Delay-robust Unbiased Extremum Seeking for 1D Static Maps via a Delay-Free Transformation

In this paper, we consider the recently introduced extremum seeking (ES) with unbiased convergence of an uncertain 1D static map with a constant and known measurement delay. We present a novel efficient constructive approach for averaging-based stability of the unbiased ES algorithm via a delay-free transformation. The transformed ES system is presented in the form of an averaged linear time-delayed system with perturbations. Then, to find a lower bound on the dither frequency for exponential stability, we employ the variation of constants formula, and exploit tight bounds on the fundamental solution of the averaged system with delay. Explicit quantitative conditions in terms of simple scalar inequalities are established which ensure the exponential unbiased convergence of the original ES system. Moreover, we obtain a simple inequality that guarantees the practical stability of the classical ES control system. The results are semi-global and ES parameters can be found for any large delay. A numerical example from the literature demonstrates the efficiency of the proposed results with the essential improvement of the previous results for practical stability.