## 3 Fault-Tolerant Aircraft Controller Design
### Design of Fault-Tolerant Aircraft Control Law  
To improve the system robustness, we introduced sliding-mode control into the nonlinear feedback component of the ADRC. For a given system model, we chose the fast-moving sliding-mode surface as follows:  
\[s(t)=e_{2}(t)-ce_{1}(t), \tag{35}\]  
where \(e_{2}(t)=\dot{e}_{1}(t)\), \(e_{1}(t)=x_{1}(t)-x_{d}(t)\), \(c>0\) satisfies the Hurwitz stability criteria.  
To simplify the calculation, according to Equations (13) and (14), we defined the nonlinear function \(\varsigma_{1}(x)=\Delta f(x)+g_{0}(x)c+\Delta g(x)c\) containing uncertain parameters of the system model and actuator deviation faults. We also defined the nonlinear function \(\varsigma_{2}(x)=\Delta g(x)-g_{0}(x)b\cdot k-\Delta g(x)b\cdot k\) containing the actuator gain faults and physical characteristic parameters of the actuator. The system was rewritten in the following form:  
\[\begin{cases}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f_{0}(x)+\varsigma_{1}(x)+g_{0}(x)u+\varsigma_{2}(x)u+d_{w}\\ y=x_{1}\end{cases}. \tag{36}\]  
Uncertain terms account for a very small proportion of system modeling. Particularly, a system with virtual control input hardly changes as long as the aircraft is not destroyed. Therefore, the following inequality holds in the uncertain terms of the system modeling:  
\[\begin{cases}-g_{0}(x)\ll\Delta g(x)\ll g_{0}(x)\\ -g(x)\ll\Delta g(x)\ll g(x)\end{cases}. \tag{37}\]It is easy to obtain \(\Delta g(x)\big{/}g(x)\in(0.5,2)\). The parameter \(\tau=\big{|}\zeta_{2}\big{/}g_{0}\big{|}\) was defined, and the following inequality was derived:  
\[\begin{split}\tau&=\left|\frac{\Delta g(x)-g_{0}(x)b \cdot v-\Delta g(x)b\cdot v}{g_{0}(x)}\right|\\ &<1-(1-b\cdot v)+\left|\frac{\Delta g(x)(1-b\cdot v)}{g_{0}(x)} \right|\end{split}. \tag{38}\]  
Because \(0<1-b\cdot v\leq 1\), \(\tau<1\). To facilitate the design of the system control rate, positive constants \(\zeta_{1}\) an \(\zeta_{2}\) were defined, and these are in the form of \(|d_{w}(t)|\leq\zeta_{1}\). The following inequalities exist for parameter \(\lambda\):  
\[\begin{cases}\lambda\geq 1,\quad\xi_{2}>0\\ \lambda\leq-1,\ \xi_{2}\leq 0\end{cases}. \tag{39}\]  
The following adaptive law was designed for some parameters involved in the control law as  
\[\left\{\begin{array}{l}\dot{\xi}_{1}=\chi_{1}|s|\\ \dot{\xi}_{2}=\chi_{2}\varpi|s|\\ \xi_{1}=\varphi_{1}s\\ \dot{\xi}_{2}=\tau\varphi_{2}|s|\end{array}\right.. \tag{40}\]  
where \(\chi_{1}\), \(\chi_{2}\), \(\varphi_{1}\), and \(\varphi_{2}\) are positive constants. The parameters \(\xi_{1}\) and \(\xi_{2}\) are the calculated outputs of the neural network and \(\xi_{1}=\xi_{1}-\xi_{1}\), \(\xi_{2}=\xi_{2}-\xi_{2}\). Let us define \(\varpi>0\), \(\varpi=\big{|}\xi_{d}+ce_{1}-f_{0}(x)-\dot{\xi}_{1}\text{sgn}(s)-\xi_{1}\big{|} -\xi_{2}\). According to Equation (28), the following control rate was defined for a nonlinear system with actuator faults  
\[u=\frac{1}{g_{0}(x)}\big{(}\ddot{x}_{d}+ce_{1}-f_{0}(x)-\dot{\xi}_{1}\text{sgn }(s)-\xi_{1}-\lambda\theta\text{sgn}(s)-\iota\text{sgn}(s)\big{)} \tag{41}\]  
where \(\vartheta=\dot{\xi}_{2}\varpi\) and \(\iota\) are positive constants.