# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 3 Design of Disturbance Observer
### Design of Disturbance Observer for the Second Subsystem  
According to (1), the second subsystem can be written as  
\[\dot{x}_{2}=f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{2} )+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right) \tag{20}\]  
It is more complicated that multiple disturbances are considered in (20). \(\Delta f_{2}(x_{1},x_{2})\) is the unmodeled system uncertainty, which is an unknown function of the system states. \(h_{2}\) is the unmeasurable unsteady moment aerodynamic coefficient vector whose model can be achieved by wind tunnel experiments. And \(d_{2}(t)\) represents the external disturbance. Similarly, different mechanisms are adopted to handle these different disturbances in this section. Meanwhile, to improve the disturbance estimation performance, the model information of the unmeasurable unsteady moment aerodynamic coefficient vector \(h_{2}\) is fully utilized. Invoking (2) and (20), the augmented second subsystem state equations can be written as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\] \[h_{2} = -B_{2}\left(\beta\right)h_{2}-A_{2}\left(\beta\right)\dot{x}_{1} \tag{21}\]  
where \(G_{2}^{{}^{\prime}}=diag\left\{G_{21},G_{22}\right\}\), and \(G_{21}>0\), \(G_{22}>0\) are constants. Similar to (13), considering the neural network approximation of \(\Delta f_{2}(x_{1},x_{2})\), (21) can be written as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+W_{2}^{*T}S_{2}\left(J _{2}\right)+G_{2}^{{}^{\prime}}h_{2}+D_{2}\] \[\dot{t}_{2} = -B_{2}\left(\beta\right)h_{2}-A_{2}\left(\beta\right)\dot{x}_{1} \tag{22}\]  
where \(D_{2}=d_{2}\left(t\right)+\iota_{2}^{*}\), \(\Delta f_{2}\left(x_{1},x_{2}\right)=W_{2}^{*T}S_{2}\left(J_{2}\right)+\iota_{2 }^{*}\). \(J_{2}=[x_{1}^{T},x_{2}^{T}]^{T}\), \(W_{2}^{*T}=\left[W_{21}^{*},W_{22}^{*}\right]^{T}\in R^{2\times p_{2}}\), \(S_{2}\left(J_{2}\right)\in R^{p_{2}}\), \(\iota_{2}^{*}\in R^{2\times 1}\). Defining \(x_{4}=G_{2}^{{}^{\prime}}h_{2}+D_{2}\) and invoking (14), the time derivative of \(x_{4}\) is given by  
\[\dot{x}_{4} = G_{2}^{{}^{\prime}}jh_{2}+\dot{D}_{2}\] \[= G_{2}^{{}^{\prime}}\left(-B_{2}\left(\beta\right)h_{2}-A_{2} \left(\beta\right)\left(f_{1}(x_{1})+g_{1}(x_{1})x_{2}+W_{1}^{*T}S_{1}\left(J _{1}\right)+D_{1}\right)\right)+\dot{D}_{2}\] \where \(x_{5}=-G_{2}^{{}^{\prime}}A_{2}\left(\beta\right)D_{1}+B_{2}\left(\beta\right)D_{2 }+\dot{D}_{2}\). Then, invoking (23), the extended system of (22) can be written as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+W_{2}^{*T}S_{2}\left(J_{2 }\right)+x_{4}\] \[\dot{x}_{4} = -B_{2}\left(\beta\right)x_{4}-G_{2}^{{}^{\prime}}A_{2}\left(\beta \right)\left(f_{1}(x_{1})+g_{1}(x_{1})x_{2}\right)-G_{2}^{{}^{\prime}}A_{2} \left(\beta\right)W_{1}^{*T}S_{1}\left(J_{1}\right)+x_{5} \tag{24}\] \[\dot{x}_{5} = \hbar_{2}\left(t\right)\]  
where \(\hbar_{2}=\left[\hbar_{21},\hbar_{22}\right]^{T}\) is an unknown bounded time derivative function vector of \(x_{5}\). Then, a linear type of the ESO can be constructed as  
\[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\]0 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\]1 \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\]2  
where \(\hat{x}_{2}\) is the estimation of \(x_{2}\); \(\hat{x}_{4}\) is the estimation of \(x_{4}\); \(\hat{x}_{5}\) is the estimation of \(x_{5}\); \(\hat{W}_{2}^{T}=\left[\hat{W}_{21},\hat{W}_{22}\right]^{T}\in R^{2\times p_{2}}\) is the estimation of \(W_{2}^{*T}\); \(\hat{x}_{2}=\hat{x}_{2}-x_{2}\); And \(l_{3f}=diag\left\{l_{3f1},l_{3f2}\right\}>0\), \(l_{4f}=diag\left\{l_{4f1},l_{4f2}\right\}>0\), \(l_{5f}=diag\left\{l_{5f1},l_{5f2}\right\}>0\) are design matrixes.