## 2 Materials and Methods
### Design of ADRC and ST-SMC Hybrid Control
#### 2.3.3 ADRC and ST-SMC Hybrid Control
The control law is as follows:  
\[u_{\text{ST-SMC}}=\frac{1}{g(x)}\left(-k_{1}|S\,^{1/2}sat(S)-k_{2}S\right) \tag{39}\]  
The final control input \(u\) in this chapter's algorithm is composed of the disturbance compensation term from ADRC and the ST-SMC:  
\[u=u_{\text{ADRC}}+u_{\text{ST-SMC}} \tag{40}\]  
Substituting \(u_{\text{ADRC}}\) and \(u_{\text{ST-SMC}}\) into Equation (41), we obtain:  
\[u=\frac{\hat{f}(x)+\hat{d}(t)}{g(x)}+\frac{1}{g(x)}\left(-k_{1}|S\,^{1/2}sat( S)-k_{2}S\right) \tag{41}\]  
After simplification, the final control input is obtained:  
\[u=\frac{1}{g(x)}\left(\hat{f}(x)+\hat{d}(t)-k_{1}|S\,^{1/2}sat(S)-k_{2}S\right) \tag{42}\]Here, we derive the dynamics of the super-twisting sliding mode surface. From the above definition of the sliding surface \(S\), the sliding surface can be expressed as:  
\[S=\dot{e}+\lambda e \tag{43}\]  
The sliding surface dynamics \(\dot{S}\) can be expressed as:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]0  
Substituting the final control input \(u\) into the equation, we obtain:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]1  
After simplification, we obtain:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]2  
Assuming the trajectory tracking error is zero, we have:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]3  
It can be further simplified as:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]4  
Through the derivation of the ADRC and the ST-SMC for the quadrotor UAV under wind disturbance conditions, the overall control law of the system was ultimately obtained. In the controller, the disturbance term and the Super-Twisting-Sliding Mode Control (ST-SMC) term play pivotal roles, working synergistically to enhance the overall system performance. In this hybrid control approach, the disturbance term is primarily generated by the Extended State Observer (ESO) within Active Disturbance Rejection Control (ADRC), whose core function is to estimate and compensate for unknown system dynamics and external disturbances (such as random airflow) in real time. By incorporating the estimated disturbance \(\hat{d}(t)\) and system dynamics \(\hat{f}(x)\) into the control law, the impact of model uncertainties on the system is significantly reduced. On the other hand, the super-twisting sliding mode control term further strengthens system robustness on the basis of disturbance compensation by designing an appropriate sliding surface \(S\) and implementing a higher-order control law. The inclusion of the sliding mode term enforces the system states to converge along the sliding surface, and even in the presence of residual estimation errors \(\widetilde{f}\) and \(\widetilde{d}\), its strong nonlinear characteristics ensure rapid response and system stability. Moreover, the use of a saturation function in the super-twisting control smooths the control signal, effectively suppressing the high-frequency chattering typically associated with conventional sliding mode control. Furthermore, the dynamic sliding surface was derived, which helped to clarify the entire control process of the controller. The control algorithm block diagram is shown below in \[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]5  
The derivative is:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]6  
Decompose \(\dot{V}(S)\) into disturbance terms and sliding mode control terms for individual discussion. First, consider the contribution of the disturbance terms. Since \(|\widetilde{f}|\leq\epsilon_{f}\) and \(|\widetilde{d}|\leq\epsilon_{d}\) are bounded, we have:  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]7  
Next, consider the contribution of the super-twisting sliding mode term.