Lie Bracket Approximation of Extremum Seeking Systems

TL;DR

This paper introduces a novel Lie bracket approximation approach to analyze extremum seeking systems, providing a new theoretical foundation that links the stability of the approximating system to the original extremum seeking behavior.

Contribution

It presents a new Lie bracket-based interpretation of extremum seeking, establishing stability results and applying them to multi-agent systems.

Findings

Lie bracket system reveals extremum seeking behavior

Stability of Lie bracket system implies stability of extremum seeking system

Proves local and semi-global stability for multi-agent extremum seeking

Abstract

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.