Systems Analysis by Valuated Matroids

本项目旨在利用求值矩阵理论，发展工程系统数学分析的代数和组合方法。得到了以下结果：(1)对构造混合矩阵组合标准形式的算法进行了实际改进，使混合矩阵的数学结果可用于系统分析。改进后的算法实现并通过网络发布。(2)有值阵的对偶定理表明，通过适当地变换变量和方程，混合多项式矩阵可以转化为标准形式（具有附加良好性质的适当有理矩阵）。对求值的矩阵对偶性的算法适合于混合多项式矩阵的标准形式。(3)详细研究了矩阵宇称问题与RCG（电）网络可解性的关系。用混合偏对称矩阵的形式表述了RCG网络的可解性，并利用一对线性三角矩阵的对偶定理，导出了RCG网络的可解性条件。这为RCG网络的可解性提供了一种有效的算法。(4)在对称的一般假设下，讨论了具有对称性的分布式控制系统的可控性。利用群表示理论的标准框架，利用拟阵理论中的Rado-Perfect定理，导出了可控性所必需的功能模块个数的界。

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.