Heuristic Order Reduction of NARX-OBF models Applied to Nonlinear System Identification

Elder Oroski (Federal University of Technology of Paraná (UTFPR), Avenida Sete de Setembro 3165, 80230-901, Curitiba-PR, Brazil.)
Beatriz Pês (Federal Institute of Paraná (IFPR), Rua Engenheiro Tourinho, 829, Campo Largo-PR, Brazil)
Adolfo Bauchspiess (University of Brasília (UnB), Campus Darcy Ribeiro, Asa Norte, Brasília-DF, Brazil.)
Marco Antonio Freitas do Egito Coelho (University of Brasília (UnB), Campus Darcy Ribeiro, Asa Norte, Brasília-DF, Brazil.)


Nonlinear system identification concerns the determination of the modelstructure and its parameters. Although the designers often seek the bestmodel for each system, it can be tricky to determine, at the same time, thebest structure and the parameters which optimize the model performance.This paper proposes the use of a Genetic Algorithm, GA, and the Levenberg-Marquardt, LM, method to obtain the model parameters, as well asperform the order reduction of the model. In order to validate the proposedmethodology, the identification of a magnetic levitator, operating in closedloop, was performed. The class NARX-OBF, Nonlinear Auto Regressivewith eXogenous input-Orthonormal Basis Function, was used. The use ofOBF functions aims to reduce the number of terms in NARX models. Oncethe model is found, the order reduction is performed using GA and LM, ina hybrid application, capable of determining the model parameters and reducing the original model order, simultaneously. The results show, considering the inherent trade-of between accuracy and computational effort, theproposed methodology provided an implementation with good mean squareerror, when compared with the full NARX-OBF model.


NARX-OBF Models;Genetic Algorithm;Levenberg Marquardt;System identification

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[1] L. Ljung. System Identification - Theory for the user, Prentice Hall - PTR, 1999.

[2] K. J. Aström, T. Hägglund. Advanced PID Control, ISA, 2011.

[3] S. A. Billings. Nonlinear System Identification: NAR-MAX Methods in the Time, Frequency and Spatio-Temporal Domains, John Wiley & Sons, 1st. ed., 2013.

[4] R. Isermann, M. Munchhof. Identification of Dynamic Systems: An Introduction with Applications. Springer-Verlag Berlin Heidelberg, 2011.

[5] S. Silva. Nonlinear Mechanical System Identification using Discrete-Time Volterra Models and Kautz Filter, 9th. Brazilian Conference on Dynamics, Control and their Ap-plications, 2010.

[6] A. Rahrooh, S. Shepard. Identification of Nonlinear Systems using NARMAX Models, Elsevier - Nonlinear Analisis, n. 71, pp. 1198-1202, 2009.

[7] O. Akanyeti, I. Rañó, U. Nehmzow, S. A. Billings. An application of Lyapunov stability analysis to improve the performance of NARMAX models, Robotics and Autonomous Systems, 2009, 58: 229-238.

[8] P. S. C. Heuberger, P. M. J. Van der Hof, B. Wahlberg. Modelling and Identification with Rational Orthogonal Basis Functions. Springer Press, 1st. Edition, 2005.

[9] D. T. Lemma, M. Ramasamy, M. Shuhaimi. Closedloop Identification of Systems with uncertain Time De-lays using ARX-OBF structure, Journal of Process Control, 2011, 21(8): 1148-1154.

[10] A. da Rosa, R. J. G. Campello, W. C. Amaral. Exact Search Directions for Optimizations of Linear and Non-linear Models based on Generalized Orthonormal Functions, IEEE Trans. on Automatic Control, 2009, 54(12): 2757-2772.

[11] E. Oroski, A. Bauchspiess, R. H. Lopes. Identification of a Magnetic Levitator using NARX-OBF Models and Genetic Algorithm, International Journal of Modeling, Identification and Control, 2017, 28(4): 307-317.

[12] H. Akaike. A new Look at the Statistical Model Identification, IEEE Trans. on Automatic Control, 1974, 19(6): 716-723.

[13] C. M. Fonseca, E. M. Mendes, P. J. Fleming, S. A. Billings. Nonlinear Model Term selection with Genetic Algorithms, IEEE Workshop on Natural Algorithms in Signal Processing, 1974,2(27): 1-8.

[14] J. Nocedal, S. Wright. Numerical Optimization, Springer, 1999.

[15] S. Boyd, L. Vandenberghe. Convex Optimization, Cambridge Press, 2009.

[16] J. Mockus, W. Eddy, G. Reklaitis. Bayesian Heuristic Approach to Discrete and Global Optimization: Algorithms, Visualization, Software and Applications, Kluwer Academic Publishers, 1st ed., 1997.

[17] J. Koza. Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, 6th ed, 1998.

[18] C. Solnon. Ant Colony Optimization and Constraint Programing, Wiley Press, 1st ed., 2010.

[19] A. E. Olssom. Particle Swarm: Theory, Techniques and Applications, New Science Publishers, 1st ed., 2011.

[20] A. D. Coley. An Introduction to Genetic Algorithms for Scientists and Engineers, Would Scientific, 1999.

[21] D. W. Marquardt. An Algorithm for Least Squares Estimation of Nonlinear Parameters, Journal of Society for Industrial and Applied Mathematics, 1963, 11(2): 431-441.

[22] M. A. E. Coelho. Permanent Magnet based Magnetic Levitation Kit, International Journal of Applied Electromagnetics and Mechanics, 2015, 47: 963-973.

[23] L. Tonks. Note on Earnshaw’s Theorem, Electrical Engineering, 1940, 59(3): 118-119.

DOI: https://doi.org/10.30564/ssid.v1i2.1341


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