QDRFormer: Quantum-Inspired Density Representation Learning with Random Forest-Guided Transformers for Early Prediction of PLL Instability and Chaos

Early prediction of nonlinear instability in phase-locked loops (PLLs) is essential for ensuring the reliability of modern power electronic and communication systems. This paper presents QDRFormer, a quantum-inspired density representation learning framework integrating density-matrix-based features, local instability descriptors, Random Forest teacher-guided augmentation, and an attention-based dual-branch Transformer. PLL trajectories generated across diverse operating conditions and initial states are labeled as Stable, Near-Critical, or Chaotic using the largest Lyapunov exponent. Sliding-window segments are transformed into quantum-inspired density representations and fused with raw state variables and engineered temporal features. On an independent trajectory-level test set, QDRFormer achieves 91.56% accuracy and a macro-AUC of 0.972, while trajectory-level five-fold Group Cross-Validation yields 84.67 ± 0.81% mean accuracy. Partial-trajectory evaluation shows that the eventual regime can be recognized from the first 10% of a trajectory with 78.21% accuracy and 92.0% Chaotic recall, with comparable discrimination when the detuning parameter Δω is excluded. Evaluation at unseen Δω operating points further assesses operating-point generalization. Robustness experiments under PLL parameter uncertainty, measurement noise, reduced sampling density, and transient disturbances identify strong tolerance to reduced sampling and mild transient disturbances but greater sensitivity to measurement noise and stronger perturbations. Interpretability analyses using principal component analysis, feature importance, and Transformer attention provide additional insight into the learned representations. Overall, QDRFormer provides an interpretable framework for early regime recognition and instability classification within the simulated PLL setting, while experimental, hardware-in-the-loop, and cross-system validation remain important future directions.