Automatica, Vol. 27, No. 2, pp. 409-412, 1991 Printed in Great Britain. 0005-1098/91 $3.00 + 0.00 Pergamon Press pie ~) 1991 International Federation of Automatic Control Brief Paper Linear Time-invariant Distributed Parameter System Identification via Orthogonal Functions* B. M. MOHANf and K. B. DATrA%$ Key Words--Parameter estimation; initial and boundary conditions estimation; distributed parameter systems; orthogonal functions. Abstract--This paper points out the mathematical inconsis- tencies traced in the literature on the identification problem of linear time-invariant distributed parameter systems via orthogonal functions, and proposes for the same problem a unified identification approach based on the concept of one shot operational matrix for repeated integration. It presents identifiability requirements for the block-pulse functions approach while suggesting a linear independence test for the full column rank of linear algebraic system arising out of the system model upon the application of orthogonal functions. Finally, it illustrates system identification with a numerical example. 1. Introduction THE AIM of this paper is to develop a general and unified identification approach using orthogonal functions (OF) for the estimation of parameters, initial conditions (ICs) and boundary conditions (BCs) of linear time-invariant single- input single-output continuous-time distributed parameter systems (DPS) modelled by a2y(x, t) a2y(x, t) . ay(x, t) au +ax~ ax 2 +a~t ax c3t ± at at +ax~+ay(x,t)=u(x,t ) (1) from its input-output records available over the region x e Ix0, xf], t e [to, tf]. Although some attempts have already been made on this problem in the past, they seem to have the following mathematical inconsistencies. As it appears, Paraskevopoulos and Bounas (1978) were the first authors to investigate this problem via Walsh functions (WF). To estimate the parameters with the ICs: g(x) = f(x) = y(x, to) , and the BCs: q(t) = y(xo, t), ay(x, t) r(t) = ~ .... ' they integrated (1) twice with respect to x and twice with respect to t to obtain an integral equation, approximated all the functions in x and/or t in finite Walsh series, and arrived at the following matrix algebraic equation from the integral equation by employing the integral operational property of WF. * Received 3 May 1989; 15 January 1990; received in final form 11 July 1990. The original version of this paper was not presented at any IFAC meeting. This paper was recom- mended for publication in revised form by Associate Editor R. Bitmead under the direction of Editor P. C. Parks. t Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721 302, India. ~tAuthor to whom all correspondence should be addressed. a.(ET)2y + a.~YE 2 + a~tETyE, + a,(ET)2yE t T 2 T2 2 T2 +axExYE , +a(E~) YE,-a,(E~) ~'~ fiAi+t, t i=0 fl--1 y--1 2 T2 - a= ~ q) AI,]+IE t - (E~) ~ h i AI+t, iE, j=O i=0 p 1 6--1 --axt(ET) 2 ~" wiAi+,.tEt-E T ~ sjA,.j+lE2t .......... i_=_o_ ........... 1=o T2 2 = (Ex) UE, with h(x) = a,g(x) + atf(x), w(x) = ay(x, to)/ax and s(t) = axq(t) +a~xr(t) +a~, St, o q(r) dr. Then they rewrote the above algebraic equation in the form of Mp=v and attempted to estimate the augmented parameter vector p from the least-square technique i.e. [~=[MTM]-I[MTv]. Here it is important to note that the columns of M corresponding to hi and a~tw~ terms (shown with a broken underline in the matrix algebraic equation) are apparently linearly dependent. Due to this linear dependence of columns of M, an inverse of [MTM] does not exist making the identification impossible. Hence, it may be concluded that the algorithm described above is not always suitable for the identication of (1). Jha and Zaman (1985) used the same algorithm for identification using Laguerre polynomials (LAP). Recently, Horng et al. (1986) investigated the identification problem of first-order DPS via shifted Chebyshev polynomials of the first kind (CP1). Most recently, Mohan and Datta (1988, 1989) have explored the potentialities of shifted Legendre polynomials (LeP) and sine-cosine functions (SCF) in DPS identification. Having seen the state of affairs, it appears to the authors that the problem of general second-order DPS identification has not yet been studied via block-pulse functions (BPF), CP1 and Chebyshev polynomials of the second kind (CP2). The concept of the one-shot operational matrix for repeated integration (OSOMRI) (Unbehauen and Rao, 1987) has recently been extended to shifted LeP and SCF and used, owing to its superiority, for the identification of DPS, by Mohan and Datta (1988, 1989). OSOMRIs of CP1 and CP2 are not yet reported. A comparative study of all OF approaches to assess the relative merits and demerits of each approach in DPS identification is also not yet reported in the literature. In this paper, by taking all the aforementioned points into consideration, an investigation is therefore made on the identification problem of DPS to eliminate the lacunae of existing methods and to introduce a more general unified identification approach based on the concept of OSOMRI. This paper, presenting some important results of this investigation, is organized as follows. In Section 2 some mathematical preliminaries including OSOMRI of CP1 and CP2 are presented. In Section 3 we present a unified identification approach for any class (elliptic, parabolic or hyperbolic) of DPS. Section 4 discusses the identifiability requirements, which are followed by a demonstration of the proposed identification scheme and a comparative study of all OF approaches by an illustrative example. AUTO 27:2-M 409