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. 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. Furthermore, exponential stability and convergence to a small neighborhood of the unknown extremum point can be obtained for locally quadratic derivatives by using backstepping transformation and averaging theory in infinite dimensions. We also give a numerical example in order to highlight the effectiveness of the proposed predictor based extremum seeking for time-delay compensation.