A SOLUTION OF DYNAMIC ONE-PARAMETRIC RICCATI-TYPE MATRIX EQUATIONS

Abstract

The Silvester-type and Riccati-type matrix equations are usually encountered in control systems synthesis problems, while Lyapunov's equations - in control systems analysis problems. Methods for solving dynamic one-parameter matrix equations of Riccati-type, based on the corresponding spectral models based on G.E. Pukhov’s differential transformations (mathematical apparatus of differential transformations of functions and equations) are proposed, which are fundamentally different from the well-known integral transformations and possess certain positive computational characteristics in comparison with the latter. Control problems and control systems observation problems in the optimization of integral quality criteria are reduced to dynamic one-parametric Riccati-type matrix equations.

Studies show that one of the relatively simple approaches to solving such problems is the use of differential transformations, leading to recurrent algebraic relations easily implemented by means of modern information technologies.

In the work, as a result of using differential transformations and algebraization of the original problem, a spectral model is obtained, generating the following computational schemes:

- sequential recurrent computational scheme (direct representation), - sequential recurrent computational scheme (kronecker representation).

When using the spectral model recurrent computational structures are obtained, which are generated at the initial X(0) matrix discrete, with further calculation of the subsequent

X(1), X(2), ... matrix discretes, by using Y(0), Y(1), Y(2), ... auxiliary matrix discretes.

A model example is considered, the solution of which is obtained by the proposed computational schemes. In both cases, it was possible to circumvent the complex operation of numerical integration of dynamic matrix equations, usually occurring in the existing methods, and obtain coinciding acceptable solutions to the problem.