## 4 Implementation  
### Predictive Control of Asymmetry of Roll Gap of the Heavy-Plate Mill Stand  
The authors developed a way of controlling the roll gap misalignment, the key idea of which consists in measuring the differential rolling force and calculation of the back pressure value used as control action. To implement the method, the following operations are conducted:  
1. The change in rolling forces (pressure in hydraulic cylinders on the sides OS and DS) and the calculation of its difference (differential rolling force). 2. The calculation of pressure values on the sides OS-DS (counter-pressures), which should be generated in hydraulic cylinders to compensate differential force. 3. The use of the signal, proportional to the counter-pressure difference, as control action in the RAC controller. A half of this value is used as additional action for controlling the SD position on each DS and OS. Received signals compensate stand transverse tension, occurring as a result of differential force action, and thus compensate gap asymmetry. In effect, the authors develop a method of direct feedback ensuring the control of the gauge profile transverse asymmetry on disturbance. Studies carried out by the authors of [24; 25] showed the difference in the output gauge on the sides of the feed to have the greatest influence on the pressure mismatch. Unless control actions are applied, this difference 'builds up' after each passage, i.e., is inherited. Tests on Mill 5000 confirm this conclusion. This is where the idea behind the predictive control of roll gap misalignment in reverse rolling came from. Figure 7 shows schematically the principle of compensating inherited feed asymmetry. The idea is to store the DS and OS force values at the end of the forward passage (Figure 7a) and force the reverse misalignment of rolls before the reverse passage, Figure 7b. During the reverse, we compute (predict) the SD positions. This is from where the name derives: the predictive method. The prototype of the developed technical solution is the method of strip hot rolling in compliance with the patent of the Russian Federation [8]. It suggests using a controlling change of roll gap in the function of temperature difference of the feed left and right edges. Temperature is measured by pyrometers installed on the sides of the stand. Below is the function that computes the roll gap asymmetry, mm:  
\[\Delta S=\frac{\Delta\sigma_{\mathrm{s}}\sqrt{R\Delta h}}{C_{\mathrm{K}}}\cdot \frac{B}{2}\cdot 10^{-3}, \tag{3}\]  
where \(R\) is the work roll radius, mm; \(\Delta h\) is the absolute compression; \(B\) is the feed width, mm; \(C_{\mathrm{K}}\) is the housing stiffness, t/mm; \(\Delta\sigma_{\mathrm{s}}=\sigma_{\mathrm{s}}(T_{\mathrm{\Delta}})-\sigma_{ \mathrm{s}}(T_{\mathrm{\Delta}})\); \(\sigma_{\mathrm{s}}(T_{\mathrm{\Delta}})\) and \(\sigma_{\mathrm{s}}(T_{\mathrm{\Delta}})\) are the strain resistances at the temperatures \(T_{1}\) and \(T_{\mathrm{\Delta}}\) of the left and right edge, kg/mm\({}^{2}\). Furthermore, it provides the equations for rolling forces \(F_{\mathrm{OS}}\) and \(F_{\mathrm{DS}}\) expressed by drafting, frame tension, frame and rolled metal stiffness:  
\[F_{\mathrm{OS}}=C_{\mathrm{K}}\cdot X_{\mathrm{OS}}=C_{\mathrm{OS}}\cdot \Delta X_{\mathrm{OS}};F_{\mathrm{DS}}=C_{\mathrm{K}}\cdot X_{\mathrm{DS}}=C _{\mathrm{DS}}\cdot\Delta X_{\mathrm{DS}}, \tag{4}\]  
where \(C_{\mathrm{K}}\) is the frame stiffness; \(C_{\mathrm{OS}}\), \(C_{\mathrm{DS}}\) is the feed stiffness \(X_{\mathrm{OS}}\), \(X_{\mathrm{DS}}\) stand tension on the corresponding sides. The difference of forces in the hydraulic cylinders is given by:  
\[\Delta F=F_{\mathrm{OS}}-F_{\mathrm{DS}}=C_{\mathrm{K}}(X_{\mathrm{OS}}-X_{ \mathrm{DS}})=C_{\mathrm{K}}(\Delta S), \tag{5}\]  
from which \(\Delta S\)= \(\Delta F/C_{\mathrm{K}}\). To compensate the asymmetry of roll gap \(\Delta S\), it is necessary to make a preliminary \"reversing\" misalignment \(\Delta S^{*}=-\Delta S\) before the feed passes to the stand.