Solution and Parameter Estimation in Linear Time-Invariant Delayed Systems Using Laguerre Polynomial Expansion

Abstract

Finite dimensional Laguerre polynomial expansion can be applied to approximate the solution of linear time-invariant systems. Here, we extend this approach to systems with small time delay by introducing a delay matrix operator to the system equations. In addition, parameters of linear unity-feedback systems with small time delay can also be estimated by using the Laguerre expansion. In this paper we provide algorithms for large scale systems to avoid direct matrix inversion and give examples to demonstrate the approximation.

Analysis of time-varying delay systems via triangular functions

“…In general, the computed response of the delay systems via orthogonal functions is not in good agreement with the exact response of the system [2]. Special attention has been given to applications of Walsh functions [3], block pulse functions [4], Laguerre polynomials [5], Legendre polynomials [6], Chebyshev polynomials [7], Haar wavelets [8], Fourier series [9] and hybrid functions [10,11].…”

Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions

“…So it is interesting to use numerical methods to solve the optimal control problem for delay systems. Many orthogonal functions or polynomials, such as block-pulse functions [1,2], Walsh functions [3], Fourier series [4], Legendre polynomials [5], Chebyshev polynomials [6] and Laguerre polynomials [7], were used to derive solutions of some systems. In recent years many authors apply the different kinds of hybrid functions [8][9][10][11].…”

Solution of time-varying delay systems by hybrid functions

“…The approach is that of converting the delay-differential equation to an algebraic form through the use of operational matrix of integration P. The matrix P can be uniquely determined based on the particular OFs. Special attention has been given to applications of Walsh functions [2], block pulse functions [3], Laguerre polynomials [4], Legendre polynomials [5], Chebyshev polynomials [6] and Fourier series [7]. The available sets of OFs can be divided into three classes.…”