## 4 Application to RTCVD System
### SPSA Learning-Based Three-Dimensional Fuzzy Modeling  
In order to obtain sufficient dynamic information of the system, interference signals with amplitudes not exceeding 10% are added to the manipulated input variables \(q_{1}\), \(q_{2}\), and \(q_{3}\), respectively. Thus, the three manipulated input variables affected by interference signals, namely, the excitation signals, can be given by the following Equation (19):  
\[ \begin{split} q_{1}(t)&=q_{1a}+\Delta_{1a}*q_{1a}* \text{rand}()*S_{r},\\ q_{2}(t)&=q_{2a}+\Delta_{2a}*q_{2a}*\text{rand}()* S_{r},\\ q_{3}(t)&=q_{3a}+\Delta_{3a}*q_{3a}*\text{rand}()* S_{r},\\ S_{r}&=\text{sign}(\text{rand}()-0.5).\end{split}  \tag{19}\]  
where \(q_{1a}\), \(q_{2a}\), and \(q_{3a}\) are steady-state inputs at a furnace temperature of 1000 K. \(\Delta_{1a}\), \(\Delta_{2a}\), and \(\Delta_{3a}\) are the disturbance amplitude of \(q_{1a}\), \(q_{2a}\) and \(q_{3a}\). In this study, \(q_{1a}\), \(q_{2a}\), and \(q_{3a}\) are 0.2028, 0.1008, and 0.2245, respectively, and \(\Delta_{1a}\), \(\Delta_{2a}\), and \(\Delta_{3a}\) are set to 10%. A sample interval of 0.5 s was established in this investigation. The entire simulation time period was 7000 s. Therefore, 14,000 samples were generated, in which 600 samples were randomly selected for training experiments, and then 300 samples were randomly selected for test experiments. In this study, for simplicity, \(J=1\) and \(K=1\) were chosen. Thus, the data set can be written as Equation (20):  
\[ \begin{split} S&=\Big{\{}\chi^{k},w_{f}^{k}\Big{\}} =\Big{\{}\Big{(}\chi_{f}^{k},\chi_{q}^{k}\Big{)},w_{f}^{k}\mid\chi_{f}^{k}\in R ^{11\times 1},\\ &\chi_{q}^{k}\in R^{1\times 3},w_{f}^{k}\in R^{11\times 1},k=1, \cdots,600\Big{\}}.\end{split}  \tag{20}\]  
where  
\[ \begin{split}\chi^{k}&=\Big{(}\chi_{f}^{k},\chi_{q}^{ k}\Big{)},\\ \chi_{f}^{k}&=[w(\bar{F},k-1)],\\ \chi_{q}^{k}&=[q_{1}(k-1),q_{2}(k-1),q_{3}(k-1)],\\ w_{f}^{k}&=w(\bar{F},k)=[w(f_{1},k),\cdots,w(f_{11},k )]^{T}.\end{split} \]  
For \(\chi^{k}\) in the data set \(S\), AP clustering is used. The AP method sets the coefficient of dampness \(\lambda\) to 0.9 and the reference degree \(P\) to 0.5. Following the process of clustering learning, a total of seven fuzzy rules and their accompanying clustering cores were successfully acquired. Equation (7) shows the breadth setting of the Gaussian spatial membership function, whereas Equation (8) shows the breadth setting of the conventional Gaussian membership function. This paper uses Fourier space base functions to represent the fuzzy rule resulting sets, and applies the SPSA method to improve the coefficients of Fourier space base functions. In the SPSA algorithm, the relevant parameter settings are as follows: \(\alpha=0.602\), \(\beta=0.101\), \(iter_{\max}=50\),000, \(A=5000\), \(d=0.003\), and \(b=0.015\). Based on the learning results of the AP algorithm, it is evident that there are a total of seven fuzzy rules, so the parameter dimension that needs to be adjusted is 28, and the optimization range control of the parameters is \([-10,10]\). The end point criterion is defined as either reaching the maximum number of iterations or when the change in iteration becomes less than 0.00001 for five consecutive occasions. The initial value of \(\hat{\theta}_{0}\) is obtained through the gradient descent algorithm as shown in Equation (21):  
\[ \begin{array}{cccc}[0.0376&29.4590&-0.0307&-1.1126\\ -0.0343&26.6246&-0.0037&0.7921\\ -0.0272&26.6246&-0.0464&0.1736\\ 0.0048&25.0165&0.0339&0.4068\\ 0.0007&18.4397&-0.0079&0.9764\\ 0.0016&18.4623&-0.0041&-0.0672\\ -0.0323&-8.0553&-0.0381&0.2669].\end{array}  \tag{21}\]  
By using the SPSA algorithm, the optimal parameters for the Fourier space base function of the fuzzy rule resulting set can be obtained. For the convenience of observation, Figure 6 shows seven space base functions.