Skip to main content
SpringerLink
Log in

Delay-Dependent H ∞ Filtering for Markovian Jump Time-Delay Systems: A Piecewise Analysis Method

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

Abstract

A delay-dependent H ∞ filtering for Markovian jump systems with time-varying delays is studied based on a piecewise analysis approach. Firstly, by exploiting delay partitioning-based Lyapunov function, a new delay-dependent criterion is derived for the H ∞ performance analysis of the filtering-error systems, which can lead to much less conservative analysis results. Secondly, based on the criterion obtained, the gain of filter can be obtained in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to demonstrate 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.

Similar content being viewed by others

References

  1. M. Basin, E. Sanchez, R. Martinez-Zuniga, Optimal linear filtering for systems with multiple state and observation delays. Int. J. Innov. Comput. Inf. Control 3(5), 1309–1320 (2007)

    Google Scholar 

  2. E. Boukas, Z. Liu, G. Liu, Delay-dependent robust stability and H ∞ control of jump linear systems. Int. J. Control 74(4), 329–340 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Chen, X.P. Liu, S.C. Tong, Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay. Fuzzy Sets Syst. 157, 2224–2240 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Elsayed, M. Grimble, A new approach to the H ∞ design of optimal digital linear filters. IMA J. Math. Control Inf. 6(2), 233–251 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Z. Fei, H. Gao, P. Shi, New results on stabilization of Markovian jump systems with time delay. Automatica 45(10), 2300–2306 (2009)

    Article  MATH  Google Scholar 

  6. H. Gao, J. Lam, C. Wang, Robust energy-to-peak filter design for stochastic time-delay systems. Syst. Control Lett. 55(2), 101–111 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. K. Gu, V. Kharitonov, J. Chen, Stability of Time-Delay Systems (Birkhäuser, Boston, 2003)

    Book  MATH  Google Scholar 

  8. L. El Ghaoui, F. Oustry, M. AitRami, A cone complementarity linearization algorithm for static output—feedback and related problems. IEEE Trans. Autom. Control 42(8), 1171–1176 (1997)

    Article  MATH  Google Scholar 

  9. L. Hu, P. Shi, Y. Cao, Delay-dependent filtering design for time-delay systems with Markovian jumping parameters. Int. J. Adapt. Control Signal Process. 21(5), 434–448 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. X.F. Jiang, Q.L. Han, Robust H ∞ control for uncertain Takagi–Sugeno fuzzy system with interval time-varying delay. IEEE Trans. Fuzzy Syst. 15, 321–331 (2007)

    Article  Google Scholar 

  11. J. Liu, W. Yu, Z. Gu, S. Hu, H ∞ filtering for time-delay systems with Markovian jumping parameters: delay partitioning approach. J. Chin. Inst. Eng. 33(3), 357–365 (2010)

    Article  Google Scholar 

  12. J. Liu, Z. Gu, E. Tian, R. Yan, New results on H ∞ filter design for nonlinear systems with time-delay through a T–S fuzzy model approach. Int. J. Syst. Sci. First published on: 16 August 2010 (iFirst)

  13. X. Mao, Exponential stability of stochastic delay interval systems with Markovian switching. IEEE Trans. Autom. Control 47(10), 1604–1612 (2002)

    Article  Google Scholar 

  14. C. Peng, Y. Tian, Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay. J. Comput. Appl. Math. 214(2), 480–494 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Peng, D. Yue, E. Tian, Y. Zhang, Improved network-based robust H ∞ filtering for uncertain linear systems. Int. J. Innov. Comput. Inf. Control 5(4), 961–970 (2009)

    MathSciNet  Google Scholar 

  16. J. Qiu, J. Chen, P. Shi, H. Yang, New stochastic robust stability criteria for time-varying delay neutral system with Markovian jump parameters. Int. J. Control. Autom. Syst. 8(2), 418–424 (2010)

    Article  Google Scholar 

  17. J. Qiu, H. Yang, P. Shi, Y. Xia, Robust H ∞ Control for class of discrete-time Markovian jump systems with time-varying delays based on delta operator. Circuits Syst. Signal Process. 27(5), 627–643 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. H. Shao, Delay-range-dependent robust H ∞ filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters. J. Math. Anal. Appl. 342(2), 1084–1095 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. C. de Souza, R. Palhares, P. Peres, Robust filter design for uncertain linear systems with multiple time-varying state delays. IEEE Trans. Signal Process. 49(3), 569–576 (2001)

    Article  MathSciNet  Google Scholar 

  20. J. Sun, J. Chen, G. Liu, D. Rees, Delay-dependent robust H ∞ filter design for uncertain linear systems with time-varying delay. Circuits Syst. Signal Process. 28(5), 763–779 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. E. Tian, C. Peng, Delay-dependent stability analysis and synthesis of uncertain T–S fuzzy systems with time-varying delay. Fuzzy Sets Syst. 157(4), 544–559 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. E. Tian, D. Yue, Y. Zhang, Delay-dependent robust H ∞ control for T–S fuzzy system with interval time-varying delay. Fuzzy Sets Syst. 160(12), 1708–1719 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Z. Wang, J. Lam, X. Liu, Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances. IEEE Trans. Circuits Syst. II, Express Briefs 51(5), 262–268 (2004)

    Article  Google Scholar 

  24. Z. Wang, Y. Liu, L. Yu, X. Liu, Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 356(4–5), 46–352 (2006)

    Google Scholar 

  25. Z. Wu, X. Xie, P. Shi, Y. Xia, Brief paper: Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching. Automatica 45(4), 997–1004 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. H.N. Wu, H.X. Li, New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay. IEEE Trans. Fuzzy Syst. 15, 482–493 (2007)

    Article  Google Scholar 

  27. Y. Xia, E. Boukas, P. Shi, J. Zhang, Stability and stabilization of continuous-time singular hybrid systems. Automatica 45(6), 1504–1509 (2009)

    Article  MATH  Google Scholar 

  28. J. Xiong, J. Lam, Robust H 2 control of Markovian jump systems with uncertain switching probabilities. Int. J. Syst. Sci. 40(3), 255–265 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Xu, T. Chen, J. Lam, Robust filtering for uncertain Markovian jump systems with mode-dependent time delays. IEEE Trans. Autom. Control 48(5), 900–907 (2003)

    Article  MathSciNet  Google Scholar 

  30. S. Xu, J. Lam, X. Mao, Delay-dependent H ∞ control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans. Circuits. Syst. I, Regul. Pap. 54(9), 2070–2077 (2007)

    Article  MathSciNet  Google Scholar 

  31. D. Yue, Q. Han, Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching. IEEE Trans. Autom. Control 50(2), 217–222 (2005)

    Article  MathSciNet  Google Scholar 

  32. D. Yue, Q. Han, Network based robust H ∞ filtering for uncertain linear systems. IEEE Trans. Signal Process. 11, 4293–4301 (2006)

    Article  Google Scholar 

  33. D. Yue, E. Tian, Y. Zhang, A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay. Int. J. Robust Nonlinear Control 19(13), 1493–1518 (2009)

    Article  MathSciNet  Google Scholar 

  34. D. Yue, E. Tian, Y. Zhang, C. Peng, Delay-distribution-dependent stability and stabilization of T–S fuzzy systems with probabilistic interval delay. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 39(2), 503–516 (2009)

    Article  Google Scholar 

  35. X. Zhang, Q. Han, A less conservative method for designing H ∞ filters for linear time-delay systems. Int. J. Robust Nonlinear Control 19(12), 1376–1396 (2008)

    Article  MathSciNet  Google Scholar 

  36. X. Zhang, M. Wu, J. She, Y. He, Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41(8), 1405–1412 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. L. Zhang, E. Boukas, Mode-dependent H ∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Automatica 45(6), 1462–1467 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Zhang, P. Shi, H. Yang, Non-fragile robust stabilization and H ∞ control for uncertain stochastic nonlinear time-delay systems. Chaos Solitons Fractals 42(5), 3187–3196 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  39. J. Zhang, Y. Xia, P. Shi, Parameter-dependent robust H ∞ filtering for uncertain discrete-time systems. Automatica 45(2), 560–565 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  40. J. Zhang, Y. Xia, R. Tao, New results on H ∞ filtering for fuzzy time-delay systems. IEEE Trans. Fuzzy Syst. 17(1), 128–137 (2009)

    Article  Google Scholar 

  41. J. Zhang, Y. Xia, P. Shi, Parameter-dependent robust H ∞ filtering for uncertain discrete-time systems. Automatica 45(2), 560–565 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. H. Zhang, H. Yan, J. Liu, Q. Chen, Robust H ∞ filters for Markovian jump linear systems under sampled measurements. J. Math. Anal. Appl. 356(1), 382–392 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Information Science and Technology, Donghua University, Shanghai, 201620, PR China

    Jinliang Liu

  2. School of Mathematics Science, Liaocheng University, Liaocheng, 252059, PR China

    Bingxue Yao

  3. Power Engineering, Nanjing Normal University, Nanjing, Jiangsu, 210042, PR China

    Zhou Gu

Authors
  1. Jinliang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bingxue Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhou Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinliang Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, J., Yao, B. & Gu, Z. Delay-Dependent H ∞ Filtering for Markovian Jump Time-Delay Systems: A Piecewise Analysis Method. Circuits Syst Signal Process 30, 1253–1273 (2011). https://doi.org/10.1007/s00034-011-9275-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-011-9275-z

Keywords

Search

Navigation