  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.
