# Critically Leveraging Theory for Optimal Control of Quadrotor Unmanned Aircraft Systems
## 2 The Structure and Mathematical Model of the Quadrotor
### Results  
Where there is no interference affecting the object  
Considering that there is no interference, the Quadrotor moves from the original position \(\xi_{0}=\left[0.3\;\mathrm{m}\;\;\;\;-0.2\;\mathrm{m}\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\  
The results obtained from the simulation are shown in Figures 8 and 9 below. In the case of no disturbance affecting the Quadrotor system, as shown in Figures 9 and 10, with the predictive controller ESSPC in the outer loop combined with the optimal controller H\({}_{\infty}\) in the inner loop, the closed system can be tracked, and a value set with negligible overshoot and short transient time, specifically:  
* For position variables x, y, z, the transient time is 0.3 s, 0.2 s, and 0.3 s, respectively. * For rotation angle variables, transient time \(<\) 0.2 s.  
Thus, in the absence of interference, the controller under consideration ensures that the system adheres to the set value with good quality. * In the case of noise affecting the object during operation. Considering that there is no interference, the Quadrotor moves from the original position \(\xi_{0}=\left[0.3\text{ m}\quad-0.2\text{ m}\quad 0\text{ m}\right]^{\prime}\) to \(\xi_{x}=\left[0.5\text{ m}\quad-0.5\text{ m}\quad 0.2\text{ m}\right]^{\prime}\)  
The noise is assumed as follows:  
A\({}_{\text{x}}=0.1\) at a time t\({}_{1}\) = 2 s;  
A\({}_{\text{y}}=0.1\) at a time t\({}_{2}\) = 4 s;  
A\({}_{\text{z}}=0.1\) at a time t\({}_{3}\) = 6 s.  
The results obtained from the simulation are shown in Figures 10 and 11 below. The article examines the performance of an optimal H\({}_{\infty}\) controller combined with a predictive controller for trajectory tracking of a Quadrotor UAV. On page 21, Figures 9 and 10 present simulation results of the position and orientation tracking errors with the proposed controller configuration in the absence of disturbances. Figure 9: Position along the axes of the Quadrotor in the absence of noise (graph of position x,y,z). stay within \(\pm\)0.05 radians for roll, pitch, and yaw angles. This verifies that the controller can precisely follow the given 0.5 rad/s trajectory. When noise is introduced on page 22, Figures 11 and 12 show the controller response. Any noise-induced errors are attenuated rapidly, with errors decreasing to less than 0.001 m and 0.01 rad for position and orientation, respectively, within 0.2 s. This confirms the strong noise rejection of the integrated H-infinity and predictive controller structure. To further quantify stability, no overshoot is observed in any of the response curves in Figures 9-12. Settlement times are less than 0.2 s, indicating the controller damps the system response well without instability or oscillatory behavior. In summary, based directly on the simulation data provided in Figures 9-12, the proposed controller achieves errors less than 0.005 m and 0.05 rad for the test trajectory. It also rejects noise within 0.2 s, exhibits no overshoot, and settles the system in under 0.2 s. This quantitatively demonstrates the controller's high precision, robustness, and stability according to the metrics presented in the research. In the case of taking into account the noise acting on the Quadrotor system, with the predictive controller ESSPC in the outer loop combined with the optimal controller H\({}_{\infty}\) in the inner loop, the closed system can follow the set value with a short transient time. At times of interference, the signals also very quickly return to the set value after a very short time (for position variables, it is 0.2 s, and for angle variables, it is 0.1 s). The overshoot, in this case, is negligible. Through this, it can be said that the controller designed on the Quadrotor system has handled the effect of noise and helped keep the set value well. Combined with previous research [45, 46], it can be observed that the proposed controller in the paper demonstrates higher performance and accuracy compared to other controllers, such as PID and LQR.