In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields. * This work was supported in part by the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft (EB 425/4-1). Corresponding author V. Grushkovskaya.

This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.

No abstract