Coupling Model and Controller Design for Four-layer Register System

A nonlinear and coupling model is established according to the multi-layer register system working principle in gravure printed electronic equipments, and then, the linear model of the register system is constructed based on one order Taylor formula. In order to improve the speed and accuracy of register control, a feedforward PID control algorithm is presented according to the linear model in multi-layer register system. This algorithm is unique in that it uses the PID to adjust the inputs of the register system, and feedforward compensation for the known disturbance, making the register control accuracy has been greatly improved. The simulation shows that the proposed register controller is able to realize a high precision control for the register system and endowed with better control performance than PID controller in register control.


Introduction
Just as the register process in traditional color printing, all parts of printed electronics can be precipitated by ink formation layer and layer correspondent to electronic materials through the method of gravure printing, and the quality of printed electronics is controlled by each layer of register accuracy. Register error, reflecting the direct index of register accuracy, comprises print-direction register and cross-direction register errors which dominated by the factors of each roller speed, web tension, web property, manufacturing and fixing errors. What's more, the cross-direction register is the key point of register system. Therefore, the thesis will discuss about it in terms of the cross-direction register, without the condition of explanation.
Kang designed his model based on the register error definition and feedforward PID controller with speed volatility of contiguous roller as the foregone interference [1,2]. But he ignore the influence of traction tension in register error. Although Li deduced detailed register model and devised feedforward controller in terms of disturbance of tension and speed, only with two layers of register system [3]. Yoshida established register error model based on web mass conservation and nonlinear control strategy, using Lyapunov's theory of stability [4]. Liu establish the relation among register error, roller speed and web tension, and design controller with method of active disturbance rejection control [5,6]. Chen put forward multi-layer register system controller strategy with the cooperation of expansion state observer and feedforward controller [7]. This thesis set up nonlinear coupling model for fourlayer register system based on the working principle of cross-direction register errors and deduces the linear model of register system. On the basis of linear model, feedforward PID register controller is designed in terms of the disturbance of tension and speed. The simulation shows that the proposed register controller is endowed with better control performance than PID controller in register control.

Four-layer register system model and linearization
As shows in Figure 1, the register system of four-layer gravure press comprise printing unit, drying mechanism and detection section. The printing roller is compelled by servo motor and works under the model of speed. Where L * is the nominal length in color cell, V i is the linear speed of the roller, T i is the tension of roller, and e ij is register error between adjacent units i and j.

V2(t) V3(t)
Tension sensor Photoelectric sensor Drying oven Refer the model of double layer register system established in reference [5], the four-layer nonlinear coupling model for register system, using the relation of tension and strain, is established as Eq. 1. Where A is cross-sectional area of web, E is elastic modulus, t T is delay time of the web transmission, V 2 (t)~V 4 (t) are system inputs, and e 12 (t)~e 34 (t) are system outputs.
The four-layer nonlinear coupling model for register system is linearized with one order Taylor formula and change it with Laplace transform as Eq. 2.
System input shows as Eq. 3 in terms of the expression transfer function G A (s) G B (s) G C (s).
The transfer functions of the disturbance V 1 (t) to each error G Di (s) are shown in Eq. 4, and the transfer functions of the disturbance T 0 (t) to each error G Ei (s) are shown in Eq. 5.
From the Eq. 2, it could read that the influence of register error e 12 e 23 e 34 are the traction tension T 0 and the speed of first roller V 1 , besides the corresponding roller speed V 2 V 3 and V 4 .

The design of feedforward PID controller
The control of register error is attained by adjusting the speed of after-layer roller in the design of feedforward PID controller. That is to say, adjusting V 2 V 3 V 4 to change e 12 e 23 e 34 respectively. So the transfer function G A (s) stands for the property of register error system with other factors as system disturbance. Therefore, PID is designed in terms of transfer function G A (s), and feedforward controller compensate other systematical influencing elements. Because T 0 V 1 V 2 V 3 can be measured, feedforward controller can be designed by using T 0 V 1 V 2 V 3 based on the invariance principle. What's more, feedforward controller can eliminate the effect of T 0 V 1 V 2 V 3 in register error. The structure of feedforward controller reveals as the Figure 2.
Based on the invariance principle, feedforward controller is devised in terms of T 0 and V 1 in E 12 . 1 1 According to Eq. 7, the feedforward controller C T1 (s) and C V11 (s) can be designed as Eq. 8.
Similarly, the feedforward controller C T2 (s) C V12 (s) and C V21 (s) can be designed as Eq. 9, and C T3 (s) C V13 (s) C V22 (s) and C V31 (s) can be designed as Eq. 10.
With this, Eq. 8, Eq. 9 and Eq. 10 constitute the feedforward controller of four-layer register system. Meanwhile, they and PID controller constitute the whole register controller, namely, feedforward PID controller.

The simulation
In order to verify the performance of the proposed feedforward PID register controller, simulation of register error with PID controller and feedforward PID controller are performed respectively. The performance of PID controller and feedforward PID controller in simulation are shown in Figure 3 when the upstream tension T 0 has a step change from 100N to 120N after 5s stable operation of the system. Compared with the register error in PID control, the peak of register error controlled by feedforward PID controller  In order to reflect the performance of different controllers in speed disturbance, simulation are performed with a man-made sinusoidal disturbance whose amplitude is 0.1r/min and frequency is 0.5rad/s after 5s stable operation of the system. As shown in Figure 4, the register error with the traditional PID control has a sinusoidal variation and the maximal register error is obviously larger than that with the proposed feedforward PID control. The results indicated that the proposed feedback controller can effectively reduce register error caused by the speed fluctuation.

Conclusion
The register error model and the control strategy of the sectional drive gravure printed electronic equipments are successfully studied in this paper. Based on the double layer register system model, a coupling mathematical model for the four-layer register error is derived, and linear model of register system is attained with one order Taylor formula. According to the established model, a feedforward PID controller is designed to alleviate the disturbance caused by the upstream tension and speed. In numerical simulation, the proposed feedforward PID controller was compared with traditional PID controller.
The results showed that with the disturbances caused by the tension and speed, the performance of the proposed feedforward PID controller is much better than that of the traditional PID controller. The proposed feedforward controller can effectively alleviate the register errors caused by the disturbances from tension and speed.