Abstract
This paper considers the parameter estimation problem of nonlinear models, which are related to the impulse or step response functions of linear time-invariant (LTI) dynamical systems, based on the response data. In terms of the nonlinear characteristic of the models, the nonlinear dynamical optimization scheme is adopted for obtaining the system parameter estimates. By constructing a gradient criterion function, a gradient recursion algorithm is derived. In order to overcome the difficulty of determining the step-size in the gradient recursion algorithm, a trying method and a numerical approach are proposed to achieve the step-size. On this basis, a stochastic gradient estimation method is presented by using a recursive step-size. Furthermore, a multi-innovation stochastic gradient method is deduced for enhancing the estimation accuracy by using the dynamical window data. Finally, a dynamical length stochastic gradient estimation technique is offered to obtain more accurate parameter estimates by using dynamical length measured data from the step response. The examples are provided to examine the algorithm performance and the simulation results indicate that the presented approaches are effective.
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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.
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, 2020.
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.
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, 2021.
Y. M. Fan and X. M. Liu, “Auxiliary model-based multiinnovation recursive identification algorithms for an input nonlinear controlled autoregressive moving average system with variable-gain nonlinearity,” International Journal of Adaptive Control and Signal Processing, vol. 36, no. 3, pp. 521–540, March 2022.
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, 2021.
J. Pan, S. D. Liu, J. Shu, and X. K. Wan, “Hierarchical recursive least squares estimation algorithm for secondorder Volterra nonlinear systems,” International Journal of Control, Automation, and Systems, vol. 20, no. 12, pp. 3940–3950, December 2022.
J. Pan, Y. Q. 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, vol. 21, no. 1, pp. 140–150, 2023.
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.
Y. Gu, Q. 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.
H. Liu, J. Wang, and Y. Ji, “Maximum likelihood recursive generalized extended least squares estimation methods for a bilinear-parameter systems with ARMA noise based on the over-parameterization model,” International Journal of Control, Automation, and Systems, vol. 20, no. 8, pp. 2606–2615, August 2022.
X. Meng, Y. Ji, and J. Wang, “Iterative parameter estimation for photovoltaic cell models by using the hierarchical principle,” International Journal of Control, Automation, and Systems, vol. 20, no. 8, pp. 2583–2593, August 2022.
G. Chen, M. Gan, S. Wang, and C. Chen, “Insights into algorithms for separable nonlinear least squares problems,” IEEE Transactions on Image Processing, vol. 30, pp. 1207–1218, 2021.
J. Ren, J. Duan, and X. Wang, “A parameter estimation method based on random slow manifolds,” Applied Mathematical Modelling, vol. 39, no. 13, pp. 3721–3732, 2015.
F. Ding, G. Liu, and X. Liu. “Parameter estimation with scarce measurements,” Automatica, vol. 47, no. 8, pp. 1646–1655, 2011.
S. Srivastava and V. S. Pandit. “A PI/PID controller for time delay systems with desired closed loop time response and guaranteed gain and phase margins,” Journal of Process Control, vol. 37, pp. 70–77, 2016.
S. Ahmed, B. Huang, and S. L. Shah, “Novel identification method from step response,” Control Engineering Practice, vol. 15, no. 5, pp. 545–556, 2007.
P. Balaguer, V. Alfaro, and O. Arrieta, “Second order inverse response process identification from transient step response,” ISA Transactions, vol. 50, no. 2, pp. 231–238, 2011.
E. Hidayat and A. Medvedev, “Laguerre domain identification of continuous linear time-delay systems from impulse response data,” Automatica, vol. 48, no. 11, pp. 2902–2907, 2012.
L. D. Tommasi, D. Deschrijver, and T. Dhaene, “Transfer function identification from phase response data,” AEU-International Journal of Electronics and Communications, vol. 64, no. 3, pp. 218–223, 2010.
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.
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, 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, 2021.
M. Kapetina, M. Rapaic, and A. Pisano, “Adaptive parameter estimation in LTI systems,” IEEE Transactions on Automatic Control, vol. 64, no. 10, pp. 4188–4195, 2019.
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.
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. 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, 2021.
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.
J. Wang, Y. Ji, and X. Zhang, “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.
H. Ma, X. Zhang, 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.
E. Carvalho, J. Martinez, and F. Pisnitchenko. “On optimization strategies for parameter estimation in models governed by partial differential equations,” Mathematics and Computers in Simulation, vol. 114, pp. 14–24, 2015.
Q. Lin, R. Loxton, C. Xu, and K. L. Teo. “Parameter estimation for nonlinear time-delay systems with noisy output measurements,” Automatica, vol. 60, pp. 48–56, 2015.
A. J. Isaksson, J. Sjöberg, D. Törnqvist, L. Ljung, and M. Kok. “Using horizon estimation and nonlinear optimization for grey-box identification,” Journal of Process Control, vol. 30, pp. 69–79, 2015.
Y. Ji and A. N. Jiang, “Filtering-based accelerated estimation approach for generalized time-varying systems with disturbances and colored noises,” IEEE Transactions on Circuits and Systems-II: Express Briefs, vol. 70, no. 1, pp. 206–210, January 2023.
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.
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.
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, 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.
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, 2021.
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, 2019.
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.
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.
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.
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.
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.
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.
J. Hou, H. Su, C. Yu, and P. Li, “Bias-correction errors-invariables Hammerstein model identification,” IEEE Transactions on Industrial Electronics, vol. 70, no. 7, pp. 7268–7279, 2023.
F. Ding, “Coupled-least-squares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, January 2013.
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.
C. Xu, Y. Qin, and H. Su, “Observer-based dynamic event-triggered bipartite consensus of discrete-time multi-agent systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 70, no. 3, pp. 1054–1058, 2023.
G. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control, Prentice Hall, Englewood Cliffs, New Jersey, 1984.
F. Ding and T. Chen. “Performance analysis of multiinnovation gradient type identification methods,” Automatica, vol. 43, no. 1, pp. 1–14, 2007.
F. Geng and X. Wu, “A novel kernel functions algorithm for solving impulsive boundary value problems,” Applied Mathematics Letters, vol. 134, 108318, 2022.
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.
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, 2013.
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.
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.
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.
T. Cui, “Moving data window-based partially-coupled estimation approach for modeling a dynamical system involving unmeasurable states,” ISA Transactions, vol. 128, pp. 437–452, 2022.
C. Wei, “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.
C. Zhang, “Gradient parameter estimation of a class of nonlinear systems based on the maximum likelihood principle,” International Journal of Control, Automation, and Systems, vol. 20, no. 5, pp. 1393–1404, 2022.
J. M. Li, “A novel nonlinear optimization method for fitting a noisy Gaussian activation function,” International Journal of Adaptive Control and Signal Processing, vol. 36, no. 3, pp. 690–707, March 2022.
X. Zhang, “Hierarchical parameter and state estimation for bilinear systems,’ International Journal of Systems Science, vol. 51, no. 2, 275–290, 2020.
H. Wang, G. Ke, J. Pan, and Q. Su, “Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system,” Scientific Reports, vol. 13, Article number 6671, 2023.
F. Ding, X. M. Liu, and H. B. Chen, “Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems,” Signal Processing, vol. 97, pp. 31–39, April 2014.
N. Zhao, A. Wu, Y. Pei, and D. Niyato, “Spatial-temporal aggregation graph convolution network for efficient mobile cellular traffic prediction,” IEEE Communications Letters, vol. 26, no. 3, pp. 587–591, 2022.
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, February 2022.
F. Ding, H. Yang, and F. Liu, “Performance analysis of stochastic gradient algorithms under weak conditions,” Science in China Series F - Information Sciences, vol. 51, no. 9, pp. 1269–1280, 2008.
H. Wang, G. Ke, and J. Pan, “Two pairs of heteroclinic orbits coined in a new sub-quadratic Lorenz-like system,” European Physical Journal B, vol. 96, no. 3, p. 28, 2023.
Y. Wang, “Recursive parameter estimation algorithm for multivariate output-error systems,” Journal of the Franklin Institute, vol. 355, no. 12, pp. 5163–5181, 2018.
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.
F. Ding, G. Liu, and X. 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. 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, vol. 18, no. 1, pp. 467–480, 2023.
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, vol. 23, no. 8, pp. 12074–12083, 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.
Y. Li, G. Yang, Z. Su, S. Li, and Y. Wang, “Human activity recognition based on multienvironment sensor data,” Information Fusion, vol. 91, pp. 47–63, March 2023.
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.
J. Lin, Y. Li, and G. C. Yang, “FPGAN: Face deidentification method with generative adversarial networks for social robots,” Neural Networks, vol. 133, pp. 132–147, January 2021.
G. C. Yang, Z. J. Chen, Y. Li, and Z. D. Su, “Rapid relocation method for mobile robot based on improved ORB-SLAM2 algorithm,” Remote Sensing, vol. 11, no. 2, 149, 2019.
F. Ding, “Least squares and multi-innovation least squares methods,” Journal of Computational and Applied Mathematics, vol. 426, p. 115107, July 2023.
F. Z. Geng and X. Y. Wu, “Reproducing kernel-based piecewise methods for efficiently solving oscillatory systems of second-order initial value problems,” Calcolo, vol. 60, no. 2, p. 20, June 2023.
X. Y. Li, and X. Y. Liu, “A hybrid kernel functions collocation approach for boundary value problems with Caputo fractional derivative,” Applied Mathematics Letters, vol. 142, 108636, 2023.
L. Xu, “Separable synthesis estimation methods and convergence analysis for multivariable systems,” Journal of Computational and Applied Mathematics, vol. 427, p. 115104, August 2023.
X. Y. Li and B. Y. Wu, “A kernel regression approach for identification of first order differential equations based on functional data,” Applied Mathematics Letters, vol. 127, p. 107832, May 2022.
F. Ding, “Filtered auxiliary model recursive generalized extended parameter estimation methods for Box-Jenkins systems for 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
Z. Shi, H. Yang, and M. Dai, “The data-filtering based bias compensation recursive least squares identification for multi-input single-output systems with colored noises,” Journal of the Franklin Institute, vol. 360, no. 7, pp. 4753–4783, 2023.
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The author declares that there is no conflict of interests regarding the publication of this paper.
Ling Xu was born in Tianjin, China. She received her master’s and Ph.D. degrees from the Jiangnan University (Wuxi, China), in 2005 and 2015, respectively. She was a Post-Doctoral Fellow at the Jiangnan University from 2016 to 2020 and is currently a Professor. She is a Colleges and Universities “Blue Project” Young Teacher (Jiangsu, China). Her research interests include process control, parameter estimation, and signal modeling.
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This work was supported by the National Natural Science Foundation of China (No. 61873111).
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Xu, L. Parameter Estimation for Nonlinear Functions Related to System Responses. Int. J. Control Autom. Syst. 21, 1780–1792 (2023). https://doi.org/10.1007/s12555-021-1028-6
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DOI: https://doi.org/10.1007/s12555-021-1028-6