# Rolling-Element Bearing Fault Diagnosis Using Advanced Machine Learning-Based Observer
## 3 Rolling-Element-Bearing Modeling
\begin{array}{l}\varepsilon_{x}(i)=\sum\limits_{n=0}^{\infty}\omega_ {n,x}\;J_{n}^{x}(i,\gamma_{x})\\ \varepsilon_{y}(i)=\sum\limits_{n=0}^{\infty}\omega_{n,y}\;J_{n}^{y}\Big{(}i, \gamma_{y}\Big{)}\end{array} \right., \tag{6}\]  
Here, \(\Big{(}\omega_{n,x},\omega_{n,y}\Big{)},\Big{(}\gamma_{x},\gamma_{y}\Big{)}\), and \(\Big{(}J_{n}^{x},J_{n}^{y}\Big{)}\) are the coefficients of the Fourier decomposition, orthonormal basis, and orthonormal function, respectively. In the next step, ARX-Laguerre system modeling is used to obtain robust system modeling. The input and output orthonormal functions are represented in Equation (7). \[\left\{ \begin{array}{l}\mathcal{O}_{n,f_{o}}(k,\gamma_{x})=\sum\limits_{i=1} ^{\infty}\;J_{n}^{x}(i,\gamma_{x})f_{o}(k-i)\\ \mathcal{O}_{n,f_{i}}\Big{(}k,\gamma_{y}\Big{)}=\sum\limits_{i=1}^{\infty}\;J _{n}^{y}\Big{(}i,\gamma_{y}\Big{)}f_{i}(k-i)\end{array} \right., \tag{7}\]Here, \(O_{n,f_{o}}\) and \(O_{n,f_{i}}\) are the filtered orthonormal functions for the output and input, respectively. The ARX orthonormal function is represented as the following equation. \[f_{o}(k)=\sum\limits_{n=0}^{\delta_{x}-1}\omega_{n,x}\;O_{n,f_{o}}(k,\gamma_{x}) +\sum\limits_{n=0}^{\delta_{y}-1}\omega_{n,y}\;O_{n,f_{i}}\Big{(}k,\gamma_{y} \Big{)}, \tag{8}\]  
In addition, the Laguerre technique is defined as Equation (9). \[\left\{ \begin{array}{l}L_{n}^{x}=\frac{\sqrt{1-\gamma_{x}{}^{2}}}{Z-\gamma_{ x}}\left(\frac{1-\gamma_{x}{}^{2}}{Z-\gamma_{x}}\right)^{n}\\ L_{n}^{y}=\frac{\sqrt{1-\gamma_{y}{}^{2}}}{Z-\gamma_{y}}\left(\frac{1-\gamma_ {y}{}^{2}}{Z-\gamma_{y}}\right)^{n}\end{array} \right., \tag{9}\]  
Here, \(L_{n}^{x}\) and \(L_{n}^{y}\) are the input and output Laguerre functions, respectively. Based on Equations (7) and (9), the input and output orthonormal functions are modified to be the following equation. \[\left\{ \begin{array}{l}\widetilde{O}_{n,f_{o}}(k,\gamma_{x})=\sum\limits_{ j=1}^{\infty}\;L_{n}^{x}(k)*f_{o}(k)\\ \widetilde{O}_{n,f_{i}}\Big{(}k,\gamma_{y}\Big{)}=\sum\limits_{j=1}^{\infty} \;L_{n}^{y}(k)*f_{i}(k)\end{array} \right., \tag{10}\]  
Here, \(\widetilde{O}_{n,f_{o}}\) and \(\widetilde{O}_{n,f_{i}}\) are the input and output modified orthonormal functions, respectively. Thus, the ARX-Laguerre system modeling and estimation is represented as the following [20]. \[ \begin{array}{l}f_{o}(k)=\sum\limits_{n=0}^{i_{x}-1}\omega_{n,x}\; \widetilde{O}_{n,f_{o}}(k,\gamma_{x})W(k)+\sum\limits_{n=0}^{i_{y}-1}\omega_{ n,y}\;\widetilde{O}_{n,f_{i}}\Big{(}k,\gamma_{y}\Big{)}I(k)\\ f_{o}(k)=\Psi\Big{(}z^{-1}\Big{)}W(z)+\Lambda\Big{(}z^{-1}\Big{)}I(z)\end{array}  \tag{11}\]  
Here, \((\Psi\Big{(}z^{-1}\Big{)}\), \(\Lambda\Big{(}z^{-1}\Big{)})\) and \((W(k)\), \(I(k))\) are the polynomial variables and filtering signals, respectively and defined as the following equations. \[ \begin{array}{l}\Psi\Big{(}z^{-1}\Big{)}=\sum\limits_{0}^{n_{x}-1}\omega_{ n,x}\;\widetilde{O}_{n,f_{o}}(k)\\ \Lambda\Big{(}z^{-1}\Big{)}=\sum\limits_{0}^{n_{y}-1}\omega_{n,y}\;\widetilde{O }_{n,f_{i}}(k)\end{array} , \tag{12}\]  
Therefore, the state-space ARX-Laguerre technique is represented as Equation (13). \[\left\{ \begin{array}{l}J_{s}(k+1)=[\sigma_{s}J_{s}(k)+\sigma_{o}J_{o}(k)+ \sigma_{i}J_{i}(k)]+\varphi(k)\\ J_{o}(k+1)=(\lambda)^{T}J_{s}(k)\end{array} \right., \tag{13}\]  
Here, \(J_{s}(k),(\sigma_{s},\sigma_{o},\sigma_{i}),J_{o}(k),J_{i}(k),\varphi(k)\), and \((\lambda)^{T}\) are the system's state, modeling coefficients, measured output, input, uncertainties and fault, and Fourier coefficients, respectively. Here, the state system modeling coefficient \(\sigma_{s}\) is represented by the following equation:  
\[\sigma_{s}=\left[ \begin{array}{cc}\sigma_{so}&O_{N_{x}N_{y}}\\ O_{N_{y}N_{x}}&\sigma_{si}\end{array} \right], \tag{14}\]where \(O_{N_{x},N_{y}}\) and \(O_{N_{y},N_{x}}\) are null matrices and \(\sigma_{so}\) and \(\sigma_{si}\) are represented in Equations (15) and (16), respectively. \[\sigma_{s_{o}}=\left[