Weighted \(\mathcal H _{\infty}\) filtering of switched systems with time-varying delay: average dwell time approach. (English) Zbl 1191.94053

Summary: This paper is concerned with the \(\mathcal H _{\infty }\) filtering problem for a continuous-time linear switched system with time-varying delay in its state. To reduce the overdesign of the quadratic framework, this paper proposes a parameter-dependent filter design procedure, which is much less conservative than the quadratic approach. By using an average dwell time approach and the piecewise Lyapunov function technique, a sufficient condition is first proposed to guarantee the exponential stability with a weighted \(\mathcal H _{\infty }\) performance for the filtering error system with the decay estimate explicitly given. Then, the corresponding solvability condition for a desired filter is established, and the filter design is cast into a convex optimization problem which can be efficiently handled by using standard numerical software. All the conditions obtained in this paper are delay dependent. Finally, a numerical example is given to illustrate the effectiveness of the proposed theory.


94A12 Signal theory (characterization, reconstruction, filtering, etc.)
93B36 \(H^\infty\)-control
93E11 Filtering in stochastic control theory
Full Text: DOI


[1] M. Basin, E. Sanchez, R. Martinez-Zuniga, Optimal linear filtering for systems with multiple state and observation delays. Int. J. Innov. Comput. Inf. Control 3(5), 1309–1320 (2007)
[2] E.K. Boukas, P. Shi, S.K. Nguang, Robust H control for linear Markovian jump systems with unknown nonlinearities. J. Math. Anal. Appl. 282, 241–255 (2003) · Zbl 1029.93064 · doi:10.1016/S0022-247X(03)00144-6
[3] J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans. Automat. Control 47(11), 1883–1887 (2002) · Zbl 1364.93559 · doi:10.1109/TAC.2002.804474
[4] C.E. de Souza, A. Trofino, An LMI approach to the design of robust H 2 filters, in Recent Advances on Linear Matrix Inequality Methods in Control, Philadelphia, PA (1999)
[5] R. DeCarlo, M. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88(7), 1069–1082 (2000) · doi:10.1109/5.871309
[6] H. Gao, C. Wang, Delay-dependent robust H and L 2-L filtering for a class of uncertain nonlinear time-delay systems. IEEE Trans. Automat. Control 48(9), 1661–1666 (2003) · Zbl 1364.93210 · doi:10.1109/TAC.2003.817012
[7] H. Gao, C. Wang, Robust L 2-L filtering for uncertain systems with multiple time-varying state delays. IEEE Trans. Circuits Syst. (I) 50(4), 594–599 (2003) · Zbl 1368.93712 · doi:10.1109/TCSI.2003.809816
[8] H. Gao, J. Lam, C. Wang, Model simplification for switched hybrid systems. Syst. Control Lett. 55(12), 1015–1021 (2006) · Zbl 1120.93311 · doi:10.1016/j.sysconle.2006.06.014
[9] J.C. Geromel, M.C. De Oliveira, H 2 and H robust filtering for convex bounded uncertain systems. IEEE Trans. Circuits Syst. (I) 50(4), 594–599 (2003) · Zbl 1368.93712 · doi:10.1109/TCSI.2003.809816
[10] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell time, in Proc. 38th Conf. Decision Control, Phoenix, AZ (1999), pp. 2655–2660
[11] J.P. Hespanha, A.S. Morse, Switching between stabilizing controllers. Automatica 38(11), 1905–1917 (2002) · Zbl 1011.93533
[12] L. Hetel, J. Daafouz, C. Iung, Stabilization of arbitrary switched linear systems with unknown time-varying delays. IEEE Trans. Automat. Control 51(10), 1668–1674 (2006) · Zbl 1366.93575 · doi:10.1109/TAC.2006.883030
[13] B. Hu, A.N. Michel, Stability analysis of digital feedback control systems with time-varying sampling periods. Automatica 36, 897–905 (2000) · Zbl 0941.93034 · doi:10.1016/S0005-1098(99)00217-4
[14] H. Ishii, B.A. Francis, Stabilizing a linear system by switching control with dwell time. IEEE Trans. Automat. Control 47(12), 1962–1973 (2002) · Zbl 1364.93641 · doi:10.1109/TAC.2002.805689
[15] S.H. Jin, J.B. Park, Robust H filter for polytopic uncertain systems via convex optimization. IEE Proc. Part D. Control Theory Appl. 148, 55–59 (2001) · doi:10.1049/ip-cta:20010237
[16] D.K. Kim, P.G. Park, J.W. Ko, Output-feedback H control of systems over communication networks using a deterministic switching system approach. Automatica 40, 1205–1212 (2004) · Zbl 1056.93527 · doi:10.1016/j.automatica.2004.01.024
[17] H. Li, M. Fu, A linear matrix inequality approach to robust H filtering. IEEE Trans. Signal Process. 45(9), 2338–2350 (1997) · doi:10.1109/78.564176
[18] D. Liberzon, Switching in Systems and Control (Birkhauser, Boston, 2003) · Zbl 1036.93001
[19] C. Meyer, S. Schroder, R.W. De Doncker, Solid-state circuit breakers and current limiters for medium-voltage systems having distributed power systems. IEEE Trans. Power Electron. 19, 1333–1340 (2004) · doi:10.1109/TPEL.2004.833454
[20] A.S. Morse, Supervisory control of families of linear set-point controllers, part I: Exact matching. IEEE Trans. Automat. Control 41(10), 1413–1431 (1996) · Zbl 0872.93009 · doi:10.1109/9.539424
[21] I.R. Petersen, A.V. Savkin, Robust Kalman Filtering for Signals and Systems with Large Uncertainties (Birkhauser, Boston, 1999) · Zbl 1033.93002
[22] P. Shi, Robust Kalman filtering for continuous-time systems with discrete-time measurements. IMA J. Math. Control Inf. 16(3), 221–232 (1999) · Zbl 0936.93048 · doi:10.1093/imamci/16.3.221
[23] P. Shi, E.K. Boukas, R.K. Agarwal, Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay. IEEE Trans. Automat. Control 44(11), 2139–2144 (1999) · Zbl 1078.93575 · doi:10.1109/9.802932
[24] X.M. Sun, J. Zhao, D.J. Hill, Stability and L 2-gain analysis for switched delay systems: a delay-dependent method. Automatica 42, 1769–1774 (2006) · Zbl 1114.93086 · doi:10.1016/j.automatica.2006.05.007
[25] R. Wang, J. Zhao, Exponential stability analysis for discrete-time switched linear systems with time-delay. Int. J. Innov. Comput. Inf. Control 3(6B), 1557–1564 (2007)
[26] 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) · Zbl 1241.93016 · doi:10.1002/acs.1030
[27] L. Wu, Z. Wang, Guaranteed cost control for linear switched systems with neutral delay via dynamic output feedback. Int. J. Syst. Sci. 40(7), 717–728 (2009) · Zbl 1291.93138 · doi:10.1080/00207720902953151
[28] L. Wu, W.X. Zheng, H model reduction for switched hybrid systems with time-varying delay. Automatica 45(1), 186–193 (2009) · Zbl 1154.93326 · doi:10.1016/j.automatica.2008.06.024
[29] M. Wu, Y. He, J.H. She, G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 40(8), 1435–1439 (2004) · Zbl 1059.93108 · doi:10.1016/j.automatica.2004.03.004
[30] L. Wu, P. Shi, H. Gao, C. Wang, H filtering for 2D Markovian jump systems. Automatica 44(7), 1849–1858 (2008) · Zbl 1149.93346 · doi:10.1016/j.automatica.2007.10.027
[31] S. Xu, Robust H filtering for a class of discrete-time uncertain nonlinear systems with state delay. IEEE Trans. Circuits Syst. (I) 49, 1853–1859 (2002)
[32] S. Xu, T. Chen, Reduced-order H filtering for stochastic systems. IEEE Trans. Signal Process. 50(12), 2998–3007 (2002) · Zbl 1369.94325 · doi:10.1109/TSP.2002.805239
[33] S. Xu, J. Lam, Improved delay-dependent stability criteria for time-delay systems. IEEE Trans. Automat. Control 50(3), 384–387 (2005) · Zbl 1365.93376 · doi:10.1109/TAC.2005.843873
[34] D. Yue, Q.L. Han, Robust H filter design of uncertain descriptor systems with discrete and distributed delays. IEEE Trans. Signal Process. 52(11), 3200–3212 (2004) · Zbl 1370.93111 · doi:10.1109/TSP.2004.836535
[35] G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Disturbance attenuation properties of time-controlled switched systems. J. Franklin Inst. 338(7), 765–779 (2001) · Zbl 1022.93017 · doi:10.1016/S0016-0032(01)00030-8
[36] G. Zhai, H. Lin, Y. Kim, J. Imae, T. Kobayashi, L 2 gain analysis for switched systems with continuous-time and discrete-time subsystems. Int. J. Control 78(15), 1198–1205 (2005) · Zbl 1088.93010 · doi:10.1080/00207170500274966
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.