# Rotating Lorentz Force Magnetic Bearings' Dynamics Modeling and Adaptive Controller Design
## 4 RLFMB Online Controller Design
### Controller Optimization Based on Current Feedback  
The third-order SISO system RLFMB can be reduced to a normal second-order system by inferring from Equation (6) that its third-order term coefficient is relatively small. To achieve the suppression of the disturbance under the influence of the disturbance torque, the system rigidity must be raised and the system damping must be enhanced. In order to avoid the peak region that is prone to resonance, it is necessary to take into account both the specified angular displacement signal frequency and the undamped natural oscillation frequency of RLFMB while constructing the closed-loop control structure. It is specifically as follows.  
For a typical second-order system, when the input signal is sinusoidal, there are  
\[\dot{\theta}(t)+2\xi\omega_{n}\dot{\theta}(t)+{\omega_{n}}^{2}\theta(t)=\frac{ {\omega_{n}}^{2}A\sin(\omega_{e}t)}{\omega_{e}k}, \tag{26}\]  
Its amplitude-frequency characteristics are  
\[H(\omega_{e})=\frac{A/k\omega_{e}}{\sqrt{\left(1-\left(\omega_{e}/\omega_{n} \right)^{2}\right)^{2}+\left(2\xi\omega_{e}/\omega_{n}\right)^{2}}}. \tag{27}\]  
\(\omega_{n}\) is the undamped natural oscillation frequency of a typical second-order system, while \(\xi\) is the damping ratio. From Equation (27), it can be seen that the closer \(\omega_{n}\) and \(\omega_{e}\) are, the higher the excitation of the input signal to the system and the greater the vibration peak.  
When RLFMB is operating, the mechanical electromagnetic parameters should be reasonably constructed so that the system's natural oscillation frequency avoids the frequency of the input signal in order to prevent excessive vibration excitation generated by the output signal to the system. In order to give the RLFMB closed-loop control system an inner loop with a faster response and wider bandwidth, more feedback channels can be added in addition to angular displacement and angular velocity feedback. This will help to address the shortcomings of narrow angular displacement and angular velocity feedback bandwidth.  
It can be seen from Equation (24) that the purpose of the velocity and displacement closed loop is to converge the velocity and displacement deviation to zero in a limited time, while at the same time improving the rapidity of the system. The three-ring control structure is shown in Figure 8.  
Figure 7: Structure of adaptive fault-tolerant control strategy.  
The current loop can be expressed as follows:  
\[\left\{\begin{array}{l}T_{y}=BiL_{a}D=J\tilde{\theta}\\ i=\frac{j\tilde{\theta}}{B\mu_{a}D}=-g(s)A\\ g(s)=K_{p}+\frac{K_{i}}{s}+K_{d}s\end{array}\right.. \tag{28}\]  
In the above formula, \(i\) is the control current of RLFMB, \(g(s)\) indicates the current control rate of RLFMB, and \(A\) is the measured value of the sensor. The controller for current feedback uses traditional PID control because the purpose of current feedback is to ensure that the actual current of the coil closely follows the reference current, thus ensuring the implementation of the given angular displacement equation. PID control is simple and easy to implement and can be placed directly on the RLFMB's drive circuit.