## 4 Simulations
### Application to Electro Hydrostatic Actuator System
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Nominal (+ Variance) Value** \\ \hline \(m\) & kg & \(9.32(+0.46)\) \\ \(A_{1}\) & m\({}^{2}\) & \(1.9635\times 10^{-3}\)\(\left(+0.0982\times 10^{-3}\right)\) \\ \(A_{2}\) & m\({}^{2}\) & \(1.2566\times 10^{-3}\)\(\left(+0.0628\times 10^{-3}\right)\) \\ \(V_{10}\) & m\({}^{3}\) & \(4.375\times 10^{-4}\)\(\left(+0.219\times 10^{-4}\right)\) \\ \(V_{20}\) & m\({}^{3}\) & \(2.88\times 10^{-4}\)\(\left(+0.144\times 10^{-4}\right)\) \\ \(\beta_{e}\) & Pa & \(5.34\times 10^{8}\) \\ \(K_{dr}\) & rad\(/\)(sV) & \(10\pi\) \\ \(D_{1}\triangleq\eta_{V}D\) & m\({}^{3}/\)rad & \(5.83\times 10^{-7}\)\(\left(+0.29\times 10^{-7}\right)\) \\ \(C_{Li}\) & m\({}^{3}/\)(sPa) & \(1.51\times 10^{-15}\) \\ \(b_{1}\) & Ns\(/\)m & 258 \\ \(b_{2}\) & N & 532 \\ \(\gamma_{2}\) & & 10 \\ \(\gamma_{3}\) & & 15 (\(+0.75\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Nominal values of the EHA system parameters. to overcome the effects of mismatched disturbances and thus cannot stabilize the target system. It is therefore excluded from the comparison. By comparison, both ESOSMC and DOBTSMC are able to force the system state to remain in the vicinity of the equilibrium. Specifically, the position state \(x_{1}\) generated by ESOSMC varies in a small, bounded region of 3.5 mm around the zero axis. In contrast, the proposed controller, DOBTSMC, provides a much higher precision of about \(\pm 0.3\) mm, which is 10 times more accurate than that of ESOSMC. The velocity trajectory of ESOSMC and the proposed controller are shown in Figure 9, which shows agreement with the position response in Figure 8. The precision of the DOBTSMC velocity signal, at \(\pm 2.3\) mm/s, is two-fold higher than that of the ESOSMC velocity signal, at \(\pm 5\) mm/s. Moreover, a larger peaking phenomenon in the transition period is seen for ESOSMC. Figure 8: Position response of the EHA system. Figure 9: Velocity response of the EHA system. Figure 10 shows the trajectory of the mismatched disturbance and the third state of the control system. It is noticeable that the mismatched disturbance is time varying and that its trajectory and \(x_{3}\) trajectories are symmetric. This phenomenon is consistent with the previous numerical study, where the system state \(x_{3}\) is controlled to cancel out the influence of the mismatched disturbance \(d_{1}\). Thus, \(x_{3}\) and \(d_{1}\) are likely to be opposite in value. The control signal generated by ESOSMC and the proposed control algorithm are recorded in Figure 11. Overall, both signals are smooth and show few chattering effects. The ESOSMC control signal oscillates inside \(\pm 0.5\,\mathrm{V}\), whereas the figure for DOBTSMC is slightly better, at \(\pm 0.3\,\mathrm{V}\). However, ESOSMC exhibits a larger peaking phenomenon than the proposed controller in the transition period. Figure 11: Control signal trajectory of ESOSMC and DOBTSMC. Figure 10: Mismatched disturbance and acceleration signal of the EHA system. Figure 12 illustrates the performance of the EOSSMC and that of the proposed controller in estimating mismatched disturbances. The estimation error of DOBTSMC, at about \(\pm 0.6\,\mathrm{m/s^{2}}\), is approximately two-fold smaller than that of the EOSSMC, at about \(\pm 1.2\,\mathrm{m/s^{2}}\). Both algorithms experience chattering phenomena in the trajectory. This can be explained as follows. Because both schemes utilize system states in observer design, measurement noise in system states has a direct impact on the performance of the observation process. Thus, chattering caused by noise appears and consequently degrades the estimation accuracy of both controllers. Finally, working pressures of the proposed controller are shown in Figure 13, which clearly implies the nonlinearity of the EHA system. Specifically, the bore chamber area \(A_{1}\) is larger than the rod chamber \(A_{2}\). Thus, to produce the same level of force, the bore chamber pressure \(P_{1}\) must be accordingly smaller than the rod chamber pressure \(P_{2}\).