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Stability analysis for Cohen–Grossberg neural networks with time-varying delays via LMI approach
Expert Systems with Applications 38 (2011) 6360–6367
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Stability analysis for Cohen–Grossberg neural networks with time-varying delays via LMI approach q Chang-Hua Lien ⇑, Ker-Wei Yu, Yen-Feng Lin, Hao-Chin Chang, Yeong-Jay Chung Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Global asymptotic stability Delay-dependent criterion Delay-independent criterion Delayed Cohen–Grossberg neural networks
a b s t r a c t The global asymptotic stability for a class of Cohen–Grossberg neural networks (CGNNs) with time-varying delays is investigated. Delay-independent and delay-dependent stability criteria are proposed to guarantee the robust stability and uniqueness of equilibrium point of CGNNs via LMI approach. Some numerical examples are illustrated to show the effectiveness of our results. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Bidirectional associative memory neural networks (BAMNNs), cellular neural networks (CNNs), Cohen–Grossberg neural networks (CGNNs), and Hopfield neural networks (HNNs) are some famous artificial neural networks (NNs). CGNNs can be used to presented other types of NNs. Hence CGNNs can be applied to more general applications and purposes. All neural networks (NNs) are designed to many applications; such as automatic control engineering, connected component detection, hole filling, image shadowing, optimization and associative memories, pattern recognition, and signal processing. BAMNNs were proposed by Kosko (1988), CNNs were proposed by Chua and Yang (1988) and Chua and Roska (2002), CGNNs were proposed by Cohen and Grossberg (1983), and HNNs were proposed by Hopfield (1982). The delayed neural networks (DNNs) are appeared in many areas including the moving images processing and pattern classification. On the other hand, artificial neural networks are usually implemented by integrated circuits. In the implementation of artificial neural networks, time delay is produced from finite switching and finite propagation speed of electronic signals. During the implementation on very large scale integrated chips, parameter perturbations and transmitting time delays will destroy the stability of the neural networks Hence the stability of Cohen–Grossberg delayed neural networks (CGDNNs) has been investigated by many researchers in recent years (Chen & Rong, 2003; Feng & Xu, 2008; Hou, Liao, & Yan, 2007; Li, Fei, Guo, & Zhu, 2009; Lien, Yu, Lin, Chung, & Chung, 2008; Wu, Cui, & Huang, 2007).
q Contract/grant sponsor: National Science Council of Taiwan, ROC; contract/ grant no.: NSC 97-2221-E-022-009-MY2. ⇑ Corresponding author. Tel.: +886 7 8100888x5223; fax: +886 7 5718302. E-mail address: [email protected] (C.-H. Lien).
0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.11.103
Depending on whether the stability criterion itself contains the size of delay, criteria for CGDNNs can be classified into two categories, namely delay-dependent criteria (Hou et al., 2007; Li et al., 2009) and delay-independent criteria (Chen & Rong, 2003; Feng & Xu, 2008; Lien et al., 2008; Wu et al., 2007). Usually the former is less conservative when the delay is small. Delay-dependent and delay-independent criteria will be developed by Lyapunov theory and Leibniz–Newton formula in this paper. Some algebraic stability criteria were proposed based on Lyapunov approach (Chen & Rong, 2003; Wu et al., 2007). It is usually difficult to use algebraic criteria to find a feasible solution. LMI approach is an efficient tool in dealing with many control problems and can be solved by using the toolbox of Matlab (Boyd, Ghaoui, Feron, & Balakrishnan, 1994). In Feng and Xu (2008), Hou et al. (2007), Li et al. (2009), Lien et al. (2008), LMI-based stability criteria for CGDNNs have been proposed. In this paper, LMI-based delay-dependent and delay-independent results are proposed by using Lyapunov approach and Leibniz–Newton formula. Some numerical examples are provided to show the improvement of our obtained results.
2. Problem formulations and preliminaries The notation that will be used throughout the paper is listed as follows: C1 AT kxk kAk kxtks diag[ai]
set of differentiable functions from [sM, 0] to R00 transpose of matrix A Euclidean norm of vector x spectral norm of matrix A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi supH6s60
_ þ sÞk2 kxðt þ sÞk2 þ kxðt
diagonal matrix with the diagonal elements ai, i = 1, 2, . . . , n
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P > 0 (resp. P < 0) P is a positive (resp., negative) definite symmetric matrix I unit matrix A B * represents the symmetric form of matrix; i.e., * = BT C n {1, 2, . . . , n}
_ xðtÞ ¼ DðxðtÞÞ ½CðxðtÞÞ þ ðA þ DAðtÞÞf ðxðtÞÞ t P 0;
t 2 ½sM ; 0;
xðtÞ ¼ /ðtÞ;
ð1bÞ
where xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ xn ðtÞ ; xðt sðtÞÞ ¼ ½ x1 ðt s1 ðtÞÞ x2 ðt s2 ðtÞÞ xn ðt sn ðtÞÞT ; n P 2 is the number of neurons in the network, 0 6 si ðtÞ 6 sM ; s_ i ðtÞ 6 sD ; i 2 n; f ðÞ is the output, J = [J1 J2 Jn]T is the external bias vector. The matrices DðxðtÞÞ ¼ diag½di ðxi ðtÞÞ; CðxðtÞÞ ¼ ½c1 ðx1 ðtÞÞ cn ðxn ðtÞÞT ; di ðÞ is positive, continuous, and bounded, ci() is differentiable with dci(xi)/dxi P di > 0, di, i 2 n, are some given constants. A and B 2 Rnn are constant matrices, and the initial vector / 2 C1. The activation functions T
f ðxðtÞÞ ¼ ½f1 ðx1 ðtÞÞ f 2 ðx2 ðtÞÞ f n ðxn ðtÞÞ
f ðxðt sðtÞÞÞ ¼ ½f1 ðx1 ðt s1 ðtÞÞÞ f 2 ðx2 ðt s2 ðtÞÞÞ f n ðxn ðt sn ðtÞÞÞT of CGNN are globally Lipschiz and satisfy one of following conditions:
ð2aÞ
t P 0;
i
i
i
i
¼ fi ðzi ðtÞ þ ~xi Þ fi ð~xi Þ; zðt sðtÞÞ ¼ ½z1 ðt s1 ðtÞÞ z2 ðt s2 ðtÞÞ zn ðt sn ðtÞÞT ¼ xðt sðtÞÞ ~x; f ðzðt sðtÞÞÞ ¼ ½f 1 ðz1 ðt s1 ðtÞÞÞ f 2 ðz2 ðt s2 ðtÞÞÞ f n ðzn ðt sn ðtÞÞÞT ; f ðz ðt s ðtÞÞÞ ¼ f ðx ðt s ðtÞÞÞ f ð~x Þ i i i i i i i i ~ ~ ¼ fi ðzi ðt si ðtÞÞ þ xi Þ fi ðxi Þ; f i ð0Þ ¼ 0: ð4cÞ From (A1) (or (A2)) and (4b), we have
ð4eÞ
where L = diag[Li], Li, i = 1, 2, . . ., n, are given in (2), S1 = diag[s1i] and S2 = diag[s2i], sji, j = 1, 2, i = 1, 2, . . . , n, are any given positive constants. By the same derivation of Lien et al. (2008), we have
zi ðtÞci ðzi ðtÞÞ P di ðzi ðtÞÞ2 ;
i ¼ 1; 2; . . . ; n;
zT ðtÞW 1 CðzðtÞÞ P zT ðtÞW 1 d zðtÞ;
ð4fÞ
ð3aÞ
where W1 = diag[w1i], w1i, i = 1, 2, . . . , n, are some any positive constants, d = diag[di], di is given in the assumption of system (1). Lemma 2.1 (Lien, Yu, Lin, Chung, and Chung, 2009). Let U, V, W and M be real matrices of appropriate dimensions with M satisfying M = MT, then
where M and Ni, i = 1, 2, are known constant matrices of appropriate dimensions, F(t) is an unknown matrix representing the parameter perturbation which satisfy
M þ UVW þ W T V T U T < 0 for all V T V 6 I;
F T ðtÞFðtÞ 6 I;
M þ e1 UU T þ e W T W < 0:
t P 0:
ð4dÞ
ð2bÞ
where Li > 0, i 2 n, are some given positive constants. DA(t) is the parametric perturbation of A, DB(t) is the parametric perturbation of B. These two perturbed matrices are bounded by
½DAðtÞ DBðtÞ ¼ MFðtÞ½N1 N2 ;
i
f T ðzðtÞÞS1 f ðzðtÞÞ 6 zT ðtÞLS1 LzðtÞ; f T ðzðt sðtÞÞÞS2 f ðzðt sðtÞÞÞ 6 zT ðt sðtÞÞLS2 Lzðt sðtÞÞ;
and
fi ðn1 Þ fi ðn2 Þ 6 Li ; n1 ; n2 2 R; n1 – n2 ; i 2 n; n1 n2 ðA2Þ jfi ðn1 Þ fi ðn2 Þj 6 Li jn1 n2 j; n1 ; n2 2 R; i 2 n;
CðzðtÞÞ ¼ CðzðtÞ þ ~xÞ Cð~xÞ; ð4bÞ
i
ð1aÞ
T
ðA1Þ 0 6
ðz ðtÞÞ ¼ DðzðtÞ þ ~xÞ; DðzðtÞÞ ¼ diag½d i i f ðzðtÞÞ ¼ f ðzðtÞ þ ~xÞ f ð~xÞ;
f ðzðtÞÞ ¼ ½f 1 ðz1 ðtÞÞ f 2 ðz2 ðtÞÞ f n ðzn ðtÞÞT ; f ðz ðtÞÞ ¼ f ðx ðtÞÞ f ð~x Þ
Consider the following CGNNs with time-varying delays:
þ ðB þ DBðtÞÞf ðxðt sðtÞÞÞ þ J;
where
ð3bÞ
Suppose that ~x ¼ ½~x1 ~x2 ~ xn 2 Rn is an equilibrium point of system (1), then we have
Dð~xÞ½Cð~xÞ þ Af ð~xÞ þ Bf ð~xÞ þ J ¼ 0; where A ¼ ðA þ DAðtÞÞ and B ¼ ðB þ DBðtÞÞ. By using the assumption Dð~xÞ > 0, we obtain
J ¼ Cð~xÞ Af ð~xÞ Bf ð~xÞ: By the following translation
if and only if there exists a scalar e > 0 such that
Lemma 2.2 (Schur complement of Boyd et al. (1994)). For a given S12 S matrix S ¼ 11 with S11 ¼ ST11 ; S22 ¼ ST22 , then the following T S12 S22 conditions are equivalent: (1) S < 0. T (2) S22 < 0; S11 S12 S1 22 S12 < 0:
3. Delay-independent results for asymptotic stability of systems
zðtÞ ¼ ½z1 ðtÞ z2 ðtÞ zn ðtÞT ¼ xðtÞ ~x; In this section, we present a delay-independent criterion for the global asymptotic stability and uniqueness of equilibrium point ~ x ðtÞ, i = 1, 2, . . . , n. for system (1) with the time delays si ðtÞ ¼ s
we can obtain the following system:
d ðzðtÞ þ ~xÞ ¼ z_ ðtÞ dt ¼ DðzðtÞ þ ~xÞ½CðzðtÞ þ ~xÞ þ Af ðzðtÞ þ ~xÞ þ Bf ðzðt sðtÞÞ þ ~xÞ þ J ¼ DðzðtÞÞ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt sðtÞÞÞ; ð4aÞ
~ of system (1) with (A1), and Theorem 3.1. The equilibrium point x ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asymptotidelays si ðtÞ ¼ s cally stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , such that the following LMI condition holds:
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2 6 6 6 6 b R¼6 6 6 6 6 4
b 11 R
0 b R 22
b 13 R
b 14 R
0
0 b 34 R
0 b 35 R
0 b 33 R
b 44 R
b 45 R b R 55
b 16 R
3
where W2 = diag[w2i] > 0. From (7) and (8), we have
7 0 7 7 b 36 7 R 7 7 < 0; 0 7 7 b 56 7 R 5 b 66 R
ð5Þ
þ ½C T ðzðtÞÞW 1 CðzðtÞÞ zT ðtÞdW 1 dzðtÞ þ ½f T ðzðtÞÞLW 2 CðzðtÞÞ þ C T ðzðtÞÞW 2 Lf ðzðtÞÞ
b 11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d; R b 13 ¼ P1 A; R b 22 ¼ ð1 sD Þ Q 1 þ LS2 L; b 16 ¼ P1 M; R R
2f T ðzðtÞÞdW 2 f ðzðtÞÞ
b 14 ¼ P1 B; R
6 Z T R Z;
ðtÞÞ f T ðzðtÞÞ f T ðzðt s ðtÞÞÞ C T ðzðtÞÞ; Z T ¼ ½zT ðtÞ zT ðt s
b 35 ¼ P3 þ AT P2 þ LW 2 ; R
b 36 ¼ P3 M; R
b 44 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N2 ; R 2
b 45 ¼ BT P2 ; R
b 55 ¼ 2P2 þ W 1 ; R
2
Proof. Define T
ðtÞÞ ¼ ½z1 ðt s ðtÞÞ z2 ðt s ðtÞÞ zn ðt s ðtÞÞ ; zðt s f ðzðt s ðtÞÞÞ f 2 ðz2 ðt s ðtÞÞÞ f n ðzn ðt s ðtÞÞÞT : ðtÞÞÞ ¼ ½f 1 ðz1 ðt s The Lyapunov functional of system is defined by
# " # n Z zi ðtÞ X p1i s p2i ci ðsÞ Vðzt Þ ¼ 2 ðsÞ ds þ 2 ðsÞ ds d d 0 0 i i i¼1 i¼1 " # Z t n Z zi ðtÞ X p3i f i ðsÞ T þ2 ðsÞ ds þ tsðtÞ z ðsÞQ 1 zðsÞds d 0 i i¼1 Z t f T ðzðsÞÞQ f ðzðsÞÞds: þ 2 zi ðtÞ
R11
6 6 6 R¼6 6 6 6 4
b 56 ¼ PT M; R 2
b 66 ¼ e1 I: R
n Z X
ð9aÞ
where
b 33 ¼ Q 2 S1 þ P3 A þ AT P 3 2 dW 2 þ e1 NT N1 ; R 1 b 34 ¼ P3 B þ e1 NT N2 ; R 1
_ t Þ þ ½zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1 f ðzðtÞÞ þ ½zT ðt Vðz ðtÞÞÞ ðtÞÞ f T ðzðt s ðtÞÞÞS2f ðzðt s ðtÞÞLS2 Lzðt s s
"
2
R22
ð6Þ
The time derivatives of V(zt) along the trajectories of system (4a) satisfy
h i _ t Þ ¼ zT ðtÞP 1 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s ðtÞÞÞ Vðz
0
R33 R34 R44
R11
0
R22
R55 3 2 2 3T 0 P1 M 7 7 6 6 7 6 0 7 0 7 6 0 7 7 7 6 6 7 7 7 6 6 T7 R35 7 þ 6 P3 M 7FðtÞ6 N1 7 7 7 6 6 7 7 7 6 6 T7 R45 5 4 0 5 4 N2 5
14 R13 R 0
0
0
R33 R34
3
7 0 7 7 R35 7 7 7 R45 7 5
0
6 6 6 6 ¼6 6 6 4
0
R13 R14
R44
3
PT2 M
R55 3T 3 2 0 P1 M 7 6 6 7 6 0 7 6 0 7 7 6 6 7 7 6 6 T7 þ 6 N1 7F T ðtÞ6 P3 M 7 ; 7 6 6 7 6 0 7 6 T7 5 4 4 N2 5 2
ðtÞ ts
0
R11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d; R22 ¼ ð1 sD Þ Q 1 þ LS2 L;
R13 ¼ P1 A;
R33 ¼ Q 2 S1 þ P3 A þ AT P3 2 dW 2 ;
ðtÞÞÞT P2 CðzðtÞÞ þ ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s ðtÞÞÞ þ f T ðzðtÞÞP ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s
R45 ¼ BT P 2 ;
ðtÞÞÞ P3 f ðzðtÞÞ þ ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s
R14 ¼ P1 B;
R35 ¼ P3 þ AT P2 þ LW 2 ;
ðtÞÞÞ; ðtÞÞÞQ 2 f ðzðt s _ ðtÞÞ f T ðzðt s ð1 s
R35 ¼ P3 þ A P2 þ LW 2 ; ð7Þ
where P1 = diag[p1i], P2 = diag[p2i], and P3 = diag[p3i]. From (4d) and (4e), we have
ð8aÞ zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1 f ðzðtÞÞ P 0; T T z ðt sðtÞÞLS2 Lzðt sðtÞÞ f ðzðt sðtÞÞÞS2 f ðzðt sðtÞÞÞ P 0: ð8bÞ From (A1) and (4f), we have
0 6 f i ðzi ðtÞÞ=zi ðtÞ 6 Li ;
0 < di 6 ci ðzi ðtÞÞ=zi ðtÞ:
R13 ¼ P1 A;
R22 ¼ ð1 sD Þ Q 1 þ LS2 L; T
2
ð8cÞ
R45 ¼ BT P 2 ;
R34 ¼ P3 B;
R44 ¼ ð1 sD Þ Q 2 S2 ;
R55 ¼ 2P2 þ W 1 :
b < 0 in (5) is equivalent to By Lemmas 2.1 and 2.2, the condition R R < 0 in (9b). By the condition (9a) with R < 0 in (9b), there exists a constant q > 0 such that
_ t Þ 6 q kzðtÞk2 : Vðz This implies that the equilibrium point x~ of system (1) with delays si ðtÞ ¼ sðtÞ; i ¼ 1; 2; . . . ; n, is globally asymptotically stable. Next we will prove the uniqueness of equilibrium point ~ x, i.e., equilibrium point ~z ¼ ½0 0T of (4a). From the system (4a) with equilibrium point ~z, we have
By some operations for (8c), we have
Dð~zÞ½Cð~zÞ þ A f ð~zÞ þ B f ð~zÞ ¼ 0:
0 < di f 2i ðzi ðtÞÞ 6 Li f i ðzi ðtÞÞci ðzi ðtÞÞ; f T ðzðtÞÞLW 2 CðzðtÞÞ þ C T ðzðtÞÞW 2 Lf ðzðtÞÞ 2f T ðzðtÞÞdW 2 f ðzðtÞÞ P 0;
Note that Dð~zÞ ¼ Dð~z þ ~xÞ > 0, we have
ð8dÞ
R34 ¼ P3 B;
R55 ¼ 2P2 þ W 1 ;
R33 ¼ Q 2 S1 þ P3 A þ AT P3 2 dW 2 ;
_ ðtÞÞ zT ðt s ðtÞÞQ 1 zðt s ðtÞÞ þ ½zT ðtÞQ 1 zðtÞ ð1 s T þ ½f ðzðtÞÞQ f ðzðtÞÞ
R14 ¼ P1 B;
R44 ¼ ð1 sD Þ Q 2 S2 ;
R11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d;
T
0
PT2 M
0
ðtÞÞÞT P1 zðtÞ þ ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s ðtÞÞÞ þ C T ðzðtÞÞP2 ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s
3
ð9bÞ
Cð~zÞ þ A f ð~zÞ þ B f ð~zÞ ¼ 0: From the above result and conditions (8a)–(8d), we can obtain
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C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
½P1 ~z þ P2 Cð~zÞ þ P3 f ð~zÞT ½Cð~zÞ þ A f ð~zÞ þ B f ð~zÞ þ ½Cð~zÞ þ A f ð~zÞ þ B f ð~zÞT ½P1 ~z þ P2 Cð~zÞ þ P3 f ð~zÞ
b
b
b 17 ¼ P1 M; R
b 22 ¼ ð1 sD Þ Q 1 þ LS2 L U T U 2 ; R 2
b
b 26 ¼ U T U 3 ; R 2
þ ð~zT LðS1 þ S2 ÞL~z f T ð~zÞðS1 þ S2 Þf ð~zÞÞ þ ½C T ð~zÞW 1 Cð~zÞ ~zT dW 1 d~z þ ½f T ð~zÞLW 2 Cð~zÞ þ C T ð~zÞW 2 Lf ð~zÞ
b
b 33 ¼ Q 2 S1 þ P3 A þ AT P3 2 dW 2 þ e1 NT N1 ; R 1 b
b 34 ¼ P3 B þ e1 NT N 2 ; R 1
2f T ð~zÞdW 2f ð~zÞ P 0: By the above condition and (9b), we have
eT RZ e P 0; Z
ð10Þ
where
b
b 37 ¼ P3 M; R
b
b 44 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N2 ; R 2
b
b 55 ¼ 2P2 þ W 1 ; R
b
b 77 ¼ e1 I: R
b
b 45 ¼ BT P2 ; R
b 66 ¼ U T U 3 ; R 3
e T ¼ ½~zT ~zT f T ð~zÞ f T ð~zÞ C T ð~zÞ: Z
b
b 35 ¼ P3 þ AT P2 þ LW 2 ; R
b
b 57 ¼ P T M; R 2
b
b < 0 in (5) is equivalent to R < 0, the following Since the condition R result is guaranteed:
Proof. From the Leibniz–Newton formula
~z ¼ f ð~zÞ ¼ ½0 0T :
Z
t
ðtÞÞ ¼ 0; z_ ðsÞds þ zðtÞ zðt s
ð12Þ
ðtÞ ts
T
Hence the equilibrium point ~z ¼ ½0 0 is unique i.e., ~x is the unique equilibrium point of CGNNs (1). This completes the proof. h In the next, we will consider the asymptotic stability of system (1) with (A1) but with different time delays.
we have
" Z
#T
t
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
ðtÞ ts
"
ðtÞÞ þ U 3 U 1 zðtÞ þ U 2 zðt s
Corollary 3.1. The equilibrium point ~ x of system (1) with (A1) is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, such that LMI condition (5) is satisfied.
Z
#
t
z_ ðsÞds
ðtÞ ts
" ðtÞÞ þ U 3 þ U 1 zðtÞ þ U 2 zðt s " Z
Z
ðtÞ ts t
#T
t
z_ ðsÞds #
ðtÞÞ ¼ 0: z_ ðsÞds þ zðtÞ zðt s
ð13Þ
ðtÞ ts
Proof. The Lyapunov functional of system is defined by
By similar derivation,
# " # n Z zi ðtÞ X p1i s p2i ci ðsÞ Vðzt Þ ¼ 2 ðsÞ ds þ 2 ðsÞ ds d d 0 0 i i i¼1 i¼1 " # n Z zi ðtÞ n Z t X X p3i f i ðsÞ þ2 ½q1i z2i ðsÞ ðsÞ ds þ d 0 ts ðtÞ n Z X
zi ðtÞ
"
i
i¼1
i¼1
_ t Þ þ ½zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1f ðzðtÞÞ þ ½zT ðt s ðtÞÞLS2 Lzðt Vðz ðtÞÞÞ þ ½C T ðzðtÞÞW CðzðtÞÞ ðtÞÞÞS f ðzðt s ðtÞÞ f T ðzðt s s 2
i
þ q2i ðsÞf 2i ðzi ðsÞÞds; where Q1 = diag[q1i] > 0, Q2 = diag[q2i] > 0, P1 = diag[p1i], P2 = diag[p2i], P3 = diag[p3i], and W1 = diag[w1i]. By the same derivation of proof of Theorem 3.1, we can complete this proof. h If we introduce some free weighting matrices to the derivation of our previous results, the following results can be obtained. Corollary 3.2. The equilibrium point ~ x of system (1) with (A1) and ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asymptotidelays si ðtÞ ¼ s cally stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , some matrices, U1, U2, U3, such that the following LMI condition holds:
2
b b 11 R
6 6 6 6 6 6 6 6 b b R ¼6 6 6 6 6 6 6 6 4
b b 12 R b b 22 R
b b 13 R
b b 14 R
b b 16 R
0
b b 26 R
0 b b
R 33
0 b b
R 34
0 b b
b b 44 R
b 45 0 R
R 35 0 b
b b 55 R
0 b b
R 66
b b 17 R
3
7 7 0 7 7 7 b b 37 7 R 7 7 7 0 7 < 0; 7 b b 57 7 7 R 7 7 0 7 5 b b 77 R
ð11Þ
b
b 11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d þ U T þ U 1 ; R 1 b
b 12 ¼ U T þ U 2 ; R 1
b
b 13 ¼ P1 A; R
b
b 14 ¼ P1 B; R
b
1
zT ðtÞdW 1 dzðtÞ þ ½f T ðzðtÞÞLW 2 CðzðtÞÞ þ CðzðtÞÞW 2 Lf ðzðtÞÞ 2f T ðzðtÞÞdW 2 f ðzðtÞÞ
b 16 ¼ U T þ U 3 ; R 1
6 Z T R Z; where "
ðtÞÞ f T ðzðtÞÞ f T ðzðt s ðtÞÞÞ CðzðtÞÞ Z ¼ z ðtÞ z ðt s T
T
T
Z
#
t
_T
z ðsÞds :
ðtÞ ts
In the similar derivation of proof of Theorem 3.1, this proof is completed. h In this condition, Corollary 3.1 can be rewritten as follows. Corollary 3.3. The equilibrium point ~ x of system (1) with (A1) is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, some matrices, U1, U2, U3, such that LMI condition (11) is satisfied. If the stability of system (1) with (A2) is considered, we choose P3 = 0 to guarantee the positive property of Lyapunov functional in (6). Corollaries 3.2–3.3 can be rewritten as follows: ~ of system (1) with (A2) and Corollary 3.4. The equilibrium point x ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asymptotidelays si ðtÞ ¼ s cally stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , some matrices, U1, U2, U3, such that LMI condition (11) with P3 = 0 is satisfied.
6364
C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
~ of system (1) with (A2) is Corollary 3.5. The equilibrium point x unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, some matrices, U1, U2, U3, such that LMI condition (11) with P3 = 0 is satisfied.
sM
Z
t
ðtÞ z_ T ðsÞR1 z_ ðsÞds 6 s
tsM
"Z 6
Z
t
z_ T ðsÞR1 z_ ðsÞds
ðtÞ ts t
#T
z_ ðsÞds
R1
"Z
ðtÞ ts
#
t
z_ ðsÞds :
ðtÞ ts
ð17aÞ 4. Delay-dependent results for asymptotic stability of system In order to derive the delay-dependent results, the following assumptions are made: (A3) The functions di(xi(t)), i = 1,2, . . . , n, in the matrix DðxðtÞÞ ¼ diag½di ðxi ðtÞÞ are bounded by
0 < di ðxi ðtÞÞ ¼ di ðzi ðtÞ þ ~xi Þ 6 gi ;
ð14aÞ
where gi, i = 1,2, . . . , n, are some given positive constants. (A4) The time derivatives of functions ci(xi(t)), i = 1,2, . . . , n, in the vector C(x(t)) = [c1(x1(t)) cn(xn(t))]T are bounded by
0 < di 6 dci ðxi Þ=dxi 6 hi ;
ð14bÞ
where di and hi, i = 1,2, . . . , n, are some given positive constants. The following result is obtained from system (4a) and condition (14a):
s2M z_ T ðtÞR1 z_ ðtÞ ¼
n X
ð15Þ
where g = diag[gi] and R1 = diag[r1i], r1i, i = 1, 2, . . . , n, are some any positive constants. Define a new Lyapunov function t
ðs ðt sM ÞÞz_ T ðsÞR1 z_ ðsÞds;
ð16aÞ
tsM
where V(zt) is defined in (6). The time derivatives of V1(zt) along the trajectories of system (4a) and condition (12) satisfy h ðtÞÞÞ V_ 1 ðzt Þ ¼ zT ðtÞP 1 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s !#
t
ðtÞ ts
ðtÞÞÞ þ CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s Z
!#T
t
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
ð17cÞ
where h = diag[hi], d = diag[di], R2 = diag[r2i], R3 = diag[r3i], r2i and r3i, i = 1, 2, . . . , n, are some any positive constants. From (12), we have
" Z
#T
t
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
ðtÞ ts
"
Z
#
t
z_ ðsÞds
" Z
Z
#T
t
z_ ðsÞds
ðtÞ ts t
#
ðtÞÞ ¼ 0: z_ ðsÞds þ zðtÞ zðt s
ðtÞ ts
From (7) and (8) and (16) and (17), we have
V_ 1 ðzt Þ þ zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1 f ðzðtÞÞ ðtÞÞÞ ðtÞÞLS2 Lzðt s ðtÞÞ f T ðzðt s ðtÞÞÞS2 f ðzðt s þ zT ð t s h i þ C T ðzðtÞÞW 1 CðzðtÞÞ zT ðtÞdW 1 dzðtÞ h i þ f T ðzðtÞÞLW 2 CðzðtÞÞ þ CðzðtÞÞW 2 Lf ðzðtÞÞ 2f T ðzðtÞÞdW 2 f ðzðtÞÞ h i þ zT ðtÞ hR2 h zðtÞ C T ðzðtÞÞR2 CðzðtÞÞ h i b T R1 Z; b þ 2 zT ðtÞ R3 CðzðtÞÞ zT ðtÞ R3 d zðtÞ 6 Z where " Z b T ¼ zT ðtÞ zT ðt s ðtÞÞ f T ðzðtÞÞ f T ðzðt s ðtÞÞÞ C T ðzðtÞÞ Z
#
t
z_ T ðsÞds ; ðtÞ ts
P 1 zðtÞ
ðtÞ ts
i ðtÞÞÞ þ C ðzðtÞÞP 2 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s T
zT ðtÞ R3 d zðtÞ 6 zT ðtÞ R3 CðzðtÞÞ;
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
h
S
C T ðzðtÞÞR2 CðzðtÞÞ 6 zT ðtÞ hR2 h zðtÞ;
ðtÞÞ þ U 3 þ U 1 zðtÞ þ U 2 zðt s
1
þ A f ðzðtÞÞ þ B f ðzðt sðtÞÞÞ;
Z
ð17bÞ
d2i ðzi ðtÞÞ2 6 c2i ðzi ðtÞÞ 6 h2i ðzi ðtÞÞ2 ;
"
6 s ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt sðtÞÞÞT gR g½CðzðtÞÞ
Z
i h i di ðzi ðtÞÞ2 6 ½zi ðtÞci ðzi ðtÞÞ 6 hi ðzi ðtÞÞ2 ; h i2 h i2 di ðzi ðtÞÞ2 6 ½zi ðtÞci ðzi ðtÞÞ2 6 hi ðzi ðtÞÞ2 ;
ðtÞ ts
s2M r1i z_ 2i
2 M
S
h
ðtÞÞ þ U 3 U 1 zðtÞ þ U 2 zðt s
i¼1
V 1 ðzt Þ ¼ Vðzt Þ þ sM
From the similar derivations in conditions (4f) and (14b), we have
h
h iT ðtÞÞÞ P 2 CðzðtÞÞ þ CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s h i ðtÞÞÞ þ f T ðzðtÞÞP 3 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s h
iT ðtÞÞÞ P 3 f ðzðtÞÞ þ CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s _ ðtÞÞ zT ðt s ðtÞÞQ 1 zðt s ðtÞÞ þ zT ðtÞQ 1 zðtÞ ð1 s _ ðtÞ f T ðzðt s ðtÞÞÞ ðtÞÞÞQ 2 f ðzðt s þ f T ðzðtÞÞQ 2 f ðzðtÞÞ 1 s Z t z_ T ðsÞR1 z_ ðsÞds ; ð16bÞ þ s2M z_ T ðtÞR1 z_ ðtÞ sM tsM
where S is an any matrix. By the inequality in Gu, Kharitonov, and Chen (2003), we have
3
2
R111 R112 R113 R114 R115 R116 6 R122 0 0 0 R126 7 7 6 6 R133 R134 R135 0 7 7 6 R1 ¼ 6 7 6 R144 0 0 7 7 6 4 R155 0 5 R166 3 3 2 0 0 T 6 0 7 6 0 7 7 7 6 6 7 6R 7 6 6 137 7 1 6 R137 7 6 7ðR1 Þ 6 7 ; 6 R147 7 6 R147 7 7 7 6 6 4 R157 5 4 R157 5 2
0
0
R111 ¼ Q 1 þ LS1 L þ hR2 h bS bS T 2 R3 d þ U T1 þ U 1 dW 1 d; R112 ¼ bS U T1 þ U 2 ;
R113 ¼ P1 A;
R114 ¼ P1 B;
ð18Þ
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C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
R115 ¼ P1 þ R3 ;
R116 ¼ bS U T1 þ U 3 ;
R122 ¼ ð1 sD Þ Q 1 þ LS2 L
U T2
U2;
T
R133 ¼ P3 A þ A P3 þ Q 2 S1 2 dW 2 ; R135 ¼ P3 þ AT P2 þ LW 2 ;
R126 ¼
U T2
Corollary 4.1. The equilibrium point ~ x of system (1) with (A1), (A3), (A4), and is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, a matrix b S 2 Rnn , such that LMI condition (19) is satisfied.
U3 ;
R134 ¼ P3 B;
R137 ¼ sM AT gR1 ;
R144 ¼ ð1 sD Þ Q 2 S2 ; R145 ¼ BT P2 ; R147 ¼ sM BT gR1 ; R155 ¼ 2P2 R2 þ W 1 ; R157 ¼ sM gR1 ;
Proof. This proof is similar to Corollary 3.1.
R166 ¼ R1 U T3 U 3 : Now we present a delay-dependent criterion for the asymptotic stability of system (1) with (A1), (A3)–(A4) and the time delays si ðtÞ ¼ sðtÞ; i ¼ 1; 2; . . . ; n. Theorem 4.1. The equilibrium point ~ x of system (1) with (A1), (A3), ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally (A4), and delays si ðtÞ ¼ s asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, R1 > 0, R2 > 0, R3 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , a matrix b S 2 Rnn , such that the following LMI condition holds:
2 6 6 6 6 6 6 6 b R1 ¼ 6 6 6 6 6 6 6 4
b 111 R
b 113 R
b 114 R
b 115 R
b 112 R b 122 R
b 116 R b 126 R
0
0
0
0
b 133 R
b 134 R b R 144
0
b 135 R b 145 R b 155 R
0
b 137 R b 147 R b 157 R
b 166 R
0
b 177 R
0
0
b 118 R
3
7 0 7 7 b 138 7 R 7 7 0 7 7 < 0; b 158 7 R 7 7 0 7 7 b 178 7 R 5 b 188 R
2
6 6 6 6 6 6 6 b R2 ¼ 6 6 6 6 6 6 6 4
b 213 R
b 214 R
b 215 R
b 212 R b 222 R
0
0
0
b 233 R
b 234 R b 244 R
b 211 R
0
0
b 235 R b 245 R b 255 R
b 237 R b 247 R b 257 R
b 266 R
0
b 277 R
b 214 ¼ P 1 B; R
b 215 ¼ P1 þ R3 ; R
b 244 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N 2 ; R 2
b 247 ¼ sM BT gR1 ; R
b 255 ¼ 2P 2 R2 þ W 1 ; R
b 134 ¼ P3 B þ e1 NT N2 ; R 1
b 266 ¼ R1 U T U 3 ; R 3
b 145 ¼ BT P 2 ; R
b 147 ¼ sM BT gR1 ; R
b 278 ¼ sM gR1 M; R
b 166 ¼ R1 U T U 3 ; R 3 b 178 ¼ sM gR1 M; R
b 177 ¼ R1 ; R
b 188 ¼ e1 I: R
Proof. Choose b S ¼ P1 S and by Lemmas 1 and 2, the condition b 1 < 0 in (19) is equivalent to R1 < 0 in (18). Hence there exists R a constant q > 0 such that 2
V_ 1 ðzt Þ 6 q kzðtÞk : This completes the proof in the similar way of Theorem 3.1.
b 257 ¼ sM gR1 ; R
b 258 ¼ PT M; R 2
b 277 ¼ R1 ; R
b 288 ¼ e1 I: R
~ of system (1) with (A2)–(A4), Corollary 4.3. The equilibrium point x and delays, i = 1, 2, . . . , n, is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, some matrices, U1, U2, U3, a matrix b S 2 Rnn , such that the LMI condition (20) is satisfied.
b 155 ¼ 2P2 R2 þ W 1 ; R
b 158 ¼ PT M; R 2
b 157 ¼ sM gR1 ; R
b 216 ¼ b R S U T1 þ U 3 ;
b 235 ¼ AT P2 þ LW 2 ; R
b 133 ¼ P3 A þ AT P3 þ Q 2 S1 2 dW 2 þ e1 NT N1 ; R 1
b 144 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N2 ; R 2
b 288 R
b 233 ¼ Q 2 S1 2 dW 2 þ e1 N T N1 ; R 1
b 245 ¼ BT P2 ; R
b 138 ¼ P3 M; R
7 0 7 7 0 7 7 7 0 7 7 < 0; b 258 7 R 7 7 0 7 7 b 278 7 R 5
b 222 ¼ ð1 sD Þ Q 1 þ LS2 L U T U 2 ; R 2
b 237 ¼ sM AT gR1 ; R
b 137 ¼ sM AT gR1 ; R
0
3
b 213 ¼ P1 A; R
R126 ¼ U T2 U 3 ;
b 135 ¼ P3 þ AT P2 þ LW 2 ; R
0
0
b 218 R
b 211 ¼ Q 1 þ LS1 L þ hR2 h b R Sb S T 2 R3 d þ U T1 þ U 1 dW 1 d;
b 234 ¼ e1 NT N2 ; R 1
b 122 ¼ ð1 sD Þ Q 1 þ LS2 L U T U 2 ; R 2
b 216 R b 226 R
ð20Þ
b 226 ¼ U T U 3 ; R 2
b 114 ¼ P1 B; R
b 116 ¼ b R S U T1 þ U 3 ;
b 115 ¼ P1 þ R3 ; R b 118 ¼ P1 M; R
b 113 ¼ P1 A; R
Corollary 4.2. The equilibrium point ~ x of system (1) with (A2)–(A4), ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asympand delays si ðtÞ ¼ s totically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , some matrices, U1, U2, U3, a matrix b S 2 Rnn , such that the following LMI condition holds:
b 218 ¼ P 1 M; R
b 111 ¼ Q 1 þ LS1 L þ hR2 h b R Sb S T 2 R3 d þ U T1 þ U 1 dW 1 d; b 112 ¼ b R S U T1 þ U 2 ;
If the delay-dependent stability of system (1) with (A2)–(A4) is considered, we choose P3 = 0 to guarantee the positive property of Lyapunov functional in (16).
b 212 ¼ b R S U T1 þ U 2 ; ð19Þ
h
5. Illustrative examples Example 5.1. Consider the delayed Cohen–Gressberg neural network in (1) under (A1) with parameters: (Example 1 of Feng & Xu (2008))
h
In the following, we will consider the delay-dependent criterion for the asymptotic stability of system (1) with (A1), (A3) and (A4) but with different time delays.
2
0:8859 0:8698 1:2909
6 A ¼ 4 0:0112
0:0621
3
7 0:5774 5;
0:4414 0:6592 0:5389
6366
C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
2
3
3:1149 0:0893 0:4332 6 7 B ¼ 4 0:0770 2:6431 0:1902 5; 0:3120 2 6 d¼4 2 6 L¼4
Table 5.1 Comparisons of the obtained results for system (1) with (21).
3:6767
0
0
0
0:1687
0
Some upper bounds on time delays to guarantee the global asymptotic stability for system (1) with (21) Feng and Xu (2008), global asymptotic stability result 0.4293 Result of this paper, global asymptotic stability result 0.4728
3 7 5;
0
0
1:1392
0:1501
0
0
0
0:0448
0
0
0
0:3178
sD
Results
0:3875 2:5820
3 7 5;
x_ 1 ðtÞ ¼ ð1 þ 0:2 cos x1 ðtÞÞ½4:2x1 ðtÞ f1 ðx1 ðtÞÞ þ 1:66f 2 ðx2 ðtÞÞ
M ¼ ½ 0:1724 0:2167 0:1021 ;
N 1 ¼ ½ 0:2987 0:1516 0:1099 ;
f1 ðx1 ðt sðtÞÞÞ; x_ 2 ðtÞ ¼ ð1 þ 0:2 sin x2 ðtÞÞ½3:8x2 ðtÞ þ f2 ðx2 ðtÞÞ þ 2:475f 1 ðx1 ðt sðtÞÞÞ f2 ðx2 ðt sðtÞÞÞ; ð22Þ
N 2 ¼ ½ 0:2984 0:2869 0:1384 :
ð21Þ
By Theorem 3.1, a feasible solution of LMI condition (5) with (21) for sD = 0.4655 is given by
2
3:0117
0
0
3
0
1:6412
0
7 5;
0
0
0:8682
6 P1 ¼ 4 2
4:1654
0
0
3
0
24:3682
0
7 5;
0
0
25:0290
0:9870
0
6 P2 ¼ 4 2 6 P3 ¼ 4
0
147:5396 0 6 S2 ¼ 4 0 138:7284 0
0
74:3435 0:1195 6 Q 2 ¼ 4 0:1195 126:6228
6 W2 ¼ 4
0:0156
23:5151
0:7158
0
0
11:4142
0
0
;
B¼
1 4:2 0 d¼h¼ ; 0 3:8
" P3 ¼
1
0
;
2:475 1 1:2 0 1 0 : L¼ ; g¼ 0 1:2 0 1
3:6254
0
0
2:4169
3:5414
0
0
1:7308
21:5891
S2 ¼
W2 ¼ 3 7 5;
R1 ¼
0
0
571:4463
0
0
0
82:7316
P2 ¼
;
S1 ¼
#
0
5:9264 0
0
5:0198
14:4127
0
"
ð23Þ
41:3764
e1 ¼ 74:4267:
By using Theorem 3.1, a comparison of the upper bounds of time delays which guarantee the global asymptotic stability of system (5) is made in Table 5.1.
0
0
1:6862
; #
18:9236
0
0
9:9559
W1 ¼
;
15:4202
0
0
14:1814
;
;
;
R2 ¼
13:4221
0
0
14:5265
4:3700
Q2 ¼
37:9113
8:3607
8:3607
11:5905
U3 ¼
;;
;
1:0547
34:1150
33:4559
0:6614
;
e1 ¼ 14:3519:
By using Theorem 4.1, comparisons of the upper bounds of time delays which guarantee the global asymptotic stability of system (22) are made in Table 5.2. Example 5.3. Consider the system (1) with (A1) and the following parameters: (Example of Lien et al. (2008))
2 1 B¼ ; 3 1 1 1 2 0 1 0 L¼ ; g¼ : 0 1 0 1 A¼
Example 5.2. Consider the delayed Cohen–Gressberg neural network in (1a) under (A1), (A3) and (A4): (Example 2 of Li et al. (2009))
#
4:3820
;
4:5098 33:4863 U2 ¼ ; 34:1013 3:3501 22:9153 27:6255 b S¼ ; 25:2540 19:1988
3 7 5;
;
; 4:3700 11:8841 23:1530 5:1185 U1 ¼ ; 9:5784 18:5116
Q1 ¼
0:6980 0
"
0 7:6793 6:5198 0 R3 ¼ ; 0 6:9170
3 0:0156 7 23:5151 5; 110:6310 3 0 7 0 5;
22:7940
0 7:8596
#
5:2066
2
2
0
P1 ¼
3
0 0
1 1:66
"
3 17:9137 0:0245 1:0841 6 7 Q 1 ¼ 4 0:0245 0:5239 0:0103 5; 1:0841 0:0103 1:3646
6 W1 ¼ 4
0 0
2
2
A¼
By Theorem 4.1, a feasible solution of LMI condition (19) with (22) for sM = 0.4, sD = 0.54 is given by
7 11:7550 0 5; 0 1:0667 2 3 56:6560 0 0 6 7 S1 ¼ 4 0 38:2833 0 5; 0 0 5:1047
2
where f1(u) = f2(u) = tanh(u), s(t) = 0.4sin2t. The following matrices can be obtained from systems (1) and (22)
3
3
;
d¼h¼
a
0
0
b
; ð24Þ
C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367 Table 5.2 Comparisons of the obtained results for system (1) with (22).
Acknowledgements sD
Results
sM
Some upper bounds on time delays to guarantee the global asymptotic stability for system (1) with (22) Li et al. (2009), global asymptotic stability 0.4 0.4 result Global asymptotic stability result (delay0.5175 Any Result of independent, Theorem 1) this sM P 0 paper Global asymptotic stability result (delay0.5249 1 dependent, Theorem 2) 0.5766 0.4
Table 5.3 Comparisons of the obtained results for system (1) with (24). Conditions
a = 6, b = 7, s1(t) = s2(t), sD = 0.24 a = 6, b = 7, s1(t) = s2(t), sD = 0.25
6367
Upper bounds of sM to guarantee asymptotic stability Wu et al. (2007)
Lien et al. (2008)
Our results
Fail
Fail
sM = 1.3
Fail
Fail
sM = 0.5
By using Theorem 4.1, comparisons of our results with (Lien et al., 2008; Wu et al., 2007), are made in Table 5.3. Our current results are less conservative than our previous ones (Lien et al., 2008) in this example.
6. Conclusions In this paper, the global asymptotic stability for uncertain Cohen–Grossberg neural networks with time-varying delays has been considered. Delay-dependent and delay-independent stability criteria have been proposed to guarantee the stability of CGNNs. Some numerical examples have been illustrated to show the improvement over other recent results.
The research reported here was supported by the National Science Council of Taiwan, ROC under Grant No. NSC 97-2221-E-022009-MY2. References Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM. Chen, T. P., & Rong, L. B. (2003). Delay-independent stability analysis of Cohen– Grossberg neural networks. Physics Letter A, 317, 436–449. Chua, L. O., & Yang, L. (1988). Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems, 35, 1257–1272. Chua, L. O., & Roska, T. (2002). Cellular neural networks and visual computing. Cambridge, UK: Cambridge University Press. Cohen, M., & Grossberg, S. (1983). Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Transactions on Systems Man and Cybernetics, 13, 815–826. Feng, J., & Xu, S. (2008). New criteria on global robust stability of Cohen–Grossberg neural networks with time-varying delays. Neurocomputing, 72, 445–457. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston, Massachusetts: Birkhauser. Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences USA, 79, 2554–2558. Hou, Y. Y., Liao, T. L., & Yan, J. J. (2007). Global asymptotic stability for a class of nonlinear neural networks with multiple delays. Nonlinear Analysis: Theory, Methods & Applications, 67, 3037–3040. Kosko, B. (1988). Bidirectional associative memories. IEEE Transactions on Systems Man and Cybernetics, 18, 49–60. Li, T., Fei, S., Guo, Y. Q., & Zhu, Q. (2009). Stability analysis on Cohen–Grossberg neural networks with both time-varying and continuously distributed delays. Nonlinear Analysis: Real World Applications, 10, 2600–2612. Lien, C. H., Yu, K. W., Lin, Y. F., Chung, Y. J., & Chung, L. Y. (2008). Stability conditions for Cohen–Grossberg neural networks with time-varying delays. Physics Letters A, 372, 2264–2268. Lien, C. H., Yu, K. W., Lin, Y. F., Chung, Y. J., & Chung, L. Y. (2009). Exponential convergence rate estimation for uncertain delayed neural networks of neutral type. Chaos, Solitons & Fractals, 40, 2491–2499. Wu, W., Cui, B. T., & Huang, M. (2007). Global asymptotic stability of Cohen– Grossberg neural networks with constant and variable delays. Chaos, Solitons and Fractals, 33, 1355–1361.
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Stability analysis for Cohen–Grossberg neural networks with time-varying delays via LMI approach q Chang-Hua Lien ⇑, Ker-Wei Yu, Yen-Feng Lin, Hao-Chin Chang, Yeong-Jay Chung Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan, ROC
a r t i c l e
i n f o
Keywords: Global asymptotic stability Delay-dependent criterion Delay-independent criterion Delayed Cohen–Grossberg neural networks
a b s t r a c t The global asymptotic stability for a class of Cohen–Grossberg neural networks (CGNNs) with time-varying delays is investigated. Delay-independent and delay-dependent stability criteria are proposed to guarantee the robust stability and uniqueness of equilibrium point of CGNNs via LMI approach. Some numerical examples are illustrated to show the effectiveness of our results. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Bidirectional associative memory neural networks (BAMNNs), cellular neural networks (CNNs), Cohen–Grossberg neural networks (CGNNs), and Hopfield neural networks (HNNs) are some famous artificial neural networks (NNs). CGNNs can be used to presented other types of NNs. Hence CGNNs can be applied to more general applications and purposes. All neural networks (NNs) are designed to many applications; such as automatic control engineering, connected component detection, hole filling, image shadowing, optimization and associative memories, pattern recognition, and signal processing. BAMNNs were proposed by Kosko (1988), CNNs were proposed by Chua and Yang (1988) and Chua and Roska (2002), CGNNs were proposed by Cohen and Grossberg (1983), and HNNs were proposed by Hopfield (1982). The delayed neural networks (DNNs) are appeared in many areas including the moving images processing and pattern classification. On the other hand, artificial neural networks are usually implemented by integrated circuits. In the implementation of artificial neural networks, time delay is produced from finite switching and finite propagation speed of electronic signals. During the implementation on very large scale integrated chips, parameter perturbations and transmitting time delays will destroy the stability of the neural networks Hence the stability of Cohen–Grossberg delayed neural networks (CGDNNs) has been investigated by many researchers in recent years (Chen & Rong, 2003; Feng & Xu, 2008; Hou, Liao, & Yan, 2007; Li, Fei, Guo, & Zhu, 2009; Lien, Yu, Lin, Chung, & Chung, 2008; Wu, Cui, & Huang, 2007).
q Contract/grant sponsor: National Science Council of Taiwan, ROC; contract/ grant no.: NSC 97-2221-E-022-009-MY2. ⇑ Corresponding author. Tel.: +886 7 8100888x5223; fax: +886 7 5718302. E-mail address: [email protected] (C.-H. Lien).
0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.11.103
Depending on whether the stability criterion itself contains the size of delay, criteria for CGDNNs can be classified into two categories, namely delay-dependent criteria (Hou et al., 2007; Li et al., 2009) and delay-independent criteria (Chen & Rong, 2003; Feng & Xu, 2008; Lien et al., 2008; Wu et al., 2007). Usually the former is less conservative when the delay is small. Delay-dependent and delay-independent criteria will be developed by Lyapunov theory and Leibniz–Newton formula in this paper. Some algebraic stability criteria were proposed based on Lyapunov approach (Chen & Rong, 2003; Wu et al., 2007). It is usually difficult to use algebraic criteria to find a feasible solution. LMI approach is an efficient tool in dealing with many control problems and can be solved by using the toolbox of Matlab (Boyd, Ghaoui, Feron, & Balakrishnan, 1994). In Feng and Xu (2008), Hou et al. (2007), Li et al. (2009), Lien et al. (2008), LMI-based stability criteria for CGDNNs have been proposed. In this paper, LMI-based delay-dependent and delay-independent results are proposed by using Lyapunov approach and Leibniz–Newton formula. Some numerical examples are provided to show the improvement of our obtained results.
2. Problem formulations and preliminaries The notation that will be used throughout the paper is listed as follows: C1 AT kxk kAk kxtks diag[ai]
set of differentiable functions from [sM, 0] to R00 transpose of matrix A Euclidean norm of vector x spectral norm of matrix A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi supH6s60
_ þ sÞk2 kxðt þ sÞk2 þ kxðt
diagonal matrix with the diagonal elements ai, i = 1, 2, . . . , n
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P > 0 (resp. P < 0) P is a positive (resp., negative) definite symmetric matrix I unit matrix A B * represents the symmetric form of matrix; i.e., * = BT C n {1, 2, . . . , n}
_ xðtÞ ¼ DðxðtÞÞ ½CðxðtÞÞ þ ðA þ DAðtÞÞf ðxðtÞÞ t P 0;
t 2 ½sM ; 0;
xðtÞ ¼ /ðtÞ;
ð1bÞ
where xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ xn ðtÞ ; xðt sðtÞÞ ¼ ½ x1 ðt s1 ðtÞÞ x2 ðt s2 ðtÞÞ xn ðt sn ðtÞÞT ; n P 2 is the number of neurons in the network, 0 6 si ðtÞ 6 sM ; s_ i ðtÞ 6 sD ; i 2 n; f ðÞ is the output, J = [J1 J2 Jn]T is the external bias vector. The matrices DðxðtÞÞ ¼ diag½di ðxi ðtÞÞ; CðxðtÞÞ ¼ ½c1 ðx1 ðtÞÞ cn ðxn ðtÞÞT ; di ðÞ is positive, continuous, and bounded, ci() is differentiable with dci(xi)/dxi P di > 0, di, i 2 n, are some given constants. A and B 2 Rnn are constant matrices, and the initial vector / 2 C1. The activation functions T
f ðxðtÞÞ ¼ ½f1 ðx1 ðtÞÞ f 2 ðx2 ðtÞÞ f n ðxn ðtÞÞ
f ðxðt sðtÞÞÞ ¼ ½f1 ðx1 ðt s1 ðtÞÞÞ f 2 ðx2 ðt s2 ðtÞÞÞ f n ðxn ðt sn ðtÞÞÞT of CGNN are globally Lipschiz and satisfy one of following conditions:
ð2aÞ
t P 0;
i
i
i
i
¼ fi ðzi ðtÞ þ ~xi Þ fi ð~xi Þ; zðt sðtÞÞ ¼ ½z1 ðt s1 ðtÞÞ z2 ðt s2 ðtÞÞ zn ðt sn ðtÞÞT ¼ xðt sðtÞÞ ~x; f ðzðt sðtÞÞÞ ¼ ½f 1 ðz1 ðt s1 ðtÞÞÞ f 2 ðz2 ðt s2 ðtÞÞÞ f n ðzn ðt sn ðtÞÞÞT ; f ðz ðt s ðtÞÞÞ ¼ f ðx ðt s ðtÞÞÞ f ð~x Þ i i i i i i i i ~ ~ ¼ fi ðzi ðt si ðtÞÞ þ xi Þ fi ðxi Þ; f i ð0Þ ¼ 0: ð4cÞ From (A1) (or (A2)) and (4b), we have
ð4eÞ
where L = diag[Li], Li, i = 1, 2, . . ., n, are given in (2), S1 = diag[s1i] and S2 = diag[s2i], sji, j = 1, 2, i = 1, 2, . . . , n, are any given positive constants. By the same derivation of Lien et al. (2008), we have
zi ðtÞci ðzi ðtÞÞ P di ðzi ðtÞÞ2 ;
i ¼ 1; 2; . . . ; n;
zT ðtÞW 1 CðzðtÞÞ P zT ðtÞW 1 d zðtÞ;
ð4fÞ
ð3aÞ
where W1 = diag[w1i], w1i, i = 1, 2, . . . , n, are some any positive constants, d = diag[di], di is given in the assumption of system (1). Lemma 2.1 (Lien, Yu, Lin, Chung, and Chung, 2009). Let U, V, W and M be real matrices of appropriate dimensions with M satisfying M = MT, then
where M and Ni, i = 1, 2, are known constant matrices of appropriate dimensions, F(t) is an unknown matrix representing the parameter perturbation which satisfy
M þ UVW þ W T V T U T < 0 for all V T V 6 I;
F T ðtÞFðtÞ 6 I;
M þ e1 UU T þ e W T W < 0:
t P 0:
ð4dÞ
ð2bÞ
where Li > 0, i 2 n, are some given positive constants. DA(t) is the parametric perturbation of A, DB(t) is the parametric perturbation of B. These two perturbed matrices are bounded by
½DAðtÞ DBðtÞ ¼ MFðtÞ½N1 N2 ;
i
f T ðzðtÞÞS1 f ðzðtÞÞ 6 zT ðtÞLS1 LzðtÞ; f T ðzðt sðtÞÞÞS2 f ðzðt sðtÞÞÞ 6 zT ðt sðtÞÞLS2 Lzðt sðtÞÞ;
and
fi ðn1 Þ fi ðn2 Þ 6 Li ; n1 ; n2 2 R; n1 – n2 ; i 2 n; n1 n2 ðA2Þ jfi ðn1 Þ fi ðn2 Þj 6 Li jn1 n2 j; n1 ; n2 2 R; i 2 n;
CðzðtÞÞ ¼ CðzðtÞ þ ~xÞ Cð~xÞ; ð4bÞ
i
ð1aÞ
T
ðA1Þ 0 6
ðz ðtÞÞ ¼ DðzðtÞ þ ~xÞ; DðzðtÞÞ ¼ diag½d i i f ðzðtÞÞ ¼ f ðzðtÞ þ ~xÞ f ð~xÞ;
f ðzðtÞÞ ¼ ½f 1 ðz1 ðtÞÞ f 2 ðz2 ðtÞÞ f n ðzn ðtÞÞT ; f ðz ðtÞÞ ¼ f ðx ðtÞÞ f ð~x Þ
Consider the following CGNNs with time-varying delays:
þ ðB þ DBðtÞÞf ðxðt sðtÞÞÞ þ J;
where
ð3bÞ
Suppose that ~x ¼ ½~x1 ~x2 ~ xn 2 Rn is an equilibrium point of system (1), then we have
Dð~xÞ½Cð~xÞ þ Af ð~xÞ þ Bf ð~xÞ þ J ¼ 0; where A ¼ ðA þ DAðtÞÞ and B ¼ ðB þ DBðtÞÞ. By using the assumption Dð~xÞ > 0, we obtain
J ¼ Cð~xÞ Af ð~xÞ Bf ð~xÞ: By the following translation
if and only if there exists a scalar e > 0 such that
Lemma 2.2 (Schur complement of Boyd et al. (1994)). For a given S12 S matrix S ¼ 11 with S11 ¼ ST11 ; S22 ¼ ST22 , then the following T S12 S22 conditions are equivalent: (1) S < 0. T (2) S22 < 0; S11 S12 S1 22 S12 < 0:
3. Delay-independent results for asymptotic stability of systems
zðtÞ ¼ ½z1 ðtÞ z2 ðtÞ zn ðtÞT ¼ xðtÞ ~x; In this section, we present a delay-independent criterion for the global asymptotic stability and uniqueness of equilibrium point ~ x ðtÞ, i = 1, 2, . . . , n. for system (1) with the time delays si ðtÞ ¼ s
we can obtain the following system:
d ðzðtÞ þ ~xÞ ¼ z_ ðtÞ dt ¼ DðzðtÞ þ ~xÞ½CðzðtÞ þ ~xÞ þ Af ðzðtÞ þ ~xÞ þ Bf ðzðt sðtÞÞ þ ~xÞ þ J ¼ DðzðtÞÞ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt sðtÞÞÞ; ð4aÞ
~ of system (1) with (A1), and Theorem 3.1. The equilibrium point x ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asymptotidelays si ðtÞ ¼ s cally stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , such that the following LMI condition holds:
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2 6 6 6 6 b R¼6 6 6 6 6 4
b 11 R
0 b R 22
b 13 R
b 14 R
0
0 b 34 R
0 b 35 R
0 b 33 R
b 44 R
b 45 R b R 55
b 16 R
3
where W2 = diag[w2i] > 0. From (7) and (8), we have
7 0 7 7 b 36 7 R 7 7 < 0; 0 7 7 b 56 7 R 5 b 66 R
ð5Þ
þ ½C T ðzðtÞÞW 1 CðzðtÞÞ zT ðtÞdW 1 dzðtÞ þ ½f T ðzðtÞÞLW 2 CðzðtÞÞ þ C T ðzðtÞÞW 2 Lf ðzðtÞÞ
b 11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d; R b 13 ¼ P1 A; R b 22 ¼ ð1 sD Þ Q 1 þ LS2 L; b 16 ¼ P1 M; R R
2f T ðzðtÞÞdW 2 f ðzðtÞÞ
b 14 ¼ P1 B; R
6 Z T R Z;
ðtÞÞ f T ðzðtÞÞ f T ðzðt s ðtÞÞÞ C T ðzðtÞÞ; Z T ¼ ½zT ðtÞ zT ðt s
b 35 ¼ P3 þ AT P2 þ LW 2 ; R
b 36 ¼ P3 M; R
b 44 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N2 ; R 2
b 45 ¼ BT P2 ; R
b 55 ¼ 2P2 þ W 1 ; R
2
Proof. Define T
ðtÞÞ ¼ ½z1 ðt s ðtÞÞ z2 ðt s ðtÞÞ zn ðt s ðtÞÞ ; zðt s f ðzðt s ðtÞÞÞ f 2 ðz2 ðt s ðtÞÞÞ f n ðzn ðt s ðtÞÞÞT : ðtÞÞÞ ¼ ½f 1 ðz1 ðt s The Lyapunov functional of system is defined by
# " # n Z zi ðtÞ X p1i s p2i ci ðsÞ Vðzt Þ ¼ 2 ðsÞ ds þ 2 ðsÞ ds d d 0 0 i i i¼1 i¼1 " # Z t n Z zi ðtÞ X p3i f i ðsÞ T þ2 ðsÞ ds þ tsðtÞ z ðsÞQ 1 zðsÞds d 0 i i¼1 Z t f T ðzðsÞÞQ f ðzðsÞÞds: þ 2 zi ðtÞ
R11
6 6 6 R¼6 6 6 6 4
b 56 ¼ PT M; R 2
b 66 ¼ e1 I: R
n Z X
ð9aÞ
where
b 33 ¼ Q 2 S1 þ P3 A þ AT P 3 2 dW 2 þ e1 NT N1 ; R 1 b 34 ¼ P3 B þ e1 NT N2 ; R 1
_ t Þ þ ½zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1 f ðzðtÞÞ þ ½zT ðt Vðz ðtÞÞÞ ðtÞÞ f T ðzðt s ðtÞÞÞS2f ðzðt s ðtÞÞLS2 Lzðt s s
"
2
R22
ð6Þ
The time derivatives of V(zt) along the trajectories of system (4a) satisfy
h i _ t Þ ¼ zT ðtÞP 1 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s ðtÞÞÞ Vðz
0
R33 R34 R44
R11
0
R22
R55 3 2 2 3T 0 P1 M 7 7 6 6 7 6 0 7 0 7 6 0 7 7 7 6 6 7 7 7 6 6 T7 R35 7 þ 6 P3 M 7FðtÞ6 N1 7 7 7 6 6 7 7 7 6 6 T7 R45 5 4 0 5 4 N2 5
14 R13 R 0
0
0
R33 R34
3
7 0 7 7 R35 7 7 7 R45 7 5
0
6 6 6 6 ¼6 6 6 4
0
R13 R14
R44
3
PT2 M
R55 3T 3 2 0 P1 M 7 6 6 7 6 0 7 6 0 7 7 6 6 7 7 6 6 T7 þ 6 N1 7F T ðtÞ6 P3 M 7 ; 7 6 6 7 6 0 7 6 T7 5 4 4 N2 5 2
ðtÞ ts
0
R11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d; R22 ¼ ð1 sD Þ Q 1 þ LS2 L;
R13 ¼ P1 A;
R33 ¼ Q 2 S1 þ P3 A þ AT P3 2 dW 2 ;
ðtÞÞÞT P2 CðzðtÞÞ þ ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s ðtÞÞÞ þ f T ðzðtÞÞP ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s
R45 ¼ BT P 2 ;
ðtÞÞÞ P3 f ðzðtÞÞ þ ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s
R14 ¼ P1 B;
R35 ¼ P3 þ AT P2 þ LW 2 ;
ðtÞÞÞ; ðtÞÞÞQ 2 f ðzðt s _ ðtÞÞ f T ðzðt s ð1 s
R35 ¼ P3 þ A P2 þ LW 2 ; ð7Þ
where P1 = diag[p1i], P2 = diag[p2i], and P3 = diag[p3i]. From (4d) and (4e), we have
ð8aÞ zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1 f ðzðtÞÞ P 0; T T z ðt sðtÞÞLS2 Lzðt sðtÞÞ f ðzðt sðtÞÞÞS2 f ðzðt sðtÞÞÞ P 0: ð8bÞ From (A1) and (4f), we have
0 6 f i ðzi ðtÞÞ=zi ðtÞ 6 Li ;
0 < di 6 ci ðzi ðtÞÞ=zi ðtÞ:
R13 ¼ P1 A;
R22 ¼ ð1 sD Þ Q 1 þ LS2 L; T
2
ð8cÞ
R45 ¼ BT P 2 ;
R34 ¼ P3 B;
R44 ¼ ð1 sD Þ Q 2 S2 ;
R55 ¼ 2P2 þ W 1 :
b < 0 in (5) is equivalent to By Lemmas 2.1 and 2.2, the condition R R < 0 in (9b). By the condition (9a) with R < 0 in (9b), there exists a constant q > 0 such that
_ t Þ 6 q kzðtÞk2 : Vðz This implies that the equilibrium point x~ of system (1) with delays si ðtÞ ¼ sðtÞ; i ¼ 1; 2; . . . ; n, is globally asymptotically stable. Next we will prove the uniqueness of equilibrium point ~ x, i.e., equilibrium point ~z ¼ ½0 0T of (4a). From the system (4a) with equilibrium point ~z, we have
By some operations for (8c), we have
Dð~zÞ½Cð~zÞ þ A f ð~zÞ þ B f ð~zÞ ¼ 0:
0 < di f 2i ðzi ðtÞÞ 6 Li f i ðzi ðtÞÞci ðzi ðtÞÞ; f T ðzðtÞÞLW 2 CðzðtÞÞ þ C T ðzðtÞÞW 2 Lf ðzðtÞÞ 2f T ðzðtÞÞdW 2 f ðzðtÞÞ P 0;
Note that Dð~zÞ ¼ Dð~z þ ~xÞ > 0, we have
ð8dÞ
R34 ¼ P3 B;
R55 ¼ 2P2 þ W 1 ;
R33 ¼ Q 2 S1 þ P3 A þ AT P3 2 dW 2 ;
_ ðtÞÞ zT ðt s ðtÞÞQ 1 zðt s ðtÞÞ þ ½zT ðtÞQ 1 zðtÞ ð1 s T þ ½f ðzðtÞÞQ f ðzðtÞÞ
R14 ¼ P1 B;
R44 ¼ ð1 sD Þ Q 2 S2 ;
R11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d;
T
0
PT2 M
0
ðtÞÞÞT P1 zðtÞ þ ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s ðtÞÞÞ þ C T ðzðtÞÞP2 ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s
3
ð9bÞ
Cð~zÞ þ A f ð~zÞ þ B f ð~zÞ ¼ 0: From the above result and conditions (8a)–(8d), we can obtain
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½P1 ~z þ P2 Cð~zÞ þ P3 f ð~zÞT ½Cð~zÞ þ A f ð~zÞ þ B f ð~zÞ þ ½Cð~zÞ þ A f ð~zÞ þ B f ð~zÞT ½P1 ~z þ P2 Cð~zÞ þ P3 f ð~zÞ
b
b
b 17 ¼ P1 M; R
b 22 ¼ ð1 sD Þ Q 1 þ LS2 L U T U 2 ; R 2
b
b 26 ¼ U T U 3 ; R 2
þ ð~zT LðS1 þ S2 ÞL~z f T ð~zÞðS1 þ S2 Þf ð~zÞÞ þ ½C T ð~zÞW 1 Cð~zÞ ~zT dW 1 d~z þ ½f T ð~zÞLW 2 Cð~zÞ þ C T ð~zÞW 2 Lf ð~zÞ
b
b 33 ¼ Q 2 S1 þ P3 A þ AT P3 2 dW 2 þ e1 NT N1 ; R 1 b
b 34 ¼ P3 B þ e1 NT N 2 ; R 1
2f T ð~zÞdW 2f ð~zÞ P 0: By the above condition and (9b), we have
eT RZ e P 0; Z
ð10Þ
where
b
b 37 ¼ P3 M; R
b
b 44 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N2 ; R 2
b
b 55 ¼ 2P2 þ W 1 ; R
b
b 77 ¼ e1 I: R
b
b 45 ¼ BT P2 ; R
b 66 ¼ U T U 3 ; R 3
e T ¼ ½~zT ~zT f T ð~zÞ f T ð~zÞ C T ð~zÞ: Z
b
b 35 ¼ P3 þ AT P2 þ LW 2 ; R
b
b 57 ¼ P T M; R 2
b
b < 0 in (5) is equivalent to R < 0, the following Since the condition R result is guaranteed:
Proof. From the Leibniz–Newton formula
~z ¼ f ð~zÞ ¼ ½0 0T :
Z
t
ðtÞÞ ¼ 0; z_ ðsÞds þ zðtÞ zðt s
ð12Þ
ðtÞ ts
T
Hence the equilibrium point ~z ¼ ½0 0 is unique i.e., ~x is the unique equilibrium point of CGNNs (1). This completes the proof. h In the next, we will consider the asymptotic stability of system (1) with (A1) but with different time delays.
we have
" Z
#T
t
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
ðtÞ ts
"
ðtÞÞ þ U 3 U 1 zðtÞ þ U 2 zðt s
Corollary 3.1. The equilibrium point ~ x of system (1) with (A1) is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, such that LMI condition (5) is satisfied.
Z
#
t
z_ ðsÞds
ðtÞ ts
" ðtÞÞ þ U 3 þ U 1 zðtÞ þ U 2 zðt s " Z
Z
ðtÞ ts t
#T
t
z_ ðsÞds #
ðtÞÞ ¼ 0: z_ ðsÞds þ zðtÞ zðt s
ð13Þ
ðtÞ ts
Proof. The Lyapunov functional of system is defined by
By similar derivation,
# " # n Z zi ðtÞ X p1i s p2i ci ðsÞ Vðzt Þ ¼ 2 ðsÞ ds þ 2 ðsÞ ds d d 0 0 i i i¼1 i¼1 " # n Z zi ðtÞ n Z t X X p3i f i ðsÞ þ2 ½q1i z2i ðsÞ ðsÞ ds þ d 0 ts ðtÞ n Z X
zi ðtÞ
"
i
i¼1
i¼1
_ t Þ þ ½zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1f ðzðtÞÞ þ ½zT ðt s ðtÞÞLS2 Lzðt Vðz ðtÞÞÞ þ ½C T ðzðtÞÞW CðzðtÞÞ ðtÞÞÞS f ðzðt s ðtÞÞ f T ðzðt s s 2
i
þ q2i ðsÞf 2i ðzi ðsÞÞds; where Q1 = diag[q1i] > 0, Q2 = diag[q2i] > 0, P1 = diag[p1i], P2 = diag[p2i], P3 = diag[p3i], and W1 = diag[w1i]. By the same derivation of proof of Theorem 3.1, we can complete this proof. h If we introduce some free weighting matrices to the derivation of our previous results, the following results can be obtained. Corollary 3.2. The equilibrium point ~ x of system (1) with (A1) and ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asymptotidelays si ðtÞ ¼ s cally stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , some matrices, U1, U2, U3, such that the following LMI condition holds:
2
b b 11 R
6 6 6 6 6 6 6 6 b b R ¼6 6 6 6 6 6 6 6 4
b b 12 R b b 22 R
b b 13 R
b b 14 R
b b 16 R
0
b b 26 R
0 b b
R 33
0 b b
R 34
0 b b
b b 44 R
b 45 0 R
R 35 0 b
b b 55 R
0 b b
R 66
b b 17 R
3
7 7 0 7 7 7 b b 37 7 R 7 7 7 0 7 < 0; 7 b b 57 7 7 R 7 7 0 7 5 b b 77 R
ð11Þ
b
b 11 ¼ 2P1 d þ Q 1 þ LS1 L dW 1 d þ U T þ U 1 ; R 1 b
b 12 ¼ U T þ U 2 ; R 1
b
b 13 ¼ P1 A; R
b
b 14 ¼ P1 B; R
b
1
zT ðtÞdW 1 dzðtÞ þ ½f T ðzðtÞÞLW 2 CðzðtÞÞ þ CðzðtÞÞW 2 Lf ðzðtÞÞ 2f T ðzðtÞÞdW 2 f ðzðtÞÞ
b 16 ¼ U T þ U 3 ; R 1
6 Z T R Z; where "
ðtÞÞ f T ðzðtÞÞ f T ðzðt s ðtÞÞÞ CðzðtÞÞ Z ¼ z ðtÞ z ðt s T
T
T
Z
#
t
_T
z ðsÞds :
ðtÞ ts
In the similar derivation of proof of Theorem 3.1, this proof is completed. h In this condition, Corollary 3.1 can be rewritten as follows. Corollary 3.3. The equilibrium point ~ x of system (1) with (A1) is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, some matrices, U1, U2, U3, such that LMI condition (11) is satisfied. If the stability of system (1) with (A2) is considered, we choose P3 = 0 to guarantee the positive property of Lyapunov functional in (6). Corollaries 3.2–3.3 can be rewritten as follows: ~ of system (1) with (A2) and Corollary 3.4. The equilibrium point x ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asymptotidelays si ðtÞ ¼ s cally stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , some matrices, U1, U2, U3, such that LMI condition (11) with P3 = 0 is satisfied.
6364
C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
~ of system (1) with (A2) is Corollary 3.5. The equilibrium point x unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, some matrices, U1, U2, U3, such that LMI condition (11) with P3 = 0 is satisfied.
sM
Z
t
ðtÞ z_ T ðsÞR1 z_ ðsÞds 6 s
tsM
"Z 6
Z
t
z_ T ðsÞR1 z_ ðsÞds
ðtÞ ts t
#T
z_ ðsÞds
R1
"Z
ðtÞ ts
#
t
z_ ðsÞds :
ðtÞ ts
ð17aÞ 4. Delay-dependent results for asymptotic stability of system In order to derive the delay-dependent results, the following assumptions are made: (A3) The functions di(xi(t)), i = 1,2, . . . , n, in the matrix DðxðtÞÞ ¼ diag½di ðxi ðtÞÞ are bounded by
0 < di ðxi ðtÞÞ ¼ di ðzi ðtÞ þ ~xi Þ 6 gi ;
ð14aÞ
where gi, i = 1,2, . . . , n, are some given positive constants. (A4) The time derivatives of functions ci(xi(t)), i = 1,2, . . . , n, in the vector C(x(t)) = [c1(x1(t)) cn(xn(t))]T are bounded by
0 < di 6 dci ðxi Þ=dxi 6 hi ;
ð14bÞ
where di and hi, i = 1,2, . . . , n, are some given positive constants. The following result is obtained from system (4a) and condition (14a):
s2M z_ T ðtÞR1 z_ ðtÞ ¼
n X
ð15Þ
where g = diag[gi] and R1 = diag[r1i], r1i, i = 1, 2, . . . , n, are some any positive constants. Define a new Lyapunov function t
ðs ðt sM ÞÞz_ T ðsÞR1 z_ ðsÞds;
ð16aÞ
tsM
where V(zt) is defined in (6). The time derivatives of V1(zt) along the trajectories of system (4a) and condition (12) satisfy h ðtÞÞÞ V_ 1 ðzt Þ ¼ zT ðtÞP 1 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s !#
t
ðtÞ ts
ðtÞÞÞ þ CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s Z
!#T
t
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
ð17cÞ
where h = diag[hi], d = diag[di], R2 = diag[r2i], R3 = diag[r3i], r2i and r3i, i = 1, 2, . . . , n, are some any positive constants. From (12), we have
" Z
#T
t
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
ðtÞ ts
"
Z
#
t
z_ ðsÞds
" Z
Z
#T
t
z_ ðsÞds
ðtÞ ts t
#
ðtÞÞ ¼ 0: z_ ðsÞds þ zðtÞ zðt s
ðtÞ ts
From (7) and (8) and (16) and (17), we have
V_ 1 ðzt Þ þ zT ðtÞLS1 LzðtÞ f T ðzðtÞÞS1 f ðzðtÞÞ ðtÞÞÞ ðtÞÞLS2 Lzðt s ðtÞÞ f T ðzðt s ðtÞÞÞS2 f ðzðt s þ zT ð t s h i þ C T ðzðtÞÞW 1 CðzðtÞÞ zT ðtÞdW 1 dzðtÞ h i þ f T ðzðtÞÞLW 2 CðzðtÞÞ þ CðzðtÞÞW 2 Lf ðzðtÞÞ 2f T ðzðtÞÞdW 2 f ðzðtÞÞ h i þ zT ðtÞ hR2 h zðtÞ C T ðzðtÞÞR2 CðzðtÞÞ h i b T R1 Z; b þ 2 zT ðtÞ R3 CðzðtÞÞ zT ðtÞ R3 d zðtÞ 6 Z where " Z b T ¼ zT ðtÞ zT ðt s ðtÞÞ f T ðzðtÞÞ f T ðzðt s ðtÞÞÞ C T ðzðtÞÞ Z
#
t
z_ T ðsÞds ; ðtÞ ts
P 1 zðtÞ
ðtÞ ts
i ðtÞÞÞ þ C ðzðtÞÞP 2 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s T
zT ðtÞ R3 d zðtÞ 6 zT ðtÞ R3 CðzðtÞÞ;
ðtÞÞ z_ ðsÞds þ zðtÞ zðt s
h
S
C T ðzðtÞÞR2 CðzðtÞÞ 6 zT ðtÞ hR2 h zðtÞ;
ðtÞÞ þ U 3 þ U 1 zðtÞ þ U 2 zðt s
1
þ A f ðzðtÞÞ þ B f ðzðt sðtÞÞÞ;
Z
ð17bÞ
d2i ðzi ðtÞÞ2 6 c2i ðzi ðtÞÞ 6 h2i ðzi ðtÞÞ2 ;
"
6 s ½CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt sðtÞÞÞT gR g½CðzðtÞÞ
Z
i h i di ðzi ðtÞÞ2 6 ½zi ðtÞci ðzi ðtÞÞ 6 hi ðzi ðtÞÞ2 ; h i2 h i2 di ðzi ðtÞÞ2 6 ½zi ðtÞci ðzi ðtÞÞ2 6 hi ðzi ðtÞÞ2 ;
ðtÞ ts
s2M r1i z_ 2i
2 M
S
h
ðtÞÞ þ U 3 U 1 zðtÞ þ U 2 zðt s
i¼1
V 1 ðzt Þ ¼ Vðzt Þ þ sM
From the similar derivations in conditions (4f) and (14b), we have
h
h iT ðtÞÞÞ P 2 CðzðtÞÞ þ CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s h i ðtÞÞÞ þ f T ðzðtÞÞP 3 CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s h
iT ðtÞÞÞ P 3 f ðzðtÞÞ þ CðzðtÞÞ þ A f ðzðtÞÞ þ B f ðzðt s _ ðtÞÞ zT ðt s ðtÞÞQ 1 zðt s ðtÞÞ þ zT ðtÞQ 1 zðtÞ ð1 s _ ðtÞ f T ðzðt s ðtÞÞÞ ðtÞÞÞQ 2 f ðzðt s þ f T ðzðtÞÞQ 2 f ðzðtÞÞ 1 s Z t z_ T ðsÞR1 z_ ðsÞds ; ð16bÞ þ s2M z_ T ðtÞR1 z_ ðtÞ sM tsM
where S is an any matrix. By the inequality in Gu, Kharitonov, and Chen (2003), we have
3
2
R111 R112 R113 R114 R115 R116 6 R122 0 0 0 R126 7 7 6 6 R133 R134 R135 0 7 7 6 R1 ¼ 6 7 6 R144 0 0 7 7 6 4 R155 0 5 R166 3 3 2 0 0 T 6 0 7 6 0 7 7 7 6 6 7 6R 7 6 6 137 7 1 6 R137 7 6 7ðR1 Þ 6 7 ; 6 R147 7 6 R147 7 7 7 6 6 4 R157 5 4 R157 5 2
0
0
R111 ¼ Q 1 þ LS1 L þ hR2 h bS bS T 2 R3 d þ U T1 þ U 1 dW 1 d; R112 ¼ bS U T1 þ U 2 ;
R113 ¼ P1 A;
R114 ¼ P1 B;
ð18Þ
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C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
R115 ¼ P1 þ R3 ;
R116 ¼ bS U T1 þ U 3 ;
R122 ¼ ð1 sD Þ Q 1 þ LS2 L
U T2
U2;
T
R133 ¼ P3 A þ A P3 þ Q 2 S1 2 dW 2 ; R135 ¼ P3 þ AT P2 þ LW 2 ;
R126 ¼
U T2
Corollary 4.1. The equilibrium point ~ x of system (1) with (A1), (A3), (A4), and is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, a matrix b S 2 Rnn , such that LMI condition (19) is satisfied.
U3 ;
R134 ¼ P3 B;
R137 ¼ sM AT gR1 ;
R144 ¼ ð1 sD Þ Q 2 S2 ; R145 ¼ BT P2 ; R147 ¼ sM BT gR1 ; R155 ¼ 2P2 R2 þ W 1 ; R157 ¼ sM gR1 ;
Proof. This proof is similar to Corollary 3.1.
R166 ¼ R1 U T3 U 3 : Now we present a delay-dependent criterion for the asymptotic stability of system (1) with (A1), (A3)–(A4) and the time delays si ðtÞ ¼ sðtÞ; i ¼ 1; 2; . . . ; n. Theorem 4.1. The equilibrium point ~ x of system (1) with (A1), (A3), ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally (A4), and delays si ðtÞ ¼ s asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, P3 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, R1 > 0, R2 > 0, R3 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , a matrix b S 2 Rnn , such that the following LMI condition holds:
2 6 6 6 6 6 6 6 b R1 ¼ 6 6 6 6 6 6 6 4
b 111 R
b 113 R
b 114 R
b 115 R
b 112 R b 122 R
b 116 R b 126 R
0
0
0
0
b 133 R
b 134 R b R 144
0
b 135 R b 145 R b 155 R
0
b 137 R b 147 R b 157 R
b 166 R
0
b 177 R
0
0
b 118 R
3
7 0 7 7 b 138 7 R 7 7 0 7 7 < 0; b 158 7 R 7 7 0 7 7 b 178 7 R 5 b 188 R
2
6 6 6 6 6 6 6 b R2 ¼ 6 6 6 6 6 6 6 4
b 213 R
b 214 R
b 215 R
b 212 R b 222 R
0
0
0
b 233 R
b 234 R b 244 R
b 211 R
0
0
b 235 R b 245 R b 255 R
b 237 R b 247 R b 257 R
b 266 R
0
b 277 R
b 214 ¼ P 1 B; R
b 215 ¼ P1 þ R3 ; R
b 244 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N 2 ; R 2
b 247 ¼ sM BT gR1 ; R
b 255 ¼ 2P 2 R2 þ W 1 ; R
b 134 ¼ P3 B þ e1 NT N2 ; R 1
b 266 ¼ R1 U T U 3 ; R 3
b 145 ¼ BT P 2 ; R
b 147 ¼ sM BT gR1 ; R
b 278 ¼ sM gR1 M; R
b 166 ¼ R1 U T U 3 ; R 3 b 178 ¼ sM gR1 M; R
b 177 ¼ R1 ; R
b 188 ¼ e1 I: R
Proof. Choose b S ¼ P1 S and by Lemmas 1 and 2, the condition b 1 < 0 in (19) is equivalent to R1 < 0 in (18). Hence there exists R a constant q > 0 such that 2
V_ 1 ðzt Þ 6 q kzðtÞk : This completes the proof in the similar way of Theorem 3.1.
b 257 ¼ sM gR1 ; R
b 258 ¼ PT M; R 2
b 277 ¼ R1 ; R
b 288 ¼ e1 I: R
~ of system (1) with (A2)–(A4), Corollary 4.3. The equilibrium point x and delays, i = 1, 2, . . . , n, is unique and globally asymptotically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, Q1 > 0, Q2 > 0, some matrices, U1, U2, U3, a matrix b S 2 Rnn , such that the LMI condition (20) is satisfied.
b 155 ¼ 2P2 R2 þ W 1 ; R
b 158 ¼ PT M; R 2
b 157 ¼ sM gR1 ; R
b 216 ¼ b R S U T1 þ U 3 ;
b 235 ¼ AT P2 þ LW 2 ; R
b 133 ¼ P3 A þ AT P3 þ Q 2 S1 2 dW 2 þ e1 NT N1 ; R 1
b 144 ¼ ð1 sD Þ Q 2 S2 þ e1 NT N2 ; R 2
b 288 R
b 233 ¼ Q 2 S1 2 dW 2 þ e1 N T N1 ; R 1
b 245 ¼ BT P2 ; R
b 138 ¼ P3 M; R
7 0 7 7 0 7 7 7 0 7 7 < 0; b 258 7 R 7 7 0 7 7 b 278 7 R 5
b 222 ¼ ð1 sD Þ Q 1 þ LS2 L U T U 2 ; R 2
b 237 ¼ sM AT gR1 ; R
b 137 ¼ sM AT gR1 ; R
0
3
b 213 ¼ P1 A; R
R126 ¼ U T2 U 3 ;
b 135 ¼ P3 þ AT P2 þ LW 2 ; R
0
0
b 218 R
b 211 ¼ Q 1 þ LS1 L þ hR2 h b R Sb S T 2 R3 d þ U T1 þ U 1 dW 1 d;
b 234 ¼ e1 NT N2 ; R 1
b 122 ¼ ð1 sD Þ Q 1 þ LS2 L U T U 2 ; R 2
b 216 R b 226 R
ð20Þ
b 226 ¼ U T U 3 ; R 2
b 114 ¼ P1 B; R
b 116 ¼ b R S U T1 þ U 3 ;
b 115 ¼ P1 þ R3 ; R b 118 ¼ P1 M; R
b 113 ¼ P1 A; R
Corollary 4.2. The equilibrium point ~ x of system (1) with (A2)–(A4), ðtÞ; i ¼ 1; 2; . . . ; n, is unique and globally asympand delays si ðtÞ ¼ s totically stable, if there exist a positive constant e1, some n n diagonal matrices P1 > 0, P2 > 0, S1 > 0, S2 > 0, W1 > 0, W2 > 0, some positive definite symmetric matrices Q 1 ; Q 2 2 Rnn , some matrices, U1, U2, U3, a matrix b S 2 Rnn , such that the following LMI condition holds:
b 218 ¼ P 1 M; R
b 111 ¼ Q 1 þ LS1 L þ hR2 h b R Sb S T 2 R3 d þ U T1 þ U 1 dW 1 d; b 112 ¼ b R S U T1 þ U 2 ;
If the delay-dependent stability of system (1) with (A2)–(A4) is considered, we choose P3 = 0 to guarantee the positive property of Lyapunov functional in (16).
b 212 ¼ b R S U T1 þ U 2 ; ð19Þ
h
5. Illustrative examples Example 5.1. Consider the delayed Cohen–Gressberg neural network in (1) under (A1) with parameters: (Example 1 of Feng & Xu (2008))
h
In the following, we will consider the delay-dependent criterion for the asymptotic stability of system (1) with (A1), (A3) and (A4) but with different time delays.
2
0:8859 0:8698 1:2909
6 A ¼ 4 0:0112
0:0621
3
7 0:5774 5;
0:4414 0:6592 0:5389
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C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367
2
3
3:1149 0:0893 0:4332 6 7 B ¼ 4 0:0770 2:6431 0:1902 5; 0:3120 2 6 d¼4 2 6 L¼4
Table 5.1 Comparisons of the obtained results for system (1) with (21).
3:6767
0
0
0
0:1687
0
Some upper bounds on time delays to guarantee the global asymptotic stability for system (1) with (21) Feng and Xu (2008), global asymptotic stability result 0.4293 Result of this paper, global asymptotic stability result 0.4728
3 7 5;
0
0
1:1392
0:1501
0
0
0
0:0448
0
0
0
0:3178
sD
Results
0:3875 2:5820
3 7 5;
x_ 1 ðtÞ ¼ ð1 þ 0:2 cos x1 ðtÞÞ½4:2x1 ðtÞ f1 ðx1 ðtÞÞ þ 1:66f 2 ðx2 ðtÞÞ
M ¼ ½ 0:1724 0:2167 0:1021 ;
N 1 ¼ ½ 0:2987 0:1516 0:1099 ;
f1 ðx1 ðt sðtÞÞÞ; x_ 2 ðtÞ ¼ ð1 þ 0:2 sin x2 ðtÞÞ½3:8x2 ðtÞ þ f2 ðx2 ðtÞÞ þ 2:475f 1 ðx1 ðt sðtÞÞÞ f2 ðx2 ðt sðtÞÞÞ; ð22Þ
N 2 ¼ ½ 0:2984 0:2869 0:1384 :
ð21Þ
By Theorem 3.1, a feasible solution of LMI condition (5) with (21) for sD = 0.4655 is given by
2
3:0117
0
0
3
0
1:6412
0
7 5;
0
0
0:8682
6 P1 ¼ 4 2
4:1654
0
0
3
0
24:3682
0
7 5;
0
0
25:0290
0:9870
0
6 P2 ¼ 4 2 6 P3 ¼ 4
0
147:5396 0 6 S2 ¼ 4 0 138:7284 0
0
74:3435 0:1195 6 Q 2 ¼ 4 0:1195 126:6228
6 W2 ¼ 4
0:0156
23:5151
0:7158
0
0
11:4142
0
0
;
B¼
1 4:2 0 d¼h¼ ; 0 3:8
" P3 ¼
1
0
;
2:475 1 1:2 0 1 0 : L¼ ; g¼ 0 1:2 0 1
3:6254
0
0
2:4169
3:5414
0
0
1:7308
21:5891
S2 ¼
W2 ¼ 3 7 5;
R1 ¼
0
0
571:4463
0
0
0
82:7316
P2 ¼
;
S1 ¼
#
0
5:9264 0
0
5:0198
14:4127
0
"
ð23Þ
41:3764
e1 ¼ 74:4267:
By using Theorem 3.1, a comparison of the upper bounds of time delays which guarantee the global asymptotic stability of system (5) is made in Table 5.1.
0
0
1:6862
; #
18:9236
0
0
9:9559
W1 ¼
;
15:4202
0
0
14:1814
;
;
;
R2 ¼
13:4221
0
0
14:5265
4:3700
Q2 ¼
37:9113
8:3607
8:3607
11:5905
U3 ¼
;;
;
1:0547
34:1150
33:4559
0:6614
;
e1 ¼ 14:3519:
By using Theorem 4.1, comparisons of the upper bounds of time delays which guarantee the global asymptotic stability of system (22) are made in Table 5.2. Example 5.3. Consider the system (1) with (A1) and the following parameters: (Example of Lien et al. (2008))
2 1 B¼ ; 3 1 1 1 2 0 1 0 L¼ ; g¼ : 0 1 0 1 A¼
Example 5.2. Consider the delayed Cohen–Gressberg neural network in (1a) under (A1), (A3) and (A4): (Example 2 of Li et al. (2009))
#
4:3820
;
4:5098 33:4863 U2 ¼ ; 34:1013 3:3501 22:9153 27:6255 b S¼ ; 25:2540 19:1988
3 7 5;
;
; 4:3700 11:8841 23:1530 5:1185 U1 ¼ ; 9:5784 18:5116
Q1 ¼
0:6980 0
"
0 7:6793 6:5198 0 R3 ¼ ; 0 6:9170
3 0:0156 7 23:5151 5; 110:6310 3 0 7 0 5;
22:7940
0 7:8596
#
5:2066
2
2
0
P1 ¼
3
0 0
1 1:66
"
3 17:9137 0:0245 1:0841 6 7 Q 1 ¼ 4 0:0245 0:5239 0:0103 5; 1:0841 0:0103 1:3646
6 W1 ¼ 4
0 0
2
2
A¼
By Theorem 4.1, a feasible solution of LMI condition (19) with (22) for sM = 0.4, sD = 0.54 is given by
7 11:7550 0 5; 0 1:0667 2 3 56:6560 0 0 6 7 S1 ¼ 4 0 38:2833 0 5; 0 0 5:1047
2
where f1(u) = f2(u) = tanh(u), s(t) = 0.4sin2t. The following matrices can be obtained from systems (1) and (22)
3
3
;
d¼h¼
a
0
0
b
; ð24Þ
C.-H. Lien et al. / Expert Systems with Applications 38 (2011) 6360–6367 Table 5.2 Comparisons of the obtained results for system (1) with (22).
Acknowledgements sD
Results
sM
Some upper bounds on time delays to guarantee the global asymptotic stability for system (1) with (22) Li et al. (2009), global asymptotic stability 0.4 0.4 result Global asymptotic stability result (delay0.5175 Any Result of independent, Theorem 1) this sM P 0 paper Global asymptotic stability result (delay0.5249 1 dependent, Theorem 2) 0.5766 0.4
Table 5.3 Comparisons of the obtained results for system (1) with (24). Conditions
a = 6, b = 7, s1(t) = s2(t), sD = 0.24 a = 6, b = 7, s1(t) = s2(t), sD = 0.25
6367
Upper bounds of sM to guarantee asymptotic stability Wu et al. (2007)
Lien et al. (2008)
Our results
Fail
Fail
sM = 1.3
Fail
Fail
sM = 0.5
By using Theorem 4.1, comparisons of our results with (Lien et al., 2008; Wu et al., 2007), are made in Table 5.3. Our current results are less conservative than our previous ones (Lien et al., 2008) in this example.
6. Conclusions In this paper, the global asymptotic stability for uncertain Cohen–Grossberg neural networks with time-varying delays has been considered. Delay-dependent and delay-independent stability criteria have been proposed to guarantee the stability of CGNNs. Some numerical examples have been illustrated to show the improvement over other recent results.
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