Extremum seeking by a dynamic plant using mixed integral sliding mode controller with synchronous detection gradient estimation

This paper presents a continuous-time optimization method for an unknown convex function restricted to a dynamic plant with an available output including a stochastic noise. For solving the problem, we propose an extremum seeking algorithm based on a modified synchronous detection method for computing a stochastic gradient descent approach. In order to reject from the beginning the undesirable uncertainties and perturbations of the dynamic plant, we employ the standard deterministic integral sliding mode control transforming the initial dynamic plant to the static one, and after (in fact, from the beginning of the process), we apply the gradient decedent technique. We consider time-decreasing parameters for compensating the stochastic dynamics. We prove the stability and the mean-square convergence of the method. To validate the exposition, we perform a numerical example simulation.