Control-Affine Extremum Seeking Control with Attenuating Oscillations: A Lie Bracket Estimation Approach

TL;DR

This paper introduces a novel control-affine extremum seeking control method that reduces oscillations using Lie bracket estimation and a geometric Kalman filter, with easier stability conditions and demonstrated effectiveness.

Contribution

It proposes a new ESC structure that attenuates oscillations via Lie bracket estimation and geometric filtering, relaxing stability conditions compared to existing methods.

Findings

Oscillations are effectively attenuated in numerical simulations.

The proposed method works under relaxed stability conditions.

It outperforms previous approaches in three benchmark problems.

Abstract

Control-affine Extremum Seeking Control (ESC) systems have been increasingly studied and applied in the last decade. In a recent effort, many control-affine ESC structures have been generalized in a unifying class and their stability was analyzed. However, guaranteeing vanishing oscillations at the extremum point for said class requires strong conditions that may not be feasible or easy to check/design by the user, especially when the gradient of the objective function is unknown. In this paper, we introduce a control-affine ESC structure that remedies this problem such that: (i) its oscillations attenuate structurally via a novel application of a geometric-based Kalman filter and a Lie bracket estimation approach; and (ii) its stability is characterized by a time-dependent (one-bound) condition that is easier to check and relaxed when compared to the generalized approach mentioned earlier. We provide numerical simulations of three problems to demonstrate the effectiveness of our proposed ESC; these problems cannot be solved with vanishing oscillations using the generalized approach in the literature.