## 2 T-S Identification Method
In addition, the multidimensional membership rules are determined based on the clustering  
Figure 1: Basic structure of fuzzy system using clustering method. method and the parameters chosen to calculate them. Thus, the form of these membership rules changes according to the methodology used. The fuzzy model of the system with \(n_{r}\) rules is represented as follows:  
\[ \begin{split} S^{i}\text{:}&\text{ If }z(k)\text{ is }M^{i}\text{ then:}\\ y_{1}(k+1)&=a_{10}^{i}+a_{11}^{i}y_{1}(k)+\dots+a_{1n _{1}}^{i}y_{1}(k-n_{1}+1)+\\ &+\sum_{j=1}^{p}b_{1j1}^{i}u_{j}(k)+\dots+b_{1 jn_{1}}^{i}u_{j}(k- n_{1}+1),\end{split}  \tag{10}\]  
which becomes  
\[ \begin{split}\hat{y}_{1}(k+1)&=\sum_{i=1}^{n_{r}} \alpha_{i}(z(k))\Big{[}a_{10}^{i}+a_{11}^{i}y_{i}(k)+\dots+a_{1n_{1}}^{i}y_{1}( k-n_{1}+1)+\\ &+\sum_{j=1}^{p}b_{1j1}^{i}u_{j}(k)+\dots+b_{1 jn_{1}}^{i}u_{j}(k- n_{1}+1)\Big{]},\end{split}  \tag{11}\]  
where  
\[\alpha_{i}(z(k))=\frac{\mu_{i}(z(k))}{\sum_{i=1}^{n_{r}}\mu_{i}(z(k))}. \tag{12}\]  
To obtain a fuzzy controller based on MDMFs, it is possible to design a controller and observer for each fuzzy rule using any controller and observer design methodology that meets the desired robustness, precision, and speed conditions. In this work, the incremental state model is used [35], as it has the advantage of introducing an integral action, which in turn allows a steady-state error of zero to be maintained in fuzzy T-S control based on MDMFs. The discrete LQR method and the optimal state observer are also applied, as defined in [35]. Nonlinear systems can be treated using the T-S fuzzy model based on MDMFs when the multidimensional fuzzy variable is known \(z=\left[ \begin{array}{cccc}z_{1}&z_{2}&\dots&z_{f}\end{array} \right]^{\dagger}\). Recalling the fuzzy model of the system with \(n_{r}\) rules previously presented in (10), the difference equation associated with each output can be transformed into a state model with an affine term, as follows:  
\[ \begin{split} x_{1}(k+1)&=\left[\begin{array}{cccc}a_{10} \cdot a_{11}\\ a_{10}\cdot a_{12}\\ \vdots\\ a_{10}\cdot a_{1n}\end{array}\right]+\left[\begin{array}{cccc}a_{11}&1&0& \dots&0\\ a_{12}&0&1&\dots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ a_{1n}&0&\dots&0&1\end{array}\right]x_{i}(k)\\ &\hskip 142.26378 pt+\left[\begin{array}{cccc}b_{111}&b_{121}&\dots&b_{1p1}\\ b_{112}&b_{122}&\dots&b_{1p2}\\ \vdots&\vdots&\ddots&\vdots\\ b_{11n}&b_{12n}&\dots&b_{1 pn}\end{array}\right]\left[\begin{array}{cccc}u_{1} (k)\\ u_{2}(k)\\ \vdots\\ u_{p}(k)\end{array}\right]\\ y_{1}(k)&=a_{10}+\left[\begin{array}{cccc}1&0&\dots&0\end{array}\right]x_{1} (k).\end{split}  \tag{13}\]  
This means that  
\[ \begin{split}& x_{1}(k+1)=x_{10}+A_{1}x_{1}(k)+B_{1}u(k)\\ & y_{1}(k)=a_{10}+C_{1}x_{1}(k),\end{split}  \tag{14}\]  
where \(A_{1}\in\mathbb{R}^{n\times n}\), \(B_{1}\in\mathbb{R}^{n\times p}\), and \(C_{1}\in\mathbb{R}^{s\times n}\). Grouping the whole system in a unique model, the following equation is obtained:  
\[\left[ \begin{array}{c}x_{1}(k+1)\\ x_{2}(k+1)\\ \vdots\\ x_{s}(k+1)\end{array} \right]=\left[ \begin{array}{c}x_{10}\\ x_{20}\\ \vdots\\ x_{s0}\end{array} \right]+\left[ \begin{array}{cccc}A_{1}&0&\cdots&0\\ 0&A_{2}&\ddots&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&0&\cdots&A_{s}\end{array} \right]\left[ \begin{array}{c}x_{1}(k)\\ x_{2}(k)\\ \vdots\\ x_{s}(k)\end{array} \right]+\left[ \begin{array}{c}B_{1}\\ B_{2}\\ \vdots\\ B_{s}\end{array} \right]\left[ \begin{array}{c}u_{1}(k)\\ u_{2}(k)\\ \vdots\\ u_{p}(k)\end{array} \right] \tag{15}\]  
\[\left[ \begin{array}{c}y_{1}(k)\\ y_{2}(k)\\ \vdots\\ y_{s}(k)\end{array} \right]=\left[ \begin{array}{c}a_{10}\\ a_{20}\\ \vdots\\ a_{s0}\end{array} \right]+\left[ \begin{array}{cccc}C_{1}&0&\cdots&0\\ 0&C_{2}&\ddots&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&0&\cdots&C_{s}\end{array} \right]\left[ \begin{array}{c}x_{1}(k)\\ x_{2}(k)\\ \vdots\\ x_{s}(k)\end{array} \right]\]  
which can be represented as follows:  
\[ \begin{array}{c}x(k+1)=a_{x}+Ax(k)+Bu(k)\\ y(k)=a_{y}+Cx(k).\end{array}