# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 4 Proposed Fault Detection and Diagnosis
### Fault Identification  
The main part of fault identification is fault estimation. The residual bank is evaluated using the fault estimation. The two blocks for fault estimation include (a) the PI observer block and (b) the observer evaluate block. In the first step, the ARX-Laguerre PI observer is designed, (17). In this step, a fault is estimated by an integral linear function, (18). To evaluate the accuracy of fault estimation, the extended-state ARX-Laguerre PI observer is considered, (21). After accurate fault estimation, the next step is fault identification. As shown in Figure 5, for fault identification, three important blocks are: (a) the threshold process block, (b) the residual generator, and (c) residual bank and logic decision. The main idea of the threshold process is generated in (22). Regarding (20) and (21), seven different faulty conditions have been defined in this research, i.e., ball, inner, outer, inner-ball, outer-ball, inner-outer, and inner-outer-ball faults. Based on Equation (22) to detect the level of faults, the following equations are represented [30]:  
\[ \begin{array}{l}\Delta_{b}=K_{b}sgn(s_{b})\\ s_{b}=\lambda_{b}e_{b}+\dot{e}_{b}.\end{array}  \tag{25}\]  
\[ \begin{array}{l}\Delta_{i}=K_{b}sgn(s_{i})\\ s_{i}=\lambda_{i}e_{i}+\dot{e}_{i}.\end{array}  \tag{26}\]  
\[ \begin{array}{l}\Delta_{o}=K_{o}sgn(s_{o})\\ s_{o}=\lambda_{o}e_{o}+\dot{e}_{o}.\end{array}  \tag{27}\]\[\Delta_{ib} =K_{ib}sgn(s_{ib}) \tag{28}\] \[s_{ib} =\lambda_{ib}e_{b}+\dot{e}_{ib}\] (29) \[\Delta_{ob} =K_{ob}sgn(s_{ob})\] (30) \[s_{ob} =\lambda_{ob}e_{ob}+\dot{e}_{ob}.\] (31) \[\Delta_{io} =K_{io}sgn(s_{io})\] \[s_{io} =\lambda_{io}e_{io}+\dot{e}_{io}.\] (32) \[\Delta_{iob} =K_{iob}sgn(s_{io})\] (33) \[s_{iob} =\lambda_{iob}e_{ib}+\dot{e}_{iob}\]  
Here, \(\Delta_{o},\Delta_{i},\Delta_{b},\Delta_{ob},\Delta_{ib},\Delta_{oi},\Delta_{ oib},(K_{i},K_{o},K_{b},K_{ib},K_{ob},K_{io},K_{iob})\) and \((\lambda_{i},\lambda_{o},\lambda_{b},\lambda_{ib},\lambda_{ob},\lambda_{io}, \lambda_{iob})\) are defined as the outer fault threshold, inner fault threshold, ball fault threshold, outer-ball fault threshold, inner-ball fault threshold, outer-inner fault threshold, outer-inner-ball fault threshold, coefficients, and surface slope coefficients, respectively. The last step for fault diagnosis is logic decision, as shown in Figure 5. This block is used for fault identification and isolation. This block has two inputs: (a) residual generation and (b) threshold process. The residual signals and threshold levels are calculated by (21) and (22), respectively. The following equation is used for the condition decision. \[ \begin{array}{l}if:r>\Gamma,r<\Delta_{b},r<\Delta_{i},r<\Delta_{o},r<\Delta_{ ib},r<\Delta_{ob},r<\Delta_{io},r<\Delta_{iob}\to r=r_{b}\\ if:r>\Gamma,r>\Delta_{b},r<\Delta_{i},r<\Delta_{o},r<\Delta_{ib},r<\Delta_{ob},r<\Delta_{io},r<\Delta_{iob}\to r=r_{i}\\ if:r>\Gamma,r>\Delta_{b},r>\Delta_{i},r<\Delta_{o},r<\Delta_{ib},r<\Delta_{ ob},r<\Delta_{io},r<\Delta_{iob}\to r=r_{o}\\ if:r>\Gamma,r>\Delta_{b},r>\Delta_{i},r>\Delta_{o},r<\Delta_{ib},r<\Delta_{ ob},r<\Delta_{io},r<\Delta_{iob}\to r=r_{ib}\\ if:r>\Gamma,r>\Delta_{b},r>\Delta_{i},r>\Delta_{o},r>\Delta_{ib},r<\Delta_{ ob},r<\Delta_{io},r<\Delta_{iob}\to r=r_{ob}\\ if:r>\Gamma,r>\Delta_{b},r>\Delta_{i},r>\Delta_{o},r>\Delta_{ib},r>\Delta_{ ob},r<\Delta_{iob}\to r=r_{io}\\ if:r>\Gamma,r>\Delta_{b},r>\Delta_{i},r>\Delta_{o},r>\Delta_{ib},r>\Delta_{ ob},r<\Delta_{iob}\to r=r_{iob}.\end{array}  \tag{34}\]  
Here, \(r_{b},r_{i},r_{o},r_{ib},r_{ob},r_{io}\) and \(r_{iob}\) are the ball fault residual signal, inner residual, outer residual, inner-ball residual, inner-outer fault residual, and inner-outer-ball residual, respectively. Based on Equation (34), faults are isolated whenever the residuals \((r_{b},r_{i},r_{o},r_{ib},r_{ob},r_{io},r_{iob})\) overshoot their corresponding thresholds \((\Gamma,\Delta_{o},\Delta_{i},\Delta_{b},\Delta_{ob},\Delta_{ib},\Delta_{oi})\).