## 2 Materials and Methods
### Modeling of Steam-Regulating Valve Opening Angle in Terms of Heat, Pressure, and Temperature
\tag{5}\]  
where \(\hat{\theta}_{N+1}\) is the updated value of the parameter estimate; \(\hat{\theta}_{N}\) is the parameter estimate; \(\varphi_{N}\) is the information vector; \(\lambda\) is the forgetting factor; \(P\) is the bias term in the parameter estimates; and \(y(N)\) is the system output. The performance of the forgetting factor recursive least squares method hinges on the value of \(\lambda\). When \(\lambda\) is excessively large, it can enhance the algorithm's robustness to noise in observations, but it can compromise its tracking capability. Conversely, a too-small \(\lambda\) results in a faster tracking speed of the algorithm, yet it becomes more sensitive to noise, leading to a decrease in the accuracy of the parameter identification. Therefore, when designing the forgetting factor \(\lambda\), a balance must be struck between the algorithmic precision and its noise resilience. The aforementioned forgetting factor recursive least squares method typically assumes a constant value for \(\lambda\). However, if the forgetting factor \(\lambda\) is be adjusted in real time based on the proximity of the identification results to the actual system model, the algorithm will be less prone to impairment in terms of its identification accuracy and tracking ability caused by overly large or small \(\lambda\) values. This dynamic adjustment allows for the continuous optimization of the algorithm's overall performance, significantly enhancing its parameter identification capabilities. The model's residual at time \(k\) is given by Equation (6), as follows:  
\[\mathrm{e}(k)=y(k)-\varphi(k)^{\mathrm{T}}\hat{\theta}_{k-1} \tag{6}\]  
where \(y(k)\) is the system output value at moment \(k\), i.e., the measured value of the system; and \(y(k)-\varphi(k)^{\mathrm{T}}\hat{\theta}_{k-1}\) is the estimated value. The presence of residuals is primarily attributed to noise and identification errors, with noise having a relatively minor impact on them. When the error in the system parameter identification is significant, resulting in large absolute values of residuals, reducing the forgetting factor can enable the system's identification results to quickly converge to parameters close to the actual model, thereby narrowing the convergence range. At this point, with smaller absolute values of residuals, increasing the forgetting factor can suppress the influence of noise on the identification results, ultimately enhancing the overall performance of the algorithm in system identification. To this end, an evaluation function is adopted, as defined in Equation (7):  
\[J_{\mathrm{F}}=\frac{1}{\mathrm{M}}\sum_{k=\mathrm{N-M}}^{\mathrm{N}}| \mathrm{e}(k)| \tag{7}\]  
Since relying solely on the absolute value of a single residual to adjust the forgetting factor leads to low accuracy, the average of residuals over a certain period was chosen as a performance indicator for convergence to the actual parameter values during the identification process. However, this indicator exhibits an inverse correlation with the degree of correction applied to the forgetting factor, making it challenging to establish a mathematical expression between the two. Therefore, fuzzy control theory was introduced to design a fuzzy controller for online modification of the forgetting factor. The change rate after the Nth observation is given by Equation (8):  
\[\Delta J_{\text{F}}(\text{N})=\frac{1}{\text{D}}(J_{\text{F}}(\text{N})-J_{ \text{F}}(N-\text{D})) \tag{8}\]  
where \(D\) is the constant-time discrete value for calculating the rate of change in \(J_{\text{F}}\); and \(J_{\text{F}}\) and \(\Delta J_{\text{F}}\) are the inputs to the fuzzy controller. The correction forgetting factor is shown in Equation (9):  
\[\lambda=\lambda_{0}+\Delta\lambda \tag{9}\]  
where \(\lambda_{0}\) is the initial forgetting factor; \(\Delta\lambda\) is the output of the fuzzy controller; and \(\lambda\) is the updated value of the forgetting factor.