广义Sylvester矩阵方程AX-XF=BY的显式解

给出了广义Sylvester矩阵方程AX-XF=BY当F为任意矩阵时的一种完全的解析通解.该通解由矩阵对(A,B)构成的能控性矩阵,一个对称算子矩阵和矩阵对(Z,F)构成的能观性矩阵组成,这里Z是一个任意的参数矩阵,用来表征该方程的解的自由度.利用著名的Levverrier算法,该解析解的一个等价形式被给出.给出的结果是参考文献[13]的推广,在[13]中F被假设为友矩阵.

Explicit solutions to the generalized Sylvester matrix equation AX- XF = BY

A complete, general and explicit solution to the generalized Sylvester matrix equation AX-XF=BY, with F being an arbitrary square matrix, is investigated. The proposed solution is in an extremely neat form represented by a controllability matrix of the matrix pair (A,B), a symmetric operator and an observability matrix of the matrix pair (Z,F), where Z is an arbitrary matrix used to denote the degree of freedom in the solution. Furthermore, based on the Faddeev-Leverrier algorithm, an equivalent form of the proposed solution is established. At the same time, an equivalent form of the solutions proposed in [13] is also induced. These results provide great convenience to the analysis and design problems in control systems. The results proposed in this note is a further discussion of the results proposed in [13].