A Time-delay Approach for Sampled-data and Delayed Scalar Extremum Seeking

This paper proposes a constructive approach for sampled-data delayed extremum seeking (ES) by using two time-delay approaches: one to averaging and another one to sampled-data control. We first investigate the continuous-time ES with square wave dithers and then expand the proposed time-delay method to its sampled-data implementation with constant delay. By transforming the ES system to a time-delay system, we have developed an improved stability analysis via a novel Lyapunov-Krasovskii functional (LKF). We derive the practical stability conditions in terms of linear matrix inequalities (LMIs) for the resulting time-delay system. Under the assumption of some known bounds on the extremum value of the map and its Hessian, the time-delay approach offers a quantitative calculation on the upper bounds of the dither-sampling period and the constant delays that the ES system is able to tolerate, as well as the ultimate bound of the extremum seeking error. The proposed method also provides a bound on the initial deviation starting from which the solution is ultimately bounded.