Gradient Extremum Seeking with Time-Varying Delays

This paper presents a gradient extremum seeking algorithm to cope with locally quadratic static maps under time-varying delays. Dealing with non constant delays has a strong impact in the predictor design in terms of the associated transport partial differential equation (PDE) with variable convection speeds as well as the conditions imposed on the delay regarding its arbitrary duration but bounded variation. A new predictor design with a perturbation-based estimate of the unknown Hessian of the map must be introduced to accommodate this variable nature of the delays, which can arise both in actuation and sensing paths. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are rigorously detailed. This result is achieved by using backstepping transformation and averaging in infinite dimensions. Numerical simulations are given to illustrate the effectiveness of the proposed predictor-based extremum seeking control for time-delay compensation.