On-line wavelet estimation of Hammerstein system nonlinearity

Przemysław Śliwiński

doi:10.2478/v10006-010-0038-y

.blurhash-client-img { display: none !important; }

On-line wavelet estimation of Hammerstein system nonlinearity

International Journal of Applied Mathematics and Computer Science

Volume 20 (2010): Issue 3 (September 2010)

By: Przemysław Śliwiński

Open Access

|Sep 2010

Abstract

A new algorithm for nonparametric wavelet estimation of Hammerstein system nonlinearity is proposed. The algorithm works in the on-line regime (viz., past measurements are not available) and offers a convenient uniform routine for nonlinearity estimation at an arbitrary point and at any moment of the identification process. The pointwise convergence of the estimate to locally bounded nonlinearities and the rate of this convergence are both established.

References

Billings, S. A. and Fakhouri, S. Y. (1978). Theory of separable processes with application to the identification of non-linear systems, Proceedings of IEE 125(10): 1051-1058.10.1049/piee.1978.0241
Search in Google Scholar
Capobianco, E. (2002). Hammerstein system representation of financial volatility processes, The European Physical Journal B—Condensed Matter 27(2): 201-211.10.1140/epjb/e20020154
Search in Google Scholar
Chen, H.-F. (2004). Pathwise convergence of recursive identification algorithms for Hammerstein systems, IEEE Transactions on Automatic Control 49(10): 1641-1649.10.1109/TAC.2004.835358
Search in Google Scholar
Chen, H.-F. (2005). Strong consistency of recursive identification for Hammerstein systems with discontinuous piecewise-linear memoryless block, IEEE Transactions on Automatic Control 50(10): 1612-1617.10.1109/TAC.2005.856658
Search in Google Scholar
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): 1873-1896.10.1080/00207178908953472
Search in Google Scholar
Coca, D. and Billings, S. A. (2001). Non-linear system identification using wavelet multiresolution models, International Journal of Control 74(18): 1718-1736.10.1080/00207170110089743
Search in Google Scholar
Daubechies, I. (1992). Ten Lectures on Wavelets, SIAM, Philadelphia, PA.10.1137/1.9781611970104
Search in Google Scholar
Donoho, D. L. (1993). Unconditional bases are optimal bases for data compression and for statistical estimation, Applied and Computational Harmonic Analysis 1(1): 100-115.10.1006/acha.1993.1008
Search in Google Scholar
Donoho, D. L., Vetterli, M., DeVore, R. A. and Daubechies, I. (1998). Data compression and harmonic analysis, IEEE Transactions on Infromation Theory 44(6): 2435-2476.10.1109/18.720544
Search in Google Scholar
Giannakis, G. B. and Serpedin, E. (2001). A bibliography on nonlinear system identification, Signal Processing 81(3): 533-580.10.1016/S0165-1684(00)00231-0
Search in Google Scholar
Greblicki, W. (1989). Nonparametric orthogonal series identification of Hammerstein systems, International Journal of Systems Science 20(12): 2355-2367.10.1080/00207728908910318
Search in Google Scholar
Greblicki, W. (2001). Recursive identification of Wiener systems, International Journal of Applied Mathematics and Computer Science 11(4): 977-991.
Search in Google Scholar
Greblicki, W. (2002). Stochastic approximation in nonparametric identification of Hammerstein systems, IEEE Transactions on Automatic Control 47(11): 1800-1810.10.1109/TAC.2002.804483
Search in Google Scholar
Greblicki, W. and Pawlak, M. (1985). Fourier and Hermite series estimates of regression function, Annals of the Institute of Statistical Mathematics 37(3): 443-454.10.1007/BF02481112
Search in Google Scholar
Greblicki, W. and Pawlak, M. (1986). Identification of discrete Hammerstein system using kernel regression estimates, IEEE Transactions on Automatic Control 31(1): 74-77.10.1109/TAC.1986.1104096
Search in Google Scholar
Greblicki, W. and Pawlak, M. (1989). Recursive nonparametric identification of Hammerstein systems, Journal of the Franklin Institute 326(4): 461-481.10.1016/0016-0032(89)90045-8
Search in Google Scholar
Greblicki, W. and Pawlak, M. (1994). Nonparametric recovering nonlinearities in block oriented systems with the help of Laguerre polynomials, Control—Theory and Advanced Technology 10(4): 771-791.
Search in Google Scholar
Greblicki, W. and Pawlak, M. (2008). Nonparametric System Identification, Cambridge University Press, New York, NY.10.1017/CBO9780511536687
Search in Google Scholar
Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression, Springer-Verlag, New York, NY.10.1007/b97848
Search in Google Scholar
Haber, R. and Keviczky, L. (1999). Nonlinear System Parameter Identification, Kluwer Academic Publishers, Dordrecht/Boston/London.
Search in Google Scholar
Hasiewicz, Z., Pawlak, M. and Śliwiński, P. (2005). Nonparametric 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): 427-442.10.1109/TCSI.2004.840288
Search in Google Scholar
Hasiewicz, Z. and Śliwiński, P. (2002). Identification of nonlinear characteristics of a class of block-oriented non-linear systems via Daubechies wavelet-based models, International Journal of Systems Science 33(14): 1121-1144.10.1080/0020772021000064171
Search in Google Scholar
Hsu, K., Poolla, K. and Vincent, T. L. (2008). Identification of structured nonlinear systems, IEEE Transactions on Automatic Control 53(11): 2497-2513.10.1109/TAC.2008.2006928
Search in Google Scholar
Jang, W. and Kim, G. (1994). Identification of loudspeaker nonlinearities using the NARMAX modelling technique, Journal of Audio Engineering Society 42(1/2): 50-59.
Search in Google Scholar
Jyothi, S. N. and Chidambaram, M. (2000). Identification of Hammerstein model for bioreactors with input multiplicities, Bioprocess Engineering 23(4): 323-326.10.1007/s004499900141
Search in Google Scholar
Kang, H. W., Cho, Y. S. and Youn, D. H. (1999). On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter, IEEE Transactions on Communications 47(4): 522-526.10.1109/26.764925
Search in Google Scholar
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): 431-444.10.1109/TBME.2004.84328615759573
Search in Google Scholar
Lang, Z. Q. (1997). A nonparametric polynomial identification algorithm for the Hammerstein system, IEEE Transactions on Automatic Control 42(2): 1435-1441.10.1109/9.633834
Search in Google Scholar
Lortie, M. and Kearney, R. E. (2001). Identification of time-varying Hammerstein systems from ensemble data, Annals of Biomedical Engineering 29(2): 619-635.10.1114/1.138042111501626
Search in Google Scholar
Mallat, S. G. (1998). A Wavelet Tour of Signal Processing, Academic Press, San Diego, CA.10.1016/B978-012466606-1/50008-8
Search in Google Scholar
Marmarelis, V. Z. (2004). Nonlinear Dynamic Modeling of Physiological Systems, IEEE Press Series on Biomedical Engineering, Wiley IEEE Press, Piscataway, NJ.10.1002/9780471679370
Search in Google Scholar
Mzyk, G. (2007). Generalized kernel regression estimate for Hammerstein system identification, International Journal of Applied Mathematics and Computer Science 17(2): 101-109, DOI: 10.2478/v10006-007-0018-z.10.2478/v10006-007-0018-z
Search in Google Scholar
Mzyk, G. (2009). Nonlinearity recovering in Hammerstein system from short measurement sequence, IEEE Signal Processing Letters 16(9): 762-765.10.1109/LSP.2009.2024795
Search in Google Scholar
Nešić, D. and Mareels, I. M. Y. (1998). Dead-beat control of simple Hammerstein models, IEEE Transactions on Automatic Control 43(8): 1184-1188.10.1109/9.704998
Search in Google Scholar
Niedźwiecki, M. (1988). Functional series modeling approach to identification of nonstationary stochastic systems, IEEE Transactions on Automatic Control 33(10): 955-961.10.1109/9.7254
Search in Google Scholar
Nordsjo, A. and Zetterberg, L. (2001). Identification of certain time-varying nonlinear Wiener and Hammerstein systems, IEEE Transactions on Signal Processing 49(3): 577-592.10.1109/78.905884
Search in Google Scholar
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): 945-961.10.1109/81.721260
Search in Google Scholar
Rutkowski, L. (1980). Sequential estimates of probability densities by orthogonal series and their application in pattern classification, IEEE Transactions on System, Man, and Cybernetics 10(12): 918-920.10.1109/TSMC.1980.4308422
Search in Google Scholar
Rutkowski, L. (1982). On-line identification of time-varying systems by nonparametric techniques, IEEE Transactions on Automatic Control 27(1): 228-230.10.1109/TAC.1982.1102833
Search in Google Scholar
Rutkowski, L. (2004). Generalized regression neural networks in time-varying environment, IEEE Transactions on Neural Networks 15(3): 576-596.10.1109/TNN.2004.82612715384547
Search in Google Scholar
Śliwiński, P. and Hasiewicz, Z. (2008). Computational algorithms for wavelet identification of nonlinearities in Hammerstein systems with random inputs, IEEE Transactions on Signal Processing 56(2): 846-851.10.1109/TSP.2007.907810
Search in Google Scholar
Śliwiński, P. and Hasiewicz, Z. (2009). Recursive nonparametric estimation of Hammerstein system nonlinearity by Haar wavelet series, IEEE Transactions on Automatic Control, (submitted for review).
Search in Google Scholar
Śliwiński, P., Hasiewicz, Z. and Wachel, P. (2007). On-line polynomial series estimates of Hammerstein system nonlinearities, Proceedings of the 13th IEEE IFAC International Conference on Methods and Models in Automation and Robotics MMAR 2007, Szczecin, Poland.
Search in Google Scholar
Śliwiński, P., Rozenblit, J., Marcellin, M. W. and Klempous, R. (2009). Wavelet amendment of polynomial models in nonlinear system identification, IEEE Transactions on Automatic Control 54(4): 820-825.10.1109/TAC.2008.2009620
Search in Google Scholar
Srinivasan, R., Rengaswamy, R., Narasimhan, S. and Miller, R. (2005). Control loop performance assessment: 2. Hammerstein model approach for stiction diagnosis, Industrial & Engineering Chemistry Research 44(17): 6719-6728.
Search in Google Scholar
Stone, C. J. (1980). Optimal rates of convergence for nonparametric regression, Annals of Statistics 8(6): 1348-1360.10.1214/aos/1176345206
Search in Google Scholar
Sureshbabu, N. and Farrell, J. A. (1999). Wavelet based system identification for non-linear control, IEEE Transactions on Automatic Control 44(2): 412-417.10.1109/9.746278
Search in Google Scholar
Unser, M. (1999). Splines. A perfect fit for signal and image processing, IEEE Signal Processing Magazine 16(6): 22-38.
Search in Google Scholar
Vörös, J. (2005). Identification of Hammerstein systems with time-varying piecewise-linear characteristics, IEEE Transactions on Circuits and Systems II: Express Briefs 52(12): 865-869.10.1109/TCSII.2005.853339
Search in Google Scholar
Walter, G. G. and Shen, X. (2001). Wavelets and other orthogonal systems with applications, 2nd Edn., Chapman & Hall, Boca Raton, FL.
Search in Google Scholar
Westwick, D. T. and Kearney, R. E. (2001). Separable least squares identification of nonlinear Hammerstein models: Application to stretch reflex dynamics, Annals of Biomedical Engineering 29(8): 707-718.10.1114/1.138580611556727
Search in Google Scholar
Westwick, D. T. and Kearney, R. E. (2003). Identification of nonlinear physiological systems, IEEE Press Series on Biomedical Engineering, Wiley/IEEE Press, Piscataway, NJ.
Search in Google Scholar
Zhao, W.-X. and Chen, H.-F. (2006). Recursive identification for Hammerstein system with ARX subsystem, IEEE Transactions on Automatic Control 51(12): 1966-1974.10.1109/TAC.2006.886502
Search in Google Scholar