This paper is concerned with the delay-dependent H∞ fuzzy static output feedback control scheme for discrete-time Takagi–Sugeno (T–S) fuzzy stochastic systems with distributed time-varying delays. To begin with, the T–S fuzzy stochastic system is transformed to an equivalent switching fuzzy stochastic system. Then, based on novel matrix decoupling technique, improved free-weighting matrix technique and piecewise Lyapunov–Krasovskii function (PLKF), a new delay-dependent H∞ fuzzy static output feedback controller design approach is first derived for the switching fuzzy stochastic system. Some drawbacks existing in the previous papers such as matrix equalities constraint, coordinate transformation, the same output matrices, diagonal structure constraint on Lyapunov matrices and BMI problem have been eliminated. Since only a set of LMIs is involved, the controller parameters can be solved directly by the Matlab LMI toolbox. Finally, two examples are provided to illustrate the validity of the proposed method.

1. ISA Transactions 51 (2012) 702–712
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Delay-dependent fuzzy static output feedback control for discrete-time fuzzy
stochastic systems with distributed time-varying delays$
ZhiLe Xia a,b,n, JunMin Li a, JiangRong Li a
a
b
School of Science, Xidian University, ShanXi, Xi’an 710071, China
School of Mathematics and Information Engineering, Taizhou University, ZheJiang, Taizhou 317000, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 18 April 2010
Received in revised form
19 June 2012
Accepted 20 June 2012
Available online 12 July 2012
This paper is concerned with the delay-dependent H1 fuzzy static output feedback control scheme for
discrete-time Takagi–Sugeno (T–S) fuzzy stochastic systems with distributed time-varying delays. To
begin with, the T–S fuzzy stochastic system is transformed to an equivalent switching fuzzy stochastic
system. Then, based on novel matrix decoupling technique, improved free-weighting matrix technique
and piecewise Lyapunov–Krasovskii function (PLKF), a new delay-dependent H1 fuzzy static output
feedback controller design approach is first derived for the switching fuzzy stochastic system. Some
drawbacks existing in the previous papers such as matrix equalities constraint, coordinate transformation, the same output matrices, diagonal structure constraint on Lyapunov matrices and BMI problem
have been eliminated. Since only a set of LMIs is involved, the controller parameters can be solved
directly by the Matlab LMI toolbox. Finally, two examples are provided to illustrate the validity of the
proposed method.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords:
Piecewise Lyapunov–Krasovskii function
(PLKF)
Linear matrix inequalities (LMIs)
Fuzzy static output feedback
Switching fuzzy stochastic system
Delay-dependent
1. Introduction
The well-known Takagi–Sugeno (T–S) fuzzy model [1] has
been recognized as a popular and powerful tool in approximating
complex nonlinear systems. As a consequence, the study of T–S
fuzzy systems has attracted an increasing interest in the past
decades (see [2–6] for more details). Moreover, system state
variables are often not fully available for practical systems. Some
state variables may be difficult/costly to measure and sometimes
have no physical meaning and thus cannot be measured at all. It
may be possible to use an observer to estimate unknown states,
but this approach not only requires more hardware resources, but
also makes the dimension of the system increase greatly. In this
situation, the static output feedback (SOF) control is more suitable
for practical application. Syrmos et al. [20] show that any
dynamic output feedback [29] problem can be transformed into
a SOF problem, but the converse is not true.
Recently, the fuzzy SOF control for T–S fuzzy systems has
drawn intensive attention [18,19,21,23–28]. As stated in
[16,17,25], the fuzzy SOF controller design for T–S fuzzy system
$
This work was supported by the National Natural Science Foundation of China
(60974139) and Fundamental Research Funds for the Central Universities
(72103676).
n
Corresponding author at: School of Mathematics and Information Engineering,
Taizhou University, ZheJiang, Taizhou 317000, China. Tel.: þ 86 13736515790.
E-mail address: zhle_xia@yahoo.com.cn (Z. Xia).
is not easy, due to the fact that many rules interference effects are
increased. Moreover, the fuzzy SOF control design becomes much
more difficult and complex than state feedback one because it
belongs to a nonlinear matrix inequalities problem. In [19], the
fuzzy SOF control law was given in terms of bilinear matrix
inequalities which cannot be solved with ease by a convex
optimization algorithm. An iterative linear matrix inequality
algorithm was proposed in [21,27] and an optimization technique
was investigated in [26]. The problem of designing robust SOF
controllers for linear discrete-time systems with time-varying
polytopic uncertainties was studied in [22]. In [23], three drawbacks existing in the previous papers such as coordinate transformation, the same output matrices and BMI problem have been
eliminated. But linear matrix equalities (LMEs) (i.e. MC yi ¼
C yi P, i ¼ 1, . . . ,r) were included in [23]. By using coordination
transformation, a less conservative result was derived in [24] by
removing the constraint that the considered Lyapunov matrix is
diagonal. In [24], all the local output matrices were assumed to be
the same (i.e. C yi ¼ C 2 , i ¼ 1, . . . ,r). If the local output matrices are
not the same, coordinate transformation is almost impossible.
Hybrid approaches for regional T–S fuzzy SOF controller design
was considered in [25]. The authors in [28] investigated the
reliable mixed L2 =H1 fuzzy SOF control for nonlinear systems
with sensor faults.
On the other hand, time delay often occurs in many dynamic
systems, such as rolling mill systems, biological systems, metallurgical processing systems, network systems, and so on. Their
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.06.011

2. Z. Xia et al. / ISA Transactions 51 (2012) 702–712
existence is frequently a cause of instability and poor performance. The T–S fuzzy model was first used to deal with the
stability analysis and control synthesis of nonlinear time delay
systems in [7]. After that, many people devoted a great deal of
effort to both theoretical research and implementation techniques for T–S fuzzy systems with time delays [8,9]. However, these
results rely on the existence of a common positive definite matrix
P for all linear models, which in general leads to a conservative
result. To reduce this conservatism, the piecewise Lyapunov
function approach [10,11] and the fuzzy Lyapunov function
approach [12,13] have been proposed. The delay-independent
SOF stabilization for T–S fuzzy system with interval time-delay
was investigated in [18] by using the common Lyapunov function
approach, which typically led to conservative results. Moreover,
as well known as, the delay-independent results [18] are generally more conservative than the delay-dependent ones [8],
especially when the size of time delay is small [10–13].
It should be noted that all the above-mentioned results are in
the deterministic setting. It is well known that stochastic disturbance often exist in many practical systems. Their existence is
a source of instability of control systems, so the investigation on
stability analysis and control design of stochastic systems has
received increasing attention in recent years [31,32,37,38]. It
should be pointed out that, so far, there have been only a few
papers that have taken the stochastic phenomena into account in
the fuzzy systems [30–37,39]. Research in this area should be
interesting yet challenging as it involves the combination of two
classes of important systems, namely, stochastic systems and
fuzzy systems [37]. The problem of delay-dependent robust fuzzy
control for stochastic fuzzy systems with parameter uncertainties
and time-varying delay was studied in [31]. Robust stability
analysis for uncertain stochastic fuzzy systems with time delay
was investigated in [32], and some delay-independent stability
conditions were derived. The authors in [33] investigated the
passivity and passification problems of the stochastic discretetime T–S fuzzy systems with delay. A sliding mode control
approach was proposed in [35] to investigate the fuzzy control
problem for uncertain stochastic systems. The problem of stabilization for T–S fuzzy stochastic delay systems was considered in
[36,39]. The filtering problem was investigated in [30,34,37]. In
addition, the distributed delays in discrete-time systems [40–43]
are seldom found in the literature when compared with the
distributed delay in continuous-time systems that are described
in the form of either finite or infinite integral [44]. As pointed in
[40,41], a large-scale system network usually has a spatial nature
due to the presence of an amount of parallel pathways of a variety
of subsystem sizes and lengths, which gives rise to possible
distributed delays for discrete-time systems. With the increasing
application of digitalization, distributed delays may emerge in a
discrete-time manner, and therefore, it becomes desirable to
study the discrete-time systems with distributed delays. The
research on discrete-time fuzzy systems with distributed delays
remains as an open topic for further investigation. To the best of
the authors’ knowledge, there have not been any reported results
on the fuzzy SOF control for discrete-time fuzzy stochastic system
with distributed time-varying delay.
Motivated by the aforementioned discussion, we investigate
the H1 fuzzy SOF control for discrete-time T–S fuzzy stochastic
systems with distributed time-varying delays. By using improved
free-weighting technique and piecewise Lyapunov–Krasovskii
function, a new delay-dependent H1 fuzzy SOF controller design
approach is derived. Adopting a new matrix decoupling technique, some drawbacks existing in the previous papers such as
coordinate transformation, the same output matrices, the diagonal constraint condition on Lyapunov matrices, matrix equalities
constraint and BMI problem have been eliminated. Since only a
703
set of LMIs is involved, the fuzzy SOF controller parameters can be
solved directly. Finally, two examples are given to illustrate the
validity of the proposed method.
Notation. For a real matrix S, He fSg denotes S þ ST . The symmetric elements of the symmetric matrix will be denoted by n.
Efng stands for the mathematical expectation of the stochastic
process or vector. I and 0 represent, respectively, identity matrix
and zero matrix.
We first introduce the following lemmas, which are crucial to
the development of our main results.
Lemma 1. For the real positive-definite symmetric matrix P1,
matrices A1, A2, B1, B2, D1 ,D2 , D3 and P2 with compatible dimensions,
the following inequalities are equivalent, where aI are extra slack
nonsingular matrix:
" T
#
A1 P 1 A1 þ BT P 1 B1 þAT A2 þ BT B2 ÀHe fAT P 2 g þ D1 D2
1
2
2
1
ðaÞ
o0,
n
D3
2
0
0
0
ÀaB1
n
P1 À2aI
0
0
ÀP 2 þ aA1
n
n
ð1À2aÞI
0
ÀaA2
n
n
n
ð1À2aÞI
n
n
n
n
ÀaB2
D1
n
ðbÞ
6
6
6
6
6
6
6
6
6
4
P1 À2aI
n
n
n
n
0
3
7
0 7
7
0 7
7
7 o0:
0 7
7
D2 7
5
D3
Proof. See the Appendix.
Remark 1. By the introduction of the auxiliary slack matrix
variable aI, the matrices P1, P2 and A1 , A2 , B1 , B2 are decoupled.
This novel technique is proposed in this paper to transform the
nonlinear matrix inequalities (30) into a set of linear matrix
inequalities (LMIs), which can be easily solved by LMI Toolbox in
Matlab.
Lemma 2 (Wei et al. [40,41]). Let M A RnÂn be a positive semidefinite matrix, xðmÞ A Rn , we have
!T
!
t
t
t
X
X
1 X
xT ðmÞMxðmÞ r À
xðmÞ M
xðmÞ :
À
t
m¼1
m¼1
m¼1
Lemma 3 (Tian et al. [46] and Lin et al. [47]). Let L, L1 A RpÂp be
symmetric matrices. Then
L þ tðtÞL1 o0
holds for all tðtÞ A ½t1 , t2 Š if and only if
L þ tn L1 o0, n ¼ 1; 2:
2. Problem formulation
Consider the following discrete-time Takagi–Sugeno (T–S)
fuzzy stochastic system with distributed time-varying delays:
Plant Rule i: IF y1 ðtÞ is mi1 and y and yp ðtÞ is mip THEN
"
#
tÀ1
X
xðnÞ þ Bi oðtÞ þB1i uðtÞ
xðt þ1Þ ¼ Ai xðtÞ þ Adi
n ¼ tÀtðtÞ
"
tÀ1
X
þ M i xðtÞ þ M di
#
xðnÞ þ B2i uðtÞ vðtÞ,
n ¼ tÀtðtÞ
"
zðtÞ ¼ C i xðtÞ þ C di
"
tÀ1
X
#
xðnÞ þ Di oðtÞ þ D1i uðtÞ
n ¼ tÀtðtÞ
þ N i xðtÞ þ Ndi
tÀ1
X
n ¼ tÀtðtÞ
#
xðnÞ þ D2i uðtÞ vðtÞ,

3. 704
Z. Xia et al. / ISA Transactions 51 (2012) 702–712
yðtÞ ¼ Gi xðtÞ,
t ¼ Àt2 ,Àt2 þ1, . . . ,0, i ¼ 1; 2, . . . ,r,
xðtÞ ¼ fðtÞ,
ð1Þ
where r is the number of IF–THEN rules; yj and mij ðj ¼ 1; 2, . . .Þ are
the premise variables and the fuzzy sets, respectively;
xðtÞ A Rn , yðtÞ A Rn1 , zðtÞ A Rn2 and fðtÞ are the state, the measured
output, the controlled output to be estimated, and the initial
condition, respectively; oðtÞ is the disturbance input which
belongs to l2 ½0,1Þ; tðtÞ denotes distributed time-varying delays
and satisfies 0 o t1 r tðtÞ r t2 ; Ai , Adi , Bi , M i , M di , C i , C di , Di , Ni ,
Ndi , B1i , B2i , D1i , D2i and Gi are all constant matrices with appropriate dimensions; v(t) is a real scalar stochastic process with
Efv2 ðtÞg ¼ 1,
EfvðtÞg ¼ 0,
EfvðlÞvðmÞg ¼ 0 ðl amÞ:
The final output of the switching fuzzy stochastic system (3) is
inferred as follows:
("
#
bðjÞ
tÀ1
X
X
xðt þ1Þ ¼
hjk Ajk xðtÞ þ Adjk
xðnÞ þ Bjk oðtÞ þ B1jk uðtÞ
"
tÀ1
X
þ M jk xðtÞ þ M djk
#
)
xðnÞ þ B2jk uðtÞ vðtÞ ,
n ¼ tÀtðtÞ
("
bðjÞ
X
zðtÞ ¼
hjk
tÀ1
X
C jk xðtÞ þ C djk
#
xðnÞ þDjk oðtÞ þ D1jk uðtÞ
n ¼ tÀtðtÞ
k¼1
"
tÀ1
X
þ N jk xðtÞ þ Ndjk
ð2Þ
Remark 2. As mentioned in introduction, the research on discrete-time fuzzy systems with distributed delays remains as an
open topic for further investigation. In [40,41,45], the authors
introduced the distributed delays in the form of constant delay
P1
PÀ1
t ¼ Àd hðxðk þ tÞÞ or infinite delay
t ¼ 1 xðtÀtÞ. The authors in
[42,43] investigated the distributed time-varying delays in the
PÀ1
form of
m ¼ ÀdðkÞ f ðxðk þmÞÞ, where nonlinear function f ðÞ was
assumed to satisfy sector-bounded condition. For the sake of
simplicity, this paper studies the distributed time-varying delay
PtÀ1
in the form of
m ¼ tÀtðtÞ xðmÞ and a new delay-dependent H 1
fuzzy static output feedback controller design approach is proposed for discrete-time T–S fuzzy stochastic system. In contrast to
aforementioned results, two new techniques, improved freeweighting technique and new matrix decoupling technique, are
developed.
n ¼ tÀtðtÞ
k¼1
#
)
xðnÞ þD2jk uðtÞ vðtÞ ,
n ¼ tÀtðtÞ
bðjÞ
X
yðtÞ ¼
hjk Gjk xðtÞ,
k¼1
xðtÞ ¼ fðtÞ,
t ¼ Àt2 ,Àt2 þ 1, . . . ,0,
yðtÞ A Oj , j ¼ 1, . . . ,s,
ð4Þ
where
Qp
mjkl ðyl ðtÞÞ
:
m ðy ðtÞÞ
l ¼ 1 jkl l
hjk ¼ hjk ðyðtÞÞ ¼ PbðjÞ l ¼ 1p
Q
k¼1
Considering the fuzzy system (3) in each subregion, we choose
the following fuzzy SOF controller:
Region Rule j:
IF yðtÞ A Oj THEN
Local Plant Rule k
IF y1 ðtÞ is mjk1 , . . . , and yp ðtÞ is mjkp THEN
uðtÞ ¼ ÀF jk yðtÞ,
k ¼ 1, . . . , bðjÞ, j ¼ 1, . . . ,s,
ð5Þ
As illustrated in [11], we will define open subregions as
Op ðp ¼ 1, . . . ,sÞ in the state-space. The corresponding close subregions are defined as O p , which satisfy
where Fjk is a local controller gain to be designed.
The final output of (5) is inferred by
O p O q ¼ @Ou , p a q, p,q ¼ 1, . . . ,s, i ¼ 1; 2, . . . ,r,
i
uðtÞ ¼ À
n ¼ tÀtðtÞ
tÀ1
X
þ M jk xðtÞ þM djk
#
tÀ1
X
#
þ Njk xðtÞ þ N djk
tÀ1
X
xðnÞ þ D2jk uðtÞ vðtÞ,
yðtÞ ¼ Gjk xðtÞ,
t ¼ Àt2 ,Àt2 þ1, . . . ,0, i ¼ 1; 2, . . . ,r,
xðt þ1Þ ¼
bðjÞ bðjÞ bðjÞ
XX X
hjk hjl hjm ½f jklm xðtÞ þ g jklm xðtÞvðtÞŠ
k¼1l¼1m¼1
9f jklm xðtÞ þ g jklm xðtÞvðtÞ,
zðtÞ ¼
bðjÞ bðjÞ bðjÞ
XX X
hjk hjl hjm ½f zjklm xðtÞ þ g zjklm xðtÞvðtÞŠ
k¼1l¼1m¼1
9f zjklm xðtÞ þg zjklm xðtÞvðtÞ,
t ¼ Àt2 ,Àt2 þ1, . . . ,0,
j ¼ 1, . . . ,s, yðtÞ A Oj ,
ð7Þ
Definition 1 (Gao et al. [38]). The closed-loop system (7) is said
to be mean-square stable if, under oðtÞ ¼ 0, for any E 40, there is
2
a dðEÞ 4 0 such that Ef9xðtÞ9 g o E,t 4 0 when supÀt2 r s r 0
2
2
Ef9xðsÞ9 g o dðEÞ. In addition, if limt-1 Ef9xðtÞ9 g ¼ 0 for any initial
conditions, then it is said to be mean-square asymptotically
stable.
#
n ¼ tÀtðtÞ
xðtÞ ¼ fðtÞ,
j ¼ 1, . . . ,s, yðtÞ A Oj :
g jklm ¼ ½M jk ÀB2jk F jl Gjm
where
f jklm ¼ ½Ajk ÀB1jk F jl Gjm Adjk Bjk Š,
Mdjk 0Š, f zjklm ¼ ½C jk ÀD1jk F jl Gjm C djk Djk Š, g zjklm ¼ ½Njk ÀD2jk F jl Gjm
PtÀ1
T
T
T
Ndjk 0Š, xðtÞ ¼ ½xT ðtÞ
n ¼ tÀtðtÞ x ðnÞ o ðtÞŠ .
xðnÞ þ Djk oðtÞ þ D1jk uðtÞ
n ¼ tÀtðtÞ
hjl hjm F jl Gjm xðtÞ,
l¼1m¼1
Applying the fuzzy SOF controller (6) to the global fuzzy
system (4), we have the following closed-loop system:
T
n ¼ tÀtðtÞ
zðtÞ ¼ C jk xðtÞ þC djk
bðjÞ bðjÞ
X X
ð6Þ
xðtÞ ¼ f ðtÞ,
xðnÞ þ B2jk uðtÞ vðtÞ,
hjl F jl yðtÞ ¼ À
l¼1
@Ou
i
¼ fyðtÞ9hi ðyðtÞÞ ¼ 1; 0 r hi ðyðtÞ þ EÞ o 1,80 o 9E9 51g, u is
where
the set of face indexes of the polyhedral hull @Oi ¼ [@Ou ,
i
P
hi ðyðtÞÞ ¼ oi ðyðtÞÞ= r ¼ 1 oi ðyðtÞÞ, oi ðyðtÞÞ ¼ Pp¼ 1 mij ðyj ðtÞÞ, yðtÞ ¼
i
j
½y1 ðtÞ, . . . , yp ðtÞŠ.
Next, we follow the idea of [14] to rewrite the system (1) to be
an equivalent discrete-time switching fuzzy stochastic system as
the following form:
Region Rule j:
IF yðtÞ A Oj THEN
Local Plant Rule k
IF y1 ðtÞ is mjk1 , . . ., and yp ðtÞ is mjkp THEN
#
tÀ1
X
xðnÞ þBjk oðtÞ þB1jk uðtÞ
xðt þ 1Þ ¼ Ajk xðtÞ þ Adjk
bðjÞ
X
ð3Þ
where Oj denotes the jth subregion, s is the number of subregions
partitioned on the state space, and bðjÞ is the number of rules in
the subregion Oj .
Definition 2 (Gao et al. [38]). Given g 40, the closed-loop system
(7) is said to be mean-square asymptotically stable with H1
performance g if it is mean-square asymptotically stable and, under
zero-initial conditions, for all nonzero disturbance oðtÞ A l2 ½0,1Þ,

4. Z. Xia et al. / ISA Transactions 51 (2012) 702–712
satisfies
EfJzðtÞJ2 g o gJoðtÞJ2 ,
ð8Þ
where
8
!1=2 9
X
=
1
,
zT ðtÞzðtÞ
EfJzðtÞJ2 g9E
: t¼0
;
f jklm ¼ ½aj Ajk ÀB1jk U jl Gjm aj Adjk aj Bjk Š,
gjklm ¼ ½aj M jk ÀB2jk U jl Gjm aj M djk aj Bjk Š,
f zjklm ¼ ½aj C jk ÀD1jk U jl Gjm aj C djk aj Djk Š,
gzjklm ¼ ½aj Njk ÀD2jk U jl Gjm aj N djk 0Š:
1
X
JoðtÞJ2 9
705
oT ðtÞoðtÞÞ1=2 :
Moreover, the switching fuzzy SOF controllers are given by
t¼0
ð9Þ
The main purpose of this paper is to design fuzzy SOF
controller of the form (6) such that the closed-loop system
(7) is mean-square asymptotically stable with an H1 performance
g.
F jl ¼ aÀ1 U jl ,
j
l ¼ 1, . . . , bðjÞ, j ¼ 1, . . . ,s:
Proof. Let
ZðnÞ ¼ xðn þ1ÞÀxðnÞ:
ð14Þ
Then,
I
ZðtÞ ¼ xðt þ 1ÞÀxðtÞ9f jklm xðtÞ þ g jklm xðtÞvðtÞ,
3. Main results
In this section, the delay-dependent H1 fuzzy SOF controller
design approach is presented for the T–S fuzzy stochastic systems
described in the previous section. Let the subregion transition
from Oj to Oi be denoted by O ¼ fðj,iÞ9yðtÞ A Oj , yðt þ 1Þ A Oi g. Here, i
may be equal to j in O, when yðtÞ and yðt þ1Þ are in the same
subregion. Consequently, we have the following result.
Theorem 1. Given a constant g 40, the closed-loop system (7) is
mean-square asymptotically stable with H1 performance g, if there exist
a set of positive-definite symmetric matrices Pj, Z 1 , Z 2 , Q 1 , Q 2 , Q 3 ,
matrices Ujl, Tj, Wj, Sj, Xj, Yj and the nonsingular matrices aj I, j¼1,y,s,
l ¼ 1, . . . , bðjÞ, such that the following LMIs are satisfied:
Pn o0, l ¼ 1, . . . , bðjÞ, ðj,iÞ A O, n ¼ 1; 2,
ijlll
Xj
Tj
n
Z1
#
Yj
Sj
n
4 0,
Z2
ð11Þ
V 2 ðtÞ ¼
t2
X
tÀ1
X
m ¼ 1 n ¼ tÀm
V 3 ðtÞ ¼
tðtÞ
X
tÀ1
X
t2
X
xT ðnÞQ 1 xðnÞ þ
ZT ðnÞZ 2 ZðnÞ,
mÀ1
X
tÀ1
X
xT ðsÞQ 1 xðsÞ,
m ¼ t1 þ 1 n ¼ 1 s ¼ tÀn
xT ðmÞQ 2 xðmÞ þ
tÀ1
X
xT ðmÞQ 3 xðmÞ,
yðtÞ A Oj ,
m ¼ tÀt2
where Pj, Q 1 ,Q 2 ,Q 3 , Z1 and Z2 are the real symmetric positive
definite matrices to be determined. Then along the solution of the
closed-loop system (7), we have
#
Xj þ Y j
Wj
Z1 þ Z2
¼ EfDV 1 ðtÞ9xðtÞg þ EfDV 2 ðtÞ9xðtÞg þ EfDV 3 ðtÞ9xðtÞg þ EfDV 4 ðtÞ9xðtÞg,
where
40,
ð13Þ
EfDV 1 ðtÞ9xðtÞg ¼ Ef½xT ðt þ 1ÞP i xðt þ 1ÞÀxT ðtÞPj xðtÞŠ9xðtÞg
T
T
T
¼ Ef½x ðtÞf jklm Pi f jklm xðtÞ þ2vT ðtÞx ðtÞg T P i f jklm xðtÞ
jklm
where
2
6
6
6
6
n
Pijklm ¼ 6
6
6
6
6
4
P1 À2aj I
n
0
P1 À2aj I
n
n
0
0
ð1À2aj ÞI
n
n
n
0
0
0
ð1À2aj ÞI
n
n
n
n
n
n
n
D1
n
Àgjklm
ÀP2 þ f jklm
Àf zjklm
Àgzjklm
n
0
0
0
0
3
7
7
7
7
7
7,
7
7
D2 7
5
D3
T
þ vT ðtÞx ðtÞg T Pi g jklm xðtÞvðtÞÀxT ðtÞPj xðtÞŠ9xðtÞg,
jklm
ð16Þ
ÈÂ
EfDV 2 ðtÞ9xðtÞg ¼ E ZT ðtÞðt2 Z 1 þ t21 Z 2 ÞZðtÞ
À
tÀ1
X
ZT ðnÞZ 1 ZðnÞÀ
n ¼ tÀt2
with
¼E
t21 ¼ t2 Àt1 , n ¼ 1; 2, P1 ¼ Pi þ t2 Z 1 þ t21 Z 2 ,
P2 ¼ ðt2 Z 1 þ t21 Z 2 Þ½I 0 0Š,
2
D11 D12
D22
D1 ¼ 6 n
4
n
n
0
3
0 7
5,
Àg2 I
0
Sj
D2 ¼ B 0
@
0
ÀW j
1
C
0 A,
0
D3 ¼ diagfÀQ 2 ,ÀQ 3 g,
D11 ¼ t2 Q 1 þ 1t21 ðt2 þ t1 À1ÞQ 1 þ Q 2 þQ 3 þ t2 X j þ t21 Y j
2
þ He fT j g þ t2 Z 1 þ t21 Z 2 ÀPj ,
1
tn
ðW j ÀT j ÀSj Þ,
D22 ¼ À
1
tn
Q 1,
tÀt1 À1
X
n ¼ tÀt2
(
D12 ¼
tÀ1
X
m ¼ t1 þ 1 n ¼ tÀm
m ¼ 1 n ¼ tÀm
tÀ1
X
t2
X
ZT ðnÞZ 1 ZðnÞ þ
J9EfVðt þ1Þ9xðtÞgÀVðtÞ ¼ Ef½Vðt þ 1ÞÀVðtÞŠ9xðtÞg
ð12Þ
n
4 0,
V 1 ðtÞ ¼ xT ðtÞPj xðtÞ,
m ¼ tÀt1
m ¼ 1, . . . , bðjÞÀ2, k ¼ m þ 1, . . . , bðjÞÀ1,
l ¼ k þ1, . . . , bðjÞ, ðj,iÞ A O, n ¼ 1; 2,
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ,
V 4 ðtÞ ¼
Pn þ Pn þ Pn þ Pn þ Pn þ Pn o 0,
ijklm
ijkml
ijlmk
ijlkm
ijmlk
ijmlk
#
where f
¼ f jklm À½I 0 0Š.
Consider a discrete-time piecewise Lyapunov–Krasovskii function candidate for the closed-loop system (7) as following:
ð10Þ
Pn þ Pn þ Pn o0, k ¼ 1, . . . , bðjÞÀ1,
ijkll
ijlkl
ijllk
l ¼ kþ 1, . . . , bðjÞ, ðj,iÞ A O, n ¼ 1; 2,
ð15Þ
I
jklm
tðtÞ
X
ZT ðtÞðt2 Z 1 þ t21 Z 2 ÞZðtÞÀ
tðtÞ
X
tÀt1 À1
X
ZT ðnÞ
m ¼ 1 n ¼ tÀt2
1
Z ZðnÞ
tðtÞ 2
ÈÂ
¼ E ZT ðtÞðt2 Z 1 þ t21 Z 2 ÞZðtÞ
tÀ1
X
ZT ðnÞ
1
tðtÞ
Z 1 ZðnÞ
)
#
xðtÞ
tðtÞ
X tÀt1 À1
X
1
1
Z 1 ZðnÞÀ
Z ZðnÞ
ZT ðnÞ
tðtÞ
tðtÞ 2
m ¼ 1 n ¼ tÀm
m ¼ 1 n ¼ tÀm
)
#
tðtÞ
X
X tÀmÀ1
1
ðZ 1 þ Z 2 ÞZðnÞ xðtÞ
ZT ðnÞ
À
tðtÞ
m ¼ 1 n ¼ tÀt2
nh
I
I
T
¼ E x ðtÞðf jklm ÞT ðt2 Z 1 þ t21 Z 2 Þf jklm xðtÞ
À
tðtÞ
X
)
m ¼ 1 n ¼ tÀt2
À
#
ZT ðnÞZ 2 ZðnÞ xðtÞ
tÀ1
X
T
ZT ðnÞ
I
þ2vT ðtÞx ðtÞg T ðt2 Z 1 þ t21 Z 2 Þf jklm xðtÞ
jklm

5. 706
Z. Xia et al. / ISA Transactions 51 (2012) 702–712
T
þ vT ðtÞx ðtÞg T ðt2 Z 1 þ t21 Z 2 Þg jklm xðtÞvðtÞ
jklm
tðtÞ
X
X tÀt1 À1
1
1
Z 1 ZðnÞÀ
Z ZðnÞ
ZT ðnÞ
tðtÞ
tðtÞ 2
m ¼ 1 n ¼ tÀm
m ¼ 1 n ¼ tÀm
)
#
tðtÞ
X
X tÀmÀ1
1
T
ð17Þ
ðZ þ Z ÞZðnÞ xðtÞ ,
Z ðnÞ
À
tðtÞ 1 2
m ¼ 1 n ¼ tÀt
À
tðtÞ
X
tÀ1
X
#
tðtÞ
tÀ1
tÀ1
X
1
1 X X
w1 ¼ 2x ðtÞT j xðtÞÀ
xðmÞÀ
ZðnÞ ¼ 0,
tðtÞ m ¼ tÀtðtÞ
tðtÞ m ¼ 1 n ¼ tÀm
T
ZT ðnÞ
ð21Þ
w2 ¼ 2xT ðtÞW j
EfDV 3 ðtÞ9xðtÞg ¼ E
ð22Þ
tðt þ 1Þ
X
t
X
T
x ðnÞQ 1 xðnÞ
tðtÞ
X
tÀ1
X
t2
X
tÀ1
X
m ¼ t1 þ 1 n ¼ tÀm þ 1
(
¼E
1
t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ
2
)
#
xT ðnÞQ 1 xðnÞ xðtÞ
tðt þ 1Þ
X
tÀ1
X
T
x ðnÞQ 1 xðnÞ þ
tðt þ 1Þ
X
m ¼ 1 n ¼ t þ 1Àm
À
tðtÞ
X
tÀ1
X
ð23Þ
On the other hand, for any appropriately dimensioned matrices
X j ¼ X T Z 0 and Y j ¼ Y T Z0, the following hold:
j
j
T
x ðnÞQ 1 xðnÞÀ
T
x ðtÞQ 1 xðtÞ
tðtÞ
X
T
x ðtÀmÞQ 1 xðtÀmÞ
xT ðnÞQ 1 xðnÞÀ
1
À
x ðmÞQ 1 xðmÞ r À
tðtÞ
m ¼ tÀtðtÞ
x ðmÞ Q 1
m ¼ tÀtðtÞ
X j xðtÞ
X j xðtÞ
tðtÞ
X
tÀt1 À1
X
tðtÞ
X
xT ðtÞ
m ¼ 1 n ¼ tÀt2
tðtÞ
X
tÀt1 À1
X
xT ðtÞ
tÀmÀ1
X
xT ðtÞ
1
tðtÞ
1
tðtÞ
1
tðtÞ
Y j xðtÞ
Y j xðtÞ
Y j xðtÞ
ð25Þ
From (16)–(25) and considering the closed-loop system (7),
we obtain
J þ EfzT ðtÞzðtÞgÀg2 wT ðtÞwðtÞ ¼ J þ EfzT ðtÞzðtÞg
ð18Þ
tÀ1
X
!
xðmÞ :
ð20Þ
From (14), we have
Àg2 wT ðtÞwðtÞ þ w1 þ w2 þ w3 þ w4 þ w5
1
T
¼ z ðtÞðXijkl þ
CÞzðtÞ
tðtÞ
!T
!
#
tðtÞ
tÀ1
xðtÞ
xðtÞ
Xj T j
1 X X
À
ZðnÞ
n Z1
tðtÞ m ¼ 1 n ¼ tÀm ZðnÞ
!T
!
#
tðtÞ tÀt1 À1
xðtÞ
xðtÞ
Y j Sj
1 X X
À
Â
ZðnÞ
ZðnÞ
n Z2
tðtÞ m ¼ 1 n ¼ tÀm
!T
!
#
tðtÞ tÀmÀ1
xðtÞ
xðtÞ
Xj þ Y j
Wj
1 X X
À
ZðnÞ
n
Z1 þ Z2
tðtÞ m ¼ 1 n ¼ tÀt2 ZðnÞ
1
T
9z ðtÞ Xijklm þ
C zðtÞÀUðxðtÞ, ZðnÞÞ
tðtÞ
ð26Þ
T
xðmÞÀ
m ¼ tÀtðtÞ
tðtÞ
X
tÀ1
X
ZðnÞ,
m ¼ 1 n ¼ tÀm
xðmÞÀtðtÞxðtÀt2 ÞÀ
tðtÞ
X
tÀmÀ1
X
ZðnÞ,
m ¼ 1 n ¼ tÀt2
m ¼ tÀtðtÞ
0 ¼ tðtÞxðtÀt1 ÞÀ
1
tðtÞ
X j xðtÞ
¼ 0:
EfDV 4 ðtÞ9xðtÞg ¼ Ef½xT ðtÞðQ 2 þQ 3 ÞxðtÞÀxT ðtÀt1 ÞQ 2 xðtÀt1 Þ
0¼
1
tðtÞ
xT ðtÞY j xðtÞÀ
m ¼ 1 n ¼ tÀt2
m ¼ tÀtðtÞ
ÀxT ðtÀt2 ÞQ 3 xðtÀt2 ÞŠ9xðtÞg:
xT ðtÞ
m ¼ 1 n ¼ tÀm
ð19Þ
tÀ1
X
xT ðtÞ
n ¼ tÀt2
À
T
tÀ1
X
1
tðtÞ
ð24Þ
tÀt1 À1
X
w5 ¼
xT ðmÞQ 1 xðmÞ
!
tÀ1
X
xT ðtÞ
¼ 0,
m ¼ tÀtðtÞ
T
0 ¼ tðtÞxðtÞÀ
tÀmÀ1
X
m ¼ 1 n ¼ tÀt2
Using Lemma 2, we have
tÀ1
X
tðtÞ
X
À
1
þ t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ
2
#
)
t2
tÀ1
X
X
xT ðnÞQ 1 xðnÞ xðtÞ
À
m ¼ t1 þ 1 n ¼ tÀm þ 1
1
¼ E t2 xT ðtÞQ 1 xðtÞ þ t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ
2
)
#
tÀ1
X
xT ðmÞQ 1 xðmÞ xðtÞ :
À
m ¼ tÀtðtÞ
tÀ1
X
tðtÞ
X
¼ t21 xT ðtÞY j xðtÞÀ
m ¼ 1 n ¼ t þ 1Àm
tÀ1
X
m ¼ 1 n ¼ tÀm
m¼1
tÀ1
X
tðtÞ
X
m ¼ 1 n ¼ tÀt2
¼ t2 xT ðtÞX j xðtÞÀ
m ¼ 1 n ¼ t þ 1Àm
À
xT ðtÞX j xðtÞÀ
n ¼ tÀt2
1
þ t21 ðt2 þ t1 À1ÞxT ðtÞQ 1 xðtÞ
2
)
#
t2
tÀ1
X
X
T
x ðnÞQ 1 xðnÞ xðtÞ
À
m ¼ t1 þ 1 n ¼ tÀm þ 1
(
t2
tÀ1
X
X
rE
xT ðnÞQ 1 xðnÞ þ t2 xT ðtÞQ 1 xðtÞ
tÀ1
X
tÀ1
X
w4 ¼
m¼1
m ¼ 1 n ¼ t þ 1Àm
t1
X
tðtÞ
tÀ1
X
X
1
1 X tÀt1 À1
xðmÞÀ
ZðnÞ ¼ 0:
tðtÞ m ¼ tÀtðtÞ
tðtÞ m ¼ 1 n ¼ tÀm
xT ðnÞQ 1 xðnÞ: þ
m ¼ 1 n ¼ tÀm
À
#
w3 ¼ 2xT ðtÞSj xðtÀt1 ÞÀ
m ¼ 1 n ¼ t þ 1Àm
À
tðtÞ
tÀ1
X
X
1
1 X tÀmÀ1
xðmÞÀxðtÀt2 ÞÀ
ZðnÞ ¼ 0,
tðtÞ m ¼ tÀtðtÞ
tðtÞ m ¼ 1 n ¼ tÀt
2
2
(
#
tÀ1
X
m ¼ tÀtðtÞ
xðmÞÀ
tðtÞ
X
tÀt1 À1
X
ZðnÞ:
m ¼ 1 n ¼ tÀm
Then, the following equations hold for any matrices T j ,W j ,Sj with
appropriate dimensions:
where zðtÞ ¼ ½x ðtÞ,xT ðtÀt1 Þ,xT ðtÀt2 ÞŠT , and
2
0
D11
T
6
B
6 X þf zjklm f zjklm þg T g zjklm þ @ n
zjklm
Xijklm ¼ 6
6
n
4
n
2
0
6n
6
6
C¼6n
6
6
4n
n
with
3
W j ÀT j ÀSj
0
0
0
ÀQ 1
0
0
n
0
0
n
n
0
n
n
n
07
7
7
0 7,
7
7
05
0
0
0
n
1
3
0 C
A
Àg2 I
D2 7
0
D3
7
7,
7
5

6. Z. Xia et al. / ISA Transactions 51 (2012) 702–712
T
T
with
X ¼ f jklm P1 f jklm þ g T P1 g jklm ÀHe ff jklm P2 g:
jklm
Then
ð28Þ
By Lemma 3, (28) is equivalent to
ð29Þ
which can be rewritten as follows:
#
T
X þf zjklm f zjklm þg T g zjklm þ D1 D2
zjklm
D3
n
Using Lemma 1, we have
2
P1 À2aj I
0
0
6
n
P1 À2aj I
0
6
6
6
n
n
ð1À2aj ÞI
6
6
6
n
n
n
6
6
n
n
n
4
n
o 0:
ð30Þ
0
Àaj g jklm
0
ÀP2 þ aj f jklm
0
ð1À2aj ÞI
Àaj f zjklm
Àaj g zjklm
n
D1
n
n
n
0
3
7
0 7
7
7
0 7
7 o 0,
0 7
7
7
D2 5
D3
hjk hjl hjm
0
n
P1 À2aj I
n
n
0
0
ð1À2aj ÞI
n
n
n
n
n
0
0
0
ð1À2aj ÞI
Àaj g jklm
ÀP2 þ aj f jklm
Àaj f zjklm
Àaj g zjklm
n
n
n
D1
n
n
n
n
0
0
0
0
7
7
7
7
7
7 o0:
7
7
D2 7
5
D3
hjk hjl hjm Pn
ijklm
k¼1l¼1m¼1
¼
bðjÞ
X
3
hjk Pn þ
ijkkk
k¼1
þ
bX
ðjÞÀ1
bðjÞ
X
2
hjk hjl ðPn þ Pn þ Pn Þ
ijkll
ijlkl
ijllk
bX bX
ðjÞÀ2 ðjÞÀ1
bðjÞ
X
hjk hjl hjm
k ¼ 1 l ¼ kþ1 m ¼ lþ1
ðPn þ Pn þ Pn þ Pn þ Pn þ Pn Þ:
ijklm
ijkml
ijlkm
ijlmk
ijmkl
ijmlk
C9wðtÞ ¼ 0 o0:
1
C9
ÞFðtÞÀUðxðtÞ, ZðnÞÞ,
J9wðtÞ ¼ 0 ¼ F ðtÞðXijkl 9wðtÞ ¼ 0 þ
tðtÞ wðtÞ ¼ 0
P
where FðtÞ ¼ ½xT ðtÞ tÀ1 tÀtðtÞ xT ðnÞxT ðtÀt1 ÞxT ðtÀt2 ÞŠT and
n¼
2
3
Sj ÀW j
6 Lijkl
7
0
0
Xijkl 9wðtÞ ¼ 0 ¼ 4
5,
T
D3
n
ð34Þ
Then, we can conclude that the closed-loop system (7) with
wðtÞ ¼ 0 is mean-square asymptotically stable by following the
same lines as in the proof of Theorem 1 in [38].
Now, to establish the H1 performance for the closed-loop
system (7), consider the following index:
(
)
1
X
T
2 T
½e ðtÞeðtÞÀg o ðtÞoðtÞŠ :
ð36Þ
J2 ¼ E
t¼0
Under zero initial condition and (27), we have
(
)
1
X
½ÀDVðtÞŠ ¼ EfÀVð1Þ þ Vð0Þg ¼ EfÀVð1Þg o0,
J2 r E
t¼0
3
W j ÀT j ÀSj
0
0
ÀQ 1
0
n
0
07
7
7:
05
n
n
0
4. Simulation
In this section, two examples are used to verify the performance of the proposed SOF controller.
Example 1. Nowadays, it is often to consider the cases of
monitoring and control through networks. Due to the existence
of transmission delay in networks for different environments.
Consider system (1) with the following parameters:
2
3
2
3
0:12 À0:01 0:03
0:11 À0:15 0:03
6 0:05 0:14 À0:08 7
6 0
À0:1 À0:07 7
A1 ¼ 4
5, A2 ¼ 4
5,
0:11
À0:6
À0:01
0
0:01
0
0:01
0:13 0:09 À0:21
2
3
0:06
0
0:05
6
7
0:02
0
Ad2 ¼ 4 0
5,
0:001
0
À0:005
3
À0:03
7
5,
ð32Þ
If (10), (11) and (12) hold, P o0 which implies that (27) holds.
Therefore, when assuming the zero disturbance input, from
(16)-(25), we obtain
n
!
0
:
0
ð35Þ
B1 ¼ ½0:14 0:25 0:31Š,
0
6n
C9wðtÞ ¼ 0 ¼ 6
6
4n
D11
7
I 5þ
n
0
J9wðtÞ ¼ 0 o 0:
0:07
2
0:02
6
Ad1 ¼ 4 0
0
k ¼ 1 l ¼ kþ1
2
3
3
Let F jl ¼ aÀ1 U jl , we obtain
j
bðjÞ bðjÞ bðjÞ
XX X
1
tðtÞ
The proof is completed.
P1 À2aj I
P¼
0
EfJeðtÞJ2 g o gJoðtÞJ2 :
k¼1l¼1m¼1
6
6
6
6
6
6
6
6
6
4
2
i I
6
40
0
which means that
which is equivalent to
2
0
Xijkl 9wðtÞ ¼ 0 þ
ð31Þ
bðjÞ bðjÞ bðjÞ
XX X
I
Xþf
T
T
zjklm f zjklm þ g zjklm g zjklm
From (13) and (34), we have
C o 0, n ¼ 1; 2,
n
!
0 h
By Schur complement and Lemma 3, LMI (30) implies
1
Xijklm þ
C o 0:
tðtÞ
tn
0
0
ð27Þ
If (13) hold and
Xijklm þ
I
Lijkl ¼
J þ EfzT ðtÞzðtÞgÀg2 oT ðtÞoðtÞ o0:
1
707
ð33Þ
D1 ¼ ½0:1 À0:3 0:4Š,
2
0:61
0:33
B2 ¼ ½0:22 0:18 0:37Š,
D2 ¼ ½0:3 À0:2 0:4Š,
3
2
0
0:37
tðtÞ ¼ 3þ ðÀ1Þt ,
3
6
7
6
7
0 5,
B11 ¼ 4 0:02 0:34 5, B12 ¼ 4 0:91
0:16 0:06
0
0:28
2
3
2
0:08
0
0:13
0
À0:31
6
7
6
0:51
0 5, M 2 ¼ 4 0
0:42
M1 ¼ 4 0
2
0
0
0:15
0
0:39
3
2
0:03
0
3
0
0:02
0:03
6
7
6
0:24 5, M d1 ¼ 4 0
D21 ¼ 4 0
0:32
0
0
2
3
0
0
0
6
7
0 5,
Md2 ¼ 4 0 À0:02
0:02
7
0 5,
0:01
0
0
0:01
0
0
3
7
0 5,
0:11

7. 708
Z. Xia et al. / ISA Transactions 51 (2012) 702–712
0:61
0:33
3
2
0
0:31
3
6
7
6
0 7
B21 ¼ 4 0:02 0:34 5, B22 ¼ 4 0:35
5,
0:16 0:06
0
0:28
2
3
2
3
0:02 0
0:1
0
0
0:03
6 0
7
6 0 0:05
0:3
0 5, C 2 ¼ 4
0 7,
C1 ¼ 4
5
0
0 0:08
0
0
À0:02
2
3
0:01
0
0:003
6
0:01
0 7
C d1 ¼ 4 0
5,
0:02
2
0:03
6
C d2 ¼ 4 0
0
2
0
À0:03
0
0
À0:03
0
0:13
3
3
2
0:11
6
D11 ¼ 4 0
0:15
0:02 7,
5
0:04
2
0:05
0
3
0
À0:12 7,
5
0:01
6
Nd2 ¼ 4
G1 ¼
0:002
0
0
0:013
0:021
0
0:16
0:31
0:19
0:31
3
0:018
0 7
5,
2
0:41
6
D22 ¼ 4 0:22
0:7
0:001
!
0:68
0:27
, G2 ¼
À0:35
0:63
h1
Membership functions
0.6
0.4
Ω2
Ω1
0
3
0
π/18
0
0:001
3
3
0
0:33 7
5,
2
0:09
0:31
0:17
0:42
0:41
!
:
1
0
−1
−2
−3
0
20
80
100
Suppose the unknown disturbance input v(t) to be random
noise, as is shown in Fig. 3. Let g ¼ 3:18, if we use a common
Lyapunov matrix P instead of piecewise Lyapunov matrix
Pi ði ¼ 1; 2,3Þ in LMIs (10)-(12), no feasible solution can be found
for switching fuzzy SOF controller design. However, based on
Theorem 1, we can obtain feasible solution as follows:
2:0863
6 0:2038
P1 ¼ 4
0:3742
0.6
0.5
0:2038
3:9144
2:7773
6
P2 ¼ 4 0:1060
1:1422
0.3
0.2
1 π/3
1.5 π/2
2:3044
6
P3 ¼ 4 0:0486
0:4124
|x1|
F 11 ¼
Fig. 1. The membership functions.
3
À0:2419 7
5,
4:0821
0:1060
1:1422
5:4118
3
À0:2770 7,
5
À0:2770
4:6235
0:0486
0:4124
2
0.1
0:3742
À0:2419
2
0.4
0.5
40
60
Time(Sec)
Fig. 3. The unknown disturbance input v(t).
2
π/18
1.5 π/2
7
5,
0.7
0
1 π/3
0.5
Fig. 2. The partition of subspaces Oi , i ¼1,2,3.
3
0
0.8
0
Ω3
|x1|
h2
0.9
∂Ω1
3
∂Ω1
2
0.2
The membership functions are shown in Fig. 1.
As illustrated in [8,11], we divide the state space into three
subregions. The membership functions hji and partition of
subregions are stated in Fig. 2. The system matrices are
A11 ¼ A21 ¼ A1 , A22 ¼ A31 ¼ A2 , Ad11 ¼ Ad21 ¼ Ad1 , Ad22 ¼ Ad31 ¼ Ad2 ,
B11 ¼ B21 ¼ B1 , B21 ¼ B31 ¼ B2 , B111 ¼ B121 ¼ B11 , B122 ¼ B131 ¼ B12 ,
M 11 ¼ M 21 ¼ M1 , M 22 ¼ M 31 ¼ M2 , M d11 ¼ M d21 ¼ Md1 , Md22 ¼
M d31 ¼ Md2 , B211 ¼ B221 ¼ B21 , B222 ¼ B231 ¼ B22 , C 11 ¼ C 21 ¼ C 1 ,
C 22 ¼ C 31 ¼ C 2 , C d11 ¼ C d21 ¼ C d1 , C d22 ¼ C d31 ¼ C d2 , D11 ¼ D21 ¼ D1 ,
D22 ¼ D31 ¼ D2 , D111 ¼ D121 ¼ D11 , D122 ¼ D131 ¼ D12 , N 11 ¼ N 21 ¼
N1 , N 22 ¼ N 31 ¼ N 2 , Nd11 ¼ N d21 ¼ Nd1 , Nd22 ¼ Nd31 ¼ N d2 , D211 ¼
D221 ¼ D21 , D222 ¼ D231 ¼ D22 , G11 ¼ G21 ¼ G1 , G22 ¼ G31 ¼ G2 . The
membership functions are h11 [ h21 ¼ h1 , h22 [ h31 ¼ h2 .
1
h31
h22
∂Ω1 ∂Ω2
1
1
0.8
0
6
0 7 N1 ¼ 6 0
0:14
0 7
D12 ¼ 4 0:2
5,
4
5,
0 0:17
0:02
0
À0:13
2
3
2
0:02
0
0:21
0:001 0:013
6
0 7, Nd1 ¼ 6 0:04 0:002
N2 ¼ 4 À0:13 0:04
5
4
0:15
0
0:11
0
0:015
2
Membership functions
0
h21
h11
1
v(t)
2
3:4077
À0:1979
0:0828
À0:9788
3
7
À0:1979 5,
3:0487
!
1:0874
,
0:0050
F 21 ¼
À0:7167
À0:4143
!
À0:0785
,
1:3926

8. Z. Xia et al. / ISA Transactions 51 (2012) 702–712
!
À0:5569
,
0:6786
0:6118
À0:9752
F 31 ¼
À1:6217
À1:3740
0.6
!
0:4802
:
1:1860
0.4
0.2
y1(t)
F 22 ¼
Based on the proposed fuzzy SOF controller (6), the state
responses of the closed-loop system (7) are stated in Figs. 4–6
with the initial conditions xðtÞ ¼ ½p=8,Àp=5, p=3ŠT ðt ¼ À4, . . . ,0Þ
and the disturbance oðtÞ ¼ 0:1 Á cos t Á eÀ0:05t . Figs. 7–11 show
the corresponding output responses.
0
−0.2
−0.4
Example 2. In this example, the proposed fuzzy SOF control
technique is applied to backing up control of a truck–trailer. We
use the following modified truck–trailer model formulated in [48]
and assume that the states are perturbed by distributed timevarying delay and stochastic disturbances.
X
tÀ1
vt
vt
x1 ðtÞ þ ð1Àl1 Þ 1À
x1 ðnÞ
x1 ðt þ 1Þ ¼ l1 1À
L
L n ¼ tÀtðtÞ
!
!
vt
vt
uðtÞ þ sinðWðtÞÞ þ
uðtÞ vðtÞ,
þ 0:01oðtÞ þ
l
l
50
Time(Sec)
100
0.4
y2(t)
0.2
0
−0.2
0.6
x1(t)
0
Fig. 7. Response curve of measured output y1 ðtÞ.
0.8
−0.4
0.4
−0.6
0.2
0
−0.2
709
0
20
40
60
Time(Sec)
80
100
Fig. 8. Response curve of measured output y2 ðtÞ.
0
50
Time(Sec)
100
0.25
0.2
Fig. 4. Response curve of state x1 ðtÞ.
0.15
Z1(t)
0.2
x2(t)
0
0.05
0
−0.2
−0.05
−0.4
−0.1
−0.6
−0.8
0.1
0
50
Time(Sec)
20
40
60
Time(Sec)
x2 ðt þ 1Þ ¼ l2 x2 ðtÞ þ ð1Àl2 Þ
tÀ1
X
x2 ðnÞ þ
n ¼ tÀtðtÞ
1.5
1
80
100
Fig. 9. Response curve of controlled output z1 ðtÞ.
100
Fig. 5. Response curve of state x2 ðtÞ.
0
x3 ðt þ 1Þ ¼ l3 x3 ðtÞ þ ð1Àl3 Þ
tÀ1
X
vt
x1 ðtÞ,
L
x3 ðnÞ þ vt sinðWðtÞÞ,
x3(t)
n ¼ tÀtðtÞ
0.5
zðtÞ ¼ À
0
vt
vt
x1 ðtÞ þsinðWðtÞÞ þ
uðtÞ,
L
l
yðtÞ ¼ GxðtÞ,
−0.5
−1
0
50
Time(Sec)
Fig. 6. Response curve of state x3 ðtÞ.
100
ð37Þ
where x1 ðtÞ is the angle difference of truck and trailer, x2 ðtÞ is the
angle of trailer, x3 ðtÞ is the vertical position of rear end of trailer,
oðtÞ ¼ 0:1eÀt þsinðtÞ is the external disturbance, and v(t) is
stochastic disturbance. l is the length of truck, L is the length
of trailer, t is sampling time, v is the constant speed of backing

9. 710
Z. Xia et al. / ISA Transactions 51 (2012) 702–712
2
0.05
6
6
A11 ¼ A22 ¼ 6
6
4
Z2(t)
0
−0.05
l1 1À vt
L
−0.15
50
Time(Sec)
l3
0
vt
L
2 2
6
Ad11 ¼ Ad12 ¼ Ad21 ¼ Ad22 ¼ 6
4
2
0.1
l3
p
2
3
7
7
0 7,
7
5
vt 10À2
100
Fig. 10. Response curve of controlled output z2 ðtÞ.
0
l2
v t 10À2
2Lp
3
7
7
0 7,
7
5
vt
−0.2
0
0
l2
2 2
v t
2L
l 1À vt
L
6 1
6
A12 ¼ A21 ¼ 6
6
4
0
vt
L
2
−0.1
l1 1À vt
L
0:01
0
0
3
0
1Àl2
0
0
0
1Àl3
Z3(t)
2
−0.1
2
vt
2L
1
6
M11 ¼ M 22 ¼ 4 0
50
Time(Sec)
100
2
6
M12 ¼ M 21 ¼ 4
0
3
0
3
0
7
0 5,
0
0
0
Fig. 11. Response curve of controlled output z3 ðtÞ.
0
vt 10À2
2Lp
10À2
0
0
p
0
3
7
0 5,
0
0
0
#
!
vt
vt10À2 vt 10À2
1 0 , C 12 ¼ C 21 ¼
À
0 ,
C 11 ¼ C 22 ¼ À
2L
L
2Lp
p
up, and
WðtÞ ¼ x2 ðtÞ þ
vt
x1 ðtÞ:
2L
D111 ¼ D112 ¼ D121 ¼ D122 ¼
The model parameters are given as l ¼ 2:8 m, L ¼ 5:5 m, v ¼
À1:0 m=s, t ¼ 2:0 s, l1 ¼ l2 ¼ l3 ¼ 0:8, tðtÞ ¼ 2:5 þ 0:5nðÀ1Þt , and
G ¼ diagfÀ1,À1,À1g.
Under the condition À179:42701 o WðtÞ o 179:42701, the nonlinear system (37) can be exactly represented by the following
switching fuzzy system:
bðjÞ
X
xðt þ 1Þ ¼
(
hjk
Ajk xðtÞ þ Adjk
tÀ1
X
xðnÞ
3
217:9
À295:5 7
5,
1016:3
F 12 ¼ ½0:4926 0:2187 À0:2072Š,
F 22 ¼ ½1:0882 À0:2040 0:0789Š:
Under the initial conditions fðÀ3Þ ¼ ½11; 4,À9ŠT , fðÀ2Þ ¼ ½12; 3,
6Š , fðÀ1Þ ¼ ½À13,À4,À3ŠT and fð0Þ ¼ ½2,0:1,À10ŠT , the simulation
results for the system (38) with stochastic disturbances are
shown in Fig. 12.
hjk ½C jk xðtÞ þ D1jk uðtÞŠ,
hjk Gjk xðtÞ,
Letting g ¼ 1:6 and using Theorem 1, we obtain
2
3
2
384:7 À35:8
151
566:5
85:3
6
7
6
193:2
P 1 ¼ 4 À35:8 202:7 À332 5, P 2 ¼ 4 85:3
151
À332 1066:7
217:9 À295:5
T
k¼1
bðjÞ
X
G11 ¼ G12 ¼ G21 ¼ G22 ¼ diagfÀ1,À1,À1g:
F 21 ¼ ½1:1493 0:0937 À0:0230Š,
!
'
þ Bjk oðtÞ þ B1jk uðtÞ þ ½M jk xðtÞ þ B2jk uðtÞŠvðtÞ ,
bðjÞ
X
vt
,
l
F 11 ¼ ½0:2721 À0:8158 0:0526Š,
n ¼ tÀtðtÞ
k¼1
yðtÞ ¼
vt
6 l 7
B111 ¼ B112 ¼ B121 ¼ B122 ¼ B211 ¼ B212 ¼ B221 ¼ B222 ¼ 4 0 5,
−0.2
zðtÞ ¼
7
7,
5
6
7
B11 ¼ B12 ¼ B21 ¼ B22 ¼ 4 0 5,
0
0
−0.3
3
ð38Þ
5. Conclusion
k¼1
where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ x3 ðtÞŠT and hjk ðj ¼ 1; 2, bð1Þ ¼ bð2Þ ¼ 2Þ are
the membership functions with
8
À2
sinðWðtÞÞÀWðtÞÁ10 =p , WðtÞ a 0,
WðtÞÁð1À10À2 =pÞ
h11 ¼ h22 ¼
: 1,
WðtÞ ¼ 0,
h12 ¼ h21 ¼
8
WðtÞÀsinðWðtÞÞ
,
WðtÞÁð1À10À2 =pÞ
: 0,
WðtÞ a 0,
WðtÞ ¼ 0,
Based on the switching fuzzy stochastic system, piecewise
Lyapunov–Krasovskii function, and state transitions between
all possible subregions, a new delay-dependent H1 fuzzy SOF
controller design approach is proposed for discrete-time
fuzzy stochastic systems with distributed time-varying
delays. In contrast to the existing results, two new techniques,
improved free-weighting technique and new matrix decoupling
technique, are developed. If the conditions are feasible, the
controller parameters can be easily constructed by solving a set
of LMIs. The theoretic results obtained in the paper have

10. 4
2
5
0
−2
0
−5
−4
−10
0
20
40
Time(Sec)
60
80
0
20
40
Time(Sec)
60
80
0
20
40
Time(Sec)
60
80
0
20
40
Time(Sec)
60
80
2
x2(t)
4
10
x1(t)
20
0
−10
−20
0
−2
0
20
40
Time(Sec)
60
−4
80
10
5
5
z(t)
10
x3(t)
711
10
u(t)
Stochastic disturbances v(t)
Z. Xia et al. / ISA Transactions 51 (2012) 702–712
0
−5
−10
0
−5
0
20
40
Time(Sec)
60
80
−10
Fig. 12. Control results of trucker–trailer.
some potential applications, such as image processing via a
large-scale system networks, which gives rise to possible
distributed time-varying delays for discrete-time systems.
Two examples are presented to demonstrate the validity of the
proposed approach.
Appendix A. Proof of Lemma 1
where
2
6
6
6
6
S0 ¼ 6
6
6
6
6
4
S0 S2
ST 0
2
S3
o 0,
S1
P1
0
0
P1
0
0
0
0
0
ÀP 2
0
0
0
0
I
0
0
I
0
0
0
ÀP T
2
0
0
D1
0
DT
2
0
0
0
ÀaI
0
0
0
ÀaI
0
6
6
6
6
S2 ¼ 6
6
6
6
4
0
0
ÀaI
0
0
0
0
0
0
0
0
0
0
3
0 7
7
7
0 7
7,
ÀaI 7
7
7
0 5
S3 ¼ diagfI,I,I,I,I,Ig:
0
Then we choose the orthogonal complement of S1 as
#
ÀBT AT ÀAT ÀBT I 0
1
2
1
2
ST ¼
,
1?
0
0
0
0
0 I
Motivated by [11,15], we can rewrite the inequality (b) as
#T
#
#
S3
S1
2
ðA:1Þ
3
0
0 7
7
7
0 7
7
,
0 7
7
7
D2 7
5
D3
which satisfies S1 S1? ¼ 0. Moreover, ½ST , S1? Š is of column full
1
rank. Then it follows that (A.1) is equivalent to the following
matrix inequality:
2
I
6
60
40
S1 ¼ 6
6
0
0
0
0
B1
I
0
0
I
0
0
ÀA1
A2
0
0
I
B2
0
3
7
07
7,
07
5
0
ST
1?
S3
S1
#T
S0 S2
ST 0
2
#
#
S3
S o 0,
S1 1?
which can be further reduced to
ST S0 S1? o 0:
1?
ðA:2Þ

11. 712
Z. Xia et al. / ISA Transactions 51 (2012) 702–712
Thus, we have shown that the inequality (b) is equivalent to
(A.2). It is also easily seen that matrix inequality (a) can also be
rewritten as (A.2).
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