Stability of extremum seeking feedback for general nonlinear dynamic systems
While the mainstream methods of adaptive control (both linear and nonlinear) deal only with regulation to known set points or reference trajectories, in many applications the set point should be selected to achieve a maximum of an uncertain reference-to-output equilibrium map. The techniques of the so-called ``extremum control'' or ``self-optimizing control'' developed for this problem in the 1950s--1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment. The main contribution of this paper is that it provides a rigorous proof of stability for an extremum seeking feedback scheme. The tools of averaging and singular perturbations are employed to show that solutions to the closed-loop system converge to a small neighborhood of the extremum of the equilibrium map. The size of the neighborhood is inversely proportional to the adaptation gain and the amplitude and the frequency of a periodic signal used to achieve extremum seeking. This analysis highlights a fundamentally nonlinear mechanism of stabilization in an extremum seeking loop.
scientific article; zbMATH DE number 1437554
scientific article