## 1 Introduction  
AMBs have been widely used in high-speed rotating machinery such as blowers, molecular pumps and flywheel energy storage due to their advantages of having no friction and active controllability [1, 2, 3, 4, 5]. However, the mass imbalance of the rotor is inevitable. Due to the existence of the above imbalance, a centrifugal force at the same frequency as the rotor speed is generated as the rotor rotates, which causes the rotor to vibrate. In high-speed rotating machinery, rotor mass imbalance is the main cause of rotor vibration [6]. Excessive vibration not only affects the suspension accuracy of the rotor but also affects the AMB system. For the rotor vibration of traditional rotating machinery, a dynamic balance machine is usually used for dynamic balance correction, and rotor vibration is reduced by reducing the unbalanced mass of the rotor. This method is offline dynamic balance, which not only makes it time-consuming to disassemble and transport the rotor but also affects the accuracy of dynamic balance correction because the dynamic balance environment is different from the working environment. To solve the vibration problem of the AMB rotor, researchers have conducted many studies and proposed many solutions. [PERSON] et al. proposed an online unbalance compensation method which integrates a least mean square (LMS) algorithm and ICM [7]. Through real-time calculation and adjustment, the method can effectively identify and compensate for the imbalance during rotor operation, thereby reducing vibration and improving system stability. The experimental results show that the method can significantly reduce rotor vibration at different rotational speeds and has strong engineering practicability. [PERSON] et al. devised a real-time polygon iterative search algorithm with a variable step size based on the rotor unbalance coefficient for an unbalanced compensator [8]. The AMB produces a controlled force equal in magnitude and opposite in phase to the rotor unbalance force. But this method requires a phase sensor to confirm the reference phase. [PERSON] et al. proposed a synchronous vibration suppression method based on phase advance compensation for variable speed maglev flywheel systems [9]. By designing a phase advance compensator, the synchronous vibration of the flywheel is solved. [PERSON] et al. designed an improved resonant controller to solve the vibration problem caused by displacement harmonics in the AMB rotor system [10]. The improved controller enhances the system's ability to suppress harmonic interference, effectively reduces vibration and improves the stability and dynamic performance of the system. [PERSON] et al. proposed a synchronous unbalanced vibration suppression method for maglev high-speed motors [11]. The polarity switching tracking filter is used to identify the synchronous vibration, and then the disturbance observer is used to observe and compensate for the disturbance in the resonance range, effectively suppressing the synchronous vibration in the whole speed range. [PERSON] et al. applied repetitive control to the AMB rotor system to suppress harmonic vibration torque, which significantly reduced the vibration level of the system under harmonic interference [12]. The effectiveness and applicability of the strategy are verified by simulation and experiment. [PERSON] et al. proposed a cascaded phase-shifted notch filter [13]. With high-speed maglev centrifugal compressor as the research object, harmonic vibration is effectively suppressed and the stability of system operation is improved through the phase compensation and notch wave property of a cascade filter. [PERSON] et al. studied the control method of a synchronous current in an active magnetic bearing system, focusing on the influence of radial displacement inconsistency on the system [14]. The control strategy effectively reduces the current fluctuation caused by displacement inconsistency and improves the stability and performance of the system. [PERSON] et al.