Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX - XF = BY andMXN - X = TY with A,M ∈ ℝ^{n × n} , B, T ∈ ℝ^{n × r} , F, N ∈ ℝ^{p × p} and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × (n + pr). The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair (A, B) and the restriction that A, F don't have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.