Logarithmic Barrier Functions for Practically Safe Extremum Seeking Control
This work addresses safety in extremum seeking control for applications like robotics or autonomous systems, but it is incremental as it builds on existing barrier function methods with specific tuning improvements.
The paper tackled the problem of optimizing unknown objective functions while enforcing safety constraints by proposing a Practically Safe Extremum Seeking method using Logarithmic Barrier Functions, resulting in rigorous proof of practical safety and local practical convergence to a modified minimizer within the safe set.
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.