# Bearing Anomaly Recognition Using an Intelligent Digital Twin Integrated with Machine Learning
## 2 Proposed Methodology
### Digital Twin
#### 2.2.1 Signal Modeling: The First Step to Design the Digital Twin
Thus, regarding (1), the state-space function for modeling the bearing in the healthy state based on the mathematical technique using 5-DOF vibration signal modeling is represented by the following definition. \[\left\{ \begin{array}{l}W_{MAT}(k+1)=\gamma_{MAT}(W_{MAT}(k),U_{in}(k))+ \Delta(W_{MAT}(k),U_{in}(k))\\ \\ Z_{MAT}(k)=(\beta_{MAT})^{T}W_{MAT}(k)\end{array} \right. \tag{10}\]  
Here, \(W_{MAT}(k),U_{in}(k),\gamma_{MAT}(W_{MAT}(k),U_{in}(k)),Z_{MAT}(k),\beta_{MAT}\), and \(\Delta(W_{MAT}(k),\)\(U_{in}(k))\) are the state of the bearing signal, the input signal that should be measured, the nonlinearity term for the bearing modeling using the 5-DOF vibration signal, the output of the signal's model using 5-DOF vibration signal modeling, the coefficient to optimize the state-space output, and finally the unknown conditions, respectively. Table 2 illustrates the bearing parameter information to the mathematical modeling of the bearing that was collected by CWRUBD. This procedure is reliable, but the complexity and uncertain condition modeling are the two main problems of the mathematical system modeling to design the digital twin. In the next step, the data-driven system identification technique is suggested to modify the accuracy of system modeling for the bearing digital twin. Thus, regarding Figure 1, first, the autoregressive technique is recommended to approximate the state-space function of the vibration signal of the bearing in the normal state. The autoregressive (AUT) technique is a linear-based signal approximation for modeling the signal in different domains such as state-space. Thus, the state-space bearing signal modeling in a healthy condition is represented as follows [42]. \[\left\{ \begin{array}{l}W_{AUT}(k+1)=\left[\gamma_{AUT}W_{AUT}(k)+\Delta_{ AUT}(k)\right]+e_{AUT}(k)\\ \\ Z_{AUT}(k)=\left(\beta_{AUT}\right)^{T}W_{AUT}(k)\\ \end{array} \right. \tag{11}\]  
Here, \(W_{AUT}(k),e_{AUT}(k),\Delta_{AUT}(k),\ Z_{AUT}(k)\), and \((\gamma_{AUT},\beta_{AUT})\) are the state of the signal modeling based on the AUT technique for the bearing's digital twin, the error of the bearing's digital twin modeling using the AUT technique, the unknown condition of the bearing's digital twin modeling that is computed using the AUT technique, the bearing's digital twin output modeling using the AUT technique, and the coefficients for the state-space bearing's digital twin tuning using the AUT method, respectively. Moreover, the unknown condition and the error of the bearing's digital twin modeling that are computed using the AUT technique are represented as follows. \[\left\{ \begin{array}{l}\Delta_{AUT}(k)=Z(k)-Z_{AUT}(k)\\ e_{AUT}(k)=Z_{AUT}(k+1)-Z_{AUT}(k)\\ \end{array} \right. \tag{12}\]  
Here, \(Z(k)\) is the bearing RAW signal in the healthy state. The AUT technique is linear. To increase the robustness and reduce the effect of uncertain conditions, which was a challenge in mathematical modeling, and the AUT method of the digital twin, the robust autoregressive-Laguerre (AUT-LAG) technique is suggested. The state-space definition of the AUT-LAG method for the bearing's digital twin modeling is represented as the following equation. \[\left\{ \begin{array}{l}W_{AUT-LAG}(k+1)=\left[\gamma_{AUT-LAG_{1}}W_{AUT- LAG}(k)+\gamma_{AUT-LAG_{2}}Z_{AUT-LAG}(k)+\Delta_{AUT-LAG}(k)\right]\\ \\ +e_{AUT-LAG}(k)\\ \\ Z_{AUT-LAG}(k)=(\beta_{AUT-LAG})^{T}W_{AUT-LAG}(k)\\ \end{array} \right. \tag{13}\]