# A Comparative Study for Control of Quadrotor UAVs
## 3 Simulation Results
### Closed-Loop System  
The complete controlled quadrotor system is proven to be able to execute complex 3D trajectories, defined by Equations (62)-(64). Both an ascent helix (trajectory 1) and a more complex ascent helix (trajectory 2) are simulated. The signals related to the controlled variables are shown in Figure 9 for trajectory 1, Figure 10 shows the corresponding quadcopter's 3D trajectory, Figure 11 shows the control command signals required for executing trajectory 1. The Euler Angles signals related to trajectory 1 are shown in Figure 12. The signals related to the controlled variables are shown in Figure 13 for trajectory 2, Figure 14 shows the corresponding quadcopter's 3D trajectory, Figure 15 shows the control command signals required for executing trajectory 2. The Euler Angles signals related to trajectory 2 are shown in Figure 16. Considering that either PID control is a model-free approach with an easy gain tuning procedure, or the outer loop controller is intended to ensure a much slower dynamic response with respect to the inner loop controller (order of a few seconds), a PID controller in the outer loop is sufficient to ensure a satisfactory dynamic response. The tracking performances are satisfactory for either the sinusoidal-based or the unitary step reference velocity signals, as shown in Figures 9 and 13, which spatially derive in the helicoidal maneuvers shown in Figures 10 and 14. Due to the inherent characteristics of PID control, a relatively small delay between the sinusoidal-based reference velocity signal and the actual quadcopter velocity is inevitable. That is reflected also in a smaller, yet present, delay in the tracking of the consequent sinusoidal-based Euler angles signals, as shown in Figures 12 and 16. Overall, the proposed architecture can effectively control the quadcopter during complex maneuvers without exceeding the saturation limits, as shown by the signals of the controller commands in Figures 11 and 15.  
Figure 8: Euler angles signals [\(\phi(t)\), \(\phi(t),\psi(t)\)] of the controlled attitude system (blue): MPC-based Attitude Stabilizer (red).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Attitude Stabilizer.** & \(\mathbf{L_{2}}^{\mathbf{u}}\) & \(\mathbf{L_{2}}^{\mathbf{x}}\) \\ \hline PID & 1.60 & 1.40 \\ \hline LQR & 4.53 & 0.41 \\ \hline FL & 5.6 & 0.64 \\ \hline SMC & 4.35 & 0.36 \\ \hline MPC & 3.70 & 0.65 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Evaluation parameters: sum of \(\mathrm{L_{2}}\) norms of the controller commands signals, \(\mathrm{L_{2}}^{\mathbf{u}}\), and sum of \(\mathrm{L_{2}}\) norms of the Euler angles error signals, \(\mathrm{L_{2}}^{\mathbf{x}}\).  
Figure 11: Control input signals \(\left[\text{u}_{1}(\text{t}),\ \text{u}_{2}(\text{t}),\ \text{u}_{3}(\text{t}),\ \text{u}_{4}(\text{t})\right]\) of the controlled quadrotor system executing trajectory 1.  
Figure 10: 3D view of trajectory 1: position trajectory of the controlled quadrotor system.  
Figure 9: Velocity signals \(\left[\text{X}^{\prime}(\text{t}),\ \text{Y}^{\prime}(\text{t}),\ \text{Z}^{\prime}(\text{t})\right]\) of the controlled quadrotor system (blue) and reference velocity signals of trajectory 1 (red).  
Figure 14: 3D view of trajectory 2: position trajectory of the controlled quadrotor system.  
Figure 12: Euler angles signals \([\phi(t),\ \vartheta(t),\psi(t)]\) of the controlled quadrotor system (blue) and reference attitude signals for trajectory 1 (red).  
Figure 13: Velocity signals \([X^{\prime}(t),\ Y^{\prime}(t),\ Z^{\prime}(t)]\) of the controlled quadrotor system (blue) and reference velocity signals of trajectory 2 (red).