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An H-infinifty variable structure control is presented for singular Markov switched systems with mismatched mismatched norm-bounded uncertainties and mismatched norm-bounded external disturbances. It is shown that the sliding mode dynamics on the given
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Journal of Control Theory and Applications 2007 5 (4) 415–419 DOI 10.1007/s11768-006-6072-5
An LMI-based variable structure control for a class
of uncertain singular Markov switched systems
Lijun GAO, Yuqiang WU
(Institute of Automation, Qufu Normal University, Rizhao Shandong 276826, China )
Abstract: An H-infinifty variable structure control is presented for singular Markov switched systems with mis-
matched norm-bounded uncertainties and mismatched norm-bounded external disturbances. It is shown that the sliding
mode dynamics on the given switching surface is regular, impulse-free, and stochastically stable and satisfies H-infinity
performance. A variable structure controller is designed to guarantee that the system trajectory converges to the linear
switching surface in some finite time. Finally, a numerical example is solved to show the effectiveness and validness of the
theoretical results.
Keywords: Markov switched systems; Variable structure; LMI; Switching surface
1 Introduction
Many practical systems that are subject to abrupt changes
such as component and/or interconnection failure or random
communication delays in automobiles can be modeled by a
class of Markov jump systems. The stability analysis and
control problems of these systems have attracted extensive
attention from many researchers [1, 2]. The singular sys-
tem represents a variety of practical systems like electri-
cal circuits, mechanical systems, robotics, etc. [3]. During
the past decades, considerable attention has been devoted
to the analysis and synthesis of linear singular systems [4].
Variable structure systems (VSS) are well known for their
robustness to system parameter variations and external dis-
turbances [5, 6]. For uncertain delay-free systems with mis-
matched uncertainties, a switching surface design method is
given [7]. An H
∞
control problem is discussed for uncertain
Markov switched time-delay systems with mismatched un-
certainties in [8]. For singular systems with Markov switch-
ing, although some results have been given [9], the H
∞
vari-
able structure control problem for the Markov switched sys-
tem is still open. This motivates our research.
The systems studied in this paper are with mis-
matched norm-bounded uncertainties and mismatched
norm-bounded external disturbances. By LMI method, a
sufficient condition is given to guarantee the existence of
linear switching surfaces such that the sliding mode dynam-
ics restricted to the surface is regular, impulse-free, and sto-
chastically stable and satisfies H
∞
performance. In addi-
tion, a variable structure controller is designed to guarantee
that the system trajectory converges to the switching sur-
face.
2 Problem statement
Consider an uncertain singular system defined in a fun-
damental probability space(Ω, F, {F
t
}
t>0
, P )
E ˙x(t) = (A
σ (t)
+∆A
σ (t)
(t))x(t)+(B+∆B
σ (t)
(t))u(t)
+ (F
σ( t)
+ ∆F
σ (t)
(t))Ω(t), (1)
z(t) = C
σ (t)
x(t),
where x(t) ∈ R
n
is state, u(t) ∈ R
m
is the control in-
put, z(t) ∈ R
q
is the output. {σ(t), t > 0} is a con-
tinuous time Markov process taking values in a finite set
N = {1, 2, · · · , N }. Ω(t) ∈ R
p
is the square-integrable
external disturbance satisfying kΩ(t)k 6 h, where h > 0.
A
i
, B and F
i
are known constant matrices with appro-
priate dimensions, rankB = m, rankE = n
E
< n.
∆A
i
(t), ∆B
i
(t), ∆F
i
(t) are real-valued time-varying ma-
trix functions representing norm-bounded parameter uncer-
tainties and satisfy
[
∆A
i
(t) ∆B
i
(t) ∆F
i
(t)
] = M
i
D
i
(t)[
N
1i
N
2i
N
3i
],
(2)
whereM
i
and N
ji
, j = 1, 2, 3 are known constant real ma-
trices with appropriate dimensions, and D
i
(t) is an un-
known time-varying matrix function with Lebesgue mea-
surable elements satisfying
D
T
i
(t)D
i
(t) 6 I. (3)
From (2) and (3), it is easily obtained that
k∆A
i
(t)k 6 kM
i
kkN
1i
kkD
i
(t)k 6 kM
i
kkN
1i
k 6 α
1
,
where α
1
= kM
i
kkN
1i
k. Similarly, we have
k∆B
i
(t)k 6 α
2
, k∆F
i
(t)k 6 α
3
. (4)
Received 11 April 2006; revised 27 April 2007.
This work was supported by the National Natural Science Foundation of China (No.60574007, 60674027).
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