A Simplified Multivariable Gradient Extremum Seeking for Distinct Input Delays with Delay-Independent Convergence Rates

In this paper, we address the design and analysis of multivariable extremum seeking for unknown static maps subject to arbitrarily long time delays. Gradient-based method is considered. Multi-input systems with different time delays in each individual input channel are dealt with. The phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation-based (averaging-based) estimate of the Hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point, even in the presence of delays. Unlike previous publications considering multiparameter extremum seeking and delays, the stability analysis is carried without using backstepping transformation, which also eliminates the complexity of the controller. In a nutshell, a simpler implementation scheme and direct analysis without invoking successive backstepping transformation can be assured. In particular, the delays in our approach are independent of the dither frequency and system's dimension such that fast convergence rates are still guaranteed. A numerical example illustrates the performance of the new delay-compensated extremum seeking scheme and its simplicity.