This paper is concerned with the robust stochastic stabilization and H∞ disturbance attenuation for uncertain discrete-time Markovian jump linear time-delay systems subject to norm-bounded parameter uncertainties. The transition of the jumping parameters is governed by a finite-state Markov process. First a sufficient condition on stochastic stability is proposed using the stochastic Lyapunov functional approach. Then sufficient conditions on robust stochastic stabilizing and γ-suboptimal H∞ memoryless state feedback controllers are presented. It is shown that a stochastic stabilizing H∞ state feedback controller can be constructed through numerical solution of a set of coupled linear matrix inequalities.
KrasovskiiN. N.LidskiiE. A.Analytical design of controllers in systems with random attributes. Automation Remote Control, 1961, 1–3 (22), 1021–1025.
2.
Abou-KandilH.FreillingG.JankG.Solution and asymptotic behavior of coupled Riccati equations in jump linear systems. IEEE Trans. Autom. Control, 1994, 39 (8), 1631–1635.
3.
BoukasE. K.Control of systems with controlled jump Markov disturbances. Control—Theory Advanced Technology, 1993, 9 (2), 577–595.
4.
De SouzaC. E.FragosoM. D.H∞ control for linear systems with Markovian jumping parameters. Control—Theory Advanced Technology, 1993, 9 (2), 457–466.
5.
FengX.LoparoK. A.JiY.ChizeckH. J.Stochastic stability properties of jump linear system. IEEE Trans. Autom. Control, 1992, 37, 38–53.
6.
JiY.ChizeckH. J.Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control. IEEE Trans. Autom. Control, 1990, 35 (7), 777–788.
7.
KushnerH.Stochastic Stability and Control, 1967 (Academic Press, New York).
8.
MaritionM.Jump Linear Systems in Automatic Control, 1990 (Marcel Dekker, New York).
9.
CostaO. L.V.do ValJ. B. R.Full information H∞-control for discrete-time infinite Markov jump parameter systems. J. Math. Analysis Applic., 1996, 202, 578–603.
10.
CostaO. L.V.FragosoM. D.Stability results for discrete-time linear systems with Markovian jumping parameters. J. Math. Analysis Applic., 1993, 179, 154–178.
11.
CostaO. L.V.MarquesR. P.Mixed H2/H∞ control of discrete-time Markovian jump linear systems. IEEE Trans. Autom. Control, 1998, 43 (1), 95–100.
12.
FragosoM. D.do ValJ. B.R.PintoD. L.Jr.Jump linear H∞ control: The discrete-time case. Control—Theory Advanced Technology, 1995, 10 (4), 1459–1474.
13.
JiY.ChizeckH. J.Controllability, observability and discrete-time Markovian jump linear quadratic control. Int. J. Control, 1988, 48 (2), 481–498.
14.
JiY.ChizeckH. J.Jump linear quadratic Gaussian control: Steady-state solution and testable conditions. Control—Theory Advanced Technology, 1990, 6 (3), 289–319.
15.
BrierleyS. D.ChiassonJ. N.LeeE. B.ZakS. H.On stability independent of delay for linear systems. IEEE Trans. Autom. Control, 1982, 27, 252–254.
16.
CaoY.-Y.SunY.-X.Robust stabilization of uncertain systems with time-varying multistate-delay. IEEE Trans. Autom. Control, 1998, 43 (10), 1484–1488.
17.
CheresE.GutmanS.PalmorZ. J.Stabilization of uncertain dynamic systems including state delay. IEEE Trans. Autom. Control, 1989, 34, 1199–1203.
18.
HaleJ.Theory of Functional Differential Equations, 1977 (Springer-Verlag, New York).
19.
VerriestE. I.IvanovA. F.Robust stability of delay-difference equations. In Proceedings of the 34th IEEEConference on Decision and Control, New Orleans, Louisiana, December 1995, pp. 386–391 (IEEE, New York).
20.
ZhouK.DoyleJ. C.GloverK.Robust and Optimal Control, 1996 (Prentice-Hall, Englewood Cliffs, New Jersey).
21.
CaoY.-Y.SunY.-X.LamJ.Delay-dependent robust H∞ control for uncertain systems with time-varying delays. Proc. Instn Electl Engrs, Part D, 1998, 145 (3), 338–344.
22.
ChoiH. H.ChungM. J.Memoryless H∞ controller design for linear systems with delayed state and control. Automatica, 1995, 31, 917–919.
23.
LeeJ. H.KimS. W.KwonW. H.Memoryless H∞ controllers for state delayed systems. IEEE Trans. Autom. Control, 1994, 39, 159–161.
24.
SongS.-H.KimJ.-K.H∞-control of discrete-time linear systems with nome-bounded uncertainties and time delay in state. Automatica, 1998, 34 (1), 137–139.
25.
MorozanT.Stochastic stability and control for discrete-time systems with jump Markov disturbances. Rev. Roum. Math. Pures Appl., 1979, 24 (1), 111–127.
26.
BoukasE. K.ShiP. H∞ control for discrete-time linear systems with Markovian jumping parameters. In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, December 1997, pp. 4134–4139 (IEEE, New York).
27.
BenjellounK.BoukasE. K.ShiP.Robust stochastic stability of discrete-time linear systems with Markovian jumping parameters. In Proceedings of the 36th IEEE Conference onDecision and Control, San Diego, California, December 1997, pp. 559–564 (IEEE, New York).
28.
BoydS.El GhaouiL.FeronE.BalakrishnanV.Linear Matrix Inequalities in System and Control Theory, 1994 (SIAM, Philadelphia, Pennsylvania).
29.
RamiM. A.GhaouiL. E.LMI optimization for non-standard Riccati equations arising in stochastic control. IEEE Trans. Autom. Control, 1996, 41 (11), 1666–1671.
30.
SworderD. D.RogersR. O.An LQG solution to a control problem with solar thermal receiver. IEEE Trans. Autom. Control, 1983, 28, 971–978.
