# Design of a Morphing Aircraft Based on Model Predictive Control
## 2 Mathematical Model of a Morphing Aircraft  
### Nonlinear Dynamic Model  
The wingspan is directly related to the aspect ratio, significantly affecting the lift and drag characteristics of the aircraft. Based on the prototype aircraft, a variable-span aircraft is designed to be capable of symmetrically extending or retracting both wings during flight to adjust the wingspan. The maximum wingspan is twice the minimum wingspan, and the specific model parameters can be found in ref. [13]. Figure 1 presents a schematic diagram of the wingspan variation of a morphing aircraft. Define the normalized description parameter of wingspan morphing for a variable-span aircraft as the wingspan morphing ratio \(\xi_{b}\), expressed as follows:  
\[\xi_{b}=\frac{b-b_{min}}{b_{max}-b_{min}}b \tag{1}\]  
where \(b\) is the wingspan, and \(b_{min}\) and \(b_{max}\) are the minimum and maximum wingspans, respectively. Obviously, \(\xi_{b}\in[0,1]\). The wingspan morphing of the aircraft is a large-scale deformation with a relatively slow deformation rate. Since the sweep angle and dihedral angle of the morphing-wing aircraft are both approximately zero, the wingspan morphing does not affect the aircraft's center of mass position, and its impact on the pitch moment of inertia is also negligible [20; 21]. For aircraft with large-scale and slow morphing, the unsteady process of structural deformation has minimal deviation in the aerodynamic parameter simulation results. Therefore, the structural morphing process can be regarded as a series of continuous quasi-steady deformation states. The aerodynamic parameters under each static quasi-steady deformation state can be calculated through DATCOM simulations [9; 22]. By selecting the altitude, Mach number, wingspan morphing rate, and pitch angle as  
Figure 1: Schematic diagram of wingspan morphing for a variable-span aircraft. operating points, the aerodynamic parameter data files obtained from DATCOM simulations for each operating point can be imported into the simulation software, allowing the visualization of phenomena through graphical representation. As shown in Figure 2, at any operating point within the small angle-of-attack range \(\alpha\in[-4^{\circ},12^{\circ}]\), a linear fit is applied to the lift coefficient \(C_{L}\) and the pitching moment coefficient \(C_{m}\), while a quadratic fit is applied to the drag coefficient \(C_{D}\), yielding the following:  
\[\left\{ \begin{array}{l}C_{L}=C_{L_{\alpha-0}}+C_{L_{\alpha}}\alpha\\ C_{D}=C_{D_{\alpha-0}}+C_{D_{\alpha}}\alpha+C_{D_{\alpha}}\alpha^{2}\\ C_{m}=C_{m_{\alpha-0}}+C_{m_{\alpha}}\alpha\end{array} \right. \tag{2}\]  
where \(C_{L_{\alpha}=0}\), \(C_{D_{\alpha}=0}\), and \(C_{m_{\alpha}=0}\) represent the lift coefficient, drag coefficient, and pitching moment coefficient at zero angle of attack, respectively. \(C_{L_{\alpha}}\), \(C_{D_{\alpha}}\), and \(C_{m_{\alpha}}\) denote the aerodynamic derivatives of lift, drag, and pitching moment with respect to the angle of attack, while \(C_{D_{\alpha}2}\) is the second-order aerodynamic derivative of drag with respect to the angle of attack. By performing a least-squares fit of the longitudinal aerodynamic coefficients \(C_{L}\), \(C_{D}\), and \(C_{m}\) at various operating points in the form of Equation (2) with respect to \(\alpha\), and incorporating the longitudinal control derivatives and dynamic derivatives obtained from DATCOM simulation calculations, the overall longitudinal aerodynamic coefficient relationships can be established as follows:  
\[\left\{ \begin{array}{l}C_{L}=C_{L_{\alpha-0}}+C_{L_{\alpha}}\alpha+C_{L_{ \alpha}}\delta_{e}+C_{L_{\alpha}}\frac{\varepsilon_{\alpha}}{2\Omega}q\\ C_{D}=C_{D_{\alpha-0}}+C_{D_{\alpha}}\alpha+C_{D_{\alpha}}\alpha^{2}\\ C_{m}=C_{m_{\alpha-0}}+C_{m_{\alpha}}\alpha+C_{m_{\alpha}}\delta_{e}+C_{m_{ \alpha}}\frac{\varepsilon_{\alpha}}{2\Omega}q\end{array} \right. \tag{3}\]  
where \(\delta_{e}\) is the elevator deflection angle, \(q\) is the pitch rate, and \(c_{A}\) is the mean aerodynamic chord of the wing.