A simple scheme for semi-recursive identification of Hammerstein system nonlinearity by Haar wavelets

• Autorzy: Przemysław Śliwiński , Zygmunt Hasiewicz , Paweł Wachel
• Języki publikacji: EN

Abstrakty

EN

A simple semi-recursive routine for nonlinearity recovery in Hammerstein systems is proposed. The identification scheme is based on the Haar wavelet kernel and possesses a simple and compact form. The convergence of the algorithm is established and the asymptotic rate of convergence (independent of the input density smoothness) is shown for piecewiseLipschitz nonlinearities. The numerical stability of the algorithm is verified. Simulation experiments for a small and moderate number of input-output data are presented and discussed to illustrate the applicability of the routine.

Słowa kluczowe

EN

Hammerstein system; non-parametric recursive identification; Haar orthogonal expansion; convergence analysis; numerical stability

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2013
• Tom: 23
• Numer: 3
• Strony: 507-520

Daty

wydano

2013

otrzymano

2012-05-09

poprawiono

2012-12-30

Twórcy

autor

Przemysław Śliwiński

• Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, Wrocław, Poland

autor

Zygmunt Hasiewicz

• Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, Wrocław, Poland

autor

Paweł Wachel

• Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, Wrocław, Poland

Bibliografia

• Capobianco, E. (2002). Hammerstein system representation of financial volatility processes, The European Physical Journal B: Condensed Matter 27(2): 201-211.
• Chen, H.-F. (2004). Pathwise convergence of recursive identification algorithms for Hammerstein systems, IEEE Transactions on Automatic Control 49(10): 1641-1649.
• Chen, H.-F. (2010). Recursive identification for stochastic Hammerstein systems, in F. Giri and E.W. Bai (Eds.), Block-oriented Nonlinear System Identification, Lecture Notes in Control and Information Sciences, Vol. 404, Springer-Verlag, Berlin/Heidelberg, pp. 69-87.
• Chen, S., Billings, S.A. and Luo, W. (1989). Orthogonal least squares methods and their application to non-linear system identification, International Journal of Control 50(5): 1873-1896.
• Chen, W., Khan, A.Q., Abid, M. and Ding, S.X. (2011). Integrated design of observer based fault detection for a class of uncertain nonlinear systems, International Journal of Applied Mathematics and Computer Science 21(3): 423-430, DOI: 10.2478/v10006-011-0031-0.
• Clancy, E.A., Liu, L., Liu, P. and Moyer, D.V.Z. (2012). Identification of constant-posture EMG-torque relationship about the elbow using nonlinear dynamic models, IEEE Transactions on Biomedical Engineering 59(1): 205-212.
• Coca, D. and Billings, S.A. (2001). Non-linear system identification using wavelet multiresolution models, International Journal of Control 74(18): 1718-1736.
• Gallman, P. (1975). An iterative method for the identification of nonlinear systems using a Uryson model, IEEE Transactions on Automatic Control 20(6): 771-775.
• Gomes, S.M. and Cortina, E. (1995). Some results on the convergence of sampling series based on convolution integrals, SIAM Journal on Mathematical Analysis 26(5): 1386-1402.
• Greblicki, W. (2002). Stochastic approximation in nonparametric identification of Hammerstein systems, IEEE Transactions on Automatic Control 47(11): 1800-1810.
• Greblicki, W. (2004). Hammerstein system identification with stochastic approximation, International Journal of Modelling and Simulation 24(2): 131-138.
• Greblicki, W. and Pawlak, M. (1986). Identification of discrete Hammerstein system using kernel regression estimates, IEEE Transactions on Automatic Control 31(1): 74-77.
• Greblicki, W. and Pawlak, M. (1987). Necessary and sufficient consistency conditions for a recursive kernel regression estimate, Journal of Multivariate Analysis 23(1): 67-76.
• Greblicki, W. and Pawlak, M. (1989). Recursive nonparametric identification of Hammerstein systems, Journal of the Franklin Institute 326(4): 461-481.
• Greblicki, W. and Pawlak, M. (2008). Nonparametric System Identification, Cambridge University Press, New York, NY.
• Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression, Springer-Verlag, New York, NY.
• Hasiewicz, Z. (1999). Hammerstein system identification by the Haar multiresolution approximation, International Journal of Adaptive Control and Signal Processing 13(8): 697-717.
• Hasiewicz, Z. (2000). Modular neural networks for non-linearity recovering by the Haar approximation, Neural Networks 13(10): 1107-1133.
• Hasiewicz, Z., Pawlak, M. and Śliwiński, P. (2005). Non-parametric identification of non-linearities in block-oriented complex systems by orthogonal wavelets with compact support, IEEE Transactions on Circuits and Systems I: Regular Papers 52(1): 427-442.
• Hasiewicz, Z. and Sliwiński, P. (2002). Identification of non-linear characteristics of a class of block-oriented non-linear systems via Daubechies wavelet-based models, International Journal of Systems Science 33(14): 1121-1144.
• Jyothi, S.N. and Chidambaram, M. (2000). Identification of Hammerstein model for bioreactors with input multiplicities, Bioprocess Engineering 23(4): 323-326.
• Krzyżak, A. (1986). The rates of convergence of kernel regression estimates and classification rules, IEEE Transactions on Information Theory 32(5): 668-679.
• Krzyżak, A. (1992). Global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation, IEEE Transactions on Information Theory 38(4): 1323-1338.
• Krzyżak, A. (1993). Identification of nonlinear block-oriented systems by the recursive kernel estimate, Journal of the Franklin Institute 330(3): 605-627.
• Krzyżak, A. and Pawlak, M. (1984). Distribution-free consistency of a nonparametric kernel regression estimate and classification, IEEE Transactions on Information Theory 30(1): 78-81.
• Kukreja, S., Kearney, R. and Galiana, H. (2005). A least-squares parameter estimation algorithm for switched Hammerstein systems with applications to the VOR, IEEE Transactions on Biomedical Engineering 52(3): 431-444.
• Kushner, H.J. and Yin, G.G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd Edn., Stochastic Modelling and Applied Probability, Springer, New York, NY.
• Lortie, M. and Kearney, R.E. (2001). Identification of time-varying Hammerstein systems from ensemble data, Annals of Biomedical Engineering 29(2): 619-635.
• Mallat, S.G. (1998). A Wavelet Tour of Signal Processing, Academic Press, San Diego, CA.
• Marmarelis, V.Z. (2004). Nonlinear Dynamic Modeling of Physiological Systems, IEEE Press Series on Biomedical Engineering, Wiley-IEEE Press, Piscataway, NJ.
• Nordsjo, A. and Zetterberg, L. (2001). Identification of certain time-varying nonlinear Wiener and Hammerstein systems, IEEE Transactions on Signal Processing 49(3): 577-592.
• Patan, K. and Korbicz, J. (2012). Nonlinear model predictive control of a boiler unit: A fault tolerant control study, International Journal of Applied Mathematics and Computer Science 22(1): 225-237, DOI: 10.2478/v10006-012-0017-6.
• Pawlak, M. and Hasiewicz, Z. (1998). Nonlinear system identification by the Haar multiresolution analysis, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 45(9): 945-961.
• Pawlak, M., Rafajłowicz, E. and Krzyżak, A. (2003). Postfiltering versus prefiltering for signal recovery from noisy samples, IEEE Transactions on Information Theory 49(12): 3195-3212.
• Rutkowski, L. (1984). On nonparametric identification with prediction of time-varying systems, IEEE Transactions on Automatic Control 29(1): 58-60.
• Rutkowski, L. (2004). Generalized regression neural networks in time-varying environment, IEEE Transactions on Neural Networks 15(3): 576-596.
• Saeedi, H., Mollahasani, N., Moghadam, M.M. and Chuev, G.N. (2011). An operational Haar wavelet method for solving fractional Volterra integral equations, International Journal of Applied Mathematics and Computer Science 21(3): 535-547, DOI: 10.2478/v10006-011-0042-x.
• Sansone, G. (1959). Orthogonal Functions, Interscience, New York, NY.
• Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York, NY.
• Skubalska-Rafajłowicz, E. (2001). Pattern recognition algorithms based on space-filling curves and orthogonal expansions, IEEE Transactions on Information Theory 47(5): 1915-1927.
• Śliwiński, P. (2010). On-line wavelet estimation of Hammerstein system nonlinearity, International Journal of Applied Mathematics and Computer Science 20(3): 513-523, DOI: 10.2478/v10006-010-0038-y.
• Śliwiński, P. (2013). Nonlinear System Identification by Haar Wavelets, Lecture Notes in Statistics, Vol. 210, Springer-Verlag, Heidelberg.
• Stone, C.J. (1980). Optimal rates of convergence for nonparametric regression, Annals of Statistics 8(6): 1348-1360.
• Szego, G. (1974). Orthogonal Polynomials, 3rd Edn., American Mathematical Society, Providence, RI.
• Van der Vaart, A. (2000). Asymptotic Statistics, Cambridge University Press, Cambridge.
• Vörös, J. (2003). Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones, IEEE Transactions on Automatic Control 48(12): 2203-2206.
• Walter, G.G. and Shen, X. (2001). Wavelets and Other Orthogonal Systems With Applications, 2nd Edn., Chapman & Hall, Boca Raton, FL.
• Westwick, D.T. and Kearney, R.E. (2003). Identification of Nonlinear Physiological Systems, IEEE Press Series on Biomedical Engineering, Wiley-IEEE Press, Piscataway, NJ.
• Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral: An Introduction to Real Analysis, Pure and Applied Mathematics, Marcel Dekker Inc., New York, NY.
• Zhou, D. and DeBrunner, V.E. (2007). Novel adaptive nonlinear predistorters based on the direct learning algorithm, IEEE Transactions on Signal Processing 55(1): 120-133.

Identyfikatory

DOI

10.2478/amcs-2013-0039