Explicit Form of Solutions of Second-Order Delayed Difference Equations: Application to Iterative Learning Control

Abstract

A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. The closed form of its solution is derived by means of a newly defined delayed matrix sine/cosine using the Z transform and determining function. This representation helps with analyzing iterative learning control by applying appropriate updating laws and ensuring sufficient conditions for achieving asymptotic convergence in tracking.