Analysis and parameter identification of time-delay systems via shifted Legendre polynomials

Abstract

The problems of analysis and parameter identification of lime-delay systems have been recently studied using rectangular functions such as Walsh and block-pulse functions. In this paper, the continuous shifted Legendre polynomials are first used to solve the above-mentioned problems. The approach followed is that of converting the delay-differential equation to an algebraic form through using the operational matrices of integration and delay. The key point is the derivation of a delay operational matrix D which relates the coefficient vector x of the shifted Legendre series of a given function x(t) with the coefficient vector x ^∗ of its delay form x(t— τ) in the form x ^∗ = Dx + α, where the elements of a are obtained from the initial function of x(t). Two computational examples are given to illustrate the utility of the method. The availability of recursive algorithms for constructing the delay operational matrix D makes the method particularly attractive to practise.