Metal fatigue cracks are one of the most common causes of structural failure in aircrafts. Accurate and effective prediction of fatigue crack propagation is crucial for developing design and maintenance strategies throughout the lifecycle of aircrafts. This issue is of significant concern to both the aerospace industry and academia. Therefore, based on the classical nonlinear grey Bernoulli model, we propose a physics-persevering cumulative sum operator, making it more interpretable. On this basis, we derive an improved form of the nonlinear grey Bernoulli model. Furthermore, separable nonlinear least squares is formulated to generate simultaneous estimates of structural parameters, power parameter and initial condition. The proposed method is validated through fatigue tests on aircraft skin lap structures and compared with other algorithms. The results demonstrate that the root mean square errors of crack propagation predictions are less than 1.9 mm for all structures, confirming the effectiveness of the proposal. This approach provides significant technical value for predicting the fatigue life of aircraft structures and offers valuable insights for predicting metal fatigue cracks in similar complex equipment.