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.
Summary
In this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions for the asymptotic stability in the sense of Lyapunov. The results generalize and extend existing results which are based on Lie bracket approximations. In particular, our approach does not rely on singular perturbation theory, as commonly used in extremum seeking of nonlinear dynamic systems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.