Non-Local Extremum Seeking Based on the Divergence Theorem

TL;DR

This paper introduces a novel extremum seeking control method using the divergence theorem to average the objective function over a sphere, helping to avoid local minima and improve global optimization in multi-dimensional systems.

Contribution

It presents a new extremum seeking approach based on spherical motion and the divergence theorem, enabling local averaging to mitigate local extrema effects.

Findings

The method effectively drives the system towards the gradient of the averaged objective.

Theoretical analysis confirms practical stability under measurement errors.

The approach can eliminate undesired local extrema for better global optimization.

Abstract

We propose a new design strategy for extremum seeking control for a multi-dimensional single-integrator system in the presence of local extrema. The proposed method employs suitably designed sinusoidal dither signals, which force the single-integrator to a spherical motion. Over time, this spherical motion gives approximate access to an integral of the objective function over a sphere. Using the divergence theorem, we identify the integral over the sphere as the gradient of an integral over the enclosed ball. This integral over the ball defines a locally averaged objective function. The proposed extremum seeking method drives the system state into the gradient direction of the averaged objective function. Such a local average of the objective function can eliminate undesired local extrema and is therefore beneficial for global optimization. Under the assumption that the averaged objective function has no undesired critical points, we prove practical asymptotic stability of the closed-loop system. Our theoretical analysis takes sufficiently small $L_\infty$-measurement errors of the objective function into account.