On Several Composite Quadratic Lyapunov Functions for Switched Systems

Three types of composite quadratic Lyapunov funtions are used for deriving conditions of stabilization and for constructing switching laws for switched systems. The three types of functions are, the max of quadratics, the min of quadratics and the convex hull of quadratics. Directional derivatives of the Lyapunov functions are used for the characterization of convergence rate. Stability results are established with careful consideration of the existence of sliding mode and the convergence rate along the sliding mode. Dual stabilization result is established with respect to the pair of conjugate Lyapunov functions: the max of quadratics and the convex hull of quadratics. It is observed that the min of quadratics, which is nondifferentiable and nonconvex, may be a more convenient tool than the other two types of functions which are convex and/or differentiable