## 2 Equations of Motion  
The considered rotor system is illustrated in Figure 1, where the rotating shaft is supported via eight electro-magnetic poles that serve as active magnetic bearings. The rotor system is assumed to be a rigid body with mass \(m\) and eccentricity \(e\) and rotates with angular speed \(v\). Accordingly, the dynamical equations that govern the system's lateral vibrations in \(X_{1}\) and \(X_{2}\) directions can be written as follows [46; 47]:  
\[m\ddot{x}_{1}-R_{X_{1}}=me\ v^{2}\cos(vt) \tag{1}\]  
\[m\ddot{x}_{2}-R_{X_{2}}=me\ v^{2}\sin(vt) \tag{2}\]  
where \(R_{X_{1}}\) and \(R_{X_{2}}\) denote the net restoring forces that are exerted on the rotating shaft due to the attractive electro-magnetic forces \(h_{j}\ (j=1,2,\dots,8)\). It is considered that the eight electro-magnetic poles are symmetric. Accordingly, the attractive magnetic force that is exerted on the rotor system due to each pole can be expressed as follows [47]:  
\[h_{j}=\frac{1}{4}\mu_{0}N^{2}A\cos(\theta)\frac{I_{j}^{2}}{g_{j}^{2}}=\Delta \frac{I_{j}^{2}}{g_{j}^{2}},\ j=1\cdots 8 \tag{3}\]  
where \(\Delta=\frac{1}{4}\mu_{0}N^{2}A\cos(\theta)\) is constant (i.e., \(\mu_{0}\) is the air permeability, \(N\) is the winding number of the electrical coil of each pole, \(A\cos(\theta)\) is the effective cross-sectional area of each pole), \(I_{j}\) is the electrical current in the \(j\) pole, and \(g_{j}\) is the air-gap size between the rotating shaft and the \(j\)th pole. For the small displacements \(x_{1}(t)\) and \(x_{2}(t)\) of the rotating shaft within the pole housing, the instantaneous air-gap size \(g_{j}\) between the rotating shaft and the \(j\) pole can be expressed as follows:  
\[ \begin{array}{l}g_{1}(x_{1},x_{2})=s_{0}+x_{1}\sin(\alpha)-x_{2}\cos(\alpha), \ g_{2}(x_{1},x_{2})=s_{0}+x_{1}\cos(\alpha)-x_{2}\sin(\alpha)\\ g_{3}(x_{1},x_{2})=s_{0}+x_{1}\cos(\alpha)+x_{2}\sin(\alpha),\ g_{4}(x_{1},x_{2}) =s_{0}+x_{1}\sin(\alpha)+x_{2}\cos(\alpha)\\ g_{5}(x_{1},x_{2})=s_{0}-x_{1}\sin(\alpha)+x_{2}\cos(\alpha),\ g_{6}(x_{1},x_{2}) =s_{0}-x_{1}\cos(\alpha)+x_{2}\sin(\alpha)\\ g_{7}(x_{1},x_{2})=s_{0}-x_{1}\cos(\alpha)-x_{2}\sin(\alpha),\ g_{8}(x_{1},x_{2}) =s_{0}-x_{1}\sin(\alpha)-x_{2}\cos(\alpha)\end{array}  \tag{4}\]  
where \(s_{0}\) is the radial clearance between the rotating shaft and each electro-magnetic pole when the geometric center of both the rotor and pole housing coincided as shown in Figure 1a, and \(2\alpha=45^{\circ}\) is the angle between every two consecutive electro-magnetic poles. Based on the system geometry shown in Figure 1a,b, the electrical currents of the eight-pole are designed based on the push-pull control mechanism as follows:  
\[I_{1}=I_{8}=I_{0}-i_{X_{2}},\ I_{2}=I_{3}=I_{0}+i_{X_{1}},\ I_{4}=I_{5}=I_{0}+i _{X_{2}},\ I_{6}=I_{7}=I_{0}-i_{X_{1}} \tag{5}\]  
where \(I_{0}\) is a per-magnetizing constant current, \(i_{X_{1}}\), and \(i_{X_{2}}\) are the control currents in \(X_{1}\) and \(X_{2}\) directions, respectively. Figure 1: (**a**) The 3-D visualization of the eight-pole electro-magnetic suspension system, (**b**) schematic diagram of the eight-pole electro-magnetic suspension system when the rotor lies within its nominal position, (**c**) schematic diagram of the eight-pole electro-magnetic suspension system when the rotor makes displacements \(x_{1}(t)\) and \(x_{2}(t)\) in \(X_{1}\) and \(X_{2}\) directions, respectively. Within this article, a new control algorithm was proposed to eliminate the non-linear oscillations of the considered system beside the conventional PD-controller. The suggested new control technique is defined as the Linear Integral Positive Position Feedback controller (LIPPF-controller). Therefore, according to the introduced control strategy, the control currents \(i_{X_{1}}\) and \(i_{X_{2}}\) are designed as follows:  
\[i_{X_{1}}=k_{1}x_{1}+k_{2}\dot{x}_{1}+k_{3}x_{3}+k_{4}x_{5},\,i_{X_{2}}=k_{1}x_{ 2}+k_{2}\dot{x}_{2}+k_{5}x_{4}+k_{6}x_{6} \tag{6}\]  
where \(k_{1}\) and \(k_{2}\) are the control gains of the conventional PD-controller.