A Constructive Approach to Multi‐Variable Extremum Seeking With Discrete‐Time Delayed Noisy Measurements

ABSTRACT In this paper, we present a constructive method for multivariable gradient‐based ES of static quadratic maps with discrete‐time delayed noisy measurements, and provide mean‐square exponential ultimate boundedness (MSEUB) analysis under essentially relaxed assumption on the unknown Hessian than the previous constructive methods. By employing the time‐delay approach to averaging, we introduce an equivalent, approximation‐free representation of the gradient‐based ES estimation error system, which is the neutral type time‐delay system with stochastic perturbations. We further present the latter system as a perturbed averaged ordinary differential equation and employ the variation of constants formula for the quantitative MSEUB analysis. The explicit condition in terms of simple scalar inequality depending on the dither frequencies and measurement noise intensity is established to guarantee the MSEUB analysis of the ES control system. The bounds on the maximum dither frequencies, measurement noise intensity tolerance margin, and the mean‐square ultimate bound of the estimation error are provided. In addition, a qualitative stability result is also presented. Examples from the literature illustrate the efficiency of the results.