Discrete-time unbiased extremum seeking with quantitative bounds

In this paper, we consider scalar quadratic maps and measurements with large delay. We extend a recently introduced unbiased extremum seeking (ES) to the discrete-time with constructive conditions. We first recover unknown Hessian of the quadratic map on the small time interval. We further propose an unbiased ES algorithm with a delay compensator that leads to a faster decay rate and simpler stability analysis. By employing a delay-free transformation, explicit quantitative conditions on the controller parameters are established for the exponential unbiased convergence of the ES system. Differently from the continuous-time, the small parameter defines the decay rate of the averaged estimation error system and a quantitative bound on this parameter is especially important in the discrete-time case. We also obtain constructive conditions for the practical stability of the classical ES system. Our results are semi-global and appropriate ES parameters can be found for any large delay. Note that there are no existing discrete-time ES results in the presence of delays. A numerical example illustrates the efficiency of the proposed method with the essential improvement of the previous non-delayed results for practical stability.