## 7 Simulation Results
### Matlab  
Figures 5-10 show the simulation results for the adaptive backstepping controller (40) applied to a simulation model of the Aerosonde UAV [1]. The controller uses perfect estimates of the matrices \(B\), \(J\), and \(D\) but no information about \(\Delta(v_{r},t)\). While the control surface deflections are controlled by the attitude controller, a PI controller is used to control airspeed using throttle [1]. The airspeed reference is constant and set to \(35\,\mathrm{m}\,\mathrm{s}^{-1}\). The attitude controller parameters are set to \(\kappa=1,k_{1}=1,K_{2}=\mathrm{diag}(7,5,7)\) and \(K_{3}=\mathrm{diag}(40,30,40)\). During the first 20 s, the reduced-attitude reference is constant, corresponding to \(\phi_{d}=60\,\mathrm{deg}\) and \(\theta_{d}=15\,\mathrm{deg}\), which might correspond to a sharp, climbing turn. During the last 20 s, we use the time-varying reference \(A\cos(2\pi f(t-20))\), with amplitude \(A\) equal to the initial 20 s, and \(f=0.1\,\mathrm{Hz}\) for roll and \(f=0.08\,\mathrm{Hz}\) for pitch. The initial conditions are set to \(\omega(0)=0,\phi(0)=-40\,\mathrm{deg}\) and \(\theta(0)=-20\,\mathrm{deg}\).  
Figure 5 shows the tracking performance in terms of roll and pitch angles, while Figure 6 shows the vector coordinates \(\eta\in\mathbb{S}^{2}\). The errors converge quickly from large initial values, and the velocity errors are kept close to zero throughout the maneuver. During the latter half of the simulated trajectory, a slightly deteriorated tracking performance is observed in pitch. This also applies to the turn rate, visualized in Figure 7. This is explained by looking at Figure 8: When stabilizing a constant reference, the assumption that aerodynamic moments are slowly-varying applies quite well, and the disturbance estimates converge towards their true values. When tracking a time-varying trajectory, however, this assumption seems to break down, which has a negative impact on tracking performance. This variation seems to be attributed to the variations in the angle of attack seen in Figure 9. Nevertheless, this simulated case study shows adequate tracking performance for both constant and time-varying reference trajectories. The effect of turn-coordination can be seen by observing the sideslip angle in Figure 9. During the first 20 s, the sideslip angle is reduced to zero. During the last 20 s, some variation is seen, but the sideslip angle is still kept at small values (less than 2 degrees). The control surface deflections are shown in the bottom half of Figure 9, and are smooth and well below the saturation limits, which in the simulation is set to \(\pm 20\,\mathrm{deg}\). Finally, the start of the maneuver is illustrated as a path on the two-sphere in Figure 10, which is an alternative to showing roll and pitch angles when visualizing reduced-attitude trajectories.  
Figure 5: Roll and pitch angles vs. references, and velocity tracking error \(e_{\omega}\).  
Figure 6: Reduced attitude \(\eta\) (blue) and the reduced-attitude reference \(\eta_{d}\) (red).  
Figure 8: Disturbance estimates (blue) vs. their true values (red).  
Figure 10: Path on the two-sphere for the 17 first seconds. Red asterisk marks the initial configuration, while the constant reference is marked green.  
Figure 9: **Top**: angle of attack \(\alpha\) and sideslip angle \(\beta\); **Bottom**: control surface deflections aileron \(\delta_{a}\), elevator \(\delta_{e}\) and rudder \(\delta_{r}\).