International Journal of Automation and Computing

9(1), February 2012, 54-62 DOI: 10.1007/s11633-012-0616-6

Dynamic Consensus of High-order Multi-agent Systems and Its Application in the Motion Control of Multiple Mobile Robots Zhong-Qiang Wu

Yang Wang

Key Laboratory of Industrial Computer Control Engineering, Yanshan University, Qinhuangdao 066004, PRC

Abstract: In this paper, the leader-following consensus problem for multi-agent linear dynamic systems is considered. All agents and leader have identical multi-input multi-output (MIMO) linear dynamics that can be of any order, and only the output information of each agent is delivered throughout the communication network. When the interaction topology is fixed, the leader-following consensus is attained by H∞ dynamic output feedback control, and the sufficient condition of robust controllers is equal to the solvability of linear matrix inequality (LMI). The whole analysis is based on spectral decomposition and an equivalent decoupled structure achieved, and the stability of the system is proved. Finally, we extended the theoretical results to the case that the interaction topology is switching. The simulation results for multiple mobile robots show the effectiveness of the devised methods. Keywords: robots.

1

Multi-agent systems, consensus problem, dynamic output feedback, linear matrix inequality (LMI), multiple mobile

Introduction

Recent technological advances have spurred a broad interest in the development of a network of unmanned autonomous systems that can operate without an extensive involvement of humans. Observations based on natural behavior of animals operating as a team have inspired scientists in different disciplines to investigate the possibilities of networking a group of systems to accomplish a given set of tasks without requiring an explicit supervisor. Some examples of such natural behaviors can be found in the migration of birds and motion of fish searching for food. Application areas of multi-agent system include automated factories, mobile robotics, unmanned aerial vehicles (UAVs), autonomous underwater vehicles (AUVs), automated highway systems (AHSs), and space explorations. Aimed at the deficiency of conventional traffic control methods, a new method based on multi-agent technology for traffic control is proposed in [1]. In the research for multi-agent systems, the main challenge is how to design simple control rules (neighbor based rules) for simple agents (with limited computing power and information interaction capability) to achieve a prescribed group behavior. In 1986, Reynolds first proposed a computer animation model to simulate collective behaviors of multiple agents. Vicsek et al.[2] proposed and analyzed a neighbor-based swarm model. Many consensus algorithms (or protocols) are introduced as interaction rules that specify the information exchange on the network between an agent and all of its neighbors in [3–6]. In [3], consensus problems are addressed under a variety of assumptions on the network topology (fixed or switching), presence or lack of communication delays, and directed or undirected network information flow. Tanner et al.[7] considered a group of mobile agents moving on the plane with double-integrator dynamics, and they proposed a set of control laws that renManuscript received November 16, 2010; revised February 21, 2011

der the group to generate a stable flocking motion. In [8], the problems of target tracking and obstacle avoidance for multi-agent systems are considered. In addition, there has been a great amount of renewed interest in self-organized groups with leaders, where the leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones[9] . Such a problem is commonly called the leader-following consensus problem in [10]. Mu et al.[11] studied the collective dynamics of a group of motile particles with a leader. Jadbabaie et al.[12] considered the coordination of a group of mobile autonomous agents following an actual leader. However, most of the leader-following consensus problems have been mainly concerned with agents that are modeled by a first-order or second-order dynamics. Recently, research interest in consensus problems has been devoted to high-order agents modeled by integrator chains of length greater than two. In [13, 14], the authors studied the consensus problem for multi-agent linear dynamic systems without a leader, and all the agents have identical multi-input multi-output (MIMO) linear dynamics, which can be of any order. Ni and Cheng[15] studied the consensus problem for high-order multi-agent systems with a leader, and linear time-invariant systems are studied in [16, 17]. But in [14–17], the consensus protocol is based upon all the states of neighboring agents. The output feedback consensus problem has been considered in [18], but all the states of the observer for each agent need to be transmitted to the neighbors, so the quantity of the transmitted information is the same as the case of state feedback. The recent work in [19] discussed various conditions for achieving consensus by output feedback, but it is limited to the cases of static output feedback. With this background, we consider a consensus problem in undirected networks with an active leader, as well as in a generalized environment that the dynamics of each agent is an n-th order MIMO linear control system, rather than one integrator or double integrators in most existing liter-

Z. Q. Wu and Y. Wang / Dynamic Consensus of High-order Multi-agent Systems and Its Application in · · ·

ature, and the leader-following consensus problem is posed under assumptions on the network topology being fixed and switching. Our main contribution is to introduce a robust H∞ output feedback control method, which was proposed by using a linear matrix inequality (LMI) approach to deal with the consensus problem, and a dynamic consensus algorithm is proposed that uses only the output information (rather than the full state) from the neighboring agents. Therefore, when the states of the systems are undetectable, we still solve the consensus problem; meanwhile, the stability of the leader is guaranteed. When the network topology is connected and fixed, we pose a method of dynamic filter design and provide a convergence analysis. The whole analysis is based on spectral decomposition to achieve an equivalent decoupled structure, so we can propose a control strategy such that the error system reaches robust H∞ stability[20] . We also extend the method to the case that the interaction topology is switching. Mobile robot examples are worked out to illustrate the effectiveness of our theoretical results. The outline of this paper is as follows. Section 2 contains the problem formulation and some preliminaries; Section 3 has the main results; some simulation results based on the mobile robot model are presented in Section 4; the conclusion is given in Section 5.

2 2.1

Problem model

statement

and

system

Communication graphs

In general, information exchange between agents in the multi-agent system can be modeled by directed/undirected graphs. Before we proceed, some basic concepts on graph theory are provided as below. Let G = (ν, ξ, A) be a weighted digraph (or directed graph) of order N where ν = {ν1 , ν2 , · · · , νN } is the set of nodes, ξ ⊆ ν × ν is the set of edges, and A is a weighted adjacency matrix with nonnegative elements aij . The node indices belong to a finite index set l = {1, 2, · · · , N }. An edge of G is denoted by eij = (vi , vj ). The adjacency elements associated with the edges of the graph are positive, i.e., eij ∈ ξ ⇔ aij > 0. Moreover, we assume there are no self-cycles, i.e., aii = 0 for all i ∈ l. The set of neighbors of vertex i is denoted by Ni = {j ∈ v; (i, j) ∈ ξ, j 6= i}. If a directed graph has the property that eij belongs to ξ for any eji ∈ ξ, the graph is called undirected. The out-degree matrix of directed graph G is ∆ = diag {d1 , · · · , dN } ∈ RN ×N , where diagonal elements di = dout (vi ) =

N X

aij ,

(i = 1, · · · , N ).

j=1

Then the Laplacian of a weighted digraph is defined as L = ∆ − A. For undirected graph, in general L is symmetric. The node of a directed graph G = (ν, ξ, A) is balanced if and only if its in-degree and out-degree are equal, i.e., din (vi ) = dout (vi ). A graph G = (ν, ξ, A) is called balanced if and only if all of its nodes are balanced, this definition is introduced in [18]. If there is a directed path from every node to every other node, the graph is said to be strongly connected (connected for undirected graph).

55

In this paper, we use an undirected and connected graph to describe the information exchange between agents and the leader, and consider a system consisting of N agents and a leader, which is denoted by a graph G. It contains N agents (related to graph G) and a leader (labeled by 0) with edges from some agents to the leader. Information can exchange between N agents and the leader in the interaction topology. If there is a path in G from every node i in G to node 0, then node 0 is called globally reachable in G. We define D = diag {β1 , · · · , βn } as the leader adjacency matrix associated with G, where βi > 0 if the leader is a neighbor of agent i and otherwise βi = 0, here we assume ( 1, follower i is connected to leader βi = 0, otherwise. The assumption will hold in the whole paper.

2.2

Systems under study and consensus protocols

We consider a multi-agent system consisting of N followers and a leader, the agents0 linear dynamics are denoted as: ( x˙ i = Axi + Bui + B 0 ωi , xi ∈ Rn , ui ∈ Rm (1) yi = Cxi , yi ∈ Rp for i = 0, 1, 2, · · · , N (when i = 0, (1) is the dynamics of the leader), where xi is the state of each agent, ui is the control vector (ui can only use local information from its neighbor agents), ωi ∈ L2 [0, ∞) is the interference signal and ω0 = 0, yi is the output, matrices A, B, B 0 , and C are of proper dimensions. Assume that the systems are stabilizable and detectable. From the system under study, all the agents have identical MIMO linear dynamics, which can be of any order, and only the output information of each agent is delivered throughout the communication network. In general, an agent refers to a dynamical system. However, in our context the term “agent” is interchangeable with “vehicle”, where a vehicle may be a mobile robot or any other ground vehicle with corresponding linear dynamics. Definition 1. The leader-following multi-agent system is said to asymptotically reach consensus if for any xi (0) , (i = 0, 1, · · · , N ), there is a dynamic output feedback ui such that the closed-loop system satisfies limt→∞ ky0 (t) − yi (t)k = 0, (i = 1, · · · , N ) for each agent in the communication network topology. In this paper we consider the problem of controlling the value ky0 (t) − yi (t)k within limits, which tends to 0 when there is an interference signal, so the high-order multi-agent system achieves leader-following consensus and can satisfy the H∞ performance index. We design the dynamic filter K (s) that u0 = −K (s) y0 for the leader where K (s) = ³ ´−1 ˆ sI − Aˆ ˆ + D. ˆ We assume that the follower i collects C B the output information of its neighboring agents by the rule zi = −

N X

aij (yi − yj ) − βi yi .

(2)

j=1

We also assume that the collected information zi is filtered by the same stable filter K (s), and is fed back to

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International Journal of Automation and Computing 9(1), February 2012

the agent i by ui = K (s) z i . Our purpose is to design the ˆ B, ˆ C, ˆ and D, ˆ so that the outputs of parameter matrices A, all the followers and the leader tends to consensus, and we can prove that the leader is stable and the whole network systems have expected characteristic of responses.

3

The stability analysis of multi-agent systems

3.1

Controller design and network dynamics

Lemma 1. If undirected graph G is connected, then the symmetric matrix H (where H = L + D) associated with G is positive definite[9] . The filter K (s) of the leader is represented in the state space form as follows ( ˆx0 − By ˆ 0 x ˆ˙ 0 = Aˆ (3) ˆx ˆ 0 u0 = C ˆ0 − Dy h iT with x ¯0 = xT , the system dynamics of the leader x ˆT 0 0 can be written as ( ¯x0 − By ¯ 0 x ˆ˙ 0 = A¯ (4) ¯x y0 = C ¯0 where " A 0

A¯ =

ˆ BC ˆ A

#

" ¯= ,B

ˆ BD ˆ B

# ¯= ,C

h

i C

0

and for the follower i, K (s) can be represented in the state space form ( ˆxi + Bz ˆ i x ˆ˙ i = Aˆ (5) ˆ ˆ i ui = C x ˆi + Dz h

iT , write the closed-loop system comxT x ˆT i i posed of (1) and (5) as ( ¯xi + Bz ¯ i + Bω e i x ˆ˙ i = A¯ (6) ¯ yi = C x ¯i with x ¯i =

"

# B0 . 0 Our purpose is to achieve that the outputs of arbitrary follower i and the leader 0 tend to consensus, so define eyi = yi −y0 , exi = xi −x0 , and then eˆxi = x ˆi −ˆ x0 , e¯xi = x ¯i −¯ x0 ( ¯exi + B ¯ (zi + y0 ) + Bω e i e¯˙xi = A¯ (7) y ¯ e¯xi e =C e= where B

i

from (2) we can obtain zi = −

N X

¢ ¡ aij eyi − eyj − βi yi

j=1

and zi + y0 = −

PN j=1

¡ ¢ aij eyi − eyj − βi eyi , we can rewrite

(7) as  N ¡ ¢  e¯˙x = A¯ ¯exi − B ¯C ¯ P aij e¯xi − e¯xj − βi B ¯C ¯ e¯xi + Bω e i i j=1  y ¯ e¯xi . ei = C (8) For simplicity, set e¯x = (¯ ex1 , e¯x2 , · · · , e¯xN )T , ey = (ey1 , ey2 , · · · , eyN )T , ω = (ω1 , ω2 , · · · , ωN )T the overall dynamics is written in the form: ¢ ¡ ¢¡ ¢  x ¡ ¯ x ¯ ¯ x ˙   e¯ =³ IN ⊗ A´ e¯ − (H ⊗ In ) IN ⊗ B IN ⊗ C e¯ + e ω IN ⊗ B  ¢  y ¡ ¯ e¯x e = IN ⊗ C (9) where H = L+D, “⊗” denotes the symbol of the Kronecker product. Starting from Lemma 1, H is a positive definite matrix. By spectral decomposition, the symmetric matrix H can be written as H = U ΛU T

(10)

where the orthogonal matrix U ∈ RN ×N is formed from the eigenvectors of H, and Λ ∈ RN ×N is a diagonal matrix formed from the eigenvalues so that Λ = diag (λ1 , λ2 , · · · , λN ). Define a coordinate transformation T : ex 7→ ξ x , ξ x := T ex , where T = U T ⊗ In , so ξ y = T ey ; and ω ¯ = T 0 ω, T 0 = U T ⊗ In . The transformation matrix T is an orthogonal transformation because U is orthogonal. Applying the transformation given in (10) to (9), after algebraic manipulations and making use of the Kronecker identities, we have the network dynamics: ³ ´ ( £ ¡ ¢¤ ¯C ¯ ξ x + IN ⊗ B e ω ξ˙x = IN ⊗ A¯ − Λ ⊗ B ¯ (11) ¡ ¢ x y ¯ ξ = IN ⊗ C ξ .

3.2

Leader-following consensus fixed interaction topology

under

In this section, we analyze the effectiveness of the controller when the interaction topology is fixed. The following theorems are given. Theorem 1. Consider the high-order MIMO multi-agent system (1), if there exist matrices X, Y, Ac , Bc , and Cc that the inequalities  T A X + XA + Bc C + C T BcT AT + Ac T  Y A + AY + BCc +  ∗  CcT B T    ∗ ∗ ∗ ∗  CT XB Y CT B    0, for a sufficiently large value of λi , ³ ´ v T Ve v = v T A¯T P + P A¯ v − λi λkvk2 > 0 ⇒ Ve > 0 which contradicts (14). Consequently, a necessary condition ¢ ¡ T T ¯ B ¯ P + PB ¯C ¯ > 0. Therefore, for (14) to hold is that C

−γ 2 I

ω ¯i

it follows that when " ¡ T T ¢ ¯ B ¯ P + PB ¯C ¯ +C ¯TC ¯ A¯T P + P A¯ − δ C T e B P

e PB 2 −γ I

# 0 I Y I Y because of the matrix Q > 0, thus (13) is true. Therefore, we can obtain I − XY > 0, and we can always get nonsingular matrices M and N that satisfy (12) by the singular value decomposition of the matrix I − XY . ¡ ¢ Pre- and post-multiplying (17) by diag ΞT and 1 , 0, 0 diag{(Ξ1 , 0, 0)} respectively, it follows Because ΞT 1 QΞ1 =

    

π11 ∗ ∗ ∗

π12 π22 ∗ ∗

CT Y CT −I ∗

XB 0 B0 0 −γ 2 I

(

³ ´ £ ¡ ¢¤ ¯C ¯ ξ x + IN ⊗ B e ω ξ˙x = IN ⊗ A¯ − Λs ⊗ B ¯ ¡ ¢ ¯ ξx ξ y = IN ⊗ C

(19)

   0 of the controller satisfy the inequality C ˙ from Theorem 1, we can obtain V < 0, and the leader system achieves asymptotic stability. ¤

3.3

are the diagonal elements of the leader adjacency matrix associated with Gs . Therefore, from (15) the network dynamics under switching interaction topology takes the following form:

Leader-following consensus under switching interaction topology

In this section, we extend the result in the last section to the case when the interaction topology is switching. Consider system (1) with the switching topology ª © Gs : s = σ (t) ∈ l0 , where l0 = {1, · · · , M } (M denotes the total number of all possible graphs) is a finite index set, and σ (t) is a switching signal that determines the network topology. Under arbitrary switching signal, the set Γ = {G1 , · · · , GM } is a finite collection of graphs with a common node set V. Ni (s) is the index set of neighbors of agent i in the graph Gs while aij (s) (i, j = 1, 2, · · · , N ) are elements of the adjacency matrix of Gs , and βi (s) (i = 1, 2, · · · , N )

where Λs is the diagonal matrix formed from the eigenvalues of Hs , (Hs = Ls + Ds , Ls is Laplacian of Gs , and Ds is the leader adjacency matrix associated with Gs ), Ni (s), aij (s), Ls and Ds are both time-varying (switched at ti , i = 0, 1, · · · ), though they are time-invariant in any interval [ti , ti+1 ). To see the stability, consider an infinite sequence of nonempty, bounded, nonoverlapping and contiguous time intervals [ti , ti+1 ) , i = 0, 1, · · · , with t0 = 0, τ 6 ti+1 − ti < T for some constants T, τ > 0, such that during each of such intervals, the interconnection topology does not change. That is during each [ti , ti+1 ), the graph Gs is fixed. With the above analysis, we get that the Laplacian matrix L and the leader adjacency matrix D are known in each time interval when the topology is fixed. If the graph Gs is jointly connected across each interval and δ = δmin (δmin is the minimum eigenvalue of Hs ) as well, the theoretical results of Theorem 1 can be applied to the case when the interaction topology is switching, and the multi-agent system can achieve leader-following dynamic consensus asymptotically under switching interaction topology. Meanwhile, similar with Theorem 2, we can get that the leader system is stable. The proving procedures are the same as in Section 3.2. Theorems 1 and 2 provide the theoretical basis of the design of dynamic output feedback controller for high-order MIMO multi-agent network systems. The whole system can achieve leader-following consensus in the situation that the performance index kξiy k2 6 γk¯ ωi k2 is satisfied. Meanwhile, the leader system is stable.

4 4.1

The application in the motion control of multiple mobile robots Mobile robot dynamical model

In this section, the agents are assumed to be specifically two-wheel independent drive mobile robot vehicles shown in Fig. 1[21] .

Fig. 1

Two-wheel independent drive mobile robot

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Z. Q. Wu and Y. Wang / Dynamic Consensus of High-order Multi-agent Systems and Its Application in · · ·

The dynamic property of the robot is given by the following equation of motion Ivi φ¨i = Dri li − Dli li M i v˙ i = Dri + Dli .

(20)

For the right- and left-wheels, the dynamic property of the driving system becomes Iωi θ¨ri + cθ˙ri = kuir − ri Dr Iωi θ¨li + cθ˙li = kuil − ri Dl

(21)

in which M i and Ivi are the mass and moment of inertia of each robot, respectively, Dli and Dri are the left and right driving forces, li is the distance between the left and right wheel, φi is the azimuth angle, v i is the velocity of each robot, Iωi is the moment of inertia of each wheel, c is the coefficient of viscous friction, k is the drive gain, ri is the radius of each wheel, θi is the angle of each wheel, uil and uir are the drive input of the left and right wheel, respectively. The geometrical relationships of φi , v i and θi are given by ri θ˙ri = v i + li φ˙ i ri θ˙li = v i + li φ˙ i .

Fig. 2

   L= 

x˙ i = Axi + Bvi + B 0 ωi yi = Cxi

0 0 0

  0 b1   1 ,B =  0 a2 b2

1 0

0 1

" C=

a1 = −

2c M i (ri )2 i

2Iωi

 b1  0 , −b2

, v i = ui + ui

, a2 = −

2cli 2Iωi (li )2

Ivi (ri )2 + i i

+ kr kr l b1 = , b2 = 2 2 i i i i i M (r ) + 2Iω Iv (r ) + 2Iωi (li )2

Ivi = 10 kg · m2 , M i = 200 kg, li = 0.3 m, ri = 0.1 m, c = 0.05 kg · m2 /s, Iωi = 0.005 kg · m2 , k = 5.

4.2

−1 0 −1 2

    

   D= 

1 0 0 0

0 0 0 0

0 0 1 0

0 0 0 0

   . 

Taking the constant γ as 0.02, the controller of the follower i can be obtained as

# 0 0

0 −1 2 −1

(23)

where a1  A= 0 0

−1 2 −1 0

and the leader adjacency matrix

Assume the state vector as xi = [v , φ˙ i , φi ]T . Furthermore, defining the input vector of leader robot as ui = [uir , uil ]T , take the output as yi = [v i , φi ]T . The system is transformed into a linear one as specified below



2 −1 0 −1

(22) i

(

The graph G of the multiple mobile robots system

Simulations under fixed topology

Suppose that a fixed communication network is given by Fig. 2, which is represented by the Laplacian



 −0.4 0 −0.2   x ˆ˙ i =  0 ˆi + −3.4 0 x 705.2 0.2 −2930.8   0 0.22    −1.23 −0.01  zi −0.8 174.25 " " # −6.564 −12.972 17.324 1 ui = x ˆi + 6.533 −12.994 −17.338 0

# 0 1

zi .

To make the velocity and the azimuth angle of robot tracking a desired trajectory, take u0 as the combination of a sinusoid signal and a step signal. See Figs. 3 and 4 for the simulation results. Fig. 3 (a) indicates that the velocity of all the followers follow the leader and change with the velocity of the leader. To observe the tracking process clearly, we give the simulation results for 0.1 s which are shown in Fig. 3 (b). Fig. 3 (c) shows that the velocity errors between each follower and the leader converges to 0 quickly. Fig. 3 (d) shows the partial enlarged detail of velocity errors. Fig. 4 indicates that the azimuth angle of each follower and the leader reach consensus asymptotically.

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International Journal of Automation and Computing 9(1), February 2012

Fig. 4 Azimuth angles tracking process under fixed topology shown in Fig. 2

4.3

Simulations under switching topology

Consider an undirected network with the switching topol© ª ogy G1 , G2 , G3 in Fig. 5.

Fig. 5

Fig. 3 Fig. 2

Velocity tracking process under fixed topology shown in

The graph Gs of the multiple mobile robots system

In this case, some of the existing communication links fail and some of them are created due to the moving of the mobile robots. We can easily see that topologies G1 , G2 , and G3 are all connected. Without loss of generality we choose the switching signal σ (t) which makes the interaction graphs switched as G1 → G2 → G3 → G1 → · · · , and each graph is active for 1/3 s. Taking the constant γ as 0.01, the controller of the follower i can be obtained as

Z. Q. Wu and Y. Wang / Dynamic Consensus of High-order Multi-agent Systems and Its Application in · · ·

61



   −0.4 0 −0.2 −0.1 7.8     ˆ˙xi =  0 ˆi +  −28.6 −0.4  zi −22 0 x 2223 5 −1582.9 −10.7 3878.7 " # " # −20.9 −52.04 71.22 1 0 ui = x ˆi + zi 20.74 −52.08 −71.3 0 1 according to Theorem 1. Choose the same initial state and input signal u0 . Simulation results are shown in Figs. 6 and 7. We can see that compared with the results under fixed topology, convergence procedure of velocity errors have greater oscillation, and the output azimuth angle errors have slow convergence rate. But the leader-following consensus can be achieved for any switching signal provided the graphs are connected.

Fig. 7 Azimuth angles tracking process under switching topology shown in Fig. 5

The above results demonstrate the effectiveness of the proposed scheme.

5

Fig. 6 Velocity tracking process under switching topology shown in Fig. 5

Conclusions

This paper addressed the consensus problem in undirected network of a multi-agent system with an active leader as well as in fixed and switching topologies, and in a more general case, the system under study is modeled as a highorder MIMO agents system and the state of each agent is undetectable. To solve the consensus problem, a dynamic output feedback strategy is proposed. The consensus problem is also posed as a stabilization problem. When the

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International Journal of Automation and Computing 9(1), February 2012

interaction topology is fixed, a dynamic filter based on the robust H∞ output feedback control method is presented so that the leader-following agent system reaches asymptotic consensus, and the result is also extended to the case that the interaction topology is switching with some constraint conditions on the controller. Moreover, mobile robot vehicle simulations verified the theoretical analysis.

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Zhong-Qiang Wu received the B. Sc. and M. Sc. degrees in automatic control from Northeast Heavy Machinery Institute, PRC in 1989 and 1992, respectively, and Ph. D. degree in control theory and control engineering from China University of Ming and Technology, PRC in 2003. He is a professor at the Institute of Electrical Engineering Yanshan, University, PRC. His research interests include robust control, fuzzy control, adaptive control, and robot control systems. E-mail: [email protected] (Corresponding author) Yang Wang received the B. Eng. degree in automation from Yanshan University, PRC in 2008. She is currently an M. Eng. candidate in intelligent control and pattern recognition at the College of Electrical Engineering, Yanshan University, PRC. Her research interests include cooperative control of multi-agent systems. E-mail: wangyang2 [email protected]