A rank-updating technique for the Kronecker canonical form of singular pencils

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2025-02-01 DOI:10.1016/j.apnum.2024.01.015

Dimitrios Christou , Marilena Mitrouli , Dimitrios Triantafyllou

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引用次数: 0

Abstract

For a linear time-invariant system

\dot{x} (t) = A x (t) + B u (t)

, the Kronecker canonical form (KCF) of the matrix pencil

(s I - A | B)

provides the controllability indices, also called column minimal indices, of the system and their sum corresponds to the dimension of the controllable subspace. In this paper we introduce a fast numerical algorithm for computing the sets of column/row minimal indices of a singular pencil

s F - G

using a rank-updating technique and the properties of piecewise arithmetic progression sequences defined by the size of the null spaces of appropriate Toeplitz matrices. The method is demonstrated and tested on various data sets.

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奇异铅笔的克罗内克典型形式的等级更新技术

对于线性时不变系统 x˙(t)=Ax(t)+Bu(t)，矩阵笔（sI-A|B）的克朗内克典型形式（KCF）提供了系统的可控性指数，也称为列最小指数，它们的总和对应于可控子空间的维度。在本文中，我们介绍了一种计算奇异铅笔 sF-G 的列/行最小指数集的快速数值算法，该算法使用了排序更新技术和由适当托普利兹矩阵的空域大小定义的分段算术级数序列的特性。该方法在各种数据集上进行了演示和测试。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.