# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 3 Proposed Control Scheme
### Disturbance Observer-Based NTSMC Design  
The nonlinear system (7) with lumped disturbances and the bias fault is first considered. Consequently, the position control design procedure for the suggested FTC is presented in this subsection. The error states are defined as follows:  
\[e_{1}=\zeta_{1}-\zeta_{1d} \tag{9}\]  
where \(\zeta_{1d}\) is the desired signal. The derivative of \(e_{1}\) can be expressed as  
\[\dot{e}_{1}=\zeta_{2}-\dot{\zeta}_{1d} \tag{10}\]  
A terminal sliding manifold is given by  
\[s=m\dot{e}_{1}+\eta e_{1}^{\lambda} \tag{11}\]  
where \(m,\eta\) are positive terms, \(\lambda=\lambda_{1}/\lambda_{2},\lambda_{1}\) and \(\lambda_{2}\) are positive odd integers satisfying \(\lambda_{1}<\lambda_{2}\). The time derivative of (11) is computed as  
\[\dot{s}=m\dot{e}_{1}+\eta\lambda e_{1}^{\lambda-1}\dot{e}_{1}=f_{2}+\zeta_{3}+ \sigma_{1}-m\dot{\zeta}_{1d}+\eta\lambda e_{1}^{\lambda-1}\dot{e}_{1} \tag{12}\]  
Let us denote the estimations of the disturbances \(\hat{\sigma}_{1},\hat{\sigma}_{2}\). The DO for mismatched disturbance \(\sigma_{1}\) is elaborated as  
\[ \begin{array}{l}\hat{\sigma}_{1}=\chi_{1}+\kappa_{1}\zeta_{2}\\ \dot{\chi}_{1}=-\kappa_{1}\chi_{1}+s-\kappa_{1}(f_{2}+g_{2}\zeta_{3}+\kappa_{ 1}\zeta_{2})\end{array}  \tag{13}\]  
where \(\chi_{1}\) is internal states, \(\kappa_{1}\) denotes the observer gain of the mismatched DO. Define the mismatched disturbance error \(\widetilde{\sigma}_{1}=\sigma_{1}-\hat{\sigma}_{1}\). Taking the derivative of \(\widetilde{\sigma}_{1}\), one yields  
\[ \begin{array}{l}\dot{\widetilde{\sigma}}_{1}=\dot{\sigma}_{1}-\dot{\sigma} _{1}=\dot{\sigma}_{1}-\dot{\chi}_{1}-\kappa_{1}\dot{\zeta}_{2}\\ =\dot{\sigma}_{1}+\kappa_{1}\chi_{1}-s+\kappa_{1}(f_{2}+g_{2}\zeta_{3}+\kappa _{1}\zeta_{2})-\kappa_{1}(f_{2}+g_{2}\zeta_{3}+\sigma_{1})\\ =\dot{\sigma}_{1}-\kappa_{1}\widetilde{\sigma}_{1}-s\end{array}  \tag{14}\]  
The exponential approach rate is used. Hence, the desired force for the mechanical subsystem is constructed as follows:  
\[\alpha_{2}=-f_{2}-\hat{\sigma}_{1}+m\dot{\zeta}_{1d}-\eta\lambda e_{1}^{ \lambda-1}\dot{e}_{1}-\gamma_{1}s-\gamma_{2}s^{\theta}-k_{1}sign(s) \tag{15}\]  
Figure 2: Sketch of the proposed control approach. where \(\gamma_{1},\gamma_{2},k_{1}\) are control gains and positive constants, and \(0<\vartheta<1\). Let the torque error be determined as follows:  
\[e_{3}=\zeta_{3}-\alpha_{2} \tag{16}\]  
The Lyapunov function candidate \(V_{1}\) is adopted as  
\[V_{1}=\frac{1}{2}s^{2}+\frac{1}{2}\widetilde{\sigma}_{1}^{2} \tag{17}\]  
Taking the derivative of \(V_{1}\) and noting (12), (14), we obtain:  
\[ \begin{array}{c}\dot{V}_{1}=s(-\gamma_{1}s-\gamma_{2}s^{\vartheta}-k_{1} sign(s)+\widetilde{\sigma}_{1}+e_{3})\\ \quad+\widetilde{\sigma}_{1}(\dot{\sigma}_{1}-\kappa_{1}\widetilde{\sigma}_{1} -s)\\ =-k_{1}|s|-\gamma_{1}s^{2}-\gamma_{2}s^{\vartheta+1}+se_{3}-\kappa_{1} \widetilde{\sigma}_{1}^{2}+\widetilde{\sigma}_{1}\dot{\sigma}_{1}\end{array}  \tag{18}\]  
Differentiating \(e_{3}\) with respect to time yields that  
\[\dot{e}_{3}=f_{3}+g_{3}u+\sigma_{2}-\dot{\alpha}_{2} \tag{19}\]  
The matched DO is constructed to approximate both matched disturbance and the bias fault \(\sigma_{2}\) as  
\[ \begin{array}{c}\dot{\sigma}_{2}=\chi_{2}+\kappa_{2}\zeta_{3}\\ \dot{\chi}_{2}=-\kappa_{2}\chi_{2}-\kappa_{2}(f_{3}+g_{3}u+\kappa_{2}\zeta_{3}) \end{array}  \tag{20}\]  
where \(\chi_{2}\) is internal states, \(\kappa_{2}\) denotes the observer gain of the matched DO. The matched disturbance error is calculated by \(\widetilde{\sigma}_{2}=\sigma_{2}-\dot{\sigma}_{2}\). Taking the derivative of \(\widetilde{\sigma}_{2}\), one obtains  
\[ \begin{array}{c}\dot{\widetilde{\sigma}}_{2}=\dot{\sigma}_{2}-\dot{\sigma} _{2}=\dot{\sigma}_{2}-\dot{\chi}_{2}-\kappa_{2}\dot{\zeta}_{3}\\ =\dot{\sigma}_{2}-\kappa_{2}\widetilde{\sigma}_{2}\end{array}  \tag{21}\]  
**Remark 3**. : _The singular problem in (15) can be overcome by using the modified terminal sliding manifold as Ref.