论文标题
传递函数矩阵的扰动理论
Perturbation theory of transfer function matrices
论文作者
论文摘要
在最小的条件下，有理传递功能矩阵$ r（λ）$的零是相关多项式系统矩阵$ p（λ）$的特征值。在本文中，我们为简单的特征值$λ_0$定义了（本地）最小值多项式系统矩阵$ P（λ）$的结构化条件编号，这反过来又是简单的零$λ_0$的传输函数矩阵$ r（λ）$。由于任何有理矩阵都可以写入多项式系统矩阵的传输函数，因此我们的分析产生了一个结构化的扰动理论，用于简单的理性矩阵$ r（λ）$。要捕获$ r（λ）$的所有零，无论它们是否是杆子，我们都会考虑词根的概念。作为主要结果的推论，我们特别注意$λ_0$的特殊情况不是$ r（λ）$的极点，因为在这种情况下，结果变得更简单，在实践中可能很有用。我们还将结构化条件编号与Tisseur的非结构化条件编号比较了基质多项式的特征值，并表明后者可以毫无根据。最后，我们通过数值实验来证实我们的分析。
Zeros of rational transfer function matrices $R(λ)$ are the eigenvalues of associated polynomial system matrices $P(λ)$, under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue $λ_0$ of a (locally) minimal polynomial system matrix $P(λ)$, which in turn is a simple zero $λ_0$ of its transfer function matrix $R(λ)$. Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yield a structured perturbation theory for simple zeros of rational matrices $R(λ)$. To capture all the zeros of $R(λ)$, regardless of whether they are poles or not, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of $λ_0$ being not a pole of $R(λ)$ since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur's unstructured condition number for eigenvalues of matrix polynomials, and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.
