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Book cover Nonlinear System Identification by Haar Wavelets

Part of the book series: Lecture Notes in Statistics ((LNS,volume 210))

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Abstract

The fundamental relation between the Hammerstein system nonlinearity and the regression function of the system output on the system input is presented. It allows recovery of the system nonlinearity with the help of the nonparametric regression function estimates. Several implications of this approach are discussed. In particular, the fact that the nonlinearity is estimated independently of the dynamics is emphasized. Some limitations, i.e., an ability of identification of the genuine nonlinear characteristic up to some system-dependent constants only and a small system-to-noise ratio of the measurement data, are also pointed out.

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Notes

  1. 1.

    The symmetric triangular pdffunction is further used in algorithms tests (see Sect. 5.1).

References

  1. Apostol, T.M.: Calculus, Volume 1. One-Variable Calculus with an Introduction to Linear Algebra, 2nd edn. Wiley, New York (1975)

    Google Scholar 

  2. Balestrino, A., Landi, A., Ould-Zmirli, M., Sani, L.: Automatic nonlinear auto-tuning method for Hammerstein modeling of electrical drives. IEEE Transactions on Industrial Electronics 48(3), 645–655 (2001)

    Google Scholar 

  3. Bayer, R.: Symmetric binary B-Trees: Data structure and maintenance algorithms. Acta Informatica 1(4), 290–306 (1972)

    Google Scholar 

  4. Bendat, J.S.: Nonlinear System Analysis and Identification. Wiley, New York (1990)

    Google Scholar 

  5. Billings, S.A.: Identification of non-linear systems—a survey. Proceedings of IEE 127(6), 272–285 (1980)

    Google Scholar 

  6. Billings, S.A., Fakhouri, S.Y.: Theory of separable processes with application to the identification of non-linear systems. Proceedings of IEE 125(10), 1051–1058 (1978)

    Google Scholar 

  7. Billings, S.A., Fakhouri, S.Y.: Non-linear system identification using the Hammerstein model. International Journal of Systems Science 10, 567–578 (1979)

    Google Scholar 

  8. Blu, T., Thévenaz, P., Unser, M.: Linear interpolation revitalized. IEEE Transactions on Image Processing 13(5), 710–719 (2004)

    Google Scholar 

  9. Blu, T., Unser, M.: Wavelets, fractals, and radial basis functions. IEEE Transactions on Signal Processing 50(3), 543–553 (2002)

    Google Scholar 

  10. Boyd, S., Chua, L.: Fading memory and the problem of approximating nonlinear operators with volterra series. Circuits and Systems, IEEE Transactions on 32(11), 1150–1161 (1985)

    Google Scholar 

  11. Buhmann, M.D.: Radial basis functions. Acta Numerica 9, 1–38 (2001)

    Google Scholar 

  12. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  13. Capobianco, E.: Hammerstein system representation of financial volatility processes. The European Physical Journal B - Condensed Matter 27(2), 201–211 (2002)

    Google Scholar 

  14. Chen, H.F.: Pathwise convergence of recursive identification algorithms for Hammerstein systems. IEEE Transactions on Automatic Control 49(10), 1641–1649 (2004)

    Google Scholar 

  15. Chen, H.F.: Strong consistency of recursive identification for Hammerstein systems with discontinuous piecewise-linear memoryless block. IEEE Transactions on Automatic Control 50(10), 1612–1617 (2005)

    Google Scholar 

  16. Cohen, A.: Numerical analysis of wavelets methods. Studies in Mathematics and Its Applications. Elsevier, Amsterdam (2003)

    Google Scholar 

  17. Cohen, A.: Theoretical, applied and computational aspects of nonlinear approximation. In: J. Bramble, A. Cohen, W. Dahmen, C. Canuto (eds.) Multiscale Problems and Methods in Numerical Simulations, Lecture Notes in Mathematics, vol. 1825/2003, pp. 1–29. Springer-Verlag, Berlin Heidelberg (2003)

    Google Scholar 

  18. Cohen, A., D’Ales, J.P.: Nonlinear approximation of random functions. SIAM Journal of Applied Mathematics 57(2), 518–540 (1997)

    Google Scholar 

  19. Cohen, A., Daubechies, I., Vial, P.: Wavelet bases on the interval and fast algorithms. Journal of Applied and Computational Harmonic Analysis 1(1), 54–81 (1993)

    Google Scholar 

  20. Cripps, S.: RF power amplifiers for wireless communications. Artech House (2006)

    Google Scholar 

  21. Dahlquist, G., Björk, A.: Numerical Methods. Prentice-Hall, Inc. Englewood Cliffs, New Jersey (1974)

    Google Scholar 

  22. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Communication on Pure and Applied Mathematics 42, 909–996 (1988)

    Google Scholar 

  23. Daubechies, I.: Ten Lectures on Wavelets. SIAM Edition, Philadelphia (1992)

    Google Scholar 

  24. Daubechies, I., Sweldens, W.: Factoring wavelet transforms into lifting steps. The Journal of Fourier Analysis and Applications 4(3), 245–267 (1998)

    Google Scholar 

  25. David, H.A., Nagaraja, H.N.: Order statistics, 3rd edn. John Wiley & Sons, Inc., Hoboken, New Jersey (2003)

    Google Scholar 

  26. Delouille, V., Franke, L., von Sachs, R.: Nonparametric stochastic regression with design-adapted wavelets. Sankhyā, Ser. A 63(3), 328–366 (2001)

    Google Scholar 

  27. Delouille, V., Simoens, J., von Sachs, R.: Smooth design-adapted wavelets for nonparametric stochastic regression. Journal of the American Statistical Association 99(467), 643–658 (2004)

    Google Scholar 

  28. Dempsey, E., Westwick, D.: Identification of Hammerstein models with cubic spline nonlinearities. IEEE Transactions on Biomedical Engineering 51(2), 237–245 (2004)

    Google Scholar 

  29. DeVore, R.A.: Nonlinear approximation. Acta Numerica 7, 51–150 (1998)

    Google Scholar 

  30. DeVore, R.A.: Optimal computation. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 (2006)

    Google Scholar 

  31. DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer-Verlag, Berlin Heidelberg New York (1993)

    Google Scholar 

  32. DeVore, R.A., Lucier, B.J.: Wavelets. Acta Numerica 1, 1–56 (1992)

    Google Scholar 

  33. Donoho, D.L.: De-noising by soft thresholding. IEEE Transactions on Information Theory 41(3), 613–627 (1995)

    Google Scholar 

  34. Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika 81, 425–455 (1994)

    Google Scholar 

  35. Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association 90, 1200–1224 (1995)

    Google Scholar 

  36. Donoho, D.L., Johnstone, I.M.: Minimax estimation via wavelets shrinkage. Annals of Statistics 26, 879–921 (1998)

    Google Scholar 

  37. Ferrari, S., Maggioni, M., Borghese, N.A.: Multiscale approximation with hierarchical radial basis functions networks. IEEE Transaction one Neural Networks 15(1) (2004)

    Google Scholar 

  38. Gallman, P.: An iterative method for the identification of nonlinear systems using a Uryson model. IEEE Transactions on Automatic Control 20(6), 771–775 (1975)

    Google Scholar 

  39. Giannakis, G.B., Serpedin, E.: A bibliography on nonlinear system identification. Signal Processing 81(3), 533–580 (2001)

    Google Scholar 

  40. Gilabert, P., Montoro, G., Bertran, E.: FPGA implementation of a real-time NARMA-based digital adaptive predistorter. IEEE Transactions on Circuits and Systems II: Express Briefs 58(7), 402–406 (2011)

    Google Scholar 

  41. Girardi, M., Sweldens, W.: A new class of unbalanced Haar wavelets that form an unconditional basis for l p on general measure spaces. The Journal of Fourier Analysis and Applications 3(4), 457–474 (1997)

    Google Scholar 

  42. Giri, F., Bai, E.W. (eds.): Block-oriented nonlinear system identification. Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin Heidelberg (2010)

    Google Scholar 

  43. Gomes, S.M., Cortina, E.: Some results on the convergence of sampling series based on convolution integrals. SIAM Journal on Mathematical Analysis 26(5), 1386–1402 (1995)

    Google Scholar 

  44. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading, Massachusetts (1994)

    Google Scholar 

  45. Gray, R.M., Neuhoff, D.L.: Quantization. IEEE Transactions on Information Theory 44(6), 2325–2383 (1998)

    Google Scholar 

  46. Greblicki, W.: Nonparametric system identification by orthogonal series. Problems of Control and Information Theory 8, 67–73 (1979)

    Google Scholar 

  47. Greblicki, W.: Nonparametric orthogonal series identification of Hammerstein systems. International Journal of Systems Science 20(12), 2355–2367 (1989)

    Google Scholar 

  48. Greblicki, W.: Nonparametric identification of Wiener systems. IEEE Transactions on Information Theory 38(5), 1487–1493 (1992)

    Google Scholar 

  49. Greblicki, W.: Nonparametric approach to Wiener system identification. IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications 44, 538–545 (1997)

    Google Scholar 

  50. Greblicki, W.: Nonlinearity recovering in Wiener system driven with correlated signal. IEEE Transactions on Automatic Control 49(10), 1805–1810 (2004)

    Google Scholar 

  51. Greblicki, W.: Nonparametric input density-free estimation of the nonlinearity in Wiener systems. Information Theory, IEEE Transactions on 56(7), 3575–3580 (2010)

    Google Scholar 

  52. Greblicki, W., Pawlak, M.: Identification of discrete Hammerstein system using kernel regression estimates. IEEE Transactions on Automatic Control 31(1), 74–77 (1986)

    Google Scholar 

  53. Greblicki, W., Pawlak, M.: Nonparametric identification of Hammerstein systems. IEEE Transactions on Information Theory 35, 409–418 (1989)

    Google Scholar 

  54. Greblicki, W., Pawlak, M.: Recursive nonparametric identification of Hammerstein systems. Journal of the Franklin Institute 326(4), 461–481 (1989)

    Google Scholar 

  55. Greblicki, W., Pawlak, M.: Dynamic system identification with order statistics. IEEE Transactions on Information Theory 40, 1474–1489 (1994)

    Google Scholar 

  56. Greblicki, W., Pawlak, M.: Nonparametric recovering nonlinearities in block oriented systems with the help of Laguerre polynomials. Control – Theory and Advanced Technology 10(4), 771–791 (1994)

    Google Scholar 

  57. Greblicki, W., Pawlak, M.: Nonparametric System Identification. Cambridge University Press, New York (2008)

    Google Scholar 

  58. Györfi, L., Kohler, M., A. Krzyżak, Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer-Verlag, New York (2002)

    Google Scholar 

  59. Haar, A.: Zur Theorie der Orthogonalen Funktionen-Systeme. Annals of Mathematics 69(1910)

    Google Scholar 

  60. Haar, A.: On the theory of orthogonal function systems. In: C. Heil, D.F. Walnut (eds.) Fundamental papers in wavelet theory, pp. 155–188. Priceton University Press, Princeton and Oxford (2006)

    Google Scholar 

  61. Haber, R., Keviczky, L.: Nonlinear System Parameter Identification. Kluwer Academic Publishers, Dordrecht-Boston-London (1999)

    Google Scholar 

  62. Hall, P., Kerkyacharian, G., Picard, D.: Block threshold rules for curve estimation using kernel and wavelet methods. The Annals of Statistics 26(3), 922–942 (1998)

    Google Scholar 

  63. Hall, P., Patil, P.: On the choice of smoothing parameter, threshold and truncation in nonparametric regression by non-linear wavelet methods. Journal of the Royal Statistical Society. Series B (Methodological) 58(2), 361–377 (1996)

    Google Scholar 

  64. Hall, P., Turlach, B.: Interpolation methods for nonlinear wavelet regression with irregularly spaced design. The Annals of Statistics 25(5), 1912–1925 (1997)

    Google Scholar 

  65. Härdle, W.: Applied Nonparametric Regression. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  66. Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.: Wavelets, Approximation, and Statistical Applications. Springer-Verlag, New York (1998)

    Google Scholar 

  67. Härdle, W., Müller, M., Sperlich, S., Werwatz, A.: Nonparametric and Semiparametric Models. Springer-Verlag, Berlin Heidelberg (2004)

    Google Scholar 

  68. Hasiewicz, Z.: Hammerstein system identification by the Haar multiresolution approximation. International Journal of Adaptive Control and Signal Processing 13(8), 697–717 (1999)

    Google Scholar 

  69. Hasiewicz, Z.: Modular neural networks for non-linearity recovering by the Haar approximation. Neural Networks 13, 1107–1133 (2000)

    Google Scholar 

  70. Hasiewicz, Z.: Wavelet network for recursive function learning. In: Neural Networks and Soft Computing, Advances in Soft Computing, pp. 710–715. 6th IEEE International Conference on Neural Networks and Soft Computing, Physica-Verlag, Springer-Verlag Company, Heidelberg, Zakopane 2002 (2003)

    Google Scholar 

  71. Hasiewicz, Z., Pawlak, M., Śliwiński, P.: 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 (2005)

    Google Scholar 

  72. Hasiewicz, Z., Śliwiński, P.: 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 (2002)

    Google Scholar 

  73. Heil, C., Walnut, D.F. (eds.): Fundamental papers in wavelet theory. Priceton University Press, Priceton and Oxford (2006)

    Google Scholar 

  74. Hunter, I.W., Korenberg, M.J.: The identification of non-linear biological systems: Wiener and Hammerstein cascade models. Biological Cybernetics 55(2–3), 135–144 (1986)

    Google Scholar 

  75. Huoa, H.B., Zhonga, Z.D., Zhua, X.J., Tua, H.Y.: Nonlinear dynamic modeling for a SOFC stack by using a Hammerstein model. Journal of Power Sources 175(1), 441–446 (2008)

    Google Scholar 

  76. Huoa, H.B., Zhua, X.J., Hub, W.Q., Tua, H.Y., Li, J., Yangd, J.: Nonlinear model predictive control of SOFC based on a Hammerstein model. Journal of Power Sources 185(1), 338–344 (2008)

    Google Scholar 

  77. Iwamoto, M., Williams, A., Chen, P.F., Metzger, A., Larson, L., Asbeck, P.: An extended Doherty amplifier with high efficiency over a wide power range. IEEE Transactions on Microwave Theory and Techniques 49(12), 2472–2479 (2001)

    Google Scholar 

  78. Jansen, M., Oonincx, P.J.: Second generation wavelets and applications. Springer-Verlag, London (2005)

    Google Scholar 

  79. Jeng, J.C., Huang, H.P.: Nonparametric identification for control of MIMO Hammerstein systems. Industrial and Engineering Chemistry Research 47(17), 6640–6647 (2008)

    Google Scholar 

  80. Jeraj, J., Mathews, V.: A stable adaptive Hammerstein filter employing partial orthogonalization of the input signals. IEEE Transactions on Signal Processing 54(4), 1412–1420 (2006)

    Google Scholar 

  81. Juditsky, A., Hjalmarsson, H., Benveniste, A., Delyon, B., Ljung, L., Sjoberg, J., Zhang, Q.H.: Nonlinear black-box models in system-identification - mathematical foundations. Automatica 31(12), 1725–1750 (1995)

    Google Scholar 

  82. Jurado, F.: A method for the identification of solid oxide fuel cells using a Hammerstein model. Journal of Power Sources 154(1), 145–152 (2006)

    Google Scholar 

  83. Jyothi, S.N., Chidambaram, M.: Identification of Hammerstein model for bioreactors with input multiplicities. Bioprocess Engineering 23(4), 323–326 (2000)

    Google Scholar 

  84. Kamiński, W., Strumiłło, P.: Kernel orthonormalization in radial basis function neural networks. IEEE Transactions on Neural Networks 8(5), 1177–1183 (1997)

    Google Scholar 

  85. Kelly, S., Kon, M., Raphael, L.A.: Pointwise convergence of wavelet expansions. Bulletin of The American Mathematical Society 30(1), 87–94 (1994)

    Google Scholar 

  86. Keys, R.: Cubic convolution interpolation for digital image processing. IEEE Transactions on Acoustics, Speech and Signal Processing 29(6), 1153–1160 (1981)

    Google Scholar 

  87. Kim, J., Konstantinou, K.: Digital predistortion of wideband signals based on power amplifier model with memory. Electronics Letters 37(23), 1417–1418 (2001)

    Google Scholar 

  88. Kim, W.J., Cho, K.J., Stapleton, S., Jong-Heon, Kim: Piecewise pre-equalized linearization of the wireless transmitter with a Doherty amplifier. IEEE Transactions on Microwave Theory and Techniques 54(9), 3469–3478 (2006)

    Google Scholar 

  89. Knuth, D.E.: The Art of Computer Programming. Volume 3. Sorting and searching. Addison-Wesley Longman Publishing Co., Inc, Boston, MA (1998)

    Google Scholar 

  90. Krim, H., Tucker, D., Mallat, S., Donoho, D.: On denoising and best signal representation. IEEE Transactions on Information Theory 45(7), 2225–2238 (1999)

    Google Scholar 

  91. Krzyżak, A., Linder, T.: Radial basis function networks and complexity regularization in function learning. IEEE Transactions on Neural Networks 9(2), 247–256 (1998)

    Google Scholar 

  92. Kukreja, S., Kearney, R., Galiana, H.: A least-squares parameter estimation algorithm for switched Hammerstein systems with applications to the VOR. IEEE Transactions on Biomedical Engineering 52(3), 431–444 (2005)

    Google Scholar 

  93. Lee, Y., Schetzen, M.: Measurement of the Wiener kernels of a non-linear system by cross-correlation. International Journal of Control 2, 237–254 (1965)

    Google Scholar 

  94. Lloyd, S.: Least squares quantization in PCM. IEEE Transactions on Information Theory 28(2), 129–137 (1982)

    Google Scholar 

  95. Lyons, R.: Understanding digital signal processing. Prentice Hall PTR (2004)

    Google Scholar 

  96. Mallat, S.G.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1998)

    Google Scholar 

  97. Marmarelis, V.Z.: Nonlinear dynamic modeling of physiological systems. IEEE Press Series on Biomedical Engineering. Wiley-IEEE Press, Piscataway, NJ (2004)

    Google Scholar 

  98. Mason, J.C., Handscomb, D.C.: Chebyshev polynomials. Chapman & Hall/CRC, Boca Raton (2003)

    Google Scholar 

  99. Max, J.: Quantizing for minimum distortion. IRE Transactions on Information Theory 6(1), 7–12 (1960)

    Google Scholar 

  100. Meijering, E.: A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE 90(3), 319–342 (2002)

    Google Scholar 

  101. Meilera, M., Schmida, O., Schudya, M., Hoferb, E.: Dynamic fuel cell stack model for real-time simulation based on system identification. Journal of Power Sources 176(2), 523–528 (2008)

    Google Scholar 

  102. Morgan, D., Ma, Z., Kim, J., Zierdt, M., Pastalan, J.: A generalized memory polynomial model for digital predistortion of RF power amplifiers. IEEE Transactions on Signal Processing 54(10), 3852–3860 (2006)

    Google Scholar 

  103. Mosteller, F., Tukey, J.W.: Data Analysis and Regression: A Second Course in Statistics. Addison-Wesley Series in Behavioral Science: Quantitative Methods. Addison-Wesley, Reading, MA (1977)

    Google Scholar 

  104. Mzyk, G.: Nonlinearity recovering in Hammerstein system from short measurement sequence. IEEE Signal Processing Letters 16(9), 762–765 (2009)

    Google Scholar 

  105. Mzyk, G.: Parametric versus nonparametric approach to Wiener systems identification. In: F. Giri, E.W. Bai (eds.) Block-oriented nonlinear system identification, Lecture Notes in Control and Information Sciences, pp. 111–125. Springer (2010)

    Google Scholar 

  106. Nadaraya, E.A.: On estimating regression. Theor. Probability Appl. 9, 141–142 (1964)

    Google Scholar 

  107. Nešić, D., Mareels, I.M.Y.: Dead-beat control of simple Hammerstein models. IEEE Transactions on Automatic Control 43(8), 1184–1188 (1998)

    Google Scholar 

  108. Nordsjo, A., Zetterberg, L.: Identification of certain time-varying nonlinear Wiener and Hammerstein systems. IEEE Transactions on Signal Processing 49(3), 577–592 (2001)

    Google Scholar 

  109. Ogden, R.: Essential Wavelets For Statistical Applications and Data Analysis. Birkhäuser, Boston (1997)

    Google Scholar 

  110. Pawlak, M., Hasiewicz, Z.: Nonlinear system identification by the Haar multiresolution analysis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 45(9), 945–961 (1998)

    Google Scholar 

  111. Pawlak, M., Hasiewicz, Z., Wachel, P.: On nonparametric identification of Wiener systems. IEEE Transactions on Signal Processing 55(5), 482–492 (2007)

    Google Scholar 

  112. Pawlak, M., Rafajłowicz, E., Krzyżak, A.: Postfiltering versus prefiltering for signal recovery from noisy samples. IEEE Transactions on Information Theory 49(12), 3195–3212 (2003)

    Google Scholar 

  113. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  114. Raab, F., Asbeck, P., Cripps, S., Kenington, P., Popovic, Z., Pothecary, N., Sevic, J., Sokal, N.: Power amplifiers and transmitters for RF and microwave. IEEE Transactions on Microwave Theory and Techniques 50(3), 814–826 (2002)

    Google Scholar 

  115. Rabiner, L.: Multirate Digital Signal Processing. Prentice Hall PTR (1996)

    Google Scholar 

  116. Rafajłowicz, E.: Nonparametric orthogonal series estimators of regression: A class attaining the optimal convergence rate in l 2. Statistic & Probability Letters 5, 219–224 (1987)

    Google Scholar 

  117. Rafajłowicz, E.: Consistency of orthogonal series density estimators based on grouped observation. IEEE Transactions on Information Theory 43, 283–285 (1997)

    Google Scholar 

  118. Roger L. Claypoole, J., Davis, G.M., Sweldens, W., Baraniuk, R.G.: Nonlinear wavelet transforms for image coding via lifting. IEEE Transactions on Image Processing 12(12), 1449–1459 (2003)

    Google Scholar 

  119. Rudin, W.: Principles of mathematical analysis, 3rd edn. McGraw-Hill, New York (1976)

    Google Scholar 

  120. Rutkowski, L.: Generalized regression neural networks in time-varying environment. IEEE Transactions on Neural Networks 15(3), 576 – 596 (2004)

    Google Scholar 

  121. Rutkowski, L., Rafajłowicz, E.: On optimal global rate of convergence of some nonparametric identification procedures. IEEE Transactions on Automatic Control 34(10), 1089–1091 (1989)

    Google Scholar 

  122. Said, A., Pearlman, W.A.: A new, fast, and efficient image codec based on set partitioning in hierarchical trees. IEEE Transactions on Circuits and Systems for Video technology 6(3), 243–251 (1996)

    Google Scholar 

  123. Schetzen, M.: Nonlinear system modeling based on the Wiener theory. Proceedings of the IEEE 69(12), 1557–1573 (1981)

    Google Scholar 

  124. Schilling, R.J., Jr., J.J.C., Al-Ajlouni, A.F.: Approximation of nonlinear systems with radial basis function neural networks. IEEE Transactions on Neural Networks 12(1), 1–15 (2001)

    Google Scholar 

  125. Shapiro, J.M.: Embedded image coding using zerotrees of wavelet coefficients. IEEE Transactions on Signal Processing 41(12), 3445–3462 (1993)

    Google Scholar 

  126. Shi, K., Zhou, G., Viberg, M.: Compensation for nonlinearity in a Hammerstein system using the coherence function with application to nonlinear acoustic echo cancellation. IEEE Transactions on Signal Processing 55(12), 5853–5858 (2007)

    Google Scholar 

  127. Sjoberg, J., Zhang, Q.H., Ljung, L., Benveniste, A., Delyon, B., Glorenec, P.Y., Hjalmarsson, H., Juditsky, A.: Nonlinear black-box modeling in system-identification - a unified overview. Automatica 31(12), 1691–1724 (1995)

    Google Scholar 

  128. Skubalska-Rafajłowicz, E.: Pattern recognition algorithms based on space-filling curves and orthogonal expansions. IEEE Transactions on Information Theory 47(5), 1915–1927 (2001)

    Google Scholar 

  129. Śliwiński, P.: Fast algorithms for non-linearity recovering in Hammerstein systems with ordered observations. In: Proceedings 10th IEEE International Conference on Methods and Models in Automation and Robotics – MMAR 2004, pp. 451–456. Institute of Control Engineering, Technical University of Szczecin, Miedzyzdroje (2004)

    Google Scholar 

  130. Śliwiński, P.: On-line wavelet estimation of Hammerstein system nonlinearity. International Journal of Applied Mathematics and Computer Science 20(3), 513–523 (2010)

    Google Scholar 

  131. Śliwiński, P., Hasiewicz, Z.: Computational algorithms for multiscale identification of nonlinearities in Hammerstein systems with random inputs. IEEE Transactions on Signal Processing 53(1), 360–364 (2005)

    Google Scholar 

  132. Śliwiński, P., Hasiewicz, Z.: Computational algorithms for wavelet identification of nonlinearities in Hammerstein systems with random inputs. IEEE Transactions on Signal Processing 56(2), 846–851 (2008)

    Google Scholar 

  133. Śliwiński, P., Rozenblit, J., Marcellin, M.W., Klempous, R.: Wavelet amendment of polynomial models in nonlinear system identification. IEEE Transactions on Automatic Control 54(4), 820–825 (2009)

    Google Scholar 

  134. Slud, E.: Entropy and maximal spacings for random partitions. Zeitschrift fúr Wahrscheinlichtstheorie und vervandte Gebiete 41(4), 341–352 (1978)

    Google Scholar 

  135. Srinivasan, R., Rengaswamy, R., Narasimhan, S., Miller, R.: Control loop performance assessment — Hammerstein model approach for stiction diagnosis. Industrial & Engineering Chemistry Research 44(17), 6719–6728 (2005)

    Google Scholar 

  136. Steinhaus, H.: Sur la division des corp materiels en parties. Bull. Acad. Polon. Sci. C1. III(IV), 801–804 (1956)

    Google Scholar 

  137. Stiles, B., Sandberg, I., Ghosh, J.: Complete memory structures for approximating nonlinear discrete-time mappings. Neural Networks, IEEE Transactions on 8(6), 1397–1409 (1997)

    Google Scholar 

  138. Stone, C.J.: Optimal global rates of convergence for nonparametric regression. Annals of Statistics 10(4), 1040–1053 (1982)

    Google Scholar 

  139. Sung, S.W.: System identification method for Hammerstein processes. Industrial and Engineering Chemistry Research 41(17), 4295–4302 (2002)

    Google Scholar 

  140. Sureshbabu, N., Farrell, J.A.: Wavelet based system identification for non-linear control. IEEE Transactions on Automatic Control 44(2), 412–417 (1999)

    Google Scholar 

  141. Sweldens, W.: The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis 3(2), 186–200 (1996)

    Google Scholar 

  142. Sweldens, W.: The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal. 29(2), 511–546 (1997)

    Google Scholar 

  143. Taubman, D., Marcellin, M.: JPEG2000. Image Compression Fundamentals, Standards and Practice, The Kluwer International Series in Engineering and Computer Science, vol. 642. Kluwer Academic Publishers (2002)

    Google Scholar 

  144. Thévenaz, P., Blu, T., Unser, M.: Interpolation revisited. IEEE Transactions on Medical Imaging 19(7), 739–758 (2000)

    Google Scholar 

  145. Unser, M., Daubechies, I.: On the approximation power of convolution-based least squares versus interpolation. IEEE Transaction on Signal Processing 45(7), 1697–1711 (1997)

    Google Scholar 

  146. Walter, G.G.: Pointwise convergence of wavelet expansions. Journal of Approximation Theory 80(1), 108–118 (1995)

    Google Scholar 

  147. Walter, G.G., Shen, X.: Wavelets and other orthogonal systems with applications, 2nd Ed. Chapman & Hall, Boca Raton (2001)

    Google Scholar 

  148. Wang, J., Wang, D., Moore, P., Pu, J.: Modelling study, analysis and robust servo control of pneumatic cylinder actuator systems. IEE Proceedings: Control Theory and Applications 148, 35–42 (2001)

    Google Scholar 

  149. Watson, G.S.: Smooth regression analysis. Sankhya, Series A 26, 355–372 (1964)

    Google Scholar 

  150. Westwick, D., Kearney, R.: Nonparametric identification of nonlinear biomedical systems, part i: Theory. Critical reviews in biomedical engineering 26(3), 153 (1998)

    Google Scholar 

  151. Westwick, D.T., Kearney, R.E.: Separable least squares identification of nonlinear Hammerstein models: Application to stretch reflex dynamics. Annals of Biomedical Engineering 29(8), 707–718 (2001)

    Google Scholar 

  152. Westwick, D.T., Kearney, R.E.: Identification of nonlinear physiological systems. IEEE Press Series on Biomedical Engineering. Wiley-IEEE Press, Piscataway (2003)

    Google Scholar 

  153. Wiener, N.: Nonlinear problems in random theory. The MIT Press, Cambridge, Massachusetts, USA (1966)

    Google Scholar 

  154. Zhao, W.X., Chen, H.F.: Recursive identification for Hammerstein system with ARX subsystem. IEEE Transactions on Automatic Control 51(12), 1966–1974 (2006)

    Google Scholar 

  155. Zhou, D., DeBrunner, V.E.: Novel adaptive nonlinear predistorters based on the direct learning algorithm. IEEE Transactions on Signal Processing 55(1), 120–133 (2007)

    Google Scholar 

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Śliwiński, P. (2013). Identification Goal. In: Nonlinear System Identification by Haar Wavelets. Lecture Notes in Statistics, vol 210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29396-2_3

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