# Optimized-Based Fault-Tolerant Control of an Electro-Hydraulic System with Disturbance Rejection
## 3 Proposed Control Scheme
### Disturbance Observer-Based NTSMC Design
[40]:_  
\[s=m\dot{e}_{1}+\varphi(e_{1}) \tag{22}\]  
_where_  
\[\varphi(e_{1})=\left\{\begin{array}{ll}\eta e_{1}^{\lambda}&if\,\overline{ s}=0\,or\,\overline{s}\
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The optimal control \(u^{*}\) can be calculated if the derivative of the HJB function with respect to \(u^{*}\) obtains:  
\[\frac{\partial H}{\partial u^{*}}=0\leftrightarrow u^{*}=-\frac{1}{2}g_{3}J_{e _{3}}^{*} \tag{26}\]  
The \(J_{e_{3}}^{*}\) function can be presented as  
\[J_{e_{3}}^{*}=2k_{3}e_{3}+2f_{3}+2\delta_{2}+J_{e_{3}}^{0} \tag{27}\]  
where \(J_{e_{3}}^{0}=J_{e_{3}}^{*}-2k_{3}e_{3}-2f_{3}-2\delta_{2}\), \(k_{3}\) is a positive scalar. By using NN approximation to construct the critic network, an unknown continuous vector function \(J_{e_{3}}^{0}\) can be estimated; it follows that:  
\[J_{e_{3}}^{0}=\phi^{*T}h(\zeta_{1},\zeta_{2},e_{3})+\varsigma \tag{28}\]  
where \(\phi^{*}\) is the ideal weight of the NN, \(\varsigma\) is the error of NN approximation, \(h(\zeta_{1},\zeta_{2},e_{3})\) is the basis function. Throughout this paper, for the simplicity of expression, we abbreviate \(h(\zeta_{1},\zeta_{2},e_{3})\) by \(h\). In general, the most popularly used Gaussian radial basis functions are employed as follows:  
\[h=\exp\left[\frac{-(\zeta_{in}-\mu_{i})^{T}(\zeta_{in}-\mu_{i})}{b_{i}^{2}}\right] \tag{29}\]  
where \(\zeta_{in}= \begin{bmatrix}\zeta_{1}&\zeta_{2}&e_{3}\end{bmatrix} ^{T}\) is the NN inputs vector, and \(\mu_{i},b_{i}\) denote the center of the neural, the Gaussian function's width, respectively. **Assumption 2**. : _There exists an ideal weight vector \(\phi^{*}\) such that \(\ \left|\varsigma\right|<\varsigma^{*}\) with constant \(\varsigma^{*}>0\) for all \(\zeta_{in}\in\Omega_{\zeta}\subset\Re^{3}\)._  
Substituting (27) and (28) to (26), one obtains  
\[u^{*}=-g_{3}\bigg{(}k_{3}e_{3}+f_{3}+\hat{\sigma}_{2}+\frac{1}{2}\phi^{*T}h+ \frac{1}{2}\varsigma\bigg{)} \tag{30}\]  
Due to the unavailable ideal weight vector \(\phi^{*}\), the optimal controller (30) cannot be implemented. Hence, the RL algorithm is introduced by utilizing the following both critic and actor NNs. The critic NN is used to evaluate the controlling performance, whilst the actor NN generates the final control law. The critic NN for evaluating the control performance is further derived as  
\[\hat{J}_{e_{3}}^{*}=2k_{3}e_{3}+2f_{3}+2\hat{\sigma}_{2}+\hat{\phi}_{c}^{T}h \tag{31}\]  
where \(\hat{\phi}_{c}\) is the estimation of NN optimal weight \(\phi^{*}\). The critic NN weight is updated by the following law as  
\[\hat{\phi}_{c}=-\beta_{c}hh^{T}\hat{\phi}_{c} \tag{32}\]  
where \(\beta_{c}\) is a positive constant.