Improved mixed-delay-dependent asymptotic stability criteria for neutral systems

Liming Ding; Yong He; Min Wu; Chongyang Ning

Improved mixed-delay-dependent asymptotic stability criteria for neutral systems

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Author(s): Liming Ding ¹ ; Yong He ² ; Min Wu ² ; Chongyang Ning ³
- Affiliations: 1: School of Information Science and Engineering, Central South University, Changsha 410083, People's Republic of China ;
  2: School of Automation, China University of Geosciences, Wuhan 430074, People's Republic of China ;
  3: School of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan 411201, People's Republic of China
Source: Volume 9, Issue 14, 17 September 2015, p. 2180 – 2187
DOI: 10.1049/iet-cta.2015.0022 , Print ISSN 1751-8644, Online ISSN 1751-8652

Received 10/08/2014, Accepted 06/05/2015, Revised 21/04/2015, Published 25/06/2015

This study concerns the problem of asymptotic stability analysis for neutral systems with mixed delays. Some simple and less conservative criteria are firstly proposed based on the Wirtinger-based integral inequality. Then, incorporating the delay-decomposition idea with the augmented Lyapunov–Krasovskii functional, a new augmented delay-decomposition Lyapunov–Krasovskii functional is constructed. In addition, the Wirtinger-based integral inequality is sufficiently utilised to cope with the derivative of the constructed functional. Those treatments lead to less conservatism, and an improved asymptotic stability condition is obtained. Finally, a numerical example is given to show the effectiveness and benefit of the proposed criteria.

References

1. 1)
  - 29. Kwon, O.-M., Park, M.-J., Park, J.-H., Lee, S.-M., Cha, E.-J.: ‘Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality’, J. Franklin Inst., 2014, 12, (351), pp. 5386–5398 (doi: 10.1016/j.jfranklin.2014.09.021).
2. 2)
  - 16. Kwon, O.-M., Park, J.-H., Lee, S.-M.: ‘Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays’, Appl. Math. Comput., 2009, 207, (1), pp. 202–212 (doi: 10.1016/j.amc.2008.10.018).
3. 3)
  - 6. He, Y., Wang, Q.-G., Lin, C., Wu, M.: ‘Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems’, Int. J. Robust Nonlinear Control, 2005, 15, (18), pp. 923–933 (doi: 10.1002/rnc.1039).
4. 4)
  - 36. Boyd, S., Ghaoui, L.-E.L., Feron, E., Balakrishnan, V.: ‘Linear matrix inequality in system and control theory’ (Studies in Applied Mathematics, SIAM, Philadelphia, 1994).
5. 5)
  - 17. Zhang, C.-K., Jiang, L., He, Y., Wu, H., Wu, M.: ‘Stability analysis for control systems with aperiodically sampled data using an augmented Lyapunov functional method’, IET Control Theory Appl., 2013, 7, (9), pp. 1219–1226 (doi: 10.1049/iet-cta.2012.0814).
6. 6)
  - 35. Hale, J.-K., Lunel, S.-V.: ‘Introduction to functional differential equations’ (Springer-Verlag, New York, 1993).
7. 7)
  - 26. Zuo, Z., Yang, C., Wang, Y.: ‘A new method for stability analysis of recurrent neural networks with interval time-varying delay’, IEEE Trans. Neural Netw., 2010, 21, (2), pp. 339–344 (doi: 10.1109/TNN.2009.2037893).
8. 8)
  - 18. Zhang, X.-M., Wu, M., She, J.-H., He, Y.: ‘Delay-dependent stabilization of linear systems with time-varying state and input delays’, Automatica, 2005, 41, (8), pp. 1405–1412 (doi: 10.1016/j.automatica.2005.03.009).
9. 9)
  - 3. Kuang, Y.: ‘Delay differential equations with applications in population dynamics’ (Academic Press, Boston, 1993).
10. 10)
  - 2. Kolmanovskii, V.-B., Nosov, V.-R.: ‘Stability of functional differential equations’ (Academic Press, London, 1986).
11. 11)
  - 23. Fridman, E.: ‘New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems’, Syst. Control Lett., 2001, 43, (4), pp. 309–319 (doi: 10.1016/S0167-6911(01)00114-1).
12. 12)
  - 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).
13. 13)
  - 1. Kolmanovskii, V.-B., Myshkis, A.: ‘Applied theory of functional differential equations’, (Kluwer Academic Publishers, Boston, 1992).
14. 14)
  - 21. Han, Q.-L.: ‘A discrete delay decomposition approach to stability of linear retarded and neutral systems’, Automatica, 2009, 45, (2), pp. 517–524 (doi: 10.1016/j.automatica.2008.08.005).
15. 15)
  - 19. Zhu, X.-L., Yang, G.-H.: ‘Jensen inequality approach to stability analysis of discrete-time systems with time-varying delay’. American Control Conf., 2008, pp. 1644–1649.
16. 16)
  - 14. He, Y., Wu, M., She, J.-H.: ‘Delay-dependent stability criteria for linear systems with multiple time delays’, IEE Proc. Control Theory Appl., 2006, 153, (4), pp. 447–452 (doi: 10.1049/ip-cta:20045279).
17. 17)
  - 25. Han, Q.-L.: ‘New delay-dependent stability criterion for linear neutral systems with norm-bounded uncertainties in all system matrices’, Int. J. Syst. Sci., 2005, 36, (8), pp. 469–475 (doi: 10.1080/00207720500157437).
18. 18)
  - 31. Liu, K., Fridman, E.: ‘Wirtinger's inequality and Lyapunov-based sampled data stabilization’, Automatica, 2012, 48, (1), pp. 102–108 (doi: 10.1016/j.automatica.2011.09.029).
19. 19)
  - 30. Lee, T.-H., Park, J.-H., Jung, H.-Y., Kwon, O.-M., Lee, S.-M.: ‘Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality’. Proc. of the 19th World Congress of the IFAC, 2014, pp. 6826–6830.
20. 20)
  - 4. Qian, W., Liu, J., Fei, S.: ‘New augmented Lyapunov functional method for stability of uncertain neutral systems with equivalent delays’, Math. Comput. Simul., 2012, 84, pp. 42–50 (doi: 10.1016/j.matcom.2012.08.003).
21. 21)
  - 12. He, Y., Wu, M., She, J.-H., Liu, G.-P.: ‘Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties’, IEEE Trans. Autom. Control, 2004, 49, (5), pp. 828–832 (doi: 10.1109/TAC.2004.828317).
22. 22)
  - 27. Kwon, O.-M., Park, M.-J., Park, J.-H., Lee, S.-M., Cha, E.-J.: ‘New delay-partitioning approaches to stability criteria for uncertain neutral systems with time-varying delays’, J. Franklin Inst., 2012, 349, (9), pp. 2799–2823 (doi: 10.1016/j.jfranklin.2012.08.013).
23. 23)
  - 34. Oliveira, M.-C., Skelton, R.-E.: ‘Stability tests for constrained linear systems’ (Springer-Verlag, Berlin, 2001).
24. 24)
  - 7. He, Y., Wu, M., She, J.-H., Liu, G.-P.: ‘Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays’, Syst. Control Lett., 2004, 51, (1), pp. 57–65 (doi: 10.1016/S0167-6911(03)00207-X).
25. 25)
  - 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).
26. 26)
  - 11. Fridman, E., Shaked, U.: ‘Delay-dependent stability and H∞ control: constant and time-varying delays’, Int. J. Control, 2003, 76, (1), pp. 48–60 (doi: 10.1080/0020717021000049151).
27. 27)
  - 8. Liu, X.-G., Wu, M., Martin, R., Tang, M.-L.: ‘Stability analysis for neutral systems with mixed delays’, J. Comput. Appl. Math., 2007, 202, (2), pp. 478–497 (doi: 10.1016/j.cam.2006.03.003).
28. 28)
  - 5. Han, Q.-L.: ‘Improved stability criteria and controller design for linear neutral systems’, Automatica, 2009, 45, (8), pp. 1948–1952 (doi: 10.1016/j.automatica.2009.03.019).
29. 29)
  - 32. Seuret, A., Gouaisbaut, F.: ‘On the use of the Wirtinger inequalities for time-delay systems’. Proc. of the 10th IFAC Workshop on Time Delay Systems, 2012 IFAC TDS’12, Boston, MA, USA.
30. 30)
  - 20. Zhang, X.-M., Han, Q.-L.: ‘Robust H∞ filtering for a class of uncertain linear systems with time-varying delay’, Automatica, 2008, 44, (1), pp. 157–166 (doi: 10.1016/j.automatica.2007.04.024).
31. 31)
  - 15. Wu, M., He, Y., She, J.-H.: ‘New delay-dependent stability criteria and stabilizing method for neutral systems’, IEEE Trans. Autom. Control, 2004, 49, (12), pp. 2266–2271 (doi: 10.1109/TAC.2004.838484).
32. 32)
  - 10. Chen, Y.-G., Fei, S.-M., Gu, Z., Li, Y.M.: ‘New mixed-delay-dependent robust stability conditions for uncertain linear neutral systems’, IET Control Theory Appl., 2014, 8, (8), pp. 606–613 (doi: 10.1049/iet-cta.2013.0569).
33. 33)
  - 22. Han, Q.-L.: ‘On stability of linear systems with mixed time-delays: a discretized Lyapunov functional approach’, Automatica, 2005, 41, (7), pp. 1209–1218 (doi: 10.1016/j.automatica.2005.01.014).
34. 34)
  - 28. Yang, B., Fan, C.-X.: ‘New stability analysis for linear systems with time-varying delay based on combined convex technique’, Math. Probl. Eng., http://dx.doi.org/10.1155/2015/425864, accessed 2015.
35. 35)
  - 9. Qian, W., Liu, J., Sun, Y., Fei, S.: ‘A less conservative robust stability criteria for uncertain neutral systems with mixed delays’, Math. Comput. Simul., 2010, 80, (5), pp. 1007–1017 (doi: 10.1016/j.matcom.2009.12.007).
36. 36)
  - 13. Wu, M., He, Y., She, J.-H., Liu, G.-P.: ‘Delay-dependent criteria for robust stability conditions of uncertain neutral systems’, Automatica, 2004, 40, (8), pp. 1435–1439 (doi: 10.1016/j.automatica.2004.03.004).

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