© 2015

The Mathematics of Networks of Linear Systems


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Paul A. Fuhrmann, Uwe Helmke
    Pages 1-20
  3. Algebraic Systems Theory: Foundations

    1. Front Matter
      Pages 21-21
    2. Paul A. Fuhrmann, Uwe Helmke
      Pages 23-83
    3. Paul A. Fuhrmann, Uwe Helmke
      Pages 85-140
    4. Paul A. Fuhrmann, Uwe Helmke
      Pages 141-206
  4. Algebraic Systems Theory: Advanced Topics

    1. Front Matter
      Pages 207-207
    2. Paul A. Fuhrmann, Uwe Helmke
      Pages 209-279
    3. Paul A. Fuhrmann, Uwe Helmke
      Pages 281-353
    4. Paul A. Fuhrmann, Uwe Helmke
      Pages 355-408
  5. Networks of Linear Systems

    1. Front Matter
      Pages 409-409
    2. Paul A. Fuhrmann, Uwe Helmke
      Pages 411-466
    3. Paul A. Fuhrmann, Uwe Helmke
      Pages 467-505
    4. Paul A. Fuhrmann, Uwe Helmke
      Pages 507-552
    5. Paul A. Fuhrmann, Uwe Helmke
      Pages 553-600
    6. Paul A. Fuhrmann, Uwe Helmke
      Pages 601-644
  6. Back Matter
    Pages 645-662

About this book


This book provides the mathematical foundations of networks of linear control systems, developed from an algebraic systems theory perspective. This includes a thorough treatment of questions of controllability, observability, realization theory, as well as feedback control and observer theory. The potential of networks for linear systems in controlling large-scale networks of interconnected dynamical systems could provide insight into a diversity of scientific and technological disciplines. The scope of the book is quite extensive, ranging from introductory material to advanced topics of current research, making it a suitable reference for graduate students and researchers in the field of networks of linear systems. Part I can be used as the basis for a first course in algebraic system theory, while Part II serves for a second, advanced, course on linear systems.

Finally, Part III, which is largely independent of the previous parts, is ideally suited for advanced research seminars aimed at preparing graduate students for independent research. “Mathematics of Networks of Linear Systems” contains a large number of exercises and examples throughout the text making it suitable for graduate courses in the area.


Algebraic Systems Theory Control Engineering Cyber-Physical Systems Functional Models and Shift Spaces Graph Theory Interconnected Dynamical Systems Linear Systems Network Science Synchronization

Authors and affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Universität Würzburg, Institute of MathematicsWürzburgGermany

About the authors

Paul A. Fuhrmann is Professor Emeritus in the Department of Mathematics at Ben-Gurion University of the Negev. His current research interests are in the following directions: Algebraic System Theory, control, observation and model reduction for networks of linear systems.

Uwe Helmke is Professor and Chair in Dynamical Systems and Control Theory at the University of Wuerzburg. His current research interests include Algebraic and Differential-Geometric Methods in Control, Linear Systems Theory, Optimization on Manifolds, Formation Control.

Bibliographic information

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“In the voluminous book under review the authors illustrate their perspective on modelling and control of networks of linear finite-dimensional systems. … This is an excellent book that can be appreciated both by readers familiar with the polynomial approach, and by readers unaccustomed to functional models. … The historical sections at the end of each chapter are helpful because they explain the broader context and suggest further reading to both types of readers.” (Paolo Rapisarda, Mathematical Reviews, March, 2016)