Maximizing Map Sensitivity and Higher Derivatives Via Extremum Seeking

Abstract

We present a generalization to the scalar Newton-based extremum seeking algorithm which, through perturbation-induced measurements of an unknown steady-state input-to-output map, maximizes the map's higher derivatives. As with other extremum seeking (ES) problems, the scheme relies on derivative estimators and learning dynamics operating in different time scales. We provide analysis for the typical deterministic (sinusoidal) perturbation of the optimal parameter estimate. Also, we alternatively include a stochastic method where the map is perturbed via the sinusoid of Brownian motion about the boundary of a circle. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the $n{\rm th}$ derivative only through measurements of the map. The Newton-based ES 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 ES. Our design stems from the existing multivariable Newton-based ES 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, whereas employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and compare the Newton-based method against the gradient-based method. We also extend this abstraction to multiplayer noncooperative games but limit our attention to a two-player game for notational simplicity and length considerations.