Global Extremum Seeking With Double Integrators in the Presence of Local Extrema

TL;DR

This paper explores two perturbation-based extremum seeking methods for global optimization involving double integrators, demonstrating conditions under which the methods achieve practical stability despite local extrema.

Contribution

It introduces a novel averaging analysis showing how local averaging transforms the objective function to reduce local extrema in extremum seeking.

Findings

Averaged system is a damped double integrator with a potential force.

Potential force is derived from the gradient of a locally averaged objective.

Provided conditions ensure semi-global practical stability of the control schemes.

Abstract

We study the problem of global extremum seeking in the presence of local extrema. We investigate two different perturbation-based methods: 1) a well-known classical extremum seeking scheme for steady-state output optimization, and 2) a source seeking scheme for a two-dimensional point mass. In each of these two scenarios, the closed-loop system involves a damped double integrator subject to an oscillatory force. An averaging analysis reveals that the respective averaged system is again a damped double integrator, but now subject to a potential force. The potential force is given by the gradient of a locally averaged objective function. Such a function is less prone to have undesired local extrema and is therefore better suited for global optimization. We provide sufficient conditions for semi-global practical uniform asymptotic stability of the closed-loop systems. The sufficient conditions only involve assumptions on the averaged objective function but not the original one.