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Research Article
Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays Yanke Du n, Rui Xu Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 January 2014 Received in revised form 26 February 2014 Accepted 11 March 2014 This paper was recommended for publication by Dr. Jeff Pieper
This paper is concerned with the global exponential synchronization for an array of hybrid coupled neural networks with leakage delay, time-varying discrete and distributed delays. By employing a novel augmented Lyapunov–Krasovskii functional (LKF), applying the theory of Kronecker product of matrices, Barbalat's Lemma and the technique of linear matrix inequality (LMI), delay-dependent sufficient conditions are obtained for the global exponential synchronization of the system. As an extension, robust synchronization criteria are derived for the corresponding system with parameter uncertainties. Some examples are given to show the effectiveness of the obtained theoretical results. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Array of neural networks Synchronization Hybrid coupling Kronecker product Lyapunov–Krasovskii functional
1. Introduction Since small-world and scale-free complex networks were proposed in [1,2], complex dynamical networks, which are a set of interconnected nodes with specific dynamics, have received increasing attention from various fields of science and engineering such as the World Wide Web, electrical power grids, communication networks, the Internet, and global economic markets. Many interesting behaviors have been observed from complex dynamical networks, e.g., synchronization, consensus, self-organization, and spatiotemporal chaos spiral waves. Synchronization as an important collective behavior of complex dynamical networks has been widely investigated in the last two decades (see, for example, [3–7]). Coupled neural networks (CNNs), as a special kind of complex networks, have been found to exhibit more complicated and unpredictable behaviors than a single neural network. Particularly, synchronization in an array of coupled neural networks, which is one of the hot research fields of complex networks, has been a challenging issue due to its potential applications in many areas such as secure communication, information science, chaos generator design, and harmonic oscillation generation. On the other hand, in the applications of neural networks, there exist unavoidably time delays due to the finite information processing speed
n
Corresponding author. E-mail address: [email protected] (Y. Du).
and the finite switching speed of amplifiers. It is well known that time delay can cause oscillation and instability of neural networks. Therefore, various synchronization criteria for CNNs with time delays have been investigated in the literature [8–21] and references therein. To mention a few representative works, the synchronization problems for an array of neural networks with hybrid coupling and interval time-varying delay were investigated in [8]. In [9], the global exponential synchronization was investigated for an array of asymmetric neural networks with time-varying delays and nonlinear coupling. Cao and Li [10] presented cluster synchronization criteria for an array of hybrid coupled neural networks with delay. Park et al. [11] obtained the delay-dependent synchronization conditions for coupled discrete-time neural networks with interval time-varying delays in network coupling. So far, very little attention has been paid to neural networks with time delay in a leakage (or “forgetting”) term (see [22–26]). This is due to some theoretical and technical difficulties. In fact, time delay in the leakage term also has great impact on the dynamics of neural networks. As pointed out by Gopalsamy [27], time delay in the stabilizing negative feedback term has a tendency to destabilize a system. On the other hand, it has been observed that neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state (see [16,17]). However, to the best of our knowledge, there are no global exponential synchronization
http://dx.doi.org/10.1016/j.isatra.2014.03.005 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
results about an array of hybrid coupled neural networks with leakage delay, time-varying discrete and distributed delays. In the real world, there exist inevitably some uncertainties caused by the existence of modeling errors, external disturbances, and parameter fluctuations, which would lead to complex dynamic behaviors (see [28–31]). Thus, a good neural network should have robustness against such uncertainties. Recently, robust synchronization problems against such uncertainties of complex neural networks were investigated in [32–34] and references therein. Motivated by the works of Zhang et al. [8], Wang et al. [17], Park et al. [22] and the discussions above, the purpose of this paper is to present some new sufficient conditions for the robust global exponential synchronization of hybrid coupled neural networks with leakage delay, time-varying delays and parameter uncertainties. To this end, we consider the following differential equation system: ^ ðt τÞ þ W ^ 1 f ðxi ðtÞÞ þ W ^ 2 f ðxi ðt hðtÞÞÞ x_ i ðtÞ ¼ Ax i Z t N ^ 3 ^ þW f ðxi ðsÞÞ ds þ uðtÞ þ ∑ g ð1Þ ij D 1 xj ðtÞ t sðtÞ
j¼1
N
N
ð3Þ ^ ^ þ ∑ g ð2Þ ij D 2 xj ðt hðtÞÞ þ ∑ g ij D 3 j¼1
j¼1
Z
t t sðtÞ
2. Preliminaries
xj ðsÞ ds;
ð1:1Þ
where i ¼ 1; 2; …; N, N is the number of coupled nodes, xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞT A Rn is the neuron state vector of the ith node, n denotes the number of neurons in a neural network, f ðxi ðÞÞ ¼ ðf 1 ðxi1 ðÞÞ; f 2 ðxi2 ðÞÞ; …; f n ðxin ðÞÞÞT A Rn
and
uðtÞ ¼ ðu1 ðtÞ;
u2 ðtÞ; …; un ðtÞÞT A Rn denote the neuron activation function vector
and the external input vector, respectively, τ, h(t) and sðtÞ denote the leakage delay, time-varying discrete delay and distributed ^ k ¼ W k þ ΔW k ðtÞ, D ^ k ¼ Dk þ delay, respectively, A^ ¼ A þ ΔAðtÞ, W ΔDk ðtÞ ðk ¼ 1; 2; 3Þ, A ¼ diagfa1 ; a2 ; …; an g A Rnn ðak 4 0; k ¼ 1; 2; …; nÞ is the self-feedback matrix, W k A Rnn ðk ¼ 1; 2; 3Þ are the connection weight matrices, Dk A Rnn ðk ¼ 1; 2; 3Þ are the constant inner-coupling matrices of nodes, which describe the individual coupling between networks, ΔAðtÞ, ΔW k ðtÞ and ΔDk ðtÞ represent the parameter uncertainties of the system, which are assumed to be of the form ½ΔAðtÞ ΔW 1 ðtÞ ΔW 2 ðtÞ ΔW 3 ðtÞ ΔD1 ðtÞ ΔD2 ðtÞ ΔD3 ðtÞ ¼ M∇ðtÞ½EA EW 1 EW 2 EW 3 ED1 ED2 ED3 ; in which M, EA, EW k and EDk ðk ¼ 1; 2; 3Þ are known constant matrices and ∇ðtÞ is an unknown matrix satisfying
For convenience, we introduce several notations. Rn is the ndimensional Euclidean space, Rmn denotes the set of m n real matrices, In represents the n-dimensional identity matrix, A 4ð Z Þ0 means that A is a symmetric positive definite (semidefinite) matrix, λmax ðAÞ denotes the largest eigenvalue of A, A B stands for the Kronecker product of matrices A and B, for B ¼ ðBij ÞðmpÞðnqÞ , Bij A Rmn ð1 r i rp; 1 rj r qÞ, A B stands for 2 3 A B11 ⋯ A B1q 6 7 ⋮ ⋱ ⋮ 4 5; A Bp1 ⋯ A Bpq for a vector x ¼ ðx1 ; x2 ; …; xn ÞT A Rn , J x J stands for the Euclidean norm, n denotes the symmetric block in a symmetric matrix. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations. Combined with the sign of the Kronecker product, (1.1) can be rewritten as ^ ^ 1 ÞFðxðtÞÞ þ ðI N W ^ 2 ÞFðxðt hðtÞÞÞ _ ¼ ðI N AÞxðt xðtÞ τÞ þ ðI N W Z t ^ 3Þ ^ 1 ÞxðtÞ þ ðI N W FðxðsÞÞ ds þ UðtÞ þ ðGð1Þ D t sðtÞ
^ 2 Þxðt hðtÞÞ þ ðGð3Þ D ^ 3Þ þ ðGð2Þ D where
xðÞ ¼ ðxT1 ðÞ; xT2 ðÞ; …; xTN ðÞÞT ,
Z
t t sðtÞ
ð2:1Þ
xðsÞ ds;
T
T
FðxðÞÞ ¼ ðf ðx1 ðÞÞ; f ðx2 ðÞÞ; …;
T
∇T ðtÞ∇ðtÞ r I; GðkÞ ¼ ðg ðkÞ ij ÞNN ðk ¼ 1; 2; 3Þ are the out-coupling matrices representing the coupling strength and the topological structure of the networks and satisfying the diffusive coupling connections: ðkÞ g ðkÞ ij ¼ g ji Z 0 ði ajÞ;
matrix. Hence, it can effectively reduce the conservatism of the synchronization criteria. (3) The proposed control method removes the traditional restriction on slowly varying delay. Using this method, even if the information about the derivative of the time-varying delay h(t) is unknown, delay-derivative-independent synchronization criteria can be derived. The organization of this paper is as follows. In Section 2, some preliminaries are given. In Section 3, synchronization criteria are derived for the CNNs without uncertainties. In Section 4, sufficient conditions are presented for the robust synchronization of system (1.1). In Section 5, some numerical examples are provided to illustrate the effectiveness of the obtained theoretical results. A brief remark is given in Section 6 to conclude this work.
g ðkÞ ii ¼
N
∑
j ¼ 1;j a i
g ðkÞ ij ði; j ¼ 1; 2; …; NÞ:
In this paper, the main contributions are given as follows. (1) It is the first time to establish the robust exponential synchronization criteria for an array of hybrid coupled neural networks with parameter uncertainties, leakage delay, timevarying discrete and distributed delays. It is worth pointing out that the addressed system includes many neural network models as its special cases (see [8,10,22,37]) and some valuable mathematical techniques have been employed, which generalize the previous results to some extent. (2) A novel augmented LKF is proposed to analyze the synchronization problem for the coupled neural networks. This method includes a new type of augmented matrices with multiple Kronecker product operations, and thus introduces more relaxed variables to alleviate the requirements of a positive definite
f ðxN ðÞÞÞT , and UðtÞ ¼ ðuT ðtÞ; uT ðtÞ; …; uT ðtÞÞT . Throughout this paper, we make the following assumptions. (H1) The time delays satisfy
τ Z 0;
0 r hm r hðtÞ rhM ;
_ rh ; hðtÞ D
0 r sðtÞ r s;
where hm, hM, hD and s are known constant scalars. þ (H2) For any y1 ; y2 A R, there exist constants δr and δr , such that the activation functions satisfy
δr r
f r ðy1 Þ f r ðy2 Þ þ r δr ; y1 y2
r ¼ 1; 2; …; n:
We denote ( n o hM if hD r 1 þ þ h¼ ; Δ1 ¼ diag δ1 δ1 ; …; δn δn ; hm if hD 4 1 ( ) δ þ δ1þ δ þ δnþ ; …; n Δ2 ¼ diag 1 : 2 2 Next, we give some useful definitions and lemmas. Definition 2.1. The CNNs (1.1) are said to be robustly globally exponentially synchronized if for all admissible uncertainties ΔAðtÞ, ΔW k ðtÞ, ΔDk ðtÞ ðk ¼ 1; 2; 3Þ, there exist η 4 0 and M0 4 0
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
such that for any initial values ϕi ðsÞ and sufficiently large T 4 0 J xi ðtÞ xj ðtÞ J rM 0 e
ηt
holds for all t 4 T, and for any i; j ¼ 1; 2; …; N. M0 and η are called the decay coefficient and the decay rate, respectively. Lemma 2.1 (Kronecker product, Graham [35]). The following properties of Kronecker product are satisfied: 1. ðαAÞ B ¼ A ðαBÞ, ðαAÞ B ¼ A ðαBÞ , where α is a constant; 2. A ðB þ CÞ ¼ A B þ A C, A ðB þ CÞ ¼ A B þA C ; 3. ðA BÞðC DÞ ¼ ðACÞ ðBDÞ.
3
Lemma 2.6 (Barbalat's Lemma, Slotine and Li [38]). Let f be a nonnegative function defined on ½0; þ 1Þ. If f is Lebesgue integrable on ½0; þ 1Þ and is uniformly continuous on ½0; þ 1Þ, then limt- þ 1 f ðtÞ ¼ 0. Lemma 2.7 (Balasubramaniam et al. [39]). Assume that Ψ ¼ Ψ , H and E are real matrices with appropriate dimensions, and F(t) is a matrix satisfying F T ðtÞFðtÞ rI. Then, Ψ þ HFðtÞE þ ½HFðtÞET o 0 holds if and only if there exists a scalar ε 4 0 satisfying Ψ þ ε 1 HH T þ εET E o 0. T
Lemma 2.8 (Schur complement, Boyd et al. [40]). Given constant
Ωi ði ¼ 1; 2; 3Þ, where Ω1 ¼ ΩT1 and Ω2 40, then Ω1 þ ΩT3 Ω2 1 Ω3 o 0 if and only if ½Ω ΩΩ o0.
matrices Lemma 2.2 (Jensen inequality, Gu et al. [36]). For any matrix W 4 0, scalars γ1 and γ2 satisfying γ 2 4 γ 1 , a vector function ω : ½γ 1 ; γ 2 -Rn such that the integrations concerned are well defined, one has "Z #T Z Z ðγ 2 γ 1 Þ
γ2
γ1
ωT ðsÞW ωðsÞ ds Z
γ2
γ1
ωðsÞ ds W
γ2
γ1
ωðsÞ ds:
According to the Jensen inequality, one can easily obtain the following lemma. Lemma 2.3. For any constant matrix M 40, the following inequality holds: " #T # " Z t xðtÞ xðtÞ M M T _ _ hðtÞ x ðsÞM xðsÞ ds r : xðt hðtÞÞ xðt hðtÞÞ n M t hðtÞ
1
T 3
2
3. Global exponential synchronization criteria In this section, we first consider the following system without uncertainties: _ ¼ ðI N AÞxðt τÞ þ ðI N W 1 ÞFðxðtÞÞ þ ðI N W 2 ÞFðxðt hðtÞÞÞ xðtÞ Z t FðxðsÞÞ ds þ UðtÞ þ ðGð1Þ D1 ÞxðtÞ þ ðI N W 3 Þ t sðtÞ
þ ðGð2Þ D2 Þxðt hðtÞÞ þ ðGð3Þ D3 Þ
Z
t t sðtÞ
xðsÞ ds;
ð3:1Þ
where the notations are the same as those in (1.1) and (2.1). Lemma 2.4 (Cao et al. [37]). Let U ¼ ðuij ÞNN , P A Rnn , x ¼ ðxT1 ; …; xTN ÞT and y ¼ ðyT1 ; …; yTN ÞT with xi ; yi A Rn ði ¼ 1; 2; …; NÞ. If U ¼ U T and each row sum of U is zero, then xT ðU PÞy ¼
∑
1riojrN
uij ðxi xj ÞT Pðyi yj Þ:
22
Lemma 2.5 (Zhang et al. [8]). For Q ¼ ðP pq ÞðnlÞðnlÞ , P pq A Rnn T T w T w T T ðwÞ ð1 r p r l; 1 r q r lÞ, xðwÞ ¼ ððxw ¼ ððyw 1 Þ ; ðx2 Þ ; …; ðxN Þ Þ and y 1Þ ;
2
Ξ 11 Ξ 12 Ξ 13 Ξ 14 6 n Ξ 22 0 0 6 6 6 n n Ξ P 33 6 6 n n n Ξ 44 6
6 6 6 6 6 6 Ωij ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0
0 0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 T
0 0
0 0
0
0
Ξ 55 Ξ 56 n Ξ 66
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Ξ 77 Ξ 78 n Ξ 88 n
n
0 0
0
0
0
0
T
0
0
0
0
Ξ 7;11 Ξ 8;11
Ξ 99
Sð3Þ 12
0
0
0
0
0
n
n
n
n
n
n
n
n
n
Sð3Þ 22
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Y
n
n
n
n
n
n
n
n
n
n
n
n
T
with
xw k,
n yw k AR
ðk ¼ 1; 2; …; N; w ¼ 1; 2…; lÞ,
N
e ¼ ð1; 1; …; 1Þ A R , EN ¼ eeT , and U ¼ NI N EN , the following equality can be obtained: 2 ð1Þ 3T 2 ð1Þ 3 3 xi xjð1Þ yi yð1Þ yð1Þ x j 6 7 6 7 6 7 7 Q 6 6 7 Q6 7: ⋮ ⋮ ∑ 4 ⋮ 5 ðU Þ4 ⋮ 5 ¼ 4 5 4 5 ðlÞ ðlÞ ðlÞ ðlÞ 1 r i o j r N ðlÞ ðlÞ x y x y y x i j i j 3 ð1Þ T
q ¼ 1; 2Þ, positive diagonal matrices J l A Rnn ðl ¼ 1; 2Þ, and real matrices Z l A Rnn ðl ¼ 1; 2Þ, such that the LMI Ωij o0 holds for all 1 r i oj r N, where
Ξ 1;11 Ξ 2;11 Ξ 3;11
0
n
T w T T ðyw 2 Þ ; …; ðyN Þ Þ
2
Theorem 3.1. Under assumptions (H1) and (H2), given constant decay rate α Z 0, the dynamical system (3.1) is globally exponentially synchronized if there exist positive definite matrices P, Q, R1, R2, T, SðkÞ SðkÞ nn ðk ¼ 1; 2; 3Þ with SðkÞ ðp ¼ 1; 2; Y A Rnn , SðkÞ ¼ n11 S12 ðkÞ pq A R
2
Ξ 11;11 Ξ 11;12
Ξ 1;13 Ξ 2;13 Ξ 3;13
3
7 7 7 7 7 P 7 7 7 0 7 7 0 7 7 7 Ξ 7;13 7 7; Ξ 8;13 7 7 7 0 7 7 7 0 7 7 7 Ξ 11;13 7 7 Ξ 12;13 7 5
ð3:2Þ
Ξ 13;13
αhm Ξ 11 ¼ ð2 αÞP R2 þ eατ Q þ τ2 eατ R1 þ eαh Sð1Þ 11 þe ð2Þ α h ð1Þ α hm ð2Þ S11 J 1 Δ1 , Ξ 12 ¼ e S12 þ e S12 þ J 1 Δ2 , Ξ 13 ¼ P þ R2 , Ξ 14 ¼ ð1Þ T T T T αh ð1Þ ð1 αÞP, Ξ 1;11 ¼ NGð1Þ ij D1 Z 1 , Ξ 1;13 ¼ P NGij D1 Z 2 , Ξ 22 ¼ e S22
in
which
T T T T 2 αs þ eαhm Sð2Þ 22 þ s e Y J 1 , Ξ 2;11 ¼ W 1 Z 1 , Ξ 2;13 ¼ W 1 Z 2 , Ξ 33 ¼ Q
R2 , Ξ 3;11 ¼ AT Z T1 , Ξ 3;13 ¼ AT Z T2 , Ξ 44 ¼ αP R1 , Ξ 55 ¼ Sð2Þ 11 þ ð2Þ αðhM hm Þ Sð3Þ , Ξ ¼ Sð2Þ þ eαðhM hm Þ eαðhM hm Þ Sð3Þ 66 11 T, Ξ 56 ¼ S12 þ e 12 22
Sð3Þ 22 ,
Ξ 77 ¼ ð1 hD ÞSð1Þ 11 2 T J 2 Δ1 ,
Ξ 78 ¼ ð1 hD ÞSð1Þ 12 þJ 2 Δ2 ,
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
ð2Þ T T ð1Þ T T Ξ 7;11 ¼ NGð2Þ ij D2 Z 1 , Ξ 7;13 ¼ NGij D2 Z 2 , Ξ 88 ¼ ð1 hD ÞS22 J 2 ,
Ξ 8;11 ¼
ð3Þ 99 ¼ S11 T, 11;11 ð3Þ T T 11;12 ¼ Z 1 W 3 , 11;13 ¼ Z 1 NGij D3 Z 2 , 2 ατ e R2 þ ðhM hm Þ2 eαhM T Z 2 Z T2 .
W T2 Z T1 ,
Ξ
T T 8;13 ¼ W 2 Z 2 ,
þDT3 Z T1 Þ, Ξ
Ξ 13;13 ¼ τ
Ξ
Ξ
Ξ
¼ NGijð3Þ ðZ 1 D3 T T 12;13 ¼ W 3 Z 2 ,
Ξ
Proof. Define zij ðÞ ¼ xi ðÞ xj ðÞ, gðzij ðÞÞ ¼ f ðxi ðÞÞ f ðxj ðÞÞ, z_ ij ðÞ ¼ x_ i ðÞ x_ j ðÞ, and denote 2 3 N1 1 ⋯ 1 6 1 N1 ⋯ 1 7 6 7 : U¼6 7 4 ⋮ ⋮ ⋯ ⋮ 5 1
1
⋯
N 1
eαt
"
6
ð3:3Þ
tτ
#T "
xðtÞ
Z ðU R1 Þ
xðuÞ du
Z "
t tτ
T Z zij ðsÞ ds R1 #T "
zij ðtÞ zij ðt τÞ "
V_ 4 ðtÞ reαðt þ hÞ
t t τ
R2
V 1 ðtÞ ¼ eαt xðtÞ Z V 2 ðtÞ ¼ V 3 ðtÞ ¼ τ
t
Z
t t τ
T Z xðsÞ ds ðU PÞ xðtÞ
t
t τ
xðsÞ ds ;
n
xðtÞ
#T
tτ t
Z
tτ T
t
FðxðtÞÞ
" þ eαðt þ hm Þ " eαt
V 6 ðtÞ ¼ s
0
t hM
Z
s
t
"
ð1 hD Þ
s
eαðu þ sÞ F T ðxðuÞÞðU YÞFðxðuÞÞ du ds:
Calculating the time derivative of Vk(t) ð1 rk r6Þ, and applying Lemmas 2.1–2.5, one can obtain T Z t Z t xðsÞ ds ðU PÞ xðtÞ xðsÞ ds V_ 1 ðtÞ ¼ αeαt xðtÞ tτ
þ2eαt xðtÞ ¼ eαt
∑
1riojrN
Z
t τ
T _ xðtÞ þ xðt τÞ xðsÞ ds ðU PÞ½xðtÞ
Z zij ðtÞ
Z ðαPÞ zij ðtÞ
t tτ
t t τ
#
FðxðtÞÞ
#9 = ; zij ðt τÞ ; zij ðtÞ
xðtÞ
ð3:6Þ
#
FðxðtÞÞ
Fðxðt hðtÞÞÞ
#T
"
xðtÞ
ð2Þ
ðU S Þ #T
"
#T
xðt hðtÞÞ
ðU
Sð1Þ
Þ
xðt hðtÞÞ
#
Fðxðt hðtÞÞÞ
#
FðxðtÞÞ "
ð2Þ
ðU S Þ
xðt hm Þ
#
" ð1 hD Þ
T zij ðsÞ ds
zij ðsÞ ds þ ð2PÞ½z_ ij ðtÞ zij ðtÞ þ zij ðt τÞ ;
xðt hðtÞÞ
#T
" Sð1Þ
xðt hðtÞÞ
#
ðU Þ Fðxðt hðtÞÞÞ Fðxðt hðtÞÞÞ " #T " # xðt hm Þ xðt hm Þ ð2Þ αðhM hm Þ Sð3Þ Þ ½U ðS e Fðxðt hm ÞÞ Fðxðt hm ÞÞ " #T " #9 xðt hM Þ xðt hM Þ = ð3Þ ðU S Þ Fðxðt hM ÞÞ Fðxðt hM ÞÞ ; 8" #T " # < zij ðtÞ zij ðtÞ α t α h ð1Þ α hm ð2Þ ∑ ðe S þ e S Þ ¼e gðzij ðtÞÞ gðzij ðtÞÞ 1 r i o j r N:
tτ
t
xðtÞ
xðt hm Þ
"
tþs
xðtÞ
Fðxðt hm ÞÞ Fðxðt hm ÞÞ " #T " # xðt hm Þ xðt hm Þ Sð3Þ α ðt þ hM hm Þ þe ðU Þ Fðxðt hm ÞÞ Fðxðt hm ÞÞ " #T " # xðt hM Þ xðt hM Þ ð3Þ ðU S Þ eαt Fðxðt hM ÞÞ Fðxðt hM ÞÞ 8" #T " # < xðtÞ xðtÞ ðeαh Sð1Þ þ eαhm Sð2Þ Þ α t ½U re FðxðtÞÞ : FðxðtÞÞ
eαðu þ τÞ ½xT ðuÞðU R1 ÞxðuÞ
s
_ du ds; þ x_ ðuÞðU R2 ÞxðuÞ " #T " # Z t xðsÞ xðsÞ ð1Þ eαðs þ hÞ ðU S Þ V 4 ðtÞ ¼ ds FðxðsÞÞ FðxðsÞÞ t hðtÞ " #T " # Z t xðsÞ xðsÞ ð2Þ eαðs þ hm Þ ðU S Þ þ ds FðxðsÞÞ FðxðsÞÞ t hm " #T " # Z t hm xðsÞ xðsÞ Sð3Þ α ðs þ hM Þ e ðU Þ þ ds; FðxðsÞÞ FðxðsÞÞ t hM Z t hm Z t _ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du ds; V 5 ðtÞ ¼ ðhM hm Þ Z
#"
U R2
#"
ð1Þ
ðU S Þ "
eαðs þ τÞ xT ðsÞðU Q ÞxðsÞ ds;
Z
xðuÞ du
zij ðsÞ ds
R2 R2
ð1 hD Þeαðt hðtÞ þ hÞ
t τ
ðU R2 Þ
k¼1
where
t
n ðU R2 Þ xðt τÞ xðt τÞ 8 < ∑ τ2 eατ ½zTij ðtÞR1 zij ðtÞ þ z_ Tij ðtÞR2 z_ ij ðtÞ ¼ eαt 1 r i o j r N:
In order to tackle the synchronization problem of (3.1), we introduce the following Lyapunov–Krasovskii functional: VðtÞ ¼ ∑ V k ðtÞ;
T
t
þ eαt
þ
NN
Z
zij ðt hðtÞÞ
#T
" ð1Þ
zij ðt hðtÞÞ
#
S gðzij ðt hðtÞÞÞ gðzij ðt hðtÞÞÞ " #T " # zij ðt hm Þ zij ðt hm Þ ðSð2Þ eαðhM hm Þ Sð3Þ Þ gðzij ðt hm ÞÞ gðzij ðt hm ÞÞ " #T " #9 zij ðt hM Þ = zij ðt hM Þ Sð3Þ ; gðzij ðt hM ÞÞ gðzij ðt hM ÞÞ ;
ð3:7Þ
ð3:4Þ V_ 2 ðtÞ ¼ eαt ½eατ xT ðtÞðU Q ÞxðtÞ xT ðt τÞðU Q Þxðt τÞ ¼ eαt
∑
1riojrN
½zTij ðtÞðeατ Q Þzij ðtÞ zTij ðt τÞQzij ðt τÞ;
ð3:5Þ
_ V_ 3 ðtÞ ¼ τ2 eαðt þ τÞ ½xT ðtÞðU R1 ÞxðtÞ þ x_ T ðtÞðU R2 ÞxðtÞ Z t T _ eαðu þ τÞ ½xT ðuÞðU R1 ÞxðuÞ þ x_ ðuÞðU R2 ÞxðuÞ du τ t τ
_ r τ2 eαðt þ τÞ ½xT ðtÞðU R1 ÞxðtÞ þ x_ T ðtÞðU R2 ÞxðtÞ
_ V_ 5 ðtÞ ¼ ðhM hm Þ2 eαðt þ hM Þ x_ T ðtÞðU TÞxðtÞ Z t _ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du þ ðhM hm Þ Z ðhM hm Þ
t hm t
t hM
_ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du
_ ¼ ðhM hm Þ2 eαðt þ hM Þ x_ T ðtÞðU TÞxðtÞ Z t hðtÞ _ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du ðhM hm Þ t hM
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Z ðhM hm Þ
t hm t hðtÞ
Applying Lemmas 2.1 and 2.4, and noting that ðU Z i ÞUðtÞ ¼ 0, and UX¼ NX for matrix X with zero column sums, we have from (3.11) that
_ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du
_ r ðhM hm Þ2 eαðt þ hM Þ x_ T ðtÞðU TÞxðtÞ Z t hðtÞ _ x_ T ðuÞðU TÞxðuÞ du ðhM hðtÞÞeαt ðhðtÞ hm Þeαt
Z
0 ¼ 2eαt
t hm
_ du x_ T ðuÞðU TÞxðuÞ
_ r ðhM hm Þ e x ðtÞðU TÞxðtÞ 8" #" # " # < xðt hðtÞÞ T ðU TÞ UT xðt hðtÞÞ þ eαt n ðU TÞ xðt hM Þ : xðt hM Þ
þ
xðt hm Þ
#T "
UT
n
ðU TÞ
xðt hðtÞÞ
#9 xðt hm Þ =
#"
ðU TÞ
xðt hðtÞÞ ;
6 4
39 zij ðt hm Þ > 0 = 6 7 U T 54 zij ðt hðtÞÞ 7 5 ; > zij ðt hM Þ ; ðU TÞ 32
ðU TÞ
UT
n
2ðU TÞ
n
n
t sðtÞ
_ U ðZ 1 AÞxðt τÞþ U ðZ 1 W 1 ÞFðxðtÞÞ xT ðsÞ ds ðU Z 1 ÞxðtÞ
þ ðNGð2Þ Þ ðZ 1 D2 Þxðt hðtÞÞ þ ðNGð3Þ Þ ðZ 1 D3 Þ
ð3:8Þ
Z
Z
¼ 2eαt
∑
1riojrN
t sðtÞ
zTij ðsÞ
Z
t t sðtÞ
ð3Þ NGð2Þ ij Z 1 D2 zij ðt hðtÞÞ NGij Z 1 D3
þ z_ Tij ðtÞ
t sðtÞ
¼ eαt
∑
T Z FðxðsÞÞ ds ðU YÞ
1riojrN
Z
t
t sðtÞ
T Z Y
t t sðtÞ
)
t
Z
V_ ðtÞ r eαt ) ð3:9Þ
According to the method of [41,42] and assumption (H2), for any diagonal matrices 0 o J i A Rnn ði ¼ 1; 2Þ, we have 8" #" # " # < zij ðtÞ T J Δ1 J Δ2 zij ðtÞ 1 1 eα t gðzij ðtÞÞ n J 1 : gðzij ðtÞÞ #" " #T " #9 zij ðt hðtÞÞ = zij ðt hðtÞÞ J 2 Δ1 J 2 Δ2 Z 0: þ gðzij ðt hðtÞÞÞ gðzij ðt hðtÞÞÞ ; n J2 ð3:10Þ On the other hand, from (3.1), it can be readily seen that the following equation always holds for any matrices Z i A Rnn ði ¼ 1; 2Þ: Z t _ 0 ¼ 2eαt xT ðsÞ dsðU Z 1 Þ þ x_ T ðtÞðU Z 2 Þ xðtÞ t sðtÞ
ðI N AÞxðt τÞ þ ðI N W 1 ÞFðxðtÞÞ þ ðI N W 2 ÞFðxðt hðtÞÞÞ Z t FðxðsÞÞ ds þ UðtÞ þ ðGð1Þ D1 ÞxðtÞ þ ðI N W 3 Þ þ ðG
ð2Þ
D2 Þxðt hðtÞÞ þ ðG
ð3Þ
Z D3 Þ
t t sðtÞ
t sðtÞ
xðsÞ ds
t
t sðtÞ
zij ðsÞ ds
t t sðtÞ
gðzij ðsÞÞ ds NGð1Þ ij Z 2 D1 xðtÞ
Z
t
t sðtÞ
zij ðsÞ ds
ð3:12Þ
:
Z
t tτ
zTij ðsÞ ds; zTij ðt hm Þ; gT ðzij ðt hm ÞÞ; zTij ðt hðtÞÞ; Z
xðsÞ ds :
Z
t
t sðtÞ
zTij ðsÞ ds;
t t sðtÞ
iT gT ðzij ðsÞÞ ds; z_ Tij ðtÞ :
From (3.2) to (3.10), (3.12) and the conditions of Theorem 3.1, we obtain that
FðxðsÞÞ ds
gðzij ðsÞÞ ds :
t sðtÞ
t
Z 2 z_ ij ðtÞ Z 2 Azij ðt τÞ þ Z 2 W 1 gðzij ðtÞÞ
s2 eαs g T ðzij ðtÞÞYgðzij ðtÞÞ
gðzij ðsÞÞ ds
Z
gðzij ðsÞÞ ds NGð1Þ ij Z 1 D1 zij ðtÞ
Z
g T ðzij ðt hðtÞÞÞ; zTij ðt hM Þ; gT ðzij ðt hM ÞÞ;
t sðtÞ
FðxðsÞÞ ds þðNGð1Þ Þ ðZ 2 D1 ÞxðtÞ
ds Z 1 z_ ij ðtÞ Z 1 Azij ðt τÞ þZ 1 W 1 gðzij ðtÞÞ
þ Z 1 W 2 gðzij ðt hðtÞÞÞ þZ 1 W 3
ξij ðtÞ ¼ zTij ðtÞ; gT ðzij ðtÞÞ; zTij ðt τÞ;
t sðtÞ
t
xðsÞ ds
Let
r eαt s2 eαs F T ðxðtÞÞðU YÞFðxðtÞÞ Z
t
t
t s
t sðtÞ
t sðtÞ
þ ðNGð2Þ Þ ðZ 2 D2 Þxðt hðtÞÞ þ ðNGð3Þ Þ ðZ 2 D3 Þ
NGð2Þ Z 2 D2 zij ðt hðtÞÞ NGð3Þ Z 2 D3 ij ij
r s2 eαðt þ sÞ F T ðxðtÞÞðU YÞFðxðtÞÞ Z t F T ðxðsÞÞðU YÞFðxðsÞÞ ds sðtÞeαt
t
_ U ðZ 2 AÞxðt τÞ þU ðZ 2 W 1 ÞFðxðtÞÞ þ x_ ðtÞ ðU Z 2 ÞxðtÞ
þ Z 2 W 2 gðzij ðt hðtÞÞÞ þZ 2 W 3
V_ 6 ðtÞ ¼ s2 eαðt þ sÞ F T ðxðtÞÞðU YÞFðxðtÞÞ Z t eαðs þ sÞ F T ðxðsÞÞðU YÞFðxðsÞÞ ds s
Z
T
þ U ðZ 2 W 2 ÞFðxðt hðtÞÞÞ þU ðZ 2 W 3 Þ
8 2 3T > zij ðt hm Þ > < 6 7 2 α hM _ T α t ∑ ðhM hm Þ e z ij ðtÞT z_ ij ðtÞ þ 4 zij ðt hðtÞÞ 5 ¼e > 1 r i o j r N> : zij ðt hM Þ 2
t
t sðtÞ
2 αðt þ hM Þ _ T
"
Z
þ U ðZ 1 W 2 ÞFðxðt hðtÞÞÞþ U ðZ 1 W 3 Þ Z t FðxðsÞÞ ds þ ðNGð1Þ Þ ðZ 1 D1 ÞxðtÞ
t hM
t hðtÞ
5
ð3:11Þ
∑
1riojrN
ξTij ðtÞΩij ξij ðtÞ o0;
ð3:13Þ
where Ωij is defined in (3.2). Denote ρ ¼ max1 r i o j r N fλmax ðΩij Þg. Obviously, ρ o 0 and it then follows from (3.13) that V_ ðtÞ r ρeαt
∑
1riojrN
ξTij ðtÞξij ðtÞ r ρeαt
Therefore, we have Z t V_ ðsÞ ds rVð0Þ þ ρ VðtÞ ¼ V ð0Þ þ 0
‖zij ðtÞ‖2 :
∑
1riojrN
Z ∑
1riojrN
0
t
eαs ‖zij ðsÞ‖2 ds: ð3:14Þ
From (3.14) and the fact that V ðtÞ Z 0 and ρ o 0, we get Z t Vð0Þ ; eαs ‖zij ðsÞ‖2 ds r ∑ jρj 1riojrN 0
ð3:15Þ
which implies that Z þ1 ∑ eαs ‖zij ðsÞ‖2 ds o þ 1: 1riojrN
0
Thus, ∑1 r i o j r N eαt ‖zij ðtÞ‖2 is Lebesgue integrable. On the other hand, d αt d ðe ‖zij ðtÞ‖2 Þ ¼ ðeαt ‖xi ðtÞ xj ðtÞ‖2 Þ dt dt ¼ αeαt ‖xi ðtÞ xj ðtÞ‖2 þ 2eαt ½xi ðtÞ xj ðtÞT ½x_ i ðtÞ x_ j ðtÞ
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Φ2 ¼ ½ NGijð1Þ ED1 EW 1 EA 0nð3nÞ NGð2Þ ij ED2 E W 2 0nð2nÞ
r αeαt ‖xi ðtÞ xj ðtÞ‖2 þ2eαt J xi ðtÞ xj ðtÞ J ð J x_ i ðtÞ J þ J x_ j ðtÞ J Þ
NGijð3Þ ED3 EW 3 0nn T :
r αeαt ‖xi ðtÞ xj ðtÞ‖2 þeαt ½‖xi ðtÞ xj ðtÞ‖2 þ ð J x_ i ðtÞ J þ J x_ j ðtÞ J Þ2 r αeαt ‖xi ðtÞ xj ðtÞ‖2 þeαt ½‖xi ðtÞ xj ðtÞ‖2 þ 2ð‖x_ i ðtÞ‖2 þ ‖x_ j ðtÞ‖2 Þ:
ð3:16Þ
From (3.15), we see that eαt ‖xi ðtÞ xj ðtÞ‖2 is bounded for all t Z 0. Moreover, we have from (3.3) and (3.14) that V 3 ðtÞ rV ðtÞ r Vð0Þ 2 _ and V 5 ðtÞ r VðtÞ r Vð0Þ, which imply that eαt ‖xðtÞ‖ is bounded. It α t 2 follows from (3.16) that ðd=dtÞðe ‖zij ðtÞ‖ Þ is bounded. Therefore, ∑1 r i o j r N eαt ‖zij ðtÞ‖2 ¼ 0 is uniformly continuous on ½0; þ 1Þ. By Lemma 2.6, we derive that lim
∑
t- þ 11 r i o j r N
eαt ‖zij ðtÞ‖2 ¼ 0;
or limt- þ 1 eαt ‖zij ðtÞ‖2 ¼ limt- þ 1 eαt ‖xi ðtÞ xj ðtÞ‖2 ¼ 0 for 1 r io j rN. Hence, system (3.1) is globally exponentially synchronized. This completes the proof. □ Remark 1. A novel LKF is proposed to remove the traditional restriction on slowly varying delay, that is, hD o 1. Concretely speaking, in the case of fast varying delay, that is, hD Z1, V 4 ðtÞ is ineffective to guarantee the synchronization of system (3.1). Hence, we resort to V 5 ðtÞ in LKF and (3.10) dealing with the time-varying delay, which can effectively solve the problem of the fast varying delay. Remark 2. If the information about hD is unknown, then by eliminating the first term of V 4 ðtÞ in the LKF, we can derive the delay-dependent and delay-derivative-independent synchronization criteria for system (3.1) based on Theorem 3.1. Remark 3. The second term of V 3 ðtÞ is to relax the restriction α o2, and thus reduces the conservative of the synchronization criteria. Remark 4. If we set α ¼ 0 in Theorem 3.1 and the LKF V(t), then the sufficient conditions of global exponential synchronization for system (3.1) reduce to that of global synchronization, which was extensively investigated (e.g., [6,8,11,13,16,18–22,32]). As known, the property of exponential synchronization is particularly important when the exponential convergence rate is used to determine the speed of neural computations. Thus, it is not only of theoretical interest, but also of practical significance to study the exponential synchronization of neural networks. In this sense, our results improve and generalize the previously published results.
In this section, based on Theorem 3.1, we have the following result of robust synchronization of system (2.1). Theorem 4.1. Under assumptions (H1) and (H2), given constant decay rate α Z 0, the dynamical system (2.1) is robustly globally
exponentially synchronized if there exist positive number λ, positive SðkÞ SðkÞ definite matrices P, Q, R1, R2, T, Y A Rnn , SðkÞ ¼ n11 S12 ðk ¼ 1; 2; 3Þ ðkÞ 22
nn SðkÞ pq A R
ðp ¼ 1; 2; q ¼ 1; 2Þ,
positive
diagonal
matrices
J l A Rnn ðl ¼ 1; 2Þ, and real matrices Z l A Rnn ðl ¼ 1; 2Þ, such that the LMI: " # Ωij þ λΦ2 ΦT2 Φ1 o0 ð4:1Þ n λI holds for all 1 r i oj r N, where
Φ1 ¼
½0nð10nÞ M T Z T1
Ωij is defined in (3.2) and
0nn M T Z T2 T ;
Ωij þ Φ1 ∇ðtÞΦT2 þ Φ2 ∇T ðtÞΦT1 o 0:
ð4:2Þ
According to Lemma 2.7, it is easy to know that (4.2) holds if there exists a number λ 4 0 such that the following inequality holds:
Ωij þ λ 1 Φ1 ΦT1 þ λΦ2 ΦT2 o 0:
ð4:3Þ
By Lemma 2.8, the inequality (4.3) is equivalent to the LMI (4.1). Therefore, we can acquire the robust synchronization criteria in Theorem 4.1. The proof is complete. □ Remark 5. In [8,10,37], the synchronization conditions for an array of neural networks with hybrid coupling and interval timevarying delay were investigated. However, the leakage delay and the distributed delay were not taken into account in the systems. In [22], the authors considered the synchronization problem for CNNs with interval time-varying delays and leakage delay, but they neglected the effects of distributed delay, hybrid coupling and the parameter uncertainties. In this paper, we presented novel delay-dependent criteria to guarantee the robust global exponential synchronization of an array of hybrid coupled neural networks with parameter uncertainties, leakage delay, time-varying discrete and distributed delays, which contains the systems studied in [8,10,22,37] as special cases. Therefore, our results effectively complement or extend the earlier ones. Moreover, the conditions are expressed in terms of LMIs. Therefore, by using the LMI Toolbox in Matlab, it is straightforward and convenient to check the feasibility of the proposed results without tuning any parameters. 5. Numerical simulations In this section, we provide several numerical examples to illustrate the feasibility of the proposed method. Example 1. Consider the following 2-neuron delayed neural network without coupling: _ ¼ Axðt τÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt hðtÞÞÞ xðtÞ Z t f ðxðsÞÞ ds þ uðtÞ; þW3 t sðtÞ
4. Robust global exponential synchronization criteria
with
Proof. Consider the same LKF in the proof of Theorem 3.1, and replace A, Wk, Dk with A þM∇ðtÞEA , W k þ M∇ðtÞEW k , Dk þ M∇ðtÞEDk ðk ¼ 1; 2; 3Þ, respectively in (3.2), the corresponding formula of (3.2) for (2.1) can be written as follows:
ð5:1Þ
where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞT A R2 is the state vector of the network, the activation function f ðxðtÞÞ ¼ ðf 1 ðx1 ðtÞÞ; f 2 ðx2 ðtÞÞÞT with f i ðxi Þ ¼ tanhðxi Þ ði ¼ 1; 2Þ, the external input vector uðtÞ ¼ ð0; 0ÞT , the delays τ ¼ 0:2, hðtÞ ¼ sðtÞ ¼ 1 0:5e t , and the connection weight matrices are as follows: 1:8 10 1 0 A¼ ; ; W1 ¼ 0:1 1:8 0 1 1:5 0:1 0:1 0:01 ; W3 ¼ : W2 ¼ 0:1 1:5 0:02 0:1 Through numerical simulations, we find that system (5.1) admits chaotic behavior. The dynamical chaotic behavior of system (5.1) with initial values ϕ1 ðsÞ ¼ 0:5, ϕ2 ðsÞ ¼ 1 is exhibited in Fig. 1. Example 2. Consider the neural network (3.1) with hybrid coupling. Choose the following matrices: 1 2 1 0:5 D1 ¼ ; D2 ¼ ; 0:5 0:5 0:3 0:6
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
x x
6
1 1
0.8
2
0.6
4
0.4
2
x2
0.2
x
0
0 −0.2
−2
−0.4
−4
−0.6
−6 −8
7
−0.8 0
100
200
300
400
−1 −8
500
−6
−4
−2
0
2
4
6
8
x1
t
Fig. 1. (a) Chaotic trajectory of (5.1) with initial values ϕ1 ðsÞ ¼ 0:5 and ϕ2 ðsÞ ¼ 1. (b) Phase portrait of the strange attractor.
10
6 4
5 2 0
−5
ij
−2
eij(t)
x (t)
0
x11 x12
−10
x22 x31
0
5
10
e22 e32
−8 −10
x32 −20
e31
−6
x21 −15
e21
−4
15
−12
0
5
10
t
15
t
Fig. 2. (a) Transient behaviors of the state variables xij(t) ði ¼ 1; 2; 3; j ¼ 1; 2Þ in Example 2. (b) Synchronization errors eij(t) ði ¼ 2; 3; j ¼ 1; 2Þ in Example 2.
D3 ¼
1
2
0:5
1:2
;
2
2
6 GðkÞ ¼ 4 1 1
1 2 1
3
2
7 1 5; 2
6 6 Sð1Þ ¼ 6 4
1
where k ¼ 1; 2; 3. The other functions and matrices are the same as those in (5.1). It is clear that hm ¼ 0:5; hM ¼ 1; hD ¼ 0:5; s ¼ 1; 0:5 0 0 0 Δ1 ¼ ; Δ2 ¼ : 0 0:5 0 0
h ¼ 1;
For a given decay rate α ¼ 0:01, using the LMI Toolbox in Matlab and solving the LMIs in Theorem 3.1, we can obtain a feasible solution 0:4771 0:6184 0:1562 0:1452 P¼ ; Q¼ ; n 0:8316 n 0:3055 5:5076 5:2839 4:0328 1:4142 ; R2 ¼ ; R1 ¼ n 8:6113 n 13:3268 0:3558 0:3137 0:0576 0:0605 ; Y¼ ; T¼ n 0:1364 n 0:7120
Z1 ¼
1:4365
0:8045
0:2651
n
4:3953
0:6892
n
n
1:5378
n
n
0:0119
0:0408
n
;
0:0132 0:1190 2 1:4957 0:5293 6 n 2:8803 6 Sð2Þ ¼ 6 4 n n
Z2 ¼
n
0:1953
n
0:0175
1:3129 0:9035 1:8536
n
n
3
0:1936 7 7 7; 0:5283 5 2:1661
3 0:6243 2:0825 7 7 7; 0:2836 5
n
2:6204
0:7272
0:2985
0:0157 0:5954 2 0:8717 0:2002 6 n 1:5306 6 Sð3Þ ¼ 6 4 n n
0:4715
;
0:4376 1:0585 n
3
1:0313 7 7 7; 0:0724 5 1:4033
J 1 ¼ diagf6:1406; 12:4643g; J 2 ¼ diagf1:3607; 4:2426g:
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
By Theorem 3.1, we see that system (3.1) can achieve global exponential synchronization. Numerical simulations illustrate the fact (Fig. 2). The states of network are shown in Fig. 2(a), and the synchronization errors are illustrated in Fig. 2(b), where eij ðtÞ ¼ xij ðtÞ x1j ðtÞ ði ¼ 2; 3; j ¼ 1; 2Þ. Example 3. Consider the robust synchronization for neural network (1.1) with parameter uncertainties and fast varying delay. Choose hðtÞ ¼ 2 e t and the following matrices: 0:1 0:02 1 0 sin t 0 M¼ ; ; ∇ðtÞ ¼ ; EA ¼ 0:01 0:1 0 1 0 cos t 0:1 0:1 0:1 0 EW i ¼ ; E Di ¼ ; i ¼ 1; 2; 3: 0 0:1 0:1 0:1 The other functions and matrices are the same as those in Examples 1 and 2. It is clear that hm ¼ 1;
hM ¼ 2;
hD ¼ 1;
h ¼ 2:
For a given decay rate α ¼ 0:01, solving the LMIs in Theorem 4.1, we can obtain a feasible solution and we omit it here. By Theorem 4.1, neural network (1.1) is robustly globally exponentially synchronized, and the synchronization performance is illustrated in Fig. 3, where Fig. 3(a) shows the time responses of state vector of
system (1.1), Fig. 3(b) depicts the synchronization errors, where eij ðtÞ ¼ xij ðtÞ x1j ðtÞ ði ¼ 2; 3; j ¼ 1; 2Þ. Example 4. In system (3.1), choose the following matrices and functions: n ¼ 3;
N ¼ 4;
τ ¼ 0:1;
hðtÞ ¼ sðtÞ ¼ 2 e t ;
2
0:2 6 W 1 ¼ 4 0:1 0:2 2
0:1
6 W 3 ¼ 4 0:2 0:3
uðtÞ ¼ ð3; 1; 2ÞT ;
f i ðxi Þ ¼ tanhðxi Þ;
0:1
3
0:1
0:2 7 5; 0:3
0:6 0:2 0:1 0:1 0:1
0:5
3
7 0:2 5;
0:2
A ¼ I3 ;
D1 ¼ 6I 3 ;
2
0:5 6 W 2 ¼ 4 0:1 0:1
0:1
2
3 6 1 6 ðkÞ G ¼6 4 1
0:1
0:3 7 5;
0:3
0:4
1
3
1
1
3
1
1
6
3
1 7 7 7: 1 5
ð5:2Þ
3
e21 e
10
31
4
e22 e
5
32
2
eij(t)
0
x ij(t)
1
For a given decay rate α ¼ 2:5, solving the LMIs in Theorem 3.1, we can obtain a feasible solution. By Theorem 3.1, neural network (3.1) with (5.2) is globally exponentially synchronized, and the synchronization performance is illustrated in Fig. 4.
15
−5
0
x11
−10
x
−2
12
−15
x21
−20
x
−25
3
0:4
1
1
D2 ¼ D3 ¼ I 3 ;
x22
−4
31
x32 0
10
20
30
40
−6
50
0
10
20
t
30
40
50
t
Fig. 3. (a) Transient behaviors of the state variables xij(t) ði ¼ 1; 2; 3; j ¼ 1; 2Þ in Example 3. (b) Synchronization errors eij(t) ði ¼ 2; 3; j ¼ 1; 2Þ in Example 3.
5
x x
4
x
x32
0
x
−1
x x
−2
41 42
x43
−3
31
e22 e
32
e42
2
e23 e33
1
e
0
43
−1 −2 −3
−4 0
33
21
e41
3 e ij(t)
x31
e
4
21
x23
1 xij(t)
12
x22
2
e
5
x13
3
−5
6 11
10
20
30 t
40
50
−4
0
10
20
30
40
50
t
Fig. 4. (a) Transient behaviors of the state variables xij(t) ði ¼ 1; 2; 3; 4; j ¼ 1; 2; 3Þ in Example 4. (b) Synchronization errors eij(t) ði ¼ 2; 3; 4; j ¼ 1; 2; 3Þ in Example 4.
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
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6. Conclusions In this paper, we incorporated leakage delay, time-varying discrete and distributed time delays into an array of neural networks with hybrid coupling and parameter uncertainties. By employing a novel augmented LKF, applying the theory of the Kronecker product of matrices, Barbalat's Lemma and the technique of LMI, delay-dependent sufficient conditions were obtained for the global exponential synchronization of system (3.1). Furthermore, the robust synchronization criteria were derived for system (2.1) with uncertainties both in coefficient matrix terms and in coupling matrix terms. Lastly, some numerical examples were given to illustrate the feasibility and effectiveness of our theoretical results. Our synchronization criteria provide a new, convenient, and efficient approach to study the synchronization for complex neural networks with hybrid coupling and mixed time delays, and some previously published results in the literature are generalized and improved. We would like to point out that one of the effective research methods in the synchronization problem is adaptive control (see [28–31]). Thus it is interesting to discuss the adaptive synchronization in an array of hybrid coupled neural networks with parameter uncertainties and mixed time delays. We leave this for our future work.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11371368, 61305076), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and the Natural Science Foundation of Young Scientist of Hebei Province (No. A2013506012). References [1] Barabási A, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. [2] Watts D, Strogatz S. Collective dynamics of ‘small-world’ networks. Nature 1998;393:440–2. [3] Zhang Y, Gu A, Xu S. Global exponential adaptive synchronization of complex dynamical networks with neutral-type neural network nodes and stochastic disturbances. IEEE Trans Circuits Syst 2013;99:1–10. [4] Yi J, Wang Y, Xiao J, Huang Y. Exponential synchronization of complex dynamical networks with Markovian jump parameters and stochastic delays and its application to multi-agent systems. Commun Nonlinear Sci Numer Simul 2013;18:1175–92. [5] Dong Y, Xian J, Han D. New condition for synchronization in complex networks with multiple time-varying delays. Commun Nonlinear Sci Numer Simul 2013;18:2581–8. [6] Mei J, jiang M, Xu W, Wang B. Finite-time synchronization control of complex dynamical networks with time delay. Commun Nonlinear Sci Numer Simul 2013;18:2462–78. [7] Sun Y, Li W, Ruan J. Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation. Commun Nonlinear Sci Numer Simul 2013;18:989–98. [8] Zhang H, Gong D, Chen B, Liu Z. Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov–Krasovskii functional method. IEEE Trans Neural Netw Learn Syst 2013;24:58–70. [9] Song Q. Synchronization analysis in an array of asymmetric neural networks with time-varying delays and nonlinear coupling. Appl Math Comput 2010;216:1605–13. [10] Cao J, Li L. Cluster synchronization in an array of hybrid coupled neural networks with delay. Neural Netw 2009;22:335–42. [11] Park MJ, Kwona OM, Park JuH, Lee SM, Cha EJ. On synchronization criterion for coupled discrete-time neural networks with interval time-varying delays. Neurocomputing 2013;99:188–96. [12] Wang H, Song Q. Synchronization for an array of coupled stochastic discretetime neural networks with mixed delays. Neurocomputing 2011;74:1572–84.
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[13] Yang X, Cao J, Lu J. Synchronization of Markovian coupled neural networks with nonidentical node-delays and random coupling strengths. IEEE Trans Neural Netw Learn Syst 2012;23:60–71. [14] Zhang W, Tang Y, Miao Q, Du W. Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects. IEEE Trans Neural Netw Learn Syst 2013;24:1316–26. [15] Wu Z, Shi P, Su H, Chu J. Sampled-data exponential synchronization of complex dynamical networks with time-varying coupling delay. IEEE Trans Neural Netw Learn Syst 2013;24:1177–87. [16] Liu Y, Wang Z, Liang J, Liu X. Synchronization of coupled neutral-type neural networks with jumping-mode-dependent discrete and unbounded distributed delays. IEEE Trans Cybern 2013;43:102–14. [17] Wang G, Yin Q, Shen Y. Exponential synchronization of coupled fuzzy neural networks with disturbances and mixed time-delays. Neurocomputing 2013;106:77–85. [18] Yang X, Cao J, Lu J. Synchronization of randomly coupled neural networks with Markovian jumping and time-delay. IEEE Trans Circuits Syst 2013;60:363–76. [19] Wang K, Teng Z, Jiang H. Adaptive synchronization in an array of linearly coupled neural networks with reaction–diffusion terms and time delays. Commun Nonlinear Sci Numer Simul 2012;17:3866–75. [20] Wang Z, Zhang H. Synchronization stability in complex interconnected neural networks with nonsymmetric coupling. Neurocomputing 2013;108:84–92. [21] Tang Y, Wong WK. Distributed synchronization of coupled neural networks via randomly occurring control. IEEE Trans Neural Netw Learn Syst 2013;24: 435–47. [22] Park MJ, Kwon OM, Park JuH, Lee SM, Cha EJ. Synchronization criteria for coupled neural networks with interval time-varying delays and leakage delay. Appl Math Comput 2012;218:6762–75. [23] Balasubramaniam P, Kalpana M, Rakkiyappan R. Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays. Math Comput Modell 2011;53: 839–53. [24] Li X, Rakkiyappan R, Balasubramaniam P. Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations. J Frankl Inst 2011;348: 135–55. [25] Gan Q. Exponential synchronization of stochastic neural networks with leakage delay and reaction–diffusion terms via periodically intermittent control. Chaos 2012;22:013124. [26] Gan Q, Liang Y. Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control. J Frankl Inst 2012;349:1955–71. [27] Gopalsamy K. Stability and oscillations in delay differential equations of population dynamics. Dordrecht: Kluwer Academic Publishers; 1992. [28] Meng D. Robust adaptive synchronization of the energy resource system with constraint. Math Probl Eng 2013, http://dx.doi.org/10.1155/2013/494218. [29] Meng D. Adaptive neural networks synchronization of a four-dimensional energy resource stochastic system, Abstr Appl Anal 2014, http://dx.doi.org/10. 1155/2014/863902. [30] Moreno-Valenzuela J. Adaptive anti control of chaos for robot manipulators with experimental evaluations. Commun Nonlinear Sci Numer Simul 2013;18: 1–11. [31] Loría A, Zavala-Río A. Adaptive tracking control of chaotic systems with applications to synchronization. IEEE Trans Circuits Syst 2007;54:2019–29. [32] Wang T, Zhou W, Zhao S. Robust synchronization for stochastic delayed complex networks with switching topology and unmodeled dynamics via adaptive control approach. Commun Nonlinear Sci Numer Simul 2013;18: 2097–106. [33] Zhang Y, Xu S, Chu Y, Lu J. Robust global synchronization of complex networks with neutral-type delayed nodes. Appl Math Comput 2010;216:768–78. [34] Yang M, Wang Y, Xiao J, Wang HO. Robust synchronization of impulsivecoupled complex switched networks with parametric uncertainties and timevarying delays. Nonlinear Anal Real World Appl 2010;11:3008–20. [35] Graham A. Kronecker products and matrix calculus: with applications. New York: John Wiley & Sons, Inc.; 1982. [36] Gu K, Kharitonov VK, Chen J. Stability of time-delay systems. Boston, MA: Birkhauser; 2003. [37] Cao J, Chen G, Li P. Global synchronization in an array of delayed neural networks with hybrid coupling. IEEE Trans Syst Man Cybern B 2008;38: 488–98. [38] Slotine JE, Li W. Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall; 1991. [39] Balasubramaniam P, Rakkiyappan R, Sathy R. Delay dependent stability results for fuzzy BAM neural networks with Markovian jumping parameters. Expert Syst Appl 2011;38:121–30. [40] Boyd S, Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM; 1994. 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Research Article
Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays Yanke Du n, Rui Xu Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 January 2014 Received in revised form 26 February 2014 Accepted 11 March 2014 This paper was recommended for publication by Dr. Jeff Pieper
This paper is concerned with the global exponential synchronization for an array of hybrid coupled neural networks with leakage delay, time-varying discrete and distributed delays. By employing a novel augmented Lyapunov–Krasovskii functional (LKF), applying the theory of Kronecker product of matrices, Barbalat's Lemma and the technique of linear matrix inequality (LMI), delay-dependent sufficient conditions are obtained for the global exponential synchronization of the system. As an extension, robust synchronization criteria are derived for the corresponding system with parameter uncertainties. Some examples are given to show the effectiveness of the obtained theoretical results. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Array of neural networks Synchronization Hybrid coupling Kronecker product Lyapunov–Krasovskii functional
1. Introduction Since small-world and scale-free complex networks were proposed in [1,2], complex dynamical networks, which are a set of interconnected nodes with specific dynamics, have received increasing attention from various fields of science and engineering such as the World Wide Web, electrical power grids, communication networks, the Internet, and global economic markets. Many interesting behaviors have been observed from complex dynamical networks, e.g., synchronization, consensus, self-organization, and spatiotemporal chaos spiral waves. Synchronization as an important collective behavior of complex dynamical networks has been widely investigated in the last two decades (see, for example, [3–7]). Coupled neural networks (CNNs), as a special kind of complex networks, have been found to exhibit more complicated and unpredictable behaviors than a single neural network. Particularly, synchronization in an array of coupled neural networks, which is one of the hot research fields of complex networks, has been a challenging issue due to its potential applications in many areas such as secure communication, information science, chaos generator design, and harmonic oscillation generation. On the other hand, in the applications of neural networks, there exist unavoidably time delays due to the finite information processing speed
n
Corresponding author. E-mail address: [email protected] (Y. Du).
and the finite switching speed of amplifiers. It is well known that time delay can cause oscillation and instability of neural networks. Therefore, various synchronization criteria for CNNs with time delays have been investigated in the literature [8–21] and references therein. To mention a few representative works, the synchronization problems for an array of neural networks with hybrid coupling and interval time-varying delay were investigated in [8]. In [9], the global exponential synchronization was investigated for an array of asymmetric neural networks with time-varying delays and nonlinear coupling. Cao and Li [10] presented cluster synchronization criteria for an array of hybrid coupled neural networks with delay. Park et al. [11] obtained the delay-dependent synchronization conditions for coupled discrete-time neural networks with interval time-varying delays in network coupling. So far, very little attention has been paid to neural networks with time delay in a leakage (or “forgetting”) term (see [22–26]). This is due to some theoretical and technical difficulties. In fact, time delay in the leakage term also has great impact on the dynamics of neural networks. As pointed out by Gopalsamy [27], time delay in the stabilizing negative feedback term has a tendency to destabilize a system. On the other hand, it has been observed that neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state (see [16,17]). However, to the best of our knowledge, there are no global exponential synchronization
http://dx.doi.org/10.1016/j.isatra.2014.03.005 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
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2
results about an array of hybrid coupled neural networks with leakage delay, time-varying discrete and distributed delays. In the real world, there exist inevitably some uncertainties caused by the existence of modeling errors, external disturbances, and parameter fluctuations, which would lead to complex dynamic behaviors (see [28–31]). Thus, a good neural network should have robustness against such uncertainties. Recently, robust synchronization problems against such uncertainties of complex neural networks were investigated in [32–34] and references therein. Motivated by the works of Zhang et al. [8], Wang et al. [17], Park et al. [22] and the discussions above, the purpose of this paper is to present some new sufficient conditions for the robust global exponential synchronization of hybrid coupled neural networks with leakage delay, time-varying delays and parameter uncertainties. To this end, we consider the following differential equation system: ^ ðt τÞ þ W ^ 1 f ðxi ðtÞÞ þ W ^ 2 f ðxi ðt hðtÞÞÞ x_ i ðtÞ ¼ Ax i Z t N ^ 3 ^ þW f ðxi ðsÞÞ ds þ uðtÞ þ ∑ g ð1Þ ij D 1 xj ðtÞ t sðtÞ
j¼1
N
N
ð3Þ ^ ^ þ ∑ g ð2Þ ij D 2 xj ðt hðtÞÞ þ ∑ g ij D 3 j¼1
j¼1
Z
t t sðtÞ
2. Preliminaries
xj ðsÞ ds;
ð1:1Þ
where i ¼ 1; 2; …; N, N is the number of coupled nodes, xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞT A Rn is the neuron state vector of the ith node, n denotes the number of neurons in a neural network, f ðxi ðÞÞ ¼ ðf 1 ðxi1 ðÞÞ; f 2 ðxi2 ðÞÞ; …; f n ðxin ðÞÞÞT A Rn
and
uðtÞ ¼ ðu1 ðtÞ;
u2 ðtÞ; …; un ðtÞÞT A Rn denote the neuron activation function vector
and the external input vector, respectively, τ, h(t) and sðtÞ denote the leakage delay, time-varying discrete delay and distributed ^ k ¼ W k þ ΔW k ðtÞ, D ^ k ¼ Dk þ delay, respectively, A^ ¼ A þ ΔAðtÞ, W ΔDk ðtÞ ðk ¼ 1; 2; 3Þ, A ¼ diagfa1 ; a2 ; …; an g A Rnn ðak 4 0; k ¼ 1; 2; …; nÞ is the self-feedback matrix, W k A Rnn ðk ¼ 1; 2; 3Þ are the connection weight matrices, Dk A Rnn ðk ¼ 1; 2; 3Þ are the constant inner-coupling matrices of nodes, which describe the individual coupling between networks, ΔAðtÞ, ΔW k ðtÞ and ΔDk ðtÞ represent the parameter uncertainties of the system, which are assumed to be of the form ½ΔAðtÞ ΔW 1 ðtÞ ΔW 2 ðtÞ ΔW 3 ðtÞ ΔD1 ðtÞ ΔD2 ðtÞ ΔD3 ðtÞ ¼ M∇ðtÞ½EA EW 1 EW 2 EW 3 ED1 ED2 ED3 ; in which M, EA, EW k and EDk ðk ¼ 1; 2; 3Þ are known constant matrices and ∇ðtÞ is an unknown matrix satisfying
For convenience, we introduce several notations. Rn is the ndimensional Euclidean space, Rmn denotes the set of m n real matrices, In represents the n-dimensional identity matrix, A 4ð Z Þ0 means that A is a symmetric positive definite (semidefinite) matrix, λmax ðAÞ denotes the largest eigenvalue of A, A B stands for the Kronecker product of matrices A and B, for B ¼ ðBij ÞðmpÞðnqÞ , Bij A Rmn ð1 r i rp; 1 rj r qÞ, A B stands for 2 3 A B11 ⋯ A B1q 6 7 ⋮ ⋱ ⋮ 4 5; A Bp1 ⋯ A Bpq for a vector x ¼ ðx1 ; x2 ; …; xn ÞT A Rn , J x J stands for the Euclidean norm, n denotes the symmetric block in a symmetric matrix. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations. Combined with the sign of the Kronecker product, (1.1) can be rewritten as ^ ^ 1 ÞFðxðtÞÞ þ ðI N W ^ 2 ÞFðxðt hðtÞÞÞ _ ¼ ðI N AÞxðt xðtÞ τÞ þ ðI N W Z t ^ 3Þ ^ 1 ÞxðtÞ þ ðI N W FðxðsÞÞ ds þ UðtÞ þ ðGð1Þ D t sðtÞ
^ 2 Þxðt hðtÞÞ þ ðGð3Þ D ^ 3Þ þ ðGð2Þ D where
xðÞ ¼ ðxT1 ðÞ; xT2 ðÞ; …; xTN ðÞÞT ,
Z
t t sðtÞ
ð2:1Þ
xðsÞ ds;
T
T
FðxðÞÞ ¼ ðf ðx1 ðÞÞ; f ðx2 ðÞÞ; …;
T
∇T ðtÞ∇ðtÞ r I; GðkÞ ¼ ðg ðkÞ ij ÞNN ðk ¼ 1; 2; 3Þ are the out-coupling matrices representing the coupling strength and the topological structure of the networks and satisfying the diffusive coupling connections: ðkÞ g ðkÞ ij ¼ g ji Z 0 ði ajÞ;
matrix. Hence, it can effectively reduce the conservatism of the synchronization criteria. (3) The proposed control method removes the traditional restriction on slowly varying delay. Using this method, even if the information about the derivative of the time-varying delay h(t) is unknown, delay-derivative-independent synchronization criteria can be derived. The organization of this paper is as follows. In Section 2, some preliminaries are given. In Section 3, synchronization criteria are derived for the CNNs without uncertainties. In Section 4, sufficient conditions are presented for the robust synchronization of system (1.1). In Section 5, some numerical examples are provided to illustrate the effectiveness of the obtained theoretical results. A brief remark is given in Section 6 to conclude this work.
g ðkÞ ii ¼
N
∑
j ¼ 1;j a i
g ðkÞ ij ði; j ¼ 1; 2; …; NÞ:
In this paper, the main contributions are given as follows. (1) It is the first time to establish the robust exponential synchronization criteria for an array of hybrid coupled neural networks with parameter uncertainties, leakage delay, timevarying discrete and distributed delays. It is worth pointing out that the addressed system includes many neural network models as its special cases (see [8,10,22,37]) and some valuable mathematical techniques have been employed, which generalize the previous results to some extent. (2) A novel augmented LKF is proposed to analyze the synchronization problem for the coupled neural networks. This method includes a new type of augmented matrices with multiple Kronecker product operations, and thus introduces more relaxed variables to alleviate the requirements of a positive definite
f ðxN ðÞÞÞT , and UðtÞ ¼ ðuT ðtÞ; uT ðtÞ; …; uT ðtÞÞT . Throughout this paper, we make the following assumptions. (H1) The time delays satisfy
τ Z 0;
0 r hm r hðtÞ rhM ;
_ rh ; hðtÞ D
0 r sðtÞ r s;
where hm, hM, hD and s are known constant scalars. þ (H2) For any y1 ; y2 A R, there exist constants δr and δr , such that the activation functions satisfy
δr r
f r ðy1 Þ f r ðy2 Þ þ r δr ; y1 y2
r ¼ 1; 2; …; n:
We denote ( n o hM if hD r 1 þ þ h¼ ; Δ1 ¼ diag δ1 δ1 ; …; δn δn ; hm if hD 4 1 ( ) δ þ δ1þ δ þ δnþ ; …; n Δ2 ¼ diag 1 : 2 2 Next, we give some useful definitions and lemmas. Definition 2.1. The CNNs (1.1) are said to be robustly globally exponentially synchronized if for all admissible uncertainties ΔAðtÞ, ΔW k ðtÞ, ΔDk ðtÞ ðk ¼ 1; 2; 3Þ, there exist η 4 0 and M0 4 0
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
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such that for any initial values ϕi ðsÞ and sufficiently large T 4 0 J xi ðtÞ xj ðtÞ J rM 0 e
ηt
holds for all t 4 T, and for any i; j ¼ 1; 2; …; N. M0 and η are called the decay coefficient and the decay rate, respectively. Lemma 2.1 (Kronecker product, Graham [35]). The following properties of Kronecker product are satisfied: 1. ðαAÞ B ¼ A ðαBÞ, ðαAÞ B ¼ A ðαBÞ , where α is a constant; 2. A ðB þ CÞ ¼ A B þ A C, A ðB þ CÞ ¼ A B þA C ; 3. ðA BÞðC DÞ ¼ ðACÞ ðBDÞ.
3
Lemma 2.6 (Barbalat's Lemma, Slotine and Li [38]). Let f be a nonnegative function defined on ½0; þ 1Þ. If f is Lebesgue integrable on ½0; þ 1Þ and is uniformly continuous on ½0; þ 1Þ, then limt- þ 1 f ðtÞ ¼ 0. Lemma 2.7 (Balasubramaniam et al. [39]). Assume that Ψ ¼ Ψ , H and E are real matrices with appropriate dimensions, and F(t) is a matrix satisfying F T ðtÞFðtÞ rI. Then, Ψ þ HFðtÞE þ ½HFðtÞET o 0 holds if and only if there exists a scalar ε 4 0 satisfying Ψ þ ε 1 HH T þ εET E o 0. T
Lemma 2.8 (Schur complement, Boyd et al. [40]). Given constant
Ωi ði ¼ 1; 2; 3Þ, where Ω1 ¼ ΩT1 and Ω2 40, then Ω1 þ ΩT3 Ω2 1 Ω3 o 0 if and only if ½Ω ΩΩ o0.
matrices Lemma 2.2 (Jensen inequality, Gu et al. [36]). For any matrix W 4 0, scalars γ1 and γ2 satisfying γ 2 4 γ 1 , a vector function ω : ½γ 1 ; γ 2 -Rn such that the integrations concerned are well defined, one has "Z #T Z Z ðγ 2 γ 1 Þ
γ2
γ1
ωT ðsÞW ωðsÞ ds Z
γ2
γ1
ωðsÞ ds W
γ2
γ1
ωðsÞ ds:
According to the Jensen inequality, one can easily obtain the following lemma. Lemma 2.3. For any constant matrix M 40, the following inequality holds: " #T # " Z t xðtÞ xðtÞ M M T _ _ hðtÞ x ðsÞM xðsÞ ds r : xðt hðtÞÞ xðt hðtÞÞ n M t hðtÞ
1
T 3
2
3. Global exponential synchronization criteria In this section, we first consider the following system without uncertainties: _ ¼ ðI N AÞxðt τÞ þ ðI N W 1 ÞFðxðtÞÞ þ ðI N W 2 ÞFðxðt hðtÞÞÞ xðtÞ Z t FðxðsÞÞ ds þ UðtÞ þ ðGð1Þ D1 ÞxðtÞ þ ðI N W 3 Þ t sðtÞ
þ ðGð2Þ D2 Þxðt hðtÞÞ þ ðGð3Þ D3 Þ
Z
t t sðtÞ
xðsÞ ds;
ð3:1Þ
where the notations are the same as those in (1.1) and (2.1). Lemma 2.4 (Cao et al. [37]). Let U ¼ ðuij ÞNN , P A Rnn , x ¼ ðxT1 ; …; xTN ÞT and y ¼ ðyT1 ; …; yTN ÞT with xi ; yi A Rn ði ¼ 1; 2; …; NÞ. If U ¼ U T and each row sum of U is zero, then xT ðU PÞy ¼
∑
1riojrN
uij ðxi xj ÞT Pðyi yj Þ:
22
Lemma 2.5 (Zhang et al. [8]). For Q ¼ ðP pq ÞðnlÞðnlÞ , P pq A Rnn T T w T w T T ðwÞ ð1 r p r l; 1 r q r lÞ, xðwÞ ¼ ððxw ¼ ððyw 1 Þ ; ðx2 Þ ; …; ðxN Þ Þ and y 1Þ ;
2
Ξ 11 Ξ 12 Ξ 13 Ξ 14 6 n Ξ 22 0 0 6 6 6 n n Ξ P 33 6 6 n n n Ξ 44 6
6 6 6 6 6 6 Ωij ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0
0 0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 T
0 0
0 0
0
0
Ξ 55 Ξ 56 n Ξ 66
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Ξ 77 Ξ 78 n Ξ 88 n
n
0 0
0
0
0
0
T
0
0
0
0
Ξ 7;11 Ξ 8;11
Ξ 99
Sð3Þ 12
0
0
0
0
0
n
n
n
n
n
n
n
n
n
Sð3Þ 22
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Y
n
n
n
n
n
n
n
n
n
n
n
n
T
with
xw k,
n yw k AR
ðk ¼ 1; 2; …; N; w ¼ 1; 2…; lÞ,
N
e ¼ ð1; 1; …; 1Þ A R , EN ¼ eeT , and U ¼ NI N EN , the following equality can be obtained: 2 ð1Þ 3T 2 ð1Þ 3 3 xi xjð1Þ yi yð1Þ yð1Þ x j 6 7 6 7 6 7 7 Q 6 6 7 Q6 7: ⋮ ⋮ ∑ 4 ⋮ 5 ðU Þ4 ⋮ 5 ¼ 4 5 4 5 ðlÞ ðlÞ ðlÞ ðlÞ 1 r i o j r N ðlÞ ðlÞ x y x y y x i j i j 3 ð1Þ T
q ¼ 1; 2Þ, positive diagonal matrices J l A Rnn ðl ¼ 1; 2Þ, and real matrices Z l A Rnn ðl ¼ 1; 2Þ, such that the LMI Ωij o0 holds for all 1 r i oj r N, where
Ξ 1;11 Ξ 2;11 Ξ 3;11
0
n
T w T T ðyw 2 Þ ; …; ðyN Þ Þ
2
Theorem 3.1. Under assumptions (H1) and (H2), given constant decay rate α Z 0, the dynamical system (3.1) is globally exponentially synchronized if there exist positive definite matrices P, Q, R1, R2, T, SðkÞ SðkÞ nn ðk ¼ 1; 2; 3Þ with SðkÞ ðp ¼ 1; 2; Y A Rnn , SðkÞ ¼ n11 S12 ðkÞ pq A R
2
Ξ 11;11 Ξ 11;12
Ξ 1;13 Ξ 2;13 Ξ 3;13
3
7 7 7 7 7 P 7 7 7 0 7 7 0 7 7 7 Ξ 7;13 7 7; Ξ 8;13 7 7 7 0 7 7 7 0 7 7 7 Ξ 11;13 7 7 Ξ 12;13 7 5
ð3:2Þ
Ξ 13;13
αhm Ξ 11 ¼ ð2 αÞP R2 þ eατ Q þ τ2 eατ R1 þ eαh Sð1Þ 11 þe ð2Þ α h ð1Þ α hm ð2Þ S11 J 1 Δ1 , Ξ 12 ¼ e S12 þ e S12 þ J 1 Δ2 , Ξ 13 ¼ P þ R2 , Ξ 14 ¼ ð1Þ T T T T αh ð1Þ ð1 αÞP, Ξ 1;11 ¼ NGð1Þ ij D1 Z 1 , Ξ 1;13 ¼ P NGij D1 Z 2 , Ξ 22 ¼ e S22
in
which
T T T T 2 αs þ eαhm Sð2Þ 22 þ s e Y J 1 , Ξ 2;11 ¼ W 1 Z 1 , Ξ 2;13 ¼ W 1 Z 2 , Ξ 33 ¼ Q
R2 , Ξ 3;11 ¼ AT Z T1 , Ξ 3;13 ¼ AT Z T2 , Ξ 44 ¼ αP R1 , Ξ 55 ¼ Sð2Þ 11 þ ð2Þ αðhM hm Þ Sð3Þ , Ξ ¼ Sð2Þ þ eαðhM hm Þ eαðhM hm Þ Sð3Þ 66 11 T, Ξ 56 ¼ S12 þ e 12 22
Sð3Þ 22 ,
Ξ 77 ¼ ð1 hD ÞSð1Þ 11 2 T J 2 Δ1 ,
Ξ 78 ¼ ð1 hD ÞSð1Þ 12 þJ 2 Δ2 ,
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
ð2Þ T T ð1Þ T T Ξ 7;11 ¼ NGð2Þ ij D2 Z 1 , Ξ 7;13 ¼ NGij D2 Z 2 , Ξ 88 ¼ ð1 hD ÞS22 J 2 ,
Ξ 8;11 ¼
ð3Þ 99 ¼ S11 T, 11;11 ð3Þ T T 11;12 ¼ Z 1 W 3 , 11;13 ¼ Z 1 NGij D3 Z 2 , 2 ατ e R2 þ ðhM hm Þ2 eαhM T Z 2 Z T2 .
W T2 Z T1 ,
Ξ
T T 8;13 ¼ W 2 Z 2 ,
þDT3 Z T1 Þ, Ξ
Ξ 13;13 ¼ τ
Ξ
Ξ
Ξ
¼ NGijð3Þ ðZ 1 D3 T T 12;13 ¼ W 3 Z 2 ,
Ξ
Proof. Define zij ðÞ ¼ xi ðÞ xj ðÞ, gðzij ðÞÞ ¼ f ðxi ðÞÞ f ðxj ðÞÞ, z_ ij ðÞ ¼ x_ i ðÞ x_ j ðÞ, and denote 2 3 N1 1 ⋯ 1 6 1 N1 ⋯ 1 7 6 7 : U¼6 7 4 ⋮ ⋮ ⋯ ⋮ 5 1
1
⋯
N 1
eαt
"
6
ð3:3Þ
tτ
#T "
xðtÞ
Z ðU R1 Þ
xðuÞ du
Z "
t tτ
T Z zij ðsÞ ds R1 #T "
zij ðtÞ zij ðt τÞ "
V_ 4 ðtÞ reαðt þ hÞ
t t τ
R2
V 1 ðtÞ ¼ eαt xðtÞ Z V 2 ðtÞ ¼ V 3 ðtÞ ¼ τ
t
Z
t t τ
T Z xðsÞ ds ðU PÞ xðtÞ
t
t τ
xðsÞ ds ;
n
xðtÞ
#T
tτ t
Z
tτ T
t
FðxðtÞÞ
" þ eαðt þ hm Þ " eαt
V 6 ðtÞ ¼ s
0
t hM
Z
s
t
"
ð1 hD Þ
s
eαðu þ sÞ F T ðxðuÞÞðU YÞFðxðuÞÞ du ds:
Calculating the time derivative of Vk(t) ð1 rk r6Þ, and applying Lemmas 2.1–2.5, one can obtain T Z t Z t xðsÞ ds ðU PÞ xðtÞ xðsÞ ds V_ 1 ðtÞ ¼ αeαt xðtÞ tτ
þ2eαt xðtÞ ¼ eαt
∑
1riojrN
Z
t τ
T _ xðtÞ þ xðt τÞ xðsÞ ds ðU PÞ½xðtÞ
Z zij ðtÞ
Z ðαPÞ zij ðtÞ
t tτ
t t τ
#
FðxðtÞÞ
#9 = ; zij ðt τÞ ; zij ðtÞ
xðtÞ
ð3:6Þ
#
FðxðtÞÞ
Fðxðt hðtÞÞÞ
#T
"
xðtÞ
ð2Þ
ðU S Þ #T
"
#T
xðt hðtÞÞ
ðU
Sð1Þ
Þ
xðt hðtÞÞ
#
Fðxðt hðtÞÞÞ
#
FðxðtÞÞ "
ð2Þ
ðU S Þ
xðt hm Þ
#
" ð1 hD Þ
T zij ðsÞ ds
zij ðsÞ ds þ ð2PÞ½z_ ij ðtÞ zij ðtÞ þ zij ðt τÞ ;
xðt hðtÞÞ
#T
" Sð1Þ
xðt hðtÞÞ
#
ðU Þ Fðxðt hðtÞÞÞ Fðxðt hðtÞÞÞ " #T " # xðt hm Þ xðt hm Þ ð2Þ αðhM hm Þ Sð3Þ Þ ½U ðS e Fðxðt hm ÞÞ Fðxðt hm ÞÞ " #T " #9 xðt hM Þ xðt hM Þ = ð3Þ ðU S Þ Fðxðt hM ÞÞ Fðxðt hM ÞÞ ; 8" #T " # < zij ðtÞ zij ðtÞ α t α h ð1Þ α hm ð2Þ ∑ ðe S þ e S Þ ¼e gðzij ðtÞÞ gðzij ðtÞÞ 1 r i o j r N:
tτ
t
xðtÞ
xðt hm Þ
"
tþs
xðtÞ
Fðxðt hm ÞÞ Fðxðt hm ÞÞ " #T " # xðt hm Þ xðt hm Þ Sð3Þ α ðt þ hM hm Þ þe ðU Þ Fðxðt hm ÞÞ Fðxðt hm ÞÞ " #T " # xðt hM Þ xðt hM Þ ð3Þ ðU S Þ eαt Fðxðt hM ÞÞ Fðxðt hM ÞÞ 8" #T " # < xðtÞ xðtÞ ðeαh Sð1Þ þ eαhm Sð2Þ Þ α t ½U re FðxðtÞÞ : FðxðtÞÞ
eαðu þ τÞ ½xT ðuÞðU R1 ÞxðuÞ
s
_ du ds; þ x_ ðuÞðU R2 ÞxðuÞ " #T " # Z t xðsÞ xðsÞ ð1Þ eαðs þ hÞ ðU S Þ V 4 ðtÞ ¼ ds FðxðsÞÞ FðxðsÞÞ t hðtÞ " #T " # Z t xðsÞ xðsÞ ð2Þ eαðs þ hm Þ ðU S Þ þ ds FðxðsÞÞ FðxðsÞÞ t hm " #T " # Z t hm xðsÞ xðsÞ Sð3Þ α ðs þ hM Þ e ðU Þ þ ds; FðxðsÞÞ FðxðsÞÞ t hM Z t hm Z t _ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du ds; V 5 ðtÞ ¼ ðhM hm Þ Z
#"
U R2
#"
ð1Þ
ðU S Þ "
eαðs þ τÞ xT ðsÞðU Q ÞxðsÞ ds;
Z
xðuÞ du
zij ðsÞ ds
R2 R2
ð1 hD Þeαðt hðtÞ þ hÞ
t τ
ðU R2 Þ
k¼1
where
t
n ðU R2 Þ xðt τÞ xðt τÞ 8 < ∑ τ2 eατ ½zTij ðtÞR1 zij ðtÞ þ z_ Tij ðtÞR2 z_ ij ðtÞ ¼ eαt 1 r i o j r N:
In order to tackle the synchronization problem of (3.1), we introduce the following Lyapunov–Krasovskii functional: VðtÞ ¼ ∑ V k ðtÞ;
T
t
þ eαt
þ
NN
Z
zij ðt hðtÞÞ
#T
" ð1Þ
zij ðt hðtÞÞ
#
S gðzij ðt hðtÞÞÞ gðzij ðt hðtÞÞÞ " #T " # zij ðt hm Þ zij ðt hm Þ ðSð2Þ eαðhM hm Þ Sð3Þ Þ gðzij ðt hm ÞÞ gðzij ðt hm ÞÞ " #T " #9 zij ðt hM Þ = zij ðt hM Þ Sð3Þ ; gðzij ðt hM ÞÞ gðzij ðt hM ÞÞ ;
ð3:7Þ
ð3:4Þ V_ 2 ðtÞ ¼ eαt ½eατ xT ðtÞðU Q ÞxðtÞ xT ðt τÞðU Q Þxðt τÞ ¼ eαt
∑
1riojrN
½zTij ðtÞðeατ Q Þzij ðtÞ zTij ðt τÞQzij ðt τÞ;
ð3:5Þ
_ V_ 3 ðtÞ ¼ τ2 eαðt þ τÞ ½xT ðtÞðU R1 ÞxðtÞ þ x_ T ðtÞðU R2 ÞxðtÞ Z t T _ eαðu þ τÞ ½xT ðuÞðU R1 ÞxðuÞ þ x_ ðuÞðU R2 ÞxðuÞ du τ t τ
_ r τ2 eαðt þ τÞ ½xT ðtÞðU R1 ÞxðtÞ þ x_ T ðtÞðU R2 ÞxðtÞ
_ V_ 5 ðtÞ ¼ ðhM hm Þ2 eαðt þ hM Þ x_ T ðtÞðU TÞxðtÞ Z t _ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du þ ðhM hm Þ Z ðhM hm Þ
t hm t
t hM
_ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du
_ ¼ ðhM hm Þ2 eαðt þ hM Þ x_ T ðtÞðU TÞxðtÞ Z t hðtÞ _ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du ðhM hm Þ t hM
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Z ðhM hm Þ
t hm t hðtÞ
Applying Lemmas 2.1 and 2.4, and noting that ðU Z i ÞUðtÞ ¼ 0, and UX¼ NX for matrix X with zero column sums, we have from (3.11) that
_ eαðu þ hM Þ x_ T ðuÞðU TÞxðuÞ du
_ r ðhM hm Þ2 eαðt þ hM Þ x_ T ðtÞðU TÞxðtÞ Z t hðtÞ _ x_ T ðuÞðU TÞxðuÞ du ðhM hðtÞÞeαt ðhðtÞ hm Þeαt
Z
0 ¼ 2eαt
t hm
_ du x_ T ðuÞðU TÞxðuÞ
_ r ðhM hm Þ e x ðtÞðU TÞxðtÞ 8" #" # " # < xðt hðtÞÞ T ðU TÞ UT xðt hðtÞÞ þ eαt n ðU TÞ xðt hM Þ : xðt hM Þ
þ
xðt hm Þ
#T "
UT
n
ðU TÞ
xðt hðtÞÞ
#9 xðt hm Þ =
#"
ðU TÞ
xðt hðtÞÞ ;
6 4
39 zij ðt hm Þ > 0 = 6 7 U T 54 zij ðt hðtÞÞ 7 5 ; > zij ðt hM Þ ; ðU TÞ 32
ðU TÞ
UT
n
2ðU TÞ
n
n
t sðtÞ
_ U ðZ 1 AÞxðt τÞþ U ðZ 1 W 1 ÞFðxðtÞÞ xT ðsÞ ds ðU Z 1 ÞxðtÞ
þ ðNGð2Þ Þ ðZ 1 D2 Þxðt hðtÞÞ þ ðNGð3Þ Þ ðZ 1 D3 Þ
ð3:8Þ
Z
Z
¼ 2eαt
∑
1riojrN
t sðtÞ
zTij ðsÞ
Z
t t sðtÞ
ð3Þ NGð2Þ ij Z 1 D2 zij ðt hðtÞÞ NGij Z 1 D3
þ z_ Tij ðtÞ
t sðtÞ
¼ eαt
∑
T Z FðxðsÞÞ ds ðU YÞ
1riojrN
Z
t
t sðtÞ
T Z Y
t t sðtÞ
)
t
Z
V_ ðtÞ r eαt ) ð3:9Þ
According to the method of [41,42] and assumption (H2), for any diagonal matrices 0 o J i A Rnn ði ¼ 1; 2Þ, we have 8" #" # " # < zij ðtÞ T J Δ1 J Δ2 zij ðtÞ 1 1 eα t gðzij ðtÞÞ n J 1 : gðzij ðtÞÞ #" " #T " #9 zij ðt hðtÞÞ = zij ðt hðtÞÞ J 2 Δ1 J 2 Δ2 Z 0: þ gðzij ðt hðtÞÞÞ gðzij ðt hðtÞÞÞ ; n J2 ð3:10Þ On the other hand, from (3.1), it can be readily seen that the following equation always holds for any matrices Z i A Rnn ði ¼ 1; 2Þ: Z t _ 0 ¼ 2eαt xT ðsÞ dsðU Z 1 Þ þ x_ T ðtÞðU Z 2 Þ xðtÞ t sðtÞ
ðI N AÞxðt τÞ þ ðI N W 1 ÞFðxðtÞÞ þ ðI N W 2 ÞFðxðt hðtÞÞÞ Z t FðxðsÞÞ ds þ UðtÞ þ ðGð1Þ D1 ÞxðtÞ þ ðI N W 3 Þ þ ðG
ð2Þ
D2 Þxðt hðtÞÞ þ ðG
ð3Þ
Z D3 Þ
t t sðtÞ
t sðtÞ
xðsÞ ds
t
t sðtÞ
zij ðsÞ ds
t t sðtÞ
gðzij ðsÞÞ ds NGð1Þ ij Z 2 D1 xðtÞ
Z
t
t sðtÞ
zij ðsÞ ds
ð3:12Þ
:
Z
t tτ
zTij ðsÞ ds; zTij ðt hm Þ; gT ðzij ðt hm ÞÞ; zTij ðt hðtÞÞ; Z
xðsÞ ds :
Z
t
t sðtÞ
zTij ðsÞ ds;
t t sðtÞ
iT gT ðzij ðsÞÞ ds; z_ Tij ðtÞ :
From (3.2) to (3.10), (3.12) and the conditions of Theorem 3.1, we obtain that
FðxðsÞÞ ds
gðzij ðsÞÞ ds :
t sðtÞ
t
Z 2 z_ ij ðtÞ Z 2 Azij ðt τÞ þ Z 2 W 1 gðzij ðtÞÞ
s2 eαs g T ðzij ðtÞÞYgðzij ðtÞÞ
gðzij ðsÞÞ ds
Z
gðzij ðsÞÞ ds NGð1Þ ij Z 1 D1 zij ðtÞ
Z
g T ðzij ðt hðtÞÞÞ; zTij ðt hM Þ; gT ðzij ðt hM ÞÞ;
t sðtÞ
FðxðsÞÞ ds þðNGð1Þ Þ ðZ 2 D1 ÞxðtÞ
ds Z 1 z_ ij ðtÞ Z 1 Azij ðt τÞ þZ 1 W 1 gðzij ðtÞÞ
þ Z 1 W 2 gðzij ðt hðtÞÞÞ þZ 1 W 3
ξij ðtÞ ¼ zTij ðtÞ; gT ðzij ðtÞÞ; zTij ðt τÞ;
t sðtÞ
t
xðsÞ ds
Let
r eαt s2 eαs F T ðxðtÞÞðU YÞFðxðtÞÞ Z
t
t
t s
t sðtÞ
t sðtÞ
þ ðNGð2Þ Þ ðZ 2 D2 Þxðt hðtÞÞ þ ðNGð3Þ Þ ðZ 2 D3 Þ
NGð2Þ Z 2 D2 zij ðt hðtÞÞ NGð3Þ Z 2 D3 ij ij
r s2 eαðt þ sÞ F T ðxðtÞÞðU YÞFðxðtÞÞ Z t F T ðxðsÞÞðU YÞFðxðsÞÞ ds sðtÞeαt
t
_ U ðZ 2 AÞxðt τÞ þU ðZ 2 W 1 ÞFðxðtÞÞ þ x_ ðtÞ ðU Z 2 ÞxðtÞ
þ Z 2 W 2 gðzij ðt hðtÞÞÞ þZ 2 W 3
V_ 6 ðtÞ ¼ s2 eαðt þ sÞ F T ðxðtÞÞðU YÞFðxðtÞÞ Z t eαðs þ sÞ F T ðxðsÞÞðU YÞFðxðsÞÞ ds s
Z
T
þ U ðZ 2 W 2 ÞFðxðt hðtÞÞÞ þU ðZ 2 W 3 Þ
8 2 3T > zij ðt hm Þ > < 6 7 2 α hM _ T α t ∑ ðhM hm Þ e z ij ðtÞT z_ ij ðtÞ þ 4 zij ðt hðtÞÞ 5 ¼e > 1 r i o j r N> : zij ðt hM Þ 2
t
t sðtÞ
2 αðt þ hM Þ _ T
"
Z
þ U ðZ 1 W 2 ÞFðxðt hðtÞÞÞþ U ðZ 1 W 3 Þ Z t FðxðsÞÞ ds þ ðNGð1Þ Þ ðZ 1 D1 ÞxðtÞ
t hM
t hðtÞ
5
ð3:11Þ
∑
1riojrN
ξTij ðtÞΩij ξij ðtÞ o0;
ð3:13Þ
where Ωij is defined in (3.2). Denote ρ ¼ max1 r i o j r N fλmax ðΩij Þg. Obviously, ρ o 0 and it then follows from (3.13) that V_ ðtÞ r ρeαt
∑
1riojrN
ξTij ðtÞξij ðtÞ r ρeαt
Therefore, we have Z t V_ ðsÞ ds rVð0Þ þ ρ VðtÞ ¼ V ð0Þ þ 0
‖zij ðtÞ‖2 :
∑
1riojrN
Z ∑
1riojrN
0
t
eαs ‖zij ðsÞ‖2 ds: ð3:14Þ
From (3.14) and the fact that V ðtÞ Z 0 and ρ o 0, we get Z t Vð0Þ ; eαs ‖zij ðsÞ‖2 ds r ∑ jρj 1riojrN 0
ð3:15Þ
which implies that Z þ1 ∑ eαs ‖zij ðsÞ‖2 ds o þ 1: 1riojrN
0
Thus, ∑1 r i o j r N eαt ‖zij ðtÞ‖2 is Lebesgue integrable. On the other hand, d αt d ðe ‖zij ðtÞ‖2 Þ ¼ ðeαt ‖xi ðtÞ xj ðtÞ‖2 Þ dt dt ¼ αeαt ‖xi ðtÞ xj ðtÞ‖2 þ 2eαt ½xi ðtÞ xj ðtÞT ½x_ i ðtÞ x_ j ðtÞ
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
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6
Φ2 ¼ ½ NGijð1Þ ED1 EW 1 EA 0nð3nÞ NGð2Þ ij ED2 E W 2 0nð2nÞ
r αeαt ‖xi ðtÞ xj ðtÞ‖2 þ2eαt J xi ðtÞ xj ðtÞ J ð J x_ i ðtÞ J þ J x_ j ðtÞ J Þ
NGijð3Þ ED3 EW 3 0nn T :
r αeαt ‖xi ðtÞ xj ðtÞ‖2 þeαt ½‖xi ðtÞ xj ðtÞ‖2 þ ð J x_ i ðtÞ J þ J x_ j ðtÞ J Þ2 r αeαt ‖xi ðtÞ xj ðtÞ‖2 þeαt ½‖xi ðtÞ xj ðtÞ‖2 þ 2ð‖x_ i ðtÞ‖2 þ ‖x_ j ðtÞ‖2 Þ:
ð3:16Þ
From (3.15), we see that eαt ‖xi ðtÞ xj ðtÞ‖2 is bounded for all t Z 0. Moreover, we have from (3.3) and (3.14) that V 3 ðtÞ rV ðtÞ r Vð0Þ 2 _ and V 5 ðtÞ r VðtÞ r Vð0Þ, which imply that eαt ‖xðtÞ‖ is bounded. It α t 2 follows from (3.16) that ðd=dtÞðe ‖zij ðtÞ‖ Þ is bounded. Therefore, ∑1 r i o j r N eαt ‖zij ðtÞ‖2 ¼ 0 is uniformly continuous on ½0; þ 1Þ. By Lemma 2.6, we derive that lim
∑
t- þ 11 r i o j r N
eαt ‖zij ðtÞ‖2 ¼ 0;
or limt- þ 1 eαt ‖zij ðtÞ‖2 ¼ limt- þ 1 eαt ‖xi ðtÞ xj ðtÞ‖2 ¼ 0 for 1 r io j rN. Hence, system (3.1) is globally exponentially synchronized. This completes the proof. □ Remark 1. A novel LKF is proposed to remove the traditional restriction on slowly varying delay, that is, hD o 1. Concretely speaking, in the case of fast varying delay, that is, hD Z1, V 4 ðtÞ is ineffective to guarantee the synchronization of system (3.1). Hence, we resort to V 5 ðtÞ in LKF and (3.10) dealing with the time-varying delay, which can effectively solve the problem of the fast varying delay. Remark 2. If the information about hD is unknown, then by eliminating the first term of V 4 ðtÞ in the LKF, we can derive the delay-dependent and delay-derivative-independent synchronization criteria for system (3.1) based on Theorem 3.1. Remark 3. The second term of V 3 ðtÞ is to relax the restriction α o2, and thus reduces the conservative of the synchronization criteria. Remark 4. If we set α ¼ 0 in Theorem 3.1 and the LKF V(t), then the sufficient conditions of global exponential synchronization for system (3.1) reduce to that of global synchronization, which was extensively investigated (e.g., [6,8,11,13,16,18–22,32]). As known, the property of exponential synchronization is particularly important when the exponential convergence rate is used to determine the speed of neural computations. Thus, it is not only of theoretical interest, but also of practical significance to study the exponential synchronization of neural networks. In this sense, our results improve and generalize the previously published results.
In this section, based on Theorem 3.1, we have the following result of robust synchronization of system (2.1). Theorem 4.1. Under assumptions (H1) and (H2), given constant decay rate α Z 0, the dynamical system (2.1) is robustly globally
exponentially synchronized if there exist positive number λ, positive SðkÞ SðkÞ definite matrices P, Q, R1, R2, T, Y A Rnn , SðkÞ ¼ n11 S12 ðk ¼ 1; 2; 3Þ ðkÞ 22
nn SðkÞ pq A R
ðp ¼ 1; 2; q ¼ 1; 2Þ,
positive
diagonal
matrices
J l A Rnn ðl ¼ 1; 2Þ, and real matrices Z l A Rnn ðl ¼ 1; 2Þ, such that the LMI: " # Ωij þ λΦ2 ΦT2 Φ1 o0 ð4:1Þ n λI holds for all 1 r i oj r N, where
Φ1 ¼
½0nð10nÞ M T Z T1
Ωij is defined in (3.2) and
0nn M T Z T2 T ;
Ωij þ Φ1 ∇ðtÞΦT2 þ Φ2 ∇T ðtÞΦT1 o 0:
ð4:2Þ
According to Lemma 2.7, it is easy to know that (4.2) holds if there exists a number λ 4 0 such that the following inequality holds:
Ωij þ λ 1 Φ1 ΦT1 þ λΦ2 ΦT2 o 0:
ð4:3Þ
By Lemma 2.8, the inequality (4.3) is equivalent to the LMI (4.1). Therefore, we can acquire the robust synchronization criteria in Theorem 4.1. The proof is complete. □ Remark 5. In [8,10,37], the synchronization conditions for an array of neural networks with hybrid coupling and interval timevarying delay were investigated. However, the leakage delay and the distributed delay were not taken into account in the systems. In [22], the authors considered the synchronization problem for CNNs with interval time-varying delays and leakage delay, but they neglected the effects of distributed delay, hybrid coupling and the parameter uncertainties. In this paper, we presented novel delay-dependent criteria to guarantee the robust global exponential synchronization of an array of hybrid coupled neural networks with parameter uncertainties, leakage delay, time-varying discrete and distributed delays, which contains the systems studied in [8,10,22,37] as special cases. Therefore, our results effectively complement or extend the earlier ones. Moreover, the conditions are expressed in terms of LMIs. Therefore, by using the LMI Toolbox in Matlab, it is straightforward and convenient to check the feasibility of the proposed results without tuning any parameters. 5. Numerical simulations In this section, we provide several numerical examples to illustrate the feasibility of the proposed method. Example 1. Consider the following 2-neuron delayed neural network without coupling: _ ¼ Axðt τÞ þ W 1 f ðxðtÞÞ þ W 2 f ðxðt hðtÞÞÞ xðtÞ Z t f ðxðsÞÞ ds þ uðtÞ; þW3 t sðtÞ
4. Robust global exponential synchronization criteria
with
Proof. Consider the same LKF in the proof of Theorem 3.1, and replace A, Wk, Dk with A þM∇ðtÞEA , W k þ M∇ðtÞEW k , Dk þ M∇ðtÞEDk ðk ¼ 1; 2; 3Þ, respectively in (3.2), the corresponding formula of (3.2) for (2.1) can be written as follows:
ð5:1Þ
where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞT A R2 is the state vector of the network, the activation function f ðxðtÞÞ ¼ ðf 1 ðx1 ðtÞÞ; f 2 ðx2 ðtÞÞÞT with f i ðxi Þ ¼ tanhðxi Þ ði ¼ 1; 2Þ, the external input vector uðtÞ ¼ ð0; 0ÞT , the delays τ ¼ 0:2, hðtÞ ¼ sðtÞ ¼ 1 0:5e t , and the connection weight matrices are as follows: 1:8 10 1 0 A¼ ; ; W1 ¼ 0:1 1:8 0 1 1:5 0:1 0:1 0:01 ; W3 ¼ : W2 ¼ 0:1 1:5 0:02 0:1 Through numerical simulations, we find that system (5.1) admits chaotic behavior. The dynamical chaotic behavior of system (5.1) with initial values ϕ1 ðsÞ ¼ 0:5, ϕ2 ðsÞ ¼ 1 is exhibited in Fig. 1. Example 2. Consider the neural network (3.1) with hybrid coupling. Choose the following matrices: 1 2 1 0:5 D1 ¼ ; D2 ¼ ; 0:5 0:5 0:3 0:6
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
Y. Du, R. Xu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
x x
6
1 1
0.8
2
0.6
4
0.4
2
x2
0.2
x
0
0 −0.2
−2
−0.4
−4
−0.6
−6 −8
7
−0.8 0
100
200
300
400
−1 −8
500
−6
−4
−2
0
2
4
6
8
x1
t
Fig. 1. (a) Chaotic trajectory of (5.1) with initial values ϕ1 ðsÞ ¼ 0:5 and ϕ2 ðsÞ ¼ 1. (b) Phase portrait of the strange attractor.
10
6 4
5 2 0
−5
ij
−2
eij(t)
x (t)
0
x11 x12
−10
x22 x31
0
5
10
e22 e32
−8 −10
x32 −20
e31
−6
x21 −15
e21
−4
15
−12
0
5
10
t
15
t
Fig. 2. (a) Transient behaviors of the state variables xij(t) ði ¼ 1; 2; 3; j ¼ 1; 2Þ in Example 2. (b) Synchronization errors eij(t) ði ¼ 2; 3; j ¼ 1; 2Þ in Example 2.
D3 ¼
1
2
0:5
1:2
;
2
2
6 GðkÞ ¼ 4 1 1
1 2 1
3
2
7 1 5; 2
6 6 Sð1Þ ¼ 6 4
1
where k ¼ 1; 2; 3. The other functions and matrices are the same as those in (5.1). It is clear that hm ¼ 0:5; hM ¼ 1; hD ¼ 0:5; s ¼ 1; 0:5 0 0 0 Δ1 ¼ ; Δ2 ¼ : 0 0:5 0 0
h ¼ 1;
For a given decay rate α ¼ 0:01, using the LMI Toolbox in Matlab and solving the LMIs in Theorem 3.1, we can obtain a feasible solution 0:4771 0:6184 0:1562 0:1452 P¼ ; Q¼ ; n 0:8316 n 0:3055 5:5076 5:2839 4:0328 1:4142 ; R2 ¼ ; R1 ¼ n 8:6113 n 13:3268 0:3558 0:3137 0:0576 0:0605 ; Y¼ ; T¼ n 0:1364 n 0:7120
Z1 ¼
1:4365
0:8045
0:2651
n
4:3953
0:6892
n
n
1:5378
n
n
0:0119
0:0408
n
;
0:0132 0:1190 2 1:4957 0:5293 6 n 2:8803 6 Sð2Þ ¼ 6 4 n n
Z2 ¼
n
0:1953
n
0:0175
1:3129 0:9035 1:8536
n
n
3
0:1936 7 7 7; 0:5283 5 2:1661
3 0:6243 2:0825 7 7 7; 0:2836 5
n
2:6204
0:7272
0:2985
0:0157 0:5954 2 0:8717 0:2002 6 n 1:5306 6 Sð3Þ ¼ 6 4 n n
0:4715
;
0:4376 1:0585 n
3
1:0313 7 7 7; 0:0724 5 1:4033
J 1 ¼ diagf6:1406; 12:4643g; J 2 ¼ diagf1:3607; 4:2426g:
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
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8
By Theorem 3.1, we see that system (3.1) can achieve global exponential synchronization. Numerical simulations illustrate the fact (Fig. 2). The states of network are shown in Fig. 2(a), and the synchronization errors are illustrated in Fig. 2(b), where eij ðtÞ ¼ xij ðtÞ x1j ðtÞ ði ¼ 2; 3; j ¼ 1; 2Þ. Example 3. Consider the robust synchronization for neural network (1.1) with parameter uncertainties and fast varying delay. Choose hðtÞ ¼ 2 e t and the following matrices: 0:1 0:02 1 0 sin t 0 M¼ ; ; ∇ðtÞ ¼ ; EA ¼ 0:01 0:1 0 1 0 cos t 0:1 0:1 0:1 0 EW i ¼ ; E Di ¼ ; i ¼ 1; 2; 3: 0 0:1 0:1 0:1 The other functions and matrices are the same as those in Examples 1 and 2. It is clear that hm ¼ 1;
hM ¼ 2;
hD ¼ 1;
h ¼ 2:
For a given decay rate α ¼ 0:01, solving the LMIs in Theorem 4.1, we can obtain a feasible solution and we omit it here. By Theorem 4.1, neural network (1.1) is robustly globally exponentially synchronized, and the synchronization performance is illustrated in Fig. 3, where Fig. 3(a) shows the time responses of state vector of
system (1.1), Fig. 3(b) depicts the synchronization errors, where eij ðtÞ ¼ xij ðtÞ x1j ðtÞ ði ¼ 2; 3; j ¼ 1; 2Þ. Example 4. In system (3.1), choose the following matrices and functions: n ¼ 3;
N ¼ 4;
τ ¼ 0:1;
hðtÞ ¼ sðtÞ ¼ 2 e t ;
2
0:2 6 W 1 ¼ 4 0:1 0:2 2
0:1
6 W 3 ¼ 4 0:2 0:3
uðtÞ ¼ ð3; 1; 2ÞT ;
f i ðxi Þ ¼ tanhðxi Þ;
0:1
3
0:1
0:2 7 5; 0:3
0:6 0:2 0:1 0:1 0:1
0:5
3
7 0:2 5;
0:2
A ¼ I3 ;
D1 ¼ 6I 3 ;
2
0:5 6 W 2 ¼ 4 0:1 0:1
0:1
2
3 6 1 6 ðkÞ G ¼6 4 1
0:1
0:3 7 5;
0:3
0:4
1
3
1
1
3
1
1
6
3
1 7 7 7: 1 5
ð5:2Þ
3
e21 e
10
31
4
e22 e
5
32
2
eij(t)
0
x ij(t)
1
For a given decay rate α ¼ 2:5, solving the LMIs in Theorem 3.1, we can obtain a feasible solution. By Theorem 3.1, neural network (3.1) with (5.2) is globally exponentially synchronized, and the synchronization performance is illustrated in Fig. 4.
15
−5
0
x11
−10
x
−2
12
−15
x21
−20
x
−25
3
0:4
1
1
D2 ¼ D3 ¼ I 3 ;
x22
−4
31
x32 0
10
20
30
40
−6
50
0
10
20
t
30
40
50
t
Fig. 3. (a) Transient behaviors of the state variables xij(t) ði ¼ 1; 2; 3; j ¼ 1; 2Þ in Example 3. (b) Synchronization errors eij(t) ði ¼ 2; 3; j ¼ 1; 2Þ in Example 3.
5
x x
4
x
x32
0
x
−1
x x
−2
41 42
x43
−3
31
e22 e
32
e42
2
e23 e33
1
e
0
43
−1 −2 −3
−4 0
33
21
e41
3 e ij(t)
x31
e
4
21
x23
1 xij(t)
12
x22
2
e
5
x13
3
−5
6 11
10
20
30 t
40
50
−4
0
10
20
30
40
50
t
Fig. 4. (a) Transient behaviors of the state variables xij(t) ði ¼ 1; 2; 3; 4; j ¼ 1; 2; 3Þ in Example 4. (b) Synchronization errors eij(t) ði ¼ 2; 3; 4; j ¼ 1; 2; 3Þ in Example 4.
Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
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6. Conclusions In this paper, we incorporated leakage delay, time-varying discrete and distributed time delays into an array of neural networks with hybrid coupling and parameter uncertainties. By employing a novel augmented LKF, applying the theory of the Kronecker product of matrices, Barbalat's Lemma and the technique of LMI, delay-dependent sufficient conditions were obtained for the global exponential synchronization of system (3.1). Furthermore, the robust synchronization criteria were derived for system (2.1) with uncertainties both in coefficient matrix terms and in coupling matrix terms. Lastly, some numerical examples were given to illustrate the feasibility and effectiveness of our theoretical results. Our synchronization criteria provide a new, convenient, and efficient approach to study the synchronization for complex neural networks with hybrid coupling and mixed time delays, and some previously published results in the literature are generalized and improved. We would like to point out that one of the effective research methods in the synchronization problem is adaptive control (see [28–31]). Thus it is interesting to discuss the adaptive synchronization in an array of hybrid coupled neural networks with parameter uncertainties and mixed time delays. We leave this for our future work.
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Please cite this article as: Du Y, Xu R. Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.005i
