Robust energy-to-peak filter design for stochastic time-delay systems. (English) Zbl 1129.93538

Summary: This paper considers the robust energy-to-peak filtering problem for uncertain stochastic time-delay systems. The stochastic uncertainties appear in both the dynamic and the measurement equations and the state delay is assumed to be time-varying. Attention is focused on the design of full-order and reduced-order filters guaranteeing a prescribed energy-to-peak performance for the filtering error system. Sufficient conditions are formulated in terms of linear matrix inequalities (LMIs), and the corresponding filter design is cast into a convex optimization problem which can be efficiently handled by using standard numerical algorithms. In addition, the results obtained are further extended to more general cases where the system matrices also contain uncertain parameters. The most frequently used ways of dealing with parameter uncertainties, including polytopic and norm-bounded characterizations, have been taken into consideration, with convex optimization problems obtained for the design of desired robust energy-to-peak filters.


93E11 Filtering in stochastic control theory
93C23 Control/observation systems governed by functional-differential equations
93E12 Identification in stochastic control theory


LMI toolbox
Full Text: DOI


[1] M. Basin, M. Skliar, Integral approach to optimal filtering and control of continuous processes with time-varying delays, in: Proceedings of 40th Conference on Decision Control, Orlando, FL, 2001, pp. 2911-2916.
[2] Boukas, E.K.; Liu, Z.K., Robust \(H_\infty\) filtering for polytopic uncertain time-delay systems with Markov jumps, Comput. elec. engng., 28, 171-193, (2002) · Zbl 0998.93041
[3] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory, (1994), SIAM Philadelphia, PA · Zbl 0816.93004
[4] Cao, Y.-Y.; Sun, Y.-X.; Lam, J., Delay-dependent robust \(H_\infty\) control for uncertain systems with time-varying delays, IEE proc. part D control theory appl., 145, 338-344, (1998)
[5] de Oliveira, M.C.; Bernussou, J.; Geromel, J.C., A new discrete-time robust stability condition, Systems control lett., 37, 261-265, (1999) · Zbl 0948.93058
[6] de Souza, C.E.; Palhares, 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, 3, 569-576, (2001) · Zbl 1369.93667
[7] El Ghaoui, L., State-feedback control of systems with multiplicative noise via linear matrix inequalities, Systems control lett., 24, 223-228, (1995) · Zbl 0877.93076
[8] Gahinet, P.; Nemirovskii, A.; Laub, A.J.; Chilali, M., LMI control toolbox User’s guide, (1995), The Math. Works Inc. Natick, MA
[9] Gao, H.; Wang, C., Robust energy-to-peak filtering with improved LMI representations, IEE proc. part J: vision image signal process., 150, 2, 82-89, (2003)
[10] Gao, H.; Wang, C., Robust \(L_2\)-\(L_\infty\) filtering for uncertain systems with multiple time-varying state delays, IEEE trans. circuits systems (I), 50, 4, 594-599, (2003) · Zbl 1368.93712
[11] Geromel, J.C.; De Oliveira, M.C., \(H_2\) and \(H_\infty\) robust filtering for convex bounded uncertain systems, IEEE trans. automat. control, 46, 1, 100-107, (2001) · Zbl 1056.93628
[12] Gershon, E.; Limebeer, D.J.N.; Shaked, U.; Yaesh, I., Robust \(H_\infty\) filtering of stationary continuous-time linear systems with stochastic uncertainties, IEEE trans. automat. control, 46, 11, 1788-1793, (2001) · Zbl 1016.93067
[13] Gershon, E.; Shaked, U.; Yaesh, I., Robust \(H_\infty\) estimation of stationary discrete-time linear processes with stochastic uncertainties, Systems control lett., 45, 257-269, (2002) · Zbl 0994.93063
[14] Grigoriadis, K.M.; Watson, J.T., Reduced order \(H_\infty\) and \(L_2\)-\(L_\infty\) filtering via linear matrix inequalities, IEEE trans. aerospace electron. systems, 33, 4, 1326-1338, (1997)
[15] Hinrichsen, D.; Pritchard, A.J., Stochastic \(H_\infty\), SIAM J. control optim., 36, 5, 1504-1538, (1998) · Zbl 0914.93019
[16] Khasminskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoffand Noordhoff Amsterdam, The Netherlands · Zbl 1259.60058
[17] Li, H.; Fu, M., A linear matrix inequality approach to robust \(H_\infty\) filtering, IEEE trans. signal process., 45, 9, 2338-2350, (1997)
[18] Liu, H.; Sun, F.; He, K.; Sun, Z., Design of reduced-order \(H_\infty\) filter for Markovian jumping systems with time delay, IEEE trans. circuits systems (II), 51, 607-612, (2004)
[19] Mao, X.; Koroleva, N.; Rodkina, A., Robust stability of uncertain stochastic differential delay equations, Systems control lett., 35, 325-336, (1998) · Zbl 0909.93054
[20] Palhares, R.M.; Peres, P.L.D., Robust filtering with guaranteed energy-to-peak performance-an LMI approach, Automatica, 36, 851-858, (2000) · Zbl 0953.93067
[21] Rotea, M.A., The generalized \(H_2\) control problem, Automatica, 29, 373-385, (1993) · Zbl 0772.93027
[22] Sun, F.; Liu, H.; He, K.; Sun, Z., Reduced-order \(H_\infty\) filtering for linear systems with Markovian jump parameters, Systems control lett., 54, 8, 739-746, (2005) · Zbl 1129.93542
[23] T. Vincent, J. Abedor, K. Nagpal, R. P. Khargonekar, Discrete-time estimators with guaranteed peak-to-peak performance, Proceedings of 13th IFAC Triennial World Congress, San rancisco, CA, USA, 1996, pp. 43-48.
[24] Wang, Z.; Burnham, K.J., Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation, IEEE trans. signal process., 49, 4, 794-804, (2001)
[25] Wang, Z.; Huang, B.; Unbehauen, H., Robust \(H_\infty\) observer design of linear state delayed systems with parametric uncertainty: the discrete-time case, Automatica, 35, 1161-1167, (1999) · Zbl 1041.93514
[26] Wilson, D.A., Convolution and Hankel operator norms for linear systems, IEEE trans. automat. control, 34, 1, 94-97, (1989) · Zbl 0661.93022
[27] Wonham, W.M., On a matrix Riccati equation of stochastic control, SIAM J. control optim., 6, 681-697, (1968) · Zbl 0182.20803
[28] Xie, L.; De Souza, C.E.; Fu, M., \(H_\infty\) estimation for discrete-time linear uncertain systems, Internat. J. robust nonlinear control, 1, 111-123, (1991) · Zbl 0754.93050
[29] Xu, S.; Chen, T., Reduced-order \(H_\infty\) filtering for stochastic systems, IEEE trans. signal process., 50, 12, 2998-3007, (2002) · Zbl 1369.94325
[30] Xu, S.; Van Dooren, P.; Stefan, R.; Lam, J., Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE trans. automat. control, 47, 1122-1128, (2002) · Zbl 1364.93723
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