## 5 Stability Analysis and Augmentation Adjustment of UAV
### Influence Analysis of Dynamic Stability
#### 5.3.1 Longitudinal Stability Augmentation Design  
The augmentation model shown in Figure 5 is adopted in this research. For convenience, the model is adjusted with an angle of attack filter and without a pitch rate filter. For the angle of attack feedback, the dynamic characteristics of the actuator and the filter are expressed as first-order inertia elements, with bandwidths of 10 rad/s and 20.2 rad/s, respectively [48], as follows:  
\[G_{\alpha}(s)=\frac{10}{s+10},\ G_{\delta}(s)=\frac{20.2}{s+20.2} \tag{13}\]  
The angle of attack filter can be taken as a part of the dynamics model, thus the state variables of the system are \(x=[V,\alpha,\theta,\alpha x_{a},x_{F}]\). Because the positive elevator deflection generates a negative pitching moment, an inverter is added between the elevator actuator and the control surface, to adopt the positive gain root locus method for design. Based on the above analysis, the augmented dynamic equation of the system can be expressed as follows:  
\[\dot{\mathbf{x}}=\left[ \begin{array}{cccc}\mathbf{A}&-\mathbf{B}&0\\ &&0\\ &&&&&0\\ 0&0&0&-20.2&0\\ 0&10&0&0&0&-10\end{array} \right]\left[ \begin{array}{c}V\\ \alpha\\ q\\ \theta\\ x_{\alpha}\\ x_{F}\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 20.2\\ 0\end{array} \right]u_{e},\ \mathbf{y}=\left[ \begin{array}{cccc}C\\ 0&0&0&0&0&1\end{array} \right]\mathbf{x} \tag{14}\]  
The angle of attack feedback gain \(k_{\alpha}=0.3\) and pitch angle velocity feedback gain \(k_{q}=0.25\) are introduced in this research to improve the flight quality of the short-period modal. With the same parameters of the stability augmentation system, the pre-selected models are adjusted under different conditions. Figure 12 shows the changes in the short-period characteristic roots, with the configuration after stability augmentation under different Mach numbers, where the arrow indicates that the sweep angle of the outer section \(\Lambda\)out, corresponding to the configuration, decreases linearly in the coordinated deformation process of the wing. When the flight speed is small, the characteristic root of the system gradually moves away from the imaginary axis. The system stability is enhanced with a decrease in \(\Lambda_{\text{out}}\), corresponding to the configurations. For the model with a relatively small wingspan proportion of the inner section, the short-period modal presents a change law that first approaches the real axis and then moves away from the real axis, with a decrease in \(\Lambda_{\text{out}}\), corresponding to the configurations. For the model with a large wingspan proportion of the inner section, the short-period mode gradually approaches the real axis. The trim angle of attack is negative and the short-period modal gradually moves away from the real axis when the flight speed is high. With an increase in the wingspan proportion of the inner section of the pre-selected models, the variation range of the characteristic roots decreases gradually, and the influence of the configuration changes on the short-period modes decreases gradually. Figures 13 and 14 show the short-period damping ratio \(\xi\) and oscillation frequency \(\omega_{n}\) of the UAV system after stability augmentation adjustment. Long-period modal characteristics are less affected. The following conclusions can be drawn:  
1. With the increase in the wingspan proportion of the inner section of the pre-selected models, the effect of the longitudinal stability augmentation system on the longitudinal short-period modal of the targeted UAV is gradually weakened. The short-period damping ratio \(\xi\) decreases with an increase in the wingspan proportion of the inner section, after the stability augmentation adjustment. The difference between the short-period damping ratio \(\zeta\) of the pre-selected model increases with the sweep angle of the outer section \(\Lambda_{\text{out}}\), corresponding to the configuration, and the \(C_{D\text{min}}\) configuration, corresponding to the minimum damping ratio \(\zeta\) of the pre-selected models. 2.