Lecture note in control and information scienses

Viterbi or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar mea

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L D Davisson • A G J MacFarlane • H Kwakernaak

J L Massey • Ya Z Tsypkin • A J Viterbi

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t r e a t m e n t 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 f r o m an input-

o u t p u t viewpoint P r i o r to t r e a t i n g the q u e s t i o n of s t a b i l i t y (and instability), we study b o t h the d e c o m p o s i t i o n and the w e l l -

p o s e d n e s s of such systems It is n o t n e c e s s a r y for the r e a d e r

to have s t u d i e d f e e d b a c k s t a b i l i t y b e f o r e t a c k l i n g this book, as

we d e v e l o p results c o n c e r n i n g f e e d b a c k systems as s p e c i a l cases

of more g e n e r a l r e s u l t s p e r t a i n i n g to l a r g e - s c a l e systems

However, the reader should k n o w some e l e m e n t a r y f u n c t i o n a l analysis (e.g L e b e s g u e spaces, c o n t r a c t i o n m a p p i n g theorem), and h a v e some general k n o w l e d g e (e.g P e r r o n - f r o b e n i u s theorem) The first c h a p t e r is introductory, and c h a p t e r s 2 and 3 c o n t a i n

b a c k g r o u n d material; a f t e r that, the r e m a i n i n g c h a p t e r s are

e s s e n t i a l l y i n d e p e n d e n t and can be r e a d in any order

I t h a n k P e t e r M o y l a n for his c a r e f u l r e a d i n g of the

m a n u s c r i p t and for s e v e r a l c o n s t r u c t i v e suggestions, and my w i f e

S h a k U n t h a l a for her support V i r t u a l l y all of my r e s e a r c h

r e p o r t e d in this b o o k was c a r r i e d out, and m o s t of the b o o k was written, w h i l e I was e m p l o y e d by C o n c o r d i a U n i v e r s i t y , Montreal

I w o u l d like to a c k n o w l e d g e r e s e a r c h s u p p o r t f r o m the N a t u r a l Sciences and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada, and to a lesser e x t e n t f r o m the U.S D e p a r t m e n t of Energy Finally, my thanks to M o n i c a E t w a r o o and J a n e S k i n n e r for t y p i n g the

m a n u s c r i p t

W a t e r l o o

S e p t e m b e r 29, 1980

M V i d y a s a g a r

TABLE OF CONTENTS CONT'D

T h r o u g h o u t t h i s b o o k , t h e e m p h a s i s is o n 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 s y s t e m , c o n -

s i s t i n g of s e v e r a l s u b s y s t e m s i n t e r a c t i n g t h r o u g h v a r i o u s i n t e r -

c o n n e c t i o n o p e r a t o r s (For a p r e c i s e d e s c r i p t i o n , see S e c t i o n 2.2) It is of c o u r s e p o s s i b l e to " a g g r e g a t e " the v a r i o u s s u b -

s y s t e m o p e r a t o r s a n d t h e v a r i o u s i n t e r c o n n e c t i o n o p e r a t o r s , so

t h a t 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 t h e f o r m o f a

" s i n g l e - l o o p " f e e d b a c k s y s t e m W i t h this r e f o r m u l a t i o n , a l l o f 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 r e s u l t s , s u c h as

introduce the concepts of t r u n c a t i o n s a n d e x t e n d e d spaces, w h i c h provide the m a t h e m a t i c a l s e t t i n g for i n p u t - o u t p u t analysis, w e then give p r e c i s e 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 stability

In C h a p t e r 3, w e i n t r o d u c e the c o n c e p t s of gain and dissipativity,

w h i c h p l a y an i m p o r t a n t role in the v a r i o u s c r i t e r i a for

stability and instability, and give e x p l i c i t m e t h o d s for com- puting gains and t e s t i n g d i s s i p a t i v i t y

In C h a p t e r 4, we p r e s e n t a few g r a p h - t h e o r e t i c tech- niques for the e f f i c i e n t d e c o m p o s i t i o n of l a r g e - s c a l e inter-

c o n n e c t e d systems S p e c i f i c a l l y , we show that by i d e n t i f y i n g the s o - c a l l e d strongly c o n n e c t e d c o m p o n e n t s (SCC's) of a g i v e n system, we can d e t e r m i n e the w e l l - p o s e d n e s s and s t a b i l i t y of the original s y s t e m by s t u d y i n g only the SCC's In C h a p t e r 5, we

p r e s e n t some s u f f i c i e n t c o n d i t i o n s for the w e l l - p o s e d n e s s of a system T h e s e criteria are g r a p h - t h e o r e t i c in nature and can be given a v e r y nice p h y s i c a l interpretation

In C h a p t e r 6, we give some g e n e r a l i z a t i o n s of the

s i n g l e - l o o p "small gain" t h e o r e m to a r b i t r a r y i n t e r c o n n e c t e d systems, w h i l e in C h a p t e r 7, we state and prove several

g e n e r a l i z a t i o n s of the s i n g l e - l o o p "passivity" theorem In

C h a p t e r 8, w e derive several L 2 - i n s t a b i l i t y c r i t e r i a for large- scale systems Finally, in C h a p t e r 9, w e show how 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 can be used to study L - s t a b i l i t y and

L - i n s t a b i l i t y u s i n g the results of C h a p t e r s 6 to 8

s u b s e t of A, a n d that if f(.) 6 LI• then IIf(.) Ill = l]f(.)llA-

M o r e o v e r • the o r d e r e d p a i r (A• II.l I A) is a B a n a c h space

In (34), one s h o u l d i n t e r p r e t

35

36

37

(t-t a) * ~(t-t b) = ~ (t-ha- ~ )

~(t-ta) * fa(t) = fa(t-ta)

i=0

i=0 then

Also, we see f r o m (39) that

H e n c e the set A is a B a n a c h a l g e b r a w i t h a unit, w i t h I I.I I A as

t h a t w e e n c o u n t e r in t h i s m o n o g r a p h (even the " u n s t a b l e " ones)

c a n b e a s s u m e d to b e of t h e f o r m (46), w h e r e the k e r n e l g(.) E

A (or, m o r e g e n e r a l l y , g(') ~ An×me , in the c a s e of a

m u l t i v a r i a b l e s y s t e m S i m i l a r l y , i t c a n b e s h o w n [Vid 4, Thrm 6.5.37] that, if G is o f the f o r m (46), t h e n G m a p s L

P (the u n e x t e n d e d space) i n t o i t s e l f V p e [I,~], if a n d o n l y if

w e a s s u m e t h a t 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

3 ÷ L z

L p e pe c a n b e r e p r e s e n t e d b y an n i x n j m a t r i x ~ij of c o n s -

t a n t r e a l n u m b e r s , i.e t h a t

n

3

(Hij yj)(t) = H ~13 yj(t) , Vt, Vyj e L p e

A c t u a l l y , this a s s u m p t i o n d o e s n o t r e s u l t in a n y loss of g e n e r - ality, b e c a u s e this a s s u m p t i o n can a l w a y s b e s a t i s f i e d b y i n c r e a s - ing the n u m b e r of s u b s y s t e m s (m) if n e c e s s a r y (If a p a r t i c u - lar o p e r a t o r H c a n n o t b e r e p r e s e n t e d b y a c o n s t a n t m a t r i x ,

7c

7d

Yl = G1 el Y2 = G2 e2

w h e r e Ul' u2' el • e2' YI' Y2 a l l b e l o n g to L ~ pe f o r s o m e f i x e d

s t a b i l i t y t h e o r y

C o m p a r i n g the g e n e r a l l a r g e - s c a l e s y s t e m d e s c r i p t i o n (1) w i t h t h e 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 (7), w e see t h a t if w e

a g g r e g a t e the e q u a t i o n s (I) i n t o the f o r m (4), t h e n (4) a n d (7) are v e r y s i m i l a r In fact, (4) is a s p e c i a l c a s e of (7), w i t h

u I = u, u 2 = 0, G 1 = G, G 2 = H, e I = e, a n d Y l = y " T h i s is

s h o w n in F i g u r e 2.3 Thus, g i v e n an L S I S , o n e c a n e i t h e r r e p -

r e s e n t it i n t h e d e c o m p o s e d f o r m (i) a n d a n a l y z e it a t t h e s u b -

s y s t e m l e v e l , or o n e c a n r e p r e s e n t it in t h e a g g r e g a t e d f o r m (4) and a n a l y z e i t as a s i n g l e - l o o p s y s t e m If o n e c h o o s e s the l a t t -

er o p t i o n • o n e c a n i m m e d i a t e l y a p p l y a l l of t h e s t a n d a r d r e s u l t s

d e r i v e d in [Des 2] a n d [Wil 2] for f e e d b a c k s y s t e m s T h u s t h e

m a i n e m p h a s i s in t h i s m o n o g r a p h is o n a n a l y z i n g a g i v e n L S I S a t the s u b s y s t e m level, t a k i n g f u l l a d v a n t a g e o f t h e f a c t t h a t t h e

s y s t e m a t h a n d is an i n t e r c o n n e c t i o n of s e v e r a l ( p r e s u m a b l y s i m p - ler) s u b s y s t e m s

* A c t u a l l y • Ul, el, Y2 a l l n e e d to b e l o n g to the s a m e s p a c e L p e

92

a n d u2' YI' e2 all n e e d to b e l o n g to the s a m e s p a c e L q e , b u t

in g e n e r a l w e c o u l d h a v e P # q' 91 ~ ~2 " T h e 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 h e r e to t h i s s i t u a t i o n is t r a n s p a r e n t

YT on u T is g l o b a l l y L i p s c h i t z c o n t i n u o u s In o t h e r w o r d s , for e a c h T < = , t h e r e e x i s t s a f i n i t e c o n s t a n t k s u c h that,

w h e n e v e r u (I) a n d u (2) are two s e t s of i n p u t s in L n a n d

pe {e(1) • y(1)} , {e(2) , y(2)} are the c o r r e s p o n d i n g s o l u t i o n sets of (i), w e h a v e

l]e(1)-e(2) ]ITp <_ kTI lu(1)-u(2) IITp

T h e a b o v e d e f i n i t i o n of w e l l - p o s e d n e s s is q u i t e b r o a d ,

as it i m p l i e s (i) 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 to the

s y s t e m e q u a t i o n s , (ii) c a u s a l d e p e n d e n c e of s o l u t i o n s o n inputs, and (iii) g l o b a l L i p s c h i t z c o n t i n u i t y of s o l u t i o n s as f u n c t i o n s

s h o w n in C h a p t e r 5 t h a t p r o p e r t i e s (Wl) - (W3) a r e p r e s e r v e d

s o l u t i o n s of (i), w e h a v e

(W2) T h e d e p e n d e n c e o f e a n d {Sij ej} o n u is

c a u s a l ; i e , w h e n e v e r u (I) a n d u (2) a r e t w o i n p u t s e t s in

L n s u c h t h a t f o r s o m e T > 0 w e h a v e

P

pe and e (I) , e (2) are the c o r r e s p o n d i n g s o l u t i o n sets of (18),

In D e f i n i t i o n (24), w e n o t o n l y r e q u i r e t h a t the e x -

p l i c i t l y d i s p l a y e d u n k n o w n s e i b e l o n g to L ~ i Vi w h e n e v e r

T h e a l t e r n a t i v e s y s t e m d e s c r i p t i o n s (i) a n d (18), to-

g e t h e r w i t h t h e a s s o c i a t e d d e f i n i t i o n s o f w e l l - p o s e d n e s s a n d

s t a b i l i t y , a r e c l o s e l y r e l a t e d T o c o n v e r t a s y s t e m d e s c r i b e d b y (18) to t h e f o r m (i), w e d e f i n e

p o n d i n g to e a c h s e t of i n p u t s u I, U 2 m , (31) m u s t h a v e a

u n i q u e set of s o l u t i o n s for e l , , e 2 m T h i s c o n d i t i o n is m o r e

r e s t r i c t i v e t h a n t h e c o r r e s p o n d i n g c o n d i t i o n r e s u l t i n g f r o m apply- ing D e f i n i t i o n (9) to the s y s t e m (I), b e c a u s e this l a t t e r o n l y

P T e l , , P T e 2 m d e p e n d in a g l o b a l l y L i p s c h i t z c o n t i n u o u s m a n n e r

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 B e -

c a u s e of all t h e s e d i f f e r e n c e s , it is c l e a r t h a t D e f i n i t i o n (19)

a p p l i e d to the s y s t e m (31) g i v e s a s t r o n g e r (i.e m o r e r e s t r i c t - ive) c o n c e p t of w e l l - p o s e d n e s s t h a n D e f i n i t i o n (9) a p p l i e d to the

s y s t e m (i) H o w e v e r , if we a s s u m e t h a t e a c h of the o p e r a t o r s Hij is causal, a n d t h a t e a c h o p e r a t o r ejT ~ (Hij ej) T has

f i n i t e i n c r e m e n t a l g a i n (see D e f i n i t i o n (3.1.i), t h e n the w e l l -

p o s e d n e s s of the s y s t e m (31) in the s e n s e of D e f i n i t i o n (19) w i t h Um+ 1 = = U 2 m = 0, is e q u i v a l e n t to the w e l l - p o s e d n e s s of the

s y s t e m (i) in the s e n s e of D e f i n i t i o n (9) A n o p e r a t o r Hij s a t -

i s f y i n g the a b o v e c o n d i t i o n s is s a i d to b e w e a k l y L i p s c h i t z (see

D e f i n i t i o n (5.1.i)) Thus, o u r c o n c l u s i o n s c a n b e s u m m a r i z e d as

f o l l o w s : 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 s y s t e m (31) is w e l l - p o s e d w i t h U m + 1 = = U 2 m = 0

in the s e n s e of D e f i n i t i o n (19) if a n d o n l y if the s y s t e m (i) is

w e l l - p o s e d in t h e s e n s e of D e f i n i t i o n (9)

S i m i l a r r e m a r k s a p p l y to t h e two s t a b i l i t y d e f i n i t i o n s (5) a n d (24), as a p p l i e d to t h e 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

D e s o e r [Cal 2]

s t a b i l i t y c r i t e r i a in the s u b s e q u e n t c h a p t e r s are c o u c h e d in

t e r m s of t h e g a i n s or d i s s i p a t i v i t y c o n s t a n t s of v a r i o u s operators,

it is i m p o r t a n t to k n o w h o w to a c t u a l l y c a l c u l a t e t h e s e c o n s t a n t s for a g i v e n o p e r a t o r Thus, the r e s u l t s of t h i s c h a p t e r are im-

p o r t a n t in o r d e r to s h o w t h a t the v a r i o u s s t a b i l i t y c r i t e r i a deriv-

ed in t h i s m o n o g r a p h c a n a c t u a l l y b e a p p l i e d to "real" s y s t e m s 3.1 GAIN, G A I N W I T H Z E R O BIAS, A N D I N C R E M E N T A L G A I N

3 1 1 D e f i n i t i o n s of V a r i o u s T y p e s of G a i n

D e f i n i t i o n S u p p o s e p 6 [i,~], t h a t n, m are p o s i t - ive i n t e g e r s , a n d t h a t G:L n ÷ L TM is a g i v e n o p e r a t o r T h e n we

pe pe

d e f i n e the v a i n of the o p e r a t o r G, d e n o t e d b y yp(G), b y

yp(G) = inf { k : ~ b < - s u c h t h a t I IGxl ITp ~ kl Ixl ITp + b,

If the s u p r e m u m in (4) does n o t exist, we set np(G) = ~

In the a b o v e d e f i n i t i o n s , w e r e c o g n i z e t h a t the con- stants yp(G), ~p(G), and ~p(G) d e p e n d n o t o n l y on the o p e r a t o r G, but a l s o on the v a l u e of p

N o t e that, in general, yp(G) ! ~p(G), and ~p(G) ~ np(G)

w h e n e v e r G(0) = 0 Also, if yp(G) is finite, then G m a p s the

u n e x t e n d e d space L n into the u n e x t e n d e d space L TM (However,

÷ L d e f i n e d b y the c o n v e r s e is not true; the o p e r a t o r G:L e ~ e

22 IT l(Gx) Ct)I at <_ IT [ IgCt,~)l t IxC~)l dr a t

2

= T [x(~)[ Igct,~)l dt ~r

t

0 and let x 6 L T h e n w e have

Now, (19) follows from (32) and (35)

Finally, to prove (20), let p • (1,-), and let

inequality [Dun I, p i19], we get

= [y (G)] p/q Yl(G) fix(.) lip

w h i c h e s t a b l i s h e s (20) o

L e m m a (15) s h o w s t h a t , f o r o p e r a t o r s of t h e f o r m (16), the g a i n s YI(G) a n d y~(G) c a n b e c a l c u l a t e d e x a c t l y , w h e r e a s

is e x a c t l y t h e s a m e as its g a i n w h e n v i e w e d as a n o p e r a t o r f r o m Lp(R+) i n t o Lp(R+)

P r o o f C l e a r l y y ~ y+ T o p r o v e the o p p o s i t e , l e t

e > 0 b e g i v e n , a n d s e l e c t 6 e (0,i) s u c h t h a t