On stability properties of a simple extremum seeking scheme

In this paper, we introduce a simple extremum seeking scheme, which is the simplest variant of schemes considered in the literature. A new definition of semi-global practical asymptotic (SPA) stability uniform in small parameters is proposed. It is shown that the closed loop system with the extremum seeking controller is SPA stable in controller parameters. Several SPA stability results that require different tuning of parameters are presented under different conditions. We note that this simple extremum seeking scheme allows for a novel interpretation of the extremum seeking control via gradient optimization algorithms.