output consensus of multi-agent systems with fixed or switching unbalanced topology

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http://www.sj-ce.org

Scientific Journal of Control Engineering April 2014, Volume 4, Issue 2, PP.34-42

Output Consensus of Multi-agent Systems with

Fixed or Switching Unbalanced Topology Shuyu Bao

#, Chen Wei, Yan Ding

School of Automatic Science and Electrical Engineering, Beihang University, Beijing 100191, China

#Email: [email protected]

Abstracts

In this paper, the output consensus problems for multi-agent systems with unbalanced topology are discussed. We proposes

weighted output consensus protocols and proves the asymptotic consensus of multi-agent systems with unbalanced topology is

reachable using Lyapunov functions with integral type. Finally, simulations are provided to demonstrate our consensus protocols

are effective for the output value consensus of multi-agent systems.

Keywords: Multi-agent System; Unbalanced Topology; Output Consensus; Lyapunov Method

1 Introduction

Consensus problems have a long history in the distributed cooperative control of multi-agent systems. In recent years,

more and more researchers both at home and aboard pay attention to the consensus problems. This is mainly due to

the broad applications of multi-agent system theory in many areas, such as formation control [5] and path following

control [10]. Vicsek presented a non-balance discrete multi-agent system model for the first time from the point of

view of statistics [1]. Then, based on matrix analysis method, Jadbabaie analysed Vicsek’s model and drew a

conclusion that as long as the topological graph is connected, the status of all the nodes in the graph achieve

consensus [2]. Olfati-Saber and Murray introduced the consensus protocols for balanced topology with and without

time-delays and provided a convergence analysis with disagreement functions [6]. Furthermore, the consensus

protocols for switching topology and nonlinear topology were given in [3]. Based on Lyapunov direct method, Liu et

al presented and rigorously proved some sufficient conditions of nonlinear protocols guaranteeing asymptotical or

exponential consensus for systems with unbalanced topology [8]. Wang introduced the notion of a “knowledge”

leader and proposed a nonlinear consensus protocol for leader-follower multi-agent systems [7].

The researches above are all confined to the consensus of the states. However, in fact, output consensus is with

greater universal significance. Consider a group of agents with complex dynamics or high order models, sometimes

we just care about some specific parts of the system or some specific output functions. Besides, in some cases, the

states of the agents may be unobservable, instead we can observe some output variables of the system.

Up to now, there have been some research achievements on the output consensus. Wang et al transformed the output

consensus problem with uncertainties to the classic adaptive tracking control problem and used some established

control methods such as backstepping to solve the adaptive output consensus tracking with mismatched uncertainties

[11]. In order to solve the output consensus problem for a class of heterogeneous uncertain linear multi-agent

systems, Kim et al embedded an internal model, which plays a role of command generator, and the command is

determined as time goes since it is the outcome of on-line consensus by communicating the outputs of the agents[12].

Yang and Liu discussed the output consensus problems of linear multi-agent systems with fixed topologies and

agents described by homogeneous or heterogeneous linear systems and provided the conditions of output consensus

with respect to a set of admissible consensus protocols [13]. Li studied nonlinear output consensus protocols for

multi-agent systems with fixed and balanced topology and proves the asymptotic consensus is reachable [9].

In this paper, we will consider the output consensus problems of networks with unbalanced topology, where network

means a weighted directed graph. Weighted output consensus protocols are proposed and convergence analysis

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based on mirror graph theory and Lyapunov function with integral form is provided.

An outline of this paper is as follows. In Section 2, we review some algebraic graph definitions and come up with the

main problems of this paper. In Section 3 and 4, we mainly discuss the output consensus problems of multi-agent

systems with fixed and switching topology. Simulation results are given in Section 5. In Section 6, this paper is

finally concluded.

2 Preliminaries and problem formulation

Let ( , , )G V A be a directed weighted graph of order n with the set of nodes is 1 2

{ , ... }n

V v v v and the set of edges

V V . Use ( , )ij i je v v to denote an edge of graph G, where 0

ije . When 0

ije it means that there exist

some information transmits from agent j to agent i, otherwise it means that there is no information transmutation

between agent j and agent i. Sometimes we need to give weights to the edges of a graph. Assume that ijw is the

weight of edge ij

e , then there exist a matrix [ ]ij

n nA a R

whose elements can be expressed as

( );

0,

,ij ij

ij

w e E Ga

other

.

Then A is called the weighted adjacent matrix. In most situations, the weighted adjacent matrix can be used to

describe the characters of a graph. Besides, another matrix which can be used to describe the characters of a graph is

the Laplacian matrix [l ] Rn n

ijL

, and the elements of L can be expressed as

=1

=- , ;

= .

ij ij

n

ii ij

j

l a i j

l a

Besides, the eigenvalues of a Laplacian matrix also can be used to reflect the system’s convergence rate. For a

connected undirected graph 'G , the following well-known property holds

220, =0

min = (L)T

T

x x

x Lx

x

1

where 1nx R

,

2(L) is the second-smallest eigenvalue of matrix L, which is also called the algebraic connectivity

of matrix L. Olfati-Saber extend the definition of algebraic connectivity to directed graphs using the concept of

mirror graph[6]

.

Consider a network of continuous-time integrator agents with dynamics

= , ,i iix t u t i j N , ( )i iy h x (2.1)

with the initial condition (s) (0), s (- ,0]i i

x x ，where { ; ( , ) }i j i j

N v V v v and

iu represents the control

protocol. Assume that, for (x)h , the following conditions hold.

Assumption 1.

1. is continuous and monotone increasing;

2.;

3.There exist a positive constant l，for any , such that

1 2

1 2

-

-

h x h xl

x x .

The definition of output consensus is given as follow.

Definition 1. If there exist iu t such that for any initial value

(0)i

x

+lim (t) - (t) =0, , 1,2,3,...,

t j ih x h x i j n

(2.2)

then we say system (2.1) achieves output consensus.

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The specific problem to be addressed in this paper is to solve the output consensus problems of multi-agent systems

with fixed and switching unbalanced topology.

3 Output consensus of systems with fixed topology

In this section, we will focus on the output consensus problems of multi-agent systems with fixed topology.

In order to make the output value of all the agents in the system (2.1) achieve consensus, design the

following consensus protocol

(t)= (t) - (t)

i

i ij j i

j N

u a h x h x

(3.1)

Given protocol (3.1), the closed-loop system can be written as

(t)= (t) - (t)

i

i ij j i

j N

a h x h xx

. (3.2)

From the preliminary analysis of (3.2), it is observed that as the decrease of (t) (t)j i

h x h x , the change rate of

(t)i

x progressively grow smaller, when (t) (t)=j i

h x h x , (t)i

x will no longer change. When the output values of

all the nodes converge to a same value, all the nodes of the system achieve stability.

We have the following theorem for output consensus under fixed topology.

Theorem 1. Consider system (2.1) whose output function h(x) satisfies assumption 1. Suppose its topology graph

( , , )G V A is fixed and has a directed spanning tree, then given feedback protocol (3.1), system (2.1) can achieve

output consensus.

Proof: Assume that 1( ( )) ( )( ( )),..., ( ( )) T

nH X t h x t h x t , then (3.2) can be rewritten as

(t) ( ( ))X LH X t . (3.3)

Assume the left eigenvector of matrix L corresponding to eigenvalue 0 is 1

ˆ ˆ ˆ=( ,..., )T

n , where ˆ>0 and

=1

ˆ =1n

ii .

Define a reference point as followˆ

=1(t)= ˆ (t)

n

i iix x

, from the properties of Laplacian matrix, we can get

ˆ

=1 =1 =1

ˆ ˆ(t)= (t)=- (t) =0n n n

i i i ij j

i i i

x x l h x

.

So, ˆ ˆ=x t x

is time-invariant.

Construct a Lyapounov function

ˆ

(t)

ˆ

=1

ˆ= (s)- ( )

ixn

i

i x

V t h h x ds

. (3.4)

Assume ( )H X t satisfy

( ) = (t) + ( )H X t a H X t1

where 1 ( ( ))=0T

H X t and 1

1( ) ( )

n

i

i

a t h xn

. Denote ˆ( )iP diag and ˆ ˆTQ P , it is obvious that

ˆ

=1

max

ˆ

1 1ˆ( ) (t)- ( ) ( ( )) ( ( )) ( ( ))= ( ( )) ( ( ))

( )( ( )) ( ( )).

(x (t))- ( )n

T T T

i i

i

Tk

iV t x x X t QH X t H X t QH X t H X t QH X tl l

QH X t H X t

l

h h x

(3.5)

where maxQ is the largest eigenvalue of matrix Q.

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Let B=PL, it is obvious that B is a balanced Laplacian matrix. We suppose that matrix B is the Laplacian matrix of

graph B

G . Then, assume that the mirror graph of B

G is ˆB

G , whose Laplacian matrix is ˆ 2T

B PL L P . From

Theorem 7 in [6], we can get that ˆˆ [ ]ij n n

B b

is symmetrical and satisfies 1

ˆ 0n

ijjb

. The eigenvalues of B̂ satisfy

that 1 2

ˆ ˆ0 B B ... ˆn

B , where 2

B̂ is the algebraic connectivity of graph ˆ

BG . Take the derivative of

( )V t , one can get

ˆ

2

ˆ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

T T T

T

V t H X t H X PLH X t H X t BH X t H X t BH X t

H X t H X t

(3.6)

From (3.5) and (3.6), one can get V t wV t , where 2 max

ˆw l B Q . Then, Owt

V t e

, i.e. V t is

exponential convergence.

4 Output consensus of systems with switching topology

In this section, we will focus on the output consensus problems of multi-agent systems with switching topology.

Consider a multi-agent system whose topology is not fixed but time-varying. Assume this system has a

time-continuous-state x and a discrete-state G that belongs to a finite collection , can be expressed as

{ , , : ( ) 1,G V A rank L G n 1 ( ) 0}T

L G (4.1)

Under switching topology, closed-loop system can be rewritten as

( ) , ( ),k K

X t L G H X t k s t G

where 0

( ) :s t R is a switching signal, N

is the index collection associated with the elements of .

Assume that [ ]k kij n n

aA

is the weighted adjacent matrix of k

G , then, consensus protocol (3.1) can be expressed as

(t)= (t) - (t)i

i kij j i

j N

u a h x h x

(4.2)

Then, we have the following theorem for output consensus under switching topology.

Theorem 2. Consider system (2.1) whose output function h(x) satisfies assumption 1. Suppose its topological graph

G is discrete and belongs to a finite collection .Given a feedback control protocol (4.2), then, system (2.1) can

reach output consensus if and only if there exist a directed spanning three in any k

G Γ .

Proof: Assume the left eigenvector of kL G corresponding to eigenvalue 0 is

1

ˆ ˆ ˆ, ...,T

k k kn , where ˆ 0

k and

=1

ˆ =1n

kii . Define a reference point ˆ

1

ˆ ( )ki

k

n

iix t x t

, whose derivative can be expressed as

ˆ

1 1 1

ˆ ˆ ( ) ( ( )) 0i

n n n

ki ki ij k j

i i j

x t x t l G h x t

.

It is obvious that ˆ ˆk k

x t x

is time-invariant when

kG is active. Then, construct the auxiliary function

ˆ

( )

ˆ1

ˆ ( ) ( ) , ( ).i

kk

x tn

k kii xV t h s h x ds k s t

Denote ˆk ki

P diag and ˆ ˆ T

k k k kQ P , then it is obvious that

ˆ ˆ ˆ

1

max

ˆ ( ) ( ( )) ( ) ( ) ( ( )) ( ) ( ) ( ( ))

( )1 1( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )).

k k k

nT T

k ki i i k k

i

T T Tk

k k

V t x t x h x t h x X t Q H X t H X X t Q H X t

QH X t Q H X t H X t Q H X t H X t H X t

l l l

(4.3)

Let k k k

B P L G , whose mirror graph’s Laplacian matrix is (ˆ ) 2k k k k

T

kP L G L G PB , it is obvious that

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http://dict.youdao.com/w/point/

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ˆ ˆ[ ]k kij n n

B b

is symmetrical and satisfy that 1

ˆ 0n

kijjb

. The eigenvalues of ˆ

kB satisfy

1 2ˆ ˆ ˆ0 ( ) ( ) ... ( )

k k n kB B B . Then

ˆ

2

ˆ( ) ( ( )) ( ) ( ( )) ( ( )) ( ( )) ( ( )) ( ( ))

ˆ ( ( )) ( ( )).

T T

k k k k k

T

k

V t H X t H X P L G H X t H X t B H X t H X t B H X t

B H X t H X t

(4.4)

From (4.3) and (4.4), one can get that ( )k

V t ( )k k

w V t , where 2 max( ) ( )

k k kw l B Q . Suppose that

,1min

k N k n kw w

,

then ( ) ( )k k

V t wV t , in other words, ( )k

V t is exponential convergence.

It is a well-known fact that, for a switched system, when all individual subsystems are stable and the switching is

sufficiently slow, the whole system is stable. But specifying a dwell time may be too restrictive, and it may be

impossible to react to possible system failures during that time interval. So, in general, the average dwell time is

enough to guarantee the astringency of the system[4]

.

From theorem 3.1 in [4], if for the Lyapunov functions of any two subsystems in system (2.1) ( )p

V t and ( )q

V t , there

exist a constant , such that

( ) ( )p q

V t V t . (4.5)

Then, the whole switching system is convergent. Now we suppose that

ˆ

ˆ

( )

ˆ

1

( )

ˆ

1

ˆ ( ) ( )

ˆ ( ) ( )

( )( )

( ).

i

p

p

i

q

q

nx t

pix

i

nx t

qix

i

p

pq

q

h s h x ds

h s h x ds

V tt

V t

(4.6)

It is obvious that ( )pq

t is continuous on R. From L'Hospital's rule, one can get that

lim ( ) lim( ) ( ) ( ( )) ( ( ))

( ) ( ) ( ( )) ( ( ))pq

T

p p p

Tt t

q q qt t

tV t V t H X t B H X t

V t V t H X t B H X t

where

2 max

ˆ ( ( )) ( ( )) ( ( )) ( ( ))( ( )) ( ( ))T T

p p

T

kB H X t H X t B H X t H X tH X t B H X t

2 max

ˆ ( ( )) ( ( )) ( ( )) ( ( ))( ( )) ( ( ))T T

q q

T

qB H X t H X t B H X t H X tH X t B H X t

So

2 max

max 2

lim ( )

ˆ( ) ( )

( ) ( )pq

p p

tq q

tB B

B B

.

which shows that ( )pq

t has a bounded limit, in other words, ( )pq

t is bounded on R. Suppose the upper bound of

( )pq

t is maxpq

, and the lower bound is minpq

, then min max

( )pq pq pq

t , which means that max( ) ( )p qpqV t V t .

Assume , max

maxp q pq

, it is obvious that

( ) ( )p q

V t V t . (4.7)

That is to say the whole switching system is convergent.

5 Simulation

Consider a multi-agent system with 4 agents whose topology graph is strongly connected. Fig.1 shows the

topological structure of it.

FIG.1. THE TOPOLOGICAL STRUCTURE OF (4.1).

http://dict.youdao.com/w/topological/

http://dict.youdao.com/w/structure/

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Assume this system has a leader and 3 followers, and the motion equation of the leader (agent 0) is

0 0

0 0

3;

3 .

p t q t

q t p t

Rewrite the states of the system into vector form as 0 0 0

T

p qx and T

i i ip qx , where i=1,2,3. Then the

equation of the leader and followers are

0 0 ,

,i i i

x f x t

x f x t u

where iu is the cooperation term. Based on theorem 1 in [7],

iu is usually constituted by consensus protocol.

Then, the whole function of the system is

0 0

0 0

,

, ( ) ( ) ( ) .i

i i ij j i i i

j N

x f x t

x f x t K h x h x K h x h x

(5.1)

First consider the linear situation. When the output function is

1

1 0( )

0 0h x x

,

which can theoretically make the leader and follows satisfy that for any i and j, i j

p p . The simulation results of

system (5.1) with output function 1

h can be seen in fig.2.

(a) (b)

FIG.2.SIMULATION RESULTS UNDER 1

h : A SHOWS THE OUTPUT VALUES; B SHOWS THE MOTION TRAILS.

Next consider another linear output function

2

1 1( )

1 1h x x

,

which can theoretically make the agents system (5.1) satisfy that for any i and j, .i i j j

p q p q Fig.3 shows the

simulation results of system (5.1) with output function 2

h .

(a) (b)



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It can be seen from the two simulations above that the output values kept consistent meanwhile the motion trails kept

consistent. It is because of that the dimension of the system is relatively low and the function on p-axis and the

function on q-axis are correlative, so when the output function is simple, the result of output consensus protocol

equate to the state consensus.

Then consider the nonlinear situation, suppose the output function is

3

1( )

1

Th x x x

.

In theory, 3

h can let the agents of system (5.1) satisfy that for any i and j, 2 2 2 2

i i j jp q p q . With output function

3h , the simulation results of system (5.1) are shown in fig.4.

(a) (b)



It can be seen from fig.4 that the output values kept consistent but the motion trails are not completely consistent.

That is because, within a certain range of error, the followers’ output value will always choose the shortest way to

catch up with the leader. But in output consensus, we just need to consider the output value, without the

consideration of the motion trails, so there are some deviations in motion trails of the agents is a normal

phenomenon.

The previous simulations are all under fixed topology, now consider the switching situation. Fig.5 shows 4 different

networks, whose topology structures are all strongly connected. A finite automaton is shown with the set of states

{ , , , }a b c d

G G G G , representing the discrete-states of a network with switching topology as a hybrid system.

(a) (b)

(c) (d)

FIG.5. FOUR STRONGLY CONNECTED GRAPHS:(a) a

G , (b) b

G , (c) cG , (d)

dG .

Fig.6 and Fig.7 shows the simulation results of system (5.1) with output functions 1

h ,2

h and

3h under the

topology shown in Fig.5.

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(a) (b)

FIG.6. SIMULATION RESULTS UNDER LINEAR OUTPUT FUNCTIONS: A UNDER 1

h ; B UNDER 2

h .

FIG.7. THE OUTPUT VALUES UNDER NONLINEAR OUTPUT FUNCTION 3

h

From fig.6 and Fig.7, we can see that the output values of the agents under switching topology are very similar to the

situation under fixed topology. These simulation results also demonstrate the availability of output consensus

protocol (4.2) under switching topology.

6 Conclusion

In this paper, output consensus problems were considered for multi-agent systems with unbalanced topology. First,

weighted output consensus protocols were proposed to extend the applicable range of the protocols from 0-1

topology to all kinds of networks. Then, based on Lyapunov method, mirror graph theory and switching system theory,

output consensus was rigorously proved for the proposed consensus protocol. Finally, simulations are proved to

demonstrate the effectiveness of this consensus protocol. The study of output consensus problems in this paper is

under the ideal situation without the consideration of time-delay and noise, so future research is needed.

REFERENCES

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951-956

[4] D. Liberzon, Switching in Systems and Control, BirkhÄauser, Boston, MA, 2003

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AUTHORS 1Shuyu Bao was born in Shenyang,

Liaoning province, China in April 12,

1988. He got his B.S. in automatic control

engineering from Beihang University in

2011.

He is currently a master students in control

theory and control engineering at the

School of Automation Science and Electrical Engineering,

Beihang University. His research interests include applied

nonlinear control, cooperative control and the consensus theory

of Multi-agent systems.

2Chen Wei was born in Shandong province, China on August,

1971. She got her Ph.D. degrees in control theory and control

engineering from Institute of Systems Science, AMSS in 1997.

From 1998 to 1999, she was a post-doctoral in Hong Kong

University of Science and Technology. She is currently a

associate professor at the School of Automation Science and

Electrical Engineering, Beihang University. Her research

interests include control theory and control engineering,

navigation and guidance, nonlinear control, fuzzy logic control

and time-delay systems.

3Yan Ding was born in Huaian, Jiangsu province, China in July

9, 1988. He got his B.S. in information science and technology

from Beijing University of Chemical Technology in 2011.

He is currently a master students in control theory and control

engineering at the School of Automation Science and Electrical

Engineering, Beihang University. His research interests include

theory of systems with time-delay, the consensus theory of

Multi-agent systems and the formation control of multiple

UAVs.