# Performance-Constraint Fault Tolerant Control to Aircraft in Presence of Actuator Deviation
## 2 A Class of Aircraft Dynamic Model Considering Actuator Deviation  
Let us consider a class of strict-feedback MIMO autonomous system with actuator deviation,  
\[ \begin{split}&\dot{\mathbf{x}}_{i}=\mathbf{f}_{i}(\overline{\mathbf{x}}_{i})+g _{i}(\overline{\mathbf{x}}_{i})x_{i+1},i=1,2,\ldots,n-1\\ &\quad\ldots\\ &\dot{\mathbf{x}}_{n}=\mathbf{f}_{n}(\overline{\mathbf{x}}_{n})+(g_{n}( \overline{\mathbf{x}}_{n})+\Delta g_{n}(\overline{\mathbf{x}}_{n},t))\mathbf{u}+\mathbf{\xi}_{ c}(\overline{\mathbf{x}}_{n},t)\\ &\mathbf{y}=\mathbf{x}_{1}\end{split}  \tag{1}\]  
where system state \(\mathbf{x}\in\mathcal{R}^{m}\), \(\overline{\mathbf{x}}_{i}=\left[x_{1},x_{2},\ldots,x_{i}\right]^{T}\), \(i=1,2,\ldots,n\). The \(\mathbf{f}_{i}(\overline{x}_{i})\in\mathcal{R}^{m}\), \(\mathbf{g}_{i}(\overline{x}_{i})\in\mathcal{R}^{m\times m}\) are known smooth function and control matrix respectively. \(\Delta g(t)\) represents unknown bounded perturbation of control matrix, which is the result of actuator failure. \(\mathbf{\xi}_{c}(\bar{\mathbf{x}}_{n},t)\in\mathcal{R}^{m}\) is the unknown time-varying disturbance. \(\mathbf{u}\in\mathcal{R}_{m}\) denotes the control input signal, \(\mathbf{y}\in\mathcal{R}^{m}\) is system output vector signal. Next, we will design a practical finite-time-convergence fault-tolerant controller to make the closed-loop system resistant to actuator losses, and system output can converge to reference signal \(\mathbf{y}_{d}\), all states of system are restricted in the predefined range \(\left|x_{i}\right|<k_{\text{bi}},i=1,2,\ldots,m\). Before controller design, some necessary lemmas about barrier-Lyapunov function and FTC theory are introduced as follows,  
**Lemma 1** ([29; 30]). : _For a class of barrier-Lyapunov function \(V(x)=\ln\frac{k^{2}}{F-x^{2}},\quad k\in\mathcal{R}^{+},\quad x\in\mathcal{R}\), and \(\left|x\right|<k\), the following inequality is true,_\[V(x)<\frac{x^{2}}{k^{2}-x^{2}}<\frac{k^{2}}{k^{2}-x^{2}} \tag{2}\]  
_further, the inequality holds for \(\mu\in(0,1)\):_  
\[V^{\mu}(x)<\left(\frac{x^{2}}{k^{2}-x^{2}}\right)^{\mu} \tag{3}\]  
**Lemma 2**. : _For \(\forall\mathbf{x},\mathbf{y}\in\mathcal{R}^{n}\), inequality \(\mathbf{x}^{T}\mathbf{y}\leq\frac{1}{2\gamma^{2}}\mathbf{x}^{T}\mathbf{x}+\frac{\gamma^{2}}{2} \mathbf{y}^{T}\mathbf{y},\gamma\in\mathcal{R}^{+}\) holds [31]._  
**Lemma 3**. : _To noiseless signal \(\alpha_{0}\), take the following slide mode differentiator [32],_  
\[\dot{z}_{0} =v \tag{4}\] \[v =-\lambda_{0}|z_{0}-\alpha_{0}|^{1/2}\operatorname{sign}(z_{0}- \alpha_{0})+z_{1}\] \[\dot{z}_{1} =-\lambda_{1}\operatorname{sign}(z_{1}-v)\]  
_and selecting properly parameters \(\lambda_{0}\) and \(\lambda_{1}\), then \(z_{0},z_{1}\) will approximate the \(\alpha_{0},\dot{\mathbf{\lambda}}\) after a certain time respectively._  
**Definition 1**. : _To \(\mu\in\mathcal{R}^{+}\), define calculation rule: \(\lceil x\rceil^{\mu}=|x|^{\mu}\operatorname{sign}(x),\forall x\in\mathcal{R}\), where the sign operator defined as \(\operatorname{sign}(x)=1,x>0;\operatorname{sign}(x)=0,x=0;\operatorname{sign} (x)=-1,x<0\)._  
**Assumption 1**. : _Actuator outputs in system (1) are bounded, \(\|\mathbf{u}\|\leqslant u_{\max}\), and the term of control matrix perturbation \(\Delta\mathbf{g}(\mathbf{\bar{x}}_{n},t)\) is also bounded, where \(\|\Delta\mathbf{g}\|\leqslant\varrho,u_{\max}\) and \(\varrho\) are known, there is an unknown upper bound of disturbance \(\xi_{c}(\bar{x}_{n},t)\in\mathcal{R}^{m}\)._  
**3. Unified Control Framework**  
A finite-time-convergence fault-tolerant controller with state constraints is designed for the system (1) based on the integral back-stepping method.