The intrinsic memory and nonlocality that allow fractional-order calculus to capture complex dynamical behaviors also pose significant challenges for accurate modeling and stable control. This article presents a unified data-driven framework that simultaneously addresses these challenges through three key innovations. First, we propose a fractional-order deep Lagrangian network (DeLaN) with a Transformer-like structure, fPLCS-DeLaN, to learn system's inherent fractional-order behaviors directly from uniformly sampled data. It not only enforces fractional-order Lagrangian structure by integrating key physical priors, but also enhances capturing ability of memory effects by incorporating long-short-term convolutional self-attention mechanism. Second, we develop a hybrid network-based disturbance observer, T2F-CRNN, which synergizes CNN's temporal feature extraction, hierarchical recurrence, and interval-based fuzzy inference to robustly estimate uncertainties with unknown nonuniform bounds and capture temporal dependencies. Third, we establish a fully fractional-order controller with practical finite-time convergence. It incorporates input saturation compensation and sliding mode constraints to ensure robustness and high performance. Simulations show that fPLCS-DeLaN achieves modeling errors at least one order of magnitude lower with less than a 15% increase in computational time. The proposed fractional-order controller reduces transient and steady-state tracking errors by 23.1% and 87.6% compared to state-of-the-art controllers, respectively. Experiments on a soft manipulator platform further demonstrate consistent superiority in model learning and tracking performance.
Shao et al. (Thu,) studied this question.