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Graduation Design

Distributed consensus of linear multi-agent systems with adaptive dynamic protocols

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Automatica 49 (2013) 1986–1995

Contents lists available at SciVerse ScienceDirect

Automatica

journal homepage: www.elsevier.com/locate/automatica

Distributed consensus of linear multi-agent systems with adaptive

dynamic protocols

✩

Zhongkui Li

a,1

, Wei Ren

, Xiangdong Liu

, Lihua Xie

State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering,

Peking University, Beijing 100871, China

Department of Electrical Engineering, University of California, Riverside, CA, 92521, USA

School of Automation, Beijing Institute of Technology, Beijing 100081, China

School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore

a r t i c l e i n f o

Article history:

Received 16 November 2011

Received in revised form

23 November 2012

Accepted 1 March 2013

Available online 23 April 2013

Keywords:

Multi-agent system

Consensus

Cooperative control

Adaptive control

Distributed tracking

a b s t r a c t

This paper considers the distributed consensus problem of multi-agent systems with general continuous-

time linear dynamics for both the cases without and with a leader whose control input might be nonzero

and time varying. For the case without a leader, based on the relative output information of neighboring

agents, two types of distributed adaptive dynamic consensus protocols are proposed, namely, the edge-

based adaptive protocol which assigns a time-varying coupling weight to each edge in the communication

graph and the node-based adaptive protocol which uses a time-varying coupling weight for each node.

These two adaptive protocols are designed to ensure that consensus is reached in a fully distributed

fashion for all undirected connected communication graphs. It is shown that the edge-based adaptive

consensus protocol is applicable to arbitrary switching connected graphs. For the case where there

exists a leader whose control input is possibly nonzero and bounded, a distributed continuous adaptive

protocol is designed to guarantee the ultimate boundedness of the consensus error with respect to

any communication graph which contains a directed spanning tree with the leader as the root and

whose subgraph associated with the followers is undirected, requiring neither global information of the

communication graph nor the upper bound of the leader’s control input. A distributed discontinuous

protocol is also discussed as a special case. Simulation examples are finally given to illustrate the

theoretical results.

1. Introduction

Consensus is an important problem in the area of cooperative

control of multi-agent systems. The main idea of consensus is

to develop distributed control policies that enable a group of

agents to reach an agreement on certain quantities of interest.

Due to its potential applications in broad areas such as spacecraft

formation flying and sensor networks, the consensus problem has

been extensively studied by numerous researchers from various

✩

This work was supported by the National Natural Science Foundation of China

under grants 61104153 and 61225013, National Science Foundation under CAREER

Award ECCS-1213291, and National Research Foundation of Singapore under grant

NRF-CRP8-2011-03. The material in this paper was partially presented at the 2012

American Control Conference (ACC2012), June 27–29, 2012, Montreal, Canada. This

paper was recommended for publication in revised form by Associate Editor Tamas

Keviczky under the direction of Editor Frank Allgöwer.

E-mail addresses: zhongkli@gmail.com (Z. Li), ren@ee.ucr.edu (W. Ren),

xdliu@bit.edu.cn (X. Liu), elhxie@ntu.edu.sg (L. Xie).

Tel.: +86 1062765037; fax: +86 1062765037.

perspectives; see Jadbabaie, Lin, and Morse (2003), Li, Fu, Xie, and

Zhang (2011), Olfati-Saber, Fax, and Murray (2007), Olfati-Saber

and Murray (2004), Ren and Beard (2005), Ren, Beard, and Atkins

(2007) and references therein. Existing consensus algorithms

can be roughly categorized into two classes, namely, consensus

without a leader (i.e., leaderless consensus) and consensus with

a leader. The case of consensus with a leader is also called

leader–follower consensus or distributed tracking.

A pioneering work on consensus is (Jadbabaie et al., 2003)

which provides a theoretical explanation for the linearized Vic-

sek model (Vicsek, Czirók, Ben-Jacob, Cohen, & Shochet, 1995)

by using tools from algebraic graph theory. In Olfati-Saber and

Murray (2004), a general framework of the consensus problem

for networks of integrators with fixed or switching topologies

is proposed. The controllability of leader–follower multi-agent

systems is considered in Rahmani, Ji, Mesbahi, and Egerstedt

(2009) from a graph-theoretic perspective. Distributed tracking

control for multi-agent consensus with an active leader is ad-

dressed in Hong, Chen, and Bushnell (2008) and Hu and Feng

(2010) by using neighbor-based state estimators. Consensus of net-

works of double- and high-order integrators is studied in Jiang

http://dx.doi.org/10.1016/j.automatica.2013.03.015

Z. Li et al. / Automatica 49 (2013) 1986–1995 1987

and Wang (2010) and Ren and Atkins (2007). Consensus algo-

rithms are designed in Carli, Bullo, and Zampieri (2009) and Li

et al. (2011) for a group of agents with quantized communi-

cation links and limited data rate. In Cao and Ren (2012), dis-

continuous controllers are studied in the absence of velocity or

acceleration measurements. The authors in Mei, Ren, and Ma

(2011) address the distributed coordinated tracking problem for

multiple Euler–Lagrange systems with a dynamic leader. The read-

ers can refer to the surveys (Olfati-Saber et al., 2007; Ren et al.,

2007) for more recent works on consensus. In most existing stud-

ies on consensus, the agent dynamics are assumed to be first-,

second-, or high-order integrators, which might be restrictive in

many cases.

This paper considers the distributed consensus problem of

multi-agent systems with general continuous-time linear dynam-

ics. Previous works along this line include Li, Duan, Chen, and

Huang (2010), Li, Liu, Lin, and Ren (2011), Ma and Zhang (2010),

Scardovi and Sepulchre (2009), Seo, Shim, and Back (2009), Tuna

(2009) and Zhang, Lewis, and Das (2011). One common feature in

Li et al. (2010, 2011), Ma and Zhang (2010), Seo et al. (2009) and

Zhang et al. (2011) is that at least the smallest nonzero eigenvalue

of the Laplacian matrix associated with the communication graph

is required to be known for the consensus protocol design. How-

ever, the smallest nonzero eigenvalue of the Laplacian matrix is

global information in the sense that each agent has to know the

entire communication graph to compute it. Therefore, the consen-

sus protocols given in Li et al. (2010, 2011), Ma and Zhang (2010),

Seo et al. (2009) and Zhang et al. (2011) cannot be implemented

by the agents in a fully distributed fashion, i.e., using only the local

information of its own and neighbors. To overcome this limitation,

distributed adaptive static consensus protocols are proposed in Li,

Liu, Ren, and Xie (2013) and Li, Ren, Liu, and Fu (in press). Similar

adaptive schemes are presented to achieve second-order consen-

sus with nonlinear dynamics in Su, Chen, Wang, and Lin (2011) and

Yu, Zheng, Lü, and Chen (2011). Note that the protocols in Li et al.

(2013, in press), Su et al. (2011) and Yu et al. (2011) rely on the

relative states of neighboring agents, which however might not be

available in many circumstances.

In this paper, we intend to present a unified framework to ad-

dress the consensus problem for both the cases without and with a

leader whose control input might be nonzero and time varying by

proposing fully distributed adaptive protocols, when the relative

state information of neighboring agents is not accessible. For the

case without a leader, based on the relative outputs of neighbor-

ing agents, two types of distributed adaptive dynamic consensus

protocols are proposed, namely, the edge-based adaptive proto-

col which assigns a time-varying coupling weight to each edge in

the communication graph and the node-based adaptive protocol

which uses a time-varying coupling weight for each node. These

two adaptive protocols are designed to ensure that consensus is

reached in a fully distributed fashion for any undirected connected

communication graph without using any global information about

the communication graph. The case with switching communi-

cation graphs is also studied. It is shown that the edge-based

adaptive consensus protocol is applicable to arbitrary switching

connected graphs. For the case where there exists a leader whose

control input is possibly nonzero and bounded, a distributed con-

tinuous adaptive protocol is designed to guarantee the ultimate

boundedness of the consensus error and the adaptive weights with

respect to any communication graph which contains a directed

spanning tree with the leader as the root and whose subgraph as-

sociated with the followers is undirected, requiring neither global

information of the communication graph nor the upper bound of

the leader’s control input. As a special case, a distributed discontin-

uous adaptive protocol is also discussed, in which case the consen-

sus error converges to zero. A sufficient condition for the existence

of these adaptive protocols is that each agent is stabilizable and

detectable.

It is worth mentioning that the dynamic consensus protocol

in Scardovi and Sepulchre (2009) does not need any global infor-

mation only when the eigenvalues of the state matrix A of each

agent lie in the closed left half-plane. As stated in Remark 1 of

Scardovi and Sepulchre (2009), global information of the com-

munication graph is required for the case where A has unstable

eigenvalues. Compared to the dynamic protocol in Scardovi and

Sepulchre (2009), the adaptive consensus protocols proposed in

this paper have advantages in three respects. First, by proposing

adaptive consensus protocols, we do not impose the assumption

that A has no unstable eigenvalues. Second, the dimension of the

node-based adaptive protocol here is lower than that of the con-

sensus protocol in Scardovi and Sepulchre (2009). Third, contrary

to the results in Scardovi and Sepulchre (2009) which are applica-

ble to only the case of a leader with zero control input, the pro-

posed framework in this paper can address the consensus problem

for the case with a leader whose control input is nonzero and time

varying. It should be mentioned that an advantage of the results

in Scardovi and Sepulchre (2009) is that they are applicable to the

case with jointly connected switching graphs.

The rest of this paper is organized as follows. Some useful pre-

liminary results are reviewed in Section 2. The leaderless consen-

sus problems under the proposed distributed adaptive protocols

are investigated in Section 3. The leader–follower consensus prob-

lem with a leader of possibly nonzero control input is studied in

Section 4. Simulation examples are presented for illustration in

Section 5. Section 6 concludes the paper.

2. Mathematical preliminaries

2.1. Notation and graph theory

Let R

n×n

be the set of n × n real matrices. The superscript T

means the transpose for real matrices. I

represents the identity

matrix of dimension p. Denote by 1 a column vector with all entries

equal to one. The matrix inequality A > (≥)B means that A − B is

positive (semi-)definite. A ⊗ B denotes the Kronecker product of

matrices A and B. For a vector x, let ∥x∥denote its 2-norm. A matrix

is Hurwitz if all of its eigenvalues have negative real parts.

A directed graph G is a pair (V, E ), where V = {v

, . . . , v

} is

a nonempty finite set of nodes and E ⊆ V × V is a set of edges, in

which an edge is represented by an ordered pair of distinct nodes.

For an edge (v

, v

), node v

is called the parent node, node v

the

child node, and v

is a neighbor of v

. A graph with the property

that (v

, v

) ∈ E implies (v

, v

) ∈ E for any v

, v

∈ V is said to

be undirected. A path from node v

to node v

is a sequence of or-

dered edges of the form (v

, v

k+1

), k = 1, . . . , l−1. An undirected

graph is connected if there exists a path between every pair of dis-

tinct nodes, otherwise it is disconnected. A directed graph contains

a directed spanning tree if there exists a node called the root, which

has no parent node, such that the node has directed paths to all

other nodes in the graph.

The adjacency matrix A = [a

] ∈ R

N×N

associated with the

directed graph G is defined by a

= 0, a

= 1 if (v

, v

) ∈ E and

= 0 otherwise. The Laplacian matrix L = [L

] ∈ R

N×N

is de-

fined as L



j=i

and L

= −a

, i = j.

Lemma 1 (Agaev & Chebotarev, 2005, Olfati-Saber & Murray, 2004,

Ren & Beard, 2005). (1) Zero is an eigenvalue of the Laplacian matrix

L with 1 as a corresponding right eigenvector and all nonzero eigen-

values have positive real parts. Furthermore, zero is a simple eigen-

value of L if and only if G has a directed spanning tree. (2) For an

undirected graph G, the smallest nonzero eigenvalue λ

of L satisfies

= min

x=0,1

x=0

1988 Z. Li et al. / Automatica 49 (2013) 1986–1995

2.2. Ultimate boundedness

Lemma 2 (Corless & Leitmann, 1981). For a system

x = f (x, t),

where f (·) is locally Lipschitz in x and piecewise continuous in t,

assume that there exists a continuously differentiable function V (x, t)

such that along any trajectory of the system,

(∥x∥) ≤ V (x, t) ≤ α

(∥x∥),

V (x, t) ≤ −α

(∥x∥) + ϵ,

where ϵ > 0, α

and α

are class K

∞

functions, and α

is a class K

function. Then, the solution x(t) of

x = f (x, t) is uniformly ultimately

bounded.

3. Leaderless consensus with undirected communication graphs

Consider a group of N identical agents with general linear

dynamics. The dynamics of the i-th agent are described by

= Ax

+ Bu

= Cx

, i = 1, . . . , N,

(1)

where x

∈ R

is the state, u

∈ R

is the control input, y

∈ R

the measured output, and A, B, C are constant matrices with com-

patible dimensions.

In this section, the communication graph among the agents is

represented by an undirected graph G. It is assumed that each

agent knows the relative outputs, rather than the relative states,

of its neighbors with respect to itself. Based on the relative output

information of neighboring agents, two novel distributed adaptive

dynamic consensus protocols are proposed, namely, the edge-

based and node-based adaptive consensus protocols.

The edge-based adaptive consensus protocol dynamically up-

dates the coupling weight for each edge (i.e., each communication

link) and is given by

˙v

= (A + BF)v

+ L



j=1

[C(v

− v

) − (y

− y

)],

= ε



− y

C(v

− v

)





− y

C(v

− v

)



= F v

, i = 1, . . . , N,

(2)

where v

∈ R

is the protocol state, i = 1, . . . , N, a

is the (i, j)-th

entry of the adjacency matrix A associated with G, c

(t) denotes

the time-varying coupling weight for the edge (i, j) with c

(0) =

(0), ε

= ε

are positive constants, and F ∈ R

p×n

, L ∈ R

n×q

, and

Γ ∈ R

2q×2q

are the feedback gain matrices.

The node-based adaptive consensus protocol assigns a time-

varying coupling weight to each node (i.e., each agent) and is

described by

˜v

= (A + BF) ˜v

+ d



j=1

[C(˜v

− ˜v

) − (y

− y

)],

= τ





j=1



− y

C(˜v

− ˜v

)









j=1



− y

C(˜v

− ˜v

)





= F ˜v

, i = 1, . . . , N,

(3)

where d

(t) denotes the coupling weight for agent i, τ

is a positive

constant, and the rest of the variables are defined as in (3).

Note that the terms



j=1

C(v

−v

) in (2) and



j=1

C(˜v

−

˜v

) in (3) imply that the agents need to use the virtual outputs of the

consensus protocols from their neighbors via the communication

topology G.

The objective in this section is to find proper feedback gain

matrices in (2) and (3) such that the N agents in (1) achieve

consensus in the sense of lim

t→∞

∥x

(t) − x

(t)∥ = 0, ∀i, j = 1,

. . . , N.

3.1. Consensus under the edge-based adaptive protocol

In this subsection, we study the consensus problem of the

agents in (1) under the edge-based adaptive protocol (2). Let z

, v

]

, e

= z

−



j=1

, z = [z

, . . . , z

]

, and e = [e

. . . , e

]

. Then, we get e = [(I

−

) ⊗ I

]z. It is easy to see

that 0 is a simple eigenvalue of I

−

with 1 as a corresponding

eigenvector and 1 is the other eigenvalue with multiplicity N −1.

Then, it follows that e = 0 if and only if z

= ··· = z

. There-

fore, the consensus problem of agents (1) under the protocol (2) is

solved if e converges to zero. Because ε

= ε

and c

(0) = c

(0),

it follows from (2) that c

(t) = c

(t), ∀t ≥ 0. It is not difficult to

obtain that e

and c

satisfy

= Me



j=1

H(e

− e

= ε

− e

)

R(e

− e

), i = 1, . . . , N,

(4)

where M =



A BF

0 A + BF



, H =



0 0

−LC LC



, and R = (I

⊗ C

)

Γ (I

⊗ C ).

The following theorem presents a sufficient condition for

designing (2) to solve the consensus problem.

Theorem 3. Suppose that the communication graph G is undirected

and connected. Then, the N agents in (1) reach consensus under the

edge-based adaptive protocol (2) with F satisfying that A +BF is Hur-

witz, Γ =



−I



, and L = −Q

−1

, where Q > 0 is a solution

to the following linear matrix inequality (LMI):

Q + QA − 2C

C < 0. (5)

Moreover, the protocol states v

, i = 1, . . . , N, converge to zero and

each coupling weight c

converges to some finite steady-state value.

Proof. Consider the Lyapunov function candidate



i=1



i=1



j=1,j=i

− α)

4ε

, (6)

where Q ,



Q + Q −Q

−Q Q



Q > 0 satisfies that

Q (A + BF) + (A +

BF)

Q < 0, and α and ς are positive constants to be determined

later. In light of the Schur Complement Lemma (Boyd, El Ghaoui,

Feron, & Balakrishnan, 1994), it is easy to know that Q > 0.

The time derivative of V

along the trajectory of (4) can be

obtained as



i=1





j=1

H(e

− e

)





i=1



j=1

− α)a

− e

)

R(e

− e

). (7)

Let

= Te

, i = 1, . . . , N, with T =



I 0

−I I



. Then, (7) can be

rewritten as



i=1









i=1



j=1



−

)





i=1



j=1

− α)a

(

−

)



−

), (8)

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