Sugeno fuzzy Hopfield neural networks with mixed ... - Semantic Scholar
Expert Systems with Applications 39 (2012) 472–481
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Delay-dependent robust asymptotic state estimation of Takagi–Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays q P. Balasubramaniam a,⇑, V. Vembarasan a, R. Rakkiyappan b a b
Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamilnadu, India
a r t i c l e
i n f o
Keywords: T–S fuzzy model Linear matrix inequality Lyapunov–Krasovskii functional Hopfield neural networks State estimation
a b s t r a c t This paper investigates delay-dependent robust asymptotic state estimation of fuzzy neural networks with mixed interval time-varying delay. In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the robust state estimation of Hopfield neural networks with mixed interval time-varying delays. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error is globally asymptotically stable. Based on the Lyapunov–Krasovskii functional which contains a triple-integral term, delay-dependent robust state estimation for such T–S fuzzy Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. The unknown gain matrix is determined by solving a delay-dependent LMI. Finally two numerical examples are provided to demonstrate the effectiveness of the proposed method. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Neural Networks (NNs) have been extensively studied and successfully applied in many areas such as combinatorial optimization, signal processing, and pattern recognition (Gupta, Jin, & Homma, 2003). Different models of neural networks such as Hopfield-type neural networks, cellular neural networks, Cohen–Grossberg neural networks, and bidirectional associative memory neural networks have been extensively investigated in the literature, see Chua and Yang (1988), Gopalsamy (2004), Hopfield (1984), and Otawara et al. (2002) and the references cited therein. Among many NNs, Hopfield neural networks (Hopfield, 1984) are the most popular. The problem of stability analysis of NNs has attracted the attention of numerous investigators. Within the electronic implementations of NNs, the finite switching speed of amplifiers and active devices as well as the inherent transmission time of neurons will incur time-delays in the interaction among the neurons. Therefore stability studies of delayed neural networks (DNNs) have also received considerable investigations (Chen & Wang, 2007; Li & Chen, 2009). Fuzzy logic theory has shown to be an appealing and an efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. Among various kinds of fuzzy methods, Takagi–Sugeno (T–S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some q The work was supported by UGC-SAP(DRS-II), Government of India, New Delhi under the grant number F.510/2/DRS/2009 (DRS-I). ⇑ Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail address: [email protected] (P. Balasubramaniam).
0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.07.038
local linear subsystems (Cao & Frank, 2000; Takagi & Sugeno, 1985; Takagi & Sugeno, 1993; Tanaka, Ikede, & Wang, 1997). These linear subsystems are smoothly blended together through fuzzy membership functions. Recently, T–S fuzzy models are used to describe delayed Hopfield neural networks. The T–S fuzzy models can be used to represent some complex nonlinear systems by having a set of delayed Hopfield neural networks as its consequent parts. With the outstanding approximation ability of the T– S fuzzy models, T–S fuzzy delayed Hopfield neural networks (Ali & Balasubramaniam, 2009; Huang, Ho, & Lam, 2005; Li, Chen, Zhou, & Qian, 2009; Sheng, Gao, & Yang, 2009) are recently recognized as an appealing and efficient tool in approximating complex nonlinear systems. Some stability problems for T–S fuzzy delayed Hopfield neural networks have been investigated in Huang et al. (2005), Ali and Balasubramaniam (2009), Li et al. (2009), and Sheng et al. (2009). On the other hand, in relatively large-scale NNs, normally only partial information about the neuron states is available in the network outputs. Therefore, in order to utilize the NNs, one would need to estimate the neuron state through available measurements. Recently, the state estimation problem for NNs has received some research interest, see He, Wang, Wu, and Lin (2006), Wang, Ho, and Liu (2005), Wang, Liu, and Liu (2009), Ahn (2010), Huang, Feng, and Cao (2008), Park and Kwon (2008), Li, Fei, and Zhu (2009), Li and Fei (2007), Lou and Cui (2008), and Wang and Song (2010). Recently, Ahn (2010) studied new delay-dependent state estimation of T–S fuzzy NNs with time-varying delays. To the best of the authors knowledge, delay-dependent robust state estimation of fuzzy NNs with mixed interval time-varying delays have
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
not been studied in the literature and it is very important in both theories and applications. In practice, the time-varying delay often arises and may vary in a range. However, in Huang et al. (2005), Ali and Balasubramaniam (2009), Li et al. (2009), Sheng et al. (2009), He et al. (2006), Wang et al. (2005), Wang et al. (2009), Ahn (2010), Huang et al. (2008), Park and Kwon (2008), Li et al. (2009), Li and Fei (2007), Lou and Cui (2008), and Wang and Song (2010), the lower bound of time-varying delay is fixed as 0, which might cause considerable conservativeness for stability region estimation. So far, removing the above conservative limitations have become urgently necessary. This situation motivates our present investigation. Recently, in the paper (Sun, Liu, Chen, & Rees, 2009), author studied Lyapunov–Krasovskii functionals containing triple integral terms for having less conservative results. Motivated by the above discussions, delay-dependent robust asymptotic state estimation of fuzzy Hopfield NNs with mixed interval time-varying delay are considered in this paper. By constructing a Lyapunov–Krasovskii functional containing tripleintegral term and by employing some analysis techniques, sufficient conditions are derived for the considered NNs in terms of LMIs, which can be easily calculated by MATLAB LMI control Toolbox. The unknown gain matrix is determined by solving a delay-dependent LMI. Numerical examples are given to illustrate the effectiveness of the proposed method. This paper is organized as follows. In Section 2, we formulate the problem. In Section 3, LMI problem for the delay-dependent asymptotic state estimation of T–S fuzzy delayed Hopfield NNs with mixed interval time varying delays are proposed. In Section 4, LMI problem for the delay-dependent robust asymptotic state estimation of T–S fuzzy delayed Hopfield NNs with mixed interval time varying delays are proposed. In Section 5, two numerical examples are given to illustrate the effectiveness of the proposed method, and finally, conclusions are presented in Section 6. Notations: Throughout this paper, Rn and Rnn denote the n-dimensional Euclidean space and the set of all n n real matrices respectively. The superscript T denotes the transposition and the notation X P Y (similarly, X > Y), where X and Y are symmetric matrices, means that X Y is positive semidefinite (similarly, positive definite). 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 Consider the following Hopfield neural networks with timevarying delay described by
_ xðtÞ ¼ AxðtÞ þ B0 gðxðtÞÞ þ B1 gðxðt sðtÞÞÞ þ B2
Z
t
gðxðsÞÞds þ J trðtÞ
ð1Þ here xðÞ ¼ ½x1 ðÞ; x2 ðÞ; . . . ; xn ðÞT 2 Rn is neuron state vector, x(t) = n(t), t 6 0; n(t) is the initial condition. A = diag{a1, . . . , an} is a diagonal matrix with ai > 0 i = 1, . . . , n, B0, B1 and B2 represent the connection weighting matrices, respectively, gðxðÞÞ ¼ ½g 1 ðx1 ðÞÞ; . . . ; g n ðxn ðÞÞT 2 Rn denotes the neuron activation function, and J ¼ ½J1 ; . . . ; Jn T 2 Rn is a constant input vector. s(t) P 0 denotes the time-varying delay and is assumed to satisfy 0 6 s(t) 6 s2 and r(t) denotes the distributed time-varying delays, and satisfies 0 6 r(t) 6 rM. To ensure the existence of a solution to (1), it is assumed that the time-varying delay s(t) and r(t) has a bounded derivative. We introduce the time-varying delays s(t) such that
0 6 s1 6 sðtÞ 6 s2 ;
s_ ðtÞ 6 l < 1; 0 6 rðtÞ 6 rM :
ð2Þ
In this paper the following neural network with mixed interval time-varying delays and network measurements equation are considered
_ ¼ AxðtÞ þ B0 gðxðtÞÞ þ B1 gðxðt sðtÞÞÞ þ B2 xðtÞ
Z
t
gðxðsÞÞds þ J
trðtÞ
yðtÞ ¼ CxðtÞ þ Df ðt;xðtÞÞ;
ð3Þ
where yðtÞ 2 Rm is the measurement output, C, D are known constant matrices with appropriate dimension. f : R Rn ! Rm is the neuron-dependent nonlinear disturbances on the network outputs and satisfies
jf ðt; xÞ f ðt; yÞj 6 jFðx yÞj;
ð4Þ
where the constant matrix F 2 Rnn is known. We assume that the neuron activation functions g() in (1) satisfy the following Lipschitz condition
jgðxÞ gðyÞj 6 jWðx yÞj;
ð5Þ
where W 2 Rnn is a known constant matrix. It is well known that the information about the neuron states is often incomplete from the network measurements (outputs) in applications and the network measurements are subject to nonlinear disturbances. For the DNNs (3), we construct the full-order state estimation as follows
^x_ ðtÞ ¼ A^xðtÞ þ B0 gð^xðtÞÞ þ B1 gð^xðt sðtÞÞÞ þ B2
Z
t
gð^xðsÞÞds
trðtÞ
þ J þ KðyðtÞ C ^xðtÞ Df ðt; ^xðtÞÞÞ
ð6Þ nm
where ^ xðtÞ is the estimation of the neuron state, and K 2 R is the estimator gain matrix to be designed. Define the error eðtÞ ¼ xðtÞ ^ xðtÞ; /ðtÞ ¼ gðxðtÞÞ gð^ xðtÞÞ, and wðtÞ ¼ f ðt; xðtÞÞ f ðt; ^ xðtÞÞ. Thus, the error-state system is given as follows:
_ eðtÞ ¼ ðA þ KCÞeðtÞ þ B0 /ðtÞ þ B1 /ðt sðtÞÞ Z t /ðsÞds KDwðtÞ þ B2
ð7Þ
trðtÞ
Obviously e(t) is bounded and continuously differentiable on [d, 0], where d = max{s2, rM}. In this section, delay-dependent state estimation of fuzzy NNs with time-varying delay will be represented by a T–S fuzzy model. The lth rule of this T–S fuzzy model is of the following form: Plant Rule l: IF h1(t) is gl1 , . . ., and hv(t) is glv , THEN
_ ¼ Al xðtÞ þ B0l gðxðtÞÞ þ B1l gðxðt sðtÞÞÞ þ B2l xðtÞ
Z
t
gðxðsÞÞds þ J
trðtÞ
yðtÞ ¼ C l xðtÞ þ Dl f ðt;xðtÞÞ
ð8Þ T
where g ¼ 1; . . . ; v Þ is the fuzzy set. h(t) = [h1(t), h2(t), . . . , hv(t)] is the premise variable vector and v is the number of IF-THEN rules. The defuzzified output system (8) is inferred as follows: l k ðk
_ xðtÞ ¼
m X l¼1
þB2l
hl ðhðtÞÞfAl xðtÞ þ B0l gðxðtÞÞ þ B1l gðxðt sðtÞÞÞ Z
)
t
gðxðsÞÞds þ J
trðtÞ
yðtÞ ¼
m X
hl ðhðtÞÞfC l xðtÞ þ Dl f ðt; xðtÞÞg
l¼1
The full-order state estimator is of the form
ð9Þ
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
^x_ ðtÞ ¼
m X l¼1
þB2l
hl ðhðtÞÞfAl ^xðtÞ þ B0l gð^xðtÞÞ þ B1l gð^xðt sðtÞÞÞ Z
c
Z
Z
c
xT ðsÞMxðsÞds P
0 t
j ½yðtÞ C l ^xðtÞ Dl f ðt; ^xðtÞÞg; gð^xðsÞÞds þ J þ K
ð10Þ
trðtÞ
j ¼ Pm hj ðhðtÞÞK j and ^ where K xðtÞ is the estimation of the neuron j¼1 state, and K j 2 Rnm is the estimation gain matrix to be designed. Let the error state be eðtÞ ¼ xðtÞ ^ xðtÞ; /ðtÞ ¼ gðxðtÞÞ gð^ xðtÞÞ, and wðtÞ ¼ f ðt; xðtÞÞ f ðt; ^ xðtÞÞ, then the error-state system can be expressed by
T
c
xðsÞds M
Z
0
c
xðsÞds :
0
Lemma 2.4 Sun, Liu, and Chen, 2009. For any constant matrix Z = ZT and a scalar s1 > 0, s2 > 0 such that the following integrations are well defined
Z
t
q ðsÞZ qðsÞds 6 tsðtÞ
Z
1
T
s2
!T
t
qðsÞds
Z
Z
tsðtÞ
!
t
qðsÞds ;
tsðtÞ
ð12Þ
_ eðtÞ ¼ ðAl þ K j C l ÞeðtÞ þ B0l /ðtÞ þ B1l /ðt sðtÞÞ Z t /ðsÞds K j Dl wðtÞ: þ B2l
ð11Þ
trðtÞ
Z
ts1
qT ðsÞZ qðsÞds 6
tsðtÞ
Z
1 ðs2 s1 Þ
!T
ts1
qðsÞds
Z
Z
tsðtÞ
!
ts1
qðsÞds ;
tsðtÞ
ð13Þ
where
h
m i X Al ðtÞ B0l ðtÞ B1l ðtÞ B2l ðtÞ ¼ hl ðhðtÞÞ½Al
B0l
B1l
B2l ;
l¼1
h
i
C l ðtÞ Dl ðtÞ K j ¼
m X
hl ðhðtÞÞ C l
Z
tsðtÞ
qT ðsÞZ qðsÞds 6
ts2
Dl
Kj ;
and
Z
Z
0
s2
M l ðhðtÞÞ hl ðhðtÞÞ ¼ Pm ; l¼1 M l ðhðtÞÞ
Ml ðhðtÞÞ ¼
v Y
glj ðhj ðtÞÞ
M l ðhðtÞÞ P 0;
m X
l ¼ 1; . . . ; m;
6
hl ðhðtÞÞ P 0;
l ¼ 1; . . . ; m;
where
hl ðhðtÞÞ ¼ 1:
l¼1
Let e(t;n) be the trajectory of system (11) under the initial condition e(h) = n(h) on s2 6 h 6 0 in LF ð½s2 ; 0; Rn Þ. It is obvious that the system (11) admits a trivial solution e(t;0) = 0. To derive our main results, the following essential Lemmas are introduced.
Z
2
s22
0
s2
X1 þ XT3 X1 2 X3 < 0; "
XT3 X2
X1
< 0;
or
X2
X3 < 0: X1
Lemma 2.2 Xie, Fu, and Souza, 1992. Let M = MT, H and E be real matrices of appropriate dimensions, with satisfying FT(t)F(t) 6 I then
M þ HFðtÞE þ ET F T ðtÞHT < 0; if and only if there exists a positive scalar
> 0 such that
1 M þ HHT þ ET E < 0:
Lemma 2.3 Gu, Kharitonov, and Chen, 2003 . For any constant matrices M 2 Rmm , scalar c > 0, vector function x : ½0; c ! Rm such that the integrations concerned are well defined, then
T Z
t
Z
0
qðsÞdsdh Z
s2
tþh
t
qðsÞdsdh ;
ð15Þ
tþh
qT ðsÞZ qðsÞdsdh
tþh
2
Z s1 Z
s212
s2
T Z s1 Z
t
t
qðsÞdsdh Z
s2
tþh
qðsÞdsdh
ð16Þ
tþh
s212 ¼ s22 s21 .
"
#
qT ðsÞZ qðsÞ qT ðsÞ P0 qðsÞ Z 1
ð17Þ
Integration of (17) from t + h to h yields,
"Rt
qT ðsÞZ qðsÞds Rt qðsÞds tþh
Rt
tþh
qT ðsÞds
# P0
hZ 1
ð18Þ
Integration of (18) from s2 to 0 yields,
2R0
s2
Rt
R0
tþh
s2
#
Z
Proof. For the proof of inequality (15), notice that
4
if and only if
ts2
t
tþh
Lemma 2.1 Boyd, Ghaoui, Feron, and Balakrishnan, 1994 , Schur Complement. Given constant matrices X1, X2 and X3 with appropriate dimensions, where XT1 ¼ X1 and XT2 ¼ X2 > 0, then
qðsÞds ;
qT ðsÞZ qðsÞdsdh
Z s1 Z
6
i¼1
m X
ts2
tsðtÞ
qðsÞds Z
t
s2
M l ðhðtÞÞ > 0
for all t. Therefore, it implies
T Z
tsðtÞ
tþh
j¼1
in which glj ðhj ðtÞÞ is the grade of membership of hj(t) in glj . According to the theory of fuzzy sets, we have
Z
ð14Þ
l¼1
1 ðs2 s1 Þ
qT ðsÞZ qðsÞdsdh
Rt
tþh
R0
s2
Rt tþh
R0
qðsÞdsdh
s2
qT ðsÞdsdh
hZ 1 dh
3 5P0
ð19Þ
Inequality (19) is equivalent to inequality (15) according to Schur complements. Similarly we can prove (16). The proof has been completed. h
3. Main results In this section, we derive a new delay-dependent criterion for global asymptotic stability of the fuzzy NNs (11) using the Lyapunov functional method combining with LMI approach. Theorem 3.1. Given scalars s2 > s1 P 0 and l, the equilibrium point of fuzzy NNs (11) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
P ¼ PT > 0; Q ¼ Q T ¼
Q 11 Q 12
Q 22
> 0; R ¼ RT ¼
R11 R12
R22
> 0;
475
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
S11 S12 > 0; Ri ¼ RTi > 0; S22 W 11 W 12 > 0; W ¼ WT ¼ W 22
S ¼ ST ¼
T
T ¼ T > 0;
T
Z¼Z ¼
Z 11
Z 12
Z 22
Proof. Choose the Lyapunov–Krasovskii functional for system (11) as
i ¼ 1; 2;
Xli;j
1515
þ
>0
Z
qT ðsÞRqðsÞds þ
ts1
< 0;
ð20Þ
Xl1;1 ¼ 2PAl 2Ll C l þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11 2s12 W 22 2R1 R2 þ aW T W þ dF T F; s2 ðs2 þ s1 Þ 1 Xl1;2 ¼ W 22 ; Xl1;3 ¼ Xl1;4 ¼ Xl1;5 ¼ Xl1;6 ¼ Xl1;7 ¼ 0; 1
s2
Xl1;8 ¼ Q 12 þ R12 þ S12 þ s12 W 22 ATl C Tl LTl ;
1 1 X ¼ R1 þ RT1 ; Xl1;10 ¼ R2 þ RT2 ; ðs2 þ s1 Þ s2 s2 1 1 l T T X1;11 ¼ R þR þ R þ R ; Xl1;12 ¼ PB2l ; s2 1 1 ðs2 þ s1 Þ 2 2 Xl1;13 ¼ PB0l ; Xl1;14 ¼ PB1l ; Xl1;15 ¼ Ll Dl ; 1 1 2 Xl2;2 ¼ ð1 lÞQ 11 W 22 W 22 Z 22 þ bW T W; 1
W T12 þ
s2
Xl2;3 ¼ ð1 lÞQ 12 ; Xl2;4 ¼
1
s12
s12
s12
Z 22 ; Xl2;6 ¼
X
l 2;7
¼X
l 2;8
1
1
s12
ðW 22 þ Z 22 Þ;
1 T ¼ X ¼ 0; X ¼ X ¼ Z ; s2 s12 12 1 Xl2;11 ¼ W T12 þ Z T12 ; Xl2;12 ¼ Xl2;13 ¼ Xl2;14 ¼ Xl2;15 ¼ 0; l 2;5
l 2;9
W T12 ;
l 2;10
s12
Xl3;3 ¼ ð1 lÞQ 22 ; Xl3;4 ¼ Xl3;5 ¼ Xl3;6 ¼ Xl3;7 ¼ Xl3;8 ¼ Xl3;9 ¼ Xl3;10 ¼ Xl3;11 ¼ Xl3;12 ¼ Xl3;13 ¼ Xl3;14 ¼ Xl3;15 ¼ 0; l 4;4
X
¼ R11
1
s12
l 4;5
Z 22 ; X
¼ R12 ;
þ
Z
Z
0
s2
þ
Z
1
s12
Z T22 ;
Xl4;11 ¼ Xl4;12 ¼ Xl4;13 ¼ Xl4;14 ¼ Xl4;15 ¼ 0; Xl5;5 ¼ R22 ; Xl5;6 ¼ Xl5;7 ¼ Xl5;8 ¼ Xl5;9 ¼ Xl5;10 ¼ Xl5;11 ¼ Xl5;12 ¼ Xl5;13 ¼ 1
s12
ðW 22 þ Z 22 Þ; Xl6;7 ¼ S12 ;
Xl6;8 ¼ Xl6;9 ¼ Xl6;10 ¼ Xl6;11 ¼ Xl6;12 ¼ Xl6;13 ¼ Xl6;14 ¼ Xl6;15 ¼ 0; Xl7;7 ¼ S22 ; Xl7;8 ¼ Xl7;9 ¼ Xl7;10 ¼ Xl7;11 ¼ Xl7;12 ¼ Xl7;13 ¼ Xl7;14 ¼ Xl7;15 ¼ 0; Xl8;8 ¼ Q 22 þ R22 þ S22 þ s2 W 22 þ s12 Z 22 þ
s22
s212
R2 2P; 2 X ¼ X ¼ X ¼ 0; X ¼ PB2l ; Xl8;13 ¼ PB0l ; 1 2 Xl8;14 ¼ PB1l ; Xl8;15 ¼ Ll Dl ; Xl9;9 ¼ W 11 2 R1 ; s2 s2 1 Xl9;11 ¼ 2 R1 þ RT1 ; Xl9;10 ¼ Xl9;12 ¼ Xl9;13 ¼ Xl9;14 ¼ Xl9;15 ¼ 0; s2 1 2 1 l X10;10 ¼ Z 11 2 R2 ; Xl10;11 ¼ 2 R2 þ RT2 ; l 8;9
l 8;10
s
l 10;12
12 l 10;13
l 8;11
2
R1 þ
l 8;12
s12
s12
¼ Xl10;14 ¼ Xl10;15 ¼ 0; 2 1 2 Xl11;11 ¼ 2 R1 ðW 11 þ Z 11 Þ 2 R2 ;
X
¼X
s2
s12
þ
s12
Xl11;12 ¼ Xl11;13 ¼ Xl11;14 ¼ Xl11;15 ¼ 0; Xl12;12 ¼ T; Xl12;13 ¼ Xl12;14 ¼ Xl12;15 ¼ 0; Xl13;13 ¼ aI þ r2M T; Xl13;14 ¼ Xl13;15 ¼ 0; Xl14;14 ¼ bI; Xl14;15 ¼ 0; Xl15;15 ¼ dI;
qT ðsÞSqðsÞds
t
qT ðsÞW qðsÞdsdh þ
Z s1 Z s2
Z
Z
0
Z s1 Z
0
Z
qT ðsÞZ qðsÞdsdh
tþh
_ T ðsÞR1 eðsÞdsdkdh _ eðsÞ
0
rM
t
_ T R2 eðsÞdsdkdh _ eðsÞ
tþk
h
Z
t
tþk
h
s2
t
Z
t
/T ðsÞT/ðsÞdsdh
ð21Þ
tþh
where qT(s) = [eT(s) e˙T(s)]. Taking the time derivative of V (t) along the trajectory of system (11) yields
_ t ; tÞ 6 2eT ðtÞPAl eðtÞ 2eT ðtÞPK j C l eðtÞ þ 2eT ðtÞPB0l /ðtÞ Vðe Z t þ 2eT ðtÞPB1l /ðt sðtÞÞ þ 2eT ðtÞPB2l /ðsÞds trðtÞ
2eT ðtÞPK j Dl wðtÞ þ qT ðtÞQ qðtÞ ð1 lÞqT ðt sðtÞÞ Q qðt sðtÞÞ þ qT ðtÞRqðtÞ qT ðt s1 ÞRqðt s1 Þ þ qT ðtÞSqðtÞ qT ðt s2 ÞSqðt s2 Þ þ s2 qT ðtÞW qðtÞ Z t þ s12 qT ðtÞZ qðtÞ qT ðsÞW qðsÞds tsðtÞ
Z
tsðtÞ
qT ðsÞðW þ ZÞqðsÞds
þ þ
s22 _ T 2
_ e ðtÞR1 eðtÞ
2
Z
Z
0
s2
s212 _ T
_ e ðtÞR2 eðtÞ
þ /T ðtÞr2M T/ðtÞ
Z
ts1
qT ðsÞZ qðsÞds
tsðtÞ t
_ e_ T ðsÞR1 eðsÞdsdh
tþh
Z s1 Z s2
Z
t
trM
t
_ e_ T ðsÞR2 eðsÞdsdh
tþh
T Z /ðsÞds T
t
/ðsÞds : ð22Þ
trM
Also add free-weighting matrices
h 2e_ T ðtÞP ðAl þ K j C l ÞeðtÞ þ B0l /ðtÞ þ B1l /ðt sðtÞÞ # Z t _ /ðsÞds K j Dl wðtÞ eðtÞ ¼0 þB2l
ð23Þ
trðtÞ
Since functions f() and g() satisfy (4) and (5), respectively, it is clear that
/ðtÞT /ðtÞ ¼ jgðxðtÞÞ gð^xðtÞÞj2 6 jWeðtÞj2 ¼ eT ðtÞW T WeðtÞ; wT ðtÞwðtÞ ¼ jf ðt; xðtÞÞ f ðt; ^xðtÞÞj2 6 jFeðtÞj2 ¼ eT ðtÞF T FeðtÞ: Then, for positive scalars a > 0, b > 0, d > 0 we have,
a½eT ðtÞW T WeðtÞ /T ðtÞ/ðtÞ P 0
ð24Þ
b½eT ðt s1 ðtÞÞW T Weðt s1 ðtÞÞ /T ðt s1 ðtÞÞ/ðt s1 ðtÞÞ P 0 ð25Þ
s12 ¼ s2 s1 : Moreover the gain matrix is given by Kj = P1Lj.
t
tþh
0
þ rM
Z
ts2
ts2
Xl4;6 ¼ Xl4;7 ¼ Xl4;8 ¼ Xl4;9 ¼ 0; Xl4;10 ¼
Xl5;14 ¼ Xl5;15 ¼ 0; Xl6;6 ¼ S11
qT ðsÞQ qðsÞds
t
s2
l 1;9
t
tsðtÞ
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
Z
Vðet ; tÞ ¼ eT ðtÞPeðtÞ þ
d½eT ðtÞF T FeðtÞ wT ðtÞwðtÞ P 0:
ð26Þ
476
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
X3;9 ¼ X3;10 ¼ X3;11 ¼ X3;12 ¼ X3;13 ¼ X3;14 ¼ X3;15 ¼ 0;
From (22)–(26) and applying Lemma 2.4, we have
_ t ; tÞ 6 f ðtÞ X fðtÞ < 0; Vðe T
l ij
ð27Þ
which is equivalent to m X
_ t ; tÞ 6 Vðe
hl ðhðtÞÞ
1
s12
Z 22 ; X4;5 ¼ R12 ;
X4;6 ¼ X4;7 ¼ X4;8 ¼ X4;9 ¼ 0; X4;10 ¼
m X
hj ðhðtÞÞfT ðtÞXlij fðtÞ < 0;
1
s12
X5;6 ¼ X5;7 ¼ X5;8 ¼ X5;9 ¼ X5;10 ¼ X5;11 ¼ X5;12 ¼ X5;13 ¼ X5;14 ¼
where
f ðtÞ ¼ eT ðtÞ eT ðt sðtÞÞ e_ T ðt sðtÞÞ eT ðt s1 Þ e_ T ðt s1 Þ !T Z t T T T eðsÞds e ðt s2 Þ e_ ðt s2 Þ e_ ðtÞ
X5;15 ¼ 0; X6;6 ¼ S11
T
1
s12
ðW 22 þ Z 22 Þ; X6;7 ¼ S12 ;
X6;8 ¼ X6;9 ¼ X6;10 ¼ X6;11 ¼ X6;12 ¼ X6;13 ¼ X6;14 ¼ X6;15 ¼ 0; X7;7 ¼ S22 ; X7;8 ¼ X7;9 ¼ X7;10 ¼ X7;11 ¼ X7;12 ¼ X7;13 ¼ X7;14 ¼ X7;15 ¼ 0;
tsðtÞ
Z
!T
ts1
Z
eðsÞds
tsðtÞ
tsðtÞ
T Z eðsÞds
ts2
T
X8;8 ¼ Q 22 þ R22 þ S22 þ s2 W 22 þ s12 Z 22 þ
!T
t
T
X8;14 ¼ PB1 ; X8;15 ¼ LD; X9;9 ¼
_ t ; tÞ < 0. Hence, Hence (27) is equivalent to (20), which implies Vðe error-system (11) is globally asymptotically stable. In the following, we will discuss the global asymptotic stability criteria for NNs without fuzzy, then (11) becomes
ð28Þ
trðtÞ
T
P ¼ P > 0; Q ¼ Q ¼
Q 11 Q 12 Q 22
T
> 0; R ¼ R ¼
R11 R12
R22
Z ¼ ZT ¼
Z 11
Z 12
Z 22
>0
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
ðXi;j Þ1515 < 0; 1
s2
W 22
2s12 1 R2 þ aW T W þ dF T F; X1;2 ¼ W 22 ; ðs2 þ s1 Þ s2 X1;3 ¼ X1;4 ¼ X1;5 ¼ X1;6 ¼ X1;7 ¼ 0; 2R1
X1;8 ¼ Q 12 þ R12 þ S12 þ s12 W 22 AT C T LT ; 1
W T12 þ
1
1
s2
R1 þ RT1 ; Xl1;10 ¼
1 R2 þ RT2 ; ðs2 þ s1 Þ R2 þ RT2 ; X1;12 ¼ PB2 ;
R1 þ RT1 þ
1 ð s2 þ s1 Þ X1;13 ¼ PB0 ; X1;14 ¼ PB1 ; X1;15 ¼ LD; 1 1 2 X2;2 ¼ ð1 lÞQ 11 W 22 W 22 Z 22 þ bW T W;
X1;11 ¼
s2
s2
X2;3 ¼ ð1 lÞQ 12 ; X2;4 ¼
s12
1
s12
X2;5 ¼ X2;7 ¼ X2;8 ¼ 0; X2;9 ¼ X2;11 ¼
1
s12
X10;10 ¼
2
s22
R1 ;
ðR1 þ RT1 Þ; X9;10 ¼ X9;12 ¼ X9;13 ¼ X9;14 ¼ X9;15 ¼ 0;
1
s12
Z 11
2
s212
R2 ; X10;11 ¼
1
s212
ðR2 þ RT2 Þ;
X11;11 ¼
2
s22
R1
1
s12
ðW 11 þ Z 11 Þ
2
s212
R2 ;
X12;13 ¼ X12;14 ¼ X12;15 ¼ 0; X13;13 ¼ aI þ r2M T; X13;14 ¼ X13;15 ¼ 0; X14;14 ¼ bI; X14;15 ¼ 0; X15;15 ¼ dI; s12 ¼ s2 s1 :
Proof. The proof immediately follows from the similar way of proof of Theorem 3.1, hence it is omitted. This completes the proof. h Remark 3.3. It is shown in Theorem 3.1 that the global asymptotic stability of the delayed fuzzy NNs (11) can be checked by examining the solvability of the LMI (20), which can be readily conducted by utilizing the Matlab LMI toolbox.
ð29Þ
X1;1 ¼ 2PA 2LC þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11
s2
1
s22
W 11
R2 2P;
> 0;
S11 S12 W 11 W 12 S¼S ¼ > 0; Ri ¼ RTi > 0; i ¼ 1; 2; W ¼ W T ¼ > 0; S22 W 22
X1;9 ¼
s212
Moreover the gain matrix is given by K = P1L.
T ¼ T T > 0;
R1 þ
X11;12 ¼ X11;13 ¼ X11;14 ¼ X11;15 ¼ 0; X12;12 ¼ T;
Corollary 3.2. Given scalars s2 > s1 P 0 and l, the equilibrium point of NNs (28) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
T
X9;11 ¼
1
s2
2
X10;12 ¼ X10;13 ¼ X10;14 ¼ X10;15 ¼ 0;
_ eðtÞ ¼ ðA þ KCÞeðtÞ þ B0 /ðtÞ þ B1 /ðt sðtÞÞ Z t /ðsÞds KDwðtÞ: þ B2
T
s22
2 X8;9 ¼ X8;10 ¼ X8;11 ¼ 0; X8;12 ¼ PB2 ; X8;13 ¼ PB0 ;
/ðsÞds
trðtÞ
/ ðtÞ / ðt sðtÞÞ w ðtÞ : T
Z T22 ;
X4;11 ¼ X4;12 ¼ X4;13 ¼ X4;14 ¼ X4;15 ¼ 0; X5;5 ¼ R22 ;
j¼1
l¼1
X4;4 ¼ R11
1
s2
1
s12
W T12 ; X2;10 ¼
4. Robust state estimation criterion
ðW 22 þ Z 22 Þ;
In this section, we will derive the delay-dependent global robust state estimation criterion for the following fuzzy NNs with mixed interval time-varying delays
1
_ eðtÞ ¼ ððAl þ DAl ðtÞÞ þ K j C l ÞeðtÞ þ ðB0l þ DB0l ðtÞÞ/ðtÞ
s12
Z 22 ; X2;6 ¼
Remark 3.4. In Ahn (2010), the author studied delay-dependent state estimation for T–S fuzzy delayed Hopfield NNs. Also in Wang and Song (2010), the author studied state estimation for neural networks with mixed interval time-varying delays. With the best of our knowledge, robust stability criteria is not studied for state estimation of fuzzy DNNs with mixed interval time-varying delays. The following section reports the robust asymptotic state estimation conditions of fuzzy DNNs with mixed interval time-varying delays.
s12
Z T12 ;
W T12 þ Z T12 ; X2;12 ¼ X2;13 ¼ X2;14 ¼ X2;15 ¼ 0;
X3;3 ¼ ð1 lÞQ 22 ; X3;4 ¼ X3;5 ¼ X3;6 ¼ X3;7 ¼ X3;8 ¼
þ ðB1l þ DB1l ðtÞÞ/ðt sðtÞÞ þ ðB2l þ DB2l ðtÞÞ Z t /ðsÞds K j Dl wðtÞ; trðtÞ
ð30Þ
477
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
The state x1 and its estimation 10
amplitude
5 0 −5 −10
True State Estimation 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
6
amplitude
4 2 0 −2 True State Estimation
−4 −6
0
1
2
3
4
5 time t The error states
6
7
8
9
10
5
amplitude
e1 e2 0
−5
0
1
2
3
4
5 time t
6
7
8
9
10
Fig. 1. The error trajectories are converging to zero for l = j = 1.
where the parametric uncertainties are assumed to be of the form
½DAl ðtÞ DB0l ðtÞ DB1l ðtÞ DB2l ðtÞ ¼ Hl F l ðtÞ½E1l
E2l
E3l
E4l
ð31Þ
in which Hl, E1l, E2l, E3l and E4l are known real constant matrices of appropriate dimensions with
F Tl ðtÞF l ðtÞ 6 I:
ð32Þ
where
b l ¼ 2PAl 2Ll C l þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11 X 1;1 2R1
1
s2
W 22
2s12 R2 þ aW T W þ dF T F þ l ET1l E1l ; ð s2 þ s1 Þ
T bl X 12;12 ¼ T þ l E4l E4l ; T 2 bl X 13;13 ¼ aI þ rM T þ l E2l E2l ;
Theorem 4.1. Given scalars s2 > s1 P 0 and l, the equilibrium point of fuzzy NNs (30) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
P ¼ PT > 0; Q ¼ Q T ¼
Q 11 Q 12 Q 22
> 0; R ¼ RT ¼
R11 R12 ast R22
T ¼ T T > 0;
Z ¼ ZT ¼
Z 11
Z 12
Z 22
>0
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
2
bl X i;j 6 4
P1l l I
P2l
3
7 0 5 < 0;
l I
iT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; 0 0 0 0 0 0 0T ;
and the other parameters are defined as in Theorem 3.1. Then the error system (30) is globally robustly asymptotically stable and the estimator gain matrix Kj can be designed as Kj = P1Lj, j = 1, 2, . . . , N.
h
P1l ¼ HTl PT
P2l ¼ ½0 0 0 0 0 0 0 HTl PT
> 0;
S11 S12 S¼S ¼ > 0; Ri ¼ RTi > 0; i ¼ 1; 2; S22 W 11 W 12 > 0; W ¼ WT ¼ W 22 T
T bl X 14;14 ¼ bI þ l E3l E3l ;
ð33Þ
Proof. Replacing Al ; B0l ; B1l ; B2l by Al þ DAl ðtÞ; B0l þ DB0l ðtÞ; B1l þ DB1l ðtÞ; B2l þ DB2l ðtÞ respectively in (11) and applying Lemma 2.2, we obtain the results which are equivalent to (33). This shows that error-system (30) is globally robustly asymptotically stable for all admissible parameter uncertainties satisfying (31) and (32). In the following, we will discuss the global robust asmptotic stability criteria for the system without fuzzy _ ¼ ððA þ DAðtÞÞ þ KCÞeðtÞ þ ðB0 þ DB0 ðtÞÞ/ðtÞ þ ðB1 þ DB1 ðtÞÞ/ðt sðtÞÞ eðtÞ Z t þ ðB2 þ DB2 ðtÞÞ /ðsÞds KDwðtÞ; ð34Þ trðtÞ
478
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
The state x1 and its estimation 10
amplitude
5 0 −5 −10
True State Estimation 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
6
amplitude
4 2 0 −2 True State Estimation
−4 −6
0
1
2
3
4
5 time t The error states
6
7
8
9
10
5
amplitude
e1 e2 0
−5
0
1
2
3
4
5 time t
6
7
8
9
10
Fig. 2. The error trajectories are converging to zero for l = j = 2.
where the parametric uncertainties are assumed to be of the form
where
½DAðtÞ DB0 ðtÞ DB1 ðtÞ DB2 ðtÞ ¼ HFðtÞ½E1
b 1;1 ¼ 2PA 2LC þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11 X
E2
E3
E4
ð35Þ
in which H, E1, E2, E3 and E4 are known real constant matrices of appropriate dimensions with
F T ðtÞFðtÞ 6 I:
P ¼ PT > 0; Q ¼ Q T ¼
S11
W ¼ WT ¼
T ¼ T T > 0;
S12
S22 W 11
Q 11 Q 12 Q 22
> 0; W 12
W 22
Z ¼ ZT ¼
> 0; R ¼ RT ¼
Ri ¼ RTi > 0;
R11 R12
R22
b i;j X
6 4
P1 I
P2
b 13;13 ¼ aI þ r2 T þ ET E2 ; X 2 M b 14;14 ¼ bI þ ET E3 ; X 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0T ;
P2 ¼ ½0 0 0 0 0 0 0 HT PT
0 0 0 0 0 0 0T
and the other parameters are defined as in Corollary 3.2. Then the error system (34) is globally robustly asymptotically stable and the estimator gain matrix K can be designed as K = P1L.
i ¼ 1; 2;
> 0; Z 11
Z 12
Z 22
>0
3
7 0 5 < 0;
I
2s12 R2 þ aW T W þ dF T F þ ET1 E1 ; ð s2 þ s1 Þ
b 12;12 ¼ T þ ET E4 ; X 4
P1 ¼ ½HT PT
> 0;
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
2
W 22
ð36Þ
Corollary 4.2. Given scalars s2 > s1 P 0 and l, the equilibrium point of NNs (34) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
S ¼ ST ¼
2R1
1
s2
ð37Þ
Proof. Replacing A, B0, B1, B2 by A + DA(t), B0 + DB0(t), B1 + DB1(t), B2 + DB2(t) respectively in (28) and applying Lemma 2.2, we obtain the results which are equivalent to (37). This shows that errorsystem (34) is globally robustly asymptotically stable for all admissible parameter uncertainties satisfying (35) and (36). h Remark 4.3. In Theorem 4.1, the matrix inequality (33) is linear on the parameters l > 0, P > 0, and Ll. Therefore, the global asymptotic convergence of the error dynamics can be readily checked by solving the LMI (33).
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
Remark 4.4. Notice that in Theorem 4.1, the robust state estimation problem is studied for delayed fuzzy NNs with mixed interval time-varying delays under the condition that 0 6 s1 6 s(t) 6 s2 where s1 and s2 are constants. For each s1 P 0 and other known model parameters, it follows from LMI (33) that corresponding s2 can be readily solved by utilizing the Matlab LMI toolbox. Furthermore, if the lower bound of time-varying delays is 0 (Li & Fei, 2007; Li et al., 2009) or another number, we just need to let s1 = 0 or an other number in our criteria.
Therefore, it follows from Theorem 3.1, that the fuzzy NNs (11) is globally asymptotically stable. The response of the state dynamics for the delayed fuzzy NNs (11) which converges to zero asymptotically is shown in Figs. 1 and 2. Example 5.2. Consider the error system (30) with parameters defined as
3 0 4 0 0:6 0:7 ; A2 ¼ ; B01 ¼ ; 0 5 0 6 0:5 0:4 0:3 0:4 0:7 0:6 ; B21 ¼ ; B11 ¼ 0:2 0:5 0:8 0:5 0:2 0:1 0:1 0:2 B02 ¼ ; B12 ¼ ; 0:1 0:3 0:2 0:1 0:3 0:1 ; C 1 ¼ C 2 ¼ D1 ¼ D2 ¼ I; B22 ¼ 0:2 0:3 " # 2cosðtÞ þ 0:03t2 ; W ¼ 0:01I; J1 ¼ J2 ¼ 2sinðtÞ 0:03t2
A1 ¼ 5. Numerical examples
Example 5.1. Consider the error system (11) with parameters defined as
4 0 0:6 0:7 A2 ¼ ; B01 ¼ ; 0 5 0 6 0:5 0:4 0:3 0:4 0:7 0:6 0:2 0:1 B11 ¼ ; B21 ¼ ; B02 ¼ ; 0:2 0:5 0:8 0:5 0:1 0:3 0:1 0:2 0:3 0:1 ; B22 ¼ ; C 1 ¼ C 2 ¼ D1 ¼ D2 ¼ I; B12 ¼ 0:2 0:1 0:2 0:3 " # 2cosðtÞ þ 0:03t2 ; W ¼ 0:1I; F ¼ 0:2I: J1 ¼ J2 ¼ 2sinðtÞ 0:03t 2
A1 ¼
3 0
;
The activation function gðxðtÞÞ ¼ 14 ½jxðtÞ þ 1j jxðtÞ 1j, and the nonlinear disturbance is of the form f(t, x(t)) = 0.4cos(x(t)), the time varying delays are chosen as s(t) = 4.4 + 0.1sin(t), which means s2 = 4.5 when s1 = 2.5, the derivative of time-varying delays s_ ðtÞ 6 l ¼ 0:1, and r(t) = 0.25 + 0.25sin(t), which means rM = 0.5 and using the Matlab LMI toolbox to solve the LMI in Theorem 3.1, we obtained the following matrices
P¼ R2 ¼
209:8522
11:0496
11:0496 100:4348 4:5224 0:3033
;
R1 ¼
0:9435
0:0604
0:0604 0:3082 90:7170 6:7743
;
; Q 11 ¼ ; 0:3033 1:3193 6:7743 56:8943 32:3014 2:4404 32:9480 3:2961 Q 12 ¼ ; Q 22 ¼ ; 5:2296 21:1137 3:2961 14:5626 88:4645 6:4540 30:6117 2:2295 R11 ¼ ; R12 ¼ ; 6:4540 55:0801 4:9163 20:4754 32:0564 3:1550 87:8479 6:4154 R22 ¼ ; S11 ¼ ; 3:1550 14:2800 6:4154 54:8988 30:6116 2:2291 32:0564 3:1549 ; S22 ¼ ; S12 ¼ 4:9163 20:4730 3:1549 14:2774 52:8603 6:2389 20:1318 2:3961 W 11 ¼ ; W 12 ¼ ; 6:2389 22:5983 3:6141 8:6170 9:7365 1:5180 91:1208 8:0340 W 22 ¼ ; Z 11 ¼ ; 1:5180 3:8981 8:0340 49:6459 31:6323 2:8135 15:9321 2:1704 Z 12 ¼ ; Z 22 ¼ ; 5:1159 18:4211 2:1704 8:2523 428:3764 205:1866 T¼ ; 205:1866 313:3236 with the gain matrices
0:7193 0:0788 K 1 ¼ P1 L1 ¼ ; 0:0427 2:4256 1:2409 0:1160 K 2 ¼ P1 L2 ¼ : 0:0393 2:3525
F ¼ 0:02I
H1 ¼ 0:1I;
E21 ¼ E22 ¼ 0:3I;
H2 ¼ 0:2I;
E11 ¼ E12 ¼ 0:4I;
E31 ¼ E32 ¼ 0:2I;
E41 ¼ E42 ¼ 0:1I;
FðtÞ ¼ tanhðtÞI: The activation function gðxðtÞÞ ¼ 14 ½jxðtÞ þ 1j jxðtÞ 1j, and the nonlinear disturbance is of the form f(t, x(t)) = 0.4cos(x(t)), the time varying delays are chosen as s(t) = 4.4 + 0.1sin(t), which means s2 = 4.5 when s1 = 2.5, the derivative of time-varying delays s_ ðtÞ 6 l ¼ 0:1, and r(t) = 0.25 + 0.25sin(t), which means rM = 0.5 and using the Matlab LMI toolbox to solve the LMI in Theorem 4.1, we obtained the following matrices
; 11:2786 100:1383 0:0612 0:2986 4:4829 0:3077 89:3435 6:9716 R2 ¼ ; Q 11 ¼ ; 0:3077 1:3142 6:9716 55:4036 32:7449 2:5569 33:1568 3:4117 ; Q 22 ¼ ; Q 12 ¼ 5:4179 21:1521 3:4117 14:6648 88:0619 6:6345 31:0491 2:3394 R11 ¼ ; R12 ¼ ; 6:6345 54:6258 5:0964 20:5204 32:2553 3:2655 87:4522 6:5960 ; S11 ¼ ; R22 ¼ 3:2655 14:3830 6:5960 54:4488 31:0489 2:3388 32:2553 3:2653 S12 ¼ ; S22 ¼ ; 5:0964 20:5181 3:2653 14:3803 51:9977 6:2913 20:1481 2:4532 ; W 12 ¼ ; W 11 ¼ 6:2913 22:2328 3:6957 8:5673 9:8527 1:5704 90:3325 8:1823 ; Z 11 ¼ ; W 22 ¼ 1:5704 3:9037 8:1823 49:0748 31:9416 2:9195 16:2151 2:2640 Z 12 ¼ ; Z 22 ¼ ; 5:2812 18:4015 2:2640 8:3111 433:1401 205:9449 T¼ ; 205:9449 317:6054
P¼
209:5866
11:2786
;
R1 ¼
0:9264
0:0612
1 ¼ 111:2686; 2 ¼ 173:5972 with the gain matrices
K 1 ¼ P1 L1 ¼ K 2 ¼ P1 L2 ¼
0:7351
0:0443 2:4377 1:2301 0:1219
0:0793
0:0485 2:3449
; :
Therefore, it follows from Theorem 4.1, that the fuzzy NNs (30) is robustly globally asymptotically stable. The response of the state
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
The state x1 and its estimation
amplitude
10 5 0 True State Estimation
−5 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
10 True State Estimation
amplitude
5 0 −5 −10
0
1
2
3
4
5 6 time t The error state e1 & e2
7
8
9
10
5
amplitude
e1(t) e2(t) 0
−5
0
1
2
3
4
5 time t
6
7
8
9
10
Fig. 3. The error trajectories are converging to zero for l = j = 1.
The state x1 and its estimation
amplitude
10 5 0 True State Estimation
−5 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
10 True State Estimation
amplitude
5 0 −5 −10
0
1
2
3
4
5 6 time t The error state e1 & e2
7
8
9
10
5
amplitude
e1(t) e2(t) 0
−5
0
1
2
3
4
5 time t
6
7
Fig. 4. The error trajectories are converging to zero for l = j = 2.
8
9
10
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
dynamics for the delayed fuzzy NNs (11) which converges to zero asymptotically is shown in Figs. 3 and 4.
6. Conclusion This paper investigated the delay-dependent robust asymptotic state estimation of fuzzy Hopfield neural networks with mixed interval time-varying delays. By constructing a new Lyapunov– Krasovskii functional containing triple-integral terms and employing Newton–Leibnitz formulation and linear matrix inequality techniques and introducing free-weighting matrices, some sufficient conditions for robust global asymptotic stability criteria has been derived in terms of linear matrix inequalities (LMIs), which can be easily calculated by MATLAB LMI control toolbox. Through the available output measurements, a state estimator is designed to estimate the neuron states and the dynamics of the estimation error is asymptotically stable. Finally, two numerical examples have been used to demonstrate the usefulness of the main results. In future, research topics would be the extension of the present results to more general cases, for example, the case that the neural network is inherently stochastic, the case that the network modes are subjected to Markovian switching, the case that the delayprobability-distribution-dependent stability and the case that the mode dependent stability. The results will appear in the near future. References Ahn, C. K. (2010). Delay-dependent state estimation for T–S fuzzy delayed Hopfield neural networks. Nonlinear Dynamics, 61, 483–489. Ali, M. S., & Balasubramaniam, P. (2009). Stability analysis of uncertain fuzzy Hopfield neural networks with time delays. Communications in Nonlinear Science and Numerical Simulation, 14, 2776–2783. Boyd, B., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia: SIAM. Cao, Y. Y., & Frank, P. M. (2000). Analysis and synthesis of nonlinear timedelay systems via fuzzy control approach. IEEE Transaction on Fuzzy Systems, 8, 200–211. Chen, T., & Wang, L. (2007). Global l-stability of delayed neural networks with unbounded time-varying delays. IEEE Transactions on Neural Networks, 18, 1836–1840. Chua, L., & Yang, L. (1988). Cellular neural networks: Theory and applications. IEEE Transactions on Circuits and Systems I, 35, 1257–1290. Gopalsamy, K. (2004). Stability of artificial neural networks with impulses. Applied Mathematics and Computation, 154, 783–813. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser.
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Gupta, M. M., Jin, L., & Homma, N. (2003). Static and dynamic neural networks. New York: Wiley. He, Y., Wang, Q.-G., Wu, M., & Lin, C. (2006). Delay-dependent state estimation for delayed neural networks. IEEE Transations on Neural Networks, 17, 1077–1081. Hopfield, J. J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences, 81, 3088–3092. Huang, H., Feng, G., & Cao, J. (2008). An LMI approach to delay-dependent state estimation for delayed neural networks. Neurocomputing, 71, 2857–2867. Huang, H., Ho, D. W. C., & Lam, J. (2005). Stochastic stability analysis of fuzzy hopfield neural networks with time-varying delays. IEEE Transactions on Circuits and systems – II: Express Briefs, 52, 251–255. Li, X., & Chen, Z. (2009). Stability properties for Hopfield neural networks with delays and impulsive perturbations. Nonlinear Analysis: Real World Applications, 10, 3253–3265. Li, H., Chen, B., Zhou, Q., & Qian, W. (2009). Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters. IEEE Transactions on Systems, Man, and Cybernetics B, 39, 94–102. Li, T., & Fei, S. (2007). Exponential state estimation for recurrent neural networks with distributed delays. Neurocomputing, 71, 28–438. Li, T., Fei, S., & Zhu, Q. (2009). Design of exponential state estimator for neural networks with distributed delays. Nonlinear Analysis: Real World Applications, 10, 1229–1242. Lou, X., & Cui, B. (2008). Design of state estimator for uncertain neural networks via the integral-inequality method. Nonlinear Dynamics, 53, 223–235. Otawara, K., Fan, L. T., Tsutsumi, A., Yano, T., Kuramoto, K., & Yoshida, K. (2002). An artificial neural network as a model for chaotic behavior of a three-phase fluidized bed. Chaos, Solitons, Fractals, 13, 353–362. Park, J. H., & Kwon, O. M. (2008). Design of state estimator for neural networks of neutral-type. Applied Mathematics and Computation, 202(1), 360–369. Sheng, L., Gao, M., & Yang, H. (2009). Delay-dependent robust stability for uncertain stochastic fuzzy Hopfield neural networks with time-varying delays. Fuzzy Sets and Systems, 160, 3503–3517. Sun, J., Liu, G. P., & Chen, J. (2009). Delay-dependent stability and stabilization of neutral time-delay systems. International Journal of Robust and Nonlinear Control, 19, 1364–1375. Sun, J., Liu, G. P., Chen, J., & Rees, D. (2009). Improved stability criteria for neural networks with time-varying delay. Physics Letters A, 373, 342–348. Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its application to modeling and control. IEEE Transactions on Systems Man and Cybernetics SMC15, 116–132. Takagi, T., & Sugeno, M. (1993). Stability analysis and design of fuzzy control systems. Fuzzy Set and Systems, 45, 135–156. Tanaka, K., Ikede, T., Wang, H.O. (1997). An LMI approach to fuzzy controller designs based on the relaxed stability conditions. In Proceedings of the IEEE international conference on fuzzy systems, Barcelona, Spain (pp. 171–176). Wang, H., & Song, Q. (2010). State estimation for neural networks with mixed interval time-varying delays. Neurocomputing, 73, 1281–1288. Wang, Z., Ho, D. W. C., & Liu, X. (2005). State estimation for delayed neural networks. IEEE Transactions on Neural Networks, 16, 279–284. Wang, Z., Liu, Y., & Liu, X. (2009). State estimation for jumping recurrent neural networks with discrete and distributed delays. Neural Networks, 22, 41–48. Xie, L., Fu, M., & Souza, CED. (1992). H1 control and quadratic stabilization of systems with parameter uncetainty via output feedback. IEEE Transactions on Automatic Control, 32, 1253–1256.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Delay-dependent robust asymptotic state estimation of Takagi–Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays q P. Balasubramaniam a,⇑, V. Vembarasan a, R. Rakkiyappan b a b
Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamilnadu, India
a r t i c l e
i n f o
Keywords: T–S fuzzy model Linear matrix inequality Lyapunov–Krasovskii functional Hopfield neural networks State estimation
a b s t r a c t This paper investigates delay-dependent robust asymptotic state estimation of fuzzy neural networks with mixed interval time-varying delay. In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the robust state estimation of Hopfield neural networks with mixed interval time-varying delays. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error is globally asymptotically stable. Based on the Lyapunov–Krasovskii functional which contains a triple-integral term, delay-dependent robust state estimation for such T–S fuzzy Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. The unknown gain matrix is determined by solving a delay-dependent LMI. Finally two numerical examples are provided to demonstrate the effectiveness of the proposed method. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Neural Networks (NNs) have been extensively studied and successfully applied in many areas such as combinatorial optimization, signal processing, and pattern recognition (Gupta, Jin, & Homma, 2003). Different models of neural networks such as Hopfield-type neural networks, cellular neural networks, Cohen–Grossberg neural networks, and bidirectional associative memory neural networks have been extensively investigated in the literature, see Chua and Yang (1988), Gopalsamy (2004), Hopfield (1984), and Otawara et al. (2002) and the references cited therein. Among many NNs, Hopfield neural networks (Hopfield, 1984) are the most popular. The problem of stability analysis of NNs has attracted the attention of numerous investigators. Within the electronic implementations of NNs, the finite switching speed of amplifiers and active devices as well as the inherent transmission time of neurons will incur time-delays in the interaction among the neurons. Therefore stability studies of delayed neural networks (DNNs) have also received considerable investigations (Chen & Wang, 2007; Li & Chen, 2009). Fuzzy logic theory has shown to be an appealing and an efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. Among various kinds of fuzzy methods, Takagi–Sugeno (T–S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some q The work was supported by UGC-SAP(DRS-II), Government of India, New Delhi under the grant number F.510/2/DRS/2009 (DRS-I). ⇑ Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail address: [email protected] (P. Balasubramaniam).
0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.07.038
local linear subsystems (Cao & Frank, 2000; Takagi & Sugeno, 1985; Takagi & Sugeno, 1993; Tanaka, Ikede, & Wang, 1997). These linear subsystems are smoothly blended together through fuzzy membership functions. Recently, T–S fuzzy models are used to describe delayed Hopfield neural networks. The T–S fuzzy models can be used to represent some complex nonlinear systems by having a set of delayed Hopfield neural networks as its consequent parts. With the outstanding approximation ability of the T– S fuzzy models, T–S fuzzy delayed Hopfield neural networks (Ali & Balasubramaniam, 2009; Huang, Ho, & Lam, 2005; Li, Chen, Zhou, & Qian, 2009; Sheng, Gao, & Yang, 2009) are recently recognized as an appealing and efficient tool in approximating complex nonlinear systems. Some stability problems for T–S fuzzy delayed Hopfield neural networks have been investigated in Huang et al. (2005), Ali and Balasubramaniam (2009), Li et al. (2009), and Sheng et al. (2009). On the other hand, in relatively large-scale NNs, normally only partial information about the neuron states is available in the network outputs. Therefore, in order to utilize the NNs, one would need to estimate the neuron state through available measurements. Recently, the state estimation problem for NNs has received some research interest, see He, Wang, Wu, and Lin (2006), Wang, Ho, and Liu (2005), Wang, Liu, and Liu (2009), Ahn (2010), Huang, Feng, and Cao (2008), Park and Kwon (2008), Li, Fei, and Zhu (2009), Li and Fei (2007), Lou and Cui (2008), and Wang and Song (2010). Recently, Ahn (2010) studied new delay-dependent state estimation of T–S fuzzy NNs with time-varying delays. To the best of the authors knowledge, delay-dependent robust state estimation of fuzzy NNs with mixed interval time-varying delays have
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
not been studied in the literature and it is very important in both theories and applications. In practice, the time-varying delay often arises and may vary in a range. However, in Huang et al. (2005), Ali and Balasubramaniam (2009), Li et al. (2009), Sheng et al. (2009), He et al. (2006), Wang et al. (2005), Wang et al. (2009), Ahn (2010), Huang et al. (2008), Park and Kwon (2008), Li et al. (2009), Li and Fei (2007), Lou and Cui (2008), and Wang and Song (2010), the lower bound of time-varying delay is fixed as 0, which might cause considerable conservativeness for stability region estimation. So far, removing the above conservative limitations have become urgently necessary. This situation motivates our present investigation. Recently, in the paper (Sun, Liu, Chen, & Rees, 2009), author studied Lyapunov–Krasovskii functionals containing triple integral terms for having less conservative results. Motivated by the above discussions, delay-dependent robust asymptotic state estimation of fuzzy Hopfield NNs with mixed interval time-varying delay are considered in this paper. By constructing a Lyapunov–Krasovskii functional containing tripleintegral term and by employing some analysis techniques, sufficient conditions are derived for the considered NNs in terms of LMIs, which can be easily calculated by MATLAB LMI control Toolbox. The unknown gain matrix is determined by solving a delay-dependent LMI. Numerical examples are given to illustrate the effectiveness of the proposed method. This paper is organized as follows. In Section 2, we formulate the problem. In Section 3, LMI problem for the delay-dependent asymptotic state estimation of T–S fuzzy delayed Hopfield NNs with mixed interval time varying delays are proposed. In Section 4, LMI problem for the delay-dependent robust asymptotic state estimation of T–S fuzzy delayed Hopfield NNs with mixed interval time varying delays are proposed. In Section 5, two numerical examples are given to illustrate the effectiveness of the proposed method, and finally, conclusions are presented in Section 6. Notations: Throughout this paper, Rn and Rnn denote the n-dimensional Euclidean space and the set of all n n real matrices respectively. The superscript T denotes the transposition and the notation X P Y (similarly, X > Y), where X and Y are symmetric matrices, means that X Y is positive semidefinite (similarly, positive definite). 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 Consider the following Hopfield neural networks with timevarying delay described by
_ xðtÞ ¼ AxðtÞ þ B0 gðxðtÞÞ þ B1 gðxðt sðtÞÞÞ þ B2
Z
t
gðxðsÞÞds þ J trðtÞ
ð1Þ here xðÞ ¼ ½x1 ðÞ; x2 ðÞ; . . . ; xn ðÞT 2 Rn is neuron state vector, x(t) = n(t), t 6 0; n(t) is the initial condition. A = diag{a1, . . . , an} is a diagonal matrix with ai > 0 i = 1, . . . , n, B0, B1 and B2 represent the connection weighting matrices, respectively, gðxðÞÞ ¼ ½g 1 ðx1 ðÞÞ; . . . ; g n ðxn ðÞÞT 2 Rn denotes the neuron activation function, and J ¼ ½J1 ; . . . ; Jn T 2 Rn is a constant input vector. s(t) P 0 denotes the time-varying delay and is assumed to satisfy 0 6 s(t) 6 s2 and r(t) denotes the distributed time-varying delays, and satisfies 0 6 r(t) 6 rM. To ensure the existence of a solution to (1), it is assumed that the time-varying delay s(t) and r(t) has a bounded derivative. We introduce the time-varying delays s(t) such that
0 6 s1 6 sðtÞ 6 s2 ;
s_ ðtÞ 6 l < 1; 0 6 rðtÞ 6 rM :
ð2Þ
In this paper the following neural network with mixed interval time-varying delays and network measurements equation are considered
_ ¼ AxðtÞ þ B0 gðxðtÞÞ þ B1 gðxðt sðtÞÞÞ þ B2 xðtÞ
Z
t
gðxðsÞÞds þ J
trðtÞ
yðtÞ ¼ CxðtÞ þ Df ðt;xðtÞÞ;
ð3Þ
where yðtÞ 2 Rm is the measurement output, C, D are known constant matrices with appropriate dimension. f : R Rn ! Rm is the neuron-dependent nonlinear disturbances on the network outputs and satisfies
jf ðt; xÞ f ðt; yÞj 6 jFðx yÞj;
ð4Þ
where the constant matrix F 2 Rnn is known. We assume that the neuron activation functions g() in (1) satisfy the following Lipschitz condition
jgðxÞ gðyÞj 6 jWðx yÞj;
ð5Þ
where W 2 Rnn is a known constant matrix. It is well known that the information about the neuron states is often incomplete from the network measurements (outputs) in applications and the network measurements are subject to nonlinear disturbances. For the DNNs (3), we construct the full-order state estimation as follows
^x_ ðtÞ ¼ A^xðtÞ þ B0 gð^xðtÞÞ þ B1 gð^xðt sðtÞÞÞ þ B2
Z
t
gð^xðsÞÞds
trðtÞ
þ J þ KðyðtÞ C ^xðtÞ Df ðt; ^xðtÞÞÞ
ð6Þ nm
where ^ xðtÞ is the estimation of the neuron state, and K 2 R is the estimator gain matrix to be designed. Define the error eðtÞ ¼ xðtÞ ^ xðtÞ; /ðtÞ ¼ gðxðtÞÞ gð^ xðtÞÞ, and wðtÞ ¼ f ðt; xðtÞÞ f ðt; ^ xðtÞÞ. Thus, the error-state system is given as follows:
_ eðtÞ ¼ ðA þ KCÞeðtÞ þ B0 /ðtÞ þ B1 /ðt sðtÞÞ Z t /ðsÞds KDwðtÞ þ B2
ð7Þ
trðtÞ
Obviously e(t) is bounded and continuously differentiable on [d, 0], where d = max{s2, rM}. In this section, delay-dependent state estimation of fuzzy NNs with time-varying delay will be represented by a T–S fuzzy model. The lth rule of this T–S fuzzy model is of the following form: Plant Rule l: IF h1(t) is gl1 , . . ., and hv(t) is glv , THEN
_ ¼ Al xðtÞ þ B0l gðxðtÞÞ þ B1l gðxðt sðtÞÞÞ þ B2l xðtÞ
Z
t
gðxðsÞÞds þ J
trðtÞ
yðtÞ ¼ C l xðtÞ þ Dl f ðt;xðtÞÞ
ð8Þ T
where g ¼ 1; . . . ; v Þ is the fuzzy set. h(t) = [h1(t), h2(t), . . . , hv(t)] is the premise variable vector and v is the number of IF-THEN rules. The defuzzified output system (8) is inferred as follows: l k ðk
_ xðtÞ ¼
m X l¼1
þB2l
hl ðhðtÞÞfAl xðtÞ þ B0l gðxðtÞÞ þ B1l gðxðt sðtÞÞÞ Z
)
t
gðxðsÞÞds þ J
trðtÞ
yðtÞ ¼
m X
hl ðhðtÞÞfC l xðtÞ þ Dl f ðt; xðtÞÞg
l¼1
The full-order state estimator is of the form
ð9Þ
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
^x_ ðtÞ ¼
m X l¼1
þB2l
hl ðhðtÞÞfAl ^xðtÞ þ B0l gð^xðtÞÞ þ B1l gð^xðt sðtÞÞÞ Z
c
Z
Z
c
xT ðsÞMxðsÞds P
0 t
j ½yðtÞ C l ^xðtÞ Dl f ðt; ^xðtÞÞg; gð^xðsÞÞds þ J þ K
ð10Þ
trðtÞ
j ¼ Pm hj ðhðtÞÞK j and ^ where K xðtÞ is the estimation of the neuron j¼1 state, and K j 2 Rnm is the estimation gain matrix to be designed. Let the error state be eðtÞ ¼ xðtÞ ^ xðtÞ; /ðtÞ ¼ gðxðtÞÞ gð^ xðtÞÞ, and wðtÞ ¼ f ðt; xðtÞÞ f ðt; ^ xðtÞÞ, then the error-state system can be expressed by
T
c
xðsÞds M
Z
0
c
xðsÞds :
0
Lemma 2.4 Sun, Liu, and Chen, 2009. For any constant matrix Z = ZT and a scalar s1 > 0, s2 > 0 such that the following integrations are well defined
Z
t
q ðsÞZ qðsÞds 6 tsðtÞ
Z
1
T
s2
!T
t
qðsÞds
Z
Z
tsðtÞ
!
t
qðsÞds ;
tsðtÞ
ð12Þ
_ eðtÞ ¼ ðAl þ K j C l ÞeðtÞ þ B0l /ðtÞ þ B1l /ðt sðtÞÞ Z t /ðsÞds K j Dl wðtÞ: þ B2l
ð11Þ
trðtÞ
Z
ts1
qT ðsÞZ qðsÞds 6
tsðtÞ
Z
1 ðs2 s1 Þ
!T
ts1
qðsÞds
Z
Z
tsðtÞ
!
ts1
qðsÞds ;
tsðtÞ
ð13Þ
where
h
m i X Al ðtÞ B0l ðtÞ B1l ðtÞ B2l ðtÞ ¼ hl ðhðtÞÞ½Al
B0l
B1l
B2l ;
l¼1
h
i
C l ðtÞ Dl ðtÞ K j ¼
m X
hl ðhðtÞÞ C l
Z
tsðtÞ
qT ðsÞZ qðsÞds 6
ts2
Dl
Kj ;
and
Z
Z
0
s2
M l ðhðtÞÞ hl ðhðtÞÞ ¼ Pm ; l¼1 M l ðhðtÞÞ
Ml ðhðtÞÞ ¼
v Y
glj ðhj ðtÞÞ
M l ðhðtÞÞ P 0;
m X
l ¼ 1; . . . ; m;
6
hl ðhðtÞÞ P 0;
l ¼ 1; . . . ; m;
where
hl ðhðtÞÞ ¼ 1:
l¼1
Let e(t;n) be the trajectory of system (11) under the initial condition e(h) = n(h) on s2 6 h 6 0 in LF ð½s2 ; 0; Rn Þ. It is obvious that the system (11) admits a trivial solution e(t;0) = 0. To derive our main results, the following essential Lemmas are introduced.
Z
2
s22
0
s2
X1 þ XT3 X1 2 X3 < 0; "
XT3 X2
X1
< 0;
or
X2
X3 < 0: X1
Lemma 2.2 Xie, Fu, and Souza, 1992. Let M = MT, H and E be real matrices of appropriate dimensions, with satisfying FT(t)F(t) 6 I then
M þ HFðtÞE þ ET F T ðtÞHT < 0; if and only if there exists a positive scalar
> 0 such that
1 M þ HHT þ ET E < 0:
Lemma 2.3 Gu, Kharitonov, and Chen, 2003 . For any constant matrices M 2 Rmm , scalar c > 0, vector function x : ½0; c ! Rm such that the integrations concerned are well defined, then
T Z
t
Z
0
qðsÞdsdh Z
s2
tþh
t
qðsÞdsdh ;
ð15Þ
tþh
qT ðsÞZ qðsÞdsdh
tþh
2
Z s1 Z
s212
s2
T Z s1 Z
t
t
qðsÞdsdh Z
s2
tþh
qðsÞdsdh
ð16Þ
tþh
s212 ¼ s22 s21 .
"
#
qT ðsÞZ qðsÞ qT ðsÞ P0 qðsÞ Z 1
ð17Þ
Integration of (17) from t + h to h yields,
"Rt
qT ðsÞZ qðsÞds Rt qðsÞds tþh
Rt
tþh
qT ðsÞds
# P0
hZ 1
ð18Þ
Integration of (18) from s2 to 0 yields,
2R0
s2
Rt
R0
tþh
s2
#
Z
Proof. For the proof of inequality (15), notice that
4
if and only if
ts2
t
tþh
Lemma 2.1 Boyd, Ghaoui, Feron, and Balakrishnan, 1994 , Schur Complement. Given constant matrices X1, X2 and X3 with appropriate dimensions, where XT1 ¼ X1 and XT2 ¼ X2 > 0, then
qðsÞds ;
qT ðsÞZ qðsÞdsdh
Z s1 Z
6
i¼1
m X
ts2
tsðtÞ
qðsÞds Z
t
s2
M l ðhðtÞÞ > 0
for all t. Therefore, it implies
T Z
tsðtÞ
tþh
j¼1
in which glj ðhj ðtÞÞ is the grade of membership of hj(t) in glj . According to the theory of fuzzy sets, we have
Z
ð14Þ
l¼1
1 ðs2 s1 Þ
qT ðsÞZ qðsÞdsdh
Rt
tþh
R0
s2
Rt tþh
R0
qðsÞdsdh
s2
qT ðsÞdsdh
hZ 1 dh
3 5P0
ð19Þ
Inequality (19) is equivalent to inequality (15) according to Schur complements. Similarly we can prove (16). The proof has been completed. h
3. Main results In this section, we derive a new delay-dependent criterion for global asymptotic stability of the fuzzy NNs (11) using the Lyapunov functional method combining with LMI approach. Theorem 3.1. Given scalars s2 > s1 P 0 and l, the equilibrium point of fuzzy NNs (11) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
P ¼ PT > 0; Q ¼ Q T ¼
Q 11 Q 12
Q 22
> 0; R ¼ RT ¼
R11 R12
R22
> 0;
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P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
S11 S12 > 0; Ri ¼ RTi > 0; S22 W 11 W 12 > 0; W ¼ WT ¼ W 22
S ¼ ST ¼
T
T ¼ T > 0;
T
Z¼Z ¼
Z 11
Z 12
Z 22
Proof. Choose the Lyapunov–Krasovskii functional for system (11) as
i ¼ 1; 2;
Xli;j
1515
þ
>0
Z
qT ðsÞRqðsÞds þ
ts1
< 0;
ð20Þ
Xl1;1 ¼ 2PAl 2Ll C l þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11 2s12 W 22 2R1 R2 þ aW T W þ dF T F; s2 ðs2 þ s1 Þ 1 Xl1;2 ¼ W 22 ; Xl1;3 ¼ Xl1;4 ¼ Xl1;5 ¼ Xl1;6 ¼ Xl1;7 ¼ 0; 1
s2
Xl1;8 ¼ Q 12 þ R12 þ S12 þ s12 W 22 ATl C Tl LTl ;
1 1 X ¼ R1 þ RT1 ; Xl1;10 ¼ R2 þ RT2 ; ðs2 þ s1 Þ s2 s2 1 1 l T T X1;11 ¼ R þR þ R þ R ; Xl1;12 ¼ PB2l ; s2 1 1 ðs2 þ s1 Þ 2 2 Xl1;13 ¼ PB0l ; Xl1;14 ¼ PB1l ; Xl1;15 ¼ Ll Dl ; 1 1 2 Xl2;2 ¼ ð1 lÞQ 11 W 22 W 22 Z 22 þ bW T W; 1
W T12 þ
s2
Xl2;3 ¼ ð1 lÞQ 12 ; Xl2;4 ¼
1
s12
s12
s12
Z 22 ; Xl2;6 ¼
X
l 2;7
¼X
l 2;8
1
1
s12
ðW 22 þ Z 22 Þ;
1 T ¼ X ¼ 0; X ¼ X ¼ Z ; s2 s12 12 1 Xl2;11 ¼ W T12 þ Z T12 ; Xl2;12 ¼ Xl2;13 ¼ Xl2;14 ¼ Xl2;15 ¼ 0; l 2;5
l 2;9
W T12 ;
l 2;10
s12
Xl3;3 ¼ ð1 lÞQ 22 ; Xl3;4 ¼ Xl3;5 ¼ Xl3;6 ¼ Xl3;7 ¼ Xl3;8 ¼ Xl3;9 ¼ Xl3;10 ¼ Xl3;11 ¼ Xl3;12 ¼ Xl3;13 ¼ Xl3;14 ¼ Xl3;15 ¼ 0; l 4;4
X
¼ R11
1
s12
l 4;5
Z 22 ; X
¼ R12 ;
þ
Z
Z
0
s2
þ
Z
1
s12
Z T22 ;
Xl4;11 ¼ Xl4;12 ¼ Xl4;13 ¼ Xl4;14 ¼ Xl4;15 ¼ 0; Xl5;5 ¼ R22 ; Xl5;6 ¼ Xl5;7 ¼ Xl5;8 ¼ Xl5;9 ¼ Xl5;10 ¼ Xl5;11 ¼ Xl5;12 ¼ Xl5;13 ¼ 1
s12
ðW 22 þ Z 22 Þ; Xl6;7 ¼ S12 ;
Xl6;8 ¼ Xl6;9 ¼ Xl6;10 ¼ Xl6;11 ¼ Xl6;12 ¼ Xl6;13 ¼ Xl6;14 ¼ Xl6;15 ¼ 0; Xl7;7 ¼ S22 ; Xl7;8 ¼ Xl7;9 ¼ Xl7;10 ¼ Xl7;11 ¼ Xl7;12 ¼ Xl7;13 ¼ Xl7;14 ¼ Xl7;15 ¼ 0; Xl8;8 ¼ Q 22 þ R22 þ S22 þ s2 W 22 þ s12 Z 22 þ
s22
s212
R2 2P; 2 X ¼ X ¼ X ¼ 0; X ¼ PB2l ; Xl8;13 ¼ PB0l ; 1 2 Xl8;14 ¼ PB1l ; Xl8;15 ¼ Ll Dl ; Xl9;9 ¼ W 11 2 R1 ; s2 s2 1 Xl9;11 ¼ 2 R1 þ RT1 ; Xl9;10 ¼ Xl9;12 ¼ Xl9;13 ¼ Xl9;14 ¼ Xl9;15 ¼ 0; s2 1 2 1 l X10;10 ¼ Z 11 2 R2 ; Xl10;11 ¼ 2 R2 þ RT2 ; l 8;9
l 8;10
s
l 10;12
12 l 10;13
l 8;11
2
R1 þ
l 8;12
s12
s12
¼ Xl10;14 ¼ Xl10;15 ¼ 0; 2 1 2 Xl11;11 ¼ 2 R1 ðW 11 þ Z 11 Þ 2 R2 ;
X
¼X
s2
s12
þ
s12
Xl11;12 ¼ Xl11;13 ¼ Xl11;14 ¼ Xl11;15 ¼ 0; Xl12;12 ¼ T; Xl12;13 ¼ Xl12;14 ¼ Xl12;15 ¼ 0; Xl13;13 ¼ aI þ r2M T; Xl13;14 ¼ Xl13;15 ¼ 0; Xl14;14 ¼ bI; Xl14;15 ¼ 0; Xl15;15 ¼ dI;
qT ðsÞSqðsÞds
t
qT ðsÞW qðsÞdsdh þ
Z s1 Z s2
Z
Z
0
Z s1 Z
0
Z
qT ðsÞZ qðsÞdsdh
tþh
_ T ðsÞR1 eðsÞdsdkdh _ eðsÞ
0
rM
t
_ T R2 eðsÞdsdkdh _ eðsÞ
tþk
h
Z
t
tþk
h
s2
t
Z
t
/T ðsÞT/ðsÞdsdh
ð21Þ
tþh
where qT(s) = [eT(s) e˙T(s)]. Taking the time derivative of V (t) along the trajectory of system (11) yields
_ t ; tÞ 6 2eT ðtÞPAl eðtÞ 2eT ðtÞPK j C l eðtÞ þ 2eT ðtÞPB0l /ðtÞ Vðe Z t þ 2eT ðtÞPB1l /ðt sðtÞÞ þ 2eT ðtÞPB2l /ðsÞds trðtÞ
2eT ðtÞPK j Dl wðtÞ þ qT ðtÞQ qðtÞ ð1 lÞqT ðt sðtÞÞ Q qðt sðtÞÞ þ qT ðtÞRqðtÞ qT ðt s1 ÞRqðt s1 Þ þ qT ðtÞSqðtÞ qT ðt s2 ÞSqðt s2 Þ þ s2 qT ðtÞW qðtÞ Z t þ s12 qT ðtÞZ qðtÞ qT ðsÞW qðsÞds tsðtÞ
Z
tsðtÞ
qT ðsÞðW þ ZÞqðsÞds
þ þ
s22 _ T 2
_ e ðtÞR1 eðtÞ
2
Z
Z
0
s2
s212 _ T
_ e ðtÞR2 eðtÞ
þ /T ðtÞr2M T/ðtÞ
Z
ts1
qT ðsÞZ qðsÞds
tsðtÞ t
_ e_ T ðsÞR1 eðsÞdsdh
tþh
Z s1 Z s2
Z
t
trM
t
_ e_ T ðsÞR2 eðsÞdsdh
tþh
T Z /ðsÞds T
t
/ðsÞds : ð22Þ
trM
Also add free-weighting matrices
h 2e_ T ðtÞP ðAl þ K j C l ÞeðtÞ þ B0l /ðtÞ þ B1l /ðt sðtÞÞ # Z t _ /ðsÞds K j Dl wðtÞ eðtÞ ¼0 þB2l
ð23Þ
trðtÞ
Since functions f() and g() satisfy (4) and (5), respectively, it is clear that
/ðtÞT /ðtÞ ¼ jgðxðtÞÞ gð^xðtÞÞj2 6 jWeðtÞj2 ¼ eT ðtÞW T WeðtÞ; wT ðtÞwðtÞ ¼ jf ðt; xðtÞÞ f ðt; ^xðtÞÞj2 6 jFeðtÞj2 ¼ eT ðtÞF T FeðtÞ: Then, for positive scalars a > 0, b > 0, d > 0 we have,
a½eT ðtÞW T WeðtÞ /T ðtÞ/ðtÞ P 0
ð24Þ
b½eT ðt s1 ðtÞÞW T Weðt s1 ðtÞÞ /T ðt s1 ðtÞÞ/ðt s1 ðtÞÞ P 0 ð25Þ
s12 ¼ s2 s1 : Moreover the gain matrix is given by Kj = P1Lj.
t
tþh
0
þ rM
Z
ts2
ts2
Xl4;6 ¼ Xl4;7 ¼ Xl4;8 ¼ Xl4;9 ¼ 0; Xl4;10 ¼
Xl5;14 ¼ Xl5;15 ¼ 0; Xl6;6 ¼ S11
qT ðsÞQ qðsÞds
t
s2
l 1;9
t
tsðtÞ
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
Z
Vðet ; tÞ ¼ eT ðtÞPeðtÞ þ
d½eT ðtÞF T FeðtÞ wT ðtÞwðtÞ P 0:
ð26Þ
476
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
X3;9 ¼ X3;10 ¼ X3;11 ¼ X3;12 ¼ X3;13 ¼ X3;14 ¼ X3;15 ¼ 0;
From (22)–(26) and applying Lemma 2.4, we have
_ t ; tÞ 6 f ðtÞ X fðtÞ < 0; Vðe T
l ij
ð27Þ
which is equivalent to m X
_ t ; tÞ 6 Vðe
hl ðhðtÞÞ
1
s12
Z 22 ; X4;5 ¼ R12 ;
X4;6 ¼ X4;7 ¼ X4;8 ¼ X4;9 ¼ 0; X4;10 ¼
m X
hj ðhðtÞÞfT ðtÞXlij fðtÞ < 0;
1
s12
X5;6 ¼ X5;7 ¼ X5;8 ¼ X5;9 ¼ X5;10 ¼ X5;11 ¼ X5;12 ¼ X5;13 ¼ X5;14 ¼
where
f ðtÞ ¼ eT ðtÞ eT ðt sðtÞÞ e_ T ðt sðtÞÞ eT ðt s1 Þ e_ T ðt s1 Þ !T Z t T T T eðsÞds e ðt s2 Þ e_ ðt s2 Þ e_ ðtÞ
X5;15 ¼ 0; X6;6 ¼ S11
T
1
s12
ðW 22 þ Z 22 Þ; X6;7 ¼ S12 ;
X6;8 ¼ X6;9 ¼ X6;10 ¼ X6;11 ¼ X6;12 ¼ X6;13 ¼ X6;14 ¼ X6;15 ¼ 0; X7;7 ¼ S22 ; X7;8 ¼ X7;9 ¼ X7;10 ¼ X7;11 ¼ X7;12 ¼ X7;13 ¼ X7;14 ¼ X7;15 ¼ 0;
tsðtÞ
Z
!T
ts1
Z
eðsÞds
tsðtÞ
tsðtÞ
T Z eðsÞds
ts2
T
X8;8 ¼ Q 22 þ R22 þ S22 þ s2 W 22 þ s12 Z 22 þ
!T
t
T
X8;14 ¼ PB1 ; X8;15 ¼ LD; X9;9 ¼
_ t ; tÞ < 0. Hence, Hence (27) is equivalent to (20), which implies Vðe error-system (11) is globally asymptotically stable. In the following, we will discuss the global asymptotic stability criteria for NNs without fuzzy, then (11) becomes
ð28Þ
trðtÞ
T
P ¼ P > 0; Q ¼ Q ¼
Q 11 Q 12 Q 22
T
> 0; R ¼ R ¼
R11 R12
R22
Z ¼ ZT ¼
Z 11
Z 12
Z 22
>0
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
ðXi;j Þ1515 < 0; 1
s2
W 22
2s12 1 R2 þ aW T W þ dF T F; X1;2 ¼ W 22 ; ðs2 þ s1 Þ s2 X1;3 ¼ X1;4 ¼ X1;5 ¼ X1;6 ¼ X1;7 ¼ 0; 2R1
X1;8 ¼ Q 12 þ R12 þ S12 þ s12 W 22 AT C T LT ; 1
W T12 þ
1
1
s2
R1 þ RT1 ; Xl1;10 ¼
1 R2 þ RT2 ; ðs2 þ s1 Þ R2 þ RT2 ; X1;12 ¼ PB2 ;
R1 þ RT1 þ
1 ð s2 þ s1 Þ X1;13 ¼ PB0 ; X1;14 ¼ PB1 ; X1;15 ¼ LD; 1 1 2 X2;2 ¼ ð1 lÞQ 11 W 22 W 22 Z 22 þ bW T W;
X1;11 ¼
s2
s2
X2;3 ¼ ð1 lÞQ 12 ; X2;4 ¼
s12
1
s12
X2;5 ¼ X2;7 ¼ X2;8 ¼ 0; X2;9 ¼ X2;11 ¼
1
s12
X10;10 ¼
2
s22
R1 ;
ðR1 þ RT1 Þ; X9;10 ¼ X9;12 ¼ X9;13 ¼ X9;14 ¼ X9;15 ¼ 0;
1
s12
Z 11
2
s212
R2 ; X10;11 ¼
1
s212
ðR2 þ RT2 Þ;
X11;11 ¼
2
s22
R1
1
s12
ðW 11 þ Z 11 Þ
2
s212
R2 ;
X12;13 ¼ X12;14 ¼ X12;15 ¼ 0; X13;13 ¼ aI þ r2M T; X13;14 ¼ X13;15 ¼ 0; X14;14 ¼ bI; X14;15 ¼ 0; X15;15 ¼ dI; s12 ¼ s2 s1 :
Proof. The proof immediately follows from the similar way of proof of Theorem 3.1, hence it is omitted. This completes the proof. h Remark 3.3. It is shown in Theorem 3.1 that the global asymptotic stability of the delayed fuzzy NNs (11) can be checked by examining the solvability of the LMI (20), which can be readily conducted by utilizing the Matlab LMI toolbox.
ð29Þ
X1;1 ¼ 2PA 2LC þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11
s2
1
s22
W 11
R2 2P;
> 0;
S11 S12 W 11 W 12 S¼S ¼ > 0; Ri ¼ RTi > 0; i ¼ 1; 2; W ¼ W T ¼ > 0; S22 W 22
X1;9 ¼
s212
Moreover the gain matrix is given by K = P1L.
T ¼ T T > 0;
R1 þ
X11;12 ¼ X11;13 ¼ X11;14 ¼ X11;15 ¼ 0; X12;12 ¼ T;
Corollary 3.2. Given scalars s2 > s1 P 0 and l, the equilibrium point of NNs (28) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
T
X9;11 ¼
1
s2
2
X10;12 ¼ X10;13 ¼ X10;14 ¼ X10;15 ¼ 0;
_ eðtÞ ¼ ðA þ KCÞeðtÞ þ B0 /ðtÞ þ B1 /ðt sðtÞÞ Z t /ðsÞds KDwðtÞ: þ B2
T
s22
2 X8;9 ¼ X8;10 ¼ X8;11 ¼ 0; X8;12 ¼ PB2 ; X8;13 ¼ PB0 ;
/ðsÞds
trðtÞ
/ ðtÞ / ðt sðtÞÞ w ðtÞ : T
Z T22 ;
X4;11 ¼ X4;12 ¼ X4;13 ¼ X4;14 ¼ X4;15 ¼ 0; X5;5 ¼ R22 ;
j¼1
l¼1
X4;4 ¼ R11
1
s2
1
s12
W T12 ; X2;10 ¼
4. Robust state estimation criterion
ðW 22 þ Z 22 Þ;
In this section, we will derive the delay-dependent global robust state estimation criterion for the following fuzzy NNs with mixed interval time-varying delays
1
_ eðtÞ ¼ ððAl þ DAl ðtÞÞ þ K j C l ÞeðtÞ þ ðB0l þ DB0l ðtÞÞ/ðtÞ
s12
Z 22 ; X2;6 ¼
Remark 3.4. In Ahn (2010), the author studied delay-dependent state estimation for T–S fuzzy delayed Hopfield NNs. Also in Wang and Song (2010), the author studied state estimation for neural networks with mixed interval time-varying delays. With the best of our knowledge, robust stability criteria is not studied for state estimation of fuzzy DNNs with mixed interval time-varying delays. The following section reports the robust asymptotic state estimation conditions of fuzzy DNNs with mixed interval time-varying delays.
s12
Z T12 ;
W T12 þ Z T12 ; X2;12 ¼ X2;13 ¼ X2;14 ¼ X2;15 ¼ 0;
X3;3 ¼ ð1 lÞQ 22 ; X3;4 ¼ X3;5 ¼ X3;6 ¼ X3;7 ¼ X3;8 ¼
þ ðB1l þ DB1l ðtÞÞ/ðt sðtÞÞ þ ðB2l þ DB2l ðtÞÞ Z t /ðsÞds K j Dl wðtÞ; trðtÞ
ð30Þ
477
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
The state x1 and its estimation 10
amplitude
5 0 −5 −10
True State Estimation 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
6
amplitude
4 2 0 −2 True State Estimation
−4 −6
0
1
2
3
4
5 time t The error states
6
7
8
9
10
5
amplitude
e1 e2 0
−5
0
1
2
3
4
5 time t
6
7
8
9
10
Fig. 1. The error trajectories are converging to zero for l = j = 1.
where the parametric uncertainties are assumed to be of the form
½DAl ðtÞ DB0l ðtÞ DB1l ðtÞ DB2l ðtÞ ¼ Hl F l ðtÞ½E1l
E2l
E3l
E4l
ð31Þ
in which Hl, E1l, E2l, E3l and E4l are known real constant matrices of appropriate dimensions with
F Tl ðtÞF l ðtÞ 6 I:
ð32Þ
where
b l ¼ 2PAl 2Ll C l þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11 X 1;1 2R1
1
s2
W 22
2s12 R2 þ aW T W þ dF T F þ l ET1l E1l ; ð s2 þ s1 Þ
T bl X 12;12 ¼ T þ l E4l E4l ; T 2 bl X 13;13 ¼ aI þ rM T þ l E2l E2l ;
Theorem 4.1. Given scalars s2 > s1 P 0 and l, the equilibrium point of fuzzy NNs (30) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
P ¼ PT > 0; Q ¼ Q T ¼
Q 11 Q 12 Q 22
> 0; R ¼ RT ¼
R11 R12 ast R22
T ¼ T T > 0;
Z ¼ ZT ¼
Z 11
Z 12
Z 22
>0
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
2
bl X i;j 6 4
P1l l I
P2l
3
7 0 5 < 0;
l I
iT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; 0 0 0 0 0 0 0T ;
and the other parameters are defined as in Theorem 3.1. Then the error system (30) is globally robustly asymptotically stable and the estimator gain matrix Kj can be designed as Kj = P1Lj, j = 1, 2, . . . , N.
h
P1l ¼ HTl PT
P2l ¼ ½0 0 0 0 0 0 0 HTl PT
> 0;
S11 S12 S¼S ¼ > 0; Ri ¼ RTi > 0; i ¼ 1; 2; S22 W 11 W 12 > 0; W ¼ WT ¼ W 22 T
T bl X 14;14 ¼ bI þ l E3l E3l ;
ð33Þ
Proof. Replacing Al ; B0l ; B1l ; B2l by Al þ DAl ðtÞ; B0l þ DB0l ðtÞ; B1l þ DB1l ðtÞ; B2l þ DB2l ðtÞ respectively in (11) and applying Lemma 2.2, we obtain the results which are equivalent to (33). This shows that error-system (30) is globally robustly asymptotically stable for all admissible parameter uncertainties satisfying (31) and (32). In the following, we will discuss the global robust asmptotic stability criteria for the system without fuzzy _ ¼ ððA þ DAðtÞÞ þ KCÞeðtÞ þ ðB0 þ DB0 ðtÞÞ/ðtÞ þ ðB1 þ DB1 ðtÞÞ/ðt sðtÞÞ eðtÞ Z t þ ðB2 þ DB2 ðtÞÞ /ðsÞds KDwðtÞ; ð34Þ trðtÞ
478
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
The state x1 and its estimation 10
amplitude
5 0 −5 −10
True State Estimation 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
6
amplitude
4 2 0 −2 True State Estimation
−4 −6
0
1
2
3
4
5 time t The error states
6
7
8
9
10
5
amplitude
e1 e2 0
−5
0
1
2
3
4
5 time t
6
7
8
9
10
Fig. 2. The error trajectories are converging to zero for l = j = 2.
where the parametric uncertainties are assumed to be of the form
where
½DAðtÞ DB0 ðtÞ DB1 ðtÞ DB2 ðtÞ ¼ HFðtÞ½E1
b 1;1 ¼ 2PA 2LC þ Q 11 þ R11 þ S11 þ s2 W 11 þ s12 Z 11 X
E2
E3
E4
ð35Þ
in which H, E1, E2, E3 and E4 are known real constant matrices of appropriate dimensions with
F T ðtÞFðtÞ 6 I:
P ¼ PT > 0; Q ¼ Q T ¼
S11
W ¼ WT ¼
T ¼ T T > 0;
S12
S22 W 11
Q 11 Q 12 Q 22
> 0; W 12
W 22
Z ¼ ZT ¼
> 0; R ¼ RT ¼
Ri ¼ RTi > 0;
R11 R12
R22
b i;j X
6 4
P1 I
P2
b 13;13 ¼ aI þ r2 T þ ET E2 ; X 2 M b 14;14 ¼ bI þ ET E3 ; X 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0T ;
P2 ¼ ½0 0 0 0 0 0 0 HT PT
0 0 0 0 0 0 0T
and the other parameters are defined as in Corollary 3.2. Then the error system (34) is globally robustly asymptotically stable and the estimator gain matrix K can be designed as K = P1L.
i ¼ 1; 2;
> 0; Z 11
Z 12
Z 22
>0
3
7 0 5 < 0;
I
2s12 R2 þ aW T W þ dF T F þ ET1 E1 ; ð s2 þ s1 Þ
b 12;12 ¼ T þ ET E4 ; X 4
P1 ¼ ½HT PT
> 0;
with appropriate dimensions, further there exist positive scalars a > 0, b > 0 and d > 0 such that the following LMI holds
2
W 22
ð36Þ
Corollary 4.2. Given scalars s2 > s1 P 0 and l, the equilibrium point of NNs (34) with mixed interval time-varying delays is globally asymptotically stable if there exist matrices
S ¼ ST ¼
2R1
1
s2
ð37Þ
Proof. Replacing A, B0, B1, B2 by A + DA(t), B0 + DB0(t), B1 + DB1(t), B2 + DB2(t) respectively in (28) and applying Lemma 2.2, we obtain the results which are equivalent to (37). This shows that errorsystem (34) is globally robustly asymptotically stable for all admissible parameter uncertainties satisfying (35) and (36). h Remark 4.3. In Theorem 4.1, the matrix inequality (33) is linear on the parameters l > 0, P > 0, and Ll. Therefore, the global asymptotic convergence of the error dynamics can be readily checked by solving the LMI (33).
479
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
Remark 4.4. Notice that in Theorem 4.1, the robust state estimation problem is studied for delayed fuzzy NNs with mixed interval time-varying delays under the condition that 0 6 s1 6 s(t) 6 s2 where s1 and s2 are constants. For each s1 P 0 and other known model parameters, it follows from LMI (33) that corresponding s2 can be readily solved by utilizing the Matlab LMI toolbox. Furthermore, if the lower bound of time-varying delays is 0 (Li & Fei, 2007; Li et al., 2009) or another number, we just need to let s1 = 0 or an other number in our criteria.
Therefore, it follows from Theorem 3.1, that the fuzzy NNs (11) is globally asymptotically stable. The response of the state dynamics for the delayed fuzzy NNs (11) which converges to zero asymptotically is shown in Figs. 1 and 2. Example 5.2. Consider the error system (30) with parameters defined as
3 0 4 0 0:6 0:7 ; A2 ¼ ; B01 ¼ ; 0 5 0 6 0:5 0:4 0:3 0:4 0:7 0:6 ; B21 ¼ ; B11 ¼ 0:2 0:5 0:8 0:5 0:2 0:1 0:1 0:2 B02 ¼ ; B12 ¼ ; 0:1 0:3 0:2 0:1 0:3 0:1 ; C 1 ¼ C 2 ¼ D1 ¼ D2 ¼ I; B22 ¼ 0:2 0:3 " # 2cosðtÞ þ 0:03t2 ; W ¼ 0:01I; J1 ¼ J2 ¼ 2sinðtÞ 0:03t2
A1 ¼ 5. Numerical examples
Example 5.1. Consider the error system (11) with parameters defined as
4 0 0:6 0:7 A2 ¼ ; B01 ¼ ; 0 5 0 6 0:5 0:4 0:3 0:4 0:7 0:6 0:2 0:1 B11 ¼ ; B21 ¼ ; B02 ¼ ; 0:2 0:5 0:8 0:5 0:1 0:3 0:1 0:2 0:3 0:1 ; B22 ¼ ; C 1 ¼ C 2 ¼ D1 ¼ D2 ¼ I; B12 ¼ 0:2 0:1 0:2 0:3 " # 2cosðtÞ þ 0:03t2 ; W ¼ 0:1I; F ¼ 0:2I: J1 ¼ J2 ¼ 2sinðtÞ 0:03t 2
A1 ¼
3 0
;
The activation function gðxðtÞÞ ¼ 14 ½jxðtÞ þ 1j jxðtÞ 1j, and the nonlinear disturbance is of the form f(t, x(t)) = 0.4cos(x(t)), the time varying delays are chosen as s(t) = 4.4 + 0.1sin(t), which means s2 = 4.5 when s1 = 2.5, the derivative of time-varying delays s_ ðtÞ 6 l ¼ 0:1, and r(t) = 0.25 + 0.25sin(t), which means rM = 0.5 and using the Matlab LMI toolbox to solve the LMI in Theorem 3.1, we obtained the following matrices
P¼ R2 ¼
209:8522
11:0496
11:0496 100:4348 4:5224 0:3033
;
R1 ¼
0:9435
0:0604
0:0604 0:3082 90:7170 6:7743
;
; Q 11 ¼ ; 0:3033 1:3193 6:7743 56:8943 32:3014 2:4404 32:9480 3:2961 Q 12 ¼ ; Q 22 ¼ ; 5:2296 21:1137 3:2961 14:5626 88:4645 6:4540 30:6117 2:2295 R11 ¼ ; R12 ¼ ; 6:4540 55:0801 4:9163 20:4754 32:0564 3:1550 87:8479 6:4154 R22 ¼ ; S11 ¼ ; 3:1550 14:2800 6:4154 54:8988 30:6116 2:2291 32:0564 3:1549 ; S22 ¼ ; S12 ¼ 4:9163 20:4730 3:1549 14:2774 52:8603 6:2389 20:1318 2:3961 W 11 ¼ ; W 12 ¼ ; 6:2389 22:5983 3:6141 8:6170 9:7365 1:5180 91:1208 8:0340 W 22 ¼ ; Z 11 ¼ ; 1:5180 3:8981 8:0340 49:6459 31:6323 2:8135 15:9321 2:1704 Z 12 ¼ ; Z 22 ¼ ; 5:1159 18:4211 2:1704 8:2523 428:3764 205:1866 T¼ ; 205:1866 313:3236 with the gain matrices
0:7193 0:0788 K 1 ¼ P1 L1 ¼ ; 0:0427 2:4256 1:2409 0:1160 K 2 ¼ P1 L2 ¼ : 0:0393 2:3525
F ¼ 0:02I
H1 ¼ 0:1I;
E21 ¼ E22 ¼ 0:3I;
H2 ¼ 0:2I;
E11 ¼ E12 ¼ 0:4I;
E31 ¼ E32 ¼ 0:2I;
E41 ¼ E42 ¼ 0:1I;
FðtÞ ¼ tanhðtÞI: The activation function gðxðtÞÞ ¼ 14 ½jxðtÞ þ 1j jxðtÞ 1j, and the nonlinear disturbance is of the form f(t, x(t)) = 0.4cos(x(t)), the time varying delays are chosen as s(t) = 4.4 + 0.1sin(t), which means s2 = 4.5 when s1 = 2.5, the derivative of time-varying delays s_ ðtÞ 6 l ¼ 0:1, and r(t) = 0.25 + 0.25sin(t), which means rM = 0.5 and using the Matlab LMI toolbox to solve the LMI in Theorem 4.1, we obtained the following matrices
; 11:2786 100:1383 0:0612 0:2986 4:4829 0:3077 89:3435 6:9716 R2 ¼ ; Q 11 ¼ ; 0:3077 1:3142 6:9716 55:4036 32:7449 2:5569 33:1568 3:4117 ; Q 22 ¼ ; Q 12 ¼ 5:4179 21:1521 3:4117 14:6648 88:0619 6:6345 31:0491 2:3394 R11 ¼ ; R12 ¼ ; 6:6345 54:6258 5:0964 20:5204 32:2553 3:2655 87:4522 6:5960 ; S11 ¼ ; R22 ¼ 3:2655 14:3830 6:5960 54:4488 31:0489 2:3388 32:2553 3:2653 S12 ¼ ; S22 ¼ ; 5:0964 20:5181 3:2653 14:3803 51:9977 6:2913 20:1481 2:4532 ; W 12 ¼ ; W 11 ¼ 6:2913 22:2328 3:6957 8:5673 9:8527 1:5704 90:3325 8:1823 ; Z 11 ¼ ; W 22 ¼ 1:5704 3:9037 8:1823 49:0748 31:9416 2:9195 16:2151 2:2640 Z 12 ¼ ; Z 22 ¼ ; 5:2812 18:4015 2:2640 8:3111 433:1401 205:9449 T¼ ; 205:9449 317:6054
P¼
209:5866
11:2786
;
R1 ¼
0:9264
0:0612
1 ¼ 111:2686; 2 ¼ 173:5972 with the gain matrices
K 1 ¼ P1 L1 ¼ K 2 ¼ P1 L2 ¼
0:7351
0:0443 2:4377 1:2301 0:1219
0:0793
0:0485 2:3449
; :
Therefore, it follows from Theorem 4.1, that the fuzzy NNs (30) is robustly globally asymptotically stable. The response of the state
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The state x1 and its estimation
amplitude
10 5 0 True State Estimation
−5 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
10 True State Estimation
amplitude
5 0 −5 −10
0
1
2
3
4
5 6 time t The error state e1 & e2
7
8
9
10
5
amplitude
e1(t) e2(t) 0
−5
0
1
2
3
4
5 time t
6
7
8
9
10
Fig. 3. The error trajectories are converging to zero for l = j = 1.
The state x1 and its estimation
amplitude
10 5 0 True State Estimation
−5 0
1
2
3
4
5 6 time t The state x2 and its estimation
7
8
9
10
10 True State Estimation
amplitude
5 0 −5 −10
0
1
2
3
4
5 6 time t The error state e1 & e2
7
8
9
10
5
amplitude
e1(t) e2(t) 0
−5
0
1
2
3
4
5 time t
6
7
Fig. 4. The error trajectories are converging to zero for l = j = 2.
8
9
10
P. Balasubramaniam et al. / Expert Systems with Applications 39 (2012) 472–481
dynamics for the delayed fuzzy NNs (11) which converges to zero asymptotically is shown in Figs. 3 and 4.
6. Conclusion This paper investigated the delay-dependent robust asymptotic state estimation of fuzzy Hopfield neural networks with mixed interval time-varying delays. By constructing a new Lyapunov– Krasovskii functional containing triple-integral terms and employing Newton–Leibnitz formulation and linear matrix inequality techniques and introducing free-weighting matrices, some sufficient conditions for robust global asymptotic stability criteria has been derived in terms of linear matrix inequalities (LMIs), which can be easily calculated by MATLAB LMI control toolbox. Through the available output measurements, a state estimator is designed to estimate the neuron states and the dynamics of the estimation error is asymptotically stable. Finally, two numerical examples have been used to demonstrate the usefulness of the main results. In future, research topics would be the extension of the present results to more general cases, for example, the case that the neural network is inherently stochastic, the case that the network modes are subjected to Markovian switching, the case that the delayprobability-distribution-dependent stability and the case that the mode dependent stability. The results will appear in the near future. References Ahn, C. K. (2010). Delay-dependent state estimation for T–S fuzzy delayed Hopfield neural networks. Nonlinear Dynamics, 61, 483–489. Ali, M. S., & Balasubramaniam, P. (2009). Stability analysis of uncertain fuzzy Hopfield neural networks with time delays. Communications in Nonlinear Science and Numerical Simulation, 14, 2776–2783. Boyd, B., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia: SIAM. Cao, Y. Y., & Frank, P. M. (2000). Analysis and synthesis of nonlinear timedelay systems via fuzzy control approach. IEEE Transaction on Fuzzy Systems, 8, 200–211. Chen, T., & Wang, L. (2007). Global l-stability of delayed neural networks with unbounded time-varying delays. IEEE Transactions on Neural Networks, 18, 1836–1840. Chua, L., & Yang, L. (1988). Cellular neural networks: Theory and applications. IEEE Transactions on Circuits and Systems I, 35, 1257–1290. Gopalsamy, K. (2004). Stability of artificial neural networks with impulses. Applied Mathematics and Computation, 154, 783–813. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser.
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