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This paper provides the first extensive comparison of NARX and NARMAX models for multi-step prediction. NARMAX models are more complex and require more sophisticated training procedures than NARX models, as described in this paper. However, for some types of data, the NARMAX models provide significantly improved forecasts for shorter prediction horizons, although the predictions converge as the horizon increases. This paper analyzes the NARX and NARMAX predictors in depth on simulated and experimental data to determine those situations where the NARMAX predictor is preferred. • This is the first comparison of NARX and NARMAX multi-step predictors. • NARMAX models should be trained to minimize multi-step prediction errors. • NARMAX models are best when systems have complex disturbances. • NARMAX improvements for multi-step predictions occur in the early steps. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new, more efficient, methods for training recurrent neural networks (RNNs) for system identification and Model Reference Control (MRC). These methods are based on a new understanding of the error surfaces of RNNs that has been developed in recent years. These error surfaces contain spurious valleys that disrupt the search for global minima. The spurious valleys are caused by instabilities in the networks, which become more pronounced with increased prediction horizons. The new methods described in this paper increase the prediction horizons in a principled way that enables the search algorithms to avoid the spurious valleys. The work also presents a novelty sampling method for collecting new data wisely. A clustering method determines when an RNN is extrapolating, which occurs when the RNN operates outside the region spanned by the training set, where adequate performance cannot be guaranteed. The new method presented in this paper uses a clustering method for extrapolation detection, and then the novel data is added to the original training set. The network performance is improved when additional training is performed with the augmented data set. The new techniques are applied to the model reference control of a magnetic levitation system. The techniques are tested on both simulated and experimental versions of the system. [ABSTRACT FROM AUTHOR]