Research ArticleMultiple Dynamic Targets Encirclement Control ofMultiagent Systems

Wenguang Zhang Jizhen Liu and Deliang Zeng

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power UniversityBeijing 102206 China

Correspondence should be addressed to Wenguang Zhang zwgbuaa126com

Received 28 September 2015 Accepted 9 November 2015

Academic Editor Peng Lin

Copyright copy 2015 Wenguang Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper develops the distributed encirclement control problemofmultiagent systems inwhich each agent tracksmultiple targetseach target can be tracked by one agent and the numbers of the agents and the targets are the same or not Firstly an encirclementcontrol protocol is proposed for multiagent systems and this protocol contains some estimators Secondly some conditions arederived under which multiagent systems can achieve encirclement control by circular formation Finally numerical simulationsare provided to illustrate the obtained results

1 Introduction

Distributed coordination control of multiagent systems hasattracted a great number of researchers from different back-grounds such as physics biology control theory roboticsand computer [1ndash17] Multiagent systems arise in wideareas including movement of flocks of birds or schools offish molecular conformation problems cooperative controlof unmanned aerial vehicles formation control of mobilerobots and power systems For instance Olfati-Saber andMurray [2] presented two consensus protocols to solveagreement problems in a network of continuous-time anddiscrete-time integrator agents and investigated a systemat-ical framework of consensus problem in networks of agentswith a simple scalar continuous-time integrator in threecases Lin and Jia [3ndash5] studied consensus problems forfirst-order or second-order multiagent systems with timevarying communication delays and switching topology In [67] 119867infin

consensus problems were respectively investigatedfor the first-order and high-order multiagent systems andthey gave the conditions of satisfying 119867

infinbased on linear

matrix inequality In [8] the constrained consensus problemof multiagent systems in dynamically changing unbalancednetworks with communication delays has been studied It has

been shown that the error auxiliary vanishes as time evolvesand the linearmain body has an exponential convergence rateto a vector as a separate system

In some situations encirclement control for multipletargets can be studied in a distributed manner However thework on this problem is rare currently In [9 10] they onlyconsidered the fixed targets In [9] a group of unmannedaerial vehicles surrounding one target by using decentralizednonlinear model predictive control was studied In [10]Chen et al used the leader-follower framework to makethe followers surround the stationary leaders with a fixedcommunication graph A multiagent cooperative controlproblem in which agents move collectively to surroundmultiple targets was studied in [11] and the proposed controllaw works not only for stationary targets but also for dynamicones But in that paper it is assumed that the numbers of theagents and the targets are the same

This paper will focus on the study of the distributedencirclement control and tracking problems of multipledynamic targets by graph theory We suppose that each agenttracks multiple targets and each target only can be trackedby one agent Firstly we design a control protocol includingsome estimators Secondly the required conditions to realize

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 467060 6 pageshttpdxdoiorg1011552015467060

2 Mathematical Problems in Engineering

encirclement are proposed by Lyapunov theory Finally weprove this theory to be effective by the simulation

The rest of this paper is organized as follows In Section 2we introduce some basic notations and some concepts ingraph theory In Section 3 the model to be researched isformulated and a distributed encirclement control protocolis proposed In Section 4 the main results are stated andderived In Section 5 numerical simulations are providedto demonstrate the effectiveness of the obtained theoreticalresults In Section 6 we conclude this paper

2 Notations and Preliminaries

Let 119866(VEA) be an undirected graph where V =

1199041 1199042 119904

119899 is the set of nodes and E isin V times V is the

set of edges The node indexes belong to a finite index setI = 1 2 119899 and119873

119894= 119904119895isin V(119904

119894 119904119895) isin E is defined as

the neighbourhood set of 119904119894A = [119886

119894119895] isin R119899times119899 is a symmetric

weighted adjacency matrix where the element 119886119894119895represents

the weight from node 119904119894to node 119904

119895 When 119904

119895isin 119873119894 then

119886119894119895gt 0 or else 119886

119894119895= 0 In the undirected graph any (119904

119894 119904119895) isin

E hArr (119904119895 119904119894) isin E The graph Laplacian with the diagraph is

defined as 119871 = [119897119894119895] where 119897

119894119894= sum119899

119895=1119886119894119895and 119897119894119895= minus119886119894119895 119894 = 119895 If

there is a path from every node to every other node the graphis said to be connected and undirected

Lemma 1 (see [18]) If the undirected graph 119866 is connectedthen its Laplacian 119871 satisfies the following

(1) Zero is a simple eigenvalue of 119871 and 1119899is the corre-

sponding eigenvector and 1198711119899= 0

(2) The remaining 119899 minus 1 eigenvalues of 119871 all have positivereal parts And 119871 is a symmetric matrix and theeigenvalues 0 = 120582min = 120582

1le 1205822sdot sdot sdot le 120582

119899= 120582max

Lemma 2 (see [19]) Suppose there is a positive definiteLyapunov function 119881(119909 119905) defined on 119880 times 119877

+ where 119880 isin

1198800is the neighbourhood of the origin There are positive real

constants 119888 gt 0 and 0 lt 120572 lt 1 such that (119909 119905) + 119888119881120572

(119909 119905) isnegative semidefinite on 119880 Then 119881(119909 119905) is locally finite-timeconvergent with a settling time

119879 le1198811minus120572

(1199090(119905))

119888 (1 minus 120572) (1)

3 Model and Problem Description

The multiagent systems under consideration comprise 119899

agents and 119898 targets Each agent is regarded as a node inan undirected graph 119866 Each edge (119904

119895 119904119894) corresponds to

an available information path from agent 119895 to 119894 Moreovereach agent updates its current state based on the informationreceived from its neighbors We suppose that the dynamic ofthe 119894th agent is

119910119894(119905) = 119906

119894(119905) 119894 isin I = 1 2 119899 (2)

where 119910119894(119905) isin 119877

2 denotes the position and 119906119894(119905) isin 119877

2 is thecontrol input of 119894th agent at time 119905

To simplify the analysis we will consider the dynamics inpolar coordinate system corresponding to system (2)

119897119894(119905) = V

119894(119905)

120579119894(119905) = 120596

119894(119905)

119894 isin I

(3)

where 119897119894(119905) isin 119877 and 120579

119894(119905) isin 119877 respectively denote the radius

and angle of the 119894th agent in the polar coordinate systemwhich regards the geometric center 119875 = (1119898)sum

119898

119894=1119903119894(119905) as

the origin 119903119894(119905) represents the position of the 119894th target at time

119905 Obviously 119910119894(119905) = 119901

119894(119905) + [119897

119894(119905)cos(120579

119894(119905)) 119897119894(119905)sin(120579

119894(119905))]119879

where 119901119894(119905) denotes the estimated value of the distance from

the 119894th agent to the geometric center 119875We say the control protocol 119906

119894(119905) can solve the distributed

encirclement problems of system (2) if the states of agentssatisfy

lim119905rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

119910119894(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

minus 119896max119894isinI

1003817100381710038171003817100381710038171003817100381710038171003817

119903119894(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

= 0

lim119905rarrinfin

[120579119894(119905) minus 120579

119895(119905) minus

2120587 (119894 minus 119895)

119899] = 0

119894 isin I

(4)

where 119896 gt 1

Assumption 3 The 119894th agent can track 119899119894ge 1 targets and each

target can only be tracked by one agentLet 119903119894(119905) = (1119899

119894) sum119899119894

119896=1119903119894119896(119905) and Ψ

119894(119905) = (119899119898)119899

119894119903119894(119905)

then (1119899)sum119899

119894=1Ψ119894(119905) = (1119898)sum

119898

119894=1119903119894(119905) To solve the dis-

tributed encirclement control problem of system (3) we canfirst estimate the distributed center of the targets and thendrive the agents encirclement and track the targets Thefollowing control protocol is

V119894(119905) = minus119896

1sign (119897

119894(119905) minus 119896120588

119894(119905))

120596119894(119905) = sum

119895isin119873119894(119905)

119886119894119895(120579119895(119905) minus 120579

119894(119905) minus

2120587 (119895 minus 119894)

119899)

119894 isin I

(5)

where 120588119894(119905) denotes the estimated value ofmaximumdistance

from 119903119894(119905) to the distributed center for the 119894th agent and 119896

1gt 0

represents the control parameterThe estimator of the 119894th agent corresponding to the dis-

tributed center positions 120588119894(119905) is given as follows

119894(119905) = 120572 sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905))

119901119894(119905) = 120593

119894(119905) + Ψ

119894(119905)

119894 isin I

(6)

Mathematical Problems in Engineering 3

where 120593119894(119905) is the intermediate variable with 120593

119894(0) = 0 and

120572 gt 0 is the control parameter

Assumption 4 The speed of all targets has the common upperbound as the definition in [17] then there exists 120573 gt 0 suchthat 119903

119894(119905) le 120573 119894 isin 1 2 119898

Noting that 119889119894(119905) = 119903

119894(119905)minus119901

119894(119905) 119894 isin I the estimator of

the 119894th agent corresponding to 120588119894(119905) is rewritten as follows

120588119894(119905) = minus119896

2sign [120588

119894(119905) minusmax

119895isinI119889119895(119905)] 119894 isin I (7)

where 1198962gt 0 represents the control parameter

To simplify analysis we transform the original system(3)ndash(5) into an equivalent system Let 120579

119894(119905) = 120579

119894(119905) minus 2119894120587119899

then the closed-loop system (3)ndash(5) is given as follows

119897119894(119905) = minus119896

1sign (119897

119894(119905) minus 119896120588

119894(119905))

120579119894(119905) = sum

119895isin119873119894(119905)

119886119894119895(120579119895(119905) minus 120579

119894(119905))

119894 isin I

(8)

Let 120579 = [1205791(119905) 1205792(119905) 120579

119899(119905)] then

120579(119905) = minus119871120579 So thecontrol protocol corresponding to system (2) is

119906119894(119905) = [

V119894(119905) cos (120579

119894(119905)) minus 119897

119894(119905) 120596119894(119905) sin (120579

119894(119905))

V119894(119905) sin (120579

119894(119905)) + 119897

119894(119905) 120596119894(119905) cos (120579

119894(119905))

]

+ 119894(119905) 119894 isin I

(9)

4 Main Results

Lemma 5 Considering estimator (6) if 120572 gt 1198992

120573 andAssumptions 3 and 4 are both satisfied there must exist 119879

1gt 0

such that lim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 for any 119894 isin I

That is to say the estimate 119901119894(119905) corresponding to distributed

center for all agents will converge to the distributed center oftargets in finite time

Proof Let 119901(119905) = [119901119879

1(119905) 119901119879

2(119905) 119901

119879

119899(119905)]119879 We define the

Lyapunov function as follows

1198811(119905)

=1

2

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

(10)

then the derivative of1198811(119905) along the trajectories of system (6)

is given by

1(119905) =

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[119894(119905)

minus1

119899

119899

sum

119896=1

119896(119905)]

=

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

sdot [

[

120572 sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905)) + Ψ

119894(119905)

minus1

119899

119899

sum

119896=1

119896(119905)]

]

(11)

For

120572

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

sdot sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905))

=120572

2

sdot

119899

sum

119894=1

sum

119895isin119873119894(119905)

[119901119894(119905) minus 119901

119895(119905)]119879

sign (119901119895(119905) minus 119901

119894(119905))

le minus120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

10038171003817100381710038171003817Ψ119894(119905)10038171003817100381710038171003817=

1003817100381710038171003817100381710038171003817

119899

119898119899119894

119903119894(119905)

1003817100381710038171003817100381710038171003817=

1003817100381710038171003817100381710038171003817100381710038171003817

119899

119898

119899119894

sum

119896=1

119903119894119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

le119899

119898119899119894120573 le 119899120573

(12)

we have

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

Ψ119894(119905)

le 119899120573

119899

sum

119894=1

1003817100381710038171003817100381710038171003817100381710038171003817

119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

le 119899120573

119899

sum

119895=1119895 =119894

max119894=12119899

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= 1198992

120573 max119894119895=12119899

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

le1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(13)

4 Mathematical Problems in Engineering

For

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[minus1

119899

119899

sum

119896=1

119896(119905)]

= [

119899

sum

119894=1

119901119894(119905) minus

119899

sum

119896=1

119901119896(119905)] [minus

1

119899

119899

sum

119896=1

119896(119905)] = 0

(14)

we get

1(119905) le minus

120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

+1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= (1198992

120573

2minus120572

2)

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(15)

Let 119898(119905) = max119894119895isinI119901119894(119905) minus 119901

119895(119905) For 119901

119894(119905) minus

(1119899)sum119899

119896=1119901119896(119905) le (1119899)sum

119899

119896=1119901119894(119905) minus 119901

119896(119905) le 119898(119905) we

obtain 1(119905) le(1198992)119898(119905)

2 Furthermore forsum119899119894=1sum119895isin119873119894(119905)

119901119894(119905) minus

119901119895(119905) ge 119898(119905) we have

1(119905) + (

120572

2minus1198992

120573

2)radic

119899

21198811(119905)12

le (1198992

120573

2minus120572

2)119898 (119905) + radic

119899

2(120572

2minus1198992

120573

2)radic

119899

2119898 (119905)

= 0

(16)

Therefore from Lemma 2 there exists 1198791

gt 0 suchthat lim

119905rarr1198791[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 and we get

lim119905rarr1198791

[119901119894(119905) minus 119901

119895(119905)] = 0

Lemma 6 Considering estimator (7) one supposes thatAssumptions 3 and 4 are both satisfied If 119896

2gt 2120573 estimator

(7) must be steady in finite time

Proof We define the Lyapunov function as follows

1198812(119905) =

119899

sum

119894=1

120576119894sign (120576

119894) (17)

where 120576119894= 120588119894(119905)minusmax

119895isinI119889119895(119905)Then the derivative of1198812(119905)

along the trajectories of system (7) is given by

2(119905) =

119899

sum

119894=1

sign (120576119894) 120576119894=

119899

sum

119894=1

sign (120576119894)

sdot [minus1198962sign(120588

119894(119905) minusmax

119895isinI119889119895(119905))

minusmax119895isinI

119889119895(119905)]

(18)

Let 119889(119905) = max119895isinI119889119895(119905) = max

119895isinI119903119895(119905) minus 119901119895(119905)

According to Lemma 5 there exists 1198791gt 0 such that 119901

119894(119905) =

(1119898)sum119898

119896=1119903119896(119905) when 119905 gt 119879

1

Then we have

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

max119895isinI

119889119895(119905)

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

max119895isinI

1003817100381710038171003817100381710038171003817100381710038171003817

1

119899119895

119899119895

sum

119896=1

119903119895119896(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

1003816100381610038161003816100381610038161003816100381610038161003816

le max119895isinI

1

119899119895

119899119895

sum

119896=1

10038171003817100381710038171003817119903119895119896(119905)10038171003817100381710038171003817+

1

119898

119898

sum

119896=1

1003817100381710038171003817119903119896(119905)1003817100381710038171003817 le 2120573

(19)

Hence

2(119905) le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] +

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816

le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] + 2120573

=

119899

sum

119894=1

minus1198962

1003816100381610038161003816sign (120576119894)1003816100381610038161003816 + 2120573

(20)

According to the condition 1198962gt 2120573 we conclude that system

(7) must be steady in finite time

Lemma 7 Considering the first equation of system (8) onesupposes that Assumptions 3 and 4 are both satisfied If 119896

2gt 2120573

and 1198961gt 2119896120573 this system can be steady in finite time

Proof It is easy to get the conclusion according to Lemma 4in [1] so we ignore the proof here

Theorem 8 Considering system (2) if the network topologyof multiagent systems is connected and Assumptions 3 and 4are both satisfied letting 120572 gt 119899

2

120573 1198961gt 2119896120573 and 119896

2gt 2120573

then protocol (9) can solve the distributed encirclement controlproblem of system (2)

Proof From Lemma 5 there exists 1198791

gt 0 such thatlim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 From Lemma 7 there

exists 1198792gt 1198791such that lim

119905rarr1198792119910119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905) minus

119896max119895isinI119903119895(119905) minus (1119898)sum

119898

119896=1119903119896(119905) = 0 As graph 119866 is

connected the Laplacian 119871 of 119866 has nonnegative real partFor

120579(119905) = minus119871120579(119905) then we have lim119905rarrinfin

[120579119894(119905) minus120579

119895(119905)] = 0 so

lim119905rarrinfin

[120579119894(119905) minus 120579

119895(119905) minus 2120587(119894 minus 119895)119899] = 0

Mathematical Problems in Engineering 5

1 2

34

Figure 1 The fixed network topology of multiagent systems

50 6010 20 30 40minus10 0minus20

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

30

20

10

0

minus10

minus20

minus30

minus40

minus50

Figure 2 The trajectories of the agents and targets of multiagentsystems with fixed topology

5 Simulation Results and Analysis

In this section the results of simulation by Matlab prove theeffectiveness of the theoretical results obtained The dynamictargets are

1199031(119905) = [

119905 + 1 minus sin 119905minus119905 + 10 minus cos 119905

]

1199032(119905) = [

119905 + 10 minus sin 119905minus119905 + 2 minus cos 119905

]

1199033(119905) = [

119905 + 2 minus sin 119905minus119905 + 3 minus cos 119905

]

1199034(119905) = [

119905 + 8 minus sin 119905minus119905 + 4 minus cos 119905

]

1199035(119905) = [

119905 + 4 minus sin 119905minus119905 + 5 minus cos 119905

]

1199036(119905) = [

119905 + 6 minus sin 119905minus119905 + 7 minus cos 119905

]

(21)

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34Ga Gb Gc Gd Ge

Figure 3 The five kinds of topologies of multiagent systems

0 10 20 30 40 50 60minus10

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

20

10

0

minus10

minus20

minus30

minus40

Figure 4 The trajectory of the agents and targets of multiagentsystems with switching topology randomly

The corresponding fixed network topology of multiagentsystems with 4 nodes is shown in Figure 1 Let 119896 = 2 120573 = 22120572 = 33 119896

1= 10 and 119896

2= 10 and the initial conditions

are (9 9) (10 9) (minus6 2) and (1 minus5) Figure 2 represents thetrajectories of the agents and targets of multiagent systemsand it shows that the multiagent systems with fixed topologycan encircle the multiple targets in the form of circularformation

Figure 3 shows the five kinds of topologies of multiagentsystems with 4 nodes and these topologies can realize therandom switch obeyed uniform distribution among themLet 119896 = 2 120573 = 22 120572 = 33 119896

1= 10 and 119896

2= 10

and the initial conditions are (1 9) (10 1) (2 2) and (1 3)Figure 4 represents the trajectories of the agents and targets ofmultiagent systems and it shows that the multiagent systemswith switching topology randomly can encircle the multipletargets in the circular formation

6 Conclusion

In this paper we investigate the distributed encirclementof multiagent systems with multiple dynamic targets withthe assumption that each agent can track multiple targetseach target only can be tracked by one agent and thenumbers of the agents and the targets are the same or not

6 Mathematical Problems in Engineering

The encirclement and tracking method in circular formationis proposed Considering that each agent can only get partialinformation of targets the target state estimators which canestimate the average position of targets are designed In finitetime every agentrsquos motion radius is locally converged tocircular formation radius of system within a settling timeAll agents can maintain the formation which can be updatedin real time according to the change of targetsrsquo state ByLyapunov function it is proved that every agent can get thewhole information of targets in finite time and meanwhile itrealizes the circular formation in finite time The simulationresults illustrate that this proposedmethod is effective for notonly multiple static targets but also multiple dynamic targets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (973 Program) (Grant no 2012CB215203)and the National Natural Science Foundation of China(Grants nos 61304155 61203080 and 61573082)

References

[1] W Ren and Y Cao Distributed Coordination of Multi-agentNetworks Springer New York NY USA 2011

[2] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[3] P Lin andY Jia ldquoAverage consensus in networks ofmulti-agentswith both switching topology and coupling time-delayrdquo PhysicaA vol 387 no 1 pp 303ndash313 2008

[4] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[5] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[6] P Lin Y Jia and L Li ldquoDistributed robust 119867infin

consensuscontrol in directed networks of agents with time-delayrdquo Systemsamp Control Letters vol 57 no 8 pp 643ndash653 2008

[7] L Mo and Y Jia ldquo119867infinconsensus control of a class of high-order

multi-agent systemsrdquo IET Control Theory amp Applications vol 5no 1 pp 247ndash253 2011

[8] P Lin and W Ren ldquoConstrained consensus in unbalancednetworks with communication delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 3 pp 775ndash781 2014

[9] A JMarasco S NGivigi andCA Rabbath ldquoModel predictivecontrol for the dynamic encirclement of a targetrdquo in Proceedingsof the American Control Conference (ACC rsquo12) pp 2004ndash2009Montreal Canada June 2012

[10] F Chen W Ren and Y Cao ldquoSurrounding control in cooper-ative agent networksrdquo Systems amp Control Letters vol 59 no 11pp 704ndash712 2010

[11] T Wei and X Chen ldquoCollective surrounding control in multi-agent networksrdquo Chinese Physics B vol 23 no 5 Article ID050201 4 pages 2014

[12] P Lin and Y Jia ldquoConsensus of second-order discrete-timemulti-agent systems with nonuniform time-delays and dynam-ically changing topologiesrdquoAutomatica vol 45 no 9 pp 2154ndash2158 2009

[13] P Lin W Ren and Y Song ldquoDistributed multi-agent optimiza-tion subject to nonidentical constraints and communicationdelaysrdquo Automatica vol 65 pp 120ndash131 2016

[14] P Lin and Y Jia ldquoMulti-agent consensus with diverse time-delays and jointly-connected topologiesrdquo Automatica vol 47no 4 pp 848ndash856 2011

[15] Y Hong L Gao D Cheng and J Hu ldquoLyapunov-basedapproach to multiagent systems with switching jointly con-nected interconnectionrdquo IEEE Transactions on Automatic Con-trol vol 52 no 5 pp 943ndash948 2007

[16] P Lin K Qin Z Li andW Ren ldquoCollective rotatingmotions ofsecond-order multi-agent systems in three-dimensional spacerdquoSystems amp Control Letters vol 60 no 6 pp 365ndash372 2011

[17] L Mo Y Niu and T Pan ldquoConsensus of heterogeneous multi-agent systems with switching jointly-connected interconnec-tionrdquo Physica A vol 427 pp 132ndash140 2015

[18] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[19] S P Bhat andD S Bernstein ldquoContinuous finite-time stabiliza-tion of the translational and rotational double integratorsrdquo IEEETransactions on Automatic Control vol 43 no 5 pp 678ndash6821998

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

encirclement are proposed by Lyapunov theory Finally weprove this theory to be effective by the simulation

The rest of this paper is organized as follows In Section 2we introduce some basic notations and some concepts ingraph theory In Section 3 the model to be researched isformulated and a distributed encirclement control protocolis proposed In Section 4 the main results are stated andderived In Section 5 numerical simulations are providedto demonstrate the effectiveness of the obtained theoreticalresults In Section 6 we conclude this paper

2 Notations and Preliminaries

Let 119866(VEA) be an undirected graph where V =

1199041 1199042 119904

119899 is the set of nodes and E isin V times V is the

set of edges The node indexes belong to a finite index setI = 1 2 119899 and119873

119894= 119904119895isin V(119904

119894 119904119895) isin E is defined as

the neighbourhood set of 119904119894A = [119886

119894119895] isin R119899times119899 is a symmetric

weighted adjacency matrix where the element 119886119894119895represents

the weight from node 119904119894to node 119904

119895 When 119904

119895isin 119873119894 then

119886119894119895gt 0 or else 119886

119894119895= 0 In the undirected graph any (119904

119894 119904119895) isin

E hArr (119904119895 119904119894) isin E The graph Laplacian with the diagraph is

defined as 119871 = [119897119894119895] where 119897

119894119894= sum119899

119895=1119886119894119895and 119897119894119895= minus119886119894119895 119894 = 119895 If

there is a path from every node to every other node the graphis said to be connected and undirected

Lemma 1 (see [18]) If the undirected graph 119866 is connectedthen its Laplacian 119871 satisfies the following

(1) Zero is a simple eigenvalue of 119871 and 1119899is the corre-

sponding eigenvector and 1198711119899= 0

(2) The remaining 119899 minus 1 eigenvalues of 119871 all have positivereal parts And 119871 is a symmetric matrix and theeigenvalues 0 = 120582min = 120582

1le 1205822sdot sdot sdot le 120582

119899= 120582max

Lemma 2 (see [19]) Suppose there is a positive definiteLyapunov function 119881(119909 119905) defined on 119880 times 119877

+ where 119880 isin

1198800is the neighbourhood of the origin There are positive real

constants 119888 gt 0 and 0 lt 120572 lt 1 such that (119909 119905) + 119888119881120572

(119909 119905) isnegative semidefinite on 119880 Then 119881(119909 119905) is locally finite-timeconvergent with a settling time

119879 le1198811minus120572

(1199090(119905))

119888 (1 minus 120572) (1)

3 Model and Problem Description

The multiagent systems under consideration comprise 119899

agents and 119898 targets Each agent is regarded as a node inan undirected graph 119866 Each edge (119904

119895 119904119894) corresponds to

an available information path from agent 119895 to 119894 Moreovereach agent updates its current state based on the informationreceived from its neighbors We suppose that the dynamic ofthe 119894th agent is

119910119894(119905) = 119906

119894(119905) 119894 isin I = 1 2 119899 (2)

where 119910119894(119905) isin 119877

2 denotes the position and 119906119894(119905) isin 119877

2 is thecontrol input of 119894th agent at time 119905

To simplify the analysis we will consider the dynamics inpolar coordinate system corresponding to system (2)

119897119894(119905) = V

119894(119905)

120579119894(119905) = 120596

119894(119905)

119894 isin I

(3)

where 119897119894(119905) isin 119877 and 120579

119894(119905) isin 119877 respectively denote the radius

and angle of the 119894th agent in the polar coordinate systemwhich regards the geometric center 119875 = (1119898)sum

119898

119894=1119903119894(119905) as

the origin 119903119894(119905) represents the position of the 119894th target at time

119905 Obviously 119910119894(119905) = 119901

119894(119905) + [119897

119894(119905)cos(120579

119894(119905)) 119897119894(119905)sin(120579

119894(119905))]119879

where 119901119894(119905) denotes the estimated value of the distance from

the 119894th agent to the geometric center 119875We say the control protocol 119906

119894(119905) can solve the distributed

encirclement problems of system (2) if the states of agentssatisfy

lim119905rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

119910119894(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

minus 119896max119894isinI

1003817100381710038171003817100381710038171003817100381710038171003817

119903119894(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

= 0

lim119905rarrinfin

[120579119894(119905) minus 120579

119895(119905) minus

2120587 (119894 minus 119895)

119899] = 0

119894 isin I

(4)

where 119896 gt 1

Assumption 3 The 119894th agent can track 119899119894ge 1 targets and each

target can only be tracked by one agentLet 119903119894(119905) = (1119899

119894) sum119899119894

119896=1119903119894119896(119905) and Ψ

119894(119905) = (119899119898)119899

119894119903119894(119905)

then (1119899)sum119899

119894=1Ψ119894(119905) = (1119898)sum

119898

119894=1119903119894(119905) To solve the dis-

tributed encirclement control problem of system (3) we canfirst estimate the distributed center of the targets and thendrive the agents encirclement and track the targets Thefollowing control protocol is

V119894(119905) = minus119896

1sign (119897

119894(119905) minus 119896120588

119894(119905))

120596119894(119905) = sum

119895isin119873119894(119905)

119886119894119895(120579119895(119905) minus 120579

119894(119905) minus

2120587 (119895 minus 119894)

119899)

119894 isin I

(5)

where 120588119894(119905) denotes the estimated value ofmaximumdistance

from 119903119894(119905) to the distributed center for the 119894th agent and 119896

1gt 0

represents the control parameterThe estimator of the 119894th agent corresponding to the dis-

tributed center positions 120588119894(119905) is given as follows

119894(119905) = 120572 sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905))

119901119894(119905) = 120593

119894(119905) + Ψ

119894(119905)

119894 isin I

(6)

Mathematical Problems in Engineering 3

where 120593119894(119905) is the intermediate variable with 120593

119894(0) = 0 and

120572 gt 0 is the control parameter

Assumption 4 The speed of all targets has the common upperbound as the definition in [17] then there exists 120573 gt 0 suchthat 119903

119894(119905) le 120573 119894 isin 1 2 119898

Noting that 119889119894(119905) = 119903

119894(119905)minus119901

119894(119905) 119894 isin I the estimator of

the 119894th agent corresponding to 120588119894(119905) is rewritten as follows

120588119894(119905) = minus119896

2sign [120588

119894(119905) minusmax

119895isinI119889119895(119905)] 119894 isin I (7)

where 1198962gt 0 represents the control parameter

To simplify analysis we transform the original system(3)ndash(5) into an equivalent system Let 120579

119894(119905) = 120579

119894(119905) minus 2119894120587119899

then the closed-loop system (3)ndash(5) is given as follows

119897119894(119905) = minus119896

1sign (119897

119894(119905) minus 119896120588

119894(119905))

120579119894(119905) = sum

119895isin119873119894(119905)

119886119894119895(120579119895(119905) minus 120579

119894(119905))

119894 isin I

(8)

Let 120579 = [1205791(119905) 1205792(119905) 120579

119899(119905)] then

120579(119905) = minus119871120579 So thecontrol protocol corresponding to system (2) is

119906119894(119905) = [

V119894(119905) cos (120579

119894(119905)) minus 119897

119894(119905) 120596119894(119905) sin (120579

119894(119905))

V119894(119905) sin (120579

119894(119905)) + 119897

119894(119905) 120596119894(119905) cos (120579

119894(119905))

]

+ 119894(119905) 119894 isin I

(9)

4 Main Results

Lemma 5 Considering estimator (6) if 120572 gt 1198992

120573 andAssumptions 3 and 4 are both satisfied there must exist 119879

1gt 0

such that lim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 for any 119894 isin I

That is to say the estimate 119901119894(119905) corresponding to distributed

center for all agents will converge to the distributed center oftargets in finite time

Proof Let 119901(119905) = [119901119879

1(119905) 119901119879

2(119905) 119901

119879

119899(119905)]119879 We define the

Lyapunov function as follows

1198811(119905)

=1

2

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

(10)

then the derivative of1198811(119905) along the trajectories of system (6)

is given by

1(119905) =

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[119894(119905)

minus1

119899

119899

sum

119896=1

119896(119905)]

=

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

sdot [

[

120572 sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905)) + Ψ

119894(119905)

minus1

119899

119899

sum

119896=1

119896(119905)]

]

(11)

For

120572

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

sdot sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905))

=120572

2

sdot

119899

sum

119894=1

sum

119895isin119873119894(119905)

[119901119894(119905) minus 119901

119895(119905)]119879

sign (119901119895(119905) minus 119901

119894(119905))

le minus120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

10038171003817100381710038171003817Ψ119894(119905)10038171003817100381710038171003817=

1003817100381710038171003817100381710038171003817

119899

119898119899119894

119903119894(119905)

1003817100381710038171003817100381710038171003817=

1003817100381710038171003817100381710038171003817100381710038171003817

119899

119898

119899119894

sum

119896=1

119903119894119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

le119899

119898119899119894120573 le 119899120573

(12)

we have

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

Ψ119894(119905)

le 119899120573

119899

sum

119894=1

1003817100381710038171003817100381710038171003817100381710038171003817

119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

le 119899120573

119899

sum

119895=1119895 =119894

max119894=12119899

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= 1198992

120573 max119894119895=12119899

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

le1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(13)

4 Mathematical Problems in Engineering

For

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[minus1

119899

119899

sum

119896=1

119896(119905)]

= [

119899

sum

119894=1

119901119894(119905) minus

119899

sum

119896=1

119901119896(119905)] [minus

1

119899

119899

sum

119896=1

119896(119905)] = 0

(14)

we get

1(119905) le minus

120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

+1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= (1198992

120573

2minus120572

2)

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(15)

Let 119898(119905) = max119894119895isinI119901119894(119905) minus 119901

119895(119905) For 119901

119894(119905) minus

(1119899)sum119899

119896=1119901119896(119905) le (1119899)sum

119899

119896=1119901119894(119905) minus 119901

119896(119905) le 119898(119905) we

obtain 1(119905) le(1198992)119898(119905)

2 Furthermore forsum119899119894=1sum119895isin119873119894(119905)

119901119894(119905) minus

119901119895(119905) ge 119898(119905) we have

1(119905) + (

120572

2minus1198992

120573

2)radic

119899

21198811(119905)12

le (1198992

120573

2minus120572

2)119898 (119905) + radic

119899

2(120572

2minus1198992

120573

2)radic

119899

2119898 (119905)

= 0

(16)

Therefore from Lemma 2 there exists 1198791

gt 0 suchthat lim

119905rarr1198791[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 and we get

lim119905rarr1198791

[119901119894(119905) minus 119901

119895(119905)] = 0

Lemma 6 Considering estimator (7) one supposes thatAssumptions 3 and 4 are both satisfied If 119896

2gt 2120573 estimator

(7) must be steady in finite time

Proof We define the Lyapunov function as follows

1198812(119905) =

119899

sum

119894=1

120576119894sign (120576

119894) (17)

where 120576119894= 120588119894(119905)minusmax

119895isinI119889119895(119905)Then the derivative of1198812(119905)

along the trajectories of system (7) is given by

2(119905) =

119899

sum

119894=1

sign (120576119894) 120576119894=

119899

sum

119894=1

sign (120576119894)

sdot [minus1198962sign(120588

119894(119905) minusmax

119895isinI119889119895(119905))

minusmax119895isinI

119889119895(119905)]

(18)

Let 119889(119905) = max119895isinI119889119895(119905) = max

119895isinI119903119895(119905) minus 119901119895(119905)

According to Lemma 5 there exists 1198791gt 0 such that 119901

119894(119905) =

(1119898)sum119898

119896=1119903119896(119905) when 119905 gt 119879

1

Then we have

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

max119895isinI

119889119895(119905)

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

max119895isinI

1003817100381710038171003817100381710038171003817100381710038171003817

1

119899119895

119899119895

sum

119896=1

119903119895119896(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

1003816100381610038161003816100381610038161003816100381610038161003816

le max119895isinI

1

119899119895

119899119895

sum

119896=1

10038171003817100381710038171003817119903119895119896(119905)10038171003817100381710038171003817+

1

119898

119898

sum

119896=1

1003817100381710038171003817119903119896(119905)1003817100381710038171003817 le 2120573

(19)

Hence

2(119905) le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] +

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816

le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] + 2120573

=

119899

sum

119894=1

minus1198962

1003816100381610038161003816sign (120576119894)1003816100381610038161003816 + 2120573

(20)

According to the condition 1198962gt 2120573 we conclude that system

(7) must be steady in finite time

Lemma 7 Considering the first equation of system (8) onesupposes that Assumptions 3 and 4 are both satisfied If 119896

2gt 2120573

and 1198961gt 2119896120573 this system can be steady in finite time

Proof It is easy to get the conclusion according to Lemma 4in [1] so we ignore the proof here

Theorem 8 Considering system (2) if the network topologyof multiagent systems is connected and Assumptions 3 and 4are both satisfied letting 120572 gt 119899

2

120573 1198961gt 2119896120573 and 119896

2gt 2120573

then protocol (9) can solve the distributed encirclement controlproblem of system (2)

Proof From Lemma 5 there exists 1198791

gt 0 such thatlim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 From Lemma 7 there

exists 1198792gt 1198791such that lim

119905rarr1198792119910119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905) minus

119896max119895isinI119903119895(119905) minus (1119898)sum

119898

119896=1119903119896(119905) = 0 As graph 119866 is

connected the Laplacian 119871 of 119866 has nonnegative real partFor

120579(119905) = minus119871120579(119905) then we have lim119905rarrinfin

[120579119894(119905) minus120579

119895(119905)] = 0 so

lim119905rarrinfin

[120579119894(119905) minus 120579

119895(119905) minus 2120587(119894 minus 119895)119899] = 0

Mathematical Problems in Engineering 5

1 2

34

Figure 1 The fixed network topology of multiagent systems

50 6010 20 30 40minus10 0minus20

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

30

20

10

0

minus10

minus20

minus30

minus40

minus50

Figure 2 The trajectories of the agents and targets of multiagentsystems with fixed topology

5 Simulation Results and Analysis

In this section the results of simulation by Matlab prove theeffectiveness of the theoretical results obtained The dynamictargets are

1199031(119905) = [

119905 + 1 minus sin 119905minus119905 + 10 minus cos 119905

]

1199032(119905) = [

119905 + 10 minus sin 119905minus119905 + 2 minus cos 119905

]

1199033(119905) = [

119905 + 2 minus sin 119905minus119905 + 3 minus cos 119905

]

1199034(119905) = [

119905 + 8 minus sin 119905minus119905 + 4 minus cos 119905

]

1199035(119905) = [

119905 + 4 minus sin 119905minus119905 + 5 minus cos 119905

]

1199036(119905) = [

119905 + 6 minus sin 119905minus119905 + 7 minus cos 119905

]

(21)

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34Ga Gb Gc Gd Ge

Figure 3 The five kinds of topologies of multiagent systems

0 10 20 30 40 50 60minus10

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

20

10

0

minus10

minus20

minus30

minus40

Figure 4 The trajectory of the agents and targets of multiagentsystems with switching topology randomly

The corresponding fixed network topology of multiagentsystems with 4 nodes is shown in Figure 1 Let 119896 = 2 120573 = 22120572 = 33 119896

1= 10 and 119896

2= 10 and the initial conditions

are (9 9) (10 9) (minus6 2) and (1 minus5) Figure 2 represents thetrajectories of the agents and targets of multiagent systemsand it shows that the multiagent systems with fixed topologycan encircle the multiple targets in the form of circularformation

Figure 3 shows the five kinds of topologies of multiagentsystems with 4 nodes and these topologies can realize therandom switch obeyed uniform distribution among themLet 119896 = 2 120573 = 22 120572 = 33 119896

1= 10 and 119896

2= 10

and the initial conditions are (1 9) (10 1) (2 2) and (1 3)Figure 4 represents the trajectories of the agents and targets ofmultiagent systems and it shows that the multiagent systemswith switching topology randomly can encircle the multipletargets in the circular formation

6 Conclusion

In this paper we investigate the distributed encirclementof multiagent systems with multiple dynamic targets withthe assumption that each agent can track multiple targetseach target only can be tracked by one agent and thenumbers of the agents and the targets are the same or not

6 Mathematical Problems in Engineering

The encirclement and tracking method in circular formationis proposed Considering that each agent can only get partialinformation of targets the target state estimators which canestimate the average position of targets are designed In finitetime every agentrsquos motion radius is locally converged tocircular formation radius of system within a settling timeAll agents can maintain the formation which can be updatedin real time according to the change of targetsrsquo state ByLyapunov function it is proved that every agent can get thewhole information of targets in finite time and meanwhile itrealizes the circular formation in finite time The simulationresults illustrate that this proposedmethod is effective for notonly multiple static targets but also multiple dynamic targets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (973 Program) (Grant no 2012CB215203)and the National Natural Science Foundation of China(Grants nos 61304155 61203080 and 61573082)

References

[1] W Ren and Y Cao Distributed Coordination of Multi-agentNetworks Springer New York NY USA 2011

[2] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[3] P Lin andY Jia ldquoAverage consensus in networks ofmulti-agentswith both switching topology and coupling time-delayrdquo PhysicaA vol 387 no 1 pp 303ndash313 2008

[4] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[5] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[6] P Lin Y Jia and L Li ldquoDistributed robust 119867infin

consensuscontrol in directed networks of agents with time-delayrdquo Systemsamp Control Letters vol 57 no 8 pp 643ndash653 2008

[7] L Mo and Y Jia ldquo119867infinconsensus control of a class of high-order

multi-agent systemsrdquo IET Control Theory amp Applications vol 5no 1 pp 247ndash253 2011

[8] P Lin and W Ren ldquoConstrained consensus in unbalancednetworks with communication delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 3 pp 775ndash781 2014

[9] A JMarasco S NGivigi andCA Rabbath ldquoModel predictivecontrol for the dynamic encirclement of a targetrdquo in Proceedingsof the American Control Conference (ACC rsquo12) pp 2004ndash2009Montreal Canada June 2012

[10] F Chen W Ren and Y Cao ldquoSurrounding control in cooper-ative agent networksrdquo Systems amp Control Letters vol 59 no 11pp 704ndash712 2010

[11] T Wei and X Chen ldquoCollective surrounding control in multi-agent networksrdquo Chinese Physics B vol 23 no 5 Article ID050201 4 pages 2014

[12] P Lin and Y Jia ldquoConsensus of second-order discrete-timemulti-agent systems with nonuniform time-delays and dynam-ically changing topologiesrdquoAutomatica vol 45 no 9 pp 2154ndash2158 2009

[13] P Lin W Ren and Y Song ldquoDistributed multi-agent optimiza-tion subject to nonidentical constraints and communicationdelaysrdquo Automatica vol 65 pp 120ndash131 2016

[14] P Lin and Y Jia ldquoMulti-agent consensus with diverse time-delays and jointly-connected topologiesrdquo Automatica vol 47no 4 pp 848ndash856 2011

[15] Y Hong L Gao D Cheng and J Hu ldquoLyapunov-basedapproach to multiagent systems with switching jointly con-nected interconnectionrdquo IEEE Transactions on Automatic Con-trol vol 52 no 5 pp 943ndash948 2007

[16] P Lin K Qin Z Li andW Ren ldquoCollective rotatingmotions ofsecond-order multi-agent systems in three-dimensional spacerdquoSystems amp Control Letters vol 60 no 6 pp 365ndash372 2011

[17] L Mo Y Niu and T Pan ldquoConsensus of heterogeneous multi-agent systems with switching jointly-connected interconnec-tionrdquo Physica A vol 427 pp 132ndash140 2015

[18] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[19] S P Bhat andD S Bernstein ldquoContinuous finite-time stabiliza-tion of the translational and rotational double integratorsrdquo IEEETransactions on Automatic Control vol 43 no 5 pp 678ndash6821998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where 120593119894(119905) is the intermediate variable with 120593

119894(0) = 0 and

120572 gt 0 is the control parameter

Assumption 4 The speed of all targets has the common upperbound as the definition in [17] then there exists 120573 gt 0 suchthat 119903

119894(119905) le 120573 119894 isin 1 2 119898

Noting that 119889119894(119905) = 119903

119894(119905)minus119901

119894(119905) 119894 isin I the estimator of

the 119894th agent corresponding to 120588119894(119905) is rewritten as follows

120588119894(119905) = minus119896

2sign [120588

119894(119905) minusmax

119895isinI119889119895(119905)] 119894 isin I (7)

where 1198962gt 0 represents the control parameter

To simplify analysis we transform the original system(3)ndash(5) into an equivalent system Let 120579

119894(119905) = 120579

119894(119905) minus 2119894120587119899

then the closed-loop system (3)ndash(5) is given as follows

119897119894(119905) = minus119896

1sign (119897

119894(119905) minus 119896120588

119894(119905))

120579119894(119905) = sum

119895isin119873119894(119905)

119886119894119895(120579119895(119905) minus 120579

119894(119905))

119894 isin I

(8)

Let 120579 = [1205791(119905) 1205792(119905) 120579

119899(119905)] then

120579(119905) = minus119871120579 So thecontrol protocol corresponding to system (2) is

119906119894(119905) = [

V119894(119905) cos (120579

119894(119905)) minus 119897

119894(119905) 120596119894(119905) sin (120579

119894(119905))

V119894(119905) sin (120579

119894(119905)) + 119897

119894(119905) 120596119894(119905) cos (120579

119894(119905))

]

+ 119894(119905) 119894 isin I

(9)

4 Main Results

Lemma 5 Considering estimator (6) if 120572 gt 1198992

120573 andAssumptions 3 and 4 are both satisfied there must exist 119879

1gt 0

such that lim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 for any 119894 isin I

That is to say the estimate 119901119894(119905) corresponding to distributed

center for all agents will converge to the distributed center oftargets in finite time

Proof Let 119901(119905) = [119901119879

1(119905) 119901119879

2(119905) 119901

119879

119899(119905)]119879 We define the

Lyapunov function as follows

1198811(119905)

=1

2

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

(10)

then the derivative of1198811(119905) along the trajectories of system (6)

is given by

1(119905) =

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[119894(119905)

minus1

119899

119899

sum

119896=1

119896(119905)]

=

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

sdot [

[

120572 sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905)) + Ψ

119894(119905)

minus1

119899

119899

sum

119896=1

119896(119905)]

]

(11)

For

120572

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

sdot sum

119895isin119873119894(119905)

sign (119901119895(119905) minus 119901

119894(119905))

=120572

2

sdot

119899

sum

119894=1

sum

119895isin119873119894(119905)

[119901119894(119905) minus 119901

119895(119905)]119879

sign (119901119895(119905) minus 119901

119894(119905))

le minus120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

10038171003817100381710038171003817Ψ119894(119905)10038171003817100381710038171003817=

1003817100381710038171003817100381710038171003817

119899

119898119899119894

119903119894(119905)

1003817100381710038171003817100381710038171003817=

1003817100381710038171003817100381710038171003817100381710038171003817

119899

119898

119899119894

sum

119896=1

119903119894119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

le119899

119898119899119894120573 le 119899120573

(12)

we have

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

Ψ119894(119905)

le 119899120573

119899

sum

119894=1

1003817100381710038171003817100381710038171003817100381710038171003817

119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

le 119899120573

119899

sum

119895=1119895 =119894

max119894=12119899

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= 1198992

120573 max119894119895=12119899

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

le1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(13)

4 Mathematical Problems in Engineering

For

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[minus1

119899

119899

sum

119896=1

119896(119905)]

= [

119899

sum

119894=1

119901119894(119905) minus

119899

sum

119896=1

119901119896(119905)] [minus

1

119899

119899

sum

119896=1

119896(119905)] = 0

(14)

we get

1(119905) le minus

120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

+1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= (1198992

120573

2minus120572

2)

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(15)

Let 119898(119905) = max119894119895isinI119901119894(119905) minus 119901

119895(119905) For 119901

119894(119905) minus

(1119899)sum119899

119896=1119901119896(119905) le (1119899)sum

119899

119896=1119901119894(119905) minus 119901

119896(119905) le 119898(119905) we

obtain 1(119905) le(1198992)119898(119905)

2 Furthermore forsum119899119894=1sum119895isin119873119894(119905)

119901119894(119905) minus

119901119895(119905) ge 119898(119905) we have

1(119905) + (

120572

2minus1198992

120573

2)radic

119899

21198811(119905)12

le (1198992

120573

2minus120572

2)119898 (119905) + radic

119899

2(120572

2minus1198992

120573

2)radic

119899

2119898 (119905)

= 0

(16)

Therefore from Lemma 2 there exists 1198791

gt 0 suchthat lim

119905rarr1198791[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 and we get

lim119905rarr1198791

[119901119894(119905) minus 119901

119895(119905)] = 0

Lemma 6 Considering estimator (7) one supposes thatAssumptions 3 and 4 are both satisfied If 119896

2gt 2120573 estimator

(7) must be steady in finite time

Proof We define the Lyapunov function as follows

1198812(119905) =

119899

sum

119894=1

120576119894sign (120576

119894) (17)

where 120576119894= 120588119894(119905)minusmax

119895isinI119889119895(119905)Then the derivative of1198812(119905)

along the trajectories of system (7) is given by

2(119905) =

119899

sum

119894=1

sign (120576119894) 120576119894=

119899

sum

119894=1

sign (120576119894)

sdot [minus1198962sign(120588

119894(119905) minusmax

119895isinI119889119895(119905))

minusmax119895isinI

119889119895(119905)]

(18)

Let 119889(119905) = max119895isinI119889119895(119905) = max

119895isinI119903119895(119905) minus 119901119895(119905)

According to Lemma 5 there exists 1198791gt 0 such that 119901

119894(119905) =

(1119898)sum119898

119896=1119903119896(119905) when 119905 gt 119879

1

Then we have

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

max119895isinI

119889119895(119905)

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

max119895isinI

1003817100381710038171003817100381710038171003817100381710038171003817

1

119899119895

119899119895

sum

119896=1

119903119895119896(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

1003816100381610038161003816100381610038161003816100381610038161003816

le max119895isinI

1

119899119895

119899119895

sum

119896=1

10038171003817100381710038171003817119903119895119896(119905)10038171003817100381710038171003817+

1

119898

119898

sum

119896=1

1003817100381710038171003817119903119896(119905)1003817100381710038171003817 le 2120573

(19)

Hence

2(119905) le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] +

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816

le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] + 2120573

=

119899

sum

119894=1

minus1198962

1003816100381610038161003816sign (120576119894)1003816100381610038161003816 + 2120573

(20)

According to the condition 1198962gt 2120573 we conclude that system

(7) must be steady in finite time

Lemma 7 Considering the first equation of system (8) onesupposes that Assumptions 3 and 4 are both satisfied If 119896

2gt 2120573

and 1198961gt 2119896120573 this system can be steady in finite time

Proof It is easy to get the conclusion according to Lemma 4in [1] so we ignore the proof here

Theorem 8 Considering system (2) if the network topologyof multiagent systems is connected and Assumptions 3 and 4are both satisfied letting 120572 gt 119899

2

120573 1198961gt 2119896120573 and 119896

2gt 2120573

then protocol (9) can solve the distributed encirclement controlproblem of system (2)

Proof From Lemma 5 there exists 1198791

gt 0 such thatlim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 From Lemma 7 there

exists 1198792gt 1198791such that lim

119905rarr1198792119910119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905) minus

119896max119895isinI119903119895(119905) minus (1119898)sum

119898

119896=1119903119896(119905) = 0 As graph 119866 is

connected the Laplacian 119871 of 119866 has nonnegative real partFor

120579(119905) = minus119871120579(119905) then we have lim119905rarrinfin

[120579119894(119905) minus120579

119895(119905)] = 0 so

lim119905rarrinfin

[120579119894(119905) minus 120579

119895(119905) minus 2120587(119894 minus 119895)119899] = 0

Mathematical Problems in Engineering 5

1 2

34

Figure 1 The fixed network topology of multiagent systems

50 6010 20 30 40minus10 0minus20

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

30

20

10

0

minus10

minus20

minus30

minus40

minus50

Figure 2 The trajectories of the agents and targets of multiagentsystems with fixed topology

5 Simulation Results and Analysis

In this section the results of simulation by Matlab prove theeffectiveness of the theoretical results obtained The dynamictargets are

1199031(119905) = [

119905 + 1 minus sin 119905minus119905 + 10 minus cos 119905

]

1199032(119905) = [

119905 + 10 minus sin 119905minus119905 + 2 minus cos 119905

]

1199033(119905) = [

119905 + 2 minus sin 119905minus119905 + 3 minus cos 119905

]

1199034(119905) = [

119905 + 8 minus sin 119905minus119905 + 4 minus cos 119905

]

1199035(119905) = [

119905 + 4 minus sin 119905minus119905 + 5 minus cos 119905

]

1199036(119905) = [

119905 + 6 minus sin 119905minus119905 + 7 minus cos 119905

]

(21)

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34Ga Gb Gc Gd Ge

Figure 3 The five kinds of topologies of multiagent systems

0 10 20 30 40 50 60minus10

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

20

10

0

minus10

minus20

minus30

minus40

Figure 4 The trajectory of the agents and targets of multiagentsystems with switching topology randomly

The corresponding fixed network topology of multiagentsystems with 4 nodes is shown in Figure 1 Let 119896 = 2 120573 = 22120572 = 33 119896

1= 10 and 119896

2= 10 and the initial conditions

are (9 9) (10 9) (minus6 2) and (1 minus5) Figure 2 represents thetrajectories of the agents and targets of multiagent systemsand it shows that the multiagent systems with fixed topologycan encircle the multiple targets in the form of circularformation

Figure 3 shows the five kinds of topologies of multiagentsystems with 4 nodes and these topologies can realize therandom switch obeyed uniform distribution among themLet 119896 = 2 120573 = 22 120572 = 33 119896

1= 10 and 119896

2= 10

and the initial conditions are (1 9) (10 1) (2 2) and (1 3)Figure 4 represents the trajectories of the agents and targets ofmultiagent systems and it shows that the multiagent systemswith switching topology randomly can encircle the multipletargets in the circular formation

6 Conclusion

In this paper we investigate the distributed encirclementof multiagent systems with multiple dynamic targets withthe assumption that each agent can track multiple targetseach target only can be tracked by one agent and thenumbers of the agents and the targets are the same or not

6 Mathematical Problems in Engineering

The encirclement and tracking method in circular formationis proposed Considering that each agent can only get partialinformation of targets the target state estimators which canestimate the average position of targets are designed In finitetime every agentrsquos motion radius is locally converged tocircular formation radius of system within a settling timeAll agents can maintain the formation which can be updatedin real time according to the change of targetsrsquo state ByLyapunov function it is proved that every agent can get thewhole information of targets in finite time and meanwhile itrealizes the circular formation in finite time The simulationresults illustrate that this proposedmethod is effective for notonly multiple static targets but also multiple dynamic targets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (973 Program) (Grant no 2012CB215203)and the National Natural Science Foundation of China(Grants nos 61304155 61203080 and 61573082)

References

[1] W Ren and Y Cao Distributed Coordination of Multi-agentNetworks Springer New York NY USA 2011

[2] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[3] P Lin andY Jia ldquoAverage consensus in networks ofmulti-agentswith both switching topology and coupling time-delayrdquo PhysicaA vol 387 no 1 pp 303ndash313 2008

[4] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[5] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[6] P Lin Y Jia and L Li ldquoDistributed robust 119867infin

consensuscontrol in directed networks of agents with time-delayrdquo Systemsamp Control Letters vol 57 no 8 pp 643ndash653 2008

[7] L Mo and Y Jia ldquo119867infinconsensus control of a class of high-order

multi-agent systemsrdquo IET Control Theory amp Applications vol 5no 1 pp 247ndash253 2011

[8] P Lin and W Ren ldquoConstrained consensus in unbalancednetworks with communication delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 3 pp 775ndash781 2014

[9] A JMarasco S NGivigi andCA Rabbath ldquoModel predictivecontrol for the dynamic encirclement of a targetrdquo in Proceedingsof the American Control Conference (ACC rsquo12) pp 2004ndash2009Montreal Canada June 2012

[10] F Chen W Ren and Y Cao ldquoSurrounding control in cooper-ative agent networksrdquo Systems amp Control Letters vol 59 no 11pp 704ndash712 2010

[11] T Wei and X Chen ldquoCollective surrounding control in multi-agent networksrdquo Chinese Physics B vol 23 no 5 Article ID050201 4 pages 2014

[12] P Lin and Y Jia ldquoConsensus of second-order discrete-timemulti-agent systems with nonuniform time-delays and dynam-ically changing topologiesrdquoAutomatica vol 45 no 9 pp 2154ndash2158 2009

[13] P Lin W Ren and Y Song ldquoDistributed multi-agent optimiza-tion subject to nonidentical constraints and communicationdelaysrdquo Automatica vol 65 pp 120ndash131 2016

[14] P Lin and Y Jia ldquoMulti-agent consensus with diverse time-delays and jointly-connected topologiesrdquo Automatica vol 47no 4 pp 848ndash856 2011

[15] Y Hong L Gao D Cheng and J Hu ldquoLyapunov-basedapproach to multiagent systems with switching jointly con-nected interconnectionrdquo IEEE Transactions on Automatic Con-trol vol 52 no 5 pp 943ndash948 2007

[16] P Lin K Qin Z Li andW Ren ldquoCollective rotatingmotions ofsecond-order multi-agent systems in three-dimensional spacerdquoSystems amp Control Letters vol 60 no 6 pp 365ndash372 2011

[17] L Mo Y Niu and T Pan ldquoConsensus of heterogeneous multi-agent systems with switching jointly-connected interconnec-tionrdquo Physica A vol 427 pp 132ndash140 2015

[18] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[19] S P Bhat andD S Bernstein ldquoContinuous finite-time stabiliza-tion of the translational and rotational double integratorsrdquo IEEETransactions on Automatic Control vol 43 no 5 pp 678ndash6821998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

For

119899

sum

119894=1

[119901119894(119905) minus

1

119899

119899

sum

119896=1

119901119896(119905)]

119879

[minus1

119899

119899

sum

119896=1

119896(119905)]

= [

119899

sum

119894=1

119901119894(119905) minus

119899

sum

119896=1

119901119896(119905)] [minus

1

119899

119899

sum

119896=1

119896(119905)] = 0

(14)

we get

1(119905) le minus

120572

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

+1198992

120573

2

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

= (1198992

120573

2minus120572

2)

119899

sum

119894=1

sum

119895isin119873119894(119905)

10038171003817100381710038171003817119901119894(119905) minus 119901

119895(119905)10038171003817100381710038171003817

(15)

Let 119898(119905) = max119894119895isinI119901119894(119905) minus 119901

119895(119905) For 119901

119894(119905) minus

(1119899)sum119899

119896=1119901119896(119905) le (1119899)sum

119899

119896=1119901119894(119905) minus 119901

119896(119905) le 119898(119905) we

obtain 1(119905) le(1198992)119898(119905)

2 Furthermore forsum119899119894=1sum119895isin119873119894(119905)

119901119894(119905) minus

119901119895(119905) ge 119898(119905) we have

1(119905) + (

120572

2minus1198992

120573

2)radic

119899

21198811(119905)12

le (1198992

120573

2minus120572

2)119898 (119905) + radic

119899

2(120572

2minus1198992

120573

2)radic

119899

2119898 (119905)

= 0

(16)

Therefore from Lemma 2 there exists 1198791

gt 0 suchthat lim

119905rarr1198791[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 and we get

lim119905rarr1198791

[119901119894(119905) minus 119901

119895(119905)] = 0

Lemma 6 Considering estimator (7) one supposes thatAssumptions 3 and 4 are both satisfied If 119896

2gt 2120573 estimator

(7) must be steady in finite time

Proof We define the Lyapunov function as follows

1198812(119905) =

119899

sum

119894=1

120576119894sign (120576

119894) (17)

where 120576119894= 120588119894(119905)minusmax

119895isinI119889119895(119905)Then the derivative of1198812(119905)

along the trajectories of system (7) is given by

2(119905) =

119899

sum

119894=1

sign (120576119894) 120576119894=

119899

sum

119894=1

sign (120576119894)

sdot [minus1198962sign(120588

119894(119905) minusmax

119895isinI119889119895(119905))

minusmax119895isinI

119889119895(119905)]

(18)

Let 119889(119905) = max119895isinI119889119895(119905) = max

119895isinI119903119895(119905) minus 119901119895(119905)

According to Lemma 5 there exists 1198791gt 0 such that 119901

119894(119905) =

(1119898)sum119898

119896=1119903119896(119905) when 119905 gt 119879

1

Then we have

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

max119895isinI

119889119895(119905)

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

max119895isinI

1003817100381710038171003817100381710038171003817100381710038171003817

1

119899119895

119899119895

sum

119896=1

119903119895119896(119905) minus

1

119898

119898

sum

119896=1

119903119896(119905)

1003817100381710038171003817100381710038171003817100381710038171003817

1003816100381610038161003816100381610038161003816100381610038161003816

le max119895isinI

1

119899119895

119899119895

sum

119896=1

10038171003817100381710038171003817119903119895119896(119905)10038171003817100381710038171003817+

1

119898

119898

sum

119896=1

1003817100381710038171003817119903119896(119905)1003817100381710038171003817 le 2120573

(19)

Hence

2(119905) le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] +

10038161003816100381610038161003816119889 (119905)

10038161003816100381610038161003816

le

119899

sum

119894=1

sign (120576119894) [minus1198962sign (120576

119894)] + 2120573

=

119899

sum

119894=1

minus1198962

1003816100381610038161003816sign (120576119894)1003816100381610038161003816 + 2120573

(20)

According to the condition 1198962gt 2120573 we conclude that system

(7) must be steady in finite time

Lemma 7 Considering the first equation of system (8) onesupposes that Assumptions 3 and 4 are both satisfied If 119896

2gt 2120573

and 1198961gt 2119896120573 this system can be steady in finite time

Proof It is easy to get the conclusion according to Lemma 4in [1] so we ignore the proof here

Theorem 8 Considering system (2) if the network topologyof multiagent systems is connected and Assumptions 3 and 4are both satisfied letting 120572 gt 119899

2

120573 1198961gt 2119896120573 and 119896

2gt 2120573

then protocol (9) can solve the distributed encirclement controlproblem of system (2)

Proof From Lemma 5 there exists 1198791

gt 0 such thatlim119905rarr1198791

[119901119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905)] = 0 From Lemma 7 there

exists 1198792gt 1198791such that lim

119905rarr1198792119910119894(119905) minus (1119898)sum

119898

119896=1119903119896(119905) minus

119896max119895isinI119903119895(119905) minus (1119898)sum

119898

119896=1119903119896(119905) = 0 As graph 119866 is

connected the Laplacian 119871 of 119866 has nonnegative real partFor

120579(119905) = minus119871120579(119905) then we have lim119905rarrinfin

[120579119894(119905) minus120579

119895(119905)] = 0 so

lim119905rarrinfin

[120579119894(119905) minus 120579

119895(119905) minus 2120587(119894 minus 119895)119899] = 0

Mathematical Problems in Engineering 5

1 2

34

Figure 1 The fixed network topology of multiagent systems

50 6010 20 30 40minus10 0minus20

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

30

20

10

0

minus10

minus20

minus30

minus40

minus50

Figure 2 The trajectories of the agents and targets of multiagentsystems with fixed topology

5 Simulation Results and Analysis

In this section the results of simulation by Matlab prove theeffectiveness of the theoretical results obtained The dynamictargets are

1199031(119905) = [

119905 + 1 minus sin 119905minus119905 + 10 minus cos 119905

]

1199032(119905) = [

119905 + 10 minus sin 119905minus119905 + 2 minus cos 119905

]

1199033(119905) = [

119905 + 2 minus sin 119905minus119905 + 3 minus cos 119905

]

1199034(119905) = [

119905 + 8 minus sin 119905minus119905 + 4 minus cos 119905

]

1199035(119905) = [

119905 + 4 minus sin 119905minus119905 + 5 minus cos 119905

]

1199036(119905) = [

119905 + 6 minus sin 119905minus119905 + 7 minus cos 119905

]

(21)

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34Ga Gb Gc Gd Ge

Figure 3 The five kinds of topologies of multiagent systems

0 10 20 30 40 50 60minus10

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

20

10

0

minus10

minus20

minus30

minus40

Figure 4 The trajectory of the agents and targets of multiagentsystems with switching topology randomly

The corresponding fixed network topology of multiagentsystems with 4 nodes is shown in Figure 1 Let 119896 = 2 120573 = 22120572 = 33 119896

1= 10 and 119896

2= 10 and the initial conditions

are (9 9) (10 9) (minus6 2) and (1 minus5) Figure 2 represents thetrajectories of the agents and targets of multiagent systemsand it shows that the multiagent systems with fixed topologycan encircle the multiple targets in the form of circularformation

Figure 3 shows the five kinds of topologies of multiagentsystems with 4 nodes and these topologies can realize therandom switch obeyed uniform distribution among themLet 119896 = 2 120573 = 22 120572 = 33 119896

1= 10 and 119896

2= 10

and the initial conditions are (1 9) (10 1) (2 2) and (1 3)Figure 4 represents the trajectories of the agents and targets ofmultiagent systems and it shows that the multiagent systemswith switching topology randomly can encircle the multipletargets in the circular formation

6 Conclusion

In this paper we investigate the distributed encirclementof multiagent systems with multiple dynamic targets withthe assumption that each agent can track multiple targetseach target only can be tracked by one agent and thenumbers of the agents and the targets are the same or not

6 Mathematical Problems in Engineering

The encirclement and tracking method in circular formationis proposed Considering that each agent can only get partialinformation of targets the target state estimators which canestimate the average position of targets are designed In finitetime every agentrsquos motion radius is locally converged tocircular formation radius of system within a settling timeAll agents can maintain the formation which can be updatedin real time according to the change of targetsrsquo state ByLyapunov function it is proved that every agent can get thewhole information of targets in finite time and meanwhile itrealizes the circular formation in finite time The simulationresults illustrate that this proposedmethod is effective for notonly multiple static targets but also multiple dynamic targets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (973 Program) (Grant no 2012CB215203)and the National Natural Science Foundation of China(Grants nos 61304155 61203080 and 61573082)

References

[1] W Ren and Y Cao Distributed Coordination of Multi-agentNetworks Springer New York NY USA 2011

[2] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[3] P Lin andY Jia ldquoAverage consensus in networks ofmulti-agentswith both switching topology and coupling time-delayrdquo PhysicaA vol 387 no 1 pp 303ndash313 2008

[4] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[5] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[6] P Lin Y Jia and L Li ldquoDistributed robust 119867infin

consensuscontrol in directed networks of agents with time-delayrdquo Systemsamp Control Letters vol 57 no 8 pp 643ndash653 2008

[7] L Mo and Y Jia ldquo119867infinconsensus control of a class of high-order

multi-agent systemsrdquo IET Control Theory amp Applications vol 5no 1 pp 247ndash253 2011

[8] P Lin and W Ren ldquoConstrained consensus in unbalancednetworks with communication delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 3 pp 775ndash781 2014

[9] A JMarasco S NGivigi andCA Rabbath ldquoModel predictivecontrol for the dynamic encirclement of a targetrdquo in Proceedingsof the American Control Conference (ACC rsquo12) pp 2004ndash2009Montreal Canada June 2012

[10] F Chen W Ren and Y Cao ldquoSurrounding control in cooper-ative agent networksrdquo Systems amp Control Letters vol 59 no 11pp 704ndash712 2010

[11] T Wei and X Chen ldquoCollective surrounding control in multi-agent networksrdquo Chinese Physics B vol 23 no 5 Article ID050201 4 pages 2014

[12] P Lin and Y Jia ldquoConsensus of second-order discrete-timemulti-agent systems with nonuniform time-delays and dynam-ically changing topologiesrdquoAutomatica vol 45 no 9 pp 2154ndash2158 2009

[13] P Lin W Ren and Y Song ldquoDistributed multi-agent optimiza-tion subject to nonidentical constraints and communicationdelaysrdquo Automatica vol 65 pp 120ndash131 2016

[14] P Lin and Y Jia ldquoMulti-agent consensus with diverse time-delays and jointly-connected topologiesrdquo Automatica vol 47no 4 pp 848ndash856 2011

[15] Y Hong L Gao D Cheng and J Hu ldquoLyapunov-basedapproach to multiagent systems with switching jointly con-nected interconnectionrdquo IEEE Transactions on Automatic Con-trol vol 52 no 5 pp 943ndash948 2007

[16] P Lin K Qin Z Li andW Ren ldquoCollective rotatingmotions ofsecond-order multi-agent systems in three-dimensional spacerdquoSystems amp Control Letters vol 60 no 6 pp 365ndash372 2011

[17] L Mo Y Niu and T Pan ldquoConsensus of heterogeneous multi-agent systems with switching jointly-connected interconnec-tionrdquo Physica A vol 427 pp 132ndash140 2015

[18] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[19] S P Bhat andD S Bernstein ldquoContinuous finite-time stabiliza-tion of the translational and rotational double integratorsrdquo IEEETransactions on Automatic Control vol 43 no 5 pp 678ndash6821998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

1 2

34

Figure 1 The fixed network topology of multiagent systems

50 6010 20 30 40minus10 0minus20

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

30

20

10

0

minus10

minus20

minus30

minus40

minus50

Figure 2 The trajectories of the agents and targets of multiagentsystems with fixed topology

5 Simulation Results and Analysis

In this section the results of simulation by Matlab prove theeffectiveness of the theoretical results obtained The dynamictargets are

1199031(119905) = [

119905 + 1 minus sin 119905minus119905 + 10 minus cos 119905

]

1199032(119905) = [

119905 + 10 minus sin 119905minus119905 + 2 minus cos 119905

]

1199033(119905) = [

119905 + 2 minus sin 119905minus119905 + 3 minus cos 119905

]

1199034(119905) = [

119905 + 8 minus sin 119905minus119905 + 4 minus cos 119905

]

1199035(119905) = [

119905 + 4 minus sin 119905minus119905 + 5 minus cos 119905

]

1199036(119905) = [

119905 + 6 minus sin 119905minus119905 + 7 minus cos 119905

]

(21)

1 2

34

1 2

34

1 2

34

1 2

34

1 2

34Ga Gb Gc Gd Ge

Figure 3 The five kinds of topologies of multiagent systems

0 10 20 30 40 50 60minus10

Target 1Target 2Target 3Target 4Target 5

Target 6Agent 1Agent 2Agent 3Agent 4

20

10

0

minus10

minus20

minus30

minus40

Figure 4 The trajectory of the agents and targets of multiagentsystems with switching topology randomly

The corresponding fixed network topology of multiagentsystems with 4 nodes is shown in Figure 1 Let 119896 = 2 120573 = 22120572 = 33 119896

1= 10 and 119896

2= 10 and the initial conditions

are (9 9) (10 9) (minus6 2) and (1 minus5) Figure 2 represents thetrajectories of the agents and targets of multiagent systemsand it shows that the multiagent systems with fixed topologycan encircle the multiple targets in the form of circularformation

Figure 3 shows the five kinds of topologies of multiagentsystems with 4 nodes and these topologies can realize therandom switch obeyed uniform distribution among themLet 119896 = 2 120573 = 22 120572 = 33 119896

1= 10 and 119896

2= 10

and the initial conditions are (1 9) (10 1) (2 2) and (1 3)Figure 4 represents the trajectories of the agents and targets ofmultiagent systems and it shows that the multiagent systemswith switching topology randomly can encircle the multipletargets in the circular formation

6 Conclusion

In this paper we investigate the distributed encirclementof multiagent systems with multiple dynamic targets withthe assumption that each agent can track multiple targetseach target only can be tracked by one agent and thenumbers of the agents and the targets are the same or not

6 Mathematical Problems in Engineering

The encirclement and tracking method in circular formationis proposed Considering that each agent can only get partialinformation of targets the target state estimators which canestimate the average position of targets are designed In finitetime every agentrsquos motion radius is locally converged tocircular formation radius of system within a settling timeAll agents can maintain the formation which can be updatedin real time according to the change of targetsrsquo state ByLyapunov function it is proved that every agent can get thewhole information of targets in finite time and meanwhile itrealizes the circular formation in finite time The simulationresults illustrate that this proposedmethod is effective for notonly multiple static targets but also multiple dynamic targets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (973 Program) (Grant no 2012CB215203)and the National Natural Science Foundation of China(Grants nos 61304155 61203080 and 61573082)

References

[1] W Ren and Y Cao Distributed Coordination of Multi-agentNetworks Springer New York NY USA 2011

[2] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[3] P Lin andY Jia ldquoAverage consensus in networks ofmulti-agentswith both switching topology and coupling time-delayrdquo PhysicaA vol 387 no 1 pp 303ndash313 2008

[4] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[5] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[6] P Lin Y Jia and L Li ldquoDistributed robust 119867infin

consensuscontrol in directed networks of agents with time-delayrdquo Systemsamp Control Letters vol 57 no 8 pp 643ndash653 2008

[7] L Mo and Y Jia ldquo119867infinconsensus control of a class of high-order

multi-agent systemsrdquo IET Control Theory amp Applications vol 5no 1 pp 247ndash253 2011

[8] P Lin and W Ren ldquoConstrained consensus in unbalancednetworks with communication delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 3 pp 775ndash781 2014

[9] A JMarasco S NGivigi andCA Rabbath ldquoModel predictivecontrol for the dynamic encirclement of a targetrdquo in Proceedingsof the American Control Conference (ACC rsquo12) pp 2004ndash2009Montreal Canada June 2012

[10] F Chen W Ren and Y Cao ldquoSurrounding control in cooper-ative agent networksrdquo Systems amp Control Letters vol 59 no 11pp 704ndash712 2010

[11] T Wei and X Chen ldquoCollective surrounding control in multi-agent networksrdquo Chinese Physics B vol 23 no 5 Article ID050201 4 pages 2014

[12] P Lin and Y Jia ldquoConsensus of second-order discrete-timemulti-agent systems with nonuniform time-delays and dynam-ically changing topologiesrdquoAutomatica vol 45 no 9 pp 2154ndash2158 2009

[13] P Lin W Ren and Y Song ldquoDistributed multi-agent optimiza-tion subject to nonidentical constraints and communicationdelaysrdquo Automatica vol 65 pp 120ndash131 2016

[14] P Lin and Y Jia ldquoMulti-agent consensus with diverse time-delays and jointly-connected topologiesrdquo Automatica vol 47no 4 pp 848ndash856 2011

[15] Y Hong L Gao D Cheng and J Hu ldquoLyapunov-basedapproach to multiagent systems with switching jointly con-nected interconnectionrdquo IEEE Transactions on Automatic Con-trol vol 52 no 5 pp 943ndash948 2007

[16] P Lin K Qin Z Li andW Ren ldquoCollective rotatingmotions ofsecond-order multi-agent systems in three-dimensional spacerdquoSystems amp Control Letters vol 60 no 6 pp 365ndash372 2011

[17] L Mo Y Niu and T Pan ldquoConsensus of heterogeneous multi-agent systems with switching jointly-connected interconnec-tionrdquo Physica A vol 427 pp 132ndash140 2015

[18] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[19] S P Bhat andD S Bernstein ldquoContinuous finite-time stabiliza-tion of the translational and rotational double integratorsrdquo IEEETransactions on Automatic Control vol 43 no 5 pp 678ndash6821998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The encirclement and tracking method in circular formationis proposed Considering that each agent can only get partialinformation of targets the target state estimators which canestimate the average position of targets are designed In finitetime every agentrsquos motion radius is locally converged tocircular formation radius of system within a settling timeAll agents can maintain the formation which can be updatedin real time according to the change of targetsrsquo state ByLyapunov function it is proved that every agent can get thewhole information of targets in finite time and meanwhile itrealizes the circular formation in finite time The simulationresults illustrate that this proposedmethod is effective for notonly multiple static targets but also multiple dynamic targets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (973 Program) (Grant no 2012CB215203)and the National Natural Science Foundation of China(Grants nos 61304155 61203080 and 61573082)

References

[1] W Ren and Y Cao Distributed Coordination of Multi-agentNetworks Springer New York NY USA 2011

[2] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions onAutomatic Control vol 49 no 9 pp 1520ndash1533 2004

[3] P Lin andY Jia ldquoAverage consensus in networks ofmulti-agentswith both switching topology and coupling time-delayrdquo PhysicaA vol 387 no 1 pp 303ndash313 2008

[4] P Lin and Y Jia ldquoFurther results on decentralised coordinationin networks of agents with second-order dynamicsrdquo IETControlTheory amp Applications vol 3 no 7 pp 957ndash970 2009

[5] P Lin and Y Jia ldquoConsensus of a class of second-order multi-agent systems with time-delay and jointly-connected topolo-giesrdquo IEEE Transactions on Automatic Control vol 55 no 3 pp778ndash784 2010

[6] P Lin Y Jia and L Li ldquoDistributed robust 119867infin

consensuscontrol in directed networks of agents with time-delayrdquo Systemsamp Control Letters vol 57 no 8 pp 643ndash653 2008

[7] L Mo and Y Jia ldquo119867infinconsensus control of a class of high-order

multi-agent systemsrdquo IET Control Theory amp Applications vol 5no 1 pp 247ndash253 2011

[8] P Lin and W Ren ldquoConstrained consensus in unbalancednetworks with communication delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 3 pp 775ndash781 2014

[9] A JMarasco S NGivigi andCA Rabbath ldquoModel predictivecontrol for the dynamic encirclement of a targetrdquo in Proceedingsof the American Control Conference (ACC rsquo12) pp 2004ndash2009Montreal Canada June 2012

[10] F Chen W Ren and Y Cao ldquoSurrounding control in cooper-ative agent networksrdquo Systems amp Control Letters vol 59 no 11pp 704ndash712 2010

[11] T Wei and X Chen ldquoCollective surrounding control in multi-agent networksrdquo Chinese Physics B vol 23 no 5 Article ID050201 4 pages 2014

[12] P Lin and Y Jia ldquoConsensus of second-order discrete-timemulti-agent systems with nonuniform time-delays and dynam-ically changing topologiesrdquoAutomatica vol 45 no 9 pp 2154ndash2158 2009

[13] P Lin W Ren and Y Song ldquoDistributed multi-agent optimiza-tion subject to nonidentical constraints and communicationdelaysrdquo Automatica vol 65 pp 120ndash131 2016

[14] P Lin and Y Jia ldquoMulti-agent consensus with diverse time-delays and jointly-connected topologiesrdquo Automatica vol 47no 4 pp 848ndash856 2011

[15] Y Hong L Gao D Cheng and J Hu ldquoLyapunov-basedapproach to multiagent systems with switching jointly con-nected interconnectionrdquo IEEE Transactions on Automatic Con-trol vol 52 no 5 pp 943ndash948 2007

[16] P Lin K Qin Z Li andW Ren ldquoCollective rotatingmotions ofsecond-order multi-agent systems in three-dimensional spacerdquoSystems amp Control Letters vol 60 no 6 pp 365ndash372 2011

[17] L Mo Y Niu and T Pan ldquoConsensus of heterogeneous multi-agent systems with switching jointly-connected interconnec-tionrdquo Physica A vol 427 pp 132ndash140 2015

[18] C Godsil and G Royle Algebraic Graph Theory vol 207 ofGraduate Texts in Mathematics Springer New York NY USA2001

[19] S P Bhat andD S Bernstein ldquoContinuous finite-time stabiliza-tion of the translational and rotational double integratorsrdquo IEEETransactions on Automatic Control vol 43 no 5 pp 678ndash6821998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of