Gradient Extremum Seeking with Delays∗

In this paper, we derive the design and analysis for scalar gradient extremum seeking control (ESC) subject to arbitrarily long input-output delays, by employing a predictor with a perturbation-based estimate of the Hessian. Exponential stability and convergence to a small neighborhood of the unknown extremum point can be guaranteed. This result is carried out using backstepping transformation and averaging in in?nite dimensions. Some simulation examples are presented to illustrate the performance of the delay-compensated ESC scheme.