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
journal contribution
posted on 2023-06-08, 17:52 authored by Peter GieslPeter Giesl, Sigurdur HafsteinGiven 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 ApplicationsISSN
1023-6198Publisher
Taylor & FrancisExternal DOI
Issue
4Volume
20Page range
610-640Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2014-07-18First Open Access (FOA) Date
2017-01-17First Compliant Deposit (FCD) Date
2017-01-17Usage metrics
Categories
No categories selectedKeywords
Licence
Exports
RefWorks
BibTeX
Ref. manager
Endnote
DataCite
NLM
DC