Abstract
This paper is concerned with the problem of reliable H ∞ filter design for a class of mixeddelay Markovian jump systems with stochastic nonlinearities and multiplicative noises. The mixed delays comprise both discrete time-varying delay and distributed delay. The stochastic nonlinearities in the form of statistical means cover several well-studied nonlinear functions. And the multiplicative disturbances are in the form of a scalar Gaussian white noise with unit variance. Furthermore, the failures of sensors are quantified by a variable varying in a given interval. A filter is designed to guarantee that the dynamics of the estimation error is asymptotically mean-square stable. Sufficient conditions for the existence of such a filter are obtained by using a new Lyapunov-Krasovskii functional and delaypartitioning method. Then a linear matrix inequality (LMI) approach for designing such a reliable H ∞ filter is presented. Finally, the effectiveness of the proposed approach is demonstrated by a numerical example.
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References
C. E. Souza, R. M. Palhares, and P. L. D. Peres, “Robust H ∞ filter design for uncertain linear systems with multiple time-varying state delays,” IEEE Trans. Signal Process., vol. 49, no. 3, pp. 569–576, 2001.
H. Gao and C. Wang, “Delay-dependent robust H ∞ and L 2 — L ∞ filtering for a class of uncertain nonlinear time-delay systems,” IEEE Trans. Autom. Control, vol. 48, no. 9, pp. 1661–1666, 2003.
N. T. Hoang, H. D. Tuan, P. Apkarian, and S. Hosoe, “Gain-scheduled filtering for time-varying discretevarying discrete systems,” IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2464–2476, 2004.
X. Q. Xiao and L. Zhou, “Delay-dependent robust L 2 — L ∞ filter design for uncertain Delta-operator time delay systems,” Int. J. Control Autom. Syst., vol. 9, no. 3, pp. 611–615, 2011.
M. Ezzine, H. S. Ali, and M. Darouach “Full order H ∞ filtering for linear systems in the frequency domain,” Int. J. Control Autom. Syst., vol. 9, no. 3, pp. 558–565, 2011.
H. L. Xu and Y. Zou, “Robust H ∞ filtering for uncertain two-dimensional discrete systems with state-varying delays,” Int. J. Control Autom. Syst., vol. 8, no. 4, pp. 720–726, 2010.
J. H. Kim, “Delay-dependent approach to robust H ∞ filtering for discrete-time singular systems with multiple time-varying delays and polytopic uncertainties,” Int. J. Control Autom. Syst., vol. 8, no. 3, pp. 655–661, 2010.
F. W. Yang, Z. D. Wang, G. Feng, and X. H. Liu, “Robust filtering with randomly varying sensor delay: The finite-horizon case,” IEEE Trans. Circuits Syst. I, Exp. Briefs, vol. 56, no. 3, pp. 664–672, 2009.
G. L. Wei, Z. D. Wang, X. He, and H. S. Shu, “Filtering for networked stochastic time-delay systems with sector nonlinearity,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 1, pp. 71–75, 2009.
H. L. Dong, Z. D. Wang, and H. J. Gao, “H ∞ filtering for systems with repeated scalar nonlinearities under unreliable communication links,” Signal Process., vol. 89, pp. 1567–1575, 2009.
Y. S. Liu, Z. D. Wang, and W. Wang, “Reliable H ∞ filtering for discrete time-delay systems with randomly occurred nonlinearities via delaypartitioning method,” Signal Process., vol. 91, pp. 713–727, 2011.
M. Liu, J. You, and X. Ma, “H ∞ filtering for sampled-data stochastic systems with limited capacity channel,” Signal Process., vol. 91, no. 8, pp. 1826–1837, 2011.
G. Wang, Q. Zhang, and C. Yang, “Exponential H ∞ filtering for time-varying delay systems: Markovian approach,” Signal Process., vol. 91, pp. 1852–1862, 2011.
X. Yao, L. Wu, and W. Zheng, “Fault detection filter design for Markovian jump singular systems with intermittent measurements,” IEEE Trans. Signal Process., vol. 59, pp. 3099–3109, 2011.
Y. Liu, Z. Wang, and W. Wang, “Reliable H ∞ filtering for discrete time-delay Markovian jump systems with partly unknown transition probabilities,” Int. J. Adapt. Control Signal Process., vol. 25, pp. 554–570, 2011.
H. Yang, H. Zhang, and H. Shi, “Robust H ∞ filtering for uncertain nonlinear stochastic systems with mode-dependent time-delays and Markovian jump parameters,” Circuits Syst. Signal Process., vol. 30, pp. 303–321, 2011.
X. Liu and N. Jiang, “Robust stability analysis of generalized neural networks with multiple discrete delays and multiple distributed delays,” Neurocomput., vol. 72, pp. 1789–1796, 2009.
L. Wu and Z. Wang, “Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay,” Signal Process., vol. 89, no. 9, pp. 1739–1753, 2009.
L. Ma, Z. Wang, J. Hu, Y. Bo, and Z. Guo, “Robust variance-constrained filtering for a class of nonlinear stochastic systems with missing measurements,” Signal Process., vol. 90, no. 6, pp. 2060–2071, 2010.
Y. L. Yaz and E. E. Yaz, “On LMI formulations of some problems arising in nonlinear stochastic system analysis,” IEEE Trans. Autom. Control, vol. 44, pp. 813–816, 1999.
G. Wei, Z. D. Wang, and H. Shu, “Robust filtering with stochastic nonlinearities and multiple missing measurements,” Automatica, vol. 45, pp. 836–841, 2009.
Z. D. Wang, F. W. Yang, D. W. C. Ho, and X. H. Liu, “Robust variance-constrained H ∞ control for stochastic systems with multiplicative noises,” J. Math. Anal. Appl., vol. 328, pp. 487–502, 2007.
E. Gershon and U. Shaked, “H ∞ output-feedback control of discrete-time systems with statemultiplicative noise,” Automatica, vol. 44, pp. 574–579, 2008.
X. Liu and J. Cao, “Exponential stability of antiperiodic solutions for neural networks with multiple discrete and distributed delays,” Proc. IMechE I: J. Syst. Control Engineer., vol. 223, pp. 299–308, 2009.
Z. D. Wang, D. W. C. Ho, Y. R. Liu, and X. H. Liu, “Robust H ∞ control for a class of nonlinear discrete time-delay stochastic systems with missing measurements,” Automatica, vol. 45, pp. 684–691, 2009.
H. Gao, C. Wang, and J. Wang, “A delay-dependent approach to robust H ∞ filtering for uncertain discrete-time state-delayed systems,” IEEE Trans. Signal Process., vol. 52, pp. 1631–1640, 2004.
Z. D. Wang, G. L. Wei, and G. Feng, “Reliable H ∞ control for discrete-time piecewise linear systems with infinite distributed delays,” Automatica, vol. 45, pp. 2991–2994, 2009.
S. Hu and D. Yue, “Event-triggered control design of linear networked systems with quantizations,” ISA Trans., vol. 51, pp. 153–162, 2012.
H. Y. Li, Q. Zhou, B. Zhou, and H. H. Liu, “Parameter-dependent robust stability for uncertain Markovian jump system with time delay,” J. Franklin Inst., vol. 348, pp. 738–748, 2011.
X. Guo and G. Yang, “Reliable H ∞ filter design for discrete-time systems with sector-bounded nonlinearities: an LMI optimization approach,” Acta Automatica Sinica, vol. 35, no. 10, pp. 1347–1351, 2009.
L. Liu, J. Wang, and G. Yang, “Reliable guaranteed variance filtering against sensor failure,” IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1402–1411, 2003.
G. Yang and D. Ye, “Adaptive reliable H ∞ filtering against sensor failures,” IEEE Trans. Signal Process., vol. 55, no. 7, pp. 3161–3171, 2007.
Z. R. Xiang, R. H. Wang, and Q. W. Chen, “Robust reliable stabilization of stochastic switched nonlinear systems under asynchronous switching,” Appl. Math. Comput., vol. 217, pp. 7725–7736, 2011.
L. Zhang, C. Wang, and L. Chen, “Stability and stabilization of a class of multi-mode linear discrete-time systems with polytopic uncertainties,” IEEE Trans. Industr. Electr., vol. 56, pp. 36984–3692, 2009.
M. Zhong, S. X. Ding, J. Lam, and H. Wang, “An LMI approach to design robust fault detection filter for uncertain LTI systems,” Automatica, vol. 39, pp. 543–550, 2003.
L. Zhang and J. Lam, “Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions,” IEEE Trans. Autom. Control, vol. 55, pp. 1695–1701, 2010.
L. Zhang and P. Shi, “Stability, L 2 — gain and asynchronous H ∞ control of discrete-time switched systems with average dwell time,” IEEE Trans. Autom. Control, vol. 54, pp. 2193–2200, 2009.
Y. Liu, Z. Wang, J. Liang, and X. Liu, “Synchronization and state estimation for discrete-time complex networks with distributed delays,” IEEE. Trans. Syst. Man Cybern. B Cybern., vol. 13, pp. 1314–1325, 2008.
Z. Wu, H. Su, and J. Chu, “H ∞ filtering for singular systems with time-varying delay,” Int. J. Robust Nonlinear Control, vol. 20, no. 10, pp. 1269–1284, 2010.
Z. Wu, H. Su, and J. Chu, “H ∞ filtering for singular Markovian jump systems with time delay,” Int. J. Robust Nonlinear Control, vol. 20, no. 8, pp. 939–957, 2010.
S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Stud. Appl. Math., Philadelphia, PA, 1994.
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This work is supported by the Natural Science Foundation of China under Grants 60974021 and 61125303, the 973 Program of China under Grant 2011CB710606, the Fund for Distinguished Young Scholars of Hubei Province under Grant 2010CDA081, National Priority Research Project NPRP 4-451-2-168, funded by Qatar National Research Fund.
Shiping Wen received his M.S. degree in Control Science and Engineering, from Wuhan University of Technology, Wuhan, China, in 2010. He is currently working toward a Ph.D. degree at Department of Control Science and Engineering, Huazhong University of Science and Technology, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. His current research interests include design and analysis of memristorbased circuits and systems, H ∞ control of networked systems.
Zhigang Zeng received his Ph.D. degree from Huazhong University of Science and Technology, Wuhan, China, in 2003. He is a professor in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. His current research interests include neural networks, switched systems, computational intelligence, stability analysis of dynamic systems, pattern recognition and associative memories.
Tingwen Huang received his B.S. degree from Southwest Normal University (now Southwest University), China, 1990, an M.S. degree from Sichuan University, China, 1993, and a Ph.D. degree from Texas A&M University, College Station, Texas, USA, 2002. He is an Associate Professor of Mathematics, Texas A&M University at Qatar. His current research interests include Dynamical Systems, Neural Networks, Complex Networks, Optimization and Control, Traveling Wave Phenomena.
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Wen, S., Zeng, Z. & Huang, T. Reliable H ∞ filter design for a class of mixed-delay Markovian jump systems with stochastic nonlinearities and multiplicative noises via delay-partitioning method. Int. J. Control Autom. Syst. 10, 711–720 (2012). https://doi.org/10.1007/s12555-012-0406-5
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DOI: https://doi.org/10.1007/s12555-012-0406-5