# Multi-Span Tension Control for Printing Systems in Gravure Printed Electronic Equipment
## 3 Design of an ADRC Decoupling Controller for Multi-Span Printing  
### Establishing the Decoupling Model  
In the printing tension-control system depicted in Figure 3, \(\omega_{p1}\), \(\omega_{p2}\), and \(\omega_{p3}\) represent the system inputs, which are the motor speeds of the respective printing units. Correspondingly, \(T_{p1}\), \(T_{p2}\), and \(T_{p3}\) are the system outputs, indicating the web tensions across each printing span. By adjusting the motor speeds \(\omega_{p1}\), \(\omega_{p2}\), and \(\omega_{p3}\) of the printing units, the tensions \(T_{p1}\), \(T_{p2}\), and \(T_{p3}\) within each span are controlled. Furthermore, the actual tension values are monitored using tension sensors strategically placed within each span. The printing tension-control system is a multi-input multi-output coupled system that requires decoupling of the model. Its mathematical model can be further rewritten as Equation (2). \[\left\{\begin{array}{l}\frac{\mathrm{d}T_{pi}(t)}{\mathrm{d}t}=-\frac{E_{D}[ AE_{E}-T_{2i}(t)]R_{pi}\omega_{p0}(t)}{\mathrm{d}t}+\frac{E_{D}[AE_{E}-T_{ p1}(t)]R_{pi}}{\mathrm{d}tE_{D}+L_{Dpi}(E_{E}-E  
Introduce the virtual control variable \(\mathbf{u}_{p}(t)= \begin{bmatrix}u_{p1}(t)&u_{p2}(t)&u_{p3}(t)\end{bmatrix} ^{T}\), and let  
\[\mathbf{u}_{p}(t)=\mathbf{B}_{p}(t)\mathbf{\omega}_{p}(t) \tag{8}\]  
Equation (3) becomes  
\[\frac{\text{d}\mathbf{T}_{p}(t)}{\text{d}t}= \mathbf{f}_{p}(t)+\mathbf{u}_{p}(t) \tag{9}\]  
In this configuration, each individual channel of the virtual control variable \(\mathbf{u}_{p}(t)\) establishes a direct, single-input single-output relationship with its corresponding channel of tension output \(\mathbf{T}_{p}(t)\). Specifically, \(u_{p1}(t)\) directly controls \(T_{p1}(t)\), \(u_{p2}(t)\) controls \(T_{p2}(t)\), and \(u_{p3}(t)\) controls \(T_{p3}(t)\), thereby creating a precise one-to-one mapping and ensuring complete decoupling between the control and output channels. The elements \(f_{p1}(t)\), \(f_{p2}(t)\), and \(f_{p3}(t)\) of \(\mathbf{f}_{p}(t)\) can be interpreted as the cumulative \"total disturbance\" arising from known factors affecting each respective channel. Notably, based on Equation (6), it becomes evident that only the first printing span, i.e., span 1, experiences disturbances, whereas spans 2 and 3 remain undisturbed. \[\big{|}\mathbf{B}_{p}(t)\big{|}=\frac{E_{D}^{3}[A\mathbf{E}_{E}-T_{p1}(t)][A\mathbf{E}_{E }-T_{p2}(t)][A\mathbf{E}_{E}-T_{p3}(t)]R_{p1}R_{p2}R_{p3}}{[L_{p1}E_{D}+L_{Dp1}(E_ {E}-E_{D})][L_{p2}E_{D}+L_{Dp2}(E_{E}-E_{D})][L_{p3}E_{D}+L_{Dp3}(E_{E}-E_{D})]} \tag{10}\]  
Due to the fact that the product of the web's cross-sectional area and the Young's modulus of elasticity is much greater than the web tension for each printing span, \(\big{|}\mathbf{B}_{p}(t)\big{|}\
eq 0\), which means that \(\mathbf{B}_{p}^{-1}(t)\) exists. \[\mathbf{\omega}_{p}(t)=\mathbf{B}_{p}^{-1}(t)\mathbf{u}_{p}(t) \tag{11}\]  
where  
\[\mathbf{B}_{p}^{-1}(t)=\begin{bmatrix}\frac{L_{p1}E_{D}+L_{Dp1}(E_{E}-E_{D})}{E[A \mathbf{E}_{E}-T_{p1}(t)]R_{p1}}&0&0\\ \frac{L_{p1}E_{D}+L_{Dp1}(E_{EThe first-order ADRC controllers, ADRCp1, ADRCp2, and ADRCp3 have the same structure, which comprises a second-order extended state observer (ESO) and a first-order nonlinear state error feedback (NLSEF) component. The ESO is the core part of the ADRC, which can track the actual tension of the system and estimate the internal and external disturbances of the controlled system. Since the tensions \(T_{rp1}\), \(T_{rp2}\), and \(T_{rp3}\) are set by the program code, the first-order ADRC controllers eliminate the tracking differentiator (TD) components. In each of the designed first-order ADRC controllers, there is no error differential processing term; thus, the NLSEF only needs to proportionally adjust the error term, further reducing the computational load.