## 2 Main Results
### Simulation Results  
A UAV yaw-channel model acquired from the helicopter-on-arm platform [45, 46] was used to confirm feasibility of the proposed control scheme.Taking input saturation and actuator faults into consideration for the UAV yaw-channel model, the non-affine nonlinear yaw dynamic model was formulated as follows:  
\[\left\{ \begin{array}{l}\phi=r\\ \dot{r}=l_{1}\varphi+l_{2}r+l_{3}\theta_{trfs}+l_{4}\theta_{trfs}^{2}+l_{5} \Omega\theta_{trfs}+D\end{array} \right. \tag{47}\]  
with \(l_{1}=-3.33\), \(l_{2}=-1.38\), \(l_{3}=63.09\), \(l_{4}=11.65\), \(l_{5}=-0.14\), and \(\Omega=1200\). \(\theta_{trfs}\) denotes the pitch angle with saturation and actuator faults. \(D=\sin t+\cos 2t+2\) is an external disturbance. Let \([\varphi,r]^{T}=[x_{1},x_{2}]^{T}\), \(\theta_{trfs}=u_{fs}\). The nonlinear controller was designed by applying the Taylor series expansion technique to Equation (47). \[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f+g\varphi u_{s}+d\end{array} \right. \tag{48}\]  
where \(f=l_{1}x_{1}+l_{2}x_{2}+l_{3}u_{fs\zeta}+l_{4}u_{fs\zeta}{}^{2}+l_{5}\Omega u _{fs\zeta}-(l_{3}+2l_{4}u_{fs\zeta}+l_{5}\Omega)u_{fs\zeta}\), \(g=l_{3}+2l_{4}u_{fs\zeta}+l_{5}\Omega\). The initial conditions of the UAV helicopter are \(x_{1}(0)=0.16\) rad and \(x_{2}(0)=-0.2\) rad/s. The desired tracking command was \(y_{d}=0.52\sin(1.2t)+0.05\cos(t)+0.05\). With \(k_{a1}=k_{a2}=0.2\), the state constraints of the system were \(|x_{1}|\leq 0.82\) rad, \(|x_{2}|\leq 0.874\) rad/s. The robust sliding mode filter parameters in Equation (6) were chosen as \(\tau=0.005,\zeta_{1}=20\), and \(\zeta_{2}=0.01\). The saturation parameter \(u_{sm}=0.055\) rad, the anti-saturation compensator parameters were designed as \(k_{3}=0.5\), \(k_{4}=1\), and the initial value of \(\zeta(0)=0\). The RBFNN was exerted to approximate the unknown nonlinear function. The neural network \(W^{*T}\Psi(Z)\) contained six units evenly distributed in the interval \([-25,25]\) with the width of each unit equal to 9. The designed adaptive parameters were taken as \(\lambda=0.2\), \(\gamma=\beta=1,\sigma_{1}=-0.3,\sigma_{2}=-0.1\), and \(a=0.1\). The initial estimated values of the adaptive parameters were \(\hat{\theta}(0)=0\) and \(\hat{\delta}(0)=-0.1\). The input controller parameters were chosen as \(k_{1}=0.5\) and \(k_{2}=1\). In the simulation scenario, the UAV helicopter suffers from actuator fault at \(t\geq 15\)s, and \(\rho=\frac{1}{0.75+e^{-\alpha s_{1}s_{2}}}\). To verify the feasibility of the proposed adaptive neural fault-tolerant control scheme, the simulation results are shown in Figures 2-9. Figure 2 shows the approximation curves of the states which illustrate the effectiveness of the approximation method (5). Figure 3 shows the good approximation performance of the robust sliding mode filter (6); the filtered value \(u_{fs\xi}\) can approximate \(u_{fs}\) with good accuracy. Figures 4 and 5 not only demonstrate the good tracking performance of system states with input saturation and an actuator fault but also meet state constraints. Figures 6 and 7 give the time trajectories of the adaptive parameters \(\hat{\theta}\) and \(\hat{\delta}\). Figure 8 shows the state \(\zeta\) change of the anti-saturation compensator. Figure 9 shows the changes of helicopter input pitch angle \(u\) and input with saturation and actuator fault \(u_{fs}\). According to the simulation, the anti-saturation compensator can accommodate input constraints and demonstrates a superior compensation characteristic. The adaptive saturation controller of Equation (36) proposed in this paper can effectively address the state constraints and input saturation of the system. We can see that when an actuator fault occurs at \(t\geq 15\) s, the system can still track the desired trajectory. Figure 4: State \(x_{1}\) tracking performance. Figure 5: State \(x_{2}\) tracking performance. Figure 3: The approximation performance of \(u_{fs}\) in system transformation. Figure 8: State of auxiliary system \(\zeta\). Figure 6: Adaptive parameter \(\hat{\theta}\).