Deterministic and Stochastic Newton-based extremum seeking for higher derivatives of unknown maps with delays

We present a Newton-based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of time delays using deterministic perturbations. Different from previous works about extremum seeking for higher derivatives, arbitrarily long input-output delays are allowed. We incorporate a predictor feedback with a perturbation-based estimate for the Hessian's inverse using a differential Riccati equation. As a bonus, the convergence rate of the real-time optimizer can be made user-assignable, rather than being dependent on the unknown Hessian of the higher-derivative map. Averaging method for arbitrary shaped derivatives under delays is presented. Exponential stability and convergence to a small neighbourhood of the unknown extremum point are achieved for locally quadratic derivatives by using a backstepping transformation and averaging theory in infinite dimensions. Furthermore, we give a brief introduction into stochastic Newton-based Extremum Seeking for constant output delays, where we show the differences and similarities with respect to the deterministic case. We also present illustrative numerical examples in order to highlight the effectiveness of the proposed predictor-based extremum seeking for time-delay compensation applying both deterministic and stochastic perturbations.