# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 5 Simulation Study
### Comparison with Traditional ESO  
In the traditional ESO design method, all the uncertainties are merged into a single compounded uncertainty. Hence, we rewrite (4) as follows  
\[\dot{x}_{1} = f_{1}(x_{1})+g_{1}(x_{1})x_{2}+\Psi_{s}\] \[\dot{x}_{2} = f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\Psi_{f} \tag{58}\]  
Considering that a linear ESO is developed in this paper, the traditional linear ESO (LESO) is considered for comparison instead of traditional nonlinear ESO. Meanwhile, the use of linear ESO offers certain advantages. Firstly, the close loop stability for nonlinear ESO is hard to established conclusively. Secondly, the LESO is easy from hardware implementation point of view.  
Figure 14: The lateral thrust vectoring angle output in attitude tracking.  
Figure 13: The aileron deflection angle output in attitude tracking.  
In [18], the authors have presented the wing rock control law based on the linear ESO. Hence, according to [18], the LESO for (58) is designed as  
\[\begin{array}{rcl}\dot{\hat{x}}_{Ls}&=&f_{1}(x_{1})+g_{1}(x_{1})x_{2}+\hat{ \Psi}_{sL}-l^{\prime}{}_{Ls1}e^{\prime}{}_{Ls}\\ \dot{\hat{\Psi}}_{sL}&=&-l^{\prime}{}_{Ls2}e^{\prime}{}_{Ls}\\ \dot{\hat{x}}_{Lf}&=&f_{2}(x_{1},x_{2})+g_{2}(x_{1},x_{2})u(v)+\hat{\Psi}_{fL}- l^{\prime}{}_{Lf1}e^{\prime}{}_{Lf}\\ \dot{\hat{\Psi}}_{fL}&=&-l^{\prime}{}_{Lf2}e^{\prime}{}_{Lf}\end{array} \tag{59}\]  
where \(\hat{x}_{Ls}\) is the estimation of \(x_{1}\); \(\hat{\Psi}_{sL}=\left[\hat{\Psi}_{sL1},\hat{\Psi}_{sL2}\right]^{T}\) is the estimation of \(\Psi_{s}\); \(e^{\prime}{}_{Ls}=\hat{x}_{Ls}-x_{1}\); \(\hat{x}_{Lf}\) is the estimation of \(x_{2}\); \(\hat{\Psi}_{fL}=\left[\hat{\Psi}_{fL1},\hat{\Psi}_{fL2}\right]^{T}\) is the estimation of \(\Psi_{f}\). \(l^{\prime}{}_{Ls1}\), \(l^{\prime}{}_{Ls2}\), \(l^{\prime}{}_{Lf1}\), \(l^{\prime}{}_{Lf2}\) are the positive observer gains.  
Similar to the developed disturbance observer in this paper, the observer gains are obtained by following the pole placement technique by placing the observer error dynamics poles at \([-15,-15]\). Hence, \(l^{\prime}{}_{Ls1}=30\), \(l^{\prime}{}_{Ls2}=225\), \(l^{\prime}{}_{Lf1}=30\), \(l^{\prime}{}_{Lf2}=225\). And initial conditions for the traditional LESO (59) are taken as \(\left[\hat{x}_{Ls}^{T}\left(0\right),\hat{\Psi}_{s}^{T}\left(0\right)\right]= \left[5.7,0,0,0\right]\), \(\left[\hat{x}_{Lf}^{T}\left(0\right),\hat{\Psi}_{f}^{T}\left(0\right)\right]= \left[0,0,0,0\right]\). The simulations are carried out by tracking the trajectory (57), and the other design parameters are the same as stated earlier. To facilitate the description, we define \(e^{\prime}{}_{\Psi_{S}}=\hat{\Psi}_{sL}-\Psi_{s}\), \(e^{\prime}{}_{\Psi f}=\hat{\Psi}_{fL}-\Psi_{f}\).  
The estimation output of the LESO (59) is presented in Figures 15-18. In (58), all the uncertainties are merged into a single compounded uncertainty, which leads to a larger value of the time derivative of the compounded uncertainty. Hence, obvious estimation errors can be observed in the output of LESO. Comparing Figures 11 and 12 with Figures 19 and 20, it is obvious that the disturbance observer proposed in this paper can achieve better estimation performance under the same conditions. Meanwhile, we can conclude that the developed disturbance observer not only has good steady-state performance, but also has good transient performance.  
Figure 16: Estimation of \(\Psi_{s2}\) using LESO.  
Figure 17: Estimation of \(\Psi_{f1}\) using LESO.  
Figure 18: Estimation of \(\Psi_{f2}\) using LESO.