Kybernetika 50 no. 4, 491-511, 2014

Exponential H_infinity filter design for stochastic Markovian jump systems with both discrete and distributed time-varying delays

Li Ma, Meimei Xu, Ruting Jia and Hui YeDOI: 10.14736/kyb-2014-4-0491

Abstract:

This paper is concerned with the exponential $H_{\infty}$ filter design problem for stochastic Markovian jump systems with time-varying delays, where the time-varying delays include not only discrete delays but also distributed delays. First of all, by choosing a modified Lyapunov--Krasovskii functional and employing the property of conditional mathematical expectation, a novel delay-dependent approach is developed to deal with the mean-square exponential stability problem and $H_{\infty}$ control problem. Then, a mean-square exponentially stable and Markovian jump filter is designed such that the filtering error system is mean-square exponentially stable and the $H_{\infty}$ performance of estimation error can be ensured. Besides, the derivative of discrete time-varying delay $h(t)$ satisfies $\dot{h}(t)\leq\eta$ and simultaneously the decay rate $\beta$ can be finite positive value without equation constraint. Finally, a numerical example is provided to illustrate the effectiveness of the proposed design approach.

Keywords:

stochastic systems, linear matrix inequality, distributed time-varying delay, $H_{\infty }$ filter

Classification:

93E03, 93B36

paper.pdf

References:

  1. P. Balasubramaniam and R. Rakkiyappan: Delay-dependent robust stability analysis for Markovian jumping stochastic Cohen-Grossberg neural networks with discrete interval and distributed time-varying delays. Nonlinear Anal. Hybrid Syst. 3 (2009), 207-214.   CrossRef
  2. Y. Y. Cao, J. Lam and L. Hu: Delay-dependent stochastic stability and $H_{\infty}$ analysis for time-delay systems with Markovian jumping parameters. J. Franklin Inst. 340 (2003), 423-434.   CrossRef
  3. W. H. Chen and W. Zheng: Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 43 (2007), 95-104.   CrossRef
  4. K. L. Chung: A Course In Probability Theory. Academic Press, London 2001.   CrossRef
  5. Y. C. Ding, H. Zhu, S. M. Zhong and Y. P. Zhang: $L_{2}-L_{\infty}$ filtering for Markovian jump systems with time-varying delays and partly unknown transition probabilities. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3070-3081.   CrossRef
  6. Y. A. Fiagbedzi and A. E. Pearson: A multistage reduction technique for feedback stabilizing distributed time-lag systems. Automatica 23 (1987), 311-326.   CrossRef
  7. T. H. Gronwall: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919), 292-296.   CrossRef
  8. K. Gu: An integral inequality in the stability problem of time-delay systems. In: Proc. 39th IEEE Conference on Decision and Control, Sydney 2000, pp. 2805-2810.   CrossRef
  9. K. Gu: An improved stability criterion for systems with distributed delays. Int. J. Robust Nonlinear Control 13 (2003), 819-831.   CrossRef
  10. K. Gu, Q. L. Han, C. J. Albert and S. I. Niculescu: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int. J. Control 74 (2001), 737-744.   CrossRef
  11. J. K. Hale: Theory Of Functional Differential Equations. Springer, New York 1977.   CrossRef
  12. J. K. Hale and S. M. V. Lunel: Introduction To Functional Differential Equations. Springer, New York 1993.   CrossRef
  13. Q. L. Han: A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 40 (2004), 1791-1796.   CrossRef
  14. Q. L. Han: A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica 45 (2009), 517-524.   CrossRef
  15. Q. L. Han: Improved stability criteria and controller design for liear neutral systems. Automatica 45 (2009), 1948-1952.   CrossRef
  16. J. Lam, H. Gao and C. Wang: $H_{\infty}$ model reduction of linear systems with distributed delay. Control Theory and Applications, IEE Proc. 152 (2005), 662-674.   CrossRef
  17. C. E. Lawrence: An introduction to stochastic differential equations. math.berkeley.edu/ evans/SDE.course.pdf.   CrossRef
  18. X. G. Li and X. J. Zhu: Stability analysis of neutral systems with distributed delays. Automatica 44 (2008), 2197-2201.   CrossRef
  19. Y. Liu, Z. Wang and X. Liu: Robust $H_{\infty}$ control for a class of nonlinear stochastic systems with mixed time delay. Int. J. Robust Nonlinear Control 17 (2007), 1525-1551.   CrossRef
  20. Y. Liu, Z. Wang and X. Liu: An LMI approach to stability analysis of stochastic high-order Markovian jumping neural networks with mixed time delays. Nonlinear Anal. Hybrid Syst. 2 (2008), 110-120.   CrossRef
  21. L. Ma and F. P. Da: Exponential $H_{\infty}$ filter design for stochastic time-varying delay systems with Markovian jumping parameters. Int. J. Robust and Nonlinear Control 20 (2010), 802-817.   CrossRef
  22. L. Ma, F. P. Da and K. J. Zhang: Exponential $H_{\infty}$ Filter Design for Discrete Time-Delay Stochastic Systems With Markovian Jump Parameters and Missing Measurements. IEEE Trans. Circuits Syst. I: Regul. Pap. 58 (2011), 994-1007.   CrossRef
  23. M. Mariton: Jump Linear Systems In Automatic Control. Marcel Dekker, New York 1990.   CrossRef
  24. X. R. Mao: Exponential stability of stochastic delay interval systems with Markovian switching. IEEE Trans. Automat. Control 47 (2002), 1604-1612.   CrossRef
  25. J. P. Richard: Time-delay systems: An overview of some recent advances and open problems. Automatica 39 (2003), 1667-1694.   CrossRef
  26. Z. Wang, S. Lauria, J. Fang and X. Liu: Exponential stability of uncettain stochastic neural networks with mixed time-delays. Chaos, Solitons Fractals 32 (2007), 62-72.   CrossRef
  27. Y. Wang and H. Zhang: $H_{\infty}$ control for uncertain Markovian jump systems with mode-dependent mixed delays. Progress Natural Sci. 18 (2008), 309-314.   CrossRef
  28. G. L. Wang, Q. L. Zhang and C. Y. Yang: Exponential $H_{\infty}$ filtering for time-varying delay systems: Markovian approach. Signal Process. 91 (2011), 1852-1862.   CrossRef
  29. G. L. Wei, Z. Wang, H. Shu and J. Fang: A delay-dependent approach to $H_{\infty}$ filtering for stochastic delayed jumping systems with sensor non-linearities. Int. J. Control 80 (2008), 885-897.   CrossRef
  30. L. Wu, P. Shi, C. Wang and H. Gao: Delay-dependent robust $H_{\infty}$ and $L_{2}-L_{\infty}$ filtering for LPV systems with both discrete and distributed delays. Control Theory and Applications, IEE Proc. 153 (2006), 483-492.   CrossRef
  31. L. Xie, E. Fridman and U. Shaked: Robust $H_{\infty}$ control of distributed delay systems with application to combustion control. IEEE Trans. Automat. Control 46 (2001), 1930-1935.   CrossRef
  32. L. Xiong, S. Zhong and J. Tian: New robust stability condition for uncertain neutral systems with discrete and distributed delays. Chaos, Solitons Fractals 42 (2009), 1073-1079.   CrossRef
  33. S. Xu and T. Chen: An LMI approach to the $H_{\infty}$ filter design for uncertain systems with distributed delays. IEEE Trans. Circuits Syst.-II: Express Briefs 51 (2004), 195-201.   CrossRef
  34. S. Xu, Y. Chu, J. Lu and Y. Zou: Exponential dynamic output feedback controller design for stochastic neutral systems with distributed delays. IEEE Trans. Systems, Man, Cybernetics - Part A: Systems and Humans 36 (2006), 540-548.   CrossRef
  35. S. Xu, J. Lam, T. Chen and Y. Zou: A delay-dependent approach to robust $H_{\infty}$ filtering for uncertain distributed delay systems. IEEE Trans. Signal Process. 53 (2005), 3764-3772.   CrossRef
  36. X. G. Yu: An LMI approach to robust $H_{\infty}$ filtering for uncertain systems with time-varying distributed delays. J. Franklin Inst. 345 (2008), 877-890.   CrossRef
  37. D. Yue and Q. L. Han: Robust $H_{\infty}$ filter design of uncertain descriptor systems with discrete and distributed delays. IEEE Trans. Signal Process. 52 (2004), 3200-3212.   CrossRef
  38. D. Yue and Q. L. Han: Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching. IEEE Trans. Automat. Control 50 (2005), 217-222.   CrossRef
  39. X. M. Zhang and Q. L. Han: A less conservative method for designing $H_{\infty}$ filters for linear time-delay systems. Int. J. Robust and Nonlinear Control 19 (2009), 1376-1396.   CrossRef
  40. X. M. Zhang and Q. L. Han: Robust $H_{\infty}$ filtering for a class of uncertain linear systems with time-varing delay. Automatica 44 (2008), 157-166.   CrossRef
  41. X. M. Zhang and Q. L. Han: Network-based $H_{\infty}$ filtering for discrete-time systems. IEEE Trans. Signal Process. 60 (2012), 956-961.   CrossRef
  42. X. M. Zhang and Q. L. Han: Network-based $H_{\infty}$ filtering using a logic jumping-like trigger. Automatica 49 (2013), 1428-1435.   CrossRef
  43. X. D. Zhao and Q. S. Zeng: New robust delay-dependent stability and $H_{\infty}$ analysis for uncertain Markovian jump systems with time-varying delays. J. Franklin Inst. 347 (2010), 863-874.   CrossRef
  44. W. Zhou and M. Li: Mixed time-delays dependent exponential stability for uncertain stochastic high-order neural networks. Appl. Math. Comput. 215 (2009), 503-513.   CrossRef