Synthesis of static output controllers based on the solving of linear matrix inequalities

One of the most practically demanded methods of control in linear systems is control in the form of a static controller. To implement this control method, it is not necessary to measure all the phase variables of the system. With this method of control, the dimension of the closed system coincides with the dimension of the original object. The problem of static controller synthesis, in general, is reduced to the search for two mutually inverse matrices that satisfy a system of linear matrix inequalities. Such a problem is nonconvex and therefore cannot be solved using the apparatus of linear matrix inequalities. The solution of such a problem is reduced to finding two mutually inverse matrices that satisfy a system of linear matrix inequalities. The article considers a special case of the problem of synthesis of static controllers, which can be reduced to solving a system of linear matrix inequalities. The conditions for the implementation of such a case are shown. Two problems of the synthesis of static controllers are considered: the synthesis of the stabilizing controller and the synthesis of the optimal controller. The obtained results are applied to the stabilization of the electromagnetic suspension when the measured variable is the vertical displacement of the rotor. Graphs of transients are presented. A comparative analysis of the quality of transients in a closed system with calculated static controllers is performed.