## 4 Numerical Simulations  
In the following simulations, the fourth-order Runge-Kutta (RK4) algorithm with a step size of 0.001 seconds is applied in the Matlab environment. The system starts from rest, and the desired flight trajectory is generated from the start point \((x_{d},z_{d})=(0,0)\) to the end point\((x_{d},z_{d})=(1,1)\) through a command pre-filter, which guarantees the continuity of the associated flight profiles. The profiles of \(x_{d}\) and its successive derivatives are shown in Figure 2, where \(z_{d}\) uses the trajectory generation scheme. The system parameters are summarized in Table 2.  
The altitude controller is constructed from (37). For comparison, three different position controllers are used:  
* _Cascade PID controller (CPID)._ The cascade PID control presented in Appendix A is used;
* _Integral backstepping controller (IBC)._ The position integral backstepping controller derived from Theorem 2 is used, where the time derivatives of the virtual controls, \(\phi_{1}\) and \(\phi_{2}\), are calculated by the approximated differentiator (APD) as illustrated in Appendix A;
* _Analytic integral backstepping controller (AIBC)._ The proposed analytic integral backstepping controller presented in Theorem 3 is applied. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(m\) & \(L\) & \(J\) & \(C_{T}\) & \(g\) & \(C_{dx}\)\(C_{dz}\) & \(C_{d\theta}\) \\ \hline Unit & kg & m & kg-m\({}^{2}\) & N/(rad/s)\({}^{2}\) & m/s\({}^{2}\) & (N/(m/s) & N.m/(rad/s) \\ \hline
0.6 & 0.25 & \(3.544\times 10^{-3}\) & \(1.339\times 10^{-6}\) & 9.8 & \(1\times 10^{-3}\) & \(1.2\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System parameters. Figure 2: Desired output \(x_{d}\) to its fourth differentiation. For CPID, the control gains are \(k_{px}=k_{pz}=24\), \(k_{ix}=k_{iz}=18\), \(k_{dx}=k_{dz}=9.5\), \(k_{p\theta}=36\), \(k_{i\theta}=42\), and \(k_{d\theta}=10.5\), respectively. The feasible solution pair \((\mathbf{P}_{\theta},\mathbf{Q}_{\theta})\) that satisfies (A13) is  
\[\mathbf{P}_{\theta}= \begin{bmatrix}2.2220&0.9223&0.0183\\ 0.9223&1.1954&0.0441\\ 0.0183&0.0441&0.0241\end{bmatrix} ,\quad\mathbf{Q}_{\theta}= \begin{bmatrix}0.77 06&0.1506&0.1413\\ 0.1506&0.6885&0.0646\\ 0.1413&0.0646&0.2089\end{bmatrix}  \tag{135}\]  
For IBC, the control gains are \(k_{z1}=10\), \(k_{z2}=5\), \(k_{px}=k_{pz}=24\), \(k_{ix}=k_{iz}=18\), and \(k_{dx}=k_{dz}=9.5\), respectively. The resulting feasible solution pair \((\mathbf{P}_{z},\mathbf{Q}_{z})\) that satisfies (43) is  
\[\mathbf{P}_{z}= \begin{bmatrix}1.3252&0.2021&0.0348\\ 0.2021&1.1050&0.0442\\ 0.0348&0.0442&0.0335\end{bmatrix} ,\quad\mathbf{Q}_{z}= \begin{bmatrix}0.6258& 0.1520&0.3657\\ 0.1520&0.8575&0.0420\\ 0.3657&0.0420&0.2742\end{bmatrix}  \tag{136}\]  
The feasible solution pair \((\mathbf{P}_{x},\mathbf{Q}_{x})\) that satisfies (50) is  
\[\mathbf{P}_{x}= \begin{bmatrix}1.3252&0.2021&0.0348\\ 0.2021&1.1050&0.0442\\ 0.0348&0.0442&0.0335\end{bmatrix} ,\quad\mathbf{Q}_{x}= \begin{bmatrix}0.6258& 0.1520&0.3657\\ 0.1520&0.8575&0.0420\\ 0.3657&0.0420&0.2742\end{bmatrix}  \tag{137}\]  
According to (53), the matrix \(\mathbf{P}_{3x}\) is  
\[\mathbf{P}_{3x}= \begin{bmatrix}1.1050&0.0442\\ 0.0442&0.0335\end{bmatrix}  \tag{138}\]  
The switching gain in IBC \(w=18>1.2\frac{\max(d_{\theta})}{J}=16.69\) is used to fulfill (60). The proposed AIBC is not sensitive to measurement noise. Therefore, higher control gains for the position loop in \(x\)-direction are considered to inhibit the effects of disturbances. The control gains are \(k_{z1}=10\), \(k_{z2}=5\), \(k_{pz}=24\), \(k_{iz}=18\), \(k_{dz}=9.5\), \(k_{px}=242\), \(k_{ix}=720\), and \(k_{dx}=27\), respectively. The corresponding feasible solution pair \((\mathbf{P}_{x},\mathbf{Q}_{x})\) is  
\[\mathbf{P}_{x}= \begin{bmatrix}15.8665&3.1698&0.0062\\ 3.1698&1.5796&0.0220\\ 0.0062&0.0220&0.0046\end{bmatrix} ,\quad\mathbf{Q}_{x}= \begin{bmatrix}4.4330& 0.7453&0.1592\\ 0.7453&2.1634&0.0629\\ 0.1592&0.0629&0.1025\end{bmatrix}