Robust Stabilizability of linear uncertain switched systems : a computational algorithm

The problem of robust stabilizability of linear uncertain systems with unstable modes has recently attracted increased interest. Given a number of unstable linear uncertain subsystems, the problem is to determine stabilizing switching sequences resulting in asymptotically stable behavior or to ascertain the absence of such laws. In a number of recent publications, necessary and sufficient conditions for the stabilizability problem have been sought. It is well known that the class of polyhedral Lyapunov functions is universal for the asymptotic stability of linear uncertain switched systems with stable subsystems. Recent results suggest that this property holds for linear uncertain switched systems with unstable subsystems, at least under some further assumptions. Motivated by these theoretical advancements, in this paper we deal with the development of a computational technique that generates polyhedral Lyapunov functions for the stabilizability problem. First, we propose an improved reliable and efficient computational algorithm for two-dimensional systems, extending the results in [11] and [14]. Second, we show how the main idea may be modified in a sufficient form to allow higher-dimensional systems to be dealt with. Further development of these computational tools for higher-dimensional systems will be the subject of future work.