Skip to main content
SpringerLink
Log in

Non-Fragile \(H_{\infty }\) Filter Design for Uncertain Stochastic Nonlinear Time-Delay Markovian Jump Systems

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

Abstract

This paper investigates the problem of non-fragile and mode-dependent \(H_{\infty }\) filter design for a class of nonlinear stochastic Markovian jump systems with mode-dependent time-varying delays and norm-bounded parameter uncertainties. The non-fragile and mode-dependent filter to be designed is assumed to include multiplicative gain variations which result from inaccuracies in filter implementation, and the desired filter ensures the filtering error system is not only exponentially mean-square stable, but also satisfies a prescribed \(H_{\infty }\)-norm level for all admissible uncertainties. A stochastic Lyapunov–Krasovskii function is proposed to reflect the information of Markovian jump modes and the mode-dependent time-varying delays, and a set of strict linear matrix inequalities are utilized to derive sufficient conditions that guarantee the desired filter can be constructed. A numerical example and a vertical take-off and landing (VTOL) helicopter system are utilized to illustrate the effectiveness and usefulness of the main results obtained.

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
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. E. Boukas, Stochastic Switching Systems: Analysis and Design (Birkhäuser, Boston, 2005)

    Google Scholar 

  2. X. Chang, G. Yang, Non-fragile fuzzy \(H_{\infty }\) filter design for nonlinear continuous-time systems with D stability constraints. Signal Process. 92, 575–586 (2012)

    Article  Google Scholar 

  3. Y. Chen, W. Zheng, Exponential \(H_{\infty }\) filtering for stochastic Markovian jump systems with time delays. Int. J. Robust. Nonlinear Control 24(4), 625–643 (2014)

  4. M. Chen, D. Wang, Z.S. Wang, Non-fragile \(H_{\infty }\) filtering for uncertain stochastic time-delay systems with Markovian jump parameters. Syst. Eng. Electr. 32(5), 1019–1023 (2010)

    MATH  Google Scholar 

  5. D. de Farias, J. Geromel, J. do Val, O. Costa, Output feedback control of Markov jump linear systems in continuous time. IEEE Trans. Autom. Control 45(5), 944–949 (2000)

    Article  MATH  Google Scholar 

  6. H. Dong, Z. Wang, H. Gao, Fault detection for Markovian jump systems with sensor saturations and randomly varying nonlinearities. IEEE Trans. Circuits Syst. Regul. Pap. 59(10), 2354–2362 (2012)

    Article  MathSciNet  Google Scholar 

  7. H. Dong, Z. Wang, D.W.C. Ho, H. Gao, Robust filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case. IEEE Trans. Signal Process. 59(7), 3048–3057 (2011)

    Article  MathSciNet  Google Scholar 

  8. H. Dong, Z. Wang, H. Gao, Distributed \(H_{\infty }\) filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks. IEEE Trans. Ind. Electron. 60(10), 4665–4672 (2013)

    Article  Google Scholar 

  9. M. Fragoso, N. Rocha, Stationary filter for continuous-time Markovian jump linear systems. SIAM J. Control Optim. 44(3), 801–815 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Gao, C. Wang, Delay-dependent Robust \(H_{\infty }\) and \(L_{2}-L_{\infty }\) filtering for a class of uncertain nonlinear time-delay systems. IEEE Trans. Autom. Control 48(9), 1661–1666 (2003)

    Article  Google Scholar 

  11. H. Gao, X. Meng, T. Chen, New design of robust \(H_{\infty }\) filter for 2-D system. IEEE Signal Process. Lett. 18, 217–220 (2008)

    Google Scholar 

  12. H. Li, Y. Shi, Robust \(H_{\infty }\) filtering for nonlinear stochastic systems with uncertainties and Markov delays. Automatica 48, 159–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Lien, Non-fragile guaranteed cost control for uncertain neutral dynamic systems with time-varying delays in state and control input. Chaos, Solitons Fractals 31(4), 889–899 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Lin, J. Lo, Robust mixed \(L2-L_{\infty }\) filtering for time-delay fuzzy systems. IEEE Trans. Signal Process. 54(8), 2897–2909 (2006)

    Article  Google Scholar 

  15. Z. Lin, Y. Lin, W. Zhang, \(H_{\infty }\) filtering for non-linear stochastic Markovian jump systems. IET Control Theory Appl. 4(12), 2743–2756 (2010)

    Article  MathSciNet  Google Scholar 

  16. H. Liu, D. Ho, F. Sun, Design of \(H_{\infty }\) filter for Markov jumping linear systems with non-accessible mode information. Automatica 44, 2655–2660 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Liu, D. Qi, S. Zhang, M. Qiu, S. Zheng, Optimal \(H_{\infty }\) fusion filters for a class of discrete-time intelligent systems with time delays and missing measurement. Neurocomputing 74, 3741–3752 (2011)

    Article  Google Scholar 

  18. R. Lu, H. Li, Y. Zhu, Quantized \(H_{\infty }\) filtering for singular time-varying delay systems with unreliable communication channel. Circuits Syst. Signal Process. 31(2), 521–538 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Ma, Z. Wang, J. Hu, Y. Bo, Z. Guo, Robust variance-constrained filtering for a class of nonlinear stochastic systems with missing measurements. Signal Process. 90, 2060–2071 (2010)

    Article  MATH  Google Scholar 

  20. X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching (Imperial College Press, London, 2006)

    Book  MATH  Google Scholar 

  21. Y. Niu, D. Ho, C. Li, Filtering for discrete fuzzy stochastic systems with sensor nonlinearities. IEEE Trans. fuzzy Syst. 18(5), 971–978 (2010)

    Article  Google Scholar 

  22. B. Shen, Z. Wang, H. Shu, G. Wei, Robust \(H_{\infty }\) finite-horizon filtering with randomly occurred nonlinearities and quantization effects. Automatica 46(11), 1743–1751 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Shi, E. Boukas, R. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters. IEEE Trans. Autom. Control 44(8), 1592–1597 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Subramanian, A. Sayed, Multi-objective filter design for uncertain stochastic time-dealy systems. IEEE Trans. Autom. Control 49(1), 149–154 (2004)

    Article  MathSciNet  Google Scholar 

  25. E. Tian, D. Yue, Reliable \(H_{\infty }\) filter design for T-S fuzzy model-based networked control systems with random sensor failure. Int. J. Robust. Nonlinear Control 23, 15–32 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Y. Wang, L. Xie, C. de Souza, Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 19, 139–149 (1992)

    Article  Google Scholar 

  27. Z. Wang, B. Shen, X. Liu, \(H_{\infty }\) filtering with randomly occurring sensor saturations and missing measurements. Automatica 48, 556–562 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Wu, D. Ho, Fuzzy filter design for nonlinear Itô stochastic systems with application to sensor fault detection. IEEE Trans. Fuzzy Syst. 17(1), 233–242 (2009)

    Article  Google Scholar 

  29. L. Xie, M. Fu, C. de Souza, \(H_{\infty }\) control and quadratic stabilization of systems with uncertainty via output feedback. IEEE Trans. Autom. Control 37(8), 1253–1256 (1992)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. S. Xu, J. Lam, G. Yang, J. Wang, Stabilization and \(H_{\infty }\) control for uncertain stochastic time-delay systems via non-fragile controllers. Asian J. Control 8(2), 197–200 (2006)

    Article  MathSciNet  Google Scholar 

  32. D. Yue, J. Lam, Non-fragile guaranteed cost control for uncertain descriptor systems with time-varying state and input delays. Optim. Control Appl. Methods 26(2), 85–105 (2005)

    Article  MathSciNet  Google Scholar 

  33. G. Yang, W. Che, Non-fragile \(H_{\infty }\) filter design for linear continuous-time systems. Automatica 44, 2849–2856 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Zhang, W. Zheng, S. Xu, On robust \(H_{\infty }\) filtering of uncertain Markovian jump time-delay systems. Int. J. Adapt. Control Signal Process. 26(2), 138–157 (2012)

    Article  MathSciNet  Google Scholar 

  35. J. Zhang, P. Shi, J. Qiu, Non-Fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems. J. Franklin Inst. 346, 676–690 (2009)

    Article  MathSciNet  Google Scholar 

  36. L. Zhang, E. Boukas, Mode-dependent \(H_{\infty }\) filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Automatica 45, 1462–1467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. X. Zhang, Q. Han, Robust \(H_{\infty }\) filtering for a class of uncertain linear systems with time-varying delay. Automatica 44(1), 157–166 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. X. Zhang, Q. Han, Network-based \(H_{\infty }\) filtering using a logic-like jumping trigger. Automatica 49(5), 1428–1435 (2013)

    Article  MathSciNet  Google Scholar 

  39. X. Zhang, Q. Han, Network-based \(H_{\infty }\) filtering for discrete-time systems. IEEE Trans. Signal Process. 60(2), 956–961 (2012)

    Article  MathSciNet  Google Scholar 

  40. F. Zheng, A robust \(L_{2}\) filtering approach and its application to equalizer design for communication systems. IEEE Trans. Signal Process. 53(8), 2735–2747 (2005)

    Article  MathSciNet  Google Scholar 

  41. S. Zhou, W. Ren, J. Lam, Stabilization for T–S model based uncertain stochastic systems. Inf. Sci. 181, 779–791 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editors and the referees for carefully reading the paper and for their comments which have helped us to greatly improve the paper. This work is supported by Liaocheng University Foundation under Grant X09005.

Author information

Authors and Affiliations

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

    Guangming Zhuang & Yunliang Wei

  2. School of Mathematical Sciences, Liaocheng University, Liaocheng , 252059, Shandong, People’s Republic of China

    Guangming Zhuang

Authors
  1. Guangming Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunliang Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangming Zhuang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhuang, G., Wei, Y. Non-Fragile \(H_{\infty }\) Filter Design for Uncertain Stochastic Nonlinear Time-Delay Markovian Jump Systems. Circuits Syst Signal Process 33, 3389–3419 (2014). https://doi.org/10.1007/s00034-014-9809-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-014-9809-2

Keywords

Search

Navigation