# Short Landing for Flying-Wing Unmanned Aircraft with Thrust Vector
## 4 Flight Control System Design  
### Angular Rate and Attitude Controller  
In an inner angular rate controller loop, the relationship between the control input and the dynamics of the UAV is obtained by the following equation:  
\[\dot{\Omega}=\underbrace{I^{-1}M_{(\cdot)}-I^{-1}\{\Omega\times I\Omega\}}_{f_{( \cdot)}}+v_{(\cdot)} \tag{15}\]  
where \(\Omega=[p,\,q,\,r]^{\mathrm{T}}\) represents the roll, pitch, and yaw angular rates in the body frame, respectively; \(v_{(\cdot)}=[v_{p},\,v_{q},\,v_{r}]^{\mathrm{T}}\) is the control acceleration input; \(I\) is the moment of the inertial matrix; and \(f_{(\cdot)}\) is the total disturbance added to the flying-wing UAV for each axis, which is composed of modeled coupling dynamics and unmodeled dynamics, \(M_{(\cdot)}\). The unmodeled dynamics include external environmental disturbance and internal coupling dynamics. Then the Equation (15) can be reformulated as follows:  
\[\left\{ \begin{array}{l}\dot{x}=f_{(\cdot)}+v_{(\cdot)}\\ \dot{f}_{(\cdot)}=w(t)\end{array} \right. \tag{16}\]  
To control the angular acceleration around the UAV's body axis, an ESO-based control law is designed, where \(w(t)\) is the derivative of \(f_{(\cdot)}\) that is unknown but bounded. An ESO is capable of estimating the lumped disturbance caused by external disturbances, unmodeled dynamics, parameter variations, and complex nonlinear dynamics [33]. Dynamic compensation is then used based on the estimated results to linearize the model. For roll, pitch, and yaw control, a second-order linear ESO is designed, and the observer is expressed as follows:  
\[\left\{ \begin{array}{l}\dot{z}_{1}=\beta_{01}(y-z_{1})+z_{2}+v_{(\cdot)}\\ \dot{z}_{2}=\beta_{02}(y-z_{1})\end{array} \right. \tag{17}\]  
In each angular rate control loop, \(y\) represents the angular rate measurements, while \(z_{1}\) and \(z_{2}\) are the estimated angular rates and disturbance. The observer gains, \(\beta_{01}\) and \(\beta_{02}\), can be determined using bandwidth methods [34]. \[\beta_{01}=2\omega_{o_{(\cdot)}},\beta_{02}=\omega_{o_{(\cdot)}}{}^{2} \tag{18}\]  
The ESO's observation bandwidth for each angular rate control loop is denoted as \(\omega_{o_{(\cdot)}}\). The error dynamics of the ESO can be calculated as follows:  
Figure 13: Trim results for short landing (angle of attack = 3 deg): (**a**) throttle and nozzle angle trim results and (**b**) throttle and lift fan trim results. \[\left\{ \begin{array}{l}\dot{e}_{1}=-\beta_{01}e_{1}+e_{2}\\ \dot{e}_{2}=-\beta_{02}e_{1}-w(t)\end{array} \right. \tag{19}\]  
where \(e_{1}\)\(=\)\(z_{1}-x\) and \(e_{2}\)\(=\)\(z_{2}-f_{(. )}\) are the estimated error. The characteristic polynomial of Equation (19) can be expressed as follows:  
\[\lambda(s)=s^{2}+\beta_{01}s+\beta_{02} \tag{20}\]  
With positive observer gains, the roots of the characteristic polynomial in Equation (20) are all in the left half plane, ensuring the ESO's bounded-input-bounded-output (BIBO) stability [35]. A proportional controller is designed, and the estimated disturbance is given as follows:  
\[v_{(.)}=k_{(.)}(\Omega_{c}-\Omega)-z_{2(.)} \tag{21}\]  
The proportional controller is designed to control the estimated disturbance in each control channel. The proportional gains or control bandwidth of the closed loop are denoted by \(k_{(.)}\in(k_{p},\;k_{q},\;k_{r})\). The estimated disturbance for each control channel is denoted by \(z_{2(. )}\), and \(\Omega_{c}=[p_{c},\;q_{c},\;r_{c}]^{\mathsf{T}}\) is the command angular rate. The stability analysis of the ESO-based controller and the relationship between the error bounds and ADRC bandwidth are presented in Reference [34]. The rate of change in attitude angles can be expressed as follows:  
\[ \begin{bmatrix}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\end{bmatrix} = \begin{bmatrix}1&\sin\phi\tan\theta&\cos\phi\tan \theta\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi/\cos\theta&\cos\phi/\cos\theta\end{bmatrix}\begin{bmatrix}p\\ q\\ r\end{bmatrix}  \tag{22}\]  
where \(\phi\), \(\theta\), and \(\psi\) are roll, pitch, and yaw angles, respectively.