A time-delay approach to stabilization by extremum seeking controller via simplified ISS analysis

We study stabilization of linear uncertain systems under unknown control directions using a bounded extremum seeking controller with a small parameter. We consider a small time-varying measurement delay. By using the recently proposed time-delay approach to Lie-Brackets-based averaging, 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. Differently from the existing analysis via Lyapunov-Krasovskii (L-K) method, we transform the neutral system to an ordinary differential equation (ODE) with delayed perturbations and employ variation of constants formula that greatly simplifies the analysis and leads to simpler stability conditions. Two numerical examples illustrate that the proposed method allows essentially larger parameter uncertainties with bounds on the small parameter and time-delay that are not too small.