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Birkhäuser

Methods of Algebraic Geometry in Control Theory: Part II

Multivariable Linear Systems and Projective Algebraic Geometry

  • Book
  • © 1999

Overview

Authors:
  1. Peter Falb

    1. Division of Applied Mathematics, Brown University, Providence, USA

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Part of the book series: Systems & Control: Foundations & Applications (SCFA)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Introduction

    • Peter Falb
    Pages 1-6

  3. Scalar Input or Scalar Output Systems

    • Peter Falb
    Pages 7-33

  4. Two or Three Input, Two Output Systems: Some Examples

    • Peter Falb
    Pages 35-57

  5. The Transfer and Hankel Matrices

    • Peter Falb
    Pages 59-77

  6. Polynomial Matrices

    • Peter Falb
    Pages 79-103

  7. Projective Space

    • Peter Falb
    Pages 105-112

  8. Projective Algebraic Geometry I: Basic Concepts

    • Peter Falb
    Pages 113-128

  9. Projective Algebraic Geometry II: Regular Functions, Local Rings, Morphisms

    • Peter Falb
    Pages 129-141

  10. Exterior Algebra and Grassmannians

    • Peter Falb
    Pages 143-160

  11. The Laurent Isomorphism Theorem: I

    • Peter Falb
    Pages 161-171

  12. Projective Algebraic Geometry III: Products, Graphs, Projections

    • Peter Falb
    Pages 173-180

  13. The Laurent Isomorphism Theorem: II

    • Peter Falb
    Pages 181-190

  14. Projective Algebraic Geometry IV: Families, Projections, Degree

    • Peter Falb
    Pages 191-200

  15. The State Space: Realizations, Controllability, Observability, Equivalence

    • Peter Falb
    Pages 201-221

  16. Projective Algebraic Geometry V: Fibers of Morphisms

    • Peter Falb
    Pages 223-230

  17. Projective Algebraic Geometry VI: Tangents, Differentials, Simple Subvarieties

    • Peter Falb
    Pages 231-242

  18. The Geometric Quotient Theorem

    • Peter Falb
    Pages 243-258

  19. Projective Algebraic Geometry VII: Divisors

    • Peter Falb
    Pages 259-270

  20. Projective Algebraic Geometry VIII: Intersections

    • Peter Falb
    Pages 271-281
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Keywords

About this book

"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).

Reviews

"The exposition is extremely clear. In order to motivate the general theory, the author presents a number of examples of two or three input-, two-output systems in detail. I highly recommend this excellent book to all those interested in the interplay between control theory and algebraic geometry." —Publicationes Mathematicae, Debrecen

"This book is the multivariable counterpart of Methods of Algebraic Geometry in Control Theory, Part I…. In the first volume the simpler single-input–single-output time-invariant linear systems were considered and the corresponding simpler affine algebraic geometry was used as the required prerequisite. Obviously, multivariable systems are more difficult and consequently the algebraic results are deeper and less transparent, but essential in the understanding of linear control theory…. Each chapter contains illustrative examples throughout and terminates with some exercises for further study." —Mathematical Reviews

Authors and Affiliations

  • Division of Applied Mathematics, Brown University, Providence, USA

    Peter Falb

Bibliographic Information

  • Book Title: Methods of Algebraic Geometry in Control Theory: Part II

  • Book Subtitle: Multivariable Linear Systems and Projective Algebraic Geometry

  • Authors: Peter Falb

  • Series Title: Systems & Control: Foundations & Applications

  • DOI: https://doi.org/10.1007/978-1-4612-1564-6

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1999

  • Hardcover ISBN: 978-0-8176-4113-9Published: 01 February 2000

  • Softcover ISBN: 978-1-4612-7194-9Published: 08 October 2012

  • eBook ISBN: 978-1-4612-1564-6Published: 01 December 2013

  • Series ISSN: 2324-9749

  • Series E-ISSN: 2324-9757

  • Edition Number: 1

  • Number of Pages: VIII, 390

  • Topics: Mathematics, general

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