Skip to main content
SpringerLink
Log in

Exponential l 2–l ∞ Control for Discrete-Time Switching Markov Jump Linear Systems

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The problem of exponential l 2–l ∞ control is considered in this paper for a class of discrete-time switching Markov jump linear systems. First, the definition of exponential l 2–l ∞ mean square stability for discrete-time switching Markov jump linear systems is introduced. Then, by resorting to the average dwell time approach, the mean square exponential stability criteria are presented with an exponential l 2–l ∞ performance index and a decay rate, and the corresponding controller is also designed. Finally, numerical and application examples are provided to demonstrate the effectiveness of the obtained results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. P. Bolzern, P. Colaneri, G.D. Nicolao, Markov jump linear systems with switching transition rates: mean square stability with dwell-time. Automatica 46(6), 1081–1088 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. O.L.V. Costa, M.D. Fragoso, R.P. Marques, Discrete-Time Markovian Jump Linear Systems (Springer, New York, 2005)

    Google Scholar 

  3. J. Daffouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans. Autom. Control 47(11), 1883–1887 (2002)

    Article  Google Scholar 

  4. J.B.R. do Val, J.C. Geromel, A.P. Goncalves, The H 2 control for jump linear systems: cluster observations of the Markov state. Automatica 38(2), 343–349 (2002)

    Article  MATH  Google Scholar 

  5. D.S. Du, B. Jiang, P. Shi, S.S. Zhou, Robust l 2–l ∞ control for uncertain discrete-time switched systems with delays. Circuits Syst. Signal Process. 25(6), 729–744 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. H.J. Gao, C.H. Wang, Robust L 2–L ∞ filtering for uncertain systems with multiple time-varying state delays. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50(4), 594–599 (2003)

    Article  Google Scholar 

  7. J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in Proceedings of 38th IEEE Conference on Decision Control, Minneapolis, MN (1999), pp. 2655–2660

    Google Scholar 

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

    Article  Google Scholar 

  9. H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)

    Article  MathSciNet  Google Scholar 

  10. S.P. Ma, E.K. Boukas, Guaranteed cost control of uncertain discrete-time singular Markov jump systems with indefinite quadratic cost. Int. J. Robust Nonlinear Control 21(9), 1031–1045 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Song, J. Fan, M.R. Fei, T.C. Yang, Robust H ∞ control of discrete switched systems with time delay. Appl. Math. Comput. 205(1), 159–169 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. A.N. Vargas, E.F. Costa, J.B.R. do Val, On the control of Markov jump linear systems with no mode observation: application to a DC Motor device. Int. J. Robust Nonlinear Control (2012). doi:10.1002/rnc.2911

    Google Scholar 

  13. L.M. Wang, C. Shao, The design of a hybrid output feedback controller for an uncertain delay system with actuator failures based on the switching method. Nonlinear Anal. Hybrid Syst. 4(1), 165–175 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Wang, J. Zhao, G.M. Dimirovski, G.P. Liu, Output feedback control for uncertain linear systems with faulty actuators based on a switching method. Int. J. Robust Nonlinear Control 19(2), 1295–1312 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. D.A. Wilson, Convolution and Hankel operator norms for linear systems. IEEE Trans. Autom. Control 34(1), 94–97 (1989)

    Article  MATH  Google Scholar 

  16. L. Wu, T. Qi, Z. Feng, Average dwell time approach to L 2–L ∞ control of switched delay systems via dynamic output feedback. IET Control Theory Appl. 3(10), 1425–1436 (2009)

    Article  MathSciNet  Google Scholar 

  17. S.Y. Xu, T.W. Chen, Robust H ∞ control for uncertain discrete-time stochastic bilinear systems with Markovian switching. Int. J. Robust Nonlinear Control 15(5), 201–217 (2005)

    Article  MATH  Google Scholar 

  18. L.X. Zhang, H ∞ estimation for discrete-time piecewise homogeneous Markov jump linear systems. Automatica 45(11), 2570–2576 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. L.X. Zhang, E.K. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica 45(2), 463–468 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. L.X. Zhang, P. Shi, E.K. Boukas, C. Wang, Robust l 2–l ∞ filtering for switched linear discrete time-delay systems with polytopic uncertainties. IET Control Theory Appl. 1(3), 722–730 (2007)

    Article  MathSciNet  Google Scholar 

  21. L.X. Zhang, E.K. Boukas, P. Shi, Exponential H ∞ filtering for uncertain discrete-time switched linear systems with average dwell time: a μ-dependent approach. Int. J. Robust Nonlinear Control 18(11), 1188–1207 (2008)

    Article  MathSciNet  Google Scholar 

  22. W. Zhang, A. Abate, J.H. Hu, M.P. Vitus, Exponential stabilization of discrete-time switched linear systems. Automatica 45(11), 2526–2536 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. G.D. Zong, L.L. Hou, S.Y. Xu, Robust l 2–l ∞ state feedback control for uncertain discrete-time switched systems with mode-dependent time-varying delays. Asian J. Control 12(4), 568–573 (2010)

    MathSciNet  Google Scholar 

  24. G.D. Zong, L.L. Hou, Y.Q. Wu, Robust l 2–l ∞ guaranteed cost filtering for uncertain discrete-time switched systems with mode-dependent time-varying delays. Circuits Syst. Signal Process. 30(1), 17–33 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under grants 61273123, 61273091, 61203013, the Natural Science Foundation of Shandong province under grant ZR2011FM033, ZR2010FQ002, the Shandong Provincial Scientific Research Reward Foundation for Excellent Young and Middle-aged Scientists of China under grant BS2010DX011, and a research grant from the Australian Research Council.

Author information

Authors and Affiliations

  1. Computer Science College, Qufu Normal University, Qufu, 273165, P.R. China

    Linlin Hou

  2. Institute of Automation, Qufu Normal University, Qufu, 273165, P.R. China

    Guangdeng Zong & Yuqiang Wu

  3. School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith South DC, NSW, 1797, Australia

    Weixing Zheng

Authors
  1. Linlin Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangdeng Zong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weixing Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yuqiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangdeng Zong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hou, L., Zong, G., Zheng, W. et al. Exponential l 2–l ∞ Control for Discrete-Time Switching Markov Jump Linear Systems. Circuits Syst Signal Process 32, 2745–2759 (2013). https://doi.org/10.1007/s00034-013-9588-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-013-9588-1

Keywords

Search

Navigation