# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 4 Simulation Results
### Evaluation Result  
Two case studies were executed to appraise the effectiveness of the suggested FTC scheme. Case study 1 considered only the lumped uncertainty, while both the internal leakage fault and lumped uncertainty were tested in Case study 2. The simulation results are described as follows. _Case study_ 1: In this case study, the electro-hydraulic system operates in normal condition with inherent lumped disturbances. Figure 3 depicts the simulation results with a ramp and a sinewave response. From Figure 3a,b, the output position of three control algorithms exhibits good tracking capability. Nevertheless, the peak value of the position tracking error using the proposed controller and BSMCDO is smaller than that of usually applied DBC, as seen in Figure 3c,d, because the DOs effectively suppress lumped disturbances, which can be achieved by Figure 4. It can be known that the combination of NTSMC and the optimized-based BC scheme can not only obtain the purpose of optimization, but also guarantee the high-accuracy tracking performance in hydraulic servosystems; then, the suggested controller outperforms the BSMCDO controllers in terms of transient and final tracking errors. To ensure the ORC's optimal performance, the critic and actor NN weight estimations \(\hat{\phi}_{c},\hat{\phi}_{a}\) of the ORC converge to the corresponding steady-state value, as shown in Figure 5. The effectiveness of comparative controllers is assessed through three performance indices and expressed in Table 2. From Table 2, the root mean square error (\(e_{RMS}\)), average tracking error (\(e_{a}\)), and maximum error (\(e_{max}\)) [7,43] of the proposed controller are, in turn, 0.0351, 0.0298, and 0.1954 mm with the desired sinewave signal, and 0.0278, 0.0201, and 0.0980 mm with the desired ramp signal, which is the smallest in comparison with other controllers. This reconfirms the perfect control performance of the proposed FTC strategy. Figure 3: Comparison of three control methods in Case study 1: (**a**) Ramp response. (**b**) VFS response. (**c**) Tracking error for ramp demand. (**d**) Tracking error for VFS demand. _Case study 2_: In this scenario, the internal leakage fault in simulation is generated after \(T_{0}=10\)s with internal leakage coefficient \(C_{f0}=3.5\times 10^{-8}m^{4}/\left(N^{1/2}s\right),\) and the fault evolution rate \(\alpha=10\). The simulation results are displayed in Figure 6, which indicates that the position errors of the DBC, BSMCDO, and the FTC proposed method are affected by the fault evolution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Controller** & **DBC** & **BSMCDO** & **Proposed** \\ \hline \multirow{3}{*}{Ramp} & \(e_{RMS}\) & 0.1027 & 0.0637 & 0.0351 \\ \cline{2-5}  & \(e_{a}\) & 0.1103 & 0.0726 & 0.0298 \\ \cline{2-5}  & \(e_{max}\) & 0.2644 & 0.2381 & 0.1954 \\ \hline \multirow{3}{*}{VPS} & \(e_{RMS}\) & 0.0986 & 0.0727 & 0.0278 \\ \cline{2-5}  & \(e_{a}\) & 0.1009 & 0.0869 & 0.0201 \\ \cline{2-5}  & \(e_{max}\) & 0.1542 & 0.1194 & 0.0980 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance indices of three controllers in simulation, Case study 1. Figure 4: Lumped disturbances estimation of the proposed method in Case study 1: (**a**) \(\hat{\sigma}_{1}\) for ramp demand. (**b**) \(\hat{\sigma}_{1}\) for VFS demand. (**c**) \(\hat{\sigma}_{2}\) for ramp demand. (**d**) \(\hat{\sigma}_{2}\) for VFS demand. Figure 5: Critic and actor NN weight estimations of the proposed method in Case study 1: (**a**) \(\hat{\phi}_{e6}\) for ramp demand. (**b**) \(\hat{\phi}_{e6}\) for VFS demand. (**c**) \(\hat{\phi}_{a19}\) for ramp demand. (**d**) \(\hat{\phi}_{a19}\) for VFS demand. by the appearance of leakage fault. However, in comparison with DBC, the tracking error of the other methods is smaller due to a disturbance compensation system of two DOs.