Non Linear Input,Output Control Volterra Synthesis

8/14/2019 Non Linear Input,Output Control Volterra Synthesis

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11N o n l i n e a r In p u t / O u tp u t

C ontrol: V olterra SynthesisP a t r i c k M . S a i nRaytheon Company,E1 Segundo, California, USA

1 1 . 1 I n t r o d u c t i o n . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . 1131 1 1 .2 P r o b l e m D e f i n i t i o n U s i n g T o t a l Sy n t h e si s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311 1 . 3 P l a n t R e p r e s e n t a ti o n . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . 11321 1 .4 Co n t r o l l e r D e s i g n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 1 1 .5 S i m p l i f ie d Pa r t ia l L i n e a r i z a t i o n Co n t r o l l e r D e s i g n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11341 1 .6 SD O F Ba s e - I so l a t e d S t r u c t u r e E x a m p l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11351 1.7 C o n c l u s i o n . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . 1138

Re f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138

1 1 .1 I n t ro d u c t i o nA power se r ie s expans ion , p rov ided i t ex is t s , o f ten p rov ides au s e f u l r e p r e s e n t a t i o n o f a n o n l i n e a r p l a n t . T h e s a m e c a n b es a id f o r th e d e s i r e d c l o s e d - l o o p i n p u t / o u t p u t m a p o f a f e e d -b a c k c o n t r o l s y s t e m . G i v e n a V o l t e r r a s er ie s r e p r e s e n t a t i o n f o re a c h o f th e s e c o m p o n e n t s , t h e n V o l t e r r a f eedba ck s y nt hes i s( V FS) c a n b e u s e d t o d e s i g n a n d r e a li z e a n o n l i n e a r c o n t r o l l e rt h a t u s e s a f i n it e c o n s e c u t i v e n u m b e r o f V o l te r r a k e rn e l s o f t h e

l a n t a n d d e s i r e d c l o s e d - l o o p i n p u t / o u t p u t m a p . A s t r i k i n gf e a t u r e o f t h i s m e t h o d i s t h a t i t p e r m i t s t h e s p e c i f i c a t io n o f an o n l i n e a r d e s i r e d c l o s e d - l o o p b e h a v i o r . T h e c o n t r o l d e s i g na k e s p l ac e i n t h e f r e q u e n c y d o m a i n a n d i s re a l i z e d a s a ni n t e r c o n n e c t e d s e t o f l in e a r s y s t e m s . T h e c o n t r o l l e r p o s s es s e sn i n t e r e s t in g r e c u r s i v e s t r u c t u r e t h a t i s re a d i l y e x p l o i t e d t oi m p l i f i e d c o n t r o l l e r i m p l e m e n t a t i o n c a n b e c o m p u t e d t o a nr b i t r a r i l y h i g h o r d e r i n a n a u t o m a t i c m a n n e r .T h e V FS a p p r o a c h i s re a s o n a b l y g e n e r a l; t h e s p e c if i c d e v e l -p m e n t p r e s e n t e d h e r e in d r a w s u p o n t h e t o t a l s yn t he s is p r o b -

( T SP) f r a m e w o r k , a n d s o t h is c h a p t e r b e g i n s w i t h c o n c i s ee s c r i p t i o n s o f t h e T SP p a r a d i g m , V o l t e r r a p l a n t r e p r e s e n t a -

n d c o n t r o l l e r s y n th e s i s. T h e l e n g t h o f t h e g e n e r a l f o r -a t th i s p o i n t i s o n l y t o b e a d m i r e d ; t h e i n t r e p i d r e a d e r

d o u b t w i t h s o m e r e li e f, th a t t h e f o l l o w i n g s e c t i o ne s c r i b e s a n e q u i v a l e n t s i m p l i f i e d r e d u c e d - o r d e r i m p l e m e n t a -

o f t h e c o n t r o l l e r s r e p r e s e n t e d b y t h e s e f o r m u l a s , c o m p l e t ep l e a p p l i c a t i o n is p r o v i d e d u s i n g

yr ight ( c ) 2005 by Academic Press .s o f r e p r o d u c t i o n i n a n y f o r m r e se r ve d .

a b a s e d - i s o l a te d s i n g le d e g r e e o f f r e e d o m s t r u c t u r e w i t h n o n -l i n e a r h y s t e r e t ic d a m p i n g .

1 1.2 P ro b l em Def i n i t i o n Us i n gTotal Synthes isBased on R ugh ' s r e su l t s (1981) us ing Vol te r r a se r ie s to r epre se n tnon l ine a r sys tems , A1-Baiyat and S a in (1986 , 1989) app l ied theu s e o f V o l te r r a o p e r a t o r s t o n o n l i n e a r r e g u l a t o r d e s i g n i nt h e c o n t e x t o f t h e t o t a l s y n t h e s i s p r o b l e m ( T S P ) i n 1 9 86 .Since then , Sa in e t a l . (1990 , 1991) have used the TSP f r ame-w o r k t o a p p l y V o l t e r ra o p e r a t o r s i n n o n l i n e a r s e r v o m e c h a n i s mdes ign , and in 1995 , Doyle e t a l . ( 1 9 95 ) c a s t t h e m e t h o d i n t o am o d e l - p r e d i c t i v e c o n t r o l d e s i g n . A 1- Bai ya t s h o w e d t h a t t h eV o l t e r r a o p e r a t o r s c o m p r i s i n g t h e c o n t r o l l e r c a n b e r e a l i z e da s i n t e r c o n n e c t i o n s o f l i n e a r s ys t e m s , a n d Sa i n ( 1 9 9 7 ) d e r i v e da n e q u i v a l e n t r e d u c e d - o r d e r r e a l i z a t i o n . Fo r b r e v i t y , a p a r t i a ll i n e a r i z a t i o n r e g u l a t o r d e s i g n ( Sa i n e t a l . , 1997b) i s p re sen te dh e r e i n . T h e a p p r o a c h e x t e n d s t o s e r v o m e c h a n i s m s , a n d t h egene ra l con t ro l le r des ign i s g iven by Sa in (1997) .

L e t R , U , a n d Y d e n o t e r e a l v e c t o r s p a c e s o f d i m e n s i o n sp , m, and p , r e sp ec t ive ly r epre se n t ing the spaces o f r eques ts ,p l a n t i n p u t s , a n d p l a n t o u t p u t s . L e t P : U --~ Y d e n o t e a n i n p u to u t p u t d e s c r i p t i o n o f a n o n l i n e a r p l a n t . D e f i n e t h e d e s i r e dc l o s e d -l o o p re s p o n s e t o a c o m m a n d b y T : R ~ Y a n d th ed e s ir e d p l a n t i n p u t f o r a c o m m a n d b y M : R - ~ U . T h e o p e r -a t o r s P , T , M , a n d E a r e a s s u m e d t o h a v e V o l t e rr a se ri es

1131


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1132 P a t r i c k M . S a i n

I MI E

FIGUR E 11 .1 TSP Regulator ConfigurationR/ \/ \/ x

M / \ T/ x/ "x/ \/ \p -.,qu ) v

F I G U R E 1 1 .2 C o m m u t a t i v e D i a g r a m

e p r e se n t a t i ons a t t he po in t o r i n t he r e g ion o f ope r a t i on . A nl lus t ra t ion o f these ope ra tor s for a r egula tor des ign i s g iven inigu r e 11 . 1 I n ge ne ra l , a pa i r ( M , T ) i s de s i r ed suc h t ha tT = P o M a nd the d i a g r a m in F igu r e 11. 2 c om m ute s . I n t heeque l , T i s g iven , and the objec t ive i s to f ind and rea l ize a

con t ro l le r G: Y --~ U such tha t M = G o E , wh ere E: R ~ Y;in th is case , Y is the sp ace of outp ut e r ror s .

1 1 . 3 P l a n t R e p r e s e n t a t i o ni t h in t he T S P f r a m e w or k , c ons ide r l i ne a r a na ly t i c p l a n t s ,

t ha t i s , p l a n t s f o r w h ic h t he s t a t e a nd ou tpu t e qua t ions a r egiven by:k = f ( x ) + g ( x ) u , x ( 0 ) = X o . ( 1 1 . 1 )y = h ( x ) . (11.2)

In equa t io ns 11 .1 and 11 .2 , x E X ; u E U ; y E Y ; f , g : X - - + X ;g( x ) : U - + X ; h : X - + Y ; and f , g , and h a re ana ly t ic in x . Suchplants can be represented us ing a so-ca l led b i l inea r approxi -m a t i o n , o b t a i n e d b y d e n o t i n g t h e a p p r o x i m a t i o n o f th e s t a tea nd ou tpu t e q ua t ions ) ~( x , u ) a nd ] ~ ( x ), r e spe c ti ve ly a nd byt a k ing a m u l t i va ri a b l e T a y lo r s e ri es e xpa ns ion a bou t t he o pe r -a t i ng po in t ( x0, u0 ) , t r unc a t ing a ll te r m s o f o r de r h ighe r t ha nn, yielding:

//~ ( X 0 , U 0 ) = ~ _ . A l i ( x o , U 0)-~ [i]i= 1

n - - 1+ ~ Dl i ( xo , U 0 ) X [ i] @ h + Dlou .i= 1

(11.3)

//J 2 ( x o ) = ~ Cli (xo , U 0 ) X I i] . ( 1 1 . 4 )i= 1

I n t h e s e e q u a t i o n s , x = x - x 0 , h = u - U o , a n d x Ill =x - . . x , t h e / - fo l d K r o n e c k e r t e n s o r p r o d u c t o f x w i t hi t se l f . To reduce nota t iona l complexi ty , is a ssumed to havep r e c e de nc e ove r m a t r i x a nd s c a la r m u l t i p l i c a t ion . S upp r e s s ingthe e xp l i c it de pe nd e nc e on ( x0 , u0 ) a nd i f f l d e no te s U f l / O x j,then the fo l lowing result s:

n n--1flJ = Z A)ix[i] Z Dj ( i -1)x[ i] @ ~"i=) i=jl l A 1 2 A I Ix[21 A22 A2n ~[2]d t " z . , . .

J L 0 A , . . , J L ~ . [~ 1+

+

D l l

i 1D 10

D 1 2 ' ' " D l ( n - 1 ) ] V ~ :

0 . . . D n ( . 1 ) J

h.

(11.5)

h (11.6)

E qua t ions . 11 .5 a nd 11 .6 a pp ly f o r 2 ~ j_ < n , o r , m or ec om pa c t ly , x = A x + D x h + B h , y = C Yc ,~(0) = x0. In these quel , t he no t a t i o n A k d e n o t e s t h e m a t r i x p a r t i ti o n c o m p o s e do f t he l e f lm os t k pa r t i ti ons o f t he t op k r ow s o f A ; C k de no te sthe l e f tm os t k pa r t i t i ons o f C ; a nd D k de no te s t he l e f tm os tk - 1 pa r t i t ion s in the top k rows of D, wi th D1 = D10.F o r a m u l t i p l e - inpu t , m u l t i p l e - ou tpu t , f i n i t e - d im e ns iona l ,c a usa l , t im e - inva r i a n t p l a n t , de f ine t he hom oge nous m u l t i -l i ne a r V ohe r r a ope r a to r a s:

t " r i - - 1P i [ u ( t ) ] = [ . , . [ p i ( T 1 . . . . . Ti)u (t-- T1) Q " " Q u ( t -T i ) d T i . . , d% ,

0 0(11.7)

where t _> "F1 ~ " ' " ~ T i Z 0 a nd w he r e P i is i ts i th Volterrakerne l . A Vol te r ra r epresen ta t ion t hen has the form (A1-Ba iya tand Sain, 1986, 1989):

o oy ( t ) = Z P i [ u ( t ) ] . (11.8)i= 1

T he c on vo lu t i ona l na tu r e o f t h i s r e p r e se n t a t i on m a ke s i te xpe d ie n t t o w or k i n t he t r a ns f o r m dom a in . D e f ine t he


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1 1 N o n l i n e a r I n p u t ~ O u t p u t C o n t r o l : V o l t er r a S y n t h e s i s 1133m ul t i d i m en s i ona l Lap lace t r ans fo rm o f t he / t h Vo l te r r a ke rne las (Bussgang e t a l . , 1974):

P i (S l . . . . s i ) . . . . p i (' rl . . . . . T i ) e ( s'~ i+ " + s i~ i) d% . . d T i . ( 1 1 . 9 )o o

M-B ai ya t and S a in (1986 ) s howed t ha t fo r a s t a t i ona ry t im einvariant , bi l inear system of the form given above, whereP ~ = ( (Sl + . ' - + s j) I - A i ) - l D i , t he t r ans fo rm s o f t he f i r stthree kernels are as fol lows:P l ( s ) = c , p ll (11.10)

- -2 - -1P 2 ( S l , S 2 ) = C 2 P 2 { P 1 @ I r a } . (11.11)- -3 - -2 - -1P 3 ( S l , s 2 , s 3 ) = C 3 P 3 { [ P 2 { P 1 @ I m } ] @ Ir a} . (11.12)

The general form i s the fo l lowing:- - " - - " - - - - 2 -- 1P j ( s , . . . . . s j) = C j P } { [F } - ]{ [ - . . P 2 { P , @ I r a } " " ] @ I m } ] @ Ir a} .

( 1 1 . 1 3 )

11 .4 Co ntro l ler D es ignNow, the goal i s to formulate an express ion for the Vol ter rake rne l s o f t he con t ro l le r G i n t he f r equency dom ai n i n t e rm s o fthe kernels of the p lant , g iven above, and those o f the des i redclosed- loop map , T. The dev elopm ent fo l lows that g iven in A1-Baiyat Sain (1986). S imi lar ly to the Vol ter ra ope rator P i [ u ( t ) ] ,def ine Vol ter ra operators associated wi th the maps T and Msuch that :

y ( t ) = ~ Tj[ r(t )] . (11.14)j = lO 0

u ( t ) = Z M k [ r ( t ) ] . (11.15)k = l

o obtain a re la t ion between the operators P i, Tj, and M k ,eplace the reques t s ignal r ( t ) by c r ( t ) , where c i s an arb i t raryons tant ; because Vol ter ra operators are mul t i l inear , equat ingexpression s for y yields:

d T j [ r ( t ) ] = Z P i d M j [ r ( t ) ] . (11.16)j = l i =1 j = l

et ri d e n o t e r ( t - T i ) and def ine:_~ " . . . ~ - " ~ J '+ ' "+ J 'P I M [ r ( t ) ] ,i d M j [ r ( t ) ] = z .. ~ i ~ j , . . . ,

i=1 j l =1 j i= lM j ~ [ r ( t ) ] ) , (11.17)

where th e fo l low ing i s true:t ' r l T i 1

P , (M j , l r ( t ) l . . . . . M j , l r / t ) l ) = J I . . . J p , < . . . . . T i ) M j , (F I )O 0 0

@ " ' " @ M j ~ ( r i ) d ' r i . . . d ' r l. (11.18)Equa t i ng ou t pu t equa t i on exp res s i ons and s upp res s i ng t heargument [ r ( t ) ] y ie lds :

c ; T ; / l - . . + j o , , ,. . . . ~ '~ i t i v i j . . . . . . M j l) j = l i =1 j l = l j i=l (11.19)

Equa t ing powers o f c on b oth s ides of equat ion 11 .19 g ives:T1 = P1M1. (11 .20 )T 2 = P 1 M 2 + P 2 ( M b M ~ ) . (11.21)T3 = P 1M 3 + P2(M b M2) + P2(M2, M1) + P3(M1, M1, M1 ). (11.22)F o r K i = ~ = 1 k j, K 0 = 0 :

~ ( i - j l i j - k l 2 i -~ j 2 - 1j = 2 k l = l k2= ' kj 1= 1

\

P j (M k ~ , M k . . . . . M k j ~, M i - . j - 1 ) ~ ./(11.23)

App l y i ng t he m u l t i d i m ens i on a l Laplace t r ans fo rm y i e ld s :r l ( s ) = P I ( S ) M I ( S ) . (11.24)

r2 (s1 , s2 ) = PI (S 1 q - $2 )M2 (s1 , s2 )+ P2(Sl, s 2 ) [ M l ( S l ) M(s2)]. (11.25)

T3(s> s2, s3) = Pl(Sl + s2 + s3)M3(s l , s2 , s3) (11.26)+ P 2 ( s l , s 2 + s 3 ) [ M I ( S l ) @ M 2 ( s 2 , s 3 )]+ P 2 (s l + 8 2 , s 3 ) [ M 2 ( s l , s 2 ) @ M I ( S 3 ) ]+ P3(s~, s2, s3)[M~(sl) M~(s2) M ~(s3)]. (11.27)

T i ( s t . . . . . s i) = P I ( S l + ' " + s i )M i ( S l . . . . . s i)~ 2 ~ / i j + l i j k l + 2 i - K) 2 - 1

+ Z Zj=2 k l= l k2=l k3_1=1P j ( S l + " " + s K ~, sK~+l + "' " + s~2 . . . . s~i_~+l + " '" + si )[ M k ~ ( S l . . . . s ~ , ) M < ( s ~ + l . . . . . sK 2) " "

~ , , (s K i 1 4 1 . . . . s i ) ] ~ .i (11.28)JObse rve th at i f the p air (M1 (s), T1 (s)) is chosen for P1 (s), the none can pro ceed to the des ign equ at ion involv ing P2(S l , s2),where the pai r (M2(Sl , s2) and T2(Sl , s2)) i s chosen and so on .To comp lete the des ign , cons ider the synthes i s of the cont ro l ler


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1134 P a t r i c k M . S a i nG tha t r e a li z es the de s i r e d m a pp ings ( M , T ) show n in F igu r e11 .1 N o te t ha t t he s im i l a ri t y o f t he r e l a ti on M = G o E pe r -i t s e xp r e s s ions f o r t he ope r a to r s M j t o be ob t a ine d i n aa nne r s im i l a r t o t he one f o r t he ope r a to r s T / . A l so f r omFigure 11 .1, no te tha t e = r - y , wh ich can be wr i t te n as:

E i [ r ( t ) ] = I [ r ( t ) ] - Z ~ [ r ( t ) ] ,i -1 i=1 (11.29)h ere I is the id en ti ty op erat or , an d, th us, E1 (s) = I - T~ (s) .

In addi t ion , for i > 1 , E l (S , . . . . . s i) = - T i ( s ~ . . . . . s i ) . A d i r e c tr e l a t ion be tw e e n the V o l t e rr a ke rne l s o f t he ope r a to r s T , M ,a nd G c a n be f oun d i f t he de s ign i s w e l l - pose d on t he f i r s tord e r or l inea r leve l, implyin g tha t El (S) has an inverse. As T i su sua l ly ope n to c ho i c e , the l a t t e r r e q u i r e m e n t i s r e a sona b le .

11.5 Sim pli f ied Partial Linearizat ionCon troll er Des ign

C ons ide r de s ign ing a f e e dba c k sy s t e m f o r a g ive n non l ine a rsys t e m so t ha t T i = O , 1 < i < n , w h e r e T i is the / th Volterrakerne l of the c losed loop. Such a des ign i s ca l led pa r t i a ll i ne a r i z a t i o n , a nd i t ha s t he de s i r ab l e a t t r ibu t e o f g r e at l yl e s se n ing t he c om ple x i ty o f t he c on t r o l l e r de s ign e qua t ions .B o th t he ge ne r a l c on t r o l l e r de s ign a n d the pa r t i a l l i ne a r iz a t i oncontro l le r des ign , a s de r ived us ing the equa t ions above , y ie ldea l iza t ions wi th repe a ted s t ruc tures . Co l lec t ing te rms wi th l ike

coef f ic ien ts y ie lds the s im pl i f ied con t ro l le r des ign . The resu l t i sp r e se n t e d he r e f o r t he pa r t i a l l i ne a r iz a t i on c on t r o l le r , w i th t hegenera l r e su l t g iven by Sa in (1997) .

S uppose f o r T t ha t t he s e c ond a n d h ighe r o r de r ke r nel s a rez e r o . T he n , i f P l ( s ) inver ts, the following results:

M a ( s ) = P ; ~ ( s ) T ~ ( s ) . (11.30)M2(s,, s2) = - P [ a ( s ~ + s 2 ) P 2 ( s ~ , s 2 ) [ M ~ ( s ~ ) Ml(S2)]. (11.31 )

M 3 ( $ 1 , s 2 , s 3 ) = - - P l l ( 5 1 - I- $ 2 + 5 3 ) { P 2 ( $ 1 , $ 2 -} - $ 3 ) [ M 1 ( s 1 ) M2 ($2 , 53 ) ] -} -P2(S l q -$2 , 53 ) [M2($1 , $2 )@ M1($3) ]+ P3(s~, s2, s 3 ) [ M ~ ( S l ) M~(s2) Ma(s3)]}. (11.3 2)

In gen era l , the fo l lowing i s t rue :M / ( sl . . . . . s i) = - P [ I ( s l + . . . + s i)~ L ~ { i j + l i- j-kl+2 i-~j 2 -1Z Z Zj = 2 k l = l k 2 = l kj l = l

P j ( s l + . . . + s n ~ , s ,q+l + . . . + &2 . . . . s~i_~+l + . . . + Si)[ m k ~ ( s ~ . . . . . s ~ , ) V k 2 (< + ~ . . . . . < ) . . . m ~ ~ ~(s~ ,+~ . . .. ~ , )] ~ . (11,33)

A n i c e f e a tu r e o f t he se e qua t ions i s t he i r r e cu r s ive na tu r e . O n c eM l ( s ) ha s be e n f ound , i t c a n be u se d t o ob t a in a n e xp r e s s ionfo r M 2 ( s > s2) and so for th . Under pa r t ia l l inea r iza t ion , theope r a to r s a r e r e p r e se n t e d by E k = 0, k > 1. As sum ing E l ( s )a n d P l ( s ) t o be i nve rt i b le , t he n t he ke r ne ls o f t he c on t r o l l e rG a re g iven by:

G1 (s ) = p ; l ( s ) T 1 ( s )E 1 1 s ) . (1 1 .3 4 )Ge (sa , s2) = - P [ ' ( s a + s 2 ) P 2 ( s > s i ) [ G ~ ( s , ) Ga(s2)]. (11.35)

G 3 ( s 1 , 5 2 , 5 3 ) = - P l l ( 5 1 - [- $ 2 ~ - $ 3 ) { P 2 ( s 1 , $ 2 -} - 5 3 ) [ G 1 ( s 1 ) G2($2, S3)] q- P2 (sl q- S2, S3)[G2(51, $2) Gl(S3)]q- P3($1 , s2 , $ 3 ) [ G I ( S l ) @ G 1 ($ 2 ) G 1 ( $3 ) ] } . ( 1 1 . 3 6 )

In general:G i( Sl . . . . . s i) = - P l l ( S l + . . . + si)~ { i j+l i-j k1+2 i / ~ j _ 2 - - 1

Z Z Zj = 2 k l = l k 2 = l k j - l= lP j ( S l + . . . + s , ~ , sn t+l + . . . + & . . . . . S / g j - l + l - ~- " ' ' - ~- S i )[ G k , ( S l . . . . . S~; @ Gk2(Stvl+l . . . . S~ )Q . . . ~ G i , ~ j - ,( s ,% ~ - I . . . . . s i) ] ~ . (11 .37)J

A ga in, no t e t he r e c u rs ive na tu r e o f t he c on t r o l l e r de s ign e qua -t i ons . O nc e G l ( s ) ha s be e n c om pu te d , i t c a n be u se d t o f i ndG2(sl , s2) and so on.

The next s tep in the rea l iza t ion process i s to subs t i tu te theexpress ion for the p lant ke rne ls P i i n t he f r e que nc y dom a in .Giv en G~ = G i ( s j , sj + l . . . . . s j + i - 1 ) , t he n :G] = -(ClP~)-1C 2~{ [P~G~] G~}. (11 .38 ) ] = _ ( c , ~ e ) - l { c a ~ { [ 8 : ] 2} + < p ~ { [ p~ ]] < ~ }

+ C3P~{ [ /)22{ P~ G~] ~}] 0~ }} . (11.39)~ _ _ ~ { i -j + l i j - ~ l + 2 i j - ~ 2 + 3 i ~ J 2 1Z Z Z ... Zj=2 k l= l k2=l k3=l k i - l = l

- ~ ; l - 1 G ~ l + l \ ]

t~,_2+1 - t~) 1+1~. (11 .40)%J J

Each of equa t ion s 11 .38 throu gh 11 .40 can be rea l ized as a se tof in te rcon nec ted l inea r subsys tems. Th ese rea l iza t ions possessa nu m b e r o f i n t e r e s ti ng f e a tu r e s tha t a r e n i c e ly i l lu s t r a te d u s ingb loc k d i a g r a m s . I n do ing so , c e r t a in no t a t i ona l c onve n ie nc e sa r e e m p loye d .

F i rs t , t he supe r sc r ip t s on t he qua n t i t i e s P /; a nd G / i n e qua -t i ons 11 .38 t h r ough 11 .40 ar e r e dund a n t a nd c a n be ne g l e ct e dwi tho ut loss (Sa in , 1997) . The pr oo f l ies in ob se rv ing tha t theo r de r ing o f t e r m s in a c on t r o l l e r de s ign e qua t ion , t oge the r


5/8

1 1 N o n l i n e a r I n p u t ~ O u t p u t C o n t r o l: V o l t e r ra S y n t h e s i s 1135i th the es tab l i shed use of pa rentheses an d bracke ts , a re suf fi -c i e n t t o de t e r m ine w h a t t he supe r sc r ip t s shou ld be .S e c ond , i f e E Y r e p r e se n ts t he ou tpu t e r r o r o f the c lo sedoop, and 811 is i t s Laplace t r ansform , th en le t 8~ i] den ote th e

/ - fo ld tensor p rod uc t o f 811 wi th i tse lf , then de f ine :

T h e m e m b e r s o f th e c o m m o n s et F 0, ( n a m e l y c~/) and [3/)),a r e c om pu te d a nd s to r e d t he f i r s t t im e the y a r e e nc oun te r e dand then reca l led as needed. In the seque l , the equa t iong /) = - p [ I c j P j [ 3 i j a nd the f o l l ow ing ide n t i f i c a t i ons a r e c on -venient :

ik = l (11 .41)

g [ 3i l t he L a p la c e t r a ns f o r m o f t he / t h o r de r c on t r o l l e ru t p u t .F ina lly , the layo ut of the b lock d iagram s fo l lows ce r ta inonve n t ions . I f t he c on t r o l l e r de s ign e qua t ions a r e f u ll ye xpa nde d a nd e a c h w r i t t e n on one l i ne , t he n t he i r a dde nds ,s r e a d f r om l e f t t o r i gh t , a ppe a r i n t he b loc k d i a g r a m s p r o -e e di n g fr o m t o p t o b o t t o m .T he f und a m e n ta l obse r va t i on be h in d t he s im p l i f i c a ti on p r o -ess is tha t quan t i t ie s presen t in the rea l iza t ion for G) reapp ear

in the rea l iza t ions for G1 , j > i. T he m o t iva t i on be h ind t hes im p l i f ic a t i on p r oc e s s i s t ha t i n c om pu t in g t he ou tpu t o f ag i ve n c o n t r o l le r c o m p o n e n t G ) , t he m os t e f f i c i e n t a lgo r i t hmis t o c o m p u te t he se r e pe a t e d qua n t i t i e s on ly onc e , s t o re t he min m e m or y , a nd t he n r e use t he m a s ne e de d . T he r e duc t ionp r oc e s s i s t hus a p r ob l e m o f i de n t i f y ing a s e t F o f e l e m e n t sc o m m o n t o a s e q u en c e o f c o n t r o ll e r c o m p o n e n t s G ] , i = 1,2 , 3 . . . . n , a nd de t e r m in ing w h ic h one s t o s a ve f o r l a t e r r e u se .I n c o m p u t i n g t h e n c o m p o n e n t s & l r ~ ( i ) ~ 1 < i < n , o f t h eou tpu t o f a n n th o r de r c on t r o l l e r , m a ny qua n t i t i e s a r e c om -pu te d r e pe at e d ly. T he r e f o re , a sy s t e m o f no t a t i o n i s now in t r o -d u c e d w i t h t h e p r i m a r y g o a l o f e l i m i n a t i n g r e d u n d a n tc a l c u la t i ons by i de n t i fy ing m e m be r s o f t he c o m m o n se t F a n dhe s e c on da r y goa l o f f u r the r s im p l i f y ing t he c on t r o l l e r ou tp u ta l c u l a t i on . T he i de n t i f i c a t i on o f F p r e se n t e d he r e i s no tn ique . T he r e f o r e , t he no t a t i on F 0 w i l l be u se d t o de no te t he

on de sc r ibe d he r e in a nd t o d i s t ingu i sh i tr o m t h e g e n e r al n o t i o n o f th e c o m m o n s et r e p re s e n te d b y F .T he f i rs t m e m b e r s o f F 0 have a l r e a dy be e n i n t r oduc e d ,

G ive n o~i j E F o and 2 < j < i de f ined by oLij =9j-113(i 1 ) ( j - l ) , then:

g i = Z - p [ 1 c j P j [ 3 / ) (11 .44)j = 2i= Z g / ) . (11 .45)

j - - 2T he a bove a lgo r i t hm f o r c a lc u l a ti ng the o u tp u t o f t he s im p l i -f ied pa r t ia l l inea r iza t ion cont ro l le r takes fu l l advantage of ther e c u rs ive na tu r e o f t he de s ign e qua t ions , r e duc ing t he num be ro f s t at e s a nd f l oa t i ng - po in t c om pu ta t i on s . F u r the r m or e , t hes im p l i fi e d f o r m i s a m e na b le t o t he c on s t r uc t i on o f ge ne r a lso f tw a r e r ou t ine s , c apa b le o f c om pu t ing t he c on t r o l l e r ou tpu tf o r a n a r b i t r a r y va lue o f t he c on t r o l l e r o r de r n .T he s im p l i f i e d b loc k d i a g r a m s f o r t he s e c ond th i r d - o r de rc on t r o l l e r c om pon e n t s a r e show n in F igu r e s 11.3 a nd 11 .4 .A ge ne r a l f o r m f o r t he i t h o r de r c om pone n t i s show n inFigure 11.5.

11 .6 SDOF Base -I so la ted Struc tureE x a m p l eC ons ide r a s ing l e de g r e e o f f r e e dom ( S D O F ) s t r uc tu r e s i t t ingon a ba se i so l a t i on sy s t e m c ons i s ti ng o f hys t er e t ic a l ly da m pe dbear ings , and le t the s t ruc ture be subjec t to ground acce le r -a t i on 2 g ( t ) a nd a c on t r o l f o r c e f ( t ) supp l i e d by a hyd r a u l i cac tua tor . Such a sys tem is shown in F igure 11 .6 The forcesexer ted by the indiv idua l bea r ings a re mode led col lec t ive ly .T h e e q u a t i o n o f m o t i o n f o r t h e h o r i z o n t al d i s p l ac e m e n t o fthe m ass m o f the s t ruc tu re i s a s fo llows:

f l l 1

i - j + l[3 i j = Z OL( i -k+ l )J @ [3k l"k = l (11 .42)

s im p l i fi e d pa r ti a l l ine a r i z a t ion n th o r d e r c on t r o l l e r ou tpu tw be c a l c u l a t e d as f o ll ow s . F ir st , c o m p u te t he ou tpu t o f

f i r s t orde r co nt ro l le r , [3 n = p[ l T 1 ( 1 - T1)-1811, and thenu tp u t o f t he n th o r de r c on t r o l l e r f o r n _> 2 i s c om pu t e d

i

i = 2 j = 2(11 .43)

FIGURE 11.3 S impl i f iedG2 Realization[5210~22 ) ~1 3 . ~ I

Pll /

FIGURE 11.4 S impl i f iedG3 Realization


6/8

1136 Patrick M . Sain1 3 ( n - ) I

13(n -2 )1? 3 2i 3 3 1O ~ ( n 2 ) 2 ~1321O ~ ( n 1 )2J 3 1 11 3 ( n 1)1 -P1-1C 2 / 5 2 g n 2O~ii +13(i) 1 ) ( ~?(i+1)i4~ ( i - 1 ) 1

( i + ) i~ 2 1( Z (n 1 ) / + ) ( ~~11 y~ ( n - ) ( i - 1 ) ~ _ P l _ l C i ~ i l g n i X[321O ~ (n 1 ) ( n - ~ )1 3 1 1 |1 3 ( n - 1 ) ( n - 2 ~ _ m - l t " , ~ I . , 21 3 1 1 q ' n ~ ' l o c ~ n 1 ~ " % C J 1 3 n ( 1 ) 1 1 " - ' n 1 n - l lg ( n _ l ) ( n _ l ) '%1 3 (n 1 ) ( n - 1 ~ _ P l . l O n ~ n g n n ) ~

FIGURE 11 .5 Genera l Form for Simplif ied Realiza tion

( f , i xp >

1 1 . 6 A Base-Isolated SDO F Structure o f Mass rnl. The structure is supported by hysteretically damped bearings bl and b 2 that areroun d acceleration ~g and control force f.

f i t ) = rn[~p( t) + ~g ( t) ] + cic;(t) + kxp(t) + Q [ x ( t ) , k ( t ) ] .(11 .46)

o f t h e v a r ia b l e s a n d p a r a m e t e r v a l u e s a r e g iv e n i nBo t h t h e b e a r i n g s a n d t h e a c t u a t o r a f f e ct t h e s t i ff -

p r o v i d e d i n T a b le 1 1 .1 a r e m e a n t t o r e p r e s e n t t h e b e h a v i o r o ft h e s y s t e m w i t h t h e a c t u a t o r a n d b e a r i n g s i n p l a c e . T h e h y s -t e r e t ic r e s t o r i n g f o r c e Q g e n e r a t e d b y t h e b a s e i s o la t i n g b e a r -ings is given by:

Q[xp( t ) , kp( t ) ] - - c ~ x p ( t ) + (1 - a)Fyz(t) , (11 .47)


7/8

1 1 N o n l i n e a r I n p u t ~ O u t p u t C o n tr o l : V o l te r r a S y n t h e s i s 1137TABLE 11 .1 Parameters for the SDOF SystemVa riab le Description Valuec Dam ping coefficient 6001T kg /sForce applied by actuator (N)k Stiffness coefficien t 3 X 1 0 6 Tr 2 kg/s2m Mass of structure 3000 kgQ Hysteretic restoring force of base-isolation Nbearingxg Horizontal ground acceleration m /s 2xp Horizontal displacement of the ma ss rn mNatural undamped frequency 5 Hz{p Dam ping ratio 1%

TABLE 11 .2 Parameters for the Hysteretically Da mp ed Base-IsolationSystemVa riab le Description ValueAtlxpYz

Y

Hysteresis oop-sha ping param eter 1Force of equivalent hysteretic dam per 40 kParameter controll ing the smoothness of the 2transition from elastic to plastic responseHorizon tal displacem ent mYield displacement of equivalent hysteretic damper 1 mmHysteretic displaceme ntPost yielding to preyielding stiffness ratio 0.2Hysteresis oop-sha ping parame ter -0.25Hysteresis oop-sha ping param eter 0.75

w h e r e z ( t ) i s a d i m e n s i o n l e s s q u a n t i t y r e p r e s e n t i n g h y s t e r e t icd i s p l a c e m e n t ( F a n e t a l . , 1988 ; S a i n e t a l . , 1 9 9 7 ) s a t i s f y i n g t h ef i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n :

Y ~ ( t ) = y l k p ( t ) l l z ( t ) l n - l z ( t) - ~ 2 p ( t ) l z ( t )l ' + A k p ( t ) . ( 1 1 . 4 8 )T a b l e 1 1 .2 l is t s d e s c r i p t i o n s o f t h e v a r i a b l e s a n d p a r a m e t e r

v a l u es . T h e a c t u a t o r f o r c e f a p p l i e d t o t h e S D O F s t r u c t u r e i sg o v e r n e d b y t he e q u a t i o n ( D y k e e t a l . , 1 9 9 5 ; D e S i l v a , 1 9 8 9 ) :

j ' ( t ) 2 B A f k q ~ l f 2 B k ~ , 2 B A }- - V [ u ( t ) - X p ( t )] - ~ - f ( t ) - V k p ( t ) .( 1 1 . 4 9 )

T h e u ( t ) i s t h e c o n t r o l s i g n a l g e n e r a t e d b y t h e c o n t r o l l e r .D e s c r i p t i o n s o f t h e q u a n t i t i e s i n e q u a t i o n 1 1. 49 a r e p r o v i d e di n T a b l e 1 1 . 3 .

D e f i n i n g A z = A f 3z = ~ , a n d % = ~ a n d u s i n g t h e f ac t t h a t] z ] = z sgn z , t hen :

~ ( t ) = [ A z - ( f 3 z + Y z s g n k p s g n z ) z ' ( t ) l k p ( t ) ( 1 1 . 5 0 )= [A ~ + a ( t ) z ~ ( t ) ] 2 p ( t ) , ( 1 1 . 5 1 )

h e r e o ' ( t ) = - [ [ 3 z + % s g n 2 p ( t ) s g n z ( t ) ] . S u b s t i t u t i o n o fh e e x p r e s s i o n f o r Q i n to t h e e q u a t i o n o f m o t i o n o f t h e

TABLE 11.3 Parameters for the ActuatorV ar ia ble Description ValueA f C r o s s - s e c t i o n a lrea of actuator m 2B Bulk modulus of hydraulic f luid N /m 2f Force applied to SDOF structure Nby actuatorg Proportional feedback gain 2.5k Flow-pressu re coefficient m 4 . s /kgkq Flow gain m 2/sxp Horizon tal displaceme nt mV Characteristic hydrau lic fluid volum e m 32 B A f k q G / V Disp lacem ent oefficient 3.6484 x 10 N /m /s2 B k c / V Force feedback coefficient 66.67 1/s2 B A 2 / V V e l o c i t yeedback coefficient 2.1891 x 107N /m

s t r u c t u re y i e ld s t h e f o l l o w i n g sy s t e m o f e q u a ti o n s . F o r c o n -v e n i e n c e, t h e e x p l i c i t d e p e n d e n c e o n t i m e i s s u p p r e s s e d :

k - - + l f ( t ) - ( 1 - o ~ ) F Y z - 2 g . ( l l . 5 2 )+ r ' x o - c 'Y m J ~ m x p m2 B A f k @ l f 2 B A } . 2 B k c 2 B A f k q ' l f ( 1 1 . 5 3 )/ = - V x p - ~ - X p - ~ - f ( t ) q ~ u .

= ( A z + a z n ) 2 p . ( 1 1 . 5 4 )G i v e n z w i t h n = 2 , w i t h z 0 = z ( t ) and Xp0 = x p ( t ) ( w h e r e t r e p r e s e n t s t h e m o s t r e c e n t p o i n t i n t i m e w h e r e o r ( t ) s w i t c h e dv a l u e y i e ld s ( f o r o - > 0 a n d o - < 0 ) c l o s e d - f o r m e x p r e s s i o n s f o rz a r e r e a d i l y o b t a i n e d ( S a i n , 1 9 9 7 ).

A n e x a m p l e a p p l i c a t i o n o f a t h i r d - o r d e r V F S p a r t i a l li n e a r -i z a t io n r e g u l a t o r d e s i g n f o r t h e S D O F s t r u c t u r e d e s c r i b e dp r e v i o u s l y is s i m u l a t e d i n e q u a t i o n s 1 1 .5 5 t o 1 1 . 5 7, u s i n g t h es i m p l i f i e d c o n t r o l l e r d e s i g n d e r i v e d a b o v e . C h o o s i n g t h e s t a t ev e c t o r [ x 1 x 2 x 3 ] I = [ X p X p f ] ' , w h e r e z c a n b e e x p r e s s e d a sp r e v i o u s l y n o t e d , d e p e n d i n g o n t h e s i g n o f , t h e r e s u l ti n gs t a te - s p a c e d e s c r i p t i o n h as t h e f o r m y = X l , a n d t h e f o l l o w i n gr e s u l t s :

21 = x2. (11 .55 )

( k . F r _~ c 1 F y ( 11 .56 )2 2 = m q - o t X l - - - x 2 - - - - x 3 - ( 1 - o r ) z .I N / / m m mB A I vk 2 B A } 2 B k c 2 B A f k q y f2 q ' Y f x l V x2 - 7 /3 - - - ~ x 3 + u. ( 1 1 .5 7)

T h i s e x a m p l e r e p r e s e n t s a p r e l i m i n a r y te s t o f t h e c o n t r o ls y s t e m t o d e m o n s t r a t e t h a t t h e a c t u a t o r c a n b e c o m m a n d e dt o m o v e t h e f i r s t f l o o r o f t h e s t r u c t u r e t o a s p e c i f i c l o c a t i o n .U n d e r t h e h y p o t h e s i s t h a t t h e V F S c o n t r o l d e si g n s w o r kt o m a k e s m a l l e r r o r s s m a l l e r , a c o m m a n d r e q u e s t r = x~ =0 . 0 0 4 m w a s s p e c i f i e d , a n d a c o n s t a n t o f f s e t o f )I0 = 0 . 0 0 1 m ,r e p r e s e n t i n g a b i a s e d p o s i t i o n s e n s o r , w a s a d d e d t o t h e m e a s -u r e d o u t p u t o f t h e p l a n t . R e g u l a t i o n a b o u t a n o n z e r o s e t p o i n t


8/8

1 1 3 8 P a t r i ck M . S a i n

0

3 .2-. 0

3 . 4 ~ i l3 .3 . . . .. . . .. . . .. . ~ . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . i . . . . . . . . . . . .. . . i . . . . . . . . . . . . . . . i . . . . . . . . .. . . . . i . . . . . . . .. . . . . . . i . . . . . . . . . . . . . . . i . . . . . . . . . . . . . .~ i f i i i i i i

/ ~ i i i i i i

. . . . . . . . . . . . . ! ~ " ~ . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . : : . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . ! . . . . . . . . . . . . . . i . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . i . . . . . . . . . . . . . .i I ~ .

. i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . ' ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 .02 0 .04 0 .06 0 .08 0 .1 0 .12 0 .14 0 .16 0 .18 0 .2T ime [sec]

U R E 11 .7 Pos i t i on o f F i r s t F loo r Fo l l ow ing In i t i a l T rans i en t f o r

b e c a u s e t h e h i g h e r o r d e r T a y l o r se r ie s c o e ff i c ie n t st h e c o n t r o l m o d e l a r e z e ro i f ~1 = 0 . P r i m a r i l y f o r c o m p u t a -

= - 1 0 0 0 i n t h i s e x a m p l e .A r e s u l t is s h o w n i n F i g u r e 1 1 .7 T h e i n i t i a l t r a n s i e n t r e -

p o n s e w a s o m i t t e d s o a s n o t t o o b s c u r e t h e e f f e ct o f t h e h i g h e rr d e r c o n t r o l le r . T h e r e s p o n s e o f t h e c l o s e d - l o o p s y s t e m ,e a n i n g t h e p o s i t i o n o f t h e f ir s t f l o o r, x l , i s s h o w n f o r th e

a g n i t u d e o f t h e o s c i ll a t io n s i n t h e r e s p o n s e a so u t p u t t e n d s t o w a r d t h e c o m m a n d e d p o s i t i o n a re m u c h

e r fo r th e t h i r d - o r d e r d e s ig n w h e n c o m p a r e d t o t h o sed u c e d b y t h e li n e a r c o n t r o l l e r d e s ig n .

1 1 . 7 C o n c l u s i o ns s h o w n t o p r o v i d e a c a p a b i l i ty f o r

f y in g n o n l i n e a r i n p u t o u t p u t b e h a v i o r f o r a c lo s e d - l o o pc p l a n t s t h a t a d m i t V o l t e r r a s e ri e s re p r e s e n t a t i o n ,a s i m p l i f ie d m e a n s o f s y s t e m a t i c a ll y r e al i z in g t h e V o l t e rr a

p u t i n g t h e o u t p u t o f t h e re s u l t in g c o n t r o l l e r is

r e c u r s iv e i n n a t u r e a n d l e n d s i ts e l f t o a n a u t o m a t e d d e s i g na n d s i m u l a t i o n p r o c e d u r e . T h e p r o c e d u r e i s a p p l i e d t o a r e g u -l a t o r d e s i g n f o r a b a s e - i s o l a t e d S D O F s t r u c t u r e w i t h h y s t e r e t icd a m p i n g .

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Non Linear Input,Output Control Volterra Synthesis

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