## 2 Materials and Methods
### Design of ADRC
#### 2.4.2 Expanded State Observer  
The expanded state observer (ESO) serves as a pioneering state estimator in active disturbance rejection control systems, underpinned by its design philosophy of real-time estimation of comprehensive state information pertaining to the control object. This estimation encompasses the system's response to both internal dynamics and external perturbations. The ESO's introduction is particularly apt for control scenarios where model accuracy is constrained or system parameters undergo significant variations, a prime example being the hot-compression molding process of bamboo-based fiber composites [26]. In this process, temperature control holds a pivotal role in determining the overall product quality.  
Within the temperature control system for the hot-pressing molding of bamboo-based fiber composites, the total disturbance, denoted as d, is treated as an unknown state variable. Based on the existing state variables of the original system, a new state is introduced, as exemplified in Equation (15).  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}=\dot{T}\\ \dot{x}_{2}=d+bu\\ \dot{x}_{3}=\dot{d}\\ y=x_{1}\end{array}\right. \tag{15}\]  
To this expanded linear system, a nonlinear state observer is constructed, as presented in Equation (16), ensuring a robust and precise estimation of the system's dynamics.  
\[\left\{\begin{array}{l}\varepsilon_{1}=z_{1}-y\\ \dot{z}_{1}=z_{2}-\beta_{01}\varepsilon_{1}\\ \dot{z}_{2}=z_{3}-\beta_{02}fal(\varepsilon_{1},\alpha_{1},\delta)+b_{0}u\\ \dot{z}_{3}=-\beta_{03}fal(\varepsilon_{1},\alpha_{1},\delta)\end{array}\right. \tag{16}\]  
In the context of the equation given above, \(z_{1}\) and \(z_{2}\) represent the observed values of the state variables \(x_{1}\) and \(x_{2}\), while \(z_{3}\) denotes the observed value of the total system disturbance. The parameters \(\alpha_{1}\), \(\delta\), \(\beta_{01}\), \(\beta_{02}\), \(\beta_{03}\), and E serve as tunable coefficients within the framework. It is noteworthy that the parameters \(\alpha_{1}\) and \(\delta\) exhibit a mutual coupling effect on the overall system performance. The tuning process of \(\beta_{01}\), \(\beta_{02}\), and \(\beta_{03}\) can be effectively transformed into a bandwidth tuning problem, which involves adjusting the controller's bandwidth to achieve desirable dynamic performance. The nonlinear function, as represented in Equation (17), plays a crucial role in estimating the state variables and the total perturbation scenario in a real-time and rapid manner.  
\[fal(\varepsilon_{i},\alpha_{i},\delta)=\left\{\begin{array}{l}|\varepsilon_ {i}|^{\alpha_{1}}\text{sgn}(\varepsilon_{i}),|\varepsilon_{i}|>\delta\\ \frac{\varepsilon_{i}}{\delta^{1-\alpha_{1}}},|\varepsilon_{i}|\leq\delta, \delta>0\end{array}\right. \tag{17}\]  
Specifically, when the error is significant, a nonlinear feedback \(|\varepsilon_{1}|^{\alpha_{1}}\text{sgn}(\varepsilon_{i})\) is provided, whereas under small error conditions, a linear feedback \(\frac{\varepsilon_{i}}{\delta^{1-\alpha_{1}}}\) is employed. This strategy ensures that the error gradually decreases over time and ultimately converges to zero, thereby facilitating the convergence of the state observer.