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Mathematical Problems in Engineering
Volume 2014, Article ID 650737, 10 pages
Research Article

On the Relation between NARX Clusters and Even/Odd Nonlinearities through Frequency-Domain Analysis

1Institute of Aeronautics and Space, Praça Marechal Eduardo Gomes 50, 12228-904 São José dos Campos, SP, Brazil
2Technological Institute of Aeronautics, Praça Marechal Eduardo Gomes 50, 12228-900 São José dos Campos, SP, Brazil

Received 22 July 2014; Revised 13 November 2014; Accepted 16 November 2014; Published 7 December 2014

Academic Editor: Evangelos J. Sapountzakis

Copyright © 2014 Alexandro G. Brito et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Although polynomial NARX models have been intensively used in nonlinear system identification, few papers discussed how to relate the inner nonlinearities to specific types of clusters and regressors. The objective of this paper is to discuss this relationship for a class of systems that contain even or odd nonlinearities. This class covers block-structured models (Hammerstein, Wiener, and others) and systems with dynamic nonlinearities. To achieve the paper’s aim, a deep frequency-domain analysis is performed. For each type of nonlinearity, all the NARX clusters are investigated and the results show that each regressor type provides specific nonlinear contribution. The investigation is based on an output power spectra analysis when a specific multisinusoidal excitation is applied. According to the spectral contributions in some of the frequency lines, the nonlinearity classification is possible. By applying the same procedure to the clusters, one interprets how these clusters can (or not) contribute to explain the system nonlinearity. The paper findings have two major impacts: (i) one gains deep knowledge on how the nonlinearities are coded by the clusters, and (ii) this information can be used, for instance, to aid a structure selection procedure (ERR, term clustering, etc.) during the discarding of the clusters which are not able to explain the system nonlinear behavior. Some practical and experimental aspects are discussed, while numerical examples are presented to show the validity of the theoretical analysis.