Distributed Consensus of High Order Multi Agents with Nonlinear Dynamics

doi:10.4236/ica.2011.21001 Published Online February 2011 (http://www.SciRP.org/journal/ica)

Distributed

H

_

Consensus of High-Order Multi-Agents

with Nonlinear Dynamics

*

Jianzhen Li

School of Automation, Nanjing University of Science and Technology, Nanjing, China

E-mail: jianzhenli1983@yahoo.com.cn

Received January 20, 2011; revised February 12, 2011; accepted February 13, 2011

Abstract

This paper deals with the distributed consensus problem of high-order multi-agent systems with nonlinear dynamics subject to external disturbances. The network topology is assumed to be a fixed undirected graph. Some sufficient conditions are derived, under which the consensus can be achieved with a prescribed H_

norm bound. It is shown that the parameter matrix in the consensus algorithm can be designed by solving two linear matrix inequalities (LMIs). In particular, if the nonzero eigenvalues of the laplacian matrix ac-cording to the network topology are identical, the parameter matrix in the consensus algorithm can be de-signed by solving one LMI. A numerical example is given to illustrate the proposed results.

Keywords:Consensus, Multi-Agent Systems, Nonlinear Dynamics, External Disturbances

1. Introduction

The consensus problem of multi-agent systems has been researched extensively in recent years. This is because of its widely application in much areas such as flocking [1-2], synchronization of coupled oscillators [3], forma-tion control of mobile robots [4-5], distributed computa-tion [6] and informacomputa-tion fusion in wireless sensor net-works [7]. The object of consensus control is to design consensus protocol such that the group of agents can asymptotically agree upon certain quantities of interest based on information received from their neighbors.

Most of the work on the consensus problem focuses on the multi-agent systems with first-order dynamics. In particular, [8] deals with the first-order multi-agent sys-tems with switching topologies and time delays in a con-tinuous setting. The fist-order multi-agent systems with switching topologies is investigated in [9] in a dis-crete-time setting. The consensus problem has also been investigated from many other aspects such as reference signals [10], asynchronous sampling time [11], and so on. Recently, the consensus problem of second-order multi- agent systems has been investigated extensively [12-14]. In particular, the consensus problem of second-order multi-agent systems with nonlinear dynamics was

inves-tigated in [15]. The nonlinear dynamics can be taken as the potential functions or the desired final dynamics of the agents. There is also some work on the consensus problems of high-order multi-agent systems [16-17].

Generally speaking, the consensus cannot be achieved accurately if there are external disturbances. To deal with this problem, the H_ consensus problem is considered [18-21]. It is shown that for undirected network topolo-gies, the desired parameter matrix in the consensus algo-rithm can be designed by solving two LMIs, which relate to the system matrix of the agents and the eigenvalues of the laplacian matrix corresponding to the network topol-ogy. The H2 consensus problem was investigated in [22].

In the aforementioned work on the H_ or H2 con-sensus problem, the nonlinear dynamics was not consid-ered. As is mentioned in [14-15] much multi-agent sys-tems have nonlinear dynamics. Motivated by this, this paper considers the H_ consensus problem of high- order multi-agent systems with nonlinear dynamics. To the best of the author's knowledge, this problem has not been considered in the literature. Some sufficient condi-tions will be derived, under which the consensus can be achieved with a prescribed H_ norm bound. It will be shown that the parameter matrix in the consensus algo-rithm can be designed by solving two LMIs, which relate to the system matrix of the agents and the smallest and

*_{This work was supported by the National Natural Science Foundation}

the biggest nonzero eigenvalue of the laplacian matrix corresponding to the network topology. In particular, if the nonzero eigenvalues of the laplacian matrix accord-ing to the network topology are identical, the parameter matrix in the consensus algorithm can be designed by solving one LMI.

2. Preliminary Notations and Problem

Formulation

Let 



  , ,



be a weighted undirected graph of o r d e r N, w h e r e  



1,,N



i s t h e n o d e s e t .

 

   is a set of unordered pairs of nodes, and  is the adjacency matrix. An undirected path is a se-quence of edges in a undirected graph of the form



v , vi1 i2



,



v , vi2 i3



,, where v vi1, i2 An

undi-rected graph is called connected if for any two nodes of the graph, there exists a path that follows the edges of the graph. The adjacency matrix is a nonnegative matrix

N N d aij

   _{ }

 satisfying a_ii 0 for any i,

aij aji 0, if

 

j i, , and aij 0 if agents j and i are not adjacent. The Laplacian matrix of the

graph is defined as N N

ij

l 

  _{ }

 with ii ij

j i

l a







and lij a ,ij  i j. We can see that  satisfies

0

 1

 and ₁T_0

where 1



1,,1



T. For matri-ces M and N, MN denotes their Kronecker product. It is well known that if the undirected network topology

 is connected, the lapalacian matrix corresponding to

 has N1 positive eigenvalues and a simple zero eigenvalue.

Consider a group of N agents with the following dy-namics:

 

1 2 ,

i i i i i

x Ax Bu B B f x (1)

where

 

m,

i

x t 

 

p

i

u t  are, respectively, the state, the control input of agent i,

 

m

i t

  is the external disturbance which belongs to 2



0,



, and

 

m i

f x 

is a nonlinear function.

Assumption 1: There exists a positive scalar  such that

 

 

 

 



 



,

, .

T

i j i j

T i j i j

m i j

f x f x f x f x

x x x x

x x 

     

   

  

 

Remark 1: Assumption 1 is similar to the Assumption 1 in [14]. It is a Lipschitz-type condition satisfied by many systems.

Definition 1: We say algorithm u_i solves the con-sensus problem if

1

0, , . N

j i

j

x

x t i

N 





    

Definition 2: We say algorithm u_i solves the con-sensus problem with H_ norm bound  if the fol-lowing two conditions are satisfied:

1) Algorithm u_i solves the consensus problem if 0;



2) If z00, the following inequality is satisfied:

2 ₂

2

0 z dt  0  dt,

 







where

1

1

1 ,

,

, N

j

i i

j

T

T T

N

T

T T

N

x z x

N

z z z

  



 

 

  

 

  







and z0 is the initial value of z.

The object of the H_ consensus control is to design consensus algorithms such that the consensus problem is solved for a prescribed H_ norm bound.

Lemma 1: (Schur complement [23]) Let S be a sym-metric matrix of partitioned form S   Sij with

11 ,

r r

S  ( )

12

r n r

S   and ( ) ( )

22

n r n r

S     Then, 0

S  if and only if S110,

1 22 21 11 12 0, S S S S  or

equivalently 1

22 0, 11 12 22 21 0.



  

S S S S S

Lemma 2: For matrices A, B, C, D with appropriate dimensions, one has



AB



T ATBT,

  

CD  AC



BD



and



AB



    C A C B C

3. Results

In this section, the H_ consensus problem of multi- agent systems with nonlinear dynamics will be investi-gated. Considers the following state feedback consensus algorithm:

 





1

N

i ij i j

j

u t K a x x







 (2)

With (2), system (1) becomes





1 2

 

1

, N

i i ij i j i i

j

x Ax BK a x x B B f x



 



  

 (3)

which can be written in a compact form as



 







1

2

,

N N

N

x I A L BK x I B

I B f



     

 



(4)

where 1 N ,

T T T

x x x _ and ( ) (1 ) .

T T

N T f  f x  f x _

3



_m



,

z HI x (5)

where HN N with

1 , 1 , . ij N i j N H i j N   _       

It can be seen that HI_N11T N, H2H, ,

T T NH  N

1 0 H1_N 0_N and H  H .

Lemma 3: There exists an orthogonal matrix N N

U  with last column 1N N H such that

1 1 1

, .

* 0 * 0

N n N

T I T

U HU _   _ U LU _  _

   

0 0

From (4) and (5) we have







 









 







1 2 1 2 . m

z H I x

H A L BK x H B

H B f

H A L BK z H B

H B f

                      (6)

Define 1

N U U N       1

 , from Lemma 3 we have

1 1 1

T

N

U HU I _ and 1 1 .

T

U LU   Define  



UTIm



z

1 , T T T N      

  we have

















1









2



1 1 1

1 1

1 2

1 1

* 0 * 0

. T

m m

T T

m m

N n N

T T

T T

N N

U I H A L BK U I z

U I H B U I H B f

I

A BK

U H U H

B B f

                                           _  _ _  _         0 0 0 0  (7)

It can be seen that  N 0. So  0 if and only if 0

i

  , i1,,N1.Define 1 1 .

T

T T

N

 _   _ _ From (7) we have







11 1

 

1 2



.

N

T T

I A BK

U H B U H B f

              (8)

Note that the eigenvalues of  are 2,,N. There exists an orthogonal matrix FN 1 N1 so

that



2, ,



.

T

N

F  F diag   

Define





1 1

T

T T T

m N

F I

    _   _ _ we have











 



1 2

1 1 1 2

, ,

.

N N

T T T T

I A diag BK

F U H B F U H B f

               _ (9)

Noting that z zT      T  T  T , we conclude that algorithm (2) solves the consensus problem with

H norm bound  if and only if system (9) is

asymp-totically stable with T_{ } , where T_{ } denotes the H_ norm of the transfer function matrix from  to .

Theorem 1: Suppose the undirected graph  is connected and the nonzero eigenvalues of  are

2, , N

   . Using algorithm (2), the consensus is

achieved with H_ norm bound  if there exists a symmetric positive definite matrix X and a matrix W

such that the LMIs

1 2 2 1 2 0 0 0, 0 0 1 0 0 1 T T m m m

B XB X

B I B I X I      _      _     _   _     (10) 2, ,

i N hold, where





.

T T T

i

AX XA  BW W B

    

In this case, the parameter matrix in (2) can be chosen as 1

.

KWX

Proof: Assume that the undirected graph  is con-nected, we have 20. Suppose there exists a symmet-ric positive definite matrix X and a matrix W such that (10) hold. Define WP X1, K WX1. Pre- and

post-multiply both sides of (10) by

0 0 0

0 0 0

0 0 0

0 0 0

m m m P I I I            

one can get

1 2 2 1 2 0 0 0, 0 0 1 0 0 1 T m T m m m m

PB B I

B P I

B I I I      _      _     _   _     (11) 2, ,

i N hold, where





T T T

i

PA A P  PBK K B P

     .

For _i, i3,,N1, there exists 0_i1 such that i2 



1  



N. It is easy to see that (11) also

holds for i3,,N1. By Lemma 1 we know that (11) holds if and only if





2 2 1

2 1 1 0 T m T m

B B I PB

B P I





    



 _ 

  (12)









 

1 1

2

1 1 1

, N

T

N N m

I PB

I PB  I

                  where  

















2 1

1 2 2 1

, , . T T N N m T T

N N m

I diag PBK K B P

I B B I PA A P I

                  

From (12) we know that  0.

Next prove that the consensus is achieved if 0. If 0, (9) becomes













1 2 1 2 , , . N N T T

I A diag BK

F U H B f

  _      

 

 _

(13)

Consider the Lyapunov function

 



1



.

T N

V   I _ P 

Because P symmetric positive definite, we have V

 



is symmetric positive definite with respect to . Taking derivative of V

 

 along (13), we have

 















































1 1 1 2

1 1 2

1

2

1 2

2

2 , ,

2 , , 2 . T N N T N N

T T T

N

T T

N

T T N

T T T

V I P I A

I P diag BK

I P F U H B f

I PA A P

diag PBK K B P

F U H B f

                                      (14)

Because U is an orthogonal matrix, one has

1 1

T

T N N

N

I U U U U

N N

   

 _{    }

   

1 1

It follows that

1 1 1. T

N

U U I _ Then we have







 



 









1 2

1 1 2 2

2

1 2 2

2

.

T T T

T T T T

T

m

T T T

N m

F U H B f

F U U F B B

f H I f

I B B f H I f

          _  _    _  _    _  _   (15) Notice that





 

 

 

 



 



1 1 1 1 T m N _T

i j i j

i j i

N _T

i j i j

i j i

f H I f

f x f x f x f x

N

x x x x

N            _  _{ }  _   









. T m T T

x H I x

z z 

  

 

  (16)

From (14)-(16) we have

 





































1 2

1 2 2

1 2 2

2 , , , , . T T N T T N

T T T

N

T T T

N m

T T N

V

I PA A P

diag PBK K B P

I B B

I PA A P B B I

diag PBK K B P

                          _  _             (17)

It follows from (12) that 2 2



1



0,

T

m

B B  I

   

which, together with (17) implies that V

 

 is negative definite with respect to . It then follows that  0 asymptotically. From the analysis above we know the consensus can be achieved.

Assume that  0. Taking derivative of V

 



along (9), we have

 



 











































1 1

2 1 1

1 2

1 2 2

2 1 1 2 , , , , 2 . T N N T T N T T

T T T

N m

T T N

T T T

V I P I A

diag BK F U H B

F U H B f

I PA A P B B I

diag PBK K B P

F U H PB

                                      (18) Assume that z00 , which implies that 00 , where 0 is the initial state of . It follows that

 

 





























2 2 2

0 0

2 2 2

0 0

2 0

1 2 2

2 1 1 0 , , 2 , T T

T T T

N m

T T N

T T T

T

z dt dt

V dt V V

I PA A P B B I

diag PBK K B P

F U H PB dt

dt                                _   _                











  (19)

where  T TT,

   V00 is the initial value of

 

V  , and











 



1 1 2 1 1 . T T T Nm

F U H PB

HU F PB  I

5

By Lemma 1 we know that  0 if and only if















 















2

1 1 1 1

2

2

1 1 1

2

, ,

1

( )

, ,

1

0.

T T N

T T T

T T N

T N

diag PBK K B P

F U H PB HU F PB

diag PBK K B P

I PB B P

 



 

 

   

 

 

 _  _{ }  _

    

 

 _  _



 (20)

Also by Lemma 1 we know that (20) is equivalent to 0

  , which has been proved in the above analysis. So we have that



is symmetric negative definite. It fol-lows from (19) that

2 2 2

0 z dt  0  dt 0.

 

 





Therefore, the consensus is achieved with H norm

bound  . The proof is completed.

Sometimes, the laplacian matrix has N1 identical nonzero eigenvalues, i. e. 02N. Take the complete graph for example. Consider the complete graph with N nodes. The laplacian matrix is chosen as

1 1 1

1 1 1

.

1 1 1

N

N N N

N

N N N

N

N N N



 _{ } 

 

 



_{ } 

 



 

 

 _ 

 

 

 





   





By some calculations we have the eigenvalues of 

are 0, 1 , , 1

1 1

N  N . In this case, we have the fol

lowing corollary.

Corollary 1: Suppose the undirected graph  is connected and the nonzero eigenvalues of  satisfy

2 N.

  Using algorithm (2), the consensus is achieved with H_ norm bound  if there exists a symmetric positive definite matrix X and a matrix W

such that the LMI

1 2

2 1

1

2

0 0

0,

0 0

1

0 0

1 T

T

m

m

m

B XB X

B I

B I

X I





 

 _ 

 



 _ 

 

 _ 

 

 





(21)

holds, where 1 .

T T T

AX XA BM M B

     In this

case, the parameter matrix in (2) can be chosen as 1_.

K WX

Proof: Assume that there exists a symmetric positive definite matrix X and a matrix W such that (21) holds.

Define

2

1 .

XM W



 It follows that

1 2

2 1

2

0 0

0

0 0

1

0 0

1 T

T

m

m

m

B XB X

B I

B I

X I





 

 _ 

 



 _ 

 

 _ 

 

 





Holds for i2. From Theorem 1 we have the con-sensus is achieved with H_ norm bound , and the parameter matrix can be chosen as KWX1.

Remark 2: From Corollary 1 one has that if the non-zero eigenvalues of the laplacian matrix are identical, the

H_ performance is determined by 2 and the system matrices of the agents. It has no relationship with the number of the agents.

4. A Numerical Example

Consider a multi-agent systems consisted of N nodes with the following second-order





, sin 0.5 ,

i i i i i i i x  v  v  u   x

where sin 1

1

i t   _ _



  is the external disturbance. This multi-agent system can be written in the form of (1) with

1 2

0 1 0 1 0 0 0

, , , .

0 0 1 0 1 0 1

A_ _ B _{ } B _ _ B _ _

       

The communication topology is given in Figure 1. The laplacian matrix is chosen as

1 1 0 0

1 2 1 0

.

0 1 2 1

0 0 1 1



 

_{ } 

 



   

 _ 

 



The eigenvalues of  are 0, 0.5858, 2 and 3.4142. Solving the LMIs in (10) with 20.5858 , 4

3.4142,  0.5 and  1.29, we can get





0.0455 0.6688

, 0.1 2605.7 .

0.6688 23.2996

X _  _ W  



 

From Theorem 1 we know that K can be chosen as





1

2844.2 193.5 .

[image:5.595.306.498.82.182.2]

KWX   

Figure 2 shows the trajectory of the external distur-bance. Figures 3 and 4 show, respectively, the position and velocity responses of nodes 1 4.

[image:5.595.107.288.585.656.2]

0.9

0 5 10 15 t(s)

E

x

te

m

al dis

tur

ba

nc

e

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

[image:6.595.60.293.64.642.2]

0

Figure 2. Positions of nodes 1-4.

–2

0 5 10 15 t(s)

Po

sit

io

n

s (m)

–3

–4

–5

–6

–7

–8

–9

Node 1 Node 2 Node 3 Node 4

Figure 3. Positions of nodes 1-4.

10

0 5 10 15 t(s)

V

elo

ci

ti

es

(m

/s

)

Node 1 Node 2 Node 3 Node 4

–15 –10 –5 5

0

Figure 4. Velocities of nodes 1-4.

5. Conclusions

The H_ consensus problem has been investigated in this paper, for the high-order multi-agent systems with nonlinear dynamics. Sufficient conditions have been

given in the forms of LMIs, under which the H_ con-sensus problem can be solved. The parameter matrix in the consensus algorithm can be designed by solving two LMIs. If the nonzero eigenvalues of the laplacian matrix according to the network topology are identical, the pa-rameter matrix in the consensus algorithm can be de-signed by solving one LMI. The numerical simulation confirmed the proposed results.

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