## 2 Materials and Methods
### Modeling of Steam-Regulating Valve Opening Angle in Terms of Heat, Pressure, and Temperature  
The opening angle of the steam conditioning valve holds a direct influence on the flow rate of hot steam, subsequently impacting the temperature of the hot press via a heat transfer mechanism. The traditional generalized representation of the second-order transfer function pertaining to this phenomenon is presented in Equation (1):  
\[G(s)=\frac{K\bullet\omega_{n}^{2}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}} \tag{1}\]  
where \(\omega_{n}\) is the natural frequency; and \(\zeta\) is the damping ratio. Conventional second-order systems have demonstrated their ability to model the oscillatory and transitional characteristics of a system. Nevertheless, acknowledging the inherent time lag associated with temperature changes, this study introduces novel modifications to the transfer function by incorporating an additional time delay term and an oscillation term. These enhancements aim to provide a more accurate representation of the system dynamics. The refined transfer function, incorporating these modifications, is presented in Equation (2):  
\[G(s)=\frac{K_{p}}{(T_{2}s^{2}+T_{1}s+1)}e^{-T_{d}s} \tag{2}\]  
where \(K_{p}\) is the gain; \(T_{1}\) is the first-order time constant; \(T_{d}\) is the second-order time constant; and is the time delay. During the heat transfer process between steam and a hot platen, the temperature change lacks a definitive analytical relationship. Consequently, the numerical solution is typically obtained through the finite element method. Therefore, in this study, we aimed to establish a correlation between the opening angle of the steam regulator valve and the temperature of the hot platen through the application of a parameter identification method. The recursive least squares method represents an efficient algorithmic approach in the domain of dynamic system parameter identification. This method facilitates the continuous real-time updating and optimization of model parameters, given a known dynamic structure of the system [22]. The formula for recursive least squares parameter estimation is presented in Equation (3). \[\left\{ \begin{array}{l}\hat{\theta}_{N+1}=\hat{\theta}_{N}+P_{N+1}\varphi_{N +1}\big{[}y(N+1)-\varphi_{N+1}^{T}\hat{\theta}_{N}\big{]}\\ \gamma_{N+1}=\left(I+\varphi_{N+1}^{T}P_{N}\varphi_{N+1}\right)^{-1}\\ P_{N+1}=\left(I-K_{N+1}\varphi_{N+1}^{T}\right)P_{N}\\ K_{N+1}=\gamma_{N+1}P_{N}\varphi_{N+1}\end{array} \right. \tag{3}\]  
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; \(P\) is the bias term in the parameter estimates; and \(y(N)\) is the system output. Based on the aforementioned information, this study employed a recursive least squares method with a dynamic forgetting factor for the parameter identification of the model. The introduction of the dynamic forgetting factor enables the model to retain useful information while reducing the accumulation of errors caused by non-stationary behaviors, thus enhancing the model's adaptability and sensitivity to new data. The performance indicator is defined as shown in Equation (4). \[J=\sum_{k=1}^{N}\lambda^{N-k}\Big{[}y(k)-\varphi^{T}(k)\hat{\theta}\Big{]}^{2} \tag{4}\]  
By applying a time-varying coefficient to the data, the older data are weighted down, while the latest data are given more weight. Consequently, this approach is also referred to as the exponential forgetting algorithm. The recursive least squares parameter estimation method with a forgetting factor is presented in Equation (5). \[\left\{ \begin{array}{l}\hat{\theta}_{N+1}=\hat{\theta}_{N+1}\varphi_{N+1} \big{[}y(N+1)-\varphi_{N+1}^{T}\hat{\theta}_{N}\big{]}\\ \gamma_{N+1}=(\lambda^{2}+\varphi_{N+1}^{T}P_{N}\varphi_{N+1})^{-1}\\ P_{N+1}=\frac{1}{\lambda^{2}}(I-K_{N+1}\varphi_{N+1}^{T})P_{N}\\ K_{N+1}=\gamma_{N+1}P_{N}\varphi_{N+1}\end{array} \right.