In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are est...In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×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 do not 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.展开更多
In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstl...In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstly, a right factorization of （sE - A）^-1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this matrix equation is established in terms of the R-controllability matrix of （E, A, B）, the generalized symmetric operator and the observability matrix associated with the Jordan matrix d and a free parameter matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.展开更多
基金This work was supported by the Chinese Outstanding Youth Foundation(No.69925308)Program for Changjiang Scholars and Innovative ResearchTeam in University.
文摘In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×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 do not 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.
基金This work was supported by the Chinese Outstanding Youth Foundation (No. 69925308)Program for Changjiang Scholars and Innovative Research Team in University.
文摘In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstly, a right factorization of （sE - A）^-1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this matrix equation is established in terms of the R-controllability matrix of （E, A, B）, the generalized symmetric operator and the observability matrix associated with the Jordan matrix d and a free parameter matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.
