## 5 Numerical Simulation
### Simulation Results  
#### 5.2.1 Slow-Motion Reference Trajectory  
In this case, the smooth reference trajectory is chosen to evaluate the effectiveness of the proposed controller as  
\[x_{1d}(t)=\frac{\pi}{8}\Big{(}1-\cos\Big{(}\frac{\pi}{2}t\Big{)}\Big{)}\big{(} 1-e^{-t}\big{)}(\mathrm{rad}) \tag{65}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Unit** & **Value** & **Parameter** & **Unit** & **Value** \\ \hline \(B_{0}\) & \(\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}\cdot\mathrm{rad}^{-1}\) & 80 & \(C_{10}\) & \(\mathrm{m}^{3}\cdot\mathrm{s}^{-1}\cdot\mathrm{Pa}^{-1}\) & \(1\times 10^{-12}\) \\ \(\beta_{e0}\) & \(\mathrm{Pa}\) or \(\mathrm{N}\cdot\mathrm{m}^{-2}\) & \(7\times 10^{8}\) & \(C_{20}\) & \(\mathrm{m}^{3}\cdot\mathrm{s}^{-1}\) & \(0\) \\ \(C_{00}\) & \(\mathrm{m}^{3}\cdot\mathrm{s}^{-1}\cdot\mathrm{Pa}^{-2}\) & 0 & \(V_{t0}\) & \(\mathrm{m}^{3}\) & \(1\times 10^{-4}\) \\ \(k_{t0}\) & \(\mathrm{m}^{3}\cdot\mathrm{s}^{-1}\cdot\mathrm{V}^{-1}\cdot\mathrm{Pa}^{-1/2}\) & \(1\times 10^{-8}\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nominal system parameters of the EHSS for backstepping control design. The tracking performances and tracking errors of all four controllers are depicted in Figures 3 and 4, respectively. As shown in Figure 3, all four controllers guarantee that the system output tracks the desired trajectory at an acceptable level. Nevertheless, from Figure 4, it is obviously seen that the proposed controller performs better than others with the smallest tracking errors, whereas the ESO-BC controller can track the reference trajectory more closely than RBF-SMC and PID controllers. For more details, the final tracking accuracies of all controllers are indicated via the three performance indexes given in Table 3. It is clear that, although it has some robustness, the PID controller can do little with uncertainties and disturbances due to the lack of model-based compensation mechanisms. Consequently, it exhibits the worst tracking performance in all performance indexes (maximal value, average, and standard deviation of the tracking error are about 0.4684 degrees, 0.2169 degrees, and 0.1326 degrees, respectively). Meanwhile, with the support of ESOs (60), both mismatched and matched disturbances are estimated and efficiently compensated; based on that, the ESO-BC controller (maximal error about 0.0315 degrees) achieves better tracking performance in comparison with the RBF-SMC controller (maximal error about 0.0358 degrees). It is noted that the maximal tracking error is only about 0.0148 degrees in the case of the proposed controller. The results demonstrate the effectiveness of the suggested control strategy in which the combination of DOBs and NNs is able to effectively deal with full unknown system dynamics and both mismatched and matched disturbances. Figure 4: The tracking errors of the considered four control strategies with slow-motion reference trajectory. Figure 3: The tracking performance of the proposed strategy compared with other control laws with slow-motion reference trajectory. Figure 5: The estimates of the mismatched and matched disturbances of the proposed method. Figure 6: The weighting values of RFB NNs employed of the proposed method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller** & \(M_{e}\)**(Degree)** & \(\mu_{e}\)**(Degree)** & \(\sigma_{e}\)**(Degree)** \\ \hline Proposed Controller & 0.0148 & 0.0097 & 0.0038 \\ RBF-SMC Controller & 0.0358 & 0.0233 & 0.0092 \\ ESO-BC Controller & 0.0315 & 0.0151 & 0.0104 \\ PID Controller & 0.4684 & 0.2169 & 0.1326 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance indexes in slow-motion reference trajectory of all considered controllersThe estimates of the mismatched and matched grouped disturbances and the weighting values of RBF NNs for approximating all unknown dynamic functions are illustrated in Figures 5 and 6, respectively. It is noted that from Figure 6 that the weighting values of RFB NNs are smooth and bounded.