## 2 T-S Identification Method
### Fuzzy Clustering
#### 3.1.3 Fuzzy Clustering GG Algorithm  
The fuzzy clustering of maximum likelihood estimates (FMLE) algorithm employs the fuzzy maximum likelihood estimation-based rule as the distance [25], as follows:  
\[D_{ik}(\mathbf{x}_{k},\mathbf{v}_{i})=\frac{\sqrt{\det(F_{wi})}}{\alpha_{i}}exp\left( \frac{1}{2}\left(x_{k},\mathbf{v}_{i}^{(l)}\right)^{T}\!F_{wi}^{-1}\!\left(\mathbf{x}_ {k},\mathbf{v}_{i}^{(l)}\right)\right). \tag{57}\]  
Note that unlike the GK algorithm, this distance involves an exponential term, meaning that it decreases faster than the induced product norm. Here, \(F_{wi}\) describes the fuzzy covariance matrix in cluster i, the calculation of which is shown in (58):  
\[F_{wi}=\frac{\sum_{k=1}^{N}\left(u_{ik}\right)^{\alpha}\left(\mathbf{x}_{k}-\mathbf{v} _{i}\right)\left(\mathbf{x}_{k}-\mathbf{v}_{i}\right)^{T}}{\sum_{k=1}^{N}\left(u_{ik} \right)^{\alpha}},1\leq i\leq c. \tag{58}\]  
The degrees of membership \(u_{ik}\) are interpreted as the a posteriori probability of selecting cluster i given sample \(x_{k}\). [PERSON] and [PERSON] determined in their research [25] that the FMLE algorithm is capable of determining clusters of different shapes, sizes, and densities. The covariance matrix of the cluster is used in conjunction with an exponential distance, and the clusters are not constrained by their volume. However, this algorithm is much less robust, requiring a good initialization. Depending on the exponential distance, it can become unstable and provide erroneous results or alternatively converge to a near-optimal local point.  
The use of a central fuzzy rule to determine a desirable working point has been considered in the same way as in common fuzzy clustering methods. The GG algorithm [25] used in this work is summarized below and illustrated in the flowchart shown in Figure 4.  
* Step 1: Given a dataset X, choose the number of \(1<c<N\) clusters, the weight exponent m \(>1\), and the termination tolerance \(\varepsilon>0\). Initialize the partition matrix U using a robust method.
* Step 2: Calculate the cluster centers as follows, where l is the current iteration number of the algorithm: \[v_{i}^{(l)}=\frac{\sum_{k=1}^{N}\left(u_{ik}^{(l-1)}\right)^{\alpha}x_{k}}{ \sum_{k=1}^{N}\left(u_{ik}^{(l-1)}\right)^{\alpha}}\quad 1\leq i\leq c.\] (59)
* Step 3: Set one of the obtained clusters as the desired fuzzy rule. This center will be the same in all iterations.
* Step 4: Calculate the fuzzy covariance matrix of the cluster: \[F_{wi}=\frac{\sum_{k=1}^{N}\left(u_{ik}\right)^{\alpha}\left(\mathbf{x}_{k}-\mathbf{v }_{i}\right)\left(\mathbf{x}_{k}-\mathbf{v}_{i}\right)^{T}}{\sum_{k=1}^{N}\left(u_{ik }\right)^{\alpha}},1\leq i\leq c.\] (60)
* Step 5: Calculate the distances \(D_{ik}\) in the following form: \[D_{ik}^{2}(x_{k},v_{i})=\frac{(2\pi)^{\frac{\pi}{2}}\sqrt{\det(F_{wi})}}{ \alpha_{i}}exp\left(\frac{1}{2}\left(x_{k},v_{i}^{(l)}\right)^{T}\!F_{wi}^{-1 }\!\left(x_{k},v_{i}^{(l)}\right)\right).\] (61) with the following a priori probability: \[\alpha_{i}=\frac{1}{N}\sum_{k=1}^{N}u_{ik}.\] (62)
* Step 6: Update the partition matrix U with the terms \[u_{ik}=\frac{1}{\sum_{k=1}^{N}\left(D_{ik}\left(x_{k},v_{i}\right)/D_{ik}\left( x_{k},v_{i}\right)\right)^{\frac{\pi}{2}}},\ \ \ 1\leq i\leq c,\ 1\leq k\leq N.\] (63)Step 7: Repeat the above steps until fulfilling \[\left\|U^{(l)}-U^{(l-1)}\right\|<\varepsilon.\] (64) After obtaining the centers, covariance matrix F, and a priori probability \(\alpha\) using algorithm, it is necessary to include a fuzzification based on the GG formulae for the T-S identification and fuzzy controller stages. The fuzzification process for each sample is as follows: \[\begin{split} D_{i}^{2}(x(k),v_{i})&=\frac{(2\pi)^{ \frac{n}{2}}\sqrt{\det(F_{wi})}}{\alpha_{i}}exp\left(\frac{1}{2}(x(k),v_{i})^{ T}F_{wi}^{-1}(x(k),v_{i})\right)\\ \mu_{i}(k)&=\frac{1}{\sum_{j=1}^{c}\left(D_{i}(x(k),v _{i})/D_{j}(x(k),v_{j})\right)^{\frac{2}{n-1}}},\ \ 1\leq i\leq c\end{split}\] (65)