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首页> 外文期刊>Applied Mathematical Modelling >An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices
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An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

机译：广义双对称矩阵上求解广义耦合Sylvester矩阵方程的迭代方法

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The generalized coupled Sylvester matrix equationsrn{AXB + CYD = M. EXF + CYH = N,rn(including Sylvester and Lyapunov matrix equations as special cases) have numerous applications in control and system theory. Annxn matrix P is called a symmetric orthogonal matrix if P = P~T = P~(-1) A matrix X is said to be a generalized bisymmetric with respect to P, if X = X~T = PXP. This paper presents an iterative algorithm to solve the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair [X, Y]. The proposed iterative algorithm, automatically determines the solvability of the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair. Due to that I (identity matrix) is a symmetric orthogonal matrix, using the proposed iterative algorithm, we can obtain a symmetric solution pair of the generalized coupled Sylvester matrix equations. When the generalized coupled Sylvester matrix equations are consistent over generalized bisymmetric matrix pair [X, Y], for any (spacial) initial generalized bisymmetric matrix pair, by proposed iterative algorithm, a generalized bisymmetric solution pair (the least Frobenius norm generalized bisymmetric solution pair) can be obtained within finite iteration steps in the absence of roundoff errors. Moreover, the optimal approximation generalized bisymmetric solution pair to a given generalized bisymmetric matrix pair can be derived by finding the least Frobenius norm generalized bisymmetric solution pair of new generalized coupled Sylvester matrix equations. Finally, a numerical example is given which demonstrates that the introduced iterative algorithm is quite efficient.
机译：广义耦合Sylvester矩阵方程rn {AXB + CYD = M. EXF + CYH = N，rn（作为特殊情况，包括Sylvester和Lyapunov矩阵方程）在控制和系统理论中有许多应用。如果P = P〜T = P〜（-1），则将Annxn矩阵P称为对称正交矩阵。如果X = X〜T = PXP，则矩阵X被称为关于P的广义双对称。本文提出了一种迭代算法来求解广义双对称矩阵对[X，Y]上的广义耦合Sylvester矩阵方程。所提出的迭代算法自动确定广义双对称矩阵对上广义耦合Sylvester矩阵方程的可解性。由于I（恒等矩阵）是一个对称正交矩阵，使用所提出的迭代算法，我们可以获得广义耦合Sylvester矩阵方程的对称解对。对于任何（空间）初始广义双对称矩阵对，当广义耦合Sylvester矩阵方程在广义双对称矩阵对[X，Y]上一致时，通过建议的迭代算法，可以得到广义双对称解对（最小Frobenius范数广义双对称解对） ）可在没有舍入误差的情况下在有限的迭代步骤中获得。此外，可以通过找到新的广义耦合Sylvester矩阵方程的最小Frobenius范数广义双对称解对来导出给定广义双对称矩阵对的最佳近似广义双对称解对。最后，给出一个数值例子，证明所引入的迭代算法是非常有效的。

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