## 3 Controller Design
### Cascade ADRC Convergence Analysis  
Considering the velocity channel subsystem (14) and the cascade NLADRC (15)-(17), the convergence of the system was analyzed. For convenience of analysis, the transformations were defined as follows:  
\[\mathrm{fal}(e_{V_{i},1},\alpha_{V_{1}},\delta_{V_{1}})=\frac{\mathrm{fal}(e_{ V_{i},1},\alpha_{V_{1}},\delta_{V_{1}})}{e_{V_{i},1}}e_{V_{i},1}=\lambda_{V_{i},1}(e_{ V_{i},1})e_{V_{i},1},(i=1,2,3) \tag{18}\]  
Then, according to Equations (15)-(17), the following is obtained:  
\[\left\{ \begin{array}{l}\ddot{z}_{V_{1},1}+\beta_{V_{1},1}\dot{z}_{V_{1},1}+ \beta_{V_{1},2}\lambda_{V_{1},1}(e_{V_{1},1})z_{V_{1},1}=\beta_{V_{1},1}\dot{x }_{V_{1}}+\beta_{V_{1},2}\lambda_{V_{1},1}(e_{V_{1},1})x_{V_{1}}+\dot{u}_{V_{1}} \\ \ddot{z}_{V_{2},1}+\beta_{V_{2},1}\dot{z}_{V_{2},1}+\beta_{V_{2},2}\lambda_{V _{2},1}(e_{V_{2},1})z_{V_{2},1}=\beta_{V_{2},1}\dot{x}_{V_{2}}+\beta_{V_{2},2 }\lambda_{V_{2},1}(e_{V_{2},1})x_{V_{2}}+\dot{u}_{V_{2}}\\ \ddot{z}_{V_{3},1}+\beta_{V_{3},1}\dot{z}_{V_{3},1}+\beta_{V_{3},2}\lambda_{V _{3},1}(e_{V_{3},1})z_{V_{3},1}=\beta_{V_{3},1}\dot{x}_{V_{3}}+\beta_{V_{3},2 }\lambda_{V_{3},1}(e_{V_{3},1})x_{V_{3}}+\dot{u}_{V}\end{array} \right. \tag{19}\]  
where \(e_{V_{1},1}=z_{V_{1},1}-x_{V_{1}}\), \(e_{V_{1},2}=z_{V_{1},2}-x_{V_{2}}\), and \(e_{V_{1},3}=z_{V_{1},3}-x_{V_{3}}\) represent the state estimation errors of each stage. According to the design idea of cascade ADRC, the following applies:  
\[\left\{ \begin{array}{l}x_{V_{2}}\to u_{V_{1}}\\ x_{V_{3}}\to u_{V_{2}}\end{array} \right. \tag{20}\]  
Substituting Equation (20) into (14), we can obtain  
\[\left\{ \begin{array}{l}\dot{u}_{V_{1}}=\ddot{x}_{V_{1}}-\dot{\dot{g}}_{V_{1 }}\\ \dot{u}_{V_{2}}=\ddot{x}_{V_{2}}-\dot{g}_{V_{2}}\\ \dot{u}_{V}=\ddot{x}_{V_{3}}-\dot{L}_{x}\end{array} \right. \tag{21}\]  
where \(L_{x}=F_{V}+g_{V_{3}}\). Then, substituting Equation (21) into (19), we have  
\[\left\{ \begin{array}{l}\ddot{z}_{V_{1},1}+\beta_{V_{1},1}\dot{z}_{V_{1},1}+ \beta_{V_{1},2}\lambda_{V_{1},1}(e_{V_{1},1})z_{V_{1},1}=\beta_{V_{1},1}\dot{ x}_{V_{1}}+\beta_{V_{1},2}\lambda_{V_{1},1}(e_{V_{1},1})x_{V_{1}}+\ddot{x}_{V_{1}}- \dot{g}_{V_{1}}\\ \ddot{z}_{V_{2},1}+\beta_{V_{2},1}\dot{z}_{V_{2},1}+\beta_{V_{2},2}\lambda_{V _{2},1}(e_{V_{2},1})z_{V_{2},1}=\beta_{V_{2},1}\dot{x}_{V_{2}}+\beta_{V_{2},2 }\lambda_{V_{2},1}(e_{V_{2},1})x_{V_{2}}+\ddot{x}_{V_{2}}-\dot{g}_{V_{2}}\\ \ddot{z}_{V_{3},1}+\beta_{V_{3},1}\dot{z}_{V_{3},1}+\beta_{V_{3},2}\lambda_{V _{3},1}(e_{V_{3},1})z_{V_{3},1}=\beta_{V_{3},1}\dot{x}_{V_{3}}+\beta_{V_{3},2 }\lambda_{V_{3},1}(e_{V_{3},1})x_{V_{3}}+\dot{u}_{V}-\dot{L}_{x}\end{array} \right. \tag{22}\]  
furthermore have  
\[\left\{ \begin{array}{l}\ddot{v}_{V_{1},1}+\beta_{V_{1},1}\dot{z}_{V_{1},1}= \dot{z}_{V_{1},2}-\dot{g}_{V_{1}}\\ \ddot{v}_{V_{2},1}+\beta_{V_{2},1}\dot{e}_{V_{2},1}=\dot{z}_{V_{2},2}-\dot{g} _{V_{2}}\\ \ddot{v}_{V_{3},1}+\beta_{V_{3},1}\dot{e}_{V_{3},1}=\dot{z}_{V_{3},2}-\dot{L}_ {x}\end{array} \right. \tag{23}\]  
If the mismatched uncertain terms \(g_{V_{i}}=\rho_{V_{i}}+d_{V_{i}}(i=1,2,3)\) in system (14) and its first-order differential are bounded, that is \(\left|g_{V_{i}}\right|\leq\delta\), \(\left|\dot{g}_{V_{i}}\right|\leq\overline{\delta}\), where \(\delta\), \(\overline{\delta}\) are known normal numbers, the total disturbance estimation errors \(e_{V_{1},2}=z_{V_{1},2}-g_{V_{1}}\), \(e_{V_{2},2}=z_{V_{2},2}-g_{V_{2}}\), and \(e_{V_{3},2}=z_{V_{3},2}-L_{x}\) for each stage in Equation (23) can be described as  
\[\left\{ \begin{array}{l}\dot{e}_{V_{1},1}+\beta_{V_{1},1}e_{V_{1},1}=e_{V_{1},2 }\\ \dot{e}_{V_{2},1}+\beta_{V_{2},1}e_{V_{2},1}=e_{V_{2},2}\\ \dot{e}_{V_{3},1}+\beta_{V_{3},1}e_{V_{3},1}=e_{V_{3},2}\end{array} \right. \tag{24}\]  
As can be seen from Equation (24), when the appropriate controller parameters are selected, the state estimation errors in each stage converge exponentially.