Extremum Seeking for the First Derivative of Nonlinear Maps with Constant Delays via a Time-Delay Approach

This paper introduces an extremum seeking (ES) scheme for the unknown map’s first derivative by tailoring a demodulation signal in which the closed-loop system is subject to constant transmission delays. Unlike most publications that manage delays using predictor-based methods, we are concerned with the delay-robustness of the introduced ES system via the newly developed time-delay approach. The original ES system is transformed to a nonlinear retarded-type plant with disturbances and the stability condition in the form of linear matrix inequalities is achieved. When the related bounds of the nonlinear map are not known, a rigorous practical stability proof is provided. Second, and more importantly, under the availability of prior knowledge about the nonlinear map, we are able to provide a quantitative calculation on the maximum allowable delay, the upper bound of the dither period, and the ultimate seeking error. Numerical examples are offered to exemplify the effectiveness of the proposed method.