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Improved Stability Criteria for Discrete-time Delay Systems via Novel Summation Inequalities

  • Shenping Xiao
  • Linxing Xu
  • Hongbing Zeng
  • Kok Lay Teo
Article
  • 13 Downloads

Abstract

This paper is concerned with the stability analysis of linear discrete time-delay systems. New discrete inequalities for single summation and double summation are presented to estimate summation terms in the forward difference of Lyapunov-Krasovskii functional (LKF), which are more general than some commonly used summation inequalities. Through the construction of an augmented LKF, improved delay-dependent stability criteria for discrete time-delay systems are established. Based on this, a time-delayed controller is derived for linear discrete time-delay systems. Finally, the advantages of the proposed criteria are revealed from the solutions of the numerical examples.

Keywords

Discrete time-delay systems Lyapunov-Krasovskii functional stability analysis summation inequality 

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References

  1. [1]
    B. L. Zhang, Q. L. Han, and X. M. Zhang, “Recent advances in vibration control of offshore platforms,” Nonlin. Dyn., vol. 89, no. 2, pp. 755–771, July 2017.CrossRefGoogle Scholar
  2. [2]
    H. B. Zeng, Y. He, M. Wu, and J. H. She, “Free-matrixbased integral inequality for stability analysis of systems with time-varying delay,” IEEE Trans. Automat. Control, vol. 60, no. 10, pp. 2768–2772, February 2015.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    X. M. Zhang and Q. L. Han, “New LyapunovKrasovskii functionals for global asymptotic stability of delayed neural networks,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 533–539, March 2009. [click]CrossRefGoogle Scholar
  4. [4]
    B. L. Zhang, Q. L. Han, and X. M. Zhang. “Event-triggered H¥ reliable control for offshore structures in network environments,” J. Sound Vib., vol. 368, pp. 1–21, April 2016.CrossRefGoogle Scholar
  5. [5]
    W. P. Luo, J. Yang, and X. Zhao, “Free-matrix-based integral inequality for stability analysis of uncertain T-S fuzzy systems with time-varying delay,” Int. J. of Control, Automation, and Systems, vol. 14, no. 4, pp. 948–956, August 2016. [click]CrossRefGoogle Scholar
  6. [6]
    Z. Zuo, Q. L Han, B. Ning, X. Ge, and X. M. Zhang, “An overview of recent advances in fixed-time cooperative control of multi-agent systems,” IEEE Trans. Ind. Inf., vol. 14, no. 6, pp. 2322–2334, June 2018. [click]CrossRefGoogle Scholar
  7. [7]
    X. M. Zhang and Q. L. Han, “Network-based H¥ filtering using a logic jumping-like trigger,” Automatica, vol. 49, no. 5, pp. 1428–1435, May 2013. [click]MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    J. Zhao and Z. Hu, “Exponential H¥ control for singular systems with time-varying delay,” Int. J. of Control, Automation, and Systems, vol. 15, no. 4, pp. 1592–1599, August 2017. [click]CrossRefGoogle Scholar
  9. [9]
    S. P. Xiao, H. H. Lian, H. B. Zeng, G. Chen, and W. H. Zheng, “Analysis on robust passivity of uncertain neural networks with time-varying delays via free-matrix-based integral inequality,” Int. J. Control Automation, and Systems, vol. 15, no. 5, pp. 2385–2394, October 2017.CrossRefGoogle Scholar
  10. [10]
    B. L. Zhang, Q. L. Han, X. M. Zhang, and X. Yu, “Sliding mode control with mixed current and delayed states for offshore steel jacket platform,” IEEE Trans. Control Syst. Technol., vol. 22, no. 5, pp. 1769–1783, May 2014. [click]CrossRefGoogle Scholar
  11. [11]
    H. Shao and Q. L. Han, “New stability criterion for linear discrete-time systems with interval-like time-varying delays,” IEEE Trans. Automat. Control, vol. 56, no. 3, pp. 619–625, November 2010. [click]CrossRefMATHGoogle Scholar
  12. [12]
    W. I. Lee, P. G. Park, S. Y. Lee, and R.W. Newcomb, “Auxiliary function-based summation inequalities for quadratic functions and their application to discrete-time delay systems,” IFAC-PapersOnLine, vol. 48, no. 12, pp. 203–208, February 2015.CrossRefGoogle Scholar
  13. [13]
    H. Huang and G. Feng, “Improved approach to delaydependent stability analysis of discrete-time systems with time-varying delay,” IET Control Theory and Applications, vol. 4, no. 10, pp. 2152–2159, October 2010. [click]MathSciNetCrossRefGoogle Scholar
  14. [14]
    X. M. Zhang and Q. L. Han, “Output feedback stabilization of network control systems with a logic zero-order-hold,” Information Science, vol. 381, pp. 78–91, March 2017.CrossRefGoogle Scholar
  15. [15]
    C. Peng, “Improved delay-dependent stabilisation criteria for discrete-time systems with a new finite sum inequlity,” IET Control Theory and Applications, vol. 6, no 3, pp. 448–453, March 2012. [click]MathSciNetCrossRefGoogle Scholar
  16. [16]
    O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “Stability and stabilization for discrete-time system with time-varying delays via augmented Lyapunov-Krasovskii functional,” J. of the Franklin Institute, vol. 350, no. 3, pp. 521–540, April 2013.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    S. H. Kim, “Relaxed inequality approach to robust H¥ stability analysis of discrete-time systems with time-varying delay,” IET Control Theory and Applications, vol. 6, no. 13, pp. 2149–2156, September 2012.MathSciNetCrossRefGoogle Scholar
  18. [18]
    C. Y. Kao, “On stability of discrete-time LTI systems with varying time delays,” IEEE Trans. Automat. Control, vol. 57, no. 5, pp. 1243–1248, November 2011. [click]MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Y. He, M. Wu, J. H. She, and G. P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Systems & Control Letters, vol. 51, no. 1, 57–65, January 2004.Google Scholar
  20. [20]
    H. Gao and T. Chen, “New results on stability of discretetime systems with time-varying state delay,” IEEE Trans. Automat. Control, vol. 52, no. 2, pp. 328–334, February 2007. [click]MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    B. Y. Zhang, S. Y. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, November 2008.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    J. Liu and J. Zhang, “Note on stability of discrete-time time-varying delay systems,” IET Control Theory and Applications, vol. 6, no. 2, pp. 335–339, January 2012. [click]MathSciNetCrossRefGoogle Scholar
  23. [23]
    K. Ramakrishnan and G. Ray, “Robust stability criteria for a class of uncertain discrete-time systems with timevarying delay,” Appl. Math. Model, vol. 37, no. 3, pp. 1468–1479, February 2013.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    F. Gouaisbaut and D. Peaucelle, “Delay-dependent stability analysis of linear time delay systems,” IFAC Proceedings Volumes, vol. 39, no. 10, pp. 54–59, November 2006.CrossRefGoogle Scholar
  25. [25]
    Z. Feng, J. Lam, and G. H. Yang, “Optimal partitioning method for stability analysis of continuous/discrete delay systems,” Int. J. Robust Nonlin. Control, vol. 25, no. 4, pp. 559–574, November 2013.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    A. Seuret, F. Gouaisbaut, and E. Fridman, “Stability of discrete-time systems with time-varying delays via a novel summation inequality,” IEEE Trans. Automat. Control, vol. 60, no. 10, pp. 2740–2745, February 2015.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    P. T. Nam, P. N. Pathirana, and H. Trinh, “Discrete Wirtinger-based inequality and its application,” J. of the Franklin Institute, vol. 352, no. 5, pp. 1893–1905, May 2015. [click]MathSciNetCrossRefGoogle Scholar
  28. [28]
    X. M. Zhang and Q. L. Han, “Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay system,” Automatica, vol. 57, pp. 199–202, July 2015.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    P. T. Nam, H. Trinh, and P. N. Pathirana, “Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems,” J. of the Franklin Institute, vol. 352, no. 12, pp. 5810–5831, December 2015. [click]MathSciNetCrossRefGoogle Scholar
  30. [30]
    P. G. Park, W. I. Lee, and S. Y. Lee, “Auxiliary functionbased integral/summation inequalities: application to continuous/discrete time-delay systems,” Int. J. of Control, Automation, and Systems, vol. 14, no. 1, pp. 3–11, 2016. [click]CrossRefGoogle Scholar
  31. [31]
    S. Y. Lee, W. I. Lee, and P. G. Park, “Polynomialsbased summation inequalities and their applications to discrete-time systems with time-varying delays,” International Journal of Robust and Nonlinear Control, vol. 27, no. 17, pp. 3604–3619, November 2017.MathSciNetMATHGoogle Scholar
  32. [32]
    H. B. Zeng, Y. He, M. Wu, and J. H. She, “New results on stability analysis for systems with discrete and distributed delays,” Automatica, vol. 60, pp. 189–192, October 2015. [click]MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    L. V. Hien and H. Trinh, “New finite-sum inequalities with applications to stability of discrete time-delay systems,” Automatica, vol. 71, pp. 197–201, September 2016.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    J. H. Kim, “Further improvement of Jensen inequality and application to stability of time-delayed systems,” Automatica, vol. 64, pp. 121–125, February 2016. [click]MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    Y. He, M. Wu, G. P. Liu, and J. H. She, “Output feedback stabilization for a discrete-time system with a time-varying delay,” IEEE Trans. Automat. Control, vol. 53, no. 10, pp. 2372–2377, November 2008.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “Improved delay-dependent stability criteria for discrete-time systems with time-varying delays,” Circuits Syst Signal Process, vol. 32, no. 4, pp. 1949–1962, 2013.MathSciNetCrossRefGoogle Scholar
  37. [37]
    X. M. Zhang and Q. L. Han, “Global asymptotic stability for a class of generalized neural networks with interval time-varying delay,” IEEE Trans. Neural Netw., vol. 22, no. 8, pp. 1180–1192, August 2011. [click]CrossRefGoogle Scholar
  38. [38]
    J. Chen, S. Y. Xu, X. L. Jia, and B. Y. Zhang, “Novel summation inequalities and their applications to stability analysis for systems with time-varying delay,” IEEE Trans. Automat. Control, vol. 62, no. 5, pp. 2470–2475, May 2017. [click]MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Shenping Xiao
    • 1
  • Linxing Xu
    • 1
  • Hongbing Zeng
    • 2
  • Kok Lay Teo
    • 3
    • 4
  1. 1.School of Electrical and Information EngineeringHunan University of Technology, and Key Laboratory for Electric Drive Control and Intelligent Equipment of Hunan ProvinceZhuzhou, HunanP. R. China
  2. 2.School of Information Science and EngineeringNortheastern UniversityShenyang, LiaoningP. R. China
  3. 3.Coordinated Innovation Center for Computable Modeling in Management ScienceTianjin University of Finance and EconomicsTianjinChina
  4. 4.Department of Mathemtics and StatsiticsCurtin UniversityPerthAustralia

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