Identification of time-varying linear and bilinear systems via Fourier series

Computers Elect. EngngVol. 17, No. 4, pp. 237-2,14, 1991 Printed in Great Britain. All rights reserved

0045-7906/91 $3.00+ 0.00 Copyright © 1991 Pergamon Press plc

IDENTIFICATION OF TIME-VARYING LINEAR AND BILINEAR SYSTEMS VIA FOURIER SERIES MOHSEN RAZZAGHI 1 a n d Sm~rr'ENG D. LIN2't ~Department of Mathematics and Statistics, Mississippi State University, MS 39762 and 2Department of Electrical Engineering, Mississippi State University, MS 39762, U.S.A.

(Received 5 December 1990; accepted in final revisedform 22 May 1991) A method for identification of time-varying linear systems and bilinear systems is proposed. The method is based upon expanding various time functions in the system as their truncated Fourier series, using the operational matrices for integration and product and hence reducing the problem into a set of algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples. Abstract

1. I N T R O D U C T I O N Recently, orthogonal functions and polynomial series have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to that of solving systems of algebraic equations thus greatly simplifying the problem and making it computationally plausible. The parameter identification of a dynamic system occurs in many diverse fields [1] and deals with the problem of determining a suitable internal structure for a system from its input-output data. Using Walsh functions, the parameter identification of a linear time-invariant system was first studied by Chen and Hsiao in 1975 [2]. Subsequently, the set of Walsh functions and block-pulse functions have been applied to parameter identification of time-invariant linear (or bilinear) systems [3-5]. Due to the nature of these functions, the solutions obtained were piecewise constant. Hwang and Shih [6] used Laguerre polynomials, Chang and Wang [7] applied Legendre polynomials, and Paraskevopoulos [8] took a Chebyshev series approach to derive continuous outcomes. Furthermore, parameter identification of time-varying linear (or bilinear) systems were then treated using shifted Legendre polynomials [9], shifted Chebyshev polynomials [10,11] and Taylor series [12]. In the present paper Fourier series is developed for parameter identification of linear time-varying systems (and bilinear systems). The method consists of reducing the identification problems into a set of linear algebraic equations by first expanding the candidate functions as a Fourier series with unknown coefficients. The operational matrices for integration and product are used to evaluate the integrals. This approach has advantages due mainly to the use of sinusoidal functions since they are widely used in applied mathematics and engineering fields and their properties are well known. Furthermore, due to the integral properties of the sine and cosine functions, the approximations involved in the operational matrix of integration are more reliable compared with other orthogonal functions. The method is suitable for digital computation, and illustrative examples are given to demonstrate the applicability of the proposed method. 2. P R O P E R T I E S

OF F O U R I E R

SERIES

A functionf(t) defined over the interval 0 to L may be expanded into a Fourier series as follows:

[ [2nnt) .[2nnt,-]

tCurrently, Researcher, Advanced Technology Center, Computer and Communication Research Laboratories, Industrial Technology Research Institute, Taiwan, R.O. China. ¢AEE17/*--A

237

238

MOHSEN RAZZAGH1and SHINFENGD. LIN

where the Fourier coefficients ai and a* are given by

ao = ~

f ( t ) dt

a. = ~

f(t)cos T

2;

(2a)

(2nnt)

a* = ~

f(t)sin T

dt n = 1,2,3 . . . .

(2b)

dt n = 1,2,3 . . . . .

(2c)

The series in equation (1) has an infinite number of terms. T o obtain an approximate expression for f ( t ) , we truncate the series at the (2r + 1)th term as

f ( t ) = aodPo(t) + ~ [a,~b,(/) + a*~b*(t)] = ~rq~(/)

(3)

tl=l

where the Fourier series coefficient vector c¢ and the Fourier series vector ~b(t) are defined as = [a0, al, a: . . . . . at, a*, a* . . . . . a*] r

(4)

¢ ( / ) = [~b0(t), ¢,(/), q~2(t) . . . . ¢,(t), q~* (t), ~b*(t) . . . . . ¢ , ( / ) ] r

(5)

with

{2nrct'~

¢ , ( t ) = cos~--~---},

• ['2nnt~

~b*(t)=s,n~---~-),

n = 0 , 1,2,3 . . . . . r n=l,Z,

3,...,r

the elements of tb(t) are orthogonal in the interval te(O,L). By integrating the elements of the vector ¢ ( t ) in equation (5) from t = 0 to t and approximating t by a truncated Fourier series, we get

~ dp(t') dt" ~ edp(t)

(6)

where P is the (2r + 1) × (2r + 1) operational matrix for integration and is given by [13,14] as

P=L 1 1

1

--1

--1 2n

"'"

-1 (r -- 1)n

-1 r~z

0

...

0

0

0

0

...

0

0

0

0

...

0

o

0

0

...

0

0

0

...

0

0

0

0

0

...

0

-1 2n

0

0

0

0

...

0

0

-1 4n

0

0

0

...

0

0

...

0

o

-1

7Z

1

2(r - t)n

0 1 2rn

(7)

Time-varyinglinear and bilinear systems

239

Furthermore, the product of ~b(t)~b r(t) with the vector 0t can be approximated by the procedure given by [15] as

~(t)¢~ r(t)~t ~, ~p(t)

(8)

where 8 is the product operational matrix for the vector ~t and is obtained as

al

a2

ar

a*

1 ao + ~ a2

1 ~ (al + a3)

at- 1

~a2

ao

1

1

1

=

1

~(aj +a3)

a,

~a,_,

,1

~a*

~(a*+a*)

--~a*-i

~(a*+a*)

~a*

I

1

1

1

~aLl

~aLl

1

1

1

1

-~a*-2

...

1

...

.

...

1 , -~a,_ z

la.

2

0

...

~a,_,

...

~ar-2

...

ao

1

-~(at--a3)

1 ~(a,-a3)

ao-~a,

1

.

~ar-l

...

1

ao--~a2

0

a* 1

~(at +a~') . . . 1

1 , - ~a,_,

ao

1

.

(a3 ,

1

5ar_2

1

~a*

ar- 2

ao+~a4

1

~a*

1

1

a2

1

I

a*

1

1

1 a,_2

(9) 3. P A R A M E T E R

IDENTIFICATION

OF TIME-VARYING

SYSTEMS

Consider a linear time-varying systems

~(t) = A(t)x(t) + B(t)u(t)

(10)

with a known initial condition x(0), and where x(t) is the n x 1 state vector, u(t) is the m x 1 control vector, and A(t) and B(t) are time-varying n x n and n x m matrices respectively. Assume that all elements of x(t), u(t), A (t) and B(t) are absolutely integrable in the interval [0, L]. Let

~,(t) = I.®dp(t)

(1 la)

~ ( t ) = I~,®d~(t)

(1 lb)

where ® stands for kronecker product [16], I. and Im are n and m dimensional identity matrices, ~,(t) and ~m(t) are (2r + 1)n x n and (2r + 1)m x m matrices respectively. We can approximate x(t) and x(0) by their truncated Fourier series as follows:

x(t) = Xo, xl . . . . , xr, x* . . . . . x*]~b(t)

(12)

x(0) = Ix(0), 0 . . . . . 0, 0 . . . . ,0]~b (t)

(13)

where xi, xj* (0 ~
(14)

u(t)

(15)

similarly =

T T q~m(t)[Uo, Ur . . . . , Ur, U*r,

..

., U'r] r.

240

MOnSEN RAZZAGFU and SrnN~r~oD. LIN

Furthermore, A (t) and

B(t) can be expressed as A ( t ) = [Ao, AI . . . . . A,, A* . . . . , A* ]q~.(t)

(16)

B(t) = [B0, Bl . . . . . Br, B* . . . . . B*lq~,,(t)

(17)

where Ai, A*, Bi and B*(0 ~
.. ,A*]:p,(t)~r,(t)[xr, x ( . . . . , x f , x *r . . . . . x*Z] r

(18)

noting that, similar to equation (8), the above equation can be approximated as A ( t ) x ( t ) = [A0, A~ . . . . . At, A * , . . . , A* ,]X4~ (t).

(19)

B(t)u(t) = [B0, B, . . . . . B,, B* . . . . , B*]0q~(t)

(20)

Also,

where .~and Uare (2r + 1)n × (2r + 1) and (2r + 1)m × (2r + 1) matrices respectively. Integrating equation (10) from t = 0 to t, using equations (12), (13), (19) and (20) together with equation (6) we obtain

[x0, x ,

.....

X,*]-- [X(0),0 . . . . .

Xr, X~' . . . . .

0, 0 . . . .

,0]

= [A0, A1 . . . . ,Ar, A ~ , . . . , A * ] . Y P + [ B o , BI . . . . , B , , B * . . . . . B,*]UP.

(El)

The parameter identification problem is to determine the unknown elements in the matrices A (t) and B(t) from equation (10) provided x(0) and input-output data are given. Now from equation (21) it can be seen that more signals must be used to obtain the parameters to be identified. Let uv(t) be the test signal and xv(t) be the response of the system to u~(t) where v = 1,2,3 . . . . , q; q >t (n + m). Then from equation (21) we get (22)

FO = G

where f = [i,®(,~p)r, i , ® ( ~ p ) r ] o = [ a o ~, , ' ' '

, a o~., A T, ,

.

.

, A. .,~,..

.

,

a *T ,

"''~

Br

(23)

a r*T n ,

,...,BOrn

r

,

B r,

...~

Br,

...,

B,r,

...,

B~r] r

(24)

and c

= [x~,

xf,

.

.• , x.o ~ , .

. ,. ~,,

x*T

,

T xv,* T ]T - [x~0(0), 0 r,

. . . .0 r, . . 0 .r,

,0r] r.

(25)

Note that, F, 0, and G are matrices of dimensions of n ( 2 r + l ) x n ( 2 r + l ) ( n + m ) , n(2r + 1)(n + m) x 1, and n(2r + 1) x 1 respectively and A0; denotes the ith column of the A0 matrix. Based on least-squares method, the parameters are obtained as (26)

0 = (F r F ) - l F r G

where F r F is assumed to be nonsingular matrix• 4. I D E N T I F I C A T I O N

OF BILINEAR

SYSTEMS

Consider bilinear systems :c(t) = A ( t ) x ( t ) + B(t)u(t) + ~ Cj(t)x(t)u,(t) j=l

x(0)

specified

(27)

Time-varying linear and bilinear systems

241

where Cj(t) is n x n matrix and u j ( t ) , j = 1, 2 , . . . , m are the elements of u(t). We now expand Cj(t) and uj(t) in their truncated Fourier series expansions as follows: Cj(t) = [CjoCj, .. . Cj, CTf .. . C * ] $ . ( t )

(28)

uj( t ) = dpr( t )[UjoUj, . . . uj, u~ . . . u* ]r

(29)

and hence, we have C j ( t ) x ( t ) u j ( t ) = ~ [CjoCjj . . . Cj, C}* . . . C * ] ) ~ b ( t ) j=[

(30)

j=l

where U/can be calculated similarly to equation (9). The parameter identification problem for the bilinear system [equation (27)] is to determine the unknown elements in A ( t ) , B ( t ) , and Cy(t) provided x(0) and input-output data are given. Let uy(t) be the test signal and Xy(t) the response of the system t o uy(t), where y = 1 , 2 , . . . , k; k > (n + m + nm). Then, similar to equation (22), we obtain (31)

FtOl = GI

where FI, 01, and Gi are matrices of dimensions of n(2r + 1) x n(2r + 1)(n + m + n m ) , n(2r + 1)(n + m + n m ) x 1 and n(2r + 1) x 1 respectively and can be obtained similar to equations (23)-(25). Further, 01 can be found similar to equation (26).

5. I L L U S T R A T I V E

EXAMPLES

5.1. E x a m p l e I

Consider the linear system with time-varying parameters (32)

g(t) = a(t)x(t) + b(t)u(t)

with initial condition x ( 0 ) = 0. Using truncated Fourier series, it is required to find 2(2r + 1) components for a ( t ) and b(t). Three different inputs, l(xo = - 1 ) , 2 - t ( x 0 = 0), and -t(Xo = 2) are used to generate 3(2r + 1) equations to determine 2(2r + 1) unknowns. The exact values for a ( t ) and b ( t ) are a(t)= l

and

b ( t ) = t.

B y using the method in Section 3, the parameters a ( t ) and b ( t ) are calculated. The computational results together with the exact solution for a ( t ) for r = 10 is given in Fig. 1 and those for b ( t ) for r = 6 and r = 10 are presented in Figs 2 and 3 respectively. Inspecting these results, we see that as r increases, the approximation error decreases. 3

-

0

I 0.2

I 0.4

I 0.6

Fig. 1. The exact and approximate solutions for

I 0.8

a(t)

I 1.0

of example I with r = 10.

I 1.2

242

MOltSEN RAZZAGHIand SHINFENGD. LIN 3 --

0

-I

1 0,2

I 0.4

I 0.6

I o.e

I 1.0

I 1.2

Fig. 2. The exact and approximate solutions for b ( t ) of example 1 with r = 6.

I 0.2

-1

I 0.4

I 0,6

1 0.8

1.O

I 1.2

Fig. 3. The exact and approximate solutions for b ( t ) of example 1 with r = 10.

5.2. Example 2 Consider a bilinear system

Yc(t) = a(t)x(t) + b(t)u(t) + c(t)x(t)u(t) with the initial condition x ( 0 ) = Xo. Four different inputs, 2t + 2(xo = 0), 2t 2 + 2t - l(x0 = 1), t: + 2t -~(Xo = 2) and 2t + l(xo = 3) are used to generate 4(2r + 1) equations to determine 3(2r + 1) unknowns. 3

-

2

-

0

-1 0

0.2

0.4

0.6

0.8

1.0

Fig. 4. The exact and approximate solutions for b ( t ) of example 2 with r = 8.

I 1.2

Time-varying linear and bilinear systems

243

2 --

0

I

0

0.2

I

I

I

I

I

0.4

0.6

0.8

1.0

1.2

%

-2

-3

L

Fig. 5. The exact and approximate solutions for c(t) of example 2 with r = 8.

T h e exact values are a ( t ) = 2t, b ( t ) = t, c ( t ) = - 1 .

By using the m e t h o d in Section 4, the p a r a m e t e r s for r = 8, are calculated. The c o m p u t a t i o n a l results for a ( t ) are nearly the same as twice those o f b ( t ) . The results for b ( t ) t o g e t h e r with the exact s o l u t i o n are given in Fig. 4 a n d those for c ( t ) are presented in Fig. 5.

6. C O N C L U S I O N S In this p a p e r the o p e r a t i o n a l p r o p e r t i e s o f F o u r i e r series together with the least-squares m e t h o d are used to estimate the u n k n o w n p a r a m e t e r s o f t i m e - v a r y i n g linear systems a n d bilinear systems. T h e i n t e g r a t i o n s are p e r f o r m e d via the o p e r a t i o n a l m a t r i x o f i n t e g r a t i o n P a n d the p r o d u c t o p e r a t i o n a l m a t r i x 8. The a d v a n t a g e o f using the F o u r i e r series as c o m p a r e d to o t h e r o r t h o g o n a l functions is t h a t in the present m e t h o d the a p p r o x i m a t i o n i n t r o d u c e d in the o p e r a t i o n a l m a t r i x o f i n t e g r a t i o n is derived only f r o m the t r u n c a t i o n in the first row o f the m a t r i x P i n t r o d u c e d in e q u a t i o n (6). This c o n s i d e r a b l y reduces the overall a p p r o x i m a t i o n error. F u r t h e r m o r e the m a t r i x P c o n t a i n s m a n y zero elements, a fact m a k i n g the m e t h o d c o m p u t a t i o n a l l y appealing. E x a m p l e s with satisfactory results are used to d e m o n s t r a t e the a p p l i c a t i o n s o f this m e t h o d . REFERENCES 1. R. Kallia and K. Spingarn, Control Identification and Input Optimization. Plenum Press, New York (1982). 2. C. F. Chen and C. H. Hsiao, Time-domain synthesis via Walsh functions. Proc. IEE 122, 565-570 (1975). 3. G. P. Rao and K. R. Palanismy, Improved algorithms for parameter identification in continuous system via Walsh function. Proc. IEE 130, 9-16 (1983). 4. W. L. Chen and N. S. Hsu, Parameter estimation of bilinear systems via Walsh functions. J. Franklin Inst. 305, 249-257 (1978). 5. B. Cheng and N. S. Hsu, Analysis and parameter estimation of bilinear systems via block-pulse functions. Int. J. Control 36, 53-65 (1982). 6. C. Hwang and Y. P. Shih, Parameter identification via Laguerre polynomials. Int. J. Syst. Sci. 13, 209-217 (1982). 7. R, Y. Chang and M. L. Wang, Parameter identification via shifted Legendre polynomials. Int. J. Syst. Sci. 13, 1125-1135 (1982). 8. P. N. Paraskevopoulos, Chebyshev series approach to system identification analysis and optimal control. J. Franklin Inst. 316, 135-157 (1983). 9. C. Hwang and T. Y, Guo, Parameter identification of a class of time-varying systems via orthogonal shifted Legendre polynomials. J. Franklin Inst. 318, 59-69 (1984). I0. Y. H. Chou and I. R. Horng, Parameter identification of lumped time-varying systems via shifted Chebyshev series. Int. J. Syst.Sci. 17, 459-464 (1986). 11. Y. H. Chou and I. R. Homg, Shifted Chebyshev-series analysis and identification of time-varying bilinear systems. Int. J. Control 43, 129-137 (1986). 12. H. Y. Chung and Y. Y. Sun, Parameter estimation of bilinear systems using Taylor operational matrices. IEEE Trans. Syst. Man. Cybern. 17, 1068-I071 (1987). 13. P. N. Paraskevopoulos, P. D. Sparis and S. G. Mouroutsos, The Fourier series operational matrix of integration. Int. J. Syst. Sci. 16, 171-176 (1985). 14. M. Razzaghi and M. Razzaghi, Fourier series direct method for variational problems. Int. J. Control48, 887-895 (1988).

244

MOHSE~ RAZZAOm and Sm~'ENO D. LIN

15. M. Razzaghi and A. Arabshahi, Analysis of linear time-varying systems and bilinearsystems via Fourier series.Int. J. Control 50, 889-898 (1989). 16. P. Lancaster, Theory of Matrices. Academic Press, New York (1969). AUTHORS' BIOGRAPHIES

Mohsen Razzaghi--Mohsen Razzaghi was born in Tehran, Iran, in 1944. He received the B.Sc. degree in Mathematics from the University of Sussex, England in 1968 and the M. Math. degree in Applied Mathematics from the University of Waterloo, Canada in 1969. He then reattended Sussex University and received the Ph.D. degree in 1972. He is currently a professor in the Department of Mathematics and Statistics at Mississippi State University, U.S.A. Dr Razzaghi has taught in the Department of Mathematics and Statistics at Shiraz University, Iran, and in the Department of Mathematics at Amir Kabir University, Tehran, Iran. From 1984 to 1986 he held visiting positions at the University of Kentucky and Auburn University, U.S.A. He has research interests and publications in the areas of solutions of the matrix Riccati equation, necessary and sufficient conditions in optimal control and applications of orthogonal functions in dynamic systems.

Shinfeng (David) Lin---Shinfeng (David) Lin was born in 1958 in Hualien, Taiwan, Republic of China. He received his B.S. degree in Automatic Control Engineering from Feng Chia University in 1980, and M.E. and Ph.D. degrees in Electrical Engineering from Mississippi State University in 1985 and 1991, respectively. From 1985 to 1987, he worked in Telecommunication Laboratory, Ministry of Communications in Taiwan. His interests include the digital signal processing and applications of orthogonal functions in dynamic systems.