# Hybrid Fault Diagnosis of Bearings: Adaptive Fuzzy Orthonormal-ARX Robust Feedback Observer
## 3 Anomaly Diagnosis Based on Proposed Hybrid Algorithm
### Fuzzy Orthonormal-ARX Adaptive Fuzzy Logic-Structure Feedback Observer for Fault Estimation  
The fuzzy orthonormal-ARX feedback observer is represented by the following definition,  
\[\left\{ \begin{array}{c}X_{\text{FO}}(k+1)=\begin{bmatrix}\sigma_{X}X_{\text{FO }}(k)+\begin{bmatrix}\sigma_{Y}Y_{\text{FO}}(k)+\sigma_{f}\hat{Y}_{f}(k)\end{bmatrix}+ \sigma_{\text{U}}U_{i}(k)\end{bmatrix}+\hat{\phi}_{\text{FO}}(k)+\sigma_{1} \times e_{\text{FO}}\\ Y_{\text{FO}}(k+1)=(\beta)^{\text{T}}X_{\text{FO}}(k)\\ e_{\text{FO}}=X_{\text{FAL}}(k)-X_{\text{FO}}(k)\\ e_{\text{r-FO}}=Y_{o}(k)-Y_{\text{FO}}(k)\\ e_{\text{f-FO}}=Y_{o}(k)-(Y_{\text{FO}}(k)+\hat{Y}_{f}(k))\end{array} \right. \tag{6}\]  
To improve the accuracy uncertainty (e.g., motor speed variation) estimation of the fuzzy orthonormal-ARX feedback observer, the following definition can be defined. \[\hat{\phi}_{\text{FO}}(k+1)=\hat{\phi}_{\text{FO}}(k)+\sigma_{2}\times e_{ \text{r-FO}}+\sigma_{3}\times e_{\text{f-FO}}. \tag{7}\]  
Here, \(X_{\text{FO}}(k),Y_{\text{FO}}(k),\hat{Y}_{f},\hat{\phi}_{\text{FO}}(k),Y_{o} (k)\), and \((\sigma_{1},\sigma_{2},\sigma_{3})\) are the fuzzy orthonormal-ARX feedback observer for estimation state of the REB model, the fuzzy orthonormal-ARX feedback observer for estimation of the REB signal, the fuzzy technique for the non-linear part of the signal estimation, the fuzzy orthonormal-ARX feedback observer for estimation the uncertainty (motor speed variation) to minimize the error in normal conditions, the REB original signals, and the coefficient of the fuzzy orthonormal-ARX feedback observer, respectively. As seen in Figure 1, the fuzzy orthonormal-ARX feedback observer suffers from poor robustness, especially in unknown (crack-variant or speed-variant) situations. Therefore, the fuzzy orthonormal-ARX structure feedback observer is developed to overcome this obstacle. The following technique is used to improve the robustness based on multi-structure algorithm. \[\left\{ \begin{array}{c}X_{\text{SFO}}(k+1)=\begin{bmatrix}\sigma_{X}X_{ \text{SFO}}(k)+\begin{bmatrix}\sigma_{Y}Y_{\text{SFO}}(k)+\sigma_{f}\hat{Y}_{f }(k)\end{bmatrix}+\sigma_{\text{U}}U_{i}(k)\end{bmatrix}+\hat{\phi}_{\text{ SFO}}(k)+\sigma_{1}\times e_{\text{SFO}}\\ Y_{\text{SFO}}(k+1)=(\beta)^{\text{T}}X_{\text{SFO}}(k)\\ e_{\text{SFO}}=X_{\text{FAL}}(k)-X_{\text{SFO}}(k)\\ e_{\text{r-SFO}}=Y_{o}(k)-Y_{\text{SFO}}(k)\\ e_{\text{f-SFO}}=Y_{o}(k)-(Y_{\text{SFO}}(k)+\hat{Y}_{f}(k))\end{array} \right. \tag{8}\]  
To improve robustness, the uncertainty can be estimated by the fuzzy orthonormal-ARX structure feedback observer based on the following definition,  
\[\hat{\phi}_{\text{SFO}}(k+1)=\hat{\phi}_{\text{SFO}}(k)+\sigma_{2}\times e_{ \text{r-SFO}}+\sigma_{3}sgn\times e_{\text{f-SFO}}. \tag{9}\]Here, \(X_{SFO}(k),Y_{SFO}(k),\hat{\varphi}_{SFO}(k)\), and \(\sigma_{3}\) are the fuzzy orthonormal-ARX structure feedback observer for estimation state of the REB model, the fuzzy orthonormal-ARX structure feedback observer for estimation of the REB signal, the fuzzy orthonormal-ARX structure feedback observer for estimation the uncertainty (motor speed variation) to minimize the error in normal conditions, and the coefficient of the fuzzy orthonormal-ARX structure feedback observer, respectively. Due to Figure 1, the TSK fuzzy technique is developed and may be defined as follows [22],  
\[if\;e_{SFO}(k)\;is\;Th\;then\;\hat{\varphi}_{f}(k+1)=\hat{\varphi}_{f}(k)+\sigma _{F}\times e_{FSFO}. \tag{10}\]  
Hence, the fuzzy orthonormal-ARX fuzzy logic-structure feedback observer can be represented by the following equation,  
\[\left\{ \begin{array}{c}X_{FSFO}(k+1)=\left[\sigma_{X}X_{FSFO}(k)+\left[\sigma _{Y}Y_{FSFO}(k)+\sigma_{f}\hat{Y}_{f}(k)\right]+\sigma_{l}U_{i}(k)\right]+\hat {\varphi}_{FSFO}(k)+\sigma_{1}\times e_{FSFO}\\ Y_{FSFO}(k+1)=(\beta)^{T}X_{FSFO}(k)\\ e_{FSFO}=X_{FAL}(k)-X_{FSFO}(k)\\ e_{r-FSFO}=Y_{o}(k)-Y_{FSFO}(k)\\ e_{f-FSFO}=Y_{o}(k)-(Y_{FSFO}(k)+\hat{Y}_{f}(k))\end{array} \right.