## 1 Introduction  
The rotor electro-magnetic suspension system has been introduced as an optimal solution to get rid of the drawbacks of the conventional bearings system such as the limited operational speed, the existence of friction between the rotor and the bearings, and the need for continuous lubrication, etc. The rotor electro-magnetic suspension system is a type of bearing that supports the rotating shafts using an electro-magnetic levitation mechanism. The main working principle of the electro-magnetic levitation is to apply controllable magnetic forces that support the rotor system in its hovering position without physical contact with the stator. The operation of the rotor with the contactless feature has earned the electro-magnetic suspension system many advantages over the other conventional bearing mechanisms such as the lack of friction between the moving parts and the stator, high-speed operation, less maintenance, clean working environment with no need for lubrication, high reliability, high durability, etc. Accordingly, the rotor electro-magnetic suspension system was and still is one of the main research subjects for scientists and engineers worldwide, where many researchers have focused their work on investigating the dynamical characteristics of the different electro-magnetic suspension configurations. On the other hand, some researchers have introduced advanced control algorithms to enhance the dynamical behaviors of electro-magnetic suspension systems regardless of their configurations. [PERSON] et al. [1] studied non-linear dynamics and the bifurcation behaviors of the four-pole electro-magnetic suspension system. They employed the asymptotic perturbation method along with the normal form technique to analyze the considered system. The authors reported that the four poles electro-magnetic suspension system may lose its stability via different types of bifurcations such as the Hopf and saddle node. [PERSON] et al. [2; 3] discussed the vibration control and bifurcation elimination of the six-pole electro-magnetic suspension system utilizing the conventional PD-control technique. The authors studied two different configurations for the introduced controller. The first PD-control strategy has been designed based on the lateral displacements and lateral velocities of the rotor system, while the second control methodology is implemented based on the rotor radial oscillations. According to the studied control techniques, they concluded that the first control method is more efficient in mitigating the system lateral vibrations, while the second technique is more robust against the system instability. [PERSON] et al. [4; 5] investigated the non-linear oscillations of the eight-pole electro-magnetic suspension system controlled via the PD-control algorithm at the primary and super-harmonic resonances. The authors illustrated that the system has both complex bifurcation behavior and multiple fixed-point solutions. [PERSON] et al. [6] introduced a non-linear position-velocity control algorithm to enhance the dynamical characteristics and mitigate the vibratory motion of the eight-pole suspension system. [PERSON] et al. [7] explored the dynamics of the eight-poles electro-magnetic suspension system when the rub and impact occur between the moving parts and the rotor housing, where the obtained results showed that the rotor system may perform periodic, quasiperiodic, or chaotic motion depending on the impact stiffness magnitude and dynamic friction coefficient. [PERSON] et al. [8; 9; 10; 11; 12; 13] extensively investigated the eight-pole suspension system with time-varying stiffness coefficients, where many non-linear phenomena have been reported such as the multi-pulse chaotic motion, saddle-node, and Hopf bifurcations. [PERSON] et al. [14] studied the 12-pole electro-magnetic suspension system utilizing the PD-controller. They investigated the effect of the different control parameters on the rotor stability charts.