## 4 Experimental Results  
To test the effectiveness of the proposed smooth sliding digital twin, the AE bearing dataset is used [9]. Figure 5 illustrates the original AE signals for normal and abnormal states. Based on this figure classification, the normal and abnormal conditions are difficult to observe. To reduce the complexity for modeling and estimation, the indirect observation algorithm is suggested in this work. To design the indirect observer, the resampled RMS feature is extracted from the original AE signals of the bearing and the RMS signal is modeled and estimated. The original AE signal is divided into 6000 windows (each group of 12,500 samples is defined as a window). Next, we obtained the RMS feature for each window. Figure 6 illustrates the resampled RMS signals for all conditions that were extracted from the original AE signals for the bearing. In the next step, the resampled RMS signal in known conditions (based on Table 2, using the normal state when the motor speed is 300 RPM) is modeled. To test the power of the indirect AE signal modeling using the proposed autoregressive fuzzy Gauss-Laguerre algorithm, this technique is compared with the autoregressive Gauss-Laguerre technique and autoregressive Gaussian method. Figure 7 illustrates the errors of RMS signal modeling using the proposed autoregressive fuzzy Gauss-Laguerre algorithm, autoregressive Gauss-Laguerre technique, and autoregressive Gaussian method. Based on this figure, the error of the proposed autoregressive fuzzy Gauss-Laguerre algorithm is less than the other two methods. Figure 5: Original AE signals for the bearing. Figure 6: RMS of original AE signals for the bearing. Figure 7: The error of RMS signal modeling using the proposed autoregressive fuzzy Gauss–Laguerre algorithm, autoregressive Gauss–Laguerre technique, and autoregressive Gaussian method for normal conditions. The other challenges of signal modeling are related to stability and robustness. To test the stability and robustness of the proposed autoregressive fuzzy Gauss-Laguerre algorithm, autoregressive Gauss-Laguerre technique, and autoregressive Gaussian method, the signal modeling errors are checked in normal conditions when the motor speeds are 300 RPM, 400 RPM, 450 RPM, and 500 RPM. Figure 8 illustrates the stability and robustness in the proposed autoregressive fuzzy Gauss-Laguerre algorithm, autoregressive Gauss-Laguerre technique, and autoregressive Gaussian method. Based on this figure the error of the proposed autoregressive fuzzy Gauss-Laguerre algorithm is less than the other two techniques. Based on Figure 8, the proposed autoregressive fuzzy Gauss-Laguerre algorithm and autoregressive Gauss-Laguerre technique are more robust than the autoregressive Gaussian method. According to the results in Figures 7 and 8, the proposed autoregressive fuzzy Gauss-Laguerre algorithm is suggested for RMS signal modeling. To test the power of RMS (indirect) signal estimation, the proposed smooth sliding digital twin is compared with the high-order sliding digital twin and sliding digital twin. Figure 9 illustrates the residual signals of the proposed smooth sliding digital twin. According to this figure, the difference between signal levels in different classes is much better than in Figure 6. This increases the anomaly detection and identification performance. Based on this figure the level of the residual signal in NS is less than the faulty conditions. Thus, the performance of fault detection in the proposed algorithm is perfect. Furthermore, the difference between residual signals in different anomaly condition is completely different. This makes the quality of the anomaly signal diagnosis excellent in this case. Figures 10 and 11 show the residual of RMS signals in the higher-order sliding digital twin and sliding digital twin, respectively. Based on Figures 10 and 11, the high-frequency oscillation in the high-order sliding digital twin is less than in the sliding digital twin. This means that the high-order technique can reduce the high-frequency oscillation.