Observer-Based Extremum Seeking Control of Static Maps with Delays

This paper develops a multivariable extremum-seeking control technique for static maps with constant and known input-output delays. The technique is based on the estimation of the gradient of the map as an unknown time-varying parameter using an infinite-dimensional observer. Un-like conventional extremum-seeking controllers, the proposed delay-compensated controller does not require the perturbation signal to be periodic, nor does it require any averaging analysis to guarantee closed-loop stability. Using Lyapunov stability analysis, we show that the observer-coupled closed-loop system exponentially converges to a small neighborhood of the origin. The effectiveness of the controller is demonstrated using a numerical simulation.