# Bearing Anomaly Recognition Using an Intelligent Digital Twin Integrated with Machine Learning
## 2 Proposed Methodology
### Digital Twin
#### 2.2.2 Signal Estimation: Second Step to Design the Digital Twin  
After the first step of the digital twin's modeling for the bearing, in the next step, the signal estimation technique is suggested to improve the accuracy and robustness of the bearing's digital twin. In this study, the hybrid robust Kalman filter, which is a combination of the Kalman filter (KF), the high-order variable structure technique (HVS), and the adaptive neural-fuzzy method (ANF) is recommended for signal estimation. Thus, first, we implement a Kalman filter for vibration signal estimation based on the MS signal modeling. The state-space KF formulation using the MS function approximation is represented as the following equation. \[\left\{ \begin{array}{l}W_{MS-KF}(k+1)=\bigg{[}\begin{array}{c}\gamma_{MS _{1}-KF}W_{MS-KF}(k)+\gamma_{MS_{2}-KF}Z_{MS-KF}(k)\\ +\Delta_{MS-KF}(k)\end{array}\bigg{]}+\\ e_{MS-KF}(k)+\partial[Z(k)-Z_{MS-KF}(k)]\\ Z_{MS-KF}(k)=(\beta_{MS-KF})^{T}W_{MS-KF}(k)\end{array} \right. \tag{23}\]  
Here, \(W_{MS-KF}(k),e_{MS-KF}(k),\Delta_{MS-KF}(k),\ Z_{MS-KF}(k),\ \partial\), and \(\big{(}\gamma_{MS_{1}-KF},\gamma_{MS_{2}-KF},\ \beta_{MS-KF}\big{)}\) are the state of the signal estimation based on the KF technique for the bearing's digital twin which was modeled by the MS technique, the error of the bearing's digital twin estimation using the KF technique which was modeled by the MS technique, the unknown condition of the bearing's digital twin estimation that was computed using the KF technique which was modeled by the MS technique, the bearing's digital twin output estimation using the KF technique which was modeled by the MS technique, the Kalman filter gain matrices, and the coefficients for the state-space bearing's digital twin tuning using the KF method which was modeled by the MS technique, respectively. Moreover, the Kalman filter gain matrices \((\partial)\) are represented as the following function. \[\partial(k)=\frac{\gamma_{MS_{1}-KF}\times\Delta_{MS-KF}(k)\times\big{(}\beta _{MS-KF}\big{)}^{T}}{\big{[}\beta_{MS-KF}\times\Delta_{MS-KF}(k)\times\big{(} \beta_{MS-KF}\big{)}^{T}+N\big{]}} \tag{24}\]Furthermore, to modify the performance of the KF estimator, the unknown condition of the bearing's digital twin estimation that was computed using the KF technique which was modeled by the MS technique is represented using the following definition. \[\Delta_{MS-KF}(k+1)= \gamma_{MS_{1}-KF}\times\Delta_{MS-KF}(k)\times\left(\gamma_{MS_{1} -KF}\right)^{T}+Q-\partial(k) \tag{25}\] \[\times[\beta_{MS-KF}\times\Delta_{MS-KF}(k)\times\left(\beta_{MS-KF }\right)^{T}+N]\times\partial^{T}(k)\]  
Here, \(N\) and \(Q\) are the sensor noise and the process noise covariance matrices. To increase the accuracy and robustness of the KF, the combination of the KF and the HVS (henceforth called RKF) is recommended in this study. The robustness of the RKF to parameters of unknown conditions is guaranteed. Thus, the state-space RKF formulation using the MS function approximation is represented as the following calculation. \[\left\{\begin{array}{c}W_{MS-RKF}(k+1)=\left[ \begin{array}{cc}\gamma_{MS_{ 1}-RKF}W_{MS-RKF}(k)+\gamma_{MS_{2}-RKF}Z_{MS-RKF}(k)+\\ \Delta_{MS-RKF}(k)\\ e_{MS-RKF}(k)+\partial[Z(k)-Z_{MS-RKF}(k)]+\gamma_{MS_{3}-RKF}\times sgn(S^{*} _{RKF})\\ Z_{MS-RKF}(k)=(\beta_{MS-RKF})^{T}W_{MS-RKF}(k)\end{array} \right]+ \tag{26}\]  
where the sign function, \((sgn(S^{*}))\), is introduced as  
\[\left\{ \begin{array}{c}sgn(S^{*})=1\\ sgn(S^{*})=-1\end{array} \right.\qquad\qquad \begin{array}{c}if\ S^{*}\geq 0\\ if\ S^{*}<0\end{array} \right.. \tag{27}\]  
Besides, the unknown condition of the bearing's digital twin estimation that was computed using the RKF technique and was modeled by the MS technique is represented using the following definition.