## 6 Predicted Class
According to the above figures, the crack size identification performance for the proposed smooth sliding digital twin is better than the other two methods; the power and accuracy of estimation in the proposed smooth sliding digital twin is much better than in the high-order sliding digital twin and sliding digital twin because in the proposed method the signals are much more recognizable for 3 mm and 6 mm in different states (faults) than the other two techniques. Based on Figures 9-11 and 15-21, the proposed smooth sliding digital twin has an excellent performance for fault diagnosis and crack size identification. These figures provide additional evidence that the crack size diagnosis performance is superior in the proposed technique. Furthermore, Table 5 compares the crack size identification accuracies for the combination of the proposed smooth sliding digital twin and SVM, combination of the high-order sliding digital twin and SVM, and combination of the sliding digital twin and SVM. Figure 21: The accuracy of crack size identification for inner-outer-ball residual of RMS signals using proposed smooth sliding digital twin, high-order sliding digital twin, and sliding digital twin. Figure 20: The accuracy of crack size identification for inner-outer residual of RMS signals using proposed smooth sliding digital twin, high-order sliding digital twin, and sliding digital twin. According to the above table, the crack size identification accuracies of the combination of the proposed smooth sliding digital twin approach and SVM, combination of the high-order sliding digital twin and SVM, and combination of the sliding digital twin and SVM are 97.78%, 90.86%, and 86.14%, respectively. Thus, the proposed smooth sliding digital twin has improved the crack size identification accuracy by 6.92 and 11.64 percentage points compared to the high-order sliding digital twin and sliding digital twin, respectively. Therefore, according to Tables 4 and 5, the accuracy of anomaly diagnosis and crack size classification in the combination of the proposed smooth sliding digital twin and SVM are 97.75% and 97.78%, respectively. These values are 91.75% and 90.86%, respectively, for the combination of the high-order sliding digital twin and SVM and 89.5% and 86.14%, respectively, for the combination of the sliding digital twin and SVM. Thus, based on these two tables, the proposed smooth sliding digital twin has improved the accuracy of anomaly diagnosis and crack size classification by 6 and 6.92 percentage points compared to the high-order sliding digital twin and 8.25 and 11.64 percentage points compared to the sliding digital twin. In the next part, to validate the effectiveness of the proposed smooth sliding digital twin, this approach is compared with following three existing methods including Kalman filter approach [8], classical feedback backstepping observer [9], and variable structure observer [17]. In [8], the authors have used the combination of mathematical-based signal modeling, support vector regression with Laguerre algorithm, and Kalman filter for bearing fault diagnosis. In this approach, the authors used the mathematical approach and identification technique for signal modeling. This approach is complex for modeling and estimation, as well. In addition, in [9], the combination of support vector regression, Gaussian Process Regression, and feedback backstepping observer was suggested for fault diagnosis in the bearing. The main challenge in this approach was stability and robustness. Furthermore, variable structure observer has used for bearing fault diagnosis in [17]. However, the nonlinear autoregression technique was used for system modeling, this technique suffers from high frequency oscillation. To validate the proposed approach further, we calculate the average diagnostic accuracy for (see Table 6) under various operating conditions. Table 6 present the diagnostic accuracy of the proposed smooth sliding digital twin algorithm, Kalman filter (KF) approach [8], classical feedback backstepping (CFB) observer [9], and