Systems Analysis by Graphs and Matroids: Structural Solvability and Controllability - Tapa blanda

Sinopsis

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.

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