# Performance-Constraint Fault Tolerant Control to Aircraft in Presence of Actuator Deviation
## 4 FTC-FT Controller Applied on the Unmanned Helicopter
The controller is designed in two steps as following. The system state \(x_{1}=\left[h,\psi\right]^{T},x_{2}=\left[w,r\right]^{T}\), errors \(\bar{h}=h-h_{d},\bar{\psi}=\psi-\psi_{d}\), bound of altitude tracking error \(k_{\bar{h}}\). Design the BL function as  
\[V_{1}(\bar{\mathbf{x}}_{1})=\frac{1}{2}\ln\frac{k_{\bar{h}}^{2}}{k_{\bar{h}}^{2}- \bar{h}^{2}}+\frac{1}{2}\bar{\psi}^{2} \tag{29}\]and define the intermediate matrix  
\[R(\tilde{\mathbf{x}}_{1})=\text{diag}\Bigg{(}\frac{1}{k_{h}^{2}-\tilde{h}^{2}},1 \Bigg{)} \tag{30}\]  
the virtual control function is designed as  
\[\mathbf{\alpha}_{1}=g_{1}^{-1}\bigg{(}\dot{y}_{d}-f_{1}-K_{1}\tilde{\mathbf{x}}_{1}-R( \tilde{\mathbf{x}}_{1})^{\mu-1}\lceil\tilde{\mathbf{x}}_{1}\rceil^{2\mu-1}-\frac{1}{2} R(\tilde{\mathbf{x}}_{1})\tilde{\mathbf{x}}_{1}\bigg{)} \tag{31}\]  
where \(\mathbf{K}_{1}=\text{diag}\big{(}k_{1,h}\), \(k_{1,\bar{\varphi}}\big{)}\). Therefore, the derivative of BL function is  
\[ \begin{split}\dot{V}_{1}(\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2})=& \mathbf{x}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_{1})(f_{1}+g_{1}(\tilde{\mathbf{x}}_{2}+\mathbf{ \alpha}_{1})-\dot{y}_{d})\\ =&-\tilde{\mathbf{x}}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_{1}) K_{1}\tilde{\mathbf{x}}_{1}-\tilde{\mathbf{x}}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_{1})^{\mu} \lceil\tilde{\mathbf{x}}_{1}\rceil^{2\mu-1}\\ &-\frac{1}{2}\left\|\tilde{\mathbf{x}}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_ {1})\right\|^{2}+\tilde{\mathbf{x}}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_{1})\mathbf{g}_{1} \tilde{\mathbf{x}}_{2}\\ \leqslant&\frac{-k_{1,h}\tilde{h}^{2}}{k_{h}^{2}- \tilde{h}^{2}}-1_{,\bar{\varphi}}\tilde{\psi}^{2}-\left(\frac{\tilde{h}^{2}}{ k_{h}^{2}-\tilde{h}^{2}}\right)^{\mu}-\left(\tilde{\psi}^{2}\right)^{\mu}+\frac{1}{2 }\|\mathbf{g}_{1}\tilde{\mathbf{x}}_{2}\|^{2}\\ \leqslant&-a_{1}V_{1}-2^{\mu}V_{1}^{\mu}+\frac{1}{ 2}\|\mathbf{g}_{1}\tilde{\mathbf{x}}_{2}\|^{2}\end{split}  \tag{32}\]  
where \(a_{1}=2\min\{k_{1,h},k_{1,\bar{\varphi}}\}\). Secondly, let us define the vertical velocity and yaw rate bounds, with \(k_{\bar{\varphi}},k_{\bar{\varphi}}\) respectively, and intermediate variable  
\[R(\tilde{\mathbf{x}}_{2})=\text{diag}\bigg{(}\frac{1}{k_{\bar{\varphi}}^{2}- \tilde{x}_{2}(1)^{2}},\frac{1}{k_{\bar{\varphi}}^{2}-\tilde{x}_{2}(2)^{2}} \bigg{)} \tag{33}\]  
and the maximum perturbation aroused by actuator is given as  
\[\Xi(\tilde{\mathbf{x}}_{2})=\varpi\delta_{\text{max}}\|R(\tilde{\mathbf{x}}_{2})\tilde {\mathbf{x}}_{2}\|. \tag{34}\]  
Design the auxiliary state of system  
\[\dot{\mathbf{\Psi}}(\tilde{\mathbf{x}}_{2})=-\frac{\mathbf{\Psi}(\tilde{\mathbf{x}}_{2})\Xi( \tilde{\mathbf{x}}_{2})}{\left\|\mathbf{\Psi}(\tilde{\mathbf{x}}_{2})\right\|^{2}+\left\| R(\tilde{\mathbf{x}}_{2})\tilde{\mathbf{x}}_{2}\right\|^{2}}-\eta\mathbf{\Psi}(\tilde{\mathbf{x}}_{2}) -\lceil\mathbf{\Psi}(\tilde{\mathbf{x}}_{2})\rceil^{2\mu-1} \tag{35}\]  
therefore, the controller is designed as  
\[ \begin{split}\delta&=g_{2}^{-1}\bigg{(}-f_{2}-K_{2} \tilde{\mathbf{x}}_{2}-R(\tilde{\mathbf{x}}_{2})^{\mu-1}\lceil\tilde{\mathbf{x}}_{2} \rceil^{2\mu-1}-\frac{1}{2}R(\tilde{\mathbf{x}}_{2})^{-1}\mathbf{g}_{1}^{T}\mathbf{g}_{1} \tilde{\mathbf{x}}_{2}\\ &\qquad-\frac{R(\tilde{\mathbf{x}}_{2})\tilde{\mathbf{x}}_{2}\Xi(\tilde{ \mathbf{x}}_{2})}{\left\|\mathbf{\Psi}(\tilde{\mathbf{x}}_{2})\right\|^{2}+\left\|R( \tilde{\mathbf{x}}_{2})\tilde{\mathbf{x}}_{2}\right\|^{2}}-\frac{\gamma^{2}}{2}\mathbf{R} (\tilde{\mathbf{x}}_{2})\tilde{\mathbf{x}}_{2}+\dot{\mathbf{x}}_{1}\bigg{)}\end{split}  \tag{36}\]  
where \(\mathbf{K}_{2}=\text{diag}(k_{2,w},k_{2,r})\), and the extended BL function is  
\[V_{2}(\tilde{\mathbf{x}}_{2})=\frac{1}{2}\ln\frac{k_{\bar{\varphi}}^{2}}{k_{\bar{ \varphi}}^{2}-\bar{w}^{2}}+\frac{1}{2}\ln\frac{k_{\bar{\varphi}}^{2}}{k_{\bar{ \varphi}}^{2}-\bar{\
u}^{2}}+\frac{1}{2}\mathbf{\Psi}(\tilde{\mathbf{x}}_{2})^{T}\mathbf{ \Psi}(\tilde{\mathbf{x}}_{2}) \tag{37}\]Take the derivative of it and put it into the controller (36) and the additional dynamic (35) to obtain  
\[