Systems Analysis by Valuated Matroids

这个项目的目的是利用赋值拟阵理论，发展用于工程系统数学分析的代数和组合方法。(1)对混合矩阵的组合标准型的构造算法进行了实用的改进，使混合矩阵的数学结果可以用于系统分析。(2)赋值拟阵的对偶定理表明，通过适当改变变量和方程，混合多项式矩阵可以转化为标准形(具有附加良好性质的适当有理矩阵)。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，并借助线性Delta矩阵对的对偶定理导出了RCG网络的可解性条件。在一般假设下，讨论了对称分布控制系统的可控性。该问题是用群表示论的标准框架来表示的，并利用拟阵理论中的Rado-Perfect定理导出了可控性所需的功能模块的个数的界。

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.