Applied Mathematics, 2011, 2, 1339-1345
doi:10.4236/am.2011.211187 Published Online November 2011 (http://www.SciRP.org/journal/am)
New Approach of Stability for Time-Delay Takagi-Sugeno
Fuzzy System Based on Fuzzy Weighting-Dependent
Lyapunov Functionals
Yassine Manai, Mohamed Benrejeb, Pierre Borne
L.A.R.A. Automatique, Ecole Nationale d’Ingénieurs de Tunis, Le Belvédère, Tunis, Tunisie
Received February 19, 2011; revised May 25, 2011; accepted June 3, 2011
Abstract
This paper deals with the stability of Takagi-Sugeno fuzzy models with time delay. Using fuzzy weighting-
dependent Lyapunov-Krasovskii functionals, new sufficient stability criteria are established in terms of Lin-
ear Matrix Inequality; hence the stability bound of upper bound delay time can be easily estimated. Finally,
numeric simulations are given to validate the developed approach.
Keywords: Takagi-Sugeno Fuzzy Models, Linear Matrix Inequalities LMIs, Fuzzy Weighting-Dependent
Lyapunov-Krasovskii Functionals
1. Introduction
Fuzzy control systems have experienced a big growth of
industrial applications in the recent decades, because of
their reliability and effectiveness. Many researches are
investigated on the Takagi-Sugeno fuzzy models [1-17]
which can associate the flexible fuzzy logic theory and
rigorous mathematical theory into a unified framework.
Thus, it becomes a powerful tool in approximating a
complex nonlinear system.
Recently, Takagi-Sugeno fuzzy model approach has
been used to examine nonlinear systems with time-delay,
and different methodologies have been proposed for ana-
lysis and synthesis of this type of systems [1,2]. Time de-
lay often occurs in many dynamical systems such as bio-
logical systems, chemical system, metallurgical process-
ing system and network system. Their existences are
frequently a cause of infeasibility and poor performan-
ces.
The developed stability approaches can be classified
into two types. The first one is called as the delay inde-
pendent stability criteria which do not include any in-
formation about the size of the time delay, whereas the
second is called as the delay dependent stability criteria,
in which the size of the time delay is taken explicitly in
the formulation. It is generally recognized that delay de-
pendent results are usually less conservative than delay
independent ones, especially when the size of delay is
small.
Two classes of Lyapunov-Krasovskii functionals are
used to analysis these systems: quadratic Lyapunov-
Krasovskii functionals and non-quadratic Lyapunov-
Krasovskii ones. The use of first class brings much con-
servativeness in the stability test. In order to reduce the
conservatism entailed in the previous results using quad-
ratic functionals, a fuzzy weighting dependent approach
is presented recently in [3] for fuzzy systems with fuzzy
weighting functions.
In this paper, a new stability conditions for time-delay
Takagi-Sugeno fuzzy systems by using fuzzy weighting-
dependent Lyapunov-Krasovskii functionals are pre-
sented. We derive delay-dependent stability conditions
using recent technique called free-weighting matrix me-
thod [4]. This criterion is expressed in terms of Linear
Matrix Inequalities (LMIs) which can be efficiently sol-
ved by using various convex optimization algorithms
[6,15]. The proposed approach improves existing ones in
[4].
The organization of the paper is as follows. In Section
2, we present the system description and problem for-
mulation and we give some preliminaries which are
needed to derive results. Section 3 will be concerned to
stability analysis for time-delay T-S fuzzy systems. Illus-
trative example is given in Section 4 for a comparison of
previous results to demonstrate the advantage of pro-
posed method. Finally Section 5 makes conclusion.

Y. MANAI ET AL. 1340
Notation: Throughout this paper, a real symmetric ma-
trix denotes being a positive definite matrix.
The superscript “T” is used for the transpose of a matrix.
And where an ellipsis “” denotes a block induced eas-
ily by symmetry.
0SS
2. System Description and Preliminaries
Consider a T-S fuzzy continuous model with time-delay
for a nonlinear system as follows:




 
 
  
11
0
If is andand is
Then ,,0
ip
ii
zt Mzt M
xtAxt A t

ip
x
t
xtt t










 (1)
where

1, 2,,,1, 2,,
ij

M
irj

m
ut
p is the fuzzy set
and r is the number of model rules; is the
state vector, is the input vector,

n
xt
i
Ann

 ,
are constant real matrices, and
are known premise variables,
nn

,,
p
z t
i
A



1
zt


t

is a
continuous vector-valued initial function on


0,0
 ;
the time-delay may be unknown but is assumed to
be smooth function of time.

t 
 
0
0, ttd   
1,
where 00

 and are two scalars. d
The final outputs of the fuzzy systems are:
      
1
r
iii
i
xth ztAxtAt

x
t





 (2)
  

0
, ,0 ,xtt t  
where
 


12
,,

 p ztz tztzt



     
1
r
ii i
i
hzt wztwzt

,
for all t.




1
p
iij
j
wztMz t



j

The term


1ij
M
zt
1i
is the grade of membership of
in

j
zt
M
Since
 
 
1
0
0, 1,2,,
i
i
i
wzt
wzti r








r

we have
 
 
1
1
0, 1,2,,
r
i
i
i
hzt
hzti r








for all t. (3)
The time derivative of premise membership functions
is given by:
 







1
dd
dd
s
i
ii
l
zt xtxt
h
hzt zt xttt




 
 

lil
(4)
We have the following property:
 
1
0.
r
k
k
hzt


 (5)
Assumption 1
The time derivative of the premises membership func-
tion is upper bounded such that k hk


, for
1, ,kr

, where, ,1,,
kkr


 are given positive
constants.
Lemma 1[4]
Under assumption 1, the time-delay Takagi-Sugeno
fuzzy system is stable if there exist positive definite
symmetric matrices 0,
j
P0,Q0,Z
j
Y and
,
j
T 1, 2,,,jr

 such that the following LMIs hold.

,1,,
kr
PP kr

1
(6)
0, ,
ij jiij

  (7)
where


1
1
00
00
0
0
1
0
ij
r
kk r
k
TTT
j
iijjijjij
T
jj
TT
j
ji
PP
PAA PPAYTAZY
YYQ
TTdQ AZT
Z
j
Z

















 


 




 













Lemma 2[4]
Under assumption1, system (2) with is asym-
ptotically stable if there exist matrices
0 d
0,
j
P
,
0,
j
W
0,
j
S0,Q,U0,R0,Z
j
Y
j
T and
,
j
N 1, 2,,j,r

 such that the following LMIs hold:
, 1,2,,1,
kk rr
kr
PW PW kr
SS
 

 

 

0,
QU
R





0, ,
ij jiij

 
where
 



11 12
00
22
00
0
0
0
,
00
i
ij
TT
ijijij jj
TT
ijj jj
j
ARZW NY
ARZS N UT
RZ
RN
Z









 



  







 








Y. MANAI ET AL.1341




1
11
1



 


r
ijk krji
k
TT T
ij jj
PP PUA
A
PUY YQ



1
12
1i
rTT
ijkkrji jjj
k
WWPUA AWYY








1
22
1
.
ii
rTT T
ijk krjjjj
k
SSTTWAAWQ




 

3. Main Results
Consider the open-loop system (2). The aim of this sec-
tion is to find conditions for the stability of the unforced
time-delay T-S fuzzy system by using the Lyapunov-
Krasovskii theory.
Theorem 1
Under assumption 1, the time-delay Takagi-Sugeno
fuzzy system is stable if there exist positive definite
symmetric matrices 0,
j
P0,Q0,Z,
T
X
X
j
Y and Tj such that the following LMIs
hold.
1, 2,j, ,r

0,1, ,
k
PX k r

 (8)
0, ,
ij jiij  (9)
where


1
00
00
0
0
1
0
ij
r
kk
k
TTT
j
iijjijjij
T
jj
TT
j
ji
PX
PAA PPAYTAZY
YY Q
TTdQ AZT
Z
j
Z













 

 


 












3
Proof
Let consider the fuzzy weighting-dependent Lyapunov–
Krasovskii functional as

12t
VxV VV (10)
where
 

,,
t
xxt 0

  , and

1
1
r
T
jj
j
Vxt htPxt




 (11)
 

2d
t
T
tt
VxsQxs
 
s (12)

0
0
3dd.
tT
t
VxsZxss



 (13)
The Newton-Leibniz formula gives
   

d
d
t
tt
which yields
   
   

2
d0
T
T
t
tt
xt YtxttTt
xtxttxss










 



(14)
where
 
1
r
j
j
j
Yth tY

 and
 
1
r
j
j
j
Tth tT


and where and
j
j
YT
r
are arbitrary matrices with appro-
priate dimensions, 1,2,,j

. With (14), the time de-
rivative of


t
Vx with respect to t along solutions to (2)
is given by:


 
  




  
 
   
 
0
11
0
11
0
2
1
d
2
1
d

 






 

 
 














 

 
t
rr
TT
jj jj
jj
T
T
t
TT
t
rr
TT
kk jj
kj
T
T
TT
t
Vx

x
th tPxtxthtPxt
x
tQxt txttQxtt
xtZxtxsZxs s
htxtPxtxthtPxt
x
tQtxtdxt tQxt t
xtZxtxsZxs s
 


   

0
2
d
















t
T
T
t
tt
xt YtxttTt
xtxttxs s
(15)
where the inequality is caused only by




0
and .td t

 

Based on (5), it follows that 10,
r
k
khX X

 where
X
symmetric matrix of proper dimension. Adding
X
to (15), and by substitution of

x
t
 with state Equation
(2), we have
    
  
   
   
  
   
  


0
1
0
1
d
2
d
rT
tk k
k
T
TTTT
T
T
t
TT
t
T
T
t
tt
VxhtxtPXxt
xtPtAtxtAtxtt
xtA txttAtPtxt
x
tQtxtdxt tQxt t
xtZxtxsZxs s
xt YtxttTt
xtxttxs s

































 
 

 
d,
x
ttxtxs


Then, based on assumption 1 and if (8) holds, (15)
yields to
s

Y. MANAI ET AL. 1342
  
 
 
0
1
0
1
d
TT
t
tTT
t
T
VxttMtZMtt
tMt xsZ
Z
Mtt Zxss




















where
  
,T
TT
txtxtt 







 
,
TTT
,

M
tYtTt




 and

(16)

 
 
 
 
 
 
 
0
0
0
1
T
T
T
T
T
T
T
PPtAtAtPt
PtA tYtTt
Yt YtQ
AtZA t
tAtZAt
Tt TtdQ
At ZAt












 







 






where

1,
r
kk
k
P
PX





 


1,
r
j
j
j
Pth ztP


 


1
r
ii
i
A
thzt

A
and
 

1
r
j
j
j
A
thztA




if (9) holds, that implies


 
 
 
 
  
  
   
1
,1
00
0
00
0
0
11 11
()
(1 )
0
10,
2
r
kk
k
r
T ji ji
ij
T
T
T
T
T
rr rr
ji ijjiijji
ji ji
PX
Pt At
hthtPA
A
tZ Yt
YtTt
YtY tQ
At ZAt
t
Tt Tt
A
tZ Tt
dQ
Z
Z
htht htht










 









 

















 


 

1
(17)
Theorem 2
which is equivalent to by
Schur complement. Therefore, from (16) we have
and (2) is stable.
 
1
00
T
tMtZMt 




0
t
Vx

Under assumption 1, and for 0


0,
j
P
the time-delay
Takagi-Sugeno fuzzy system is stable if there exist posi-
tive definite symmetric matrices
11
0,Q0,Z
,
T
X
X 22 ,
T
X
X Yj and ,
j
T
such
that the following LMIs hold.
1, 2,,jr,
4. Further Fuzzy Weighting-Dependent
Lyapunov-Krasovskii Functional Methods

12
0,1,,
k
PX Xkr   


r
(18)
10, 1,2,, j
PX j


 (19)
In this section, we give a less conservative fuzzy weight-
ing-dependent Lyapunov-Krasovskii functional. The main
development is stated as follows.
0, ,
ij jiij

  (20)
where



12
1
1110
00
0
0
1
0
r
kk
k
TT
0
T
j
iijjijjij
T
jj
ij
TT
j
ji
PX X
PXAAPX PXAYTAZY
YY Q
TTdQAZ T
Z
j
Z



 











 















Y. MANAI ET AL. 1343
Proof
Let consider the fuzzy weighting-dependent Lyapunov-
Krasovskii functional as


12t
VxV VV
3
(21)
where
 

,,
t
xxt 0

  , and
 


11
1
r
T
jj
j
VxthtP Xxt 





 (22)
 

2d
t
T
tt
VxsQxs
 
s (23)
 
0
0
3dd.
tT
t
VxsZxss



 (24)
The Newton-Leibniz formula gives
   

d,
t
tdt
x
tdt xtxss



which yields
 


 



2
d0
T
T
t
tt
xt YtxttTt
xtxttxss









 



(25)
where
 
1
r
j
j
j
Yth tY

 and
 
1
r
j
j
j
Tth tT


and where and
j
j
YT
1, 2,j

are arbitrary matrices with appro-
priate dimensions, . With (14), the time de-
rivative of
,r

t
Vx with respect to t along solutions to (2)
is given by:
    
 


  




 
 


  

 
0
1
1
1
1
0
1
1
1
1
2
1
d
2
r
T
tjj
j
r
T
jj
j
T
T
t
TT
t
rT
kk
k
r
T
jj
j
T
Vxxth tPXxt
xthtPXxt
x
tQxttxttQxtt
xtZxtxsZxs s
htxtPXxt
xthtPXxt
xtQtxt



 





























 


 


 




  
   
   

0
0
1
d
2
d
T
t
TT
t
T
T
t
tt
dxttQxtt
xtZxtxsZxs s
xt YtxttTt
xtxttxs s








 











 

(26)
where the inequality is caused only by




0
and .td t

 

Based on (5), it follows that

11
110,
r
k
khXX




 and 22
10,
r
k
khX X





where 12
X
and X symmetric matrices of proper dimen-
sion. Adding 12
X
and X to (15), and by substitution of


x
t
 with state Equation (2), we have

 

  
 


 


  
   
 



0
12
1
0
1
d
2
d
rT
tk k
k
T
TTTT
T
T
t
TT
t
T
T
t
tt
Vxh txtPXX
xtPtAt xtAtxtt
xtA txttAtPtxt
x
tQtxtdxttQxtt
xtZxtxsZxs s
xt YtxttTt
xtxttxs s


































 


 


where
 



1
1,
r
kk
k
Pth ztPX 




  
1,
r
ii
i
A
thztA


  
1.
r
and
j
j
j
A
thztA




Then, based on assumption 1 and if (8) holds, (15) yields
to
  
 
 
0
1
0
1
d
TT
t
tTT
t
T
VxttMtZMtt
tMt xsZ
Z
Mtt Zxss





















(27)
where
  
,T
TT
txtxtt 






,
 
,
TTT
M
tYtTt




 and


 
 
 
 
 
 
 
0
0
0
1
T
T
T
T
T
T
T
t
PPtAtAtPt
PtAtYtTt
Yt YtQ
AtZA t
At ZAt
Tt TtdQ
At ZAt
















 










 









 

where

12
1
r
kk
k
PPX





X
if (9) holds, that implies

Y. MANAI ET AL. 1344

 


 
 
  
 
 
 

1
,1
00
0
00
0
0
11
11
()
(1 )
0
1
2
0,
r
ji jiT
ij
T
T
T
T
T
rr
ji ij
ji
rr
j iijji
ji
PhthtPXA
Pt At
A
tZ Yt
YtT t
YtY tQAtZAt
Tt Tt
t
A
tZ Tt
dQ
Z
Z
htht
htht
















 























(28)
which is equivalent to by
 
1
00
T
tMtZMt 



Schur complement. Therefore, from (16) we have
and (2) is stable.

0
t
Vx

7. Numerical Examples
In order to show the improvements of proposed approa-
ches over some existing results, in this section, we pre-
sent two numerical examples, which concern the feasi-
bility of a time delay T-S fuzzy system. Indeed, we com-
pare our fuzzy weighting-dependent Lyapunov-Krasov-
skii approach (Theorem 1 and Theorem 2) with the
Lemma 1 in [2].
Example 1. Consider the following T-S fuzzy system
with time-delay:
   

 
  
1
0
, ,0,
r
iii
i
xth ztAxtAt
x
t
xtt t








(29)
with: 2r
1
3.2 0.6
02.1
A




2
0.9 0
11.6
A




,, ,
.
2
10
13
A




1
10.9
02
A




The purpose is to compute the maximum value of 0

such that the fuzzy system is stable. Table 1 lists the
computation results using delay-dependent conditions in
Theorem1, and in Theorem 2 (0.1

) compared with
the existing delay-dependent method in Lemma1 in [2],
for different values of d. It is seen from Table 1 that the
largest values of 0

are obtain by using our methods.
Example 2. Consider the time delay T-S fuzzy system
(29).
with: 2r
1
54
12
A










,2
24
20 2
A









, ,
1
0.5 0.6
10.4
A




2
0.3 0
0.5 0
A

.4






.
This example shows that the stability cannot be tested
by quadratic methods while can be verified by our fuzzy
weighting-dependent methods in Theorems.
The purpose is to compute the maximum value of 0

such that the fuzzy system is asymptotically stable.
Table 2 lists the computation results for different val-
ues of d under the same upper bound 0.8
i

 . It
reveals from Table 2 that Theorem 1 and Theorem 2
with 0.1


produces better results than Lemma 1 and
Lemma 2.
8. Conclusions
This paper provided new conditions for Delay-dependent
Table 1. Comparison results of maximum τ0 for Example 1.
Methods d = 0 d = 0.02 d = 0.1d = 0.9
Lemma 10.61850.5618 0.48090.4513
Theorem 10.61850.5618 0.48100.4530
β = 1
Theorem 2+∞ +∞ 0.55000.4689
Lemma 10.62480.5630 0.48140.4537
Theorem 10.62480.5630 0.48170.4562 β = 0.5
Theorem 2+∞ +∞ 0.55000.4695
Table 2. Comparison results of maximum τ0 for Example 2.
Methods d = 0 d = 0 .5 d = 0.9
Lemma 1 0.3883 0.3225 0.2518
Lemma 2 +∞ 0.5221 0.2844
Theorem 1 +∞ 0.6933 0.3058
β = 0.8
Theorem 2 +∞ +∞ 0.8133

Y. MANAI ET AL.1345
stability problems of time-delay Takagi-Sugeno fuzzy
systems in terms of a combination of the LMI approach
and the use of fuzzy weighting-dependent Lyapunov-
Krasovskii functionals.
The stability conditions proposed in this note are less
conservative than some of those in the literature, which
has been illustrated via examples.
9. References
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