# Multi-Span Tension Control for Printing Systems in Gravure Printed Electronic Equipment
## 1 Introduction
The web tension is not only related to the printing speed and roll diameter changes but also to the mechanical accuracy, materials, temperature, humidity, and other factors. With the increasing demand for printing accuracy, these influencing factors need to be considered when modeling tension systems. [PERSON] proposed a fast stabilization method for web tension in the drying process to reduce the time required for the stabilization of the tension [13]. [PERSON] conducted a dynamic study on the film and guide rollers in the film transport system of gravure printing [14]. He comprehensively considered the influence of temperature and established the most basic single-span tension model [15; 16]. Based on this, He explored the coupling relationship of multi-span tension systems [8]. In terms of tension control, employing the correct tension model and adopting appropriate control strategies can essentially achieve effective tension control. Simply relying on PID control in industrial applications proves difficult for accommodating the nonlinearity, strong coupling, and time-varying parameters characteristic of tension systems. Consequently, some scholars have employed PID control as the core and integrated it with other control algorithms to devise composite control methods specifically for tension control. [PERSON] designed a variable-gain PID to control the tension of the unwinding system [17]. Figure 1: Diagram of a roll-to-roll gravure printed electronic machine [8]. [PERSON] devised a feedforward control strategy to improve the control accuracy for known disturbances in the system [18]. He also designed and discussed two adaptive PI control schemes for the control of web tension in R2R manufacturing systems [19]. In recent years, with the improvements in device computing power, modern control methods, such as robust control, fuzzy control, and neural network control, have developed rapidly, and these control methods have also been studied and applied in tension systems. [PERSON] adopted the H\(\infty\) robust control algorithm for the tension control of the unwinding system, which has demonstrated a significant interference suppression effect in simulation results [20]. [PERSON] applied regularized variable learning-rate backpropagating artificial neural networks for the control of a roll-to-roll system [21]. [PERSON] proposed a fuzzy-logic-based hybrid approach specifically tailored for tension systems, enabling the automatic adjustment of control parameters for adapting to variations in system parameters and structural uncertainties, as reported in [22]. [PERSON] devised a backstepping control strategy specifically for a tension-control system, with simulation results demonstrating its capability of achieving high accuracy in tension control [23]. [PERSON] successfully achieved robust control of the unwinding tension subsystem through the innovative design of a disturbance observer, as described in [24]. He designed a high-precision reduced-order observer-based LQR control method for roll-to-roll systems [25]. [PERSON] introduced an advanced observer-based single-loop feedback system for roll-to-roll machine tension-control applications, and the simulation results numerically validated the feasibility of the proposed solution [26]. The modern control methods also have some problems in practical applications, such as fuzzy control requiring rich manual experience to formulate logical rules, robust control relying heavily on models, and neural network control requiring a lot of training. For systems with high complexity, the workload associated with implementing these modern control methods multiplies. Active disturbance rejection control (ADRC) technology has evolved from PID control and proposes several improvement methods to address the shortcomings of PID control. The main methods encompass: (1) Incorporating a tracking differentiator (TD) to arrange the transition process and extract error signals more effectively.