## 4 Numerical Simulation and Discussion
### Case Study 2: Smooth Reference Trajectory  
To further examine the effectiveness of the proposed control method, a sinusoidal reference trajectory is employed as \(x_{1d}(t)=\frac{\pi}{4}\mathrm{sin}(\pi t/2)\) (rad). The time-varying prescribed performance function is kept as same as in the above case study. As shown in Figure 11, by virtue of disturbance compensations, all control approaches can track the reference trajectory consistently in spite of the effects of disturbances on the system dynamics. Nonetheless, the DESO-BC and SESO-BC controllers violate the predefined output tracking error constraints as illustrated in Figure 12. Meanwhile, the proposed controller with the PPC mechanism is still able to guarantee the smallest tracking error and satisfy these constraints. It is worth noting that the mismatched and matched disturbances on the control system are estimated more exactly by the combination of the ESMO and ESO and then effectively suppressed by the proposed method, consequently.  
Figure 10: Lumped matched uncertainty estimation performances of all control approaches in case of step reference trajectory.  
Figure 9: Lumped mismatched uncertainty estimation performances of all control approaches in case of step reference trajectory.  
The tracking performance indexes of all controllers are provided in Table 3. Similar to the step reference trajectory case study, the suggested control strategy performs better in terms of all performance indexes with smaller values of the maximal absolute tracking error, average tracking error, and standard deviation compared to the remainder control approaches. The results indicate tracking ability and robustness of the proposed method in the presence of disturbances and the nonexistence of velocity measurement mechanism.  
Through the two case studies, it can be observed that the proposed control method can be considered as a valuable solution to achieve the predefined performance for tracking  
Figure 11: Tracking performances of all controllers in case of sinusoidal reference trajectory.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller** & \(\mathbf{M_{\epsilon}}\) **(rad)** & \(\mathbf{\mu_{\epsilon}}\) **(rad)** & \(\mathbf{\sigma_{\epsilon}}\) **(rad)** \\ \hline Proposed & \(7.7322\times 10^{-4}\) & \(5.0029\times 10^{-4}\) & \(2.2030\times 10^{-4}\) \\ DESO-BC & \(27.1048\times 10^{-4}\) & \(16.4598\times 10^{-4}\) & \(7.7958\times 10^{-4}\) \\ SESO-BC & \(67.3774\times 10^{-4}\) & \(39.7416\times 10^{-4}\) & \(18.6279\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performance indexes of all controllers in the case of the smooth reference trajectory.  
Figure 12: Tracking errors of all controllers in case of sinusoidal reference trajectory.  
control of the EHSs subject to parameter deviations, modeling errors, and unknown external disturbances. In this control scheme, the trajectory planner can be separately designed to ensure that the filtered trajectory is feasible since it takes the kinematic constraints into account. Based on that the input saturation problem is actively avoided and the high-accuracy tracking performance can be achieved. In terms of estimation qualification, compared to the well-known ESO design which is widely adopted in the literature, the ESMO exhibits a higher estimation accuracy. Besides, the PPC is skillfully integrated into the CF-BSC control framework to guarantee that the system output is able to track the desired trajectory with the prescribed convergence rate, overshoot, and steady-state error. However, the predefined performance should be carefully selected to avoid the singularity problem which may cause unexpected issues and should be sufficiently considered in real applications.