On Function Extrapolation by Fixed Point Iteration for Time-Delayed Systems

In practical applications the problem of numerical extrapolation of the value of a function over a discrete time-grid so that the previous values can be only “experimentally” observed, often arises. For instance, the extrapolation of the response of a dynamic system that is controlled on the basis of an available approximate dynamic model in adaptive control is a typical example. A quite wide class of controllers e.g. the Optimal Controllers realized by Nonlinear Programming over a discrete time-grid as Receding Horizon Controllers are good examples, too. In 2009 a novel adaptive controller design method was suggested that in the first step transforms the control problem into a fixed point iteration task then it so solves this problem that during one digital control step only one step of the iteration can be realized. This controller adaptively learns from the past. The method widely was investigated for various systems that were free from considerable time-delay, but only preliminary investigations were done how the delay in using the observed data and in exerting the calculated control signal concerns this method. In the present paper systematic investigations restricted to the approximation of known functions are presented in order to better reveal the approximation issues. On the basis of these investigations the usefulness of the suggested iterative method is concluded.