# Design of a Morphing Aircraft Based on Model Predictive Control
## 2 Mathematical Model of a Morphing Aircraft
### Nonlinear Dynamic Model
\(C_{L_{\delta_{e}}}\) and \(C_{m_{\delta_{e}}}\) are the aerodynamic derivatives of the lift and pitching moment with respect to elevator deflection, while \(C_{L_{\delta_{e}}}\) and \(C_{m_{\eta}}\) are the aerodynamic derivatives of the lift and pitching moment with respect to the pitch rate. A variable-span aircraft follows a general morphing pattern, allowing the establishment of a conventional six-degree-of-freedom dynamic model. Therefore, the longitudinal nonlinear dynamic model of the variable-span aircraft is given as follows:  
Some large-deformation aircraft, such as folding-wing and variable-sweep aircraft, require a multibody dynamics model to reasonably describe the effects of structural changes. However, conventional six-degree-of-freedom (6-DOF) dynamic models can still be used  
Figure 2: This illustrates the relationship between longitudinal aerodynamic coefficients and angle of attack under different wingspan deformation rates for a morphing wing aircraft at an altitude of 5000 m and a Mach number of \(\text{Ma}=0.12\). (**a**), (**b**), and (**c**), respectively, describe the relationships between the pitch angle and the lift coefficient, drag coefficient, and pitching moment coefficient. for general morphing configurations [9; 23]. Based on the research objectives of this study, the following assumptions are made:  
1. The curvature of the Earth's surface is neglected, adopting the so-called \"flat Earth assumption\". 2. The ground coordinate system is considered an inertial reference frame. 3. The gravitational acceleration \(g\) is treated as a constant, ignoring variations with altitude. 4. The morphing aircraft is assumed to be a rigid body, and its mass \(m\) remains constant throughout the entire flight process, including the structural morphing phase. 5. The engine of the morphing aircraft is installed along the body's longitudinal axis, and the thrust \(T\) passes through the aircraft's center of mass with zero offset angle. This means the thrust acts only along the longitudinal axis and does not generate thrust-induced moments. 6. The morphing aircraft performs level, sideslip-free flight, satisfying \(\phi=0\) and \(\beta=0\), where \(\phi\) is the roll angle, \(\beta\) is the sideslip angle, and \(p\) and \(r\) are the roll rate and yaw rate, respectively. Therefore, the longitudinal nonlinear dynamic model of the morphing aircraft is formulated as follows:  
\[\left\{ \begin{array}{l}\dot{V}=\frac{1}{m}T\cos\alpha-\frac{1}{m}D(\xi_{b}) -g\sin(\theta-\alpha)\\ \dot{\alpha}=-\frac{1}{m\sqrt{T}}\sin\alpha-\frac{1}{m\sqrt{T}}L(\xi_{b})+q+ \frac{1}{\sqrt{T}}g\cos(\theta-\alpha)\\ \dot{\theta}=q\\ \dot{q}=\frac{1}{t_{y}}M(\xi_{b})\\ \dot{h}=V\sin(\theta-\alpha)\end{array} \right. \tag{4}\]  
where \(L\), \(D\), and \(M\) represent the aerodynamic lift, drag, and pitching moment, respectively. \(m\) is the aircraft mass, \(g\) is the gravitational acceleration, and \(I_{y}\) is the pitch moment of inertia. The thrust \(T\) is described by a simple linear relationship \(T=T_{\delta_{t}}\delta_{t}\), where \(T_{\delta_{t}}\) is the thrust coefficient, and \(\delta_{t}\) is the throttle setting.The expressions for aerodynamic forces and moments are given as follows:  
\[\left\{ \begin{array}{l}L(\xi_{b})=QS_{w}C_{L}(\xi_{b},h,M_{a},\alpha,\delta _{e},q)\\ D(\xi_{b})=QS_{w}C_{D}(\xi_{b},h,M_{a},\alpha)\\ M(\xi_{b})=QS_{w}C_{m}(\xi_{b},h,M_{a},\alpha,\delta_{e},q)\end{array} \right. \tag{5}\]  
where \(\delta_{e}\) represents the elevator deflection angle. \(Q=0.5\rho V^{2}\) is the dynamic pressure, \(\rho\) is the air density, and \(S_{w}\) is the reference wing area.