Stabilization under unknown control directions and disturbed measurements via a time-delay approach to extremum seeking

We study stabilization of linear uncertain systems under unknown control directions using a bounded extremum seeking controller in the presence of a small time-varying measurement delay. We assume that the measurements are subject to discontinuous disturbances. The main novelty is that these disturbances possess not only the constant part as in the existing results, but also small discontinuous part that may appear due to quantization. We consider two types of measurements: the state measurements and the state quadratic norm ones. In the latter case the constant part of the disturbances may be arbitrary large. 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 with no terms depending on the disturbance derivative, 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. We further transform the neutral system to an ordinary differential equation (ODE) with delayed perturbations and employ variation of constants formula leading to explicit conditions in terms of simple inequalities with less conservative results in the most of numerical examples. Two numerical examples are provided to illustrate the efficiency of the method.