# Multi-Span Tension Control for Printing Systems in Gravure Printed Electronic Equipment
## 2 Design of the Control Structure for the Printing Tension System
### Control Structure Design for the Printing Tension System  
Equation (1) represents the universal mathematical model of a multi-span printing tension system. For the convenience of research, the number of spans in this multi-span tension system is chosen to be 3. The control structure of the printing tension system, shown in Figure 3, depicts a three-span coupling system comprising four identical printing units. Adjacent spans are coupled through a shared printing unit and substrate.  
For the three-span printing tension system, the inputs are the web tension of the unwinding traction span \(T_{u2}\); the motor speeds of each printing unit are \(\omega_{p0}\), \(\omega_{p1}\), \(\omega_{p2}\), and \(\omega_{p3}\); and the outputs are the web tensions of each printing span \(T_{p1}\), \(T_{p2}\), and \(T_{p3}\). Due to the fact that \(T_{u2}\) is the tension output of the unwinding traction system, it can only be used as a reference or interference for the printing tension-control system. Moreover, the motor speed of printing unit 1, \(\omega_{p0}\), is the speed reference of the entire system. Therefore, the actual input control quantities of the printing tension system are the motor speeds \(\omega_{p1}\), \(\omega_{p2}\), and \(\omega_{p3}\), and the actual output quantities are the web tensions \(T_{p1}\), \(T_{p2}\), and \(T_{p3}\). The purpose of the printing tension system is to maintain a constant tension on the web of each span, making the web tension of each span a constant tension-control type.  
There are complex multi-span coupling relationships in the printing tension system. For adjacent spans, the web tension of the previous span and the rotational speed of the shared printing unit in the middle are coupled physical variables of the two spans. When designing a printing tension controller, it is necessary to decouple this coupling relationship. Since there are no time-varying parameters in the physical quantities of the printing tension system, the robustness performance requirements for the controller are not high. However, there are drying devices in various printing spans that can affect the tension of the web. The mathematical model of the tension system used in designing the tension controller should include the factor of web temperature changes. According to the mechanism of web tension generation in a single span, the web tension is very sensitive to the roller speed at the front and rear ends of the span, and small speed changes can cause significant tension changes. Therefore, when controlling the web tensions of the printing system, it should be  
Figure 3: Control structure of the printing tension system.  
noted that the changes in roller speed should not be too large; that is, when adjusting the rotation speeds of the printing units in a steady state, only minor adjustments can be made.  
Based on the characteristics of the printing tension system, a three-span tension-control structure was designed that comprises three controllers, each dedicated to one span, as shown in Figure 3. This three-span tension-control structure can be easily extended to multi-span printing tension systems. The input of the tension controller for printing span i is the set tension \(T_{rpi}\), and the web tension \(T_{pi}\) is controlled by the motor speed \(\omega_{pi}\). The actual web tension \(T_{pi}\) is fed back to the controller to form a closed-loop control system. At the same time, the web tension of the previous span and the motor speed \(\omega_{p(i-1)}\) of printing unit i - 1 are input as disturbances into the controller for calculation.  
The controllers for each span of the printing tension system employ active disturbance rejection control technology in conjunction with the static decoupling model of the printing tension system to achieve ADRC decoupling control of the multi-span printing tension system. This control structure effectively addresses control challenges such as decoupling and interference mitigation in the printing tension system.