# Critically Leveraging Theory for Optimal Control of Quadrotor Unmanned Aircraft Systems
## 2 The Structure and Mathematical Model of the Quadrotor
### Evaluating Quadrotor Controller Robustness to Disturbances in Maritime Environments  
Controller Evaluation Methodology: To evaluate robustness, simulations are run with the controller tracking a reference trajectory under different wind disturbances. Disturb  
Figure 12: Quadrotor phase angles in the presence of noise (phase angle graph [\(\varphi\)\(\emptyset\)\(\blacktriangledown\)]).  
bances are produced by the above models configured to ISO sea states [47], generating turbulence intensities and gust scales as per Table 2.  
Quantitative metrics are used to compare controller performance across sea states. These include the Integral of Time-multiplied Absolute value of Error (ITAE), Integral of Squared Error (ISE), and maximum position/attitude errors. Trends in these metrics indicate the controller's ability to reject disturbances of increasing magnitude. The following is a simulation of the disturbance and the proposed controller when operating under the influence of the proposed disturbance (Figures 12 and 13). The simulation presented was conducted with the input signal set to 1, and the parameters of the LQR controller were referenced from document [48].  
The dots in Figure 13 represent the effects of signal noise, and the different colors make it easier for readers to see and read the diagram. Through the simulation presented in Figure 13, it can be observed that within the first 10 s of the simulation duration, the noise signal exerts a substantial impact, resulting in an expanded and uneven operational range position. Regarding Figure 14, it is discernible that the proposed controller achieves stability marginally sooner, albeit insignificantly, in terms of the time required to attain a steady state. However, in terms of amplitude, the control oscillations exhibit a diminished amplitude error, which is readily noticeable.  
\begin{table}
\begin{tabular}{c c c} \hline
**Sea State** & **Turbulence Intensity (\%)** & **Gust Scales (m/s)** \\ \hline Calm & 5 & \(\pm 1\) \\ Moderate & 10 & \(\pm 2\) \\ Rough & 15 & \(\pm 3\) \\ \hline \end{tabular}
\end{table}
Table 2: Simulated sea states and disturbance parameters.  
Figure 13: Simulation of the impact of interference for a simulation time of 10 s in high wind conditions.  
Figure 14: Comparison of the control of velocities of two LQR controllers and proposed controllers. (**a**) Control signal (\(\chi\)); (**b**) control signal (\(y\)); (**c**) control signal (\(z\)); (**d**) control signal (phi); (**e**) control signal (theta); (**f**) control signal (psi).  
This research paper's analysis highlights the broad applicability of advanced UAV control technology in various industries. Its precision and robustness against disturbances make it ideal for precision agriculture, enhancing crop monitoring, and pesticide application. In disaster management, its reliable data collection capabilities are crucial for effective rescue operations. The technology's stable and precise flight patterns are essential for surveillance tasks, including border and sensitive area monitoring. In urban planning, its utility extends to high-precision mapping and infrastructure inspection, while in environmental monitoring, its ability to navigate challenging terrains is invaluable. These applications underscore the technology's enhanced precision, stability, and reliability, demonstrating its significant potential for industries that demand high accuracy and adaptability.  
In this research, computational time is not a significant concern with the model predictive control (MPC) technique despite it being a common issue in Quadrotor control methods. This can be attributed to advancements in computational power and algorithms, making MPC more efficient for real-time applications. The integration of advanced predictive algorithms possibly optimizes computation, allowing for quick response without compromising the control system's accuracy or stability. Thus, the innovative approach effectively manages the computational demands typically associated with MPC in Quadrotor UAV control.