Switching-Controlled Invariance of Polytopic Sets for Switched Affine Systems with Dwell Time

Switching among affine systems under dwell-time constraints in order to steer the state as close as possible to a desired point involves the consideration of a nontrivial region into which the state is eventually driven and maintained. Suitable regions for this aim have the following property: For every point within the region, a switching law satisfying the dwell-time constraint exists so that the trajectory beginning at that point remains within the region for all future times. This paper focuses on addressing and characterizing the latter property, which is here called switching-controlled invariance (SCI) with dwell time, through a convexity-based approach. It is shown that SCI of a set S is equivalent to the existence of some cover of S, where each subset in the cover can be selected to share specific properties of S: convexity, closedness, boundedness, and, most important, polytopic structure. Our theoretically most important result is to establish that if an SCI with dwell time set S is a (closed and convex) polytope and the switched affine system has Nsubsystems, then S can be covered by N subpolytopes so that the state trajectories beginning within each subpolytope are kept within S during at least one dwell time without switching. This result is highly nontrivial, is not true for other geometric structures such as ellipsoids, and further motivates the consideration of polytopic sets in the control of switched affine systems with dwell time.