# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 5 Conclusions  
The optimal FTC strategy was successfully applied to the EHA system with lumped disturbances and internal leakage faults via the cascade control using NTMSC and a simplified RL algorithm based on the optimized technology. The lumped uncertainties including matched, mismatched disturbances and the internal leakage fault are catered for simultaneously. Two DOs were deployed for estimations of them. A novel controller-based NTSMC, and an optimal force tracking control approach is further constructed for the EHA system, which can not only obtain the purpose of optimization, but also guarantee the high-accuracy tracking performance in hydraulic servosystems. The simulation results were implemented to verify the superior performance and system stability of the suggested strategy, even in faulty conditions.  
Nevertheless, the presented strategy still has some disadvantages. First, the loss of effectiveness fault was ignored in this work. Second, verifying the proposed algorithm in actual experiments has not yet been performed. In future works, we will tend to develop the FTC scheme for practical EHAs with some types of actuator faults, including bias fault and loss of effectiveness fault.  
Figure 8: Critic and actor NN weight estimations of the proposed method in Case study 2: (a) \(\hat{\phi}_{\text{c}6}\) for ramp demand. (b) \(\hat{\phi}_{\text{c}6}\) for VFS demand. (c) \(\hat{\phi}_{\text{a19}}\) for ramp demand. (d) \(\hat{\phi}_{\text{a19}}\) for VFS demand.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Controller** & **DBC** & **BSMCDO** & **Proposed** \\ \hline \multirow{3}{*}{Ramp} & \(e_{RMS}\) & 0.1190 & 0.1024 & 0.0838 \\ \cline{2-5}  & \(e_{a}\) & 0.1201 & 0.1106 & 0.0778 \\ \cline{2-5}  & \(e_{max}\) & 0.4198 & 0.2684 & 0.2106 \\ \hline \multirow{3}{*}{VPS} & \(e_{RMS}\) & 0.1595 & 0.0857 & 0.0442 \\ \cline{2-5}  & \(e_{a}\) & 0.1768 & 0.0912 & 0.0357 \\ \cline{2-5}  & \(e_{max}\) & 0.1986 & 0.1458 & 0.1260 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performance indices of three controllers in simulation, Case study 2.  
**Author Contributions:** [PERSON] was the supervisor providing funding and administrating the project, and he reviewed and edited the manuscript. [PERSON] carried out the investigation, methodology, analysis, validation, made the MATLAB simulation, and wrote the original manuscript. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported in part by Basic Science Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science and ICT, South Korea, under Grant NRF 2020R1A2B5B03001480, and in part by the Regional Innovation Strategy (RIS) through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (MOE) under Grant 2021 RIS-003.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.