Construction of a Complete Lyapunov Function using Quadratic Programming

Peter Giesl, Carlos Argáez, Sigurdur Hafstein, Holger Wendland

2018

Abstract

A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is split into the chain-recurrent set, where the function is constant, and the part characterizing the gradient-like flow, where the function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about the stability of connected components of the chain-recurrent set and the basin of attraction of attractors therein. In a previous method, a complete Lyapunov function was constructed by approximating the solution of the PDE V0(x) = −1, where 0 denotes the orbital derivative, by meshfree collocation. We propose a new method to compute a complete Lyapunov function: we only fix the orbital derivative V0(x0) = −1 at one point, impose the constraints V0(x) ≤ 0 for all other collocation points and minimize the corresponding reproducing kernel Hilbert space norm. We show that the problem has a unique solution which can be computed as the solution of a quadratic programming problem. The new method is applied to examples which show an improvement compared to previous methods.

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Paper Citation


in Harvard Style

Giesl P., Argáez C., Hafstein S. and Wendland H. (2018). Construction of a Complete Lyapunov Function using Quadratic Programming.In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics - Volume 1: CTDE, ISBN 978-989-758-321-6, pages 560-568. DOI: 10.5220/0006944305600568


in Bibtex Style

@conference{ctde18,
author={Peter Giesl and Carlos Argáez and Sigurdur Hafstein and Holger Wendland},
title={Construction of a Complete Lyapunov Function using Quadratic Programming},
booktitle={Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics - Volume 1: CTDE,},
year={2018},
pages={560-568},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006944305600568},
isbn={978-989-758-321-6},
}


in EndNote Style

TY - CONF

JO - Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics - Volume 1: CTDE,
TI - Construction of a Complete Lyapunov Function using Quadratic Programming
SN - 978-989-758-321-6
AU - Giesl P.
AU - Argáez C.
AU - Hafstein S.
AU - Wendland H.
PY - 2018
SP - 560
EP - 568
DO - 10.5220/0006944305600568