IEEE TRANSACTIONS ON CIRCUITS AND …aerocontrol/duanzs/论文/10－tcsi...observed in [43] by using graph theory. In [23], a general framework of the consensus problem for networks

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 1, JANUARY 2010 213

Consensus of Multiagent Systems andSynchronization of Complex Networks: A

Unified ViewpointZhongkui Li, Zhisheng Duan, Guanrong Chen, Fellow, IEEE, and Lin Huang

Abstract—This paper addresses the consensus problem of multi-agent systems with a time-invariant communication topology con-sisting of general linear node dynamics. A distributed observer-type consensus protocol based on relative output measurementsis proposed. A new framework is introduced to address in a uni-fied way the consensus of multiagent systems and the synchroniza-tion of complex networks. Under this framework, the consensus ofmultiagent systems with a communication topology having a span-ning tree can be cast into the stability of a set of matrices of thesame low dimension. The notion of consensus region is then intro-duced and analyzed. It is shown that there exists an observer-typeprotocol solving the consensus problem and meanwhile yieldingan unbounded consensus region if and only if each agent is bothstabilizable and detectable. A multistep consensus protocol designprocedure is further presented. The consensus with respect to atime-varying state and the robustness of the consensus protocol toexternal disturbances are finally discussed. The effectiveness of thetheoretical results is demonstrated through numerical simulations,with an application to low-Earth-orbit satellite formation flying.

Index Terms—Consensus, graph theory, linear matrix inequality(LMI), multiagent system, synchronization, control.

I. INTRODUCTION

I N RECENT years, the coordination problem of multiagentsystems has received compelling attention from various sci-

entific communities due to its broad applications in such areasas satellite formation flying [4], cooperative unmanned air vehi-cles[2], scheduling of automated highway systems[3], and airtraffic control[38]. One critical issue arising from multiagentsystems is to develop distributed control policies based on localinformation that enables all agents to reach an agreement oncertain quantities of interest, which is known as the consensusproblem.

Consensus problem has a long-standing tradition in computerscience. In the context of multiagent systems, recent years have

Manuscript received October 30, 2008; revised March 05, 2009. First pub-lished June 02, 2009; current version published January 27, 2010. This work wassupported in part by the National Natural Science Foundation of China underGrant 60974078,Grant 10832006 and Grant 60674093 and in part by the KeyProjects of Educational Ministry under Grant 107110. This paper was recom-mended by Associate Editor M. Di Marco.

Z. Li, Z Duan, and L. Huang are with the State Key Laboratory for Tur-bulence and Complex Systems, Department of Mechanics and Aerospace En-gineering, College of Engineering, Peking University, Beijing 100871, China(e-mail: [email protected]; [email protected]).

G. Chen is with the State Key Laboratory for Turbulence and Complex Sys-tems, Department of Mechanics and Aerospace Engineering, College of Engi-neering, Peking University, Beijing 100871, China, and also with the Centre forChaos and Complex Networks and the Department of Electronic Engineering,City University of Hong Kong, Kowloon, Hong Kong.

Digital Object Identifier 10.1109/TCSI.2009.2023937

witnessed dramatic advances of various distributed strategiesthat achieve agreements. Reference[43] proposed a simplemodel for phase transition of a group of self-driven particlesand numerically depicted the complexity of the model. Refer-ence[15] provided a theoretical explanation for the behaviorobserved in [43] by using graph theory. In [23], a generalframework of the consensus problem for networks of dynamicagents with fixed or switching topologies and communicationtime delays was established. The conditions derived in[23] werefurther relaxed in[29]. Reference[1] proposed a passivity-baseddesign framework to deal with the group coordination problem,where both fixed and time-varying communication structureswere considered. References[12] and [13] considered trackingcontrol for multiagent consensus with an active leader anddesigned a local controller together with a neighbor-basedstate-estimation rule. Predictive mechanisms were introducedin [44] to achieve ultrafast consensus. Reference[19] investi-gated the consensus problem for directed networks of agentswith external disturbances and model uncertainties on fixedand switching topologies. Reference[7] characterized a dis-tributed algorithm that asymptotically achieved consensus, andprovided two discontinuous distributed algorithms that achievemax and min consensus, respectively, in finite time. For a rela-tively complete coverage of the literature on consensus, readersare referred to the recent surveys [24], [30]. One well-knownproblem in most existing works is that the agent dynamics areoften restricted to be single or double integrators or structuralhigh-order linear systems. Another common problem is thatmost proposed distributed consensus protocols are based onthe relative states between neighboring agents, which, in manycases, is not available.

Another topic that is closely related to the consensus of mul-tiagent systems is the synchronization of coupled nonlinear os-cillators. In the pioneering work [26], the synchronization phe-nomenon of two master–slave chaotic systems was observedand applied to secure communications. References [22] and [27]addressed the synchronization stability of a network of oscilla-tors by using the master stability function method. Recently, thesynchronization of complex dynamical networks, such as small-world and scale-free networks, has been widely studied (see[5],[9],[17],[21], [28], [40], [42] and the references therein). Dueto nonlinear node dynamics, usually, only sufficient conditionscan be given for verifying the synchronization.

This paper is concerned with the consensus problem of mul-tiagent systems under a fixed (time-invariant) communicationtopology, where each agent has general linear dynamics, whichmay also be considered as the linearized model of a nonlinear

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214 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 1, JANUARY 2010

network. A distributed observer-type consensus protocol basedon relative output measurements is proposed, which can beregarded as an extension of the traditional observer-basedcontroller for a single system. A new unified framework is in-troduced, which converts the consensus problem of multiagentsystems with a communication topology having a spanning treeinto the stability of a set of matrices with the same low dimen-sion (as a single agent). The proposed framework is, in essence,consistent with the master stability function method used inthe synchronization of complex networks and yet presents aunified viewpoint to both the consensus of multiagent systemsand the synchronization of complex networks. The notion ofconsensus region is then introduced and analyzed with the helpof the stability of matrix pencils. It is shown that there existsan observer-type protocol solving the consensus problem andmeanwhile yielding an unbounded consensus region if andonly if each agent is stabilizable and detectable. A constructivethree-step consensus protocol design procedure is further pro-posed. The first two steps deal with only the agent dynamicsand feedback gain matrices of the consensus protocol, whilein the third step, the effect imposed by the communicationtopology of the agents on the consensus stability is managed byadjusting the coupling strength. One favorable feature of thisapproach is that the protocol designed to achieve consensusfor one given communication topology can be directly used tosolve the consensus problem for any other topology containinga spanning tree, where the only task is to select a suitablecoupling strength.

Roughly speaking, there are two fundamental questions aboutthe consensus problem of multiagent systems: “how to reachconsensus” and “consensus on what.” They are both addressedin this paper. The beginning part of this paper is mainly de-voted to the first question. To that end, the second question, theconsensus with respect to a time-varying state, is brought for-ward and discussed under the proposed framework in the lastpart of this paper. The robustness of consensus protocols to ex-ternal disturbances will also be studied, where the communica-tion topology is restricted to be undirected. As mentioned ear-lier, this paper intends to relate the consensus of multiagent sys-tems and the synchronization of complex networks. Pertinentworks along this line include [39] and [37], where synchroniza-tion problems of identical discrete- and continuous-time linearsystems are studied. Differing from the protocols proposed inthis paper, the coupling law used in [39] is essentially relativestate feedback, and the dynamic coupling law in[37] requiresthe absolute output measurement of each agent, which is im-practical in many cases. A typical instance is deep-space for-mation flying, where the measurement of the absolute positionof the spacecraft is accurate only on the order of kilometers,which is thus useless for control purposes [35]. Other kinds ofdynamic protocols have been proposed in different scenarios,e.g., in[10], [36]. To briefly highlight the main differences, thispaper generalizes the existing results on consensus of multia-gent systems in at least two aspects. First, the agent dynamicsare extended to be in a general linear form, without limiting tointegrators. The agent dynamics are excluded from having polesin the open right-half plane in order to reach a nontrivial con-sensus value. Second, observer-type consensus protocols basedon relative output measurements between neighboring agents

are proposed, which is more general than most existing modelssuch as the protocol based on relative states and the static pro-tocol used in [39]. For completeness, relative-state consensusprotocols are also considered here as a special case of the presentunified framework.

The rest of this paper is organized as follows. Section II in-troduces some basic notation and reviews some useful results ofalgebraic graph theory.Section III presents a new framework totackle the consensus problem of multiagent systems and inves-tigates possible applications of consensus algorithms to satelliteformation flying in the low Earth orbit. The consensus problemwith respect to a reference state is considered in Section IV. Therobustness of the consensus protocol to external disturbances isformulated and analyzed in Section V.Section VI concludes thispaper.

A. Notation and Preliminaries

Let and be the sets of real and complexmatrices, respectively. The superscript means transpose forreal matrices, and means conjugate transpose for complex ma-trices. represents the identity matrix of dimension , and

denotes the identity matrix of an appropriate dimension. Letdenote the vector with all entries equal to one. Matrices,

if not explicitly stated, are assumed to have compatible dimen-sions. For , denote by its real part and by itsimaginary part. represents a block-diagonalmatrix with matrices , on its diagonal. The ma-trix inequality means that and are square Hermitianmatrices and that is positive definite. For ,

denotes its maximal singular value. A matrixis Hurwitz (or stable) if all of its eigenvalues have strictly nega-tive real parts. A matrix is irreducible if there does not exist apermutation matrix such that is block triangular. TheKronecker product of matrices and isdefined as

.... . .

...

which satisfies the following properties:

A directed graph consists of a node set and an edge set, in which an edge is represented by a pair of distinct

nodes of : , where is the parent node, is the childnode, and is neighboring to . A graph with the property that

implies that is said to be undirected. A pathon from nodes to is a sequence of ordered edges in theform of , . A directed graph is saidto be strongly connected if, for any pair of distinct nodes, thereexists a path between them. A directed graph has or contains adirected spanning tree if there exists a node called root such thatthere exists a directed path from this node to every other node.

Suppose that there are nodes in a graph. The adjacencymatrix is defined by ,if and 0 otherwise. Thus, represent both


LI et al.: CONSENSUS OF MULTIAGENT SYSTEMS AND SYNCHRONIZATION OF COMPLEX NETWORKS 215

nodes and indices, which should not cause confusion from thecontext. The Laplacian matrix is defined as

, for . Clearly, matrix is symmetricif the graph is undirected.

Lemma 1: All the eigenvalues of have nonnegative realparts. Zero is an eigenvalue of , with as the correspondingright eigenvector [34].

Lemma 2: Zero is a simple eigenvalue of if and only ifgraph has a directed spanning tree [18], [29].

Lemma 3: If graph contains a directed spanning tree, then,with proper permutation, can be reduced to the Frobeniusnormal form [41]

......

. . ....

where , , are irreducible, each has atleast one row with positive row sum, and is irreducible oris a zero matrix of dimension one.

II. UNIFIED APPROACH TO THE CONSENSUS PROBLEM

Consider a group of identical agents with general lineardynamics, which may be regarded as the linearized model ofsome nonlinear systems. The dynamics of the th agent are de-scribed by

(1)

where is the state, is thecontrol input, and is the measured output. It is assumedthat is stabilizable and detectable.

The communication topology among agents is represented bya directed graph , where is the setof nodes (i.e., agents) and is the set of edges. Anedge in graph means that agent can obtain informationfrom agent but not conversely.

The relative measurements of other agents with respect toagent are synthesized into a single signal in the following way:

(2)

where denotes the coupling strength, , andif agent can obtain information from agent but 0 otherwise.An observer-type consensus protocol is proposed as

(3)

where is the protocol state, , andand are the feedback gain matrices to be

determined. The term in (3) denotes theinformation exchanges between the protocol of agent and thoseof its neighbors. Note that protocol (3) is distributed, since it isbased only on the relative information of neighboring agents.

With (3), system (1) can be written as

(4)

where

is the Laplacian matrix of , is the aug-mented node dynamics, and denotes the inner linking matrix.System (4) may be viewed as the linearized model of a complexnonlinear network.

One says that (3) solves the consensus problem if the statesof system (4) satisfy

(5)

Let be the left eigenvector ofassociated with zero eigenvalue, satisfying . Introducethe following variable:

(6)

where , and satisfies. Similar to[23], is referred to as the disagreement

vector. It can be verified that evolves according to the fol-lowing so-called disagreement dynamics:

(7)

The following presents a necessary and sufficient conditionfor the consensus problem under dynamic protocol (3).

Theorem 1: For a directed network of agents with communi-cation topology that has a directed spanning tree, protocol (3)solves the consensus problem if and only if all matrices ,

, , are Hurwitz, where , ,are the nonzero eigenvalues of the Laplacian matrix of .

Proof: First, it is to show that the consensus problem ofnetwork (4) is equivalent to the asymptotical stability problemof the disagreement dynamics (7). Rewrite (6) as

(8)

where

......

. . ....

By the definition of , it is easy to see that zero is a simpleeigenvalue of , with as the corresponding right eigenvector,and one is another eigenvalue with multiplicity . Then, itfollows from (8) that if and only if . Thatis, the consensus problem is solved if and only if , as

.Next, the stability of system (7) is discussed, which

will solve the consensus problem indirectly. Let



, , , and uppertriangular be such that

(9)where the diagonal entries of are the nonzero eigenvalues of

. Introduce the state transformation , with. Then, (7) can be represented in terms of

as follows:

(10)

On the other hand, it can be seen from (6) that

(11)

Note that the elements of the state matrix of (10) are either blockdiagonal or block upper triangular. Hence, , ,converge asymptotically to zero if and only if the sub-systems along the diagonal, i.e.,

(12)

are asymptotically stable. It is easy to check that matricesare similar to

Therefore, the stability of matrices , ,, is equivalent to the case in which the state of (7)

converges asymptotically to zero, i.e., the consensus problem issolved.

Remark 1: The importance of this theorem lies in the factthat it converts the consensus problem of a large-scale multia-gent network under the observer-type protocol (3) into the sta-bility of a set of matrices with the same dimension as a singleagent, thereby significantly reducing the computational com-plexity. The observer-type consensus protocol (3) can be seenas an extension of the traditional observer-based controller for asingle system to the one for multiagent systems. The separationprinciple of traditional observer-based controllers still holds inthe multiagent setting. Communication topology is directedand only assumed to have a directed spanning tree. Such an as-sumption is quite general and weak, as it is intuitively clear thatconsensus is impossible to reach if has disconnected compo-nents. The effects of the communication topology on the con-sensus problem are characterized by the eigenvalues of the cor-responding Laplacian matrix , which may be complex, ren-dering the matrices complex-valued in Theorem 1.

Remark 2: Theorem 1 generalizes the existing results onthe consensus problem in at least two aspects. First, the agentdynamics are extended to be general linear but not limited tosingle-integrator, double-integrator, or structural high-orderlinear systems, as usually assumed in most existing papers[23], [29], [31]. Here, once again, the linear dynamics can beconsidered as the linearized dynamics of some originally non-linear systems. Second, an observer-type consensus protocol isproposed, which is based on relative output measurements be-tween neighboring agents, in contrast to [39] where a full-statecoupling law is used and to [37] where the dynamic protocol

requires the absolute output of each agent to be available. Com-pared to that of existing consensus protocols, a unique featureof the consensus protocol (3) is that a positive scalar called thecoupling strength is introduced, similar to the complex networkmodels studied in [8], [27], and[40]. With this parameter, thenotion of consensus region can be brought forward, as detailedin the following section. It is also worth noticing that the methodleading to Theorem 1 is, in essence, consistent with the masterstability function method proposed in [22] and[27]. Therefore,one can deal with the consensus problem of multiagent systemsand the synchronization problem of complex dynamic networksin a unified way.

Theorem 2: Consider the multiagent network (4) whose com-munication topology has a directed spanning tree. If protocol(3) satisfies Theorem 1, then

...

(13)

where is such that and .Proof: From (4), the network dynamics can be rewritten in

compact form as

(14)

where . The solution of (14) can be obtainedas

(15)

where matrices , , and are defined in (9). By Theorem1, is Hurwitz. Thus

as

It then follows from (15) that

as

implying that

... as (16)

for . Because is Hurwitz,(16) directlyleads to the assertion.

Remark 3: Some algebraic characteristics of the agent dy-namics implied by Theorems 1 and 2 are now briefly discussed.First, the agent dynamics (1) are excluded from having polesin the open right-half plane; otherwise, the consensus valuereached by the agents will tend to infinity exponentially. There-fore, it is assumed hereinafter that matrix has no eigenvalueswith positive real parts. Furthermore, if system (1) is asymp-totically stable, i.e., if is Hurwitz, then it follows from (13)



that the consensus value reached by the agents is zero. Thus,the matrix in (1) having eigenvalues along the imaginaryaxis is critical for the agents to reach consensus at a nonzerovalue under protocol (3). Typical examples of this case includethe single and double integrators considered in the existingliterature [23],[29], [31], [32]. It should be pointed out thatsimilar results have been obtained for some special cases of thistheorem, e.g., in[32] where the agent dynamics are assumed tobe double integrators and in [39] where the consensus protocolis static.

A. Consensus Region Analysis

Given a protocol in the form of (3), the consensus problemcan be cast into analyzing the following system:

(17)

where and .The stability of systems(17) depends on parameter . The re-

gion of complex parameter , such that (17) is asymptoti-cally stable, is called the consensus region of network (4) in thispaper. It follows from Theorem 1 that consensus is reached ifand only if

where , , and . For anundirected communication graph, its consensus region is aninterval or a union of several intervals on the real axis. However,for a directed graph, where the eigenvalues of are generallycomplex numbers, its consensus region is a region or a unionof several regions on the complex plane.

It is worth noting that consensus region can be seen as thestability region of the matrix pencil with respect tocomplex parameter . Thus, tools from the stability of matrixpencils will be utilized to analyze the consensus region problem.Before moving on, the following lemma is needed.

Lemma 4: Given a complex-coefficient polynomial [25]

(18)

where is stable if and only if and.

In the aforementioned lemma, only second-order polyno-mials are considered. Similar results for high-order complex-co-efficient polynomials can also be given (see [25]). However, inthe latter case, the analysis will be more complicated.

The following example has a bounded consensus region.Example 1: The agent dynamics and consensus protocol are

given by (1) and (3), respectively, with

Obviously, is Hurwitz. The characteristic polynomialof is

Fig. 1. (a) Bounded consensus region. (b) Communication topology.

By Lemma 4, is stable if and only ifand . The consensusregion in this case is shown in Fig. 1(a), which is bounded.Assume that the communication graph is given by Fig. 1(b), sothe corresponding Laplacian matrix is

with nonzero eigenvalues . It canbe verified that consensus is achieved if and only if

. Figs. 2 and 3 show the states of network (4) with dif-ferent coupling strengths ’s for this example.

Remark 4: The consensus region issue discussed in this sec-tion is similar to the synchronization region issue studied in[8],[17], [20], and [27]. The consensus region serves in a certainsense as a measure for the robustness of protocol (3) to para-metric uncertainty. An example can be easily constructed suchthat the bound of the consensus region on the real axis for a cer-tain protocol is rather narrow, e.g., [1, 1.005], and all the eigen-values of are one. Assume that the coupling strength in (2)is subjected to multiplicative uncertainties, e.g.,(2) is changedto , where denotes the uncer-tainty. Clearly, if , then this protocol fails to solvethe consensus problem. Therefore, given a consensus protocol,the consensus region should be large enough for the protocol tomaintain a desirable robustness margin. As a matter of fact, theconsensus region analysis given in this section provides a cru-cial basis for the consensus protocol design, which is the topicof the next section.

The following example has a disconnected consensus region.1) Example 2: The agent dynamics and consensus protocol

are given by(1) and (3), respectively, with

Obviously, is Hurwitz. The characteristic polynomialof is



Fig. 2. States of six-agent network (4) when � � ��.

By Lemma 4, is stable if and only if. The consensus region in this case is

shown in Fig. 4, which is composed of two disjoint subregions,both with unbounded real part and bounded imaginary part. Ifthe communication graph is still given by Fig. 1(b), then con-sensus is achieved if and only if . If couplingstrength can be negative, then consensus will be achieved alsowhen .

B. Consensus Protocol Design

In many cases, the protocols have to be designed so as tosolve the consensus problem for various given communicationtopologies.

From the previous section, it can be seen that the cases withbounded consensus regions are more complicated than the caseswith unbounded consensus regions. Hence, it is convenient todesign a protocol such that the consensus region is unbounded.

Proposition 1: Given the agent dynamics (1), there exists amatrix such that is Hurwitz for alland if and only if is detectable.

Proof: (Necessity) It is trivial by letting and .

Fig. 3. States of six-agent network (4) when � � ��.

Fig. 4. Disconnected consensus region.

(Sufficiency) Because is detectable, there exists a ma-trix such that is Hurwitz, i.e., there exists a matrix

such that

Let . Then, the aforesaid inequality becomes



By Finsler’s lemma [14], there exists a matrix satisfying theaforementioned inequality if and only if there exists a scalar

such that

(19)

Because matrix is to be determined, let without loss ofgenerality. Then

(20)

Obviously, when (20) holds, for any , (19) holds. Take, i.e., . By the previous inequalities,is Hurwitz for all . Thus, one has

for all and .Theorem 1 and the aforementioned proposition together lead

to the following result.Theorem 3: For the multiagent network (4) with containing

a directed spanning tree, there exists a distributed protocol in theform of (3) that solves the consensus problem and meanwhileyields an unbounded consensus regionif and only if is stabilizable and detectable.

A multistep consensus protocol design procedure based onthe consensus region notion is now presented.

Algorithm 1: Given that is stabilizable and de-tectable and containing a directed spanning tree, a protocolin the form of (3) solving the consensus problem can be con-structed according to the following steps.

1) Choose matrix such that is Hurwitz.2) Solve the linear matrix inequality (LMI)(20) to get one

solution . Then, choose the feedback gain matrix.

3) Select a coupling strength that is larger than the thresholdvalue , which is given by

(21)

where , , denotes the nonzero eigenvaluesof .

Remark 5: One distinct feature of Algorithm 1 is that it de-couples the effects of the agent and protocol dynamics on theconsensus stability from that of the communication topology.More specifically, steps 1) and 2) deal only with the agent dy-namics and feedback gain matrices of the consensus protocol,leaving the communication topology of the multiagent networkto be handled in step 3) by manipulating the coupling strength.One favorable consequence of this decoupling property is thatthe protocol so designed to achieve consensus for one givencommunication graph can be used directly to solve the con-sensus problem for any other graphs containing a spanning tree,with the only different task of appropriately selecting the cou-pling strength as in step 3). This feature will be more desirablefor the case when agent number is large, for which the eigen-values of the corresponding Laplacian matrix are hard to deter-

Fig. 5. Unbounded consensus region.

mine or even troublesome to estimate. Here, in this case, oneonly needs to choose the coupling strength to be large enough.

Now, Example 1 in the previous section is revisited.Example 3: The agent dynamics and feedback gain matrix

of protocol (3) remain the same as in Example 1, while matrixwill be redesigned via Algorithm 1. By solving LMI (20), a

feasible solution is obtained as . Differing

from Example 1, an unbounded consensus region in the form ofcan be obtained here. This can be verified

in another way by noticing that the characteristic polynomial ofbecomes

Thus, is Hurwitz if and only if

The consensus region in this case is the right-half plane by thevertical line , except the white area shown inFig. 5, which obviously contains the region .Protocol (3) with feedback gains and (the same as that inthe previous) and any will solve the consensus problemfor any communication graph containing a spanning tree.

C. Relative-State Consensus Protocol

In this section, a special case when the relative states betweenneighboring agents are available is considered. For this case, adistributed static protocol is proposed as

(22)

where and are the same as that defined in (2), andis the feedback gain matrix. With (22), system (1)

becomes

(23)

where is defined in (4).Corollary 1: For a directed network of agents with com-

munication topology that has a directed spanning tree, pro-



tocol (22) solves the consensus problem if and only if all ma-trices , , are Hurwitz.

Corollary 2: Consider the agent network (23) whose com-munication topology has a directed spanning tree. If protocol(22) satisfies Corollary 1, then the states , as ,for , where is defined in (13).

It is observed by comparing Theorem 2 and Corollary 2 thateven if the consensus protocol takes the dynamic form (3) orthe static form (22), the final consensus value reached by theagents will be the same, which relies only on the communicationtopology, the initial states, and the agent dynamics.

Similar to that in previous section, the consensus region ofprotocol(22) corresponds to the stability region of system

with respect to , where .The dual of Proposition 1 is presented as follows.Proposition 2: Given the agent dynamics(1), there exists a

matrix such that is Hurwitz for alland if and only if is stabilizable.

Algorithm 2: Given that is stabilizable, a protocolin the form of(22) solving the consensus problem can be con-structed according to the following steps.

1) Solve the following LMI:

(24)

to get one solution . Then, choose the feedback gainmatrix .

2) Select the coupling strength , with being givenin (21).

D. Application to Spacecraft Formation Flying

In this section, a possible application of the consensus al-gorithm developed in the previous sections to spacecraft for-mation flying in the low Earth orbit is considered. A pertinentwork is[33], which addresses the formation keeping problem ofdeep-space spacecraft whose translational dynamics are mod-eled as double integrators. In order to simplify the analysis, as-sume that a virtual reference spacecraft is moving in a circularorbit of radius . The relative dynamics of the other spacecraftwith respect to the virtual spacecraft will be linearized in the fol-lowing coordinate system, where the origin is on the mass centerof the virtual spacecraft, the -axis is along the velocity vector,the -axis is aligned with the position vector, and the -axis com-pletes the right-hand coordinate system.

The linearized equations of the relative dynamics of the thsatellite with respect to the virtual satellite are given by Hill’sequations, which are

(25)

where , , and are the position components of the th satel-lite in the rotating coordinate; , , are the control in-puts; and denotes the angular rate of the virtual satellite. Themain assumption inherent in Hill’s equations is that the distancebetween the th and virtual satellites is very small in comparisonto orbital radius .

Denote the position vector by and the con-trol vector by . Then, (25) can be rewrittenas

(26)

where

Satellites are said to achieve formation flying if their velocityvectors converge to the same value and their positions maintaina prescribed separation, i.e., , , as

, where denotes the desired constantseparation between satellites and .

Represent the communication topology among the satel-lites by a directed graph . Assume that measurements of bothrelative positions and relative velocities between neighboringsatellites are available. The control input to satellite is pro-posed here as

(27)where ; and are the constant feedback gainmatrices to be determined; and and if satellitecan obtain information from satellite but 0 otherwise. If satel-lite receives no information from any other satellite, then theterm is set to zero. With (27), (26) can be reformulated as

(28)

The following result is a direct consequence of Corollary 1.Corollary 3: Assume that graph has a directed spanning

tree. Then, protocol (27) solves the formation flying problem

if and only if the matrices are

Hurwitz for , where , , denote thenonzero eigenvalues of the Laplacian matrix of .

The feedback gain matrices and satisfying Corollary 3can be designed by following Algorithm 2. Because system (26)is stabilizable, they always exist.

Example 4: Consider the formation flying of four satelliteswith respect to a virtual satellite that moves in a circular orbitat rate . Assume that the communication topologyis given by Fig. 6(a), from which it is observed that satellite 1plays the leader’s role. The nonzero eigenvalues of the Lapla-cian matrix in this case are 1, 1, and 2. Select in (27) to be1 for simplicity. One can solve the formation flying problem inthe following steps: 1) Select properly the initial state of leadingsatellite 1 such that it moves in a spatial circular relative orbitwith respect to the virtual satellite; 2) design the consensus pro-tocol in the form of(27) such that the four satellites togethermaintain a desired formation flying situation.



Fig. 6. (a) Communication topology. (b) Relative velocity. (c) Relative posi-tions�� .

Select the initial state of satellite 1 as , ,, , , . In such a case, satellite 1 maintains

a circular relative orbit with respect to the virtual satellite, ofradius , and with the tangent angle to the orbitalplace . Suppose that the four satellites will maintain asquare shape with a separation of 500 m in a plane tangent tothe orbit of the virtual satellite by an angle . Let

, , , and. By Algorithm 2, the feedback gain

matrices solving the formation flying problem can be obtainedby using the LMI Toolbox [11] as

The simulation result is shown in Fig. 6.

III. CONSENSUS WITH RESPECT TO A REFERENCE STATE

In the previous section, the consensus problem of multia-gent systems with consensus protocols (3) and (22) has beenstudied. It is worth noting that the final consensus value, whichdepends on the initial values and agent dynamics, might be un-known a priori. However, in many practical cases, it is desirablethat the agents’ states asymptotically approach a reference state,which can also be time varying. This is called a model-refer-ence consensus problem in [31] and[32], where only integratordynamics are discussed. If the reference model is taken as avirtual leader, the model-reference consensus problem is actu-ally the leader–follower consensus problem, as studied in [12],[13], and [15]. The model-reference consensus problem is, insome sense, related to the target acquisition problem concerned

in [10]. In this section, the problem is extended to the generalframework proposed in the last section.

The agent’s dynamics are still described by (1). The referencetrajectory , which the agents’ states will follow,satisfies

and (29)

where , , and matrices , , and are thesame as those defined in (1). It is assumed that only a subset ofagents have access to the output variable of the reference model(29), whereas all the agents in the network have access to thereference input. In this case, the information available to agent

is given by

(30)

where , is the same as that defined in (3), andif agent has access to the reference model (29) but zero other-wise. A distributed protocol based on(30) is proposed as

(31)

where , , and are the feedbackgain matrices to be determined and is the state ofsystem .

With protocol (31), the model-reference consensus problemis said to be solved if

Let , ,, , . Then, the closed-

loop network dynamics can be written in the following form, asis the case in the last section:

(32)

where and are defined in (4), , and is theLaplacian matrix of communication topology among theagents.

It is easy to see that the model-reference consensus problemis solved if and only if all the states of (32) converge to zero.

Lemma 5: Suppose that the directed communication graphhas a spanning tree and that the root agent of such a tree has

access to the reference model. Then, all the eigenvalues of thematrix defined in (32) have positive real parts.

Proof: Without loss of generality, one can rearrange if nec-essary the order of the nodes in the network such that Laplacianmatrix takes the Frobenius normal form. Because the rootagent has access to the reference model, in light of Lemma 3,all the submatrices along the diagonal of are irreducible, andthus, it has at least one row with positive row sum. Then, by fol-lowing the steps in proving Lemma 1 in [6], one can easily showthat all the eigenvalues of have positive real parts.



Corollary 4: Assume that communication topology has adirected spanning tree and that the root agent of such a tree hasaccess to the reference model. Then, protocol (31) solves themodel-reference consensus problem if and only if all matrices

, , , are Hurwitz, where ,, are the eigenvalues of .

Proof: It follows from the proof of Theorem 1 that the sta-bility of matrices is equivalent to the stability of (32),which implies that , , as . There-fore, , , , as , i.e., themodel-reference consensus problem is solved.

Remark 6: Similar to Theorem 1, this corollary casts the con-sensus problem with respect to a reference state equivalentlyinto the stability of matrices that all have the same dimen-sion (equal to that of a single agent). A distinct feature of thiscorollary is that matrix here is allowed to be unstable for themodel-reference consensus problem, as opposed to Theorem 1where matrix is required to have no eigenvalues with posi-tive real parts. According to Lemma 5, the reference consensuscan possibly be reached by using protocol (31), even when onlythe root agent has access to the output variable of the referencemodel.

Similar toSection III, both the consensus region analysis andconsensus protocol design can be discussed correspondinglywith very little modification, so they are omitted here for brevity.

IV. CONSENSUS WITH PERFORMANCE SPECIFICATION

Practically, an agent itself may be subjected to external distur-bances. It is interesting to study the robustness of consensus pro-tocols to external disturbances, which is formulated as an issueof additional performance specification. A related workis [16], where the gain of multivehicle formations with adouble-graph strategy was investigated. For simplicity, the con-nections among agents are assumed to be undirected, keepingsymmetric Laplacian matrices with real eigenvalues.

The agent dynamics perturbed by external disturbances aredescribed by

(33)

where is the state of agent , is the controlinput, is the external disturbance, andis the measured output.

First, consider the robustness of protocol(31) to external dis-turbances. Because the agent states are desired to converge tothe reference state satisfying (29), it is natural to define theperformance variables as

(34)

Let and . Then,the closed-loop network dynamics resulting from (31), (33),and(34) are written as

(35)

where , , , and are the same as that in(32) and

Denote by the transfer function matrix from to ofsystem (35).

The model-reference consensus with the desired perfor-mance specification is stated as follows. For a given ,find an appropriate protocol in the form of(31) such that thefollowing are achieved: 1) The model-reference consensus issolved with , i.e., the state of (35) with convergesto zero, and 2) , where is the normof , defined by [45].

Theorem 4: For an undirected network of agents describedby (33), with at least one agent having access to the referencemodel, the model-reference consensus is solved along with

if and only if there exists a protocol (31) suchthat the following systems are asymptotically stable and,moreover, the norms of their transfer function matrices areall less than :

(36)

where denotes the eigenvalues of .Proof: Because graph is assumed to be undirected and

at least one agent has access to the reference model, it followsfrom Lemma 5 that matrix is positive definite. Let be sucha unitary matrix in that .Introduce the following sate transformations: ,with . Then, network (35) can be rewrittenas

(37)

Furthermore, reformulate disturbance variable and perfor-mance variable via

(38)

where and . Then, sub-stituting (38) into (37) gives

(39)

It is worth noting that (39) is composed of independent sys-tems as in (36). Denote by and the transfer functionmatrices of systems (39) and (36), respectively. Then, it followsfrom (36)–(39) that

(40)

By observing that both and are unitary ma-trices, from (40), one obtains the relationships between thenorms of , , and as follows:

(41)



Therefore, the model-reference consensus problem withis reduced to designing a common controller

in the form of (31) such that all the systems in (36) aresimultaneously asymptotically stable, with .

Remark 7: Similar to Theorem 1, the previous theorem con-verts the model-reference consensus problem with prescribed

performance specification into the control problem ofa set of independent systems, each of which has a dimensionthat is equal to that of a single agent. Here, the communicationgraph is assumed to be undirected and connected, which is morestringent than the assumptions in Theorem 1. It is worth notingthat the key to this theorem relies heavily on the input and outputtransformations in addition to the state transformation.

Next, consider the robustness of protocol(3) to external dis-turbances. In this case, define the performance variable as

, . The closed-loop network dynamics can bewritten as

(42)

where , , , and aredefined in (4). Denote by the transfer function matrix from

to of system (42).Corollary 5: For an undirected network of agents de-

scribed by (33), protocol (3) solves the consensus problemwith if and only if the following systems areasymptotically stable and, moreover, the norms of theirtransfer function matrices are all less than :

(43)

where are the eigenvalues of .Remark 8: Because of the singularity of Laplacian matrix ,

the state matrix of system (1) is required to be Hurwitz soas to guarantee the existence of the norm from the distur-bance to performance variables of the resulting network. Thisgives rise to the following conflict: Matrix is desired to haveeigenvalues on the -axis in order to reach nonzero consensusvalues, whereas is not allowed to have eigenvalues on the

-axis in order to validate the performance specification.It should be noted that the transfer function of the system in(43) corresponding to is the same as that of an isolatedagent from to . Therefore, an interesting consequence isthat the performance of the agent network(42) (if it exists)will never be better than that of an isolated agent, implying thatprotocol(3) cannot render the agent network any better distur-bance rejection level as compared to that of an isolated agent.On the contrary, it is possible for the model-reference consensusprotocol(31) to be so (see Theorem 4).

V. CONCLUSION

This paper has studied the consensus problem of multiagentsystems under a time-invariant communication topology, witheach agent having a general form of linear dynamics. An ob-server-type consensus protocol based on relative output mea-surements between neighboring agents has been proposed andanalyzed. A novel framework has been introduced, which can

describe in a unified way both the consensus of multiagent sys-tems and the synchronization of complex dynamic networks.The notion of consensus region has been introduced and an-alyzed by using tools from the stability of matrix pencils, onwhich a multistep consensus protocol design procedure has alsobeen presented. Finally, two important issues of consensus withrespect to a time-varying state and the robustness of consensusprotocols to external disturbances have been addressed. Gener-alizing the results of this paper to the case where the commu-nication topology is dynamically evolving or has time delays isan important yet challenging topic for future research.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor and allthe anonymous reviewers for their careful reading of this paperand their constructive suggestions, which have led to the im-provement of the presentation of this paper.

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Zhongkui Li received the B.S. degree in aerospaceengineering from the National University of DefenseTechnology, Changsha, China, in 2005. He is cur-rently working toward the Ph.D. degree in the StateKey Laboratory for Turbulence and Complex Sys-tems, Department of Mechanics and Aerospace Engi-neering, College of Engineering, Peking University,Beijing, China.

From November 2008 to February 2009, he was aResearch Assistant with the City University of HongKong, Kowloon, Hong Kong. His research interests

include nonlinear control, complex networks, and cooperative control of multi-agent systems with applications to aerospace engineering.

Zhisheng Duan received the M.S. degree in math-ematics from Inner Mongolia University, Hohhot,China, and the Ph.D. degree in control theory fromPeking University, Beijing, China, in 1997 and 2000,respectively.

From 2000 to 2002, he was a Postdoctor withPeking University. Since 2008, he has been aProfessor with the Department of Mechanics andAerospace Engineering, College of Engineering,Peking University, where he is also currently with theState Key Laboratory for Turbulence and Complex

Systems. His research interests include robust control, stability of intercon-nected systems, flight control, and analysis and control of complex dynamicalnetworks.

Dr. Duan was the recipient of the 2001 Chinese Control Conference GuanZhao-Zhi Award.

Guanrong Chen (M’87–SM’92–F’97) received theM.Sc. degree in computer science from Zhongshan(Sun Yat-Sen) University, Guangzhou, Taiwan, in1981 and the Ph.D. degree in applied mathematicsfrom Texas A&M University, College Station, in1987.

He was a tenured Full Professor with the Univer-sity of Houston, Houston, TX. He is currently a ChairProfessor with the Department of Electronic Engi-neering, City University of Hong Kong, Kowloon,Hong Kong, where he has been the founding Director

of the Centre for Chaos and Complex Networks since January 2000. He isalso with the State Key Laboratory for Turbulence and Complex Systems, De-partment of Mechanics and Aerospace Engineering, College of Engineering,Peking University, Beijing, China. His research interests include chaotic dy-namics, complex networks, and nonlinear control.

Lin Huang received the B.S. and M.S. degrees inmathematics and mechanics from Peking University,Beijing, China, in 1957 and 1961, respectively.

In 1961, he joined the Department of Mechanicsand Aerospace Engineering, College of Engineering,Peking University, where he is currently a Professorof control theory and is with the State Key Labo-ratory for Turbulence and Complex Systems. Hisresearch interests include the stability of dynamicalsystems, robust control, and nonlinear systems. Hehas authored three books and coauthored more than

150 papers in the fields of stability theory and control systems.Prof. Huang is a member of the Chinese Academy of Sciences.


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IEEE TRANSACTIONS ON CIRCUITS AND …aerocontrol/duanzs/论文/10－tcsi...observed in [43] by using graph theory. In [23], a general framework of the consensus problem for networks