Ensuring the stability of the vertical position of plasma is a paramount task of magnetic control for modern D-shaped tokamaks. The introduction of an additional horizontal field coil located near the vacuum vessel provides more than an order of magnitude larger vertical controllability domain than the use of a pair of P6&P12 poloidal field coils in the IGNITOR tokamak project. Two robust systems for controlling the vertical position of plasma are synthesized by the \( H_\infty \)-optimization theory method and by tuning PID controllers in a cascade system by the method of linear matrix inequalities. The results of mathematical modeling of control systems show that the modernization of the poloidal system of the IGNITOR tokamak project leads to an increase in the stability margin of the plasma vertical position control system, the quality of control when rejecting disturbances, and the reliability of the tokamak.