Analysis of Discrete-Time Linear Switched Systems: A Variational Approach

A powerful approach for analyzing the stability of continuous–time switched systems is based on using tools from optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. More generally, this so–called variational approach was successfully applied to derive nice–reachability–type results for both linear and nonlinear continuous–time switched systems. Motivated by this, we develop in this paper an analogous approach for discrete–time linear switched systems. We derive and prove a necessary condition for optimality of the “most unstable” switching law. This yields a type of discrete–time maximum principle (MP). We demonstrate using an example that 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 for the auxiliary system imply nice–reachability results for the original discrete– time linear switched system. Using this approach, we derive several new Lie–algebraic conditions guaranteeing nice–reachability results. These results, and their proofs, turn out to be quite different from their continuous–time counterparts.