Generalized kernel regression estimate for the identification of Hammerstein systems

Mzyk, G.

International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 2, pp. 189-197

Résumé

A modified version of the classical kernel nonparametric identification algorithm for nonlinearity recovering in a Hammerstein system under the existence of random noise is proposed. The assumptions imposed on the unknown characteristic are weak. The generalized kernel method proposed in the paper provides more accurate results in comparison with the classical kernel nonparametric estimate, regardless of the number of measurements. The convergence in probability of the proposed estimate to the unknown characteristic is proved and the question of the convergence rate is discussed. Illustrative simulation examples are included.

Détail

Keywords: Hammerstein system, nonparametric regression, kernel estimation
Mots-clés : system Hammersteina, regresja nieparametryczna, estymacja jądra

Bibliographie

[1] Bai E.W. (2003): Frequency domain identification of Hammerstein models. - IEEE Trans. Automat. Contr., Vol. 48, No. 4, pp. 530-542.

[2] Bai E.W. and Li D. (2004): Convergence of the iterative Hammerstein system identification algorithm. - IEEE Trans. Automat. Contr., Vol. 49, No. 11, pp. 1929-1940.

[3] Billings S.A. and Fakhouri S.Y. (1982): Identification of systems containing linear dynamic and static nonlinear elements. - Automat., Vol. 18, No. 1, pp. 15-26.

[4] Chang F.H.I. and Luus R. (1971): A non-iterative method for identification using Hammerstein model. - IEEE Trans. Automat. Contr., Vol. AC-16, No. 4, pp. 464-468.

[5] Chen H.F. (2005): Strong consistency of recursive identification for Hammerstein systems with discontinuous piecewiselinear memoryless block. -IEEE Trans. Automat. Contr., Vol. 50, No. 10, pp. 1612-1617.

[6] Giannakis G.B. and Serpedin E. (2001): A bibliography on nonlinear system identification. - Signal Process., Vol. 81, No. 3, pp. 533-580.

[7] Giunta G., Jacovitti G. and Neri A. (1991): Bandpass nonlinear system identification by higher cross correlation. - IEEE Trans. Signal Process., Vol. 39, No. 9, pp. 2092-2095.

[8] Gomez J.C. and Basualdo M. (2000): Nonlinear model identification of batch distillation process. - Proc. Int. IFAC Symp. Advanced Control of Chemical Processes, ADCHEM, Pisa, Italy, pp. 953-959.

[9] Greblicki W. (1989): Nonparametric orthogonal series identification of Hammerstein systems. - Int. J. Syst. Sci., Vol. 20, No. 12, pp. 2355-2367.

[10] Greblicki W. (2001): Recursive identification of Wiener systems. -Int. J. Appl. Math. Comp. Sci., Vol. 11, No. 4, pp. 977-991.

[11] Greblicki W., Krzyżak A. and Pawlak M. (1984): Distribution free pointwise consistency of kernel regression estimate.- Ann. Stat., Vol. 12, No. 4, pp. 1570-1575.

[12] Greblicki W. and Pawlak M. (1986): Identification of discrete Hammerstein systems using kernel regression estimates.- IEEE Trans. Automat. Contr., Vol. 31, No. 1, pp. 74-77.

[13] Greblicki W. and Pawlak M. (1989): Nonparametric identification of Hammerstein systems. - IEEE Trans. Inf. Theory, Vol. 35, No. 2, pp. 409-418.

[14] Greblicki W. and Pawlak M. (1994): Cascade non-linear system identification by a non-parametric method. - Int. J. Syst. Sci., Vol. 25, No. 1, pp. 129-153.

[15] Haber M. and Keviczky L. (1999): Nonlinear System Identification - Input-Output Modeling Approach. - Dordrecht: Kluwer.

[16] Haber R. and Zeirfuss P. (1988): Identification of an electrically heated heat exchanger by several nonlinear models using different structures and parameter estimation methods. - Tech. Rep., Inst. Machine and Process Automation, Technical University of Vienna, Austria.

[17] Hannan E.J. and Deistler M. (1998): The Statistical Theory of Linear Systems. -New York: Wiley.

[18] Hasiewicz Z. and Mzyk G. (2004a): Combined parametricnonparametric identification of Hammerstein systems. - IEEE Trans. Automat. Contr., Vol. 49, No. 8, pp. 1370-1376.

[19] Hasiewicz Z. and Mzyk G. (2004b): Nonparametric instrumental variables for Hammerstein system identification. -Int. J. Contr., (submitted).

[20] Hasiewicz Z., Pawlak M. and Śliwiński P. (2005): Nonparametric identification of nonlinearities in block-oriented systems by orthogonal wavelets with compact support. - IEEE Trans. Circ. Syst. I: Fund. Theory Applic., Vol. 52, No. 2, pp. 427-442.

[21] Härdle W. (1990): Applied Nonparametric Regression. - Cambridge: Cambridge University Press.

[22] Janczak A. (1999): Parameter estimation based fault detection and isolation in Wiener and Hammerstein systems. - Int. J. Appl. Math. Comput. Sci., Vol. 9, No. 3, pp. 711-735.

[23] Jang W. and Kim G. (1994): Identification of loudspeaker nonlinearities using the NARMAX modelling technique. - J. Audio Eng. Soc., Vol. 42, No. 1/2, pp. 50-59.

[24] Krzyżak A. (1990): On estimation of a class of nonlinear systems by the kernel regression estimate.-IEEE Trans. Inf.Theory, Vol. IT-36, No. 1, pp. 141-152.

[25] Krzyżak A., Sąsiadek J. and Kégl B. (2001): Identification of dynamic nonlinear systems using the Hermite series approach. - Int. J. Syst. Sci., Vol. 32, No. 10, pp. 1261-1285.

[26] Latawiec K.J. (2004): The Power of Inverse Systems in Linear and Nonlinear Modeling and Control. - Opole: Opole University of Technology Press.

[27] Ljung L. (1987): System Identification: Theory for the User. - Englewood Cliffs, NJ: Prentice Hall.

[28] Narendra K.S. and Gallman P.G. (1966): An iterative method for the identification of nonlinear systems using the Hammerstein model. - IEEE Trans. Automat. Contr., Vol. 11, No. 3, pp. 546-550.

[29] Pawlak M. and Hasiewicz Z. (1998): Nonlinear system identification by the Haar multiresolution analysis. - IEEE Trans. Circ. Syst., Vol. 45, No. 9, pp. 945-961.

[30] Söderström T. and Stoica P. (1982): Instrumental-variable methods for identification of Hammerstein systems. - Int. J. Contr., Vol. 35, No. 3, pp. 459-476.

[31] Söderström T. and Stoica P. (1989): System Identification. - Englewood Cliffs, NJ: Prentice Hall.

[32] Van den Hof P., Heuberger P. and Bokor J. (1995): System identification with generalized orthonormal basis functions. - Automatica, Vol. 31, No. 12, pp. 1821-1834.

[33] Vörös J. (1999): Iterative algorithm for identification of Hammerstein systems with two-segment nonlinearities.-IEEE Trans. Automat. Contr., Vol. 44, No. 11, pp. 2145-2149.

[34] Wand M.P. and Jones H.C. (1995): Kernel Smoothing. - London: Chapman and Hall.

[35] Zhang Y.K. and Bai E.W. (1996): Simulation of spring discharge from a limestone aquifer in Iowa. - Hydrogeol. J., Vol. 4, No. 4, pp. 41-54.

[36] Zhu Y. (2000): Identification of Hammerstein models for control using ASYM. - Int. J. Contr., Vol. 73, No. 18, pp. 1692-1702.