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On pinning synchronization of complex dynamical networks
Automatica 45 (2009) 429–435
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Brief paper
On pinning synchronization of complex dynamical networksI Wenwu Yu a,∗ , Guanrong Chen a , Jinhu Lü b a
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
b
Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
article
info
Article history: Received 17 March 2008 Received in revised form 22 May 2008 Accepted 21 July 2008 Available online 17 December 2008 Keywords: Complex network Pinning control Synchronization Adaptive tuning
a b s t r a c t There exist some fundamental and yet challenging problems in pinning control of complex networks: (1) What types of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What kinds of controllers may be designed to ensure the network synchronization? (3) How large should the coupling strength be used in a given complex network to achieve synchronization? This paper addresses these technique questions. Surprisingly, it is found that a network under a typical framework can realize synchronization subject to any linear feedback pinning scheme by using adaptive tuning of the coupling strength. In addition, it is found that the nodes with low degrees should be pinned first when the coupling strength is small, which is contrary to the common view that the most-highly-connected nodes should be pinned first. Furthermore, it is interesting to find that the derived pinning condition with controllers given in a high-dimensional setting can be reduced to a lowdimensional condition without the pinning controllers involved. Finally, simulation examples of scale-free networks are given to verify the theoretical results. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Many large-scale systems in nature and human societies, such as biological neural networks, ecosystems, metabolic pathways, the Internet, the WWW, electrical power grids, etc., can be described by networks with the nodes representing individuals in the system and the edges representing the connections among them. Recently, the study of various complex networks has attracted increasing attention from researchers in various fields of physics, mathematics, engineering, biology, and sociology. In the early 1960s, Erdös and Rényi (1959, 1960) proposed a random-graph model, which had laid a solid foundation of modern network theory. In a random network, each pair of nodes is connected with a certain probability. To describe a transition from a regular network to a random network, Watts and Strogatz
I This work was supported by the NSFC-HKRGC Joint Research Scheme under Grant N-CityU107/07, the Hong Kong Research Grants Council under the GRF Grant CityU 1117/08E, the National Natural Science Foundation of China under Grants, No. 60221301 and No. 60772158, the National Basic Research (973) Program of China under Grant 2007CB310805, the Important Direction Project of Knowledge Innovation Program of Chinese Academy of Sciences under Grant KJCX3-SYW-S01, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen. ∗ Corresponding author. Tel.: +852 27887791; fax: +852 21942553. E-mail addresses: [email protected] (W. Yu), [email protected] (G. Chen), [email protected] (J. Lü).
0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.07.016
(1998) proposed an interesting small-world network model. Then, Newman and Watts (1999) modified it to generate another variant of the small-world model. Later, Barabási and Albert (1999) proposed a scale-free network model, in which the degree distribution of the nodes follows a power-law form. Thereafter, small-world and scale-free networks have been extensively investigated. Synchronization, on the other hand, is a typical collective behavior in nature. Since the pioneering work of Pecora and Carroll (1990), chaos control and synchronization have received a great deal of attention (Yu & Cao, 2007; Yu, Cao, Wong, & Lü, 2007; Yu, Chen, Cao, Lü, & Parlitz, 2007) due to their potential applications in secure communications, chemical reactions, biological systems, and so on. Typically, there are large numbers of nodes in realworld complex networks. Therefore, a large amount of work has been devoted to the study of synchronization in various largescale complex networks (Cao, Li, & Wang, 2006; W. Lu & T. Chen, 2004; Wang & Cao, 2006; Wang & Chen, 2002a,b; Yu, Cao, & Lü, 2008). In Wang and Chen (2002a,b), local synchronization was investigated by the transverse stability to the synchronization manifold, where synchronization was discussed on small-world and scale-free networks. In Wu (2005) and Wu and Chua (1995), a distance from the collective states to the synchronization manifold was defined, based on which some results were obtained for global synchronization of coupled systems (Cao et al., 2006; W. Lu & T. Chen, 2004; Wang & Cao, 2006). A general criterion was given in Yu et al. (2008), where the network sizes can be extended to be much larger compared to those very small ones studied in Cao
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W. Yu et al. / Automatica 45 (2009) 429–435
et al. (2006), W. Lu and T. Chen (2004) and Wang and Cao (2006). However, it is still very difficult to ensure global synchronization in very large-scale networks due to the computational complexity. In the case where the whole network cannot synchronize by itself, some controllers may be designed and applied to force the network to synchronize. However, it is literally impossible to add controllers to all nodes. To reduce the number of controlled nodes, some local feedback injections may be applied to a fraction of network nodes, which is known as pinning control. In Grigoriev, Cross, and Schuster (1997), pinning control of spatiotemporal chaos, and later in Parekh, Parthasarathy, and Sinha (1998) global and local control of spatiotemporal chaos in coupled map lattices, were discussed. Very recently, in Wang and Chen (2002c), both specific and random pinning schemes were studied, where specific pinning of the nodes with large degrees is shown to require a smaller number of controlled nodes than the random pinning scheme. However, there exist some fundamental and challenging problems which have not been solved to date. A key problem is how the local controllers on the pinned nodes affect the global network synchronization. This paper aims to address the following questions: (1) What kinds of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What types of controllers may be designed to ensure the synchronization? (3) How large should the coupling strength be used for a network with a fixed topological structure to effectively achieve global network synchronization? The rest of the paper is organized as follows. In Section 2, some preliminaries are briefly outlined. The main theorems and corollaries for pinning synchronization on complex networks are given in Section 3. In Section 4, the pinning schemes on some scale-free networks are simulated to verify the theoretical analysis. Conclusions are finally drawn in Section 5. 2. Preliminaries Consider a complex dynamical network consisting of N identical nodes with linearly diffusive coupling (Cao et al., 2006; W. Lu & T. Chen, 2004; J. Lü & G. Chen, 2005; Wang & Chen, 2002a,b; Yu et al., 2008), described by x˙ i (t ) = f (xi (t ), t ) + c
N X
(1)
where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t ))T ∈ Rn is the state vector of the ith node, f : Rn × R+ −→ Rn is a continuously differentiable vector function, c is the coupling strength, Γ ∈ Rn×n is the inner coupling matrix, G = (Gij )N ×N is the coupling configuration matrix representing the topological structure of the network, where Gij are defined as follows: if there exists a connection between node i and node j, then Gij = Gji > 0; otherwise, Gij = Gji = 0 (j 6= i); and the diagonal elements of matrix G are defined by N
X
Gij ,
N X
Gij Γ xj (t ) + ui ,
i = 1, 2, . . . , l,
j =1
x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t ),
j =1
i = l + 1, 2, . . . , N ,
(5)
where ui = −cdi Γ (xi − s(t )) ∈ Rn ,
i = 1, 2, . . . , l,
(6)
are n-dimensional linear feedback controllers with all the control gains di > 0. The objective of control here is to find some appropriate controllers (6) such that the solutions of the controlled network (5) synchronize with the solution of (4), in the sense that lim kxi (t ) − s(t )k = 0,
t →∞
i = 1, 2, . . . , N .
(7)
When the controlled complex network (5) achieves synchronization, the coupling terms and control inputs will automatically vanPN ish due to the diffusive condition j=1 Gij = 0. This indicates that any solution xi (t ) of any single node is also a solution of the synchronized coupled network. Next, the pinning synchronization of network (5) is investigated. Subtracting (4) from (5) yields the following error dynamical network: N X
Gij Γ ej (t )
j =1
j=1,j6=i
Gii = −
x˙ i (t ) = f (xi (t ), t ) + c
e˙ i (t ) = f (xi (t ), t ) − f (s(t ), t ) + c
Gij Γ (xj (t ) − xi (t )),
i = 1, 2, . . . , N ,
In the following, the synchronization of complex network model (3) is investigated. To realize the synchronization of network (3), some pinning controllers will be added to part of its nodes if the network is not self-synchronized. Here, the pinning strategy is applied on a small fraction δ (0 < δ < 1) of the nodes in network (3). Suppose that the nodes i1 , i2 , . . . , il are selected, where l = xδ N y represents the integer part of the real number δ N. Without loss of generality, rearrange the order of the nodes in the network, and let the first l nodes be controlled. Thus, the pinning controlled network can be described by
(2)
j=1,j6=i
− cdi Γ ei (t ),
i = 1, 2, . . . , l,
e˙ i (t ) = f (xi (t ), t ) − f (s(t ), t ) + c
N X
Gij Γ ej (t ),
j =1
i = l + 1, 2, . . . , N ,
(8)
where ei (t ) = xi (t ) − s(t ), i = 1, 2, . . . , N. Throughout the rest of the paper, the following assumption is needed: Assumption. There exists a constant matrix K such that
(x − y)T (f (x, t ) − f (y, t )) ≤ (x − y)T K Γ (x − y), ∀x, y ∈ Rn .
(9)
(4)
Note that Assumption (9) is very mild. For example, all linear and piecewise linear functions satisfy this condition. In addition, if ∂ fi /∂ xj (i, j = 1, 2, . . . , n) are bounded and Γ is positive definite, the above condition is satisfied. So, it includes many wellknown systems, such as the Lorenz system (Lorenz, 1963), Chen system (Chen & Ueta, 1999), Lü system (Lü & Chen, 2002), recurrent neural networks (Yu, Cao, & Chen, 2008), Chua’s circuit, and so on, some of which will be used for simulation in Section 4 below.
Here, s(t ) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. In this paper, the undirected networks are considered.
Hereafter, suppose that network (3) is connected, therefore there are no isolated clusters. From Theorem 6.2.24 (Horn & Johnson, 1985), a connected network is equivalent to an irreducible
PN
which ensures the diffusion that j=1 Gij = 0. Equivalently, network (1) can be rewritten in a simpler form as follows: x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t ),
i = 1, 2, . . . , N .
(3)
j =1
Note that a solution s(t ) of an isolated node satisfies s˙(t ) = f (s(t ), t ).
W. Yu et al. / Automatica 45 (2009) 429–435
network. Thus, only connected networks are considered throughout the paper. Let λmax (F ) be the largest eigenvalue of matrix F , and F T be the transpose of matrix F . For matrices e A and e B with the same order, e A >e B denotes that e A −e B is a positive definite matrix. Lemma 1 (Schur complement (Boyd, Ghaoui, Feron, & Balakrishnan, 1994)). The following linear matrix inequality (LMI)
Q(x) S (x)T
S ( x) R ( x)
> 0,
where Q(x) = Q(x)T , R(x) = R(x)T , is equivalent to one of the following conditions: (i) Q(x) > 0, R(x) − S (x)T Q(x)−1 S (x) > 0; (ii) R(x) > 0, Q(x) − S (x)R(x)−1 S (x)T > 0. Lemma 2 (Chen, Liu, & Lu, 2007). If G is irreducible, Gij = Gji ≥ 0 for i 6= j, and j=1 Gij = 0, for all i = 1, 2, . . . , N, then all eigenvalues of the matrix
PN
G11 − ε G21
G12 G22
.. .
.. .
GN1
GN2
... ... .. . ...
431
From (10), it is easy to see that (7) is satisfied, so the undirected network (5) is globally synchronized under the given linear feedback pinning controllers. The proof is completed. Note that the dimension of the matrix condition in (10) is N × n, so it is not easy to verify this condition. Let Γ be a positive definite T
matrix. If K and Γ are commutable and let θ = λmax ( K +2K ), then it can be considered as the passivity degree (Xiang & Chen, 2007), where θ > 0 means that each node needs energy from outside to stabilize the network, while θ < 0 means that each node itself is already stable. Here, only θ = kK k > 0 is considered throughout the paper for simplicity; otherwise, the network is synchronized even without pinning control. Corollary 1. Suppose that Assumption (9) holds and Γ is a positive definite matrix. The controlled undirected network (5) is globally synchronized if the following condition is satisfied: C < 0,
(13)
where C = θ IN + c (G − D) with θ = kK k and K defined in (9).
G1N G2N
Proof. First, one has
.. .
IN ⊗ (K Γ ) + c (G − D) ⊗ Γ ≤ (θ IN + cG − cD) ⊗ Γ .
(14)
If θ IN + cG − D < 0 in (13), and Γ > 0, then one can easily show (Lütkepohl, 1996) that
GNN
are negative for any positive constant ε .
(θ IN + cG − cD) ⊗ Γ < 0.
3. Pinning synchronization criteria for complex networks
So, condition (10) is satisfied, completing the proof.
In this section, some pinning criteria are established to ensure the global synchronization of complex dynamical networks. 3.1. General pinning synchronization criteria In this subsection, some general pinning synchronization criteria are derived. Theorem 1. Suppose that Assumption (9) holds. The controlled undirected network (5) is globally synchronized if the following condition is satisfied: IN ⊗ (K Γ ) + c (G − D) ⊗ Γ < 0,
(10)
where ⊗ is the Kronecker product, D
=
diag(d1 , . . . , dl ,
| 0, . . . , 0), and IN is the N-dimensional identity matrix. | {z }
{z l
}
N −l
Condition (13) is a general criterion to ensure the network pinning synchronization (Chen et al., 2007; Li, Wang, & Chen, 2004; Wang & Chen, 2002c). Remark 1. Though the condition C < 0 in (13) is very simple, it is still difficult to choose an appropriate gain matrix D. One may attempt to use the MATLAB LMI Toolbox to solve (13). In the previous works (Cao et al., 2006; W. Lu & T. Chen, 2004; Wang & Cao, 2006), LMI approach was used to study the global synchronization of linearly coupled networks with delays, and it is solvable only for networks of extremely small sizes. However, in Yu et al. (2008), global synchronization of linearly hybrid coupled networks with time-varying delays was investigated, where the network size can be extended to be much larger. If the network size N is very large, the computation of LMI conditions is difficult in finite time. Therefore, it is still a challenging problem to find some appropriate pinning controllers to achieve global network synchronization with low computational complexity.
Proof. Consider the Lyapunov functional candidate: V (t ) =
N 1X
2 i=1
3.2. Coupling strength in pinning control eTi (t )ei (t ).
(11)
The derivative of V (t ) along the trajectories of (8) gives V˙ =
N X
eTi
In this subsection, some criteria for choosing an appropriate coupling strength are established. Corollary 2. Suppose that Assumption (9) holds and Γ is a positive definite matrix. The controlled undirected network (5) is globally synchronized if the following condition is satisfied:
(t )˙ei (t )
i =1
=
N X
" eTi
(t ) f (xi (t ), t ) − f (s(t ), t ) + c
i =1
N X
#
c>
Gij Γ ej (t )
j =1 l
−c
X
di eTi
(t )Γ ei (t )
i=1
≤
N X
" eTi
(t ) K Γ ei (t ) + c
i =1
N X
# Gij Γ ej (t ) − c
j=1
l X
di eTi (t )Γ ei (t )
i=1
= e (t ) [(IN ⊗ K Γ ) + c (G ⊗ Γ ) − c (D ⊗ Γ )] e(t ), T
where e(t ) = ( (t ), eT1
eT2
(t ), . . . ,
eTN
(t )) . T
(12)
θ . |λmax (G − D)|
(15)
Proof. By Lemma 2, it is easy to see that G − D∗ is negative definite, where D∗ corresponds to only one controller, i.e., l = 1. It is easy to see that λmax (G − D) ≤ λmax (G − D∗ ) < 0. From (15), one has c (G − D) + θ IN ≤ c λmax (G − D) + θ < 0, so (13) is satisfied. The proof is completed. Remark 2. In Chen et al. (2007), pinning complex networks by a single controller, i.e., l = 1, was studied. However, it requires a very large coupling strength in general, which may not be very practical.
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W. Yu et al. / Automatica 45 (2009) 429–435
It is easy to see that the theoretical coupling strength given in (15) is too conservative, usually much larger than the needed value. Clearly, it is desirable to make the coupling strength as small as possible. Here, the adaptive technique (Chen et al., 2007; Yu, Cao, & Chen, 2007; Yu, Cao, Wong et al., 2007; Yu, Chen et al., 2007; Zhou, Lu, & Lü, 2006) is adopted to achieve this goal. The selected pinning controllers in (5), associated with the adaptive coupling law, lead to N X
x˙ i (t ) = f (xi (t ), t ) + c (t )
Gij Γ xj (t )
j =1
− c (t )di Γ (xi (t ) − s(t )), N X
x˙ i (t ) = f (xi (t ), t ) + c (t )
3.3. Selective pinning scheme In this subsection, the pinning scheme is designed when the network structure and coupling strength are fixed. PN Let C = (Cij )N ×N in (13) and ki = j=1,j6=i Gij be the total weights between node i and all the other nodes, where ki is the degree of node i. It is easy to see that Cij = cGij (i 6=j), Cii = θ −cki −cdi (1 ≤ i ≤ l), and Cii = θ − cki (l + 1 ≤ i ≤ N), where θ = kK k with K defined in (9). Corollary 3. Suppose that Assumption (9) holds and Γ is a positive definite matrix. To satisfy condition (13), it is necessary that
i = 1, 2, . . . , l,
Cii = θ − cki − cdi < 0,
Gij Γ xj (t ),
Cii = θ − cki < 0,
j =1
i = l + 1, 2, . . . , N ,
1 ≤ i ≤ l,
l + 1 ≤ i ≤ N,
(19)
where C = θ IN + cG − cD, and θ = kK k with K defined in (9).
N X c˙ (t ) = α (xj (t ) − s(t ))T Γ (xj (t ) − s(t )),
(16)
Proof. To ensure C = θ IN + cG − D < 0, it is necessary that Cii < 0. This completes the proof.
j =1
where α is a small positive constant. Theorem 2. Suppose that Assumption (9) holds and Γ is a positive definite matrix. Then, the adaptively controlled undirected network (16) is globally synchronized for a small constant α > 0. Proof. Consider the Lyapunov functional candidate: V (t ) =
N 1X
2 i =1
eTi (t )ei (t ) +
β (c (t ) −e c )2 , 2α
(17)
where β and e c are positive constants to be determined below. The derivative of V (t ) along the trajectories of (16) gives V˙ =
N X
eTi (t )˙ei (t ) + β(c (t ) −e c)
N X
i=1
=
X
Up to this point, it is still not very clear as how to determine the control gains. Next, a theorem is given this concern. to address
ej (t )T Γ ej (t )
Let C = θ IN + cG − cD =
j=1
"
N
i=1
+ c (t )
# Gij Γ ej (t ) − c (t )
j =1
+ β(c (t ) −e c)
l X
ej (t )T Γ ej (t )
e C < 0.
≤ eT (t )[(θ IN + c (t )(G − D + β IN ) − βe c ) ⊗ Γ ]e(t ),
(18)
where e(t ) = ( (t ), (t ), . . . , (t )) . From Lemma 2, one can see that G − D + β IN is negative definite when β is sufficiently small. If one chooses a large initial condition c (0) and let α be small enough, then c (t ) increases very slowly. In this case, the synchronization can be achieved by Corollary 2 when c is very large. After reaching synchronization, c (t ) converges to a constant and is bounded. Since c (t ) is bounded, one can always choose a sufficiently largee c such that θ IN +c (t )(G−D+β IN )−βe c< 0. Then, in the whole time evolution process, V˙ is always negative definite. Consequently, the network is synchronized, so the third equation in (16) will vanish, leading to c (t ) → constant. This completes the proof. eT2
eTN
B
, where A and B are e C
Theorem 3. Suppose that Assumption (9) holds and Γ is a positive definite matrix. If the control gains di can be sufficiently large, condition (13), namely, C < 0, is equivalent to
j=1
eT1
BT
matrix C . di eTi (t )Γ ei (t )
i =1 N X
A −e D
matrices with appropriate dimensions, e D = diag(d1 , . . . , dl ), and e C is obtained by removing the 1, 2, . . . , l row–column pairs of
eTi (t ) f (xi (t ), t ) − f (s(t ), t ) N X
Remark 4. From (19), one can see that for the nodes without pinning controllers, it is necessary that their degrees ki > θ /c under condition (13). When the coupling strength c is very large, the degree of each node satisfies (19). In Chen et al. (2007), Li et al. (2004), Wang and Cao (2006), Xiang and Chen (2007, in press), Xiang, Liu, Chen, Chen, and Yuan (2007) and Zhou, Lu, and Lü (2008), condition (19) is assumed to be satisfied in the proposed pinning schemes, namely, the coupling strength c is very large therein. As seen here, interestingly, the nodes with low degrees should be controlled first, if c is not very large. It is common that the nodes with low degrees receive very little information from other nodes which can not be used to synchronize with the network due to the nonlinearity of the single node.
T
Remark 3. It is noted that the condition in Theorem 2 is very mild, which is independent of the network structure (G) and the fraction of the nodes (l). Under any fixed network structure and pinning controllers in the form of (6), the coupling strengths can be selfadaptively determined according to (16) by Theorem 2 to realize network synchronization.
(20)
For simplicity, one can select e D > λmax (A − Be C −1 BT )Il . Proof. It is easy to see that if C < 0 then e C < 0. So, one only needs to prove that if e C < 0 then C < 0. Choose e D > A − Be C −1 BT . Then, by Lemma 1, one indeed has C < 0. The proof is completed. From the definition of e C , one has e C = θ IN −l + ce G, where e Gij = Gl+i,l+j for i, j = l + 1, . . . , N. Corollary 4. Suppose that Assumption (9) holds and Γ is a positive definite matrix. The controlled undirected network (5) is globally synchronized if e D > λmax (A − Be C −1 BT )Il and one of the following conditions holds:
(i) c λmax (e G) + θ < 0, e (ii) λmax (G) < −θ /c , θ (iii) c > . |λmax (e G)|
(21)
Remark 5. Condition (20) is very simple since the pinning controllers are not involved, which provides some guidance to choose the control gains in e D and therefore is very useful.
W. Yu et al. / Automatica 45 (2009) 429–435
Condition (i) in (21) provides a criterion for ensuring the network synchronization with fixed network structure, coupling strength and pinning scheme in the form of (6); (ii) shows that one can choose some appropriate pinning controllers to achieve network synchronization with fixed network structure and coupling strength; (iii) proposes a way to choose the coupling strength with fixed network structure and pinning scheme (6). Remark 6. In Chen et al. (2007), Li et al. (2004), Wang and Cao (2006), Xiang and Chen (2007, in press), Xiang et al. (2007) and Zhou et al. (2008), similar pinning control results as Corollaries 1 and 2 were given. In this paper, conditions (i) and (iii) are considered with a fixed pinning scheme in the form of (6). However, sometimes condition (i) is not satisfied under some fixed pinning schemes when the coupling strength is not very large. In this case, the pinning scheme will be very important. In Li et al. (2004) and Wang and Chen (2002c), some pinning schemes, such as selective pinning scheme (nodes with degrees in order), and random pinning scheme, are proposed. Nevertheless, it is still too difficult to choose a minimum number of controllers to achieve network synchronization in general. A lower bound for the minimum number of pinning controllers is given in Xiang and Chen (2007) as follows: Proposition 1. If the matrix θ IN + cG has m non-negative eigenvalues, and if C = θ IN + c (G − D) < 0, then the number of nodes to be selected for control cannot be less than m. The theoretical analysis of this proposition is not difficult. However, we are interested in the question if these m controllers exist and how to find them. From simulation results, we found that in many cases these m controllers do not exist, namely, this necessary condition may not be very useful. If there exist some nodes with degrees ki ≤ θ /c, and the network cannot achieve synchronization without controlling these nodes by using criterion (13) or (20), then this means that when the coupling strength is very small, the nodes with lower degrees should be controlled first. Some detailed analysis will be given through simulations in Section 4. Similarly, the gain matrix e D > λmax (A − Be C −1 BT )Il in (20) may be much larger than the needed value, so the adaptive control approach is applied here (Chen et al., 2007; Yu, Cao & Chen, 2007; Yu, Cao, Wong et al., 2007; Yu, Chen et al., 2007; Zhou et al., 2006). The pinning controllers selected by (20) yield the following controlled network: x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t )
j=1
− cdi (t )Γ (xi (t ) − s(t )), x˙ i (t ) = f (xi (t ), t ) + c
N X
i = 1, 2, . . . , l,
Gij Γ xj (t ),
i = l + 1, 2, . . . , N ,
j=1
d˙ i (t ) = qi ei (t )T Γ ei (t ),
i = 1, 2, . . . , l,
(22)
where qi are positive constants. Corollary 5. Suppose that Assumption (9) holds and Γ is a positive definite matrix. If the condition in (20) is satisfied, then the undirected network (22) is globally synchronized under the adaptive scheme (22).
433
Taking the derivative of V (t ) along the trajectories of (22) gives V˙ =
N X
eTi (t )˙ei (t ) + c
l X (di (t ) − d)ei (t )T Γ ei (t )
i=1
=
N X
i=1
" eTi
(t ) f (xi (t ), t ) − f (s(t ), t ) + c
i=1
−c
N X
# Gij Γ ej (t )
j =1 l X
di (t )eTi (t )Γ ei (t ) + c
i=1
l X (di (t ) − d)ei (t )T Γ ei (t ) i =1
≤ e (t )[(θ IN + c (G − de IN )) ⊗ Γ ]e(t ), T
where e(t ) = ( (t ), eT1
eT2
(t ), . . . ,
eTN
(24)
(t )) and e IN = diag(1, . . . , 1, | {z } T
l
0, . . . , 0).
| {z } N −l
From Theorem 3, it follows that e C < 0, and by choosing d > λmax (A − Be C −1 BT ), network (22) is globally synchronized. This completes the proof.
4. Simulation examples In this section, some simulation examples are given to verify the criteria established above. In 1963, Lorenz found the first chaotic system (Lorenz, 1963). Then, in 1999, Chen and Ueta found the dual of the Lorenz system (Chen & Ueta, 1999). Later in 2002, Lü and Chen discovered another new chaotic system (Lü & Chen, 2002), which bridges the gap between the Lorenz system and the Chen system. Generally, chaotic systems are more difficult to synchronize than non-chaotic systems. Here, consider network (3) that consists of N identical Chen systems, described as follows: x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t ),
i = 1, 2, . . . , N ,
(25)
j=1
where Γ = diag(1, 2, 1), and
( f (xi , t ) =
35(xi2 − xi1 ), −7xi1 − xi1 xi3 + 28xi2 , xi1 xi2 − 3xi3 .
(26)
From numerical simulation, it is found that there exist some constants M1 = 23, M2 = 32, and M3 = 61, such that the chaotic attractor (s1 , s2 , s3 ) of the Chen system satisfies |s1 | < M1 , |s2 | < M2 , and |s3 | < M3 . By (4) and Lemma 2, one has
(xi − s)T (f (xi , t ) − f (s, t )) = eTi (35ei2 − 35ei1 , −7ei1 + 28ei2 − xi1 xi3 + s1 s3 , −3ei3 + xi1 xi2 − s1 s2 ) = −35e2i1 + (28 + M3 )|ei1 ei2 | + 28e2i2 − 3e2i3 + M2 |ei1 ei3 | 28 + M3 M2 ≤ −35 + ρ +η e2i1 2 2 28 + M3 M2 + 28 + e2i2 + −3 + e2i3 2ρ 2η ≤ θ (e2i1 + 2e2i2 + e2i3 ).
(27)
Choosing ρ = 1.3139 and η = 0.4715, one obtains θ = 30.9342. 4.1. Pinning scheme on scale-free networks
Proof. Consider the Lyapunov functional candidate: V (t ) =
N 1X
2 i=1
eTi
(t )ei (t ) +
l X c i=1
2qi
(di (t ) − d) , 2
where d is a positive constant to be determined below.
(23)
In the simulated scale-free network, N = 1000, m0 = m = 3, which contains about 3000 connections. We performed a simulation-based analysis on the controlled scale-free complex network of Chen systems by using lowdegree, high-degree, and random pinning schemes, respectively.
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W. Yu et al. / Automatica 45 (2009) 429–435
use the high-degree pinning scheme when a small fraction δ is controlled since some particular nodes in the scale-free network have very large degrees, which is known to be fragile to selective attack (Wang & Chen, 2002a). 4.2. Adaptive coupling strength
Fig. 1. Orbits of λmax (e G) as functions of the fraction δ by low-degree, high-degree, and random pinning schemes on a scale-free network of Chen systems.
In this simulation, the coupling strength is designed by Theorem 2 based on the adaptive technique for a scale-free network of Chen systems, where N = 100, and m0 = m = 3. Clustering coefficient is a characteristic property of complex networks, which reflects the group behavior in the network. In this example, the pinning controllers are added to some nodes (δ = 0.2) with the largest clustering coefficients. The states of error ei (t ) and the coupling strength are illustrated in Figs. 2 and 3, respectively, which show that the controlled network is globally synchronized and the designed coupling strength c = 6.1197 is very effective. 5. Conclusions
Fig. 2. Error states ei (t ) in a scale-free network of Chen systems.
In this paper, pinning synchronization of a class of complex dynamical networks has been investigated in detail. A general criterion for ensuring network synchronization has been derived. Some analytical and adaptive techniques have been proposed to obtain appropriate coupling strengths for achieving network synchronization. It is surprising to find that a network can realize synchronization under any linear feedback pinning scheme by adaptively adjusting the coupling strength. Furthermore, some effective pinning schemes have been designed for networks with fixed structure and coupling strength, where the original condition is shown to be equivalent to a simple condition without controls. It is found that the nodes with low degrees should be first pinned when the coupling strength is very small, which is contrary to the common belief that most-highly-connected nodes should be pinned first. Finally, some examples on scale-free networks have been simulated, which verify well the theoretical analysis. References
Fig. 3. Coupling strength c (t ) in a scale-free network of Chen systems.
The orbits of λmax (e G) as functions of the fraction δ by lowdegree, high-degree, and random pinning schemes are illustrated in Fig. 1, respectively. There are 392 nodes with degree 3, and 188 nodes with degree 4. By controlling these nodes, the network synchronizability is greatly improved. However, it is better to
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435 Wenwu Yu received the B.S. degree in information and computing science and M.S. degree in applied mathematics from the Department of Mathematics, Southeast University, Nanjing, China, in 2004 and 2007, respectively. Currently, he is working towards the Ph.D. degree at the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. He held several visiting positions in Hong Kong, USA, and China. His research interests include multi-agent systems, nonlinear dynamics and control, complex networks and systems, neural networks, cryptography, and communications. Guanrong Chen received his M.Sc. degree in computer science from Zhongshan (Sun Yat-sen) University, China in Fall 1981 and Ph.D. degree in applied mathematics from Texas A & M University in Spring 1987. He is currently a chair professor and the founding director of the Centre for Chaos and Complex Networks at the City University of Hong Kong, prior to which he was a tenured Full Professor at the University of Houston, Texas, USA. He is a Fellow of the IEEE (Jan. 1997), with research interests in chaotic dynamics, complex networks and nonlinear controls.
Jinhu Lü received the Ph.D degree in applied mathematics from the Chinese Academy of Sciences, Beijing, China in 2002. Currently, he is an Associate Professor with the Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China. He held several visiting positions in Australia, Canada, France, Germany, Hong Kong and USA, and was a Visiting Fellow in Princeton University, USA from 2005 to 2006. He is the author of two research monographs and more than 70 international journal papers published in the fields of nonlinear circuits and systems, complex networks and complex systems. He served as a member in the Technical Committees of several international conferences and is now serving as a member in the Technical Committees of Nonlinear Circuits and Systems and of Neural Systems and Applications in the IEEE Circuits and Systems Society. He is also an Associate Editor of IEEE Transactions on Circuits and Systems II, Journal of Systems Science and Complexity and DCDIS-A. Dr. Lü received the prestigious Presidential Outstanding Research Award from the Chinese Academy of Sciences in 2002, the National Best Ph.D. Theses Award from the Office of Academic Degrees Committee of the State Council and the Ministry of Education of China in 2004, the First Prize of Natural Science Award from the Ministry of Education of China in 2007 and the Lu Jiaxi Youth Talent Award from the Chinese Academy of Sciences in 2008. He is the co-author of the Most Cited SCI Paper of Chinese Scholars in the field of mathematics during the periods of 2001–2005 and 2002–2006. He is also an IEEE Senior Member.
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Brief paper
On pinning synchronization of complex dynamical networksI Wenwu Yu a,∗ , Guanrong Chen a , Jinhu Lü b a
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
b
Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
article
info
Article history: Received 17 March 2008 Received in revised form 22 May 2008 Accepted 21 July 2008 Available online 17 December 2008 Keywords: Complex network Pinning control Synchronization Adaptive tuning
a b s t r a c t There exist some fundamental and yet challenging problems in pinning control of complex networks: (1) What types of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What kinds of controllers may be designed to ensure the network synchronization? (3) How large should the coupling strength be used in a given complex network to achieve synchronization? This paper addresses these technique questions. Surprisingly, it is found that a network under a typical framework can realize synchronization subject to any linear feedback pinning scheme by using adaptive tuning of the coupling strength. In addition, it is found that the nodes with low degrees should be pinned first when the coupling strength is small, which is contrary to the common view that the most-highly-connected nodes should be pinned first. Furthermore, it is interesting to find that the derived pinning condition with controllers given in a high-dimensional setting can be reduced to a lowdimensional condition without the pinning controllers involved. Finally, simulation examples of scale-free networks are given to verify the theoretical results. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Many large-scale systems in nature and human societies, such as biological neural networks, ecosystems, metabolic pathways, the Internet, the WWW, electrical power grids, etc., can be described by networks with the nodes representing individuals in the system and the edges representing the connections among them. Recently, the study of various complex networks has attracted increasing attention from researchers in various fields of physics, mathematics, engineering, biology, and sociology. In the early 1960s, Erdös and Rényi (1959, 1960) proposed a random-graph model, which had laid a solid foundation of modern network theory. In a random network, each pair of nodes is connected with a certain probability. To describe a transition from a regular network to a random network, Watts and Strogatz
I This work was supported by the NSFC-HKRGC Joint Research Scheme under Grant N-CityU107/07, the Hong Kong Research Grants Council under the GRF Grant CityU 1117/08E, the National Natural Science Foundation of China under Grants, No. 60221301 and No. 60772158, the National Basic Research (973) Program of China under Grant 2007CB310805, the Important Direction Project of Knowledge Innovation Program of Chinese Academy of Sciences under Grant KJCX3-SYW-S01, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen. ∗ Corresponding author. Tel.: +852 27887791; fax: +852 21942553. E-mail addresses: [email protected] (W. Yu), [email protected] (G. Chen), [email protected] (J. Lü).
0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.07.016
(1998) proposed an interesting small-world network model. Then, Newman and Watts (1999) modified it to generate another variant of the small-world model. Later, Barabási and Albert (1999) proposed a scale-free network model, in which the degree distribution of the nodes follows a power-law form. Thereafter, small-world and scale-free networks have been extensively investigated. Synchronization, on the other hand, is a typical collective behavior in nature. Since the pioneering work of Pecora and Carroll (1990), chaos control and synchronization have received a great deal of attention (Yu & Cao, 2007; Yu, Cao, Wong, & Lü, 2007; Yu, Chen, Cao, Lü, & Parlitz, 2007) due to their potential applications in secure communications, chemical reactions, biological systems, and so on. Typically, there are large numbers of nodes in realworld complex networks. Therefore, a large amount of work has been devoted to the study of synchronization in various largescale complex networks (Cao, Li, & Wang, 2006; W. Lu & T. Chen, 2004; Wang & Cao, 2006; Wang & Chen, 2002a,b; Yu, Cao, & Lü, 2008). In Wang and Chen (2002a,b), local synchronization was investigated by the transverse stability to the synchronization manifold, where synchronization was discussed on small-world and scale-free networks. In Wu (2005) and Wu and Chua (1995), a distance from the collective states to the synchronization manifold was defined, based on which some results were obtained for global synchronization of coupled systems (Cao et al., 2006; W. Lu & T. Chen, 2004; Wang & Cao, 2006). A general criterion was given in Yu et al. (2008), where the network sizes can be extended to be much larger compared to those very small ones studied in Cao
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W. Yu et al. / Automatica 45 (2009) 429–435
et al. (2006), W. Lu and T. Chen (2004) and Wang and Cao (2006). However, it is still very difficult to ensure global synchronization in very large-scale networks due to the computational complexity. In the case where the whole network cannot synchronize by itself, some controllers may be designed and applied to force the network to synchronize. However, it is literally impossible to add controllers to all nodes. To reduce the number of controlled nodes, some local feedback injections may be applied to a fraction of network nodes, which is known as pinning control. In Grigoriev, Cross, and Schuster (1997), pinning control of spatiotemporal chaos, and later in Parekh, Parthasarathy, and Sinha (1998) global and local control of spatiotemporal chaos in coupled map lattices, were discussed. Very recently, in Wang and Chen (2002c), both specific and random pinning schemes were studied, where specific pinning of the nodes with large degrees is shown to require a smaller number of controlled nodes than the random pinning scheme. However, there exist some fundamental and challenging problems which have not been solved to date. A key problem is how the local controllers on the pinned nodes affect the global network synchronization. This paper aims to address the following questions: (1) What kinds of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What types of controllers may be designed to ensure the synchronization? (3) How large should the coupling strength be used for a network with a fixed topological structure to effectively achieve global network synchronization? The rest of the paper is organized as follows. In Section 2, some preliminaries are briefly outlined. The main theorems and corollaries for pinning synchronization on complex networks are given in Section 3. In Section 4, the pinning schemes on some scale-free networks are simulated to verify the theoretical analysis. Conclusions are finally drawn in Section 5. 2. Preliminaries Consider a complex dynamical network consisting of N identical nodes with linearly diffusive coupling (Cao et al., 2006; W. Lu & T. Chen, 2004; J. Lü & G. Chen, 2005; Wang & Chen, 2002a,b; Yu et al., 2008), described by x˙ i (t ) = f (xi (t ), t ) + c
N X
(1)
where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t ))T ∈ Rn is the state vector of the ith node, f : Rn × R+ −→ Rn is a continuously differentiable vector function, c is the coupling strength, Γ ∈ Rn×n is the inner coupling matrix, G = (Gij )N ×N is the coupling configuration matrix representing the topological structure of the network, where Gij are defined as follows: if there exists a connection between node i and node j, then Gij = Gji > 0; otherwise, Gij = Gji = 0 (j 6= i); and the diagonal elements of matrix G are defined by N
X
Gij ,
N X
Gij Γ xj (t ) + ui ,
i = 1, 2, . . . , l,
j =1
x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t ),
j =1
i = l + 1, 2, . . . , N ,
(5)
where ui = −cdi Γ (xi − s(t )) ∈ Rn ,
i = 1, 2, . . . , l,
(6)
are n-dimensional linear feedback controllers with all the control gains di > 0. The objective of control here is to find some appropriate controllers (6) such that the solutions of the controlled network (5) synchronize with the solution of (4), in the sense that lim kxi (t ) − s(t )k = 0,
t →∞
i = 1, 2, . . . , N .
(7)
When the controlled complex network (5) achieves synchronization, the coupling terms and control inputs will automatically vanPN ish due to the diffusive condition j=1 Gij = 0. This indicates that any solution xi (t ) of any single node is also a solution of the synchronized coupled network. Next, the pinning synchronization of network (5) is investigated. Subtracting (4) from (5) yields the following error dynamical network: N X
Gij Γ ej (t )
j =1
j=1,j6=i
Gii = −
x˙ i (t ) = f (xi (t ), t ) + c
e˙ i (t ) = f (xi (t ), t ) − f (s(t ), t ) + c
Gij Γ (xj (t ) − xi (t )),
i = 1, 2, . . . , N ,
In the following, the synchronization of complex network model (3) is investigated. To realize the synchronization of network (3), some pinning controllers will be added to part of its nodes if the network is not self-synchronized. Here, the pinning strategy is applied on a small fraction δ (0 < δ < 1) of the nodes in network (3). Suppose that the nodes i1 , i2 , . . . , il are selected, where l = xδ N y represents the integer part of the real number δ N. Without loss of generality, rearrange the order of the nodes in the network, and let the first l nodes be controlled. Thus, the pinning controlled network can be described by
(2)
j=1,j6=i
− cdi Γ ei (t ),
i = 1, 2, . . . , l,
e˙ i (t ) = f (xi (t ), t ) − f (s(t ), t ) + c
N X
Gij Γ ej (t ),
j =1
i = l + 1, 2, . . . , N ,
(8)
where ei (t ) = xi (t ) − s(t ), i = 1, 2, . . . , N. Throughout the rest of the paper, the following assumption is needed: Assumption. There exists a constant matrix K such that
(x − y)T (f (x, t ) − f (y, t )) ≤ (x − y)T K Γ (x − y), ∀x, y ∈ Rn .
(9)
(4)
Note that Assumption (9) is very mild. For example, all linear and piecewise linear functions satisfy this condition. In addition, if ∂ fi /∂ xj (i, j = 1, 2, . . . , n) are bounded and Γ is positive definite, the above condition is satisfied. So, it includes many wellknown systems, such as the Lorenz system (Lorenz, 1963), Chen system (Chen & Ueta, 1999), Lü system (Lü & Chen, 2002), recurrent neural networks (Yu, Cao, & Chen, 2008), Chua’s circuit, and so on, some of which will be used for simulation in Section 4 below.
Here, s(t ) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. In this paper, the undirected networks are considered.
Hereafter, suppose that network (3) is connected, therefore there are no isolated clusters. From Theorem 6.2.24 (Horn & Johnson, 1985), a connected network is equivalent to an irreducible
PN
which ensures the diffusion that j=1 Gij = 0. Equivalently, network (1) can be rewritten in a simpler form as follows: x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t ),
i = 1, 2, . . . , N .
(3)
j =1
Note that a solution s(t ) of an isolated node satisfies s˙(t ) = f (s(t ), t ).
W. Yu et al. / Automatica 45 (2009) 429–435
network. Thus, only connected networks are considered throughout the paper. Let λmax (F ) be the largest eigenvalue of matrix F , and F T be the transpose of matrix F . For matrices e A and e B with the same order, e A >e B denotes that e A −e B is a positive definite matrix. Lemma 1 (Schur complement (Boyd, Ghaoui, Feron, & Balakrishnan, 1994)). The following linear matrix inequality (LMI)
Q(x) S (x)T
S ( x) R ( x)
> 0,
where Q(x) = Q(x)T , R(x) = R(x)T , is equivalent to one of the following conditions: (i) Q(x) > 0, R(x) − S (x)T Q(x)−1 S (x) > 0; (ii) R(x) > 0, Q(x) − S (x)R(x)−1 S (x)T > 0. Lemma 2 (Chen, Liu, & Lu, 2007). If G is irreducible, Gij = Gji ≥ 0 for i 6= j, and j=1 Gij = 0, for all i = 1, 2, . . . , N, then all eigenvalues of the matrix
PN
G11 − ε G21
G12 G22
.. .
.. .
GN1
GN2
... ... .. . ...
431
From (10), it is easy to see that (7) is satisfied, so the undirected network (5) is globally synchronized under the given linear feedback pinning controllers. The proof is completed. Note that the dimension of the matrix condition in (10) is N × n, so it is not easy to verify this condition. Let Γ be a positive definite T
matrix. If K and Γ are commutable and let θ = λmax ( K +2K ), then it can be considered as the passivity degree (Xiang & Chen, 2007), where θ > 0 means that each node needs energy from outside to stabilize the network, while θ < 0 means that each node itself is already stable. Here, only θ = kK k > 0 is considered throughout the paper for simplicity; otherwise, the network is synchronized even without pinning control. Corollary 1. Suppose that Assumption (9) holds and Γ is a positive definite matrix. The controlled undirected network (5) is globally synchronized if the following condition is satisfied: C < 0,
(13)
where C = θ IN + c (G − D) with θ = kK k and K defined in (9).
G1N G2N
Proof. First, one has
.. .
IN ⊗ (K Γ ) + c (G − D) ⊗ Γ ≤ (θ IN + cG − cD) ⊗ Γ .
(14)
If θ IN + cG − D < 0 in (13), and Γ > 0, then one can easily show (Lütkepohl, 1996) that
GNN
are negative for any positive constant ε .
(θ IN + cG − cD) ⊗ Γ < 0.
3. Pinning synchronization criteria for complex networks
So, condition (10) is satisfied, completing the proof.
In this section, some pinning criteria are established to ensure the global synchronization of complex dynamical networks. 3.1. General pinning synchronization criteria In this subsection, some general pinning synchronization criteria are derived. Theorem 1. Suppose that Assumption (9) holds. The controlled undirected network (5) is globally synchronized if the following condition is satisfied: IN ⊗ (K Γ ) + c (G − D) ⊗ Γ < 0,
(10)
where ⊗ is the Kronecker product, D
=
diag(d1 , . . . , dl ,
| 0, . . . , 0), and IN is the N-dimensional identity matrix. | {z }
{z l
}
N −l
Condition (13) is a general criterion to ensure the network pinning synchronization (Chen et al., 2007; Li, Wang, & Chen, 2004; Wang & Chen, 2002c). Remark 1. Though the condition C < 0 in (13) is very simple, it is still difficult to choose an appropriate gain matrix D. One may attempt to use the MATLAB LMI Toolbox to solve (13). In the previous works (Cao et al., 2006; W. Lu & T. Chen, 2004; Wang & Cao, 2006), LMI approach was used to study the global synchronization of linearly coupled networks with delays, and it is solvable only for networks of extremely small sizes. However, in Yu et al. (2008), global synchronization of linearly hybrid coupled networks with time-varying delays was investigated, where the network size can be extended to be much larger. If the network size N is very large, the computation of LMI conditions is difficult in finite time. Therefore, it is still a challenging problem to find some appropriate pinning controllers to achieve global network synchronization with low computational complexity.
Proof. Consider the Lyapunov functional candidate: V (t ) =
N 1X
2 i=1
3.2. Coupling strength in pinning control eTi (t )ei (t ).
(11)
The derivative of V (t ) along the trajectories of (8) gives V˙ =
N X
eTi
In this subsection, some criteria for choosing an appropriate coupling strength are established. Corollary 2. Suppose that Assumption (9) holds and Γ is a positive definite matrix. The controlled undirected network (5) is globally synchronized if the following condition is satisfied:
(t )˙ei (t )
i =1
=
N X
" eTi
(t ) f (xi (t ), t ) − f (s(t ), t ) + c
i =1
N X
#
c>
Gij Γ ej (t )
j =1 l
−c
X
di eTi
(t )Γ ei (t )
i=1
≤
N X
" eTi
(t ) K Γ ei (t ) + c
i =1
N X
# Gij Γ ej (t ) − c
j=1
l X
di eTi (t )Γ ei (t )
i=1
= e (t ) [(IN ⊗ K Γ ) + c (G ⊗ Γ ) − c (D ⊗ Γ )] e(t ), T
where e(t ) = ( (t ), eT1
eT2
(t ), . . . ,
eTN
(t )) . T
(12)
θ . |λmax (G − D)|
(15)
Proof. By Lemma 2, it is easy to see that G − D∗ is negative definite, where D∗ corresponds to only one controller, i.e., l = 1. It is easy to see that λmax (G − D) ≤ λmax (G − D∗ ) < 0. From (15), one has c (G − D) + θ IN ≤ c λmax (G − D) + θ < 0, so (13) is satisfied. The proof is completed. Remark 2. In Chen et al. (2007), pinning complex networks by a single controller, i.e., l = 1, was studied. However, it requires a very large coupling strength in general, which may not be very practical.
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It is easy to see that the theoretical coupling strength given in (15) is too conservative, usually much larger than the needed value. Clearly, it is desirable to make the coupling strength as small as possible. Here, the adaptive technique (Chen et al., 2007; Yu, Cao, & Chen, 2007; Yu, Cao, Wong et al., 2007; Yu, Chen et al., 2007; Zhou, Lu, & Lü, 2006) is adopted to achieve this goal. The selected pinning controllers in (5), associated with the adaptive coupling law, lead to N X
x˙ i (t ) = f (xi (t ), t ) + c (t )
Gij Γ xj (t )
j =1
− c (t )di Γ (xi (t ) − s(t )), N X
x˙ i (t ) = f (xi (t ), t ) + c (t )
3.3. Selective pinning scheme In this subsection, the pinning scheme is designed when the network structure and coupling strength are fixed. PN Let C = (Cij )N ×N in (13) and ki = j=1,j6=i Gij be the total weights between node i and all the other nodes, where ki is the degree of node i. It is easy to see that Cij = cGij (i 6=j), Cii = θ −cki −cdi (1 ≤ i ≤ l), and Cii = θ − cki (l + 1 ≤ i ≤ N), where θ = kK k with K defined in (9). Corollary 3. Suppose that Assumption (9) holds and Γ is a positive definite matrix. To satisfy condition (13), it is necessary that
i = 1, 2, . . . , l,
Cii = θ − cki − cdi < 0,
Gij Γ xj (t ),
Cii = θ − cki < 0,
j =1
i = l + 1, 2, . . . , N ,
1 ≤ i ≤ l,
l + 1 ≤ i ≤ N,
(19)
where C = θ IN + cG − cD, and θ = kK k with K defined in (9).
N X c˙ (t ) = α (xj (t ) − s(t ))T Γ (xj (t ) − s(t )),
(16)
Proof. To ensure C = θ IN + cG − D < 0, it is necessary that Cii < 0. This completes the proof.
j =1
where α is a small positive constant. Theorem 2. Suppose that Assumption (9) holds and Γ is a positive definite matrix. Then, the adaptively controlled undirected network (16) is globally synchronized for a small constant α > 0. Proof. Consider the Lyapunov functional candidate: V (t ) =
N 1X
2 i =1
eTi (t )ei (t ) +
β (c (t ) −e c )2 , 2α
(17)
where β and e c are positive constants to be determined below. The derivative of V (t ) along the trajectories of (16) gives V˙ =
N X
eTi (t )˙ei (t ) + β(c (t ) −e c)
N X
i=1
=
X
Up to this point, it is still not very clear as how to determine the control gains. Next, a theorem is given this concern. to address
ej (t )T Γ ej (t )
Let C = θ IN + cG − cD =
j=1
"
N
i=1
+ c (t )
# Gij Γ ej (t ) − c (t )
j =1
+ β(c (t ) −e c)
l X
ej (t )T Γ ej (t )
e C < 0.
≤ eT (t )[(θ IN + c (t )(G − D + β IN ) − βe c ) ⊗ Γ ]e(t ),
(18)
where e(t ) = ( (t ), (t ), . . . , (t )) . From Lemma 2, one can see that G − D + β IN is negative definite when β is sufficiently small. If one chooses a large initial condition c (0) and let α be small enough, then c (t ) increases very slowly. In this case, the synchronization can be achieved by Corollary 2 when c is very large. After reaching synchronization, c (t ) converges to a constant and is bounded. Since c (t ) is bounded, one can always choose a sufficiently largee c such that θ IN +c (t )(G−D+β IN )−βe c< 0. Then, in the whole time evolution process, V˙ is always negative definite. Consequently, the network is synchronized, so the third equation in (16) will vanish, leading to c (t ) → constant. This completes the proof. eT2
eTN
B
, where A and B are e C
Theorem 3. Suppose that Assumption (9) holds and Γ is a positive definite matrix. If the control gains di can be sufficiently large, condition (13), namely, C < 0, is equivalent to
j=1
eT1
BT
matrix C . di eTi (t )Γ ei (t )
i =1 N X
A −e D
matrices with appropriate dimensions, e D = diag(d1 , . . . , dl ), and e C is obtained by removing the 1, 2, . . . , l row–column pairs of
eTi (t ) f (xi (t ), t ) − f (s(t ), t ) N X
Remark 4. From (19), one can see that for the nodes without pinning controllers, it is necessary that their degrees ki > θ /c under condition (13). When the coupling strength c is very large, the degree of each node satisfies (19). In Chen et al. (2007), Li et al. (2004), Wang and Cao (2006), Xiang and Chen (2007, in press), Xiang, Liu, Chen, Chen, and Yuan (2007) and Zhou, Lu, and Lü (2008), condition (19) is assumed to be satisfied in the proposed pinning schemes, namely, the coupling strength c is very large therein. As seen here, interestingly, the nodes with low degrees should be controlled first, if c is not very large. It is common that the nodes with low degrees receive very little information from other nodes which can not be used to synchronize with the network due to the nonlinearity of the single node.
T
Remark 3. It is noted that the condition in Theorem 2 is very mild, which is independent of the network structure (G) and the fraction of the nodes (l). Under any fixed network structure and pinning controllers in the form of (6), the coupling strengths can be selfadaptively determined according to (16) by Theorem 2 to realize network synchronization.
(20)
For simplicity, one can select e D > λmax (A − Be C −1 BT )Il . Proof. It is easy to see that if C < 0 then e C < 0. So, one only needs to prove that if e C < 0 then C < 0. Choose e D > A − Be C −1 BT . Then, by Lemma 1, one indeed has C < 0. The proof is completed. From the definition of e C , one has e C = θ IN −l + ce G, where e Gij = Gl+i,l+j for i, j = l + 1, . . . , N. Corollary 4. Suppose that Assumption (9) holds and Γ is a positive definite matrix. The controlled undirected network (5) is globally synchronized if e D > λmax (A − Be C −1 BT )Il and one of the following conditions holds:
(i) c λmax (e G) + θ < 0, e (ii) λmax (G) < −θ /c , θ (iii) c > . |λmax (e G)|
(21)
Remark 5. Condition (20) is very simple since the pinning controllers are not involved, which provides some guidance to choose the control gains in e D and therefore is very useful.
W. Yu et al. / Automatica 45 (2009) 429–435
Condition (i) in (21) provides a criterion for ensuring the network synchronization with fixed network structure, coupling strength and pinning scheme in the form of (6); (ii) shows that one can choose some appropriate pinning controllers to achieve network synchronization with fixed network structure and coupling strength; (iii) proposes a way to choose the coupling strength with fixed network structure and pinning scheme (6). Remark 6. In Chen et al. (2007), Li et al. (2004), Wang and Cao (2006), Xiang and Chen (2007, in press), Xiang et al. (2007) and Zhou et al. (2008), similar pinning control results as Corollaries 1 and 2 were given. In this paper, conditions (i) and (iii) are considered with a fixed pinning scheme in the form of (6). However, sometimes condition (i) is not satisfied under some fixed pinning schemes when the coupling strength is not very large. In this case, the pinning scheme will be very important. In Li et al. (2004) and Wang and Chen (2002c), some pinning schemes, such as selective pinning scheme (nodes with degrees in order), and random pinning scheme, are proposed. Nevertheless, it is still too difficult to choose a minimum number of controllers to achieve network synchronization in general. A lower bound for the minimum number of pinning controllers is given in Xiang and Chen (2007) as follows: Proposition 1. If the matrix θ IN + cG has m non-negative eigenvalues, and if C = θ IN + c (G − D) < 0, then the number of nodes to be selected for control cannot be less than m. The theoretical analysis of this proposition is not difficult. However, we are interested in the question if these m controllers exist and how to find them. From simulation results, we found that in many cases these m controllers do not exist, namely, this necessary condition may not be very useful. If there exist some nodes with degrees ki ≤ θ /c, and the network cannot achieve synchronization without controlling these nodes by using criterion (13) or (20), then this means that when the coupling strength is very small, the nodes with lower degrees should be controlled first. Some detailed analysis will be given through simulations in Section 4. Similarly, the gain matrix e D > λmax (A − Be C −1 BT )Il in (20) may be much larger than the needed value, so the adaptive control approach is applied here (Chen et al., 2007; Yu, Cao & Chen, 2007; Yu, Cao, Wong et al., 2007; Yu, Chen et al., 2007; Zhou et al., 2006). The pinning controllers selected by (20) yield the following controlled network: x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t )
j=1
− cdi (t )Γ (xi (t ) − s(t )), x˙ i (t ) = f (xi (t ), t ) + c
N X
i = 1, 2, . . . , l,
Gij Γ xj (t ),
i = l + 1, 2, . . . , N ,
j=1
d˙ i (t ) = qi ei (t )T Γ ei (t ),
i = 1, 2, . . . , l,
(22)
where qi are positive constants. Corollary 5. Suppose that Assumption (9) holds and Γ is a positive definite matrix. If the condition in (20) is satisfied, then the undirected network (22) is globally synchronized under the adaptive scheme (22).
433
Taking the derivative of V (t ) along the trajectories of (22) gives V˙ =
N X
eTi (t )˙ei (t ) + c
l X (di (t ) − d)ei (t )T Γ ei (t )
i=1
=
N X
i=1
" eTi
(t ) f (xi (t ), t ) − f (s(t ), t ) + c
i=1
−c
N X
# Gij Γ ej (t )
j =1 l X
di (t )eTi (t )Γ ei (t ) + c
i=1
l X (di (t ) − d)ei (t )T Γ ei (t ) i =1
≤ e (t )[(θ IN + c (G − de IN )) ⊗ Γ ]e(t ), T
where e(t ) = ( (t ), eT1
eT2
(t ), . . . ,
eTN
(24)
(t )) and e IN = diag(1, . . . , 1, | {z } T
l
0, . . . , 0).
| {z } N −l
From Theorem 3, it follows that e C < 0, and by choosing d > λmax (A − Be C −1 BT ), network (22) is globally synchronized. This completes the proof.
4. Simulation examples In this section, some simulation examples are given to verify the criteria established above. In 1963, Lorenz found the first chaotic system (Lorenz, 1963). Then, in 1999, Chen and Ueta found the dual of the Lorenz system (Chen & Ueta, 1999). Later in 2002, Lü and Chen discovered another new chaotic system (Lü & Chen, 2002), which bridges the gap between the Lorenz system and the Chen system. Generally, chaotic systems are more difficult to synchronize than non-chaotic systems. Here, consider network (3) that consists of N identical Chen systems, described as follows: x˙ i (t ) = f (xi (t ), t ) + c
N X
Gij Γ xj (t ),
i = 1, 2, . . . , N ,
(25)
j=1
where Γ = diag(1, 2, 1), and
( f (xi , t ) =
35(xi2 − xi1 ), −7xi1 − xi1 xi3 + 28xi2 , xi1 xi2 − 3xi3 .
(26)
From numerical simulation, it is found that there exist some constants M1 = 23, M2 = 32, and M3 = 61, such that the chaotic attractor (s1 , s2 , s3 ) of the Chen system satisfies |s1 | < M1 , |s2 | < M2 , and |s3 | < M3 . By (4) and Lemma 2, one has
(xi − s)T (f (xi , t ) − f (s, t )) = eTi (35ei2 − 35ei1 , −7ei1 + 28ei2 − xi1 xi3 + s1 s3 , −3ei3 + xi1 xi2 − s1 s2 ) = −35e2i1 + (28 + M3 )|ei1 ei2 | + 28e2i2 − 3e2i3 + M2 |ei1 ei3 | 28 + M3 M2 ≤ −35 + ρ +η e2i1 2 2 28 + M3 M2 + 28 + e2i2 + −3 + e2i3 2ρ 2η ≤ θ (e2i1 + 2e2i2 + e2i3 ).
(27)
Choosing ρ = 1.3139 and η = 0.4715, one obtains θ = 30.9342. 4.1. Pinning scheme on scale-free networks
Proof. Consider the Lyapunov functional candidate: V (t ) =
N 1X
2 i=1
eTi
(t )ei (t ) +
l X c i=1
2qi
(di (t ) − d) , 2
where d is a positive constant to be determined below.
(23)
In the simulated scale-free network, N = 1000, m0 = m = 3, which contains about 3000 connections. We performed a simulation-based analysis on the controlled scale-free complex network of Chen systems by using lowdegree, high-degree, and random pinning schemes, respectively.
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W. Yu et al. / Automatica 45 (2009) 429–435
use the high-degree pinning scheme when a small fraction δ is controlled since some particular nodes in the scale-free network have very large degrees, which is known to be fragile to selective attack (Wang & Chen, 2002a). 4.2. Adaptive coupling strength
Fig. 1. Orbits of λmax (e G) as functions of the fraction δ by low-degree, high-degree, and random pinning schemes on a scale-free network of Chen systems.
In this simulation, the coupling strength is designed by Theorem 2 based on the adaptive technique for a scale-free network of Chen systems, where N = 100, and m0 = m = 3. Clustering coefficient is a characteristic property of complex networks, which reflects the group behavior in the network. In this example, the pinning controllers are added to some nodes (δ = 0.2) with the largest clustering coefficients. The states of error ei (t ) and the coupling strength are illustrated in Figs. 2 and 3, respectively, which show that the controlled network is globally synchronized and the designed coupling strength c = 6.1197 is very effective. 5. Conclusions
Fig. 2. Error states ei (t ) in a scale-free network of Chen systems.
In this paper, pinning synchronization of a class of complex dynamical networks has been investigated in detail. A general criterion for ensuring network synchronization has been derived. Some analytical and adaptive techniques have been proposed to obtain appropriate coupling strengths for achieving network synchronization. It is surprising to find that a network can realize synchronization under any linear feedback pinning scheme by adaptively adjusting the coupling strength. Furthermore, some effective pinning schemes have been designed for networks with fixed structure and coupling strength, where the original condition is shown to be equivalent to a simple condition without controls. It is found that the nodes with low degrees should be first pinned when the coupling strength is very small, which is contrary to the common belief that most-highly-connected nodes should be pinned first. Finally, some examples on scale-free networks have been simulated, which verify well the theoretical analysis. References
Fig. 3. Coupling strength c (t ) in a scale-free network of Chen systems.
The orbits of λmax (e G) as functions of the fraction δ by lowdegree, high-degree, and random pinning schemes are illustrated in Fig. 1, respectively. There are 392 nodes with degree 3, and 188 nodes with degree 4. By controlling these nodes, the network synchronizability is greatly improved. However, it is better to
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435 Wenwu Yu received the B.S. degree in information and computing science and M.S. degree in applied mathematics from the Department of Mathematics, Southeast University, Nanjing, China, in 2004 and 2007, respectively. Currently, he is working towards the Ph.D. degree at the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. He held several visiting positions in Hong Kong, USA, and China. His research interests include multi-agent systems, nonlinear dynamics and control, complex networks and systems, neural networks, cryptography, and communications. Guanrong Chen received his M.Sc. degree in computer science from Zhongshan (Sun Yat-sen) University, China in Fall 1981 and Ph.D. degree in applied mathematics from Texas A & M University in Spring 1987. He is currently a chair professor and the founding director of the Centre for Chaos and Complex Networks at the City University of Hong Kong, prior to which he was a tenured Full Professor at the University of Houston, Texas, USA. He is a Fellow of the IEEE (Jan. 1997), with research interests in chaotic dynamics, complex networks and nonlinear controls.
Jinhu Lü received the Ph.D degree in applied mathematics from the Chinese Academy of Sciences, Beijing, China in 2002. Currently, he is an Associate Professor with the Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China. He held several visiting positions in Australia, Canada, France, Germany, Hong Kong and USA, and was a Visiting Fellow in Princeton University, USA from 2005 to 2006. He is the author of two research monographs and more than 70 international journal papers published in the fields of nonlinear circuits and systems, complex networks and complex systems. He served as a member in the Technical Committees of several international conferences and is now serving as a member in the Technical Committees of Nonlinear Circuits and Systems and of Neural Systems and Applications in the IEEE Circuits and Systems Society. He is also an Associate Editor of IEEE Transactions on Circuits and Systems II, Journal of Systems Science and Complexity and DCDIS-A. Dr. Lü received the prestigious Presidential Outstanding Research Award from the Chinese Academy of Sciences in 2002, the National Best Ph.D. Theses Award from the Office of Academic Degrees Committee of the State Council and the Ministry of Education of China in 2004, the First Prize of Natural Science Award from the Ministry of Education of China in 2007 and the Lu Jiaxi Youth Talent Award from the Chinese Academy of Sciences in 2008. He is the co-author of the Most Cited SCI Paper of Chinese Scholars in the field of mathematics during the periods of 2001–2005 and 2002–2006. He is also an IEEE Senior Member.