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PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R
Author(s) -
Zhou Bin,
Duan GuangRen
Publication year - 2007
Publication title -
asian journal of control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.769
H-Index - 53
eISSN - 1934-6093
pISSN - 1561-8625
DOI - 10.1111/j.1934-6093.2007.tb00437.x
Subject(s) - sylvester equation , mathematics , matrix (chemical analysis) , matrix function , sylvester's law of inertia , sylvester matrix , observability , parametric statistics , riccati equation , symmetric matrix , pure mathematics , algebra over a field , mathematical analysis , physics , polynomial matrix , matrix polynomial , differential equation , eigenvalues and eigenvectors , materials science , statistics , quantum mechanics , polynomial , composite material
ABSTRACT In this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX ‐ XF = BY and the regulator matrix equation AX ‐ XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair ( A, B ), a symmetric operator and the generalized observability matrix of matrix pair ( Z, F ) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.