## 2 Dynamics Modelling and Problem Formulation  
### Aircraft Dynamics and Actuator Modelling  
The dynamic model of a civil aircraft can be built via Newton law, and in normal operating conditions, such a model can be decoupled into longitudinal motion and lateral motion, as they have a slight impact on each other. For the purpose of autolanding, the longitudinal motion model is employed here ([PERSON] et al., 2019; [PERSON] et al., 2020), which is  
\[ \begin{array}{l}\dot{V}=\frac{1}{m}[T\cos(\alpha+\varepsilon_{T})-D-mg\sin( \gamma)]\\ \dot{\gamma}=\frac{1}{mV}[T\sin(\alpha+\varepsilon_{T})+L-mg\cos(\gamma)]\\ \dot{q}=\frac{M}{l_{yy}}\\ \dot{\theta}=q\end{array}  \tag{1}\]  
where \(m\) is the aircraft mass, \(V\) is the longitudinal speed, \(\alpha\) is the angle of attack, \(q\) is the pitch rate, \(\theta\) is the pitch rate, \(\gamma\) is the flight path angle, \(\varepsilon_{T}\) is the thrust inclination angle, \(I_{yy}\) is the principal moment of inertia in pitch axis, \(M\) is the pitching moment, and \(T\), \(L\), and \(D\) are the thrust, lift and drag, respectively. The aerodynamics forces and moment in Equation (1) can be described, as follows:\[ \begin{array}{l}L=\frac{1}{2}\rho V^{2}SC_{L},D=\frac{1}{2}\rho V^{2}SC_{D}\\ M=\frac{1}{2}\rho V^{2}SC_{M},T=T_{0}+\frac{\partial T}{\partial\phi_{t}}\delta_{t }\end{array}  \tag{2}\]  
where the aerodynamic coefficients can be described, as follows:  
\[ \begin{array}{l}C_{L}=C_{L0}+C_{L}^{\alpha}\alpha+C_{L}^{\delta_{L}}\delta_{ e}+(C_{L}^{\dot{\alpha}}\dot{\alpha}+C_{L}^{l}q)\frac{\dot{\gamma}}{2\dot{V}_{ 0}}\\ C_{D}=C_{D0}+C_{D}^{\alpha}\alpha+C_{D}^{\dot{\gamma}}\delta_{e}+(C_{D}^{\dot {\alpha}}\dot{\alpha}+C_{D}^{d}q)\frac{\dot{\gamma}}{2\dot{V}_{0}}\\ C_{M}=C_{M0}+C_{M}^{\alpha}\alpha+C_{M}^{\dot{\gamma}}\delta_{e}+(C_{M}^{ \dot{\alpha}}\dot{\alpha}+C_{M}^{d}q)\frac{\dot{\gamma}}{2\dot{V}_{0}}\end{array}  \tag{3}\]  
Table 1 provides the corresponding parameters of the civil aircraft Boeing 747 [24], which are used for simulations and analysis. Equation (1) can be trimmed and linearized according to small perturbation linearization theory. Table 2 lists the trim conditions. The elevator actuator and engine models are listed, as follows:  
\[\frac{\delta_{e}}{\delta_{ec}}=\frac{10}{s+10} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Description** \\ \hline \(m\) & 250,000 & kg & Mass \\ \(S\) & 510 & m\({}^{2}\) & Wing reference area \\ \(\bar{c}\) & 8.3 & m & Mean aerodynamic chord \\ \(I_{yy}\) & \(41.35\times 10^{6}\) & kg\(\cdot\)m\({}^{2}\) & Pitch axis inertia \\ \(\varepsilon_{T}\) & 0.044 & rad & Thrust inclination angle \\ \(C_{L0}\) & 1.71 & & \\ \(C_{L}^{\dot{\alpha}}\) & 5.67 & 1/rad & Angle of attack derivative for lift \\ \(C_{L}^{\dot{\alpha}}\) & 0.36 & 1/rad & Elevator variation derivative for lift \\ \(C_{L}^{\dot{\alpha}}\) & 6.7 & s/rad & Angle of attack rate derivative for lift \\ \(C_{L}^{\dot{\mu}}\) & 5.65 & s/rad & Pitch rate derivative for lift \\ \(C_{L}^{\dot{\mu}}\) & 0.263 & & \\ \(C_{D}^{\dot{\alpha}}\) & 1.13 & 1/rad & Angle of attack derivative for drag \\ \(C_{D}^{\dot{\alpha}}\) & 0 & s/rad & Angle of attack rate derivative for drag \\ \(C_{D}^{\dot{\alpha}}\) & 0 & s/rad & Angle of attack rate derivative for drag \\ \(C_{D}^{\dot{\alpha}}\) & 0 & s/rad & Pitch rate derivative for drag \\ \(C_{D}^{\dot{\alpha}}\) & 0 & s/rad & Pitch rate derivative for drag \\ \(C_{M}^{\dot{\alpha}}\) & \(-\)0.093 & & \\ \(C_{M}^{\dot{\alpha}}\) & \(-\)1.45 & 1/rad & Angle of attack derivative for moment \\ \(C_{M}^{\dot{\alpha}}\) & \(-\)1.40 & 1/rad & Elevator variation derivative for moment \\ \(C_{M}^{\dot{\alpha}}\) & \(-\)3.3 & s/rad & Angle of attack rate derivative for moment \\ \(C_{M}^{\dot{\alpha}}\) & \(-\)21.4 & s/rad & Pitch rate derivative for moment \\ \(T_{0}\) & 382.572 & kN & \\ \(\frac{dT}{\delta\phi_{t}}\) & 7801.63 & kN/rad & Throttle variation derivative for thrust \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The Boeing 747 aircraft parameters.