The minimum achievable stability radius of switched linear systems with feedback

We present a scheme for estimating the minimum achievable decay rate of a switched linear system via path-dependent feedback control, when the switching signal belongs to a language generated by a strongly connected graph. This growth rate is characterized by the constrained joint spectral radius (CJSR) of the system, a generalization of the joint spectral radius to account for the switching constraint. Our key tool in analyizing the CJSR is the multinorm, a collection of mode-indexed norms which demonstrate contractiveness along admissible modal trajectories. We may approximate the CJSR to any desired accuracy by computing quadratic multinorms as solutions to a system of LMIs, using an estimation scheme presented in [15]. These LMIs are of similar form to those which characterize the stability of the switched system in [6]. The feasiblity of any one of these LMIs allows the construction of a suitable controller; we use the infeasibility of such an LMI to provide a lower bound on the closed-loop decay rate achieved by any path-dependent controller.