# Rolling-Element Bearing Fault Diagnosis Using Advanced Machine Learning-Based Observer
## 3 Rolling-Element-Bearing Modeling
\begin{array}{ccc}\gamma_{x}&\cdots&0\\ \vdots&\ddots&\vdots\\ (-\gamma_{x})^{N_{x}-1}\big{(}1-{\gamma_{x}}^{2}\big{)}&\cdots&\gamma_{x}\\ \end{array} \right], \tag{15}\]  
\[\sigma_{s_{i}}=\left[ \begin{array}{ccc}\gamma_{y}&\cdots&0\\ \vdots&\ddots&\vdots\\ (-\gamma_{y})^{N_{y}-1}\big{(}1-{\gamma_{y}}^{2}\big{)}&\cdots&\gamma_{y}\\ \end{array} \right], \tag{16}\]  
The output system modeling coefficient \(\sigma_{o}\) is represented in Equation (17). \[\sigma_{o}=\sqrt{1-\gamma_{x}}^{2}\left[ \begin{array}{c}1\\ -\gamma_{x}\\ -\gamma_{x}^{2}\\ \vdots\\ (-\gamma_{x})^{N_{x}-1}\\ \end{array} \right], \tag{17}\]  
In addition, the input system modeling coefficient \(\sigma_{i}\) can be represented in Equation (18). \[\sigma_{i}=\sqrt{1-\gamma_{y}}^{2}\left[ \begin{array}{c}1\\ -\gamma_{y}\\ -\gamma_{y}^{2}\\ \vdots\\ (-\gamma_{y})^{N_{y}-1}\\ \end{array} \right] \tag{18}\]  
To improve the ARX-Laguerre system modeling accuracy, the fuzzy technique is recommended in this research. The fuzzy ARX-Laguerre (FAL) system modeling is defined as follows. \[\left\{ \begin{array}{c}J_{s}(k+1)=\begin{bmatrix}\sigma_{s}J_{s}(k)+\sigma_{o }J_{o}(k)+\sigma_{i}J_{i}(k)+\sigma_{f}J_{f}(k)\end{bmatrix}+\varphi(k)\\ J_{o}(k+1)=(\lambda)^{T}J_{s}(k)\end{array} \right., \tag{19}\]  
Here, \(\sigma_{f}\) and \(J_{f}\) are the fuzzy coefficient and fuzzy function for system estimation, respectively. The fuzzy if-then rule in this research is defined by the following rule. \[ \begin{array}{c}\textit{If }\langle\textit{ first input }\rangle\textit{ is }\langle\textit{ linguistic variable for first input }\rangle\textit{ and }\langle\textit{ second input }\rangle\textit{ is }\\ \langle\textit{ linguistic variable for second input }\rangle\textit{ then }\langle\textit{ output }\rangle\textit{ is }\langle\textit{ linguistic variable for output }\rangle\end{array}  \tag{20}\]  
The membership functions of a fuzzy set for the system estimation of error \((e)\) in the interval of \(\left[ \begin{array}{c}-0.3,\quad 0.3\end{array} \right]\) are the Gaussian and the linguistic variables, which are defined by negative high (NH), negative medium (NM), negative low (NL), zero (Z), positive low (PL), positive medium (PM), and positive high (PH). The fuzzy membership functions for the system estimation of the change of error \((\dot{e})\) in the interval of \(\left[ \begin{array}{c}-0.1,\quad 0.1\end{array} \right]\) are the Gaussian and the linguistic variables, which are defined by NH, NM, NL, Z, PL, PM, and PH. In addition, the fuzzy linguistic variables for \(J_{f}\) in the interval of \(\left[ \begin{array}{c}-0.08,\quad 0.08\end{array} \right]\) are the Gaussian and the fuzzy sets, which are defined by NH, NM, NL, Z, PL, PM, and PH. Table 2 illustrates the fuzzy rule table for the FAL system estimation technique. According to this table, the error has seven linguistic variables, the change of error has seven linguistic variables and the fuzzy system estimation has seven linguistic variables. Therefore, the fuzzy technique to improve system modeling has 49 rule-bases. Based on these 49 rule-bases, the fuzzy technique has improved the accuracy of system modeling to achieve the minimum estimation error. Figures 3 and 4 show the estimation accuracy and errors of REB modeling for the normal and faulty conditions based on the proposed FAL technique, ARX-Laguerre system modeling technique, and ARX system estimation method. Based on these figures, the system modeling accuracy in the proposed fuzzy ARX-Laguerre system modeling is higher than the ARX-Laguerre technique and ARX system modeling techniques. Regarding Figure 3, the error rate in the proposed fuzzy ARX-Laguerre technique is close to zero.