This work introduces a novel control framework using the Caputo fractional derivative (CFD) with respect to another function—a derivative that has not been thoroughly treated in control theory. By extending the widely recognized Caputo-Hadamard (CH) fractional-order derivative, we address its utility in nonlinear systems. The core of our contribution is the practical stability for systems governed by this derivative, which ensures convergence toward a bounded region around the origin. Additionally, we extend the Lipschitz condition (LC) to the one-sided Lipschitz (OSL) condition for observer design and observer based-control design in fractional-order systems, ensuring its practical stability. Finally, three numerical examples validate the effectiveness of our proposed framework, providing practical insights for control theory advancements.

This work presents a generalization and a comparative study of the recursive maximum likelihood estimation algorithm for large-scale interconnected nonlinear systems, extending existing integer-order frameworks to fractional-order dynamics. While prior research introduced Mamdani fuzzy-based parameter estimators for networked integer-order interconnected nonlinear autoregressive moving average with exogenous input (INARMAX) models, this study addresses the challenges of time-varying fractional-order systems with memory-dependent behavior and stochastic disturbances. A novel recursive estimator is developed by integrating the Grünwald–Letnikov fractional difference operator into the prediction error framework, coupled with a Mamdani fuzzy supervisor to dynamically tune the forgetting factors. The proposed method is rigorously validated through simulations on interconnected subsystems with time-varying coefficients and nonlinear couplings. The results demonstrate a 30%–50% reduction in steady-state prediction errors and 40%–60% faster convergence compared with integer-order benchmarks, alongside superior robustness to noise ( ${\sigma }_i^2\le 0.25$) and abrupt parameter changes. This work establishes the first fuzzy-augmented fractional Maximum Likelihood Estimation (MLE) framework for large-scale systems, offering theoretical guarantees and empirical validation. Applications in power networks, biomedical systems, and industrial processes with hereditary dynamics are highlighted. The study underscores the necessity of fractional-order modeling in complex systems and provides a scalable solution for real-world deployment.

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.