Stability analysis of hybrid systems with a linear performance index

We consider the constrained finite time optimal control (CFTOC) problem for the class of discrete-time linear hybrid systems. For a linear performance index the solution to the CFTOC problem is a time-varying piecewise affine function of the state. However, when a receding horizon control strategy is used stability and/or feasibility of the closed-loop system is not guaranteed. In this paper we present an algorithm that by analyzing the CFTOC solution a posteriori extracts regions of the state-space for which closed-loop stability and feasibility can be guaranteed. The algorithm computes the maximal positively invariant set and stability region of a piecewise affine system by combining reachability analysis with some basic polyhedral manipulation. The simplicity of the overall computation stems from the fact that in all steps of the algorithm only linear programs need to be solved.