## 2 Main Results
### Controller Design and Stability Analysis
At this time, the state \(\zeta\) is generated by the anti-saturation compensator, which compensates for the deviation caused by input saturation until \(\Delta u=0\). Furthermore, the larger the values of parameters \(k_{3}\) and \(k_{4}\) are, the faster the auxiliary system's compensation rate is. Therefore, with the anti-saturation auxiliary system, this paper's design effectively deals with the adverse effects created by input saturation and guarantees helicopter's yaw angle true tracking to the desired trajectory under the input constraints._  
**Theorem 1**. : _Consider the non-affine nonlinear system (4), affine-form system (5), and Assumptions 1-4. By establishing an adaptive saturation controller (36), a virtual control signal (21), and by designing the adaptation law (32), with the initial conditions bounded and all signals of the closed-loop system uniformly bounded, the proposed scheme can guarantee that the helicopter's yaw channel system tracking errors \(z_{1}\) and \(z_{2}\) converge to small compact sets around the origin._  
Proof. : We can get the following inequality from (41):  
\[\dot{V}\leq-kV+C \tag{42}\]  
where \(k=min\{2k_{1},-k_{2}G,-\beta\sigma_{2},-\gamma\sigma_{1},k_{3}\}\) and \(C=\frac{\alpha^{2}}{2}+\frac{\Delta u^{2}}{2k_{3}}+0.2785\lambda\delta\delta+ \frac{\hat{C}\sigma_{2}}{2}\delta^{2}+\frac{C\sigma_{1}}{2}\delta^{2}\). At the same time, integrating both sides of (42) over \([0,t]\), we get  
\[0\leq V(t)\leq\left[V(0)-\frac{C}{k}\right]e^{-kt}+\frac{C}{k} \tag{43}\]  
where \(V(0)\) is the initial value of \(V\). Equation (43) shows that \(z_{1},z_{2},\tilde{\delta},\tilde{\theta}\), and \(\zeta\) are bounded. Then, because \(x_{1}=z_{1}+y_{d}\) and \(|y_{d}|\leq\beta_{1}\), we have \(|x_{1}|\leq|z_{1}|+|y_{d}|<k_{a1}+\beta_{1}\leq\alpha_{1}\). From the design of \(\alpha\) in (21), we know the function \(\alpha\) contains \(x_{1},y_{d}\) and \(\dot{y}_{d}\), and that the maximum value of \(\alpha\) exists, namely \(\alpha\leq\alpha_{m}\). Considering that \(x_{2}=z_{2}+\alpha\), we get \(|x_{2}|\leq|z_{2}|+|\alpha|<k_{a2}+\alpha_{m}\leq\alpha_{2}\). Since \(\delta\), \(\tilde{\delta}\), \(\theta\), and \(\tilde{\theta}\) are bounded, \(\dot{\delta}=\delta-\tilde{\delta}\) and \(\dot{\theta}=\theta-\tilde{\theta}\) are also bounded. The designed actual control input \(u\) from (36) verifies that \(u\) is bounded. Therefore, all the signals in the closed-loop system remain uniformly bounded. Further analysis shows that  
\[ \begin{split}&\ln\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}}\leq 2 [V(0)-\frac{C}{k}]e^{-kt}+\frac{2C}{k}\\ &\ln\frac{k_{a2}^{2}}{k_{a2}^{2}-z_{2}^{2}}\leq 2[V(0)-\frac{C}{k}]e^{ -kt}+\frac{2C}{k}\end{split}  \tag{44}\]  
Taking exponentials on both sides of (44) leads to  
\[ \begin{split}&\frac{k_{a1}^{2}}{k_{a1}^{2}-z_{1}^{2}}\leq e^{2[V( 0)-\frac{C}{k}]e^{-kt}+\frac{2C}{k}}\\ &\frac{k_{a2}^{2}}{k_{a2}^{2}-z_{2}^{2}}\leq e^{2[V(0)-\frac{C}{k }]e^{-kt}+\frac{2C}{k}}\end{split}  \tag{45}\]  
Since \(k_{a1}^{2}-z_{1}^{2}>0\) and \(k_{a2}^{2}-z_{2}^{2}>0\), multiplying by \(k_{a1}^{2}-z_{1}^{2}\) and \(k_{a2}^{2}-z_{2}^{2}\) in (45) results in  
\[ \begin{split}&|z_{1}|\leq k_{a1}\sqrt{1-e^{-2[V(0)-\frac{C}{k}]e^{ -kt}-\frac{2C}{k}}}\\ &|z_{2}|\leq k_{a2}\sqrt{1-e^{-2[V(0)-\frac{C}{k}]e^{-kt}-\frac{ 2C}{k}}}\end{split}  \tag{46}\]It can be seen that given any \(\Delta_{1}>k_{a1}\sqrt{1-e^{-2\frac{\widetilde{C}}{k}}}\) and \(\Delta_{2}>k_{a2}\sqrt{1-e^{-2\frac{\widetilde{C}}{k}}}\), there is a \(T\) such that for any \(t>T\), \(|z_{1}|\leq\Delta_{1}\) and \(|z_{2}|\leq\Delta_{2}\). \(|z_{1}|\leq k_{a1}\sqrt{1-e^{-2\frac{\widetilde{C}}{k}}}\) and \(|z_{2}|\leq k_{a2}\sqrt{1-e^{-2\frac{\widetilde{C}}{k}}}\) along with \(t\rightarrow\infty\). It can be concluded that \(|z_{1}|\leq k_{a1}\sqrt{1-e^{-2\frac{\widetilde{C}}{k}}}\) and \(|z_{2}|\leq k_{a2}\sqrt{1-e^{-2\frac{\widetilde{C}}{k}}}\) as \(t\rightarrow\infty\).