Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
stochastic systems; distributed time-varying delay; $H_{\infty }$ filter; linear matrix inequality
stochastic systems; distributed time-varying delay; $H_{\infty }$ filter; linear matrix inequality
Summary:
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.
References:
[1] Balasubramaniam, P., Rakkiyappan, R.: 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. MR 2535910 | Zbl 1184.93093
[2] Cao, Y. Y., Lam, J., Hu, L.: Delay-dependent stochastic stability and $H_{\infty}$ analysis for time-delay systems with Markovian jumping parameters. J. Franklin Inst. 340 (2003), 423-434. DOI 10.1016/j.jfranklin.2003.09.001 | MR 2034548 | Zbl 1040.93068
[3] Chen, W. H., Zheng, W.: Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 43 (2007), 95-104. DOI 10.1016/j.automatica.2006.07.019 | MR 2266774 | Zbl 1140.93466
[4] Chung, K. L.: A Course In Probability Theory. Academic Press, London 2001. MR 1796326 | Zbl 0980.60001
[5] Ding, Y. C., Zhu, H., Zhong, S. M., Zhang, Y. P.: $L_{2}-L_{\infty}$ filtering for Markovian jump systems with time-varying delays and partly unknown transition probabilities. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3070-3081. DOI 10.1016/j.cnsns.2011.11.033 | MR 2880476 | Zbl 1243.62118
[6] Fiagbedzi, Y. A., Pearson, A. E.: A multistage reduction technique for feedback stabilizing distributed time-lag systems. Automatica 23 (1987), 311-326. DOI 10.1016/0005-1098(87)90005-7 | MR 0899830 | Zbl 0629.93046
[7] Gronwall, T. H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919), 292-296. DOI 10.2307/1967124 | MR 1502565
[8] Gu, K.: 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.
[9] Gu, K.: An improved stability criterion for systems with distributed delays. Int. J. Robust Nonlinear Control 13 (2003), 819-831. DOI 10.1002/rnc.847 | MR 1998314 | Zbl 1039.93031
[10] Gu, K., Han, Q. L., Albert, C. J., Niculescu, S. I.: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int. J. Control 74 (2001), 737-744. DOI 10.1080/00207170010031486 | MR 1826755 | Zbl 1015.34061
[11] Hale, J. K.: Theory Of Functional Differential Equations. Springer, New York 1977. MR 0508721 | Zbl 0662.34064
[12] Hale, J. K., Lunel, S. M. V.: Introduction To Functional Differential Equations. Springer, New York 1993. MR 1243878 | Zbl 0787.34002
[13] Han, Q. L.: A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 40 (2004), 1791-1796. DOI 10.1016/j.automatica.2004.05.002 | MR 2155472 | Zbl 1075.93032
[14] Han, Q. L.: A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica 45 (2009), 517-524. DOI 10.1016/j.automatica.2008.08.005 | MR 2527352 | Zbl 1158.93385
[15] Han, Q. L.: Improved stability criteria and controller design for liear neutral systems. Automatica 45 (2009), 1948-1952. DOI 10.1016/j.automatica.2009.03.019 | MR 2879521
[16] Lam, J., Gao, H., Wang, C.: $H_{\infty}$ model reduction of linear systems with distributed delay. Control Theory and Applications, IEE Proc. 152 (2005), 662-674.
[17] Lawrence, C. E.: An introduction to stochastic differential equations. math.berkeley.edu/ evans/SDE.course.pdf. MR 3154922
[18] Li, X. G., Zhu, X. J.: Stability analysis of neutral systems with distributed delays. Automatica 44 (2008), 2197-2201. DOI 10.1016/j.automatica.2007.12.009 | MR 2531353 | Zbl 1283.93212
[19] Liu, Y., Wang, Z., Liu, X.: Robust $H_{\infty}$ control for a class of nonlinear stochastic systems with mixed time delay. Int. J. Robust Nonlinear Control 17 (2007), 1525-1551. DOI 10.1002/rnc.1185 | MR 2356998 | Zbl 1128.93015
[20] Liu, Y., Wang, Z., Liu, X.: 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. MR 2381041 | Zbl 1157.93039
[21] Ma, L., Da, F. P.: 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. DOI 10.1002/rnc.1477 | MR 2657281
[22] Ma, L., Da, F. P., Zhang, K. J.: 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. DOI 10.1109/TCSI.2010.2089554 | MR 2827933
[23] Mariton, M.: Jump Linear Systems In Automatic Control. Marcel Dekker, New York 1990.
[24] Mao, X. R.: Exponential stability of stochastic delay interval systems with Markovian switching. IEEE Trans. Automat. Control 47 (2002), 1604-1612. DOI 10.1109/TAC.2002.803529 | MR 1929934
[25] Richard, J. P.: Time-delay systems: An overview of some recent advances and open problems. Automatica 39 (2003), 1667-1694. DOI 10.1016/S0005-1098(03)00167-5 | MR 2141765 | Zbl 1145.93302
[26] Wang, Z., Lauria, S., Fang, J., Liu, X.: Exponential stability of uncettain stochastic neural networks with mixed time-delays. Chaos, Solitons Fractals 32 (2007), 62-72. DOI 10.1016/j.chaos.2005.10.061 | MR 2271102
[27] Wang, Y., Zhang, H.: $H_{\infty}$ control for uncertain Markovian jump systems with mode-dependent mixed delays. Progress Natural Sci. 18 (2008), 309-314. MR 2419784
[28] Wang, G. L., Zhang, Q. L., Yang, C. Y.: Exponential $H_{\infty}$ filtering for time-varying delay systems: Markovian approach. Signal Process. 91 (2011), 1852-1862. Zbl 1217.93170
[29] Wei, G. L., Wang, Z., Shu, H., Fang, J.: A delay-dependent approach to $H_{\infty}$ filtering for stochastic delayed jumping systems with sensor non-linearities. Int. J. Control 80 (2008), 885-897. DOI 10.1080/00207170701203608 | MR 2334740 | Zbl 1124.93056
[30] Wu, L., Shi, P., Wang, C., Gao, H.: 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. MR 2351871
[31] Xie, L., Fridman, E., Shaked, U.: Robust $H_{\infty}$ control of distributed delay systems with application to combustion control. IEEE Trans. Automat. Control 46 (2001), 1930-1935. DOI 10.1109/9.975483 | MR 1878215 | Zbl 1017.93038
[32] Xiong, L., Zhong, S., Tian, J.: New robust stability condition for uncertain neutral systems with discrete and distributed delays. Chaos, Solitons Fractals 42 (2009), 1073-1079. DOI 10.1016/j.chaos.2009.03.002 | MR 2554818 | Zbl 1198.93170
[33] Xu, S., Chen, T.: 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. DOI 10.1109/TCSII.2003.822432
[34] Xu, S., Chu, Y., Lu, J., Zou, Y.: 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. DOI 10.1109/TSMCA.2006.871648
[35] Xu, S., Lam, J., Chen, T., Zou, Y.: A delay-dependent approach to robust $H_{\infty}$ filtering for uncertain distributed delay systems. IEEE Trans. Signal Process. 53 (2005), 3764-3772. DOI 10.1109/TSP.2005.855109 | MR 2239897
[36] Yu, X. G.: An LMI approach to robust $H_{\infty}$ filtering for uncertain systems with time-varying distributed delays. J. Franklin Inst. 345 (2008), 877-890. DOI 10.1016/j.jfranklin.2008.05.003 | MR 2478513 | Zbl 1201.93126
[37] Yue, D., Han, Q. L.: Robust $H_{\infty}$ filter design of uncertain descriptor systems with discrete and distributed delays. IEEE Trans. Signal Process. 52 (2004), 3200-3212. DOI 10.1109/TSP.2004.836535 | MR 2095601
[38] Yue, D., Han, Q. L.: Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching. IEEE Trans. Automat. Control 50 (2005), 217-222. DOI 10.1109/TAC.2004.841935 | MR 2116427
[39] Zhang, X. M., Han, Q. L.: A less conservative method for designing $H_{\infty}$ filters for linear time-delay systems. Int. J. Robust and Nonlinear Control 19 (2009), 1376-1396. DOI 10.1002/rnc.1407 | MR 2537820 | Zbl 1169.93418
[40] Zhang, X. M., Han, Q. L.: Robust $H_{\infty}$ filtering for a class of uncertain linear systems with time-varing delay. Automatica 44 (2008), 157-166. DOI 10.1016/j.automatica.2007.04.024 | MR 2530479
[41] Zhang, X. M., Han, Q. L.: Network-based $H_{\infty}$ filtering for discrete-time systems. IEEE Trans. Signal Process. 60 (2012), 956-961. DOI 10.1109/TSP.2011.2175224 | MR 2919490
[42] Zhang, X. M., Han, Q. L.: Network-based $H_{\infty}$ filtering using a logic jumping-like trigger. Automatica 49 (2013), 1428-1435. DOI 10.1016/j.automatica.2013.01.060 | MR 3044024
[43] Zhao, X. D., Zeng, Q. S.: 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. DOI 10.1016/j.jfranklin.2010.03.009 | MR 2645396 | Zbl 1286.93199
[44] Zhou, W., Li, M.: Mixed time-delays dependent exponential stability for uncertain stochastic high-order neural networks. Appl. Math. Comput. 215 (2009), 503-513. DOI 10.1016/j.amc.2009.05.025 | MR 2561507 | Zbl 1206.65025
Search
Partner of
