## 1 Introduction
In addition, research on the roll-to-roll printing process also refers to the control methods of pressure uniformization [28], adaptive feedforward control [29], multi-roll coordinating optimal control [30] and sensorless tension estimation and disturbance compensation [31]. Analysis of these research results shows that these register control methods mainly depend on the substrate tension coupling model and multi-axis synchronous control technology [32; 33]. The processing of decoupling calculations is complex and time-consuming. However, register error detection accuracy on the printed matter is basically achieved by detecting the register color mark, so the key point is understanding how to reliably locate the register color mark. Traditional control schemes usually use photoelectric sensors to detect the register color mark; however, detection results are easily affected by various factors, such as printing quality, color, tension fluctuation, mechanical vibration, mechanical and electrical parameters of photoelectric sensors, detection speed variation and so on. For the research issue of the register color-mark-detection mechanism, there are few research results. Some researchers have tried to solve printing control problems using machine vision technology, which has been widely used in the printing industry, especially in the application of online printing defect detection [34; 35]. In ref. [36], a machine-vision-based register control system was designed for use in chromatic printing machines. The detection algorithm obtains the registration color mark pattern through the CCD (charge-coupled device) and calculates the center of gravity of the color mark, which is compared with the standard position to show whether registration deviation exists or not. In ref. [37], a register error detection method based on the control strip was proposed to realize real-time and online detection of register errors. The translations of the solid block's edge from the reference block edge in the horizontal and vertical directions were calculated based on the characteristics of the control strip. Lastly, the register error of plates from the reference plate can be obtained. The authors of ref. [38] put forward a relatively effective and discriminative feature region searching algorithm that can automatically detect shapes, such as quasi-rectangular, oval, and so on, as feature regions. The entire image and sub-regional experimental results show that the proposed method can be used to extract ideal shape regions, which can be used as characteristic shape regions for image registration in printing pattern detection systems. However, the real-time reliability of the machine vision register control system need to be further studied and verified in applications of industrial automation. In summary, traditional register control methods based on decoupling control cannot effectively solve the problem of color mark detection and positioning, and it is difficult to balance the relationship between error compensation speed and register accuracy. The more innovative methods, such as machine vision and machine learning, tend to focus on printing quality monitoring, such as defect detection or fault warning. However, the implementation of image processing combined with motion control systems is expensive, complex and difficult. Moreover, various research limitations and model assumptions make it difficult for many register error models to be applied in engineering practice. In this paper, a new register control algorithm is proposed to improve register color-mark-detection accuracy and error compensation performance without increasing hardware configuration using the pulse-width subdivision detection algorithm. The optimized error compensation strategy provides better control performance than traditional algorithms.