Refined Jensen-based inequality approach to stability analysis of time-delay systems

Le Van Hien; Hieu Trinh

Refined Jensen-based inequality approach to stability analysis of time-delay systems

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Author(s): Le Van Hien ¹ and Hieu Trinh ²
- Affiliations: 1: Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam ;
  2: School of Engineering, Deakin University, Geelong, VIC 3217, Australia
Source: Volume 9, Issue 14, 17 September 2015, p. 2188 – 2194
DOI: 10.1049/iet-cta.2014.0962 , Print ISSN 1751-8644, Online ISSN 1751-8652

Received 08/09/2014, Accepted 28/05/2015, Revised 28/05/2015, Published 24/06/2015

In this study, the authors derive some new refined Jensen-based inequalities, which encompass both the Jensen inequality and its most recent improvement based on the Wirtinger integral inequality. The potential capability of this approach is demonstrated through applications to stability analysis of time-delay systems. More precisely, by using the newly derived inequalities, they establish new stability criteria for two classes of time-delay systems, namely discrete and distributed constant delays systems and interval time-varying delay systems. The resulting stability conditions are derived in terms of linear matrix inequalities, which can be efficiently solved by various convex optimisation algorithms. Numerical examples are given to show the effectiveness and least conservativeness of the results obtained in this study.

References

1. 1)
  - 19. Chen, W.H., Zheng, W.X.: ‘Delay-dependent robust stabilization for uncertain neutral systems with distributed delays’, Automatica, 2007, 43, (1), pp. 95–104 (doi: 10.1016/j.automatica.2006.07.019).
2. 2)
  - 15. Souza, F.O., Palhares, R.M.: ‘New delay-interval stability condition’, Int. J. Syst. Sci., 2014, 45, (3), pp. 300–306 (doi: 10.1080/00207721.2012.720297).
3. 3)
  - 24. Seuret, A., Gouaisbaut, F.: ‘Complete quadratic Lyapunov functionals using Bessel–Legendre inequality’. European Control Conf. (ECC), Strasbourg, France, 2014, pp. 448–453.
4. 4)
  - 21. Meng, X.Y., Lam, J., Du, B.Z., Gao, H.J.: ‘A delay-partitioning approach to the stability analysis of discrete-time systems’, Automatica, 2010, 46, (3), pp. 610–614 (doi: 10.1016/j.automatica.2009.12.004).
5. 5)
  - 20. Hien, L.V., Phat, V.N.: ‘Exponential stability and stabilization of a class of uncertain linear time-delay systems’, J. Franklin Inst., 2009, 346, (6), pp. 611–625 (doi: 10.1016/j.jfranklin.2009.03.001).
6. 6)
  - 6. Li, Y., Gu, K., Zhou, J., Xu, S.: ‘Estimating stable delay intervals with a discretized Lyapunov–Krasovskii functional formulation’, Automatica, 2014, 50, (6), pp. 1691–1697 (doi: 10.1016/j.automatica.2014.04.019).
7. 7)
  - 27. Lee, W.I., Park, P.G.: ‘Second-order reciprocally convex approach to stability of systems with interval time-varying delays’, Appl. Math. Comput., 2014, 229, pp. 245–253 (doi: 10.1016/j.amc.2013.12.025).
8. 8)
  - 14. Shao, H.: ‘New delay-dependent stability criteria for systems with interval delay’, Automatica, 2009, 45, (3), pp. 744–749 (doi: 10.1016/j.automatica.2008.09.010).
9. 9)
  - 4. Jiang, X., Han, Q.-L.: ‘New stability criteria for linear systems with interval time-varying delay’, Automatica, 2008, 44, (10), pp. 2680–2685 (doi: 10.1016/j.automatica.2008.02.020).
10. 10)
  - 13. Peng, C., Tian, Y.: ‘Improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay’, IET Control Theory Appl., 2008, 2, (9), pp. 752–761 (doi: 10.1049/iet-cta:20070362).
11. 11)
  - 10. Hien, L.V., Trinh, H.M.: ‘A new approach to state bounding for linear time-varying systems with delay and bounded disturbances’, Automatica, 2014, 50, (6), pp. 1735–1738 (doi: 10.1016/j.automatica.2014.04.025).
12. 12)
  - 11. Hien, L.V., An, N.T., Trinh, H.: ‘New results on state bounding for discrete-time systems with interval time-varying delay and bounded disturbance inputs’, IET Control Theory Appl., 2014, 8, (14), pp. 1405–1414 (doi: 10.1049/iet-cta.2013.0980).
13. 13)
  - 7. Zhang, X.-M., Han, Q.-L.: ‘New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities’, IET Control Theory Appl., 2014, 8, (12), pp. 1054–1061 (doi: 10.1049/iet-cta.2013.0840).
14. 14)
  - 2. Fridman, E.: ‘Introduction to time-delay systems: analysis and control’ (Birkhäuser, 2014).
15. 15)
  - 26. Fridman, E., Shaked, U., Liu, K.: ‘New conditions for delay-derivative-dependent stability’, Int. J. Control, 2009, 45, (11), pp. 2723–2727.
16. 16)
  - 1. Sipahi, R., Niculescu, S.-I., Abdallah, C.T., Michiels, W., Gu, K.: ‘Stability and stabilization of systems with time delay’, IEEE Control Syst., 2011, 31, (1), pp. 38–65 (doi: 10.1109/MCS.2010.939135).
17. 17)
  - 19. He, Y., Wang, Q., Xie, L., Lin, C.: ‘Further improvement of free-weighting matrices technique for systems with time-varying delay’, IEEE Trans. Autom. Control, 2007, 52, (2), pp. 293–299 (doi: 10.1109/TAC.2006.887907).
18. 18)
  - 35. Park, P., Ko, J.W., Jeong, C.: ‘Reciprocally convex approach to stability of systems with time-varying delays’, Automatica, 2011, 47, (1), pp. 235–238 (doi: 10.1016/j.automatica.2010.10.014).
19. 19)
  - 22. Zhu, S., Li, Z., Zhang, C.: ‘Delay decomposition approach to delay-dependent stability for singular time-delay systems’, IET Control Theory Appl., 2010, 4, (11), pp. 2613–2620 (doi: 10.1049/iet-cta.2009.0426).
20. 20)
  - 17. Seuret, A., Gouaisbaut, F.: ‘Wirtinger-based integral inequality: application to time-delay systems’, Automatica, 2013, 49, (9), pp. 2860–2866 (doi: 10.1016/j.automatica.2013.05.030).
21. 21)
  - 3. Fridman, E., Shaked, U.: ‘A descriptor system approach to H∞ control of time-delay systems’, IEEE Trans. Autom. Control, 2002, 47, (2), pp. 253–270 (doi: 10.1109/9.983353).
22. 22)
  - 18. Seuret, A., Gouaisbaut, F., Fridman, E.: ‘Stability of systems with fast-varying delay using improved Wirtinger's inequality’. IEEE Conf. on Decision and Control, Florence, Italy, 2013, pp. 946–951.
23. 23)
  - 9. Hien, L.V.: ‘An explicit criterion for finite-time stability of linear nonautonomous systems with delays’, Appl. Math. Lett., 2014, 30, pp. 12–18 (doi: 10.1016/j.aml.2013.12.005).
24. 24)
  - 25. Park, M.J., Kwon, O.M., Park, J.H., Lee, S.M., Cha, E.J.: ‘Stability of time-delay systems via Wirtinger-based double integral inequality’, Automatica, 55, 2015, pp. 204–208 (doi: 10.1016/j.automatica.2015.03.010).
25. 25)
  - 23. Gu, K., Kharitonov, V.L., Chen, J.: ‘Stability of time-delay systems’ (Birkhäuser, 2003).
26. 26)
  - 5. Sun, J., Liu, G.P., Chen, J., Rees, D.: ‘Improved delay-range-dependent stability criteria for linear systems with time-varying delays’, Automatica, 2010, 46, (2), pp. 466–470 (doi: 10.1016/j.automatica.2009.11.002).
27. 27)
  - 8. Feyzmahdavian, H.R., Charalambous, T., Johansson, M.: ‘Exponential stability of homogeneous positive systems of degree one with time-varying delays’, IEEE Trans. Autom. Control, 2014, 59, (6), pp. 1594–1599 (doi: 10.1109/TAC.2013.2292739).

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