# Bearing Fault Identification Using Machine Learning and Adaptive Cascade Fault Observer
## 3 Normal Signal of Bearing Approximation Using Fuzzy Orthonormal Regressive Technique  
Designing the observer for nonlinear and nonstationary signals is a vital challenge. Therefore, to develop an observer, normal signal of bearing approximation using the time-series identification technique is the first step. In this work, we develop the fuzzy orthonormal regressive procedure. Based on Figure 1, the orthonormal regressive technique is implemented to approximate and extract the state-space equation from the bearing signals. Moreover, the orthonormal regressive approach is selected to improve the robustness of the state-space bearing function approximation. Finally, the error of the state-space bearing model can be reduced using the fuzzy orthonormal regressive technique. The regressive (R) algorithm for approximate the REB vibration and AE normal signals is [36; 45]:  
\[Y_{R}(k)=\sum\
olimits_{i=1}^{\delta_{Y}}\mathbb{C}_{Y}(i)Y_{R}(k-i)+\sum \
olimits_{i=1}^{\delta_{U_{i}}}\mathbb{C}_{U_{i}}(i)U_{i}(k-i), \tag{1}\]  
where \(Y_{R}(k),U_{i}(k),\left(\mathbb{C}_{x}(i),\mathbb{C}_{y}(i)\right)\), and \(\left(\delta_{U_{i}},\delta_{Y}\right)\) are the normal bearing signal approximation using the R-technique, the uncertainties for REB signal approximation, parameters to tune the function approximation, and the order of the function approximation technique, respectively. To increase the resistance of the modeled regressive function against uncertainties and disturbance, the orthonormal regressive (OR) procedure is used. The state-space equation using the OR technique to model the REB is:  
\[Y_{OR}(k)=\sum\limits_{n=0}^{i_{V}-1}\mathbb{C}_{Y}(i)\ \overline{O}_{n,Y_{OR}}(k, \gamma_{Y})Y_{OR}(k-i)+\sum\limits_{n=0}^{i_{U_{i}}-1}\mathbb{C}_{U_{i}}(i)\ \overline{O}_{n,U_{i}}\big{(}k,\gamma_{U_{i}}\big{)}U_{i}(k-i) \tag{2}\]  
Here, \(Y_{OR}(k),\overline{O}_{n,Y_{OR}}(k,\gamma_{Y})\), and \(\overline{O}_{n,U_{i}}\big{(}k,\gamma_{U_{i}}\big{)}\) are the normal bearing signal approximation using the OR technique, the function of orthonormal, and the orthonormal of uncertain condition for REB, respectively. The accuracy of the normal signal approximation is an important factor for function approximation. To reduce the signal approximation error and increase the nonlinear approximation accuracy, the fuzzy technique is recommended. Generally, the fuzzy algorithm can be introduced using the following definition:  
\[if\langle I_{1}\rangle\text{s}\langle LV_{1}\rangle\text{and}\langle I_{2} \rangle\text{s}\langle LV_{2}\rangle\text{then}(O)\text{s}\langle LV_{O}\rangle. \tag{3}\]  
Here, \(\langle I_{1}\rangle\text{and}\langle I_{2}\rangle\), \(\langle LV_{1}\rangle\text{and}\langle LV_{2}\rangle\), \(\langle O\rangle\), and \(\langle LV_{O}\rangle\) are fuzzy logic input signals, linguistic variables for input signals, the fuzzy logic output signal, and the linguistic variable for the fuzzy logic output \begin{table}
\begin{tabular}{c c c c} \hline
**Datasets** & **Faults** & **Motor (RPM)** & **Crack Sizes (mm)** \\ \hline \multirow{8}{*}{C} & NRM & 450 & \multirow{8}{*}{3 and 6} \\  & IRF & 450 & \\  & ORF & 450 & \\  & BLF & 450 & \\  & IRF-ORF & 450 & \\  & IRF-BLF & 450 & \\  & ORF-BLF & 450 & \\  & IRF-ORF-BLF & 450 & \\  & NRF-ORF-BLF & 450 & \\  & NRM & 500 & \\  & IRF & 500 & \\  & ORF & 500 & \\  & BLF & 500 & \\  & IRF-ORF & 500 & \\  & IRF-BLF & 500 & \\  & ORF-BLF & 500 & \\  & IRF-ORF-BLF & 500 & \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._signal, respectively. Thus, the state-space function approximation for normal bearing signal using fuzzy orthonormal regressive (FOR) is represented using Equation (4). \[\left\{ \begin{array}{c}X_{FOR}(k+1)=[\sigma_{X}X_{FOR}(k)+\sigma_{Y}Y_{FOR}(k)+ \sigma_{U}U_{i}(k)]+\sigma_{F}\varnothing_{F}(k)\\ Y_{FOR}(k)=(\beta)^{T}X_{FOR}(k)\end{array} \right.