SYNTHESIS OF DIGITAL CONTROLS BY SETTING THE STABILITY AND OSCILLATION DEGREES OF AUTOMATED CONTROL SYSTEMS

Abstract

The synthesis of digital controllers, which are characterized by a gain and have one zero and one pole, is considered. Using the example of a typical second-order control object, it is shown that such controllers can be synthesized in such a way that the resulting automated control system has degrees of stability and oscillation no worse than the specified ones. This is achieved by placing the three roots of the characteristic equation of such a system on the zplane so that they do not go beyond the zones bounded by the lines of constant degree of stability and constant degree of oscillation. The transient process in such a system, obtained as a result of modeling in the Simulink package, showed its compliance with the specified degrees of stability and oscillation. The border of the zone, which is a line of constant degree of stability on the z-plane, is a circle – with an increase in the degree of stability, the radius of such a circle decreases. The border of the zone, which is a line of constant degree of oscillation on the z-plane, is a spiral – with a decrease in the degree of oscillation, the dimensions of such a spiral also decrease. Thus, the z-plane is conventionally divided into four zones: zone I, in which the conditions of neither a given degree of stability, nor a given degree of oscillation are satisfied; zone II, in which the condition of a given degree of stability is satisfied; zone III, in which the condition of a given degree of oscillation is satisfied; zone IV, in which the conditions of both a given degree of stability and a given degree of oscillation are satisfied. Such an arrangement of the roots of the characteristic equation was achieved by solving a system of three equations, which included the gain of the digital controller, its one zero and its one pole as unknowns. It should be noted that, on the one hand, the presence of specified degrees of stability and oscillation does not exclude, for example, the presence of a static error in the automated control system. But at the same time, on the other hand, it is possible to choose a digital controller with a more complex structure and, having performed similar calculations, make such a system astatic, as a result of which the static error will be very small or absent altogether.