Skip to main content
SpringerLink
Log in

Maximum likelihood gradient-based iterative estimation algorithm for a class of input nonlinear controlled autoregressive ARMA systems

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

Abstract

This paper considers the parameter estimation problem for an input nonlinear controlled autoregressive ARMA model. The basic idea is to combine the maximum likelihood principle and the gradient search and to present a maximum likelihood gradient-based iterative estimation algorithm. The analysis and simulation results show that the proposed algorithm can effectively estimate the parameters of the input nonlinear controlled autoregressive ARMA 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
Fig. 4
Fig. 5

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. Spall, J.: Identification for systems with binary subsystems. IEEE Trans. Autom. Control 59(1), 3–17 (2013)

    Article  MathSciNet  Google Scholar 

  4. Dai, J.Y., Tan, C., Ying, J., Wu, G.H.: Fuzzy multi-model switching H-infinity for helicopters in a full envelope. Circuits Syst. Signal Process. 32(5), 2185–2197 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Elliott, J., Kuen, S., Fung, S.: Filtering a nonlinear stochastic volatility model. Nonlinear Dyn. 67(2), 1295–1313 (2012)

    Article  MATH  Google Scholar 

  8. 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 

  9. Södersöm, T., Hong, M., Schoukens, J., Pintelon, R.: Accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors-in-variables and output error identification. Automatica 46(4), 721–727 (2010)

    Article  MathSciNet  Google Scholar 

  10. 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  MathSciNet  MATH  Google Scholar 

  11. Vörös, J.: Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones. IEEE Trans. Autom. Control 48(12), 2203–2206 (2003)

    Article  Google Scholar 

  12. Li, H., Shi, Y.: Robust H-infinity filtering for nonlinear stochastic systems with uncertainties and random delays modeled by Markov chains. Automatica 48(1), 159–166 (2012)

  13. 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 

  14. Shi, Y., Yu, B.: Robust mixed \(\text{ H }\_2/\text{ H }\_\)infinity control of networked control systems with random time delays in both forward and backward communication links. Automatica 47(4), 754–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, H., Shi, Y.: State-feedback H-infty control for stochastic time-delay nonlinear systems with state and disturbance-dependent noise. Int. J. Control 85(10), 1515–1531 (2012)

    Article  MATH  Google Scholar 

  16. 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 

  17. Zhang, Y., Cui, G.M.: Bias compensation methods for stochastic systems with colored noise. Appl. Math. Model. 35(4), 1709–1716 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kulikova, V.: Maximum likelihood estimation of linear stochastic systems in the class of sequential square-root orthogonal filtering methods. Autom. Remote Control 72(4), 766–786 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, J.H., Gu, J.P., Ma, W.G., Ding, R.: Maximum likelihood forgetting stochastic gradient estimation algorithm for Hammerstein CARARMA systems. In: The 2012 24th Chinese Control and Decision Conference (2012 CCDC), May 23–25, Taiyuan, China, pp. 2545–2550 (2012)

  21. Wu, A.G., Lv, L.L., Hou, M.Z.: Finite iterative algorithms for extended Sylvester-conjugate matrix equations. Math. Comput. Model. 54(9–10), 2363–2384 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dehghan, M., Hajarian, M.: Iterative algorithms 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 

  23. Wu, A.G., Lv, L.L., Duan, G.R.: Iterative algorithms for solving a class of complex conjugate and transpose matrix equations. Appl. Math. Comput. 217(21), 8343–8353 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dehghan, M., Hajarian, M.: Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation \(A_1X_1B_1+A_2X_2B_2=C\). Math. Comput. Model. 49(9–10), 1937–1959 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xie, L., Liu, Y.J., Yang, H.Z.: Gradient based and least squares based iterative algorithms for matrix equations \(AXB+CX^TD=F\). Appl. Math. Comput. 217(5), 2191–2199 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ding, F., Zhang, M.: Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems. IET Control Theory Appl. 8. doi:10.1049/iet-cta.2013.1044 (2014)

  27. Liu, Y.J., Ding, F., Shi, Y.: An efficient hierarchical identification method for general dual-rate sampled-data systems. Automatica 50(3), 962–970 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ding, F.: State filtering and parameter identification for state space systems with scarce measurements. Signal Process. 104, 369–380 (2014)

    Article  Google Scholar 

  29. Hu, Y.B.: Iterative and recursive least squares estimation algorithms for moving average systems. Simul. Model. Pract. Theory 34, 12–19 (2013)

    Article  Google Scholar 

  30. Wang, C., 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)

  31. Gu, Y., Ding, F., Li, J.H.: State filtering and parameter estimation for linear systems with d-step state-delay. IET Signal Process. 8(6), 639–646 (2014)

  32. Xie, L., Yang, H.Z.: Gradient based iterative identification for nonuniform sampling output error systems. J. Vib. Control 17(3), 471–478 (2011)

  33. Ding, J., Lin, J.X.: Modified subspace identification for periodically non-uniformly sampled systems by using the lifting technique. Circuits Syst. Signal Process. 33(5), 1439–1449 (2014)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Vörös, J.: Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities. IEEE Trans. Autom. Control 44(11), 2145–22149 (1999)

    Article  MATH  Google Scholar 

  36. Gu, Y., Ding, F., Li, J.H.: States based iterative parameter estimation for a state space model with multi-state delays using decomposition. Signal Process. 106, 230–294 (2015)

  37. 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  MathSciNet  MATH  Google Scholar 

  38. Wang, L.Y., Xie, L., Wang, X.F.: The residual based interactive stochastic gradient algorithms for controlled moving average models. Appl. Math. Comput. 211(2), 442–449 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Jiang, X.Z., Jian, J.B.: A sufficient descent Dai-Yuan type nonlinear conjugate gradient method for unconstrained optimization problems. Nonlinear Dyn. 72(1–2), 101–112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, Y.J., Ding, R.: Consistency of the extended gradient identification algorithm for multi-input multi-output systems with moving average noises. Int. J. Comput. Math. 90(9), 1840–1852 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Aladag, C.H., Egrioglu, E., Kadilar, C.: Forecasting nonlinear time series with a hybrid methodology. Appl. Math. Lett. 22(9), 1467–1470 (2009)

    Article  MATH  Google Scholar 

  43. Wang, C., Tang, T.: Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique. Nonlinear Dyn. 77(3), 769–780 (2014)

  44. Xie, L., Yang, H.Z.: Interactive parameter estimation for output error moving average systems. Trans. Inst. Meas. Control 35(1), 34–43 (2013)

    Article  Google Scholar 

  45. Bai, E.W.: An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems. Automatica 34(3), 333–338 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  46. 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)

  47. Li, J.H.: Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration. Appl. Math. Lett. 26(1), 91–96 (2013)

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

  49. Li, J.H., Ding, R.: Maximum likelihood gradient based identification algorithm for output error systems with colored noises. In: The 32nd Chinese Control Conference (2013 CCC), July 26–28, Xi’an, China, pp. 1968–1973 (2013)

  50. 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)

    Article  MATH  Google Scholar 

  51. Ding, F.: Hierarchical parameter estimation algorithms for multivariable systems using measurement information. Inf. Sci. 277, 396–405 (2014)

    Article  Google Scholar 

  52. Luan, X.L., Zhao, S.Y., Liu, F.: H-infinity control for discrete-time Markov jump systems with uncertain transition probabilities. IEEE Trans. Autom. Control 58(6), 1566–1572 (2013)

    Article  MathSciNet  Google Scholar 

  53. Zhu, D.Q., Huang, H., Yang, S.X.: Dynamic task assignment and path planning of multi-AUV system based on an improved self-organizing map and velocity synthesis method in 3D underwater workspace. IEEE Trans. Cybern. 43(2), 504–514 (2013)

    Article  Google Scholar 

  54. Zhu, D.Q., Liu, Q., Hu, Z.: Fault-tolerant control algorithm of the manned submarine with multi-thruster based on quantum behaved particle swarm optimization. Int. J. Control 84(11), 1817–1829 (2012)

    Article  MathSciNet  Google Scholar 

  55. Sun, B., Zhu, D.Q., Yang, S.X.: A bio-inspired filtered backstepping cascaded tracking control of 7000m manned submarine vehicle. IEEE Trans. Ind. Electron. 61(7), 3682–3692 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61273194, 61403217) and the PAPD of Jiangsu Higher Education Institutions.

Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi, 214122, People’s Republic of China

    Feiyan Chen & Feng Ding

  2. School of Electrical Engineering, Nantong University, Nantong, 226019, People’s Republic of China

    Junhong Li

Authors
  1. Feiyan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junhong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Ding.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, F., Ding, F. & Li, J. Maximum likelihood gradient-based iterative estimation algorithm for a class of input nonlinear controlled autoregressive ARMA systems. Nonlinear Dyn 79, 927–936 (2015). https://doi.org/10.1007/s11071-014-1712-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1712-7

Keywords

Search

Navigation