# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 3 Proposed Control Scheme
### Disturbance Observer-Based NTSMC Design
\begin{array}{l}\dot{V}\leq-\Big{(}\gamma_{1}-\frac{1}{2}\Big{)}s^{2}-\big{(}k_ {3}g_{3}^{2}-2-\frac{7}{4}g_{3}^{4}\big{)}e_{3}^{2}\\ -\sum\limits_{i=1}^{2}\Big{(}\kappa_{i}-\frac{1}{2}\Big{)}\widehat{\sigma}_{i}^ {2}+\frac{1}{2}\hat{\sigma}_{1}^{2}+f_{3}^{2}+\sigma_{2}^{2}+\frac{1}{2}\hat{ \alpha}_{2}^{2}+\frac{1}{4}\hat{\phi}_{a}^{T}hh^{T}\hat{\phi}_{a}\\ -\beta_{a}\widehat{\phi}_{a}^{T}hh^{T}\widehat{\phi}_{a}+(\beta_{a}-\beta_{c}) \widehat{\phi}_{a}^{T}hh^{T}\widehat{\phi}_{c}-\beta_{c}\widehat{\phi}_{c}^{T} h^{T}\widehat{\phi}_{c}\end{array}  \tag{43}\]  
Applying [PERSON]'s inequality and according to the equivalent transformation (i.e., \(\widehat{\phi}_{a}^{T}hh^{T}\widehat{\phi}_{a}+2\widehat{\phi}_{a}^{T}hh^{T} \phi^{*}+\phi^{*}hh^{T}\phi^{*}=\widehat{\phi}_{a}^{T}hh^{T}\widehat{\phi}_{a}\)), one obtains  
\[ \begin{array}{l}\widehat{\phi}_{a}^{T}hh^{T}\hat{\phi}_{a}=\frac{1}{2} \widehat{\phi}_{a}^{T}hh^{T}\widehat{\phi}_{a}+\frac{1}{2}\hat{\phi}_{a}^{T} hh^{T}\widehat{\phi}_{a}-\frac{1}{2}\phi^{*}hh^{T}\phi^{*}\\ \widehat{\phi}_{a}^{T}hh^{T}\hat{\phi}_{c}=\frac{1}{2}\widehat{\phi}_{a}^{T}hh^ {T}\widehat{\phi}_{c}+\frac{1}{2}\widehat{\phi}_{c}^{T}hh^{T}\widehat{\phi}_{ c}-\frac{1}{2}\phi^{*}hh^{T}\widehat{\phi}^{*}\\ \widehat{\phi}_{a}^{T}hh^{T}\hat{\phi}_{c}\leq\frac{1}{2}\widehat{\phi}_{a}^{T} hh^{T}\widehat{\phi}_{a}+\frac{1}{2}\widehat{\phi}_{c}^{T}hh^{T}\widehat{\phi}_{c} \end{array}  \tag{44}\]  
Combined with (43) and (44), we obtain  
\[ \begin{array}{l}\dot{V}\leq-\Big{(}\gamma_{1}-\frac{1}{2}\Big{)}s^{2}-\big{(} k_{3}g_{3}^{2}-2-\frac{7}{4}g_{3}^{4}\big{)}e_{3}^{2}\\ -\sum\limits_{i=1}^{2}\Big{(}\kappa_{i}-\frac{1}{2}\Big{)}\widehat{\sigma}_{i} ^{2}-\frac{\beta_{c}}{2}\widehat{\phi}_{a}^{T}hh^{T}\widehat{\phi}_{a}-\frac{ \beta_{c}}{2}\widehat{\phi}_{c}^{T}hh^{T}\widehat{\phi}_{c}\\ +\frac{1}{2}\hat{\sigma}_{1}^{2}+\varphi_{3}^{2}+\sigma_{2}^{2}+\frac{1}{2} \hat{\alpha}_{2}^{2}+\frac{(\beta_{a}+\beta_{c})}{2}\phi^{*}hh^{T}\phi^{*}\\ -\Big{(}\frac{\beta_{a}}{2}-\frac{1}{4}\Big{)}\hat{\phi}_{a}^{T}hh^{T}\widehat {\phi}_{a}-\frac{(-\beta_{a}+2\beta_{c})}{2}\hat{\phi}_{c}^{T}hh^{T}\hat{ \phi}_{c}\end{array}  \tag{45}\]  
If the inequality \(\frac{1}{2}<\beta_{a}<\frac{1}{2}\beta_{c}\), holds, there exist the upper bounds of the nonlinear function \(g_{3},f_{3},\dot{\alpha}_{2}\) (i.e., \(g_{3}<\Lambda_{3}f_{3}\leq\sigma_{3m},\dot{\alpha}_{2}\leq\alpha_{2m}\)) [25,35], and \(\sigma_{2}\leq d_{2m},\dot{\sigma}_{1}\leq D_{1m}\) under Assumption 1. Thus, the time derivative of \(V\) can be given as  
\[\dot{V}\leq-\Gamma V+\psi \tag{46}\]  
where  
\[ \begin{array}{l}\Gamma=\min\biggl{(}\begin{array}{l}\lambda_{\min}(2 \gamma_{1}-1),\lambda_{\min}\big{(}2k_{3}\Lambda_{3}^{2}-4-\frac{7}{4}\Lambda_ {3}^{4}\big{)},\\ \lambda_{\min}(2\kappa_{1}-1),\lambda_{\min}(2\kappa_{2}-1),\lambda_{\min} \big{(}\beta_{c}hl^{T}\big{)}\end{array}\biggr{)}\\ \psi=\frac{1}{2}D_{1m}^{2}+\varphi_{3m}^{2}+d_{2m}^{2}+\frac{1}{2}\mathbf{a}_{2m} ^{2}+\frac{(\beta_{a}+\beta_{c})}{2}\phi^{*}hh^{T}\phi^{*}\end{array} \]  
**Theorem 1**. : _With the EHA system (7), if the two DOs are designed as (13), (20), the position control law is presented in (15), the optimal force controller is indicated by (33). The adaptive law of the critic and actor NN weights is determined by (32) and (34). Then, all closed-loop signals, i.e., tracking errors, disturbance errors, and critic and actor NN weight errors are uniformly ultimately bounded (UIUB) in the face of actuator failures and lumped disturbances._  
**Proof.