## 2 Equations of Motion
\(k_{3}\) and \(k_{4}\) are the control signal gains of the LIPPF-controller that is connected to the system oscillation mode in the \(X_{1}\) direction. \(k_{5}\) and \(k_{6}\) are the control signal gains of the LIPPF-controller that is connected to the system oscillation mode in the \(X_{2}\) direction. The LIPPF-controller is a combination of both first-order and second-order filters that are excited linearly by the rotor system displacement, where the equations of motion of the proposed LIPPF-controller are given as follows:  
\[\ddot{x}_{3}+c_{1}\dot{x}_{3}+\lambda_{1}x_{3}=\delta_{1}x_{1} \tag{7}\]  
\[\ddot{x}_{4}+c_{2}\dot{x}_{4}+\lambda_{2}x_{4}=\delta_{2}x_{2} \tag{8}\]  
\[\dot{x}_{5}+\lambda_{3}x_{5}=\delta_{3}x_{1} \tag{9}\]  
\[\dot{x}_{6}+\lambda_{4}x_{6}=\delta_{4}x_{2} \tag{10}\]  
where \(c_{1}\), \(c_{2}\) represent the linear damping coefficients, \(\lambda_{1}\), \(\lambda_{2}\) denote the natural frequency, and \(\lambda_{3}\), \(\lambda_{4}\) are the internal loop feedback gain of the suggested controller (i.e., LIPPF-controller). In addition, \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), and \(\delta_{4}\) represent the feedback signal gains of the LIPPF-controller. The engineering implementation of the suggested control method is illustrated in Figure 2, where the instantaneous Cartesian displacements \(x_{1}(t)\) and \(x_{2}(t)\) of the rotor system are measured using proximity sensors. The measured signals are fed into a digital computer to manipulate them according to the proposed control strategy to form the control currents \(i_{X_{1}}=k_{1}x_{1}+k_{2}\dot{x}_{1}+k_{3}x_{3}+k_{4}x_{5}\) and \(i_{X_{2}}=k_{1}x_{2}+k_{2}\dot{x}_{2}+k_{5}x_{4}+k_{6}x_{6}\). Then, the electrical currents of the eight-pole (i.e., \(I_{1}\), \(I_{2}\ldots,I_{8}\)) are applied via a power amplifier network as illustrated in Figure 2 in detail. Now to obtain the full mathematical model that governs the rotor and the controller dynamics, let us substitute Equations (4)-(6) into Equation (3), we get the restoring forces of the eight-pole as follows:  
\[h_{1}=\Delta\bigg{(}\frac{I_{0}-k_{1}x_{2}-k_{2}\dot{x}_{2}-k_{5}x_{4}-k_{6}x _{6}}{s_{0}+x_{1}\sin(\alpha)-x_{2}\cos(\alpha)}\bigg{)}^{2} \tag{11}\]  
\[h_{2}=\Delta\bigg{(}\frac{I_{0}+k_{1}x_{1}+k_{2}\dot{x}_{1}+k_{3}x_{3}+k_{4}x_{ 5}}{s_{0}+x_{1}\cos(\alpha)-x_{2}\sin(\alpha)}\bigg{)}^{2} \tag{12}\]  
\[h_{3}=\Delta\bigg{(}\frac{I_{0}+k_{1}x_{1}+k_{2}\dot{x}_{1}+k_{3}x_{3}+k_{4}x_{ 5}}{s_{0}+x_{1}\cos(\alpha)+x_{2}\sin(\alpha)}\bigg{)}^{2} \tag{13}\]  
Figure 1:0  
Figure 1:1  
Figure 1:2  
Figure 1:3According to the orientations of the eight attractive forces \(h_{j}\) (\(j=1,2,\ldots,8\)) shown in Figure 1, the net restoring forces \(R_{X_{1}}\) and \(R_{X_{2}}\) in \(X_{1}\) and \(X_{2}\) directions can be expressed as:  
Figure 1:4  
Figure 1:5  
Substituting Equations (11)-(20) into Equations (1) and (2), with introducing the dimensionless variables and parameters: \(y_{1}=\frac{\chi_{1}}{\omega_{7}}\), \(y_{2}=\frac{\chi_{2}}{\omega_{7}}\), \(y_{3}=\frac{\chi_{3}}{\omega_{7}}\), \(y_{4}=\frac{\chi_{4}}{\omega_{7}}\), \(y_{5}=\frac{\chi_{5}}{\omega_{6}}\), \(y_{6}=\frac{\chi_{6}}{\omega_{7}}\), \(\tau=\omega_{n}t\), \(\omega_{n}=\sqrt{\Delta/ms_{0}^{3}}\), \(p=\frac{\varsigma_{0}}{\omega_{1}}k_{1}\), \(d=\frac{\varsigma_{0}\omega_{3}}{\omega_{1}}k_{2}\), \(\mu_{1}=\frac{\varsigma_{1}}{\omega_{6}}\), \(\mu_{2}=\frac{\varsigma_{2}}{\omega_{2}}\), \(\omega_{1}=\sqrt{\frac{\lambda_{1}}{\omega_{6}^{2}}}\), \(\omega_{2}=\sqrt{\frac{\lambda_{2}}{\omega_{7}^{2}}}\), \(\rho_{1}=\frac{\lambda_{3}}{\omega_{7}}\), \(\rho_{2}=\frac{\lambda_{4}}{\omega_{7}}\), \(\eta_{1}=\frac{8\omega_{6}}{\omega_{1}}\cos(\alpha)k_{3}\), \(\eta_{2}=\frac{8\omega_{6}}{\omega_{1}}\cos(\alpha)k_{4}\),  
Figure 1:6 The schematic diagram shows the interconnection between the proposed LIPPF-controller and the electro-magnetic suspension system.