## 3 Design of High-Gain Observer-Based Sliding-Mode Dynamic Surface Controller
### Designs of High-Gain Observer  
To estimate the derivative of the system input signal for feedback to the controller, high-gain observer is used. \(x_{1}\) is the input into the high-gain observer, and the output of the observer \(\hat{x}_{1}\), \(\hat{x}_{2}\), and \(\hat{x}_{3}\) is put into the controller instead of \(x_{1}\), \(x_{2}\), and \(x_{3}\). To analyze the convergence of observer, the system model given in Equation (16) is considered:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}\\ \dot{x}_{3}=F(x,u,D)\\ y=x_{1}\end{array} \right. \tag{17}\]  
where \(F(x,u,D)=f(x)+0.02899u+D\), satisfying Lipschitz condition. The high-gain observer is designed as follows:  
\[\left\{ \begin{array}{l}\dot{\hat{x}}_{1}=\hat{x}_{2}+h_{1}(y-\hat{x}_{1})\\ \dot{\hat{x}}_{2}=\hat{x}_{3}+h_{2}(y-\hat{x}_{1})\\ \dot{\hat{x}}_{3}=F(\hat{x},u)+h_{3}(y-\hat{x}_{1})\end{array} \right. \tag{18}\]  
where \(\hat{F}(\hat{x},u)\) is the nominal model of \(F(x,u,D)\). \[\text{Defining observation error}\ \widetilde{x}=\left[ \begin{array}{l} \widetilde{x}_{1}\\ \widetilde{x}_{2}\\ \widetilde{x}_{3}\end{array} \right]=\left[ \begin{array}{l}x_{1}-\hat{x}_{1 }\\ x_{2}-\hat{x}_{2}\\ x_{3}-\hat{x}_{3}\end{array} \right],\ \text{considering (\ref{eq:17}) and (\ref{eq:18}), we know that  
Figure 3: Configuration of the compound controller. \[\left\{ \begin{array}{l}\dot{\widetilde{x}}_{1}=-h_{1}\widetilde{x}_{1}+ \widetilde{x}_{2}\\ \dot{\widetilde{x}}_{2}=-h_{2}\widetilde{x}_{1}+\widetilde{x}_{3}\\ \dot{\widetilde{x}}_{3}=-h_{3}\widetilde{x}_{1}+\delta(x,\widetilde{x},d) \end{array} \right. \tag{19}\]  
where \(\delta(x,\widetilde{x},D)=F(x,u,D)-\hat{F}(\hat{x},u)\) is modeling uncertainty. Equation (17) can be written as follows:  
\[\dot{\widetilde{x}}=A\widetilde{x}+\left[ \begin{array}{c}0\\ 0\\ \delta\end{array} \right] \tag{20}\]  
where \(A=\left[ \begin{array}{ccc}-h_{1}&1&0\\ -h_{2}&0&1\\ -h_{3}&0&0\end{array} \right]\), \(\widetilde{x}=\left[ \begin{array}{c}\widetilde{x}_{1}\\ \widetilde{x}_{2}\\ \widetilde{x}_{3}\end{array} \right]\)  
When \(\delta\) is not considered, the eigenvalue is negative if \(A\) is Hurwitz. Then, the observation \(\widetilde{x}\) is asymptotically convergent. Therefore, we need to choose the suitable \(h_{1}\), \(h_{2}\), and \(h_{3}\) to make \(A\) satisfy [PERSON]. As for the effect of \(\delta\) on \(\widetilde{x}\), from Equation (19), it can be written as follows:  
\[\left\{ \begin{array}{l}s\widetilde{x}_{1}=-h_{1}\widetilde{x}_{1}+ \widetilde{x}_{2}\\ s\widetilde{x}_{2}=-h_{2}\widetilde{x}_{1}+\widetilde{x}_{3}\\ s\widetilde{x}_{3}=-h_{3}\widetilde{x}_{1}+\delta\end{array} \right.\]  
Therefore  
\[\left\{ \begin{array}{l}\widetilde{x}_{1}=\delta/(s^{3}+h_{1}s^{2}+h_{2}s+h_{ 3})\\ \widetilde{x}_{2}=\widetilde{x}_{1}(s+h_{1})\\ \widetilde{x}_{3}=\widetilde{x}_{1}(s^{2}+h_{1}s+h_{2})\end{array} \right.\]  
and we obtain the following transfer function:  
\[ \begin{array}{l}G_{10}(s)=\frac{\widetilde{x}_{1}(s)}{\delta(s)}=\frac{1}{s ^{3}+h_{1}s^{2}+h_{2}s+h_{3}}\\ G_{20}(s)=\frac{\widetilde{x}_{2}(s)}{\delta(s)}=\frac{(s+h_{1})}{s^{3}+h_{1}s^ {2}+h_{2}s+h_{3}}\\ G_{30}(s)=\frac{\widetilde{x}_{3}(s)}{\delta(s)}=\frac{(s^{2}+h_{1}s+h_{2})}{s^ {3}+h_{1}s^{2}+h_{2}s+h_{3}}\end{array}