## 5 Numerical Simulation
### Simulation Results
#### 5.2.2 Fast-Motion Reference Trajectory  
For further capability investigation of the suggested control algorithm, a smooth fast-motion reference trajectory is chosen as  
\[x_{1d}(t)=\frac{\pi}{16}(1-\cos(2\pi t))\big{(}1-e^{-t}\big{)}(\text{rad}) \tag{66}\]  
The tracking characteristics of all controllers with the fast-motion reference trajectory are illustrated in Figure 8. As in the above case, all controllers still follow the desired trajectory. Due to the bandwidth limitations of EHSSs, the tracking errors of all control approaches depicted in Figure 9 rise remarkably along with the increase in the frequency of the reference trajectory. The tracking accuracies of all controllers in terms of the three performance indexes are illustrated in Table 4. As shown, the tracking performances of the PID controller and the RBF-SMC controller significantly degrade with the maximal final tracking errors, which are 0.5364 degrees and 0.0621 degrees, respectively. Meanwhile, the ESO-BC controller demonstrates robustness against uncertainties and disturbances based on the support of ESOs and known nominal system dynamics. However, it is worth noting that the proposed controller tracks the reference trajectory more accurately than others. This once again demonstrates the effectiveness of the recommended method in dealing with full model uncertainties and disturbances.  
Figure 7: The control signals of all controllers with slow-motion reference trajectory.  
The simulation results of the two case studies demonstrate the superiorities and robustness against disturbances and uncertainties of the proposed control mechanism compared with others in achieving high-accuracy tracking performance for the EHSS without a chattering phenomenon. Although the use of the RBF NNs to cope with unstructured uncertainties is efficient, a number of parameters need to be carefully tuned to obtain better performance, which impedes its applications. Additionally, the suitability of the selection of radial basis function type when employing the RBF NNs to approximate unknown functions requires further careful investigation. Moreover, expert knowledge about the EHSS dynamics is also needed when designing the structure of NNs. Finally, the effectiveness of the adopted DOBs to compensate for the effects of disturbances on the controlled systems depends on the accuracy of the system state derivative calculation based on the Levant's differentiator,  
\begin{table}
\begin{tabular}{c c c c} \hline
**Controller** & \(\mathbf{M_{e}}\) **(Degree)** & \(\mathbf{\mu_{e}}\) **(Degree)** & \(\mathbf{\sigma_{e}}\) **(Degree)** \\ \hline Proposed Controller & 0.0263 & 0.0138 & 0.0086 \\ RBF-SMC Controller & 0.0621 & 0.0331 & 0.0203 \\ ESO-BC Controller & 0.0360 & 0.0151 & 0.0117 \\ PID Controller & 0.5364 & 0.2287 & 0.1843 \\ \hline \end{tabular}
\end{table}
Table 4: Performance indexes in fast-motion reference trajectory of all considered controllers  
Figure 8: The tracking performance of the proposed strategy compared with other control laws in fast-motion reference trajectory.  
Figure 9: The tracking errors of the four controllers in fast-motion reference trajectory.  
whose parameters are required to be meticulously selected to make the tradeoff between the computational precision and robustness against measurement noises.