Maximizing higher derivatives of unknown maps with stochastic extremum seeking

We present a stochastic generalization to the scalar Newton-based extremum seeking algorithm which through measurements of an unknown map, maximizes the map's higher derivatives. The proposed method perturbs the estimate of the optimal parameter with the sinusoid of Brownian motion about the boundary of a circle. Then by properly demodulating the map output of the randomly perturbed estimate, the extremum seeking algorithm maximizes the nth derivative only through measurements of the map. The Newton-based extremum seeking approach removes the dependence of the convergence rate on the unknown Hessian of the higher derivative, an effort to improve performance over standard gradient-based extremum seeking. Our design stems from the existing multivariable Newton-based extremum seeking algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, where-as employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear static maps via stochastic averaging theory developed for extremum seeking.