Logarithmic Barrier Functions for Practically Safe Extremum Seeking Control

TL;DR

This paper introduces a safe extremum seeking control method using logarithmic barrier functions to ensure safety constraints are strictly enforced while optimizing unknown functions.

Contribution

It provides a rigorous proof of practical safety and convergence for the proposed barrier-based extremum seeking approach, validated by numerical simulations.

Findings

Trajectories remain within a safety margin, ensuring safety.

The method guarantees local practical convergence within the safe set.

Numerical simulations validate the theoretical safety and convergence results.

Abstract

This paper presents a methodology for Practically Safe Extremum Seeking (PSfES), designed to optimize unknown objective functions while strictly enforcing safety constraints via a Logarithmic Barrier Function (LBF). Unlike traditional safety-filtered approaches that may induce chattering, the proposed method augments the cost function with an LBF, creating a repulsive potential that penalizes proximity to the safety boundary. We employ averaging theory to analyze the closed-loop dynamics. A key contribution of this work is the rigorous proof of practical safety for the original system. We establish that the system trajectories remain confined within a safety margin, ensuring forward invariance of the safe set for a sufficiently fast dither signal. Furthermore, our stability analysis shows that the model-free ESC achieves local practical convergence to the modified minimizer strictly within the safe set, through the sequential tuning of small parameters. The theoretical results are validated through numerical simulations.