Skip to main content
SpringerLink
Log in

A Complement on Elimination and Realization in Rational Representations

  • Conference paper
  • First Online:
Mathematical Control Theory I

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 461))

  • 2283 Accesses

Abstract

In this paper we study a number of problems in the context of rational representations of behaviors. In that context, a given proper real rational matrix can represent three behaviors. In the first place it can represent an input–output behavior. Second, it can represent the kernel behavior of the rational ‘differential operator’ associated with the rational matrix. Third, it can represent the image behavior asociated with the rational matrix. On the other hand, every proper real rational matrix admits a realization as a finite-dimensional linear state-space system. Such realization can represent three system behaviors: an input-state-output behavior, an output nulling behavior, or a driving variable behavior. In this paper we will study the relation between the three external behaviors of these state representations, and the behaviors given by the three rational representations associated with the underlying rational matrix. Preliminary results from [5] will be complemented to obtain necessary and sufficient conditions such that the respective external behaviors are equal.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H. Aling, J.M. Schumacher, A nine-fold canonical decomposition for linear systems. Int. J. Control 39(4), 779–805 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. P.J. Antsaklis, A.N. Michel, A Linear Systems Primer. (Birkhäuser Boston, 2007)

    Google Scholar 

  3. L. Ehrenpreis, Fourier Analysis In Several Complex Variables, Pure and Applied Mathematices, vol. XVII (Wiley-Interscience Publishers A Division of John Wiley and Sons, New York, 1970)

    Google Scholar 

  4. S.V. Gottimukkala, S. Fiaz, H.L. Trentelman, Equivalence of rational representations of behaviors. Syst. Control Lett. 60, 119–127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. S.V. Gottimukkala, H.L. Trentelman, S. Fiaz, Realization and elimination in rational representations of behaviors. Syst. Control Lett. 62, 708–714 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Kuijper, First-Order Representations of Linear Systems (Birkhäuser, 1994)

    Google Scholar 

  7. H.B. Minh, Model Reduction in a Behavioral Framework, Doctoral Dissertation (Rijksuniversiteit Groningen, 2009). http://dissertations.ub.rug.nl/faculties/science/2009/h.b.minh

  8. M.E.C. Mutsaers, S. Weiland, Rational representations and controller synthesis of \({\cal L}\) behaviors. Automatica 48, 1–14 (2012)

    Google Scholar 

  9. M.E.C. Mutsaers, Control Relevant Model Reduction and Controller Synthesis for Complex Dynamical Systems, Doctoral dissertation, Technische Universiteit Eindhoven, 2012. http://repository.tue.nl/734624

  10. V.P. Palamodov, Linear Differential Operators With Constant Coefficients, Translated from the Russian by A.A. Brown. Die Grundlehren der mathematischen Wissenschaften, Band 169, (Springer, New York, 1970)

    Google Scholar 

  11. J.W. Polderman, J.C. Willems, Introduction to Mathematical Systems Theory: a Behavioral Approach (Springer, Berlin, 1997)

    MATH  Google Scholar 

  12. H.L. Trentelman, A.A. Stoorvogel, M.L.J. Hautus, Control Theory Linear System (Springer, London, 2001)

    Book  Google Scholar 

  13. M.E. Valcher, A Note on the Driving Variable Realization of Behaviors, in Proceedings of 42nd IEEE Conference on Decision and Control, pp. 1627–1632, 9–12, Maui, Hawai (USA), Dec. (2003)

    Google Scholar 

  14. S. Weiland, Theory of Approximation and Disturbance Attenuation for Linear Systems (Doctoral Dissertation, Rijksuniversiteit Groningen, 1991)

    Google Scholar 

  15. S. Weiland, A.A. Stoorvogel, Rational representations of behaviors: interconnectability and stabilizability. Math. Control Signal Syst. 10, 125–164 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.C. Willems, Input-output and state-space representations of finite-dimensional linear time-invariant systems. Linear Algebra Appl. 50, 581–608 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.C. Willems, Y. Yamamoto, Behaviors defined by rational functions. Linear Algebra Appl. 425, 226–241 (2007)

    Article  MathSciNet  Google Scholar 

  18. K. Zhou, J.C. Doyle, Essentials of Robust Control (Prentice Hall, 1998)

    Google Scholar 

  19. E. Zerz, Topics in Multidimensional Linear Systems Theory, Springer Lecture Notes in Control and Information Sciences 2564 (Springer, London, 2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P. O Box 800, 9700 AV, Groningen, The Netherlands

    Harry L. Trentelman & Tjerk W. Stegink

  2. Hightech Solutions B.V. Apeldoorn, Apeldoorn, The Netherlands

    Sasanka V. Gottimukkala

Authors
  1. Harry L. Trentelman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tjerk W. Stegink
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sasanka V. Gottimukkala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harry L. Trentelman .

Editor information

Editors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    M. Kanat Camlibel

  2. Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA

    A. Agung Julius

  3. Department of Electrical Engineering, Indian Institute of Technology, Chennai, Tamil Nadu, India

    Ramkrishna Pasumarthy

  4. Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Trentelman, H.L., Stegink, T.W., Gottimukkala, S.V. (2015). A Complement on Elimination and Realization in Rational Representations. In: Camlibel, M., Julius, A., Pasumarthy, R., Scherpen, J. (eds) Mathematical Control Theory I. Lecture Notes in Control and Information Sciences, vol 461. Springer, Cham. https://doi.org/10.1007/978-3-319-20988-3_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20988-3_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20987-6

  • Online ISBN: 978-3-319-20988-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation