Stochastic Multivariable Extremum Seeking Control Considering Input and Output Delays

This paper introduces a design and analysis framework for a multivariable gradient-based stochastic extremum-seeking control system that considers delays. The study addresses multi-input systems with time delays in both input and output channels. The introduced new predictor feedback mechanism utilizes stochastic sinusoidal perturbation-based estimates of the Hessian’s matrix in order to compensate the delay effects in the real-time optimizer. The incorporation of this feedback into the closed-loop system ensures exponential stability for the average dynamics and convergence to a small neighborhood around the unknown extremum point. The derived results are rigorously established through the application of backstepping transformation and stochastic averaging techniques in infinite dimensions. The effectiveness of the proposed predictor-based stochastic extremum-seeking approach for delay compensation is illustrated through a numerical example.