This paper considers the H control problem for a class of singular Markovian jump systems (SMJSs), where the jumping signal is not always available. The main contribution of this paper introduces a new approach to a mode-independent (MI) H controller by exploiting the nonfragile method. Based on the given method, a unified control approach establishing a direct connection between mode-dependent (MD) and mode-independent controllers is presented, where both existence conditions are given in terms of linear matrix inequalities. Moreover, another three cases of transition probability rate matrix (TRPM) with elementwise bounded uncertainties, being partially unknown and to be designed are analyzed, respectively. Numerical examples are used to demonstrate the effectiveness of the proposed methods.

References

References
1.
Xu
,
S. Y.
,
Chen
,
T. W.
, and
Lam
,
J.
,
2003
, “
Robust H∞ Filtering for Uncertain Markovian Jump Systems With Mode-Dependent Time Delays
,”
IEEE Trans. Autom. Control
,
48
(
5
), pp.
900
907
.10.1109/TAC.2003.820138
2.
Xiong
,
J. L.
, and
Lam
,
J.
,
2006
, “
On Robust Stabilization of Markovian Jump Systems With Uncertain Switching Probabilities
,”
Automatica
,
41
(
5
), pp.
897
903
.10.1016/j.automatica.2004.12.001
3.
Zhang
,
L. X.
, and
Lam
,
J.
,
2010
, “
Necessary and Sufficient Conditions for Analysis and Synthesis of Markov Jump Linear Systems With Incomplete Transition Descriptions
,”
IEEE Trans. Autom. Control
,
55
(
7
), pp.
1695
1701
.10.1109/TAC.2010.2046607
4.
Zhang
,
Y.
,
He
,
Y.
,
Wu
,
M.
, and
Zhang
,
J.
,
2011
, “
Stabilization for Markovian Jump Systems With Partial Information on Transition Probability Based on Free-Connection Weighting Matrices
,”
Automatica
,
47
(
1
), pp.
79
84
.10.1016/j.automatica.2010.09.009
5.
Wang
,
G. L.
, and
Zhang
,
Q. L.
,
2012
, “
Robust Control of Uncertain Singular Stochastic Systems With Markovian Switching Via Proportional-Derivative State Feedback
,”
IET Control Theory Appl.
,
6
(
8
), pp.
1089
1096
.10.1049/iet-cta.2011.0549
6.
Shu
,
Z.
,
Lam
,
J.
, and
Xiong
,
J. L.
,
2010
, “
Static Output-Feedback Stabilization of Discrete-Time Markovian Jump Linear Systems: A System Augmentation Approach
,”
Automatica
,
46
(
4
), pp.
687
694
.10.1016/j.automatica.2010.02.001
7.
Wu
,
Z. G.
,
Shi
,
P.
,
Su
,
H. Y.
, and
Chu
,
J.
,
2011
, “
Passivity Analysis for Discrete-Time Stochastic Markovian Jump Neural Networks With Mixed Time Delays
,”
IEEE Trans. Neural Netw.
,
22
(
10
), pp.
1566
1575
.10.1109/TNN.2011.2163203
8.
Xu
,
J.
,
Su
,
J. T.
, and
Yue
,
D.
,
2012
, “
Stochastic Finite-Time Stability of Nonlinear Markovian Switching Systems With Impulsive Effects
,”
ASME J. Dyn. Syst., Meas., Control
,
134
(
1
), p.
011011
.10.1115/1.4005359
9.
Guo
,
C. Y.
, and
Zhang
,
W. D.
,
2013
, “
H∞ Estimation for Stochastic Time Delays in Networked Control Systems by Partly Unknown Transition Probabilities of Markovian Chains
,”
ASME J. Dyn. Syst., Meas., Control
,
135
(
1
), p.
014508
.10.1115/1.4007973
10.
de Souza
,
C. E.
,
Trofino
,
A.
, and
Barbosa
,
K. A.
,
2006
, “
Mode-Independent H∞ Filters for Markovian Jump Linear Systems
,”
IEEE Trans. Autom. Control
,
51
(
11
), pp.
1837
1841
.10.1109/TAC.2006.883060
11.
de Souza
,
C. E.
,
2006
, “
Robust Stability and Stabilization of Uncertain Discrete-Time Markovian Jump Linear Systems
,”
IEEE Trans. Autom. Control
,
51
(
5
), pp.
836
841
.10.1109/TAC.2006.875012
12.
Xu
,
S. Y.
, and
Lam
,
J.
,
2006
,
Control and Filtering of Singular Systems
,
Springer-Verlag
,
Berlin
.
13.
Boukas
,
E. K.
,
2008
,
Control of Singular Systems With Random Abrupt Changes
,
Springer-Verlag
,
Berlin
.
14.
Zames
,
G.
,
1981
, “
Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses
,”
IEEE Trans. Autom. Control
,
26
(
2
), pp.
301
320
.10.1109/TAC.1981.1102603
15.
Zhang
,
H.
,
Shi
,
Y.
, and
Liu
,
M. X.
,
2013
, “
H∞ Step Tracking Control for Networked Discrete-Time Nonlinear Systems With Integral and Predictive Actions
,”
IEEE Trans. Ind. Inf.
,
9
(
1
), pp.
337
345
.10.1109/TII.2012.2225434
16.
Zhang
,
H.
,
Shi
,
Y.
, and
Liu
,
M. X.
,
2012
, “
Observer-Based H∞ Feedback Control for Time-Varying Discrete-Time Systems With Intermittent Measurements and Input Constraints
,”
ASME J. Dyn. Syst., Meas., Control
,
134
(
6
), p.
061008
.10.1115/1.4006070
17.
Zhang
,
H.
, and
Shi
,
Y.
,
2012
, “
Robust H∞ PID Control for Multivariable Networked Control Systems With Disturbance/Noise Attenuation
,”
Int. J. Rob. Nonlinear Control
,
22
(
2
), pp.
183
204
.10.1002/rnc.1688
18.
Zhang
,
H.
,
Shi
,
Y.
, and
Mehr
,
A. S.
,
2012
, “
On H∞ Filtering for Discrete-Time Takagi-Sugeno Fuzzy Systems
,”
IEEE Trans. Fuzzy Syst.
,
20
(
2
), pp.
396
401
.10.1109/TFUZZ.2011.2175933
19.
Wang
,
G. L.
,
Zhang
,
Q. L.
, and
Yang
,
C. Y.
,
2012
, “
Dissipative Control for Singular Markovian Jump Systems With Time Delay
,”
Opt. Control Appl. Methods
,
33
(
4
), pp.
415
432
.10.1002/oca.1004
20.
Wang
,
G. L.
,
Zhang
,
Q. L.
, and
Yang
,
C. Y.
,
2013
, “
Exponential H∞ Filtering for Singular Systems With Markovian Jump Parameters
,”
Int. J. Rob. Nonlinear Control
,
23
(
7
), pp.
792
806
.10.1002/rnc.2805
21.
Boukas
,
E. K.
,
Shi
,
P.
, and
Benjelloun
,
K.
,
1999
, “
On Robust Stabilization of Uncertain Linear Systems With Jump Parameters
,”
Int. J. Control
,
72
(
9
), pp.
842
850
.10.1080/002071799220768
22.
Mahmound
,
M. S.
, and
Shi
,
P.
,
2003
, “
Robust Stability, Stabilization and H∞ Control of Time-Delay Systems With Markovian Jump Parameters
,”
Int. J. Rob. Nonlinear Control
,
13
(
8
), pp.
755
784
.10.1002/rnc.744
23.
Xiong
,
J. L.
, and
Lam
,
J.
,
2006
, “
Fixed-Order Robust H∞ Filter Design for Markovian Jump Systems With Uncertain Switching Probabilities
,”
IEEE Trans. Signal Process.
,
54
(
4
), pp.
1421
1430
.10.1109/TSP.2006.871880
24.
Feng
,
J. E.
,
Lam
,
J.
, and
Shu
,
Z.
,
2010
, “
Stabilization of Markovian Systems Via Probability Rate Synthesis and Output Feedback
,”
IEEE Trans. Autom. Control
,
55
(
3
), pp.
773
777
.10.1109/TAC.2010.2040499
You do not currently have access to this content.