This paper is concerned with exponential estimates and H∞ control for a class of singular Markovian jump systems with interval time-varying delay. By constructing an appropriate Lyapunov-Krasovskii functional and using some algebraic techniques, a sufficient condition, which does not only guarantee the regularity, absence of impulses, and mean square exponential stability but also gives the estimates of decay rate and decay coefficient, is derived in terms of linear matrix inequalities (LMIs). A bounded real lemma is also developed. Based on these, a state-feedback controller which makes the closed-loop system stochastically exponentially admissible with a guaranteed H∞ noise-attenuation performance and a prescribed lower bound of the decay rate is designed by employing the LMI technique. Numerical examples are provided to illustrate the effectiveness of the proposed design methods.
MaritonM.Jump linear systems (Marcel Dekker, New York1990)
2.
ShiP.BoukasE. K.‘H∞ control for Markovian jumping linear systems with parametric uncertainty.’J. Optim. Theory Appl.95 (1) (1997): 75–99
3.
De FariasD. P.GeromelJ. C.Do ValJ. B. R.CostaO. L. V.‘Output feedback control of Markov jump linear systems in continuous-time.’IEEE Trans. Autom. Control45 (10) (2000): 944–949
4.
XiongJ.LamJ.GaoH.HoD. W. C.‘On robust stabilization of Markovian jump systems with uncertain switching probabilities.’Automatica41 (5) (2005): 897–903
5.
BoukasE. K.Stochastic switching systems, analysis and design (Birkhauser, Boston, Massachusetts2006)
6.
DongJ.YangG.‘Robust H2 control of continuous-time Markov jump linear systems.’Automatica44 (5) (2008): 1431–1436
7.
ZhangL.BoukasE. K.‘Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities.’Automatica45 (2) (2009): 463–468
8.
HaleJ. K.LunelVerduyn S. M.Introduction to functional differential equations (Springer-Verlag, New York1993)
9.
CaoY.LamJ.‘Robust H∞ control of uncertain Markovian jump systems with time-delay.’IEEE Trans. Autom. Control45 (1) (2000): 77–83
10.
MaoX.‘Exponential stability of stochastic delay interval systems with Markovian switching.’IEEE Trans. Autom. Control47 (10) (2002): 1604–1612
11.
YuanC.MaoX.‘Robust stability and controllability of stochastic differential delay equations with Markovian switching.’Automatica40 (3) (2004): 343–354
12.
YueD.HanQ.‘Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching.’IEEE Trans. Autom. Control50 (2) (2005): 217–222
13.
WuJ.ChenT.WangL.‘Delay-dependent robust stability and H∞ control for jump linear systems with delays.’Syst. Control Lett.55 (11) (2006): 939–948
14.
ZhanS.LamJ.XuS.‘Robust stabilization of Markovian delay systems with delay-dependent exponential estimates.’Automatica42 (11) (2006): 2001–2008
15.
XuS.LamJ.MaoX.‘Delay-dependent H∞ control and filtering for uncertain Markovian jump systems with time-varying delays.’IEEE Trans. Circuits Syst. I: Fundam. Theory Appl.54 (9) (2007): 2070–2077
16.
KangY.ZhangJ.GeS. S.‘Robust output feedback H∞ control of uncertain Markovian jump systems with mode-dependent time-delays.’Int. J. Control81 (1) (2008): 43–61
17.
ShaoH.‘Delay-range-dependent robust H∞ filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters.’J. Math. Anal. Appl.342 (2) (2008): 1084–1095
18.
SathananthanS.BeaneC.KeelL. H.‘Delay-dependent criteria for robust stabilization of Markovian switching networks with time-varying delay.’Stoch. Anal. Appl.27 (4) (2009): 694–712
19.
WangG.ZhangQ.SreeramV.‘Robust delay-range-dependent stabilization for Markovian jump systems with mode-dependent time delays and nonlinearities.’Optim. Control Appl. Meth.31 (3) (2010): 249–264
20.
CampbellS.‘Singular linear systems of differential equations with delays.’Appl. Anal.2 (1980): 129–136
21.
FridmanE.ShakedU. ‘H∞-control of linear state-delay descriptor systems: an LMI approach.’ Linear Algebr. Appl. 351/352 (2002): 271–302
22.
KolmanovskiiV. B.NosovV. R.Stability of functional differential equations (Academic Press, New York1986)
23.
RasvanV.NiculescuS.‘Oscillations in lossless propagation models: A Liapunov-Krasovskii approach.’IMA J. Math. Control Inf.19 (2002): 157–172
24.
HaidarA.BoukasE. K.‘Exponential stability of singular systems with multiple time-varying delays.’Automatica45 (2) (2009): 539–545
25.
BoukasE. K.XuS.LamJ.‘On stability and stabilizability of singular stochastic systems with delays.’J. Optim. Theory Appl.127 (2) (2005): 249–262
26.
HaidarA.BoukasE. K.‘Robust stability criteria for Markovian jump singular systems with time-varying delays.’ In Proceedings of the 47th IEEE Conference onDecision and control, Cancun, Mexico, 9-11 December 2008, pp. 4657–4662 (IEEE Press, Piscataway, New Jersey).
27.
WuZ.SuH.ChuJ.‘Delay-dependent H∞ control for singular Markovian jump systems with time delay.’Optim. Control Appl. Meth.30 (5) (2009): 443–461
28.
ZhouW.FangJ.‘Delay-dependent robust H∞ admissibility and stabilization for uncertain singular system with Markovian jumping parameters.’Circuits Syst. Signal Process.28 (3) (2009): 433–450
29.
WuZ.SuH.ChuJ.‘H∞ filtering for singular Markovian jump systems with time delay.’Int. J. Robust Nonlinear Control20 (8) (2010): 939–957
MaS.BoukasE. K.ChinniahY.‘Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay.’Int. J. Robust Nonlinear Control20 (2010): 531–543
32.
MaS.ZhangC.‘Robust stability and H∞ control for uncertain discrete Markovian jump singular systems with mode-dependent time delay.’Int. J. Robust Nonlinear Control19 (9) (2009): 965–985
33.
ZhouW.LuH.DuanC.LiM.‘Delay-dependent robust control for singular discrete-time Markovian jump systems with time-varying delay.’Int. J. Robust Nonlinear Control (2009): in press
34.
TianE.YueD.ZhangY.‘Delay-dependent robust H∞ control for T-S fuzzy system with interval time-varying delay.’Fuzzy Set Syst.160 (12) (2009): 1708–1719
35.
HeY.WangQ.LinC.WuM.‘Delay-range-dependent stability for systems with time-varying delay.’Automatica43 (2) (2007): 371–376
36.
ShaoH.‘New delay-dependent stability criteria for systems with interval delay.’Automatica45 (3) (2009): 744–749
37.
LiT.SongA.FeiS.GuoY.‘Synchronization control of chaotic neural networks with time-varying and distributed delays.’Nonlinear Anal. Theory Methods Appl.71 (5/6) (2009): 2372–2384
38.
ZhanS.LamJ.‘Exponential estimates and stabilization of uncertain singular systems with discrete and distributed delays.’Int. J. Control81 (6) (2008): 865–882
39.
GohK. C.SafonovM. G.PapavassilopoulosG. P.‘A global optimization approach for the BMI problem.’ In Proceedings of the 33rd Conference onDecision and control, Lake Buena Vista, Florida, 14-16 December, 1994, pp. 2009–2014 (IEEE Press, Piscataway, New Jersey).
40.
KharitonovV.MondieS.ColladoJ.‘Exponential estimates for neutral time-delay systems: An LMI approach.’IEEE Trans. Autom. Control50 (5) (2005): 666–670
41.
YueD.HanQ.PengC.‘State feedback controller design of networked control systems.’IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process.51 (11) (2004): 640–644
