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Delays and Interconnections: Methodology, Algorithms and Applications pp 187–197Cite as

Delay-Dependent Reciprocally Convex Combination Lemma for the Stability Analysis of Systems with a Fast-Varying Delay

Part of the Advances in Delays and Dynamics book series (ADVSDD,volume 10)

Abstract

This chapter deals with the stability analysis of linear systems subject to fast-varying delays. The main result is the derivation of a delay-dependent reciprocally convex lemma allowing a notable reduction of the conservatism of the resulting stability conditions with the introduction of a reasonable number of decision variables. Several examples are studied to show the potential of the proposed method.

This work was partially supported by the ANR project SCIDIS, contract number 15-CE23-0014.

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  • DOI: 10.1007/978-3-030-11554-8_12
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References

  1. Fridman, E.: Introduction to Time-Delay Systems: Analysis and Control. Springer, Berlin (2014)

    CrossRef  Google Scholar 

  2. Fridman, E., Shaked, U., Liu, K.: New conditions for delay-derivative-dependent stability. Automatica 45(11), 2723–2727 (2009)

    MathSciNet  CrossRef  Google Scholar 

  3. Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the IEEE Conference on Decision and Control (2000)

    Google Scholar 

  4. He, Y., Wang, Q., Lin, C., Wu, M.: Delay-range-dependent stability for systems with time-varying delay. Automatica 43(2), 371–376 (2007)

    MathSciNet  CrossRef  Google Scholar 

  5. Hien, L., Trinh, H.: Refined Jensen-based inequality approach to stability analysis of time-delay systems. IET Control. Theory Appl. 9(14), 2188–2194 (2015)

    MathSciNet  CrossRef  Google Scholar 

  6. Moon, Y., Park, P., Kwon, W., Lee, Y.: Delay-dependent robust stabilization of uncertain state-delayed systems. Int. J. Control. 74(14), 1447–1455 (2001)

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  9. Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: application to time-delay systems. Automatica 49(9), 2860–2866 (2013)

    MathSciNet  CrossRef  Google Scholar 

  10. Seuret, A., Gouaisbaut, F.: Hierarchy of LMI conditions for the stability of time delay systems. Syst. Control. Lett. 81, 1–7 (2015)

    MathSciNet  CrossRef  Google Scholar 

  11. Seuret, A., Gouaisbaut, F.: Stability of linear systems with time-varying delays using Bessel-Legendre inequalities. IEEE Trans. Autom. Control. (2017, to appear)

    Google Scholar 

  12. Seuret, A., Gouaisbaut, F., Fridman, E.: Stability of systems with fast-varying delay using improved Wirtinger’s inequality. In: IEEE Conference on Decision and Control, pp. 946–951, Florence, Italy, December 2013

    Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  14. Su, H., Ji, X., Chu, J.: New results of robust quadratically stabilizing control for uncertain linear time-delay systems. Int. J. Syst. Sci. 36(1), 27–37 (2005)

    MathSciNet  CrossRef  Google Scholar 

  15. Xu, S., Lam, J.: A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39(12), 1095–1113 (2008)

    MathSciNet  CrossRef  Google Scholar 

  16. Zeng, H., He, Y., Wu, M., She, J.: Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Trans. Autom. Control. 60(10), 2768–2772 (2015)

    MathSciNet  CrossRef  Google Scholar 

  17. Zhang, C.-K., He, Y., Jiang, L., Wu, M., Wang, Q.-G.: An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay. Automatica (2017)

    Google Scholar 

  18. Zhang, X.-M., Han, Q.-L., Seuret, A., Gouaisbaut, F.: An improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional for stability of linear systems with time-varying delay. Automatica (2017, to appear)

    Google Scholar 

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Authors and Affiliations

  1. LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France

    Alexandre Seuret & Frédéric Gouaisbaut

Authors
  1. Alexandre Seuret
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  2. Frédéric Gouaisbaut
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Corresponding author

Correspondence to Alexandre Seuret .

Editor information

Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Prof. Giorgio Valmorbida

  2. Laboratoire d’Analyse et d’Architecture de Systèmes-CNRS, Toulouse, France

    Dr. Alexandre Seuret

  3. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Islam Boussaada

  4. Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Prof. Rifat Sipahi

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Seuret, A., Gouaisbaut, F. (2019). Delay-Dependent Reciprocally Convex Combination Lemma for the Stability Analysis of Systems with a Fast-Varying Delay. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_12

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