Delay dependent robust stability of T-S fuzzy systems with additive time varying delays

Abstract

This paper investigates the problem of robust stabilization for uncertain Takagi-Sugeno (T-S) fuzzy systems with additive time varying delays. An appropriate Lyapunov-Krasovskii function is considered for solving this problem, and obtains considerably less conservative results than existing methods. The proposed approach constructs a new Lyapunov-Krasovskii functional using two additive delay components, and no free weighting matrices are employed in the theoretical result derivation. This reduces the number of scalar decision variables in linear matrix inequalities. The fuzzy state feedback gain is derived through the numerical solution of a set of linear matrix inequalities (LMIs). Finally, numerical examples are provided to illustrate the effectiveness of the proposed method, and to allow comparison with previous works. © 2017 International Journal of Automation and Smart Technology.

Full text

Applied Mathematical Sciences, Vol. 6, 2012, no. 1, 1 - 12
Delay Dependent Robust Stability of T-S Fuzzy
Systems with Additive Time Varying Delays
Idrissi Said
LESSI. Department of Physics, Faculty of Sciences B.P. 1796 Fès-Atlas
Said_idrissi09@yahoo.fr
El Houssaine Tissir
LESSI. Department of Physics, Faculty of Sciences B.P. 1796 Fès-Atlas
elh_tissir@yahoo.fr
Abstract
This paper presents delay dependent stability conditions of T-S fuzzy systems
with additive time varying delays. The approach is based on constructing a new
Lyapunov-Krasovskii functional, and Finsler’s lemma. The perturbations
considered are norm bounded and the results are expressed in terms of LMIs.
Numerical examples are provided to show the effectiveness of the present
technique, compared to some recent results.
Keywords: Additive time varying delays, T-S fuzzy system, Liner matrix
inequality (LMI), Robust stability
1. Introduction
During the past two decades the stability analysis for Takagi-Sugeno (T-S) fuzzy
systems [23] has been studied extensively. Lots of stability criteria of T-S fuzzy
systems have been expressed in linear matrix inequality LMIs via different
approaches [6, 24, 26]. These fuzzy systems are described by a family of fuzzy
IF–THEN rules. However, all the aforementioned methods are proposed for time-
delay free T–S fuzzy systems. In practice time delay often appears in many
practical systems such as chemical processes, metallurgical process, long
transmission lines in pneumatic, mechanics, and communications networks, etc.
[1,9]. Since time delay, is usually a source of instability and degradation of

2 Idrissi Said and El Houssaine Tissir
systems performance, the analysis and synthesis issues of fuzzy systems with time
delay has received more attention in recent years [13, 5, 16, 19].
Some approaches developed for general delay systems have been applied to deal
with fuzzy systems with time delays, e.g., the Lyapunov–Krasovskii functional
approach [2, 3], Li et al. [13]. Applying the model transformation also called
Moon’s inequality [18] for bounding cross terms, Guan and Chen [4] have studied
the delay-dependent robust stability and guaranteed cost control of the time-delay
fuzzy systems. Recently a free weighting matrix approach has been employed in
[5, 16, 11, 15]. In Liu et al. [17], the problem of stability for uncertain T-S fuzzy
systems with time varying delay has been studied by employing a further
improved free weighting matrix method. The free weighting matrix approach has
been shown to be less conservative than the previous approaches.
In the literature the fuzzy systems with time-varying delay have been modelled as
a system with a single delay term in the state. Recently in [8, 12, 22], it was noted
that in networked controlled system, successive delays with different properties
are introduced in the transmission of signals between different points through
different segments of networks. Thus it is appropriate to consider different time-
delays )(
1t
τ
and )(
2t
τ
in the same state where, )(
1t
τ
is the time-delay induced
from sensor to controller and )(
2t
τ
is the delay induced from controller to the
actuator.
In this work, motivated by the above idea, we derive a new and improved delay-
dependent condition for asymptotic stability of T-S fuzzy system with two
additive delay components. The condition is extended to cover systems with norm
bounded uncertainties. The sufficient conditions for asymptotic stability and
robust stability analysis are derived by using Lyapunov-Krasovskii functional
method and making use of improved technique and Finsler’s lemma. By solving a
set of LMIs, the upper bounds of the time delays can be obtained. We provide two
illustrative examples to show that the new stability conditions proposed in this
paper are less conservative.
Lemma 1 [7]: Consider a vector n
R
χ
∈, a symmetric matrix nn
QR
×
∈ and
matrix mn
R
×
Β∈ , such that ()rank n
Β
<. The following statements are equivalent:
i. 0
TQ
χχ
<,
χ
∀ such that 0
χ
Β
=, 0
χ
≠
ii. 0
TQ
¬¬
ΒΒ<
iii.
R
μ
∃∈ : 0
T
Q
μ
−ΒΒ<
iv. :0
nm T T
FR QF F
×
∃∈ +Β+Β <
Where ¬
Β denotes a basis for the null-space of
Β
Lemma 2 [14]: for any constant matrix Tnn
MM R
×
=
∈, M0>, scalar
(t) 0γ≥η > , vector function
[
]
n
:0, Rωγ→ such that the integrations in the
following are well defined, then:

Delay dependent robust stability 3
T
(t) (t) (t)
T
000
(t) ( )M ( )d ( )d M ( )d
ηηη
⎡
⎤⎡ ⎤
η ωβ ωββ≥ ωββ ωββ
⎢
⎥⎢ ⎥
⎣
⎦⎣ ⎦
∫∫∫
Lemma 3 [20]: Let T
QQ =, H, E, and )t(F satisfying I)t(F)t(FT≤ are
appropriately dimensioned matrices, the following inequality :
0H)t(FEE)t(HFQ TTT <++
Is true, if and only if the following inequality holds for any matrix 0Y >,
0YEEHHYQ TT1 <++ −.
2. System description
Consider a T–S fuzzy time-varying delay system, which can be described by a T–
S fuzzy model, composed of a set of fuzzy implications, and each implication is
expressed by a linear system model. The ith rule of the T–S fuzzy model is
described by following IF – THEN form:
Plant Rule i:
IF )t(z1 is i
1
W and … and )t(zg is i
g
W THEN
⎩
⎨
⎧
=−∈φ=
−−Δ++Δ+=
r,...,2,1i],0,h[t),t()t(x
))t(h)t(ht(x))t(AA()t(x))t(AA()t(x 21didii0i0
&
(1)
where )t(z1, )t(z 2, … , )t(z g are the premise variables, and i
j
W , g,....,2,1j =
are fuzzy sets, n
R)t(x ∈ is the state variable,
r
is the number of if-then rules,
)t(
φ is a vector-valued initial condition, )t(h1and )t(h 2 is the time-varying
delays satisfying
11 h)t(h0 ≤≤ , 11 d)t(h ≤
&, 22 h)t(h0 ≤≤ , 22 d)t(h ≤
&,21 hhh += and
21 ddd += (2)
The parametric uncertainties )t(A i0
Δand )t(A di
Δ
are time-varying matrices with
appropriate dimensions, which can be described as :
[]
[
]
dii0iidii0 EE)t(FD)t(A)t(A
=
ΔΔ , r,...,2,1i
=
(3)
Where i
D,i0
E, di
E are known constant real matrices with appropriate dimensions
and )t(Fiare unknown real time-varying matrices with Lebesgue measurable
elements bounded by:
I)t(F)t(F i
T
i≤, r,...,2,1i
=
(4)
By using the center-average deffuzzifier, product inference and singleton
fuzzifier, the global dynamics of T-Z fuzzy system (1) can be expressed as

4 Idrissi Said and El Houssaine Tissir
))]t(h)t(ht(x))t(AA()t(x))t(AA))[(t(z(µ)t(x 21didi
r
1i
i0i0i −−Δ++Δ+= ∑
=
& (5)
Where,
∑
=
ωω= r
1i
iii ))t(z(/))t(z())t(z(µ , ∏
=
=ω
g
1j
j
i
ji ))t(z(W))t(z(
And ))t(z(W j
i
jis the membership value of )t(z j in i
j
W, some basic properties of
))t(z(µi are 0))t(z(µi≥, 1))t(z(µ
r
1i
i=
∑
=
.
3. Main results
In this section, we will obtain the stability criteria for T-S fuzzy time-varying
delay systems with two additive time varying delay based on a new Lyapunov-
Krasovskii functional approach. First the following nominal system of system (5)
will be considered:
⎩
⎨
⎧
−∈φ=
−−+=
]0,h[t),t()t(x
))t(h)t(ht(xA)t(xA)t(x 21d0
&
(6)
Where i0
r
1i
i0 A))t(z(µA ∑
=
= and di
r
1i
id A))t(z(µA ∑
=
=
Theorem 1: The system described by (6) and satisfying conditions (2) is
asymptotically stable if there exist symmetric positive definite matrices P , 1
Q,
2
Q, 1
R, 2
R and any appropriately dimensioned matrices, 0
F, 1
F, 2
F, such that
R1-R2>0 and the following LMIs are feasible for r,...,2,1i
=
0
***
FAF**
0Q
h
1
*
FAFPFAAFQ
h
1
44
T
2
T
di133
2
2
22
T
2
T
i00
T
1
T
i0di01
1
11
i<
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
Φ
+−Φ
Φ
+−+Φ
=Φ (7)
Where,
T
O
T
i0i0011
1
11 FAAFRQ
h
1+++−=Φ

Delay dependent robust stability 5
)RR)(d1(Q
h
1
Q
h
1
2112
2
1
1
22 −−−−−=Φ
T
1
T
didi12212
2
33 FAAFR)dd1(Q
h
1++−−−−=Φ
T
22221144 FFQhQh −−+=Φ
Proof: Define the following Lyapunov–Krasovskii functional
)x(V)x(V)x(V)x(V t3t2t1t
+
+
= (8)
Where,
)t(Px)t(x)x(V T
t1 = (9)
θ+θ= ∫∫∫∫
−
−− θ+
−θ+
dsd)s(xQ)s(xdsd)s(xQ)s(x)x(V
1
211
h
hh
t
t
2
T
0
h
t
t
1
T
t2 &&&& (10)
∫∫
−
−−−
+=
)t(ht
)t(h)t(ht
2
T
t
)t(ht
1
T
t3
1
211
ds)s(xR)s(xds)s(xR)s(x)x(V (11)
Computing the time derivative of (9)-(11) one obtain,
)t(Px)t(x2)x(V T
t1 &
&= (12)
[][]
θθ+θ+−+θθ+θ+−= ∫∫
−
−−−
d)t(xQ)t(x)t(xQ)t(xd)t(xQ)t(x)t(xQ)t(x)x(V
1
211
h
hh
2
T
2
T
0
h
1
T
1
T
t2 &&&&&&&&
&
∫∫
−
−−
−+−=
1
1
ht
ht
2
T
2
T
2
t
ht
1
T
1
T
1ds)s(xQ)s(x)t(xQ)t(xhds)s(xQ)s(x)t(xQ)t(xh &&&&&&&& (13)
For any symmetric positive definite matrices 1
Qand 2
Qthe following inequalities
always hold, see [22].
∫∫ −
−
−≤−
t
)t(ht
1
T
t
ht
1
T
11
ds)s(xQ)s(xds)s(xQ)s(x &&&&
∫∫
−
−
−
−
−≤−
)t(ht
)t(ht
2
T
ht
ht
2
T
11
ds)s(xQ)s(xds)s(xQ)s(x &&&&
Where )t(h)t(h)t(h 21 +=
Applying the above inequalities to the integral terms in (13) one obtain,
∫∫
−
−−
−+−≤
)t(ht
)t(ht
2
T
2
T
2
t
)t(ht
1
T
1
T
1t2
1
1
ds)s(xQ)s(x)t(xQ)t(xhds)s(xQ)s(x)t(xQ)t(xh)x(V &&&&&&&&
&(14)
By using lemma 2 we obtain:

6 Idrissi Said and El Houssaine Tissir
[][]
[][]
)t(ht(x))t(ht(xQ)t(ht(x))t(ht(x
h
1
)t(xQ)t(xh
)t(ht(x)t(xQ)t(ht(x)t(x
h
1
)t(xQ)t(xh)x(V
12
T
1
2
2
T
2
11
T
1
1
1
T
1t2
−−−−−−−+
−−−−−≤
&&
&&
&
[]
[
)t(ht(xQ))t(ht(x2))t(ht(xQ))t(ht(x
h
1
)t(xQ)t(xh
)t(ht(xQ)t(ht(x)t(ht(xQ)t(x2)t(xQ)t(x
h
1
)t(xQ)t(xh
21
T
121
T
2
2
T
2
111
T
11
T
1
T
1
1
T
1
−−−−−−+
−−+−−−≤
&&
&&
]
)t(ht(xQ)t(ht(x 1
T−−+ (15)
))t(ht(xR))t(ht(x))t(h)t(h1())t(ht(xR))t(ht(x))t(h1(
))t(ht(xR))t(ht(x))t(h1()t(xR)t(x)x(V
2
T
21121
T
1
111
T
11
T
t3
−−−−−−−−+
−−−−=
&&&
&
&
))t(ht(xR))t(ht(x)dd1(
))t(ht(x)RR))(t(ht(x)d1()t(xR)t(x
2
T
21
1211
T
11
T
−−−−−
−−−−−≤ (16)
Where we assume that R1>R2. Let
()
T
TT
1
TT )t(x))t(ht(x))t(ht(x)t(x)t( &
−−=ξ
Taking account of (12), (15) and (16), and letting
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
+
−−−−
−−−−−
+−
=Ψ
2211
2212
2
2
2
2112
2
1
1
1
1
11
1
QhQh***
0R)dd1(Q
h
1
**
0Q
h
1
)RR)(d1(Q
h
1
Q
h
1
*
P0Q
h
1
RQ
h
1
(17)
We obtain:
)t()t()x(V T
tξΨξ≤
& (18)
Now let
[
]
IA0AB
~
d0 −= and
[
]
T
T
2
T
1
T
0FF0FF =. Then, since
()
1)t(zµ
r
1i i=
∑
=
we can verify that 0B
~=ξ , 0
≠
ξ
∀
. Since condition (7) holds, it
follows that the matrices 0FB
~
B
~
FTT <++Ψ and therefore by lemma 1 we have
0)t()t(
T<ξΨξ which implies that 0)x(V t<
&. This completes the proof. 

Delay dependent robust stability 7
Theorem 2: The uncertain system (5) satisfying conditions (2) is robustly stable if
there exist symmetric positive definite matrices P , 1
Q, 2
Q, 1
R, 2
R, Y and any
appropriately dimensioned matrices, 0
F, 1
F, 2
F, such that R1-R2>0 and the
following LMIs are feasible for i=1, …,r.
0
Y****
DF***
DFFAFYEE**
00Q
h
1
*
DFFAFPYEEFAAFQ
h
1
YEE
i244
i1
T
2
T
di
1di
T
di
33
2
2
22
i0
T
2
T
i0
0di
T
i0
T
1
T
i0
di01
1
i0
T
i0
11
<
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
Φ
+−+Φ
Φ
+−+++Φ
(19)
Where 11
Φ, 22
Φ, 33
Φand 44
Φ are defined in (7).
Proof: Replacing i0
A and di
A by i0iii0 E)t(FDA
+
and diiidi E)t(FDA
+
in (7),
respectively, the corresponding formula of (7) for system (5) can be rewritten as
follows:
0H)t(FEE)t(HF TT
i
T
ii <++Φ (20)
Where
[
]
T
2
T
i
T
1
T
i
T
0
T
i
TFDFD0FDH = and
[
]
0E0EE dii0
=
. According to
Lemma 3, (20) is true If there exist 0Y >, such that the following inequality
holds:
0YEEHHY TT1
i<++Φ − (21)
By Schur complement, (21) is equivalent to (19). This completes the proof.
Remark 1. To the best of our knowledge, all the results studying T-S fuzzy
systems with time delay consider systems with single delay term as:
IF )t(z1 is i
1
W and … and )t(zg is i
g
W THEN
⎩
⎨
⎧
=−∈φ=
−Δ++Δ+=
r,...,2,1i],0,h[t),t()t(x
))t(ht(x))t(AA()t(x))t(AA()t(x didii0i0
&
Where h)t(h0 ≤≤ and d)t(h ≤
&, and there is no results dealing with additive
time varying delay.
Remark 2. For time delay systems with single delay term, free weighting
matrices approach has been used in [5, 16, 15, 17, 25] and less conservative
results have been established compared with Moon’s inequality approach
employed in [4, 10]. In [19] stability criteria for T-s Fuzzy systems with delay

8 Idrissi Said and El Houssaine Tissir
have been developed by employing neither free weighting matrices nor model
transformation and derived less conservative results than those in the above
references. In this paper, we use a new Lyapunov Krasovskii functional and our
method is based on Finsler’s Lemma. Comparing theorem 1 with corollary 2 of
[19], concerning the stability of system (6) with )t(h)t(h)t(h 21 =
+
and
h)t(h0 ≤≤ , the numbers of variables required in theorem 1 and corollary 2 are
2
)1n(n5
n3 2+
+ and n2n6 2+ respectively. It can be seen that theorem 1
requires less number of variable that is 2
)1n(n
−
. Consequently, with our results
the computational demand on searching for the solution of stability conditions can
be alleviated. This advantage can be revealed especially for systems with large
dimension n. A second advantage of our approach is that we expect a reduced
conservatism. This is illustrated in the examples
4. Numerical examples
In this section, we aim to demonstrate the effectiveness of the proposed approach
presented in this paper by theorem 1 and theorem 2.
Example 1: Consider a system with the following rules:
Rule 1: If )t(z1is 1
W, then
))t(h)t(ht(xA)t(xA)t(x 211d01
−
−+=
&
If )t(z 2is 2
W, then
))t(h)t(ht(xA)t(xA)t(x 212d02
−
−+=
&
And the membership functions for rule 1 and rule 2 are
))t(z2exp(1
1
))t(z(
1
11 −+
=μ , ))t(z(1))t(z( 1112
μ
−
=
μ
Where,
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=9.00
02
A01 , ⎥
⎦
⎤
⎢
⎣
⎡
−−
−
=11
01
A1d , ⎥
⎦
⎤
⎢
⎣
⎡
−
−
=10
5.01
A02 , ⎥
⎦
⎤
⎢
⎣
⎡
−
−
=11.0
01
A2d
Applying Theorem 1, we fix different values of 1
h and search for the corresponding
upper bounds 2
h of )t(h 2. Hence we fix different values of 2
h and search for the
corresponding upper bounds 1
h of )t(h1. In order to compare with the literature

Delay dependent robust stability 9
results, since to our knowledge, there is no results dealing with fuzzy systems
with additive delays, we let )t(h)t(h)t(h 21
=
+
such that h)t(h0 ≤≤ . Then we
apply the literature conditions. From our results we compute h as in (2). The
results are summarized in table 1.
Table 1: Upper bound h for invariant delays
Method Upper bound h
Wang et al.[21] 1.597
Tian and Peng
[25] Corollary 1
1.597
Chen et al. [5] 1.597
Fan et al. [17] 1.597
This paper upper bound 2
hfor given 1
h upper bound
1
hfor given 2
h
Theorem 1 of
this paper 1
h= 1 1
h= 1,2 1
h= 1,5 2
h= 0,2 2
h= 0,3 2
h= 0,5
0,897 0,667 0,208 1,504 1,449 1,323
We can see that the maximum allowable upper bound, 21 hhh += obtained by
theorem 1 is greater than the bound h obtained by the results of [21, 25, 5, 17].
Our approach leads to less restrictive condition although computing 21 hhh +=
may be conservative, in fact the delays h1(t) and h2(t) may have sharply different
properties and when )t(h)t(h 21 + reaches its maximum, we do not necessarily
have both h1(t) and h2(t) reach their maximum at the same time.
Example 2: Consider the following uncertain fuzzy system with two additive
time varying delay:
[]
∑
=
−−Δ++Δ+μ= 2
1i
21didii0i01i ))t(h)t(ht(x))t(AA()t(x))t(AA())t(z()t(x
&
where,
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=11.0
12
A01 ,⎥
⎦
⎤
⎢
⎣
⎡
−−
−
=11
5.01
A1d , ⎥
⎦
⎤
⎢
⎣
⎡
−
−
=10
02
A02 , ⎥
⎦
⎤
⎢
⎣
⎡
−
−
=10
06.1
A2d
⎥
⎦
⎤
⎢
⎣
⎡
=05.00
06.1
E01 , ⎥
⎦
⎤
⎢
⎣
⎡
=3.00
03.0
E1d , ⎥
⎦
⎤
⎢
⎣
⎡
−
=05.00
5.06.1
E02 , ⎥
⎦
⎤
⎢
⎣
⎡
−
−
=3.00
1.01.0
E2d
⎥
⎦
⎤
⎢
⎣
⎡
−
=05.00
005.0
Di

10 Idrissi Said and El Houssaine Tissir
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
π−−+
×
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
π−−+
−=μ ))2/5.0/)t(z(3exp(1
1
))2/5.0/)t(z(3exp(1
1
1))t(z(
11
11 ,
))t(z(1))t(z( 1112 μ−=μ
Employing Theorem 2 of this paper, we calculate the upper bound 1
hof )t(h1or
2
hof )t(h 2, when the other is known. The upper bound of h is obtained by
summing the two delay bounds 1
h and 2
h . In order to compare with the literature
results, we consider the above system as an uncertain fuzzy systems with a single
delay term )t(h , i.e., )t(h)t(h)t(h 21
=
+ satisfying h)t(h0 ≤≤ . Then we apply
the literature conditions. The results are presented in table 2.
Table 2: Upper bound h for invariant delay
Method Upper bound h
Chen et al. [5] 1.431
Fan et al. [17] 1.439
upper bound 2
hfor given 1
h upper bound
1
hfor given 2
h
Theorem 2 of this
paper 1
h= 1 1
h= 1.1 1
h= 1.3 2
h= 0.6 2
h=0.7 2
h=0.8
0.615 0.495 0.220 1.013 0.924 0.828
The results guarantee the stability of uncertain fuzzy system for invariant delays.
It is shown that the upper bound h obtained by theorem 2 is better then those
obtained by single delay approach in [5, 17].
5. Conclusion
We have established delay dependent conditions for asymptotic stability of T-S
fuzzy systems with two additive time varying delays. The results are extended to
cover the class of uncertain T-S fuzzy systems. The perturbations considered are
assumed to be norm-bounded. The LMIs proposed have been obtained by utilizing
a new Lyapunov Krasovskii functional and Finsler’s lemma. The less
conservativeness of the results is shown by two numerical examples in which we
obtained a large delay upper bound as shown in table 1 and table 2.
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Received: May, 2011