## 3 Design of the Control Scheme
### High-Speed Train Tracking Control  
**Example 1**. : _This part is used to verify the control performance of ALADRC when the HST has no coefficient time-varying during operation. The HST with ALADRC and LADRC operates in the same environment, and the control performance of the two controllers is reflected through the tracking of the ideal speed by the HST. ALADRC and LADRC have the same initial parameters, and their parameters come from the values obtained by IPSO optimization in Section B, as shown in Table 3. Other parameters are set as follows: \(\varphi_{\xi}=\varphi_{p}=\varphi_{d}=1\), \(\alpha=50\), and \(\rho=0.5\)._  
Figure 5: LADRC parameters change curve. Figure 6: Objective function value change curve. The speed and displacement tracking curves of the four carriages are shown in Figures 7 and 8. It can be seen that both ALADRC and LADRC can make HSTs operate stably according to the ideal speed and displacement curves. When the ideal speed suddenly changes, the speed and displacement tracking errors of both are relatively small. They can meet the requirement that the maximum speed deviation of the train operation does not exceed \(\pm\)2 km/h. Compared with LADRC, ALADRC obviously has better control effect. The speed and displacement tracking errors of the four carriages are shown in Figures 9 and 10. It can be seen from Figure 9 that the control performance of ALADRC and LADRC is almost the same at the beginning of the acceleration stage and the final braking stage. However, ALADRC has better control performance during the HST operation. When the HST encounters additional resistance disturbance and a sudden change of the ideal speed signal, ALADRC can suppress the speed fluctuation better and re-track the ideal speed signal in a shorter time. The speed tracking error always exists during HST operation, which will lead to the accumulation of HST displacement error. Therefore, the displacement tracking error of HSTs with ALADRC is significantly smaller than LADRC. However, both controllers have a high parking accuracy. The adaptive values of \(k_{p}\) and \(k_{d}\) are shown in Figure 11. The results show that parameters \(k_{p}\) and \(k_{d}\) in LADRC can be adjusted in real-time according to the dynamic characteristics and tracking errors. Through calculation, the speed mean absolute errors of ALADRC and LADRC are 0.0096 km/h and 0.0271 km/h, respectively. This shows that the control performance of LADRC can be effectively improved by setting adaptive laws for PD controller parameters and LESO. This also proves that ALADRC is more suitable as a HST speed tracking controller. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameters** & \(\mathbf{k_{p0}}\) & \(\mathbf{k_{d0}}\) & \(\mathbf{w}\) & \(\mathbf{b}\) \\ \hline First carriage & 4969 & 268 & 200 & 0.111 \\ Fourth carriage & 4991 & 200 & 200 & 0.113 \\ \hline \end{tabular}
\end{table} 
Table 3: Controller parameters of ALADRC and LADRC. Figure 7: Speed tracking curve of the four carriages: (**a**) Speed tracking curve of the first two carriages; (**b**) Speed tracking curve of the last two carriages. Figure 8: Displacement tracking curve of the four carriages: (**a**) Displacement tracking curve of the first two carriages; (**b**) Displacement tracking curve of the last two carriages. Figure 10: Displacement tracking error of the four carriages: (**a**) Displacement tracking error of the first two carriages; (**b**) Displacement tracking error of the last two carriages. Figure 9: Speed tracking error of the four carriages: (**a**) Speed tracking error of the first two carriages; (**b**) Speed tracking error of the last two carriages. **Example 2**. : _This part is used to verify the control performance of ALADRC when HST coefficients change with time. The coefficients of the Davis equation and the mass of the carriage may change during the HST operation, and periodic functions are added to the above coefficients as time-varying perturbations [37].