## 6 Conclusions  
Within this article, an advanced control technique (i.e., ALIPPF-controller) was presented to eliminate the non-linear vibrations of the rotor electro-magnetic suspension system. The introduced controller is a combination of both first-order and second-order filters that were coupled linearly to the electro-magnetic suspension system in order to reduce the undesired oscillations. Based on the suggested control method, the equations of motion that govern the dynamics of both the electro-magnetic suspension system and the proposed controller were derived as six non-linear ordinary differential equations. Then, an analytical approximate solution for the established dynamical model was obtained using the multiple time-scales perturbation procedure. The performance of the ALIPPF-controller in eliminating the lateral vibrations of the considered system was explored via plotting the different bifurcation diagrams. According to the introduced investigations given above, we can conclude the following important remarks:  
1. The uncontrolled eight-pole electro-magnetic suspension system may respond as a linear system at the small rotor eccentricity (\(f\)) (i.e., \(f<0.01\)).
2. At the large rotor eccentricity, the electro-magnetic suspension system may lose its stability to perform quasi-periodic or chaotic oscillations thanks to the complex bifurcation behaviors.
3. Integrating the conventional PPF-controller into the electro-magnetic suspension system can eliminate the rotor vibration amplitudes at the perfect resonance conditions (i.e., \(\sigma=0\)). 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.
4. The coupling of the LIR-controller to the eight-pole electromagnetic suspension system can eliminate the non-linear bifurcation and force the rotor to respond as a linear dynamical system. However, the controlled system may exhibit high oscillation at the perfect resonance conditions (i.e., \(\sigma=0\)).
5. The integration of the LIPPF-controller to the considered system can eliminate rotor oscillations at the perfect resonance conditions as well as suppress the non-linear vibrations to very small magnitudes if the resonance conditions have been lost.
6. The ALIPPF-controller (i.e., when setting \(\omega_{1}=\omega_{2}=\Omega\)) can eliminate the undesired vibrations of the electro-magnetic suspension system to zero regardless of the angular speed and eccentricity of the rotating shaft.  
Conceptualization, N.A.S., J.A. and M.S.M.; data curation, N.A.S.; formal analysis, N.A.S. and M.S.M.; funding acquisition, J.A. and M.S.M.; investigation, N.A.S. and M.S.M.; methodology, N.A.S.; project administration, N.A.S. and J.A.; resources, N.A.S., A.A.A.M. and M.S.M.; software, N.A.S. and A.A.A.M.; supervision, J.A.; validation, N.A.S. and J.A.; visualization, N.A.S. and A.A.A.M.; writing--original draft, N.A.S. and J.A.; writing--review and editing, N.A.S., J.A. and M.S.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Researchers Supporting Project, Taif University, Taif, Saudi Arabia, under grant number TURSP-2020/160. This work was also supported by the National Science Centre, Poland, under grant number OPUS 14 No. 2017/27/B/ST8/01330.  
Not applicable.  
The authors would like to thank Taif University, where this work was supported by the Taif University Researchers Supporting Project number (TURSP-2020/160), Taif, Saudi Arabia.  
The authors declare no potential conflict of interest with respect to the research, authorships, and/or publication of this article.