Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 659251, 10 pages.

Exponential L 2 - L Filtering for a Class of Stochastic System with Mixed Delays and Nonlinear Perturbations

Zhaohui Chen and Qi Huang

Full-text: Open access

Abstract

The delay-dependent exponential L 2 - L performance analysis and filter design are investigated for stochastic systems with mixed delays and nonlinear perturbations. Based on the delay partitioning and integral partitioning technique, an improved delay-dependent sufficient condition for the existence of the L 2 - L filter is established, by choosing an appropriate Lyapunov-Krasovskii functional and constructing a new integral inequality. The full-order filter design approaches are obtained in terms of linear matrix inequalities (LMIs). By solving the LMIs and using matrix decomposition, the desired filter gains can be obtained, which ensure that the filter error system is exponentially stable with a prescribed L 2 - L performance γ . Numerical examples are provided to illustrate the effectiveness and significant improvement of the proposed method.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 659251, 10 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394806097

Digital Object Identifier
doi:10.1155/2013/659251

Mathematical Reviews number (MathSciNet)
MR3147895

Zentralblatt MATH identifier
06950807

Citation

Chen, Zhaohui; Huang, Qi. Exponential ${L}_{2}\text{-}{L}_{\infty }$ Filtering for a Class of Stochastic System with Mixed Delays and Nonlinear Perturbations. J. Appl. Math. 2013, Special Issue (2013), Article ID 659251, 10 pages. doi:10.1155/2013/659251. https://projecteuclid.org/euclid.jam/1394806097


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References

  • K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Sys-tems, Birkhäauser, Boston, Mass, USA, 2003.
  • E. Fridman and U. Shaked, “A descriptor system approach to ${H}_{\infty }$ control of linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 47, no. 2, pp. 253–270, 2002.
  • F. Gouaisbaut and D. Peaucelle, “A note on stability of time delaysystems,” in Proceedings of the IFAC Symposium on Robust Con-trol Design, Toulouse, France, 2006.
  • X. M. Zhang and Q. L. Han, “A delay decomposition approach to delay-dependent stability for linear systems with time-varying delays,” International Journal of Robust and Nonlinear Control, vol. 19, no. 17, pp. 1922–1930, 2009.
  • T. L. Hsien and C. H. Lee, “Robust stability of discrete bilinear uncertain time-delay systems,” Circuits, Systems, and Signal Processing, vol. 30, no. 6, pp. 1417–1443, 2011.
  • P. Shi, R. K. Agarwal, E. K. Boukas, and Y. Shi, “Optimal guaranteed cost control of uncertain discrete time-delay systems,” Jour-nal of Computational and Applied Mathematics, vol. 157, no. 2, pp. 435–451, 2003.
  • S. Ma and E. K. Boukas, “Stability and robust stabilisation for uncertain discrete stochastic hybrid singular systems with time delay,” IET Control Theory & Applications, vol. 3, no. 9, pp. 1217–1225, 2009.
  • L. Wu and D. W. C. Ho, “Reduced-order ${L}_{2}$$-$${L}_{\infty }$ filtering for aclass of nonlinear switched stochastic systems,” IET Control The-ory & Applications, vol. 3, no. 5, pp. 493–508, 2009.
  • H. Gao and C. Wang, “Robust ${L}_{2}$$-$${L}_{\infty }$ filtering for uncertain systems with multiple time-varying state delays,” IEEE Transactions on Circuits and Systems I, vol. 50, no. 4, pp. 594–599, 2003.
  • C. Lin, Q. G. Wang, and T. H. Lee, “A less conservative robust stability test for linear uncertain time-delay systems,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 87–91, 2006.
  • H. Zhang and Z. Liu, “Stability analysis for linear delayed sys-tems via an optimally dividing delay interval approach,” Automatica, vol. 47, no. 9, pp. 2126–2129, 2011.
  • Q. L. Han, “A discrete delay decomposition approach to stability of linear retarded and neutral systems,” Automatica, vol. 45, no. 2, pp. 517–524, 2009.
  • B. Zhang and Y. Li, “Exponential ${L}_{2}-{L}_{\infty }$ filtering for distributed delay systems with Markovian jumping parameters,” Signal Processing, vol. 93, no. 1, pp. 206–216, 2013.
  • Z. Feng and J. Lam, “Integral partitioning approach to robust stabilization for uncertain distributed time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 22, no. 6, pp. 676–689, 2012.
  • Y. Liu, Z. Wang, and X. Liu, “Robust ${H}_{\infty }$ control for a classof nonlinear stochastic systems with mixed time delay,” International Journal of Robust and Nonlinear Control, vol. 17, no. 16, pp. 1525–1551, 2007.
  • Z. Wu and W. Zhou, “Delay-dependent robust stabilization for uncertain singular systems with discrete and distributed delays,” Journal of Control Theory and Applications, vol. 6, no. 2, pp. 171–176, 2008.
  • J. Wu, G. Lu, S. Wo, and X. Xiao, “Exponential stability and sta-bilization for nonlinear descriptor systems with discrete and distributed delays,” International Journal of Robust and Nonlinear Control, vol. 23, no. 12, pp. 1393–1404, 2013.
  • M. S. Mahmoud, A. Y. Al-Rayyah, and Y. Xia, “Quantised feedback stabilization of interconnected discrete-delay systems,” IET Control Theory & Applications, vol. 5, no. 6, pp. 795–802, 2011.
  • L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability ana-lysis and stabilization of discrete-time T-S fuzzy time-varying delay systems,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41, no. 1, pp. 273–286, 2011.
  • Y. E. Wang, X. M. Sun, and J. Zhao, “Stabilization of a class ofswitched stochastic systems with time delays under asynchronous switching,” Circuits, Systems, and Signal Processing, vol. 32, no. 1, pp. 347–360, 2013.
  • S. Xie and L. Xie, “Stabilization of a class of uncertain large-scale stochastic systems with time delays,” Automatica, vol. 36, no. 1, pp. 161–167, 2000.
  • D. Yue and S. Won, “Delay-dependent robust stability of sto-chastic systems with time delay and nonlinear uncertainties,” Electronics Letters, vol. 37, no. 15, pp. 992–993, 2001.
  • D. Yue and Q. L. Han, “Delay-dependent exponential stability ofstochastic systems with time-varying delay, nonlinearity, and Markovian switching,” IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 217–222, 2005.
  • S. Xu, J. Lam, X. Mao, and Y. Zou, “A new LMI condition fordelay-dependent robust stability of stochastic time-delay systems,” Asian Journal of Control, vol. 7, no. 4, pp. 419–423, 2005.
  • R. Yang, P. Shi, and H. Gao, “New delay-dependent stability cri-terion for stochastic systems with time delays,” IET Control The-ory & Applications, vol. 2, no. 11, pp. 966–973, 2008.
  • P. Cheng, F. Deng, and Y. Peng, “Delay-dependent exponential stability of impulsive stochastic systems with time-varying delay,” Journal of Systems Engineering and Electronics, vol. 22, no.5, pp. 799–809, 2011.
  • S. Xu and T. Chen, “Robust ${H}_{\infty }$ control for uncertain stochastic systems with state delay,” IEEE Transactions on Automatic Con-trol, vol. 47, no. 12, pp. 2089–2094, 2002.
  • S. Xu and T. Chen, “${H}_{\infty }$ output feedback control for uncertain stochastic systems with time-varying delays,” Automatica, vol. 40, no. 12, pp. 2091–2098, 2004.
  • E. K. Boukas and Z. K. Liu, “Robust \emphH$_{\infty }$ filtering for polytopic uncertain time-delay systems with Markov jumps,” Computers and Electrical Engineering, vol. 28, no. 3, pp. 171–193, 2002.
  • W. Zhang, B. S. Chen, and C. S. Tseng, “Robust ${H}_{\infty }$ filtering fornonlinear stochastic systems,” IEEE Transactions on Signal Proc-essing, vol. 53, no. 2, pp. 589–598, 2005.
  • W. Zhang, B. S. Chen, L. Sheng, and M. Gao, “Robust ${H}_{2}/{H}_{\infty }$ filter design for a class of nonlinear stochastic systems with state-dependent noise,” Mathematical Problems in Engineering, vol. 2012, Article ID 750841, 16 pages, 2012.
  • H. Gao, J. Lam, and C. Wang, “Robust energy-to-peak filter design for stochastic time-delay systems,” Systems & Control Letters, vol. 55, no. 2, pp. 101–111, 2006.
  • Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust \emphH$_{\infty }$ filtering for stochastic time-delay systems with missing measurements,” IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2579–2587, 2006.
  • J. Xia, S. Xu, and B. Song, “Delay-dependent ${L}_{2}$$-$${L}_{\infty }$ filter design for stochastic time-delay systems,” Systems & Control Letters, vol. 56, no. 9-10, pp. 579–587, 2007.
  • Y. Chen, A. Xue, and S. Zhou, “New delay-dependent ${L}_{2}-{L}_{\infty}$filter design for stochastic time-delay systems,” Signal Processing, vol. 89, no. 6, pp. 974–980, 2009.
  • R. Yang, H. Gao, and P. Shi, “Delay-dependent ${L}_{2}-{L}_{\infty }$ filter design for stochastic time-delay systems,” IET Control Theory & Applications, vol. 5, no. 1, pp. 1–8, 2011.
  • R. Yang, H. Gao, P. Shi, and L. Zhang, “Delay-dependent energy-to-peak filter design for stochastic systems with time delay: a delay partitioning approach,” in Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Con-trol Conference (CDC/CCC '09), pp. 5472–5477, Shanghai, China, December 2009.
  • L. Li and Y. Jia, “Robust ${L}_{2}$$-$${L}_{\infty }$ filtering for stochastic systems with discrete and distributed time-varying delays,” Asian Journal of Control, vol. 14, no. 4, pp. 1047–1058, 2012.
  • Z. Wang, Y. Liu, and X. Liu, “Exponential stabilization of a classof stochastic system with Markovian jump parameters and mode-dependent mixed time-delays,” IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1656–1662, 2010.
  • K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the 39th IEEE Confernce onDecision and Control, pp. 2805–2810, Sydney, Australia, December 2000. \endinput