Extremum Seeking is Stable for Scalar Maps that are Strictly but Not Strongly Convex

TL;DR

This paper demonstrates that Newton-based extremum seeking control (NESC) remains practically stable for scalar maps that are strictly convex but not strongly convex, even when the Hessian is zero, by leveraging average Hessian estimates.

Contribution

It extends the stability analysis of NESC to cases where the Hessian is zero, showing practical asymptotic stability without requiring strong convexity.

Findings

NESC guarantees practical stability for strictly convex but not strongly convex maps.

Average Hessian estimates remain positive despite zero actual Hessian.

Model-free perturbation-based extremum seeking converges exponentially in practice.

Abstract

For a map that is strictly but not strongly convex, model-based gradient extremum seeking has an eigenvalue of zero at the extremum, i.e., it fails at exponential convergence. Interestingly, perturbation-based model-free extremum seeking has a negative Jacobian, in the average, meaning that its (practical) convergence is exponential, even though the map's Hessian is zero at the extremum. While these observations for the gradient algorithm are not trivial, we focus in this paper on an even more nontrivial study of the same phenomenon for Newton-based extremum seeking control (NESC). NESC is a second-order method which corrects for the unknown Hessian of the unknown map, not only in order to speed up parameter convergence, but also (1) to make the convergence rate user-assignable in spite of the unknown Hessian, and (2) to equalize the convergence rates in different directions for multivariable maps. Previous NESC work established stability only for maps whose Hessians are strictly positive definite everywhere, so the Hessian is invertible everywhere. For a scalar map, we establish the rather unexpected property that, even when the map behind is strictly convex but not strongly convex, i.e., when the Hessian may be zero, NESC guarantees practical asymptotic stability, semiglobally. While a model-based Newton-based algorithm would run into non-invertibility of the Hessian, the perturbation-based NESC, surprisingly, avoids this challenge by leveraging the fact that the average of the perturbation-based Hessian estimate is always positive, even though the actual Hessian may be zero.