Extremum Seeking Control with an Adaptive Gain Based On Gradient Estimation Error

TL;DR

This paper introduces an extremum seeking control algorithm with an adaptive gain that adjusts based on gradient estimation quality, ensuring stability and improved performance in various scenarios.

Contribution

The paper proposes a novel adaptive step-size mechanism for extremum seeking control based on a new gradient estimation error metric, enhancing stability and robustness.

Findings

Ensures input-to-state stability with respect to dither signals.

Demonstrates effectiveness through five numerical examples.

Provides a bounded estimation error analysis considering cost function curvature.

Abstract

This paper presents an extremum seeking control algorithm with an adaptive step-size that adjusts the aggressiveness of the controller based on the quality of the gradient estimate. The adaptive step-size ensures that the integral-action produced by the gradient descent does not destabilize the closed-loop system. To quantify the quality of the gradient estimate, we present a batch least squares estimator with a novel weighting and show that it produces bounded estimation errors, where the uncertainty is due to the curvature of the unknown cost function. The adaptive step-size then maximizes the decrease of the combined plant and controller Lyapunov function for the worst-case estimation error. We prove that our ESC is input-to-state stable with respect to the dither signal. Finally, we demonstrate our proposed ESC through five numerical examples; one illustrative, one practical, and three benchmarks.