Second-order consensus of multi-agent systems with nonlinear dynamics via impulsive control

Neurocomputing 125 (2014) 142–147

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Second-order consensus of multi-agent systems with nonlinear dynamics via impulsive control Yufeng Qian a, Xiaoqun Wu a, Jinhu Lu¨ b,n, Jun-An Lu a a b

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China LSC, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 April 2012 Received in revised form 21 August 2012 Accepted 1 October 2012 Available online 28 February 2013

In many real-world multi-agent systems, the intrinsic dynamics of velocity for each agent is usually nonlinear dynamic rather than static. Moreover, it is often difficult to obtain the continuous velocity information of multi-agent systems. To overcome the above essential difficulties, this paper aims at investigating the second-order consensus problem of multi-agents systems with nonlinear dynamics by using impulsive control signal protocol. In detail, by using the impulsive signals from agents and virtual leaders, several impulsive control protocols are designed for reaching the second-order consensus of multiagent systems with fixed or switching topologies. The theoretical analysis is also given to guarantee the second-order consensus based on algebraic graph theory and stability theory of impulsive differential equations. Finally, two typical examples are used to validate the above developed theoretical results. & 2013 Elsevier B.V. All rights reserved.

Keywords: Second-order consensus Multi-agent systems Impulsive control Virtual leader

1. Introduction Over the last decade, the consensus problem for multi-agent systems has attracted increasing attention due to its extensive applications in real-world distributed computation, rendezvous tasks, flocking, swarming, biological systems, sensor networks, and so on [2,11,18–20]. In many cooperative multi-agent systems, a group of agents usually share local information with their neighbors so as to reach agreement on some certain states asymptotically or in finite time. Many existing literatures focus on the first-order consensus [1,4,6,15,26,31]. Recently, the second-order consensus has received increasing attention due to various real-world applications [5,7,8,14,16], where all the agents are governed by secondorder dynamics, such as the position and velocity states. Note that there are few results reported on second-order consensus with time-varying velocities. Moreover, in many real-world multiagent systems, the agents usually have a time-varying intrinsic velocities rather than constant, even after a velocity consensus has been reached. During the process of consensus, each agent will automatically adjust its own dynamics based on the information of its neighbors or some leaders. However, it is very difficult for the agents to obtain the continuous velocity information. Therefore, it is very important to further investigate second-order

n

Corresponding author. E-mail addresses: [email protected] (Y. Qian), [email protected] (X. Wu), ¨ [email protected] (J.-A. Lu). [email protected] (J. Lu), 0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.10.027

consensus of multi-agent systems with time-varying intrinsic velocities by using some suitable techniques. The impulsive phenomena are ubiquitous in nature. It is well known that the impulsive control is very effective, robust, and lost-cost. Over the last few decades, it has been widely applied into the consensus and synchronization problems of various complex networks [3,13,20,23–25,28,30,31]. In [12], several fundamental criteria were introduced for the impulsive synchronization of complex dynamical systems. It indicates that the impulsive synchronization depends on the eigenvalues and eigenvectors of corresponding coupling matrices. Following this line, some basic synchronous or consensus criteria were established for various complex systems via impulsive control approaches, such as the multi-agent systems with switching topology [22]. In [29,32,33], it is shown that the consensus algorithms based on the first-order impulsive control have the much faster convergence speeds than the standard consensus algorithms [21,25]. However, the conditions for impulsive synchronization or consensus are rather conservative based on algebraic graph theory and stochastic matrix theory [9,10]. This is because the impulsive intervals are often very narrow and the impulsive gains usually have very strict constraints. In [27], Lu and his colleagues investigated the synchronization of impulsive dynamical networks using the concept of ‘‘average impulsive interval’’. This paper aims to further investigate the second-order consensus problem of multi-agents systems with nonlinear dynamics by using impulsive control signal protocol. Several fundamental consensus criteria are obtained based on algebraic graph theory and stability theory of impulsive differential equations by designing the suitable impulsive control protocols.

Y. Qian et al. / Neurocomputing 125 (2014) 142–147

This paper is organized as follows. Some basic preliminaries are introduced in Section 2. Section 3 further investigates the second-order consensus of multi-agent systems with fixed or switching topologies. Numerical examples are given to show the effectiveness of theoretical results in Section 4. Finally, the concluding remarks are also given in Section 5.

2. Preliminaries In this section, an undirected graph will be used to characterize the multi-agent systems with N agents. To begin with, some basic concepts and lemmas on algebraic graph theory [17] are briefly introduced. The following notations are used throughout the paper. The superscripts ‘‘T’’ means the transpose of a matrix,  denotes the Kronecker product of matrices, J  J indicates the Euclidean norm, IN ðON Þ is the N-dimensional identity (zero) matrix, 1N ð0N Þ denotes the N-dimensional column vector whose elements are all ones (zeros), and N þ denotes the set of positive integers. Let G ¼ ðV,E,AÞ be a weighted undirected graph with order N, V ¼ fv1 ,v2 , . . . ,vN g be the set of all nodes, and E DV  V be the set of all edges. A ¼ ½aij NN is the adjacency matrix with nonnegative elements aij, where aii ¼ 0 for i ¼ 1,2, . . . ,N. The undirected edge is denoted by eij ¼ ðvi ,vj Þ, which means that node i and node j of the graph can exchange information with each other. The neighboring set of node i is defined by Ni ¼ fvj A V : ðvj ,vi Þ A Eg, where the edge eij exists if and only if aij ¼ aji 40. A path of length r from i to j is defined by a sequence of edges ðvi ,vi1 Þ,ðvi1 ,vi2 Þ, . . . ,ðvir1 ,vj Þ. An undirected graph G is connected if and only if there exists an undirected path between any two vertices in G. The Laplacian matrix L ¼ ðlij ÞNN of graph G is defined by 8 i a j, aij , > > < N X lij ¼  l , i ¼ j: > > : j ¼ 1,j a i ij It means that all the row sums of L are zero, that is, 1N ¼ ½1,1, . . . ,1T A RN is a right eigenvector of L associated with the eigenvalue l ¼ 0. Lemma 1 (Horn and Johnson [9]). The Laplacian matrix L of an undirected graph G is semi-positive definite. It has a simple zero eigenvalue and all the other eigenvalues are positive if and only if the graph G is connected, that is, all the eigenvalues of L satisfy 0 ¼ l1 ðLÞ o l2 ðLÞ r    lN ðLÞ. Lemma 2 (Horn and Johnson [9]). The second smallest eigenvalue l2 ðLÞ of the Laplacian matrix L of an undirected graph G satisfies

l2 ðLÞ ¼

xT 1

xT Lx : T N ¼ 0,x a 0N x x min

ðaAÞ  B ¼ A  ðaBÞ, ðA þ BÞ  C ¼ A  C þ B  C,

ð3Þ

ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ,

ð4Þ

ðA  BÞT ¼ AT  BT :

3. Main results The main results are given in this section.

The general second-order consensus protocol is described by x_ i ðtÞ ¼ vi ðtÞ, v_ i ðtÞ ¼ ui ðtÞ, ui ðtÞ ¼ a

N X

aij ðxj ðtÞxi ðtÞÞ þ b

j A Ni

N X

aij ðvj ðtÞvi ðtÞÞ,

ð3Þ

j A Ni

where xi A Rn and vi A Rn are the position and velocity of the ith agent, respectively. a and b are the coupling strengths. A ¼ ½aij NN is the symmetric adjacency matrix characterizing the topology structure of undirected network. Definition 1. The multi-agent system (3) is said to achieve second-order consensus, if for any initial conditions lim Jxj ðtÞxi ðtÞJ ¼ 0,

t-1

lim Jvj ðtÞvi ðtÞJ ¼ 0,

t-1

8i, j ¼ 1,2 . . . ,N:

ð4Þ

As we know now, the second-order consensus can be reached if the coupling strengths and spectra of Laplacian matrix satisfy some suitable conditions. However, for the real-world multiagent systems, the intrinsic dynamics of velocity for each agent is often nonlinear. Furthermore, it is much more difficult to obtain the continuous velocity information compared with the position information. To cope with the above essential difficulty, an impulsive control technique is introduced, where each agent can update its position and velocity states at impulsive instants. Therefore, the multi-agent system with impulsive control signals is described by 8 x_ i ðtÞ ¼ vi ðtÞ, > > > P > > aij ðxj ðtÞxi ðtÞÞ, t a t k , v_ ðtÞ ¼ f ðvi ,tÞ þ a > > > i < j A Ni  ð5Þ xj ðt kþ Þxi ðt kþ Þ ¼ Bk ðxj ðt  > k Þxi ðt k ÞÞ, > > > v ðt þ Þv ðt þ Þ ¼ B ðv ðt  Þv ðt  ÞÞ, > j k i k i k k j k > > > : 8i, j ¼ 1,2 . . . ,N, where f ðvi ,tÞ is a nonlinear continuously differentiable vectorvalued function and Bk is the impulsive gain at time tk. Assume that xi ðtÞ, vi ðtÞ are left continuous at tk. That is, xi ðt k Þ ¼ xi ðt  k Þ and vi ðt k Þ ¼ vi ðt  k Þ. The time sequence ft k g satisfies 0 r t 0 ot 1 o    o t k ot k þ 1 o    and limk- þ 1 t k ¼ þ1 with tk ¼ t k þ 1 t k . Remark 1. It should be pointed out that the impulsive signal does not have any effect on the consensus of multi-agent systems for JBk J ¼ 1. When JBk J o 1, the impulsive signal has positive effect on the consensus. However, it has negative effect on the consensus for JBk J 4 1. In the following, some sufficient conditions will be derived for the consensus of multi-agent systems via impulsive control.

ð1Þ

Lemma 3 (Horn and Johnson [10]). For matrices A, B, C, and D with appropriate dimensions, the Kronecker product  has the following properties: ð1Þ ð2Þ

143

Remark 2. According to (5), for any impulsive signals x~ k and v~ k , þ  ~ ~ ~ one gets xi ðt kþ Þx~ k ¼ Bk ðxi ðt  k Þx k Þ and vi ðt k Þv k ¼ Bk ðvi ðt k Þv k Þ. þ þ  That is, xi ðt k Þ ¼ Bk xi ðt k Þ þðIn Bk Þx~ k and vi ðt k Þ ¼ Bk vi ðt  k Þþ ðIn Bk Þv~ k . Therefore, for each agent, the corresponding position and velocity can be adjusted at any impulsive instant. Moreover, the impulsive signals are not necessary to be identical at different impulsive instants. Proposition 1. There exists a nonnegative constant y satisfying

ð2Þ

~ ~ Jf ðv,tÞf ðv,tÞJ r yJvvJ

8v, v~ A Rn :

Remark 3. It is easy to verify that some classical chaotic systems satisfy the above assumption, including Lorenz system, Chen system, Lu¨ system, Chua’s circuit, and so on.

144

Y. Qian et al. / Neurocomputing 125 (2014) 142–147

3.1. Consensus with fixed network topology

From (10) and (11), one gets

This section focuses on the consensus of multi-agent systems with fixed network topology. b i ðtÞ ¼ vi ðtÞvðtÞ, where xðtÞ ¼ Denote b x i ðtÞ ¼ xi ðtÞxðtÞ and v P PN ð1=NÞ k ¼ 1 xi ðtÞ and vðtÞ ¼ ð1=NÞ N k ¼ 1 vi ðtÞ are the average values of positions and velocities for all the agents, T T T b ¼ ðv b T1 , v b T2 , . . . , v b TN ÞT , and x2, . . . ,b x N ÞT , v respectively. Define b x ¼ ðb x1,b T

T

VðtÞ rVðt 0þ Þe2yðtt0 Þ for t A ðt 0 ,t 1 . It follows that Vðt 1 Þ rVðt 0þ Þe2yðt1 t0 Þ and Vðt 1þ Þ r l1 Vðt0þ Þe2yðt1 t0 Þ . In general, for any t A ðtk ,t k þ 1 ,kA N þ , one has VðtÞ r l1 l2 . . . lk Vðt 0þ Þe2yðtt0 Þ :

T

Fðv,tÞ ¼ ðf ðv1 ,tÞ, f ðv2 ,tÞ, . . . ,f ðvN ,tÞÞT . Thus the error dynamics can be described by 8 > b b ðtÞ, x_ ðtÞ ¼ v > > ! ! > > > > 1N 1TN >v _ ðtÞ ¼ > b I  x ðtÞ,  I > n Fðv,tÞaðL  In Þb N < N > t a tk , > > > > > x ðt kþ Þ ¼ ðIN  Bk Þb x ðt  >b k Þ, > > > :v b ðt þ Þ ¼ ðIN  Bk Þv b ðt Þ:

Since o 41, one obtains lnðolk Þ þ 2yðt k t k1 Þ r 0, olk e2yðtk tk1 Þ r1. It follows from (12) that

ð6Þ ¼

It follows that the second-order consensus of multi-agent systems (5) can be achieved if and only if the impulsive dynamical system (6) is asymptotically stable. Theorem 1. Suppose that Proposition 1 holds and graph G is connected. Then the impulsive dynamical system (5) can achieve second-order consensus if there exists a constant oðo 41Þ such that ln olk þ2yðt k þ 1 t k Þ r0,

kANþ ,

where lk is the largest eigenvalue of

ð7Þ BTk Bk .

Proof. Consider the following Lyapunov candidate: T

T

b ðtÞ: b ðtÞv VðtÞ ¼ 12 b x ðtÞ þ 12v x ðtÞðaL  In Þb

ð8Þ

Obviously, one has VðtÞ Z

a 2

1 2

T

i.e.,

VðtÞ r l1 l2 . . . lk e2yðtt0 Þ Vðt 0þ Þ

k

k

ð12Þ

r

1

ok 1

ok

Vðt 0þ Þe2yðttk Þ olk e2yðtk tk1 Þ    ol1 e2yðt1 t0 Þ Vðt 0þ Þe2yðttk Þ :

b ðtÞ ¼ 0. That is, the error Thus, VðtÞ-0 as t-1, i.e., b x ðtÞ ¼ 0 and v dynamical system (6) is asymptotically stable at the origin. It means that the multi-agent system (5) can achieve second-order consensus. The proof is completed. & If the impulsive gain matrix Bk is time-invariant and the impulsive interval tk ¼ t k þ 1 t k is a positive constant for any k, one obtains the following corollary. Corollary 1. Suppose that Proposition 1 holds and the graph G is connected. Let Bk ¼ B and tk ¼ t for k A N þ . Then the impulsive dynamical system (5) can realize second-order consensus if there exists a constant oðo 4 1Þ such that lnðolÞ þ 2yt r 0,

T

b ðtÞv b ðtÞ Z 0, l2 ðLÞb x ðtÞb x ðtÞ þ v

b ðtÞ ¼ 0. The derivative of where VðtÞ ¼ 0 if and only if b x ðtÞ ¼ 0 and v V(t) along the trajectories of (6) for t A ðt k1 ,t k  is given by ! 1N 1TN T T _ b ðtÞ þ v b ðtÞ IN  V ðtÞ ¼ b x ðtÞðaL  In Þv N ! !

ð13Þ

ð14Þ

where l is the largest eigenvalue of BT B. Remark 4. For a constant y, the impulsive interval t can be chosen from a wide interval with a suitable impulsive gain matrix B. Therefore, the above conditions are rather simple and can also be easily satisfied for the second-order consensus of multi-agent systems.

In Fðv,tÞaðL  In Þb x ðtÞ T

b ðtÞ ¼v

IN 

1N 1TN N

!

!

!

3.2. Consensus with switching network topology

 In Fðv,tÞ

b ðtÞ b T ðtÞv r yv r2yVðtÞ:

ð9Þ

It implies that þ Þe2yðttk1 Þ , VðtÞ r Vðt k1

t A ðt k1 ,t k , k A N þ :

ð10Þ

On the other hand, for t ¼ t k ,ðk A N þ Þ, one obtains T

T

b ðt kþ Þ b ðt kþ Þv Vðt kþ Þ ¼ 12 b x ðt kþ Þ þ 12v x ðt kþ ÞðaL  In Þb T

T b  ¼ 12b x ðt  k ÞðIN  Bk ÞðaL  In ÞðIN  Bk Þx ðt k Þ T b  b T ðt  þ 12v k ÞðI N  Bk ÞðIN  Bk Þv ðt k Þ T T b  ¼ 12b x ðt  k Þð L  Bk Bk Þx ðt k Þ T T  b ðt k ÞðIN  Bk Bk Þv b ðt  þ 12v kÞ  1 bT  b  b T ðt  x ðt k Þ þ 12v r lk ð2 x ðt k Þð L  In Þb k Þv ðt k ÞÞ  ¼ lk Vðt k Þ:

a

a

ð11Þ

This section focuses on the second-order consensus of multiagent systems with switching topology. The switching impulsive dynamical system is described by 8 x_ i ðtÞ ¼ vi ðtÞ, > > P > > > _ i ðtÞ ¼ f ðvi ,tÞ þ a aij ðsðtÞÞðxj ðtÞxi ðtÞÞ, v > > > j A N i ðsðtÞÞ > > < t a tk , þ þ   > > x > j ðt k Þxi ðt k Þ ¼ Bk ðxj ðt k Þxi ðt k ÞÞ, > > > þ þ   > > vj ðt k Þvi ðt k Þ ¼ Bk ðvj ðt k Þvi ðt k ÞÞ > > : 8i, j ¼ 1,2 . . . ,N,

ð15Þ

where sðtÞ : R þ -f1,2, . . . ,lg is a piecewise function with l different network topology structures. In detail, for t A ðt k1 ,t k , sðtÞ ¼ sðkÞ A f1,2, . . . ,lg, which means that the sðkÞth network topology is activated. Assume that the network topology Gs switches in a set of connected graphs with order N. That is, Gs A cN ¼ fG : G ¼ ðV,E, AÞ is connectedg. With the same notations as stated in the previous section, the error dynamics of system (15) can be rewritten into the following

Y. Qian et al. / Neurocomputing 125 (2014) 142–147

compact form: 8 > b_ b > > x ðtÞ ¼ v ðtÞ, ! ! > > > > 1N 1TN > _ ðtÞ ¼ > b v I  x ðtÞ,  I > n Fðv,tÞaðLsðkÞ  In Þb N < N

ð16Þ

> t a tk , > > > > >b x ðt kþ Þ ¼ ðIN  Bk Þb x ðt  > k Þ, > > > þ :v b ðt Þ ¼ ðIN  Bk Þv b ðt  Þ,

b ðtÞ ¼ 0. Thus, the derivative and VðtÞ ¼ 0 if and only if b x ðtÞ ¼ 0 and v of V(t) along the trajectories of (16) for t A ðt k1 ,t k  is given by ! 1N 1TN T b ðtÞ þ v b T ðtÞ IN  V_ ðtÞ ¼ b x ðtÞðaLsðkÞ  In Þv N ! !   x ðt Þ In Fðv,tÞa LsðkÞ  In b

k

k

ln olk þ2yðt k þ 1 t k Þ r 0,

kANþ ,

ð17Þ

Proof. Construct the Lyapunov candidate T b T ðtÞv b ðtÞ: x ðtÞðaLsðkÞ  In Þb x ðtÞ þ 12v VðtÞ ¼ 12 b

ð18Þ

It follows that:

a bT b 1 gx ðtÞx ðtÞ þ vb T ðtÞvb ðtÞ Z0, 2

2

!

!

!

 In Fðv,tÞ

b ðtÞ b T ðtÞv r yv r 2yVðtÞ:

ð19Þ

It implies that þ VðtÞ rVðt k1 Þe2yðttk1 Þ ,

t A ðt k1 ,t k , k A N þ :

ð20Þ

Similar to the proof of Theorem 1, one can prove that the error dynamical system (16) is asymptotically stable, i.e., the multiagent system (15) can achieve second-order consensus. The proof is thus completed. &

where lk is the largest eigenvalue of BTk Bk .

VðtÞ Z

1N 1TN IN  N

b T ðtÞ ¼v

where LsðkÞ is the Laplacian matrix for the sðkÞth network topology. Theorem 2. Suppose that Proposition 1 holds and the graph Gs is connected. Then the impulsive dynamical system (15) can achieve second-order consensus if there exists a constant oðo 41Þ such that

145

Remark 5. Theorem 2 indicates us that the multi-agent system with switching topology can also achieve second-order consensus based on some suitable nonconservative conditions, so long as the graph connectivity is maintained. For simplicity, one obtains the following corollary.

where

g ¼ min l2 ðGs Þ

Corollary 2. Suppose that Proposition 1 holds and the graph Gs is connected. Assume that Bk ¼ B and tk ¼ t. Then, the impulsive dynamical system (15) can achieve second-order consensus if there exists a constant oðo 4 1Þ such that

G s A cN

lnðolÞ þ 2yt r 0,

10

ð21Þ T

where l is the largest eigenvalue of B B.

4. Numerical simulations

0 −5

In this section, two numerical examples are given to illustrate the theoretical results obtained in the previous section.

−10 1

Example 1. Let the nonlinear function f to be Chua’s circuit [13]. Then f is described by 0 1 zðvi1 þ vi2 lðvi1 ÞÞ B C vi1 vi2 þvi3 f ðvi ,tÞ ¼ @ ð22Þ A, Rvi2

0.5 0 −0.5 −1

−2

−4

4

2

0

vi1

where lðvi1 Þ ¼ bvi1 þ ððabÞ=2Þð9vi1 þ199vi1 19Þ is a piecewiselinear function. System (22) has a typical Chua’s double-scroll

Fig. 1. Chua’s double-scroll chaotic attractor.

15

5 4

vij(t)−v1j(t),i=1,2,...,N;j=1,2,...,n

vi2

xij(t)−x1j(t),i=1,2,...,N;j=1,2,...,n

vi3

5

3 2 1 0 −1 −2 −3 −4 −5

0

1

2

3

t

4

5

10 5 0 −5 −10 −15

0

1

2

3

t

Fig. 2. (a) Position and (b) velocity errors of agents in scale-free network.

4

5

Y. Qian et al. / Neurocomputing 125 (2014) 142–147

35

5

30

4

vij(t)−v1j(t),i=1,2,...,N;j=1,2,...,n

xij(t)−x1j(t),i=1,2,...,N;j=1,2,...,n

146

25 20 15 10 5 0 −5 −10 −15

0

1

2

3

4

5

3 2 1 0 −1 −2 −3 −4 −5

0

t

1

2

3

4

5

t

Fig. 3. (a) Position and (b) velocity errors of agents in switching small-world network.

chaotic attractor for z ¼ 10, R ¼ 18, a ¼ 4=3 and b ¼ 3=4, as shown in Fig. 1. In the following, one considers the BA scale-free network with 100 nodes. It can be easily verified that Proposition 1 holds for y ¼ 4:3871. Without loss of generality, add impulsive signal into the first agent. Let Bk ¼ diagf0:6,0:6,0:6g, tk ¼ t k þ 1 t k ¼ 0:1 and o ¼ 1:1. Then, one has lk ¼ 0:36 and lnðolk Þ þ 2yðtk þ 1 tk Þ ¼ 0:0489 r0. It indicates that the condition (14) holds. According to Corollary 1, the impulsive dynamical system (5) can achieve second-order consensus. Fig. 2 shows the position and velocity errors of agents in scale-free network. From Fig. 2, all the agents reach second-order consensus. Example 2. Similarly, let the nonlinear function f to be Chua’s circuit [13]. Suppose that the network topology of 100-agent system switches among three different NW small-world networks with m ¼ 2, p ¼ 0:1, m ¼ 2, p ¼ 0:05, and m ¼ 4, p ¼ 0:1, respectively, where 2m is the number of neighboring nodes of node i and p is the probability of edge-adding. Moreover, the switching signal is defined by sðtÞ : ðt k1 ,t k -sðkÞ A f1,2,3g, sðtÞ ¼ ððk1Þ mod 3Þ þ 1. Without loss of generality, the impulsive signal is randomly chosen. Let Bk ¼ diagf0:8,0:8,0:8g, tk ¼ t k þ 1 t k ¼ 0:05 and s ¼ 1:001. Then, one has lk ¼ 0:64 and lnðslk Þ þ 2yðtk þ 1 tk Þ ¼ 0:0066 r 0. Therefore, the condition (21) holds. From Corollary 2, the switching impulsive dynamical system (15) can achieve second-order consensus. Fig. 3 shows the position and velocity errors of agents in switching small-world network. According to Fig. 3, all the agents reach second-order consensus. 5. Conclusion This paper has further investigated the second-order consensus of multi-agents systems with nonlinear dynamics by using impulsive control signal protocol. Moreover, several impulsive control protocols are then designed for achieving the second-order consensus of multiagent systems with fixed and switching topologies. Finally, two numerical examples are also given to verify the proposed theoretical results. It sheds some light on the future real-world applications.

Acknowledgment This work was supported by the National Natural Science Foundation of China under Grants 61025017, 11072254, 61174028, 11172215, and 61203148.

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¨ received his Ph.D degree in applied mathematics Jinhu Lu from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in 2002. Currently, he is a Professor of the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He was a Visiting Fellow in Princeton University, USA from 2005 to 2006. He is the Author of two research monographs and more than 100 international journal papers published in the fields of complex networks and systems, nonlinear circuits and systems, with more than 5500 SCI citations and h-index 38. He also has two authorized patents. He is now serving as the Chair of Technical Committees of Neural Systems and Application and Secretary of Technical Committees of Nonlinear Circuits and Systems in the IEEE Circuits and Systems Society. Prof. Lu¨ served and is serving as Editors in various ranks for 11 SCI journals including the IEEE Transactions on Circuits and Systems I: Regular Papers, IEEE Transactions on Circuits and Systems II: Brief Papers, IEEE Transactions on Neural Networks, IEEE Transactions on Industrial Informatics, PLoS Computational Biology, International Journal of Bifurcation and Chaos, and Asian Journal of Control. Prof. Lu¨ received the prestigious National Natural Science Award twice from the Chinese government, the First Prize of Science and Technology Award from the Beijing City of China and the First Prize of Natural Science Award from the Ministry of Education of China, the 11th Science and Technology Award for Youth of China and the Australian Research Council Future Fellowships Award. Moreover, Prof. Lu¨ attained the National Natural Science Fund for Distinguished Young Scholars and 100 Talents Program from the Chinese Academy of Sciences. He is also the Fellow of IEEE.

Yufeng Qian received the B.S. degree in information and computing science from Central China Normal University, Wuhan, China and the M.S. degree in computational mathematics from Wuhan University, Wuhan, China, in 2008 and 2010, respectively. Currently, he is working toward the Ph.D. degree in the School of Mathematics and Statistics, Wuhan University, Wuhan, China. His research interests include multi-agent systems, complex networks, and nonlinear dynamics.

Junan Lu graduated from the department of geophysics, Beijing University, Beijing, China, in 1968 and received the M.Sc. degree in applied mathematics from Wuhan University, Wuhan, China, in 1982, respectively. Currently, he is a full Professor in the School of Mathematics and Statistics, Wuhan University, Wuhan, China. His research interests include nonlinear dynamical systems, complex networks, and scientific and engineering computing. He has published more than 180 journal papers in these fields. Prof. Lu received the prestigious National Natural Science Award from the Chinese government in 2008 and First Prize of Natural Science Award from the Ministry of Education of China in 2007.

Xiaoqun Wu received the B.Sc. degree in applied mathematics and the Ph.D in computational mathematics both from Wuhan University, Wuhan, China, in 2000 and 2005, respectively. Currently, she is an Associate Professor in the School of Mathematics and Statistics, Wuhan University, Wuhan, China. She held several visiting positions in Hong Kong and Australia over the last few years. Her research interests include nonlinear dynamics, complex networks, and multiagent systems. She has published more than 30 SCI journal papers in the fields of complex systems and complex networks.