Extremum seeking subject to multiple and distinct input delays

After considering the multivariable extremum seeking problem for output-delay systems in a previous work, here we address a more general class of multi-input systems with distinct time delays in each individual input channel. Analogously to the earlier results, the delay compensation is achieved by employing predictor feedback with a perturbation-based estimate of the Hessian. The predictor design for the Newton-based extremum seeking can be derived with the standard backtepping transformation and exploring the decoupling property in the average model of the control channels under different delays. On the other hand, a new backstepping transformation must be applied to obtain the predictor for the Gradient-based extremum seeking and to prove its exponential stability. Another basic difference is that the phase compensation is now carried out in the additive dither unlike the previous framework, where the demodulating signals should be modified. The properties of the proposed delay-compensated extremum seeking approaches are illustrated by some numerical simulations.