## 3 Controller Design  
### Generalized Super-Twisting Disturbance Observers  
Based on the model (6), two GSTDOs are constructed to estimate mismatched uncertainty and matched uncertainty, respectively.  
For the mismatched uncertainty, the errors are defined as  
\[\left\{\begin{array}{l}e_{1}=\hat{x}_{2}-x_{2}\\ e_{2}=\hat{d}-d\end{array}\right. \tag{8}\]  
where \(\hat{x}_{2}\) and \(\hat{d}\) are the observed values of \(x_{2}\) and \(d\), respectively.  
The GSTDO is designed to  
\[\left\{\begin{array}{l}\dot{\hat{x}}_{2}=-\alpha_{1}\gamma_{1}\Psi_{1}(e_{1} )+x_{3}+f_{1}(x_{2})+\hat{d}\\ \dot{\hat{d}}=-\alpha_{2}\gamma_{1}{}^{2}\Psi_{2}(e_{1})\end{array}\right. \tag{9}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are positive constants, \(\gamma_{1}>0\) is the gain of the observer, and \(\Psi_{1}(e_{1})\) and \(\Psi_{2}(e_{1})\) are nonlinear stabilization terms, expressed as follows:  
\[\left\{\begin{array}{l}\Psi_{1}(e_{1})=\mu_{d1}|e_{1}|^{\frac{1}{2}}\text{ sign}(e_{1})+\mu_{d2}|e_{1}|^{\rho_{1}}\text{sign}(e_{1})\\ \Psi_{2}(e_{1})=\frac{\mu_{d1}^{2}}{2}\text{sign}(e_{1})+\mu_{d1}\mu_{d2}\left( \rho_{1}+\frac{1}{2}\right)|e_{1}|^{(\rho_{1}-\frac{1}{2})}\text{sign}(e_{1}) \\ \hskip 14.226378 pt+\mu_{d2}^{2}|e_{1}|^{(2\rho_{1}-1)}\text{sign}(e_{1})\end{array}\right. \tag{10}\]  
where \(\mu_{d1}\) and \(\mu_{d2}\) are nonnegative constants and are not zero at the same time, and \(\rho_{1}>\frac{1}{2}\) is a real number. \(\Psi_{1}(e_{1})\) and \(\Psi_{2}(e_{1})\) are related and monotone increasing functions of \(e_{1}\); since \(\Psi_{2}(e_{1})=\Psi_{1}^{\prime}(e_{1})\Psi_{1}(e_{1})\), \(\Psi_{1}(e_{1})\) is continuous while \(\Psi_{2}(e_{1})\) is discontinuous at \(e_{1}=0\).  
The estimation errors equations are as follows  
\[\left\{\begin{array}{l}\dot{e}_{1}=-\alpha_{1}\gamma\psi_{1}(e_{1})+e_{2}\\ \dot{e}_{2}=-\alpha_{2}\gamma^{2}\psi_{2}(e_{1})-h_{1}\end{array}\right. \tag{11}\]  
where \(h_{1}\) is the change rate of mismatched uncertainty.  
For the matched uncertainty, the errors are defined as  
\[\left\{\begin{array}{l}e_{3}=\hat{x}_{3}-x_{3}\\ e_{4}=\hat{q}-q\end{array}\right. \tag{12}\]  
where \(\hat{x}_{3}\) and \(\hat{q}\) are the observed values of \(x_{3}\) and \(q\), respectively.  
The GSTDO is designed as  
\[\left\{\begin{array}{l}\dot{\hat{x}}_{3}=-\alpha_{3}\gamma_{2}\Psi_{3}(e_{3}) +g(u,x_{3})u+f_{2}(x_{2},x_{3})+\hat{q}\\ \dot{\hat{q}}=-\alpha_{4}\gamma_{2}{}^{2}\Psi_{4}(e_{3})\end{array}\right. \tag{13}\]  
where \(\alpha_{3}\) and \(\alpha_{4}\) are positive constants, \(\gamma_{2}>0\) is the gain of the observer and \(\Psi_{3}(e_{3})\) and \(\Psi_{4}(e_{3})\) are nonlinear stabilization terms, expressed as follows:  
\[\left\{\begin{array}{l}\Psi_{3}(e_{3})=\mu_{q1}|e_{3}|^{\frac{1}{2}}\text{ sign}(e_{3})+\mu_{q2}|e_{3}|^{\rho_{2}}\text{sign}(e_{3})\\ \Psi_{4}(e_{3})=\frac{\mu_{q1}^{2}}{2}\text{sign}(e_{3})+\mu_{q1}\mu_{q2}\Big{(} \rho_{2}+\frac{1}{2}\Big{)}|e_{3}|^{(\rho_{2}-\frac{1}{2})}\text{sign}(e_{3}) \\ \quad\quad\quad\quad+\mu_{q2}^{2}|e_{3}|^{(2\rho_{2}-1)}\text{sign}(e_{3}) \end{array}\right. \tag{14}\]  
where \(\mu_{q1}\) and \(\mu_{q2}\) are nonnegative constants and are not zero at the same time, and \(\rho_{2}>\frac{1}{2}\) is a real number. \(\Psi_{3}(e_{3})\) and \(\Psi_{4}(e_{4})\) are related, just like \(\Psi_{1}(e_{1})\) and \(\Psi_{2}(e_{1})\).  
The estimation errors equations are as follows:  
\[\left\{\begin{array}{l}\dot{e}_{1}=-\alpha_{1}\gamma\psi_{1}(e_{1})+e_{2}\\ \dot{e}_{2}=-\alpha_{2}\gamma^{2}\psi_{2}(e_{1})-h_{2}\end{array}\right. \tag{15}\]  
where \(h_{2}\) is the change rate of matched uncertainty.  
**Lemma 1**.: _The GSTDO can converge to zero in finite time when there are unknown inputs and no unknown inputs. When the initial error between the estimated and the actual state is very large, the convergence time of the GSTDO is basically the same [36]._