Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 691650, 23 pages
http://dx.doi.org/10.1155/2013/691650
Research Article

On Exponential Stability for a Class of Uncertain Neutral Markovian Jump Systems with Mode-Dependent Delays

Department of Auto, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China

Received 10 May 2013; Revised 10 July 2013; Accepted 10 July 2013

Academic Editor: Elena Braverman

Copyright © 2013 Xinghua Liu and Hongsheng Xi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1992.
  2. S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, vol. 269 of Lecture Notes in Control and Information Sciences, Springer, London, UK, 2001. View at MathSciNet
  3. P. L. Liu, “Further results on the exponential stability criteria for time delay singular systems with delay-dependence,” International Journal of Innovative Computing, Information and Control, vol. 8, no. 6, pp. 4015–4024, 2012. View at Google Scholar
  4. P. L. Liu, “Improved delay-dependent robust exponential stabilization criteria for uncertain time-varying delay singular systems,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 1, pp. 165–178, 2013. View at Google Scholar
  5. C.-H. Lien and J.-D. Chen, “Discrete-delay-independent and discrete-delay-dependent criteria for a class of neutral systems,” Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 125, no. 1, pp. 33–41, 2003. View at Publisher · View at Google Scholar · View at Scopus
  6. J. Tian and S. Zhong, “Improved delay-dependent stability criterion for neural networks with time-varying delay,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10278–10288, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Tian and X. Zhou, “Improved asymptotic stability criteria for neural networks with interval time-varying delay,” Expert Systems with Applications, vol. 37, no. 12, pp. 7521–7525, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. J. Tian and X. Xie, “New asymptotic stability criteria for neural networks with time-varying delay,” Physics Letters A, vol. 374, no. 7, pp. 938–943, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. N. A. Parlakçı, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed time-varying discrete and neutral delays,” Asian Journal of Control, vol. 9, no. 4, pp. 411–421, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. O. M. Kwon and J. H. Park, “Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 58–68, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Xu and J. Lam, “A survey of linear matrix inequality techniques in stability analysis of delay systems,” International Journal of Systems Science, vol. 39, no. 12, pp. 1095–1113, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. C.-C. Shen and S.-M. Zhong, “New delay-dependent robust stability criterion for uncertain neutral systems with time-varying delay and nonlinear uncertainties,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2277–2285, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. F. Qiu and B.-T. Cui, “Improved exponential stability criteria for uncertain neutral system with nonlinear parameter perturbations,” International Journal of Automation and Computing, vol. 7, no. 4, pp. 413–418, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. G. Zhang, C. Han, Y. Guan, and L. Wu, “Exponential stability analysis and stabilization of discrete-time nonlinear switched systems with time delays,” International Journal of Innovative Computing, Information and Control, vol. 8, no. 3A, pp. 1973–1986, 2012. View at Google Scholar · View at Scopus
  15. H. R. Karimi, M. Zapateiro, and N. Luo, “Stability analysis and control synthesis of neutral systems with time-varying delays and nonlinear uncertainties,” Chaos, Solitons & Fractals, vol. 42, no. 1, pp. 595–603, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. H. Park, “Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 413–421, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Zhang, P. Shi, and J. Qiu, “Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties,” Chaos, Solitons & Fractals, vol. 38, no. 1, pp. 160–167, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. He, Q.-G. Wang, L. Xie, and C. Lin, “Further improvement of free-weighting matrices technique for systems with time-varying delay,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 293–299, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. Tanikawa, “On new smoothing algorithms for discrete-time linear stochastic systems with unknown disturbances,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 1, pp. 15–24, 2008. View at Google Scholar · View at Scopus
  20. B. Chen, H. Li, P. Shi, C. Lin, and Q. Zhou, “Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays,” Information Sciences, vol. 179, no. 16, pp. 2851–2860, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. Liu, Z. Gu, and S. Hu, “H filtering for Markovian jump systems with time-varying delays,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 3, pp. 1299–1310, 2011. View at Google Scholar · View at Scopus
  22. J. Xiong and J. Lam, “Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers,” Automatica, vol. 42, no. 5, pp. 747–753, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. Wu, T. Chen, and L. Wang, “Delay-dependent robust stability and H control for jump linear systems with delays,” Systems & Control Letters, vol. 55, no. 11, pp. 939–948, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. P. Shi, Y. Xia, G. P. Liu, and D. Rees, “On designing of sliding-mode control for stochastic jump systems,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 97–103, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  25. S. Xu, J. Lam, and X. Mao, “Delay-dependent H control and filtering for uncertain Markovian jump systems with time-varying delays,” IEEE Transactions on Circuits and Systems I, vol. 54, no. 9, pp. 2070–2077, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  26. J. Tian, Y. Li, J. Zhao, and S. Zhong, “Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5769–5781, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. L. L. Xiong, X. B. Zhou, J. Qiu, and J. Lei, “Stability analysis for Markovian jump neutral systems with mixed delays and partially known transition rates,” Abstract and Applied Analysis, vol. 2012, Article ID 450168, 22 pages, 2012. View at Publisher · View at Google Scholar
  28. L. Zhang and J. Lam, “Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions,” IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1695–1701, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Q. Ma, S. Xu, and Y. Zou, “Stability and synchronization for Markovian jump neural networks with partly unknown transition probabilities,” Neurocomputing, vol. 74, no. 17, pp. 3404–3411, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. X. Luan, F. Liu, and P. Shi, “Finite-time filtering for non-linear stochastic systems with partially known transition jump rates,” IET Control Theory & Applications, vol. 4, no. 5, pp. 735–745, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  31. Y. Zhang, Y. He, M. Wu, and J. Zhang, “Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices,” Automatica, vol. 47, no. 1, pp. 79–84, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. View at MathSciNet
  33. X.-M. Sun, J. Zhao, and D. J. Hill, “Stability and L2-gain analysis for switched delay systems: a delay-dependent method,” Automatica, vol. 42, no. 10, pp. 1769–1774, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  34. X. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic Processes and Their Applications, vol. 79, no. 1, pp. 45–67, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. H. B. Ji, Algebra Foundation of Control Theory, University of Science and Technology of China Press, Hefei, China, 2008.
  36. Y. G. Sun, L. Wang, and G. Xie, “Exponential stability of switched systems with interval time-varying delay,” IET Control Theory & Applications, vol. 3, no. 8, pp. 1033–1040, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  37. J. Sun, G. P. Liu, and J. Chen, “Delay-dependent stability and stabilization of neutral time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 19, no. 12, pp. 1364–1375, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. L. Xie, “Output feedback H control of systems with parameter uncertainty,” International Journal of Control, vol. 63, no. 4, pp. 741–750, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. L. Xiong, J. Tian, and X. Liu, “Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities,” Journal of the Franklin Institute, vol. 349, no. 6, pp. 2193–2214, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  41. S. Mondié and V. L. Kharitonov, “Exponential estimates for retarded time-delay systems: an LMI approach,” IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 268–273, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  42. P.-L. Liu, “Exponential stability for linear time-delay systems with delay dependence,” Journal of the Franklin Institute, vol. 340, no. 6-7, pp. 481–488, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. S. Xu, J. Lam, and M. Zhong, “New exponential estimates for time-delay systems,” IEEE Transactions on Automatic Control, vol. 51, no. 9, pp. 1501–1505, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  44. Y. Chen, A. Xue, R. Lu, and S. Zhou, “On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2464–2470, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. D. Wang, W. Wang, and P. Shi, “Exponential H filtering for switched linear systems with interval time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 19, no. 5, pp. 532–551, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. D. Wang, W. Wang, P. Shi, and X.-M. Sun, “Controller failure analysis for systems with interval time-varying delay: a switched method,” Circuits, Systems, and Signal Processing, vol. 28, no. 3, pp. 389–407, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. S. He and F. Liu, “Exponential stability for uncertain neutral systems with Markov jumps,” Journal of Control Theory and Applications, vol. 7, no. 1, pp. 35–40, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet