An explicit solution to the matrix equation AX − XF = BY
An explicit solution to the matrix equation AX − XF = BY
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DOI：
10.1016/j.laa.2005.01.018
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发表时间：
2005-06
影响因子：
1.1
通讯作者：
Bin Zhou;G. Duan
中科院分区：
文献类型：
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作者：
Bin Zhou;G. Duan
给出了广义西尔维斯特矩阵方程AX−XF=BY的一个完整的、一般的和显式的解，其中矩阵F是伴随形式的.该解是一个非常简洁的形式，由对称算子矩阵，Hankel矩阵和矩阵对（A，B）的可控性矩阵表示。此外，还给出了该解的几种等价形式。在此基础上，建立了正常西尔维斯特方程和著名的李雅普诺夫矩阵方程的显式解.这些结果为方程解的分析提供了极大的方便，在控制系统理论中的许多分析和设计问题中具有重要的作用。作为示范，一个简单而有效的方法参数极点配置。
A complete, general and explicit solution to the generalized Sylvester matrix equation AX−XF=BY, with the matrix F in a companion form, is proposed. The solution is in an extremely neat form represented by a symmetric operator matrix, a Hankel matrix and the controllability matrix of the matrix pair (A,B). Furthermore, several equivalent forms of this solution are also presented. Based on these presented results, explicit solutions to the normal Sylvester equation and the well-known Lyapunov matrix equation are also established. The results provide great convenience to the analysis of the solution to the equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, a simple and effective approach for parametric pole assignment is proposed.