Less Conservative Consensus of Multi-agent Systems with Generalized Lipschitz Nonlinear Dynamics

International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 3 Issue 7 ǁ July 2015 ǁ PP.15-21
www.ijres.org 15 | Page
Less Conservative Consensus of Multi-agent Systems with
Generalized Lipschitz Nonlinear Dynamics
Jian Li1
, Zhiyang Wu1
, Fang Song1,2
1.Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science,
Shanghai, 201620, P. R. China
2. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, 150001, P. R. China
Abstract : This paper addresses the problem of consensus of multi-agent systems with general Lipschitz
nonlinear dynamics. The goal of this note was to introduce a design approach. We take advantage of the
structure of the part of the non-linearity and inequality scaling technology. The advantage of the developed
approach was that it was significantly less conservative than other previously published results for Lipschitz
systems. A numerical example was presented to show the superiority of this letter.
Keywords- Consensus, Less Conservative Condition, Lipschitz Nonlinear Dynamics, Multi-agent Systems
I. Introduction
Recently, consensus for a group of agents has became an important problem in the area of cooperative
control of multi-agent systems and has been widely investigated by numerous researchers, due to its potential
applications in such broad areas as satellite formation flying, sensor networks, and cooperative surveillance[1,2].
In [3] a distributed observer-type consensus protocol based on relative output measurements was proposed. In
[4], a general framework of the consensus problems for networks of dynamic agents with fixed or switching
topologies and communication time delays was established. Relative-state distributed consensus protocols was
established in [5]. A distributed algorithm was proposed in [6] to achieve consensus in finite time. However the
provided synthesis conditions which infeasible for systems with big Lipschitz constants restrict all these
approaches, such as [6],[7],[8]. Then in [18] the authors introduced a generalized version of the Lipschitz
condition which includes some structural knowledge of the system non-linearity.
This paper considers the problem of consensus of multi-agent systems with general Lipschitz nonlinear
dynamics under a fixed topology. Consensus of multi-agent systems with general linear dynamics was researched
in [3],[5],[14],[15],[19]. Especially, different static and dynamic consensus protocols are established in
[3],[5],[14] which is difficult to tackle and implement by each agent in distributed fashion. Then, we design a
distributed consensus protocol for the relative states and an adaptive law for adjusting the coupling weights
between neighboring agents which were first proposed in [6]. One contribution of this paper is to extend the
result of [3],[5],[6],[14],[20] to a generalized Lipschitz nonlinear system which includes the classical Lipschitz
system as a special case. The goal of this note was to introduce a design approach. We take advantage of the
structure of the part of the non-linearity and inequality scaling technology. The advantage of the developed
approach was that it was significantly less conservative than other previously published results for Lipschitz
systems.
The rest of this paper was organized as follows. Some useful preliminaries results were reviewed in Section 2.
The consensus problems of multi-agent systems with generalized Lipschitz non-linear dynamics using distributed
adaptive protocols were investigated in Section 3. Extensions to Lipschitz non-linearity dynamics were studied in
Section 4. In Section 5, we use a simulative example to illustrate the applications of our consensus algorithm.
Section 6 concludes the paper.
II. Preliminaries
A. Graph theory notions
Let n n
R  be the set of n n real matrices, n n
C 
be the set of n n complex matrices. Denote by
1 ( P
R1 ) a column vector with all entries equal to one. The superscript T means transpose for real
matrices. PI represents the identity matrix of dimension P . For any matrices, if not explicitly stated, were
assumed to have compatible dimensions. Diag 1( ,..., )nA A represents a block-diagonal matrix with
matrix , 1,...,iA i n , on its diagonal .The matrix inequality ( )A B  means that A and B were square
Hermitian matrices and that A B was positive (semi-) definite. A B denotes the Kronecker product of

Less Conservative Consensus of Multi-agent Systems with Generalized Lipschitz Nonlinear Dynamics
matrices A and B . For a vector x , let x  denote its 2-norm. An undirected graph ( , )G   ,where
1{ ,..., }Nv v  was the set of nodes( i .e ,agents),and     was the set of edges( i .e,
communication links).An edge ( , )( )i jv v i j means that agents iv and jv can obtain information from each
other. A path between distinct nodes iv and lv was a sequence of edges of the form 1( , ), ,..., 1k kv v k i l   .
An undirected graph was connected if there exists a path between every pair of distinct nodes, otherwise was
disconnected. A directed graph G was a pair ( , )  , where 1{ ,..., }Nv v  was a non-empty finite set of notes
and     was the set of edges. For an edge ( . )i jv v , node iv was called the parent node ,node jv was
the child node, .i jv v were adjacent. 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 a directed path to every other node in the graph.
A directed graph was strongly connected if there was a directed path from every node to every other node. A
directed graph has a directed spanning tree if it was strongly connected , but not vice versa.
The adjacency matrix [ ] N N
i jA a R 
  associated with the directed graph G was defined by 0, 1ii i ja a 
if ( , )j iv v  and 0i ja  otherwise. Adjacency matrix A associated with the undirected graph was defined
by 0, 1ii ji i ja a a   if ( , )j iv v  and 0ji i ja a  otherwise. Laplacian matrix [ ] N N
ijL L R 
  was
defined as ii ijj i
L a
  and ,ij ijL a i j   .
B. Lemmas:
Lemma 1[8,9]: Zero was an eigenvalue of L with 1 as a right eigenvector and all non-zero eigenvalues
have positive real parts .Furthermore ,zero was a simple eigenvalue of L if and only if the graph G has a
directed spanning tree .
Lemma2[17]: For any vectors , n
a b R and scalar 0  ,we have 1
2 T T T
a b a a b b 
  .
III. Consensus of multi-agent systems with generalized Lipschitz non-linear dynamics .
Consider a group of N identical agents with generalized Lipschitz nonlinearity dynamics. The dynamics of
the i -th agent were described by
( )i i i i ix A x Df Hx Bu   1, . . . ,i N (1)
where
n
ix R was the state,
p
iu R was the control input, and , , ,A B D H were constant matrices with
compatible dimensions, and nonlinear function ( )iDf Hx was assumed to satisfy the following general form
Lipschitz condition which was defined in [18] as follows
T T
f Wf x Rx    (2)
where the W and R were positive definite symmetric matrices . ( ) ( ),i j i jf f Hx f Hx x x x     . It can
be seen that any non-linear function ( )Df Hx was a Lipschitz function with max min( )/ ( )R W   which
called Lipschitz constant. Any Lipschitz non-function ( )Df Hx can satisfy condition (2) with 2
,W I R I  .
(2) can also be rewritten as a generalized form W RK f K x    , WK 、 RK were two positive definite matrices.
,W RK W K R  .
The communication topology among the agents was represented by an undirected graph ( , )G   , where
{1,..., }N  was the set of nodes( i .e ,agents), and     was the set of edges( i .e, communication links).
An edge ( , )( )i jv v i j means that agents iv and jv can obtain information from each other. A path between
distinct nodes 1v and lv was a sequence of edges of the form 1( , ), 1,..., 1k kv v k l   . An undirected graph
was connected if there exists a path between every pair of distinct nodes , otherwise was disconnected.
A variety of static and dynamic consensus protocols have been proposed to reach consensus for agents with
dynamics given by (1) e.g, in [1]–[5]. For instance, a dynamic consensus protocol based on the relative states
between neighboring agents was given in [6] as follows
(3)
1
( )
N
i i j i j i j
j
u F c a x x

 ( ) ( )T
ij i j i j i j i jc k a x x x x   

where i ja was ( ,i j )-th entry of the adjacency matrix A associated with G , ij jik k were positive
constants, i jc denotes the time-varying coupling weight between agents iv and jv with
(0) (0)i j i jc c ,and p n
F R 
 and n n
R 
 were the feedback gain matrices.
Theorem 1 :Solve the LMI :
(4)
to get a matrix 0P  and a scalar 0  .Then the N agent described by (1) reach global consensus under the
protocol (3) with 1T
F B P
  and 1 1T
P BB P 
  for any connected communication graph G .
Proof: As argued in the Section 2 ,it follows that ( ) ( ), 0.i j jic t c t t   Using (3) for (1),we obtain the
closed-loop network dynamics as
( ) ( )T
ij i j i j i j i jc k a x x x x    1,...,i N (5)
Letting and 1[ ,..., ] ,T T T
Ne e e we get [( (1/ )11 ) ]T
N ne I N I x   .We can see that e satisfies the following
dynamics:
( ) ( )T
ij i j i j i j i jc k a e e e e    1,...,i N (6)
where ij ijc c   and  was a positive constant. Consider the Lyapunov function candidate
(7)
the time derivative of V along the trajectory of (8) as follows:
(8)
According to the equivalent form of Lipschitz condition (2) and Lemma 2 we obtain
(9)
. .1
1
1 1 1,
2
N N N
i j i jT
i i
i i j j i i j
c c
V e P e
k

   
   
 
 
1
1
0
T T T T
AP PA DW D BB PH
HP R
 



   
 
 
1
( ) ( )
N
i i i i j i j i j
j
x Ax Df Hx BF c a x x

   
1 1
1
( ) ( ) ( ) ( )
N N
i i i i ij i j i j
j j
e Ae Df Hx Df Hx c a BF e e
N

 
       
. .1
1
1 1 1,
2
N N N
i j i jT
i i
i i j j i i j
c c
V e P e
k

   
   
 
 
1 1 1 1
1 1 1 1 1
1
2 2 2 [ ( ) ( ) ( ) ( )]
N N N N N
T T T T
i i ij i j i i i i j
i i j i j
e P Ae L e P BB P e e P D f Hx f Hx f Hx f Hx
N
   
    
        
1 1
1 1 2 2
1 1 1
2 [ ( ) ( )] 2 [ ( ) ( )]
1
[ ( ) ( )] [ ( ) ( )]
T T
i i i i i i
T T T
i i i i i i
e P D f Hx f Hx e P DW W f Hx f Hx
e P DW D P e f Hx f Hx W f Hx f Hx


 
  
  
   
1 1 1 1T T T T
i i i ie P DW D P e e H RHe

  
 
1 1 1 1
[ ] , 0T T T
i ie P DW D P H RH e 

  
   

As
1
0
N
T
i
i
e

 , then
(10)
Set up 1
ˆi ie P e
 , and ˆ [ ,..., ]T T T
i Ne e e , based on (9)and(10)we can draw from (8)that
(11)
Set up N N
U R 
 be the unitary matrix satisfying 2(0, ,..., )T
NU LU diag     ,set up
1
ˆ[ ,..., ] ( )T T T T
N NU I e     , then 1
1 [(1 / ) ] 0T N P e 
   ,from (11)we have
（12）
We choose sufficiently big  so that 2 , 2,...,i i N   , then
(13)
We get inequality (4) from (14) by using the Schur complement Lemma[12]. we know that ( ) 0.S  
1 0V  , so 1( )V t and every ijc were bounded. And from (5) we know that ijc was monotonically increasing,
so ijc converge to finite value and ijc as well. By using LaSalle-Yoshizawa Theorem [13] we
get lim ( ) 0t S   ,so lim 0, 2,...,t i i N   , 1 0  . In conclusion, lim , ( ) 0t e t  . so, we achieve
the proof .
Remark 1: Equation (2) was a generic form Lipschitz condition. Any Lipschitz non-function ( )f Hx can
satwasfy equation (2) with 2
,W I R I  which was first define in [18] includes some structural knowledge of
the system non-linearity for observer design. Then, it was used to consensus of multi-agent systems with
non-linear dynamics
Deduction to Lipschitz non-linearity dynamics
This subsection considers the consensus problem of the agents in (1) under the adaptive protocol (3). The
communication topology among the agents was represented by an undirected graph. The nonlinear function
( )iDf Hx was assumed to satisfy the Lipschitz condition with a Lipschitz constant 0  , i.e.
(14)
Theorem 2 :Solve the LMI :
(15)
To get a matrix 0P  and a scalar 0  .Then the N agent described by (1) reach global consensus under
the protocol (3) with 1T
F B P
  and 1 1T
P BB P 
  given as in Theorem 1 for any connected
communication graph G .The proof of Theorem 1can be shown by following similar steps in proving Theorem 2
Remark 2:Theorem 2 tends Theorem 1 to the case with Lipschitz non-linear dynamics. When
2
,W I R I  , Theorem 1will reduce to Theorem 2.Theorem 2 shows that the agents in (1) under the adaptive
protocol (3) can also get consensus with the nonlinear function satisfying the Lipschitz condition.
1
2
1
[ 2 ] ( )
N
T T T T T
i i i
i
AP PA DW D PH RHP BB S    



     
1 1
ˆ ˆ[ ( ) 2 ]T T T T T
i N ie I AP PA DW D PH RHP L BB e 


      
1
1 1
1
[ ( ) ( )] 0
N N
T
i i j
i j
e P f Hx f Hx
N

 
  
( ) ( ) ( )f Hx f Hy H x y     
0
T T T T
AP PA DD BB PH
HP I
  
 
   
 
 
1
1
1
[ ( ) 2 ]T T T T T
NV I AP PA DW D PH RHP BB   


       
1
2
1
[ 2 ] ( )
N
T T T T T
i i i
i
AP PA DW D PH RHP BB S    



     
1 1
2T T T T
AP PA DW D PH RHP BB 


   
1 1
0T T T T
AP PA DW D PH RHP BB 


     
, n
x y R 

IV. Numerical comparisons and simulation examples
In order to evaluate our design methodology, we present in this Section a numerical example to validate the
effectiveness of the theoretical results. In fact, we show through this example that our approach was the best
method that takes into account the structure of the nonlinearity in detail.
Consider a network of single-link manipulators with revolute joints actuated by a DC motor. The dynamics
of the i-th manipulator was described by (3) with
[0 21.6 0 0]T
B 
[0 0 0 0.333]T
D  
[0 0 1 0]H 
3( ) [0 0 0 0.333sin( )]T
i iDf Hx x 
Clearly, ( )if Hx satisfies (4) with a Lipschitz constant 0.333  , solving the LMI (9) by using the LMI
toolbox of Matlab gives the feasible solution as
Fig 1 .Communication topology.
Fig 1 .Communication topology.
1
2
3
4
i
i
i
i
i
x
x
x
x
x
 
 
 
 
 
 
0 1 0 0
48.6 1.25 48.6 0
0 0 0 10
1.95 0 1.95 0
A
 
   
 
 
 
3239 500 2579 6459
500 77 398 998
2579 398 2054 5144
6459 998 5144 12881
 
   
   
 
 
[ 56.68 8.767 45.15 113.1]F    
0.8088, 0.3634  
Xi1
Xi3
Xi4Xi2

Fig 2. States of the eight manipulators under (3)
Fig 3 . Coupling weights i jc .
In order to illustrate Theorem 2 ,we let the communication graph G as in fig 1. G was connected and
undirected. There were eight manipulators. In other words, , 1,...,8i j  in (1) , 1ij jik k  , (0) (0)ij jic c .
The coupling weighs were as Fg 2 . They converge to steady values. The states of the manipulators which
satisfies (3) were as Fg 3. They also get consensus.
Table1 .Comparwason of maximum Lipschitz constant for various LMI design techniques
Method LMI(14)of L. Zhongkui(2013) LMI(21)
max 0.34
6
10
We focus our study on the conservation of the states LMI approach. The comparison between different LMI
conditions in Table 1 shows that our approach is significantly less conservative than other previously published
results for Lipschitz systems.
V. Conclusions
We have studied the adaptive consensus problem of multi-agent systems with generalized Lipschitz
nonlinear dynamics. A less conservative adaptive consensus condition has been proposed by carefully
considering the structure information of nonlinearities and using a sharp inequality to deal with the generalized
Lipschitz condition. It can significantly reduce the conservatism in some existing adaptive consensus results for
Lipschitz nonlinear system，which is verified through a numerical example.
VI. Acknowledgement
This work was partly supported by the State Key Laboratory of Robotics and System (HIT) under Grant
SKLRS-2014-MS-10, the Jiangsu Provincial Key Laboratory of Advanced Robotics Fund Projects under Grant
JAR201401, and the Foundation of Shanghai University of Engineering Science under Grant nhky-2015-06,
14KY0130 and 15KY0117.
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