## 2 Materials and Methods
### Register Control Algorithm Based on Pulse Width Subdivision
#### 2.2.2 Optimized Feedforward Register Control  
After obtaining the accurate and reliable register error data, the controller then outputs the compensation signal to the printing roller servo drive mechanism in real-time according to the error detection results; the error compensation operation is performed by increasing or decreasing the running speed of the printing roller in a limited time so as to align the color mark patterns. In Figure 6, the register error compensation output block diagram is shown. The PD controller adaptively adjusts the optimal output parameters according to the output strategy, including the register error size, running speed, filter period and other parameters, outputs the compensation signal to the printing roller servo motor and finally, achieves the purpose of the printing register by superposing the printing roller speed through the mechanical transmission system. Figure 5: Pulse-width subdivision detection model. The output compensation formula of register error is expressed as:  
\[e_{r}=\varepsilon*\Delta V_{i}*t_{output},t_{output}<\frac{L_{c}^{*}}{V_{i}^{*}}, \tag{5}\]  
where \(\varepsilon\) is the output strength influence factor or speed superposition influence factor, which is a constant determined by the ratio of the servo driver electronic gearbox and the mechanical transmission ratio, and \(t_{output}\) represents the output time in milliseconds. Generally, the execution time is linear with the size of the error, and \(\Delta V_{i}\) is the printing roller's superimposed speed, which is a constant in normal operations. According to Formula (5), the larger \(\varepsilon\), \(\Delta V_{i}\) and \(t_{output}\) are, the faster the error compensation speed. However, it is also more prone to error oscillation. On the contrary, when error adjustment accuracy is higher, execution speed is slower. When there is a large register error, such as a joint of two webs, fast adjustment can complete error compensation in a short time, but this can easily cause error oscillation due to an adjustment resolution that is too large. Therefore, the adaptive adjustment of influence factors and output period according to the size of the register error is an effective method to improve control system performance. The printing roller of the output time is the printing roller one-cycle operation time \(\mathrm{T}=L_{c}^{*}/V_{i}^{*}\), as shown in Figure 7, while the register error output time-segment model ignores the influence of time \(L_{c}^{*}/\Delta V_{i}\). Then, each printing cycle time can be divided into three sections, namely, detection time \(t_{u}\), output delay \(t_{g}\) and adjustable time \(t_{w}\). Obviously, the executable time decreases with an increase in running speed, and the adjustable amount of error also decreases. In order to avoid the output execution time exceeding one printing cycle, the maximum compensation limit at one time can be set to \(e_{max}=\varepsilon*\Delta V_{i}*T\). The new compensation error is obtained from the next detection result, and the register errors shall not be accumulated and stacked. The optimized feedforward error compensation strategy is expressed as  
\[\left\{ \begin{array}{l}\varepsilon=\varepsilon^{*}\\ \varepsilon=p_{0}[\varepsilon(t)+0.9]\varepsilon^{*},\\ \varepsilon=p_{1}\varepsilon^{*}\end{array} \right.,\qquad \begin{array}{l}e(t) <=0.1\;\mathrm{mm}\\ 0.1\;\mathrm{mm}<e(t)<=1.1\;\mathrm{mm},\\ e(t)>1.1\;\mathrm{mm},\end{array} \qquad \begin{array}{l}0<p_{0}<=1\\ p_{1}>1\end{array} \right.. \tag{6}\]  
We define the output intensity influence factor as constant \(\varepsilon^{*}\). When the register error \(e(t)\leq 0.1\;\mathrm{mm}\), the output mode remains unchanged; when \(0.1\;\mathrm{mm}<e(t)\leq 1.1\;\mathrm{mm}\), the output intensity influence factor changes linearly according t  
Figure 6: Block diagram of register error compensation output. Figure 7: Time-segmented model of registration error output. when \(e(t)>1.1\) mm, the output intensity influence factor is directly scaled to accelerate the compensation speed.