# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 4 Design of Robust Backstepping Attitude Control Based on Disturbance Observer  
At high AOA, the aerodynamic characteristics of the flow field around an aircraft is quite complicated. Any small changes of the attitude of an aircraft may cause drastic changes of the aerodynamic force and aerodynamic moment, which will severely degrade the aircraft flight performance. Hence, the robust backstepping attitude tracking control law should be further developed for the aircraft at high AOA. To compensate for the matched and unmatched disturbances, the ESOs proposed in Section 3 are employed to estimate them. And the disturbance estimations will be introduced into the design of the virtual control law and the practical control law. Invoking (15) and (24), the system (1) can be written as  
\[\dot{x}_{1} = f_{1}(x_{1})+g_{1}(x_{1})x_{2}+{W_{1}^{*}}^{T}S_{1}\left(J_{1} \right)+x_{3}\] \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]  
Considering the input saturation problem, the following auxiliary system is constructed:  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\] \[\dot{\xi}_{2} = -k_{aux2}\xi_{2}+g_{2}\left(x_{1},x_{2}\right)\Delta u \tag{30}\]  
where \(\xi_{1},\xi_{2}\in R^{2\times 1};\Delta u=u(v)-v;k_{aux1}=k_{aux1}^{T}>0\), \(k_{aux2}=k_{aux2}^{T}>0\) are design matrixes. Considering the backstepping control law design method, we define  
\[e_{1} = x_{1}-y_{d}-\xi_{1}\] \[e_{2} = x_{2}-x_{2c}-\xi_{2} \tag{31}\]  
where \(x_{2c}\) is the virtual control law which will be designed later. In the following, we consider the robust backstepping attitude control design for the wing-rock motion. In the attitude control design, the adaptive RBFNN and the ESO are combined, and the detailed design process is appended as follows. **Step 1:** Considering the first equation in (29) and differentiating \(e_{1}\) with respect to time yields  
\[\dot{e}_{1} = \dot{x}_{1}-\dot{y}_{d}-\dot{\xi}_{1} \tag{32}\] \[= f_{1}(x_{1})+g_{1}(x_{1})x_{2}+{W_{1}^{*}}^{T}S_{1}\left(J_{1} \right)+x_{3}-\dot{y}_{d}-g_{1}\left(x_{1}\right)\xi_{2}+k_{aux1}\xi_{1}\] \[= f_{1}(x_{1})+g_{1}(x_{1})\left(e_{2}+x_{2c}+\xi_{2}\right)+{W_{ 1}^{*}}^{T}S_{1}\left(J_{1}\right)+x_{3}-\dot{y}_{d}-g_{1}\left(x_{1}\right) \xi_{2}+k_{aux1}\xi_{1}\] \[= f_{1}(x_{1})+g_{1}(x_{1})\left(e_{2}+x_{2c}\right)+{W_{1}^{*}}^ {T}S_{1}\left(J_{1}\right)+x_{3}-\dot{y}_{d}+k_{aux1}\xi_{1}\]  
The ideal virtual control law \(\bar{x}_{2c}\) for \(x_{2}\) is designed as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]0  
where \(c_{1}=c_{1}^{T}>0\) is a matrix to be designed. And the weight value of RBFNN is updated by  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]1  
where \(\Lambda_{1}=\Lambda_{1}^{T}>0\), \(\rho_{1}>0\) are the design parameters. The dynamic surface control (DSC) technique technique is adopted to obtain the approximation of the time derivative of \(x_{2c}\). Introducing a first-order filter with time constant \(\Gamma_{1}=diag\left\{\tau_{11},\tau_{12}\right\}\), and passing \(\bar{x}_{2c}\) through it yields  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]2  
Defining \(\varepsilon_{f1}=x_{2c}-\bar{x}_{2c}\) and invoking Equation (35), we have  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]3 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]4 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]5 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]6  
where \(M_{1}(\bullet)\) is the sufficiently smooth function vector about \(\Pi_{0}\) and \(\Pi_{1}:(x_{1},\hat{W}_{1},\hat{x}_{3},\xi_{1},\xi_{2})\).