Extremum seeking of general nonlinear static maps: A time-delay approach

In this paper, we extend a newly developed time-delay approach of extremum seeking (ES) from quadratic maps to general nonlinear maps, which substantially expands the scope of application of the time-delay method. The gradient-based ES in the case of single-variable and multi-variable nonlinear static maps are considered. Different from the recent literature, the time-delay method in this paper is applicable to a big family of nonlinear maps satisfying a couple of mild assumptions. By transforming the original ES system into a kind of time-delay system of neutral type, and further transforming the neutral delay differential equation into the perturbed ordinary differential equation (ODE), we provide simple inequalities that guarantee practical stability of the ES closed-loop system. With a prior knowledge about the upper bounds of the nonlinear map and its gradient and Hessian, the time-delay approach suggests a quantitative calculation on the lower bound of dither frequency and the ultimate upper bound of estimation error, which is difficult to achieve by the classical averaging method.