Skip to main content
SpringerLink
Log in

An Improved Reciprocally Convex Inequality and its Application to Time-Varying Delay Systems

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper studies the stability of systems with a time-varying delay. Firstly, an improved reciprocally convex inequality is introduced which includes some existing inequalities as special cases. Secondly, a new stability criterion is obtained in terms of linear matrix inequalities (LMIs) by using the improved reciprocally convex inequality. Finally, two numerical examples are provided to show the effectiveness of the presented 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

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Fridman, E.: Introduction to time-delay systems Analysis and control. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  2. Shao, H.Y.: New delay-dependent stability criteria for systems with interval delay. Automatica 45(3), 744–749 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Sun, J., Liu, G.P., Chen, J., Rees, D.: Improved delay-range-dependent stability criteria for linear systems with time-varying delay. Automatica 46(2), 466–470 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Qian, W., Liu, J.: New stabiility analisis for systems with interval time-varying delay. J. Frankl. Inst. 350(4), 890–897 (2013)

    Article  MATH  Google Scholar 

  5. Ding, L.M., He, Y., Wu, M., Zhang, X.M.: A novel delay partitioning method for stability analysis of interval time-varying delay systems. J. Frankl. Inst. 354, 1209–1219 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zeng, H.B., Zhai, Z.L., He, Y., Teo, K.L., Wang, W.: New insights on stability of sampled-data systems with time-delay. Appl. Math. Comput. 404, 125041 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jin, L., He, Y., Jiang, L.: A novel integral inequality and its application to stability analysis of linear system with multiple time delays. Appl. Math. Lett. 124, 107648 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tian, J.K., Ren, Z.R., Zhong, S.M.: A new integral inequality and application to stability of time-delay systems. Appl. Math. Lett. 101, 106058 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhao, N., Lin, C., Chen, B., Wang, Q.G.: A new double integral inequlity and application to stability test for time-delay systems. Appl. Math. Lett. 65, 26–31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tian, Y.F., Wang, Z.S.: A new multiple integral inequality and its application to stability analysis of time-delay systems. Appl. Math. Lett. 105, 106325 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gu, K.: An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE conference on decision and control. Sydney, Australia 2805-2810 (2010)

  12. Kim, J.H.: Further improvement of Jensen inequality and application to stability of time-delayed systems. Automatica 64, 121–125 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: application to time-delay systems. Automatica 49, 2860–2866 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Park, P.G., Lee, W.I., Lee, S.Y.: Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J. Frankl. Inst. 352(4), 1378–1396 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, K., Seuret, A., Xia, Y.Q.: Stability analysis of systems with time-varying delays via the second-order Bessel-Legendre inequality. Automatica 76, 138–142 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zeng, H.B., He, Y., Mu, M., She, J.: Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Trans. Automat. Control 60, 2768–2772 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zeng, H.B., Liu, X.G., Wang, W.: A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems. Appl. Math. Comput. 354, 1–8 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tan, G.Q., Wang, Z.S.: Stability analysis of systems with time-varying delay via a delay-product-type integral inequality. Math. Meth. Appl. Sci. 45(11), 6535–6545 (2022)

    Article  MathSciNet  Google Scholar 

  19. Park, P., Ko, J., Jeong, C.: Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, X.M., Han, Q.L.: State estimation for static neural networks with time-varying delays based on an improved reciprocally convex inequality. IEEE Trans. Neural Netw. Learn. Syst. 29, 1376–1381 (2018)

    Article  Google Scholar 

  21. Lin, H.C., Zeng, H.B., Zhang, X.M., Wang, W.: Stability analysis for delayed Neural Networks via a generalized reciprocally convex inequality. IEEE Trans. Neural Netw. Learn. Syst. (2022). https://doi.org/10.1109/TNNLS.2022.3144032

    Article  Google Scholar 

  22. Chen, J., Zhang, X.M., Park, J.H., Xu, S.: Improved stability criteria for delayed neural networks using a quadratic function negative-definiteness approach. IEEE Trans. Neural Netw. Learn. Syst. 33, 1348–1354 (2022)

    Article  MathSciNet  Google Scholar 

  23. Tan, G.Q., Wang, Z.S.: Reachable set estimation of delayed markovian jump Neural Networks based on an improved reciprocally convex inequality. IEEE Trans. Neural Netw. Learn. Syst. 33(6), 2737–2742 (2022)

    Article  MathSciNet  Google Scholar 

  24. Seuret, A., Liu, K., Gouaisbaut, F.: Generalized reciprocally convex combination lemmas and its application to time-delay systems. Automatica 95, 488–493 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zeng, H.B., Lin, H.C., He, Y., Teo, K.L., Wang, W.: Hierarchical stability conditions for time-varying delay systems via an extended reciprocally convex quadratic inequality. J. Frankl. Inst. 357, 9930–9941 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin, H.C., Zeng, H.B., Wang, W.: New Lyapunov-Krasovskii Functional for Stability Analysis of Linear Systems with Time-Varying Delay. J. Syst. Sci. Complex. 34, 632–641 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zeng, H.B., Zhai, Z.L., Wang, W.: Hierarchical stability conditions of systems with time-varying delay. Appl. Math. Comput. 404, 126222 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zeng, H.B., Lin, H.C., He, Y., Zhang, C.K., Teo, K.L.: Improved negativity condition for a quadratic function and its application to systems with time-varying delay. IET Control Theory Appl. 14, 2989–2993 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to show their gratitude to the editor and reviewers for valuable advices in helping to improve the manuscript. This work is supported by Basic Research Program of Guizhou Province (grant number Qian Ke He JiChu [2021] YiBan 005); Project of Youth Science and Technology Talents of Guizhou Province(grant number Qian Jiao He KY Zi [2020] 095).

Author information

Authors and Affiliations

  1. School of Mathematics, Zunyi Normal University, Zunyi, 563006, Guizhou, People’s Republic of China

    Zerong Ren & Junkang Tian

Authors
  1. Zerong Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junkang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors studied and prepared the manuscript. ZR analyzed the results, and JT made necessary improvements. ZR is the major contributor in writing the paper. All authors studied the results together, and they read and approved the final manuscript.

Corresponding author

Correspondence to Junkang Tian.

Ethics declarations

Conflict of interest

The authors declare that there are no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ren, Z., Tian, J. An Improved Reciprocally Convex Inequality and its Application to Time-Varying Delay Systems. Qual. Theory Dyn. Syst. 21, 119 (2022). https://doi.org/10.1007/s12346-022-00651-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00651-5

Keywords

Mathematics Subject Classification

Search

Navigation