Skip to main content
SpringerLink
Log in

Exponential Convergence of Time-Delay Systems in the Presence of Bounded Disturbances

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, the problem of control design for exponential convergence of state/input delay systems with bounded disturbances is considered. Based on the Lyapunov–Krasovskii method combining with the delay-decomposition technique, a new sufficient condition is proposed for the existence of a state feedback controller, which guarantees that all solutions of the closed-loop system converge exponentially (with a pre-specified convergence rate) within a ball whose radius is minimized. The obtained condition is given in terms of matrix inequalities with one parameter needing to be tuned, which can be solved by using a one-dimensional search method with Matlab’s LMI Toolbox to minimize the radius of the ball. Two numerical examples are given to illustrate the superiority of the proposed method.

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

Similar content being viewed by others

References

  1. Kofman, E., Seron, M.M., Haimovich, H.: Control design with guaranteed ultimate bound for perturbed systems. Automatica 44, 1815–1821 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Corless, M.: Guaranteed rates of exponential convergence for uncertain systems. J. Optim. Theory Appl. 64, 481–494 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Corless, M., Leitmann, G.: Bounded controllers for robust exponential convergence. J. Optim. Theory Appl. 76, 1–12 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyd, S., Ghaoui, El., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities and Control Theory. Philadelphia, SIAM (1994)

    Book  MATH  Google Scholar 

  5. Khalil, H.: Nonlinear Systems. Prentice-Hall, New Jersey (2002)

    MATH  Google Scholar 

  6. Gu, K., Kharitonov, V.L., Chen, L.: Stability of Time-delay Systems. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  7. Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gouaisbaut, F., Peaucelle, D.: Delay-dependent stability analysis of linear time delay systems. IFAC Workshop Time Delay Systems, Aquila, Italy (2006)

  9. Sun, J., Liu, G.P., Chen, J., Rees, D.: Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica 46, 466–470 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Phat, V.N., Ha, Q.P., Trinh, H.: Parameter-dependent H ∞ control for time-varying delay polytopic systems. J. Optim. Theory Appl. 147, 58–70 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Udwadia, F.E., Von Bremen, H., Phohomsiri, P.: Time-delayed control design for active control of structures: Principles and applications. Struct. Control Health Monit. 14, 27–61 (2007)

    Article  Google Scholar 

  12. Kwon, O.M., Park, J.H.: Exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations. J. Optim. Theory Appl. 139, 277–293 (2008)

    Article  MathSciNet  Google Scholar 

  13. Oucheriah, S.: Exponential stabilization of linear delayed systems using sliding-mode controllers. IEEE Trans. Circuits Syst. I 50, 826–830 (2003)

    Article  MathSciNet  Google Scholar 

  14. Oucheriah, S.: Robust exponential convergence of a class of linear delayed systems with bounded controllers and disturbances. Automatica 42, 1863–1867 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fridman, E., Dambrine, M.: Control under quantization, saturation and delay: A LMI approach. Automatica 45, 2258–2264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Han, X., Fridman, E., Spurgeon, S.K.: Sliding-mode control of uncertain systems in the presence of unmatched disturbances with applications. Int. J. Control 83, 2413–2426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, X.M., Wu, M., She, J.H., He, Y.: Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41, 1405–1412 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, X.M., Li, M., Wu, M., She, J.H.: Further results on stability and stabilisation of linear systems with state and input delays. Int. J. Syst. Sci. 40, 1–10 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nam, P.T., Phat, V.N.: Robust stabilization of linear systems with delayed state and control. J. Optim. Theory Appl. 140, 287–299 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Du, B., Lam, J., Shu, Z.: Stabilization for state/input delay systems via static and integral output feedback. Automatica 46, 2000–2007 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers and the editor for their very helpful comments and suggestions, which have improved our paper. This work is supported by the National Foundation for Science and Technology Development, Vietnam under grant 101.01.2011.51 and the Australian Research Council under the Discovery grant DP0667181.

Author information

Authors and Affiliations

  1. Department of Mathematics, Quynhon University, Binhdinh, Vietnam

    P. T. Nam

  2. School of Engineering, Deakin University, Geelong, VIC, 3217, Australia

    P. N. Pathirana & H. Trinh

Authors
  1. P. T. Nam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. N. Pathirana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Trinh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. T. Nam.

Additional information

Communicated by Martin Corless.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nam, P.T., Pathirana, P.N. & Trinh, H. Exponential Convergence of Time-Delay Systems in the Presence of Bounded Disturbances. J Optim Theory Appl 157, 843–852 (2013). https://doi.org/10.1007/s10957-012-0240-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0240-1

Keywords

Search

Navigation