## 2 Fuzzy T-S Monodimensional Modeling with Classic Formulation  
The proposed method utilizes generalized T-S identification to identify the nonlinear system in [5; 6]. Identification is based on input-output data. The method estimates the parameters of the nonlinear system by minimizing a quadratic performance index using a parameter weighting method [2]. It models nonlinear functions as a set of difference equations with IF-THEN rules for an \(n\)-th-order system. The nomenclature \(u_{i}\) is used for the inputs and \(y_{i}\) for the outputs, considering \(p\) inputs. The method is based on identifying functions of the following form:  
\[f:\mathbb{R}^{p}\longrightarrow\mathbb{R}\] \[y=f(u_{1},u_{2},\cdots,u_{p})\]  
For identification purposes, we will also have \(m\) measurable variables \((z_{1},z_{2},\cdots,z_{m})\), which can be inputs, outputs, or intermediate variables. Each IF-THEN rule \(S^{(i_{1}\cdots i_{m})}\), for an \(n\)-th order multivariable system, with \(p\) inputs and \(q\) outputs, can be written as follows for each output:  
\[ \begin{split}&\text{S}^{(i_{1}\cdots i_{m})}\text{: If }z_{1}(k)\text{ is }M_{1}^{i_{1}}\text{ and }z_{2}(k)\text{ is }M_{2}^{i_{2}}\text{ and }\dots\\ &\text{and }z_{m}(k)\text{ is }M_{m}^{i_{m}}\text{ then:}\\ & y_{j}(k+1)=a_{i0}^{(i_{1}\cdots i_{m})}+a_{i1}^{(i_{1}\cdots i_{ n})}y_{j}(k-1)+a_{i2}^{(i_{1}\cdots i_{m})}y_{j}(k-2)\\ &\quad+\cdots+a_{ii_{i}}^{(i_{1}\cdots i_{m})}y_{j}(k-n_{i})+ \sum_{j=1}^{p}\left(b_{ij1}^{(i_{1}\cdots i_{m})}u_{j}(k-1)+\right.\\ &\quad+b_{i2}^{(i_{1}\cdots i_{m})}u_{j}(k-2)+\cdots+b_{i\
eq i _{i}}^{(i_{1}\cdots i_{m})}u_{j}(k-n_{i})\left.\right)\end{split}  \tag{1}\]  
where \(y_{1}(k)\), \(y_{2}(k)\), \(\cdots y_{q}(k)\) are the system outputs, and \(u_{1}(k)\), \(u_{2}(k)\), \(\cdots u_{p}(k)\) are the system inputs. The parameters \(a_{i0},a_{i1},\cdots a_{i_{n_{i}}}\) are related to the system outputs, and the parameters \(b_{ij_{1}},b_{ij_{2}},\cdots b_{ij_{i}}\) are related to the system inputs. As for the notation, \(j\) is the fuzzy variable index and \(i_{j}\) is the fuzzy rule index associated with the fuzzy variable. The fuzzy estimation of the output is as follows:  
\[ \begin{split}&\hat{y}_{j}(k+1)=\left(\sum_{i_{1}=1}^{r_{1}}\cdots \sum_{i_{n}=1}^{r_{m}}w^{(i_{1}\cdots i_{m})}(z(k))\Big{[}a_{i0}^{(i_{1}\cdots i _{m})}+\right.\\ &\left.a_{i1}^{(i_{1}\cdots i_{m})}y_{j}(k-1)+\cdots+a_{ii_{i}}^{ (i_{1}\cdots i_{m})}y(k-n_{i})\right.\\ &\left.+\sum_{j=1}^{p}\left(b_{ij1}^{(i_{1}\cdots i_{m})}u_{j}(k- 1)+\cdots+b_{i\
eq i_{i}}^{(i_{1}\cdots i_{m})}u_{j}(k-n_{i})\left.\right) \right]\right)\\ &\left.\left.\left/\left(\sum_{i_{1}=1}^{r_{1}}\cdots\sum_{i_{n}=1 }^{r_{m}}w^{(i_{1}\cdots i_{m})}(z(k))\right)\right.\right.\end{split}  \tag{2}\]  
where  
\[w^{(i_{1}\cdots i_{m})}(z(k))=\prod_{s=1}^{m}\mu_{s}^{i_{s}}(z_{s}(k)) \tag{3}\]  
and the fuzzy rules weights \(w^{(i_{1}\cdots i_{m})}\), and \(\mu_{s}^{i_{s}}(z_{s}(k))\) is a membership function of the fuzzy set \(M_{j}^{i_{j}}\). Let \(n_{s}\) be a set of input-output system samples \([x_{1k},x_{2k},...,x_{nk},y_{k}]\). The parameters of the fuzzy system can be calculated by minimizing the following quadratic performance index:  
\[J=\sum_{k=1}^{n_{s}}(y_{k}-\hat{y}_{k})^{2}=\left\|Y-XP\right\|^{2} \tag{4}\]  
where \(Y\) is an output vector, \(X\) is the input-output T-S matrix, and \(P\) is the T-S parameters vector. If \(X\) is a matrix of complete rank, the parameters of the fuzzy system are obtained as follows:  
\[J=\left\|Y-XP\right\|^{2}=(Y-XP)^{T}(Y-XP) \tag{5}\]\[\
abla J=X^{T}(Y-XP)=X^{T}Y-X^{T}XP=0 \tag{6}\]  
\[P=\left(X^{T}X\right)^{-1}X^{T}Y \tag{7}\]  
The problem exists if the membership functions are defined as triangular ones overlapped by pairs. In this case, the matrix \(X\) is not of full rank and thus is not invertible. An effective approach for solving this problem with low computational effort, based on the well-known parameters weighting method, is presented in [6].