Lecture Notes in
Control and
Information Sciences
Edited by A.V. Balakrishnan and M, Thoma
29
M. Vidyasagar
InputOutput Analysis of
LargeScale
Interconnected Systems
Decomposition, WelIPosedness and Stability
SpringerVerlag
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 3540105018 SpringerVerlag Berlin Heidelberg NewYork
ISBN 0387105018 SpringerVerlag 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, reuse 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.
© SpringerVedag Berlin Heidelberg 1981
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2061/3020543210
This book is intended to be a fairly comprehensive
treatment of largescale 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 largescale systems.
However,
the reader should know some elementary
analysis
(e.g. Lebesgue spaces,
and have some general knowledge
(e.g. Perronfrobenius
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 WellPosedness
and Stability
Weakly Lipschitz, Smoothing and Strictly
Causal Operators . . . . . . . . . . . . .
SingleLoop Systems . . . . . . . . . . .
ContinuousTime
Systems . . . . . . . . .
DiscreteTime
Systems . . . . . . . . . .
s7
57
.
73
81
88
88
94
95
103
SingleLoop 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
DISSIPATIVITYTYPE CRITERIA FOR L2STABILITY . 133
7.1
7.2
7.3
CHAPTER 8:
12
SMALLGAINTYPE CRITERIA FOR LpSTABILITY.. • 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
. . . . . . . . . . . . . .
WELLPOSEDNESS OF LARGESCALE I~TERCO~NECTED
SYSTEMS. . . . . . . . . . . . . . . . . . . .
5.1
CHAPTER 6:
Truncations, Extended Spaces,
Causality
. . . . . . . . . . . . . . . .
Definitions of WellPosedness
and
Stability . . . . . . . . . . . . . . . .
DECOMPOSITION OF LARGESCALE INTERCONNECTED
SYSTEMS. . . . . . . . . . . . . . . . . . . .
4.1
CHAPTER5:
4
GAIN AND DISSIPATIVITY . . . . . . . . . . . .
3.1
CHAPTER 4.
. . . . . . . . . . . . . . . . .
SingleLoop Systems . . . . . . . . . . .
134
General DissipativityType
C r i t e r i a . . . 139
Special Cases:
SmallGain and
PassivityType
Criteria . . . . . . . . .
144
L21NSTABILITY CRITERIA. . . . . . . . . . . .
164
8.1
8.2
8.3
164
168
175
SingleLoop Systems . . . . . . . . . . .
Criteria of the SmallGain Type .....
DissipativityType
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"
largescale
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,
wellposedness,
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
"singleloop"
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 "singleloop"
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 "singleloop"
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 inputoutput analysis, we
then give precise definitions of wellposedness
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 graphtheoretic
niques for the efficient decomposition of largescale
connected systems.
Specifically,
tech
inter
we show that by identifying
the socalled strongly connected components
(SCC's) of a given
system, we can determine the wellposedness
and stability of the
original system by studying only the SCC's.
present some sufficient conditions
system.
These criteria are graphtheoretic
given a very nice physical
In Chapter 5, we
for the wellposedness
interpretation.
In Chapter 6, we give some generalizations
singleloop
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 singleloop
"passivity"
Chapter 8, we derive several L2instability
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
III 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
ntuples
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 IIp
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
{ IIf
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 II 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 discretetime
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
III Ip :
£p
~
in
R+
S
For p 6
.
by
Ix(i) Im) I/p
llxllp = ¢
18
sequences
i=0
We also define
III 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
ntuples
6 S
{x} (i)
~
(resp.
=
Zp)
[x~ i)
Vj
.
,
x 2(i)
•
.... x n(i)] ,
The norm
Iflip
{" 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
i1
sup I Ix(i~ll
iz
p < 
if
p = 
i
I II ]
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,
timeinvari
ant operators.
31
Definition
f(.)
The set
A
consists
of all distributions
of the form
f(t)
32
=~
0,
[
t < 0
fi 6 (tt 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(tT)
g(T)
dT =
A, and that if
pair
(Jtf(T)
g(tT)
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
(tt a) * ~(tt b) = ~ (tha ~ )
36
Thus,
if
~(tta)
* fa(t) = fa(tta)
f
g
and
are of the form
37
f(t) =
~ fi 6(tti)
i=0
38
g(t) =
~
i=0
+ fa (t)
gi ~(tTi)
+ ga (t)
then
39
(f,g) (t) =
+
~
~
i=0 3
fi gj ~(ttiTj)
~ gj fa(t~j)
j=0
+
fa(tT)
and right
IIf*gltA
40
Also, we see from
41
!
~ fi ga
i=0
ga(T)
(tti)
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 welldefined 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 sdomain,
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 ith 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
wellposedness.
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
69, 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
largescale
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