SOLVING THE SYLVESTER MATRIX EQUATION BY THE SPECTRAL METHOD
DOI:
https://doi.org/10.20998/2079-0023.2019.02.02Keywords:
matrix equations, spectral decomposition of matrices, eigenvalues, eigenvectors, linear operator, quasibiorthogonality of bases, adjoint operatorAbstract
The matrix linear equations of Sylvester and Lyapunov are widely used in the control theory and theory of movements sustainability, as well as in solving the Riccati equation in the problem of the analytical construction of optimal controllers. The problem of solving the Sylvester equation has gained particular relevance in connection with the solution of problems of synthesis of low-dimensional Luenberger observers and problems of modal synthesis of control systems for linear automatic systems. The following paper analyzes the existing methods for solving the Sylvester matrix equation. It was justified the limitedness of the basic methods for the numerical solution of matrix equations, as well as the lack of analytical methods for solving. In this paper we propose quite simple method for solving the linear matrix Sylvester equation, which is a generalization of the widely known in the theory of stability of the Lyapunov matrix equation. The method is based on the spectral decomposition of the matrix linear operator in its eigenvectors, which are the matrices formed by the multiplication of the matrices` eigenvectors of the linear and conjugate operators. As a result, an analytical solution of the Sylvester matrix equation is obtained. We consider the cases of both real and complex conjugate roots of the characteristic equations of the matrices of Sylvester equations. In order to solve the Sylvester matrix equation of a large dimension the algorithm and software have been developed. For the method implementation the standard procedures of solving the complete eigenvalue problem for real matrices are used. After conducting a great number of experiments it was confirmed the high efficiency of the proposed method both in terms of time costs and the accuracy of the results obtained when solving the matrix equations of Sylvester and Lyapunov of large dimension.References
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