Precise Solution of Stability Problem for Linear Plane Switched Systems

We consider second-order switched dynamical systems consisting of a family of subsystems. The problem is to find conditions guaranteeing exponential stability of the system for any switching sequence. The most of the results on the problem are obtained by the Lyapunov function method which provides sufficient conditions for system stability. The known necessary and sufficient conditions are too complicated and can hardly be used for actual check of stability. The checkable precise stability results are found for particular second-order systems with two subsystems. In this paper simple explicit necessary and sufficient conditions for exponential stability of a general second-order system with an arbitrary number of subsystems are obtained. It is shown that the boundary of a stability region correspond to either infinitely fast switching (chattering) or a periodic switching law. For the last case, a precise upper bound for the number of switching points is found.