SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial Life
Vol. 9(1); June (2022), pp. 72–76
ISSN (Online): 2352-6386; ISSN (Print): 2405-9021
https://alife-robotics.org/jrnal.html

Research Article

Bipartite Consensus of Linear
Systems with Encoding-Decoding

Discrete-time

Multi-agent

Jintao Hu, Yingmin Jia, Yaxin Li

The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang
University (BUAA), 37 Xueyuan Road, Haidian District, Beijing, 100191, China

ARTIC LE INFO
Article History
Received 20 November 2020
Accepted 10 August 2021

Keywords
Multi-agent systems
signed undirected graphbipartite consensus
structural balance/unbalance
encoding and decoding

ABS TRACTIn this paper, the consensus problem of general linear discrete-time multi-agent systems with thecooperative-antagonistic interactions and limited communication bandwidth is investigated, anda bipartite consensus con-trol protocol is proposed for each agent, which is given in terms of thestate of its dynamic encoder and decod-ers with uniform quantizer. It’s demonstrated that thebipartite consensus can be realized, if the signed undi-rected graph is connected and structurallybalanced/unbalanced. Furthermore, the explicit form of the conver-gence rate is given. Numerous
simulations are presented to illustrate the feasibility of the control scheme used on this system.

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

and sends code to neighbors. The receiver of theneighbors gets the code and decodes it to estimate thestate of the sender2-4.
In addition, most of the works in the literatureassume that the relationship between agents iscooperative. However, cooperation often coexists
with antagonism actually. In this case, the concept ofstructural balance/unbalance is given. By utilizing theproperty of structural balance/unbalance ofcommunication network, the bipartite consensus isintroduced where all gents attain agreement
concerning a value, which is same for all in modulus
but not in sign5-7. However, there is no research dateon the bipartite consensus of MASs with the encoding
and decoding mechanism.
Motivated by the above discussion, we consider aconsensus control protocol which is designed basedon a dynamic encoding-decoding mechanism. Theagents can achieve bipartite consensus under thecontrol scheme. Moreover, the clear form of theconvergence rate is given.
The following notations will be used in this paper:

1. Introduction
Recently, the consensus problem in the field of multiagent systems (MASs) has received much attention
due to its broad application prospects, including
attitude synchronization of spacecraft1, distributed
computation, market and economic theory in thecomplex environment, self-driven motion ofbiological particles and wireless multi-sensor network,etc.
In many relevant works, it’s assumed that theagents can obtain exact information of the states of theneighbors by local communication. This means thatthe communication bandwidth is infinite. However, inpractice, the communication ability of the network islimited. Thus, the encoding and decoding scheme isintroduced to deal with this problem. At each stage,
the sender of the agents encodes the quantized state

Corresponding author’s E-mail: hujintao@buaa.edu.cn, ymjia@buaa.edu.cn www.buaa.edu.cn

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1nn-dimensional column vector whoseelements
are all 1
Euclidean norm of a vector or matrix ∞Infinite norm of a vector or matrix
ρ ()Spectral radius of a matrix
sgn()
Sign functionthat the information transmitted between agents only
can be binary number, but not exact value. Thus, thecorresponding encoder and decoder are designed. The
encoder of agent i is proposed as follows:φˆi (k ) =0l (k − 1) Z i (k ) + Aφˆi (k − 1), φˆi (0) =ˆxi (k ) − Aφi (k − 1))
 Z i (k ) = Q(l (k − 1)(3)Where φˆi (k ) and Z i (k ) are the internal variable andoutput of the encoder, respectively. Apart from this,Q(⋅)= [q (⋅), , q (⋅)]T represents the product quantizer.
l (k ) = l0 r k is the scaling function, where l0 ∈  andr ∈ (0,1) , which prevents the quantizer fromsaturation. The decoder of agent j which receives theinformation from agent i is designed as follows:ϕˆ ji (0) = 0ˆ(4)l (k − 1) Z i (k ) + Aϕˆ ji (k − 1)
ϕ ji (k ) =2. Preliminaries and Problem Formulation
2.1 Signed Graph
Let  (  ,  ,  ) be a signed undirected graph, where = {v1 , v2 , , vN } is a set of N agents with idenoting the ith agent,  ⊆  ×  is the edge set ofagents, and = [aij ] ∈  N × N with the signed elements
is the weighted adjacency matrix of  . The set of
agents who can communicate with agent i is denoted{ j ∈  | (i, j ) ∈  } . The Laplacian matrix of=by  i is represented by =−  , where = diag (∑ j∈1 a1 j ,  , ∑ j∈ N aNj ) . A signedgraph  is structurally balanced if the nodes in  are
divided into two parts {1 , 2 } , where 1  2 =  and1  2 = ∅ , bring aij ≥ 0 ∀i, j ∈ p ( p ∈ {1, 2}) ,aij ≤ 0 ∀vi ∈ ,q , p ≠ q ( p , q ∈ {1, 2})
p , v j ∈ otherwise, the signed graph  is structurally
unbalanced.

Where ϕˆ ji (k ) which is connected with agent i is theinternal variable of the decoder of agent j .
The objective of this paper is to propose a
distributed control scheme based on dynamic
encoding and decoding mechanism, such that themulti-agent system realizes bipartite consensus whencommunicationgraph
isstructurally
balanced/unbalanced.
For this purpose, we consider the following
protocolui (k ) K ∑ j∈ aij (φˆi (k ) − sgn(aij )ϕˆij (k )) (5)=Where K ∈  1×n is the control gain to be designedlater.
Definition 1. The consensus of multi-agent systems
can be achieved, if there exists the control law suchthatlim xi (k ) − sgn(aij ) x j (k ) =0

2.2. Problem FormulationiIn this paper, we consider the bipartite consensus
control for a multi-agent system with the discrete-time
dynamics
xi (k +=1) Axi (k ) + Bui (k )(1)
Where xi (k ) ∈  n and ui (k ) ∈  , respectively,
represent the state and input of the agent i at time k .
A ∈  n×n and B ∈  n×1 denote the state and inputmatrix.
A uniform quantizer is utilized to quantize the stateof agent. The quantizer q ( x) is defined as follows:−1 / 2 ≤ x ≤ 1 / 20,
b,(2b − 1) / 2 < x ≤ (2b + 1) / 2,
q( x) = =b 1, , L − 1
 L,x > (2 L − 1) / 2
−q (− x), x < −1 / 2k →∞For any initial state xi (0) , ∀i ∈ [1, N ] .
Assumption 1. xi (0) ∞ ≤ Cx , ∀i ∈ [1, N ] and Cx canbe obtained by all agents.
Assumption 2. There exists K ∈  1×n such that theA − λi () BK , i =2, , Neigenvalues
ofaredistributed inside the open unit circle of complex
plane.
3. Main Results(2)
Obviously, the level of the quantizer is 2 L + 1 .
When x ≤ L + 1/ 2 , the quantization error of theuniform quantizer satisfies x − q ( x) ≤ 1/ 2 . Assume

3.1. Structurally balanced graphTo get the main result, we need divide the agents into
two groups d i ∈ {1, −1} i = 1, , N , where the agents73

4. Simulation and analysis

are in the same group when they are cooperating, i.e.,
aij > 0 .
Lemma 1. For a connected signed and structurally
balanced graph, there always exists a diagonal matrix
D = diag (d1 , , d N ) such that all elements of DD0 λ1 () < λ2 () ≤  ≤ λN () .
are nonnegative, and =Lemma 2. For any P ∈  n×n and ε > 0 , we can
P k ≤ M η k , ∀k ≥ 0obtainthat,
wheren −1=η ρ ( P) + ε P .
=Mn (1 + 2 / ε ) andTheorem 1. Consider the multi-agent system (1) withthe structurally balanced and connected signed
undirected communication network. Suppose that
Assumption 1 and 2 hold. Select appropriate controlgain K
such that ρ ( J ( K )) < 1 , whereJ ( K ) =diag ( A, A + λ2 BK , A + λN BK ) . For anyr ∈ ( ρ ( J ( K )),1) and σ ∈ (0, (r − ρ ( J ( K ))) / J ( K ) ) ,let=M (K , r)

A ∞ + 2d * BK ∞2rConsider a signed connected undirected andstructurally balanced communication graph  with 4
nodes and the adjacency matrix
0.2 −0.2 −0.2  0 0.20−0.10 = −0.2 −0.1 00.1 00.10  −0.2The state and input matrix are assumed to be
A= B= 1 . The initial states are chosen asxi (0)= i,=i 1, , 4 .The control gain K , the initialvalue l0 and factor r of scaling function are taken as
2, 4 and 0.995, respectively. Besides, we take L = 1 .
The evolution of the states of these multi-agents ispresented in Fig.1. It can be seen that the bipartite
average consensus is achieved. Then, we chooser = 0.99 , and the corresponding result is shown inFig.2. Combine these two figures, we can find that thesmaller the factor r , the faster the state converges.

2+n3/2 λN2 M N BK ∞r 2 (1 − η )and for any L > M ( K , r ) − 1/ 2 , let
A Cx 4r 2 Cx (1 − η )l0 > max{ ∞ ,}L + 1/ 2 nλN BK ∞Then under the proposed control scheme (5) given by(2), (3) and (4), the closed-loop system realizes thebipartite average consensus=x (k ) d=d x (0), i 1, , Nlimi ∑ j =1 j j→∞ iNkProof. For the specific proof process, please refer toTheorem 1 in [8].
3.2. Structurally unbalanced graphFig.1. States of the MASs with structurally balanced
communication network and r = 0.995 .

In order to deal with the condition of MASs withstructurally unbalanced communication graph, weneed the following lemma.
Lemma 3. If a connected signed graph is structurally
unbalanced, the corresponding Laplacian matrix satisfies 0 < λ1 () ≤ λ2 () ≤  ≤ λN () .
The main result in this subsection is similar toTheorem 1. The key difference is that the signed
average state error ∆(k ) is substituted by the state
X (k ) of MASs, and the conclusion turns into
lim k →∞ xi (=k ) 0=, i 1, , N . Besides, because of thestructurally unbalanced communication graph, theblock diagonal matrix J ( K ) is changed withdiag ( A + λ1 BK , A + λ2 BK , , A + λN BK ) . Pleaserefer to theorem 1 for specific proof.

Fig.2. States of the MASs with structurally balanced
communication network and r = 0.99 .

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With regard to structurally unbalanced graph  ,the adjacency matrix is chosen as2.

 0 0.2 -0.2 -0.2  0.2 0 0.10 
= 
-0.2 0.1 00.1 
0.2
00.10 
3.

The evolution of the states of these multi-agents which
eventually converge to zero is shown in Fig.3.

4.

5.
6.

7.
Fig.3. States of the MASs with structurally unbalanced
communication network and r = 0.995 .

8.

5. Conclusion
The bipartite consensus control problem of multiagent systems with limited communication rate basedon the signed connected undirected network topology
is solved in this paper. For structurally
balanced/unbalanced topology, it’s demonstrated thatthe consensus can be realized by employing theproposed control scheme. In addition, the consensus
convergence rate is given explicitly. Future research
topics about the coding-decoding-based informationtransmission problem may include these aspects ofcommunication
delay,
unmeasurable
state
information, event-triggered sampling and packet loss,
etc.

Authors IntroductionMr. Jintao Hu

He received the B.S. degree inautomation
fromChongqing
University, China, in 2019. He iscurrently pursuing the Ph.D. degreewith the Seventh Research Divisionand the Center for Information andControl, School of Automation
Science and Electrical Engineering,
Beihang University, China. His
research interests include consensus control of multiagent systems and event-triggered control.

Acknowledgements
This work was supported by the NSFC (62133001,61520106010) and the National Basic Research
Program of China (973 Program: 2012CB821201).
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Prof. Yingmin Jia

He received the B.S. degree incontrol theory from Shandong
University, China, in 1982, and theM.S. and PhD degrees both incontrol theory and applications fromBeihang University, China, in 1990
and 1993, respectively. Then, hejoined the Seventh Research
Division at Beihang University where he is currently
Professor of automatic control. His research interests
include robust control, intelligent control and theirapplications in robot systems.
Mr. Yaxin Li

He received the B.S. degree inautomation
fromShandong
University, China, in 2018. He iscurrently pursuing the Ph.D. degreewith the Seventh Research Divisionand the Center for Information andControl, School of Automation
Science and Electrical Engineering,
Beihang University, China. His
research interests include distributed control andformation control of multi-agent systems.

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