1999 Fiscal Year Final Research Report Summary
Systems Analysis by Valuated Matroids
| Project/Area Number |
09450042
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Engineering fundamentals
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
MUROTA Kazuo Research Institute for Mathematical Sciences, KYOTO UNIVERSITY Professor, 数理解析研究所, 教授 (50134466)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIHARA Masaaki Nagoya University, School of Engineering Professor, 工学部, 教授 (80154483)
FURIHATA Daisuke Research Institute for Mathematical Sciences, KYOTO UNIVERSITY Research Associate, 数理解析研究所, 助手 (80242014)
OKAMOTO Hisashi Research Institute for Mathematical Sciences, KYOTO UNIVERSITY Professor, 数理解析研究所, 教授 (40143359)
IWATA Satoru Osaka University, Faculty of Engineering Science Associate Professor, 基礎工学部, 助教授 (00263161)
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| Project Period (FY) |
1997 – 1999
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| Keywords | valuated matroid / mixed polynomial matrix / combinatorial canonical form / transfer function matrix / RCG network / group symmetry / distributed system |
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| Research Abstract |
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.
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