A Time-delay Approach for Extremum Seeking of Scalar Dynamical Systems

Recently a new time-delay approach was introduced for extremum seeking (ES), but the results were confined to static maps. In the present paper, the time-delay approach is extended for the 1st time to ES of dynamic maps in the case of scalar plants. We not only provide a precise perturbed system of ES without any approximation, but also suggest a direct Lyapunov-Krasovskii method for the transformed time-delay plant to find efficient stability conditions for the closed-loop ES system in the form of linear matrix inequalities (LMIs). Differently from the approaches based on classical averaging and Lie brackets, we provide the quantitative bounds on the frequency and the resulting extremum seeking error, under assumption that the extremum point, the extremum value and the Hessian are uncertain from some known intervals.