Stabilization of Switched Systems via Composite Quadratic Functions

This paper investigates stabilization of switched systems by using a Lyapunov function composed of a family of continuously differentiable functions. Switching laws are constructed by using the directional derivatives along the trajectories of the subsystems. Conditions for stabilization are established with careful consideration of sliding modes and directional derivatives along sliding modes. Three types of composite quadratic Lyapunov functions, the max of quadratics, the min of quadratics and the convex hull of quadratics are used for deriving matrix conditions of stabilization and for constructing switching laws. Dual stabilization results are established with respect to the pair of conjugate Lyapunov functions: the max of quadratics and the convex hull of quadratics. Conditions of stabilization are derived as bilinear matrix inequalities and solved with LMI-based numerical tools. Relationship between the newly derived conditions and some existing conditions are investigated. 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. Numerical examples are used to demonstrate the synthesis results and the advantage of the composite quadratic functions over existing multiple Lyapunov functions. In particular, better results have been obtained when the number of quadratic functions is greater than the number of subsystems.