最小多项式矩阵与线性多变量系统(Ⅰ)

黄琳; 于年才

最小多项式矩阵与线性多变量系统(Ⅰ)

黄琳,
于年才

北京大学力学系

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出版历程
- 收稿日期: 1984-05-02
- 刊出日期: 1985-07-15

Minimal Polynomial Matrix and Linear Multivariable System (Ⅰ)

Department of Mechanics, Peking University, Beijing

摘要

摘要: 本文分为两部分:(Ⅰ)为关于最小多项式矩阵的理论:(Ⅱ)为最小多项式矩阵理论在线性多变量系统中的应用.在(Ⅰ)中,我们给出了线性变换在向量组的消失多项式矩阵与最小多项式矩阵的概念,给出了不变子空间的生成组与最小生成组的概念.在讨论了这些概念的基本性质之后,我们研究了它们与线性变换在任何不变子空间上诱导算子对应的特征矩阵之间的关系,给出了向量组的最小多项式矩阵类的特征,并给出了有相同生成空间的生成组之间的充分必要条件.利用这些结果,对于给定的矩阵A,给出了能使系统x=Ax+Bu完全可控的矩阵B的全体的集合的表达式.

Abstract: Part (Ⅰ) of this work is on the theory of minimal polvnomial matrix and Part (Ⅱ) on the applications of this theory to linear multivariable systems.In Part (Ⅰ), concepts of annihilating polvnomial matrix and the minimal polynomial matrix of a given linear transformation in a vector group are given and the concepts of the generating system and minimal generating system of an invariant subspace for a given linear transformation are given as well. After discussing the basic properties of these concepts the relations between them and the characteristic matrix corresponding to an induced operator of a given linear transformation in any of its invariant subspace are studied in detail. The characteristics of the minimal polynomial matrix for a given vector group and the necessary and sufficient condition for the two generating systems to have the same generating suhspace is given. Using these results we can give the expression for the set of all B which makes the system x=Ax+Bu a complete controllable system for a given A.

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参考文献(7)

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[2]	黄琳,《系统与控制理论中的线性代数》,科学出版社,北京〔1983).
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[7]	Hwang Ling.Generating element and controllability.Proceeding of the Bilateral Meeting on Control Systems.(P.R.C and U.S.A.)Scienific Press.Beijing(1981).