# A Parameter Self-Tuning Decoupling Controller Based on an Improved ADRC for Tension Systems
## 2 Modeling and Analysis of Tension Systems for Roll-to-Roll Coating Machines  
Figure 1 illustrates the structure of the tension system of a roll-to-roll coating machine, which mainly consists of three major subsystems: the unwinding tension subsystem, the coating tension subsystem, and the rewinding tension subsystem. Among these, the unwinding tension subsystem consists of an unwinding and an unwinding infeed unit, the coating tension subsystem consists of a coating unit, and the rewinding tension subsystem consists of a rewinding and a rewinding outfeed unit. Additionally, there is the dancer roll mechanism set up in the unwinding, rewinding outfeed, and the rewinding unit, the tension sensor set up in the unwinding outfeed and the coating unit, and an oven set up in the coating unit. According to the literature [16], combined with the specific structure of the tension system in Figure 1, the global tension system coupling model of the roll-to-roll coater is established as follows in Equation (1), and the meanings of the parameters in Equation (1) are shown in Table 1.  
\[\left\{ \begin{array}{l}\left[\begin{array}{c}\frac{\mathrm{d}T_{1}(t)}{ \mathrm{d}t}\\ \frac{\mathrm{d}T_{2}(t)}{\mathrm{d}t}\\ \frac{\mathrm{d}T_{3}(t)}{\mathrm{d}t}\\ \frac{\mathrm{d}T_{4}(t)}{\mathrm{d}t}\\ \frac{\mathrm{d}T_{5}(t)}{\mathrm{d}t}\end{array}\right]=\left[\begin{array}[ ]{ccc}\frac{[AE-T_{1}(t)]}{L_{1}(t)}\frac{\mathrm{d}L_{1}(t)}{\mathrm{d}t}\\ 0\\ \frac{[AE-T_{2}(t)]}{L_{3}(t)}\frac{[AE-T_{4}(t)]}{\mathrm{d}t}-\frac{[AE-T_{2} (t)]R_{3}\omega_{3}}{L_{4}(t)}\\ \frac{[AE-T_{5}(t)]}{L_{5}(t)}\frac{\mathrm{d}L_{5}(t)}{\mathrm{d}t}\end{array} \right]+\\ \left\{\begin{array}{ccc}-\frac{[AE-T_{5}(t)]R_{0}}{L_{1}(t)}&\frac{[AE-T_{ 1}(t)]R_{1}}{L_{1}(t)}&0&0&0\\ 0&-\frac{[AE-T_{2}(t)]R_{2}}{L_{2}}&0&0\\ 0&0&-\frac{[AE-T_{2}(t)]R_{2}}{L_{3}}&0&0\\ 0&0&0&\frac{[AE-T_{4}(t)]R_{4}}{L_{4}(t)}&0\\ 0&0&0&-\frac{[AE-T_{5}(t)]R_{4}}{L_{5}(t)}&\frac{[AE-T_{5}(t)]R_{5}(t)}{L_{5} (t)}\end{array}\right]\left[\begin{array}{c}\omega_{0}(t)\\ \omega_{1}(t)\\ \omega_{2}(t)\\ \omega_{4}(t)\\ \omega_{5}(t)\end{array}\right]\\ L_{3}=\left[L_{3}^{*}+(L_{c1}-1)\frac{E}{E_{c1}}+(L_{c2}-1)\frac{E}{E_{c2}} +\cdots+(L_{cn}-1)\frac{E}{E_{cn}}\right.\end{array}  \tag{1}\]  
The sensor should be placed close to the driving roll that controls the substrate tension in the unwinding infeed and coating unit. This ensures accurate measurement of tension [17]. The dancer roll mechanism is located in the unwinding, rewinding outfeed, and rewinding unit. It consists of the dancer roll, dancer rod, and cylinder that provides thrust to the dancer roll [18] and is shown in Figure 2. When the cylinder thrust torque and the substrate tension torque are in equilibrium, the dancer roll remains in a stable position. However, when the substrate tension changes, the dancer roll deflects and produces a deflection angle of \(\theta\). The substrate tension can be indirectly calculated through the dancer roll mechanism tension detection model. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Meaning** \\ \hline \(T_{1}\sim T_{5}\) & The tension of the substrate in each unit \\ \(L_{1}\sim L_{5}\) & The length of the substrate in each unit \\ \(\omega_{0}\sim\omega_{2}\) & The angular speed of each motor \\ \(R_{0}\sim R_{5}\) & The radius of the unwinding, drive rolls, and rewinding \\ \(L_{c1}\sim L_{cn}\) & Length of each section of the drying mechanism \\ \(E_{c1}\sim E_{cn}\) & [PERSON]’s modulus of elasticity of substrate at different temperatures \\ \(L_{3}^{*}\) & Length of the substrate without drying mechanism \\ \(A\) & The cross-sectional area of the substrate \\ \(E\) & [PERSON]’s modulus of elasticity of substrate \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Meaning of coupled model parameters. Figure 1: The roll-to-roll coating machine structure of the tension system. In addition, when the dancer roll mechanism deflects, the lengths of \(L_{1}\), \(L_{4}\), and \(L_{5}\) will also change.