# Hybrid Fault Diagnosis of Bearings: Adaptive Fuzzy Orthonormal-ARX Robust Feedback Observer
## 2 Rolling-Element Bearing Modeling Using Fuzzy Orthonormal-ARX Technique  
System modeling is the main argument in complicated systems, especially in uncertain and unknown conditions. Mathematical-based bearing modeling is reliable and stable but is limited in uncertain and unknown conditions. Therefore, to decrease these limitations, the fuzzy orthonormal-ARX technique is prescribed in this work. As shown in Figure 1, the REBs modeling technique based on the fuzzy orthonormal-ARX technique is classified into three main steps: (a) ARX technique for REBs modeling, (b) ARX technique robustness improvement using the orthonormal-ARX procedure, and (c) non-linear system modeling accuracy improvement using the fuzzy orthonormal-ARX algorithm. The ARX technique for vibration and AE signals can be developed using Equation (1) [11; 12]. \[Y_{M_{A}}(k)=\sum\
olimits_{i=1}^{\delta_{Y_{M_{i}}}}\mathbb{C}_{Y_{M}}(i)Y_{M _{A}}(k-i)+\sum\
olimits_{i=1}^{\delta_{U_{i}}}\mathbb{C}_{U_{i}}(i)U_{i}(k-i), \tag{1}\]  
where \(Y_{M_{L}}(k),U_{i}(k),\left(\mathbb{C}_{x}(i),\mathbb{C}_{Y_{M}}(i)\right)\), and \(\left(\delta_{U_{i}},\delta_{Y_{M_{i}}}\right)\) are the REB model based on the ARX technique, the reference signal for the REB model, REB model parameters, and the order of the REB model, respectively. The REB model parameters can be represented as follows:  
\[\left\{ \begin{array}{c}\mathbb{C}_{U_{i}}(i)=\sum_{n=0}^{\infty}\alpha_{n, U}\,\mathbb{\mathcal{G}}_{n}^{U_{i}}\big{(}i,\partial_{U_{i}}\big{)}\\ \mathbb{C}_{Y_{M}}(i)=\sum_{n=0}^{\infty}\alpha_{n,Y_{M}}\,\mathbb{\mathcal{G }}_{n}^{Y_{M}}\big{(}i,\partial_{Y_{M}}\big{)}\end{array} \right. \tag{2}\]  
Here, \(\left(\alpha_{n,U_{i}},\alpha_{n,Y_{M}}\right)\),\(\left(\partial_{U_{i}},\partial_{Y_{M}}\right)\), and \(\left(\partial_{n}^{U_{i}},\partial_{n}^{Y_{M}}\right)\) are the Fourier decomposition coefficients, orthonormal basis, and orthonormal function, respectively. The REB signals are non-linear, non-stationary, and complex, but the ARX technique is a linear algorithm. This technique cannot accurately estimate the REB signals. Therefore, to improve robustness, an orthonormal-ARX procedure is developed for REB modeling. \[Y_{M_{\text{O}A}}(k)=\sum\
olimits_{n=0}^{i_{Y}-1}\mathbb{C}_{Y}(i)\ \alpha_{n,Y_{M_{A}}}\big{(}k,\partial_{Y_{M}}\big{)}Y_{M_{A}}(k-i)+\sum \
olimits_{m=0}^{i_{U_{i}}-1}\mathbb{C}_{U_{i}}(i)\ \alpha_{n,U_{i}}\big{(}k,\partial_{U_{i}}\big{)}U_{i}(k-i). \tag{3}\]  
Figure 1: Proposed hybrid bearing fault diagnosis approach. Here, \(Y_{M_{\rm OA}}(k),\alpha_{n,Y_{M_{\rm AL}}}\left(k,\partial_{Y_{M}}\right)\), and \(\alpha_{n,U_{i}}\left(k,\partial_{U_{i}}\right)\) are the REB model based on the orthonormal-ARX technique, the REB orthonormal function for measurement output, and the REB orthonormal function for measurement input, respectively. To address the issues of non-linearity in the REB model and to increase modeling accuracy, a fuzzy technique is developed. To modeling the non-linearity in the REB, a two-input fuzzy technique is developed based on the following equation. \[if\langle F.I\rangle\textit{is}\langle F.LV\rangle\textit{and}\langle S.I \rangle\textit{is}\langle S.LV\rangle\textit{then}(\textit{out})\textit{is}( \textit{out}.LV). \tag{4}\]  
Here, \(\langle F.I\rangle\), \(\langle F.LV\rangle\), \(\langle S.I\rangle\), \(\langle S.LV\rangle\), \(\langle out\rangle\), and \(\langle out.LV\rangle\) are the first fuzzy input, the first fuzzy linguistic variable, the second fuzzy input, the second fuzzy linguistic variable, the output of the fuzzy, and the linguistic variable for the output of the fuzzy, respectively. Thus, the state-space fuzzy orthonormal-ARX equation for modeling the REB is generated as follows:  
\[\left\{ \begin{array}{c}X_{FAL}(k+1)=\left[\sigma_{x}X_{FAL}(k)+\sigma_{y}Y_{ FAL}(k)+\sigma_{f}Y_{f}(k)+\sigma_{U}U_{i}(k)\right]\\ Y_{FAL}(k+1)=(g)^{T}X_{FAL}(k)\end{array} \right.