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. This work was supported by the Chinese Outstanding Youth Foundation (No.69925308) and Program for Changjiang Scholars and Innovative Research Team in University. Bin ZHOU was born in HuBei Province, China, in 1981. He received the Bachelor’s degree from the Department of Control Science and Engineering at Harbin Institute of Technology, Harbin, China, in 2004. He is now a graduate student in the Center for Control Systems and Guidance Technology in Harbin Institute of Technology. His current research interests include linear systems theory and constrained control systems. Guang-Ren DUAN was born in Heilongjiang Province, 1962. He received his BSc. degree in Applied Mathematics, and both his M.S. and Ph.D. degrees in Control Systems Theory. He is currently the Director of the Center for Control Systems and Guidance Technology at Harbin Institute of Technology. His main research interests include robust control, eigenstructure assignment and descriptor systems.