On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays

Abstract

This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressions are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.