Skip to main content
SpringerLink
Log in

Stabilization of Discrete-Time Switched Linear Systems: A Control-Lyapunov Function Approach

  • Conference paper
Hybrid Systems: Computation and Control (HSCC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5469))

Included in the following conference series:

Abstract

This paper studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. A number of versions of converse control-Lyapunov function theorems are proved and their connections to the switched LQR problem are derived. It is shown that the system is exponentially stabilizable if and only if there exists a finite integer N such that the N-horizon value function of the switched LQR problem is a control-Lyapunov function. An efficient algorithm is also proposed which is guaranteed to yield a control-Lyapunov function and a stabilizing strategy whenever the system is exponentially stabilizable.

This work was partially supported by the National Science Foundation under Grant CNS-0643805.

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 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. Liberzon, D., Morse, A.S.: Basic problems in stability and design of switched systems. IEEE Control Systems Magazine 19(5), 59–70 (1999)

    Article  Google Scholar 

  2. DeCarlo, R., Branicky, M., Pettersson, S., Lennartson, B.: Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of IEEE, Special Issue on Hybrid Systems 88(7), 1069–1082 (2000)

    Google Scholar 

  3. Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control 43(4), 475–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Skafidas, E., Evans, R.J., Savkin, A.V., Petersen, I.R.: Stability results for switched controller systems. Automatica 35(12), 553–564 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pettersson, S.: Synthesis of switched linear systems. In: IEEE Conference on Decision and Control, Maui, HI, pp. 5283–5288 (December 2003)

    Google Scholar 

  6. Pettersson, S.: Controller design of switched linear systems. In: Proceedings of the American Control Conference, Boston, MA, pp. 3869–3874 (June 2004)

    Google Scholar 

  7. Hai, L., Antsaklis, P.J.: Switching stabilization and L 2 gain performance controller synthesis for discrete-time switched linear systems. In: IEEE Conference on Decision and Control, San Diego, CA, pp. 2673–2678 (December 2006)

    Google Scholar 

  8. Hai, L., Antsaklis, P.J.: Hybrid H  ∞  state feedback control for discrete-time switched linear systems. In: IEEE 22nd International Symposium on Intelligent Control, Singapore, pp. 112–117 (October 2007)

    Google Scholar 

  9. Wicks, M.A., Peleties, P., DeCarlo, R.A.: Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. European J. Control 4(2), 140–147 (1998)

    Article  MATH  Google Scholar 

  10. Pettersson, S., Lennartson, B.: Stabilization of hybrid systems using a min-projection strategy. In: Proceedings of the American Control Conference, Arlington, VA, pp. 223–228 (June 2001)

    Google Scholar 

  11. Albertini, F., Sontag, E.D.: Continuous control-Lyapunov functions for asymptotically controllable time-varying systems. Internat. J. Control 72(18), 1630–1641 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kellett, C.M., Teel, A.R.: Discrete-time asymptotic controllability implies smooth control-Lyapunov function. Systems & Control Letters 52(5), 349–359 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, W., Hu, J.: On Optimal Quadratic Regulation for Discrete-Time Switched Linear Systems. In: International Workshop on Hybrid Systems: Computation and Control, St Louis, MO, USA, pp. 584–597 (2008)

    Google Scholar 

  14. Zhang, W., Hu, J.: On the value functions of the optimal quadratic regulation problem for discrete-time switched linear systems. In: IEEE Conference on Decision and Control, Cancun, Mexico (December 2008)

    Google Scholar 

  15. Zhang, W., Hu, J., Abate, A.: Switched LQR problem in discrete time: Theory and algorithms. Submitted IEEE Transactions on Automatic Control (available upon request) (2008)

    Google Scholar 

  16. Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Englewood Cliffs (2002)

    MATH  Google Scholar 

  17. Zhang, W., Hu, J.: Stabilization of switched linear systems with unstabilizable subsystems. Technical report, Purdue University, TR ECE 08-28 (2008)

    Google Scholar 

  18. Bertsekas, D.P.: Dynamic Programming and Optimal Control, 2nd edn., vol. 2. Athena Scientific (2001)

    Google Scholar 

  19. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical and Computer Engineering, Purdue University, IN 47907, USA

    Wei Zhang & Jianghai Hu

  2. Department of Aeronautics and Astronautics, Stanford University, CA 94305, USA

    Alessandro Abate

Authors
  1. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Abate
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianghai Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of California, 4531E Boelter Hall, CA 90095, Los Angeles, USA

    Rupak Majumdar

  2. Department of Electrical Engineering, University of California, 66-147F Engineering, IV Building, CA 90095-1594, Los Angeles, USA

    Paulo Tabuada

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Zhang, W., Abate, A., Hu, J. (2009). Stabilization of Discrete-Time Switched Linear Systems: A Control-Lyapunov Function Approach. In: Majumdar, R., Tabuada, P. (eds) Hybrid Systems: Computation and Control. HSCC 2009. Lecture Notes in Computer Science, vol 5469. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00602-9_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-00602-9_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00601-2

  • Online ISBN: 978-3-642-00602-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation