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The stability and robust state feedback stabilisation problems for a class of non-linear discrete-time descriptor Markov jump systems with parameter uncertainties are investigated. The non-linear function satisfies a quadratic constraint. First, based on Lyapunov theory and the implicit function theorem, a linear matrix inequality (LMI) sufficient condition is developed, which guarantees that the non-linear discrete-time descriptor Markov jump systems are regular, causal, and have unique solution in a neighbourhood of the origin, and are stochastically stable. With this condition, another LMI condition is developed based on the singular value decomposition approach, which guarantees that the systems are regular, causal, have unique solution in a neighbourhood of the origin, and are stochastically stable. Then, the design method of robust state feedback controllers is given. Last, numerical examples are provided to illustrate the effectiveness of the proposed methods.
References
-
-
1)
-
W.J. Mao ,
J. Chu
.
Robust decentralised stabilisation of interval discrete-time singular large-scale systems.
IET Control Theory Appl.
,
2 ,
244 -
252
-
2)
-
L. Xie ,
C.E. de Souza
.
Robust H∞ control for linear systems with norm-bounded time-varying uncertainty.
IEEE Trans. Autom. Control
,
8 ,
1188 -
1191
-
3)
-
S. Xu ,
J. Lam
.
(2006)
Robust control and filtering of singular systems.
-
4)
-
S. Ma ,
E.K. Boukas
.
A singular system approach to robust sliding mode control for uncertain Markov jump systems.
Automatica
,
11 ,
2707 -
2713
-
5)
-
L. Dai
.
(1989)
Singular control systems.
-
6)
-
J. Zhang ,
Y. Xia ,
E.K. Boukas
.
New approach to H∞ control for Markovian jump singular systems.
IET Control Theory Appl.
,
2273 -
2284
-
7)
-
S. Ma ,
E.K. Boukas
.
Stability and robust stabilisation for uncertain discrete stochastic hybrid singular systems with time delay.
IET Control Theory Appl.
,
9 ,
1217 -
1225
-
8)
-
E.K. Boukas
.
(2008)
Control of singular systems with random abrupt changes.
-
9)
-
G. Lu ,
D.W.C. Ho ,
L. Zhou
.
A note on the existence of a solution and stability for Lipschitz discrete-time descriptor systems.
Automatica
,
1525 -
1529
-
10)
-
J. Lam ,
Z. Shu ,
S. Xu ,
E.K. Boukas
.
Robust H∞ control of descriptor discrete-time Markovian jump systems.
Int. J. Control
,
3 ,
374 -
385
-
11)
-
Z. Shu ,
J. Lam ,
J. Xiong
.
Static output-feedback stabilization of discrete-time Markovian jump linear systems: a system augmentation approach.
Automatica
,
4 ,
687 -
694
-
12)
-
N. Yi ,
D. Yang
.
H∞ control for nonlinear discrete singular systems.
J. Northeastern Univ. (Natural Science)
,
6 ,
773 -
776
-
13)
-
Y. Cao ,
J. Lam
.
Robust H∞ control of uncertain Markovian jump systems with time-delay.
IEEE Trans. Autom. Control
,
1 ,
77 -
83
-
14)
-
M. Abbaszadeh ,
H.J. Marquez
.
LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems.
Int. J. Robust Nonlinear Control
,
3 ,
313 -
340
-
15)
-
S. Xu ,
J. Lam
.
Robust stability and stabilization of discrete-time singular systems: an equivalent characterization.
IEEE Trans. Autom. Control
,
4 ,
568 -
574
-
16)
-
Z. Wu ,
P. Shi ,
H. Su ,
J. Chu
.
l2-l∞ filter design for discrete-time singular Markovian jump systems with time-varying delays.
Inf. Sci.
,
24 ,
5534 -
5547
-
17)
-
Y. Zhang ,
J. Yang ,
Q. Zhang
.
Robust stabilization of a class of Markovian jump systems.
Int. J. Inf. Syst. Sci.
,
1 ,
141 -
154
-
18)
-
Z. Wu ,
H. Su ,
J. Chu
.
H∞ filtering for singular Markovian jump systems with time delay.
Int. J. Robust Nonlinear Control
,
8 ,
939 -
957
-
19)
-
Y. Xia ,
E.-K. Boukas ,
P. Shi ,
J. Zhang
.
Stability and stabilization of continuous-time singular hybrid systems.
Automatica
,
6 ,
1504 -
1509
-
20)
-
P. Shi ,
E.K. Boukas ,
K. Agarwal
.
Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay.
IEEE Trans. Autom. Control
,
11 ,
2139 -
2144
-
21)
-
S. Ma ,
C. Zhang ,
S. Zhu
.
Robust stability for discrete-time uncertain singular Markov jump systems with actuator saturation.
IET Control Theory Appl.
,
2 ,
255 -
262
-
22)
-
L.X. Zhang ,
E.K. Boukas
.
Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities.
Automatica
,
463 -
468
-
23)
-
G. Lu ,
D.W.C. Ho
.
Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation.
IEEE Trans. Autom. Control
,
5 ,
818 -
823
-
24)
-
S. Ma ,
C. Zhang ,
X. Liu
.
Robust stability and H∞ control for uncertain discrete-time Markovian jump singular systems.
J. Control Theory Appl.
,
2 ,
133 -
140
-
25)
-
S. Xu ,
T. Chen ,
J. Lam
.
Robust H∞ filtering for a class of nonlinear discrete-time Markovian jump systems.
J. Optim. Theory Appl.
,
3 ,
651 -
668
-
26)
-
L. Ma ,
Z. Wang ,
Y. Bo ,
Z. Guo
.
Finite-horizon H2/H∞ control for a class of nonlinear Markovian jump systems with probabilistic sensor failures.
Int. J. Control
,
11 ,
1847 -
1857
-
27)
-
I.R. Petersen
.
A stabilization algorithm for a class of uncertain linear systems.
Syst. Control Lett
,
4 ,
351 -
357
-
28)
-
D.W.C. Ho ,
G. Lu
.
Robust stabilization for a class of discrete-time non-linear systems via output feedback: the unified LMI approach.
Int. J. Control
,
2 ,
105 -
115
-
29)
-
E. Fridman ,
U. Shaked
.
An improved stabilization method for linear time-delay systems.
IEEE Trans. Autom. Control
,
11 ,
1931 -
1937
-
30)
-
H. Dong ,
Z. Wang ,
D.W.C. Ho ,
H. Gao
.
Robust H∞ filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case.
IEEE Trans. Signal Process.
,
7 ,
3048 -
3057
-
31)
-
Y. Xia ,
P. Shi ,
G. Liu ,
Rees D. David
.
Robust mixed H2/H∞ state-feedback control for continuous-time descriptor systems with parameter uncertainties.
Circuits Syst. Signal Process.
,
4 ,
431 -
443
-
32)
-
O.L.V. Costa ,
M.D. Fragoso ,
R.P. Marques
.
(2005)
Discrete-time Markovian jump linear systems.
-
33)
-
G. Lu ,
D.W.C. Ho
.
Generalized quadratic stabilization for discrete-time singular systems with time-delay and nonlinear perturbation.
Asian J. Control
,
3 ,
211 -
222
-
34)
-
V.A. Yakubovich
.
The S-procedure in nonlinear control theory.
Vestnik Leningrad Univ. Math.
,
73 -
93
-
35)
-
Y. Xia ,
J. Zhang ,
E.K. Boukas
.
Control for discrete singular hybrid systems.
Automatica
,
2635 -
2641
-
36)
-
W. Zhou ,
H. Lu ,
C. Duan ,
M. Li
.
Delay-dependent robust control for singular discrete- time Markovian jump systems with time-varying delay.
Int. J. Robust Nonlinear Control
,
10 ,
1112 -
1128
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