## 3 Results and Discussion  
The step-response performance is an important indicator for measuring the system. In order to evaluate the effectiveness of the proposed approach, step-response experiments in 1% and 10% of the whole stroke are conducted in this section (compared with the indexes (Table 1) obtained by the traditional control method), and the steady-state error of the system is evaluated. According to the actual machining demand and [27], the reference performance indices used in the research are listed in Table 1.  
Using the torque-equilibrium method to obtain the torque coefficient of the motor and the identification method for motor parameters proposed by [PERSON] et al. [15], the relevant parameters of the experimental platform were obtained (Table 2). (Five repeated experiments were conducted, and the results were found to be similar. Therefore, this paper takes the results of one of them as an example.) \begin{table}
\begin{tabular}{c c} \hline \hline
**Performance Index** & **Value** \\ \hline Step-response time for 1\% of the whole stroke (ms) & 1.3 \\ Step-response time for 10\% of the whole stroke (ms) & 4.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reference performance indices for the motor of the GS. Figure 3: The experimental platform. By substituting the data in Table 1 into Equation (9), the following equation is obtained:  
\[X(k+1)=\left[ \begin{array}{cc}1&2.499917e^{-5}\\ 0&0.999934\end{array} \right]X(k)+\left[ \begin{array}{cc}0.000211\\ 16.891609\end{array} \right](u(k)+d(k))\]  
The parameters selected for the DSVC+DDC are listed in Table 3.  
Figures 4 and 5 respectively show the step-response results in 1% and 10% of the whole stroke of the GS controlled by the DSVC+DDC. The solid black line is the expected position, and the solid red line is the actual position of the rotor. The overshoot of the two red solid lines is less than 5%. Figure 4 and Table 1 show that the DSVC+DDC reduces the rise time of step-responses in 1% of the whole stroke from 1.3 ms to 1 ms (\(T_{r1}\)), a reduction of 23%. Figure 5 and Table 1 show that the DSVC+DDC reduces the rise time of step-responses in 10% of the whole stroke from 4.8 ms to 2 ms (\(T_{r}\)), reducing it by 58%. Because small-angle swings are dominant in the operation of the motor, the steady-state error in the step-response in 1% of the whole stroke is the primary factor influencing the machining accuracy (Figure 6). As shown in Figure 6, the steady-state error of the GS small-angle position tracking is within 20 urad, that is, less than 1.2\({}^{\circ}\)%, which meets the accuracy required in processing. In the experimental process, there are two main factors affecting the tracking error. One is that accurate dynamic parameters such as inertia and damping are the premise for realizing the proposed control method. The error of dynamic parameter identification results will affect the model's precision, which will affect the control's precision; the other factor is that the chattering phenomenon is inevitable due to the nature of the sliding mode variable structure control itself [13], which will have a negative impact on the control accuracy. To further reduce tracking errors, our future work will focus on these factors. Figure 4: Step-responses in 1% of the whole stroke (0.0038 rad). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(Ce^{T}\) & \([80\ 1]^{T}\) \\ \(\alpha\) & 0.99 \\ \(\beta\) & 0.002 \\ \(g\) & 0.005 \\ \(m\) & \(5\times 10^{-6}\) \\ \(T_{s}/(\mathrm{s})\) & \(25\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the DSVC+DDC. Compared with the reference performance index (Table 1), the experimental results show that the proposed method significantly improves the step-response speed, especially for the step-responses of 10% of the whole stroke, under the condition of ensuring a certain accuracy and overshoot.