Skip to main content
SpringerLink
Log in

Outlier robust stochastic approximation algorithm for identification of MIMO Hammerstein models

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper considers the robust recursive stochastic gradient algorithm for identification of multivariable Hammerstein model with a static nonlinear block in polynomial form and a linear block described by output-error model. The algorithm is designed for unknown parameters in vector form. It is assumed that there is a priori information about a distribution class to which a real disturbance belongs. Such class of distributions describes the presence of outliers in observations. The main contributions of the paper are: (i) design of robust stochastic approximation algorithm for MIMO Hammerstein models using robust statistics (Huber’s theory); (ii) design of general form of nonlinear block; (iii) a strong consistency of estimated parameter whereby proof is based on martingale theory, generalized strictly positive real condition and persistent excitation condition. The properties of algorithm are illustrated by simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Kushner, K.J., Yin, G.G.: Stochastic Approximation Algorithms and Applications. Springer, Berlin (2003)

    MATH  Google Scholar 

  2. Chen, H.F.: Stochastic Approximation and its Applications. Kluwer, New York (2002)

    MATH  Google Scholar 

  3. Benveniste, A., Metivier, M., Priouret, P.: Adaptive Algorithms and Stochastic Approximations. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  4. Chen, H.F., Zhao, W.: Recursive Identification and Parameter Estimation. CRC Press, Boca Raton (2014)

    Book  MATH  Google Scholar 

  5. Ding, F., Chen, T.: Identification of Hammerstein nonlinear ARMAX systems. Automatica 41, 1479–1489 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wang, F., Ding, F.: Gradient—based iterative identification methods for multivariate pseudo-linear mobbing average systems using the data filtering. Nonlinear Dyn. 84, 2003–2015 (2016)

    Article  MATH  Google Scholar 

  7. Ma, L., Liu, X.: A nonlinear recursive instrumental variable identification method of Hammerstein ARMAX system. Nonlinear Dyn. 79, 1601–1613 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  8. Voros, J.: Recursive identification of discrete-time nonlinear cascade systems with time-varying output hysteresis. Nonlinear Dyn. (2016). doi:10.1007/s11071-0163124-3

    Google Scholar 

  9. Abu-Arqub, O., Abo-Hammour, Z.: Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf. Sci. 279, 396–415 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Zhang, Y.Q., Wang, X.Y.: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci. 273, 329–351 (2014)

    Article  Google Scholar 

  11. Zhang, Y.Q., Wang, X.Y.: A new image encryption algorithm based on non-adjacent coupled map lattice. Appl. Soft Comput. 26, 10–20 (2015)

    Article  Google Scholar 

  12. Zhang, Y.Q., Wang, X.Y., Liu, J., Chi, Z.L.: An image encryption scheme based on the MLNCML system using DNA sequences. Opt. Lasers Eng. 82, 95–103 (2016)

    Article  Google Scholar 

  13. Giri, E., Bai, E. (eds.): Block-Oriented Nonlinear System Identification. Springer, Berlin (2010)

    MATH  Google Scholar 

  14. Mzyk, G.: Combined Parametric-Nonparametric Identification of Block-Oriented Systems. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  15. Sliwinski, P.: Nonlinear Systems Identification by Haar Wavelets. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  16. Narendra, K., Galmann, P.: An iterative method for the identification of nonlinear systems using a Hammerstein model. IEEE Trans. Autom. Control 11, 546–550 (1966)

    Article  Google Scholar 

  17. Yan, J., Bernstein, S.: Minimum modeling retrospective cost adaptive control of uncertain Hammerstein system using auxiliary nonlinearities. Int. J. Control 87, 483–505 (2014)

    Article  MATH  Google Scholar 

  18. Fabri, S.G., Wittenmark, B., Bugeja, M.K.: Dual adaptive extremum control of Hammerstein systems. Int. J. Control 88, 1271–1286 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  19. Barnett, V., Lewis, T.: Outliers in Statistical Data. Wiley, New York (1994)

    MATH  Google Scholar 

  20. Pearson, R.K.: Mining Imperfect Data. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  21. Pearson, R.K.: Exploring Data in Engineering, The Sciences and Medicine. Oxford University Press, Oxford (2011)

    Google Scholar 

  22. Huber, P.: Robust estimation of a location parameter. Ann. Math. Stat. 35, 73–101 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  23. Huber, P., Ronchetti, E.: Robust Statistics. Wiley, London (2009)

    Book  MATH  Google Scholar 

  24. Tsypkin, Y.Z.: On the Foundation of Information Identification Theory. Nauka, Moscow (1984). (in Russian)

    MATH  Google Scholar 

  25. Filipovic, V.Z.: Consistency of the robust recursive Hammerstein model identification algorithm. J. Frankl. Inst. 352, 1932–1945 (2015)

    Article  MathSciNet  Google Scholar 

  26. Filipovic, V.Z.: Recursive identification of multivariable ARX models in the presence of a priori information: robustness and regularization. Signal Process. 116, 68–77 (2015)

    Article  Google Scholar 

  27. Filipovic, V.Z.: Stochastic multivariable self-tuning tracker for non-Gaussian systems. Int. J. Appl. Math. Comput. Sci. 15, 351–357 (2005)

    MATH  MathSciNet  Google Scholar 

  28. Filipovic V.Z.: A global convergent outlier robust adaptive predictor for MIMO Hammerstein models. Int. J. Robust. Nonlinear Control. (2016). doi:10.1002/rnc:3705

  29. Greblicki, W.: Stochastic approximation in nonparametric identification of Hammerstein systems. IEEE Trans. Autom. Control 47, 1800–1810 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hersh, M., Zarrop, M.: Strong consistency of a class of recursive stochastic gradient algorithms. Int. J. Control 43, 1115–1123 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  31. Stout, W.: Almost Sure Convergence. Academic Press, New York (1974)

    MATH  Google Scholar 

  32. Desoer, C., Vidyasagar, M.: Feedback Systems: Input–Output Properties. Academic Press, New York (1975)

    MATH  Google Scholar 

  33. Kottenstette, N., McCourt, M., Xia, M., Gupta, A., Antsaklis, P.: On relationships among positivity, positive realness, and dissipativity. Automatica 50, 1003–1016 (2014)

    Article  MATH  Google Scholar 

  34. Ljung, L., Soderstrom, T.: Theory and Practice of Recursive Identification. MIT Press, Massachusetts (1983)

    MATH  Google Scholar 

  35. Papoulis, A.: Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  36. Caines, P.: Linear Stochastic Systems. Wiley, New York (1988)

    MATH  Google Scholar 

  37. Tukey, J.: A survey for sampling from contaminated distributions. In: Olkin, I. (ed.) Contributions for Probability and Statistics, pp. 448–485. Stanford University Press, Stanford (1960)

    Google Scholar 

  38. Bromwich, T.: An Introduction to the Theory of Infinite Series. MacMilan, London (1947)

    MATH  Google Scholar 

  39. Ding, F., Chen, T.: Performance analysis of multi-inovation gradient type identification methods. Automatica 43, 1–14 (2007)

    Article  MATH  Google Scholar 

  40. https://www.mathworks.com/matlabcentral/fileexchange/50608-counting-the-floating-point-operations--flops

Download references

Acknowledgements

This work was supported by the Republic of Serbia, Ministry of Education and Science, through project No. 33026-TR. The author thanks to the referees for their constructive comments that have helped me to improve the article significantly.

Author information

Authors and Affiliations

  1. Department for Automatic Control, Robotic and Fluid Technique, Faculty of Mechanical and Civil Engineering, University of Kragujevac, Dositejeva 19, Kraljevo, 36000, Serbia

    Vojislav Z. Filipovic

Authors
  1. Vojislav Z. Filipovic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vojislav Z. Filipovic.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Filipovic, V.Z. Outlier robust stochastic approximation algorithm for identification of MIMO Hammerstein models. Nonlinear Dyn 90, 1427–1441 (2017). https://doi.org/10.1007/s11071-017-3736-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3736-2

Keywords

Search

Navigation