## 4 Controller Design
### Design of Fuzzy Sliding Mode Control (FSMC) for Path-Positioning Controller  
The FSMC consists of two parts, a sliding surface estimator and a fuzzy logic controller [18; 19; 20]. Subtracting the target value from the feedback output signal produces the error \(e\) and the error rate \(\dot{e}\), multiplies the error term by the gain \(\alpha\), and combines them into a sliding surface \(\sigma\) as:  
\[\sigma=\alpha\cdot e+\dot{e} \tag{1}\]  
The traditional fuzzy controller adopts a two-dimensional fuzzy rule base whose control targets are \(e\to 0\) and \(\dot{e}\to 0\). After employing the sliding surface estimator, the two-dimensional input is replaced by the plane equation \(\sigma=\alpha\cdot e+\dot{e}\), the error \(e\) and the error rate \(\dot{e}\) are reduced to an input value \(\sigma\), and the control target becomes \(\sigma=\alpha\cdot e+\dot{e}\to 0\), that is, the final output can also reach the set reference target. Combining the above two points, we can know that in the fuzzy sliding mode control, the two-dimensional input of the fuzzy rule base can be judged as \(e=0\) and \(\dot{e}=0\) becomes \(\sigma=\alpha\cdot e+\dot{e}=0\), and the simplified fuzzy rule base will reduce its complexity and reduce the computer calculation in real time. As long as the sliding surface is maintained at \(\sigma=\alpha\cdot e+\dot{e}=0\), the system will eventually converge to the set ideal target value [21]. The value of \(\alpha\) is the slope of the fuzzy sliding surface \(\sigma\). The key of the fuzzy sliding surface planning is the selection of the value of \(\alpha\). Therefore, the selection of the value of \(\alpha\) will directly affect the convergence speed of the control system and the performance of the controller, and determine the stability of the system. The determination of the value of \(\alpha\) must depend on the system characteristics of the control system. The value of \(\Phi\) is the convergence boundary of the fuzzy sliding surface \(\sigma=\alpha\cdot e+\dot{e}=0\), as shown in Figure 8. The selection of the value of \(\Phi\) is determined according to the control input of the fuzzy sliding surface \(\sigma=\alpha\cdot e+\dot{e}\). Therefore, the value of \(\Phi\) must be selected suitably according to the control input and the actual system operation. The control parameters such as \(\alpha,\Phi\), \(u_{0}\), \(G_{u}\) have certain influence on the dynamic response of the control system. Therefore, the selection of parameters must be adjusted according to the actual needs and characteristics of the system. The establishment of the membership function first sets the membership function of the input variable \(\sigma\) is \(T(\sigma)=\{NB,NM,NS,ZR,PS,PM,PB\}\), and the membership function of the output variable \(u\) is \(T(u)=\{NB,NM,NS,ZR,PS,PM,PB\}\), according to the established fuzzy set combined with the sliding mode control theory, the relationship between the fuzzy sliding surface \(\sigma\) and the control input \(u\) can be obtained. According to the output state of the system, we can set the following fuzzy rules shown in Table 3. NB, NM, NS, ZR, PS, PM, and PB are all labels of the fuzzy set, and the membership function of the input variable \(\sigma\) of the fuzzy sliding surface is shown in Figure 9, the membership function of the output variable \(u\) of the fuzzy sliding surface is shown in Figure 10.  
Figure 8: Fuzzy sliding surface. Therefore, the relationship between the fuzzy sliding surface \(\sigma\) and the control input \(u_{f}\) is expressed by the following Equation (2). \[ \begin{array}{c}u_{f}=u_{0}-u_{d}\\ u_{d}=G_{u}(x,t)\cdot\textit{sig}(w)\\ w=\frac{\sigma}{\Phi}\end{array}  \tag{2}\]  
where \(u_{0}\) can be regarded as the offset of the control signal input, and sigmoid function \(\textit{sig}(w)\) can be described as:  
\[