Skip to main content
SpringerLink
Log in

Stability Analysis and Stabilization of Discrete-Time 2D Switched Systems

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

Abstract

This paper is concerned with stability analysis and stabilization problems for two-dimensional (2D) discrete switched systems represented by a model of Roesser type. First, sufficient conditions for the exponential stability of the 2D discrete switched system are derived via the average dwell time approach. Then, based on this result, a state feedback controller is designed to achieve the exponential stability of the corresponding closed-loop system. All the results are presented in linear matrix inequalities (LMIs) form. A numerical example is given to illustrate the effectiveness of the proposed method.

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
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. B. Anderson, P. Agathoklis, E.I. Jury, M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems. IEEE Trans. Circuits Syst. 33(3), 261–267 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Benzaouia, A. Hmamed, F. Tadeo, Stability conditions for discrete 2D switching systems, based on a multiple Lyapunov function, in European Control Conference, August, Budapest, Hungary (2009), pp. 23–26

    Google Scholar 

  3. A. Benzaouia, A. Hmamed, F. Tadeo, A.E. Hajjaji, Stabilisation of discrete 2D time switching systems by state feedback control. Int. J. Syst. Sci. 42(3), 479–487 (2011)

    Article  MATH  Google Scholar 

  4. J. Bochniak, K. Galkowski, E. Rogers, Multi-machine operations modelled and controlled as switched linear repetitive processes. Int. J. Control 81(10), 1549–1567 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bochniak, K. Galkowski, E. Rogers, D. Mehdi, O. Kummert, A. Bachelier, Stabilization of discrete linear repetitive processes with switched dynamics. Multidimens. Syst. Signal Process. 17(2/3), 271–295 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. S.P. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  7. M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 186–200 (1998)

    Article  MathSciNet  Google Scholar 

  8. S.F. Chen, Stability analysis for 2-D systems with interval time-varying delays and saturation nonlinearities. IEEE Trans. Signal Process. 90(7), 2265–2275 (2010)

    MATH  Google Scholar 

  9. C. Du, L. Xie, Control and Filtering of Two-Dimensional Systems (Springer, Berlin, 2002)

    MATH  Google Scholar 

  10. E. Fornasini, G. Marchesini, Stability analysis of 2-D systems. IEEE Trans. Circuits Syst. 33(3), 1210–1217 (1986)

    MathSciNet  Google Scholar 

  11. K. Galkowski, E. Rogers, S. Xu, J. Lam, D.H. Owens, LMIs-a fundamental tool in analysis and controller design for discrete linear repetitive processes. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(6), 768–778 (2002)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  13. K. Hu, J. Yuan, Improved robust H ∞ filtering for uncertain discrete-time switched systems. IET Control Theory Appl. 3(3), 315–324 (2009)

    Article  MathSciNet  Google Scholar 

  14. J. Lodge, M. Fahmy, Stability and overflow oscillations in 2-D state-space digital filters. IEEE Trans. Acoust. Speech Signal Process. 29(6), 1161–1171 (1981)

    Article  MathSciNet  Google Scholar 

  15. W.S. Lu, Two-Dimensional Digital Filters (Marcel Dekker, New York, 1992)

    MATH  Google Scholar 

  16. W.S. Lu, E. Lee, Stability analysis for two-dimensional systems via a Lyapunov approach. IEEE Trans. Circuits Syst. 32(1), 61–68 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Kaczorek, New stability tests of positive standard and fractional linear systems. Circuits Syst. 2(4), 261–268 (2011)

    Article  MathSciNet  Google Scholar 

  18. T. Kaczorek, Two-Dimensional Linear Systems (Springer, Berlin, 1985)

    MATH  Google Scholar 

  19. H. Kar, V. Singh, Stability of 2-D systems described by the Fornasini–Marchesini first model. IEEE Trans. Signal Process. 51(6), 1675–1676 (2003)

    Article  MathSciNet  Google Scholar 

  20. J. Klamka, Controllability of infinite-dimensional 2-D linear systems. Adv. Syst. Sci. Appl. 1(1), 537–543 (1997)

    MathSciNet  Google Scholar 

  21. J. Klamka, Controllability of 2-D nonlinear systems. Nonlinear Anal. 30(5), 2963–2968 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Klamka, Controllability of 2-D systems: a survey. Appl. Math. Comput. Sci. 7(4), 101–120 (1997)

    MathSciNet  Google Scholar 

  23. J. Klamka, Constrained controllability of positive 2-D systems. Bull. Pol. Acad. Sci., Tech. Sci. 46(1), 95–104 (1998)

    MATH  Google Scholar 

  24. J. Klamka, Controllability of 2-D continuous-discrete systems with delays in control. Bull. Pol. Acad. Sci., Tech. Sci. 46(3), 363–373 (1998)

    MathSciNet  MATH  Google Scholar 

  25. P. Peleties, R.A. DeCarlo, Asymptotic stability of m-switched systems using Lyapunov functions, in Proceedings of the 31th IEEE Conference on Decision and Control, (1992), pp. 3438–3439

    Google Scholar 

  26. E. Rogers, K. Galkowski, D.H. Owens, Control Systems Theory and Applications for Linear Repetitive Processes. LNCIS, vol. 349 (Springer, Berlin, 2007)

    Google Scholar 

  27. Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and stabilization for discrete-time switched systems with mode-dependent time-varying delays. Appl. Math. Comput. 180(2), 428–435 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and H ∞ control for uncertain discrete-time switched systems with mode-dependent time delays. Appl. Math. Comput. 187(2), 1228–1237 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. M.E. Valcher, On the internal stability and asymptotic behavior of 2D positive systems. IEEE Trans. Circuits Syst. 44(7), 602–613 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y.J. Wang, Z.X. Yao, Z.Q. Zuo, H.M. Zhao, Delay-dependent robust H ∞ control for a class of switched systems with time delay, in IEEE International Symposium on Intelligent Control (2008), pp. 882–887

    Chapter  Google Scholar 

  31. H. Xu, X. Liu, K.L. Teo, A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays. Nonlinear Anal. Hybrid Syst. 2(1), 38–50 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Ye, W. Wang, Stability analysis and stabilisation for a class of 2-D nonlinear discrete systems. Int. J. Syst. Sci. 42(5), 839–851 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach, in Proceedings of the American Control Conference (2000), pp. 200–204

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their valuable comments which helped to significantly improve the quality and presentation of this paper.

This work was supported by the National Natural Science Foundation of China under Grant No. 60974027 and NUST Research Funding (2011YBXM26).

Author information

Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Zhengrong Xiang & Shipei Huang

Authors
  1. Zhengrong Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shipei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhengrong Xiang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiang, Z., Huang, S. Stability Analysis and Stabilization of Discrete-Time 2D Switched Systems. Circuits Syst Signal Process 32, 401–414 (2013). https://doi.org/10.1007/s00034-012-9464-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-012-9464-4

Keywords

Search

Navigation