## 2 Materials and Methods
### Design of ADRC and ST-SMC Hybrid Control
#### 2.3.3 ADRC and ST-SMC Hybrid Control
To address this issue, a new higher-order sliding mode control method based on sliding mode control has been developed. This method not only incorporates the many advantages of sliding mode control but also suppresses the high-frequency chattering caused by switching control. This method is known as ST-SMC. The core idea of this method is to optimize the higher-order sliding mode algorithm by designing a suitable sliding surface that satisfies the Lyapunov stability conditions. By integrating smoothing function approximations, filters, and other optimization algorithms, the dynamic response of the control signal is further smoothed. This approach effectively enhances the steady-state accuracy and stability performance of the controller. Therefore, for the design of an ST-SMC controller, the design of the sliding surface is crucial. The following will provide a detailed analysis of the basic principles and methods for designing the sliding surface. (1) Sliding Surface Design  
The sliding surface is defined as:  
\[S=\dot{e}+\lambda e \tag{29}\]  
where \(e=x-x_{ref}\) represents the tracking error, and \(\lambda\) is the sliding surface gain. (2) ST-SMC Law  
The super-twisting control incorporates higher-order terms to reduce chattering:  
\[u_{\rm ST-SMC}=\frac{1}{g(x)}\Big{(}-k_{1}|S\,{}^{1/2}{\rm sign}(S)-k_{2}S \Big{)} \tag{30}\]  
where \(k_{1}\) and \(k_{2}\) are the sliding mode gains. To ensure the stability of the ST-SMC, the sliding mode gains must satisfy the following conditions:  
\[k_{1}>\frac{2\varepsilon}{\alpha},k_{2}>\frac{\varepsilon}{\alpha^{2}} \tag{31}\]  
Here, \(\varepsilon\) is the upper bound of the system's uncertainty, and \(\alpha\) is a positive number, typically set to 1. These conditions ensure the robustness and stability of the sliding mode control. To further suppress the chattering phenomenon in sliding mode control, a saturation function is incorporated into the control law, resulting in the control system as follows:\[u_{\text{ST-SMC}}=\frac{1}{g(x)}\left(-k_{1}|S\,^{1/2}sat(S)-k_{2}S\right) \tag{32}\]  
where \(sat(S)\) is the saturation function, defined as:  
\[sat(S)=\left\{ \begin{array}{cc}sign(S)&if\ |S|>\delta\\ \frac{S}{\delta}&if\ |S|>\delta\end{array} \right. \tag{33}\]  
Here, \(\delta\) in the saturation function is a positive number used to control the smoothness of the saturation function. * Design of ADRC and ST-SMC Hybrid Control  
The Extended State Observer (ESO) from ADRC is used to observe the system states and disturbances in real time. The state equation is:  
\[\dot{z}_{1}=z_{2}-\beta_{1}(z_{1}-y) \tag{34}\]  
\[\dot{z}_{2}=z_{3}-\beta_{2}(z_{1}-y)+g\cdot u \tag{35}\]  
\[\dot{z}_{3}=-\beta_{3}(z_{1}-y) \tag{36}\]  
where \(z_{1}\), \(z_{2}\), and \(z_{3}\) correspond to the observed variable's value, its first derivative, and its second derivative, respectively. \(y\) is the measured system output, and \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) are the gain coefficients of the ESO. Using the state observer in ADRC, the system states and disturbances can be estimated as:  
\[u_{\text{ADRC}}=\frac{\hat{f}(x)+\hat{d}(t)}{g(x)} \tag{37}\]  
By employing disturbance compensation, the effects of system dynamics and external disturbances can be eliminated. Introducing the disturbance from the state observer into the control law for compensation, the disturbance compensation term in ADRC is:  
\[\hat{d}(t)=z_{3},\ \hat{f}(x)=z_{2} \tag{38}\]  
By incorporating a saturation function into the original control law, the chattering phenomenon in sliding mode control is suppressed.