Finite-Time Control for Markov Jump Systems with Partly Known Transition Probabilities and Time-Varying Polytopic Uncertainties

  • Chen ZhengEmail author
  • Xiaozheng Fan
  • Manfeng Hu
  • Yongqing Yang
  • Yinghua Jin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9377)


In this paper, the finite-time control problem for Markov systems with partly known transition probabilities and polytopic uncertainties is investigated. The main result provided is a sufficient conditions for finite-time stabilization via state feedback controller, and a simpler case without controller is also considered, based on switched quadratic Lyapunov function approach. All conditions are shown in the form of LMIs. An illustrative example is presented to demonstrate the result.


finite-time stabilization Markov systems polytopic uncertainties partly known transition probabilities linear matrix inequalities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zhang, L., Boukas, E.K., Lam, J.: Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities. IEEE Trans. Auto. Con. 53, 2458–2464 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Zhang, L., Boukas, E.K., Baron, L.: Fault detection for discrete-time Markov jump linear systems with partially known transition probabilities. Int. J. Con. 83, 1564–1572 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Braga, M.F., Morais, C.F., Oliveira, R.C.L.F.: Robust stability and stabilization of discrete-time Markov jump linear systems with partly unknown transition probability matrix. In: American Control Conference (ACC), pp. 6784–6789. IEEE Press, Washington (2013)Google Scholar
  4. 4.
    Tian, E., Yue, D., Wei, G.: Robust control for Markovian jump systems with partially known transition probabilities and nonlinearities. J. Fran. Ins. 350, 2069–2083 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Shen, M., Yang, G.H.: H 2 state feedback controller design for continuous Markov jump linear systems with partly known information. Inter. J. Sys. Sci. 43, 786–796 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Tian, J., Li, Y., Zhao, J.: Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates. Ap. Mathe. Com. 218, 5769–5781 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rao, R., Zhong, S., Wang, X.: Delay-dependent exponential stability for Markovian jumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partially known transition rates via a differential inequality. Ad. Differ. Equa. 2013, 1–14 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Auto. 37, 1459–1463 (2001)CrossRefGoogle Scholar
  9. 9.
    Zhang, X., Feng, G., Sun, Y.: Finite-time stabilization by state feedback control for a class of time-varying nonlinear systems. Auto. 48, 499–504 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hu, M., Cao, J., Hu, A.: A Novel Finite-Time Stability Criterion for Linear Discrete-Time Stochastic System with Applications to Consensus of Multi-Agent System. Cir. Sys. Sig. Pro. 34, 1–19 (2014)Google Scholar
  11. 11.
    Zhou, J., Xu, S., Shen, H.: Finite-time robust stochastic stability of uncertain stochastic delayed reaction-diffusion genetic regulatory networks. Neu. 74, 2790–2796 (2011)Google Scholar
  12. 12.
    Zuo, Z., Li, H., Wang, Y.: Finite-time stochastic stabilization for uncertain Markov jump systems subject to input constraint. Trans. Ins. Mea. Con. 36, 283–288 (2014)CrossRefGoogle Scholar
  13. 13.
    Yin, Y., Liu, F., Shi, P.: Finite-time gain-scheduled control on stochastic bioreactor systems with partially known transition jump rates. Cir. Sys. Sig. Pro. 30, 609–627 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bhat, S.P., Bernstein, D.S.: Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Trans. Auto. Con. 43, 678–682 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Luan, X., Liu, F., Shi, P.: Neural-network-based finite-time H  ∞  control for extended Markov jump nonlinear systems. Inter. J. Adap. Con. Sig. Pro. 24, 554–567 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

<SimplePara><Emphasis Type="Bold">Open Access</Emphasis> This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (, which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. </SimplePara> <SimplePara>The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.</SimplePara>

Authors and Affiliations

  • Chen Zheng
    • 1
    Email author
  • Xiaozheng Fan
    • 1
  • Manfeng Hu
    • 1
    • 2
  • Yongqing Yang
    • 1
    • 2
  • Yinghua Jin
    • 1
    • 2
  1. 1.School of ScienceJiangnan UniversityWuxiChina
  2. 2.Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education)Jiangnan UniversityWuxiChina

Personalised recommendations