Veretennikova M.P.; Lebedev S.K.; Kulenko M.S.; Koltsova E.A.

(Ivanovo State Power Engineering University, Russia)

POSITIONING SYSTEM WITH PI2I(D) POSITION CONTROLLER

This article addresses the issue of electromechatronic positioning module intended to change the position of the executive part of technological installation. Modern electric drives are integrated into modern equipment and are becoming its essential part. In many cases, the electromechanic system contains modules which main purpose is positioning. The aim of the article is to develop and compare the characteristics of the control system of the electromechatronic positioning module with the PI2I(D) position controller with different options of accounting for the inertia of the torque contour using the MathCAD software package.

In this work we employed the mathematical method, the comparative method and the method of observation. Virtual models of the investigated systems with torque contour and without torque contour were created, and then compared with each other. The employed methods allowed simplifying the research and increasing the accuracy of the result.

The article is divided into 5 parts. In the first part, the equations of state and the equations of output for the systems under study were found.
Functional diagrams of positioning systems were compiled and systems of differential equations were obtained according to thеsе diagrams. The next step was to reduce the resulting systems to the equation of state.  For this aim, the state vectors and input vectors were set, and then the matrices of the parameters of the equation of state and the equation of output were compiled. As a result, we obtained the equation of state and the equation of the output for the electromechanic positioning module with a torque contour and without a torque contour.

In the second part, the regulator parameters which ensure the Bessel positioning dynamics were calculated. The Bessel root distribution used in this study ensures the monotony of the processes, minimal overshoot and constancy of the group delay time. In calculating the regulator parameters, the coefficients of the Bessel polynomials and the coefficients of the characteristic polynomials of the systems were equated. As a result, the regulator parameters for the systems under study were deduced from the equations obtained.

In the third part, ‘input-output’ or equivalent matrices for the considered systems were determined. The components of equivalent matrix are transfer functions between input and output variables. Therefore, equivalent matrices allow obtaining all transfer functions of systems, and consequently all the time characteristics and frequency characteristics of the given systems. The denominator of the transfer function corresponds to the Bessel dynamics, but the presence of a non-unit numerator worsens the dynamics of the system, which will further be proved. Thus, to compensate the influence of the numerator polynomial on dynamic processes, dynamic links with an inverse numerator transfer function were used. This transfer functions were serially connected at the regulator input.

In the fourth part, the time characteristics of the studied systems were obtained and analyzed. The time characteristics are system reactions to typical impacts under zero initial conditions. They allow evaluating the dynamic properties of the positioning system. The time characteristics of the systems were determined from their transfer functions for a single step action, linear growth of the task, and quadratic growth of the task. The results showed a significant increase in overshoot, a longer regulation time and an increase in oscillation in case of a system without an input filter. The results obtained allow us to conclude that there is a dynamic error with a single step action, with a constant signal and a linearly increasing signal, there is no steady-state error, and with a quadratic signal growth, a constant steady-state error arises.

In the fifth part, frequency characteristics were obtained and analyzed.
The frequency characteristics of the system is the response of the system to harmonic typical effects under zero initial conditions. It shows how many times the system operating in steady state changes the amplitude of the input sinusoid of frequency and at what angle the phase of the input sinusoid shifts. The results of analyzing the frequency characteristics of the system with a torque circuit and system without a torque circuit have demonstrated that for systems with a torque circuit a wider bandwidth and a larger slope of the asymptote in the suppression band is an advantage.

To recap, the functional diagram of the electromechanic positioning module with the PI2ID regulator has been generated, which made it possible to form the structural diagrams of the positioning systems. Then, the equations of state and the equations of output for the systems under study have been obtained as well as the parameters of the position controllers, which ensure the achievement of the Bessel dynamics. Further, the time characteristics and frequency characteristics of the systems were obtained and analyzed. Analysis of the characteristics showed that the system with a torque circuit allows providing Bessel dynamics with a bandwidth exceeding the preset value, which is an undoubted advantage, since it increases the speed of the electromechanic positioning module and, ultimately, the productivity of the technological equipment.

The results of this work can be used to conduct research work on the topics of automated electric drive, robotics and mechatronics, and can also be applied in the educational process for studying the electric drive and automation of industrial installations.