No abstract

Matrix equations and systems of matrix equations are widely used in control system optimization problems. However, the methods for their solving are developed only for the most popular matrix equations – Riccati and Lyapunov equations, and there is no universal approach for solving problems of this class. This paper summarizes the previously considered method of solving systems of algebraic equations over a field of real numbers [1] and proposes a scheme for systems of polynomial matrix equations of the second degree with many unknowns. A recurrent formula for fractionalization a solution into a continued matrix fraction is also given. The convergence of the proposed method is investigated. The results of numerical experiments that confirm the validity of theoretical calculations and the effectiveness of the proposed scheme are presented.

In view of the growing role of unmanned aerial vehicles in the military sphere and the increasing automation of enterprises, the problem of improving control quality is urgent. Matrix equations and systems of matrix equations are widely used in some applied disciplines, in particular, in optimization problems of control systems. However, there is no universal approach to solving all problems of this class: only methods of solving the most popular matrix equations, such as the Sylvester, Riccati, and Lyapunov equations, have been developed. There is an opportunity to explore this topic in more detail in the papers of Beavers A.N., Denman E.D., Boichuk A.A., Krivosheya S.A. and Kramer K. In one of our previous articles, a method for solving systems of algebraic equations over the eld of real numbers was proposed. In this paper, the previously considered approach is generalized, and an approximate solution scheme for systems of polynomial matrix equations of the second degree with many unknowns is presented. A recurrent formula for calculating the approximate solution of systems in the form of a continued matrix fraction is also given. The convergence of the proposed method is investigated based on Vorpitsky's sucient condition. The results of numerical experiments that conrm the validity of theoretical calculations and the eectiveness of the proposed scheme for the approximate solution of matrix equations are presented.