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Automatica, Vol. 17, No , 3, pp. 541 -544 , 1981 0005-1098/81/030541-04502.00/0P r i n t e d i n G r e a t B r i t a i n P e r g a m o n P r e s s L t d .,~ 1 9 8 1 I n t e r n a t i o n a l F e d e r a t i o n o f A u t o m a t i c C o n t r o l

Brief PaperMul t ivar iab le Sys tem Struc ture and ParameterIdent i f i cat ion Us ing the Corre lat ion Method*

H . E L - S H E R I E F t

Key Word s--ldentifi cation; structure and parameter estimation; discrete time systems; multivariablesystems; least squares method; state space methods.Abstract --This paper describes an algorithm for the structuredetermination and parameter identification of linear discrete-time multivariable systems from input-output measurements.The algorithm starts by determining the structure parametersof a certain canonical state space representation from anestimate of the correlation functions of the output sequence.Then the parameters of the A matrix are estimated from theestimated correlation functions using the recursive leastsquares method. Finally a normalized stochasticapproximation algorithm is used for the estimation of theparameters of the B matrix from inpu t-out put measurements.1 . I n t r o d u c t i o nIN RECENT years considerabl e work has been done in the fieldof multivariable system identification from input-outputmeasurement s (EI-Sherief and Sinha, 1979a). Due to thepractical importance of the state space representation,especially in cont rol and filtering theory (El-Sherief andSinha, 1979b; Mehra, 1970; Saridis, 1977), much of this workhas been done on the problem of identification ofmultivariable systems in the state space representation (Budin,1971; Lobb ia and Saridis, 1973; Blessing, 1977; Sinha a ndKwong, 1979; E1-Sherief, 1980). In general this problem canbe divided into two main steps: structure determination andparameter estimation of system matrices. The first step ismore difficult, especially for the case of noisy data, and severalpapers have been published on this subject (Guidorzi, 1975;Suen and Liu, 1978; EI-Sherief and Sinha, 1979c). It is wellknown that the representation of linear multivariable systemsin the state space form is not unique and any non-singulartransformation of the state vector will lead to a similar statespace representati on. Because of this non-un iquenes s severalcanonical state space forms have been developed for theidentification of multivariable systems (Weinert and Anton,1972; Mayne, 1972; El-Sherief and Sinha, 1979d) which willsimplify the structure determination problem and reduce thenumber of parameters to be estimated in the system matrices.In this paper an algorithm is proposed for the structuredetermination and parameter estimation of linear discrete-time multivariable systems from input-output measurements.The system is identified in a certain canonical state spacerepresentation (Blessing, 1977). The algorithm starts bycalculating recursively the correlation functions of the outputsequence. Then the structural parameters of the canonicalstate representation are determined by testing the determinantof a certain matrix whose entries are the correlation functions.The parameters of the A matrix are also estimated from thecalculated correlation functions using a recursive least squaresmethod. Finally the parameters of the B matrix are estimatedrecursively from the input-output data using a normalizedstochastic approximation algorithm. A similar algorithm hasbeen proposed by Tse and Weinert (1975) but t he controlsignal has been assumed to be identically zero.

*Received May 20 1980; revised October 21 1980 . Theoriginal version of this paper was not presented at any IFACMeeting. This paper was recommended for publication inrevised form by Associate Editor M. Younis."t'Department of Electrical Engineering, Universit y ofPetroleum and Minerals, Dhahran, Saudi Arabia.541

The proposed algorithm is applied to estimate the structureand parameters of a simulated example and the results ofsimulation indicate that it works quite well.2 . P r o b l e m f o r m u l a t i o nA linear time-invariant discrete-time multivariable systemcan be described by the following state-space equations

x * ( k + 1 ) = A * x * ( k ) + B * u ( k ) (11y ( k ) = C * x * ( k ) + v ( k )

where x * ( k ) is the n-dimensional state vector, u ( k ) is the p-dimensional input vector, y(k) is the m-dimensional noisyoutput vector and v ( k ) is the m-dimensional output noisevector. The sequences u ( k ) and r(k) are assumed to beGaussian with the following statisticsE { u ( k )} = E { v ( k ) } =0E { u ( k ) u T ( j ) } = U c5 - j ( 2 )E { v ( k ) v r ( j )} = V 3 k _ ~E { u ( k ) v r ( j ) } =0

where E{ - } denotes the expectati on, T denotes transpose and6k-j is the Kronecker delta function.Assuming that the system given in equation (1) iscompletely observable and let the matrix C* be partitioned asc * = l c r c r . . c . I T (3)

then construct the following vector sequences* * T *c ! , A c l , . . ., (4 )tc ~ , A * r c * . . . .

and select them in the following manner to construct thetransformation matrixT * * T * . . . . . . . t,~2 . . . . ~ ~ - i (5 )

where a vector A* r 'c .*, is retained if and only if it isindependent of all previously selected ones. Let nl, n 2 , . . . , n mbe the number of vectors selected from the first,second ... .. ruth sequence in equation (4); these are called thestructural parameters of the system and because of thecomplete observability of the system they satisfy the followingrelationn l+ n 2+ ... +n,~=n. (6)

Using the following equivalence transformation x ( k )= S x * ( k ) , system (1) can be transformed to the followingcanonical state space representation (Tse and Weinert. 1975;Sinha and Rozsa, 1976; Blessing, 1977)

x ( k + l ) = A x ( k ) + B u ( k ) (7)y ( k ) = C x ( k ) + v ( k ) ,

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5 4 2 B r i e f P a p e rw h e r e A = S A * S -1 , B = S B * , C = C * S - ~ a n d t h e m a t r i c e s Aa n d C h a v e t h e f o l lo w i n g c a n o n i c a l f o r m

Al lA = :

A m t0

A . = i(a , O )

0Al l = i(i > j O)i jc,~=10 . .0 1

c r = l a . ( O ). .. a . ( n t - l ) . . .

. . A ~ , .

1. . . a , ( n i - 1 )

.. . o i i )

. . . a i j ( n j - -

0 . . . 0 1 i f n ~ > 0a i . i - l ( n i - - 1 ) 0 . . . 0 1

( 8 )

i f n , = 0 ( 9)

t h e 1 i n c r i s i n c o l u m n l + n ~ + . . . + n ~ _ t . T h e m a t r i x B h a sn o s p e c i a l f o r m . W e c a n s e e f r o m e q u a t i o n s ( 7 - 9 ) t h a t s y s t e m( 1 ) h a s b e e n p a r t i t i o n e d t o a n u m b e r o f s u b s y s t e m s e q u a l t ot h e n u m b e r o f o u t p u t s ( m ).T h e i d e n ti f ic a t io n p r o b l e m c a n b e s u m m a r i z e d s o a s t oe s t i m a t e t h e s tr u c t u r e p a r a m e t e r s n~, i = l , 2 , . . . , m a n d t h ep a r a m e t e r s o f th e s y s t e m m a t r i c e s A a n d B f r o m t h e g i v e ni n p u t - - o u t p u t m e a s u r e m e n t s .3 . S t r u c t u r a l i d e n t i f i c a t i o nF r o m t h e d e f i n i ti o n o f th e s t r u c t u r a l p a r a m e t e r s n ~ , i= 1 ,2 . . . . m a n d t h e s p e c i a l c a n o n i c a l f o r m o f t h e m a t r i c e s Aa n d C ( e q u a t i o n s 8 a n d 9 ) , t h e f o l l o w i n g r e l a t i o n s c a n b eo b t a i n e d f o r t h e i t h s u b s y s t e m ( o u t p u t )

a n de r A ' = y ~ a , f l l) c f A ' i f n / > 0

j = l I = 0

i - - l n 2 - 1c r iA ' ~ = ~ ~ a o ( l ) c f A t if n ~ = O . ( 1 0 )j = l I = 0

L e t P d e n o t e t h e c o v a r i a n c e m a t r i x o f t h e s t a t e s o f s y s t e m1 7). T h e n f r o m e q u a t i o n s ( 7) a n d u s i n g r e l a t i o n s ( 2 ) w e g e t

P = A P A r + B U B r . ( 1 1 )L e t R d e n o t e t h e c o r r e l a t i o n m a t r i x o f th e o u t p u t s e q u e n c e

d e f i n e d a sR ( a ) = E [ y ( k + a ) y r ( k ) ] , ( 1 2 )

t h e n f r o m e q u a t i o n s ( 7 ) w e g e t f o r a > 0R [ a ) = E [ C A x ( k + a - 1 ) x r ( k ) C r + C A x ( k + a - 1 ) v r ( k )

+ C B u l k + a _ l ) x r ( k ) C r + C B u { k + a _ l ) v r ( k ) ( 1 3 )+ v ( k + o ) x r ( k ) c r + v ( k + a )v T (k ) ],

t h e n u s i n g r e l a t i o n s ( 2 ) a n d t h e f o l l o w i n g r e l a ti o nE [ x (k + a - l ) x r ( k ) ] = A " - t P

e q u a t i o n ( 13 ) is r e d u c e d t oR ( a ) = C A P C r. ( 1 4 )

D e f i n i n g t h e c o r r e l a t i o n f u n c t i o n r # l a ) a s t h e i , j t h e n t r y o ft h e c o r r e l a t i o n m a t r i x R ( a ) , w e g e t f r o m e q u a t i o n ( 14 )r o ( n i + r ) = c r A " ~ A ~ P c j , ( 1 5 )

w h e r e i = l , 2 . . . . . m a n d r = l , 2 . . . .

S u b s t i t u t i n g f r o m e q u a t i o n { 10 } w e g e ti nf - 1% ( n , + r ) = ~ . Y" a , , ( l l c r A ' . 4 ' Pc ~ n , > 0

t = l 1= 0i - 1 n~ - I

= Z Z a i ,( I )c r tAIA~PcJ n / = O .t = I /= o

(16}

S u b s t i t u t i n g a g a i n f r o m e q u a t i o n ( 1 5) in t o e q u a t i o n ( 1 6 ) w eg e t

= a . l l ) r o ( l + r ) n i > Ot = l I = O

i - 1 . , - ~ ( 1 7 )= ~ ~ a l t i l ) r , f l l + r ) n i = O .

t = l 1= o

F o r r = 1 , 2 . . . , K w h e r e K i s a l a r g e v a l u e a n d a s s u m i n g t h ev a l u e o f n ~ i s l~ , e q u a t i o n ( 1 7 ) c a n b e r e w r i t t e n i n a m a t r i xf o r m f o r t h e i t h s u b s y s t e m a s f o l lo w s :

w h e r er i (K ) = H t (K )0

r i (K ) = Ir l s ( l + 1 )ro( l i + 2 ) . . . rij(l i + K )I tOi = l a , ( 0 ) . . . a , ( n l - 1 ) . . . a i s (O) . . , a i s ( l i - 1 )1

Ht ,(K ) =r t j ( 1 ) " " r l j ( n l ) i !! r i i l l )

I r i s ( K ) . . . r l~ ( K + n t - l ) r i~ (K )D e f i n e t h e m a t r i x S ( l i ) a s f o l l o w s :

S(l~) = H r , ,(K )H, , (K ) .

( 1 8 )

"' " %(ll)i 1 ). . . r O(K + I -

{19)B e c a u s e o f t h e l i n e a r c o n s t r a i n t o f e q u a t i o n ( 17 ) w e h a v e

d e t S ( l , ) > 0 i f l ~ < n i ( 2 0 )= 0 l i > n iH e n c e t h e t r u e v a l u e o f n ~, i = 1 , 2 . . .. m c a n b e e s t i m a t e d o n eb y o n e a s f o l l o w s :E s t i m a t i o n r u l eS t e p 1 . S e t i = I a n d c o n s t r u c t t h e m a t r i x S ( l ~ ) o fd i m e n s i o n l ~ w h e r e n I i s a s s u m e d t h e v a l u e 1 1. I n c r e a s e t h ev a l u e o f I t ( I 1 = 1 , 2 . . . ) a n d p l o t d e t S ( l 1 ) v e r s u s 1 1 u n t i l d e tS ( I * ) = 0 i n w h i c h c a s e n 1 i s f o u n d t o b e n 1 = l * - 1 .S t e p 2 . S e t i = 2 c o n s t r u c t t h e m a t r i x S(12) o f d i m e n s i o n n~+ l 2 u s i n g t h e v a l u e o f n 1 e s t i m a t e d i n s t e p 1 a n d a s s u m i n g n 2= l 2 . In c r e a s e t h e v a l u e o f 12(12= 1 , 2 . . . ) a n d p l o t d e t S ( l 2 )v e r s u s n 2 u n t il d e t S ( I * ) = 0 , i n w h i c h c a s e n 2 is f o u n d t o b en 2 = 1~ - I.S t e p 3 . S e t i = 3 , . . . , m a n d r e p e a t s t e p 2 f o r e a c h v a l u e o f i.W e c a n n o t i c e t h a t t h e s t r u c t u r a l i d e n t if i c a t io n t e s t c a n b ec a r r i e d o u t d i r e c t l y o n t h e v e c t o r s o f t h e m a t r i x H ~ , (K ) b yc h e c k i n g i ts r a n k . B u t i t i s n o t e f f i ci e n t t o d o s o b e c a u s e o ft h e l a r g e a m o u n t o f s t o r a g e n e c e s s a r y , a n d t h i s w i ll b e v e r ys l o w o n a c o m p u t e r . I n s t e a d t h e t e s t is d o n e o n t h e m a t r i xS(l~) w h i c h h a s d i m e n s i o n s m u c h l e ss th a n H t (K ) .T h e c o r r e l a ti o n f u n c t i o n m a t r i x R ( a ) u s e d ' i n e q u a t i o n ( 18 )a n d d e f i n e d i n e q u a t i o n ( 12 ) c a n b e c o m p u t e d r e c u r s i v e l y b yt h e f o ll o w i n g a l g o r i t h m ( S i n h a , M a h a l a n a b i s a n d E I - Sh e r ie f ,1 9 7 8 )

1t~k+, i~l = ,~ d~ )- ~+ f I ,~kl~)- .vlk + ~)fl ' ik) I 1211w h e r e / ~ k ~ d a ) i s t h e e s t i m a t e o f R ( a ) o b t a i n e d f r o m k i n p u t -o u t p u t s a m p l e s .W e m a y a d d t h a t i n p r a c ti c e d e t S ( l ~ ) f o r I~>n~ m a y n o t b ee q u a l t o z e r o a s s h o w n i n e q u a t i o n ( 2 0) b u t t o ~ w h i c h i s as m a l l e r r o r . I n t h i s c a s e d e t S ( l ~ ) w i ll be p l o t t e d ve r s u s l~ u n t i ld e t S(1")

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4 . P a r a m e t r i c id en t ! f i ca t io n o f th e "A" m a t r i xA f t e r d e t e r m i n i n g t h e s t r u c t u r e p a r a m e t e r s n~ , i = 1 , 2 . . . . mt h e p a r a m e t e r s o f t h e A m a t r i x c a n b e e s t i m a t e d f r o m t h ec o r r e l a t i o n f u n c t i o n s a s f o l lo w s .F o r a l a r g e v a lu e o f K a n d a s s u m i n g t h e s t r u c t u r ep a r a m e t e r s n ~ , i = 1 , 2 . . . . . m a r e k n o w n , a n e s t i m a t e o f th ep a r a m e t e r v e c t o r 0~ ( e q u a t i o n 1 8 ) c a n b e o b t a i n e d b y t h ef o l l o w i n g l e a s t s q u a r e s f o r m u l a

O , ( K ) = [ H ~ ( K ) H . ( K } ] - t H ~ I ( K ) r ( K ) , (2 2 )w h e r e O d K ) i s th e e s t i m a t e o f 0 i f r o m K c o r r e l a t i o n f u n c t i o n s .A r e c u r s i v e v e r s i o n o f e q u a t i o n ( 22 ) f o r e s t i m a t i n g O AK ) i sg i v e n b y

Q~IK )hAK + l ) lro(K + 1 ) - h~ (K + l )O~(K !/~,(K + I ) = O,(K 1+ 1 + hT (K + 1 )Q~(K )hi(K + I )

Q , ( K + I ) = Q ~ I K I - Q ' ( K ) h A K + I k' -" [O IK ~ I KI +i. 1 )] 71 + J ' 7 ( K + 1 ) Q i ( K ) t l I ( K + I )

w h e r eH , ( K )H , ( K + l ) = h r l K + 1 )

a n d

123)

) ' i (k + n ~)= Y T ( k } q ~ i + v i t k + n l ) ,

h~(K + 1 ) = l r o ( K + l ) . . . r i ; iK + n t ) . . . r O ( K + l ) . . . r , j (K + n~)[r .5 . Pa r a m e t r ic" id en t i f i ca t io n o f t h e "B" m a t r i xA s s u m i n g t h a t t h e i n i t i a l s t a t e s o f e q u a t i o n ( 7) a r e ze r o s .T h e n f r o m e q u a t i o n ( 7) t h e i t h o b s e r v e d o u t p u t o f t h e s y s t e ma t t h e k t h s a m p l i n g i n s t a n t i s g i v e n b y

y i ( k ) = ~ C A ~ - I B u ( j - 1 } + v ~ ( k ) , (2 4 )j = l

B e c a u s e o f t h e s p e c ia l s t r u c t u r e o f t h e m a t r i c e s A a n d C( e q u a t i o n s 8 a n d 9 ) , e q u a t i o n ( 2 4 ) c a n b e r e d u c e d t o

yi(nl) ~ b ( j ) u (j - - I ) v~(n , (25)j= 1w h e r e t h e m a t r i x B h a s b e e n p a r t i t i o n e d a s f o l l o w s :

B = t b ~ ( l ) b ~ ( 2 ) . . . b ~ O h ) b 2 ( 1 ) . . . b = ( n = ) [ L (2 6)I t m a y b e n o t i c e d t h a t t h e p a r a m e t e r s b~(j) 's r e p r e s e n t t h eM a r k o v p a r a m e t e r s o f th e s y s te m .F o r k s a m p l e s e q u a t i o n ( 25 ) c a n b e r e w r i t te n i n a v e c t o rf o r m a s

(27)

"T _

u'J

w h e r eY i ( k ) = t u r ( k ) u T ( k + l ) . . . u T ( k + n 1 - 1 ) I T

(k~ = IbT(1 )bT (2) .. . bT(ni)l r .T h e p a r a m e t e r v e c t o r ~b~ o f e q u a t l o n ( 27 ) c a n b e e s t i m a t e dr e c u r s i v e l y b y t h e f o l l o w i n g n o r m a l i z e d s t o c h a s t i ca p p r o x i m a t i o n a l g o r i t h m ( E I -S h e r i ef a n d S i n h a , 1 9 7 9 e) :

Y [ ( k ) r49 ~(k + l ) = 6 i l k ) + v A k ~ l y A k + n ~ ) - Y ~ ( k ) ~ ( k )l(2 8 )

w h e r e ( ~ ( k ) i s t h e e s t i m a t e o f qS~ a t t h e k t h s a m p l i n g i ns ta ~ nta n d t h e s e q u e n c e s v i ( k ) s a t i s f y D v o r e t z k y ' s c o n d i t i o n s( D v o r e t z k y , 1 9 5 6) . W e c a n s e e f r o m e q u a t i o n ( 27 ) th a t t h ee r r o r t e r m r A k + , i ) i s u n c o r r e l a t e d w i t h t h e f o rc i n g f u n c t i o n} ' , rl k i : h e n c e i t c a n b e p r o v e d t h a t t h e e s t i m a t e o f t h ep a r a m e t e r v e c t o r o f e q u a t i o n ( 28 ! is c o n s i s t e n t a n d u n b i a s e d(E I - S he r i e f an d S in ha . 1 9 7 9 1 ).

6 . S im u la t io n r e s u l t sI n t h i s s e c t io n t h e a p p l i c a t i o n o f t h e p r o p o s e d i d e n t i fi c a t i o na l g o r i t h m t o a s i m u l a t e d f o u r t h o r d e r 2 - i n p u t 2 - o u t p u t s y s t e mi s p r e s e n t e d . T h e s y s t e m i s d e s c r i b e d b y t h e f o l l o w i n ge q u a t i o n s :

x ( k + l l :0 I 0 (l

- 0. I 0.65 0 00 0 0 I0 .6 7 1 . 6 7 - 0 . 2 5 l

0 20.25 0.8x ( k ) + u ( k )0 0I I

1 0 0 0v ( k ) = 0 0 x ( k t + v ( k ). 0 lT h e a b o v e s y s t e m w a s s i m u l a t e d w i t h z e r o i n i ti a l s t a t es a n d

t h e i n p u t v e c t o r u ( k ) w a s t a k e n a s a z e r o - m e a n u n c o r r e l a t e ds e q u e n c e w i t h u n i t v a r i a n c e . T h e n o i s e v e c t o r v ( k ) w a s t a k e na s a z e r o - m e a n w h i t e n o i s e w i th s t a n d a r d d e v i a t i o n o f 0 .1 .F i r s t t h e c o r r e l a t i o n f u n c t i o n s w e r e c a l c u l a t e d f r o m 1 0 00s a m p l e s o f th e o u t p u t s e q u e n c e T h e n d e t S(I~) w a s e s t i m a t e df o r i = 1 , 2 f r o m 1 2 c o r r e l a t i o n f u n c t i o n s a n d i s p l o t t e d i n F i g .1 . W e c a n s e e f r o m F i g . 1 t h a t t h e v a l u e s o f n l = 2 a n d n 2 = 2 .

O I

15

I0

5

2

I

0 5

1"12

Brief Paper 543

o.ol L I 1I 2 3L

F I G . 1 . S t r u c t u r e d e t e r m i n a t i o n t e s t f o r s u b s y s t e m s 1 a n d 2 .

A f t e r e s t i m a t i n g t h e s t r u c t u r e p a r a m e t e r s o f t h e s y s t e m t h ep a r a m e t e r s o f th e A m a t r i x w e r e e s t i m a t e d f r o m t h ec a l c u l a t e d c o r r e l a t i o n f u n c t i o n s u s i n g t h e m e t h o d o f s e c ti o n 4 .T h e f i na l e s t i m a t e o f m a t r i x A a f t e r 4 0 0 i t e r a t io n s i s o b t a i n e da s

,~=0 1 0 0

- 0 . 1 6 7 0 . 78 0 0 00 0 0 1

- 0 . 5 3 0 0 . 6 70 - 0 . 3 9 9 1 .1 10

F i n a l l y t h e p a r a m e t e r s o f t h e B m a t r i x w e r e e s t i m a t e d u s i n gt h e m e t h o d o f s e c t i o n 5 a n d t h e f i n a l e s t i m a t e f r o m 5 0 0

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544 Brief Papersamples of the input--output data is obtained as

0.009 1.988O.259 0.796B = 0.007 0.0370.953 0.966

We can add that the estimate of the matrices A and B can beimproved by increasing the number of iterations.7. ConclusionIn this paper an algorithm has been developed for bothstructure determination and parameter identification of lineardiscrete-time multivariable systems from noisy data. Theproposed algorithm is similar to Tse and Weinert's algorithmbut with the following improvements. First, the algorithm hasbeen extended to handle the case where the control signalexists. Second, the structure para mete rs of the system wereestimated from more correlation functions which gives a goodestimate. Finally, a recursive least squares was used toestimate the parameters of the A matrix from morecorrelation functions which improves the obtained estimate.The proposed algorithm was applied to a simulated fourthorder 2-input 2-output system. The results of simulationindicate that the proposed structure and parameter estimationalgorithm works quite well. When the order of the system isincreased this will not create any computational difficulty asthe identification process is done separately on the msubsystems.

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