Synchronization between two general complex networks with time ...
Neurocomputing 144 (2014) 215–221
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Synchronization between two general complex networks with time-delay by adaptive periodically intermittent pinning control Mo Zhao a,b, Huaguang Zhang a,n, Zhiliang Wang a, Hongjing Liang a a b
College of Information Science and Engineering, Northeastern University, Shenyang 110004, China College of Air Traffic Management, Civil Aviation University of China, Tianjin, 300300, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2014 Received in revised form 11 April 2014 Accepted 23 April 2014 Communicated by S. Arik Available online 6 June 2014
This letter investigates the problem of synchronization between two general delayed dynamical complex networks. Based on the Lyapunov stability theory and the adaptive periodical pinning intermittent control technique, some criteria are obtained for the synchronization between two general dynamical complex networks. Moreover, two typical BA scale complex networks with time-delayed Chua oscillator are chosen to illustrate the effectiveness of the theoretical results. & 2014 Elsevier B.V. All rights reserved.
Keywords: Complex networks Synchronization Pinning control Intermittent control
1. Introduction A complex network is a large set of interconnected nodes and has many applications in almost all the fields of the real world, such as the neural networks, the cellular and metabolic networks, the Word Wide Web, the electrical power grids and the social networks [1–4]. In the former literature, several kinds of network models have been proposed for the purpose of describing the real world more realistic [5,6], such as regular networks, random networks [6], small world networks [1] and scale free networks [3]. Besides these studies of topology structures, synchronization is a significant dynamical behavior of the dynamical elements in the complex networks and has been widely investigated [7–9]. In the coupling networks, there are several kinds of synchronization, for example, complete synchronization [10], phase synchronization [11], cluster synchronization [12,13], partial synchronization [14] and so on. Different from those categories, the phenomenon of synchronization can also be classified into “inner synchronization” [7,8] and “outer synchronization” [15–18]. “Inner synchronization” means a collective behavior of all the nodes within a network, while “outer synchronization” refers to the synchronization occurring between two or more coupled networks regardless of happening of the inner synchronization. In Nature, there are so many examples that can be taken to illustrate the phenomenon of outer synchronization, for instance, the infectious disease spreads between different communities, the avian influenza spreads among domestic
n
Corresponding author. E-mail address: [email protected] (H. Zhang).
http://dx.doi.org/10.1016/j.neucom.2014.04.052 0925-2312/& 2014 Elsevier B.V. All rights reserved.
and wild birds, and the different species development in balance. All these challenging topics show the great importance of researching outer synchronization between coupled networks. Thus, the outer synchronization has attracted more and more attention. In some cases, different complex networks can achieve outer synchronization by themselves. But, there also exists the situation that the networks with identical system parameters cannot synchronize with each other by themselves for different initial values. Thus, many kinds of control techniques have been adopted to make the networks achieve synchronization [19–25]. Among these methods, a pinning control is a special control method of adding controllers to partial of the nodes rather than all of the nodes [26]. The pinning control not only simplifies the coupling topological structure, but also saves the cost [16] through reducing the number of controllers. Thus, its status in engineering application has become more and more important. In the process of controlling, the signal will become weak due to diffusion, so it needs some external control until the strength of the signal reaches an upper level. After that, the external control can be removed in order to reduce the cost. This kind of discontinuous control method is different from the continuous control and is named as an intermittent control. Usually, the control time of intermittent is periodic, and in every period, the time with a controller is denoted as work time and the rest is denoted as rest time (see Fig. 1) [27]. From the perspective of economic costs, the method of pinning control will reduce the number of controllers in the complex networks and the periodically intermittent control will decrease the working time. Therefore, if we combine two kinds of control methods together, the control cost will greatly be saved [28–33].
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M. Zhao et al. / Neurocomputing 144 (2014) 215–221
One Period T
In this paper, we assume that ‖A‖2 ¼ α 4 0 and ρmin denotes
One Period T
the minimum eigenvalue of matrix ðA þ AT Þ=2. G^ is a modifying matrix of G via replacing the diagonal elements gii by ðρmin =αÞg ii . s
h
work time
rest time
work time
T
Choose the matrix G^ ¼ ðG^ þ G^ Þ=2 and its eigenvalues are expressed as λ1 Z λ2 Z …λN . In the following, some basic definition, lemmas, and assumption will be given.
h
rest time
Definition 1. The drive system (1) and the response system (2) are defined to achieve synchronization, if xi ðtÞ yi ðtÞ-0;
Fig. 1. Sketch map of intermittent control.
In this paper, we focus on the problem of outer synchronization between two generalized complex dynamical networks with time delay. By using the adaptive pinning periodically intermittent control, some novel synchronization criteria will be obtained by the Lyapunov stability theory. Numerical simulations are also presented to show the effectiveness of the proposed method. The rest of this paper is organized as follows. In Section 2, some general driver and response complex dynamical network models are introduced, and some necessary preliminaries are given. In Section 3, based on the Lyapunov stability theorem, some pinning controllers are designed to ensure that the driver and response systems with time delay achieve outer synchronization under different situations. In Section 4, some numerical simulations are given to verify the effectiveness of proposed theoretical results and Section 5 gives the conclusion of the paper.
t-1; i ¼ 1; …; N:
ð3Þ
Assumption 1. If there exists constants krs 4 0, the nonlinear function f r ðt; xðtÞ; xðt τÞÞ satisfies the uniform Lipschitz condition with respect to time t, jf r ðt; xi ðtÞ; xi ðt τÞÞ f r ðt; yi ðtÞ; yi ðt τÞÞj n
r ∑ krs ðjxis ðtÞ yis ðtÞj þ jxis ðt τÞ yis ðt τÞjÞ
ð4Þ
s¼1
where 1 rr rn. Lemma 1 (Schur complement, Boyed et al. [34]). The following linear matrix inequality (LMI) ! AðxÞ BðxÞ 4 0; ð5Þ ðBðxÞÞT CðxÞ where AðxÞ ¼ ðAðxÞÞT ; CðxÞ ¼ ðCðxÞÞT is equivalent to one of the following conditions: (a) AðxÞ 4 0 and CðxÞ BðxÞT AðxÞ 1 BðxÞ 4 0; (b) CðxÞ 4 0 and AðxÞ BðxÞCðxÞ 1 BðxÞT 4 0.
2. Network model and preliminaries Here we consider a complex network consisting of N identical linearly and diffusively coupled nodes, and every node in the network is an n dimensional dynamical unit. Then the network model of the drive system is denoted as N
x_ i ðtÞ ¼ Fðt; xi ðtÞ; xi ðt τÞÞ þ c ∑ g ij Axj ðtÞ þ ui ðtÞ; j¼1
i ¼ 1; 2; …; N;
ð1Þ
where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞT A Rn is the state variable of the ith node, Fðt; xi ðtÞ; xi ðt τÞÞ ¼ ðf 1 ðt; xi ðtÞ; xi ðt τÞÞ; f 2 ðt; xi ðtÞ; xi ðt τÞÞ; …; f n ðt; xi ðtÞ; xi ðt τ ÞÞÞT A Rn , Fðt; xi ðtÞ; xi ðt τÞÞ : Rn ⟶Rn is a nonlinear vector valued function describing the dynamics of nodes and c 4 0 is the coupling strength of the whole network. The matrix G ¼ ðg ij Þ A RNN is the outer coupling configuration matrix, in which g ij A R is defined as follows: if there is a coupling from node i to node j ði ajÞ, g ij 4 0; otherwise, g ij ¼ 0. At the same time, the diagonal elements of G are defined as g ii ¼ ∑N j ¼ 1;j a i g ij . The inner coupling matrix A denotes the inner coupling relationship between every two nodes. Compared with the drive system mentioned above, the response complex network is denoted as N
y_ i ðtÞ ¼ f ðt; yi ðtÞ; yi ðt τÞÞ þc ∑ g ij Ayj ðtÞ; j¼1
i ¼ 1; 2; …; N;
ð2Þ
where yi ðtÞ ¼ ðyi1 ðtÞ; yi2 ðtÞ; …; yin ðtÞÞT A Rn denotes the state variables of the response system, and other parameters involved in the system (2) all have the same meanings with the corresponding parameters in system (1). Remark 1. In the drive and response systems above, the outer coupling configuration matrix G need not to be symmetric or irreducible.
Lemma 2 (Halanay [35]). Let w : ½μ τ ; 1Þ⟶½0; 1Þ be a continuous function such that _ r awðtÞ þ b max wt wðtÞ holds for t Z μ. If a 4 b 40, then wðtÞ r ½max wμ expf γ ðt τ Þg;
t Zμ
where max wt ¼ supt τ r θ r t wðθÞ, and γ 4 0 is the smallest real root of the equation a γ b expfγτg ¼ 0: Lemma 3 (Xia and Cao [31]). Let w : ½μ τ; 1Þ⟶½0; 1Þ be a continuous function such that _ r awðtÞ þ b max wt wðtÞ holds for t Z μ. If a 4 b 40, then wðtÞ r max wt r½max wμ expfða þ bÞðt τÞg;
t Zμ
where max wt ¼ supt τ r θ r t wðθÞ. 3. Main results In this part, in order to realize outer synchronization between the drive and response systems by adaptive pinning periodically intermittent control, some controllers are needed to add on partial nodes of the network. Denote the error expression as ei ðtÞ ¼ xi ðtÞ yi ðtÞ;
i ¼ 1; 2; …; N:
ð6Þ
Here, select the first l nodes to be pinned and the adaptive pinning periodically intermittent control ui is designed as follows: 8 t A ½mT; mT þhÞ; > < ki ðtÞei ðtÞ; 1 ri r l; l þ 1 ri r N; t A ½mT; mT þhÞ; ui ðtÞ ¼ 0; ð7Þ > : 0; 1 ri r N; t A ½mT þ h; ðm þ 1ÞTÞ
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
the updating laws ( αi expða1 tÞ‖ei ðtÞ‖22 ; k_ i ðtÞ ¼ 0;
t A ½mT; mT þ hÞ; t A ½mT þ h; ðm þ1ÞTÞ
ð8Þ
N
e_ i ðtÞ ¼ F ðt; ðt τÞÞ þ c ∑ g ij Aej ðtÞ; j¼1
i ¼ 1; 2; …; N:
ð10Þ
n
n
ε ð1 εÞ
i¼1r ¼1s¼1 N
N
n
n
ε ð1 εÞ
eis ðtÞ þ ∑ ∑ ∑ eir ðtÞkrs krs i¼1r ¼1s¼1
N
l
eis ðt τÞ þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ i¼1j¼1
T
i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi N
n
n
1
r ∑ ∑ ∑
i¼1r ¼1s¼12 N
n
n
1
þ ∑ ∑ ∑
i¼1r ¼1s¼12 N
2ε
2ð1 εÞ 2 eis ðtÞÞ
2ε
2ð1 εÞ 2 eis ðt
ðkrs e2ir ðtÞ þ krs
ðkrs e2ir ðtÞ þ krs
N
l
τÞÞ
T
þc ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ i¼1j¼1
j¼1
N
l a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i þ ∑ ðki ðtÞ kÞ‖ei ðtÞ‖2 2 i¼1 αi i¼1
r ∑ ∑ ∑ eir ðtÞkrs krs
where αi ði ¼ 1; 2; …; NÞ and a1 are positive constants, ki ð0Þ 40 ði ¼ 1; 2; …; lÞ are initial value and ki ððm þ1ÞTÞ ¼ ki ðmT þhÞ, m ¼ 0; 1; 2…: T 4 0 denotes the control period and the work time h satisfies 0 o h oT. Then, based on the adaptive pinning periodically intermittent controllers (7) and the error expression (6), the error systems can be written as when t A ½mT; mT þ hÞ, 8 N > > > > F ðt; ðt τÞÞ þ c ∑ g ij Aej ðtÞ ki ðtÞei ðtÞ; 1 r ir l; < j¼1 ð9Þ e_ i ðtÞ ¼ N > > > l þ1 r ir N: > : F ðt; ðt τÞÞ þ c ∑ g ij Aej ðtÞ; where F ðt; t τÞ ¼ Fðt; xi ðtÞ; xi ðt τÞÞ Fðt; yi ðtÞ; yi ðt τÞÞ. When t A ½mT þ h; ðm þ 1ÞTÞ, the error systems can be expressed as
217
N
i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi n
1 n 2ε 2ð1 εÞ 2 2ð1 εÞ 2 ∑ ½2krs e2ir ðtÞ þ krs eis ðtÞ þ krs eis ðt τÞ i¼1r ¼12s¼1
¼ ∑ ∑ N
N
l
T
þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ i¼1j¼1
i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi n N n 1 1 n 2ð1 εÞ 2 2ε 2ð1 εÞ 2 ∑ ð2krs þ ksr ¼ ∑ ∑ Þeir ðtÞ þ ∑ ksr eir ðt τÞ 2s¼1 i¼1r ¼1 2s¼1
Theorem 1. Suppose that τ r h and τ r T h, and choose h ¼ R1 T; τ ¼ R2 T. Then, if there exist the positive constants a1 ; a2 ; and k such that s 1 Q ¼ p þ a1 I N þ cαG^ D r 0; 2 1 p ða2 a1 Þ þ cαλ1 r 0 2 γ ðR1 R2 Þ ða2 a1 þqÞð1 R1 Þ 4 0;
ð11Þ
N
∑
N
i ¼ 1 j ¼ 1;j a i
Nl
2ε
2ð1 εÞ
p ¼ max1 r r r n pr ¼ max1 r r r n ð1=2Þ∑ns ¼ 1 ð2krs þ ksr
Þ,
2ð1 εÞ max1 r r r n ∑ns ¼ 1 ksr ,
q ¼ max1 r r r n
qr ¼ a2 4 a1 4q, and γ 4 0 is the smallest real root of the equation a1 γ q expðγτÞ ¼ 0. Then, the drive system (1) and the response system (2) achieve outer synchronization under the adaptive pinning periodically intermittent controllers (7). Proof. In order to verify the conclusion of Theorem 1, it is necessary to construct a Lyapunov function as 1 N T 1 l ðk ðtÞ kÞ2 ∑ ei ðtÞei ðtÞ þ ∑ expð a1 tÞ i : 2i¼1 2i¼1 αi
ð12Þ
T
i¼1
N l 1 N T r p ∑ eTi ðtÞei ðtÞ þ q ∑ eTi ðt τÞei ðt τÞ ∑ kei ðtÞei ðtÞ 2 i¼1 i¼1 i¼1 N
l
l
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi
þc ∑
D ¼ diagðk; …; k ; 0; …; 0 Þ; |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl}
N
i¼1j¼1
where
VðtÞ ¼
N
þc ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ
αgij ‖ei ðtÞ‖2 ‖ej ðtÞ‖2 þ c ∑ g ii ρmin eTi ðtÞei ðtÞ i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi 2
s a1 ðk ðtÞ kÞ ∑ expð a1 tÞ i ¼ eT ðtÞðpI N þcαG^ DÞeðtÞ 2 i¼1 αi 1 T þ qe ðt τÞeðt τÞ 2 s 1 1 T ¼ e ðtÞ p þ a1 I N þ cαG^ D eðtÞ þ qeT ðt τÞeðt τÞ 2 2 " # l a1 N T ðki ðtÞ kÞ2 ∑ e ðtÞeðtÞ þ ∑ expð a1 tÞ 2 i¼1 αi i¼1 l
r a1 VðtÞ þ qVðt τÞ;
ð13Þ
where eðtÞ ¼ ð‖e1 ðtÞ‖2 ; ‖e2 ðtÞ‖2 ; …; ‖eN ðtÞ‖2 ÞT , p ¼ max1 r r r n fð1=2Þ When mT r t o mT þ h, using the Cauchy inequality and the conditions in Theorem 1, the derivative of V(t) with respect to time t along the trajectory of the error system (9) can be calculated as follows: N
V_ ðtÞ ¼ ∑
i¼1
a1 eTi ðtÞe_ i ðtÞ 2
l
þ ∑ expð a1 tÞ i¼1
l
∑ expð a1 tÞ
i¼1
ðki ðtÞ kÞ2
αi
ðki ðtÞ kÞ _ k i ðtÞ
αi
N
¼ ∑ eTi ðtÞðFðt; xi ðtÞ; xi ðt τÞÞ Fðt; yi ðtÞ; yi ðt τÞÞÞ i¼1
N
N
l
þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ ki ðtÞeTi ðtÞei ðtÞ i¼1j¼1
i¼1
2ε
2ð1 εÞ
2ð1 εÞ
∑ns ¼ 1 ð2krs þksr Þg, and q ¼ max1 r r r n f∑ns ¼ 1 ksr g. Since a1 4 q, Vðt τÞ r supt τ r θ r t VðtÞ, thus, according to Lemma 2, we have VðtÞ r
max
mT τ r θ r mT
VðθÞ expf γ ðt mTÞg;
ð14Þ
where γ is the smallest real root of the equation a1 γ q expfγτg ¼ 0. When mT þ h r t oðm þ 1ÞT, for m ¼ 0,1,2,… from the conditions in Theorem 1, we have a1 ðk ðtÞ kÞ V_ ðtÞ ¼ ∑ eTi ðtÞe_ i ðtÞ ∑ expð a1 tÞ i 2 αi i¼1 i¼1 N
N
l
2
¼ ∑ eTi ðtÞðFðt; xi ðtÞ; xi ðt τÞÞ Fðt; yi ðtÞ; yi ðt τÞÞÞ i¼1
218
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
N
N
þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ i¼1j¼1
N
rp ∑
i¼1
1 eTi ðtÞei ðtÞ þ q 2
N
N
∑
þc ∑
i ¼ 1 j ¼ 1;j a i
N
∑
i¼1
r
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi
eTi ðt
þ γ ðR1 R2 Þhg:
τÞei ðt τÞ
VðtÞ r
N
αgij ‖ei ðtÞ‖2 ‖ej ðtÞ‖2 þ c ∑ gii ρmin eTi ðtÞei ðtÞ
ð19Þ T l So, from VðtÞ ¼ ð1=2Þ∑N i ¼ 1 ei ðtÞei ðtÞ þ ð1=2Þ∑i ¼ 1 expð a1 tÞðki ðtÞ kÞ2 =αi , we have
‖eðtÞ‖2 r 2
mT þ h τ r θ r mT þ h
ð15Þ
VðθÞ expfβ ðt mT hÞg
VðtÞ r max VðθÞ expð γ tÞ: When h r t o T,
τrθr0
VðθÞ expf γ ðt TÞg
When T þh r t o 2T, max
VðθÞ expfβ ðt T hÞg
r max VðθÞ expfβðt T hÞ þ βðT hÞ 2γ ðh τÞg: τrθr0
By the above analogy, we could estimate the value of V(t) for the integer m. When mT rt o mT þh, VðtÞ r max VðθÞ expf γ ðt mTÞ þnβ ðT hÞ mγ ðh τÞg: τrθr0
ð16Þ
As h ¼ R1 T, and τ ¼ R2 T, we have r max VðθÞ expfβð1 R1 Þt þ γ ðR1 R2 Þð t þ hÞg
When mT þh r t o ðm þ 1ÞT, VðtÞ r max VðθÞ expfβðt mT hÞ þmβ ðT hÞ ðm þ 1Þγ ðh τÞg: τrθr0
ð17Þ
τrθr0
2ð1 εÞ
Þ,
ð21Þ
q ¼ max1 r r r n
denotes the maximum eigenvalue of matrix
Letting Γ ¼ p þ cαλ1 and selecting a positive constant a2 which satisfy the condition of a2 ¼ 2Γ þ a1 4 0, then the second inequation of (21) in Corollary 1 holds. The fourth inequation of (21) can be reduced γ ðR1 R2 Þ ð2Γ þqÞð1 R1 Þ 4 0. Thus, we can obtain the following corollary. Corollary 2. Under Assumption 1, given a positive constant a1 4 q. If there exist following conditions 2p þ a1 γ ðR1 R2 Þ ð2Γ þ qÞð1 R1 Þ 4 0 2cα
2ð1 εÞ , max1 r r r n qr ¼ max1 r r r n ∑ns ¼ 1 ksr s imum eigenvalue of matrix G^ ij , 40 is
τrθr0
r max VðθÞ expfβt β R1 t tðγ R1 γ R2 Þg
λ
^s max ðG ij Þ
ð22Þ 2ε
max V ðθÞ expf ½γ ðR1 R2 Þ βð1 R1 Þt þ γ ðR1 R2 Þhg:
τrθr0
2ε
p ¼ max1 r r r n ð1=2Þ∑ns ¼ 1 ð2krs þ ksr
2ð1 εÞ
where p ¼ max1 r r r n pr ¼ max1 r r r n ð1=2Þ∑ns ¼ 1 ð2krs þ ksr
τrθr0
VðtÞ r max VðθÞ expfβt ðm þ 1ÞβR1 T ðm þ 1ÞTðγ R1 γ R2 Þg
N l. Then, we obtain the following corollary.
s
τrθr0
Substitute h ¼ R1 T, and τ ¼ R2 T into (17), we obtain
B Þ, we have Q o 0. If the
s s s Q ¼ ðp þð1=2Þa1 ÞI N l þcαG^ , where G^ ij ¼ G^ l þ i;l þ j for i; j ¼ 1; …;
λmax ðG^ ij Þ r
VðtÞ r max VðθÞ expfmβð1 R1 ÞT mγ ðR1 R2 ÞTg
¼
1 T
γ the smallest real root of the equation a1 γ q expðγτÞ ¼ 0. Then the drive system (1) is synchronous with the response system (2) under the adaptive pinning periodically intermittent controllers (7).
τrθr0
T þhτrθrT þh
when Q o 0 and k 4 λmax ðE BQ
2ð1 εÞ ∑ns ¼ 1 ksr , s G^ ij , 4 0 is
r max VðθÞ expf γ ðt TÞ þ βðT hÞ γ ðh τÞg:
VðtÞ r
matrix Q can be obtained by removing the first 1, 2, …, l rowcolumn pairs of matrix Q. According to Lemma 1, it is clear that
where
When T rt r T þ h, max
matrices with appropriate dimensions, D ¼ diagðk; …; k Þ, and the |fflfflffl{zfflfflffl}l
s 1 1 p þ a1 þ cαλmax ðG^ ij Þ r 0; p ða2 a1 Þ þ cαλ1 r 0a2 4 a1 4q 2 2 γ ðR1 R2 Þ ða2 a1 þ qÞð1 R1 Þ 4 0;
VðθÞ exp½βðt hÞ
r max VðθÞ½β ðt hÞ γ ðh τÞ:
T τ rθrT
Corollary 1. Suppose that τ r h and τ r T h, and h ¼ R1 T; τ ¼ R2 T. If there exist positive constants a1 and a2, such that
τrθr0
VðtÞ r
½γ ðR1 R2 Þ βð1 R1 Þ γ ðR1 R2 Þh tþ 2 2
parameters k is big enough, Q o 0 is equivalent to Q o 0. Here,
In the following, we will estimate the value of V(t) from above. When 0 rt o h,
max
exp
Q can be expressed as Q ¼ ðE BTD QB Þ. In this expression, E and B are
Let β ¼ a2 a1 þ q. Then, according to Lemma 3, we have the following inequality, max
1=2
According to the conditions in Theorem 1, we could obtain the conclusion, and the proof is completed. From the conclusion in Theorem 1, the conditions can be simplified by the Schur complement. Based on the characters of s the parameters in matrix Q ¼ ðp þ ð1=2Þa1 ÞI N þ cαG^ D, the matrix
1 a2 l ðk ðtÞ kÞ2 ∑ expð a2 tÞ i þ a2 eT ðtÞeðtÞ þ 2 2 i¼1 αi
r ða2 a1 ÞVðtÞ þ qV ðt τÞ:
max V ðθÞ
τrθr0
ð20Þ
1 a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i a1 eT ðtÞeðtÞ 2 2 i¼1 αi
hτrθrh
max VðθÞ expf γ ðR1 R2 Þ β ð1 R1 Þt þ γ ðR1 R2 Þhg:
τrθr0
i¼1
1 ^ ¼ eT ðtÞðpI N þ cαGÞeðtÞ þ qeT ðt τÞeðt τÞ 2 a1 l ðki ðtÞ kÞ2 ∑ expð a1 tÞ 2 i¼1 αi s 1 1 r eT ðtÞ p ða2 a1 Þ I N þ cαG^ eðtÞ þ qeT ðt τÞeðt τÞ 2 2
VðtÞ r
ð18Þ
Therefore, for any t Z 0, we have
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi
VðtÞ r
max VðθÞ expf ½γ ðR1 R2 Þ βð1 R1 Þt
τrθr0
Þ, q ¼
s λmax ðG^ ij Þ denotes the max-
γ the smallest real root of the equation a1 γ q expðγτÞ ¼ 0. Then the drive system (1) can synchronize with the response system (2) by the adaptive pinning periodically intermittent controllers (7) and the updating laws (8). 4. Numerical simulation In this section, the Chua oscillator with time delay is used as an uncoupled node in the drive and response systems to show the
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
219
effectiveness of the proposed control scheme. A single Chua oscillator is given as follows: x_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt τÞÞ ¼ Cxi ðtÞ þ g 1 ðxi ðtÞÞ þ g 2 ðxi ðt τÞÞ
1 0.5
x2
ð23Þ
where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; xi3 ðtÞÞ A R , g 1 ðxi ðtÞÞ ¼ ð ð1=2Þαðm1 m2 Þðjxi1 ðtÞ þ 1j jxi1 ðtÞ 1jÞ; 0; 0ÞT A R3 , g 2 ðxi ðt τÞÞ ¼ ð0; 0; βϱ sin ðνxi1 ðt τÞÞÞT A R3 , 0 1 α 0 αð1 þ m2 Þ B 1 1 1 C C¼@ A; 0 β w T
0 −0.5 −1 10 5
5 0
0
−5
x3
−10
−5
x1
Fig. 2. Chaotic behavior of time delayed Chua attractors.
3
and α ¼ 10, β ¼19.53, w ¼0.1636, m1 ¼ 1.4325, m2 ¼ 0.7831, ν ¼ 0.5, ϱ ¼0.2, and τ ¼0.02. It has been known that the Chua oscillators exhibit chaotic behavior, and Fig. 2 shows it clearly. Based on the parameters mentioned above, it is easy to verify that f ðt; xi ðtÞ; xi ðt τÞÞ f ðt; y ðtÞ; y ðt τÞÞj 1 1 i i r ½αð1 þ m2 Þ þ 1 αðm2 m1 Þ xi1 ðtÞ yi1 ðtÞ þ α xi2 ðtÞ yi2 ðtÞj; 2
jf 2 ðt; xi ðtÞ; xi ðt τÞÞ f 2 ðt; yi ðtÞ; yi ðt τÞÞj r jxi1 ðtÞ yi1 ðtÞj þ jxi2 ðtÞ yi2 ðtÞj þ jxi3 ðtÞ yi3 ðtÞj; 1 0.8
0.6
0.6
0.4
T=2
0.2 0 −0.2
0 −0.2 −0.4
T=0.2
−0.4 −0.6
T=0.2
0.2
ei3(t)
ei1(t)
0.4
T=2
0
0.5
1
1.5
2
2.5
−0.6
3
Time t
−0.8
Fig. 3. Synchronous errors ei1 ðtÞ; i ¼ 1; 2; …10 between the drive and response systems for 0 r t r 1:5; T ¼ 0:2, and T ¼ 2.
0
0.5
1
1.5
2
2.5
3
Time t Fig. 5. Synchronous errors ei3 ðtÞ; i ¼ 1; 2; …10 between the drive and response systems for 0r t r 1:5; T ¼ 0:2, and T ¼ 2.
1
300
0.8 0.6
250
T=2
0.4
200
0
ki(t)
ei2(t)
0.2
150
−0.2 T=0.2
−0.4
100
−0.6
50
−0.8 −1
0
0.5
1
1.5
2
2.5
3
Time t Fig. 4. Synchronous errors ei2 ðtÞ; i ¼ 1; 2; …10 between the drive and response systems for 0 r t r 1:5; T ¼ 0:2, and T ¼ 2.
0
0
0.5
1
1.5
2
Time t Fig. 6. Adaptive control gain of controllers.
2.5
3
220
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
and jf 3 ðt; xi ðtÞ; xi ðt τÞÞ f 3 ðt; yi ðtÞ; yi ðt τÞÞj
r βϱνjxi1 ðt τÞ yi1 ðt τÞj þ βjxi2 ðtÞ yi2 ðtÞj þ wjxi3 ðtÞ yi3 ðtÞj:
Calculate the value of the parameters, then we have f 11 ¼ α ð1 þ m2 Þ þð1=2Þαðm2 m1 Þ ¼ 5:416, f 12 ¼ α ¼ 10, f 13 ¼ 0, f 21 ¼ f 22 ¼ f 23 ¼ 1, f 31 ¼ βϱν ¼ 1:953, f 32 ¼ β ¼ 19:53, f 33 ¼ w ¼ 0:1636, and ϱ ¼ 1=2. Thus, the value of p and q are p¼22.2284 and q¼30.53. The coupling configuration matrices of the drive and response systems are chosen as a BA scale model. The parameters of BA model are given by m0 ¼m ¼3, N ¼10. In the simulations, we add the adaptive periodical intermittent controllers to the first six nodes. According to the coupling matrix, we known that λ1 ¼0.6790, α ¼1.2 and λmax ¼ 1:4999, then we choose the parameters T ¼0.2, a1 ¼ 60, a2 ¼ 162, the value of work time h will be obtained as h¼ 0.18 and the conditions in Corollary 1 are satisfied. Use the same method, we could obtain the result that the value of work time h will become larger when control period T ¼2. Therefore, there would be some effect on control cost when the control period becomes larger. Select the initial values of the nodes in the complex networks randomly, then the errors of the corresponding nodes in the drive and response systems are shown in Figs. 3, 4 and 5. From the simulations, we could see that there is nearly no change for the time of synchronization errors that tend to zero between drive and response systems when T ¼0.2 and T ¼2. Remark 2. In the former literatures [31,32], Xia and Cai studied the inner synchronization of delayed dynamical nodes via linear periodically pinning intermittent control. In that case, the control 1 gain k obtained in [31,32] needed to satisfy k 4 λmax ðE BQ BT Þ which will be larger than the needed values for practical problems. If we adopt the control method in [31,32], the control gain should satisfy k 4 459:4634. While, in this paper, the numerical simulation of adaptive control gain in Fig. 6 illustrates that our results are less conservative and more practically applicable than it in [31,32]. 5. Conclusions In this paper, we have investigated the problem of outer synchronization between the drive and response systems with time delay dynamical nodes by means of adaptive pinning periodically intermittent controllers. Based on the Lyapunov stability theory, the adaptive control technique and the differential inequality method, some synchronous criteria have been derived analytically. At last, both the theoretical and numerical analysis illustrate the effectiveness of the proposed control methodology.
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 60804006, 71103126 and the Northeastern University Fundamental Research (No. N110404023).
[7] L.M. Pecora, T.L. Carrol, Master stability function for synchronization coupled systems, Phys. Rev. Lett. 80 (10) (1998) 2109–2112. [8] X.F. Wang, G.R. Chen, Synchronization in small-world dynamical networks, Int. J. Bifurc. Chaos 12 (2002) 187–192. [9] X.F. Wang, G.R. Chen, Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits Syst. I 49 (2002) 54–62. [10] Y.Z. Xiao, W. Xu, X.C. Li, Adaptive complete synchronization of chaotic dynamical networks with unknown and mismatched parameters, Chaos 17 (2007) 033118. [11] C.A.S. Batista, A.M. Batista, J.A.C. de Pontes, R.L. Viana, S.R. Lopes, Chaotic phase synchronization on scale-free networks of bursting neurons, Phys. Rev. E 76 (2007) 016218. [12] T. Wang, L. Tao, X. Yang, S.M. Fei, Cluster synchronization for delayed Lure dynamical networks based on pinning control, Neurocomputing 83 (2001) 72–82. [13] Q. Ma, J.W. Lu, Cluster synchronization for directed complex dynamical networks via pinning control, Neurocomputing 101 (2013) 354–360. [14] D.J. Wagg, Partial synchronization of nonidentical chaotic systems via adaptive control, with applications to modeling coupled nonlinear systems, Int. J. Bifurc. Chaos 12 (2002) 561. [15] X.J. Wu, H.T. Lu, Outer synchronization of uncertain general complex delayed networks with adaptive coupling, Neurocomputing 82 (2012) 157–166. [16] C.X. Fan, G.P. Jiang, F.H. Jiang, Synchronization between two complex dynamical networks using scalar signals under pinning control, IEEE Trans. Circuits Syst. I 57 (11) (2011) 2991–2998. [17] C.P. Wang, C.X. Xu, W.G. Sun, J. Xu, J. Kurths, Outer synchronization of coupled discrete-time networks, Chaos 19 (2009) 013106. [18] C.P. Li, W.G. Sun, J. Kurths, Synchronization between two coupled complex networks, Phys. Rev. E 76 (2007) 046204. [19] Y. Liang, X.Y. Wang, J. Eustace, Adaptive synchronization in complex networks with non-delay and variable delay couplings via pinning control, Neurocomputing 123 (2014) 292–298. [20] J. Zhou, J.A. Lu, J.H. Lü, Adaptive synchronization of an uncertain complex dynamical network, IEEE Trans. Autom. Control 51 (4) (2006) 652–656. [21] T. Liu, J. Zhao, J.H. David, Exponential synchronization of complex delayed dynamical networks with switching topology, IEEE Trans. Circuits Syst. I 57 (2010) 2080–2967. [22] J. Zhou, J.A. Lu, J.H. Lü, Pinning adaptive synchronization of a general complex dynamical networks, Automatica 44 (2008) 996–1003. [23] W.W. Yu, G.R. Chen, J.H. Lü, On pinning synchronization of complex dynamical networks, Automatica 45 (2009) 429–435. [24] J.J. Xing, H.J. Jiang, C. Hu, Exponential synchronization for delayed recurrent neural networks via periodically intermittent control, Neurocomputing 113 (2013) 122–129. [25] X.J. Wu, H.T. Lu, Hybrid synchronization of the general delayed and non-delayed complex dynamical networks via pinning control, Neurocomputing 89 (2012) 168–177. [26] G. Hu, Z.L. Qu, Controlling spatiotemporal chaos in coupled map lattice systems, Phys. Rev. Lett. 72 (1994) 68–71. [27] X.W. Liu, T.P. Chen, Cluster synchronization in directed networks via intermittent pinning control, IEEE Trans. Neural Netw. 22 (7) (2011) 1009–1020. [28] N. Li, H.Y. Sun, X.J., Q.L. Zhang, Exponential synchronization of united complex dynamical networks with multi-links via adaptive periodically intermittent control, IET Control Theory Appl. http://dx.doi.org/10.1049/iet-cta.2013.0159, 2013. [29] N. Li, J.D. Cao, Periodically intermittent control on robust exponential synchronization for switched interval coupled networks, Neurocomputing (2014), http://dx.doi.org/10.1016/j.neucom.2013.10.040. [30] S.M. Cai, Z.R. Liu, F.D. Xu, J.W. Shen, Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit, Phys. Lett. A 373 (2009) 3846–3854. [31] W.G. Xia, J.D. Cao, Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos 19 (2009) 013120. [32] S.M. Cai, J.J. Hao, Q.B. He, Z.R. Liu, Exponential synchronization of complex delayed dynamical networks via pinning periodically intermittent control, Phys. Lett. A 375 (2011) 1965–1971. [33] Y. Liang, X.Y. Wang, Synchronization in complex networks with non-delay and delay couplings via intermittent control with two switched periods, Phys. A 395 (2014) 434–444. [34] S. Boyed, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix in System and Control Theory, Philadelphia, PA, SIAM, 1994. [35] A. Halanay, Different Equations: Stability, Oscillations, Time Lags, Academic, New York, 1966.
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Mo Zhao was born in Liaoning, China, in 1984. She received her B.S. and M.S. degree in faculty of electrical and control engineering of Liaoning Technical University, Fuxin, China, in 2006 and 2009. She is now a Ph.D candidate at Northeastern University. Her main research interests include synchronization of complex networks, multi-agent system and its applications.
M. Zhao et al. / Neurocomputing 144 (2014) 215–221 Huaguang Zhang was born in Jilin, 1959. He received the BS and MS degrees in control engineering from Northeast Dianli University, Jilin City, China, in 1982 and 1985, respectively, and the PhD degree in thermal power engineering and automation from Southeast University, Nanjing, China, in 1991. In 1992, he joined the Department of Automatic Control, Northeastern University, University, Shenyang, China, as a Postdoctoral Fellow for two years, where, since 1994, he has been a Professor and the Head of the Institute of Electric Automation. He has authored or co-authored more than 100 SCI papers. He is an Associate Editor of Automatica and Neurocomputing. His current research interests include fuzzy control, neural-network-based control, non-linear control, and their applications. Prof. Zhang serves as an Associate Editor of the IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, IEEE Transactions on Fuzzy Systems and IEEE Transactions on Neural Networks. He was awarded a Nationwide Excellent Post-doctor, and the recipient of the Outstanding Youth Science Foundation Award from the National Natural Science Foundation Committee of China in 2003. He was named a Cheung Kong Scholar by the Education Ministry of China in 2005. He is a Deputy Director for Intelligent System Engineering Committee of CAAI.
Zhiliang Wang received the B.S. degree in applied mechanics and the M.S. degree in computational mechanics from Jilin University, Changchun, China, in 1997 and 2000, respectively, and the Ph.D.degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2004. He is currently as an Associate Professor with the School of Information Science and Engineering, Northeastern University. His research interests include nonlinear and adaptive control, and chaos theory and its applications.
221 Hongjing Liang was born in Liaoning, China, in 1986. He received his B.S. degree in mathematics from Bohai University, Jinzhou China, in 2009, and M.S. degree in Fundamental Mathematics from Northeastern University, Shenyang, China, in 2011. He is now a Ph.D candidate at Northeastern University. His main research interests include multi-agent systems, complex systems and output regulation.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Synchronization between two general complex networks with time-delay by adaptive periodically intermittent pinning control Mo Zhao a,b, Huaguang Zhang a,n, Zhiliang Wang a, Hongjing Liang a a b
College of Information Science and Engineering, Northeastern University, Shenyang 110004, China College of Air Traffic Management, Civil Aviation University of China, Tianjin, 300300, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2014 Received in revised form 11 April 2014 Accepted 23 April 2014 Communicated by S. Arik Available online 6 June 2014
This letter investigates the problem of synchronization between two general delayed dynamical complex networks. Based on the Lyapunov stability theory and the adaptive periodical pinning intermittent control technique, some criteria are obtained for the synchronization between two general dynamical complex networks. Moreover, two typical BA scale complex networks with time-delayed Chua oscillator are chosen to illustrate the effectiveness of the theoretical results. & 2014 Elsevier B.V. All rights reserved.
Keywords: Complex networks Synchronization Pinning control Intermittent control
1. Introduction A complex network is a large set of interconnected nodes and has many applications in almost all the fields of the real world, such as the neural networks, the cellular and metabolic networks, the Word Wide Web, the electrical power grids and the social networks [1–4]. In the former literature, several kinds of network models have been proposed for the purpose of describing the real world more realistic [5,6], such as regular networks, random networks [6], small world networks [1] and scale free networks [3]. Besides these studies of topology structures, synchronization is a significant dynamical behavior of the dynamical elements in the complex networks and has been widely investigated [7–9]. In the coupling networks, there are several kinds of synchronization, for example, complete synchronization [10], phase synchronization [11], cluster synchronization [12,13], partial synchronization [14] and so on. Different from those categories, the phenomenon of synchronization can also be classified into “inner synchronization” [7,8] and “outer synchronization” [15–18]. “Inner synchronization” means a collective behavior of all the nodes within a network, while “outer synchronization” refers to the synchronization occurring between two or more coupled networks regardless of happening of the inner synchronization. In Nature, there are so many examples that can be taken to illustrate the phenomenon of outer synchronization, for instance, the infectious disease spreads between different communities, the avian influenza spreads among domestic
n
Corresponding author. E-mail address: [email protected] (H. Zhang).
http://dx.doi.org/10.1016/j.neucom.2014.04.052 0925-2312/& 2014 Elsevier B.V. All rights reserved.
and wild birds, and the different species development in balance. All these challenging topics show the great importance of researching outer synchronization between coupled networks. Thus, the outer synchronization has attracted more and more attention. In some cases, different complex networks can achieve outer synchronization by themselves. But, there also exists the situation that the networks with identical system parameters cannot synchronize with each other by themselves for different initial values. Thus, many kinds of control techniques have been adopted to make the networks achieve synchronization [19–25]. Among these methods, a pinning control is a special control method of adding controllers to partial of the nodes rather than all of the nodes [26]. The pinning control not only simplifies the coupling topological structure, but also saves the cost [16] through reducing the number of controllers. Thus, its status in engineering application has become more and more important. In the process of controlling, the signal will become weak due to diffusion, so it needs some external control until the strength of the signal reaches an upper level. After that, the external control can be removed in order to reduce the cost. This kind of discontinuous control method is different from the continuous control and is named as an intermittent control. Usually, the control time of intermittent is periodic, and in every period, the time with a controller is denoted as work time and the rest is denoted as rest time (see Fig. 1) [27]. From the perspective of economic costs, the method of pinning control will reduce the number of controllers in the complex networks and the periodically intermittent control will decrease the working time. Therefore, if we combine two kinds of control methods together, the control cost will greatly be saved [28–33].
216
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
One Period T
In this paper, we assume that ‖A‖2 ¼ α 4 0 and ρmin denotes
One Period T
the minimum eigenvalue of matrix ðA þ AT Þ=2. G^ is a modifying matrix of G via replacing the diagonal elements gii by ðρmin =αÞg ii . s
h
work time
rest time
work time
T
Choose the matrix G^ ¼ ðG^ þ G^ Þ=2 and its eigenvalues are expressed as λ1 Z λ2 Z …λN . In the following, some basic definition, lemmas, and assumption will be given.
h
rest time
Definition 1. The drive system (1) and the response system (2) are defined to achieve synchronization, if xi ðtÞ yi ðtÞ-0;
Fig. 1. Sketch map of intermittent control.
In this paper, we focus on the problem of outer synchronization between two generalized complex dynamical networks with time delay. By using the adaptive pinning periodically intermittent control, some novel synchronization criteria will be obtained by the Lyapunov stability theory. Numerical simulations are also presented to show the effectiveness of the proposed method. The rest of this paper is organized as follows. In Section 2, some general driver and response complex dynamical network models are introduced, and some necessary preliminaries are given. In Section 3, based on the Lyapunov stability theorem, some pinning controllers are designed to ensure that the driver and response systems with time delay achieve outer synchronization under different situations. In Section 4, some numerical simulations are given to verify the effectiveness of proposed theoretical results and Section 5 gives the conclusion of the paper.
t-1; i ¼ 1; …; N:
ð3Þ
Assumption 1. If there exists constants krs 4 0, the nonlinear function f r ðt; xðtÞ; xðt τÞÞ satisfies the uniform Lipschitz condition with respect to time t, jf r ðt; xi ðtÞ; xi ðt τÞÞ f r ðt; yi ðtÞ; yi ðt τÞÞj n
r ∑ krs ðjxis ðtÞ yis ðtÞj þ jxis ðt τÞ yis ðt τÞjÞ
ð4Þ
s¼1
where 1 rr rn. Lemma 1 (Schur complement, Boyed et al. [34]). The following linear matrix inequality (LMI) ! AðxÞ BðxÞ 4 0; ð5Þ ðBðxÞÞT CðxÞ where AðxÞ ¼ ðAðxÞÞT ; CðxÞ ¼ ðCðxÞÞT is equivalent to one of the following conditions: (a) AðxÞ 4 0 and CðxÞ BðxÞT AðxÞ 1 BðxÞ 4 0; (b) CðxÞ 4 0 and AðxÞ BðxÞCðxÞ 1 BðxÞT 4 0.
2. Network model and preliminaries Here we consider a complex network consisting of N identical linearly and diffusively coupled nodes, and every node in the network is an n dimensional dynamical unit. Then the network model of the drive system is denoted as N
x_ i ðtÞ ¼ Fðt; xi ðtÞ; xi ðt τÞÞ þ c ∑ g ij Axj ðtÞ þ ui ðtÞ; j¼1
i ¼ 1; 2; …; N;
ð1Þ
where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞT A Rn is the state variable of the ith node, Fðt; xi ðtÞ; xi ðt τÞÞ ¼ ðf 1 ðt; xi ðtÞ; xi ðt τÞÞ; f 2 ðt; xi ðtÞ; xi ðt τÞÞ; …; f n ðt; xi ðtÞ; xi ðt τ ÞÞÞT A Rn , Fðt; xi ðtÞ; xi ðt τÞÞ : Rn ⟶Rn is a nonlinear vector valued function describing the dynamics of nodes and c 4 0 is the coupling strength of the whole network. The matrix G ¼ ðg ij Þ A RNN is the outer coupling configuration matrix, in which g ij A R is defined as follows: if there is a coupling from node i to node j ði ajÞ, g ij 4 0; otherwise, g ij ¼ 0. At the same time, the diagonal elements of G are defined as g ii ¼ ∑N j ¼ 1;j a i g ij . The inner coupling matrix A denotes the inner coupling relationship between every two nodes. Compared with the drive system mentioned above, the response complex network is denoted as N
y_ i ðtÞ ¼ f ðt; yi ðtÞ; yi ðt τÞÞ þc ∑ g ij Ayj ðtÞ; j¼1
i ¼ 1; 2; …; N;
ð2Þ
where yi ðtÞ ¼ ðyi1 ðtÞ; yi2 ðtÞ; …; yin ðtÞÞT A Rn denotes the state variables of the response system, and other parameters involved in the system (2) all have the same meanings with the corresponding parameters in system (1). Remark 1. In the drive and response systems above, the outer coupling configuration matrix G need not to be symmetric or irreducible.
Lemma 2 (Halanay [35]). Let w : ½μ τ ; 1Þ⟶½0; 1Þ be a continuous function such that _ r awðtÞ þ b max wt wðtÞ holds for t Z μ. If a 4 b 40, then wðtÞ r ½max wμ expf γ ðt τ Þg;
t Zμ
where max wt ¼ supt τ r θ r t wðθÞ, and γ 4 0 is the smallest real root of the equation a γ b expfγτg ¼ 0: Lemma 3 (Xia and Cao [31]). Let w : ½μ τ; 1Þ⟶½0; 1Þ be a continuous function such that _ r awðtÞ þ b max wt wðtÞ holds for t Z μ. If a 4 b 40, then wðtÞ r max wt r½max wμ expfða þ bÞðt τÞg;
t Zμ
where max wt ¼ supt τ r θ r t wðθÞ. 3. Main results In this part, in order to realize outer synchronization between the drive and response systems by adaptive pinning periodically intermittent control, some controllers are needed to add on partial nodes of the network. Denote the error expression as ei ðtÞ ¼ xi ðtÞ yi ðtÞ;
i ¼ 1; 2; …; N:
ð6Þ
Here, select the first l nodes to be pinned and the adaptive pinning periodically intermittent control ui is designed as follows: 8 t A ½mT; mT þhÞ; > < ki ðtÞei ðtÞ; 1 ri r l; l þ 1 ri r N; t A ½mT; mT þhÞ; ui ðtÞ ¼ 0; ð7Þ > : 0; 1 ri r N; t A ½mT þ h; ðm þ 1ÞTÞ
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
the updating laws ( αi expða1 tÞ‖ei ðtÞ‖22 ; k_ i ðtÞ ¼ 0;
t A ½mT; mT þ hÞ; t A ½mT þ h; ðm þ1ÞTÞ
ð8Þ
N
e_ i ðtÞ ¼ F ðt; ðt τÞÞ þ c ∑ g ij Aej ðtÞ; j¼1
i ¼ 1; 2; …; N:
ð10Þ
n
n
ε ð1 εÞ
i¼1r ¼1s¼1 N
N
n
n
ε ð1 εÞ
eis ðtÞ þ ∑ ∑ ∑ eir ðtÞkrs krs i¼1r ¼1s¼1
N
l
eis ðt τÞ þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ i¼1j¼1
T
i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi N
n
n
1
r ∑ ∑ ∑
i¼1r ¼1s¼12 N
n
n
1
þ ∑ ∑ ∑
i¼1r ¼1s¼12 N
2ε
2ð1 εÞ 2 eis ðtÞÞ
2ε
2ð1 εÞ 2 eis ðt
ðkrs e2ir ðtÞ þ krs
ðkrs e2ir ðtÞ þ krs
N
l
τÞÞ
T
þc ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ i¼1j¼1
j¼1
N
l a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i þ ∑ ðki ðtÞ kÞ‖ei ðtÞ‖2 2 i¼1 αi i¼1
r ∑ ∑ ∑ eir ðtÞkrs krs
where αi ði ¼ 1; 2; …; NÞ and a1 are positive constants, ki ð0Þ 40 ði ¼ 1; 2; …; lÞ are initial value and ki ððm þ1ÞTÞ ¼ ki ðmT þhÞ, m ¼ 0; 1; 2…: T 4 0 denotes the control period and the work time h satisfies 0 o h oT. Then, based on the adaptive pinning periodically intermittent controllers (7) and the error expression (6), the error systems can be written as when t A ½mT; mT þ hÞ, 8 N > > > > F ðt; ðt τÞÞ þ c ∑ g ij Aej ðtÞ ki ðtÞei ðtÞ; 1 r ir l; < j¼1 ð9Þ e_ i ðtÞ ¼ N > > > l þ1 r ir N: > : F ðt; ðt τÞÞ þ c ∑ g ij Aej ðtÞ; where F ðt; t τÞ ¼ Fðt; xi ðtÞ; xi ðt τÞÞ Fðt; yi ðtÞ; yi ðt τÞÞ. When t A ½mT þ h; ðm þ 1ÞTÞ, the error systems can be expressed as
217
N
i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi n
1 n 2ε 2ð1 εÞ 2 2ð1 εÞ 2 ∑ ½2krs e2ir ðtÞ þ krs eis ðtÞ þ krs eis ðt τÞ i¼1r ¼12s¼1
¼ ∑ ∑ N
N
l
T
þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ i¼1j¼1
i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi n N n 1 1 n 2ð1 εÞ 2 2ε 2ð1 εÞ 2 ∑ ð2krs þ ksr ¼ ∑ ∑ Þeir ðtÞ þ ∑ ksr eir ðt τÞ 2s¼1 i¼1r ¼1 2s¼1
Theorem 1. Suppose that τ r h and τ r T h, and choose h ¼ R1 T; τ ¼ R2 T. Then, if there exist the positive constants a1 ; a2 ; and k such that s 1 Q ¼ p þ a1 I N þ cαG^ D r 0; 2 1 p ða2 a1 Þ þ cαλ1 r 0 2 γ ðR1 R2 Þ ða2 a1 þqÞð1 R1 Þ 4 0;
ð11Þ
N
∑
N
i ¼ 1 j ¼ 1;j a i
Nl
2ε
2ð1 εÞ
p ¼ max1 r r r n pr ¼ max1 r r r n ð1=2Þ∑ns ¼ 1 ð2krs þ ksr
Þ,
2ð1 εÞ max1 r r r n ∑ns ¼ 1 ksr ,
q ¼ max1 r r r n
qr ¼ a2 4 a1 4q, and γ 4 0 is the smallest real root of the equation a1 γ q expðγτÞ ¼ 0. Then, the drive system (1) and the response system (2) achieve outer synchronization under the adaptive pinning periodically intermittent controllers (7). Proof. In order to verify the conclusion of Theorem 1, it is necessary to construct a Lyapunov function as 1 N T 1 l ðk ðtÞ kÞ2 ∑ ei ðtÞei ðtÞ þ ∑ expð a1 tÞ i : 2i¼1 2i¼1 αi
ð12Þ
T
i¼1
N l 1 N T r p ∑ eTi ðtÞei ðtÞ þ q ∑ eTi ðt τÞei ðt τÞ ∑ kei ðtÞei ðtÞ 2 i¼1 i¼1 i¼1 N
l
l
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi
þc ∑
D ¼ diagðk; …; k ; 0; …; 0 Þ; |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl}
N
i¼1j¼1
where
VðtÞ ¼
N
þc ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ kei ðtÞei ðtÞ
αgij ‖ei ðtÞ‖2 ‖ej ðtÞ‖2 þ c ∑ g ii ρmin eTi ðtÞei ðtÞ i¼1
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi 2
s a1 ðk ðtÞ kÞ ∑ expð a1 tÞ i ¼ eT ðtÞðpI N þcαG^ DÞeðtÞ 2 i¼1 αi 1 T þ qe ðt τÞeðt τÞ 2 s 1 1 T ¼ e ðtÞ p þ a1 I N þ cαG^ D eðtÞ þ qeT ðt τÞeðt τÞ 2 2 " # l a1 N T ðki ðtÞ kÞ2 ∑ e ðtÞeðtÞ þ ∑ expð a1 tÞ 2 i¼1 αi i¼1 l
r a1 VðtÞ þ qVðt τÞ;
ð13Þ
where eðtÞ ¼ ð‖e1 ðtÞ‖2 ; ‖e2 ðtÞ‖2 ; …; ‖eN ðtÞ‖2 ÞT , p ¼ max1 r r r n fð1=2Þ When mT r t o mT þ h, using the Cauchy inequality and the conditions in Theorem 1, the derivative of V(t) with respect to time t along the trajectory of the error system (9) can be calculated as follows: N
V_ ðtÞ ¼ ∑
i¼1
a1 eTi ðtÞe_ i ðtÞ 2
l
þ ∑ expð a1 tÞ i¼1
l
∑ expð a1 tÞ
i¼1
ðki ðtÞ kÞ2
αi
ðki ðtÞ kÞ _ k i ðtÞ
αi
N
¼ ∑ eTi ðtÞðFðt; xi ðtÞ; xi ðt τÞÞ Fðt; yi ðtÞ; yi ðt τÞÞÞ i¼1
N
N
l
þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ ∑ ki ðtÞeTi ðtÞei ðtÞ i¼1j¼1
i¼1
2ε
2ð1 εÞ
2ð1 εÞ
∑ns ¼ 1 ð2krs þksr Þg, and q ¼ max1 r r r n f∑ns ¼ 1 ksr g. Since a1 4 q, Vðt τÞ r supt τ r θ r t VðtÞ, thus, according to Lemma 2, we have VðtÞ r
max
mT τ r θ r mT
VðθÞ expf γ ðt mTÞg;
ð14Þ
where γ is the smallest real root of the equation a1 γ q expfγτg ¼ 0. When mT þ h r t oðm þ 1ÞT, for m ¼ 0,1,2,… from the conditions in Theorem 1, we have a1 ðk ðtÞ kÞ V_ ðtÞ ¼ ∑ eTi ðtÞe_ i ðtÞ ∑ expð a1 tÞ i 2 αi i¼1 i¼1 N
N
l
2
¼ ∑ eTi ðtÞðFðt; xi ðtÞ; xi ðt τÞÞ Fðt; yi ðtÞ; yi ðt τÞÞÞ i¼1
218
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
N
N
þ c ∑ ∑ g ij eTi ðtÞAej ðtÞ i¼1j¼1
N
rp ∑
i¼1
1 eTi ðtÞei ðtÞ þ q 2
N
N
∑
þc ∑
i ¼ 1 j ¼ 1;j a i
N
∑
i¼1
r
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi
eTi ðt
þ γ ðR1 R2 Þhg:
τÞei ðt τÞ
VðtÞ r
N
αgij ‖ei ðtÞ‖2 ‖ej ðtÞ‖2 þ c ∑ gii ρmin eTi ðtÞei ðtÞ
ð19Þ T l So, from VðtÞ ¼ ð1=2Þ∑N i ¼ 1 ei ðtÞei ðtÞ þ ð1=2Þ∑i ¼ 1 expð a1 tÞðki ðtÞ kÞ2 =αi , we have
‖eðtÞ‖2 r 2
mT þ h τ r θ r mT þ h
ð15Þ
VðθÞ expfβ ðt mT hÞg
VðtÞ r max VðθÞ expð γ tÞ: When h r t o T,
τrθr0
VðθÞ expf γ ðt TÞg
When T þh r t o 2T, max
VðθÞ expfβ ðt T hÞg
r max VðθÞ expfβðt T hÞ þ βðT hÞ 2γ ðh τÞg: τrθr0
By the above analogy, we could estimate the value of V(t) for the integer m. When mT rt o mT þh, VðtÞ r max VðθÞ expf γ ðt mTÞ þnβ ðT hÞ mγ ðh τÞg: τrθr0
ð16Þ
As h ¼ R1 T, and τ ¼ R2 T, we have r max VðθÞ expfβð1 R1 Þt þ γ ðR1 R2 Þð t þ hÞg
When mT þh r t o ðm þ 1ÞT, VðtÞ r max VðθÞ expfβðt mT hÞ þmβ ðT hÞ ðm þ 1Þγ ðh τÞg: τrθr0
ð17Þ
τrθr0
2ð1 εÞ
Þ,
ð21Þ
q ¼ max1 r r r n
denotes the maximum eigenvalue of matrix
Letting Γ ¼ p þ cαλ1 and selecting a positive constant a2 which satisfy the condition of a2 ¼ 2Γ þ a1 4 0, then the second inequation of (21) in Corollary 1 holds. The fourth inequation of (21) can be reduced γ ðR1 R2 Þ ð2Γ þqÞð1 R1 Þ 4 0. Thus, we can obtain the following corollary. Corollary 2. Under Assumption 1, given a positive constant a1 4 q. If there exist following conditions 2p þ a1 γ ðR1 R2 Þ ð2Γ þ qÞð1 R1 Þ 4 0 2cα
2ð1 εÞ , max1 r r r n qr ¼ max1 r r r n ∑ns ¼ 1 ksr s imum eigenvalue of matrix G^ ij , 40 is
τrθr0
r max VðθÞ expfβt β R1 t tðγ R1 γ R2 Þg
λ
^s max ðG ij Þ
ð22Þ 2ε
max V ðθÞ expf ½γ ðR1 R2 Þ βð1 R1 Þt þ γ ðR1 R2 Þhg:
τrθr0
2ε
p ¼ max1 r r r n ð1=2Þ∑ns ¼ 1 ð2krs þ ksr
2ð1 εÞ
where p ¼ max1 r r r n pr ¼ max1 r r r n ð1=2Þ∑ns ¼ 1 ð2krs þ ksr
τrθr0
VðtÞ r max VðθÞ expfβt ðm þ 1ÞβR1 T ðm þ 1ÞTðγ R1 γ R2 Þg
N l. Then, we obtain the following corollary.
s
τrθr0
Substitute h ¼ R1 T, and τ ¼ R2 T into (17), we obtain
B Þ, we have Q o 0. If the
s s s Q ¼ ðp þð1=2Þa1 ÞI N l þcαG^ , where G^ ij ¼ G^ l þ i;l þ j for i; j ¼ 1; …;
λmax ðG^ ij Þ r
VðtÞ r max VðθÞ expfmβð1 R1 ÞT mγ ðR1 R2 ÞTg
¼
1 T
γ the smallest real root of the equation a1 γ q expðγτÞ ¼ 0. Then the drive system (1) is synchronous with the response system (2) under the adaptive pinning periodically intermittent controllers (7).
τrθr0
T þhτrθrT þh
when Q o 0 and k 4 λmax ðE BQ
2ð1 εÞ ∑ns ¼ 1 ksr , s G^ ij , 4 0 is
r max VðθÞ expf γ ðt TÞ þ βðT hÞ γ ðh τÞg:
VðtÞ r
matrix Q can be obtained by removing the first 1, 2, …, l rowcolumn pairs of matrix Q. According to Lemma 1, it is clear that
where
When T rt r T þ h, max
matrices with appropriate dimensions, D ¼ diagðk; …; k Þ, and the |fflfflffl{zfflfflffl}l
s 1 1 p þ a1 þ cαλmax ðG^ ij Þ r 0; p ða2 a1 Þ þ cαλ1 r 0a2 4 a1 4q 2 2 γ ðR1 R2 Þ ða2 a1 þ qÞð1 R1 Þ 4 0;
VðθÞ exp½βðt hÞ
r max VðθÞ½β ðt hÞ γ ðh τÞ:
T τ rθrT
Corollary 1. Suppose that τ r h and τ r T h, and h ¼ R1 T; τ ¼ R2 T. If there exist positive constants a1 and a2, such that
τrθr0
VðtÞ r
½γ ðR1 R2 Þ βð1 R1 Þ γ ðR1 R2 Þh tþ 2 2
parameters k is big enough, Q o 0 is equivalent to Q o 0. Here,
In the following, we will estimate the value of V(t) from above. When 0 rt o h,
max
exp
Q can be expressed as Q ¼ ðE BTD QB Þ. In this expression, E and B are
Let β ¼ a2 a1 þ q. Then, according to Lemma 3, we have the following inequality, max
1=2
According to the conditions in Theorem 1, we could obtain the conclusion, and the proof is completed. From the conclusion in Theorem 1, the conditions can be simplified by the Schur complement. Based on the characters of s the parameters in matrix Q ¼ ðp þ ð1=2Þa1 ÞI N þ cαG^ D, the matrix
1 a2 l ðk ðtÞ kÞ2 ∑ expð a2 tÞ i þ a2 eT ðtÞeðtÞ þ 2 2 i¼1 αi
r ða2 a1 ÞVðtÞ þ qV ðt τÞ:
max V ðθÞ
τrθr0
ð20Þ
1 a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i a1 eT ðtÞeðtÞ 2 2 i¼1 αi
hτrθrh
max VðθÞ expf γ ðR1 R2 Þ β ð1 R1 Þt þ γ ðR1 R2 Þhg:
τrθr0
i¼1
1 ^ ¼ eT ðtÞðpI N þ cαGÞeðtÞ þ qeT ðt τÞeðt τÞ 2 a1 l ðki ðtÞ kÞ2 ∑ expð a1 tÞ 2 i¼1 αi s 1 1 r eT ðtÞ p ða2 a1 Þ I N þ cαG^ eðtÞ þ qeT ðt τÞeðt τÞ 2 2
VðtÞ r
ð18Þ
Therefore, for any t Z 0, we have
a1 l ðk ðtÞ kÞ2 ∑ expð a1 tÞ i 2 i¼1 αi
VðtÞ r
max VðθÞ expf ½γ ðR1 R2 Þ βð1 R1 Þt
τrθr0
Þ, q ¼
s λmax ðG^ ij Þ denotes the max-
γ the smallest real root of the equation a1 γ q expðγτÞ ¼ 0. Then the drive system (1) can synchronize with the response system (2) by the adaptive pinning periodically intermittent controllers (7) and the updating laws (8). 4. Numerical simulation In this section, the Chua oscillator with time delay is used as an uncoupled node in the drive and response systems to show the
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
219
effectiveness of the proposed control scheme. A single Chua oscillator is given as follows: x_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt τÞÞ ¼ Cxi ðtÞ þ g 1 ðxi ðtÞÞ þ g 2 ðxi ðt τÞÞ
1 0.5
x2
ð23Þ
where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; xi3 ðtÞÞ A R , g 1 ðxi ðtÞÞ ¼ ð ð1=2Þαðm1 m2 Þðjxi1 ðtÞ þ 1j jxi1 ðtÞ 1jÞ; 0; 0ÞT A R3 , g 2 ðxi ðt τÞÞ ¼ ð0; 0; βϱ sin ðνxi1 ðt τÞÞÞT A R3 , 0 1 α 0 αð1 þ m2 Þ B 1 1 1 C C¼@ A; 0 β w T
0 −0.5 −1 10 5
5 0
0
−5
x3
−10
−5
x1
Fig. 2. Chaotic behavior of time delayed Chua attractors.
3
and α ¼ 10, β ¼19.53, w ¼0.1636, m1 ¼ 1.4325, m2 ¼ 0.7831, ν ¼ 0.5, ϱ ¼0.2, and τ ¼0.02. It has been known that the Chua oscillators exhibit chaotic behavior, and Fig. 2 shows it clearly. Based on the parameters mentioned above, it is easy to verify that f ðt; xi ðtÞ; xi ðt τÞÞ f ðt; y ðtÞ; y ðt τÞÞj 1 1 i i r ½αð1 þ m2 Þ þ 1 αðm2 m1 Þ xi1 ðtÞ yi1 ðtÞ þ α xi2 ðtÞ yi2 ðtÞj; 2
jf 2 ðt; xi ðtÞ; xi ðt τÞÞ f 2 ðt; yi ðtÞ; yi ðt τÞÞj r jxi1 ðtÞ yi1 ðtÞj þ jxi2 ðtÞ yi2 ðtÞj þ jxi3 ðtÞ yi3 ðtÞj; 1 0.8
0.6
0.6
0.4
T=2
0.2 0 −0.2
0 −0.2 −0.4
T=0.2
−0.4 −0.6
T=0.2
0.2
ei3(t)
ei1(t)
0.4
T=2
0
0.5
1
1.5
2
2.5
−0.6
3
Time t
−0.8
Fig. 3. Synchronous errors ei1 ðtÞ; i ¼ 1; 2; …10 between the drive and response systems for 0 r t r 1:5; T ¼ 0:2, and T ¼ 2.
0
0.5
1
1.5
2
2.5
3
Time t Fig. 5. Synchronous errors ei3 ðtÞ; i ¼ 1; 2; …10 between the drive and response systems for 0r t r 1:5; T ¼ 0:2, and T ¼ 2.
1
300
0.8 0.6
250
T=2
0.4
200
0
ki(t)
ei2(t)
0.2
150
−0.2 T=0.2
−0.4
100
−0.6
50
−0.8 −1
0
0.5
1
1.5
2
2.5
3
Time t Fig. 4. Synchronous errors ei2 ðtÞ; i ¼ 1; 2; …10 between the drive and response systems for 0 r t r 1:5; T ¼ 0:2, and T ¼ 2.
0
0
0.5
1
1.5
2
Time t Fig. 6. Adaptive control gain of controllers.
2.5
3
220
M. Zhao et al. / Neurocomputing 144 (2014) 215–221
and jf 3 ðt; xi ðtÞ; xi ðt τÞÞ f 3 ðt; yi ðtÞ; yi ðt τÞÞj
r βϱνjxi1 ðt τÞ yi1 ðt τÞj þ βjxi2 ðtÞ yi2 ðtÞj þ wjxi3 ðtÞ yi3 ðtÞj:
Calculate the value of the parameters, then we have f 11 ¼ α ð1 þ m2 Þ þð1=2Þαðm2 m1 Þ ¼ 5:416, f 12 ¼ α ¼ 10, f 13 ¼ 0, f 21 ¼ f 22 ¼ f 23 ¼ 1, f 31 ¼ βϱν ¼ 1:953, f 32 ¼ β ¼ 19:53, f 33 ¼ w ¼ 0:1636, and ϱ ¼ 1=2. Thus, the value of p and q are p¼22.2284 and q¼30.53. The coupling configuration matrices of the drive and response systems are chosen as a BA scale model. The parameters of BA model are given by m0 ¼m ¼3, N ¼10. In the simulations, we add the adaptive periodical intermittent controllers to the first six nodes. According to the coupling matrix, we known that λ1 ¼0.6790, α ¼1.2 and λmax ¼ 1:4999, then we choose the parameters T ¼0.2, a1 ¼ 60, a2 ¼ 162, the value of work time h will be obtained as h¼ 0.18 and the conditions in Corollary 1 are satisfied. Use the same method, we could obtain the result that the value of work time h will become larger when control period T ¼2. Therefore, there would be some effect on control cost when the control period becomes larger. Select the initial values of the nodes in the complex networks randomly, then the errors of the corresponding nodes in the drive and response systems are shown in Figs. 3, 4 and 5. From the simulations, we could see that there is nearly no change for the time of synchronization errors that tend to zero between drive and response systems when T ¼0.2 and T ¼2. Remark 2. In the former literatures [31,32], Xia and Cai studied the inner synchronization of delayed dynamical nodes via linear periodically pinning intermittent control. In that case, the control 1 gain k obtained in [31,32] needed to satisfy k 4 λmax ðE BQ BT Þ which will be larger than the needed values for practical problems. If we adopt the control method in [31,32], the control gain should satisfy k 4 459:4634. While, in this paper, the numerical simulation of adaptive control gain in Fig. 6 illustrates that our results are less conservative and more practically applicable than it in [31,32]. 5. Conclusions In this paper, we have investigated the problem of outer synchronization between the drive and response systems with time delay dynamical nodes by means of adaptive pinning periodically intermittent controllers. Based on the Lyapunov stability theory, the adaptive control technique and the differential inequality method, some synchronous criteria have been derived analytically. At last, both the theoretical and numerical analysis illustrate the effectiveness of the proposed control methodology.
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 60804006, 71103126 and the Northeastern University Fundamental Research (No. N110404023).
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Mo Zhao was born in Liaoning, China, in 1984. She received her B.S. and M.S. degree in faculty of electrical and control engineering of Liaoning Technical University, Fuxin, China, in 2006 and 2009. She is now a Ph.D candidate at Northeastern University. Her main research interests include synchronization of complex networks, multi-agent system and its applications.
M. Zhao et al. / Neurocomputing 144 (2014) 215–221 Huaguang Zhang was born in Jilin, 1959. He received the BS and MS degrees in control engineering from Northeast Dianli University, Jilin City, China, in 1982 and 1985, respectively, and the PhD degree in thermal power engineering and automation from Southeast University, Nanjing, China, in 1991. In 1992, he joined the Department of Automatic Control, Northeastern University, University, Shenyang, China, as a Postdoctoral Fellow for two years, where, since 1994, he has been a Professor and the Head of the Institute of Electric Automation. He has authored or co-authored more than 100 SCI papers. He is an Associate Editor of Automatica and Neurocomputing. His current research interests include fuzzy control, neural-network-based control, non-linear control, and their applications. Prof. Zhang serves as an Associate Editor of the IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, IEEE Transactions on Fuzzy Systems and IEEE Transactions on Neural Networks. He was awarded a Nationwide Excellent Post-doctor, and the recipient of the Outstanding Youth Science Foundation Award from the National Natural Science Foundation Committee of China in 2003. He was named a Cheung Kong Scholar by the Education Ministry of China in 2005. He is a Deputy Director for Intelligent System Engineering Committee of CAAI.
Zhiliang Wang received the B.S. degree in applied mechanics and the M.S. degree in computational mechanics from Jilin University, Changchun, China, in 1997 and 2000, respectively, and the Ph.D.degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2004. He is currently as an Associate Professor with the School of Information Science and Engineering, Northeastern University. His research interests include nonlinear and adaptive control, and chaos theory and its applications.
221 Hongjing Liang was born in Liaoning, China, in 1986. He received his B.S. degree in mathematics from Bohai University, Jinzhou China, in 2009, and M.S. degree in Fundamental Mathematics from Northeastern University, Shenyang, China, in 2011. He is now a Ph.D candidate at Northeastern University. His main research interests include multi-agent systems, complex systems and output regulation.
