## 2 Materials and Methods
### Design of ADRC
#### 2.4.1 Tracking Differentiator  
As an integral component of ADRC, the tracking differentiator enhances the system's responsiveness and stability by delicately smoothing and filtering noise from the input signals, providing the controller with denoised and continuously differentiable signals [25]. This mechanism not only optimizes the quality of the control inputs, mitigating the adverse effects of measurement errors and external disturbances, but it also furnishes crucial dynamic information for the design of control laws. In dynamic processing environments, the presence of the tracking differentiator significantly elevates the adaptability of the temperature control system, enabling it to promptly and accurately reflect the actual conditions during the hot-pressing process of bamboo-based fiber composites and achieve precise temperature regulation. Conventional differentiators typically approximate delay elements using first-order inertial links, yet they possess the limitation of amplifying random noise. Therefore, the classical differentiator was refined into a second-order nonlinear form, as depicted in Equation (10):  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-\mathrm{r}\;\mathrm{sgn}\Big{(}x_{1}-\mathrm{v}(t)+\frac{x_{2} \left|x_{2}\right|}{\mathrm{r}}\Big{)}\end{array} \right. \tag{10}\]  
\(\dot{x}_{1}=x_{2}\) signifies that the rate of change in the state quantity \(x_{1}\) is equivalent to the state quantity \(x_{2}\). In the context of a universal differentiator, \(x_{1}\) denotes the position, while \(x_{2}\) represents the velocity. The symbol \(\dot{x}_{2}\) denotes the rate of change in \(x_{2}\), expressed as a nonlinear function that depends on \(x_{1}\) and the control input. The term -rsgn(\(x_{1}-\mathrm{v}(t)\)) serves as feedback to the error signal (\(x_{1}-\mathrm{v}(t)\)), where \(r\) is the gain coefficient, also known as the speed factor, which governs the tracking speed. The tracking speed is directly proportional to the value of \(r\), resulting in an improved speed compared to the linear tracking differentiator. However, in cases where the input signal contains high-frequency noise, the output differential signal may still exhibit significant noise. The function sgn is utilized to determine the tracking speed based on the deviation from the target trajectory \(\mathrm{v}(t)\), thereby dictating the direction of the feedback. \(\frac{x_{2}\left|x_{2}\right|}{\mathrm{r}}\) introduces nonlinear damping to the state quantity \(x_{2}\). The fastest tracking algorithm is incorporated into the nonlinear tracking differentiator, and the discretization of the aforementioned equation is presented in Equation (11):  
\[\left\{ \begin{array}{l}x_{1}(k+1)=x_{1}(k)+\mathrm{h}x_{2}(k)\\ x_{2}(k+1)=x_{2}(k)+\mathrm{h}u,\left|u\right|\leq\mathrm{r}\end{array} \right. \tag{11}\]  
where \(h\) is the step size, and the expression for \(u\) is shown in Equation (12):  
\[\mathrm{u}=-r\;\mathrm{sgn}\bigg{(}x_{1}(k)-\mathrm{v}(k)+\frac{x_{2}(k)|x_{2 }(k)|}{2\mathrm{r}}\bigg{)} \tag{12}\]  
The introduction of the than function into the control law, as detailed in Equation (13), incorporates a crucial parameter \(a\), which serves as an accuracy factor that dictates the precision of the differential operation. Figure 4: Basic structure of ADRC. \[\left\{\begin{array}{l}d=rh\\ d_{0}=hd\\ a_{0}=\sqrt{d^{2}+8r|y|}\\ a=\left\{ \begin{array}{l}x_{2}+\frac{(a_{0}-d)}{2}\mathrm{sgn}(y),|y|>d_{0}\\ x_{2}+\frac{y}{h}\left|y\right|\leq d_{0}\\ \mathrm{fham}=-\left\{\begin{array}{l}r\mathrm{sgn}(a),|a|>d\\ r\mathrm{\hat{a}}_{d}^{\prime},|a|\leq d_{0}\end{array}\right.\end{array} \right. \tag{13}\]  
Then, the expression of the final tracking differentiator is shown in Equation (14):  
\[\left\{ \begin{array}{l}\mathrm{e}(k)=x_{1}(k)-v_{1}(k)\\ x_{1}(k+1)=x_{1}(k)+\mathrm{h}v_{2}(k)\\ x_{2}(k+1)=x_{2}(k)+\mathrm{h}\mathrm{fham}(\mathrm{e}(k),x_{2}(k),r,h_{0}) \end{array} \right.