The multidimensional n-th order heavy ball method and its application to extremum seeking

In this paper the extension of the heavy ball method to n-th order integrator dynamics is considered. We propose a gradient based controller that achieves to find the extremum of a function depending on multiple variables and prove asymptotic stability for all functions from the set of strongly convex functions. Furthermore, we propose a gradient-free extremum seeking controller that approximates the proposed gradient-based controller and prove practical asymptotic stability of the extremum using Lie bracket averaging techniques. The result does not rely on singular perturbation methods and provides a new approach to extremum seeking for dynamic maps.