Extremum Seeking (ES) is Practically Stable Whenever Model-Based ES is Stable

TL;DR

This paper proves that the practical stability of model-free extremum seeking control can be guaranteed if the corresponding model-based system is stable, simplifying stability analysis and dither signal selection.

Contribution

It establishes a direct link between the stability of model-based and model-free extremum seeking control systems, reducing the complexity of stability analysis.

Findings

Model-free ESC inherits practical stability from stable model-based ESC.

Stability of the average system depends on the model-based ESC's stability.

Guidelines for selecting dither amplitudes to ensure stability.

Abstract

Extremum seeking control (ESC) are optimization algorithms in continuous time, with model-based ESCs using true derivative information of the cost function and model-free ESCs utilizing perturbation-based estimates instead. Stability analysis of model-free ESCs often employs the associated average system, whose stability is dependent on the selection of the dither signal. We demonstrate first the challenge of this analysis approach by showing selections of relative dither amplitudes and rates at different ESC inputs which result in the average system always having an unstable equilibrium. Then we go on to show that, if the model-based ESC is globally asymptotically stable (GAS), then the average system is semiglobally practically asymptotically stable (sGPAS), and the model-free ESC is semiglobally practically uniformly asymptotically stable (sGPUAS). Thus, we free the system analyst from analyzing the stability of the average system with various dither signals, as it is sufficient to analyze the stability of the model-based ESC. The result for the original model-free ESC also provides a guideline for the user for how to select the dither amplitudes to ensure sGPUAS.