## 5 Numerical Validations and Temporal Oscillations
The performance of the ALIPPF-controller (i.e., \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\), \(\sigma_{1}=\sigma_{2}=\sigma=0.15\)), LIPPF-controller (i.e., \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\)), and the conventional PPF-controller (i.e., \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\), \(\eta_{2}=\eta_{4}=\eta_{7}=\eta_{8}=0\)) in eliminating the non-linear oscillation of the considered system was explored in Figures 11 and 12 utilizing \(f\) as the main bifurcation parameter at two different values of the rotor angular speed \(\Omega\) (i.e., when \(\Omega=\omega+\sigma\), \(\sigma=0,\ 0.15\)). Figure 11 shows that the three controllers (i.e., PPF, LIPPF, and ALIPPF) can suppress the rotor oscillation amplitudes (\(a_{1}\) and \(a_{2}\)) close to zero at \(\sigma=0.0\), which agrees with Figures 9 and 10. It is clear from Figure 11 that the three control techniques have the same performance in eliminating the rotor oscillation when \(\sigma=0.0\) along the \(f-\) axis. On the other hand, Figure 12 shows that the conventional PPF-controller failed to suppress the rotor lateral oscillations when \(\sigma=0.15\), where the system may lose its stability at the large eccentricity magnitudes. In addition, the figure demonstrates that LIPPF-controller can mitigate the rotor vibrations to a very small level along the \(f-\) axis. Moreover, the figure confirms that the ALIPPF can eliminate the rotor system regardless of the eccentricity magnitude. To investigate the transient and steady-state oscillatory motions of both the rotor system and the connected controllers, the system equations of motion (i.e., Equations (21)-(26)) were solved numerically using the ODE45 MATLAB solver according to Figure 12 (i.e., when \(\sigma=0.15\)) at three different values of the rotor eccentricity \(f\) (i.e., \(f=0.02,\ 0.05\), and \(0.08\)) as shown in Figures 13-15. Figure 13 compares the performance of the PPF-controller (i.e., \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=0.2\), \(\eta_{2}=\eta_{4}=\eta_{7}=\eta_{8}=0\)), LIPPF-controller (i.e., \(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\)), and the ALIPPF-controller (\(\eta_{1}=\eta_{3}=\eta_{5}=\eta_{6}=\eta_{2}\eta_{7}=\eta_{4}\eta_{8}=0.2\), \(\sigma_{1}=\sigma_{2}=\sigma=0.15\)) when \(f=0.02\) and \(\sigma=0.15\). Figures 14 and 15 are repetitions of Figure 13, but when \(f=0.05\) and \(0.08\), respectively. Comparing Figures 13-15, one can conclude that the conventional PPF-controller has failed to suppress the rotor vibrations in the three studied cases, where the system performs high amplitude periodic motion (at \(f=0.02\)), quasiperiodic motion (at \(f=0.05\)), and chaotic oscillation (at \(f=0.08\)). On the other hand, Figures 13-15 demonstrate the high vibration suppression efficiency of the LIPPF-controller and ALIPPF-controller. Figure 13: _Cont._  
Figure 13: Rotor electro-magnetic suspension time response, Poincare map, and the frequency spectrum with the PPF-controller, LIPPF-controller, and ALIPPF-controller at \(\sigma=0.15\) and \(f=0.02\): (**a,d**) the temporal oscillations of the rotor system in \(X_{1}\) and \(X_{2}\) directions, (**b,e**) the corresponding Poincare map of the rotor system, (**c,f**) the corresponding frequency spectrum of the rotor, and (**g–j**) the temporal oscillations of the controllers. Figure 14: Rotor electro-magnetic suspension time response, Poincare map, and the frequency spectrum with the PPF-controller, LIPPF-controller, and ALIPPF-controller at \(\sigma=0.15\) and \(f=0.05\): (**a**,**d**) the temporal oscillations of the rotor system in \(X_{1}\) and \(X_{2}\) directions, (**b**,**e**) the corresponding Poincare map of the rotor system, (**c**,**f**) the corresponding frequency spectrum of the rotor, and (**g**–**j**) the temporal oscillations of the controllers.