Skip to main content
SpringerLink
Log in

Iterative Algorithm for Feedback Nonlinear Systems by Using the Maximum Likelihood Principle

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

Abstract

This paper aims to find a maximum likelihood least squares-based iterative algorithm to solve the identification issues of closed-loop input nonlinear equation-error systems. By adopting the key term separation technique, the parameters of the forward channel are identified separately from the parameters of the feedback channel to address the cross-product terms. The hierarchical identification principle is introduced to decompose the original system into two subsystems for reduced computational complexity. The iterative estimation theory and the maximum likelihood principle are applied to design a new least-squares algorithm with high estimation accuracy by taking full use of all the measured input-output data at each iterative computation. Compared with the recursive least-squares (RELS) method. The simulation results verify theoretical findings, and the proposed algorithm can generate more accurate parameter estimates than the RELS algorithm.

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. 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, 2022.

    Article  Google Scholar 

  2. 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 

  3. J. Pan, S. Liu, and J. Shu, “Hierarchical recursive least squares estimation algorithm for secondorder Volterranonlinear systems,” International Journal of Control, Automation, and Systems, vol. 20, no. 12, pp. 3940–3950, 2022.

    Article  Google Scholar 

  4. F. Ding, X. G. Liu, and J. Chu, “Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle,” IET Control Theory and Applications, vol. 7, no. 2, pp. 176–184, January 2013.

    Article  MathSciNet  Google Scholar 

  5. 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  Google Scholar 

  6. J. W. Wang and Y. Ji, “Two-stage gradient-based iterative algorithms for the fractional-order nonlinear systems by using the hierarchical identification principle,” International Journal of Adaptive Control and Signal Processing, vol. 36, no. 7, pp. 1778–1796, July 2022.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  8. Y. Ji, Z. Kang, and X. M. 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 

  9. R. Uematsu, S. Masuda, and M. Kano, “Closed-loop identification of plant and disturbance models based on data-driven generalized minimum variance regulatory control,” Journal of Process Control, vol. 115, pp. 197–208, July 2022.

    Article  Google Scholar 

  10. K. Li, H. Luo, and S. Yin, “A novel bias-eliminated subspace identification approach for closed-loop systems,” IEEE Transactions on Industrial Electronics, vol. 68, no. 6, pp. 5197–5205, 2021.

    Article  Google Scholar 

  11. 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 

  12. F. Ding, G. Liu, and X. P. Liu, “Partially coupled stochastic gradient identification methods for non-uniformly sampled systems,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1976–1981, August 2010.

    Article  MathSciNet  Google Scholar 

  13. S. Sharma and B. Verma, “Closed-loop identification of stable and unstableprocesses with time-delay,” Journal of the Franklin Institute, vol. 359, pp.3313–3332, 2022.

    Article  MathSciNet  Google Scholar 

  14. J. Chen, J. Ma, M. Gan, and Q. Zhu, “Multidirection gradient iterative algorithm: A unified framework for gradient iterative and least squares algorithms,” IEEE Transactions on AutomaticControl, vol. 67, no. 12, pp. 6770–6777, 2022.

    MathSciNet  Google Scholar 

  15. J. Chen, M. Hu, Y. Mao, and Q. Zhu, “Modified multi-direction iterative algorithm for separable nonlinear models with missing data,” IEEE Signal Processing Letters, vol. 29, pp. 1968–1972, 2022.

    Article  Google Scholar 

  16. J. Chen, B. Huang, M. Gan, and C. Chen, “A novel reduced-order algorithm for rational model based on Arnoldi process and Krylov subspace,” Automatica, vol. 129, pp. 109663, July 2021.

    Article  MathSciNet  Google Scholar 

  17. J. Chen, Q. Zhu, 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.

  18. H. F. Xia, Y. Ji, and T. Hayat, “Maximum likelihood-based gradient estimation for multivariable nonlinearsystems using the multi-innovation identification theory,” International Journal of Robust and Nonlinear Control, vol. 30, no.14, pp. 5446–5463, September 2020.

    Article  MathSciNet  Google Scholar 

  19. C. Yu and C. Zhang, “A new deterministic identification approach to Hammerstein systems,” IEEE Transaction on Signal Processing, vo. 62, no. 1, pp. 131–140, 2014.

    Article  MathSciNet  Google Scholar 

  20. E. W. Bai and D. Li, “Convergence of the iterative Hammerstein system identification algorithm,” IEEE Transactions on Automatic Control, vol. 49, no. 11, pp. 1929–1940, November 2004.

    Article  MathSciNet  Google Scholar 

  21. Q. Zhang, H. Wang, and C. Liu, “MILM hybrid identification method of fractional order neural-fuzzy Hammerstein model,” Nonlinear Dynamics, vol. 108, no. 3, pp.2337–2351, May 2022.

    Article  Google Scholar 

  22. V. Raghuraman and J. P. Koeln, “Hierarchical MPC for coupled subsystems using adjustable tubes,” Automatica, vol. 143, pp. 110435, September 2022.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. J. M. Li, “A novel nonlinear optimization method for fitting a noisy Gaussian activationfunction,” International Journal of Adaptive Control and Signal Processing, vol. 36, no. 3, pp. 690–707, March 2022.

    Article  Google Scholar 

  25. J. Vörös, “Parameter identification of discontinuous Hammerstein systems,” Automatica, vol. 33, no. 6, pp. 1141–1146, June 1997.

    Article  MathSciNet  Google Scholar 

  26. J. Vörös, “Recursive identification of Hammerstein systems with discontinuous nonlinearitiescontaining dead-zones,” IEEE Transactions on AutomaticControl, vol. 48, no. 12, pp. 2203–2206, December 2003.

    Google Scholar 

  27. X. Xu, F. Wang, G. Liu, and F. Qian, “Identification of Hammerstein systems using key-termseparation principle, auxiliary model and improved particle swarm optimisation algorithm,” IET Signal Processing, vol. 7, no. 8, pp. 766–773, October 2013.

    Article  Google Scholar 

  28. X. J. Ping, K. Zhang, S. Y. Zhao, X. L. Luan, and F. Liu, “Multitask maximum likelihood identificationfor ARX model with multisensor,” IEEE Transactions on Instrumentation and Measurement, vol. 71, 2509710, June 2022.

    Article  Google Scholar 

  29. M. H. Li and X. M. Liu, “Maximum likelihood least squares based iterative estimation for aclass 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 

  30. F. Ding, H. Ma, J. Pan, and E. F. Yang, “Hierarchical gradient- and least squares-based iterative algorithms for inputnonlinear output-error systems using the key term separation,” Journal of the Franklin Institute, vol. 358, no. 9, pp. 5113–5135, June 2021.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  32. 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, 2020.

    Article  MathSciNet  Google Scholar 

  33. L. Xu, “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, 2021.

    Article  MathSciNet  Google Scholar 

  34. 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. 11, pp. 2056–2074, 2021.

    Article  MathSciNet  Google Scholar 

  35. Y. Ji and Z. Kang, “Model recovery for multi-input signaloutput nonlinear systems based on thecompressed sensing recovery theory,” Journal of the Franklin Institute, vol. 359, no. 5, pp. 2317–2339, March 2022.

    Article  MathSciNet  Google Scholar 

  36. F. Ding and T. Chen, “Parameter estimation of dual-rate stochastic systems by using an output error method,” IEEE Transactions on Automatic Control, vol. 50, no. 9, pp. 1436–1441, September 2005.

    Article  MathSciNet  Google Scholar 

  37. Y. Wang and S. Tang, “Parameter estimation for nonlinear Volterra systems by using the multi-innovation identification theory and tensor decomposition,” Journal of the FranklinInstitute, vol. 359, no. 2, pp. 1782–1802, 2022.

    MathSciNet  Google Scholar 

  38. F. Ding and T. Chen, “Combined parameter and output estimation of dual-rate systems using an auxiliary model,” Automatica, vol. 40, no. 10, pp. 1739–1748, 2004.

    Article  MathSciNet  Google Scholar 

  39. Y. Wang and L. Yang, “An efficient recursive identification algorithm for multilinear systems based ontensor decomposition,” International Journal of Robust and Nonlinear Control, vol. 31, no. 16, pp. 7920–7936, November 2021.

    Article  MathSciNet  Google Scholar 

  40. L. Xu, “Auxiliary model multiinnovation stochastic gradient parameter estimation methods for nonlinear sandwich systems,” International Journal of Robust and Nonlinear Control, vol. 31, no. 1, pp. 148–165, 2021.

    Article  MathSciNet  Google Scholar 

  41. Y. Gu, Q. M. Zhu, and H. Nouri, “Identification and U-control of a state-space system with time-delay,” International Journal of Adaptive Control and Signal Processing, vol. 36, no. 1, pp. 138–154, January 2022.

    Article  MathSciNet  Google Scholar 

  42. 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 

  43. H. Ma, J. Pan, and W. Ding, “Partially-coupled least squares basediterative parameter estimation for multivariable output-error-like autoregressivemoving average systems,” IET Control Theory and Applications, vol. 13, no. 18, pp. 3040–3051, December 2019.

    Article  MathSciNet  Google Scholar 

  44. J. Pan, H. Ma, and J. Sheng, “Recursive coupled projection algorithms for multivariable output-error-like systems with coloured noises,” IET Signal Processing, vol. 14, no. 7, pp. 455–466, September 2020.

    Article  Google Scholar 

  45. T. Cui, “Moving data window-based partially-coupled estimation approach for modelinga dynamical system involving unmeasurable states,” ISA Transactions, vol. 128, Part B, pp. 437–452, September 2022.

    Article  Google Scholar 

  46. 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.

    Article  Google Scholar 

  47. 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, pp. 3727–3752, 2020.

    Article  MathSciNet  Google Scholar 

  48. F. Ding, “Coupled-least-squares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, January 2013.

    Article  MathSciNet  Google Scholar 

  49. X. Liu, “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, 2021.

    Article  MathSciNet  Google Scholar 

  50. M. H. Li, “The filtering-based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle,” International Journal of Adaptive Control and Signal Processing, vol. 33, no. 7, pp. 1189–1211, July 2019.

    Article  MathSciNet  Google Scholar 

  51. Y. Fan and X. Liu, “Two-stage auxiliary model gradient-based 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  Google Scholar 

  52. X. Zhang, “Recursive parameter estimation methods and convergence analysis for a special class of nonlinear systems,” International Journal of Robust and Nonlinear Control, vol. 30, no. 4, pp. 1373–1393, 2020.

    Article  MathSciNet  Google Scholar 

  53. 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 

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

    Article  MathSciNet  Google Scholar 

  55. 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, pp. 871–987, 2021.

    Article  MathSciNet  Google Scholar 

  56. Y. J. Wang, “Recursive parameter estimation algorithm for multivariateoutput-error systems,” Journal of the Franklin Institute, vol. 355, no. 12, pp.5163–5181, 2018.

    Article  MathSciNet  Google Scholar 

  57. S. Zhao, Y. Shmaliy, and F. Liu, “Batch optimal FIR smoothing: Increasing state informativity in nonwhite measurement noise environments,” IEEE Transactions on IndustrialInformatics, 2023. doi: https://doi.org/10.1109/tii.2022.3193879.

  58. S. Zhao, K. Li, C. Ahn, B. Huang, and F. Liu, “Tuning-free Bayesian estimation algorithms for faulty sensor signals in state-space,” IEEE Transactions Industrial Electronics, vol. 70, no. 1, pp. 921–929, 2023.

    Article  Google Scholar 

  59. S. Zhao, J. Wang, Y. Shmaliy, and F. Liu, “Discrete time q-lag maximum likelihood FIR smoothing and iterative recursive algorithm,” IEEE Transactions on Signal Processing, vol. 69, pp. 6342–6354, 2021.

    Article  MathSciNet  Google Scholar 

  60. T. Zhang, S. Zhao, X. Luan, and F. Liu, “Bayesian inference for state-space models with student-t mixture distributions,” IEEE Transactions on Cybernetics, vol. 53, no. 7, pp. 4435–4445, 2023.

    Article  Google Scholar 

  61. S. Zhao, B. Huang, and C. Zhao, “Online probabilistic estimation of sensor faulty signal in industrial processes and its applications,” IEEE Transactions Industrial Electronics, vol. 68, no. 9, pp. 8858–8862, 2021.

    Article  Google Scholar 

  62. S. Zhao, Y. Shmaliy, J. A. Lucio, and F. Liu, “Multipass optimal FIR filtering for processes with unknown initial states and temporary mismatches,” IEEE Transactions on Industrial Informatics, vol. 17, no. 8, pp. 5360–5368, 2020.

    Article  Google Scholar 

  63. S. Zhao, Y. Shmaliy, C. Ahn, and F. Liu, “Self-tuning unbiased finite impulse response filtering algorithm for processes with unknown measurement noise covariance,” IEEE Transactions on Control Systems Technology, vol. 29, no. 3, pp. 1372–1379, 2021.

    Article  Google Scholar 

  64. L. Xu, “Separable synchronous multi-innovation gradient-based iterative signal modeling from on-line measurements,” IEEE Transactions on Instrumentation and Measurement, vol. 71, p. 6501313, 2022.

    Google Scholar 

  65. J. Pan, Y. Liu, and J. Shu, “Gradient-based parameter estimation for an exponential nonlinear autoregressive timeseries model by using the multi-innovation,” International Journal of Control, Automation, and Systems, vol. 21, no. 1, pp.140–150, January 2023.

    Article  Google Scholar 

  66. J. Pan, H. Zhang, H. Guo, S. Liu, and Y. Liu, “Multivariable CAR-like system identification with multi-innovation gradient and least squares algorithms,” International Journal of Control, Automation, and Systems, vol. 21. no. 5, pp. 1455–1464, 2023.

    Article  Google Scholar 

  67. X. Zhang, “State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors,” International Journal of Adaptive Control and Signal Processing, vol. 33, no. 7, pp. 1157–1173, July 2019.

    Article  MathSciNet  Google Scholar 

  68. J. Hou, H. Su, C. Yu, and P. Li, “Bias-correction errors-in-variables Hammerstein model identification. IEEE Transactions on Industrial Electronics, vol. 70. no. 7, pp. 7268–7279, 2023.

    Article  Google Scholar 

  69. J. Hou, H. Su, C. Yu, and T. Li, “Consistent subspace identification of errors-in-variables Hammerstein systems,” IEEE Transactions on Systems Man and Cybernetics: Systems, vol. 53, no. 4, pp. 2292–2303, 2023.

    Article  Google Scholar 

  70. L. J. Wan, “Decomposition- and gradient-based iterative identification algorithms for multivariable systems using the multi-innovation theory,” Circuits Systems and Signal Processing, vol. 38, no. 7, pp. 2971–2991, 2019.

    Article  Google Scholar 

  71. J. Xiong, J. Pan, and G. Chen, “Sliding mode dual-channel disturbance rejection attitude control for a quadrotor,” IEEE Transactions on Industrial Electronics, vol. 69, no. 10, pp. 10489–10499, October 2022.

    Article  Google Scholar 

  72. S. Y. Liu, “Hierarchical principle-based iterative parameter estimation algorithm for dual-frequency signals,” Circuits Systems and Signal Processing, vol. 38, no. 7, pp. 3251–3268, July 2019.

    Article  Google Scholar 

  73. J. Pan, Q. Chen, J. Xiong, and G. Chen, “A novel quadruple boost nine level switched capacitor inverter,” Journal of Electrical Engineering & Technology, vol. 18, no. 1, pp. 467–480, January 2023.

    Article  Google Scholar 

  74. F. Ding, “Least squares and multi-innovation least squares-methods,” Journal of Computational and Applied Mathematics, vol. 426, p. 115107, July 2023.

    Article  Google Scholar 

  75. S. Zhao and B. Huang, “Trial-and-error or avoiding a guess? Initialization of the Kalman filter,” Automatica, vol. 121, p. 109184, 2020.

    Article  MathSciNet  Google Scholar 

  76. F. Ding, L. Xu, X. Zhang, and Y.H. Zhou, “Filtered auxiliary model recursive generalized extended parameter estimation methods for Box-Jenkins systemsfor Box-Jenkins systems by means of the filtering identification idea,” International Journal of Robust and Nonlinear Control, vol. 33, 2023. doi:https://doi.org/10.1002/rnc.6657

  77. S. Zhao, Y. Shmaliy, C. Ahn, and L. Luo, “An improved iterative FIR state estimator and its applications,” IEEE Transactions on Industrial Informatics, vol. 16, no. 2, pp. 1003–1012, 2020.

    Article  Google Scholar 

  78. L. Xu, “Separable synthesis estimation methods and convergence analysis for multivariable systems,” Journal of Computational and Applied Mathematics, vol. 427, p. 115104, August 2023.

    Article  MathSciNet  Google Scholar 

  79. S. Zhao, Y. Shmaliy, C. Ahn, and C. Zhao, “Probabilistic monitoring of correlated sensors for nonlinear processes in state space,” IEEE Transactions Industrial Electronics, vol. 67, no. 3, pp. 2294–2303, 2020.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Taizhou Electric Power Conversion and Control Engineering Technology Research Center, Taizhou University, Taizhou, 225300, China

    Huafeng Xia

Authors
  1. Huafeng Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huafeng Xia.

Ethics declarations

The author declares no conflict of interest.

Additional information

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

This work was supported by the Natural Science Foundation of Colleges and Universities of Jiangsu Province (Grant No.23KJA120003), the Talent Start-up Fund of Taizhou University under Grants TZXY2020QDJJ007, Taizhou Science and Technology Support Plan (Social Development) Project under Grants SSF20210004, the “333” Project of Jiangsu Province (Grant No. BRA202237031), the Talent Start-up Fund of Binjiang College of Nanjing University of Information Science and Technology under Grant 550220019.

Huafeng Xia received her B.Sc. degree from the Jiangsu University of Technology (Changzhou, China) in 2003, an M.Sc. degree from Wuhan University of Science and Technology (Wuhan, China) in 2008, and a Ph.D. degree from Jiangnan University, Wuxi, China, in 2020. From 2003 to 2005, she was a teacher in Jiangyin Huazi Vocatinal School, Jiangyin, Jiangsu Province. From 2008 to now, she is a teacher in Taizhou University, Taizhou, Jiangsu Province. She has been an associate professor in Taizhou University since 2018. Her research interests include system identification.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, H. Iterative Algorithm for Feedback Nonlinear Systems by Using the Maximum Likelihood Principle. Int. J. Control Autom. Syst. 22, 1409–1417 (2024). https://doi.org/10.1007/s12555-022-1002-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-022-1002-y

Keywords

Search

Navigation