## 2 Methodology
### Fuzzy Systems Description  
In this section, a concise explanation of the fuzzy systems considered in this paper is presented, which perform a mapping from \(U\subset\mathbb{R}^{2}\) to \(V\subset\mathbb{R}\). Also we recall an outstanding result in function approximation of fuzzy systems. Consider the fuzzy systems \(\hat{f}_{i}\) and \(\hat{g}_{i}\) (\(i=1,2\)) for the case of two inputs \(\phi_{ij}\) (\(j=1,2\)), i.e., \(\hat{f}_{i}(\phi_{i1},\phi_{i2})\) and \(\hat{g}_{i}(\phi_{i1},\phi_{i2})\), where \(\left[ \begin{array}{cc}\phi_{i1}&\phi_{i2}\end{array} \right]\in U_{f_{i}} \subset\mathbb{R}^{2}\) and \(\left[ \begin{array}{cc}\phi_{i1}&\phi_{i2}\end{array} \right]\in U_{g_{i}} \subset\mathbb{R}^{2}\), and one output \(\zeta_{f_{i}}\in V_{f_{i}}\subset\mathbb{R}\) and \(\zeta_{g_{i}}\in V_{g_{i}}\subset\mathbb{R}\) respectively. Each input variable \(\phi_{ij}\) is defined by \(p_{ij}\) fuzzy sets \(A^{l_{ij}}_{ij}\) (\(l_{ij}=1,2,\cdots,p_{ij}\)), each one of them characterized by a membership function \(\mu_{\hat{u}_{ij}^{l_{ij}}(\phi_{ij})}\), and each one of the output membership functions \(\zeta_{f_{i}}^{l_{ij}l_{2}}\), \(\zeta_{g_{i}}^{l_{ij}l_{2}}\) is corresponded by a singleton function type. Such systems \(\hat{f}_{i}\) and \(\hat{g}_{i}\) are constructed from the set of \(\prod_{j=1}^{2}p_{ij}\) fuzzy IF-THEN rules in the forms  
\[\text{IF }\phi_{i1}\text{ is }A^{l_{11}}_{i1}\text{ \emph{AND }}\phi_{i2}\text{ is }A^{l_{2}}_{i2}\text{ \emph{THEN }}\hat{f}_{i}\text{ is }\zeta_{f_{i}}^{l_{ij}l_{2}}\] \[\text{IF }\phi_{i1}\text{ is }A^{l_{i1}}_{i1}\text{ \emph{AND }}\phi_{i2}\text{ is }A^{l_{2}}_{i2}\text{ \emph{THEN }}\hat{g}_{i}\text{ is }\zeta_{g_{i}}^{l_{ij}l_{2}}\]  
Moreover, if specifically product inference engine and center average defuzzifier are employed, the fuzzy system \(\hat{f}_{i}(\phi_{i1},\phi_{i2})\) can be represented as  
\[\hat{f}_{i}=\frac{\sum_{l_{i}=1}^{p_{i1}}\sum_{l_{i2}=1}^{p_{i2}}\zeta_{f_{i}}^ {l_{i1}l_{i2}}\left(\prod_{j=1}^{2}\mu_{A^{l_{ij}}_{ij}}(\phi_{ij})\right)}{ \sum_{l_{i}=1}^{p_{i1}}\sum_{l_{i2}=1}^{p_{i2}}\left(\prod_{j=1}^{2}\mu_{A^{l_ {ij}}_{ij}}(\phi_{ij})\right)}. \tag{6}\]  
and \(\hat{g}_{i}(\phi_{i1},\phi_{i2})\) as  
\[\hat{g}_{i}=\frac{\sum_{l_{i}=1}^{p_{i1}}\sum_{l_{i2}=1}^{p_{i2}}\zeta_{g_{i}}^ {l_{i1}l_{i2}}\left(\prod_{j=1}^{2}\mu_{A^{l_{ij}}_{ij}}(\phi_{ij})\right)}{ \sum_{l_{i}=1}^{p_{i1}}\sum_{l_{i2}=1}^{p_{i2}}\left(\prod_{j=1}^{2}\mu_{A^{l_ {ij}}_{ij}}(\phi_{ij})\right)}. \tag{7}\]Let \(\xi_{(\cdot)_{i}}^{l_{1}l_{2}}\) be the free parameters or adjustable parameters, which can be collected into a vector \(\boldsymbol{\theta}_{(\cdot)_{i}}\in\mathbb{R}^{\prod_{j=1}^{2}p_{ij}}\), i.e., \(\boldsymbol{\theta}_{(\cdot)_{i}}=\left[ \begin{array}{cccc}\xi_{(\cdot)_{i}}^{ 11}&\cdots&\xi_{(\cdot)_{i}}^{p_{1}p_{2}}\end{array} \right]^{T}\), then (6) and (7) can be rewritten respectively as  
\[\hat{f}_{i}(\phi_{11},\phi_{i2}|\boldsymbol{\theta}_{fi})=\boldsymbol{\theta}_ {fi}^{T}\boldsymbol{\zeta}_{i}(\phi_{i1},\phi_{i2}) \tag{8}\]  
\[\hat{g}_{i}(\phi_{i1},\phi_{i2}|\boldsymbol{\theta}_{gi})=\boldsymbol{\theta}_ {gi}^{T}\boldsymbol{\eta}_{i}(\phi_{i1},\phi_{i2}) \tag{9}\]  
where \(\boldsymbol{\zeta}_{i}=\left[ \begin{array}{cccc}\xi_{i}^{11}&\cdots&\xi_{i} ^{p_{1}p_{2}}\end{array} \right]^{T}\in\mathbb{R}^{\prod_{j=1}^{2}p_{ij}}\) and \(\boldsymbol{\eta}_{i}=\left[ \begin{array}{cccc}\eta_{i}^{11}&\cdots&\eta_{ i}^{p_{1}p_{2}}\end{array}