## 2 T-S Identification Method  
The identification methodology for T-S fuzzy models is formulated as an optimization problem in which the nonlinear system parameters are estimated by minimizing a quadratic performance index [2]. However, the conventional T-S identification approach becomes ineffective when triangular membership functions overlap in pairs, leading to a T-S matrix that is rank-deficient and consequently non-invertible [6]. To overcome this limitation, a generalized T-S identification framework incorporating a parameter weighting strategy was proposed in [6; 7] to enhance the robustness of the estimation process. The above method is based on the identification of nonlinear functions that can be expressed as a set of difference equations by the following IF-THEN rules for an \(n\)th order system:  
\[ \begin{split}\text{S}^{(i_{1}\ldots i_{m})}&\text{: If }z_{1}(k)\text{ is }M_{1}^{i_{1}}\text{ and }\ldots\text{ and }z_{m}(k)\text{ is }M_{m}^{i_{m}}\text{ then:}\\ y_{i}(k+1)&=a_{i0}^{(i_{1}\ldots i_{m})}+a_{i1}^{(i_ {1}\ldots i_{m})}y_{i}(k)+\cdots+a_{in_{i}}^{(i_{1}\ldots i_{m})}y_{i}(k-n_{i} +1)+\\ &+\sum_{j=1}^{p}b_{i\bar{j}1}^{(i_{1}\ldots i_{m})}u_{j}(k)+\cdots +b_{i\bar{j}n_{i}}^{(i_{1}\ldots i_{m})}u_{j}(k-n_{i}+1)\end{split}  \tag{1}\]  
where \(\{y_{1}(k),y_{2}(k),\ldots y_{s}(k)\}\) are the system outputs, \(\{u_{1}(k),u_{2}(k),\ldots u_{p}(k)\}\) are the system inputs, and \(\{z_{1}(k),z_{2}(k),\ldots z_{m}(k)\}\) are measurable variables, hereinafter referred to as fuzzy variables. The term \(\text{S}^{(i_{1}\ldots i_{m})}\) denotes the \((i_{1}\ldots i_{m})\)th fuzzy rule, while the parameters \(a_{i0\ldots i\bar{n}_{i}}^{(i_{1}\ldots i_{m})}\) and \(b_{i\bar{j}1\ldots i\bar{n}_{i}}^{(i_{1}\ldots i_{m})}\) are coefficients associated with the output and input values, respectively. In this fuzzy framework, \(j\) designates the fuzzy variable index and \(i_{j}\) denotes the fuzzy rule index associated with the corresponding fuzzy variable. Under this denotation, the fuzzy estimation of the output is provided by  
\[ \begin{split}&\hat{y}_{i}(k+1)=\sum_{i_{1}=1}^{r_{1}}\cdots \sum_{i_{m}=1}^{r_{m}}\beta^{(i_{1}\ldots i_{m})}\Big{(}z_{(i_{1}\ldots i_{m} )}(k)\Big{)}\left[a_{i0}^{(i_{1}\ldots i_{m})}+a_{i1}^{(i_{1}\ldots i_{m})}y_{ i}(k)+\cdots+\right.\\ &+a_{in_{i}}^{(i_{1}\ldots i_{m})}y_{i}(k-n_{i}+1)+\sum_{j=1}^{p} b_{i\bar{j}1}^{(i_{1}\ldots i_{m})}u_{j}(k)+\cdots+b_{i\bar{j}n_{i}}^{(i_{1} \ldots i_{m})}u_{j}(k-n_{i}+1)\Big{]}.\end{split}  \tag{2}\]where  
\[\beta^{(i_{1}\ldots i_{m})}\left(z_{(i_{1}\ldots i_{m})}(k)\right)=\frac{\prod_{j= 1}^{m}\left(\mu_{ij_{j}}(z_{j}(k))\right)}{\sum_{i_{1}=1}^{r_{1}}\cdots\sum_{i_ {m}=1}^{r_{m}}\left(\prod_{j=1}^{m}\left(\mu_{ji_{j}}(z_{j}(k))\right)\right)}, \tag{3}\]  
in which \(\mu_{ij_{j}}(z_{j})\) represents the membership function associated with the fuzzy set \(M_{j}^{ij}\). In the context of the parameter weighting method, it is assumed that an initial affine linear estimation model is available, i.e., that a preliminary estimation of the following parameters is known:  
\[P_{0}=\Big{[} \begin{array}{cccccccc}a_{i0}^{0}&\ldots&a_{in_{i}}^{0}&b_{i1 1}^{0}&\ldots&b_{i1n_{i}}^{0}&\ldots&b_{ip1}^{0}&\ldots&b_{ipn_{i}}^{0}\end{array}  \Big{]}. \tag{4}\]  
To obtain this first estimation, the classical least squares method can be applied to the set of data in the system. This estimation can then be utilized as reference parameters for all the T-S fuzzy model subsystems:  
\[p_{0}=\underbrace{\Big{[} \begin{array}{cc}P_{0}&P_{0}&\ldots&P_{0}\end{array}  \Big{]}^{t}}_{r_{1}\cdot r_{2}\cdots r_{m}}. \tag{5}\]  
Then, the fuzzy model parameters can be obtained by minimizing  
\[