Computation of Lyapunov functions for nonlinear discrete time systems by linear programming

Abstract

Given a nonautonomous discrete 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 fulfills certain properties, then such a linear programming problem possesses a feasible solution. We suggest 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 system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibrium s domain of attraction. The system is assumed to have a C 2 right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type like linear, piecewise affine, etc. Our approach is a non-trivial adaption of the CPA method to compute Lyapunov functions for continuous systems to discrete systems.

Computation and Verification of Lyapunov Functions

“…[9,10]. We start by defining general triangulations and CPA functions, then we define the triangulations which we use in this paper and derive their basic properties.…”

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

“…For simplices with the origin as a vertex, however, this problem requires a much more involved solution that only works if the origin is exponentially stable, cf. [7].…”

“…For instance, collocation methods were presented in [5] and [6], graph algorithms are used to compute complete Lyapunov functions in [3] and [14], and continuous piecewise ane (CPA) Lyapunov functions for discrete-time systems were computed as the solution of a linear programming problem in [7].…”

“…Values are dened at each vertex of the triangulation and a continuous and piecewise ane function is then dened via a convex interpolation of these values. In [7], the vertex values are obtained by solving a linear programming problem that incorporates error estimates.…”

An alternative approach for stability analysis of discrete time nonlinear dynamical systems

“…With the best of our acknowledge, the efforts that have been made up to the present time in the framework of discrete-time systems are limited only to some special classes of systems (linear, polynomial systems [2], [13], homogeneous [15], exponential stable systems [3], [4]). Generally speaking, for a given discrete time nonlinear system one has little hope to find the Lyapunov function unless the system belongs to one of the special classes.…”