A geometrical treatment for obtaining necessary and sufficient conditions for joint quadratic Lyapunov function existence for state-dependent, switched systems: A two-dimensional case

Abstract

The question of existence of joint quadratic Lyapunov functions (QLFs) for state-dependent, switched dynamical systems is given a preliminary geometrical treatment in this paper. The joint QLF problem for a switched system and a collection of regions defined by state vectors that determine when switching occurs consists of finding nonempty intersections of convex sets of QLFs. The existence of a joint QLF guarantees switched system stability. Necessary and sufficient conditions for the existence of a joint QLF are obtained for a two-dimensional problem.