Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl, Peter A and Hafstein, Sigurdur F (2014) Revised CPA method to compute Lyapunov functions for nonlinear systems. Journal of Mathematical Analysis and Applications, 410 (1). pp. 292-306. ISSN 0022-247X

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Abstract

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium’s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C² and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

Item Type: Article
Keywords: Lyapunov function, nonlinear system, exponential stability, basin of attraction, CPA function, piecewise linear function, linear programming
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Peter Giesl
Date Deposited: 16 Jun 2015 16:25
Last Modified: 10 Mar 2017 17:21
URI: http://sro.sussex.ac.uk/id/eprint/54583

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