Abstract
In this paper, advanced concepts for the identification of complex nonlinear systems are discussed. Three major problems are addressed: The nonlinearity of the system, noise in the data upon which the model has to be built, and the potential to incorporate qualitative and quantitative prior knowledge about the system. As an integrated solution approach, local model networks (LMNs) with appropriate parameter estimation schemes are proposed. LMNs generally offer a versatile structure for the identification of nonlinear dynamic systems. In order to account for a realistic situation when noise is present both in input and output data, an equality constrained generalised total least squares algorithm for the local model parameter estimation of the LMN is presented; the incorporation of equality constraints allows to mathematically enforce desired system properties. As an application and benchmark problem, the vertical dynamics of a vehicle is considered. After training the LMN on a rough road, excellent predictions of the behaviour of the vehicle at crossing a single obstacle are obtained, thus proving the effectiveness of the proposed algorithm. It is illustrated how both the application of a proper parameter estimation scheme and the integration of system constraints systematically improve the performance of the model.
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Sjoberg J., Zhang Q., Ljung L., Benveniste A., Deylon B., Glorennec P.-Y., Hjalmarsson H., Juditsky A.: Nonlinear black-box modeling in system identification: a unified overview. Automatica 31, 1691–1724 (1995)
Ljung L.: Perspectives on system identification. Ann. Rev. Control 34, 1–12 (2010)
Murray-Smith R., Johansen T.: Multiple Model Approaches to Modelling and Control. Taylor & Francis, London (1997)
Gao, R.: Local model network application in control. PhD thesis, Dublin Institute of Technology (2004)
Gregorcic G., Lightbody G.: Local model network identification with Gaussian processes. IEEE Trans. Neural Netw. 18, 1404–1423 (2007)
Novak, J., Bobal, V.: Predictive control of the heat exchanger using local model network. In: Proceedings of the 17th Mediterranean Conference on Control and Automation, pp. 657–662 (2009)
Nelles O.: Nonlinear System Identification. Springer, Berlin (2002)
Gregorcic G., Lightbody G.: Nonlinear system identification: from multiple-model networks to Gaussian processes. Eng. Appl. Artif. Intell. 21, 1035–1055 (2008)
Soderstrom T.: Errors-in-variables methods in system identification. Automatica 43, 939–958 (2007)
Van Huffel S., Zha H.: The restricted total least squares problem: formulation, algorithm, and properties. SIAM J. Matrix Anal. Appl. 12, 292–309 (1991)
Nayak A., Trucco E., Thacker N.A.: When are simple LS estimators enough? An empirical study of LS, TLS, and GTLS. Int. J. Comput. Vis. 68, 203–216 (2006)
Paige C.C., Wei M.: Analysis of the generalized total least squares problem AX = B when some columns of A are free of error. Numer. Math. 65, 177–202 (1993)
Van Huffel S., Vandewalle J.: Analysis and properties of the generalized total least squares problem AX = B when some or all columns of A are subject to error. SIAM J. Matrix Anal. Appl. 10, 294–315 (1989)
Milanese M., Novara C.: Set membership identification of nonlinear systems. Automatica 40, 957–975 (2004)
Sun, F., Cui, Y.: Influence of parameter variations on system identification of full car model. In: Proceedings of the International MultiConference of Engineers and Computer Scientists 2011, pp. 839–843 (2011)
Milanese, M., Novara, C., Mastronardi, F., Amoroso, D.: Experimental modeling of vertical dynamics of vehicles with controlled suspensions. SAE Paper Nr. 2004-01-1546. doi:10.4271/2004-01-1546 (2004)
Breiman L., Friedman J., Stone C.J., Olshen R.: Classification and Regression Trees. Chapman and Hall/CRC, London (1984)
Cherkassky V., Ma Y.: Multiple model regression estimation. IEEE Trans. Neural Netw. 16, 785–798 (2005)
Hametner, C., Jakubek, S.: Neuro-fuzzy modelling using a logistic discriminant tree. In: Proceedings of the American Control Conference 2007, pp. 864–869 (2007)
Fletcher R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (1987)
Gill P.E., Murray W., Wright M.H.: Practical Optimization. Academic Press, London (1981)
Dempster A.P., Laird N.M., Rubin D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39, 1–38 (1977)
Hametner, C., Jakubek, S.: Nonlinear identification with local model networks using GTLS techniques and equality constraints. Neural Netw. IEEE Trans. 22, 1406–1418. doi:10.1109/TNN.2011.2159309. URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5958611&isnumber=6006653 (2011)
Mitschke M., Wallentowitz H.: Dynamik der Kraftfahrzeuge, 4. Auflage. Springer, Berlin (2004)
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Dedicated to Professor Hans Irschik on the occasion of his 60th birthday.
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Hametner, C., Edelmann, J., Jakubek, S. et al. An advanced algorithm for partitioning and parameter estimation in local model networks and its application to vehicle vertical dynamics. Acta Mech 223, 1693–1706 (2012). https://doi.org/10.1007/s00707-012-0638-8
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DOI: https://doi.org/10.1007/s00707-012-0638-8