Abstract
This paper considers the \(H_\infty \) filtering problem for stochastic singular fuzzy systems with time-varying delay. We assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by Ito-type stochastic differential equations. Based on an auxiliary vector and an integral inequality, a set of delay-dependent sufficient conditions is established, which ensures that the filtering error system is \({\mathrm{e}^{\lambda t}}\)-weighted integral input-to-state stable in mean (iISSiM). A fuzzy filter is designed such that the filtering error system is impulse-free, \({\mathrm{e}^{\lambda t}}\)-weighted iISSiM and the \(H_\infty \) attenuation level from disturbance to estimation error is below a prescribed scalar. A set of sufficient conditions for the solvability of the \(H_\infty \) filtering problem is obtained in terms of a new type of Lyapunov function and a set of linear matrix inequalities. Simulation examples are provided to illustrate the effectiveness of the proposed filtering approach developed in this paper.
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Souley, A.H., Zasadzinski, M., Rafaralahy, H., Darouach, M.: Robust \(H_\infty \) reduced order filtering for uncertain bilinear systems. Automatica 42, 405–415 (2006)
Kim, J.H.: Reduced-order delay-dependent \(H_\infty \) filtering for uncertain discrete-time singular systems with time-varying delay. Automatica 47, 2801–2804 (2011)
Mahmoud, S., Almutairi, N.B.: Stability and implementable \(H_\infty \) filters for singular systems with nonlinear perturbations. Nonlinear Dyn. 57, 401–410 (2009)
Su, Y.K., Chen, B., Lin, C., Zhang, H.G.: A new fuzzy \(H_\infty \) filter design for nonlinear continuous-time dynamic systems with time-varying delays. Fuzzy Set. Syst. 160, 3539–3549 (2009)
Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Cybern. 15(1), 116–132 (1985)
Huang, H., Ho, D.W.C.: Delay-dependent robust control of uncertain stochastic fuzzy systems with time-varying delay. IET Control Theory A 1, 1075–1085 (2007)
Taieb, N.H., Hammami, M.A., Delmotte, F., Ksontini, M.: On the global stabilization of Takagi–Sugeno fuzzy cascaded systems. Nonlinear Dyn. 67, 2847–2856 (2012)
Yang, C., Zhang, Q.: Multiobjective control for T–S fuzzy singularly perturbed systems. IEEE Trans. Fuzzy Syst. 17, 104–115 (2009)
Assawinchaichote, W., Nguang, S.K.: \(H_\infty \) fuzzy control design for nonlinear singularly perturbed systems with pole placement constraints: an LMI approach. IEEE Trans. Syst. 1(34), 579–588 (2004)
Benzaouia, A., Oubah, R.: Stability and stabilization by output feedback control of positive Takagi–Sugeno fuzzy discrete-time systems with multiple delays. Nonlinear Anal. Hybrid. Syst. 11, 154–170 (2014)
Kim, S.H.: T–S fuzzy control design for a class of nonlinear networked control systems. Nonlinear Dyn. 73, 17–27 (2013)
Balasubramaniam, P., Krishnasamy, R., Rakkiyappan, R.: Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays. Commun. Nonlinear Sci. 17, 3612–3627 (2012)
Lin, K.: Stabilization of uncertain fuzzy control systems via a new descriptor system approach. Comput. Math. Appl. 64, 1170–1178 (2012)
Liu, P., Yang, W.T., Yang, C.: Robust observer-based output feedback control for fuzzy descriptor systems. Expert Syst. Appl. 40, 4503–4510 (2013)
Zhang, H., Shen, Y., Feng, G.: Delay-dependent stability and \(H_\infty \) control for a class of fuzzy descriptor systems with time-delay. Fuzzy Set. Syst. 160, 1689–1707 (2009)
Han, C., Wu, L., Shi, P., Zeng, Q.S.: On dissipativity of Takagi–Sugeno fuzzy descriptor systems with time-delay. J. Franklin I 349, 3170–3184 (2012)
Long, S.H., Zhong, S.M., Liu, Z.J.: \(H_\infty \) filtering for a class of singular Markovian jump systems with time-varying delay. Sign. Process. 92, 2759–2768 (2012)
Masoud, A., Horacio, J.M.: A generalized framework for robust nonlinear \(H_\infty \) filtering of Lipschitz descriptor systems with parametric and nonlinear uncertainties. Automatica 48, 894–900 (2012)
Peng, T., Han, C.S., Xiong, Y.Y., Wu, L.G., Pang, B.J.: Filter design for discrete-time singularly perturbed T–S fuzzy systems. J. Franklin I 350, 3011–3028 (2013)
Zeng, Y., Cheng, J., Zhong, S., Dong, X.: Delay-dependent finite-time \(H_{\infty }\) filtering for Markovian jump systems with different system modes. J. Appl. Math. 2013, 1–13 (2013)
Zhang, X., Han, Q.: Robust \(H_\infty \) filtering for a class of uncertain linear systems with time-varying delay. Automatica 44, 157–166 (2008)
Wu, Z., Su, H., Chu, J.: \(H_\infty \) filtering for singular Markovian jump systems with time delay. Int. J. Robust Nonlinear Control 20, 939–957 (2010)
Wei, G., Wang, Z., Shu, H., Feng, J.: A delay-dependent approach to \(H\infty \) filtering for stochastic delayed jump systems with sensor non-linearities. Int. J. Control 80, 885–897 (2007)
An, J.Y., Wen, G.L., Xu, W.: Improved results on fuzzy \(H_\infty \) filter design for T–S fuzzy systems. Discrete Dyn. Nat. Soc. ID 392915 (2010)
Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34, 435–443 (1989)
Zhao, P., Kang, Y., Zhai, D.: On input-to-state stability of stochastic nonlinear systems with Markovian jumping parameters. Int. J. Control 85, 343–349 (2012)
Ning, C., He, Y., Wu, M., Liu, Q., She, J.: Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function. Syst. Control Lett. 61, 1254–1259 (2012)
Sontag, E.D.: Comments on integral variants of ISS. Syst. Control Lett. 34, 93–100 (1998)
Vu, L., Chatterjee, D., Liberzon, D.: Input-to-state stability of switched systems and switching adaptive control. Automatica 43, 639–646 (2007)
Xu, S., Lam, J.: Robust Control and Filtering of Singular Systems. Springer, Berlin (2006)
Xu, S., Lam, J., Zou, Y.: \(H_\infty \) filtering for singular systems. IEEE Trans. Automat. Control 12(48), 2217–2222 (2003)
Xia, J.W., Xu, S.Y., Song, B.: Delay-dependent \(l_{2}-l_{\infty }\) filter design for stochastic time-delay systems. Syst. Control Lett. 56, 579–587 (2007)
Cao, Y., Frank, P.M.: Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach. IEEE Trans. Fuzzy Syst. 2(8), 200–211 (2000)
Lu, C.Y., Tsai, J.S.H., Jong, G.J., Su, T.J.: An LMI-based approach for robust stabilization of uncertain stochastic systems with time-varying delays. IEEE Trans. Automat. Control 48(2), 286–289 (2003)
Chen, W.H., Guan, Z.H., Lu, X.M.: Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach. Syst. Control Lett. 54(6), 547–555 (2005)
He, Y., Liu, G., Rees, D., Wu, M.: Stability analysis for neural networks with time-varying interval delay. IEEE Trans. Neural Netw. 6(18), 1850–1854 (2007)
Wu, H.N., Wang, J.W., Shi, P.: A delay decomposition approach to \({L_2}-{L_\infty }\) filter design for stochastic systems with time-varying delay. Automatica 47, 1482–1488 (2011)
Zhao, F., Zhang, Q.L., Yan, X.G., Cai, M., Ren, J.C.: Stochastic input-to-state stability and \(H_\infty \) filtering for a class of stochastic nonlinear systems. Appl. Math. Comput. 227, 672–686 (2014)
Mahmoud, M.S., Almutairi, N.B.: Stability and implementable \(H_\infty \) filtering for singular systems with nonlinear perturbations. Nonlinear Dyn. 57, 401–410 (2009)
Wu, M., He, Y., She, J.H.: Stability analysis and robust control of time-delay systems. Science Press, Springer, Beijing and Heidelberg (2010)
Liao, X., Mao, X.: Exponential stability and instability of stochastic neural nerworks. Stochast. Anal. Appl. 14(2), 165–185 (1996)
Blythe, S., Mao, X., Liao, X.: Stability of stochastic delay neural networks. J. Franklin I 338, 481–495 (2001)
Mao, X., Yuan, C., Zou, J.: Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl. 304, 296–320 (2005)
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Zhao, F., Zhang, Q., Yan, X. et al. \(H_\infty \) filtering for stochastic singular fuzzy systems with time-varying delay. Nonlinear Dyn 79, 215–228 (2015). https://doi.org/10.1007/s11071-014-1658-9
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DOI: https://doi.org/10.1007/s11071-014-1658-9