Advertisement

Robust Stability Analysis of Time-varying Delay Systems via an Augmented States Approach

  • Chao-Yang Dong
  • Ming-Yu Ma
  • Qing Wang
  • Si-Qian Ma
Regular Papers Control Theory and Applications
  • 14 Downloads

Abstract

This paper focuses on the robust stability analysis of linear systems with time varying delays. The second derivative of the state is used to construct a novel Lyapunov-Krasovskii functional (LKF). Then, both the integrals and derivatives of the states along with the delayed states could be taken into consideration in handling of the LKF. We introduce these augmented variables and establish correlations between them. Moreover, our approach makes it available that quadratic convex combination can be applied not only to the time delay, but also to the derivative of it. Robust stability criteria are presented, and numerical examples illustrate that the methods are less conservative.

Keywords

Augmented states Lyapunov-Krasovskii functional quadratic convex method robust stability time varying delay 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. C. Hua, S. X. Ding, and X. P. Guan, “Robust controller design for uncertain multiple-delay systems with unknown actuator parameters,” Automatica, vol. 48, no. 1, pp. 211–218, January 2012.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R. Saravanakumar, M. Syed Ali, and H. R. Karimi, “Robust H control of uncertain stochastic Markovian jump systems with mixed time-varying delays,” Int. J. System Science, vol. 48, no. 4, pp. 862–872, February 2017.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    J. Qiu, H. Gao, and S. X. Ding, “Recent advances on fuzzymodel-based nonlinear networked control systems: a survey,” IEEE Trans. Industrial Electronics, vol. 63, no. 2, pp. 1207–1217, February 2016.CrossRefGoogle Scholar
  4. [4]
    X.-P. Chen and H. Dai, “Stability analysis of time-delay systems using a contour integral method,” Appl. Math. Comput., vol. 273, pp. 390–397, January 2016.MathSciNetGoogle Scholar
  5. [5]
    M. Lyu and Y. Bo, “Variance-constrained resilient H¥ filtering for time-varying nonlinear networked systems subject to quantization effects,” Neurocomputing, vol. 267, pp. 283–294, December 2017.CrossRefGoogle Scholar
  6. [6]
    H. Shao, H. Li, and C. Zhu, “New stability results for delayed neural networks,” Appl. Math. Comput., vol. 311, pp. 324–334, October 2017.MathSciNetCrossRefGoogle Scholar
  7. [7]
    M.-Y. Ma, C.-Y. Dong, Q. Wang, and M. Ni, “Robust stability analysis for systems with time-varying delays: a delay function approach,” Asian J. Control, vol. 18, no. 6, pp. 1923–1933, September 2016.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    S. B. Stojanovic, D. Lj. Debeljkovic, and M. A. Misic, “Finite-time stability for a linear discrete-time delay systems by using discrete convolution: an LMI approach,” Int. J. Control, Autom., Systems, vol. 14, no. 4, pp. 1144–1151, August 2016.CrossRefGoogle Scholar
  9. [9]
    W. Qian, L. Wang, and Y. Sun, “Improved robust stability criteria for uncertain systems with time-varying delay,” Asian J. Control, vol. 13, no. 6, pp. 1043–1050, November 2011.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Y. S. Moon, P. Park, H K. Kwon, and Y. S. Lee, “Delaydependent robust stabilization of uncertain state-delayed systems,” Int. J. Control, vol. 74, no. 14, pp. 1447–1455, September 2001.CrossRefMATHGoogle Scholar
  11. [11]
    M. Wu, Y. He, J. H. She, and H. P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, vol. 40, no. 8, pp. 1435–1439, August 2004.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Y. He, Q. G. Wang, L. Xie, and C. Lin, “Further improvement of free-weighting matrices for systems with timevarying delay,” IEEE Trans. Automatic Control, vol. 52, no. 2, pp. 293–299, February 2007.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Z. Lian, Y. He, C.-K. Zhang, and M. Wu, “Stability analysis for T-S fuzzy systems with time-varying delay via freematrix-based integral inequality,” Int. J. Control, Autom., Systems, vol. 14, no. 1, pp. 21–28, February 2016.CrossRefGoogle Scholar
  14. [14]
    C.-K. Zhang, Y. He, L. Jiang, W.-J. Lin, and M. Wu, “Delay-dependent stability analysis of neural networks with time-varying delay: A generalized free-weightingmatrix approach,” Appl. Math. Comput., vol. 294, pp. 102–120, February 2017.MathSciNetGoogle Scholar
  15. [15]
    P. Park and J.W. Ko, “Stability and robust stability for systems with a time-varying delay,” Automatica, vol. 43, no. 10, pp. 1855–1858, October 2007.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. H. Kim, “Note on stability of linear systems with timevarying delay,” Automatica, vol. 47, no. 9, pp. 2118–2121, September 2011.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    J. H. Kim, “Further improvement of Jensen inequality and application to stability of time-delayed systems,” Automatica, vol. 64, pp. 121–125, February 2016.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    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, January 2011.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    W. I. Lee and P. G. Park, “Second-order reciprocally convex approach to stability of systems with interval timevarying delays,” Appl. Math. Comput., vol. 229, pp. 245–253, February 2014.MathSciNetMATHGoogle Scholar
  20. [20]
    A. Seuret and F. Gouaisbaut, “Delay-dependent reciprocally convex combination lemma,” Rapport LAAS n16006. 2016. http://hal.archives-ouvertes.fr/hal-01257670/. Google Scholar
  21. [21]
    C.-K. Zhang, Y. He, L. Jiang, M. Wu, and Q.-G. Wang, “An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay,” Automatica, vol. 85, pp. 481–485, November 2017MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Z. Zhang, C. Lin, and B. Chen, “New stability criteria for linear time-delay systems using complete LKF method,” Int. J. System Science, vol. 46, no. 2, pp. 377–384, January 2015.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    C. K. Zhang, Y. He, L. Jiang, W. J. Lin, and M. Wu, “Delay-dependent stability analysis of neural networks with time-varying delay: A generalized free-weightingmatrix approach,” Appl. Math. Comput., vol. 294, pp. 102–120, February 2017.MathSciNetGoogle Scholar
  24. [24]
    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. Control, Autom., Systems, vol. 14, no. 4, pp. 958–956, August 2016.Google Scholar
  25. [25]
    E. Gyurkovics, G. Szabo-Varga, and K. Kiss, “Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities,” Appl. Math. Comput., vol. 311, pp. 164–177, October 2017.MathSciNetGoogle Scholar
  26. [26]
    L. V. Hien and H. Trinh, “Exponential stability of timedelay systems via new weighted integral inequalities,” Appl. Math. Comput., vol. 275, pp. 335–344, February 2016.MathSciNetGoogle Scholar
  27. [27]
    R. Saravanakumar and M. S. Ali, “Robust H¥ control for uncertain Markovian jump systems with mixed delays,” Chinese Phys. B, vol. 25, no. 7, 070201, July 2016.Google Scholar
  28. [28]
    A. Seuret and F. Gouaisbaut, “Wirtinger-based integral inequality: Application to time-delay systems,” Automatica, vol. 49, no. 9, pp. 2860–2866, September 2013.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    Y. Liu and M. Li, “Improved robust stabilization method for linear systems with interval time-varying input delays by usingWirtinger inequality,” ISA T., vol. 56, pp. 111–122, June 2015.CrossRefGoogle Scholar
  30. [30]
    C. K. Zhang, Y. He, L. Jiang, M. Wu, and H. B. Zeng, “Stability analysis of systems with time-varying delay via relaxed integral inequalities,” Syst. Control Lett., vol. 92, pp. 52–61, June 2016.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    S. Y. Lee, W. I. Lee, and P. G. Park, “Improved stability criteria for linear systems with interval time-varying delays: Generalized zero equalities approach,” Appl. Math. Comput., vol. 292, pp. 336–348, January 2017.MathSciNetGoogle Scholar
  32. [32]
    C. Hua, S. Wu, X. Yang, and X. Guan, “Stability analysis of time-delay systems via free-matrix-based double integral inequality,” Int. J. System Science, vol. 48, 2, pp. 257–263, January 2017.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    X.-M. Zhang, Q.-L. Han, A. Seuret, and F. Gouaisbaut, “An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay,” Automatica, vol. 84, pp. 221–226, October 2017.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Y. Ariba and F. Gouaisbaut, “Delay-dependent stability analysis of linear systems with time-varying delay,” Proc. of the 46th IEEE Conf. Decision and Control, pp. 2053–2058, December 2007.Google Scholar
  35. [35]
    M. J. Park, O. M. Kwon, J. H. Park, and S. M. Lee, “A new augmented Lyapunov-Krasovskii functional approach for stability of linear systems with time-varying delays,” Appl. Math. Comput., vol. 217, pp. 7197–7209, May 2011.MathSciNetMATHGoogle Scholar
  36. [36]
    Y. Ariba and F. Gouaisbaut, “An augmented model for robust stability analysis of time-varying delay systems,” Int. J. Control, vol. 82, no. 9, pp. 1616–1626, July 2009.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    W. Qian, S. Cong, T. Li, and S. Fei, “Improved stability conditions for systems with interval time-varying delay,” Int. J. Control, Autom., Systems, vol. 10, no. 6, pp. 1146–1152, December 2012.CrossRefGoogle Scholar
  38. [38]
    F. Liao, J. Wu, and M. Tomizuka, “An improved delaydependent stability criterion for linear uncertain systems with multiple time-varying delays,” Int. J. Control, vol. 87, no. 4, pp. 861–873, April 2014.CrossRefMATHGoogle Scholar
  39. [39]
    M. Syed Ali, and R. Saravanakumar, “Novel delaydependent robust H¥ control of uncertain systems with distributed time-varying delays,” Appl. Math. Comput., vol. 249, pp. 510–520, December 2014.MathSciNetMATHGoogle Scholar
  40. [40]
    Y.-L. Zhi, Y. He, and M. Wu, “Improved free matrixbased integral inequality for stability of systems with timevarying delay,” IET Control Theory Appl., vol. 11, no. 10, pp. 1571–1577, June 2017.MathSciNetCrossRefGoogle Scholar
  41. [41]
    C. Ge, C.-C. Hua, and X.-P. Guan, “New delay-dependent stability criteria for neutral systems with time-varying delay using delay-decomposition approach,” Int. J. Control, Autom., Systems, vol. 12, no. 4, pp. 786–793, August 2014.CrossRefGoogle Scholar
  42. [42]
    Z. Feng, J. Lam, and G. H. Yang, “Optimal partitioning method for stability analysis of continuous/discrete delay systems,” Int. J. Robust Nonlinear Control, vol. 25, no. 4, pp. 559–574, March 2015.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    P. L. Liu, “A delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay,” ISA T., vol. 51, no. 6, pp. 694–701, November 2012.CrossRefGoogle Scholar
  44. [44]
    J. An, “Improved delay-derivative-dependent stability criteria for linear systems using new bounding techniques,” Int. J. Control, Autom., Systems, vol. 15, no. 2, pp. 939–946, April 2017.CrossRefGoogle Scholar
  45. [45]
    H. Zhang and Z. Liu, “Corrigendum to ‘Stability analysis for linear delayed systems via an optimally dividing delay interval approach’ [Automatica vol. 47 (2011) 2126-2129],” Automatica, vol. 50, no. 6, pp. 1739–1740, June 2014.MathSciNetCrossRefGoogle Scholar
  46. [46]
    H. Xia, P. Zhao, L. Li, Y. Wang, and A. Wu, “Improved stability criteria for linear neutral time-delay systems,” Asian J. Control, vol. 17, no. 1, pp. 1–9, January 2015.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    Y. Du, W. Wen, S. Zhong, and N. Zhou, “Complete delaydecomposing approach to exponential stability for uncertain cellular neural networks with discrete and distributed time-varying delays,” Int. J. Control, Autom., Systems, vol. 14, no. 4, pp. 1012–1020, August 2016.CrossRefGoogle Scholar
  48. [48]
    X. Zhang, Y. Wang, and X. Fan, “Stability analysis of linear systems with an interval time-varying delay-a delayrange-partition approach,” Int. J. Control, Autom., Systems, vol. 15, no. 2, pp. 518–526, April 2017.CrossRefGoogle Scholar
  49. [49]
    X.-M. Zhang and Q.-L. Han, “A delay decomposition approach to delay-dependent stability for linear systems with time-varying delays,” Int. J. Robust Nonlinear Control, vol. 19, no. 17, pp. 1922–1930, November 2009.MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    S. Duan, J. Ni, and A. G. Ulsoy, “An improved LMI-based approach for stability of piecewise affine time-delay systems with uncertainty,” Int. J. Control, vol. 85, no. 9, pp. 1218–1234, April 2012.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    J. Qiu, H. Tian, Q. Lu, and H. Gao, “Nonsynchronized robust filtering design for continuous-time T-S fuzzy affine dynamic systems based on piecewise Lyapunov functions,” IEEE Trans. Cybernetics, vol. 43, no. 6, pp. 1755–1766, December 2013.CrossRefGoogle Scholar
  52. [52]
    J. Qiu, Y. Wei, and L. Wu, “A novel approach to reliable control of piecewise affine systems with actuator faults,” IEEE Trans. Circuits and Systems-II: Express Briefs, vol. 64, no. 8, pp. 957–961, August 2017. CrossRefGoogle 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

  • Chao-Yang Dong
    • 1
  • Ming-Yu Ma
    • 1
  • Qing Wang
    • 2
  • Si-Qian Ma
    • 1
  1. 1.School of Aeronautic Science and EngineeringBeihang UniversityBeijingChina
  2. 2.School of Automation Science and Electrical EngineeringBeihang UniversityBeijingChina

Personalised recommendations