# Performance-Constraint Fault Tolerant Control to Aircraft in Presence of Actuator Deviation
## 2 A Class of Aircraft Dynamic Model Considering Actuator Deviation
\begin{split}\dot{V}_{n}=&\sum_{i=1}^{m}\frac{\mathbf{ \hat{x}}_{ni}\dot{\delta}_{ni}}{k_{ni}^{2}-\mathbf{\hat{x}}_{ni}^{2}}+\mathbf{\Psi}(\bm {\hat{x}}_{n})^{T}\dot{\mathbf{\Psi}}(\mathbf{\hat{x}}_{n})=\mathbf{\hat{x}}_{n}^{T}\mathbf{ R}(\mathbf{\hat{x}}_{n})[\mathbf{f}_{n}(\mathbf{\hat{x}}_{n})\\ &+(\mathbf{g}_{n}(\mathbf{\hat{x}}_{n})+\Delta\mathbf{g}_{n}(\mathbf{\hat{x}}_{n},t))\mathbf{u}+\mathbf{\xi}_{\mathbf{c}}(\mathbf{\hat{x}}_{n},t)-\mathbf{\hat{x}}_{n-1}]+\mathbf{\Psi}( \mathbf{\hat{x}}_{n})^{T}\dot{\mathbf{\Psi}}(\mathbf{\hat{x}}_{n})\\ \leqslant&-\mathbf{\hat{x}}_{n}^{T}\mathbf{R}(\mathbf{\hat{x}}_{n} )\mathbf{K}_{n}\mathbf{\hat{x}}_{n}-\mathbf{\hat{x}}_{n}^{T}\mathbf{R}(\mathbf{\hat{x}}_{n})^{\mu }[\mathbf{\hat{x}}_{n}\rceil^{2\mu-1}-\frac{1}{2}\|\mathbf{g}_{n-1}\mathbf{\hat{x}}_{n}\|^ {2}\\ &+\left\|\mathbf{\hat{x}}_{n}^{T}\mathbf{R}(\mathbf{\hat{x}}_{n})\right\|\!\! \left\|\mathbf{\hat{u}}_{\max}-\frac{\left\|\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{ n}\right\|^{2}\Xi(\mathbf{\hat{x}}_{n})}{\left\|\mathbf{\Psi}(\mathbf{\hat{x}}_{n}) \right\|^{2}+\left\|\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{n}\right\|^{2}}\\ &-\frac{\gamma^{2}}{2}\|\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{n}\|^ {2}+\mathbf{\hat{x}}_{n}^{T}\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{\xi}}_{\mathbf{c}}(\mathbf{ \hat{x}}_{n},t)+\mathbf{\Psi}(\mathbf{\hat{x}}_{n})^{T}\mathbf{\Psi}(\mathbf{\hat{x}}_{n})\\ \leqslant&-\sum_{j=1}^{m}K_{n,jj}\frac{\mathbf{\hat{x}}_{ nj}^{2}}{k_{nj}^{2}-\mathbf{\hat{x}}_{nj}^{2}}-\sum_{j=1}^{m}\left(\frac{\mathbf{\hat{x}}_{nj}^{2}}{k_{nj}^ {2}-\mathbf{\hat{x}}_{nj}^{2}}\right)^{\mu}+\frac{1}{2\gamma^{2}}\|\mathbf{\xi}_{\mathbf{c }}(\mathbf{\hat{x}}_{n},t)\|^{2}\\ &-\frac{1}{2}\|\mathbf{g}_{n-1}\mathbf{\hat{x}}_{n}\|^{2}+\frac{\left\|\mathbf{ \hat{x}}(\mathbf{\hat{x}}_{n})\right\|^{2}\Xi(\mathbf{\hat{x}}_{n})}{\left\|\mathbf{\Psi}( \mathbf{\hat{x}}_{n})\right\|^{2}+\left\|\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{n} \right\|^{2}}+\mathbf{\Psi}(\mathbf{\hat{x}}_{n})^{T}\mathbf{\Psi}(\mathbf{\hat{x}}_{n})\\ \leqslant&-\lambda_{\min}(\mathbf{K}_{n})\sum_{j=1}^{m}\ln \left(\frac{k_{nj}^{2}}{k_{nj}^{2}-\mathbf{\hat{x}}_{nj}^{2}}\right)-\left[\sum_{j= 1}^{m}\ln\left(\frac{\mathbf{\hat{x}}_{nj}^{2}}{k_{nj}^{2}-\mathbf{\hat{x}}_{nj}^{2}} \right)\right]^{\mu}\\ &-\eta_{1}\|\mathbf{\Psi}(\mathbf{\hat{x}}_{n})\|^{2}-\|\mathbf{\Psi}(\mathbf{\hat {x}}_{n})\|^{2\mu}-\frac{1}{2}\|\mathbf{g}_{n-1}\mathbf{\hat{x}}_{n}\|^{2}+\frac{1}{2 \gamma^{2}}\|\mathbf{\hat{\xi}}_{\mathbf{c}}(\mathbf{\hat{x}}_{n},t)\|^{2}\\ \leqslant&-a_{n}V_{n}-b_{n}V_{n}^{\mu}-\frac{1}{2}\|\mathbf{g }_{n-1}\mathbf{\hat{x}}_{n}\|^{2}+\frac{1}{2\gamma^{2}}\|\mathbf{\xi}_{\mathbf{c}}(\mathbf{ \hat{x}}_{n},t)\|^{2}\end{split} \]  
where \(a_{n}=\min\{2\lambda_{\min}(\mathbf{K}_{n}),2\eta_{1}\},b_{n}=2^{\mu},\gamma>0\). Define the system Lyapunov function as  
\[V(\mathbf{\hat{x}}_{1},\mathbf{\hat{x}}_{2},\ldots,\mathbf{\hat{x}}_{n},\mathbf{\Psi}(\mathbf{\hat{x}}_ {n}))=\sum_{k=1}^{n}V_{k} \tag{22}\]based on above analysis, its derivative satisfies the following condition  
\[ \begin{split}\dot{V}&\leqslant-\sigma\sum_{i=1}^{n}V_{i }-\varrho\sum_{i=1}^{n}V_{i}^{\mu}+\frac{1}{2\gamma^{2}}\|\xi_{c}(\overline{x} _{n},t)\|^{2}\\ &\leqslant-\sigma V-\varrho V^{\mu}+\varepsilon\end{split}  \tag{23}\]  
where \(\sigma=\min\{a_{1},a_{2},\ldots,a_{n}\},\varrho=\min\{b_{1},b_{2},\ldots,b_{n }\},\varepsilon=\frac{1}{2\gamma^{2}}\|\xi_{c}(\overline{x}_{n},t)\|^{2}\). According to the theorem in [33; 34], it can be concluded that system state tracking error \(\tilde{x}_{i},i=1,2,\ldots,n\) and the extended state \(\boldsymbol{\Psi}(\tilde{x}_{n})\) will achieve finite time stability, if the initial time \(t_{0}=0\), the stabilizing time is  
\[T_{r}\leq\max\biggl{(}\frac{1}{\theta_{0}\sigma(1-\mu)}\ln\frac{\theta_{0} \sigma V^{1-\mu}(0)+\varrho}{\varrho},\frac{1}{\sigma(1-\mu)}\ln\frac{\sigma V ^{1-\mu}(0)+\theta_{0}\varrho}{\theta_{0}\varrho}\biggr{)} \tag{24}\]  
and \(0<\theta_{0}<1\).