## 2 Materials and Methods
### Function Approximation Using IT2 Fuzzy Logic Systems
 \tag{13}\]  
where \(\[PERSON]=[\[PERSON]_{l}^{T}\ \ \[PERSON]_{r}^{T}]^{T}\in R^{M\times n}\) and \(\[PERSON]=\left[\frac{1}{2}\[PERSON]_{l}^{T}\ \ \frac{1}{2}\[PERSON]_{r}^{T}\right]^{T}\in R^{M\times 1}\) are defined as follows:  
\[\[PERSON]_{l}^{i}=\dfrac{h_{l}^{i}}{\sum_{i=1}^{L}\overline{\ell}+\sum_{l=1}^{M} \underline{\ell}^{i}},\Xi_{r}^{i}=\dfrac{h_{r}^{i}}{\sum_{i=1}^{R}\overline{ \ell}+\sum_{l=R+1}^{M}\underline{\ell}^{i}}, \tag{14}\]  
with the relationships as follows:  
\[h_{l}^{i}=\left\{ \begin{array}{ll}\overline{\ell}^{i},\ i=1,\ldots,L\\ \underline{\ell}^{i},\ i=L+1,\ldots,M\end{array} \right.,\ h_{r}^{i}=\left\{ \begin{array}{ll}\overline{\ell}^{i},\ i=1,\ldots,R\\ \underline{\ell}^{i},\ i=R+1,\ldots,M\end{array} \right.. \tag{15}\]  
The function \(f(x)\) can be approximated as [40]  
\[f(x\Big{|}\Theta)=\Theta^{*\mathsf{T}}\Xi(x)+\varepsilon^{*}, \tag{16}\]  
where \(|\varepsilon^{*}|\leq\varepsilon_{m}\), \(\Theta^{*}\) is the optimized value for \(\Theta\), and \(\varepsilon^{*}\) denotes the approximation error. The optimized fuzzy output is given as  
\[\Theta^{*\mathsf{T}}\Xi=\hat{\Theta}^{T}\Xi+\varepsilon_{\theta}, \tag{17}\]  
where \(\varepsilon_{\theta}=\widetilde{\Theta}^{T}\Xi\) and \(\widetilde{\Theta}=\Theta^{*}-\hat{\Theta}\). where \(\lambda_{ij,i,1}\) and \(\lambda_{i,ij,2}\) are positive constants, \(0<\gamma_{ij,i,2}<1\), and \(0<\gamma_{ij,i,3}<1\) are constants. It is noted that the symbols \(f_{i,j_{i}}\) and \(g_{i,j_{i}}\) denotes \(f_{i,j_{i}}(\overline{x}_{i,1},\ldots,\overline{x}_{i,n_{i}-1},x_{i,1},\ldots, x_{i,n_{i}-1},\)\(u_{1},\ldots,u_{i-1})\) and \(g_{0i,j_{i}}(\overline{x}_{i,1},\ldots,\overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}-1},u_{1},\ldots,u_{i-1})\), respectively. Based on (16), we can assume that the nonlinear terms in (1) can be approximated by  
\[f_{i,j_{i}}(x_{i,j_{i}}\big{|}\Theta_{i,j_{i}})=\Theta_{i,j_{i}}^{\ast T} \Xi_{i,j_{i}}+\epsilon_{i,j_{i}}^{\ast}. \tag{25}\]  
Using the time derivative of (24) and applying \(D^{-\gamma_{ij,i^{2}}}\) to both sides of (24), we obtain  
\[ \begin{array}{ll}D^{\overline{\gamma}_{i,i,2}}s_{i,j_{i}}&=\dot{\epsilon}_{ i,j_{i}}+D^{\overline{\gamma}_{i,i,3}}[\lambda_{i,j_{i},1}\epsilon_{i,j_{i}}+ \lambda_{i,j_{i},2}sig^{1/2}(\epsilon_{i,j_{i}})]\\ &=\Theta_{i,j_{i}}^{\ast T}\Xi_{i,j_{i}}+g_{0i,j_{i}}x_{i,j_{i}+1}+\rho_{i,j_{ i}}-\dot{x}_{j}f_{i,j_{i}}+\phi_{i,j_{i}},j_{i}=1,\ldots,n_{i},\ i=1,\ldots,m,\end{array}  \tag{26}\]  
where \(D^{\overline{\gamma}_{i,i,2}}\) and \(D^{\overline{\gamma}_{i,i,3}}\) are represented by \(D^{1-\gamma_{i,i,2}}\) and \(D^{\gamma_{i,i,3}-\gamma_{i,i,2}}\), respectively, \(\phi_{i,j_{i}}=D^{\overline{\gamma}_{i,i,j_{i}}}[\lambda_{i,j_{1}}\epsilon_{ i,j_{i}}+\lambda_{i,j_{i}2}sig^{1/2}(\epsilon_{i,j_{i}})]\), and \(\rho_{i,j_{i}}=l_{i,j_{i}}+\epsilon_{i,j_{i}}^{\ast}\). **Lemma 3**. : _The following condition is satisfied as follows:_  
\[\left|\sum_{k=1}^{\infty}\frac{\Gamma(1+\overline{\gamma}_{i,j_{i},1})}{ \Gamma(1+k+\overline{\gamma}_{i,j_{i},1})\Gamma(1+k)}D^{k}s_{i,j_{i}}D^{ \overline{\gamma}_{i,i,1}-k}s_{i,j_{i}}\right|\leq\beta_{s,i,j_{i}}|s_{i,j_{i} }|^{\epsilon_{i,j_{i}}}, \tag{27}\]  
_where \(\beta_{s,i,j_{i}}>0\) and \(\epsilon_{s,i,j_{i}}>1\) are positive constants._  
Proof.