## 3 Design of High-Gain Observer-Based Sliding-Mode Dynamic Surface Controller
### Designs of Sliding-Mode Dynamic Surface Controller  
For the system shown in Equation (16), the detailed design steps of the controller are as follows:  
**Step 1. **  
Define the error of the first surface:  
\[z_{1}=x_{1}-y_{d} \tag{23}\]  
where \(y_{d}\) is the given tracking signal and its derivative is known. The time derivative of \(z_{1}\) is given as follows:  
\[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]  
To make \(z_{1}\) converge to 0, design virtual control variable:  
\[\overline{x}_{2}=-c_{1}z_{1}+\dot{y}_{d} \tag{25}\]  
where \(c_{1}\) is the parameter to be designed. Unlike the backstepping control method, put \(\overline{x}_{2}\) into a low-pass first-order filter and get the output \(x_{2d}\):  
\[x_{2d}+\tau_{2}\dot{x}_{2d}=\overline{x}_{2} \tag{26}\]  
where \(\tau_{2}\) is a time constant of the filter. The initial condition of Equation (26) is as follows:  
\[\overline{x}_{2}(0)=x_{2d}(0) \tag{27}\]  
**Step 2. **  
Define the error of the second surface:  
\[z_{2}=x_{2}-x_{2d}, \tag{28}\]  
and its derivative is as follows:  
\[\dot{z}_{2}=\dot{x}_{2}-\dot{x}_{2d}=x_{3}-\dot{x}_{2d} \tag{29}\]To make \(Z_{2}\) converge to 0, design virtual control variable:  
\[\overline{x}_{3}=-c_{2}z_{2}+\dot{x}_{2d} \tag{30}\]  
where \(c_{2}\) is design parameter. Again, put \(\overline{x}_{3}\) into a low-pass first-order filter and get the output \(x_{3d}\):  
\[x_{3d}+\tau_{3}\dot{x}_{3d}=\overline{x}_{3} \tag{31}\]  
where \(\tau_{3}\) is a time constant of the filter. The initial condition of Equation (31) is as follows:  
\[\overline{x}_{3}(0)=x_{3d}(0) \tag{32}\]  
**Step 3. **  
Define the error of the third surface:  
\[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]0  
and its derivative is as follows:  
\[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]1  
To make \(z_{3}\) converge to 0, design the control law:  
\[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]2  
where \(c_{3}\) is also the parameter to be designed. Due to \(|D|\leq\Phi\), design \(\eta\geq\Phi\). The flowchart of the proposed control algorithm is shown in \[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]3  
Define the filtering error \(y_{2}\), \(y_{3}\):  
\[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]4  
Considering position tracking, virtual control, and filtering error, define Lyapunov function:  
\[\dot{z}_{1}=x_{2}-\dot{y}_{d} \tag{24}\]5  
Differentiating Equation (36) with respect to time:  
\[ \begin{array}{rcl}\dot{V}&=&z_{1}\dot{z}_{1}+z_{2}\dot{z}_{2}+z_{3}\dot{z}_{ 3}+y_{2}\dot{y}_{2}+y_{3}\dot{y}_{3}\\ &=&z_{1}(x_{2}-\dot{y}_{d})+z_{2}(x_{3}-\dot{x}_{2d})+z_{3}(f(x)+gu+D-\dot{x}_{ 3d})\\ &&+y_{2}(\dot{x}_{2d}-\dot{\overline{x}}_{2})+y_{3}(\dot{x}_{3d}-\dot{\overline {x}}_{3})\end{array} \]  
Invoking Equations (26), (28), (30), (33), and (35), we get the following:  
\[ \begin{array}{rcl}\dot{V}&=&z_{1}(z_{2}+y_{2}-c_{1}z_{1})+z_{2}(z_{3}+y_{3}- c_{2}z_{2})-c_{3}z_{3}^{2}\\ &&+z_{3}(D-\eta\text{sgn}(z_{3}))+y_{2}(T_{2}-\frac{y_{2}}{\tau_{2}})+y_{3}(T_ {3}-\frac{y_{3}}{\tau_{3}})\\ &=&z_{1}(z_{2}+y_{2})+z_{2}(z_{3}+y_{3})-c_{1}z_{1}^{2}-c_{2}z_{2}^{2}-c_{3}z_{ 3}^{2}\\ &&+z_{3}(D-\eta\text{sgn}(z_{3}))+y_{2}(T_{2}-\frac{y_{2}}{\tau_{2}})+y_{3}(T_ {3}-\frac{y_{3}}{\tau_{3}})\end{array}  \tag{37}\]where  
\[T_{2}=c_{1}(z_{2}+y_{2}-c_{1}z_{1})-\ddot{y}_{d}\]  
\[T_{3}=c_{2}(z_{3}+y_{3}-c_{2}z_{2})-\ddot{x}_{2d}\]  
are all bounded. Define the upper bound of them as \(M_{2}\) and \(M_{3}\), respectively, yielding \(\frac{T_{l}^{2}}{M_{i}^{2}}-1\ \leq\ 0\,(i\ =\ 1,2)\).