Constructive method for unbiased extremum seeking of static maps in the presence of delays

In this paper, we study the unbiased extremum seeking (ES) algorithm for n-dimensional uncertain quadratic static maps in the presence of time-varying measurement delays. For the first time, we present a quantitative analysis of the unbiased ES. We consider delays with a large known constant part and a small time-varying uncertainty. Such delays may arise when measurements together with a time stamp are transmitted to ES controller via communication network. For the quantitative bounds, we assume that the Hessian is uncertain from a known range. By applying a delay-free transformation, explicit quantitative conditions in terms of simple scalar inequalities depending on the tuning parameters are established which ensure the exponential unbiased convergence of the ES system. Moreover, the corresponding results for the classical ES are presented. For globally quadratic maps, our results are semi-global, whereas for locally quadratic static maps, we provide a bound for the region of convergence. Appropriate ES parameters can be found for any large known delay and small enough delay uncertainty. Two numerical examples from the literature illustrate the efficiency of the proposed method.