Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Properties of storage functions and applications to nonlinear stochastic H ∞ control

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

Abstract

This paper discusses the properties of the storage functions for a class of nonlinear stochastic systems. Some necessary and sufficient conditions for a function to be a storage function are derived. As applications, the finite and infinite horizon nonlinear stochastic H ∞ controls for systems with state, control, and external disturbance dependent noise are investigated, which generalize the previous results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. J. C. Willems, Dissipative dynamic systems, Part I: general theory, Arch. Rat. Mech. Anal, 1972, 45: 321–351.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. A. Ball and J. W. Helton, Viscosity solutions of Hamilton-Jacobi equations arising in nonlinear H ∞-control, Journal of Mathematical Systems, Estimation, and Control, 1996, 6(1): 1–22.

    MathSciNet  Google Scholar 

  3. W. Zhang and B. S. Chen, State feedback H ∞ control for a class of nonlinear stochastic systems, SIAM J. Control Optim., 2006, 44(6): 1973–1991.

    Article  MathSciNet  MATH  Google Scholar 

  4. Z. Lin, Y. Lin, and W. Zhang, H ∞ controller design for a class of SISO nonlinear stochastic systems without solving HJIs, International Journal of Innovative Computing, Information and Control, 2009, 5(11A): 3839–3848.

    Google Scholar 

  5. P. V. Pakshin and V. A. Ugrinovskii, Stochastic problems of absolute stability, Automation and Remote Control, 2006, 67(11): 1811–1846.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. G. Crandall, H. Ishii, and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 1992, 27: 1–67.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. W. Helton and M. R. James, Extending H ∞ Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives, PA: SIAM, Philadelphia, 1999.

    Google Scholar 

  8. J. Yong and X. Y. Zhou, Stochastic Controls, Springer-Verlag, New York, 1998.

    Google Scholar 

  9. N. Berman and U. Shaked, H ∞ for nonlinear stochastic systems, Proc. 42nd IEEE Conference on Decision and Control, 2003: 5025–5030.

  10. H. Shu and G. Wei, H ∞ analysis of nonlinear stochastic time-delay systems, Chaos, Solitions and Fractals, 2005, 26: 637–647.

    Article  MathSciNet  MATH  Google Scholar 

  11. Z. Wang and D. W. C. Ho, Filtering on nonlinear time-delay stochastic systems, Automatica, 2003, 39(1): 101–109.

    Article  MathSciNet  MATH  Google Scholar 

  12. W. Zhang, H. Zhang, and B. S. Chen, Stochastic H 2/H ∞ control with (x, u, v)-dependent noise: Finite horizon case, Automatica, 2006, 42: 1891–1898.

    Article  MATH  Google Scholar 

  13. M. James, A partial differential inequality for dissipative nonlinear systems, Systems and Control Letters, 1993, 21: 315–320.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Z. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphenaan den Rijn-Germantown, MD, 1980.

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Qixia Zhang

  2. College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, 266510, China

    Weihai Zhang

Authors
  1. Qixia Zhang
    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

Corresponding author

Correspondence to Qixia Zhang.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant Nos. 10921101 and 60874032, the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904, the Key Project of Natural Science Foundation of Shandong Province under Grant No. ZR2009GZ001, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20103718110006.

This paper was recommended for publication by Editor Jifeng ZHANG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Q., Zhang, W. Properties of storage functions and applications to nonlinear stochastic H ∞ control. J Syst Sci Complex 24, 850–861 (2011). https://doi.org/10.1007/s11424-011-9117-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-011-9117-x

Key words

Search

Navigation