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Computation of Lyapunov functions for nonlinear discrete systems by linear programming.pdf (2.53 MB)

Computation of Lyapunov functions for nonlinear discrete systems by linear programming

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posted on 2023-06-08, 17:52 authored by Peter GieslPeter Giesl, Sigurdur Hafstein
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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Journal of Difference Equations and Applications

ISSN

1023-6198

Publisher

Taylor & Francis

Issue

4

Volume

20

Page range

610-640

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2014-07-18

First Open Access (FOA) Date

2017-01-17

First Compliant Deposit (FCD) Date

2017-01-17

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