# Rolling-Element Bearing Fault Diagnosis Using Advanced Machine Learning-Based Observer
## 3 Rolling-Element-Bearing Modeling  
In this research, the hybrid technique is proposed for FEDI. The model-based approach is the core of the proposed method. First, the Lagrangian formulation based on potential energy, kinetic energy, and generalized forces can be expressed as the following equation [41]. \[\frac{d}{dt}\bigg{(}\frac{\partial K}{\partial q_{i}}\bigg{)}-\frac{\partial K }{\partial q_{i}}+\frac{\partial p}{\partial q_{i}}=Q_{i},i=1,2,3,4,\ldots \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Dataset** & **Conditions** & **Load (hp)** & **Crack Sizes (in)** \\ \hline \multirow{3}{*}{Dataset 1} & Normal & 0 & \\  & BF & 0 & \\  & IF & 0 & 0.007, 0.014, and 0.021 \\  & OF & 0 & \\ \hline \multirow{3}{*}{Dataset 2} & Normal & 1 & \\  & BF & 1 & \\  & IF & 1 & \\  & OF & 1 & \\ \hline \multirow{3}{*}{Dataset 3} & Normal & 2 & \\  & BF & 2 & \\  & IF & 2 & \\  & OF & 2 & \\ \hline \multirow{3}{*}{Dataset 4} & Normal & 3 & \\  & BF & 3 & \\ \cline{1-1}  & IF & 3 & \\ \cline{1-1}  & OF & 3 & \\ \hline \end{tabular}
\end{table} 
Table 1: The detailed description of the Case Western Reserve University (CWRU) datasets in normal and abnormal conditions. Figure 2: The system simulator’s block [22]. Here, \(K,P,Q_{i}\), and \(q_{i}\) are kinetic energy, potential energy, generalized force, and generalized coordinate, respectively. The energy equation can be obtained by the derivative of the Equation (1) and is expressed as follows [6; 41]:  
\[F_{(q)}=(A+\Delta A)(q)\Big{[}\ddot{q}\Big{]}+(B+\Delta B)(q)[\dot{q}\Big{]}+(C+ \Delta C)(q)[q]+\varphi, \tag{2}\]  
where \(F_{(q)},A(q),B(q),C(q),\varphi\), and \((\Delta A,\Delta B,\Delta C)\) are the force vector, mass vector, time-variant stiffness matrix, time-variant damping matrix, fault (IF, OF, BF) vectors, and unknown modeling parameters for mass, stiffness, and damping matrix, respectively. If \(\Delta=\Delta A(q)\Big{[}\ddot{q}\Big{]}+\Delta B(q)[\dot{q}\Big{]}+\Delta C(q )[q]\), the dynamic equation of the bearings can be represented as follows:  
\[F_{(q)}=A(q)\Big{[}\ddot{q}\Big{]}+B(q)\Big{[}\dot{q}\Big{]}+C(q)[q]+\Delta+\varphi, \tag{3}\]  
If \(\omega\Big{(}q,\dot{q}\Big{)}=B(q)\Big{[}\dot{q}\Big{]}+C(q)[q]\) and \(\psi\Big{(}q,\dot{q}\Big{)}=A^{-1}(q)\left(\Delta+\varphi\right)\) represent the uncertainties and faults, the bearing dynamic equation is rewritten as follows:  
\[\Big{[}\ddot{q}\Big{]}=A(q)^{-1}\Big{\{}F_{(q)}-\omega\Big{(}q,\dot{q}\Big{)} \Big{\}}-\psi\Big{(}q,\dot{q}\Big{)}. \tag{4}\]  
Apart from several advantages of mathematical-based system modeling, in most complicated systems, such as bearing systems, the precise mathematical formulation of energy and force in the bearing is nonlinear and complicated. In addition, mathematical modeling is not accurate in the presence of uncertainty. Moreover, the dynamic behavior of the bearing in theoretical and practical applications may be different, which causes challenges in system modeling for FEDI. Therefore, the fuzzy ARX-Laguerre (FAL) technique is represented for REB modeling. This system modeling technique has three main steps. In the first step, the ARX system modeling is defined. To improve the robustness and reliability of the system modeling, the ARX-Laguerre technique is represented in the second step. In addition, to improve the system's modeling accuracy for the ARX-Laguerre technique, the fuzzy technique is represented. The mathematical formulation for an ARX system model is represented as [20; 21; 22]:  
\[f_{o}(k)=\sum\limits_{i=1}^{\delta_{x}}\varepsilon_{x}(i)f_{o}(k-i)+\sum \limits_{i=1}^{\delta_{y}}\varepsilon_{y}(i)f_{i}(k-i), \tag{5}\]  
where \(f_{o}(k),\Big{(}\varepsilon_{x}(i),\varepsilon_{y}(i)\Big{)},f_{i}(k),\) and \(\Big{(}\delta_{x},\delta_{y}\Big{)}\) are the output, model parameters, input, and order of the system, respectively. To represent the model parameters \(\Big{(}\varepsilon_{x},\varepsilon_{y}\Big{)}\), the following equation is presented. \[\left\{