SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 1(1); June (2020), pp. 35–39
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

No Free Lunch Principle in Agent Swarm Systems: One CaseStudy
Yunzhong Song1, Fengzhi Dai2, Huimin Xiao3, Shumin Fei4

School of Electrical Engineering and Automation, Henan Polytechnic University, 2001 Century Avenue, Jiaozuo, 454003, P.R. China
School of Electronic Information and Automation, Tianjin University of Science and Technology, 1038 Dagu Nanlu Tianjin, 300222, P.R. China
3
School of Computer and Information Engineering, Henan University of Economics and Law, 180 Jinshui Donglu Zhengzhou, 450046, P.R. China
4
School of Automation, Southeast University, 2 Sipai Lou, Nanjing, 210096, P.R. China
12ARTICLE INFO
Article History
Received 01 October 2019
Accepted 15 July 2020
Keywords
Free lunch principle
Agent swarm systems

ABSTRACTThis note comes with information flooding of the control action of the leader in leader follower
framework. In this situation, even observer for the leader velocity is built in a distributed style,the agent swarm system in a whole could not be classified into distributed one. One case study
example will be borrowed to demonstrate the truth of the no free lunch principle in agent swarmsystems.

© 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/).

Distributed observer
Leader-follower framework

While besides that, the moving agents, especially theactive leader one, must have its own velocity and added
control action. For second order leader-follower agent
swarms, the velocity of the leader agent was paid much
more attention and its observers design is necessarily
executed for its unavailable peculiarity for not all thefollower agents. To be distinct, the added control actionof the leader agent is assumed possible for all thefollower agents, and this assumption is overly strong to
be possible, for the availability of the added control
action of the leader node is always impossible unlessthere is some broadcasting mechanism about that signal.
This note will concentrate on the complexity balancebetween information and control of the agent swarm
systems, the simplicity of one side will deteriorate on theother side. That is no free lunch principle in agent swarm
systems. According to our knowledge, there is so little
papers on agent systems concentrate on no free lunch
principle.

1. Introduction
The agent swarm systems are recognized widely and thistrend is still flooding [1]. Networks of networks forleader-follower system was explored by Song [2], whichemphasizes interdependence among the agents. After that,the root node agents and the leaf node agents, could flock
into each other, were touched upon further [3]. For agent
swarm systems, on account of the interwoven between
information and control, complexities are enlarged
unprecedented [4],[5]. Leader-follower agent swarms
tracking control, was focused upon by several scientists
and thereafter [6],[7],[8],[9].
To be set apart from the main results anddemonstrate its importance, this note will contribute to
the neglected part of the dual roles of information andcontrol. It is well known that the distributed control isoverly dependent on the neighbor based informationexchange among agents, local control based regulation isalways the main theme of the agent swarm systems.

Corresponding author’s E-mail: songhpu@126.com, daifz@tust.edu.cn, xiaohm@huel.edu.cn, smfei@seu.edu.cn

35

The note will be arranged as follows. At start,
problem formulation will be proposed, then after themain results will be presented in the second part.
Followed will be some conclusions.

 x0 v0 , x0 ∈ R m ,=m=v0 u0 , u0 ∈ R ,y = x0 ,Where x0 , v0 and y are position velocity and the only
measurable variable, respectively. The dynamic of thefollower agents is described as:

2. Distributed Observers Design for LeaderFollower Agent Swarm systems
This part will begin with distributed observers design forleader-follower agent swam systems.

m= xi vi , xi ∈ R ,mvi = ui , ui ∈ R , i = 1,..., n,2.1. Problem formulation(2)

The distributed observer demonstrated was in thefollowing form:

Assume the agent swarm system is composed with one
leader agent and n follower agents. Here, the graph G isused to describe the graph of the follower agents, whichis undirected graph. We use G denotes the graph with theleader agent. Some agents in the follower ones willconnect the leader one with directed arcs. Obviously, theG can be named as connected if at least one of thefollower agents is connected with the leader one. To
highlight the main theme of this note, the topology of theagent system is assumed to be static. One form of theleader-follower agent topology of this note can be listed
as Figure 1. In Figure 1, we have two differentconnections from the leader agent to the follower agents,
one is characterized as solid line with arrow, the other arecharacterized as discontinuous lines with arrows. Here,the solid line is employed to represent the physicalproximity of the interacting agents, while thediscontinuous lines are for information flooding from theleader to the followers, broadcasting can be one kind ofmechanisms. There are other illustrated examples of
network topology of agent systems, we just give one ofthem in case of simplicity.

xi − x j , xi 0 =xi − x0 , xij =
ui =u0 − k (vi − vˆi )ui = −l ( ∑ aij xij + ∑ bi xi 0 ),i∈N 0 ( t )j∈ N i ( t )2vˆ =u0 − (l / k )( ∑ aij xij + ∑ bi xi 0 ),
 ii∈N 0 ( t )j∈ N i ( t )(3)Where N i (t ), N 0 (t ) denotes the neighbour agents of thefollower agent i and the leader agent. aij , bi is relatedwith the topology of G. If agent j is in theneighbourhood of agent i , then aij = 1 . Similarly, if
the follower agent i is connected with the leader agent
directly, then bi > 0 .

Comment 1: The protocol of (3) for (1) and (2) was put
forward by Hong et al in [7], concentrated by Sarras in[8], and replied by Hong et al again in [9]. To be noticed,
the team of Professor Hong has contributed a lot in agent
systems research, especially in new mechanism design ofcollaboration of agent individuals.
Comment 2: The seminal work of Hong et al in [7]declare that their protocol to be superior to the other ones
in its simplicity in observer structure, while the dynamicof the system is in second order, the observer in
distributed form can be in first order. That avoids theintricacy of the full order observer.
Comment 3: The communication between Hong et al
and Sarras et al was only constrained in the coefficient of
the protocol, without doubt, the selection of thecoefficient is vitally important, and to do that, theresearch was promoted a lot, that can be seen clearly in[8] and [9].
Comment 4: To be set apart from the already existed
results, no free lunch principle in leader follower agent
swarms would be introduced, and without losing thesimplicity of (3).

130(1)

2
Fig.1 Network consists of 4 nodes

2.2. The Main Results
Leader agent is assumed to be active. The dynamic of theleader agent is expressed as:36

Comment 5: That availability u0 for all the followerξ=η，
η = - l ( L + B)ξ - kη + ζ + u ,lζ = ( L + B )ξ + u , kagents was too strong to be true. In fact, some extra
mechanism must be taken to guarantee its availability,
broadcasting can be one of them, but broadcastingeasily suffer from information leakage and adversarial
attacks.
Comment 6: Control simplicity and information
complexity are always stand by each other. The gainof the control simplicity comes along with information
intricacy. That is the no free lunch principle in leaderfollower agent swarm systems. Our improved protocol
is like:xi − x j , xi 0 =xi − x0 , xij =
ui =u0 − k (vi − vˆi )ui = −l ( ∑ aij xij + ∑ bi xi 0 ) + u ,j∈ N i ( t )i∈N 0 ( t )2vˆ =u0 − (l / k )( ∑ aij xij + ∑ bi xi 0 ) + u , ij∈ N i ( t )i∈N 0 ( t )ξ  
Assume z = η  , then in a compact form, (7) can beζ  casted into
 0I0z =Fz = −lH −kI I  + δ , l− H0 0 kWhere H =+L B, δ =(0, u , u , 0,..., 0)T . For system (7),a control Lypunov function (CLF) can be constructedas V ( z ) = z T (t ) Pz (t ),(4)Where we assume that u= u0 + u , and u0 is virtuallypossible to drive the controller and the observer of thefollower agents, u is added in to compensate thisvirtually dealt scheme, u satisfies u ≤ M ≥ 0 .
Comment 7: The suggested protocol for the follower
agents about the controller function of the leader agent,
which is in the closed control loop, is feasible. This canbe said in the following reasons: At first the on board
sensor, actuator for the second integrator in the sameseries can be guessed on some sense, though with someerrors, and second, the error induced disturbance canagain be estimated by customized observer.
Lemma 1: For leader (1) and followers (2), with u = 0 ,k − I2
1 − I  , which can beI2
1k − II 22 
guaranteed positive if parameters l , k are properly


 kI

where P  I=
− k I 2t →∞Consequently,

(5)

t →∞

Ichosen. To be detail, they satisfy the following
inequalitiesl ≥ 2 / λmin , k ≥ 4 + λmax , where λmin , λmax are theminimum and maximum eigenvalues of positivematrix H .

the controller-observer pair (3) can produce

lim xi (t ) − x0=(t ) 0, lim vi (t ) − v0=(t ) 0;(7)

V ( z (t )) ≤ e − β tV ( z (0)) + β 0 M 2 .

Theorem 1: There exists a constant cM > 0 with

(8)

This implies (6).

lim cM = 0 , such thatM →0

lim xi (t ) − x0 (t ) ≤ cM , lim vi (t ) − v0 (t ) ≤ cM ;(6)

Proof:In
caseofconvenience,TT,η (v1 ,..., vn ) − v0 1,=
ξ ( x1 ,.., xn ) − x0 1=

sett →∞

t →∞

1and=
ζ k (v1 ,..., vn ) − kv0 1, where=
1 (1,...,1) ∈ R . ThenTTm0

the closed system can be turned into:

2Fig.2 Case study example

37

References

Here, we use Figure 2 as the case study example,
parameters are selected as:

1.
R. Olfati, R. Murray, Consensus problems in
networks of agents with switching topology and timedelays, IEEE Transactions on Automatic Control,
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hybrid agents via root block, Journal of Robotics,
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discrete dynamics, International Journal of Control,
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WSEAS Transactions on Systems and Control,9(2014),526-532
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Y. Hong, J. Hu, L. Gao, Tracking control for
multi-agent consensus with an active leader and variable
topology, Automatica, 42(2006) 1177-1182.
7.
Y. Hong, G. Chen, L. Bushenell, Distributed
observers design for leader-following control of multiagent netowrks, Automatica, 44(2008) 846-850.
8.
Sarras, P. Lahanier, J. Marzat, S. Bertrand,
Comments on ‘distributed observers design for leaderfollowing control of multi-agent netowrks, Automatica, 44:846-850, 2008’, Automatica, 105(2019)453-454.
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Y. Hong, G. Chen, L. Bushenell, Authors’ reply
to comments on “distributed observers design for leaderfollowing control of multi-agent netowrks”, Automatica,105(2019) 455-455.
10.
Y. Jia, Robust control with decoupling
performance for steering and traction of 4WS vehicles
under velocity-varying motion, IEEE Transactions on
Control Systems Technology, 8(2000) 554-569.
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Y. Jia, Alternative proofs for improved LMI
representations for the analysis and the design ofcontinuous-time systems with polytopic type uncertainty:
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u=u0 + u , u0 =cos t , u =0.02 cos t , l =40, b =0.2, k =
200.
40

30
2520
150

1

1

201

1x -x ,x -x ,v -v ,v -v

2

35

1050-50

100

200

300

400

500

600

700

800

900

1000

t/s

Fig.3 Position and velocity tracking errors of two followers

The corresponding results are demonstrated as Figure 3.
From simulation results, we can say the suggested
scheme is effective, which is with the virtual control
function estimation and perturbation term of amendment.
2.3. Further Comments
Inspirations can be always possible if we pay much more
attention to the interplay of information and control inagent swarm systems. No free lunch principle can be one
example of them, though sometimes it is subtle.
3. Conclusion

No free lunch principle can be a big help for us
to promote the research about leader-follower
agent swarm systems.
System intricacy, could not be easilymastered if we are kept away from the concretesystems. Case study like high performancevehicle driving [10],[11], can be the milestonesystems. Time varying delays and othernegative points, can also be the adversary
factors. Adaptive scheme in Jia [12] provided agood example. Refined control of leader
follower system according their roles of theleader agent, was recommended also7. So,balance between information and control can be
an eternal topic in agent swarm systems.
Acknowledgements
This work is partially supported by NSFC Grant(61340041 and 61374079) and the Project-sponsored by
SRF for ROCS, SEM to Yunzhong Song as well as
NaturalScience
FundofHenanProvince(182300410112). Special thanks for Professor Yingmin
Jia of Beihang University and Professor Masanori
Sugisaka for their encouragement and recommendation.

38

Authors Introduction
Dr. Yunzhong Song

Dr. Fengzhi Dai

Dr. Huimin Xiao

Dr. Shumin Fei

He received his PhD degree from theDepartment of Systems Science,
Zhejiang University, China in 2006.
He is currently a full professor inHenan Polytechnic University, China.

He received his PhD degree in Oita
University in Japan in 2004. He iscurrently an associate professor in
Tianjin University of Science andTechnology, China.

He is a full Professor of Henan
University of Economics and Law inChina. His main interests in research ison hybrid control, switching control
and large scale systems.

He is a full Professor of Southeast
University. He focuses himself oncomplex systems control.

39