# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 2 Problem Formulation and Preliminaries  
### Problem Statement  
According to the typical characteristics of wing rock motion, the following nonlinear MIMO lateral attitude motion model is considered [40]:  
\[\dot{x}_{1} = f_{1}(x_{1})+g_{1}(x_{1})x_{2}+\Delta f\left(x_{1}\right)+d_{1} \left(t\right)\] \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Delta f_{2}(x_{1},x_{ 2})+G_{2}^{{}^{\prime}}h_{2}+d_{2}\left(t\right)\] \[y = x_{1} \tag{1}\]  
where \(x_{1}=\left[\beta,\mu\right]^{T}\in R^{2}\) is an attitude angle vector which includes sideslip angle and flight-path roll angle, respectively; \(x_{2}=\left[p,r\right]^{T}\in R^{2}\) is the body-axis angular rate vector; \(y\) is the output; \(f_{1}(x_{1})\in R^{2}\), \(f_{2}(x_{1},x_{2})\in R^{2}\) are known nonlinear function vectors; \(g_{1}(x_{1})\in R^{2\times 2}\), \(g_{2}(x_{1},x_{2})\in R^{2\times 2}\), \(G_{2}^{\prime}\in R^{2\times 2}\) are known invertible function matrices; \(\Delta f_{1}(x_{1})\) and \(\Delta f_{2}(x_{1},x_{2})\) are the system uncertainties which include the modeling errors and the perturbation of aerodynamic coefficients; \(d_{1}(t)\), \(d_{2}(t)\) are the unknown external disturbance vectors; \(v=\left[v_{1},v_{2}\right]^{T}\) is the designed control input; \(u\left(v\right)=\left[u_{1}\left(v_{1}\right),u_{2}\left(v_{2}\right)\right]^{T} =\left[\delta_{a},\delta_{y}\right]^{T}\) denotes the practical control input; \(\delta_{a}\) is the aileron deflection angle and \(\delta_{y}\) is the lateral thrust vectoring angle; \(h_{2}=\left[h_{l_{l}},h_{n}\right]^{T}\in R^{2}\) is the unsteady moment aerodynamic coefficient vector, and \(h_{l},h_{n}\) are unsteady rolling moment aerodynamic coefficient and unsteady yawing moment aerodynamic coefficient, respectively. On the basis of work in [41], the unsteady aerodynamic coefficient can be modeled by first-order linear differential equation which is given by  
\[\dot{h}_{2}=-B_{2}\left(\beta\right)h_{2}-A_{2}\left(\beta\right)\dot{x}_{1} \tag{2}\]  
where \(B_{2}\left(\beta\right)=diag\left\{b_{21}\left(\beta\right),b_{22}\left(\beta \right)\right\}\), \(A_{2}\left(\beta\right)=\left[ \begin{array}{cc}a_{21}\left(\beta\right)&0\\ a_{22}\left(\beta\right)&0\end{array} \right]\), and \(b_{21}\left(\beta\right)>0,b_{22}\left(\beta\right)>0,a_{21}\left(\beta\right)\), \(a_{22}\left(\beta\right)\) are known bounded function variables of \(\beta\). At high AOA, the lateral attitude motion of (1) becomes unstable. When system (1) suffers no disturbances, simulation results for an angle of attack of \(55^{\circ}\) with initial conditions of \(\left[\beta\left(0\right),\mu\left(0\right)\right]^{T}=\left[5^{\circ},0^{ \circ}\right]^{T}\), \(\left[p\left(0\right),r\left(0\right)\right]^{T}=\left[0^{\circ}\!/_{s}\!\!,0^{ \circ}\!/_{s}\!\!\right]^{T}\) are given in Figures 1 and 2. From Figures 1 and 2, it can be observed that the roll angle exhibits large amplitude oscillatory motion, and the sideslip angle motion also exhibits considerable oscillation. Meanwhile, the maximum amplitude of the sideslip motion is closed to \(20^{\circ}\), which is very dangerous for aircrafts at high AOA. Figure 1: Roll angle oscillation without disturbances. When system (1) suffers the disturbances, simulation results for an angle of attack of \(55^{\circ}\) with initial conditions of \(\left[\beta\left(0\right),\mu\left(0\right)\right]^{T}=\left[5^{\circ},0^{ \circ}\right]^{T}\), \(\left[p\left(0\right),r\left(0\right)\right]^{T}=\left[0^{\circ}/\!s,0^{\circ}/ \!s\right]^{T}\) are given in Figures 3 and 4. And the equations of the system uncertainty, unsteady aerodynamic disturbance and external disturbance will be given in details in the simulation section. In Figure 3 and 4, the states of the aircraft diverge quickly, then stable wing rock appears after 50 s. Comparing Figures 3 and 4 with Figures 1 and 2, it can be concluded that the disturbances will further aggravate the oscillation.