## 5 Conclusions  
In this research, a proposed smooth sliding digital twin was used for anomaly diagnosis and crack size identification in bearings based on the AE signals. First, the RMS feature signal was extracted from the AE signals to increase the signal modeling accuracy. Then, the normal RMS feature signal was modeled using the autoregressive fuzzy Gauss-Laguerre approach. This technique is robust and can model the normal RMS signal with high accuracy. Next, a smooth digital twin was proposed for RMS signal estimation in different conditions. This method was designed by combining the proposed modeling algorithm with the proposed estimation scheme. Apart from stability and robustness of the sliding mode digital twin, it has a high-frequency oscillation challenge. To overcome this challenge, a high-order and super-twisting technique was suggested. Furthermore, the fuzzy technique was used with the high-order sliding digital twin to improve the flexibility of signal estimation. Moreover, the combination of the smooth sliding digital twin and SVM was proposed for anomaly diagnosis and crack size identification. The power of the proposed smooth sliding digital twin for anomaly diagnosis and crack size identification was compared with those of the high-order sliding digital twin and sliding digital twin. The proposed smooth sliding digital twin was shown to improve the accuracy of anomaly diagnosis by 6 and 8.25 percentage points compared to the high-order sliding digital twin and sliding digital twin, respectively. Furthermore, the proposed algorithm has enhanced the accuracy of crack size identification by 6.92 and 11.64 percentage points compared to the high-order sliding digital twin and sliding digital twin, respectively. Although the proposed method has a reliable response under unknown conditions, it can be improved in future work by utilizing the parallel combination of the digital twin and deep-learning approach. Moreover, the robust algorithms for modeling and estimation are used to reduce the effect of noise and uncertain conditions. With this combination, the performance of signal estimation and reliability can be improved.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**States**} & \multicolumn{2}{c}{**Proposed Smooth Sliding**} & \multirow{2}{*}{**KF [8] (\%)**} & \multirow{2}{*}{**CFB [9] (\%)**} & \multirow{2}{*}{**VS [17] (\%)**} \\  & **Digital Twin and SVM (\%)** & & & \\ \hline NS & 100 & 100 & 100 & 100 \\ BS & 99 & 88 & 90 & 91 \\ IS & 97 & 89 & 88 & 90 \\ OS & 98 & 89 & 92 & 87 \\ IBS & 96 & 90 & 90 & 88 \\ IOS & 96 & 92 & 92 & 92 \\ OBS & 98 & 90 & 91 & 93 \\ IOBS & 98 & 94 & 95 & 89 \\ \hline Average accuracy & 97.75 & 91.5 & 92.25 & 91.25 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison between the proposed smooth sliding digital twin, Kalman filter (KF) [8], classical feedback backstepping (CFB) observer [9], and variable structure (VS) observer [17] for anomaly diagnosis.  
**Author Contributions:** Conceptualization, F.P., C.-H.K., and J.-M.K.; data curation, F.P.; formal analysis, F.P. and C.-H.K.; funding acquisition, J.-M.K.; methodology, F.P., C.-H.K., and J.-M.K.; software, F.P.; supervision, J.-M.K.; validation, F.P., C.-H.K., and J.-M.K.; visualization, F.P.; writing--original draft, F.P.; writing--review and editing, J.-M.K. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the Technology Development Program (S3126818) funded by the Ministry of SMEs and Startups (MSS, Korea). This work was also supported by a Korea Intelligent Information Society Promotion Agency grant funded by the Ministry of Science and ICT (2022 5G-based Digital Twin Public Leading Project).  
**Data Availability Statement:** The data are available in the UIAI Lab.  
**Conflicts of Interest:** The authors declare no conflict of interest.