Nonlinear Dynamics

, Volume 74, Issue 1–2, pp 21–30 | Cite as

Newton iterative identification for a class of output nonlinear systems with moving average noises

  • Feng Ding
  • Junxia Ma
  • Yongsong Xiao
Original Paper


This paper discusses iterative identification problems for a class of output nonlinear systems (i.e., Wiener nonlinear systems) with moving average noises from input–output measurement data, based on the Newton iterative method. The basic idea is to decompose a nonlinear system into two subsystems, to replace the unknown variables in the information vectors with their corresponding estimates at the previous iteration, and to present a Newton iterative identification method using the hierarchical identification principle. The numerical simulation results indicate that the proposed algorithms are effective.


Parameter estimation Iterative identification Newton method Nonlinear system 



This work was supported by the National Natural Science Foundation of China (No. 61273194, 61203111), the Natural Science Foundation of Jiangsu Province (China, BK2012549), the Priority Academic Program Development of Jiangsu Higher Education Institutions and the 111 Project (B12018).


  1. 1.
    Ding, F.: System Identification—New Theory and Methods. Science Press, Beijing (2013) Google Scholar
  2. 2.
    Shi, Y., Yu, B.: Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Autom. Control 54(7), 1668–1674 (2009) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Shi, Y., Yu, B.: Robust mixed H-2/H-infinity control of networked control systems with random time delays in both forward and backward communication links. Automatica 47(4), 754–760 (2011) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Li, H., Shi, Y.: Robust H-infty filtering for nonlinear stochastic systems with uncertainties and random delays modeled by Markov chains. Automatica 48(1), 159–166 (2012) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Zhang, Q.J., Luo, J., Wan, L.: Parameter identification and synchronization of uncertain general complex networks via adaptive-impulsive control. Nonlinear Dyn. 71(1–2), 353–359 (2013) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Qin, P., Nishii, R., Yang, Z.J.: Selection of NARX models estimated using weighted least squares method via GIC-based method and l1-norm regularization methods. Nonlinear Dyn. 70(3), 1831–1846 (2012) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Olson, C.C., Nichols, J.M., Virgin, L.N.: Parameter estimation for chaotic systems using a geometric approach: theory and experiment. Nonlinear Dyn. 70(1), 381–391 (2012) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hizir, N.B., Phan, M.Q., Betti, R., Longman, R.W.: Identification of discrete-time bilinear systems through equivalent linear models. Nonlinear Dyn. 69(4), 2065–2078 (2012) CrossRefGoogle Scholar
  9. 9.
    Ding, F., Chen, T.: Identification of Hammerstein nonlinear ARMAX systems. Automatica 41(9), 1479–1489 (2005) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ding, F., Shi, Y., Chen, T.: Gradient-based identification methods for Hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45(1–2), 31–43 (2006) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bai, E.W.: A blind approach to the Hammerstein–Wiener model identification. Automatica 38(6), 967–979 (2002) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Wang, D.Q., Ding, F.: Extended stochastic gradient identification algorithms for Hammerstein–Wiener ARMAX systems. Comput. Math. Appl. 56(12), 3157–3164 (2008) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wang, D.Q., Ding, F.: Hierarchical least squares estimation algorithm for Hammerstein–Wiener systems. IEEE Signal Process. Lett. 19(12), 825–828 (2012) CrossRefGoogle Scholar
  14. 14.
    Wang, D.Q., Ding, F.: Least squares based and gradient based iterative identification for Wiener nonlinear systems. Signal Process. 91(5), 1182–1189 (2011) CrossRefMATHGoogle Scholar
  15. 15.
    Li, J.H., Ding, F.: Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique. Comput. Math. Appl. 62(11), 4170–4177 (2011) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, J.H., Ding, F., Yang, G.W.: Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems. Math. Comput. Model. 55(3–4), 442–450 (2012) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Wang, W., Ding, F., Dai, J.Y.: Maximum likelihood least squares identification for systems with autoregressive moving average noise. Appl. Math. Model. 36(5), 1842–1853 (2012) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dehghan, M., Hajarian, M.: An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Appl. Math. Model. 34(3), 639–654 (2010) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dehghan, M., Hajarian, M.: Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl. Math. Model. 35(7), 3285–3300 (2011) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wu, A.G., Li, B., Zhang, Y., Duan, G.R.: Finite iterative solutions to coupled Sylvester-conjugate matrix equations. Appl. Math. Model. 35(3), 1065–1080 (2011) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Liu, X.G., Lu, J.: Least squares based iterative identification for a class of multirate systems. Automatica 46(3), 549–554 (2010) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ding, F., Liu, Y.J., Bao, B.: Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng. 226(1), 43–55 (2012) CrossRefGoogle Scholar
  23. 23.
    Lorentzen, R.J., Naevdal, G.: An iterative ensemble Kalman filter. IEEE Trans. Autom. Control 56(8), 1990–1995 (2011) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ding, F., Liu, X.P., Liu, G.: Gradient based and least-squares based iterative identification methods for OE and OEMA systems. Digit. Signal Process. 20(3), 664–677 (2010) CrossRefGoogle Scholar
  25. 25.
    Wang, D.Q., Yang, G.W., Ding, R.F.: Gradient-based iterative parameter estimation for Box–Jenkins systems. Comput. Math. Appl. 60(5), 1200–1208 (2010) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Xie, L., Yang, H.Z.: Gradient based iterative identification for non-uniform sampling output error systems. J. Vib. Control 17(3), 471–478 (2011) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sun, Z., Zeng, J.P., Xu, H.R.: Generalized Newton-iterative method for semismooth equations. Numer. Algorithms 58(3), 333–349 (2011) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Arnal, J., Migallón, V., Penadés, J.: Parallel Newton two-stage multisplitting iterative methods for nonlinear systems. BIT Numer. Math. 43(5), 849–861 (2003) CrossRefMATHGoogle Scholar
  29. 29.
    Xiong, W.L., Ma, J.X., Ding, R.F.: An iterative numerical algorithm for modeling a class of Wiener nonlinear systems. Appl. Math. Lett. 26(4), 487–493 (2013) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Liu, M.M., Xiao, Y.S., Ding, R.F.: Iterative identification algorithm for Wiener nonlinear systems using the Newton method. Appl. Math. Model. 37(9), 6584–6591 (2013) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ding, F., Chen, T.: Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41(2), 315–325 (2005) MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ding, F., Chen, T.: Hierarchical least squares identification methods for multivariable systems. IEEE Trans. Autom. Control 50(3), 397–402 (2005) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ding, F., Chen, T.: Hierarchical identification of lifted state-space models for general dual-rate systems. IEEE Trans. Circuits Syst. I, Regul. Pap. 52(6), 1179–1187 (2005) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ding, J., Ding, F., Liu, X.P., Liu, G.: Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data. IEEE Trans. Autom. Control 56(11), 2677–2683 (2011) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Liu, Y.J., Ding, F., Shi, Y.: Least squares estimation for a class of non-uniformly sampled systems based on the hierarchical identification principle. Circuits Syst. Signal Process. 31(6), 1985–2000 (2012) MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wang, D.Q., Ding, R., Dong, X.Z.: Iterative parameter estimation for a class of multivariable systems based on the hierarchical identification principle and the gradient search. Circuits Syst. Signal Process. 31(6), 2167–2177 (2012) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ding, F., Shi, Y., Chen, T.: Auxiliary model based least-squares identification methods for Hammerstein output-error systems. Syst. Control Lett. 56(5), 373–380 (2007) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Han, H.Q., Xie, L., Ding, F., Liu, X.: Hierarchical least squares based iterative identification for multivariable systems with moving average noises. Math. Comput. Model. 51(9–10), 1213–1220 (2010) CrossRefMATHGoogle Scholar
  39. 39.
    Zhang, Z.N., Ding, F., Liu, X.G.: Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems. Comput. Math. Appl. 61(3), 672–682 (2011) MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Ding, F.: Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling. Appl. Math. Model. 37(4), 1694–1704 (2013) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ding, F., Liu, X.P., Liu, G.: Identification methods for Hammerstein nonlinear systems. Digit. Signal Process. 21(2), 215–238 (2011) CrossRefGoogle Scholar
  42. 42.
    Ding, F., Liu, X.G., Chu, J.: Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle. IET Control Theory Appl. 7(2), 176–184 (2013) CrossRefMathSciNetGoogle Scholar
  43. 43.
    Zhang, Y.: Unbiased identification of a class of multi-input single-output systems with correlated disturbances using bias compensation methods. Math. Comput. Model. 53(9–10), 1810–1819 (2011) CrossRefMATHGoogle Scholar
  44. 44.
    Ding, F., Liu, G., Liu, X.P.: Parameter estimation with scarce measurements. Automatica 47(8), 1646–1655 (2011) MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Ding, F.: Two-stage least squares based iterative estimation algorithm for CARARMA system modeling. Appl. Math. Model. 37(7), 4798–4808 (2013) MathSciNetCrossRefGoogle Scholar
  46. 46.
    Liu, Y.J., Xiao, Y.S., Zhao, X.L.: Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model. Appl. Math. Comput. 215(4), 1477–1483 (2009) MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Liu, Y.J., Sheng, J., Ding, R.F.: Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems. Comput. Math. Appl. 59(8), 2615–2627 (2010) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Ding, F.: Coupled-least-squares identification for multivariable systems. IET Control Theory Appl. 7(1), 68–79 (2013) CrossRefMathSciNetGoogle Scholar
  49. 49.
    Ding, F.: Decomposition based fast least squares algorithm for output error systems. Signal Process. 93(5), 1235–1242 (2013) CrossRefGoogle Scholar
  50. 50.
    Wang, D.Q.: Least squares-based recursive and iterative estimation for output error moving average systems using data filtering. IET Control Theory Appl. 5(14), 1648–1657 (2011) MathSciNetCrossRefGoogle Scholar
  51. 51.
    Li, J.H., Ding, R.F., Yang, Y.: Iterative parameter identification methods for nonlinear functions. Appl. Math. Model. 36(6), 2739–2750 (2012) MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Ding, L., Han, L.L., Chen, X.M.: Time series AR modeling with missing observations based on the polynomial transformation. Math. Comput. Model. 51(5–6), 527–536 (2010) MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Chen, J., Zhang, Y., Ding, R.F.: Auxiliary model based multi-innovation algorithms for multivariable nonlinear systems. Math. Comput. Model. 52(9–10), 1428–1434 (2010) MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Chen, J., Wang, X.P., Ding, R.F.: Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities. Appl. Math. Model. 36(1), 238–243 (2012) CrossRefMATHGoogle Scholar
  55. 55.
    Chen, J., Lv, L.X., Ding, R.F.: Multi-innovation stochastic gradient algorithms for dual-rate sampled systems with preload nonlinearity. Appl. Math. Lett. 26(1), 124–129 (2013) MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Li, J.H.: Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration. Appl. Math. Lett. 26(1), 91–96 (2013) MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Hu, P.P., Ding, F.: Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle. Nonlinear Dyn. 73(1–2), 583–592 (2013) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education)Jiangnan UniversityWuxiP.R. China
  2. 2.Control Science and Engineering Research CenterJiangnan UniversityWuxiP.R. China

Personalised recommendations