Consensus and swarming behaviors for a proportional-derivative system with a cut-off interaction

Jun Wu; Yicheng Liu

doi:10.1515/auto-2020-0068

Published by De Gruyter (O) May 27, 2021

Consensus and swarming behaviors for a proportional-derivative system with a cut-off interaction

Konsens-Schwarmverhalten für ein proportional-derivatives System mit einer Cut-off Interaktion

Jun Wu
Jun WU received her M. S. and Ph. D. degrees from the School of Mathematics and Econometrics, Hunan University, Changsha, Hunan. She has visited Canada as a visiting scholar at 2016. Her research areas include Fixed point theory, Particle system, Cybernetics, Artificial intelligence, Functional Differential equation and so on. Now she is an associated professor of Changsha University of Science and Technology.
and Yicheng Liu
Yicheng Liu received his M. S. and Ph. D. degrees from the Department of Mathematics, National University of Defense Technology, Changsha, Hunan. He has visited Canada as a visiting scholar. His research areas include Multi-system, Cluster, Cybernetics, Artificial intelligence, Differential equation dynamical system and so on. Now he is a professor of National University of Defense Technology.

From the journal at - Automatisierungstechnik

https://doi.org/10.1515/auto-2020-0068

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Abstract

This paper presents a proportional-derivative protocol for the consensus problem of a class of linear second-order multi-agent systems with local information transmission. The communication topology among the agents is switching and agents receive information within a critical bounded distance. As new observations, we show that the desired protocol system undergoes consensus and swarming behaviours when 1 is a simple eigenvalue of the adjacency matrix. In this case, both final velocity and final relative position are formulated. Simulation results show the effectiveness of the proposed protocol.

Zusammenfassung

In diesem Beitrag werden für lineare Multiagentensysteme zweiter Ordnung mit lokaler Informationsübertragung Lösungen für das Konsens-Schwarmproblem vorgestellt. Die Kommunikationstopologie zwischen den Agenten ist schaltend und die Agenten empfangen Informationen innerhalb einer kritischen beschränkten Distanz. Für den Fall, dass 1 ein einfacher Eigenwert der Adjazenzmatrix ist, werden Bedingungen für das Konsens-Schwarmverhalten abgeleitet. Dafür werden sowohl die Endgeschwindigkeit als auch die Endposition angegeben. Simulationsergebnisse zeigen die Leistungsfähigkeit des vorgeschlagenen Verfahrens.

Keywords: proportional-derivative system; consensus emergence; swarming

MSC 2010: 34A36; 34K06; 34M03

Schlagwörter: Proportional-derivatives System; Konsensfindung; Schwarm

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11671011

Funding statement: This work was supported by the National Natural Science Foundation of China (11671011).

About the authors

Jun Wu

Jun WU received her M. S. and Ph. D. degrees from the School of Mathematics and Econometrics, Hunan University, Changsha, Hunan. She has visited Canada as a visiting scholar at 2016. Her research areas include Fixed point theory, Particle system, Cybernetics, Artificial intelligence, Functional Differential equation and so on. Now she is an associated professor of Changsha University of Science and Technology.

Yicheng Liu

Yicheng Liu received his M. S. and Ph. D. degrees from the Department of Mathematics, National University of Defense Technology, Changsha, Hunan. He has visited Canada as a visiting scholar. His research areas include Multi-system, Cluster, Cybernetics, Artificial intelligence, Differential equation dynamical system and so on. Now he is a professor of National University of Defense Technology.

Appendix A

Proof of Lemma 3.1.

To formulate the fundamental solution of (3.3), we consider the following three cases:

Case I.λ2(1−μ2)>4σ. In this case, each factor in (3.4) has two distinct negative real roots corresponding to μi (i=2,3,⋯,m0), denoted as zi1, zi2 and zi1<zi2. Let

(A.1)φi(t)=zi2ezi1t−zi1ezi2tzi2−zi1,ψi(t)=ezi2t−ezi1tzi2−zi1,i=2,3,⋯,m0,

where zi1=−λ(1−μi)−λ2(1−μi)2−4σ(1−μi)2 and zi2=−λ(1−μi)+λ2(1−μi)2−4σ(1−μi)2.

Case II.λ2(1−μm0)<4σ. In this case, each factor in (3.4) has two imaginary roots corresponding to μi (i=2,3,⋯,m0), denoted as αi±iβi. Let

(A.2)φi(t)=eαit[cos(βit)−sin(βit)βi],ψi(t)=eαitsin(βit)αiβi,i=2,3,⋯,m0,

where αi=−λ(1−μi)2 and βi=4σ(1−μi)−λ2(1−μi)2.

Case III. There is k0 such that λ2(1−μk0)=4σ. In this case,

(A.3)φk0(t)=1+λ(1−μk0)2te−λ(1−μk0)2t,ψk0(t)=te−λ(1−μk0)2t.

In summary, we take

(A.4)φi(t)=zi2ezi1t−zi1ezi2tzi2−zi1,ifλ2(1−μi)>4σ,cos(βit)−sin(βit)βie−λ(1−μi)2t,ifλ2(1−μi)<4σ,1+λ(1−μi)2te−λ(1−μi)2t,ifλ2(1−μi)=4σ,

and

(A.5)ϕi(t)=ezi2t−ezi1tzi2−zi1,ifλ2(1−μi)>4σ,sin(βit)αiβie−λ(1−μi)2t,ifλ2(1−μi)<4σ,te−λ(1−μi)2t,ifλ2(1−μi)=4σ.

Then

(A.6)Φ(t)=φ2(t)Ip20⋯00φ3(t)Ip3⋯0⋮⋮⋱⋮00⋯φm0(t)Ipm0,

and

(A.7)Ψ(t)=ψ2(t)Ip20⋯00ψ3(t)Ip3⋯0⋮⋮⋱⋮00⋯ψm0(t)Ipm0

are two fundamental solution matrices of the second equation in (3.3). Thus

X(t)=T0100Φ(t)T0−1X0+T0t00Ψ(t)T0−1V0,V(t)=T0000Φ˙(t)T0−1X0+T0100Ψ˙(t)T0−1V0,

is the fundamental solution of the homogeneous system (3.1).

Appendix B

Proof of Lemma 3.2.

Consider the relationship of λ and σ undergoing three cases: λ2(1−μ2)>4σ, λ2(1−μm0)<4σ and λ2(1−μk0)=4σ. We will finish the proof by next three steps.

Step I. If λ2(1−μ2)>4σ, from (A.1), we see that, for all i=2,3,⋯,m0,

|φi(t)|≤λ(1−μi)λ2(1−μi)2−4σ(1−μi),|ψi(t)|≤2λ2(1−μi)2−4σ(1−μi)

and

|φ˙i(t)|≤2σ(1−μi)λ2(1−μi)2−4σ(1−μi),|ψi˙(t)|≤λ(1−μi)λ2(1−μi)2−4σ(1−μi).

Let c=λ(1−μi)−λ2(1−μi)2−4σ(1−μi)2,

K=maxi=2,⋯,m0{max{λ,2σ}λ2−4σ1−μi,2λ2(1−μi)2−4σ(1−μi)},

then

max{‖Φ(t)‖,‖Φ˙(t)‖,‖Ψ(t)‖,‖Ψ˙(t)‖}≤Ke−ct,t>0.

Step II. If λ2(1−μm0)<4σ, let αi=−λ(1−μi)2 and βi=4σ(1−μi)−λ2(1−μi)2. From (A.2), we have

|φi(t)|≤1+βi−2,|ψi(t)|≤1αiβ1,|ψi˙(t)|≤αi−2+βi−2

and

|φ˙i(t)|≤(1+αi)2+(βi+αiβi)2.

Let c=mini=2,⋯,m0λ(1−μi)2,

K=maxi=2,⋯,m0{1+βi−2,1αiβ1,αi−2+βi−2,(1+αi)2+(βi+αiβi)2},

then

max{‖Φ(t)‖,‖Φ˙(t)‖,‖Ψ(t)‖,‖Ψ˙(t)‖}≤Ke−ct,t>0.

Step III. If there is k0 such that λ2(1−μk0)=4σ, let c0=λ(1−μk0)4, from (A.3),

|φk0(t)|≤1+2e,|ψk0(t)|≤4eλ(1−μk0),|ψk0˙(t)|≤1+2e

and

|φ˙k0(t)|≤λ(1−μk0)e.

Let c0=mini=2,⋯,m0λ(1−μi)4,

K0=max{1+2e,4eλ(1−μk0),λ(1−μk0)e},

then

max{‖Φ(t)‖,‖Φ˙(t)‖,‖Ψ(t)‖,‖Ψ˙(t)‖}≤Ke−ct,t>0.

For i>k0, by Step I, there are constants c1 and K1 to make the lemma holding. Also, for i<k0, by Step II, there are constants c2 and K2 to make the lemma holding.

Finally, let c=min{c0,c1,c2} and K=max{K0,K1,K2}, then

max{‖Φ(t)‖,‖Φ˙(t)‖,‖Ψ(t)‖,‖Ψ˙(t)‖}≤Ke−ct,t>0.

This completes the proof.

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Received: 2020-04-25

Accepted: 2020-09-11

Published Online: 2021-05-27

Published in Print: 2021-06-25

Consensus and swarming behaviors for a proportional-derivative system with a cut-off interaction

Abstract

Zusammenfassung

About the authors

Proof of Lemma 3.1.

Proof of Lemma 3.2.

References

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