Average Consensus in Networks of Multi Agent with Multiple Time Varying Delays

doi:10.4236/ijcns.2010.32028 Published Online February 2010 (http://www.SciRP.org/journal/ijcns/).

Average Consensus in Networks of Multi-Agent with

Multiple Time-Varying Delays

Tiecheng ZHANG, Hui YU

Institute of Nonlinear and Complex Systems, China Three Gorges University, Yichang, China Email: [email protected], [email protected]

Received November 19, 2009; revised December 15, 2009; accepted January 21, 2010

Abstract

The average consensus in undirected networks of multi-agent with both fixed and switching topology cou-pling multiple time-varying delays is studied. By using orthogonal transformation techniques, the original system can be turned into a reduced dimensional system and then LMI-based method can be applied veniently. Convergence analysis is conducted by constructing Lyapunov-Krasovskii function. Sufficient con-ditions on average consensus problem with multiple time-varying delays in undirected networks are obtained via linear matrix inequality (LMI) techniques. In particular, the maximal admissible upper bound of time-varying delays can be easily obtained by solving several simple and feasible LMIs. Finally, simulation examples are given to demonstrate the effectiveness of the theoretical results.

Keywords: Average Consensus, Multi-Agent System, Multiple Time-Varying Delays, Linear Matrix Inequality

1. Introduction

Recently, more and more researchers have paid a great deal of attention on distributed coordinated control of networks of dynamic agents within the control com-munity. Especially the consensus problem was dis-cussed widely, which can be attributed to the broad applications of multi-agent systems in many areas, including cooperative control of unmanned air vehicles, formation control of multi-robot, flocking, swarming, distribution sensor fusion, attitude alignment and con-gestion control in communication networks, etc. In cooperative control of multi-agent system, a critical issue is to design appropriate protocol and algorithms such that all agents can reach a common consensus value. This problem is called consensus problem.

In the past decades, some theoretical results have been established in [1–11], to name a few. In [1], Vicsek et al. proposed a simple model but interesting discrete-time model of autonomous agents all moving in the plane with the same speed but with different headings. Simulation results provided in [1] show that all agents can eventually move in the same direction without centralized coordination. The first paper pro-viding a theoretical explanation for these observed be-haviors in Vicsek model is [2]. The theoretical results

in [2] are extended to the case of directed graph by Ren et al. [3], in which matrix analysis and algebraic graph theory were used. Moreau [4] used a set-valued Lyapunov approach to study consensus problem with unidirectional time-dependent communication links. Saber et al. [5] discussed average consensus problem. When network communication is affected by time de-lay, the consensus problem is investigated in [5–7]. In [8], Saber provided a theoretical framework for analy-sis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed in-formation flow, robustness to changes in network to-pology due to link/node failures, time-delays, and per-formance guarantees. Ren et al. provided a tutorial overview of information consensus in multivehicle cooperative control in [9]. Yu et al. [10] proposed weighted average consensus in direction networks and unidirectional networks with time-delay. Multi-vehicle consensus with time-varying reference state was dis-cussed in [11]. Some other issues on consensus prob-lem can be found in [12–16].

Currently, consensus problem for multi-agent net-works with time delay was studied using linear matrix inequality method, for example [17–19] and [20]. In time delay systems of multi-agent, the network topol-ogy of multi-agent is a key factor in the analysis of stability of multi-agent system. Average consensus problem for multi-agent networks with both constant







and time-varying delay has been extensively consid-ered for the cases of multi-agent undirected and/or di-rection networks with fixed and/or switching topology [17,18] and [19]. However, the studies on consensus problem of multi-agent networks with multiple time- varying delays are still sparse. In this case, theoretical analysis is more challenging. In [20], the authors pro-posed a consensus protocol for multi-agent undirected network with multiple time-varying delays based on LMI method.

In this paper, we study the average consensus prob-lem for continuous time undirected networks of multi- agent, coupling multiple time-varying delays. Because the closed-loop system matrix is singular and tradi-tional LMI-based control theory is invalid. Therefore, theoretical analysis for this case is a challenging task. By using orthogonal transformation techniques, suffi-cient conditions based on LMI for multi-agent achiev-ing average consensus is obtained. Compared to [20], a different approach is used in this paper. First of all, the original system is turned into a reduced dimensional system by orthogonal transformation; Secondly, via constructing a different Lyapunov-Krasovskii function, a sufficient condition expressed as LMI is proposed to guarantee all agents reach average consensus in fixed and switching network. Finally, the maximal admissi-ble upper bound of multiple time-varying delays can be easily obtained by solving several simple and feasible LMIs.

This paper is organized as follows. Section 2 is the notation and formally states the problem. Section 3 contains our main results. Simulation results are pre-sented in Section 4. The concluding remarks are made in Section 5.

2. Problem Statement and Preliminaries

In this section, we provide a brief introduction about algebraic graph theory [24] and state the problem.

2.1. Algebraic Graph Theory

Let be a weighted undirected graph of

or-der , where is the set of

nodes, is the set of edges, and

is a weighted adjacency matrix. The node indexes belong to a finite index set



, ,

G V E A



2



n n

E V

n n ij A_{ }a _ 

 



1, ,2 3 , n

V  v v v v

V  



1, 2, ,

I  n . In undirected graph, . if and only if

. Moreover, we assume for all i

ij ji

e e e_ijE 0 ij  0

ij

a  a I. A

undirected graph is always connected. The set of neighbors of nodes vi is denoted by







: ,



i j i j

N  v V v v E .

The out-degree of node vi is defined as follows:

1

deg ( )_{out i} n _ij

i

v a







. The degree matrix of graph is a

diagonal matrix

G

ij

D_{  } d , where dij 0 for all i j and d_ii deg (_out v_i). The Laplacian matrix associated with the graph is defined as

1,

, ,

, . n

ij j j i ij

ij

a j i

L l D A

a j

 

 _

  

_{    }

  





i

ar

An important fact of L is that all the row sums of L

e zero and thus 1_{n }



1,1, ,1_



T_n _{is an eigenvector} of L associated with the eigenvalue  0.

Lemma 1 [5]. If the undirected graph G is con-nected, then its Laplacian matrix L satisfies:

1) L is symmetric and rank ( )L  n 1;

2) zero is one eigenvalue of , and and are the corresponding left and right eigenvector respectively, then,

L 1T

n 1n

1T 0

n L and 1L n 0;

3) the rest n1 eigenvalues are all positive and real.

2.2. Consensus Problem on Network

Consider a group of

n

agents with dynamics given by

 

( )

i i

x t u t , i I (1)

where n

i

x  is the state of the th agent at time , which might represent physical quantities such as atti-tude, position, temperature, voltage, and so on, and

i

t

( ) i

u t  is the control input (or protocol) at time .

t

We say protocol asymptotically solves the con-sensus problem, if and only if the states of agents satisfy

i

u

lim x t_i( ) x t_j( ) 0

t    , for all i,j I .Furthermore, if

 

1

1

lim ( )_t i n i 0 ( (0))

i

x t x Ave x n

 



_  ,

We say protocol asymptotically solves the aver-ag

2.3. Control Protocol for Time Delay

Let

i

u

e consensus problem.

ij

 notes the time delay for information

communi-s

cated from agentjto agent

i

. Because of time delay, two difficult consensu protocols have been studied. One is

[ ( ( )) ( ( ))]

u  a x t t x t t

j i

i ij j ij i ij

v



N

, i I (2)

t i I . (3)

Literature [5] have taken the protocol in (2) w

[ ( ( )) ( )]

j i

ij j ij i

v N

a x t  t x



  ,

i

u 



In the undirected network with switching topology, we

ad

ith_ij  into account, and obtained that it can as-ymptotically solves the average consensus problem in the fixed and undirected network topology if and only if

 

max

0, 2 L

 _   _. Some conclusions also were

inves-For consensus pr tigated in [16-18].

otocol in (3), communication delay on

er, we are interested in discussing the aver-ly affects the agents who are transmitted. Literature [4] has established the consensus results in the directed and dynamically switching graph. Some results also appeared in [21–23].

In this pap

age consensus problem in network of dynamic agents with fixed and switching topology coupling multiple time-varying delays, where the information passes through different edge with different time-varying delays. To solve such a problem, we assume the time delay sat-isfies ijji in undirected graph, i.e., the delays in transmission from xi to xj and xj to xi coincide. So we use the follow g pro col:

[ ( ( )) (

u 



a x t t x t

in to

( ))]

j i

i ij j k i k

v N

t





,

i I



(4)

With (4), (1) can be written in matrix form:

k

 

N ))

1

( (

i k

k

x t L x t



 





  t , (5)

where ( )



1

   

, 2 , ,

 



,



1 2



T n

x t  x t x t  x t Nn n , is the matrix defined by

t and L

 k    lkij

1

, ( ) ( ), 0 , ( ) ( ),

.

ij k ij

kij k ij

N

ij j

a i j t t

l i j t

a i j

 

 



  



_  



 _





Because ofAis symmetric and _ij _ji is a sy

in the undi-re

The tim ing dela

cted network we can obtain L_k mmetric and N

L L



.

e-vary ,

1

k k

y ( )_k t ,k1, 2, , Nis assumed to satisfy the follow inequations:

0_k( )t d_k,_k( )t h_k  h 1, k1, 2, , N (6) where

d

_k and are constants. and

bou

k

h

k

d

d

h

are the

upper nd of and

k

h

, nam ly,e d ax

 

d_k ,

 

max k

h h .

1m k N

1 k N

dress the follow hybrid system:

 

N ks ( k( ))

1

k

x t  



L x t t ,



( )

s t I. (7)

where the map ( ) :[0,t  ) I_ 



1, 2, , M



is a at determines the network topo

switching signal th logy,

and M_{ }_{denotes the total number of all possible}

switc directed graphs.



N L_ks L G_s( )_s is the

Laplacian matrix of the graph



hing un

1

k



, ,

s V E As s s that b

G elongs

to a set 



G k_k: I_



, which finite.

Lemm mplement [25]). Let S₁₁,

is obviously

a 2 (Schur co S₁₂,

0  22 be given symmetric matrices such that 22 ,

11 12 ₀

S S

 

S

then

S

12T 22

S S 

 _  S S S 1S T 0

   11 12 22 12 

Le n

.

n L_

s an orth mma 3. For any Laplacian matrix of un-directed connected network, there exist ogonal matrix

W

such that

 _{ }

 

1 1

1 1 0

0 0

n

n

L _{ }

 

 T

W LW 

 

 

where the last column of matrix W is 1n n ,

(n 1) (n1)

L_    _.

use

Proof: Beca and are the corresponding left

an tor respec

rovide the convergence analysis of

is 1T

n 1n

d right eigenvec tively. The proof of this lemma is straightforward.

3. Main Results

In this section, we p

the average consensus problem in undirected network with fixed and switching topology coupling multiple time-varying delays. Sufficient conditions expressed as LMI are presented for undirected networks of multi- agent with fixed and switching topology, respectively.

3.1. Networks with Fixed Topology

If the fixed communication topology G V E A



, ,



nd 1L_{k n} 0 kept connected, we have 1T 0

n Lk  , a  , which imply n x_i n u_i 0

1 1

i i

 







. Then,  Ave( (0))x

an invariant we have t sed

equation ( ) 1n ( )

is

quantity. Thus he decompo

x t   t , where



1, ,2 3 ,



T n       n

 , sat

_{ }

i i t

)

By Lemma 3, we have

)) 

then

)) 

Let , wher

 

t L t)

  ₍₈₎

1 ( ( N k k k t    



.

 

k t WW 1 ( ( N T T k k

L WW t t

    



  _,

 

1 ( ( N

T T T

k k

k

W  t W L WW  t t



 





 _.

 

 

,0 T

T T

W  t  _ t _ e_

 

_{t }n1 ng equation:

. Then (8) can be transformed into the followi

 

1

( ( ))

k k

k

t L t  t



  



N   (9)

whereLksatisfies

 1 1

0_{n }

  1 1 0 0 k T k n L W L W

           .

Lemma 4. If li

 

tm  t 0, then limt 

 

t 0.

Proof: From _WT_

 

_{t  } T

 

_t ,0_T

 

T

  . Therefore, when

, we have

 

_{t W} T

 

_t , 0

  _ _ li

 

tm  t 0,

we havelim

 

0. This completes the proof. t  t

In the following secti he

conver-dynamical sy .

u e

tim

on, we will discuss t gence of stem (9), that is lim

 

0

t t 

Theorem 1. Consider an undirected network of multi-agent with fixed topology coupling m ltipl

e-varying delays satisfies (6). Assume the communi-cation topology G is kept connected. Then, system (5) asymptotically solves the average consensus problem, if there exist positive definite matrices P Q R, k, k

n  1 n 1

 , satisfying

1

 

 1 ( 1) ( 1)

( 1) ( 1)

1 1

( 1) ( 1) ( 1) ( 1) 2

0

0 0

0 0

k n n

k

N N

T k

k n n

k k k

n n n n

PL R L P d                                







(10)

where N  





1 1 N T

k k k

k

PL L P Q



  



  ，





2 [ ( )1 1 1 1 1 1 1, ,

T

diag t L R L h Q

   





( ) T 1 ].

N t L R LN N N h QN N

  

Proof: Define a Lyapunov-Krasovskii fun n for

system (9) as follows: ctio

1 2 3

( ) ( ) ( ) ( )

V t V t V t V t ,

1( ) ( ) ( )

T

V t   t P t , N

2 _{( )}

1 ( ) ( ) ( ) k t _T k t t k

V t _ s Q s

 





  ds,

0 3 1 ( ) ( ) ( ) k N _t T k d t k

V t _{ } s R s dsd





 

 

where P, Q_k and n 1 n1

k

R     are positive definite matrix. Along the trajectory of system (9), we have

1

1

) 2 _k ( _k( ))

k

t t  t



   ,

( N T( )

V



 t PL 

2 1

( ) N { ( )T ( )

k k

V t t Q t







 



(1 ( )) T( ( )) ( ( ))}

k t t k t Qk t k t

         1 { ( ) ( ) N T k k

t Q t







 

(1 ) T( ( )) ( ( ))}

k k k k

h t  t Q t  t

      ,

3

1

( )

N

{

T

(

( ))

T

(

( ))

k k

k k k k

k

V t

d

t



t L R L

t



t











 



( ) ( ) } k t _T k t s R s ds



_

  _.

By Newton-Leibniz formula

s an ( ) ( ( )) ( ) ) k t

k _{t t}

t  t t _ s d



    

_

₍

d note that _{2x y}T _{x F x}T 1 _{y Fy}T definite matrix

F

, we

hold for any

ap-propriate positive have:

1

( ))

k

t



2 N T( ) (

k k

t PL t 





  

( ) 1

{ 2 ( ) ( ) 2[ ( )] ( ) }

k

N _t

T T T

k _{t t} k

k

t PL t _ L P t s ds







   

_

 T

1 1

{ 2 ( ) ( ) ( ) ( )

N

T T T

k k k k k

k

t PL t d t PL R L P t

 T 



      ( ) ( ) ( ) }. t _T t k

s R s ds

  

_

  Consequently, 1 1 k

( ) N{ 2 T( ) ( ) T( ) T T ( )

k k k k k

V t_{ }



_{ } t PL_{  }t d t PL R L P _ t

( ( )) T ( ( )) T( ) ( )

k k k k k k k

d T t _{ t L} R L t _ t t Q t

       

(1 ) T( ( )) ( ( ))}

k k k k

h t  t Q t  t

     



1



1

{ ( ) ( )

N

T T T

k k k k k k k

k

t PL L P Q d PL R L P t







     

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

T T

k k k k k k k k

t  t d L R L h Q t  t

     



1



1

{ ( ) ( 0,

N

T T T

k k k k k k k

k

t PL L P Q d PL R L P t

       



(11) and )} 1

{ ( ( ))( (1 ) ) ( ( ))} 0.

N

T T

k k k k k k k k

t  t d L R L h Q t  t

     



(12) As a result, the matrix inequalities (11) hold, if and only if

k



1



1

0

k k k k k k k

k

PL L P Q d PL R L P



    



Then N

T  T

, by Schur complement formula the matrix ine-quality (13) is equivalent to

(13) 1 1 k k N N PL       



1 1 0 N T k k

k k k

R L P d          





 (14)

where is defined in (10).

Therefore, from Lemma 4, average consensus can be achieved if the matrix inequality (6) holds. This

com-he

witching communication topology following disagreement switching system:

1 

pletes t proof.

3.2. Networks with Switching Topology

Considering the s

( )

s

G s I _ with system (7), we can obtain the

 

1 ( ( )), ( ) N ks k k

L t  t s  t I_







   

 _{, (15)}

where

t

 

ks

L satisfies

    1 1 1 1 0 0 0 ks n T ks n L

W L W  

           .

Theorem 2. Consider a undirected network of multi-agent with switching topology coupling multiple time-varying delays satisfies (6). Assume the communi-cation topology G s Is(  ) is kept connected. Then, system (7) asymptotically solves the average consensus problem, if there exist positive definite matrices

 1  

, , n

k k

P Q R   n1 _{, satisfy}ing

1 ( 1) ( 1)

1

0

N

ks n n

k PL R      _  



 ( 1) 1 1

( 1) ( 1) ( 1) ( 1) 2

0 0

0 0

N N

T k

ks n n

k k k

n n n n

L P

d  

                        





, (16)

where ₁





1

N

T

ks ks k

k

PL L P Q



  



  ,





2 [ ( )1 1 1 1 1 1 1,

T

s s

diag t L R L h Q

    ,





( ) T 1 ].

N t LNs R LN Ns h QN N

  

Proof: The proof of theorem 2 is similar to that of theorem 1, so it is omitted here.

ark: When

Rem h1

can obtai 1( )t V3

of the condition (6) is replaced with , we n corresponding results if choosi

1 h

ng ( )V t V  ( )t .

4.

n. Con

1 shows four examples of undirected graph, which are all connected, and the corresponding

adja-e limitadja-ed to 0-1 matricadja-es. Ladja-et thadja-e is



Simulation

In this section, simulation examples will be given to validate the theoretical results obtained in the previous sectio sider a group of 10 agents labeled 1 through 10. Figure

cency matrices ar thogonal matrix W

0.0275 0.0303 0.3594 0.1500 0.4082 0.0275 0.0303 0.3594 0.1500 0.4082 0.0275 0.0303 0.3594 0.1500 0.8165 0.0594 0.0294 0.3782 0.1390 0.0000 0.0594 0.0294                       

 0.2404 0.1390 0.0000

0.1323 0.5742 0.2482 0.5742 0.0000

   

0.4914 0.6506 0.2653 0.2557 0.0000 0.8162 0.3793 0.2234 0.1447 0.0000 0.2559 0.3149 0.4793 0.6714 0.0000 0.0000 0.0000 0.0000 0.1615 0.0000

  _{  }       _{   }      

0.7071 0.0809 0.1749 0.2062 0.3162 0.7071 0.0809 0.1749 0.2062 0.3162 0.0000 0.0809 0.1749 0.2062 0.3162 0.0000 0.1310 0.6023 0.5946 0.3162 0.0000 0.1310 0.6694 0.5946 0.3162 0.0000 0.2061 0.1207 0.3231 0.3162 0.0000               10 10 .

0.1511 0.1290 0.2450 0.3162 0.0000 0.0820 0.1087 0.0230 0.3162 0.0000 0.0090 0.0206 0.0206 0.3162 0.0000 0.9348 0.0000 0.0000 0.3162 _

            _         _     _

where the last column of matrix W is 1n 10. For

4.1. Examples of Networks with Fixed Topology

Consider an undirected network with fixed topology in Figure 1. Employing Theorem 1, we have:

1) for , i.e.,

0 5 10 15 20 25 30

−20 −15 −10 −5 0 5 10 15 20

time(s)

error of system

Time−Delay=0.17 |cos 0.51234t|

1

G

1 0

h   ₁ d₁, . Figure 2

shows the corresponding error system converges zero asymptotically.

2) for Figure 3 shows the

corre-sponding error system converges zero asymptotically. 1 0.25

d 

1 0.5

h  , d₁0.17.

G1

2

1 3 4 5

6 7

8 9

10

G2

G3

G4

2

1 3 4 5

6 7

8 9

10

2

1 3 4 5

6 7

8 9

10

2

1 3 4 5

6 7

8 9

[image:6.595.316.527.78.253.2]

10

Figure 1. Examples of connected undirected graph.

0 5 10 15 20 25 30

−25 −20 −15 −10 −5 0 5 10

time(s)

error of system

[image:6.595.73.269.199.469.2]

Time−Delay=0.25

Figure 2. Error system with fixed topology and cons nta t time-delay d10.25s with h10 converges to zero

as-ymptotically.

Figure 3. E tem wit ed topology and time-

varying delay

rror sys h fix

1( ) 0.17 cos0.51234t t s h1 0.5

  with 

converges to zero asymptotically.

4.2. Examples of Networks with Switching Topology

A finite automation with set of states



_{G G G, ,}



is shown in Figure 4, which represents the of a network with switching topology an

hybrid system. It starts at the disc and switches every simulation time step to t

cording to the state machine in Figure

system with time-varying communication ti we take the derivative of delay h₁0.5. Fr 2,

the feasible maximum is

2 3 4 discrete states d time delay as a rete state G₂

he next state ac-4. For multi-agent

me delay, om theorem of the system delay bound

1 0.12

d  s. And the corresponding feasible solution P, e obtained by employing

1

QandR₁ can b the LM

box in Matlab.

Assume the time-varying dela

I To

e error system ol

is y of th

1( ) 0.12 cos 0.51234t t s

  . Figure 5 shows the

corre-sponding error system converges zero asymptotically.

G2

G3

G4

t=0

[image:6.595.66.277.497.677.2] [image:6.595.341.506.578.696.2]

0 5 10 15 20 25 30 −20

−15 −10 −5 0 5 10 15

time(s)

error of system

[image:7.595.79.269.76.231.2]

Time−Delay=0.12 |cos 0.51234t|

Figure 5. Error system with switching topology and time-varying delay ₁( ) 0.12 cos0.51234t  ts converges to z asymptotically.

5. Conclusions

This paper addresses an average consensus problem of multiagent systems. Undirected networks with fixed switching network topology coupling multiple tim varying communication delays are considered in this paper. An orthogonal matrix is introduced and the origi-nal system is turned into a reduced dimensioorigi-nal sy At the same time, a Lyapunov-Krasovskii function is constructed in the stable analysis. Sufficient conditions in terms of LMI are given to guarantee the system reach average consensus. Moreover, numerical simulation amples are shown to verify the theoretical analysis.

supported by National Natural Science a under the grant 60604001 and 606-

incial Natural Science Found under the grant 2008CDB316 and 2008CDZ046, and

neighbor rules,” IEEE Transactions on Automatic Control, 001, 2003.

[3] W. Ren, and R. W. Beard, “Consensus seeking in m

“Consensus problems in topology and EEEE Transactions on Automatic Control, pp. 1520–1533, 2004.

eighted average

th a time-varying

93–3398, December 2006.

. 303–313,

ime-delay,”

sus

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[9] W. Ren, R. W. Beard, and E. M. Atkins, “Information con

and

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[11] W. Ren, “Multi-vehicle consensus wi

stem. reference state,” Systems & Control Letters, Vol. 56, No. 7-8, pp. 474–483, July 2007.

[12] D. P. Spanos, R. O. Saber, and R. M. Murray, “Dynamic

ex-consensus on mobile networks,” In IFAC World Congr., Prague, Czech Republic, 2005.

[13] Y. Hatano and M. Mesbahi, “Agreement over random networks,” IEEE Transactions on Automatic Control, Vol. 50, No. 11, pp. 1867–1872,

6. Acknowledgment

This work was

Nov

Foundation of Chin

74081, Hubei Prov ation [15] A. T. Salehi and A. Jadbabaie, “A necessary and suffi-_{cient condition for consensus over random networks,”} IEEE Transactions on Automatic Control, vol. 53, No. 4, pp. 791–795, April 2008.

Scientific Innovation Team Project of Hubei Provincial College under the grant T200809.

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