Estimating the region of attraction using polynomial optimization

A converse Lyapunov result

Morgan Jones, Hesameddin Mohammadi, Matthew Peet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

In this paper, we propose an iterative method for using SOS programming to estimate the region of attraction of a polynomial vector field, the conjectured convergence of which necessitates the existence of polynomial Lyapunov functions whose sublevel sets approximate the true region of attraction arbitrarily well. The main technical result of the paper is the proof of existence of such a Lyapunov function. Specifically, we use the Hausdorff distance metric to analyze convergence and in the main theorem demonstrate that the existence of an n-times continuously differentiable maximal Lyapunov function implies that for any ϵ > 0, there exists a polynomial Lyapunov function and associated sub-level set which together prove stability of a set which is within e Hausdorff distance of the true region of attraction. The proposed iterative method and probably convergence is illustrated with a numerical example.

Original languageEnglish (US)
Title of host publication2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1796-1802
Number of pages7
Volume2018-January
ISBN (Electronic)9781509028733
DOIs
StatePublished - Jan 18 2018
Event56th IEEE Annual Conference on Decision and Control, CDC 2017 - Melbourne, Australia
Duration: Dec 12 2017Dec 15 2017

Other

Other56th IEEE Annual Conference on Decision and Control, CDC 2017
CountryAustralia
CityMelbourne
Period12/12/1712/15/17

Fingerprint

Lyapunov functions
Converse
Lyapunov Function
Lyapunov
Polynomials
Polynomial
Hausdorff Distance
Optimization
Polynomial function
Iterative methods
Iteration
Polynomial Vector Fields
Hausdorff Metric
Maximal Function
Distance Metric
Continuously differentiable
Computer programming
Level Set
Programming
Imply

ASJC Scopus subject areas

  • Decision Sciences (miscellaneous)
  • Industrial and Manufacturing Engineering
  • Control and Optimization

Cite this

Jones, M., Mohammadi, H., & Peet, M. (2018). Estimating the region of attraction using polynomial optimization: A converse Lyapunov result. In 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017 (Vol. 2018-January, pp. 1796-1802). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2017.8263908

Estimating the region of attraction using polynomial optimization : A converse Lyapunov result. / Jones, Morgan; Mohammadi, Hesameddin; Peet, Matthew.

2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. Vol. 2018-January Institute of Electrical and Electronics Engineers Inc., 2018. p. 1796-1802.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Jones, M, Mohammadi, H & Peet, M 2018, Estimating the region of attraction using polynomial optimization: A converse Lyapunov result. in 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. vol. 2018-January, Institute of Electrical and Electronics Engineers Inc., pp. 1796-1802, 56th IEEE Annual Conference on Decision and Control, CDC 2017, Melbourne, Australia, 12/12/17. https://doi.org/10.1109/CDC.2017.8263908
Jones M, Mohammadi H, Peet M. Estimating the region of attraction using polynomial optimization: A converse Lyapunov result. In 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. Vol. 2018-January. Institute of Electrical and Electronics Engineers Inc. 2018. p. 1796-1802 https://doi.org/10.1109/CDC.2017.8263908
Jones, Morgan ; Mohammadi, Hesameddin ; Peet, Matthew. / Estimating the region of attraction using polynomial optimization : A converse Lyapunov result. 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. Vol. 2018-January Institute of Electrical and Electronics Engineers Inc., 2018. pp. 1796-1802
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