A Time-Delay Approach to Multi-Variable Extremum Seeking with Measurement Noise

For multi-variable static quadratic map, we present a time-delay approach to gradient-based extremum seeking (ES) with measurement noise, and provide a mean-square exponential ultimate boundedness (MSEUB) analysis. We consider the uncertain map where the Hessian matrix
$H$
has a nominal known part and norm-bounded uncertainty, the extremum point belongs to a known ball, and the extremum value to a known interval. By applying a time-delay approach to the resulting stochastic ES system, we arrive at the neutral type time-delay system with stochastic perturbations. We further present the latter system as a retarded one and employ the variation of constants formula for the MSEUB analysis. Under the assumption that the upper bound of the 6th moment of the estimation error is a known arbitrarily large constant L, explicit condition in terms of simple scalar inequality depending on the bound L, tuning parameters and the intensity of measurement noise is established to guarantee the MSEUB analysis of the ES control systems. Example from the literature illustrates the efficiency of the new approach.