# A Parameter Self-Tuning Decoupling Controller Based on an Improved ADRC for Tension Systems
## 3 Decoupled Controller Design for Global Tension System  
### Tension System Coupled Model Decoupling Control  
The tension system global coupling model of a roll-to-roll coating machine is a typical multi-input and output system. To transform it into a single-input single-output system, virtual control quantities \(\mathbf{u}(t)\) are introduced, followed by the use of the static decoupling model to determine the actual control quantities \(\mathbf{\omega}(t)\), resulting in the decoupling control of the tension system. Equation (1) is written in the form of a matrix:  
\[\frac{\text{d}\mathbf{T}(t)}{\text{d}t}=\mathbf{f}(t)+\mathbf{B}(t)\mathbf{\omega}(t) \tag{3}\] \begin{table}
\begin{tabular}{l l} \hline
**Parameters** & **Meaning** \\ \hline \(l_{D}\) & The equivalent rotational inertia of the pendulum arm \\ \(F_{P}\) & The cylinder thrust \\ \(B\) & The friction coefficient \\ \(r_{1},r_{2}\) & The size of the radius of the guiding roller in the dancer roll mechanism \\ \(L_{1}^{*}\) & The length of substrate in the unwinding unit in the initial steady state \\ \(L_{5}^{*}\) & The length of substrate in the rewinding unit in the initial steady state \\ \(l_{1}\) & Distance between the center of rotation of the dance roll and the connection point \\ \(l_{2}\) & The length of the pendulum rod \\ \(k\) & Spring force coefficient of the spring in the cylinder \\ \hline \end{tabular}
\end{table} 
Table 2: Meaning of the parameters of the auxiliary equation of tension. Figure 2: The structure of dancer roll mechanism. Among them:  
\[\mathbf{T}(t)=\left[ \begin{array}{cccc}T_{1}(t)&T_{2}(t)&T_{3}(t)&T_{4}(t)&T_{5}(t) \end{array} \right]^{T} \tag{4}\]  
\[\mathbf{\omega}(t)=\left[ \begin{array}{cccc}\omega_{0}(t)&\omega_{1}(t)&\omega_{2 }(t)&\omega_{4}(t)&\omega_{5}(t)\end{array} \right]^{T} \tag{5}\]  
\[f(t)=\left[ \begin{array}{c}f_{1}(t)\\ f_{2}(t)\\ f_{3}(t)\\ f_{4}(t)\\ f_{5}(t)\end{array} \right]=\left[ \begin{array}{c}\frac{[AE-T_{1}(t)]}{L_{1}( t)}\frac{\mathrm{d}L_{1}(t)}{\mathrm{d}t}\\ 0\\ \frac{[AE-T_{2}(t)]}{L_{4}(t)}\frac{\mathrm{d}L_{4}(t)}{\mathrm{d}t}-\frac{[ AE-T_{2}(t)]R_{3}\omega_{3}}{L_{4}(t)}\\ \frac{[AE-T_{5}(t)]}{L_{5}(t)}\frac{\mathrm{d}L_{5}(t)}{\mathrm{d}t}\end{array} \right] \tag{6}\]  
\[\mathbf{B}(t)=\left[ \begin{array}{cccc}-\frac{[AE-T_{s}(t)]R_{0}}{L_{1}(t)}& \frac{[AE-T_{1}(t)]R_{1}}{L_{1}(t)}&0&0&0\\ 0&-\frac{[AE-T_{1}(t)]R_{1}}{L_{2}}&\frac{[AE-T_{2}(t)]R_{2}}{L_{2}}&0&0\\ 0&0&-\frac{[AE-T_{2}(t)]R_{2}}{L_{3}}&0&0\\ 0&0&0&\frac{[AE-T_{4}(t)]R_{4}}{L_{4}(t)}&0\\ 0&0&0&-\frac{[AE-T_{4}(t)]R_{4}}{L_{5}(t)}&\frac{[AE-T_{5}(t)]R_{5}(t)}{L_{5}( t)}\end{array} \right] \tag{7}\]  
\(\mathbf{T}(t)\), \(\mathbf{\omega}(t)\) is the tension system output and input, \(\mathbf{f}(t)\) is the tension system control quantities of the outer part, known as the \"dynamic coupling\" part; \(\mathbf{B}(t)\mathbf{\omega}(t)\) is the \"static coupling\" part of the system. Introducing the virtual control quantities \(\mathbf{u}(t)=[u_{1}(t)u_{2}(t)u_{3}(t)u_{4}(t)u_{5}(t)]^{T}\), so that \(\mathbf{u}(t)=\mathbf{B}(t)\mathbf{\omega}(t)\), then Equation (3) becomes:  
\[\frac{\mathrm{d}\mathbf{T}(t)}{\mathrm{d}t}=\mathbf{f}(t)+\mathbf{u}(t) \tag{8}\]  
From Equation (8), \(\mathbf{u}(t)\) and \(\mathrm{d}\mathbf{T}(t)/\mathrm{d}t\) form a single-input single-output system to realize the tension system decoupling. \(\mathbf{f}(t)\) is the sum of perturbations of each unit when applying the ADCR controller to control substrate tension, the disturbance sum \(\mathbf{f}(t)\) can be estimated and compensated, while the virtual control quantities \(\mathbf{u}(t)\) can be obtained. Then, the actual input quantities \(\mathbf{\omega}(t)\) can be obtained using Equation (9), which realizes decoupling control of the tension system. \[\mathbf{\omega}(t)=\mathbf{B}^{-1}(t)\mathbf{u}(t) \tag{9}\]  
\[|\mathbf{B}(t)|=-\frac{[AE-T_{s}(t)][AE-T_{1}(t)][AE-T_{3}(t)][AE-T_{4}(t)][AE-T_{ 5}(t)]R_{0}(t)R_{1}R_{2}R_{4}R_{5}(t)}{L_{1}(t)L_{2}L_{k}L_{4}(t)L_{5}(t)} \tag{10}\]  
\[\mathbf{B}^{-1}(t)=\left[