## 4 Implementation
### Reconfiguring the RAC Controller  
To reduce the transverse asymmetry of gauge profile applying a proportional and differential (PD) RAC controller is suggested which, in fact, controls a filtered derivative of the input signal. The reason for selecting such a controller is that the controllers of such a type provide for the formation of control action with the account of both the control error and its change rate. This conclusion is clarified by time diagrams, given in Figure 8[26]. Solid lines show the signals at the output of a closed loop for controlling the parameter \(x_{\text{out}}\) at the linear change of the input signal shown by dashed lines. The dynamic control error, and output signal delay in relation to the input signal, occurs in the system of automatic controlling with a proportional controller (Figure 8a) On the contrary, in the system with a PD controller (Figure 8b), the output signal surpasses the task signal by the acceleration value. This is possible due to control on the input signal derivative. When setting controller parameters, one should take into account two conditions:  
* amplification factor (proportional part) should be equal to 1 to fully compensate misalignment;
* its differential part should be adjusted for predictive signal tracing, this will make the system response faster;
* as clear differentiation leads to the reduction of the system interference resistance, the controller input should have the operating filter. The transfer function of the PD controller with a filter with a transfer function \(W_{\text{f}}=\frac{K\cdot p}{T_{\text{f}}\cdot p+1}\) is shown in the structure in Figure 5 as a separate block. The transfer function has the following designations: \(p\)--Laplace operator; \(K\) and \(T_{\text{f}}\)--coefficient of the amplification of the RAC controller and the filter time constant. Also, Figure 5 shows the time constant of the RAC controller \(T_{\text{d}}\). The proposed solution is the development of the known principle of predictive control, which is implemented by means of introducing additional connection on the input signal derivative [27]. In this case the principle was applied in the new technical method for the systems of automatic control of auxiliary mechanisms, linked by the metal strip, of the reversing stand at the heavy-plate rolling mill. PD controllers, which are a subset of PID controllers, are difficult to synthesize, as the authors [28; 29; 30; 31] underline. PD RAC synthesis herein was adjusted for the fact that formula-calculated parameters do not provide an optimal configuration. The reason is that analytically derived values are based on simplified models of the object. Paper [32] details the known synthesis methods. Those methods were analyzed by the authors here; as a result, we used a method that combined analytical (manual) calculations and virtual  
Figure 8: Time diagrams of the output signal of a closed loop with a proportional controller—(**a**) and proportional and differential controller—(**b**). controller synthesis in Matlab. Commissioning electrical systems with the help of virtual models (digital twins) is the topic of the authors' paper [33]. The method follows these steps:  
1. First, we used analytical dependencies to find the controller coefficients. This part of the method is based on analytical synthesis in frequency domain [34; 35; 36]. 2. Synthesis was run in interactive software on a computer temporarily connected to the control circuit. We generated control signals and monitored response to them in order to evaluate the system's response to a change in control signals or external influences or noise. Rules applied were based on the authors' experience, theoretical analysis, and numerical experiments: * an increase in the proportional coefficient would improve performance and reduce stability; * an increase in the differential component would increase the margin of stability and improve performance. 3. Next, the computer was disconnected, and the controller parameters were copied to the controller onsite.