Abstract
This paper focuses on the maximum likelihood estimation for bilinear systems in the presence of colored noise. The state variables in the model is eliminated and an input-output expression is provided. The input-output data of the system is filtered by an estimated noise transfer function, and the system is transformed into two subsystems. A filtering based maximum likelihood recursive least squares algorithm is proposed to strengthen the identification accuracy and improve computational efficiency. The superior performance of the developed methods are demonstrated by numerical simulations.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
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.
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.
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.
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. 971–987, February 2021.
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.
J. Adánez, B. Al-Hadithi, and A. Jiménez, “Multidimensional membership functions in T-S fuzzy models for modelling and identification of nonlinear multivariable systems using genetic algorithms,” Applied Soft Computing, vol. 75, pp. 607–615, February 2019.
N. Vafamand, M. Arefi, and A. Khayatian, “Nonlinear system identification based on Takagi-Sugeno fuzzy modeling and unscented Kalman filter,” ISA Transactions, vol. 74, pp. 134–143, March 2018.
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.
X. Zhang and E. Yang, “State filteringbased least squares parameter estimation for bilinear systems using the hierarchical identification principle,” IET Control Theory & Applications, vol. 12, no. 12, pp. 1704–1713, August 2018.
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, July 2020.
J. Pan, X. Jiang, X. 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, May 2017.
H. Ma, J. Pan, and W. Ding, “Partially-coupled least squares based iterative parameter estimation for multivariable output-error-like autoregressive moving average systems,” IET Control Theory and Applications, vol. 13, no. 18, pp. 3040–3051, December 2019.
J. Pan, H. Ma, X. Zhang, and J. Sheng, “Recursive coupled projection algorithms for multivariable output-errorlike systems with coloured noises,” IET Signal Processing, vol. 14, no. 7, pp. 455–466, September 2020.
H. Ma, X. Zhang, Q. Liu, and T. Hayat, “Partially-coupled gradient-based iterative algorithms for multivariable output-error-like systems with autoregressive moving average noises,” IET Control Theory and Applications, vol. 14, no. 17, pp. 2613–2627, November 2020.
F. Ding, Y. Liu, and B. Bao, “Gradient based and least squares based iterative estimation algorithms for multi-input 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, 2012.
X. Liu and J. Lu, “Least squares based iterative identification for a class of multirate systems,” Automatica, vol. 46, no. 3, pp. 549–554, March 2010.
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.
J. Wang, Y. Ji, X. Zhang, and L. Xu, “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, 2022.
F. Ding, L. Lv, J. Pan, 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.
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.
C. Guo, L. Wang, and F. Deng, “The auxiliary model based hierarchical estimation algorithms for bilinear stochastic systems with colored noises,” International Journal of Control, Automation, and Systems, vol. 18, no. 3, pp. 650–660, March 2020.
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.
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, 2021.
F. Ding, “Combined state and least squares parameter estimation algorithms for dynamic systems,” Applied Mathematical Modelling, vol. 38, no. 1, pp. 403–412, 2014.
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, October 2004.
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, November 2021.
F. Ding, “Coupled-least-squares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, January 2013.
M. Li and X. Liu, “Filtering-based maximum likelihood gradient iterative estimation algorithm for bilinear systems with autoregressive moving average noise,” Circuits Systems and Signal Processing, vol. 37, no. 11, pp. 5023–5048, November 2018.
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.
Y. Wang, “Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model,” Automatica, vol. 71, pp. 308–313, September 2016.
L. Wang and Y. He, “Recursive least squares parameter estimation algorithms for a class of nonlinear stochastic systems with colored noise based on the auxiliary model and data filtering,” IEEE Access, vol. 7, pp. 181295–181304, November 2019.
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, March 2021.
D. Verbeke and M. Khorasani, “Frequency domain maximum likelihood identification with Gaussian input-output uncertainty,” IEEE Control Systems Letters, vol. 4, no. 1, pp. 109–114, 2019.
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, December 2020.
M. Li and X. Liu, “The least squares based iterative algorithms for parameter estimation of a bilinear system with autoregressive noise using the data filtering technique,” Signal Processing, vol. 147, pp. 23–34, 2018.
L. Wang, Y. Ji, and H. Yang, “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.
L. Wang, Y. Ji, L. Wan, and N. Bu, “Hierarchical recursive generalized extended least squares estimation algorithms for a class of nonlinear stochastic systems with colored noise,” Journal of the Franklin Institute, vol. 356, no. 16, pp. 10102–10122, November 2019.
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.
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.
Y. Zhou, “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, 2021.
Y. Ji, Z. Kang, and X. Zhang, “Model recovery for multi-input signal-output nonlinear systems based on the compressed sensing recovery theory,” Journal of the Franklin Institute, vol. 359, no. 5, pp. 2317–2339, March 2022.
L. Xu and F. Chen, “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.
J. Chen, B. Huang, 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.
J. Ding, “Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2677–2683, November 2011.
Y. Wang, S. Tang, and X. Gu, “Parameter estimation for nonlinear Volterra systems by using the multi-innovation identification theory and tensor decomposition,” Journal of the Franklin Institute, vol. 359, no. 2, pp. 1782–1802, 2022.
Y. Wang and L. Yang, “An efficient recursive identification algorithm for multilinear systems based on tensor decomposition,” International Journal of Robust and Nonlinear Control, vol. 31, no. 16, pp. 7920–7936, November 2021.
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.
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.
X. Zhang, “Optimal adaptive filtering algorithm by using the fractional-order derivative,” IEEE Signal Processing Letters, vol. 29, pp. 399–403, 2022.
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.
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.
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.
X. Zhang, “Highly computationally efficient state filter based on the delta operator,” International Journal of Adaptive Control and Signal Processing, vol. 33, no. 6, pp. 875–889, June 2019.
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.
X. Zhang, “Adaptive parameter estimation for a general dynamical system with unknown states,” International Journal of Robust and Nonlinear Control, vol. 30, no. 4, pp. 1351–1372, March 2020.
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, March 2020.
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.
J. Hou, H. Su, C. Yu, and P. Li, “Bias-correction errors-invariables Hammerstein model identification,” IEEE Transactions on Industrial Electronics, pp. 1–11, 2022. DOI: https://doi.org/10.1109/TIE.2022.3199931
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, pp. 1–12, 2022. DOI: https://doi.org/10.1109/TSMC.2022.3213809
C. Xu, H. Xu, Z. Guan, and Y. Ge, “Observer-based dynamic event-triggered semi-global bipartite consensus of linear multi-agent systems with input saturation,” IEEE Transactions on Cybernetics, pp. 1–14, 2022. DOI: https://doi.org/10.1109/TCYB.2022.3164048
Y. Cao, Y. Yang, and J. Wen, “Research on virtual coupled train control method based on GPC & VAPF,” Chinese Journal of Electronics, vol. 31, no. 5, pp. 897–905, 2022.
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, 2022.
Y. Cao, J. Wen, A. Hobiny, and T. Wen, “Parameter-varying artificial potential field control of virtual coupling system with nonlinear dynamics,” Fractals, vol. 30, no. 2, 2240099, 2022.
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.
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, 2021.
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.
L. Xu, “Decomposition strategy-based hierarchical least mean square algorithm for control systems from the impulse responses,” International Journal of Systems Science, vol. 52, no. 9, pp. 1806–1821, 2021.
J. Pan, S. Liu, J. Shu, and X. Wan, “Hierarchical recursive least squares estimation algorithm for second-order Volterra nonlinear systems,” International Journal of Control, Automation, and Systems, pp. 1–11, 2022. DOI: https://doi.org/10.1007/s12555-021-0845-y
J. Pan, Y. Liu, and J. Shu, “Gradient-based parameter estimation for an exponential nonlinear autoregressive time-series model by using the multi-innovation,” International Journal of Control, Automation, and Systems, pp. 1–11, 2023. DOI: https://doi.org/10.1007/s12555-021-1018-8
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.
Y. Wang, S. Tang, and M. Deng, “Modeling nonlinear systems using the tensor network B-spline and the multiinnovation identification theory,” International Journal of Robust and Nonlinear Control, vol. 32, no. 13, pp. 7304–7318, September 2022.
L. Xu, W. Xiong, 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, 2018.
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.
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.
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.
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, 2022.
J. Pan, Q. Chen, J. Xiong, and G. Chen, “A novel quadruple boost nine level switched capacitor inverter,” Journal of Electrical Engineering & Technology, 2022. DOI: https://doi.org/10.1007/s42835-022-01130-2
Author information
Authors and Affiliations
Corresponding author
Additional information
Longjin Wang received his Ph.D. degree in control science and engineering from Harbin Engineering University in 2009. From 2009 to 2013, he was an engineer at China Shipbuilding heavy Industry Corporation. Since 2013, he has been an associate professor in control science and engineering at Qingdao University of Science and Technology. His current research interests include system identification and motion control of marine crafts.
Shun An received his B.E. degree from the College of Mechanical and Electrical Engineering, Qingdao University of Science and Technology, Qingdao, China, in 2019, where he is currently pursuing a Ph.D. degree in mechanical engineering. His current research interests include system identification and nonlinear control.
Yan He received her Ph.D. degree in power and engineering thermophysics from Huazhong University of Science and Technology in 2005. She has been a professor at Qingdao University of Science and Technology. She is an academic leader in the first-tier discipline of power engineering and engineering thermophysics in Qingdao University of Science and Technology. Her current research interests include thermal transformation, energy development and utilization, and Industrial process identification.
Jianping Yuan received his M.E. degree from the College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China, where he is currently pursuing a Ph.D. degree in ocean engineering. His current research interests include system identification, intelligent ship equipment and control technology, fault tolerant control, and intelligent control and hydrodynamic technology of marine vehicle.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the Taishan Scholar Project of Shandong Province (ts20190937), National Natural Science Foundation of China (52176076, 52101401), Guangdong Province in 2019 Ordinary University Key Areas Special Project (2019KZDZX1024) and State Administration of Science, Technology and Industry for National Defense (JCKYS2021SXJQR-02).
Rights and permissions
About this article
Cite this article
Wang, L., An, S., He, Y. et al. The Filtering Based Maximum Likelihood Recursive Least Squares Parameter Estimation Algorithms for a Class of Nonlinear Stochastic Systems with Colored Noise. Int. J. Control Autom. Syst. 21, 151–160 (2023). https://doi.org/10.1007/s12555-021-0923-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-021-0923-1