Identification of nonlinear differential equations via Fourier series operational matrix for repeated integration

N O R ~ - ~

Identification of Nonlinear Differential Equations via Fourier Series Operational Matrix for Repeated Integration M. Razzaghi

Department of Mathematics and Statistics Mississippi State University Mississippi State, Mississippi 39762 A. Arabshahi

Computational Fluid Dynamic Laboratory NSF-ERC for Computational Field Simulation Mississippi State, MS 39762 and S. D. Lin

Computer Center, National Dong-Hwa University Jyh-Shyue Tsuen, Show-Fing Shiang, Hualien Taiwan, Republic of China Transmitted by Melvin Scott

ABSTRACT A method for the parameter identification of nonlinear differential equations using Fourier series is discussed. Operational matrix of integration, as well as one shot operational matrix for repeated integration is used to reduce the identification problem to that of solving systems of algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples.

1.

INTRODUCTION

Recently, orthogonal functions and polynomial series have received considerable attention in dealing with various control problems. The main

APPLIED MATHEMATICSAND COMPUTATION68:189-198 (1995) ¢~) Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 1O010

0096-3003/95/$9.50 SSDI 0096-3003(94)00093-J

190

M. RAZZAGHI,A. ARABSHAHI,AND S. D. LIN

characteristic of this technique is that of reducing these problems to that of solving a system of algebraic equations thus greatly simplifying the problem and making it computationally plausible. The approach is based on converting the underlying differential equations into integral equations through integration, approximating various signals involved in the equation by truncated orthogonal series and using the operational matrix or integration P to eliminate the integral operations. Clearly, the form of P depends on the particular choice for the orthogonal functions. Special attention has been given to applications of Walsh functions [1], block pulse functions [2], Laguerre polynomials [3], Legendre series [4], Chebyshev polynomials [5], and Fourier series [6]. The numerical methods for the identification of nonlinear systems has been presented, among others, by [7, 8]. Sinha et al. [7] applied Walsh functions and Hsu and Cheng [8] introduced block pulse functions for the identification problems. Due to the nature of these functions, the approximate outcomes were piecewise constants. Furthermore, Zaman and Jha [9] employed Laguerre polynomials, Chou and Horng [10] used Chebyshev polynomials and Horng and Chou [11] applied Legendre polynomials to derive continuous outcomes for the identification of a nonlinear system. In the present paper the Fourier series is used for the identification of nonlinear first-order and second-order differential equations. 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 matrix for integration, and one shot operational matrix for repeated integration (OSOMRI) are then 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. The method is suitable for digital computation, and illustrative examples are given to demonstrate the applicability of the proposed method. In the following section, the properties of the Fourier series are briefly discussed.

2. P.1.

PROPERTIES OF FOURIER SERIES Fourier series

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

Parameter Identification

191

where the Fourier coefficients ai and a~ are given by

(2a)

ao = -~

f(t)dt

an :

-~

f ( t ) cos

dt

n = 1,2,3,...

(2b)

a n = -~

f ( t ) sin

dt

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

(2c)

.

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

f ( t ) ~ ao¢0(t) ~- y~[anCn(t) + a~¢*(t)] = ~T¢(t),

(3)

n~l

where the Fourier series coefficient vector a and the Fourier series vector ¢(t) are defined as c~ = [a0, al, a 2 , . . . , at, aa, a 2 , . . . , a*] T

¢(t) =

[¢0(~),¢l(t),¢~(~),...,¢~(t),¢~(t),¢~(t),...,¢;(~)]

(4)

r

(5)

with ( ~ n ----- COS

- -

n = 0,1,2,3,... n = 1,2,3, . . . .

The elements of ¢(t) are orthogonal in the interval t E (0, L).

2.2.

Operational mat~q~ of integration

B y integrating the elements of the vector ¢(t) from t = 0 to t and approximating t by a truncated Fourier series we get

f0 t ¢(v)dv ~- PC(t),

(6)

192

M. R A Z Z A G H I , A. A R A B S H A H I ,

A N D S. D. L I N

where P is the (2r + 1) x (2r + 1) operational matrix for integration and is given by [12]

0

0

0

...0

0

0

0

...0

0

:

i P=L

:

--I

--I

1

0

--I

"~-2-'-~'"-('7-~ '''

--I

-~

0

0

1 2-(;=i~

0

:

. o °

. o •

0

0

0

...0

0

0

0

'

0

0

0

...0

0

0

0

...

0

2~

1

-1

0

...0

0

0

0

...

0

0

1

0

0

0

0

-..

0

0

:

:

:

:

0

0

0

0

-~-~10 ... 0

:

(7)

. • o

1

0

0

... 0 2--~

.--

It is noted that the approximation introduced in (7) is arrived at only from the truncation of Fourier series for t in the first row and all other integrations involved are exact.

2. 3.

One shot operational matrix for repeated integration

Using (6), for k times repeated integration of ¢(t) in (5) we have

fo f • -.

¢ ( r ) d r k --~ Pk¢(t).

(8)

In order to reduce the error in approximating the repeated integration, we use the concept of one shot operational matrix for repeated integration (OSOMRI) originally introduced by Mohan and D a t t a [13]. According to this, one first integrates k times every element of ¢(t) between 0 and t and then expressing the result in truncated Fourier series• We denote this by

/0'/0 • ""

¢ ( r ) d r k -~ Pk¢(t),

(9)

where Pt¢ is the (2r + 1) × (2r + 1) OSOMRI whose elements can be derived using a procedure given in [13].

Parameter Identification

3.

3.1.

193

IDENTIFICATION OF THE NONLINEAR DIFFERENTIAL EQUATION

First-order system Consider a nonlinear differential equation with constant coefficients c2

~

.

dy(t)

. cl--~

(10)

= u(t)

in which u(t) and y(t) are, respectively, the input and output of the system, assumed known and Cl, c2 are the parameters of the system to be identified. Let the initial condition of (10) be y(0) --- bl

and

y2(0) = b~ = b2.

(11)

Even though bl and b2 are actually known from the output and input records, they will be treated as unknown as they may not represent the true values in noisy situation [13]. The identification problem is thus to determine the unknown parameters Cl,C2, bl, and b2 by using the inputoutput data. We first write y(0) and y2(0) as follows:

y(O) = blET ¢(t) y2(0) = b2ET ¢(t),

(12) (13)

where E is a (2r + 1) x 1 vector having its first element unity and all other elements zero. Integrating (10) with respect to t and using (12) and (13), we obtain c2y2(t) + cly(t) - (b2c2 + blcx)ET ¢(t) =

~0t u(t)dt.

(14)

We also approximate y2(t), y(t), and u(t) by a finite Fourier series as r

y~(t) = z0¢0(t) + ~E][z~¢~(t) + z;¢;(t)] = z r ¢ ( t )

(15)

i-----1 f

y(t) = yo¢o(t) + ~E][y,¢~(t) + y;¢;(tl] = Yr¢(t)

(16)

i=1 r

u(t) = uo¢0(t) + y~'[~,¢,(t) + ~;¢;(tll = gr¢(t). i=l

(17)

194

M. RAZZAGHI,A. ARABSHAHI, AND S. D. LIN

Substituting (15)-(17) in (14) and using (6) we obtain c2Z T q- e l y T - (b2c2 + blCl)E T = u T p.

(18)

Equation (18) can be written as FO = G,

(19)

where z0 y0 1 Zl Yl 0

F =

(20)

zryrO

~ y; o z;y~0 G = [(uTp)I(uTp)2""(uTp)2r+I] T

(21)

8 = [c2 Cl a~T

(22)

d = -(b2c2 + bzCl)

(23)

and with and ( u T p ) k , k -- 1 , . . . , 2r + 1, indicates the kth column of U T p . Applying the least-squares method, (19) yields the estimate 8 as 0 = (FTF)-IFTG

(24)

provided F T F is invertible. Using the values of c2,cl, and d in (23), the values for bl and b2 = b~ can be obtained. It is noted that (23) gives two roots for bl; once bl is obtained, the solution which is closer to the value of y(0) is the required solution and the other is discarded. 3.2.

Second-order system

Consider a nonlinear equation described by g d2y2(t)

dy

(25)

Parameter Identification

195

with initial conditions given by y(0) = hi,

y2(0) = h 2 = h2.

(26)

Integrating (25) twice with respect to t and approximating t by a truncated Fourier series as t ---- [tO,t l , . . . , tr, t ; , . . . , t*]¢(t) = TT¢(t)

(27)

we obtain similarly to (17)

g2Z T + g l y T p + dlT T + d2E T = uTp2,

(28)

where

dl = - h l g l

d2 = -h292

(29)

and P2 is the OSOMRI defined in (9). Equation (27) can also be written as F101 = G1,

(30)

where z0 Y0 to 1 zl Yl tl 0

F~

=

(31)

Zr Yr tr 0 z~ y~ t~ 0

zr Yr tr 0

G1 = [(uTp2)I(UTp2)2""(uTp2)2r+I] T

(32)

~1 = [g2 gl

(33)

dl

d2]T

and (uTp2)k, k = 1 , . . . , 2r + 1 indicates the kth column of uTp2. Based on least-squares method, the parameters are obtained as

O1 = (FT F1)-IFTG1,

(34)

196

M. RAZZAGHI, A. ARABSHAHI, AND S. D. LIN

TABLE 1. ESTIMATED AND EXACT VALUES FOR PROBLEM 1

Parameters c2 cx d

r ----10

r = 15

2.000061 0.999954 -0.260704

2.000014 0.999983 -0.271082

Exact 2 1 -.28

TABLE2. ESTIMATED AND EXACT VALUES FOR PROBLEM 2

Parameters 91 g2

r = 10

r = 15

Exact

1.018370 0.478638

1.001630 0.491508

1 1/2

w h e r e FTF1 is a s s u m e d t o b e n o n s i n g u l a r m a t r i x . B y using Pk in (9) t h i s m e t h o d c a n b e e x t e n d e d t o a n y o r d e r of n o n l i n e a r differential e q u a t i o n s .

4.

4.1.

ILLUSTRATIVE EXAMPLES

Example 1

C o n s i d e r t h e first-order differential e q u a t i o n a n d initial c o n d i t i o n s given b y (10) a n d (11). T h e p r o b l e m is t o e s t i m a t e t h e values of cl,c2 and d = - ( b 2 c 2 + blcl) for given values of y(t) a n d u(t). B y using t h e m e t h o d in Section 3, for y(t) = t + 0 . 2 a n d u(t) = 4 t + 1.8, t h e e s t i m a t e d results a r e found. T h e c o m p u t a t i o n a l results t o g e t h e r w i t h t h e e x a c t values for Cl, c2, a n d d w i t h r = 10 a n d r = 15 w i t h L = 1 a r e p r e s e n t e d in T a b l e 1.

4.2.

Example 2

C o n s i d e r t h e s e c o n d - o r d e r differential e q u a t i o n in (25) w i t h t h e i n i t i a l c o n d i t i o n s given by

y(o) = o, F o r y(t) -- s i n t a n d u(t) w i t h t h e e x a c t values for shown in T a b l e 2.

y2(o) = o.

(35)

= cos2t + cost, the estimated results together gl,g2 w i t h r -- 10 a n d r -- 15 w i t h L -- 1 a r e

Parameter Identification 5.

197

CONCLUSION

In this paper the operational properties of Fourier series together with the least-squares method are used to estimate the unknown parameters for first-order and second-order nonlinear differential equations. The integrations are performed via the operational matrix of integration P and (OSOMRI) Pk. This method can also be extended to any order of nonlinear differential equations. The matrices P and Pk contain many zero elements and hence making the method computationally attractive. Furthermore, due to the integral properties of the sine and cosine functions, the approximations involved in P and Pk are more reliable compared with other orthogonal functions. Examples with satisfactory results are used to demonstrate the application of this method.

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M. RAZZAGHI, A. ARABSHAHI, AND S. D. LIN

12 P. N. Paraskevopoulos, P. D. Sparis, and S. G. Mouroutsos, The Fourier series operational matrix of integration, Internat. J. Systems Sci. 16:171176 (1985). 13 B.M. Mohan and K. B. Katta, Identification via Fourier series for a class of lumped and distributed parameter systems, IEEE Trans. Circuits Systems 36(11):1254-1458 (1989).