## 3 Controller Design
### Switch Control Strategy Based on the Interval Type-2 Fuzzy Logic System
\begin{table}
\begin{tabular}{c c c c} \hline \(\underline{|e_{gi}|}\) & \(\mathbf{S}\) & \(\mathbf{M}\) & \(\mathbf{B}\) \\ \hline S & \(Y_{L}^{1}=\left[\underline{y}_{L}^{1},\overline{y}_{L}^{1}\right]=[0,0.2]\) & \(Y_{L}^{4}=\left[\underline{y}_{L}^{4},\overline{y}_{L}^{4}\right]=[0.6,0.4]\) & \(Y_{L}^{7}=\left[\underline{y}_{L}^{7},\overline{y}_{L}^{7}\right]=[0.6,0.4]\) \\ M & \(Y_{L}^{2}=\left[\underline{y}_{L}^{2},\overline{y}_{L}^{2}\right]=[0.2,0.4]\) & \(Y_{L}^{5}=\left[\underline{y}_{L}^{5},\overline{y}_{L}^{5}\right]=[0.45,0.55]\) & \(Y_{L}^{8}=\left[\underline{y}_{L}^{8},\overline{y}_{L}^{8}\right]=[0.8,0.6]\) \\ B & \(Y_{L}^{3}=\left[\underline{y}_{L}^{3},\overline{y}_{L}^{3}\right]=[0.4,0.6]\) & \(Y_{L}^{6}=\left[\underline{y}_{L}^{6},\overline{y}_{L}^{6}\right]=[0.6,0.8]\) & \(Y_{L}^{9}=\left[\underline{y}_{L}^{9},\overline{y}_{L}^{9}\right]=[1,0.8]\) \\ \hline \end{tabular}
\end{table} 
Table 1: The rule base and corresponding results for LADRC. \begin{table}
\begin{tabular}{c c c c} \hline \(\underline{|e_{gi}|}\) & \(\mathbf{S}\) & \(\mathbf{M}\) & \(\mathbf{B}\) \\ \hline S & \(Y_{N}^{1}=\left[\underline{y}_{N}^{1},\overline{y}_{N}^{1}\right]=[1,0.8]\) & \(Y_{N}^{4}=\left[\underline{y}_{N}^{4},\overline{y}_{N}^{4}\right]=[0.4,0.6]\) & \(Y_{N}^{7}=\left[\underline{y}_{N}^{7},\overline{y}_{N}^{7}\right]=[0.4,0.6]\) \\ M & \(Y_{N}^{2}=\left[\underline{y}_{N}^{2},\overline{y}_{N}^{2}\right]=[0.8,0.6]\) & \(Y_{N}^{5}=\left[\underline{y}_{N}^{5},\overline{y}_{N}^{5}\right]=[0.55,0.45]\) & \(Y_{N}^{8}=\left[\underline{y}_{N}^{8},\overline{y}_{N}^{8}\right]=[0.2,0.4]\) \\ B & \(Y_{N}^{3}=\left[\underline{y}_{N}^{3},\overline{y}_{N}^{3}\right]=[0.6,0.4]\) & \(Y_{N}^{6}=\left[\underline{y}_{N}^{6},\overline{y}_{N}^{6}\right]=[0.4,0.2]\) & \(Y_{N}^{9}=\left[\underline{y}_{N}^{9},\overline{y}_{N}^{9}\right]=[0,0.2]\) \\ \hline \end{tabular}
\end{table} 
Table 2: The rule base and corresponding results for NLADRC. Figure 3: Three-dimensional graph of the Gaussian interval type-2 membership function. \[Y_{\text{cos}}(x_{i})=\bigcup\limits_{ \begin{subarray}{c}j^{i}\in\bar{\Gamma}^{i} \\ y^{j}\in\bar{\Gamma}^{i}\end{subarray} }\frac{\sum\limits_{i=1}^{n}f^{i}y^{i}}{ \sum\limits_{i=1}^{n}f^{i}}=[y_{l},y_{r}] \tag{31}\]  
\[y_{l}=\min\limits_{k\in[1,N-1]}\frac{\sum\limits_{n=1}^{k}\underline{\underline{ \Gamma}^{n}\overline{\Gamma}^{n}+\sum\limits_{n=k+1}^{N}\underline{\underline{ \Gamma}^{n}\underline{\Gamma}^{n}}}}{\sum\limits_{n=1}^{k}\overline{\overline{ \Gamma}^{n}}+\sum\limits_{n=k+1}^{N}\underline{\underline{\Gamma}^{n}}} \tag{32}\]  
\[y_{r}=\max\limits_{k\in[1,N-1]}\frac{\sum\limits_{n=1}^{k}\overline{\underline {\Gamma}^{n}}\underline{\underline{\Gamma}^{n}}+\sum\limits_{n=k+1}^{N} \overline{\underline{\Gamma}^{n}}}{\sum\limits_{n=1}^{k}\underline{\underline {\Gamma}^{n}}+\sum\limits_{n=k+1}^{N}\overline{\underline{\Gamma}^{n}}} \tag{33}\]  
\[y=\frac{y_{l}+y_{r}}{2} \tag{34}\]  
Considering n-dimensional input vector \(X(t)=[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)]\), the type-2 fuzzy set is described as \(\bar{\Gamma}=\left[\bar{\Gamma}_{1}^{j},\bar{\Gamma}_{2}^{j},\cdots,\bar{ \Gamma}_{n}^{j}\right]\) and the number of fuzzy rules as \(j=1,2,\cdots,m\). The defuzzification by central descent method, and the output of the type-2 fuzzy logic controller is \(y=(y_{l}+y_{r})/2\). Then, taking \(y_{l}\) as an example, the algorithm can be described as follows:  
Step 1: Calculate the relationship between each input variable \(x_{i}(t)(i=1,2,\cdots,n)\) and the fuzzy set \(\bar{\Gamma}=\left[\bar{\Gamma}_{1}^{j},\bar{\Gamma}_{2}^{j},\cdots,\bar{ \Gamma}_{n}^{j}\right]\), that is, calculate the intersection point between the input variable \(x_{i}(t)\) and the upper and lower boundaries of the fuzzy set \(\bar{\Gamma}\).