## 5 Simulation and Experiment Study  
### Simulation Study  
In this section, the proposed PACFTB control strategy was simulated in MATLAB/Simulink environment to examine its effectiveness. All the parameters in simulation are listed in Table 1 based on finite element analysis. The values of parameters of the proposed controller were: \(k_{1}=40\), \(k_{2}=1000\), \(k_{3}=10,\)000, \(k_{d}=1\), \(k_{q}=1\), \(\tau_{1}=0.1\), \(\tau_{2}=0.1\), \(\tau_{3}=0.1\), \(\tau_{4}=5\times 10^{6}\), \(m_{1}=0.001\), \(m_{2}=0.001\), \(m_{3}=0.001\), \(m_{4}=0.001\), \(p_{1}=p_{2}=5\) and \(q_{1}=q_{2}=3\). The values of parameters in command filter were chosen as: \(\xi=0.707\), \(\omega_{n}=300\), the magnitude and rate constraints were 200 A and 20,000 A/s. As shown in Figure 3, the designed fuzzy membership functions were: \(\mu_{i}^{j}=\exp\left[-(x+l\times 2)^{2}/7\right]\), where \(l\in Z\) and \(l\in[-2,\ 2]\). The initial values of vector \(W_{i}(i=1,2,3)\) were defined as \(W_{i}=(\underbrace{0.1,0.1,\ldots,0.1}_{S^{2}})\). Notice that all the state variables input to the FLS should be included in fuzzy membership functions. That is to say, all the state variables input to the FLS must be normalized. In order to test the effectiveness of the proposed PACFTB, the conventional command-filtered backstepping controller (CBC) and PI controller were applied as comparisons to demonstrate the better performance of the designed controller. In simulation, the given d-axis current is 80 A, and the given velocity was set as follows:  
\[v_{1}^{d}=\left\{ \begin{array}{ll}4,&0\leq t<3\\ 10,&3\leq t<8\\ 0,&8\leq t<14\end{array} \right.\]  
Meanwhile, external disturbances \(F_{L}=200\sin\left(\pi t\right)\) were added in 6 s. The signal curves in the simulation are shown in Figures 4-9. By comparing the diagrams of Figures 4 and 5, the PACFTB had faster convergence than CBC and PI controller, which declared the better dynamic performance of PACFTB. In addition, it can be found that PACFTB had better static and dynamic performance than CBC and PI controllers when external load disturbance changed. From the point of view of indices, the response time of our designed controller was 87.5% shorter than that of PI controller, and 83.3% shorter than that of traditional CBC controller. Moreover, the chattering rate of PI caused by external \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(R_{s}(\Omega)\) & 0.0709 & \(R_{r}(\Omega)\) & 0.1311 \\ \(L_{s}\)(mH) & 4.8 & \(L_{r}\)(mH) & 4.8 \\ \(L_{m}\)(mH) & 3.9 & \(M\)(kg) & 351.264 \\ \(D\)(kg/s) & 40.95 & \(h\)(m) & 0.2 \\ \(p\) & 4 & \(l\)(m) & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the investigated LIM. Figure 3: The fuzzy membership functions. Figure 6 shows the d-axis current, which can imply that the proposed controller could stabilize magnetic field. The input and output signals of the command filter are shown in Figure 7, which shows that the output signal \(i_{qs}^{c}\) could effectively track the input signal \(i_{qs}^{d}\) availably and we could get the differential of the input signal directly from the output of the filter. At the same time, by the use of constraint command filter, the output signal was well bounded. The compensator signal is shown in Figure 8. The filter compensating signal. Figure 8: The filter compensating signal. Figure 7: The q-axis current without filtering and with filtering. Figure 9: Adaptive estimation for unknown parameters. (**a**) The estimation of \(f_{1}(X)\); (**b**) The estimation of \(f_{2}(X)\); (**c**) The estimation of \(f_{3}(X)\); (**d**) The estimation of load disturbance. Figure 8, which compensated for the inherent error of the command filter so that the input saturation problem was figured out.