Nonsmooth Extremum Seeking Control With User-Prescribed Fixed-Time Convergence

Abstract

This article introduces a new class of nonsmooth extremum seeking controllers (ESCs) with convergence bounds given by class- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {K}\mathcal {L}$</tex-math></inline-formula> functions that have a uniformly bounded settling time. These ESCs are characterized by nominal average systems that render <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniformly globally fixed-time stable</i> (UGFxTS), the set of minimizers of the response map of a stable nonlinear plant. Given that, under suitable tuning of the parameters of the controllers, the ESCs inherit the convergence properties of their average systems, the proposed dynamics can achieve a better transient performance compared to the traditional ESCs based on gradient descent or Newton flows. Moreover, for the case when the plant is a static map, the convergence time of the proposed algorithms can be prescribed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> by the users for all initial conditions without the need of retuning the gain of the learning dynamics of the ESC. Since autonomous feedback controllers with fixed-time convergence properties are necessarily non-Lipschitz continuous, standard averaging and singular perturbation tools, traditionally used in ESC, are not applicable anymore. We address this issue by using averaging and singular perturbation tools for nonsmooth and set-valued systems, which further allows us to consider ESCs modeled by discontinuous vector fields that are typical in fixed-time and finite-time optimization problems.