# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 4 Proposed Fault Detection and Diagnosis
### ARX-Laguerre System Modeling
\begin{array}{l}X(k+1)=\big{[}AX(k)+b_{y}y(k)+b_{u}u(k) \big{]}+F_{d}(k)+\delta(k)\\ Y(k+1)=(K)^{T}X(k+1)\end{array} \right. \tag{11}\]  
where \(X(k),\big{(}A,b_{y},b_{u}\big{)},Y(k),u(k),F_{d}(k),\delta(k)\) and \((K)^{T}\) are the system state, coefficient matrices, measured output, control input, uncertainty and disturbance, faults, and the Fourier coefficient, respectively. Based on Figure 3, seven types of faults are defined as inner fault \(\delta_{i}(k)\), outer fault \(\delta_{o}(k)\), ball fault \(\delta_{b}(k)\), inner-outer fault \(\delta_{io}(k)\), inner-ball fault \(\delta_{ib}(k)\), outer-ball fault \(\delta_{ob}(k)\), and inner-outer-ball fault \(\delta_{iob}(k)\). \(A,b_{y},b_{u}\), and \(K\) are presented in Equations (12), (13), (14), (15) and (16), respectively. \[A=\left[ \begin{array}{cc}A_{o}&O_{N_{\alpha},N_{b}}\\ O_{N_{b},N_{a}}&A_{i}\end{array} \right], \tag{12}\]Here, \(A_{0}\) and \(A_{i}\) are defined in Equations (13) and (14), respectively  
\[A_{i}=\left[ \begin{array}{cccc}\zeta_{\delta}&0&...&0\\ 1-\zeta_{\delta}{}^{2}&\zeta_{\delta}&...&0\\ -\zeta_{\delta}(1-\zeta_{\delta}{}^{2})&1-\zeta_{\delta}{}^{2}&...&0\\...&...&...&0\\...&...&...&0\\ (-\zeta_{\delta})^{N_{\delta}-1}(1-\zeta_{\delta}{}^{2})&...&...&\zeta_{ \delta}\end{array} \right]\text{ and } \tag{13}\]  
\[A_{i}=\left[ \begin{array}{cccc}\zeta_{\beta}&0&...&0\\ 1-\zeta_{\beta}{}^{2}&\zeta_{\delta}&...&0\\ -\zeta_{\beta}(1-\zeta_{\beta}{}^{2})&1-\zeta_{\delta}{}^{2}&...&0\\...&...&...&0\\...&...&...&0\\ (-\zeta_{\beta})^{N_{\delta}-1}(1-\zeta_{\beta}{}^{2})&...&...&\zeta_{\beta} \end{array} \right] \tag{14}\]  
where \(O_{N_{u},N_{b}}\) and \(O_{N_{b},N_{a}}\) are null matrices of the dimensions \(N_{a}\times N_{b}\) and \(N_{b}\times N_{a}\). \(b_{y}\) and \(b_{u}\) are defined as follows:  
\[(b_{y}=\sqrt{1-\zeta_{\delta}^{2}}\left[ \begin{array}{c}1\\ -\zeta_{\delta}{}^{2}\\..\\..\\..\\ (-\zeta_{\delta})^{N_{\delta}-1}\end{array} \right])\text{, and }(b_{u}=\sqrt{1-\zeta_{\beta}^{2}} \left[ \begin{array}{c}1\\ -\zeta_{\beta}\\ (-\zeta_{\beta})^{2}\\..\\..\\ (-\zeta_{\beta})^{N_{\delta}-1}\end{array} \right]) \tag{15}\]  
Here, \(a_{1}=\frac{\sqrt{1-\zeta_{\delta}}}{Z-\zeta_{\delta}}\), \(a_{2}=\frac{1-\zeta_{\delta}z}{Z-\zeta_{\delta}}\), \(a_{n}=\frac{1-\zeta_{\delta}z}{Z-\zeta_{\delta}}\), \(b_{1}=\frac{\sqrt{1-\zeta_{\delta}}}{Z-\zeta_{\delta}}\), \(b_{2}=\frac{1-\zeta_{\delta}z}{Z-\zeta_{\delta}}\), \(b_{2}=\frac{1-\zeta_{\delta}z}{Z-\zeta_{\delta}}\), and \(K=\left[C_{0,A},...,C_{N_{a}-1,A},C_{0,\beta},...,C_{N_{\beta}-1,\beta}\right]\). \(\left(K\right)^{T}\) is optimized via a gradient descent (GD) optimization method using the following equations:  
\[(\hat{K})=(\hat{K})-\left[P(k-1)-\frac{P(k-1)X(k)X^{T}(k)P(k-1)}{1+X^{T}(k)P(k- 1)X(k)}\right][Y(k)-(\hat{K})^{T}(k-1)X(k)] \tag{16}\]  
Figure 6: ARX-Laguerre Orthonormal Filter Technique. Here, \([P(k-1)-\frac{P(k-1)X(k)X^{T}(k)P(k-1)}{1+X^{T}(k)P(k-1)X(k)}]\) is the covariance matrix and \([Y(k)-\big{(}\hat{K}\big{)}^{T}(k-1)X(k)\big{]}\) is the error prediction. Figures 7 and 8 illustrate the estimation accuracy for the healthy and faulty conditions using the ARX-Laguerre method. For the healthy state in Figure 7, the sensitivity of the torque estimation is exceptionally high, and the rate of error is close to zero. In the faulty condition, the error rate increases, which causes an increase in the residual signal which is generated by the sliding mode technique. Figure 8 illustrates the power of the estimated signal and error rate in the faulty condition.