frequency domain system identification

8/14/2019 Frequency Domain System Identification

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6Frequency D om ain

System IdentificationG a n g J i nord Motor Company,Dearborn, Michigan, USA

6 .1 I n t r o d u c t i o n . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . 10696 .2 F r e q u e n c y D o m a i n C u r v e -F i t ti n g . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . 1069

6 .2 .1 M a t r i x F r a c t i o n P a r a m e t e r i z a t i o n 6 . 2 .2 P o l y n o m i a l M a t r i x P a r a m e t e r i z a t i o n 6 . 2. 3 L e a s t S q u a r e s O p t i m i z a t i o n A l g o r i t h m s6 .3 S t a t e -Sp a c e Sy s t e m Re a l i z a t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10746 .3 .1 M a r k o v P a r a m e t e r s G e n e r a t i o n 6 . 3 . 2 T h e E R A M e t h o d

6 .4 A p p l i c a t i o n S t u d i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10756 . 4 .1 I d e n t i f i c a t i o n o f a 1 6 - S t o r y S t r u c t u r e 6 . 4 .2 I d e n t i f i c a t i o n o f t h e S e i s m i c - A c t i v e M a s sD r i v e r B e n c h m a r k S t r u c t u r e6 .5 C o n c l u s i o n . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . 1078Re f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

6 .1 Introduct ionA g e n e r a l p r o c e d u r e f o r t h e f r e q u e n c y d o m a i n i d e n t i f i c a t io n o fm u l t i p l e i n p u t s / m u l t i p l e o u t p u t s ( M I M O ) l i n e a r t i m e i n v a r i -ant systems is i l lustra ted in Figure 6.1. Typically, one star tsw i t h t h e e x p e r i m e n t a l fr e q u e n c y r e s p o n s e f u n c t i o n ( FRF) o ft h e t e s t s y s t e m . T h e s e FRF d a t a m a y e i t h e r b e c o m p u t e d f r o mt h e s a v e d i n p u t / o u t p u t m e a s u r e m e n t d a t a o r m e a s u r e d d i r -e c t l y o n l i n e b y a s p e c t r u m a n a ly z e r . Ba s e d o n t h e s e d a t a , t h em a t r i x f r a c t i o n ( M F ) o r t h e p o l y n o m i a l m a t r i x ( P M ) c u r v e -f i t t i n g t e c h n i q u e i s a p p l i e d t o f i n d a t r a n s f e r f u n c t i o n m a t r i x(TFM ) tha t c lose ly f it s in to the FR F da ta . De ta i led a lgor i thm so r t h e c u r v e - f i t ti n g a re i n t r o d u c e d i n Se c t i o n 6 .2 . F r e q u e n t l y ,o r t h e p u r p o s e s o f s i m u l a t i o n a n d c o n t r o l , o n e n e e d s a s t at e -p a c e r e a l i z a t i o n o f t h e s y s t e m . T h i s m a y b e a c h i e v e d b yh e e i g e n s y s t e m r e a l i z a t i o n a l g o r i t h m ( E RA ) i s p r e s e n t e d i n

Sec t ion 6 .3 fo r th i s purpose , thanks to i t s many successes inr e v i o u s a p p l i c a t i o n s t u d i e s . T h e M a r k o v p a r a m e t e r s , b a s e dn t h e p a r a m e t e r s f r o m w h i c h t h e s t a t e - s p a c e m o d e l w i l l b ee r i v e d , c a n b e e a s il y g e n e r a t e d f r o m t h e i d e n t i f i e d t r a n s f e ru n c t i o n m a t r i x . F in a ll y, a s a m e a s u r e o f p e r f o r m a n c e , t h eo d e l F R F is c o m p u t e d a n d i s c o m p a r e d t o t h e e x p e r i m e n t a l

FRE T h i s i s il l u s tr a t e d i n Se c t i o n 6 .4 b y m e a n s o f tw o e x p e r i -e n t a l a p p l i c a t i o n e x a m p l e s .

opyright 2005 by AcademicPress.form reserved.

6 .2 Fr e que nc y D o m a in C ur v e - F i tt ingF r e q u e n c y d o m a i n c u r v e - f i t t i n g i s a t e c h n i q u e t o f i t a T F Mc l o s e ly i n t o t h e o b s e r v e d FRF d a t a . L i k e o t h e r s y s t e m i d e n t i f i-c a t i o n t e c h n i q u e s , t h i s i s a t w o - s t e p p r o c e d u r e : m o d e l s t r u c -t u r e s e l e c t i o n a n d m o d e l p a r a m e t e r o p t i m i z a t i o n . I n t h i sc o n t e x t , t h e f i r s t s t e p i s t o p a r a m e t e r i z e t h e T FM i n s o m es p e c ia l f o r m s . T w o s u c h f o r m s a r e i n t r o d u c e d i n t h e f o ll o w i n g:t h e m a t r i x f r a c t io n ( M F ) p a r a m e t e r i z a ti o n a n d t h e p o l y n o m i a lma t r ix (PM) pa rame te r iza t ion . Th is i s a lways a c r i t ica l s tep inthe ide n t i f ica t io n because i t w i ll gene ra l ly lead to qu i te d i f f e r -e n t p a r a m e t e r o p t i m i z a t i o n a l g o r i t h m s a n d r e s u l t i n g m o d e lp r o p e r t i e s . I n p a r t i c u l a r , t h i s s e c t i o n s h o w s t h a t f o r t h e M Ff o r m , t h e p a r a m e t e r s c a n b e o p t i m i z e d b y m e a n s o f li n e a r le a sts q u a r e s ( L L S) s o l u t i o n s . A s f o r t h e PM p a r a m e t e r i z a t i o n ,one has to r e sor t to some nonl inea r techn iques ; spec i f ica l ly ,t h i s s e c t i o n i n t r o d u c e s t h e c e l e b r a t e d G a u s s - N e w t o n ( G N )m e t h o d . O n t h e o t h e r h a n d , t h e P M p a r a m e t e r i z a t i o n o f f e r sm ore f lex ib i l i ty in the sense tha t i t a l lows the des ign e r tospec i fy ce r ta in p ro pe r t ie s o f the iden t i f ied m od e l ( e .g. , f ixedz e r o s i n a n y i n p u t / o u t p u t c h a n n e l s ) . T h i s f e a t u r e m a y b e q u i t ed e s i r a b le a s s h o w n b y t h e a p p l i c a t i o n s t u d ie s i n Se c t i o n 6 .4 .

Be f o r e s t a r t in g t h e d i s c u s s io n , i t is i m p o r t a n t t o m a k e c l e a rt h e n o t a t i o n s t h a t w i l l b e u s e d t h r o u g h o u t t h i s s e c t i o n .A s s u m e t h e t e s t s y s t e m h a s r i n p u t e x c i t a t i o n c h a n n e l s a n d

1069


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1070 Gang f in/ \

/ \/ E x p e r i m e n t a l \ I --I i n p u t - o u t p u t ~ - -- . .~ . 1 F R F, ,measurement/ I CalculationL_ _\ /\ \ \

X M o d e le v a l u a t i o n

@ ~ ~ M o d e l RFg e n e r a t i o n

M F / P MC u r v e - f i t t i n g

E R A S y s t e mr e a l i z a t i o n

M a r k o v p a r a m -e t e r s g e n e r a t i o n

FIGURE 6.1 Ge neral Procedure o f Frequency Domain System Identification. Bolded compon ents imply critical steps; dashed com ponentsimply steps may be excluded.m o u t p u t m e a s u r e m e n t c h a n n e ls . U s e {~((Oi)}i_l,. . .1 t o deno t et he obs e rved F R F da t a based on wh i ch t he TF M G ( z 1 ) will bees t imated . To evaluate G ( z 1) at d iscrete fre quen cies coi , useh e m a p z ( c o i ) = e % ~ # w ' , with ws being the sampl ing f re-uency. The curve-f i t t ing error i s measured by Frobeniuso rm ]]- ]]F for m at r ices and by Z-n orm ]]. ][2 for vectors .s e I , t o d eno t e an i de n t i t y m a t r i x o f d i m ens i ons n n .

1 M a t r i x F r a c t i o n P a r a m e t e r i z a t i o nhe m a t r i x f r a ct i o n ( M F ) p a r a m e t e r i z a t i o n o f a TF M t akeshe fo l lowing form:G ( z - 1 ) = Q l ( z - 1 ) R ( z 1 ), (6.1)

Subs t i tu t ing the Q(z -1) and R ( z 1 ) po l ynom i a l s i n equa t i ons6 .2 and 6 .3 and vector iz ing the summat ion , equat ion 6 .4 i schanged i n t o t he fo rm :G*(z 1) = a rg m i n l la n o , I I F , ( 6 . 5 )G=Q-IR

where:

~ =-Z I(OJ1)GT(o)I) .. z P(COl)GT((DI) -- Ir --z-l(oJ1)Ir .. . --Z P(COI)/r

/i i i i-Z-I(o ~i)~T( "~t) z-P(to I){~T(to I) -It --Z-l(tOl)Ir ... --Z P((ol)Ir(6.6)

T T = [G(tOl)..-G(o~l)]. (6.7) T = [ Q , . . . q p Ro R I ' " R p ] . (6.8)

Q ( z - 1 ) = I m + Q I z 1 + Q z z - 2 + . . . + Q q z - q , (6 .2)R ( z 1 ) = R o + R 1 z - 1 + R 2 z - 2 + - ' ' + R p z - p . (6 .3)

Q 1 . . . . . Q q and R0, R1 . . . . . Rp are re-

ta t ion , w i tho ut loss of general i ty , assume p = q .To f i t the T FM G ( z 1 ) as in equat ion 6 .1 in to the observedata {G(coi) i=1,.. . , l , one ma y so lve the fo l lowing param eter

1G*(z 1 ) = a r g m i n Z I lQ ( z - l (~ ' ) ) G ( ~ i ) - R(z l ( c o , ) ) l l ~ . ( 6 . 4 )G=Q- R i=1

Thus , the MF curve-f i t t ing has been reduced to a s tandardLLS problem, which can be so lved by var ious ef f ic ient a lgo-r i thms (e .g . , the QR factor izat ion approach quoted inalgor i thm of equat ions 6 .29 through 6 .31) .6 . 2. 2 P o l y n o m i a l M a t r ix P a r a m e t e r i z a ti o nT h e p o l y n o m i a l m a t r i x ( P M ) p a r a m e t e r i z a t i o n of a TF M hast he fo rm :

C ( z _ l ) _ B ( z - ' )~x( z-') ' (6.9)where:

B ( Z - 1 ) = Bo + B 1 z - 1 B zZ -2 " ' " - }- B pZ -p . (6.10)o t ( z - 1 ) = 1 + a l z - 1 a2 z- 2 + .. . + aqz -q . (6.11)


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F r e q u e n c y D o m a i n S y s te m I d e n t if ic a t io n 1071i o n s a r e th e n u m e r a t o r p o l y n o m i a l m a t r i x a n d t h e

s o f th e n u m e r a t o r a n d d e n o m i n a t o r a r e e qu a li. e. , p = q) . The goa l of pa ra me te r op t imi za t io n i s to f indG * ( z - ~ ) w i th a p r e spe c i f i e d o r de r p suc h t ha t t he e s t im a t ion

lG * ( z 1 ) = a r g m i n ~ ' w 2 ( o 3 i ) l l G ( t o i ) _ G ( z 1 ( o ) ) 1 l 2 (6.12)G z . . .. .ai= 1a dd i t i ona l t e r m w ( ) is i nc lude d to a l l ow de s i ra b l ey w e igh t ing on t he e s t im a t ion e r r o r. I n t he f o l l ow ing ,( z - 1 ) i n e q ua t ion 6 .12 w i l l be pa r a m e t i z e d i n a w a y tha t

t o t he M I M O c ase.L e t t he num e r a to r po lynom ia l o f t he S I S O sys t e m be t he

B ( z - 1 ) = B ( z 1 ) . ~ ( z - 1 ) (6.13)= t~Tb t~Tb, (6.14)

h e r e b = [ b 0 , b l . . . . . [J ps ]w i s t h e n u m e r a t o r p a r a m e t e re c to r c o r r e spon d ing t o t he f i xe d z e ro s ; b - - [ b0 , b l . . . . .~ p _ p , ] T i s t h e t o - b e - e s t i m a t e d n u m e r a t o r p a r a m e t e r v e c t o r ;and ~ = [1, z 1 . . . . z - P s ] T an d t~ = [1, z - 1 . . . . . z (P p, )] T

a r e t he c o r r e spond ing z - ve c to r s . S im i l a r ly , t he de nom ina to rpoly nom ia l i s a s fol lows:

cx(z 1) = 1 + ~ b T a , (6.15)he r e a = [ al . . . . . at ,] w and ~b = [z -1 . . . . . z - P ] T . T he e s t i -

m a t i o n e r r o r t o b e m i n i m i z e d i s t h e n:

- 1 + ~ b T ( c o i ) a / (6.16)

= / _ ~ l w ( t i ) lqbr(eoi)a ( ~ ( ~ i ) - [ ~ r ( ~ i ) b ' ~ T ( ~ i )

_ _ ~ ( ( o i ) . (b T ( 0 3 i )] i i l ) 2, (6.17)h ere for s imp l ic i ty t~ and ~ a re wr i t te n as func t io ns o f coi .For the MIMO case , equa t ion 6 .17 i s genera l ized to ( reca l lh a t m a n d r d e n o t e t h e n u m b e r s o f o u t p u t a n d i n p u t c h a n n e ls

respec t ive ly) :~ - ~ l 1 W ( ~ i ) (F = ~ + +~(~ oi)a Gik(~o~)j = l k = l i = 1 2

- % k ( o ~ i ) - .*jk(~Oi )b j k Gjk(O~i ) ~bT(~ol)] (6.18)

F ina l ly , the r ight -hand s ide of equa t ion 6 .18 i s vec tor ized fors t a nda r d op t im iz a t i ons :

F ( 0 ) = I I W ( a ) ( y - H o ) l l ~ , (6.19)whe re 0 [b~ b~ , -W, . . . . b m r , a T ] T , y i s a ve c to r c on t a in ingthe m e a su r e d F R F da t a , a nd W ( a ) i s a w e igh t ing f unc t ionwi th va r ia te a .Readers should have no d i f f icu l t ie s to de r ive the de ta i lede xp r e s s ions o f W ( a ) a nd H . F o r t he spe c ia l c a se w he n the r e i sno f ixed zeros ( i.e ., t~Yb = 1 in eq ua tio n 6.14) , the re sults areg ive n i n B a ya r d (1992 ). I t is im po r t a n t t o po in t ou t t ha t W ( a )a nd H ha ve the f o l l ow ing s t r uc tu r e :

W ( a ) =

H =

- W ( a )

0

0- X It l 1

0

0

0W ( a )

0~t12

. . 0

" 0

" . 0o W ( a )

0 ( D l l: (I)12

o i

~ m r qbm r

(6.20)

(6.21)

T he se s t r uc tu r e s e na b l e t he de s ign o f a n e f f ic i e n t op t im iz a t i ona lgor i thm. This wi l l be d iscussed in the next sec t ion .

6 .2.3 Leas t Squares O pt i m i za t i on Al gor i t hm sThis sec t ion se rves two purposes . F i r s t , i t g ives a br ie f (butge ne ra l ) a c c oun t on a f e w o f t he m o s t im p or t a n t pa r a m e te rop t im iz a t i on a lgo r i t hm s , na m e ly t he linear east squares (LLS)m e thod , t he N e w ton ' s m e thod , a nd t he G a uss - N e w ton ' s ( G N )m e th od . S e c ond , t he d i s c us s ion a pp l i e s som e o f t he m e tho ds t othe pa r a m e te r op t im iz a t i on p r ob l e m s a r i s i ng f r om the c u r ve -f i t t ing process . In pa r t icu la r , th is sec t ion presents a f a s t a lgo-r i t h m b a s e d o n t h e G N m e t h o d t o t h e m i n i m i z a t i o n o f e q u a t i o n6.19 . Exce l len t tex tbooks in th is f ie ld a re abundant , and th isd iscuss ion only re fe r s the read er to a f ew of them : Gi l l e t a l . ,(1981) , Dennis and Schnabe l (1996) , and S tewar t (1973) .General Algorithm D evelopmentLet F (O) be the sca la r -va lued mul t iva r ia te objec t ive fun c t io n tobe m inimized I f the f i r s t and secon d de r iva t ives of F a reavai lab le , a loca l qua dra t ic mo de l o f the o bjec t ive fun c t ionm a y be ob t a ine d by t a k ing t he f i rs t t h r e e t e rm s o f t he T a y lo r-s e r i e s e xpa ns ion a bou t a po in t O k in t he pa r a m e te r ve c to rspace:


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1072 Gang ]inF(Ok + p) ~ Fk + g~p + ~pTG kp, (6.22)

where p denotes a s tep in the parameter space and whereFk, gk, an d Gk denote the value, gradient , and Hess ian of theobject ive func t ion a t Ok. Equ at ion 6 .22 indicates that to f ind alocal m ini m um of the objective funct ion , an i t era tive searchingp rocedu re i s r equ i r ed . The ce l eb ra t ed Newt on ' s m e t hod i sde f i ned by choos i ng t he st ep p = p k s o t ha t O k + P k is as t a t i ona ry po i n t o f the q uad ra t i c m o de l i n 6 .22 . Th is am oun t sto so lv ing the fo l lowing l inear eq uat ion:

G k P k = - - g k . (6.23)In th e sys tem ide nt i f i cat ion case , the object ive func t ion F(O) iso f t en i n t he s um s o f s quares fo rm , a s i n equa t i on 6 .19 :

1 nF ( O ) -- ~ Z ) ~ ( 0 ) 2 = l l f ( O ) l l , (6.24)i=1

where 1~ i s the i th component of the vector f i To implementt he N ewt on ' s m e t hod , t he g rad i en t an d H es s ian o f F a re ca lcu -lated as:g ( O ) = l ( O ) T f ( o )G ( O ) = J ( O ) r J ( O ) + Q ( O ) ,

(6.25)(6.26)

where l (0 ) i s t he J acob ian m at r i x o f f and where Q (0 )=}-~=ly~(0)Gi(0), with Gi(O) be i ng t he Hes s i an o f )q (0 ) . TheNew ton ' s equat ion 6 .23 thu s becom es as fol lows:

( l ( O k ) ) T j ( O k ) + Q ( O k ) ) P k = - - J ( O k ) T f ( O k ) . (6.27)

m i t ted f r om equa t ion 6 .27 . I f th i s i s the case , then so lv ing Pkf rom 6 .27 i s equivalent to so lv ing the fo l lowing l inear l eas tsquares (LLS) problem :Pk = a rg rn~ n I I J ( 0 k / P + f(Ok/l122 (6.28)

Equa t i on 6 .28 g i ves t he Gaus s -Newt on m et hod . The LLSrob l em i s o f t en s o l ved by t he QR fac t o r i za t i on m et hodiven in the fo l lowing algor i thm.1 (QR F actorization and LLS So lution)

et A E R mxn have fu ll co l um n rank . Th en A can be un i que l yactor ized in to the form:A = QR, (6.29)

where Q has o r t hono rm al co l um ns and where R i s uppert r iangular wi th pos i t ive d iagonal e lements . The unique so lu-t i on o f t he LLS p rob l em :

is given by:m i n ]lAx 1 1 2 2 , ( 6 . 3 0 )xERn

x* = R 1QTy. (6.31)Application to the Curve-Fit t ing ProblemsThi s d i s cus s i on now re t u rns t o t he P M cu rve- f it t ing p rob l em i nequa t ion 6 .19 . By le t ting f = W ( a ) ( y - H O ) , the Gauss-New t on m et hod i n equa t i on 6 .28 m ay be app li ed . I t t u rn s

FIGURE 6.2 Picture of the 16-Story Structure


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F r e q u e n c y D o m a i n S y s te m I d e n t i fi c a t io n6.1 Key Identification Param eters

Curve-fitting System realizationstructure Type p q a ~ n

6-Sto ry MF 16 16 32 32 10PM 8 8 12 30 8

t h a t t h e ] a c o b i a n m a t r i x I ( 0 k) o f f h a s a n i d e n t ic a l s t r u c t u r eH in eq ua t io n 6 .21. Thu s , in s tead o f so lv ing equa t ion 6 .28e c tl y , t h e f o l l o w i n g d e c o m p o s i t i o n o f L L S p r o b l e m s m a y b e

2 ( D e c o m p o s it io n o f L L S P r o b l e m )e t A E R mn h a v e f u l l c o l u m n r a n k . D e c o m p o s e A i n t o

w i t h A l C R m x p a n d A2 C R m x q . L e t th e u n i q u eQR fac to r iza t ion o f A1 be A1 = Q1R~. Th en the un iq ue so lu -

t h e L L S p r o b l e m o f e q u a t i o n 6 .3 0 h a s t h e f o r m* = [x~ T, x~ T] r , wi th xq and x~ be ing the un iqu e so lu t ionso f t h e f o l l o w i n g L L S p r o b l e m s :

1073m i n I l A . 2 x 2 y l 1 2 ~ . (6 .32)x2ERqm i n I l A l X l ( y - A = . ~ ) I I = = .x ] c RP (6 .33)

In these equ at ion s , ]~2 = (I - Q1QT)A2.In p rac t i ce , ](Ok) i s d iv ided in to ] (O k ) = [ h ( 0 k ) , h ( 0 k ) ] ,

w i t h l l ( O k ) c o r r e s p o n d i n g t o t h e b l o c k - d i a g o n a l t e r m s a n d1 2( 0k ) t h e l a s t b l o c k m a t r i x c o l u m n i n ] (O k ) . So lv ing the LLSpro b lem s w i th 12 (0 k ) and 11 (0 k ) thus co r re spond s to upd a t ingt h e d e n o m i n a t o r a a n d t h e n u m e r a t o r s bjk e s t i m a t i o n s . M o r e -o v e r, d u e t o i ts b l o c k - d i a g o n a l s t r u c t u r e , t h e L L S p r o b l e m w i t h11 (Ok) shou ld be fu r ther dec om pos ed ( i. e. , the LLS so lu t ions o fbjk are indepe nde n t ly so lved fo r each j and k ). Thus , thec o m p u t a t i o n a l c o s t o f t h e o p t i m i z a t i o n a l g o r i t h m i s s i gn i fi -can t ly reduced .

T o c o m p l e t e t h e d i s c u s si o n o f t h e a l g o r i t h m , n o t e t h a t t h ei n it ia l v a lu e s f o r t h e G a u s s - N e w t o n i t e r a t i o n m a y b e g e n e r a t e db y t h e c la s si c al S a n a t h a n a n - K o e r n e r ( SK ) i t e r a t i o n c o m p o s e do f a s e q u e n c e o f r e w e i g h t e d L L S p r o b l e m s :

f 1 6 f 1 2- 4 0 : . ! . . . . . 4 0 [ ~ - - 1- 6 0 : : . . . . : - 6 08 o- ,O O l y : V f , 1 1 - ,o o

- 1 2 0 L . . . . . . " . . . . " . . . . . . . ~ - 1 2 02 0 4 0 6 0 2 0 4 0 6 0

-60 -60- 8 0 - 8 0

- 1 0 0- 1 0 0 - 1 2 0- 1 2 0 2 0 4 0 6 0 2 0 4 0 6 0

f8 f 4-4 0 : : I [ z I- 6 0 : i - 6 0- 8 o ~ - 8 0

- 1 0 0 - 1 0 0_ 1 2 0 i : , \ f j - 1 2 0

2 0 4 0 6 0 2 0 4 0 6 0

- 6 0 : i : . . . . 6 0 ~ ~ ". . . : i- 8 0 . . . . : : - 8 0 i : . . .o o k ,- 1 2 o r r y . . . . Y

- 1 4 0 F ; F . . . . . . . - 1 2 0 I " " " ' , - - " i l ~ 12 0 4 0 6 0 2 0 4 0 6 0

- 6 o . i . . . . . 6 0 : .. . : :x0- ' I , . . . . V - Y 1 : . . . . 1- ' ~ v t ] - 1 4 0 t . . . . J2 0 4 0 6 0 2 0 4 0 6 0

- 6 o i i . . . . i . . . - 6 o . : . . ~ . :- 8 0 i . i . . - 8 0 i . i . . .,oo

_ 1 2 o t , ~ , , , j - , 2 O l , i i t _ - v2 0 4 0 6 0 2 0 4 0 6 0

- 6 0 [ i t ' ~ 1 - 6 0 ' . . . . - 6 0 t- 8 0 i i - 8 0

x - 1 0 0 - 1 0 0_ 1 o o _ 1 2 o | - , . , ~ ~ Y 1 - 1 2 o i- 1 2 0 - 1 4 0 ~ , . . . . . ~ , ~2 0 4 0 6 0 2 0 4 0 6 0

F r e q u e n c y ( H z ) F r e q u e n c y ( H z )2 0 4 0 6 0F r e q u e n c y ( H z )

- 6 0 " ! i . . . . i - 8 0 . . . . .

- 1 2 I ~. " ~ - : ' , ' 1- 1 4 0 t - - ~ - " , ; . 1

2 0 4 0 6 0F r e q u e n c y ( H z )

FIGURE 6.3 Com parison of Experimental and Model FRF for the 16-Story Structure: Magnitude Plot. ~ denotes the input force on the jthfloor; xj denotes the o utput displacement of the jth floor; dotted lines are for mea surem ent data; solid lines are for mod el output.


6/10

10740k+a = arg rnoin][W(ak)(y-- H0) [ [ ~ ,

w i t h i n i t i a l c o n d i t i o n 0 0 = 0 .

( 6 . 34)

6 . 3 S t a t e - S p a c e S y s t e m R e a l i z a t i o nSystem realization is a technique to d e t e r m i n e a n i n t e r n a ls t a t e - sp a c e d e s c r i p t i o n f o r a s y s t e m g i v e n w i t h a n e x t e r n a ld e s c r ip t i o n , t y p i c a l ly it s T F M o r i m p u l s e r e s p o n s e . T h e n a m er e f lec t s the f ac t tha t i f a s t a te - spac e de sc r ip t ion i s ava i lab le , ane l e c t r o n i c c i r c u i t c a n b e b u i l t i n a s t r a i g h t f o r w a r d m a n n e r t or e a li z e th e s y s t e m r e s p o n s e . T h e r e i s a g r e a t a m o u n t o f l i t e ra -t u r e o n t h i s s u b j e c t b o t h f r o m a s y s t e m t h e o r e t i c a l p o i n t o fv i e w ( A n t s a k l i s a n d M i c h e l , 1 9 9 7 ) a n d f r o m a p r a c t i c a l s y s t e mi d e n t i f i c a t i o n p o i n t o f v i e w ( ] u a n g , 1 9 9 4 ) . I n t h e f o l l o w i n g , aw e l l - d e v e l o p e d m e t h o d i n t h e s e c o n d c a t e g o r y , t h e e i g e n s y s -

Gang fint e m r e a l i z a t i o n a l g o r i t h m ( E R A ) , i s s e l e c t e d t o c o n s t r u c t am o d e l i n t h e s t a t e s p a c e f o r m . F i r s t p r e s e n t e d a r e t h e f o r m u l a st o g e n e r a te t h e M a r k o v p a r a m e t e r s f r o m t h e T F M , w h i c h a r et h e s t a r ti n g p o i n t f o r t h e E R A m e t h o d .

6 .3 .1 M a r k o v P a r a m e t e r s G e n e r a t i o nT o c a l c u l a t e t h e M arkov parameters Y0, Y1, Y2 . . . f r o m t h es y s t e m T F M , f i r s t n o t e t h a t :

G(Z-1) = ~ y i Z i .i=o ( 6 . 35)

F o r t h e c a s e w h e n G (z -1 ) i s p a r a m e t e r i z e d i n t h e M F f o r m(i.e., G (z 1 ) = Q - l ( z 1 ) R ( z - 1 ) ) ' t h e s y s t e m M a r k o v p a r a m -e t e r s c a n b e d e t e r m i n e d f r o m :

f16 f12

- 5 0 . . . . . 100 t . . . . . . . . . . . . . . . . . . . . . . . . . . .- 1 0 0 . . . . 200- 1 5 0 I U - 30 0

20 40 60 20 40 60

0

-200 ' .L . .#

-400 . . . . .

f8 f4

20 40 60

0- 2 0 0- 4 0 0- 6 0 0

20 40 60

- i o o,i o o l= - q N

- 3 0 0 [ . . . . . ~ U - U L20 40 60

. . . . n50 . . . . . .-1 O0 . . . . . . . .- + O l L j L20 40 60

l O O ~ - . ! . . . . . . . . i . . . . . . . . i . . . .o~ i - i . . . i . -- l O O L ~ . . . . . .-2vqF 73 iN- 3 0 0 I . . . . ~ - - - - U L _

20 40 60

0

- 2 0 0

- 4 0 0

20 40 60

o + : : -20 0 i .- 4 o 0 I . . . .. . . .. . . .. . . .. .~ - 6 o o : :+ . . + . . . i+

20 40 60 20 40 60

0 ... ~.......: ~ i ....... 0 ~-50 . . . . . . . i ...i . .. . -200-lOO . . . . . . i . . . . i . . . . . -4oo- 1 5 0 - 6 0 0

20 40 60 20 40 60O '

- 2 0 0=g -400

- 6 0 020 40 60

F r e q u e n c y ( H z )

0 ~ . . . . . . ; . . . . . ~ . . . . . . ~ . . . . .- 2 0 0 1 ~ ,-400 i : : : ,- 6 0 0 '- 8 0 0

-1 O0- 2 0 0

- 1 0- 2 0 0- 3 0 0

20 40 60 20 40 60 20 40 60Frequency (Hz) Frequenc y (Hz) Frequen cy (Hz)

FIGU RE 6.4 Co mp ar ison o f Experimental and M odel FRF for the 1 6-Story Structure: Phase Plot. Same notat ions are used as in Figure 6.3.


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6 F r e q u e n c y D o m a i n S y s t e m I d e n t i fi c a t io n 1075T A B L E 6 . 2 P M I t e r a t io n R e c o r dI t e r a t i o n I t e r a t i o n F R F c~ S t e p 13 S t e pt y p e i n d e x r e s i d ue n o r m n o r m

1 6 5 1 . 4 7 1 0 0 % 1 0 0 %S 2 1 5 9 . 4 6 1 . 9 1 % 1 4 . 8 %K 3 1 5 6 . 7 5 0 . 1 6 % 1 . 3 6 %

4 1 5 6 . 7 4 0 . 2 4 % 0 . 8 4 %1 1 3 7 . 0 3 1 0 . 9 % 2 1 . 8 %2 1 3 3 . 4 7 4 . 2 3 % 8 . 3 1 %

G 3 1 2 8 . 03 0 . 5 6 % 2 . 1 5 %N 4 1 2 3 . 8 1 0 . 2 1 % 0 . 5 0 %

5 1 2 3 . 3 4 0 . 4 0 % 1 . 0 7 %6 1 2 3 . 48 0 . 1 5 % 0 . 3 7 %

FIGUR E 6.5 Picture of the Seismic-AM D Benchm ark Structure

Q i= ~O Q i Z i ) ( i = ~ o y i Z i ) P i ~ _ o R i Z i , (6.36)fo l lowing i te r a t ive ca lcu la t ions s ta r t ing f rom Y0 = R0:

R k - ~ - - 1 Q i Y k i ,gk = --~Pi 1 Qi gk- i ,

f or k = 1 . . . . p .fo r k = p + 1 . . . . o c . (6.37)

I f t h e T FM i s p a r a m e t e r i z e d i n t h e PM f o r m , t h e d e r i v a t i o nf th e s y s t e m M a r k o v p a r a m e t e r s i s a lm o s t t h e s a m e : o n e s t a r t s

Y0 = B0, a n d c o n t i n u e s w i t h t h e f o l l o w i n g i te r a t i v e p r o -e d u r e :

f Bk - 2 ~ = 1 aigk i ,Yk f or k = 1 . . . . p.f or k = p + l . . . . . o c.

(6.38)

6 .3 .2 T h e E R A M e t h o dTo so lve fo r a s ta te - space mode l (A , B, C, D) us ing the ER Am e t h o d , f i r s t f o r m t h e g e n e r a l i z e d H a n k e l m a t r i c e s :

H ( k - 1) =Yk Yk+ l

Yk+l Yk+2

Yk+~ ~ Yk+~

. Y k + f 3 1 ]! Y k + f 3 ] .

r Y k + c ~ + 1 3 - 2 _ l(6.39)

No te th a t in gene ra l , c~ and [3 a re cho sen to b e the sma l le s tn u m b e r s s u c h t h a t H ( k ) h a s a s l a r g e r o w a n d c o l u m n r a n k sa r e p o ss i b le . A d d i t i o n a l s u g g e s t io n s t o d e t e r m i n e t h e i r o p t i m a lv a l u e s a re g i v e n i n J u a n g ( 1 9 9 4 ). L e t t h e s i n g u l a r v a l u e d e c o m -p o s i t io n o f H ( 0) b e H ( 0 ) = U ~ , V r , a n d l e t n d e n o t e t h e i n d e xw h e r e t h e s i n g u l a r v a lu e s h a v e t h e l ar g e s t d r o p i n m a g n i t u d e .T h e n , H ( 0 ) c a n b e a p p r o x i m a t e d b y :

H ( 0 ) ~ U ~ V T , ( 6. 40 )w h e r e U , a n d V , a r e t h e f ir s t n c o l u m n s o f U a n d V, r e s p e c -t ive ly , and 1 i;~ i s the d iago na l m a t r ix co n ta in ing the la rges t ns ingu la r va lues o f H(0 ) . F ina l ly , an n th o r de r s ta te - space r ea l -i z a t i o n (A , B , C , D) c a n b e c a l c u l a te d b y :

a = ~ 1 / 2 U r N ( l ) g n ~ n l / 2 ,C = E r U , ;1 /2 D = Y o ,m n ~ n '

X ' I / 2 v T EB = "-~n n r , (6.41)

w h e r e E r a n d E ~ a r e t h e e l e m e n t a r y m a t r i c e s t h a t p i c k o u tt h e f i rs t r ( t h e n u m b e r o f s y s t e m i n p u t s ) c o l u m n s a n d f i r st m( t h e n u m b e r o f s y s t e m o u t p u t s ) r o w s o f t h e i r m u l t i p l i c a n d s ,respectively.

6 . 4 A p p l i c a t io n S t u d i e sT h i s s e c t i o n p r e s e n t s t w o e x p e r i m e n t a l l e v e l a p p l i c a t i o ns t u d i e s c o n d u c t e d i n t h e S t r u c t u r a l D y n a m i c s a n d C o n t r o l /


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1076 G a n g f i nU m

. ~ 20-o 0"( 3

- 2 0- 4 0 , 5 1 0 1 5 2 0 2 5 3 0 3 5

4 0 . . . . ' " i ' i . . . . i . . . . '. '

- 4 0 i i i i i i i i i i i i i I I Y I I I I I I I I I I I I I I I I I I5 1 0 1 5 2 0 2 5 3 0 3 5

" 5" 0"( 3x~

~ 5" 0"( 3EX

EX

0- 2 0- 4 0 5 1 0 1 5 2 0 2 5 3 0 3 5

5 1 0 1 5 2 0 2 5 3 0 3 510

05 1 0 1 5 2 0 2 5 3 0 3 5

F r e q u e n c y ( H z )

4 02 0

0- 2 0

4 02 0

0- 2 0- 4 0

2 00

- 2 0- 4 0

2 00

- 2 0- 4 0

- 5 0

- 1 0 0

X g d d O t

5 1 0 1 5 2 0 2 5 3 0 3 5

5 1 0 1 5 2 0 2 5 3 0 3 5

5 1 0 1 5 2 0 2 5 3 0 3 5

5 1 0 15 2 0 2 5 3 0 3 5

5 1 0 1 5 2 0 2 5 3 0 3 5F r e q u e n c y ( H z )

Com parison of Experimental and Model FRF for the Seismic-AM D Benchmark Structure: Magnitude Plot. Um denotes the inputomm and to the AMD; xg ddo t denotes the inp ut g round acceleration to the structure; x jd d o t denotes the o utpu t acceleration o f the jth floor;m d d o t denotes the outpu t acceleration of the AM D; Xm denotes the outpu t displacement of the AM D; dotted lines are for measur em ent data;olid lines are for mo del outp ut.

a r t h q u a k e E n g i n e e r i n g L a b o r a t o r y ( S D C / E E L ) a t U n i v e r s i t yo f N o t r e D a m e . T h i s s e c t i o n o n l y p r e s e n t s t h e r e s u l ts p e r t i -

en t to iden t i f i ca t ion s tud ies d i scussed so fa r in th i s chap te r .F o r d e t a i l e d i n f o r m a t i o n a b o u t t h e s e e x p e r i m e n t s , i n c l u d -n g e x p e r i m e n t a l s e t u p s a n d / o r c o n t r o l d e v e l o p m e n t s , t h er e a d e r m a y r e f e r t o J i n e t a l . (2000) , J in (2002) , and Dykee t a L ( 1 9 9 4 ) , respect ively .

. 4. 1 I d e n t i f i c a t i o n o f a 1 6 - S t o r y S t r u c t u r eh e f i r s t iden t i f i ca t ion t a rge t i s a 16 -s to ry stee l s t ruc tu re m ode l

s h o w n i n F i g u r e 6 . 2 . T h e s y s t e m i s e x c i t e d b y i m p u l s ef o r c e p r o d u c e d b y a P C B h a m m e r a n d a p p l i e d i n d i v i d u a l l ya t the 16 th , 12 th , 8 th , and 4 th f loo rs . The acce le ra t ions o ft h e s e f l o o r s a r e s e l e c t e d a s t h e s y s t e m m e a s u r e m e n t o u t p u t sa n d a r e s e n s e d b y P C B a c c e l e r o m e t e r s . T h e g o a l o f t h e i d e n t i -

f i ca t ion i s to c ap tu re accu ra te ly the f i r s t f ive pa i rs o f thec o m p l e x p o le s o f t h e s t r u c t u r e . F o r t h i s p u r p o s e , a D S P TS i g l a b s p e c t r u m a n a l y z e r i s u s e d t o m e a s u r e t h e F R F d a t a .T h e s a m p l i n g r a t e i s s e t a t 2 5 6 H z , a n d t h e f r e q u e n c y r e s o -lu t ion i s se t a t 0 .125 Hz . The exper imen ta l FRF i s p recond i -t i o n e d t o e l i m i n a t e t h e s e c o n d o r d e r d i r e c t c u r r e n t ( d c ) z e r o sf r o m a c c e l e r a t i o n m e a s u r e m e n t . T h e M F p a r a m e t e r i z a t i o ni s c h o s e n f o r t h e c u r v e - f i t t i n g , w h i c h i s c o m p l e m e n t e d b yt h e E R A m e t h o d f o r s t a t e - s p a c e r e a l i z a t i o n . T h e k e y i d e n t i f i -ca t ion parameters a re g iven in Tab le 6 .1 . The f ina l d i sc re te -t ime s ta te -space rea l i za t ion has 10 s ta tes . The magn i tudea n d p h a s e p l o t s o f i ts t r a n s f e r fu n c t i o n s a r e c o m p a r e d t othe exper imen ta l FRF da ta in F igu res 6 .3 and 6 .4 . Exce l l en ta g r e e m e n t s a r e f o u n d i n a ll b u t t h e v e r y h ig h f r e q u e n c yr a n g e . T h e m i s m a t c h t h e r e i s p r i m a r i l y d u e t o t h e u n m o d e l e dh i g h - f r e q u e n c y d y n a m i c s .


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F r e q u e n c y D o m a i n S y s t e m I d e n t i f i c a t i o n 1077

0"o%x

Um

- 1 O 0- 2 0 0- 3 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5

t1 0 1 5 2 0 2 5 3 0 3 50

- 2 0 0- 4 0 0- 6 0 0

5 1 0 1 5 2 0 2 5 3 0 3 5

- lOO- 2 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5

- 1 2 0 I . . . . . . . . .

0- 2 0 0- 4 0 0

x g d d o t

! i i . . . . i ! l5 1 0 1 5 2 0 2 5 3 0 3 5

- 1 0 0- 2 0 0- 3 0 0

5 1 0 1 5 2 0 2 5 3 0 3 5

- 1 0 05 1 0 1 5 2 0 2 5 3 0 3 5

_ 4 o o t . . . . . . . . . . . t5 1 0 1 5 2 0 2 5 3 0 3 5

. . . . . . . i i i i i i i i i i i ii i ii l l- 2 0 0 L ~ , ; . . . . " :-4 0 0 I - U . . . . : . ~ '~w~- ,,~ ., ,,~Z.~- 6 0 0 t . . . . ~ r . "- 8 0 0 ~ : . . . . . . . . . ~ ,5 1 0 1 5 2 0 2 5 3 0 3 5 5 1 0 1 5 2 0 2 5 3 0 3 5

F r e q u e n c y ( H z ) F r e q u e n c y ( H z )Com parison of Experimental and Model FRF for the Seismic-AM D Benchmark Structure: Phase Plot. Same notations are used

I d e n t i f i c a t i o n o f t h e S e i s m i c - Ac t i v eM a s s D r i v e r B e n c h m a r k S tr u c t u r e

secon d iden t i f i ca t ion t a rge t o f th i s d i scuss ion i s a th ree-e e l s t r u c t u r e m o d e l , w i t h a n a c t iv e m a s s d r i v e r ( A M D )

a l l e d o n t h e t h i r d f l o o r t o r e d u c e t h e v i b r a t i o n o f t h et o s i m u l a t e d e a r t h q u a k e s . A p i c t u r e o f t h e

ys tem i s g iven in F igu re 6 .5 . Th is sy s tem has been used

o n s : t h e v o lt a g e c o m m a n d s e n t to t h e A M D b y t h e c o n t r ole r . T h e s y s t e m r e s p o n s e s a r e m e a s u r e d b y f o u r a c c e l e r o m -

o r t h e t h r e e f l o or s a n d t h e A M D a n d o n e l i n e a r v a r ia b l eD . T h e s a m p l i n g r a t e is 25 6 H z , a n d t h e f r e q u e n c y

e s o l u t i o n i s 0 . 0 6 2 5 H z . D u e t o n o i s e a n d n o n l i n e a r i t y ,n l y t h e f r e q u e n c y r a n g e o f 3 to 3 5 H z o f t h e F R F d a t a i sons idered to be accu ra te and , thus , th i s range i s u sed fo r theden t i f i ca t ion .

A p r e l i m i n a r y c u r v e - f i t t i n g i s c a r r i e d o u t u s i n g t h e M Fa r a m e t e r i z a t i o n . T h e i d e n t i f i e d m o d e l m a t c h e s t h e e x p e r i -

m e n t a l d a t a a c c u r a t e l y i n a l l b u t t h e l o w - f r e q u e n c y r a n g e o fc h a n n e l s c o r r e s p o n d i n g t o t h e A M D c o m m a n d i n p u t a n d t h ea c c e l e r a t io n o u t p u t s . A d e t a i l e d a n a l y ti c a l m o d e l i n g o f t h esys tem revea l s tha t the re a re fou r ( respec t ive ly two) f ixed dcz e ro s f r o m A M D c o m m a n d i n p u t t o t h e s t r u c tu r e ( r es p e ct iv e lyA M D ) a c c e l e r a t i o n o u t p u t s :

l i ra G x ~ u m ( S ) - - k i , i = 1 , 2 , 3 . (6 .42)s-e0 S4

l im G ~ u m ( S ) - k i n . (6 .43)s-*O S2

T h e G ~ , ~ a n d G ~ , , u , ~ a r e t h e t r a n s f e r f u n c t i o n s f r o mA M D c o m m a n d i n p u t u m t o s t r u c t u r e a n d A M D a c c e l e r a t i o nou tpu t s , respec t ive ly . The k i a n d k m are the s t a t i c ga in s o fthese t rans fe r func t ions wi th the f ixed dc ze roesr e m o v e d . T h e s e f i x e d ze r o s d i c ta t e t h e u s e o f t h e P M c u r v e -f i t t ing t echn ique to exp l ic i t ly inc lude such a p r io r i in fo rma-t ion .

A g a i n , k e y i d e n t i f i c a t i o n p a r a m e t e r s a r e p r e s e n t e d i n T a b l e6 .1 . T h e o u t p u t s o f t h e p a r a m e t e r o p t i m i z a t i o n i t e r a t i o n s a r e


10/10

1078 G a n g J i nd o c u m e n t e d i n T a b l e 6 .2 . T h e f i n a l d i s c r e t e - t i m e s t a t e - s p a c er e a l i z a ti o n h a s e i g h t s t a t e s a s p r e d i c t e d b y t h e a n a l y t ic a l m o d -e l in g . T h e m a g n i t u d e a n d p h a s e p l o t s o f it s tr a n s f e r f u n c t i o n sa r e c o m p a r e d t o t h e e x p e r i m e n t a l FRF d a t a i n F i g u r e s 6 .6a n d 6 .7 . A l l t h e i n p u t o u t p u t c h a n n e l s a r e i d e n t if i e d a c c u r a t e l ye x c e p t f o r t h e ( 5 ,2) e l e m e n t , w h i c h c o r r e s p o n d s t o t h e g r o u n da c c e le r a ti o n i n p u t a n d t h e d i s p la c e m e n t o u t p u t o f th e A M D .T h e p o o r f i t t in g t h e r e i s c a u s e d b y t h e e x t r e m e l y lo w s i g n a l - t o -no ise r a t io .

6 . 5 C o n c l u s i o nT h i s c h a p t e r d i s cu s s es t h e i d e n t i f i c a t i o n o f l i n e a r d y n a m i cs y st e m s u s in g f r e q u e n c y d o m a i n m e a s u r e m e n t d a ta . A f t e r

u t l i n i n g a g e n e r a l m o d e l i n g p r o c e d u r e , t h e t w o m a j o ro m p u t a t i o n s te p s, f r e q u e n c y d o m a i n c u r v e -f i tt i n g a n d

. T h e a l g o r i t h m s e m p l o y T F M m o d e l s i no f m a t r i x f ra c t i o n o r p o l y n o m i a l m a t r i x a n d r e -y l i n e a r o r n o n l i n e a r p a r a m e t e r o p t i m i z a -

d a t e d t h r o u g h t h e m o d e l i n g o f t w o e x p e r i m e n t a l t e st

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unconstrained optimization and nonlinear equations. Philadelphia:SIAM Press.Dyk e, S., Spenc er, Jr., B., Belknap , A., Fe rrell, K., Quast, E, and Sain,M. (1994). Absolute acceleration feedback control strategies for theactive mass driver. Proceedings of the World Conference on StructuralControl 2, TPh51-TPI :60 .Gill, EE., M urray, W., and W right, M.H. (1981). Practical optimiza-tion. New Y ork: Academ ic Press.Jin, G., Sain, M.K., and Spencer, Jr., B.E (2000). Frequency domainidentification with fix ed ze ros : First generation seismic-AMDbenchmark. Proceedings of the American Control Con ference,981-985.Jin, G. (2002). System identi f ication for controlled structures in civilengineering application: Algorithm development and experimentalverification. Ph.D. Dissertation, Un iversity of No tre Dam e.Juang, J.N. (1994). Applied system identi f ication. Englewood Cliffs, NJ:Prentice Hall.Stewart, G.W . (1973). Introduction to matrix computations. New York:Academ ic Press.

frequency domain system identification

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