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Lie bracket approximation of extremum seeking systems
KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0001-9940-5929
2013 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 49, no 6, 1538-1552 p.Article in journal (Refereed) Published
Abstract [en]

Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced system-theoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing behavior of the extremum seeking system. Furthermore, we establish a theoretical foundation and prove that uniform asymptotic stability of the Lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system. We use the established results in order to prove local and semi-global practical uniform asymptotic stability of the extrema of a certain map for multi-agent extremum seeking systems.

Place, publisher, year, edition, pages
2013. Vol. 49, no 6, 1538-1552 p.
Keyword [en]
Extremum seeking, Approximation methods, Lie brackets, Practical stability
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:kth:diva-124280DOI: 10.1016/j.automatica.2013.02.016ISI: 000319540500003Scopus ID: 2-s2.0-84877579601OAI: oai:DiVA.org:kth-124280DiVA: diva2:634147
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation
Note

QC 20130628

Available from: 2013-06-28 Created: 2013-06-28 Last updated: 2017-12-06Bibliographically approved

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Johansson, Karl Henrik

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CiteExportLink to record
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  • apa
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