Fixed-Time Seeking and Tracking of Time-Varying Extrema

Motivated by recent (semi-global practical) fixed-time convergence results in time-invariant model-free optimization problems, in this paper we introduce new tracking bounds and guidelines for the design of extremum seeking controllers in model-free optimization problems with dynamic cost functions. Using semi-global practical input-to-state stability characterizations, we show that the proposed non-smooth ES dynamics are able to significantly reduce the tracking error compared to the traditional smooth algorithms studied in the literature. Moreover, under a suitable tuning of the gains of the algorithm, the nominal average dynamics of the controller are able to achieve global fixed-time tracking for a general class of dynamic cost functions. For tuning parameters that do not completely eliminate the tracking error in the nominal average dynamics, but which preserve the continuity of the vector field, we show that "almost complete" error rejection is achieved whenever the gain of the algorithm exceeds a particular threshold. Numerical results are presented to illustrate the performance of the algorithms.