# Rotating Lorentz Force Magnetic Bearings' Dynamics Modeling and Adaptive Controller Design
## 3 RLFMB Rotational Dynamics Modeling
### RLFMB Rotor Dynamics Model  
After being activated, the coil in rotating Lorentz force magnetic bearings is exposed to the Lorentz force in the magnetic field. The coil travels through two working air gaps in series between the inner and outer magnetic steel and is subjected to two equal-sized and opposite-direction Lorentz forces that produce torque. When a torque is generated, the coil bracket's output shaft is driven to rotate, achieving fixed shaft rotation. The coil bracket and output shaft both use mechanical transmission structures. This method is described as follows:  
\[\left\{\begin{array}{l}F=\frac{H_{e}(2I_{pl}+2I_{g2})}{(\frac{I_{1}}{s_{1} \rho_{0}W}+\frac{I_{2}}{s_{2}\rho_{0}W}+\frac{I_{2}}{s_{0}}+\frac{I_{2}}{s_{1} \rho_{0}W}+\frac{I_{2}}{s_{2}\rho_{0}W}})S_{a}i\\ T=\frac{H_{e}(2I_{pl}+2I_{g2})}{(\frac{I_{1}}{s_{1}\rho_{0}W}+\frac{I_{2}}{s_{2} \rho_{0}W}+\frac{I_{2}}{s_{0}}+\frac{I_{2}}{s_{1}\rho_{1}\rho_{0}W}+\frac{I_{ 2}}{s_{2}\rho_{0}W}})S_{a}Li\\ T=Ja+K_{f}\omega+K_{e}\theta\end{array}\right. \tag{4}\]  
In the above formula, \(J\) is the moment of inertia of rotating Lorentz force bearings' rotor assembly and load, \(K_{f}\) is the frictional moment coefficient, \(K_{e}\) is the elastic moment coefficient, \(\alpha\) is the angular acceleration of the rotor, and \(\omega\) is the angular velocity of the rotor.  
The main part of the circuit of RLFMB is a coil, which is expressed by the following formula:  
\[\left\{\begin{array}{l}u=Ri+L\frac{di}{dt}+e\\ e=BL_{a}\frac{d\theta}{dt}\\ \end{array}\right. \tag{5}\]  
In the above formula, \(u\) is the operating voltage, \(R\) is the equivalent resistance of RLFMB, and \(L\) is the equivalent inductance of RLFMB, that is, the inductance of the coil. \(e\) is the back EMF, generated by the motion of the coil cutting the magnetic inductive lines in a magnetic field. \(L_{a}\) is the working length of the coil, \(\theta\) is the angular displacement of the coil in RLFMB, \(B\) is the magnetic induction intensity, and \(i\) is the operating current in the equivalent circuit.  
Simultaneously, (4) and (5) derive the transfer function from the input voltage to the rotor displacement:  
\[\frac{\theta}{u}=\frac{BL_{a}D}{(JR+Lk_{f})s^{3}+(k_{f}R+k_{e}R+B^{2}{L_{a}}^{ 2}D)s^{2}+k_{e}Ls} \tag{6}\]  
This can be rewritten into state-space form:  
\[\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx\end{array} \tag{7}\]  
In the above formula, \(x\) is the rotor state vector, \(u\) is the control input, and \(y\) is the rotor output. And  
\[A=\begin{bmatrix}0&1&0\\ 0&0&1\\ \frac{-k_{e}}{J}&\frac{-(k_{f}R+k_{e}R+B^{2}{L_{a}}^{2}D)}{LJ}&\frac{-(JR+Lk_{ f})}{LJ}\end{bmatrix}\]  
\[B=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}\]  
\[C=\begin{bmatrix}\frac{BL_{a}D}{LJ}&0&0\end{bmatrix}\]  
Consider that when the stable platform is working, it is affected by the environmental torque, and the attitude actuator such as the thruster and flywheel excites the spacecraft and stabilizes the platform to generate vibration when working, which adds unknowndisturbance term \(f(x_{1},x_{2})\). Take the RLFMB rotor displacement and angular velocity as the controlled quantity at the same time, so that the vector  
\[x_{1}=\begin{bmatrix}\theta\\ \omega\end{bmatrix} \tag{8}\]  
The following is derived:  
\[\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=bu+f(x_{1},x_{2})\end{array}\]  
In the above formula, \(u\) is the RLFMB output.  
RLFMB is a third-order system, as shown by Equation (6), and it can be controlled in accordance with the controllability condition. When designing the RLFMB structure and choosing magnetic steel, the best course of action can be planned to make the magnetic field at the air gap of the magnet uniform and to have a large phase margin and amplitude margin at the same time because each parameter is an actual physical quantity combined with Formula (3).