Extremum Seeking for the First Derivative of Nonlinear Maps: A Time-delay Approach

In this paper, an extremum seeking (ES) scheme for the first derivative of an unknown nonlinear map is carried out by tailoring a demodulation signal. We further extend the time-delay approach to the stability analysis of the ES system. Specifically, a precise time-delay model is attained via the time-delay approach to averaging the original ES system. By utilizing the direct Lyapunov method, we obtain the stability conditions in the form of linear matrix inequalities (LMIs). The proposed method offers the quantitative bounds on the dither frequency and ultimate seeking error, under the postulation of a prior knowledge about the nonlinear map. Numerical simulations are provided to exemplify the effectiveness of the proposed method.