Extremum Seeking of Static Maps in the Presence of Unknown Large Time-Varying Delays

TL;DR

This paper introduces a robust discrete-time extremum seeking algorithm for static quadratic maps that effectively handles unknown large time-varying delays, ensuring unbiased exponential convergence.

Contribution

It presents the first extremum seeking method that is resilient to large, unknown, time-varying delays with proven convergence guarantees.

Findings

Achieves unbiased exponential convergence despite large delays.

Provides quantitative bounds on controller parameters for convergence.

Demonstrates effectiveness through a numerical example.

Abstract

In this paper, we present the discrete-time unbiased extremum seeking (ES) algorithm for n-dimensional (nD) static quadratic maps in the presence of unknown time-varying measurement delays bounded by known constants which can be large. The existing ES results in the presence of large delays are usually confined to known constant or slowly-varying delays, which is restrictive. We provide the first ES algorithm, which is robust with respect to unknown large time-varying delays. Moreover, we achieve the unbiased exponential convergence. We manage with such delays by choosing dithers with frequencies of the order of \sqrt{\epsilon}, where the small parameter {\epsilon} > 0 appears in the dynamics of the real-time estimator. As expected, larger delays lead to a slower convergence. We provide qualitative and quantitative results based on the averaging analysis via delay-free transformation. For the quantitative bounds on the controller parameters that ensure the exponential unbiased convergence of the ES system, we assume that the Hessian of the map is uncertain and lies within a known range. Differently from its continuous-time counterpart, the small parameter in the discrete-time case defines the decay rate of the estimation error system, making a quantitative bound on this parameter particularly important. We present also constructive conditions for the practical stability of the classical ES system. Our results are semi-global for globally quadratic maps, while for locally quadratic static maps, we provide a bound on the region of convergence. Our analysis shows that appropriate ES parameters can be found for any large unknown time-varying bounded delay. A numerical example highlights the efficiency of the method.