Development of control system based on digital twins of physical processes is a promising area of research in the rolling industry. Closed-loop control systems are developed to control the coordinates of two-mass electromechanical systems in order to limit the dynamic loads on the equipment of main rolling lines. These control systems are based on observers (digital shadows) that indirectly detect (reconstruct) the roll speed and the elastic torque of the shaft (spindle) in real time. Notably, observers are required to work fast in order to reconstruct transients attributable to shock (impact) loads. Literature review shows that the known observers, which use complex algorithms to compute coordinates, do not respond fast enough. The paper analyzes the kinematic diagram of Mill 5000, a plate rolling mill. It presents oscillograms that prove that the elastic torque does oscillate as the rolls grip the strip dynamically. The authors hereof have developed an observer that reconstructs the coordinates of the uncontrolled mass (the shaft) and the spindle torque from the parameters of the controlled mass, namely the torque and speed of the motor. The paper further rationalizes an approach that consists of simulating the processes on a model to further directly configure them on the object. The authors analyze the transients of the reconstructed two-mass system coordinates, which are associated with the rolls gripping the strip. The paper compares data against oscillograms recorded on the mill itself. The accuracy is satisfactory. The proposed observer has been used to developed a three-loop automatic speed control system for the uncontrolled mass. Controller configurations are substantiated. The paper shows coordinates obtained by simulation modeling as functions of time. It further presents experiments run on Mill 5000; the conclusions are that the amplitude and oscillations of the elastic torque drop significantly. The paper concludes with recommendations on industrial adoption of the observer and the novel electric drive coordinate control system. Study presented herein substantiates and implements a concept of developing algorithms that solve specific problems and are readily implementable on the existing equipment without need for additional computing devices. The contribution of the paper consists of stating and solving the problem of developing and testing an automatic elastic torque control system for the shaft of a heavy-duty rolling mill. This system has been implemented in the form of algorithms that run in the software of the existing industrial controllers (PLCs). It is simple and performs well. It does not need additional sensors or computers to be implemented, nor does it rely on complex computational algorithms. Such algorithms are based on computational tables that require a priori data on numerous process parameters. In our literature review, we have not come across any industrial implementation of such algorithms on hot-rolling mills.