Probabilistic sorting and stabilization of switched systems

In this paper, we consider Lyapunov stability of switched linear systems whose switching signal is constrained to a subset of indices. We propose a switching rule that chooses the most stable subsystem among those belonging to the subset. This rule is based on an ordering of the subsystems using a common Lyapunov function. We develop randomized algorithms for finding the ordering as well as for finding a subset of systems for which a common Lyapunov function exists. It is shown that the class of randomized algorithms known as the Las Vegas type is useful in the design procedure. A third-order example illustrating the efficacy of the approach is presented.