## 2 Material and Methods
### Methods
#### 2.2.2 Takagi-Sugeno Type-2 Fuzzy Inference  
The major difference between the type-1 and type-2 Takagi-Sugeno fuzzy models obviously lies in the incorporation of type-2 fuzzy sets in the inference process. This requires corresponding fuzzy rule interpretation and type-reducer mechanism. A type-1 fuzzy rule in terms of the Takagi-Sugeno model, takes the following form:  
\[R\ :_{\text{df}}\ \text{ IF }(x_{\text{i}}\ \text{is}\ X_{\text{1}})\text{o} \ldots\text{o}(\text{x}_{\text{i}}\ \text{is}\ X_{\text{i}})\ \text{THEN f}(x_{\text{1}},\ldots x_{\text{i}})\]  
where \(X_{\text{i}}\) (i = 1,..., I) are type-1 fuzzy sets defined over corresponding domains. The operator 'o' is assumed as a \(\otimes\), \(\oplus\) : \([0,1]^{2}\rightarrow[0,1]\) are the t- and s-norms [33], respectively. These binary operators are applied in fuzzy logic as generalizations of the conjunction and disjunction Boolean logic operators. The Zadeh's t- and s- norms defined by the min and max operators, were applied, respectively. The rule consequents are multivariable  
Figure 1: An interval type-2 fuzzy set \(\widetilde{\text{F}}\) determined by membership functions of the type-1 fuzzy sets: \(\widetilde{\text{E}}\) and \(\widetilde{\text{F}}\). functions, defined as combinations of polynomials. The most popular solution, i.e., first-order Takagi-Sugeno model, assumes combinations of linear functions. Therefore, any rule consequent takes the form:  
\[\text{Consequent: f}(x_{1},\ \ldots,x_{i})=_{\text{df}}\sum_{k=1}^{i}\left(a_{k}x_{k}+b_{ k}\right)=\sum_{k=1}^{i}a_{k}x_{k}+\sum_{k=1}^{i}b_{k},\]  
where \(a_{k}\) and \(b_{k}\) are the parameters of the corresponding linear functions, defined with respect to each \(x_{i}\). It is interesting to note that in the Takagi-Sugeno model, there is no need to define the consequents as fuzzy sets and therefore, there is no need to define any membership function in the consequents as it should be done using, for example, the Mamdani model [30]. The final output of the system is the weighted over all rule outputs, as defined in Equation (6):  
\[\text{Final Output}=\ \frac{\sum_{k=1}^{N}w_{k}f_{k}}{\sum_{k=1}^{N}w_{k}}, \tag{6}\]  
where N is the number of rules, \(w_{k}\) is the rule-firing value derived from the \(k^{th}\) rule antecedent and \(f_{k}\) is the value of the kth rule consequent. Considering type-2 fuzzy sets, for an input vector \(\overline{x}{=\{x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{i}^{\prime}\}}\), typical computations of an \(\Pi\Pi\) fuzzy system consist of the following steps:  
1. Compute the membership intervals of \(x_{i}^{\prime}\) for each \(\widetilde{x_{i}^{\prime}}\), \(\left[\mu_{\widetilde{x_{i}^{\prime}}}(x_{i}^{\prime}),\mu_{\widetilde{x_{i}^ {\prime}}}(x_{i}^{\prime})\right],\) i = 1,..., I; \(n=1,\)..., N (N is the number of rules),
2. Compute the firing interval of the \(n^{th}\) rule: \[F^{n}\left(\widetilde{x}\right)=_{\text{df}}\left[\mu_{\widetilde{x_{i}^{ \prime}}}(x_{i}^{\prime})o\ldots o\ \mu_{\widetilde{x_{i}^{\prime}}}(x_{i}^{\prime}),\mu_{\widetilde{x_{i}^{ \prime}}}(x_{i}^{\prime})o\ldots o\ \mu_{\widetilde{x_{i}^{\prime}}}(x_{i}^{\prime})\right],\]
3. Use type reduction to combine \(F^{n}\left(\widetilde{x}\right)\) with corresponding rule consequents. The most popular one is the center-of-set (COS) type reducer [34] using the Karnik-Mendel algorithm [34; 35] or their modifications [36; 37]. In order to apply the Karnik-Mendel algorithm, a single type-2 fuzzy set must be derived using the rules outputs. For a Takagi-Sugeno type-2 fuzzy system, the aggregate set is generated using the following procedure:  
1. Sort the rule outputs from all rules into ascending order,
2. For each output, define the UMF value using the maximum firing range of the considered rule,
3. For each output, define the LMF value using the minimum firing range of the considered rule. As an example, consider the data shown in Table 2. Let us assume five rules with output values and firing range limits.