## 4 Simulations  
This section introduces a trajectory tracking simulation in MATLAB to verify the performance of TSMCSPO. The desired trajectory used in this simulation is shown in Figure 6. The desired trajectory includes all possible response inputs (i.e., ramp, sinusoidal, and step). It increased until it reached 45 degrees at 19.7 s. At the start, the trajectory gradually increased by the ramp input until 5 s. Then, it increased by a mix of sinusoidal input (3 rad/s, 0.477 Hz) and ramp input until it reached its maximum value (i.e., 45 degrees). It then dropped until it reached 15 degrees at 25 s. From 25 s, the reference input was designed as a step input. During all the movement, the external force is set as:  
\[\Psi=\mathrm{h}\cdot x_{1}. \tag{35}\]  
where \(x_{1}\) is the real trajectory, \(\mathrm{h}\) is a positive number, and the maximum frequency of perturbation is considered as 3 rad/s, which is assumed as the real operation frequency. Table 1 reports the parameters used in the simulations. \(\alpha_{3}\), \(\frac{k_{1}}{\varepsilon_{01}}\) and \(k_{2j}\) can be determined by \(\lambda_{d}\) shown in Equation (15). Figure 6: Desired trajectory. The fractional power term \(\dot{\tilde{e}}|\tilde{e}|^{r-1}\) in the control law of TSMCSPO can be seen as the term that bridges between linear control (\(r\to 1\)) and nonlinear control (\(r\to 0\)). If we increase the ratio (\(r\to 0\)), the convergence time is reduced. Otherwise, the chattering will be much more severe. Moreover, in the simulation, the parameter \(\eta_{1}\) is selected to be larger than the boundary of the expected perturbation, which is proposed in the previous section. Because the maximum frequency of the disturbance is given as 3 rad/s (oscillation part in the desired trajectory) \(\lambda_{d}\) can be selected as 10 when the bandwidth is near 5 rad/s in the sliding perturbation observer. Figure 7 shows the simulation result of the error trajectory comparison between the proposed control scheme and other three controllers. The other three controllers were: (a) SMC designed as \(u_{smc}=-J_{1}K_{1}sat(s)+D_{1}\theta_{12}+J_{1}\tilde{\theta}_{1d}-J_{1} \dot{c}\) with parameters \(K_{1}=10\) and \(c=10\); (b) SMCSPO used in Equation A6 from Reference [32] with same parameters in SPO and \(K_{1}=10\); and (c) conventional TSMC described in [28]\(u_{tsmc}=-\eta_{2}J_{1}sign(s)-\eta_{3}J_{1}|s|^{r}sign(s)+D_{1}\theta_{12}+J \tilde{\theta}_{1d}-J_{1}\beta r|e^{r-1}\Big{(}\dot{c}\Big{)}\) with parameters \(\eta_{2}=0.6\) and \(\eta_{3}=10\). Figure 8 gives the control input result of SMCSPO and TSMCSPO. The blue line presents the control input of TSMCSPO, and the line shows that of SMCSPO. An enlarged part of the interval from 11.5 to 11.6 s result shows that the control input of TSMCSPO had more chattering than SMCSPO, but not too great. TSMC also had more chattering than SMC; their results almost overlapped--similar to the result of TSMCSPO and SMCSPO. The overlapping phenomenon was caused by the huge control input needed for moving the hydraulic system. The chattering resulted from the nonlinear terminal sliding surface. \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameters** & **Values** \\ \hline \(\beta_{1},\lambda_{d},\eta_{1}\) & 10 \\ \hline \(\varepsilon_{c}\), h & 1 \\ \hline \(r\) & 0.7 \\ \hline \(\Delta\) & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter selections of simulation and experiment. Figure 7: Trajectory Tracking Error Comparisons. Comparing the error trajectories of SMC, TSMC, TSMCSPO, and SMCSPO is discussed in Table 2 which includes the maximum error, steady-state error, undershoot, and settling time. TSMC has a faster convergence time with less steady state error in the final step input section. This means TSMC has more attenuation than SMC. However, the results of both SMCSPO and TSMCSPO showed less error and 0 steady state error, which benefited from the SPO. In both SMCSPO and TSMCSPO, an undershoot error existed, which was caused by the time delay.