Analysis of linear time-invariant time-delay systems via orthogonal functions

Abstract

This paper first critically reviews the up-to-date literature available on the problem of analysis of linear time-invariant dynamical systems containing time delay(s) and discusses the practical limitations and shortcomings of existing orthogonal functions techniques. Then it presents some mathematical preliminaries very important for the analysis of delay systems. As a part of this, it introduces: the derivative operational matrix of Legendre, Laguerre, Hermite and Fourier orthogonal systems; the delay-integration operational matrix of orthogonal polynomial systems; a new integration operational matrix of Fourier orthogonal system; and the error analysis of arbitrary function in terms of these orthogonal functions and some approximate tools. Next, it proposes delay operational matrix approach and time-partition technique for the analysis of delay systems. It shows that recursive algorithms are possible if block pulse functions are used in the proposed methods whose practical limitations are also pointed out. At the end of the paper the analogy between the well known single-term PCBF approach and the proposed time-partition technique is given.