# Bearing Fault Identification Using Machine Learning and Adaptive Cascade Fault Observer
## 4 Adaptive Cascade Observer with Machine Learning for Fault Identification
### Adaptive Cascade Observer for Signal Estimation
Therefore, to modify the efficiency in the structure PI observer with FOR normal signal approximation, the fuzzy logic approach is developed here. First, the T-S fuzzy logic approach is defined as follows. \[\textit{if }e_{FOR-fSPIO}(k)\textit{ is Th then }\hat{\varphi}_{f}(k+1)=\hat{ \varphi}_{f}(k)+\sigma_{f}\times e_{FOR-fSPIO}. \tag{10}\]  
Here, \(e_{FOR-fSPIO}(k)\), \(\textit{Th},\hat{\varphi}_{f}(k)\), and \(\sigma_{f}\) are the error of estimation based on the cascade observer with FOR normal signal approximation, the threshold level for tuning, the T-S fuzzy estimation function, and fuzzy logic tuning parameter, respectively. Based on Equation (10), the cascade observer with FOR normal signal approximation is developed by Equations (11) and (12). \[\left\{ \begin{array}{c}\hat{X}_{\text{FOR-}fSPIO}(k+1)=\left[ \sigma_{X}\hat{X}_{\text{FOR-}CO}(k)+\sigma_{Y}\hat{Y}_{\text{FOR-}CO}(k)+ \sigma_{U}U_{i}(k)\right]+\hat{\varphi}_{\text{FOR-}CO}(k)+\\ \sigma_{1}\times e_{FOR-CO}\\ \hat{Y}_{\text{FOR-}CO}(k+1)=(\beta)^{T}\hat{X}_{\text{FOR-}CO}(k)\\ e_{FOR-CO}=X_{A-L}(k)-\hat{X}_{\text{FOR-}CO}(k)\\ e_{r-FOR-CO}=Y_{o}(k)-\hat{Y}_{\text{FOR-}CO}(k)\\ \end{array} \right.. \tag{11}\]  
That, the \(\hat{\varphi}_{\text{FOR-}CO}(k)\) is defined as follows. \[\hat{\varphi}_{\text{FOR-}CO}(k+1)=\hat{\varphi}_{\text{FOR-}CO}(k)+\sigma_{2 }\times e_{r-FOR-CO}+\sigma_{3}\textit{sgn}\times e_{\text{FOR-}r-CO}+\sigma_ {4}\times\hat{\varphi}_{f}(k). \tag{12}\]  
Here, \(\hat{X}_{\text{FOR-}CO}(k)\), \(\hat{Y}_{\text{FOR-}CO}(k)\), \(\hat{\varphi}_{\text{FOR-}CO}(k)\), and \(\sigma_{4}\) are state estimation using the fuzzy logic structure PI (cascade) observer with FOR normal signal approximation, the estimation of the unknown signals using the cascade observer with FOR normal signal approximation, unknown condition (motor speed variant) estimation using the cascade observer with FOR normal signal approximation, and the constant for tuning the fuzzy parameters of function, respectively. To increase the reliability and accuracy of bearing signal estimation in normal condition, the adaptive (online tuning) technique is developed to auto-tune the important coefficient. For auto-tuning the coefficient of robust function, the fuzzy logic algorithm is recommended. This definition is provided below. \[\sigma_{3-\textit{adaptive}}=\sigma_{3}\times\theta. \tag{13}\]  
Therefore, the improved cascade observer using adaptive technique is expressed as follows:  
\[\left\{ \begin{array}{c}\hat{X}_{\text{FOR-}ACO}(k+1)=\left[ \sigma_{X}\hat{X}_{\text{FOR-}ACO}(k)+\sigma_{Y}\hat{Y}_{\text{FOR-}ACO}(k)+ \sigma_{U}U_{i}(k)\right]+\hat{\varphi}_{\text{FOR-}ACO}(k)+\\ +\sigma_{1}\times e_{\text{FOR-}ACO}\\ \hat{Y}_{\text{FOR-}ACO}(k+1)=(\beta)^{T}\hat{X}_{\text{FOR-}ACO}(k)\\ \varepsilon_{\text{FOR-}ACO}=X_{A-L}(k)-\hat{X}_{\text{FOR-}ACO}(k)\\ e_{r-FOR-ACO}=Y_{o}(k)-\hat{Y}_{\text{FOR-}ACO}(k)\\ \end{array} \right., \tag{14}\]  
\[ \begin{array}{c}\hat{\varphi}_{\text{FOR-}ACO}(k+1)=\hat{\varphi}_{\text{ FOR-}ACO}(k)+\sigma_{2}\times e_{r-\text{FOR-}ACO}+\\ \sigma_{3-\textit{adaptive}}\textit{sgn}\times e_{\text{FOR-}r-ACO}+\sigma_{4} \times\hat{\varphi}_{f}(k).\\ \end{array}  \tag{15}\]  
Here, \(\hat{X}_{\text{FOR-}ACO}(k)\), \(\hat{Y}_{\text{FOR-}ACO}(k)\), \(\hat{\varphi}_{\text{FOR-}ACO}(k)\), \(\theta\), and \(\sigma_{3-\textit{adaptive}}\) state estimation using the adaptive cascade observer with FOR normal signal approximation, the estimation of the unknown signals using the adaptive cascade observer with FOR normal signal approximation, unknown condition (motor speed variant) estimation using the adaptive cascade observer with FOR normal signal approximation, the adaptive auto-tuning parameter using the fuzzy algorithm, and the auto-tuned coefficient using the adaptive technique, respectively. Therefore, due to Figure 1 and Equations (14) and (15), the adaptive cascade observer is developed to estimate the unknown vibration and AE signals of a bearing. The stability and robustness for proposed algorithm are proof in the Appendix A.