# Research on Backstepping Linear Active Disturbance Rejection Control of Hypersonic Vehicle
## 3 Proposed Methods
### Design of Backstepping Linear Active Disturbance Rejection Controller
#### 3.2.2 Design of Height Subsystem Controller  
(1) Design of Height Loop Controller  
For Equation (19), take the height error: \(e_{h}=h-h_{d}\), where \(h_{d}\) is the reference height trajectory, and \(h\) is the actual height trajectory, and then derive the height tracking error:  
\[\dot{e}_{h}=\dot{h}-\dot{h}_{d} \tag{27}\]  
Bringing Equation (19) into Equation (27),  
\[\dot{e}_{h}=V\gamma-\dot{h}_{d} \tag{28}\]  
The controllable rate of the combination of Formulas (27) and (28) is as follows:  
\[\gamma_{d}=\frac{\dot{h}_{d}-c_{2}e_{h}}{V} \tag{29}\]  
where \(c_{2}\) is an adjustable normal number. (2) Design of Track Angle Loop Controller  
For Equation (18), take the track angle error: \(e_{\gamma}=\gamma-\gamma_{d}\), where \(\gamma_{d}\) is the reference track angle trajectory, and \(\gamma\) is the actual track angle trajectory, and then derive the track angle tracking error:  
\[\dot{e}_{\gamma}=\dot{\gamma}-\dot{\gamma}_{d} \tag{30}\]  
Bringing Equation (18) into Equation (30),  
\[\dot{e}_{\gamma}=g_{\gamma}\theta+f_{\gamma}-\dot{\gamma}_{d} \tag{31}\]  
The controllable rate of the combination of Formulas (19) and (20) is as follows:  
\[\theta_{d}=\frac{1}{g_{\gamma}}\big{(}\dot{\gamma}_{d}-f_{\gamma}-c_{3}e_{ \gamma}\big{)} \tag{32}\]  
where \(c_{3}\) is an adjustable normal number. The track angle subsystem represented by the definition \(x_{1\gamma}=\gamma\) and \(x_{2\gamma}=f_{\gamma}\) in Equation (18) can be expressed in the form of the following state space expression:  
\[\left\{ \begin{matrix}\dot{x}_{1\gamma}=x_{2\gamma}+g_{\gamma}\theta\\ \dot{x}_{2\gamma}=\dot{f}_{\gamma}\\ y_{\gamma}=x_{1\gamma}\end{matrix} \right. \tag{33}\]  
The state variables of the system represented by Equation (33) are observed using the LESO. The observation equation of the LESO is shown by the following Equation (34):  
\[\left\{ \begin{matrix}\varepsilon_{\gamma}=z_{1\gamma}-y_{\gamma}\\ \dot{z}_{1\gamma}=z_{2\gamma}-\beta_{01}^{\gamma}\varepsilon_{\gamma}+g_{ \gamma}\theta\\ \dot{z}_{2\gamma}=-\beta_{02}^{\gamma}\varepsilon_{\gamma}\end{matrix} \right. \tag{34}\]  
The tracking of the state variables of the original system (33) can be realized when the system reaches stability through the observation of the LESO. According to Equations (32) and (34), the controllable rate is as follows:  
\[\theta_{d}=\frac{1}{g_{\gamma}}\big{(}\dot{\gamma}_{d}-z_{2\gamma}-c_{3}e_{ \gamma}\big{)} \tag{35}\]  
(3) Design of Pitch Angle Loop Controller  
For Equation (20), take the pitch angle error: \(e_{\theta}=\theta-\theta_{d}\), where \(\theta_{d}\) is the reference pitch angle trajectory, and \(\theta\) is the actual pitch angle trajectory, and then derive the pitch angle tracking error:  
\[\dot{e}_{\theta}=\dot{\theta}-\dot{\theta}_{d} \tag{36}\]  
Bringing Equation (20) into Equation (36),  
\[\dot{e}_{\theta}=Q-\dot{\theta}_{d} \tag{37}\]  
The controllable rate of the combination of Formulas (36) and (37) is as follows:  
\[Q_{d}=\dot{\theta}_{d}-c_{4}e_{\theta} \tag{38}\]  
where \(c_{4}\) is an adjustable normal number. (4) Design of Pitch Angle Rate Loop Controller  
For Equation (20), take the pitch angle rate error: \(e_{Q}=Q-Q_{d}\), where \(Q_{d}\) is the reference pitch angle rate trajectory, and \(Q\) is the actual pitch angle rate trajectory, and then derive the pitch angle rate tracking error:  
\[\dot{e}_{Q}=\dot{Q}-\dot{Q}_{d} \tag{39}\]  
Bringing Equation (20) into Equation (39),  
\[\dot{e}_{Q}=g_{Q}\dot{e}_{e}+f_{Q}-\dot{Q}_{d} \tag{40}\]  
The controllable rate of the combination of Formula (39) and Formula (40) is as follows:  
\[\delta_{e}=\frac{\dot{Q}_{d}-f_{Q}-c_{5}e_{Q}}{g_{Q}} \tag{41}\]  
where \(c_{5}\) is an adjustable normal number. The pitch angle rate subsystem represented by the definition \(x_{1Q}=Q\) and \(x_{2Q}=f_{Q}\) in Equation (20) can be expressed in the form of the following state space expression:  
\[\left\{ \begin{aligned} \dot{x}_{1Q}=x_{2Q}+g_{Q}\delta_{e}\\ \dot{x}_{2Q}=\dot{f}_{Q}\\ y_{Q}=x_{1Q}\end{aligned} \right.