ~i3 and and for every
on
0eR ni
of
and n o n - n e g a t i v e
•
5
l)
an
,
~i
where
N i x C R ni
~i
for every
Vi~(t)
2)
Xi#0
and
, it is p o s i t i v e
such that the set
for every
the
define
(i).
, functions
, ~i1
(6) for which
These assumptions
Niy O = { y i : Y i e N i y , Y i ~ 0 }
and
there is tractive
subsystems
will be accepted.
N i x O : {Xi:xieNix, Xi~0}
Let
I)
theory
~i>O
on
,
definite
Niy } , such that the set
for every
~e]O,~i[
Via(t)
is a s y m p t o t i c a l l y
con-
,
5)
Viy i T ( t , x i , Y i ) g i ( t , x i , y i , 0 , 0 , 0 ) _<-~i(t,Yi) and, r e s p e c t i v e l y • {v(t,x,y)eRxNx×Ny} ,
4)
Viy iT
(t,×i,Yi)[gi(t,xi,yi,Mi,pr+i,Sr+i)
~ i l P i ~ i ( t , x i ) + ~ i 2 ~ i ( t , Y i ) , Yi #0 ,
, yi# 0 ,
V(t,x•y)ERXNxXNy ,
_ gi( t , x i , y i , 0 , 0 , 0 ) ] _<
V(t,x,y,Mi,p,s)ER×NxxNy×MixPxSs •
Aggregation and stability criteria
and r e s p e c t i v e l y , where
235
{v(t,x,y,Mi,p,$)
M i = diag{0,...,0,#i,0,...,0}
m RxNxXNyXMixpxSs } ,
, M ± = {Mi:0<#±~l}
T v i t ( t , × i , Y i) + V i x i ( t , × i , Y i ) f i ( t , × i , y Z , P i , S
5)
+ ~i4~(t,Yi)
, xi~0
,
i) ~ ~ i 3 ~ i ( t , X i)
,
and v i t ( t , 0 , y i) ~ ~i4~i(t,Yi ) , V(t,x,y,P,S) ~ R X N x X N y X p X S s , or r e s p e c t i v e l y , {V(t,x,y,P,S) e R x N x X N y X P X S s } . • .Assumption that 1)
3.
There
for every
exist
i=1,2,...,r
real
number
~ij
, ~ij~0
for
i~j
, such
,
#i[ui(t,×i)+vi(t,xi,Yi)]li[fi(t,x,y,pi,Si)-fi(t,xi,yi,pi,Si)
]
+ ViYi T [gi(t,x,y,M" Pr+i~ Sr+i)_gi( t 'xl'Yl' . . Mi 'Pr+i ,St+i) ] r ~i ~ ~ i j ~ j ( t , x j , Y j ) , V ( t , x , y , M , P , S ) 6 ~XNxXNyX~XPxS x , Xi~0 j=l Yii0 2)
,
,
T i ( t , x i , Y i ) [ g i ( t , X , y , M , P r + i , S r + i ) _ g i ( t , 0 , y i M i ,Pr+i,Sr+i)] viy r ~i E ~ij~j(t,xj,yj) , V ( t , x , y , M , P , S ) e RXNxXNyXMXPxSs , xi=O , j=1 Yi#O
3)
,
and
[~i(t,×i)+vi(t,xi,Yi)]li[fi(t,×,Y,Pi,Si)-fi(t,xi,O,Pi,Si)] r ~ ~ij~j(t,xj,yj) , V ( t , x , y , M , P , S ) ~ R X N x X N y X M X p X S s , Xi~O , Y i : O j=l
,
where ~i(t,×i,Yi)
= (1-~il-~i1~C-1-~i3)~i(t,xi)+(~i2+~i4)(Fi-~i)~i(t,Yi)
~i = ( i - ~ i 2 ) ( ~ i 2 + ~ i 4 ) - I Notice
that
~i
admissible
~i
is the r e q u i r e d so that
The p r e c e d i n g
' and
~i e ]0,Fi[ estimate
,
. •
of the upper bound
~im
of
M i = {M:0
assumptions
introduce
the a g g r e g a t i o n
matrix
A , AeR rxr,
A=(aij) ,aij = - ~ i j + ~ i j The order (m+n)
r
of the a g g r e g a t i o n
of the whole
Theorem i. the system
system
(a) Assumptions
~xPxS s
less
than the order
and
of
1-3 hold; and A
is stable.
vi
to (I),
Nix=R ni , Niy=R mi , @i=+~ ,
are radially unbounded for every
of the system
the whole over
(XT,FT)T:0
it is sufficient that both
(ZI) If, in addition (XT,yT)T=o
is much
(I) For structural asymptotic stability of (1) over
(b) the matrix
~i=+ ~ , ~
matrix
(i).
~xpxS s
i=i,2,...,r , then
(7) is structuraZly asymptoticaZZy
stable in
236
Oh. IV
Paoo~.
v = (VI,V2,...,Vr)T , V i = ~ + v i , Vi=l,2,...,r , and
(i) Let
where
bERr
Singularly perturbed large-scale systems
is d e t e r m i n e d
for a r b i t r a r y
CER r , c>0
v = bTv
elementwise,
from
b = -(AT)-IC Since
all
aij ~ 0
sis of A s s u m p t i o n tion
for
(b) it results
the vector
b
Assumption function
due to the c o n d i t i o n A = (aij)
(M. F i e l d e r
is p o s i t i v e
is p o s i t i v e
the set
on
(~)
Chapter
III.4.1.i) that
the c o n d i t i o n 2 prove
and that
contractive
i of
that
there
is
the ~>0
for every
:
T (t, xi,Yi) fi(t,x,y,pi,Si + Vixi
i) +vit(t,xi,Y
i)
) +~ T ~i ViYi(t'xi'Yi)gi(t'x'y'M'Pr+i'Sr+iJ
= Pit(t,xi ) + pTi x i ( t , x i ) f i ( t , x Z ,' 0 , 0 , 0 i
1962,
result,
N = NxXNy
is a s y m p t o t i c a l l y of
Vi = P i t ( t , X i ) +pTzX i ( t ,x i ) f i ( t , x , y , p i , S
•
This
due to the condi-
1 and 2 of A s s u m p t i o n
definite
Vf(t)
. A l o n g motions
(a) and the h y p o t h e -
is stable
and V. Ptak
elementwise.
1 and the c o n d i t i o n s v
such that ~6]0,@[
i~j
3, and since
) + pTi x i ( t , X i ) [ f i ( t , x Z ," y Z ,"p i , $
T
i)
•
- fi(t'xX'0"0'0) ] +~ii V i Y i ( t ' x i ' Y i ) g i ( t ' x Z ' y i ' 0 ' 0 ' 0 ) T + P_~i i ViYi [gi(t'xi' yi 'Mi' Pr+i' Sr+i)_gi( t 'xi ,yi 0,0,0)] _7_i T + P i ViYi [gi (t' x'Y' M' Pr+i ' Sr+i)-gi (t' xi'yz' Mi' Pr+i " Sr+i) ] + Vit(t,xi~Y i )
+
T i (t,xi,Yi)fi(t,xi Vix
yi,pi,Si)
+ [ Pi(t,X i) + vi(t,xi,Yi)] Tx i [ f i ( t , x , y , p i , S i ) - f i ( t for
Xi#0
and
Yi#0
, Vi=1,2,. ,r , V ( t , x , y , M , P , S )
Now,
4 and 6 of A s s u m p t i o n
i, 5-5 of A s s u m p t i o n
.X. i. . vi Pi,Si)]
,
6R×Nxxt4yXA4XP×S s
2 5 and
1 of A s s u m p t i o n
3 imply r
vi < -~i+gil~i+~i2~i-~i+~il~i
i+~i
~i2~i + X ~ijnj j=l
+~i3~i
r --- [(l -~il - ~il~ip-i )~i + (_! @i Xi#0 , Yi#0 , V i : l , 2 , . . . , r This result, and the
r Vi -< Z aij~j j=1 or for
, V(t,X,y,M,P,S)
the d e f i n i t i o n
fact that
1 ~i2 - ~ 1 2 -~i4)(~i] + j=l Z (Xijnj ;/i
0<#±<1
of
6RXNxXNyXMXpxS
H i , 2± , ~i
yield
for
xi~0
(Assumption and
s
,
.
3) and
ai 9 ,
Yi¢0
, Vi=1,2,...,r , V ( t , X , y , M , P , S ) E R X N x X N y X ~ X P X S s ,
e = (~1,~2,...,~r)T
,
@(t,x,y,M,P,S) ~Ae(t,x,y) ×i~O
, V ( t , x , y , M , P , S ) eRxNx×NyxMxP×S~ ,
, Yi#O , vi=1,2,...,r
(14)
Aggregation and stability criteria
When
xi(t) = 0
Xi(Oil+eil)
~ 0
237
and
yi(t) ~ 0
iff
eil<+~
for all
, ell>0
I T +~i viYi(t'Xi'Yi)gi(t'xi'yi'0'0'0)
= vit(t'Xi) + vit(t'xi'Yi)
i T " +~i Viyi(t'xi'Yi)[gi(t'xi'yl'M i
on
'Pr+i'Sr+i)
T
This result, vi
i
ViYi(t~Xi'Yi)[~i(t~X~y~M'Pr+i'Sr+i)
Vt E [oil~oil+Eil[
Nix
and
T (t ,xi,Yi) gi (t, x,y ~ M, Pr+i, Sr+i ) + ~ iI ViYi
Vi = Pit(t,xi) + vit(t,xi'Yi)
+~i
t • [Oii,Oii+6ii[
then,
- gi(t'xi'yi'0'0'0)]
-gi(t'
X i
i
'
,Jl ~ M ~ P r + i ~ S r + i
)]
.
5 of Assumption
1 together
, 5 of A s s u m p t i o n
with
positive
2, 2 of Assumption
~i , ~i • ~i (Assumption 3) and A = (aij) r Vi ~< Z aij~j , Vt • [all,all+ell [ j=l
definiteness
of
5, the definition
of
imply
Hence, 9(t,x,y,M,P,S)
v(x,y,M,P,S) When
, vte[oil,oi1+6i1[
eNxXNyxMXpXS
xi(t) f 0
and
Yi(t) = 0
yi((;i2+6i2) ~ 0
iff
el2<+ ~ , el2>0
s , x=x(t)
for all
, vi:1,2,...,r
,
, y=y(t)
(15)
t 6 [oi2,0i2+6i2 [
and
, then
¢i = ~it(t'Xi ) + P iTX i (t, xi,Yi ) fi (t,x,y, pi, Si ) + vit(t,xl,Yi) T + Vi×i (t, Xi ,yi ) fi (t, x,Y, pi, Si) T = Pit(t,xi,Yi) + P i x i ( t , x i , Y i ) f i ( t , x i , 0 , 0 , 0) + P iTx i ( t , x i ) [ f i ( + [ 5(t'xi)
t ,xl,0,Pi,Si ) - fi(t,xZ,0,0,0)]
+ vi(t ,xi,Yi)] Tx i [ f i ( t , x , P i , S
T i (t,xi,Yi)fi (t,xi,0,Pi,Si) + vi× This result,
4 of A s s u m p t i o n
~i~0 , the definition of A s s u m p t i o n 3 yield Vi(t,x,y,M,P,S)
of
~
i ) - f i ( t , x ~ ,•0 , p i , S i ) ]
, Vt 6 [ai2,oi2+ei2 [
i, 5 of Assumption
Ni , ~i , ~i
r ~ aij~ j j=1
+ vit(t,xi,Yi)
2,
(Assumption
~il~0
5),
Vt • [oi2,0i2+6i2[
,
, ~i>0
A=(aij)
Vi=l,2,...,r
,
and 3
.
Hence, 9(t,x,y,M,P,S) ~ Ae(t,x,y)
, vt • [oi2,oi2+6i2[
V ( x , y , M , P , S ) eNxXNyXMXPXSs , x : x ( t ) When
xi(t):0
and
Yi(t)=0
, Vt • [ui3,~i3+ei3[
, Vi=1,2,...,r ,
, y=y(t) and
0<ei3~+~
(16) , then
238
Ch. IV
Singularly perturbed large-scale systems
evidently, Vi[t,xi(t),Yi(t)] Now,
(i4)-(16)
= const.
, Vt E [ o i 3 , o i 3 + e i 3
[ , vi:i,2,...,r
. (17
imply V(t,x,y,M,P,S) and
•RXNxXNyXMXpxS
xi~0 , Yi~0
s
, Vi:l,2,...,r or
v(t,x,y,M,P,S)
~ -cTe(t,x,y)
, X=X(t)
, y=y(t)
Vj=1,2,3
, Vt • [ a i j , a i j + e i 3 [
, Vi=&,2,...,r
,
,
V(x.y.M.P.S) eNxXNy×~XPXSs Following and
the
ally rem
now
the
proved
13 of
radially
are
i)
which
There
N i y ~ R mi ~ii
(xT,yT) T : 0
M×P×S s
to
(I),
results
of
all
(i)
from
(18
is
structur-
conditions
of T h e o -
N i x : R ni , N i y : R mi , ~i:+ = , ~ i : + ~ ,
then
completes
the
of the
sets
Nx=R n , N y : R m , ~ : + ~ proof.
Vi~(t)
of and
Pi(t,xi) 6
exist
06R mi fi2
connected
and
v
is
•
need
not
;
~i2 , ~il
C(1'I) (R×Nixo,R+) on
neighbourhoods
, functions
Nix
vi , ~ i , so that
, it is
decrescent
2)
~i
is p o s i t i v e
definite
on
Nix
,
3)
~i
is p o s i t i v e
definite
on
Nix
,
4)
vit(t,Xi ) +
5)
(i.e.
unbounded
and
xi#O
it
be r e q u e s t e d
as
N i x C R ni
OER ni
follows.
Assumption 4.
numbers
that
over
in a d d i t i o n
contraction
in w h a t
v
13 of 1 - 3 . 2 . 5 ,
satisfied).
radially
unbounded,
Asymptotic shown
are
When,
vi
of
stable
1-5.2.5
(II)
of T h e o r e m
properties
asymptotically
and
and
proof
(18
both
and
~i
and
for
every
positive
of
non-negative
i=l,2,...,r definite
,
on
Nix
,
Ti x i ( t , x i ) f i ( t , X Z , 0 , 0 , 0 )
~-~i(t,Xi)
, V(t,xi) E RXNix
,
,
g 0 , VteR
Vit(t,0)
and
6)
, T i x i ( t . x i ) [ f i ( t . X i yi Pi.S± ) _ f i ( t . x i +~i2~i(t,Yi
Assumption ~i2 , ~ i 3
5. and
) , xi#O There ~i4
, V(t,X,Y,P,S)
exist and
functions
a positive
0.0.0) ] ~ g i l ~ i ( t . x i ) ÷
6RXNxXNyXpxS
s
v i , non-negative integer
p
such
that
• numbers ~i1
~ii , , ~i2
A~gregation and stability criteria
and for every i)
i:l,2,...,r
239
,
vi(t,xi,Yi) 6 C(l'l'l)(RXNix0XNiy0,R+) definite
and d e c r e s c e n t
on
Nix×Niy
{Vi(b,yi) 6 c ( l , l ) ( R X N i y 0 , R + ) and d e c r e s c e n t 2)
on
Niy}
and
and
vi
is both p o s i t i v e
, or, vi
is both p o s i t i v e
definite
,
V iT ~ i ( t ~ x i ~ Y i ) g i ( t ~ x i , y l , 0 , 0 ~ 0 ) S - ~ i ( t , Y i )
,
V(t,X,y,P,S) e RXNxXNyxPXS s , 5)
T ViYi(t,xi,Yi)[gi(t,xi,yi,Ml,Pr+i,Sr+i)-gi(t,xi,yi,0,0,0)] ~il@~i(t~Xi)+~i2~i(t,Yi) and r e s p e c t i v e l y ,
• ~i #0 ~ V ( t , x , y , M i , p , S ) e R × N x × N y × M i × p × S s
{V(t,×,y,Mi,p,$) • RX~x×Ny×Mi×PxSs } ,
and ~)
v i t ( t , x i , ~ i) + v Ti x i ( t , x i , ~ i ) f i ( t , x i ~ y i , p i ~ S i) ~ i 3 ~ i ( t = x i ) + ~ i 4 ~ i ( t , Y i ) • xi~0 and
vit(t,0,Y i ) s ~i4~i(t,Yi ) , V(t,x,y,P,S) 6 RXNxXNyXP×S s ,
and r e s p e c t i v e l y ,
Theorem 2.
(I) For s t r u c t u r a ~
(xT,yT) T = 0 both
{v(t,x,y,P,S) • R X N x × N y x P × S s}
of the s y s t e m
uniform asymptotic
(1) over
M×P×S s
.
•
stability
of
it is s u f f i c i e n t
that
N i x = R ni , Niy=R mi , ~
and
:
(a) A s s u m p t i o n s
3-5 hold
and (b) the m a t r i x
A
is stable.
(II) If, w£
are r a d i a l l y
of the s y s t e m the whole PrOOf.
in a d d i t i o n
to
(I) is s t r u c t u r a l l y
over
3), and hence,
(b) for a r b i t r a r y
C>0
positive
b , the f u n c t i o n
the c o n d i t i o n s 2. Hence, stability
of
~ijA0 for
is both p o s i t i v e
b ,
stable
in
(xT,yT) T : 0 w h i c h proves
of
b : -(AT)-IC
definite
, is also
For such a v e c t o r
and d e c r e s e e n t
on
5. The c o n d i t i o n s
2-6 of A s s u m p t i o n
i and
5 are the same as 3-5 of A s s u m p t i o n
the p r o o f of T h e o r e m satisfies
, (the h y p o t h -
, and the c o n d i t i o n
1962).
4 are the same as c o n d i t i o n s
v
, i#j
i~j
4 and i of A s s u m p t i o n
2-4 of A s s u m p t i o n s
The f u n c t i o n
(a),
aij~0
, cER r , the v e c t o r
v:bTv
by f o l l o w i n g
tion 1-3.2.4),
(xT,yT) T = 0
asymptotically
(M. F i e d l e r and V. Ptak,
due to i of A s s u m p t i o n
2-6 of A s s u m p t i o n
hold.
uniformly
(I) In view of the c o n d i t i o n
elementwise
i=l,2,...,r , then
M×F×S s
esis of A s s u m p t i o n
Nx×Ny
(I),
u n b o u n d e d for every
1 it r e s u l t s
all c o n d i t i o n s
(i) over
MxpxS s
the a s s e r t i o n
under
that
(14)-(18)
for u n i f o r m a s y m p t o t i c (Theorem (I).
7 of the Sec-
,
240
Ch. IV
(II) are
When,
radially
also
radially
1-3.2.4
in a d d i t i o n
unbounded,
the
which
proof.
IV. 3.2.2. Applioation A singularly
(i),
Vi=l,2,...,r
unbounded,
completes
to
Singularly
in v i e w
Nx=R n , Ny=R m
of T h e o r e m
Lur'e
like
absolute
system
r
1
r 1 ~ I/iSr+i,jBijxj j=l
aij ~ 0
-S I
i,l
of the
are
diagonal
I
1
(19a)
) m
(19b)
T 3 + cj6Sr+i,jyj)
, j--1,2,5
appropriate
order,
matrices.
,'", S ii , i - i
,Si,2
1
,
vi-- i, 2 ,...,r
, i=i,2,...,r
are
i2Y
+ qi3~i3(°i3
T 2 Z ~i(cjsSr+i,jXj 9=i
e [0,gij]
m a t r i c e s and v e c t o r s 1 2 3 far Sij , Sij and Sij
Si
:
j~i
All
section
CT
= cilx +
+ BiYi + q i 2 ~ i 2 ( ° i 2 )
:
oil0ij(Oij)
is
by
T
,ail
r
for
vi
v
stability analysis
is d e s c r i b e d
+ ~ SijAijYj j=1
T oi2 : ~/ici3xi + c iT 4 Y i • ui3
where
and
and
8 of the
systems
•
Xi = A i × i + q i l ~ i l ( a i l )
I/iY i :
large-soale
N i x = R ni , N i y = R mi , 5
, then
to the structural
perturbed
perturbed
I
,
(19d)
.
and
in p a r t i c u -
Let
, S iI , i + l
1
(19c)
, ...,S il,r
1
l
Sr+i, l,Sr+i,2,-..,Sr+i,i_ 1 , I , Sr+i,i+l•...,Sr+i, r 2 2 2 2 Sr+i, l,Sr+i,2,...~Sr+i,i_ 1 0 , S r2 + i , i + l , ' " , S r + i , r
=
LS
3 3 r+i,l,Sr+i,2,'",Sr+i,i-i
S = block diag
0
,
3 3 ~ Sr+i,i+1~'",Sr+i, r
(S1,S2,...,S r}
and S s : {S:OsS where The by
I
is,
as
usually•
disconnected setting
×i
the
singularly arid
yi
stability of that
subsystems x
and
) + AiiYi
subsystems
will
of T h e o r e m
20 of
the
section
(20).
means
that
both
This
of the
,
(20)
be u s e d 1.5.3
by
are
Assumption
appropriate
are
obtained
T
from
'
Yil = Cil Xz + c I 2 Y z , ,
(19)
was
studied
satisfied 4 and
(20a)
Vi=l,2,...,r
supposing
for
Assumption
(20b)
in 1.5.4.
that
all
the
The
conditions i-th
5 are
system true
and
that i > ~il+~il
, Vi=l,2,...,r
,
,
order.
y , respectively,
+ BiYi + q i 2 ~ i 2 ( ° i 2 )
of the
section
of
, Vm=2,3,Vi,j=l,2,...,r,Vk=i,2,3}
matrix
perturbed
instead
#i#i : ~iBiixi Absolute
, Sr+i,r=O
identity
Xi = Ai×i + q i l ¢ i l ( g i l
results
m
jsl , S i:l , S $ + i , r = l
(21a)
Aggregation and stability criteria
and
241
~ie]0,~i [ , ~i - 1 - ~ i 2
The comparison ~i(×i,Yi) Let
~i
(Assumption
=~i3(i-~i1-~il)llxiU
c~jcR nk
xi~O
function
and
be the
Yi~O
k-th
i:1,2,...,r
.
3) is obtained
+~i2(~i-~i)Pi311Yill
vector
(21b)
component
in the
, Vi=i,2,...,r
of vector
Cij
[H i
.
(22)
. Then,
for
,
T i [ gi (x,y,M, Si)-gi (Xi ~i pTiXi (Xi)[ fi(x,y,Si)-fi (xi,yi,Si)] + v iy T + # i V i x i ( Y i) [fi(x,y,Si)-fi(xi,yi,Si )] T Xi = ~i Pi(Xi)
form
,Yi ,Mi ~Si)
~il (u[1) i i T 0il o Cil(Cil) ]{qil[¢il((;il)-~il(Fil)] Gil T r r + ~ S lijAijy j } + Yi [~i 7, Srl+i .Bi Xj + q i 3 ~ i B ( U i 3 ) ] vi(Yi) j=l ,3 3 j=l j~i j#i -i r r pi{~il~i2[ Z [[Aij[[.[[yj[[ +Killlqiln Z ([]CJl[[.[[XjU+[[CiJ2[[.[[yj[])] + j=l j=l j~i r r + Z [[Bij[[.[[xj[[ + [[qi3[tki3 X ([[Cj5[[.[[XjI[+][Cj6[[.]]yj[[) } j=l j=l j~i j~i 1
+.1 2
]
+
25)
Let
24a)
7ij = kilKqill['[ICiJlB~i~i
2 + (U Bij II + K i 3 [lqi3g'n cj5 11)(1-6 ij ) , -i 6ij = (HAij[[ + gil[lqilll'llc~2H)-~il~i2 + llqi31lgi3[lcj6[l(l-~ij) •
Now,
the following
inequality
will
be used
7ijIlXjII+6ijiIyjII~ijWj(xj,y provided
24b)
j)
(25a)
that
~ij = max { 7iJ ~j3(i-~jl-~jl) (23)-(25)
eiJ } , " ~j2(~j-~j)Pj3
i,j=1,2,
,r .
(25b)
yield
T ' #iVixi[fi(x,y,Si)-fi(xi,yZ,Si
T )] + V i Y i [ g i ( x , y , M , S i ) - g i ( x i , y i , M i , S i ) ] r + PiV~xi[fi(x'y'Si)-fi(xi:yi'Si )] ~ ~i ~ ~ij~j (Xj,yj) , xi#0 , Yi~0 j=1
Vi=l,2,...,r Repeating
, V(x,y,M,S) • the p r e c e d i n g
Rn×RmxM×Ss
procedure
,
(26a)
it is obtained r
vTy[gi(x,y,M,Si)-gi(0,yi,Mi,Si Vi=l,2,...,r
'
) ] _< I/i
V(x,y,M,S) •RnxRm×MxS$
~ (~ijnj(xj,yj) j=l
, xi=0 , Yi #0 , (26b)
242
Ch. IV
Singularly perturbed large-scale systems
and r
[ u i ( x i ) + v i ( Y i ) ] ~ z •[ f i ( x , y , S i ) - f i ( x i , y i , S i ) ] , Yi:O
xi~O
(26a)-(26c) matrix
fulfil
A=(aij)
(19)
then
is s t r u c t u r a l l y
the
Definition
lutely
equilibrium
Example
i.
absolutely
on
Let
[OxK] the
(19)
(n:m:6)
three
tems.
They
are d e t e r m i n e d
(q=r:3)
Ai =
(24)
and
(25).
of the
[0,K]×M×F×S s
,
system
in the
'
Ci2
of the
[0,K]×M×P×S s
(M,P,S) e MxP×S s
be n u m e r i c a l l y singularly
Ci3
:
,
=
, '
disconnected
in E x a m p l e
is o b t a i n e d
:
defined
perturbed
as follows.
Lur'e
singularly
:71 ,j¢i ,7
il
aij
Bij
j Cj5
singularly
15 of the
calculating
,
8ii--10-3I
~ Ci4
2000
'
:7I
system
perturbed
" qil
i2
1.5.4.
according
to
,j#i
:
(24)
the
and
decom-
subsys-
0.012 ,
0.012 , - 0 . 0 9 4
,
(20)
results
(25)
0.0i2] 0.012|
0.012 , -0.0941J
,
1/10
'
~
'
qi3
=
were
[:] ,
•
considered
of that
example
the a g g r e g a t i o n
!
0.012 ,
[0I
, ~i2:gi3:1
as --0.944 ,
:
:
subsystems
Using
:
'
, qi2
, Cj6
perturbed
section
A =
(19)
. •
[i [0:] [:] [:] [:] ,
system
iff it is a b s o -
by
-
zl
(xT,yT) T : 0
interconnected
, Aii : I -1
A
aggregation
by
(×T yT)T : 0
over
stable over
system
12th order
into
and
stable
for every
posed
The
state
The equilibrium state
I.
stable
It is the
--
3. The
(26c)
of
is structurally
Bi
of A s s u m p t i o n
, is now d e t e r m i n e d
absolutely
•
v(x,y~M,S) 6 ~nx£mxMxSs
,
all c o n d i t i o n s
, aij : - S i j + ~ i j
If it is stable
sense
, Vi=l,2,...,r
g # i j=1 ~ ~ijNj(xj,yj)
matrix
Aggregation and stability criteria
The stability
property
Kotelyanskii-Hicks)
243
is tested by v e r i f y i n g
conditions
(-i) lalll > 0 ,
a21 , a22 case
(-i)(-0.944) >0
,
take the
-0.944 ,
I
Hence,
all
state
(xT,yT) T : 0
over
[0,K]×MxS
~a31 , a32 ,
-0.944 ,
0.0i2
0.012 ,
of Theorem
, where
2 have
time
analysis
levels.
mutual
perturbed
of such systems
On the first
level
was
subsystems
was tested.
properties
of interactions
hierarchical
level
q:r
disconnected
general
for c o n s t r u c t i o n
of the a g g r e g a t i o n
singularly
m+n
perturbed
of the whole
analysis
of singularly
large-scale
stability
of the twelfth
stability
was verified
of the
verification
matrix
fourth
order
is equal
order.
of stability
about
On the
On the
second
A .
absolute
level,
singularly
illus-
absolute absolute
perturbed
as a final
third
of the
stability
The example
first
of
less than
application
structural
level,
of the a g g r e g a t i o n
matrix
to the number
systems.
used for testing system.
sin-
that and
and it is much
off structural
for three d i s c o n n e c t e d
out on two
was used on the second
The conceptua]
Lur'e
systems
parameters
of d i s c o n n e c t e d
Information
subsystems
system.
which were
carried
of the a g g r e g a t i o n
was to the analysis
perturbed
trated the results,
systems
stable
,
of small
stability
perturbed
the order
The e q u i l i b r i u m
absolutely
large-scale
independence
about q u a l i t a t i v e
The order
verified.
test,
subwas
order matrix.
IV.3.3. Uniform time-scaling Uniform t i m e - s c a l i n g both
(3) and
utilizing
.
, Vi:1,2,5}
of singularly
by complete
stability
hierarchical gularly
i
>0
0.012 , -0.944
K = diag{2,1,1,2,1,1,2,1,1}
scaling
is characterized • The
0.0121 0.0i2
Conclu8ion
Non-uniform
~i
been
system is structurally
: {M:M : diag{#l,~2,p 3} , ~iE]0,0.447[ IV. 3 . 2 . $ .
0.012 ,
0.012 , -0.944 ,
0.012 , - 0 . 9 4 4
of the
33
form
> 0 , (-1)3
conditions
(Sevastyanov-
fall , a 1 2 , a131 > 0 , (-1)31a211 ' a22 ' a23 > 0
(-l)21ali ' ai21
which in this
the
(111.4.1.1)
is defined
(4) hold.
Lur'e
The
by
stability
like a g g r e g a t i o n
! , ~ie]0, +~[ analysis
, and
~i
will be carried
form and quadratic
such that out by
aggregation
form.
244
Ch. IV
IV. 3.3.1. Lur'e The Lur'e
like aggregation form and stability criteria
like a g g r e g a t i o n
Assumption
6.
There
exist
Nixc_R ni
, functions
1)
Pi
is d i f f e r e n t i a b l e
is
Pi>0
~i
3)
~iEC(RxR,R)
a connected
is p o s i t i v e
definite
[0,Ki[
4) ~ij,~ijEC(R,R) and
Assumption
7.
There
and for
exist
i)
is d i f f e r e n t i a b l e
is p o s i t i v e
, ~i6]0,+o~[
on
-< ~ i ( t , x i )
R×Nix
5)
~iEC(RxR, R) , ~i(t,0) - 0
is
definite
and
q
on
for
, V(t,X,y,M,P,S)
i)
There
definite the set
E RXNx×NyxpxS s ,
on
Niy
Vi~(t)
xeN x , Vi:l,2,..,r
, K q + i e ] 0 , + °°] , V i : 1 , 2 , _ , r
e RXNxXNyX~X})XSs
pl,p2E]0,+~[
,
,
: ,
, Vi:i,2,...,r
. •
such that
q T i=l~l l U i x i ( t ' x i ) [ f i ( t ' x ' y ' P i ' S i ) - f ± ( t ' x ' 0 ' 0 ' 0 ) ] U V(L,X,y,P,S)
and
exist
Yi:0
, vi=l,2,...,p ,
for every ~i #0
of
+
such that
such that
v~e]0,q,i[ ~iy
and
~ij
Niy
and p o s i t i v e
~i>0
e [0 ~Kq+i [ • V(t,oJi)6RxR
8.
contractive
, Vi=1,2,...,q ,
4) 7 i 3 , ~ i j e C ( R , R ) , Vi,j=l,2,...,r , and T 5) riViYi (t, x,Yi ) ~i (t, x,Y, M, Pq+i, Sq+i) P -< ~oi(t,x,y i ) ~ [Tij(t)¢oj(t,X,y j ) + ~ i j ( t ) ~ j ( t , f ~ j ) ] j=1
Assumption
and there
,
neighbourhood
R×Nx×Niy
contractive,
~i
VTiE[Ti,Ti[
,
,
vi , t°i , ~i , 7ij
2)
coi
on
, Vi=l,2,...,q
a connected
x6N x , and there
is a s y m p t o t i c a l l y
Xi:0
~ [~ij(t)'~j(t,xj) j=l , Vi:l,2,...,q , V ( t , x ) e R × N x •
, functions
for every
of
~i ~ 0 :
, V(t,~i)ERxR
NiyC_R mi vi
Nix
such that
is a s y m p t o t i c a l l y
R×Nix
on
vit(t,x i) + V i x i ( t , × i ) f i ( t , x , 0 , 0 , 0 ) +~ij(t)~j(t,~j)]
definite
Vi~(t)
, Vi,j:l,2,...,q
T
~ij
assumptions.
, Vi=1,2,...,q ,
, ~i(t,0) - 0
~i
by the next
neighbourhood and
and p o s i t i v e the set
~E]0,Pi[
~i(t'nl) e
5)
form is d e f i n e d
vi ,~i , fi , ~ij
such that
for every 2)
Singularly perturbed large-scale systems
both r ~
Pl i=1~ 2 i ( t , X , Y i )
,
Aggregation and stability criteria
P Z [vlt(t,x,y
2)
245
~ i) + Vix(t,x,Yi)f(t,X,y,P,S)]
P w~(t , ) ~ P2 ~ ,x Yi ,
i=l
i=l
V(t,X,y,P,S) where
e RX~xX~yXPXS s •
T T)T f : (fT,f2,...,fq
The a g g r e g a t i o n to the
o f the
functional
• system
aggregation
(i) d e f i n e d matrices
by A s s u m p t i o n s
Ai , Bi
and
6 and
7 leads
C i , i:i,2
, in
the f o r m Al(t)=(~ij(t))
, Bi(t)=(~ij(t))
, Cl(t)~I
,
(27a)
A2(t)=(Tij(t))
, B2(t)=(nij(t))
, C2(t)~I
.
(275)
Let O I = diag{01,02,...,0 q}
, 82 = diag{Sq+1,0q+2,...,0q+ r}
,
(28a)
K I = diag{kl,k2,...jk q}
, K 2 = diag{kq+l,kq+2,...,kq+ r}
,
(28b)
Li = 2K~l-(OiBi+B~Oi ) , i = 1 , 2 ,
(29)
Qi =-71 [ A ~ + A i + ( A T S i + I i ~ B i ) L ~ I ( A ~ ® i + I i + B i ) T ] , i = 1 , 2 In the
case
Qi
is p o s i t i v e
definite
then
Qi(t)-eil i > 0 , vteR For s u c h bound
(preferably
~im
the
greatest)
of a d m i s s i b l e
~i
e2
will
e2max ~i : m i n {i , ~ _7i }
there
is
.
(30)
ciE]0,+~[
obeying
, i=1,2 the
(31)
estimate
be d e t e r m i n e d
~i
of the u p p e r
by
, i:l,2,...,r ,
(32)
where = Plmax{(l+ei~i):i6[l,q]}
Theorem 3.
(I)
(xT,yT) T = 0
+P2max{(l+Si~i):ie[q+l,q+r]}
For structural asymptotic stability over
of the system
(a) Assumptions
MxpxSs
(33) of
(1) it is sufficient that
6-8 hold,
(b) there exist non-negative diagonal matrices the functional matrices every
•
i=1,2
Li
and
qi
0 i , i=1,2 , such that
are positive definite for
,
(o) the functions
~i
and
~i , i=l,2,...,q , and
wi
and
~i •
i=1,2,...,r , are all time independent, and {d) ~i , i=l,2,...,p
are d e t e r m i n e d by
(31)-(33).
(II) If, in addition to (I), unbounded and
Pi=+~
in Assumption
radially unbounded for every every
i=l,2,...,r , then
asymptotically (Ill)
xER n
(xT,yT) T = 0
NxXNy =Rm+n
6 for every and
~i=+ ~
Pi
is radially
i=l,2,...,q , in Assumption
of the system
stable in the whole over
,
vi
is
7 for
(1) is structurally
~xP×S s
For structural asymptotic stability over
~xpxS s
of
246
Ch. IV
(×T,yT)T = 0
of the system
(e) Assumptions
(I) it is also sufficient
that
6-8 hold,
(f) the functional every
Singularly perturbed large-scals systems
i=1,2
matrices
Li
~ and for
and
Q£
®I=0, ~2=0
are positive
definite
for
,
and (g) ~ i , i=l,2,...,r
are determined by
(IV) If, in addition unbounded and radially every
pi=+ ~
i=1,2,...,r , then
asymptotically
to (III),
in Assumption
unbounded for every
(31)-(33). Nx×Ny=R m+n ,
6 for every
×6R n
and
(xT,yT) T = 0
¢i=+ ~
:
q {Pi(t'xi
Z
all
The c o n d i t i o n s
(a) and
vi(t'x'Yi)
function
tion 6 and
(1) of A s s u m p t i o n on a c o n n e c t e d
is
Vfe]0,@[
~>0
'~i
"~i
and
~i
are
+
[
vz ~i[~i(O) ] do ~i(O )
7 prove set
derivative
both that N
V~(t) of
system
of
v
(i) of A s s u m p -
is p o s i t i v e
(xT,yT) T = 0
is a s y m p t o t i c a l l y v
t
(i).
5, the c o n d i t i o n
neighbourhood
such that the
. The E u l e r i a n
in the form
+Sq+i
of the whole
(b) of T h e o r e m
decrescent there
wi
~i(o)
r + i=l~ Liapunov
is
7 for
(I) is structurally
fi[ni(a)] de
) +@i
i:1
be a tentative
vi
M×P×S s
Paoo~. (I) In view off (c) of the theorem, time independent. Let v(t,x,y)
is radially
in Assumption
of the system
stable in the whole over
vi
i=1,2,...,q ,
a l o n g motions
definite
and that contractive
of (i) is found
r " : Zq (Pi+@i ~ ii~IPi) + Z (Vr+@q+i~iwilvi) " - " i=I i=1
Now we use Pi(t,Xi) : Pit(t,Xi) + P ~ x i ( t , X i ) f i ( t , X , 0 , 0 , 0 ) + T + ViX i" [fi(t,x,y,pi,Si)-fi(t,X,0,0,0)] vi(t,x,Yi) : Vit(t~x,Yi)
+ V TiX i ( t ~ x , Y i ) f i ( t ~ x ~ y , P , S )
,
+
__ T (tjx,yi) ~i (t,x 'Y'M~pq+i ,Sq+i) i riViYi +~I and A s s u m p t i o n s
6-8 in order
to evaluate
v
'
from above,
v(t,x,y,M,P,S) q Z {ni(Xi)[~ij(t)%(X $,j=l
j) + ~ i j ( t ) { j ( % ) ] [ l
+@i~i(ei)nil(xi)]}
+
Aggregation and stability criteria
247 r
+ pl m a x { ( l + ~ i ~ i ) : i E [ l • q ] }
~ ~ 2 ( x , y i) + i=1 r
+ p2 m a x { ( i + 0 q + i ~ q + i ) : i ~ [ l , r ] }
Z
~2(x,Yi)
+
£=i r
1
I~
+ ~i
{ooi ( x , y [ ) [1, i j ( t ) ~ j ( X , y j ) +
i,j=l
+nij(t)~j(°~j)][1+0q+i~i(c°i)cJil(X•Yi)]}
,
V(t,x,y,M•P,$) •RxNxxNyxMxFxSs After
simple
inequality
algebraic
and u s i n g T
manipulations (27)-(33)
T
on the r i g h t - h a n d
side of this
it is o b t a i n e d
T
T
s WlAlW I +WlBlZ 1 + ZlOIAlW I + Z l ® i B i z I + @Uw2112
+
(34)
i T T + TI(W2A2w 2 + w2B2z2 + z~O2A2w2 + z 2T0 2 B 2 z 2 where
Wl = (~l,~2•'",~q)T
Z 2 = (~l,~2,...,~r)T side
of the
last
Zl = (~I
. We now add to and inequality
Ci=I
Di = ~i (A~Oi+I)
subtract
,
together
positive
condition definite.
As a result• rem and
(b) of the Hence•
it follows
(34)-(36)
the r i g h t - h a n d
are
,
(36a)
,
(36b)
(360)
.
theorem
there
from
(35)
with
• i=i•2
-Qi : A~+Ai+XiX~ ' i : i , 2 the
from
XiY i : HiBi+D i , i=1,2
F i = K~ i - ~i( S i B i + B i OT i )
Under
W2 = (~i •~2,'",~r )T
T T -I o 2 = (w2-z2K 2 )z 2 z 0 ,
and
i H i =~ I
and define
•
beth
T T -I) O I = (Wl-ZlK i z i ~ 0 employ
~2,'"" ~q)T
)
it r e s u l t s
ei e ]0•+~[
(31)-(33),
the
that such
condition
Qi:Q~ that
(30)
(31)
is
holds.
(d) of the theo-
that
~ ( t • x , y , M • P , S ) ~ -~illWl(X)U z- ( ~~ i2 - ~ ) t l w 2 ( x , y ) U2 • V ( t , x , y • M , P , S ) eRxNxXNyX~XpxSs . The r i g h t - h a n d
side
of this
inequality
for every
#i~]0,~i[
due
el>0
conditions
of T h e o r e m
. All
verified
on
(II)
to the
MxpxS s , w h i c h In this
case
ally u n b o u n d e d
implying
ative
on
definite
asymptotically
stable
then
definite
2 of A s s u m p t i o n
13 of the s e c t i o n (I) of T h e o r e m
NxXNy : R m+n
~xPxS s
is n e g a t i v e
condition
proves
radial
(37)
and all
unboundedness (xT•yT) T : 0
in the whole
over
1.3.2.5
of
Mxpx$ s
NxXWy
have
7 and been
3.
vi
of
on 6 and
v
and
vi
. Since
(i)
are r a d i v
is neg-
is s t r u c t u r a l l y
248
Ch. IV
(III) 82:0
Singularly perturbed large-scale systems
In this case i n s t e a d of
(37) it is o b t a i n e d
for
01:0
and
that e2
~ ( t , x , y , M , P , S ) ~ - etflwt(t,x)H 2 - (~-t-1-@)Ilw2(t'x'Y)U2 ' v ( t , x , y , M , P , S ) • RxNxXNyXMXpxS s The r i g h t - h a n d for every el>0
side of this
~IE]0,~I[
inequality
V~(t)
of T h e o r e m
~×PxS s , which proves totically
the a s s e r t i o n u n d e r
v
q ~ i=l
V = This
(xT,yT) T = 0
the p r o o f
of
of
(II) and
3 provides
constructed function
on
NxXNy
are s a t i s f i e d
ai(°)
asymp-
(III)
that
N i x ~ R ni
, functions
i)
is d i f f e r e n t i a b l e ,
it is e v i d e n t
the proof.
•
level by u s i n g form defined
1979b),
r + i =~ i
dg
stability
There exist
v i + 8q+ i
vl ~i[~i(a) ] do ~i(o ) 0
conditions
for s t r u c t u r a l
under the f o l l o w i n g a s s u m p t i o n s
a connected
Pi ,~i • ~ij
neighbourhood
and
positive
~ij
definite
Nix
:
of
xi:0
~
3)
~i 6 C ( R x R , R )
is p o s i t i v e
definite
, ~i(t,O)£O
on
Nix
and d e c r e s c e n t
on
, Vi=l,2,...,q ,
and for
n i ~0
:
~±(t,n~) E [O,Mi[ ~)
~ij,~ij 6 C(R,R)
, V(t,~i)ERxR
, Vi,j=l,2,._,q
,
Ki6]O,+~]
, Vi=l,2,...,q ,
,
and 5)
v i t ( t , x i) + P ~ x i ( t ~ x i ) ~ i ( t , x , O , O , O ) q ~i(t,Xi) ~ [~ij(t)ep(t,xj) +~ij(t)~j(t,nj)] j=l Vi:l,2,...,q , V(t,x) E R x N x .
•
,
such that
Vi=l,2,...,q , 2)
on
(i) is s t r u c t u r a l l y
and new a g g r e g a t i o n
can be also used to e s t a b l i s h
9.
6 and 7 and
VCe]0,@[
0
Assumption
Nx×Ny
a new form of a s y s t e m s c a l a r Lia-
concept
[i[~i(a)]
~i
uniform asymptotic
definite
on a h i g h e r h i e r a r c h i c a l
6 and 7. (Lj.T. Gruji~,
Yi + 8 i
function
Pi
contractive,
(IV) holds, w h i c h c o m p l e t e s
the v e c t o r L i a p u n o v by A s s u m p t i o n s
2 of A s s u m p t i o n
is p o s i t i v e
on
~xFxS s .
The p r o o f of the T h e o r e m function
v
definite
13 of the s e c t i o n 1 . 3 . 2 . 5
that
stable over
(IV) F o l l o w i n g
punov
(I),
is a s y m p t o t i c a l l y
All c o n d i t i o n s
is n e g a t i v e
due to the c o n d i t i o n
As it is showed u n d e r
and
(38)
,
RXNix
Aggregation and stability criteria
Assumption
I0.
There
exist
NiyC_R mi
, functions
1)
is d i f f e r e n t i a b l e
vi
decrescent 2)
wi
is p o s i t i v e
5)
~iEC(R×R,R) 00i
4)
a connected
vi , wi , ~i , 7ij
Niy
on
249
on
and
RXNxXNiy
on
, ~i(t,O)=O
Niy
such
of
Yi:O
,
that
and b o t h p o s i t i v e
for e v e r y
and for
6 [0,Kq+ i [ , V(t,O)i) 6 ~ x R
7ij,~ij 6C(R,R)
~ij
Niy
definite
and
XENx , Vi:l,2,...,r ,
for e v e r y
definite
neighbourhood
, Vi,j:l,2,...,r
w i fO
xEN x , Vi=l,2,...,r ,
:
, K q + i 6 ] O , + ~ [ , Vi:l,2,...,r ,
,
and 5)
T r i V i Y i (t ~ x,Yi ) ~i (t, x,Y, M, Pq+i, Sq+i) P 7~ [Tij(t)toj(t,x,yj) j=1
~i(t,x,Yi).
VTie[_ri,? i ] ,
Theorem 4. ~xpxS s
(I)
V(t,x,y,M,P,S)eRxNxxNyxMxpxSs
For structural
(xT,yT)T:0
of
(a) Assumptions
, Vi:l,2,...,r
,
•
uniform asymptotic
stability
over
of the system (I) it is sufficient that
8-10 held,
(b) there exist non-negative the functional every
+nij(t) ~j(t,~j)]
i:i,2
matrices
diagonal matrices LI
and
0i , i : 1 , 2
are positive
Qi
, such that
definite
fer
,
(c) the functions
~i
and
~i , i=1,2,...,r , and
wi
and
~i ,
i=1,2,...,r , are all time independent,
and (d) ~£ , i=l,2,..,r , are determined by (31)-(33). (If) If, in addition
to (I),
N x X N y X R m+n
unbounded for every i=1,2,...,q , and vi every x6R n and every i=l,2,...,r , then is structurally
uniformly asymptotically
,
p£
is radially
is radially unbounded fer ( x T , y T ) T = 0 of the system (I) stable in the whole over
~×P×S~ .
(Ill)
For structural
of the system
(xT, y T ) T = 0
(e) Assumptions
i=1,2 , and for
that
LI
and
Qi
(IV) If, in addition
definite
for
(31)-~3).
to (III),
unbounded for every
radially unbounded for every
are positive
01=0 , ~2=0 and
(g) ~i ' i=l,2,...,p , are determined by
are radially
stability of
it i8 also sufficient
~xP×S s
8-10 hold,
(f) the functional matrices every
uniform asymptotic
(1) over
XER n
N x X N y = R m+n , functions
i=l,2,...,q , and functions
and every
vi
i=1,2,...,r , then
vi are
250
Ch. IV
(xT, ym)m:O
of the system (1) is uniformly structurally asymptotically
stable in the whole over Proof.
Let
~×P×S s .
v be defined by (39). The condition
and 10 implies that scent on
Singularly perturbed large-scale systes~s
NxXNy
v
is differentiable,
Nx×Ny=R m+n
~ and when
8 we get
is negative
definite on
i0, which completes Example
2.
(II), and
due to (2) of Assumptions
1 (l+t+2t2)(Xll+Xl2)2
2
-12(1+t2)(Xll+X12
9 and
(Theorem 7 and Theorem 8 of 1.3.2.4).
An eight order non-linear non-stationary
=
dt
9 and 10 and Assumption
(38) for (III) and (IV). Hence,
NxXNyXMXpxS s
the proof
of two fourth order interconnected
dXt
Following the proof of
(3)-(5) of Assumptions
(37) for (I) and
•
system is composed
subsystems described by
~(Xll+2X12 ) 2
) -2(3Xll+4X12
Y2
9
then it is also radially un-
bounded in view of (II) or (IV) of the theorem. the theorem 3 and using
(1) of Assumptions
positive definite and decre-
+ ~SllXl2
2
)
i/5
= f1(t,x,y,Sl)
,
0 Xl+
- 0.i (1 + sin t) sl2x~l
•
• 2sI4YI2 3
2
2
2
8 + 0.60#ls31Y22 -~(Yll+?Y12)
ff'l~
dX2 dt
:
2
Yl : gI(X,y,M,S3)
2
--13x21 + 0.05(1.5+ cost) s 2 1 x l l - 0 . 5 ( x 2 12 + 6 x 2 22 ) x 2 1 ] +
-
s23Y22 0
dw =
I L
where
2
, 0
, -r-0.40s32y~1-2(Sy11+sx12) +.2x11]
-10×22 + O.lO s22x12 -
~2 ~
+#1x21
,
Yl
-17Y21
+ 0.10#2s41Yll
_10Y22
1 +~s42x12Y12
,
sij
2
2(7x12+x22)x22
= f2(t,x,y,S2
)
•
, s24Y21
M:diag{#l,#2}
r2e[l,l]
]
0
2
22
- (y21+6Y22)Y2
:I
= g2(x,y,M,S4)
, ~ : {M:0<#i
denotes
,
. 2 2 -2(4Y21+3Y22)Y2
sij(t)
, s13-s12
, s14 HSll
r2:1 and
so that
,
Aggregation and stability criteria
:
251
1
• S2 =
'
o
' S22 '
' S23 '
o
' S24 '
S,
, Sll , 0
• S12
, S31
, S21 • 0
,
0
, S22 , 0
, S32
, $41
•
0
, S23 ,
o]
,
0 , S24
, S42
so that
ft(t,x,y,St)
= 2
2
2
1
2
2
i/5
[-11(l+t+2t )(Xll+Xl2 ) -~(Xll+2X12 )
7
I 13 ] ~ X22
]XllJ
2Y12Y22
+ S1 • 2 2 2 2 i/5 [-12(i+t2)(Xll+X12 ) -2(3Xll+4X12 ) ]x12
-YlIY21 - O . l ( l + s i n t) x$1
-0.05 ( 1 . 5 + c o s t ) x 1 1 0 0
2 2 -13x21 - 0 . 5 ( x 2 1 + 6 x 2 2 ) x 2 1 f2(t,x,y,S2)
0.10 X12 +S 2 •
=
2
YIIY22 0
2
-iOx22 -2(Tx12+x22)X22
0 YI2Y21
s3.[o.i2y111 6o
l~ 2 _ 2 .2 ] [-8 - ~ k Y l I + Y Y I 2 ) + ~ i x 2 1 ] Y l l
gl(x,y,M,S3)
=
2 +
2
[-7 -2 (8YII 5Y12)
2
2+
~2Xll ]YI2
L- o.,o~ , ~ j
2
-17Y21 -(Y21+6Y22)Y21 g2(x,y,M,S4)
Let
Nix:R 2
Ny={y:llyII
=
2 2 -10Y22 -2(4Y21+3Y22)Y22
and
Niy:{Yi:IlyiU<1}
. All
these
sets
are
[Sx12y12
, Vi:1,2 open,
. Hence,
connected,
Nx:R 4 xieNix
and and
yi6Niy ,
.
The
f o r m of t h e
the
functions
system Pi
and
nonlinearities vi :
suggests
the
following
choice
of
252
Ch. I V
Pi(Xi) =llxill 2 Now,
Assumptions
and
Singularly
perturbed
vi(Yi) =liYill z
large-scale
Vi=1
•
)2
systems
•
8-i0 can be v e r i f i e d .
V z r i f i c a t i o n of Assumption 9. i) The f u n c t i o n scent on
ui
Nix
2) The f u n c t i o n s definite
on
~i
positive
definite
and d e c r e -
.
and
Nix
by the a n a l y s i s
~2
and
,
N2x
~[=Pl
and
, respectively.
~2=v~/2
, are p o s i t i v e
There
form is s u g g e s t e d
presented
3) The f u n c t i o n s ~i6C(R,R)
is d i f f e r e n t i a b l e ,
, Vi=l,2
fl
in the sequel u n d e r the i t e m 5. 3 f2 , ~i = ~ i + ~ / 5 and ~2 = ~2 ' obey
and
, ~i(0)=0
, and for
~i ~ 0
:
~i(ni) ~ i The f o r m of
~i
~ /0,+~[
is s u g g e s t e d
4) In this e x a m p l e
all
~ij
, V~igR.,
vi=i,2
by the a n a l y s i s
and
~ij
.
p r e s e n t e d in the i t e m 5.
are real n u m b e r s
duc to the next
analysis. 5) It n o w r e s u l t s
that
T I ( x l ) f 1 ( t , x , O , O , O ) s ~ i ( - 2 0 ~ 1 - ~ i + 7i ~2) , VxeR4 P l t ( X l ) + PlX i u2t(x1) +P~x2(X2)f2(t,X,O,O,O) K ~ 2 ( ~ i -
1 2 0 ~ 2 + ~ 1 - ~2) :
VxER4 ,
which yield -20
,
0
-1 , ~
AI=
, ,
1
BI:
,
-20
,
K
:0
-I
V e r i f i c a t i o n of Assumption 10. i) The f u n c t i o n definite 2) The
vi
functions
and
N2y
is d i f f e r e n t i a b l e
and d e c r e s c e n t ~i=v~/2
on and
, respectively,
Niy
RxNxxNiy
on
for e v e r y
~2=v~/2
for e v e r y
and both p o s i t i v e
xEN x , Vi=l,2
, are p o s i t i v e xeN x . T h e i r
.
definite
on
form results
Nly from
i t e m 5. 3) The f u n c t i o n s ~i(0):0
~i
and for
and
~2
~i ~ 0
:
- - E
,
[0,+~[
~i
The a n a l y s i s
~I : ~
V~iER '
all
7ij
and
~2=~2
Vi:l,2
, obey
~iEC(R,R)
.
the p r e c e d i n g
Wij
choice
are real n u m b e r s
of
~i
•
due to the next
analysis. 5) It is now o b t a i n e d
that
(notice that
~i=~i=1
T
VlYl(Yl)g1(x,y,M,S 3 ) K ~ I ( - I O ~ I + 2 ~ 2 - ~ I ) T
,
'
f r o m i t e m 5 suggests
4) In this e x a m p l e
and
2V2F2(F2)g2(x,y,M,S 4) ~2(2~1-i0~2-~2)
~ i.e.
, V(x,y,M,S)
rl=l ) ,
eR4XNyXMXSs,
, V(X,F,M,S) • R4×Ny×~XSs .
Aggregation and stability criteria
253
Hence,
A2
0] [0 I , B2
: ,
do not depend
on small
:
-i0
parameters
,
#I
and
#2
-i
•
V e r i f i c a t i o n o~ Assumption 8. The A s s u m p t i o n
8 is verified
as follows
:
2
II[grad vi (yi) ]T [ fi (t, x,y, Si) -fi (t,x, 0,0) ] D i=l 2 -< ~- llgradvi(Yi)llllfi(t,x,y,Si)-fi(t,x,0,0)ll i=I
-< 8 {o~2[Vl(Y1)]+c~2[v2(Y2)]}
V(t,x,y,S) eRxR4xNyXSs Hence, Pl:8 Assumptions8-10 hold. The condition (a) of T h e o r e m 4 is valid. In order to test the condition (b) of T h e o r e m 4 let 81 =20-I [2 . Then,
L1
= 2K[ 1 - (®IBI+BT01) = 10- I .
> 0
-½,
and
T ) -I(AT®I+I2+BI)T ] Q1 = - 2 - I [ A IT +At + ( ATeI+I2+B I ) (2K~I-®i BI-BI®I : [11.996 • 375 Let
O2:0
, 2.575] ,
> 0 .
15
so that L 2 : 2K~I-(O2B2+B[02)
= 212
> 0
,
and
Q2:
- 2 - i [A[ + A2 + (A[e2+I~+B 2) (2K-I-®2B2-B[02)-I (A[e2+I2+B2)T ]
:
> 0 . -1
This result e2E[0,8[ Hence,
,
shows
that
Q2-e212
is positive
definite
for every
. #I = min {i , --~--} Cmax = 1 , ~2 ~ = rain { i , ¢2max]~ = I
I2¢
which Hencej fied.
admit
#i6]0,1[
the conditions
, i=1,2 (b) and
, together
with
#2=T21~i
(d) have been verified.
, ~2E[2-I,I] They are satis-
.
254
Ch. IV
The f u n c t i o n s
~i
time i n d e p e n d e n t .
and
~i
Hence,
In view of I of T h e o r e m uniformly with
Singulerly perturbed large-scale systems
as well as
~i
the c o n d i t i o n 4,
asymptotically
(xT,yT) T = 0
stable over
and
~i
, i=1,2
(b) of T h e o r e m
, are all
4 is satisfied.
of the s y s t e m is s t r u c t u r a l l y
MxpxSs
and
~2 = ~ - i ~ I
together
r2e[2-1,1] .
Notice eight
that the s y s t e m c o n s i d e r e d order.
Its s t a b i l i t y
in the p r e c e d i n g
ness of two s e c o n d order m a t r i c e s . tual and n u m e r i c a l
example
is of the
p r o p e r t y was t e s t e d via p o s i t i v e
advantage
b e t w e e n the v e c t o r L i a p u n o v
This
illustrates
of the s i m u l t a n e o u s function
concept
definite-
the great
usage
concep-
of and link
and the s i n g u l a r
pertur-
bation approach.
I V . 3 . 3 . 2 . The q~adr~tie ~ g r e g a t i o n The f o r m of i n t e r a c t i o n s often influence son functions.
often
s h o u l d be t a k e n
the form of the a g g r e g a t i o n In order to a c h i e v e
o u s l y the n u m b e r of s t a b i l i t y sition-aggregation reduction
form
tests,
we shall p r o p o s e
analysis
connected
Ii.
For e v e r y
neighbourhoods
functions
vi
, ~i
and
N i x C R ni ~i
and/or
and every
of
compari-
simultane-
another
stability
hierarchical
i=l,2,...,q
It s h o u l d
decompo-
Such a f o r m will p e r m i t
to a single
s h o u l d be c a r r i e d out on the h i g h e s t Assumption
functions
this goal and to r e d u c e
f o r m of the systems.
of the s t a b i l i t y
into account.
O~R ni
test that
level. j=l,2,...,r
and
, and real n u m b e r s
the
N j y C R mj
Dij
t h e r e exist of
, kij
and
OER mj Aij
such that 1)
vi(t,x i
e C ( I ' I ) ( R × N i x , R+)
radially
unbounded
that the set ~e]O,Pi[ 2)
~i(t,xi):O
and
~i~(t)
is a s y m p t o t i c a l l y
i=i,2,...,q
iff Yi:0
, Vi:l,2,-.,r
4)
vit(t,xi ) + T
(t,xi)fi(t,x,0,0,0)
iXi V(t,x) E R x N x , Vi:l,2,...,q , v Ti x i ( t , X i ) [ f i ( t , x , y , p i , S i ) q
j=l
definite
and there
is
contractive
Nix
~i>O
, such
for every
, g
q ~
j=l
~ij~i(t,xi)~j(t,xj ) ,
- fi(t,x,O,O,O)]s r
Aij~i(t,xi)~j(t,Xj)+
on
,
, Vi:l,2,...,q
~i(t,Yi):0
Z
Nix=R ni
iff xi:O
3)
5)
it is p o s i t i v e
,
as soon as
~
Aij~i(t,xi)~j(t,yj)
j=l
V ( t , x , y , P , 5 ) q R x N x × N y X P X S s , Vi=l,2,...,q .
•
,
,
Aggregation and stability criteria
Assumptlon
IZ.
For every
exist c o n n e c t e d
i:1,2,...,q
neighbourhoods
06R mj , functions ~i and @ij such that i)
255
j=i,2,...,r
0@R ni
and
and real numbers
vj(t~x~yj) e C(I'I'I)(RXNxXNjy,R+)
, it is positive
(on
unbounded ~j>0
Njy
and
of
~j
Nx×Njy
• v3
, and every
~ixSR ni
as soon as
uniformly
in
X6N x
such that the set
every
~E]0,@j [
and
vj
does not d e p e n d
as soon as
Yj~(t)
there
NjySR mj
of
7ijk , ~iJk • ~ij
definite on
~jy=~mj
is a s y m p t o t i c a l l y
on
x ), r a d i a l l y
and there
is
contractive
for
j =1,2,...,r ,
T + Vjx(t,x,yj)f(t,x,y,P,S
2)
Vjt(t,x~yj)
)
5)
q q,r ~ 7jik~i(t'Xi)~k(t'Xk ) + ~ ~jik ~ i ( t , x i ) ~ k ( t , Y k ) , i,k=l i,k=l T T T V(t,X,y,P,S) 6 ~×~xX~yX~×Ss , where f=(fl,f2,...,fq) , T rjVjyj(t,x,yj)gj(t,x,y,M~pq+j,Sq+j) g
@i
q ~ Pji~i(t,xi)~j(t,yj) i=l
V(t,×,y,~I,M,P,S ) e Under A s s u m p t i o n one of w h i c h particular be positive depend on
12 the function vj
definite t
and/or
Let the elements D=(dij)
= 2
yj
[+~ij+~ji÷
r Z ~kij k=l
= 2 - I (~ij+~ji)
#i
and
A(B)
by
X
and a n o t h e r only on
yj
one
in
in the case that
7jik=0
and/or
matrices
~jik=0
B=(bij)
yj
. In a
and then it should vj
does
, C=(oij)
and
r Z (Tkij+Tkji)] k=1
, vi,j=l,2,...,q
,
' Vi=l,2,...,q , vj=l,2,...,r ,
(by A(D)
(42)
~I ) the upper a d m i s s i b l e the maximal
eigenvalues
, respectively
ilCll2 so that Pi
~I~]0,~[
. The required and
vj
may be selected. stability
bound
of the small
of the m a t r i c e s
B
and introduce
(h3) '
Evidently,
criterion
which are not decrescent
(40) (41)
~1 = 4A(B)A(D)
~i>0
not
, respectively.
, vi,j=l,2,_.,r
, we denote and
in
by
-i [~ij+~ j
parameter
•
may be the sum of two functions,
. Besides,
then all
of a g g r e g a t i o n
In order to estimate D
in x
ci j z A i j + # i j + dij
definite
.
) '
vj
may be dependent
be d e t e r m i n e d
bij
~X~xX~yX]O,~im[×~mxP×Ss
is positive
case,
r + ~ ~ji~i(t'Yi)~j(t'yj i=l
A(B)A(D) > 0
can be now stated
implies
in terms
in view of the p r o p e r t y
of
of the
256
Ch. IV
sets
@i~(t)
ii and
and
Vj~(t)
Singularly perturbed large-scale systems
expressed
in the c o n d i t i o n
i of A s s u m p t i o n s
12, r e s p e c t i v e l y .
Theorem 5.
(I) For ~tructur~l
(xT, yT) T = 0
of the system
(a) Assumptions
stability
(1) it is sufficient
M×P×S s
over
of
that
11 and 12 hold,
(b) the aggregation negative
asymptotic
matrices
B=(bij )
(40)
and
D=(dij)
(42)
are both
definite,
and (c)
~I
is determined by
Vi=l,2,.-,r
(43),
~IE]0,~[
(II) For structural ~xP×S
(xT,yT) T = 0
of
, MIE]0,#I[
and
~i=Pl~
1
.
(d) all the conditions
asymptotic
of the system under
stability in the whole over
(1) it is sufficient
(I) hold for
Nx=R n
and
that both
My=R m ,
and (e) that al~ P~oof.
pi=+=
and
(I) Let
is
function ~>0
every
in Assumptions
q
v(t,x,y) This
~j=+=
r
Z Pi(t,x i ) + Z vj(t,x,yj) . i=l j=l c(l'l'l)(RX~xXNy) and positive definite,
is in
=
such that the set
V~(t)
is a s y m p t o t i c a l l y
~e]0,~[
due to the c o n d i t i o n
the c o n d i t i o n s
4 and 5 of A s s u m p t i o n
Assumption
12 and
11 and 12, respectively.
(40)-(42)
contractive
i of A s s u m p t i o n s
il and
i1 and the c o n d i t i o n s
it results
and there for
12. Using
2 and 3 of
that
v ( t , x , y , M , P , S ) ~ A(B)llwi(t,x)uZ+llCIlUwi(t,x)Ullw2(t,y)II + ~ii A(D)llW2(t,y)l t
[llwi(t'x)ll'llw2(t'Y)ll]
=
I?
i
U CII
A(D)] [llw2(t,y)ll
V~ic]O,~i[ , ~ i 6 ] O , ~ i [ , V(t,x,y,M,P,S) e RXNxXNyXMmXPXSs , where ditions
WI=(~I,~2,...,~q)T 2 and
(II) When every
~6]0,+~[
structural ~×F×S s
Wp=(~i,~2,...,~r )T . This result,
5 of A s s u m p t i o n
13 of the section is true.
ally u n b o u n d e d
and
1.3.2.5
Nx=R n
on
and
and the set
11 complete
(Corollary
Ny=R m Vf(t)
stability
stability
then the f u n c t i o n is a s y m p t o t i c a l l y
of
(xT,yT) T = 0 1.3.2.5
property
v
under
contractive under
is s a t i s f i e d
(I)
is also radifor
(I) prove
of the system
is r e q u i r e d
the con-
of T h e o r e m
the statement
with the c o n d i t i o n s
2 of the section
In the case a u n i f o r m
all r e q u i r e m e n t s
~xpxS s . Hence,
, which together
asymptotic
(44)
over
(i) over ~×P×S s ).,
then the condi-
Aggregation and stability criteria
tions
for a s y m p t o t i c
should be r e p l a c e d
257
contraction
by a n o t h e r
of the sets
one.
rem 6 for w h i c h the f o l l o w i n g
This
~i~(t)
and
is p r e c i s e l y
assumptions
Vj~(t)
explained
Assumpti0n 13. For every i=i,2,...,q and every j=l,2,...,p connected n e i g h b o u r h o o d s Nix~R ni of 0eR ni and Njy~RmJ functions
~,
~i
and
~j
in Theo-
are needed.
, and real numbers
~ij • kij
there exist of 0eRmJ ,
and
Aij
such
that
i)
Pi(t~Xi) 6 C(I'I)(R×Nix,R+) on
Nix
2)
~i
is positive
3)
~j
is p o s i t i v e
4)
the conditions
Assumption connected
14.
, Vi=l,2,...,q ,
definite
on
Njy
, Vj=1,2,...,r ,
4 and 5 of A s s u m p t i o n i:l,2,...,q
and
Nix~R ni ~j
on
NxXNjy
(on
the c o n d i t i o n s
(I) For structural
~xP×S s
(xT,yT) T = 0
of
• j=l,2,...,r
and
NjyCRmJ
, it is p o s i t i v e in
of
0E~ mj and
definite
vj
xeN x
j
Pij
and de-
does not depend as soon as
12 are valid.
uniform asymptotic
of the system
there exist
~ijk ~ ~ijk , ~ij
only as soon as uniformly
, vi=l,2,...,q ,
•
stability over
(1) it is sufficient
that
13 and 14 hold,
(b) the aggregation negative
0eR ni
2 and 5 of A s s u m p t i o n
Theorem 6.
(a) Assumptions
and every
of
Njy
on x ) and r a d i a l l y u n b o u n d e d Njy=RmJ ~ vj=l,2,...,r , 2)
iI hold.
and real numbers
and d e c r e s c e n t
~ix=~ ni
Nix
vj(t,x,yj) 6 C(I'I'I)(R×Nx×Njy,R+) crescent
definite
as soon as
on
For every
~i , vj
• it is p o s i t i v e
unbounded
definite
neighbourhoods
functions such that I)
, and r a d i a l l y
matrices
B=(bij)
(40)
and
D=(dij)
(42)
are both
definite,
and (c)
~I is determined by (40)-(43) ~i~]0,~i [ , ~IE]0,~I [ Vi=l,2,...,r . (II) For structural
whole over
MxpxS s
of
that all the conditions P~OOf.
This
function
crescent
under
(I) Let V(t~x~y) is in
=
(44) holds.
(I) hold for
C(I'I'I)(RXNx×Ny~R+)
4 of A s s u m p t i o n The conditions
stability
of the system
~1=~i~i
in the
(1) it is sufficient
Nx=R n
and
Ny=R m
q r Z ~i(t,Xi) + ~ vj(t~x,yj) i=i j=l
due to the c o n d i t i o n
conditions
uniform asymptotic
(xT,£T) T = 0
and
, positive
i of A s s u m p t i o n s
definite
and de-
15 and 14. Using the
15 and 2 of A s s u m p t i o n 2 and 5 of A s s u m p t i o n
14 i% follows
that
15, the p r o p e r t i e s
,
258
of
Ch. IV
v
and
(44) prove
(xT, yT) T = 0 Theorem
of the system
7 of the section (II) When
under
uniform
(i) over
1.3.2.4
Nx=R n
and
over
then structurally .
MxpxS s
stability
because
of T h e o r e m
of
MxpxS s .
to the conditions
8 of the section
(XT, yT) T = 0
asymptotically
of
all conditions
over
in addition
Ny=R m
M×PxS s . Hence, uniformly
asymptotic
are satisfied
(I) then all the conditions
are satisfied MxPxS s
structural
Singularly p e r t u r b e d large-scale systems
1.3.2.4
of the system
stable
in the whole
(i) is over
•
I V . 3 . 3 . 2 . A p p l i c a t i o n to t h e s t r u c t u r a l a b s o l u t e ~ t a b i l i t y The p r e c e d i n g of the Lur'e
result type
can be applied
singularly
i + ~P SijAijYj 9=I
Xi = Aixi + qil~i1(Oil) q
to the absolute
perturbed
large-scale
analysis
stability
analysis
system
-T Xi+612y , ail = Cil
Vi=1,2,...,q
j
1
~i9 i = ai2
Z ~iSq+i,j Bijx j + BiY i + qi2~i2(oi2 ) + qi3~i3(oi3 ) , Vi=i,2,...,r j=l -T aT q aT 2 P - T $3 = Migi3Xi + Ci4Y i , oi3 = Z ~icj5Sq+i,j Xj + ~ Cj6 q+i,j 9j , j=l j=l
,
Vi=i,2,...,r
,
@ij(uiJ) e [ 0 ~ K i j ] c R + °ij
{i=l,2,...~q
when
j=l
i=l,2,...,r
when
j=2,5
,
, 1
2
3
The structural matrices Sij , Sij , Sij are defined in the section IV.3.2.2. It is assumed
that the numbers
fi
oij ~0
.
(45)
and
S , as well
the set
Ti
are given.
The functions
and
Ss ,
Pi ,~i ,vj and ~j and numbers ~ij , ~ i j , Aij ,7ijk , ~ i j k , ~i9 and #ij are to be d i s c o v e r e d so that Assumptions 13 and 14 are satisfied. A solution and vj
to be presented
will be relied
on a special
form of
vi
V e r i f i c a t i o n of Assumption 13. It is supposed that A i is a stable matrix, the pair controllable, and that there exist numbers 8ie[0,+~[ such that - (Ai-J ~ ) - i Ki~ + Re (i+j ~i~) C~l
V~e[0,+~]
qi i - C i l q iTl (A~+j~ii)-i
(Ai_J~li)-i qil > 0
.
o
Then, Pi(Xi) is a Liapunov
function
f°il = xlHixi + e i L ~il(°>d°
of
xi = Aixi + qil~il(°il ) , °il = CilXi
(Ai,qi I) is and eile]0,+~[
,
Aggregation and stability criteria
provided
that
~i
259
is the matrix
A~Hi + HiAi
+ gigl
solution
= - eilli
of the Lur'e
,
hi+Hiqi
equations
(46),
: -~igi
(46)
for
7i : K i- l - @ i C i-m lqi In order
to determine
Assumption
15 which
tion 11. Since
the n u m b e r s requires
~ij
test
~i(Xi)=llxill
and
, we test
first
of the conditions
is independent
Vi
vTXi(xi)fi(t,X,0,0,0 Hence,
hi : 2-i (@i AICil + Cil )
'
of
t
then
condition
4 of
4 and 5 of Assumpand
Pit-0
) -<-6ilIIxi ~2 , VxieR ni ~ vi,j=l,2,...,q
~ij : - @ i l ~ i j
.
The numbers ~ij and Aij are determined from condition tion ii, T v i X i ( x i ) [ f i ( t , X , Y , P i , S i) - fi(t,x,0,0,0)] _<
5 of Assump-
- -T ^Tl+6i T < 211xill llHi+2~--@iMilCilCiln[2-1KilA(qilci lqil)Uxill + r -T + X (~illIqilci2II+l[Aijll)llyjil] j=1 Vi:l,2,.,,q The n u m b e r s
kij
e RnxRm×p×Ss
, V(x,y,P,S)
and
Aij
,
are now determined
-T -T Xij :MilllHi+2-1@iKilCilCilllA(qiiCil
+Cil
T qil)~ij
Ai j : 211Hi+2-1@i~il- CilCil ~T ~(~iiiIq i ici21J+lIAijII) ~T
Vzrifiaation of Assumption Let
vi(Yi):llYi[l 2
7ijk:0
and
Assumption
~ijk:0
by , Vi,j:l,2,.-,q , Vi:l,2,...,q
, Vj:i,2,...,r
;4.
be chosen.
Then,
. Further,
14 it is necessary
vit~0
in order to test
and
vix~0
to verify
so that
It now results Pij
that
, v(x,y,M,p,S) ~i(Yi)={}yi~
: 2(llBijli +~i211qi2cl311~ij
~ij : 2{[2T-iA(Bi+BI) vi,j =i, 2,...,r .
e~nxRmx~xPxSs
-
all
the condition
the condition
2 of
5 of A s s u m p t i o n
rj Vjyj (t°x,y 9 )gj (t,X,y,M,pq+ i ,Sq+i) _< q -T ~T < ~i ~ 2([18jili +~j211qj2cj3116ij +Kj311qj3ci5N)llxill'nyjn i=l r -T -T + Z 2{[2-1_rjA(Bj+B3) + 2 i g j 2 f j A ( q j 2 C j 4 + c j 4 q j 2 ) ] 6 i j + i=l +gj3rjllqj3£16 u } {lyin-llyjll , V~:l,2,...,r
,
12,
+
.
and
~T +Ki311qi3cj5U) ^T
-
T
+Ki2TiA(qi2Ci4+Ci4qi2)]
, Vi=l,2,...,r -
, vj=I,2 ,...,q , T
~ij + Ki3Tillqi3cj6U}
,
.
260
ch. IV
The elements and
D
bij ~ ci9
and
dij
Singularly perturbed large-scale ~ystems
of the a g g r e g a t i o n matrices
have been obtained via (40)-(42) bij = -(eil-kii)Sij oij = Aij+#ij
B
is diagonal.
of T h e o r e m 6 is that all
, Vi,j=l,2,...,q
,
, Vi=l,2,...,q , Vj=1,2~...~r
dij = 2~l(~ij+~ji) The matrix
B , C
in the form
~ Vi,j=l,2~...,r .
The necessary condition for the fulfilment bii<0
, i.e.
eil > ~ii • Vi=l,2,...,q . If it is satisfied,
then negative definiteness of
tural absolute stability of M6~ , #i@]0,~i[ every
, ~16]0,~i[
(xT, yT) T = 0 , where
~i
D
implies the struc-
of the system
(45) for every
is defined by (45), and for
S6S s . This result illustrates the essential r e d u c t i o n of the
stability analysis. We should verify only simple inequalities
(47) and
negative definiteness of the symmetric matrix
r.
However,
D
of the order
the consequence of such a strong order p r o b l e m r e d u c t i o n can
be a smaller estimate
~I
of the upper allowable bound of
~i
than
that obtained via the multilevel hierarchical analysis.
IV.3.3.3.
ConaZ~sion
U n i f o r m time scaling requires that all time scales be mutually related in certain bounds.
In the case this requirement
is satisfied the sta-
bility test is reduced to v e r i f i c a t i o n of simple algebraic conditions imposed on the system on the highest
structural level only. The form
of the conditions depend on the a g g r e g a t i o n form, which was shown via Lur'e-like and quadratic-like a g g r e g a t i o n forms with conceptual applications to the absolute stability analysis of large-scale singularly perturbed Lur'e systems.
IV.4. COMMENTS The d e c o m p o s i t i o n - a g g r e g a t i o n approach to the stability analysis appears attractive when it is based on the vector Liapunov function.
It
can be more effective for singularly perturbed systems than for those which are not singularly perturbed.
A greater order p r o b l e m reduction
can be achieved for the former. The main problems
for effective a p p l i c a t i o n of the method to larger
class of the systems are the following
:
Co~nent
261
- the form of aggregation
functions
tially to few known applicable
vi
and
forms.
mental stability problem-construction - If
vi
and
from the
vj
i-th
j-th
tive of the resulting
v
bersome.
vi
subsystem,
However,
if
and
and possibly
quence of it, it appears ness of the derivative - Vector Liapunov
on
respectively,
depend only on
at best as neutral
difficult
majorizations,
v
of the whole
yj
cum,
moreover,
on disconnected
v
As a consedefinite-
system.
so far has led to various,
of interactions
or destabilizing
and
often to rough majoriza-
over subspaces
are treated
over the same subspaces.
The ben-
and its drawback is too strin-
subsystems.
The approach to singularly perturbed
systems
and in this Chapter is directly applicable nuous but non-differentiable
and
to assure the negative
efit of this is simple stability test, gent requirement
×i
by the form of interactions.
of the function
Stabilizing actions
system appears
t , then the form of the resulting
function aggregation
sometimes numerous,
different
then their deriva-
the sign test of the deriva-
for the whole
vj
essen-
of a system Liapunov function.
and, hence,
function
function is not influenced
tions.
is restricted
depend on states of the other subsystems and
tives become very complex,
respectively,
vi
This is related to the funda-
developed
in Chapter I
to such systems with conti-
nonlinearities
(Gruji~,
Fiedler, M., and V. Ptak (1962), On matrices diagonal elements and positive principal 12, No.87, 382-400.
1982a,b).
with non-positive off minors. Czheah. Nat. J.,
Gruji6, Lj.T. (1976a), Vector Liapunov functions and singularly perturbed large-scale systems. Proc. 1976 JACC, Lafayette, 408-416. Gruji6, Lj.T. (1976b), General stability analysis of large-scale systems. IFAC Symp. on Larch-scale systems ~heory and appliaations, Udine, 203-213. Qruji~, Lj.T. (1977), Converse lemma and singularly perturbed scale systems. 1977 JACC, 1107-1112.
large-
Gruji~, Lj.T. (1978), Singular perturbations, uniform asymptotic stability and large-scale systems . Proc. 1978 JACC, ISA, New York, 339-347. GrujiE, Lj.T. (1979a), Singular perturbations, large-scale systems and asymptotic stability of invariant sets. Int. J. SyGtems Sci., 10, No.12, 1323-1341. Grujie, Lj.T. (1979b), Singular perturbations Int. J. Control, 29, No.I, 159-169.
and large-scale
Grujie, Lj.T. (1979c), Sets and singularly perturbed Science, 5, No.4, 327-338.
systems.
systems.
Systems
Grujie, Lj.T. (1981), Uniform asymptotic stability of non-linear singularly perturbed general and large-scale systems. Int. J. Control, 33, No.3, 481-504.
262
Ch. IV
Singularly perturbed large-scale systems
Gruji6, Lj.T. (1982a), Asymptotic stability conditions for singularly perturbed systems with non-dlfferentiable non-linearities. Automatika, Zagreb, 23, No ~-4, 8~-84. Gruji6, Lj.T. (1982b), On asymptotic stability of large-scale singularly perturbed systems with non-differentiable nonlinearities. Automatika, Zagreb, 23, No. 3-4, 85-86; Proc. of IXth Int. Conf. on Non-Zinear Oscillations, 2~ Naukova Dumka, Kiev (1984), 95-97.
CHAPTER V
LARGE-SCALEPOWERSYSTEMSSTABILITY
NOTATION
N.B. B e s i d e s t h e 9 e n e r a l n o t a t i o n used t h r o u g h o u t t h e book, t h e f o l l a w ing symbols are s p e c i f i c to t h i s Chapter. number of system's generators
N n
,
n=N-1
(or machines)
contrary to the previous chapters
n does not
denote here the dimension of system's state Aij : EjEj Yij D.i
mechanical damping coefficient of
i-th
gen-
erator Dij
electromagnetic damping coefficient between i-th
Ei
and
j-th
modulus of motive force
i-th
generators generator's internal electro-
(voltage)
k i : Mi 1 i-th
generator
Mi
inertia coefficient of
Pmi
mechanical power delivered to the
i-th
gen-
i-th
gen-
erator from its turbine Pei
electrical power delivered by the erator to the network
Pmi
variation of the mechanical power of generator
pO. ml o
Pmi
steady state value of
P m±.
steady state value of
Pmi
i-th
264
Ch. V
Large-scale
power
systems
stability
o
Pi : Pmi -Pmi PiN = Pi -PN admittance matrix of the network reduced at
Y
the internal generator nodes modulus of the
Yij
Y;
Yij :Yji
gain of the first order proportional
regulator
of
i-th
ij-th
element of
generator
~ij :AijM~"I n
F i : h i +XNi + Z kij j:l j/l rotor angle of
1
i-th
generator relative
to a
reference ~.. : ~ . - ~ . i3
l
]
5°
the equilibrium under consideration
i
of
i-th
generator o
8iN
value of
~iN
0ij
argument
of the
h i : DiMi I
if
h i :X
at the equilibrium ij-th
, constant
element
for
of
state ¥;
0ij:$ji
i : 1,2,...,N , one
speaks of "uniform"
(mechanical)
damping
ij : D~j M~ ~ Aij : Aij-}'Nj ANi : ~N- Xi+ XiN -i
Vi = 1,2,...,n
time constant of the first order proportional speed regulator of
i-th
generator
O
aiN : 6iN - SiN
aij
ajN
: ai~-
rotor speed of chronous
speed
the value of
l
i-th : ~i
generator above the syn-
~i : ~i at the steady state opera-
tion called "equilibrium nij : n i- nj ~i = ~ i - ~
l
state"
Introduction
265
coij : coi-Coj
5,0,~
without subscript
(~,n)
these letters denote vectors
state vector with variables
6i ,~i
,
i = 1,2,...,n
(6s,0) (6ui, 0) CCT
or
value of the state vector at the SEP value of the state vector at the
i-th
UEP
Critical Clearing Time
tc
COA
Center Of Angles
SEP
Stable E q u i l i b r i u m Point of the system in its post-fault
configuration
UEP
Unstable E q u i l i b r i u m Point surrounding SEP
SDE
Stability Domain Estimate
PSDE
Practical SDE
V.1.
INTRODUCTION
Modern electric power systems are a typical example of complex largescale systems. An electric power system is composed of several generation stations ("generators") which convert electricity,
fuel, water or other types of energy into
substations that distribute power to consumers
and transmission lines that tie them together.
("loads"),
The power system is re-
quested to operate without interruption and under random energy supply requirements,
despite electric power cannot be stored.
The nowadays continuously increasing complexity and interdependence of power systems make their stability assessment increasingly important and difficult. The difficulty increases even more with the economic requirement to operate them closer and closer to their capacity limits; at the same time, guaranteeing their stability becomes vital. Note that the concept of "stability" used here receives many interpretations. One of them corresponds to the so-called
"transient stability",
con-
sidered in this Chapter. From a system theory point ofview, and for the purpose of their transient stability analysis, highiy nonlinear
electric systems belong to the class of large-scale,
systems operating under random perturbations and
randomly varying parameters. Their transient
stability assessment
266
Ch. V
Large-scale power systems stability
appears to be an extremely challenging problem, tional efficiency is of paramount importance.
inasmu0h as computa-
Indeed, their proper
m o n i t o r i n g imposes stringent computing requirements with respect to core memory and execution time. Direct methods suitably adapted show particularly appealing and promising.
This quite naturally explains
the intensive research effort which has been devoted to this area for almost twenty-five years. The present Chapter attempts to give an overview of how direct methods solve the above important practical problem, obtained results~
to bring out essential
to pinpoint the difficulties
still existing,- and to
discuss some current and future research directions. Two different methodologies are considered, namely the "scalar" and Belman's
"vector" Liapunov approaches.
ing attracted researchers' where in the sixties,
attention.
The former is the first to havIts beginning,
situated some-
is marked by the construction of (scalar) Liapu-
nov functions for power system dynamics.
Several methods thus start
being explored,
(Zubov,
rion
(Popov,
such as
: Zubov's method
1962), Moore and Anderson's theorem
1961), Popov's crite(Moore and Anderson,
1968), or simply calculation of a first integral of the equations of motion. Quite early, however,
it appears that, whatever the method,
constructing practical Liapunov functions imposes stringent simplifying assumptions for the system modelling.
It also appears that in order
to make direct methods practically attractive it is necessary to define "suitable"
stability domain estimates,
i.e. estimates combining accu-
racy and computational efficiency. Thus, by the m i d - s e v e n t i e s research efforts concentrate on the definition of Stability Domain Estimates, "theoretical" one.
more workable and reliable than the
Indeed, the latter imposes considerable computing
burden, besides providing quite conservative practical results, have moreover an uncertain degree of conservativeness.
which
Section V.3
examines this question and shows that it has by now received satisfactory solutions. This is a crucial step towards the effective application of direct methods to power system stability analysis.
Relaxing the
direct methods as much as possible from simplifying assumptions is another important issue discussed in Section V.3. Promising approaches recently proposed are indicated. Also by the mid-seventies, vestigated
another research direction starts being in-
: Belman's approach and the derived "vector" Liapunov
func-
tions. This methodology aims essentially at tackling the problem of "how to relax the construction of Liapunov functions from severe simpli-
Physical description - Modelling
267
lying assumptions".
As will be shown
proach has actually
reached
unacceptably
this objective
high conservativeness;
sults and improvements
in Section V.4,
achieved,
the
"vector"
but at the expense
and despite
this drawback
ap-
of an
nice theoretical
re-
has not as yet been
overcome.
N.B. This C h a p t e r i s
not a s u r v e y .
It
does not i n t e n d
to d e s c r i b e
all
c o n t r i b u t i o n s i n t h e area and t k e r e f e r e n c e s g i v e n a t i ~ s end do n o t p r e t e n d do be c o m p l e t e , nor do t h e y r e f l e c t t h e abondenc~ of p u b l i c a t i o n s d e v o t e d t o t h e s u b j e c t . For a more c o m p r e h e n s i v e a c c o u n t , t h e r e a d e r may r e f e r t o e x i s t i n g s u r v e y p u b e i c a t i o n s ( R i b b c n s - P a v e l l a , 1971b; Fouad, 1975; Evans, 1978; Pai, 1981; R i b b e n s - P a v e l l a and Evans, 1985a; V a r a i y a e t a t . , 1985).
V.2. THE PHYSICAL PROBLEM AND ITS MATHEMATICALMODELLING N.B. The thrust Evans
of this Section
is taken
from R i b b e n s - P a v e l l a
and
(1985a).
V.2.1. Problem d e f i n i t i o n
This paragraph aims a t g i v i n g t h e r e a d e r who i s
not f a m i l i a r ~ i t h
power
s y s t e m dynamics t h e f u n d a m e n t a l d e s c r i p t i o n of t h e p h y s i c a l s y s t e m and of t h e c o n v e n t i o n a l t r a n s i e n t s t a b i l i t y a n a l y s i s . The power s y s t e m e n g i n e e r may go d i r e c t l y hereafter nationale
to paragraph
V.2.2.
The d e f i n i t i o n s
of IEEE Committee R e p o r t on Terms and D e f i n i t i o n s For the purposes
of its transient
system may be regarded chronous
machines)
lines
(i.e.
Fig.l
gives
generators "behind",
stability
a schematic
representation
are c h a r a c t e r i z e d i.e.
in series
force of say the
where
6i
is its "rotor
rotating with the
i-th
All generators
characterized
by its
including
by their
with, i-th
of an
through
N-machine
angle",
i.e.
the angle
and an axis
are i n t e r c o n n e c t e d Y matrix,
i.e.
reactance;
is r e p r e s e n t e d formed
rotating
through
by
equipment. The
force
the electroE i = El/6 i
by an axis at synchronous
the network,
its admittance
(syn-
system.
electromotive
their transient
generator
units
power
transmission
the a s s o c i a t e d
internal
generator
an electric
generating
i n t e r c o n ne c t e d
lines or transformers),
(1982).
analysis,
as a set of electrical
and of loads
motive
speed.
given
comply w i t h t h e r e c o m m a n d a t i o n s of CIGRE ( " C o n f e r e n c e I n ~ # r des Grands R~seaux E l e c t r i q u e s " , B a r b i e r e t a l . , 1978), and
matrix
which
reduced
is at
268
Ch. V
Large-scale power systems stability
a o r o Note
that
generator I ~
Y accounts transient
in a d d i t i o n
l 01 02 ~eli o o ~ 1
for the
reactances,
to the line and
transformer m o d e l l i n g and to the loads, m o d e l l e d here as
k:Y
constant
admittances.
Figure 1 For i l l u s t r a t i o n ing nine
purposes
generating
its electric
units.
components
responding
network
Definition
i.
we have
chosen
Fig. 2a
a real power
describes
and their values;
equivalent
this
Fig. 2b
system
compris-
system along with represents
its cor-
representation.
A power system is said to be in a synchronous operation
if all its i n t e r c o n n e c t e d synchronous machines are in synchronous operation with the network and with each other. All the
system m a c h i n e s
of its rotor angular synchronous ical
input
speed.
During
2.
Definition
by each generator
the electrical
equilibrium
and the number
output
speed
(product
of its pole pairs),
such a "steady-state"
power received
its losses, This
run then at the same electrical
velocity
operation~
equals,
power d e l i v e r e d
is upset when a large and sudden
the
the mechan-
when n e g l e c t i n g
by it to the network. disturbance
occurs.
A ~arge disturbance is a disturbance for which the equa-
tions that describe the dynamics of the power system cannot be linearized for the purpose of analysis. Such
disturbances
as short
are
circuits,
erator busbars,
: loss of loads,
particularly
and even more
severe
falling out of synchronism)
the former
large
the mechanical
input power
power.
The generator
other;
the motion
the Newton's equation"
of the
of,
rotors say,
law equation,
(see Notation)
Y matrix.
the
then
i-th
referred
changes
such
at the gen-
loss of generators
of large d i s t u r b a n c e s
It causes
to each generator start
circuits
(or
which may be caused by one of
disturbances 2 . Occurrence
changes
significant
short
than that,
generators
significant
network
three-phase
an imbalance
and its electrical
"swinging"
generator in the
with respect
is then
literature
output to each
described as the
imply
between
by
"swing
:
l These are fictitious nodes taken behind the generator transient reactances. 2 In the sequel the term "large" will be sometimes omitted.
Physical
description
A
'!
~
- Modelling
B
. ~,o ' L ~ ~
"J.7.5.jgLI__.'~ "
59.'~
~
IV
..Lj413
~
l
~
°
269
I- - ' v ' ' - "
~
,,=.A~
I .....
j25
--
I--""'-~o
"
75
I ~
.J
. ~
":~'~%
..L
~1
,oo
,,o,
,,_o,,~.,,o
/
Figure 2.a
-~i-7R~8-i~7 I~.074SJ'~O02 ~ 6
-j5
d
e
--Eq ~
u X~a
~N j0.0206
0,,688-)Ct349
-q--
,.
o
c
Figure 2.b
o
"-6""
-E"
"7-
-3-
-i-"
6o,m
270
Ch. V
Large-scale power systems stability
N
Mi~ i +Di~ i + E D i j ( ~ i - ~ j) = Pmi - P e i (8)
(2.1)
j=1
Roughly speaking,
the transient stability problem is the study of this
dynamic system's capacity to recover,
after a disturbance,
a synchron-
ous steady-state equilibrium condition. More precisely we give the following
Definition 3.
A power system is transientty stable for a particular
(pre-fault) steady-state operatCng condition and for a particular distur~ance if. following
that disturbance,
it reaches an acceptable
steady-state operatin~ condition. This definition suggests that the transient
stability analysis implies
consideration of three different operating phases fault
(before occurrence of a disturbance),
(Figs.3)
the fault-on
: the pre-
(during dis-
turbance action; one speaks of "faulted system"), and the post-fault one. Note also that the post-fault
steady-state
(synahronous) operat-
ing conditions that the system may reach long after clearing the disturbance do not necessarily ones
(and generally)
coincide with the pro-fault
(see Fig. 3c
~o|
, ~ 3~:.-~-
Figure 3.a
Pre-fautt
J .Fautt-on I.
Post- fauH
.....
Figure 3. b
.
Pre-fault .
.
-ii
to~-'lte Fault-on I .
0 i
.
Post-fault
.
.
.
t
.
,
,8~
0;J
--
,, Figure 3.c
.d
/~: I 'o
_ ;'~
P,e-,ou. 1F~_~u.-onJ
~
-
~
,
Post-foott
,.... i
L
Physical
description
- Modelling
271
V,2.2. Conventional problem formulation Transient
stability
well as in
analysis
everyday
extremely
complicated
equations
of m o t i o n
ber o f p a r a m e t e r s usual
is very
important
system operation problem,
would
which
in e n g i n e e r i n g
for the r i g o r o u s
necessitate
vary
in p l a n n i n g
and monitoring.
in t i m e
problems,
taking
solution
into
studiQs
It is also
of the a b o v e
account
a large
and are i n t e r d e p e n d e n t .
one t h e n
is led to m a k e
as
an
num-
As is
simplifying
assumptions. Whatever
the a p p r o x i m a t i o n s ,
b l e m is f o r m u l a t e d More p r e c i s e l y , its t r a n s i e n t simulations
in t e r m s m a y be
- what
: given
critical
?
an8
position
transient
stability
system
adisturbance
initially
a p p e a r s at t o :
system
after
the dis-
if yes,
clearing
time
may remain without
b a c k to a s t e a d y - s t a t e
the c a p a b i l i t y
from
disturbance
of the d i s t u r b a n c e
a power
for the
pro-
t e ).
robustness
through
evaluation
conventional
as f o l l o w s
equilibrium
is c l e a r e d
is the
CCT i . T h i s
and
stability
(CCT or
of a s y s t e m ' s
o p e r a t i o n and a s s u m i n g t h a t
a stable
disturbance come
of its
Time
is p e r f o r m e d
one at a time)
formulated
in a s t e a d y - s t a t e
of v i e w
transient
Clearing
exploration
point
(generally
severity
turbance
of C r i t i c a l
the s t a n d a r d
stability
analysis
- is t h e r e
the c o n v e n t i o n a l
in t e r m s
the
?
i.e.
system
operation;
the m a x i m u m losing
and
this
time
time must
of s y s t e m p r o t e c t i o n s and c i r c u i t
that
its c a p a b i l i t y
breakers
the to
lie w i t h i n to c l e a r
the fault So, w h a t e v e r has two - S£ep
the m e t h o d ,
steps A which
studies
to the c l e a r i n g - Step
B which
(final)
• if the
from
system
critical
transient
stability
analysis
system
the e v o l u t i o n te
onwards
is f o u n d
clearing
time
is f o u n d
by n u m e r i c a l l y
o f the
(faulted)
of the
system
system
from
t
o
te ;
Both of the above two s t e p s method
conventional
the e v o l u t i o n
time
studies
phase,
• if the
the
(figs. 3) :
are
in its p o s t - f a u l t
:
to be u n s t a b l e ,
te
is g r e a t e r
than
the
te , to be s t a b l e , investigated
integrating
t e
the s y s t e m
s~ep-by-step
s w i n g equations of type (2.1): 2
Admittedly this conventional "measure" of a system's robustness from its transient stability point of view is arguable for practical applications. It shows, however, extremely useful for the purpose of developing appropriate new theoretical methodologies; in a subsequent stage of the investigations, the new methods developed may be adapted Zo other, more appropriate stability "measures". 2 Indeed, eq.(2.1) describes the generator motion for both the fault-on and postfault periods, but with different parameters.
272
Ch. V
upon to
fixing to
rotor tion
a clearing
(step A), angles
values
of of
able
Remark i. system fying
whether
kind
of " s t a b i l i z a t i o n "
upon
the
3.i b e l o w
and
system
angles
Curves
example,
with
"uniform"
to p o s s e s s
(see the
line
of dampings are n e g l e c t e d ,
V.2.3.
rotor
motion
our a t t e n t i o n
the n o t i o n s
"domain"
Referring nected
or all
3b);
expect
(see the
However,
set,
and
will
Santalo
to four d i f f e r e n t described
if,
in S e c t i o n
expect
moreover,
the simple
line
simpli-
curves
from rotor
all
kinds
stability
in Fig.
3b).
that,
of the o r i g i n
in this
sense. the
The
domain
system
on the
difference
between
as the
(1961)
a "region"
defined
"domain"
union
is an open as an open
of a d o m a i n w i t h
some,
conand none,
points. for s t a b i l i t y
a neighbourhood
is that
be f o c u s e d
(1976)
"region"
to e m p h a s i z e
only
closed)
(either
open
is of interest
reason
for u s i n g
of a s y m p t o t i c
can be closed
analysis
herein.
or n e i t h e r
Hahn
(1967)
a neighbourhood
stability
(Gruji~,
of n o n - l i n e a r
or closed
that
used
sys-
o p e n nor "domain"
can be
of an e q u i l i b r i u m
closed
of a non-
1976).
X E ~ M , f : R M ~ R M and : £(X)
Following
Definition state
of a power
the r e l a t e d
can only
but
dotted
step-by-
"region".
and L e f s c h e t z
of its b o u n d a r y
We want
Let
and
to La S a l l e
set.
connected
linear
cannot
stabiliza-
of its relative
stability
in Fig.
swing being
process.
require
]c r e f e r
one
of the before
standard
modelling
the
Determina-
of a s s u m e d
towards
with
to
D e f i n i t i o n s of s t a b i l i t y domains and t h e i r estimates
For a while
tems,
curves one
should
damping,
the a s y m p t o t i c
solid
of the relative
3b and
simplified
mechanical
time
by the
along
from
describing
trials
and t e d i o u s
one
modelling
of Figs. the
a long
a tendency
a long
The
first
t e , computation
shows
to be quite
systems stability
(step B).
3 or more)
computation
depends
For
onwards
for quite
system
system
the p r o g r a m
te
(generally
the CCT
power
"swing curves",
the
from
for a g i v e n
the
appears
assumptions.
cases.
time
be p u r s u e d
Obviously,
procedure
computes
with
t e . Moreover,
to decide
step
one
t e , one runs
several
step B must
t i o n or not.
the
then
evolution
of CCT r e q u i r e s
curves
time
Large-scale
x:O
Oruji6 4.
(1975,1976),
A set
defined
Ds
we accept
, Ps ~ R M
(2.2) :
, is the domain
by o
Ps : U[Ps(~): eeR+]
,
of the equilibrium
Physical description - Modelling
273
where
Ps(C)
is such a n e i g h b o u r h o o d of
Vt e R ~
, holds provided only that
x:0
that
IIX(t;O,Xo)ll <e o
Xo e D s ( e )
The next definition has been commonly used 1967; La Salle and Lefschetz,
for every
(Krasovskii,
e eR+ . i963; Hahn,
1961).
Definition 5. A set Pa , ~ E RM , is the domain of attraction of the equilibrium state X = 0 of the system (2.1) if and only if it is such a neighbourhood of
×=0
that
lim [lIX(t;0,Xo)H :t ÷ + ~ ] holds provided only that
: 0
XoeD a . •
It is now natural to accept the definition of the domain of asymptotic stability of
x:0
Definition 6. X:O
as proposed by Gruji6
A set
of the system
D , D cMR_
(1975,1976,1981)
, is the domain
:
of asymptotic stability of
(2.2) iff it is both a n e i g h b o u r h o o d of
x:0
and
the intersection of its domain of stability and domain of attraction, that is, that
D :Dsn ~
is a n e i g h b o u r h o o d of
x:0
. •
The exact determination of the domain of asymptotic stability has great engineering and theoretical importance.
Unfortunately,
we can
realize it only in special cases. For these reasons we investigate its estimate Pavella
E defined as follows by referring to Gruji@ and Ribbens(1977,1978),
Definition 7.
A set
Grujid et al.
(1979), Grujid
E , ECR M , is an estimate
mate) of the asymptotic stability domain
P of
(1981)
:
set (in brief, estix=0
of the system
(2.2) if and only if (i)
(ii)
E
is a neighbourhood of
x=0
,
~$
and (iii)
E
is positively invariant set of the system
that
XoeE
implies
X(t;O,x o) • E
for every
(2.2), that is, teR+
.
V.2.4. Liapunov's method applied to conventional transient s t a b i l i t y analysis The principle consists in replacing the computations implied by step B defined in Section V.2.2 and their repetitions
by a time domain test.
This in turn consists in comparing an initial Liapunov function value (Liapunov function value at the initial state) with its critical value (its value on the boundary of the stability domain).
It thus requires
274
Ch. V
construction domain
of a Liapunov
surrounding
its post-fault applied state
space the domain
vectors
let
of (asymptotic)
let
ity domain boundary
V
be a Liapunov
estimation
clearing
of
V
times
V~
for
t =te3
anymore;
for
function
SEP
in its of sub-
rotor velocities
(asymptotic) that
i.e.
theorems
and sufficient
stabil-
it takes
of CCT amounts
direct
to computing
instability
character
lies
conditions
clearing
criterion.
does not guarantee
of the Liapunov
on the
for successive
: the c o r r e s p o n d i n g
to the Liapunov
of its conservative
necessary
system
be composed
one sees that the CCT corresponds
the criterion
fact that the stability
Liapunov
V£
but it does not guarantee
the main sources
provide
used
evaluation
tirae will be the CCT according
Note that
in the
for the post-fault
(some)
be the value
, until r e a c h i n g
this procedure;
in
method
to d e f i n i n g
o£ the power
and to
function
Then,
describes
bility
amounts
along the system trajectory, tel
of the system
= [~T~T]T
and let
of this domain.
the values
stability
the Liapunov
stability
rotor angles
X Further,
(SEP)
and let the state vector
to (some)
of the
of system trajectory
be the state model
configuration,
related
stability
the intersection
(2.2)
Point
Stated differently,
system transient
and to d e t e r m i n i n g
post-fault
and d e t e r m i n a t i o n
the Stable E q u i l i b r i u m
configuration.
to power
To fix ideas,
function
Large-scale power systems stability
t e2 " system's sta-
either.
One of
in the well
direct
Fig. 4
to
method
known does not
for the form of the system
'
/
/
..4" .~.-'-~'-.
", 5 M R~gio~ of ~ .,' ~"--lasymptot,c stability
Figure 4 The practical
procedure
for computing
method
is illustrated
in Fig. 5
steps.
For a given
function
at
:
to
V
CCT according
and summarized and disturbance
to the Liapunov
in the f o l l o w i n g three assumed
to start
acting
I Even though the existence of a Liapunov function is necessary and sufficient for uniform (asymptotic) stability (in the large) (Chapter I).
Physical description - Modelling
275
(i) compute the limit value,
V£
, that
its Stability Domain Estimate
V
takes on the boundary of
(SDE);
(ii) for successive clearing times
tel
(>t o ) , compute the corre-
sponding values of the state variables;
let
[8(tei),~(tel)]
denote symbolically these latter; (iii) compute the c o r r e s p o n d i n g value until reaching
VE
V [~(tei),~(tei)]
at
to+,tel,...
:
tcZ ~ CCT
for
V [6(tc~),~(tc£)]
: V~
.
V
v~
Figure 5
tcJ Remarks 2.
te
a) It is obvious that this CCT is obtained with just one
pass through integration of the faulted system equations. b) It is also obvious that the success of the above approach depends essentially upon
:
- the "quality" of the Liapunov function the most
"suitable" Liapunov
i.e. that form of
V
function,
: it is important to construct V,
for the physical system,
which allows as refined as possible m a t h e m a t i c a l
description while p o s s e s s i n g the largest possible SDE; - the speed of the numerical computation of this SDE for the computation of the
V
(the time required
function being negligible
in general).
V.2.5. System modelling The m o d e l l i n g process has been dealt with e x t e n s i v e l y Shackshaft,
1972; Dandeno,
(Chorlton and
1977). Models of (almost) any desired degree
of precision can be specified for the generator and its controls. The equations of motion may be written in a number of reference
frames -
for example in terms of absolute angles, relative angles between machines, or relative to a "center of angle" for the system Pavella, 1979a).
i97±b; Tavora and Smith,
1972; Evans,
(Ribbens-
1978; Athay et al.,
276
Ch. V
The swing equation of the expressed by eq.
i-th
Large-scale power systems stability
generator of an
(2.1). The expressions for
N-machine
system is
Pei are nonlinear func-
tions of the g e n e r a t o r angles and represent the internai transient behaviour of the synchronous machines, damping effects
their mutual interactions,
their
(which are much more complex than velocity damping)
and their response to e x c i t a t i o n control action.
M o d e l l i n g with full
accuracy becomes very complex. It is certainly traditional for power system engineers to make "simplifying assumptions" according to the "need" of the problem and according to the time scale of interest. scale aspect is given in Quazza
A good perspective on the time
(1976). The effects of these assump-
tions have been well studied and compared with full scale tests (Chorlton and Shackshaft,
1972; Dandeno,
1977). For example,
an indi-
cation of "f£rst sw£ny" 8tab£~£ty ' can still be got from the simplest assumption~
(constant voltage behind transient reactance),
mainly be-
cause the effects of initial a c c e l e r a t i o n and inertia tend to be dominant d u r i n g this short period stability of subsequent
swings
(up to about
i/2 second). However,
("resultant stability")
the
is far more
complex. For everyday security m o n i t o r i n g purposes, essential. Thus,
e.g.,
reactance,
is expressed by
Pei
simple assumptions become
for constant generator voltage behind transient :
N
Pei(~)
: Re [Ei[~] : Re Ei ~ Yij-~Ej-~ j:l N
: E~Yii cos eii + j ~ i EiE jYi~
oo~ (~.-~.- ~..) l ] i]
j~i
(2.3)
N
= E~Gii + ~ j=l Note that matrix
E i E j B i j s i n ( $ i - ~ j) + E i E j G i j c o s ( ~ - 8 ] )
Y = [Yij ] = [ Y i j ~ i j ]
changes with network topology
and in p a r t i c u l a r during a fault, and for each different
fault.
Another example is the c o n s i d e r a t i o n of first-order p r o p o r t i o n a l governors;
in this case
Pmi
present
in eq. (2.1)
is expressed by
:
I The indication of "first swing" stability is available from examination of the swing curves of the most perturbed machines during say the first quarter cycle of their (pseudo-) sinusoidal oscillations, Note that the first swing stability is an essential consideration. In a survey conducted by the Power System Planning and Operation within the EPRI (Electric Power Research Institute) Gelopoulos and Laurent (1980) indicate that at least 85 % of the respondents look for first swing stability in more than 95 % of the cases they run.
Physical description - Modelling
277
0
Pmi : Pmi + P m i (2.4) Pmi : - ~ i P m i - ~ i ~ i
"
V.2.6. Mathematical formulation Many state m o d e l s
may be built,
made and the r e s u l t i n g p h y s i c a l salient
intrinsic
ticated
state d e s c r i p t i o n s ,
In eq. (2.1)
depending
characteristics
describing
upon the kind of a s s u m p t i o n s
modelling.
Nevertheless,
are p r e s e n t
as those
even in the less
considered
the m o t i o n of the
(some of the)
i-th
sophis-
below. generator,
let us
assume that -
-
Pmi
and
Pmi
implies
Mi
are c o n s t a n t
d u r i n g the t r a n s i e n t s * ; c o n s t a n c y
that eqs. (2.4)
degenerate
Dij : 0 ; Pei is e x p r e s s e d by eq. (2.3)
with
to
Pmi : P ~ i
E i , Ej
of
;
constant
during
the
transients l D e r i v i n g the a p p r o p r i a t e
state e q u a t i o n s
choice o f state v a r i a b l e s . ous ways by m a n y and p h y s i c a l energetical 1971b); 1972a,
authors;
and m a t h e m a t i c a l
1972b);
minimal
They all a r r i v e
If, m o r e o v e r ,
argumentation
and o b s e r v a b i l i t y realization
aspects
damping or
then the state v a r i a b l e s velocities
as well.
1969,
(Pal and Murthy,
1974). of c o n c e r n
r o t o r speeds.
is e i t h e r u n i f o r m or t o t a l l y
neglect-
DiMil= k , Vi = 1,2,...,N ,
are r e l a t i v e
Thus,
sible r e f e r e n c e
rotor angles
the
and r e l a t i v e
to a r e f e r e n c e
N-th
generators;
generator
rotor
frame.
is to refer the s y s t e m ' s m o t i o n
"Center Of A n g l e "
(COA) d e f i n e d by
for example,
one An-
to the s o - c a l l e d
(Tavora and Smith,
N -I N 80 = ( Z Mi) ~ Mi~ i i=l i=l
sys-
S u c h a pos-
to be this r e f e r e n c e .
other p o s s i b i l i t y
1979a)
(2.5)
in b o t h cases the m o t i o n of the o v e r a l l
is one of the s y s t e m ' s
may choose a r b i t r a r i l y
.
1970);
(Sastry and Murthy,
that the state v a r i a b l e s
tem has to be d e s c r i b e d w i t h r e s p e c t
|
1969,
if Di = 0
al.,
in vari-
considerations
(Ribbens-Pavella, concept
rotor a n g l e s and a b s o l u t e
mechanical
been e x p l o r e d
: energy
(Di C a p r i o and S a c c o m a n n o ,
at the c o n c l u s i o n
here are r e l a t i v e
ed, i.e.
let us quote a few
system reduction
controllability
implies m a k i n g the p r o p e r
This has e x t e n s i v e l y
1972; A t h a y
et
(2.6)
i.e. constant during the fault-on and post-fault phases and equal to their corresponding pre-fault values.
278
Oh. V
From a theoretical of r e f e r e n c e
In the case w h e r e (2.1) and ~:
similarly,
one is in p r i n c i p l e
from a p r a c t i c a l
conditions
the choice b e t w e e n
is i m m a t e r i a l ;
of the r e f e r e n c e be true
viewpoint
these a b o v e two frames
the c h o i c e a m o n g the
indifferent.
However,
N generators
this may not
viewpoint.
the m e c h a n i c a l
(2.5) and w h e r e
N
dampings
do not obey the p a r t i c u l a r
acts as the r e f e r e n c e
(2.3) y i e l d the f o l l o w i n g ~i-
L a r g e - s c a l e power systems stability
state e q u a t i o n s
generator,
eqs.
:
~N • (i
= i,2,...,N)
N
hi
= - Dini
{Pmi -
+ Mi I
E[aci -
z
j=l
[EiE j B i j
sin (SiN -~N )
+ EiE j Gij c o s ($iN-~jN)]} . (2.7) The above denoting
s y s t e m of e q u a t i o n s by
X
the
(2N-i)
state c o m p o s i t e
6
[(
:
6~N
is the
Point ) ,
one
arrives
i974)
Indeed,
vector
s
~]N-$IN )
,'",
6s
(6nN- nN)
]T
[~I,~2,..,~N ]T
:
and w h e r e
form.
: [~T~T]T
X where
may also be set in a m a t r i x
i-th
component
at the
of the SEP
following
(Stable E q u i l i b r i u m
formulation
(e.g.
see Henner,
:
(2.8) a
In the above
(Henner,
~k f(a)
•
fk(~k)
,
z]
is a v e c t o r - v a l u e d
component
Cx
1974)
-8~.
: 6i3
=
k : (i-1) N
i(i+1)2 +j
f u n c t i o n with d i m e n s i o n
types
m =Nn/2
,
, whose
k-th
EiE j [Bij sin (o k + ~ .i] - sin ~sij) +Gij sin (o k -6~j _ c o s 6 isj ) ] , (k : i,2,-.,m)
are a p p r o p r i a t e
The above
i <j
is :
A ,B ,C
,
equivalent
of L i a p u n o v
for the system
constant
matrices.
formulations
(2.7) and
function
of first i n t e g r a l s
constructions,
of the eqs~ (2.7),
(2.8).
(2.9)
(2.8)
namely
suggest
viz.
to construct
neglect
eonductances
two different
: analytic
calculation
and Lur'e type L i a p u n o v
But both face the same d i f f i c u l t y
it in the same way, of t r a n s f e r
,
a Liapunov
of the p h y s i c a l
and circumvent
function system.
function
they require
Scalar a p p r o a c h
279
Formally, this means that the coefficients
Gij
present in (2.7) and
(2.8) have to be set equal to zero. Indeed, as soon as (2.9), the function
fk(Ok)
fk(Ok) o k > 0 for some values
~k
about
~k :0
Ykm i ~ Hence, eqs. (2.8)
Gij : 0
in eq.
obeys the sector condition k = 1,2,-.,m
(2.10)
:
< Yk < Jk ~ ×
;
take on the standard Lur'e - P o s t n i k o v type form with
multiple nonlinearities and satisfy the generalized Popov criterion of Moore and Anderson readily
(1968). A Lur'e type Liapunov function derives then
:
~
V(x) : xT Px +]0 f(~) Qd~ .
(2.11)
A-similar reasoning, developed below, shows that neglect of transfer conductances seems necessary for devising any energy type Liapunov function whatever the construction procedure used.
V.3. SCALAR LIAPUNOV APPROACH V.3.1,
Preliminaries
As has been mentioned,
from the very beginning of their development,
direct methods have exhibited attractive features but also serious difficulties. Their attractiveness
lies mainly in the possibility to pro-
vide a synthetic answer to the stability problem without simulating the whole transient. The difficulties are of two types
:
(i) The first is implied by the very construction of Liapunov functions which imposes quite simplified system description
(at least
in the multimachine case). The most stringent among them h a v e been mentioned in Section V.2.5. (ii) The second is related to the inability of the classical stability theory to provide reliable and eomputationally efficient SDEs; better means for evaluating SDEs are needed
:
- in order to overcome the conservativeness that the theoretical SDE confers on the Liapunov criterion; - in order to speed up significantly its computation which, otherwise counterbalances the computational advantage of the direct criterion over the numerical integration procedure. Both of the above two types of difficulties along w i t h possible means
280
Ch. V
to a l l e v i a t e
t h e m are e x a m i n e d
a p p e a r below,
the s o - c a l l e d
Large-scale power systems stability
in this Section.
"energy type"
b a s i s of our r e a s o n i n g and is s t u d i e d
V
For r e a s o n s w h i c h will
f u n c t i o n is chosen as the
first.
V.3,2. The "energy type" Liapunov function This f u n c t i o n are a s s u m e d
is d e v i s e d
constant,
magnetic
dampings
V.3.2.I.
State
and electro-
as the r e f e r e n c e ,
we get the
model the
N-th
generator
[~IN,62N,...,~nN ,~IN,~2N,...,~nN]T : [~T~T]T
:
We shall s y m b o l i c a l l y equations KiN
~iN
denote
derive
it by
then r e a d i l y
solutions
(3.1)
(8,~) from
(2.1) and
(2.3)
:
~iN
:
: - ~ n i N + [Pml - P e i (~)] M[ I -
[PmN - P e N (8)] MN 1
: - k ~ i N + fi(8) The
Ei • Pmi • Mi
obey eq. (2.5),
vector X
State
dampings
are zero.
Choosing arbitrarily state
in the most s i m p l e case w h e r e
mechanical
of
(3.2)
i = 1,2,...,n
(3.2) are given by iH : 0
(a) i = 1,2,...,n
(3.3)
fi(~ ° ) = 0 T h e y are t h e r e f o r e
(b)
o b t a i n e d by s o l v i n g the
n
nonlinear
algebraic
eqs.
(3.3.b). In what
follows,
we w i l l be led to c o n s i d e r two types of s o l u t i o n s
( e q u i l i b r i u m points)
of the s y s t e m in its p o s t - f a u l t
- the stable e q u i l i b r i u m S
s
s
[$1N,~2N,...,6nN,0,0,...,0 ally d e n o t e d by - unstable
point
IT
(ss,0)
(SEP) of c o n c e r n
be its c o m p o n e n t s ;
- if any;
:
let
they w i l l be s y m b o l i c -
;
e q u i l i b r i u m points
(UEP),
surrounding
a m o n g these UEPs will be s y m b o l i c a l l y
V.3.2.2.
configuration
the SEP; the
d e n o t e d by
(~ui,0)
i-th
.
Liapunov function
It is p o s s i b l e integrating
to c o n s t r u c t
a Liapunov
the state e q u a t i o n s
below are s a t i s f i e d
function
(5.2), p r o v i d e d
for s y s t e m
(5.2), by
that c o n d i t i o n s
(5.4)
Scalar approach
281
af i 8f~ a~--~ : 8~i
i,~ : 1,2,...,n
This in turn implies neglecting
(3.4)
(some of) the transfer conductances,
i.e. setting equal to zero (some of) the expressions
Gij
:
Gij = Yij COS 8ij = 0 . Of course,
conditions
(5.4) are always verified when setting
e±j : ~ With this assumption, manipulations
(i~j)
(5.6)
integration of eqs. (3.2)
(e.g. see Ribbens-Pavella, n
N
v(~,n) : ~
z
yields after some
1969, 1971a)
{~MiMjaij2
i=l j:i+l : V~(n) +Vp(8) The derivative of
(3.5)
V($,~)
_(PiMj _PjMi)(~ij N - ( Z Mi) EiE j Bij (cos ~ij i:l
is then expressed by n
-8~j) _
ys
cos
ij )} (3.7)
:
N
i:l j:i+l One easily verifies that this
:
V
3
function is indeed a Liapunov function
in a certain domain surrounding the SEP
(~s,0)
, since it verifies the
conditions
(a)
v(~S,0) :0 ;
(b)
V(~,~)
is positive definite in this domain;
(c)
V(~,~)
is negative semi-definite everywhere for
or
9 -0
for
Hence, the above when
V
function guarantees the system asymptotic
A >0 , or the simple stability when
~ =0
(~ >0)
,
l) The above
V
stability
. On the other hand,
the "size" of the stability domain does not depend on Remarks 3.
~0
~:0
V.
expression contains two groups of terms
:
one of the "kinetic energy" type, and one of the "potential energy" type. 2) With the above construction, presence of uniform damping reinforces the system stability
(asymptotic instead of simple) but does
not allow enlarging the stability domain, as might be physically expected. 3) The
V
function
(3.7) may be considered as "optimal" in
the sense that it determines the largest stability domain estimate for the considered system modelling. 4) One of the prices to be paid for the advantage of the
282
Ch. V
Liapunov's
"global"
stability
od to account
for t r a n s f e r
effect.
influence
Their
Large-scale power systems stability
assessment,
conductances
is the i n a b i l i t y
w h i c h may have a s i g n i f i c a n t
comes from the fact that the a d m i t t a n c e
used here is the one r e d u c e d at the g e n e r a t o r amounts
to i n t r o d u c i n g
o t h e r than g e n e r a t o r (e.g.
see A p p e n d i x
(some of the)
nodes
(3.3) but
effects
in R i b b e n s - P a v e l l a
implies
nodes;
Gij
does not m o d i f y
replacing
therein
Pei
loads c o n n e c t e d
terms of the
and Evans,
matrix
this r e d u c t i o n
of the
in the o f f - d i a g o n a l
5) N e g l e c t i n g and
of the meth-
to
Y matrix
1985a). the form of eqs. (3.2)
(2.3) by
N
2 Pei : EiGii + ~ EiE ] Bij sin (~i-~j) j:i If the terms gral t e r m computable
I(6)
containing
(3.9) would (Evans,
Gij
w e r e not neglected,
have a p p e a r e d
1978)
in
(3.8)
an a d d i t i o n a l
(]-7), w h i c h
into-
is not a n a l y t i c a l l y
:
n N [~ij N E ~ 1 {Mj [ ~ EiE k G i k cos 6ik] k#i i : 1 j : i+l j6~j
:
(3.9)
N
- Mi [ ~ E k E j G j k cos Bjk]} d6ij k~j
V.3.2.3.
Stability
domain e s t i m a t e s
V.3.2.3.1. Theoretical determination of determining
a SDE has been i n v e s t i g a t e d
A b i a d and N a g a p p a n , 1974).
2966;
Here we shall
(3.7) and for
9 =0
values of
For small C
increases,
surface
C,
1970,
reasoning.
V(B,~) = C
(>0)
closed hypersurfaces,
the h y p e r s u r f a c e s
The p r o b l e m
in v a r i o u s ways
and Willems,
~ , the curves
.
"swell" while
(e.g.
With
V
of
see EI-
Willems,
1971, given by
in the state surrounding
remaining
stops b e i n g closed as soon as it meets a m u l t i p l e
is a point
space
the SEP.
closed; point,
the
that
for w h i c h BV
OV
~
Now,
Willems
follow a g e o m e t r i c a l
are, As
Vli m :VTh
this m u l t i p l e
= a-~i N-
point
=
0
(i
=
1,2,...,n)
is by c o n s t r u c t i o n
a UEP,
solution
of eqs.
(3.3). The p r a c t i c a l (a) compute
procedure
aZ~ UEPs
for c o m p u t i n g
surrounding
VTh
is s u m m a r i z e d
the SEP of c o n c e r n
for the s y s t e m in
its final c o n f i g u r a t i o n ; i }'or V
as follows
negative semi-definite, the same conclusion is reached.
:
Scalar approach
(b)
define of
283
VTh
the
as
closed
being
the
value
hypersurface VTh
where of The
(sui,0)
computations
tremely known
heavy,
- and
character
of a
to
is
eqs. "first
ful
i
The
practical
the
the
by
surrounding
function
SEP
on
the
boundary
: (3.10)
"closest"
"theoretical"
of
UEPs
location.
And
(3.3b)
which
guess"
procedure
V
A = rain v ( ~ U i , 0 ) i the so-called
this
number
their
the
(in
the
sense
of
SEP.
implied for
so
providing
these
represents
(3.10)) U E P
of
have
about
this
suggested
to
be
because
to
be
SDE
explored of
ex-
a priori
un-
highly
any may
is a n o t h e r
obviously
is
the
solved,
information
below
are
nonlinear
means be
capable
extremely
means
to
of use-
circumvent
difficulties.
Fig. 6 illustrates
the above considerations
on the basis of a realistic
system.(It derives from the reduction of a real power system.) have been plotted in the state plane they are organized around
300 210170130110
ss
(613,623)
for
3-machine
The " e q u i - V "
curves
~13 : [~23 : 0 . Observe that
(i.e. the SEP). Incidentally,
VTh :96
.
130 170 200
Figure
6
l From physical arguments one can propose as many as 2 n starting points for solving eqs. (3.3b) . This may also be inferred from exploration of the sector conditions (2.1o).
284
Oh. V
V.3.2.3.2. Praa£iaa£ aomNatation of Pavella
Large-scale power systems stability
Vli m :V£
.
Suggested by Ribbens-
(1975), it is summarized as follows.
i °) Consider all "type 1" starting points;
these are points correspond-
ing to the physical situation where one machine goes unstable with respect to the remaining system. For instance, the starting point is expressed by 8sj
:
s
s
[~IN'~2N '''''
2 °) A good a p p r o x i m a t i o n of V~ :
6~
]-I,N
VTh
j-th
type I
:
'~-
~s
iN'
~s
s
j+t,N'""6nN
]
T
(].21)
will then be the value
min v(&SJ,0) j:l,2,...,n
(3.i2)
J u s t i f i c a t i o n of the above procedure is based on physical argumentation
(Ribbens-Pavella,
1975). Its use contributes to speed up consider-
ably the computation of an approximate value of
VTh
Obviously,
how-
ever, it does not alleviate its conservative character.
V . 3 . 2 . 4 . A££avia£ing t h z aonservativenes~ of Liapunov'~ a r i £ e r i o n PSDE~ Among the many approaches proposed within this context,
those recogniz-
ing the "relevant" UEP as related to the fault location and to the practical SDE
(PSDE) have certainly contributed to make the Liapunov
method truly reliable and effective. They came about in the late seventies. In what
follows, we describe two of them which are based on sound
justifications,
while exhibiting interesting practical features.
V.3.2.4.J. The Kyoto approach.
For
k:0
, V(~,~)
is identically zero.
C o n j e c t u r i n g that the trajectory of a sustained disturbance coincides with the critically cleared one, Kakimoto et al.
"~]most" (1978,1980,
1981) propose to identify the crossing of the faulted trajectory with the PSDE. Thus, they come up with the following procedure. (i) At successive clearing times, V[~(te),~(te)]
t e , compute the value of
and of its potential part
Vp[~(te)]
, until reaching
a m a x i m u m of the latter. This will be considered as the limit value, V k , taken by
V
on the boundary of the PSDE V k = Vpmax(~)
(ii) The CCT, V[6(te),~(te)]
;
: (3.13)
t k , will accordingly be found at the intersection of with
Vk : V[6(tk),~(tk)]
Fig. 7 illustrates this procedure.
= Vk
(3.14)
Scalar approach
285
vli 9 -lV'~90.35v0.36
/
II
Sustained fault CGT
"
0.,
a2
023
0.3550:,-"%
V.3.Z.4.2.
The
0.6
should
requires
is described denote
below.
(3.11),
(i) At tion,
approach
to+
, i.e.
determine
respectively (3.11)
by
i
and compute
"fastest" consists
"fastest" and
N,
and
let us assume
following
"slowest"
define
accordingly
be sustained.
the
known what
Va
Then
This we
the
:
disturbance
machines.
value
func-
machines.
two steps
"interesting"
the limit
V
So, this method
machines.
after the c o n s i d e r e d and
is
first
of the
"relevant"
"slowest"
the
to its c o r r e s p o n d i n g
the value
of some
of the
immediately
the
disturbance
so that
identification
by
conjecture
which
of its value at UEP.
For the time being
symbolical]y
acceleration
The u n d e r l y i n g
to be very close
tion at SP is a good a p p r o x i m a t i o n essentially
-- "-- : a t 0 . 3 6 8.
to the machine
the considered
(SP) of form
= 0 . 3 5 5 s. fault cleared
te
approach.
this UEP is assumed
starting point
a5
UEP is related
would go out of step, Moreover,
o.a
aacegaration
that the i n t e r e s t i n g
: - : a t 0 . 3 5 s.
Figure 7 Principle of the Kyoto's method for assessing CCT. Inspired by Kakimoto and Hayashi, 1981, and applied to a 3~SC at GB f[5 of the 7-machine system.
Vk
0.2 0.3
fault cleared
Denoting
incepthem
UEP to be given
v a : v(~ s i , O ) (ii) At successive V[~(te),~(te)]
clearing
times,
, until reaching
(3.15)
t e , compute
V a ; the CCT,
the value of
V,
t a , is given by
V[~ (ta),~(t a) ] : V a Remarks
4.
shown to be
1) When a cluster (almost)
equally
no longer the SP of form than 1" a p p r o p r i a t e l y 2) The machines tions
(3.11),
the
a disturbance
(3.16)
instead
of a single
the i n t e r e s t i n g
of "type
one,
UEP of concern
1", but rather
are is
a "type higher
expressed.
identification
rely on i n f o r m a t i o n
that
of machines, "fast",
by
:
system m a c h i n e s inception.
rules
provided acquire
The details
of the
"fastest"
by the values at
to+
and
"slowest"
of the accelera-
, i.e.
of the procedure
immediately are given
in
after
286
Ch. V
Toumi et al.
Large-scale power systems stability
(1985). Let us only mention here that, roughly speaking,
the "fastest" machine corresponds to that p o s s e s s i n g the largest magnitude of initial accelerations, whereas the "slowest" to that with the m i n i m u m initial acceleration.
V.~.2.5.
Practiaag improvements of the energy t y p e function
V.3.2.5.1.
Approximate ac~aunt of t r a n s f e r conduetances.
An approxi-
mate way of accounting for transfer conductances consists of adding to the eq. (3.7)
the n o n - a n a l y t i c a l l y computable term (3.9), numerically
evaluated along the fault trajectory. The obtained function is no longer a real L i a p u n o v function.
From a practical point of view, however,
this so-called "Transient energy function" (Athay et al.,
V.3.2.5.2.
1979a,
A s p e e d i n g up p r o c e d u r e .
1978,
8(re) , ~(t e)
and
ineeption,
and hence of
over the fault trajectory, by using a Taylor series
expansion instead of the step-by-step method. @i(t e)
1980).
Under certain conditions it is
possible to speed up the computation of V/@(te),~(te) ]
shows very interesting
1979b; Kakimoto et al.,
~i(te) one gets
about
to+
, i.e.
Indeed,
expressing
immediately after a disturbance
(Ribbens-Pavella et al.,
1976, 1977)
6i(te) = ~o 1 t2 1 " 4 i + ~.7i e + 2~TT[te +"" (3.17) ai(te) where
7i
= 7it e + ~ T i t ~ +-..
denotes the a c c e l e r a t i o n of machine
i at
to+
.
This procedure may be shown quite interesting in practice.
V.5.2.6.
The i n d i v i d u a l energy t y p e funation
Rather than assessing system's stability through global information contained in the
V
function,
the lowa group proposes
formation contained in an individual
attribute the system separation which would cause
(if ever)
ity, to the motion of a certain "critical" machine, Thus, Michel et al.
to rely on in-
function. The leading idea is to
(1983), Fouad et al.
say the
instabili-th.
(1984) derive the individual
energy type function by taking that part of functions
(3.7) + (3.9)
which corresponds to this machine. Initially devised with respect to COA, the individual function can also be constructed with
N
as the reference machine.
One gets
Scalar approach
287
1
2
s
ViN : ~ MiMN~i -PiN(~iN-~iN)
N
[~iN
+ MNE i
Z | i=l J
- MiE N
~ | Ej [BNj sin BNj +GNj cos ~ j ] d 6iN j:l ~&s iN
N
V.3.2.7. Taylor
Ej [Bij sin $ij +Gij cos ~ij] d6iN
[~iN
based energy function~
Interesting analytic
developments
of the above Taylor series expansion
are obtained when truncating eqs. hand term for
+
~i
(3.17) after the second right-
and consequently
after the first one for
~i
Indeed, this allows relating linearly rotor angle differences;
• one gets
readily (6ik In turn,
_~o
this approximation
(3.9) which account tions
o
ik) = (~k~-Bk~) ~k~Tik
;
~ik : ~k£ ?k~
~ik
makes analytically
for transfer conductance
integrable
effects.
(3.7) + (3.9) and (3.18) may now be formulated
single state variable.
Further,
(Toumi et al.,
The above approximation
shows extremely
range,
for small
essentially
the terms
Moreover,
func-
in terms of any
evaluation of the PSDEs becomes ex-
tremely flexible and straightforward
i.e.
(3.19)
te's
1985).
interesting within its validity (see also the Closure of the
above reference).
V. 3 . 2 . 8 .
Direct
criteria
for r e a £ t i m e o p e r a t i o n
The combination of the above four types of functions energy type
V
function
(3.9), the individual
(3.7), the transient
type
along with the two PSDEs proaches),
(viz. the pure
energy function
(3.18) and the Taylor-based
(viz. the Kyoto and the acceleration
yields a large number of variants.
Details
(3.7) +
function), ap-
of the many con-
tributions along with their pros and cons on the basis of their performances with respect quirements
to accuracy,
are discussed
sions are brought
reliability
in Toumi et al.
out and summarized
V.3.3. Family of the "energy type"
Liapunov functions
re-
conclu-
in Section V.3.5.
V functions
In this Section we give a very short account to construct
and computational
(1985). Some salient
of some methodologies
used
of the energy type or similar to it.
288
Ch. V
Large-scale power systems stability
V.3.3.1. Calculating a n a l y t i c a l l y f i r s t i n t e g r a l s of t h e ~ t a t z z~aatlon~ Two different Liapu n o v
approaches
technique
stricto
sound one e v o l v i n g son
(1947),
authors
tem "kinetic"
computational
succeeding
thoro u g h
second
through
mentioned
remain
and the
technique Indeed,
consists
these
calculation
and consideration
of the sys-
of their
analysis
tools and oF
today,
prevented
these
effective
practical
stability
exemplary
School
for clarity,
pioneers
from
tools.
and give a
insight. concerns
first
the c o n s t r u c t i o n
integrals
in Section V.3.2
(3.7) + (3.9),
by Magnus-
Information
method.
through
energies
in elaborating
approach
definite
direct
criterion
available
their works
physical
physically
Initiated
1971) i , this
The lack of the numerical
facilities
Nevertheless,
and T e c h n o l o g i c a l
the L i a p u n o v
"potential"
signs.
is not a
and later on by Gorev and the Soviet
the direct
and
and
considerations.
of the U.S.S.R.,
implicitly
The first
sensu but rather intuitive, energy
of Scientific
"rebuild"
variations
The
(1958)
of Sciences
in applying
fully
purely
Aylet
(Union Institute Academy
may be distinguished.
individual
of the
: function
type
of L i a p u n o v
system
(3-7),
function
state
transient
(3.18),
functions
equations, energy
Taylor based
as
function
type and
their variants.
V.3.3.2. This
The g e n e r a l i z e d Popov a r i t z r i o n
criterion
Liapu n o v
is certainly
functions.
Credit
an attractive for applying
multimachine
case goes to J.L.
J.C.
(1970).
Willems
for g e n e r a l i z i n g
These
the Popov
Willems
authors
technique
it for the (1970b),
for g e n e r a t i n g first
time to the
and to J.L.
used Moore and Anderson's
frequency
criterion
Willems
and
theorem
to feedback with multi-
ple nonlinearities. The
Popov c r i t e r i o n
integral" stability Henner
V
and by Pai and Murthy
once again
uniform mechanical
allow c o n c l u d i n g
in the uniform
damping
This has theoretical
mechanical
to Lur'e
for studying
type
the
The
damping
same
case.
has also been envisaged
function
difficult
since
(3.7)
of non-
by the above
appealing,
are generally
V
system
in 1974 by
Inclusion
size of the SDE; m o r e o v e r
constants
form plus multi-
- multi-output
simultaneously
(1974).
more than practical
about
damping
"quadratic
multi-input
has also been exploited a l m o s t
(1974)
is devised
leading
functions
authors.
it does not
the values
of
to assess.
I Western scientists are not very familiar with Russian works in the field : apart from some private translations, these works are generally published in Russian only.
Scalar approach
289
Extension of the Moore
and Anderson
ion has been used by Kakimoto multi-argument second)
order
is i n c o r p o r a t e d type
the effects
of this
instability
arises
flux decay, regulators
V
model
The authors
suggests
or perhaps
constant
which case the generated
Liapunov
specify
AVRs only when the time
lie within
transfer
Kitamura
et al.
use the p e r t u r b a t i o n
parameters tematic
function
reflect
(1983)
These
(i981).
transfer
preserving
of absorbed
leading where
neglecting
ing the resistances
of values.
in order
to incor-
conductance
The
the small
effects.
Its sys-
as "external
feedback
of the main part of the sysand for which a LiaFurther
investigations
contribution.
moda£
allowing
in the admittance
In this case,
method
conductances
inclusion
conductances
model"
range
for
zffz~t~
in the usual way.
The 6 t r ~ c t ~ r e
ture p r e s e r v i n g
a limited
transfer ~ o n d u c t a n c e s
this original
The i n t e r e s t i n g
(3.7).
accounting
difficult.
effective
of the system transfer
after all - in
of terms where
is constructed
is
behind
of small parameters.
not contain
approach
forces
a series
act as sort of disturbances
Another original
voltage
if flux decay
by means
tem which does
V.3.3.4.
of automatic
picture
effects
is however regard
due to
contains
punov function could make
conductance
some of the transfer
application
Saeki et al. loops".
conductance
Liapunov
region
comes back to the form
Approximating
porate transfer
term r e p r e s e n t i n g
assumption
V.5.3.5.
obtained
The
show that a new type of
that their method allows
constants
than
flux decay.
electromotive
function
systems
(rather
of the stability
may be a satisfactory
(~977)
field
that the inclusion
that
criter-
to encompass
function has an additional
would give a more realistic
the authors
of Popov's
of third
so as to consider
with the vanishing
reactances
Moreover,
(1978,1980)
Machine
flux decay.
but this (AVRs)
to be modelled~ transient
et al.
nonlinearities.
resulting Lur'e
generalization
of the most
was proposed
important
by Bergen
idea is the c o n s i d e r a t i o n
the loads matrix
are explicitly
reduced
the transfer
of the t r a n s m i s s i o n
and Hill of a "struc-
retained
at the generator
conductances
part
amounts
instead nodes.
to neglect-
lines only, w h i c h are general-
ly small compared with reactances. To preserve
the unreduced
quency dependent frequency network"
load
network
is taken to be linear thus
topology,
(FDL) modelling,
resulting
about
comprises
the authors
where nominal
(N+no)
consider
the load variation frequency.
buses,
where
a frewith
The
"augmented
no
is the
290
Ch. V
total number variables
of the network
for describing
For the above function
"structure
network
to incorporate
derivative
ever, The
preserving
nonlinear
In fact,
of voltage
Liapunov
by Athay and Su
load modelling.
(1981)
A new TE function
is
it is a positive
definite
function
negative
(semi-)
numerically
definite
but
its
: the sign of the
along the trajectory.
This,
how-
computations.
(1984)
and local
type
is a pseudo-Liapunov
system's
and Musavi
an energy
by Bergen and Hill (1981).
however,
is not a priori
same unreduced
murthi
model",
state
dampirlg case.
model which,
has to be examined may impose heavy
(2N-l+no)
in the non-uniform
idea has been exploited
static,
for this interesting
function only.
latter
it necessitates
its dynamics
is once again constructed
The unreduced
derived
buses;
Large-scale power ~ystems stability
idea has recently
who model
frequency
been used by Narasimha-
the active
power
and the reactive
load as a function
power load as a func-
tion of voltage. Overall, receive
this nice
idea shows potential.
many d e v e l o p m e n t s
resting practical
which
achievements
(see also
V.3.3.5.
Thz Hamiltonian approach
Proposed
by Kumagai
namic
circuit
transient
and Wu
model
at the structure loads which,
(1982),
and derives
stability
analysis
preserving
however,
FDL r e p r e s e n t a t i o n
Very
likely,
sooner or later will
this approach
systems.
model with
are r e p r e s e n t e d
and constitutes
at inte-
the survey by Varaiya et al., 1985).
a Hamiltonian
of power
it is going to arrive
uses a nonlinear
formulation
for the
The method
arrives
(N+no)
buses
by a power
dy-
again
so as to keep the
demand with or without
an interesting
contribution.
V.3.4. The Zubov method This method
is a priori
to generate
Liapunov
boundaries.
Applied
relaxation trans f e r
functions to power
of current
conductances.
prevailed
appealing
together
systems,
simplifying However,
over the above
if not inapplicable
because
Indeed,
assumptions,
its inherent
advantages
to large-scale
in the conventional function
corresponding
boundary
SDE
in addition
such as accounting
drawbacks
have
restricting
for
till now
unattraetive~ its interest
system. (Yu and Vongasuriya,
in a (truncated)
is estimated.
it offers
corresponding
allows
and made the method systems,
Zubov method
is constructed
with their
the method
to the case of " o n e - m a c h i n e - i n f i n i t e - b u s "
Liapu n o v
of the p o s s i b i l i t y
}Iowever,
this
series latter
1967),
a
form and its does not ap-
Scalar a p p r o a c h
291
preach the true boundary the series' boundary.
monotonically;
terms does not guarantee
This
disadvantage
used by Prabhakara
et al.
tained in a closed
form.
however,
related
variables.
(1974) Another
And apparently,
increasing
improvement
is cleared
to the choice
in the m u l t i m a c h i n e
thus,
the number
of the stability
in the generalized
where
the stability
difficulty
arises
of an appropriate
this difficulty
of
Zubov method
boundary
in this
is ob-
latter
transformation
case,
of the
has not yet been overcome
case.
V.3.5. Numerical simulations The s i m u l a t i o n s r e p o r t e d the Liege
group.
(1985),
some others
trical"
details
V.3.5.7.
hereafter
are a sample
of those performed
Some of them have been published by R i b b e n s - P a v e l l a
et al.
by Toumi
(1985b).
by
et al.
For more
"elec-
the reader may refer to the above publications.
Simulation dasaription
The s y s t e m s .
Eight
reduced versions
systems
have been
of) power networks
investigated
along with
comprising
real
some test-systems
to cover a wide range of system characteristics.
(or
so as
They are identified
as follows *
3-machine
test
*
6-machine
T u n i s i a n EHV
7-machine
CIGRE
system
*
9-machine
power
system
* 14-machine
system simplified
Greek EHV simplified
used here corresponds
power
power
system
to its situation
* 15-machine
test
* 17-machine
lowa system,
system
(N.B. The c o n f i g u r a t i o n
in the early
seventies)
system reduced
version
of the network
of the State
of lowa * 40-machine Information
Belgian about
EHV simplified
data of the above
power
system.
systems
may be obtained
upon re-
quest.
Thz d i s t u r b a n a a ~ . three-phase
short
always Generators'
The
fir.st group of d i s t u r b a n c e s
circuit
(30SC)
Bus-bars
to Other than generators'
type.
(0B),
Bus-bars
A second group
of disturbances
line trippings
and load
Their
apart
is of the standard
location
eoneerns
from a few e×ceptions
relative
(OB).
has also been considered;
sheddings.
almost
it concerns
292
Ch. V
T~z rzsagts.
The
results
in Section V.3.5.2 to ot h e r types The
standard
parisons.
and s u m m a r i z e d
of disturbances
step-by-step
Its CCTs
the generators' pressed
swing
in sec. They
of w h i c h appear The simulation
by-step ble,
with the
Table
i:
methods
sought
relative
V.3.5.3. in our com-
integration
of
All CCTs are ex-
(lower bound)
have been
case.
investigated,
seven
hereafter. to explore
two essential
effi~iency.
methods'
The accuracy
of CCTs provided
of a method
by it and by the step-
conditions
Evaluating
an a p p r o x im a t e
quali-
and, w h e n e v e r
computational
first
possi-
efficien-
estimation
is only
follows.
obtained with
It collects
3@SC t y p e
of T a b l e s
I,
information in the
disturbance
location
Generators'
Busbars
identification
relative
columns
: unless
in Toumi
identification
specified
method
:
"fastest" et al.
(see column
it concerns
in column
tc ;
machine,
(1985);
this
9 below)
in the a c c e l e r a t i o n
the case is specified in Table
and accuracy
i to ii as follows.
(GBs);
method
1 UEPs are chosen,
to r e l i a b i l i t y
numbered
otherwise
of the relevant
in the individual
disturbances
II and I l l .
by the step-by-step
by the test proposed
4:
stable
same simulation
It is o r g a n i z e d
scribed
up to 1.50see.
to the
comparisons
Organization
wise,
Results
in Section
through numerical
or variants
more haza~,dous;
# 2 : CCT provided 3:
pursued
and computational
Results
i.
aspects.
I to III.
same system modelling.
given in what
V.3.5.2.1.
in Tables
are commented
is u s e d as the b e n c h m a r k
correspond
under the
cy is somewhat
V.5.5.2.
curves
results
through
one,
simulations
are reported
method
in the Tables
: sccuraey
is assessed
to 3@SC
have been a s s e s s e d
A dozen of praetieal
ties
relative
Large-scale power systems stability
as is provided is the one used
and~
approach
whenever
as well.
type
Other-
6 below and further
de-
II;
of the relevant
"slowest"
machine,
as is provided
by the test; 5:
CCT provided (5.12))
:
by the eonventional
t¢~
6 : CCT provided
by the a c c e l e r a t i o n
by eq. (3.7),
i.e.
Va
by eq. (3.16);
expressed
those of columns dicated
(~).
Liapunov
criterion
approach
:
(see eq.
;
without
3 and
accounting
function
for transfer
the considered
4, except
V
expressed
conductances;
relevant
for "other than type
machines
are
i" UEPs in-
Scalar approach
# 7:
293
CCT p r o v i d e d pressed
ductances; 8:
:
by the
tkC
Table
If.
machine
V
by the
(3.18)
by eq.
:
accounting
(3.14)
"Global"
Kyoto
expressed
: t~G
approach
V
function
ex-
for t r a n s f e r
con-
;
with
transfer
by a d d i n g
eqs.
conduc-
(3.7)
and
:
It c o l l e c t s
type
Kyoto
approach;
Vin
is e x p r e s s e d
;
by the T a y l o r
approach
than
approach
function
based
function
combined
with
the
accel-
tT .
all
cases
is to be c o n s i d e r e d
ing " o t h e r
Kyoto
without
individual
; tkl
CCT p r o v i d e d eration
i.e.
;
CCT p r o v i d e d by eq.
"Global"
expressed
included;
(3.9)
#10:
Vk
CCT p r o v i d e d tances
9:
by the
by eq. (3.7),
where
a cluster
in the a c c e l e r a t i o n
1" UEPs
correspond
of,
rather
approach.
to the m a c h i n e s
than
The
one
result-
indicated
in
this Table. Table
Ill.
The first 1:
It a s s e m b l e s concerns
number
three
groups
investigated
of s y s t e m
systems'
generators
total
number
of s y s t e m
buses
# 3:
total
number
of s y s t e m
branches
methods
group
summarizes
are r e p o r t e d
the
in c o l u m n s
(load
4 and
5 as
of the K y o t o
"Global"
5:
accuracy
of the K y o t o
"Individual"
each the
- sub-column t c
,
of the a b o v e following
I
data
:
columns
generator
nodes)
( N + N L)
;
of T a b l e follows
method,
listed
method,
I. The
in ~ 8
listed
is s u b d i v i d e d
compared
: of T a b l e
in # 9
in three
I;
of Table I.
sub-columns
:
: maximum
discrepancy
of m e t h o d ' s
CCT w i t h
respect
to
in see.;
- sub-column
limits
II
over
- sub-column limits The t h i r d methods'
: number
total III
over
number
total
group
of T a b l e
the
times
fied
in i t a l i c s
compares needed
II gives
time
t c ; three , te3 > t c
in T a b l e
respectively
CCTs
within
t c ±0.02
3@SCs;
of s i m u l a t e d
yielding
CCTs
within
t c ±0.01
3@SCs.
a first
in % of CPU
approximately
(T b) m e t h o d s ;
of s i m u l a t e d
computing
, te2 : t c
yielding
of s i m u l a t i o n s
number
for e v a l u a t i n g tel < t c
of s i m u l a t i o n s
: number
computing
# 6 : compares
7:
plus
information
accuracy
providing
topological
(B) .
# 4:
Moreover,
essential
(N) ;
# 2:
The second
of i n f o r m a t i o n .
approximate
evaluation
of the
:
required trials
by the
have
; the
tc
been
step-by-step assumed
considered
method
necessary
here
are
:
identi-
I;
the
by the
ratios
in ~ of the
"Global"
(T G)
computing
and by the
times
step-by-step
294
Ch. V
0"~.~
O0 CO e4
h_<. ~
0 0 0
o.~. ~
"
-
systems
stability
0 0 0 0 0 0 0
0 0 0 ~ 0 0
O O O O O O O
0 0 0 0 0 0 0 0 0 0 0 0
O O O ~ O O
~
0 0 0 0 0 0 0 0 0 0 0 0
6
o
o
~
~oooooooooo
~ o o
~ o ~
~CO~OCO~ 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 ~ 0 0
A
b
,--I
power
~
o "~ "~ ~1 on : ~ o,:m'~ ~ "~1 6 o o .g ~ '~
Large-scale
~= 0 0 0 ~ 0 0 u
=
~-~
Z
~.~..
0 0 0 0 0 0 ~
u~
O O O O O O O O O O O ~
A
OCD Ug~
uets Luni auLq~- 9
3~913 au .Lq~u- L
au .Lq~ W - 6
approach
Scalar
295
~ O O O O O O ~ O O O O O O
O
O
~
N
~
~
O
~
~ O O O O O O ~ O O O O O O
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
~ M ~ N M Q N
~
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o
o
#
~
...........
~ Q N M N ~ M N N N M 0
0
0
0
0
0
0
~
0
0
0
0
0
0
0 0 0 0 0 0 0 ~ 0 0 0 0 0 0
0 0 0 0 0 0 0 0
O O O O O O O ~ O O O O O O
0 0 0 0 0 0 0 0
~
o
~
~
N ~ N N ~ N O
o
M N N ~
0 0 0 0 0 ~ 0 ~ 0 0 0 0 0 0 0 0 0 ~ 0 ~
~
~ N ~ N O O
O O O O O O O ~ O O O O O O
0 0 0 0 0 0 0 0
N N
0 0 0 0 ~ 0 0
~ M
O O O O O O O O
A
0 0 0 0 0 0
:~aa,x9 au .tqoe"'-tTI
N N
Q ~
~ 0 0 0 0 0 0 0
auL qo~'"-Sl
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
et4Ol ~u .Lq~'"-LI
296
ch. V
Large-scale power systems stability
0 ~.~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
•
°
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o%~ CO
0
0.~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0
-2_
55 B v
U
U
F-
CC
o~o
~
O0
0
0
~ 0 0 0 ~
~
>
0 ~ 0
-
O~
=
~-~
.
L~
N N M N ~ 0
0
0
0
M M N 0
0
0
0
0
M M M 0
0
0
0
~
M
M
N
0
0
0
0
N 0
M 0
N
M 0
N 0
N ue~6le8 euLq3em-O#
0
0
0
M
~
0
0
N 0
M 0
Q 0
M
N 0
M 0
M 0
N 0
0
Scalar approach
297
TABLE II Exploring the acceleration ap4~roach. Identification of other than Type I UEPs and of t h e i r related machines SYSTEM
Fault location
Involved machines' identification
3-machine
I
(1,3) : Type 2 UEP
6-machine (Tunisian)
4
(5,4,3,6,1) : Type 5 UEP
7-machine (CIGRE}
4
(4,5) : Type 2 UEP
9-machine
15
(3,6) : Type 2 UEP
14-machine (Greek)
0 17 19
(0,21,19,17) : Type 4 UEP (17,19) : Type 2 UEP (19,17) : Type 2 UEP
15-machine
QI 01
(Q1,01) : Type 2 UEP (OI,QI) : Type 2 UEP
40-machine (Belgian)
DROG3 FARC3 KALO3B LAN3B LANO3A MERC1 PTRE4 RODH3A STAL3
(DORG3 ,PTBR3) (FARC3 ,FARC4) (SCHE3B ,KALO3B) Type 2 (RODH3A, LANB3) UEP (STAL3, EISD3) (DOELIA,DOELIB) , (BRES4 ,SCLE4,PTRE4) : T.3 (RODH3A,LANB3) : T.2 (STAL3 ,EISD3) : 1.2 TABLE I I I
11213
4
SYSTEM DATA N +N L
B I
6 7 9 14
20 10 34 92
15
44
17 40
162 163
8,9:
CPUCOMPUTINGTIMES(T). RATIOS IN %
INDIVIDUAL
GLOBAL N
I'1819110
5 METHOD ACCURACY
II
Ill
26 13 36 119
.16 .01 .07 .04
99
.04
618
6/8
281 269
.03 .01
9/10 10/10
8/10 10/10
4/5 4/5 7/7 7/7 1 1 / 1 2 10/12 9/10 9/10
III
s.b.s. (Tb)
TG Tb
TI TG
TT TG
Tr Tr+TI
I
II
.14 .01 .03 .03
4/5 7/7 11/12 9/10
4/5 7/7 9/12 9/10
15
11
60
8
9
9 18 40
14 9 14
66 54 41
4 5 1.7
3 8 B
:03 .02 .01
7/8 6/8 10/10 9/10 1 0 / 1 0 10/10
32 147 100
22 8 95
41
1.4
9
37 19
0.2 0.8
64 37
compare the computing times needed respectively by the "Indivi-
dual approach" respect to
(TI) , and by the "Taylor approach"
(TT) , all with
T G ; the ratios are expressed in %; note also that
these comparisons do not take into account the computing time needed for the computation of the admittance matrix reduced at the generators'
nodes;
~i0: ratio of the CPU time needed to compute
Yr (Tr)
with respect to
the overall CPU time needed by the "Individual approach".
298
ch. V
V.5.5.2.2.
(i)
Discussion
Accuracy evaluation.
of T a b l e
l).
Notice
• InfZ~ence
of tranafer generally
are
eonductanaesallows ~ 7 and
servation
holds
not r e p o r t e d
for the
tion
of t r a n s f e r
raey
of
• Globa~ methods of the
vs.
individual
latter
• Tay~or's exceeding
loads
eonduetances
improves.
behave
the
almost over
functions
equally
the
with,
(see also
For a limited
0.45 see , it shows
according
columns
Computing proceed
Times
(CT)
similar
step-by-step
CCTs
periods.
parts
and
cerns
the r e d u c t i o n
the
fault-on
through Evans
zhe
period
Tb
choice 7:
the
9:
notice
The
was
say
two
advantage
of T a b l e
of CCTs,
III).
for CCTs not
various
hand,
corresponding admittance at GBs
for this
type
this
used
Indeed,
split
CCTs
the
separately; the
other
is why we
rapidity and
the
admittance
into
one
matrix
two
conof the
matri×
of
one
in R i b b e n s - P a v e l l a
of d i s t u r b a n c e s ~
of
the post-
the rest
of the p r e - f a u l t
suggested
our
the
simulations
the p r o c e d u r e s
the r e d u c e d that
During
Therefore,
in the f a u l t - o n
matrix,
from
technique
we have
systems.
we have
N.B.
sought.
~ndication;
valid,
is d i f f e r e n t
5~SC
not
a crude
them
may be d e r i v e d
ratios
generally
is o b v i o u s l y
gain
and
reduction
is
substantial
the
system
size.
The
i00 %
rapidly
gain
with
the
system
size;
in CT for m e d i u m - s i z e d
and e v e n
more
~ystems.
group (LS).
inc,'ease w i t h
arbitrary;
in CT i n c r e a s e s
the
Other t y p e s
second
Sheddings
for
I). The
for all.
for large
V.3.5.3.
their
superposition
once
the
other
of the
that
(1985a); thus,
computed 6:
Note
give
To make in the
On the
estimated
procedure.
efficiency
integration
fault
approxima-
accurate.
can only
by c o m p a r i s o n s .
implying the
computational
ob-
and the aecu-
a slight
( i i ) Estimating computational efficiency (Table I I I ) . simulations,
the
of T a b l e
# 4,5
same
to s i m u l a t i o n s
justified
~ 8,9
range
The
at GBs,
however,
I). The
transfer
seen by c o m p a r i n g
criteria.
is b e t t e r
(columns
well,
former
approximation.
as it may be global
localized
neglect
of T a b l e
t c . Including
criteria
are
column.
~ 6~7~8
with
to the Kyoto
acceleration
: when
ta
(columns
agreement
(column ~ 5
criterion
of this
improvements,
relative
here
direct
results
conductanc~s
in good
further
8
Conventional
*
the erratic
results columns
Large-scale power systems stability
of d i s t u r b a n c e s
of d i s t u r b a n c e s Table
IV lists
concerns some
Line
Trippings
of our i n v e s t i g a t i o n
(LT)
and Load
results.
Scalar approach
299
TABLE IV
Dist Dist Dist Dist Dist
Obviously,
Disturb. identif,
tc
tc£
ta
~I ~2 ~3 ~4 ~5
t' kG
tkG
tkl
i
0.32
0.08
0.35
0.35
0.32
0.33
2
>i
>i
>I
0.38
0.38
0.66
3
>I
>1
>I
0.33
0.33
0.51
4
>1
>1
>I
0.28
0.28
0.35
5
>I
>1
>I
0.44
0,43
0.42
: : : : :
14-machine Greek system : 14-machine Greek system : 14-machine Greek system : 40-machine Belgian system 40-machine Belgian system
the Kyoto
and
LT of the double line 32-72 LS of the load 191.7 MWat GB ~! LS of the load 344.3 MWat OB ~30 : LS of the load 267 MW at GB ~MOL3 : LS of the load 855 MWat OB I~AVELI
the acce]eratior~
approaches
behave
quite
dif-
ferently.
The a c c e ] e r a t i o n by the
V
methodo]ogy.
function
justified
and
to n e g l e c t
reasonable
accuracy
It is b o u n d
its a c c u r a c y
transfer limits
to the
depends
conductances.
and
it never
assumptions
upon
which
Yet,
it always
fails
implied
extent
to depict
it is
lies
the
within
severity
of a d i s t u r b a n c e .
The Kyoto m e t h 0 d o ] 0 g y .
It g e n e r a l l y
ever,
exceptions.
the
following
(i) In p r e s e n c e curves Kyoto
two
of "mild"
oscillations method
will
"stop"
system
unstable
whereas
CCT may
be larger
0.3sec
).
(ii)
Fig.8a
Conversely,
that
Vp
larger
chronism Kyoto
(apart
from
CCT over
pose
For each ~max
the a b o v e
to c o m b i n e te
: max ~ij
(i) if
Vpmax
serious
of very
maximum
small
of
Vs
swing
: the
and d e c l a r e
the
from
instability
(e.g.
the K y o t o ' s
CCT
severe
for r o t o r very
provided
disturbances,
angular
the
:
close
it may h a p p e n
differences
to s y s t e m ' s
exceptions).
by Kyoto
much loss
In this
overoptimistic.
discrepancies, approach
the m a x i m u m
= max(~i-~j) is r e a c h e d
that
maxima
how-
(E.g.
of syn-
case,
the
the ratio
may be as low as
30 %.)
a situation.
the Kyoto
compute
happen
with,
, whereas
few p e c u l i a r
that
accurate
case.
unacceptably
such
local
it is far
is in fact
very
it may
1.5sec this
its m a x i m u m
becomes
Fig. Sb i l l u s t r a t e s To a v o i d
than
very
irrelevant,
first
actually
i80 °, w h i c h
of the a c t u a l
at the
in p r e s e n c e
criterion
local,
illustrates
reaches
than
disturbances,
produce
actual
proves
with rotor
Van the
angular
(i,j = 1,2,...,m) before
~ma×
Cutsem
et al.
following
difference
denote
reaches
(1986)
practical
it.
: let
Then
90 ° , d e c l a r e
pro-
rule.
: that
CCT
300
Ch. V
is very (ii)
large
(larger
conversely,
if
than
6max
Large-scale
power
systems
stability
i see ); reaches
180 ° b e f o r e
Vp
reaches
its actual
!
maximum, rather (iii)
declare
than
Vpmax
otherwise,
the s t a n d a r d
!
Vma x = V p
(6 = ~max : 180°)
and
apply
Kyoto
with
Vma x
;
i.e.
Kyoto
if
Vpmax
is r e a c h e d
for
180 ° > S m a × > 9 0 ° , use
procedure.
Angle (deg.) 300 Ang[e(deg.) js'" js
15
200
sS s
10
I00
~T"
t
t (s)
(s)
m.
0
o
0.1 0.2 0.3 0.~ 0.5 0.6 0.7 0.8 0,9 1.0 3~SC at bus ~9
o.1
Figure 8a
of the v a r i a n t s
pros
With
proposed and
respect
(1) The
o6
Figure 8b Swing curve of some r o t o r angles o f the Belgian system
Conalu6ion
provement Their
o'.5
: sustained fault : fault (critically) c l e a r e d at .36see.
Time e v o l u t i o n of Bmax =632 o f the CIGRE system
Some
0.3
3~SC at T R I V 3
C l e a r i n g time : t e = O . 2 8 s e c . A c t u a l C C T > lsec. (see in T a b l e I)
V.3.5.4.
o2
Taylor
function
cons to
may
in T o u m i
Cutsem
be
series-based
proves
the r e m a i n i n g
criteria,
(i.e.
equally
in fact
very
have
along been
with
the
assessed
im-
here.
:
(combination
- with
those
(1985)
as follows.
either
to be the most
among
- and
al. (1986)
z~iaieney
criterion
or i n d i v i d u a l
PSDE),
et
et al.
summarized
cemp~ta£iena£
- global
acceleration
devised
by V a n
which
- fast)
of the
energy-type
of the K y o t o
and
the
straightforward; show e q u a l l y
are
:
straightforward
Scalar approach
301
(2) the Kyoto individual TE function, i.e. conjunction
with the Kyoto
the individual
function
With respect criterion
in conjunction
in the case of very lies almost
with
the a c c e l e r a t i o n
to aaa~raa~, criteria
(2) whatever
always
the severity
severe within
in
approach;
(3) the acceleration global energy type function, i.e. energy
function,
(1) and
(2) are very accurate
(see below).
whereas
standard
approach.
of the disturbance,
disturbances ± 5%,
the
criterion
:
criterion
Their
(1)
accuracy
(3) is less accurate
(± 15 to 20 %). , With respect (i) and used.
to p o s s i b i l i t i e s
(3) show very
Indeed,
in order to assess
for a given
clearing
limit
of
value
V
the Kyoto m e t h o d
(generally
whereas
a priori rather
criterion the
time,
be
to instability"
one has to know the
computation
step-by-step
criteria
(2) cannot
"distance
short)
prior to a step-by-step necessitates
6tabili~y margin~,
of devising
interesting,
up to CCT;
computation
for even
and larger
time than CCT. Hence,
three
direct
- the T a y l o r stability may
series-based margins,
individual
not appropriate
tions,
criterion
0.45 sec,
the disturbance
criterion
and belong
cern is "how to know indeed
is valid
the
to the type
at the instant
for a given power related strong
line tripping
of the
induce machines'
large
disturbances,
:
for it
it is concept;
than the
margin
defini-
disturbances"
short
But then,
ones
for which
up to,
circuit"
(3@SC)
lying
accelerations.
initial Indeed,
is typically
Table V illustrates on the initial
: one can see that
(which
is electrically
in quite
be-
?
accelerations
severity
of con-
disturbance
by the ranking of the machines' inception)
say,
the question
the considered
of the disturbance.
accelerations
margin
accurate
to CCTs ranking
the range of these
at AVEL1
stability
"very severe"
system machines
(855 MW)
appropriate
less
severe
buses.
of the disturbance
tions of the Belgian
fast,
for stability
"very
of disturbance
to the severity dependence
needs
severity.
for sure whether
system,
with
fast but
"three-phase
generators'
to the class
different
fast and very accurate,
correspond
A sound answer may be provided (i.e.
for very
it is appropriate
acting at "important"
longs
: extremely
: very
: very
In the above discussion,
the Taylor
to meet
ones;
criterion
criteria,
whatever
criterion
called
for use in conjunction
the a c c e l e r a t i o n previous
emerge,
and very accurate
fail for less severe
the Kyoto
N.B.
criteria
different
the
accelera-
the
3~SC and a
close
to DOEL1)
ranges.
302
Ch. V
Large-scale power systems stability
TABLE V
I
2
3
4
Bus
30SC at AVEL1 7
Bus
LT at AVEL1 7
DOELIA DOELIB CLIO02 SCHE3A SERA2 DROG3 PTBR3 COUG4 MERK4 CO01 TRIV3 FARC3 MONC3 PTRE4 BAUD3 FARC4 AWIR3 MOL3 SCIIE3B DAMP3 GRAMI VILG4 AWIR4 8RES4 KALO3A RODH3B KAL03B SCLE4 LANB3 RODH3A LAN03B REVII EISD3 RUIE3 FV~ASI STAL3 ROMK1 LANO3A
3.44 3.27 2.65 1.90 1.88 1,77 1.72 1.55 1.28 1.24 1.13 1.04 1.01 0.98 0,95 0.94 0.93 0.91 0.89 0,83 0.81 0.75 0.74 0.73 0.67 0.64 0.64 0,61 0.59 0.53 0.50 0.41 0.39 0.37 0.29 0,27 0.17 0.14
DOELIA DOELIB SCHE3A AWIR3 SERA2 PTBR3 COUG4 C001 DROG3 AW[R4 SCHE3B CHO02 GRAMI VILG4 MERK4 TRIV3 FARC3 MONC3 FARC4 DAMP3 BRES4 MOL3 BAUD3 KALO3A PTRE4 SCLE4 KAL03B EISD3 RODH3A RODH3B SIAL3 LANB3B LAN03B LANO3A RULE3 REVI1 MAAS] ROMKI
V.3.6.
Direct criteria
for
19.20 17,86 10.95 10,63 9.93 9.71 9.12 9.01 8.72 8.71 7.65 7.49 7,35 7.27 7,11 7.00 6,96 6.94 6.92 6.82 6.47 6,30 5.18 5.13 5.01 5.01 4.87 4.83 4.50 4.42 4.42 4.40 3.42 2.57 2,27 1.08 0.41 0.38
real-time operation.
Prospectives and p e r s p e c t i v e s I) The above criteria solve efficiently the real-time transient
sta-
bility problem, provided that the very simplified system modelling that they impose is deemed satisfactory enough. 2) Within the applicability
limits determined above, their reliability
and accuracy are not affected by the system size. 3) The only lengthy computation they require is the reduction of the admittance matrix at the generators' improvements along this direction.
nodes; there is still room for
Vector approach
303
4) With the exception smoothly tion,
of the above,
with the system
(whenever)
needed
inception
and clearance,
(Taylor's
expansion,
5) A timely
question
al research
indeed,
there
varies quite
the step-by-step
the short period
between
computa-
disturbance
short-cuts
may be used to speed
variable
step size,
etc.).
is whether
seems
purposes,
and comparisons.
as regards
time
many
in terms of disturbance not. The answer
size; during
the computation
"No"
the conventional
Critical
Thus,
of other,
or
"no".
a handy way of e×plorations
it allows
effective
other,
it i~s very
"Yes",
assessment
is adequate
to be "yes" and
for actual
the outburst
(CCT)
since
are certainly
to consider.
stability
Clearing Time
it up
for m e t h o d o l o g i c -
real-time
more physically
likely
applications;
sound,
parameters
that the coming years will
more adequate,
alternative
stability
see
measures
and assessment. 6) Another
short-term
and use of the Hill
(198~).
admittance
perspective
"structure
In essence,
matrix
this model
at all the system
ard load flow calculation, nodes.
Relaxation
computationally makes
matri×,
demanding
of transfer
acceptable;
the unreduced
in its non-diagonal tained more
rather
of this matrix
most
the neglect
and p e r f e c t l y
terms.
system model
advantages
will be obtained
variables
probably
predominate
nodes,
by Bergen
to k e p p i n g
and
unreduced
as they appear
reduction
amounts
But much more
unlike
the
in a stand-
the
importantly,
assumption
part
it
very realistic
of the losd effects
refinements
than with the reduced see De Marco
~t the expense burden.
to a v o i d i n g
the redu('ed admittance
not include
Other model
(e.g.,
and implied
proposed
amounts
conductance
one does
easily with this
the practi~,al applic:~tion
model"
than redu~,ing it at the generators'
step.
indeed,
equivalent
state
concerns
preserving
and Bergen, of a grester
But overall,
over this relatively
may also be obat the Kenerators'
minor
1984).
Those
number of
the advsnL~{ges will inc~)nvenience.
V.4. VECTOR LIAPUNOV APPROACH V.4.1.
Introduction
Attempbs tions,
to overcome
have
position-aggregation speaking,
some of the difficulties
led to a new approach method
this approach
based
and the vector
consists
met with
the scalar
upon the concept Lia~unov
in d e c o m p o s i n g
func-
of the decom-
functions.
Roughly
the multimachine
system
304
Ch. V
into a number tions
of i n t e r c o n n e c t e d
and of suitably
latter,
and for evaluating
tructed which V
involves
functions,
SDEs and certain
metho d o l o g y
appears
analysis
special
quite
simple
low-order
features
to the sensitivity
In the studies
devoted
and developed.
They mainly
(i) the power
properties
parts;
for these
for c o n s t r u c t i n g
function
model
scalar
of their
based on the subsys-
of a power system;
: besides models
it offers
and hence
using
interconnections.
mathematical
subsystems,
V
is cons-
system is d e t e r m i n e d
for many reasons
expressions
analysis
of interconnee-
an aggregate
Liapunov
sophisticated
analytical
structural
horizons
Then,
attractive
of more
of disconnected
to obtain
(free)
techniques
SDEs.
a vector
composed
and a SDE of the overall
subsystem's
ing c o n s i d e r a t i o n
subsystems,
disconnected
use is made of standard
functions
tems'
chosen
Large-scale power systems stability
This
allow-
for the
possibilities
information
about the
this in turn opens new
problem.
to this area,
various
aspects
have been examined
deal with
system modelling;
(ii) the system equations
decomposition, of the overall
each subsystem
into
i.e.
the d e c o m p o s i t i o n
system into
subsystems
its disconnected
(free) part
of the state and
further
of
and its inter-
connections; (iii)
the subsystem functions
stability
for the
analysis,
free subsystems
i.e.
the c o n s t r u c t i o n
and the evaluation
of
V
of their
SDEs; (iv)
the system aggregation, function
ru~e relating inition
i.e.
for the overall
the subsystems'
of an a g g r e g a t i o n
the choice
system, V
matrix
of a scalar L i a p u n o v
resulting
from the a g g r e g a t i o n
functions
and evolving
which accounts
the def-
for the influence
of the interconnections; (v) the d e t e r m i n a t i o n Concerning
developments
that all studies so - transfer non-uniform. posit i o n
take
of the
(asymptotic)
of system m o d e l l i n g
into account
conductances
and mechanical
(ii),
one can succeed
in getting
Lur'e-Postnikov
V functions
and their SDEs are readily
composition
plays
consideration,
an important
but also
the c o n s t r u c t i o n
influences
(i),
damping
considerably
because
free
type
be able
of the a g g r e g a t i o n the degree
system
subsystem's
Note that
to the above
(iv) which
of conservativeness
suitable
system de-
of i n t e r ~ o n n e c t i o n s matrix
or
decom-
representa-
for which
role not only with respect the choice
to do
- either uniform
system,
available.
system.
let us at once say
- or at least woul~
This is owing to the fact that by a suitable
tion of the w e l l - k n o w n
ences
SDE of the overall
influ-
in turn
of the overall
305
vector approach
method. In fact, the conservativeness, tutes the weak point of the method,
which as we shall see consti-
depends essentially upon above
steps (iv) and (v). In what follows we develop at first a thorough "classical" analysis based on Bellman's d e c o m p o s i t i o n - a g g r e g a t i o n method modified and adapted to the power systems p r o b l e m (Gruji6 and Ribbens-Pavella,
1977). Then we briefly outline the main feautres of
other a g g r e g a t i o n - d e c o m p o s i t i o n
forms of power systems. This review
merely aims to bring out the essential features of the vector L i a p u n o v function approach at the present
stage of progress, with the ultimate
goal to pinpoint its specific potential field of practical use.
V.4.2.
large-scale systems decompositions and aggregations in general
Stationary
V.4.Z.?. Gznzral comparison of ~ca~ar and v z a ~ o r
Liap~nov fa~ction~
In the framework of Liapunov stability considerations there exist different approaches to power systems aggregation,
which are not completely
unrelated but can lead to different stability requirements imposed on the systems. They were considered in general in the preceding chapters and outlined in the sequel to the extent suitable for power systems. One is Liapunov's classical aggregation of dynamic systems If the system is described by an tor differential equation
M-th
(Chapter I).
order nonlinear autonomous vec-
(2.2) denoted herein as
(4.1),
: f(x) then, by following Liapunov tion
v : RM~R
(1892) a scalar positive definite func-
is used as an aggregation function for the system.
the sake of clearness, Its eulerian
(4.1)
let
v e C ( I ~ R M)
(total time) derivative
be differentiable v along motions
in
X(t;xo)
For
×eR M ,
X(0;Xo) ~ x o , of (4.1) is obtained in the form v(X) : [grad v(x)] Tf(x)
,
(4.2)
where av
grad v(X) Here,
(a~ '
Ov
OV
ax 2 '""
T
' a~ )
(4.3)
X = (XI,X2,...,XM)T
Notice that for the autonomous can for instance be
2N-1
N - m a c h i n e power system the dimension
, M : 2N-i
,
(4.4)
and f(x)
: PX + h ( x )
Use of a scalar positive definite aggregation function
v allows the
M
306
Ch. V
M-th
order system
Large-scale power systems stability
(4.1) to be aggregated into a first order equation
(4.2), which can be set in the form v(x) : ~(X)
,
~ : (grad v)Tf
.
(4.5)
For Liapunov stability analysis it is necessary and sufficient to prove an appropriate sign property of X
of
x=0
¢ on a certain connected neighbourhood
. For asymptotic stability analysis of
should be chosen so that
x=0
x=0
, the set
X
be the unique equilibrium state of (4.1)
in it. Liapunov's aggregation of (4.1) into
(4.2) is called also 8caZar Lia-
punov function ayyreyation of (4.1). In general it is extremely difficult to find a suitable function (4.1) such that a required sign property of ined on the set
v for
~ can be effectively exam-
X. This problem is fundamental
for the Liapunov sta-
bility theory and its applications, but it has not yet been solved either in general or for power systems in particular. That is why different majorization approaches were proposed. When they can be applied, they yield exact problem solutions rather than approximate. One of them is based on the comparison principle i950; Matrosov,
a scalar function
(i)
(Kamke, 1932; Wa~ewski,
1968; and Chapter II). It is effective if we can find @i
such that
@(X) ~ @ l [ v ( x ) ]
(or
@(X) ~ ~ l [ v ( x ) ]
)
,
VX e X
,
(4.6)
and
(ii) the required stability property of (4.1) can be effectively proved for the comparison system : @I(W)
(4.7)
We can note that a stability property of (4.7) can be in special cases concluded from the motions of (4.7) obtained by solving it, or by repeating the whole above procedure applied now to (4.7) rather than to
(4.5). Another majorization approach is based on the assumption that we can find a scalar function
(i)
¢(x)
@2
~ ¢2(x)
such that
:
(or
~ ¢2(x) )
¢(x)
,
vx•X
,
(4.8)
and
(2) the required sign property of
X (vx e X )
@2
can be effectively verified on
.
From the technical viewpoint,
suitable form of
~2(X) = uT(x) A(X)w(x)
@2
,
is the following :
(4.9)
Vector approach
A :R M ~ R s x s
where s
307
is a p o s i t i v e
sign p r o p e r t y
, U,W :R M ~ R s
integer.
on
X.
on
are d e p e n d e n t
T h e y are r e q u i r e d
X in g e n e r a l
to p o s s e s s
The test of these p r o p e r t i e s
and
the a d e q u a t e
is s i m p l i f i e d
if
~2
can be found such that e i t h e r (a)
U(x) E w ( x )
and
A
is
(elementwise)
U(X) e b
(elementwise)
constant,
or (b)
is
The L i a p u n o v
aggregation
system s t r u c t u r e . procedure Bellman
link b e t w e e n
V
function
and the s y s t e m a g g r e g a t i o n ,
proposed
their approaches
(Vl,V2,..-,Vs)
functions
V and
stability property
of
From the p u r e l y
T
two o t h e r a p p r o a c h e s .
we i n t r o d u c e
V :
,
V are
The
(1976). s
V :~M
scalar Rs
functions
, v : RM~R s ,
In fact,
(4.10)
v e c t o r Lfapunov f u n c t i o n s of
verification
elementwise
mathematical
(Vl,V2,...,Vs)T
called
allows
(4.1).
sign p r o p e r t y
(4.1)
construction
(1962)
(4.1) iff t h e i r a p p l i c a t i o n
with
the
,
V :
a certain III).
does not r e c o g n i z e
RM~R and v e c t o r f u n c t i o n s
, Vi :
vector
(4.1)
A is also constant.
them can be made by f o l l o w i n g La S a l l e
ml,m2,...,m s : M
The
of the s y s t e m
and M a t r o s o v
In o r d e r to e x p l a i n V i : Rmi~R
v e c t o r and
F o r this r e a s o n and in o r d e r to find a more e f f e c t i v e
for b o t h
(1962)
constant
V and
(La Salle,
point of view,
of a r e q u i r e d
v are r e q u i r e d 1976,
V,
or
Liapunov to p o s s e s s
and C h a p t e r s
v,
II,
is a s s o c i a t e d
so that
V(X) = ~(X)
,
•
:R M~R s
(4.11)
or 9(X) From the s y s t e m s t r u c t u r a l into
s
interconnected Xi : fi (X)
If the s u b s y s t e m s h i : R M ~ R mi
and f u n c t i o n s
viewpoint,
(4.12)
the s y s t e m
(4.1)
is d e c o m p o s e d
subsystems
• fi : R M ~ Rmi
are l i n e a r l y
describe
= ~(×)
their
gi : R m i ~ m i
, x i e R mi
, Vi : 1,2,...,s .
interconnected
then the f u n c t i o n s
i n t e r a c t ions a n d are c a l l e d interactions, describe
free (disconnected) subsystems
xi : g i ( x l ) , Vi : 1,2,...,s so that
(4.13)
(4.14)
fi : gi + hi ;
interconnected subsystems are t h e n g i v e n by Xl
The f u n c t i o n s (4.15),
vi
in general.
:
gi(xi
) + hi(x)
,
vi
are a s s o c i a t e d w i t h the Then,
their
: 1,2,.,.,s
i-th
(4.15)
subsystem
form is i n f l u e n c e d
(4.13),
by the f o r m of
i.e. fi
308
Ch. V
(of both
gi
and
by the form of
hi
respectively).
Large-scale power systems stability
The form of
Vi
is governed only
gi , and the form of the dependence of
hi
not taken into consideration. The a g g r e g a t i o n of the system posed by Bellman decomposition
(1962) and M a t r o s o v
(4.13) in general,
c o n s t r u c t i o n of tively. Then,
Vi
for
on
x is
(4.1) pro-
(1962) is based on the system
or on (4.15) in particular,
(4.14) and
vi
for
and the
(4.13) or (4.15), respec-
(4.11) or (4.12), respectively,
is the aggregated opera-
tion related to (4.1). The B e l l m a n - M a t r o s o v a g g r e g a t i o n
(4.11) or
(4.12) of (4.1) is called
the veotor Liapunov function a~gregation of (4.1). Notice that Bellman
(1962) proposed the use of
(4.10) was i n t r o d u c e d by La Salle
V
(4.10) whereas
v
(1976) and d e v e l o p e d in Chapter II.
Further p r o c e d u r e is the same for both
(4.11) and
(4.12). Hence,
it
will be explained for (4.11). The a g g r e g a t e d equation
(4.11) can be used in two conceptually differ-
ent ways which are analogous to those explained gated e q u a t i o n
(4.5).
One p o s s i b i l i t y is realizable that
for the scalar aggre-
if there exists a vector
w : R s ~ R s such
(elementwise) • (X) ~ w [ V ( x ) ]
Then, p r o v i d e d the function
(or
~(x)
~ w[V(x)] )
w : (Wl,W2~...,ws)
T
, vx e X
(4.16)
satisfies
' V 'i+l,'",Vs' ) -< wi(Vl, " V "2,'",V "i-l,Vi,Vi+l " wi(VI,V~,'",V~-I,Vi, ,''',V "s ) ,
v] ~v~
,
vi,j
:1,2,...,s
,
i~j
,
vVcR s
,
(4.17)
the stability property of (4.1) is deduced from that of the comparison system (Kamke,
1932; WaBewski,
1950; Matrosov,
:w(u)
,
1968, and Chapter II)
u eR s .
:
(4.18)
S t a b i l i t y test of the comparison system is generally not available when W
is a n o n l i n e a r function.
Often, the test requires application of the
(second) L i a p u n o v method at the last step. Another p o s s i b i l i t y
for stability analysis based on the use of the
a g g r e g a t e d e q u a t i o n is to employ a scalar function where
° s
b e R+
vI ,
vi(x)
= bTV(x)
,
(4.19)
vl(x)
: bT@(x)
,
(4.20)
. Then,
due to (4.11).
In fact, we arrive at the p r o b l e m of testing appropriate
sign properties of
vI
The approach of M a t r o s o v
(4.19) and
@ = bT~
(4.20).
(1962) supposes existence of
v i's
such that
vector approach
309
v = v I + v 2 +...+v s
and
These c o n s i d e r a t i o n s aimed for practical
~ =v
possess
show that
the a p p r o p r i a t e
crucial
applications
steps
in the
sign properties.
stability
are the f o l l o w i n g
analysis
:
A. The system decomposition. B. The choice
of the a g g r e g a t i o n
Liapunov
v.
function
C. Sign tests D. Useful
of
v
and
estimation
The first two
steps
(C and D) provide
¢ =v
functions
system
.
of the asymptotic
stability
(A and B) considered
a number
and of the overall
of different
domain.
separately
from the last two
possitibilities.
Essentially,
the problem consists in the choice of such an aggregation-decomposition form that sign tests of with practically This p r o b l e m scale power
is treated systems
-
in general
aggregation
over the scalar one
use of a more
the overall
and in the
domain.
framework
of
system,
Vi's
of the system
for the f o l l o w i n g
refined m a t h e m a t i c a l
system via
- possible
herein
function
- easier c o n s t r u c t i o n
whole
~ can be effectively performed together of large-
in particular.
The vector L i a p u n o v advantageous
v and
useful estimation of the asymptotic stability
:
model of the system;
for subsystems
and hence,
is c o n c e p t u a l l y
easier
than of
construction
v
of
directly v
for
for the
V;
reduction
of the order
of the a g g r e g a t i o n
matrix
to the num-
ber of the subsystems; - derivation -
of effective
condideration variations
easier p a r a m e t e r
The drawback
estimation
of the vector
property
subsystems
sensitivity
- easier analytical
of the
Liapunov
in it. However,
function a g g r e g a t i o n Majorizations follows
which
hence also on these Majorizations
(asymptotic)
function
application
used in a g g r e g a t i o n
: interactions
of the overall
and therefore
system
from
possibility
stability
aggregation
state of the r e s e a r c h
can also require
state space are considered
of
of
analysis;
tions, which are at the up-to-date sarily involved
of the system and influence
on its stability;
to infer stability
of d i s c o n n e c t e d
criteria;
properties
of its structure
possibility stability
stability
of structural
of the
different
scalar
domain.
are m a j o r i z a almost
neces-
Liapunov
majorizations.
of the system can be i n t e r p r e t e d
are stabilizing
as d e s t a b i l i z i n g
on suhspaces on the whole
as
of the system state
space;
subspaces.
can prevent
effective
application
of the vector
Liapunov
310
Cb. v
function aggregation;
this
Large-scale power systems stability
is a t e c h n i c a l
rather than a conceptual
draw-
b a c k of the a p p r o a c h .
V.4.2.2. $aJ.tab£a aggregation form~ ~nd ¢omp~ri~on ~na£i~n~ The preceding
exposition
ine e f f e c t i v e
possibilities
The procedure
o f the v e c t o r L i a p u n o v
a
suitable
decomposition
With each subsystem tiable
requires
function Vi(x)
and d e t e r m i n e
let
four d i f f e r e n t of
starts with (4.15).
definite
differen-
, vi = 1 , 2 ....,s .
(4.21)
, Vi : 1,2,...,s .
(4.22)
•
forms w h i c h are b a s e d on suit-
-
One is b a s e d on the a s s u m p t i o n , functions
(4.20).
subsystems
a positive
:RM~
aggregation
@i's
and
,
= [grad V i ( x i ) ] T [ g i ( X i ) ÷ hi(x)]
able majorizations
xi = 0
(4.1) into
Vi
@i
(4.12)
function aggregations
(4.15} we a s s o c i a t e
Vi
in o r d e r to exam-
(4.11),
= [grad V i ( x i ) ] T [ g i ( x i ) + h i ( x ) ]
~i(x)
i #j
(4.9),
of the s y s t e m
F o r the sake o f c o n c i s e n e s s
T h e i r exist
deeper elaboration
of u s i n g
u i : Rmi~R
that there exist n u m b e r s
~ij
, ~ij a0 ,
and o p e n c o n n e c t e d n e i g h b o u r h o o d
Xi
of
such that S
@i(X)
Here,
~
~
~ijuj(Xj)
j:l
X :XIXX2x...XXs
form
of Gruji6
(1973),
ui(xi)
and S i l j a k increasing
: ¢i3(llxill)
for c h o s e n totic
+ integral Another Grujic
, Vi : 1,2,...,s
Chapter
comparison
analysis
IIl. All
functions
, determined
of all
aggregation
~i3
ui's in
of power" systems w h e n
. This ag-
generalizes
that
are r e q u i r e d to be lJxill ,
by the d i s c o n n e c t e d form a p p e a r s
Xi's
form)
(4.25)
subsystems
too s t r i n g e n t Vi
(4.14)
for asymp-
is " q u a d r a t i c
form
of n o n l i n e a r i t y " .
form is of the L u r ' e - P o s t n i k o v et al.
functions III)
(in short,
V i . This a g g r e g a t i o n
stahility
VX EX
is the c a r t e s i a n p r o d u c t
gregation-comparison
strictly
,
(1976),
Chapter
and real n u m b e r s
III.
~ij
type p r o p o s e d
It a s s u m e s
, and
~ij
by Gruji6
e x i s t e n c e of
such that e i t h e r
(1974),
ui
and
fi
(Chapter
: S
*i(x)
~ ui(x i)
Z
{~ijuj
j=l
(Xj
+ #ijfj
[uj(xj)]}
,
vxeX
, Vi
=l,2,...,s
,
(4.24)
or (Chapter III) S
{~ijuj(Xj)
*i(x) j~1
+ ~ijfj[uj(xj)]}
, VX e X
~ Vi :1,2,...,s
, (4.25)
Vector approach
for
~ij a0
Vu i > 0
3! I
, ~ j a0
, fi(0) =0
suitable
, i ~j
and
. The non-linearities
fi(ui) • C ( R )
for global asymptotic
fi
obey
. This aggregation
stability analysis,
fi(ui) z0
form appears
which is not the
case for power systems. The third possible aggregation of functions
~j : RM~R
and
of the system is based on the existence ui
such that
(Chapter III)
:
S
~i(x) ~ u~(x i)
~ ~ j ( x ) uj(x]) j=l
,
vx~x
, vi :1,2,...,s (4.26)
In the case of power systems this form appears most suitable for estimating the asymptotic
stability domain.
In special cases, when Michel
Then,
ui(x i) =¢i3(|Xi|)
~ij(X) z~ij =constant
, the aggregation
(1974) results.
Another suitable form for asymptotic tems is obtained from Gruji6 S
(1974)
stability analysis of power sys(Chapter III)
:
S
¢i(Xi) s
Z ~ij(X) ui(Xi)uj(xj) i~j=l
i:l For power systems
~ij(x) z~ij
,
Vx6X
is obtained, where then
Since this research
.
A,
is aimed to establish both asymptotic
criteria and asymptotic
stability domain estimates,
(4.27)
~ij
element of the overall power system aggregation matrix
form
is
i~-th
A = (~ij) stability
the aggregation
(4.26) will be only used.
Seeking possibilities
for the relaxation of the stability
conditions
and for the best stability domain estimate we considered different composition-aggregation al.
.
form of
(1977,1978)
The aggregation
in terms of
Some of them were proposed by Joci~ et
and by Gruji6 et al.
(1977) and others were new.
form (4.26) appeared the most suitable
system aggregation. system without
forms.
de-
It enables one-shot
requiring majorizations
for the power
stability analysis of the whole of the comparison
functions
Vi .
The investigations ui(xi) =llxi]i
were done with the following comparison ,
ui(x i) : ~wi(xi)ll
or
functions
ui(×i) = (w~Wiwi)I/2
for wi(x i) = [¢i(Xil) xi2] T
,
wi(xi) : [~(I¢i(xil)I + Ibihl2~i(xil)I)IX~I{ i Xi2] ~ Or
wi(xi) : [V~i(xi~) xil xi2] T
u.1
312
Ch. V
F r o m the v i e w p o i n t
wi(x i ) seems
of the r e l a x a t i o n
estimate totic
But,
of the d o m a i n of a s y m p t o t i c
stability.
of o t h e r
conditions
its use e s s e n t i a l l y stability.
seems to e n a b l e the best Its use r e l a x e s
forms of
the p r o p o s e d
estimate
the s t a b i l i t y
u i . For this r e a s o n ,
decomposition
The c o m p a r i s o n
function
of the d o m a i n of asympconditions
we accept
and a g g r e g a t i o n
r e d u c e d the
more than use
ui(Xi) =UXill
of the p o w e r
and
system.
General stability analysis of stationary large-sca]e systems
V.4.3.
V.4.3.1.
Stabili£y
Let a l a r g e - s c a l e be d e c o m p o s e d denoted
every
xo ER M
Let
Xi
Xi's
,
criterion system described
into
(4.15)
s
ty
Xi(t;Xo)
subsystems
, X i ( 0 , × o) e x l o
a n d to be c o n t i n u o u s
be a subset
of
R mi
and
in
X
equation
(4.15).
, is s u p p o s e d
teR+
to exist
for e v e r y
the c a r t e s i a n
(4.1)
A m o t i o n of for
XoeR M
product
of all
X :XlxX2x...XXs .
Assumption
1.
functions
V i : Rmi~R
There
, satisfying
(i)
by the d i f f e r e n t i a l
interconnected
In the sequel the f o l l o w i n g
i ~j
of the s t a b i l i t y
i [~/(l¢i(Xil)l +Ibih12@i(xil)l)]xil xi2] T
:
to be the best a m o n g them.
ui(x i) =IIxiH
Large-scale power systems stability
x =0
a s s u m p t i o n w i l l be u s e d
:
exist a c o n n e c t e d n e i g h b o u r h o o d and
u i : Rmi~R
(i)-(iv)
Xi
, and real n u m b e r s
of
xi=O
,
eij
'eij
z0
:
is the unique
equilibrium
state of the s y s t e m
(4.15)
in
X, (ii)
Vi
is p o s i t i v e
definite
and d i f f e r e n t i a b l e
(iii)
ui
is p o s i t i v e
definite
on
(iv)
the t o t a l time d e r i v a t i v e tions of
Notice
that
(4.15)
(4.21),
(4.22),
links those of M i c h e l Michel
(1974)
functions
obeys
(4.21)
of
(4.26).
•
XC
functions
rather
and r e l a t e d to s c a l a r L i a p u n o v (4.14) and s i m u l t a n e o u s l y tem aggregation aggregation
(4.2i),
(1974). ui
Vi
taken a l o n g mo-
functions (4.26)
and s t a b i l i t y a n a l y s i s
that
In c o m p a r i s o n w i t h
to be any p o s i t i v e
than to be those d e p e n d e n t
to be s t r i c t l y
(4.22),
X I , Vi = i,2,...,s ,
define a s y s t e m a g g r e g a t i o n
(1974) and Gruji6
it a l l o w s the
on the s e t
Vi
(4.22), (4.26)
on
X i , Vi = 1,2,...,s ,
of d i s c o n n e c t e d increasing.
on
~XiU
subsystems
The p r o p o s e d
a l l o w s also an o n e - s h o t
of the o v e r a l l
definite
system.
sys-
both
In c o m p a r i s o n
Vector approach
with Gruji6
313
(1974), it provides a larger estimate set of the asymptotic
stability domain. For stability analysis we need Assumption 2. on an
The function
ei - n e i g h b o u r h o o d
Vi
:
of
Vi(~iX i) < Vi(~2X i) V~kX i e Ni(Xi,~i)
1 is radially
of Assumption
Ni(Xi,ei)
increasing
X i , that is
iff
0 s~ 1 <~2
,
, Vi : 1,2,...,s , Vk : 1,2 ,
where
N i ( X i , [ i)
: {x : i n f ( H x - y l ; : y e X i )
A criterion for asymptotic stability of in this section by referring to Gruji6 rem 23 of Section III.4.2.5 in which M(L,e)
:
i.
Theorem
matrix inite;
A T B + BA
×:0
'
~i >0 . •
of (4.1) is established
(1974), Michel L =ATB +BA
(1974) and Theo-
, e =0
so that
.
If Assumption
B with positive then
~ ~i}
x=O
1 holds,
A = (aij)
diagona~ such that
of the system
This theorem is also valid when
and there is a diagonal ATB + BA
is negative
(4.15) is asymptotically ~ij ~O
, i #j
def-
stable.
, is not satisfied in
Assumption I. In particular cases determined by negative definiteness of is sufficient to set La Salle
s×s
matrix
ATB + B A
A : (~ij)
obeys
AT + A
it
matrix.
(1976) stated that there is a diagonal matrix
diagonal such that sxs
B : I - the identity
B with positive
is negative definite as soon as t h e M e t z l e r (4.28)
(see Conjecture
1 in SeCtion
V.4.3.2 for possible link between this statement and d) of Proposition 13 in Section 11.3.4) I~ii
: ~12
"'" ~ikl
1
> 0
,
Vk : 1,2,...,s
If Assumption I holds and A obeys (4.15) is asymptotically stable.
Corollary I.
the system
Positive result on asymptotic stability of cient for e n g i n e e r i n g applications,
X:0
(4.28)
then
(4.28)
x=O
of
often is not suffi-
e.g. for power systems stability
analysis. K n o w l e d g e , or at least an estimate, of the domain of asymptotic stability is needed.
314
Ch. V
Large-scale power systems stability
V.4.$.Z. Estima£zs of thz asymN£otic ~tabZ£Zty demaLn In o r d e r to define t e n t a t i v e (4.10),
vI
(4.19)
estimates
7 1 : min Weissenberger
(1973)
:
is the
i-th
rain
of
subset of
the b o u n d a r y
Now we can d e f i n e
V°
and the c l o s u r e
of E1
and c o n t a i n s
x i =0
is bounded.
and
of
[i
Vi
}
v
is the largest
:
vl×U2×
.
possible
× v .~
estimate
D we i n t r o d u c e
~ij
[ui(xi)
by r e f e r r i n g
2.
of
will
D
the e x c e l l e n t
symbol,
x=O
, xeX}
,
that we can d e r i v e under which
£ : (eij)
all
,
of (4.]5).
(4.54)
for c h o s e n
V
V is an esti-
by Vi,j : 1,2,...,s .
Vi's
(4.55)
are a s s u m e d b o u n d e d and
sup ,
to Gruji6
(4.55)
,
(1975) p r e s e n t e d
:xi e S V i ]
[ui(xi)
ri2 =
: xi E V i l
o Vi
and R i b b e n s - P a v e l l a
(4.36)
(1977),
Gruji6
et al.
EV ° < 0
then the set
at
If Assumptions
(4.34) i8 an estimate of (Gruji£
aV i
£i
conditions
V?
(1979), one a r r i v e s
:1,2
{x . : O <. V ( x )
=.
the m a t r i x
is the K r o n e c k e r inf
Vi
: [ril6ij + r i 2 ( l - 6 i j ) ] ~ij
ril :
. Besides,
s t a b i l i t y d o m a i n e s t i m a t e of
V ° . In o r d e r to e s t a b l i s h
eij
,
is d e n o t e d by
V ,
Proof
of
E2 ,
The
The0rem
V.
(4.32)
whenever
of the e x p o n e n t i a l
Now,
such that the c l o s u r e
Vi
It is to be n o t e d that W e i s s e n h e r g e r
Here
(4.31)
.
Xi
analysis
and
(4.3o)
: {×i : 0
[i = {x : v i ( × ) < ~
mate of
1,2,...,s]
s] ,
l
is c o n n e c t e d
set
V
D we use
(4.29)
: i = 1,2,..,
ivy: i :
component
h denote
of
(1973) as w e l l as
introduced
rVi(*a bi
V ° i
E2
and
[biV ~ :i =l,2,...,s]
v2(x ) : max u
Here
EI
by f o l l o w i n g G r u j i ~ and S i l j a k
I and 2 hold and if D
of
×:0
and R i b b e n s - P a v e l l a ,
of the ~ystem
1977;
Gruji6
Pl~ : {V : 0 _< bTv <7}
,
V
(4.15).
et al.,
1979).
Let (4.37)
Vector
approach
3] 5
P2~ : {V : 0 < vi~ < ~ bi
, V i :l,2,...,s}
,
(4.38)
and
Ei7 = {x :o ~ v i ( x ) be c o n n e c t e d
sets c o n t a i n i n g
Ei7 : {x : V(x) 6 P i 7
~ 7}
,
the origins.
' x ~X}
vi =1,2
,
(4.39)
Hence,
and
Ei7 i = ~i
,
i : 1,2
,
(4.40)
as w e l l as E2 = V Let
a+Pl?
and
iff
a+P27i
V~bi I = 72
be d e f i n e d
, Vi =1,2,...,s
.
(4.41)
by
~+PI7 : {V :bTV : ~ , V E P I 7 ~
(4.42)
a+P271 : {V : V i b i I : 7 , V ~P27 } .
(4.43)
and
Let aE27 i : {x : V(X) eB+P2~i , x EE27 } Under Assumption of
El7
is o b v i o u s
for
8E17
- the b o u n d a r y
:
Statement Let
2 the f o l l o w i n g
(4.44)
aE2~
I.
If
x 6 El7
then
denote the b o u n d a r y
Statement I I .
x~BE27
iff
V(x) e ~+P17 of
E27
iff
x E BE17
.
. Now,
~ i 6 {1,2,-.,s}
X e a E 2 7 i , i.e.
such that
s
~E27
Statement
III.
Under A s s u m p t i o n
V(x) ~V(y)
, VxeEi7
y =XlX Statements Lemma tions
I. vi
If there are
The
Assumption
i, all
em : min
~i
of the s y s t e m , vxe~£i7
,
, Vi : 1,2
Xi
of
x i :0
s u c h that A s s u m p t i o n s (4.15)
, 7e]0,~
invariant
i] , i e { 1 , 2 }
set of the s y s t e m
function
v2
of e q u a t i o n
on
N(X,~)
and A s s u m p t i o n
, func-
I and 2 hold
:
increasing b i >0
{9i: i:l,2,...,s}
1.
numbers
is a p o s i t i v e l y
Statement IV.
2 and for
: a compact n e i g h b o u r h o o d
and p o s i t i v e
nite and r a d i a l l y
.
, Vye~E17 , v z e N ( E i T , e m ) \ Ei7 , V7 e ] 0 , 7 1 [
I-III imply L e m m a
~£(X) < 0 £i7
U ~E2~ i i:1
, z =X2x , ~ k e R , Vk = i , 2
and'if a l o n g m o t i o n s
then
:
(4.31)
for some 2.
(4.45) (4.15).
is p o s i t i v e ¢>0
due to
defi(ii) of
316
Ch. v
Assumption
1 and the d e f i n i t i o n Au(x)
of
Y
Large-scale power systems stability
(4.35),(4.36)
imply
-< £V ° , V × e ~ V
(4.46)
where U : [Ul,U2,...,Us ]T From equation
(4.28)
applied
to
£ and
(4.47)
(iii) of A s s u m p t i o n
1 it results
that 0 sign Vi(x)
if
xi : 0 ,
:
s
sign E
(4.48)
Z ~ijuj(×j)~
if
× i ~0
, Vi :1,2,-.,s
sign 0 : 0
. Equations
, VxeX
,
j-i where
sign ~ = ~ I~ I-I
(4.48)
and
£V ° < 0
T h i s result,
for
~ ~0
(the c o n d i t i o n
V(x) < 0
, v×e~V
equation
(4.48),
is a p o s i t i v e l y
and
,
of T h e o r e m ×i # 0
(4.46)
2) imply
, vi :1,2,...,s .
Assumption
1 and L e m m a
invariant
set of the s y s t e m
, V ° >0
and
and
(4.49)
1 prove
that
(4.15).
Further, EV ° < 0 imply
stability
(4.28) Let
E,
that
is, that
, i ~j E = (elj)
15 in C h a p t e r
, Vi,j :1,2,...,s
,
obeys r e l a t i o n s h i p
11.3.4).
-i ~i : rilri2 , Vi : 1,2,...,s . Now,
Statement. V. Let
of
(see P r o p o s i t i o n
eli->0
@ie]O,l[
, vi =1,2,...,s
A@ : (Gi~j) , ~i~j : @iGii~ij + G i j ( l - 6 i j )
Statement
VI.
A -
as soon as all
, Vi,j :l,2,...,s
~ii -<0
and
~ij >-0 , i ~ j ,
Vi,j = 1,2,...,s . When
E obeys r e l a t i o n s h i p
due to the d e f i n i t i o n s that m u l t i p l i c a t i o n does not i n f l u e n c e Statement
VII.
is M e t z l e r i a n that
of any
and stable,
A@
E and the p r o p e r t y
and
(row)
also obeys r e l a t i o n s h i p
c o l u m n by a p o s i t i v e
of
E implies
there is
13,c in S e c t i o n
gether with Statement
VI,
that
stability
b~R+ II.3.4)
A5 < 0
Hence
and
of
(4.28)
of d e t e r m i n a n t s -i n u m b e r (e.g. ri2)
: A @ . Because
that
VIII.
There
(ATB + B A )
is d i a g o n a l is n e g a t i v e
matrix definite.
A@
, b : [bl,b2,...,bs ]T , such A@b < 0
, w h i c h implies,
(Proposition
to-
13,d in S e c t i o n
:
Statement such
(4.28), A@
the sign of a d e t e r m i n a n t .
Stability
(Proposition
TI.3.4)
of
B with positive
diagonal
Vector approach
317
In the r e f e r e n c e s , (1979),
Gruji~
we a c c e p t e d
and used
bi's
and R i b b e n s - P a v e l l a
B = diag{bl,b2,...,b s}
for
b :
the
following
Statement Since
IX.
V
+BA) u(x)
is o b v i o u s
Both
vI
satisfying
et al.
Statement
VIII
< 0
,
V(x#O) e~
.
:
and
is p o s i t i v e l y
Gruji~
b = [bl,b2,...,bs] T . Hence,
uT(x)(ATB Now,
(1977),
(-vl)
invariant
are p o s i t i v e
set
definite
and S t a t e m e n t
on
IX holds,
V. it r e s u l t s
that lira Hence
[llx(t;Xo)ll
:t~+~]
: 0
, VXoeV
.
(4.50)
:
Statement
X.
the s y s t e m
V
is an e s t i m a t e
of the
domain
of a t t r a c t i o n
x:0
of
of
(4.15).
O
Let
e~R+
be a r b i t r a r y
Be = {x : llxll < e} due to the Notice
positive
that
and that
CElT * CBe.. ve • [e*,+=[
Therefore, . Hence
Xl.
V
test
then
EV°< 0
tion
II.3.4).
condition V
and
quite
of
E
construct
kij
and
,
, Vt ~ R +
,
IX.
increasing
e*
in
e 6R+
, Vt • R +
,
,
satisfying
x ( t ; X o ) • BE
D
be b e t t e r
matrix
under than
numbers
< - Vi(xi)
domain
in v i e w
However, E £
the best
conditions
of
positive
N i l U i2(Xi) then we
stability
the
cannot
strictly
conditions
of
estimates
exist
simple
is r e q u i r e d .
we get
estimates
and S t a t e m e n t
7*
theorem
Hence,
V ° . When
El7 c B c
of s t a b i l i t y
of
of D e f i n i t i o n s
x=0
4-7
of
(Sec-
•
iff
EV°< 0
and
of the
can be s a t i s f i e d
for o t h e r
If there
the
Chapter).
2 presents
V
guarantees
is an e s t i m a t e
X and XI prove
stability
of
that
X(t;×o) EB e
:
Statements
Theorem
such
implies
numbers
Xo 6 V
(4.15).
2.3 of this
be a n u m b e r
is c o n t i n u o u s
are p o s i t i v e
the s y s t e m
tion
?>0
x ° 6£17 nV
invariability
7 =7(e)
there
Statement
and
. Then,
is s t a b l e
of T h e o r e m relaxed V
Nil
2 _< N i 2 u i ( X i ) K : (kij)
£
required.
estimate
and
of
2 are not
conditions. estimate Ni2
of
0.
is a M e t z l e r
(Proposition
is i m p l i c i t l y
possible
the
for e s t i m a t i o n since
met
D
No matrix
13, SecUnder
the
for c h o s e n then we
Certainly,
look
these
(4.34). such
that
, VX i e X i , Vi : 1,2,...,s , (4.51)
,
: [~i~/2.~ij + ~ ; ~ / 2 ( 1 - ~ i j ) ]
~ij
, V i , j =1,2,...,s
,
(4.52)
3i8
Ch. V
and propose
the next t h e o r e m b y
(v)l/2 Theorem 3. (4.28)
E2
Proo~
KV°I/2
then
V
Zs
(4.51) and e q u a t i o n s
j:l Further,
~ijUj(Xj)
s Z
g
l]
and
x=O
K fulfiZe
of the ~ystem
1977;
Gruji~
et al.,
(4.15). P.
i979).
In-
]
,
VxEX
, Vi : 1,2,.-,s
- 3
(4.53)
"
b :(bl,b2,...,bs )T , b i : b^2i , Vi : i 2,...,S ,
for
:-(K)-Ic
(4.28) and
which
(4.52) y i e l d
k..Vl~(x.)
j:l
stability
V1/2]T
D of
~ 71/2
VX E ~£2~ i , V7 e ]0,72] b
in
(4.34) i8 an estimate of
s kij V ~ 2 ( X j ) j:l
Let
(1979)
systems
1 and 2 hold and the matrix
(Gruji~ and R i b b e n s - P a v e l l a ,
equalities
power
"--2 "-'" s
is an estimate of
If in addition
et al.
[. 112 v1/'2
: "Vl
If Assumptions
then
Gruji6
Large-scale
oS
for
C 6R+
. Now,
, (4.54)
, Vi : ],2,...,s .
b>0
K is a M e t z l e r matrix.
(4.54) and A s s u m p t i o n
Xs kijb j j:l
because
K obeys
This c h o i c e of
b,
relationship
equations
(4.53)
i yield
s
Vi(X)
s 7 1 / 2 u i ( x i ) Z kijb j < 0 , VX 6 a E 2 7
i
, Vi = ],2,...,s , V7 • ]0,72 ] •
j=1 The
following
(4.55)
statements
result
from e q u a t i o n s
(4.40),
(4.42) and
(4.44).
Statement XII exist
~>0
•
In fact,
7(Y)
Statement
XIII.
Now,
For
, and
,
y
i 6 {l,2,...,s}
is p a r t i t i o n e d such that
The
function
7 : R M ~R+
is c o n t i n u o u s
XII a n d XIII t o g e t h e r w i t h e q u a t i o n
, 3iE{l,2,...,s}
s u c h that
y EaE27(y)i
as
x,
there
y E aE27(y)i
= max {Vi(Yi)(b~) -I :i = 1,2,...,s} , i.e.
statements
VY6£272
(y#O) •R M
every
, 7 =7(Y)
7(Y) = v2(Y) :
(4.55)
7(x) 6 c ( R M ) . imply
and
s
Vi(Y ) s 7 i / 2 ( y ) u i ( Y i ) Let Statement continuity
7[X(t;Xo)] XIV. of
Statement P
in
tER+
Let it be s u p p o s e d that 7 o • ]0,72[
such that
Z
j=l
kijb.
(4.56)
3
= P ( t ; x o)
XII! and c o n t i n u i t y
(4.57) of
X
in
t~R÷
guarantee
, VX o 6 R M
for a r b i t r a r y [270 c £ 2 r ( t , X o )
Xo • £ 2 , VtTR+
• Xo ~ 0 , there
exists
Vector
approach
319
Then, there exist ~o < Fl/2(t,Xo ) of S t a t e m e n t
~o : ~ ( X o ) e ] 0 , + ~ [
and
and
~o : ~ ( X o ) e l 0 , + = [
u i [ x i ( t ; X o ) ] > Go , Vt 6 R +
X I V and e q u a t i o n
(4.57).
s u c h that
, vi : 1,2,...,s , b e c a u s e
Hence,
s
V2[X(t;Xo)] The r i g h t - h a n d and
C ~
< ~o~o max
side is n e g a t i v e
s . Inequality
{ Z
kljbj
j:i
:i :l,2,...,s} < 0 .
due to the c h o i c e of
(4.58)
can be i n t e g r a t e d
b,
from
(4.58)
b =-(KT)-Ic
to=0
to
,
tER+
,
s
v2[x(t;xo)] This implies
-< v20 + ~o~o t max { Z
j:i
the f o l l o w i n g a b s u r d v2[x(t;Xo)] <0
where
: v20[-~o~° max which is a c o n s e q u e n c e true v a l u e
7 ° :0
:
, Vt e ] T , + ~ [
, V(Xo~0) e £ ? ,
s ~ -i { Z kijb j :i : 1 , 2 , . . . , s }3 e ] O , + ~ [ , j--i
of the a s s u m p t i o n
. But
kijb j :i : 1,2,...,s}
7 o :0
implies
that
7 o >0
i n s t e a d of the
lira [E2F(t ) :t ~ + ~ ]
: {0}
.
This r e s u l t p r o v e s lim Further,
let
[UX(t;Xo)~
6 6 ]0,+=[
:t ~ + ~] : 0
be a r b i t r a r y
and
, VXo E E 2 . 76 6 ]0,72]
(4.59)
such that
E27 e _C B6 n E 2
Now, we d e f i n e
6 : 8 ( e ) e]O,+~[
(4.60)
so t h a t
B8 c E27e
Statement
XV.
The m o t i o n
passes t h r o u g h
BE27
B r E [0,+~[ Assumption Statement
XVI.
Equations
can leave
E27
, r: T(Xo,7 ) , X ( T ; X o ) e aE27
The
, xo •E27
, iff it
:
2, S t a t e m e n t s
tem (4.15),
x ( t ; x o)
(4.61)
set
, V7 • ]0,72 ]
II and XV, and (4.55) p r o v e : E27
is a p o s i t i v e l y
invariant
set of the sys-
V7 ~ ] 0 , 7 2 ]
(4.60) and
(4.61) and S t a t e m e n t
XVI imply
(4.62), o
X ( t ; X o ) EBe , Vt 6R+ , VX° 6B~ , r e • R + Moreover,
Statement
XVI implies
X ( t ; X o) • B e
where
e*
is a p o s i t i v e
,
.
(4.62)
(4.63),
V t ER+
,
VX ° 6 E
n u m b e r such that
2
,
re•
E 2 CBe.
[e*,+~[
,
(4.63)
320
Ch. V
Now, t h e r e s u l t s ( 4 . 5 9 ) - ( 4 . 6 3 ) domain
of a s y m p t o t i c
K~l/2
addition, E2 = V
because
the p r o o f
then we can
of e q u a t i o n
of T h e o r e m
2.
Theorem
i and T h e o r e m
matrix,
respectively.
which
is s h a r p e r
teresting
prove
stability
to d i s c o v e r
that
of
E2
X=0
set
bi
(4.4i),
is an e s t i m a t e
of the
of the s y s t e m (4.15). If, in 1 : V oi.[V Oj) , - i , w h i c h implies
and c o m p l e t e s
the p r o o f
by
following
•
2 impose
than
Large-scale power systems stability
conditions
In essence,
these
to r e q u i r e
A
under
what
on s y s t e m s
matrices
to obey
should
(4.28).
conditions
A
through obey
Hence,
matrix
£ and (4.28),
it is in-
can be used
the a s y m p t o t i c s t a b i l i t y d o m a i n estimate. One case h a p p e n s w h e n i/2 u i =V i , Vi : 1,2,...,s . T h e n ~il = ~ i 2 = 1 , Vi = i,2,...,s , and, fore,
K =A
are more Another
in T h e o r e m
relaxed
than
case w h e n
A
3. In such
those
a case,
the
of W e i s s e n b e r g e r
can be e m p l o y e d
conditions
K
for
there-
of T h e o r e m
3
(1973).
for e s t i m a t i n g
D
is d i s c o v e r e d
by
Theorem 4 .
If Assumptions
1 and 2 are fulfilled and the matrix
is negative definite for a diagonal matrix
ATB + B A
diagonal then the set
El
is an estimate of
~ of
B with po6itive of the system
x=O
(4.15). Proo~.
Along
motions
of e q u a t i o n
vl(x) because ness
of A s s u m p t i o n
of all
1 prove
ui
that
V7 E ]0,71]
on
El7
last
. Hence,
result
e 6 ]0,+~[
Let
~ =
the
,
vxeX
Assumption
definiteness invariant
(4.63),
being
the d e r i v a t i v e
+BA] U(X)
result,
equation bi
(4.63)
2, p o s i t i v e - d e f i n i t e of
ATB + B A
and Lemma
set of the s y s t e m
positive
i-th
,
definiteness
diagonal
element
of
of
vI B,
[ilX(t;Xo)ll
be a r b i t r a r y
:t ~ +~]
and
:
0
7 =7(~)
,
VX ° E E 1
Positive (4.66)
for
and
such
(4.64)
that
EZ7 CB e n [ 1 .
5(e)
(4.15),
prove lim
Let
i. This
X i , negative is a p o s i t i v e l y
b = [bl,b2,-.,bs] T , the
~ uT(x)[ATB
(4.i5)
B~
obey
invariability
of
El7
eB~
, vteR+
(4.65)
cE17
, V7 • ]0,71 ]
(4.66) and e q u a t i o n s
(4.65)
and
prove X(t;Xo)
Further,
for
e*
such
X(t;Xo)
that
E I c BE.
eB E , vteR+
, VXoeB 5
, Vee]0,+oo[
, e* e ] o , + ~ [ , VXoeE 1 , w
.
, we have e]e~,+~[
(4.67) :
.
(4.68)
Vector approach
Equations V
321
(4.64),
(4.67) and (4.68) prove that
in view of Definitions
This criterion requires knowledge of completely determined.
El
is an estimate of
4-7 (Section 2.3 of this Chapter). B in order the set
E1
tive diagonal for which
ATB + BA
B can be with
B = d i a g { b I b 2 ... b s}
and
for A
bi -
c • R~ . Then,
B with posi-
is negative definite. The first trial
for determination of b :-(AT)-Ic
to be
For the asymptotic stability domain estimate it
is not enough to know that there exists a diagonal matrix
for the matrix
•
B = I . The second one is with
i-th
element of
conditions
b determined by
(4.28) are to be verified
(Proposition 13 in Section II.3.4)
for which the
following conjecture can be useful. C0,jecture
i.
If a Metzler matrix
A obeys
(4.28) and
b : [b I b2,..bs ]T
is determined from (4.69), b
for
c ~R~
then
ATB + B A
:
-(A)Tc
(4.69)
is negative definite as soon as
B = d i a g { b I b 2 ... b s} Remark 5.
The Conjecture I fails for the following
A
:
,
20
A
obeys
C
oounterezample.
Let
E R+
:
-201
(4.28) and
F221] o2 L 11j R+ .
b. : - ( A T ) - I c Hence,
B : diag {221
11}
.
However, -442 ATB + B A = [2430
24301 -4422]
is not negative definite. Remark 6.
If two theorems or all three, among Theorems 2-4, are satis-
fied then the final estimate of
P
is the union of estimates determined
by the theorems.
V.4.4. Power systems modelling
V.4.4.1. Uniform dampi.9 Consider an
N-machine
power system. With the simplifying assumptions
322
Ch. V
a d o p t e d in S e c t i o n s tem s t r u c t u r e ) ,
V . 2 . 1 and V.2.5
the m o t i o n of the
e x p r e s s e d by eqs.
Large-scale power systems stability
(constant i-th
machine
electromagnetic
taken
and we choose
damping,
generator,
then we get the state v a r i a b l e s (Section V . 3 . 2 . 1 ) ,
N-th
damping,
mechanical (3.2)
E1 , P m i , Mi
and sys-
separately
is
(2.1),(2.3).
W h e n m o r e o v e r we n e g l e c t
tions
:
machine
we c o n s i d e r u n i f o r m
to act as the r e f e r e n c e
(3.~) and the state equa-
r e p e a t e d as
(4.70)
:
giN : ~ i N
kin
(4.70
= -~niN+MZI(pmi-Pei)--M~I(PmN-PeN)
In o r d e r that the o r i g i n of the state space equilibrium V.3.2),
solutions
we adopt
of (4.70), w h i c h
, Vi=1,2,..,n coincides
T (4.28) may now be w r i t t e n
as follows
(3.3)
(Section
and state vector
:
R2n
XT : [OlN~IN~2N~2N~...,anN~nN] System
w i t h one of the
is e x p r e s s e d by
s u i t a b l e n e w state v a r i a b l e s
.
E
(4.71
:
°iN : ~iN'
(4.72 N
&iN
= - X~iN -Mi I
n
Z
fij(oij)+MN 1
Z
j:z
fNj(aNj)
, vi =1,2,...,n ,
j:1
where fij(aij)
: Aij
[ cos (aij +
~?. l]
0 ij )] - @ i j ) - cos (~?.l]
' Vi,j
: 1,2,...,N
(4.73) and w h e r e we have set compatibility
°iN
0
: ~iN - ~ N
w i t h the n o n - u n i f o r m
and
~iN ~ ~iN
for the sake of
d a m p i n g case.
V.4.4.2. Non-~niform damping With both mechanical the a b s o l u t e V.2.1),
where
T h e r e are
and e l e c t r o m a g n e t i c
m o t i o n of Pei
2N-I
i-th
machine
is still e x p r e s s e d
state v a r i a b l e s
dampings
t a k e n into account,
is d e s c r i b e d by by
(2.3)
in the p r e s e n t
(2.1)
(Section
(Section V.2.5). case,
namely
~IN,~2N,---,~nN,~I,~2,...,~N . From
(2.1) we set the state e q u a t i o n s
~iN = ~i - ~ N ~i
, i = 1,2,.,.,n = N - 1 N
= -~i~i-
The e q u i l i b r i u m
Z ~ij(~i-~j)
j:l
states
of the s y s t e m
(see N o t a t i o n )
:
, (4.74)
+Mii[Pmi-Pei(B)]
of (4.74) are g i v e n by
= fi(B,~) :
, i =i,2,...,N .
Vector approach
323
N [Pmi ~[
~
.~on
:
= ~
us m a k e
the
equilibrium aiN
:
origin
(4.75)
,
~i
(4.?5)
N Z i=l
= 0
of the
solutions
-Pei (5°)]
i:l
: "" :
f i ( 6 ° , n °) Let
: ~o
,
state
by
i :I,2,...,N space
choosing
= ~i - o
Di
.
coincide
the
new
with
one
variables
of the ~i
and
, i :1,2,...,N o
aiN The
state
vector
is n o w
equations aiN
' i : 1,2,...,n
, i = 1,2,-.,n
= - Ai~i
-
Rn +N
into
:
,
N
~i
.
:
(4.?4) are t r a n s f o r m e d
of m o t i o n
: ~i-~N
6iN
[alN,a2N,...,OnN,~I,~2,...,~N ]T E
x =
The
= 6iN-
(4.76)
N
~ Aij(~i-~j)-Mi j:l
I
Z fij(oij) j:l
, i :i,2,...,N
,
j¢i where
fij
are
still
expressed
V . 4 . 4 . 3 . Non-uniform damping With the
governors equations
(2.1), The
~iN
of the
(2.4)
overall
equations : ~i-
as
first
system's
V.2.1
motion
(4.73).
and governors aa£ion
absolute
(Sections
order
motion
porportional
of the
i-th
speed machine
regulators, are
given
by
and V.2.5). is g o v e r n e d
by the
(n+2N)
: (3N-I)
state
: ~N
hi
= -~i~i-
Pmi
: - ~iPmi
whose
modelled
by
steady
' i = 1,2,...,n : N - I N Z ~ij(~i-~j j:l - ~ i ~i state
'
) +
, M-I pO _ i Pmi + M ~ I[ mi Pei (s)]
i : 1,2,...,N
solutions
are
O
O
O
: ~2
: "" : ~ N
:
: ~o
, fi (6'~'p)
(4.77)
N ~I
:
:
[P~i - P e i (6°)] i:l N -i) Y~ (D i + ~ i ~ i i=l
fi(~°~fZ°,p °) : 0
324
Ch. V
o
By a d o p t i n g
the new
state
aiN
variables
aiN
8iN _ 6 °iN
°iN ,~i
and
o
= Pmi - P m i
, i : 1,2,...,N
,
N
-~.(.o. - Z Xij(coi-oaj)+Milpi-MZ i i j:l
the X
,
, i : 1,2,-.,n
Pi = - ~ i p i -~ico i Obviously,
Pi
, i = 1,2,...,n ,
N
:
stability
equations
: ~i - ~ N
coi
systems
, i : 1,2,...,N
state
Pi we get the
: -p-~.
:
power
o
ai
Pmi
Large-scale
state
1 ~ fij(oij j:i
)
(4.78)
, i : i,2,...,N .
vector
is e x p r e s s e d
in t h i s
case
by
[alN,O2N,...,gnN,~l,~2,...,~N,Pl,P2,...,p N] 6 R n + 2 N
:
V.4.5. Power Systems decompositions and aggregations
V.4.5.1.
Uniform damping c a s e
V.4.5. I.I.
Form
I.
Darwish
and Fantin
F o r the
sake
ri ,¢ij
, @ij
This
decomposition
further
and p a r a m e t e r s
analysis
~i , T i ,Pij
we d e f i n e
as f o l l o w s
o o = COS (aiN + 6iN ; @iN ) - COS (6iN + @iN )
Pi ,
,
i = i,2,...,n
(4.79)
¢iJ (aiJ) } : c°s(~ij ~ij(Oij)
+
6°
ij ~ e i j )
#ij 7i model
13
"
: AijMi I
,
: AiNMN I
,
(4.72)
takes ~iN
i,j =1,2,-.,N
- cos(6?.
~i = A i N M i I
mathematical
functions :
ri(aiN )
The
by G r u j i ~ ,
(1977).
of s i m p l i f y i n g
Pi(aiN ) I
f o r m was p r o p o s e d
i,j
the
: ~iN
@ij )
i = 1,2,...,n
,
:1,2~...,n
o
i : 1,2,...,n
.
form ,
:
"
(4.80)
Vector
approach
325
n
~iN
: -k~iN-~iPi(°iN)
+Tiri(SiN)
+
Z {~Nj~Nj(aNj)-~ij¢ij(aij)} j:l
(4.81)
i = i,2,...,n The a b o v e s y s t e m by c h o o s i n g
(4.81) may be r e a d i l y
the state v e c t o r of the X i : [aiN
~iN ]
T
decomposed
i-th
= [Xil
,
xi2]
into
subsystems
s:n
s u b s y s t e m to be T
,
i : i,2,...,s
It comes X i = gi(xi) + hi(x)
(4.82)
, i = 1,2,...,s ,
where gi(xi)
Pc
: Pixi + f i P i ( O i)
:
,
o i : e~x i
[00:] [0] :
fi
;
;
-~i
,
e i : [i
hi(x)
:
0] T
[0] xi(x)
;
n
Xi(X ) = 7iTi(Xil ) + Z {~Nj@Nj(Xil ) -~ij¢ij(xil-Xjl)} j:l Thus,
the s u b s y s t e m
(4.82)
is c o m p o s e d
of the d i s c o n n e c t e d
.
(4.83)
subsystems
Xi : gi(xi ) : Pixi + f i P i ( ° i ) and of the The
interconnections
hi(x)
state v e c t o r of the w h o l e xT
we n o t e that subsystem Notice
x i :0
(4.84),
that
(4.83).
s y s t e m is :
[x T x T
xTl
• 2,-.-• s J ;
is an i s o l a t e d
equilibrium
solution
of the
i-th
i : i,2,...,s .
o i =aiN
Let the d i s c o n n e c t e d t i o n of the L u r ' e
(4.84)
holds. subsystem
form
Vi(×i)
(4.84) be a g g r e g a t e d
by u s i n g
Vi
func-
: : xiHixi + ~i
(4.85)
Pi(°i ) dai i
where
~i
is a n o n - n e g a t i v e
number.
S i = { aiN : 2 ( S i n - ~ n ) Vi
is a p o s i t i v e
matrix
Hi
i
p r o v i d e d that
the
(symmetric)
i
(h~2)2
i.e.
h12
h22
Si ,
a aiN ~ - 2 ( ~ - @ i n +~°in ) } •
function,
definite•
[i i] Lh~2 hll
Hi
definite
is p o s i t i v e
In the s e c t o r
such that
i
hll > 0
i
; hllh22 >
(4.86)
326
Ch. V
However,
because
of the structure
be p o s i t i v e
definite;
So,
: P~Hi + HiPi =
for the stability
under which part
Gi
of the d e r i v a t i v e
of
defined
by
i i hll-Ahl2
0
(4.87)
i i hil - k h l 2
analysis
is positive
at most negative
Gi
indeed we find I
-Gi
Pi , matrix
i : P~H i + H i P i
-G
cannot
of
Large-scale power systems stability
we can only ask for conditions
semi-definite;
V.
therefore
along the d i s c o n n e c t e d
1
semi-definite.
From
(4.87)
on
Hi
the quadratic subsystem
it follows
that
will be
this is
true iff
i
i
hll = Hhl2 Notice Then,
i hi2 > 0
that
'
2(h
2
-Hh
is r e q u i r e d
along motions
of (4.82)
i
2 ) : -g22
'
Hi > 0
due to
for
the total
>0
g22
derivative
(4.88)
i hll > 0 of
Vi
and
k>0
(4.85)
.
is
Vi = (grad Vi )T (gi + hi) (4.89) = _ g 2 2i~ i N 2
We choose
fi
i ~i = 2~ih22 Further,
_ 2 ~ i P i GiNhl2 i +~iNPi(~i so that
the term
the largest
compact
so that
GiNPi(OiN) c I e ]0,+~[ follows.
Choosing
exists
Smaller
the a s y m p t o t i c ditions,
but i g22
from
interval
+ h 2i2 ~ i N )
is c a n c e l l e d
:
stability
iff
(4.86)
that
domain
VaiN
iN) > 0
can be a c h i e v e d i h22
that
Xi
i h22
is p o s i t i v e
(Section
and b e t t e r under
conditions
, as follows
"
.
; this is a c c e p t e d
Li
in
estimate
appropriate
of
con-
to b e satisfied.
:
1 + ~iei i ~ h12
-
definite
i 2 i Vi ~ -2~ieih1211xill +2Xi(hl2°iN It now results
E Li
~o
interval
for these
i ~ie ihl2 Hi
,
sin (SIN-
, and t h e r e f o r e : 2
t i of aiN , i i = [£im,ZiM] , e i 6 ]0,+~[ the f o l l o w i n g is sa-
2 ~ 6iOiN
e i , larger
less p o s s i b i l i t y
i g22 we get
~iNPi(f i -2~ih$2)
for certain
tisfied Such
+ 2 x i ( h li2 G i N
"
is to be chosen
what
_2~ih~2)
V.4.3.1)
and from
i +h22°~iN)
•
is given by
:
(4.89)
(4.90)
I
2
X i = {X i : OiN e i i ,COIN e [-(Ah~2 + 2 ~iEi) £im ,
2 +i
2 1}
~ ~ici ) Zim
Vector
approach
327
o
where
~il :
Moreover,
using majorizations
JPil
sin (SiN-~iN) on
Li :
~ @ill °iNI
lql-< L21%NI
;
o
~i2 : sin (~iN+0iN) Ti]" : Isin(~°'-0±] ij)l
~i3
+~
= min ~i
~i4 :
1 #iel)]
+ max (1 , - -
+~i6i ~, ~
I + ~iei k 7i~i2
i + ~iei
~ij
iff
sin (0ij-~°j)
iff
sin ( i j - ~ i j )
~ 0
?ij ~i3rij
o
~ 0
and i -¢ij(hl2OiN
i ~ i 2 +h22 iN ) -< hl2(TijlI×ill +~i3rijllXiIIIIxjll)
We obtain after trivial ~il >0
calculations
and under the assumption
that
, n
Vi ~ - 2 h ~ 2 ( ~ i e i -~i4 - ~
j:i
~ijTij) UXi ~2 +
j~i n
+2h~2~i3
Z
j:1
(~Nj~j2 +~ijTij)
~xjll Dxill
,
vXeX
.
(4.91)
j#i i i Let h12 = 7 " Comparison yields
of (4.91) with ui(xi)
(4.21),
(4.22) and
(4.26)
: lIxill
and n
-(~iei - ~i4 - Z ~ijTij ) , j:l j~i
(~ij :
~i3(~Nj ~j2 +~ij rlj)
,
i=j (4.92) i ~j
Matrix A : (~ij) of Assumption I (Section V.4.3.1) is now completely determined by (4.92)• Notice that e i 6 ]0,~il [ is only meaningful, and ei 6 [~il,+~[ is meaningless• The set
X
V.4.5.1.2.
define
(Assumption Form 2.
functions
i) is completely
Following ¢i ' @i
determined
via (4•90).
Gruji~ and R i b b e n s - P a v e l l a
and parameters
ai
and
bi
(1977) we as follows
:
328
Ch. V
Large-scale power systems stability
o
¢i(OiN)
: sin(oiN+~iN)
~i(OiN)
= cos (~iN+SiN)
The m a t h e m a t i c a l
~o
-sin
o
,
iN 6o iN '
-cos
ai : (7i+ ~i
sin 0iN
,
b i : (7i-~ i
cos 0iN
•
model
takes
i : 1,2,...,n ,
i : 1,2,...,n .
(4.72
the
form
:
aiN : ~iN ~iN
: -X~iN-ai@i(°iN
) +bi~i(°iN ) + ~ j:l
{~Nj*Nj(aNj)
j~i The above
system
by c h o o s i n g
(4.95)
the state
for the a g g r e g a t i o n Xi Eq. (4.82)
:
of the
decomposition
lOiN
is valid
may be readily
vector
~iN IT
=
i : 1,2,...,n .
decomposed
i-th
form
[Xil
-~ij%ij(oij)}
into
subsystem
s=n
,
(4.93) subsystems
to he the same as
1,
xi2
]T
,
i : 1,2,...,s
for gi(Xi)
: Pixi + fi¢i(o i) ,
0 ;
hi(x)
]
:
bi~i(~ i) +×i(×) n
Xi(X ) :
Thus,
Z {~NjCNj(ONj ) - ~ i j ¢ i j ( a i j ) } j:l
the i n t e r c o n n e c t e d
subsystems
subsystem
(4.82)
is c o m p o s e d
of d i s c o n n e c t e d
(4.95), Xi : gi(xi ) : Pixi + fi¢i(ai )
and i n t e r c o n n e c t i o n s We accept
(4.94)
Vi
hi(x )
,
i : 1,2,...,s ,
(4.95)
(4.94).
in the form
l ain
T Vi = xiHixi + ~i ~ Then,
along m o t i o n s
found easily We choose
~i
of (4.82)
the total
along the same lines so that term
¢i (eiN) daiN time d e r i v a t i v e
as for the Form
¢i(ai)
is c a n c e l l e d
i ~i = 2aih22 Further,
we select
both a s u f f i c i e n t l y
of
Vi
is
I. in
Vi
•
small
e i >0
and a compact
Vector
approach
interval
Li
329
of
OiN , i i = [~im,£iM] aiN¢i(aiN)
The c o n d i t i o n
e i • ]0,+~[
larger interval Then, c h o o s i n g
>- eia2i N
Now,
Xi
Assumption
e i e ]0,+~[
'
is based
o
on
cos 8iN > 0
. Smaller
ei ,
L i but less p o s s i b i l i t y for A m a t r i x to be stable. i , and therefore i g22 h22 , as follows : i " g22 : 2 a i e i h ~ 2
we get that
on w h i c h
i h22
'
:
i + ale i i k h12
'
H i >0
is d e f i n e d
by
(4.90),
which
is the
i-th
component
of
X
in
I. o
Let
~il : cos ~iN
" It is a s s u m e d l¢i(aiN)l
that
so that
~il >0
<~ilfoiNl
on
Li
Let
~i2 : sin ( ~ N + eiN) , TiJ = isin (6oj _eij)l
,
~i3 = rain
' v~
X
i + a l e i 7.. z3 "[ij :
~i4
~ ~i3rlj
sin
sin(Sij-&~, j) _>0
iff
o sin(#ij-6ij)
6iNI
o
iff
b i sin 6iN ~ 0
iff
b i sin 6 °iN ~ 0
1 calculations
we get
n
Vi -< - l [ x l I l 2 ( a i e i - ~ i ~ -
of
(4.96)
n
~ ~ijTij ) + j=i j~i
with
(4.21), ui(xi)
and
<0
:
After some trivial
Comparison
(l , i + aki e i
iff
1 +aie i o : { ----~b i Sino ~iN ~i31bi
i h12
max
~i3[tXi[[
X j=~
(~Nj~j2 + ~ij~ij )
j~i
(4.22)
and
: llXill
(4.26)
HXjH . (4.96)
shows
that
330
Ch. V
Large-scale
power
systems
stability
n
-(aiei-~i~g~..
~ ~ijTij ) j=l
, i=j
,
(4.97)
=
13 ~i3(~Nj~jg+ The matrix
A : (~ij)
that ei e ]0,~il [ meaningless. F.4.9.1.3. the whole
is now completely
is only meaningful.
Commznt~.
, i Cj
determined The
between
them lies
into
by
choice
In the two d e c o m p o s i t i o n
system was d e c o m p o s e d
The d i f f e r e n c e
~ijrij)
s=n
of
(4.97).
e i e [~il,+~[
forms above
second
order
in the d e f i n i t i o n
Notice is
considered,
subsystems.
of subsystems
and
interactions. In the first
case,
free s u b s y s t e m s
gi(Xi ) =
and interactions
Xl +
[o]
Pi
are d e s c r i b e d
by
'
[ 0 ] hi(x ~ ) :
n
7i~- i + Z (~Nj~Nj - ~ i j ~ i j ) j=l
while
in the second
case they are d e s c r i b e d
gi(Xi ) :
by
[00] [01 Xi +
¢i
•
-a i
-
0
hi(x i )
Hence tively. second first
-~iPi
and
If
7i : ~i
case, w h i c h case,
both
71T i then
bi,i+ j=(~Nj*Nj Z -~ij~ij) are r e p l a c e d bi:0
-~iPi
and For
by
-ai~ i
and the term
is important
one is to be majorized. decomposition
n
:
for a v o i d i n g
7iT i 7i ~ Pi
is more suitable.
appear
and
bi~ i
bi~ i , respecvanishes
its majorization.
for
7i : ~i
it is not evident
in the In the
and the latter a priori which
Vector approach
V.4.5.2.
331
Non-uniform damping and governoas a c t i o n
Various
decomposition-aggregation
for p o w e r
systems
without
Ribbens-Pavella, 1984,
1985).
1977-1979;
rather
than
were
governors
Gruji@
T w o of t h e m will
and d r a w b a c k s
forms
and w i t h
et al.,
be only
considered actions
1979;
Shaaban
presented
to try g i v i n g
final
to
system,
with mechanical
represented with
by a constant
a Ist-order
(2.4)
(Sections
the a b s o l u t e
voltage
proportional V.2.1 °°
i.e.
N
an
N-machine
dampings. of t h e
is g i v e n by
,
show p o s s i b i l i t i e s
its t r a n s i e n t
governor,
and V.2.5),
Mi6 i = -Di~ i -
motion
behind
and Gruji~,
Consider
and e l e c t r o m a g n e t i c
ferential equation describing
authors
and
results.
V.4.5.2.1. Pairwisz dzaompo~iZion-aggrzsation. power
by the
(Gruji~
by
The
i-th
dif-
machine,
reactance
(2.1),
and
(2.3)
and
(4.98)•
.
Z Dij(Bi-Bj) j=l
+Pmi +Pmi -Pei
= fi(~,~,p~)
Pmi
= - ~iPmi - ~ i ~ i
where
= j=l ~ giEjYij
Pei(~iN'~jN) The m o t i o n (3N-1) ing
(4.98)
N
of the
state
(3N-1)
system
equations state
taken
as a w h o l e
derived
variables
cos($ij-Sij)
(4.99)
is d e s c r i b e d
from e q u a t i o n
(4.98)
(Ribbens-Pavella,
1971b)
61N,~l,Pml•~2N,~2,Pm2,...•$nN,~n,Pmn,~N,PmN Here,
the
in o r d e r librium
N-th
machine
to make of the
the
acts
origin
system,
as the of the
we adopt
comparison state
the new
space state
in terms
with
the
of
follow-
: -
machine.
Furthermore,
coincide
with
an equi-
variables
O
aiN = SiN - S i N where
(Ribbens-Pavella,
o Pmi
~i ~o = --#i
and w h e r e
. ,
~o
~o :
PmN - P e N (~°'6° :
6 °iN • i = 1,2•...,n , are
the o v e r a l l
Pi
= Pmi - P moi
,
Vi
= 1,2,...,N
aN D N + -~N
fi(6
equations
= ~ i _~o
1971b)
O
Hence,
~i
O
,~
system's
solutions
, (4.1oo)
of
O
,pm ) = 0
,
motion
Vi = 1,2,...,n .
is g o v e r n e d
by the
: aiN = ~i - ~ N
,
i = l•2,...,n ,
(4.1Ol) (3N-1)
state
332
Ch. V
Large-scale power systems stability
N
~i
= -Xi~i-
N
Z
Xij(~i-~j)
+ki(P i - Z
j:l Pi Obviously,
: - ~ i p i -~io)i
the
state
x
vector
of m o s t
interesting
ing
derived
from
= PiN
,
i : 1,2,...,n,N:n+l
is n o w
decomposition
(4.102)
(in w h i c h
, vi =1,2,-.,n
aiN
-- W i N
~iN
= -PlaiN
(4.102)
,
.
[OlN,COl,Pl,a2N,CO2,P2,..-,anN,~n,Pn,~N,PN
:
One
Pi-PN
Aij¢ij)
j:l
forms
the
, are
appeared
differences
used)
IT
(4.103)
to be the ~i-¢~N
follow-
= ¢~iN
:
• +MilPiN
-(hi-AN)
~ N + (M~"I - M N I) PN - a i ~ i + b i @ i
n
+
~
(Aiju;jN + ~ N j ~ N j
j:l
- ~ij¢ij)
,
Vi : 1,2,...,n
j~i PiN
, (4.104
: - ~i¢°iN - ~ i P i N
+ (~N-~i)
O)N + (~N-~i)
PN
"
where a i : (~iN+~Ni)
sin 8iN
,
b i : (~Ni-~iN)
cos @iN
,
and n
Note
that
last
one
Thus,
the
~N
: - XNa~N + M N I p N
PN
= - ~N°JN - #NPN
first
n
is of o r d e r
by d e f i n i n g
+
Z (XNj°~jN - ~NjCNj ) ' j=l
(4.1o5
"
interconnected
subsystems
are
of
order
3, the
2•
the
subvectors
X i = [Xil,Xi2,xis]
x i , XN
as
T = [aiN,~iN,PiN
]T
follows ,
:
Vi = 1,2,...,n
, (4 106
X N : [ X N I , X N 2 IT : [ ~ N , P N IT we d e c o m p o s e nected
the
overall
subsystems
xi
system
o f the
= gi(xi ) + hi(x)
general
,
of equation form
(4.102)
into
N
intercon-
:
= Pi×i + ~ i ¢ i ( X i l )
+ hi(x)
,
Vi = 1,2,...,N
•
(4.107) Here, vector
the £i
state
vector
and
£i
hi
:
x
and
[0
still
has
matrices
-a i
0] T
the Pi
,
form are
Pi
~
of
equation
defined
as
(4.103);
follows
[iio] -F i
Mi 1
-~i
- ~i
:
Vector approach
333
n
(kN-ki) X N I + hi(x)
(MiI-MNI)XN2
+bi~ i + Z (Aijxj2 +~NjCNj j:l
:
(CZN-~ i) XNI + (#N-#i)
-~ij¢ij)
XN2
Vi : 1,2,...,n ,
£N
[0 O] T
PN :
[
-xN -eN
hN(X)
(4.108)
] -#NJ
[ (ANjXj2-~NjCNj)J ~ j:l
=
(4.i09)
0
We can now decompose disconnected tions.
(free)
The
i-th
the part
free
i-th
subsystem
of e q u a t i o n
of the L u r ' e - P o s t n i k o v
subsystem
(4.107)
type,
has the general
form
into a
and i n t e r c o n n e c :
Xi = PiXi + ~i¢i(°iN ) " i :i,2,...,n,N ,
(4.if0)
aiN = e~x i • where e i = [i 0 0] T For the free s u b s y s t e m
of e q u a t i o n
tion of the L u r ' e - P o s t n i k o v linearity"
;
, Vi = 1,2,...,n
type
(4.1i0),
eN :
[o
we accept
"quadratic
(4.111)
o] T
a Liapunov
form + integral
func-
of the non-
:
Vi(×i)
T [ °iN : XiHiXi + ~i ~i(OiN) dOiN
,
Vi : 1,2,...,n , (4.112)
VN(XN) In these
~i
expressions,
are p o s i t i v e
Lur'e
:
T XNHNX N •
sector
Hi
are symmetric
positive-definite
T R3x3 H i : (h j) : Hj 6
,
HN :
,
(h~j)
numbers,
condition
= HN 6 R 2 x 2
(4.113)
~i(OiN)
sector
o
obeys
S i , defined
the by
:
o
S i : {OiN : ~ - 2 6 i N provided
:
Vi : 1,2,...,n ,
and the n o n l i n e a r i t y
only on the b o u n d e d
matrices
~OiN ~-~-26iN
that o
~ > SiN > - ~
"
} ,
(4.i14)
334
The
Ch. V
appeared In w h a t
to be the follows,
determination
: Ilxill
In o r d e r symmetric
most
we
of the
A
i : 1,2,...,n,N
for
equation
systems
the
and
~i
matrix
the q u a d r a t i c t e r m of i 6 i = (gkj) d e f i n e d by :
Vi
and
(4.113),
we
stability
i
i
i
to the
calculate
structure
6i
Gi
:
i
i i 2 (kih23-~ih33)
.
cannot
(4.116) be n e g a t i v e - d e f i n i t e .
[!001
:
i -hll
2
0
0
2b 2
i ~ i + rihl2 + ihl3
+ (Pi+~i)
(4.117)
h i _M-lh i 13 i 12 : 0 , i i -i i h23 + ~ i h 3 3 - M i h22 : 0 ,
(4.118)
- h 1i2
(4.120)
expressions
i
of the Mi i
i
(4.119)
+ Fih~ 2 + (xih23 = 1 , #i
the
hi
33
h .~] i
_M[Ii
h23
versus
= b2 hl2i
: h ~R+
i h13
-
and
i
i h22
=
b2 4.122) 4. 123)
,
Mi 1 i ~i h12
4.12~)
"
o
h12
Let
: 0 ,
~i
yields
the
(kih~2-gih~3)
2 (h12-Fih22-~h23) i i
Vi : 1,2,.,.,n
i -h13
to the
(h~3_( Fi+ #i )h23+kih22i i i Gih 33 )
i
(h13- (ri+ #i) h23+klh22- ~zh23 )
very
analysis.
further
therefore
, we
find
i i i (hllqh~2%h13)
by its
(4.115)
G i : P~H i + HiP i •
(4.108)
i i i (hll- ~hl2-~ihl3) i i (klhl2-gihl3)
,
proceed
~ and
to d e t e r m i n e
on e x p r e s s i o n s
that
system
(4.115)
H i , numbers
0
which
stability
matrix.
_
Based
,
convenient
accept
of m a t r i c e s
construction
Note
power
functions u i ( × i)
_G i =
Large-scale
4.124)
'
ff ~i(Fi+;~i ) h12
+ ~ b 2 + ~iMi I + ~ i ( r i + ~ i )
( ri+~ i) (~iMi I + #iri)
4.125)
Vector approach
335
i + #iM71 - b 2 ~ i F i M~-l(ri+ALi) hi2
i h23 :
i + ~ - +b ~ i ~ i - ~ +b2Fi( r i + , i ) Mi2(i + ri,Z l) hj_2 ( ri+ ~i ) (~LMi L + ri ~i)
i
h33 : Using equations
(4.117)-(4.121),
along the motion ¢i
(4.126)
( q + ~ i ) (~iMi i + r i l l )
of the whole
the d e r i v a t i v e
system
of
Vi
(4.127)
is o b t a i n e d
:
: _ 2~iN2 -b2piN2 _2hi2~iNaiN¢iN + ~ i N ¢ i N ( ~ i _ 2 a i h ~ 2 ) _ 2 a l h ~ 3 P i N ¢ i •
'
i
"
"
i
i
+X~(2h~2aiN + 2h22~iN +2h~3PLN) +X~(2h#3OLN +2h23~iN +2h33PiN) i
where i
Xl
i
and
X2
are the f o l l o w i n g
=
:
rl
X1
-(Tii-TIN) ~ON+ (M~-I-MN l) pN + bi@ i + Z (Aij~jN + ~NjCNj - ~ i j ¢ i j ) j:i j~i
i X2 : -(°~i-°~N) ~ N Let us choose (~i
functions
~i : 2 a i h ~ 2
is p o s i t i v e
(4.125)
(~i-~N) PN
whenever
of the p o s i t i v e n e s s i we choose h12 >0 ) .
Let the interval
h 2i 3 : 0
the f o l l o w i n g
and
{¢il
we eliminate
relationship
ensures
Moreover, which
> eilaiN{
h$2
the term i h12 and : ~i
i h12
so that
:
,
b2
q b2 - ~ i M i I
the p o s i t i v e - d e f i n i t e n e s s
imposing
disappears
in e q u a t i o n
Vain e l i (4.128) i - 2 a i h 2 3 P i N ¢ i . This implies
between
Mil(Fi+~i ) h 1i2 which
~iN¢iN
of
L i be L i = {aiN :ZiM a aiN a Zim }
OiN¢ i ~ O By setting
, so that the term in
because
of
Hi .
1 : eiai
implies b2 =
Mi~(h+,i) hi2 + ,iM~ ~
(4.129)
we get Vi ~ - 2 m i n
(1,b2)llxill 2 + I×~1 i • (2hl2aiN + 2h~2~iN + 2 h ~ 3PiN ) + + Ix~l(2h~3~i.
Using the f o l l o w i n g
•
i
+ 2h~3~i~ + 2b33Pi.)
: [~aiN + ~ i N [
-< Z2(~,~)II×NII
,
I~¢°iN +TfiNl
g Z3((~,~,T)Ji×NJl
,
•
Ch. V
336
Large-scale power systems stability
o
Ihl <- Isin(a~)lloi~l ~ij
= Isin
6°jl
'
loi~l -< loial + I%~I
(4.130)
,
where
z2(~,#) z3(~,~,7)
: min { v ' - ~ m a x ( l ~ l , l # l ) , l ~ 1 +
: min {2 max ( l ~ l , l # l , l ~ l ) , v ~
)+1~1 ,
max(l~l,l~
we o b t a i n i " i -2 min (l,b2)llxi ~2 +211xil I Z3(h12,h~2,h23 )
× n
×{lh-xNl I'~NI
+
IMII-M-~NI lPN] +b~h~l°iNl
+ e I [IAijl
I~jN]
+
2=
+ (~Nj~Nj * ~ij ~ij){ ajNl + ~ i j B i j l O i N l ] } +
.h23.h33)
I~i-~NII~NI+I~i-~NIIPNI
•
This y i e l d s 4 i i , , 2 m i n - - ( i ,2) b + 2Z3(h72,h22,h23)(Ibil~.. ~ i N +
i i i 2Z3(h12,h22,h23) :
z2(Aij,~Nj~Nj
i
i
i
i
i
i
2Z 2 {Z3(hi2,h22,h2s)iXi_XN
~ij
+~ij#ij) i
n Z
j#± for
i
i
i
the
first
pressed tion
by e q u a t i o n
(4.107);
(4.112)
n
j =i,
, j ~N ,
i
I ,
i-
+ Za(h13.h23.h3a)l,i-,~l} for
that in o r d e r to d e t e r m i n e
i /i
l +Z3(hls,h23,hss)l~i_~N
Za
This determines
~i4#i4)jj for
rows of the a g g r e g a t i o n
the last row of and c a l c u l a t e
j =N .
(4.131)
matrix
A.
Notice
A we start w i t h
VN
ex-
VN
along motions
of equa-
hence T VN = x~(P~HN+ H N P N ) XN+ 2XNHNhN(X)
Let
HN
be e x p r e s s e d
We can c a l c u l a t e
ON
by
a l o n g the o v e r a l l
system's n
motion n
~N : - 2 xN~ N - 2k~Np~ + 2 (M~ I - k~ N) ~NPN + 2~N ( ~ l xNj ~jN -~=~Z ~N~ CNj ) • Choosing
k
so as to c a n c e l
the t e r m
in
~Np N
Vector approach
337
k -- MNI~N I and using the majorizations
in equation
(4.130)
and
O
ICNj I ~ I sin(~Nj-6Nj)l
laNj I = TNj lajNl
we get VN ~ - 2 min (~N,~NIMNI~N)IIxNII2 +211XNII j:l~ [~Nj I¢°jNl + ~Nj TNj I°jN I] Comparing
the latter expression
with equations
(4.21),
(4.22),
(4.26)
and use of (4.115) yield - 2 rain (~N,~NIM~I~N)
for
j :N
2Z2(ANj,~NjTNj)
for
j ~N .
~Nj =
(4.152)
For a chosen e i , the limits solutions of the equation
Rim
:
and
tiM
~O
~i%ij
sin(Zij+
of
Li
are the nonzero
O
•
i N ) - s i n 6iN
•
j :m,M
(4 133)
Notice that £im < 0 and £iM > 0 . Smaller e i , larger interval ii and better estimate of the asymptotic stability domain can be achieved under appropriate ditions
conditions
but with less possibility
Following
Walker and McClamrock
(1967)
o
and Siljak
[ °iN
ei , eN
In the present
(1969) we get
aiN : {%iM are defined by equation case,
and for
:
71
~iN where
for these con-
to be satisfied.
u i = U×ill
(4.154)
(4.111). we get
2 A 2 AmiUi ~ Vi(X i) < MiUi ,
(4.155)
where kmi -- Xmi(H i) denotes
the minimum
eigenvalue AMi
denotes
and
- AMi(H i + } {i~ilei eT)
the maximum eigenvelue
of the corresponding
~il :~mi Numbers
ril , rj2
(4.36)
(4.136)
•
(4.137) matrix.
Hence
~i2 :AMi
are now found to be expressed
:
(4.138) by
:
i T V~. ] - 1 / 2 ril = [AM(Hi + 2 ~i~ileiei) ri2 : [km(Hi ) V°]-i/2i
(4.159)
338
Ch. V
Large-scale power systems stability
t
[COMPUTEflij, ~ij, °~ij ,V~I This cycle is to be donewhen E2 _c_p is aimed
No
@°o I
COMPUTEeij,ki,,
i,j=l,2,...,s
~:(e"ij) i ~=(kij)
v
J
I
Yes T
~ K
stable~ Yes .......
I RESULT
INo
I CHOOSEB>O I
JRe~It E2-~Pl JRe~olt:El~OJ JResult:E]c_PsJ J
Yes
J Result
Fc_I) J
t t I Resu]t: 6] UE2_CP } T
Figure 9.
..m
l
L
1 Result : E2 C_Ds o
°
E1 _cO
J
Algorithm for power system s t a b i l i t y analysis
V-CPs J
Vector approach
339
We e a s i l y
find in v i e w of
matrices
E and
(4.135)-(4.i39)
K are related
by
that
in the p r e s e n t
case,
:
K : E ( V ° ) 1/2
where
(v°)l/2
= [V~ i / 2 " V ~ t / 2 " ' ' ' ' V ° i l 2 s
We n o w p o s s e s s The power
all e l e m e n t s
system
the a l g o r i t h m
stability
in F i g u r e
Numerical
example
stability
analysis,
system,
I.
to c o m p u t e
and to c o m p a r e derived
• columns Gruji~
analysis
can be p e r f o r m e d
than performing
we h a v e p r e f e r r e d
values
by
cases.
in T a b l e Vl w h e r e
(1977)
vector
These
3-machine approach,
cases
a n d the
: cases
or by B o u f f i o u x
to the f o r m a l i s m
conventional
a simple
f r o m the
I V and V deal w i t h p a r t i c u l a r
I! is r e l a t i v e
I-4.
as i n d i c a t e d
an a c t u a l
to c o n s i d e r
parameters
in d i f f e r e n t
indicated
and R i b b e n s - P a v e l l a
• column
to a p p l y T h e o r e m s
9.
interesting
ate
I, III,
necessary
Rather
their
results
]T
treated
by
(1978);
established
by J o c i 6
et al.
(1978); • columns
VI a n d VII are r e l a t i v e
to the g e n e r a l
case
treated
in this
section. The
general
lowing
d a t a of the
(Joci~
et al.,
3-machine
1978)
E i = i.o17 Yi2
E 2 = 1.005
= 0.98 x 10 -3
812 : 86 o 8i
ent
cases,
E 3 = 1.033
@13
: 88
823 : 89
(degrees)
8 o3 : 0
(degrees)
= 3
M 2 = 0.01 compute
the a g g r e g a t i o n
data corresponding
information
matrix,
or o t h e r a n a l o g u e • the m i n i m u m matrix;
matrix
(p.u.) in the
differ-
to e a c h
. case are r e p o r t e d
in the
1.
provided
• the a g g r e g a t i o n
A
(p.u.)
adopted
first r o w o f T a b l e The
= 0.106
M 3 : 0.0101
~i = ~2 = 0.i The p a r t i c u l a r
fol-
(p.u.)
Y23
= -2
we h a v e
are the
= 0.114
8o2
in o r d e r to
here
YI3
M I : 0.01 Moreover,
system considered
:
by this
according
expressions
and maximum
table
concerns
:
to e x p r e s s i o n s
adapted
eigenvalues
(4.131)
and
to the c o r r e s p o n d i n g
o f the
corresponding
(4.132)
case;
aggregation
340
Ch. V
• the limit
values
systems,
V~ s,
responding We o b s e r v e matrix
according
that
in all cases
point.
We also
tially
of
A
observe
in the case w h e r e that
to the
implies
that
the
(4.i54)
the
for that
corollary
stability values
governors
the subsystems'
of
disconnected
sub-
or its a n a l o g u e
cor-
case.
- except
According
stability
functions
to e x p r e s s i o n
to each p a r t i c u l a r
is stable.
V.4.5.I,
imply
of the L i a p u n o v
Large-scale power systems stability
to T h e o r e m
of the
of the
are
stability
of c o l u m n
taken
II - the
i of S e c t i o n
system's
V~ s
A
equilibrium
increase
into account;
substan-
this
could
is reinforced.
TABLE VI II
I
Data
i = 1,...,3
Xi = 10
x~(A)
^.(A) v~
-1.767
0.348
-
; XS : 9 " 9
i = I,-.,3
ei=O
i=1,..,3
e i =0
i=i,...,3
i =i,...,3
~i:o
i : I,_.,3
~ i =0
i=1,_.,3
i~j
-1.770 0.348 0.120
0.089 17.4501 18.9 0 0.089]
0.3151
-1.761]
-
18.i0
2.028] 2.190|
-19.800]
[
-1.769
0.313
-
- 19.870 I. 366 -
44.33 44.58
43.43 43.63
5.35 5.22
20171
0.346-1.763 2.178 0.311 0.311-19.800]
19.830 1.409
-
18.27
43.43 43.63
0.315 -1.761 0.110
i}~j
i i = ; ~ 2 = l O ; A3=9.9 kij:O.ll i,j=l,...,3 i~j
i , j = i,...,3
2. 095 1,434
-
;xI=)~2 =10
I V
Lij=O
i , j = I,..,3
xlj =o ~i=0 ~i =0 Aggregation matrix A
III
VII Data
;%1 =A2 = i0 AI2:0.i0 k31 =0. i i
; A3 =9.90 i.A.21 : O. 12 ; k13:0.12
~-39 = O. Ii ; X23 =O.lO ~i=O i = I,...,3 ~i=O i = i,...,3 Aggregation matrix A
"-1.769 O. 489 0.311
X~(A) AM(A)
0.442 - I . 763 0.311 -
v~
The
i = I,...,3
/~i : 1
i = i,-.,3
2.017]
-1.788 0.317 0.120
19.870 1.230
£
-
£
matrix
of c o l u m n
=
0.286 -1.783 0.112 -
of the
in the case
i , j : I,...,3
¢~i = 25
2. 178 / - 19.800]
VII
Xl:A2 : i0 ; 13:9.90 LI2 = 0.i0 ; ~21:0.12 A31 =0.ii ; A13 =0.12
i = 1,-.,3
k l j =0.11
44.33 44.58
computation
(4.56);
k i = ]0
i #j
k32 =0~11 ; k23 =0. I0 ~ i = 25 i = i,...,3 Pl = 1
1.846" 1,994 -7.920
'-1.788 0.448 0.311
0.305 -1.783 0.311
7. 988 1.417
-8.116 1.219
2.851 2.915
2.851 2.915
derives
from
2.031" 2.194 -7.920
-
expressions
we c o m p u t e
-!.32 × 10 -4
1.88 x 10 -2
0
32 x l O -3
-6.92 × 1 0 -4
0
94 × 10 -2
1.92 x 10 -2
0
I
i = 1,...,3
(4.55)
and
Vector approach
Matrix
341
K is r e a d i l y
c o m p u t e d via e x p r e s s i o n
(4.51) and
44 x l 0 -I
-3.74 x l 0 -2
1.04 Neither
E nor
estimate
of
1.04
K is stable;
therefore,
-3 98
E2
cannot
be used as an
D.
Let us now examine the m a t r i x matrix.
we find
.03 2 19
-
K =
(4.52);
AT8 + 8A , w i t h
B = B T = [ , the identity
In the case of column VII we get
AT+A
Obviously,
--3,576
0.751
0.751
-3,564
2.342
2.505
=
this is a stable matrix;
2.342]
2.5051 -15.84
]
a c c o r d i n g to T h e o r e m
4, we c o n c l u d e
that
[vlx1 ]
E l = {x :vl(×~ ~ i } where
VI(X)
= [1
1 1]
V2(x 2
,
V3(x 3 0
71 = is an e s t i m a t e The t r a n s i e n t performed
of
by c o m p u t i n g
analysis
of the 3 - m a c h i n e
Vl(t e) , V2(t e) ,V3(t e)
Vi
function
ing times,
t e . The critical
Vl(Xc)
•
= 71
0
(i = 1,2,3)
c l e a r i n g time,
contributed
by S h a a b a n '
(Shaaban,
s y s t e m s h o u l d now be
, where
Vi(te)
denotes
takes at d i f f e r e n t
clear-
t e , is the one for w h i c h
V.4.5,2.2. T r / p Z z ~ i 6 £ dzcompo~iZion-aggrzgation. is m a i n l y
,
N.
stability
the v a l u e that the
0
min [VI,V2,V 3] = 2.851
The p r e s e n t
section
1983; S h a a b a n and GrujiE,
1985). An
N-machine
power
s y s t e m is d e c o m p o s e d
into s u b s y s t e m s ,
ing of two m a c h i n e s
in a d d i t i o n to the c o m p a r i s o n
is d e c o m p o s e d
(N-1)/2
N
into
. Considering
transfer
interconnected
conductances,
machine.
subsystems
mechanical
each c o n s i s t The s y s t e m
for odd n u m b e r
damping,
electro-
J H. Shaaban was on leave from the University of Menoufia~ Ei-Kom, Egypt, at the University of Belgrade where his research (Shaaban, 1983) was guided by Lj .T. Gruji~.
342
Ch. V
Large-scale power systems stability
magnetic damping and speed governor action, the mathematical model of the system is derived, and it is decomposed into and one second-order interconnected subsystems. system is decomposed in
(N-2)/2
sixth-order,
(N-i)/2 If
N
sixth-order
is even then the
one third order and one
second order subsystems. Each of these systems is decomposed into a free tem and interconnections.
(disconnected)
assumed tO include the largest number of nonlinearities, linearities.
subsys-
Each of the sixth-order free subsystems is i.e. six non-
For this subsystem we adopt a scalar Liapunov function in
the form "quadratic form + sum of
the integrals of the six nonlineari-
ties". A vector Liapunov function, whose components are Liapunov functions of the free subsystems,
is constructed, and used for the system
aggregation.
A square aggregation matrix of the order
(N+I)/2
tained when
N
N
is odd, and of the order
(N+2)/2
when
is ob-
is even.
Stability of this matrix implies asymptotic stability of the system equilibrium. In this section
N
is decomposed into
is odd, without losing generality. (N-I)/2
The system (4.102)
interconnected subsystems, each consisting
of two machines and the comparison machine, using the triplewise decomposition.
It is to be noted that none of the system machines
the comparison machine)
Now, by introducing the set tors
xI
and
×N
Jl :{il,iI +I}
as follows
and defining the state vec-
:
=
×I
(except
can be included in more than one subsystem.
,
[ailN • OiI+I,N '~ilN
]T
~il+l,N ' PilN ~ PiI+I,N
: [Xil XI 2 xI 8 XI 4 xI 5 xI 8
]T
,
and ×N : [~N
PN IT : [XNI
XN2]T •
(4.140)
we can decompose the system mathematical model
(4.103) into
s = (N-1)/2
sixth-order interconnected subsystems and the second-order interconnected subsystem, which has the general form xN
:
where
PNxN
+ hN(X)
[
,
-x~ M~1]
PN
:
uNJ
-~N
and {XNj~jN - M N I A N j O N j ( ° N j ) }
hN : |- j : 1
[
0
(4.141)
Vector approach
343
E a c h of the s i x t h - o r d e r
subsystems
xI : P I x I + B I l l ( e l ) and it can be d e c o m p o s e d
may be w r i t t e n
+ hi(x)
i n t o the
for
free
in the
general
form
I : 1,2,.-,s ,
(disconnected)
(4.i42)
subsystems
given
by XI : PIXI + B I f I ( a I )
,
(4.i43)
where a I = CTx I and the In
interconnections
(4.&43),
fl(al) define
hi(x)
the m a t r i c e s
is a n o n l i n e a r the m a t r i x
PI , B I
vector
ties
that
given
the
I = 1,2,...,s
and
T CI
(4.i44)
.
function.
are
constant
Referring
to
PI
as
-0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
- rI
AI
-1 Mi I
o
o
~
-~-
o
MZ i~-+ l
0
0
-aii
0
-~ii
0
0
0
0
-~ii,l
0
Pi :
Assuming
for
free
subsystem
matrices,
(4.103),
and
we can
(4. 145)
(4.143)
-~ii+l
contains
the
six n o n l i n e a r i -
by O
o
¢ll(ell)
:
COS (ail N + 6ii N - ~ i l N) - cos (5ii N - ~ i i N )
,
¢i2(012)
:
cos ( o i i + l , N + 5 ~ I + I , N - S i l + l , N ) - COS (5 °i i + l , N - 0 i I + I , N )
¢13(ai3)
:
¢i4(014)
=
Cos (ei I • ii+l + 5oi i , i l + 1 - ~ i l , i l + l ) - COS ( S oi l , i l + 1 - ~ i l , i l + I 6° cos ( a i i + l , i I + i i + l , i I - 8ii+l,i I) - COS (5 °i I + l , i I - 8 i i . l , i i
¢i5(ai5)
=
cos (aNi I + 6Ni I - 0iiN)
O
- cos (5 NiI ° - 8iiN)
6o
¢i6(ai6 ) =
cos ( O N , i i + l +
o
N,ii+l-@ii+l,N)-
COS ( S N , i i ¢ l - 8ii÷l,N ) , (4.146
we can d e f i n e
the
following
matrices
fi(-i ) : [ ¢ i i ( 0 i i ) , ¢ i 2 ( ~ i 2 ) , ¢ i 3 ( a i 3
:
),¢I4(014),¢I5(015),¢I6(°I6)]
T ,
(4.147
~
I
LJ.
+
N
H
l
i~ .
~
"I"
~.
~
::~
~
~,
~-~.
I
~
I
~'
P°
I
~
~.
~
I~
~
II
I
Il ~
+
;~
I-~
I ~
H.
~'~"
II
~
i-~
I~ •
~
O~
+
I~ " H
Z
II
I-,,
F-I
H
H
H
~
I~ •
]I
I- ~
H
H
~ % ~
+
~o
~'°
Q
II
H.
I
0 v I
H + ~
•°
t_l.
• ¢~
~
h-~
I
UI
"-~
LI.
i.--t
+
!
+
Q
r'.
LJ.
FH U.
. . . . .
H>~
~°
.-I-
..::l
.-.,'l I
ixJ
+
to
"I i-~ X
+
"~I I.-1
I.J.
0
I~ -
0
o
f~
o
v
v
o
o
I
o
o
o
o
o
o
o
o
I
0
o
0
o
I I
o
o
o
o
,g
o
0
0
0
H ~
I
o
0
0
0
0
O
(;3
O
C)
O
o
(23
~t
'~"r
b-~
ZJ
I.--i h -I +
I
O
H
~-+ k's
I
o
II
~-~
~t
~ t
t,-I
~l
0
H. I H ~
!
O
H
H
~
O
o
O
C:,
o
o
o
"1
0
Vector approach
345
N F I : kii + ~Ni I +
~
N kii j
FI : kil+l +kN,il+l +
;
J~il+l
It is obvious
~I = ~i I -~N
;
~I : Bil+l - g N
,
~I = ~i I -~N
;
~I : ~il+l - ~ N
'
~I : MTl zi -M~ I
;
~i
that the state vector
by
T x
:
For each of the s i x t h - o r d e r
system
is given now
×T,×~]T
[Xl,X2,...
,
,
: Mi 11+ 1 -MN I
of the whole
T
kii+l,j
(4.151)
S
free subsystems,
we accept
a
VI
function
of the form VI(XI) where
HI
T 6 I~l~ : XiHiXi + Z @ I ~ ¢li(ali) dal~ £=i 0
is a s i x t h - o r d e r
are a r b i t r a r y given by
positive
I : 1,2,...,s , (4.152)
positive-definite
and the n o n l i n e a r i t i e s
matrix,
@I£
¢iz(ai£)
are
(4.146).
For the last Liapunov
symmetric
numbers,
for
(S+l)-th
second-order
free subsystem,
we construct
the
function :
XN
(4.153)
1 where
C is an arbitrary
positive
number.
A l o n g the motion of the free s u b s y s t e m
(4.145)
we compute
Vif
,
6 VI(XI) f
: XTI ( - G I ) XI + 2 f~(@1)
B~HIXI+
VI :l,2,...,s where
GI
is a symmetric
matrix,
defined
Z @I~fI~( ~I£) °I~ "
I:l
,
(4.154)
as
-G I : P~H I + H I P I Substituting
the matrix
is computed,
and it is found that
inite Nowj
(the first
two diagonal
under the condition
in the form
PI
from
(4.145)
(4.155) in (4.155),
this m a t r i x
elements
AI =AI =0
cannot
the matrix be p o s i t i v e
GI def-
are zeros).
, and by c h o o s i n g
the matrix
HI
346
Ch. V
-h~l
HI
h~3
0
0 I h22
h~3
0
:
Large-scale power systems stability
h~5
0
0 I h24
0
0 I h26
h~3
0
0
0
0
h~6
I
,
I
0 I h15
h24
0
h44
0
0
0
0 i h55
0
h~6
0
0
0
where I hll
-i Hi I i I ) h 1!3 : (P I + ~i I
I h15
-i I : MilhlS / ~il
I h22
:
-1 I hi6 : Mil+lh24
0
,
,
-i Mil+l~il+l + #ii+ l
(~I
(4.156
/gil+l
) h2I q ,
,
I i +K I I h33 : ~ h13 hl 4 : 1 + K I hl 4 , -1
:
1 ~ii
hI 66 :
1
hi5
we obtain
the matrix
G I
ber)
~ii+i
Mil I + -#i - I h13 )
-i I (Mil+lh44
-i Mil+l +
~ii*i
,
I h24)
'
in the form
61
0
0
0
0
0
0
0
@
0
0
0
0
0
0
I 2Klhl3.
0
0
0
0
0
0
I 2K1h24
0
0
0
0
0
O
2~ilh~5
0
0
O
0
0
O
2~il+lh~ $
:
It is to be noted equation
-i I (Hiih33
(4.156)
that positive
definiteness
can be g u a r a n t e e d
only under the two conditions
(K I
of the matrix
is an a r b i t r a r y
h13 >0
(4. 157
and
h24 >0
HI
of
positive
num-
Vector approach
Now,
347
substituting
into equation
from equations
(4.154),
(4.147)-(4.149),
and selecting
the positive
-i I ~ii : 2MiiAlh33 -i~ I ~13 : 2MilAlh33
h~
(4.156)
and
(4.157)
numbers
-i
-
I
~i2 : 2Mil+iAih44 ~;i4 -- 2N.~I+ i~Ih414 -iI ~16 = 2M N Alh44
: 2M~iAih~
we obtain Vl(Xl)f
I 2 I 2 -2Klh13x13 -2Klh24x14 -i I -2MiiAihl3Xil¢iI(Oil)
:
,I x 2 I 2 -2~iln55 15 - 2 ~ i l , l h 6 6 X l 6 -i - I -2Mii+iAih24xi2¢i2(oi2) -
- 2Mi~1¢13(°i3)(~3x11
+ h33xi4)~
I )(h24x I 2 + h44XI3 ) + I I +2MNIAI@Is(aIs)(h~3xII + h 2 4 X i 2 + h44xi4 ) + I
- 2 M i-i I * i A~I ¢ I 4 ( a I 4
+2M~l~i¢i6(ai6)(h~3Xi Let us now introduce = 1,2,5,6
,
~Iz
the positive
is satisfied
on a compact
and
0i£
> eI~ I~ interval
interval estimate
ii~
: [~I£
£ : 1,2,5,6
Oil]
oil ,
~I3
and
~i4
of
~i3 : sin (Sii,ii+ I
eI
respec-
.
(4.162) smaller,
and so we obtain
-12 ) -~13~13(a13)] -i ~ I +2Mii+iAih24[oI4~i4(ai4)
il,il+l)
solutions,
is taken
from the right-hand
are determined - ~°
(4.161)
~ = 1,2,5,6
L I given by (4.161) becomes larger, for the system stability domain.
-i~I 2MilAlh13[°I3%I3(OI3
(4.160)
, i.e.
and positive
for
(4.159)
el£ E ]0,~i£ [ , for ~o sin (8i£- iZ) so that
of
if the value
Now, by adding and subtracting non-negative expression
where
for
: el£al£
to note that
:
~i£
02
are the negative
¢i~(ai£) It is important
by
_
fl~ where ~I~ tively, of
constants
are determined
ali¢Iz(~l£)
I + h24Xi2I +h~3xi3)
the
a larger
side of (4.159)
the
(4.163) _~-1.214914['aI4)]" ,
as
and
~14 : sin (0il,il+ I +
6~
~i,il+ I) (4.164)
we can "majorize" calculations, as
the right-hand
side of (4.159)
VI(XI) f ~ -XIIIXIII 2
,
after
VI : 1,2,...,s
trivial
,
(4.165)
348
Ch. V
where 61
kI
is the m i n i m a l
, which
is g i v e n
(positive)
eigenvalue
of the
symmetric
matrix
as
g[1
61
Large-scale power systems stability
o
gI
o
o
g~8 0
g~ 3
0
0
0
I g27
I
0
0
0
0
0
g44
0
0
g47
0
O
0
0
0
I g12
I g22
0
g23
g33
g 4
0
I
o
=
, (4.166)
I
0
0
0
0
g55
0
0
0
0
0
0
g66
I
0 I g18
g27
I
0 I
0
I
g38
I
I
g47
0
0
g77
0
0
0
0
0
I g98
g38
where
g~1 : 2AIh~3<M i ~
+M~
) ,
i : 2~h~4(M[l1+1¢12
g22 I g44
I = 2Klh24
I g66
:
I g55
I = 25ilh55
I g12
: -MN I (Aih~4 ~Nil + A-l h l I3 ~ N , i l + l )
I g14
-i I : -MN Alh44[N,ii
g23
= -~h
I g38
-i ~ I = -Mil+iAlh44
I g77
-i~ I : 2MilAIhl3 :
~Ni I
: el6
be c h o s e n
~il tlh
~6 , (4.167)
, -i ~ I = -Mil+iAlh24
I g27
-i~ I : -MilAlhl3
•
I g47
:
/ ~I 3 '
I g88
-i -- I = 2Mil+iAlh24
33 ~ N , i l + l
to n o t e
-- 0 . 0 0 1 so that
2
'
that
-i~ I -MilAlh33
, ' ' /~I 4
' O
and
~N,il+l
= sin(@il+l,N
the n u m b e r s
are c h o s e n
M~ii6) ,
'
I g18
,
~o sin (@il N + ilN)
It is i m p o r t a n t (615
,
+
K I , hl13 , hi4 , EI I
to be s l i g h t l y
we c a n g u a r a n t e e
+ ~il+l,N)
positive
larger
than
definiteness
and
zero)
el9 should
of the m a t r i x
6I
Now,
we p r o c e e d
to m a j o r i z e
in the
I : 2 (hll3Xll + h 3 3 x 1 3 )
[grad V I ( X l ) ] T h I ( X ) N-I
×[~xN~+~xN2+
right-hand
Z
side
of what
follows
×
-i
{A~j~-Mi~AiIj¢i~j(oiIj)+M~A~j~Nj(oNj)}]
J~JI + 2(hI4xi2
+h414x14 ) x
:
Vector approach
349
N-I × [~IXNI + ~IXN 2 +
+ ~ (~x~
JgJ~
+h~x~)(-~x~
-BIXN2 ) + 2 (h~6x12 + h ~ s x i 6 ) ( - ~ i X N l
-~IXN2 ) (4.168)
N-I
S
Noting that
Z is equivalent J~JI following ma~orizations :
, and introducing Z Z K~I jeJ k
to
~ilJ(lXxll+[x~11 ) , ~iij
lfiIj(aiIJ )I ~
I¢il,l,j(oii+l,j)l ~ ~ii+l,j(Ixi21 + IxJiI)
,
= sin (~iI j - ~ i j )
the
,
~ii,l,j : sin (@ii+l,j -6~i+l,j)
,
(4.169) we can majorize
the right-hand
side of (4.168)
in the form s
~ 2ZINIIXIIII×NH +~illxi g2 +2
[ g r a d VI(XI)]ThI(X) where
VI :l,2,...,s
ZIN: Z2(ZlNa,ZlNb)
Z ZIKIIxI~IIxKII , K~I (4.170)
,
(4.171)
I
ZINb = Z2 EZ3{(NIh~3 - ~ h f s ) N,I h 3 3
-
I
I , ~Ih~6}] , "Ih[5} ; Z3{(NIh24-~Ihfs) , -NIh44
and kl is the maximal eigenvalue QI , whose elements are given as
of the fourth-order
symmetric
,
matrix
I I I I I1 I I I qll = q22 = q33 = q44 : q 2 : q14 : q23 : q34 = 0 , I -1 I q13 : Milh33
S
~ (AiiiK~iiiK + A i l , i K + l ~ i i , i K + I) , K~I
I : M-1 hI q24 iI+l 44 In (4.170),
ZIK
(4.172)
S
E (Aii,l,iK~ii+l,i K + A i i + l , i K + l ~ i i * l , i K + l ) K/I
is defined as
ziK
:
Z2EZ2(ZIK,ZIK);Z2(~K,~IK)]
(4.173)
,
where
-i ZI K = Z2 [(MN ANiK~NiK + M i-i iAiiiK~iiiK) (MNIANiK~NiK ~IK
:
I I Z2(h13,h33) ;
+ -i I I Mii+iAii+l,iK~ii+l,i K) Z 2 ( h 2 4 , h 4 4 ) ]
Z2 [ ( M N I A N , i K + I ~ N , i K + I
(M IAN,i +I N,iK+I
+M-I iiAii,iK+l~ii,iK+l) + M -I
,
I I Z2(h13Jh33);
Z2(h 4,h 4)] ,
350
Ch. V
I
Large-scale power systems stability
I
I
ZIK : Z 2 [I/iIiK-XNiK I Z2(hls,h33) ; IXiI+I,iK-XNiK[ ZIK : Z 2 [ I k i I , i K + t - X N , i K + t l We can define A : [~IK ]
the first
as follows
Z2(h13,h33) ; t k i i + l , i K + l (N-I)/2
I
Z2(h24,h44)] ,iK+lt
rows of the aggregation
, Z2(h23,h44 ~ • matrix
: -(II-XI)
K =I
2ZIK ~IK :
K ~I N-1 K :-~-
2ZIN for
(4.174)
I = 1,2,..., S
N-1 2
=----
To define the last ((N+l)/2th) row of the matrix A, the total time derivative of V N , along the motion of the subsystem (4.141) is computed
as VN(X)
2
= - 2CXNX~I - 2 gNXN2 + 2 XNlXN2
(cM~
-aN)
(4.175)
N-1
- 2~N1 .z (XNj~Nj +M~IANn~Nj(ONj)) 3:1
Choosing the constant C : MN~N , we c a n e l i m i n a t e and then equation (4.175) is rewritten as VN(X)
:
2 • - 2 (MN~NkNX~I + PNXN2 ) - 2XN1 + ~N,iK+I~N,iK+I
The left-hand
the
(4.176)
in
XN1XN2 ,
s -i g ~ {kNiK~NiK + MN ANiKCNiK ( N i K ) K:I
+ MNIAN,iK+ICN,iK+I(ON,iK+I
side of equation
term
)}
may be directly
(4.176) majorized
as
%(X) : -2 m i n ( M N ~ N k N , #N)IIXNII 2 + S
+2 This
K=I
z2 [M] l z2 (A.iK~iK,AN,iK+I[N,iK+I) ; Z2(XNiK ,XN,iK+I) ] llXNlJ fiXKil .
implies I - 2min
~s+l,K
:
2Z2
(MN~NXN
,~N)
{MNIZ2(ANiK~NiK
K : s+l ,AN,iK+I~N,iK+I~ K :I
for
; Z2(XNi K , ~N,iK+I)} I : 1,2,...,s -
N-1 2 (4.177)
The
system
(4.177).
aggregation
matrix
is completely
defined
now by (4.t74)
and
•
Vector approach
351
Numerical example 2. in Fig. as the
In this
10 is c o n s i d e r e d comparison
connected
machine,
subsystems,
example,
as a s i m p l e
as
the
the
system
indicated
seven-machine
power
system.
is d e c o m p o s e d
in Fig.
into
11.
E:i.~S/0" M . lD.O
Pm=3.87
/,.zl+j O.BgS
t
o ,.olio.,,,-,.I IIo., l\ E:1.30/9" M:o3o~
Pm:~ls'3
/
/
t
/
I1
tl
/(~
E=i.3s/Ir I
l
J
m>-r ~q /
B-
o.~<:j'o.,,
Pm:0"117 ~0.75/-82"
q
~ E:,
/
0~
- - ~ \ F~
.
'~'J°-~' _ k l ~:'~J
_
0.55/-82" ~Pm:O,252
o.oo,~
O.BO~jO
~E:,.~/io'
H~\ hI ~,:o.~, I, , ' ~ ' ~ ~Pm=O.786
/
g:g.~~'1-/"
~o.,o~
o.7o/-8r L
%°17, / ~ V / k m= .
11
0.65 +jO
Figure 10
s,/'/ ~ \ ' \
s3
/
®
'\ ®
\-.... /
\
/
Figure 11 The
following
parameters
are
h i = 8.0 Xij
;
, i ~j
ejl
:Ej2:0.70
;
;
~7
: 8.1
;
, i,j =1,2,...,7
~i --25 , i = 1,2,-..,7
hl] 3 =hip4 = 1.0 K I :2.0
:
, i :1,2,...,6
= 0.10
#i : 2 0
selected
, j = 1,2,3
; ;
;
K 2:2.6
;
K 3:2.1
, j : 1,2
;
e31
system
Choosing
;
:e32:0.65
shown
machine
three
7
inter-
352
Ch. V
Then the matrices of the three matrices (4.177)
G*
subsystems,
A =
This matrix
B : BT
k*
and
the a g g r e g a t i o n
for each
k for these
matrix
of ( 4 . 1 7 4 ) a n d
0.6213
0.5688
9.5094]
-2.0747
0.5727
12 99611
0.6880
0.6278
-1.5126
11
0.4839
0.4480
0.4051
-40
the conditions asymptotic
I to T h e o r e m
(4.28)
stability
.
96911 00O0J
and thus
it is a stable
of the system e q u i l i b r i u m
1.
to determine
Therefore
in the form
negative
are c o n s t r u c t e d
0.6841
implies
system.
(4.172)
-1.5013
We can now go further, the whole
Finally,
I
due to Corollary
Q
as
satisfies
This
and
and the e i g e n v a l u e s
are determined. is computed
matrix.
(4.166)
Large-scale power systems stability
a stability
the matrix
B : diag{l
1 1
domain
ATB + BTA
estimate
, with
20 } , is computed
for
the matrix and found to be
definite.
According
to T h e o r e m
4 we conclude EI
: {x
that
v(x)-<71}
:
,
(4.178)
where V(X)
= VI(XI) + V 2 ( X 2 )
and 71 : min is an estimate
where
equation
and
Xim
V I : 10.37
In terms
stability
(see A p p e n d i x
,
of the o r i g i n a l
tem,
the estimate
El :
{(~,~,P)
ofShaaban,
E1
: (VI(~,~,P)
and
, the constant
physical
is w r i t t e n
i983)
:
¢I£(°I£ ) d°I£
of
Xim
, we get
O
V 2 : 21.12
(V~ =~)
domain.
(xI)THIxI + ~ ~I£ £:i
O
values
+ 2 0 V4(X4) , Q
are lower and upper bounds
O
Using these
0
(Vl,V 2 ,V3,2 0 V 4 ) ,
min min m:l,2 ,...,6 XimE{Xim, ~im} Xim
+V3(X3) 0
of the system asymptotic
U s i n g the following
VI :
0
variables
V 3 : 13.82 71 : 10.37 ~ ,¢~
and
is determined. P of the sys-
as
+V2(6,~,P)
+ V 3 ( S , ~ , P ) + 2 0 V 4 ( ~ 7 , P T ) ) ~ 10.3y}
where Vl (6,~, P ) :
: 11.7(817 + 0 . 0 9 ) 2 + 1 1 . 8 ( 6 2 7
-0.19) 2 + 0 . 3 5 { ( ~ i - ~ 7
)2
+ (~2-~7)2} +
Conclusion
353
+ O.04{(P1-P7 )2 + (P2-P7)2} +2(617 + 0 . 0 9 ) { ( ~ 1 - ~ 7 ) + 2(627 - 0 . 1 9 ) { ( ~ 2 - ~ 7 )
+ O . 1 3 ( P 2 - P 7 ) } +2.0
+ 2.03 sin (627 -1.8) +2.25 + 2.36 sin {(627-617)
sin{(~17-627)
-1.71}
+0.12 (P1-P7)} +
sin (617 -1.74) -1.71}
+
+
- 0.08 { sin (617 + 1.74) + sin (627 + 1.8)}
+
+ 1.08 617 -1.205 627 + 8.85 ,
V2(~,~,P)
=
= 12.8 (637 - 0 . 1 6 ) 2 + 12.7 (647 - 0 . 2 4 ) 2 + 0.41 { ( ~ 3 - ~ 7 )2 + (~4-~7)2 } + + O.06{(P3-P7 )2 + (P4-P7)2} + 2(647 - 0 . 2 4 ) { ( ~ 4 - ~ 7 ) + 3.19 sin (647 -1.75) + 3.34 sin {(647-637)
+ 2(637 - 0 . 1 6 ) { ( ~ 3 - ~ 7 )
+0.16 (P4-P7)}
-1.73}
+
+ +
,
=
= 12.3(657 - 0 . 1 7 ) 2 + 12.6 (667 + 0 . 1 1 ) 2 +
0.36{(~5-~7)2 + (~6-~7)2 }
+ 0.05 {(P5-P7 )2 + (P6-P7)2} + 2 (657 - 0 . 1 7 ) { ( ~ 5 - ~ 7 ) + 2 (667 + 0 . 1 1 ) { ( ~ 6 - ~ 7) +0.16 (P6-P7)} + 2.26 sin (667 -1.75)
+ 2.2 sin
+ 2.4 s i n { ( 6 6 7 - 6 5 7 ) - 1 . 7 1 } -
+
- 0.10 { sin (637 + 1.83) + sin (647 + 1.75)}
+ 0.904 637 -0.863 647 + 13.395 V3(6,~,P)
+ 3.31 sin (~37 -1-83)
+ 3.44 sin {(637-647) -1.73}
+0.16 (Pa-P7)}
+0.14 (Ps-P7)} +
+ 2.07 sin (657 - 1.82)
(657-667)
-1.71
- 0.072{ s i n ( 6 5 7
+
+
+
+1.82)
+ sin (667 + 1 . 7 5 ) }
+
1.167 657 + 1.90 667 + 9.144
and
V4(~7~P 7 ) : 250 ~
2 + P7
V.5. CONCLUSION Two direct
methods
for transient
stability
have been developed approaches.
and examined
The scalar
seems
efforts
approach
produced
the numerous moving
the necessity
: the scalar
carried
handicap
of adopting
out
i.e.
during
in this
maturity,
system
systems Liapunov
owing
the last twenty-five
field have
its conservative
simplified
of power
and the vector
by now reached
by many researchers
studies
the major
to have
analysis
succeeded
character.
representation,
to the years
in re-
Concerning it may not
:
354
Ch. V
Large-scale power systems stability
be - and in our opinion is not - a drawback of the method, when used for first contingency dynamic security monitoring. nov or Liapunov-like
In fact, the Liapu-
criteria should be considered as complementary,
rather than competitive with the standard numerical integration procedure, each method possessing its own field of application. For the vector Liapunov approach it is still too early to tell how interesting it could become for practical applications.
A priori,
its
essential characteristics make this approach likely to be more appropriate for decentralized control purposes, rather than for global stability investigations.
Once again, by their very nature, the scalar
and the vector approaches appear to be complementary rather than competitive. Whatever the evolution in this field, one by now may claim that power system transient stability has been a very difficult and extremely challenging domain of research; yet one of the most efficient and fruitful applications of the Liapunov stability theory.
Araki, M. (1975), Application of M - m a t r i c e s to the stability problems of composite dynamical systems. J. Math. Anal. a~d Appl., 52, No.3, 309-321. Araki, M., M. Saeki, and B. Kondo (1980), Application of a new stability criterion of composite systems to multimachine power systems. IEEE Trans. Autom. ControIj AC-25, 4807483. Athay, T., R. Podmore, and S. Virmani (1979) , A practical method for the direct analysis of transient stability. IEEE Trans PAS, PAS-98, 575. Athay, T., and I. Sun (1981), A structure preserving model for power system stability analysis. IEEE Trans. on PAS, PAS-IO0, No.l, 25. Aylett, P.D. (1958), The energy-integral criterion of transient stability limits of power systems. IEE Proc., 105-C, No.8, 527. Barbier, C., L. Carpentier, and F. Saccomanno (1978), Tentative classification and terminologies relating to stability problems of power systems, Eleatra, No.56. Bellman, R. (1962), Vector Liapunov functions. No.l, 32-34.
J. SIAM Control,
I,
Bergen, A.R., and D.J. Hill (1981), A structure preserving model for power system stability analysis. IEEE Trans. on PAS, PAS-IO0, No.l, 25. Bouffioux, A. (1978), Fonctions vectoriell~s de Liapunov pour syst~mes ~nerg~tiques. Travail de fin d'6tudes, University of Liege, 1-95. Chen, Y.K., and R. Schinzinger (1980), Lyapunov stability of multimachine power systems using d e c o m p o s i t i o n - a g g r e g a t i o n method. IEEE PES Winter Meetiny, Paper A 80 036-4.
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Differencial-
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Direct
The Math.
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U.S.
POSTFACE
System non-stationariness can be originated by its time-varying physical nature. An adequate system mathematical model is then also timevarying. However,
the system description via a state deviation is non-
stationary despite the system is physically time-invariant as soon as it is non-linear and its reference motion is time varying. For such a physical and/or mathematical reason it is important to study non-stationary non-linear systems. Their greater complexity than that of stationary non-linear systems results in a great variety of stability properties. They depend in general on the initial moment
t o . The class-
ical stability theory tackles mainly the stability problems whose solutions are independent of
t o , which has been a basic assumption at the
very beginning of Liapunov's dissertation
(Liapunov,
1892) and for the
development of the stability theory for decades. The stability theory for n o n - s t a t i o n a r y non-linear systems in general, taking into account the influence of
t o , is presented in Chapter I.
It shows that stability properties should be tested on time-varying sets rather than on time-invariant sets due to non-stationariness of the system and/or its mathematical model expressed in terms of the state deviation. The general absolute stability results p r e s e n t i n g necessary and sufficient Liapunov like conditions are used as a basis for new linear systems like absolute stability criteria related to nonstationary Lur'e systems with multiple nonlinearities.
Besides, new
relaxed general asymptotic stability conditions are established for time-varying non-linear singularly perturbed systems. Liapunov like stability conditions are expressed in terms of existence of a
v
function
(or, functional family) with specified features called
a system Liapunov function tively. Hence,
(or, L i a p u n o v functional family),
respec-
a fundamental scientific problem and the greatest obsta-
362
Postface
cle for wider effective practical application of Liapu~ov's
(direetj
second) method is the problem of a Liapunov function construction for a given system (or for a class of systems). Higher the system dimension, more difficult the problem. more complex the problem.
More complex the system structure,
Greater the system dynamical complexity
(number and/or form of nonlinearities), more severe the problem. order to simplify the problem,
Bellman
In
(1962) proposed the vector Lia-
punov function concept for a Liapunov aggregation of a (complex, interconnected)
large-scale system decomposed into interconnected subsys-
tems. In Chapter II the concept of matrix Liapunov functions is proposed and developed. This new mathematical tool together with a novel development of the comparison principle presented therein constitute a powerful basis for a new stability theory direction. This is shown via new general stability criteria. One-shot L i a p u n o v function construction appears more real for subsystems than for the overall system. A number of different aggregation forms are developed in Chapter III. They are used to derive simple overall system algebraic
stability conditions via the Liapunov func-
tions of the subsystems. They lead to new forms of Liapunov functions for the whole system even if it is subject to arbitrary structural and/or parameter variations.
Hence, both structural and/or parameter
robustness is assured for various stability properties of non-stationary non-linear large-scale systems. Moreover, their robustness ranteed for cone variations of nonlinearities.
is gua-
An intrinsic advantage
of the results is a qualitative simplification of the stability conditions reduced to a stability test of an aggregation matrix with constant entries and with the dimension inherently reduced to the number of the subsystems. The stability theory of non-stationary non-linear large-scale systems with multiple time-scales and subject to singular perturbations as well as to arbitrary structural variations is presented in Chapter IV. Both non-uniform and uniform time scaling are treated. The aggregation forms established therein are accompanied by simple algebraic conditions expressed via Liapunov functions of the subsystems.
Hence,
all preceding
types of stability robustness are also assured for n o n - s t a t i o n a r y nonlinear large-scale singularly perturbed systems. Chapter V presents an analysis of stability achievements via both scalar and vector Liapunov functions.
Postface
363
Within the vector Liapunov concept framework, of the asymptotic
three different estimates
stability domain of the equilibrium state can be gua-
ranteed under simple and relaxed algebraic conditions based on knowledge of subsystems Liapunov functions. The vector Liapunov function approach appears also attractive
for possible use of a more refined
mathematical model of a power system,
simple stability test, and easy
asymptotic stability domain estimation. This is shown in general by developing various aggregation forms in terms of subsystems Liapunov functions. However,
a number of majorizations used for construction of
the overall system aggregation matrix can impose too sharp conditions on the system. Effective applications of the large-scale systems stability theory is crucially dependent on solving the next three qualitative problems
:
Problem 1.
How to construct a Vector Liapunov function for a given
large-scale
system
?
This problem essentially means how to select Liapunov functions for subsystems,
i.e. for lower order
(non-stationary non-linear)
systems.
Several important contributions to solving this problem have appeared since the time this book was completed Michel et al.
(1982)
(Gruji~ et al., 1984).
(1982) developed a methodology for computer construction
of Liapunov functions. Vanelli and V i d y a s a g a r
(1985) proved necessary
and sufficient conditions for a set to be the domain of attraction of the origin. The conditions are expressed in terms of existence of a suitable Liapunov function. They also presented a new iterative procedure for the construction of a system Liapunov function. resting new procedures to Genesio et al.
Other inte-
for the Liapunov function construction are due
(1985) and Chin
(1986). Gruji~
sary and sufficient conditions for asymptotic
(1986b) proved neces-
stability of the origin,
for its domain and for a Liapunov function construction.
The preceding
results are promising for both effective determination of Liapunov functions of the subsystems and effective estimation of the asymptotic stability domains of their equilibria.
Hence, they constitute a strong
new basis for effective solutions of Problem i in general and advanced application of results presented in Chapters II-V. These results can be directly combined with those of Chapter V for their wider applications.
ProbZem 2.
Which comparison
the aggregation procedure, conditions
on the overall
functions mostly relax majorisations
requirements
on interactions
system by assuring
of
and stability
simultaneously
the order
364
Postface
reduction of its a@gregation systems
matrix at Zeast to the number of its ~ub-
?
Michel and Miller
(1983) proposed use of stability preserving mappings
for stability analysis.
It is interesting to explore deeper whether
stability preserving mappings Another new promising
can provide a tool for solving Problem 2.
mathematical tool for solving Problem 2 is the
concept of the matrix Liapunov functions by M a r t y n y u k which has been further developed by himself dently in another original way by Djordjevi~
(Chapter II),
(19~-1986),
and indepen-
(1983,1986). A link among
the concepts of scalar, vector and matrix Liapunov functions is shown by Gruji~
(1986a).
All these and other approaches
should reduce the majorizations stiff-
ness so that stabilizing actions of interactions over subspaces are adequately preserved in the aggregation matrix.
Problem
3.
What ~8 the best possible
attraction and/or asymptotic Bondi and Gambardella
effective
e~timation
of the
stability domain of an equilibrium
state
?
(1986) contributed to solving the attraction
domain estimation in the framework of n o n - s t a t i o n a r y non-linear systems without requiring positive definiteness of Liapunov functions of subsystems.
Gruji~
(1986b) established conditions for exact determination
of the asymptotic stability domain of the origin in the framework of stationary non-linear systems. These results linked with those solving Problems i and 2 open new horizons for the asymptotic stability domain estimations specified in Chapter V. Shaaban and Gruji~
(1984-1986) discovered new relaxed decomposition-
aggregation forms for power systems, which provide a basis for more effective application of the vector Liapunov function concept to more refined mathematical models of power systems. Arbitrary structural variations of power systems are also incorporated in their analysis, which is a further development of the methodology presented in Chapter V. With regards to the power system stability problem, our final qualitative conclusion is the following has potential
: vector or matrix Liapunov approach
: it simplifies the overall system Liapunov function
c o n s t r u c t i o n and the asymptotic stability domain estimation;
it could
cope with the problem of testing asymptotic stability under arbitrary structural perturbations;
it enables a c c o u n t i n g for more refined math-
ematical models than those amenable to one-shot application of a scalar Liapunov function. However,
the latter imposes less
stringent conditions on the system.
(if not much less)
Called to meet urgent practical
References
365
needs, it has been continually developing. No doubt, by the time this book appears, the situation will have further changed. Things change faster in engineering sciences than in the book writing process !
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