## 2 Materials and Methods
### Design of ADRC
#### 2.4.3 Nonlinear State Error Feedback  
Nonlinear state error feedback assumes a pivotal role in managing systems with intricate dynamic characteristics. Specifically, nonlinear functions are leveraged to devise a feedback control law for state errors, aiming to bolster the system's robustness against a diverse array of operating conditions and perturbations [27]. The crux of this approach lies in the design of the nonlinear function, which must aptly capture the magnitude and direction of the error, offer suitable control effects, and adapt to the system's nonlinear characteristics across various operating regions [28]. The state error is expressed in Equation (18):  
\[ \begin{array}{l}\text{e}_{1}=\
u_{1}-\text{z}_{1}\\ \text{e}_{2}=\
u_{2}-\text{z}_{2}\end{array}  \tag{18}\]  
where \(e_{1}\) and \(e_{2}\) represent the state errors; \(\
u_{1}\) is the tracking signal for the reference input signal; \(\
u_{2}\) denotes the differential signal of the tracking signal; and \(z_{1}\) and \(z_{2}\) are the state outputs of the state observer. Subsequently, by combining \(e_{1}\) and \(\text{e}_{2}\), the expansion state quantities are compensated using an independent nonlinear method. This nonlinear combination is formulated in Equation (19), ensuring a precise and adaptive control response to the system's dynamic behavior. \[\text{u}_{0}=\beta_{1}fal(\text{e}_{1},\alpha_{1},\delta)+\beta_{2}fal(\text {e}_{2},\alpha_{2},\delta) \tag{19}\]  
The nonlinear function serves as the cornerstone of the nonlinear state error feedback mechanism [29]. It typically exhibits a segmented definition, wherein the gain of the control law is modulated in diverse manners based on the error magnitude, thereby enabling precise regulation of the \(\beta_{1}fal(\text{e}_{1},\alpha_{1},\delta)\) system's state. When the error surpasses a predetermined threshold \(\delta\), the fal() function manifests as a symbolic function of \(e_{1}\), and the control system incorporates a hard saturation feature to guarantee a robust response to significant deviations. Conversely, for errors below the threshold, the function assumes a form proportional to \(e_{1}\) and \(\alpha_{1}\), ensuring a smooth and continuous control action that avoids over-amplification in scenarios of minor errors. This approach leverages the mathematical advantage of \"large error, small gain; small error, large gain\", reducing the steady-state duration of high-frequency oscillations while exhibiting a simple structure, high control efficiency, and robust characteristics [30]. Within the given formulation, the parameters \(\alpha_{1}\), \(\delta\), \(\beta_{1}\), \(\beta_{2}\), \(\alpha_{2}\), and \(\delta\) require adjustment. These parameters exhibit a mutual coupling effect on the system's performance, and \(\beta_{1}\) and \(\beta_{2}\) can be fine-tuned using bandwidth-based methods. The nonlinear regulation mechanism assumes paramount importance in the temperature control of hot-pressing molding for bamboo-based fiber composites [31]. Specifically, the fal() function empowers the control system to provide apt control inputs under diverse operating conditions, encompassing rapid temperature gradient changes and external perturbations [32]. By optimizing the control strategy, the hot-pressing temperature can be stabilized within an optimal range, thereby ensuring the consistency and stability of the processing quality for bamboo-based fiber composites. The intermediate control law \(u_{0}\) and the extended state \(z_{3}\) collectively constitute the final control law, as depicted in Equation (20). \[\mathrm{u}=\frac{\mathrm{u}_{0}-\mathrm{z}_{3}}{\mathrm{b}_{0}} \tag{20}\]  
Equation (21) can be obtained from Equation (18):  
\[ \begin{array}{l}\dot{\mathrm{e}}_{1}=\mathrm{v}_{1}-\dot{\mathrm{z}}_{1}\\ \dot{\mathrm{e}}_{2}=\mathrm{v}_{2}-\dot{\mathrm{z}}_{2}\end{array}