## 5 Stability Analysis and Augmentation Adjustment of UAV
### Establishment and Linearization of Dynamic Equations  
The dynamics equations of the multi-section variable-sweep wing UAV are established based on [PERSON]'s method. The influence of the centroid offset, caused by the wing structure deformation, is considered to modify the relative terms of the wing sections in the dynamics model. The dynamics model of the targeted morphing UAV system can then be simplified as follows:  
\[J_{E}E+J_{A}\left[ \begin{array}{c}L_{w}\\ \mathbf{D}_{w}\\ L_{t}\\ D_{t}\\ Y_{t}\\ D_{f}\end{array} \right]+J_{T}T+E_{G}\left[ \begin{array}{c}\mathbf{G}_{f}\\ \mathbf{G}_{w}\\ \end{array} \right]+\left[ \begin{array}{c}0\\ \vdots\\ L_{p}\overline{p}\\ \vdots\\ 0\end{array} \right]=K_{V}\left[ \begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array} \right]+K_{\Omega}\left[ \begin{array}{c}\dot{p}\\ \dot{q}\\ \dot{r}\end{array} \right]+K_{\eta}\dot{\eta}+\mathbf{N} \tag{9}\]  
where \(J_{E}\) represents the coefficient matrix related to engine thrust; \(J_{A}\) represents the coefficient matrix related to the aerodynamics of each wing section, V-tail, and fuselage; \(J_{T}\) represents the coefficient matrix related to the torque at each wing section; \(E_{G}\) represents the coefficient matrix related to the mass in which the tail and fuselage are treated as a body; \(L_{p}\ \overline{p}\) represents the effect of roll damping; \(K_{V}\) represents the coefficient matrix related to velocity; \(K_{\Omega}\) represents the coefficient matrix related to the angular velocity of the fuselage; \(K_{\eta}\) represents the coefficient matrix related to the angular acceleration of sweep change; and \(\mathbf{N}\) contains the square term and the cross term of the generalized velocity of the UAV. Among them, the correction term of the centroid offset appears in the relevant positions of \(J_{A},K_{V},K_{\Omega}\), and \(K_{\eta}\), respectively, to achieve the modification of aerodynamic force, velocity, angular velocity, and wing deformation. Considering the difference between the centroid of the wing with the actual structure, and that of the wing with uniform distribution, and the movement of the centroid of the wing in the \(Ox_{y}y_{y}\) plane, caused by the deformation of the structure, the modified values of the centroid offset of the inner and outer sections are introduced into the model, which are defined as (\(\triangle x_{\text{in}}\),\(\triangle y_{\text{in}}\),\(0\)) and (\(\triangle x_{\text{out}}\),\(\triangle y_{\text{out}}\),\(0\)). The modified values are functions related to the actual structure distribution and deformation mode. The modified offset of the centroid can be expressed as a quadratic function of the sweep angle of the section when the wingspan ratio of the inner and outer sections is determined. Table 7 lists the correction functions of the centroid offset of the right-wing sections, considering the influence of the actual structure deformation of the wing, in which the independent variable is the sweep angle corresponding to the wing sections of each configuration. The correction functions of the centroid offset of the inner and outer wings of the left wing can be obtained using the corresponding coordinate transformation. In this analysis, the thrust of the engine is along the axis of the fuselage. According to the previous analysis, it can be observed that the trimming angle of attack is small, so it can be considered that \(\alpha^{*}=0^{\circ}\). Based on the small disturbance hypothesis [38], the linearized results of the lift, drag, and lateral forces can be expressed as follows:  
\[ \begin{array}{l}\Delta D=D_{V}\Delta V+D_{\alpha}\Delta\alpha+D_{\beta}\Delta \beta+D_{\delta_{a}}\Delta\delta_{a}+D_{\delta_{c}}\Delta\delta_{e}+D_{\Lambda} \Delta\Lambda\\ \Delta L=L_{V}\Delta V+L_{\alpha}\Delta\alpha+L_{\delta}\Delta\dot{\alpha}+L_{ \beta}\Delta\beta+L_{q}\Delta q+L_{\delta_{a}}\Delta\delta_{a}+L_{\delta_{c}} \Delta\delta_{e}+L_{\Lambda}\Delta\Lambda\\ \Delta Y=Y_{\beta}\Delta\beta+Y_{p}\Delta p+Y_{\gamma}\Delta\gamma+Y_{\delta_{ r}}\Delta\delta_{r}\end{array}