Multivariable linear systems and projective algebraic geometry
著者
書誌事項
Multivariable linear systems and projective algebraic geometry
（Methods of algebraic geometry in control theory / Peter Falb, pt. 2）
Birkhäuser, c1999
- : us
- : gw
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注記
Includes bibliographical references (p. [375]-379) and index
内容説明・目次
- 巻冊次
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: us ISBN 9780817641139
内容説明
"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is quite satisfactory and natural for scalar systems, the study of multi-input, multi-output linear time invariant control systems requires projective algebraic geometry. Thus, this second volume deals with multi-variable linear systems and pro jective algebraic geometry. The results are deeper and less transparent, but are also quite essential to an understanding of linear control theory. A review of * From the Preface to Part 1. viii Preface the scalar theory is included along with a brief summary of affine algebraic geometry (Appendix E).
目次
1 Scalar Input or Scalar Output Systems.- 2 Two or Three Input, Two Output Systems: Some Examples.- 3 The Transfer and Hankel Matrices.- 4 Polynomial Matrices.- 5 Projective Space.- 6 Projective Algebraic Geometry I: Basic Concepts.- 7 Projective Algebraic Geometry II: Regular Functions, Local Rings, Morphisms.- 8 Exterior Algebra and Grassmannians.- 9 The Laurent Isomorphism Theorem: I.- 10 Projective Algebraic Geometry III: Products, Graphs, Projections.- 11 The Laurent Isomorphism Theorem: II.- 12 Projective Algebraic Geometry IV: Families, Projections, Degree.- 13 The State Space: Realizations, Controllability, Observability, Equivalence.- 14 Projective Algebraic Geometry V: Fibers of Morphisms.- 15 Projective Algebraic Geometry VI: Tangents, Differentials, Simple Subvarieties.- 16 The Geometric Quotient Theorem.- 17 Projective Algebraic Geometry VII: Divisors.- 18 Projective Algebraic Geometry VIII: Intersections.- 19 State Feedback.- 20 Output Feedback.- Appendices.- A Formal Power Series, Completions, Regular Local Rings, and Hubert Polynomials.- B Specialization, Generic Points and Spectra.- C Differentials.- D The Space.- E Review of Affine Algebraic Geometry.- References.- Glossary of Notations.
- 巻冊次
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: gw ISBN 9783764341138
内容説明
This monograph is an introduction to the ideas of algebraic geometry written for graduate tudents in systems, control and applied mathematics. The work presents core ideas in the algebro-geometric treatment of multivariable linear system theory.
目次
- Scalar input or scalar output systems
- two or three input, two output systems - some examples
- the transfer and Hankel matrices
- polynomial matrices
- projective space
- projective algebraic geometry I - basic concepts
- projective algebraic geometry II - regular functions, local rings, morphisms
- exterior algebra and grassmannians
- the Laurent isomorphism theorem I
- projective algebraic geometric III - products, projections, degree
- the Laurent isomorphism theorem II
- projective algebraic geometry IV - families, projections, degree
- the state space - realizations, controllability, observability, equivalence
- projective algebraic geometry V - fibres of morphisms
- projective algebraic geometry VI - tangents, differentials, simple subvarieties
- the geometry quotient theorem
- projective algebraic geometry VII -divisors
- projective algebraic geometry VIII - intersections
- state feedback
- output feedback
- formal power series, completions, regular local rings, and Hilbert polynomials
- specialization, generic points and spectra
- differentials
- the space nm
- review of affine algebraic geometry.
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