# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 5 Results and Analysis  
The effectiveness of the proposed extended-state ARX-Laguerre PI observer for the identification of various conditions (e.g., normal, inner, outer, ball, inner-ball, outer-ball, inner-outer, and inner-outer-ball) is compared to the ARX-Laguerre PI observer. The raw signal for different faulty conditions is illustrated in Figure 10. Regarding (32), the absolute of threshold values for dataset 4 for normal, ball, inner, outer, inner-ball, outer-ball, and inner-outer conditions are 0.5, 1, 2, 3, 4, 6, and 8, respectively. The absolute of residual signal for dataset 4 for normal, ball, inner, outer, inner-ball, outer-ball, inner-outer, and inner-outer-ball conditions are \(|r_{n}|<0.5,0.5<|r_{b}|<1,1<|r_{i}|<2,2<|r_{o}|<3\), \(3<|r_{ib}|<4,4<|r_{ob}|<6,6<|r_{oi}|\langle 8\), and \(|r_{iob}|\rangle 8\), respectively. Figure 11 illustrates the accuracy of the fault detection using dataset 4. As shown in Figure 11, the faulty signal can be detected easily. Figure 9: Block diagram of the proposed method for fault detection and diagnosis. Figures 12-19 illustrate fault diagnosis for various conditions. Figure 12 illustrates the normal signal identification for dataset 4. Figure 13 shows ball fault identification for dataset 4. In this figure, the ball residual signal is lower than the ball threshold value. Figures 14-19 illustrate the fault diagnosis for outer, inner-ball, outer-ball, inner-outer, and inner-outer-ball, respectively. As shown in Figure 14, in some samples, the inner residual signal level is higher than the ball threshold value, which can be used for fault diagnosis and fault isolation. In the experiment, we have 120,000 samples for each condition for all datasets. Dataset 4 is used as a training dataset to adjust the threshold. The other datasets (dataset 1, dataset 2, and dataset 3) are used as a test. Figure 12 shows normal condition, and level of the residual signal is less than the normal threshold value. It means that in this state, the proposed method accurately detects the normal condition. For the faulty ball condition, the level of a residual signal is more prominent than the normal threshold and lower than ball fault threshold. Therefore, for the faulty ball condition, the residual signal should be between normal threshold and  
Figure 11: Residual signal for the normal condition, faulty condition, and the normal threshold value for fault detection. Figure 10: Original raw signals for various faulty conditions. Figure 16: Residual signal for the inner-ball faulty condition, normal, ball, inner, outer, and inner-ball thresholds with a crack width of 6 mm. Figure 17: Residual signal for the outer-ball faulty condition, normal, ball, inner, outer, inner-ball, and outer-ball thresholds with a crack width of 6 mm. Figure 18: Residual signal for the inner-outer faulty condition, normal, ball, inner, outer, inner-ball, outer-ball, and inner-outer thresholds with a crack width of 6 mm. Figure 15: Outer residual signal, normal threshold, ball threshold, inner threshold, and outer threshold signal for the outer faulty condition with a crack width of 6 mm. Tables 2-5 illustrate the accuracy of fault diagnosis using the proposed extended-state ARX-Laguerre PI observer and ARX-Laguerre PI observer for the normal condition, faulty ball state, inner fault, outer fault, inner-outer fault, inner-ball fault, outer-ball fault, and inner-outer-ball fault, respectively. The diagnostic accuracy is reported as a percentage of correct fault identification in all data. As shown in Tables 2-5, the average rate of failure identification is 98.8% for the proposed extended-state ARX-Laguerre PI and 80.5% for the ARX-Laguerre PI observer. The proposed extended-state ARX-Laguerre PI observer fault diagnosis method outperforms the state-of-the-art ARX-Laguerre PI observer method, yielding an average performance improvement of 17.82%, and 16.625% for 3 mm and 6 mm cracks, respectively. Overall, the proposed extended-state