## 1 Introduction  
The quality of an ultra-high vacuum environment is increasingly becoming a limiting condition in the advancement of cutting-edge scientific research. Active magnetic bearings (AMBs), with the merits of being free of oil pollution, friction, micro-vibration, etc. [1; 2; 3; 4], is particularly suitable for application scenarios which require high-speed, vacuum, and an ultra-clean environment [5; 6; 7; 8]. As one kind of high-performance pumping equipment, the magnetically suspended turbo molecular pump (MSTMP) has become a key component for obtaining an ultra-high vacuum environment  
In pursuit of high pumping speed and high compression ratio, the turbo rotor of the MSTMP is typically designed with multi-stage blades [9]. The dominant and prevalent vibrations of the turbo rotor are rotor mass imbalance vibration (co-frequency synchronous vibration) and multi-stage-blade modal vibration (multi-frequency mode vibration) [10], both of which significantly threat the stability of the magnetically suspended rotor (MSR) system. Notably, consideration of the phase stability margin should not be neglected while performing vibration suppression. Otherwise, the impact of the high-speed AMB rotor on the auxiliary bearing during an unstable state will lead to severe wear and deformation of blades, which will result in irreversible mechanical structural damage to the turbo rotor [11; 12], and will inevitably result in substantial economic losses. The various disturbance factors that influence the stability of an MSTMP originate from different sources and exert different impacts. These disturbance factors are complicatedly related to electromagnetism, machinery, processing and assembly, and control methods [13]. For instance, even though complete balancing can theoretically be achieved through offline and online dynamic balancing corrections, residual imbalanced mass cannot be eliminated in practice [14]. In particular, the multi-stage blades of the turbo rotor are fixed at a single end of the elongated spindle, presenting a cantilevered structure. Such a cantilevered turbo rotor is highly susceptible to multi-stage-blade modal vibration when subjected to a current step signal during suspension or an imbalanced excitation force during high-speed rotation. Fortunately, the stiffness and damping of the AMB-rotor system are adjustable [15]. So, by improving the structure and optimizing the multi-parameters of the AMB controller, the balance between phase stability margin and notch vibration performance can be achieved. In fact, the interplay between mechanical design and control methods in an MSTMP is complex. Mechanical design can provide the essential guidance for control methods, but its problem-solving scope is inherently limited, necessitating effective control strategies to address various vibration issues. The continuous improvement of the controller structure and the continuous optimization of the multi-parameters are thus of paramount importance. For instance, [PERSON] et al. investigated a feedforward approach to compensate for the displacement stiffness of high-speed magnetically suspended centrifugal compressors, which was used to suppress rotor mass imbalance vibration [16]. [PERSON] et al. designed two-stage notch filters at low speed and high speed to suppress synchronous vibration caused by the gyroscopic effect [17]. An approach that involved adjusting the angular acceleration of the turbo rotor to make the rotor speed skip the natural frequency in a brief timeframe was proposed in [18] to avoid resonance. A current vibration suppression strategy based on the phase shift generalized integrator (PSGI) was proposed to filter out synchronous components in the displacement signal of the AMB-rotor [19]. However, it was complicated to select the appropriate phase shift angle according to the pole deviation angle of the imaginary axis. [PERSON] et al.