Semi-Global Practical Extremum Seeking with Practical Safety

TL;DR

This paper presents a safe extremum seeking controller that ensures practical safety and stability while optimizing an unknown objective, extending quadratic program-based safety filters to unknown barrier functions.

Contribution

It introduces a semi-global practical extremum seeking algorithm with safety guarantees for unknown barrier functions, including non-convex problems, using Lyapunov methods.

Findings

Proves semi-global practical asymptotic stability of the controller.

Establishes conditions under which non-convex problems are solvable.

Demonstrates the algorithm's effectiveness through an example.

Abstract

We introduce a type of safe extremum seeking (ES) controller, which minimizes an unknown objective function while also maintaining practical positivity of an unknown barrier function. We show semi-global practical asymptotic stability of our algorithm and present an analogous notion of practical safety. The dynamics of the controller are inspired by the quadratic program (QP) based safety filter designs which, in the literature, are more commonly used in cases where the barrier function is known. Conditions on the barrier and objective function are explored showing that non convex problems can be solved. A Lyapunov argument is proposed to achieve the main results of the paper. Finally, an example is given of the algorithm which solves the constrained optimization problem.