Finite Gain lp Stabilization Is Impossible by Bit-Rate Constrained Feedback

  • Nuno C. Martins
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)


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.


External Excitation Quantization Scheme Uniform Quantizer Small Gain Theorem Logarithmic Quantizer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brocket, R.W., Liberzon, D.: Quantized Feedback Stabilization of Linear Systems. IEEE Transaction on Automatic Control 45(7), 1279–1289 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Elia, N., Mitter, S.K.: Stabilization of Linear Systems With Limited Information. IEEE Transaction on Automatic Control 46(9), 1384–1400 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Delchamps, D.: Stabilizing a Linear System with Quantized State Feedback. IEEE Transaction on Automatic Control 35(8), 916–924 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 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)CrossRefMATHGoogle Scholar
  5. 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Tatikonda, S., Mitter, S.: Control under communication constraints. IEEE Transaction on Automatic Control 49(7), 1056–1068 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Liberzon, D., Hespanha, J.P.: Stabilization of Nonlinear Systems with Limited Information Feedback. IEEE Transaction on Automatic Control (to appear)Google Scholar
  10. 10.
    Liberzon, D., Nesic, D.: Input-to-state stabilization of linear systems with quantized feedback. Submitted to the IEEE Transactions on Automatic ControlGoogle Scholar
  11. 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. 12.
    Jiang, Z.P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Sontag, E.D.: Input to state stability: Basic concepts and results. In: Lecture Notes in Mathematics, Springer, Heidelberg (2005): CIME Course, Cetraro, June 2004Google Scholar
  14. 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. 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. 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. 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 TACGoogle Scholar
  18. 18.
    Martins, N.C.: Finite Gain lp Stabilization Requires Analog Control, Institute for Systems Research Technical Report TR 2005-111Google Scholar
  19. 19.
    Pinsker, M.S.: Information and Information Stability of Random Variables and Processes, Holden Day (1964)Google Scholar
  20. 20.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-Iterscience Publication, Chichester (1991)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nuno C. Martins
    • 1
  1. 1.ISR and ECE Dept.University of MarylandCollege ParkUSA

Personalised recommendations