MIMO连续系统中可交换矩阵的充要条件

The sufficient and necessary conditions of commutative matrices for MIMO linear system

首先对ＭＩＭＯ或ＳＩＳＯ连续闭合线性系统在状态矩阵可交换下，给出了反馈阵存在的充要条件，即ＳＩＳＯ闭合系统归结到矩阵Ａ的特征值与特征根的情形，ＭＩＭＯ闭环系统归结到矩阵Ａ的特征子空间的不变性。而对于ＳＩＳＯ及ＭＩＭＯ开环系统情形时，以Ｋｒｏｎｅｃｋｅｒ积作为工具，将状态矩阵集可交换时反馈阵的存在性等价于Ｌｙａｐｕｎｏｖ方程解的存在性问题，同时给出了Ｌｙａｐｕｎｏｖ方程的具体形式。最后，通过几个具体数字例子来说明所得的结论。

Commutative matrices of MIMO linear system are considered. The existence of the feedback matrices of commutative state matrix set in the SISO closed-loop systems is reduced to the eigenvalue and eigenvector of matrices A and the existence of feedback matrices in the MIMO closed-loop systems is reduced to invariance subspace of matrix A, and the existence of feedback matrices in the open-loop systems is equivalent,to the existence of the solution of Lyapunov matrix equation. Kronecker product is used as the tool in dealing with commutative state matrix set. Results are demonstrated by two material examples.