Skip to main content
SpringerLink
Log in

Finite Gain l p Stabilization Is Impossible by Bit-Rate Constrained Feedback

  • Conference paper
Hybrid Systems: Computation and Control (HSCC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3927))

Included in the following conference series:

Abstract

In this paper, we show that the finite gain (FG) l p stabilization, with 1 ≤ p ≤ ∞, of a discrete-time, linear and time-invariant unstable plant is impossible by bit rate constrained feedback. In addition, we show that, under bit rate constrained feedback, weaker (local) versions of FG l p stability are also impossible. These facts are not obvious, since recent results have shown that input to state stabilization (ISS) is viable by bit-rate constrained control. We establish a comparison with existing work, leading to two conclusions: (1) in spite of ISS stability being attainable under bit rate constrained feedback, small changes in the amplitude of the external excitation may cause, in relative terms, a large increase in the amplitude of the state (2) FG l p stabilization requires logarithmic precision around zero, implying that even without bit-rate constraints FG l p stabilization is impossible in practice. Since our conclusions hold with no assumptions on the feedback structure, they cannot be derived from existing results. We adopt an information theoretic viewpoint, which also brings new insights into the problem of stabilization.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brocket, R.W., Liberzon, D.: Quantized Feedback Stabilization of Linear Systems. IEEE Transaction on Automatic Control 45(7), 1279–1289 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Elia, N., Mitter, S.K.: Stabilization of Linear Systems With Limited Information. IEEE Transaction on Automatic Control 46(9), 1384–1400 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Delchamps, D.: Stabilizing a Linear System with Quantized State Feedback. IEEE Transaction on Automatic Control 35(8), 916–924 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fagnani, F., Zampieri, S.: Stability Analysis and Synthesis for Linear Systems with Quantized State Feedback. IEEE Transaction on Automatic Control 48(9), 1569–1584 (2003)

    Article  MATH  Google Scholar 

  5. Nair, G.N., Evans, R.J.: Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates. SIAM Journal Control and Optim. 43(2), 413–436 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sarma, S., Dahleh, M., Salapaka, S.: On time-varying Bit-Allocation Maintaining Stability:A convex parameterization. In: Proceedings of the IEEE CDC (2004)

    Google Scholar 

  7. Phat, V.N., Jiang, J., Savkin, A.V., Petersen, I.: Robust Stabilization of Linear Uncertain Discrete-Time Systems Via a Limited Capacity Communication Channel. Systems and Control Letters 53, 347–360 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tatikonda, S., Mitter, S.: Control under communication constraints. IEEE Transaction on Automatic Control 49(7), 1056–1068 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liberzon, D., Hespanha, J.P.: Stabilization of Nonlinear Systems with Limited Information Feedback. IEEE Transaction on Automatic Control (to appear)

    Google Scholar 

  10. Liberzon, D., Nesic, D.: Input-to-state stabilization of linear systems with quantized feedback. Submitted to the IEEE Transactions on Automatic Control

    Google Scholar 

  11. Liberzon, D.: On Quantization and Delay Effects in Nonlinear Control Systems. In: Proceedings of the Workshop on Netorked Embeded Sensing and Control, University of Notre Dame (2005) (to appear at the Springer LNCS series)

    Google Scholar 

  12. Jiang, Z.P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sontag, E.D.: Input to state stability: Basic concepts and results. In: Lecture Notes in Mathematics, Springer, Heidelberg (2005): CIME Course, Cetraro, June 2004

    Google Scholar 

  14. Baillieul, J.: Feedback designs in information-based control. In: Proceedings of the Workshop on Stochastic Theory and Control, NY, pp. 35–57 (2001)

    Google Scholar 

  15. Vidyasagar, M.: Nonlinear Systems Analysis (Classics in Applied Mathematics, 42). In: Society for Industrial and Applied Mathematic, 2nd edn. (October 2002)

    Google Scholar 

  16. Martins, N.C., Dahleh, M.A.: Fundamental Limitations in The Presence of Finite Capacity Feedback. In: Proceedings of the ACC (2005)

    Google Scholar 

  17. Martins, N.C., Dahleh, M.A., Elia, N.: Feedback Stabilization of Uncertain Systems inthe Presence of a Direct Link. Accepted for publication in the IEEE TAC

    Google Scholar 

  18. Martins, N.C.: Finite Gain lp Stabilization Requires Analog Control, Institute for Systems Research Technical Report TR 2005-111

    Google Scholar 

  19. Pinsker, M.S.: Information and Information Stability of Random Variables and Processes, Holden Day (1964)

    Google Scholar 

  20. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-Iterscience Publication, Chichester (1991)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ISR and ECE Dept., University of Maryland, College Park, USA

    Nuno C. Martins

Authors
  1. Nuno C. Martins
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Center for Control, Dynamical Systems and Computation, University of California, 93106, Santa Barbara, CA, USA

    João P. Hespanha

  2. SRI International, 333 Ravenswood Ave, Menlo Park, CA, USA

    Ashish Tiwari

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Martins, N.C. (2006). Finite Gain l p Stabilization Is Impossible by Bit-Rate Constrained Feedback. In: Hespanha, J.P., Tiwari, A. (eds) Hybrid Systems: Computation and Control. HSCC 2006. Lecture Notes in Computer Science, vol 3927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11730637_34

Download citation

  • DOI: https://doi.org/10.1007/11730637_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-33170-4

  • Online ISBN: 978-3-540-33171-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation