## 7 Experimental Results
The performance of signal pattern recognition in these techniques appears to be satisfactory. However, to increase the performance of fault diagnosis, pattern identification and crack size detection, the AMRXU-AVSPMI is suggested in the paper. Figure 6 shows the effect of the SNFS technique and training the  
Figure 4: Error in AR, ARXU, and AMRXU techniques from normal resampled RMS signal modeling. Figure 5: Training, validation, and testing MSE curves during RMS resampled normal signal modeling using AMRXU. inputs on the RMS error of signal estimation. In this case, the SNFS algorithm has 34 nodes, 32 linear parameters, 18 nonlinear parameters, and eight fuzzy rules. The elapsed time in this SNFS design is 0.223 s. To have the minimum RMS error, the first, second, and third inputs need to have 15-, 12-, and 9-times training, respectively. Figure 7 shows the power of estimation algorithms for signal classification using the proposed scheme (AMRXU-AVSPMI), the AMRXU-VSPMI technique, and the AMRXU-PMI method. As seen in Figure 7, with the proposed AMRXU-AVSPMI technique, which is an adaptive hybrid-nonlinear observer, the RMS signal amplitude for the four classes is detectable. In the AMRXU-PMI method, which is a linear observer, the distinction between the inner and outer faults is either very low or indistinguishable. These conditions improved somewhat with the AMRXU-VSPMI technique, which is a robust observer, but were less detectable than with the proposed AMRXU-AVSPMI observer. To test the accuracy of each state's identification, a combination of the proposed scheme (AMRXU-AVSPMI) with an SVM (AMRXU-AVSPMI+SVM) is compared against AMRXU-VSPMI+SVM and AMRXU-PMI+SVM. The fault identification and classification performance for the methods mentioned above are assessed by using important metrics [46], such as averaged recall (\(Rec_{av}\)), averaged precision (\(Prec_{av}\)), averaged F1-score (\(F1_{av}\)), and total fault identification and classification accuracy (FICA). We also use these metrics to test the robustness of the proposed methods. These metrics are defined as the following equations:  
\[Rec_{av}=\frac{\sum_{k=1}^{K}TP_{k}}{\sum_{k=1}^{K}(TP_{k}+FN_{k})}\times 100; \tag{46}\]  
Figure 6: Effect of the SNFS on reducing the signal estimation error. Figure 7: Resampled RMS residual signals under normal and abnormal conditions using AMRXU-PMI, AMRXU-VSPMI, and the proposed AMRXU-AVSPMI observers. Figure 3.0  
Figure 3.1  
Figure 3.2  
where, \(TP_{k}\), \(FP_{k}\), \(FN_{k}\), \(N\) are the true-positive value for the class of \(k\), false-positive value for the class of \(k\), and false-negative value for the class of \(k\), and the data samples, respectively. Also, \(K\) is the number of classes, and \(k\) is the index of class such that \(k=1\),..., \(K\). To test the robustness of the above algorithms, the results are tested in 10 experiments and summarized in Figure 3.3  
Figure 8 shows the boxplots of the FICA metric over 10 experiments for AMRXU-AVSPMI+SVM, AMRXU-VSPMI+SVM, and AMRXU-PMI+SVM. From Figure 8, it is observed that the classification accuracy of the proposed method (AMRXU-AVSPMI+SVM) is the most robust because it does not deviate meaningfully from the average of FICA, which validates the repeatability of the test. For AMRXU-VSPMI+SVM, the deviation of FICA is not very high; however, the average accuracy is lower than that of the proposed method. Unlike the proposed method, the AMRXU-PMI+SVM boxplot has a significant deviation from the average of the FICA. From Figure 8, we know that the combination of the AVSPMI observation technique, the modeling algorithm using AMRXU, and the SVM classifier can improve the classification and identification performance and fault diagnosis stability and robustness. Figures 9-12 show the average of confusion matrices for 10 experiments to observe the specific details about the fault diagnosis performance when the torque load varies from 0 hp to 3 hp.