## 1 Introduction  
The application of fuzzy logic [1] has significantly advanced artificial intelligence by incorporating human-like uncertainty into computational decision-making. This approach has been particularly integrated in the modeling and control of nonlinear systems with the Takagi-Sugeno (T-S) fuzzy model [2]. By decomposing a complex nonlinear system into a collection of linearized subsystems interconnected through fuzzy inference, the T-S model provides a structured framework for system identification and control. Recent studies [3, 4, 5] have explored and extended the capabilities of T-S fuzzy models in recent years. The approach developed in [6] can be considered as a generalized version of the T-S method. This technique is designed to be computationally efficient, realizing enhanced model accuracy on the basis of parameter weighting. In [7], the method was further developed to accommodate multivariable systems. Compared to the original T-S model [2], these approaches demonstrate high accuracy in modeling nonlinear systems. On the other hand, the authors of [8] introduced an enhanced T-S fuzzy modeling approach that uses multidimensional membership functions (MDMFs) to improve system identification. Their method eliminates a limitation of the conventional fuzzy inference method based on 1 DMFs that can lead to suboptimal placement of fuzzy rules when modeling nonlinear multivariable systems. In contrast, the adoption of MDMFs enhances system identification while requiring fewer fuzzy rules. A genetic algorithm (GA) is integrated to optimize the MDMFs in combination with the T-S approach for modeling and system identification. Experimental results demonstrate that this method achieves lower identification errors compared to the traditional T-S model while reducing the number of fuzzy rules. Despite advances in the use of MDMFs in fuzzy models, their application in nonlinear multivariable systems remains limited. One major limitation is the challenge of properly tuning membership functions in complex systems, which can lead to reduced accuracy of system identification and control. In this paper, we address this gap by proposing an approach based on MDMFs defined through fuzzy clustering algorithms. Other algorithms have been developed to facilitate this task, such as the one proposed in [9], which optimizes the placement of MDMFs using a new algorithm based on the inertia matrix. In [10], collaborative fuzzy clustering (CFC) was applied to the identification of multiple-input multiple-output (MIMO) systems. Different adjustment algorithms have been explored through several methods in order to determine the parameters of MDMFs (antecedents) and the T-S model (consequents). One such approach is the application of GAs [8,11,12]; in this approach, the GA is used for optimizing the T-S antecedents and the T-S identification method [2] is used to estimate the consequents. In addition to the optimization of MDMFs, fuzzy clustering algorithms have been developed to obtain fuzzy rules for the best separation of data in clustering problems. Examples of these algorithms are fuzzy C-means (FCM) [13, 14, 15], fuzzy possibilistic c-means [16], Gustafson-Kessel (GK) [17, 18, 19], Gath-Geva (GG) [20], and other approaches [21]. The K-means approach to clustering was initially proposed by [PERSON] in 1957 [22] as a technique for modulation via coded impulses. However, it was [PERSON] [23] who finished defining it and assigned the current name ten years later. The fuzzy K-means clustering algorithm was created by [PERSON] and his team [24], who developed it in 1984. In 1978, [PERSON] and [PERSON] [17] proposed a fuzzy clustering algorithm with fuzzy covariances. This algorithm can better estimate the precision of mixed densities. In 1989, [PERSON] and [PERSON] presented their own fuzzy clustering algorithm, which aims to perform fuzzy clustering without a priori knowledge of the number of clusters necessary to classify the set of samples.