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Nonlinear Dynamics

, Volume 45, Issue 1–2, pp 31–43 | Cite as

Gradient-Based Identification Methods for Hammerstein Nonlinear ARMAX Models

  • Feng Ding
  • Yang Shi
  • Tongwen Chen
Article

Abstract

Two identification algorithms, an iterative gradient and a recursive stochastic gradient based, are developed for a Hammerstein nonlinear ARMAX model, a linear dynamical block following a memoryless nonlinear block. The basic idea is to use the gradient search principle, to replace unmeasurable noise terms in the information vectors by their estimates, and to compute iteratively or recursively the noise estimates based on the obtained parameter estimates. Convergence analysis of the recursive stochastic gradient algorithm indicates that the parameter estimation error consistently converges to zero under certain conditions. The simulation results show the effectiveness of the proposed algorithms.

Key words

convergence properties Hammerstein models least squares martingale convergence theorem parameter estimation recursive identification stochastic gradient Wiener models 

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References

  1. 1.
    Narendra, K. S. and Gallman, P. G., ‘An iterative method for the identification of nonlinear systems using a Hammerstein model’, IEEE Transactions on Automatic Control 11(3), 1966, 546–550.CrossRefGoogle Scholar
  2. 2.
    Stoica, P., ‘On the convergence of an iterative algorithm used for Hammerstein system identification’, IEEE Transactions on Automatic Control 26(4), 1981, 967–969.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Rangan, S., Wolodkin, G., and Poolla, K., ‘Identification methods for Hammerstein systems’, in Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 697–702.Google Scholar
  4. 4.
    Haist, N. D., Chang, F., and Luus, R., ‘Nonlinear identification in the presence of correlated noise using a Hammerstein model’, IEEE Transactions on Automatic Control 18(5), 1973, 553–555.CrossRefGoogle Scholar
  5. 5.
    Cerone, V. and Regruto, D., ‘Parameter bounds for discrete-time Hammerstein models with bounded output errors’, IEEE Transactions on Automatic Control 48(10), 2003, 1855–1860.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bai, E. W., ‘An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems’, Automatica 34(3), 1998, 333–338.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bai, E. W., ‘Identification of linear systems with hard input nonlinearities of known structure’, Automatica 38(5), 2002, 853–860.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bai, E. W., ‘A blind approach to the Hammerstein–Wiener model identification’, Automatica 38(6), 2002, 967–979.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Wigren, T. and Nordsjö, A. E., ‘Compensation of the RLS algorithm for output nonlinearities’, IEEE Transactions on Automatic Control 44(10), 1999, 1913–1918.CrossRefzbMATHGoogle Scholar
  10. 10.
    Chang, F. and Luus, R., ‘A noniterative method for identification using Hammerstein model’, IEEE Transactions on Automatic Control 16(5), 1971, 464–468.CrossRefGoogle Scholar
  11. 11.
    Nešić, D. and Mareels, I. M. Y., ‘Dead-beat control of simple Hammerstein models’, IEEE Transactions on Automatic Control 43(8), 1998, 1184–1188.CrossRefGoogle Scholar
  12. 12.
    Ding, F. and Chen, T., ‘Identification of Hammerstein nonlinear ARMAX systems’, Automatica 41(9), 2005, 1479–1489.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Bai, E. W., ‘Decoupling the linear and nonlinear parts in Hammerstein model identification’, Automatica 40(4), 2004, 671–676.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Pawlak, M., ‘On the series expansion approach to the identification of Hammerstein system’, IEEE Transactions on Automatic Control 36(6), 1991, 763–767.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Ninness, B. and Gibson, S., ‘Quantifying the accuracy of Hammerstein model estimation’, Automatica 38(12), 2002, 2037–2051.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Vörös, J., ‘Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones’, IEEE Transactions on Automatic Control 48(12), 2003, 2203–2206.CrossRefGoogle Scholar
  17. 17.
    Gallman, P. G., ‘A comparison of two Hammerstein model identification algorithms’, IEEE Transactions on Automatic Control 21(1), 1976, 124–126.CrossRefzbMATHGoogle Scholar
  18. 18.
    Ljung, L., System Identification: Theory for the User, 2nd edn., Prentice-Hall, Englewood Cliffs, NJ, 1999.Google Scholar
  19. 19.
    Goodwin, G. C. and Sin, K. S., Adaptive Filtering, Prediction and Control, Prentice-Hall, Englewood Cliffs, NJ, 1984.Google Scholar
  20. 20.
    Ding, F. and Chen, T., ‘Combined parameter and output estimation of dual-rate systems using an auxiliary model’, Automatica 40(10), 2004, 1739–1748.CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Ding, F. and Chen, T., ‘Hierarchical gradient-based identification of multivariable discrete-time systems’, Automatica 41(2), 2005, 315–325.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ding, F., Shi, Y., and Chen, T., ‘Performance analysis of estimation algorithms of non-stationary ARMA processes’, IEEE Transactions on Signal Processing, in press.Google Scholar
  23. 23.
    Lai, T. L. and Wei, C. Z., ‘Extended least squares and their applications to adaptive control and prediction in linear systems’, IEEE Transactions on Automatic Control 31(10), 1986, 898–906.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Guo, L. and Chen, H. F., ‘The {Å}ström–Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers’, IEEE Transactions on Automatic Control 36(7), 1991, 802–812.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ren, W. and Kumar, P. K., ‘Stochastic adaptive prediction and model reference control’, IEEE Transactions on Automatic Control 39(10), 1994, 2047–2060.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Control Science and Engineering Research CenterSouthern Yangtze UniversityWuxiChina
  2. 2.Department of Mechanical EngineeringUniversity of SaskatchewanSaskatoonCanada
  3. 3.Department of Electrical and Computer EngineeringUniversity of AlbertaEdmontonCanada

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