Hierarchical neural identification approach for Hammerstein large-scale stochastic systems: A simulation study of hydraulic process

Rihab Issaoui; Mourad Elloumi; Imed Bouzida; Omar Naifar; Rihab Issaoui; Mourad Elloumi; Imed Bouzida; Omar Naifar

doi:10.3934/math.2026498

AIMS Mathematics

2026, Volume 11, Issue 4: 12132-12154. doi: 10.3934/math.2026498

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Research article

Hierarchical neural identification approach for Hammerstein large-scale stochastic systems: A simulation study of hydraulic process

1.
Control and Energy Management Laboratory, National School of Engineering, Sfax University, Sfax, Tunisia
2.
Faculty of Sciences of Gafsa, University of Gafsa, Gafsa, Tunisia
3.
Laboratory of Sciences and Technology of Automatic Control and Computer Engineering, National School of Engineering of Sfax, Sfax University, P.O. Box 1173, Sfax 3038, Tunisia
4.
Higher Institute of Applied Science and Technology of Kairouan, University of Kairouan, Kairouan, Tunisia
5.
Department of Mathematics and Statistics, College of Engineering, Abu Dhabi University, Abu Dhabi, United Arab Emirates
6.
Laboratory of Probability and Statistics, Sfax University, Sfax, Tunisia

Received: 17 February 2026 Revised: 23 March 2026 Accepted: 30 March 2026 Published: 29 April 2026
MSC : 68T07, 93A30, 93B30, 93C56

This paper proposes an interconnected Hammerstein neural network (IHNN)-based hybrid identification method for large-scale interconnected Hammerstein systems subject to stochastic disturbances. In the proposed method, the static nonlinear blocks are approximated by neural networks, while the linear dynamic parameters are recursively estimated using a recursive least-squares scheme with forgetting and covariance adaptation. The proposed identification framework preserves the block-oriented Hammerstein structure and is designed to handle strong subsystem interconnections and noisy operating conditions. A Lyapunov-based analysis is further developed to establish convergence and stability conditions for the overall learning algorithm, which combines backpropagation for the neural-network parameters and recursive estimation for the linear dynamics. The effectiveness of the proposed IHNN identification method is validated through a benchmark interconnected system and a hydraulic-process case study. The simulation results show consistent improvements over a conventional recursive extended least squares (RELS) baseline, including root mean square error (RMSE) reductions of about 35–38% and prediction-error variance reductions of about 60%, at the expense of increased computational time. These results demonstrate that the proposed IHNN approach provides an accurate and practical solution for identifying noisy large-scale interconnected Hammerstein systems.
- system identification,
- Hammerstein model,
- neural networks,
- large-scale systems,
- stochastic systems,
- hydraulic process
Citation: Rihab Issaoui, Mourad Elloumi, Imed Bouzida, Omar Naifar. Hierarchical neural identification approach for Hammerstein large-scale stochastic systems: A simulation study of hydraulic process[J]. AIMS Mathematics, 2026, 11(4): 12132-12154. doi: 10.3934/math.2026498

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Abstract

This paper proposes an interconnected Hammerstein neural network (IHNN)-based hybrid identification method for large-scale interconnected Hammerstein systems subject to stochastic disturbances. In the proposed method, the static nonlinear blocks are approximated by neural networks, while the linear dynamic parameters are recursively estimated using a recursive least-squares scheme with forgetting and covariance adaptation. The proposed identification framework preserves the block-oriented Hammerstein structure and is designed to handle strong subsystem interconnections and noisy operating conditions. A Lyapunov-based analysis is further developed to establish convergence and stability conditions for the overall learning algorithm, which combines backpropagation for the neural-network parameters and recursive estimation for the linear dynamics. The effectiveness of the proposed IHNN identification method is validated through a benchmark interconnected system and a hydraulic-process case study. The simulation results show consistent improvements over a conventional recursive extended least squares (RELS) baseline, including root mean square error (RMSE) reductions of about 35–38% and prediction-error variance reductions of about 60%, at the expense of increased computational time. These results demonstrate that the proposed IHNN approach provides an accurate and practical solution for identifying noisy large-scale interconnected Hammerstein systems.

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