Multiple Components Fixed Point Iteration in the Adaptive Control of Single Variable 2nd Order Systems

In various fields of applications as robotics, life sciences, chemistry, typical nonlinear systems are controlled for the description of which only imprecise and often partial system models are available and the model-based approaches have to be completed by either robust or adaptive solutions for the compensation of the effects of the models' imperfectness and imprecisions. An alternative approach transformed the control task into a fixed point iteration and tried to solve it in real-time on the basis of Banach's fixed point theorem. This approach does not require complete state estimation. However, it needs feeding back noisy signals that may corrupt the quality of such adaptive controllers. In the preliminary investigations in [1] this multiple variable solution was utilized for the control of a single variable 1st order system so that it yielded some noise filtering possibility. In the present paper this method is extended for the control of single input - single output 2nd order dynamical systems by applying it to shorter time series of the state variable. The operation of this method is investigated via numerical simulations for the control of a strongly nonlinear, oscillating dynamical system, the van der Pol oscillator that further was modified with the introduction of time-delay effects. The simulations are promising.