Skip to main content
SpringerLink
Log in

On the observability and detectability of linear stochastic systems with Markov jumps and multiplicative noise

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper presents the notions of exact observability and exact detectability for Markov jump linear stochastic systems of Itô type with multiplicative noise (for short, MJLSS). Stochastic Popov-Belevith-Hautus (PBH) Criterions for exact observability and exact detectability are respectively obtained. As an application, stochastic H 2/H ∞ control for such MJLSS is discussed under exact detectability.

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.

Similar content being viewed by others

References

  1. R. Z. Has’minskii, Stochastic Stability of Differential Equations, Sijtjoff and Noordhoff, Alphen, 1980.

    Google Scholar 

  2. R. S. Liptser and A. N. Shiryayev, Statistics of Random Processes, Springer-Verlag, New York, 1977.

    MATH  Google Scholar 

  3. J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999.

    MATH  Google Scholar 

  4. V. A. Ugrinovskii, Robust H ∞ control in the presence of stochastic uncertainty, Int. J. Contr., 1998, 71: 219–237.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley, 1974.

  6. E. F. Costa and J. B. R. do Val, On the observability and detectability of continuous-time Markov jump linear systems, SIAM J. Control Optim., 2005, 41: 1295–1314.

    Article  Google Scholar 

  7. Y. Huang, W. Zhang, and G. Feng, Infinite horizon H 2/H ∞ control for stochastic systems with Markovian jumps, Automatica, 2008, 44: 857–863.

    Article  MathSciNet  Google Scholar 

  8. M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, New York, 1990.

    Google Scholar 

  9. V. Dragan and T. Morozan, Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise, Stoch. Anal. Appl., 2002, 20(1): 33–92.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Caines and J. F. Zhang, On the adaptive control of jump parameter systems via nonlinear filtering, SIAM J. Control Optim., 1995, 33: 1758–1777.

    Article  MATH  MathSciNet  Google Scholar 

  11. X. Mao, G. G. Yin, and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 2007, 43: 264–273.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. Li and V. A. Ugrinovskii, On necessary and sufficient conditions for H-infinity output feedback control of Markov jump linear systems, IEEE Transactions on Automatic Control, 2007, 52(7): 1287–1292.

    Article  MathSciNet  Google Scholar 

  13. X. Li, X. Y. Zhou, and M. A. Rami, Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of Global Optimization, 2003, 27: 149–175.

    Article  MATH  Google Scholar 

  14. T. Damm, On detectability of stochastic systems, Automatica, 2007, 43: 928–933.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Guatteri and F. Masiero, Infinite horizon and ergodic optimal quadratic control for an affine equation with stochastic coefficients, SIAM J. Control Optim., 2009, 48(3): 1600–1631.

    Article  MathSciNet  Google Scholar 

  16. W. Zhang and B. S. Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica, 2004, 40: 87–94.

    Article  MATH  Google Scholar 

  17. W. Zhang, H. Zhang, and B. S. Chen, Generalized Lyapunov Equation Approach to State-Dependent Stochastic Stabilization/Detectability Criterion, IEEE Trans. Automat. Contr., 2008, 53: 1630–1642.

    Article  Google Scholar 

  18. S. He, J. Wang, and J. Yan, Semimartingale and Stochastic Analysis, Science Press and CRC Press Inc, 1995.

  19. L. E. Ghaoui and M. A. Rami, Robust state-feedback stabilization of jump linear systems via LMIs, International Journal of Robust and Nonlinear Control, 1996, 6: 1015–1022.

    Article  MATH  MathSciNet  Google Scholar 

  20. V. M. Ungureanu, Stochastic uniform observability of linear differential equations with multiplicative noise, J. Math. Anal. Appl., 2008, 343: 446–463.

    Article  MATH  MathSciNet  Google Scholar 

  21. T. Damm, Rational Matrix Equations in Stochastic Control, Lecture notes in Control and Information Sciences, Berlin-Heidelberg, Springer, 2004, Vol. 297.

    MATH  Google Scholar 

  22. J. W. Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Syst., 1978, 25: 772–781.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice Hall, Englewood Cliffs, New Jersey, 1989.

    Google Scholar 

  24. B. S. Chen and W. Zhang, Stochastic H 2/H ∞ control with state-dependent noise, IEEE Trans. Automat. Contr., 2004, 49: 45–57.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yuanhua Ni & Haitao Fang

  2. College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, China

    Weihai Zhang

Authors
  1. Yuanhua Ni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weihai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haitao Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanhua Ni.

Additional information

This work is supported by National Natural Science Foundation of China under Grant Nos 60774020, 60736028, and 60821091.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ni, Y., Zhang, W. & Fang, H. On the observability and detectability of linear stochastic systems with Markov jumps and multiplicative noise. J Syst Sci Complex 23, 102–115 (2010). https://doi.org/10.1007/s11424-010-9270-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-010-9270-7

Key words

Search

Navigation