# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 5 Simulation Study  
In this section, the simulation results are given to verify the effectiveness of the proposed robust control scheme in suppression of wing rock motion at high AOA. In this section, the aircraft model in [40] and the unsteady model in [41] are adopted to demonstrate the attitude control performance and the disturbance rejection performance of the above designed controller. In simulations, initial conditions for the aircraft are take as \(\alpha=55^{\circ}\), \(x_{1}\left(0\right)=\left[5^{\circ},0^{\circ}\right]^{T}\), \(x_{2}\left(0\right)=\left[0^{\circ}\!/_{s},0^{\circ}\!/_{s}\right]^{T}\), whereas initial conditions for the proposed ESO are taken as \(\left[\hat{x}_{1}^{T}\left(0\right),\hat{x}_{3}^{T}\left(0\right)\right]= \left[5.7,0,0,0\right]\), \(\left[\hat{x}_{2}^{T}\left(0\right),\hat{x}_{4}^{T}\left(0\right),\hat{x}_{5}^ {T}\left(0\right)\right]=\left[0,0,0,0,0\right]\). And the initial condition for the auxiliary system is chosen as \(\left[\hat{x}_{1}^{T},\hat{x}_{2}^{T}\right]=\left[0,0,0,0\right]\). The time constant for (35) is designed as \(\Gamma_{1}=diag\left\{0.08,0.08\right\}\). It is very important to choose suitable parameters for the ESO. And the observer gains of the error dynamics (18) and (27)are designed by placing the observer error dynamics poles around \(\left[-15,-15\right]\) and \(\left[-15,-15,-15\right]\), respectively. Hence, \(\left[k_{s1},k_{s2}\right]=\left[15,15\right]\), \(\left[k_{f1},k_{f2}\right]=\left[15,15\right]\), \(\left[\omega_{11},\omega_{12}\right]=\left[6,6\right]\), \(\left[\omega_{21},\omega_{22}\right]=\left[6,6\right]\). Large \(k_{si}\), \(k_{fi}\), \(\omega_{1i}\), \(\omega_{2i},i=1,2\) will improve the disturbance performance, and make the disturbance estimation errors arbitrarily small. Meanwhile, large \(k_{si}\) and \(k_{fi}\) will reduce the conservatium of the controller proposed in this paper. The parameters for the controller (41) are chosen as \(\left[c_{1}^{T},c_{2}^{T}\right]=\left[6,6,10,10\right]\). Large control parameters \(\left[c_{1}^{T},c_{2}^{T}\right]\) will improve the control performance and reduce steady-state tracking errors. However, large control parameters will also induce serious input saturation. The parameters for the auxiliary system (30) is designed as \(\left[k_{aux1}^{T},k_{aux2}^{T}\right]=\left[6,6,60,60\right]\). The parameters for the adaptive RBFNNs (34) and (42) are chosen as \(\Lambda_{1}=40\), \(\rho_{1}=0.5\), \(\Lambda_{2}=1\), \(\rho_{2}=0.8\). In the simulation study, the system uncertainties \(\Delta f_{1}\left(x_{1}\right)\) and \(\Delta f_{2}\left(x_{1},x_{2}\right)\) are assumed as \(+40\%\) variation of the force and moment aerodynamic coefficients. In addition, we assume that the unknown time-varying disturbances imposed on the aircraft are  
\[\left[ \begin{array}{c}h_{l}\\ h_{n}\end{array} \right]=-\left[ \begin{array}{cc}5&0\\ 0&5\end{array} \right]\left[ \begin{array}{c}h_{l}\\ h_{n}\end{array} \right]-\left[ \begin{array}{c}-0.02\\ -0.05\end{array} \right]\hat{\beta} \tag{52}\]  
\[d_{1}=\left[ \begin{array}{c}d_{11}\\ d_{12}\end{array} \right]=\left[ \begin{array}{c}0.01\left(\sin\left(0.5\pi t \right)+\sin\left(0.2\pi t\right)+0.1\right)\\ 0.02\left(\sin\left(0.5\pi t\right)+\sin\left(0.2\pi t\right)+0.1\right)\end{array} \right] \tag{53}\]  
\[d_{2}=\left[ \begin{array}{c}d_{21}\\ d_{22}\end{array} \right]=\left[ \begin{array}{c}0.01\left(\sin\left(5t \right)+\sin\left(2t\right)+0.3\right)\\ 0.01\left(\sin\left(5t\right)+\sin\left(2t\right)+0.3\right)\end{array} \right] \tag{54}\]  
According to Figures 1-4, the states of the aircraft diverge rapidly when the wing rock motion appears. Hence, it is necessary to adopt appropriate control scheme immediately after the wing rock phenomenon is detected. In the following simulations, we assume that the wing rock motion is detected at 5 s, and then the controller (41) is executed immediately.