Polytopic Lyapunov Functions are not Straightforward for Minimum Dwell-Time Switched Affine Systems

This paper addresses stabilization of switched affine systems relying on polytopic Lyapunov functions. Several methods have been developed for this family of systems, most of them based on quadratic Lyapunov functions for an average system. An immediate conjecture is that replacing such a function with one of polytopic type should also achieve stable behavior. We disprove this conjecture on a strategy that employs the hybrid systems formalism, by means of examples. Specifically, we show that for a given switching strategy based on a quadratic Lyapunov function, naïvely replacing this function by its polytopic counterpart does not guarantee the same stability properties, and that this happens even in the case of dwell-time switching. We deeply investigate such a problem, identify the conditions that prevent asymptotic stability, and give some directions on how to avoid them to ensure stability. These results are finally illustrated in simulation.