| Title:
|
Exponential $H_{\infty }$ filter design for stochastic Markovian jump systems with both discrete and distributed time-varying delays (English) |
| Author:
|
Ma, Li |
| Author:
|
Xu, Meimei |
| Author:
|
Jia, Ruting |
| Author:
|
Ye, Hui |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
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0023-5954 (print) |
| ISSN:
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1805-949X (online) |
| Volume:
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50 |
| Issue:
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4 |
| Year:
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2014 |
| Pages:
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491-511 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
stochastic systems |
| Keyword:
|
distributed time-varying delay |
| Keyword:
|
$H_{\infty }$ filter |
| Keyword:
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linear matrix inequality |
| MSC:
|
93B36 |
| MSC:
|
93E03 |
| idZBL:
|
Zbl 06386423 |
| idMR:
|
MR3275081 |
| DOI:
|
10.14736/kyb-2014-4-0491 |
| . |
| Date available:
|
2014-11-06T14:53:28Z |
| Last updated:
|
2016-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143980 |
| . |
| Reference:
|
[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. Zbl 1184.93093, MR 2535910 |
| Reference:
|
[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. Zbl 1040.93068, MR 2034548, 10.1016/j.jfranklin.2003.09.001 |
| Reference:
|
[3] Chen, W. H., Zheng, W.: Delay-dependent robust stabilization for uncertain neutral systems with distributed delays..Automatica 43 (2007), 95-104. Zbl 1140.93466, MR 2266774, 10.1016/j.automatica.2006.07.019 |
| Reference:
|
[4] Chung, K. L.: A Course In Probability Theory..Academic Press, London 2001. Zbl 0980.60001, MR 1796326 |
| Reference:
|
[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. Zbl 1243.62118, MR 2880476, 10.1016/j.cnsns.2011.11.033 |
| Reference:
|
[6] Fiagbedzi, Y. A., Pearson, A. E.: A multistage reduction technique for feedback stabilizing distributed time-lag systems..Automatica 23 (1987), 311-326. Zbl 0629.93046, MR 0899830, 10.1016/0005-1098(87)90005-7 |
| Reference:
|
[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. MR 1502565, 10.2307/1967124 |
| Reference:
|
[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. |
| Reference:
|
[9] Gu, K.: An improved stability criterion for systems with distributed delays..Int. J. Robust Nonlinear Control 13 (2003), 819-831. Zbl 1039.93031, MR 1998314, 10.1002/rnc.847 |
| Reference:
|
[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. Zbl 1015.34061, MR 1826755, 10.1080/00207170010031486 |
| Reference:
|
[11] Hale, J. K.: Theory Of Functional Differential Equations..Springer, New York 1977. Zbl 0662.34064, MR 0508721 |
| Reference:
|
[12] Hale, J. K., Lunel, S. M. V.: Introduction To Functional Differential Equations..Springer, New York 1993. Zbl 0787.34002, MR 1243878 |
| Reference:
|
[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. Zbl 1075.93032, MR 2155472, 10.1016/j.automatica.2004.05.002 |
| Reference:
|
[14] Han, Q. L.: A discrete delay decomposition approach to stability of linear retarded and neutral systems..Automatica 45 (2009), 517-524. Zbl 1158.93385, MR 2527352, 10.1016/j.automatica.2008.08.005 |
| Reference:
|
[15] Han, Q. L.: Improved stability criteria and controller design for liear neutral systems..Automatica 45 (2009), 1948-1952. MR 2879521, 10.1016/j.automatica.2009.03.019 |
| Reference:
|
[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. |
| Reference:
|
[17] Lawrence, C. E.: An introduction to stochastic differential equations..math.berkeley.edu/ evans/SDE.course.pdf. MR 3154922 |
| Reference:
|
[18] Li, X. G., Zhu, X. J.: Stability analysis of neutral systems with distributed delays..Automatica 44 (2008), 2197-2201. Zbl 1283.93212, MR 2531353, 10.1016/j.automatica.2007.12.009 |
| Reference:
|
[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. Zbl 1128.93015, MR 2356998, 10.1002/rnc.1185 |
| Reference:
|
[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. Zbl 1157.93039, MR 2381041 |
| Reference:
|
[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. MR 2657281, 10.1002/rnc.1477 |
| Reference:
|
[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. MR 2827933, 10.1109/TCSI.2010.2089554 |
| Reference:
|
[23] Mariton, M.: Jump Linear Systems In Automatic Control..Marcel Dekker, New York 1990. |
| Reference:
|
[24] Mao, X. R.: Exponential stability of stochastic delay interval systems with Markovian switching..IEEE Trans. Automat. Control 47 (2002), 1604-1612. MR 1929934, 10.1109/TAC.2002.803529 |
| Reference:
|
[25] Richard, J. P.: Time-delay systems: An overview of some recent advances and open problems..Automatica 39 (2003), 1667-1694. Zbl 1145.93302, MR 2141765, 10.1016/S0005-1098(03)00167-5 |
| Reference:
|
[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. MR 2271102, 10.1016/j.chaos.2005.10.061 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 1124.93056, MR 2334740, 10.1080/00207170701203608 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 1017.93038, MR 1878215, 10.1109/9.975483 |
| Reference:
|
[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. Zbl 1198.93170, MR 2554818, 10.1016/j.chaos.2009.03.002 |
| Reference:
|
[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. 10.1109/TCSII.2003.822432 |
| Reference:
|
[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. 10.1109/TSMCA.2006.871648 |
| Reference:
|
[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. MR 2239897, 10.1109/TSP.2005.855109 |
| Reference:
|
[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. Zbl 1201.93126, MR 2478513, 10.1016/j.jfranklin.2008.05.003 |
| Reference:
|
[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. MR 2095601, 10.1109/TSP.2004.836535 |
| Reference:
|
[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. MR 2116427, 10.1109/TAC.2004.841935 |
| Reference:
|
[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. Zbl 1169.93418, MR 2537820, 10.1002/rnc.1407 |
| Reference:
|
[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. MR 2530479, 10.1016/j.automatica.2007.04.024 |
| Reference:
|
[41] Zhang, X. M., Han, Q. L.: Network-based $H_{\infty}$ filtering for discrete-time systems..IEEE Trans. Signal Process. 60 (2012), 956-961. MR 2919490, 10.1109/TSP.2011.2175224 |
| Reference:
|
[42] Zhang, X. M., Han, Q. L.: Network-based $H_{\infty}$ filtering using a logic jumping-like trigger..Automatica 49 (2013), 1428-1435. MR 3044024, 10.1016/j.automatica.2013.01.060 |
| Reference:
|
[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. Zbl 1286.93199, MR 2645396, 10.1016/j.jfranklin.2010.03.009 |
| Reference:
|
[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. Zbl 1206.65025, MR 2561507, 10.1016/j.amc.2009.05.025 |
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