## 7 Discussion
### Theoretical Contributions  
The following are the theoretical contributions of the proposed method:  
* Introduction of Multidimensional Membership Functions (MDMFs)
* The proposed method introduces a novel approach to design multidimensional membership functions using the inertia matrix derived from the system's point cloud data. This approach ensures that the membership functions are well positioned within the system's operational range, reducing the likelihood of generating unnecessary rules and enhancing the accuracy of the model. The use of inertia axes to subdivide the space into regions of equal \"weight\" and assigning membership functions to the centers of gravity of these regions are significant theoretical advancements. * By employing multidimensional membership functions, the proposed method significantly reduces identification errors compared to traditional one-dimensional membership functions. This is achieved through a more accurate representation of the system's behavior, capturing the complexities of multivariable nonlinear systems more effectively. The proposed method demonstrates a notable reduction in identification errors for various test cases, including the multivariable thermal mixing process and the binary distillation column. * Efficiency in Computational Complexity
* The proposed method offers a computationally efficient solution for system identification. The design of the membership functions using moments of inertia and the subsequent optimization process ensures that the computational cost remains low, which is crucial for real-time applications. The comparative analysis shows that the multidimensional identification algorithm has a shorter execution time than the traditional T-S identification algorithm, making it suitable for online control applications. * The multidimensional membership functions designed using the proposed method exhibit robust performance across different scenarios. The ability to generalize well in dense input regions enhances the control performance and robustness of the fuzzy models. This generalization capability is particularly beneficial for handling complex nonlinear dynamics in various engineering applications. \begin{table}
\begin{tabular}{c c c} \hline
**Method/Reference** & **Processing Time (\%)** & **Computational Resources Required** \\ \hline
**Current Proposal (Multidimensional T-S)** & 0.426 & Lower, which is due to the simplicity of multidimensional membership functions \\ UKF and Dual Estimation [3] & 0.65 & High, which is due to intensive matrix \\  & & calculations and nonlinear transformations \\ Quadratic Cost Minimization [4] & 0.60 & Medium, which is due to the need for quadratic \\  & & optimization \\ Parameter Weighting [5] & 0.55 & Medium, since weighting requires additional but \\  & & not excessive calculations \\ Evolving T-S Models [7] & 0.58 & Medium, as there is efficiency in input space \\  & & decomposition but requires fine-tuning \\ State Subspace [8] & 0.67 & High, since subspace techniques are complex \\  & & and computationally costly \\ \hline \end{tabular}
\end{table} 
Table 11: Processing time and computational resources. * The integration of the proposed multidimensional membership functions with the Takagi-Sugeno (T-S) identification framework is a key theoretical contribution. This integration leverages the strengths of both approaches, combining the interpretability and adaptability of T-S models with the precision and efficiency of the proposed membership functions. The resulting hybrid model provides a powerful tool for identifying and controlling nonlinear multivariable systems. * Improvement in Control Precision
* The proposed method's ability to reduce decomposition errors typically introduced by conventional methods leads to improved model accuracy and control precision. This is particularly important for applications requiring high precision and reliability, such as thermal mixing processes and distillation columns.