Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Mean-Square Exponential Stability and Stabilisation of Stochastic Singular Systems with Multiple Time-Varying Delays

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

Abstract

The stability and stabilisation problems for a series of continuous stochastic singular systems with multiple time-varying delays are studied in this paper. First, a useful lemma is proposed and a delay-distribution-dependent Lyapunov functional is constructed. Then, a novel delay-distribution-dependent condition is given to ensure the unforced stochastic singular systems to be regular and impulse-free. The mean-square exponential stability of the whole system is guaranteed under the proposed lemma. As a result, a suitable feedback controller is designed via strict linear matrix inequality such that the system’s stabilisation problem is guaranteed. Finally, numerical examples are illustrated to show the proposed result are less conservative than the existing ones and the potential of such technology.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. E.K. Boukas, Control of Singular Systems with Random Abrupt Changes (Springer, Berlin, 2008)

    MATH  Google Scholar 

  2. E.K. Boukas, S. Xu, J. Lam, On stability and stabilizability of singular stochastic systems with delays. J. Optim. Theory Appl. 127(2), 249–262 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Chen, S. Peng, Design of a sliding mode control system for chemical process. J. Process Control 15, 515–530 (2005)

    Article  Google Scholar 

  4. H. Chen, P. Hu, New result on exponential stability for singular systems with two interval time-varying delays. IET Control Theory Appl. 7(15), 1941–1949 (2013)

    Article  MathSciNet  Google Scholar 

  5. L. Dai, Singular Control Systems, Lecture Notes in Control and Information Science (Springer, New York, 1989)

    Book  MATH  Google Scholar 

  6. Z. Feng, J. Lam, H. Gao, Delay-dependent robust \(H_{\infty }\) controller synthesis for discrete singular delay systems. Int. J. Robust Nonlinear Control 21(16), 1880–1902 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Haidar, E.K. Boukas, Exponential stability of singular systems with multiple time-varying delays. Automatica 45, 539–545 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Hale, J.K. Veruyn, S.M. Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)

    MATH  Google Scholar 

  9. H. Huang, J. Cao, Exponential stability analysis of uncertain stochastic neural networks with multiple delays. Nonlinear Anal. Real World Appl. 8, 646–653 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. X. Ji, H. Su, J. Chu, Robust state feedback \(H_{\infty }\) control for uncertain linear discrete singular systems. IET Control Theory Appl. 1(1), 195–200 (2007)

    Article  MathSciNet  Google Scholar 

  11. F.A. Khasawneh, B.P. Mann, A spectral element approach for the stability analysis of time-periodic delay equations with multiple delays. Commun. Nonlinear Sci. Numer. Simul. 18, 2129–2141 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. J.H. Kim, Delay-dependent robust \(H_{\infty }\) filtering for uncertain discrete-time singular systems with interval time-varying delay. Automatica 46, 591–597 (2010)

    Article  MATH  Google Scholar 

  13. F.L. Lewis, A survey of linear singular systems. Circuits Syst. Signal Process. 22, 3–36 (1986)

    Article  Google Scholar 

  14. J. Li, H. Su, Z. Wu, J. Chu, Robust stabilization for discrete-time nonlinear singular systems with mixed time delays. Asian J. Control 14(5), 1411–1421 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Li, H. Su, Y. Zhang, Z. Wu, J. Chu, Chattering free sliding mode control for uncertain discrete time-delay singular systems. Asian J. Control 15(1), 260–269 (2013)

    Article  MathSciNet  Google Scholar 

  16. J. Li, H. Su, Z. Wu, J. Chu, Less conservative robust stability criteria for uncertain discrete stochastic singular systems with time-varying delay. Int. J. Syst. Sci. 44(3), 432–441 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Li, H. Su, Z. Wu, J. Chu, Modelling and control of Zigbee-based wireless networked control system with both network-induced delay and packet dropout. Int. J. Syst. Sci. 44(6), 1160–1172 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Lu, X. Dai, H. Su, J. Chu, A. Xue, Delay-dependent robust stability and stabilization condition for a class of Lur’e singular time-delay systems. Asian J. Control 10(4), 462–469 (2008)

    Article  MathSciNet  Google Scholar 

  19. S. Ma, C. Zhang, \(H_{\infty }\) control for discrete-time singular Markov jump systems subject to actuator saturation. J. Frankl. Inst. 349, 1011–1029 (2012)

    Article  MATH  Google Scholar 

  20. S. Ma, C. Zhang, Z. Cheng, Delay-dependent robust \(H_{\infty }\) control for uncertain discrete-time singular systems with time-delays. J. Comput. Appl. Math. 217, 194–211 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. P.M. Nia, R. Sipahi, Controller design for delay-independent stability of linear time-invariant vibration systems with multiple delays. J. Sound Vib. 332, 3589–3604 (2013)

    Article  Google Scholar 

  22. C. Peng, D. Yue, E. Tian, Z. Gu, A delay distribution based stability analysis and synthesis approach for networked control systems. J. Frankl. Inst. 346, 349–365 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Wang, A. Xue, R. Lu, Absolute stability criteria for a class of nonlinear singular systems with time delay. Nonlinear Anal. 70, 621–630 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. L. Wu, W. Zheng, Passivity-based sliding mode control of uncertain singular time-delay systems. Automatica 45, 2120–2127 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Wu, D.W.C. Ho, Sliding mode control of singular stochastic hybrid systems. Automatica 46, 779–783 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. L. Wu, P. Shi, H. Gao, State estimation and sliding mode control of Markovian jump singular systems. IEEE Trans. Autom. Control 55(5), 1213–1219 (2010)

    Article  MathSciNet  Google Scholar 

  27. Z. Wu, H. Su, J. Chu, Delay-dependent \(H_{\infty }\) filtering for singular Markovian jump time-delay systems. Signal Process. 90, 1815–1824 (2010)

    Article  MATH  Google Scholar 

  28. Z. Wu, H. Su, J. Chu, \(H_{\infty }\) filtering for singular systems with time-varying delay. Int. J. Robust Nonlinear Control 20, 1269–1284 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Z. Wu, P. Shi, H. Su, J. Chu, Delay-dependent stability analysis for switched neural networks with time-varying delay. IEEE Trans. Syst. Man Cybern. B 41(6), 1522–1530 (2011)

    Article  MathSciNet  Google Scholar 

  30. Z. Wu, J.H. Park, H. Su, J. Chu, Stochastic stability analysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilities. J. Frankl. Inst. 349, 2889–2902 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Z. Wu, J.H. Park, H. Su, J. Chu, Admissibility and dissipativity analysis for discrete-time singular systems with mixed time-varying delays. Appl. Math. Comput. 218, 7128–7138 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. J. Xia, J.H. Park, H. Zeng, H. Shen, Delay-difference-dependent robust exponential stability for uncertain stochastic neural networks with multiple delays. Neurocomputing 140, 210–218 (2014)

    Article  Google Scholar 

  33. S. Xu, C. Yang, \(H_{\infty }\) state feedback control for discrete singular systems. IEEE Trans. Autom. Control 45(7), 1405–1409 (2000)

    Article  MATH  Google Scholar 

  34. S. Xu, J. Lam, Y. Zou, An improved characterization of bounded realness for singular delay systems and its applications. Int. J. Robust Nonlinear Control 18, 263–277 (2008)

    Article  MathSciNet  Google Scholar 

  35. S. Xu, J. Lam, Y. Zou, J. Li, Robust admissibility of time-varying singular systems with commensurate time delays. Automatica 45, 2714–2717 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. D. Yue, Q.L. Han, Robust \(H_{\infty }\) filter design of uncertain descriptor systems with discrete and distributed delays. IEEE Trans. Signal Process. 52(11), 3200–3212 (2004)

    Article  MathSciNet  Google Scholar 

  37. D. Yue, J. Lam, D.W.C. Ho, Reliable \(H_{\infty }\) control of uncertain descriptor systems with multiple time delays. IEE Porc. Control Theory Appl. 150(6), 557–564 (2003)

    Article  Google Scholar 

  38. D. Yue, E. Tian, Z. Wang, J. Lam, Stabilization of systems with probabilistic interval input delays and its applications to networked control system. IEEE Trans. Syst. Man Cybern. A 39, 939–945 (2009)

    Article  Google Scholar 

  39. I. Zamani, M. Shafiee, A. Ibeas, Exponential stability of hybrid switched nonlinear singular systems with time-varying delay. J. Frankl. Inst. 350, 171–193 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  40. D. Zhang, L. Yu, Q. Wang, C. Ong, Z. Wu, Exponential \(H_{\infty }\) filtering for discrete-time switched singular systems with time-varying delays. J. Frankl. Inst. 349, 2323–2342 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper. This work was also supported by the National Natural Science Foundation of China (No. 61403113), by the Zhejiang Provincial Natural Science Foundation of China (No. LQ14F030010) and by the Scientific Research Foundation of Hangzhou Dianzi University (No. KYS065613036).

Author information

Authors and Affiliations

  1. Institute of System Science and Control Engineering, School of Automation, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

    Jian-Ning Li

  2. Institute of Automation, Zhejiang Sci-Tech University, Hangzhou, 310018, Zhejiang, People’s Republic of China

    Yibo Zhang

  3. Department of Mechanical Engineering, Dalhousie University, P.O. Box 15000, Halifax, NS, B3H 4R2, Canada

    Ya-Jun Pan

Authors
  1. Jian-Ning Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yibo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ya-Jun Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-Ning Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, JN., Zhang, Y. & Pan, YJ. Mean-Square Exponential Stability and Stabilisation of Stochastic Singular Systems with Multiple Time-Varying Delays. Circuits Syst Signal Process 34, 1187–1210 (2015). https://doi.org/10.1007/s00034-014-9893-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-014-9893-3

Keywords

Search

Navigation