Extremum seeking for static map with measurement noise and delays using a one-stage sequential predictor

Extremum seeking (ES) for static quadratic maps, considering the effects of measurement noise, distinct input delays in the ES, and output delay, rather than focussing on distinct delays in the map, is proposed. The modified ES based on the one-stage sequential predictor is proposed. By utilising a constructive time-delay approach to averaging, the estimation error of ES is transformed into the classical averaged system with perturbations, providing an approximation-free representation of the estimation error. Subsequently, the analysis of quantitative mean-square exponential ultimate boundedness (MSEUB) of estimation error is presented. Explicit conditions, presented as simple scalar inequalities, are provided to ensure the MSEUB for ES, along with quantitative bounds on noise intensity tolerance, precise guidance for selecting the dither frequency for ensuring MSEUB, and the mean square ultimate bound (UB) of estimation error. An example from the literature demonstrates the effectiveness of the proposed results.