# Rotating Lorentz Force Magnetic Bearings' Dynamics Modeling and Adaptive Controller Design
## 5 Simulation Results and Analysis
### Angular Displacement Signal Tracking Error Comparison  
The traditional PID controller is straightforward and simple to use, but it has a lot of empirical components and complex percentage, integral, and differential coefficients to change. The PID settings are optimized using a genetic algorithm. The absolute value of the error time integral performance index is employed as the minimal objective function of parameter selection in order to achieve satisfactory dynamic features of the transition process. The penalty function is used, and the overshoot is used as the term of the ideal index in order to minimize the overshoot as much as is feasible.  
In our work, the crossover probability was set to 0.9, the mutation probability to 0.03, and the initial population size was set at 30. The proportional coefficient was 9.7528, the integral coefficient was 2.2521, and the differential coefficient was 0.0817 after iteratively optimizing PID parameters.  
Because the robust sliding mode controller needs to define the upper and lower bounds of the disturbance, the upper bound is 10% of the input signal, and the lower bound is 0.  
Figure 8: Structure of RLFMB three-loop control.  
In the approach law, the coefficient of the sign function \(\varepsilon\) is 0.1 and the coefficient of the power function \(k\) is 1.  
Table 1 shows the simulation parameters of the adaptive controller. Table 2 gives the structural parameters of RLFMB, and the parameters of Table 2 are brought into Equation (26). The undamped natural oscillation frequency of the system is 0.0768, and when the input sinusoidal signal meets \(\omega_{\varepsilon}>>\omega_{n}\), the amplitude of the excited vibration is approximately equal to 0.  
In summary, let the given angular displacement signal be \(\theta(t)=\sin t\). In the legend of the simulation diagram, Adaptive refers to the adaptive controller, PID refers to the PID controller, and Robust refers to the robust sliding mode controller. The simulation results are shown in Figure 9.  
\begin{table}
\begin{tabular}{c c} \hline
**Items** & **Value** \\ \hline \(k\) & 5 \\ \(\gamma\) & 10 \\ \(\eta\) & 0.1 \\ \(c_{i}\) & \([-2,-1,0,1,2]\) \\ \(b_{i}\) & 3.0 \\ Initial value of the grid weight element & 0.1 \\ \(\overset{\wedge}{\lambda}(0)\) & 1.0 \\ \(\varepsilon\) & 0.1 \\ \(k\) & 1 \\ \hline \end{tabular}
\end{table}
Table 1: Controller simulation parameters.  
\begin{table}
\begin{tabular}{c c} \hline
**Items** & **Value** \\ \hline \(B/(\mathrm{T})\) & 0.5 \\ \(L_{a}/(\mathrm{m})\) & 4 \\ \(D/(\mathrm{m})\) & 0.1 \\ \(L/(\mathrm{H})\) & 0.1 \\ \(R/(\mathrm{\Omega})\) & 3.0 \\ \(J/\left(\mathrm{kg}\cdot\mathrm{m}^{2}\right)\) & 0.425 \\ \hline \end{tabular}
\end{table}
Table 2: RLFMB structure parameters.  
Figure 9: _Cont._  
As can be seen in Figure 9, the output of the adaptive controller, for a given sinusoidal signal, almost never overshoots and can converge the tracking error to the minimum the first time, whereas the static error of the PID controller and the robust sliding mode controller cannot be eliminated even after many oscillations. As a result, the RLFMB can quickly cause the load compartment to rotate at a specified angle and prevent repeated swing thanks to the adaptive controller. This not only enhances the satellite load's capacity to maneuver with agility but also prevents the recurrent oscillations that cause other components to vibrate.