# Nonlinear Extended-state ARX-Laguerre PI Observer Fault Diagnosis of Bearings
## 3 Problem Statements and Proposed Method Objectives  
The principal target of this paper is to detect and estimate an induction motor's mechanical faults based on the extended-state ARX-Laguerre PI observer. The foremost issue of the induction motor is mathematical modeling in the presence of uncertainties and faults. Thus, this paper utilizes a state-of-the-art induction motor experiment for modeling based on system identification. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Fault Types** & **Rotational Speed (RPM)** & **Fault Crack Size (mm)** \\ \hline \multirow{7}{*}{**Dataset 1**} & **Normal States** & 300 & \\  & **IR Fault** & 300 & \\  & **OR Fault** & 300 & \\  & **Ball Fault** & 300 & \\  & **Inner-Outer Fault** & 300 & \\  & **Inner-Ball Fault** & 300 & \\  & **Unter-Ball Fault** & 300 & \\  & **Inner-Outer-Ball Fault** & 300 & \\ \hline \multirow{7}{*}{**Dataset 2**} & **Normal States** & 400 & \\  & **IR Fault** & 400 & \\  & **OR Fault** & 400 & \\  & **Ball Fault** & 400 & \\  & **Inner-Outer Fault** & 400 & \\  & **Inner-Ball Fault** & 400 & \\  & **Unter-Ball Fault** & 400 & \\ \hline \multirow{7}{*}{**Dataset 3**} & **Normal States** & 450 & \\  & **IR Fault** & 450 & \\  & **OR Fault** & 450 & \\  & **Ball Fault** & 450 & \\  & **Inner-Outer Fault** & 450 & \\  & **Unter-Ball Fault** & 450 & \\  & **Inner-Outer-Ball Fault** & 450 & \\ \hline \multirow{7}{*}{**Dataset 4**} & **Normal States** & 500 & \\  & **IR Fault** & 500 & \\  & **OR Fault** & 500 & \\  & **Ball Fault** & 500 & \\  & **Inner-Outer Fault** & 500 & \\  & **Inner-Ball Fault** & 500 & \\  & **Unter-Ball Fault** & 500 & \\  & **Inner-Outer-Ball Fault** & 500 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed information of the datasets. Figure 3: Different fault conditions in a bearing: (**a**) outer raceway fault, (**b**) inner raceway fault, (**c**) ball raceway fault, (**d**) inner-outer raceway fault, (**e**) inner-ball raceway fault, (**f**) outer-ball raceway fault, and (**g**) inner-outer-ball raceway fault. The three-phase space vector of induction motor formulation based on the stator and rotor voltage, current, and flux are presented below [1; 27]:  
\[\left\{ \begin{array}{l}\left[V_{ stator}\right]=\left[Z_{ stator}\right]\times\left[i_{ stator}\right]+\left[\hat{\varphi}_{ stator}\right]\\ \left[V_{rotor}\right]=\left[Z_{rotor}\right]\times\left[i_{rotor}\right]+ \left[\hat{\varphi}_{rotor}\right]\end{array} \right., \tag{1}\]  
where \(V_{ stator},V_{rotor},[Z_{ stator}],[Z_{rotor}],i_{ stator},i_{rotor},\hat{\varphi}_{ stator}\) and \(\hat{\varphi}_{rotor}\) are the three phase stator voltage, three phase rotor voltage, stator impedance, rotor impedance, three phase stator current, three phase rotor current, change of stator flux, and change of rotor flux, respectively. The flux linkage can be written as follows:  
\[\left\{ \begin{array}{l}\left[\varphi_{ stator}\right]=\left[L_{ stator}\right]\times\left[i_{ stator}\right]+\left[M_{ stator.rotor}\right]\times\left[i_{rotor}\right]\\ \left[\varphi_{rotor}\right]=\left[L_{rotor}\right]\times\left[i_{rotor} \right]+\left[M_{ stator.rotor}\right]\times\left[i_{ stator}\right]\end{array} \right. \tag{2}\]  
Here, \(\varphi_{ stator},\varphi_{rotor},L_{ stator},L_{rotor}\) and \(M_{ stator.rotor}\) are the stator flux, rotor flux, stator inductance matrix, rotor inductance matrix, and stator-rotor mutual inductance, respectively. Based on (1) and (2), the stator and rotor currents are calculated as follows:  
\[ \begin{array}{l}\frac{d}{dt}\left[\begin{array}{c}i_{ stator}\\ i_{rotor}\end{array}\right]=\left[\begin{array}{cc}L_{ stator}&M_{ stator.rotor}\\ M_{ stator.rotor}&L_{rotor}\end{array}\right]^{-1}\times\\ \left(\left[\begin{array}{c}V_{ stator}\\ V_{rotor}\end{array}\right]-\left[\begin{array}{cc}Z_{ stator}&\omega_{r}\frac{d}{dt}M_{ stator.rotor}\\ \omega_{r}\frac{d}{dt}M_{ stator.rotor}\end{array}\right]\left[\begin{array}{c}i_{ stator}\\ i_{rotor}\end{array}\right]\right)\end{array}