Synthesis of controllers on the basis of a solution of linear matrix inequalities and a search algorithm for reciprocal matrices

It is shown how the synthesis of stabilizing, $H^{\infty}$-optimal, and robust controllers of the prescribed order in the output for linear continuous and discrete systems reduces to a solution of linear matrix inequalities relative to two reciprocal matrices. To verify the solvability of such inequalities and to find their solutions, an algorithm realizable in the MATLAB package is proposed. An example of stabilization of an inverted pendulum with the aid of a controller of the reduced order in the output is given.