## 5 Result and Discussion
### Summary  
In the previous subsection, the controllers' simulation result in all four motions, roll, pitch, yaw, and altitude motions, was presented and discussed. A comparative study was undertaken with several conventional controllers to demonstrate the proposed controller's effectiveness, namely PD, PID, LQR, P-LQR, PD-LQR, and PD2-LQR. Moreover, since the proposed controller here uses an improved version of the Grey Wolf Optimization algorithm (IGWO), we also compare our result with the original GWO using the PD2-LQR controller.  
We also compared our controller's performance with 20 different models from the references cited in Table 1. Overall, we can see that mostly the IGWO PD2-LQR controller has a superior performance compared to the other controllers. It is essential to mention that the GWO PD2-LQR and IGWO PD2-LQR controllers' performance show less variance. This is because both controllers use an optimization algorithm to find an optimal controller's parameter; therefore, both controllers can have equally good performance compared to  
Figure 6: The step response of the system in the presence of unknown external disturbance, (**a**) roll, (**b**) pitch, (**c**) yaw, and (**d**) altitude motions.  
the others. However, the IGWO PD2-LQR controller still shows supremacy against the GWO PD2-LQR controller regarding the rise time, settling time, percentage overshoot, steady-state error, and RMSE in most of the motion.  
Figure 7: The response of the altitude and attitude error of the system, (a) roll, (b) pitch, (c) yaw, and (d) altitude motions.  
Finally, to test the proposed controller's robustness, a simulation subjected to the unknown external disturbance and sensor noise was conducted to simulate a real-world application. The external disturbance was modeled using the sinusoidal wave function with an amplitude of 1 and a frequency of 8 Hz. The sensor noise was modeled using white Gaussian noise with a zero mean value and variance of 0.01. The external disturbance and the sensor noise were applied for the simulation period. The simulation result shows that the proposed controller can still work effectively even under these circumstances with minimal error produced.  
This study only conducted a simulation work to prove the IGWO PD2-LQR controller's superior performance. However, according to several studies that compare simulation and experimental work, it was found that the performance is quite different in a real implementation. Nonetheless, the differences between these two works are not too great. Therefore, we present a few references that work in both the simulation and experimental studies of the quadrotor control and stabilization to support our findings.  
[PERSON] and [PERSON] [29] presented their findings that the pitch and roll angle are within [\(-\)3, +4]\({}^{\circ}\) for simulation while [\(-\)8, +7] and [\(-\)6, +12]\({}^{\circ}\) in pitch and roll, respectively, for experimental work. [PERSON] and [PERSON] [86] found that the quadrotor's attitude angle fluctuated at \(\pm\)5\({}^{\circ}\) from a 0.1\({}^{\circ}\) reference angle. In work by [PERSON], [PERSON] [9], the stabilization time for the attitude angle was around 1.3 s for simulation, while for an experiment, it took 2.2 s to stabilize. The results of [PERSON], [PERSON] [87] show that simulation and experimental work are almost the same, with approximately less than 10 s to settle at the desired altitude. [PERSON] and [PERSON] [88] undertook the position control of the quadrotor using a gain-scheduling PD controller. They found that the time taken to reach the desired reference was 9.55 s for simulation while 17.5 s for experimental.  
We can conclude that simulation and experimental work performance differs only a few degree/meters based on these references. Thus, our result is still good even if we consider these differences.  
Figure 8: Sensor noise signal.  
Figure 9: The step response of the system in the presence of sensor noise, (**a**) roll, (**b**) pitch, (**c**) yaw, and (**d**) altitude motions.