Extremum Seeking Control With an Adaptive Gain Based on Gradient Estimation Error

This article presents an extremum-seeking control (ESC) algorithm with an adaptive step size that adjusts the aggressiveness of the controller based on the quality of estimates obtained using a gradient estimator, which is intrinsic to many ESC algorithms. The adaptive step size ensures that the integral-action produced by the ESC control law does not destabilize the closed-loop system. To quantify the quality of the gradient estimate, we present a batch least-squares (BLS) estimator with a novel weighting term and guarantee that the gradient estimation error is bounded. The adaptive step size then maximizes the decrease of the combined plant and controller Lyapunov function for the worst-case estimation error. We also ensure that our ESC controller is input-to-state stable with respect to a class of dither signals. Finally, we demonstrate our ESC controller through benchmark examples and a practical application: leak detection with drones.