Systems Analysis by Graphs and Matroids

Recent technology involves large-scale physical or engineering systems consisting of thousands of interconnected elementary units. This monograph illustrates how engineering problems can be solved using the recent results of combinatorial mathematics through appropriate mathematical modeling. The structural solvability of a system of linear or nonlinear equations as well as the structural controllability of a linear time-invariant dynamical system are treated by means of graphs and matroids. Special emphasis is laid on the importance of relevant physical observations to successful mathematical modelings. The reader will become acquainted with the concepts of matroid theory and its corresponding matroid theoretical approach. This book is of interest to graduate students and researchers.

Contenu

1. Preliminaries.- 1. Convention and Notation.- 2. Algebra.- 2.1. Algebraic independence.- 2.2. Rank, term-rank and generic-rank.- 3. Graph.- 3.1. Directed graph.- 3.2. Bipartite graph.- 3.3. Network.- 4. Matroid.- 4.1. Basic concepts.- 4.2. Examples.- 4.3. Linking system.- 4.4 Decomposition principle for submodular functions.- 4. 5. Independent-flow problem.- 2. Graph-Theoretic Approach to the Solvability of a System of Equations.- 5. Structural Solvability of a System of Equations.- 5.1. Introductory comments.- 5.2. Formulation of structural solvability.- 6. Representation Graph.- 7. Graphical Conditions for Structural Solvability.- 7.1. Generality of elements.- 7.2. Graphical conditions.- 8. Decompositions of a Graph by Menger-type Linkings.- 8.1. L-decomposition.- 8.2. Decomposition of a network by minimum cuts.- 8.3. M-decomposition.- 8.4. Relation among various decompositions.- 9. Decompositions and Reductions of a System of Equations.- 9.1. Problem decomposition by the L-decomposition.- 9.2. Problem decomposition by the M-decomposition.- 9.3. Cycles on the representation graph.- 9.4 Decomposition of inconsistent parts.- 9.5 Amount of numerical computation.- 10. Application of the Graphical Technique.- 11. Examples.- 3. Graph-Theoretic Approach to the Controllability of a Dynamical System.- 12. Descriptions of a Dynamical System.- 13. Controllability of a Dynamical System.- 13.1. Controllability of a system in standard form.- 13.2. Controllability of a system in descriptor form.- 13.3. Necessary conditions for the controllability.- 14. Graphical Conditions for Structural Controllability.- 14.1. Structural controllability.- 14.2. Structural controllability of a descriptor system.- 14.3. Structural controllability of a system in extended form.- 15. Discussions.- 15.1. Dynamic graph.- 15.2. Combinatorial analogue of Kalman's canonical decomposition.- 15.3. Greatest common divisor of minors of modal controllability matrix.- 4. Physical Observations for Faithful Formulations.- 16. Mixed Matrix for Modeling Two Kinds of Numbers.- 16.1. Two kinds of numbers.- 16.2. Generality assumptions.- 16.3. Mixed matrix.- 17. Algebraic Implication of Dimensional Consistency.- 17.1. Introductory comments.- 17.2. Dimensioned matrix.- 17.3. Total unimodularity of a dimensioned matrix.- 18. Physical Matrix.- 18.1. Physical matrix.- 18.2. Physical matrices in a dynamical system.- 5 Matroid-Theoretic Approach to the Solvability of a System of Equations.- 19. Rank of a Mixed Matrix.- 19.1. Rank Identity for a mixed matrix.- 19.2. Rank Identity in matroid-theoretic terms.- 19.3. Another identity.- 20. Algorithm for Computing the Rank of a Mixed Matrix.- 20.1. Algorithm for the rank of a layered mixed matrix.- 20.2. Algorithm for the rank of a mixed matrix.- 21. Matroidal Conditions for Structural Solvability.- 22. Combinatorial Canonical Form of a Layered Mixed Matrix.- 23. Relation to Other Decompositions.- 23.1. Introductory comments.- 23.2. Decomposition by L(p?) and the DM-decomposition.- 23.3. Decomposition for electrical networks with admittance expression.- 23.4. Extensions and remarks.- 24. Block-Triangularization of a Mixed Matrix.- 24.1. LU-decomposition of an invertible mixed matrix.- 24.2. Block-triangularization of a general mixed matrix.- 25. Decomposition of a System of Equations.- 26. Miscellaneous Notes.- 26.1. Eigenvalues of a mixed matrix.- 26.2. Hybrid immittance matrix of multiports.- 6. Matroid-Theoretic Approach to the Controllability of a Dynamical System.- 27. Dynamical Degree of a Dynamical System.- 27.1. Introductory comments.- 21.2. Dynamical degree of a descriptor system.- 27.3. An algorithm for determining the dynamical degree.- 28. Matroidal Conditions for Structural Controllability.- 29. Algorithm for Testing the Structural Controllability.- 30. Examples.- 31. Discussions.- 31.1. Relation to other formulations.- 31.2. Greatest common divisor of minors of modal controllability matrix.- Conclusion.- References.