Skip to main content
SpringerLink
Log in

Stochastic Stabilization of Discrete-Time Markov Jump Systems with Generalized Delay and Deficient Transition Rates

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

Abstract

In this paper, the problem of mode-dependent state feedback controller design is studied for discrete-time Markov jump systems with generalized delay and deficient transition rates. The time delay under consideration is subject to mode-dependent and time-varying, and the transition probabilities of the jumping process are assumed to be partially accessible. By utilizing some novel summation inequalities, and by constructing an improved Lyapunov–Krasovskii functional which comprises a mode-dependent quadratic functional and some single, double and triple summation terms, delay-dependent stabilization conditions are derived in terms of tractable linear matrix inequalities. Two numerical examples are given to illustrate the effectiveness of the proposed conditions.

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

Similar content being viewed by others

References

  1. W.H. Chen, Z.H. Guan, P. Yu, Delay-dependent stability and \(H_\infty \) control of uncertain discrete-time Markovian jump systems with mode-dependent time delays. Syst. Control Lett. 52(5), 361–376 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. M.C. de Oliveira, R.E. Skelton, Stability test for constrained linear systems. In: Perspectives in Robust Control. Lecture Notes in Control and Information Sciences, vol. 268 (Springer, London, 2001)

  4. A.R. Fioravati, A.P.C. Gonşalves, G.S. Deaecto, H.J.C. Geromel, Obtaining alternative LMI constraints with applications to discrete-time MJLS and switched systems. J. Frankl. Inst. 350(8), 2212–2228 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. J.C. Geromel, A.P. Gonçalves, A.R. Fioravanti, Dynamic output feedback control of discrete-time Markov jump linear systems through linear matrix inequalities. SIAM J. Control Optim. 48(2), 573–593 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. W.S. Gray, O.R. González, M. Doğan, Stability analysis of digital linear flight controllers subject to electromagnetic disturbances. IEEE Trans. Aero. Electron. Syst. 36(4), 1204–1218 (2000)

    Article  Google Scholar 

  7. L.V. Hien, N.T. Dzung, H.B. Minh, A novel approach to state bounding for discrete-time Markovian jump systems with interval time-varying delay. IMA Math. Control Info. 33(2), 293–307 (2016)

    Article  MathSciNet  Google Scholar 

  8. L.V. Hien, N.T. Dzung, H. Trinh, Stochastic stability of nonlinear discrete-time Markovian jump systems with time-varying delay and partially unknown transition rates. Neurocomputing 175, 450–458 (2016)

    Article  Google Scholar 

  9. L.V. Hien, H. Trinh, New finite-sum inequalities with applications to stability of discrete time-delay systems. Automatica 71, 197–201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Huang, S.K. Nguang, State feedback control of uncertain networked control systems with random time delays. IEEE Trans. Autom. Control 53(3), 829–834 (2008)

    Article  MathSciNet  Google Scholar 

  11. O.C. Imer, S. Yüksel, T. Başar, Optimal control of LTI systems over unreliable communication links. Automatica 42(9), 1429–1439 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. O.M. Kwon, M.J. Park, J.H. Park, S.M. Lee, E.J. Cha, Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov-Krasovskii functional. J. Frankl. Inst. 350(3), 521–540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. T.H. Lee, S. Lakshmanan, J.H. Park, P. Balasubramaniam, State estimation for genetic regulatory networks with mode-dependent leakage delays, time-varying delays, and Markovian jumping parameters. IEEE Trans. Nanobiosci. 12(4), 363–375 (2013)

    Article  Google Scholar 

  14. J.N. Li, W.D. Bao, S.B. Li, C.L. Wen, L.S. Li, Exponential synchronization of discrete-time mixed delay neural networks with actuator constraints and stochastic missing data. Neurocomputing 207, 700–707 (2016)

    Article  Google Scholar 

  15. J.N. Li, L.S. Li, Mean-square exponential stability for stochastic discrete-time recurrent neural networks with mixed time delays. Neurocomputing 151, 790–797 (2015)

    Article  Google Scholar 

  16. Z. Li, G. Sun, H. Gao, Guaranteed cost control for discrete-time Markovian jump linear system with time delay. Int. J. Syst. Sci. 44(7), 1312–1324 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.N. Li, Y. Zhang, Y.J. Pan, Mean-square exponential stability and stabilisation of stochastic singular systems with multiple time-varying delays. Circuits Syst. Signal Process. 34(4), 1187–1210 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Lian, J. Liu, Y. Zhuang, Mean stability of positive Markov jump linear systems with homogeneous and switching transition probabilities. IEEE Trans. Circuit. Syst. II: Express Briefs 62(8), 801–805 (2015)

    Google Scholar 

  19. J. Liang, Z. Wang, X. Liu, Distributed state estimation for uncertain Markov-type sensor networks with mode-dependent distributed delays. Int. J. Robust Nonlinear Control 22(3), 331–346 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Lin, S. Fei, J. Shen, Delay-dependent \(H_\infty \) filtering for discrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilities. Signal Process. 91(2), 277–289 (2011)

    Article  MATH  Google Scholar 

  21. M. Liu, D.W.C. Ho, Y. Niu, Stabilization of Markovian jump linear system over networks with random communication delay. Automatica 45(2), 416–421 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Long, S. Zhong, Mean-square exponential stability for a class of discrete-time nonlinear singular Markovian jump systems with time-varying delay. J. Frankl. Inst. 351(10), 4688–4723 (2014)

    Article  MathSciNet  Google Scholar 

  23. S. Ma, E.K. Boukas, Y. Chinniah, Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay. Int. J. Robust Nonlinear Control 20(5), 531–543 (2010)

    MathSciNet  MATH  Google Scholar 

  24. M. Mariton, Jump Linear Systems in Automatic Control (Marcel Dekker, New York, 1990)

    Google Scholar 

  25. N. Meskin, K. Khorasani, Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels. Automatica 45(9), 2032–2040 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. P.T. Nam, P.N. Pubudu, H. Trinh, Discrete Wirtinger-based inequality and its application. J. Frankl. Inst. 352, 1893–1905 (2015)

    Article  MathSciNet  Google Scholar 

  27. W. Qi, X. Gao, \(L_1\) Control for positive Markovian jump systems with time-varying delays and partly known transition rates. Circuits Syst. Signal Process. 34(8), 2711–2726 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. W. Qi, X. Gao, Y. Kao, L. Lian, J. Wang, Stabilization for positive Markovian jump systems with actuator saturation. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-016-0307-6

    Google Scholar 

  29. A. Seuret, F. Gouaisbaut, E. Fridman, Stability of discrete-time systems with time-varying delays via a novel summation inequality. IEEE Trans. Autom. Control 60(10), 2740–2745 (2015)

    Article  MathSciNet  Google Scholar 

  30. H. Shen, L. Su, J.H. Park, Extended passive filtering for discrete-time singular Markov jump systems with time-varying delays. Signal Process. 128, 68–77 (2016)

    Article  Google Scholar 

  31. P. Shi, F. Li, A survey on Markovian jump systems modeling and design. Int. J. Control Autom. Syst 13(1), 1–16 (2015)

    Article  Google Scholar 

  32. V.A. Ugrinovskii, H.R. Pota, Decentralized control of power systems via robust control of uncertain Markov jump parameter systems. Int. J. Control 78(9), 662–677 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Wang, Q. Zhang, V. Sreeram, Design of reduced-order \(H_\infty \) filtering for Markovian jump systems with mode-dependent time delays. Signal Process. 89(2), 187–196 (2009)

    Article  MATH  Google Scholar 

  34. J. Xiong, J. Lam, Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers. Automatica 42(5), 747–753 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Y. Zhang, Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities. Nonlinear Dyn. 75(1), 101–111 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. L. Zhang, E.K. Boukas, Stability and stabilization of Markovian jump linear systems with partially unknown transition probability. Automatica 45(2), 463–468 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Zhang, E.K. Boukas, L. Baron, H.R. Karimi, Fault detection for discrete-time Markov jump linear systems with partially known transition probabilities. Int. J. Control 83(8), 1564–1572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. L. Zhang, E.K. Boukas, J. Lam, Analysis and synthesis of Markov jump linear systems with time-varying delay and partially known transition probabilities. IEEE Trans. Autom. Control 53(10), 2458–2464 (2008)

    Article  MathSciNet  Google Scholar 

  39. H. Zhang, W.S. Gray, O.R. González, Performance analysis of digital flight control systems with rollback error recovery subject to simulated neutron-induced upsets. IEEE Trans. Control Syst. Tech. 16(1), 46–59 (2008)

    Article  Google Scholar 

  40. J. Zhang, Z. Han, F. Zhu, Stochastic stability and stabilization of positive systems with Markovian jump parameters. Nonlinear Anal. Hybrid Syst. 12, 147–155 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Editors and anonymous Reviewers for their constructive comments and helpful suggestions. This work was supported by the NAFOSTED of Vietnam (Grant No. 101.01-2014.35).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hanoi Pedagogical University No. 2, Phuc Yen, Vinh Phuc, Vietnam

    Nguyen Trung Dzung

  2. Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam

    Le Van Hien

Authors
  1. Nguyen Trung Dzung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Le Van Hien
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Le Van Hien.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dzung, N.T., Hien, L.V. Stochastic Stabilization of Discrete-Time Markov Jump Systems with Generalized Delay and Deficient Transition Rates. Circuits Syst Signal Process 36, 2521–2541 (2017). https://doi.org/10.1007/s00034-016-0410-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-016-0410-8

Keywords

Search

Navigation