Stabilization of switched systems via optimal control

We consider switched systems composed of linear time invariant unstable dynamics and we deal with the problem of computing an appropriate switching law such that the controlled system is globally asymptotically stable. On the basis of our previous results in this framework, we first present a method to design a feedback control law that minimizes a linear quadratic (LQ) performance index when an infinite number of switches are allowed and at least one dynamics is stable. Then, we show how this approach can be useful when dealing with the stabilization problem of switched systems characterized by unstable dynamics, by applying the proposed procedure to a “dummy” system, augmented with a stable dynamics. If the system with unstable dynamics is globally exponentially stabilizable, then our method provides the feedback control law that minimizes the chosen quadratic performance index, and that guarantees the closed loop system to be globally asymptotically stable.