Lecture Notes in Control and Information Sciences Edited by A.V. Balakrishnan and M, Thoma
29 M. Vidyasagar
Input-Output Analysis of Large-Scale Interconnected Systems Decomposition, WelI-Posedness and Stability
Springer-Verlag Berlin Heidelberg New York1981
Series Editors
A. V. Balakrishnan - M. Thoma Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya. Z. Tsypkin • A. J. Viterbi Author
Prof. M. Vidyasagar Dept. of Electrical Engineering University of Waterloo Waterloo, Ontario Canada
ISBN 3-540-10501-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10501-8 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to 'Verwertungsgesellschaft Wort', Munich. © Springer-Vedag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2061/3020-543210
This book is intended to be a fairly comprehensive treatment of large-scale interconnected
systems from an input-
output viewpoint.
Prior to treating the question of stability
(and instability),
we study both the decomposition
posedness of such systems.
It is not necessary
and the well-
for the reader
to have studied feedback stability before tackling this book, as we develop results concerning feedback systems as special cases of more general results pertaining to large-scale systems. However,
the reader should know some elementary
analysis
(e.g. Lebesgue spaces,
and have some general knowledge
(e.g. Perron-frobenius
The first chapter is introductory, background material;
after that,
functional
contraction mapping theorem), and chapters
theorem).
2 and 3 contain
the remaining chapters are
essentially independent and can be read in any order. I thank Peter Moylan for his careful reading of the manuscript and for several constructive ShakUnthala
for her support.
suggestions,
and my wife
Virtually all of my research
reported in this book was carried out, and most of the book was written, while I was employed by Concordia University,
Montreal.
I would like to acknowledge research support from the Natural Sciences and Engineering Research Council of Canada, lesser extent from the U.S. Department of Energy.
and to a
Finally,
thanks to Monica Etwaroo and Jane Skinner for typing the manuscript.
Waterloo September 29, 1980
M. Vidyasagar
my
TABLE OF CONTENTS
PAGE
PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . .
v
CHAPTER
1
i:
CHAPTER 2:
INTRODUCTION
~THEMATICAL PRELIMINARIES . . . . . . . . . . 2.1 2.2
CHAPTER 3:
3.2 3.3
4.2 4.3
5.2 5.3 5.4
2~ 26 42 46
Some Results From the Theory of Directed Graphs . . . . . . . . . . . . . Decomposition into Strongly Connected Components . . . . . . . . . . . . . . . . Results on Well-Posedness and Stability
Weakly Lipschitz, Smoothing and Strictly Causal Operators . . . . . . . . . . . . . Single-Loop Systems . . . . . . . . . . . Continuous-Time Systems . . . . . . . . . Discrete-Time Systems . . . . . . . . . .
s7 57
.
73 81
88 88 94 95 103
Single-Loop Systems . . . . . . . . . . . Criteria Based on a Test Matrix ..... C r i t e r i a B a s e d o n an E s s e n t i a l S e t Decomposition . . . . . . . . . . . . . .
105 107 126
DISSIPATIVITY-TYPE CRITERIA FOR L2-STABILITY . 133 7.1 7.2 7.3
CHAPTER 8:
12
SMALL-GAINTYPE CRITERIA FOR Lp-STABILITY.. • lO5 6.1 6.2 6.3
CHAPTER 7:
4
Gain, Gain with Zero Bias, and Incremental Gain . . . . . . . . . . . . . Dissipativity and Passivity . . . . . . . Conditional Gain and Conditional Dissipativity . . . . . . . . . . . . . .
WELL-POSEDNESS OF LARGE-SCALE I~TERCO~NECTED SYSTEMS. . . . . . . . . . . . . . . . . . . . 5.1
CHAPTER 6:
Truncations, Extended Spaces, Causality . . . . . . . . . . . . . . . . Definitions of Well-Posedness and Stability . . . . . . . . . . . . . . . .
DECOMPOSITION OF LARGE-SCALE INTERCONNECTED SYSTEMS. . . . . . . . . . . . . . . . . . . . 4.1
CHAPTER5:
4
GAIN AND DISSIPATIVITY . . . . . . . . . . . . 3.1
CHAPTER 4.
. . . . . . . . . . . . . . . . .
Single-Loop Systems . . . . . . . . . . . 134 General Dissipativity-Type C r i t e r i a . . . 139 Special Cases: Small-Gain and Passivity-Type Criteria . . . . . . . . . 144
L2-1NSTABILITY CRITERIA. . . . . . . . . . . .
164
8.1 8.2 8.3
164 168 175
Single-Loop Systems . . . . . . . . . . . Criteria of the Small-Gain Type ..... Dissipativity-Type Criteria . . . . . . .
Vl
TABLE OF CONTENTS CONT'D. . . . .
CHAPTER 9:
L~-STABILITY AND L~-INSTABILITY USING EXPONENTIAL WEIGHTING. . . . . . . . . . . . .
189
9.1 9.2 9.3
190 198 205
General Special General
Stability Result . . . . . . . . . Cases . . . . . . . . . . . . . . Instability Result . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . .
213
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . .
218
CIIAPTER i: INTRODUCTION D u r i n g the p a s t decade or so, there has b e e n a great deal of i n t e r e s t in the study of l a r g e - s c a l e systems as a s e p a r a t e d i s c i p l i n e in itself. many factors,
p h y s i c a l systems circuits,
This i n t e r e s t is t r a c e a b l e to
i n c l u d i n g the g r o w i n g r e a l i z a t i o n that many
etc.)
(e.g. power networks,
several s i m p l e r subsystems, and "structure"
large-scale
integrated
can in fact be v i e w e d as i n t e r c o n n e c t i o n s of and that m u c h v a l u a b l e
information
is lost if the m e t h o d of a n a l y s i s does not take
into a c c o u n t the i n t e r c o n n e c t e d nature of the s y s t e m at hand. Moreover,
several s u b j e c t s d e a l i n g w i t h
reached m a t u r i t y ,
"small"
systems have
so that in order to expand the h o r i z o n s of
k n o w l e d g e by t a c k l i n g new and c h a l l e n g i n g p r o b l e m areas,
re-
searchers have set their sights on l a r g e - s c a l e
Some
systems.
prime e x a m p l e s of this are o p t i m a l c o n t r o l theory, s t a b i l i t y t h e o r y of s i n g l e - l o o p
and the
f e e d b a c k systems.
It is as yet too soon to c l a i m that there e x i s t s a comprehensive
theory of l a r g e - s c a l e systems.
stability theory of l a r g e - s c a l e
Nevertheless,
systems is a w e l l - d e v e l o p e d
in w h i c h a large v a r i e t y of results is available. effect two m e t h o d o l o g i e s
in s t a b i l i t y theory,
methods and i n p u t - o u t p u t methods.
While
the area
T h e r e are in
namely Lyapunov
there are some con-
n e c t i o n s b e t w e e n L y a p u n o v s t a b i l i t y and i n p u t - o u t p u t stability, the actual t e c h n i q u e s used to e s t a b l i s h the two types of s t a b i l i t y are r a t h e r different; of l a r g e - s c a l e systems.
Lyapunov
systems are w e l l - d o c u m e n t e d Miller [Mic.
this is e s p e c i a l l y methods
so in the case
for l a r g e - s c a l e
in the r e c e n t books by M i c h e l and
i] and S i l j a k [Sil.
i] .
contains come i n p u t - o u t p u t results,
However,
though [Mic.
i]
there is not at p r e s e n t a
c o m p r e h e n s i v e book on the i n p u t - o u t p u t a n a l y s i s of l a r g e - s c a l e systems.
In the same vein,
Desoer and V i d y a s a g a r [Des.
the books by W i l l e m s [Wil.
2] and
2] cover f e e d b a c k systems quite
t h o r o u g h l y from an i n p u t - o u t p u t viewpoint,
and it is n a t u r a l to
attempt a s i m i l a r t r e a t m e n t of l a r g e - s c a l e
systems.
This b o o k is i n t e n d e d to be a h i g h - l e v e l r e s e a r c h m o n o g r a p h t h a t sets forth m o s t of the a v a i l a b l e results on the decomposition,
well-posedness,
s t a b i l i t y and i n s t a b i l i t y of large-
scale systems,
that can be o b t a i n e d by i n p u t - o u t p u t methods.
Since m a n y r e s u l t s
for f e e d b a c k systems can be o b t a i n e d as
special cases of those given here for l a r g e - s c a l e systems, not n e c e s s a r y to have read [Wil. book.
2] or [Des. 2|
it is
to follow this
T h o u g h the e m p h a s i s h e r e is on i n p u t - o u t p u t stability, we
note that i n p u t - o u t p u t m e t h o d s can be u s e d to e s t a b l i s h L y a p u n o v s t a b i l i t y as well. is i n p u t - o u t p u t stable,
In particular,
also g l o b a l l y a s y m p t o t i c a l l y (see [Wil.
3] , [Moy.
if a n o n l i n e a r system
r e a c h a b l e and detectable,
then it is
stable in the sense of L y a p u n o v
4] ) .
T h r o u g h o u t this book,
the e m p h a s i s
is on t r e a t i n g the
l a r g e - s c a l e s y s t e m at h a n d as an i n t e r c o n n e c t e d system, sisting of several s u b s y s t e m s c o n n e c t i o n operators. 2.2).
con-
i n t e r a c t i n g through various inter-
(For a p r e c i s e d e s c r i p t i o n ,
It is of course p o s s i b l e to "aggregate"
s y s t e m o p e r a t o r s and the v a r i o u s
see S e c t i o n
the v a r i o u s sub-
i n t e r c o n n e c t i o n operators,
so
that the l a r g e - s c a l e s y s t e m at h a n d is r e c a s t in the f o r m of a "single-loop"
f e e d b a c k system.
W i t h this r e f o r m u l a t i o n ,
the s t a n d a r d s i n g l e - l o o p f e e d b a c k s t a b i l i t y results, those in [Des.
2] and [Wil.
2] b e c o m e applicable.
w h e t h e r a given s y s t e m is a "single-loop" connected"
all of
such as
Therefore,
s y s t e m or an "inter-
s y s t e m depends on the m e t h o d of a n a l y s i s u s e d to
tackle it.
However,
it can be e a s i l y shown that c o n v e r t i n g the
s y s t e m into a "single-loop" conservative
f o r m u l a t i o n gives u n n e c e s s a r i l y
s t a b i l i t y c r i t e r i a and w e l l ' p o s e d n e s s
Therefore,
criteria.
in this b o o k we only p r e s e n t results that
p e r t a i n to i n t e r c o n n e c t e d systems, w h e r e b y the a n a l y s i s
is
c a r r i e d out in terms of the s u b s y s t e m o p e r a t o r s and the interc o n n e c t i o n operators;
we avoid t r e a t i n g the s y s t e m as a w h o l e .
For this reason, we e x c l u d e linear t i m e - i n v a r i a n t systems f r o m our study.
The r e a s o n is that,
and s u f f i c i e n t c o n d i t i o n s interconnected conditions
though one can derive n e c e s s a r y
for the s t a b i l i t y and w e l l - p o s e d n e s s
linear t i m e - i n v a r i a n t systems,
(of necessity)
of
the n e c e s s a r y
involve t a c k l i n g the s y s t e m as a whole.
A s u b s y s t e m level a n a l y s i s can p r o d u c e s u f f i c i e n t c o n d i t i o n s s t a b i l i t y and s u f f i c i e n t c o n d i t i o n s n e c e s s a r y and s u f f i c i e n t conditions.
for instability, b u t not
for
The book is organized as follows:
In Chapter 2, we
introduce the concepts of truncations and extended spaces, which provide the mathematical
setting for input-output analysis, we
then give precise definitions of well-posedness
and stability.
In Chapter 3, we introduce the concepts of gain and dissipativity, which play an important role in the various criteria for stability and instability,
and give explicit methods for com-
puting gains and testing dissipativity. In Chapter 4, we present a few graph-theoretic niques for the efficient decomposition of large-scale connected systems.
Specifically,
tech-
inter-
we show that by identifying
the so-called strongly connected components
(SCC's) of a given
system, we can determine the well-posedness
and stability of the
original system by studying only the SCC's. present some sufficient conditions system.
These criteria are graph-theoretic
given a very nice physical
In Chapter 5, we
for the well-posedness
interpretation.
In Chapter 6, we give some generalizations single-loop
of a
in nature and can be
of the
"small gain" theorem to arbitrary interconnected
systems, while in Chapter generalizations
7, we state and prove several
of the single-loop
"passivity"
Chapter 8, we derive several L2-instability scale systems.
Finally,
theorem.
In
criteria for large-
in Chapter 9, we show how the technique
of exponential weighting can be used to study L -stability and L -instability using the results of Chapters
6 to 8.
CHAPTER 2: MATHEMATICAL PRELIMINARIES 2.1
TRUNCATIONS,
In this notation
section,
and terminology
particular
notation
Let functions
X
R+ =
here
and
As
introduce
the m a t h e m a t i c a l
is f r o m
this book.
[Vid.
4] and
the set of all r e a l - v a l u e d
into
[0,~),
measure.
we briefly
employed
R+
SPACES r CAUSALITY
t h a t is u s e d t h r o u g h o u t
denote
mapping
numbers, Lebesgue X
EXTENDED
R, w h e r e
R
denotes
the m e a s u r a b i l i t y
is c u s t o m a r y ,
The
[Des.
measurable
the s e t of r e a l
is w i t h r e s p e c t
we define
2].
various
to the
subsets
of
as f o l l o w s :
1
Definition
For
p 6
[i,~),
the s e t
L P
notes
the s e t of all
functions
tion
t +
is i n t e g r a b l e
f(.)
E L
[If(t) I]P
for a f i x e d
P
2
p e
f(.)
[i,~)
in
over
X
such
[0,~).
if a n d o n l y
= L [0,~) deP t h a t the f u n c -
In o t h e r w o r d s , if
If(t) Ip dt < 0
Similarly, in
X
[0,-)
L
= L
such that •
If
p 6
[0,~) f(.)
[i,~)
denotes
the
set of all
is e s s e n t i a l l y we d e f i n e
,
bounded
the f u n c t i o n
functions
over I'
.
f(.)
the i n t e r v a l
Ilp : Lp
÷
R+
by
I tfI1p = [
If(t) lp dt] 1/p , vf e Lp 0
If
p = -
, we define
II-I I~ : L
I IfEl. = e s s ° t 6
= inf
where
p e
~[.]
[1,~],
space.
denotes
sup
÷ R+
by
IfCt) j
[0,~)
{r
: ~ [ t : I f ( t ) I > r] = 0}
the Lebesgue
measure
~f 6 L
of a set.
I t is w e l l - k n o w n
[Dun.
i, p. 146]
the o r d e r e d
(Lp
, I.I I . , Ip) .
pair
,
t h a t for e a c h
constitutes
a Banach
In o r d e r can
study
to h a v e
"unstable"
the c o n c e p t
of
as w e l l
truncated
Definition is d e f i n e d
a mathematical as
"stable"
functions
and
T < ~
; then
Let
For b r e v i t y , refer
the
we use
the
XT(.)
as
to
interval
sense
that
introduce
spaces.
the o p e r a t o r
PT
: X + X
Vx•
Note
that
that
PT
denotes
fT(. ) e L p
belong
to
of the
space
X
t > T notation the
xT
to d e n o t e
truncation
the
of a g i v e n
function
function
PT x, x(.)
[0,T].
the
PT
For
the YT
Lp).
operator
" PT =
Definition L pe [0,~)
we
we
t • [0,T]
0
to the
systems,
extended
whereby
by setting
(PTx)(t) = { x(t)
and
framework
a fixed
s e t of all < ~
The
PT
p •
Lpe
[i,~],
functions
(though
space
is a p r o j e c t i o n
on
X
in
symbol
L
=
"
f(.)
the
f(.) itself
is r e f e r r e d
in may
pe such
X
or m a y
to as the
not
extension
L P
Example e_~d s p a c e s
Lpe
The
for
the u n e x t e n d e d
spaces
tan
t
does
C X
.
Moreover,
all
finite
T
is t h e
Then
for
p • Lp
not belong
It is c l e a r
Lle
function
all
for
that, Lp
, it is c l e a r
Definition every
set
that
p E
, the
The
unextended
fixed
Lpe
[i,-]
c Lle to u s e
be
truncated
to
the e x t e n d -
not belong
to an[
function spaces
f2(t) L
of Vp •
[i,~].
in this
fixed, norm
L c L p pe [0,T] for
L1
and
Thus
book.
let
IIfl ITp
T < is d e f i n e d
IIf11 p = IIfTlIp= llpTfllp Let
p = 2, a n d
truncated
inner
let
T < ~
product
Then
T
for e v e r y
is d e f i n e d
=
pe
by
i0
of
p, we h a v e
is a s u b s e t
that we need
Let
f • Lpe
for e a c h
belongs
does
[i,~].
the
[0,T]
= t
but
p •
to a n y of
since
largest
fl(t)
[i,-],
f, g E L 2 e by
, the
T
ii
T
I
= =
f(t)
g(t)
p E
[I,~]
dt
0
Note the q u a n t i t y every
for e v e r y
I If| |Tp
T < ~
belongs
that,
is a w e l l - d e f i n e d
, though
I Ifl Ip
to the u n e x t e n d e d
and every finite
is d e f i n e d
space
L
only
f E L
real number if
Moreover,
f
pe ' for
actually
we have
P 12
Lemma
Let
is a n o n d e c r e a s i n g unextended c
as
P
IIfl |Tp
In o r d e r
Then
n-tuples
the s e t
f(.)~ =
is d e f i n e d
[fl(.)
Lpe)
¥i
.
! |" | 1
L
I
L n2
systems
I]fIITp +
having multiple
Ln p
[1,~]
and
and
inputs
Ln pe
let
n > 1
Lpe) n
b e an in-
consists
... fn(.)] ' , w h e r e
of the v e c t o r
space,
with
I Ip d t } i / P
ass.
sup
t e
[O,~) n o r m on
of all
fi(.) f(.)
• Lp 6 L nP
is
p < - ,
if
p :-
I I-|Ip
to d e n o t e
Ln . P
This usage
is
any c o n f u s i o n .
norm
of the n o r m { I-If
the c h o i c e
by
if
Rn
as the n o r m on
in
on (14),
and the i n n e r p r o d u c t
is d e f i n e d
]If(t) IE
the same s y m b o l
as w e l l
P to c a u s e
However,
product in
In this case,
0
In the d e f i n i t i o n choice
(the P constant
a n d is l e f t to the r e a d e r .
i o I If(t)
t h a t w e use
the n o r m on
not expected
ry.
f e L
a finite
T h e n o r m of a f u n c t i o n
is the E u c l i d e a n
Note both
I If| |Tp
by
I lf(_) t l p :
where
exists
(respectively
f2(.)
{
14
p 6
L np
Then
from below.
the s p a c e s
Let
f C Lp
Furthermore,
< -
to d e a l w i t h
we introduce
(respectively
VT
is o b v i o u s
Definition teger.
T
if t h e r e
, monotonically
The proof
13
_< c
and let
of
if a n d o n l y
T ÷ ~
and o u t p u t s ,
[i,~]
function
space)
such that
IIfll
p 6
I I-I Ip
Rn
on
L np,
is e s s e n t i a l l y
the s p a c e
L n2
the arbitra-
is an inner-
of two e l e m e n t s
f,g
~
15
5
=
0
where and
fi(.) g(.)
and
f, (t) g(t) ~
gi(.)
are the c o m p o n e n t
, respectively.
~
The
truncated
and the t r u n c a t e d
inner product
fined
analogous
in a m a n n e r
to
(i0)
16
S
Definition {x(i)}~= 0 . in
S
For
The
set
set
: L n ÷ R+ pe are de-
we introduce
the
subsets.
consists
1 ~ p < - , the
I I.I IT
f(.)
(ii), r e s p e c t i v e l y .
systems,
S
of
: L ~e × L~ e ÷ R
and
and its v a r i o u s
functions
norm
<. , ">T
To study discrete-time s p a c e of s e q u e n c e s
n [ i=l
dt =
£p
of all
sequences
consists
of all {x (i) }
such that
I
17
Ix(i) Ip < "
i=O
The
set
[i,~)
,
£
consists
we define
of all b o u n d e d
the
function
II-I Ip :
£p
~
in R+
S
For p 6
.
by
Ix(i) Im) I/p
llxllp = ¢
18
sequences
i=0 We also define
II-I I. : £= + R+
[Ixllo--
19
by
sup Ix¢i>l i
W e can a l s o d e f i n e present
Definition is d e f i n e d
For each
i ~ 0 , the o p e r a t o r
Finally,
x (j)
0 < j < i
0
j > i
Sn
(rasp.
£~)
-
we define
Definition set
in the
Pi
: S + S
by
(Pix) (j) = {
22
of t r u n c a t i o n s
context.
20
21
the c o n c e p t
Let
n
-
the s p a c e s
S n a n d £n P
be a p o s i t i v e
is d e f i n e d
integer.
Then
as the set o f all s e q u e n c e s
the of
n-tuples 6 S
{x} (i) ~
(resp.
=
Zp)
[x~ i) Vj
.
,
x 2(i)
•
.... x n(i)] ,
The norm
If-lip
{" x ij~ ( ) "
such t h a t
: %n ÷ R+ P
is d e f i n e d
by
( ~
r I EXllp = 4 [
23
T1~(i) IIP) I/p
i--1 sup I Ix(i~ll
iz
p < -
if
p = -
i
I I-I ]
where
denotes
We next
24
the E u c l i d e a n
introduce
Definition causal
An operator
PT G = PT G P T
of c a u s a l i t y .
G :Lle
÷Lle
is s a i d
to be
'
~T <
equivalently,
26
(Gf) T =
27
Lemma
(GfT) T
whenever gT
f
and
for some
operator
28
(24)
g
, we have
Proof
For
G :Lle
+Lle
÷Lle
is c a u s a l
if the f o l l o w i n g
(Gf) T =
in
Lle
has property
(Gf)T =
and that
in the
is t r u e
Such t h a t
fT =
let us say t h a t an
(s)
if
(Gg)T in the s e n s e of D e f i n i t i o n
(s)
T o s h o w this,
fT = gT
for s o m e
T
suppose
(24)
first
Then by
is that
(25), w e
have
29
(Gf) T = so t h a t property
G
(GfT) T =
has property (s)
Since
(GgT) T =
(s) fT =
:
(Gg) T
the sake of c l a r i t y ,
to p r o p e r t y
is c a u s a l ,
Vf E Lle
G :Lle
if and o n l y
show that causality
equivalent
YT < - ,
are two f u n c t i o n s
T < ~
fT = gT ~ We must
,
An operator
s e n s e of D e f i n i t i o n
G
the c o n c e p t
Rn
if
25 or,
n o r m on
(Gg) T
Conversely, (fT)T
suppose
%~f, w e h a v e f r o m
G (28)
has that
30
(Gf) T = (GfT) T so that
G
is causal. It is clear
ators on
Lle
well define Lqm e
that there
is nothing
as far as causality
causality
or from
Sn
with respect
to
Sm
where
•
goes,
special
about oper-
and that one can equally
to operators p, q 6
[i•~]
from
L pe n
to
and
n,m
are
•
positive
integers. We conclude
which plays
this section by introducing
an important
the set
role in the study of linear
A,
time-invari-
ant operators. 31
Definition f(.)
The set
A
consists
of all distributions
of the form f(t)
32
=~
0,
[
t < 0 fi 6 (t-t i) + fa(t)
,
t >_ 0
i=0 where
6(.)
< ...
are real constants,
norm
denotes
If. If A
33
on
the unit impulse
A
is defined
I If(.) I IA =
The product
~ i=0
-
(f,g)
distribution,
{fi } q £i '
f(.)
f0
Remarks by delayed
subset of Moreover•
and
g (.)
in
A
is defined
i.e.,
(t) =
()tf(t-T)
g(T)
dT =
A, and that if pair
(Jtf(T)
g(t-T)
dT
0
Basically, impulses.
the ordered In
The
Ifa(t) I dt
0
mented
0 ~ tO < t1
fa(. ) G L 1 .
by
Ifil +
of two elements
as their convolution;
34
and
the set
A
consists
It is easy tO see that f(.) (A•
(34), one should
6 LI•
then
II.l IA) interpret
of L1
L1
aug-
is a
IIf(.) Ill = l]f(.)llA-
is a Banach
space.
10 35
(t-t a) * ~(t-t b) = ~ (t-ha- ~ )
36 Thus,
if
~(t-ta)
* fa(t) = fa(t-ta)
f
g
and
are of the form
37
f(t) =
~ fi 6(t-ti) i=0
38
g(t) =
~ i=0
+ fa (t)
gi ~(t-Ti)
+ ga (t)
then 39
(f,g) (t) =
+
~ ~ i=0 3
fi gj ~(t-ti-Tj)
~ gj fa(t-~j) j=0
+
fa(t-T)
and right-
IIf*gltA
40
Also, we see from 41
!
~ fi ga i=0 ga(T)
(t-ti)
dT
0
It is routine to verify from commutative, leftition, and that
+
(39) that convolution
is
distributive with respect to add-
IIfllA • IlglIA
(39) that
f*~ = ~*f = f ,
Vf • A
Hence the set A is a Banach algebra with a unit, with the norm, * as the product, and ~ as the unit. Given any
I I.I IA as
f(.) 6 A, the integral ~
f(s)
42
=
f
f(t)
~st dt
0
is well-defined whenever
Re s > 0,
and in fact,
43 where Laplace
C+ = {s: Re s ~ 0}. transformable,
Thus every element
f(.)
and the region of convergence
of
A
of the
is
Laplace transform C+
f(.)
i n c l u d e s the c l o s e d r i g h t h a l f - p l a n e
For n o t a t i o n a l c o n v e n i e n c e ,
44
Definition
The set
forms of the e l e m e n t s of
we i n t r o d u c e the set
A .
A c o n s i s t s of the L a p l a c e
trans-
A .
Since c o n v o l u t i o n in the time d o m a i n is e q u i v a l e n t to p o i n t w i s e m u l t i p l i c a t i o n in the s-domain, p r o d u c t s of e l e m e n t s of
A
can be shown q u i t e e a s i l y that any every
s E C+
f 6 A
, and a n a l y t i c at e v e r y
{s: Re s > 0}
A
A .
Also,
is c o n t i n u o u s
s ~ C+o
(where
C+).
Finally,
d e n o t e s the interior of
that e v e r y e l e m e n t of
we see that sums and
once again b e l o n g to
is b o u n d e d over
it
at
C+o
=
(43) shows
C+
^
A n×m of
A, d e n o t e d
45
by
such that
The set
fT(.) ~ A,
A
e VT ~ 0
N o t e that D e f i n i t i o n inition
We next define
the extension
c o n s i s t s of all d i s t r i b u t i o n s
(45) is e n t i r e l y a n a l o g o u s
to Def-
(7).
The set G
A , we can also d e f i n e
A e
Definition f(.)
if
and
Once we have d e f i n e d A and ~nxm in an o b v i o u s way.
Ae
is i m p o r t a n t b e c a u s e
it can be shown that,
is a linear c o n v o l u t i o n o p e r a t o r of the type (Gf) (t) = J'g(t-~) f
46
f(T)
dT
Lpe
into itself
0 then
G
is causal and m a p s
and only if the k e r n e l
(or "impulse response")
yp 6
[1,-],
if
g(.)
e Ae .
The
proof of this i m p o r t a n t f a c t can be o b t a i n e d by a d a p t i n g [Des. 2, T h e o r e m IV.7.5]. that we e n c o u n t e r
Thus,
Ae
(or, m o r e generally,
multivariable Thrm. 6.5.37]
g(')
system. that,
if
(the u n e x t e n d e d space) g(.) e A .
This
all linear c o n v o l u t i o n o p e r a t o r s
in this m o n o g r a p h
can be a s s u m e d to be of the form
(even the "unstable"
(46), w h e r e
the k e r n e l
ones) g(.)
E
~ An×me , in the case of a
Similarly, G
that of
it can be shown
is of the f o r m
into itself
shows that the set
Vp e A
(46), then [I,~],
[Vid. 4, G
maps L
if and o n l y if
e s s e n t i a l l y c o n s i s t s of
P
12
all "stable"
2.2
impulse r e s p o n s e s
(see D e f i n i t i o n 3.1.1).
D E F I N I T I O N S OF W E L L - P O S E D N E S S AND S T A B I L I T Y
In this section, we d e l i n e a t e interconnected
the class of l a r g e - s c a l e
systems u n d e r study in this book,
and we give pre-
c i s e d e f i n i t i o n s of w h a t is m e a n t by such a s y s t e m b e i n g w e l l p o s e d or stable.
T h r o u g h o u t this book, we shall be c o n c e r n e d w i t h analysis of a l a r g e - s c a l e
interconnected system
(LSIS)
d e s c r i b e d by the
set of e q u a t i o n s m
la
ei = ui -
[ j =i
H
ij
yj i = l,...,m
ib
Yi = Gi ei n.
where
ui' ei' Yi
fixed
p 6
[1,-]
all b e l o n g
Lpel
to the e x t e n d e d space
and some p o s i t i v e integer
n i , the o p e r a t o r G i
n.
maps n. l
n.
L i pe
into itself,
and the o p e r a t o r
H.. 13
maps
L 3 pe
into
.
Lpe
We can refer to
and output,
y
ui' ei' Yi
respectively.
to d e n o t e the m - t u p l e and
for a
to d e n o t e
(Ul,
(YI'
as the i-th input, error,
W h e r e convenient, ..., Um),
..., ym ) .
e
we use the symbol u
to d e n o t e
N o t e that
m Ln , where n = [ n. pe i=l i spirit, we s o m e t i m e s use the symbols G and H
to the p r o d u c t space
ators f r o m
Ln pe
G =
(el,
u, e, y
..., em),
all b e l o n g
In the same to d e n o t e o p e r -
into itself d e f i n e d by
I°J i.
G
*To a v o i d a p r o l i f e r a t i o n of symbols, we a s s u m e that the s y s t e m Gi
has an equal n u m b e r of inputs and outputs.
is e n t i r e l y d i s p e n s a b l e .
This a s s u m p t i o n
13
H =
IHll Hml
W i t h these definitions,
the system e q u a t i o n s
(1) can be c o m p a c t l y
e x p r e s s e d as
4a
e = u - Hy
4b
y = Ge
The system d e s c r i p t i o n able of r e p r e s e n t i n g think of
several
(i) as r e p r e s e n t i n g
subsystems,
(1) is quite g e n e r a l and is cap-
types of p h y s i c a l systems. several
"isolated"
c o r r e s p o n d i n g to the o p e r a t o r s
One can
or "decoupled"
GI,...,G m
, such that
the input to ui
G. is a linear c o m b i n a t i o n of an e x t e r n a l i n p u t l and several "interaction" signals Hij yj This is d e p i c t e d
in Figure
2.1
.
Yi
Gi
Hil Yl
Him Ym F I G U R E 2.1
For this reason, we refer G I, .....G m
to
m
as the n u m b e r of subsystems,
as the s u b s y s t e m operators,
and
Hll,...,Hmm
as the
i n t e r c o n n e c t i o n operators.
In some cases, p a r t i c u l a r l y
in p r o v i n g d i s s i p a t i v i t y -
type theorems for s t a b i l i t y and instability,
(Chapters 7 and 8)
we assume that for all i,j, the i n t e r c o n n e c t i o n o p e r a t o r Hij: n. n. Lpe3 ÷ L pez can be r e p r e s e n t e d by an nixn j m a t r i x ~ij of c o n s tant real numbers,
i.e.
that
14 n.
(Hij yj)(t) Actually, ality,
= H..~13 yj(t)
this a s s u m p t i o n
because
,
Vt,
Vyj e Lpe3
does not result in any loss of gener-
this a s s u m p t i o n
ing the number of subsystems
can always be satisfied by increas(m)
if necessary.
(If a particu-
lar o p e r a t o r
H.. cannot be r e p r e s e n t e d by a c o n s t a n t matrix, 13 m by one and include H.. among the operators 13 If all i n t e r c o n n e c t i o n operators can be r e p r e s e n t e d by
then increase G i) .
c o n s t a n t matrices,
then we refer
to the c o n s t a n t
n×n
matrix
H
defined by
H
l
=
LEml as the i n t e r c o n n e c t i o n
mmj
matrix.
uI
u2
FIGUR~ The standard 2.2
and studied
2.2
feedback
in detail
in
configuration,
[Des. i] and
is a special case of the system d e s c r i p t i o n system of Figure
2.2 is d e s c r i b e d by
7a
el = Ul - Y2
7b
e2 = u2 + Yl
shown in Figure
[Wil. i] among others, (I)
The feedback
15 7c
Yl = G1 el
7d
Y2 = G2 e2 where p •
Ul' u2' el
[i,~]
e2' YI' Y2
•
and some p o s i t i v e integer .
into itself.
To put the s y s t e m
(two subsystems),
H
where
0 ~%)
order
~×9
all b e l o n g to
L~ pe
~ , and
GI,G 2
(7) in the form
n I = n 2 = ~, n = 2~, Gl~ 2
for some fixed map
(i), let
as in
m = 2
(7), and
=
and
I~ ~
denote
respectively.
the null m a t r i x and i d e n t i t y m a t r i x of N o t e that the i n t e r - c o n n e c t i o n opera-
tors can be r e p r e s e n t e d by c o n s t a n t m a t r i c e s in this case• that the i n t e r c o n n e c t i o n m a t r i x ible.
L pe ~
H
and
is s k e w - s y m m e t r i c and invert-
T h e s e p r o p e r t i e s are i m p l i c i t y u s e d in m u c h of f e e d b a c k
s t a b i l i t y theory.
Comparing
the g e n e r a l l a r g e - s c a l e
(1) w i t h the f e e d b a c k s y s t e m d e s c r i p t i o n a g g r e g a t e the e q u a t i o n s are v e r y similar.
(I) into the form
In fact,
system description
(7), we see that if we (4), then
(4) is a s p e c i a l case of
(4) and
(7), w i t h
u I = u, u 2 = 0, G 1 = G, G 2 = H, e I = e, and Yl = y " shown in F i g u r e 2.3
.
Thus,
g i v e n an LSIS,
r e s e n t it in the d e c o m p o s e d form system level,
(7)
T h i s is
one can e i t h e r
rep-
(i) and a n a l y z e it at the sub-
or one can r e p r e s e n t it in the a g g r e g a t e d
and a n a l y z e it as a s i n g l e - l o o p system.
form
(4)
If one chooses the latt-
er option•
one can i m m e d i a t e l y apply all of the s t a n d a r d r e s u l t s
d e r i v e d in
[Des.
main emphasis
2] and
[Wil.
2] for f e e d b a c k systems.
in this m o n o g r a p h is on a n a l y z i n g a g i v e n LSIS at
the s u b s y s t e m level,
taking full a d v a n t a g e of the fact that the
system at h a n d is an i n t e r c o n n e c t i o n of several ler)
(presumably simp-
subsystems.
*Actually• and
T h u s the
Ul, el, Y2
u2' YI' e2
all n e e d to b e l o n g to the same space
all need to b e l o n g to the same space
in g e n e r a l we could have
P # q' 91 ~ ~2
"
92 Lqe
Lpe , but
The e x t e n s i o n of the
r e s u l t s p r e s e n t e d here to this s i t u a t i o n is transparent.
16
u
y
FIGURE
W i t h regard tions
(i)
to the system d e s c r i b e d by the set of equa-
(or, e q u i v a l e n t l y ,
pes of questions.
2.3
(4)), one can ask b a s i c a l l y two ty-
The first type of q u e s t i o n takes the following
form: Does the s y s t e m
(1) h a v e a u n i q u e set of s o l u t i o n s
e,y
in
Ln c o r r e s p o n d i n g to each set of inputs u e L n ? If so, is pe pe the d e p e n d e n c e of e,y on u causal, and g l o b a l l y L i p s c h i t z continuous? the s y s t e m
The d e f i n i t i o n and study of the w e l l - p o s e d n e s s of (i) takes into a c c o u n t such c o n s i d e r a t i o n s .
second type of q u e s t i o n takes the f o l l o w i n g form:
The
G i v e n a set of
inputs
u • Ln (the u n e x t e n d e d space) and a s s u m i n ~ that the P s y s t e m e q u a t i o n s (i) have one or m o r e s o l u t i o n s for e,y in L pe' n do these s o l u t i o n s in fact b e l o n g to L n ? If so, does the relaP tion m a p p i n g u into (e,y) have "finite gain"? The d e f i n i tion and study of the s t a b i l i t y of the s y s t e m a c c o u n t such c o n s i d e r a t i o n s
as the above.
(1) takes into
The r e a s o n for sep-
a r a t i n g the two types of q u e s t i o n s is that u s u a l l y the c o n d i t i o n s t h a t imply w e l l - p o s e d n e s s n a t u r e from the c o n d i t i o n s seen b y c o m p a r i n g C h a p t e r
are quite d i s t i n c t and d i f f e r e n t in that imply stability.
This can be
5 w i t h C h a p t e r s 6 to 9
We now turn to the d e f i n i t i o n s .
Definition
The s y s t e m
the f o l l o w i n g c o n d i t i o n s hold:
(i) is said to be w e l l - p o s e d
if
17
u e Ln there exists a pe ' unique set of errors e e Ln and a set of outputs y E L n such pe pe that the system equations (i) are satisfied.
i e.
(WI)
For each set of inputs
(W2)
The d e p e n d e n c e
whenever
u (I)
and
of
u (2)
e
and
y
on
u
is causal;
are two input sets in
L n such pe
•
that for some
T > 0
10
we have
:
then the c o r r e s p o n d i n g Y (2) }
solution
sets
, y(1) }
{e (I)
and
{e (2)
satisfy
ii
=
y(1)
12
(2) YT
=
(W3) YT
on u T
For each finite
for each
T < = , there exists
whenever
u (I)
{e(1)
• y(1)}
sets of
T, the d e p e n d e n c e
is g l o b a l l y L i p s c h i t z and
continuous.
a finite constant
, y(2)}
and
such that, Ln and pe solution
(i), we have l]e(1)-e(2) ]ITp <_ kTI lu(1)-u(2) IITp
14
fly(1)-y(2)[ITp<_kT[ lu(1)-uC2~lITp The above d e f i n i t i o n as it implies
(i) e x i s t e n c e
system equations, (iii)
however,
Note that
we list
light it. served w h e n
and u n i q u e n e s s
(W2)
continuity (W2)
Gi, Hij
Gi, Hij
of solutions
of solutions
as functions
implied by
condition
of w e l l - p o s e d n e s s
are p e r t u r b e d
in [Wil.2]
slightly. (Wl)
(W3) ;
in order
to high-
requires be pre-
We do not m a k e
of w e l l - p o s e d n e s s .
5 that p r o p e r t i e s
to the
on inputs,
n a m e l y that all of the above p r o p e r t i e s
this a p a r t of the d e f i n i t i o n shown in C h a p t e r
is quite broad,
of solutions
is a c t u a l l y
as a separate
The d e f i n i t i o n
something more:
of w e l l - p o s e d n e s s
(ii) causal d e p e n d e n c e
global L i p s c h i t z
of the input.
if each
in
are the c o r r e s p o n d i n g
13
and
eT
kT
u (2) are two sets of inputs
, {e(2)
of
In other words,
- (W3)
is r e p l a c e d by another o p e r a t o r
However,
it is
are p r e s e r v e d of the same or
18
higher
"class"
Notice assume
that in a d o p t i n g
that each
subsystem
of a m u l t i - v a l u e d output
Yi
from
equations.
The
ator m e a n s
that this
in a n o t h e r
sense.
questions. nection
Now,
Definition
Rather,
we a s s u m e
for i n s t a n c e
the error
ei
fact
G i : e i ~ Yi
that
system
(1), we
equations
subsystem
instead
that
by s o l v i n g
(9) is not c o n c e r n e d that each
by an operator,
of the o v e r a l l
description
by an o p e r a t o r ,
suppose
set of d i f f e r e n t i a l
is r e p r e s e n t e d
posedness
is r e p r e s e n t e d
relation.
is o b t a i n e d
differential
the s y s t e m
the
a set of is an oper-
is w e l l - p o s e d with
such
and i n t e r c o n -
and ask a b o u t
from an i n p u t - o u t p u t
the w e l l point
of
view.
It is i m p o r t a n t posed
in the sense
finite over,
escape
are L i p s c h i t z
for all ion
time b e c a u s e
in a w e l l - p o s e d
PTy
finite
(15)
and
However,
defined
Definition merely
stable
if
ing p r o p e r t i e s
to
e,y
in
from
stable
this does
L n [0,T] into itself, P in the sense of D e f i n i t -
not mean
the w e l l - p o s e d n e s s
The
that the maps
u ~ e
(15).
of the s y s t e m
(1),
to stability.
system
is o b v i o u s
For each Ln pe
(1)
is said
to be L - s t a b l e
from the context)
if the
(or
follow-
such
set of inputs that
u 6 L n , we have that P (i) is satisfied, a c t u a l l y b e l o n g
Ln P ($2)
that,
There
whenever
solutions
16
p
is w e l l -
from h a v i n g
hold:
(SI) any
maps
in the sense of D e f i n i t i o n
we n o w turn our a t t e n t i o n
15
that
is p r e c l u d e d
u 6 Ln implies e,y 6 L n . Morepe pe the m a p s PT u ~ PT e and PT u ~
and are h e n c e
are stable
Having
that a s y s t e m (9)
system,
continuous
T,
below.
u ~ y
to note
of D e f i n i t i o n
of
exist
u 6 Ln and P (i), we have
Ileilp
~
finite
constants
e,y E L n P
Ypiluilp + bp
are
yp
and
bp
such
some c o r r e s p o n d i n g
19
IIyilp pllullp÷bp
17
If
b
= 0, we say that the s y s t e m
(1) is ~ - s t a b l e
w i t h zero
P bias.
It is i m p o r t a n t to note that in D e f i n i t i o n
(15), we do
not a s s u m e e x i s t e n c e and/or u n i q u e n e s s of s o l u t i o n s to
(i).
If
for a p a r t i c u l a r satisfied,
u 6 L n , no e,y 6 L n e x i s t such t h a t (i) is p pe then c o n d i t i o n s (Sl) and (S2) are s a t i s f i e d v a c u o u s l y .
In this way,
the s t a b i l i t y issue is d i v o r c e d from the issue of
well-posedness.
Also,
note that the s y s t e m
and only if the r e l a t i o n m a p p i n g in the sense of D e f i n i t i o n
u
(3.1.1).
into
(i) is L p - s t a b l e
(e,y)
Similarly,
if
has finite gain, the s y s t e m
(i) is
L - s t a b l e w i t h zero b i a s if and only if the r e l a t i o n m a p p i n g u P into (e,y) has finite g a i n w i t h zero bias, the sense of Definition(3.1.1).
Note that,
in order for the system
in the sense of D e f i n i t i o n either
(i) to be L -unstable p (15), one of two things m u s t h a p p e n :
(i) there exist a set of inputs
u 6 L n and a set of P e,y • L n such t h a t (1) is satisfied, but e i t h e r P does not b e l o n g to L pn ' or (ii) there exists a sequ-
errors/outputs e
or
y
i
ence of input sets
u (j) • L n and a c o r r e s p o n d i n g s e q u e n c e of P e r r o r / o u t p u t sets e ( ~ ) E L n , y ( J ) e L n such that one of the sequence P P {I le(J)llp/l Is(J) llp} or {I IY(J)IIp/l lu(J) lip} is unbounded. (In this case, in t h a t fy
the s y s t e m
u 6 LP n
~
(i) m a y still have a f o r m of "stability"
e 6 LP n , y e L Pn ; however,
it does not satis-
(16) - (17)).
We now p r e s e n t and d i s c u s s an a l t e r n a t i v e of the s y s t e m d e s c r i p t i o n and s t a b i l i t y definition. that,
in the s y s t e m d e s c r i p t i o n
operators,
(Hij).
It is clear
(i), there are two types of
n a m e l y the s u b s y s t e m o p e r a t o r s
nection operators
formulation
(G i) and the i n t e r c o n -
This d i s t i n c t i o n serves a u s e f u l
p u r p o s e in d e r i v i n g the results of C h a p t e r s
6-9, w h e r e
the k i n d s
of c o n d i t i o n s i m p o s e d on the s u b s y s t e m o p e r a t o r s are q u i t e d i f f e r e n t in n a t u r e from those i m p o s e d on the i n t e r c o n n e c t i o n operators. techniques
However,
in C h a p t e r 4 and 5, w h e r e we apply some
from g r a p h theory to study the d e c o m p o s i t i o n and w e l l -
20
posedness
of
large-scale
between
subsystem
irely.
In these
and
system
description
system
and
interconnection
situations, that
described
this
systems, operators
it is t h e r e f o r e
also makes
interconnection
To m e e t systems
interconnected
the d i s t i n c t i o n disappears
logical
no d i s t i n c t i o n
ent-
to a d o p t
between
a
sub-
operators.
objective,
in C h a p t e r
4 and
5 we
study
by m
18
e. = u. - ~ S.. e. , l l j=l 13 3
i=l,...,m
n.
where
ei,
u i 6 L p el
for some
n., and S.. : L nj + L ni . l 13 pe pe m summing junctions, where consist other
of an e x t e r n a l
summing
Figure
junction
fixed
We
see
the
input
that
(u i)
outputs
this
positive
system
to e a c h
and
(Sij
some
consists
summing
interaction
ei).,
This
integer of
junction
signals
depicted
from
in
2.4
19
Definition following (W1)
ui
+f--~
Sil
eI
The
For
each
set of e r r o r s
(18)
satisfied.
are
(W2) causal; such
i.e., that
The
2.4 (18)
is s a i d
to be w e l l - p o s e d
hold: set
of i n p u t s
e 6 Ln pe
dependence
whenever
u (I)
for
T > 0
some
ei
Sire em
system
conditions
a unique
Ln P
and
inputs
FIGURE
if the
p
such
of and
that
e u (2)
we h a v e
u e L pe n the
and are
' there
system
{Sij two
ej} input
exists
equations
on
u
sets
is in
21
UT(1) =
20
u (2)
then the c o r r e s p o n d i n g
solution
e (I)
sets
and
e (2)
of
(18)
eT
and
satisfy 21
eT(1) =
22
(Sij e j(I)) T = (Sij e j(2) )T ' (w3) Sij ejT words,
on
For each finite
uT
for each
that, w h e n e v e r and
eT(2)
Vi,j
T, the d e p e n d e n c e
is g l o b a l l y Lipschitz T < ~ u (I)
e (I) , e (2)
there exists and
u (2)
of
continuous.
In other
a finite constant
k T such
are two sets of inputs
are the c o r r e s p o n d i n g
solution
in
sets of
Ln pe (18),
we have 23a
I Ie(1)-e(2) IITp <_ kTI [u(1)-u (2) I ITp
23b
l lSij
24
Definition the following
e (i) - S e (2) lu (I) u (2) j ij j l!Tp <- kTl l lTp
The system
properties
(18) is said to be L p -stable . - -
if
hold:
(S1)
For each set of inputs
u 6 L n , we have that any P e in L n such that (18) is satisfied, actually b e l o n g to L n pe n. P ' and moreover, we also have that S. • e. 6 L i Vi,j . ±3 ] P ($2)
There exist finite constants
that, w h e n e v e r solution of
u e L n and P (18), we have
e 6 Ln P
25
IlelIp ! y p I I U I l p + bp
26
I ISij ejl Ip ~ y p I l U I l p + bp If
b = 0, we say that the system
yp
and
bp
such
is some c o r r e s p o n d i n g
,
(18)
¥i,j
.
is --p L -stable with
zero
bias. In D e f i n i t i o n plicitly
displayed
(24), we not only require
unknowns
ei
belong
to
L~ i P
that the Vi
ex-
whenever
22
n,
the i n p u t
set
whenever
u 6
u q L n , but also p
that
S.. e. e L i 13 3 P
¥i,j
n.
m ~ j=l
then
(24)
Ln
Clearly
•
p
n. S.. e. 6 L z 13 3 P
requires
something
if
Vi
more,
e.
l
6 L 1
p
whenever
namely
gi
whenever
u e Ln . P
that
each
u 6 Ln
p
t
But Definition
of
the
terms
n.
Sij
ej i n d i v i d u a l l y
belong
The alternative gether with stability, (18)
the a s s o c i a t e d are closely
to the f o r m
27
(19)
Lp 1
whenever
u e Ln P
system descriptions definitions
related.
(i) a n d
(18),
of w e l l - p o s e d n e s s
To c o n v e r t
to-
and
a system described
by
(i), we d e f i n e
Yij = Sij ej Then
to
can b e r e c a s t
,
Vi,j
as
m
28a
ei = ui -
28b
[
Yij
j=l
Yij = Sij ej which
is of the f o r m (i). M o r e o v e r ,
tion
(19)
for the w e l l - p o s e d n e s s
l e n t to D e f i n i t i o n Definition
(24)
to D e f i n i t i o n
(9) a p p l i e d
applied
To convert
of the s y s t e m
to the s y s t e m
for the s t a b i l i t y
(15)
it can be v e r i f i e d
of s y s t e m
to the s y s t e m
a system described
(28). (18)
u. = 0 1
for
30
e i = Gi_ m ei_ m
for
Then
by
(i) b e c o m e s
Similarly,
is e q u i v a l e n t
(I) to the f o r m
i = m+l,...,2m
the system
is e q u i v a -
(28).
define
29
that Defini-
(28)
i = m+l,...,Im
(19),~
23 m ei -- ui
31a
-
S
j~=l
ij ej+m I
31b
i = 1,...,m
ei+ m = ui+ m + G i e i
In order for the system sense of D e f i n i t i o n ponding unique
set of solutions
el,...,e2m
(9) to the system
existence
= U2m = 0
for
(ii)
in the
(i) C o r r e s -
u I, .... U2m , (31) must have a
than the c o r r e s p o n d i n g
ing D e f i n i t i o n requires
to be w e l l - p o s e d
(19), we m u s t have the following:
to each set of inputs
restrictive
(31)
.
This c o n d i t i o n
condition
resulting
(I), b e c a u s e
and u n i q u e n e s s The q u a n t i t i e s
is more
from apply-
this latter only
of solutions w h e n el,...,e2m
Um+ 1 = ..°
as well
as
Hllem+l,...,Hmme2m must depend in a causal m a n n e r on Ul,...,U2m. If D e f i n i t i o n (9) is applied to the system (i), then we only require that
el,...,e2m
mention
Hllem+l,...,Hmme2m
of
depend c a u s a l l y on .
U l , . . . , u m , with no
(iii) The q u a n t i t i e s
PTel,...,
PTe2m as well as P T H l l e m + l , . . . , P T H m m e 2 m must depend in a globally L i p s c h i t z continuous m a n n e r on PTUl,...,PTU2m. If D e f i n i t i o n (9) is applied PTel,...,PTe2m
(i), we only require
to the system depend
in a g l o b a l l y L i p s c h i t z
that q u a n t i t i e s
continuous
manner
on P T U l , . . . , P T U m w i t h no m e n t i o n of P T H l l e m + l , ° . . , P T H m m e 2 m . Because of all these differences, it is clear that D e f i n i t i o n (19) applied ive)
system Hij
to the system
(31) gives a stronger
c o n c e p t of w e l l - p o s e d n e s s (i).
However,
is causal,
gain
of the system
Um+ 1 = ... = U2m = 0, system
if we assume
(i) in the sense of D e f i n i t i o n
follows:
(5.1.i)).
Thus,
(Hij ej) T
(3.1.i),
to the
has
then the w e l l -
in the sense of D e f i n i t i o n
is e q u i v a l e n t
isfying the above conditions Definition
ejT ~
(see D e f i n i t i o n (31)
(9) applied
that each of the operators
and that each o p e r a t o r
finite i n c r e m e n t a l posedness
(i.e. more r e s t r i c t -
than D e f i n i t i o n
(19) w i t h
to the w e l l - p o s e d n e s s (9).
An o p e r a t o r
Hij sat-
is said to be w e a k l y L i p s c h i t z our c o n c l u s i o n s
of the (see
can be s u m m a r i z e d
as
If
H.. is a w e a k l y L i p s c h i t z o p e r a t o r for all i,j , ~3 then the system (31) is w e l l - p o s e d w i t h Um+ 1 = ... = U2m = 0
in the sense of D e f i n i t i o n well-posed
(19)
if and only if the system
in the sense of D e f i n i t i o n
(9).
(i) is
24
S i m i l a r remarks apply to the two s t a b i l i t y d e f i n i t i o n s (5) and
(24), as a p p l i e d to the a p p r o p r i a t e s y s t e m d e s c r i p t i o n s .
In order for the system (31) to be L -stable in the sense of P D e f i n i t i o n (24), two c o n d i t i o n s m u s t hold: (i) w h e n e v e r n. ui e Lpl for i = l , . . . , 2 m , any c o r r e s p o n d i n g e I ,...,e2m m u s t n. nm b e l o n g to L P l ,...,Lp , r e s p e c t i v e l y ; moreover, the r e l a t i o n s taking the input set into (as in
(25)).
(el,...,e2m)
If we r e s t r i c t
Um+ 1
m u s t have finite g a i n
through
U2m
to be zero,
then this is the same c o n d i t i o n as is o b t a i n e d by a p p l y i n g Definition
(15) to the s y s t e m
i = l,...,2m,
(I) .
(ii)
any c o r r e s p o n d i n g
Whenever
for u i 6 Lni P
Hllem+l,...,Hmme2m
must belong
to the a p p r o p r i a t e L -space; m o r e o v e r , the r e l a t i o n taking the P input set into (Hllem+l,...,Hmme2m) m u s t have finite gain. This c o n d i t i o n does not a r i s e at all w h e n D e f i n i t i o n to the s y s t e m
(i).
Thus Definition
(15)
is a p p l i e d
(24) as a p p l i e d to the system
(31) gives a s t r o n g e r c o n c e p t of s t a b i l i t y than D e f i n i t i o n a p p l i e d to the s y s t e m finite gains, U2m = 0 tem
(i).
However,
then the s y s t e m
(31)
in the sense of D e f i n i t i o n
if all o p e r a t o r s
is L p - S t a b l e w i t h (24)
(16)
Hij
have
Um+ 1 = ...
if and o n l y if the sys-
(i) is L p - s t a b l e in the s e n s e of D e f i n i t i o n
(15).
N o w that we have e s t a b l i s h e d the i n t e r r e l a t i o n s h i p s b e t w e e n the two a l t e r n a t i v e
system descriptions
(1) and
(19),
h e r e a f t e r we m a k e use of w h i c h e v e r system d e s c r i p t i o n is a p p r o p r i a t e to the t e c h n i q u e b e i n g used, w i t h o u t f,u r t h e r comment. Specifically,
in C h a p t e r s
ition and w e l l - p o s e d n e s s ,
4 and 5, w h e r e we study the d e c o m p o s respectively,
of l a r g e - s c a l e
c o n n e c t e d systems, we use the d e s c r i p t i o n that the results
in C h a p t e r s
(18).
4 and 5 involve
inter-
The r e a s o n is
the c o n s t r u c t i o n of
a d i r e c t e d g r a p h a s s o c i a t e d w i t h the g i v e n system,
and in this
process the d i s t i n c t i o n b e t w e e n the s u b s y s t e m o p e r a t o r s and the interconnection operators in C h a p t e r s
is lost completely.
6-9, w e use the d e s c r i p t i o n
On
the other hand,
(i), b e c a u s e the r e s u l t s
there do m a k e a d i s t i n c t i o n b e t w e e n the s u b s y s t e m o p e r a t o r s and the i n t e r c o n n e c t i o n operators.
NOTES AND R E F E R E N C E S
The use of e x t e n d e d spaces to give a g e n e r a l f o r m u l a t i o n of the i n p u t - o u t p u t s t a b i l i t y p r o b l e m is due to S a n d b e r g [San.
1,2,3],
and Zames [Zam.
p i o n e e r i n g w o r k on f e e d b a c k stability. at g r e a t e r length by Saeks [Sae. by D e s o e r and Wu [Des.
i] .
i] .
1,2,3], w h o also did C a u s a l i t y is d i s c u s s e d The set A was i n t r o d u c e d
The d e f i n i t i o n of w e l l - p o s e d n e s s
used h e r e d i f f e r s s o m e w h a t from that of W i l l e m s [Wil. f e e d b a c k systems,
but is in the same spirit.
the s t a b i l i t y d e f i n i t i o n Desoer [Cal.
2].
(2.2.24)
2] for
The f o r m u l a t i o n of
is due to Callier,
Chan and
CHAPTER 3: GAIN AND DISSIPATIVITY In this chapter,
we define various
used throughout
the rest of the book.
with zero bias,
incremental
dissipativity,
conditional
gain and c o n d i t i o n a l pertain
ly e n c o u n t e r e d
namely
types of operators,
ors and m e m o r y l e s s
nonlinear
stability
in the s u b s e q u e n t
operators.
terms of the gains or d i s s i p a t i v i t y it is i m p o r t a n t
for a given operator.
3.1.1
GAIN,
Definitions
integral
of v a r i o u s
of this chapter stability
to "real"
AND I N C R E M E N T A L
Suppose
~T >- 0, in
p 6
[i,~],
~pCG~ = sup ~ 0
incremental
GAIN
that
n, m
are posit-
G
sup
Then we
I IGxl ITp ~ kl Ixl ITp + b,
Vx 6 Lne } p we set
G, d e n o t e d by
sup xT ~ 0
is d e n o t e d by
sup
T ~ 0 x T ~ YT
yp(G)
= ®
We define
~p(G), by
{IGxIIT~ llx11Tp
supremum does not exist,
vain of
np(G) =
such that
(2) is empty,
the vain with zero bias of
If the indicated
deriv-
systems.
of V a r i o u s Types of Gain
= inf {k:~b < -
k
are im-
criteria
is a given operator. G:L n ÷ L TM pe pe the vain of the operator G, denoted by yp(G), by
If the set of
operators,
these constants
and that
yp(G)
operat-
are couched in
calculate
can a c t u a l l y be applied
Definition ive integers, define
linear
constants
the results
GAIN WITH ZERO BIAS,
dissipativity.
Since almost all of the
to show that the various
ed in this m o n o g r a p h 3.1
Thus,
gain
passivity,
to the two m o s t common-
chapters
to k n o w how to a c t u a l l y
portant in order
that are
These include gain,
gain, passivity, strict
We also discuss how these concepts
criteria
concepts
we set
np(G),
~p(G)
= ~ . The
and is defined by
IIGx-GYIITp_ IIx-yIITp
R7 If the supremum in (4) does not exist, we set In the above definitions, stants yp(G),
~p(G), and ~p(G)
np(G)
we recognize
= ~
that the con-
depend not only on the operator G,
but also on the value of p . Note that, in general,
yp(G)
! ~p(G), and
~p(G)
~ np(G)
whenever G(0) = 0. Also, if yp(G) is finite, then G maps the unextended space L n into the unextended space L TM . (However, P P the converse is not true; the operator G:L e ÷ L ~e defined by (Gx) (t) = x2(t) maps
L
is easy to show that if
into G
L
, but
is linear,
y (G) = ~).
then
yp(G)
Finally,
it
= ~p(G) =
np(G). It is routine to verify that, given operators G2
defined on appropriate
G1
and
spaces, we have
yp(G 1 G 2) < yp(G I)
yp(G 2)
~p(G 1 G 2) <_ ~p(G I)
~p(G 2)
~p(G 1 G 2) <_ ~p(G I)
~p(G 2)
In the above definitions, we assume that G is an opermapping L pe n into L TM pc; i.e., we assume that, corresponding to every x in L n there is a unique element Gx in L TM pc' pc" However, even if G is a relation (i.e., G is a subset of L npe × Lpe)' m one can still define yp(G), ~p(G), and np(G), ator
by making only slight modifications Before proceeding
to (2),
yp(G)
Suppose
=
(4).
to the actual computation of the con-
yp(G), gp(G), and ~p(G) for stants we present a useful general result. Lemma
(3), and
G:L~e ÷ Lmpe
inf {k:~b < ® Yx E L~}
various types of o p e r a t o r s ,
is causal.
such that
Then
llGXIIp _< k 1 1 X l l p + b ,
28
i0
~p(G)
=
I GxIlp
sup x~0
IXllp
x 6 Ln P
ii
Up(G)
=
I Gx-Gyl tP Kx-yl Ip
sup x ~ y x,y 6 L n P
Proof
We prove only
are e n t i r e l y similar. G
(9);
To p r o v e
(i0) and
(ll)
(9), it is enough to p r o v e
the p r o o f s of
that
satisfies
IIGXlIp _< kIIxll p + b,
12
¥x 6 L np
for a p a r t i c u l a r choice of the c o n s t a n t s
k
and
b, if and only
if it s a t i s f i e s
13
IIGXIITp < k I I X I I T p + b -
for the same T ÷ ~
in
k
and b.
part,
x e Ln pe
.
-
Vx E L pe n
The "if" p a r t follows r e a d i l y by letting
(13), and is true e v e n if
the "only if" and let
YT > 0, '
suppose
G
G
is not causal.
To prove
is causal and s a t i s f i e s
(12),
T h e n we have
lIGxIITp = lIGxTIITp ~ IIGxTIl p ~ kllxT!1 p + b
14
=kilxEITp+ b This proves
(9). []
The v a l u e Lmpe
is causal,
of L e m m a
(8) is in showing that if
then the c o n s t a n t s
yp(G) , ~p(G)
be c a l c u l a t e d by e x a m i n i n g only the b e h a v i o u r of tended space
3.1.2
and G
G:L pe n ÷ np(G)
on the unex-
Ln P
C a l c u l a t i o n of G a i n for L i n e a r I n t e g r a l O p e r a t o r s
In this subsection,
can
we show how to c a l c u l a t e
~p(G)
29
for a given is linear,
linear
integral
we have
15
Lemma
16
(Gx) (t) =
yp(G)
Consider
operator
= ~p(G)
G .
an operator
[tg(t,T)
Note
that,
since
G
defined
by
= np(G)
x(T)
G:LIe ÷ L l e
dr
d
0 where
g(.,.)
is measurable
and satisfies
the condition
T
17
I
sup ~E [0,T]
Ig(t,~) I dt ~ C T < -,
~T >_ 0
r
Then we have 18
sup
Y1 (G) =
Igct,~)l
3>0
19
y (G) =
yp(G)
20
T
sup t > 0 -
dt
J
--
i
t Ig(t, T) I dr
0
<_ [YI(G)] I/p
IT (G)] I/q
yp 6
(i,~), where
q = p/(p-l) Proof
First of all, we show that the condition
enough to guarantee that G does indeed map Accordingly, suppose x 6 Lle, n so that
I
T
21
IX(r) I dr < ~,
~
Lle
into
>_ 0
0 Then,
22
from
(17)
IT
and
(21), we get
l(Gx) Ct)I
at <_ IT
0
IxC~)l
dr at
IT [x(~)[ ;TIgct,~)l
dt ~r
0
=
[t IgCt,~)l
0
2 0
T
(17) Lle.
is
30 T
I
<_ C T
Ix(~) I d~ < -,
WT > 0
0 where the interchange Fubini's Lle
theorem
of the order of i n t e g r a t i o n
[Dun.
i, p. 190].
Hence
G
is justified by
maps
Lle
into
• To prove
23
sup ~ >-0
(18), suppose
first of all that
Ig(t,T) I dt <
Then,
by letting
into
LI, and that
T = =
in
(22),
it is clear
that
G
maps
L1
~
24
>l(~)
<_
T h e proof that e q u a l i t y "hand-waving"
5
sup z >-- 0
Ig(t,~)i
d= < ®
T
holds in
(24)
is f a c i l i t a t e d by a bit of
using the unit impulse d i s t r i b u t i o n
the basic idea of the proof
is demonstrated,
that the a r g u m e n t can be made m a t h e m a t i c a l l y sequence ~(.)
25
of
Ll-functions
x(t)
for a fixed
26
Once
it will become precise,
Ll-norm
clear
by using a
and converge
Suppose we let
= ~ ( t - T 0)
T0 ~ 0 .
Then we have
(Gx) (t) = g(t,T 0) Since
~(t-T 0)
can be a r b i t r a r i l y
sense of distributions, we conclude
27
that have unit
in the sense of distributions.
6(.)
that
g(.,~0 ) • L 1 and m o r e o v e r
by an
closely approximated,
Ll-function
having u n i t
in the Ll-norm,
31 ~
28
S
yl(G) = ~l(C) _> Ilg(-,T0) ll i =
l g ¢l t ,,T o ) ,
~t =
0
=
Ig(t,~o) I dt T0
where
in the last step we use the fact that
t < •
Since
(28) holds
for
every
g(t,T)
= 0 whenever
T O ~ 0 , we have
~
29
Yl(G)
Now,
(24) and
~
(29)
Next,
f
sup • 0~ 0
Ig(t,T0) I dt
TO
together
we prove
prove (19).
(18). Suppose
first of all that
t 30
I
sup t and let
Ig(t,T)l
x 6 L
£
d <-
Then we have
l cG~)ct)l
31
dT
0
i
<
t
Igct,~)l
txc~)l
d~ <_ a
llx(.)11®
0
which
shows
32
y.(G)
To prove
33
that <_
sup t
the opposite
x(T)
s
t Ig(t,T) I dT
0
inequality,
= sign g(t0,T)
={
fix
t O ~ 0 , and let
-i Then clearly
x(.) e L
and
if
g(t0,T)
~ 0
if
g(t0,T)
< 0
1
fix(.) If. = 1 .
Thus
t 34
y.¢s) >_ ll¢Gx)(.)ll® = sup I t t 0
>-
I 0
g(t0,T)
x(T)
g(t,T) x(T) aT1 0
dr
t 0
=
I 0
,g(t0,~),
dT
82 from the way in which every t 0 , we have 35
y.(G) ~
Now,
x(.)
is chosen.
Since (34) holds for
rt0 J ]g(t0,T) I d~ 0
sup t0
(19) follows from (32) and (35).
Finally, to prove (20), let p • (1,-), and let x(.) E Lp Then, letting q = p/(p-l) and applying HOlder's inequality [Dun. I, p. i19], we get
36
I(cx) (t) l <_
f
t
Ix(~)l
lg(t,~)I
d~
0
=
Ig(t, T) II/q + I/Plx(T)
I
[g(t,r) I d~ }i/q {
ig(t,T) i
dT
0 <_ { 0
0 Ix(T) Ip aT} I/p
37
I (Gx) (t) Ip <_ [y (G)] p/q
38
I0+
I (Gx) (t) Ip
dt
I t Ig(t,T) I Ix(T) Ip dT 0
<_ lye(G) ]p/q
t i+I 0 0
Ig(t,T) 1
Ix(T) Ip dT
<_ [y (G)] p/q
IxCT) I p dT
Yl(G) 0
= [y (G)] p/q Raising both sides of (38) to the power
39
] I (Gx) (.) I Ip <_ [y~(G)] I/q
Yl(G) i/p
P fix(.) lip
gives
[Yl(G)] I/p
Ilx(.) lip
S3 which establishes
the g a i n s the g a i n p . while
(20). o
Lemma
(15)
YI(G)
and
yp(G)
shows
In p r o v i n g fact that
G
bound
for
40
maps
(18)
scalar
G maps YI(G),
L nle
though
Lemma
and
side of
(20)
(19), w e m a d e
into
form
exactly,
(16),
whereas
values
of
a p p r o a c h e s y~(G),
Yl(G).
inputs
into L mle
scalar
essential
u s e of the
outputs.
If,
' we can only obtain
it is s t i l l p o s s i b l e
G : L nle + L mle
Suppose
of the
for a l l i n t e r m e d i a t e
, the r i g h t
p ~ I , it a p p r o a c h e s
o t h e r hand,
for o p e r a t o r s
can be calculated
can be estimated
N o t e t h a t as p + ~ for
that,
y~(G)
on the
an u p p e r
to c a l c u l a t e y~(G).
is of the f o r m
t
41
s
(Gx) (t) =
G(t,T)
X(~)
dr
0
where
G(.,.)
is m e a s u r a b l e
and satisfies
T
42
I
sup ~e[0'T] where
ll.
norm.
T h e n we h a v e
43
II
llGCt,~)ll
denotes
Y1 ¢c) !
the m a t r i x
sup T > -O -
44
~(G)
=
sup
t > 0 -
yp(G)
45
at
Proof
proceed we have
,
~
norm induced
m
0
]lG(t,~)ll
dt
I IG(t,~)ll
d~
s
t
0
<_ [YI(G)] I/p
The proofs (24)
and
b y the E u c l i d e a n
T
[~
(G)] I / q
where
same as t h o s e of
< ®
t
of
(20),
j u s t as in the p r o o f
of
V p ~
q = p/(p-l)
(43) a n d
(45)
respectively. (19).
(i,~),
are e s s e n t i a l l y the To p r o v e
F i r s t of all,
(44), w e
as in
(32),
34
46
i
v (G) <_ sup t -> 0 To prove e(T)
the reverse
t
l JG(t,~)II d~
0
inequality,
be any unit vector
let
such that
t0 ~ 0
be arbitrary,
I IG(t0,T)
let
e(T) If =
fiG(to,T) II , and let 47
x(T) Then
x(.)
48
= e(T)
,
VT
• L n.
and
I Ix(.) II~ = 1 .
Moreover,
~(G)
~ II (Gx)(.)II® ~ II (Gx)(t0) If =
f- lIG(t0,~) I[ d~ 0
Since and
(48) holds
(44)
for every
In the special one can obtain and
t o Z 0 , we have
the reverse
of
(46),
follows.
(40).
somewhat
case of linear
better
estimates
One can also study a wider
those of the form
(16).
where
is defined
the set
A
convolution than those
operators, in Lemmas
class of operators
To do this, we recall D e f i n i t i o n
(15)
than (2.1.31),
as the set of all distributions
of
the form I
O,
t < 0
49 f(t)
where
~(.)
=
~
denotes
are real constants,
50
fi 6(t-ti)
1
+ fa (t)'
the unit delta distribution, {fi } 6 ~i ' and
I If(.) ]I A =
51
Lemma
52
(GX) (t) =
~ i=0
Consider
Itg(t-~)
Then
• L 1 ; we also define
Ifa(t) I dt
an operator
X(~)
dT=
G : Lle
÷ Lle of the form
~ gi x(t-ti) i=0
ga(t-~) 0
• A .
0 ! t O < tl...
0
+ g(.)
fa(.)
f~
Ifi I +
0
where
t ~ 0
x(T)
dT
85
53
yl(G) = ~®(G) = l lg(.) II a
54
yp(G)
] l g ( . ) II A
<_
Proof
,
Vp £
F i r s t of all,
a c t u a l l y b e l o n g s to
L1,
then
if
[i,-] gi = 0
(53) and
(54)
¥i,
i.e.,
if
g(.)
f o l l o w d i r e c t l y from
L e m m a (15). The m o r e g e n e r a l case c o n s i d e r e d here is p r o v e d in [Des. 2. p. 247]. []
We now p r e s e n t a s e q u e n c e of three lemmas, w h o s e p u r pose is to p r e s e n t an e x p l i c i t e x p r e s s i o n for
55
Lemma
Suppose
G : Lp(R)
~ Lp(R)
Y2(G).
is a c o n v o l u t i o n
o p e r a t o r d e f i n e d by
56
(Gx) (t)
where
i
=
t g(t-T)
g(.) 6 A , and suppose
x(~)
dT
p < -
Then
y -- y+
57
where
ft Gx lip y =
58
sup
x ~ Lp(r)/{0}
i lxl Ip II Gx
y+= sup x e Lp(R+)/{0}
59
Remarks the g a i n of
G
IP
IIxl p
The p u r p o s e of this lemma is to show that
v i e w e d as an o p e r a t o r from
Lp(R)
into
Lp(R)
is e x a c t l y the same as its gain w h e n v i e w e d as an o p e r a t o r Lp(R+)
into
Proof e > 0
60
from
Lp(R+)
be given,
Clearly
y ~ y+
and s e l e c t
c C1 - y -
~
l/p)
6 e
/Cl-~)
i/p
To p r o v e the opposite, (0,i)
>_ 1
such that
-
let
36
This
can
(60)
approaches
that such
always
because
and
as
6 ÷
> 1 - e . [ IGXllp
[
0,
Next,
the
left
select
[l-(E/2)]y
side
of
x ~ L
; then
(R) such P select T < -
that
I
ix(t)I p
T
can
tinuous T +-~
always
dt
1
vanishes
=
b e done,
nondecreasing
, and
xl(t) Then
done,
l-(e/2)
Ilxl Ip = 1
61
This
be
as
because
function T + ~
for
the
of
Now
t < T
and
left
side
of
(61)
T
that
approaches
let
x(t)
= xl(t)
x2(t)
vanishes
is a con0
as
+ x2(t) , ~ahere
for
t > T
.
clearly
llx11~= IlXlll P + ]1~211~
62
so t h a t
I IXllI~ = 1-~ , I Ix211pP
63
Next,
we
= 6
have
IGXlII p = EIGx - Gx211 p ~ tlGxEIp - IfGx211 p
64
¢ (i - ~)
IGXll 3e
>
1 -
i/p y - ~
_
y =
Y
Now,
observe
that
c 2
61/p) Y
~ I/p
65
JXll Ip
(i
>
(I-E) 7
(1-~) 1/p Xl,
Gx I
actually
belong
to
L2[T,-).
Define
66
y(t)
Then
y 6 Lp(R+),
time-invariant
67
= xl(t-T)
and
,
V t 6 R+
l[ylIp= [IXlI1e
operator,
(Gy) (t)
=
(GXl) (t-T)
,
¥ t 6 R+
Also,
since
G
is a
37 so t h a t
Gy E L p ( R + ) ,
IIGyIIp
68
and
> (1-~)x
ll~Ilp In effect, construct proves
w h a t we h a v e a function
that
69 where
shown
such
given
any
that
(68)
~ > 0, holds.
we can This
y+ = y . [] Lemma
Suppose
6 A
Then
g(.)
70
G : L2(R)
IIGxtlz
sup x E ~2(~)/{o}
Proof 71
is that,
y 6 Lp(R+)
For
÷ L2(R)
is d e f i n e d
=
lg(J~)l
sup
by
(56),
llxl12
the sake
of c o n v e n i e n c e ,
define
sup l&cj~l I
c 2
b3
Given
any
E > 0 , select
c 2 - (e/2) .
Since
w e also h a v e
where
(t) = exp
a > 0 .
Then
x
=
73 where
(j~) (j~)
74
such
(~/2).
Now,
(-t2/4g 2) cos xa(.)
that
is the c o m p l e x
Ig(-j~o) I [ c 2 -
x
72
~o 6 R
g(-j~o )
Ig(j~o) I >
conjugate
of
g(j~o ),
let
(~ot)/(2~a2) I/4
~ L2(R) , fix
(.)If2
= i, and
[~ ¢jm-j~o) + ~ ¢ j ~ + j ~ o ) ]/2
= (8~ 2) 1/4
e_~2 ~2
Also (Gx a)
75 Now,
by l e t t i n g
concentrate frequencies
76
(j~)
=
g(j~)
x
(j~)
~ + ~ , it is c l e a r
the f r e q u e n c y ~ = Z~o
spectrum
By Parseval's
1 I IG~oll22 - 2-v
S~
t h a t one can e s s e n t i a l l y
^
Ixa(j~) I
around
equality,
the two
we h a v e
Ig(j~) I2 IXo.(j~)I 2 d~
38
Hence,
as
o ~ ® , I IGxo112 ÷
So we can select a finite this can be done
~
for any
[I;(j~o) I + such that
I;(-j~o) I]/2~c2-(c/2). IIGxoll
[ c2-E
since
z , we see that
IIGxEi2
77
sup
~
x • L2(R)/{0}
To prove ity.
the opposite
This gives,
78
{IGx1122
inequality,
for all
lfo
:
2~
IGx(j~) 12
1
<-
shows
79
Sup
]IGx{12 llx] 12
(79)
together
Lemma
81
(Gx) (t) = ~|t g(t-T)
Consider
Y2(G) =
Proof to
L2(R)
follows
Ix(j~) 12
of
readily
G
x(T)
sup
(70).
an operator
d0
• A .
<_
establish
80
82
f~
~1
d~
that
(77) and
g(.)
d~
x(j~)12
l;(J'") I2
x • L2CR)/{0}
where
equal-
d~
^
{g(jm)
sup
Now,
we gain use Parseval's
x • L2(R)
2~
which
c2
{{xl[ 2
l;(j~) I : c 2
[]
G : L2 ÷ L2
of the form
dT
Then
sup
l;(j~) I
Clearly,
G
defined
by
from Lemmas
defined by (81).
(69)
and
(56)
is an extension
The equality (55). D
(82) now
39
W e c l o s e o u t the d i s c u s s i o n integral output
operators
by
convolution
presenting
of the g a i n s of
linear
a r e s u l t on m u l t i - i n p u t - m u l t i -
operators.
83
Lemma
Suppose
84
(Gx) (t) =
G
: L nle ÷ L mle
G(t-~)
x(~)
is of the f o r m
dT
0 where
G(.)
85
~ .mxn
~l(G)
Then
IollG(t)
~ y (G) =
ll dt
0 86
Y2(G)
where
'+'
the l a r g e s t
=
denotes
(44).
imax
[6+(jm)~(j~)]l/2
the c o n j u g a t e
eigenvalue
Proof and
sup
and
denotes
max
of a m a t r i x .
The condition
To prove
transpose,
~ c2
(86),
(85)
observe
follows
that,
routinely
f r o m (43) n x 6 L 2 , we
for a n y
have
lIGxII22 --
87
_
1 2=
f~
[Gx(je) ]+[Gx(j~)]
1 2~
d~
x+ (j,,,)G+ (j~) G (j~) x (j~) d~
i <--YF c22
I
2
2
x+(j~) x(j~)
d~
-- c 2 llxIl2 which
shows
follows
that
along
Y2(G)
Calculation
In t h i s of the f o r m
The proof
the s t e p s of L e m m a s
Remark
3.1.3.
~ c2 .
Note
that
(55)
(86) h o l d s
and
72(G)
= c2
now
(69). a
e v e n if
of G a i n for M e m o r y l e s s
subsection,
that
G(.) E A m × n
Nonlinearities
we study operators
G : L nle ÷ L TM le
40
88
(Gx) (t) = G(t,x(t)) where
G(.,.) : R + x R n + R m
89
Lemma
90
I IG(t,v) I [ ~ k where
I I -I I
91
in
~p(G) Proof
Suppose
there
exists
I Ivl ],
(90)
denotes
< k,
Vp 6
For any
ITI IG(t,x(t))
92
is c o n t i n u o u s .
v v E R n,
so that
(91)
93
such
that
norm.
Then
[i,~] p E
[i,~),
I Ip dt <_ k p
we have
IT [ Ix(t)
I[ p dt
,
%rf >_ 0
0 follows
i/p
readily
Next,
ess. sup t 6 [0,T ] so that
k
Vt [ 0
the E u c l i d e a n
0
the p o w e r
a constant
for
by raising
both
sides
of
(92)
to
p = ~ ,
I ]G(t,x(t)) I I ~ k
ess.sup [ Ix(t) I I t E [0,T]
~ (G) < k . []
94
Lemma
Suppose
95
IIG(t,u)-G(t,v)
there
exists
a constant
[ ] ~ a I lu-vl I,
~
such
vu,v E R n ,
that
Vt ~ 0
Then 96
np(G) Proof Next,
97
we p r e s e n t Lemma
that
<_ ~ ,
Vp e
Similar
[i,-]
to t h a t of L e m m a
a specialized Suppose
there
result exist
(89).
[]
for the c a s e
p =
constants
and
c
b
such
41
98
~
lJG(t,v)]I
c
IIv]I
+ b
,
vv
E Rn
,
¥t
[
0
Then
y® (G)
99
<_ c
Proof
i00
Clearly,
proves
i01
to t h e
that
(98)
does
not
A function
sector
[~,B]
G
Remarks [s,B]
if,
for
shaded
region
Ilx(t) l
ess.sup
+ b
[0,T]
In the
each shown
case
that
yp(G)
: R+xR n ÷ R n
< 0
Vv 6 R n
n = i, G
t > 0 , the in F i g u r e
imply
<_ c
is s a i d
, for
to
if
[G(t,v)-sv] ' [G(t,v)-Bv]
102
C
o
Definition belong
~
, then
t 6
(99).
Notice
x 6 L e
IIG(t,x(t))II
ess.sup t e [0,T] which
if
graph
of
,
belongs G(t,.)
Vt
> 0
to t h e lies
3.1.
SLOPE B
G (t x)
SLOPE
FIGURE
3.1
sector in t h e
e
42
Lemma
103 [~,8],
and
(88).
Then
Suppose
let
104
G
denote
72(G-cI)
where
I denotes
105
c =
equivalently
! r
the
Vp •
identity
By
,
operator
r =
where
ll.II
readily
from
3.2
the
(106).
for
which
all values
of
concepts applied
assume
to h a v e
Q
and
dissipative
(102)
can be
{ 0
The bound
have
(104)
now
In this
a truncated
inner
To keep m = n
,
introduced
to operators
: Ln 2 e + L m2e
that
Vt
PASSIVITY
G
section,
and dissipativity, G
"
we
required
define
which
can only
In other
words,
product
structure
the e x p o s i t i o n i.e.
the conL n L mp e : pe ÷
that
G
to h a n d l e
:
L
the
be i t is on
the
simple, we n 2e + L n2e
the case
m ~ n
apparent.
Definition with
' and
square,"
we
applied
the modifications
readily
L n2e
norm.
AND
[i,®].
spaces.
throughout
However, are
p •
of passivity
and output
on
Vv • R n ,
section,
can be
to operators
essential input
by
o
DISSIPATIVITY
of gain,
defined
[8-u]/2
the Euclidean
In the previous cept
operator
sector
as
denotes
follows
to the
[I,®]
"completing
expressed
belongs
corresponding
I I G ( t , v ) - c v l ]2 ! r 2 v ' v
106
R with
Let
G
symmetric. respect
(Q,R,S)-dissipative
2
: R+xR n + R n
the
,
[8+u]/2
Proof
G
: Ln + n and let Q , R , S e R n×n 2e L2e' say that the operator G is
We
to t h e
triplet
(Q,R,S) , o r
simply
if
< G x , Q G x > T + < x , R X > T + T > 0 , ~ T -
> 0 , Yx 6 L n 2e
43
Before we
first
state
to L e m m a
discussing
a general
Lemma
and only
Suppose
G
semidefinite.
for causal
of d i s s i p a t i v i t y ,
operators,
analogous
: Ln ÷ Ln is causal, and that Q 2e 2e G is (Q,R,S)-dissipative if
Then
if
Proof
+ <x,Rx>
First
(Q,R,S)-dissipative), T ~ m
Then
"
~Gx,Sx>
of all,
and
let
n Yx 6 L 2
>_ 0 ,
suppose
(2) h o l d s
x E L n2 .
since
suppose
xT 6 L 2
However,
+
Then
(4)
(i.e.,
G
follows
by
is
.
Conversely, x 6 Ln 2e
result
implications
(3.1.8).
is n e g a t i v e
letting
the
Q
G
satisfies
for all
+ <XT,RXT>
is n e g a t i v e
T
(4),
and by
and
let
(4) w e h a v e
+ > 0
semidefinite
and
G
is c a u s a l ,
we have
< G X T , Q G X T > <_ < G X T , Q G X T > T
Similarly,
by
the
causality
of
G
=
T
, we have
t
= T
Clearly,
Substituting
<XT,RXT>
= <x,Rx> T
into
gives
(5)
T + <x,Rx> T + T ~ 0
which
establishes
(2).
As with lies
in the
fact
[]
Lemma
that,
(3.1.8),
the
for a causal
significance
operator
G
of L e m m a
, one
can
(3)
44 determine
whether
is n e g a t i v e
or n o t
G
semidefinite)
o n the u n e x t e n d e d
is
(Q,R,S)-dissipative
by examining n L2 .
space
o n l y the b e h a v i o u r
W e n o w t u r n to the i m p l i c a t i o n s Note
that,
denote
in the i n t e r e s t s
il.ll 2
i0
Lemma where
Q
Proof
ll
of
~ =
.
Then
12
-
12
l lGxJ IT which
shows
13
that
respect
Lemma
15 respect
to
J IxJ IT < =
<
~2 (G)
in d e t a i l .
dissipative
<
the l a r g e s t
(least n e g a t i v e )
J [Gxl IT
VT
> 0
Vx6
-
the square,
L n2e
(ii)
to
< ~
n×n
> 0
-
.
and
on n o t i n g
'
Then
I
¥ x 6 L n2e
G
is d i s s i p a t i v e
n
denotes
0n
denotes
the i d e n t i t y
G
that
ilxII
V T >- 0 ,
2e
: Ln + Ln is d i s s i p a t i v e w i t h 2e 2e ~2(G) < ~ . T h e n it is a l s o d i s s -
(Q-~In,R+~
Obvious,
~
'
.
2(G) In,S) , for a l l 2
by combining
The next definition
studied
(Q,R,S)
(-In,~2(G) In,0n), w h e r e
a n d the
which was historically
to
. o
E 2(G) 3 2
Suppose
respect
Proof
n×n
-
(Q,R,S)~
ipative with
~
Obvious
lIGxli
14
is ~2(G)
By completing
of d i m e n s i o n
Proof
Then
denote
Suppose
of d i m e n s i o n
J I-I I
(2) b e c o m e s
to the t r i p l e t
the null matrix matrix
<
-
G
the f o r m
~2(G)
Lemma with
G : L n2 ~ L n2
I ISI I
into
of
of d i s s i p a t i v i t y .
we use
8J IxJ 12 -< ~I Jxl IT
J IRI I, e =
can be manipulated
Q
section.
definite.
Let -I
Q
I I IGxl
where
this
Suppose
is n e g a t i v e
eigenvalue
of b r e v i t y ,
throughout
(where
inequalities
introduces
the f i r s t
an i m p o r t a n t
type of d i s s i p a t i v i t y
~ _ > 0 •
(2) and
(14).0
concept to b e
45
16
Definition
An o p e r a t o r
G : Ln ÷ L n is said to be 2e 2e passive if it is d i s s i p a t i v e w i t h r e s p e c t to (0n,0n,In), i.e. if
<x,Gx> T > 0 ,
17
¥T > 0 ,
-
G
Vx 6 L n 2e
-
is said to be s t r i c t l y p a s s i v e if it is d i s s i p a t i v e w i t h
r e s p e c t to
(0n,-ZIn,I n) <x,Gx> T ->
18
for some
Ilxll~
~
¢ > 0 , i.e., VT
' ' ' T
'
> 0 -
if
VX E L n 2e
'
We now c o n s i d e r how these v a r i o u s c o n c e p t s apply to linear c o n v o l u t i o n o p e r a t o r s and m e m o r y l e s s n o n l i n e a r operators. Due to shortage of space, we do not d i s c u s s the d i s s i p a t i v i t y of i n t e g r a l o p e r a t o r s of the type is r e f e r r e d
to
[Wil.
o p e r a t o r s of the type
Lemma
19
(3.1.16).
Instead,
the r e a d e r
4] for a d i s c u s s i o n of the p a s s i v i t y of (3.1.16).
Consider a convolution operator
G : Ln2e ÷ Ln2e
of the form
Gx(t)
20
f
=
t G(t-T)
X(T)
dT
0 where
G(.) 6 A n×n
Then
G
is d i s s i p a t i v e w i t h r e s p e c t to
(0n,-~In,I n) , w h e r e 21
= Proof Lemma
22
(3).
Let
inf ~ 6R
Imi n
[G+(j~)
Clearly n
x e L2
<x,Gx> = - 2~ -
1 2~
2~
G
+ G(j~)]/2
is causal.
; then
by
H e n c e we can a p p l y
Parseval's
theorem,
Re
I" ~+
(jm)
Ix(je)
[G+(J~) + G ( ~ ) ] 2
d~
=
~
Iixll
x(J~)
2
d~
46
Hence G
(4) h o l d s
is
with
R = -6In,
By Lemma
S = In
(3) ,
(0n,-~In,In)-dissipative.
23
Lemma form
(3.1.85),
G(.,.) to
Q = 0n,
Suppose
where
satisfies
G
: L n2e + L n2e
G
belongs
(3.1.99).
is an o p e r a t o r
to the s e c t o r
Then
G
[e,8],
is d i s s i p a t i v e
of the
i.e.,
with
respect
( - I n , - ~ S I n , (~+B)In) . Proof
24
From
(3.1.99),
one
can easily
show
that
T ~ 0
which
can be
3.3
readily
manipulated
CONDITIONAL
GAIN
In this operators,
i.e.
AND
section,
operators •
CONDITIONAL
we
M
(G) =
The
conditional
to
Suppose
~ain
of
result.
D
DISSIPATIVITY
be
concerned
with
"unstable"
: Ln ÷ Ln w h i c h do n o t m a p pe pe i n t r o d u c e the b a s i c d e f i n i t i o n s , and
{x e L n P
P
shall
the d e s i r e d
G
Ln into L m . We first P P then discuss how they pertain
Definition
to y i e l d
G
linear
G
convolution
: L n ÷ Lm pe pe
operators.
Then we
define
: Gx • L n} P , denoted
by
~cp(G),
is d e f i n e d
by
II Gx IEp ~cp (G) =
If
n = m,
x E ~p(G)/{o}
p = 2, the
(Q,R~S)-dissipative
that
~cp(G)
with
zero
bias."
use
Ycp (G)
operator
with
Note
sup
respect
+ <x,Rx>
should However,
G
[I x lip is s a i d
+
properly this
to b e c o n d i t i o n a l l y
to a g i v e n
is
be too
triplet
(Q,R,S)
>_ 0 ,
Vx e M2(G )
called
"conditional
cumbersome,
if
gain
a n d we n e v e r
47 The f o l l o w i n g simple lemmas b r i n g out some r e l a t i o n ships b e t w e e n the c o n c e p t s "unconditional"
concepts
Lemma ~cp(G)
i n t r o d u c e d in D e f i n i t i o n
Suppose
~p(G)<~
Suppose
G
; then
is causal and that
d i t i o n a l l y d i s s i p a t i v e w i t h r e s p e c t to (i.e.,
Q
is
(Q,R,S)
is n e g a t i v e s e m i d e f i n i t e ) ; suppose
= L np
and
G
is
where
M2(G)
con-
Q < 0
= L n2 .
Then
(Q,R,S)-dissipative.
Proof
The h y p o t h e s e s
imply that
+ <x,Rx> +
Since
Mp(G)
= ~p(G) Lemma
G
(1) and the
i n t r o d u c e d earlier.
G
is causal,
n Vx E L 2
~ 0
it follows by L e m m a
(3.2.3)
that
G
is
(Q,R,S)-dissipative.o
Lemma ally
Suppose
G
is causal,
( Q , R , S ) - d i s s i p a t i v e b u t not n Then M2(G) ~ L 2 .
that
G
is c o n d i t i o n -
( Q , R , S ) - d i s s i p a t i v e , and suppose
Q ~ 0 .
ally
Proof
T h i s is just the c o n t r a p o s i t i v e of L e m m a
Lemma
Suppose
G
is causal,
(Q,R,S)-dissipative where
definite);
suppose
M2(G)
Q < 0
~ Ln 2 .
Then
that
(i.e., G
G Q
is not
(6).o
is c o n d i t i o n is n e g a t i v e (Q,R,S)-dissipa-
tiv~.
Proof (3.2.10), w e h a v e Lemma
(5).
Since
If
G
is
(Q,R,S)-dissipative,
~2(G)< ~ , w h i c h shows that n M2(G) ~ L 2 , G c a n n o t be
then by L e m m a = Ln 2 , by
M2(G)
(Q,R,S)-dissipative.o
We next i n t r o d u c e a class of o p e r a t o r s t h a t are u n s t a b l e in a p a r t i c u l a r way.
Such o p e r a t o r s play an i m p o r t a n t
role in the i n s t a b i l i t y theorems of C h a p t e r s
i0
Definition b e l o n g to class U if
An o p e r a t o r
8 and 9.
G : Ln + Ln 2e 2e
is said to
48
such
(i)
G
(ii)
There
is l i n e a r is an u n b o u n d e d
family
of c o n s t a n t s
aT
that
ii
I IGxl IT2 <- a T
I Ixl IT2
~ x 6 L n2e
'
n
12
(iii)
M2(G)
(iv)
~c2 (G)
Lemma
Suppose
Then
M2(G)
is a p r o p e r
L2n .
It is a l s o
Proof
that remains
~ L2
<
Since
G
a proper
to be
: L n2e + L n2e
G
closed
shown
subspace
is linear,
subspace is t h a t
belongs n L2
of
M2(G)
L n2
of
it is a c l o s e d
x
suppose {xi} ~ is a s e q u e n c e in M2(G) * n 6 L 2 ; we must show that x 6 M2(G ) . Let
y*
= Gx* e Ln2e"
,
~c2(G)
< ~
z E L 2n
that
for e a c h
On the other
for x
denote
hand,
T
since
Lemma
14 some
is
a
ization a linear
of t h e
of a r i g h t - c o p r i m e
a Cauchy from
sequence.
(II),
z
, we
see
shows
that
w
*
y
= Gx
we
see,
to
YT
"
that YT = ZT
E L n2
and
G
: Ln ÷ Ln 2e 2e
belongs
to c l a s s
U.
rest
= 0
for
all
x 6 M2(G ) }
elements.
MI(G)
sets
since
e M2(G ) . D
n
convolution
and
converges
to
This
Since
= {z E L 2 :
closed
The
x
Now,
{(Gxi) T}
.
To do
to n Yi = Gxi 6 L 2 ,
by
nonzero
proper
zT
subspace. converging
is a l s o
converges
to
Suppose
Proof which
{yi }
of
M 2 ( G ) ~ L n2 • All
sequence,
{yi } .
sequence
that
defined
MI(G) contains
{yi }
of
y * = z 6 L2n
n
MI(G),
that
limit
, the
6 L 2 , it f o l l o w s
Then
is a C a u c h y
converges
, i.e.,
13
seen
the
T < ~
= {(Gxi) T}
all
{x i}
, it c a n b e
Let
{YiT }
Since
U.
is a s u b s p a c e
since
this,
to c l a s s
is the subspace
of this Mp(G)
orthogonal n
of
section
and
operator. factorization.
MI(G) T o do
L2
.
complement
of
D
is d e v o t e d
to a c h a r a c t e r -
in the c a s e w h e r e this,
M2(G),
we
need
the
G
is
concept
49 15
Definition G(.)
of dimension
Given a Laplace
right-coprime
factorization
(i)
N, D E ~nxn
(ii) ~n×n
N and 6 such that
P, 6
in
(r.c.f.)
~(s~ ~(s~ +6¢s) ~(s) : ~
17
(iii)
G(s) = N(s)
Right-coprime
of
,
[D¢s)]-i
factorizations
G(.)
18
if
i.e.,
there exist
v s ~ c+ ,
Vs e C+
have several useful proper-
some of which are brought out below.
[Vid. i] and
distribution
(N,D) is said to be a
are ri~ht-coprime,
~6
ties,
transformable
n×n, the ordered pair
Others can be found in
[Vid. 5].
Lemma
If
G(s)
is the sum of an
s
and an element of
rational functions of
n×n
matrix
of
then
G(.)
~n×n,
has
an r.c.f. For a proof of this result,
see [Vid. 5].
shows that the most commonly encountered tion matrices" 19
Lemma
(18)
type of "transfer func-
have r.c.f.'s.
Lemma Let (NI,DI), (N2,D2) be two different r.c.f.'s Then there is an R(.) 6 ~n×n such that ~-i E ~nxn , ^
of
G(.).
~2 = ~i ~' ~2 = ~i~" Remarks
Lemma
(19) shows that an r.c.f, of
unique to within a "regular"
element
to within an element ~nxn
~nxn whose inverse also belongs to
Proof
If
R
in
(NI,DI)
, (N2,D2)
R of the ring
G is
are both r.c.f.'s of A
~n×n , then by assumption, there exist such that
~n×n , i.e.,
^
~I,el,P2,Q2
in ~nxn
50 ^
21
^
P2N2 + Q2D 2 = I ^
Multiplying
both
sides of
(20) by ^
22
_
D1 1 gives
_
~l&lel-i ÷ 61 = D1 1 B y assumption,
23
G = NIDI -I = &262 -1
Hence
~1&2~2 -1 ÷ 61 : ~1-1 Multiplying
both
sides of
^
24
(23) by ^ ^
D1-162-(24)
since
shows
that
A ~e
' which
Also,
(24)
ly, an entirely analogous argument (20) shows that ~-i 6 ~nxn . [] Lemma e ~nxn
.
Suppose
Then
Proof
~nxn
,
^
D2 = DIR
&lSl -I = N2D2 -I
25
gives
~l&2 +QID2 : ^
NOW,
62
(N,D)
(N + HD, D)
implies
starting
is an r.c.f, there exist
(21)
of
of
N2 = N1 ~ '
R e ~nxn
from
is an r.c.f,
By assumption,
that
shows that
. Final-
instead of
G , and suppose G + H
P,Q
in
~nxn
such
that ^ ^
26
A ^
PN + QD = I Hence
27 which shows that & + HD,D are right coprime. evident that G + H = (N + HD).~-i s Lemma tion m a t r i x
(25)
G1
states
which
Finally,
it is
that if we are given a transfer func-
can be expressed
as the sum of a "stable"
^
part
H
and an "unstable"
find an r.c.f, ^
part
of the unstable
G , then it is only necessary to part
G , because,
once we have
^
r.c.f.
(N,D)
for
namely
(N + HD,D).
G , we readily
have an r.c.f,
for
G + H ,
51
28
Lemma
A point
a n d o n l y if d e t
D(So)
Proof
[D(s) ]-l
s o 6 C+
= 0 , where
"if" S u p p o s e
becomes
is a s i n g u l a r i t y
unbounded
(N,D)
of
G(.)
is a n y r . c . f ,
of G.
d e t D ( s o) = 0.
as
s + so
Then
because
if
s o m e e l e m e n t of
otherwise
^
det P,Q
[ D ( s ) ] - i = i / d e t D(s) in ~n×n such t h a t
remains
bounded
,
=z
both
s ÷ s
Now
select
O
29 Multiplying
as
sides by
[D(s)] -I
vs
c+
gives
5(s) G(s) + O(s) = [D(s)]-I
30
As
s + s o , the r i g h t
must
the l e f t side.
bounded s ÷ sO
as
shows
" o n l y if"
G(s)
where
Adj D(s)
Hence
that
sO
since
= N(s).
of
of
A , it f o l l o w s
Adj D(s)
6 ~n×n
i.e.,
Adj D(s)
of
unbounded
as
G(.) .
, we can write
of c o f a c t o r s
is o b t a i n e d
a n d is b o u n d e d
over
of
D(s)
by multiplying
t h a t Adj D(s)
E ~nxn
e+^.
so,
s ÷ s O , w e m u s t h a v e d e t D(s)
. Since
or a d d i n g
, whence if
+ 0
G(s) as
N(s). becomes
s ÷ so ,
a c t D ( s o) = 0 . [] With
the s e t
this background
Mp (G)
"transfer
32
must become
so
remain
/ d e t D(s)
the matrix
each element
as
G(s)
whence
and Q(s)
is a s i n g u l a r i t y
Adj D(s)
denotes
unbounded,
, P{s)
6 = ~ ~-i
elements
unbounded
(30) b e c o m e s
p,Q 6 ~n×n
s ÷ s o 6 C+
, which
31
side of
Since
whenever
G
o n r . c . f . 's , w e c a n c h a r a c t e r i z e
is a c o n v o l u t i o n
function m~trix"
G(.)
Theorem
n G : Ln pe ÷ L pe
Suppose
, which
operator
h a s an r . c . f .
with^ ^ (N,D).
is a c o n v o l u t i o n
oper-
a t o r of the f o r m
(Gx) (t) =
33
I
t G(t-T)
x(T)
dT =
(G,x) (t)
0
where
G(.)
is a L a p l a c e
has t h e r . c . f .
(N,D)
transformable
Then,
for e v e r y
distribution, p 6
[I,®],
and
G(.)
52
34
M
(G)
p
=
{x
=
: x = D~z
{x
: x(t)
for
some
z 6 L n}
p
~t ~ D(t-T)
=
Z(~)
dT
for
some
o
0 z(.)
Proof into
D*z
image
of
D(L~):
.
Let
Then
: L pn e
D
the
right
÷ L pn e
side
be
of
the
(34)
operator
is
D(Ln);
suppose
x E ^
some
z 6 Ln Then P E L n , which P opposite containment, ^
D(L~),
^ ^
shows
i.e.,
F
x = Dz
Gx
, and
that
suppose
suppose
Gx
= Gx
x 6 M
P (G),
x 6 M
=
^
^
^
PGx+6x
35 ^
Since
so
To
= Nz
prove
that
, so
the
x 6 Ln
and
^
^
( a n d t h ePr e f o r e ^ x gives
^
-ix
; then,
in
P*Gx
+ Q*x
z
and
Gx
x
that
x
x
is
D ( L ~) P
=
both
of
the
the
time
belong
form
domain,
P Ln
to
D,z
,
(35)
(36)
where
z E
becomes
shows Ln P
.
that This
z E shows
L np ' that
.
Remarks every
for
^
z = ~-i
36
x 6
the
x = D*z
-luz
(G) .
• Ln . Select P,Q in ~nxn s u cPh t h a t (29) P (30)) h o l d s . Multiplying b o t h s i d e s o f (30) b y
so
z
D(L~),
^ ^
Gx
Let
mapping
simply
Ln under the operator D . We claim that M (G) = P P To prove this claim, observe first of all that Mp(G)
contains
that
6 L n} P
value
of
p 6
(ii)
In
(i)
(32)
characterizes
Mp(G)
for
[i,~].
In
the
Theorem
(34),
case
one
where
can
use
any
p =
2
, we
be
as
r.c.f.
can
(N,D)
also
of
G(.).
characterize
M I (G).
37
Theorem (N,D)
38
be
any
Mi
Let
r.c.f,
(G)
=
of
G
: L n2e G(.)
Then
{ x e L 2n :
D'(T) 0
in Theorem
X(T+t)
d~
=
(32)
and
0
>_ 0}
~t
let
53
Proof
39
By Theorem
M2(G)
(32),
we have
= { x 6 L n2 : x = D 2
for
some
z • L2}
Hence
40
n
M1
(G) = {x e L 2 : < x , D z > n = {x e L 2 : D * x
where
D*
41
: L n2 ÷ L n2
<x#Dz>
In t h e p r e s e n t
42
is t h e
=
z e L
}
= 0}
adjoint
operator
of
D
defined
by
we have
I°
=
It
x'(t)
0
=
for all
case,
<x,Dz>
= 0
D(t-T)
z(T)
dT d t
0
I"
I
z' (T)
0
D' (t-T)
x(t)
d t dT
x(T)
dT = 0
~t
>_ 0}
x(r+t)
dr = 0
Vt
>_ 0}
so t h a t
(D'x) (T) =
43
Hence,
by
44
D'(t-T)
x(t)
dt
(40) ,
M1
(G) = {x 6 L n2 :
D' (T-t) t
= { x 6 L 2n :
D'(T) 0
which
proves
(38).
a
Theorems characterizations somewhat which
(32) of
abstract.
are especially
the The
and
(37),
sets next
useful
though
Mp(G) result
they give
and gives
in C h a p t e r s
M1 more
(G),
complete are
concrete
8 a n d 9.
still results,
54
45 has
Theorem Suppose G : Ln + Ln ^ ~ 2e 2e an r . c . f . (N,D), and b e l o n g s to c l a s s
a pole
of
G(.).
(i)
Under
if
so
such
these
that
(ii)
D' (s O)
if
is real,
D' (sb)
belongs
Proof by Lemma
Hence NOW
there
46
so
x(t)
Clearly
x(.)
a nonzero
= v exp(-Sot)
~ D'(~)
a vector
Then
x(t)
, where
let
v
.
to
be
M1
x(t)
Cn
=
*
denotes
a vector
Then
in
(G);
in
Rn
such
= v exp(-Sot)
s o • C +o
is a p o l e
= 0 , so t h a t d e t vector then
v
such
D + ( s O)
of
D ' ( s O)
that
G(), = 0 also.
D ' ( s o)
v* = 0.
v = 0. Let
+ v* e x p ( - s ~ t )
because
x(~+t)
Let
be
belongs
v = 0
is c o m p l e x ;
I
47
.
M I (G)
d e t D ( s o)
• L n2 ,
v
v = 0 .
of all , if
First
(28),
exists
suppose
to
let
exp(-s~t)
conjugate,
that
then
+ v*
complex
So^
U
(32),
s o • C+o be
conditions.
is c o m p l e x ,
v exp(-Sot)
is as in T h e o r e m
so E C + °
(i.e.
Re
So>0) .
Also,
dT
0
=
D'(~)
v exp(-SoT),
D'(T)
V*
exp(-So£)
d~
0 ~
S
+
exp(-s~r),
exp(-s:t)
dT
0
=
D' ( S o ) V e x p ( - S o t )
+ D+ (So)V *e x p ( - s * t )
Wt
which s
o
shows
is r e a l
± space
M
that
x • M 1 (G),
follows
Theorem (G),
similarly.
(45)
in the
by Theorem
The
> 0
case
where
[]
demonstrates special
(37).
= 0 ,
some
case where
elements G
of
belongs
the
sub-
to
U
and
55
^ G(.)
C +o •
has a pole in
e l e m e n t s of
M I (G)
However,
it does not say that all
are of the f~rm
(46).
W e close out this s e c t i o n by giving a l t e r n a t e characteri z a t i o n s of c o n d i t i o n a l g a i n and c o n d i t i o n a l d i s s i p a t i v i t y
for a
linear convolution operator.
48
Lemma
Suppose
49
~c2(G)
=
Proof
The proof is e n t i r e l y a n a l o g o u s to that of
(3.1.83). whenever
50
In fact
G
is as in T h e o r e m
sup [Ima x
(3.1.84)
[G+(j~)
applies,
(45).
Then
~(j~)]}i/2
n Gx ~ L 2
if we note that
x e M2(G ) . D
Lemma be any r.c.f,
Suppose
G
G(.).
Then
of
is as in T h e o r e m G
w i t h r e s p e c t to a g i v e n t r i p l e t
(45), and let
(N,D)
is c o n d i t i o n a l l y d i s s i p a t i v e (Q,R,S)
if and only if the
m a t r ix 51
P(~)
=
[D+(j~)S N(j~)
+ N+(jm) S ' D ( 9 ~ ) ] / 2
+ D + ( j ~ ) R D(j~) is
positive
Proof dissipative
By D e f i n i t i o n
(1), G is
(Q,R,S)
conditionally
if and only if
52
+ N + ( j ~ ) Q N(j~)
s e m i d e f i n i t e for all ~ .
+ <x,Rx> +
~ 0
Vx 6 M 2 ( G ) ^
By T h e o r e m
(32),
x 6 M
(G)
if and o n l y if
^^
x = Dz for some
2^^
z E L n2 , in w h i c h case ly by u s i n g and d e f i n e
53
D
Gx =
Nz
If we abuse n o t a t i o n slight-
to d e n o t e the o p e r a t o r m a p p i n g
Nz = N,z
, then
z
into
D~z
(52) is e q u i v a l e n t to
+ +
>_ 0 ,
n Yz 6 L 2
i.e., w
54
,
~
n
where
* d e n o t e s the a d j o i n t operator.
(54) holds all
>_ 0
~ .
if and only if
P(~)
Vz 6 L 2
By P a r s e v a l ' s
is p o s i t i v e
equality,
semidefinite
for
,
56
NOTES AND R E F E R E N C E S The utility of the c o n c e p t of gain in studying stability [Zam.
arises from the work of S a n d b e r g
3].
The gain calculations
many sources,
including
[San.
of Section
3.1 are taken from
the work of S a n d b e r g
D e s o e r [Wu l] and W i l l e m s
[Wil.
of course w e l l - e s t a b l i s h e d
i].
[San.
The c o n c e p t
of p a s s i v i t y
c o n c e p t of d i s s i p a t i v i t y
[Wil.
3,4,5]
and e x p l o i t e d by Hill and M o y l a n [Hil.
[Moy.
2].
The concept of c o n d i t i o n a l and Bergen [Tak.
pativity was introduced M o y l a n [Hil.
3] , [Moy.
are i d e n t i f i e d
in ]Tak.
i], while c o n d i t i o n a l
operator
calculation
can be found in [Vid.
in the dissi ~
by Hill and
of a class U operator
The c a l c u l a t i o n
range of an unstable of M±(G)
1,2],
gain is implicit
The properties
i].
is
The
was i n t r o d u c e d by Willems
(under a d i f f e r e n t name) 3].
3], Wu and
in c i r c u i t and s y s t e m theory.
more general
w o r k of Takeda
feedback
2] and Zames
of the domain and
are found in [Vid. 7].
5], while
the
CHAPTER 4: DECOMPOSITION OF LARGE-SCALE INTERCONNECTED SYSTEMS In this chapter,
we show how the study of a g e n e r a l
large-scale interconnected
system
first d e c o m p o s i n g the LSIS
into its s o - c a l l e d s t r o n g l y c o n n e c t e d
components
(SCC's).
(LSIS) can be s i m p l i f i e d by
By using this procedure,
one n e v e r has to
deal w i t h a m o r e c o m p l e x s y s t e m than the one w i t h w h i c h he began, and in m a n y cases,
the o r i g i n a l s y s t e m is r e p l a c e d by several
smaller systems w h i c h t o ~ e t h e r are s i m p l e r than the o r i g i n a l system.
T h e p r i m a r y tool u s e d to e f f e c t this d e c o m p o s i t i o n
the t h e o r y of d i r e c t e d graphs,
or digraphs,
is
and we b e g i n w i t h a
d e v e l o p m e n t of the results that we need from this theory.
It
should be m e n t i o n e d at the o u t s e t that the f o l l o w i n g d i s c u s s i o n is h i g h l y selective;
the reader is r e f e r r e d to
[Baa. i] for a
m o r e c o m p l e t e discussion.
4.1 4.1.1
SOME RESULTS F R O M THE THEORY OF D I R E C T E D GRAPHS Basic Concepts Definition
(V,E), w h e r e subset of vertices,
V × V and
A directed ~raph
V = { V l , . . . , v n}
E
.
The set
(or digraph)
is a finite set V
is r e f e r r e d
and
is a pair E
is r e f e r r e d to as the set of to as the set of edges.
(vi,vj) e E , then we say that there is an edge from
A p a r t from the above a b s t r a c t definition, a p i c t o r i a l r e p r e s e n t a t i o n of a d i g r a p h in the plane are l a b e l l e d as from
vi
to
vj
is a
If vi
we o f t e n use
(V,E), w h e r e b y
w i t h an a r r o w h e a d d i r e c t e d towards
Consider
n
points
v i , . . . , v n , and an arc is d r a w n vj
(vi,vj) 6 E . Example
t_~o vj .
the d i g r a p h
(V,E), w h e r e
V = {Vl, v2, v3, v 4}
E = {(Vl,V2),(Vl,V4),(V2,V2),(V2,V3),(V3,Vl), (V3,V4),(V4,V2)} The p i c t o r i a l r e p r e s e n t a t i o n of this d i g r a p h is shown in
whenever
S8
Figure
4.1
.
vI
V3 v
~
v
FIGURE
Definition from
vi 6 V
with
A vi0 = v i
to
Given
vj 6 V and
4.1
V4
a digraph
(V,E),
is a n o r d e r e d
Vik
= vj
set
a p a t h of l e n g t h k
{ v i 0 , V l.l.,. .
, such that
(vi£,vi£+l)
,Vik } e E
for
£ = 0,...,k-i
equivalent
One can
state
the a b o v e
manner:
A p a t h of l e n g t h
sequence
of v e r t i c e s
V i k = vj
, such that there
£ = 0,...,k-i
is
from
vi
to
vj
is a
vi0 = v i
vi£
for all
from
from
of F i g u r e
to e ~ e r y o t h e r
Given
reachable
is a p a t h
vi R vi
is an e d g e
In the d i g r a p h
Definition
there
from
{vi0,Vil,...,Vik) , with
from every vertex
vj E V
k
in the f o l l o w i n g
and
__t° v.l£+l
for
.
Example path
definition
vi
a digraph vi E V
to
vj
4.1,
(V,E),
(denoted by
~
i , e v e n if t h e r e
there
is a
vertex.
we
say t h a t v i R vj)
By c o n v e n t i o n , is no p a t h
if
we t a k e
from
vi
to
itself.
Note obvious imply vi from
to
that
that vj
R
that
R
defines
is t r a n s i t i v e ,
v i R vk
a binary i.e.,
(in o t h e r w o r d s ,
and a path
to
relation
v i R vj
on
and
V
if t h e r e is a p a t h
from
vj
v k , then there
Given
a digraph
.
It is
vj R v k from
is a p a t h
v i to Vk). Definition
of v e r t i c e s
(vi,v j)
is s t r o n g l y
(V,E), w e say t h a t a pair
connected
if
v i R vj
and
58
vj R v i . If
i ~ J , then
there is a p a t h from By convention,
vi
a pair
(vi,v j)
to
vj
(vi,v i)
Definition
4.1.2.
vi
A digraph
(vi,vj) 6 V × V
vj
to
vi
is taken to be s t r o n g l y c o n n e c t e d
e v e n if t h e r e _ i s no p ~ t h frn~
every pair
is s t r o n g l y c o n n e c t e d if
and a path from
to itself.
(V,E)
is s t r o n g l y c o n n e c t e d if
is s t r o n g l y connected.
T e s t i n g f o r Strong C o n n e c t i v i t y
We n e x t d i s c u s s two c o m p u t e r a l g o r i t h m s
for d e t e r m i n i n g
w h e t h e r or not a g i v e n d i g r a p h is s t r o n g l y connected.
N o t e that
the a l g o r i t h m s g i v e n here are o n l y two of several p o s s i b l e ones; we s e l e c t these p a r t i c u l a r a l g o r i t h m s b e c a u s e they r e q u i r e v e r y little b a c k g r o u n d . is ~ e f e r r e d to
For a m o r e c o m p l e t e discussion,
the r e a d e r
[Baa. 1].
The a l g o r i t h m s g i v e n here m a k e use of B o o l e a n v a r i a b les, w h i c h are defined next.
Definition
i0
{o,1}
The B o o l e a n set
(B,+,-)
is the set
t o g e t h e r w i t h the b i n a r y o p e r a t i o n s + and
B =
" on B d e f i n e d
by
Ii
0+0
=
0,
12
0'0
=
i'0
0+i
=
i+0
=
0"i
=
=
I+i
=
1
0,
i'i
=
i
T h o s e f a m i l i a r w i t h a b s t r a c t a l g e b r a can v e r i f y that (B,+,-)
is a c o m m u t a t i v e algebra.
For the rest,
note that the b i n a r y o p e r a t i o n s + and commutative,
it s u f f i c e s to
• are a s s o c i a t i v e and
and that + is left - and r i g h t - d i s t r i b u t i v e over • ;
in o t h e r words,
for all
13
a+b = b+a
14
a+ (b+c) =
a,b,c
(a+b) +c
in
B, we have
60
15
a-b = b-a
16
a-(b-c)
=
(a.b).c
17
(a+b)-c
=
(a.c)+(b-c)
18
a- (b+c)
=
(a-b)+(a-c)
Sometimes "and".
+ is r e f e r r e d
Also,
(B,+,'). rows
and
this
by
defined also
we
can
Specifically, m
columns
D 6 B n×m in the
belongs
19
.
B n×m
"or",
of
and
Boolean
"-"
such
is r e f e r r e d
matrices
if D is an a r r a y
over
consisting
to as
the
algebra
of
n
that
Matrix
usual
to
to as
speak
way; and
d.. • B for all i,j, we d e n o t e 13 addition and multiplication are
i.e.,
E • B nxm
is d e f i n e d
by
,
then
F = D + E ~
~
f.. = d.. + e.. 13 13 13 where and
Similarly, a d d i t i o n is a c c o r d i n g to (ii). ×£ B n×£ and E 6 B , then F = D'E b e l o n g s to the
if
D 6 B n×m
is
defined
by m
20
bij
21
Definition Vn} , we d e f i n e
=
[ k=l
its
dik'ekj
Given
a digraph
adjacency
matrix
(V,E)
A 6 B nxn
1
if
(vi,vj)
0
if
(vi,v j) ~ E
with
V = {v I, ....
by
E E
22 aij
23
=
Definition Vn} , we d e f i n e
rij
Given
a digraph
its r e a c h a b i l i t y
=
{ 1
if
matrix
(V,E)
R E B nxn
v i R vj
24 0
Note
that,
otherwise
for
any
digraph,
with
we have
by
V = {Vl,...,
61
25
r.. = 1
for a l l
ll
i
It is clear that a d i g r a p h is s t r o n g l y c o n n e c t e d and o n l y i f 1 .
a l l e l e m e n t s of its t e a c h a b i l i t y m a t r i x are equal to
It is also c l e a r that the a d j a c e n c y m a t r i x of a d i g r a p h can
be formed by inspection.
In order to d e v e l o p an a l g o r i t h m to
test for s t r o n g - c o n n e c t e d n e s s , R
given
A
26
.
A
we d e r i v e a m e t h o d
for c o m p u t i n g
T h i s is the p u r p o s e of the next s e v e r a l lemmas.
Lemma ing
if
C o n s i d e r the m a t r i x
by itself
£
times
(for
A £ , o b t a i n e d by m u l t i p l y -
£ ~ i)
Then
if there is a p a t h of l e n g t h £ from Z ij
{ 1
v i to vj
27 0
Proof
otherwise
The proof
is by induction.
note that a path of length (vi,vj).
Hence
(27)
true for
£ = k-i
1
T h e n for
£ = i,
v. to v. is just an edge i ] £ = 1 . N o w suppose (27) is
is true for
.
First, w h e n
from
£ = k, we have
n
28
(Ak) ij = "
~
(Ak-l)im
m=l
F r o m the d e f i n i t i o n s of + and to see that
29
(Ak).. = 1 13 m
such that
" amj
"
(namely
(ii) and
(12)),
it is e a s y
if and o n l y if
(Ak-l) im = 1
By the i n d u c t i v e hypothesis,
(27)
and
is true
amj = 1
for
£ = k-i
.
Thus
(29) is e q u i v a l e n t to
30
m to
vm
Clearly
31
such t h a t there is a path of length k-I from
and an edge
(Vm,V j) .
(30) is e q u i v a l e n t to
T h e r e is a path of length k from
vi
to
vj
.
v.
l
62
Thus
(Ak) o. = 1
if a n d o n l y
if
for
~ = k
, and
completes
32
Lemma
Given
a digraph,
33
R =
13
(27)
is t r u e
~
Proof identity
i ~ j, some
we
Since
matrix),
see
that
Z)
from
Lemma
35
R =
shows by
that
induction. 0
we have
R = I + A + A2 +
we
vi
34
This
the p r o o f
see t h a t -
ro.
...
= 1
ll
(where
for a l l
I
denotes
i .
For
r.. = 1 if a n d o n l y if (A£).. = 1 for 13 13 if and o n l y if t h e r e is a p a t h (of some
£ h 1 , i.e.,
length
holds.
Ai ~
i=0
the
(31)
to
vj
Given
n-i ~
.o
a digraph
with
n
vertices,
we have
if w e c a n
show
Ai
i=O Proof whenever of
there
length
The
is a p a t h
at m o s t
n-i
there
is a p a t h
{viol
v i , ...,Vlk_l.
defining
this
of
sequence
of v e r t i c e s
latter
vj
to
quantity
argument to
vi
until
at
two
0 < ~ < m -
larger
vj
, and
sequence
the l i s t elements < k -
.
length
than
of
to
k+l
that
are
we
vertices the
clearly
k-(m-£). can
at m o s t
suppose
of v e r t i c e s
of
Then
n-l,
length
is a path
let
. ,...,Vik} a l s o ,VXm+l
it has
a path
~ v i , there Accordingly,
the
, then
{vi0 ,.. .,vi
find
vi be
least
, and
vj vj
that,
same.
the
defines If this
repeat
n-i
the
from
. []
36
Lemma
37
R =
that
Giwen
(7 + A) S
Proof clear
from
k [ n
is a g a i n we
to to
=~ vj}
where
vj
vi vi
k
, Vik
contains
v. = v. i£ im
from
from
If
Suppose
is e s t a b l i s h e d
from
length
path.
(vi0,...,Vik)
path
lemma
in f a c t
First,
a digraph
fox
from
with
every
n
vertices,
we have
s >_ n-I
the p r o o f
of L e m m a
(35),
it is
vi
a
63
s
R =
38
~ i=0
~
We
now claim
Ai
for
every
s >_ n - i
that S
Ai = "
39 i=0
Clearly,
if
We prove
(39)
since true
both for
(39)
(I + A ) s ~
for e v e r y
can be established,
by induction.
sides
equal
I
then
First, in t h i s
s = £, a n d o b s e r v e
s >
(39)
the lemma is t r u e
case.
that
0
Next,
M + M = M
is p r o v e d .
for
s = 0,
suppose
(39)
is
for all matrices
M.
Thus °
(I + A) £ + I =
40
(I + A) £
(I + A)
[ Ai÷ i=0
=
~+i ~
=
( [
A I)
(I + A)
[ A i÷l
"
i=0
"
Ai
i=O This
proves
the
The algorithm
for
lemm~, o
discussion computing
Algorithm
41
t
2s
M2s
M
.
given
suggests A
for computing
Given
A
step
2
select
an integer
Step
3
Calculate
Note
that step
M2s
, let
R
the
Reachability
M = I + A
as
3 can be
t
M 2t
such that
for
some
s < t,
then
R
2 t [ n-i
.
.
accomplished
are
Matrix
.
(M.M = M 2, M 2 . M 2 = M 4 , e t c . ) . multiplications
following
.
The
1
matrix =
R
to n o w
Step
multiplications than
up
by
t
matrix
Sometimes
fewer
required.
For
instance,
clearly
R = M
. Thus
if
if
64
2S
the squaring new matrix
process at
some
(M
. M 2s = M 2s+l
stage,
then
It is o b v i o u s log 2 n
matrix
Boolean
matrices
row additions Thus,
of o r d e r
(i.e.,
standard
42
of
R
Now,
n×n
can
for
R
computing and
notation of t h i s
is g i v e n
.
Warshall's
This
Algorithm
for
R + A
Step
2
For
k +
1
to n, do
For
i ÷
1
to n,
If
rik
matrix
two
(n2/log2 n)
226-231].
[Baa.
as W a r s h e l l ' s 2 requires n
t h a t we u s e ~i'"
i, pp.
the
The
222-223].
R
do
then r o w of
R i = R i + Rk R)
R + R+I
Example adjacency
replaces
in
computing
= 1
also
"~2
1
3
known
Note
found
that
using
proof.
Step
Step
requires
row additions.
algorithm
to m e a n
c a n be
2
in a
.
shown
I, pp.
algorithm,
(R i = i - t h
43
n
without
"~I ÷ ~2"
algorithm
algorithm
[Baa.
result R
be m u l t i p l i e d
requires
another
not
it c a n be
operations)
We now present
row additions,
details
"or"
does
is the m a t r i x
the a b o v e
multiplications.
computation
algorithm,
that
that
)
Consider
the
digraph
of Figure
4.2
.
Its
is
VI
V7
FIGURE
4.2
65
44
A
=
0
1
0
1
0
0
0
0
0
1
0
1
0
0
1
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
1
0
Using either A l g o r i t h m
(41) or A l g o r i t h m
(42), we can c o m p u t e
R
as
45
R
Since
R
=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
0
0
0
1
1
1
1
0
0
0
1
1
1
1
0
0
0
1
1
1
1
does not c o n t a i n all l's,
1
1
the d i g r a p h is not s t r o n g l y
conneeted~
4.1.3.
D e c o m p o s i t i a n into S t r o n g l y C o n n e c t e d C o m p o n e n t s S u p p o s e that a g ~ v e n d i g a a p h is not s t r o n g l y connected.
Can we d e d u c e some f u r t h e r i n f o r m a t i o n a b o u t its s t r u c t u r e ? answer is p r o v i d e d by i d e n t i f y i n g the s o - c a l l e d components
46
(SCC's)
of t_he digraph.
Definition S
on
Given a digraph
V is d e f i n e d by
is s t r o n g l y c o n n e c t e d
v i S vj
V
.
{i.e., v i R vj
Since
(without ambiguity)
S
(V,E), the b i n a r y r e l a t i o n
if and o n l y if the pair
It is e a s y to s h o w that r e l a t i o n on
and
S
(vi,v j)
vj R vi).
is in fact an e q u i v a l e n c e
is an e q u i v a l e n c e r e l a t i o n we can say
that "a set of v e r t i c e s
is s t r o n g l y connected,"
w h e n w h a t we m e a n is that the v e r t i c e s are p a i r w i s e connected.
The
s t r o n g l y connected
strongly
66
Since partition digraph
V
S
is an e q u i v a l e n c e
into
(V,E)
its e q u i v a l e n c e
is s t r o n g l y class under
classes
under
of
V
reachability 48
matrix
Lemma denote
S R
Proof
(V,E)
Otherwise,
matrix.
"if"
rij
Clearly V
a
itself
is
the e q u i v a l e n c e
identified
Then
the
v i S vj
J-th
Suppose
row
Ri = Rj . .
since
S if
be a g i v e n d i g r a p h
denmtes
rii = I, so the h y p o t h e s i s Similarly,
under
V, we c a n
using
the
.
its r e a c h a b i l i t y Ri
S .
on
if and o n l y
c a n be e a s i l y
Let
R i = R j , where
classes
connected
an e q u i v a l e n c e
relation
implies
a n d let
R
if a n d o n l y
of
if
R .
By definition,
~
that
rji = i, i.e.,
= rjj = I, w e h a v e
v i R vj
vj R v i .
Hence
v i S vj "only vj R v i .
Now,
rik = i; this together
if"
Suppose
whenever
is b e c a u s e
with
v i R vj
implies
argument, Ri = Rj .
Example From S
are
{ V l , V 2 , V 3}
and
In g e n e r a l ,
classes
(41) or
under
i n g rows.
S
This
be efficiently
S , and
can order v b • Vj need
some
once
comparison
R
is c o m p u t e d
it is e a s y
, which
i.e., k
rik = 1 .
implies
of F i g u r e
is c a l l e d
V
rjk =i.
4.2
classes
R
either
under
see
~ith
[Ba~.
of the
the e q u i v a l e n c e its
succeed-
"string matching,"
and can
I, Ch. 4].
i n t o its e q u i v a l e n c e
Vl,...,V k . classes
using
to d e t e r m i n e
e a c h r o w of
w e par_tition
equivalence
in s u c h a w a y t h a t if
(Vb,V a) g E.
classes
We n o w s h o w t h a t w e
In o r d e r
v a•
to do this,
Vl, we
concepts.
Definition of l e n g t h g r e a t e r
for some
o u t by c o m p ~ t e r ~
i < j , then
further
vj R v k
v i R Vk,
{v4,v5,v6,v7}.
l a b e l t h e m as
these and
and
k, w e a l s o h a v e
a g a i n the d i g r a p h
by comparing
Suppase under
some
see t h a t t h e e q u i v a l e n c e
(42),
carried
that
rik = 1
Consider
(45), w e i m m e d i a t e l y
Algorithms
50
rjk = 1 implies
In o t h e r
49
i.e., v i R vj
for
By a s y m m e t r i c a l words,
v i S vj,
rjk = 1
Given a digraph
than one
from a vertex
(V,E),
a cycle
is a p a t h
to itself. A n e d g e of the
87 form
(vi,vi)
cycle.
is called a s e l f - l o o ~ and is not c o n s i d e r e d
(Note that
itself).
(vi,v i)
The digraph
contain any cycles
(V,E)
is a p r e d e c e s s o r if
v i + vj
,
and
vj 6 V , and (vi,vj)
Suppose
there exists a v e r t e x Proof
acyclic
Given a d i g r a p h
of
Lemma
52
is a path of length one from is
in
Assume
has a predecessor,
to
self-loops).
(V,E), we say that v i e V
vj
is a successor
of
vi ,
6 E .
the d i g r a p h V
vi
if it is does not
(it m a y however c o n t a i n
Definition
51
to be a
(V,E)
is acyclic.
Then
that does not have a predecessor.
the contrary,
i.e.,
suppose every vertex
and c o n s t r u c t a sequence in V
as follows:
Select
v. e V arbitrarily, and select v. to be a prede±0 ik+l cessor of v. for k > 0 . By assumption, this sequence can be ik c o n s t r u c t e d indefinitely, and since V contains only a finite number of elements, sequence.
...,v.1£+m = vi£ } Clearly
an e l e m e n t of
V
must occur
In other words we can c o n s t r u c t such that
V.lk+l
twice in the
a sequence
is a p r e d e c e s s o r
{v i
{vi£,vi£+l , of
Vik Vk.
,v i ,. . . . ~+m £+m-i "''v1£+l'V1~ (V,E), which c o n t r a d i c t s the h y p o t h e s i s
v. } is a cycle in l£+m that (V,E) is acyclic.
Hence the original
[]
53
a s s u m p t i o n is false.
Proposition digraph
(V,E).
Let
{Vil .... ,vi£ = Vil}
Then the set of v e r t i c e s
be a cycle
{Vil,...,vi£_l}
in a is
strongly connected. The proof 54
is obvious.
Definition denote
the e q u i v a l e n c e
ivity).
such that
of
Then the reduced d i g r a p h
The vertex an edge
Given a d i g r a p h classes
set
V = {Vl,...,Vk},
(Vi,V j) (Va,Vb)
V
(V,E), under
(V,E)
let
S
VI,...,V k
(strong c o n n e c t -
is defined
and the edge set
if and only if there exist
E
as follows: contains
Va e v i , v b 6 Vj
e E
The reduced digraph has a very simple
interpretation.
88 Suppose we modify the original digraph vertices in
V.
(V,E)
into a single vertex, for
resulting digraph is the reduced digraph. strongly connected,
by collapsing
i = l,...,k . Note that if
all
The (V,E) is
then its reduced digraph consists of a single
vertex and a self-loop.
55
Lemma
For any digraph
(V,E), its reduced digraph is
Proof
Assume the contrary, namely that
acyclic.
Vim,Vim+l = Vil}
is a cycle in
contains the edges of
(V,E).
This means that
(Vi2'Vil)''''' (Vim'Vil)"
By the definition
E , this implies that there exist vertices
v!ljI) ' v!lj2) in
Vi. , j = l,...,m , having the following property. edge set
E
contains the edges
(v(2) (i) , (v(2),v(1) im_l ,Vim im il ). class under
(v! 2) ± l 'v
Now, since
Vij
S , there is a path from
J = l,...,k .
(Vil'Vi2'''''
The original
)),(v(2) (i)) ..... i 2 'vi 3 is an equivalence
vlj ~i) to
v(2) ij
, for
So what we have shown is that there is a cycle in
the original digraph
(V,E)
containing the vertices
{v!l),v (2) lI iI '
"''' v(1) i ,v.~2) }. By Proposition (53), this implies that all these m m vertices are strongly connected. However, this is a contradiction, because from
{v!l),v! 2)} belongs to a distinct equivalence class 11 l1
(1),v(2)} {vi2 i2
(for example).
This contradiction shows that
our original assumption is false, and that the reduced digraph is acyclic. D We can now state a procedure for renumbering the equivalence classes VI,...,V k in the manner described before Definition (50). Given a digraph (V,E), first construct its reduced digraph
(V,E).
Now, identify all vertices in
do not have pre~ecessors~ and ]abe] this set as the vertices in
V1
as
all vertices in
91
and all edges leaving vertices in
all edges of the form
WI,W2,...,Wnl
Vl "
(Vi,.)
with
9
that
Renumber
Next, remove from (V,E)
V i e ~i).
91 (i.e.,
The resulting
69
digraph
is a g a i n a c y c l i c .
Identify
t h a t do n o t h a v e p r e d e c e s s o r s , to see t h a t Renumber
until
V
Vi
WI,...W k
of s y m b o l s ,
lie a m o n g
i < j, t h e n
original
digraph
then
set
V (i)
as
It is e a s y to
.
To
now denote
i.e.,
{ V l , . . . , V i _ I } , for ~ E
equivalence
Vl "
i = 2,..,k
t h a t if
form
of
avoid a the v e r t i c e s
such that all predecess-
With reference
this m e a n s
classes)
t h a t all p r e d e c e s s o r s
i = 2,...,k
Vl,...,V k
Given
a digraph
Vl,...,V k
VI,...,V k under
(ii)
"
belong
.
v a 6 V.3
the d e f i n i t i o n
This means
to the and
i < j,
of E).
Thus
the f o l l o w i n g
Theorem vertex
(i.e.,
in s u c h a w a y
(this f o l l o w s
V2 must
,... W Repeat this nl+ 1 ' n2 V are e x h a u s t e d . A t this
in
let
it as
of v e r t i c e s
W
the v e r t i c e s
(Vj,Vi)
(V,E),
(Vb,V a) ~ E
56
as
in the p r o p e r o r d e r ,
that if
we have proved
92
and l a b e l
its p r e d e c e s s o r
{ W I , . . . , W i _ I } , for
numbered
ors of
then
all v e r t i c e s
as
lie a m o n g
proliferation of
in
we h a v e r e n u m b e r e d
Vl,...,V k Wi
V i 6 V2'
the v e r t i c e s
procedure stage,
if
its c o l l e c t i o n
if
are
strong
(V,E), o n e c a n p a r t i t i o n
the
in s u c h a w a y t h a t the e q u i v a l e n c e
classes
of
V
connectedness;
v a e V i , v b • Vj
, and
(vb,v a) • E , t h e n
i > j
't/
v2
FIGURE
4.3
v4
v
v7
%
v5
70
57
Example adjacency
58a
matrix
A
=
Consider
the digraph
of Figure
4.3,
whose
is
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
1
0
1
0
1
0
0
0
0
0 i
Using
either
58b
R
Using
Lemma
vertices
59
Algorithm
=
(48),
(41)
or
(42),
we can compute
1
1
1
1
1
0
i"
0
1
1
1
1
0
1
0
1
1
1
1
0
1
0
0
0
1
0
0
0
0
1
1
1
1
0
1
0
1
1
1
1
1
1
0
1
1
1
1
0
1
we can determine
that
the reduced
digraph
is a s
shown
v
digraph and
V4
V 3 = {v4} , V 4 = { v 6}
in F i g u r e
4.4
.
v4 FIGURE
V1
classes
½
v3 This
the equivalence
are
V 1 = {Vl} , V 2 = { v 2 , v 3 , V s , V 7 } , Hence
that
4.4
is a c y c l i e ,
as e x p e c t e d .
Further,
do not have
predecessors.
So we
the v e r t i c e s
let
W1 = V1 ,
of
71 W2 = V4 .
If we remove these vertices,
ing these vertices, Clearly
V2
we get the digraph
does not have a predecessor,
V3
W4 = V3
appropriate 60a
so we let
leav-
.
W3 = V2 .
4.5
Hence the e q u i v a l e n c e
order,
4.5
V2 FIGURE
Finally,
as well as the edges shown in Figure
classes,
numbered
in the
are
V 1 = {Vl}, V 2 = {v6} , V 3 = {v2,v3,v5,v7}, Note that this o r d e r i n g
is not unique;
V 4 = {v 4}
for instance,
we can also
take 60b
v I = {v6}, V 2 = {Vl} , V 3 = {v2,v3,v5,v7},
V 4 = {v 4}
We close
introducing
out this s u b s e c t i o n by formally
concepts of a strongly
connected
component
and an i n t e r c o n n e c t i n g
subgraph. 61
Definition denote
the e q u i v a l e n c e
way that connected (V i U Vj subgraph
classes of
(Vb,V a) ~ E
Then the digraph
63
Given a digraph whenever
(vi,
component , (Vi×Vj) (IS), for Example
(Vi×Vi)
(SCC), N E)
(V,E),
V
under
let
Vl,...,V k
S, ordered
Va E V i , V b E Vj, and n E)
for
in such a i < j
is called the i-th s t r o n g l y
i = l,...,k
is called the
.
The digraph
ij-th i n t e r - c o n n e c t i n g
1 ~ i ~ j ~ k . Consider
once again the digraph of Figure
Its strongly c o n n e c t e d c o m p o n e n t s 63a
SCCI
~
({Vl},
(Vl,Vl) }
63b
scc2
=
({v6},
~)
63c
SCC3
=
({V2,V3,V5,V7}, (v7,v2))
4.3
are
(v2,v 3) , (v3,v 5) , (v5,v 7) ,
.
72 63d
SCC4 where
@
=
({v4} , ~)
denotes
the empty set.
Its i n t e r c o n n e c t i n g
64a
ISl2 =
({Vl,V6} , @)
64b
IS13 =
({Vl,V2,V3,V5,V7},
64c
IS14 =
({Vl,V 4},
64d
IS23 =
({v6,v2,v3,v5,v7},
64e
IS24 =
({v6,v4},
64f
IS34 =
({v2,v3,v5,v7,v4},
subgraphs
are
The i m p o r t a n t SCC's
a proper
~) (v3,v4),
connected, E .
(v5,v4))
(V,E) ; that is, the v e r t e x E .
the union of all the
subgraph of
subset of
SCC's of
(V,E),
in that its edge set is a for the digraph of Figure
SCC's
has only five edges.
the p r o b l e m of analyzing
simpler
than analyzing
4.1.4.
D i r e c t e d Trees Definition
the original
to be a m a x i m a l
(V,E)
that contains
neither
cycles
nor self-loops.
graph
(V,E t)
creates
66
in
a directed
tree of
neither cycles
that the i n c l u s i o n
(V,E)
(V,E),
subgraph
of
is a sub-
nor self-loops, of any edge from
with E-E t
either a cycle or a self-loop.
Example (v5,v 7)
the
c o n n e c t e d digraph
(V,E)
property
4.3,
digraph.
Given a strongly
that contains
is
Thus,
tree of
In other words,
(V,E)
(V,E)
all of the SCC's is still
we define a d i r e c t e d
the a d d i t i o n a l
set of the Unless
For instance,
union of the four
65
(v6,v 3) , (v6,v 5) , (v6,v7))
V, and its edge set is a subset of
is strongly
general,
(Vl,V4))
point to note is that the union of all the
is a subgraph of
union is
(Vl,V2) , (Vl,V3))
If we remove
the edges
from the digraph of Figure
Alternatively,
we can remove
4.2
(Vl,V 2)
(v3,vl) , (v4,v I)
, we get a d i r e c t e d and
(v7,v 6)
and get
and tree.
73 another directed 67
Lemma (V,E s)
tree. Given a strongly c o n n e c t e d
be a subgraph of
self-loops. (V,E t)
Then
(V,E)
there is a s u p e r s e t
is a d i r e c t e d
tree of
This lemma states
prove
DECOMPOSITION
system
arrangement with those discuss and
(LSlS)
"below"
it.
whereby
in S e c t i o n
tree. to
5 . SUBSYSTEMS inter-
into a h i e r a r c h i c a l
each s u b s y s t e m
The advantages
b e g i n n i n g with
Consider
in a d i r e c t e d
we show how a given large-scale
in turn each of a l t e r n a t i v e
(2.2.18),
such that
This lemma is needed
can be d e c o m p o s e d
are d i s c u s s e d
let
cycles nor
that contains
INTO STRONGLY C O N N E C T E D
of subsystems,
decomposition
Es
can be imbedded
results of C h a p t e r
In this section, connected
of
that any subgraph
and is omitted.
the w e l l - p o s e d n e s s
4.2
Et
(V,E),
neither
(V,E).
neither cycles nor self-loops The proof is obvious
digraph
that contains
interacts
only
of carrying out such a 4.3.
In what follows,
system d e s c r i p t i o n s
we
(2.2.1)
(2.2.1).
a large-scale
interconnected
system d e s c r i b e d
by m la
ei = ui = j~l Hij yj
} i = l,...,m
ib
Yi = Gi ei n,
where
ui' ei' Yi
belong
to
L pe~
for some fixed n.
some p o s i t i v e
integer
Given the system as follows: (vj,v i)
then it means
m
Hij ~ 0 that
joint subsystems
vertices
as
Vl,...,v m ,
If the r e s u l t i n g
(i) actually
H.. 13
: L pe3 + L pe" l
constructed
is not connected,
a collection
that do n o t i n t e r a c t with each other.
case each c o n n e c t e d
component
can be a n a l y z e d
we can safely assume that the d i g r a p h
n.
and draw an edge
digraph
represents
and
n.
and
(i), we associate with it a digraph
Label
if
[i,-]
n.
: Lpei ÷ L pei '
ni ' Gi
p •
of disIn this
separately.
associated with
Hence
(i) (referred
74 to h e r e a f t e r
as the s y s t e m
digraph)
If the s y s t e m d i g r a p h there
is n o t h i n g
not strongly vertices
as i n d i c a t e d
a renumbering into
further
connected,
to be done.
in S e c t i o n
l+l,...,Vn.}, --
no e d g e Now,
(Va,V b)
(uj, vj e Vi)
zi =
(yj, vj 6 Vi)
di =
(ej, vj 6 Vi)
(Gj,
t h i s is done, set
is the
w e have
V = { V l , . . . , v m}
, in s u c h a w a y
that
v a 6 V i , v b 6 Vj
and
i > j.
(i) in a c o r r e s p o n d i n g
z. = F .
xi,
d.
l
=
R.. = 0 w h e n e v e r 13 into SCC's.
the s y s t e m e q u a t i o n s
i < j.
(i) can
as i-i ~ R.. z. j=l 13 3
1
z i E L p e1 %).
Rij
that
d.
%).
and
v s 6 Vj)
we have
R.. z . 11 1
I
di,
i = l,...,k
definitions,
expressed
7b
(Uni_l +I' .... Un')l
of the d e c o m p o s i t i o n
the a b o v e
d. = x . i i
I
=
j E Vi)
of t h e r e n u m b e r i n g ,
is t h e o b j e c t i v e
7a
8a
i = l,...,k
(Hrs , v r e Vi,
be e q u i v a l e n t l y
L peI '
Once
the s y s t e m e q u a t i o n s
xi =
With
where
4.1.
and number
then
Define
Ri j = Because
If the s y s t e m d i g r a p h the S C C ' s
of the v e r t e x
exists whenever
F i = Diag
This
connected,
1
we partition
manner.
is a l s o s t r o n g l y
then identify
and partitioning
V i = {Vni
is c o n n e c t e d .
: Lp
x.
1
-
for s o m e p o s i t i v e
integer
%).
÷
R..
ll
pe
z.
1
.
L e t us d e f i n e
~i' Fi
: L pel ÷
75 8b
z. = F . d. 1 1 1 as
(Si), or the i-th isolated
equivalently isolated that
subsystems
(S i)
isolated
(S I) thru
system interacts
are a r r a n g e d
(Sj)
In this connection, system,
approach on a given system,
subscript.
has not lost anything.
On the other hand,
connected,
The and the
it is i m p o r t a n t is in general
if one tries this
then the w o r s t that can happen
is strongly
is not strongly
each
all the i n t e r c o n n e c t i o n
Therefore,
that the system digraph
k
Thus the
whereby
subsystems
because
have been omitted.
(or
property
system can be deduced by studying
alone.
than the original Rij
(i)
of the
is that the w e l l - p o s e d n e s s
to note that the union of the isolated simpler
i > j
only with those having a higher
subsystems
operators
if
in a hierarchy,
of such an a r r a n g e m e n t
stability of the overall isolated
Then the LSIS
(Sk) but with the a d d i t i o n a l
does not interact with
subsystems
advantage
subsystem.
(7)) can be viewed as an i n t e r c o n n e c t i o n
connected,
is
in w h i c h case one
if the system digraph
then c o n s i d e r a b l e
savings
in complex-
ity can result.
Example (i.e.,
Consider
5 subsystems),
a system of the form
and the following
(i), w h e r e
interconnection
m = 5
operators
are nonzero: H21, H25 , H32 , H42 , H43 , H51 , H53 . The r e m a i n i n g H.. , i ~ j are assumed to be zero. (Note that we need not a3 bother about operators of the form Hii , b e c a u s e they r e p r e s e n t self-loops
in the system digraph,
the d e t e r m i n a t i o n Figure
v1
4.6
of the SCC's).
and therefore do not enter into The system digraph
is shown in
One can easily verify that there are three SCC's,
v2
~
FIGURE
_
4.6
v
5
76 and
that their vertex
V 1 = {vi} , V 2 = {v2,
sets
(arranged
in the p r o p e r
v3, v5} , V 3 = {v4}.
Accordingly,
i0
x I = Ul, x 2 =
[u 2, u 3, u5]'
, x3 = u4
ii
d I = el,
d2 =
[e2, e3,
e5]'
, d3 = e4
12
Zl = YI'
z2 =
[Y2' Y3'
Y5 ] ' ' z3 = Y4
El 0 G2
13
F1 = G1 , F2
=
G3 0
14a
Rll = Hll
14b
R22
, RI2
= H12
I~ 22
=
LH52
The new system description
It is c l e a r
t h a t is u s e d
is w h e t h e r
can cause unnecessarily Example
(9) above,
ily c h e c k
if
o t h e r hand,
Chapter
H41
to s a f e l y
H23 H33
0 51 H3
0
H55j
' ~3=
[~4H3~ 0] ~ = •
"'33
(7).
decomposition
is a
t h a t the o n l y i n f o r m a t i o n
H.. is z e r o or n o n z e r o . S o m e t i m e s • this z3 conservative results. F o r i n s t a n c e , in were
would
to b e n o n z e r o ,
t h e n o n e c a n eas-
be s t r o n g l y
connected,
r e s u l t b y this p r o c e d u r e .
be very
ignore
it.
"small" Such
a n d it m a y
issues
O n the
therefore
are discussed
in
6.
Now we consider
a system described
equations m
15
F3 = G4
, RI3 = 0
in the s e n s e
H41
might
we d e f i n e
G
t h a t the s y s t e m d i g r a p h w o u l d
a n d no s i m p l i f i c a t i o n possible
,
t h a t the p r o p o s e d
decomposition,
are
ii
is n o w g i v e n b y
"structural"
order)
e. = u. - [ S.. e. , 1 ~ j=l 13 3
i
~
l•°,,•m
b y the s e t of
be
77 n,
where
ei,
u i 6 L p ez
for some f i x e d n,
integer
ni ,
associated manner
with
entirely
Vl,...,v m done,
and
i > j.
m
in a
vertices
Sij M 0 .
components
as
(Va,V b)
(uj
, vj • V i)
17
di =
(ej
, vj 6 Vi)
18
z.. z] =
(Sk£ e£
19
Rij =
(Sk£
Once
of the s y s t e m
V i = {Vni_l+l ,...,vni} exists whenever
to n o t e
m.m.xl
v a • V.l ,
z
]
that the
vector,
m.m. c o m p o n e n t s of z.. z 3 z] w h e r e a s the m.m. components z 3 matrix. With these definit-
R.. a r e a r r a n g e d in an m . x m 0 z3 z 3 ions, the s y s t e m e q u a t i o n s c a n be r e w r i t t e n
as
i d. = x. - ~ R.. d. l 1 j=l z3 3
20
because,
by construction,
Rij = 0
whenever
i < j.
We refer
to t h e s y s t e m
21
d. = x. - R.. 1
1
ll
as the i - t h i s o l a t e d
d.
1
subsystem.
We can further modify From
(18) w e see t h a t
zij = Rij
is
, v k • V i , v~ • V~)3
of
22
as
this
, Vk 6 V i , v£ • Vj)
it is i m p o r t a n t in an
s[stem digraph is c o n s t r u c t e d
We then define
xi =
are a r r a n g e d
if
connected
16
where
and some positive
We label
(vj,v i)
the v e r t i c e s
t h a t no e d g e
, and
(15)
to b e f o r e .
the s t r o n g l y
and r e n u m b e r
in s u c h a w a y v b ~ Vj
The
the s e t of e q u a t i o n s analogous
[1,~]
n.
: L p e3 ÷ L pez
Sij
a n d d r a w an e d g e
we find
digraph,
p •
dj
the s y s t e m d e s c r i p t i o n
(21).
78
where 23
Rij = Diag [Columns of
Rij]
Moreover, we have 24
Rij = Kij Rij where the operator
25
K.. ~3
is represented by the
m.×m.m, l • 3
matrix
Kij = [Im. I --IIm.] 1 l and
Im. denotes the identity matrix of dimension mi×m i. If 1 (I+Rii)-l exists for all i , we can express the system equations (20) solely in terms of the z. 's , as follows: 13
26
zij = Rij dj = Rij (I+Rjj)-I "
) -i
-- Rij(I+Rjj
=
Rij (I+Rjj)-I
(xj - k=l j~l Rjk dk)
j-i
(xj (xj
~ Kjk Rjk d k) k=l
j[l - k=l Kjk zjk)
Hence the final form of the system equations is 27
zij = Rij(I+Rjj) -I (xj - 311 Kjk Zjk) k=l where it is important to observe that the operator represented by a constant matrix.
28
Example
Kik
is
Consider a system described by (15), where
the following operators are nonzero: Sll, S13, $21, S22, $32, $41, $43, $46, $52 , $53, $54 , $67 , $75 , S77 The associated system digraph is shown in Figure 4.7. (Note that the self-loops corresponding to the operators SII , $22 , and $77 are not shown in Figure 4.7 because self-loops do not figure in the determination of the strongly connected subsystems).
79
VI
V7
FIGURE
From Figure of v e r t i c e s ,
4.7, w e see t h a t t h e e q u i v a l e n c e
dI
=
order,
Accordingly,
e2
,
d2
classes
[e4]
in the a p p r o p r i a t e
V 2 = {v 4, v5, v6, VT).
29
4.7
=
are
V 1 = {v l, v2,
v3},
we define
e5
e3
e6 e7
ii° !i
30
R I I --
[~
21
-0
R22 =
a n d of c o u r s e
s41° s4
$22
'
R21 =
$32
0
$46
0
S54
0
0
0
0
0
0
S6.
0
$75
0
$77
R I 2 = 0.
Moreover,
we have
0
$52
0 0
0 0
5
80
~ii
el ~
$21
e1
IS31 e 1 iSl2 e 2 31
Zll = !$22 e 2 i
$32 e 2 IS13 e 3 iS23 e 3
~ and
33 e3
z12 , z21, z22
are similarly defined.
In (31), we display
e.g. S31 e I instead of 0 (note that~ S31 = 0) to make the pattern clearer. Next, the operator Rll is defined by
32
Rll
while
R21 , R22
KII RII,
33
=
Sll
0
0
S21
0
0
s31
0
0
0
S12
0
0
$22
0
0
S32
0
0
0
S13
0
0
S23
0
0
S33
are similarly defined.
Note that
i
0
0
1
0
0
1
0
)I
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
where
Kn
=
I
;J
RII =
81 The m a t r i c e s
~21
' ~22
are o b t a i n e d
The operators column subsystems
Rij
(I+Rjj) -I
corresponding
As shown in the next section, in the d e c o m p o s i t i o n 4.3
system
(LSIS)
only the isolated
AND S T A B I L I T Y
subsystems,
system can be a s c e r t a i n e d The advantages
and
by studying
of such results
because
the union of all isolated
is in general
simpler
than the original
we relate the p r o p e r t i e s
inter-
into an i n t e r c o n n e c t -
the w e l l - p o s e d n e s s
apparent,
TO b e g i n with,
(27).
play a key role
we show that once a large-scale
subsystems.
the
systems.
has been d e c o m p o s e d
stability of the overall
systems
to the system d e s c r i p t i o n
RESULTS ON W E L L - P O S E D N E S S
ion of s t r o n g l y c o n n e c t e d
are readily
are said to r e p r e s e n t
column subsystems
of l a r g e - s c a l e
In this section, connected
similarly.
sub-
system.
of the system
i-i la
d i = x i - Rii z i -
j=l
R.. z. ~3 3 I
ib
(S)
i = l,...,k
zi = F i d i to those of the systems
2a
d i = x i - Rii z.1
2b
z. = F . d. l l l Theorem i > j , (i) finite
Consider
the operator
T , there exists
incremental
(Si)
gain of
the system
(S).
Suppose
that for all
R.. is causal, and (ii) for each ~3 a finite constant kij T such that the
PT Rij
is less than or equal to
kij T .
Under these conditions,
the system
(S) is w e l l - p o s e d
if each of the isolated
subsystems
(S i) is well-posed.
Proof posed. set
"if"
Suppose
each of the systems
We show first of all that,
x = (Xl,...,x k) E L n pe
'
corresponding
there exist a unique
if and only
(S i) is w e l l -
to each input d e Ln pe
and
82 n z £ Lpe
a unique actually tion.
a
such that
collection
First of all,
of
(I) is satisfied.
k
for
equations,
j = i,
Since
we prove
(i) is
this by induct-
(i) is
zI = F I d I which is the same as the isolated and u n i q u e n e s s
subsystem
follows by the h y p o t h e s i s
are well-posed.
N o w suppose
exactly one solution
for
that for
(S l)
; hence e x i s t e ~ e
that all subsystems
j = l,...,i-l,
dl,...,d j
and
(Si)
(i) has
Zl,...,z j .
For
j =i,
we have i-i 5a
di = xl•
5b
zi = F i d i
-
Ril.
z I. -
j~
=i
R..
z.
13
3
:
x'
-
i
R..
ix
z.
i
where i-i x! A x i - ~ R.. z. i = j=l 13 I
N o w note that
x~ l
is u n i q u e l y determined,
hypothesis,
and that
x! E L i i pe
determines
d i 6 Lpei
and
the inductive exhibits and
z
process
existence dj,
zj,
and u n i q u e n e s s
X l , . . . , x i , because on
Rij
x I! and hence on
ity of way.
(6) that
(S i)
PT d
and
PT z
Hence the system
well-posed
(5)
This shows that
Hence the system
of solutions.
(i)
To prove that
depend causally on x!
depends
d
x l,...,x i . as functions
di,
Xl,...,xi_ 1 .
c a u s a l l y on
is causal w h e n e v e r
i > j zi
Next,
depend causally
The global Lipschitz of
PT u
continu-
is p r o v e d in the same
(1) is well-posed.
"only if"
Suppose is not.
not well-posed.
uniquely.
is well-posed,
but
(S i)
is well-posed,
x , we again proceed by induction.
j = l,...,i-I
Then it is clear form
since the system
z i E Lpei
(S i)
can be continued.
depend c a u s a l l y on
Suppose
Since
by the inductive
that systems
(Sl),...(Si_l)
We show that the system
are
(i) is als0
83
To be specific, because
suppose that
it v i o l a t e s c o n d i t i o n
solution.
If
(S i)
is not w e l l - p o s e d
(WI) of D e f i n i t i o n
(2.2.9),
i.e.,
xi0 • LpeI , (2) does not have a
that for some s p e c i f i c i n p u t unique
(S i)
v i o l a t e s either
a r g u m e n t s b e l o w are e a s i l y modified.
(W2) or
(W3), the
x. = 0 for j = i,..., 3 i-l; by the "if" part of the proof above and the a s s u m p t i o n that (SI),...,(Si_ I)
dl,...,di_ 1
are w e l l - p o s e d ,
and
i = l,...,i-i
.
Zl,...,zi_ 1
W i t h this input,
such that
(I) is s a t i s f i e d
for
i-I [ Rij z. j=l 3
the i-th e q u a t i o n in
8a
d i = xi0 +
8b
z. = F . d. 1 1 1
(i) b e c o m e s
i-I i-i [ R.. z. - R.. z. j=l z3 3 ii 1 j=l Rij
(8) does not have a u n i q u e solution.
(i) is not well-posed.
Remarks that,
we see that there e x i s t u n i q u e
N o w let
x i = xi0 +
By assumption,
Let
(i)
zj = xi0 - R i i z i
Hence
system
D
It is clear from
the proof of T h e o r e m
of s o l u t i o n s to
(I), we can state the f o l l o w i n g result:
"(S)
e x h i b i t s e x i s t e n c e and u n i q u e n e s s of s o l u t i o n s c o r r e s p o n d i n g each
(3)
if we are o n l y i n t e r e s t e d in the e x i s t e n c e and u n i q u e n e s s
x 6 Ln pe
if and o n l y if e a c h s y s t e m
(S i)
to
has e x a c t l y one
n. s o l u t i o n c o r r e s p o n d i n g to e a c h x i e L i ,, In o t h e r words, we pe do not n e e d to m a k e any a s s u m p t i o n s about Rij if we are o n l y i n t e r e s t e d in e x i s t e n c e and u n i q u e n e s s of solutions.
(ii) that
Rij
Definition
The h y p o t h e s i s on
Rij
in T h e o r e m
is a w e a k l y L i p s c h i t z o p e r a t o r w h e n e v e r
(3) imply i > j
(see
(5.1.1)).
Theorem
(i) s o l u t i o n for
W i t h r e s p e c t to the s y s t e m
For each input set d, z
in
Ln pe
x e Ln pe
'
(S), suppose t h a t
(i) has a u n i q u e
84
(ii) Under
Y(Ris)
< ~
Vi > j
these conditions,
the system
if each of the i s o l a t e d
subsystems
Proof (I) to denote
"if"
To avoid confusion,
quantities
associated
Suppose
each of the subsystems
ki, b i
are finite
constants
xi' d(I)i , zi(I)
arbitrary
input to the system
from
subsystems.
and suppose
< -
ki!Ixill p + b. 1
satisfy
(2) .
(S).
Now let
Then
x e L np
d I = d I)
be an =
Zl
(I) Zl
(10), we get
Ii
lldlIl p, For
is Lp-Stable,
such that
whenever
Hence
(3.1.1)).
we use the superscript
with the isolated
(S i)
I d(I) l!p,!Iz(I) I I i i ''p
i0
(recall Definition
(S) is L -stable if and only P (S i) is Lp-Stable.
]]ZlI] p
i = 2, we have
from
~
(i) that
l!d21rp, T1~211p
12
k I llxl!l p + b I
_< ~2(Irx2- R21 h(I) l!p) ÷ b 2 k2CI!x2[rp ÷ k I lIXlllp + b 1) + b 2
The proof by induction "only if" theme exist
finite
is obvious. Suppose
com_s%ants
the system
(1) is Lp-stable.
k
such that
b
11dIIp , IIZllp <_k ! X I 1 p ÷ b
13 whenever
x, d, z
satisfy
(i)
and let d,z
denote
Then
(13)
that
implies
(1).
(2), and let
i
[ R.13 ~ 3
j=l
x =
x = 0 of
in
(i).
Now let x i 6 Lp
1
(0,0,0,...,xi,0,...,0)
input to the system
i-I
+
let
solutions
IId! Ip ~ b , IlZ!Ip ~ b.
be the "corresponding"
~l =x
In particular,
the corresponding
be any input to the system • Ln P 14
and
Then
(i), where
85
By the h i e r a c h i c a l
arrangement
that. with the input dj , zj = zj
for
x
of the subsystems
to the system
j = i, .... i-i
{{diI{p ~ l{d!{p ~
i5
.
in
(i), we see
(i), we still have
Moreover,
by
II~Ilp+ b = k
dj =
(13) we have
l{~iI{p+ b
k {Ixi{l p + b 0 where i-i b0 ~ b + k I! [ R.. z'I! j =i lJ 3 P
16
S i m i l a r l y we can show that
I!~iIIp~ k !Ixil{p+ bo
17
Now, with the input
xo
to the system
(2), d! I)
and
z! I)
l
1
l
satisfy 18a
d (I) xi z (1) i = - Rii i
18b
z.(1) = F. 1
However,
x.
1
d i(I) = di
1
and
z!l I) = z.z
because
i-i 19a
dm. . . .~i
19b
z. = F . 1
Hence from
1
(15) and
~ j=l
Ri" 3
z.
3
-
R..
J.m
z.
m
=
x.
1
-
R..
mz
z.
J.
d. 1
(16) we get
I d!Z){Ipl ' {!z(~)I{pi~k IIxiTIp + b
20 which
shows that the s y s k e m Theorem
(S i)
is L2=stabl~. n
(3) states that if each i n t e r c o n n e c t i o n o p e r a t o r
Rij is w e a k l y Lipschitz,
then the LSIS
and only if each of the subsystems
(S i)
(S) is w e l l - p o s e d is well-posed.
if
Theorem
86
(9) shows that, gain,
if each interconnection
and the overall system
system
(S i)
exhibits
then the overall subsystem
(S)
existence
system
operator R.. has finite 13 (or, equivalently, each sub-
(S)
(S)
and uniqueness
is Lp-stable.
one only needs to study t h e _subsystems strongly connected
components,
ness and stability. we assume withQut have strongly
of solutions,
is L -stable if and only if each P In both cases, it is clear that corresponding
to the
.in order to determine well-posed-
Therefore,
throughout
the rest of this book,
loss of generality that all LSIS's we study
connected s y s t e m d i g r a p h s .
It is clear that,
the union of the isolated subsystems
is no more complex
original
all the isolated
LSIS,
individually
the task of analyzing
is no more complex,
than analyzinq
and is in general
since
than the
subsystems
less complex,
the originaL_LSIS.
We now turn to the study of systems of the form 21
zij = Rij The results
(I+Rjj) -I (uj - 311 Kjk Zjk) k=l
in this case are far nearer than Theorems
(3) and
(9) . Theorem Consider
22
system is well-posed operators
by (21).
This
defined by Rij (I+Rjj)-I
23
is well-posed;
i.e., n. Lpe3
operator mapping "
PT Rij (I+Rjj
)-i
Proof
(i) Rij (I+Rjj)-I into
"if"
Then,
a well-posed
operator
is a well-defined
causal
n.n. L pei 3 ' (ii) the incremental
is finite for all
well-posed.
Rjj(I+Rjj) -I 24
a system described
if and only if each of the column subsystem
gain of
T .
Suppose each of the column subsystem
in particular, for all
is a well-posed
we have that
j , and hence operator
for all
Rjj(I+Rjj) -I
is is
Kjj Rjj(I+Rjj) -I = j
Since
I = (I+Rjj)(I+Rjj) -I = (I+Rjj) -I + Rjj(I+Rjj) -I
87 we have
(I+Rjj) -I = I - Rjj (I+Rjj) -I
25
w h i c h shows that
(I+Rjj) -I
of the proof e s s e n t i a l l y "only if"
Parallel
In c o m p a r i n g Theorem
is w e l l - p o s e d
follows
(3) and
Theorem
26
insures that theorem
is played by
by constant m a t r i c e s Similar
Kij
, and therefore
remarks apply to the
Consider
a system d e s c r i b e d
is L p - s t a b l e
(I+Rjj) -I
(21), and
subsystem o p e r a t o r s
for all
i,j
Note that the a s s u m p t i o n
is now obvious
by
Then this system is L p - s t a b l e
of w e l l - p o s e d n e s s
exists as a causal operator. from T h e o r e m
The
(9). o
NOTES AND REFERENCES Much of the b a c k g r o u n d follows the d e v e l o p m e n t decomposition Callier,
(21), ,
pmoved next.
if and only if each of the column
Proof
on the
In the system d e s c r i p t i o n
suppose this system is well-posed. Rij(I+Rj.j)-I -
(22), we note that in
operators
have a host of nice properties. stability results,
The rest
I
(3). []
to impose some h y p o t h e s e s
interconnection operators R.. 13 the role of the i n t e r c o n n e c t i o n w h i c h can be r e p r e s e n t e d
j
(3).
to that of T h e c m e m
Theorem
(3), we are o b l i g e d
for all
that of Theorem
material
of Baase [Baa.
into strongly
connected
Chan and Desoer [Cal.
1,2].
i|.
on d i r e c t e d
graphs
The results
components
on
are due to
CHAPTER 5: WELL-POSEDNESS OF LARGE-SCALE INTERCONNECTED SYSTEMS In this chapter, ditions tems.
The
conditions
subsystem
operators
by no m e a n s
into
necessary;
if these
to d e t e r m i n e
two types.
however,
"compensation"
of the s y s t e m
in order
ately
from c o n t i n u o u s - t i m e
In this
chapter,
though
the w e l l - p o s e d n e s s
cases,
the d e t a i l s
trast,
we do not e x p l i c i t l y
Chapters
needs
criteria
be satisfied.
Therefore
systems
is that,
time
AND
those
STRICTLY
the same
in both
In
systems
con-
in
results
for
for c o n t i n u o u s -
CAUSAL
OPERATORS
given here depend
or s m o o t h i n g
separ-
even
different.
and i n s t a b i l i t y
results
of a w e a k l y L i p s c h i t z operator.
The reason
parallel
SMOOTHING
The w e l l - p o s e d n e s s
ly c a u s a l
in various
conditions
discrete
essentially
concepts
to be i n t r o d u c e d
mention
systems
LIPSCHITZ,
it is easy
somewhat
discrete-time
WEAKLY
to verify. More-
are e s s e n t i a l l y
the s t a b i l i t y
5.1
easy
are
satisfied,
are
because
systems.
conditions
these
we treat d i s c r e t e - t i m e
of the proof
in that
As such,
t h a t these
6-9,
time
in nature,
sys-
the various
are not
systems.
s u f f i c i e n t con-
on c l a s s i f y i n g
they are v e r y
conditions
parts
some
interconnected
are s t r u c t u r a l
and d e p e n d
sufficient
what
and prove
of l a r g e - s c a l e
given here
they are g r a p h - t h e o r e t i c ,
over,
we state
for the w e l l - p o s e d n e s s
operator,
we b e g i n by d e f i n i n g
on the
and a stric~
these
concepts
precisely.
Definition
1 tegers.
We
KT
incremental
exists
2
a
[i,=]
such that
and
R:L
and if for e v e r y np(PTR)
gain of an o p e r a t o r
that an o p e r a t o r Lipschitz,
p E
say that an o p e r a t o r
if it is causal, constant
Let
R : L ~ e ÷ Lmpe
in addition,
~ = ~(~,T)
in
np[Pt+~(RPt+6-RPt)]
n,m
e ÷ L mpe
~ kT
, where
such
<_ u
,
exists
np(.)
(see D e f i n i t i o n is s m o o t h i n ~
in-
Lipschitz a finite
denotes
(3.1.1)).
the
We say
if it is w e a k l y
T > 0
and
that
Vt E
be g i v e n
is w e a k l y
T > 0 , there
for every (0,T)
and let
[0,T-~]
~ > 0 , there
89 Remarks
i) Vt •
[0,T].
as f o l l o w s :
Hence
pendence made
If
R
~ k T Vt e is c a u s a l ,
= 0 Y s e [0,t].
s ~
of c a u s a l i t y ,
we can also define
np(PtR)
2)
ever
because
a weakly
np(PtR)
~ ~ p ( P T R)
Lipschitz operator
R : L n + Lm is w e a k l y L i p s c h i t z if it pe pe a n d for e v e r y T > 0 there exists a finite constant
such that
RPt)x](s)
that,
an o p e r a t o r
is c a u s a l , kT
Note
[t,t+~]. of
(Rx)
Thus, (s),
small
It,t+6]
small
Suppose
For every
R
on
is s m o o t h i n g x(s),
incremental
(see E x a m p l e s ( 9 )
Example ing c o n d i t i o n :
it is e a s y to see t h a t
[(RPt+ ~ -
{[Pt+~(RPt+~-RPt)]x}(s)
an o p e r a t o r
s •
to h a v e a r b i t r a r i l y
sufficiently
Hence
[0,T].
and
s •
= 0
when-
if the de-
[t,t+~]
gain by making
can be 6
(21) b e l o w .
# : R+xR n ÷ R n
satisfies
T > 0, t h e r e e x i s t s
the f o l l o w -
a constant
kT
such
that
l]¢(t,x)
Then
the m e m o r y l e s s
(RlX) (t)
is w e a k l y y(.)
e Ln pe
Lipschitz.
- ¢(t,y) I I _< k T
operator
R1
I Ix-yl I, V x , y E R n , Y t E [0,T]
n ÷ L pe n : L pe
defined
by
= ¢ (t,x(t)) T O see this,
observe
that,
whenever
, we have
IIRlX-h YlITp ~ kT llx-ylITp , This
shows that
n p ( P T R I) ~ k T
Example R2 : Ln pe
Ln pe
Let
defined
f
, VT.
8 [ 0.
Then
the d e l a y o p e r a t o r
by
0 ,
t < 8
x(t-%)
t > 8
(R2x) (t)
is w e a k l y
Lipschitz,
because
np(PTR)
~ 1
%r9 .
x(.),
90
Example Then
the o p e r a t o r
Suppose
F : R+×R+ + R m x n
R3 : L n ÷ Lm pe pe
is continuous.
defined by
t i0
(R3x) (t) =
F(t,T)
x(T)
dT
0 is smoothing.
To see this,
observe
that
IS ii
I F(S,T) t
( [Pt+~ (R3Pt+6-R3Pt) Ix} (S) =
0
Hence,
whenever
12
t 6
[0, T-6],
dr, s 6
X(T)
[t,t+~]
otherwise
we have
I IPt+~ (R3Pt+ ~-R3P t) x] ]p
II
= { t Now,
following
(3.1.15), 13
F(S,T)
x(~)
dT I Ip ds}i/P
t
the same r e a s o n i n g
as in the proof of Lemma
we get
] IPt+6 (R3Pt+~-R3Pt) x] I p ~
C1l/p
c~/q I]xll
p
where t+~
14a
cI =
sup T 6
[t,t+6]
IT
l]F(s,~)ll
ds
rs 14b
c~ =
sup s E
[t,t+~]
j . II F(s,T ) II dr t
If we now define
15
--
sup 0 < s, • < T
I fF(s,~) f
it is easy to see that
c I ~ MT~
in (13), and o b s e r v i n g
that since
its i n c r e m e n t a l
, c® ~ MT~.
Substituting
Pt+6(R3Pt+~-R3Pt)
gain is the same as its gain,
gives
this
is linear,
gl
16 If we let
17
~p[Pt+~(R3Pt+~-R3Pt)]
<_ ~
t = 0
(16), we get
and
~ = T
[0,T-6]
np(PTR 3) ~ T Now,
(17)
that
R3
shows that satisfies
R3
is weakly
(2).
Hence
It is immediate are both weakly Lipschitz if HG. 18
in
, Vt E
G
and
H
Finally,
Lipschitz,
R3
from Definition (smoothing),
H
be specified. finite,
Clearly
GH
are w e a k l y Lipschitz.
satisfies
shows
(1) that if
G
and Also,
then so are
GH
and
is smoothing
and that
(2).
H
So it only remains suppose
is weakly
np(PtH)
! nT
is smoothing.
is weakly Lipschitz,
Accordingly,
Since
and also
GH
T > 0
Lipschitz,
Vt 6
since both to show that
and let np(PTH)
[0,T].
Now pick
~ > 0
~ nT 6 E
is (0,T)
such that 19
np[Pt+ ~ (GPt+~-GP t) ] < (~/n T) , Vt E This
is always possible
repeated 20
H
we have
Proof and
(16)
then so are G ± H.
are both weakly Lipschitz,
Lemma Suppose G : L n ÷ Lm L£ Ln pe pe H : ÷ is weakly Lipschitz. Then pe pe
GH
while
is smoothing.
because
use of causality,
G
[0,T-~]
is smoothing.
we get
~p [Pt+~ (GHPt+~ -GHPt) ] = ~p[Pt+~ (GPt+~HPt+6-GPtHPt)
]
= ~p[Pt+6 (GPt+~HPt+$-GPtHPt+6)
]
= np[{Pt+ ~ (GPt+6-GP t) }. Pt+6HPt+6] <_ np[Pt+ ~(GPt+d-GP t) ] • np(Pt+6 H) <-
n% " nT = ~ '
Yt 6
[0,T-6]
Now,
making
G
92
where
in the t h i r d
P t = Pt
21
" Pt+~
step we also use Thus
GH
the f a c t s
is s m o o t h i n g .
that
Pt+6
2 = Pt+~ '
D
Lm be s m o o t h i n g , a n d let G : Ln pe pe b e a d e l a y of the f o r m (8). Then HG is s m o o t h i n g .
22
Lemma
Let
Proof
Clearly
HG
Pt+$(HGPt+~-HGPt) Now,
23
is w e a k l y
Lipschitz.
Also,
= Pt+~_s(GPt+~-GPt).
by causality,
n p [ P t + ~ _ 8 ( G P k + ~ - G P t) ] < n p [ P t + ~ ( G P t + ~ - G P t) ], Y t 6 Since
G
is s m o o t h i n g •
can be made smoothing.
the q u a n t i t y
arbitrarily
on the r i g h t
small by making
6
[0,T-~]
side of
small.
(23)
Hence
HG
is
[]
Example
24 #
H
satisfies
(4)
,
F
Suppose and t h a t
p 6
: R + x R + + R n x n is c o n t i n u o u s , that n ÷ Ln de[I,~). Then R 4 : L pe pe
fined by
25
i
(R4x) (t) =
t F(t,T)
~ (x(T))
dT
0
This
is s m o o t h i n g . Lemma
follows
In s t u d y i n g operator
operator,
Pi G = Pi GPi Pi+l G Vi[
discrete-time
is n a t u r a l l y
(9) and
G : Sn ÷ Sn
V i > 0, and is s t r i c t l y
u
and
we have
v
(Gu) (1)
, then
1
P..~e± G u . = P i + l if
states
are t w o i n p u t s
P . u = P. v 1
operator
the c o n c e p t
of a
b y t h a t of a s t r i c t l y
causal
is c a u s a l
if
if
Pi+l GPi =
0.
--
if
systems,
replaced
An operator
The above definition whenever i > 0
(3) a n d
de£inecL.next.
Definition
26
causal
from Examples
(18).
smoothing causal
readily
that
in
Sn
P. G u
depends
if,
t h a t for some G is s t r i c t l y
1
It is c l e a r o n l y on
is c a u s a l
= P. Gv;
1
Gv.
G such
that
u (0) , u (l)
G ,---
is a c a u s a l ,u (i)
•
93
whereas
G
is a strictly
u (0),...,u (i-l)
causal
if
In particular,
(Gu) (0) is a c t u a l l y In the
independent
(Gu) (i)
if
G
case of c o n t i n u o u s - t i m e
systems,
6
~
arbitrarily
can only assume
define a smoothing o p e r a t o r tion, we go back ator
R4
ul(t)
= u2(t)
of
we still have "smoothing if
ul(J)
Let
for
t e
Ul(.),
u2(.)
= u2(J)
for
in d i s c r e t e - t i m e
To get a suitable definicase,
and study the oper-
be two inputs
j = 0, .... i
such that In this case,
(GUl) (J) =
in the d i s c r e t e - t i m e but
(Gu2) (J)
u l ( i + ~ ~ u2(i+l) for
case, , we
j = 0,...,i+i.
This
the c o n c e p t of strict causality.
stability
are d i s c u s s e d
The following Lemma
27 H : Sn + Sn
let
is a
so that one c a n n o t
Ul(T +) ~ u2(T+).
Note that strict c a u s a l i t y
28
However,
in this way.
[0,T], but
G
(R4Ul) (T +) = (R4u2) (T+), and hence that name
should still have
feedback
systems,
integer values,
operator". Correspondingly,
is e x a c t l y
then
can be made a r b i t r a r i l y
small.
to the c o n t i n u o u s - t i m e
(25).
only on causal,
of u.
smoothing o p r e a t o r i f np[Pt+~(GPt+~-GPt)] small by m a k i n g
depends
is strictly
in
and its implications on
[Sae. i].
lemmas are easy to pro~e.
Suppose
G
:
Sn
is s t r i c t l y causal.
+
Sn
Then
is causal and HG
is strictly
causal.
Lemma Suppose G : S n ÷ S n is s t r i c t l y causal and S n S n Di : ÷ be a delay of i units, i ~ 0. Then Di G
is strictly causal. Lemma
29
Suppose
i > i, be a d e l a y of
i
G : Sn ÷ Sn units
Note that Lemma
Then
is causal, D.G
and
is strictly
(29) does not have continuous
Di , causal time
counterpart. TO m o t i v a t e
the use of the p r e c e d i n g
analyzing well-posedness, omitted, presented
we p r e s e n t two lemmas.
as these lemmas are a c t u a l l y subsequently
(Lemma
concepts
in
The proofs
are
special cases of results
(30) is a special case of L e m m a
94
(5.3.1) w h i l e L e m m a Theorem
(31) can be p r o v e d by a d a p t i n g the p r o o f of
(5.4.1)).
30
Lemma (I+G) -I
Suppose
n ÷ L pe n G : L pe
is smoothing.
Then
exists and is w e a k l y L i p s c h i t z .
Lemma
31
(I+G) -i exists
Suppose
G : Sn + Sn
is s t r i c t l y causal.
Then
(as an o p e r a t o r ) .
5.2
S I N G L E - L O O P SYSTEMS
In this section• we state two simple s u f f i c i e n t cond i t i o n s for the w e l l - p o s e d n e s s of f e e d b a c k systems. Theorem
Consider a single-loop
feedback system
d e s c r i b e d by
2a
el = Ul - Y2
2b
e2 = u2 + Yl
2c
Yl = G1 el
2d
Y2 = G2 e2 where p 6
Ul' u2' el
[1,~]
into itself. if b o t h
•
e2' YI' Y2
and some p o s i t i v e
G1
all b e l o n g to integer
U n d e r these conditions, and
G2
Ln pe
n, and
the s y s t e m
are w e a k l y L i p s c h i t z ,
for some fixed
GI, G 2
map L pe n (2) is well-posed
and at least one of
them is smoothing. Theorem d e s c r i b e d by Sn
(2), w h e r e
Ul' u2' el' e2' YI' Y2
for some p o s i t i v e integer
itself. both
Consider a single-loop feedback system
G1
causal.
n, and
U n d e r these conditions• and G 2
are causal,
GI, G 2
the s y s t e m
all b e l o n g map
Sn
to
into
(2) is w e l l - p o s e d if
and at least one of them is strictly
95
The
proofs
of t h e s e
special
cases
o f the
stating
these
theorems
results
are
natural
Note similar iously
one
results
generalizations
in
[Des.
2, p.
in t h i s
5.3
In this
systems.
2b
z = Gd
some
system
(2)
we
we
while
the
are
purpose
single-loop
result (3)
results.
than
has
a
not
prev-
the m a i n
large-scale
on
the
interconnected
a p~eliminaxy
a system
described
for
integers
n, r I, r2;
H
and
r2 : L prl e ÷ L pe
G
result
by
some
the
equations
fixed
p E
[1,~]
: Ln × L r2 ÷ L rl pe pe pe is s m o o t h i n g .
Then
the
is w e l l - p a s e d .
Proof
We
first
establish
u e Ln , (2) h a s a u n i q u e s o l u t i o n pe (2) c a n b e r e w r i t t e n a s
that, for
corresponding (d,z)
to e a c h
rl L × pe
in
Lr2 pe
Now,
z = GH(u,z)
For
fixed
smoothing,
u E Ln , H(u,.) pe so t h a t b y L e m m a
is s m o o t h i n g . GH(u,.)
is
of
SYSTEMS
r2 z E Lpe
Lipschitz
The
they
"large-scale"
Theorem
present
require
rI d e Lpe '
positive
is w e a k l y
48],
of
the
as
form.
Consider
d -- H(u,z)
section.
is a b e t t e r
continuous-time
this,
u E Lnpe '
next
CONTINUOUS-TIME
2a
and
(i)
section,
of
T o do
Lemma
where
in t h e
are o m i t t e d ,
is to s h o w h o w
that Theorem
appeared
well-posedness
here
theorems
Let
T > 0
smoothing,
we
is w e a k l y
Lipschitz
(4.2.18),
the operator
be can
arbitrary find
a
~ E
and
let
(0,T)
and
G
u < 1 such
is
z ~ GH(u,z) .
that
Since
96
~p[Pt+~(GH(u,Pt+~.) Now,
by causality,
-GH(u,Pt.))]
(3) is e q u i v a l e n t
<_ ~ , V t E
[0,T-~]
to
Pt z = P t G H ( u , P t z ) First
of all,
that the m a p P6z ~ r2 , is a c o n t r a c t i o n on the space Lp [0,6]. H e n c e the
P~GH(u,P~z)
setting
t = 0
in
(4) shows
equation P~z = P6GH(u,P6z) has a u n i q u e
solution
for
Pdz,
has a u n i q u e
solution
over
[0,5].
of
[0,T].
over
so that the s y s t e m
equation
(3)
We n o w show t h a t this
solution
can be e x t e n d e d
to all
Suppose
solution
to
found
that a unique
the i n t e r v a l
[0,t] w h e r e
ion can b e e x t e n d e d
Pt+6z
to
(3) has b e e n
t < T - 6; we s h o w
[0,t+~].
From
(3) and
that
this solut-
(5) we g e t
= Pt+~GH(u,Pt+6z) = P t + 6 G H (u,Pt+~z) -Pt+~ GH (u,Ptz) + P t + ~ G H ( u , P t z) = Pt+~[GH(u'Pt+~z)-GH(u'Ptz)
N o w note With
that
P t + ~ G H ( u , P t z)
this d e f i n i t i o n ,
Pt+6z Now,
because
GH(u,Ptz)]
of
*Note
(7) is of the
the m a p p i n g
is a c o n t r a c t i o n
t h a t if
quantity
the
t = 0, t h e n
~p[P~ ( G H ( u , P ~ . ) - G H ( u , 0 ) )
Pt+~z
GH(u,0)
it
vt .
+ vt
~ Pt+6[GH(u,Pt+~z)
o n the space
contraction
- call
form
= Pt+6[GH(u,Pt+~z)-GH(u,Ptz)] (4),
again by applying
is a k n o w n
] + P t + 6 G H ( u ' P t z)
mapping
r2 Lp [0,t+6]. theorem,
is i n d e p e n d e n t
] = n p [ P 6 G H ( u , P ~ . ) ] <_ ~ .
So once
we see that
of
z; hence
97
(8) d e t e r m i n e s
Pt+~z
uniquely.
ion can be e x t e n d e d to all of it follows that
In this way,
[0,T].
Since
the u n i q u e solut-
T
s p o n d i n g to each
u 6 Ln pe
(2a) d e t e r m i n e s
d
Once
z e L r2 correpe is u n i q u e l y determined,
z
uniquely.
Thus we have shown that the system dition
(WI) of D e f i n i t i o n
e a s i l y verified,
(1.3.9).
(2.2.9)
in
u .
(2) s a t i s f i e s
Conditions
if one o b s e r v e s that
and w e a k l y L i p s c h i t z Definition
is arbitrary,
(3) has a unique s o l u t i o n for
(W2) and
H(u,z)
con-
(W3) are
is c a u s a l in
u
T h u s all the c o n d i t i o n s of
are satisfied,
and the s y s t e m
(2) is w e l l -
posed. D Remarks still smoothing, Lipschitz.
By L e m m a
while
Thus,
H
(5.1.18), G f o l l o w e d by a delay is
f o l l o w e d by a delay is still w e a k l y
the system
delays are i n t r o d u c e d a f t e r
(2) r e m a i n s w e l l - p o s e d even if G
and
H
.
We now come to the m a i n r e s u l t of this section.
Theorem
Consider a large-scale
interconnected
system
d e s c r i b e d by m
i = 1,...,m
e. = u. - [ S.. e. , 1 l j=l z3 3
10
where
n. e i, u i 6 Lpel
integer
n i , and
digraph
D
for some fixed Sij
nj ÷ Lni : Lpe pe
corresponding
that all o p e r a t o r s
Sij
to
p ~
[i, -]
and some p o s i t i v e
S u p p o s e that the s y s t e m
(i0) is s t r o n g l y connected,
and
are either w e a k l y L i p s c h i t z or smoothing.
D e l e t e all edges c o r r e s p o n d i n g to s m o o t h i n g o p e r a t o r s
from the
s y s t e m d i g r a p h D, and label the r e s u l t i n g d i g r a p h as the r e d u c e d digraph.
U n d e r these conditions,
the s y s t e m
(10)
is w e l l - p o s e d
if the r e d u c e d d i g r a p h c o n t a i n s n e i t h e r cycles nor self-loops. Remarks
The a s s u m p t i o n that the s y s t e m d i g r a p h
strongly connected may appear fact it r e s u l t s
to be o v e r l y r e s t r i c t i v e ,
in no loss of g e n e r a l i t y .
is not s t r o n g l y connected,
F i r s t of all,
if
we can a n a l y z e the w e l l - p o s e d n e s s
each s t r o n g l y c o n n e c t e d c o m p o n e n t of
D
individually
D is
but in
(see
D of
98
Theorem
(4.3.22)).
Alternatively,
to the zero operator, connected;
as required,
if this p r o c e d u r e
that the zero o p e r a t o r
Proof cycles tree;
this
to s m o o t h i n g
Since
the
digraph
D
strongly
be remembered
contains
it can be i n c o r p o r a t e d all
cotree
set of edges
edges
into
neither
a directed
(or links)
correspond
in the c o t r e e be d e n o t e d
the input v e r t i c e s vertices
the input v e r t e x
of the edges
of the edges
in
of an edge, we m e a n
and by the o u t p u t
to w h i c h
to m a k e
it s h o u l d
operators.
and the o u t p u t
leaves,
the r e d u c e d
is done,
Let L e t us label
in order
is followed,
corresponding
is smoothing.
nor self-loops, when
we can a d d e d g e s
the edge
vertex
is directed.
loop,
di = zi
E
as
we m e a n
some of the
also,
if an edge
for some
in
i .
E .
dl,...,d k ,
Zl,...,z k
the v e r t e x f r c m w h i c h
Now,
zi's;
have
in as
of an edge,
the same as some of the then we will
E
by
.
By
the edge
the v e r t e x
d.'s 1 E
All
m a y be is a self-
this does not
matter.
L e t us now go b a c k union
of the d i r e c t e d
ing o p e r a t o r s , L e t us c o n s i d e r Figure shown
5.1.
and a c o t r e e a typical
Suppose
in F i g u r e
5.2,
to the d i g r a p h
tree c o n t a i n i n g containing
cotree
we b r e a k by
edge
the edge
introducing
!
Gj FIGURE
5.1
weakly only ni ni
D
, which
Lipschitz smoothing
in
E
,
operators.
as shown in
into three
two new v e r t i c e s
is the
or smooth-
edges
as
d i' and zi'.
9g
DIRECT'~DD TREE~~
i'9 !
--
z i
G i,
FIGURE
Since
ni
is a o o t r e e
tree
creates
a unique
edge, cycle.
-d'
i
5.2
the a d d i t i o n Hence
there
of
ni
to the d i r e c t e d
is a p a t h
from
zi
d. c o n s i s t i n g o n l y of edges from the d i r e c t e d tree. With i e r e n c e to F i g u r e 5.2, this m e a n s t h a t there is a path from
to ref-
z~ 1 d~ This is true of all edges in E . M o r e o v e r , it is clear 1 that the p a t h from z! to d~ does not pass t h r o u g h d! for l i 3 j ~ i T h u s we can e x p r e s s the q u a n t i t y d! in terms of the to
•
vector
1
z' =
(z~ ..... z~)
L e t this r e l a t i o n s h i p
lla
and
the o p e r a t o r
Lipschitz
Fi
operators,
the digraph,
llb
u =
(u I ..... u m)
be d e n o t e d
d'l = Fi(u,z')
Now,
the i n p u t v e c t o r
,
i = l,...,k
consists and
is thus
of sums
and c o m p o s i t i o n s
a weakly
Lipschitz
operator•
'
i = l,...,k
where G~ is a s m o o t h i n g o p e r a t o r . L o o k i n g at e q u a t i o n s 1 we can see that we h a v e s u c c e e d e d in r e w r i t i n g the s y s t e m in the form
From
we also h a v e
z!1 = G!(dl)l
posed.
of w e a k l y
(2).
Hence by Lemma
(i),
the s y s t e m
(10)
(ii), equations
is w e l l -
100
Remarks digraph
Basically,
the condition that the reduced
(obtained by removing all edges corresponding to smooth-
ing operators)
is acyclic insures that all cycles contain at
least one edge corresponding
to a smoothing operator.
The method
of proof makes it clear that if the hypotheses of Theorem
(9) are
satisfied,
then one can find two sets of variables d~ and z? l l such that the system equations can be rearranged in the form (ll).
It is also clear that, hypotheses of Theorem
if a particular system
(9) and is thus well-posed,
perturb each of the operators be well-posed
(10) satisfies the
G i , Hij
and if we
, the system continues to
so long as each smoothing operator is perturbed into
another smoothing operator,
and each weakly Lipschitz operator is
perturbed into another weakly Lipschitz or smoothing operator.
~e4
u1
S 12
S34
u2
u3
u4
FIGURE 5.3
12
Example suppose that
S12
Consider the system of Figure 5.3, and thru
$43
are all weakly Lipschitz operators;
the question is, which of these operators need to be smoothing in order for Theorem
(9) to apply.
possibilities.
For instance,
then we select
d{
and
d~
if
Clearly, $21
and
13a
d{ = ui+Sl2(Z~÷u2) + Sl4(Z~÷u 4)
13b
d~ = u3+S32(z{+u 2) + s34(z~÷u 4) =
13d
=
s21
are smoothing,
as shown in Figure 5.4, and the
system equations become
13c
there are many $43
101
uI --
r
SI2 FIGURE
$34 5.4
u2 On the other hand,
suppose
u3
S12,
$34
d~ , d 2I , d 3! , z~ , z 2! , z~
Then
and
u4 S14
are smoothing.
w o u l d be chosen as in Figure
5.5, and the system equations w o u l d become !
I
Z3
SI4
dS i
S21/••
.T-
uI
$32
k z2
z1
FIGURE
$34
!
I
5.5
u2
u4
u3
14a
d~ = u 2 ÷ s21(Ul+S~+z ~)
14b
d½ = u 4 + S 1 3 ( u 3 + z ~ + S 3 2 ( u 2 + S 2 1 ( U l + Z { + Z ~ ) ) )
14c
d~ : d~
14d
z I = Sl2d I , z 2 = S34d ~ Once again,
15
the system e q u a t i o n s Example
loops.
z~ = Sl4d~
are of the form
In this example,
we study a system with self-
Let the system under c o n s i d e r a t i o n
It is easy to v e r i f y that if
SII,
(ii).
$21,
be as in Figure
and $22
5.6
are s m o o t h i n g
102
S II
|
$22
.,..~,~/~
~i s2. z~i ~ ' , ~
%2
, _.~
U,
FIGURE
5.6
S13 operators
and the rest are w e a k l y L i p s c h i t z
reduced digraph Choosing
d{
operators,
then the
is acyclic and does not contain any self-loops.
thru
z~
as indicated
in Figure
5.6, we get the
system equations 16a
d~ = u I + S12(u2+z½+z~)+z { + S 1 3 ( U 3 + S 3 2 ( z ~ + u 2 + z ; ) )
16b
d~ : d{
16c
d 3,
16d
Z{ = S l l d ~ , z~ : S21d~ , z~ = $22d ~
=
z 2, + U 2
which are of the form
(ii).
It is clear from the above examples satisfy the h y p o t h e s e s operators
Sii
of T h e o r e m
m u s t be smoothing.
N o w we turn to s y s t e m s d e s c r i b e d by
17a
ei = ui -
m [ Hij Yj j=l i = l,...,m
17b
Yi = Gi ei
that,
in order to
(10), all s e l f - i n t e r a c t i o n
103
n,
where
ui' ei' Yi E L a pe
for some fixed n.
positive
integer
AS shown in Section
[i,~]
n.
Gi : L pei ~ Lpex
hi'
p E
2.2, the system
,
and
(17)
H ij
and for some
n. : Lp ~ -~
L
n. pe1
can be e q u i v a l e n t l y
d e s c r i b e d by m
iBa
e i = u i - j=l ~ H.lj ej+m I i = 1,...,m
18b
ei+ m = G i e i Suppose now that all operators Then the system (2.2.19)
(18)
Gi, Hij
is w e l l - p o s e d
if and only if the system
sense of D e f i n i t i o n can a p p l y T h e o r e m described
(2.2.9).
(See
is w e l l - p o s e d
(2.2.31)
et seq.) .
in the Hence we
of systems
(17).
5.4
DISCRETE-TIME
The g r a p h - t h e o r e t i c of d i s c r e t e - t i m e
continuous-time
(17)
(i) to study the w e l l - p o s e d n e s s
in the form
well-posedness
are w e a k l y Lipschitz.
in the sense of D e f i n i t i o n
systems.
SYSTEMS
c o n d i t i o n p r e s e n t e d b e l o w for the systems
is the same as that for
But the m e t h o d
of proof
is quite
different. Theorem
Consider
a discrete-time
system d e s c r i b e d by
m
ei = u i -
where and
ei, u i Sij
causal,
n. S i
b e l o n g to
n. n. : S 3 ÷ S ~ that
[ Sij ej j=l
PkSij
Suppose has finite
,
i = 1 ..... m
for some positive that for all incremental
k , and that some of these o p e r a t o r s A s s o c i a t e d with the system (5.3.9).
well-posed
if the reduced digraph,
edges c o r r e s p o n d i n g digraph,
ni ,
i, j, Sij
is
gain for all integers
are s t r i c t l y
(2), c o n s t r u c t
in T h e o r e m
integer
causal.
a reduced digraph
Under these conditions,
the s y s t e m
o b t a i n e d by removing
to strictly causal operators
as
(2) is all
from the system
does not contain either cycles or self-loops.
104
Proof the h y p o t h e s e s two
Just
as in the p r o o f
of the t h e o r e m
equations
d' = H(u,z')
3b
z' = G d'
where Pk H
H
is c a u s a l
have
theorem
finite
and
G
if we
gains
(5.3.9)•
if
then we can find
{z I' .... ,z~}
such that
into the form
is s t r i c t l y
incremental
is p r o v e d
and
can be r e w r i t t e n
3a
of T h e o r e m
satisfied•
' ,d.'} {dl,... ~
sets of v a r i a b l e s
the s y s t e m
are
causal,
and b o t h
for all i n t e g e r s
k
Pk G,
.
show t h a t a s y s t e m of the form
So the
(3) is
well-posed.
First u 6 S n • (3) has G
is s t r i c t l y
hence
z '(0)
a unique
d '(0)
so on.
we prove
causal,
Thus
condition
Conditions
(2) is w e l l - p o s e d ,
(W2)
-
(W3)
that
contains
free
It is w e l l - k n o w n
that if a filter
operator,
in the proof
NOTES
The c o n t e n t s Well-posedness while ness
[Des.
of s o l u t i o n s
analogs
AND
that
loops•
This
u (0), find
(2.2.20)
z '(I) #
is satis-
Hence
digraph
the
be a c y c l i c
there
corresponding
are no
"delay
of d i g i t a l
then
filters
it can be
is e s s e n t i a l l y
imple-
w h a t we do
REFERENCES
of this C h a p t e r systems
closely
is s t u d i e d
some r e s u l t s
in the case
of the results
since
, and
(i) .
of f e e d b a c k
2] c o n t a i n s
d' and
to verify.
in the t h e o r y
fashion.
of T h e o r e m
z '(0)
at least one edge
i.e.
has no d e l a y - f r e e
in a r e c u r s i v e
Now,
of
(3b) we can
of D e f i n i t i o n
the r e d u c e d
causal
mented
from
are easy
to a s t r i c t l y loops".
and
Knowing
to any
z'
o
condition
that each cycle
(WI)
d'
for
Then
•
corresponding
is i n d e p e n d e n t
(3b).
(3a)
fied.
implies
solution
from
from
system
The
that,
(Gd') (0)
is k n o w n
we can find and
of all,
on the e x i s t e n c e
of f e e d b a c k
given here
follow
systems.
can be found
[Vid.
bY W i l l e m s
9].
[Wil.
2],
and unique-
State-space
in [Moy.
4] .
CHAPTER 6: SMALL-GAIN TYPE CRITERIA FOR Lp-STABILITY In this Chapter,
we state and p r o v e s e v e r a l s u f f i c i e n t
c o n d i t i o n s of the "small-gain"
type for a l a r g e - s c a l e
c o n n e c t e d system to be L p - S t a b l e . be p l a c e d into two categories: matrix,
and
(ii) c o n d i t i o n s
inter-
The results p r e s e n t e d here can
(i) c o n d i t i o n s
i n v o l v i n g a test
involving a minimal essential
d e c o m p o s i t i o n of the s y s t e m digraph. that b o t h types of results reduce,
set
It is i n t e r e s t i n g to note
in the s i n g l e - l o o p
case,
to
the s t a n d a r d small gain t h e o r e m s of S e c t i o n 6.1.
The r e s u l t s of this c h a p t e r can be u s e d to e s t a b l i s h L - s t a b i l i t y for any v a l u e of p in [i,®]. In contrast, the P r e s u l t s of C h a p t e r 7 r e q u i r e the input space to have an inner p r o d u c t structure.
In this respect,
h a v e a w i d e r scope of a p p l i c a t i o n the other hand,
the p r e s e n t results,
interconnection" conservative
or "worst-case"
the r e s u l t s of this cahpter
than those in C h a p t e r in m o s t cases,
type,
s t a b i l i t y criteria.
7.
On
are of "weak
and can lead to o v e r l y
T h i s p o i n t is e l a b o r a t e d in the
sequel.
6.1
SINGLE-LOOP
SYSTEMS
In this section, we p r e s e n t the s t a n d a r d s m a l l - g a i n t h e o r e m s for s i n g l e - l o o p systems.
The proofs are omitted,
as the
t h e o r e m s here are special cases of s u b s e q u e n t results.
Theorem
Consider a single-loop
f e e d b a c k system des-
c r i b e d by the e q u a t i o n s
2a
el = Ul - Y2
2b
e2 = u2 + Yl
2c
Yl = G1 el
2d
Y2 = G2 e2
w h e r e Ul' u2' el' e2' YI' Y2 E L 9 pe and some p o s i t i v e integer ~ and '
for some fixed GI, G 2
map
L9 pe
p e
[i,-]
into
108 itself. ators
Let G1
yp(G1)
and
and
yp(G 2)
G 2 , as d e f i n e d
these conditions,
the system
denote the gains of the oper-
in D e f i n i t i o n
(3.1.1).
(2) is L -stable P
Under
if
7p(G I) yp(G 2) < 1 Theorem
Suppose
that either
Under these conditions,
linear.
and
yp(G I)
7p(G 2)
u2 = 0
the system
are finite,
or that
(2) is
G2
is
L -stable if P
and further,
7 p ( G 2 G I) < 1 Remarks referred either
l) Both conditions
to as "small-gain"
the gain of an o p e r a t o r
to be less than one. d i t i o n than
(5) is a less r e s t r i c t i v e
sufficient
condition
(i), in cases where either
In other words,
operators generally,
-G 1 if
G 2 , then
and/or
(3) holds
(3) and
(3) holds
(5) are "sign-insensit-
is replaced by
~2 G2
= le] 7p(G)
interchanged
than
in linear.
for a p a r t i c u l a r
for a p a r t i c u l a r
is r e p l a c e d by
3) T h e o r e m
G2
(3) is still s a t i s f i e d G2
follow from the obvious yp(eG)
or
con-
Hence, Theorem
for L p - s t a b i l i t y
u2 = 0
if
-G 2 .
pair of G1
is
More
pair of o p e r a t o r s
(3) is still satisfied w h e n e v e r
Ul G1 ' G2 remarks
if
G 1 , G 2 , then
r e p l a c e d by
(of operators)
yp(G2G1) ~ 7p(G2) yp(G1) .
2) Both the conditions ive."
(5) are usually as they r e q u i r e
or a p r o d u c t gains
Note that
(3), since
(4) gives a weaker Theorem
(3) and
type conditions,
' and
G1 ~i
G1 ,
is replaced by
' s2 E
[-i,i].
These
identity
, ~
(4) holds if the subscripts
1 and 2
are
throughout.
The f o l l o w i n g
results
use what is known as a "Loop
transformation." Theorem
Consider
be a linear operator inverse
(I+KGI)-I
the system
such that
: L~ ~ L9 pe pe
7p(K) "
(2), and let < = , and
9 ÷ L pe u K : L pe
(I+KG I)
Under these conditions,
has an the
107
system
(2) is Lp-stable
if
8
7p(G2-K)
. 7p(GI(I+KGI )-I)
9
Theorem
Consider
Theorem Under
(7).
Suppose
i0
the system
that either
these conditions, yp[(G2-K)
GI(I+KGI )-I]
To apply any
of the above
system described
be as in
is linear. if
theorems
to a specific
example,
the gain of a given operator.
BASED
In this section,
G2
(2) is L -stable P
by the results
CRITERIA
or
K
< 1 .
one has to be able to calculate
6.2
(2) and let
u2 = 0
the system
This task is facilitated
< 1
of Section
3.1.
ON A TEST MATRIX
we study a large-scale
interconnected
by m
la
ei = ui -
~
Hij yj
j=l
I i = 1,...,m
ib
Yi = Gi ei where
ni ui' ei' Yi • L pe
positive
integer
The stability
for some fixed
ni' Gi
criteria
so-called
"test matrix"
principal
minors
presented
RESULTS
We first present
A
radius
checking
a
all of its leading
ON N O N N E G A T I V E
MATRICE~q
a few preliminary
results
on non-
matrices. Fact
of
Hij
n. n. : Lp ~ ÷ L pe1
here all involve
to verify whether
and some
[i,~]
arepositive.
6.2.1.
negative
p 6
n. n. : L pei ÷ L pei ' and
(i.e., p(A)
Given
A 6 R nxn
, let
the set of eigenvalues of
A
by
sp(A)
of A),
denote
and define
the spectrum the spectral
108
p (A) =
Suppose
A • R nxn
Proof
Fact elements the
max ~e Sp (A)
1I 1
has all n o n n e g a t i v e elements.
See
[Gan. i, p..66].
Let
(i.e.b..
i]
T h e n p(A)• Sp(A).
B E R n×n have all n o n p o s i t i v e o f f - d i a g o n a l < 0 -
whenever
i # j) .
U n d e r these conditions,
f o l l o w i n g s t a t e m e n t s are all equivalent:
(i)
All e i g e n v a l u e s
of
B
have p o s i t i v e real parts.
(ii)
The leading p r i n c i p a l m i n o r s of
B
are all
positive. (iii) T h e r e exist p o s i t i v e c o n s t a n t s that
D B
D = D i a g {dl,...,dn}.
n
dl,...,d n
such
where
In other words,
exist positive constants
there
such that
n
.. > [ dj b33 i=l
Id i bij I = -
i@j One can also express
d I ..... d n
is s t r i c t l y r o w - d o m i n a n t ,
[ d. b.. i=l 1 x3 '
Yj
i~j
(5) e q u i v a l e n t l y as
n
i=l (iv)
d i b.. > 0 13 '
Vj
There exist positive constants that
B C
Cl,...,c n
is s t r i c t l y c o l u m n - d o m i n a n t ,
C = Diag {c l, .... Cn}. exist positive constants
In other words, Cl,...,c n
such
where there
such that
n
[
bij cj > 0 ,
Vi
j=l In effect, sufficient conditions
(ii)-(iv) for
(i) .
are all e q u i v a l e n t n e c e s s a r y and The p r o o f of F a c t
(4), plus
several o t h e r e q u i v a l e n t 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 for (i), can be found in p. 396].
[Fie. i] or m o r e a c c e s s i b l y in
[Sil. i,
109
Lemma
Suppose
Then
p(A)
I -A n
are all positive.
< 1
.
Hence,
positive, p(A)
"if"
By Fact
(2),
l-p(A)
entries.
it follows
from Fact
(4) that
of
is an e i g e n v a l u e
if the leading p r i n c i p a l m i n o r s of
In-A
of
are all
l-p(A)
> 0 , i.e.
Then
IRe X I < 1
< 1 . "only if"
ever
X • Sp(A),
fore
1-Re X > 0
have p o s i t i v e
Suppose
p(A)
by the d e f i n i t i o n YX • Sp(A),
real parts.
Lemma (In-A)-i
Suppose
Proof positive, inverse,
p(A)
Fact elements. X n} with
whenThereIn-A that all
are p o s i t i v e . o
has all n o n n e g a t i v e elements
elements.
if the leading
are all positive.
< 1
by Lemma
(8).
Hence
In-A
In-A
are all
has an
is given by
which shows that ii
A E R nxn
In-A
(in_A)-i =
i0
In-A
of
(4), it follows
If the leading p r i n c i p a l minors of
then which
of
by Fact
has all n o n n e g a t i v e
p r i n c i p a l minors of
< 1 .
of the spectral radius.
so that all eigenvalues
Hence,
the leading p r i n c i p a l - m i n o r s
Then
has all n o n n e g a t i v e
if and only if the leading p r i n c i p a l minors
Proof In-A
A 6 R nxn
~ i=0
(A) i
(In-A) -I Suppose
has all n o n n e g a t i v e
B E R n×n has all n o n p o s i t i v e
T h e n there exists a diagonal m a t r i x Xi > 0 V i
elements,
such that
B' A + A B
if and only if the leading p r i n c i p a l minors of
off-diagonal
A = Diag
is p o s i t i v e B
o
{XI,...,
definite,
are all
positive. Proof
See [Ara.
6.2.2
BASIC
i].
"TEST-M/~TRIX"
TYPE C R I T E R I A
With the aid of these results, stability criteria
for the system
(i)
we can derive
several
110
Theorem
12
for all
Suppose all operators
i , and define the test matrix
13
Gi
have finite gains
Q1 e R m×m
by
qlij = 7p(Hij Gj) Assume that
qlij
is finite for all
i,j
Then the system
is Lp-stable
if all the leading principal minors of
Im-Q 1
(i) are
positive. Remarks equivalent
By Lemma
to requiring Proof
(8), the hypotheses on
that
bij
are
p(Ql ) < 1 .
By the definition of
finite constants
Q1
7p(Hij
Gj)
, there exist
such that n,
14
I Ixl ITp+bij , YT > 0, Vx E Lpe3
I IHijGjxl ITp < 7p(HijGj) Substituting equations
from
(ib) into
(la), we can recast the system
in the form m
15
ei = ui -
~ H.. G. e. , j=l a3 3 3
Taking norms in (15) and applying
i = l,...,m
(14), we get m
16
IIeilIT p ~ IluillT p + j=l [
IIH i3.G.e 3 J IITp m
<_ d i + IIuiIITp +
[
7p(HijG j)
IIejIITp
j=l
where m
17
di =
[ b.. j=l ~3
Suppose now that n. Lp ~ Vi Since
upper bound
n. u i E Lp a Vi ; we first show that ei' Yi ~ n. u.l E Lpl , we can replace IIuiIIT p by its
lluill p
in (16)
This gives
111
m
18
IIeilITp ~ d i + IIuiIlp +
[
j=l
~p(Hij Gj)
I lejllTp
m
= di + IIuiIIp +
~
j=l
qlij
IIejIITp
m
19
j~'l(~ij-qlij ) =
IIejIITp <_ d i + IIuill p
where 8ij denotes the Kronecker delta. VT E Rm and w ~ R m by 20
v T = [I lel[IT -.. IIemIIT ]'
21
w
Let us define vectors
= [d i + IIUll I ... d m + ] lUmlI]'
Then (19) is equivalent to 22
[(Im-Ql) VT]i <- (w) i Since the leading principal minors of Im-Q 1 are all positive, we have by Lemma (9) that (Im-Ql)-I contains all nonnegative elements. Hence we can multiply both sides of (22) by (Im-Ql)-l; this gives
23
(VT) i = IIeiIITp ~ [(Im-Ql )-I w] i Since the right side of (23) is independent of from Lemma (2.1.12) that e i E Lni p ¥i . Since
T , it follows 7p(G i) < ® ¥i
it follows that Yi E L~i ¥i . Finally, it is clear from (23) and the finiteness of 7p(G i) ¥i that there exist finite constants k, b such that 24a
II [Ip
k J.I Ip÷b
24b
ilyllp k flu Ip+b where e = (el,...,e m) and u,y the system (i) is Lp-stable. o
are defined similarly.
Hence
112
The above proof makes Theorem yp(Hij
(12), one can take Gj),
Theorem
(6.1.1).
(6.1.2), w i t h 25
qlij
if the c a l c u l a t i o n
In the single-loop
it clear that, of
case,
To see this,
in applying
to be any upper bound yp(Hij Theorem
Gj)
(12) reduces
apply T h e o r e m
for
is too cumbersome.
(12)
to
to the system
n = 2 , and
Hll = H22 = 0 ; H12 =-H21 = I where
26
I
denotes
the identity operator
on
Lv pe
Then we get
7(HIIG I) = Y(H22G 2) = 0 ; Y(HI2G 2) = Y(G 2)
;
Y(H21G I) = Y(G I)
01Lo G2 I
27
(G I)
Thus the condition are positive 28
0
that the leading p r i n c i p a l minors
reduces
of
I2-Q 1
to
7(G2) • y(G I) < 1 which
is the same as
(6.1.3)
It is possible
to "aggregate"
into the form 29a
e = u - Hy
29b
y = Ge where
30
e = ( e l , . . . , e m)
31
G = Diag {G I ..... G m}
the system equations
(i)
113
I
HII
32
H
~
Hlm
---
"
°
LHml and
u,y
(6.1.4)
Hram
are d e f i n e d
analogously
to the a g g r e g a t e d
to
(30).
"single-loop"
Applying
system
Theorem
description
(29),
we g e t
33
yp(HG)
as a s u f f i c i e n t
< 1
condition
for the L p - S t a b i l i t y
It is i n t e r e s t i n g (33)
with
is p o s s i b l e
to d e f i n e
in an i n f i n i t e on
to compare
t h a t of T h e o r e m
(12).
the
stability
F i r s t of all,
the n o r m on the input
number
of ways.
of the s y s t e m
(i).
criterion
observe
t h a t it
space
Ln = ~ L ni P i=l P c h o i c e s of the n o r m
For d i f f e r e n t
Lp n
yp(HG) w i l l be d i f f e r e n t . Now, in o r d e r ' the n u m b e r for the s y s t e m (i) to be stable, it is s u f f i c i e n t for yp(HG)
be less
than one
represents defines
for some c h o i c e
a very
flexible
the n o r m on
then the c r i t e r i o n that of T h e o r e m advantage
(12).
On the o t h e r
s h o w n to c o n t a i n T h e o r e m
34
we
f i r s t give
Definition norm
stability
I]-Ildp
on
criterion.
(12)
hand,
offerred
by
if one (33),
as a special
takes
then
case.
To
full
(33)
can be
illustrate
a
Given positive
constants
dl,...,d m
m n. Ln = ~ L i is d e f i n e d by P i=l p m
11XlIdP=i=l [ di llxiilP'
35
Given
any
F
n + L pe n : L pe
' its g a i n
Yx
=
ydp(F)
to
Ln Thus, (33) P If one s i m p l y
Ln in the u s u a l w a y ( D e f i n i t i o n 2.1.13), P (33) n e i t h e r c o n t a i n s nor is c o n t a i n e d by
of the f l e x i b i l i t y
this point,
of the n o r m on
[Xl...Xm] ' E L n P
is d e f i n e d
by
, the
114
36
Ydp(F)
= inf {k: B b < -
such that
5 k
IlXTIIdp+b,
In other words, norm on
L np , given by
gain of the operator
l l-Ildp
(35), and
L n . The relationship P brought out in the next 37
Lemma in
(13).
of
Under
Im-Q 1
Given these
dl,...,d m
such that Remarks
the system
Lemma
(37)
states
a suitably
norm on
Theorem (33).
Theorem
(12)
choices
of
(12)
that,
is
the matrix
Q1
principal
as
minors
constants
if the stability
by Theorem
(12),
the criterion
(33)
of
then it can together
with
L n . In this sense, and in this P is a special case of the "small gain"
However,
it is clearly more easy to apply
than it is to calculate dl,...,d m
as the norm (12)
< 1
by applying
selected
(i), define
if the leading
(i) can be established
sense only,
I I-IIdp
then there exist positive
Ydp(HG)
type of
is the corresponding
using
(33) and Theorem
the system conditions,
also be established
criterion
Ydp(F)
between
are all positive,
VT { 0, Vx ~ L~e}
is just a particular
F , calculated
on
II(Fx)TIId p
and seeing
Ydp(HG)
for several
if any of these
is less than
one. Proof
Note
that all elements
so that all the off-diagonal Hence, ive,
if all the leading
then there exist
elements
principal
constants
i=l
di
> 0
(~iJ-qlij)
Yj
or, equivalently. m
39
I
i=l
di qlij
Now pick a number
< dj
s < 1
Vj such that
m
40
di qlij i--1
~ edj
Vj
minors
dl,...,d m
m 38
of
of
Q1 Im-Q 1 of
are nonnegative, are nonpositive. Im-Q 1
such that
are posit-
115
This is clearly possible, 7dp(HG) 41
_< s < 1 .
II (HGX)TIIdp =
in view of
To show this,
(39).
let
We now claim that
x = [Xl...Xm]'
m ~ d i I I(HGx) iIIT p i=l
m
(from
~ L npe . Then (35))
m
(from(31)and(32)) [ d i If( ~ Hij Gjxj) IITp i=l j =i m m <_ ~ d i ( ~ 7p(HijG j) I IxjIITp+bij ) (from (14)) i=l j =i m m
=
=
~-
[
di
i=l j=l
qlij
<_ ~
m ~ dj j=l
=
IIXTildp ÷b
I Ixj I ITp +b
I IxjIITp +b
(from
(40))
where 42
b =
m m [ [ di b i=l j=l 13
The extreme inequalities in (41) show that which proves the lemma. [] 43 matrix 44
Theorem Consider Q2 • Rmxm by
q2ij = yp(Hij)
the system
Ydp(HG)
~ s < 1 ,
(1) , and define
the test
7p(Gj)
Assume that q2ij is finite for all i,j Under these conditions, the system (i) is L -stable if the leading principal P minors of Im-Q 2 are all positive. Proof
Obvious
yp(HijG j) ~ yp(Hij) As before, any upper bound for yp (Gj)
from Theorem
yp(Sj)
(12), if one observes
that
. []
one can take yp(Hij) and
q2ij ~ aij bj , where aij is bj is any upper bound for
116
Now we consider the stability of the system some of the operators
Gi
and
results also show why Theorem "weak interconnection" 45
Theorem
Consider the system (1), and suppose that all
~ij Hij
Proof
(12) are satisfied.
Under these con-
for some
eij e [-i,i]
Hij
is
, ¥i,j
By assumption, we have that all the leading
principal minors of each Hi3. by be defined by
These
(43) are called
the system (1) remains Lp-stable whenever
replaced by
46
are perturbed.
type.
the hypotheses of Theorem ditions,
Hij
(12) and Theorem
(i) when
Im-Q 1
are positive.
aij H..13 , where
Now suppose we replace
ei3. 6 [-i,i]
Let
Q1 6 Rm×m
qlij = Yp(eij Hij Gj) Then, since
47
yp(SijHijGj)
= leijl yp(HijG j)
we have 48
0 <_ qlij
Since
qlij
<- qlij ' Vi,j
is an u ~ e r
bound for
leading principal minors of
Im-Q 1
qlij' the fact that all the are positive insures the
Lp-Stability of the perturbed system as well. Theorem
49
Consider the system
the hypotheses of Theorem
(43) are satisfied.
remains Lp-stable whenever each each
Gi
is replaced by Proof
(i), and suppose that all
Hij
8iGi, and
Then the system (i)
is replaced by
eij Hi j, sij 6 [-i,i], 8 i 6 [-i,i].
Entirely similar to that of Theorem
The properties brought out in Theorems are usually referred to as connective stability Theorem
(45) states that if the system
of satisfying the conditions of Theorem
(45). []
(45) and (49) [Sil. i].
(i) is Lp-stable by virtue (12), then it remains
117
stable if each i n t e r c o n n e c t i o n "weaker"
interconnection
In particular, previously i,j).
present
the c o n d i t i o n
operator
interconnections
that all the
(leading)
automatically
(43) requires
given any satisfy
GI,...,G m
requires
if the system
it remains Vi,j
that
said to contain of s t a b i l i t y Corollary
whenever
(43)
and
Q2
¥i,j Hij = 0
This means ¥i,j,
is s u f f i c i e n t l y
(12) and c o r o l l o r y
then small
(43) can be
or "weak coupling"
the criteria
(43) are "sign-insensitive", Q1
note that,
(and hence of T h e o r e m
yp(Hij),
Theorem Also,
Im-Q 1
i.e., that
(similarly
Finally,
small
"weak interconnection"
criteria.
test m a t r i c e s
qlii < 1 ¥i,
(i) is L p - S t a b l e with
Lp-stable
for some
note that
yp(G i) < = Vi, one can always
sufficiently
For this reason,
Also,
p r i n c i p a l minors of that
q2ii < i ¥i).
of c o r o l l a r y
yp(Hij)
eij = 0
(49).
must be less than one
such that
the criteria
(12)) by m a k i n g that,
(i.e. let
remarks apply to T h e o r e m
all the "selfinteractions" Corollary
H.. is r e p l a c e d by a z3 =.. H.. where e.. E [-i,i]. • 3 z3 z3 stable even if we remove some
the system remains
Similar
are p o s i t i v e
operator
of T h e o r e m
type
(12) and
in the sense that the
are u n a f f e c t e d
even if we replace
H.. by -H.. for some i,j It is i n t u i t i v e l y clear that any z3 z3 stability c r i t e r i a that can insure such a sturdy type of stability m u s t n e c e s s a r i l y
be e x t r e m e l y
conservative.
In the case of L2-stability, a "sign-sensitive" special case. Theorem
50 p = 2
Suppose
(not n e c e s s a r i l y 51
<ei'HijGjej>T
criterion
Consider that
~-sij
that contains
the system
Y2(Gi)
positive)
it is possible Theorem
to o b t a i n
(12) as a
(i), and suppose
that
< = ¥i, and that there exist finite
constants
~ij'
IIeiI!T211ejIIT2-Bij
8ij
such that
I eiI!T2,
YT ~ 0 ,
n. n Ye • 6 L l , Ye. 6 L 3 z pe 3 pe Under these conditions, exist p o s i t i v e constants S e Rm×m d e f i n e d by
52
the system Ii,...,I m
(i) is L 2 - s t a b l e
if there
such that the m a t r i x
sij = li ~ij - (li sij + lj ~ji )/2
118
is positive
definite.
The proof depends Lemma
53
Suppose
b • R m , and that c , depending
on the following
A • R m×m
d • R .
only on
A
is p o s i t i v e
definite,
Then there exist constants
and
d, such that w h e n e v e r
that
7
and
x • Rm
satisfies 54
x' A x < b' x + d x
also satisfies
Ilxli
55 where
]I-I]
+
in
(55) denotes
Proof of L e m m a of
A
.
Then,
c
whenever
the E u c l i d e a n
(53) Let x • Rm
~
norm on
Rm
be the s m a l l e s t eigenvalue
satisfies
(54),
x
also satis-
fies
~1I~1I 2 ~
56 Completing
b'
x +
the square in
d
lIbll
~
llxlI
(56) readily yields
Proof of T h e o r e m
(50)
+
d
(55). 0
From the system equations
(i),
we get
m 57
flail 12 = <ei,ei> T = <ei,ui> T - 3[i.= < e i ' H i j G j e j > T m
<- I Iei I IT2
I Iui I IT2 +
~
j=l
I lei]IT21 lejl IT2 + 8ij
where Now,
in the last step we use Schwarz's choose p o s i t i v e
is p o s i t i v e
definite.
constants Then from
~ij
I leiIIT2
inequality
X1,...,A m
and
(52).
such that m a t r i x
(57), we get
S
119
m
[2
m
58 i:l
T2 m
m
ei{IT2(i[ui{
T2+8i )
m
+ i=l ~
59
i
i:l
[ li ~ z3 I{ei{ IT2 1 leJ j =i
{T2
m
[ i=l j=l
(Ii6ij-~i~ij)
l lei{ IT2 l lej { {T2
m
<_ [ l leil {T2(li I lui{ IT2 + lib i) i--i where m
60
8i =
[
j--1
8ij
n.
Suppose now that and
M e R mxm
u i • L2~ Vi , and define
v T = [IIelIIT2
62
b
= [llull
M =
I
...I leml IT2 ]'
2 + Sl " "
ll(l-all)
IlUmll2 + 8m]'
-llal2
"'"
-llalm
I
-ImUm2
...
Im(l-umm ~
:
I
L- t m a m l Then
b • Rm ,
by
61
63
v T • Rm ,
(59) is equivalent
to
V TI M v T <_ b' v T
64
NOW note that vT S vT .
S
Hence
is the symmetric part of (64) is equivalent
M , so that
v T'
to
v T' S v _< b' v T
65 Since
S
is positive definite,
by Lemma
(53) there exists a
M
VT=
120
constant
y
such that
IIVTil ~
66
llb11 n.
It is clear from it follows
that
(66)
that e i 6 L21 ¥i ni Yi 6 L 2 ¥i . Finally, Y2(Gi)
case.
(I) is L2-stable. o
Note that the c r i t e r i o n sensitive",
Vi
in that the test m a t r i x
H.. by -H. • Also, 13 13 (49) et seq.) does not n e c e s s a r i l y
Theorem
(50)
(50) gives
is satisfied.
This
less c o n s e r v a t i v e
case.
S
is "sign-
is in g e n e r a l
altered
stability"
leads one to b e l i e v e than T h e o r e m
(50) contains
observe
then there exists a finite c o n s t a n t
67
(50)
that if
8ij
I IHijGjejl IT 2 ~ 72(HijGj)
from
in this
hold if the c r i t e r i o n
criteria
As a prelude,
< ® ¥i ,
(24) holds
"connective
We now show that T h e o r e m a special
, that
of T h e o r e m
we replace
Y2(Gi)
it is easy to see,
(66) and the fact that Hence the system
< ~
Since
if
(see of
that Theorem
(12).
Theorem
(12) as
72(HijGj)
< ~ ,
such that
I lejl IT 2 + 8ij
, YT ~ 0 ,
n0
Yej 6 L
So by Schwarz's
68
inequality,
<ei'HijGjej>T
3 2e
we have
> - I leiIIT2
- 7 2 ( H i j G j)
IIHi3. G .3e J lIT2 Ileil IT2
IIejIIT2
- 8ij
I leillT2
n.
YT > 0 -
Hence 69
(51)
is s a t i s f i e d w i t h Lemma
Consider
7 2 ( H i j G j) < ~ , ¥i,j
.
2e
Ye "
j
6 L 3 2e
sij = 72(HijGj)"
the system
Under
n.
ye i 6 L i "
,
(I), and suppose
these conditions,
72(G i) < -,
if the leading
121
principal minors of the matrix
Im-Q 1
are all positive
(recall
(13)), then there exist positive constants ll,...,lm such that the matrix S of (52) is positive definite, with aij = Y2(HijGj).
70
Proof
If we define
S
(Im-Q I) + (Im-Q1) ' A]/2
=
[A
A = Diag {ll,...,im}
The lemma now follows readily from Fact
, then
(ii) . D
We conclude this subsection by presenting a result similar to Corollary (43), but with a slightly different test matrix. 71 7p(Hij)
72
Theorem Consider the system (i), and suppose < ® , ¥i,j . Define the matrix Q3 E R m×m by
q3ij = 7p(Gi)
7p(Gi)<-,
yp(Hij)
Under these conditions,
the system (i) is L -stable if the leadP ing principal minors of Im-Q 3 are all positive. Proof
73
Let the constants
IIGjx[ IT ~ yp(Gj)
b. , d.. i 13
be chosen such that
[Ixl IT + bj
nj VT >_ 0 , Vx G Lpe
74
IIHijXIIT ~ 7p(Hij) From
75
[IXIIT + dij
(i), we get
IIyiIITp ~ ~p(Gi) llei[ITp + bi <_ ~p(Gi) +
[lluiJITp
m I 7p(Hij) j=l
I IYjl [Tp + dij] + b i
m
76 j=l
(6ij-q3i j) [ IyjIITp ~ 7p(G i) I Iui] ITp + c i
122
where m
77
c i = b i + T p ( G i) The i n e q u a l i t y lity
(76)
~ dij j=l
is similar to
(19), and the p r o o f of Lp-stabi-
now follows along the same lines as that of T h e o r e m
(12).
The d e t a i l s are left to the reader, m
Theorem because
(71) is not as g o o d a r e s u l t as T h e o r e m
the latter i n v o l v e s
terms such as
the former i n v o l v e s terms such as g e n e r a l larger than
Tp(GiHij) .
c o m p a r a b l e to C o r o l l a r y
7p(Hij
(12),
Gj), w h e r e a s
7 p ( G i) Tp(Hij) , w h i c h is in In this respect,
Theorem
(71) is
(43).
We leave the p r o o f of the f o l l o w i n g
" c o n n e c t i v e stab-
ility" r e s u l t to the reader• b e c a u s e it is e n t i r e l y s i m i l a r to that of T h e o r e m
78
(45).
Theorem
C o n s i d e r the s y s t e m
the h y p o t h e s e s of T h e o r e m
(71) are satisfied.
(i) r e m a i n s L p - s t a b l e w h e n e v e r each each
Gi
is r e p l a c e d b y
6.2.3
(i), and
~i Gi
Hij
' and
suppose t h a t all T h e n the system
is r e p l a c e d by
~ij 6
[-i,i],
generalize Theorem
(6.1.7).
79
Lemma
80
e = u - Hy
C
and
r e s u l t s that
To do this, we n e e d a
C o n s i d e r a s y s t e m d e s c r i b e d in a g g r e g a t e form by
, y = Ge
e,u,y 6 L n and pe ' D
8 i • ~-i,i].
C R I T E R I A B A S E D ON LOOP T R A N S F O R M A T I O N S
W e now p r e s e n t some " l o o p - t r a n s f o r m a t i o n "
where
uij Hij'
H,G
map
Ln pe
into itself.
are linear o p e r a t o r s on
Ln pe
such t h a t
Suppose
CD = DC = 0,
and d e f i n e
81
f
Suppose
H
=
e
+
Cy
,
is linear,
z
=
y
-
and that
De
(I + HD) -I • (I + CG) -I
exist
123
as
operators
ions
(79)
on
can
variables
Ln . Under pe equivalently
be
z
and
f
z =
(G-D) (I+CG)-If
82b
f =
(I+HD)-I
u-
Proof
First
of
z =
-
Ge
Next,
84
e
85
e =
Combining
86
=
f
is
87
88
is
f
(85)
all,
Next,
from
(81)
the
new
(81)
and
(80)
we
get
that
f
we
have
- Hy
Cy
=
u
C)(z
u
-
(H
-
=
u
-
(H
- C) z
=
u
-
(H-C) z
= u
-
(H
Theorem
of
gives
=
(82b).
terms
equat-
- CGe
(G-D) (I+CG)-I
(I+HD)-I
in
system
(I+HD)-I(H-C)z
from
(I+CG) -I
e +
expressed
the
(G-D) e
f
and
f =
which
=
get
- Cy
(82a).
f =
De
=
(83)
z =
which
89
we
conditions,
as
82a
83
these
+
u-
Cy
=
u
-
(H-C)y
De)
- HDe
- HDf
- C)z
+
+
(because
CD
=
0)
(because
DC
=
0)
HDCy
- HDf
(I+HD) -I
(H-C) z
[]
Consider
a
system
described
by
124
m
90a
ei = ui - j~l Hij
90b
Yi = Gi ei
yj
n.
where p e G
ei'
ui' Yi e L 1 for some i n t e g e r n i and some fixed pe n. n. n. n. H : L 3 ÷ L z Define G i : L p eI ÷ L pei ' a n d pe pe ij
[I,-], and
H
as in
(31)
and
(32).
91a
C = Diag
{ C l , . . . , C m}
91b
D = Diag
{ D I , . . . , D m}
s u c h t h a t a l l the h y p o t h e s i s
Select
of L e m m a
92
F =
(G-D) (I+CG) -I = D i a g
93
R =
(I+HD) -I
and partition yp(F)
R
Q4 e R m x m
94
, and
= yp[Rij
these conditions,
ing p r i n c i p a l
minors
The proof by applying
Theorem
Note ption
that
(79)
D
of the f o r m
are s a t i s f i e d .
{(Gi-Di) ( I + C i G i ) - I
Define
i=l .....
,
compatible y p [ ( I + H D ) -I]
with
t h a t of
H
Suppose
< - , and d e f i n e
by
q4ij Under
< ~
and
(H-C)
in a m a n n e r
< ~ , yp(R)
C
that,
(I+CjGj) -I]
the s y s t e m
of
is L - s t a b l e P are all p o s i t i v e .
Im-Q 4
is o m i t t e d , (12)to
with
CD = DC = 0
this a s s u m p t i o n
(Gj-Dj)
(90)
as the t h e o r e m
the modified
C means
and
D
that
can be satisfied
if the lead-
follows readily
system description
of the f o r m C i D i = 0 ¥i
by setting
either
(91),
(82).
the assum-
In particular, Ci = 0
or
D. = 0 for e a c h i . In this case, the t r a n s f o r m a t i o n (81) means I t h a t we e i t h e r p l a c e a " f e e d - f o r w a r d " of D i or a " f e e d b a c k " of Ci
around
the s u b s y s t e m
Gi, b u t n e v e r both.
125
To show that T h e o r e m Theorem
(6.1.7)
is a special case of
(89), c o n s i d e r once again the s y s t e m
linear o p e r a t o r
K
: L~ ÷ L9 pe pe
with
yp(K)
(6.1.2),
and let a
< ~ be given.
Select
95
since
96
11°I
G =
,
H =
G
it is r o u t i n e to v e r i f y that
G 97
(G-D) (I+CG)
-i
(I+KG I)
=
i 0I
0
98
(I+HD) -I
99
(I+HD) -i
G2-K
(H-C) = H
E K31
Thus,
all the c o n d i t i o n s n e e d e d to a p p l y T h e o r e m
fied,
and the test m a t r i x
i00
Q3
(89) are satis-
becomes
Q4 = Yp [G 1 (I+KG I) -i]
From Theorem
(89), the s y s t e m
ing p r i n c i p a l m i n o r s of case if
I2-Q 4
(6.1.2)
0
is L p - s t a b l e
if the lead-
are b o t h positive,
w h i c h is the
126
i01
yp(G2-K)
y p [ G I ( I + K G I )-I < 1
T h i s is w h a t T h e o r e m
6.3
(6.1.7)
states.
C R I T E R I A B A S E D ON AN E S S E N T I A L SET D E C O M P O S I T I O N
In this section, ility b a s e d g r a p h - t h e o r e t i c that of an e s s e n t i a l generalize
we p r e s e n t some c r i t e r i a for L -stabP techniques. The key c o n c e p t here is
set of a graph.
The c r i t e r i a p r e s e n t e d here
, in a n a t u r a l way, T h e o r e m
loop-transformation
theorems
(6.1.7)
T h r o u g h o u t this section, m ~ Sij ej j=l
ei = ui -
,
(6.1.4)
and
as well as the
(6.1.9).
we study systems d e s c r i b e d by
i = 1 ..... m
n,
where
el, u i ~ L i pe
integer
n i , and
for some fixed Sij
n.
n.
: Lpe ÷
pe
p 6
[i,~]
For a d i s c u s s i o n of the
r e l a t i o n s h i p b e t w e e n the s y s t e m d e s c r i p t i o n s S e c t i o n 2.2.
A s s o c i a t e d w i t h the system
system d i g r a p h
D as
follows:
and c o n s t r u c t an edge
(vi,v j)
and some positive
Label if
m
(i) and
(6.2.1),
see
(i), we c o n s t r u c t the v e r t i c e s as V l , . . . , v m ,
S~i. ~ 0
If
D
is not
s t r o n g l y connected,
we can find the s t r o n g l y c o n n e c t e d components
of
in S e c t i o n 4.2
~ , as d e t a i l e d
loss of g e n e r a l i t y that
.
We now d e f i n e an e s s e n t i a l
Definition and
E
is
Let
E A
~V2
D
D .
d e n o t e the v e r t e x set
G i v e n any subset
V2
of
V ,
is d e f i n e d as the d i g r a p h w h o s e edge set
[ (V-V2)x (V-V2) ].
In o t h e r words, by
set of
V = { V l , . . . , v m}
the set of edges of
the s e c t i o n g r a p h
H e n c e we can a s s u m e without
~ is s t r o n g l y connected.
(i) r e m o v i n g from
removing from
E
the section g r a p h V
all the v e r t i c e s
all the edges t h a t either
DV2 in
is o b t a i n e d from V 2 , and
(ii)
leave or enter a
127 v e r t e x in
V2 .
Definition essential
set of
A set of v e r t i c e s
Let d e f i n e d by
A • Rm × m
aij
(vi,v j) • E if A
to contain
DV2
is c a l l e d an is a c y c l i c
self-loops).
d e n o t e the a d j a c e n c y m a t r i x of
if
(vi,vj) 6 E
Similarly,
Now, then
= 1
V 2 C V
if the s e c t i o n g r a p h
I~V2 i s p e r m i t t e d
(however,
DV 2 set),
?
let
A
and
aij = 0
0 ,
if
d e n o t e the a d j a c e n c y m a t r i x of
is a c y c l i c (i.e. if V 2 is an e s s e n t i a l ~V 2 can be m a d e into a lower t r i a n g u l a r m a t r i x by a
s y m m e t r i c p e r m u t a t i o n of rows and c o l u m n s r e n u m b e r i n g the v e r t i c e s
in
(which c o r r e s p o n d s
V-V2).
To see how this c o n c e p t can be used in a n a l y z i n g system
(i),
diagraph
let
V2 c V
D , and let
be an e s s e n t i a l V 1 = V-V 2 .
W i t h this r e n u m b e r i n g ,
V2
the
set of the s y s t e m
N u m b e r the v e r t i c e s
such a way that the a d j a c e n c y m a t r i x of and number e v e r y v e r t e x in
to
DV2
in
is lower-triangular,
higher than e v e r y v e r t e x
the a d j a c e n c y m a t r i x
V 1 in
A
of
~
in
V1
is of the
form
k
m-k
IAll
A12 1
k
LA21
A22 ~
m-k
A =
where
k
is the number of v e r t i c e s
triangular. way,
in
V 1 , and
If we r e n u m b e r the u n k n o w n s
the system e q u a t i o n s
All is lower-
el,...,e m
in the same
(i) b e c o m e
5a
i m e. = u. - [ S.. e. - [ l z j -i 13 ] j=k+l
5b
m e. = u. - [ S.. e. , l i j =i l] 3
S.. e. , 13 ]
i = k+l,...,m
i = 1 ..... k
128
Let
us a g g r e g a t e
the e q u a t i o n s
(5) by d e f i n i n g
6a
RII
=
(Sij)
,
i = 1,...,k
; j = l,...,k
6b
RI2
=
(Sij)
,
i = 1,...,k
; j = k+l,...,m
6c
R21
=
(Sij)
•
i = k + l ..... m
; j = 1 .... ,k
6d
R22
=
(Sij)
,
i = k+l, .... m
; j = k+l,...,m
7a
dI =
(e i) ,
i = 1,...~k
7b
d2 =
(e i) ,
i = k+l,...,m
8a
vI =
(v i) ,
i = i, .... k
8b
v2 =
(v i) ,
i = k+l,...,m
With
these
definitions,
the
system
9a
d I = v I - R I I d I - RI2
d2
9b
d 2 = v 2 - R21
d2
where
the o p e r a t o r
"triangular"
d I - R22
I+RII
nature
of
is e a s y RII
.
equations
(5) b e c o m e
to
because
This
invert
is the p o i n t
of
of
the
the
renumbering.
From like
to m a k e
possible, minimum set.
~y
the
k
as
of v e r t i c e s , finding
of k n o w n
than
digraph).
(9), w e
the
in
can derive
V I) set
a so-called essential and solve
(the n u m b e r there
equations
small-gain and
that we would as l a r g e
as
containing minimum
a
essential
set of a g i v e n
the c o m p u t a t i o n a l
to
essential
(6.1.1)
set
m
however,
system
Theorems
the
a minimum
"near-minimal"
Once
generalize
i.e.
algorithms
In p r a c t i c e ,
it is c l e a r
an e s s e n t i a l
problem,
any polynomial
finding
(i.e. V2
is a n N P - c o m p l e t e
complexity
form
discussion,
number
In g e n e r a l ,
faster
above
choosing
number
digraph
for
the
this
are
problem
increases
of vertices
efficient
of t h e
algorithms
sets.
have type
been
rearranged
stability
(6.1.4).
in the
criteria
that
129
l0
Theorem
Consider
is well-defined. < ~
Suppose
the system yp(Rii)~
Under these conditions,
ii
yp(R21(I+Rll )-l) Proof
• 7p(R12)
12
dI =
13
d 2 = v2-R21(I+RII )-l Suppose norms
now that
< ~ Vi,j
the system
We can rewrite
(I+Rll)-I
(9), and suppose , and
(I+RII)
-i
7p((I+RII )-I)
(9) is L -stable P
if
< 1
+ yp(R22)
(9) in the form
(Vl-R12d 2)
(ll) holds,
(Vl-R12d2)
and that
- R22d 2
Vl, v 2 e L
Then,
P
taking
in (13), we get
14
lld211Tp
~ IIv21[ p + 7p(R21(I+RII )-I)
lid211Tp where
8
is a finite
+
constant.
7p(R22) lid211Tp Rearranging
< lid2 I ITp - [I-yp(R21(I+RII)-I).
15
(IlVlIlp
+ 7p(R12)
+ S
(13), we get
Yp(RI2)-Yp(R22)
]-i
[l Iv211p + 7p(R21(I-Rll )-1 I1Vll Ip + B] It is now routine Remarks dI
and
original
d2
to verify
set of unknowns
to any arbitrary el,...,e m
to an essential
easy to verify whether
or not
if Sii = 0 in (i) whenever automatically satisfied. Theorem
16 (I+R22)-I
L -stable P
Consider
if
< ~ ¥i
.
.
(10), we can take partitioning
However,
set of vertices, (I+RII)-I
of the
by choosing we make
exists.
d2
it very
For instance,
v i @ V 1 , then this assumption
the system
are well-defined.
yp((I+Rii )-I)
(9) is L -stable.[] P
In order to apply T h e o r e m
to correspond
to correspond
that the system
Under
Suppose
is
(9), and suppose
(I+RII)-I,
yp(Rij)
, and
these conditions,
< ~ ¥i,j
the system
(9) is
130
17
7p((I+R22 )-I) . 7p(R21(I+RII )-I) . 7p(R12 ) < 1 Proof
18
d2 =
In this case,
(I+R22)-I
The rest of the proof Theorems
(13)
is equivalent
[v2-R21 (I+RII)-l
is similar
(10)
and
to
(Vl-R12d2)]
to that of Theorem
(16) both reduce
(10). []
to Theorem
(6.1.1)
in the case where RII = R22 = 0 , RI2 = - G 2 , R21 = G 1 , which corresponds to the single-loop system (6.1.2). The next two results
generalize
they are quite
Theorem
Theorem
19 Suppose linear. 20
Suppose
in addition Under
Their proofs
the conditions
that either
v2 = 0
these conditions,
Theorem Suppose linear.
22
Suppose
in addition Under
(6.1.7)
Theorem is a w e l l - d e f i n e d
Consider linear
R21(I+RII )-I
of Theorem
the system
some generalizations and
the system
(9) is L -stable P
K
is
if
< 1 of the loop-transfo~
(i), and suppose
Suppose
7p(Rij)
(I+RII)-I
< = ~i,j
is linear,
< 1 we have from
,
the system with
7p[(I+Rll )-I RI2+K(R22+I)] + yp(R22)
Since
(16) hold.
R21(I+RII)-I
and that 7~(I+RII )-I) < ~ Under these conditions, (i) is L -stable if we can find a linear operator K P yp(K) < ~ such that (I+KR21)-I exists, and
Proof
is
if
(6.1.9).
operator.
7p(R21(I+KR21)-I).
(I0) hold.
(9) is L -stable P
or that
7p(R21(I+RII )-I RI2)
We now present
24
as
< 1
v2 = 0
these conditions,
theorems
or that
the conditions
that either
7p((I+R22)-l).
mation
are omitted
of Theorem
the system
7p[R21(I+RII )-I R12 - R22]
21
23
(6.1.4).
straightforward.
(9b)
that
131
25
Kd 2 = Kv 2 - KR21 d I - KR22 d 2 Substituting (i+Rll) -i
26
from
(25) into
(12), and using the linearity of
gives d I = (I+RII)-I Vl-(I+Rll )-I RI2 d 2 - K d 2 + K d 2 = (I+RII)-I Vl-[(I+Rll )-l RI2 +K] d 2 + Kv 2 - KR21 d I - KR22 d 2
27
d I = (I+KR21)-I
[(I+RII )-I Vl+KV 2 - ((I+RII)-IRI2
+ K(R22+I))d 2] = (I+KR21)-I
[W-((I+RII)-IRI2
where W = (I+RII)-I v I + Kv 2 E L P Substituting (27) into (9b) gives 28
+ K(R22+I))d 2]
whenever
Vl, v 2 E L
d 2 = v2-R21(I+KR21 )-I [W-((I+Rll)-lRl2+I))d2 ] - R22d 2 The conclusion of the proof is now straight-forward. norms in (28), and assuming
29
P
that
Taking
v 2 , w E Lp , we get
Ild2llTp ~ llv211P+~p(R2x(x+~R2x )-l) [IIwll + 7p((I+RII)-IRI2+K(R22+I))
÷ ~p(R22) lid211Tp SO the system is L -stable if (24) holds. P
lld21[Tp]
NOTES
The Sandberg scale
small
[San.
systems
The
connection
was
established
matrices
are
and Porter
gain
theorem
2] a n d
Zames
[Zam.
by Cook
[Coo.
i]
between
these
by A r a k i
due
for 3].
results 3].
[Por.
i] .
[Moy.
i] , a n d
The
stability
criteria
are
due
to C a l l i e r , for
some
based
results
the
results
[Fie.
is d u e
extended
small
to
gain
on n o n n e g a t i v e [Ara.
results
c a n be
[Sil.
i] .
to t h a t
found
in [Sil.
presented
i] .
here.
i].
theorem
Siljak
[Cal.
to
large-
[Por.
i] , A r a k i
on an e s s e n t i a l
and Desoer not
and
systems
and Michel
Additional
is s i m i l a r
Chan
It w a s
The
and Ptak
of M o y l a n
stability
feedback
and P o r t e r
[Ara.
to F i e d l e r
of c o n n e c t i v e
i]
REFERENCES
and Michel
in the w o r k
[Las.
AND
The
i] found concept I].
set d e c o m p o s i t i o n See
[Mic.
i]
and
CHAPTER 7: DISSIPATIVITY-TYPE CRITERIA FOR L2-STABILITY
In this Chapter, vity-type" connected
important,
These c r i t e r i a differ
type of c r i t e r i a c o n t a i n e d
in C h a p t e r 6.
(6.2.12)
and
(6.3.10)
Most
c r i t e r i a are "sign-
w h e r e a s the b a s i c s m a l l - g a i n type c r i t e r i a
whereas
inter-
in several ways from
the b a s i c d i s s i p a t i v i t y - t y p e
sensitive", Theorems
"dissipati-
c r i t e r i a for the s t a b i l i t y of a l a r g e - s c a l e system.
the "small-gain"
result,
we state and prove several
(e.g.
are " s i g n - i n s e n s l t i v e " .
As a
s m a l l - g a i n type c r i t e r i a can be t h o u g h t of as
"weakinterconnection"
conditions,
dissipativity-type
criteria
are in g e n e r a l less c o n s e r v a t i v e and can be i n t e r p r e t e d as "beneficial interconnection" dissipativity-type errors to belong
conditions.
outputs,
and
to an e x t e n d e d inner p r o d u c t space, w h e r e a s
c r i t e r i a r e q u i r e all the inputs,
the
small-qain criteria require only a normed consequence,
On the o t h e r hand,
space setting.
As a
s m a l l - g a i n c r i t e r i a can be d i r e c t l y a p p l i e d to study
L - s t a b i l i t y for all v a l u e s of p , w h e r e a s d i s s i p a t i v i t y - t y p e P c r i t e r i a can be d i r e c t l y applied to study o n l y L 2 - s t a b i ~ i t y . In order to use d i s s i p a t i v i t y m e t h o d s
to d e r i v e
L -stability
criteria,
one has to r e s o r t to the t e c h n i q u e of e x p o n e n t i a l
weighting
(see C h a p t e r
9).
Finally,
it is i n t e r e s t i n g to note
that there are m a n y m o r e d i s t i n c t g e n e r a l i z a t i o n s of the singleloop p a s s i v i t y t h e o r e m
than there are of the s i n g l e - l o o p
small-
gain theorem.
T h r o u g h o u t this chapter,
we a s s u m e that each of the
i n t e r c o n n e c t i o n o p e r a t o r s can be r e p r e s e n t e d by a c o n s t a n t m a t r i x of a p p r o p r i a t e dimension, the i n t e r c o n n e c t i o n matrix, (see S e c t i o n 2.2), generality,
as in
and use the symbol (2.2.6).
to d e n o t e
this a s s u m p t i o n does not r e s u l t in any loss of
and can always be s a t i s f i e d by i n c r e a s i n g
of s u b s y s t e m s if necessary.
Actually,
the number
we can r o u t i n e l y e x t e n d
all of the results of this C h a p t e r to the case w h e r e linear operator, but not n e c e s s a i r l y a matrix. case,
H
As n o t e d p r e v i o u s l y
Hij
However,
the s t a b i l i t y "criteria" r e q u i r e a linear o p e r a t o r
strongly passive "criterion"
(see D e f i n i t i o n
(3.2.16)).
is a in this to be
S i n c e such a
i n v o l v e s v e r i f y i n g a p r o p e r t y of the o v e r a l l
system,
134 it w o u l d go against the g e n e r a l p h i l o s o p h y is to check the s t a b i l i t y of the c o m p l e x e x a m i n i n g various it is p o s s i b l e
subsystems
linearities Hij
nonlinear Instead,
and dynamics
definite
criteria
If.If 2
we use
If.If T
and
in place of
If.lIT2
the basic
E v e n though these theorems
for large-scale
systems,
both for their h i s t o r i c a l
physical
this section,
and
SYSTEMS
systems.
results
of interest,
in this
of brevity.
we state and prove
well as for their a p p e a l i n g
they
value as
interpretation.
we consider
a single-loop
system d e s c r i b e d by
la
el = Ul - Y2
ib
e2 = u2 + Yl
ic
Yl = G1 el
id
Y2 = G2 e2 where
If.If
for s i n g l e - l o o p
are subsumed by s u b s e q u e n t
feedback
our
being positive
study L 2 - s t a b i l i t y
SINGLE-LOOP
In this section,
Throughout
As a result,
test m a t r i c e s
in the interests
7.1
are n e v e r t h e l e s s
i,j
and assume that
semi~efinite..
respectively,
dissipativity
but we do not do this,
we prefer to absorb all non-
for all
Since we e x c l u s i v e l y chapter,
operators;
involve certain
or p o s i t i v e
Similarly,
here to the case where the
into the subsystems,
is a constant matrix,
stability
system by
and interconnections.
to extend the results
H..'s are m e m o r y l e s s 13 for the same reason.
adopted here, which
overall
Ul' u2'
some positive
el' e2' Yl' Y2 integer
~
and
all b e l o n g GI, G 2
'
Recall now the following
to the space
map
L~ 2e
LU2e
for
into itself.
r e s u l t from Chapter
6.
135
Lemma
Suppose
and t h a t
d e R .
Then
depending
o n l y on
A
A e R nxn is p o s i t i v e there
and
exist constants
d , such
definite, y
and
that whenever
b • Rn ,
c ,
x 6 Rn
satisfies
x' A x < b' x + d
x
also
where
satisfies
II-II
denotes
Proof
the E u c l i d e a n
See Lemma
We now present
Theorem i = 1,2,
n o r m on
(6.2.53).
the m o s t g e n e r a l
Suppose
Rn
there
exist
s u c h t h a t the o p e r a t o r s
G1
r e s u l t of this
real
constants
and
G2
~ T >- 0
, V x e L ~x e
section.
ci'
satisfy
6i' Sl'
the
conditions
<x,GiX>T Then
illxll
the s y s t e m
÷
~i + ~2 > 0
7b
a2 + ~i > 0
T
F r o m the s y s t e m e q u a t i o n s
+ T
= T
= T
+ T
Cl]lelllT2 + ~211Y211~ Substituting
2 + si
for
'
(i) is L 2 - s t a b l e , p r o v i d e d
7a
Proof
6iltGiXllT
eI
(i), w e h a v e
+ T
+ 6 1 ] l Y l l i 2T + ~i + £ 2 1 1 e 2 1 1 ~ + ~2
from
(la)
and e 2 from
(ib) g i v e s
i=l,2
136
T + T ~ ~l]lUx-Y211~ + ~ l l l Y l l [ ~
+ ~I+~211Y2+YllI#
+ 6211Y2[IT 2 + s 2 =
~l~II~lII# - 2T * IIy211~l + ~lIIYlIIT2 + 2T + IIYzlIT]2 + 8211Y2IIT2
+ ~l[llu21l~
+
~i
+ ~2
Applying Schwarz's inequality Suppose now that Ul, u 2 e L 2 repeatedly in (9), collecting all the quadratic terms in yl,y 2 on the left, l0
and the remaining
terms on the right,
gives
(~2+~l) llyzll~ + (~x+~2) llY211# ~ IlYxtlT(tluzll
+21~21[Iu21II
+ l lw211T([lu2II
+ 2f~zlllUlIl)
+ 1~lI llulll 2 + 1~21 Ilu21I 2 + Sl + ~2 where we have replaced bounds
[lUlII
of
is a positive
(i0)
IIy211T and
and
whenever
IIy211T
llUll IT flu211
are bounded
Ilu211T
respectively,
definite
(7) holds,
and
quadratic
it follows
independently
by their upper
since
the left side
form in
I[ylIIT
by Lemma
(2) that
of
, IIylllT
T'ie'YI' Y2 E L 2
From (la) and (ib), this implies that el, e 2 e L 2 . Furthermore, it is clear from (10) that the coefficients of the linear terms in I lylIIT , fly21 IT are themselves linear in Thus, with reference to Lemma (2), the vector b that are linear argument
in
that,
there exist
IIUlII,
not only do
finite
constants
flu211 yl,y 2 y
Now, belong
and
6
IIUlIl, I I~211. has components
it is clear from these to L v2 , but also
such that
I ly211 ~ y([IUll I + flu21 I) + ~ similar remarks e I , e 2 . Thus the system (i) is L2-stable. D Several
interesting
cases of Theorem
(5).
which
known as the passivity
is usually
systems):
Chief
criteria
flY 1 I, apply
can be obtained
as special
among them is the following theorem
to
result,
(for feedback
137
Corollary
ii such
Suppose
there
exist
real
constants
that
12
< x , G 1 x > T ~ ~ llxl
2 T
'
VT >_ 0
,
Vx e
L~ 2e
13
!IGlX'! T -< E IIxl
T
'
~
'
vx~
n ~2 e
14
<x'G2X>T
T2
,
%~2
,
Vx
Then
the
system
15
~ ~ IIxl
(I)
is L 2 - s t a b l e ,
> 0
-
> -
0
E
L ~
2e
provided
e + 6 > 0
Proof
16
Since
(15)
holds,
pick
8 > 0
such
that
e - e + ~ > 0
NOW,
17
e, ~,
(12)
and
<~,GI~> T _
>
(13)
together
~ I1~11~ = (~-e)
apply
that
IlxllT
IT,,x~, T2 + (e/~2) r[
(~-e) Now
imply
Theorem
(5) w i t h
2 + e Ilxl IIGlXlIT
e I = e-e,
2 IT
2
VT
61 = e/~
2 ,
> 0
,
-
~xE
L~ 2e
e 2 = 0 , and
~2 = 6 .o It is a l s o relaxed
to a n o t h e r
_ <
IIGIXlIT
18
However, (5) w o u l d Since only
in this be
in no w a y
study
the
first
version
of
to p r o v e the
is,
(ii),
following
the p a s s i v i t y
(ll)
with
(13)
version
speaking,
slight
assume
result,
that
which
theorem.
of T h e o r e m
as T h e o r e m
of C o r o l l a r y
we
L ~2e
VX 6
general
elegant
this
the usefulness
,
> 0
generally
L -stability,
the
VT
more
is n o t as b
Corollary
form:
,
+ b
a slightly
which
in C o r o l l a r y
6 = 0 , then we get
of
IlXllT
constant of
diminishes
If,
>
case,
needed,
the a d d i t i v e in the
possible
condition
loss
(5). present
of g e n e r a l i t y
(ii).
e
>
0
and
is h i s t o r i c a l l y
138 19
Corollary and
~
S u p p o s e there exist p o s i t i v e c o n s t a n t s
c
such that i
20
<X,Gl.>
T
>-
~
i
i
i ~
,,~llxltZ
vT
'
>-
0
'
L~
vx
2e
21 -
Suppose
G2
22
'
-
'
28
satisfies
<x,G2x> T > 0 , ~T > 0 , Vx 6 L v 2e U n d e r these conditions,
the s y s t e m
An o p e r a t o r operator,
G2
(i) is L 2 - s t a b l e .
satisfying
w h e r e a s an o p e r a t o r
G1
is c a l l e d a s t r i c t l y p a s s i v e operator. states the following:
Suppose
finite gain w i t h zero bias, system
is called a passive (28) w i t h
Hence Corollary
G2
is s t r i c t l y p a s s i v e and has
and that
G2
both Corollaries
i n t e r c h a n g e d throughout.
"symmetric"
is passive;
c o n d i t i o n s on
G1
and
(ii) and
are w o r t h m e n t i o n i n g .
23
Corollary passive;
N o t e that T h e o r e m
i.e.,
(5) imposes
G2
(5) that
Both are easy to prove.
Suppose both
G1
and
G2
are s t r i c t l y
suppose there e x i s t p o s i t i v e c o n s t a n t s
<x'G~X>T~ >- eiIIxIl~" • T Then the s y s t e m
25
26
G1
eI
and
such that
24
~2
'
%~f > 0 , Yx 6 L v2e ' i = 1,2
(i) is L 2 - s t a b l e .
Corollary and
then the
(19) hold w i t h
T h e r e are two o t h e r c o r o l l a r i e s of T h e o r e m
~2
e > 0 (19)
(i) is L2-stable.
Actually, and
G1
(22)
satisfying
Suppose
there e x i s t p o s i t i v e c o n s t a n t s
such that
<x,Gix> T ~ ~±lIGixll
v T2 , WT > _ 0 , VX E L2e
, i = 1,2
~i
139
T h e n the system
(i) is L 2 - s t a b l e .
N o t e that there is no a s s u m p t i o n of finite g a i n in Corollary
(23), and that there is no a s s u m p t i o n of strict passivity
in C o r o l l a r y
(25).
Also,
the results of s e c t i o n 3.2 are v e r y
useful for d e t e r m i n i n g the v a r i o u s c o n s t a n t s c o r r e s p o n d i n g to a g i v e n o p e r a t o r
7.2
E
and
6
G .
GENERAL D I S S I P A T I V I T Y - T Y P E C R I T E R I A
In this section we state and p r o v e some g e n e r a l d i s s i p a t i v i t y - t y p e c r i t e r i a for the L 2 - s t a b i l i t y of l a r g e - s c a l e i n t e r c o n n e c t e d systems.
S p e c i a l cases of the g e n e r a l results
g i v e n here, w h i c h are easier to apply,
are p r o v e d in the n e x t
section.
T h r o u g h o u t this section,
and i n d e e d t r o u g h o u t the rest
of this chapter, we shall be c o n c e r n e d w i t h a s y s t e m d e s c r i b e d by m
la
ei = ui -
~.
j=l
Hij Yj i = l,...,m
ib
Yi = Gi ei where Gi
u i , ei" Yi
maps
dimension
all b e l o n g to
n. L 2 e l into itself, nixn j .
and
Then matrix
n. L 2ez Hi3.
=
[zlil
is c a l l e d the i n t e r c o n n e c t i o n matrix.
We b e g i n w i t h an obvious
ni
is a c o n s t a n t m a t r i x of
H 6 R n×n
defined by
H
for some i n t e g e r
(where
n =
m ~ n i) i=l
140
3
Lemma i = l,...,m
.
4
Suppose
Then
is
Q = Diag
{QI'''''Qm }
5b
R = Diag
{R 1 ..... R m}
5c
S = Diag
{S I , . . . , S m}
Apply
Definition
The next result t h a t of the i n t e r c o n n e c t e d
Lemma (4) a n d Then
Consider
(3.2.1).
relates system
the dissipativity
Suppose
(1) is d i s s i p a t i v e
of
G
with
(1).
the s y s t e m
(2), r e s p e c t i v e l y .
the s y s t e m
by
where
5a
6
defined
for
{G 1 ..... G m}
(Q,R,S)-dissipative,
Proof
(Qi,Ri,Si)-dissipative
G : L n2e + Ln2e
G = Diag is
Gi
(i), a n d d e f i n e G
with
is
G,H
by
(Q,R,S)-dissipative.
respect
to the t r i p l e t
(Q,R,K) , w h e r e
7a
Q = Q + H'RH
7b
R = R
7c
S = S - 2H 'R
Proof written
9
The system equations
(i) c a n b e c o m p a c t l y
as
8
e
Since
1 - ~ (SH + H'S')
G
is
=
u
-
Hy
,
y
Ge
(Q,R,S)-dissipative,
T + <e'Re>T
Substituting
=
for
e
from
we have
+ T
(8) i n t o
> 0 , -
(9) g i v e s
%~f >- 0
,
V e 6 L n2e
141
i0
T + < u - H y ' R u - R H Y > T
+ T
> 0 -
or, after e x p a n d i n g
ii
and c o l l e c t i n g
VT > 0 '
-
Vu 6 L n2e
'
terms,
T + T + T >_ 0 , VT >_ 0 , Vu 6 L n 2e Since
u
is the input and
y
(ii)
shows that the system
that
Q
is the symmetric
is the output of the system
(i) is
the basic d i s s i p a t i v i t y - t y p e
for L2-stability. Theorem
12 (Q,R,S)
(Note
Q+K'RH-SH).
The next theorem gives criterion
(Q,R,S)-dissipative.
part of
(1),
Consider
dissipative.
the system
Then the system
(Ta) is negative
definite.
Proof
By Lemma
(3.2.10),
then the r e l a t i o n m a p p i n g
u
into
(i), and suppose (i) is L 2 - s t a b l e
if y
Q
G if
is Q
of
is n e g a t i v e definite,
has finite gain with
zero
bias. [] Remarks
As was pointed out in Chapter
of gain and d i s s i p a t i v i t y in Chapter
can be defined
3 they are defined
3, the concepts
for relations,
for o p e r a t o r s
though
to keep the exposition
simple. The next result gives a n e c e s s a r y condition
for the r e l a t i o n b e t w e e n
gain with zero bias. testing procedure, Theorem
but is useful
Theorem
Consider
denote the r e l a t i o n b e t w e e n these conditions, Q,R,S
such that
negative
definite.
~2(F) G
and
y
to have finite
The n e c e s s i t y part is of little value as a
(12) is a r e a s o n a b l e
13
u
and s u f f i c i e n t
is
< ®
to show that the c o n d i t i o n
of
one. the system
(1) , and let
F C L2en ×L2en
u
as defined by
(i).
and
y
if and only if there exist
(Q,R,S)-dissipative
and
Q
of
Under matrices
(7a) is
142 Proof
"if" i m m e d i a t e
"only if"
14
~2(F)
-T + < u ' ~ ( F ) U > T However,
since
u
e+Hy
(14).
This gives
in
15
and
y
and c o l l e c t i n g
T
(12).
Then
Yu, y e F
(1), we can replace
(e+Hy)> T
terms in
+ <e,~
< ~
~ 0 ,
satisfy
-T + ~ e + H y , ~ ( F ) Expanding
16
Suppose
from T h e o r e m
> -
0 ,
u
~e • L n2 e
by
' y
=Ge
(15) gives
2(F)H'e>T (F)e>T+2
> 0 -
Ve • L n2e ' y = Ge This
17
shows that
Q = ~(F) Also,
G
from
that
Q = -I n , w h i c h is negative
to give some f l e x i b i l i t y
(12), we can introduce
some auxiliary
constants. This
type,
and all s u b s e q u e n t c r i t e r i a are special cases of it. Theorem
stability
in the application
the m o s t general
Consider
(Qi,Ri,Si)-dissipative
for
criterion of the d i s s i p a t i v i t y
the system i = l,...,m
the system
(i) is L 2 - s t a b l e
ll,...,l m
such that the test m a t r i x M = - ~AQ+H'ARH]
is p o s i t i v ~ definite,
20
(14)
H'
gives
18
19
with
[] In order
of T h e o r e m
(Q,R,S)-dissipative
H'H-I n , R = ~ ( F ) I n , ~ = 2 ~ ( F )
it is obvious
definite.
is
.
Under
+ (ASH+H'S'A)/2
Inl''''Xm Inm}
Gi
is
these conditions,
if there exist positive
where
A = Diag {Al
(i), and suppose
constants
143
Proof
21
Since
Gi
T
is
(Qi,Ri,Si)-dissipative,
+ <xi,RiXi>T
we have
+ T [ 0 , n. > 0 , Vx i E L a 2e
VT
-
Multiplying m
shows
triplet applying case
-M
(21)
that
by
G
(AQ,AR,AS) Theorem .
definite,
Xi
of
the
ASH
23
is p o s i t i v e
the
ASH
.
Then
the
constants
is p o s i t i v e
(5).
Now,
is in
if
M
to
this is p o s i t i v e
three
useful
corollaries
(i)
for
some
v' ASH
choice
v >_ 0
is L 2 - s t a b l e
if
of
¥u E R the
A , the n * . Under
test marrix
is p o s i t i v e , so is
Suppose system
H
(i)
XI,...,X m
then
M
is n o n s i n g u l a r ,
that
+ AR]
+
(P'AS
that
N = P'MP
so t h a t
. []
is L 2 - s t a b l e such
M { M o,
the
and
if t h e r e
let exist
test matrix
+ S'AP)/2
definite.
Proof M > 0
we r e s e r v e matrices.
with
that,
i.e.,
definite,
N = -[P'AQP
if
by
(7a)
1
to the
+ H'ARH]
If
is p o s i t i v e
25
only
of
is L 2 - s t a b l e
section
system
Corollary
positive
Q
from
definite.
Proof
24
this
Suppose
M o = -[AQ
P = H -I
varies respect
are defined
that
(i)
is p o s i t i v e ,
conditions,
Mo
i with
(18).
Corollary
22
if
Q,R,S
we get
system
We conclude
these
as
o
of T h e o r e m
matrix
summing
is d i s s i p a t i v e
where (12),
Hence
, and
(4)
the
Note
, so t h a t
N > 0
if a n d
.D
term
"positive
semidefinite"
for
symmetric
144 26
Corollary A, and that
H
S u p p o s e that
is nonsingular.
ASH
is p o s i t i v e
T h e n the system
for some
(i) is L2-stable
if the test m a t r i x
27
N
= -[P'AQP + AR]
o
is positive.
P
Proof
o n l y if
7.3
Note that
N o = P'MoP,
so that
No > 0
if and
M > 0 .~
S P E C I A L CASES:
Theorem
S~LL-GAIN
(7.12.18)
AND P A S S I V I T Y - T Y P E C R I T E R I A
is a p o w e r f u l general r e s u l t that can
be applied
to a wide v a r i e t y of situations,
operators
Gi
are s t r i c t l y passive,
etc.
can o b t a i n several useful, criteria. always,
e.g.
w h e n some of the
have finite gain, others are passive, By s p e c i a l i z i n g T h e o r e m and r e a d i l y applicable,
Some such r e s u l t s are p r e s e n t e d
still others (7.2.18), one
stability
in this section.
As
we study a system d e s c r i b e d by m
la
e i = u i = j=l [
H ij Yi } i = l,°..,m
Ib
Yi = Gi e i
where and
ui' ei Hij
'
n. Yi e L 2 ei
for some integer
is a c o n s t a n t m a t r i x of d i m e n s i o n
First, we p r e s e n t a "small gain"
n. n. 1 ÷ L 1 n i ' G i : L 2e 2e
'
n i x nj
type c r i t e r i o n based
on the d i s s i p a t i v i t y approach.
Theorem ¥i
By
.
C o n s i d e r the s y s t e m
Define the test m a t r i x
A [ B
W 6 R n×n
(A > B), we m e a n that
(positive definite)
A-B
(I), and suppose ~2(Gi)<~ by
is p o s i t i v e
semi-definite
145
Wll W =
•.. W l m
[ 1 Wml
4
Wij Under
these
= ~2(Gi)
conditions,
find p o s i t i v e
5
•
M
is p o s i t i v e
the s y s t e m
constants
=
ll,...,lm
(i) is L 2 - s t a b l e
=
Diag
Proof
s u c h t h a t the m a t r i x
where
{AI In2
Since
..... Am I n m }
~2(Gi)
< = , we c a n w r i t e
2
< - G i x , G i x > T + <x,~ i x> T ~ 0, where
Gi
we use
~i
= ~2(Gi)
is dissipative
Theorem
=
~
~ 0,
in the i n t e r e s t s
with respect
(7.2.18),
M
if o n e c a n
A - W'AW
definite,
A
Hij
to
of b r e v i t y .
2
(-In.,g i In ,0). 1 1
we g e t t h a t the t e s t m a t r i x
A
-
n. l
Nx e L2e
M
of
Thus
Applying (7.2.19)
H'ARH
where
R = Diag N o w n o t e that,
since
2 {~i In I'
2 "'~m I n m }
R is d i a g o n a l ,
we h a v e
AR = R I / 2 A
R I/2
Hence M
i0
Thus
=
the s y s t e m
is p o s i t i v e
A - H ' R I/2 A R I/2 H = A- W ' A W =
(i) is L 2 - s t a b l e definite,
o
if
A
c a n be f o u n d
such that
is
146
N o t e that, theorem
[Fre. I,
if and o n l y if
by the d i s c r e t e - t i m e v e r s i o n of Liapunov's
p.166],
one can find a
p(W)< i.
However,
not be able to find a d i a @ o n a l
A > 0
even if
A > 0
such that
p (W) < 1 ,
such that M > 0 .
try to give a m o r e e x p l i c i t c r i t e r i o n than T h e o r e m
M > 0
one may If we
(2), i.e.,
one that does not d e p e n d on b e i n g able to find some u n k n o w n constants
(with no s y s t e m a t i c p r o c e d u r e
r e s u l t that is very similar 11
Theorem ¥i.
to T h e o r e m
Consider
the s y s t e m
D e f i n e the test m a t r i x
12
nij = where
I IHijl I
N E Rm × m
~2(Gi ) IIHijll is the
to find them), we get a (6.2.71).
(i), and suppose ~2(Gi)<~ by
= ~i
llHijll
£ 2 - i n d u c e d n o r m of the m a t r i x
Under these conditions,
the system
(i) is L 2 - s t a b l e
leading p r i n c i p a l m i n o r s of the m a t r i x
I-N
Hij
if all
are positive.
The proof d e p e n d s on the f o l l o w i n g lemma, w h i c h is p r o v e d in
[Ara. 2].
13
Lemma
Suppose a m a t r i x
n e g a t i v e elements, Im-A
are all positive.
diagonal matrix
A 6 Rm × m
Under these conditions,
A > 0
such that
Im-A
there exists a
A - A'A A > 0
As b r o u g h t out in Lemma aij [ 0
has all non-
and that the leading p r i n c i p a l m i n o r s of
(6.2.8),
the fact that
and the n o n n e g a t i v i t y of the leading p r i n c i p a l m i n o r s of
implies that
there exists a p r o p e r t y of
p(A)
~ > 0
< 1 .
Thus,
by L i a p u n o v ' s
such that ~ - A ' A A > 0 .
A, n a m e l y
a.. > 0 13 -
¥i,j,
theorem,
The s p e c i a l
allows us to select
to be d i a g o n a l as well.
Proof of T h e o r e m h y p o t h e s e s of T h e o r e m diagonal
~ > 0
(ll)
By L e m m a
(II) are satisfied,
such that
(13), if the then there exists a
~ - N' A N > 0 .
the d i a g o n a l e l e m e n t s of this
A , and define
claim that, w i t h this choice,
M
of
To e s t a b l i s h this claim,
Let A by
ll,...,Xm (6).
We
(5) is p o s i t i v e definite.
note that
A-N'AN
> 0
be
147
implies
14
that there
exists
an
e > 0
such that
m
m
m
;. ~ v~ <_
~ ~i v ~ -
~ ~i~Y-
i=l
i=l
m
i=Z
N O W let x e R n , and partition n. x. 6 R i Then 1
15
x' M x
~ij vj ~ , ~ v ~ m
j=l
x
as
[x~ ...x~],
m m m [ lil!xill 2 [ lil I [ i=l i=l j=l
=
m
m
[ ~ilIxi li2i=l
~iHij
xj'.l 2
m
[ ~i (I ~iIIHijl7 llxjI1)2 i=l j=l
m 12 m [ I i llxil - ~ Ii i=l i=l
=
where
_ (~ nij
I IxJI
)2
m
>_
~ ~
Jlxill 2--~ x,x, by Cz4~
i=l
This
shows
that
M > 0.
Thus,
by Theorem
(2),
the s y s t e m
(i) is
L 2 - s t a b l e . [] The r e s t of this passivity-type
stability
t h a t e a c h of the o p e r a t o r s following
conditions:
section
criteria. Gi
is d e v o t e d In w h a t
satisfies
(i) t h e r e e x i s t
to d e r i v i n g
follows,
we a s s u m e
o n e or the o t h e r
constants
ci
and
of the ~i
such that
16
2
Ixr T
< x , G i x > T ~ ci
n.
VT > 0 , Vx 6 L 1 2e
lIGiXllT ~ ~i Ix'
17
or
(ii)
there
exists
T
a constant
~i
such t h a t n.
18
<x'GiX>T
Throughout, positive,
-> ~i
I IGi xl 12 ' %~f -> 0 , V x 6 L2el
we a l s o a s s u m e
i.e.,
t h a t the i n t e r c o n n e c t i o n
matrix
H
is
148
x' Hx > 0
19 If
H
is positive,
then from (1) we get
m [ T = i=l
20
Vx ~ R n
m ~ T i=l
m ~ i=l
m ~ T j=l
m
<_
[ T i=l
Thus the assumption that energy absorbed by the
H m
is positive implies that the total subsystems is no larger than the total
energy entering through the inputs.
In this case, by a slight
abuse of language one can say that the system interconnections.
Further,
holds with an equality,
H
(i) has passive
is skew-symmetric,
and we say that the system
lossless interconnections. feedback system
if
then (20)
(i) has
Note that the standard single-loop
(7.1.1) has lossless interconnections.
We first prove two lemmas concerning matrices. Lemma
21 let
A22 • R
Let
~2x~2
A e R £×£
be positive semi-definite,
be positive definite, where
£i = IAII
£2 AI2 1
£i
A22
£2
A
22
LA'I2
Let
A E R
£1×£1
be positive definite.
Then the matrix
defined by
r j [0I All + A
23
and
B =
AI2
=A+
L A ' 12
A22
B 6 R £x£
149
is p o s i t i v e
definite.
Proof since
it
Hence, show x'
is
to
the
show
that
B
two
B
x'
B x = 0 = x =
.
Then
x'
denotes
the
x~ A22 of
x
.
x2 = 0
Since .
A22
Hence
24
is e n o u g h
it
x 6 R£
and
x' 1 A x I = 0,
~i
x~
components
denotes
is p o s i t i v e and
p,
and
x
definite,
the
Q E
lemma
R £×£
where .
Now,
Hence
the
last
this
to
suppose
of
A xI 0 = xI = 0
x 2 6 R ~2
Let
definite,
Let
x = 0
Lemma
semidefinite,
matrices.
0.
first
, where
positive
semidefinite
A x = 0
definite,
x2 = 0
least
is p o s i t i v e
that
is p o s i t i v e
is a t
positive
B x = 0
xI 6 R A
Clearly s u m of
since
x' A x =
£2
components
implies
that
is p r o v e d .
be
positive
that
P - uQ
semidefinite,
and
suppose
25
rank
Then
there
P = rank
exists
an
[PIQ]
~ > 0
such
is p o s i t i v e
semidefinite.
Proof
The
is c o n t a i n e d
in
space
of
contains
R £x~
as
P
Q
a direct
, which
matrix
the
are
of
the
sum
of
implies
, or space
range
of
subspaces.
of
k
(9) P
null the
orthogonal
representation
26
condition
range
P
that
the
equivalently, of
P P
With
range
that
.
Now,
we
and
the
null
this
new
of
the
can
express
space
basis,
Q
null
of
the
becomes
~-k
P ÷ 0
where Now,
k since
is
the
representation form
the
rank
null of
of
space Q
with
£-k
P
, and
of
Q
P1
is p o s i t i v e
contains
respect
to
the
definite.
that
of
same
basis
P
, the matrix is o f
the
150
27
Since also
P1
is p o s i t i v e
positive
is p o s i t i v e
stability
the
some ~i'
it is c l e a r
sufficiently
for
a i d of
sufficiently
these
two
that
small
~
small
lemmas,
we
Pl-UQ1 .
Hence
~
is P-=Q
.
can
now
state
criteria.
Theorem
28
for
semidefinite
With the
definite,
definite
integer
k < m
i=l,...,k
, such
Consider
the
,
exist
there
system
suppose
(i).
constants
Ei
that
for
and
that n,
29
VT>0,
< x , G i x > T > ~i I Ixl 12 ,
Yx6L
1 2e
-
i=l ..... k 30
l IGixl i~ <_ ~i {IxIIT Suppose
there
exist
constants
%."12 [ 0
,
ni vx 6 L2e
,
6i , i=k+l,...,m
, such
that
n.
31
<x,G=x>~
>
6iI{Gix{i~
,
~
l VX 6 L2e
>- 0
'
i=k+l,...,m
m
Let
r =
~ nj j=l
,
and
partition
the
interconnection
matrix
H
as f o l l o w s :
32
H
Define
=
diagonal
r
n-r
Haa
Hab
Hba
Hbb
I
matrices
n-r
E a q R r×r
and
D b E R (n-r)×(n-r)
by
151
I E
33
Ea
=
Inl ..
0
ek
Db
=
1
Ink
[i °J k+l
34
0
Ink+ 1
"..
8m In m
Under these conditions, (i)
35
the system
the matrix
(i) is L2-stable if
M 1 • R nxn defined by
IN'aa Ea Haa
H'aa Ea
Hab
H'ab Ea Haa
H'ab Ea
Hab
bl
Ml=
+D
is positive semidefinite.
(ii)
the matrix definite.
(iii)
we have
H'ab Ea Hab + D b
H I
36
37
rank
M 1 = rank
(iv)
the m&trix
Proof
Let
Q =
[M 1 I--~. ]aa ab H
Q E R nxn
[" al
is positive. be defined by
E"aa "obl
is positive
152
Then
condition
of
(36)
,
states
that
the r a n g e
or e q u i v a l e n t l y ,
of
M1
t h a t the n u l l
contains
space
of
(37)
that
that
[HaaIHab]
abJ contains space
that
of
implies
Q
of
M1
is the
that
the
Now, same
null
by Lemma
(24),
positive
semidefinite.
there
Now is p o s i t i v e
exist
pick
+ Db
is p o s i t i v e of c o n d i t i o n
that
for
that
space
of an
~>0
and
(ii).
[HaalHab]
a>0
The
From
from
contains
such
definite.
i=l,...,k
of Q
such
sufficiently
semidefinite
view
38
as
it is c l e a r
that
such
is a l w a y s
and
(30),
of
(36)
M1
.
M 1 - ~Q
the m a t r i x
latter
(29)
Hence
that
small
that
the null
Hence
is
that
MI-~ Q
H' (Ea-~I)H ~ ab p o s s i b l e in
it r e a d i l y
follows
,
<x,cix> T ~
~illxl[~
2 + ~ llxl 2 = (~l-~) IlxlIT iT
(ci~ llxI1~+ (~z~ llGixll~, ~ 0 , n.
Vx6
Hence,
for
respect
i=l,...,k
, the o p e r a t o r
Gi
L ~ 2e
is d i s s i p a t i v e
39
Qi =
-
(~/~2)
in . , Ri
=_
(ei_~)
in . , Si
1
Similarly, operator
40
from Gi
Let
is
41
(31),
we
case,
us n o w
see
applicable.
M0 = -
that
for
with
respect
, R . = 0 ni
apply
ASH = H
corollary
is p o s i t i v e ,
Forming
[Q + H'
in .
=
1
is d i s s i p a t i v e
Qi = - 61• In i
In t h i s
with
to
the
RH]
i=k+l,...,m
1
, the
to
, S i = I ni
(7.1.22) so t h a t
test matrix
with
corollary
M 0 , we get
A = In
.
(7.1.22)
153
=
+ 0
=
H'
Db
MI-
0
A
0
0
0
~Q +
where 42
= Diag
{a/~
Inl,''-,u/~
We claim that this m a t r i x the first term, Next,
namely
its s u b m a t r i x
Finally,
A
is positive
Gi
satisfying
Theorem
l)
that
Ei
passive
is positive).
Similarly,
condition
One of the i n t e r e s t i n g
This trade-off
p o s i t i v e as possible
(iii)
~i
(Theorem
Note
and
in order
the s t a b i l i t y 3)
and
an o p e r a t o r Clearly
passive
operators
aspects of T h e o r e m
between
it is clear from T h e o r e m
choose the c o n s t a n t s satisfying
theorem
there is
operators.
the constants
is one of the n o t e w o r t h y
passivity
G i satisfy-
for the L 2 - s t a b i l i t y
of some p s e u d o - s t r i c t l y
is that it allows a "trade-off"
context,
Mo
the system
("pseudo!' b e c a u s e
w i t h finite gain and some p s e u d o - p a s s i v e
~i "
(7.2.22),
Let us refer to an o p e r a t o r
(28) gives a s u f f i c i e n t
single-loop
(21),
(31) is said to be pseudo-passive.
of an i n t e r c o n n e c t i o n
2)
is p o s i t i v e definite.
Hence by Lemma
whence by C o r o l l a r y
(29) as p s e u d o - s t r i c t l y
no a s s u m p t i o n
Hab + D b
definite.
F i r s t of all, semidefinite.
[]
Remarks ing
H'ab (Ea-eIr)
definite,
definite.
M 1 - u Q , is p o s i t i v e
is positive
(i) is L2-stable.
is positive
Ink}
features
(7.1.2)).
(28)
Ei
and
of the
In this
(28) that one should try to
6i
(as appropriate)
to be as
to have the b e s t chance of
conditions.
that if
are a u t o m a t i c a l l y
M1
is positive
satisfied.
definite
then
(ii)
154 4) the
Note that the h y p o t h e s e s
submatrix
H'aa E a H aa
5)
to be p o s i t i v e
The submatrices
the stability
conditions
(iv)
the r e q u i r e m e n t
H
(namely,
be positive).
stability
Thus,
Hab
and
Hbb
do not figure in
(28), except in condition
that the i n t e r c o n n e c t i o n
for all p r a c t i c a l
purposes,
among the p s e u d o - s t r i c t l y
(the i n t e r a c t i o n
t~o the p s e u d o - s t r i c t l y thru
Hba
of T h e o r e m
(iii) of T h e o r e m
passive
subsystems).
(28) are satisfied,
and if
Hba
Hbb
H
subsystems G1
Consider
represented
pseudo-strictly
are
0 H
H aa is to adjust
=
of three
G1 , G2 , G3 ,
and has finite gain,
Let the i n t e r c o n n e c t i o n
1
0
0
-i
1
0
el =ir-]li
and
where
G2 , G3
m a t r i x be
Y5
FIGURE
This system is shown in Figure matrix
M1
(i)
and thereby
an interconnection
by the o p e r a t o r
passive
pseudo~passive.
44
subsystems)
subsystems
the system. Example
43
positive,
(the
If conditions
add "compensation"
so as to make
the
passive
then one can always
and
matrix
Haa
from the p s e u d o - p a s s i v e
positive, stabilize
(28) require
semidefinite.
criteria depend only on the submatrices
"self-interaction" and
of T h e o r e m
in
(35) is
7.1
.
For this system,
7.1
the
155
45
M1 =
I! ° 1 el+62 0
Hence all the conditions system at hand
46
63
of Theorem
is L2-stable
el+~ 2 > 0 ,
(28) are satisfied
if
63 > 0
We now show that Theorem a special
case.
and suppose
Consider
G1 , G2
'
48
llGlXTl T -< ~ IIXIIT
'
49
<x,G2x> T -> 6 TllG2xt12
Then the interconnection
Hence
matrix
M1
C0~ M1
=
It is easy to verify satisfied
if
system
(7.1.2) (7.1.1),
'
~
> 0
~
-
'
> 0
~
'
> 0
-
'
vx ~
~
vxS
~ 2e
2e
vx ~
~2e
is
=
the test m a t r i x
51
Theorem
feedback
satisfy
< x , % x > T -> ~ Irxll~
H
(28) contains
the single-loop
47
50
and the
of
0~ (e+6)
(35) becomes
~I I
that all conditions
e+6 > 0 .
of Theorem
(12) are
156
We now present another stability criterion, which also contains the single-loop Theorem (7.1.2) as a special case, but is in general more restrictive than Theorem (28). However, it has the advantage that it has an instability counterpart (see Theorem (8.3.42)). 52
Theorem Consider the system (i), and suppose G 1 .... ,Gm satisfy (29)-(31), as appropriate. Suppose H is nonsingular and positive, and let P = H -I . Partition P as
r aa
n-r Pa bl
r
LPba
PbaJ
n-r
IP p =
53
Define E a and D b as in (33) and (34), respectively. Under these conditions, the system (i) is L2-stable if (i)
The matrix =FEa + Pba Db Pba
54
N1
ba Pbb P' Db 1
! Db Pbb
LPbb Db Pba
is positive definite. (ii)
we have
55
rank
N 1 = rank
56
rank
Pab = n-r
Proof
By Le~ma
an
e>0
I [N 1 I r Qnn_r ]
(24), if (55) holds,
such that the matrix
then there exists a
157
57
N2 =A NI
is p o s i t i v e
0 ]
0
0
semidefinite.
(30) we have,
58
-I 2~I r
as before,
Let
~>0
that for
be so chosen.
From
(29) -
i=l,...,k
<x, Gix> T _> ¢~i-oo
I lxl I~ + (o~/~.)
I IGixl I~
>_ ¢~i-oo
I lxll~ + ¢~/~2)
i iGixll~
,
~_>
n.
Vx 6 L 1 2e where = max i
59
Hence we define,
~ J-
as in the proof of T h e o r e m
A = Im e
(28),
S = I n , and
60
61
R = -
ITa r
QI If we apply C o r o l l a r y
:] :I
(7.2.26),
the test m a t r i x
No
becomes
o
158
-~I 62 N
o
= - (P'QP+R)
0
Pba Db Pba
Pba Db Pbb
0
+ L P'bb Db Pba
Pbb Db Pbb
= L 0a
I! PeaPab] aa Paa
+ "~--
Pab PabJ
ab Pea
i PP ill J aa
= N2 +
aa
aa Pab
Pab Paa
P'ab Pab
r
0
N3
Now,
N3
is clearly
positive
semidefinite.
Also,
the lowest
(n-r)×(n-r) submatrix of N 3 is positive definite, because P'ab Pab is positive definite (recall that Pab 6 R r×(n-r) has rank n-r). Hence by Lemma (21) , N o is positive definite, whence by Corollary
(7.2.26),
Theorem Theorem
r > n-r
have rank
, i.e. r > n/2
subsystems
not; moreover, (7.1.1)
Theorem
(49).
system of
In this case,
is nonsingular,
and
the latter (6.1.1),
implies
claim,
consider
the interconnection
that
Theorem
(28) does
the single-loop
as does Theorem
and suppose
than
is
that
However,
which Theorem
(52) also contains case,
H
at least half of the
passive.
counterpart,
matrix
the requirement
; in other words,
as a special
To prove feedback
(ii)
n-r automatically
m u s t be p s e u d o - s t r i c t l y
(52) has an instability Theorem
and
D
scope of application
(i~ the interconnection
to be non-singular,
Pab 6 R r×(n-r)
(i) is L2-stable.
(52) has a narrower
(28), because
required
the system
the single-loop
G1 , G2 matrix
result
(28).
satisfy H
(47)
given by
-
(50)
159
63
I9 The test matrix
N1
O,a of (54) is given by
64
It is easy to verify that all conditions of Theorem (52) are satisfied if e+d > 0 . Thus Theorem (52) contains Theorem (7.1.2) as a special case. Next, we further specialized the results of Theorem (28), and in the process, obtain some generalizations of Corollary (7.1.7). Though the results that follows are quite conservative, they have the advantage that they involve only the interconnection matrix H , and are therefore quite easy to apply. 65
Corollary some integer and
Suppose that for
k ~ m , there exist positive constants
~l,...,~k
66
Consider the system (i).
el,...,e k
such that
<x, T . ,Gix> ..
_> Eillxl ~ . , . .I
,
VT _> 0
,
Wx 6 Lnin.2e I i=l,...,k
67
IIGixl IT _< ~il Ixll T ,
Suppose
Gk+l,...,Gm
VT _> 0
,
Vx 6 L2el
satisfy n,
68
<x,Gix> T -> 0
,
%~f -> 0
, Vx 6 L 2el
'
i=k+l,...,m
k Let
r =
Z
i=l follows:
ni
and partition the interconnection matrix
H
as
160
r
69
H
Under
n-r
Haa
Hab
Hab
HbbJ
=
these conditions, (i)
H
(ii)
rank
the system
is positive,
Proof
n-r
(1) is L2-stable
if
and
Hab = n-r
Apply Theorem
(28) with
Then we have
Db = 0
[HAa] 70
M1
=
Fa
[Haa
Hab]
L";bJ Since
Ea
is positive
is positive Finally,
to verify
Hab
n-r and
is
H'ab Ea Hab
special
is positive
is positive
case.
it is clear
from
(36) holds,
by inspection.
Also,
(ii) of Theorem Ea
Remarks matrix
definite,
semidefinite.
(28), observe definite,
(54) that
M1
that since rank
we have
that
definite.
l) Corollary
To see this,
for the single-loop
(65)
observe system
contains
Corollary
(7.1.7)
that the interconnection
(7.1.1)
is
71 I~
which
satisfies 2)
questions:
09
all the hypotheses Corollary
(65) provides
under what conditions
passive
subsystems
finite
gain result
of Corollary
to the following
does an interconnection
and some strictly in an overall
an answer
(65).
passive
system
subsystems
of some with
that is L2-stable?
The
161
answer
is that the o v e r a l l system is L 2 - s t a b l e p r o v i d e d
i n t e r c o n n e c t i o n m a t r i x is positive, from the p a s s i v e s u b s y s t e m s have the f o l l o w i n g property: signals from the p a s s i v e subsystems,
and
(i) the
(ii) the i n t e r c o n n e c t i o n s
to the s t r o n g l y p a s s i v e s u b s y s t e m s If we k n o w the i n t e r c o n n e c t i n g
s u b s y s t e m s to the s t r o n g l y p a s s i v e
then we can u n i q u e l y d e t e r m i n e the o u t p u t s of the
p a s s i v e subsystems.
Roughly
speaking,
condition
(ii) m e a n s
that any e r r a t i c b e h a v i o u r at the o u t p u t s of the p a s s i v e s u b s y s t e m s can be d e t e c t e d t h r o u g h the i n t e r c o n n e c t i o n
signals
at the s t r o n g l y p a s s i v e subsystems.
3) have rank Hence,
Since
n-r
Hab • R r×(n-r)
implies that
, the r e q u i r e m e n t
r ~ n-r
in o r d e r to a p p l y C o r o l l a r y
, i.e.
that
that
Hab
r ~ n/2
(65), at least h a l f the sub-
systems m u s t be s t r o n g l y p a s s i v e w i t h finite gain.
Theorem
(72)
S u p p o s e all the h y p o t h e s e s of C o r o l l a r y
(65)
b e l o w r e m o v e s this r e s t r i c t i o n . 72
Theorem hold,
e x c e p t that
(ii')
(ii) is r e p l a c e d by the f o l l o w i n g condition:
whenever
v • R (n-r)
is a n o n z e r o
s o l u t i o n of
Hab v = 0 , we have
73
v' Hbb v > 0 Under these conditions,
Proof
74
for
Since
the system
(66) and
i=l,...,k
(67) hold,
that
e < ci
Then,
u s i n g the f a m i l i a r argument,
<X,~lX> T ~
(~i-~)
,
(i) is L2-Stable.
and let
choose
~ = max
~ > 0
such
{ ~ l , . . . , ~ k }.
we have
IlxllT 2 + (~/~2)
llGixll~
,
n,
VT > 0 , -
NOW apply Theorem
(7.1.18), w i t h
Vx6L
l 2e
i=l,...,k '
A = I n , S = In
162
(e/~ 2 ) I r Q = -
75
where in
E a is defined (7.1.19) becomes
(u/~2)
76
M
=
Then the test m a t r i x
in (33).
ir
L
0
R
denote
(Ea-eIr)Haa
H'aa (Ea-~Ir)Hab
ab
(Ea-eIr)Haa
H ab' (Ea-UIr) Hab
+ 0
(H + H')/2
the sum of the last two matrices.
positive
semidefinite,
positive
definite
defined
Ii ia
+
Let
M
we have by Lemma
if the b o t t o m
Since
(21) that
(n-r) x(n-r)
M
R
is
is
submatrix of
R ,
i.e. 77
Rbb =A H'ab is positive have
definite.
v' Rbb v ~ 0
is p o s i t i v e implies
(E a - ai r ) Hab + Rbb
Vv E R n-r
definite,
v = 0 .
Since
(Hbb + H'bb )/2
is p o s i t i v e So,
in order
semi-definite, to prove that
it is e n o u g h to show that
Accordingly,
suppose
we ~b
v' Rbb v = 0
v' Rbb v = 0 .
Then from
(77) we get 78
v' Rbb v = (Hab v)' Since (78)
E a - si r implies
is p o s i t i v e
that
79
Hab v = 0
80
V'
Hbb V
= 0
(E a - ai r ) (Hab v) + v' Hbb v = 0
definite
and
Hbb
is p o s i t i v e ,
163
N o w by c o n d i t i o n Hence Lemma
Rbb
(ii'),
(79) and
(80) t o g e t h e r
is p o s i t i v e definite,
M
imply t h a t
v = 0 .
is p o s i t i v e d e f i n i t e by
(21), and the system at hand is
L 2 - s t a b l e by T h e o r e m
(7.1.18). []
81
Example m = 3, and and
ni = 1
for
(67), and suppose
82
H
Then
H
.
.
Suppose
satisfy
but C o r o l l a r y
r < n-r , so that However,
Theorem
i=i,2,3 G2 , G3
G1
(68).
(I), w i t h
satisfies
(66)
Suppose
=
is positive,
because n-r
C o n s i d e r a system of the form
Theorem
Hab
(65) can not be applied,
can not p o s s i b l y h a v e rank
(72) has no such r e s t r i c t i o n .
Applying
(72) to the system at hand, we note first of all that
is positive. satisfied.
Next,
it is e a s y to v e r i f y that c o n d i t i o h
H
(ii')
is
Thus the g i v e n s y s t e m is L 2 - s t a b l e .
NOTES AND REFERENCES
The p a s s i v i t y t h e o r e m for f e e d b a c k systems was given by S a n d b e r g [San.
2] and Zames [Zam.
3].
The c r i t e r i o n a l l o w i n g a
"trade-off" b e t w e e n the forward and f e e d b a c k s u b s y s t e m s is due to Cho and N a r e n d r a [Cho 1 and 2].
The g e n e r a l d i s s i p a t i v i t y
c r i t e r i a are due to M o y l a n and Hill [Moy. e a r l i e r w o r k in [Sun.
I] and [Vid.
3].
general r e s u l t s can be found in [Vid.
2]; these g e n e r a l i z e
The s p e c i a l i z a t i o n s of the 8].
A n o t h e r e x t e n s i o n of
the p a s s i v i t y t h e o r e m to l a r g e - s c a l e systems is g i v e n by S a n d b e r g [San.
4].
As yet,
there are no s a t i s f a c t o r y g e n e r a l i z a t i o n s
the "multiplier" m e t h o d s [Zam.
4] to l a r g e - s c a l e
systems.
of
CHAPTER 8: L2-1NSTABILITY CRITERIA In this Chapter,
we p r e s e n t several c r i t e r i a
l a r g e - s c a l e i n t e r c o n n e c t e d s y s t e m to be L 2 - u n s t a b l e .
for a These
c r i t e r i a c o n t a i n the i n s t a b i l i t y c o u n t e r p a r t s of both the "small gain"
type s t a b i l i t y c r i t e r i a of C h a p t e r 6 and the "dissipativit~'
type s t a b i l i t y c r i t e r i a of C h a p t e r 7.
M a n y of the results here
are b a s e d on an o r t h o g o n a l d e c o m p o s i t i o n of the input space.
In
C h a p t e r 9, we show how these results can be e x t e n d e d to L i n s t a b i l i t y u s i n g the t e c h n i q u e of e x p o n e n t i a l w e i g h t i n g .
We b e g i n by d i s c u s s i n g the s i n g l e - l o o p case in Section 8.1.
In S e c t i o n 8.2, we p r e s e n t i n s t a b i l i t y c r i t e r i a of the
"small gain"
type, w h i c h are the i n s t a b i l i t y c o u n t e r p a r t s of the
results in C h a p t e r 6.
Finally,
in S e c t i o n 8.3, we p r e s e n t
i n s t a b i l i t y c r i t e r i a of the " d i s s i p a t i v i t y " instability yield both
type, w h i c h are the
c o u n t e r p a r t s of the results of C h a p t e r 7. "small gain"
as special cases.
and "passivity"
Throughout,
U.I 2 and U.IT2 , r e s p e c t i v e l y ,
8.1
These
type i n s t a b i l i t y criteria
we use a.~ and 11.1]T to denote b e c a u s e we deal only w i t h L2-spaces.
SINGLE-LOOP
SYSTEMS
In this section, we p r e s e n t the b a s i c results concerning the L 2 - i n s t a b i l i t y of s i n g l e - l o o p f e e d b a c k systems.
To f a c i l i t a t e the discussion, d e f i n i t i o n s and facts from S e c t i o n Definition
we restate here some
3.3.
An o p e r a t o r G: L v + L ~ is said to belong 2e 2e
to class U if
(i)
G is linear.
(ii)
The set M(G)
M(G)
v c L2 d e f i n e d by
= {x • L 2 : Gx e L }
is a p r o p e r subset of L 2.
185
(iii) 3 (iv)
There exists
a finite constant
UGxB
llxil, Yx e M(G)
~ ~c(G)
There exists (~T,T6R+)
4
"(GX)~T
~c(G)
a family of finite constants
such that
-< s T [[X[[ T , YX 6 L ~ 2e
A useful property of class U operators in the following 5
lemma
Lemma
such that
(see Lemma
(3.3.12)
Let G: L ~ L v belong 2e ÷ 2e
is b r o u g h t out
for the proof).
to class U
Then M(G)
v is a closed subspace of L 2. As stated in Lemma the property
that its "set of s t a b i l i z i n g
proper closed subspace c o m p l e m e n t MI(G) 6
(5), an operator
MI(G)
of L 2.
M(G)
is a
its o r t h o g o n a l
defined by = {z6L~:
= 0
VxeM(G)}
at least one nonzero element.
transfer
function m a t r i x G(-),
Recall
~n e x p l i c i t
inputs"
As a result,
contains
factorization
G of class U has
that,
if G is a linear c o n v o l u t i o n
(N(-), D(-))
and if G(.)
in ~n×n,
characterization
o p e r a t o r with
has a r i g h t - c o p r i m e
then it is p o s s i b l e
of the set M(G) (Theorem
to give
(3.3.32).
^
Further,
if G(-)
has a pole in the open right half-plane,
one can d e m o n s t r a t e Theorem instability
and ~c2(G)
of Ml(G)
(7) below is the basic
theorem
Theorem
* Throughout
some elements
for single-loop Consider
this chapter,
then
(Theorem 3.3.45).
"small gain"
type L 2-
systems.
a system d e s c r i b e d by
we use M(G)
in the interests
and
of brevity.
~c(G)
instead of M2(G)
166
8a
el = Ul - Y2
8b
e2 = u2 + Yl
8c
Yl = G1 el
8d
Y2 = G2 e2
where
Ul, u2' el' e2' Yl ' Y2 6 L 2e v for some positive
and G I, G 2 map LV2e into itself. and ~(G 2) < ~.
Suppose,
(G2) In particular,
~,
to Class U,
to each Ul, u 2 in L~
in L2e for e I , e 2, Yl' Y2"
the system
~c(GI)
G 1 belongs
that corresponding
(8) has at least one solution these conditions,
Suppose
integer
(8) is L2-unstable
Under
if
-< 1
we have that Yl g L2 whenever
u 2 = 0, and
u I E M±(GI)/{0}.
Note
that,
u I is a nonzero It is shown adapted
element
in Chapter
to encompass Note
Theorem
(6.1.1).
Roughly
the gain of the stable
theorem
belongs
system
system
(7) states
an unstable
is itself
unstable,
gain of the unstable
feedback
that,
if
forward provided system and
instability
systems.
the system
Suppose
of
one.
is the basic passivity-type
Consider
to Class U.
Theorem
counterpart
around
system does not exceed
for single-loop Theorem
10
MI(GI ) .
can be readily
(7) is the stability speaking,
feedback
(10)
whenever
L -instability.
of the conditional
Theorem
results
complement
9 that such results
then the overall
the product
(7), instability
of the orthogonal
that Theorem
we place a stable system,
in Theorem
that,
(8), and suppose
G1
for each u I, u 2 ~ L~,
~8) has
167
Suppose
at least one s o l u t i o n in L v 2 for el, e2, YI' Y2 .
in
a d d i t i o n that
(i)
T h e r e exists a c o n s t a n t e such that
ii
<X,Sl,X> ~ e Ilxll2 ,
(ii)
¥x E M(GI)
There exists a c o n s t a n t ~ such that
12
<x,G2x> ~ 6 IIG2xll2, (iii)
¥x • L 2
We have
13
G2x = 0
U n d e r these conditions,
14
= x = 0
the s y s t e m
(8) is L 2 - u n s t a b l e
if
~+~ > 0
Specifically,
if u I = 0 and u 2 • M I ( G I ) / { 0 } ,
we have that e i t h e r
Yl or Y2 does not b e l o n g to L 2. An i n t e r e s t i n g feature of T h e o r e m n o n z e r o input is applied,
not to the u n s t a b l e s y s t e m G I, but to
the p o s s i b l y stable s y s t e m G 2. w h e t h e r the s y s t e m
In T h e o r e m
It is still an o p e n q u e s t i o n
(8) can be m a d e L 2 - u n s t a b l e w i t h u 2 con-
s t r a i n e d to be zero,
inputs,
(I0) is that the
if
(14) holds.
(i0), we show that a p a r t i c u l a r c h o i c e of
e i t h e r Yl or Y2 does not b e l o n g to L 2.
By a d d i n g an
e x t r a assumption, we can show that Yl ~ L2 for a p a r t i c u l a r choice of inputs. 15
Corollary hold,
S u p p o s e all the h y p o t h e s e s
and that in addition,
conditions,
of T h e o r e m
G 2 maps L 2 into itself.
we have that Yl g L2 w h e n e v e r u I = 0,
u 2 • M I(G I)/{0}.
(i0)
U n d e r these
168
8.2
C R I T E R I A OF THE SMALL GAIN TYPE
In this section,
we present several L 2 - i n s t a b i l i t y
c r i t e r i a of the "small gain" systems.
These results
those of Chapter
6.
type for large-scale
are the instability
Note that all of the criteria
are based on selecting
some nonzero elements
c o m p l e m e n t MI(Gi) , and can therefore stability
(see Chapter Throughout
interconnected
counterparts
of
given here
from an orthogonal
be extended
to L=-in-
9).
this section,
we c o n s i d e r
systems described
by m
la
ei = ui -
lb
Yi = Gi ei
z Hij yj j=l i = l,...,m
n,
n.
where u i, e i, Yi 6 L2el for some positive n.
n.
integer n i , Gi:
L
2el
n,
L z and H : L 3 + L i 2e ' ij 2e 2e" time, we assume that,
Without
stating
corresponding
it e x p l i c i t y
every
to every set of inputs
n. U i
6 L2z Vi,
the system equations
(i) have at least one solution
n.
for ei, Yi in L2e.Z the system
This is a w e a k e r a s s u m p t i o n
(i) to be well-posed,
the system
(i) is well-posed.
conditions
for the w e l l - p o s e d n e s s
given in Chapter
and is certainly
Hij(0)
if
of systems of the form
(i) are
5. counterpart
of
(6.2.50). Theorem
belongs
satisfied
Recall that some sufficient
The first result is an i n s t a b i l i t y Theorem
than requiring
Consider
the s y s t e m
to class U for all i, and = 0) ¥i,j.
negative)
constants
(i), and suppose
(ii) H.. zj is unbiased
Suppose there exist
(not n e c e s s a r i l y
(i) G i (i.e. non-
sij such that n.
<ei'-HijGjej>
< ~ij
]leill Ilejll Ye i ~ L21, ¥ej 6 M(Gj)
169
Under these conditions, can find positive
the system
constants
(i) is L 2 - u n s t a b l e
ll,...,lm
if one
such that the m a t r i x
S E R m x m defined by
sij = li$ij
is positive
-
definite.
(liuij + ljuji)/2'
In particular,
6ij = K r o n e c k e r
delta
whenever
u i E MI(Gi ) Vi n. n. and u i ~ 0 for some i, we have that either e i ~ L21or Yi ~ L21 for some i. Proof contradiction
Let u i 6 MI(Gi) Vi, and suppose by way of n. n. that e i 6 L21 , Yi 6 L21 Vi. Since G i belongs
class U for all i, this implies the system equations
that e i 6 M(Gi)
vi.
Now,
to
from
(i), we get
m ei = ui -
Z H. • G. e. j=l 13 3 3
m <ei,ei > = <ei,ui>
m = - j = l~
where we use
<e i, Hij Gj ej>
j=l
m <ei, H..x3 G.3 e.>3 -< j=lZ ~ij IIeill IIejII
(3) and the fact that <ei,ui > = 0 because
u i ~ Ml(Gi ) . positive
-
Next,
definite.
m
let l l , . . . , I m be chosen such that S is Then
(6) implies
m
m
z li IIei{I2 z z i=l i=lj=l
m
However,
li ~ij
IIeill IIejll -< 0
m
E z i=l j=l sij
IIeill IIejll ~ 0
since S is p o s i t i v e
e i = 0 Vi.
that
definite,
G i e i = 0 Vi, and since Hij is unbiased, ¥i.
(8) implies
Since G i (being in class U) is linear,
This leads to a contradiction,
that we have
we have H i j Gj e 3. = 0
since this
implies
170
m
u. = e. + Z H.. G. e. = 0, Vi l l J -"=i 13 ] ]
whereas
u. was chosen to be n o n z e r o for at least some i. This 1 assumption is false, i.e. c o n t r a d i c t i o n shows that our o r i g i n a l n. ni either e i ~ L21 or Yi ~ L2 for some i .
i0
Corollary hold,
Suppose
and in addition,
conditions,
the system
that the hypotheses of T h e o r e m (2) n. n. Hij maps L21 into L21 , vi,j. Under these (2) is L2-unstable.
In particular,
w h e n e v e r u i q MI(Gi ) ¥i and u.l ~ 0 for some i, we have that n. Yi g L21 for some i. Proof contradiction we have
n. (2a) that e i e L21 Vi.
from
contradiction, original
Let u i q MI(G i) Vi, and suppose by way of n. nj n. that Yi 6 L21 Vi. Then, since Hij(L 2 ) c L21 ¥i,j, This now leads to a
just as in the proof of T h e o r e m (i). Hence our ni is false, i.e. Yi ~ L2 for some i.
assumption
The d i f f e r e n c e
between Theorem
(2) and C o r o l l a r y
(I0)
is that in T h e o r e m
produce errors n. not in L21. n.
In Corollary
Corollary
~(Hij)
is an i n s t a b i l i t y
Consider
the system
to class U for all i. < ~ ¥i,j.
L2-unstable
12
condition
that
counterpart
of
(6.2.43). Theorem
belongs
by a d d i n g t h e
C L21 ¥i,j, we draw a better conclusion: n a m e l y that n. i n p u t s p r o d u c e o u t p u t s t h a t a r e n o t i n L21. The next result
ii
(10),
n.
Hij(L23) certain
(2), we show that c e r t a i n inputs either ni t h a t a r e n o t i n L2 o r p r o d u c e o u t p u t s t h a t a r e
Suppose
(I), and suppose G i in addition
Under these conditions,
if the test m a t r i x P E R m x m defined by
Pij = ~(Hij ) ~c (Gj)
that
the s y s t e m
(i) is
171
has a spectral
radius
less than or equal to one.
ui • MI(Gi)/{0}
whenever
Proof
¥i,
In particular, n. Yi g L2 ~ f o r some i .
we h a v e t h a t
Let u i e M ± ( G i ) /
0
Vi, and suppose by way of
n. c o n t r a d i c t i o n that Yi 6 L21 Vi. Since ~(Hij) < - Vi,j, this n. n. n. implies by (la) that e i e L21 Vi. Next, e i E L21 , Yi ~ L21 implies
that e i E M(Gi)
~i.
N o w define
m
13
zi = u i - e i =
E Hij yj j=l
Since u i E M±(Gi ) and e i E M(Gi),
we have
llzill2 = lluill2 + IIeill2 > lleill2 Vi since u01 @ 0
14
On the other hand,
since
m 15
• =
zz
m
E
j=l
Hi"3
y "3
=
E j=l
H..
z3
G.
3
e.
3
we also get 16
IlzilI _<
=
m m E IIHijGjej II < E ~(Hij) j=l j=l m Z j=l Pij
llej II
Let p (P) denote
the spectral
0(P)
since P has all n o n n e g a t i v e
S i.
Frobenius
Now,
t h e o r e m [Gan.
radius of P; by assumption,
I, p. 66]
value of P, and the c o r r e s p o n d i n g choosen to have all n o n n e g a t i v e least one positive
element).
elements,
states that p(P) row e i g e n v e c t o r
elements
m m Z v i llzilI > E v. IIeill i=l i=l l
the Perron-
is an eigenv can be
(and of course,
Let v be so chosen.
we get 17
~c(Gj ) llejll
Then
at from
(14)
172
On the other hand,
from
(16) we get
m 18
m
m
m
i=iE vi IIzill ~ i=lj=l~ l v i Pij
llejll = p(P)
where we use the fact that v is a row eigenvector since p(P) original
E i,
(17) and
assumption
of P.
Now,
(18) contradict
is false,
The following (2) and
j=iz v.3 fleeII
each other. Hence our ni i.e. Yi ~ L2 for some i.
result
is a corollary
of both Theorems
(Ii).
19
Theorem
Consider
the system
belongs
to class U for all i.
~(Hij)
< ~ Vi,j.
L2-unstable
Suppose
(1), and suppose in addition
Under these conditions,
if p(P)
that
the system
< I, where P • R mxm is defined
G. 1
by
(i) is (12).
particular,
whenever u i • Ml(G i) ¥i and u. ~ 0 for some i l n. have that Yi ~ L2Z for some i.
In we
Proof
Let u i 6 MI(Gi ) Vi, and suppose by w a y of n. contradiction that Yi e L21 ¥i. Then (la) implies that n. ni n. e i 6 L2z Vi; also, e i e L 2 , Yi e L21 implies that e i 6 M(Gi). Now define
z i as in
(13).
Since
u i 6 Ml(Gi ) and e i 6 M(Gi) ,
we have 20
IIzilI > IIeilI vi,
because
IIzilI > IIeilI for some i
u. ~ 0 for some i. 1
Also,
as in
(16), we get
m
21
llzill ~ j=iE Pij
By assumption,
p(P)
IIejll
< i, and P has all nonnegative
Also note that the gains that they can therefore increase
the gains
~(Hij)
and ~c(Gi)
be increased.
~(Hij)
and ~c(Gi)
elements.
are upper bounds,
So it is possible such that
and
to
(i) all elements
173
of P are p o s i t i v e ,
and
(ii)
t h e o r e m [Gan.
i, p.
corresponding
row e i g e n v e c t o r
positive
66]
p (P) < i.
elements.
p(P)
is an e i g e n v a l u e v c a n be c h o s e n
L e t v be so c h o s e n .
m
22
By the P e r r o n - F r o b e n i u s
Then
of P, a n d the such that v has from
all
(14) we g e t
m
Z v. llzill > Z v i lleilI i=l i i=l
because
of
(20) and the
j u s t as in
m
23
i.e.
As m e n t i o n e d (2) as w e l l
24
previously,
as T h e o r e m
of T h e o r e m
a corollary
so t h a t
25
if p(P)
To see t h a t T h e o r e m
~c(Gj)
(19)
< ~ Vi,j,
of is
then
lleill llejll
(6.2.8));
constants
definite,
A = Diag
ll,...,Im
Observe
minors
by F a c t
of I m - P a r e
(6.2.11)
s u c h t h a t A(Im-P)
+
there
(Im-P) 'A
where
is r e q u i r e d
is a c o r o l l a r y
the h y p o t h e s i s
therefore
{ll ..... Im }
is e x a c t l y w h a t (19)
~c(Gj ) = Pij
(by L e m m a
exist positive is p o s i t i v e
Theorem
is a c o r o l l a r y
t h a t if ~(Hij)
< i, t h e n the l e a d i n g p r i n c i p a l
all positive
This
(19)
is false,
(3) is s a t i s f i e d w i t h
~ij = ~(Hij)
26
(ii) o
< ~(Hij)
assumption
Theorem
(2), o b s e r v e
<ei,-HijGjej>
Thus,
7 v. llejll ==i j x 3
c o n t r a d i c t s (22). Hence our original ni Yi ~ L2 for s o m e i. o
Theorem
hand,
m
7. v. llzilI _< p(P) i i=l
which
On the o t h e r
fact t h a t v. > 0 Vi. l
(18) we g e t
for T h e o r e m
of T h e o r e m
(2) to apply.
Thus
(2).
a l s o t h a t if p (P) < 1 t h e n p (P) -< i, so t h a t
of T h e o r e m
(ii)
is s a t i s f i e d .
In this
sense,
174
Theorem
(19) is a c o r o l l a r y
of inputs that produce in the two theorems.
"unstable" In T h e o r e m
I, then instability
p(P) Theorem
results
then i n s t a b i l i t y
(ii).
outputs (ii),
However,
is slightly
the set different
it is shown that if
if u i 6 MI(Gi)/{0}
(19), it is shown that if p(P)
condition), inputs,
of T h e o r e m
¥i.
In
< 1 (which is a stronger
is p r o d u c e d by a b r o a d e r
class of
namely
other words,
6 MI(Gi ) ¥i and u i ~ 0 for at least one i; in ui instability can be e x c i t e d from anyone of the
inputs. Up to now, we have only c o n s i d e r e d of several
subsystems
of class U.
an interconnection
We now study interconnections
of some systems of class U and other systems h a v i n g Theorem
27
each i, either
Consider
the system
~(G i) < ~ or G i belongs
least one of the G.'s belongs 1 ~(Hij)
< ~ for all i,j.
28
Pij = ~(Hij)
that for
to class U.
Also,
suppose
~c(Gj )' ¥i,j the system
In particular,
(I) is L 2 - u n s t a b l e
if
u i e MI(Gi ) for all i and ni u i ~ 0 for some i, we have that Yi g L2 for some i.
M(Gi)
< i.
(i), and suppose
to class U, and that at
Define the test m a t r i x PqR m x m by
under these conditions, p(P)
finite gain.
whenever
Remarks Note that, if ~(G i) < ~ for a certain i, then ni = L 2 and ~c(Gi) = ~(Gi). Also, for this value of i,
MI(Gi ) = {0}, so that u i e MI(Gi ) implies u i = 0 w h e n e v e r ~(Gi)
< ~.
in T h e o r e m
Hence
the set of "destabilizing"
(27) consists
of:
inputs d e m o n s t r a t e d
u i = 0 if ~(G i) < ~, u i ~ MI(Gi )
if G. b e l o n g s to class U, and u. ~ 0 for some i. In 1 1 particular, i n s t a b i l i t y can be e x c i t e d from anyon e of the inputs to the class U subsystems. Proof can be increased
Since
~(Hii),
~c(Gi)
are all upper bounds,
in such a way that P.. > 0 ~i,j 13
and p(P)
they < i.
Let the inputs u. be selected as above, and suppose hy way of l ni c o n t r a d i c t i o n that Yi E L 2 ¥i. Then (la) implies that
175 n.
n0
e i E L21 ¥i, which e i E M(Gi)
together
whenever
with Yi • L21 implies
G i belongs
to class U.
that
As before,
define
m
29
zl• Now,
=
E j=l
H.
=
lj Yi
u-
-
1
i = l,...,m
e-,
1
if ~(G i) < -, then u i = 0 so that
belongs
to class U, then
llzill2 = lluill2 + IIeill2 since
ui • MI(Gi ) and e.l • M(G i) . 30
llzilI = IIeill; if G i
Hence
Ilzill >_ lleill ¥i Also, since u i ~ 0 for some i, strict inequality for some i. Now let v 6 R m be a row eigenvector corresponding v. > 0 ¥i.
to the eigenvalue
p(P),
Then on the one hand,
1
m
chosen
we have
holds of P
in
(30)
in such a way that
from
(30) that
m
31
v i IlzilI > i=l
E V i IleilI i=l
On the other hand, m 32
m
m
i=l ~ vi IIzill ~ i=l~ v.l P''l 3 lle911 = p (P) i=lZ v.3 llejll
Hence
(31) and
original
(32) contradict
assumption
8.3
is false;
DISSIPATIVITY-TYPE
In this section, criteria
the general
decomposition
parts of those
dissipativity
while
7.
type
the rest make use of
of the input space.
The results in Chapter
several
Some of these are based on the
dissipativity,
dissipating-type
type criteria.
CRITERIA
we derive
for L2-instability.
idea of conditional an orthogonal
each other, which shows that our ni i.e., Yi g L2 for some i. []
criteria,
By specializing
we can obtain passivity-
given here are instability
counter-
176
T h r o u g h o u t this section, we shall be c o n c e r n e d with systems d e s c r i b e d by m
la
ei = ui -
z
i=l
Hij Yj i -- i, . . . ,m
lb
Yi = Gi ei n. n. n. w h e r e ei, ui' Yi E L i for some i n t e g e r ni, Gi: L i ÷ L z and 2e 2e 2e' Hij
is a c o n s t a n t m a t r i x of d i m e n s i o n ni×n''3
assume that,
ThroughoUt,n. we
c o r r e s p o n d i n g to each set of inputs u i • L21 Vi,
the s y s t e m e q u a t i o n s
n. (i) have a unique s o l u t i o n set e i • L l 2e '
n. Yi • L2el ¥i, w h i c h depends in a causal way on the input set {Ul,...,Um}. well-posed
This is c e r t a i n l y the case if the s y s t e m
(i) is
(see C h a p t e r 5).
8.3.1
CONDITIONAL DISSIPATIVITY APPROACH
We b e g i n by r e c a l l i n g a d e f i n i t i o n from S e c t i o n
Definition
Suppose G: Ln2e ÷ L2e'n and let Q , R , S e R n×n
be given, w i t h Q and R symmetric.
Then G is said to be
c o n d i t i o n a l l y d i s s i p a t i v e w i t h r e s p e c t to
+ <x,Rx> + ~ 0, It is shown in S e c t i o n 3.3 that,
(Q,R,S)
if
YxEM(G) if G is a linear
c o n v o l u t i o n o p e r a t o r and its t r a n s f e r f u n c t i o n m a t r i x G(-) right-coprime
3.3.
factorization,
then it is p o s s i b l e to give a
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 for G to be conditionally dissipative
has a
(see Lemma
(Q,R,S)
-
(3.3.50)).
We lead up to the m a i n r e s u l t of the s u b s e c t i o n t h r o u g h a series of lemmas~ Lemma form as
C o n s i d e r the s y s t e m
(i), d e s c r i b e d in aggregated
177
e
where
=
u
-
Hy,
y
e, u, y 6 L2e ,n
Ge
=
G: L n2e ÷ L2e,n a n d H e R n n
conditionally
dissipative,
then the system
conditionally
dissipative,
where
6a
= Q + H'RH
-
(SH + H ' S ' ) / 2
We wish
to s h o w that,
(5) is
If G is
(Q,R,S)
(Q,R,S)
-
6b
6c
= S - 2H'R
Proof
whenever
u 6 L n2 a n d
n y ~ L2, we h a v e
Now,
+
+
~ 0
n if u e L n2 a n d y e L n2, t h e n e e L2,
s i n c e G is whenever
(Q,R,S) - c o n d i t i o n a l l y n n e 6 L2, y 6 L2,
If w e r e p l a c e details (7.2.6).
+ <e,Re>
+
e by u- Hy in
are entirely
Further,
w e h a v e that,
~ 0
(8) b e c o m e s
to t h o s e
(7).
in the p r o o f
The
of L e m m a
D
Lemma
Consider
the
system
(i), and s u p p o s e
(Q,R,S)
- conditionally
dissipative,
(Q,R,S)
- dissipative.
Then the system
Q-<
(5) .
dissipative,
(8), t h e n
analagous
from
b u t the s y s t e m
G is
(1) is n o t
(i) is L 2 - u n s t a b l e
if
0. Proof
corresponding
By a s s u m p t i o n ,
to e a c h u, w h i c h
F: L2en ÷ Ln2e is c a u s a l . dissipative,
we have
Also,
from Lemma
(i) has a u n i q u e we c a n w r i t e s i n c e G is
solution
y
as y = Fu w h e r e
(Q,R,S)
[4) t h a t F is
- conditionally
(~,R,S)-condition~
ally dissipative. By L e m m a (3.3.8), if 9_<0, a n d F is n o t (Q,R,S) - d i s s i p a t i v e , t h e n M ( F ) ~ L ~ , i . e . , the s y s t e m (i) is L 2 - u n s t a b l e .
178
The next L2-instability,
10
based
Lemma (Qi,Ri,Si)
these
the s y s t e m
- conditionally
(i.e.G.
is g i v e n by
(i).
Suppose
for all
- dissipative,
(0) = 0 for all o t h e r
conditions,
approach
the system
to
results.
dissipative
(Qi,Ri,Si) z
a general
on the p r e c e d i n g
Consider
of the G i is n o t unbiased
lemma gives
i.
t h a t G i is Suppose
one
a n d t h e r e s t are
values
of i).
(i) is L 2 - u n s t a b l e
Under
if Q s 0, w h e r e
(6a), a n d
Q = Diag
{QI ..... Q m }, R = D i a g
S = Diag
{ S l , . . . , S m}
{R 1 ..... R m}
ll
Proof Lemma
(4), F is
L e t us w r i t e (Q,R,S)
S 0 by hypothesis.
Thus,
(Q,R,S)- d i s s i p a t i v e , Lemma
y = Fu, w h e r e
- conditionally
F is c a u s a l .
dissipative.
if we c a n s h o w t h a t F is n o t
then L2-instability
would
follow
from
(9).
To s h o w t h a t F is n o t as f o l l o w s : assumption,
12
since there
(Q,R,S)- d i s s i p a t i v e ,
we p r o c e e d
G i is n o t
(Qi,Ri,Si) - d i s s i p a t i v e by ni an x i E L 2 e a n d a T < = s u c h t h a t
exist
T
+ <xi,RiXi>T
+ T
< 0
L e t us n o w c h o o s e
13
e = [0,
0,
...,
x i, 0,
14
y = [0,
0,
...,
G i x i, 0,
15
u = e + Hy Then
By
Also
this
Moreover,
set of u , e , y
satisfies
...,0]
...,0]
the s y s t e m e q u a t i o n s
(i).
179 16
T + T + T = T + <e'Re>T + T
= T
+ <xi,RiXi>T
+ T
<0 Hence F is not
(Q,R,S)
Remark suppose
- dissipative.o
Instead of r e q u i r i n g
G i to be unbiased,
that T ~ 0 for all T.
condition
is satisfied Lemma
L2-instability instability whether
(i0) gives a general
criterion,
dissipative specific
but not
instability
repetition,
approach
conditionally
because
operator
it is not yet clear how to verify G i is
(Qi,Ri,S i) - c o n d i t i o n a l l y
(Qi,Ri,Si)
- dissipative.
criteria,
based on Lemma
dissipative
TO avoid
G2(0)
= 0, i.e., G 2 is unbiased. are quite natural.
it is a s s u m e d that one of the o p e r a t o r s loss of generality,
we can renumber
GI; we can then "aggregate"
into G2, which must satisfy
In Lemma
to
(Qi,Ri,Si).
the Gi's so that this the r e m a i n i n g
(A2) in order for Lemma
(i0) to apply. Theorem
17 (A2) hold. system
C o n s i d e r the system (i), and suppose (AI), n1 Suppose N(G I) ~ L 2 . Under these conditions, the
(i) is L 2 - u n s t a b l e
(I0),
G i is c o n d i t i o n a l l y
but not d i s s i p a t i v e with r e s p e c t
o p e r a t o r becomes
(Qi,Ri,Si)-
for i = 1,2
The above a s s u m p t i o n s
operators
We now give a few (i0).
m = 2, and the o p e r a t o r G i is
(A2)
Without
to d e r i v i n g
but is itself not a specific
we state below two s t a n d i n g assumptions:
(AI)
dissipative
This latter
if Gi0 = 0 or if Qi S 0.
criteria,
a particular
we can
if Q ~ 0 and Q1 < 0.
180
Proof
The theorem
follows
from Lemma
that G 1 is not (QI,RI,SI) - dissipative. nI n1 pick x • L 2 such that G 1 x • L 2 . Then unbounded
as T + ~.
Now,
(10)
if we show n1 Since M(G I) ~ L 2 ,
fIG1 xllT becomes
since Q1 < 0, it is clear
that as
T + ~, the quantity 18
T + <X,RlX> T + T eventually
becomes
negative.
(QI,RI,SI)
- dissipative, o Theorem
Consider
This
shows
the system
that G 1 is not
(I), and suppose
well-posed. Suppose (AI) and (A2) hold, and suppose n1 M(G I) M L 2 , but ~c(Gl) < ~. under these conditions, system negative
(I) is L2-unstable
20
since
So suppose (QI,RI,SI)
~c(GI)
(20) and
+ <X,RlX>
+
+ <x,e~ c2(Gl)X>
By
dissipative, ~ 0,
i.e.,
VxTM(G I)
for all m > 0, that
> 0,
V x • M ( G I)
+ <x, ( R I + ~ 2 (G l)Inl)x> _> 0,
23
from
(21) gives
Hence G 1 is also
follows
that Q < 0 and Q1 ~ 0.
- conditionally
< ~, we also have,
Adding
22
G 1 is
21
if Q s 0, Q1 5 0, and one of them is
If Q f 0 and Q1 < 0, the result
(17) above.
assumption,
Now,
the
definite. Proof
Theorem
it is
V x • M(G I)
(QI,RI,S I) - conditionally
61 = QI-~InI'
RI=RI +e~2(Gl)In I'
+
dissipative,
S1 : S1
where
181
However,
since Q1 < 0, G 1 is not
the original
"test matrix"
we replace QI,RI,SI
Q is Q + H'RH -
by QI,RI,SI,
Now,
(QI,RI,S I) - dissipative*. (SH + H'S')/2.
the new "test matrix"
If
is
24
Since Q < 0, the new test m a t r i x Q ~ 0 for s u f f i c i e n t l y Hence the system
(I) is L 2 - u n s t a b l e , by T h e o r e m
Both Theorems
(17) and
(17). o
(19) permit G 1 to be nonlinear,
in c o n t r a s t with results that require linear).
small e.
G 1 to be class U
(and hence
Moreover,
none of the G. need be unstable. However, 1 since they do not c o n s t r u c t a d e s t a b i l i z i n g input in terms of an
orthogonal extended
decomposition
of the input space,
to L -instability.
8.3.2
CRITERIA
BASED ON O R T H O G O N A L
In this subsection, L2-instability space.
they can not be
These criteria
stability
Throughout observe
By Lemma
we assume
in-
that the inter-
To r a t i o n a l i z e
that if H is nonsingular,
from
for
of the input
cases.
this subsection,
two statements
follows readily
some c r i t e r i a
decomposition
several p a s s i v i t y - t y p e
m a t r i x H is nonsingular.
assumption, following
contain
c r i t e r i a as special
connection
we p r e s e n t
based on an o r t h o g o n a l
DECOMPOSITION
are e q u i v a l e n t
this
then the
(their e q u i v a l e n c e
(la)):
if G 1 is (QI,RI,SI) - d i s s i p a t i v e with n1 Q1 < 0, then M(G I) = L 2 . Since this is not the case, G 1 is
not
(3.2.10),
(QI,RI,S l) - dissipative.
182
(i) (ii)
Also,
u E L n2 ~ e
e L n2
n = E n u ~ L2 y L2
if H is nonsingular,
the following
two statements
are
equivalent: (iii)
The relation b e t w e e n
u and e has finite gain
(iv)
The r e l a t i o n b e t w e e n
u and y has finite gain.
In other words,
the n o n s i n g u l a r i t y
g e n e r a c y assumption,
w h i c h allows us to determine
of the system by e x a m i n i n g
the b e h a v i o u r
y, w i t h o u t having to examine We now p r e s e n t 25
Theorem hold.
Suppose
in addition
nI 26
the other.
the system
(i) , and suppose
that G 1 belongs
Let
n2
7
P = H -I = I=II
PI21
LP21
P22J
n1
!
Under these conditions, (i) (ii) (iii)
n2
the s y s t e m
(I) is L 2 - u n s t a b l e
S 1 is nonsingular;
Q1 ~ 0 The
"test matrix"
A d e f i n e d by
P'S+S'R
27
Q2 [ P21
is p o s i t i v e
definite.
P22 ] +
(AI) ,(A2)
to class U, and that
m a t r i x H is nonsingular:
r~
the stability
of only e, or of only
the basic criterion.
Consider
the i n t e r c o n n e c t i o n
of H is a kind of nonde-
2
if
183
In p a r t i c u l a r ,
28
whenever
u I = H I I ( S I I) 'v, u 2 = H 2 1 ( S l I) 'v, for s o m e v 6 M I ( G I ) / { 0 }
we have
that the corresponding
Proof
Select
e or y b e l o n g s
u I a n d u 2 as in
of c o n t r a d i c t i o n t h a t the c o r r e s p o n d i n g nI n1 S i n c e Yl • L2 a n d e I • L 2 , it f o l l o w s definition
of M ( G I).
Now,
e,y both belong
(28), a n d s u p p o s e
by way n to L 2.
that e I • M(GI),
by the
since
[uI E
29
n n to L 2 e / L 2.
u2
0
a n d s i n c e P = H -I, w e h a v e
30
Also,
from
(la), w e h a v e y = H - l ( u - e )
31
= <e,S'y>
= P(u-e),
= <e,S'Pu>
so t h a t
- <e,S'Pe>
= -<e,S'Pe>
w h e r e we use the f a c t t h a t
32
<e,S'Pu>
On t h e o t h e r h a n d , dissipative,
since
G. is 1
~ -<e,Re>
<e,Re> since Q1 34
0.
= 0 because
eleM(Gl),
(Qi,Ri,Si)
veMl(Gl )
-conditionally
we have
33
= <el,v>
Note
-
-
n o w that,
since y = P(u-e),
we have
Y2 = P 2 1 ( u l - e l ) + P 2 2 ( u 2 - e 2 ) = - ( P 2 1 e l + P 2 2 e 2 )
184
because
P21Ul
35
+ P22u2
Combining
(31)
= 0 from
>_ - < e , R e >
and
(35)
36
-<e,S'Pe>
37
0 >_ - < e , R e >
A
implies
from
(34).
this
contradicts
This
contradiction
i.e.,
that
>_ - < e , R e >
either
38
=
is r e p l a c e d
)>
-
(37)
implies
= 0
that
that
(because
S'Pu
because
shows
+
e = 0.
In turn,
G 1 is linear)
= S'P(e
+ Hy)
v is a n o n z e r 0
= 0.
Theorem
element
of M I ( G I ) .
our
(25)
remains
valid
and Y 2 = 0 However,
original assumption n [] n o t b e l o n g to L 2.
e or y d o e s
)>
<e,Ae>
we have
(30),
Theorem
becomes
-
Yl = G l e l
Finally,
(33)
-
> 0 by h y p o t h e s i s ,
this
Hence
gives
+ <e,S'Pe>
Since
(29).
is false;
if c o n d i t i o n
(iii)
by I
39
(iv)
40
(v)
be easily el=0.
The
rest
Just
shown,
it f o l l o w s
from
e 2 = 0.
It is of
(19). assumed ~c(Gl) Theorem
Form
in T h e o r e m < ~. (19).
Also,
from
we
n1 [Al--x--] = r a n k A P ~
this
follows
the
(19),
we arrive (37)
we
to c o m p a r e
assume
be both
that
that
e 2 = 0 as w e l l .
the
(25). o
constructive
criterion ~c(Gl)
(19)
criterion
of T h e o r e m
< ~, w h i c h
u Implies
in T h e o r e m
Theorems
as c a n and
= - P 2 2 e 2 ' i.e.,
G 1 e class
linear
Now,
that Ae=0
t h a t of T h e o r e m
because
G 1 must
Similarly,
(37).
(39)
implies
nonconstructive
(25),
at
and
g e t Y2 = G 2 e 2
(40),
interest
(25) w i t h
In T h e o r e m
as b e f o r e ,
(34),
of the p r o o f
of T h e o r e m
rank
( G 2 + P 2 2 ) e 2 = 0 ~ e 2 = 0.
Proof
Hence,
(G2+P22)
A ~ 0, a n d
and
(25) (25)
is a l s o
that but
not
assume
in
185
that Q1 ~ 0.
Thus,
the only extra a s s u m p t i o n s
are that S 1 and H are n o n s i n g u l a r . been discussed previously. a t e c h n i c a l condition,
(25)
The n o n s i n g u l a r i t y of H 1 has
The n o n s i n g u l a r i t y of S 1 seems to be
whose significance
note that the h y p o t h e s i s
in T h e o r e m
is not clear.
"A > 0" in T h e o r e m
s t r o n g e r than "Q < 0" in T h e o r e m
Finally,
(25) is s l i g h t l y
(19) because,
as can be e a s i l y
verified,
41
A = -P' (Q -
In o t h e r words,
[:001 )P
if A > 0, then Q < 0; also,
A > 0 if and only if Q < 0.
To summarize,
if Q1 = 0, then it is not n e c e s s a r y
to tack on too m a n y u n n a t u r a l c o n d i t i o n s to turn the nonconstructive
c r i t e r i o n of T h e o r e m
c r i t e r i o n of T h e o r e m
(25).
(19) into the c o n s t r u c t i v e
One only has to
(i) a s s u m e that
G e class U, i n s t e a d of just s a t i s f y i n g
~c(GI)
(ii) s t r e n g t h e n "Q < 0" to "A > 0", and
(iii) assume that SI, H
< ~,
are nonsingular.
By s p e c i a l i z i n g T h e o r e m "passivity-type"
Theorem
(25), one can o b t a i n various
i n s t a b i l i t y criteria.
(42) b e l o w is the m o s t general i n s t a b i l i t y
r e s u l t of the p a s s i v i t y type, and is an i n s t a b i l i t y c o u n t e r p a r t of T h e o r e m
42
(7.3.52).
Theorem
C o n s i d e r the s y s t e m
(i), and suppose that
G I , . . . , G k b e l o n g to class U for some integer k S m. there e x i s t real c o n s t a n t s
Suppose
el,...,e k and ~ k + l , . . . , ~ m such that
43
<x,G.x>l > Ei NxlI2 YX 6 M(Gi) , i = l,...,k
44
<x,Gix> ~ 6 i llGixll2, Vx 6 L 2e l ' i = k+l,...,m
n.
S u p p o s e H is n o n s i n g u l a r , conditions,
the s y s t e m
(i)
and let P = H -I
(i) is L 2 - u n s t a b l e
the m a t r i x
U n d e r these if
186
45
Al
is p o s i t i v e
= E + P'
D P
semidefinite,
where
46
E = Diag
_{~IInl , . . . , e k I n k , 0, .... 0}
47
D = Diag
{0,...,0,
(ii)
48
r =
I [M 21~]
k Z n ii=l
(iii) is p a r t i t i o n e d
Let in
P = H -I be p a r t i t i o n e d
(41).
m G.e. + ~ i i j=k+l
49
Ink+l'''''~mIm}
M 2 satisfies
rank M 2 = rank
where
6k+l
(iv) In p a r t i c u l a r ,
in the
same way
as H
Then
P..e. = 0 for 13 3
i = k+l,...,m
for
i = k+l,...,m
~ e. = 0 l
H is p o s i t i v e . if we
select
inputs
of the
form
k 50
Z
Hi
vj E MI(Gj) n. yj g L 2 1 for
for
ui where then
=
j=l
Proof
51
conditions
satisfied
because
i
=
1
,m
j = 1 ..... k a n d vj ~ 0 for at l e a s t
some
Apply
Q = -D,
Then
v
J j. . . . .
i.
Theorem
R = -E,
(i) and
(25) w i t h
S = I
(ii)
Q1 = 0, a n d
n
of T h e o r e m
(25)
S 1 = I r.
Also,
are the
A becomes
52
A = E + P'DP
one
+
(P'+P)/2
= A1 +
(P'+P)/2
automatically "test
matrix"
J,
187
If P is p o s i t i v e , positive Theorem
t h e n A ~ A I.
definite, (25).
we have
Everything
However,
to s l i g h t l y is the
same
since
modify unt~l
A 1 may
n o t be
the p r o o f (37),
which
of now
becomes
53
0 a <e,Ae>
Now,
(53)
and
show
that
e2 = 0
rest
of the p r o o f
54
(48)
~ <e,Ale>
show
(just as follows
Corollary GI,...,G k belong there
exists
55
positive
Next,
in the p r o o f
the
U for
constants
_> eiIIxll2,
system
some
e I = 0 plus
of C o r o l l a r y
t h a t of T h e o r e m
Consider
to c l a s s
<X,GlX>
and
t h a t e I = 0.
(25).
(I),
integer
V x 6 M(Gi) ,
(38)).
The
o
and
suppose
k ~ m.
Cl,...,c k such
(49)
Suppose
that
i = 1 ..... k
suppose n.
56
<x,Gix ~
Under
these
0,
conditions,
YT 6 L2e1 ,
i = k+l,...,m
the
(i)
system
is L 2 - u n s t a b l e
if
(49)
holds.
Proof
In t h i s
case,
we have
D = 0, so t h a t A 1 of
becomes
57
A1 =
0a
where
58
E a = Diag
Clearly
{ ~
A 1 is p o s i t i v e
Inl ..... ek Ink}
semidefinite
and
satisfies
(48).
x
(45)
188
NOTES AND R E F E R E N C E S
The o r t h o g o n a l d e c o m p o s i t i o n a p p r o a c h to the L 2i n s t a b i l i t y of f e e d b a c k systems was i n t r o d u c e d by T a k e d a and Bergen [Tak. [Vid.
i] , and e x t e n d e d to l a r g e - s c a l e
2,3,7,10].
The n o n d i s s i p a t i v i t y a p p r o a c h is g i v e n by
M o y l a n and Hill [Moy. [Vid.
10].
in [Mic.
systems in
2] and some further r e s u l t s are in
A d d i t i o n a l results on L 2 - i n s t a b i l i t y can be found
i] .
CHAPTER 9: L.-STABILITY AND L.-I!!STABILITY USING EXPntIE~ITIAL WEIGHTItIG In Chapters 6 and 7, we have presented two distinct approaches to obtaining stability criteria for large-scale interconnected
systems,
"dissipativity"
namely the "small gain" approach and the
approach.
Certain features of these two
approaches can now be stated.
The small gain approach can be
applied to systems set in an arbitrary space Lpe, p • [i,~].
In contrast,
for @ny
the dissipativity approach requires an
inner product structure on the input spaces, be directly used only to study L2-stability. favour of the small gain approach.
and can therefore This is a point in
On the other hand, the small
gain approach invariably leads to "weak interaction" results, which are sign-insensitive vative.
In contrast,
and are therefore conser-
the dissipativity
results that are sign sensitive,
type of
approach leads to
and less conservative than
those obtained by the small gain approach.
Unfortunately,
the
dissipativity approach of Chapter 7 can not be directly applied to study L -stability,
because L
is not an inner product space.
So, what we propose in this chapter is a method of studying the L -stability of a system by studying instead a property resembling the L2-stability of an associated system.
Since both
dissipativity as well as small gain methods can be used to study this property of the associated system,
it is clear that the
results given here in effect extend the scope of applicability of the dissipativity approach to include L -stability studies. There is also another point worth mentioning. Generally speaking,
the criteria for L -stability obtained by
directly using the small gain approach involve calculations the time domain.
in
The reason for this is that the L -induced
norm of a convolution operator is calculated in the time domain (see Lemma
(3.1.51)).
In contrast,
the L2-stability criteria
obtained by using either the small gain or dissipativity approaches usually involve frequency domain calculations.
The
reason is that the L2-induced norm of a convolution operator is calculated in the frequency domain,
and the matrices Q,R, and S
encountered throughout Chapter 7 are also calculated in the frequency domain
(see (3.1.55)
and Section 3.2).
Thus, as a
190 consequence obtain are
of
the r e s u l t s
frequency
very
useful
experimental frequency
domain because
data,
structure
on
the
"exponential system
the
input
they
space. we
readily often
is able
Such
applied
obtained
results
too make
to
results
to
by m e a n s
of C h a p t e r
use
By m a k i n g
show
whenever
GENERAL
general
"exponential
of
of
chapter
an a s s o c i a t e d
8 pertain
an i n n e r
appropriate
in this
STABILITY
result
weighting",
Definition n, :
one
product
use of
that
system
a given
is
L 2-
.
The
W
quite
instability
weighting",
9.1
of
chapter,
for L - s t a b i l i t y .
c a n be
are
because
is L - u n s t a b l e
unstable
in this
tests.
Finally, to L 2 - i n s t a b i l i t y ,
they
which
response
given
criteria
given which
Given
RESULT
here
depends
is d e s c r i b e d
a real
number
on the
technique
briefly.
=, the
operator
n.
L i + L i is d e f i n e d 2e 2e
by n.
(W x)(t)
= x(t)
exp(~t)
Yx 6 L l 2e
•
It is c l e a r
that W
merely
"weights"
each
function
n.
x(')
E L 1 by the 2e ~, W~ 1 = W _ ~ m a p s
exponential L ni 2e o n t o
If ~ is c l e a r by x
.
Similarly,
if
w
G. = W 1
Note
that,
from
* Yi = We Gi el• W G.
1
the
n. L 2e I
context,
n. _~ L 2 e 1
. W -I
* e.
1
=
Also,
then
W
~
G.
l
we
for
usually
, we e m p l o y
l
if Yi = Gi ei'
=
exp(~t).
each
itself.
Gi:
. G. ~
term
W -I e* ~ l
the
denote notation
W x
191
In p a r t i c u l a r , with
impulse response
suppose
matrix
G i is a c o n v o l u t i o n
Gi(-);
i.e.,
operator
suppose
t Yi(t)
= I o
Gi(t-T)
ei(~)dT
Then t Yi(t)
=
exp(~t)
Gi(t-T)
ei(~)dT
o t = I Gi(t-~)
exp(~(t-T))
. ei(~)
exp(~)
dr
o t =
where
the a s t e r i x
I G ; ( t - ~ ) e * i (3) dT O denotes
e x p o n e n t i a l w e i g h t i n g and not m a t r i x , transposition. I n o t h e r w o r d s , G. i s a l s o a c o n v o l u t i o n o p e r a t o r 1, w h o s e i m p u l s e r e s p o n s e m a t r i x i s G. ( . ) . Equivalently, 1 ^, G (j~) : G(~ + j~). In the same vein, linearity
in the sector
Yi(t)
where
¢i:
=
suppose
G i is a m e m o r y l e s s
[a,b] ; i.e.,
(Gie i) (t) = #i(t'
suppose
e i(t))
n. n. R+ x R i ÷ R i and
[ ¢i(t,a)
- aa] ' [ ¢i(t,u)
- b~]
_< 0, Yt>_0, Vc 6 R
Then Yi(t)
= ~i(t,ei(t))
where i0
¢i(t,~)
= exp(st)
~i(t,
exp(-~t) u)
Hence ii
non-
[ ¢i(t,o)-a~]
' [ ~i(t,~)
- bs]
n. 1
192 = [exp
at ¢ i ( t , e x p ( - e t ) ~ )
[ e x p at ¢i(t,
- ac] '.
exp(-~t)~)
= exp(2et)[¢i(t,exp(-at)o)
[ ~i(t'exp(-~t]°)
- bo]
- a exp(-at) a]'
- b e x p (-at) o]
n°
-< 0, ¥t >_ 0, %; o 6 R 1
which
shows
that
~i a l s o b e l o n g s
In a d d i t i o n
to the c o n c e p t
we a l s o n e e d t h a t of " d e c a y i n g 12
Definition decaying
Ll-memory
nonincreasing
An operator
if t h e r e
function
of e x p o n e n t i a l
Gi:
exists
m i(-)
[a,b].
weighting,
Ll-memory". n. n. L 2ei ÷ L 2ei is s a i d to have
a nonnegative-valued,
e L 1 such t h a t
t II(Gix) (t) ll2 < [ mi(t-~)
13
to t h e s e c t o r
Ilx(T)ll2 at
-
¥t>0 '
'
n. Vx e L 1 2e
o 14
L e m.... ma with
impulse
decaying
n. ni L 2ez ÷ L 2 e be a c o n v o l u t i o n
L e t Gi:
operator
response
Ll-memory
m a t r i x G. (.). T h e n the o p e r a t o r G. has 1 i n. ×n i if the f u n c t i o n t + G i(t) exp(~t) 6 L2X
for some e > 0.
Proof
We h a v e t
15
(Gix) (t) = I Gi(t-T)
X(T)
dT
o =
It
Gi (t--T) exp (~ (t-T))
x (T) e x p (-a (t--T)) d~
O Hence
by Schwarz's i n e q u a l i t y , t
16
II(Gix) (t)}12 _ < I o
[IGi(t-T)11 2 e x p ( 2 a ( t - T ) ) a t
193
t I Ilx(~)II2 exp(-2~(t-T))d~ o t
=
II~i (~) II2 exp ( 2 ~ ) d T o
fix (~) 112exp (-2c~ ( t - T ) ) d~ o
t --- I mi0 e x p ( - 2 ~ ( t - T ) ) H x ( T ) 1 1 2 O
where
17
llGi(r)l) 2 exp(2sr)d~
mio
<
o
Hence
(13)
18
is satisfied m i(t)
and the proof
with
= mi0 exp(-2~t)
is complete,
o n.
Lemma
19
Suppose n,
Gi:
n.
L 2ei ~ L 2ei has decaying
V e L , and that e i • L i has the property 2e we have 20
that,
Ll-memory,
for some ~ > 0
llW~e illT2 -< ))vII~ exp(~T) , ¥T TM_ 0 n,
Then G.e.ll • L~I,
and
IIGieilI~ is bounded
by a constant
times
IIvlI.. Proof
(20) can be rewritten
as
t 21
I IIei(~)II 2 exp(-2~(t-T))
dT <_ r 2,
¥t
O
where we let r denote
IIVII~ for brevity.
Define
t f(t) = I
22
llei(~)ll2 exp(-2a(t-T))d~
o Then
f(t)
5 r 2 ¥t,
by
(21).
Moreover,
we h a v e f r o m
(22)
that
194
23
f(t)
Now,
+ 2sf(t)
Ilei(t)
=
ll 2
since G i has decaying Ll-memory,
we have by
(13) that
t 24
II(Giei) (t) l12 -< I mi(t-T)
llei(~)ll2 aT
°t = I mi(t-~)
[ f(~) + 2~f(T)]dT
O
Using integration by parts in (24), we get t 25
II(Gie i) (t)II2 _< mi(O)f(t)
- mi(t)f(o)+ If(T)mi(t-~)d~ o
t + I 2~f{T) mi(t-T) O However,
since m. (.) is nonnegative-valued
dT
and nonincreasing,
1
(25)
implies
that t
26
II(Gie i) (t) N2 ~ [m i(0) +
I 2smi(T)
d~]r 2
O
where we also use the fact that 0 s f(T) s r2VT. m l ( . ) e L1, we g e t
27
from
(26)
Finally,
since
that
ll(giei) (t) II2_< r2.[mi(0)
+ 2~ llmi(-)IIl]
n.
Hence Gie i e L ~ i , and in fact llGieill= is bounded by a constant times r, i.e., a constant times
IIVII~. [] n.
Remark
Suppose e i 6 L l; then it is easy to see that,
for any s > 0, we have
llerl~ II W e lIT2
28
<-
exp(sT) ,
~
WT >- 0
(2~) n.
Hence,
if Gi:
n°
L 2ez + L 2ei
has decaying Ll-memory,
~ ( G i) < ~, in view of Lemma
(19).
then
195
We now present L=-stability deduced
of a given
by studying
associated
result,
large-scale
a property
whereby
interconnected
resembling
the s y s t e m can be
the L 2 - s t a b i l i t y
of an
system.
Theorem
29
a general
Consider
a system
described
by
m
30a ei = ui 30b
j--[lHij yj
}
i = 1 ..... m
Yi = G i e i n.
ui'
where
ei'
Yi a l l
b e l o n g t o L2eZ f o r some i n t e g e r
n. n. L i + L I , a n d H..: 2e 2e l]
Gi:
associated
n. n. L 3 + L l 2e 2e
D
n i,
Now consider
the
system m
31a
d. = u. [ H.. z. z i j=l z3 z
31b
z. = G . d. I 1 l
i = l,...,m
where
G I. W_u
Gi = W
Suppose number
~(Hij)
and H.. 13 = W u H ij W_s,
< = for a l l
u > 0 a n d an i n t e g e r
(i) (ii) (iii) (iv)
For i = l,...,k,
k s m such that
we have
H.. = 0 for k + l z3
~ i,j
the s y s t e m
satisfies,
(31)
previously.
i,j, and s u p p o s e we c a n find a
G i has d e c a y i n g
For i = k+l,...,m,
as d e f i n e d
Ll-memory.
~(Gi)<=.
~ m. for s o m e c o n s t a n t
t 0 < ~,
the c o n d i t i o n m
32
IIdilIT2 -< ~0
Under
these
conditions,
Proof the f o l l o w i n g
n.
[ IIvjlIT2, Y T > 0, V v j 6 L 2 e~ , j=l the s y s t e m
Multiplying
system
(30)
i=l ..... k
is L = - s t a b l e .
both
sides
of
description,
which
is e q u i v a l e n t
(30) by W u, we g e t to
(30):
196
m
e:l = u*l
33a
H* * ij Yj
j=l i=l .... ,m
33b
Yi* Now,
(33)
=
G *i
ei*
is of the form
of d i , vi,
and
zi
(31), w i t h e i, u i, Yi p l a y i n g
the roles
~ ¥i. respectively. S u p p o s e now that uieL ni n. We w i l l show t h a t ei, Y i a L 1 Vi, and that [leiH~ , IIyill~ are m b o u n d e d by a c o n s t a n t t i m e S n ~ =I'[ lluill . * To nid° this, o b s e r v e ,
f i r s t of all that
if u i • L
' , then
u i 6 L2e
, and m o r e o v e r ,
T 34
IIUiHT2 =
-2
IIIui(t) II2
exp(2~t)
dt
o
T -< IluiIl2 I exp(2~t)
dt
o _< l]uill2 e x p ( 2 ~ T ) / 2 a so that 35
]]Ui]]~
l]uilIT2 -< c
where
c =
(i/2a) I'2. /
exp(eT),
Since
(32)
¥i
is a s s u m e d
to hold,
it f o l l o w s
that ,
36
m
[lujlI=l e x p (sT) , i=l ..... k
lleiIIT2 -< C [ j=l Since Lemma exists
G i has a d e c a y i n g L l - m e m o r y for i=l ..... k, we h a v e from n. (19) that Yi E L 1 for i=l ..... k, and m o r e o v e r , t h e r e a finite
constant
6
0
such
that
m
]]Yi ]]- -< ~0
37
Since
38
3
[
]]uj[]~,
i=l .....
H.. = 0 for i,j = k + l , . . . , m , 13
ei = ui -
k [ Hij yj, j=l
k
we have
i = k + l ..... m
197
n.
Since
~ ( H i j) < ~
(38) shows that e i 6 L i for
¥i,j,
i = k+l ..... m, and that there exists a finite c o n s t a n t
61 such
that lleiN~ ~ ~i j=l llujll. ,
39
i=k+l ..... m n.
Since
~ ( G i) < ~ Vi, we next have that Yi e L I for i=k+l ..... m,
and also the finite gain p r o p e r t y from
analogous
to
(37).
Finally,
(30a), we get m
40
ei = ui -
Since
~(Hij)
~ Hij yj, j=l
i=l ..... k
n. (40) shows that e i e L l for i=l,...,k,
< ~,
we also have the finite gain p r o p e r t y the system
(29).
First of all, consider
Conditions
and i n t e r c o n n e c t i o n
(i) and
operators
when viewed as o p e r a t o r s Condition
don't directly crucial
(iii)
"L2-stable"
the h y p o t h e s e s
have finite gain with
Hence
of
zero bias
on L -spaces of a p p r o p r i a t e
dimension,
have d e c a y i n g
states that the r e m a i n i n g
interact among themselves.
condition.
subsystems
Condition
It states that the a s s o c i a t e d
L l-
(iv) is the
system
(31) is
in a special
It is clear from
sense, namely that it satisfies (32). ni (32) that if v i E L~ i Vi, then d i e L 2 for
i=l,...,k.
However,
requiring
the relation
L2-stable.
and
(39).
(ii) state that all s u b s y s t e m
and that some of the s u b s y s t e m operators memory.
to
(30) is L -stable. [] Remarks
Theorem
analogous
in general,
(32) is not the same as
from v I ..... v m to dl,...,d k to be
To e s t a b l i s h
this equivalence,
we require
a few
extra conditions. 41
Lemma corresponding
With regard to the system. (31), suppose
to each set of input v i e L2~
that,
, there exists
a
n.
unique set d i e L2el • i=l,...,k satisfied. set
Suppose
(Vl,...,v m) into
further
such that the eauations ( 3 1 ) _
that the operator m a p p i n g
(dl,...,d k) is casual.
and only if the operator has finite L2-gain with
taking
Then
(Vl,...,v m) into
zero bias.
are
the input
(32) holds (dl,...,d k)
if
198
Proof
The "only if" part is obvious.
To prove the
"if" part, let (vl,...,v m) be a set of inputs in L n2e, and let T < = be specified. Consider the system of equations 42a
d0i = ViT - j~l= Hij z0i
42b
z0i = G i d0i
}
i=l,...,m
.
By uniqueness and causality, we have (d0i) T = diT, for i=l,...,k. Also, since the operator taking (vI .... ,vm) into (dl,...,d k) has finite L2-gain with zero bias, there exists a finite constant ~ such that m
43
Ifd0ifl~ -< ~
X
llvjTJl2 ,
i=l .....
k
j=l Finally,
from (43) we get m
44
;IvjlIT2,
IIdillT2 = IId0irrT2 ~ IId0ill2 ~
i=l .....
k
j=l and the lemma is proved. D 9.2
SPECIAL CASES
In this section, we present some specific criteria for L -stability, based on exponential weighting. obtained by applying Theorem
These criteria are
(9.1.29) in conjunction with the
results of Chapters 6 and 7. First, we consider a system described by m
la ib
e i = u i - HiY i -
~
j=l
Bij yj } i = l , . . . , m
Yi = G i e i n. where for all i we have ui,ei,Y i 6 L2em ' and moreover (i)
G.x is a linear convolution operator with
impulse r e s p o n s e m a t r i x Gi ( . ) .
199
(ii)
H. is a m e m o r y l e s s z [c i -
(l-~i)ri,
6i,ri.
For some ~ > 0, the function n.xn. E L21 l , and
(iv)
in the sector
ci+(l-~i)r i] for some real c i
and some p o s i t i v e (iii)
nonlinearity
sup Ema x [F~l (J~) F i(j~)l
t ÷ Gi(t)
exp(~t)
~ r?21
W
where Emax(-) + denotes
denotes the largest
eigenvalue,
the conjugate transpose,
and
A
Fi(J~)
= [I + c i Gi
One can think of the system loop feedback
systems
(i) as a c o l l e c t i o n
that are i n t e r a c t i n g
Bij, which may be n o n l i n e a r Our o b j e c t i v e for the L = - s t a b i l i t y
(~ + j~)]-i Gi(~+j~) "
and time-varying.
is to derive some s u f f i c i e n t
of the s y s t e m
that if B.. = 0 ¥i,j, x3
collection
of m isolated
it is i n t u i t i v e l y (17) below makes
then the system
this idea precise.
Suppose
Then
llqllT2 < ni llfIIT2 where
=
(9.1.29).
Hence
B.. are z3 (i) is L -stable. Theorem
Before p r o c e e d i n g
f, q ~ L2e satisfy
q = f - Hi Gi q
ni
(i) is just a
each of which can be shown
(2) and T h e o r e m
we give a p r e l i m i n a r y Lemma
(9.1.29)
As a first step,
clear that if the i n t e r a c t i o n s
"sufficiently weak", theorem,
6.
then the system
subsystems,
to be L -stable by virtue of
conditions
(i), using T h e o r e m
in c o n j u n c t i o n w i t h the results of Chapter observe
of m single-
through the operators
(Ici}ri I + l)/~i
to the
200
Proof
(4) is e q u i v a l e n t
q = f - H*i ql
to
ql = Gi* q
"
N o w define
q2 = q + Then
in terms
c.a. l"l
of q2'
(7) b e c o m e s W
q2 = f -
W
(Hi -
ciI)ql'
ql
*
= Giq2
-
ciGiql
or, e q u i v a l e n t l y , *
10
q2 = f where
(Hi - c.I) l
the i n v e r t i b i l i t y
of I + c.G. 1
that ^
Gi
g(~+s)).
is
a linear
Now
from
ql =
ql'
convolution
-1
(I + ciG i)
is g u a r a n t e e d
*
Giq 2 by
(2)
(note
1
operator
with
transfer
function
(10) we get
ii
Ilq211T2 ~< IIfIIT2 + ~2 [ (Hi-c;I)]
12
IIqlllT2 ~< ~2 [ (I + ciG
IIqlIIT2
w
Next,
observe
transfer
that
function
)-i Gi ] ilq211T2
. is a linear
G ~ 1
matrix
convolution
is G i(~+j~) ; hence
operator (2) implies
whose that
*-i * -< r -I (see L e m m a (3.1.83)) ~2 [ (I+ciGi) Gi] l , Also, H i b e l o n g s to the same s e c t o r as H i , n a m e l y
[c i -
(l-6i)r i, c i +
H i - ciI b e l o n g s
(l+~i)ri],
to the sector
~2(Hi
- ciI)
-< (i + ~i)ri
these
bounds
into
(ii) and
because [-(l~i)ri,
(see Lemma (12)
of
and d o i n g
Ilq211T2 ~< [IfIIT2 +
(l-~i)
Ilq211T2
hence
(l+~i)r i] , and
(3.1.103)).
we get
13
(9.1.11);
a ifttle
Substituting manipulation,
201
14
Nq211T2 ~ ~l j]fllT2
15
ilq2;IT2 -~ r~.l~iI JlfllT2 where the last step follows by substituting Finally,
16
(14) into
(12).
we have
llqllT2 ~ IIq211~2 + Icil IIq111T2 S (6~l+Ici 16ilr~ I) which is the same as
llfllT2
(5). []
Now we present the main stability criterion system
Theorem
17
for the
(i). Consider
the system
(i), and define the test
matrix P 6 Rm×m by 18
Pij = ~ij - ni 52 (Bij Gj) where
19
~ij denotes the Kronecker delta,
, , ~2(BijGj)
ni is given by
(5), and
IIBijGjXlIT2 = sup
Under these conditions,
sup
XT~0
the system
IIxIIT2 (i) is L -stable if the
leading principal minors of P are all positive. Proof
In order to apply Theorem
the system equations
(9.1.29), we rewrite
(i) as m
20a
ei = ui
20b
ei+m = Yi
20c
Yi = Gi ei
j~l Bij yj - Yi+m
~i=l,...,m
202
20d
Yi+m = Hi ei+m It is now routine to verify that the system conditions
(i)-(iii)
"associated
of Theorem
(9.1.29).
system" corresponding
to
(20) satisfies Further,
the
(20), as given by
(9.1.31),
is m 21a
d±•
21b
di+ m = z i
=
v I.
911=
-
Bi~ j
Zj - zi+ m
i=l,...,m 21c
Z.
1
=
G,
1
d.
l
w
21d
zi+ m = H i di+ m
It is easy to eliminate this gives m di = vi -j[l=
22
Applying Lemma
(3) to
z I ..... Z2m and dm+l,...,d2m
B..G.d.1] ] 3 - H.G.d.,I i i
from
(21);
i = 1 ..... m
(22) gives m
23
lldillT2
~
~iY(BijGj ) HdjlIT2,
i=l,...,m
j=l Hence
(23) implies that m
24
Pijlldjl;T 2 -< ~i;Ivil;T2,
i=l ..... m
j=l Since P has all nonpositive
off-diagonal
elements,
the condition
that the leading principal minors of P are all positive by Lemma elements.
implies,
(6.2.9), that p-i exists and has all nonnegative Hence from
(24) we get
203
m
25
lldillT2-< j=[l (P-l)ij
Thus
(9.1.32)
by T h e o r e m
i=l,...,m
BJ llvJlIT2 '
is satisfied, w h e n c e the system
(9.1.29).
(i) is L -stable,
o
Next, we p r e s e n t a typical p a s s i v i t y - t y p e c r i t e r i o n for L -stability. (9.1.29)
Clearly,
stability
by a p p l y i n g T h e o r e m
in c o n j u n c t i o n w i t h each of the v a r i o u s results
in
C h a p t e r 7, it is p o s s i b l e to g e n e r a t e a large n u m b e r of such criteria.
In the i n t e r e s t s of brevity,
such result, w h i c h c o r r e s p o n d s
we p r e s e n t only one
to T h e o r e m
(7.3.28).
We b e g i n
w i t h some p r e l i m i n a r y results.
26
Lemma
Let G:
L2e + L2e be a c o n v o l u t i o n o p e r a t o r
w i t h impulse r e s p o n s e g(.), some ~ > 0.
27
and suppose t + g(t)
e x p ( ~ t ) e L 2 for
If we denote W GW -1 by G , then we have
<x,G x> T _> c llx
2' ¥T _> 0, V x e L 2 e
where ^
28
= min Re g(~+j~) ~ER
Proof
This is a special case of L e m m a
29
Lemma
Let ¢:
30
0 ~ ~¢(~)
and let G: 31
(3.2.19).
R ÷ R b e l o n g to the sector [ 0,k] , i.e.
~ k~ 2,
V~ E R
L2e + L2e be d e f i n e d by (Gx) (t) = # (x(t)) ,
L e t ~ > 0 and let G
32
<x,G x> T ~
= W G W -I.
(l/k)
IIGx
Then we have
2 '
%~f ~ 0,
Vx ~ L2e
204
Proof
We have T
< x , G x> T =
33
x(t)
exp(c~t)
~[x(t)
exp(-c~t)]dt
OT P
= ~ exp(2at)
• x(t)
(-at) ] dt
exp(-c~t)~[x(t)exp
o T >
(l/k)
I exp(2~t) ( ¢ [ x ( t ) e x p ( - a t ) ] ) 2 d t o
=
T
(l/k)
J [ (G'x) (t)] 2at o
= so that
(32)
(l/k)
is proved.
IIG*XlIT2
[]
34
Theorem
Consider
the s y s t e m
35a
e i = u i - j=l hiJYJ
m i=l,...,m 35b
Yi = Gi ei
where Gi:
ui, ei, Yi • L2e for all i, hij L2e + L2e.
convolution
operator
in addition, invariant
Suppose with
that
for some
impulse
for i = k + l , . . . , m ,
nonlinearity
are real c o n s t a n t s , integer
response
k 5 m, G i is a
gi (-) for i=l,...,k;
G i is a m e m o r y l e s s
in the s e c t o r suppose
[0,ki]. that
for k+l ~ i,j ~ m.
Finally,
functions
t + qi(t)
e x p ( ~ t ) 6 L 2 for i=l,...,k.
36
E. = rain 1 ~ER
gi(~+j~),
37
~i = i/ki"
38
E a = Diag
{e 1 .... ,e k}
39
D b = Diag
{dk+l,...,6m}
^
i=l .... ,k
i = k+l ..... m
and
time-
Suppose
for some
hij = 0 f
~ > 0, the
Now define
205
Haa 40
H =
(hij) =
M =
Under
m-k
Haa
H'aa
Ea
Hab
Hab
Ea
H'aa
H'ab
Ea
Hab
(i)
the system
M is positive
(iii) (iv)
Proof
9.3
(35) is L -stable
Db
if
semidefinite
H'ab Ea Hab + D b is positive
definite
H is positive
Apply
Theorem
GENERAL
(9.1.29)
INSTABILITY we derive
one to deduce
system by establishing Theorem
+
H' [M I aa] I H' ab
In this section, that permits
1
we have
r a n k M = rank
In this sense,
m-k
Ea
(ii)
sult,
0
IH'aa
these conditions,
42
k
Hba k
41
Hab
(7.3.28).
a general
instability
the L -instability
re-
of a given
of a associated
is an instability
counterpart
system. of
(9.1.29). We begin by establishing Lemma
Suppose
G(.)
a preliminary
result.
is an nxn-dimensional
transformable
distribution
with
corresponding
operator
L n2e + L n2e by
G:
o
RESULT
the L 2 - i n s t a b i l i t y
this result
with Theorem
support
in [0,~),
Laplace
and define
the
206
rt (Gx) (t) =
~G(t--T)
x(r)
dz
O
Suppose
G belongs
to c l a s s
U
(see D e f i n i t i o n
^
(3.3.10)),
has a right-coprime
Definition D a(t)
,
(3.3.15))
D a(.)
in
(ii) o C+ of
G(-)
has
poles that
sup
factorization
where
e L- n1 x n
(i)
D(.)
Under
D(.))
fact t h a t
of G.
Otherwise, whence
= Do~(t)+
singularities
singularities
in
(i) f o l l o w s
of G(.)
Moreover, the
readily
in C+ are
there
strip
~c2(G)
< ~.
To p r o v e
0 ~ Re s ~ ~
from Lemma (ii),
let
isolated
is a ~ o > 0 such O
(3.3.48) , a n d ^
(N,D) be the r.c.f.
Then
inf
I d e t D(J~)I
there exists
G ( j ~ i) b e c o m e s
(3.3.28)).
Now,
f(a,M)
(4), f(0,M)
> 0
a sequence
unbounded,
consider
Idet D
= sup
> 0.
Since
> 0
{~i } s u c h t h a t D e t
in v i o l a t i o n
f(.,M)
is a Oo(M)
(3)
D(J~i)÷0, (see L e m m a
Also,
by the R i e m a n n - L e b e s g u e
(a+j~) I
is c o n t i n u o u s ,
s u c h t h a t f(o,M)
I~I ÷ ~; m o r e o v e r ,
of
the f u n c t i o n
there
and
(see
conditions
^
By
that
IIG[j~)II < ~
The only
no
(N(-),
is of the f o r m D(t)
these
f i n i t e order.
Proof the
and
in A n × n
^
G(')
Lemma,
d e t D O ~ 0 by
we see t h a t t h e r e e x i s t s a ~
for e v e r y M
> 0 whenever
D(~+j~) (4).
+ D
o
0~O~Oo(M).
whenever
Combining
these
> 0 such that O
^
inf
inf
Idet D(a+je) I > 0
0 ~ 0
*
Recall
t h a t C+ = {s:
Re s >- 0} and C +° = {s%
Re s > 0}
~0 facts,
207
This shows that G(.)
has no singularities in the strip ^ 0 ~ Re s S o O, i.e. that all singularities^ of G must lie in the
half-plane
Re s > ~o"
singularities
Since D is analytic
o all of these in C+,
must be isolated poles of finite order. []
Lemma
(i) shows that if G belongs
to class U and G has
an r.c.f,
in A n×n , then it is reasonable to assume that G has a ^ o Further, if G has a pole at s o 6 C +° ' pole at some s O 6 C+. then one can e x p l i c i t l y
calculate
Theorem
is the key p o i n t
(3.3.45).
(13) below,
This
some elements
of M±(G),
using
in the proof of T h e o r e m
w h i c h is the main result of this section.
C o n s i d e r now a large-scale
system described
by
m 7a ei = ui - j~l HiJYJ 7b
}
i=l ..... m
Yi = Gi ei n. n. n. L 2ez + L 2eI ' and where e i, ui' Yi 6 L2el for some integer ni, Gi: n. n. Hi3.: L 23e + L2el , ¥i,j. To set the stage for a general L~instability
result
of L2-instability. instability
There features:
8.1,
if a p p r o p r i a t e
instability
occurs
criteria
for L 2-
8.2, and 8.3.2, but they all
(i) they assume that some of the
say G I, .... G k, b e l o n g
instability
(7), we begin with a d i s c u s s i o n
are several diverse
given in Sections
share two common operators,
for the system
to class U;
criteria
whenever
(ii) they state that,
are satisfied,
the input u. in 1
then L 2-
(7) are s e l e c t e d to
be of the form k ui = j~l Fijvj'
vj 6 MI(Gj) /{0}
where the F.. are c o n s t a n t matrices 13 criteria used. We now introduce
.
n.
i.e.,
that depend on the p a r t i c u l a r
the e x p o n e n t i a l
weighting
n.
either e i g L21 or Yi g L2Z for some i.
used to
208
establish
the L - i n s t a b i l i t y
L2-instability Definition
of the s y s t e m
of an a s s o c i a t e d
(9.1.1)
system.
t h a t the o p e r a t o r
(7) by s t u d y i n g
Given
WI:
the
I > 0, r e c a l l
from
Ln ÷ Ln is d e f i n e d 2e 2e
by
(WAx) (t) = x(t) clearly system
exp(At)
W~ 1 = W_A a l s o m a p s L n into itself. 2e (7), w e c a n r e w r i t e (7) as
Now,
g i v e n the
m
10a
W_A e i = W_A u.l - j=l [ (W - AHijWA) . W _ l y j i=l,...,m
10b
W-k Yi =
Note
that,
weighting the
(W-IGiWI)
if i > 0, t h e n
ii
technique
of t h a t p r o p o s e d
of e x p o n e n t i a l
in S e c t i o n
9.1.
For
let us d e f i n e
G# i = W - xGiWx Then
ei
the a b o v e
is the o p p o s i t e
sake of b r e v i t y ,
W-I
the " a s s o c i a t e d
'
H# lj = W _ x H i j W ~
system"
corresponding
to
(7) is
m 12a
d.
12b
z. = G# d. 1 l 1
l
=
r.
l
-
H#. z. m]
[
j=l
i=l,...,m
where new
r i is the n e w
"output".
a n d H l#j• :
Moreover
n. n. L2 e + L 2ei
L -instability associated
following
system
2e
•i,j.
~heorem system
" e r r o r " , and z i is the G# : n. n. L I ÷ L I and ' l 2e 2e
(13) b e l o w
relates
the
(7) to t h a t of the
(12). Consider
conditions
(cl) F o r satisfy
'
d i is the n e w n. ri, di, z i e L i
of the original
Theorem
13
"input",
the s y s t e m
(7), and s u p p o s e
the
are satisfied:
some integer
the h y p o t h e s e s
that
k ~ m,
the o p e r a t o r s
(i) G i is a c o n v o l u t i o n
G i, i = l , . . , k operator
209
~n. ×n. l 1 , and G. in A l
^
belonging
to class U,
(ii) G i has an r.c.f,
has a pole at some s i e C+o , for i=l,. .. ,k . (c2) Select I > 0 such that ^
14
sup fIGi(l+j~)II < ~,
i=l ..... k
and such that G i has a pole Sio = ~io + J~io with ~io < 2~ Remark
(i) below).
associated
For this choice of l, c o n s i d e r the
system
and further,
(12).
Suppose
the system
there exist matrices
that L 2 - i n s t a b i l i t y i
(see
results
(12) is L2-unstable,
i=l .... ,m, j=l .... ,k such
Fi~,
i.e. d i ~g L 2ni or z i ~ L 2ni for some
whenever k
15
ri =
• Fij vj, j=l
~ J vj 6 MI(G~)/{0},
Under these conditions,
j=l
the original
.... k, i=l ..... m
system
(7) is
L -unstable. Remarks to satisfy
(i) It is always possible
(14).
In view of
(cl) and Lemma
to select
I so as
(i), there exists a
^
$o > 0 such that none of the functions Gi(-) in the strip [0,~o/2].
has a s i n g u l a r i t y
0 ~ Re s ~ ~o; one can choose any I in the interval
However,
other choices
for ~ may also be possible.
^
Note that Gi(-)
is only r e q u i r e d to have one pole with r e a l ^ p a r t
greater than 2~; there is no r e g u i r e m e n t have real parts
greater than 2~.
(ii) instability
As m e n t i o n e d
of
(12) results
not at all restrictive, Sections
8.1,
that a l l poles of Gi(-)
8.2,
Proof
earlier,
the r e q u i r e m e n t
for inputs r i of the form
since all of the i n s t a b i l i t y
that L 2(15) is criteria
and 8.3.2 meet this requirement.
F i r s t of all,
note that if G. is a c o n v o l u t i o n l
operator w i t h kernel Gi(.), operator with Gi(.) Also,
G~ belongs
that G~Xis
then G#l is also a c o n v o l u t i o n
exp(-l(.)),
so that
to class U, for i=l,...,k.
linear,
of
and satisfies
(3.3.11)
(s) = Gi(s+l). To see this, (because W_l,
note
G i, Wl
210
individually ~c (G[)
satisfy
(3.3.11)) ; finally,
(14) guarantees
that
< ~. ^
It is easy to see that if
^
^
(Ni,D i) is an r.c.f,
of G i,
^
then
(Ni(s+l) , Di(s+l)
Gi has a pole at Sio, (3.3.45),
is an r.c.f,
Similarly,
then G.# has a pole at s. -I. 1 io
MI(G #) contains
exp[ (-Oio+l)t]
of Gi(s+l).
a nonzero
cos(eiot).
element
if
By Theorem
of the form v i
Now, by condition
(c2), if we select
k 16
r i(t)
in
= j=l~ F.13. v.3 exp[ (-Ojo+~)t]cos(~jo t), i=l ..... m
(12), then either
(i0) and
n. n. z i ~ L21 or d i ~ L21 for some i.
Comparing
(12), we see that if we select k
17
W_lu i (t) =
i.e.,
j=l
F..v. expl (-~-'o+~)3 t] cos(~_.o t ) ] , i=l ..... m 13 J
if we select
18
ui(t)
k ~ F..v. exp[ (-~jo+2~)t] j=l 13 ]
=
cos
n,
in
(~jo t)
i=l ..... m
n.
(7), then either W_ly i ~ L21 or W_le i g L21 for some i.
In p a r t i c u l a r , t h i s means t h a t , w i t h t h e c h o i c e of i n p u t s u i ( . ) n. n. Yi g L ~ l o r e . ~ L 1 f o r some i ( n o t e t h a t , i f of ( 1 8 ) ' n e i t h e r f(.) E L i, then W _ X f ( . ) e L21 Zn' ) . S i n c e t h e i n p u t u i d e f i n e d b y n.
(18) belongs
to L 1 , we see that the s y s t e m
It is obvious obtain L -instability the L2-instability is unnecessary "small gain" 19
20
criteria
criteria
using Theorem corresponding
of sections
8.1,
to list of all of these.
type result
Theorem described
that,
for the purposes
Consider
a single-loop
(7)
is L -unstable.m
(9.3.13), to almost 8.2,
However,
e2 = u2+Yl'
all of
and 8.3.2. we state a
of illustration. feedback
system
by
el = ul-Y2'
one can
Yl = Glel'
Y2 = G2e2
It
211
where
G 1 is a c o n v o l u t i o n
in An×n;
operator
G 2 is a m e m o r y l e s s
21
of class
operator
U, a n d has an r.c.f.
of the f o r m
(G2e 2) (t) = ~(t,e 2(t)) and ~ belongs
22
to the s e c t o r [~,8],
[ ~(t,v)
under
- ~v] ' [ ~(t,v)
these conditions,
i.e.,
- 8v]
the s y s t e m
~ 0,
(20)
c a n find a I > 0 s u c h t h a t the f o l l o w i n g
~ t ~ 0, ¥ v • R n
is L - u n s t a b l e conditions
if one
are
satisfied:
23
(i)
24
(ii)
sup E m a x [ H% (~+j~)
H(-)
H (l+j~) ] < 6 -1
has at l e a s t one p o l e w i t h r e a l p a r t g r e a t e r
t h a n 21, w h e r e
25
^ (s) = G(s)
26
y = [ S+~]/2,
a n d Emax(-) % denotes
the
the
the c o n j u g a t e
former
involves
denotes
^ [I + yG(s)] -I
~ = [ 8-~]/2
largest
is s i m p l e
Comparing
Theorem
Gi(-~+j~)
of a m a t r i x ,
transpose.
The p r o o f
involves
eigenvalue
a n d is t h e r e f o r e
(19) w i t h T h e o r e m
G i ( l + j ~ ) for s o m e for s o m e ~ > 0.
omitted.
(9.2.17),
I > 0, w h i l e
the
we
see
latter
212
NOTES AND R E F E R E N C E S
The e x p o n e n t i a l w e i g h t i n g a p p r o a c h to study L s t a b i l i t y is due to S a n d b e r g [San.
2] and Zames [Zam.
R e l a t e d results can be found in [Ber. e x t e n d e d to l a r g e - s c a l e
i] .
2].
This a p p r o a c h was
systems by L a s l e y and M i c h e l [Las.
2].
The g e n e r a l s t a b i l i t y t h e o r e m given here is taken from [Vid. w h i l e s p e c i f i c r e s u l t s are f r o m [Las. instability
is c o n t a i n e d in [Vid.
8].
2].
6],
The a p p r o a c h to L -
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INDEX A A,9 Ae,ll A,II Acyclic digraph,67 Adjacency matrix,58
C Causality,8 for discrete-time systems,9,92 strict...,92 Class U operator,47 Connective stability,l16,122 Coprime factorization,49 Cycle,66
D Decaying Ll-memory,192 Directed graph,57 acyclic...,67 strongly connected...,59 system...,73,77 Directed tree,72 Dissipativity,42 conditional...,46 of a convolution operator,45 verification of conditional...,55 Dissipativity theorems for single-loop systems,135 for instability,179,182 for interconnected systems,141,142
E Essential set,127 Exponential weighting,190 instability theorems using...,208,210 stability theorems using .... 195,201,204 Extended space,5
219
G Gain,26 conditional...,46 incremental...,26 of a convolution
operator,34,38,39
of a linear integral operator,29,33 of a memoryless
nonlinearity,40
With zero bias,26
! Instability
theorems
for single-loop
systems,165,166
of dissipativity of passivity of small-gain Interconnection Isolated
type,179,182
type,185 type,165,168,170,172,174 matrix,14
subsystem,75,77
t Loop transformation for single-loop
systems,106,107
for interconnected Lossless
systems,123
interconnections,148
Lpe,5 L -stability,18,21 P with zero bias,18,21
M Mp (G) ,45 characterization M 1 (S)
of...,51
characterization
of...,52
some elements
in...,54
N Nonnegative theorems
matrices for...,108
P Passive
interconnections,148
220
Passivity,45 of a convolution operator,45 strict...,45 Passivity theorems for instability,185 for interconnected systems,150,156,159,161 for single-loop systems,137,138 Perron-Frobenius theorem,107
R Reachability,58 algorithms for testing .... 63,64 matrix,60
S Section graph,126 Sector,41 Self-loop,67 Small-gain type theorems for instability 165,168,170,172,174 for interconnected systems,l10,115,117,121,129,146 for single-loop systems,105,106,107 Smoothing operator,88 Stability,definition of,18,21 Strict passivity,45 Strong connectedness algorithms for testing...,63,64 of a digraph,59 of a pair of vertices,58 Strongly connected component,71 d~graph,59 System digraph,73,77
T Tree,directed,72 Truncated inner product,5 norm, 5 Truncation,5
221
W Warshall's algorithm, 64 Weak interaction,ll7 Weakly Lipschitz operator,88 Well-posedness definition,16,20 of continuous time systems,97 of decomposed systems,81,86 of discrete-time systems,103
