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DOI:
10.1109/TFUZZ.2017.2752723
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Citation for published version (APA):
Wang, L., & Lam, H. K. (2017). A new approach to stability and stabilization analysis for continuous-time Takagi-
Sugeno fuzzy systems with time delay. IEEE Transactions on Fuzzy Systems, (99).
https://doi.org/10.1109/TFUZZ.2017.2752723
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Download date: 09. Aug. 2022
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. , NO. , JANUARY 1
A new approach to stability and stabilization
analysis for continuous-time Takagi-Sugeno fuzzy
systems with time delay
Likui Wang and Hak-Keung Lam Senior Member, IEEE
Abstract—Till now, there are lots of stability and stabilization
results about T-S (Takagi-Sugeno) fuzzy systems with time delay,
but most of them are independent of the analysis of membership
functions. Since the membership functions are an essential
component to make a fuzzy system different from others, the
conditions without its information are conservative. In this brief
paper, a new Lyapunov-Krasovskii functional is designed to
investigate the stability and stabilization of continuous-time T-
S fuzzy systems with time delay. Different from the existing
results in the literature, the integrand of the Lyapunov-Krasovskii
functional in this paper depends not only on the integral variable
but also on the membership functions, and thus, the information
of the time-derivative of membership functions can also be
used to reduce the conservativeness of finding the maximum
delay bounds. Utilizing the information of the time-derivative
of membership, a bunch of controllers are designed according to
their sign, and then a switching idea is applied to stabilize the
fuzzy system. In the end, two examples are given to illustrate the
feasibility and validity of the design and analysis.
Keywords: Takagi–Sugeno’s fuzzy model, Time delay, Par-
allel distributed compensation law, Membership dependent
Lyapunov-Krasovskii function.
I. INTRODUCTION
In the recent decades, the stability and stabilization analysis
for fuzzy systems ([24]-[30]) especially for Takagi-Sugeno
fuzzy model [1] with time delay has been a hot topic (see
[2]-[13], [22], [23] and the references therein) and the focus
is on how to maximize the tolerance of time delay. All
kinds of Lyapunov-Krasovskii functional have been developed
to reduce the conservativeness of finding the maximum de-
lay bounds, for example, an augmented Lyapunov-Krasovskii
functional is proposed in [5] and an improved Jensen’s
inequality which is a more general and tighter bounding
technology is used to deal with the cross product terms. The
results in [5] are improved by [8] where a fuzzy weighting-
dependent Lyapunov-Krasovskii functional is designed for
L.K. Wang is with the School of Mathematics and Information Science,
Nanchang Hangkong University, Nanchang, 330063, China and also with the
Department of Informatics, King’s College London, Strand, London, WC2R
2LS, United Kingdom. e-mail:wlk0228@163.com.
H.K. Lam is with the Department of Informatics, King’s College
London, Strand, London, WC2R 2LS, United Kingdom. e-mail:hak-
keung.lam@kcl.ac.uk.
This work has been done during the one year visit of Likui Wang at the
Department of Informatics, King’s College London, Strand, London under
the China Scholarship Council Visiting Scholar Program. This paper is also
supported in part by the National Natural Science Foundation of China
under Grants 61463036 and Natural Science Foundation of Jiangxi Education
Program GJJ160695.
uncertain T-S fuzzy systems with time-varying delays. The
results in [8] are less conservative than [5] because of an
input-output approach and the fuzzy weighting-dependent
Lyapunov-Krasovskii functional. A discretized Lyapunov-
Krasovskii functional is proposed in [2] where the number of
free variables increases as the discretization level increases.
These variables help to reduce the conservativeness but in-
crease the computing burden heavily. For the case of constant
time delay, the results in [5], [8] and [2] are further improved
by [11] where new simple and effective stabilization conditions
are proposed by applying the Wirtinger inequality [14] and
fuzzy line-integral Lyapunov functional [15]. The results in
[11] are improved by [17] where a new augmented Lyapunov-
Krasovskii functional is constructed and some new cross terms
are included. For the first time, a distributed fuzzy optimal
control law relied on actual physical meaning by adaptive
dynamic programming algorithm is proposed in [20] which
gives a new thought for the optimization of multi-agent system.
However, all of the results mentioned above are independent
of the analysis of membership functions which will lead
to conservativeness because the membership functions are
an essential component to make a fuzzy system different
from other systems. More recently, a membership function
dependent Lyapunov-Krasovskii functional is designed in [18]
and some less conservative results are obtained by analyzing
the time derivative of the membership function, however the
drawback is that the local stabilization region becomes smaller
as the delay increases.
Based on the above discussions, the contributions of this
paper are as follows: 1), A new membership-function de-
pendent Lyapunov-Krasovskii functional is designed in this
paper which is different from the existing ones such as [11],
[17] where the integrands are independent of the membership
functions. 2), A new method is used to deal with the time
derivative of the Lyapunov-Krasovskii functional. Since the
integrand of the Lyapunov-Krasovskii functional depends not
only on the integral variable but also on the membership
functions, a switching idea is applied to ensure that the time
derivative of the Lyapunov-Krasovskii functional is negative.
Due to the above contributions, the obtained stability and
stabilization criteria improve the existing results. In the end,
two examples are provided for verification. One example is for
stability and the other one is for stabilization. Both examples
show that less conservative results can be obtained in this
paper than the existing ones in the literature.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. , NO. , JANUARY 2
II. PRELIMINARIES AND BACKGROUNDS
Consider the following nonlinear model
˙x (t) = f
1
(z (t)) x (t) + f
2
(z (t)) x (t − τ )
+f
3
(z (t)) u (t) , (1)
x (t) = φ (t) , t ∈
£
−τ 0
¤
where x(t) ∈ R
n
is the state, u (t) ∈ R
m
is the input,
z(t) ∈ R
p
is known premise variables. f
1
(·), f
2
(·) and
f
3
(·) are nonlinear functions or matrix functions with proper
dimensions, φ (t) is the initial condition and the time delay
τ is assumed to be constant. Applying the sector nonlinearity
method or local approximation method in [16], one has the
following well known time delay T-S fuzzy model:
˙x (t) =
r
X
i=1
h
i
(z (t)) A
i
x (t) +
r
X
i=1
h
i
(z (t)) A
τ i
x (t − τ)
+
r
X
i=1
h
i
(z (t)) B
i
u (t) , (2)
x (t) = φ (t) , t ∈
£
−τ 0
¤
,
where A
i
∈ R
n×n
, A
τ i
∈ R
n×n
, B
i
∈ R
n×m
are known
matrices and h
i
(z(t)) are membership functions. To lighten
the notation, we will drop the time t, for instance, we will use
x instead of x (t). For simplicity, single sums are written as
X
h
=
r
P
i=1
h
i
X
i
. For any matrix X, He(X) = X + X
T
.
In this paper, the following controller (3) is used to stabilize
the fuzzy system
u = K
h
x (t) + K
τ h
x (t − τ) . (3)
The following Lemma 1 is useful in this paper.
Lemma 1: (see [19]) For a function F (t) =
α
2
(t)
R
α
1
(t)
f (s, t) d
s
,
if α
1
(t), α
2
(t) are differentiable for t and f (s, t) is contin-
uous for s and patrially derivative for t, we have
dF (t)
dt
= ˙α
2
(t) f (α
2
(t) , t) − ˙α
1
(t) f (α
1
(t) , t)
+
α
2
(t)
Z
α
1
(t)
∂f (s, t)
∂t
d
s
. (4)
In the following, we will discuss how to ensure
˙
X
h
≤ 0,
˙
Y
h
≤ 0 and
˙
Z
h
≤ 0 where X
i
> 0, Y
i
> 0 and Z
i
> 0. Note
˙
X
h
=
r
X
i=1
˙
h
i
X
i
=
r−1
X
k=1
˙
h
k
(X
k
− X
r
) , (5)
˙
Y
h
=
r
X
i=1
˙
h
i
Y
i
=
r−1
X
k=1
˙
h
k
(Y
k
− Y
r
) , (6)
˙
Z
h
=
r
X
i=1
˙
h
i
Z
i
=
r−1
X
k=1
˙
h
k
(Z
k
− Z
r
) , (7)
where
˙
h
k
are the time-derivative of membership functions and
are negative or positive as time goes by. Since X
i
, Y
i
and Z
i
are variables to be designed, we can use a switching idea to
ensure
˙
X
h
≤ 0,
˙
Y
h
≤ 0 and
˙
Z
h
≤ 0 as follows:
(
if
˙
h
k
≤ 0, then X
k
− X
r
≥ 0, Y
k
− Y
r
≥ 0, Z
k
− Z
r
≥ 0,
if
˙
h
k
> 0, then X
k
− X
r
< 0, Y
k
− Y
r
< 0, Z
k
− Z
r
< 0.
(8)
There are 2
r−1
possible cases in (8). Let H
l
, l =
1, 2, · · · , 2
r−1
be the set that contains the possible permu-
tations of
˙
h
k
and C
l
be the set that contains the constraints of
X
i
, Y
i
and Z
i
, (8) can be presented as
if H
l
, then C
l
. (9)
For example, if r = 3, we have
˙
X
h
=
˙
h
1
(X
1
− X
3
) +
˙
h
2
(X
2
− X
3
) ,
˙
Y
h
=
˙
h
1
(Y
1
− Y
3
) +
˙
h
2
(Y
2
− Y
3
) ,
˙
Z
h
=
˙
h
1
(Z
1
− Z
3
) +
˙
h
2
(Z
2
− Z
3
) .
There are 2
2
constraints C
l
, l = 1, 2, 3, 4 to ensure
˙
X
h
≤ 0,
˙
Y
h
≤ 0,
˙
Z
h
≤ 0 and (9) is expressed as following
If H
1
, then C
1
; If H
2
, then C
2
;
If H
3
, then C
3
; If H
4
, then C
4
,
where
H
1
:
˙
h
1
≤ 0,
˙
h
2
≤ 0; H
2
:
˙
h
1
≤ 0,
˙
h
2
> 0;
H
3
:
˙
h
1
> 0,
˙
h
2
≤ 0; H
4
:
˙
h
1
> 0,
˙
h
2
> 0,
C
1
:
(
X
1
≥ X
3
, X
2
≥ X
3
, Y
1
≥ Y
3
,
Y
2
≥ Y
3
, Z
1
≥ Z
3
, Z
2
≥ Z
3
.
)
,
C
2
:
(
X
1
≥ X
3
, X
2
< X
3
, Y
1
≥ Y
3
,
Y
2
< Y
3
, Z
1
≥ Z
3
, Z
2
< Z
3
.
)
,
C
3
:
(
X
1
< X
3
, X
2
≥ X
3
, Y
1
< Y
3
,
Y
2
≥ Y
3
, Z
1
< Z
3
, Z
2
≥ Z
3
.
)
,
C
4
:
(
X
1
< X
3
, X
2
< X
3
, Y
1
< Y
3
,
Y
2
< Y
3
, Z
1
< Z
3
, Z
2
< Z
3
.
)
.
Based on the above discussion, we get the following lemma.
Lemma 2: For some membership function dependent ma-
trices X
h
, Y
h
and Z
h
where X
i
> 0, Y
i
> 0 and Z
i
> 0 are
free variables, we have
˙
X
h
≤ 0,
˙
Y
h
≤ 0 and
˙
Z
h
≤ 0, if the
switching rules (9) are satisfied where l = 1, 2, · · · , 2
r−1
.
If Lemma 2 is used to get some stabilization conditions,
for different C
l
and H
l
, l = 1, · · · , 2
r−1
, the corresponding
controller is
u
l
= K
l,h
x (t) + K
l,τ h
x (t − τ) , (10)
K
l,h
=
r
X
i=1
h
i
K
l,i
, K
l,τ h
=
r
X
i=1
h
i
K
l,τ i
.
The final controller (3) becomes a switching controller below
u :
u
1
= K
1,h
x (t) + K
1,τ h
x (t − τ) for H
1
,
.
.
.
u
l
= K
l,h
x (t) + K
l,τ h
x (t − τ) for H
l
.
(11)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. , NO. , JANUARY 3
III. MAIN RESULTS
In this section, the stability and stabilization problem for
T-S fuzzy system with time delay is considered by utilizing
new Lyapunov-Krasovskii functional and a switching control
method.
A. Stability
Theorem 1: For a given scalar τ > 0, the closed-loop T-
S fuzzy system (2) is globally asymptotically stable if there
exist matrices
"
P
11i
P
12i
∗ P
22i
#
> 0, Q
i
> 0, Z
i
> 0, M
1i
,
M
2i
, M
3i
, M
4i
, such that the following inequalities hold for
i, j = 1, · · · , r,
˙
Q
h
≤ 0,
˙
Z
h
≤ 0,
˙
P
h
≤ 0, (12)
P
h
=
"
P
11h
P
12h
∗ P
22h
#
,
Θ
ij
+ Θ
ji
≤ 0, (13)
Θ
ij
=
Θ
11ij
Θ
12ij
Θ
13ij
Θ
14ij
∗ Θ
22ij
Θ
23ij
Θ
24ij
∗ ∗ −12τ
−2
Z
i
Θ
34ij
∗ ∗ ∗ Θ
44ij
,
Θ
11ij
= He (P
12i
+ M
1i
A
j
) + Q
i
− 4Z
i
,
Θ
12ij
= −P
12i
− 2Z
i
+ M
1i
A
τ j
+ A
T
j
M
T
3i
,
Θ
13ij
= P
22i
+ 6τ
−1
Z
i
+ A
T
j
M
T
4i
,
Θ
14ij
= P
11i
− M
1i
+ A
T
j
M
T
2i
,
Θ
22ij
= −Q
i
− 4Z
i
+ He (M
3i
A
τ j
) ,
Θ
23ij
= −P
22i
+ 6τ
−1
Z
i
+ A
T
τ j
M
T
4i
,
Θ
24ij
= A
T
τ j
M
T
2i
− M
3i
,
Θ
34ij
= P
T
12i
− M
4i
, Θ
44ij
= τ
2
Z
i
− He (M
2i
) .
Proof: Choose the Lyapunov-Krasovskii functional as
V (x
t
) = V
1
(x
t
) + V
2
(x
t
) + V
3
(x
t
) , (14)
V
1
(x
t
) = ρ
T
P
h
ρ, ρ
T
=
·
x (t)
T
t
R
t−τ
x (s)
T
ds
¸
,
V
2
(x
t
) =
t
Z
t−τ
x (s)
T
Q
h
x (s) ds,
V
3
(x
t
) = τ
0
Z
−τ
t
Z
t+θ
˙x (s)
T
Z
h
˙x (s) d
s
d
θ
.
It follows that
˙
V
1
(x
t
) = 2ρ
T
P
h
˙ρ + ρ
T
˙
P
h
ρ.
Applying Lemma 1, we have
˙
V
2
(x
t
) = x
T
Q
h
x − x (t − τ )
T
Q
h
x (t − τ)
+
t
Z
t−τ
x (s)
T
˙
Q
h
x (s) ds, (15)
˙
V
3
(x
t
) = τ
2
˙x
T
Z
h
˙x − τ
t
Z
t−τ
˙x (s)
T
Z
h
˙x (s) d
s
+τ
0
Z
−τ
t
Z
t+θ
˙x (s)
T
˙
Z
h
˙x (s) d
s
d
θ
. (16)
Considering the constraints in (12), we have
˙
V (x
t
) ≤ 2ρ
T
P
h
"
˙x (t)
x (t) − x (t − τ )
#
+x (t)
T
Q
h
x (t) − x (t − τ )
T
Q
h
x (t − τ)
+τ
2
˙x (t)
T
Z
h
˙x (t) − τ
r
X
i=1
h
i
t
Z
t−τ
˙x (s)
T
Z
i
˙x (s) d
s
.
Using the zero equation
2M × (A
h
x + A
τ h
x (t − τ) − ˙x) = 0, (17)
M = x
T
M
1h
+ ˙x
T
M
2h
+x (t − τ )
T
M
3h
+
t
Z
t−τ
x (s)
T
d
s
M
4h
,
and the results in [14] to deal with the term
−τ
t
R
t−τ
˙x (s)
T
Z
i
˙x (s) d
s
, we have
˙
V (x
t
) < η
T
Ωη which can
be ensured by (13), where
η
T
=
·
x (t)
T
x (t − τ)
T
t
R
t−τ
x (s)
T
d
s
˙x (t)
T
¸
,
Ω =
Ω
11
Ω
12
Ω
13
Ω
14
∗ Ω
22
Ω
23
Ω
24
∗ ∗ −12τ
−2
Z
h
Ω
34
∗ ∗ ∗ Ω
44
,
Ω
11
= He (P
12h
+ M
1h
A
h
) + Q
h
− 4Z
h
,
Ω
12
= −P
12h
− 2Z
h
+ M
1h
A
τ h
+ A
T
h
M
T
3h
,
Ω
13
= P
22h
+ 6τ
−1
Z
h
+ A
T
h
M
T
4h
,
Ω
14
= P
11h
− M
1h
+ A
T
h
M
T
2h
,
Ω
22
= −Q
h
− 4Z
h
+ He (M
3h
A
τ h
) ,
Ω
23
= −P
22h
+ 6τ
−1
Z
h
+ A
T
τ h
M
T
4h
,
Ω
24
= A
T
τ h
M
T
2h
− M
3h
,
Ω
34
= P
T
12h
− M
4h
, Ω
44
= τ
2
Z
h
− He (M
2h
) .
Remark 1: Note, the Lyapunov-Krasovskii functional used
in Theorem 1 is different from the one used in the liter-
ature such as [11], [17], [18]. In V
2
(x
t
) and V
3
(x
t
) of
this paper, Q
h
and Z
h
are independent of the integral vari-
able s but on the time t, so the information of the time-
derivative of the membership function can be used to obtain
new less conservative stability conditions, at the same time,
there comes a problem that how to deal with the integral
part
t
R
t−τ
x (s)
T
˙
Q
h
x (s) ds and τ
0
R
−τ
t
R
t+θ
˙x (s)
T
˙
Z
h
˙x (s) d
s
d
θ
. A
simple and direct method is letting
˙
Q
h
≤ 0,
˙
Z
h
≤ 0 and
˙
P
h
≤
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. , NO. , JANUARY 4
0. Since
˙
Q
h
=
r−1
P
k=1
˙
h
k
(Q
k
− Q
r
),
˙
Z
h
=
r−1
P
k=1
˙
h
k
(Z
k
− Z
r
),
˙
P
h
=
r−1
P
k=1
˙
h
k
(P
k
− P
r
) , then
˙
Q
h
≤ 0,
˙
Z
h
≤ 0 and
˙
P
h
≤ 0
can be ensured by the switching rules (9). Then, based on
Lemma 2, applying (13) with each C
l
, we obtain a maximum
delay denoted as τ
l
, and the final delay obtained by Theorem 1
is τ = min
1≤l≤2
r−1
(τ
l
). Since Theorem 1 does not depend on the
initial conditions of the fuzzy systems, the switching method is
feasible. In this paper,
˙
X
h
,
˙
Y
h
and
˙
Z
h
are presented as (5), (6)
and (7).Since
r
P
i=1
˙
h
i
= 0, there are some other expressions, for
example,
r−l−1
P
i=1
˙
h
i
+
r
P
j=r−l+1
˙
h
j
= −
˙
h
r−l
, l = 1, 2, · · · r − 1.
We can also use these expressions to derive the results.
B. Stabilization
Theorem 2: For a given scalar τ > 0 and parameters
λ
1
, λ
2
, the closed-loop T-S fuzzy system (2) is asymptoti-
cally stabilized by the controller (3), if there exist matrices
"
¯
P
11j
¯
P
12j
∗
¯
P
22j
#
> 0,
¯
Q
i
> 0,
¯
Z
i
> 0,
¯
M, such that (12) and
the following LMIs hold for i, j = 1, · · · , r,
¯
Ω
ij
+
¯
Ω
ji
≤ 0, (18)
¯
Ω
ij
=
¯
Ω
11ij
¯
Ω
12ij
¯
Ω
13ij
¯
Ω
14ij
∗
¯
Ω
22ij
¯
Ω
23ij
¯
Ω
24ij
∗ ∗
¯
Ω
33ij
¯
P
T
12i
∗ ∗ ∗
¯
Ω
44ij
,
¯
Ω
11ij
= He
¡
¯
P
12i
+
¯
A
¢
+
¯
Q
i
− 4
¯
Z
i
,
¯
Ω
12ij
= −
¯
P
12i
− 2
¯
Z
i
+
¯
A
τ
+ λ
2
¯
A
T
,
¯
Ω
13ij
=
¯
P
22i
+ 6τ
−1
¯
Z
i
,
¯
Ω
14ij
=
¯
P
11i
−
¯
M + λ
1
¯
A
T
,
¯
Ω
22ij
= −
¯
Q
i
− 4
¯
Z
i
+ He
¡
λ
2
¯
A
τ
¢
,
¯
Ω
23ij
= −
¯
P
22i
+ 6τ
−1
¯
Z
i
,
¯
Ω
24ij
= λ
1
¯
A
T
τ
− λ
2
¯
M,
¯
Ω
33ij
= −12τ
−2
¯
Z
i
,
¯
Ω
44ij
= τ
2
¯
Z
i
− He
¡
λ
1
¯
M
¢
,
¯
A=A
j
¯
M + B
j
¯
K
i
,
¯
A
τ
= A
τ j
¯
M + B
j
¯
K
τ i
,
and the controller gains are K
i
=
¯
K
i
¯
M
−1
, K
τ i
=
¯
K
τ i
¯
M
−1
.
Proof: The extension from stability to stabilization is
straightforward. In order to avoid too many parameters, let
¯
M = M
−T
1
, M
2h
= λ
1
M
1
, M
3h
= λ
2
M
1
, M
4h
= 0, using
the controller (3) and replacing A
h
and A
τ h
with A
h
and A
τ h
respectively where
A = A
h
+ B
h
K
h
, A
τ
= A
τ h
+ B
h
K
τ h
.
Pre-and post-multiplying both sides of Ω < 0 with diag{M
−1
1
,
M
−1
1
, M
−1
1
, M
−1
1
} and its transpose respectively and defin-
ing K
i
¯
M =
¯
K
i
, K
τ i
¯
M =
¯
K
τ i
,
¯
M
T
P
11h
¯
M =
¯
P
11h
,
¯
M
T
P
12h
¯
M =
¯
P
12h
,
¯
M
T
P
22h
¯
M =
¯
P
22h
,
¯
M
T
Q
h
¯
M =
¯
Q
h
,
¯
M
T
Z
h
¯
M =
¯
Z
h
, we get (18).
Remark 2: Similar to the analysis in Remark 1, applying
(18) with C
l
, we obtain a maximum delay denoted as τ
l
,
l = 1, · · · , 2
r−1
and the corresponding controller denoted as
(10) The final maximal delay is τ = min
l=1,··· ,2
r−1
(τ
l
) and the
final controller is (11). For any initial states φ (0), it can be
driven to the origin by a sequence of controller u
l
activated
by H
l
. Note, there are 2
r−1
constraints in C
l
and we need
compute 2
r
− 1 times to get the maximum delay bound. For
each time, the number of LMIs is 3×2
r−1
+3+r(r+1)/2 and
the computation burden may increase as the number of rules
increase. The number of decision variables of Theorem 1 is
n(3n+2)r+4n
2
r and Theorem 2 is n(3n+2)r+n
2
r+2mnr.
Remark 3: Comparing with the existing results in the liter-
ature, the merit of this paper is that the Lyapunov-Krasovskii
functional is dependent on the membership functions and
the time derivatives of the membership functions are also
considered. If we let Q
h
, Z
h
and P
h
be membership function
independent as Q, Z, P and the slack variables be M
3h
= 0,
M
4h
= 0, Theorem 1 and Theorem 2 in this paper will
become the results in [11], so this brief paper contains [11]
as a special case. Another possible choice is letting V
2
(x
t
) =
t
R
t−τ
x (s)
T
Q
h(s)
x (s) ds where Q
h(s)
=
r
P
i=1
h
i
(z(s)) Q
i
is de-
pendent on the integral variable s, but it has been shown in [18]
that this choice could not help reduce the conservativeness.
Remark 4: The differences between this paper and [18] are
as follows: 1), The chosen Lyapunov-Krasovskii functional is
different. In this paper, the integrand of V
2
(x
t
) and V
3
(x
t
)
depends not only on the integral variable s but also on the
membership functions, while in [18], V
2
(x
t
) and V
3
(x
t
) are
independent of the membership function. 2), The method to
deal with the time derivatives of the membership function is
completely different. In this paper, a novel switching idea
is applied to ensure the time-derivative of the Lyapunov-
Krasovskii functional is negative, while in [18], the upper
bounds on the time-derivative of the membership functions
defined a priori by using the mod function and floor function
b·c. 3), The applicable scope is different. In this paper, Theo-
rem 1 is global stability and Theorem 2 is global stabilization,
while in [18], the results are only local and the obtained local
stabilization region becomes smaller as the delay increases.
IV. NUMERICAL EXAMPLES
In this section, two examples are presented to demonstrate
the effectiveness of the proposed method. Example 1 is for
stability and Example 2 is for stabilization.
Example 1: Consider the following two-rule fuzzy system
that has been studied in [2], [5], [8], [11], [17],
A
1
=
"
−2 0
0 −0.9
#
, A
2
=
"
−1 0.5
0 −1
#
,
A
τ 1
=
"
−1 0
−1 −1
#
, A
τ 2
=
"
−1 0
0.1 −1
#
.
This open loop fuzzy system has been studied extensively in
the literatures and the goal is to compute the maximum delay
τ under which the fuzzy system is still stable. Table 1 shows
the maximum delay τ obtained by different methods. From
this table, we can see the best result is obtained by applying
