# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## Appendix B  
The detailed analysis of the time derivative of Lyapunov function \(V_{2}\) is given as following:  
The Lyapunov function \(V_{2}\) is given by  
\[V_{2}=\frac{1}{2}e_{2}^{T}e_{2}+\frac{1}{2}\sum_{i=1}^{2}\varepsilon_{245i}^{T} P_{fi}\varepsilon_{245i}+\frac{1}{2}tr\left(\tilde{W}_{2}^{T}\Lambda_{2}^{-1} \tilde{W}_{2}\right)+\frac{1}{2}\xi_{2}^{T}\xi_{2} \tag{12}\]  
Similar to (10), considering the following facts:  
\[tr\left(\tilde{W}_{2}^{T}\Lambda_{2}^{-1}\hat{W}_{2}\right) =tr\left(\tilde{W}_{2}^{T}S_{2}\left(J_{2}\right)e_{2}^{T}-\rho_ {2}\tilde{W}_{2}^{T}\hat{W}_{2}\right)\leq e_{2}^{T}\tilde{W}_{2}^{T}S_{2} \left(J_{2}\right)-\frac{\rho_{2}}{2}\big{\|}\tilde{W}_{2}\big{\|}^{2}+\frac{ \rho_{2}}{2}\|W_{2}^{*}\|^{2}\] \[\quad-\sum_{i=1}^{2}e_{2i}\tilde{x}_{4  
\[\dot{V}_{2}= -e_{2}^{T}S_{1}^{T}\left(x_{1}\right)e_{1}-e_{2}^{T}\tilde{W}_{2}^{T}S _{2}\left(J_{2}\right)-\sum_{i=1}^{2}e_{2i}\delta_{ii}-c_{2}e_{2}^{T}e_{2}+tr \left(\tilde{W}_{2}^{T}\Lambda_{2}^{-1}\dot{W}_{2}\right)\] \[+\sum_{i=1}^{2}\left( \begin{array}{c}-\omega_{2i}\varepsilon_{245 i}^{T}I\varepsilon_{245i}+\varepsilon_{245i}^{T}P_{fi}B_{1 fi}\tilde{W}_{2i}^{T}S_{2} \left(J_{2}\right)-\frac{1}{\omega_{2i}^{2}}\varepsilon_{245i}^{T}P_{fi}B_{3 fi} \hbar_{2i}\left(t\right)\\ -\frac{1}{\omega_{2i}}\varepsilon_{245i}^{T}P_{fi}B_{2 fi}G_{2i}a_{2i}\left( \beta\right)\tilde{W}_{11}^{T}S_{1}\left(I_{1}\right)\end{array} \right)\] \[-\xi_{2}^{T}\tilde{T}_{kaux2}\xi_{2}+\xi_{2}^{T}\tilde{S}_{2} \left(x_{1},x_{2}\right)\Delta u \tag{100}\] \[\leq -e_{2}^{T}g_{1}^{T}\left(x_{1}\right)e_{1}-e_{2}^{T}\tilde{W}_{2} ^{T}S_{2}\left(J_{2}\right)+\sum_{i=1}^{2}\left(\frac{\omega_{2i}}{4} \varepsilon_{245i}^{T}\varepsilon_{245i}\right)+\max\left\{\omega_{2i}\right\} e_{2}^{T}e_{2}-c_{2}e_{2}^{T}e_{2}\] \[+\sum_{i=1}^{2}\left( \begin{array}{c}-\omega_{2i}\varepsilon_{245 i}^{T}I\varepsilon_{245i}+\varepsilon_{245i}^{T}P_{fi}B_{1 fi}\tilde{W}_{2i}^{T}S_{2} \left(J_{2}\right)-\frac{1}{\omega_{2i}^{2}}\varepsilon_{245i}^{T}P_{fi}B_{3 fi }\hbar_{2i}\left(t\right)\\ -\frac{1}{\omega_{2i}}\varepsilon_{245i}^{T}P_{fi}B_{2 fi}G_{2i}a_{2i}\left( \beta\right)\tilde{W}_{11}^{T}S_{1}\left(I_{1}\right)\end{array} \right)\] \[+e_{2}^{T}\tilde{W}_{2}^{T}S_{2}\left(J_{2}\right)-\frac{\rho_{2 }}{2}\left\|\tilde{W}_{2}\right\|^{2}+\frac{\rho_{2}}{2}\left\|W_{2}^{*} \right\|^{2}-\xi_{2}^{T}k_{aux2}\xi_{2}+\frac{1}{2}\lambda_{2}\xi_{2}^{T} \xi_{2}+\frac{1}{2}\Delta u^{T}\Delta u\]  
Furthermore, considering the following facts:  
\[\varepsilon_{245i}^{T}P_{fi}B_{1 fi}\tilde{W}_{2i}^{T}S_{2}\left(J_ {2}\right)\leq\left\|\varepsilon_{245i}^{T}\right\|\left\|P_{fi}B_{1 fi}\right\| \left\|\tilde{W}_{2i}\right\|\left\|S_{2}\left(J_{2}\right)\right\|\] \[\qquad\leq\frac{\tau}{2}M_{PBS2}^{2}\varepsilon_{245i}^{T}\varepsilon _{245i}+\frac{1}{2\tau}\left\|\tilde{W}_{2}\right\|^{2}\] \[-\frac{1}{\omega_{2i}}\varepsilon_{245i}^{T}P_{fi}B_{2 fi}G_{2i}a_ {2i}\left(\beta\right)\tilde{W}_{11}^{T}S_{1}\left(I_{1}\right)\leq\frac{1}{ \omega_{2i}}\left\|\varepsilon_{245i}^{T}\right\|\left\|P_{fi}B_{2 fi}G_{2i}a_ {2i}\left(\beta\right)\right\|\left\|\tilde{W}_{11}^{T}\right\|\left\|S_{1} \left(I_{1}\right)\right\|\] \[\qquad\leq\frac{\tau}{2\omega_{2i}^{2}}M_{PBS2}^{2}\varepsilon_{245 i}^{T}\varepsilon_{245i}+\frac{1}{2\tau}\left\|\tilde{W}_{11}\right\|^{2} \tag{101}\] \[-\frac{1}{\omega_{2i}^{2}}\varepsilon_{245i}^{T}P_{fi}B_{3 fi} \hbar_{2i}\left(t\right)\leq\frac{1}{\omega_{2i}^{2}}\left\|\varepsilon_{245i}^ {T}\right\|\left\|P_{fi}B_{3 fi}\right\|\left\|\hbar_{2i}\right\|\] \[\qquad\leq\frac{M_{PB_{fi}}}{4\tau}\varepsilon_{245i}^{T}\varepsilon _{245i}+\frac{M_{PB_{fi}}}{\omega_{2i}^{4}}M_{\hbar_{2i}}^{2}\]  
Then, (100) can be written as  
\[\dot{V}_{2}\leq -e_{2}^{T}g_{1}^{T}\left(x_{1}\right)e_{1}-e_{2}^{T}\tilde{W}_{2}^ {T}S_{2}\left(J_{2}\right)+\sum_{i=1}^{2}\left(\frac{\omega_{2i}}{4}\varepsilon_ {245i}^{T}\varepsilon_{245i}\right)+\max\left\{\omega_{2i}\right\}e_{2}^{T}e_{2} -c_{2}e_{2}^{T}e_{2}\] \[+\sum_{i=1}^{2}\left(