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Reduced-order $$H_{\infty}$$ filtering for linear systems with Markovian jump parameters. (English) Zbl 1129.93542
Summary: This paper addresses the reduced-order $$H_{\infty}$$ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order $$H_{\infty}$$ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order $$H_{\infty}$$ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.

##### MSC:
 93E11 Filtering in stochastic control theory 93B36 $$H^\infty$$-control
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##### References:
 [1] Cao, Y.Y.; Lams, J., Robust $$H_\infty$$ control of uncertain Markovian jump systems with time-delay, IEEE trans. automatic control, 45, 1, 77-83, (2002) [2] Costa, O.L.V.; Guerra, S., Robust linear filtering for discrete-time hybrid Markov linear sytems, Int. J. control, 75, 10, 712-727, (2002) · Zbl 1018.93028 [3] Faris, D.P.; Geromel, J.C.; Val, J.B.R.; Costa, O.L.V., Output feedback control of Markovian jump linear systems in continous-time, IEEE trans. automatic control, 45, 5, 944-949, (2000) · Zbl 0972.93074 [4] Gao, H.J.; Wang, C.H., Robust $$L_2 - L_\infty$$ filtering for uncertain systems with multiple time-varying state delays, IEEE trans. circuits syst.—ifundament. theory appl., 50, 4, 594-599, (2003) · Zbl 1368.93712 [5] Gao, H.J.; Wang, C.H., Delay-dependent robust $$H_\infty$$ and $$L_2 - L_\infty$$ filtering for a class of uncertain nonlinear time-delay systems, IEEE trans. automatic control, 48, 9, 1661-1666, (2003) · Zbl 1364.93210 [6] Gao, H.J.; Wang, C.H., A delay-dependent approach to robust $$H_\infty$$ filtering for uncertain discrete-time state-delayed systems, IEEE trans. signal process., 52, 6, 1631-1640, (2004) · Zbl 1369.93175 [7] Ghaoui, L.E.; Rami, M.A., Robust state-space stabilization of jump linear systems via lmis, Int. J. robust nonlinear control, 6, 1015-1022, (1996) · Zbl 0863.93067 [8] Grigoriadis, K.M.; Skelton, R.E., Low-order control design for LMI problems using alternating projection methods, Automatica, 32, 8, 1117-1125, (1996) · Zbl 0855.93026 [9] Grigoriadis, K.M.; Watson, J.T., Reduced-order $$H_\infty$$ and $$L_2 - L_\infty$$ filtering via linear matrix inequalities, IEEE trans. aerospace electronic syst., 33, 4, 1326-1338, (1997) [10] Iwasaki, T.; Skelton, R.E., All controllers for the general $$H_\infty$$ control problemslmi existence conditions and state space formulas, Automatica, 30, 1307-1317, (1994) · Zbl 0806.93017 [11] Li, H.; Fu, M., A linear matrix inequality approach to robust $$H_\infty$$ filtering, IEEE trans. signal process., 45, 2338-2350, (1997) [12] Mahmoud, M.S.; Shi, P., Robust control for Markovian jump linear discrete-time systems with unknown nonlinearities, IEEE trans. circuits syst.—ifundament. theory appl., 49, 4, 538-542, (2002) · Zbl 1368.93154 [13] Mahmoud, M.S.; Shi, P., Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters, IEEE trans. circuits syst.—ifundament. theory appl., 50, 1, 98-105, (2003) · Zbl 1368.93725 [14] Miller, B.M.; Runggaldier, W.J., Kalman filtering for linear systems with coeffients driven by a hidden Markov jump process, Systems and control letters, 31, 93-102, (1997) · Zbl 0901.93069 [15] Nagpal, K.; Helmick, R.E.; Sims, C.S., Reduced-order estimation, Internat. J. control, 45, 1867-1888, (1987) · Zbl 0624.93064 [16] Nagpal, K.; Helmick, R.E.; Sims, C.S., Filtering and smoothing in an $$H_\infty$$ setting, IEEE trans. automatic control, 36, 152-166, (1991) [17] Pan, Z.G.; Basar, T., $$H_\infty$$ control of large scale jump linear systems via averaging and aggregation, (), 2574-2579 [18] Shaked, U.; Theodor, Y., $$H_\infty$$ optimazation estimationa tutorial, () [19] Skelton, R.; Iwasaki, T.; Grigoriadis, K., A unified algebraic approach to linear control design, (1997), Taylor & Francis London [20] Shi, P.; Boukas, E.K.; Agarwal, R.K., Robust control for Markovian jumping discrete-time system, Internat. J. syst. sci., 30, 8, 787-797, (1999) · Zbl 1112.93313 [21] deSouza, C.E.; Fragoso, M.D., $$H_\infty$$ filtering for Markovian jump linear systems, Internat. J. syst. sci., 33, 11, 909-915, (2002) · Zbl 1045.93045 [22] Wang, Z.; Yang, F., Robust filtering for uncertain linear systems with delayed states and outputs, IEEE trans. circuits systems—ifundament. theory appl., 49, 125-130, (2002) [23] Watson, J.T.; Grigoriadis, K.M., Optimal unbiased filtering via linear matrix inequalities, Systems and control letters, 35, 111-118, (1998) · Zbl 0909.93069 [24] Xie, L.H.; de Souza, C.E., On robust filtering for linear systems with parameter uncertainty, (), 2072-2087 [25] Xu, S.Y., Robust $$H_\infty$$ filtering for a class of discrete-time uncertain nonlinear systems with state delay, IEEE trans. circuits syst.—ifundament. theory appl., 49, 1853-1859, (2002) [26] Xu, S.Y.; Chen, T.W., Reduced-order $$H_\infty$$ filtering for stochastic systems, IEEE trans. signal process., 50, 12, 2998-3007, (2002) · Zbl 1369.94325 [27] Yang, F.W.; Hung, Y.S., Robust mixed $$H_\infty / H_2$$ filtering with regional pole assignment for uncertain discrete-time systems, IEEE trans. circuits syst.—ifundament. theory appl., 49, 1236-1241, (2002) · Zbl 1368.93733 [28] Zhou, K.M.; Chen, X., Design of optimal reduced order $$H_2$$ filters, Systems and control letters, 38, 135-138, (1999) · Zbl 1043.93555
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