## 3 Controller Design
### Stable Analysis of LADRC/NLADRC Switching  
The switching between LADRC and NADRC is essentially the switching between LESO and NLESO. Therefore, a simple switching stability analysis method between LESO and NLESO is given as follows.  
Consider the following system  
\[\left\{\begin{array}{l}\varepsilon=z_{1}-y\\ \dot{z}_{1}=z_{2}-\beta_{1}e\\ \dot{z}_{2}=z_{3}-\beta_{2}\text{fal}(e,\alpha_{2},\delta_{2})+bu\\ \dot{z}_{3}=-\beta_{3}\text{fal}(e,\alpha_{3},\delta_{3})\end{array}\right. \tag{25}\]  
Define  
\[\left\{\begin{array}{l}\text{fal}(e,\alpha_{2},\delta_{2})=\frac{\text{fal} (e,\alpha_{2},\delta_{2})}{e}e=\lambda_{2}(e)e\\ \text{fal}(e,\alpha_{3},\delta_{3})=\frac{\text{fal}(e,\alpha_{3},\delta_{3} )}{e}e=\lambda_{3}(e)e\end{array}\right. \tag{26}\]  
Then, the Equation (25) can be rewrite as  
\[\left\{\begin{array}{l}e=z_{1}-y\\ z_{1}=z_{2}-\beta_{1}e\\ \dot{z}_{2}=z_{3}-\beta_{2}\lambda_{2}(e)e+bu\\ \dot{z}_{3}=-\beta_{3}\lambda_{3}(e)e\end{array}\right. \tag{27}\]  
Both \(\beta_{2}\lambda_{2}(e)\) and \(\beta_{3}\lambda_{3}(e)\) in Equation (23) can be regarded as the gain coefficient of the error \(e=z_{1}-y\). Because the error gain coefficients \(\beta_{2}\lambda_{2}(e)\) and \(\beta_{3}\lambda_{3}(e)\) are the function of the error, it can be understood that the variation range of the gain coefficient is within a perturbation range related to the error. Then, when an equivalent perturbation range independent of error is selected to replace the gain coefficient, the system (27) can be regarded as a linear extended state observer with variable gain. Its transfer function model can be described as  
\[\left\{\begin{array}{l}z_{1}=\frac{(\beta_{1}s^{2}+\beta_{2}\lambda_{2}s+ \beta_{3}\lambda_{3})\cdot y+bu\cdot s}{s^{3}+\beta_{1}s^{2}+\beta_{2}\lambda_ {2}s+\beta_{3}\lambda_{3}}\\ z_{2}=\frac{(\beta_{2}\lambda_{2}s+\beta_{3}\lambda_{3})\cdot y\cdot s+(s+ \beta_{1})\cdot bu\cdot s}{s^{3}+\beta_{1}s^{2}+\beta_{2}\lambda_{2}s+\beta_{ 3}\lambda_{3}}\\ z_{3}=\frac{\beta_{3}\lambda_{2}\cdot y\cdot s^{2}-\beta_{3}\lambda_{3}\cdot bu -\beta_{3}\lambda_{3}\cdot f(s)}{s^{3}+\beta_{1}s^{2}+\beta_{2}\lambda_{2}s+ \beta_{3}\lambda_{3}}\end{array}\right. \tag{28}\]  
As can be seen from Equation (28), the sufficient and necessary condition for the stability of the system (25) is \(\beta_{1}\beta_{2}\lambda_{2}>\beta_{3}\lambda_{3}\). When \(\text{fal}(e,\alpha_{2},\delta_{2})=\text{fal}(e,\alpha_{3},\delta_{3})\), we can obtain \(\lambda_{2}=\lambda_{3}\) and then \(\beta_{1}\beta_{2}>\beta_{3}\).  
For stable NLESO, the poles of the transfer function model (28) are located in a region of the left half plane. When switching between LESO and NLESO, in order to ensure the stability of switching, the poles of LESO also need to be in the same area of the left half plane [31; 32]. Therefore, the sufficient and necessary condition for the switching stability between LESO and NLESO is that the parameters of LESO should meet.  
\[\left\{\begin{array}{l}\beta_{2}(\lambda_{2}(e))_{\text{min}}<\beta_{2}^{*} <\beta_{2}(\lambda_{2}(e))_{\text{max}}\\ \beta_{3}(\lambda_{3}(e))_{\text{min}}<\beta_{3}^{*}<\beta_{3}(\lambda_{3}(e)) _{\text{max}}\end{array}\right. \tag{29}\]According to the above analysis, reasonable selection parameters for ESO can ensure the stability of switching between LESO and NLESO.