Piecewise linear dynamic systems with time delays

Abstract

A new method of constructing periodic solutions for piecewise linear dynamic systems with time delays is investigated. Although the existence and the uniqueness of the periodic solution are guaranteed by well-known theorems, existing schemes for actually constructing the periodic solution are either purely formal or approximate. The idea of constructing the exact solution is first pursued with the linear delay systems. The formal representation of the solution to the linear problem is viewed as a system of Fredholm integral equations of the second kind. Since the matrix kernel for this system of integral equations is separable, the integral equation can be reduced to a system of algebraic equations involving certain integral moments of the initial function. These observations lead to a transfer relationship between two state vectors in the form of a matrix equation. Then the problem can be posed as either an initial value problem (if one is seeking the transient solution), or a periodic solution problem (if one is seeking the unknown initial data). This Fredholm Integral Equation Method is used effectively to construct periodic solutions to piecewise linear differential-difference equations. The periodic solutions are constructed from a cascaded product of matrix equations derived for each linear region. The stability of the periodic solution is determined by solving an associated eigenvalue problem. The periodic solution and its stability analysis are exact in the sense that the error induced by the truncation process in the Fredholm Integral Equation Method can be made exponentially small as the size of the transfer matrix is increased.