This paper considers the class of discrete-time jump linear systems with time-delay and polytopic uncertain parameters. The problems of delay-independent robust stability, stabilization and $H∞$ control are cast into the framework of linear matrix inequality (LMI) and thus solved by LMI Toolbox of Matlab. By extending the system state, the system with time-delay is converted into a higher dimension Markov jump system without time-delay, and thus can be handled as a standard jump linear system with uncertain parameters. Numerical examples are provided to show the usefulness of the theoretical results.

1.
Costa
,
O. L. V.
, and
Fragoso
,
M. D.
,
1993
, “
Stability Results for Discrete-Time Markovian Jump Linear Systems With Markovian Jumping Parameter Systems
,”
J. Math. Anal. Appl.
,
179
, pp.
154
178
.
2.
Wonham, W. M., 1971, “Random Differential Equations in Control Theory,” Probabilistic Methods in Applied Mathematics, 2, A. T. Bharucha-reid, ed., Academic Press, New York.
3.
Sworder
,
D. D.
, and
Robinson
,
V. G.
,
1974
, “
Feedback Regulators for Jump Parameter Systems With State and Control Dependent Transition Rates
,”
IEEE Trans. Autom. Control
,
18
, pp.
355
359
.
4.
Ji
,
Y.
, and
Chizeck
,
H. J.
,
1988
, “
Controllability, Observability, and Discrete-Time Markovian Jump Linear Quadratic Control
,”
Int. J. Control
,
48
(
2
), pp.
481
498
.
5.
Ji
,
Y.
,
Chizeck
,
H. J.
,
Feng
,
X.
, and
Loparo
,
K. A.
,
1991
, “
Stability and Control of Discrete-Time Jump Linear Systems
,”
,
7
(
2
), pp.
247
270
.
6.
de Souza
,
C. E.
, and
Fragoso
,
M. D.
,
1993
, “
H Control for Linear Systems With Markovian Jumping Parameters
,”
,
9
(
2
), pp.
457
466
.
7.
Mariton, M., 1990, Jump Linear Systems in Automatic Control, Marcel Dekker, New York.
8.
Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM.
9.
Boukas
,
E. K.
,
1993
, “
Control of System With Controlled Jump Markov Disturbances
,”
,
9
(
2
), pp.
577
597
.
10.
Boukas
,
E. K.
, and
Yang
,
H.
,
1995
, “
Stability of Discrete-Time Linear Systems With Markovian Jumping Parameters
,”
Mathematics of Control, Signals and Systems
,
8
, pp.
390
402
.
11.
Costa
,
O. L.
, and
Boukas
,
E. K.
,
1998
, “
Necessary and Sufficient Condition for Robust Stability and Stabilizability of Continuous-Time Linear Systems With Markovian Jumps
,”
J. Optim. Theory Appl.
,
99
(
2
), pp.
75
99
.
12.
Shi
,
P.
, and
Boukas
,
E. K.
,
1997
, “
H Control for Markovian Jumping Linear Systems With Parametric Uncertainty
,”
J. Optim. Theory Appl.
,
95
(
2
), pp.
359
381
.
13.
Jeung
,
E. T.
,
Kim
,
J. H.
, and
Park
,
H. B.
,
1998
, “
H-Output Feedback Controller Design for Linear Systems With Time-Varying Delayed State
,”
IEEE Trans. Autom. Control
,
43
(
7
), pp.
971
974
.
14.
Benjelloun
,
K.
, and
Boukas
,
E. K.
,
1998
, “
Mean Square Stochastic Stability of Linear Time-Delay System with Markovian Jumping Parameters
,”
IEEE Trans. Autom. Control
,
43
(
10
), pp.
1456
1459
.
15.
Benjelloun
,
K.
,
Boukas
,
E. K.
, and
Yang
,
H.
,
1996
, “
Robust Stabilizability of Uncertain Linear Time-Delay Systems With Markovian Jumping Parameters
,”
ASME J. Dyn. Syst., Meas., Control
,
118
(
4
), pp.
776
783
.
16.
Boukas
,
E. K.
, and
Liu
,
Z. K.
,
2001
, “
Robust H of Discrete-Time Markovian Jump Linear Systems With Mode-Dependent Time-Delays
,”
IEEE Trans. Autom. Control
,
46
, pp.
1918
1924
.
17.
Mahmoud
,
M. S.
, and
Al-Muthairi
,
N. F.
,
1994
, “
Design of Robust Controllers for Time-Delay Systems
,”
IEEE Trans. Autom. Control
,
39
(
5
), pp.
995
999
.