Skip to main content

Inverse Radial Matrices and Maximal Stability Robustness

  • Chapter
Book cover Control of Uncertain Systems

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 6))

Abstract

In this work properties of inverse radial matrices and their relations to generalized stability radii are discussed. A characterization of an inverse radial matrix is introduced. The connection between these matrices and the stability radii is presented. The case of equality between the complex and the real stability radii is characterized.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.

    Book  Google Scholar 

  2. G. H. Golub and C. F. Van - Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore Maryland, 1983.

    Google Scholar 

  3. D. Hinrichsen, A. J. Pritchard, “Stability Radii of Linear Systems”, System 4 Control Letters, Vol. 8, pp. 1–10, 1986.

    Article  Google Scholar 

  4. J. Kautsky, N. K. Nichols, P. Van Dooren, “Robust Pole Assignment in Linear State Feedback”, Int. J. Contr., Vol. 41, pp. 1129–1155, 1985.

    Article  Google Scholar 

  5. C. F. Van - Loan, “How Near is a Stable Matrix to an Unstable Matrix” in Linear Algebra and its Role in Systems Theory, in Proc. AMS-IMSSIAM Conf. held July 29 - Aug. 4, 1984. R. A. Brualdi et al. Eds., Contemporary Mathematics, American Mathematical Society, Vol. 47, 1985.

    Google Scholar 

  6. L. Qiu, E. J. Davison, “New Perturbation Bounds for the Robust Stability of Linear State Space Models”, Proc. C.D.C. Athens,pp. 751–755, 1986.

    Google Scholar 

  7. J. M. Martin, “State Space Measures for Stability Robustness”, IEEE Trans. Auto. Cont., Vol. AC-32, pp. 509–512, June 1987.

    Google Scholar 

  8. W. Rudin, Functional Analysis, McGraw - Hill, 1973.

    Google Scholar 

  9. W. Rudin, Real and Complex Analysis, McGraw - Hill, third edition, 1987.

    Google Scholar 

  10. M. Goldberg, G. Zwass, “On Matrices Having Equal Spectral Radius and Spectral Norm”, Linear Algebra and its Applications, Vol. 8, pp. 427–434, 1974.

    Article  Google Scholar 

  11. I. Lewkowicz, “Remarks on Inequality in Johnson’s Lower Bound for the Smallest Singular Value”, Linear Algebra and its Applications, Vol. 120, pp. 39–46, Aug. 1989.

    Article  Google Scholar 

  12. D. Hinrichsen, A. J. Pritchard, “A note on some differences between real and complex stability radii”, Report Nr. 215, Inst. fuer Dynamische Systeme, Universität Bremen 1989, to appear in Systems and Control Letters.

    Google Scholar 

  13. A. J. Pritchard, S. Townley, “Robustness of Linear Systems”, Journal of Differential Equations, Vol 77, No. 2, pp. 254–286, February 1989.

    Article  Google Scholar 

  14. D. Hinrichsen, A. Ilchmann, A J Pritchard, “Robustness of Stability of Time Varying Linear Systems”, Journal of Differential Equations, Vol. 82, No 2, pp. 219–250, December 1989.

    Article  Google Scholar 

  15. P. P. Khargonekar, I. R. Petersen, K. Zhou, “Robust Stabilization of Uncertain Linear Systems: Quadratic Stability and H, Theory”, to appear in IEEE Trans. Auto. Control

    Google Scholar 

  16. I. Lewkowicz, R. Sivan, “Maximal Stability Robustness for State Equations”, IEEE Trans. Auto. Control, Vol. AC–33, pp. 297–300, March 1988.

    Google Scholar 

  17. D. Hinrichsen, A. J. Pritchard, “New Robustness Results for Linear Systems under Real Perturbations”, Proc. 27th C.D.C., Austin Texas, 1988, pp. 1375–1379.

    Google Scholar 

  18. L. Qiu and E. J. Davison, “A New Method for the Stability Robustness Determination of State Space Models with Real Perturbations”, Proc. 27th C.D.C., Austin Texas, 1988, pp. 538–543.

    Google Scholar 

  19. D. Hinrichsen, M. Motscha, “Optimization Problems in Robustness Analysis of Linear State Space Systems”, in Approximation and Optimization Proceedings, Havana 1987, Lecture Notes in Mathematics, pp. 54–77, No. 1354, Springer Verlag 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lewkowicz, I., Sivan, R. (1990). Inverse Radial Matrices and Maximal Stability Robustness. In: Hinrichsen, D., Mårtensson, B. (eds) Control of Uncertain Systems. Progress in Systems and Control Theory, vol 6. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-2108-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-2108-9_9

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4757-2110-2

  • Online ISBN: 978-1-4757-2108-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics