group consensus for heterogeneous multi-agent systems with...

Group consensus for heterogeneous multi-agent systemswith parametric uncertainties

Hong-xiang Hu a,n, Wenwu Yu b,c,d, Qi Xuan e, Chun-guo Zhang a, Guangming Xie f

a Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, Chinab School of Electrical and Computer Engineering, RMIT University, Melbourne VIC3001, Australiac Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabiad Department of Mathematics, Southeast University, Nanjing, 210096, Chinae Zhejiang Provincial United Key Laboratory of Embedded Systems, Department of Automation, Zhejiang University of Technology, Hangzhou, 310023, Chinaf Center for Systems and Control, State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University, Beijing, 100871, China

a r t i c l e i n f o

Article history:Received 29 September 2013Received in revised form10 February 2014Accepted 14 April 2014Communicated by Y. Liu

Keywords:Heterogeneous agentsGroup consensusDouble-integrator systemEuler–Lagrange system

a b s t r a c t

In this paper, a group consensus problem is investigated for the heterogeneous agents that are governedby the Euler–Lagrange system and the double-integrator system, respectively, and the parameters of theEuler–Lagrange system are uncertain. To achieve group consensus, a novel group consensus protocol anda time-varying estimator of the uncertain parameters are proposed. By combining algebraic graph theorywith the Barbalat lemma, several effective sufficient conditions are obtained. It is found that the time-delay group consensus can be achieved provided that the inner coupling matrices are equal in thedifferent sub-networks. Besides, the switching topologies between homogeneous agents are alsoconsidered, with the help of the Barbalat-like lemma, and some relevant results are also obtained.Finally, these theoretical results are demonstrated by the numerical simulations.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Over the past few years, extensive research has been conductedon distributed coordination of multi-agent systems from variousscientific communities [1–8]. Applications of this research includeformation control of mobile robots [9–12], design of sensor net-works [13–16], optimization-based distributed control [17–20],and so on. Please be referred to [21,22] for a more comprehensiveoverview of the field.

As one type of distributed coordination problems, group con-sensus has many civil and military applications in surveillance,reconnaissance, battle field assessment, etc. Research on the groupconsensus not only helps better understand the mechanisms ofnatural collective phenomena, but also provides some useful ideasfor distributed cooperative control. In the group consensus problem,the whole network is divided into multiple sub-networks withinformation exchanges between them, and the aim is to designappropriate protocol such that agents in the same sub-networksreach the same consistent states. In [23], Yu and Wang studied thegroup consensus problem in a multi-agent network with time-varying topologies, and introduced a double-tree-form transforma-tion under which the dynamic equation of agents was transformedinto a reduced-order system. In that work, it was assumed that the

channels between different groups must exist continuously, thusthe protocols proposed in [23] are purely continuous-time ones.Considering that the information exchange between differentgroups may be intermittent in practice, Hu et al. [24] investigatedthe group consensus problem with discontinuous informationtransmissions among different groups, and designed the hybridprotocol to solve it. Recently, Su et al. [25] considered the pinningcontrol problem for cluster synchronization of undirected complexdynamical networks, and proposed a novel decentralized adaptivepinning-control scheme on both coupling strengths and feedbackgains. Furthermore, in [26], Su et al. investigated the clustersynchronization of coupled harmonic oscillators with multipleleaders in an undirected fixed network, and it was shown that alloscillators in the same group could asymptotically synchronize withthe corresponding leader even when only one oscillator in eachgroup has access to the information of the corresponding leader.

Generally, the dynamics of agents is an important part ofa multi-agent system, which will largely influence their consensusstates. However, almost all the aforementioned results were onlyconcerned with distributed coordination of homogeneous multi-agent systems, i.e. all the agents have the same dynamics, whichmight not be realistic in nature where individual heterogeneity isubiquitous [27]. In [28], Zheng et al. considered the consensusproblem of heterogeneous multi-agent system composed of first-order and second-order agents, for which the consensus protocolshave both position and velocity information. It should be noted

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Please cite this article as: H.-x. Hu, et al., Group consensus for heterogeneous multi-agent systems with parametric uncertainties,Neurocomputing (2014), http://dx.doi.org/10.1016/j.neucom.2014.04.021i

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that all the agents achieve static consensus [29] asymptoticallydue to the existence of first-order agents. Based on the aboveresults, the finite-time consensus problem of heterogeneousmulti-agent systems was proposed in [30]. Liu and Hill [31]investigated the global consensus problem between a multi-agent system and a known objective signal by designing an impulsiveconsensus control scheme. In their framework, the consensus criteriain the multi-agent system with the uncertainties, heterogeneousagents, and coupling time-delays were studied in terms of linearmatrix inequalities (LMIs) and algebraic inequalities. More recently,Liu et al. [32] studied the quasi-synchronization issue of linearlycoupled networks with discontinuous nonlinear functions in eachheterogeneous node. Here, quasi-synchronization means the syn-chronization with an error level. By introducing a virtual target, somesufficient quasi-synchronization conditions were presented andexplicit expressions of error levels were derived to estimate thesynchronization error in [33].

In this paper, we investigate a new group consensus of hetero-geneous multi-agent systems, where two types of agents aregrouped as two sub-networks. The agents in one sub-networkare governed by the Euler–Lagrange system with parametricuncertainties, and those in another sub-network are describedby the double-integrator dynamics. Meanwhile, there are time-delays when transmitting information between two sub-networks.Note that the group consensus protocol in [23] cannot be directlyapplied here due to the inherent nonlinearity of the Euler–Lagrangesystem. Instead, we design a novel group consensus protocol anda time-varying estimator of the uncertain parameters for the agentsgoverned by the Euler–Lagrange system to achieve group consensus.Furthermore, considering that the interaction topology betweenagents may change dynamically in practice, we also investigate thegroup consensus with switching topologies. It is worth noting thatthere are such heterogeneous multi-agent systems in reality, such asthe satellite-based vehicle positioning system inwhich the dynamicsof satellite is described by the Euler–Lagrange system, while thedynamics of vehicle is governed by the double-integrator dynamics.

The rest of the paper is organized as follows. In Section II, somebasic definitions in graph theory and mathematical preliminaryresults are provided. In Section III, a novel group consensusprotocol and a time-varying estimator of the uncertain parametersare proposed. Some group consensus results are established, andthese results are validated by the simulation experiments inSection IV. Finally, the paper is concluded in Section V.

Notations. Throughout this paper, ðU ÞT and ðU Þ�1 denotetranspose and inverse, respectively. � denotes the Kroneckerproduct, and J J is the Euclidean norm. If ω is a column vector,then diagðωÞ denotes the diagonal matrix with the ith diagonalentry being the ith element of vector ω.

2. Preliminaries

In this section, some basic definitions in graph theory andpreliminary mathematical results are firstly introduced for thesubsequent use.

2.1. Topology description

For a multi-agent system, information exchange betweenagents can be modeled by directed or undirected graphs [2,22].Let G¼ ðV ; ζ;AÞ be a weighted directed graph of n agents with a setof agents V ¼ fπ1; π2;…πng, a set of edges ζDV � V , and a non-negative adjacency matrix A¼ ½aij�, which is used to represent thenetwork topology. The agent indexes belong to a finite index setI¼ f1;2;…ng, and an edge between two nodes is denoted byeij ¼ ðπi; πjÞ. Moreover, the adjacency elements associated with

the edges of the graph are positive, i.e., ðπi; πjÞAζ3aij40, andthe neighbor set of node πi is denoted by Ni ¼ fπjjðπj; πiÞAζg. Here,we assume that aii ¼ 0 for all iA I, which means i=2Ni for each agent i.The elements in the corresponding Laplacian matrix L¼ ½lij�ARn�n

are defined as

lij ¼ �aij; ia j;

lii ¼ ∑jANi

aij; ð1Þ

satisfying that ∑nj ¼ 1lij ¼ 0; 8 iA I. A directed path in G from πi to πj

is a sequence of distinct vertices starting from πi and ending to πjwith the consecutive vertices being adjacent. Meanwhile, a directedgraph is strongly connected, if there is a directed path between anytwo distinct nodes.

The following lemma gives a property of strongly connectedgraph.

Lemma 1. ([34]). Suppose that GðAÞ represents a directed graph andL is the corresponding Laplacian matrix, then GðAÞ is stronglyconnected if and only if there exists a positive column vector ω¼½ω1;⋯;ωn�T , such that ωTL¼ 0. Furthermore, the matrix ððdiagðωÞLþLTdiagðωÞÞ=2Þ is positive semi-definite.

2.2. System model

In our framework, there are two kinds of dynamics in theheterogeneous multi-agent systems.

Firstly, the agents in one sub-network are described by theEuler–Lagrange dynamics, which has the form

MiðqiÞ €qiþCiðqi; _qiÞqiþgiðqiÞ ¼ τi; iA I1 ¼ f1;…;ng; ð2Þwhere qiARp is the vector of generalized coordinates, MiðqiÞARp�p

is the symmetric positive-definite inertia matrix, Ciðqi; _qiÞqiARp isthe vector of Coriolis and centrifugal torques, giðqiÞ is the vector ofgravitational torque, and τiARp is the group consensus protocol onthe i th agent. Owing to the structure of Euler–Lagrange systems[35], Eq. (2) exhibits certain fundamental properties as follows:

(P1). For any iAf1;…;nþmg, there are positive constants kM ,

kM , kC , and kg such that kM IprMiðqiÞrkMIp, JCiðqi; _qiÞJrkC J _qi J ,and JgiðqiÞJrkg .

(P2). The matrix _MiðqiÞ�2Ciðqi; _qiÞ is skew symmetric.(P3). The Euler–Lagrange systems can be linearly parameter-

ized, that is

MiðqiÞxþCiðqi; _qiÞyþgiðqiÞ ¼ Yiðqi; _qi; x; yÞθi; 8x; yARp; ð3Þwhere Yiðqi; _qi; x; yÞ is the regression matrix and θi is an unknownvector associated with the i th agent.

According to (P3), θi should be estimated by using the informa-tion from the regression matrix Yi.

Then, the agents in another sub-network are governed by thedouble-integrator dynamics:

_xiðtÞ ¼ viðtÞ;_viðtÞ ¼ uiðtÞ�Λ2viðtÞ;

(8 iA I2 ¼ fnþ1;⋯;nþmg; ð4Þ

where xiðtÞ; viðtÞARp, and the matrix �Λ2ARp�p is Hurwitz. Herethe term �Λ2vi represents the velocity damping term [29]. Itshould be especially noted that in [29], the damping force is inproportion to the magnitude of velocity, which is not requiredhere and thus make our results more general. Besides, uiðtÞARp isthe control input or the group consensus protocol.

The following lemmas will be used to derive the main results ofthis paper.

Lemma 2. ([36]). Let ϕ : R-R be a uniformly continuous function on½0;1Þ. Suppose that lim

t-1R t0 ϕðτÞdτ exists and is finite, then ϕðtÞ-0 as

t-1.

H.-x. Hu et al. / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





Lemma 3. ([37]). Given the system_x¼ Axþu; ð5Þsuppose that the system _x¼ Ax is asymptotically stable and uAL1.Then the solution xðtÞ of system (5) and its derivative _xðtÞ belong toL1. In addition, xðtÞ is uniformly continuous, where L1 is defined asthe set ff : RZ0-RnjJ f J1o1g with J f J1 ¼ sup

tZ0J f ðtÞJ .

3. Main results

In this section, the group consensus problem of the hetero-geneous multi-agent systems (2) and (4) is investigated. A novelgroup consensus protocol and a time-varying estimator of theuncertain parameters θi, iA I1 are successfully applied to theanalysis of consensus. Furthermore, the group consensus withswitching topology is also considered.

In the heterogeneous multi-agent systems, time delay ofinformation exchange between agents of different sub-networksis usually unavoidable with practical reasons such as the commu-nication congestion of the channels, the spreading speed of thehardware implementation, and thus is considered in our frame-work, as shown in Fig. 1. Furthermore, let a network topology ðG; xÞconsist of nþm ðn;m41Þ agents. The first n agents constitutea sub-network G1 with the Laplacian matrix L1, and the rest magents constitute the other sub-network G2 with the Laplacianmatrix L2. Meanwhile, it is assumed that the sub-networks G1 andG2 are both strongly connected, and the information exchangebetween G1 and G2 is undirected. Such assumption indicates thatthe information may exchange more frequently between homo-geneous agents than between heterogeneous agents, which con-forms to the law of nature [27].

Denote I1 ¼ f1;2;…ng, I2 ¼ fnþ1;nþ2;…nþmg, V1 ¼ fv1; v2;…vng, V2 ¼ fvnþ1;…vnþmg, N1i ¼ fvjAV1jðvj; viÞAξg, N2i ¼ fvjAV2jðvj; viÞAξg, and denote by ST ¼ fTijjTij40; 8 iA I1; jA I2 or8 iA I2; jA I1g the set of time delay. The following definitionsare needed in order to obtain the main results.

Definition 1. The protocols τ and u are considered to asymptoti-cally solve the group consensus problem, if for any initial states ofsystems (2) and (4), the states of agents satisfy lim

t-1Jqi�qj J ¼

limt-1

J _qi� _qj J ¼ 0, 8 i; jA I1, limt-1

Jxi�xj J ¼ limt-1

Jvi�vj J ¼ 0, 8 i; jA I2.

Furthermore, the protocols τ and u are considered to asymptoti-cally solve the time-delay group consensus problem, if for any

iA I1, jA I2, and TAST , the states of agents satisfy one of thefollowing conditions:

limt-1

Jqiðt�TÞ�xjðtÞJ ¼ limt-1

J _qiðt�TÞ�vjðtÞJ ¼ 0;

limt-1

JqiðtÞ�xjðt�TÞJ ¼ limt-1

J _qiðtÞ�vjðt�TÞJ ¼ 0:

To solve the group consensus problem, a novel group consen-sus protocol is defined as follows:

τi ¼ Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθ̂iþωð1Þi Λ1 ∑

jAN1i

aijðqj�qiÞþωð1Þi ∑

jAN1i

aijð _qj� _qiÞ

þ ∑jAN2i

aij½Λ2xjðt�TijÞþ _xjðt�TijÞ�Λ1qiðtÞ� _qiðtÞ�; 8 iA I1; ð6Þ

ui ¼ ωð2Þi Λ2 ∑

jAN2i

aijðxj�xiÞþωð2Þi ∑

jAN2i

aijðvj�viÞ

þ ∑jAN1i

aij½Λ1qjðt�TijÞþ _qjðt�TijÞ�Λ2xiðtÞ�viðtÞ�; 8 iA I2; ð7Þ

where Λ1; Λ2ARp�p, and �Λ1; �Λ2 are both Hurwitz, θ̂i is theestimation of the unknown parameter θi, the vectors ωð1Þ ¼½ωð1Þ

1 ;⋯;ωð1Þn �T and ωð2Þ ¼ ½ωð2Þ

nþ1;⋯;ωð2Þnþm�T satisfy LT1ω

ð1Þ ¼ 0 and

LT2ωð2Þ ¼ 0, respectively, and the matrix Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞ is

defined in Eq. (3). According to (6) and (7), time delay only affectsthe information exchange between different sub-networks, and itis not assumed that Tij ¼ Tji, 8 iA I1, jA I2.

Remark 1. The matrices Λ1 and Λ2 are called the inner couplingmatrices [38] describing the interactions between different statecomponents of agents, and are always assumed to be symmetric ordiagonal in the most of the former works. However, in this paper,it is only assumed that �Λ1 and �Λ2 are both Hurwitz. Moreover,the protocol (6) includes the term Yiðqi; _qi; �Λ _qi; �ΛqiÞθ̂i, whichprovides an effective mechanism to cope with uncertainties. Incontrast to the protocol in [39], the protocol (6) does not requirethe existence of the damping term in the Euler–Lagrange system.

Then, the main result of the paper is given by the followingtheorem:

Theorem 1. Consider a network with n Euler–Lagrange agents andm double-integrator agents governed by the form (2) and (4),

…… ……

i jTime delay

1G 2G

Fig. 1. The structure of the heterogeneous multi-agent systems.

H.-x. Hu et al. / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





respectively, let the protocols be defined by (6) and (7), together withthe estimation law_̂θi ¼ �Γ�1

i YTi ðqi; _qi; �Λ1 _qi; �Λ1qiÞ½ _qiþΛ1qi�; 8 iA I1; ð8Þ

where ΓiARp�p is a positive definite matrix, then the protocols (6)and (7) solve the group consensus problem. Furthermore, if thematrices Λ1; Λ2 satisfy Λ1 ¼ Λ2, then the protocols (6) and (7) solvethe time-delay group consensus problem.

Proof. Firstly, denote the auxiliary variables by si ¼ _qiþΛ1qi,8 iA I1, rj ¼ vjþΛ2xj, 8 jA I2, and ~θ i ¼ θ̂i�θi. According to (P3), itfollows that:

MiðqiÞ½�Λ1 _qi�þCiðqi; _qiÞ½�Λ1qi�þgiðqiÞ ¼ Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθi;ð9Þ

Hence, from Eqs. (2) and (7), it implies

MiðqiÞ½€qiþΛ1 _qi�þCiðqi; _qiÞ½ _qiþΛ1qi� ¼ τi�Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθi;ð10Þ

i.e.,

MiðqiÞ_siþCiðqi; _qiÞsi ¼ τi�Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθi; ð11ÞNext, we consider the following auxiliary function

VðtÞ ¼ 12

∑n

i ¼ 1sTi MiðqiÞsiþ ~θ

Ti Γi

~θ ih i

þ12

∑nþm

j ¼ nþ1rTj rj

þ12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij

Z t

t�Tji

sTi ðτÞsiðτÞdτ

þ12

∑nþm

j ¼ nþ1∑n

i ¼ 1aji

Z t

t�Tij

rTj ðτÞrjðτÞdτ; ð12Þ

and we have

_VðtÞ ¼ ∑n

i ¼ 1sTi MiðqiÞ_siþ

12sTi _MiðqiÞsiþ ~θ

Ti Γi

_~θ i

� �þ ∑

nþm

i ¼ nþ1rTj _rj

þ ∑n

i ¼ 1∑

nþm

j ¼ nþ1

aij2½sTi ðtÞsiðtÞ�sTi ðt�TjiÞsiðt�TjiÞ�

þ ∑nþm

j ¼ nþ1∑n

i ¼ 1

aji2½rTj ðtÞrjðtÞ�rTj ðt�TijÞrjðt�TijÞ�: ð13Þ

Considering the term sTi MiðqiÞ_siþð1=2ÞsTi _MiðqiÞsiþ ~θTi Γi

_~θ i,according to (P2), Eqs. (8) and (11), we have

sTi MiðqiÞ_siþ12sTi _MiðqiÞsiþ ~θ

Ti Γi

_~θ i

¼ sTi τi�Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθi�Ciðqi; _qiÞsi� �þ1

2sTi _MiðqiÞsi

� ~θTi Y

Ti ðqi; _qi; �Λ1 _qi; �Λ1qiÞsi

¼ sTi τi�Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθi� ��sTi Ciðqi; _qiÞsiþ

12sTi _MiðqiÞsi

� ~θTi Y


¼ sTi τi�Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθi� �� ~θ

Ti Y


¼ sTi ωð1Þi ∑

jAN1i

aijðsj�siÞ" #

þsTi ∑jAN2i

aijðrjðt�TijÞ�siðtÞÞ" #

: ð14Þ

Then, according to Eqs. (4) and (7), it follows that:

rTj _rj ¼ rTj ½_vjþΛ2 _xj�¼ rTj ½uj�Λ2vjþΛ2vj�

¼ rTj ωð2Þj ∑

iAN2j

ajiðri�rjÞ" #

þrTj ∑iAN1j

ajiðsiðt�TjiÞ�rjðtÞÞ" #

: ð15Þ

Substituting Eqs. (13) and (14) into Eq. (15), we obtain

_VðtÞ ¼ ∑n

i ¼ 1sTi ωð1Þ

i ∑jAN1i

aijðsj�siÞ" #

þ ∑n

i ¼ 1sTi ∑

jAN2i

aijðrjðt�TijÞ�siðtÞÞ" #

þ ∑nþm

j ¼ nþ1rTj ωð2Þ

j ∑iAN2j

ajiðri�rjÞ" #

þ ∑nþm

j ¼ nþ1rTj ∑

iAN1j

ajiðsiðt�TjiÞ�rjðtÞÞ" #

þ ∑n

i ¼ 1∑

nþm

j ¼ nþ1

aij2

sTi ðtÞsiðtÞ�sTi ðt�TjiÞsiðt�TjiÞ� �

þ ∑nþm

j ¼ nþ1∑n

i ¼ 1

aji2

rTj ðtÞrjðtÞ�rTj ðt�TijÞrjðt�TijÞh i

¼ ∑n

i ¼ 1sTi ωð1Þ

i ∑jAN1i

aijðsj�siÞ" #

þ ∑nþm


j ∑iAN2j

ajiðri�rjÞ" #

þ ∑n

i ¼ 1∑

nþm

j ¼ nþ1aijsTi rjðt�TijÞ�siðtÞ

� �þ ∑n

i ¼ 1∑

nþm

j ¼ nþ1aijrTj siðt�TjiÞ�rjðtÞ

� �

þ ∑n

i ¼ 1∑

nþm

j ¼ nþ1

aij2

sTi ðtÞsiðtÞ�sTi ðt�TjiÞsiðt�TjiÞ� �

þ ∑nþm

j ¼ nþ1∑n

i ¼ 1

aij2

rTj ðtÞrjðtÞ�rTj ðt�TijÞrjðt�TijÞh i

¼ ∑n

i ¼ 1sTi ωð1Þ

i ∑jAN1i

aijðsj�siÞ" #

þ ∑nþm


j ∑iAN2j

ajiðri�rjÞ" #

�12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij½siðtÞ�rjðt�TijÞ�T ½siðtÞ�rjðt�TijÞ�

�12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij½siðt�TjiÞ�rjðtÞ�T ½siðt�TjiÞ�rjðtÞ�: ð16Þ

By using vector notations, we can rewrite Eq. (16) with a morecompact form as follows:

_VðtÞ ¼ �½sT1;…; sTn� diagðωð1ÞÞL1 � Ip� s1

⋮sn

264

375

�½rTnþ1;…; rTnþm� diagðωð2ÞÞL2 � Ip� rnþ1

⋮rnþm

264

375

�12

∑n

i ¼ 1∑

nþm


�12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij½siðt�TjiÞ�rjðtÞ�T ½siðt�TjiÞ�rjðtÞ�

¼ �½sT1 ;…; sTn�diagðωð1ÞÞL1þLT1diagðωð1ÞÞ

2

" #� Ip

( ) s1⋮sn

264

375

�½rTnþ1;…; rTnþm�diagðωð2ÞÞL2þLT2diagðωð2ÞÞ

2

" #� Ip

( ) rnþ1

⋮rnþm

264

375

�12

∑n

i ¼ 1∑

nþm


�12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij½siðt�TjiÞ�rjðtÞ�T ½siðt�TjiÞ�rjðtÞ�; ð17Þ

where Ip denotes the p� p identity matrix. According to Lemma 1,it follows _VðtÞr0, indicating that VðtÞ is not increasing with t.Since VðtÞZ0, 8 tA ½0; þ1Þ, the function VðtÞ is bounded, and thisyields that SAL1, ~θAL1, and RAL1, where S¼ ½sT1;…; sTn�T ,~θ ¼ ½ ~θT1 ;…; ~θ

Tnþm�T , and R¼ ½rTnþ1;…; rTnþm�T . By the definition of si,

rj, and Lemma 3, we can get that qiðtÞ; _qiðtÞAL1, and xiðtÞ;viðtÞAL1, then, Λ1qiðtÞ, Λ1 _qiðtÞ and Λ2viðtÞ are also bounded.According to (P1), the matrix Ciðqi; _qiÞ is bounded, which impliesthat the regression matrix Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞ is also boundedfrom Eq. (9). Hence, according to Eq. (6) and (P1), we obtain that_S is bounded. Since _riðtÞ ¼ €xiðtÞþΛ2 _xiðtÞ ¼ _viðtÞþΛ2viðtÞ ¼ uiðtÞ,according to Eq. (7), _R is also bounded. Therefore, we have that€VðtÞ is a bounded function and _VðtÞ is a uniformly continuous






function on ½0;1Þ. Since Lemma 2 indicates limt-1

_VðtÞ ¼ 0, due to the

assumption that sub-networks G1 and G2 are strongly connected,we can conclude that

limt-1

Jsi�sj J ¼ 0; 8 i; jA I1; ð18Þ

and

limt-1

Jri�rj J ¼ 0; 8 i; jA I2: ð19Þ

From the definition of si and ri, we have

si�sj ¼ _qi� _qjþΛ1ðqi�qjÞri�rj ¼ vi�vjþΛ2ðxi�xjÞ: ð20Þ

Because the matrices �Λ1 and �Λ2 are Hurwitz, we obtain that

limt-1

Jqi�qj J ¼ limt-1

J _qi� _qj J ¼ 0; 8 i; jA I1;

limt-1

Jxi�xj J ¼ limt-1

Jvi�vj J ¼ 0; 8 i; jA I2: ð21Þ

Thus the protocols (6) and (7) solve the group consensus problemof the heterogeneous multi-agent systems. Furthermore, if Λ1 ¼ Λ2,we have

siðtÞ�rjðt�TijÞ ¼ _qiðtÞþΛ1qiðtÞ� _xjðt�TijÞ�Λ1xjðt�TijÞ¼ ð_qiðtÞ� _xjðt�TijÞÞþΛ1ðqiðtÞ�xjðt�TijÞÞ; ð22Þand

siðt�TjiÞ�rjðtÞ ¼ _qiðt�TjiÞþΛ1qiðt�TjiÞ� _xjðtÞ�Λ1xjðtÞ¼ ð_qiðt�TjiÞ� _xjðtÞÞþΛ1ðqiðt�TjiÞ�xjðtÞÞ: ð23ÞHence, according to Eq. (17), we can conclude that if there existsthe information exchange between agents i and j in the differentsub-networks,

limt-1

JqiðtÞ�xjðt�TijÞJ ¼ limt-1

J _qiðtÞ�vjðt�TijÞJ ¼ 0;

limt-1

Jqiðt�TjiÞ�xjðtÞJ ¼ limt-1

J _qiðt�TjiÞ�vjðtÞJ ¼ 0: ð24Þ

Moreover, for any TAST , there exist indexes i0A I1 and j0A I2 suchthat T ¼ Ti0j0 or T ¼ Tj0 i0 . Without loss of generality, it is assumedthat T ¼ Ti0 j0 , and it follows Eq. (24) that

limt-1

Jqi0 ðtÞ�xj0 ðt�Ti0 j0 ÞJ ¼ limt-1

J _qi0 ðtÞ�vj0 ðt�Ti0j0 ÞJ ¼ 0: ð25Þ

Therefore, for any iA I1 and jA I2, we have

JqiðtÞ�xjðt�TÞJ ¼ JqiðtÞ�qi0 ðtÞþqi0 ðtÞ�xj0 ðt�Ti0 j0 Þþxj0 ðt�Ti0 j0 Þ�xjðt�Ti0j0 ÞJ

r JqiðtÞ�qi0 ðtÞJþ Jqi0 ðtÞ�xj0 ðt�Ti0 j0 ÞJþ Jxj0 ðt�Ti0j0 Þ�xjðt�Ti0 j0 ÞJ : ð26Þ

then

limt-1

JqiðtÞ�xjðt�TÞJ ¼ 0; ð27Þ

Thus the protocols (6) and (7) solve the time-delay group con-sensus problem of the heterogeneous multi-agent systems. Thiscompletes the proof.

Remark 2. In Theorem 1, by the protocols (6) and (7), we solve thegroup consensus problem asymptotically. Note that the protocolsin [23,24] cannot be directly applied to the heterogeneous system,since they only consider the single-integrator dynamics in theirmodels. Furthermore, we give the definition of the time-delaygroup consensus, and it is proved that the protocols (6) and (7) cansolve the time-delay group consensus problem provided that theinner coupling matrices are equal in the different sub-networks.

In practice, the relationships between homogeneous agentsmay vary over time [27], and their interconnection topology mayalso be dynamically changing. Consequently, switching topologiesbetween homogeneous agents should be taken into account in

some cases. Suppose that there are N topologies G1, …, GN , and foreach complex network ðGρ;VÞ, its sub-networks Gρ

1 and Gρ2 are

strongly connected. Then, a switching topology is defined bya switching signal sðtÞ : ½0; þ1Þ-ℑ, which is represented by apiecewise constant function fskg : tk�1; tkð �-sk with the timesequence ftkg satisfying: 0¼ t0ot1ot2o…otko…; lim

k-1tk ¼

1, and there is a constant μ (called dwell time) with tkþ1�tkZμ

for all kZ0, which helps to avoid the case of infinitely fastswitching (chattering). Moreover, sðtÞ ¼ ρ means that the topologyGρ is adopted at time t. Note that the interconnection topologybetween heterogeneous agents does not change in this process.

Then, a group consensus protocol with switching topologiesbetween homogeneous agents is defined as follows: 8 tA tk�1; tkð �,τi ¼ Yiðqi; _qi; �Λ1 _qi; �Λ1qiÞθ̂iþωsðtÞð1Þ

i Λ1 ∑jAN1i

asðtÞij ðqj�qiÞ

þωsðtÞð1Þi ∑

jAN1i

asðtÞij ð _qj� _qiÞ

þ ∑jAN2i

aij½Λ2xjðt�TijÞþ _xjðt�TijÞ�Λ1qiðtÞ� _qiðtÞ�; 8 iA I1; ð28Þ

ui ¼ ωsðtÞð2Þi Λ2 ∑

jAN2i

asðtÞij ðxj�xiÞþωsðtÞð2Þi ∑

jAN2i

asðtÞij ðvj�viÞ

þ ∑jAN1i

aij½Λ1qjðt�TijÞþ _qjðt�TijÞ�Λ2xiðtÞ�viðtÞ�; 8 iA I2;

ð29Þ

where the vectors ωð1ÞsðtÞ ¼ ½ωð1Þ

1sðtÞ;…;ωð1ÞnsðtÞ�T and ωð2Þ

sðtÞ ¼ ½ωð2Þnþ1sðtÞ;…;

ωð2ÞnþmsðtÞ�T satisfy LTsðtÞ1ω

ð1ÞsðtÞ ¼ 0 and LTsðtÞ2ω

ð2ÞsðtÞ ¼ 0, respectively.

Remark 2. Due to the switching topologies, Lemma 2 cannot bedirectly used to establish group consensus. In fact, the function_VðtÞ in Theorem 1 is piecewise continuous, which does not satisfythe condition of Lemma 2. In this paper, the Barbalat-like lemmaoriginally established in [40] is adopted to solve the groupconsensus problem with switching topologies.

The following Theorem shows that the group consensus problemcould be solved even when the communication graph is dynamic.

Theorem 2. Consider that the complex network ðG;VÞ has switchingtopologies between homogeneous agents, whose dynamics are gov-erned by either Eqs. (2) or (4). Provided by the estimation law_̂θi ¼ �Γ�1

i YTi ðqi; _qi; �Λ1 _qi; �Λ1qiÞ½ _qiþΛ1qi�; iA I1; ð30Þ

where ΓiARp�p is a positive definite matrix, the protocols (28) and(29) solve the group consensus problem. Furthermore, if thematrices Λ1; Λ2 satisfy Λ1 ¼ Λ2, then the protocols (28) and (29)solve the time-delay group consensus problem.

Proof. Denote the auxiliary variables by si ¼ _qiþΛ1qi, 8 iA I1,rj ¼ vjþΛ2xj, 8 jA I2, and ~θ i ¼ θ̂i�θi. Consider the following aux-iliary function

VðtÞ ¼ 12

∑n

i ¼ 1½sTi MiðqiÞsiþ ~θ

Ti Γi

~θ i�þ12

∑nþm

j ¼ nþ1rTj rjþ

12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij

Z t

t�Tji

sTi ðτÞsiðτÞdτþ12

∑nþm

j ¼ nþ1∑n

i ¼ 1aji

Z t

t�Tij

rTj ðτÞrjðτÞdτ; ð31Þ

Then, 8 tA tk�1; tkð �, the derivative of VðtÞ along the trajectory ofEqs. (2) and (4) is given by

_VðtÞ ¼ �½sT1 ;…; sTn�diagðωð1Þ

sðtÞÞL1sðtÞ þLT1sðtÞdiagðωð1ÞsðtÞÞ

2

" #� Ip

( ) s1⋮sn

264

375

�½rTnþ1;…; rTnþm�diagðωð2Þ

sðtÞÞL2sðtÞ þLT2sðtÞdiagðωð2ÞsðtÞÞ

2

" #� Ip

( )






rnþ1

⋮rnþm

264

375�1

2∑n

i ¼ 1∑

nþm


�12

∑n

i ¼ 1∑

nþm

j ¼ nþ1aij½siðt�TjiÞ�rjðtÞ�T ½siðt�TjiÞ�rjðtÞ�r0; ð32Þ

Since V ðtÞZ0, 8 tA ½0;1Þ, it yields that V ðtÞ is bounded, whichimplies that SAL1, RAL1, and ~θAL1, with S¼ ½sT1;…; sTn�T ,R¼ ½rTnþ1;…; rTnþm�T , and ~θ ¼ ½ ~θT1 ;⋯; ~θ

Tnþm�T .

According to the assumption on the switching time sequence,there exists a subsequence of switching times ftwk g such that thetime intervals twkþ 1 �twk Zτ, k¼ 1;2;…, apparently, τ4μ, andsðtÞ ¼ ρ on these time intervals. Denote the union of these timeintervals by Ξ, and construct the auxiliary function as follows:

xΞðtÞ ¼� _V ðtÞ; tAΞ

0; t=2Ξ

(: ð33Þ

Obviously, xΞðtÞ is piecewise continuous, and xΞðtÞZ0. FromEq. (32), we obtain that 8 tZ0,Z t

0xΞðsÞdsrVðtw1 Þ�VðtÞrV ðtw1 Þ: ð34Þ

Thus,R10 xΞðsÞds exists and is finite. Next, we proceed to prove that

limt-1

xΞðtÞ ¼ 0. Suppose that it is not true, then there existsa constant ε40 and an infinite sequence of times ft0kg � Ξ suchthat xΞðt0kÞZε, 8k. Due to the fact that €VðtÞ is bounded on theinterval twk ; twk þ1

� �, the function xΞðtÞ is uniformly continuous on

twk ; twk þ1� �

. Therefore, there exists a constant η40 such that forany t belonging to time interval of length η, one has xΞðtÞZðε=2Þ,which results in a contradiction. So, we have lim

t-1xΞðtÞ ¼ 0, and

limt-1

_VðtÞ ¼ 0. Due to the assumption that sub-networks Gρ1 and Gρ

2are strongly connected, we can conclude that

limt-1

Jsi�sj J ¼ 0; 8 i; jA I1; ð35Þ

and

limt-1

Jri�rj J ¼ 0; 8 i; jA I2: ð36Þ

Similar to Theorem 1, it follows that the protocols (28) and (29)solve the group consensus problem. Moreover, if Λ1 ¼ Λ2, theprotocol (28) and (29) also solve the time-delay group consensusproblem. This completes the proof.

4. Numerical simulation

In this section, two numerical examples are provided toillustrate the effectiveness of the proposed theoretical results.

In the simulation, the dynamics of Euler–Lagrange system isdescribed as

Mi11 Mi

12

Mi21 Mi

22

" #€qð1Þi

€qð2Þi

24

35þ

�hi _qð2Þi �hið _qð1Þi þ _qð2Þi Þ

hi _qð1Þi 0

24

35 _qð1Þi

_qð2Þi

24

35¼

τð1Þi

τð2Þi

24

35;

iA I1; ð37Þwhere qi ¼ ½qð1Þi ; qð2Þi �T , τi ¼ ½τð1Þi ; τð2Þi �T , andMi

11 ¼ a1þ2a3 cos ðqð2Þi Þþ2a4 sin ðqð2Þi ÞMi

12 ¼Mi21 ¼ a2þa3 cos ðqð2Þi Þþa4 sin ðqð2Þi Þ

Mi22 ¼ a2

hi ¼ a3 sin ðqð2Þi Þ�a4 cos ðqð2Þi Þ; ð38Þ

with

a1 ¼ d1þm1l21þd2þm2l

22þm2l

23

a2 ¼ d2þm2l22

a3 ¼m2l3l2 cos ðδÞa4 ¼m2l3l2 sin ðδÞ: ð39ÞHere the parameters are chosen asm1 ¼ 1,m2 ¼ 2, l1 ¼ 0:5, l2 ¼ 0:6,l3 ¼ 1, d1 ¼ 0:12, d2 ¼ 0:25 and δ¼ ðπ=6Þ. Moreover, according toEq. (3), Yiðqi; _qi; x; yÞAR2�4 is defined as

Yi11 ¼ x1; Yi

12 ¼ x2; Yi21 ¼ 0; Yi

22 ¼ x1þx2;

Yi13 ¼ ð2x1þx2Þ cos ðqð2Þi Þ�ð _qð2Þi y1þ _qð1Þi y2þ _qð2Þi y2Þ sin ðqð2Þi Þ

Yi14 ¼ ð2x1þx2Þ sin ðqð2Þi Þþð_qð2Þi y1þ _qð1Þi y2þ _qð2Þi y2Þ cos ðqð2Þi Þ

Yi23 ¼ x1 cos ðqð2Þi Þþ _qð1Þi y1 sin ðqð2Þi Þ

Yi24 ¼ x1 sin ðqð2Þi Þ� _qð1Þi y1 cos ðqð2Þi Þ; ð40Þ

where x¼ ½x1; x2�T and y¼ ½y1; y2�T .

Example 1. Consider ten agents in the heterogeneous multi-agentsystems, with their interaction topology presented in Fig. 2.

From Fig. 2, the whole network G is divided into two sub-networks G1 and G2, and information exchanges exist between the

Fig. 2. The interaction topology of ten agents.






two sub-networks. It should be noted that the agents in G1 aregoverned by the Euler–Lagrange system, and the agents in G2 aredescribed by the double-integrator dynamics. Furthermore, theLaplacian matrixes of G1 and G2 can be easily obtained as follows:

L1 ¼

0:2 0 �0:2 0 0�0:3 0 0:3 0 00 0 0:4 0 �0:40 �0:5 �0:8 1:3 00 0 0 �0:7 0:7

26666664

37777775;

L2 ¼

0:6 0 �0:6 0 0�0:9 1:4 0 �0:5 00 �0:3 0:3 0 00 0 0 0:4 �0:40 0 �0:5 0 0:5

26666664

37777775: ð41Þ

Then the adjacency matrix of information exchanges betweenG1 and G2 is given by

B¼

0 0:3 0 0 00 0 0:5 0 00:7 0 0 0 00 0 0 0 00 0:6 0 0 0

26666664

37777775: ð42Þ

Note that the element 0.3 in B represents the weight betweenagent 1 and agent 7. According to Eq. (41), sub-networks G1 and G2

are strongly connected. Next, the initial values of qi, _qi, xi and vi arechosen randomly within ½�0:5;0:5� � ½�0:5;0:5�. For simplicity,we choose the identical inner coupling matrices for the sub-networks G1 and G2, i.e.,

Λ1 ¼ Λ2 ¼1:5819 �1:2273�0:6546 1:4181

� �: ð43Þ

Meanwhile, the time-delays between different sub-networks arealso identical, i.e., Tij ¼ 2. According to Eqs. (2), (4), (6) and (7), theevolutions of qi and _qi are shown in Fig. 3 and 4, while theevolutions of xi and vi are shown in Fig. 5 and 6, respectively.Apparently, the protocols (6) and (7) solve the group consensusproblem. Moreover, the error between agents 1 and 6 is defined asfollows:

e1ðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑2

i ¼ 1½qðiÞ1 ðt�2Þ�xðiÞ6 ðtÞ�2

s;

e2ðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑2

i ¼ 1½ _qðiÞ1 ðt�2Þ�vðiÞ6 ðtÞ�2

s; ð44Þ

Fig. 3. The evolution of qiðtÞ with fixed topology.

Fig. 4. The evolution of _qiðtÞ with fixed topology.

Fig. 5. The evolution of xiðtÞ with fixed topology.

Fig. 6. The evolution of viðtÞ with fixed topology.

Fig. 7. The evolution of eðtÞ with fixed topology.






where x6 ¼ ½xð1Þ6 ; xð2Þ6 �T , v6 ¼ ½vð1Þ6 ; vð2Þ6 �T . From Theorem 1, the proto-cols (6) and (7) could solve the time-delay group consensusproblem provided that Λ1 ¼ Λ2, which is shown in Fig. 7.

Example 2. Consider ten agents with switching topologies shownin Fig. 8. Here, the dynamics of each agent in G1 is also describedby Euler–Lagrange systems (37), and the initial conditions andother parameters are all set the same as those in Example 1.

In this case, there are two different interaction topologies. For(a), the Laplacian matrixes and the adjacency matrix B are chosen

the same as those in Example 1, while for (b), the Laplacianmatrixes of G1 and G2 are as follows:

L1 ¼

0:5 0 �0:5 0 0�0:8 1:5 0 �0:7 00 �0:8 0:8 0 00 0 0 0:7 �0:70 0 �0:3 0 0:3

26666664

37777775;

Fig. 8. Two different interaction topologies of ten agents.

Fig. 9. The evolution of qiðtÞ with switching topologies.

Fig. 10. The evolution of _qiðtÞ with switching topologies.

Fig. 11. The evolution of xiðtÞ with switching topologies.

Fig. 12. The evolution of viðtÞ with switching topologies.






L2 ¼

0:5 0 �0:5 0 0�0:6 0 0:6 0 00 0 0:8 0 �0:80 �0:8 �0:6 1:4 00 0 0 �0:3 0:3

26666664

37777775: ð45Þ

And the adjacency matrix of information exchanges betweenG1 and G2 is the same as that in Example 1.

Moreover, systems (2) and (4) starts at (a), and switches everyT ¼ 20 s between (a) and (b). According to Eqs. (2), (4), (28) and(29), the evolutions of qi and _qi for these ten agents are shown inFigs. 9 and 10 while the evolutions of xi and vi are shown in Figs. 11and 12, respectively. Therefore, the protocols (28) and (29) solvethe group consensus problem with switching topologies. Further-more, the evolution of eðtÞ is shown in Fig. 13, which implies thatthe protocols (28) and (29) can also solve the time-delay groupconsensus problem. It should be noted that the dotted linerepresents the switching time.

5. Conclusion

This paper discussed the group consensus problem of theheterogeneous agents which are governed by either the Euler–Lagrange system or the double-integrator system. In this study, theparameters of the Euler–Lagrange system are uncertain. In order toachieve group consensus, a novel group consensus protocol anda time-varying estimator of the uncertain parameters have beenproposed. By combining algebraic graph theory with the Barbalatlemma, several effective sufficient conditions were obtained.Compared with the existent works in the literature, we proposedthe definition of the time-delay group consensus, and proved thatthe time-delay group consensus can be achieved suppose that theinner coupling matrices are equal in the different sub-networks.Besides, the paper also considered switching topologies betweenhomogeneous agents. With the help of the Barbalat-like lemma, itwas proved that the group consensus problem with switchingtopologies can be solved. Finally, two numerical simulations werepresented to illustrate the theoretical results. In the future, time-delay needs to be considered in the information exchangesbetween homogeneous agents and the corresponding group con-sensus protocols need to be designed.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under the Grant nos. 60974017, 61273212,

61104145, 61322302, 61374096 and 61004097, Zhejiang ProvincialNational Natural Science Foundation of China under Grant no.LQ14F030011, the Natural Science Foundation of Jiangsu Provinceof China under Grant no. BK2011581, the Research Fund for theDoctoral Program of Higher Education of China under Grant no.20110092120024, the Fundamental Research Funds for the CentralUniversities of China, and the Information Processing and Automa-tion Technology Prior Discipline of Zhejiang Province-OpenResearch Foundation under Grant nos. 20120801 and 20120802respectively.

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Hong-xiang Hu received his B.S. degree in ElectronicEngineering from Zhejiang Gongshang University in2005, an M.S. degree in Mathematics from HangzhouDianzi University in 2009, and a Ph.D. degree in ControlTheory and Control Engineering from Zhejiang Universityof technology. At present, he is a lecturer in HangzhouDianzi University, Hangzhou, China. His current researchinterests include multi-agent systems, collective dynamics,complex networks, hybrid and switched systems.

Wenwu Yu received the B.Sc. degree in informationand computing science and the M.Sc. degree in appliedmathematics from the Department of Mathematics,Southeast University, Nanjing, China, in 2004 and2007, respectively, and the Ph.D. degree from theDepartment of Electronic Engineering, City Universityof Hong Kong, Kowloon, HongKong, in 2010. He iscurrently a full Professor with the Research Center forComplex Systems and Network Sciences, Departmentof Mathematics, Southeast University. He is also withthe School of Electrical and Computer Engineering,Royal Melbourne Institute of Technology University,Melbourne, Australia. He held several visiting positions

in Australia, China, Germany, Italy, The Netherlands, and the US. His researchinterests include multiagent systems, nonlinear dynamics and control, complexnetworks and systems, neural networks, cryptography, and communications. Dr. Yuis the recipient of the Best Master Degree Theses Award from Jiangsu ProvinceChina, in 2008, DAAD Scholarship from Germany in 2008, TOP 100 Most CitedChinese Papers Published in International Journals in 2008, the Best Student PaperAward in the 5th Chinese Conference on Complex Networks in 2009, the First Prizeof Scientific and Technological Progress Award of Jiangsu Province in 2010, GoldenAward for Scopus Young Scientific Stars in Information Science in 2012, The BestPaper Award in 9th Asian Control Conference in 2013, and The Best Theory PaperAward in Neural Systems and Applications Technical Committee of IEEE Circuitsand Systems Society in 2013. Moreover, he was awarded a National Natural ScienceFund for Excellent Young Scholars in 2013.

Qi Xuan received the B.Eng. degree and Ph.D. degree incontrol theory and engineering from Zhejiang Univeristy,Hangzhou, China in 2003 and 2008, respectively. He wasa researcher in the Centre for Chaos and Complex Net-works, City University of Hongkong in China and theDepartment of Computer Science, University of CaliforniaDavis in USA. At present, he is an associate professor inZhejiang University of technology, Hangzhou, China. Hiscurrent research interests include social network datamining, social synchrony, reaction–diffusion dynamics,and open sourece software engineering.

Chun-guo Zhang received his B.S. degree in Mathe-matics from Xiamen University in 1986, an M.S. degreein Mathematics from Zhejiang University in 2001. Now heis a Professor in Hangzhou Dianzi University, Hangzhou,China. His current research interests include linear sys-tem control theory, distributed parameter control system.

Guangming Xie received his B.S. degrees in AppliedMathematics and Computer Science and Technology, anM.E. degree in Control Theory and Control Engineering,and a Ph.D. degree in Control Theory and ControlEngineering from Tsinghua University, Beijing, Chinain1996, 1998 and 2001, respectively. Then he worked asa postdoctoral research fellow in the Center for Systemsand Control, Department of Mechanics and EngineeringScience, Peking University, Beijing, China from July2001 to June 2003. In July 2003, he joined the Centeras a lecturer. Now he is a Professor of Dynamics andControl. He is also a Guest Professor of East ChinaJiaotong University. He is an Editorial Advisory Board

member of the International Journal of Advanced Robotic Systems and an EditorialBoard Member of the Open Operational Research Journal. His research interestsinclude hybrid and switched systems, networked control systems, multiagentsystems, multirobot systems, and biomimetic robotics.




































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