# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 4 Proposed Fault Detection and Diagnosis
### ARX-Laguerre System Modeling  
As shown in Figure 5, the first step to design a model-reference observer is system modeling. This step is important to design an accurate observer for fault diagnosis. Based on the proposed method, this system is modeled by the ARX-Laguerre technique. The proposed method is robust in the presence of motor speed variation. When the motor speed changes, the system's model is changed, and the proposed observer detects the model's change. The filter network ARX-Laguerre orthonormal technique for modeling an induction motor is presented based on the system input and output, Fourier coefficients, and Laguerre-based orthonormal function as follows [28]:  
\[ \begin{array}{l}\begin{array}{l}\mathcal{O}(k)=\sum\limits_{0}^{N_{R}-1}K_{ n,\alpha}(\sum\limits_{j=1}^{\infty}\frac{\sqrt{1-\zeta_{s}^{2}}}{Z-\zeta_{ \phi}}(\frac{1-\zeta_{\phi,\pi}}{Z-\zeta_{\phi}})^{n}*Q(k)).S_{n,O}(k)+\\ \sum\limits_{0}^{N_{R}-1}K_{n,\beta}(\sum\limits_{j=1}^{\infty}\frac{\sqrt{1- \zeta_{s}^{2}}}{Z-\zeta_{\phi}}(\frac{1-\zeta_{\phi,\pi}}{Z-\zeta_{\phi}})^{n}* I(k)).S_{n,I}(k)\end{array}\end{array}  \tag{4}\]  
where \(\mathcal{O}(k),(K_{n,\alpha}\&K_{n,b}),(N_{a},N_{b}),\left(\left(\frac{\sqrt{1 -\zeta_{s}^{2}}}{Z-\zeta_{\phi}}\frac{1-\zeta_{\phi,\pi}}{Z-\zeta_{\phi}} \right)\&\left(\frac{\sqrt{1-\zeta_{\phi}^{2}}}{Z-\zeta_{\phi}}\frac{1-\zeta_ {\phi,\pi}}{Z-\zeta_{\phi}}\right)\right),\left(\zeta_{a}\&\zeta_{\phi}),*,I(k ),S_{n,O}(k)\), and \(S_{n,I}(k)\) are the system output, coefficients of Fourier, order of system, functions of the Laguerre-based orthonormal, Laguerre poles, convolution product, system input, output signal  
Figure 5: Block diagram of the sliding mode extended-state ARX-Laguerre PI observer for fault diagnosis in an induction motor. filtered, and filtered input signal, respectively. To calculate the state-space equation, the following variables are defined in Equations (5) and (6). \[\left\{ \begin{array}{l}L_{n}^{\alpha}=\frac{\sqrt{1-\zeta_{\alpha} ^{2}}}{Z-\zeta_{\alpha}}(\frac{1-\zeta_{\alpha},z}{Z-\zeta_{\alpha}})^{n}\\ L_{n}^{\beta}=\frac{\sqrt{1-\zeta_{\beta}^{2}}}{Z-\zeta_{\beta}}(\frac{1- \zeta_{\beta},z}{Z-\zeta_{\beta}})^{n}\\ \end{array} \right. \tag{5}\]  
\[\left\{ \begin{array}{l}\chi_{n,\rho}(k,\zeta_{\alpha})=\sum L_{n}^{\alpha}* O(k)\\ \chi_{n,I}\big{(}k,\zeta_{\beta}\big{)}=\sum L_{n}^{\beta}*I(k)\\ \end{array} \right. \tag{6}\]  
Based on Equations (4)-(6) the ARX-Laguerre technique is represented as  
\[ \begin{array}{l}O(k)=\sum\limits_{0}^{N_{\alpha}-1}K_{n,\alpha}\chi_{n,O}(k )S_{n,O}(k)+\sum\limits_{0}^{N_{\beta}-1}K_{n,\beta}\chi_{n,I}(k)S_{n,I}(k)=\\ A\big{(}z^{-1}\big{)}S_{n,O}(z)+B\big{(}z^{-1}\big{)}S_{n,I}(z)\\ \end{array}  \tag{7}\]  
Here, \(A\big{(}z^{-1}\big{)}\) and \(B\big{(}z^{-1}\big{)}\) are two polynomials with degrees \(N_{\alpha}\) and \(N_{\beta}\), respectively. Based on Equation (7) the transfer function is represented as follows:  
\[H(z)=\frac{S_{n,O}}{S_{n,I}}=\frac{B\big{(}z^{-1}\big{)}}{1-A(z^{-1})} \tag{8}\]  
The identification between Equations (6) and (7) provides (9). \[A\big{(}z^{-1}\big{)}=\sum\limits_{0}^{N_{\alpha}-1}K_{n,\alpha}\chi_{n,O}(k),\quad B\big{(}z^{-1}\big{)}=\sum\limits_{0}^{N_{\beta}-1}K_{n,\beta}\chi_{n,I}(k) \tag{9}\]  
Based on Equations (5), (6), (7), and (9) the optimization problem is defined as follows [28]:  
\[\underset{\zeta_{\alpha}}{min}\|A\big{(}z^{-1}\big{)}-\sum\limits_{0}^{N_{ \alpha}-1}K_{n,\alpha}\chi_{n,O}(k)\|^{2},\quad\underset{\zeta_{\beta}}{min}\| B\big{(}z^{-1}\big{)}-\sum\limits_{0}^{N_{\beta}-1}K_{n,\beta}\chi_{n,I}(k)\|^{2} \tag{10}\]  
From the relations Equations (5), (6), and (9), the minimization technique Equation (10) is used to decompose the polynomials \(A\big{(}z^{-1}\big{)}\) and \(B\big{(}z^{-1}\big{)}\) on the Laguerre orthonormal bases, and the \(\zeta_{\alpha}\) and \(\zeta_{\beta}\) are optimized. A block diagram of the ARX-Laguerre technique is illustrated in Figure 6. The ARX-Laguerre state-space equation is written as follows:  
\[\left\{