# Hybrid Fault Diagnosis of Bearings: Adaptive Fuzzy Orthonormal-ARX Robust Feedback Observer
## 3 Anomaly Diagnosis Based on Proposed Hybrid Algorithm
### Fault Detection and Fault Classification  
Figure 1 shows that the proposed method minimizes the error in REB signal estimation in normal conditions. In this case, the residual signal \((r_{P}(k))\) is minimized in normal conditions and is defined as follows:  
\[r_{P}(k)=Y_{o}(k)-Y_{PM}(k). \tag{16}\]  
Thus, a fault can be identified based on the differences between residual signals in different conditions. Figure 3 illustrates the residual signals in normal and anomalous conditions based on the proposed method. Due to the Figure 3, the fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback observer is a suitable technique for detection and classification the signals. Moreover, to improve the fault  
Figure 3: Residual signals in normal and abnormal conditions using fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback (PM) observer. detection and classification performance, the energy feature \(\left(\overline{E}\right)\) is extracted from the residual signal and is introduced using the following equation,  
\[\overline{E}=\sum_{i=1}^{K}{r_{P_{i}}}^{2}. \tag{17}\]  
Here, \(r_{P_{i}}\) is the \(i^{th}\) sample of the residual signal obtained by the fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback observer and \(K\) is the sample number. Figure 4 shows the energy of the bearing signals in normal and anomalous conditions, such as a ball fault, an inner fault, and outer degradation. The figure shows that the proposed method, residual signal generation, and energy feature extraction are outstanding techniques for fault detection and identification. Besides, the decision-making unit is developed to detect and classify the faults based on the difference between the amplitude of the residual signal's energy. Therefore, to develop a decision-making unit, a support vector machine (SVM) that is a powerful mathematical technique is presented in this work. The soft margin SVM can be defined by the following equation [20; 23; 24]. \[w_{i}\left(\varnothing^{T}\rho(q_{i})+c\right)\geq w_{i}-v_{i}, \tag{18}\]  
where \((q_{i},\ w_{i}),(\varnothing,c),\rho(q_{i})\), and \(v_{i}\) are the inputs, outputs, features, and maximum distance, respectively. The proposed technique to estimate, detect, and identify faults in the bearing is detailed in Algorithm 1. ```
1:Model the systems using the fuzzy orthonormal-ARX procedure (5)
2:Develop a fuzzy orthonormal-ARX feedback observer for signal estimation (6,7)
3:Develop a fuzzy orthonormal-ARX structure feedback observer to improve the robustness (8,9)
4:Improve the fault estimation accuracy using the fuzzy orthonormal-ARX fuzzy logic-structure feedback observer (11,12)
5:Develop the fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback observer to increase fault estimation accuracy, reliability, and robustness (14,15)
6:Generate the residual signal (16)
7:Run residual signal characterization by energy (17)
8:Apply the classification technique using SVM for fault diagnosis (18)

Figure 4: Energy features in normal and anomalous conditions based on the fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback (PM) observer. ## 4 Datasets and Results

In this work, two different datasets are used to evaluate the proposed algorithm. ### Datasets

The Case Western Reserve University (CWRU) benchmark bearing dataset and the Smart HSE (SHSE) Lab bearing dataset are used validate the single- and multi-type fault diagnosis based on the SVM-based fuzzy orthonormal-ARX adaptive fuzzy logic-structure feedback observer (proposed method), the SVM-based fuzzy orthonormal-ARX structure feedback observer (SFO), and the SVM-based fuzzy orthonormal-ARX feedback observer (FO). #### 4.1.1 Case Western Reverse University (CWRU) Bearing Dataset

The CWRU dataset is used to test the effectiveness of the proposed algorithm using a vibration single crack fault. The vibration sensor is used to collect vibration signals in various conditions such as normal faults, ball faults, inner (IR) faults, and outer (OR) faults.