## 3 AMB Rigid Rotor System Based on Fuzzy Control Strategy  
The dynamic performance of AMBs exhibits strong nonlinearity. During the high-speed operation of AMB motors, the vibration frequency varies with the rotational speed, making traditional fixed-parameter control methods inadequate to adapt to these dynamic changes. However, the AFC strategy can adjust parameters in real time to accommodate changes in the rotational speed, optimizing the system's dynamic response, thereby enhancing systemic stability.  
The AMB high-speed rigid rotor system itself is an unstable strong nonlinear system, and its mathematical model is not easy to accurately obtain; thus, when fuzzy control is used as a classical nonlinear control method, the combination of the traditional PID control and fuzzy control is utilized to achieve the fuzzy adaptive PID control of vibrations of a AMB high-speed rigid rotor system. In the control system depicted in Figure 3, the fuzzy PID controller structure is based on the traditional PID control, with the fuzzy  
Figure 1: Structure of AMBs-rigid rotor system.  
Figure 2: Schematic diagram of rotor imbalance.  
controller correcting the proportional, integral, and derivative systems to achieve the goal of cooperative control. The schematic diagram illustrating the structure of fuzzy PID control is provided in Figure 3. Typically, the implementation of fuzzy control mainly includes fuzzification, knowledge base, logical judgment, and defuzzification. Its working principle is as follows: the filtered rotor vibration displacement signal is compared with the reference displacement signal to generate the error signal and its rate of change, which serve as the input variables for the fuzzy controller. Based on a predefined fuzzy rule base, the fuzzy controller performs fuzzification, rule evaluation, and defuzzification using the Mamdani inference method to generate a continuous control signal. This control signal acts on the rigid rotor system, driving the system output to track the reference signal. Throughout the control process, the fuzzy controller dynamically adjusts the amplitude and direction of the control signal to achieve robust control against the system's nonlinear characteristics and external disturbances. Simultaneously, the notch filter effectively suppresses periodic interference components, further enhancing the system's control accuracy and stability.  
The design of the fuzzy PID control in the AMB rigid rotor system is shown in Figure 3. Due to its four-degree-of-freedom AMBs and rotor model of internal coupling, the four fuzzy control rules are the same as those of four-degree-of-freedom AMBs.  
Using a dual-input and three-output fuzzy controller, the PID control parameters are corrected according to the different variations in the error \(E\) and its derivative \(E_{C}\), and the updated values are replaced by the former PID parameter values; then, the updated formula is calculated as follows:  
\[\left\{\begin{aligned} & K_{p}=K_{p0}+\Delta K_{p}\\ & K_{i}=K_{i0}+\Delta K_{i}\\ & K_{d}=K_{d0}+\Delta K_{d}\\ \end{aligned}\right., \tag{2}\]  
Figure 3: Schematic diagram of AMBs-rigid rotor system with fuzzy control.  
The parameters \(K_{p}\), \(K_{i}\), and \(K_{d}\) are the actual parameters of the controller; \(K_{p0}\), \(K_{d0}\), and \(K_{d0}\) are the PID preset tuning parameters; \(\Delta K_{p}\) is the proportional coefficient correction; \(\Delta K_{i}\) is the integral coefficient correction; and \(\Delta K_{d}\) is the differential coefficient correction.