Switching Law Construction for Discrete-Time Systems Via Composite Quadratic Functions

Three composite quadratic Lyapunov functions are used for the construction of stabilizing laws for discrete-time switched systems. The three functions include the max of quadratics, the min of quadratics and the convex hull of quadratics. Conditions for stabilization are derived as bilinear matrix inequalities and the convergence rate is optimized via linear matrix inequality (LMI) based tools. Numerical examples show the accuracy of the characterization of the convergence rate via the matrix inequalities and the improvement of using nonquadratic functions over quadratic functions. Among the three Lyapunov functions, the min of quadratics, which is not convex and not differentiable, turns out to be the most effective and easiest to handle.