Using artificial delays for stabilization of linear second-order systems under unknown control directions by extremum seeking controller

We consider a linear second-order system subject to unknown control directions under the measurements of the position whereas the velocity is not available for measurements. Such system can be stabilized by an extremum seeking (ES) controller using the position and velocity. In this paper, the velocity is approximated via a finite difference leading to a delay-dependent ES controller. By applying the recently proposed time-delay approach to Lie-Brackets-based averaging method, we transform the closed-loop system to a time-delay (neutral type) one, which has a form of perturbed Lie brackets system. The input-to-state stability (ISS) of the time-delay system guarantees the same for the original one. Then, by employing variation of constants formula we derive explicit conditions in terms of simple inequalities for finding the quantitative bounds on the dither period and delay that ensure the regional ISS. An example is provided to illustrate the efficiency of the results.