event-triggered multitarget formation control for multiagent...

Research ArticleEvent-Triggered Multitarget Formation Control forMultiagent Systems

Lingmin Zhang12 Xinbin Li1 Jing Yan1 and Xinping Guan3

1 Institute of Electrical Engineering Yanshan University Qinhuangdao 066004 China2Institute of Mathematics and Information Technology Hebei Normal University of Science and TechnologyQinhuangdao 066004 China3School of Electronic and Electric Engineering Shanghai Jiaotong University Shanghai 200240 China

Correspondence should be addressed to Lingmin Zhang lingmin9999163com

Received 23 June 2017 Revised 22 October 2017 Accepted 22 October 2017 Published 5 December 2017

Academic Editor Leonid Shaikhet

Copyright copy 2017 Lingmin Zhang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problems of multitarget selection and formation for multiagent systems are considered First of all an improved multitargetselection method based on auction algorithm is proposed such that each agent can automatically choose its target and during theprocess of choosing targets we apply an event-triggered mode to reduce the communication links between agents Second in viewof the fact that all agents with the same target need to form a desired formation shape we provide an event-triggered formationcontroller for each agent Finally we carry out the simulation experiment of the algorithm and the simulation results have illustratedthe effectiveness of it

1 Introduction

In recent years the problem of multiagent system (MAS) hasbeenwidely studied bymany researchers [1ndash4]These systemscan potentially consist of a great number of agents suchas unmanned aerial vehicles (UAG) unmanned underwatervehicles (UUV) and unmanned ground vehicles (UGA)MASs provide many applications in various practical fieldssuch as intelligent transportation systems building automa-tion underwater exploration and surveillanceAdvantages ofMASs over single agent include cost reduction efficiency androbustness improvement

One interesting issue of multiagent system is formationcontrol [5ndash7] Its objective is to design algorithms tomotivateagents to form a desired formation Meanwhile study onformationwith a single target has become one of the hot spotsissues in research of multiagent systems formation control[8ndash11] In [8] a methodology for group coordination andcooperative control of 119899 agents to achieve a target-capturingtask in 3D space was studied and the proposed approachwas based on a cyclic pursuit strategy where agent 119894 simplypursued agent 119894 + 1 modulo 119899 In [9] the cooperative targetpursuit problem by multiple agents based on directed acyclic

graph was investigated The target appeared at a randomlocation and moved only when sensed by the agents andagents pursued the target once they detected its existence In[10] the problem of flockingmotion combinedwith topologyoptimization for mobile multiagent systems was consideredand a distributed multiflocking method was designed basedon the partial information exchange In [11] the cooperativecontrol of a team of robots to estimate the position of amoving target using onboard sensing was investigated Theabove works are all based on the common assumption that agroup of agents pursue the same target that is it is supposedthat there is only one target in the workplace However thisassumption is strict in certain situations For instance whenmore than one target is considered in the workplace agentswill face a dilemma in choosing their targets Thus someresearchers studied multitarget formation [12ndash14] In [12]a team of agents who can accomplish multitarget pursuitformation by using a developed leader-follower strategywas designed In [13] a flocking algorithm with multitargettracking formultiagent systems was adopted It was supposedthat each target could accept a certain number of agentsWhich target would be chosen by an agent was determinedby the distances from the agent to the targets In [14] to

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1318376 8 pageshttpsdoiorg10115520171318376

2 Mathematical Problems in Engineering

solve this problem a distributed multiflocking method wasadopted based on the partial information exchange Butin the above existing multitarget selection algorithms thetargets assignment is static that is each agent selects aninvariable target to pursue However as the system evolveseach agent may select a different target to pursue accordingto certain optimal objectives Therefore considering systemswith dynamic targets assignment will be more significantThus how to design a dynamic multitarget selection is aproblem to be solved

In addition in the existing algorithms of multitargetselection and multiagent systems formation control time-triggered control mode is widely adopted Time-triggeredcontrol mode is simple but it may cause large amount ofbandwidth and communication Event-triggered control isan alternative to time-triggered control [15 16] The distinctfeature of event-triggered control is that control action isupdated only when some specific event occurs Comparedwith time-triggered control mode event-triggered controlmode has the often cited advantages on communicationreduction and energy saving It has been studied exten-sively in network control systems and decentralized systems[15 17 18] In many cases such as formation control pursuitcontrol and path planning it outperforms the traditionaltime-triggered control [16 19ndash22] Thus how to apply event-triggered mode to multiagent formation control is anotherproblem to be solved For the above two problems this paperfocuses on the dynamic multitarget selection and formationof multiagent systems and applies event-triggered controlmode to multitarget selection and formation The maincontributions of this paper are as follows

(1) Unlike in most of the existing multitarget selectionalgorithms the targets assignment is static and eachagent selects an invariable target to pursue we con-sider systems with dynamic targets assignment aseach agent may select a different target to pursueaccording to certain optimal objectives In additionan improved dynamic selection method based onauction algorithm is adopted and the event-triggeredcontrol mode is applied to the multitarget selection toreduce the communication links between agents

(2) In the process of multitarget formation control formultiagent systems we adopt the event-triggeredcontrol mode instead of time-triggered-controlmode When event-triggered control mode is appliedto multiagent systems the stability of the system canbe maintained and compared with time-triggeredcontrol mode it has the advantages of reducingthe number of information updating and savingbandwidth resources and energy

The rest of this paper is organized as follows In Section 2system modeling and problem formulation are presented InSection 3 we apply the event-triggered mode to dynamic tar-get selection and the formation strategy Simulation studiesare provided to illustrate the effectiveness of our method inSection 4 Conclusions are given in Section 5

2 Preliminaries and Problem Formulation

21 Graph Theory A graph 119866 is a pair that consists of a setof vertices 119881 = 1 2 119873 and edges 119864 sube (119894 119895) 119894 119895 isin119881 119895 = 119894 The graph is said to be undirected if (119894 119895) isin119864 hArr (119895 119894) isin 119864 And in order to ensure cooperation andcoordinating among agents each agent has to know the statesof other agents Therefore agents have to communicate witheach other Given an agent 119894 the set of agents from which itcan receive information is called a neighbor set119873119894 that is119873119894 = 119895 isin (1 2 119873) (119894 119895) isin 119864 119894 = 1 2 119873 (1)

A graph is connected if any two vertices can be joinedwith an edge It is assumed that the graph describing theinformation structure is connected A graph also admitsmatrix representations Some of these matrices such asthe adjacency matrix the degree matrix and the Laplacianmatrix will be reviewed subsequently

The adjacency matrix 119860(119866) encoding of the adjacencyrelationship in the graph 119866 is defined as

119886119894119895 =

1 (119894 119895) isin 1198640 (119894 119895) notin 119864

(2)

where 119886119894119895 is the (119894 119895) entry of the adjacency matrix 119860(119866) isin119877119899times119899 The degree matrix 119863(119866) for an undirected graph 119866is a diagonal matrix diag 1198891 1198892 119889119899 where 119889119894 is thecardinality of neighbor set119873119894 of agent 119894The adjacencymatrixof undirected graph is symmetric because 119886119894119895 = 119886119895119894 for 119894 =119895 The Laplacian matrix 119871(119866) associated with an undirectedgraph 119866 is defined as 119871(119866) = 119863(119866) minus 119860(119866) where 119863(119866)and 119860(119866) are degree matrix and adjacency matrix of graph119866 respectively

22 System Modeling and Problem Formulation

Multiagent Systems Define a set of agents as Δ = 1 2 3 119873 where119873 is the number of agents For the agent 119894with two-dimensional coordinate the position and input vectors aredenoted by 119901119894 isin 1198772 and 119906119894 isin 1198772 respectively The dynamicsof agent 119894 at time 119905 are described by the following continuous-time equation

119901119894 (119905) = 119906119894 (3)

For the dynamic system the following assumptions aremade

Assumption 1 Initially it is assumed that targets and agentsdisperse randomly in the workplace Meanwhile agents candetect the state information about the target at the initial time

Assumption 2 Each agent can only obtain the state infor-mation of its neighbors and each target can only accept acertain number of agents In the following we will give animproved auction algorithmbased on event-triggered controlto complete the target selection then an event-triggeredcontroller for each agent will be given to form a desiredformation shape

Mathematical Problems in Engineering 3

Input coordinate of agent 119894 119909119894 119910119894 coordinate of target 119895 119909119895 119910119895 and the cycle index 119899(1) for 119905 = 1 119899(2) Calculate the distance function 119871 119894119895 = radic(119909119894 minus 119909119895)2 + (119910119894 minus 119910119895)2(3) Calculate the value of 119862119894119895= 119871119894119895(4) for 119904 = 1 119894(5) Calculate the value of V119895 (119895 = 1 2 119899)(6) Calculate 119903119894119895 and 119887119894(7) While 119887119894 ge 119901(8) Agent 119894 will choose target 119895(9) Inform the information to other agents(10) 119873119895 = 119873119895 minus 1(11) 119901 = 119901 minus 01(12) end(13) Output the position coordinate of each agent(14) end(15) end

Algorithm 1 Dynamic selecting algorithm based on auction

3 Event-Triggered Multitarget Formation

In this section we will design a team of agents whocan accomplish multitarget formation by using an event-triggered formation method First we will present an event-triggered dynamic strategy for choosing a target Second wewill provide an event-triggered controller for each agent toform a desired multiagent systems formation

31 Strategy for Choosing a Target In the dynamic systemeach target is considered as a commodity and we define thevalue of the target 119894 as 119881119894 forall119894 = 119895 119881119894 = 119881119895 = 119881 at the initialtime and the price to catch up with 119895 for 119894 is 119862119894119895 The systemdesigns an open platform in which all the 119898 targets havean auction and all the 119873 agents are involved in the auctionIn this mode of auction the auction platform begins witha given price and all agents are aware of the current priceannounced The price is gradually reduced until some agentselects it In the designed auction algorithm all targets aresimultaneously on auctionWhen an agentrsquos income is greaterthan or equal to the outcry of the current system the agentselects the corresponding target If target 119895 is chosen by 119873119895agents the value obtained by agent 119894 is

119881119894 = 119881119873119895

(4)

When 119873119895 increases the value obtained from target 119895 will bedecreased gradually and thus the agent will tend to choosethe target that is chosen by fewer agents In this way we caneffectively avoid the problem of selecting the same target formany agents

Building the proceeds functions as follows

119903119894119895 = 119881119895 minus 119862119894119895 (5)

where 119862119894119895 = 119871 119894119895 is the cost function for agent 119894 to selecttarget 119895 119871 119894119895 = 119875119894 minus 119890119895 and 119875119894 and 119890119895 are positions of agent119894 and target 119895 respectively After calculating its proceeds for

selecting each target the agent will choose the one with thebiggest proceeds as its target In this way each agent will tendto choose target relatively close to it in order to get the finalrate of exchange

119887119894 = max 1199031198941 1199031198942 119903119894119899 (6)

When the current bid price 119901 is less than 119887119894 or equal to 119887119894agent 119894 will select the corresponding target and drop out ofthe auctionThe remaining agentswill continue to achieve theselection until the last agent accomplishes the target selectionAfter selecting the targets agents will continue to move andwill update the data at the next time node in order to achievethe dynamic selection and make a response to the changes ofthe sceneThepseudocode of the auction algorithm is showedin Algorithm 1

In the design of the algorithm if the 119895th target has beenselected 119873119895 = 119873119895 + 1 the 119894th agent may select the furthertarget In the next choice as the agentrsquos position has beenchanged and the value of 119873119895 may have been changed agent119894 may select more nearer target and then the target replace-ment phenomenon occurs In addition as the time-triggeredcontrolmode is adopted in themultitarget dynamic selectionthe agent will update the selection data at each sampling timeandmake a target selection It will lead to frequent calculationof the agentrsquos own proceeds frequent replacement of thetarget and large amount of communication among agents Inaddition it takes up a lot of bandwidth and will result in a lotof unnecessary energy consumption For solving the aboveproblems in the improved algorithm we define the targetvalue as follows

119881119894119895 = 119881 + 120576 (120576 gt 0) (7)

where 119881 is the initial value of the target in the system and 120576is the value increment of target 119895 relative to agent 119894 In thisway when the target proceeds119881119894119895 are changed the gain of theagent selecting the original target will be increased and thus


it will tend to select the target selected last time In additionwe introduce an event-triggered function [22]

1003817100381710038171003817119890119894 (119905119896 + 119897ℎ)100381710038171003817100381722 le 1003817100381710038171003817119911119894 (119905119896 + 119897ℎ)100381710038171003817100381722 (8)

where ℎ is the sampling period for all agents synchronized bya clock 119890119894(119905119896 + 119897ℎ) is defined as the position difference at thelast event time and the currently sampled time

119890119894 (119905119896 + 119897ℎ) = 119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ) (9)

and 119911119894(119905119896+119897ℎ) is the comparison of the position of agent 119894withall its neighbors

119911119894 (119905119896 + 119897ℎ) = sum119909119894 (119905119896) minus 119909119895 (119905119896 + 119897ℎ) (10)

At each sampling instant each agent broadcasts its state infor-mation to the neighbors and also receives state informationfrom its neighbors for event detection If the condition in(8) is satisfied the agent remains selecting the same targetotherwise a new round of target selection will be carriedout This process is defined as an event trigger and thesystem updates the input information and the deviation value119890119894(119905119896 + 119897ℎ) to continue to begin the next triggering By addingan increment 120576 in the value function the agents can keepselecting the fixed target and when the number of the targetsis changed in the scene they will response to the changingand change the target in time

Corollary 3 When119881 = 4119871119898119886119909+120572 where119881 is the initial valueof each agent 119871119898119886119909 is the maximum of 119871 119894119895 and 0 lt 120576 lt 1198812the agent will respond to the state changing of the scene in thedynamic selection and will not be affected by the increment

Proof (proof by contradiction) Suppose that there appears anew target 119905 in the system and the number of targets is lessthan that of the agents the new target 119905 can not be selectedby any agent Since the new target is not chosen by any agent119903119894119905 lt 119903min where 119903min is the minimal proceeds of 119894 and thenew target 119905 is given the initial value 119881 As the target has notbeen selected by any agent119873119905 = 1 then the agent calculatesthe proceeds of choosing target 119905 And

119903min lt119881 + 120576max

119873119895minus 119871 119894119895 = 3119881

2119873119895minus 119871 119894119895 119873119895 ge 2 (11)

Thus 119903min lt 31198814And because 119903119894119905 = 119881 minus 119871 119894119905 ge 119881 minus 119871max ge 31198814 we can

obtain that 119903119894119905 gt 119903min

32 The Event-Triggered Multitarget Formation

Remark 4 In this paper there are 119873 agents and 119898 targetsNamely 119873 agents will be divided into 119898 groups based onthe strategy of choosing targets Agents with the same targetwill be in the same group For simplicity we only provide thecontrolmethod to one group of agentsThe proposedmethodcan then be extended to the remaining 119898 minus 1 groups byupdating the number of agents in each group In the specifiedgroup 119899 (119899 lt 119873) agents are considered

119909119894 (119905) = 119906119894 119894 = 1 2 119899 (12)

where 119909119894 and 119906119894 are state and controlled input of agent 119894respectively Then the governing equations can be describedas

119906119894 (119905) = minussum119895isin119873119894

119886119894119895 (119909119894 (119905) minus 119909119895 (119905)) (13)

where 119886119894119895 is the (119894 119895) entry of the adjacency matrix 119873119894 is theneighbor set of agent 119894 at time 119905 and 119906119894 is the control input Inthis design the control equation of (13) can be changed to

119906119894 (119905) = minussum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (14)

where 120601 = 119886119894119895 And the formation control method adoptedin this design is the one with an offset As the formationcontrol method adopts a static formation keeping mode thedeviation is a constant The state with offset is

119909119894 (119905) = 119909119894 minus 120574119894 (15)

As 120574119894 is a constant we can obtain that

119909119894 (119905) = 119894 (16)

The equation with offset can also be applied to the controlequation of (14) and then the equation with offset is

119906119894 (119905) = sum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (17)

By applying the control equation the neighboring agentsof the system can be given the desired formation In theformation problem the essence of event-triggered controlmode is to propose an event-triggered control mechanismin order to reduce the communication between neighboringagents and the energy consumption of event detection foreach agentThe event condition for agent 119894 has the form of (8)The event-triggered control method is applied to the agentsrsquoformation control with an offset and then (17) is as follows

119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896) (18)

When the deviation 119890119894(119905119896 + 119897ℎ) in (9) is added to (18)

119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896)

= minussum119895isin119873119894

119909119894 (119905119896 + 119897ℎ) minus 119909119895 (119905119896 + 119897ℎ)

minus sum119895isin119873119894

119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ)

+ sum119895isin119873119894

119909119895 (119905119896) minus 119909119895 (119905119896 + 119897ℎ)

(19)

By [22] we know that when the event-triggered controlmode is applied to multiagent systems the stability of thesystem can bemaintained and comparedwith time-triggeredcontrol mode the event-triggered control mode has theadvantage of saving energy


Table 1 Positions of the agents and targets

Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Target 1 Target 2 Target 3119883 (m) 04 04 09 09 14 14 94 48 87119884 (m) 04 14 01 20 04 14 31 92 87

Table 2 Final results of the targets allocation in Figure 1(d)

Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 60 lt 119905 lt 200 Target 1 Target 2 Target 1 Target 2 Target 2 Target 1119905 ge 200 Target 3 Target 2 Target 1 Target 3 Target 2 Target 1

Table 3 The original positions of the agents

119883 coordinate (m) 119883 offset (m) 119884 coordinate (m) 119884 offset (m)Agent 1 323 0 461 05Agent 2 567 0 231 minus05Agent 3 981 05 111 0

4 Simulation Results

This section presents the simulation of the proposed multi-targets selection and formation method Initially the agentsand targets are randomly dispersed in the workplace and thescene size is 10 times 10

41 Simulation of the Target Selection The original value ofthe target is 119881 = 56 and there are 6 agents and 2 targets(Target 1 and Target 2) In Figure 1(d) when 119905 = 200 thereappears a new target (Target 3) The initial positions of theagents and targets are in Table 1

Comparing Figure 1(a) with 1(b) it can be seen thatwhen applying the improved algorithm in the target selectionthe agents do not exchange the target frequently and theagents do not appear to get together to choose the sametarget In addition comparing Figure 1(b) with 1(c) it canbe seen that when applying the event-triggered control modeinstead of time-triggered control mode we can not onlyobtain the expected results but also reduce communicationamong agents thus reducing the energy consumption

Based on Table 2 we can see that when the systemincreases a new target two agents abandon the original targetand choose the new one It can be seen that the incrementalvalue 120576 does not affect the response of the system to thechanging of the target number

42 Simulation of the Event-Triggered Formation ControlBased on Figures 1(a) 1(b) 1(c) and 1(d) we can see thatstate of the agents tend to be consistent due to presence ofthe offset and the input value tends to be zero Thus weknow that by applying offset in the formation themultiagentsystem can form a desired formation

Furthermore we choose agents with the original posi-tions as in Table 3 Based on Figures 1(e)ndash1(k) we can seethat when applying event-triggered control mode in theformation control the result of the system basically has nodifference with that of the time-triggered control mode andthus the feasibility of event-triggered control mode in the

formation control is verified Meanwhile it can reduce thetimes of updating information and lengthen the intervaltime of updating and thus can reduce the consumption ofresources and energy greatly

The essence of event-triggered control is to control theevent condition so the deviation 119890 and event-triggered value119911 are used to define the event occurring Based on Figure 1(l)we can see that when the system is running the amount ofdeviation 119890 is greater than that of the comparison value 119911

5 Conclusions

The multitarget dynamic selection method and the event-triggered formation control strategy of multiagent systemsare presented in this paper First as in most of the existingmultitarget selection algorithms the targets assignment isstatic and each agent selects an invariable target to pursuehowever inmany practical application as the system evolveseach agentmay select a different target to pursueThuswe givethe dynamicmultitarget selection algorithmbased on auctionto solve this problem and apply event-triggered control modeto it Second as the event-triggered control mode has theadvantages of reducing the number of information updatingand saving bandwidth resources and energy while givingthe formation of agents that choose the same target inthe formation process the event-triggered control mode isadopted instead of the time-triggered control mode

In our futurework for taking full advantages of the event-triggered control mode we will apply it to more multiagentformation problems

Conflicts of Interest

The authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Lingmin Zhang and Xinbin Li carried out the proof of thetheorems and gave the simulation and Jing Yan and Xinping


Agent 1Agent 2Agent 3


Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(a)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(b)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(c)



Target

0123456789

10Y

(m)

2 4 6 8 100X (m)

(d)

x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(e)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(f)

Figure 1 Continued


x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)

minus10minus8minus6minus4minus2

02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)

Target 1Target 2Target 3

1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)

Comparative value z(t)Deviation e(t)

0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)

Figure 1 (a) Selecting result of the original algorithm (b) Selecting result of the improved algorithm with an increment 120576 (c) Selectingresult of the improved algorithm based on event-triggered mode (d) Response of the system when increasing a new target (e)119883 under time-triggered control model (f) 119884 under time-triggered control model (g) 119883 under event-triggered control model (h) 119884 under event-triggeredcontrol model (i) Input under time-triggered control model (j) Input under event-triggered control model (k) Construction of formationcontrol (l) Deviation value 120576 and event-triggered value 119911


Guan carried out the check of the manuscript All authorsread and approved the final manuscript

Acknowledgments

The work was partially supported by NSF of China under61503320 61571387 and 61172095 by China PostdoctoralScience Foundation Funded Project under 2015M570235 byYouth Foundation of Hebei Educational Committee underQN2015187 by Postdoctoral Science Foundation FundedProject of Hebei Province under B2015003018 by the OpenProject Program of Key Laboratory of System Controland Information Processing Ministry of Education underScip201501 and by the Self-Determined Project of YanshanUniversity under 14LGA010 and 14LGA020

References

[1] G Antonelli F Arrichiello and S Chiaverini ldquoFlocking formulti-robot systems via the Null-space-based behavioral con-trolrdquo Swarm Intelligence vol 4 no 1 pp 37ndash56 2010

[2] Y Liu and Y Jia ldquoAn iterative learning approach to formationcontrol of multi-agent systemsrdquo Systems amp Control Letters vol61 no 1 pp 148ndash154 2012

[3] Y Zhao ZDuanGWen andGChen ldquoDistributed finite-timetracking for a multi-agent system under a leader with boundedunknown accelerationrdquo Systems amp Control Letters vol 81 pp8ndash13 2015

[4] YDai Y Kim SWeeD Lee and S Lee ldquoA switching formationstrategy for obstacle avoidance of amulti-robot system based onrobot priority modelrdquo ISA Transactions vol 56 pp 123ndash1342015

[5] J-L Wang and H-N Wu ldquoLeader-following formation controlof multi-agent systems under fixed and switching topologiesrdquoInternational Journal of Control vol 85 no 6 pp 695ndash705 2012

[6] X Lu F Austin and S Chen ldquoFormation control for second-order multi-agent systems with time-varying delays underdirected topologyrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1382ndash1391 2012

[7] X Chen and FHao ldquoEvent-triggered average consensus controlfor discrete-time multi-agent systemsrdquo IET Control Theory ampApplications vol 6 no 16 pp 2493ndash2498 2012

[8] T-H Kim and T Sugie ldquoCooperative control for target-capturing task based on a cyclic pursuit strategyrdquo Automaticavol 43 no 8 pp 1426ndash1431 2007

[9] J Yan X-P Guan and X-Y Luo ldquoConsensus pursuit ofheterogeneous multi-agent systems under a directed acyclicgraphrdquo Chinese Physics B vol 20 no 4 Article ID 048901 2011

[10] X Luo D Liu X Guan and S Li ldquoFlocking in target pursuit formulti-agent systems with partial informed agentsrdquo IET ControlTheory amp Applications vol 6 no 4 pp 560ndash569 2012

[11] K Hausman J Muller A Hariharan N Ayanian and GS Sukhatme ldquoCooperative control for target tracking withonboard sensingrdquo International Symposium on ExperimentalRobotics 2014

[12] J Yan X-P Guan and X-Y Luo ldquoMulti-target pursuit forma-tion of multi-agent systemsrdquo Chinese Physics B vol 20 no 1Article ID 018901 2011

[13] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[14] H Pei S Chen and Q Lai ldquoMulti-target consensus circlepursuit for multi-agent systems via a distributed multi-flockingmethodrdquo International Journal of Systems Science vol 47 no 16pp 3741ndash3748 2016

[15] T Henningsson E Johannesson and A Cervin ldquoSporadicevent-based control of first-order linear stochastic systemsrdquoAutomatica vol 44 no 11 pp 2890ndash2895 2008

[16] J Lunze and D Lehmann ldquoA state-feedback approach to event-based controlrdquo Automatica vol 46 no 1 pp 211ndash215 2010

[17] J Mazo and P Tabuada ldquoDecentralized event-triggered controlover wireless sensoractuator networksrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 56 no 10 pp 2456ndash2461 2011

[18] M Mazo and M Cao ldquoAsynchronous decentralized event-triggered controlrdquo Automatica vol 50 no 12 pp 3197ndash32032014

[19] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of The Franklin Institute vol352 no 3 pp 1080ndash1098 2015

[20] H Zhang R Yang H Yan and Q Chen ldquoDistributed event-triggered control for consensus of multi-agent systemsrdquo Journalof The Franklin Institute vol 352 no 9 pp 3476ndash3488 2015

[21] B Watkins S Al-Areqi S Reimann and S Liu ldquoEvent-basedcontrol of constrained discrete-time linear systems with guar-anteed performancerdquo International Journal of Sensors WirelessCommunications and Control vol 5 no 2 pp 72ndash80 2015

[22] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquoAutomatica vol 49 no 7 pp 2125ndash21322013

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solve this problem a distributed multiflocking method wasadopted based on the partial information exchange Butin the above existing multitarget selection algorithms thetargets assignment is static that is each agent selects aninvariable target to pursue However as the system evolveseach agent may select a different target to pursue accordingto certain optimal objectives Therefore considering systemswith dynamic targets assignment will be more significantThus how to design a dynamic multitarget selection is aproblem to be solved

In addition in the existing algorithms of multitargetselection and multiagent systems formation control time-triggered control mode is widely adopted Time-triggeredcontrol mode is simple but it may cause large amount ofbandwidth and communication Event-triggered control isan alternative to time-triggered control [15 16] The distinctfeature of event-triggered control is that control action isupdated only when some specific event occurs Comparedwith time-triggered control mode event-triggered controlmode has the often cited advantages on communicationreduction and energy saving It has been studied exten-sively in network control systems and decentralized systems[15 17 18] In many cases such as formation control pursuitcontrol and path planning it outperforms the traditionaltime-triggered control [16 19ndash22] Thus how to apply event-triggered mode to multiagent formation control is anotherproblem to be solved For the above two problems this paperfocuses on the dynamic multitarget selection and formationof multiagent systems and applies event-triggered controlmode to multitarget selection and formation The maincontributions of this paper are as follows

(1) Unlike in most of the existing multitarget selectionalgorithms the targets assignment is static and eachagent selects an invariable target to pursue we con-sider systems with dynamic targets assignment aseach agent may select a different target to pursueaccording to certain optimal objectives In additionan improved dynamic selection method based onauction algorithm is adopted and the event-triggeredcontrol mode is applied to the multitarget selection toreduce the communication links between agents

(2) In the process of multitarget formation control formultiagent systems we adopt the event-triggeredcontrol mode instead of time-triggered-controlmode When event-triggered control mode is appliedto multiagent systems the stability of the system canbe maintained and compared with time-triggeredcontrol mode it has the advantages of reducingthe number of information updating and savingbandwidth resources and energy

The rest of this paper is organized as follows In Section 2system modeling and problem formulation are presented InSection 3 we apply the event-triggered mode to dynamic tar-get selection and the formation strategy Simulation studiesare provided to illustrate the effectiveness of our method inSection 4 Conclusions are given in Section 5

2 Preliminaries and Problem Formulation

21 Graph Theory A graph 119866 is a pair that consists of a setof vertices 119881 = 1 2 119873 and edges 119864 sube (119894 119895) 119894 119895 isin119881 119895 = 119894 The graph is said to be undirected if (119894 119895) isin119864 hArr (119895 119894) isin 119864 And in order to ensure cooperation andcoordinating among agents each agent has to know the statesof other agents Therefore agents have to communicate witheach other Given an agent 119894 the set of agents from which itcan receive information is called a neighbor set119873119894 that is119873119894 = 119895 isin (1 2 119873) (119894 119895) isin 119864 119894 = 1 2 119873 (1)

A graph is connected if any two vertices can be joinedwith an edge It is assumed that the graph describing theinformation structure is connected A graph also admitsmatrix representations Some of these matrices such asthe adjacency matrix the degree matrix and the Laplacianmatrix will be reviewed subsequently

The adjacency matrix 119860(119866) encoding of the adjacencyrelationship in the graph 119866 is defined as

119886119894119895 =

1 (119894 119895) isin 1198640 (119894 119895) notin 119864

(2)

where 119886119894119895 is the (119894 119895) entry of the adjacency matrix 119860(119866) isin119877119899times119899 The degree matrix 119863(119866) for an undirected graph 119866is a diagonal matrix diag 1198891 1198892 119889119899 where 119889119894 is thecardinality of neighbor set119873119894 of agent 119894The adjacencymatrixof undirected graph is symmetric because 119886119894119895 = 119886119895119894 for 119894 =119895 The Laplacian matrix 119871(119866) associated with an undirectedgraph 119866 is defined as 119871(119866) = 119863(119866) minus 119860(119866) where 119863(119866)and 119860(119866) are degree matrix and adjacency matrix of graph119866 respectively

22 System Modeling and Problem Formulation

Multiagent Systems Define a set of agents as Δ = 1 2 3 119873 where119873 is the number of agents For the agent 119894with two-dimensional coordinate the position and input vectors aredenoted by 119901119894 isin 1198772 and 119906119894 isin 1198772 respectively The dynamicsof agent 119894 at time 119905 are described by the following continuous-time equation

119901119894 (119905) = 119906119894 (3)

For the dynamic system the following assumptions aremade

Assumption 1 Initially it is assumed that targets and agentsdisperse randomly in the workplace Meanwhile agents candetect the state information about the target at the initial time

Assumption 2 Each agent can only obtain the state infor-mation of its neighbors and each target can only accept acertain number of agents In the following we will give animproved auction algorithmbased on event-triggered controlto complete the target selection then an event-triggeredcontroller for each agent will be given to form a desiredformation shape







119881119894 = 119881119873119895

(4)



119903119894119895 = 119881119895 minus 119862119894119895 (5)



119887119894 = max 1199031198941 1199031198942 119903119894119899 (6)



119881119894119895 = 119881 + 120576 (120576 gt 0) (7)




1003817100381710038171003817119890119894 (119905119896 + 119897ℎ)100381710038171003817100381722 le 1003817100381710038171003817119911119894 (119905119896 + 119897ℎ)100381710038171003817100381722 (8)


119890119894 (119905119896 + 119897ℎ) = 119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ) (9)


119911119894 (119905119896 + 119897ℎ) = sum119909119894 (119905119896) minus 119909119895 (119905119896 + 119897ℎ) (10)




119903min lt119881 + 120576max

119873119895minus 119871 119894119895 = 3119881

2119873119895minus 119871 119894119895 119873119895 ge 2 (11)





119909119894 (119905) = 119906119894 119894 = 1 2 119899 (12)


119906119894 (119905) = minussum119895isin119873119894

119886119894119895 (119909119894 (119905) minus 119909119895 (119905)) (13)


119906119894 (119905) = minussum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (14)


119909119894 (119905) = 119909119894 minus 120574119894 (15)


119909119894 (119905) = 119894 (16)


119906119894 (119905) = sum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (17)


119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896) (18)


119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896)

= minussum119895isin119873119894

119909119894 (119905119896 + 119897ℎ) minus 119909119895 (119905119896 + 119897ℎ)

minus sum119895isin119873119894

119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ)

+ sum119895isin119873119894

119909119895 (119905119896) minus 119909119895 (119905119896 + 119897ℎ)

(19)


















5 Conclusions










Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(a)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(b)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(c)



Target

0123456789

10Y

(m)

2 4 6 8 100X (m)

(d)

x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(e)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(f)

Figure 1 Continued


x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)


1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)


0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)




Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of










119881119894 = 119881119873119895

(4)



119903119894119895 = 119881119895 minus 119862119894119895 (5)



119887119894 = max 1199031198941 1199031198942 119903119894119899 (6)



119881119894119895 = 119881 + 120576 (120576 gt 0) (7)




1003817100381710038171003817119890119894 (119905119896 + 119897ℎ)100381710038171003817100381722 le 1003817100381710038171003817119911119894 (119905119896 + 119897ℎ)100381710038171003817100381722 (8)


119890119894 (119905119896 + 119897ℎ) = 119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ) (9)


119911119894 (119905119896 + 119897ℎ) = sum119909119894 (119905119896) minus 119909119895 (119905119896 + 119897ℎ) (10)




119903min lt119881 + 120576max

119873119895minus 119871 119894119895 = 3119881

2119873119895minus 119871 119894119895 119873119895 ge 2 (11)





119909119894 (119905) = 119906119894 119894 = 1 2 119899 (12)


119906119894 (119905) = minussum119895isin119873119894

119886119894119895 (119909119894 (119905) minus 119909119895 (119905)) (13)


119906119894 (119905) = minussum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (14)


119909119894 (119905) = 119909119894 minus 120574119894 (15)


119909119894 (119905) = 119894 (16)


119906119894 (119905) = sum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (17)


119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896) (18)


119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896)

= minussum119895isin119873119894

119909119894 (119905119896 + 119897ℎ) minus 119909119895 (119905119896 + 119897ℎ)

minus sum119895isin119873119894

119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ)

+ sum119895isin119873119894

119909119895 (119905119896) minus 119909119895 (119905119896 + 119897ℎ)

(19)


















5 Conclusions










Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(a)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(b)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(c)



Target

0123456789

10Y

(m)

2 4 6 8 100X (m)

(d)

x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(e)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(f)

Figure 1 Continued


x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)


1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)


0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)




Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






1003817100381710038171003817119890119894 (119905119896 + 119897ℎ)100381710038171003817100381722 le 1003817100381710038171003817119911119894 (119905119896 + 119897ℎ)100381710038171003817100381722 (8)


119890119894 (119905119896 + 119897ℎ) = 119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ) (9)


119911119894 (119905119896 + 119897ℎ) = sum119909119894 (119905119896) minus 119909119895 (119905119896 + 119897ℎ) (10)




119903min lt119881 + 120576max

119873119895minus 119871 119894119895 = 3119881

2119873119895minus 119871 119894119895 119873119895 ge 2 (11)





119909119894 (119905) = 119906119894 119894 = 1 2 119899 (12)


119906119894 (119905) = minussum119895isin119873119894

119886119894119895 (119909119894 (119905) minus 119909119895 (119905)) (13)


119906119894 (119905) = minussum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (14)


119909119894 (119905) = 119909119894 minus 120574119894 (15)


119909119894 (119905) = 119894 (16)


119906119894 (119905) = sum119895isin119873119894

120601 (119909119894 (119905) minus 119909119895 (119905)) (17)


119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896) (18)


119909119894 (119905) = sum119895isin119873119894

119909119894 (119905119896) minus 119909119895 (119905119896)

= minussum119895isin119873119894

119909119894 (119905119896 + 119897ℎ) minus 119909119895 (119905119896 + 119897ℎ)

minus sum119895isin119873119894

119909119894 (119905119896) minus 119909119894 (119905119896 + 119897ℎ)

+ sum119895isin119873119894

119909119895 (119905119896) minus 119909119895 (119905119896 + 119897ℎ)

(19)


















5 Conclusions










Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(a)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(b)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(c)



Target

0123456789

10Y

(m)

2 4 6 8 100X (m)

(d)

x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(e)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(f)

Figure 1 Continued


x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)


1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)


0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)




Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




















5 Conclusions










Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(a)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(b)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(c)



Target

0123456789

10Y

(m)

2 4 6 8 100X (m)

(d)

x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(e)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(f)

Figure 1 Continued


x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)


1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)


0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)




Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(a)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(b)



Target

0123456789

10

Y (m

)

2 4 6 8 100X (m)

(c)



Target

0123456789

10Y

(m)

2 4 6 8 100X (m)

(d)

x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(e)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(f)

Figure 1 Continued


x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)


1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)


0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)




Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





x1 (t)

x2 (t)

x3 (t)

3

4

5

6

7

8

9

10

x (m

)

2 4 6 8 100Time t (s)

(g)

y1 (t)

y2 (t)

y3 (t)

1

15

2

25

3

35

4

45

5

Y (m

)

2 4 6 8 100Time t (s)

(h)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(i)

u1 (t)

u2 (t)

u3 (t)


02468

10

u (t

)

2 4 6 8 100Time t (s)

(j)


1

15

2

25

3

35

4

45

5

Y (m

)

4 5 6 7 8 9 103X (m)

(k)


0

01

02

03

04

05

06

Stat

e of d

evia

tion

(m)

1 2 3 4 5 6 70Time t (s)

(l)




Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




event-triggered multitarget formation control for multiagent...

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