This paper presents the robust stability condition of uncertain TakagiSugeno(TS) fuzzy systems with timevarying delay. New augmented LyapunovKrasovskii function is constructed to ensure that the system with timevarying delay is globally asymptotically stable. Also, less conservative delaydependent stability criteria are obtained by employing some integral inequality, reciprocally convex approach and new delaypartitioning method. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.
1. Introduction
Since TakagiSugeno(TS) fuzzy model was first introduced in
[1]
, the stability and design conditions for TS fuzzy systems have been paid much attention. The main advantage of TS fuzzy model is that it can combine the exibility of fuzzy logic theory and rigorous mathematical theory of linear system into a unified framework to approximate complex nonlinear systems
[2

4]
. On the other hand, time delays often appears in many dynamical systems such as metallurgical processes, biological systems, neural networks, networked control systems and so on. The existence of time delay may cause poor performance or instability. Hence, the stability of TS fuzzy systems with time delay has been studied by many researchers
[5

24
,
30
,
31]
.
It is well known that the delaydependent stability criteria are less conservative that delayindependent ones especially that the timedelay is small. The main issue of delaydependent stability criteria is to find a maximum delay bounds to guarantee the asymptotic stability of the considered systems. Therefore, the study of increasing the maximum delay bounds in delaydependent stability criteria for fuzzy systems is an important topic and have been investigated by many researchers. In [5], the delay dependent stability problem for TS fuzzy systems with time varying delay was investigated. Some stability criteria or stabilization of delayed TS fuzzy systems were derived by employing freeweighting matrix [6,9,12]. Furthermore, the results was further studied by using delaypartitioningbased approach
[13
,
17
,
19
,
23]
. Recently, in
[18]
, an augmented LyapunovKrasovskii functional approach that introduces a triple integral and some augmented vectors was employed to investigate the stability problem of TS fuzzy systems with timevarying delay. In
[20]
, the improved results was obtained by quadratically convex approach. The results was further improved in
[24]
by employing the delaypartitioning method and reciprocally convex approach. However, though these results and analytic methods are elegant, there still exist some rooms for further improvements. First, in
[8
,
11
,
20
,
18]
, Jensen's inequality, freeweighting matrix and quadratically convex combination approach are used to derive the stability condition. However, reciprocally convex approach
[25]
, which can play an important role in reducing conservatism of the stability condition, is not used in
[8
,
11
,
20
,
18]
. Second, though the reciprocal convex approach, delaypartitioning and integral inequalities method are combined to obtain some less conservative results in
[24]
, it still needs some improvements since it only used the improved inequality in constant delay, not employed in timevarying delay. Furthermore, it can be predicted that delaypartitioning approach can provide tighter upper bounds than the results without delaypartitioning approach. However, as delaypartitioning number increases, matrix formulation becomes complex and time consuming and computational burden grow bigger. Therefore, there are rooms for further improvement in stability analysis of TS fuzzy systems with timevarying delay.
In this paper, the stability analysis conditions for uncertain TS fuzzy systems with timevarying delay are proposed. By construction of a modified augmented LyapunovKrasovskii functional approach, an improved stability criterion for guaranteeing the asymptotically stable is derived by using Wirtingerbased integral inequality
[26]
, reciprocally convex approach
[25]
, and new delaypartitioning method. It should be pointed out that different with delaypartitioning method used in
[24]
, we only divide the time interval into two subintervals, and consider two different cases of delaypartitioning method. Moreover, some robust stability criteria of uncertain TS systems with time varying delay is provided. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed method.
Notation: Throughout the paper,
R
^{n}
denotes the
n
dimensional Euclidean space,
R
^{m×n}
denotes the set of
m
by
n
real matrix. For symmetric matrices
X
,
X
>0 and
X
<0, mean that
X
is a positive/negative definite symmetric matrix, respectively.
I
and 0 denote the identity matrix and zero matrix with appropriate dimension. ⋆ represents the elements below the main diagonal of a symmetric matrix.
diag
... denotes the diagonal matrix.
2. Problem Statements
Consider the following nonlinear system which can be modeled as TF fuzzy model type subject to timevarying delay:
where
θ
_{1}
(
t
),
θ
_{2}
(
t
),...,
θ_{n}
(
t
) are the premise variables,
M_{ij}
is fuzzy set,
i
=1,2,...,
r
,
j
=1,2,...,
n
r
is the index number of fuzzy rules, and
x
(
t
)∈
R^{n}
denotes the state of the system.
A_{i}
and
A_{di}
are the known system matrices and delayedstate matrices with appropriate dimensions, respectively.
φ
(
t
) is a continuously realvalued initial function vector. we assume that
h
(
t
) is a timevarying delay satisfying
where
h_{M}
,
μ
are known constants.
The uncertainties satisfy the following condition:
where
D
,
E_{i}
,
E_{di}
are known constant matrices;
F
(
t
)∈
R
^{n×n}
is the unknown real timevarying matrices with Lebesgue measurable elements bounded by
Using singleton fuzzifier, product inference, and centeraverage defuzzifier, the global dynamics of the delayed TS system (1) is described by the convex sum form
where
p_{i}
(
θ
(
t
)) denotes the normalized membership function satisfying
where
M_{ij}
(
θ_{i}
(
t
)) is the grade of membership of
θ_{i}
(
t
) in
M_{ij}
It is assumed that
Then, we have the following condition
For the sake of simplicity, let us define
Now, the system (5) can be rewritten as
In what follows, some essential lemmas are introduced.
Lemma 1
[26]
For a given matrix R>0, the following inequality holds for all continuously differentiable function
x
(
t
) in [
a
,
b
]∈
R^{n}
:
where
Lemma 2
[28]
For a given matrix M>0,
h_{m}
≤
h
(
t
)≤
h_{M}
, and any appropriate dimension matrix
X
, which satisfies
Then, the following inequality holds for all continuously differentiable function
x
(
t
)
where
Lemma 3
(Fisher's Lemma
[27]
) Let
ξ
∈
R^{n}
,
Φ
=
Φ
^{T}
∈
R
^{n×n}
, and
B
∈
R
^{m×n}
such that
rank
(
B
) ≤
n
. The following statements are equivalent

(i)ξTBξ<0,∀Bξ=0,ξ≠0,

(ii)whereB⊥is a right orthogonal complement ofB.

(iii)
3. Main Results
In this section, we first propose a stability criterion for delayed TS fuzzy systems without uncertainties, and the following nominal system will be considered:
For the sake of simplicity of matrix and vector representations,
e_{i}
∈
R
^{8n ×n}
(
i
=1,2,...,8) are defined as block entry matrices (for example (
e
_{4}
= [000
I
0000]
^{T}
). The other notations are defined as :
Now we have the following Theorem.
Theorem 1
For given scalars
h_{M}
>0,0 <α<1,
μ
, the system (11) is globally asymptotically stable if there exist symmetric positive matrices
P
∈
R
^{3n × 3n}
,
Q
_{1}
,
Q
_{2}
,
Q
_{3}
,
R
_{1}
,
R
_{2}
, and any matrix
S_{j}
(
j
=1,2)∈
R
^{2n × 2n}
such that the following LMIs hold
where
Proof
: Let us consider the following LyapunovKrasovskii functional candidate as
where
Depending on whether the timevarying delay
h
(
t
) belongs the interval 0≤
h
(
t
) ≤
h_{M}
or
αh_{M}
≤ h(t) ≤
h_{M}
, different upper bound of the
V_{i}
(
i
= 1,4,5) can be obtained as two cases:
When 0≤
h
(
t
) ≤
αh_{m}
, the timederivative of
V_{i}
(
i
= 1,2,3) can be calculated as
By applying Lemma 2, an upper bound of
is obtained as
where
Note that when
h
(
t
) = 0 or
h
(
t
) =
h_{M}
, we have
β
_{1}
(
t
) =
β
_{2}
(
t
) = 0 or
β
_{3}
(
t
) =
β
_{4}
(
t
) = 0 Then (19) still holds.
Next, an upper bound of
can be derived by Lemma 1,
where
Therefore, in the case of 0≤
h
(
t
) ≤
h_{M}
, form Eqs. (16)(20), an upper bound of
can be given as
Based on Lemma 3,
ξ^{T}
(
t
)
Y
_{1}
ξ(
t
) < 0
with
is equivalent to
Furthermore, the above condition is affinely dependent on
h
(
t
). Hence, (12) and (14) imply
Next, when
αh_{M}
≤
h
(
t
) ≤
h_{M}
, the timederivative of
V
_{1}
is
Based on Eq. (17) and (18), an derivative of
V_{i}
(
i
= 2,3) can be calculated as
By Lemma 1,
where
Also, an upper bound of
can be obtained by utilizing Lemma 2
where
Note that when
h
(
t
) =
αh_{M}
or
h
(
t
) =
h_{M}
, we have
γ
_{1}
(
t
) =
γ
_{2}
(
t
) = 0 or
γ
_{3}
(
t
) =
γ
_{4}
(
t
) = 0,
β
(
t
) = 0. Thus, Eq. (25) still holds.
Therefore, from Eqs. (22)(25), an upper bound of
in the case of
αh_{M}
≤
h
(
t
) ≤
h_{M}
can be given as
Based on Lemma 3,
with
is equivalent to
Furthermore, the above condition is affinely dependent on
h
(
t
). Hence, (13) and (14)
This completes the proof. ■
Remark 1.
Unlike in
[24]
, the proposed LyapunovKrasovskii functional in (15) are divided the time delay interval [0,
h
] into different size because of introducing parameter
α
. When
α
= 0.5, it can be reduced to the ones employed in
[24]
, which divides the time delay interval into the same size, that is,
In other words, based on two delay decomposing approach, the LyapunovKrasovskii functional constructed in this paper is more general than the ones used in
[24]
. When α= 0.5, constructing the following Lyapunov functional candidate as
where
the others are the same with the ones in (15).
Remark 2.
It should be pointed out that the proposed delaypartitioning method is different from existing ones
[24
,
29]
. In
[29]
, by using nonuniform decomposition method that the whole delay interval is nonuniformly decomposed into multiple subintervals. In
[24]
, uniform decomposition method is used, which divides the delay interval into the same size. While the conventional method use preknown constant value to divide the delay interval, a new nonuniform delaypartitioning method is proposed by introducing parameter
α
, that is, delay interval is divided as [0,
h_{M}
]=[0,
αh_{M}
]∪[
αh_{M}
,
h_{M}
].
Based on Eq. (27) with
α
= 0.5, the following Corollary can be obtained from Theorem 1.
Corollary 1.
For given scalars
h_{M}
>0,
α
= 0.5,
μ
, the system (11) is globally asymptotically stable if there exist symmetric positive matrices
P
∈
R
^{3n × 3n}
,
Q
∈
R
^{2n × 2n}
,
Q
_{1}
,
R
_{1}
,
R
_{2}
, and any matrix
S_{j}
(
j
= 1,2) ∈
R
^{2n × 2n}
such that the following LMIs hold
where
Remark 3.
Unlike the constructed LyapunovKrasovskii functional in (15), the cross term of the state
x
(
t
) and
x
(
t
−
αh_{M}
) in (15) are considered, which may provide improved stability condition.
For uncertain TS fuzzy system (10), since
P^{T}
(
t
)
p
(
t
) ≤
q^{T}
(
t
)
q
(
t
), there exists a positive scalar
ε
satisfying the following inequality:
ε
[
q^{T}
(
t
)
q
(
t
) 
P^{T}
(
t
)
p
(
t
) ≥ 0. Define
and
(
i
= 1,2,...,9), and the other notations are given as follows:
Now we have the following Corollary 2 and Corollary 3.
Corollary 2.
For given scalars
h_{M}
> 0.0 <
α
< 1.
μ
, the system (10) is globally asymptotically stable if there exist symmetric positive matrices
P
∈
R
^{3n × 3n}
,
Q
_{1}
,
Q
_{2}
,
Q
_{3}
,
R
_{1}
,
R
_{2}
, and any matrix
S_{j}
(
j
= 1,2) ∈
R
^{2n × 2n}
, and a positive scalar
ε
such that the following LMIs hold
Corollary 3.
For given scalars
h_{M}
>0,
α
=0.5,
μ
, the system (10) is globally asymptotically stable if there exist symmetric positive matrices
P
∈
R
^{3n × 3n}
,
Q
∈
R
^{2n × 2n}
Q
_{1}
,
R
_{1}
,
R
_{2}
, and any matrix
S_{j}
(
j
=1,2)∈
R
^{2n × 2n}
, and a positive scalar
ε
such that the following LMIs hold
where
4. Numerical Examples
In this section, two numerical examples are given to show the effectiveness of the proposed method.
Example 1
Consider the system with the following parameters:
For different
μ
, the upper bounds of the timevarying delay computed by the proposed method and those in
[8
,
11
,
20
,
18
,
24]
are listed in
Table 1
. It is easy to know that the proposed method in this paper is less conservative than those in the existing results.
다른μ값에 대한 상한유계지연hMTable 1Upper delay boundhMfor differentμ
다른 μ값에 대한 상한유계지연 h_{M} Table 1 Upper delay bound h_{M} for different μ
Example 2
Consider the system with the following parameters:
For different
μ
, the upper bounds of the timevarying delay computed by the proposed method and those in
[5
,
9
,
12
,
24]
are listed in
Table 2
. It can be concluded that the result proposed in this paper is better than the existing ones.
다른μ값에 대한 상한유계지연hMTable 2Upper delay boundhMfor differentμ
다른 μ값에 대한 상한유계지연 h_{M} Table 2 Upper delay bound h_{M} for different μ
5. Conclusions
The robust stability for uncertain TS fuzzy systems with timevarying delay has been investigated. Based on a modified LyapunovKrasovskii functional, some less conservative criteria have been obtained by employing new delaypartitioning technique, integral inequality and reciprocally convex approach. It should be worthwhile pointed out that different case of delaypartitioning method is used in this paper, that is, the delay interval is divided into even and not even. Two numerical examples have been given to demonstrate the effectiveness of the proposed method.
BIO
유 아 연(Yajuan Liu)
She received her B.S degree in mathematics and applied mathematics from shanxi nominal university, Linfen, China, in 2010, and M.S degree in applied mathematics from University of Science and Technology Beijing, Beijing, China, in 2012. She is currently working toward the Ph.D degree in Electronic Engineering from Daegu University, Korea.
이 상 문(Sangmoon Lee)
1973년 6월 15일생. 1999년 경북대학교 전자공학과 졸업(공학). 2006년 포항공과대학교 전기전자공학부 졸업(공박). 현재 대구대학교 전자전기공학부 부교수.
Email : moony@daegu.ac.kr
권 오 민(Ohmin Kwon)
1974년 7월 13일생. 1997년 경북대학교 전자공학과 졸업(공학). 2004년 포항공과대학교 전기전자공학부 졸업(공박). 현재 충북대학교 전기공학부 부교수.
Email : madwind@chungbuk.ac.kr
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