# Performance Evaluation of a Quadcopter by an Optimized Proportional-Integral-Derivative Controller
## 4 Detailed Discussions and Results
The comparative results are demonstrated in Table 2 where the three parameters of \((k_{P}\), \(k_{i}\), \(k_{d})\) are assumed as \((1,\ 0.0,\ 0.0)\), \((1,\ 1/300,\ 0.0)\), and \((0.1,\ 1/100,\ 0)\), respectively. Finally, it is obvious to understand that the \"Rise time\" of the three different PID models can be adjusted and obtained as corresponding to values of 0.0180, 0.0180, and 0.0352 s. Actually, in Table 2, the \"rise time\" generated from the proposed \"Optimized PID\" with three different PID models is compared with the outcomes from article [15]. Figure 8: The reading data from the quadcopter of yaw, pitch, and roll. Figure 7: The flight controller outputs a response signal. Eventually, the validation of the performance evaluated on the proposed optimized PID could be certified with quantification. A feasible first-order continuous system with time delay and the transfer function of \(1/(60\text{s}+1)\) is applied to the explain the behavior for the proposed optimized PID shown in Figure 1. The characteristic equation can be calculated by making the denominator expressed in the transfer function equal to zero, that is, \((60\text{s}+1)=0\). According to the previous discussion, the PID can mainly provide feedback on the changing signal depending on the error values measured from the output of the controller. Typically, the order of the control steps is as follows: adjust the \(k_{P}\) value, then the \(k_{i}\) factor, and finally the parameter of \(k_{P}\). There are 12 figures included in Table 3 where the simulation of the optimal PID controller proposed in this article is carried out by adjusting different PID parameter values. In order to limit the size of the text, only some of the plots obtained in the simulation section will be explained in this article. For example, in graph (a) shown in Table 3, the yellow and green curves denote the control signal as \"after PID regulation\" and \"without PID regulation\", respectively. It is clear to see that the stability of the controller can be quickly researched under conditions without \(k_{i}\) values. On the other hand, the speed of stability for a PID controller is definitely not dominated by the value of \(k_{d}\). Additionally, plot (c) shown in Table 3 tells that \(k_{i}\) should decrease since the larger value of \(k_{i}\) will lead to divergence of the control signal. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Articles** & **Parameters** & **PID-1** & **PID-2** & **PID-3** \\ \hline \multirow{3}{*}{[15]} & (\(k_{P},k_{i},k_{d}\)) & (4.1302, 89.8994, 0.0511) & (2.9055, 22.8899, 0.0208) & (2.5865, 20.9157, 0.0106) \\ \cline{2-5}  & “Rise Time” (s) & 0.0228 & 0.0415 & 0.0479 \\ \hline \multirow{3}{*}{Proposed} & (\(k_{P},k_{i},k_{d}\)) & (1, 0.0, 0.0) & (1, 1/300, 0.0) & (0.1, 1/100, 0.0) \\ \cline{1-1} \cline{2-5}  & “Rise Time” (s) & 0.180 & 0.180 & 0.352 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparative results of “Rise time” for three different PID models. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Fixed Value** & \multicolumn{3}{c}{**Variated Values**} \\ \hline \hline \multirow{4}{*}{PID-1} & \multicolumn{2}{c}{\(k_{i}=0\) (a)} & \(k_{i}=5\) (b) & \(k_{i}=1/10\) (c) \\ \cline{2-4}  & \multicolumn{2}{c}{\(k_{d}=0\)} & \(k_{d}=1/10\) (e) & \(k_{d}=10\) (f) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The simulation results from the optimal PID controller proposed in this article.