Skip to main content
SpringerLink
Log in

Hierarchical Recursive Least Squares Estimation Algorithm for Secondorder Volterra Nonlinear Systems

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

This paper considers the parameter identification problems of a Volterra nonlinear system. In order to overcome the excessive calculation amount of the Volterra systems, a hierarchical least squares algorithm is proposed through combining the hierarchical identification principle. The key is to decompose the Volterra systems into three subsystems with a smaller number of parameters and to estimates the parameters of each subsystem, respectively. The calculation analysis indicates that the proposed algorithm has less computational cost than the recursive least squares algorithm. Finally, the simulation results indicate that the proposed algorithm are effective for identifying Volterra 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

Similar content being viewed by others

References

  1. J. Wang, Y. Ji, and C. Zhang, “Iterative parameter and order identification for fractional-order nonlinear finite impulse response systems using the key term separation,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 8, pp. 1562–1577, August 2021.

    Article  MathSciNet  Google Scholar 

  2. Y. Ji, Z. Kang, and X. Liu, “The data filtering based multiple-stage Levenberg-Marquardt algorithm for Hammerstein nonlinear systems,” International Journal of Robust and Nonlinear Control, vol. 31, no. 15, pp. 7007–7025, October 2021.

    Article  MathSciNet  Google Scholar 

  3. M. Li and X. Liu, “Maximum likelihood least squares based iterative estimation for a class of bilinear systems using the data filtering technique,” International Journal of Control, Automation, and Systems, vol. 18, no. 6, pp. 1581–1592, June 2020.

    Article  Google Scholar 

  4. M. Li and X. Liu, “Maximum likelihood hierarchical least squares-based iterative identification for dual-rate stochastic systems,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 2, pp. 240–261, February 2021.

    Article  MathSciNet  Google Scholar 

  5. M. Li and X. Liu, “Iterative identification methods for a class of bilinear systems by using the particle filtering technique,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 10, pp. 2056–2074, October 2021.

    Article  MathSciNet  Google Scholar 

  6. Y. Ji, C. Zhang, Z. Kang, and T. Yu, “Parameter estimation for block-oriented nonlinear systems using the key term separation,” International Journal of Robust and Nonlinear Control, vol. 30, no. 9, pp. 3727–3752, June 2020.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Ma and L. Wang, “Filtering-based recursive least squares estimation approaches for multivariate equation-error systems by using the multiinnovation theory,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 9, pp. 1898–1915, September 2021.

    Article  MathSciNet  Google Scholar 

  8. Y. Mao, S. Liu, and J. Liu, “Robust economic model predictive control of nonlinear networked control systems with communication delays,” International Journal of Adaptive Control and Signal Processing, vol. 34, no. 5, pp. 614–637, May 2020.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Ji, Z. Kang, and C. Zhang, “Two-stage gradient-based recursive estimation for nonlinear models by using the data filtering,” International Journal of Control, Automation, and Systems, vol. 19, no. 8, pp. 2706–2715, August 2021.

    Article  Google Scholar 

  10. Y. Ji, X. Jiang, and L. Wan, “Hierarchical least squares parameter estimation algorithm for two-input Hammerstein finite impulse response systems,” Journal of the Franklin Institute, vol. 357, no. 8, pp. 5019–5032, May 2020.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Ding, Y. Liu, and B. Bao, “Gradient based and least squares based iterative estimation algorithms for multiinput multi-output systems,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 226, no. 1, pp. 43–55, February 2012.

    Google Scholar 

  12. G. Chen, M. Gan, C. Chen, and H. Li, “A regularized variable projection algorithm for separable nonlinear leastsquares problems,” IEEE Transactions on Automatic Control, vol. 64, no. 2, pp. 526–537, February 2019.

    MathSciNet  MATH  Google Scholar 

  13. L. Xu and G. Song, “A recursive parameter estimation algorithm for modeling signals with multi-frequencies,” Circuits Systems and Signal Processing, vol. 39, no. 8, pp. 4198–4224, August 2020.

    Article  MATH  Google Scholar 

  14. L. Xu, “Separable multi-innovation Newton iterative modeling algorithm for multi-frequency signals based on the sliding measurement window,” Circuits Systems and Signal Processing, vol. 41, no. 2, pp. 805–830, February 2022.

    Article  Google Scholar 

  15. Y. Fan and X. Liu, “Two-stage auxiliary model gradientbased iterative algorithm for the input nonlinear controlled autoregressive system with variable-gain nonlinearity,” International Journal of Robust and Nonlinear Control, vol. 30, no. 14, pp. 5492–5509, September 2020.

    Article  MathSciNet  MATH  Google Scholar 

  16. X. Liu and Y. Fan, “Maximum likelihood extended gradient-based estimation algorithms for the input nonlinear controlled autoregressive moving average system with variable-gain nonlinearity,” International Journal of Robust and Nonlinear Control, vol. 31, no. 9, pp. 4017–4036, June 2021.

    Article  MathSciNet  Google Scholar 

  17. C. Cheng, Z. Peng, X. Dong, W. Zhang, and G. Meng, “A novel damage detection approach by using Volterra kernel functions based analysis,” Journal of the Franklin Institute, vol. 352, no. 8, pp. 3098–3112, August 2015.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Ding, L. Lv, and X. Jin, “Two-stage gradient-based iterative estimation methods for controlled autoregressive systems using the measurement data,” International Journal of Control, Automation, and Systems, vol. 18, no. 4, pp. 886–896, April 2020.

    Article  Google Scholar 

  19. Y. Ji and Z. Kang, “Three-stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems,” International Journal of Robust and Nonlinear Control, vol. 31, no. 3, pp. 871–987, February 2021.

    Article  MathSciNet  Google Scholar 

  20. L. Xu, “Separable Newton recursive estimation method through system responses based on dynamically discrete measurements with increasing data length,” International Journal of Control, Automation, and Systems, vol. 20, no. 2, pp. 432–443, February 2022.

    Article  Google Scholar 

  21. L. Xu and E. Yang, “Auxiliary model multi-innovation stochastic gradient parameter estimation methods for nonlinear sandwich systems,” International Journal of Robust and Nonlinear Control, vol. 31, no. 1, pp. 148–165, January 2021.

    Article  MathSciNet  Google Scholar 

  22. L. Xu and E. Yang, “Separable recursive gradient algorithm for dynamical systems based on the impulse response signals,” International Journal of Control, Automation, and Systems, vol. 18, no. 12, pp. 3167–3177, December 2020.

    Article  Google Scholar 

  23. G. Zhao, T. Cao, Y. Wang, and C. X. Wan, “Optimal sizing of isolated microgrid containing photovoltaic/photothermal/wind/diesel/battery,” International Journal of Photoenergy, vol. 2021, p. 5566597, 2021.

    Article  Google Scholar 

  24. X. Wang, M. Zhao, Y. Zhou, Z. Wan, and W. Xu, “Design and analysis for multi-disc coreless axial-flux permanentmagnet synchronous machine,” IEEE Transactions on Applied Superconductivity, 31, no. 8, p. 5203804, November 2021.

    Google Scholar 

  25. X. Wang, Z. Wan, L. Tang, W. Xu, and M. Zhao, “Electromagnetic performance analysis of an axial flux hybrid excitation motor for HEV drives,” IEEE Transactions on Applied Superconductivity, vol. 31, no. 8, p. 5205605, November 2021.

    Article  Google Scholar 

  26. Z. Zhao, Y. Zhou, X. Wang, Z. Wang, and Y. Bai, “Water quality evolution mechanism modeling and health risk assessment based on stochastic hybrid dynamic systems,” Expert Systems with Applications 193, p. 116404, 2022.

    Article  Google Scholar 

  27. Q. Chen, Z. Zhao, X. Wang, K. Xiong, and C. Shi, “Microbiological predictive modeling and risk analysis based on the one-step kinetic integrated Wiener process,” Innovative Food Science & Emerging Technologies, vol. 75, p. 102912, January 2022.

    Article  Google Scholar 

  28. N. Zhao, A. Wu, Y. Pei, Y. Liang, and D. Niyato, “Spatialtemporal aggregation graph convolution network for efficient mobile cellular traffic prediction,” IEEE Communications Letters, vol. 26, no. 3, pp. 587–591, 2022.

    Article  Google Scholar 

  29. Y. Chen, C. Zhang, C. Liu, Y. Wang, and X. Wan, “Atrial fibrillation detection using feedforward neural network,” Journal of Medical and Biological Engineering, vol. 242, no. 1, pp. 63–73, 2022.

    Article  Google Scholar 

  30. J. Wang, C. Ding, M. Wu, and G. Chen, “Lightweight multiple scale-patch dehazing network for real-world hazy image,” KSII Transactions on Internet and Information Systems, vol. 15, no. 12, pp. 4420–4438, 2022.

    Google Scholar 

  31. X. Zhang, “Optimal adaptive filtering algorithm by using the fractional-order derivative,” IEEE Signal Processing Letters, vol. 29, pp. 399–403, 2022.

    Article  Google Scholar 

  32. Y. Zhou, “Modeling nonlinear processes using the radial basis function-based state-dependent autoregressive models,” IEEE Signal Processing Letters, vol. 27, pp. 1600–1604, 2020.

    Article  Google Scholar 

  33. J. Hou, F. Chen, P. Li, and Z. Zhu, “Gray-box parsimonious subspace identification of Hammerstein-type systems,” IEEE Transactions on Industrial Electronics, vol. 68, no. 10, pp. 9941–9951, October 2021.

    Article  Google Scholar 

  34. F. Ding, H. Chen, 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.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Ding, Z. Cao, J. Z. Chen, and G. Jiang, “Weighted parameter estimation for Hammerstein nonlinear ARX systems,” Circuits Systems and Signal Processing, vol. 39, no. 4, pp. 2178–2192, April 2020.

    Article  Google Scholar 

  36. L. Feng, J. Ding, and Y. Han, “Improved sliding mode based EKF for SOC estimation of lithium-ion batteries,” Ionics, vol. 26, no. 6, pp. 2875–2882, June 2000.

    Article  Google Scholar 

  37. J. Ding, L. Chen, Z. Cao, and H. Guo, “Convergence analysis of the modified adaptive extended Kalman filter for the parameter estimation of a brushless DC motor,” International Journal of Robust and Nonlinear Control, vol. 31, no. 16, pp. 7606–7620, November 2021.

    Article  MathSciNet  Google Scholar 

  38. Z. Li, J. Ding, and J. Lin, “Discrete fractional order PID controller design for nonlinear systems,” International Journal of Systems Science, vol. 52, no. 15, pp. 3206–3213, November 2021.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Chen and C. Chen, “A novel reduced-order algorithm for rational models based on Arnoldi process and Krylov subspace,” Automatica, vol. 129, Article Number: 109663, July 2021.

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Chen and Y. Liu, “Modified Kalman filtering based multi-step-length gradient iterative algorithm for ARX models with random missing outputs,” Automatica, vol. 118, Article Number: 109034, August 2020.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Chen and Y. Liu, “Stochastic average gradient algorithm for multirate FIR models with varying time delays using self-organizing maps,” International Journal of Adaptive Control and Signal Processing, vol. 34, no. 7, pp. 955–970, July 2020.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Pan, X. Jiang, 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.

    Article  Google Scholar 

  43. Y. Wang and M. Wu, “Recursive parameter estimation algorithm for multivariate output-error systems,” Journal of the Franklin Institute, vol. 355, no. 12, pp. 5163–5181, August 2018.

    Article  MathSciNet  MATH  Google Scholar 

  44. L. Xu, F. Chen, and T. Hayat, “Hierarchical recursive signal modeling for multi-frequency signals based on discrete measured data,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 5, pp. 676–693, May 2021.

    Article  MathSciNet  Google Scholar 

  45. J. Ding and W. Zhang, “Finite-time adaptive control for nonlinear systems with uncertain parameters based on the command filters,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 9, pp. 1754–1767, September 2021.

    Article  MathSciNet  Google Scholar 

  46. J. Pan, W. Li, and H. 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.

    Article  Google Scholar 

  47. X. Zhang, “Hierarchical parameter and state estimation for bilinear systems,” International Journal of Systems Science, vol. 51, no. 2, pp. 275–290, 2020.

    Article  MathSciNet  MATH  Google Scholar 

  48. Y. Cao, Y. Sun, G. Xie, and P. Li, “A sound-based fault diagnosis method for railway point machines based on two-stage feature selection strategy and ensemble classifier,” IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 8, pp. 12074–12083, 2022.

    Article  Google Scholar 

  49. Y. Cao, J. Wen, A. Hobiny, P. Li, and T. Wen, “Parametervarying artificial potential field control of virtual coupling system with nonlinear dynamics,” Fractals, vol. 30, no. 2, p. 2240099, 2022.

    Article  MATH  Google Scholar 

  50. S. Su, T. Tang, J. Xun, F. Cao, and Y. Wang, “Design of running grades for energy-efficient train regulation: A case study for Beijing Yizhuang line,” IEEE Intelligent Transportation Systems Magazine, vol. 13, no. 2, pp. 189–200, February 2021.

    Article  Google Scholar 

  51. Y. Cao, Z. Wang, F. Liu, and G. Xie, “Bio-inspired speed curve optimization and sliding mode tracking control for subway trains,” IEEE Transactions on Vehicular Technology, vol. 68, no. 7, pp. 6331–6342, July 2019.

    Article  Google Scholar 

  52. Y. Cao, Y. Sun, G. Xie, and T. Wen, “Fault diagnosis of train plug door based on a hybrid criterion for IMFs selection and fractional wavelet package energy entropy,” IEEE Transactions on Vehicular Technology, vol. 68, no. 8, pp. 7544–7551, August 2019.

    Article  Google Scholar 

  53. Y. Cao, L. Ma, S. Xiao, and W. Xu, “Standard analysis for transfer delay in CTCS-3,” Chinese Journal of Electronics, vol. 26, no. 5, pp. 1057–1063, September 2017.

    Article  Google Scholar 

  54. Y. Cao, J. Wen, and L. Ma, “Tracking and collision avoidance of virtual coupling train control system,” Alexandria Engineering Journal, vol. 60, no. 2, pp. 2115–2125, April 2021.

    Article  Google Scholar 

  55. S. Su, X. Wang, Y. Cao, and J. Yin, “An energy-efficient train operation approach by integrating the metro timetabling and eco-driving,” IEEE Transactions on Intelligent Transportation Systems, vol. 21, no. 10, pp. 4252–4268, October 2020.

    Article  Google Scholar 

  56. S. Su, J. She, K. Li, and Y. Zhou, “A nonlinear safety equilibrium spacing based model predictive control for virtually coupled train set over gradient terrains,” IEEE Transactions on Transportation Electrification, vol. 8, no. 2, pp. 2810–2824, 2022.

    Article  Google Scholar 

  57. C. Yin and J. Zhao, “Nonexponential asymptotics for the solutions of renewal equations with applications,” Journal of Applied Probability, vol. 43, no. 3, pp. 815–824, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  58. C. Yin and K. Yuen, “Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs,” Journal of Industrial and Management Optimization, vol. 11, no. 4, pp. 1247–1262, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  59. C. Yin and K. Yuen, “Optimality of the threshold dividend strategy for the compound Poisson model,” Statistics & Probability Letters, vol. 81, no. 12, pp. 1841–1846, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  60. H. Dong, C. Yin, and H. Dai, “Spectrally negative Levy risk model under Erlangized barrier strategy,” Journal of Computational and Applied Mathematics, vol. 351, pp. 101–116, May 2019.

    Article  MathSciNet  MATH  Google Scholar 

  61. X. Sha, Z. Xu, and C. Yin, “Elliptical distribution-based weight-determining method for ordered weighted averaging operators,” International Journal of Intelligent Systems, vol. 34, no. 5, pp. 858–877, May 2019.

    Article  Google Scholar 

  62. C. Yin and Y. Wen, “An extension of Paulsen-Gjessing's risk model with stochastic return on investments,” Insurance Mathematics & Economics, vol. 52, no. 3, pp. 469–476, May 2013.

    Article  MathSciNet  MATH  Google Scholar 

  63. Y. Zhao, P. Chen, and H. Yang, “Optimal periodic dividend and capital injection problem for spectrally positive Levy processes,” Insurance Mathematics & Economics, vol. 74, pp. 135–146, May 2017.

    Article  MathSciNet  MATH  Google Scholar 

  64. X. Zhao, H. Dong, and H. Dai, “On spectrally positive Levy risk processes with Parisian implementation delays in dividend payments,” Statistics & Probability Letters, vol. 140, pp. 176–184, September 2018.

    Article  MathSciNet  MATH  Google Scholar 

  65. Y. Zhao and C. Yin, “The expected discounted penalty function under a renewal risk model with stochastic income,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 6144–6154, January 2012.

    MathSciNet  MATH  Google Scholar 

  66. Y. Zhou, “Partially-coupled nonlinear parameter optimization algorithm for a class of multivariate hybrid models,” Applied Mathematics and Computation, vol. 414, p. 126663, Februray 2022.

    Article  MathSciNet  MATH  Google Scholar 

  67. Y. Zhou and X. Zhang, “Hierarchical estimation approach for RBF-AR models with regression weights based on the increasing data length,” IEEE Transactions on Circuits and Systems-II: Express Briefs, vol. 68, no, 12, pp. 3597–3601, December 2021.

    Article  Google Scholar 

  68. H. Wang, H. Fan, and J. Pan, “A true three-scroll chaotic attractor coined,” Discrete and Continuous Dynamical Systems-Series B, vol. 27, no. 5, pp. 2891–2915, 2022.

    Article  MathSciNet  MATH  Google Scholar 

  69. L. Wang, Y. Ji, H. Yang, and L. Xu, “Decomposition-based multiinnovation gradient identification algorithms for a special bilinear system based on its input-output representation,” International Journal of Robust and Nonlinear Control, vol. 30, no. 9, pp. 3607–3623, June 2020.

    Article  MathSciNet  MATH  Google Scholar 

  70. S. Y. Liu, “Expectation-maximization algorithm for bilinear systems by using the Rauch-Tung-Striebel smoother,” Automatica, vol. 142, p. 110365, 2022.

    Article  MathSciNet  MATH  Google Scholar 

  71. C. Wei, X. Zhang, and E. Yang, “Overall recursive least squares and overall stochastic gradient algorithms and their convergence for feedback nonlinear controlled autoregressive systems,” International Journal of Robust and Nonlinear Control, vol. 32, no. 9, pp. 5534–5554, 2022.

    Article  MathSciNet  Google Scholar 

  72. Z. Kang, Y. Ji, and X. Liu, “Hierarchical recursive least squares algorithms for Hammerstein nonlinear autoregressive output-error systems,” International Journal of Adaptive Control and Signal Processing, vol. 35, no. 11, pp. 2276–2295, 2021.

    Article  Google Scholar 

  73. J. Lin, Y. Li, and G. Yang, “FPGAN: Face de-identification method with generative adversarial networks for social robots,” Neural Networks, vol. 133, pp. 132–147, January 2021.

    Article  Google Scholar 

  74. G. Yang, Z. Chen, Y. Li, and Z. Su, “Rapid relocation method for mobile robot based on improved ORB-SLAM2 algorithm,” Remote Sensing 11, no. 2, Article Number: 149, January 2019.

    Article  Google Scholar 

  75. Y. Wang and G. Yang, “Arrhythmia classification algorithm based on multi-head self-attention mechanism,” Biomedical Signal Processing and Control, vol. 79, p. 104206, 2023.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical and Electronic Engineering, Hubei University of Technology, Wuhan, 430068, China

    Jian Pan, Sunde Liu, Jun Shu & Xiangkui Wan

Authors
  1. Jian Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sunde Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiangkui Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Pan.

Additional information

Jian Pan was born in Wuhan, China. He received his B.Sc. degree from Hubei University of Technology (Wuhan, China) in 1984. He has been a Professor in the School of Electrical and Electronic Engineering, Hubei University of Technology. His research interests include control science and engineering, computer control systems, and power electronics.

Sunde Liu was born in Anqing, Anhui, China. He received his B.Sc. degree from the Jiangxi University of Science and Technology (Ganzhou, China) in 2018. He is now a master student at the Huibei University of Technology, Wuhan, China. His research interests include system identification and control theory.

Jun Shu was born in Wuhan, China. He received his B.Sc. degree in 2009 and an M.Sc. degree from Hubei University of Technology (Wuhan, China). He has been an Associate Professor at Hubei University of Technology. His research interests include intelligent control and process control.

Xiangkui Wan received his M.S. and Ph.D. degrees in mechatronic engineering from Chongqing University, in 2002 and 2005, respectively. He is currently a professor in the School of Electrical and Electronic Engineering, Hubei University of Technology, Wuhan, China. His research interests include digital signal processing, biomedical signal processing and analysis, and biomedical modeling and simulation.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Natural Science Foundation of China (No. 61571182, 61273192).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, J., Liu, S., Shu, J. et al. Hierarchical Recursive Least Squares Estimation Algorithm for Secondorder Volterra Nonlinear Systems. Int. J. Control Autom. Syst. 20, 3940–3950 (2022). https://doi.org/10.1007/s12555-021-0845-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-021-0845-y

Keywords

Search

Navigation