## 7 Discussion
### Practical Contributions  
The proposed multidimensional Takagi-Sugeno (T-S) identification technique offers several practical applications and potential impacts across various fields. Below are some of the key contributions and benefits of implementing this method:  
* Enhanced System Modeling Accuracy
* **Application in Control Engineering:*
* The improved accuracy in system identification can lead to more precise control strategies in industrial automation and robotics. For instance, in the control of multivariable thermal processes or distillation columns, accurate modeling ensures better performance and stability of the control systems. * This results in increased efficiency, reduced operational costs, and enhanced safety in industrial operations. * Computational Efficiency
* **Real-Time Applications:*
* The proposed method's lower computational cost, compared to traditional Gaussian radial basis functions, makes it suitable for real-time applications. For example, in adaptive control systems or online fault
detection, where quick response times are crucial, the proposed method can significantly improve performance. * **Benefit:** This allows for faster system responses and real-time adjustments, improving the overall system reliability and robustness. * Robustness in Nonlinear System Identification
* **Adaptive Systems:*
* The method's robustness in identifying nonlinear systems makes it ideal for adaptive control systems in aerospace, automotive, and consumer electronics, where systems often encounter varying operational conditions. * Enhances the adaptability and resilience of systems to changing environments and unforeseen disturbances. * Reduced Identification Errors
* **Process Industries:*
* In industries such as chemical processing, where the precise control of process variables is essential, the reduction in identification errors can lead to higher quality products and less waste. * Contributes to sustainability and cost savings by optimizing resource use and minimizing errors in production processes. * Improved Control Performance
* **HVAC Systems:*
* The application of this method in heating, ventilation, and air conditioning (HVAC) systems can improve energy efficiency and occupant comfort by providing more accurate system models for control design. * Leads to significant energy savings and improved environmental conditions within buildings. * Versatility and Applicability
* **Multivariable Systems:*
* The method's ability to handle multivariable systems makes it applicable in a wide range of fields including biomedical engineering (e.g., modeling physiological processes), finance (e.g., economic forecasting), and environmental systems (e.g., climate modeling). * Its versatility ensures broad applicability, allowing for improvements in various complex systems that require accurate and efficient modeling. However, despite the advantages of the proposed method, there are also limitations, which are discussed below:  
Limitations  
While the proposed method for the multidimensional identification of nonlinear multivariable systems offers significant improvements in reducing identification errors and computational efficiency, there are certain limitations that must be considered. * Complexity of membership function design:
* The design of multidimensional membership functions (MDMFs) involves complex mathematical computations, such as the calculation of the inertia matrix and eigenvectors. This can be computationally intensive, particularly for systems with a large number of variables or high-dimensional datasets. * Applicability to highly dynamic systems
* The proposed method may not perform as well in systems with highly dynamic or rapidly changing behaviors. In such cases, the static nature of the calculated membership functions might not capture the system dynamics accurately, leading to suboptimal results. * Dependency on point cloud data
* The accuracy of the method is highly dependent on the quality and representativeness of the point cloud data used to design the membership functions.