Two-stage Recursive Least Squares Parameter Estimation Algorithm for Multivariate Output-error Autoregressive Moving Average Systems

  • Yunze Guo
  • Lijuan Wan
  • Ling Xu
  • Feng DingEmail author
  • Ahmed Alsaedi
  • Tasawar Hayat
Regular Papers Control Theory and Applications


This paper focuses on the parameter estimation problem of multivariate output-error autoregressive moving average (M-OEARMA) systems. By applying the auxiliary model identification idea and the decomposition technique, we derive a two-stage recursive least squares algorithm for estimating the M-OEARMA system. Compared with the auxiliary model based recursive least squares algorithm, the proposed algorithm possesses higher identification accuracy. The simulation results confirm the effectiveness of the proposed algorithm.


Auxiliary model decomposition technique least squares parameter estimation recursive identification 


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  1. [1]
    Y. Gu, J. Liu, X. Li, Y. Chou, and Y. Ji, “State space model identification of multirate processes with time-delay using the expectation maximization,” Journal of the Franklin Institute, vol. 356, no. 3, pp. 1623–1639, February 2019.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Y. Gu, Y. Chou, J. Liu, and Y. Ji, “Moving horizon estimation for multirate systems with time-varying time-delays,” Journal of the Franklin Institute, vol. 356, no. 4, pp. 2325–2345, March 2019.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Y. Cao, P. Li, and Y. Zhang, “Parallel processing algorithm for railway signal fault diagnosis data based on cloud computing,” Future Generation Computer Systems, vol. 88, pp. 279–283, November 2018.CrossRefGoogle Scholar
  4. [4]
    Y. Z. Zhang, Y. Cao, Y. H. Wen, L. Liang, and F. Zou, “Optimization of information interaction protocols in cooperative vehicle-infrastructure systems,” Chinese Journal of Electronics, vol. 27, no. 2, pp. 439–444, March 2018.CrossRefGoogle Scholar
  5. [5]
    Y. Cao, L. C. Ma, S. Xiao, X. Zhang, and W. Xu, “Standard analysis for transfer delay in CTCS-3,” Chinese Journal of Electronics, vol. 26, no. 5, pp. 1057–1063, September 2017.CrossRefGoogle Scholar
  6. [6]
    Y. Cao, Y. Wen, X. Meng, and W. Xu, “Performance evaluation with improved receiver design for asynchronous coordinated multipoint transmissions,” Chinese Journal of Electronics. vol. 25, no. 2, pp. 372–378, March 2016.CrossRefGoogle Scholar
  7. [7]
    F. Liu, “Rough maximal functions supported by subvarieties on Triebel-Lizorkin spaces,” Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-Matematicas, vol. 112, no. 2, pp. 593–614, April 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    F. Liu, Z. Fu, and S. Jhang, “Boundedness and continuity of Marcinkiewicz integrals associated to homogeneous mappings on Triebel-Lizorkin spaces,” Frontiers of Mathematics in China, vol. 14, no. 1, pp. 95–122, January 2019.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    X. S. Zhan, L. L. Cheng, J. Wu, Q. S. Yang, and T. Han, “Optimal modified performance of MIMO networked control systems with multi-parameter constraints,” ISA Transactions, vol. 84, no. 1, pp. 111–117, January 2019.CrossRefGoogle Scholar
  10. [10]
    M. Gan, C. L. P. Chen, G. Y. Chen, and L. Chen, “On some separated algorithms for separable nonlinear squares problems,” IEEE Transactions on Cybernetics, vol. 48, no. 10, pp. 2866–2874, October 2018.CrossRefGoogle Scholar
  11. [11]
    M. Gan, H. X. Li, and H. Peng, “A variable projection approach for efficient estimation of RBF-ARX model,” IEEE Transactions on Cybernetics, vol. 45, no. 3, pp. 462–471, March 2015.CrossRefGoogle Scholar
  12. [12]
    A. Brouri, F. Giri, F. Ikhouane, F. Z. Chaoui, and O. Amdouri, “Identification of Hammerstein-Wiener systems with backlask input nonlinearity bordered by straight lines,” Proc. of 19th IFAC World Congress, vol. 47, no. 3, pp. 475–480, August 2014.Google Scholar
  13. [13]
    C. P. Yu, L. H. Xie, and Y. C. Soh, “Blind channel and source estimation in networked systems,” IEEE Transactions on Signal Processing, vol. 62, no. 17, pp. 4611–4626, September 2014.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    R. N. Mahia, M. Singh, and D. M. Fulwani, “Identification of optimal set of driver nodes in complex networked systems using region of attraction,” International Journal of Control Automation and Systems, vol. 16, no. 1, pp. 97–107, February 2018.CrossRefGoogle Scholar
  15. [15]
    L. Xu, “The parameter estimation algorithms based on the dynamical response measurement data,” Advances in Mechanical Engineering, vol. 9, no. 11, pp. 1–12, November 2017. doi: Google Scholar
  16. [16]
    Y. J. Wang, F. Ding, and M. H. Wu, “Recursive parameter estimation algorithm for multivariate output-error systems,” Journal of the Franklin Institute, vol. 355, no. 12, pp. 5163–5181, August 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    G. Y. Chen, M. Gan, C. L. P. Chen, and H. X. Li, “A regularized variable projection algorithm for separable nonlinear least-squares problems,” IEEE Transactions on Automatic Control, vol. 64, no. 2, pp. 526–537, February 2019.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Z. P. Zhou and X. F. Liu, “State and fault estimation of sandwich systems with hysteresis,” International Journal of Robust and Nonlinear Control, vol. 28, no. 13, pp. 3974–3986, September 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    J. Pan, X. Jiang, X.K. Wan, and W. Ding, “A filtering based multi-innovation extended stochastic gradient algorithm for multivariable control systems,” International Journal of Control Automation and Systems, vol. 15, no. 3, pp. 1189–1197, June 2017.CrossRefGoogle Scholar
  20. [20]
    A. Brouri, L. Kadi, and S. Slassi, “Frequency identification of Hammerstein-Wiener systems with backlash input nonlinearity,” International Journal of Control Automation and Systems, vol. 15, no. 5, pp. 2222–2232, October 2017.CrossRefGoogle Scholar
  21. [21]
    A. Brouri, O. Amdouri, F. Z. Chaoui, and F. Giri, “Frequency identification of Hammerstein-Wiener systems with piecewise affine input nonlinearity,” Proc. of 19th IFAC World Congress, vol. 47, no. 3, pp. 10030–10035, August 2014.Google Scholar
  22. [22]
    J. Na, J. Yang, X. Wu, and Y. Guo, “Robust adaptive parameter estimation of sinusoidal signals,” Automatica, vol. 53, pp. 376–384, March 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    L. Xu, “Application of the Newton iteration algorithm to the parameter estimation for dynamical systems,” Journal of Computational and Applied Mathematics, vol. 288, pp. 33–43, November 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    L. Xu and F. Ding, “Parameter estimation for control systems based on impulse responses,” International Journal of Control Automation and Systems, vol. 15, no. 6, pp. 2471–2479, December 2017.CrossRefGoogle Scholar
  25. [25]
    J. L. Ding, “Recursive and iterative least squares parameter estimation algorithms for multiple-input-output-error systems with autoregressive noise,” Circuits Systems, and Signal Processing, vol. 37, no. 5, pp. 1884–1906, May 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    X. Zhang, L. Xu, F. Ding, and T. Hayat, “Combined state and parameter estimation for a bilinear state space system with moving average noise,” Journal of the Franklin Institute, vol. 355, no. 6, pp. 3079–3103, April 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    J. Chen, B. Jiang, and J. Li, “Missing output identification model based recursive least squares algorithm for a distributed parameter system,” International Journal of Control Automation and Systems, vol. 16, no. 1, pp. 150–157, February 2018.CrossRefGoogle Scholar
  28. [28]
    H. Cho and S. C. Yu, “Variable data-window-size recursive least-squares algorithm for dynamic system identification,” Electronics Letters, vol. 51, no. 4, pp. 341–343, February 2015.CrossRefGoogle Scholar
  29. [29]
    Y. J. Wang and F. Ding, “A filtering based multi-innovation gradient estimation algorithm and performance analysis for nonlinear dynamical systems,” IMA Journal of Applied Mathematics, vol. 82, no. 6, pp. 1171–1191, November 2017.MathSciNetCrossRefGoogle Scholar
  30. [30]
    X. Zhang, F. Ding, L. Xu, and E. F. Yang, “State filtering-based least squares parameter estimation for bilinear systems using the hierarchical identification principle,” IET Control Theory and Applications, vol. 12, no. 12, pp. 1704–1713, August 2018.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Z. W. Ge, F. Ding, L. Xu, A. Alsaedi, and T. Hayat, “Gradient-based iterative identification method for multivariate equation-error autoregressive moving average systems using the decomposition technique,” Journal of the Franklin Institute, vol. 356, no. 3, pp. 1658–1676, February 2019.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    L. Xu, W. L. Xiong, A. Alsaedi, and T. Hayat, “Hierarchical parameter estimation for the frequency response based on the dynamical window data,” International Journal of Control Automation and Systems, vol. 16, no. 4, pp. 1756–1764, August 2018.CrossRefGoogle Scholar
  33. [33]
    L. Xu and F. Ding, “Iterative parameter estimation for signal models based on measured data,” Circuits Systems and Signal Processing, vol. 37, no. 7, pp. 3046–3069, July 2018.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Y. P. Pan, X. Li, and H. Y. Yu, “Least-squares learning control with guaranteed parameter convergence,” Proc. of 22nd International Conference on Automation and Computing (ICAC), Colchester, UK, 2016.Google Scholar
  35. [35]
    Y. P. Pan and H. Y. Yu, “Composite learning from adaptive dynamic surface control,” IEEE Transactions on Automatic Control, vol. 61, no. 9, pp. 2603–2609, September 2016.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Q. Y. Liu and F. Ding, “Auxiliary model-based recursive generalized least squares algorithm for multivariate output-error autoregressive systems using the data filtering,” Circuits Systems and Signal Processing, vol. 38, no. 2, pp. 590–610, February 2019.CrossRefGoogle Scholar
  37. [37]
    T. L. Lai and C. Z. Wei, “least squares estimates in stochastic regression models with applications to identification and control of dynamic systems,” The Annals of Statistics, vol. 10, no. 1, pp. 154–166, January 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    F. Ding, “Coupled-least-squares identification for multi-variable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, January 2013.MathSciNetCrossRefGoogle Scholar
  39. [39]
    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, September 2016.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Y. J. Wang and F. Ding, “The auxiliary model based hierarchical gradient algorithms and convergence analysis using the filtering technique,” Signal Processing, vol. 128, pp. 212–221, November 2016.CrossRefGoogle Scholar
  41. [41]
    Y. J. Wang, F. Ding, and L. Xu, “Some new results of designing an IIR filter with colored noise for signal processing,” Digital Signal Processing, vol. 72, pp. 44–58, January 2018.MathSciNetCrossRefGoogle Scholar
  42. [42]
    Y. Ji and F. Ding, “Multiperiodicity and exponential attractivity of neural networks with mixed delays,” Circuits Systems and Signal Processing, vol. 36, no. 6, pp. 2558–2573, June 2017.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    F. Ding, Y. J. Wang, J. Y. Dai, Q. S Li, and Q. J. Chen, “A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input-output data filtering,” Journal of the Franklin Institute, vol. 354, no. 15, pp. 6938–6955, October 2017.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    H. B. Chen, Y. S. Xiao, and F. Ding, “Hierarchical gradient parameter estimation algorithm for Hammerstein nonlinear systems using the key term separation principle,” Applied Mathematics and Computation, vol. 247, pp. 1202–1210, November 2014.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    Y. W. Mao and F. Ding, “A novel parameter separation based identification algorithm for Hammerstein systems,” Applied Mathematics Letters, vol. 60, pp. 21–27, October 2016.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    Y. Ji and X. M. Liu, “Unified synchronization criteria for hybrid switching-impulsive dynamical networks,” Circuits Systems Signal Processing, vol. 34, no. 5, pp. 1499–1517, May 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    L. Xu, “A proportional differential control method for a time-delay system using the Taylor expansion approximation,” Applied Mathematics and Computation, vol. 236, pp. 391–399, June 2015.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    L. Xu, L. Chen, and W. L. Xiong, “Parameter estimation and controller design for dynamic systems from the step responses based on the Newton iteration,” Nonlinear Dynamics, vol. 79, no. 3, pp. 2155–2163, February 2015.MathSciNetCrossRefGoogle Scholar
  49. [49]
    L. Xu, “The damping iterative parameter identification method for dynamical systems based on the sine signal measurement,” Signal Processing, vol. 120, pp. 660–667, March 2016.CrossRefGoogle Scholar
  50. [50]
    L. Xu and F. Ding, “The parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the hierarchical principle,” IET Signal Processing, vol. 11, no. 2, pp. 228–237, April 2017.CrossRefGoogle Scholar
  51. [51]
    L. Xu, F. Ding, and Q. M. Zhu, “Hierarchical Newton and least squares iterative estimation algorithm for dynamic systems by transfer functions based on the impulse responses,” International Journal of Systems Science, vol. 50, no. 1, pp. 141–151, January 2019.MathSciNetCrossRefGoogle Scholar
  52. [52]
    F. Liu, “Boundedness and continuity of maximal operators associated to polynomial compound curves on Triebel-Lizorkin spaces,” Mathematical Inequalities & Applications, vol. 22, no. 1, pp. 25–44, January 2019.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    L. Feng, Q. X. Li, and Y. F. Li, “Imaging with 3-D aperture synthesis radiometers,” IEEE Transactions on Geoscience and Remote Sensing, vol. 57, no. 4, p. 2395–2406, April 2019.CrossRefGoogle Scholar
  54. [54]
    W. X. Shi, N. Liu, Y. M. Zhou, and X. A. Cao, “Effects of postannealing on the characteristics and reliability of polyfluorene organic light-emitting diodes,” IEEE Transactions on Electron Devices, vol. 66, no. 2, pp. 1057–1062, February 2019.CrossRefGoogle Scholar
  55. [55]
    N. Zhao, R. Liu, Y. Chen, M. Wu, Y. Jiang, W. Xiong, and C. Liu, “Contract design for relay incentive mechanism under dual asymmetric information in cooperative networks,” Wireless Networks, vol. 24, no. 8, pp. 3029–3044, November 2018.CrossRefGoogle Scholar
  56. [56]
    J. Pan, W. Li, and H. P. Zhang, “Control algorithms of magnetic suspension systems based on the improved double exponential reaching law of sliding mode control,” International Journal of Control Automation and Systems, vol.16, no. 6, pp. 2878–2887, December 2018.CrossRefGoogle Scholar
  57. [57]
    Y. Wang, Y. Si, B. Huang, and S. X. Ding, “Survey on the theoretical research and engineering applications of multivariate statistics process monitoring algorithms: 2008–2017,” The Canadian Journal of Chemical Engineering, vol. 96, no. 10, pp. 2073–2085, October 2018.CrossRefGoogle Scholar
  58. [58]
    X. Y. Li, H. X. Li, and B. Y. Wu, “Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments,” Applied Mathematics and Computation, vol. 349, pp. 3043–313, May 2019.MathSciNetGoogle Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of Internet of Things EngineeringJiangnan UniversityWuxiP. R. China
  2. 2.College of Automation and Electronic EngineeringQingdao University of Science and TechnologyQingdaoP. R. China
  3. of Internet of Things TechnologyWuxi Vocational Institute of commerceWuxiP. R. China
  4. 4.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

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