## 1 Introduction
The proposed approach was tested on an industrial multi-color gravure printing production line, and the fluctuation range of the testing register errors data can be controlled within \(+/-0.1\) mm. To study the fluctuation of control signals, a novel idea using the biological functioning of cells communicating was considered to recognize the optimal register output signal [17]. In ref. [18], a mathematical model of a machine directional (MD) register was derived, where the compensation method was proposed to cancel out the upstream disturbance of the MD register. The proposed MD register model and compensator could be used to improve the performance of the MD register controller in multi-layer roll-to-roll printed electronics. The authors of ref. [19] presented a control strategy for the speed-up phase in the R2R printing system, particularly for printing systems with electronic line shafts. The tension fluctuation caused by torque variation during acceleration is the main cause of register error generation. A mechanical model of R2R printing systems in the speed-up phase was developed based on the principle of mass conservation and torque balance, and a model-based feedforward proportion derivative controller was designed to reduce register errors caused by tension fluctuations. In ref. [20], a feedforward controller based on an active disturbance rejection control (ADRC) strategy was presented to strengthen the accuracy and stability of the register system in multi-color shaftless gravure printing machines. The controller used feedforward control to compensate for the known interference, and ADRC was used to adjust the inputs of the register system and actively estimate and compensate the unknown disturbances in real-time, which greatly improved the accuracy of the register controller. In ref. [21], it was demonstrated by experimental verification that measurement error is generated by the widening and agglomeration of the register mark. The error was shown to differ with the size and shape of the mark under identical printing conditions. The results illustrated the importance of improving the printing quality of the register mark, selecting the desired geometry for register marks with regard to printability and utilizing an edge-detection algorithm in the control program for high-precision register control. The authors of ref. [22] estimated the register errors according to tension fluctuation of the substrate and achieved microscale resolution register control. The experimental results indicate that applying the proposed register algorithm achieved a resolution below \(+/-30\) mu m. In ref. [23], the authors eliminated register coupling between printing units using a fully decoupled proportional-derivative (FDPD) control algorithm, which was effective in processing the impact of different printing roll lengths on register error fluctuations. The authors of ref. [24] studied three different register control algorithms, i.e., model-based feedforward proportional differential (MFPD) and direct-decoupling proportional differential (DDPD). In addition, the performance indicators and industrial implementation difficulty of these algorithms were evaluated and tested. In the field of flexible electronic production, ref. [25] employed the response acceleration input to achieve rapid response in the register control of flexible substrates with a high-precision register control below \(+/-10\) mu m in industrial-scale R2R manufacturing systems. In ref. [26], the engraved marks in the printing cylinder were measured to an accuracy of \(\pm 10\)\(\upmu\)m. In the field of roll-to-roll additive manufacturing [27], the authors investigated the relationship between smearing defects and register error, which can be improved by adjusting the parameters of the control system.