## 1 Introduction
### Multidimensional Membership Functions, Multivariable Modeling and Identification  
The design of membership functions is the most crucial step in T-S identification for multivariable nonlinear systems. In [4; 5; 27; 28], membership functions are designed using histograms, but this method can place fuzzy rules in inappropriate locations, whereas multidimensional membership functions are designed using point mapping, thus reducing identification errors. Multidimensional membership functions are designed in [29] using normalized histograms. The transformation from probability to possibility is resolved using two methods: the bijective transformation method [30] and the uncertainty conservation method [31]. This method assumes that the degree of membership is the same as the frequency of occurrence. A bidimensional synthetic Gaussian distribution is used for the probability-to-possibility transformation. In [32], multilayer feedforward neural networks are used to generate membership functions from labeled training data. One disadvantage of this method is that the shape of the membership function is unpredictable in regions where there are no training data. Membership functions are defined by linearly interpolating the membership degrees of characteristic points [33]. To achieve efficient interpolation, Delaunay triangulation of characteristic points is used, and interpolation is performed using barycentric coordinates. Composite and non-composite multidimensional fuzzy sets are proposed in [34]. A composite multidimensional fuzzy set is typically the result of combining two one-dimensional membership functions through union or intersection. Optimization techniques are also introduced to fine-tune parameterized membership functions for improved performance. In [35], multidimensional membership functions are presented in T-S fuzzy models for the modeling and identification of multivariable nonlinear systems using a genetic algorithm. This allows parameter convergence within a reasonable computation time and reduces identification errors with a smaller number of fuzzy rules. In [36], the enhancement of fuzzy modeling by directly utilizing multidimensional fuzzy membership functions to model complex, nonlinear systems is addressed. The authors demonstrate that this approach reduces the decomposition errors typically introduced by conventional methods, thereby improving model accuracy and performance. They employ radial basis function (RBF) networks to model the membership functions, showing significant improvements in transparency and interpretability. A novel method for identifying nonlinear systems using kernel functions is presented in [37]. The approach combines hierarchical identification principles with stochastic gradient algorithms, leading to improved parameter estimation accuracy. The study demonstrates that the proposed kernel functions, based on Gaussian membership functions, effectively handle the fuzzification and defuzzification processes, enhancing the overall performance of fuzzy logic control systems. The use of multidimensional membership functions in the fuzzy modeling of nonlinear systems is investigated in [38]. The authors propose a novel approach that maintains high model accuracy while reducing computational complexity. The results demonstrate significant improvements in model performance, particularly in handling complex nonlinear dynamics. An enhanced control strategy for nonlinear systems utilizing multidimensional membership functions is proposed in [39]. The authors show that this method significantly improves control accuracy and robustness by providing a more precise representation of the system's dynamics. The study highlights the practical applications of this approach in various engineering fields. The application of multidimensional fuzzy membership functions in the identification of complex systems is explored in [40]. It is shown that this approach provides a more accurate and reliable model by capturing the intricate relationships within the data.