## 1 Introduction
The acquired results showed that the system stability can be controlled using the proportional and derivative gain of the applied controller. [PERSON] and [PERSON] [15] introduced both the radial and cartesian proportional-derivative control methods to stabilize the 16-pole electro-magnetic suspension system. They derived the equations of motion according to the suggested control methodologies. In addition, perturbation analysis was employed to explore the system's steady-state oscillation. Based on the obtained results, the authors confirmed that the radial control technique is more robust than the Cartesian one. In addition, extensive investigations of the 16-pole electro-magnetic suspension system with time-varying stiffness have been presented by [PERSON] and his coworkers [16; 17; 18; 19; 20]. Due to the flexibility of the electro-magnetic suspension system in adjusting its dynamical behavior according to the applied control algorithm, many researchers employed this suspension system as an active actuator to reshape the dynamics of some rotating machinery [21; 22; 23; 24]. [PERSON] and [PERSON] [21] implemented a non-linear magnetic absorber based on the push-pull control mechanism to mitigate the lateral vibrations of a non-linear Jeffcott rotor system. [PERSON] et al. [22; 23; 24] applied a non-linear position-velocity control algorithm to enhance the dynamical characteristics of an asymmetric rotor system utilizing the four-pole electro-magnetic suspension system as an actuator. One of the feasible control techniques that are applied extensively to suppress the non-linear oscillations of different dynamical systems is known as Positive Position Feedback (PPF) controller [25; 26; 27; 28; 29]. The PPF-controller is a second-order filter that couples to the targeted vibratory system linearly to suppress its undesired oscillation. [PERSON] et al. [29] applied the PPF-controller to mitigate the non-linear oscillation of the eight-pole electro magnetic suspension system. The authors reported that the PPF-controller can eliminate the lateral oscillation of the rotor system at the perfect resonance conditions (i.e., when the rotor angular velocity is equal to the natural frequency of the suspension system); however, the PPF-controller can add more excessive vibratory energy to the rotor system rather than suppress it, if the perfect resonance conditions have been lost. Besides the PPF-controller, there is another efficient control algorithm known as the Linear Integral Resonant (LIR) controller that has been utilized to dampen the non-linear vibration of many dynamical systems [30; 31; 32; 33; 34; 35; 36; 37; 38]. The LIR-controller is a first-order filter that has been coupled to the targeted system to modify its linear damping coefficients, which ultimately mitigates the undesired oscillation. [PERSON] et al. [38] studied the vibration control of the electro-magnetic suspension system utilizing the LIR-controller. The authors reported that the integration of the LIR-controller to the rotor system can eliminate the non-linear bifurcations and mitigate the system vibration to respond as a linear system. With this work, a modified version of the control algorithms that have the advantage of both the PPF-controller and LIR-controller has been investigated for the first time. The proposed control strategy is a combination of first-order and second-order filters that have been couped to the considered suspension system to reduce the undesired lateral oscillation. The main task of the second-order filter is to behave as a PPF-controller to eliminate the catastrophic oscillations of the considered system at the perfect resonance conditions, while the essential role of the first-order filter is to work as an LIR-controller that increases the controlled system linear damping, which ultimately removes the drawback of the PPF-controller when the resonance condition is lost. According to this philosophy, the whole system mathematical model was derived.