The use of Mann iteration in the stable inversion of NARX models that have been converted to state space form is investigated to either recover the convergence or improve the accuracy of the best approximate solution under conditions when Picard iteration fails to converge. Attention is given to the use of filtering and time-varying iteration gains. The results are potentially of use in response reconstruction for fatigue testing purposes where the inverse of a NARX model, obtained by system identification, may be used to achieve the reconstruction.
In this paper, we study convergence and data dependence of SP and normal-S iterative methods for the class of almost contraction mappings under some mild conditions. The validity of these theoretical results is confirmed with numerical examples. It has been observed that a special case of SP iterative method, namely, normal-S iterative method, performs better and so the latter is implemented in the stable inversion of nonlinear discrete time dynamical systems to yield convergence results when Picard iterative method diverges. This is also illustrated with a numerical example. Our work extends and improves upon many results existing in the literature.
The conventional Iterative Learning Control (ILC) algorithm for model-based ILC of nonlinear systems is presented with use of a nonlinear inverse model as ILC compensator. The nonlinear inverse model is solved with stable inversion. In addition an alternative ILC algorithm for model-based ILC of nonlinear systems is developed, also with using a nonlinear inverse model as ILC compensator. Some connections between the conventional and alternative ILC algorithms and Picard, Mann and Ishikawa iteration are explored. In a number of theoretical examples the conventional and alternative algorithms are compared.
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