Shifted Legendre series solution and parameter estimation of linear delayed systems

Abstract

The finite-dimensional shifted Legendre polynomials expansion is applied to approximate the solution of linear time-invariant systems with time delay. An integration matrix and a delay matrix for the shifted Legendre vector are derived so that the solution of a linear time-delay state equation is reduced to the solution of a linear algebraic matrix equation. In addition, parameters of the delayed state equation are also estimated by using the shifted Legendre expansion and the least-squares method. Two examples are given to demonstrate the accuracy of this approach.