An algorithmic approach to stability verification of polyhedral switched systems

Abstract

We present an algorithmic approach for analyzing Lyapunov and asymptotic stability of polyhedral switched systems. A polyhedral switched system is a hybrid system in which the continuous dynamics is specified by polyhedral differential inclusions, the invariants and guards are specified by polyhedral sets and the switching between the modes do not involve reset of variables. The analysis consists of first constructing a finite weighted graph from the switched system and a finite partition of the state space, which represents a conservative approximation of the switched system. Then, the weighted graph is analyzed for certain structural properties, satisfaction of which implies stability. However, in the event that the weighted graph does not satisfy the properties, one cannot, in general, conclude that the system is not stable due to the conservativeness of the graph. Nevertheless, when the structural properties do not hold in the graph, a counterexample indicating a potential reason for the failure is returned. Further, a more precise approximation of the switched system can be constructed by considering a finer partition of the statespace in the construction of the finite weighted graph. We present experimental results on analyzing stability of switched systems using the above method.