Mamdani fuzzy parameter estimation of fractional-order large-scale interconnected systems

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.