Skip to main content
SpringerLink
Log in

Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique

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

Abstract

This paper considers iterative identification problems for a class of nonlinear systems with colored noises, which can be described by a linear-in-parameters output error autoregressive model. A gradient-based iterative (GI) algorithm, a filtered GI algorithm, and a filtered three-stage GI algorithm are developed using the decomposition technique and filtering technique, and their computational efficiencies are analyzed and compared. The simulation results indicate that the proposed algorithms can estimate effectively the parameters of nonlinear systems.

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

Similar content being viewed by others

References

  1. Ding, F.: System Identification-New Theory and Methods. Science Press, Beijing (2013)

    Google Scholar 

  2. Ding, F.: System Identification-Performances Analysis for Identification Methods. Science Press, Beijing (2014)

    Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. Alarcin, F.: Nonlinear modelling of a fishing boat and fuzzy logic control design for electro-hydraulic fin stabilizer system. Nonlinear Dyn. 61(1–2), 29–41 (2010)

    Google Scholar 

  6. Togun, N., Baysec, S.: Nonlinear modeling and identification of a spark ignition engine torque. Mech. Syst. Signal Process. 26, 294–304 (2012)

    Article  Google Scholar 

  7. Ding, F., Liu, X.P., Liu, G.: Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises. Signal Process. 89(10), 1883–1890 (2009)

    Article  MATH  Google Scholar 

  8. Ding, J., Fan, C.X., Lin, J.X.: Auxiliary model based parameter estimation for dual-rate output error systems with colored noise. Appl. Math. Model. 37(6), 4051–4058 (2013)

    Article  MathSciNet  Google Scholar 

  9. Chen, J., Ding, R.: An auxiliary-model-based stochastic gradient algorithm for dual-rate sampled-data Box–Jenkins systems. Circuits Syst. Signal Process. 32(5), 2475–2485 (2013)

    Article  MathSciNet  Google Scholar 

  10. Ding, F., Chen, H.B., Li, M.: Multi-innovation least squares identification methods based on the auxiliary model for MISO systems. Appl. Math. Comput. 187(2), 658–668 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, J., Zhang, Y., Ding, R.F.: Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model. Nonlinear Dyn. 72(4), 865–871 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Ding, F.: Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling. Appl. Math. Model. 37(4), 1694–1704 (2013)

    Article  MathSciNet  Google Scholar 

  14. Wang, D.Q., Ding, F.: Hierarchical least squares estimation algorithm for Hammerstein–Wiener systems. IEEE Signal Process. Lett. 19(12), 825–828 (2012)

    Article  Google Scholar 

  15. 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)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ding, F., Liu, G., Liu, X.P.: Partially coupled stochastic gradient identification methods for non-uniformly sampled systems. IEEE Trans. Autom. Control 55(8), 1976–1981 (2010)

    Article  MathSciNet  Google Scholar 

  17. Ding, F.: Coupled-least-squares identification for multivariable systems. IET Control Theory Appl. 7(1), 68–79 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Ding, F., Liu, X.P., Liu, G.: Multi-innovation least squares identification for linear and pseudo-linear regression models. IEEE Trans. Syst. Man Cybernet. Part B: Cybernetics 40(3), 767–778 (2010)

    Google Scholar 

  20. Ding, F.: Several multi-innovation identification methods. Digit. Signal Process. 20(4), 1027–1039 (2010)

    Article  Google Scholar 

  21. Zhang, J.B., Ding, F., Shi, Y.: Self-tuning control based on multi-innovation stochastic gradient parameter estimation. Syst. Control Lett. 58(1), 69–75 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, Y.J., Yu, L., Ding, F.: Multi-innovation extended stochastic gradient algorithm and its performance analysis. Circuits Syst. Signal Process. 29(4), 649–667 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. 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)

    Article  MATH  MathSciNet  Google Scholar 

  24. 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)

    Google Scholar 

  25. Sun, J.L., Liu, X.G.: A novel APSO-aided maximum likelihood identification method for Hammerstein systems. Nonlinear Dyn. 73(1–2), 449–462 (2013)

    Article  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  Google Scholar 

  27. Liu, Y.J., Ding, F., Shi, Y.: An efficient hierarchical identification method for general dual-rate sampled-data systems. Automatica 50(3), 962–973 (2014).

    Google Scholar 

  28. 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)

    Article  Google Scholar 

  29. Liu, Y.J., Wang, D.Q., Ding, F.: Least squares based iterative algorithms for identifying Box–Jenkins models with finite measurement data. Digit. Signal Process. 20(5), 1458–1467 (2010)

    Article  Google Scholar 

  30. Bai, E.W., Li, K.: Convergence of the iterative algorithm for a general Hammerstein system identification. Automatica 46(11), 1891–1896 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Dehghan, M., Hajarian, M.: Iterative algorithmsx for the generalized centro-symmetric and central anti-symmetric solutions of general coupled matrix equations. Eng. Comput. 29(5), 528–560 (2012)

    Article  Google Scholar 

  32. Dehghan, M., Hajarian, M.: Fourth-order variants of Newton’s method without second derivatives for solving non-linear equations. Eng. Comput. 29(4), 356–365 (2012)

    Article  Google Scholar 

  33. Ding, F.: Two-stage least squares based iterative estimation algorithm for CARARMA system modeling. Appl. Math. Model. 37(7), 4798–4808 (2013)

    Article  MathSciNet  Google Scholar 

  34. Ding, F., Ma, J.X., Xiao, Y.S.: Newton iterative identification for a class of output nonlinear systems with moving average noises. Nonlinear Dyn. 74(1–2), 21–30 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  35. Li, J.H., Ding, F., Hua, L.: Maximum likelihood Newton recursive and the Newton iterative estimation algorithms for Hammerstein CARAR systems. Nonlinear Dyn. 75(1–2), 235–245 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  36. Shen, Q.Y., Ding, F.: Iterative estimation methods for Hammerstein controlled autoregressive moving average systems based on the key-term separation principle. Nonlinear Dyn. 75(4), 709–716 (2014)

    Article  MATH  Google Scholar 

  37. Wang, D.Q., Ding, F.: Least squares based and gradient based iterative identification for Wiener nonlinear systems. Signal Process. 91(5), 1182–1189 (2011)

    Article  MATH  Google Scholar 

  38. Liu, Y., Bai, E.W.: Iterative identification of Hammerstein systems. Automatica 43(2), 346–354 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wang, D.Q., Ding, F., Liu, X.M.: Least squares algorithm for an input nonlinear system with a dynamic subspace state space model. Nonlinear Dyn. 75(1–2), 49–61 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  40. 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)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ding, F.: Decomposition based fast least squares algorithm for output error systems. Signal Process. 93(5), 1235–1242 (2013)

    Article  Google Scholar 

  42. Ding, F., Duan, H.H.: Two-stage parameter estimation algorithms for Box–Jenkins systems. IET Control Theory Appl. 7(8), 646–654 (2013)

    MathSciNet  Google Scholar 

  43. Wang, D.Q., Ding, F.: Input–output data filtering based recursive least squares parameter estimation for CARARMA systems. Digit. Signal Process. 20(4), 991–999 (2010)

    Article  Google Scholar 

  44. Shi, Y., Fang, H.: Kalman filter based identification for systems with randomly missing measurements in a network environment. Int. J. Control 83(3), 538–551 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kohli, A.K., Amrita, R.: Numeric variable forgetting factor RLS algorithm for second-order volterra filtering. Circuits Syst. Signal Process. 32(1), 223–232 (2013)

    Article  Google Scholar 

  46. 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)

    Article  MathSciNet  Google Scholar 

  47. Wang, D.Q., Shan, T., Ding, R.: Data filtering based stochastic gradient algorithms for multivariable CARAR-like systems. Math. Model. Anal. 18(3), 374–385 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  48. Wang, Z.Y., Shen, Y.X., Ji, Z.C., Ding, R.: Filtering based recursive least squares algorithm for Hammerstein FIR-MA systems. Nonlinear Dyn. 73(1–2), 1045–1054 (2013)

    Google Scholar 

  49. Wang, D.Q., Ding, F., Chu, Y.Y.: Data filtering based recursive least squares algorithm for Hammerstein systems using the key-term separation principle. Inf. Sci. 222, 203–212 (2013)

    Google Scholar 

  50. Wang, W., Tang, T.: Recursive least squares estimation algorithm applied to a class of linear-in-parameters output error moving average systems. Appl. Math. Lett. 29, 36–41 (2014)

    Article  MathSciNet  Google Scholar 

  51. Goodwin, G.C., Sin, K.S.: Adaptive Filtering Prediction and Control. Prentice-hall, Englewood Cliffs (1984)

    MATH  Google Scholar 

  52. Ding, F., Liu, X.M., Chen, H.B., Yao, G.Y.: Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems. Signal Process. 97, 31–39 (2014)

    Article  Google Scholar 

  53. Ding, F., Liu, X.P., Liu, G.: Identification methods for Hammerstein nonlinear systems. Digit. Signal Process. 21(2), 215–238 (2011)

    Article  Google Scholar 

  54. Ding, F.: Combined state and least squares parameter estimation algorithms for dynamic systems. Appl. Math. Model. 38(1), 403–412 (2014)

    Article  MathSciNet  Google Scholar 

  55. Ding, F.: Hierarchical parameter estimation algorithms for multivariable systems using measurement information. Inf. Sci. (2014). http://dx.doi.org/10.1016/j.ins.2014.02.103

  56. Ding, F., Deng, K.P., Liu, X.M.: Decomposition based Newton iterative identification method for a Hammerstein nonlinear FIR system with ARMA noise. Circ. Syst. Signal Process. 33, (2014). doi:10.1007/s00034-014-9772-y

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61103153) and the National 863 Program (2011AA110502).

Author information

Authors and Affiliations

  1. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing , 100044, People’s Republic of China

    Cheng Wang & Tao Tang

  2. School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing , 100044, People’s Republic of China

    Cheng Wang & Tao Tang

Authors
  1. Cheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cheng Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, C., Tang, T. Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique. Nonlinear Dyn 77, 769–780 (2014). https://doi.org/10.1007/s11071-014-1338-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1338-9

Keywords

Search

Navigation