## 2 Materials and Methods
### Design of ADRC and ST-SMC Hybrid Control
#### 2.3.1 Active Disturbance Rejection Control  
The ADRC architecture consists of three components: the Tracking Differentiator (TD), the Extended State Observer (ESO), and the Nonlinear State Error Feedback (NLSEF), as shown in Figure 2. These components work collaboratively to achieve effective disturbance suppression and precise control. The TD smooths the input signal and extracts its differential information, preventing abrupt control commands and balancing system responsiveness with stability. The ESO models the system's internal nonlinear dynamics and external disturbances as a unified \"total disturbance\", which is estimated and compensated in real time, significantly enhancing the system's disturbance rejection capability. The NLSEF generates the control signal by nonlinearly combining the state errors, replacing the linear weighted approach of traditional PID control, thereby optimizing dynamic response and robustness. The core strength of this architecture lies in its dynamic compensation capability: the ESO eliminates the effects of disturbances in real time, reducing the dependence on an accurate system model. Meanwhile, the synergy between TD and NLSEF effectively suppresses high-frequency oscillations and minimizes chattering. With strong adaptability, ADRC is particularly suitable for nonlinear and strongly coupled systems, such as UAV attitude  
Figure 2: Schematic diagram of ADRC. control. Through a closed-loop mechanism of \"observation-compensation-feedback\", it achieves high-precision and highly robust control objectives [26; 27]. 1. Tracking Differentiator (TD)  
The TD, also known as the transition process regulator, is designed to smooth the desired external input signal, enabling a gradual transition. This helps avoid abrupt control actions caused by a large discrepancy between the desired and actual values, thus balancing the precision and speed of the system's response. Unlike traditional differentiators, which may face issues such as local non-differentiability, noise interference, or signal distortion during signal extraction, TD ensures a smooth transition of the input signal while effectively extracting its differential information, thereby improving control performance. The expression for the TD is as follows:  
\[\left\{ \begin{aligned} \begin{aligned} v_{1}&=v_{1}+ hv_{2}\\ v_{2}&=v_{2}+hf_{h}\\ f_{h}&=\textit{than}(v_{1}-v_{0},v_{2},r_{0},h_{0}) \end{aligned}\end{aligned} \right. \tag{13}\]  
where the function _flan_ is designed to construct a nonlinear error feedback term, enabling smooth tracking and finite-time convergence, \(h\) is the sampling step size, and the velocity factor and filter factor are denoted as \(r_{0}\) and \(h_{0}\), respectively, \(v_{1}\) is the smooth tracking value of the reference signal, and \(v_{2}\) is the differential estimation of the reference signal. 2. Extended State Observer (ESO)  
Compared to traditional observers, ESO does not require an explicit form of disturbances to estimate them. Its principle is to unify both internal and external disturbances into a total disturbance and extend it as a new state variable. ESO estimates the total disturbance in real time by analyzing the system's control inputs and outputs and then compensates for it. As the core component of the ADRC, ESO exhibits strong adaptability, effectively addressing complex factors, such as strong coupling, nonlinearity, model uncertainties, and external disturbances, enabling precise observation and dynamic compensation. Typically, the ESO is constructed using a common second-order system, with the second-order system form assumed to be as follows:  
\[\left\{ \begin{aligned} \dot{x}_{1}&=x_{2}\\ \dot{x}_{2}&=f(x_{1},x_{2})+d(t)+bu\\ y&=x_{1}\end{aligned} \right. \tag{14}\]  
The above equation is expanded to the following form:  
\[\left\{ \begin{aligned} \dot{x}_{1}&=x_{2}\\ \dot{x}_{2}&=x_{3}+bu\\ \dot{x}_{3}&=\dot{f}(x_{1},x_{2},\omega)\\ y&=x_{1}\end{aligned} \right.