# Performance-Constraint Fault Tolerant Control to Aircraft in Presence of Actuator Deviation
## 2 A Class of Aircraft Dynamic Model Considering Actuator Deviation
**Step 1**: Define the tracking error vector,  
\[\tilde{\mathbf{x}}_{1}=\mathbf{x}_{1}-\mathbf{y}_{d},\tilde{\mathbf{x}}_{i}=\mathbf{x}_{i}-\mathbf{ \alpha}_{i-1},i=2,3,\ldots,n \tag{5}\]  
where state \(\mathbf{x}_{i}=\left[x_{i1},x_{i2},\ldots,x_{im}\right]^{T},i=1,2,\ldots,n\), and a following barrier-Lyapunov function as  
\[V_{1}(\tilde{\mathbf{x}}_{1})=\frac{1}{2}\sum_{i=1}^{m}\ln\left(\frac{k_{1i}^{2}} {k_{1i}^{2}-\tilde{\mathbf{x}}_{1i}^{2}}\right) \tag{6}\]  
Its derivative along the trajectory of the system is obtained. \[\dot{V}_{1}=\sum_{i=1}^{m}\frac{\tilde{\mathbf{x}}_{1i}\dot{\mathbf{x}}_{1i}}{k_{1i}^{ 2}-\tilde{\mathbf{x}}_{1i}^{2}} \tag{7}\]  
Define the following intermediate variables  
\[\mathbf{R}(\tilde{\mathbf{x}}_{1})=\operatorname{diag}\left(\left[\frac{1}{k_{11}^{2} -\tilde{\mathbf{x}}_{11}^{2}},\frac{1}{k_{12}^{2}-\tilde{\mathbf{x}}_{12}^{2}},\ldots, \frac{1}{k_{1m}^{2}-\tilde{\mathbf{x}}_{1m}^{2}}\right]\right) \tag{8}\]  
then, the Equation (7) can be represented as  
\[\dot{V}_{1}=\tilde{\mathbf{x}}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_{1})\dot{\mathbf{x}}_{1}= \tilde{\mathbf{x}}_{1}^{T}\mathbf{R}(\tilde{\mathbf{x}}_{1})(f_{1}+\mathbf{g}_{1}(\mathbf{\alpha}_ {1}+\tilde{\mathbf{x}}_{2})-\dot{\mathbf{y}}_{d}) \tag{9}\]  
Define the virtual control function \(\alpha_{1}\)\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]0  
where \(\mu\in(0,1)\), and \(K_{1}\) is a diagonal positive definite matrix, further, we can obtain  
\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]1 \[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]2 \[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]3  
Based on Lemma 1, the following inequalities hold,  
\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]4 \[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]5  
therefore, inequality (11) can be rewritten as  
\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]6 \[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]7  
where \(\lambda_{\min}(\mathbf{K}_{1})\) denotes the minimum eigenvalue of matrix \(\mathbf{K}_{1}\), and \(b_{1}=2^{\mu}\). **Step i**: Define the barrier-Lyapunov function as  
\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]8  
and also the virtual function \(\mathbf{\alpha}_{i}\) as  
\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]9 (15) \[\dot{z}_{0} =v \tag{4}\]0  
Design system extended dynamic as follows  
\[\dot{z}_{0} =v \tag{4}\]1  
where \(\eta_{1}>0,\mathbf{\Psi}=\left[\Psi_{1},\Psi_{2},\ldots,\Psi_{m}\right]^{T}\). Finally, a finite-time-convergence fault-tolerant controller for system (1) is designed as  
\[ \begin{split}\mathbf{u}=& g_{n}^{-1}(\mathbf{\hat{x}}_{n}) \bigg{(}-\mathbf{f}_{n}(\mathbf{\hat{x}}_{n})-\mathbf{K}_{n}\mathbf{\hat{x}}_{n}-\mathbf{R}(\mathbf{ \hat{x}}_{n})^{\mu-1}\lceil\mathbf{\hat{x}}_{n}\rceil^{2\mu-1}-\frac{\gamma^{2}}{ 2}\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{n}\\ &-\frac{1}{2}\mathbf{R}(\mathbf{\hat{x}}_{n})^{-1}\mathbf{g}_{n-1}^{T}\mathbf{g}_ {n-1}\mathbf{\hat{x}}_{n}-\frac{\mathbf{R}(\mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{n}\Xi(\mathbf{ \hat{x}}_{n})}{\left\|\mathbf{\Psi}(\mathbf{\hat{x}}_{n})\right\|^{2}+\left\|\mathbf{R}( \mathbf{\hat{x}}_{n})\mathbf{\hat{x}}_{n}\right\|^{2}}+\mathbf{\hat{x}}_{n-1}\bigg{)}\end{split}  \tag{19}\]  
further, given an extended barrier-Lyapunov function as  
\[V_{n}(\mathbf{\hat{x}}_{n},\mathbf{\Psi}(\mathbf{\hat{x}}_{n}))=\frac{1}{2}\sum_{j=1}^{m} \ln\left(\frac{k_{nj}^{2}}{k_{nj}^{2}-\mathbf{\hat{x}}_{nj}^{2}}\right)+\frac{1}{ 2}\mathbf{\Psi}(\mathbf{\hat{x}}_{n})^{T}\mathbf{\Psi}(\mathbf{\hat{x}}_{n}) \tag{20}\]  
thus, its derivative can be obtained by substituting the proposed controller (19),  
\[