## 4 Application to RTCVD System
### SPSA Learning-Based Three-Dimensional Fuzzy Modeling
The seven three-dimensional fuzzy rules derived from the data are listed as follows:  
\(\overline{\mathbf{V}}_{1}:\) if \(w(\bar{F},k-1)\) is very Positive Large and \(q_{1}(k-1)\) is Positive Small and \(q_{2}(k-1)\) is Positive Large and \(q_{3}(k-1)\) is very Positive Large, then \(w(\bar{F},k)\) is [0.8147 0.9058 0.1270 0.9134 0.6324 0.0975 0.2785 0.5469 0.9575 0.9649 0.1576]. \(\overline{\mathbf{V}}_{2}:\) if \(w(F,k-1)\) is more than Zero and \(q_{1}(k-1)\) is less than Positive Medium and \(q_{2}(k-1)\) is Positive Small and \(q_{2}(k-1)\) is Positive Small and \(q_{1}(k-1)\) is less than Positive Medium and \(q_{2}(k-1)\) is more than Zero and \(q_{3}(k-1)\) is very Positive Large, then \(w(\bar{F},k)\) is [12.7369 \(-\)14.0099 \(-\)10.1810 \(-\)11.3661 \(-\)11.1470 \(-\)5.8834 \(-\)9.8322 \(-\)2.5678 \(-\)10.5907 \(-\)0.4775 \(-\)4.1538]. \(\overline{\mathbf{V}}_{4}:\) if \(w(\bar{F},k-1)\) is very Positive Large and \(q_{1}(k-1)\) is Positive Large and \(q_{2}(k-1)\) is slightly Positive Small and \(q_{3}(k-1)\) is Positive Large, then \(w(\bar{F},k)\) is [0.4617 0.9713 8.2346 6.9483 3.130 9.5022 0.3445 4.3874 3.8156 7.6552 7.9520]. \(\overline{\mathbf{V}}_{5}:\) if \(w(\bar{F},k-1)\) is more than Positive Medium and \(q_{1}(k-1)\) is Positive Large and \(q_{2}(k-1)\) is more than Zero and \(q_{3}(k-1)\) is Positive Medium, then \(w(\bar{F},k)\) is [1.8687 4.8976 4.4559 6.4631 7.0936 7.5469 2.7603 6.7970 6.5510 1.6261 1.1900]. \(\overline{\mathbf{V}}_{6}:\) if \(w(\bar{F},k-1)\) is more than Positive Medium and \(q_{1}(k-1)\) is Positive Medium and \(q_{2}(k-1)\) is slightly Positive Small and \(q_{3}(k-1)\) is very Positive Large, then \(w(\bar{F},k)\) is [\(-2.4918 -4.7987 -1.7019 -2.9263 -1.1191 -3.7563 -1.2755 -2.5298 -2.5673 -2.3571 \(-\)3.125]. \(\overline{\mathbf{V}}_{7}:\) if \(w(\bar{F},k-1)\) is Positive Large and \(q_{1}(k-1)\) is Positive Small and \(q_{2}(k-1)\) is Positive Small and \(q_{3}(k-1)\) is less than Positive Medium, then \(w(\bar{F},k)\) is [\(-1.12263\) \(-0.4110\) \(-1.1889 -0.7443 -1.0026 -0.4407 -1.3535 -1.6484 -2.6840 -1.4376 -0.6583]. Finally, the complete three-dimensional fuzzy modeling of RTCVD is achieved. The abbreviation for the suggested three-dimensional modeling technique, which combines AP clustering with Fourier space base functions, is AP-Fourier-SPSA-3D fuzzy modeling. The simulation results of AP-Fourier-SPSA-3D are shown in the following figures. The predicted result of the model in the training data is displayed in Figure 7, while the discrepancies between the expected and actual values are illustrated in Figure 8. Figure 9 displays the anticipated outcome of the procedure in the test data, whereas Figure 10 illustrates the flaws in the predictions. The root mean square error (RMSE) performance indices are shown in Table 1.  
Figure 6: Seven space base functions learned by the SPSA algorithm. Figure 7: The predicted process output of the AP-Fourier-SPSA-3D model in the training data. Figure 8: The predicted errors of the AP-Fourier-SPSA-3D model in training data. Figure 9: The predicted process output of the AP-Fourier-SPSA-3D model in test data.