The Use of Multiple Components Fixed Point Iteration in the Adaptive Control of Single Variable Systems

Abstract

In the practice as e.g., in robotics, life sciences, chemistry, normally strongly nonlinear systems have to be controlled on the basis of imprecise and often partial system models that -in spite of their imperfectness- reveal significant features of the systems under control. In these cases the use of the completely data-driven, “model-free; approach is not exclusively advantageous, since the a priori information encoded in the available model can be utilized, too. The consequences of the modeling imprecisions can be compensated by the use of either robust or adaptive techniques. The prevailing adaptive approach uses Lyapunov functions that often need complete state estimation that in the lack of the necessary sensors- cannot be executed because of missing information. An alternative approach transformed the control task into a fixed point iteration that can be solved real-time without the need of complete state estimation. It is a mathematical framework based on Banach's fixed point theorem, and it can be realized by the combination of various components. A solution to multiple variable system was recently published. In the present paper this method is used for single input - single output systems by applying it to shorter time series of the state variable. This structure allows the application of a simple noise filtering technique that is investigated by numerical simulations in the case of the adaptive control of a nonlinear paradigm also containing delay in its model. The results support the idea that this simple noise filtering technique can be useful.