## 2 Fuzzy T-S Monodimensional Modeling with Classic Formulation
This method can also be used for parameter tuning of the T-S model from local parameters obtained through the identification of a system in an operating region or from any physical input/output data. It is supposed that in this case, a first estimation of the parameters is available, which is obtained by the well-known least squares method. \[P_{0}=\left[ \begin{array}{cccc}p_{0}^{0}&p_{1}^{0}&p_{2}^{0}&\ldots&p_{n}^{0 }\end{array} \right]^{T} \tag{8}\]  
This first approximation can be utilized as reference parameters for all the subsystems. Then, the parameters vector of the fuzzy model can be obtained minimizing the following:  
\[J=\sum_{k=1}^{n_{k}}(y_{k}-\hat{y}_{k})^{2}+\gamma^{2}\sum_{i_{1}=1}^{r_{1}} \cdots\sum_{i_{m}=1}^{r_{m}}\sum_{j=0}^{n}\left(p_{j}^{0}-p_{j}^{(i_{1}\ldots i _{m})}\right)^{2} \tag{9}\]  
\[J=\|Y-XP\|^{2}+\gamma^{2}\|p_{0}-p\|^{2} \tag{10}\]  
\[J=\left\|\left[ \begin{array}{c}Y\\ \gamma p_{0}\end{array} \right]-\left[ \begin{array}{c}X\\ \gamma I\end{array} \right]p\right\|^{2} \tag{11}\]  
\[J=\|Y_{a}-X_{a}p\|^{2} \tag{12}\]  
where  
\[p_{0}=\underbrace{\left[ \begin{array}{cc}P_{0}&P_{0}&\cdots&P_{0}\\ \end{array} \right]^{T}}_{r_{1}\cdot r_{2}\cdots r_{m}} \tag{13}\]  
In this case, the factor \(\gamma\) represents the degree of confidence of the parameters initially estimated. In a similar way to the previous equation, different weight factors of \(\gamma_{j}^{(i_{1}\ldots i_{m})}\) can be used to each one of the parameters \(p_{j}^{(i_{1}\ldots i_{m})}\) depending on the reliability of the initial parameter \(p_{j}^{0}\) in the specific rule. The fuzzy discrete system can be described by the following IF-THEN rules for an \(n\)-th order system as a state model, as shown in [28]. \[ \begin{split}\text{S}^{(i_{1}\ldots i_{m})}&:\text{If $z_{1}(k)$ is $M_{1}^{i_{1}}$ and $z_{2}(k)$ is $M_{2}^{i_{2}}$ and $\ldots$}\\ &\text{and $z_{m}(k)$ is $M_{m}^{i_{m}}$ then:}\\ x(k+1)&=A_{o}^{(i_{1}\ldots i_{m})}+A^{(i_{1}\ldots i_{m})}x(k)+B^{(i_ {1}\ldots i_{m})}u(k)\\ y(k)&=C_{o}^{(i_{1}\ldots i_{m})}(k)+C^{(i_{1}\ldots i_{m})}x(k) \end{split}  \tag{14}\]  
The discrete model of the fuzzy multivariable nonlinear system is represented as shown below:  
\[ \begin{split} x(k+1)&=A_{o}(k)+A(k)\cdot x(k)+B(k) \cdot u(k)\\ y(k)&=C_{o}(k)+C(k)\cdot x(k)\end{split}  \tag{15}\]where the matrices \(A_{0}(k)\), \(C_{0}(k)\), \(A(k)\), and \(B(k)\) are the fuzzy blends of the matrices \(A_{0}^{(i_{1}\dots i_{n})}\), \(C_{0}^{(i_{1}\dots i_{n})}\), \(A_{0}^{(i_{1}\dots i_{n})}\), \(A_{0}^{(i_{1}\dots i_{m})}\), and \(B^{(i_{1}\dots i_{m})}\) for a specific point. The effect of fuzzy rules on controller performance is related to the identification of the nonlinear system. By making the rules more fuzzy and choosing the appropriate universe of discourse, a precise identification of the system will be obtained, which will closely approximate the dynamic behavior of the nonlinear system. This, in turn, leads to the design of a robust and well-damped fuzzy controller that will meet the system requirements. However, increasing the number of fuzzy rules increases computational cost, so it is necessary to maintain a reasonable number of fuzzy rules. The process for determining the membership functions, and thus the number of fuzzy rules and fuzzy subsystems, is carried out through an iterative trial and error process. Generalized identification is tested with a fuzzy system configuration, and the identification error value is checked:  
\[e(k)=y(k)-\hat{y}(k) \tag{16}\]  
where \(\hat{y}_{k}\) is the estimated output of the fuzzy system described in (2). Ultimately, the resulting fuzzy system configuration is a trade-off between reducing the T-S identification error and maintaining a reasonable number of fuzzy rules. Equation (17) will be used to calculate the relative modeling error:  
\[erm(y_{i})=\sqrt{\frac{\sum_{k}(y_{i}(k)-\hat{y_{i}}(k))^{2}}{\sum_{k}(y_{i}(k ))^{2}}}=\frac{\|y-\hat{y}\|}{\|y\|} \tag{17}\]