## 2 Design of a New Adaptive Fuzzy Controller  
### Robustness of Input Controls  
In the proposed control, the fuzzy model is applied as the first module. The fuzzy model is well explained in [15; 16; 17; 18]. Among many types of the fuzzy models, the interval type 2 fuzzy neural networks (IT2 FNN) model is chosen in this work. To fasten the calculation progress, the clustering method is used to decrease the volume of data. Then, the rule base of fuzzy model is defined and the \(j\)th If-Then rule can be described as follows:  
\[R_{f}^{j}\text{: If }h_{1}\text{ is }\mathbf{H}_{f1}^{j}\text{ and }\dots\text{ and }h_{n}\text{ is }\mathbf{H}_{f\mu}^{j}\text{ Then }g \text{ is }a_{0}^{j}+\sum\limits_{i=1}^{n}a_{i}^{j}h_{i} \tag{1}\]  
where, \(\mathbf{H}_{f1}^{j}(i=1,\dots,n;j=1,\dots,m)\) are fuzzy sets, \(m\) is the number of rules, and \(a_{i}^{j}\) are interval sets. The output is found by:  
\[g_{f}=\frac{g_{l}+g_{r}}{2}=\frac{\Xi_{l}^{T}\Xi_{l}^{f}+\Xi_{r}^{T}\Xi_{r}^{ f}}{2} \tag{2}\]  
In Equation (2), \(\Xi_{l}^{T}=\left[w_{1}^{l}\;w_{2}^{l}\;w_{3}^{l}\dots w_{n}^{l}\right]\) and \(\Xi_{r}^{T}=\left[w_{1}^{r}\;w_{2}^{r}\;w_{3}^{r}\dots w_{n}^{r}\right]\) are the weighting vectors, which symbolize the relation between the rule layer and type-reduction. The weights firing strength vectors in Equation (2) are given by:  
\[\mathbf{\xi}_{l}^{f}=\left[ \begin{array}{ccc}\frac{f_{1}}{n}&\frac{f_{2}}{n}& \frac{f_{3}}{n}&\frac{f_{4}}{n}\\ \sum\limits_{i=1}^{n}\frac{f_{i}}{f_{i}}&\sum\limits_{i=1}^{n}\frac{f_{i}}{f_{ i}}&\sum\limits_{i=1}^{n}\frac{f_{i}}{f_{i}}\end{array} \right]^{T}\text{, }\mathbf{\xi}_{r}^{f}=\left[ \begin{array}{ccc}\frac{f_{1}}{n}&\frac{f_{2}}{n}& \frac{f_{3}}{n}&\frac{f_{3}}{n}&\dots&\frac{f_{n}}{n}\\ \sum\limits_{i=1}^{n}\frac{f_{i}}{f_{i}}&\sum\limits_{i=1}^{n}\frac{f_{i}}{f_{ i}}&\sum\limits_{i=1}^{n}\frac{f_{i}}{f_{i}}&\dots&\frac{f_{n}}{n}\\ \end{array} \right]^{T}\]  
In this study, the \(n\)th system is used which belongs to a single-input single-output (SISO in short) system. The governing equation of the system is defined as follows:  
\[\dot{\mathbf{x}}_{\
abla}=\mathbf{f}_{\
abla}(\mathbf{x}_{\
abla})+\mathbf{g}_{\
abla}(\mathbf{x }_{\
abla})u_{\
abla}(t)+\mathbf{d}_{\
abla}(\mathbf{t}) \tag{3}\]  
where, \(\mathbf{f}_{\
abla}(\mathbf{x}_{\
abla})\in R^{n}\) and \(\mathbf{g}_{\
abla}(\mathbf{x}_{\
abla})\in R^{n}\) are two unknown non-linear function vectors, \(u_{\
abla}(t)\in R^{1}\) is control input, \(\mathbf{d}_{\
abla}(\mathbf{t})\in R^{n}\) is an external disturbance vector, \(|\mathbf{d}_{\
abla}(\mathbf{t})|\leq\delta\mathbf{d}\), where the value \(\delta\mathbf{d}\in R^{n}\) is an upper bound of \(\mathbf{d}_{\
abla}(\mathbf{t})\) and \(\mathbf{x}_{\
abla}=\left[x_{1\
abla},x_{2\
abla},\dots,x_{n\
abla}\right]^{T}= \left[x_{1\
abla},\dot{x}_{1\
abla},\dots,x_{1\
abla}(n-1)\right]^{T}\in R^{n}\) is the state vector. Let a new vector \(\tilde{\mathbf{x}}_{\
abla}\) be \(\tilde{\mathbf{x}}_{\
abla}=\left[x_{1\
abla}\;x_{2\
abla}\;x_{3\
abla}\;\dots\;x_ {(n-1)\
abla}\right]^{T}\). Then, the system (3) can be rewritten as follows:  
\[\frac{\dot{\mathbf{x}}_{\
abla}}{\dot{\mathbf{x}}_{\
abla}}=S_{1\
abla}\tilde{\mathbf{x}}_ {\
abla}+S_{2\
abla}^{T}s_{\
abla} \tag{4}\]In the above,  
\[S_{1\
abla}=\left[ \begin{array}{ccccc}0&1&0&\dots&0\\ 0&0&1&\dots&0\\.&.&.&\dots&.\\ -k_{1}&-k_{2}&-k_{3}&\dots&-k_{n-1}\end{array} \right],\ S_{2\
abla}=\left[ \begin{array}{c}0\\ 0\\.\\ 1\end{array} \right]\]  
The tracking error is then defined as \(e_{\
abla}(t)=x_{1\
abla}-x_{d\
abla}\), where \(x_{d\
abla}\) is the desired value. It is noted that \(x_{1\
abla}\) is the same as \(x_{\
abla}\) in Equation (4). The second (prescribed) sliding surface is defined as follows [16]:  
\[\sigma_{s\
abla}=\dot{\varphi}+c_{s}\
abla\varphi \tag{5}\]  
In the above, \(c_{s\
abla}>0\). The constant \(c_{s\
abla}\) is always positive because its design is derived from the sliding surface of the conventional sliding mode control.