Minimum achievable decay rates of the discrete linear inclusion

In this paper we provide a scheme for estimating the minimum achievable exponential decay rate of a switched linear system via output feedback control. The output feedback stabilization of switched linear systems using path-dependent controllers is characterized exactly in [8] in terms of an increasing family of linear matrix inequalities; the feasibility of any one of these inequalities allows for the construction of a stabilizing controller, while infeasible conditions are disregarded. In this paper we use the infeasibility of these conditions to provide a lower bound on the achievable decay rate of the closed-loop system. We consider the special case of the discrete linear inclusion - e.g., a system with unrestricted switching - and develop a family of estimators of the minimum achievable decay rate for the system. For controllers of a fixed path-dependence, the resulting bounds provide an interval containing the minimum performance level whose length may be made as small as desired. The lower bound of these intervals is used as a nondecreasing estimation of the achievable decay rate. A simple, physically motivated example is provided to demonstrate the practical application of this result.