## 2 T-S Identification Method
\begin{split}& J=\sum_{k=1}^{n_{n}}(y(k)-\hat{y}(k))^{2}+\gamma^{2} \sum_{i_{1}=1}^{r_{1}}\cdots\sum_{i_{m}=1}^{r_{m}}\sum_{j=0}^{n}\left(p_{j}^{ 0}-p_{j}^{(i_{1}\ldots i_{m})}\right)^{2}=\\ &=\|Y-XP\|^{2}+\gamma^{2}\|p_{0}-P\|^{2}=\left\|\left[\begin{array}{c}Y \\ \gamma p_{0}\end{array}\right]-\left[\begin{array}{c}X\\ \gamma I\end{array}\right]P\right\|^{2}=\|Y_{a}-X_{a}P\|^{2},\end{split}  \tag{6}\]  
where \(Y\) represents the output data, \(X\) represents the fuzzy input/output data, and \(P\) stands for the parameters of the fuzzy model. As explained in [6; 7], the matrix \(X^{t}X\) is non-invertible due to the linear dependency among the columns of \(X\), which makes direct computation of the parameter vector P via the classical expression \(P=(X^{t}X)^{(-1)}X^{t}Y\) infeasible. To overcome this limitation, a factor \(\gamma\) is introduced. This parameter quantifies the confidence degree associated with the initially estimated parameters; its inclusion ensures that the extended matrix \(X_{a}\) maintains full rank even if the triangular membership functions overlap in pairs. Thus, the vector \(P\) can be determined by isolating it from Equation (6):  
\[ \begin{split}& J=\|Y_{a}-X_{a}P\|^{2}=(Y_{a}-X_{a}P)^{t}(Y_{a}-X_ {a}P)\\ &\
abla J=X_{a}^{t}(Y_{a}-X_{a}P)=X_{a}^{t}Y_{a}-X_{a}^{t}X_{a}P=0\\ & P=\left(X_{a}^{t}X_{a}\right)^{-1}X_{a}^{t}Y_{a}.\end{split}  \tag{7}\]  
In the case where the state variables are measurable [7], Equation (1) can also be represented by a set of discrete state-space models:  
\[ \begin{split}&\text{S}^{(i_{1}\ldots i_{m})}\text{: If }z_{1}(k)\text{ is }M_{1}^{i_{1}}\text{ and }\ldots\text{ and }z_{m}(k)\text{ is }M_{m}^{i_{m}},\text{ then:}\\ & x(k+1)=a_{0}^{(i_{1}\ldots i_{m})}+A^{(i_{1}\ldots i_{m})}x(k)+B^ {(i_{1}\ldots i_{m})}u(k).\end{split}  \tag{8}\]  
Thus, fuzzy identification can be accomplished with state models [7] using similar formulations to Equations (2) and (3). Note that in the T-S identification method the fuzzy rules weights \(w^{(i_{1}\ldots i_{m})}=\prod_{j=1}^{m}\left(\mu_{ij_{j}}(z_{j})\right)\) are inherently determined by the inference of 1 DMFs \(\mu_{ij_{j}}(z_{j})\). In this work, we propose MDMFs to solve this problem. Fuzzy Control Based on Multidimensional Membership Functions Adjusted by Fuzzy Clustering Algorithms  
This work presents a new method that for the identification, modeling, and control of nonlinear systems described by the T-S model. The proposed method uses MDMFs and different clustering methods to adjust these functions, achieve a better fit of these functions, and even reduce the number of membership functions necessary to achieve adequate control of the nonlinear system. Figure 1 shows the control process using clustering techniques, illustrating the basic structure of a fuzzy system. The approach proposed in this work for multidimensional fuzzy sets \(M_{i}\) is described below. The fuzzy variable is defined as follows:  
\[ \begin{split} z&=\left[\begin{array}{cccc}z_{1}&z_ {2}&\cdots&z_{f}\end{array}\right]^{t}\\ z&\in\mathbb{R}^{f}\end{split}  \tag{9}\]  
and the MDMF is defined as \(\mu_{i}(z)\). For each multidimensional fuzzy set, a central point \(z_{i}e\mathbb{R}^{f}\) must be designed. This central point is defined by clustering methods in order to select points that produce a greater separation between the datasets. In this work, the dataset used for the clustering process corresponds to the set of measured input-output samples collected from the system under study. The concept of distance should also be defined, as there are several ways to define it [34]. The distance is needed in order to establish the proximity of a certain sample with respect to the central points of the membership functions, which is related to the degree of membership regarding each of the rules. Clustering techniques further determine and define this distance depending on which technique was selected during the design phase.