## 2 Main Results
### Controller Design and Stability Analysis  
In this section, we present the adaptive neural network fault-tolerant control using the backstepping technique for helicopter yaw control system. The recursive design process contains two steps in the system of Equation (5).The design procedure is as follows:  
Step 1: The tracking error is defined as \(z_{1}=x_{1}-y_{d}\) and its time derivative is  
\[\dot{z}_{1}=\dot{x}_{1}-\dot{y}_{d}=x_{2}-\dot{y}_{d} \tag{18}\]  
Choose the barrier Lyapunov function candidate as  
\[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]  
where \(k_{a1}\) is a positive scalar. By introducing \(z_{2}=x_{2}-\alpha\), where \(\alpha\) is the designed virtual control signal to be defined later, the derivative of \(V_{1}\) is given by  
\[\dot{V}_{1}=\frac{z_{1}\dot{z}_{1}}{k_{a1}^{2}-z_{1}^{2}}=\frac{z_{1}(z_{2}+ \alpha-\dot{y}_{d})}{k_{a1}^{2}-z_{1}^{2}} \tag{20}\]  
Design the virtual control \(\alpha\) as  
\[\alpha=-k_{1}z_{1}+\dot{y}_{d}-\frac{z_{1}}{2(k_{a1}^{2}-z_{1}^{2})} \tag{21}\]  
with \(k_{1}>0\). Based on Equations (20) and (21), then, one has  
\[\dot{V}_{1}=\frac{-k_{1}z_{1}^{2}}{k_{a1}^{2}-z_{1}^{2}}-\frac{z_{1}^{2}}{2(k _{a1}^{2}-z_{1}^{2})^{2}}+\frac{z_{1}z_{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{22}\]  
Then from Lemma 2, we obtain  
\[\frac{z_{1}z_{2}}{k_{a1}^{2}-z_{1}^{2}}\leq\frac{z_{1}^{2}}{2(k_{a1}^{2}-z_{1 }^{2})^{2}}+\frac{1}{2}z_{2}^{2} \tag{23}\]  
Figure 1: The adaptive neural fault-tolerant control structure for UAV helicopter yaw control system with input saturation and full-state constraints. By substituting (23) into (22), we have  
\[\dot{V}_{1}\leq\frac{-k_{1}z_{1}^{2}}{k_{a1}^{2}-z_{1}^{2}}+\frac{1}{2}z_{2}^{2} \tag{24}\]  
Step 2: In this step, the actual control input will be derived. Selecting the barrier Lyapunov function as  
\[V_{2}=V_{1}+\frac{1}{2}\ln\frac{k_{a2}^{2}}{k_{a2}^{2}-z_{2}^{2}} \tag{25}\]  
where \(k_{a2}\) is a positive number, we define the variable \(\Delta u=u_{s}-u\). The time derivative of the error variable \(z_{2}\) is  
\[\dot{z}_{2} =\dot{x}_{2}-\dot{\alpha} \tag{26}\] \[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]0 \[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]1  
Invoking (24), (25), and (26),  
\[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]2 \[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]3 \[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]4  
where \(\dot{\alpha}\) can be estimated by Lemma 4. The unknown nonlinear function is defined as \(F(Z)=f+g\rho\Delta u-\dot{\alpha}+\frac{z_{2}}{2}(k_{a2}^{2}-z_{2}^{2})\). The RBFNN is used to approximate the unknown function \(F(Z)\) as  
\[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]5  
where \(Z=[x_{1},x_{2},y_{d},\dot{y}_{d}]^{T}\in\Omega_{Z},l(Z)\) is the approximation error. To reduce the computational burden, the variable \(\theta=\hat{G}^{-1}\|W^{*}\|^{2}\) is applied. Substituting (28) into (27), we have  
\[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]6  
By using [PERSON]'s inequality, the second term of Equation (29) can be rewritten as  
\[V_{1}=\frac{1}{2}\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}} \tag{19}\]7  
where \(a\) is a positive design parameter. A new variable is defined as \(\delta=\hat{G}^{-1}(d_{m}+  
The adaptive law for the unknown parameters \(\theta\) and \(\delta\) is given by  
\[\left\{ \begin{array}{l}\dot{\theta}=\gamma\left[\sigma_{1}\dot{\theta}-\frac{{z_ {2}}^{2}}{{2a^{2}(k_{a2}^{2}-z_{2}^{2})}^{2}}\Psi^{T}\Psi\right]\\ \dot{\delta}=\beta\left[\sigma_{2}\dot{\delta}-\frac{{z_{2}}}{{k_{a2}^{2}-z_{2 }^{2}}}tanh(\frac{{z_{2}}}{{\lambda(k_{a2}^{2}-z_{2}^{2})}})\right]\end{array} \right. \tag{32}\]  
where \(\dot{\theta}\) and \(\dot{\delta}\) are the estimated values of the unknown parameters \(\theta\) and \(\delta\), respectively. \(\gamma>0\), \(\beta>0\), \(\lambda>0\), \(\sigma_{1}<0\) and \(\sigma_{2}<0\) are parameters to be designed.