Skip to main content
SpringerLink
Log in

Noninteraction and Triangular Decoupling Using Geometric Control Theory and Transfer Matrices

  • Conference paper
  • First Online:
Mathematical Control Theory II

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 462))

Abstract

In this chapter we consider linear systems that in addition to a control input and a measurement output also have \(\mu \) exogenous inputs and \(\mu \) exogenous outputs. The main topic is then the design of measurement feedback controllers such that in the closed-loop system the transfer matrix between certain specified pairs of exogenous inputs and outputs is zero. Two cases are considered in particular. First, the case that the off-diagonal blocks of the closed-loop system transfer matrix are zero, resulting in a noninteracting behavior. And second, the case that only the blocks above the main diagonal in the closed-loop system transfer matrix are zero, resulting in triangular decoupling. The techniques in this chapter to derive solvability conditions and measurement feedback controllers are based on transfer matrices and the celebrated geometric approach toward system theory. The main results are necessary and sufficient conditions in geometric or transfer matrix terms for the noninteracting control problem and the triangular decoupling problem. Also variations of these problems are treated, like the version with additional stability requirements, or the ‘almost’ version of the two problems.

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. Hautus, M.L.J.: (A, B)-invariant and stabilizability subspaces, a frequency domain description. Automatica 16, 703–707 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hautus, M.L.J., Heymann, M.: Linear feedback - an algebraic approach. SIAM J. Contr. Optimiz. 16, 83–105 (1979)

    Article  MathSciNet  Google Scholar 

  3. Morse, A.S., Wonham, W.M.: Status of noninteracting control. IEEE Trans. Autom. Control AC-16, 568–581 (1971)

    Google Scholar 

  4. Schumacher, J.M.: (C, A)-invariant subspaces: Some facts and uses. Wiskundig Seminarium Vrije Universiteit Amsterdam, The Netherlands, Report 110 (1979)

    Google Scholar 

  5. Schumacher, J.M.: Compensator synthesis using (C, A, B) pairs. IEEE Trans. Automat. Control 25, 1133–1138 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Trentelman, H.L.: Almost Invariant Subspaces and High Gain Feedback. CWI tracts 29, Amsterdam (1986)

    Google Scholar 

  7. Trentelman, H.L., van der Woude, J.W.: Almost invariance and noninteracting control : a frequency domain analysis. Linear Algebra Appl. 101, 221–254 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Willems, J.C.: Almost nointeracting control design using dynamic state feedback. In: Proceeding of 4th International Conference Analysis and Optique Systems Versailles. Lecture notes in Control and Information Sciences, vol. 28, pp. 555–561. Springer, Berlin (1980)

    Google Scholar 

  9. Willems, J.C.: Almost invariant subspaces: an approach to high feedback design: part I: almost controlled invariant subspaces. IEEE Trans. Automat. Control AC-26, 232–252 (1981)

    Google Scholar 

  10. Willems, J.C.: Almost invariant subspaces: an approach to high feedback design: part II: almost conditionally invariant subspaces. IEEE Trans. Autom. Control AC-27, 1071–1085 (1982)

    Google Scholar 

  11. Willems, J.C., Commault, C.: Disturbance decoupling by measurement feedback with stability or pole-placement. SlAM J. Control Optim. 19, 490–504 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wonham, W.M.: Linear Multivariable Control: A Geometric Approach, 2-nd edn. Springer, Verlag (1979)

    Book  MATH  Google Scholar 

  13. van der Woude, J.W.: Feedback decoupling and stabilization for linear systems with multiple exogenous variables. Ph.D. thesis, Eindhoven University of Technology (1987)

    Google Scholar 

  14. van der Woude, J.W.: Almost disturbance decoupling by measurement feedback : a frequency domain analysis. IEEE Trans. on Automatic Control 35, 570–573 (1990)

    Article  MATH  Google Scholar 

  15. van der Woude, J.W.: On the existence of a common solution \(X\) to the matrix equations \(A_iXB_j=C_{ij}, (i, j) \in \Gamma \). Linear Algebra Appl. 375, 135–145 (2003)

    Article  MathSciNet  Google Scholar 

  16. van der Woude, J.W., Trentelman, H.L.: Non interacting control with internal and input/output stability. In: Proceedings 25th IEEE Conference on Decision and Control, Athens, Greece, pp. 701–702 (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherland

    Jacob van der Woude

Authors
  1. Jacob van der Woude
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacob van der Woude .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology, Mumbai, India

    Madhu N. Belur

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

    M. Kanat Camlibel

  3. Department of Electronics and Computer Science, University of Southampton, Southampton, United Kingdom

    Paolo Rapisarda

  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

van der Woude, J. (2015). Noninteraction and Triangular Decoupling Using Geometric Control Theory and Transfer Matrices. In: Belur, M., Camlibel, M., Rapisarda, P., Scherpen, J. (eds) Mathematical Control Theory II. Lecture Notes in Control and Information Sciences, vol 462. Springer, Cham. https://doi.org/10.1007/978-3-319-21003-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21003-2_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21002-5

  • Online ISBN: 978-3-319-21003-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation