# 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
### Traditional Sliding-Mode Control (SMC)
#### 2.1.2 Analysis of Traditional SMC Reaching Law Problems  
Here, a three-axis motion platform driven by a permanent magnet synchronous linear motor (PMLSM) is used as the control object, and the motor position, \(x_{1}\), and motor speed, \(x_{2}\), are selected as the system state variables. The system model of the three-axis motion platform driven by a PMLSM can be expressed using the following differential equations:  
\[\begin{cases}\begin{aligned} \dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-A\dot{x}_{1}+Bu+f\end{aligned}\end{cases} \tag{1}\]  
where \(A=k_{e}/M\), \(B=k_{f}/M\), \(k_{e}\), and \(k_{f}\) are the motor thrust coefficient and electromagnetic thrust coefficient of the PMLSM, respectively, and \(f\) is the external disturbance. The differential sliding surface is selected as shown in Equation (2), which can ensure the asymptotic stability of the sliding mode [32], and its asymptotic convergence rate is related to the value of \(C\).  
\[s=Ce+\dot{e} \tag{2}\]  
where \(e=x_{d}-x_{1}\), and \(x_{d}\) and \(x_{1}\) are the reference position and actual position of the motor, respectively.  
The reaching law is selected as the classical exponential-reaching law (ERL), proposed and designed by academician [PERSON] [33], which can be expressed as:  
\[\dot{s}=-k\cdot\text{sgn}(s)-g\cdot s \tag{3}\]  
where \(k\cdot\text{sgn}(s)\) is the constant-velocity-reaching term, \(g\cdot s\) is the exponential-reaching term, and \(s\) is the sliding-mode surface function. The traditional exponential approach  
Figure 1: State trajectory of SMC system.  
law cannot reach the sliding-mode surface within a limited time. Therefore, the constant velocity arrival term, \(k\cdot\text{sgn}(s)\), is added to ensure that when \(s\) approaches 0, the arrival velocity is \(k\) instead of 0. However, there is still a problem that the speed approaching the sliding-mode surface is related to the parameter \(k\) in the exponential approach law, making it difficult to achieve a balance between increasing the approach velocity and reducing sliding flutter [24; 25; 26]. Next, a detailed analysis will be conducted.  
Take Equation (2) into Equation (3), we obtain:  
\[-k\cdot\text{sgn}(s)-g\cdot s=C\dot{e}+\ddot{e} \tag{4}\]  
According to Equations (1), (2), and (4):  
\[-k\cdot\text{sgn}(s)-gs=C\dot{e}+\ddot{x}_{1d}+A\dot{x}_{1}-Bu+f \tag{5}\]  
Then, according to Equation (5), the control input \(u\) can be easily expressed as:  
\[u=B^{-1}\big{[}C\dot{e}+\ddot{x}_{1d}+Ax_{2}+f+k\cdot\text{sgn}(s)+g\cdot\big{(} Ce+\dot{e}\big{)}\big{]} \tag{6}\]  
Equation (6) shows that the control input contains discontinuous terms, \(B^{-1}[k\cdot\text{sgn}(s)]\), which is the key to causing the flutter, and the flutter level directly reaches the value of \(k\).  
In Equation (3), when \(S>0\), we can obtain:  
\[\dot{s}=-k-g\cdot s \tag{7}\]  
The time required to reach the sliding surface can be obtained by integrating Equation (7):  
\[t^{*}=\frac{1}{g}\bigg{\{}\ln\bigg{[}s(0)+\frac{k}{g}\bigg{]}-\ln\frac{k}{g} \bigg{\}} \tag{8}\]  
From Equation (8), it can be seen that the time to approach the sliding-mode surface is related to \(k\). As the value of \(k\) increases, the flutter during the approach will increase. If the value of \(k\) decreases, it will increase the time required for the system state to stabilize, which is not conducive to system control. Therefore, it is not easy to balance the approaching speed and the size of the flutter. The following section will propose an improved S-type function sliding-mode approaching law.