Solution of time-varying delay systems by hybrid functions

A method for finding the solution of time-delay systems using a hybrid function is proposed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrices of product and delay are introduced. These matrices together with the operational matrix of integration are utilized to reduce the solution of time-delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.