## 1 Introduction
### Fuzzy T-S Identification and Modeling  
In the literature, various approaches have been proposed for T-S fuzzy model identification. For instance, ref. [3] presents two novel learning algorithms for online T-S fuzzy model identification based on the Unscented Kalman Filter (UKF) and the dual estimation concept. These algorithms provide a more applicable parameter identification method by using the unscented transform instead of linearization, resulting in a more accurate propagation of the mean and covariance in highly nonlinear systems. Another implementation of T-S fuzzy model identification is found in [4], where parameter estimation of the nonlinear system is achieved by minimizing a quadratic cost. The study demonstrates that the system exhibits stable behavior and good transient response by designing a generalized T-S identification method along with an optimal state controller and an observer in each fuzzy rule. Similarly, the work by [5] employs a parameter weighting approach to optimize both local and global approximation in a T-S fuzzy model identification method. This methodology is pertinent to our study because it demonstrates how parameter weighting can significantly enhance the dynamic response and robustness of the system, ensuring zero steady-state error even under disturbances and modeling errors. Advanced methodologies for the identification and optimal control of nonlinear systems using generalized Takagi-Sugeno (T-S) models are presented in [5; 6]. Both studies focus on precise parameter estimation by minimizing a quadratic performance index and employing parameter weighting techniques. These methods aim to optimize both local and global model approximation, resulting in systems that exhibit robust dynamic responses and effective damping. The application of evolving Takagi-Sugeno fuzzy models for nonlinear system identification is discussed in [7], showcasing the model's ability to decompose input space into fuzzy subspaces, enhancing identification accuracy and computational efficiency. In [8], three subspace state-space algorithms are tested to select the appropriate algorithm that faithfully reproduces the dynamics of the real system with minimal estimation error. By combining T-S identification with enhancements in state-space techniques, a robust controller is achieved. Further advancements in T-S identification are illustrated in [9], where a tool is proposed for constructing T-S systems based on fuzzy c-regression models (FCRMs) and fuzzy clustering Gath-Geva. This method focuses on data clustering, particularly useful when domain expert data contain many errors, to construct fuzzy systems from experimental data. A novel data-driven methodology for constructing Takagi-Sugeno fuzzy models from experimental data is presented in [10], demonstrating the model's effectiveness in approximating unknown nonlinear systems with fewer linearizations and higher control precision. Additionally, ref. [11] proposes a modified Gath-Geva fuzzy clustering algorithm for T-S model identification, optimizing model construction and achieving higher accuracy for nonlinear system identification. The use of fuzzy clustering techniques for system identification based on T-S fuzzy inference introduces innovative identification techniques. For example, ref. [12] defines a unidimensional Gaussian function as the antecedent of the rule, which showed better performance in all experiments and is attractive for areas such as data prediction. Another instance is the study presented in [13], which offers an extensive overview of evolving fuzzy and neuro-fuzzy methodologies in clustering, regression, identification, and classification. The researchers explore how these approaches can adapt and learn in real-time settings, progressively enhancing their understanding from incoming data. Notable applications in adaptive control and identification systems are discussed, illustrating the potential of these techniques to boost accuracy and efficiency across various practical scenarios.