Meixner models of linear discrete systems (Q914602)

Orthogonal polynomials of a discrete variable are used in various problems of filtering [see e.g. \textit{V. P. Perov}, Autom. Remote Control 39, 199-206 (1978); ibid. 38, 1480-1487 (1978); ibid. 37(1976), 1517-1522 (1977). Russian originals in Avtom. Telemekh. 1978, No.2, 53-61 (1978; Zbl 0421.93054); ibid. 1977, No.10, 65-73 (1977; Zbl 0421.93053); ibid. 1976, No.10, 58-65 (1976; Zbl 0421.93052); resp.], identification [see \textit{P. R. Clement}, J. Franklin Inst. 313, 85-95 (1982; Zbl 0481.93025); the author, Autom. Remote Control 48, No.3, 346-352 (1987); translation from Avtom. Telemekh. 1987, No.3, 88-96 (1987; Zbl 0625.93020)], and approximation (see the work of the author, loc. cit.). Expansion in orthogonal polynomials ensures compression of information, i.e., long time series are transformed into a shorter spectrum of the expansion. This reduces the computational costs. Exponential power polynomials appear to be the best for the representation of linear discrete systems (see, e.g., Petrov, loc. cit.). In this paper, we investigate orthogonal Meixner polynomials. We give a shift formula in the time domain and use it to derive the state equation for the expansion coefficients of the input and output processes in both unnormalized and normalized polynomials. The normalized Meixner model has the same qualitative properties (stability, controllability, observability) as the initial system. The unnormalized Meixner model of an unstable system may prove to be asymptotically stable under certain conditions. The normalized Meixner model is a generalization of the Laguerre model [see the author and \textit{Yu. Yaaksoo}, ``Laguerre state equations of a multivariable discrete time system'', Proc. 8th IFAC World Congr., Kyoto/Jap. 1981, Vol. 2, 1153-1158 (Oxford/Engl. 1982)], and the unnormalized Meixner polynomials are a generalization of the exponential power series proposed by Petrov, loc. cit. Thus, Meixner models may be used to solve filtering, identification, and approximation problems. In this paper, we present a scheme for the identification of a linear dynamic system using a Meixner model.