# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 3 Design of Disturbance Observer
### Design of Disturbance Observer for the Second Subsystem
Invoking (24) and (25), we have  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\]3 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\]4 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\]5  
Defining \([\varepsilon_{21},\varepsilon_{22}]^{T}=[\hat{x}_{21},\hat{x}_{22}]^{T}\), \([\varepsilon_{41},\varepsilon_{42}]^{T}=\left[\frac{\hat{x}_{41}}{\omega_{21} ^{2}},\frac{\hat{x}_{22}}{\omega_{22}}\right]^{T}\), \([\varepsilon_{51},\varepsilon_{52}]^{T}=\left[\frac{\hat{x}_{51}}{\omega_{21} ^{2}},\frac{\hat{x}_{52}}{\omega_{22}^{2}}\right]^{T}\), we have  
\[\left[ \begin{array}{c}\dot{\hat{x}}_{2i}\\ \dot{\hat{x}}_{4i}\\ \dot{\hat{x}}_{5i}\end{array} \right]=\omega_{2i}A_{fi}\left[ \begin{array}{c} \varepsilon_{2i}\\ \epsilon_{4i}\\ \varepsilon_{5i}\end{array} \right]+B_{1 fi}\hat{W}_{2i}^{T}S_{2}\left(J_{2} \right)-B_{2 fi}\frac{1}{\omega_{2i}}G_{2i}a_{2i}\left(\beta\right)\hat{W}_{11} ^{T}S_{1}\left(J_{1}\right)-B_{3 fi}\frac{1}{\omega_{2i}^{2}}\hbar_{2i}\left(t\right) \tag{27}\]  
where \(A_{fi}=\left[ \begin{array}{cc}\frac{-l_{3 fi}}{\omega_{2i}^{2}}&1&0\\ \frac{-l_{4 fi}}{\omega_{2i}^{2}}&-\frac{b_{2i}}{\omega_{2i}}&1\\ \frac{-l_{5 fi}}{\omega_{2i}^{2}}&0&0\end{array} \right]\), \(B_{1 fi}=\left[ \begin{array}{c}1\\ 0\\ 0\end{array} \right]\), \(B_{2 fi}=\left[ \begin{array}{c}0\\ 1\\ 0\end{array} \right]\), \(B_{3 fi}=\left[ \begin{array}{c}0\\ 0\\ 1\end{array} \right]\), \(i=1,2\); \(\omega_{2i}>0\) is a design constant. According to the Routh-Hurwitz criterion, it can be easily proved that if the parameters of (25) can be chosen as \(\left[l_{3f1},l_{3f2}\right]=\left[3k_{f1}\omega_{21},3k_{f2}\omega_{22}\right]\), \(\left[l_{4f1},l_{4f2}\right]=\left[3k_{f1}^{2}\omega_{21}^{2},3k_{f2}^{2} \omega_{22}^{2}\right]\), \(\left[l_{5f1},l_{5f2}\right]=\left[k_{f1}^{3}\omega_{21}^{3},k_{f2}^{3}\omega_{2 2}^{3}\right]\), and \(k_{f1}>0\), \(k_{f2}>0\) are design constants, \(A_{fi}\) will be Hurwitz. Thus, there exists a positive definite matrix \(P_{fi},i=1,2\) satisfying the following Lyapunov equation:  
\[A_{fi}^{T}P_{fi}+P_{fi}A_{fi}=-2I \tag{28}\]  
In this subsection, the disturbance observer (25) for the second subsystem (20) has been proposed. Selecting the parameters of (25) by the given method, \(A_{fi}\) has been proved to be Hurwitz. And the positive definite matrix \(P_{fi}\), \(i=1,2\) will be used in the design of the Lyapunov function (44), which is very important in the stability analysis of the proposed observer (25) and the closed-loop system. Now, we have illustrated the design process of disturbance observers (16) and (25). However, according to (18) and (27), we still can't prove their stability due to the coupling between the observersand the RBFNN which will be design later. Hence, the stability of the disturbance observers will be proved in the stability analysis of the closed-loop system.