Skip to main content
SpringerLink
Log in

Finite-Time Stabilization of Switching Markov Jump Systems with Uncertain Transition Rates

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

Abstract

In this paper, the problems of finite-time stability and stabilization for continuous-time switching Markov jump systems with uncertain transition rates are investigated, where some elements in the transition rate matrix are not known precisely, but their bounds are known or completely unknown. The main results are achieved by piecewise integrating the Markov processes and allowing the stochastic multiple Lyapunov function to increase at every switching instant with a limited switching rate such that the state trajectories are kept within the prescribed bound in a short time interval. Finally, two examples illustrate the validity of the obtained results.

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. H. Bo, G. Wang, General observer-based controller design for singular Markovian jump systems. Int. J. Innov. Comput. Inf. Control 10(5), 1897–1913 (2014)

    Google Scholar 

  2. P. Bolzern, P. Colaneri, G.D. Nicolao, Markov jump linear systems with switching transition rates: mean square stability with dwell-time. Automatica 46, 1081–1088 (2010)

    Article  MATH  Google Scholar 

  3. P. Bolzern, P. Colaneri, G.D. Nicolao, Almost sure stability of Markov jump linear systems with deterministic switching. IEEE Trans. Autom. Control 58(1), 209–213 (2013)

    Article  Google Scholar 

  4. E.K. Boukas, Stochastic Switching Systems: Analysis and Design (Birkhauser, Basel, 2005)

    Google Scholar 

  5. P. Dorato, Short time stability in linear time-varying systems. Proceedings of the IRE International Convention Record, Part 4. (New York, 1961), pp. 83–87

  6. J.C. Geromel, P. Colaneri, Stability and stabilization of continuous-time switched linear systems. SIAM J. Control Opt. 45(5), 1915–1930 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. L.L. Hou, G.D. Zong, W.X. Zheng, Exponential \(l_{2}-l_{\infty }\) control for discrete-time switching Markov jump linear systems. Circuit Syst. Signal Process. 32(6), 2745–2759 (2013)

    Article  MathSciNet  Google Scholar 

  8. C.A. Ibanez, M.S. Suarez-Castanon, O. Gutierrez-Frias, A switching controller for the stabilization of the damping inverted pendulum cart system. Int. J. Innov. Comput. Inf. Control 9(9), 3585–3597 (2013)

    Google Scholar 

  9. M. Karan, P. Shi, C.Y. Kaya, Transition probability bounds for the stochastic stability robustness of continuous-and discrete-time Markovian jump linear systems. Automatica 42(12), 2159–2168 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. H. Liu, Y. Shen, \(H_{\infty }\) finite-time control for switched linear systems with time-varying delay. Intell. Control Autom. 2(2), 203–213 (2011)

    Article  Google Scholar 

  11. X.L. Luan, F. Liu, P. Shi, Robust finite-time \(H_{\infty }\) control for nonlinear jump systems via neural networks. Circuit Syst. Signal Process. 29(3), 481–498 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Y. Wang, X. Shi, G. Wang, Z. Zuo, Finite-time stability for continuous-time switched systems in the presence of impulse effects. IET Control Theory Appl. 6(11), 1741–1744 (2011)

    Article  MathSciNet  Google Scholar 

  14. L. Wu, X. Su, P. Shi, Output feedback control of markovian jump repeated scalar nonlinear systems. IEEE Trans. Autom. Control 59(1), 199–204 (2014)

    Article  MathSciNet  Google Scholar 

  15. J.L. 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 

  16. R. Yang, G.P. Liu, P. Shi, C. Thomas, M.V. Basin, Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropouts. IEEE Trans. Ind. Electron. 61(1), 512–520 (2014)

    Article  Google Scholar 

  17. R. Yang, P. Shi, G.P. Liu, Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropouts. IEEE Trans. Autom. Control 11(56), 2655–2660 (2011)

    Article  MathSciNet  Google Scholar 

  18. Y. Yin, F. Liu, P. Shi, Finite-time gain-scheduled control on stochastic bioreactor systems with partially known transition jump rates. Circuits Syst. Signal Process. 30(3), 609–627 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Int. J. Syst. Sci. 32(8), 1055–1061 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Zhang, J. Cao, G. Wang, Mode-independent guaranteed cost control of singular Markovian delay jump systems with switching probability rate design. Int. J. Innov. Comput. Inf. Control 10(4), 1291–1303 (2014)

    Google Scholar 

  21. L.X. Zhang, E.K. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica 45(2), 463–468 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Z. Zuo, H. Li, Y. Liu, Y. Wang, On finite-time stochastic stability and stabilization of Markovian jump systems subject to partial information on transition probabilities. Circuits Syst. Signal Process. 31(6), 1973–1983 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi, 214122, China

    Xiaoli Luan, Changzhong Zhao & Fei Liu

Authors
  1. Xiaoli Luan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changzhong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoli Luan.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61473137 and 61203126).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luan, X., Zhao, C. & Liu, F. Finite-Time Stabilization of Switching Markov Jump Systems with Uncertain Transition Rates. Circuits Syst Signal Process 34, 3741–3756 (2015). https://doi.org/10.1007/s00034-015-0034-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-015-0034-4

Keywords

Search

Navigation