# 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  
To modify the bearing signal estimation accuracy, the adaptive cascade observer is recommended. The cascade observer is developed using the fuzzy-structure PI observer and the fuzzy orthonormal regressive technique is used to approximate the normal signal. Based on Figure 1, the adaptive cascade observer follows four steps: (a) establish a linear observer using an PI observer and FOR normal signal approximation, (b) the structure fault observer is used to increase the resistance of the PI observer against uncertainties such as rotational speed variation or torque load variation, (c) the fuzzy logic algorithm is used to improve the accuracy of the structure PI observer and develop cascade observer, and (d) the adaptive algorithm is used for auto tuning and important parameter of cascade observer using fuzzy logic technique. The first step of cascade observer using PI observation technique and FOR normal signal approximation is represented using the Equations (6) and (7):  
\[\left\{ \begin{array}{c}\hat{X}_{\text{FOR-}PIO}(k+1)=\left[\sigma_{X}\hat{X}_ {\text{FOR-}PIO}(k)+\sigma_{Y}\hat{Y}_{\text{FOR-}PIO}(k)+\sigma_{U}U_{i}(k) \right]+\hat{\phi}_{\text{FOR-}PIO}(k)+\\ \sigma_{1}\times e_{\text{FOR-}PIO}\\ \hat{Y}_{\text{FOR-}PIO}(k)=\left(\beta\right)^{T}\hat{X}_{\text{FOR-}PIO}(k) \\ e_{\text{FOR-}PIO}=X_{\text{FOR}}(k)-\hat{X}_{\text{FOR-}PIO}(k)\\ e_{\text{r-FOR-}PIO}=Y_{o}(k)-\hat{Y}_{\text{FOR-}PIO}(k)\end{array} \right. \tag{6}\]  
To improve the uncertainties accuracy in PI observer with FOR normal signal approximation, the following technique is developed. \[\hat{\phi}_{\text{FOR-}PIO}(k+1)=\hat{\phi}_{\text{FOR-}PIO}(k)+\sigma_{2} \times e_{\text{r-FOR-}PIO}. \tag{7}\]  
Here, \(\hat{X}_{\text{FOR-}PIO}(k)\), \(\hat{Y}_{\text{FOR-}PIO}(k)\), \(\hat{\phi}_{\text{FOR-}PIO}(k)\), \(Y_{o}(k)\), and \((\sigma_{1},\sigma_{2})\) are the state estimation using the PI observer with FOR normal signal approximation, the estimation of the unknown signals using the PI observer with FOR normal signal approximation, unknown condition (the speed or torque variant) observation using the PI observer with FOR normal signal approximation, the original signal, and the coefficients, respectively. Notwithstanding, this technique is easy to implement, in unknown conditions the accuracy is reduced. Thus, the structure fault observer is established to increase the resistance of the PI observer against uncertainties such as rotational speed variation or torque load variation. Moreover, the structure PI observer with FOR normal signal approximation is defined as follows. \[\left\{ \begin{array}{c}\hat{X}_{\text{FOR-}PIO}(k+1)=\left[\sigma_{X}\hat{X }_{\text{FOR-}SPIO}(k)+\sigma_{Y}\hat{Y}_{\text{FOR-}SPIO}(k)+\sigma_{U}U_{i }(k)\right]+\hat{\phi}_{\text{FOR-}SPIO}(k)+\\ \sigma_{1}\times e_{\text{FOR-}SPIO}\\ \hat{Y}_{\text{FOR-}SPIO}(k)=\left(\beta\right)^{T}\hat{X}_{\text{FOR-}SPIO}( k)\\ e_{\text{FOR-}SPIO}=X_{\text{FOR}}(k)-\hat{X}_{\text{FOR-}SPIO}(k)\\ e_{\text{r-FOR-}SPIO}=Y_{o}(k)-\hat{Y}_{\text{FOR-}SPIO}(k)\end{array} \right. \tag{8}\]  
That, the \(\hat{\phi}_{\text{FOR-}SPIO}(k)\) is defined as follows. \[\hat{\phi}_{\text{FOR-}SPIO}(k+1)=\hat{\phi}_{\text{FOR-}SPIO}(k)+\sigma_{2} \times e_{\text{r-FOR-}SPIO}+\sigma_{3}.sgn\times e_{\text{r-FOR-}SPIO} \tag{9}\]  
Here, \(\hat{X}_{\text{FOR-}SPIO}(k)\), \(\hat{Y}_{\text{FOR-}SPIO}(k)\), \(\hat{\phi}_{\text{FOR-}SPIO}(k)\), and \(\sigma_{3}\) are the state estimation using the structure PI observer with FOR normal signal approximation, the estimation of the unknown signals using the structure PI observer with FOR normal signal approximation, unknown condition (the speed or torque variant) observation using the structure PI observer with FOR normal signal approximation, and the variable structure surface coefficient, respectively. Next, the fuzzy logic algorithm is used to improve the accuracy of the structure PI observer and develop cascade observer.