Fixed-Time Newton-Like Extremum Seeking

In this paper, we present a novel Newton-based extremum seeking controller for the solution of multivariable model-free optimization problems in static maps. Unlike existing asymptotic and fixed-time results in the literature, we present a scheme that achieves (practical) fixed time convergence to a neighborhood of the optimal point, with a convergence time that is independent of the initial conditions and the Hessian of the cost function, and therefore can be arbitrarily assigned a priori by the designer via an appropriate choice of parameters in the algorithm. The extremum seeking dynamics exploit a class of fixed time convergence properties recently established in the literature for a family of Newton flows, as well as averaging results for perturbed dynamical systems that are not necessarily Lipschitz continuous. The proposed extremum seeking algorithm is model-free and does not require any explicit knowledge of the gradient and Hessian of the cost function. Instead, real-time optimization with fixed-time convergence is achieved by using real time measurements of the cost, which is perturbed by a suitable class of periodic excitation signals generated by a dynamic oscillator. Numerical examples illustrate the performance of the algorithm.