# Bearing Fault Identification Using Machine Learning and Adaptive Cascade Fault Observer
## 4 Adaptive Cascade Observer with Machine Learning for Fault Identification
### Fault Detection and Classification Using SVM  
In the previous section, the signal was estimated using the adaptive cascade observer. Additionally, the residual signals for the adaptive cascade observer with FOR normal signal approximation \((r_{FOR-r-ACO}(k))\) is defined by the following functions, respectively. \[r_{FOR-r-ACO}(k)=Y_{o}(k)-\hat{Y}_{FOR-ACO}(k). \tag{16}\]  
Figure 3 illustrates the residual signals for the REB vibration dataset using the adaptive cascade observer. Regarding this figure, it is clear that the adaptive cascade observation technique for signal estimation obtained differentiable signals for fault diagnosis. After calculating the residual signals, in the next step, the energy signals are extracted from the residual signals. These signals are used for classification algorithm using SVM. Therefore, the energy signals \(\left(\overline{E}\right)\) extracted from the residual signals is calculated as follows. \[\overline{E}=\sum_{i=1}^{K}r_{FOR-r-ACO-i}^{2}. \tag{17}\]  
Here, \(r_{FOR-r-ACO-i}\) is the _ith_ sample of the residual signal obtained by the adaptive cascade observer and \(K\) is the number of samples. The features of energy that are extracted from the residual signals for the CWRU dataset using adaptive cascade observer are shown in Figure 4. Since the adaptive cascade observer is tuned for working in the normal condition, the amplitude of the energy of the residual signal is reduced. However, in the abnormal condition, the accuracy of signal estimation is reduced. Therefore, the energy of residual signal in the abnormal condition is increased. Thus, after determining the energy of residual signals using Equations (16) and (17) as a high discriminative feature, the machine learning algorithm using support vector machine (SVM) is selected for identification the faults in the  
Figure 3: The residual signals for the CWRU dataset using the FOR (fuzzy orthonormal regressive)-adaptive cascade observation procedure. bearing. This technique is a mathematical-based algorithm and used for classification. The soft margin SVM is selected according the following definition [34; 50]. \[w_{i}\big{(}\varnothing^{T}\rho(q_{i})+c\big{)}\geq w_{i}-v_{i}. \tag{18}\]  
Here, \((q_{i},\ w_{i}),\ (\varnothing,c),\ \rho(q_{i})\), and \(v_{i}\) are the SVM inputs, the SVM outputs, features that are used in SVM, and maximum distance for boundary, respectively. The primal algorithm is used to solve Equation (18) and defined as follows. \[ \begin{array}{rr}min&\frac{1}{2}\ \varnothing^{T}\varnothing+\rho\sum\limits_{ i}v_{i}\\ s.t.&w_{i}\big{(}\varnothing^{T}\rho(q_{i})+c\big{)}\geq w_{i}-v_{i}&v_{i}\geq 0 \end{array}  \tag{19}\]  
Here, \(\rho\) is a penalty to tuning the boundary of SVM. To solve this equation, the minmax (saddle) point is defined based on the following equation. \[MM_{p}=\frac{1}{2}\varnothing^{T}\varnothing+\rho\sum\limits_{i}v_{i}-\sum \limits_{i}\partial_{i}\Big{[}w_{i}\big{(}\varnothing^{T}\rho(q_{i})+c\big{)}- w_{i}+v_{i}\Big{]}-\sum\limits_{i}\sigma_{i}v_{i} \tag{20}\]  
where \(MM_{p}\) and \((\partial_{i},\sigma_{i})\) are the minmax point and the saddle function coefficients, respectively. To maximize the saddle point, \((\partial_{i},\sigma_{i})\) and to minimize it \((\varnothing,\ v_{i},c)\) have an essential role. To solve the minmax challenge the quadratic programming is represented. Therefore, the \(\partial_{i}\) is represented as follows:  
\[ \begin{array}{rr}min&\frac{1}{2}\partial^{T}K\partial+H^{T}\partial\\ s.t.&\sum\limits_{i}\partial_{i}w_{i}=0\\ &0\leq\partial_{i}\leq\rho&\forall i\end{array}  \tag{21}\]  
Figure 4: The energy extracted from residual signals of the CWRU dataset using the Adaptive Cascade Observer. \[\text{Here},H=\left[ \begin{array}{c}-1\\ -1\\ \vdots\\ -1\end{array}