# Research on Multiple-Axis Contour Error Suppression Method Based on Composite Layered Control
## 2 Design of a Single-Axis Controller Based on Sliding-Mode Disturbance Observer
### Design and Analysis of Sliding-Mode Approach Law Based on Improved S-Type Function (ISSMC)
#### 2.2.2 Performance Analysis of ISSMC  
According to Equation (2), the selected sliding surface is used to obtain the position tracking error and error derivative as follows:  
\[\begin{cases}e(t)=x_{d}-x_{1}\\ \dot{e}(t)=\dot{x}_{d}-\dot{x}_{1}\end{cases} \tag{10}\]  
The simultaneous Equations (1), (2), and (10) can obtain:  
\[\begin{split}\dot{s}(t)=&\ \dot{c}\dot{e}(t)-\ddot{e}(t)=c \big{[}\dot{x}_{d}-\dot{x}_{1}\big{]}-\big{[}\ddot{x}_{d}-\ddot{x}_{1}\big{]} \\ &=c\big{[}\dot{x}_{1d}-\dot{x}_{1}\big{]}+\big{[}\ddot{x}_{d}+A \dot{x}_{1}-Bu\big{]}\end{split} \tag{11}\]  
Figure 2: Function curve under different \(a\) values.  
Connecting Equation (10) with the new sliding-mode reaching law proposed in Equation (9), the improved sliding-mode reaching law is shown below:  
\[u(t)=\frac{1}{B}\Bigg{[}\begin{array}{c}\varepsilon|x|^{c}f(s)-k|s|^{df(s)}s\\ +c\big{(}\dot{x}_{d}-\dot{x}_{1}\big{)}+\ddot{x}_{d}+A\dot{x}_{1}\end{array} \Bigg{]} \tag{12}\]  
To verify the stability of the system, the Lyapunov function is established as follows [22]:  
\[V_{1}=\frac{1}{2}s^{2} \tag{13}\]  
The derivative of \(V\) is obtained by combining Equation (9) as follows:  
\[\begin{array}{l}\dot{V}_{1}=s\dot{s}=s\Big{\lfloor}-\varepsilon|x|^{a}f(s)- k|x|^{bf(s)}s\Big{\rfloor}\\ =-s\varepsilon|x|^{a}f(s)-k|x|^{bf(s)}s^{2}\end{array} \tag{14}\]  
where \(\varepsilon>0\), \(K>0\), \(|x|^{a}>0\), \(|x|^{b}>0\), and \(f(s)\) is an odd function. Therefore, there is \(\dot{V}_{1}\leq 0\) for any synovial surface. Based on the Lyapunov stability theory, the motion system proposed in this paper can converge to a stable state in a limited time.  
When \(s>1\), Equation (9) can be expressed as:  
\[\begin{array}{l}\frac{ds}{dt_{1}}=-\varepsilon|x|^{a}-k|s|^{b+1}\\ dt_{1}=\Big{(}-\varepsilon|x|^{a}-ks^{b+1}\Big{)}^{-1}ds\end{array} \tag{15}\]  
When \(s(t_{1})=0\), integrating \(t=0\) to \(t=t_{1}\) for Equation (15) can lead to the following:  
\[\begin{array}{l}\int_{0}^{t_{1}}d_{t1}=\int_{0}^{t_{1}}\big{(}\varepsilon|x| ^{a}\big{)}^{-1}\Big{[}1+\big{(}ks(0)^{b+1}/\varepsilon|x|^{a}\big{)}\Big{]}^ {-1}ds\end{array} \tag{16}\]  
Because \(\varepsilon>0\), \(K>0\), \(0<b<1\), and \(0<a<1\), Equation (16) can be written as:  
\[\begin{array}{l}\int_{0}^{t_{1}}d_{t1}\leq\int_{0}^{t_{1}}\big{(}- \varepsilon|x|^{a}\big{)}^{-1}\big{[}1+\big{(}ks(0)/\varepsilon|x|^{a}\big{)} \big{]}^{-1}ds\\ t_{1}\leq\big{(}-\varepsilon|x|^{a}\big{)}^{-1}\text{arctan}(ks(0)/ \varepsilon|x|)\end{array} \tag{17}\]  
When \(s>1\), Equation (9) can be expressed as:  
\[\begin{array}{l}\frac{ds}{dt_{2}}=\varepsilon|x|^{a}-k|s|^{b-1}\\ d_{t_{2}}=\Big{(}\varepsilon|x|^{a}-ks^{b-1}\Big{)}^{-1}ds\end{array} \tag{18}\]  
When \(s(t_{2})=0\), integrating \(t=0\) to \(t=t_{2}\) for Equation (18) can result in the following:  
\[\begin{array}{l}\int_{0}^{t_{2}}d_{t_{2}}=\int_{0}^{t_{2}}\Big{(}\varepsilon| x|^{a}-ks(0)^{b-1}\Big{)}^{-1}ds\end{array} \tag{19}\]  
Because \(0<b<1\), Equation (19) can be written as:  
\[\begin{array}{l}\int_{0}^{t_{2}}dt_{2}\leq\int_{0}^{t_{2}}\big{(}\varepsilon| x|^{a}\big{)}^{-1}ds\\ t_{2}\leq s(0)\big{(}\varepsilon|x|^{a}\big{)}^{-1}\end{array} \tag{20}\]  
In summary, using the improved S-type-function-based sliding-mode control (ISSMC) can enable the system state outside the stable point to reach the stable point in a finite time,and the required time to reach it will be less than \([t_{1}+t_{2}]\). The specific calculation is shown in Equation (21).  
\[\begin{split} t=t_{1}+t_{2}&\leq\big{(}-\epsilon|x|^{ a}\big{)}^{-1}\text{arctan}(ks(0)/\epsilon|x|)\\ &+s(0)\big{(}\epsilon|x|^{a}\big{)}^{-1}\end{split} \tag{21}\]