Systems Analysis by Valuated Matroids

通过评估拟阵进行系统分析

• 批准号: 09450042
• 负责人: MUROTA Kazuo
• 金额: $ 3.65万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09450042/
• 关键词: valuated matroid; mixed polynomial matrix; combinatorial canonical form; transfer function matrix; RCG network; group symmetry; distributed system; マトロイド; システム解析; 混合行列; 多項式行列; M凸関数; 離散双対定理

This project aims at developing algebraic and combinatorial methods for mathematical analysis of engineering systems by means of valuated matroid theory. The following results have been obtained.(1) Practical improvements are made on the algorithm for constructing the combinatorial canonical form of mixed matrices, so that the mathematical results on mixed matrices can be utilized in systems analysis. The improved algorithm is implemented and made available through internet.(2) The duality theorem for valuated matroids implies that a mixed polynomial matrix can be brought into a canonical form (a proper rational matrix with additional nice properties) by a suitable change of variables and equations. The algorithm for the valuated matroid duality is tailored to the canonical form of a mixed polynomial matrix.(3) The relationship between the matroid parity problem and the solvability of RCG (electrical) networks is investigated in detail. The solvability of RCG networks is formulated in terms of mixed skew-symmetric matrices, and the solvability condition is derived with the aid of the duality theorem for a pair of linear delta matroids. This leads to an efficient algorithm for the solvability of RCG networks.(4) Controllability of distributed control systems with symmetry is discussed under the genericity assumption under symmetry. The problem is formulated using the standard framework of group representation theory and a bound on the number of functioning modules necessary for controllability is derived by means of the Rado-Perfect theorem in matroid theory.

本计画的目的是利用赋值拟阵理论，发展工程系统数学分析的代数与组合方法。取得了以下结果。(1)对构造混合矩阵组合标准形的算法进行了改进，使有关混合矩阵的数学结果能用于系统分析。改进后的算法已经实现并通过互联网发布。(2)赋值拟阵的对偶定理意味着一个混合多项式矩阵可以通过变量和方程的适当改变而变成一个标准形（一个具有额外好性质的适当有理矩阵）。赋值拟阵对偶的算法适合于混合多项式矩阵的标准形。(3)研究了拟阵奇偶问题与RCG（电）网络可解性之间的关系。用混合反对称矩阵表示RCG网络的可解性，并利用线性Delta拟阵对的对偶定理导出RCG网络的可解性条件。这导致一个有效的算法的RCG网络的可解性。(4)在对称分布控制系统的通有性假设下，讨论了对称分布控制系统的能控性。这个问题是制定使用的标准框架的群表示理论和约束的功能模块的数量所需的可控性是通过在拟阵理论的Rado-Perfect定理。

项目成果

K.Murota: "Matrices and Matroids for Systems Analysis (Algorithms and Combinatorics 20)"Springer-Verlag. 483 (2000)

K.Murota：“系统分析的矩阵和拟阵（算法和组合学 20）”Springer-Verlag。

Kiyohiro Ikeda: "Mode switching and recursive bifurcation in granular materials" Journal of Mechanical Physical Solids. 45・11/12. 1929-1953 (1997)

Kiyohiro Ikeda：“粒状材料中的模式切换和递归分叉”《机械物理固体杂志》45・11/12 1929-1953（1997）。

R.Tanaka and K.Murota: "Quantitative Analysis for Controllability of Symmetric Control Systems"International Jounal of Control. 73・3. 254-264 (2000)

R. Tanaka 和 K. Murota：“对称控制系统可控性的定量分析”《国际控制杂志》73・3（2000 年）。

R. Tanaka and K. Murota: "Quantitative Analysis for Controlla-bility of Symmetric Control Systems"International Journal of Control. 73-3. 254-264 (2000)

R. Tanaka 和 K. Murota：“对称控制系统可控性的定量分析”国际控制杂志。

K. Murota, Discrete Convex Analysis: "Discrete Structures and Algorithms, V S. Fujishige, ed."Kindai-Kagaku-sha (Chap. 2,). 51-100 (1998)

K. Murota，离散凸分析：“离散结构和算法，V S. Fujishige 编辑”Kindai-Kagaku-sha（第 2 章）。