## 5 Result and Discussion  
In this section, the performance of the proposed controller using a nature-inspired optimization algorithm called GWO is evaluated by utilizing a linear simulation of a quadrotor in terms of the trajectory tracking problem for altitude (z) and attitude (roll, pitch, and yaw) motion. An improved version of the GWO called IGWO was proposed and applied instead of the original one. The simulation was conducted in a MATLAB/SIMULINK simulation environment. When the simulation was conducted, the only response that we considered optimizing was the system's transient response. Therefore, the control input value was set to be in the range of [0, 300]. This setting was made with the assumption that the controller directly controls the state of the system. Hence, no additional simulation concerning control effort was investigated in this work. The optimal parameters of the proposed controller obtained by the IGWO algorithm and the gain \(K\) with the \(Q\) and \(R\) matrices of the LQR controller are tabulated in Table 2 below. A comparative study was undertaken for two cases to verify the effectiveness of the proposed controller. The comparison was to realize the optimization algorithm's effectiveness in tuning the controller parameter compared to traditional tuning methods such as manual or ZN methods. Moreover, we also wanted to show how much was the improvement achieved by the IGWO compared to the conventional GWO. Therefore, the PD2-LQR controller was optimized using the GWO and IGWO algorithm, while the other controller was optimized using the manual tuning method. First, we discuss the performance of the IGWO based PD2-LQR controller with the conventional controller. After that, we discuss the performance improvement made by the IGWO based PD2-LQR controller compare to GWO based PD2-LQR controller. Figure 3: Pseudo-code of improved Grey Wolf Optimization (IGWO) algorithm. In the first part, since the proposed controller shows the best performance compared to the other controllers in every model simulated, the average value is presented for the discussion. For the second part, the best controller, the IGWO PD2-LQR controller, is assessed using 20 different models. The assessment is based on the performance comparison between the proposed controller with the existing result from those journals where the quadrotor's parameters are obtained. For both cases, the controller's performance will be evaluated in terms of the rise time, settling time, percentage overshoot, steady-state error, and RMSE. The overall result is summarized in Tables 3 and 4 for part one and Table 5 for part two in each section. Finally, in the robustness test section, only the proposed controller will be used for simulating the effect of the system's performance in the presence of a time-varying unknown external disturbance in the system and the sensor noise in the feedback loop. Performance Comparison between Improved Grey Wolf Optimization (IGWO) Proportional Double Derivative and Linear Quadratic Regulator (PD2-LQR) Controller with a Conventional Controller and Original GWO-Based Controller  
In the first study, the performance of the PD2-LQR controller using IGWO was analyzed by comparing with the conventional controllers, namely PD, PID, LQR, P-LQR, PD-LQR, and PD2-LQR. Then, we separately discuss the performance of the optimization-based controller (GWO PD2-LQR and IGWO PD2-LQR). One of these studies aims to realize this nature-inspired optimization algorithm's ability to obtain an optimal controller's parameter to control the quadrotor with improved performance. The reference step input given to the quadrotor is (1,1,1,1) with the initial condition of (0,0,0,0) for roll, pitch, yaw, and altitude. The step response of the system in the time domain for roll, pitch, yaw, and altitude motion are plotted as shown in Figure 4a-d, respectively. The figure shows that all motion controllers have successfully driven the quadrotor to reach and stabilize it at the desired reference altitude and angle.