Iterative Identification Algorithms for Bilinear-in-parameter Systems by Using the Over-parameterization Model and the Decomposition

  • Mengting Chen
  • Feng DingEmail author
  • Ahmed Alsaedi
  • Tasawar Hayat
Regular Papers Control Theory and Applications


This paper focuses on the identification problem for a class of bilinear-in-parameter systems with an additive noise modeled by an autoregressive moving average process. By using the over-parameterization model, the special form of the bilinear term can be obtained by the model equivalent transformation. Then, we use a decomposition of the model into two synthetic models in order to separate the effect of the two sets of parameters, i.e., the coefficients of the nonlinear basis functions from the parameters of the colored noise. Moreover, two decomposition based iterative algorithms are proposed to identify the unknown parameters. A numerical example is presented to confirm the effectiveness of the proposed methods.


Bilinear-in-parameter system decomposition iterative identification over-parameterization parameter estimation 


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  1. [1]
    D. Q. Zhu, X. Cao, B. Sun, and C. M. Luo, “Biologically inspired self-organizing map applied to task assignment and path planning of an AUV system,” IEEE Transactions on Cognitive and Developmental Systems, vol. 10, no. 2, pp. 304–313, June 2018.Google Scholar
  2. [2]
    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.Google Scholar
  3. [3]
    P. Li, R. X. Li, Y. Cao, and G. Xie, “Multi-objective sizing optimization for island microgrids using triangular aggregation model and Levy-Harmony algorithm,” IEEE Transactions on Industrial Informatics, vol. 14, no. 8, pp. 3495–3505, August 2018.Google Scholar
  4. [4]
    P. Li, R. Dargaville, Y. Cao, D. Y. Li, and J. Xia, “Storage aided system property enhancing and hybrid robust smoothing for large-scale PV systems,” IEEE Transactions on Smart Grid, vol. 8, no. 6, pp. 2871–2879, November 2017.Google 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.Google Scholar
  6. [6]
    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.Google Scholar
  7. [7]
    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.MathSciNetGoogle Scholar
  8. [8]
    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.Google Scholar
  9. [9]
    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, 2018.Google Scholar
  10. [10]
    P. Ma, F. Ding, and Q. M. Zhu, “Decomposition-based recursive least squares identification methods for multivariate pseudolinear systems using the multi-innovation,” International Journal of Systems Science, vol. 49, no. 5, pp. 920–928, April 2018.MathSciNetGoogle Scholar
  11. [11]
    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.MathSciNetzbMATHGoogle Scholar
  12. [12]
    J. Y. Zhai and Z. B. Song, “Global finite-time stabilization for a class of switched nonlinear systems via output feedback,” International Journal of Control, Automation, and Systems, vol. 15, no. 5, pp. 1975–1982, October 2017.Google Scholar
  13. [13]
    K. H. Yang and H. B. Ji, “Hierarchical control for nonlinear systems by partial-state feedback,” International Journal of Control, Automation, and Systems, vol. 15, no. 2, pp. 595–602, April 2017.Google Scholar
  14. [14]
    A. Zribi, M. Chtourou, and M. Djemel, “Multiple model reduction approach using gap metric and stability margin for control nonlinear systems,” International Journal of Control, Automation, and Systems, vol. 15, no. 1, pp. 267–273, February 2017.Google Scholar
  15. [15]
    Y. Y. Wang, Y. Q. Xia, H. Shen, and P. F. Zhou, “SMC design for robust stabilization of nonlinear Markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 63, no. 1, pp. 219–224, January 2018.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Y. Y. Wang, H. Shen, H. R. Karimi, and D. P. Duan, “Dissipativity-based fuzzy integral sliding mode control of continuous-time T-S fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 3, pp. 1164–1176, June 2017.Google Scholar
  17. [17]
    Y. Y. Wang, Y. B. Gao, H. R. Karimi, H. Shen, and Z. J. Fang, “Sliding mode control of fuzzy singularly perturbed systems with application to electric circuit,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, August 2017.Google Scholar
  18. [18]
    D. Q. Wang, Z. Zhang, and J. Y. Yuan, “Maximum likelihood estimation method for dual-rate Hammerstein systems,” International Journal of Control, Automation, and Systems, vol. 15, no. 2, pp. 698–705, April 2017.Google Scholar
  19. [19]
    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 inputoutput data filtering,” Journal of the Franklin Institute, vol. 354, no. 15, pp. 6938–6955, October 2017.MathSciNetzbMATHGoogle Scholar
  20. [20]
    M. T. Chen, F. Ding, L. Xu, T. Hayat, and A. Alsaedi, “Iterative identification algorithms for bilinear-in-parameter systems with autoregressive moving average noise,” Journal of the Franklin Institute, vol. 354, no. 17, pp. 7885–7898, November 2017.MathSciNetzbMATHGoogle Scholar
  21. [21]
    F. Ding and X. H. Wang, “Hierarchical stochastic gradient algorithm and its performance analysis for a class of bilinear-in-parameter systems,” Circuits, Systems and Signal Processing, vol. 36, no. 4, pp. 1393–1405, April 2017.MathSciNetzbMATHGoogle Scholar
  22. [22]
    F. Ding, L. Xu, and Q. M. Zhu, “Performance analysis of the generalised projection identification for time-varying systems,” IET Control Theory and Applications, vol. 10, no. 18, pp. 2506–2514, December 2016.MathSciNetGoogle Scholar
  23. [23]
    L. Xu and F. Ding, “Recursive least squares and multiinnovation stochastic gradient parameter estimation methods for signal modeling,” Circuits, Systems and Signal Processing, vol. 36, no. 4, pp. 1735–1753, April 2017.zbMATHGoogle Scholar
  24. [24]
    L. Xu and F. Ding, “The parameter estimation algorithms for dynamical response signals based on the multiinnovation theory and the hierarchical principle,” IET Signal Processing, vol. 11, no. 2, pp. 228–237, April 2017.Google Scholar
  25. [25]
    L. Xu, F. Ding, Y. Gu, A. Alsaedi, and T. Hayat, “A multiinnovation state and parameter estimation algorithm for a state space system with d-step state-delay,” Signal Processing, vol. 140, pp. 97–103, November 2017.Google Scholar
  26. [26]
    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.Google Scholar
  27. [27]
    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.MathSciNetGoogle Scholar
  28. [28]
    X. Zhang, F. Ding, A. Alsaadi, and T. Hayat, “Recursive parameter identification of the dynamical models for bilinear state space systems,” Nonlinear Dynamics, vol. 89, no. 4, pp. 2415–2429, September 2017.MathSciNetzbMATHGoogle Scholar
  29. [29]
    F. Ding, L. Xu, F. E. Alsaadi, and T. Hayat, “Iterative parameter identification for pseudo-linear systems with ARMA noise using the filtering technique,” IET Control Theory and Applications, vol. 12, no. 7, pp. 892–899, May 2018.MathSciNetGoogle Scholar
  30. [30]
    F. Ding, H. B. Chen, L. Xu, J. Y. Dai, Q. S. Li, and T. Hayat, “A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation,” Journal of the Franklin Institute, vol. 355, no. 8, pp. 3737–3752, May 2018.MathSciNetzbMATHGoogle Scholar
  31. [31]
    M. H. Li and X. M. Liu, “The least squares based iterative algorithms for parameter estimation of a bilinear system with autoregressive noise using the data filtering technique,” Signal Processing, vo. 147, pp. 23–34, June 2018.Google Scholar
  32. [32]
    F. Ding, D. D. Meng, J. Y. Dai, Q. S. Li, A. Alsaedi, and T. Hayat, “Least squares based iterative parameter estimation algorithm for stochastic dynamical systems with ARMA noise using the model equivalence,” International Journal of Control, Automation and Systems, vol. 16, no. 2, pp. 630–639. April 2018.Google Scholar
  33. [33]
    E. W. Bai, “A blind approach to the Hammerstein-Wiener model identification,” Automatica, vol. 38, pp. 967–979, June 2002.MathSciNetzbMATHGoogle Scholar
  34. [34]
    X. Zhang, F. Ding, L. Xu, and E. F. Yang, “State filteringbased 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.Google Scholar
  35. [35]
    F. Liu, Q. Y. Xue, and K. Yabuta, “Rough maximal singular integral and maximal operators supported by subvarieties on Triebel-Lizorkin spaces,” Nonlinear Analysis, vol. 171, pp. 41–72, June 2018.MathSciNetzbMATHGoogle Scholar
  36. [36]
    F. Liu, “Continuity and approximate differentiability of multisublinear fractional maximal functions,” Mathematical Inequalities & Applications, vol. 21, no. 1, pp. 25–40, January 2018.MathSciNetzbMATHGoogle Scholar
  37. [37]
    F. Liu and H. X. Wu, “Singular integrals related to homogeneous mappings in triebel-lizorkin spaces,” Journal of Mathematical Inequalities, vol. 11, no. 4, pp. 1075–1097, December 2017.MathSciNetzbMATHGoogle Scholar
  38. [38]
    F. Liu and H. X. Wu, “Regularity of discrete multisublinear fractional maximal functions,” Science China–Mathematics, vol. 60, no. 8, pp. 1461–1476, August 2017.MathSciNetzbMATHGoogle Scholar
  39. [39]
    F. Liu and H. X. Wu, “On the regularity of maximal operators supported by submanifolds,” Journal of Mathematical Analysis and Applications, vol. 453, no. 1, pp. 144–158, September 2017.MathSciNetzbMATHGoogle Scholar
  40. [40]
    F. Liu, “On the triebel-lizorkin space boundedness of marcinkiewicz integrals along compound surfaces,” Mathematical Inequalities & Applications, vol. 20, no. 2, pp. 515–535, April 2017.MathSciNetzbMATHGoogle Scholar
  41. [41]
    C. C. Yin and J. S. Zhao, “Nonexponential asymptotics for the solutions of renewal equations, with applications,” Journal of Applied Probability, vol. 43, no. 3, pp. 815–824, September 2008.zbMATHGoogle Scholar
  42. [42]
    H. L. Gao and C. C. Yin, “The perturbed sparre Andersen model with a threshold dividend strategy,” Journal of Computational and Applied Mathematics, vol. 220, no. 1–2, pp. 394–408, October 2008.MathSciNetzbMATHGoogle Scholar
  43. [43]
    C. C. Yin and C. W. Wang, “The perturbed compound Poisson risk process with investment and debit interest,” Methodology and Computing in Applied Probability, vol. 12, no. 3, pp. 391–413, September 2010.MathSciNetzbMATHGoogle Scholar
  44. [44]
    C. C. Yin and K. C. Yuen, “Optimality of the threshold dividend strategy for the compound Poisson model,” Statistics & Probability Letters, vol. 81, no. 12, pp. 1841–1846, December 2011.MathSciNetzbMATHGoogle Scholar
  45. [45]
    C. C. Yin and Y. Z. Wen, “Exit problems for jump processes with applications to dividend problems,” Journal of Computational and Applied Mathematics, vol. 245, pp. 30–52, June 2013.MathSciNetzbMATHGoogle Scholar
  46. [46]
    C. C. Yin and Y. Z. Wen, “Optimal dividend problem with a terminal value for spectrally positive Levy processes,” Insurance Mathematics & Economics, vol. 53, no. 3, pp. 769–773, November 2013.MathSciNetzbMATHGoogle Scholar
  47. [47]
    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.MathSciNetGoogle Scholar
  48. [48]
    N. Zhao, Y. Chen, R. Liu, M. H. Wu, and W. Xiong, “Monitoring strategy for relay incentive mechanism in cooperative communication networks,” Computers & Electrical Engineering, vol. 60, pp. 14–29, January 2017.Google Scholar
  49. [49]
    X. F. Li, Y. D. Chu, and Y. T. Andrew, “Synchronization of uncertain chaotic systems via complete-adaptive-impulsive controls,” Chaos Solitons & Fractals, vol. 100, pp. 24–30, July 2017.MathSciNetzbMATHGoogle Scholar
  50. [50]
    Z. H. Rao, C. Y. Zeng, M. H. Wu, Z. F. Wang, N. Zhan, M. Liu, and X. K. Wan, “Research on a handwritten character recognition algorithm based on an extended nonlinear kernel residual network,” KSII Transactions on Internet and Information Systems, vol. 12, no. 1, pp. 413–435, January 2018.Google Scholar
  51. [51]
    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.Google Scholar
  52. [52]
    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.Google Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Mengting Chen
    • 1
  • Feng Ding
    • 1
    • 2
    Email author
  • Ahmed Alsaedi
    • 3
  • Tasawar Hayat
    • 3
    • 4
  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. 3.NAAM Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  4. 4.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan

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