Variational Bayesian identification for Hammerstein large-scale stochastic systems: A comparative simulation study of hydraulic process
Keywords:
System identification, Hammerstein model, Variational Bayesian Hammerstein Identification, Large-scale interconnected systems, Uncertainty quantification, Evidence Lower Bound, Hydraulic processAbstract
Large-scale interconnected Hammerstein systems operating under stochastic disturbances present persistent challenges for identification, particularly when coupling between subsystems is strong and operating noise is non-negligible. This paper develops a Variational Bayesian Hammerstein Identification (VBHI) framework for such systems, where static nonlinear blocks are modelled by feedforward neural approximators and the linear dynamic parameters are estimated through a structured mean-field variational posterior. By treating all identifiable parameters as latent random variables and maximising a closed-form Evidence Lower Bound (ELOB), the proposed method delivers calibrated posterior uncertainty on every parameter block simultaneously, without recourse to expensive Markov Chain Monte Carlo sampling. The VBHI framework inherits a block-oriented Hammerstein structure and is designed to accommodate strong subsystem interconnections, time-varying coefficients, and measurement noise. Convergence of the variational update recursion is established through a Lyapunov-type descent argument applied to the negative ELOB. The method is evaluated on a two-subsystem numerical benchmark and a three-tank hydraulic-process case study, and compared against a conventional Recursive Extended Least Squares (RELS) baseline. Across both experiments, VBHI achieves Root Mean Square Error (RMSE) reductions of approximately 37–41%, prediction-error variance reductions of roughly 62–65%, and provides explicit confidence intervals that RELS cannot supply, at the cost of a moderate increase in per-sample computation. These results confirm that the proposed VBHI approach is a principled and practically effective solution for identifying noisy large-scale interconnected Hammerstein systems with built-in uncertainty quantification.
References
[1] Gustavo E. Ceballos Benavides, Manuel A. Duarte-Mermoud, and Lisbel Bárzaga Martell, Control error convergence using Lyapunov direct method approach for mixed fractional order model reference adaptive control, Fractal Fract. 9 (2025), no. 2, Art. 98, https://doi.org/10.3390/fractalfract9020098.
[2] Saurav Gupta, Subhransu Padhee, and Libor Pekar, Recursive least squares identification of heat exchanger system using block-structured models, Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 236 (2022), no. 4, 870–879, https://doi.org/10.1177/09596518211054921.
[3] Rabab Hamza, Ali Zribi, and Yassin Farhat, A novel neural emulator identification of nonlinear dynamical systems using Lyapunov stability theory, Trans. Inst. Meas. Control 45 (2023), no. 14, 2802–2811, https://doi.org/10.1177/01423312231174956.
[4] Yongjun He, Lin Xiao, Fuchun Sun, and Yaonan Wang, A variable-parameter ZNN with predefined-time convergence for dynamic complex-valued Lyapunov equation and its application to AOA positioning, Appl. Soft Comput. 130 (2022), 109703, https://doi.org/10.1016/j.asoc.2022.109703.
[5] Alireza Hosseinnajad and Navid Mohajer, Barrier Lyapunov function-based homogeneous fixed-time controller design for a double integrator system, ISA Trans. 152 (2024), 68–80, https://doi.org/10.1016/j.isatra.2024.06.005.
[6] Hui Huang, Jinniao Qiu, and Konstantin Riedl, On the global convergence of particle swarm optimization methods, Appl. Math. Optim. 88 (2023), no. 2, Art. 45, https://doi.org/10.1007/s00245-023-09983-3.
[7] Yuzhu Huang, Zhaoyan Zhang, and Shuo Zhao, Neural robust optimal tracking control for nonlinear state-constrained systems with input saturation and external disturbances, Appl. Intell. 55 (2025), no. 11, https://doi.org/10.1007/s10489-025-06682-0.
[8] Hongjun Lang, Yiqun Bi, Meihang Li, and Yan Ji, A novel filtering-based recursive identification method for a fractional-order Hammerstein state space system with piecewise nonlinearity, Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 239 (2025), no. 6, 1048–1065, https://doi.org/10.1177/09596518241309132.
[9] Yanan Li, Junhong Li, Fuchao Li, and Yaqi Duan, Recursive parameter estimation of fractional order Hammerstein output error autoregressive model, Circuits Systems Signal Process. 44 (2025), no. 7, 4828–4846, https://doi.org/10.1007/s00034-025-03016-w.
[10] Zhe Li, Li Zhu, Ching Wen Chung, and Junghui Chen, Recursive identification of kernel-based Hammerstein model for online energy efficiency estimation of large-scale chemical plants, Energy 309 (2024), 132946, https://doi.org/10.1016/j.energy.2024.132946.
[11] Jianqiang Lin and Shing Chow Chan, Recursive identification of nonlinear Hammerstein systems with FIR linear parts under impulsive noise with model selection and order determination, IEEE Access 12 (2024), 138702–138715, https://doi.org/10.1109/access.2024.3433583.
[12] Mingxuan Mao and Diyu Gui, Enhanced adaptive-convergence in Harris’ hawks optimization algorithm, Artificial Intelligence Rev. 57 (2024), no. 7, Art. 182, https://doi.org/10.1007/s10462-024-10802-6.
[13] Takayasu Matsuo, Shun Sato, and Kansei Ushiyama, A unified discretization framework for differential equation approach with Lyapunov arguments for convex optimization, Advances in Neural Information Processing Systems 36, Neural Information Processing Systems Foundation, Inc., 2023, pp. 26092–26120.
[14] Mohammad Jahani Moghaddam, Non-integer order system identification with time delays and disturbance rejection, Int. J. Dyn. Control 13 (2025), no. 1, https://doi.org/10.1007/s40435-024-01531-3.
[15] Omkar Sudhir Patil, Duc M. Le, Max L. Greene, and Warren E. Dixon, Lyapunov-derived control and adaptive update laws for inner and outer layer weights of a deep neural network, IEEE Control Syst. Lett. 6 (2022), 1855–1860, https://doi.org/10.1109/lcsys.2021.3134914.
[16] Chafea Stiti, Mohamed Benrabah, Abdelhadi Aouaichia, Adel Oubelaid, Mohit Bajaj, Milkias Berhanu Tuka, and Kamel Kara, Lyapunov-based neural network model predictive control using metaheuristic optimization approach, Sci. Rep. 14 (2024), no. 1, Art. 18721, https://doi.org/10.1038/s41598-024-69365-9.
[17] Chao Sun, Pengfei Liu, Haoran Guo, Yinlu Di, Qingquan Xu, and Xiaochen Hao, Control of precalciner temperature in the cement industry: A novel method of Hammerstein model predictive control with ISSA, Processes 11 (2023), no. 1, Art. 214, https://doi.org/10.3390/pr11010214.
[18] Manu Upadhyaya, Sebastian Banert, Adrien B. Taylor, and Pontus Giselsson, Automated tight Lyapunov analysis for first-order methods, Math. Program. 209 (2025), no. 1, 133–170, https://doi.org/10.1007/s10107-024-02061-8.
[19] Vandana Rani Verma, Dinesh Kumar Nishad, Vishnu Sharma, Vinay Kumar Singh, Anshul Verma, and Dharti Raj Shah, Quantum machine learning for Lyapunov-stabilized computation offloading in next-generation MEC networks, Sci. Rep. 15 (2025), no. 1, https://doi.org/10.1038/s41598-024-84441-w.
[20] M. Vidyasagar, Convergence of stochastic approximation via martingale and converse Lyapunov methods, Math. Control Signals Systems 35 (2023), no. 2, 351–374, https://doi.org/10.1007/s00498-023-00342-9.
[21] Guancheng Wang, Zhihao Hao, Bob Zhang, and Long Jin, Convergence and robustness of bounded recurrent neural networks for solving dynamic Lyapunov equations, Inform. Sci. 588 (2022), 106–123, https://doi.org/10.1016/j.ins.2021.12.039.
[22] Jingwen Wang, Jiawei Tian, Xia Zhang, Bo Yang, Shan Liu, Lirong Yin, and Wenfeng Zheng, Control of time delay force feedback teleoperation system with finite time convergence, Front. Neurorobot. 16 (2022), 877069, https://doi.org/10.3389/fnbot.2022.877069.
[23] Yuanda Yue, Ling Mi, Chuan Chen, and Yanqing Yang, Predefined-time stability-based zeroing neural networks and their application in solving the Lyapunov equation, Neural Process. Lett. 56 (2024), no. 1, https://doi.org/10.1007/s11063-024-11470-x.
[24] Yinyan Zhang, Improved GNN method with finite-time convergence for time-varying Lyapunov equation, Inform. Sci. 611 (2022), 494–503, https://doi.org/10.1016/j.ins.2022.08.061.
[25] Ping Zhou, Xikui Hu, Zhigang Zhu, and Jun Ma, What is the most suitable Lyapunov function?, Chaos Solitons Fractals 150 (2021), 111154, https://doi.org/10.1016/j.chaos.2021.111154.
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