The solution of time-delay systems is obtained by using a hybrid function. The properties of the hybrid functions consisting of block-pulse functions and Chebyshev polynomials are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrix of delay is introduced. The operational matrices of integration and delay 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.

M. Razzaghi, H. R. Marzban.
A hybrid domain analysis for systems with delays in state and control,
Mathematical Problems in Engineering,
DOI: 10.1155/S1024123X01001673