Lie bracket approximation of extremum seeking systems

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...

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Bibliographic Details
Published in:Automatica (Oxford) 2013-06, Vol.49 (6), p.1538-1552
Main Authors: Dürr, Hans-Bernd, Stanković, Miloš S., Ebenbauer, Christian, Johansson, Karl Henrik
Format: Article
Language:English
Subjects:
Applied sciences
Approximation
Approximation methods
Asymptotic properties
Brackets
Computer science
control theory
systems
Control theory. Systems
Dynamical systems
Dynamics
Exact sciences and technology
Extremum seeking
Lie brackets
Mathematical analysis
Practical stability
Stability
System theory
Trajectories
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Summary: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.
ISSN:0005-1098
1873-2836
1873-2836
DOI:10.1016/j.automatica.2013.02.016