Nice-reachability results for discrete-time linear switched systems with applications to stability under arbitrary switching laws

A powerful approach for analyzing the stability under arbitrary switching of continuous-time switched systems is based on analyzing stability for the “most unstable” switching law. This approach has been successfully applied to derive nice-reachability-type results for both linear and nonlinear continuous-time switched systems. We develop an analogous approach for discrete-time linear switched systems. We first prove a necessary condition for the “most unstable” switching law in the form of a discrete-time maximum principle (MP). This MP is in fact weaker than its continuous-time counterpart. To overcome this, we introduce the auxiliary system of a discrete-time linear switched system, and show that regularity properties of time-optimal controls (TOCs) for the auxiliary system imply nice-reachability results for the original discrete-time linear switched system. We derive several new Liealgebraic conditions guaranteeing nice-reachability results. These results, and their proofs, turn out to be quite different from their continuous-time counterparts.

1. Introduction

Let denote the set of natural numbers. Consider the discrete-time (DT) linear switched system $$\eqalignno{ x(k+1) & =A_{\sigma(k)}x(k),\cr x(0) & =x_{0}, & \hbox{(1)}}$$ where , and is the switching law. This models a system that can switch between the linear subsystems: with the switching law determining which system is active at each time step.