Skip to main content
SpringerLink
Log in

A necessary and sufficient stabilization condition for discrete time-varying stochastic systems with multiplicative noise

  • Published:
Control Theory and Technology Aims and scope Submit manuscript

Abstract

This paper investigates a fundamental problem of stabilization for time-varying multiplicative noise stochastic systems. A necessary and sufficient stabilization condition is presented based on the receding horizon approach. The explicit time-varying controller is designed if the condition is satisfied. The presented results are new to the best of our knowledge.

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

Similar content being viewed by others

References

  1. X. R. Mao. Stochastic Differential Equations and Applications. 2nd ed. Chichester, U.K.: Horwood Publication, 2007.

    MATH  Google Scholar 

  2. L. Li, H. Zhang. Linear quadratic regultion for discrete-time systems with state delays and multiplicative noise. Control Theory and Technology, 2015, 13(4): 348–359.

    Article  MathSciNet  Google Scholar 

  3. J. L. Willems, J. C. Willems. Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica, 1976, 12(3): 277–283.

    Article  MATH  MathSciNet  Google Scholar 

  4. U. Haussmann. Stability of linear systems with control dependent noise. SIAM Journal on Control, 1973, 11(2): 382–394.

    Article  MATH  MathSciNet  Google Scholar 

  5. T. Morozan. Stabilization of some stochastic discrete-time control systems. Stochastic Analysis and Applications, 1983, 1(1): 89–116.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Yaz. Stabilization of discrete-time systems with stochastic parameters. Systems & Control Letters, 1985, 5(5): 321–326.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Yaz. Certainty equivalent control of stochastic systems: stability property. IEEE Transactions on Automatic Control, 1986, 3(12): 178–180.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. A. Rami, X. Zhou. Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Transactions on Automatic Control, 2000, 45(6): 1131–1143.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. E. Ghaoui. State-feedback control of systems with multiplicative noise via linear matrix inequalities. Systems & Control Letters, 1995, 24(3): 223–228.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. Zhang, H. Zhang, B. Chen. Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion. IEEE Transactions on Automatic Control, 2008, 53(7): 1630–1642.

    Article  MathSciNet  Google Scholar 

  11. N. Elia. Remote stabilization over fading channels. Systems & Control Letters, 2005, 54(3): 237–249.

    Article  MATH  MathSciNet  Google Scholar 

  12. K. Y. You, L. H. Xie. Minimum data rate for mean square stabilization of discrete LTI systems over lossy channels. IEEE Transactions on Automatic Control, 2010, 55(10): 2373–2378.

    Article  MathSciNet  Google Scholar 

  13. S. Aberkane, V. Dragan. Robust stability and robust stabilization of a class of discrete-time time-varying linear stochastic systems. SAIM Journal on Control and Optimization, 2015, 53(1): 30–57.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. Sheng, W. Zhang, M. Gao. Mixed H2/H8 control of time-varying stochastic discrete-time systems under uniform detectability. IET Control Theory and Application, 2014, 8(17): 1866–1874.

    Article  MathSciNet  Google Scholar 

  15. W. H. Kwon, A. E. Pearson. A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Transactions on Automatic Control, 1977, 22(5): 838–842.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. W. Lee, W. H. Kwon, J. H. Choi. On stability of constrained receding horizon control with finite terminal weighting matrix. Automatica, 1998, 34(12): 1607–1612.

    Article  MATH  Google Scholar 

  17. W. H. Kwon, K. B. Kim. On stabilizing receding horizon controls for linear continuous time-invariant systems. IEEE Transactions on Automatic Control, 2000, 45(7): 1329–1334.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. D. Nicolao, L. Magni, R. Scattolini. Stabilizing receding horizon control of nonlinear time-varying systems. IEEE Transactions on Automatic Control, 1998, 43(7): 1030–1036.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. A. Mohammadkhani, F. Bayat, A. A. Jalali. Design of explicit model predictive control for constrained linear systems with disturbances. International Journal of Control, Automation, and Systems, 2014, 12(2): 294–301.

    Article  Google Scholar 

  20. J. A. Primbs, C. H. Sung. Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise. IEEE Transactions on Automatic Control, 2009, 54(2): 221–230.

    Article  MathSciNet  Google Scholar 

  21. D. Bernardini, A. Bemporad. Stabilizing model predictive control of stochastic constrained linear systems. IEEE Transactions on Automatic Control, 2012, 57(6): 1468–1480.

    Article  MathSciNet  Google Scholar 

  22. H. Zhang, H. Wang, L. Li. Adapted and casual maximum principle and analytical solution to optimal control for stochastic multiplicative noise systems with multiplt inputdelays. Proceedings of the 51st IEEE Annual Conference on Decision and Control, Hawaii: IEEE, 2012: 2122–2177.

    Google Scholar 

  23. L. Chen, Z. Wu. Maximum principle for the stochastic optimal control problem with delay and application. Automatica, 2010, 46(6): 1074–1080.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Niwa, M. Hayase, I. Sugiura. Stability of linear time-varying systems with state dependent noise. IEEE Transactions on Automatic Control, 1976, 21(5): 775–776.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Tan, L. Li, H. Zhang. Stabilization of networked control systems with both network-induced delay and packet dropout. Automatica, 2015, 59: 194–199.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan Shandong, 250061, China

    Rong Gao & Huanshui Zhang

  2. School of Mathematics and Statistics Science, Ludong University, Yantai Shandong, 264025, China

    Rong Gao & Xiaohua Liu

Authors
  1. Rong Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaohua Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huanshui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huanshui Zhang.

Additional information

This work was supported by the Taishan Scholar Construction Engineering by Shandong Government and the National Natural Science Foundation of China (Nos. 61120106011, 61203029).

Rong GAO received the B.Sc. degree in Mathematics from Yantai University, Yantai, China, 1999, and the M.Sc. degree in Operational Research and Control Theory from Ludong University, Yantai, China, in 2005. He is currently pursuing the Ph.D. degree in Control Theory and Engineering at Shandong University. His research interests include receding horizon control, time-delay systems and stochastic systems.

Xiaohua LIU was born in Yantai, Shandong, China, in 1959. He received his B.Sc. degree in Mathematics from Shandong Normal University, Jinan, China, in 1982, and his M.Sc. and Ph.D. degrees in Operational Research and Control Theory from Qufu Normal University, Qufu, China, in 1989 and from Dalian University of Technology, Dalian, China, in 2003, respectively. He is currently a professor with College of Mathematics and Statistics Science, Ludong University, Yantai, China. His current research interests include stochastic system control and predictive control theory.

Huanshui ZHANG graduated in Mathematics from the Qufu Normal University in 1986 and received his M.Sc. and Ph.D. degrees in Control Theory from Heilongjiang University, China, and Northeastern University, China, in 1991 and 1997, respectively. He worked as a postdoctoral fellow at Nanyang Technological University from 1998 to 2001 and Research Fellow at Hong Kong Polytechnic University from 2001 to 2003. He is currently a Changjiang Professorship at Shandong University, China. He held Professor in Harbin Institute of Technology from 2003 to 2006. He also held visiting appointments as Research Scientist and Fellow with Nanyang Technological University, Curtin University of Technology and Hong Kong City University from 2003 to 2006. His interests include optimal estimation and control, time-delay systems, stochastic systems, signal processing and wireless sensor networked systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, R., Liu, X. & Zhang, H. A necessary and sufficient stabilization condition for discrete time-varying stochastic systems with multiplicative noise. Control Theory Technol. 14, 151–158 (2016). https://doi.org/10.1007/s11768-016-5078-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-016-5078-x

Keywords

Search

Navigation