Distributed consensus of multi-agent systems using distributed time delayed protocols

Accepted Manuscript Title: Distributed consensus of multi-agent systems using distributed time delayed protocols Author: Chengrong xie Yuhua Xu Shengyin Meng Dongbing Tong Anding Dai PII: DOI: Reference:

S0030-4026(15)00628-2 http://dx.doi.org/doi:10.1016/j.ijleo.2015.07.076 IJLEO 55791

To appear in: Received date: Accepted date:

12-6-2014 14-7-2015

Please cite this article as: C. xie, Y. Xu, S. Meng, D. Tong, A. Dai, Distributed consensus of multi-agent systems using distributed time delayed protocols, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.07.076 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Distributed consensus of multi-agent systems using distributed time delayed protocols Chengrong xie a, c, 

College of Science, Nanjing Audit University, Jiangsu 211815, China

b

School of Finance, Nanjing Audit University, Jiangsu 211815, China

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a

d

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Department of Mathematics and Finance, Yunyang Teachers’ College, Hubei, Shiyan 442000, China College of Electronic and Electrical Engineering, Shanghai University of Engineering Science,

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c

Yuhua Xu b,c Shengyin Meng c Dongbing Tong d Anding Dai e

Shanghai 201620, China

College of mathematics and Compute Science, Hunan City University, Hunan 413000, China

an

e

Abstract: This paper considers the first order consensus problem of multi-agent systems. A new kind of

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distributed control protocol is constructed for achieving distributed consensus. Based on linear matrix

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inequalities and the Lyapunov function method, the sufficient conditions for the achievement of the

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distributed consensus are obtained. Illustrative examples are provided to show the effectiveness of this method.

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Keywords: Consensus; Time-delay; Multi-agent systems 1. Introduction

The consensus problem of multi-agent networks has been extensively investigated due to its widespread applications in satellite formation flying, design of sensor networks, air traffic control, etc[18]. An important problem in cooperative control of multi-agent systems is the consensus problem. The key issue in consensus problem is to design decentralized controllers such that all agents can reach a consensus value [9-13]. In the past years, the first order consensus problem of multi-agent systems has been intensively studied [14-25]. In [26], Olfati-Saber and Murray proposed a systematic framework to study the



E-mail addresses: [email protected]. 1

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first-order consensus and showed that the consensus can be achieved if the digraph is strongly connected. In [27], Ren et al. proposed some improved conditions for state agreement under dynamically changing directed topology. By using a linear matrix inequality method. Sun et al. discussed average consensus

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problem in undirected networks of dynamic agentswith fixed and switching topologies as well as multiple time-varying communication delays in [28].

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In recent years, the first order consensus problem of multi-agent systems have been further

following consensus protocol based on delay feedback:

 aij ( x j (t   )  xi (t   )),

jNi

i V ,

(1)

an

xi 

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investigated in different situations, for example, in [26], Olfati-Saber R and Murray R M proposed the

where   0 represents communication delay.

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In [29], Wu and Fang discussed the following consensus protocol using the delay-state derivative feedback:

d

 aij  x j (t   )   x j (t   )    xi (t   )   xi (t   ), i  V ,

jNi

(2)

te

xi 

where 
is the intensity of delay-state-derivative feedback and satisfies   (0, min{ ,1 N ( L)}) .

Ac ce p

In [30], Meng and Li considered the following consensus protocol with the delay induced consensus algorithm:

n

n

j 1

j 1

xi  K1  aij ( xi (t )  x j (t ))  K 2  aij ( xi (t   )  x j (t   )), i  1, 2, , n ,

(3)

where K1 , K 2  R gm .

In this paper, the new consensus protocol using distributed time delayed protocols is designed as follows: xi (t )  ui (t ),

i  1,, N N

(4) 

ui  di ( xi (t )  x0 )   aij  p(s)( x j (t  s)  xi (t  s))ds , i  1, 2, , N . 0 j 1

(5)

where xi  ( xi1 , xi 2 , , xin )T  R n is the state, ui  R P is the control input,   0 , the inner coupling matrix 2

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  R nn

and   0 , aij is the (i, j )-th entry of the adjacency matrix A  R N  N , di  0 ,   0 is time

delay and the continuous density function p : [0, ]  [0, ] is prescribed, which satisfies x0  R n



0 p( s)ds  1 .

denotes the desired equilibrium state.

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Remark 1. Dynamical networks with constant discrete time delays have been used to approximate the real systems. However, many systems have distribution of transmission delays and cannot be effectively

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modeled by networks with discrete time delays. Therefore, dynamical networks with distributed time

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delays are proposed to model these systems.

Throughout this paper, the following notations will be used: The information exchange among the

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nodes in a multiagent network can be described by an interaction graph. Let G  {V ,  , A} be a diagraph, in which V  {1, 2, , N } is the node set,   V  V is the edge set, and A  (aij ) N N is the associated

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weighted adjacency matrix. An edge of G is denoted by eij  (vi , v j ) , which means that node i can

d

receive information from node j . The entry aij  0 if eij   ; otherwise aij =0 . In this paper, it is

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always assumed that aii =0 for all i  V . The in-degree and out-degree of the node i are defined as follows: degin (i )   Nj 1 a ji , deg out (i )   Nj 1 aij , respectively. I represents the identity matrix of dimension

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The Laplacian matrix L  (lij ) N  N associated with the adjacency matrix A is defined as follows: lij   aij , i  j ;

lii 

N



j 1, j  i

aij ,

(6)

Generally speaking, the Laplacian matrix of a digraph is asymmetric. 2. Main results

In this section, we study the consensus problem for a group of identical agents. The objective of the consensus is to design controllers for network (4) to achieve limt  xi (t )  x0  0, i  1, 2, , N .

To obtain the main results, we need the following preliminaries. Lemma 1 [31] For matrices A, B, C , D with compatible dimensions, the following equations can be

3

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held: i) ( A  B)T  AT  BT ; ii) ( A  B)  C  A  C  B  C ; iii) ( A  B)(C  D)  ( AC )  ( BD) . Lemma 2 [32] Let Q be a symmetric N N real matrix and Q  0 . Then

0

T

  p(s)eT (t  s)Qe(t  s)ds    p(s)e(t  s)ds    p(s)e(t  s)ds   0   0 

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Now, we give some criterions to reach distributed consensus of the agents.

(7)

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Theorem 1: Solve the following LMI: 1   Q   D   2  L    0   1  L   I  2 

ip t



to get a  . Then, the agents described by (4) reach consensus under the protocol (5), where the matrix D  diag{d1 , , d N } .

M

Q 0,

Proof: Let ei  xi (t )  x0 , then, it follows that ei satisfies the following dynamics: N



d

ei  di ei + aij  p(s)(e j (t  s)  ei (t  s))ds .

te

j 1

0

(8)

in view of Eq. (6), we can easily obtain that

N



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ei  di ei  lij  p(s)e j (t  s)ds j 1

0

(9)

Let e (t )  (e1T (t ), e2T (t ), , eTN (t ))T , by Lemma 1 and rewrite Eq. (9) in the matrix form as  e     D  e   L   p(s)e(t  s)ds . 0

(10)

Consider the Lyapunov function candidate  t 1 V1  eT e+ p(s)   eT ( )Qe( )d  ds , t 0 2  s 

where the matrix Q  0 . The time derivative of V1 along the trajectory of (10) is given by   V1  eT e   p(s)eT (t )Qe(t )ds   p(s)eT (t  s)Qe(t  s)ds 0

0



 eT e  eT (t )Qe(t )   p(s)eT (t  s)Qe(t  s)ds 0

4

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   eT    D  e   L   p(s)e(t  s)ds   eT (t )Qe(t )   p(s)eT (t  s)Qe(t  s)ds   0 0 



0

0

 eT   D  e  eT  L   p(s)e(t  s)ds  eT (t )Qe(t )   p(s)eT (t  s)Qe(t  s)ds .

Using lemma 2, it follows that T

ip t

   V1  eT   D  e  eT  L   p(s)e(t  s)ds  eT (t )Qe(t )    p(s)e(t  s)ds    p(s)e(t  s)ds  0 0 0    

e e        ,      p(s)e(t  s)ds   p(s)e(t  s)ds  0  0 

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1   Q   D   2  L       1  L   I  2 

an

where

cr

T

By choosing compatible  such that   0 . Therefore, V1  0 . Therefore, the multi-agent network (4)

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can globally achieve consensus under the given distributed time delayed protocols. The proof is

d

completed.

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In the following case, the distributed adaptive consensus protocol is proposed: Theorem 2: Solve the following LMI:

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1   * Q   D   2  L     0   1  L   I  2 

(11)

to get a  . Then, the agents described by (4) reach consensus under the following adaptive the protocol (12-13), where the matrix Q  0 , D*  diag{d1* , , d N* } , d i*  0 . N



ui  di ( xi (t )  x0 )   aij  p(s)( x j (t  s)  xi (t  s))ds ,

(12)

di  i eiT ei .

(13)

0

j 1

Proof: Let ei  xi (t )  x0 , then, it follows that ei satisfies the following dynamics: N



ei  di ei + aij  p(s)(e j (t  s)  ei (t  s))ds . j 1

0

(14)

Consider the Lyapunov function candidate 5

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V2 

t  1N T 1N 1 ei ei + p(s)   eT ( )Qe( )d  ds   (di  d * )2 .  0 2 i1 2 i 1 i  t s 

Then, similar to the proofs of Theorem 1, it follows that



0 p(s)e

T

ip t

N N     V2  eiT  di ( xi (t )  x0 )   aij  p(s)( x j (t  s)  xi (t  s))ds    p(s)eT (t )Qe(t )ds  0   0 i 1 j 1   N

(t  s)Qe(t  s)ds  (di  d * )di i 1









us

cr

   eT   D*  e   L   p(s)e(t  s)ds   eT (t )Qe(t )   p(s)eT (t  s)Qe(t  s)ds   0 0  T

an

    eT   D*  e   L   p(s)e(t  s)ds  eT (t )Qe(t )    p(s)e(t  s)ds    p(s)e(t  s)ds    0  0   0 

e e        , =   p(s)e(t  s)ds   p(s)e(t  s)ds  0  0 

d

where

1   * Q   D   2  L   .   1  L   I  2 

M

T

te

By choosing compatible  such that   0 . Therefore, V2  0 . Therefore, the error system (14) is

Ac ce p

globally stable at the origin. Then it follows that the agents networks (4) can asymptotically achieve adaptive consensus to the desired equilibrium state x0 . Remark 2. Solvability of (7) and (11) holds for any  , this implies that the consensus can be achieved with an arbitrary speed.

3. Simulation and results

In this section, computer simulations are used to verify and demonstrate the effectiveness of the above method. Example 1. In this example, we consider the multi-agent network (4) with 10 nodes, the initial states of the agents are [0.4 1.2 0.5 1.2;1.7 1.1 0.9 1.4;-1.0 1.3 -1.5 1.0;-1.3 1.5 -0.3 0.1;-0.3 0.6 -0.1 1.3;-0.6 1.2 -0.6 0.2;-1.2 1.4 -0.7 1.9;-1.4 1.6 -0.8 0.4;-0.64 0.12 -0.25 1.32;-0.17 1.31 -0.19 0.34;],  =1 ,  =I , and

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the value of the desired equilibrium state is set to be 1.0, i.e., x0  1.0 . The values of L, A, D, Q are set to be the following values. The parameter  =6.3939 , which is the solution of LMIs(7) by computing LMIs(7) with matlab toolbox. The state trajectories of all the agents are shown in Fig. 1. 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  1 0 0 0 0 0 0 1 0 0 1 0  1 0 1 1 0 0 , 0 1 0 0 0 0  0 1 0 0 0 0  1 0 0 0 0 1 0 0 0 0 1 0 

cr

0 0 0

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0 2 0 0

0 0 1 0

0 0 0 2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

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1  0 0  0 0 D 0 0  0  0 0 

0 0 0

an

0 0

0 0 0

0 1   0 0 0 0   0 0  0 0  Q 0 , 0  0 0   0 0   0 0  0 1 

M

0 0 0

0 0 0

d

0 0

 0 0 1 0    0 0 1 0  1 1 0 1   1 2 1 0 0 0 0 0  0 0 1 0  0 0 0 1 0 1 3 1 0 0 1 0  A 0 0 1 3 1 1 0 0  , 0 0 0 0  0 0 0 0 0 0 0 1 1 0 0 0   0 0 0 0 0 0 0 1 0 1 0 0    0 0 1 0 0 0 2 1 0 0 0 0  0 0 0 0 0 0 0 0 0 0 1 1   0 0 0

ip t

0 1 0 0 1 1 0 0 1 3 1 0

te

1  0  1  0 0 L 0 0  0  0 0 

0 1 0 0

0 0 2 0

0 0 0 2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0

0  0 0  0 0  0 . 0  0  0 2 

Fig. 1. Consensus tracking with x0  1.0 Example 2. In this example, similar to the computer simulations of Example 1, let  =2 , i  1 , and the value of the desired equilibrium state is set to be 1.5, i.e., x0  1.5 . The initial states of the agents are

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set to be the case of Example 1. The initial states of di are 0. The values of L, A, D, Q are set to be the following values. The parameter  =5.8753 , which is the solution of LMIs(11) by computing LMIs(11) with matlab toolbox. The state trajectories of all the agents are shown in Fig. 2. The coupling weights

0 0 0

0 0 0

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1

0  0 0  0 0  0 , 0  1  1 0 

0 0 0 0 0 0 0 0 0  2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0  0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0  0 0 0 0 2 0 0 0 0 . 0 0 0 0 0 1 0 0 0  0 0 0 0 0 0 2 0 0  0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 

d

M

0 0 0 0 0 0 0 0 0 1   2 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0   0 0 2 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0  Q 0 0 0 0 3 0 0 0 0 , 0  0 0 0 0 0 0 2 0 0 0   0 0 0 0 0 0 0 4 0 0   0 0 0 0 0 0 0 3 0 0  0 0 0 0 0 0 0 0 0 2 

Ac ce p

2  0 0  0 0 D 0 0  0  0 0 

0 0 0 1

cr

0 0 0

 0 1 1 0    1 0 1 0  1 1 0 1    0 0 1 0 0 0 0 1 1 3 1 0 0 1 0   A 0 1 3 1 1 0 0  , 0 0 0 0  0 0 0 0 0 0 1 2 1 0 0   0 0 0 0 0 0 1 1 4 1 1   0 1 0 0 1 3 1 0 0 0 0  0 0 0 0 0 0 0 0 1 1 2  

0 0 1 2

us

1 1 3 1

an

1 2 1 0

te

2   1  1  0 0 L 0 0  0  0 0 

ip t

are shown in Fig. 3, which converge to finite steady-state values.

Fig. 2. Consensus tracking with x0  1.5

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ip t cr

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Fig. 3. Coupling weights di

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4. Conclusions

The distributed consensus of multi-agent systems using distributed time delayed protocols has been

M

studied in this paper. To realize distributed consensus, a new class of distributed control protocol is designed and utilized. We have obtained the sufficient condition for the distributed consensus based on

te

Acknowledgments

d

LMIs and Lyapunov function method.

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