## 7 Simulation Verification  
The simulation was carried out using MATLAB(2022b)/Simulink software. The performance of the proposed control strategy was verified by the comparative analysis of the PI control, ADRC, and proposed FTSMC-ESO control strategy.  
The PID parameters were obtained through trial and error. Adjusted the parameters of the three components of ADRC separately. First, the parameters of the differential tracker were adjusted until it could track the input signal and its differential signal well. Then, the parameters of the extended state observer were adjusted in sequence until the observer performed well in terms of observing the respective states. Finally, adjusted the parameters of the nonlinear combined controller in sequence until the system output error was minimized. The parameters of the three components of FTSMC-ESO were also adjusted separately. The adjustment steps and values for the parameters of the differential tracker and the extended state observer were consistent with those of ADRC. Finally, the parameters of the fast terminal sliding mode controller were adjusted sequentially based on experience until the system control performance reached an ideal state. Ultimately, the parameter values for each controller were presented in Table 5.  
The sine wave signal was selected for the loading simulation to verify the dynamic performance of the controllers, and the simulation results are shown in Figure 11. The system tracks the signal well under the three control strategies, and the error under FTSMC-ESO is smaller than that under PI control and ADRC.  
A half-sine wave loading simulation was carried out, and the simulation results are shown in Figure 12. There was almost no difference in the effect of the three control strategies in the output curve (a), and the signal was well tracked. In the error curve (b), it can be clearly seen that the errors under the three control strategies increased sequentially from the PI control to the ADRC, and to the FTSMC-ESO.  
The anti-disturbance ability of the proposed strategy was verified by applying external disturbance to the electro-hydraulic servo loading system; step disturbance \(15 step(t-4)KN\) was applied to the system at the fourth second. The simulation results are shown in Figure 13.  
From Figure 13, it can be seen that the three control strategies can better track the ramp signal. After the external disturbance was applied to the system, the PI control did not have the ability to resist the disturbance; ADRC and FTSMC-ESO effectively suppressed  
Figure 11: Simulation of sine wave loading.  
Figure 12: Simulation of half-sine wave loading.  
\begin{table}
\begin{tabular}{l l} \hline
**Controller** & **Parameters** \\ \hline PI & \(k_{p}=1.25,k_{i}=0.001\) \\ \hline \multirow{3}{*}{ADRC} & Tracking differentiator: \(h=1\times 10^{-5}\), \(\varphi=2.95\times 10^{5}\) \\  & ESO: \(b=175785\), \(\beta_{1}=5\times 10^{4}\), \(\beta_{2}=9\times 10^{6}\), \(\beta_{3}=4\times 10^{8}\), \(\delta=0.0001\) \\  & Nonlinear feedback controller: \(k_{p}=155,k_{d}=1.5,\alpha_{3}=0.98,\alpha_{4}=1.1\) \\ \hline \multirow{3}{*}{FTSMC-ESO} & Tracking differentiator: \(h=1\times 10^{-5}\), \(\varphi=2.95\times 10^{5}\) \\  & ESO: \(b=175785\), \(\beta_{1}=5\times 10^{4}\), \(\beta_{2}=9\times 10^{6}\), \(\beta_{3}=4\times 10^{8}\), \(\delta=0.0001\) \\ \cline{1-1}  & Fast terminal sliding mode \\ \cline{1-1}  & controller:\(c_{1}=8,c_{2}=2730,c_{3}=900,\sigma=\frac{3}{5},k_{1}=5020,k_{2}=0.8,\Delta=0.05\) \\ \hline \end{tabular}
\end{table}
Table 5: Controller parameters.  
Figure 13: Simulation of ramp loading under external disturbance.  
the disturbance and returned to the equilibrium state. Under the proposed FTSMC-ESO, the fluctuation of the system after external disturbance was smaller than that in ADRC, and the time required to return to the equilibrium state was shorter than that in ADRC; hence, the control system has good robustness to disturbance.