Parameter identification of linear distributed systems via Taylor ...

413

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 35, NO. 3, AUGUST 1988

Parameter Identification of Linear Distributed Systems Via Taylor Operational Matrix HUNG-YUAN CHUNG

Absirucf-This paper gives a simple and effective approach to deal with problems in the parameter identification of linear distributed systems. Due to the simple structure of Taylor operational matrices, a computationally efficient algorithm for the parameter estimation of linear distributed systems is presented. Further, a numerical example is demonstrated. A satisfactory result is obtained.

Indexing terms-Identification,

AND

YORK-YIH SUN

An analytic function y = y ( t ) can be expanded about the point t = 0 into the Maclaurin form of the Taylor series as

t t2 y ( t )=y‘O)(O)+ - y(I)(O)+- y q o ) + . . . l! 2! tin

algorithm, control systems.

1

~

I

+-y@-1)(0)

(rn - l)!

(1)

1. INTRODUCTION

I

N RECENT studies, many reports have been devoted to the development of algebraic methods for the analysis, identification, and optimal observer design of control systems. The aim of these studies has been to obtain effective algorithms that are suitable for the digital computer. Their major effort has been concentrated on the methods of the orthogonal polynomials and functions. Typical examples are the application of Walsh functions [ 11, [2], block-pulse functions [3], [4],Laguerre polynomials [5], Legendre polynomials [6], and Chebyshev polynomials [7]. More recently, the Taylor polynomials have been used in the analysis and optimal control of the linear time-invariant system [8], as well as those of time-varying systems [9], but not extensively. Due to the simple structure of the Taylor operational matrix of integration introduced by Mouroutsos et al. [8], it leads to significant computational advantages for a wide variety of control problems. In many industrial applications, we often encounter the models of distributed parameter systems. However, it is difficult to identify or estimate the parameters of those systems. It is feasible to obtain the approximate solution by using this approach. In this presentation, this approach is extended for the linear time-invariant distributed systems. The result shows that the Taylor polynomial provides a higher computational efficiency and an excellent agreement with the true value as compared with other polynomial series. 11. TAYLOR POLYNOMIALS

using rn terms, withy@) = dny/dtn. Equation (1) may be written as a product form of two vectors, that is

Y ( t >= y ‘fct) where y- is the coefficient vector

(3) andf(t) is the function basis vector

The Taylor series basis functions satisfy the following recurrence relation: (5)

fn(f)=tfn-,(f).

Again, one can easily show that

The simplicity of (5) and (6) gives a computational matrix integration P - that has the following simple form [8].

In this section, some important background on the Taylor expansion of an analytic function is formulated for the development of the method. Manuscript received February 24, 1987; revised November 16, 1987. This work was supported by the National Science Council of the Republic of China. H.-Y. Chung is with the Department of Electrical Engineering, National Central University, Chung-Li 32054, Taiwan, Republic of China. Y.-Y. Sun is with the Department of Electrical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, Republic of China. IEEE Log Number 8821786.

(2)

0 0

Ol

0

(7)

0 Thus, the n-multiple integral of the function y ( t ) may be

0278-0046/88/0800-0413$01.00 O 1988 IEEE

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414

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 35. NO. 3, AUGUST 1988

Next, arranging all the functions as bivariate polynomials, one can have

approximated by using the algorithm

i' 1' -

y ( 7 ) (d7)" z g T f n-f ( t ) .

* * *

(8)

M-1 N - l

C r , r r ( t ) ~ , , ( x ) = ' L ( ~ ) C ~(14) ~J~(f)

Y(X, t)=

n times

J=o

In view of the simple structure of the operational matrix of integration f,the computation of the powers of f is very easy. This elegant operational property will be used in the next section for the simplifications of the problem.

l=O

M-I N-

rr,rI(t)t,(x)='L(~)~MN:N(f)

k- I

111. PARAMETER IDENTIFICATION

ii

i i y ( x , t ) dx dx+a4

+a3

1' 1' 0

k- I

drSL(x)CY;:I:N(t)(k

=

< N,

(16)

r=O

m- I

m- I

u(x)=

uitr(x)= r=O

1' 1' y ( x , t ) dt dx 0

drrr(t) i=O

Let us consider a linear time-invariant distributed parameter system characterized by the following second-order differential equation:

a5

(15)

,=o r=o

Y(0, t)=

Now, integrating (9) twice w. r. t. t and twice w. r. t. x, one obtains

I

O=

Y(X,

u,[L(x)C?C(lIN(t)(m