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7Modeling Interconnected Systems:

A Functional Perspect ivePeoria, Ill inois, USA

7 .1 I n t r o d u c t i o n . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . 10797 .2 T h e C o m p o n e n t C o n n e c t i o n M o d e l . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. 1079

7.2.1 Detailed Insight to Comp one nt Connect ion Philosophy7 .3 Sy s t e m I d e n t i f i c a t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827 .4 S i m u l a t i o n . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . 10827 .5 Fa u l t A n al ys is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827 .6 Co n c l u d i n g Re m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083

Re f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083

.1 Introductionhis chap te r , ada p ted f rom L ibe r ty and Saeks (1973 , 1974),o n t a i n s a p a r t i c u l a r v i e w p o i n t o n t h e m a t h e m a t i c a l m o d e l i n gf s y s te m s t h a t f o c u s e s o n a n e x p l i c it a l g eb r a ic d e s c r i p t i o n o f

c o m p o n e n t c o n n e c t i o n m o d e l . T h ep o n e n t c o n n e c t i o n a p p r o a c h t o s y s t em m o d e l i n g p r o v i d esf i c a n t c o n c e p t u a l v a l u e . T h e g e n e r a l a p p l i c a b i l i ty o f t h i sp o i n t t o s o m e o f th e s e s y s t e m p r o b l e m s i s p r e s e n t e d h e r e

u s t r a ti n g t h e c o m p o n e n t c o n n e c t i o n m o d e l s h o u l d f i rs t r e f e re c a r l o a n d Sa ek s ( 1 9 8 1 ). A d d i t i o n a l r e f e r e n c e s a r e p r o -s p ec i fi c a p p l ic a t io n s o f t h e c o m p o n e n t c o n n e c t i o n

f t h e g e n e r a l c o n c e p t o f fu n c t i o n s h o u l d h a v e n oc u l t y c o m p r e h e n d i n g t h e m a t e r i a l i n t h is s e c t i o n .

. 2 T he C om ponent C onnec t i on Mode ln s i m p l i s ti c m a t h e m a t i c a l t e r m s , i s a m a p p i n g f r o m

set o f inp uts (signals) to a se t of outp ut s (signals) ( i.e ., ann p u t / o u t p u t r e l a ti o n w i t h a u n i q u e o u t p u t c o r r e s p o n d i n g t oa c h i n p u t ) . ( I n u s i n g t h i s e x t e r n a l s y s t e m d e s c r i p t i o n , t h eo n t e x t o f a n i n t e r n a l s y s t e m s t a te h a s b e e n d e l i b e r a t e l y s u p -opyright 2005 by AcademicPress.any form reserved.

p r e s s e d f o r s i m p l i c i t y o f p r e s e n t a t i o n i n t h i s c h a p t e r . ) T h i sm a y b e a b s t r a c t l y n o t a t e d b y :

y = Su, (7 .1 )w h e r e S i s t h e s y m b o l f o r t h e i n p u t / o u t p u t r e l a t i o n ( t h es y s te m ) . O n t h e o t h e r h a n d , o n e m a y t h i n k o f t h e p h y si c als y s te m s y m b o l i z ed b y S a s a n i n t e r c o n n e c t i o n o f c o m p o n e n t s .E n g i n e e r i n g e x p e r i e n c e e x p l a i n s t h a t a p a r t i c u l a r S i s d e t e r -m i n e d b y tw o f a c t o r s : t h e t y p e s o f c o m p o n e n t s i n t h e s y s t e ma n d t h e w a y i n w h ic h t h e c o m p o n e n t s a r e i n te r c o n n e c t e d . T h isl a t t e r o b s e r v a t i o n o n t h e p h y s i c a l s t r u c t u r e o f a s y s t e m g i v e sr i se to the c o m p o n e n t c o n n e c t i o n v i e w p o i n t .A g a i n , t h i n k i n g o f a p h y s ic a l s y s te m , t h e s y s te m ' s c o n n e c t i o ns t r u c t u r e c a n b e h e l d f i x e d w h i l e th e t y p e s o f s y s t e m c o m p o -n e n t s o r t h e i r v a l u e s c h a n g e . E x p e r i e n c e e x p la i n s t h a t f o r e a c hd i s ti n c t s e t o f c o m p o n e n t s o r c o m p o n e n t v a lu e s, a u n i q u ei n p u t / o u t p u t r e l a t i o n i s d e t e r m i n e d . F r o m a m a t h e m a t i c a lp e r s p e c t i v e , a g iv e n c o n n e c t i o n d e s c r i p t i o n d e t e r m i n e s a m a p -p i n g o f th e i n t e r n a l c o m p o n e n t p a r a m e t e r s t o t h e i n p u t /o u t p u t r e l a t i o n .

N o w , o n e c a n g e n e r a l i z e t h is s l i g h t l y b y t h i n k i n g o f t h ep h y s i c a l s y s t e m a s a n i n t e r c o n n e c t i o n o f su b s y s te m s , e a c hw i t h i t s o w n i n p u t / o u t p u t r e l a t i o n . N o t e t h a t a s u b s y s t e mm a y b e e i t h e r a d i s c r e t e c o m p o n e n t , a n i n t e r c o n n e c t i o n o fc o m p o n e n t s , o r a n i n t e r c o n n e c t i o n o f " s m a l l e r" s u b sy s te m s .T h i s h i e r a r c h i c a l s t r u c t u r e i s g e r m a n e t o t h e c o m p o n e n t c o n -

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S t a n l e y R . L i b e r t yn d i s p rec i se l y t h e fu n d am en t a l co n cep trge-scale s ystem theo ry.Think ing in th i s way , one no tes tha t a g iven connect ion

i o n d e t e rm i n es a s y s t em -v a lu ed fu n c t i o n o f a s y s t em -ab le . Because th i s fun ct ion i s en t i re ly determ ined by

S = f ( Z ) , (7.2)

n e n t p a r a m e t e rs a n d w h e r e f s y m b o l i z e s t h e c o n n e c t i o nthe specia l case where the subsys tems represen teday b e i n t e rp re t ed a s ap o n en t t r an s fe r fu n c t i o n s , g a in s, o r i m m i t an ces

s imp or ta n t to po in t ou t tha t , even when a l l o f the subsys-n ea r, t h e co n n e c t i o n fu n c t i o n f i s n o n l in ea r .W h a t m a y b e s u r pr is in g i s t h a t f o r m o s t c o m m o n l y e n c o u n -em s , th e n o n l i n ea r fu n c t i o n f can b e co m p l e t e l yb ed b y fo u r m a t r ice s t h a t p e rm i t t h e co n n ec t i o n fu n c t i o nm anip u la te d qu i te read i ly . Ind eed , feasib le ana ly t i ca l and

p u t a t i o n a l t ech n i q u es b as ed o n t h is p o i n t o f v iew h av eTraub oth , 1973; Prasad and Reiss , 1970; Trau bothPrasad , 1970) , sens i t iv i ty s tud ies (Ran som, 1973 , 1972;nsom and Saeks, 1975; DeC ar lo , 1983), syn thes is (Ransom ,

1973) , s t ab i l ity analys i s (Lu and Liu , 19 72) , op t im izat ionRi ch a rd so n , 1 9 6 9 ; R i ch a rd s o n e t a l . , 1969), wel l-posednessRan som, 1972; Singh , 1972), an d fau l t analysi s (Ransom,1973, 1 972 ; Saeks, 197 0; Saeks e t a L , 1 9 7 2 ; R a n s o m a n daeks , 19 73a, 1973b; DeC ar lo and Rapisarda , 1988; Rei sig

DeC ar lo , 19 87; Rapisarda a nd D eCar lo , 1 983; Garz ia ,1971) . The reason for th i s core o f l inear i ty in the fdes cr ip t iont h a t co n n ec t i o n i n fo rm a t i o n i s g en e ral ly co n t a i n ed i n " co n -b les , y ie ld l inear "co nne ct ion " equat ions .T o o b t a i n an i n t u it i v e fee l f o r w h a t t y p es o f m an i p u l a t i o n s

one performs on f to so lve the sys tems prob lems prev ious lyent ioned, consider, for example, a classical passive networksynthes i s p rob lem. In th i s p rob lem, one i s g iven the inpu t /o u t p u t r e l a t i o n S , an d o n e k n o w s w h a t t y p e o f n e t w o rk s t ruc -ure i s des i red . Fo r example , a l adder syn thes i s p rob le m s ta te-

en t cons i s t s o f a t rans fer funct ion speci f i ca t ion and theco n n ec t i o n i n fo rm a t i o n " o f lad d e r s t ru ct u re . T h e S an d f a r epeci f i ed , and i t i s des i red to determine Z ( the components ) .l ear ly , to f ind Z , one m us t inver t f . In th i s case, one seeks a

r igh t inverse o f f because the un iquen ess o f Z i s no t a con cernand f i s v iewed as "on to" the c lass o f t rans fer funct ions o finterest .I f , o n t h e o t h e r h an d , o n e i s a tt em p t i n g t o i d en t i fy t h e v a l u eo f a c o m p o n e n t i n a n i n te r c o n n e c te d s y s te m o f c o m p o n e n t sg i v en ex t e rn a l m eas u rem en t s an d k n o w l ed g e o f t h e co n n ec -

t ions , then one seeks a l e f t inverse o f f . Here , the concern mayb e w i t h u n i q u en ess , an d f i s v i ew ed as o n e - t o -o n e an d i n t o t h es e t o f i n p u t / o u t p u t r e la t io n s .Another c lass ica l sys tem prob lem encountered in des ignanalys i s i s tha t o f de termin ing the sens i t iv i ty o f the inpu t /o u t p u t r e l a ti o n S t o v a r i a t io n s i n ce r t a in c o m p o n en t v a lu es.In t h is ca se , d i f f e r en ti a t io n o f th e co n n ec t i o n fu n c t i o n fw i t hre sp ec t to t h e co m p o n en t p a ram e t e r s o f i n te r e s t is t h e m a j o rcon s t i tuen t o f the sens i t iv ity analys is . Speci fi c app l ica t ions o fthe co mp one nt co nnec t ion mod el to sens i t iv i ty analys is arecon ta ined in Ran som ( 1973, 1972), R anso m and Saeks ( 1975) ,and DeCar lo (1983) .The concep tual descr ip t ion jus t descr ibed should prov idet h e r ead e r w i t h a " f ee l " fo r t h e co m p o n en t co n n ec t i o n p h i -losophy . This chap ter now prov ides more deta i l ed ins igh t .7 .2 .1 D e t a i l e d I n s i g h t t o C o m p o n e n tC o n n e c t i o n P h i l o so p h yA l t h o u g h an ab s t r ac t f u n c t i o n a l i n t e rp re t a t i o n o f t h e co n -nect ions in a sys tem i s concep tual ly val id , it i s o f no prac t i ca lvalue un less one has a specif ic and com puta t iona l ly v iab lerepresen ta t ion of the funct ion . For tunate ly , there i s sucha r ep re s en t a t i o n , an d t h e co n cep t u a l p ro b l em o f i n v e r t i n go r d i f f e r en t i a t i n g t h e co n n ec t i o n fu n c t i o n m ay ac t u a l l y b ecarr i ed ou t .Class ica l ly , in c i rcu i t and sys tem theory , connect ions arerepresen ted by some ty pe of g raph (e .g ., l inear g raph , bo ndgraph , s ignal f low graph) o r b lock d iag ram. This g raphica ld ep i c t i o n o f co n n ec t i o n i n fo rm a t i o n can r ead i l y b e co n v e r t edin to a se t o f linear a lgebra ic cons t ra in t s o n th e c i rcu i t o r sys temvariables for sub sequ ent analysis . As such , i t is na tura l to ado ptan a lgebraic m ode l o f the con nect ions f ro m the s tar t .

T h e p rec is e fo rm o f t h e co m p o n e n t co n n ec t i o n m o d e l w i t hi t s a lgebra ic descr ip t ion of a sys tem 's con nec t ion in fo rma t ionmay be seen by examin ing Figure 7 .1 . Figure 7 .1 (A) shows ab o x d ep i c t io n o f a s y st em w i t h i n p u t s u an d o u t p u t s y . T h ei n n e r b o x l ab el ed Z r ep re s en ts t h e s y s tem co m p o n en t s , an d t h eo u t e r d o n u t s h ap ed a r ea r ep re s en t s t h e s y s t em co n n ec t i o n s(e.g. , scalers , adde rs , K irch hof f law con straints , etc.) . Now , i fZ r ep re s en ts a m a t r i x o f co m p o n en t i n p u t / o u t p u t r e l a ti o n s,the fo l lowing ma y be abs t rac t ly wr i t ten :

b = Z a , (7.3)w h ere b d en o t e s t h e v ec t o r o f co m p o n e n t o u t p u t v a riab le s an da t h e v ec t o r o f co m p o n en t i n p u t v a ri ab le s. T h e c o m p o n e n tco n n ec t i o n m o d e l can b e o b t a i n ed b y r e& aw i n g t h e s y s tem o fFigure 7 .1 (A) as in Figure 7 .1 (B) , where the components andconnect ions have been separa ted . Thus , the overa l l sys tem i sco m p o s ed o f t w o i n t e r co n n ec t ed bo x es . O n e b o x co n t a i n s t h eco m p o n en t s a s d es c r i b ed b y eq u a t i o n 7 . 3 , an d t h e s eco n dd o n u t - s h ap ed b o x co n t a i n s t h e co n n ec t i o n s . T h e d o n u t -shaped box has inpu ts u and b and ou tpu t s a and y . Final ly ,

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7 M ode ling In terconnected Systems: A Fun ctiona l Perspective 1081

(A) System with Inputs / OutputsF I G U R E 7 . 1

y = u

( B ) S e pa r at e d C om pone nt s a nd C onne c t ionsA System as an Interconnection of Comp onents

s i n c e t h e c o n n e c t i o n s a r e c h a r a c t e r i z e d e n t i r e l y b y l i n e a r a lg e -b r a ic c o n s t ra i n ts , t h e d o n u t - s h a p e d b o x m a y b e m a t h e m a t i -c a l ly m o d e l e d ( i. e. , t h e c o n n e c t i o n s ) b y t h e m a t r i x e q u a t i o n :

[ ; ] = r L l l L 1 2 1L L 2 1 L 2 2 J [ : ]T h e c o m p o n e n t c o n n e c t i o n m o d e l i s t h u s a p a i r o f e q ua t io n s ,7 .3 a n d 7 .4 , w i t h t h e m a t r i x Z r e p r e s e n t i n g t h e c o m p o n e n t s ,a n d t h e s e t o f f o u r L m a t r i c e s r e p r e s e n t i n g t h e c o n n e c t i o n s .

A n i n t e r e s t i n g o b s e r v a t i o n c a n b e m a d e h e r e . I n t h e v e r ys p e c i a l c a s e w h e r e e a c h c o m p o n e n t i s a n i n t e g r a t o r , t h efo l lowing i s t rue :

b=a.Subs t i tu t ing equa t ion 7 .5 in to 7 .4 y ie lds :

= L l l b q - L 1 2 u .y = L 2 1 b q - L 2 2 u .

r e a d e r w i l l re c o g n i z e t h e se e q u a t i o n s a s t h e f a m i l i a r " s t a t ed e l " o f l i n e a r d y n a m i c a l s y s t e m t h e o r y . T h u s , i n t u i ti v e ly , th ep o n e n t c o n n e c t i o n m o d e l m a y b e v ie w e d a s a g e n e ra l iz a -o f t h e l i n e a r s t a t e m o d e l . I n f a c t, t h e L m a t r i c e s d e s c r i b et o i n t e r c o n n e c t s u b s y s te m s t o f o r m a la r g e -s c a l e s y s t e m S ,

p u t e r t o s i m u l a t e a s y s t em . I f o n e v ie w s t h e L m a t r i c e s a s at h e s t a t e m o d e l , w h e r e t h e i n t e g r a t o r s h a v e

r e p l a c e d b y g e n e r a l c o m p o n e n t s , a b e n e f i c ia l c ro s s f e rt i l-

u t c o n t r o l l a b l e o r o b s e r v a b l e c o n n e c t i o n s ( S i n gh , 1 9 72 ).T h e c o m p o n e n t c o n n e c t i o n m o d e l w a s f ir st u s e d i n t u it i v el y

r a u b o t h ( S i n g h a n d T r a u b o t h , 1 9 7 3 ; P r a s a d970; Trau both an d Prasad , 1970) and fo rm a l ized by

e t a l . , 1972) fo r app l ica t ion in the

f a u l t is o l a t i o n p r o b l e m . E x i s t e n c e c o n d i t i o n s f o r t h e L m a t r i c e swere f i rs t s tud ie d by Prasad (P ra sad and Re iss , 1970) and la te rb y S i n g h ( 1 9 7 2 ) w h o h a v e s h o w n t h a t t h e e x i s t e n c e o f Lmatr ice s i s a r ea sonab le a ssumpt ion . Th is i s e ssen t ia l ly thes a m e t y p e o f a s s u m p t i o n t h a t o n e m a k e s w h e n a s s u m i n g t h a t

(7 .4 ) the s ta te equ a t ion s o f a sys tem ex is t. As seen above , i f thec o m p o n e n t s i n t h e s y s t e m a r e i n te g r a t o r s , t h e n t h e L m a t r i c e sa re p rec ise ly the s ta te ma t r ice s .

T h e a d v a n t a g e o f t h e c o m p o n e n t c o n n e c t i o n m o d e l o v e r t h ec la ss ic a l g r a p h ic a l a n d d i a g r a m m a t i c c o n n e c t i o n m o d e l s i st h a t , b e i n g i n h e r e n t ly a lg e b ra i c, t h e c o m p o n e n t c o n n e c t i o nm o d e l is r ea d i ly m a n i p u l a t e d a n d a m e n a b l e t o n u m e r i c a lt e c h n i q u e s . M o r e o v e r , t h i s m o d e l u n i f i e s th e v a r i o u s g r a p h i c a la n d d i a g r a m m a t i c m o d e l s . I n f a c t , e l e c t r o n i c c i r c u it s , w h i c h

( 7 .5 ) a r e c o m m o n l y d e s c r i b e d b y h y b r i d b l o c k d i a g r a m - l i n e a r g r a p hmode ls , a r e hand led a s r ead i ly a s pass ive c i r cu i t s o r ana logc o m p u t e r d i a g r a m s [ s e e D e Ca r l o a n d Sa e k s ( 1 9 8 1 ) f o r s o m es i m p l e e x a m p l e s ] .

U s i ng t h e c o m p o n e n t c o n n e c t i o n m o d e l , t h e d e s ir e d r e p re -( 7 .6 ) s e n t a t i o n f o r t h e c o n n e c t i o n f u n c t i o n is o b t a i n e d . T h i s i s

i l l u s t r a t e d b y a l i n e a r e x a m p l e s i n c e t h e c o n c e p t c a n b e m o r ei n t u i t i v e l y u n d e r s t o o d i n t h e l i n e a r c a se . I t is i m p o r t a n t t op o i n t o u t a g a i n , h o w e v e r , th a t m o r e g e n e r a l i t y i s a c t u a l l yp r e s e n t . V i e w i n g Z a s a m a t r i x o f t r a n s f e r f u n c t i o n s , t h e o v e r a l ls y s t e m t r a n s f e r f u n c t i o n i s d e s i r e d . S i m u l t a n e o u s s o l u t i o n o fequations 7.1, 7 .3, and 7.4 yie lds:

S = L 2 2 q - L2 1 1 - ZLH) 1 Z L 1 2 = f ( Z ) , ( 7 . 7 )a n d w e h a v e a r e p r e s e n t a ti o n o f t h e c o n n e c t i o n f u n c t i o n i nt e r m s o f t h e f o u r L m a t r ic e s e v e n t h o u g h t h e c o n n e c t i o nf u n c t i o n i s n o n l i n e a r i n Z . T h i s m a t r i x r e p r e s e n t a t i o n f a c i l i -t a te s m a n i p u l a t i o n o f th e c o n n e c t i o n f u n c t i o n a s sh o w n l a te r(Ransom, 1973 , 1972; Saeks , 1970; Ransom and Saeks , 1973b ,1 9 7 3 a ) . T h e f i r s t o b s e r v a t i o n t h a t t h e c o n n e c t i o n f u n c t i o nc o u l d b e s o r e p r e s e n t e d w a s m a d e b y Sa e k s ( 1 9 7 0 ) , w h i l e t h ef i rs t e x p l o i t a t i o n o f th e c o n c e p t i s a t t r i b u t a b l e t o R a n s o m(1973, 1972) and Saeks (1973b, 1973a) .

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1082 S t a n l e y R . L i b e r t y7.3 System Iden t i f icat ionIn a s y s t e m i d e n t i f ic a t i o n p r o b l e m , o n e i s n o r m a l l y a s ke d t oi d e n t i fy th e m a t h e m a t i c a l i n p u t / o u t p u t r e l a ti o n S fr o m i n p u t /o u t p u t d a t a . I n t h e o r y , th i s i s s tr a i g h t f o r w a r d i f t h e s y s t e m i st r u l y l i n e a r ( Ch a n g , 1 9 73 ), b u t i n p r a c t i c e t h i s m a y b e d i f fi c u l td u e t o n o i s y d a t a a n d f i n i t e p r e c i s i o n a r i t h m e t i c . T h e n o n -l inea r ca se i s d i f f icu l t in gene ra l because se r ie s approx ima t ionsm u s t b e m a d e t h a t m a y n o t c o n v e rg e ra p i d l y e n o u g h f o r l a rg es igna l mode l ing (Be l lo e t a l . , 1 9 7 3 ) . I n b o t h l i n e a r a n d n o n -l i n e a r i d e n t i f i c a t io n t e c h n i q u e s , r a n d o m i n p u t s a r e c o m m o n l yused a s p ro bes (Be llo e t a l . , 1 9 73 ; Co o p e r a n d M c G i l l e m , 1 9 7 1;L e e a n d Sc h e t z e n , 1 9 6 1) . T h e i d e n t i f i c a t i o n o f S is t h e n c a r r i e do u t b y m a t h e m a t i c a l o p e r a t i o n s o n t h e s y s t e m i n p u t a n do u t p u t a u t o c o r r e l a t i o n s a n d t h e c r o s s - a u t o c o r r e l a t i o n s b e -t w e e n i n p u t a n d o u t p u t .

T h i s s e c t i o n d o e s n o t c o n t a i n a d i s c u s s io n o f s u c h t e c h -n i q u e s , b u t s o m e o b s e r v a t i o n s a r e p r e s e n t e d . I t s h o u l d b en o t e d t h a t i f o n l y i n p u t / o u t p u t i n f o r m a t i o n i s a s s u m e dk n o w n ( a s i s t h e c a s e i n s t a n d a r d i d e n t i f i c a t i o n t e c h n i q u e s ) ,h e n s y s te m i d e n t i fi c a t io n d o e s n o t s u p p l y a n y i n f o r m a t i o n o n

t h e t y p e s o r v a lu e s o f c o m p o n e n t s . I nu c h c as es , t h e c o m p o n e n t c o n n e c t i o n m o d e l c a n n o t b e u s e d .

I n m a n y a p p l i c a t i o n s , h o w e v e r , c o n n e c t i o n i n f o r m a t i o n a n dc o m p o n e n t t y p e i n f o r m a t i o n a r e a v a i l a b l e . I n s u c h c a s e s , t h ec o m p o n e n t c o n n e c t i o n m o d e l m a y b e a p p l i c a b l e . I n d e e d , f o ri d e n t i f i c a t io n s c h e m e s u s i n g r a n d o m i n p u t s , i t is e a si ly s h o w nt h a t t h e c o m p o n e n t c o r r e l a t i o n f u n c t i o n s a n d t h e o v e r a l ls y s t e m c o r r e l a t i o n f u n c t i o n s a r e r e l a te d b y :

Ryy = L2 1 a b b LT 1 + Ryu LT2 2 + L2 2 Ru y - - L2 2 Ruu L2 T2 ( 7 . 8 )R a a = L l lR b b L 1 T 1 + L l l R b u L ~ + L 1 2 R u b L T l l + L 1 2 R uu L 1T 2 ( 7 . 9 )

L 2 1 R b u = [ R y u - L 2 2 R u u ] . (7 .10)R a b = L l l R b b + L 1 2 R u b . (7 .11)

h e R j k i s the c ro sscor re la t ion o f s igna l j an d s igna l k . I f the le f te o f L21 e x is ts , t h e n t h e s u b s y s t e m c o r r e l a t i o n f u n c t i o n s

a n b e d e t e r m i n e d a n d u s e d t o i d e n t i f y t h e s u b s y s t e m s v ia

L21 m u s t b e u s e d t o y i e l d s u b s y s t e m

s o l u t i o n o f l a r g e n u m b e r s o f d if f e r e n ti a l e q u a t i o n s . T h ein so lv ing a s ing le d i f f e ren t ia l equ a t ion i s en t i r e ly

i n e d b y t h e n u m e r i c a l t e c h n iq u e s e m p l o y e d . H o w e v e r,a l a rg e - s c a le s y s te m f r o m t h e c o m p o n e n t c o n n e c -

t i o n v i e w p o i n t , t h e s t a r t i n g p o i n t i s a s p e c i f i e d c o l l e c t i o n o fd e c o u p l e d c o m p o n e n t d i f f e r e n ti a l e q u a t i o n s a n d a s e t o fc o u p l e d a l g e b r a i c c o n n e c t i o n e q u a t i o n s . T h u s , it is possiblet o t a k e a d v a n t a g e o f t h e s p e c i a l f o r m o f th e e q u a t i o n s c h a r a c -t e r i z i n g a l a r g e - s c a l e d y n a m i c a l s y s t e m a n d t o f o r m u l a t ea n a ly s is p r o c e d u r e s t h a t a r e m o r e e f f i c i e n t t h a n t h o s e w h i c hm i g h t r e su l t f ro m a p u r e l y n u m e r i c a l s t u d y o f t h e c o u p l e dd i f fe r e n t i a l e q u a t i o n c h a r a c t e r i z i n g t h e e n t i r e s y s t e m .S u c h a p r o c e d u r e , n a m e l y a r e l a x a t io n s c h e m e , w h e r e i n o n ei n t e g ra t e s e a c h o f t h e c o m p o n e n t d i f fe r e n t i a l e q u a t i o n s b ys e p a r a t e ly i t e r a t in g t h r o u g h t h e c o n n e c t i o n e q u a t i o n s t oo b t a i n t h e s o l u t i o n o f t h e o v e r a l l s y s te m , w a s i m p l e m e n t e do v e r 3 0 y e a rs a g o a t t h e M a r s h a ll Sp a c e F l i g h t Ce n t e r ( P r a s a dand Re iss , 1970; Tra ubo th and Prasad , 1970; Saeks, 1969). T hef e a s ib i l it y o f t h e s c h e m e w a s v e r i f ie d , a n d s i g n i fi c a n t c o m p u t e rm e m o r y s a v i n gs r e su l t e d . T h e k e y t o t h i s a p p r o a c h o f t h enum er ica l ana lys is o f a la rge - sca le sys tem is tha t n um er ica lr o u t i n e s a r e a p p l i e d t o t h e d e c o u p l e d c o m p o s i t e c o m p o n e n td i f f e r e n t i a l e q u a t i o n s r a t h e r t h a n a n o v e r a l l e q u a t i o n f o r t h ee n t ir e s y s te m . T h u s , t h e c o m p l e x i t y o f t h e n u m e r i c a l c o m p u -t a t io n s i s d e t e r m i n e d b y t h e c o m p l e x i t y o f t h e l a rg e s t c o m p o -n e n t i n t h e s y s t e m . In c o n t r a s t , i f t h e n u m e r i c a l p r o c e d u r e s a r ea p p l i e d d i r e c t l y t o a s e t o f d i f f e r e n t ia l e q u a t i o n s c h a r a c t e r i z i n gt h e e n t i r e s y st e m , th e c o m p l e x i t y o f t h e n u m e r i c a l c o m p u t a -t i o n s i s d e t e r m i n e d b y t h e c o m p l e x i t y o f t h e e n t i r e s y s te m ( t h es u m o f t h e c o m p l e x it ie s o f a ll t h e c o m p o n e n t s ) .

S i n c e 1 97 0 , c o m p u t e r s i m u l a t i o n s o f t w a r e h a s e v o l v e d s u b -s tan t ia l ly , and packages tha t dea l wi th la rge - sca le sys temse i t h e r e x p li c i tl y o r i m p l i c i tl y e x p l o i t t h e c o m p o n e n t c o n -n e c t i o n p h i l o s o p h y . O n e o f t h e m o r e r e c e n t so f t w a r e d e -v e l o p m e n t s f o r m o d e l i n g a n d s i m u l a t i o n o f la r g e- s c al eh e t e r o g e n e o u s s y s t e m s i s a n o b j e c t - o r i e n t e d l a n g u a g e c a l l e dM o d e l i c a ; s ee M a t t s o n e t a l. ( 1 9 9 8 ) f o r a d d i t i o n a l i n f o r m a t i o n .

7.5 Fault An alys isF a u l t a n a l y s i s i s s imi la r to sys tem iden t i f ica t ion in tha t thes y s t e m is re i d e n t i f ie d to d e t e r m i n e i f t h e c o m p o n e n t i n p u t /o u t p u t r e l a t i o n s h a v e c h a n g e d . I t s h o u l d b e c l e a r t h a t c o n n e c -t i o n i n f o r m a t i o n i s e s se n t ia l to d e t e c t i o n o f a f a u l ty c o m p o -nen t . As be fore , no te tha t f au l t ana lys is i s no t r e s t r ic ted tol i n e a r s y s te m s . T o i l lu s t ra t e s o m e o f t h e t e c h n i q u e , t h e d i s c u s -s i o n w i ll f o c u s o n a l i n e a r t i m e - i n v a r i a n t s i t u a t i o n .

To a t tack the f au l t ana lys is p rob lem, a ssume tha t the re i s ag i v e n se t o f e x t e rn a l s y s te m p a r a m e t e r s m e a s u r e d a t a f i n it e s e to f f r e q u e n c i e s S ( ~ o i ) , i = 1 , 2 . . . . . n a n d t h a t t h e r e a r e t h ec o n n e c t i o n m a t r i c e s , L l l , L 12 , L 2 ~, a n d L 2 2. Co m p u t e o r a p -p r o x i m a t e Z ( o ~) . Fo r s i m p l i c i t y o f p r e s e n t a t i o n , a s s u m e t h a tL22 = 0 . T h e r e i s n o l o ss o f g e n e r a l i ty w i t h t h i s a s s u m p t i o ns ince one can a lways r ep lace S ( ~ o i ) b y S ( c o i ) = S ( o ~ i ) - L22 an dt h e n w o r k w i t h t h e s y s t e m w h o s e m e a s u r e d e x t e r n a l p a r a m -e t e r s a r e g i v e n b y S a n d w h o s e c o n n e c t i o n m a t r i c e s a r eL l l = L l l , L 1 2 = L 1 2 , L 2 1 = L 2 1 , a n d L 2 2 = 0 .

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M o d e l i n g I n t e r c o n n e c t e d S y s t e m s : A F u n c t i o n a l P e r s p e c ti v e 1083I n t h e m o s t e l e m e n t a r y f o r m o f t h e f a u lt a n a ly s is p r o b l e m ,

t h a t S (~ o) i s g iv e n a t o n l y a s i n g l e f r e q u e n c y a n dp u t e Z ( ~ o) e x ac t ly . T h i s f o r m o f t h e p r o b l e m w a s f ir s t

e t a l . , 1972). Itsf c a n b e d e c o m p o s e d i n t o t w o f u n c t io n s ( u n d e r t h e

f = h o g . (7.12)

e t e r m a t r i x Z i n t o a n i n t e r m e d i a r y m a t r i x R v ia :R = g ( Z ) = (1 - Z L l l ) - l Z , (7.13)

e t e r m a t r i x S v ia :S = h (R ) = L 2 1 R L 1 2 . (7.14)

i n v e r se o f f i s g i v e n in t e r m s o f g a n d h v i a:f - L = g - % h - L . (7.15)

e a lgebra wi l l revea l tha t g - L a lways ex is t s and i s g iven by

Z = g - L ( R ) = (1 + R L H ) - L R . (7.16)f a u l t i s o l a ti o n p r o b l e m is r e d u c e d t o t h a t o f f in d i n gnverse of the l inea r fu nc t io n h . Thi s i s mo s t eas i ly do ne

i t h t h e m a t r i x r e p r e s e n t a t i o n o f h as :h = [ LT @ L21], (7.17)

972; Saeks e t a l . , 1 9 72 ; R a o a n d M i t r a , 1 9 7 1) . S t a n d a r d m a t r i xt e c h n i q u e s c a n b e u s e d t o c o m p u t e h - L . I n d e e d , t h e

h c a n e a c h b e i d e n t if i e d w i t h e x t e r n a l i n p u t / o u t p u t. " T h u s , t h e p r o b l e m o f c h o o s i n g a m i n i m a l s e t o f e x te r -

g a m i n i m a l s e t o f r o w s in h t h a t r e n d e r i t s c o l u m n s

a c h e s h a v e b e e n d e v e l o p e d f o r a ll e v i at i n g t h e s e ( R a n s o m ,973, 1972; Ransom and Saeks , 1973b) .

6 Con cluding Remarkst h is c h a p t e r h a s n o t p r o v i d e d a c o m p l e t e o r d e t a il e d

c o m p o n e n t c o n n e c t i o n p h i l o so p h y , t h e re a d e rf u n d a m e n t a l u n d e r s t a n d i n g o f t h e p o w e r

a n d u t i l it y o f t h e p h i lo s o p h y . E v e n t h o u g h t h e c o m p o n e n tc o n n e c t i o n a p p r o a c h t o m o d e l i n g l a r g e - s c a l e s y s t e m s o r i g i -na ted over three decades ago , i t i s l i ke ly tha t i t s fu l l po tent i a lhas n ot b een rea l ized . Th ere a re s evera l a reas of resea rch inw h i c h t h e c o m p o n e n t c o n n e c t i o n a p p r o a c h m i g h t l e a d t o n e wr e su l ts . O n e o f t h e se i s t h e a r e a o f o p t i m a l d e c e n t r a l i z e dc o n t r o l o f l a r g e - sc a l e s y s t e m s .

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1 0 8 4 Stan ley R . L iber tyan som , M.N., and Saek s, R.E . (1973b). Fault isolation via termexpansion. Proceedings of the Third Pittsburgh Symposium on Mo d-eling and Simulation. 224-228.ansom , M.N., an d Saeks , R.E. (1975) . The connec t ion fun ct io n-theory and applicat ion. IEEE Transactions on Circuit Theory andApplications 3, 5- 21 .ao, C.R ., and Mitra, S.K. (1971). Generalized inverse matrices a nd itsapplications. New York: Joh n W iley & Sons.apisarda, L. , and DeCarlo, R.A. (1983). Ana log mulitifrequ ency faultdiagnosis. 1EE E Transactions on Circuits and Systems CAS-30(4),223-234.eisig, D., and DeCarlo, R.A . (1987). A me tho d o f analog-digitalmultiple fault diagnosis. IEEE Transactions on Circuit Theory andApplications 15, 1-22.ich ardson , M .H. (1969). O ptim izatio n o f large-scale discrete-timesystems by a comp on ent pro blem meth od. Ph.D. Thesis , Universi tyof Not r e Dam e.Richardson, M .H., Leake, R.J., an d Saeks, R.E. (1969). A com po ne ntcon nec tion formu latio n for large-scale discrete-time system opti -

mization. Proceedings of the Th ird Asilomar Conference on Circuitand Systems. 665-670.Saeks, R.E . (1969). Studies in system simulation. NASA TechnicalMemorandum 53868 . George Marshall Space Flight Center, Hunts-ville.Saeks, R.E. (1970). Fault isolat ion, com pon ent decoupling, an d theconnec t ion g r oupoid . NAS A-AS EE Sum mer Facul ty Fel lowship Pro-gram Research Reports. 505-534. Auburn University.

Saeks, R. E., Singh, S .P. , and Liu, R .W . (1972). Fault isolationvia components s imulat ion. IEEE Transactions on Circuit TheoryCT-19, 634-660.Singh, S.P. (1972). Structural prope rties o f large-scale dyn amic alsystems. Ph.D. Thesis, University o f No tre Dam e.Singh, S.E, and Trauboth, H. (1973). MARSYAS. 1EEE Circuits an dSystems Society Newsletter ZTraub oth, H., and Prasad, N.S. (1970). MA RSYA S---A oftware systemfor the digital simula tion of physical systems. Proceedings of theSpring Joint Computing Conference. 223-235.