Filterless Fixed-Time Extremum Seeking for Scalar Quadratic Maps

In this paper, we study a novel fixed-time extremum-seeking algorithm that eliminates the need for filters to obtain an appropriate estimation of the gradient of a static map for optimization problems where the cost function is available only via measurements or evaluations. Previous research leveraged these filters to facilitate the application of averaging theory in analyzing the stability properties of the system. Specifically, they were employed to separate, using multi-time scale techniques, the non-smooth terms of the algorithms from the rapidly fluctuating oscillatory terms associated with periodic dithers. This separation was achieved through a singular perturbation argument, where the filter acted as boundary layer system with a sufficiently fast transient. However, since in many practical applications such transient cannot be made arbitrarily fast, and since classic extremum-seeking algorithms are also known to be stable even in the absence of filters, it is natural to ask whether the fixed-time extremum-seeking dynamics can also be simplified by removing the filters while achieving semi-global practical fixed-time convergence properties. This paper addresses this question for scalar quadratic cost functions, providing positive and negative answers depending on the structure of the cost. Additionally, we demonstrate that removing the filters results in average dynamics distinct from the conventional fixed-time gradient flow dynamics found in existing literature. Furthermore, we provide numerical examples to illustrate our findings.