矩阵方程AX-EXF=By的通解及其应用

General solutions to the matrix equation AX-EXF=BY and their applications

给出矩阵方程AX—EXY=BY的一个完全解析的、具有显式表达式和完全自由度的参数解(X,Y)．这里假设矩阵束(E,A B)为R-能控的,F为任意的方阵．相比于现有结论,求解算法不要求矩阵A和F具有特殊的形式,且对它们的特征值没有任何的限制,此外,本文给出的通解还具有结构简洁的特点,作为一个应用,给出了广义系统正常Luenberger函数观测器的一种参数化的设计方法,算例证明了方法的有效性．

The problem of solving the matrix equation AX - EXY = BY is studied in this paper. Firstly, a complete general parametric expression for （X, Y） satisfying this equation is obtained. Here matrix triple （E, A, B） is R-controllable and F is an arbitrary matrix. Compared with existing results, the algorithm does not need the matrices F and A to be in any canonical forms and the knowledge of their eigenvalues. Furthermore, this type of solution has a very neat form. As a demonstration, the Luenberger function observer design of descriptor linear system is considered and a parametric design procedure is established. Finally, a numerical example shows the effectiveness of the proposed approach.