Zhou Zhongli, Huang Guangxin
Geomathematics Key Laboratory of Sichuan Province, College of Management Science, Chengdu University of Technology, Chengdu 610059, China.
ScientificWorldJournal. 2013 Nov 11;2013:952974. doi: 10.1155/2013/952974. eCollection 2013.
The general coupled matrix equations (including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations over reflexive matrix solution. When the general coupled matrix equations are consistent over reflexive matrices, the reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix is chosen. Furthermore, the unique optimal approximation reflexive solution to a given matrix group in Frobenius norm can be derived by finding the least-norm reflexive solution of the corresponding general coupled matrix equations. A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.
一般耦合矩阵方程（包括广义耦合西尔维斯特矩阵方程作为特殊情况）在控制和系统理论中有许多应用。本文构造了一种迭代算法来求解自反矩阵解上的一般耦合矩阵方程。当一般耦合矩阵方程在自反矩阵上是相容的时，在没有舍入误差的情况下，自反解可以通过迭代算法在有限的迭代步骤内自动确定。当选择合适的初始矩阵时，可以导出一般耦合矩阵方程的最小Frobenius范数自反解。此外，通过求解相应一般耦合矩阵方程的最小范数自反解，可以得到给定矩阵组在Frobenius范数下唯一的最优逼近自反解。给出了一个数值例子来说明所提出迭代算法的有效性。
