Computation of Lyapunov functions for nonlinear discrete systems by linear programming

Given an autonomous discrete time system with an equilibrium at the origin and a hypercube D containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function V : D -> R for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfils certain properties, then such a linear programming problem possesses a feasible solution. We present an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibrium's basin of attraction. The system is assumed to have a right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type such as linear, piecewise affine and so on. Our approach is a non-trivial adaptation of the CPA method to compute Lyapunov functions for continuous time systems to discrete time systems

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 S-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