# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 4 Design of Robust Backstepping Attitude Control Based on Disturbance Observer
Since the set \(\Pi_{0}\) and \(\Pi_{1}\) are compacts, \(\Pi_{0}\times\Pi_{1}\) is also compact. Therefore, \(M_{1}(\bullet)\) has a maximum \(\bar{M}_{1}\) on \(\Pi_{0}\times\Pi_{1}\). Then, we have  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]7  
Choose the Lyapunov function candidate as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]8  
Define \(\varepsilon_{13i}=\left[\varepsilon_{1i},\varepsilon_{3i}\right]^{T}\), then the time derivative of \(V_{1}\) can be written as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+{W_{2}^{*}}^{T}S_{2} \left(J_{2}\right)+x_{4} \tag{29}\]9 \[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]0 \[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]1  
where \(\|P_{si}C_{si}\|=M_{PC_{si}}\), \(\|\bar{\eta}_{1i}\|\leq M_{\bar{W}_{1i}}\), \(\|P_{si}B_{si}\|\left\|S_{1}\left(I_{1}\right)\right\|\leq M_{PBS1}\), and \(\tau>0\) is a design constant. The detailed analysis of the time derivative of \(V_{1}\) is shown in Appendix A. **Step 2:** Invoking (29), differentiating \(e_{2}\) yields  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]2 \[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]3 \[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]4  
The control law \(v\) is designed as  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]5  
where \(c_{2}=c_{2}^{T}>0\) is a design matrix, and \(\hat{W}_{2}\) is updated by  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]6where \(\rho_{2}>0\), \(\Lambda_{2}=\Lambda_{2}^{T}>0\) are design parameters. Substituting the controller \(v\) into Equation (40), we obtain  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]7  
where \(\tilde{W}_{2}=\tilde{W}_{2}-W_{2}^{*}\), \(\tilde{x}_{4}=\hat{x}_{4}-x_{4}\) are the estimation errors. Define \(\varepsilon_{245i}=\left[\varepsilon_{2i},\varepsilon_{4i},\varepsilon_{5i} \right]^{T}\), \(i=1,2\), and consider the Lyapunov function candidate as  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]8  
The time derivative of \(V_{2}\) is given by  
\[\dot{\xi}_{1} = g_{1}\left(x_{1}\right)\xi_{2}-k_{aux1}\xi_{1}\]9 \[\dot{\xi}_{2} = -k_{aux2}\xi_{2}+g_{2}\left(x_{1},x_{2}\right)\Delta u \tag{30}\]0 \[\dot{\xi}_{2} = -k_{aux2}\xi_{2}+g_{2}\left(x_{1},x_{2}\right)\Delta u \tag{30}\]1 \[\dot{\xi}_{2} = -k_{aux2}\xi_{2}+g_{2}\left(x_{1},x_{2}\right)\Delta u \tag{30}\]2  
where \(\left\|P_{fi}B_{1 fi}\right\|\left\|S_{2}\left(J_{2}\right)\right\|\ \leq\ M_{PBS2}\), \(\left\|P_{fi}B_{2 fi}G_{2i}a_{2i}\left(\beta\right)\right\|\left\|S_{1}\left(J_ {1}\right)\right\|\ \leq\ M_{PBASi}\), \(\left\|P_{fi}B_{3 fi}\right\|\ \leq\ M_{PB_{fi}}\), \(\left\|\hbar_{2i}\right\|\ \leq\ M_{\hbar_{2i}}\). The detailed analysis of the time derivative of \(V_{2}\) is shown in Appendix B. Then, the above robust backstepping wing rock control design procedure can be summarized in the following theorem. **Theorem 1**. : _Considering the nonlinear MIMO lateral model (1) in the presence of unsteady aerodynamics disturbances, external disturbances, system uncertainties, and input saturation, the disturbance observers are designed as (16), (25), the auxiliary system is designed as (30), the filter is designed as (35), and the robust backstepping attitude control law is proposed as (41). Then, the attitude tracking error is convergent and bounded under the proposed adaptive robust backstepping control scheme._  
Proof.