Robust recursive identification of interconnected fractional Hammerstein systems under impulsive disturbances

Slim Dhahri; Mourad Elloumi; Hend Aljahani; Salem Albalawi; Sahar Almashaan; Hatem Alwardi; Foued Mtiri; Slim Dhahri; Mourad Elloumi; Hend Aljahani; Salem Albalawi; Sahar Almashaan; Hatem Alwardi; Foued Mtiri

doi:10.3934/math.20261124

AIMS Mathematics

2026, Volume 11, Issue 9: 28202-28253. doi: 10.3934/math.20261124

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Research article

Robust recursive identification of interconnected fractional Hammerstein systems under impulsive disturbances

1.
Department of Computer Engineering and Networks, College of Computer and Information Sciences, Jouf University, Sakaka 72341, Al-Jouf, Saudi Arabia
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.
Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia
5.
Department of Mathematics, College of Science, Jouf University, Sakaka, Saudi Arabia
6.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia

Received: 23 May 2026 Revised: 16 July 2026 Accepted: 31 July 2026 Published: 03 September 2026
MSC : 26A33, 62F35, 62L20, 93E12, 93E24

Interconnected fractional Hammerstein models are useful for nonlinear processes with hereditary local dynamics and cross-subsystem interactions, but ordinary fractional recursive extended least squares (FRELS) estimators remain fragile under impulsive measurements. This paper develops Huber and maximum correntropy criterion (MCC) robust stochastic FRELS (RS-FRELS) recursions for finite-memory fractional interconnected Hammerstein regressions. The key point is not a new robust score but a careful frozen-weight interpretation: The online update exactly solves a weighted quadratic surrogate whose weights are evaluated at prior innovations, rather than the full nonlinear robust empirical risk. This framing yields explicitly conditional guarantees. On the algebraic side, these cover surrogate optimality, covariance positivity under either a deterministic or an in-expectation weighted-excitation condition, bounded single-sample influence, and an estimator-side saturation variant for leverage control. On the statistical side, they cover Huber population consistency under score orthogonality, a local orthogonality-defect bias bound, MCC stationarity without global uniqueness, local stochastic approximation convergence in general, global convergence for projected Huber RS-FRELS on a compact convex projection set under uniform Huber curvature, and conservative tracking under slow drift. The assumptions are connected to checkable diagnostics for weighted information, score defects, saturation activation, and regressor norms, together with a step-by-step verification workflow for practitioners. Simulations include 30-seed Monte Carlo comparisons, paired significance tests, heavy-tailed and leverage stress tests, MCC kernel and impulse severity sweeps, same-regressor algorithm baselines including a redescending recursive least M-estimate and adaptively tuned robust variants, model structure ablations, and scalability timing up to 50 subsystems. The synthetic study is complemented by a real-data validation on the DaISy (Database for the Identification of Systems) liquid–saturated steam heat exchanger benchmark, in which the robust recursions match ordinary FRELS on the raw record and improve held-out prediction accuracy under injected impulsive sensor faults. The results support the bounded score mechanism while clearly separating what the experiments validate from unresolved colored output error consistency questions.
- fractional-order systems,
- Hammerstein systems,
- interconnected systems,
- recursive identification,
- robust RLS,
- Huber M-estimation,
- maximum correntropy,
- impulsive noise,
- frozen-weight IRLS,
- scalability diagnostics
Citation: Slim Dhahri, Mourad Elloumi, Hend Aljahani, Salem Albalawi, Sahar Almashaan, Hatem Alwardi, Foued Mtiri. Robust recursive identification of interconnected fractional Hammerstein systems under impulsive disturbances[J]. AIMS Mathematics, 2026, 11(9): 28202-28253. doi: 10.3934/math.20261124

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Abstract

Interconnected fractional Hammerstein models are useful for nonlinear processes with hereditary local dynamics and cross-subsystem interactions, but ordinary fractional recursive extended least squares (FRELS) estimators remain fragile under impulsive measurements. This paper develops Huber and maximum correntropy criterion (MCC) robust stochastic FRELS (RS-FRELS) recursions for finite-memory fractional interconnected Hammerstein regressions. The key point is not a new robust score but a careful frozen-weight interpretation: The online update exactly solves a weighted quadratic surrogate whose weights are evaluated at prior innovations, rather than the full nonlinear robust empirical risk. This framing yields explicitly conditional guarantees. On the algebraic side, these cover surrogate optimality, covariance positivity under either a deterministic or an in-expectation weighted-excitation condition, bounded single-sample influence, and an estimator-side saturation variant for leverage control. On the statistical side, they cover Huber population consistency under score orthogonality, a local orthogonality-defect bias bound, MCC stationarity without global uniqueness, local stochastic approximation convergence in general, global convergence for projected Huber RS-FRELS on a compact convex projection set under uniform Huber curvature, and conservative tracking under slow drift. The assumptions are connected to checkable diagnostics for weighted information, score defects, saturation activation, and regressor norms, together with a step-by-step verification workflow for practitioners. Simulations include 30-seed Monte Carlo comparisons, paired significance tests, heavy-tailed and leverage stress tests, MCC kernel and impulse severity sweeps, same-regressor algorithm baselines including a redescending recursive least M-estimate and adaptively tuned robust variants, model structure ablations, and scalability timing up to 50 subsystems. The synthetic study is complemented by a real-data validation on the DaISy (Database for the Identification of Systems) liquid–saturated steam heat exchanger benchmark, in which the robust recursions match ordinary FRELS on the raw record and improve held-out prediction accuracy under injected impulsive sensor faults. The results support the bounded score mechanism while clearly separating what the experiments validate from unresolved colored output error consistency questions.

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