系统工程数学方法.docVIP

课程编号：S209G027 系统工程数学方法 学时：32 学分：2 开课时间：春、秋 授课单位：管理学院 任课教师：唐万生 一、课程内容简介 本课程的目的是使系统工程专业的硕士研究生熟练掌握系统科学与控制科学中常用的数学基础知识和基本技能。本课程由以下三部分内容组成。 第一部分讲授线性空间的基本知识，包括向量、向量的线性相关与线性无关的概念、有限维线性空间的基底与坐标、坐标变换、子空间、线性映射的定义及性质、线性映射的值域与核、线性映射的不变子空间、线性映射的特征值与特征向量、线性映射在不同坐标系下的表示与矩阵的相似性、矩阵的秩及其计算、矩阵的特征多项式、求特征多项式的Leverrier-Faddeeva递推算法、哈密顿-凯拉定理、矩阵的自然标准形、矩阵的广义逆、矩阵分析及应用。 第二部分讲授度量空间的基础知识，包括距离、范数及内积的定义、非线性泛函的连续性、非线性泛函的可微性、非线性泛函的Gateaux导数及Frecht导数、非线性泛函的极值、非线性泛函取得极值的必要性条件。 第三部分讲授Laplace变换和Z-变换及应用，包括Laplace变换的定义、Laplace变换的性质、Laplace变换的逆变换、Laplace变换在连续动态系统中的应用、Z-变换定义及性质，Z-变换的逆变换、Z-变换在离散系统中的应用。 Mathematical Methods in Systems Engineering The purpose of this course is to introduce the fundamental knowledge of mathematical methods in systems engineering. It is used to enable the master students of systems engineering to advanced study. It is assumed that the students taking the course had also taken, as a prerequisite, an undergraduate course in calculus and also were familiar with element linear algebra as well as the theory of ordinary differential equations. The content of this course contains the three following major parts. The first part outlines the basic theory of linear vector space. We begin by formally defining and illustrating the concept of a linear vector space over a number field. In this way, we are able to introduce the notions of linearly independent and dependent vectors, a basis for a finite dimensional vector space, and the use of matrices to represent a change of basis. We introduce linear mapping and illustrate the use of matrices to represent the finite dimensional linear mapping in a given basis as well as an altered basis. A number of important and useful definitions and results involving matrices are presented. In particular, matrix rank is formally defined, and various tests for rank determination are outlined. The notions of eigenvalues and characteristic polynomial of matrices with element in the complex number field are also presented along with the very importan