In this paper, the problem on simultaneous estimation of the actuator and sensor faults is first addressed for a class of Markovian jump systems with mixed mode-dependent time-varying delays. By using a generalized system technique, the original system is first transformed into a descriptor one; its states consist of original states and sensor fault. Then, a Markovian adaptive observer is designed for the descriptor system to provide simultaneous estimations of the state, actuator fault, and sensor fault. In the light of online acquired information, a state-feedback-based fault-tolerant controller is constructed to stabilize the closed-loop system in the presence of the actuator fault. Using the Lyapunov–Krasovskii functions, sufficient and necessity conditions for the existence of designed observer and controller are derived in terms of linear matrix inequalities, which can be solved readily through efficient mathematical tools. Finally, numerical and practical examples are given to validate the effectiveness of the proposed method.

References

References
1.
Alwi
,
H.
,
Edwards
,
C.
, and
Tan
,
C. P.
,
2000
, “
Sliding Mode Observers for Fault Detection
,”
Automatica
,
36
(
4
), pp.
541
553
.
2.
Gao
,
Z.
, and
Ding
,
S. X.
,
2007
, “
Actuator Fault Robust Estimation and Fault-Tolerant Control for a Class of Nonlinear Descriptor Systems
,”
Automatica
,
43
(
5
), pp.
912
920
.
3.
Li
,
X. J.
, and
Yang
,
G. H.
,
2012
, “
Robust Adaptive Fault-Tolerant Control for Uncertain Linear Systems With Actuator Failures
,”
Control Theory Appl. IET
,
6
(
10
), pp.
1544
1551
.
4.
Zhang
,
K.
,
Jiang
,
B.
, and
Cocquempot
,
V.
,
2009
, “
Fast Adaptive Fault Estimation and Accommodation for Nonlinear Time-Varying Delay Systems
,”
Asian J. Control
,
11
(
6
), p.
643C652
.
5.
Cheng
,
J.
,
Zhu
,
H.
,
Zhong
,
S.
,
Zeng
,
Y.
, and
Dong
,
X.
,
2013
, “
Finite-Time H∞ Control for a Class of Markovian Jump Systems With Mode-Dependent Time-Varying Delays Via New Lyapunov Functionals
,”
ISA Trans.
,
52
(
6
), pp.
768
774
.
6.
Krasovskii
,
N. N.
,
Lidskii
,
E. A.
,
Krasovskii
,
N. N.
, and
Lidskii
,
E. A.
,
1961
, “
Analysis Design of Controller in Systems With Random Attributes-Part I
,”
Autom. Remote Control
,
22
, pp. 1021–1025.
7.
Boukas
,
E. K.
,
2005
, “
Stochastic Switching Systems: Analysis and Design
,”
IEEE Trans. Autom. Control
,
52
(
4
), p.
764
.
8.
Zhao
,
X.
, and
Tian
,
E.
,
2013
, “
Stability and Stabilization for Discrete System With Probabilistic Nonlinearities
,”
ASME J. Dyn. Syst. Meas. Control
,
135
(
5
), pp.
826
826
.
9.
He
,
S.
, and
Liu
,
F.
,
2010
, “
Stochastic Finite-Time Stabilization for Uncertain Jump Systems Via State Feedback
,”
ASME J. Dyn. Syst. Meas. Control
,
132
(
3
), pp.
333
342
.
10.
Liu
,
M.
,
Shi
,
P.
,
Zhang
,
L.
, and
Zhao
,
X.
,
2011
, “
Fault-Tolerant Control for Nonlinear Markovian Jump Systems Via Proportional and Derivative Sliding Mode Observer Technique
,”
IEEE Trans. Circuits Syst.
,
58
(
11
), pp.
2755
2764
.
11.
Wu
,
Z.-G.
,
Shi
,
P.
,
Su
,
H.
, and
Chu
,
J.
,
2014
, “
Asynchronous l2-l∞ Filtering for Discrete-Time Stochastic Markov Jump Systems With Randomly Occurred Sensor Nonlinearities
,”
Automatica
,
50
(
1
), pp.
180
186
.
12.
Gao
,
L.
,
Wang
,
D.
, and
Wu
,
Y.
,
2014
, “
Non-Fragile Observer-Based Sliding Mode Control for Markovian Jump Systems With Mixed Mode-Dependent Time Delays and Input Nonlinearity
,”
Appl. Math. Comput.
,
229
(
6
), pp.
374
395
.
13.
Kao
,
Y.
,
Xie
,
J.
,
Wang
,
C.
, and
Karimi
,
H. R.
,
2015
, “
A Sliding Mode Approach to H∞ Non-Fragile Observer-Based Control Design for Uncertain Markovian Neutral-Type Stochastic Systems
,”
Automatica
,
52
, pp.
218
226
.
14.
Zhu
,
J.
,
Yu
,
X.
,
Zhang
,
T.
,
Cao
,
Z.
,
Yang
,
Y.
, and
Yi
,
Y.
,
2016
, “
Sliding Mode Control of MIMO Markovian Jump Systems
,”
Automatica
,
68
, pp.
286
293
.
15.
Shi
,
P.
,
Zhang
,
Y.
, and
Agarwal
,
R. K.
,
2015
, “
Stochastic Finite-Time State Estimation for Discrete Time-Delay Neural Networks With Markovian Jumps
,”
Neurocomputing
,
151
(Pt. 1), pp.
168
174
.
16.
Wu
,
Z.-G.
,
Shi
,
P.
,
Su
,
H.
, and
Chu
,
J.
,
2013
, “
Stochastic Synchronization of Markovian Jump Neural Networks With Time-Varying Delay Using Sampled Data
,”
IEEE Trans. Cybern.
,
43
(
6
), pp.
1796
1806
.
17.
Xu
,
S.
,
Lam
,
J.
,
Zhang
,
B.
, and
Zou
,
Y.
,
2013
, “
New Insight Into Delay-Dependent Stability of Time-Delay Systems
,”
Int. J. Robust Nonlinear Control
,
25
(
7
), pp.
961
970
.
18.
Chen
,
Y.
, and
Zheng
,
W. X.
,
2014
, “
Exponential H∞ Filtering for Stochastic Markovian Jump Systems With Time Delays
,”
Int. J. Robust Nonlinear Control
,
24
(
4
), pp.
625
643
.
19.
Chen
,
L.
,
Huang
,
X.
, and
Fu
,
S.
,
2016
, “
Fault-Tolerant Control for Markovian Jump Delay Systems With an Adaptive Observer Approach
,”
Circuits Syst. Signal Process.
,
35
, p.
4290
.
20.
Chen
,
W. H.
,
Xu
,
J. X.
, and
Guan
,
Z. H.
,
2004
, “
Guaranteed Cost Control for Uncertain Markovian Jump Systems With Mode-Dependent Time-Delays
,”
IEEE Trans. Autom. Control
,
48
(
12
), pp.
2270
2277
.
21.
Wang
,
Z.
,
Liu
,
Y.
, and
Liu
,
X.
,
2010
, “
Exponential Stabilization of a Class of Stochastic System With Markovian Jump Parameters and Mode-Dependent Mixed Time-Delays
,”
IEEE Trans. Autom. Control
,
55
(
7
), pp.
1656
1662
.
22.
Li
,
F.
,
Wu
,
L.
, and
Shi
,
P.
,
2014
, “
Stochastic Stability of Semi-Markovian Jump Systems With Mode-Dependent Delays
,”
Int. J. Robust Nonlinear Control
,
24
(
18
), pp.
3317
3330
.
23.
Shu
,
Z.
, and
Lam
,
J.
,
2008
, “
Exponential Estimates and Stabilization of Uncertain Singular Systems With Discrete and Distributed Delays
,”
Int. J. Control
,
81
(
81
), pp.
865
882
.
24.
Liu
,
Y.
,
Wang
,
Z.
,
Liang
,
J.
, and
Liu
,
X.
,
2009
, “
Stability and Synchronization of Discrete-Time Markovian Jumping Neural Networks With Mixed Mode-Dependent Time Delays
,”
IEEE Trans. Neural Networks
,
20
(
7
), pp.
1102
1116
.
25.
Liu
,
Y.
,
Liu
,
W.
,
Obaid
,
M. A.
, and
Abbas
,
I. A.
,
2016
, “
Exponential Stability of Markovian Jumping Cohen–Grossberg Neural Networks With Mixed Mode-Dependent Time-Delays
,”
Neurocomputing
,
177
, pp.
409
415
.
26.
Muthukumar
,
P.
, and
Subramanian
,
K.
,
2016
, “
Stability Criteria for Markovian Jump Neural Networks With Mode-Dependent Additive Time-Varying Delays Via Quadratic Convex Combination
,”
Neurocomputing
,
205
, pp.
75
83
.
27.
Wang
,
Y.
,
Shi
,
P.
,
Wang
,
Q.
, and
Duan
,
D.
,
2013
, “
Exponential Filtering for Singular Markovian Jump Systems With Mixed Mode-Dependent Time-Varying Delay
,”
IEEE Trans. Circuits Syst.
,
60
(
9
), pp.
2440
2452
.
28.
Liu
,
Y.
,
Wang
,
Z.
, and
Liu
,
X.
,
2008
, “
State Estimation for Discrete-Time Markovian Jumping Neural Networks With Mixed Mode-Dependent Delays
,”
Phys. Lett. A
,
372
(
48
), pp.
7147
7155
.
29.
He
,
S. P.
,
2014
, “
Fault Estimation for T-S Fuzzy Markovian Jumping Systems Based on the Adaptive Observer
,”
Int. J. Control Autom. Syst.
,
12
(
5
), pp.
977
985
.
30.
Tao
,
F.
, and
Zhao
,
Q.
,
2007
, “
Synthesis of Active Fault-Tolerant Control Based on Markovian Jump System Models
,”
IET Control Theory Appl.
,
1
(
4
), pp.
1160
1168
.
31.
Liu
,
M.
, and
Ho
,
D. W. C.
,
2014
, “
Adaptive Backstepping Control of Markovian Jump Systems With Actuator Faults
,”
33rd Chinese Control Conference (CCC)
, pp. 4300–4305.
32.
Shi
,
P.
,
Liu
,
M.
, and
Zhang
,
L.
,
2015
, “
Fault-Tolerant Sliding-Mode-Observer Synthesis of Markovian Jump Systems Using Quantized Measurements
,”
IEEE Trans. Ind. Electron.
,
62
(
9
), pp.
5910
5918
.
33.
Liu
,
M.
,
Ho
,
D. W. C.
, and
Shi
,
P.
,
2015
, “
Adaptive Fault-Tolerant Compensation Control for Markovian Jump Systems With Mismatched External Disturbance
,”
Automatica
,
58
, pp.
5
14
.
34.
Li
,
H.
,
Shi
,
P.
,
Yao
,
D.
, and
Wu
,
L.
,
2016
, “
Observer-Based Adaptive Sliding Mode Control for Nonlinear Markovian Jump Systems
,”
Automatica
,
64
, pp.
133
142
.
35.
Li
,
H.
,
Gao
,
H.
,
Shi
,
P.
, and
Zhao
,
X.
,
2014
, “
Fault-Tolerant Control of Markovian Jump Stochastic Systems Via the Augmented Sliding Mode Observer Approach
,”
Automatica
,
50
(
7
), pp.
1825
1834
.
36.
He
,
S.
, and
Liu
,
F.
,
2016
, “
Resilient Fault Detection Observer Design of Fuzzy Markovian Jumping Systems With Mode-Dependent Time-Varying Delays
,”
J. Franklin Inst.
,
35
(13), pp. 2943–2965.
37.
Boukas
,
E.-K.
,
2007
,
Stochastic Switching Systems: Analysis and Design
,
Springer Science & Business Media
, Berlin.
You do not currently have access to this content.