# A Comparative Study for Control of Quadrotor UAVs
## 3 Simulation Results
### Inner Loop
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Inner Loop Controller** & **Control Parameter** & **Value** \\ \hline \multirow{4}{*}{PID} & \(\text{K}_{\text{P}}^{\text{t}}=\text{K}_{\text{P}}^{\text{t}}=\text{K}_{\text{P}}^ {\text{t}}\) & 0.56 \\  & \(\text{K}_{\text{J}}^{\text{t}}=\text{K}_{\text{J}}^{\text{t}}=\text{K}_{\text{J}}^ {\text{t}}\) & 0.28 \\  & \(\text{K}_{\text{J}}^{\text{t}}=\text{K}_{\text{J}}^{\text{t}}=\text{K}_{\text{J}}^ {\text{t}}\) & 0.23 \\  & Derivative Filter Coefficient & 49.74 \\ \hline \multirow{4}{*}{LQR} & [q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\)] & [0, 0, 0, 1, 1, 1] \\  & \(\text{r}_{\text{1}}=\text{r}_{\text{2}}=\text{r}_{\text{3}}\) & 0.0001 \\ \hline \multirow{4}{*}{FL} & [D\({}_{\text{J}}\),P\({}_{\text{1}}\)] & [\(-\)120, 38] \\  & [D\({}_{\text{J}}\),P\({}_{\text{2}}\)] & [\(-\)120, 38] \\  & [D\({}_{\text{3}}\),P\({}_{\text{3}}\)] & [\(-\)50, 20] \\ \hline \multirow{4}{*}{SM} & [K\({}_{\text{d1}}\),\(\beta_{\text{1}}\)] & [0.2, 20] \\  & [K\({}_{\text{d2}}\),\(\beta_{\text{2}}\)] & [0.2, 20] \\  & [K\({}_{\text{d3}}\),\(\beta_{\text{3}}\)] & [0.05, 5] \\  & a & 0.1 \\ \hline \multirow{4}{*}{MPC} & \(\text{T}_{\text{s}}\) & 0.01 s \\  & \(\text{T}_{\text{P}}\) & 0.1 s \\ \cline{1-1}  & \(\text{T}_{\text{s}}\) & 0.1 s \\ \cline{1-1}  & \(\text{T}_{\text{s}}\) & 0.05 s \\ \cline{1-1}  & \(\text{[q}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\),q\({}_{\text{J}}\)] & [10, 10, 1, 0, 0, 0] \\ \cline{1-1}  & \(\text{r}_{\text{1}}=\text{r}_{\text{2}}=\text{r}_{\text{3}}\) & 0.0001 \\ \cline{1-1}  & [p\({}_{\text{J}}\),P\({}_{\text{2}}\),P\({}_{\text{3}}\),P\({}_{\text{4}}\),P\({}_{\text{5}}\),P\({}_{\text{6}}\)] & [1, 1, 1, 0, 0, 0] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Attitude stabilization controllers’ parameters. uncertainties, as the performances significantly decrease when the controller is applied to the nonlinear attitude system. As far as the classical PID control approach is concerned, which is the approach adopted in this work, a systematic procedure exists for tuning the controller's gains; however, a highly time-consuming offline gains optimization is required when dealing with nonlinear systems, as for quadrotor UAVs. The amount of computation related to the online optimization of the MPC-based attitude stabilizer is such that the optimal solution is computed, on average, within 0.01 s. A control horizon \(\mathrm{T_{z}}=0.05\mathrm{s}\) is chosen in order to guarantee the optimal controller command at each step. A trial-and-error tuning procedure is adopted for the choice of the control parameters of the other attitude stabilizers, which show faster dynamics and greater robustness to model uncertainties. The sum of the \(\mathrm{L_{2}}\) norms of the Euler angles' tracking error signals, \(\mathrm{L_{2}}^{\mathbf{x}}\), is chosen as the approximative evaluation parameter of the controllers' tracking performance. Analogously, the sum of the \(\mathrm{L_{2}}\) norms of the controller command signals, \(\mathrm{L_{2}}^{\mathbf{u}}\), represents the approximative evaluation parameter of the expected command activity of an on-board hardware implementation of the controller. Since the \(\mathrm{L_{2}}\) norm of a signal represents the \"energy\" associated with the signal itself, it can be considered as an approximative quantitative estimation of both the overall tracking error and the controller command effort. Table 3 shows the controllers' evaluation parameters referring to the attitude stabilization simulations presented in this section. The proposed attitude stabilizers are ranked from best to worse as SMC, LQR, FL, MPC, PID for \(\mathrm{L_{2}}^{\mathbf{x}}\) and as PID, MPC, SMC, LQR, FL for \(\mathrm{L_{2}}^{\mathbf{u}}\).