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 (A(i), B-i, C-i). 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 (A(i), B-i, C-i) for fixed i, there exist suitable realizations for each of these compensators, which assure stability under arbitrary switching.