# 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
\[\Delta_{MS-RKF}(k+1) =\gamma_{MS_{1}-RKF}\times\Delta_{MS-RKF}(k)\times\left(\gamma_{ MS_{1}-RKF}\right)^{T}+Q-\partial(k) \tag{28}\] \[\times[\beta_{MS-RKF}\times\Delta_{MS-RKF}(k)\times\left(\beta_ {MS-RKF}\right)^{T}+N]\times\partial^{T}(k)\] \[+\gamma_{MS_{4}-RKF}\times sgn\|(S^{*}_{RKF})\|^{0.5}-\varphi_{ RKF}\]  
and  
\[\dot{\varphi}_{RKF}=\gamma_{MS_{5}-RKF}\times sgn(S^{*}) \tag{29}\]  
Moreover, the surface of the variable structure method in the RKF, \((S^{*}_{RKF})\), is represented as the following explanation. \[S^{*}_{RKF}=Z-Z_{MS-RKF}(k) \tag{30}\]  
Here, \(W_{MS-RKF}(k),e_{MS-RKF}(k),\Delta_{MS-RKF}(k),\ Z_{MS-RKF}(k),S^{*}_{RKF}\), and \(\left(\gamma_{MS_{i}-RKF},\ \beta_{MS-RKF}\right)\) are the state of the signal estimation based on the RKF 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 RKF technique which was modeled by the MS technique, the RKF technique which was modeled by the MS technique, the bearing's digital twin output estimation using the RKF technique which was modeled by the MS technique, the surface of the variable structure method in the RKF, and the coefficients for the state-space bearing's digital twin tuning using the RKF method which was modeled by the MS technique, respectively. However, the RKF improves the robustness of the KF to increase the flexibility and accuracy of the bearing's digital twin. The combination of the RKF and the ANF technique (henceforth called HRKF) modeled by the MS approach is recommended. The ANF approach is an artificial intelligence-based technique that is a combination of two approaches, a fuzzy logic algorithm and a neural network. Hence, the state-spaceHRKF formulation using the MS function approximation is represented as the following definition. \[\left\{ \begin{array}{c}W_{MS-HRKF}(k+1)=\left[\begin{array}{c}\gamma_{MS_{1}- HRKF}W_{MS-HRKF}(k)+\\ \gamma_{MS_{2}-HRKF}Z_{MS-HRKF}(k)+\\ \Delta_{MS-HRKF}(k)\end{array}\right]+\\ \epsilon_{MS-HRKF}(k)+\partial[Z(k)-Z_{MS-HRKF}(k)]+\gamma_{MS_{3}-HRKF}\times s \hat{g}n(S^{*}_{HRKF})\\ Z_{MS-HRKF}(k)=(\beta_{MS-HRKF})^{\intercal}W_{MS-HRKF}(k)\end{array} \right. \tag{31}\]  
Above and beyond, the unknown condition of the bearing's digital twin estimation that was computed using the HRKF technique and was modeled by the MS technique is represented using the following definition. \[ \begin{array}{c}\Delta_{MS-HRKF}(k+1)=\gamma_{MS_{1}-HRKF}\times\Delta_{MS- HRKF}(k)\times(\gamma_{MS_{1}-HRKF})^{\text{T}}\\ +Q-\partial(k)\times\left[\beta_{MS-HRKF}\times\Delta_{MS-HRKF}(k)\times( \beta_{MS-HRKF})^{\text{T}}+N\right]\times\partial^{\text{T}}(k)\\ +\gamma_{MS_{4}-HRKF}\times s\hat{g}n\left\|(S^{*}_{HRKF})\right\|^{0.5}-\phi_ {HRKF}\\ +\gamma_{MS_{6}-HRKF}\times Z_{ANF}(k)\end{array}  \tag{32}\]  
and  
\[\hat{\phi}_{HRKF}=\gamma_{MS_{5}-RKF}\times s\hat{g}n(S^{*}_{HRKF}) \tag{33}\]  
Moreover, the surface of the variable structure method in the HRKF, \((S^{*}_{HRKF})\), is represented as the following explanation. \[S^{*}_{HRKF}=Z-Z_{MS-HRKF}(k) \tag{34}\]  
Here, \(W_{MS-HRKF}(k),e_{MS-HRKF}(k),\Delta_{MS-HRKF}(k),\ Z_{MS-HRKF}(k),S^{*}_{HRKF},Z_{ANF}(k)\), and \(\left(\gamma_{MS_{i}-HRKF},\beta_{MS-HRKF}\right)\) are the state of the signal estimation based on the HRKF 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 HRKF technique which was modeled by the MS technique, the unknown condition of the bearing's digital twin estimation that was computed using the HRKF technique which was modeled by the MS technique, the bearing's digital twin output estimation using the HRKF technique which was modeled by the MS technique, the surface of the variable structure method in the HRKF, the estimation compensator using the ANF method, and the coefficients for the state-space bearing's digital twin tuning using the HRKF method which was modeled by the MS technique, respectively.