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arXiv · 2401.11319

Initialization-Free Lie-Bracket Extremum Seeking in $\mathbb{R}^n$

Mahmoud Abdelgalil·Jorge Poveda
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Abstract

Stability results for extremum seeking control in $\mathbb{R}^n$ have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with \emph{global} (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all our results via different analytical and/or numerical examples.

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BibTeXRIS

Mahmoud Abdelgalil, Jorge Poveda. 2024-01-25. Initialization-Free Lie-Bracket Extremum Seeking in $\mathbb{R}^n$. https://arxiv.org/abs/2401.11319

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