## 1 Introduction  
As an advanced bearing technology, active magnetic bearings (AMBs) are widely used in high-speed rotating machines, such as high-speed motors, flywheel energy storage systems, centrifugal pumps, and other high-speed rotating machines due to their significant advantages of being contactless and frictionless and having high precision and a long life [1, 2, 3]. The AMB system is a typical electromechanical coupled system with complex dynamic characteristics, strong nonlinearities, and parameter uncertainties. During the operation of high-speed AMB motors, significant vibrations are generated due to factors such as rotor imbalance and external disturbances, particularly during high-speed and acceleration phases. These vibrations not only degrade the motor's performance but may also lead to system instability or damage. Therefore, it is necessary to maximize system stability and control rotor vibration. Currently, fuzzy control [4; 5; 6], as a modern control strategy, does not rely on precise mathematical models, and it has good intelligence and robustness for both linear and nonlinear systems [7; 8; 9; 10]. Fuzzy control, as a control method that combines fuzzy control with classical PID control algorithms, significantly improves the system's dynamic responsiveness and operational stability by virtue of its intelligence and robustness. [PERSON] [11] proposed a model based on a fuzzy controller for active electromagnetic bearings to achieve fast and stable levitation. [PERSON] [12] proposed a fuzzy adaptive control strategy to improve rotor speed recovery. [PERSON] [8] proposed a filter-based fuzzy adaptive control scheme to improve the convergence and stability of the system. [PERSON] [13] proposed a suspension control strategy based on adaptive nonsingular terminal sliding mode control. By integrating adaptive control with terminal sliding mode control, this strategy effectively reduces system chattering and significantly enhances the dynamic performance of the suspension system. AFC is capable of handling nonlinear systems and does not rely on precise mathematical models. The dynamic characteristics of AMB systems are complex and highly nonlinear, making traditional PID control inadequate to meet the systems' requirements. In contrast, the AFC dynamically adjusts control parameters in real time through fuzzy rules, effectively addressing the nonlinear characteristics of the system. In general, in order to ensure the stability of the system, it is also necessary to suppress the vibration of the rotor due to unbalance and other factors in high-speed operation. The ANF is capable of monitoring rotor vibration signals in real time and adjusting the center frequency and bandwidth based on changes in rotational speed, effectively suppressing vibration components that are synchronous with the rotational frequency. [PERSON] [14] addresses the problem of the rotor unbalanced vibration of active magnetic bearings by estimating the rotational speed via the bi-directional filtering of noise and generating synchronized signals to eliminate the periodic unbalance effect. [PERSON] [15] improves the vibration suppression performance of magnetic bearings. An adaptive trap based on the minimum mean square error is used to extract vibration signals from rotor displacement signals. [PERSON] [16] proposed a variable-angle compensation algorithm based on an adaptive notch filter to suppress synchronous unbalance vibration in finite iterative searches in an AMB rigid rotor system. This algorithm directly achieves unbalance compensation based on real-time rotor positions. [PERSON] [17] proposes a speed-adaptive trap-based phase-locked loop for sensorless permanent magnet synchronous motors to suppress fluctuations induced by mid-frequency harmonics that is robust to perturbations in the full frequency range.