## 5 Illustrative Examples
### Multidimensional T-S Identification Using Five Rules
#### 5.4.1 Multivariable Thermal Mixing Process  
In Figure 26a, the assignment of the central point of the central rule is obtained, and the inertia axes are shown in Figure 26b.  
In Figures 27 and 28, the subdivision of the point map into four zones delimited by the inertia axes is carried out.  
Figure 26: Centers of gravity and level curves. (**a**). Center of gravity of the central rule. (**b**) Principal axes of inertia.  
Figure 25: Central points of fuzzy rules using 1 DMF fuzzy inference of binary distillation column \(x_{\text{D}}\).  
Then, the central points of the four zones are obtained (see Figure 29).  
Figure 28: Case (a): zones 3–4 of the subdivision of the data points map.  
Figure 27: Case (a): zones 1–2 of the subdivision of the data points map.  
Figure 29: Subdivision of the data points map.  
The centers of gravity of the four regions formed by the intersection of the inertia axes shown in Figure 30 are considered.  
The centers of gravity of each zone are obtained (see Figure 31).  
Various tests are conducted by modifying the formula assigning central points to the new rules (see Table 3).  
Figure 31: Centers of gravity of the 4 intersecting regions.  
Figure 30: Four regions—intersection.  
In Figure 32b, contour curves are generated. It can be observed that there are zones that depend on a single rule and other zones that are outside the influence area of the rules.  
The multidimensional membership functions of the five rules are observed according to Figure 33.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(Q_{m}\) & \(T_{s}\) & **Equation** & **Subdivision Point Map** \\ \hline (a) & \(r_{i}=c_{i}\) & (25) & Figure 29 \\ (b) & \(r_{i}=2c_{i}-r_{1}\) & (26) & Figure 31 \\ (c) & \(r_{i}=c_{i}\) & (25) & Figure 31 \\ (d) & \(r_{i}=2c_{i}-r_{1}\) & (26) & Figure 31 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Design criteria of the multivariable thermal mixing process.  
Figure 32: Case (a): centers of gravity and level curves. (**a**) Centers of gravity of the rules. (**b**) Level curves of the 5 rules.  
Figure 33: Case (a): obtained membership functions.  
Using for \(T_{s}\) and \(Q_{m}\longrightarrow r_{i}=2c_{i}-r_{1}\)  
In this way, it is proposed to work with one central rule and four lateral rules located in the described zones. The centers of gravity of each rule are obtained in Figure 34, which are calculated according to (26).  
The contour plots of the five rules are generated as shown in Figure 34b. In this figure, it can be observed that there are no longer points outside the influence area of the rules and that the majority of points are within the area of at least two rules.  
Finally, the multidimensional membership functions of the five rules are observed according to Figure 35.  
2. Using for \(Q_{m}\) and \(T_{s}\longrightarrow r_{i}=c_{i}\)  
Using the design criterion of (25), the centers of gravity and the level curves of the five rules are obtained as shown in Figure 36.  
In Figure 37, the multidimensional membership functions are shown.  
Figure 34: Case (b): centers of gravity and level curves. (**a**) Centers of gravity of the rules. (**b**) Level curves of the 5 rules.  
Figure 35: Case (b): multidimensional membership functions designed.  
3. Using for \(Q_{m}\) and \(T_{s}\longrightarrow r_{i}=2c_{i}-r_{1}\) In this test, the design criterion of (26) is considered for both the flow rate and temperature. Thus, the centers of gravity of the five rules are obtained, as shown in Figure 38.  
Figure 38: Case (d): centers of gravity and level curves. (**a**) Centers of gravity of the rules. (**b**) Level curves of the 5 rules.  
Figure 37: Case (c): multidimensional membership functions.  
The multidimensional membership functions designed in this test are detailed in Figure 39.