Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations * *This work is supported in part by the Alexander von Humboldt Foundation, the Deutsche Forschungsgemeinschaft (EB 425/4-1), and the State Fund for Fundamental Research of Ukraine (F63-726).

Abstract

Abstract The paper presents a control algorithm that steers a system to an extremum point of a time-varying function. The proposed extremum seeking law depends on values of the cost function only and can be implemented without knowing analytical expression of this function. By extending the Lie brackets approximation method, we prove the local and semi-global practical uniform asymptotic stability for time-varying extremum seeking problems. For this purpose, we consider an auxiliary non-autonomous system of differential equations and propose asymptotic stability conditions for a family of invariant sets. The obtained control algorithm ensures the motion of a system in a neighborhood of the curve where the cost function takes its minimal values. The dependence of the radius of this neighborhood on the bounds of the derivative of a time-varying function is shown.