# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 4 Proposed Fault Detection and Diagnosis
### Extended-State ARX-Laguerre PI Observer  
The proposed methodology comprises of six major parts, as shown in Figure 5: (a) system modeling using the ARX-Laguerre technique, (b) proportional-integral (PI) observer, (c) observer evaluator, (d) residual generator, (e) threshold process, and (f) residual bank and logic decision. Figure 8: ARX-Laguerre orthonormal method for system identification/estimation in the faulty condition: (**a**) desired and estimated torque and (**b**) error. Figure 7: ARX-Laguerre orthonormal method for system identification/estimation in the normal condition: (**a**) desired and estimation torque and (**b**) error. The classical ARX-Laguerre PI observer offers a linear approach to find an optimized estimation of the system and fault. This technique is stable, but it has an issue regarding accurate fault estimation and robustness. To evaluate the ARX-Laguerre PI observer, a sliding mode extended-state observer is considered. This technique improves the robustness and precision of fault estimation. Using ARX-Laguerre state space system modeling, the ARX-Laguerre PI observer method is defined to estimate the system modeling output and fault. Therefore, the block of the PI observer is defined as the following equation. \[\left\{ \begin{array}{l}\hat{X}(k+1)=A\hat{X}(k)+b_{y}\hat{Y}(k)+b_{u}u(k)+F_ {d}(k)+\hat{\delta}(k)+K_{p}\big{(}Y(k)-\hat{Y}(k)\big{)}\\ \hat{Y}(k+1)=(K)^{T}\hat{X}(k+1)\end{array} \right. \tag{17}\]  
where \(\hat{X}(k),\hat{Y}(k),\hat{\delta}(k)\) and \(K_{p}\) are the state estimated, output estimated, faults estimated, and the proportional coefficient, respectively. Based on bearing conditions, the fault estimation can be estimated in seven different conditions such as inner fault \(\hat{\delta}_{i}(k)\), outer fault \(\hat{\delta}_{o}(k)\), ball fault \(\hat{\delta}_{b}(k)\), inner-outer fault \(\hat{\delta}_{io}(k)\), inner-ball fault \(\hat{\delta}_{ib}(k)\), outer-ball fault \(\hat{\delta}_{ob}(k)\), and inner-outer-ball fault \(\hat{\delta}_{ib}(k)\). The future state of PIO is a function of the current state, current output, current input, faults, disturbance, and current error of the signal. For fault estimation using the ARX-Laguerre PIO technique, the integral term is used to reduce the fault estimation error. The integral term of fault estimation in the ARX-Laguerre PI observer is given in Equation (12). \[\hat{\delta}(k+1)=\hat{\delta}(k)+K_{I}\big{(}Y(k)-\hat{Y}(k)\big{)} \tag{18}\]  
Here, \(K_{I}\) is the integral term coefficient. Using Equation (18), the inner, outer, ball, inner-outer, inner-ball, outer-ball, and inner-outer-ball faults are estimated based on the integral term. Using this technique, the future faults are estimated by the current fault and error of the signal. This technique has been used in several applications [28], but it has a drawback in the presence of uncertain and noisy conditions. To address this issue, an observer evaluator is considered, as shown in Figure 5. The extended-state ARX-Laguerre PI observer is designed to improve the performance of faults estimation. Therefore, the observer evaluator block, as shown in Figure 5, is defined as the following two Equations, (13) and (14):  
\[\left\{ \begin{array}{l}\hat{X}(k+1)=A\hat{X}(k)+b_{y}\hat{Y}(k)+b_{u}u(k)+F_ {d}(k)+\hat{\delta}(k)+K_{p}\big{(}Y(k)-\hat{Y}(k)\big{)}\\ \hat{Y}(k)=(K)^{T}\hat{X}(k)\end{array} \right. \tag{19}\]  
\[\hat{\delta}(k+1)=2\times\big{(}\hat{\delta}(k)+K_{I_{a}}\big{(}Y(k)-\hat{Y}(k )\big{)}\big{)}+2\times\big{(}\hat{\delta}(k)+K_{I_{b}}\big{(}Y(k)-\hat{Y}(k) \big{)}\big{)}^{2} \tag{20}\]  
Here, \(K_{I_{a}}\) and \(K_{I_{b}}\) are the extended-state coefficients. Based on bearing conditions, the fault estimation can be estimated in seven different conditions such as inner fault \(\hat{\delta}_{i}(k)\), outer fault \(\hat{\delta}_{o}(k)\), ball fault \(\hat{\delta}_{b}(k)\), inner-outer fault \(\hat{\delta}_{io}(k)\), inner-ball fault \(\hat{\delta}_{ib}(k)\), outer-ball fault \(\hat{\delta}_{ob}(k)\), and inner-outer-ball fault \(\hat{\delta}_{ibb}(k)\).