# Short Landing for Flying-Wing Unmanned Aircraft with Thrust Vector
## 4 Flight Control System Design
### Control Allocation  
In the UAV configuration depicted in Figure 1, both the roll and yaw control are achieved by the aileron and SSD. The control allocation process in this configuration can utilize a pseudo-inverse method, which is outlined below:  
\[ \begin{bmatrix}\delta_{ac}\\ \delta_{SDD_{c}}\end{bmatrix} = \begin{bmatrix}L_{\delta_{x}}&0\\ 0&N_{\delta_{SDD}}\end{bmatrix} ^{-1} \begin{bmatrix}v_{p}\\ v_{r}\end{bmatrix}  \tag{26}\]  
where \(\delta_{ac}\) and \(\delta_{SDD_{c}}\) are the aileron and SSD command values, and the roll and yaw control effectiveness parameters are as follows:  
\[ \begin{split} L_{\delta_{x}}&=\frac{1/2\rho V^{2}\delta C_{ \delta_{x}}}{I_{xx}}\\ N_{\delta_{SDD}}&=\frac{1/2\rho V^{2}\delta C_{\delta_{SDD}}}{I_{xz}}\end{split}  \tag{27}\]  
where \(C_{\delta_{x}}\) is the control derivative of the aileron. The control allocation, Equation (26), does not consider the coupling effect between the SDD and aileron, as it is treated as a disturbance that can be estimated by the ESO. The control allocation for the pitch channel involves both aero surfaces and lift fans, which differ from the roll and yaw channels. Due to the distinct response characteristics of the elevator and lift fans, both a high-frequency and low-frequency control allocation for the pitch channel are carried out, as illustrated in Figure 15.  
To allocate the control inputs for the pitch channel, the desired pitch angular acceleration, \(v_{q}\), is first filtered using a first-order lag transfer function with a time constant of 1 s (\(\tau_{r}\)= 1 s). The resulting desired pitch control value is then split into two components: a low-frequency part (LFP), \(v_{q}\); and a high-frequency part (HFP), \(v_{qh}\). The LFP is allocated using a daisy-chaining method [32], which involves calculating the elevator command value as follows:  
\[\delta_{\text{acs}}=\max\left(\min\left(v_{ql}/M_{\delta_{r}},\delta_{\text{ max}}\right),\delta_{\text{emin}}\right) \tag{28}\]  
Figure 14: Complete control diagram for inner and outer loop control. Figure 15: Pitch channel control allocation. where \(M_{\delta_{g}}\) is the control effectiveness parameter of the elevator and can be calculated as follows:  
\[M_{\delta_{g_{e}}}=\frac{1/2\rho V^{2}ScC_{m\delta_{g}}}{I_{yy}} \tag{29}\]  
where \(C_{m\delta_{e}}\) is the control derivative of the elevator. If the elevator deflection angle reaches its limits, the lift fan is utilized to compensate for the LFP. The left LFP, \(u_{g}\), is then obtained through the lift fan:  
\[ \begin{array}{c}v_{gg}=v_{ql}-M_{\delta_{g}}\delta_{ecs}\\ \omega_{fcs}=v_{qg}/M_{fan}\end{array}  \tag{30}\]  
where \(\omega_{fcs}\) is the command value of the lift fan, and \(M_{fan}\) is the control effectiveness parameter of the lift fan, as follows:  
\[M_{fan}=\frac{2n\rho C_{t}\omega_{f}D_{f}^{4}R_{f}}{I_{yy}} \tag{31}\]  
Secondly, the HFP is allocated via parallel allocation. The elevator's and lift fan's residual control powers are denoted as \(u_{nle}\) and \(u_{rfm}\), respectively. When \(v_{qh}\geq 0\), the output of the parallel output is given by the following:  
\[\left\{ \begin{array}{c}k=v_{qh}/\left(u_{nle}+u_{rfan}\right)\\ \omega_{fcp}=k\left(\omega_{fmax}-\omega_{fcs}\right)\\ \delta_{ecp}=k(\delta_{\text{emax}}-\delta_{ecs})\end{array} \right. \tag{32}\]  
otherwise, we have the following:  
\[ \begin{array}{c}k=v_{qh}/\left(u_{nle}+u_{rfan}\right)\\ \omega_{fcp}=k\left(\omega_{fmin}-\omega_{fcs}\right)\\ \delta_{ecp}=k(\delta_{\text{emin}}-\delta_{ecs})\end{array}  \tag{33}\]  
where we have the following:  
\[ \begin{array}{c}u_{nle}=\left\{\begin{array}{c}M_{\delta_{g}}(\delta_{ \text{emax}}-\delta_{ecs}),v_{qh}<0\\ M_{\delta_{g}}(\delta_{\text{emin}}-\delta_{ecs}),v_{qh}\geq 0\\ \delta_{cep}=k(\delta_{\text{emin}}-\delta_{ecs})\end{array}\right.\end{array}  \tag{34}\]  
In Equation (34), \(\omega_{f\text{ max}}\) and \(\omega_{f\text{ min}}\) represent the maximum and minimum rotation speeds of the lift fans, respectively.