# Short Landing for Flying-Wing Unmanned Aircraft with Thrust Vector
## 4 Flight Control System Design
### Sink Rate and Velocity Control  
According to the analysis in Section 3, the 3 BSD's nozzle pitch angle is utilized to control the UAV's velocity, and its sink rate is controlled by using the turbojet engine throttle. For velocity control, a simple proportional-integral (PI) controller is applied to obtain the desired tangential acceleration, as the following equation:  
\[k_{vi}\int_{0}^{t}{(V_{c}-V)dt}-k_{cp}V=\dot{V}_{des} \tag{41}\]  
where we have the following:  
\[k_{cp}=2\zeta_{T}\omega_{T} \tag{42}\] \[k_{pi}=\omega_{T}^{2}\]  
Figure 16: Diagram of the angle of attack controller. where \(\omega_{T}\) is the closed-loop bandwidth, and \(\zeta_{T}\) is the damping ratio which is set to 0.8 to reduce overshoot. The command value of the nozzle pitch angle is achieved using the following equation:  
\[\delta_{vqc}=\max\left(\min\left(\frac{-m\Big{(}\dot{V}_{des}-g\sin\gamma\Big{)} }{\delta_{vTc}T_{vmax}\cos\delta_{vqb}}\delta_{vqb},\delta_{vqmax}-\delta_{vqb} \right),-\delta_{vqb}\right) \tag{43}\]  
where \(\delta_{vTc}\) is the throttle command value, \(\delta_{vqb}\) is the baseline nozzle pitch angle given by the setting at the beginning of short landing, and \(\delta_{vq\text{ max}}\) is the maximum nozzle pitch angle. The output command of the nozzle pitch angle is equal to the sum of \(\delta_{vqc}\) and \(\delta_{vqb}\). For sink rate control, two controllers are used for different short-landing stages. The first uses the pitch angle to control the sink rate [22], and the controller is activated in the deceleration period. The closed-loop dynamics of the pitch angle are regarded as a first-order low-pass filter, as shown in Figure 17a. The closed-loop transfer function from \(\gamma_{c}\) to \(\gamma\) is as follows:  
\[\frac{\gamma(s)}{\gamma_{c}(s)}=\frac{k_{\theta}k_{\gamma}Z_{\alpha}}{s^{2}+k _{\theta}s+k_{\theta}k_{\gamma}Z_{\alpha}} \tag{44}\]  
To reduce overshoot, the damping ratio of the second-order system in Equation (44) is selected to be 0.8. The controller gain, \(k_{\gamma}\), can then be calculated through pole placement:  
\[k_{\gamma}=k_{\theta}/\left(1.6^{2}Z_{\alpha}\right) \tag{45}\]  
The other sink rate controller uses the throttle to control the sink rate and is activated during the approach stage, as depicted in Figure 17b. For sink rate control in the approach stage, the vertical dynamics of the UAV are given by the following equation:  
\[-\dot{V}_{gz}/\cos\gamma=Z_{\alpha}\omega V\cos\alpha-g\cos\gamma+\frac{1/2 \rho V^{2}SC_{D}\sin\alpha}{m}+\frac{\delta_{vT}T_{vmax}\sin\delta_{vq}}{m}+ \frac{n\rho C_{t}\omega_{f}{}^{2}D_{f}^{4}}{m} \tag{46}\]  
where we have the following:  
\[\dot{V}_{gz}=-\dot{h} \tag{47}\]  
Equation (46) can be rewritten as follows:  
Figure 17: Sink rate control diagram: (**a**) control sink rate obtained using pitch attitude angle and (**b**) control sink rate obtained using the throttle. \[k_{cp}=2\zeta_{T}\omega_{T} \tag{42}\]0  
where \(A_{1}\) is the known part of the sink rate control, \(b_{0}\) is the control effectiveness parameter for the throttle, and \(f_{a}(\cdot)\) is the lumped disturbance, which includes the unmodeled dynamics and external disturbances. An NDI controller is designed as a baseline sink rate controller, as shown in Figure 17b. The output of the baseline controller is formulated as follows:  
\[k_{cp}=2\zeta_{T}\omega_{T} \tag{42}\]1  
where \(k_{sr}\) is the designed sink rate control bandwidth. Similar to the approach in Section 4.1, to improve the tracking performance, a second-order ESO is designed to estimate the disturbance,\(f_{a}(\cdot)\), as follows:  
\[\left\{ \begin{array}{l}\dot{z}_{1,sr}=\beta_{01,sr}(y-z_{1,sr})+z_{2,sr}+a_{ zE,d}\\ \dot{z}_{2,sr}=\beta_{02,sr}(y-z_{1,sr})\end{array} \right.