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JACIII Vol.27 No.4 pp. 683-690
doi: 10.20965/jaciii.2023.p0683
(2023)

Research Paper:

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A Sufficient Condition on Polynomial Inequalities and its Application to Interval Time-Varying Delay Systems

Meng Liu*1,*2,*3 ORCID Icon, Yong He*1,*2,*3,† ORCID Icon, and Lin Jiang*4 ORCID Icon

*1School of Automation, China University of Geosciences
388 Lumo Road, Hongshan District, Wuhan, Hubei 430074, China

*2Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems
388 Lumo Road, Hongshan District, Wuhan, Hubei 430074, China

*3Engineering Research Center of Intelligent Technology for Geo-Exploration, Ministry of Education
388 Lumo Road, Hongshan District, Wuhan, Hubei 430074, China

*4Department of Electrical Engineering and Electronics, University of Liverpool
Brownlow Hill, Liverpool, Merseyside L 3, United Kingdom

†Corresponding author

Received:
November 23, 2022
Accepted:
April 11, 2023
Published:
July 20, 2023
Creative Commons License
Keywords:
stability, time-varying delay, polynomial inequality, negative-definiteness condition
Abstract

This article examines the stability problem of systems with interval time-varying delays. In the derivation of Lyapunov–Krasovskii functional (LKF), non-convex higher-degree polynomials may arise with respect to interval time-varying delays, making it difficult to determine the negative definiteness of LKF’s derivative. This study was conducted to obtain stability conditions that can be described as linear matrix inequalities (LMIs). By considering the idea of matrix transition and introducing the delay-dependent augmented vector, a novel higher-degree polynomial inequality is proposed under the condition that the lower bound of the polynomial function variable is non-zero, which encompasses the existing lemmas as its special cases. Then, benefiting from this inequality, a stability criterion is derived in terms of LMIs. Finally, several typical examples are presented to verify the availability and strength of the stability condition.

Cite this article as:
M. Liu, Y. He, and L. Jiang, “A Sufficient Condition on Polynomial Inequalities and its Application to Interval Time-Varying Delay Systems,” J. Adv. Comput. Intell. Intell. Inform., Vol.27 No.4, pp. 683-690, 2023.
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References
  1. [1] K. Gu, V. L. Kharitonov, and J. Chen, “Stability of time-delay systems,” Birkhäuser, 2003.
  2. [2] X.-C. Shangguan et al., “Robust load frequency control for power system considering transmission delay and sampling period,” IEEE Trans. on Industrial Informatics, Vol.17, No.8, pp. 5292-5303, 2021. https://doi.org/10.1109/TII.2020.3026336
  3. [3] Z.-H. Pang et al., “An event-triggered networked predictive control method using an allowable time delay,” J. Adv. Comput. Intell. Intell. Inform., Vol.26, No.5, pp. 768-775, 2022. https://doi.org/10.20965/jaciii.2022.p0768
  4. [4] E. Fridman, “Introduction to time-delay systems: Analysis and control,” Birkhäuser, 2014.
  5. [5] Y. Liu, J. H. Park, and B.-Z. Guo, “Results on stability of linear systems with time-varying delay,” IET Control Theory & Applications, Vol.11, No.1, pp. 129-134, 2017. https://doi.org/10.1049/iet-cta.2016.0634
  6. [6] J.-A. Wang, X.-H. Ma, and X.-Y. Wen, “Less conservative stability criteria for neural networks with interval time-varying delay based on delay-partitioning approach,” Neurocomputing, Vol.155, pp. 146-152, 2015. https://doi.org/10.1016/j.neucom.2014.12.038
  7. [7] J. Chen, J. H. Park, and S. Xu, “Stability analysis of continuous-time systems with time-varying delay using new Lyapunov–Krasovskii functionals,” J. of the Franklin Institute, Vol.355, No.13, pp. 5957-5967, 2018. https://doi.org/10.1016/j.jfranklin.2018.05.054
  8. [8] J. Sun et al., “Improved delay-range-dependent stability criteria for linear systems with time-varying delays,” Automatica, Vol.46, No.2, pp. 466-470, 2010. https://doi.org/10.1016/j.automatica.2009.11.002
  9. [9] T. H. Lee and J. H. Park, “A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function,” Automatica, Vol.80, pp. 239-242, 2017. https://doi.org/10.1016/j.automatica.2017.02.004
  10. [10] T. H. Lee and J. H. Park, “Improved stability conditions of time-varying delay systems based on new Lyapunov functionals,” J. of the Franklin Institute, Vol.355, No.3, pp. 1176-1191, 2018. https://doi.org/10.1016/j.jfranklin.2017.12.014
  11. [11] F. Long et al., “Stability analysis of systems with time-varying delay via improved Lyapunov–Krasovskii functionals,” IEEE Trans. on Systems, Man, and Cybernetics: Systems, Vol.51, No.4, pp. 2457-2466, 2021. https://doi.org/10.1109/TSMC.2019.2914367
  12. [12] C.-K. Zhang et al., “Delay-variation-dependent stability of delayed discrete-time systems,” IEEE Trans. on Automatic Control, Vol.61, No.9, pp. 2663-2669, 2016. https://doi.org/10.1109/TAC.2015.2503047
  13. [13] C.-K. Zhang et al., “Notes on stability of time-delay systems: Bounding inequalities and augmented Lyapunov–Krasovskii functionals,” IEEE Trans. on Automatic Control, Vol.62, No.10, pp. 5331-5336, 2017. https://doi.org/10.1109/TAC.2016.2635381
  14. [14] X. Wang et al., “Improved stability conditions for time-varying delay systems via relaxed Lyapunov functionals,” Int. J. of Control, Vol.96, No.6, pp. 1568-1581, 2023. https://doi.org/10.1080/00207179.2022.2056716
  15. [15] M. Wu et al., “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, Vol.40, No.8, pp. 1435-1439, 2004. https://doi.org/10.1016/j.automatica.2004.03.004
  16. [16] F. Liu et al., “Robust H∞ damping control of multi-FACTS devices for stability enhancement of power systems with signal’s time delay,” J. Adv. Comput. Intell. Intell. Inform., Vol.19, No.2, pp. 239-246, 2015. https://doi.org/10.20965/jaciii.2015.p0239
  17. [17] Q.-L. Han, “Absolute stability of time-delay systems with sector-bounded nonlinearity,” Automatica, Vol.41, No.12, pp. 2171-2176, 2005. https://doi.org/10.1016/j.automatica.2005.08.005
  18. [18] A. Seuret and F. Gouaisbaut, “Wirtinger-based integral inequality: Application to time-delay systems,” Automatica, Vol.49, No.9, pp. 2860-2866, 2013. https://doi.org/10.1016/j.automatica.2013.05.030
  19. [19] P. Park, W. I. Lee, and S. Y. Lee, “Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems,” J. of the Franklin Institute, Vol.352, No.4, pp. 1378-1396, 2015. https://doi.org/10.1016/j.jfranklin.2015.01.004
  20. [20] A. Seuret and F. Gouaisbaut, “Hierarchy of LMI conditions for the stability analysis of time-delay systems,” Systems & Control Letters, Vol.81, pp. 1-7, 2015. https://doi.org/10.1016/j.sysconle.2015.03.007
  21. [21] H.-B. Zeng et al., “Free-matrix-based integral inequality for stability analysis of systems with time-varying delay,” IEEE Trans. on Automatic Control, Vol.60, No.10, pp. 2768-2772, 2015. https://doi.org/10.1109/TAC.2015.2404271
  22. [22] H.-B. Zeng et al., “New results on stability analysis for systems with discrete distributed delay,” Automatica, Vol.60, pp. 189-192, 2015. https://doi.org/10.1016/j.automatica.2015.07.017
  23. [23] J. Chen et al., “Two general integral inequalities and their applications to stability analysis for systems with time-varying delay,” Int. J. of Robust and Nonlinear Control, Vol.26, No.18, pp. 4088-4103, 2016. https://doi.org/10.1002/rnc.3551
  24. [24] P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, Vol.47, No.1, pp. 235-238, 2011. https://doi.org/10.1016/j.automatica.2010.10.014
  25. [25] C.-K. Zhang et al., “An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay,” Automatica, Vol.85, pp. 481-485, 2017. https://doi.org/10.1016/j.automatica.2017.07.056
  26. [26] A. Seuret, K. Liu, and F. Gouaisbaut, “Generalized reciprocally convex combination lemmas and its application to time-delay systems,” Automatica, Vol.95, pp. 488-493, 2018. https://doi.org/10.1016/j.automatica.2018.06.017
  27. [27] K. Liu, A. Seuret, and Y. Xia, “Stability analysis of systems with time-varying delays via the second-order Bessel–Legendre inequality,” Automatica, Vol.76, pp. 138-142, 2017. https://doi.org/10.1016/j.automatica.2016.11.001
  28. [28] X.-M. Zhang et al., “Overview of recent advances in stability of linear systems with time-varying delays,” IET Control Theory & Applications, Vol.13, No.1, pp. 1-16, 2019. https://doi.org/10.1049/iet-cta.2018.5188
  29. [29] X.-M. Zhang and Q.-L. Han, “Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach,” Neural Networks, Vol.54, pp. 57-69, 2014. https://doi.org/10.1016/j.neunet.2014.02.012
  30. [30] J.-H. Kim, “Further improvement of Jensen inequality and application to stability of time-delayed systems,” Automatica, Vol.64, pp. 121-125, 2016. https://doi.org/10.1016/j.automatica.2015.08.025
  31. [31] F. Long et al., “Stability analysis of linear systems with time-varying delay via a quadratic function negative-definiteness determination method,” IET Control Theory & Applications, Vol.14, No.11, pp. 1478-1485, 2020. https://doi.org/10.1049/iet-cta.2019.0471
  32. [32] C.-K. Zhang et al., “A relaxed quadratic function negative-determination lemma and its application to time-delay systems,” Automatica, Vol.113, Article No.108764, 2020. https://doi.org/10.1016/j.automatica.2019.108764
  33. [33] F. Liu et al., “Two relaxed quadratic function negative-determination lemmas: Application to time-delay systems,” Automatica, Vol.147, Article No.110697, 2023. https://doi.org/10.1016/j.automatica.2022.110697
  34. [34] F. Long et al., “A sufficient negative-definiteness condition for cubic functions and application to time-delay systems,” Int. J. of Robust and Nonlinear Control, Vol.31, No.15, pp. 7361-7371, 2021. https://doi.org/10.1002/rnc.5682
  35. [35] F. S. S. de Oliveira and F. O. Souza, “Further refinements in stability conditions for time-varying delay systems,” Applied Mathematics and Computation, Vol.369, Article No.124866, 2020. https://doi.org/10.1016/j.amc.2019.124866
  36. [36] X.-M. Zhang, Q.-L. Han, and X. Ge, “Novel stability criteria for linear time-delay systems using Lyapunov–Krasovskii functionals with a cubic polynomial on time-varying delay,” IEEE/CAA J. of Automatica Sinica, Vol.8, No.1, pp. 77-85, 2021. https://doi.org/10.1109/JAS.2020.1003111
  37. [37] X. Zhang, P. Tsiotras, and T. Iwasaki, “Lyapunov-based exact stability analysis and synthesis for linear single-parameter dependent systems,” Int. J. of Control, Vol.83, No.9, pp. 1823-1838, 2010. https://doi.org/10.1080/00207179.2010.497544
  38. [38] L. Jiang et al., “Delay-dependent stability for load frequency control with constant and time-varying delays,” IEEE Trans. on Power Systems, Vol.27, No.2, pp. 932-941, 2012. https://doi.org/10.1109/TPWRS.2011.2172821

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Last updated on Sep. 19, 2024