Abstract
This paper is concerned with the reliable H ∞ filtering problem against sensor failures for a class of discrete-time switched singular systems with time-varying delay. A practical sensor failure model, which consists of a scaling factor with upper and lower bounds to the output measuring is considered. The purpose is to design a switched full-order H ∞ filter such that, for all possible sensor failures, the resulting filtering error system is regular, causal, and uniformly asymptotically stable with a guaranteed H ∞ performance index under arbitrary switching signals. By using the switched Lyapunov function approach and establishing a finite sum equality, a delay-dependent bounded real lemma (BRL) for the filtering error systems is derived via linear matrix inequality (LMI) formulation. An improved BRL is also established by introducing additional slack matrix variables to realize the decoupling between the filtering error system matrices and the Lyapunov matrices. Then, based on the decoupled BRL, the existence criterion of the desired filter is obtained by employing the LMI technique. Some numerical examples are provided to illustrate the effectiveness and the potential of the proposed methods.
Similar content being viewed by others
References
M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998)
J. Daafouz, R. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans. Autom. Control 47(11), 1883–1887 (2002)
L. Dai, Singular Control Systems (Springer, Berlin, 1989)
M.C. de Oliverrira, J.C. Geromel, J. Bernussu, Extended H ∞ and H 2 norm characterizations and controller parameterizations for discrete-time systems. Int. J. Control 75(9), 666–679 (2002)
D. Du, H ∞ filter for discrete-time switched systems with time-varying delays. Nonlinear Anal. Hybrid Syst. 4(4), 782–790 (2010)
D. Du, B. Jiang, P. Shi, S. Zhou, H ∞ filtering of discrete-time switched systems with state-delays via switched Lyapunov function approach. IEEE Trans. Autom. Control 52(8), 1520–1525 (2007)
X. Guo, G. Yang, Reliable H ∞ filter design for a class of discrete-time nonlinear systems with time-varying delay. Optim. Control Appl. Methods 31(4), 303–322 (2010)
A. Halanay, V. Rasvan, Stability radii for some propagation models. IMA J. Math. Control Inf. 14(1), 95–107 (1997)
J.P. Hespanha, A.S. Morse, Switching between stabilising controllers. Automatica 38(11), 1905–1917 (2002)
K. Hu, J. Yuan, Finite sum equality approach to H ∞ output-feedback control for switched linear discrete-time systems with time-varying delay. IET Control Theory Appl. 3(8), 1006–1016 (2009)
J.H. Kim, Delay-dependent robust H ∞ filtering for uncertain discrete-time singular systems with interval time-varying delay. Automatica 46(3), 591–597 (2010)
S. Kim, Stability of switching systems with delay. Ph.D. dissertation, University of Waterloo, Waterloo, ON, Canada (2005)
D. Koenig, B. Marx, H ∞-filtering and state feedback control for discrete-time switched descriptor systems. IET Control Theory Appl. 3(6), 661–670 (2009)
C.M. Lee, I.K. Fong, H ∞ filter design for uncertain discrete-time singular systems via normal transformation. Circuits Syst. Signal Process. 25(4), 525–538 (2006)
H. Li, M. Fu, A linear matrix inequality approach to robust H ∞ filtering. IEEE Trans. Signal Process. 45(9), 2338–2350 (1997)
D. Liberzon, Switching in Systems and Control (Birkhauser, Boston, 2003)
D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19(5), 59–70 (1999)
H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)
J. Liu, J. Wang, G. Yang, Reliable guaranteed variance filtering against sensor failures. IEEE Trans. Signal Process. 51(5), 1403–1411 (2003)
Y. Liu, Z. Wang, W. Wang, Reliable H ∞ filtering for discrete time-delay systems with randomly occurred nonlinearities via delay-partitioning method. Signal Process. 91(4), 713–727 (2011)
R. Lu, Y. Xu, A. Xue, H ∞ filtering for singular systems with communication delays. Signal Process. 90(4), 1240–1248 (2010)
S. Ma, E.K. Boukas, Robust H ∞ filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay. IET Control Theory Appl. 3(3), 351–361 (2009)
S. Ma, C. Zhang, Z. Wu, Delay-dependent stability and H ∞ control for uncertain discrete switched singular systems with time-delay. Appl. Math. Comput. 206(1), 413–424 (2008)
M.S. Mahmoud, Delay-dependent H ∞ filtering of a class of switched discrete-time state delay systems. Signal Process. 88(11), 2709–2719 (2008)
K.M. Nagpal, P.P. Khargonekar, Filtering and smoothing in an H ∞ setting. IEEE Trans. Autom. Control 36(2), 152–166 (1991)
K.S. Narendra, J. Balakrishnan, Adaptive control using multiple models. IEEE Trans. Autom. Control 42(2), 171–187 (1997)
S. Shi, Z. Yuan, Q. Zhang, Fault-tolerant H ∞ filter design of a class of switched systems with sensor failures. Int. J. Innov. Comput. Inf. Control 5(11), 3827–3838 (2009)
M.S. Silva, T.P. de Lima, Looking for nonnegative solutions of a Leontief dynamic model. Linear Algebra Appl. 364, 281–316 (2003)
Z. Sun, S.S. Ge, Switched Linear Systems: Control and Design (Springer, New York, 2005)
X. Sun, J. Zhao, D.J. Hill, Stability and L 2-gain analysis for switched delay systems: a delay-dependent method. Automatica 42(10), 1769–1774 (2006)
D. Wang, W. Wang, P. Shi, Exponential H ∞ filtering for switched linear systems with interval time-varying delay. Int. J. Robust Nonlinear Control 19(5), 532–551 (2009)
F. Wang, Q. Zhang, LMI-based reliable H ∞ filtering with sensor failure. Int. J. Innov. Comput. Inf. Control 2(4), 737–748 (2006)
L. Wu, J. Lam, Sliding mode control of switched hybrid systems with time-varying delay. Int. J. Adapt. Control Signal Process. 22(10), 909–931 (2008)
L. Wu, J. Lam, Weighted H ∞ filtering of switched systems with time-varying delay: average dwell time approach. Circuits Syst. Signal Process. 28(6), 1017–1036 (2009)
Z. Wu, H. Su, J. Chu, Delay-dependent H ∞ filtering for singular Markovian jump time-delay systems. Signal Process. 90(6), 1815–1824 (2010)
Z. Wu, H. Su, J. Chu, H ∞ filtering for singular systems with time-varying delay. Int. J. Robust Nonlinear Control 20(11), 1269–1284 (2010)
S. Xu, J. Lam, Robust stability and stabilization of discrete singular systems: an equivalent characterization. IEEE Trans. Autom. Control 40(4), 568–574 (2004)
S. Xu, J. Lam, C. Yang, Robust H ∞ control for discrete singular systems with state delay and parameter uncertainty. Dyn. Contin. Discrete-Time Impuls. Syst. Ser. B 11(3), 497–506 (2002)
S. Xu, J. Lam, Y. Zou, H ∞ filtering for singular systems. IEEE Trans. Autom. Control 48(12), 2217–2222 (2003)
G. Yang, D. Ye, Adaptive reliable H ∞ filtering against sensor failures. IEEE Trans. Signal Process. 55(7), 3161–3171 (2007)
D. Yue, Q.-H. Han, Robust H ∞ filter design of uncertain descriptor systems with discrete and distributed delays. IEEE Trans. Signal Process. 52(11), 3200–3212 (2004)
L. Zhang, H. Gao, Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010)
B. Zhang, S. Xu, D. Du, Robust H ∞ filtering of delayed singular systems with linear fractional parametric uncertainties. Circuits Syst. Signal Process. 25(5), 627–647 (2006)
W. Zhang, L. Yu, Stability analysis for discrete-time switched time-delay systems. Automatica 45(10), 2265–2271 (2009)
X. Zhu, Y. Wang, Y. Gan, H ∞ filtering for continuous-time singular systems with time-varying delay. Int. J. Adapt. Control Signal Process. 25(2), 137–154 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, J., Fei, S. & Wu, Q. Reliable H ∞ Filtering for Discrete-Time Switched Singular Systems with Time-Varying Delay. Circuits Syst Signal Process 31, 1191–1214 (2012). https://doi.org/10.1007/s00034-011-9361-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-011-9361-2