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An LMI approach for global robust dissipativity analysis of T–S fuzzy neural networks with interval time-varying delays
Expert Systems with Applications 39 (2012) 3345–3355
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An LMI approach for global robust dissipativity analysis of T–S fuzzy neural networks with interval time-varying delays S. Muralisankar a,⇑, N. Gopalakrishnan a, P. Balasubramaniam b a b
School of Mathematics, Madurai Kamaraj University, Madurai 625 021, Tamilnadu, India Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India
a r t i c l e
i n f o
Keywords: Global dissipativity T–S fuzzy model Neural networks Linear matrix inequality Time-varying delays
a b s t r a c t Takagi–Sugeno (T–S) fuzzy models are often used to represent complex nonlinear systems by means of fuzzy sets and fuzzy reasoning applied to a set of linear sub-models. In this paper, the global robust dissipativity of T–S fuzzy neural networks with interval time-varying delays are investigated. By constructing a proper Lyapunov–Krasovskii functional and using linear matrix inequality (LMI) technique, delaydependent criteria for checking the global dissipativity and global exponential dissipativity of fuzzy neural networks have been derived in terms of LMI, which can be solved numerically using LMI toolbox in MATLAB. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results. 2011 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, dynamics of neural networks (NNs) have been widely studied due to their extensive applications in aerospace, defense, robotic, telecommunications, signal processing, pattern recognition etc. (Feng, Yang, & Wu, 2009; Haykin, 1998; Li, 2010a, 2010b). Fuzzy logic theory was shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. Takagi and Sugeno (1985) proposed an effective way to transform a nonlinear dynamic system to a set of linear sub-models via some fuzzy models by defining a linear input/output relationship as its consequence of individual plant rule. In Cao and Frank (2000), the standard T–S fuzzy model was extended to one with time delays and some stability conditions were presented in terms of LMIs. Recently, the Lyapunov– Krasovskii approach and the Lyapunov–Razumikhin method have been used to study the stability of delayed fuzzy systems (Cao & Frank, 2000, 2001). Moreover, the concept of incorporating fuzzy logic into NNs has grown into a popular research topic (Huang, 2006; Liu & Tang, 2004). The notion of dissipative dynamical system was first introduced by Willems (1972), and it is subsequently generalized in Zhang, Yan, and Chen (2010) via various approaches. Dissipative systems theory has wide ranging implications and applications in control theory. Applications of dissipativeness in the stability analysis of linear systems with certain nonlinear feedback were first discussed in Willems (1972). In addition, dissipativeness was crucially used in the stability analysis of nonlinear systems (Hill & Moylan, ⇑ Corresponding author. Tel.: +91 452 2452371; fax: +91 451 2453071. E-mail address: [email protected] (S. Muralisankar). 0957-4174/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.09.021
1976). The theory of dissipative systems generalizes basic tools including the passivity theorem, bounded real lemma, Kalman Yakubovich lemma and the circle criterion (Tan, Soh, & Xie, 1999). It is well known that the stability problem is central to the analysis of a dynamic system, where various types of stability of an equilibrium point have captured the attention of researchers. Nevertheless, from a practical point of view, it is not always the case that every NN has its orbits approaching a single equilibrium point. It is possible that there is no equilibrium point in some situations. Therefore, the concept on dissipativity has been introduced in Hale (1988). The concept of dissipativity in dynamical systems is a more general concept and it has found applications in various areas such as stability theory, chaos and synchronization theory, system norm estimation and robust control. Recently, many interesting results have been proposed for the dissipativity of delayed NNs (Arik, 2004; Cao, Yuan, Ho, & Lam, 2006; Huang, Xu, & Yang, 2007; Liao & Wang, 2003; Lou & Cui, 2008; Masubuchi, 2006; Song & Cao, 2008, 2010; Song & Zhao, 2005; Wang, Cao, & Wang, 2009; Zhang et al., 2010). Motivated by the above discussions, we shall generalize the ordinary dissipativity analysis on uncertain NNs to express dissipativity of T–S fuzzy uncertain NNs with interval time-varying delays. The main purpose of this paper is to study the global robust dissipativity of T–S fuzzy NNs with interval time-varying delays. To the best of authors knowledge, there were no results for global robust dissipativity analysis of T–S fuzzy NNs with mixed interval timevarying delays in terms of LMIs, which can be easily solved by MATLAB LMI toolbox. The main advantage of the LMI based approaches is that the LMI conditions can be solved numerically using the effective interiorpoint algorithms. We also provide numerical examples to demonstrate the effectiveness of the proposed dissipativity results.
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Notations: throughout this paper, Rn and Rnn denote, respectively, the n-dimensional Euclidean space and the set of all n n real matrices. The superscript T denotes the transposition and the notation X P Y (respectively, X > Y), where X and Y are symmetric matrices, means that X Y is positive semi-definite (respectively, positive definite). In is the n n identity matrix. k k is the Euclidean norm in Rn . The notation ⁄ always denotes the symmetric block in one symmetric matrix. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise. 2. Problem description and preliminaries In this paper, we consider the following neural network with mixed time-varying delays
dxðtÞ ¼ AxðtÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt sðtÞÞÞ þ W 3 dt Z t f ðxðsÞÞds þ u
ð1Þ
trðtÞ
for t P 0, where xðtÞ ¼ ½x1 ðtÞ; . . . ; xn ðtÞT 2 Rn is the state vector of the network at time t, n corresponds to the number of neurons; A = diag(a1, a2, . . . , an) > 0 is a positive diagonal matrix; W1 = (aij)nn, W2 = (bij)nn and W3 = (cij)nn represent the connection weight matrix, the discretely delayed connection weight matrix and the distributively delayed connection weight matrix, respectively; f(x(t)) = [f1(x1(t)), . . . , fn(xn(t))]T denotes the neuron activation function at time t; u ¼ ðu1 ; . . . ; un ÞT 2 Rn is a constant external input vector; and s(t) and r(t) denote the discrete and distributed timevarying delays, respectively. Assumption (H1). The time-varying delays s(t) and r(t) satisfy
0 6 h1 6 sðtÞ < h2 ;
0 6 rðtÞ 6 r;
½DAk ðtÞ DW 1k ðtÞ DW 2k ðtÞ DW 3k ðtÞ h i 1 2 3 EW EW ; ¼ Gk F k ðtÞ EAk EW k k k
W2 1 3 where Gk ; EAk ; EW and EW are known real constant matrices k ; Ek k with appropriate dimensions, and Fk(t) is the time-varying uncertain matrix which satisfies that
F Tk ðtÞF k ðtÞ 6 I:
ð4Þ
Remark 2.1. In Song and Cao (2008), authors studied the problem of global dissipativity analysis on uncertain NNs with mixed timevarying delays. This paper attempts at the first time to dealt with T–S fuzzy NNs with interval time-varying delays. In this paper, the global robust dissipativity problem is investigated for T–S fuzzy NNs with interval time-varying delays. Thus, our results extend the existing ones. The defuzzified output of the T–S fuzzy system (2) is represented as follows
( r dxðtÞ X ¼ xk ðhðtÞÞ ðAk þ DAk ðtÞÞxðtÞ þ ðW 1k þ DW 1k ðtÞÞf ðxðtÞÞ dt k¼1 þðW 2k þ DW 2k ðtÞÞ f ðxðt sðtÞÞÞ ) Z t f ðxðsÞÞds þ u ; þðW 3k þ DW 3k ðtÞÞ where p Y t ðhðtÞÞ ; tk ðhðtÞÞ ¼ xk ðhðtÞÞ ¼ Pr k gkj ðhj ðtÞÞ j¼1 tj ðhðtÞÞ j¼1
in which gkj ðhj ðtÞÞ is the grade of membership of hj ðtÞ in gkj . According to the theory of fuzzy sets, we have
s_ ðtÞ 6 l;
tk ðhðtÞÞ P 0; k ¼ 1; 2; . . . ; r; Assumption (H2). For any j 2 {1, 2, . . . , n}, fj(0) = 0 and there exist þ constants lj and lj such that
fj ða1 Þ fj ða2 Þ þ 6 lj a1 a2
tk ðhðtÞÞ > 0 for all t:
Therefore, it implies r X
xk ðhðtÞÞ ¼ 1 for all t:
k¼1
Plant Rule k: IF h1(t) is gk1 and . . . and hp(t) is gkp THEN
dxðtÞ ¼ ðAk þ DAk ðtÞÞxðtÞ þ ðW 1k þ DW 1k ðtÞÞf ðxðtÞÞ dt þ ðW 2k þ DW 2k ðtÞÞf ðxðt sðtÞÞÞ Z t f ðxðsÞÞds þ u; þ ðW 3k þ DW 3k ðtÞÞ
ð2Þ
trðtÞ
t 2 ½q; 0;
r X k¼1
xk ðhðtÞÞ P 0; k ¼ 1; 2; . . . ; r;
for all a1 – a2. The kth rule of the T–S fuzzy neural network with parameter uncertainties is of the following form:
xðtÞ ¼ uðtÞ;
ð5Þ
trðtÞ
where h1, h2, r and l are constants.
lj 6
ð3Þ
q ¼ maxðh2 ; rÞ; k ¼ 1; 2; . . . ; r;
where gki ði ¼ 1; 2; . . . ; pÞ is the fuzzy set, h(t) = [h1(t), . . . , hp(t)]T is the premise variable vector, r is the number of IF–THEN rules. The norm is _ defined by kxkq ¼ maxfsupq6t60 kxðtÞk; suph6t60 kxðtÞkg. Ak, W1k, W2k and W3k are constant known real matrices. DAk(t), DW1k(t), DW2k(t) and DW3k(t) denote the time-varying parameter uncertainties. Assumption (H3). The parameter uncertainties DAk(t), DW1k(t), DW2k(t) and DW3k(t) are of the form:
Definition 2.2. A neural network (5) is said to be globally dissipative if there exists a compact set S # Rn , such that 8 x0 2 Rn ; 9 Tðx0 Þ > 0, where t P t0 + T(x0), x(t,t0,x0) # S, where x(t,t0,x0) denotes the solution of Eq. (5) from initial state x0 and initial time t0. In this case, S is called a globally attractive set. A set S is called a positive invariant if "x0 2 S implies x(t,t0,x0) # S for t P t0. Definition 2.3. Let S be a globally attractive set of neural networks [Eq. (5)]. A neural network (5) is said to be globally exponentially dissipative system if there exist a compact set S⁄ S in Rn such that 8x0 2 Rn n S , there exists a constant M(x0) > 0 and a > 0 such that
inf fkxðt; t 0 ; x0 Þ ~xk : ~x 2 S g 6 Mðx0 Þeaðtt0 Þ :
x2Rn nS
Set S⁄ is globally exponentially attractive set, where x 2 Rn n S . The following Lemmas are using to prove our theorems.
Lemma 2.4 Rakkiyappan and Balasubramaniam (2010). Let D and N be real constant matrices of appropriate dimensions, matrix F(t) satisfies FT(t)F(t) 6 I. Then (i) for any scalar > 0, DF(t)N + NTFT(t)DT 6 1D DT + NTN. (ii) For any P > 0, 2aTb 6 aTP1a + bTPb.
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Lemma 2.5 Gu (1994). For any constant matrix M 2 Rnn ; M ¼ MT > 0, scalar g > 0, vector function x : ½0; g ! Rn such that the integrations are well defined, the following inequality holds
Z
g
T Z xðsÞds M
0
g
Z xðsÞds 6 g
0
g
Xk2;3 ¼ NT5 þ N 6 ;
Xk2;7 ¼ ð1 lÞS2 þ L2 W; Xk2;9 ¼ Xk2;10 ¼ Xk2;11 ¼ 0; Xk2;14 ¼ N6 ; Xk2;15 ¼ 0;
xT ðsÞMxðsÞds: 0
k 3;4
X Lemma 2.6 Boyd, Ghoui, Feron, and Balakrishnan (1994). Let M, P, Q be the given matrices such that Q > 0, then
"
T
P
M
M
Q
Xk2;4 ¼ N3 þ NT4 ;
k 3;5
¼X
k 3;6
¼X
¼X
Xk2;8 ¼ 0; Xk2;12 ¼ N 2 ;
Xk3;3 ¼ Q 1 þ Q 2 þ 2N 5 ;
Xk3;14 ¼ N5 ;
Xk4;10 ¼ Xk4;11 ¼ Xk4;12 ¼ 0; k 5;5
3. Main result
X
For convenience, we set r X
W 2 þ DW 2 ðtÞ ¼
xk ðhðtÞÞðW 1k þ DW 1k ðtÞÞ;
W 3 þ DW 3 ðtÞ ¼
Xk5;7 ¼ M 1 W 2k ;
Xk4;14 ¼ Xk4;15 ¼ 0;
Xk5;8 ¼ Xk5;9 ¼ Xk5;10 ¼ Xk5;11
1 Xk6;6 ¼ Y 1 þ Z 1 þ r2 Z 2 2V þ k EW k
T 1 EW ; k
Xk6;7 ¼ Xk6;8 ¼ Xk6;9 ¼ Xk6;10 ¼ 0;
xk ðhðtÞÞðW 2k þ DW 2k ðtÞÞ; Xk6;11 ¼ Xk6;12 ¼ Xk6;13 ¼ Xk6;14 ¼ Xk6;15 ¼ 0;
k¼1 r X
Xk4;13 ¼ N4 ;
Xk5;15 ¼ M1 W 3k ;
k¼1 r X
Xk4;4 ¼ 2N4 ;
¼ Xk5;12 ¼ Xk5;13 ¼ Xk5;14 ¼ 0;
k¼1
W 1 þ DW 1 ðtÞ ¼
Xk3;15 ¼ 0;
¼ h2 R1 þ ðh2 h1 ÞR2 2M 1 ;
Xk5;6 ¼ M1 W 1k ;
xk ðhðtÞÞðAk þ DAk ðtÞÞ; r X
Xk3;8 ¼ S1 þ S2 ;
¼ 0;
Xk4;5 ¼ Xk4;6 ¼ Xk4;7 ¼ Xk4;8 ¼ Xk4;9 ¼ 0;
< 0 () P þ M T Q 1 M < 0:
A þ DAðtÞ ¼
Xk2;13 ¼ N3 ;
Xk3;9 ¼ Xk3;10 ¼ Xk3;11 ¼ 0; Xk3;12 ¼ Xk3;13 ¼ 0;
#
k 3;7
Xk2;5 ¼ Xk2;6 ¼ 0;
2 Xk7;7 ¼ ð1 lÞY 2 2W þ k EW k
xk ðhðtÞÞðW 3k þ DW 3k ðtÞÞ;
T 2 EW ; k
k¼1
Xk7;8 ¼ Xk7;9 ¼ Xk7;10 ¼ Xk7;11 ¼ Xk7;12 ¼ Xk7;13 ¼ Xk7;14 ¼ Xk7;15 ¼ 0;
then system (5) can be rewritten as
dxðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðW 1 þ DW 1 ðtÞÞf ðxðtÞÞ þ ðW 2 dt þ DW 2 ðtÞÞf ðxðt sðtÞÞÞ þ ðW 3 þ DW 3 ðtÞÞ Z t f ðxðsÞÞds þ u:
Xk8;8 ¼ Y 1 þ Y 2 ; Xk8;9 ¼ 0; Xk8;10 ¼ Xk8;11 ¼ Xk8;12 ¼ Xk8;13 ¼ Xk8;14 ¼ Xk8;15 ¼ 0; X ð6Þ
trðtÞ
Theorem 3.1. Suppose that (H1)–(H3) hold. If there exist a symmetric positive definite matrices P > 0, Qi > 0 (i = 1, 2, 3, 4), Rj, Yj, Zj > 0 (j = 1, 2), and Q > 0, positive diagonal matrices V > 0 and W > 0, a matrices S1 S2, M1 and Nl (l = 1, . . . , 6) and a positive diagonal constant k > 0 such that the following LMI
k ¼
Xk
MGk
k I
< 0;
ð7Þ
hold for k = 1, 2, . . . , r, where Xk ¼ Xki;j
1515
¼X
k 9;12
¼X
T EAk
2
þ 2N1 þ 2Q ;
Xk1;2 ¼ N 1 þ NT2 ;
Xk1;5 ¼ P ATk M T1 ;
Xk9;9 ¼ Z 1 ;
¼ 0;
Xk10;10 ¼ Q 3 ;
Xk10;11 ¼ Xk10;12 ¼ Xk10;13 ¼ Xk10;14 ¼ Xk10;15 ¼ 0; 1 Q ; Xk11;12 ¼ Xk11;13 ¼ Xk11;14 ¼ Xk11;15 ¼ 0; h2 h1 4 1 ¼ R1 ; Xk12;13 ¼ 0; h2
Xk11;11 ¼ Xk12;12
1 ðR1 þ R2 Þ; h2 h1 1 ¼ R2 ; h2 h1
Xk12;14 ¼ Xk12;15 ¼ 0; Xk13;13 ¼ Xk13;14 ¼ Xk13;15 ¼ 0;
Xk14;14
with
Xk1;1 ¼ Q 1 þ h1 Q 3 þ ðh2 h1 ÞQ 4 2L1 V þ k EAk
Xk1;3 ¼ Xk1;4 ¼ 0;
k 9;11
Xk9;13 ¼ Xk9;14 ¼ Xk9;15 ¼ 0;
Now, we discuss the globally dissipative of neural network model (6) as follows.
k 9;10
Xk1;6 ¼ S1 þ L2 V;
3 Xk15;15 ¼ Z 2 þ k EW k
Xk14;15 ¼ 0;
T 3 EW ; k
M ¼ ½0 0 0 0 M 1 0 0 0 0 0 0 0 0 0 0T the neural network (6) is globally dissipative, and S ¼ n o uk x : kxk 6 kkM1ðQ is a positive invariant and globally attractive set. Þ min
Xk1;7 ¼ Xk1;8 ¼ Xk1;9 ¼ Xk1;10 ¼ Xk1;11 ¼ 0; Xk1;12 ¼ N1 ;
Xk1;13 ¼ Xk1;14 ¼ Xk1;15 ¼ 0;
Xk2;2 ¼ ð1 lÞQ 2 2L1 W 2N2 þ 2N3 2N6 ;
Proof. Consider the following positive radially unbounded Lyapunov–Krasovskii functional candidate for model (6) as
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ;
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where
From assumption (H2), we have
þ fi ðxi ðtÞÞ li xi ðtÞ fi ðxi ðtÞÞ li xi ðtÞ 6 0;
T
V 1 ðtÞ ¼ x ðtÞPxðtÞ; V 2 ðtÞ ¼
T Q 1 S1 xðsÞ xðsÞ ds ST1 Y 1 f ðxðsÞÞ th1 f ðxðsÞÞ Z th1 xðsÞ T Q 2 S2 xðsÞ þ ds; T S2 Y 2 f ðxðsÞÞ f ðxðsÞÞ tsðtÞ Z
t
V 3 ðtÞ ¼ h1
Z
Z
0
t
xT ðsÞQ 3 xðsÞds dh þ
Z
tþh
h1
Z
h1
which are equivalent to
2
t
xT ðsÞQ 4 xðsÞds dh;
V 4 ðtÞ ¼
Z
0
V 5 ðtÞ ¼
_ x_ T ðsÞR1 xðsÞds dh þ
tþh
h2
Z
t
Z
Z
h1
n X
Z
t
f T ðxðsÞÞZ 1 f ðxðsÞÞds þ r
tr
0
Z
r
_ x_ T ðsÞR2 xðsÞds dh;
t
f T ðxðsÞÞZ 2 f ðxðsÞÞds dh:
dV 1 ðtÞ _ ¼ 2xT ðtÞPxðtÞ; dt
ð8Þ
f ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ x ðt h1 ÞQ 2 xðt h1 Þ þ 2xT ðt h1 ÞS2 f ðxðt h1 ÞÞ þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ ð1 lÞf T ðxðt sðtÞÞÞY 2 f ðxðt sðtÞÞÞ; T Z xðsÞds Q 3
th1
t
ð9Þ xðsÞds
th1
þ ðh2 h1 Þx ðtÞQ 4 xðtÞ !T ! Z th1 Z th1 1 xðsÞds Q 4 xðsÞds ; h2 h1 th2 th2
th2
1 h2
t
tsðtÞ
!T _ xðsÞds
R1
tsðtÞ
2
T
xðtÞ f ðxðtÞÞ
T
xðtÞ f ðxðtÞÞ
f ðxðtÞÞ
6 0 i ¼ 1; 2; . . . ; n;
2 4
þ
li li ei eTi
þ l i þli 2
þ l i þli 2
ei eTi
ei eTi
ei eTi
3 xðtÞ 5 6 0; f ðxðtÞÞ
L1 V
L2 V
L2 V
V
xðtÞ f ðxðtÞÞ
ð13Þ
6 0;
þ þ þ l þl þ n is approwhere L1 ¼ diag l1 l1 ; . . . ; ln ln ; L2 ¼ diag 1 2 1 ; . . . ; ln þl 2 priate dimension. Similarly one has,
2
xðt sðtÞÞ f ðxðt sðtÞÞÞ
T
L1 W
L2 W
L2 W
W
xðt sðtÞÞ f ðxðt sðtÞÞÞ
6 0:
ð14Þ
ð15Þ
t
ð10Þ
ð16Þ
tsðtÞ
tsðtÞ
_ xðsÞds ;
0 ¼ 2ðxT ðt h1 ÞN 5 þ xT ðt sðtÞÞN6 Þ xðt h1 Þ xðt sðtÞÞ
ð17Þ
Z
th1
! _ xðsÞds :
ð18Þ
tsðtÞ
It follows from the inequalities (8)–(14) and using assumption (H3) and Lemma 2.4 in (15)–(18), that
dVðtÞ _ þ xT ðtÞQ 1 xðtÞ xT ðt h1 ÞQ 1 xðt h1 Þ 6 2xT ðtÞPxðtÞ dt
! _ xðsÞds þ ðh2
þ 2xT ðtÞS1 f ðxðtÞÞ 2xT ðt h1 ÞS1 f ðxðt h1 ÞÞ þ f T ðxðtÞÞY 1 f ðxðtÞÞ
tsðtÞ
dV 5 ðtÞ 6 f T ðxðtÞÞZ 1 f ðxðtÞÞ f T ðxðt rÞÞZ 1 f ðxðt rÞÞ dt þ r2 f T ðxðtÞÞZ 2 f ðxðtÞÞ !T ! Z t Z t f ðxðsÞÞds Z 2 f ðxðsÞÞds : trðtÞ
0 ¼ 2ðxT ðtÞN1 þ xT ðt sðtÞÞN2 ÞðxðtÞ xðt sðtÞÞ Z t _ xðsÞdsÞ;
T _ ðR1 xðsÞds
_ h1 Þx_ T ðtÞR2 xðtÞ !T ! Z th1 Z th1 1 _ _ xðsÞds xðsÞds ; R2 h2 h1 tsðtÞ tsðtÞ
trðtÞ
xðtÞ
th2
th2
Z
0 ¼ 2ðxT ðt sðtÞÞN3 þ xT ðt Z h2 ÞN4 Þ xðt sðtÞÞ xðt h2 Þ
T
Z
5
trðtÞ
2ð1 lÞxT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ
Z
ei eTi
3
þ ðW 2 þ DW 2 ðtÞÞf ðxðt sðtÞÞÞ Z t f ðxðsÞÞds þ u; þ ðW 3 þ DW 3 ðtÞÞÞ
ð1 lÞxT ðt sðtÞÞQ 2 xðt sðtÞÞ
dV 4 ðtÞ 1 _ 6 h2 x_ T ðtÞR1 xðtÞ dt h2 h1 Z tsðtÞ _ þ R2 Þ xðsÞds
ei eTi
ei eTi
_ ðA þ DAðtÞÞxðtÞ þ ðW 1 þ DW 1 ðtÞÞf ðxðtÞÞ 0 ¼ 2ðx_ T ðtÞM1 ÞðxðtÞ
T
t
2
þ l i þli 2
We can see that following equations hold for any matrices M1, N1, N2, N3, N4, N5 and N6 with appropriate dimensions
dV 2 ðtÞ 6 xT ðtÞQ 1 xðtÞ xT ðt h1 ÞQ 1 xðt h1 Þ þ 2xT ðtÞS1 f ðxðtÞÞ dt 2xT ðt h1 ÞS1 f ðxðt h1 ÞÞ þ f T ðxðtÞÞY 1 f ðxðtÞÞ
Z
þ l i þli
that is
Calculating the time derivative of V(t), we have
dV 3 ðtÞ 2 6 h1 xT ðtÞQ 3 xðtÞ dt
4
þ
li li ei eTi
t
tþh
T
vi
i¼1
tþh
h2
2
where er denotes the unit column vector having one element on its rth row and zeros elsewhere. Let V = diag{v1, v2, . . . , vn}, W = diag{w1, w2, . . . , wn}, then
2 Z
T
xðtÞ f ðxðtÞÞ
tþh
h2
i ¼ 1; 2; . . . ; n;
f T ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ xT ðt h1 ÞQ 2 xðt h1 Þ ð1 lÞxT ð11Þ
ðt sðtÞÞQ 2 xðt sðtÞÞ þ 2xT ðt h1 ÞS2 f ðxðt h1 ÞÞ 2ð1 lÞxT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ ð1 lÞf T 2
ðxðt sðtÞÞÞY 2 f ðxðt sðtÞÞÞ þ h1 xT ðtÞQ 3 xðtÞ
Z
t
T Z xðsÞds Q 3
th1
ð12Þ
t
th1 T
þ ðh2 h1 Þx ðtÞQ 4 xðtÞ
xðsÞds
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
Z
1 h2 h1
!T
th1
xðsÞds th2
1 h2
Z
1 h2 h1 !T Z _xðsÞds R1
t
tsðtÞ
xðsÞds
tsðtÞ
T Z _ ðR1 þ R2 Þ xðsÞds
th2
_ _ xðsÞds þ ðh2 h1 Þx_ T ðtÞR2 xðtÞ
!T
th1
_ xðsÞds
Z
R2
tsðtÞ
!
th1
T
_ xðsÞds þ f ðxðtÞÞZ 1 f ðxðtÞÞ
tsðtÞ
T
Z
2 T
f ðxðt rÞÞZ 1 f ðxðt rÞÞ þ r f ðxðtÞÞZ 2 f ðxðtÞÞ
!T
t
f ðxðsÞÞds
trðtÞ
!
Z
t
Corollary 3.3. Suppose that (H1)–(H3) hold. If there exist symmetric positive definite matrices P > 0, Qi > 0 (i = 1, 2, 3, 4), Rj, Yj, Zj > 0 (j = 1, 2) and Q > 0, positive diagonal matrices V > 0 and W > 0, a matrices S1, S2, M1 and Nl (l = 1, . . . , 6) such that the following LMI holds
þ 2x_ T ðtÞM 1 W 2 f ðxðt sðtÞÞÞ þ f T ðxðt sðtÞÞÞðEW 2 ÞT
Z
ðEW 3 ÞT ðEW 3 Þ
trðtÞ
!
t
!T f ðxðsÞÞds
xðtÞ xðt sðtÞÞ
Z
!
t
tsðtÞ
tsðtÞ
ð21Þ
invariant and globally attractive set. In the following theorem, we extend the above result to global exponential dissipativity of neural network (6) as follows.
_ xðsÞds þ 2ðxT ðt sðtÞÞN3
þ xT ðt h2 ÞN 4 Þ Z xðt sðtÞÞ xðt h2 Þ
X < 0;
where X = (Xi,j)15 15 are defined as in Theorem 3.1, the neural netn o 1 uk is a positive work (20) is globally dissipative and S ¼ x : kxk 6 kkM min ðQ Þ
f ðxðsÞÞds þ 2x_ T ðtÞM 1 u þ 2ðxT ðtÞN1 þ xT ðt sðtÞÞN2 Þ
trðtÞ
ð20Þ
trðtÞ
þ f T ðxðtÞÞðEW 1 ÞT ðEW 1 Þf ðxðtÞÞ
trðtÞ
Rt _ is Remark 3.2. In this paper, the integral term th2 x_ T ðsÞR1 xðsÞds Rt R tsðtÞ _ _ tsðtÞ x_ T ðsÞR1 xðsÞds divided into two parts are th2 x_ T ðsÞR1 xðsÞds; R th _ and the integral term th21 x_ T ðsÞR2 xðsÞds is divided into two parts as R tsðtÞ T R th1 T _ _ _ _ th2 x ðsÞR2 xðsÞds; tsðtÞ x ðsÞR2 xðsÞds, which may lead to less con-
dxðtÞ ¼ AxðtÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt sðtÞÞÞ dt Z t f ðxðsÞÞds þ u: þ W3
f ðxðt sðtÞÞÞ L2 W W f ðxðt sðtÞÞÞ _ 2x_ T ðtÞM 1 AxðtÞ þ 1 x_ T ðtÞM1 GGT M T1 xðtÞ _ 2x ðtÞM 1 xðtÞ A T A T T þ x ðtÞðE Þ ðE ÞxðtÞ þ 2x_ ðtÞM 1 W 1 f ðxðtÞÞ
t
when x 2 Rn n S. Therefore, neural network (6) is a globally dissipative system and the set S is a positive invariant and globally attractive set as LMI (7) hold. This completes the proof. h
In the following, we consider the neural network without fuzzy and uncertainties then (5) becomes
_T
ðEW 2 Þf ðxðt sðtÞÞÞ ! Z t Z T _ f ðxðsÞÞds þ þ 2x ðtÞM 1 W 3
dVðtÞ 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM 1 uÞ 6 2kxkðkmin ðQÞkxk þ kM 1 ukÞ dt < 0;
servative results.
xðtÞ T L1 V L2 V xðtÞ f ðxðsÞÞds 2 f ðxðtÞÞ L2 V V f ðxðtÞÞ trðtÞ T L1 W L2 W xðt sðtÞÞ xðt sðtÞÞ
Z2 2
_ xðsÞds
th2
!
t
tsðtÞ
tsðtÞ
Z
1 h2 h1
!
th1
th2
Z
_ þ h2 x_ T ðtÞR1 xðtÞ
Z
Q4
_ xðsÞds
Theorem 3.4. Under the conditions of Theorem 3.1, neural network n o uk (6) is globally exponentially dissipative and S ¼ x : kxk 6 kkM1ðQ is Þ
th
min
2 þ 2 xT ðt h1 ÞN5 þ xT ðt sðtÞÞN6 ! Z th1 _ xðsÞds xðt h1 Þ xðt sðtÞÞ
a positive invariant and globally attractive set. Further, the exponential dissipativity rate index k = /2 can be estimated by the following inequality
tsðtÞ
dVðtÞ nðtÞ; 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM 1 uÞ þ nT ðtÞt dt
ð19Þ
where,
T
T
T
T
_T
T
nðtÞ ¼ x ðtÞ x ðt sðtÞÞ x ðt h1 Þ x ðt h2 Þ x ðtÞ f ðxðtÞÞ Z t T f T ðxðt sðtÞÞÞf T ðxðt h1 ÞÞ f T ðxðt rÞÞ xðsÞds Z
!T
th1
xðsÞds !T
th1
_ xðsÞds tsðtÞ
Z
!T Z
t
th1
tsðtÞ
_ xðsÞds
tsðtÞ
th2
Z
Z
t
trðtÞ
T _ xðsÞds
Wk
MGk
k I
< 0;
ð22Þ
hold for k = 1, 2, . . . , r, where Wk ¼ Wki;j
1515
W k1;1 ¼ P þ eh1 Q 1 þ þ
A T A k ðEk Þ ðEk Þ
2 h1 eh1 Q 3
!T 3T f ðxðsÞÞds 5
with
þ ðh2 h1 Þeh2 Q 4 2L1 V
þ 2N1 þ 2Q ;
Wk1;2 ¼ N1 þ NT2 ; Wk1;3 ¼ Wk1;4 ¼ 0;
th2
Wk1;5 ¼ P ATk MT1 ;
Wk1;6 ¼ eh1 S1 þ L2 V; Wk1;7 ¼ 0; Wk1;8 ¼ Wk1;9 ¼ Wk1;10 ¼ Wk1;11 ¼ 0;
and
t ¼ ðti;j Þ
Nk ¼
Wk1;12 ¼ N1 ;
Wk1;13 ¼ Wk1;14 ¼ Wk1;15 ¼ 0;
1
¼ ðXi;j Þ þ diag 0; 0; 0; 0; M1 GG
T
M T1 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0
Wk2;2 ¼ ð1 lÞQ 2 2L1 W 2N2 þ 2N3 2N6 ; :
Considering xk(h(t)) P 0 (k = 1, 2, . . . , r) and k < 0 (k = 1, 2, . . . , r) in Pr Theorem 3.1, we have Noting that k¼1 xk ðhðtÞÞ k < 0. Pr k¼1 xk ðhðtÞÞ ¼ 1, we obtain k < 0 by using Lemma 2.6. Thus from condition (7) and inequality (19), we get
Wk2;3 ¼ NT5 þ N6 ;
Wk2;4 ¼ N 3 þ NT4 ; Wk2;5 ¼ Wk2;6 ¼ 0;
Wk2;7 ¼ ð1 lÞS2 þ L2 W; Wk2;8 ¼ Wk2;9 ¼ Wk2;10 ¼ Wk2;11 ¼ 0; Wk2;12 ¼ N2 ; Wk2;13 ¼ N 3 ;
3350
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
Wk2;14 ¼ N6 ;
Wk3;3 ¼ Q 1 þ eðh2 h1 Þ Q 2 þ 2N5 ;
Wk2;15 ¼ 0;
V 3 ðtÞ ¼ h1
Wk3;4 ¼ Wk3;5 ¼ Wk3;6 ¼ Wk3;7 ¼ 0; k 3;8
W
ðh2 h1 Þ
¼ S1 þ e
Wk3;14 ¼ N5 ; k 4;4
W
¼ 2N4 ; W ¼W
W
k 3;10
¼W
k 3;11
¼W
k 3;12
¼W
k 3;13
¼W
k 4;6
¼W
k 4;13
¼ 0;
W
¼W
¼W
k 4;8
k 4;9
¼W
k 4;10
¼W
W
k 5;7
¼ M 1 W 1k ; W
¼W
þ
¼ N 4 ;
W
k 5;9
Z
¼W
¼W
1
1 ÞT ðEW k Þ;
¼W
k 5;12
¼W
k 5;13
¼W
k 5;14
¼W
¼ 0;
Wk6;6 ¼ eh1 Y 1 þ er Z 1 þ r2 er Z 2 2V
Wk5;15 ¼ M 1 W 3k ;
k ðEWk
k 5;11
Wk6;7 ¼ Wk6;8 ¼ Wk6;9 ¼ Wk6;10 ¼ Wk6;11 ¼ 0;
Wk6;12 ¼ Wk6;13 ¼ Wk6;14 ¼ Wk6;15 ¼ 0; k 7;7
W
k 7;9
W
W2 T W2 k ðEk Þ ðEk Þ;
¼ ð1 lÞY 2 2W þ k 7;10
¼W
¼W
k 7;11
¼W
k 7;12
k 7;13
¼W
W
k 7;14
k 7;8
eðsþh2 Þ xT ðsÞQ 4 xðsÞds dh;
_ eðsþh2 Þ x_ T ðsÞR1 xðsÞds dh
Z
h1
t
Z
t
_ eðsþh2 Þ x_ T ðsÞR2 xðsÞds dh;
tþh
eðsþrÞ f T ðxðsÞÞZ 1 f ðxðsÞÞds Z
0
r
Z
t
eðsþrÞ f T ðxðsÞÞZ 2 f ðxðsÞÞds dh:
tþh
Calculating the time derivative of V(t), we have
dV 1 ðtÞ _ ¼ et xT ðtÞPxðtÞ þ 2et xT ðtÞPxðtÞ; dt
þ 2eðtþh1 Þ xT ðtÞS1 f ðxðtÞÞ 2et xT ðt h1 ÞS1 f ðxðt h1 ÞÞ
¼W
¼ 0;
Wk8;8 ¼ Y 1 þ eðh2 h1 Þ Y 2 ; Wk8;9 ¼ 0;
þ eðtþh1 Þ f T ðxðtÞÞY 1 f ðxðtÞÞ et f T ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ eðtþh2 h1 Þ xT ðt h1 ÞQ 2 xðt h1 Þ
Wk8;10 ¼ Wk8;11 ¼ Wk8;12 ¼ Wk8;13 ¼ Wk8;14 ¼ Wk8;15 ¼ 0;
et ð1 lÞxT ðt sðtÞÞQ 2 xðt sðtÞÞ
Wk9;9 ¼ Z 1 ; Wk9;10 ¼ Wk9;11 ¼ Wk9;12 ¼ 0;
þ 2eðtþh2 h1 Þ xT ðt h1 ÞS2 f ðxðt h1 ÞÞ
Wk9;13 ¼ Wk9;14 ¼ Wk9;15 ¼ 0;
2ð1 lÞet xT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ
Wk10;10 ¼ Q 3 ;
þ eðtþh2 h1 Þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ
Wk10;11 ¼ Wk10;12 ¼ Wk10;13 ¼ Wk10;14 ¼ Wk10;15 ¼ 0; Wk11;11 k 12;12
W
ð1 lÞet f T ðxðt sðtÞÞÞ Y 2 f ðxðt sðtÞÞÞ;
1 ¼ Q ; Wk ¼ Wk11;13 ¼ Wk11;14 ¼ Wk11;15 ¼ 0; h2 h1 4 11;12 1 ¼ R1 ; Wk12;13 ¼ 0; h2
Wk12;14 ¼ Wk12;15 ¼ 0;
3 Wk14;15 ¼ 0; Wk15;15 ¼ Z 2 þ k EW k
T 3 EW ; k
M ¼ ½0 0 0 0 M 1 0 0 0 0 0 0 0 0 0 0 : Proof. Consider the following positive radially unbounded Lyapunov–Krasovskii functional candidate for model (6) as
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ;
V 1 ðtÞ ¼ et xT ðtÞPxðtÞ; t
" eðsþh1 Þ
xðsÞ
#T "
Q1
S1
#"
xðsÞ
# ds
ST1 Y 1 f ðxðsÞÞ f ðxðsÞÞ " # " # " # T Z th1 Q 2 S2 xðsÞ xðsÞ ðsþh2 Þ þ e ds; tsðtÞ ST2 Y 2 f ðxðsÞÞ f ðxðsÞÞ th1
ð24Þ
xðsÞds
th1
ð25Þ
dV 4 ðtÞ _ 6 h2 eðtþh2 Þ x_ T ðtÞR1 xðtÞ dt Z tsðtÞ T Z tsðtÞ et _ _ ðR1 þ R2 Þ xðsÞds xðsÞds h2 h1 th2 th2 !T ! Z t Z t et _ _ R1 xðsÞds xðsÞds h2 tsðtÞ tsðtÞ _ þ ðh2 h1 Þeðtþh2 Þ x_ T ðtÞR2 xðtÞ !T ! Z Z th1 th1 et _ _ R2 xðsÞds xðsÞds ; h2 h1 tsðtÞ tsðtÞ
where
Z
t
þ ðh2 h1 Þeðtþh2 Þ xT ðtÞQ 4 xðtÞ !T ! Z th1 Z th1 et xðsÞds Q 4 xðsÞds ; h2 h1 th2 th2
T
V 2 ðtÞ ¼
dV 3 ðtÞ 2 6 h1 eðtþh1 Þ xT ðtÞQ 3 xðtÞ dt Z t T Z et xðsÞds Q 3 th1
1 ðR1 þ R2 Þ; h2 h1 1 ¼ R2 ; h2 h1
Wk13;13 ¼
Wk13;14 ¼ Wk13;15 ¼ 0; Wk14;14
ð23Þ
dV 2 ðtÞ 6 eðtþh1 Þ xT ðtÞQ 1 xðtÞ et xT ðt h1 ÞQ 1 xðt h1 Þ dt
¼ 0; k 7;15
¼W
t
tþh
tr
¼ M 1 W 2k ;
k 5;10
t
h2
V 5 ðtÞ ¼
eðsþh1 Þ xT ðsÞQ 3 xðsÞds dh
tþh
Z
0
þr k 5;8
Z
h1
h2
k 4;11
Wk4;14 ¼ Wk4;15 ¼ 0; Wk5;5 ¼ h2 eh2 R1 þ ðh2 h1 Þeh2 R2 2M 1 ; k 5;6
Z
t
tþh
h2
V 4 ðtÞ ¼ k 4;7
Z
0
h1
Z
¼ 0;
Wk3;15 ¼ 0; k 4;5
k 4;12
S2 ;
k 3;9
þ
Z
ð26Þ
dV 5 ðtÞ 6 eðtþrÞ f T ðxðtÞÞZ 1 f ðxðtÞÞ et f T ðxðt rÞÞZ 1 f ðxðt rÞÞ dt þ r2 eðtþrÞ f T ðxðtÞÞZ 2 f ðxðtÞÞ !T ! Z t Z t f ðxðsÞÞds Z 2 f ðxðsÞÞds : ð27Þ et trðtÞ
trðtÞ
3351
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
It follows from inequalities (13) and (14) and using assumption (H3), Lemma 2.4 in (15)–(18) and (23)–(27) that
dVðtÞ t T _ þ eh1 xT ðtÞQ 1 xðtÞ 6e x ðtÞPxðtÞ þ 2xT ðtÞPxðtÞ dt xT ðt h1 ÞQ 1 xðt h1 Þ þ 2eh1 xT ðtÞS1 f ðxðtÞÞ
þ2eðh2 h1 Þ xT ðt h1 ÞS2 f ðxðt h1 ÞÞ 2ð1 lÞxT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ þ eðh2 h1 Þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ ð1 lÞf T 2 xðt sðtÞÞÞY 2 f ðxðt sðtÞÞÞ þ h1 eh1 xT ðtÞQ 3 xðtÞ Z t T Z t xðsÞds Q 3 xðsÞds þ ðh2 h1 Þ eh2 xT ðtÞQ 4 xðtÞ
th1
1 h2 h1
!T
th1
xðsÞds
Q4
Z
th2
Z
tsðtÞ
th1
! _ xðsÞds þ h2 eh2 x_ T ðtÞR1 xðtÞ
th2
T Z _ xðsÞds ðR1 þ R2 Þ
Z
f ðxðsÞÞds
1=2 Vðxð0ÞÞ eð=2Þt ; kmin ðPÞ
which means that neural network (6) is globally exponentially dissipative and the set S is a positive invariant and globally attractive set as LMI (22) hold. This completes the proof. h In the following, we consider the neural network without fuzzy and uncertainties then (5) becomes
dxðtÞ ¼ AxðtÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt sðtÞÞÞ dt Z t f ðxðsÞÞds þ u: þ W3
ð32Þ
Corollary 3.5. Under the conditions of Corollary 3.3, neural network n o 1 uk is (32) is globally exponentially dissipative and S ¼ x : kxk 6 kkM min ðQ Þ a positive invariant and globally attractive set. Further, the exponential dissipativity rate index k = /2 can be estimated by the following inequality
W < 0;
ð33Þ
where W = (Wi,j)15 15 are defined as in Theorem 3.4.
trðtÞ
!#
t
ð31Þ
trðtÞ
þer f T ðxðtÞÞZ 1 f ðxðtÞÞ f T ðxðt rÞÞZ 1 f ðxðt rÞÞ !T Z t þr2 er f T ðxðtÞÞZ 2 f ðxðtÞÞ f ðxðsÞÞds Z 2
kxk 6
tsðtÞ
1 _ xðsÞds h2 h1 th2 th2 ! ! T Z t Z t 1 _ _ _ xðsÞds xðsÞds þ ðh2 h1 Þeh2 x_ T ðtÞR2 xðtÞ R1 h2 tsðtÞ tsðtÞ !T ! Z th1 Z th1 1 _ _ xðsÞds xðsÞds R2 h2 h1 tsðtÞ tsðtÞ
From the definition of V(x(t)), we know that
From Eqs. (30) and (31), we get
þeh1 f T ðxðtÞÞY 1 f ðxðtÞÞ f T ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ eðh2 h1 Þ xT t h1 ÞQ 2 xðt h1 Þ ð1 lÞxT ðt sðtÞÞ Q 2 xðt sðtÞÞ
Z
ð30Þ
VðxðtÞÞ P et xT ðtÞPxðtÞ:
2xT ðt h1 ÞS1 f ðxðt h1 ÞÞ
th1
VðxðtÞÞ 6 Vðxð0ÞÞ:
2
trðtÞ
xðtÞ
T
f ðxðtÞÞ
L1 V L2 V L2 V
V
xðtÞ
f ðxðtÞÞ
dVðtÞ 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM1 uÞ þ nT ðtÞnðtÞ; dt where,
¼ ði;j Þ ¼ ðWi;j Þ þ diag 0; 0; 0; 0; 1 M1 GGT M T1 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0 < 0 ð28Þ and
Remark 3.6. In Liao and Wang (2003), Arik (2004), Song and Zhao (2005), Cao et al. (2006), Masubuchi (2006), Huang et al. (2007), Lou and Cui (2008), Song and Cao (2008), Wang et al. (2009), Zhang et al. (2010) and Song and Cao (2010), authors studied the various types of neural networks and few of them have proposed results in LMI approach. In this paper, the problem of global robust dissipativity is derived for T–S fuzzy neural network using LMI approaches. Therefore, our results are less conservative than those given in the previous literature.
4. Numerical examples In this section, we will give examples showing the effectiveness of established theoretical results.
" nðtÞ ¼ xT ðtÞ xT ðt sðtÞÞ xT ðt h1 Þ xT ðt h2 Þ
Example 1. Consider the system (6) with parameters defined as
x_ T ðtÞ f T ðxðtÞÞ f T ðxðt sðtÞÞÞf T ðxðt h1 ÞÞ f T ðxðt rÞÞ !T !T Z t T Z th1 Z t _ xðsÞds xðsÞds xðsÞds Z
tsðtÞ
th2
tsðtÞ
th2
th1
T _ xðsÞds
Z
th1
A1 ¼
!T _ xðsÞds
tsðtÞ
Z
t
trðtÞ
!T 3T f ðxðsÞÞds 5 :
Considering xk(h(t)) P 0 (k = 1, 2, . . . , r) and Nk < 0 (k = 1, 2, . . . , r) in Pr Theorem 3.4, we have Noting that k¼1 xk ðhðtÞÞNk < 0. Pr k¼1 xk ðhðtÞÞ ¼ 1. From inequality (28) we get
dVðtÞ 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM1 uÞ dt 6 2kxkðkmin ðQ Þkxk þ kM 1 ukÞ < 0;
ð29Þ
when x 2 RnnS. Integrating two sides of Eq. (29) from 0 to an arbitrary t > 0, we have
W 31
2:5 0:2 W 11 ¼ ; 0 2 4:5 3:5 0:2 0:05 ¼ ; 0:35 0:8
u¼
2 0
0:02 0:04
W3 2 EW 1 ¼ E1 ¼
W 22 ¼
;
;
G1 ¼
0
0
0:2
0:01
0
0
0:01
2:5 0:6 0:4 2:5
;
0:3
; A2 ¼
W 32 ¼
;
1 EA1 ¼ EW ¼ 1
2:5 0 0 2
W 21 ¼
1
;
2
0:4
;
0:5 2:8
; W 12 ¼
0:5 0:4 0:8
0:03
0
0
0:03
2 0:4 5 3:5
G2 ¼
;
;
0:4
0
0
0:2
;
3352
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
1 EA2 ¼ EW ¼ 2
0:02 0 ; 0 0:02
2 3 ¼ EW ¼ EW 2 2
0:01 0 ; 0 0:01
4
f 1 ðxÞ
¼ f2 ðxÞ ¼ tanhðxÞ;
3
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 0:66; h2 ¼ 0:9928; r ¼ 0:3; li ¼ 0; li ¼ 2; li ¼ 0; li ¼ 2. Thus,
0 0 0 0
;
L2 ¼
1 0
:
0 1
1
state
L1 ¼
2
0
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (7) as follows
0:0018 0:0000
P¼
#
"
;
0:0000 0:0002 " # 6:1341 0:7844 Q2 ¼ ; 0:7844 6:0528 "
Q3 ¼
4:8497
0:0000
Q1 ¼
16:89027
8:3005
8:3005
27:9421
#
" Q4 ¼
2:0756
0:0000
0:0000 4:8497 0:0000 " # 0:3807 0:2000 ; R1 ¼ 106 0:20000 0:1269 "
0:0010 0:0001
#
Y2 ¼
5:4128 3:7775
"
Z1 ¼
3:7199 1:6957
# Z2 ¼
1:6957 3:5925 # 2:3772 3:7126 ; 1:6271 4:8341
Q¼
−0.4
−0.2
0
time (sec)
0.2
0.4
0.6
0.8
1
Fig. 1. The transient responses of time and state for k = 1 for Example 1.
4
9:1717 5:4144
#
3
; 2
5:9714
1:7470
1:7470
8:0115
1
#
0
;
−2
0 0:0033 0 W¼ ; 0 1:0303 0 0:0033 5:7573 2:7534 ; S1 ¼ 2:4063 6:0477
V¼
−0.6
−1
;
1:7403
2:0756
5:4144 9:2464
"
−0.8
;
;
;
"
#
#
3:7775 5:5331 "
Y1 ¼
;
0:0001 0:0011 "
−2
;
−3 −1
;
R2 ¼
−1
#
state
"
−3 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (sec) Fig. 2. The transient responses of time and state for k = 2 for Example 1.
S2 ¼
3:9688 2:2482
;
0:3353 0:0188
M1 ¼ ; 2:0564 4:1238 0:0189 0:3510 0:1829 0:1241 ; N1 ¼ 106 0:0121 0:0102 6
0:3825
0:2015
N2 ¼ 10 0:2015 0:1267 0:0030 0:0002 N4 ¼ ; 0:0002 0:0032
0:0034 0:0051 N5 ¼ ; 0:0052 0:0107 e1 ¼ 0:0246;
;
N3 ¼
0:0030 0:0002 0:0002 0:0032
Example 2. Consider the system (20) with parameters defined as
" A¼
;
0:0030 0:0002 N6 ¼ ; 0:0002 0:0032
e2 ¼ 0:0370:
Therefore, the neural network (6) is globally dissipative and the positive invariant and global attractive set are S = {x : kxk 6 0.0175}. Figs. 1 and 2 depicts the time and state of the considered network with initial conditions x1(t) = 0.3, x2(t) = 0.2 and x1(t) = 0.4, x2(t) = 0.3, t 2 [1, 0].
W3 ¼
u¼
4 0
#
" W1 ¼
;
0 3 " 1
0:4
0:6
1
0:03
;
0:04
3:2 0:4 4
3:6
#
" ;
W2 ¼
2:2 1:2 1:2
4
# ;
# ;
f 1 ðxÞ ¼ f2 ðxÞ ¼ tanhðxÞ:
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 0:8; h2 ¼ 1:0173; r ¼ 0:3; li ¼ 0; li ¼ 2; li ¼ 0; li ¼ 2. Thus,
L1 ¼
0 0 0 0
;
L2 ¼
1 0 0 1
:
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (21) as follows
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
0:4687 0:2161 139:3402 ; Q1 ¼ 0:2161 0:1121 104:1924 42:2231 2:0579 52:0426 Q2 ¼ ; Q3 ¼ 2:0579 35:4697 0:0000 18:5802 0:0000 0:0134 ; R1 ¼ Q4 ¼ 0:0000 18:5802 0:0067
P¼
0:1534 0:0663 ; R2 ¼ 0:0663 0:0536 32:7623 12:0209 ; Y2 ¼ 12:0209 32:1703
; 12:2526 36:8993 2:6908 1:3524 Q¼ ; 1:8003 2:5689 38:0998 12:2526
Z1 ¼
V¼
1:2916
0
S2 ¼
13:2857
Z2 ¼
6:4957
0:0135 0:0067 ; 0:0067 0:0034 0:7409 0:3227 N4 ¼ ; 0:3227 0:2552
N2 ¼
N5 ¼
0:0566 0:2528 0:2529
0:3244
;
75:8144
26:3889
26:3889
81:0423
u¼
2:5
0:03
0:0948
0
0
0:0944
;
;
G1 ¼
;
3
0:8
0:4
0
0
0:3
;
1 EA1 ¼ EW ¼ 1
A2 ¼
W 32 ¼
;
0:5 2:8
1 EA2 ¼ EW ¼ 2
W 21 ¼
;
0:03 0 ; 0 0:03
0
0
2:2
0:5
0:6
1
2 3 EW ¼ EW ¼ 2 2
0:4
3
0:01
0
0
0:01
;
;
;
;
G2 ¼
0:3
0
0
0:4
0:02 0 ; 0 0:02
;
f 1 ðxÞ
M1 ¼
0:0566
0:0256
0:0260
0:0172
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 1; h2 ¼ 1:3953; r ¼ 0:3; li ¼ 0; li ¼ 1:4; li ¼ 0; li ¼ 1:4. Thus,
;
0 0
L1 ¼
N3 ¼
0 0
"
0:6876
0:2960
0:2961
0:2420
;
L2 ¼
1:2483
0:7
0
0
0:7
:
1:2660
#
"
4:5796 2:1274
3
; Q 1 ¼ 10 1:2660 2:0244 " # 780:2072 78:4188 Q2 ¼ ; 78:4188 809:5559
P¼
N6 ¼
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (22) as follows
0:7331 0:3188 ; 0:3188 0:2532
:
Q3 ¼
0:0000
0:0000 895:1100 0:0121 0:0178 ; R1 ¼ 0:0178 0:0261
0:3703
1.5
895:1100
Y2 ¼
1
Z1 ¼
0:1403
Q4 ¼
0:1403 0:6408 944:1951 642:3267 642:3267 928:3844
Y 1 ¼ 103
;
807:6208 323:8715
323:8715 795:7156 3:6310 0:9571 Q¼ ; 0:9563 2:9879
0.5
;
2:1274 3:0179
−0.5
V¼
−1
S1 ¼
−1.5
2:5433
0:0000
443:1641
0
time (sec)
0.5
Fig. 3. The transient responses of time and state for Example 2.
1
;
1:6935
0:9540
0:9540 1:6687
;
;
;
;
W¼
Z 2 ¼ 103
0 3:5203 718:8600 465:1139 382:3011 665:2249
S2 ¼
0
;
0:0000
0
#
443:1641
R2 ¼
−0.5
2:5 0:5
2:5
1
¼ f2 ðxÞ ¼ tanhðxÞ:
;
0:02
W 22 ¼
0
0:02 0 2 3 EW ¼ EW ¼ ; 1 1 0 0:02 2:2 0:5 W 12 ¼ ; 4 3:7
2
state
2 0:4 W 11 ¼ ; 0 1:5 4 3:8 0:4 0:3 ¼ ; 0:6 0:6
W 31
Therefore, the neural network (20) is globally dissipative and the positive invariant and global attractive set are S = {x : kxk 6 0.0028}. Fig. 3 depicts the time and state of the considered network with initial conditions x1(t) = 0.3, x2(t) = 0.5, t 2 [1, 0].
−2 −1
A1 ¼
65:5176 24:0352 Y1 ¼ ; 24:0352 64:3342
;
3:8269 12:8548 0:0011 0:0005 ; N1 ¼ 0:0004 0:0002
Example 3. Consider the system (6) with parameters defined as
W¼ 0 1:3905 26:3861 12:8220 ; S1 ¼ 7:2846 25:3714
104:1924 ; 289:2376 0:0000 ; 52:0426 0:0067 ; 0:0033
479:5892 343:6215
290:2815 443:6318 0:0017 0:0025 ; N1 ¼ 0:0021 0:0031
2:1559
1:1890
1:1890
2:0490
1:4185
0
0
2:2452
;
;
;
;
M1 ¼
0:1825
0:1022
0:1025
0:3284
;
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
0:0088 0:0129 N2 ¼ ; 0:0129 0:0189 0:9651 0:3975 N4 ¼ ; 0:3975 1:6805 N5 ¼
2:3964
2:0070
1:8188 1:1410
e1 ¼ 214:4287;
;
N3 ¼
N6 ¼
0:9649 0:3972 ; 0:3972 1:6800
0:9355
0:3531
0:3539
1:6177
3:8 0 2:6 0:6 ; W1 ¼ ; 0 2:6 3:5 4 1 0:8 W3 ¼ ; 0:6 1
A¼
;
u¼
e2 ¼ 8:7415:
Therefore, the neural network (6) is globally exponentially dissipative. Moreover, from Eq. (22), we can get that the exponential dissipativity rate index k = /2 = 0.0131. It is easy to compute that the positive invariant and globally attractive set are S = {x : kxk 6 0.0053}. Figs. 4 and 5 depicts the time and state of the considered network with initial conditions x1(t) = 0.3, x2(t) = 0.5,t 2 [1, 0].
0 0
;
L2 ¼
0:6
0
0
0:6
:
1:5600 0:9497 Q 1 ¼ 103 ; 1:6037 1:1272 0:9497 2:3336 409:7816 27:0418 ; Q2 ¼ 27:0418 350:2791 2:8398
1:6037
549:2708
;
0:0000
0:0000 549:2708 0:0293 0:0191 ; R1 ¼ 0:0191 0:0124
;
Q4 ¼
286:5514
0:0000
0:0000
286:5514
R2 ¼ state
f 1 ðxÞ ¼ f2 ðxÞ ¼ tanhðxÞ:
;
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (33) as follows
1
0
0:3179
Y2 ¼
0:1127
0:1127 0:1814 362:5386 153:1137 153:1137 343:8646
Z1 ¼
−2
−3 −0.5
0
0.5
time (sec)
Fig. 4. The transient responses of time and state for k = 1 for Example 3.
S1 ¼
2.5
S2 ¼
1:3162
0
;
795:4553 321:9837 321:9837 756:6649
Z 2 ¼ 103
W¼
0 2:3333 294:3487 144:4242 88:9780
240:3678
140:6995
70:9736
43:4687 114:0745 0:0064 0:0042 ; N1 ¼ 0:0040 0:0026
1.5
1
N2 ¼
0:0282
0:0183
1:1373
0
0
0:8509
0.5 0 −0.5
N5 ¼
−1 −1.5 −2 −0.6
−0.4
−0.2
0
time (sec)
0.2
0.4
0.6
Fig. 5. The transient responses of time and state for k = 2 for Example 3.
0.8
0:1809 0:6614 0:7086 0:7503
;
;
;
M1 ¼
;
; 0:0183 0:0119 0:9798 0:3714 ; N4 ¼ 0:3714 0:5476
;
1:1376 0:6761 ; 0:6761 1:3358
2
;
;
V¼
Y1 ¼
;
456:6791 144:5897 ; 144:5897 443:3150 1:7808 1:5789 Q¼ ; 1:5784 2:3721
−1
state
0 0
L1 ¼
Q3 ¼
−2.5 −0.8
0:02
3
2
2:8 0:6 ; 0:8 3:2
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 0:7; h2 ¼ 1:0529; r ¼ 0:3; li ¼ 0; li ¼ 1:2; li ¼ 0; li ¼ 1:2. Thus,
P¼ Example 4. Consider the system (32) with parameters defined as
0:03
W2 ¼
N3 ¼
N6 ¼
0:2639
0:1160
0:1182
0:1368
0:9793
0:3711
0:3711
0:5474
0:8987
0:3185
0:3183
0:5132
;
;
:
Therefore, the neural network (32) is globally exponentially dissipative. Moreover, from Eq. (33), we can get that the exponential dissipativity rate index k = /2 = 0.2261. It is easy to compute that the positive invariant and globally attractive set are S = {x : kxk 6 0.0255}. Fig. 6 depicts the time and state of the considered network with initial conditions x1(t) = 0.4, x2(t) = 0.5, t 2 [1, 0]. It should be pointed out that Theorems 1 and 2, Corollaries 1 and 3 in Song and Cao (2008) cannot be applied to this example since
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
0.8 0.6
state
0.4 0.2 0 −0.2 −0.4 −0.6 −0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
time (sec) Fig. 6. The transient responses of time and state for Example 4.
the inequalities in Song and Cao (2008) do not have feasible solutions. Remark 5. From the above examples, we can see that the proposed results in this paper improve and generalize than those in Song and Cao (2008).
5. Conclusion In this paper, we have studied the global robust dissipativity of T–S fuzzy neural networks with interval time-varying delays. By a proper Lyapunov–Krasovskii functional and employing analytic techniques, some sufficient conditions ensuring the global dissipativity and global exponential dissipativity of T–S fuzzy neural networks have been derived in terms of LMIs. The new results given in this paper to improve the earlier dissipativity results. The numerical examples are provided to demonstrate the effectiveness of the proposed results. References Arik, S. (2004). On the global dissipativity of dynamical neural networks with time delays. Physics Letters A, 326, 126–132. Boyd, B., Ghoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM.
3355
Cao, Y. Y., & Frank, P. M. (2000). Analysis and synthesis of nonlinear time delay systems via fuzzy control approach. IEEE Transactions on Fuzzy Systems, 8, 200–211. Cao, Y. Y., & Frank, P. M. (2001). Stability analysis and synthesis of nonlinear timedelay systems via linear Takagi–Sugeno fuzzy models. IEEE Transactions on Fuzzy Systems, 124, 213–229. Cao, J., Yuan, K., Ho, D. W. C., & Lam, J. (2006). Global point dissipativity of neural networks with mixed time-varying delays. Chaos, 16, 013105. Feng, W., Yang, S. X., & Wu, H. (2009). On robust stability of uncertain stochastic neural networks with distributed and interval time-varying delays. Chaos, Solitons and Fractals, 42, 2095–2104. Gu, K. (1994). Integral inequality in the stability problem of Time-Delay Systems. In Proceeding 39th IEEE CDC, Sydney, Philadelphia. Hale, J. K. (1988). Asymptotic behavior of dissipative systems. Mathematical surveys and monographs (Vol. 25). Providence, RI, USA: American Mathematical Society. Haykin, S. (1998). Neural networks: A comprehensive foundation. NJ: Prentice Hall. Hill, D. J., & Moylan, P. J. (1976). Stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control, 21, 708–711. Huang, T. (2006). Exponential stability of fuzzy cellular neural networks with distributed delay. Physics Letters A, 351, 48–52. Huang, Y., Xu, D., & Yang, Z. (2007). Dissipativity and periodic attractor for nonautonomous neural networks with time-varying delays. Neurocomputing, 70, 16–18. Li, X. (2010a). Existence and global exponential stability of periodic solution for delayed neural networks with impulsive and stochastic effects. Neurocomputing, 73, 749758. Li, X. (2010b). Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type. Applied Mathematics and Computation, 215, 4370–4384. Liao, X., & Wang, J. (2003). Global dissipativity of continuous-time recurrent neural networks with time delay. Physical Review E, 68, 1–7. Liu, Y., & Tang, W. (2004). Exponential stability of fuzzy cellular neural networks with constant and time-varying delays. Physics Letters A, 323, 224–233. Lou, X. Y., & Cui, B. T. (2008). Global robust dissipativity for integro-differential systems modeling neural networks with delays. Chaos, Solitons and Fractals, 36, 469–478. Masubuchi, I. (2006). Dissipativity inequalities for continuous-time descriptor systems with applications to synthesis of control gains. Systems & Control Letters, 55, 158–164. Rakkiyappan, R., & Balasubramaniam, P. (2010). Delay-probability-distributiondependent stability of uncertain stochastic genetic regulatory networks with mixed time-varying delays: An LMI approach. Nonlinear Analysis: Hybrid Systems, 4, 600–607. Song, Q., & Cao, J. (2008). Global dissipativity analysis on uncertain neural networks with mixed time-varying delays. Chaos, 18, 043126. Song, Q., & Cao, J. (2010) Global dissipativity on uncertain discrete-time neural networks with time-varying delays. Discrete Dynamics in Nature and Society doi:10.1155/2010/810408. Song, Q., & Zhao, Z. (2005). Global dissipativity of neural networks with both variable and unbounded delays. Chaos, Solitons and Fractals, 25, 393–401. Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics, 15, 116–132. Tan, Z. Q., Soh, Y. C., & Xie, L. H. (1999). Dissipative control for linear discrete-time systems. Automatica, 35, 1557–1564. Wang, G., Cao, J., & Wang, L. (2009). Global dissipativity of stochastic neural networks with time delay. Journal of the Franklin Institute, 346, 794–807. Willems, J. C. (1972). Dissipative dynamical system – Part 1: General theory. Archive for Rational Mechanics and Analysis, 45, 321–351. Zhang, H., Yan, H., & Chen, Q. (2010). Stability and dissipative analysis for a class of stochastic system with time-delay. Journal of the Franklin Institute, 347, 882–893.
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
An LMI approach for global robust dissipativity analysis of T–S fuzzy neural networks with interval time-varying delays S. Muralisankar a,⇑, N. Gopalakrishnan a, P. Balasubramaniam b a b
School of Mathematics, Madurai Kamaraj University, Madurai 625 021, Tamilnadu, India Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India
a r t i c l e
i n f o
Keywords: Global dissipativity T–S fuzzy model Neural networks Linear matrix inequality Time-varying delays
a b s t r a c t Takagi–Sugeno (T–S) fuzzy models are often used to represent complex nonlinear systems by means of fuzzy sets and fuzzy reasoning applied to a set of linear sub-models. In this paper, the global robust dissipativity of T–S fuzzy neural networks with interval time-varying delays are investigated. By constructing a proper Lyapunov–Krasovskii functional and using linear matrix inequality (LMI) technique, delaydependent criteria for checking the global dissipativity and global exponential dissipativity of fuzzy neural networks have been derived in terms of LMI, which can be solved numerically using LMI toolbox in MATLAB. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results. 2011 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, dynamics of neural networks (NNs) have been widely studied due to their extensive applications in aerospace, defense, robotic, telecommunications, signal processing, pattern recognition etc. (Feng, Yang, & Wu, 2009; Haykin, 1998; Li, 2010a, 2010b). Fuzzy logic theory was shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. Takagi and Sugeno (1985) proposed an effective way to transform a nonlinear dynamic system to a set of linear sub-models via some fuzzy models by defining a linear input/output relationship as its consequence of individual plant rule. In Cao and Frank (2000), the standard T–S fuzzy model was extended to one with time delays and some stability conditions were presented in terms of LMIs. Recently, the Lyapunov– Krasovskii approach and the Lyapunov–Razumikhin method have been used to study the stability of delayed fuzzy systems (Cao & Frank, 2000, 2001). Moreover, the concept of incorporating fuzzy logic into NNs has grown into a popular research topic (Huang, 2006; Liu & Tang, 2004). The notion of dissipative dynamical system was first introduced by Willems (1972), and it is subsequently generalized in Zhang, Yan, and Chen (2010) via various approaches. Dissipative systems theory has wide ranging implications and applications in control theory. Applications of dissipativeness in the stability analysis of linear systems with certain nonlinear feedback were first discussed in Willems (1972). In addition, dissipativeness was crucially used in the stability analysis of nonlinear systems (Hill & Moylan, ⇑ Corresponding author. Tel.: +91 452 2452371; fax: +91 451 2453071. E-mail address: [email protected] (S. Muralisankar). 0957-4174/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.09.021
1976). The theory of dissipative systems generalizes basic tools including the passivity theorem, bounded real lemma, Kalman Yakubovich lemma and the circle criterion (Tan, Soh, & Xie, 1999). It is well known that the stability problem is central to the analysis of a dynamic system, where various types of stability of an equilibrium point have captured the attention of researchers. Nevertheless, from a practical point of view, it is not always the case that every NN has its orbits approaching a single equilibrium point. It is possible that there is no equilibrium point in some situations. Therefore, the concept on dissipativity has been introduced in Hale (1988). The concept of dissipativity in dynamical systems is a more general concept and it has found applications in various areas such as stability theory, chaos and synchronization theory, system norm estimation and robust control. Recently, many interesting results have been proposed for the dissipativity of delayed NNs (Arik, 2004; Cao, Yuan, Ho, & Lam, 2006; Huang, Xu, & Yang, 2007; Liao & Wang, 2003; Lou & Cui, 2008; Masubuchi, 2006; Song & Cao, 2008, 2010; Song & Zhao, 2005; Wang, Cao, & Wang, 2009; Zhang et al., 2010). Motivated by the above discussions, we shall generalize the ordinary dissipativity analysis on uncertain NNs to express dissipativity of T–S fuzzy uncertain NNs with interval time-varying delays. The main purpose of this paper is to study the global robust dissipativity of T–S fuzzy NNs with interval time-varying delays. To the best of authors knowledge, there were no results for global robust dissipativity analysis of T–S fuzzy NNs with mixed interval timevarying delays in terms of LMIs, which can be easily solved by MATLAB LMI toolbox. The main advantage of the LMI based approaches is that the LMI conditions can be solved numerically using the effective interiorpoint algorithms. We also provide numerical examples to demonstrate the effectiveness of the proposed dissipativity results.
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
Notations: throughout this paper, Rn and Rnn denote, respectively, the n-dimensional Euclidean space and the set of all n n real matrices. The superscript T denotes the transposition and the notation X P Y (respectively, X > Y), where X and Y are symmetric matrices, means that X Y is positive semi-definite (respectively, positive definite). In is the n n identity matrix. k k is the Euclidean norm in Rn . The notation ⁄ always denotes the symmetric block in one symmetric matrix. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise. 2. Problem description and preliminaries In this paper, we consider the following neural network with mixed time-varying delays
dxðtÞ ¼ AxðtÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt sðtÞÞÞ þ W 3 dt Z t f ðxðsÞÞds þ u
ð1Þ
trðtÞ
for t P 0, where xðtÞ ¼ ½x1 ðtÞ; . . . ; xn ðtÞT 2 Rn is the state vector of the network at time t, n corresponds to the number of neurons; A = diag(a1, a2, . . . , an) > 0 is a positive diagonal matrix; W1 = (aij)nn, W2 = (bij)nn and W3 = (cij)nn represent the connection weight matrix, the discretely delayed connection weight matrix and the distributively delayed connection weight matrix, respectively; f(x(t)) = [f1(x1(t)), . . . , fn(xn(t))]T denotes the neuron activation function at time t; u ¼ ðu1 ; . . . ; un ÞT 2 Rn is a constant external input vector; and s(t) and r(t) denote the discrete and distributed timevarying delays, respectively. Assumption (H1). The time-varying delays s(t) and r(t) satisfy
0 6 h1 6 sðtÞ < h2 ;
0 6 rðtÞ 6 r;
½DAk ðtÞ DW 1k ðtÞ DW 2k ðtÞ DW 3k ðtÞ h i 1 2 3 EW EW ; ¼ Gk F k ðtÞ EAk EW k k k
W2 1 3 where Gk ; EAk ; EW and EW are known real constant matrices k ; Ek k with appropriate dimensions, and Fk(t) is the time-varying uncertain matrix which satisfies that
F Tk ðtÞF k ðtÞ 6 I:
ð4Þ
Remark 2.1. In Song and Cao (2008), authors studied the problem of global dissipativity analysis on uncertain NNs with mixed timevarying delays. This paper attempts at the first time to dealt with T–S fuzzy NNs with interval time-varying delays. In this paper, the global robust dissipativity problem is investigated for T–S fuzzy NNs with interval time-varying delays. Thus, our results extend the existing ones. The defuzzified output of the T–S fuzzy system (2) is represented as follows
( r dxðtÞ X ¼ xk ðhðtÞÞ ðAk þ DAk ðtÞÞxðtÞ þ ðW 1k þ DW 1k ðtÞÞf ðxðtÞÞ dt k¼1 þðW 2k þ DW 2k ðtÞÞ f ðxðt sðtÞÞÞ ) Z t f ðxðsÞÞds þ u ; þðW 3k þ DW 3k ðtÞÞ where p Y t ðhðtÞÞ ; tk ðhðtÞÞ ¼ xk ðhðtÞÞ ¼ Pr k gkj ðhj ðtÞÞ j¼1 tj ðhðtÞÞ j¼1
in which gkj ðhj ðtÞÞ is the grade of membership of hj ðtÞ in gkj . According to the theory of fuzzy sets, we have
s_ ðtÞ 6 l;
tk ðhðtÞÞ P 0; k ¼ 1; 2; . . . ; r; Assumption (H2). For any j 2 {1, 2, . . . , n}, fj(0) = 0 and there exist þ constants lj and lj such that
fj ða1 Þ fj ða2 Þ þ 6 lj a1 a2
tk ðhðtÞÞ > 0 for all t:
Therefore, it implies r X
xk ðhðtÞÞ ¼ 1 for all t:
k¼1
Plant Rule k: IF h1(t) is gk1 and . . . and hp(t) is gkp THEN
dxðtÞ ¼ ðAk þ DAk ðtÞÞxðtÞ þ ðW 1k þ DW 1k ðtÞÞf ðxðtÞÞ dt þ ðW 2k þ DW 2k ðtÞÞf ðxðt sðtÞÞÞ Z t f ðxðsÞÞds þ u; þ ðW 3k þ DW 3k ðtÞÞ
ð2Þ
trðtÞ
t 2 ½q; 0;
r X k¼1
xk ðhðtÞÞ P 0; k ¼ 1; 2; . . . ; r;
for all a1 – a2. The kth rule of the T–S fuzzy neural network with parameter uncertainties is of the following form:
xðtÞ ¼ uðtÞ;
ð5Þ
trðtÞ
where h1, h2, r and l are constants.
lj 6
ð3Þ
q ¼ maxðh2 ; rÞ; k ¼ 1; 2; . . . ; r;
where gki ði ¼ 1; 2; . . . ; pÞ is the fuzzy set, h(t) = [h1(t), . . . , hp(t)]T is the premise variable vector, r is the number of IF–THEN rules. The norm is _ defined by kxkq ¼ maxfsupq6t60 kxðtÞk; suph6t60 kxðtÞkg. Ak, W1k, W2k and W3k are constant known real matrices. DAk(t), DW1k(t), DW2k(t) and DW3k(t) denote the time-varying parameter uncertainties. Assumption (H3). The parameter uncertainties DAk(t), DW1k(t), DW2k(t) and DW3k(t) are of the form:
Definition 2.2. A neural network (5) is said to be globally dissipative if there exists a compact set S # Rn , such that 8 x0 2 Rn ; 9 Tðx0 Þ > 0, where t P t0 + T(x0), x(t,t0,x0) # S, where x(t,t0,x0) denotes the solution of Eq. (5) from initial state x0 and initial time t0. In this case, S is called a globally attractive set. A set S is called a positive invariant if "x0 2 S implies x(t,t0,x0) # S for t P t0. Definition 2.3. Let S be a globally attractive set of neural networks [Eq. (5)]. A neural network (5) is said to be globally exponentially dissipative system if there exist a compact set S⁄ S in Rn such that 8x0 2 Rn n S , there exists a constant M(x0) > 0 and a > 0 such that
inf fkxðt; t 0 ; x0 Þ ~xk : ~x 2 S g 6 Mðx0 Þeaðtt0 Þ :
x2Rn nS
Set S⁄ is globally exponentially attractive set, where x 2 Rn n S . The following Lemmas are using to prove our theorems.
Lemma 2.4 Rakkiyappan and Balasubramaniam (2010). Let D and N be real constant matrices of appropriate dimensions, matrix F(t) satisfies FT(t)F(t) 6 I. Then (i) for any scalar > 0, DF(t)N + NTFT(t)DT 6 1D DT + NTN. (ii) For any P > 0, 2aTb 6 aTP1a + bTPb.
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
Lemma 2.5 Gu (1994). For any constant matrix M 2 Rnn ; M ¼ MT > 0, scalar g > 0, vector function x : ½0; g ! Rn such that the integrations are well defined, the following inequality holds
Z
g
T Z xðsÞds M
0
g
Z xðsÞds 6 g
0
g
Xk2;3 ¼ NT5 þ N 6 ;
Xk2;7 ¼ ð1 lÞS2 þ L2 W; Xk2;9 ¼ Xk2;10 ¼ Xk2;11 ¼ 0; Xk2;14 ¼ N6 ; Xk2;15 ¼ 0;
xT ðsÞMxðsÞds: 0
k 3;4
X Lemma 2.6 Boyd, Ghoui, Feron, and Balakrishnan (1994). Let M, P, Q be the given matrices such that Q > 0, then
"
T
P
M
M
Q
Xk2;4 ¼ N3 þ NT4 ;
k 3;5
¼X
k 3;6
¼X
¼X
Xk2;8 ¼ 0; Xk2;12 ¼ N 2 ;
Xk3;3 ¼ Q 1 þ Q 2 þ 2N 5 ;
Xk3;14 ¼ N5 ;
Xk4;10 ¼ Xk4;11 ¼ Xk4;12 ¼ 0; k 5;5
3. Main result
X
For convenience, we set r X
W 2 þ DW 2 ðtÞ ¼
xk ðhðtÞÞðW 1k þ DW 1k ðtÞÞ;
W 3 þ DW 3 ðtÞ ¼
Xk5;7 ¼ M 1 W 2k ;
Xk4;14 ¼ Xk4;15 ¼ 0;
Xk5;8 ¼ Xk5;9 ¼ Xk5;10 ¼ Xk5;11
1 Xk6;6 ¼ Y 1 þ Z 1 þ r2 Z 2 2V þ k EW k
T 1 EW ; k
Xk6;7 ¼ Xk6;8 ¼ Xk6;9 ¼ Xk6;10 ¼ 0;
xk ðhðtÞÞðW 2k þ DW 2k ðtÞÞ; Xk6;11 ¼ Xk6;12 ¼ Xk6;13 ¼ Xk6;14 ¼ Xk6;15 ¼ 0;
k¼1 r X
Xk4;13 ¼ N4 ;
Xk5;15 ¼ M1 W 3k ;
k¼1 r X
Xk4;4 ¼ 2N4 ;
¼ Xk5;12 ¼ Xk5;13 ¼ Xk5;14 ¼ 0;
k¼1
W 1 þ DW 1 ðtÞ ¼
Xk3;15 ¼ 0;
¼ h2 R1 þ ðh2 h1 ÞR2 2M 1 ;
Xk5;6 ¼ M1 W 1k ;
xk ðhðtÞÞðAk þ DAk ðtÞÞ; r X
Xk3;8 ¼ S1 þ S2 ;
¼ 0;
Xk4;5 ¼ Xk4;6 ¼ Xk4;7 ¼ Xk4;8 ¼ Xk4;9 ¼ 0;
< 0 () P þ M T Q 1 M < 0:
A þ DAðtÞ ¼
Xk2;13 ¼ N3 ;
Xk3;9 ¼ Xk3;10 ¼ Xk3;11 ¼ 0; Xk3;12 ¼ Xk3;13 ¼ 0;
#
k 3;7
Xk2;5 ¼ Xk2;6 ¼ 0;
2 Xk7;7 ¼ ð1 lÞY 2 2W þ k EW k
xk ðhðtÞÞðW 3k þ DW 3k ðtÞÞ;
T 2 EW ; k
k¼1
Xk7;8 ¼ Xk7;9 ¼ Xk7;10 ¼ Xk7;11 ¼ Xk7;12 ¼ Xk7;13 ¼ Xk7;14 ¼ Xk7;15 ¼ 0;
then system (5) can be rewritten as
dxðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðW 1 þ DW 1 ðtÞÞf ðxðtÞÞ þ ðW 2 dt þ DW 2 ðtÞÞf ðxðt sðtÞÞÞ þ ðW 3 þ DW 3 ðtÞÞ Z t f ðxðsÞÞds þ u:
Xk8;8 ¼ Y 1 þ Y 2 ; Xk8;9 ¼ 0; Xk8;10 ¼ Xk8;11 ¼ Xk8;12 ¼ Xk8;13 ¼ Xk8;14 ¼ Xk8;15 ¼ 0; X ð6Þ
trðtÞ
Theorem 3.1. Suppose that (H1)–(H3) hold. If there exist a symmetric positive definite matrices P > 0, Qi > 0 (i = 1, 2, 3, 4), Rj, Yj, Zj > 0 (j = 1, 2), and Q > 0, positive diagonal matrices V > 0 and W > 0, a matrices S1 S2, M1 and Nl (l = 1, . . . , 6) and a positive diagonal constant k > 0 such that the following LMI
k ¼
Xk
MGk
k I
< 0;
ð7Þ
hold for k = 1, 2, . . . , r, where Xk ¼ Xki;j
1515
¼X
k 9;12
¼X
T EAk
2
þ 2N1 þ 2Q ;
Xk1;2 ¼ N 1 þ NT2 ;
Xk1;5 ¼ P ATk M T1 ;
Xk9;9 ¼ Z 1 ;
¼ 0;
Xk10;10 ¼ Q 3 ;
Xk10;11 ¼ Xk10;12 ¼ Xk10;13 ¼ Xk10;14 ¼ Xk10;15 ¼ 0; 1 Q ; Xk11;12 ¼ Xk11;13 ¼ Xk11;14 ¼ Xk11;15 ¼ 0; h2 h1 4 1 ¼ R1 ; Xk12;13 ¼ 0; h2
Xk11;11 ¼ Xk12;12
1 ðR1 þ R2 Þ; h2 h1 1 ¼ R2 ; h2 h1
Xk12;14 ¼ Xk12;15 ¼ 0; Xk13;13 ¼ Xk13;14 ¼ Xk13;15 ¼ 0;
Xk14;14
with
Xk1;1 ¼ Q 1 þ h1 Q 3 þ ðh2 h1 ÞQ 4 2L1 V þ k EAk
Xk1;3 ¼ Xk1;4 ¼ 0;
k 9;11
Xk9;13 ¼ Xk9;14 ¼ Xk9;15 ¼ 0;
Now, we discuss the globally dissipative of neural network model (6) as follows.
k 9;10
Xk1;6 ¼ S1 þ L2 V;
3 Xk15;15 ¼ Z 2 þ k EW k
Xk14;15 ¼ 0;
T 3 EW ; k
M ¼ ½0 0 0 0 M 1 0 0 0 0 0 0 0 0 0 0T the neural network (6) is globally dissipative, and S ¼ n o uk x : kxk 6 kkM1ðQ is a positive invariant and globally attractive set. Þ min
Xk1;7 ¼ Xk1;8 ¼ Xk1;9 ¼ Xk1;10 ¼ Xk1;11 ¼ 0; Xk1;12 ¼ N1 ;
Xk1;13 ¼ Xk1;14 ¼ Xk1;15 ¼ 0;
Xk2;2 ¼ ð1 lÞQ 2 2L1 W 2N2 þ 2N3 2N6 ;
Proof. Consider the following positive radially unbounded Lyapunov–Krasovskii functional candidate for model (6) as
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ;
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
where
From assumption (H2), we have
þ fi ðxi ðtÞÞ li xi ðtÞ fi ðxi ðtÞÞ li xi ðtÞ 6 0;
T
V 1 ðtÞ ¼ x ðtÞPxðtÞ; V 2 ðtÞ ¼
T Q 1 S1 xðsÞ xðsÞ ds ST1 Y 1 f ðxðsÞÞ th1 f ðxðsÞÞ Z th1 xðsÞ T Q 2 S2 xðsÞ þ ds; T S2 Y 2 f ðxðsÞÞ f ðxðsÞÞ tsðtÞ Z
t
V 3 ðtÞ ¼ h1
Z
Z
0
t
xT ðsÞQ 3 xðsÞds dh þ
Z
tþh
h1
Z
h1
which are equivalent to
2
t
xT ðsÞQ 4 xðsÞds dh;
V 4 ðtÞ ¼
Z
0
V 5 ðtÞ ¼
_ x_ T ðsÞR1 xðsÞds dh þ
tþh
h2
Z
t
Z
Z
h1
n X
Z
t
f T ðxðsÞÞZ 1 f ðxðsÞÞds þ r
tr
0
Z
r
_ x_ T ðsÞR2 xðsÞds dh;
t
f T ðxðsÞÞZ 2 f ðxðsÞÞds dh:
dV 1 ðtÞ _ ¼ 2xT ðtÞPxðtÞ; dt
ð8Þ
f ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ x ðt h1 ÞQ 2 xðt h1 Þ þ 2xT ðt h1 ÞS2 f ðxðt h1 ÞÞ þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ ð1 lÞf T ðxðt sðtÞÞÞY 2 f ðxðt sðtÞÞÞ; T Z xðsÞds Q 3
th1
t
ð9Þ xðsÞds
th1
þ ðh2 h1 Þx ðtÞQ 4 xðtÞ !T ! Z th1 Z th1 1 xðsÞds Q 4 xðsÞds ; h2 h1 th2 th2
th2
1 h2
t
tsðtÞ
!T _ xðsÞds
R1
tsðtÞ
2
T
xðtÞ f ðxðtÞÞ
T
xðtÞ f ðxðtÞÞ
f ðxðtÞÞ
6 0 i ¼ 1; 2; . . . ; n;
2 4
þ
li li ei eTi
þ l i þli 2
þ l i þli 2
ei eTi
ei eTi
ei eTi
3 xðtÞ 5 6 0; f ðxðtÞÞ
L1 V
L2 V
L2 V
V
xðtÞ f ðxðtÞÞ
ð13Þ
6 0;
þ þ þ l þl þ n is approwhere L1 ¼ diag l1 l1 ; . . . ; ln ln ; L2 ¼ diag 1 2 1 ; . . . ; ln þl 2 priate dimension. Similarly one has,
2
xðt sðtÞÞ f ðxðt sðtÞÞÞ
T
L1 W
L2 W
L2 W
W
xðt sðtÞÞ f ðxðt sðtÞÞÞ
6 0:
ð14Þ
ð15Þ
t
ð10Þ
ð16Þ
tsðtÞ
tsðtÞ
_ xðsÞds ;
0 ¼ 2ðxT ðt h1 ÞN 5 þ xT ðt sðtÞÞN6 Þ xðt h1 Þ xðt sðtÞÞ
ð17Þ
Z
th1
! _ xðsÞds :
ð18Þ
tsðtÞ
It follows from the inequalities (8)–(14) and using assumption (H3) and Lemma 2.4 in (15)–(18), that
dVðtÞ _ þ xT ðtÞQ 1 xðtÞ xT ðt h1 ÞQ 1 xðt h1 Þ 6 2xT ðtÞPxðtÞ dt
! _ xðsÞds þ ðh2
þ 2xT ðtÞS1 f ðxðtÞÞ 2xT ðt h1 ÞS1 f ðxðt h1 ÞÞ þ f T ðxðtÞÞY 1 f ðxðtÞÞ
tsðtÞ
dV 5 ðtÞ 6 f T ðxðtÞÞZ 1 f ðxðtÞÞ f T ðxðt rÞÞZ 1 f ðxðt rÞÞ dt þ r2 f T ðxðtÞÞZ 2 f ðxðtÞÞ !T ! Z t Z t f ðxðsÞÞds Z 2 f ðxðsÞÞds : trðtÞ
0 ¼ 2ðxT ðtÞN1 þ xT ðt sðtÞÞN2 ÞðxðtÞ xðt sðtÞÞ Z t _ xðsÞdsÞ;
T _ ðR1 xðsÞds
_ h1 Þx_ T ðtÞR2 xðtÞ !T ! Z th1 Z th1 1 _ _ xðsÞds xðsÞds ; R2 h2 h1 tsðtÞ tsðtÞ
trðtÞ
xðtÞ
th2
th2
Z
0 ¼ 2ðxT ðt sðtÞÞN3 þ xT ðt Z h2 ÞN4 Þ xðt sðtÞÞ xðt h2 Þ
T
Z
5
trðtÞ
2ð1 lÞxT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ
Z
ei eTi
3
þ ðW 2 þ DW 2 ðtÞÞf ðxðt sðtÞÞÞ Z t f ðxðsÞÞds þ u; þ ðW 3 þ DW 3 ðtÞÞÞ
ð1 lÞxT ðt sðtÞÞQ 2 xðt sðtÞÞ
dV 4 ðtÞ 1 _ 6 h2 x_ T ðtÞR1 xðtÞ dt h2 h1 Z tsðtÞ _ þ R2 Þ xðsÞds
ei eTi
ei eTi
_ ðA þ DAðtÞÞxðtÞ þ ðW 1 þ DW 1 ðtÞÞf ðxðtÞÞ 0 ¼ 2ðx_ T ðtÞM1 ÞðxðtÞ
T
t
2
þ l i þli 2
We can see that following equations hold for any matrices M1, N1, N2, N3, N4, N5 and N6 with appropriate dimensions
dV 2 ðtÞ 6 xT ðtÞQ 1 xðtÞ xT ðt h1 ÞQ 1 xðt h1 Þ þ 2xT ðtÞS1 f ðxðtÞÞ dt 2xT ðt h1 ÞS1 f ðxðt h1 ÞÞ þ f T ðxðtÞÞY 1 f ðxðtÞÞ
Z
þ l i þli
that is
Calculating the time derivative of V(t), we have
dV 3 ðtÞ 2 6 h1 xT ðtÞQ 3 xðtÞ dt
4
þ
li li ei eTi
t
tþh
T
vi
i¼1
tþh
h2
2
where er denotes the unit column vector having one element on its rth row and zeros elsewhere. Let V = diag{v1, v2, . . . , vn}, W = diag{w1, w2, . . . , wn}, then
2 Z
T
xðtÞ f ðxðtÞÞ
tþh
h2
i ¼ 1; 2; . . . ; n;
f T ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ xT ðt h1 ÞQ 2 xðt h1 Þ ð1 lÞxT ð11Þ
ðt sðtÞÞQ 2 xðt sðtÞÞ þ 2xT ðt h1 ÞS2 f ðxðt h1 ÞÞ 2ð1 lÞxT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ ð1 lÞf T 2
ðxðt sðtÞÞÞY 2 f ðxðt sðtÞÞÞ þ h1 xT ðtÞQ 3 xðtÞ
Z
t
T Z xðsÞds Q 3
th1
ð12Þ
t
th1 T
þ ðh2 h1 Þx ðtÞQ 4 xðtÞ
xðsÞds
3349
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
Z
1 h2 h1
!T
th1
xðsÞds th2
1 h2
Z
1 h2 h1 !T Z _xðsÞds R1
t
tsðtÞ
xðsÞds
tsðtÞ
T Z _ ðR1 þ R2 Þ xðsÞds
th2
_ _ xðsÞds þ ðh2 h1 Þx_ T ðtÞR2 xðtÞ
!T
th1
_ xðsÞds
Z
R2
tsðtÞ
!
th1
T
_ xðsÞds þ f ðxðtÞÞZ 1 f ðxðtÞÞ
tsðtÞ
T
Z
2 T
f ðxðt rÞÞZ 1 f ðxðt rÞÞ þ r f ðxðtÞÞZ 2 f ðxðtÞÞ
!T
t
f ðxðsÞÞds
trðtÞ
!
Z
t
Corollary 3.3. Suppose that (H1)–(H3) hold. If there exist symmetric positive definite matrices P > 0, Qi > 0 (i = 1, 2, 3, 4), Rj, Yj, Zj > 0 (j = 1, 2) and Q > 0, positive diagonal matrices V > 0 and W > 0, a matrices S1, S2, M1 and Nl (l = 1, . . . , 6) such that the following LMI holds
þ 2x_ T ðtÞM 1 W 2 f ðxðt sðtÞÞÞ þ f T ðxðt sðtÞÞÞðEW 2 ÞT
Z
ðEW 3 ÞT ðEW 3 Þ
trðtÞ
!
t
!T f ðxðsÞÞds
xðtÞ xðt sðtÞÞ
Z
!
t
tsðtÞ
tsðtÞ
ð21Þ
invariant and globally attractive set. In the following theorem, we extend the above result to global exponential dissipativity of neural network (6) as follows.
_ xðsÞds þ 2ðxT ðt sðtÞÞN3
þ xT ðt h2 ÞN 4 Þ Z xðt sðtÞÞ xðt h2 Þ
X < 0;
where X = (Xi,j)15 15 are defined as in Theorem 3.1, the neural netn o 1 uk is a positive work (20) is globally dissipative and S ¼ x : kxk 6 kkM min ðQ Þ
f ðxðsÞÞds þ 2x_ T ðtÞM 1 u þ 2ðxT ðtÞN1 þ xT ðt sðtÞÞN2 Þ
trðtÞ
ð20Þ
trðtÞ
þ f T ðxðtÞÞðEW 1 ÞT ðEW 1 Þf ðxðtÞÞ
trðtÞ
Rt _ is Remark 3.2. In this paper, the integral term th2 x_ T ðsÞR1 xðsÞds Rt R tsðtÞ _ _ tsðtÞ x_ T ðsÞR1 xðsÞds divided into two parts are th2 x_ T ðsÞR1 xðsÞds; R th _ and the integral term th21 x_ T ðsÞR2 xðsÞds is divided into two parts as R tsðtÞ T R th1 T _ _ _ _ th2 x ðsÞR2 xðsÞds; tsðtÞ x ðsÞR2 xðsÞds, which may lead to less con-
dxðtÞ ¼ AxðtÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt sðtÞÞÞ dt Z t f ðxðsÞÞds þ u: þ W3
f ðxðt sðtÞÞÞ L2 W W f ðxðt sðtÞÞÞ _ 2x_ T ðtÞM 1 AxðtÞ þ 1 x_ T ðtÞM1 GGT M T1 xðtÞ _ 2x ðtÞM 1 xðtÞ A T A T T þ x ðtÞðE Þ ðE ÞxðtÞ þ 2x_ ðtÞM 1 W 1 f ðxðtÞÞ
t
when x 2 Rn n S. Therefore, neural network (6) is a globally dissipative system and the set S is a positive invariant and globally attractive set as LMI (7) hold. This completes the proof. h
In the following, we consider the neural network without fuzzy and uncertainties then (5) becomes
_T
ðEW 2 Þf ðxðt sðtÞÞÞ ! Z t Z T _ f ðxðsÞÞds þ þ 2x ðtÞM 1 W 3
dVðtÞ 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM 1 uÞ 6 2kxkðkmin ðQÞkxk þ kM 1 ukÞ dt < 0;
servative results.
xðtÞ T L1 V L2 V xðtÞ f ðxðsÞÞds 2 f ðxðtÞÞ L2 V V f ðxðtÞÞ trðtÞ T L1 W L2 W xðt sðtÞÞ xðt sðtÞÞ
Z2 2
_ xðsÞds
th2
!
t
tsðtÞ
tsðtÞ
Z
1 h2 h1
!
th1
th2
Z
_ þ h2 x_ T ðtÞR1 xðtÞ
Z
Q4
_ xðsÞds
Theorem 3.4. Under the conditions of Theorem 3.1, neural network n o uk (6) is globally exponentially dissipative and S ¼ x : kxk 6 kkM1ðQ is Þ
th
min
2 þ 2 xT ðt h1 ÞN5 þ xT ðt sðtÞÞN6 ! Z th1 _ xðsÞds xðt h1 Þ xðt sðtÞÞ
a positive invariant and globally attractive set. Further, the exponential dissipativity rate index k = /2 can be estimated by the following inequality
tsðtÞ
dVðtÞ nðtÞ; 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM 1 uÞ þ nT ðtÞt dt
ð19Þ
where,
T
T
T
T
_T
T
nðtÞ ¼ x ðtÞ x ðt sðtÞÞ x ðt h1 Þ x ðt h2 Þ x ðtÞ f ðxðtÞÞ Z t T f T ðxðt sðtÞÞÞf T ðxðt h1 ÞÞ f T ðxðt rÞÞ xðsÞds Z
!T
th1
xðsÞds !T
th1
_ xðsÞds tsðtÞ
Z
!T Z
t
th1
tsðtÞ
_ xðsÞds
tsðtÞ
th2
Z
Z
t
trðtÞ
T _ xðsÞds
Wk
MGk
k I
< 0;
ð22Þ
hold for k = 1, 2, . . . , r, where Wk ¼ Wki;j
1515
W k1;1 ¼ P þ eh1 Q 1 þ þ
A T A k ðEk Þ ðEk Þ
2 h1 eh1 Q 3
!T 3T f ðxðsÞÞds 5
with
þ ðh2 h1 Þeh2 Q 4 2L1 V
þ 2N1 þ 2Q ;
Wk1;2 ¼ N1 þ NT2 ; Wk1;3 ¼ Wk1;4 ¼ 0;
th2
Wk1;5 ¼ P ATk MT1 ;
Wk1;6 ¼ eh1 S1 þ L2 V; Wk1;7 ¼ 0; Wk1;8 ¼ Wk1;9 ¼ Wk1;10 ¼ Wk1;11 ¼ 0;
and
t ¼ ðti;j Þ
Nk ¼
Wk1;12 ¼ N1 ;
Wk1;13 ¼ Wk1;14 ¼ Wk1;15 ¼ 0;
1
¼ ðXi;j Þ þ diag 0; 0; 0; 0; M1 GG
T
M T1 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0
Wk2;2 ¼ ð1 lÞQ 2 2L1 W 2N2 þ 2N3 2N6 ; :
Considering xk(h(t)) P 0 (k = 1, 2, . . . , r) and k < 0 (k = 1, 2, . . . , r) in Pr Theorem 3.1, we have Noting that k¼1 xk ðhðtÞÞ k < 0. Pr k¼1 xk ðhðtÞÞ ¼ 1, we obtain k < 0 by using Lemma 2.6. Thus from condition (7) and inequality (19), we get
Wk2;3 ¼ NT5 þ N6 ;
Wk2;4 ¼ N 3 þ NT4 ; Wk2;5 ¼ Wk2;6 ¼ 0;
Wk2;7 ¼ ð1 lÞS2 þ L2 W; Wk2;8 ¼ Wk2;9 ¼ Wk2;10 ¼ Wk2;11 ¼ 0; Wk2;12 ¼ N2 ; Wk2;13 ¼ N 3 ;
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Wk2;14 ¼ N6 ;
Wk3;3 ¼ Q 1 þ eðh2 h1 Þ Q 2 þ 2N5 ;
Wk2;15 ¼ 0;
V 3 ðtÞ ¼ h1
Wk3;4 ¼ Wk3;5 ¼ Wk3;6 ¼ Wk3;7 ¼ 0; k 3;8
W
ðh2 h1 Þ
¼ S1 þ e
Wk3;14 ¼ N5 ; k 4;4
W
¼ 2N4 ; W ¼W
W
k 3;10
¼W
k 3;11
¼W
k 3;12
¼W
k 3;13
¼W
k 4;6
¼W
k 4;13
¼ 0;
W
¼W
¼W
k 4;8
k 4;9
¼W
k 4;10
¼W
W
k 5;7
¼ M 1 W 1k ; W
¼W
þ
¼ N 4 ;
W
k 5;9
Z
¼W
¼W
1
1 ÞT ðEW k Þ;
¼W
k 5;12
¼W
k 5;13
¼W
k 5;14
¼W
¼ 0;
Wk6;6 ¼ eh1 Y 1 þ er Z 1 þ r2 er Z 2 2V
Wk5;15 ¼ M 1 W 3k ;
k ðEWk
k 5;11
Wk6;7 ¼ Wk6;8 ¼ Wk6;9 ¼ Wk6;10 ¼ Wk6;11 ¼ 0;
Wk6;12 ¼ Wk6;13 ¼ Wk6;14 ¼ Wk6;15 ¼ 0; k 7;7
W
k 7;9
W
W2 T W2 k ðEk Þ ðEk Þ;
¼ ð1 lÞY 2 2W þ k 7;10
¼W
¼W
k 7;11
¼W
k 7;12
k 7;13
¼W
W
k 7;14
k 7;8
eðsþh2 Þ xT ðsÞQ 4 xðsÞds dh;
_ eðsþh2 Þ x_ T ðsÞR1 xðsÞds dh
Z
h1
t
Z
t
_ eðsþh2 Þ x_ T ðsÞR2 xðsÞds dh;
tþh
eðsþrÞ f T ðxðsÞÞZ 1 f ðxðsÞÞds Z
0
r
Z
t
eðsþrÞ f T ðxðsÞÞZ 2 f ðxðsÞÞds dh:
tþh
Calculating the time derivative of V(t), we have
dV 1 ðtÞ _ ¼ et xT ðtÞPxðtÞ þ 2et xT ðtÞPxðtÞ; dt
þ 2eðtþh1 Þ xT ðtÞS1 f ðxðtÞÞ 2et xT ðt h1 ÞS1 f ðxðt h1 ÞÞ
¼W
¼ 0;
Wk8;8 ¼ Y 1 þ eðh2 h1 Þ Y 2 ; Wk8;9 ¼ 0;
þ eðtþh1 Þ f T ðxðtÞÞY 1 f ðxðtÞÞ et f T ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ eðtþh2 h1 Þ xT ðt h1 ÞQ 2 xðt h1 Þ
Wk8;10 ¼ Wk8;11 ¼ Wk8;12 ¼ Wk8;13 ¼ Wk8;14 ¼ Wk8;15 ¼ 0;
et ð1 lÞxT ðt sðtÞÞQ 2 xðt sðtÞÞ
Wk9;9 ¼ Z 1 ; Wk9;10 ¼ Wk9;11 ¼ Wk9;12 ¼ 0;
þ 2eðtþh2 h1 Þ xT ðt h1 ÞS2 f ðxðt h1 ÞÞ
Wk9;13 ¼ Wk9;14 ¼ Wk9;15 ¼ 0;
2ð1 lÞet xT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ
Wk10;10 ¼ Q 3 ;
þ eðtþh2 h1 Þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ
Wk10;11 ¼ Wk10;12 ¼ Wk10;13 ¼ Wk10;14 ¼ Wk10;15 ¼ 0; Wk11;11 k 12;12
W
ð1 lÞet f T ðxðt sðtÞÞÞ Y 2 f ðxðt sðtÞÞÞ;
1 ¼ Q ; Wk ¼ Wk11;13 ¼ Wk11;14 ¼ Wk11;15 ¼ 0; h2 h1 4 11;12 1 ¼ R1 ; Wk12;13 ¼ 0; h2
Wk12;14 ¼ Wk12;15 ¼ 0;
3 Wk14;15 ¼ 0; Wk15;15 ¼ Z 2 þ k EW k
T 3 EW ; k
M ¼ ½0 0 0 0 M 1 0 0 0 0 0 0 0 0 0 0 : Proof. Consider the following positive radially unbounded Lyapunov–Krasovskii functional candidate for model (6) as
VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ þ V 3 ðtÞ þ V 4 ðtÞ þ V 5 ðtÞ;
V 1 ðtÞ ¼ et xT ðtÞPxðtÞ; t
" eðsþh1 Þ
xðsÞ
#T "
Q1
S1
#"
xðsÞ
# ds
ST1 Y 1 f ðxðsÞÞ f ðxðsÞÞ " # " # " # T Z th1 Q 2 S2 xðsÞ xðsÞ ðsþh2 Þ þ e ds; tsðtÞ ST2 Y 2 f ðxðsÞÞ f ðxðsÞÞ th1
ð24Þ
xðsÞds
th1
ð25Þ
dV 4 ðtÞ _ 6 h2 eðtþh2 Þ x_ T ðtÞR1 xðtÞ dt Z tsðtÞ T Z tsðtÞ et _ _ ðR1 þ R2 Þ xðsÞds xðsÞds h2 h1 th2 th2 !T ! Z t Z t et _ _ R1 xðsÞds xðsÞds h2 tsðtÞ tsðtÞ _ þ ðh2 h1 Þeðtþh2 Þ x_ T ðtÞR2 xðtÞ !T ! Z Z th1 th1 et _ _ R2 xðsÞds xðsÞds ; h2 h1 tsðtÞ tsðtÞ
where
Z
t
þ ðh2 h1 Þeðtþh2 Þ xT ðtÞQ 4 xðtÞ !T ! Z th1 Z th1 et xðsÞds Q 4 xðsÞds ; h2 h1 th2 th2
T
V 2 ðtÞ ¼
dV 3 ðtÞ 2 6 h1 eðtþh1 Þ xT ðtÞQ 3 xðtÞ dt Z t T Z et xðsÞds Q 3 th1
1 ðR1 þ R2 Þ; h2 h1 1 ¼ R2 ; h2 h1
Wk13;13 ¼
Wk13;14 ¼ Wk13;15 ¼ 0; Wk14;14
ð23Þ
dV 2 ðtÞ 6 eðtþh1 Þ xT ðtÞQ 1 xðtÞ et xT ðt h1 ÞQ 1 xðt h1 Þ dt
¼ 0; k 7;15
¼W
t
tþh
tr
¼ M 1 W 2k ;
k 5;10
t
h2
V 5 ðtÞ ¼
eðsþh1 Þ xT ðsÞQ 3 xðsÞds dh
tþh
Z
0
þr k 5;8
Z
h1
h2
k 4;11
Wk4;14 ¼ Wk4;15 ¼ 0; Wk5;5 ¼ h2 eh2 R1 þ ðh2 h1 Þeh2 R2 2M 1 ; k 5;6
Z
t
tþh
h2
V 4 ðtÞ ¼ k 4;7
Z
0
h1
Z
¼ 0;
Wk3;15 ¼ 0; k 4;5
k 4;12
S2 ;
k 3;9
þ
Z
ð26Þ
dV 5 ðtÞ 6 eðtþrÞ f T ðxðtÞÞZ 1 f ðxðtÞÞ et f T ðxðt rÞÞZ 1 f ðxðt rÞÞ dt þ r2 eðtþrÞ f T ðxðtÞÞZ 2 f ðxðtÞÞ !T ! Z t Z t f ðxðsÞÞds Z 2 f ðxðsÞÞds : ð27Þ et trðtÞ
trðtÞ
3351
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
It follows from inequalities (13) and (14) and using assumption (H3), Lemma 2.4 in (15)–(18) and (23)–(27) that
dVðtÞ t T _ þ eh1 xT ðtÞQ 1 xðtÞ 6e x ðtÞPxðtÞ þ 2xT ðtÞPxðtÞ dt xT ðt h1 ÞQ 1 xðt h1 Þ þ 2eh1 xT ðtÞS1 f ðxðtÞÞ
þ2eðh2 h1 Þ xT ðt h1 ÞS2 f ðxðt h1 ÞÞ 2ð1 lÞxT ðt sðtÞÞS2 f ðxðt sðtÞÞÞ þ eðh2 h1 Þ f T ðxðt h1 ÞÞY 2 f ðxðt h1 ÞÞ ð1 lÞf T 2 xðt sðtÞÞÞY 2 f ðxðt sðtÞÞÞ þ h1 eh1 xT ðtÞQ 3 xðtÞ Z t T Z t xðsÞds Q 3 xðsÞds þ ðh2 h1 Þ eh2 xT ðtÞQ 4 xðtÞ
th1
1 h2 h1
!T
th1
xðsÞds
Q4
Z
th2
Z
tsðtÞ
th1
! _ xðsÞds þ h2 eh2 x_ T ðtÞR1 xðtÞ
th2
T Z _ xðsÞds ðR1 þ R2 Þ
Z
f ðxðsÞÞds
1=2 Vðxð0ÞÞ eð=2Þt ; kmin ðPÞ
which means that neural network (6) is globally exponentially dissipative and the set S is a positive invariant and globally attractive set as LMI (22) hold. This completes the proof. h In the following, we consider the neural network without fuzzy and uncertainties then (5) becomes
dxðtÞ ¼ AxðtÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt sðtÞÞÞ dt Z t f ðxðsÞÞds þ u: þ W3
ð32Þ
Corollary 3.5. Under the conditions of Corollary 3.3, neural network n o 1 uk is (32) is globally exponentially dissipative and S ¼ x : kxk 6 kkM min ðQ Þ a positive invariant and globally attractive set. Further, the exponential dissipativity rate index k = /2 can be estimated by the following inequality
W < 0;
ð33Þ
where W = (Wi,j)15 15 are defined as in Theorem 3.4.
trðtÞ
!#
t
ð31Þ
trðtÞ
þer f T ðxðtÞÞZ 1 f ðxðtÞÞ f T ðxðt rÞÞZ 1 f ðxðt rÞÞ !T Z t þr2 er f T ðxðtÞÞZ 2 f ðxðtÞÞ f ðxðsÞÞds Z 2
kxk 6
tsðtÞ
1 _ xðsÞds h2 h1 th2 th2 ! ! T Z t Z t 1 _ _ _ xðsÞds xðsÞds þ ðh2 h1 Þeh2 x_ T ðtÞR2 xðtÞ R1 h2 tsðtÞ tsðtÞ !T ! Z th1 Z th1 1 _ _ xðsÞds xðsÞds R2 h2 h1 tsðtÞ tsðtÞ
From the definition of V(x(t)), we know that
From Eqs. (30) and (31), we get
þeh1 f T ðxðtÞÞY 1 f ðxðtÞÞ f T ðxðt h1 ÞÞY 1 f ðxðt h1 ÞÞ þ eðh2 h1 Þ xT t h1 ÞQ 2 xðt h1 Þ ð1 lÞxT ðt sðtÞÞ Q 2 xðt sðtÞÞ
Z
ð30Þ
VðxðtÞÞ P et xT ðtÞPxðtÞ:
2xT ðt h1 ÞS1 f ðxðt h1 ÞÞ
th1
VðxðtÞÞ 6 Vðxð0ÞÞ:
2
trðtÞ
xðtÞ
T
f ðxðtÞÞ
L1 V L2 V L2 V
V
xðtÞ
f ðxðtÞÞ
dVðtÞ 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM1 uÞ þ nT ðtÞnðtÞ; dt where,
¼ ði;j Þ ¼ ðWi;j Þ þ diag 0; 0; 0; 0; 1 M1 GGT M T1 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0 < 0 ð28Þ and
Remark 3.6. In Liao and Wang (2003), Arik (2004), Song and Zhao (2005), Cao et al. (2006), Masubuchi (2006), Huang et al. (2007), Lou and Cui (2008), Song and Cao (2008), Wang et al. (2009), Zhang et al. (2010) and Song and Cao (2010), authors studied the various types of neural networks and few of them have proposed results in LMI approach. In this paper, the problem of global robust dissipativity is derived for T–S fuzzy neural network using LMI approaches. Therefore, our results are less conservative than those given in the previous literature.
4. Numerical examples In this section, we will give examples showing the effectiveness of established theoretical results.
" nðtÞ ¼ xT ðtÞ xT ðt sðtÞÞ xT ðt h1 Þ xT ðt h2 Þ
Example 1. Consider the system (6) with parameters defined as
x_ T ðtÞ f T ðxðtÞÞ f T ðxðt sðtÞÞÞf T ðxðt h1 ÞÞ f T ðxðt rÞÞ !T !T Z t T Z th1 Z t _ xðsÞds xðsÞds xðsÞds Z
tsðtÞ
th2
tsðtÞ
th2
th1
T _ xðsÞds
Z
th1
A1 ¼
!T _ xðsÞds
tsðtÞ
Z
t
trðtÞ
!T 3T f ðxðsÞÞds 5 :
Considering xk(h(t)) P 0 (k = 1, 2, . . . , r) and Nk < 0 (k = 1, 2, . . . , r) in Pr Theorem 3.4, we have Noting that k¼1 xk ðhðtÞÞNk < 0. Pr k¼1 xk ðhðtÞÞ ¼ 1. From inequality (28) we get
dVðtÞ 6 2ðxT ðtÞQxðtÞ þ x_ T ðtÞM1 uÞ dt 6 2kxkðkmin ðQ Þkxk þ kM 1 ukÞ < 0;
ð29Þ
when x 2 RnnS. Integrating two sides of Eq. (29) from 0 to an arbitrary t > 0, we have
W 31
2:5 0:2 W 11 ¼ ; 0 2 4:5 3:5 0:2 0:05 ¼ ; 0:35 0:8
u¼
2 0
0:02 0:04
W3 2 EW 1 ¼ E1 ¼
W 22 ¼
;
;
G1 ¼
0
0
0:2
0:01
0
0
0:01
2:5 0:6 0:4 2:5
;
0:3
; A2 ¼
W 32 ¼
;
1 EA1 ¼ EW ¼ 1
2:5 0 0 2
W 21 ¼
1
;
2
0:4
;
0:5 2:8
; W 12 ¼
0:5 0:4 0:8
0:03
0
0
0:03
2 0:4 5 3:5
G2 ¼
;
;
0:4
0
0
0:2
;
3352
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
1 EA2 ¼ EW ¼ 2
0:02 0 ; 0 0:02
2 3 ¼ EW ¼ EW 2 2
0:01 0 ; 0 0:01
4
f 1 ðxÞ
¼ f2 ðxÞ ¼ tanhðxÞ;
3
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 0:66; h2 ¼ 0:9928; r ¼ 0:3; li ¼ 0; li ¼ 2; li ¼ 0; li ¼ 2. Thus,
0 0 0 0
;
L2 ¼
1 0
:
0 1
1
state
L1 ¼
2
0
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (7) as follows
0:0018 0:0000
P¼
#
"
;
0:0000 0:0002 " # 6:1341 0:7844 Q2 ¼ ; 0:7844 6:0528 "
Q3 ¼
4:8497
0:0000
Q1 ¼
16:89027
8:3005
8:3005
27:9421
#
" Q4 ¼
2:0756
0:0000
0:0000 4:8497 0:0000 " # 0:3807 0:2000 ; R1 ¼ 106 0:20000 0:1269 "
0:0010 0:0001
#
Y2 ¼
5:4128 3:7775
"
Z1 ¼
3:7199 1:6957
# Z2 ¼
1:6957 3:5925 # 2:3772 3:7126 ; 1:6271 4:8341
Q¼
−0.4
−0.2
0
time (sec)
0.2
0.4
0.6
0.8
1
Fig. 1. The transient responses of time and state for k = 1 for Example 1.
4
9:1717 5:4144
#
3
; 2
5:9714
1:7470
1:7470
8:0115
1
#
0
;
−2
0 0:0033 0 W¼ ; 0 1:0303 0 0:0033 5:7573 2:7534 ; S1 ¼ 2:4063 6:0477
V¼
−0.6
−1
;
1:7403
2:0756
5:4144 9:2464
"
−0.8
;
;
;
"
#
#
3:7775 5:5331 "
Y1 ¼
;
0:0001 0:0011 "
−2
;
−3 −1
;
R2 ¼
−1
#
state
"
−3 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (sec) Fig. 2. The transient responses of time and state for k = 2 for Example 1.
S2 ¼
3:9688 2:2482
;
0:3353 0:0188
M1 ¼ ; 2:0564 4:1238 0:0189 0:3510 0:1829 0:1241 ; N1 ¼ 106 0:0121 0:0102 6
0:3825
0:2015
N2 ¼ 10 0:2015 0:1267 0:0030 0:0002 N4 ¼ ; 0:0002 0:0032
0:0034 0:0051 N5 ¼ ; 0:0052 0:0107 e1 ¼ 0:0246;
;
N3 ¼
0:0030 0:0002 0:0002 0:0032
Example 2. Consider the system (20) with parameters defined as
" A¼
;
0:0030 0:0002 N6 ¼ ; 0:0002 0:0032
e2 ¼ 0:0370:
Therefore, the neural network (6) is globally dissipative and the positive invariant and global attractive set are S = {x : kxk 6 0.0175}. Figs. 1 and 2 depicts the time and state of the considered network with initial conditions x1(t) = 0.3, x2(t) = 0.2 and x1(t) = 0.4, x2(t) = 0.3, t 2 [1, 0].
W3 ¼
u¼
4 0
#
" W1 ¼
;
0 3 " 1
0:4
0:6
1
0:03
;
0:04
3:2 0:4 4
3:6
#
" ;
W2 ¼
2:2 1:2 1:2
4
# ;
# ;
f 1 ðxÞ ¼ f2 ðxÞ ¼ tanhðxÞ:
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 0:8; h2 ¼ 1:0173; r ¼ 0:3; li ¼ 0; li ¼ 2; li ¼ 0; li ¼ 2. Thus,
L1 ¼
0 0 0 0
;
L2 ¼
1 0 0 1
:
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (21) as follows
3353
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
0:4687 0:2161 139:3402 ; Q1 ¼ 0:2161 0:1121 104:1924 42:2231 2:0579 52:0426 Q2 ¼ ; Q3 ¼ 2:0579 35:4697 0:0000 18:5802 0:0000 0:0134 ; R1 ¼ Q4 ¼ 0:0000 18:5802 0:0067
P¼
0:1534 0:0663 ; R2 ¼ 0:0663 0:0536 32:7623 12:0209 ; Y2 ¼ 12:0209 32:1703
; 12:2526 36:8993 2:6908 1:3524 Q¼ ; 1:8003 2:5689 38:0998 12:2526
Z1 ¼
V¼
1:2916
0
S2 ¼
13:2857
Z2 ¼
6:4957
0:0135 0:0067 ; 0:0067 0:0034 0:7409 0:3227 N4 ¼ ; 0:3227 0:2552
N2 ¼
N5 ¼
0:0566 0:2528 0:2529
0:3244
;
75:8144
26:3889
26:3889
81:0423
u¼
2:5
0:03
0:0948
0
0
0:0944
;
;
G1 ¼
;
3
0:8
0:4
0
0
0:3
;
1 EA1 ¼ EW ¼ 1
A2 ¼
W 32 ¼
;
0:5 2:8
1 EA2 ¼ EW ¼ 2
W 21 ¼
;
0:03 0 ; 0 0:03
0
0
2:2
0:5
0:6
1
2 3 EW ¼ EW ¼ 2 2
0:4
3
0:01
0
0
0:01
;
;
;
;
G2 ¼
0:3
0
0
0:4
0:02 0 ; 0 0:02
;
f 1 ðxÞ
M1 ¼
0:0566
0:0256
0:0260
0:0172
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 1; h2 ¼ 1:3953; r ¼ 0:3; li ¼ 0; li ¼ 1:4; li ¼ 0; li ¼ 1:4. Thus,
;
0 0
L1 ¼
N3 ¼
0 0
"
0:6876
0:2960
0:2961
0:2420
;
L2 ¼
1:2483
0:7
0
0
0:7
:
1:2660
#
"
4:5796 2:1274
3
; Q 1 ¼ 10 1:2660 2:0244 " # 780:2072 78:4188 Q2 ¼ ; 78:4188 809:5559
P¼
N6 ¼
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (22) as follows
0:7331 0:3188 ; 0:3188 0:2532
:
Q3 ¼
0:0000
0:0000 895:1100 0:0121 0:0178 ; R1 ¼ 0:0178 0:0261
0:3703
1.5
895:1100
Y2 ¼
1
Z1 ¼
0:1403
Q4 ¼
0:1403 0:6408 944:1951 642:3267 642:3267 928:3844
Y 1 ¼ 103
;
807:6208 323:8715
323:8715 795:7156 3:6310 0:9571 Q¼ ; 0:9563 2:9879
0.5
;
2:1274 3:0179
−0.5
V¼
−1
S1 ¼
−1.5
2:5433
0:0000
443:1641
0
time (sec)
0.5
Fig. 3. The transient responses of time and state for Example 2.
1
;
1:6935
0:9540
0:9540 1:6687
;
;
;
;
W¼
Z 2 ¼ 103
0 3:5203 718:8600 465:1139 382:3011 665:2249
S2 ¼
0
;
0:0000
0
#
443:1641
R2 ¼
−0.5
2:5 0:5
2:5
1
¼ f2 ðxÞ ¼ tanhðxÞ:
;
0:02
W 22 ¼
0
0:02 0 2 3 EW ¼ EW ¼ ; 1 1 0 0:02 2:2 0:5 W 12 ¼ ; 4 3:7
2
state
2 0:4 W 11 ¼ ; 0 1:5 4 3:8 0:4 0:3 ¼ ; 0:6 0:6
W 31
Therefore, the neural network (20) is globally dissipative and the positive invariant and global attractive set are S = {x : kxk 6 0.0028}. Fig. 3 depicts the time and state of the considered network with initial conditions x1(t) = 0.3, x2(t) = 0.5, t 2 [1, 0].
−2 −1
A1 ¼
65:5176 24:0352 Y1 ¼ ; 24:0352 64:3342
;
3:8269 12:8548 0:0011 0:0005 ; N1 ¼ 0:0004 0:0002
Example 3. Consider the system (6) with parameters defined as
W¼ 0 1:3905 26:3861 12:8220 ; S1 ¼ 7:2846 25:3714
104:1924 ; 289:2376 0:0000 ; 52:0426 0:0067 ; 0:0033
479:5892 343:6215
290:2815 443:6318 0:0017 0:0025 ; N1 ¼ 0:0021 0:0031
2:1559
1:1890
1:1890
2:0490
1:4185
0
0
2:2452
;
;
;
;
M1 ¼
0:1825
0:1022
0:1025
0:3284
;
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S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
0:0088 0:0129 N2 ¼ ; 0:0129 0:0189 0:9651 0:3975 N4 ¼ ; 0:3975 1:6805 N5 ¼
2:3964
2:0070
1:8188 1:1410
e1 ¼ 214:4287;
;
N3 ¼
N6 ¼
0:9649 0:3972 ; 0:3972 1:6800
0:9355
0:3531
0:3539
1:6177
3:8 0 2:6 0:6 ; W1 ¼ ; 0 2:6 3:5 4 1 0:8 W3 ¼ ; 0:6 1
A¼
;
u¼
e2 ¼ 8:7415:
Therefore, the neural network (6) is globally exponentially dissipative. Moreover, from Eq. (22), we can get that the exponential dissipativity rate index k = /2 = 0.0131. It is easy to compute that the positive invariant and globally attractive set are S = {x : kxk 6 0.0053}. Figs. 4 and 5 depicts the time and state of the considered network with initial conditions x1(t) = 0.3, x2(t) = 0.5,t 2 [1, 0].
0 0
;
L2 ¼
0:6
0
0
0:6
:
1:5600 0:9497 Q 1 ¼ 103 ; 1:6037 1:1272 0:9497 2:3336 409:7816 27:0418 ; Q2 ¼ 27:0418 350:2791 2:8398
1:6037
549:2708
;
0:0000
0:0000 549:2708 0:0293 0:0191 ; R1 ¼ 0:0191 0:0124
;
Q4 ¼
286:5514
0:0000
0:0000
286:5514
R2 ¼ state
f 1 ðxÞ ¼ f2 ðxÞ ¼ tanhðxÞ:
;
By the MATLAB LMI Control Toolbox, we find solution to the LMI in Eq. (33) as follows
1
0
0:3179
Y2 ¼
0:1127
0:1127 0:1814 362:5386 153:1137 153:1137 343:8646
Z1 ¼
−2
−3 −0.5
0
0.5
time (sec)
Fig. 4. The transient responses of time and state for k = 1 for Example 3.
S1 ¼
2.5
S2 ¼
1:3162
0
;
795:4553 321:9837 321:9837 756:6649
Z 2 ¼ 103
W¼
0 2:3333 294:3487 144:4242 88:9780
240:3678
140:6995
70:9736
43:4687 114:0745 0:0064 0:0042 ; N1 ¼ 0:0040 0:0026
1.5
1
N2 ¼
0:0282
0:0183
1:1373
0
0
0:8509
0.5 0 −0.5
N5 ¼
−1 −1.5 −2 −0.6
−0.4
−0.2
0
time (sec)
0.2
0.4
0.6
Fig. 5. The transient responses of time and state for k = 2 for Example 3.
0.8
0:1809 0:6614 0:7086 0:7503
;
;
;
M1 ¼
;
; 0:0183 0:0119 0:9798 0:3714 ; N4 ¼ 0:3714 0:5476
;
1:1376 0:6761 ; 0:6761 1:3358
2
;
;
V¼
Y1 ¼
;
456:6791 144:5897 ; 144:5897 443:3150 1:7808 1:5789 Q¼ ; 1:5784 2:3721
−1
state
0 0
L1 ¼
Q3 ¼
−2.5 −0.8
0:02
3
2
2:8 0:6 ; 0:8 3:2
It is easy to check that assumptions (H1) and (H3) are satisfied and þ þ h1 ¼ 0:7; h2 ¼ 1:0529; r ¼ 0:3; li ¼ 0; li ¼ 1:2; li ¼ 0; li ¼ 1:2. Thus,
P¼ Example 4. Consider the system (32) with parameters defined as
0:03
W2 ¼
N3 ¼
N6 ¼
0:2639
0:1160
0:1182
0:1368
0:9793
0:3711
0:3711
0:5474
0:8987
0:3185
0:3183
0:5132
;
;
:
Therefore, the neural network (32) is globally exponentially dissipative. Moreover, from Eq. (33), we can get that the exponential dissipativity rate index k = /2 = 0.2261. It is easy to compute that the positive invariant and globally attractive set are S = {x : kxk 6 0.0255}. Fig. 6 depicts the time and state of the considered network with initial conditions x1(t) = 0.4, x2(t) = 0.5, t 2 [1, 0]. It should be pointed out that Theorems 1 and 2, Corollaries 1 and 3 in Song and Cao (2008) cannot be applied to this example since
S. Muralisankar et al. / Expert Systems with Applications 39 (2012) 3345–3355
0.8 0.6
state
0.4 0.2 0 −0.2 −0.4 −0.6 −0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
time (sec) Fig. 6. The transient responses of time and state for Example 4.
the inequalities in Song and Cao (2008) do not have feasible solutions. Remark 5. From the above examples, we can see that the proposed results in this paper improve and generalize than those in Song and Cao (2008).
5. Conclusion In this paper, we have studied the global robust dissipativity of T–S fuzzy neural networks with interval time-varying delays. By a proper Lyapunov–Krasovskii functional and employing analytic techniques, some sufficient conditions ensuring the global dissipativity and global exponential dissipativity of T–S fuzzy neural networks have been derived in terms of LMIs. The new results given in this paper to improve the earlier dissipativity results. The numerical examples are provided to demonstrate the effectiveness of the proposed results. References Arik, S. (2004). On the global dissipativity of dynamical neural networks with time delays. Physics Letters A, 326, 126–132. Boyd, B., Ghoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM.
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