Maximum likelihood estimation method for dual-rate Hammerstein systems

  • Dong-Qing Wang
  • Zhen Zhang
  • Jin-Yun Yuan
Regular Papers Control Theory and Applications


For a dual-rate sampled Hammerstein controlled autoregressive moving average (CARMA) system, this paper uses the polynomial transformation technology to obtain its dual-rate bilinear identification model which is suitable for the available dual-rate sampled-data, uses the maximum likelihood principle to construct a unified parameter vector of all parameters and an information vector formed by the derivative of the noise variable to the unified parameter vector, and directly identifies the parameters of the linear block and the nonlinear block for the dual-rate Hammerstein CARMA system. The unified parameter vector contains the minimum number of the unknown parameters, and the proposed maximum likelihood estimation algorithm has higher computational efficiency than the over-parameterization model based least squares algorithm.


Dual-rate sampled system Hammerstein system maximum likelihood polynomial transformation system identification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Kim and S. J. Kwon, “Dynamic modeling of a twowheeled inverted pendulum balancing mobile robot,” International Journal of Control, Automation, and Systems, vol. 13, no. 4, pp. 926–933, 2015. [click]CrossRefGoogle Scholar
  2. [2]
    P. Shi and F. B. Li, “A Survey on Markovian Jump Systems: Modeling and Design,” International Journal of Control, Automation, and Systems, vol. 13, no. 1, pp. 1–16, 2015. [click]CrossRefGoogle Scholar
  3. [3]
    Y. Shi, J. Huang, and B. Yu, “Robust tracking control of networked control systems: Application to a networked DC motor,” IEEE Transactions on Industrial Electronics, vol. 60, no. 12, pp. 5864–5874, 2013.CrossRefGoogle Scholar
  4. [4]
    Y. J. Wang and F. Ding, “Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model,” Automatica, vol. 71, pp. 308–313, 2016. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    F. Ding, X. M. Liu, and Y. Gu, “An auxiliary model based least squares algorithm for a dual-rate state space system with time-delay using the data filtering,” Journal of the Franklin Institute, vol. 353, no. 2, pp. 398–408, 2016.MathSciNetCrossRefGoogle Scholar
  6. [6]
    X. G. Liu and J. Lu, “Least squares based iterative identification for a class of multirate systems,” Automatica, vol. 46, no. 3, pp. 549–554, 2010. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Chen, “Several gradient parameter estimation algorithms for dual-rate sampled systems,” Journal of the Franklin Institute, vol. 351, no. 1, pp. 543–554, 2014. [click]CrossRefzbMATHGoogle Scholar
  8. [8]
    Y. Liu and E. W. Bai, “Iterative identification of Hammerstein systems,” Automatica, vol. 43, no. 2, pp. 346–354, 2007. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Salimifard, M. Jafari, and M. Dehghani, “Identification of nonlinear MIMO block-oriented systems with moving average noises using gradient based and least squares based iterative algorithms,” Neurocomputing, vol. 94, pp. 22–31, 2012. [click]CrossRefGoogle Scholar
  10. [10]
    G. Q. Li and C. Y. Wen, “Convergence of fixed-point iteration for the identification of Hammerstein and Wiener systems,” International Journal of Robust and Nonlinear Control, vol. 23, no. 13, pp. 1510–1523, 2013. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    B. Zhang and Z. Z. Mao, “Consistent parameter estimation and convergence properties analysis of hammerstein output-error models,” International Journal of Control, Automation, and Systems, vol. 13, no. 2, pp. 302–310, 2015. [click]MathSciNetCrossRefGoogle Scholar
  12. [12]
    F. Ding, Y. Shi, and T. Chen, “Auxiliary model based leastsquares identification methods for Hammerstein outputerror systems,” Systems & Control Letters, vol. 56, no. 5, pp. 373–380, 2007.MathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Jafari, M. Salimifard, and M. Dehghani, “Identification of multivariable nonlinear systems in the presence of colored noises using iterative hierarchical least squares algorithm,” ISA Transactions, vol. 53, no. 4, pp. 1243–1252, 2014. [click]CrossRefGoogle Scholar
  14. [14]
    Y. W. Mao and F. Ding, “Adaptive filtering parameter estimation algorithms for Hammerstein nonlinear systems,” Signal Processing, vol. 128, pp. 417–425, 2016. [click]CrossRefGoogle Scholar
  15. [15]
    J. Vörös, “Identification of nonlinear cascade systems with time-varying backlash,” Journal of Electrical Engineering, vol. 62, no. 2, pp. 87–92, 2011. [click]CrossRefGoogle Scholar
  16. [16]
    Q. Y. Shen and F. Ding, “Iterative identification methods for input nonlinear multivariable systems using the keyterm separation principle,” Journal of the Franklin Institute–Engineering and Applied Mathematics, vol. 352, no. 7, pp. 2847–2865, 2015.MathSciNetCrossRefGoogle Scholar
  17. [17]
    X. H. Wang and F. Ding, “Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle,” Signal Processing, vol. 117, pp. 208–218, 2015. [click]CrossRefGoogle Scholar
  18. [18]
    F. Ding, X. H. Wang, Q. J. Chen, and Y. S. Xiao, “Recursive least squares parameter estimation for a class of output nonlinear systems based on the model decomposition,” Circuits, Systems and Signal Processing, vol. 35, no. 9, pp. 3323–3338, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Q. Wang, “Hierarchical parameter estimation for a class of MIMO Hammerstein systems based on the reframed models,” Applied Mathematics Letters, vol. 57, pp. 13–19, 2016. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Q. Wang and F. Ding, “Parameter estimation algorithms for multivariable Hammerstein CARMA systems,” Information Sciences, vol. 355-356, pp. 237–248, 2016. [click]MathSciNetCrossRefGoogle Scholar
  21. [21]
    J. H. Li, “Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration,” Applied Mathematics Letters, vol. 26, no. 1, pp. 91–96, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    L. Ma and X. G. Liu, “Recursive maximum likelihood method for the identification of Hammerstein ARMAX system,” Applied Mathematical Modelling, vol. 40, no. 13–14, pp. 6523–6535, 2016. [click]MathSciNetCrossRefGoogle Scholar
  23. [23]
    D. Q. Wang and W. Zhang, “Improved least squares identification algorithm for multivariable Hammerstein systems,” Journal of the Franklin Institute, vol. 352, no. 11, pp. 5292–5307, 2015. [click]MathSciNetCrossRefGoogle Scholar
  24. [24]
    D. Q. Wang, H. B. Liu, and F. Ding, “Highly efficient identification methods for dual-rate Hammerstein systems,” IEEE Transactions on Control Systems Technology, vol. 23, no. 5, pp. 1952–1960, 2015.CrossRefGoogle Scholar
  25. [25]
    X. H. Wang and F. Ding, “Convergence of the recursive identification algorithms for multivariate pseudo-linear regressive systems,” International Journal of Adaptive Control and Signal Processing, vol. 30, no. 6, pp. 824–842, 2016. [click]MathSciNetCrossRefGoogle Scholar
  26. [26]
    Y. J. Wang and F. Ding, “The filtering based iterative identification for multivariable systems,” IET Control Theory and Applications, vol. 10, no. 8, pp. 894–902, 2016.MathSciNetCrossRefGoogle Scholar
  27. [27]
    F. Ding, “Hierarchical estimation algorithms for multivariable systems using measurement information,” Information Sciences, vol. 277, pp. 396–405, 2014. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    D. Q. Wang, L. Mao, and F. Ding, “Recasted models based hierarchical extended stochastic gradient method for MIMO nonlinear systems,” IET Control Theory and Applications, vol. 11, no. 4, pp. 476–485, 2017.CrossRefGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Automation and Electrical EngineeringQingdao UniversityQingdaoP. R. China

Personalised recommendations