A separation principle for linear switching systems and parametrization of all stabilizing controllers

In this paper, we investigate the problem of designing a switching compensator for a plant switching amongst a (finite) family of given configurations (Ai, Bi, Ci). We assume that switching is uncontrolled, namely governed by some arbitrary switching rule, and that the controller has the information of the current configuration i.As a first result, we provide necessary and sufficient conditions for the existence of a family of linear compensators, each applied to one of the plant configurations, such that the closed loop plant is stable under arbitrary switching. These conditions are based on a separation principle, precisely, the switching stabilizing control can be achieved by separately designing an observer and an estimated state (dynamic) compensator. These conditions are associated with (non-quadratic) Lyapunov functions. In the quadratic framework, similar conditions can be given in terms of LMIs which provide a switching controller which has the same order of the plant. As a second result, we furnish a characterization of all the stabilizing switching compensators for such switching plants. We show that, if the necessary and sufficient conditions are satisfied then, given any arbitrary family of compensators K i,(s), each one stabilizing the corresponding LTI plant (Ai, Bi, Ci) for fixed i, there exist suitable realizations for each of these compensators, which assure stability under arbitrary switching.