On Infinity Norms as Lyapunov Functions for Piecewise Affine Systems

This paper considers off-line synthesis of stabilizing static feedback control laws for discrete-time piecewise affine (PWA) systems. Two of the problems of interest within this framework are: (i) incorporation of the j-procedure in synthesis of a stabilizing state feedback control law and (ii) synthesis of a stabilizing output feedback control law. Tackling these problems via (piecewise) quadratic Lyapunov function candidates yields a bilinear matrix inequality at best. A new solution to these problems is proposed in this work, which uses infinity norms as Lyapunov function candidates and, under certain conditions, requires solving a single linear program. This solution also facilitates the computation of piecewise polyhedral positively invariant (or contractive) sets for discrete-time PWA systems.