Newton-based extremum seeking of higher-derivative maps with time-varying delays

We present a Newton-based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of time-varying delays. Dealing with time-varying delays has impact in the predictor design in terms of the transport PDE with variable convection speed functions, the backstepping transformation as well as the conditions imposed on the delay. First, the delay can grow at a rate strictly smaller than one but not indefinitely, the delay must remain uniformly bounded. Second, the delay may decrease at any uniformly bounded rate but not indefinitely, that is, it must remain positive. We incorporate a filtered predictor feedback with a perturbation-based estimate for the Hessian's inverse using a differential Riccati equation, where 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 are achieved for locally quadratic derivatives by using backstepping transformation and averaging theory in infinite dimensions.