共轭积框架下多项式矩阵理论研究及在系统设计中的应用

项目名称： 共轭积框架下多项式矩阵理论研究及在系统设计中的应用

项目编号： No.61273094

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 自动化技术、计算机技术

项目作者： 吴爱国

作者单位： 哈尔滨工业大学

项目金额： 82万元

中文摘要： 申请者在研究系统设计中出现的Sylvester方程时提出了多项式矩阵的共轭积的概念。当多项式矩阵为实时，共轭积就退化为正常的乘积。对共轭积的研究可能会为控制系统的设计提供更好的工具。从数学的角度看，共轭积丰富了多项式矩阵理论。本项目拟在共轭积框架下对多项式矩阵进行研究，并将部分结果用于控制系统设计，主要内容包括：（1）研究共轭积框架下的多项式向量空间，多项式矩阵的秩；（2）以多项式向量空间的结果为基础，研究共轭积框架下的多项式矩阵方程解的存在性，建立通解的表达式；通过简化这些结果并将其用于广义线性系统的设计；（3）构造共轭积框架下的有理分式矩阵，并考虑有理分式矩阵的标准型和互质分解；将获得的结果用于复系数系统分析和设计；（4）以共轭积为工具研究合相似变换下的Jordan标准型及合不变空间。 此项目属于由控制系统的研究带来的新的理论课题，其成果可能会给控制系统设计提供新的研究思路。

中文关键词： 共轭积；多项式矩阵；互质性；有理分式；反线性系统

英文摘要： The concept of conjugate product for polynomial matrices is firstly proposed based on investigation on Sylvester matrix equations appearing in control systems design. The conjugate product is reduced to the ordinary product when the considered polynomial matrices are real. From the mathematical point of view, the conjugate product enriches the theory of polynomial matrices. In this project, the polynomial matrices are first investigated in the framework of conjugate product, and some results are then utilized to control systems design. The main contents in this project are as follows. (1) The polynomial vector space in the framework of conjugate product is investigated, and then the rank of polynomial matrices in the framework of conjugate product is characterized; (2) Based on obtained results on polynomial vector space, the existence of solutions to polynomial matrix equations are investigated in the framework of conjugate product, and general expressions of the solutions are established; descriptor linear systems are investigated based on reduced version of the obtained results; (3) Rational fractional matrices in the framework of conjugate product are firstly constructed, and then some canonical forms of rational fractional matrices are provided; In addition, some results on rational fractional matrices are

英文关键词： conjugate product；polynomial matrices；coprimeness；rational fraction；antilinear systems