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Robust $$H_{\infty }$$-filter design for neutral stochastic uncertain systems with time-varying delay. (English) Zbl 1161.93025
Summary: The robust $$H_{\infty }$$ filtering problem for a class of neutral stochastic systems is discussed. The system under consideration contains parameter uncertainties, Itô-type stochastic disturbances, and time-varying delay. The parameter uncertainties are assumed to be time-varying norm-bounded. Using the stochastic Lyapunov stability theory and Itô’s differential rule, a full-order filter is designed for all admissible uncertainties and time-varying delay, which is expressed in the form of linear matrix inequality. The dynamics of the filtering error systems are guaranteed to be robust stochastically mean square asymptotically stable, while achieving a prescribed stochastic robust $$H_{\infty }$$ performance level. At the end of this paper, a numerical example is given to demonstrate the usefulness of the proposed method.

##### MSC:
 93E11 Filtering in stochastic control theory 34K40 Neutral functional-differential equations 15A39 Linear inequalities of matrices 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory
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##### References:
 [1] Kushner, H., Stochastic stability and control, (1967), Academic Press New York · Zbl 0183.19401 [2] Mao, X., Stochastic differential equations and applications, (2007), Horwood [3] Boukas, E.K., Stabilization of stochastic nonlinear hybrid systems, Int. J. innov. comput. inf. control, 1, 131-141, (2005) · Zbl 1085.93026 [4] Mao, X., Robustness of exponential stability of stochastic differential delay equations, IEEE trans. automat. control, 41, 442-447, (1996) · Zbl 0851.93074 [5] Choi, H.H.; Chung, M.J., An LMI approach to $$H_\infty$$ controller design for linear time-delay systems, Automatica, 33, 737-739, (1997) · Zbl 0875.93104 [6] Hinrichsen, D.; Pritchard, A.J., Stochastic $$H_\infty$$, SIAM J. control optim., 36, 1504-1538, (1998) · Zbl 0914.93019 [7] Xu, S.; Chen, T., Reduced-order $$H_\infty$$ filtering for stochastic systems, IEEE trans. signal process., 50, 2998-3007, (2002) · Zbl 1369.94325 [8] Xu, S.; Chen, T., Robust $$H_\infty$$ control for uncertain systems with state delay, IEEE trans. automat. control, 47, 2089-2094, (2002) · Zbl 1364.93755 [9] Xu, S.; Chen, T., Robust $$H_\infty$$ filtering for uncertain stochastic time-delay systems, Asian J. control, 5, 364-373, (2003) [10] Gershon, E.; Limebeer, D.J.N.; Shaked, U., Robust $$H_\infty$$ filtering of stationary continuous time linear systems with stochastic uncertainties, IEEE trans. automat. control, 46, 1788-1793, (2001) · Zbl 1016.93067 [11] Chen, B.S.; Zhang, W., Stochastic $$H_2 / H_\infty$$ control with state-dependent noise, IEEE trans. automat. control, 49, 45-57, (2004) · Zbl 1365.93539 [12] Zhang, W.; Chen, B.S.; Tseng, C.S., Robust $$H_\infty$$ filtering for nonlinear stochastic systems, IEEE trans. signal process., 53, 589-598, (2005) · Zbl 1370.93293 [13] Limebeer, D.J.N.; Anderson, B.D.O.; Hendel, B., A Nash game approach to mixed $$H_2 / H_\infty$$ control, IEEE trans. automat. control, 39, 69-82, (1994) · Zbl 0796.93027 [14] Xu, S.Y.; Chen, T.W., $$H_\infty$$ output feedback control for uncertain stochastic systems with time-varying delays, Automatica, 40, 2091-2098, (2004) · Zbl 1073.93022 [15] Xu, S.Y.; Shi, P.; Chu, Y.M.; Zou, Y., Robust stochastic stabilization and $$H_\infty$$ control of uncertain neutral stochastic time-delay systems, J. math. anal. appl., 314, 1-16, (2006) · Zbl 1127.93053 [16] Xu, S.Y.; Chen, T.W., Robust $$H_\infty$$ filtering for uncertain impulsive stochastic systems under sampled measurements, Automatica, 39, 509-516, (2003) · Zbl 1012.93063 [17] Xu, S.Y.; Lam, James, Exponential $$H_\infty$$ filtering design for uncertain takagi – sugeno fuzzy systems with time delay, Eng. appl. artif. intell., 17, 645-659, (2004) [18] Xia, J.W.; Xu, S.Y.; Song, B., Delay-dependent $$\mathcal{L}_2 - \mathcal{L}_\infty$$ filter design for stochastic time-delay systems, Systems control lett., 56, 579-587, (2007) · Zbl 1157.93530 [19] Wang, Z.D.; Liu, Y.R.; Liu, X.H., $$H_\infty$$ filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica, 44, 1268-1277, (2008) · Zbl 1283.93284 [20] Liu, Y.R.; Wang, Z.D.; Liu, X.H., Robust $$H_\infty$$ filtering for discrete nonlinear stochastic systems with time-varying delays, J. math. anal. appl., 341, 318-336, (2008) · Zbl 1245.93131 [21] Bernstein, D.B.; Haddad, W.M., Steady-state Kalman filtering with $$H_\infty$$ error bound, Systems control lett., 21, 9-16, (1989) · Zbl 0684.93081 [22] Peterson, I.R.; Savkin, A.V., Robust Kalman filtering for signals and systems with large uncertainties, (1999), Birkhäuser Boston [23] Geromel, J.C.; Bernussou, J.; Garcia, G.; de Oliveira, M.C., $$H_2$$ and $$H_\infty$$ filtering for discrete-time linear systems, SIAM J. control optim., 38, 1353-1368, (2000) · Zbl 0958.93091 [24] Fridman, E.; Shaked, U.; Xie, L., Robust $$H_\infty$$ filtering of linear systems with time-varying delay, IEEE trans. automat. control, 48, 159-165, (2003) · Zbl 1364.93797 [25] Bittanti, S.; Cuzzola, F.A., Continuous-time periodic $$H_\infty$$ filtering via LMI, Eur. J. control, 7, 2-16, (2001) · Zbl 1293.93734 [26] de Souza, C.E.; Phlhares, R.M.; Peres, P.L.D., Robust $$H_\infty$$ filter design for uncertain linear systems with multiple time-varying state delays, IEEE trans. signal process., 49, 569-576, (2001) · Zbl 1369.93667 [27] Lam, J.; Gao, H.; Xu, S.Y.; Wang, C., $$H_\infty$$ and $$\mathcal{L}_2 / \mathcal{L}_\infty$$ model reduction for system input with sector nonlinearities, J. optim. theory appl., 125, 137-155, (2005) · Zbl 1062.93020 [28] Wang, Y.; Xie, L.; de Souza, C.E., Robust control of a class of uncertain nonlinear systems, Systems control lett., 19, 139-149, (1992) · Zbl 0765.93015
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