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Nonlinear Dynamics

, Volume 98, Issue 3, pp 1629–1643 | Cite as

Interval variable step-size spline adaptive filter for the identification of nonlinear block-oriented system

  • Liangdong Yang
  • Jinxin LiuEmail author
  • Zhibin Zhao
  • Ruqiang Yan
  • Xuefeng Chen
Original paper
  • 77 Downloads

Abstract

In order to improve the convergence speed of the nonlinear spline adaptive filter (SAF) in the identification of block-oriented systems, an interval variable step-size algorithm is proposed. Traditional SAF algorithm uses constant step size during iteration, leading to a contradiction between convergence speed and steady-state accuracy. In this paper, a new kind of variable step-size algorithm is proposed, fully considering the particularity of spline interpolation in the nonlinear part of the block-oriented model. The step size of each interpolation interval is independent from that of other intervals, and it is dominated by the correlated squared error which is evaluated by an exponential-weighted averaging (EWA) process. In this paper, the independent step size in each interpolation interval is also updated through an EWA process of the correlated error. The effects of the parameters on the convergence performance of the proposed strategy have been theoretically analyzed and verified by simulations. Finally, some numerical simulations have confirmed that the proposed interval variable step-size approach can significantly improve the convergence speed as well as reduce the steady-state error compared with the traditional SAF and the existing variable step-size SAF algorithms.

Keywords

Spline adaptive filter Block-oriented model Nonlinear identification Variable step size 

Notes

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos. 51705396, 51835009) and the Postdoctoral Science Foundation of China (No. 2018T111047).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

References

  1. 1.
    Billings, S.A.: Identification of nonlinear systems—a survey. In: IEE Proceedings D (Control Theory and Applications), vol. 127, pp. 272–285. IET (1980)Google Scholar
  2. 2.
    Comminiello, D., Principe, J.C.: Adaptive Learning Methods for Nonlinear System Modeling. Butterworth-Heinemann, Oxford (2018)zbMATHGoogle Scholar
  3. 3.
    Haykin, S.S.: Adaptive Filter Theory. Pearson Education India, London (2005)zbMATHGoogle Scholar
  4. 4.
    Lesiak, C., Krener, A.: The existence and uniqueness of Volterra series for nonlinear systems. IEEE Transactions on Automatic Control 23(6), 1090–1095 (1978)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lee, J., Mathews, V.J.: A fast recursive least squares adaptive second order Volterra filter and its performance analysis. IEEE Transactions on Signal Processing 41(3), 1087–1102 (1993)CrossRefGoogle Scholar
  6. 6.
    Ogunfunmi, T.: Adaptive Nonlinear System Identification: The Volterra and Wiener Model Approaches. Springer, Berlin (2007)CrossRefGoogle Scholar
  7. 7.
    Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall PTR, Upper Saddle River (1994)zbMATHGoogle Scholar
  8. 8.
    Rubaai, A., Kotaru, R.: Online identification and control of a DC motor using learning adaptation of neural networks. IEEE Transactions on Industry Applications 36(3), 935–942 (2000)CrossRefGoogle Scholar
  9. 9.
    Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dynamics 89(3), 1569–1578 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pao, Y.: Adaptive Pattern Recognition and Neural Networks. Addison-Wesley Publishing Co., Inc., Reading (1989)zbMATHGoogle Scholar
  11. 11.
    Comminiello, D., Scarpiniti, M., Azpicueta-Ruiz, L.A., Arenas-Garcia, J., Uncini, A.: Functional link adaptive filters for nonlinear acoustic echo cancellation. IEEE Transactions on Audio, Speech, and Language Processing 21(7), 1502–1512 (2013)CrossRefGoogle Scholar
  12. 12.
    Scardapane, S., Wang, D., Panella, M., Uncini, A.: Distributed learning for random vector functional link networks. Information Sciences 301, 271–284 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zhao, H., Zeng, X., He, Z., Yu, S., Chen, B.: Improved functional link artificial neural network via convex combination for nonlinear active noise control. Applied Soft Computing 42, 351–359 (2016)CrossRefGoogle Scholar
  14. 14.
    Comminiello, D., Scarpiniti, M., Scardapane, S., Parisi, R., Uncini, A.: Improving nonlinear modeling capabilities of functional link adaptive filters. Neural Networks 69, 51–59 (2015)CrossRefGoogle Scholar
  15. 15.
    Liu, W., Principe, J.C., Haykin, S.: Kernel Adaptive Filtering: A Comprehensive Introduction, vol. 57. Wiley, New York (2011)Google Scholar
  16. 16.
    Chen, B., Liang, J., Zheng, N., Principe, J.C.: Kernel least mean square with adaptive kernel size. Neurocomputing 191, 95–106 (2016)CrossRefGoogle Scholar
  17. 17.
    Li, K., Principe, J.C.: Transfer learning in adaptive filters: the nearest instance centroid-estimation kernel least-mean-square algorithm. IEEE Transactions on Signal Processing 65(24), 6520–6535 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Scarpiniti, M., Comminiello, D., Parisi, R., Uncini, A.: Nonlinear spline adaptive filtering. Signal Processing 93(4), 772–783 (2013)CrossRefGoogle Scholar
  19. 19.
    Scarpiniti, M., Comminiello, D., Parisi, R., Uncini, A.: Hammerstein uniform cubic spline adaptive filters: learning and convergence properties. Signal Processing 100, 112–123 (2014)CrossRefGoogle Scholar
  20. 20.
    Scarpiniti, M., Comminiello, D., Parisi, R., Uncini, A.: Novel cascade spline architectures for the identification of nonlinear systems. IEEE Transactions on Circuits and Systems I: Regular Papers 62(7), 1825–1835 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Scarpiniti, M., Comminiello, D., Parisi, R., Uncini, A.: Nonlinear system identification using IIR spline adaptive filters. Signal Processing 108, 30–35 (2015)CrossRefGoogle Scholar
  22. 22.
    Scarpiniti, M., Comminiello, D., Scarano, G., Parisi, R., Uncini, A.: Steady-state performance of spline adaptive filters. IEEE Transactions on Signal Processing 64(4), 816–828 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Scardapane, S., Scarpiniti, M., Comminiello, D., Uncini, A.: Diffusion spline adaptive filtering. In: 2016 24th European Signal Processing Conference (EUSIPCO), pp. 1498–1502. IEEE (2016)Google Scholar
  24. 24.
    Sersour, L., Djamah, T., Bettayeb, M.: Nonlinear system identification of fractional Wiener models. Nonlinear Dynamics 92(4), 1493–1505 (2018)CrossRefGoogle Scholar
  25. 25.
    Zhang, J., Chin, K.S., Lawrynczuk, M.: Nonlinear model predictive control based on piecewise linear Hammerstein models. Nonlinear Dynamics 92(3), 1001–1021 (2018)CrossRefGoogle Scholar
  26. 26.
    Lawrynczuk, M.: Nonlinear predictive control of dynamic systems represented by Wiener–Hammerstein models. Nonlinear Dynamics 86(2), 1193–1214 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, Y., Ding, F.: Recursive least squares algorithm and gradient algorithm for Hammerstein–Wiener systems using the data filtering. Nonlinear Dynamics 84(2), 1045–1053 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Giri, F., Bai, E.W.: Block-Oriented Nonlinear System Identification, vol. 1. Springer, Berlin (2010)CrossRefGoogle Scholar
  29. 29.
    Peng, S., Wu, Z., Zhang, X., Chen, B.: Nonlinear spline adaptive filtering under maximum correntropy criterion. In: TENCON 2015-2015 IEEE Region 10 Conference, pp. 1–5. IEEE (2015)Google Scholar
  30. 30.
    Liu, C., Zhang, Z., Tang, X.: Sign normalized spline adaptive filtering algorithms against impulsive noise. Signal Processing 148, 234–240 (2018)CrossRefGoogle Scholar
  31. 31.
    Guan, S., Li, Z.: Normalised spline adaptive filtering algorithm for nonlinear system identification. Neural Processing Letters 46(2), 595–607 (2017)CrossRefGoogle Scholar
  32. 32.
    Patel, V., George, N.V.: Nonlinear active noise control using spline adaptive filters. Applied Acoustics 93, 38–43 (2015)CrossRefGoogle Scholar
  33. 33.
    Patel, V., Comminiello, D., Scarpiniti, M., George, N.V., Uncini, A.: Design of hybrid nonlinear spline adaptive filters for active noise control. In: 2016 International Joint Conference on Neural Networks (IJCNN), pp. 3420–3425. IEEE (2016)Google Scholar
  34. 34.
    Yang, Y., Yang, B., Niu, M.: Spline adaptive filter with fractional-order adaptive strategy for nonlinear model identification of magnetostrictive actuator. Nonlinear Dynamics 90(3), 1647–1659 (2017)CrossRefGoogle Scholar
  35. 35.
    Catmull, E., Rom, R.: A class of local interpolating splines. In: Barnhill, R.E., Riesenfeld, R.F. (eds.) Computer Aided Geometric Design, pp. 317–326. Academic Press, Cambridge, MACrossRefGoogle Scholar
  36. 36.
    Huang, H.C., Lee, J.: A new variable step-size nlms algorithm and its performance analysis. IEEE Transactions on Signal Processing 60(4), 2055–2060 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang, S., Zhang, J., Han, H., Zhang, S., Zhang, J., Han, H.: Robust variable step-size decorrelation normalized least-mean-square algorithm and its application to acoustic echo cancellation. IEEE/ACM Transactions on Audio, Speech and Language Processing (TASLP) 24(12), 2368–2376 (2016)CrossRefGoogle Scholar
  38. 38.
    Bismor, D., Czyz, K., Ogonowski, Z.: Review and comparison of variable step-size LMS algorithms. International Journal of Acoustics and Vibration 21(1), 24–39 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory for Manufacturing Systems EngineeringXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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