group consensus for discrete-time heterogeneous...

Research ArticleGroup Consensus for Discrete-Time Heterogeneous MultiagentSystems with Input and Communication Delays

Yiliu Jiang12 Lianghao Ji 12 Xingcheng Pu 3 and Qun Liu12

1Chongqing Key Laboratory of Computational Intelligence Chongqing University of Posts and TelecommunicationsChongqing 400065 China2School of Computer Science and Technology Chongqing University of Posts and Telecommunications Chongqing 400065 China3School of Science Chongqing University of Posts and Telecommunications Chongqing 400065 China

Correspondence should be addressed to Lianghao Ji lianghaojigmailcom

Received 11 January 2018 Revised 24 August 2018 Accepted 26 September 2018 Published 17 October 2018

Academic Editor Marcio Eisencraft

Copyright copy 2018 Yiliu Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Group consensus seeking is investigated for a class of discrete-time heterogeneous multiagent systems composed of first-orderand second-order agents with both communication and input time delays Considering two types of system topologies novelprotocols based on the competitive and cooperative relationships among the agents are presented respectively By matrix theoryand frequency domain analysis method the sufficient conditions solving consensus problem are obtained The results show thatthe achievement of group consensus is bound up with the input time delays coupling weights between the agents and the systemrsquoscontrol parameters but it is irrelevant to the communication delays Finally numerical simulations are presented to illustrate thecorrectness of the theoretical results

1 Introduction

Multiagent systems (MASs) have recently attracted wide-spread concern It not only has extensive application indistributed sensor networks formation control and artificialintelligence but also contains great potential value for home-land security and military affairs Consensus a typical andcrucial issue of the coordination of MASs refers to that allthe agents asymptotically achieve certain global agreementonly via the information interactions between them As anextended problem of consensus group consensus means thatdifferent consensus states can be reached by the differentsubgroups of a complex system while there is no consensusbetween those subgroups It has been widely used to handlemultitasking problems and the decomposition of large-scaletasks So far there has been a lot of research on consensus andgroup consensus of MASs such as [1ndash5]

Note that most of the existing work is established mainlyon homogeneous agents given that all agents in one complexsystem possess identical dynamics However the differenceof the agent dynamics is ubiquitous in practical applications

which is the result of external impacts or interaction restric-tions of system Furthermore we usually have the demandof using multiple agents with different dynamics to reducecontrol costs Thus the consensus problem of heterogeneousMASs is increasingly being studied which is also a morecomplex and challenging task such as [6] where certainconditions were presented to guarantee the consensus ofheterogeneous MASs under fixed and switching topology In[7] Sun considers the consensus tracking control problem forgeneral linear multiagent systems with unknown dynamicsJiang [8] investigated the consensus of the heterogeneousnetworks with the influence of time delays In [9] fordiscrete-time heterogeneous systems Sun discussed both thesufficient conditions to ensure consensus and the consensusequilibrium point based on the graph theory and the matrixtheory Liu [10] proposed some sufficient conditions forheterogeneous multiagent systems to achieve consensus withsampled bounded communication delays In [11] Bernardoet al discussed the consensus of time-varying heterogeneouscommunication delays and applied this strategy for platoon-ing of vehicles

HindawiComplexityVolume 2018 Article ID 8319537 12 pageshttpsdoiorg10115520188319537

2 Complexity

At the same time due to the application demands ofmultitasking the group consensus problem of MASs is nowoccupying research hotspots [12ndash14] Currently methods forachieving group consensus include common cooperation-based strategies [15ndash19] and fewer competition-based strate-gies [20] Since the analysis of the group consensus problemof heterogeneous systems is a more complex work thanthat of homogeneous system there are few related articlesThis incites us to study the group consensus problem forheterogeneous multiagent systems The main contributionsof this paper are as follows First a heterogeneous systemis established which is composed of agents with first-orderand second-order dynamics And the sufficient algebraicconditions are correspondingly deduced to ensure the con-sensus of the system The dynamics of the first-order agentsin our system model does not include the virtual velocitywhich exists in [15 17 19] but reduces the flexibility of thesystem Second utilizing the characteristics of two types ofcommon topologies effective control algorithms to achievegroup consensus are proposed based on competitive andcooperative relationships respectively Finally note that thereexists communication delay when the channel is congestedand the input delaywhen the sensor is aging or the computingpower is insufficient while most of articles like [10 1518] did not consider any or just consider one of themThe two kinds of delays are reasonably considered in theagentsrsquo dynamics of our system Utilizing frequency domainanalysis and Gerschgorin circle theorem we derived theupper bound of input time delay to guarantee the consen-sus of system Meanwhile discrete-time based systems arecommon in the real world and current related research isfocused on continuous systems [10ndash19] Hence it is alsoof practical values to investigate the consensus problem ofthe discrete-time systems The above innovations make oursystem model closer to reality and having a wider scope ofapplication

The remaining of this paper is organized as followsIn Section 2 we first list some preliminaries about graphtheory lemmas and problem statements Section 3 containsthe main analysis of the group consensus of discrete-timeheterogeneous MASs with multiple time delays In Section 4several simulations are demonstrated to validate the theoret-ical results Finally some conclusions are drawn based on ourwork

Note Throughout this paper we use R R119899 R119899times119898 denotingthe one-dimensional Euclidean space the n-dimensional realvector space and the 119899times119898 dimensional real matrices respec-tively 119868119873 represents the identity matrix with 119873-dimensionC indicates complex numbers sets the modulus and the realparts of forall119911 isin C are represented as |119911| and Re(119911) det(Λ)and 120582119894(Λ) indicate the determinant and the 119894th eigenvalue ofmatrix Λ2 Preliminaries and Problems Statements

To assist our work some definitions lemmas and statementsare first introduced

21 Graph eory and Interconnection Topology Based ongraph theory in a MAS with 119899 agents the agents and theinformation exchange between them can be described by aweighted directed graph (digraph) 119866 = (119881 119864 119860) where 119881 =V1 V2 V119899 denotes the node set 119864 = 119881 times 119881 denotes theedge set and 119860 = (119886119894119895)119899times119899 isin R119899times119899 denotes the adjacencymatrix 119890119894119895 = (119894 119895) isin 119864 denotes that there exists a directededge from node 119895 to node 119894 in the digraph 119866 so that node 119894can receive information from node 119895 119886119894119895 = 0 if 119890119894119895 isin 119864 inaddition we assume 119886119894119894 = 0 for all 119894 isin 1 119899

Meanwhile the neighbor set and the in-degree of 119894th nodeare represented as 119873119894 = 119895 isin 119881 119890119894119895 isin 119864 and 119889119894 =degin(119894) = sum119895isin119873119894 119886119894119895 respectively Denote in-degree matrix as119863 = diag1198891 1198892 119889119899 then Laplacian matrix 119871 = 119863 minus 119860

A directed spanning tree consists of nodes with one andonly one directed edge for receiving information from othernodes more than a node called the root A graph which has adirected spanning tree means that there exists such a tree thatis composed of several edges and all the nodes of this graph

Definition 1 There is a graph 119866 = (119881 119864) which can becalled bipartite graph when the following two conditions aresatisfied

(i) The vertex 119881 can be completely divided into twodisjoint subsets 1198811 and 1198812

(ii) The two vertexes 119894 119895 associated with each edge 119890 =(119894 119895) of the graph belong to different subset 119894 isin 1198811119895 isin 1198812Lemma 2 (see [21]) Consider a complex network with bipar-tite digraph topology which has a directed spanning tree it thenhasRe(120582119894(119863+119860)) gt 0when 120582119894(119863+119860) = 0 and 119903119886119899119896(119863+119860) =119899 minus 1Lemma 3 (see [20]) Suppose that 119911 = [1199111 1199112 119911119899] and 119871 isinR119899times119899 is a Laplacianmatrixen the following four conditionsare equivalent

(i) 120582119894(119871) 119894 isin 119899 all have positive real parts except a simplezero eigenvalue

(ii) 119871119911 = 0 indicates that 1199111 = 1199112 = sdot sdot sdot = 119911119899(iii) the consensus is asymptotically achieved for a system119911 = minus119871119911(iv) the directed graph represented by 119871 has at least one

directed spanning tree

22 Discrete-Time Heterogeneous Multiagent System Con-sider a discrete-time heterogeneous MASs including 119899 +119898 agents For the purposes of discussion we assume thesecond-order agents are 1 119899 and the first-order agents arerepresented by the remaining 119899 + 1 119899 + 119898 Hence thedynamics of the system are as follows119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896)

V119894 (119896 + 1) = V119894 (119896) + 119906119894 (119896) 119894 isin 1205901 (1)

Complexity 3

119909119897 (119896 + 1) = 119909119897 (119896) + 119906119897 (119896) 119897 isin 1205902 (2)

where1205901 = 1 2 119899 1205902 = 119899+1 119899+2 119899+119898120590 = 1205901 cup1205902 using 119906119894(119896) 119909119894(119896) V119894(119896) isin R to represent control inputposition state and velocity state of 119894th agent respectively

For heterogeneous MASs each agent 119894may have second-order and first-order neighbors which can be described as119873119894119904 and 119873119894119891 separately Accordingly all the neighbors of 119894are denoted as 119873119894 = 119873119894119891 cup 119873119894119904 Based on the differentorders of the dynamics we partition the adjacency matrixof the system 119860 as 119860 = ( 119860119904 119860119904119891119860119891119904 119860119891

) where 119860119904 isin R119899times119899 and119860119891 isin R119898times119898 denote adjacency matrix consisting only ofsecond-order and first-order nodes 119860 119904119891 is composed of theadjacency weights among the nodes that are from the second-order to first-order and 119860119891119904 indicate the opposite meaningMeanwhile we rewritten the Laplacian matrix 119871 as 119871 = 119863 minus119860 = ( 119871119904+119863119904119891 minus119860119904119891minus119860119891119904 119871119891+119863119891119904

) where 119871 119904 and 119871119891 indicate the Laplacianmatrix considering second-order and first-order nodes withtheir connections 119863119904119891 = diagsum119895isin119873119894119891 119886119894119895 119894 isin 1205901 and 119863119891119904 =diagsum119895isin119873119894119904 119886119894119895 119894 isin 1205902 Thus we can rewrite matrix119863+119860 as119863 + 119860 = ( 119863119904+119860119904+119863119904119891 119860119904119891

119860119891119904 119863119891+119860119891+119863119891119904)

Definition 4 The discrete-time heterogeneous MAS (1) and(2) can asymptotically reach consensus if and only if thefollowing two conditions hold for any initial states

(i) lim119896997888rarrinfin119909119894(119896) minus 119909119895(119896) = 0 for 119894 119895 isin 120590119896 (119896 = 1 2)(ii) lim119896997888rarrinfinV119894(119896) minus V119895(119896) = 0 for 119894 119895 isin 1205901

Lemma 5 (see [22]) e inequality sin(((2119863 + 1)2)120596)sin(1205962) le 2119863 + 1 120596 isin [minus120587 120587] holds for all nonnegativeintegers 119863

3 Main Results

Next we will investigate the group consensus problem for thediscrete-time heterogeneous MASs with communication andinput time delays under two typical topologies

31 Couple-Group Consensus for Discrete-Time HeterogeneousMASs with Multiple Time Delays under Bipartite DigraphMost of the existing works such as [15ndash19] are based onthe agentsrsquo cooperative relationship From a different pointof view a novel group consensus protocol was proposed in[9] referring the characteristic of the bipartite digraph whichis based on the competitive relationship between the agentsInspired by their work two novel delayed protocols based onthe competitive relationship are designed which are listed asfollows

119906119894 (119896) = minus120572[[sum119895isin119873119894

119886119894119895 [119909119895 (119896 minus 120591119894119895) + 119909119894 (119896 minus 120591)]]]minus 120573V119894 (119896 minus 120591) 119894 isin 1205901 (3)

and

119906119897 (119896) = minus120574[[sum119895isin119873119897

119886119897119895 [119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591)]]] 119897 isin 1205902 (4)

where 120591119894119895 denotes the communication time delays from theagent 119895 to agent 119894 and 120591 denotes the input time delays 120572 120573 120574 gt0 are the control parameters of the systems And we assume119886119894119895 gt 0 if there exists an interaction among nodes 119894 and 119895 Theclosed form of the heterogeneous systems (1) and (2) with theprotocols (3) and (4) is shown as119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896)

V119894 (119896 + 1) = V119894 (119896)minus 120572[[sum

119895isin119873119894

119886119894119895 (119909119895 (119896 minus 120591119894119895) + 119909119894 (119896 minus 120591))]]minus 120573V119894 (119896 minus 120591) 119894 isin 1205901(5)

and119909119897 (119896 + 1) = 119909119897 (119896)minus 120574[[sum119895isin119873119897

119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591))]] 119897 isin 1205902

(6)

Theorem 6 Couple-group consensus for heterogeneous MASs(5) and (6) under bipartite digraph topology with a spanningtree can be achieved asymptotically if the following conditionsare satisfied

(i) ere exists a real number 1205961198940 ge 0 satisfying the equa-tion 120572119889119894 = 4 sin(12059611989402) cos((1 + 2120591)12059611989402) tan(12059611989402)so that the inequality 2120573 minus 120572119889119894 gt 4 sin(12059611989402) sin((1 +2120591)12059611989402) holds

(ii) 1205732120572(120573 + 2) gt max119889119894 120574 lt 1max119889119894 120591 isin [0min12120573 minus 12 12120574max119889119894 minus 12) where 119889119894 =sum119895isin119873119894 119886119894119895 119894 isin 1205901 and 119889119894 = sum119895isin119873119894 119886119894119895 119894 isin 1205902

Proof Let (5) and (6) do 119911-transformation Assuming 119909119894(0) =0 it yields that119911119909119894 (119911) = 119909119894 (119911) + V119894 (119911) 119911V119894 (119911) = V119894 (119911)minus 120572[[ sum

V119895isin119873119894

119886119894119895 (119909119895 (119911) 119911minus120591119894119895 + 119909119894 (119911) 119911minus120591)]]minus 120573V119894 (119911) 119911minus120591 119894 isin 1205901

4 Complexity

119911119909119897 (119911) = 119909119897 (119911)minus 120574[ sum

V119894isin119873119897

119886119897119895 (119909119895 (119911) 119911minus120591119897119895 + 119909119897 (119911) 119911minus120591)] 119897 isin 1205902

(7)

where 119909119894(119911) and V119894(119911) represent 119909119894(119896) and V119894(119896) after 119911-transformation

Rewriting (7) to vector form let119909119904 (119911) = (1199091 (119911) 119909119899 (119911))119879 119909119891 (119911) = (119909119899+1 (119911) 119909119899+119898 (119911))119879 119863 + 119860 =

119911minus120591119894119895119886119894119895 119894 = 119895sum119895isin119873119894

119886119894119895119911minus120591 119894 = 119895(8)

Based on the equivalent representation mentioned above wecan rewrite matrix 119863+119860 = ( 119904+119860119904+119904119891 119860119904119891

119860119891119904 119891+119860119891+119891119904) Then we

get (119911 minus 1)2 119909119904 (119911) = minus120572 (119863119904 + 119860119904 + 119863119904119891) 119909119904 (119911)minus 120572119860119904119891119909119891 (119911)minus (119911 minus 1) 120573119890minus120591119909119904 (119911) (119911 minus 1) 119909119891 (119911) = minus120574119860119891119904119909119904 (119911)minus 120574 (119863119891 + 119860119891 + 119863119891119904) 119909119891 (119911) (9)

Define 119909(119911) = (119909119904(119911) 119909119891(119911))119879 then (9) can be rewritten as(119911 minus 1) 119909 (119911) = Φ (119911) 119909 (119911) (10)

whereΦ (119911)= (minus (119911 minus 1)2 119868 minus 120572 (119863119904 + 119860119904 + 119863119904119891)120573119911minus120591 minus120572119860 119904119891120573119911minus120591minus120574119860119891119904 minus120574 (119863119891 + 119860119891 + 119863119891119904)) (11)

The characteristic equation of system (10) is given by

det ((119911 minus 1) 119868 minus Φ (119911)) = 0 (12)

According to Lyapunov stability theory we know that ifthe roots of (12) are either at 119911 = 1 or in the unit circle ofthe complex plane the group consensus of the system can berealized Next we discuss these two cases separately

When 119911 = 1 det((119911 minus 1)119868 minus Φ(119911)) = (120572120573)119899(minus120574)119898 det(119863 +119860) Based on Lemma 2 we know that zero is a simpleeigenvalue of119863 + 119860 Then the roots of (12) is at 119911 = 1

When 119911 = 1 define det((119911 minus 1)119868 minus Φ(119911)) = det(119868 + 119866(119911))where 119866(119911) = minusΦ(119911)(119911 minus 1) It follows that119866 (119911) = minusΦ (119911)119911 minus 1= ((119911 minus 1)2 119868 + 120572 (119863119904 + 119860119904 + 119863119904119891)(119911 minus 1) 120573119890minus120591 120572119860 119904119891(119911 minus 1) 120573119911minus120591120574119860119891119904119911 minus 1 120574 (119863119891 + 119860119891 + 119863119891119904)119911 minus 1 ) (13)

where 119911 = 119890119895119908 Based on general Nyquist criteria one canknow that the roots of (12) can be located in the unit circleof the complex plane if and only if the point (minus1 1198950) is notenclosed by the Nyquist curve of 119866(119890119895119908)

According to Gerschgorin circle theorem an area con-taining all the eigenvalues of the matrix (the union of a set ofcircles 119866119894) is described as 120582(119866(119890119895120596)) isin 119866119894 119894 isin 1205901 cup 119866119894 119894 isin1205902 Next we separately discuss the two cases of the second-order (119894 isin 1205901) and the first-order (119894 isin 1205902) nodes

When 119894 isin 1205901 the circles are identified as

119866119894= 119909 119909 isin 119862 10038161003816100381610038161003816100381610038161003816100381610038161003816119909 minus 120572(119890119895120596 minus 1) 120573 sum

119895isin119873119894

119886119894119895 minus 119890119895120596 minus 1120573 11989011989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

100381610038161003816100381610038161003816100381610038161003816 120572119886119894119895(119890119895120596 minus 1) 120573119890minus119895120596(120591119894119895minus120591)100381610038161003816100381610038161003816100381610038161003816 (14)

Define 119889119894 = sum119895isin119873119894 119886119894119895 119894 isin 1205901 and119863119894 (120596) = 120572(119890119895120596 minus 1) 120573 sum

119895isin119873119894

119886119894119895 + 119890119895120596 minus 1120573 119890119895120596120591 (15)

119863119894(120596) denotes the center of the disc119866119894 Suppose the first crossof the curve119863119894(120596) on the real axis locates at 1205961198940 According to(15) the following equation is established

120572119889119894 = 4 sin 12059611989402 cos (1 + 2120591) 12059611989402 tan 12059611989402 (16)

Since the point (minus119886 1198950) 119886 ge 1 cannot be enclosed in119866119894 119894 isin 1205901 we obtain100381610038161003816100381610038161003816100381610038161003816minus119886 minus 120572119889119894(1198901198951205961198940 minus 1) 120573 minus 1198901198951205961198940 minus 1120573 1198901198951205961198940120591100381610038161003816100381610038161003816100381610038161003816gt sum119895isin119873119894


Based on Eulerrsquos formula and from (17) we have

Complexity 5

10038161003816100381610038161003816100381610038161003816minus119886 + 1205721198891198942120573 minus 1120573 (cos1205961198940 (1 + 120591) minus cos1205961198940120591) + 119895 (minus1205721198891198942120573 sdot sin12059611989401 minus cos1205961198940 + 1120573 (sin1205961198940 (1 + 120591) minus sin1205961198940120591))10038161003816100381610038161003816100381610038161003816gt sum119895isin119873119894

10038161003816100381610038161003816100381610038161003816100381610038161003816120572119886119894119895120573 minus12 cos1205961198940 (120591 minus 120591119894119895) minus sin1205961198940 sin1205961198940 (120591 minus 120591119894119895)2 (1 minus cos1205961198940) +119895( sin1205961198940 (120591 minus 120591119894119895)2 minus sin1205961198940 cos1205961198940 (120591 minus 120591119894119895)2 (1 minus cos1205961198940) )10038161003816100381610038161003816100381610038161003816100381610038161003816 (18)

After some manipulation we get

1198862 minus 119886(120572119889119894120573 minus 2 sdot cos1205961198940 (1 + 120591) minus cos1205961198940120591120573 ) + 2sdot 1 minus cos12059611989401205732 minus 2 sdot 120572119889119894 cos1205961198940 (1 + 120591)1205732 gt 0 (19)

It is easy to see that (19) holds for 119886 gt 1 if and only if thefollowing two inequalities are satisfied

2120573 minus 120572119889119894 gt 4 sin 12059611989402 sin (1 + 2120591) 12059611989402 (20)

1 minus 120572119889119894120573 + 2 sdot cos1205961198940 (1 + 120591) minus cos1205961198940120591120573 + 2sdot 1 minus cos12059611989401205732 minus 2 sdot 120572119889119894 cos1205961198940 (1 + 120591)1205732 gt 0 (21)

As 120573 gt 0 we can obtain1205732 minus 120572120573119889119894 + 2120573 [cos1205961198940 (1 + 120591) minus cos1205961198940120591] + 2sdot (1 minus cos1205961198940) minus 2120572119889119894 cos1205961198940 (1 + 120591) gt 0 (22)

The inequation (22) can be satisfied if the following twoinequations hold1205732 minus 120572120573119889119894 minus 2120572119889119894 cos1205961198940 (1 + 120591) gt 0 (23)120573 [cos1205961198940 (1 + 120591) minus cos1205961198940120591 + 1 minus cos1205961198940 gt 0 (24)

It is easy to know that cos1205961198940(1+120591) le 1 and from (23)we have1205732(120573 + 2) gt 120572119889119894 According to (24) and based on Lemma 5we have 120591 lt 12120573 minus 12

In addition since 120591 gt 0 we get1205732120572 (120573 + 2) gt max 119889119894 120591 isin [0 12120573 minus 12] (25)

When 119894 isin 1205902 the following inequality can also beobtained by Gerschgorin circle theorem

119866119894 = 119909 119909 isin | 10038161003816100381610038161003816100381610038161003816100381610038161003816119909 minus 120574119890119895120596 minus 1 sum119895isin119873119894

119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (26)

Define 119889119894 = sum119895isin119873119894 119886119894119895 119894 isin 1205902 If the point (minus119886 1198950) 119886 ge 1is not enclosed in 119866119894 119894 isin 1205902 it follows that1003816100381610038161003816100381610038161003816100381610038161003816minus119886 minus 120574119889119894119890119895120596 minus 1119890minus1198951205961205911003816100381610038161003816100381610038161003816100381610038161003816 gt sum

119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (27)

After some manipulations we have

1198862 minus 119886120574119889119894 cos120596120591 minus cos120596 (1 + 120591)1 minus cos120596 gt 0 (28)

From (28) we get

120574 lt 1max 119889119894

120591 isin [0 12120574max 119889119894 minus 12] (29)

Combining the analysis aforementioned Theorem 6 isproved

In Theorem 6 agents with identical input time delay anddifferent communication time delay are investigated Nextwe will extend our work to the case of different input andcommunication delays Based on (5) and (6) considering thefollowing systems listed as (30) and (31)119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

minus 120572[[sum119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591119897))]] 119897 isin 1205902

(31)

where 120591119894119895 denotes the communication time delay from the119895th agent to the 119894th agent 120591119894 denotes the input time delay ofthe 119894th agent 120572 120573 120574 gt 0 are the control parameters of thesystems

6 Complexity

Corollary 7 Couple-group consensus for the heterogeneousMASs (30) and (31) under bipartite digraph topology with aspanning tree can be achieved if the following conditions aresatisfied

(i) there exists a real number 1205961198940 ge 0 satisfying the equa-tion 120572119889119894 = 4 sin(12059611989402) cos((1 + 2120591)12059611989402) tan(12059611989402)so that the inequality 2120573 minus 120572119889119894 gt 4 sin(12059611989402) sin((1 +2120591)12059611989402) holds

(ii) 1205732120572(120573 + 2) gt max119889119894 120574 lt 1max119889119894(iii) 120591119894 isin [0 12120573 minus 12] 119894 isin 1205901 and 120591119894 isin [0 12120574119889119894 minus12] 119894 isin 1205902 where 119889119894 = sum119895isin119873119894 119886119894119895 119894 isin 1205901 and 119889119894 =sum119895isin119873119894 119886119894119895 119894 isin 1205902 respectivelyBased on Lemmas 2 and 5 as the proof is similar to

Theorem 6 we omit it here

Remark 8 The results inTheorem 6 and Corollary 7 indicatethat the control parameters of the system 120572 120573 120574 andthe coupling weight of the agent interaction 119889119894 are criticalfactors for achieving group consensus Furthermore thereduction in either control parameters or coupling weightscan make the system more tolerant of input time delaysHowever communication time delays do not play a key rolein achieving the group consensus

Remark 9 Corollary 7 shows that the maximum input timedelay that each node with respective dynamics can tolerate isdifferent it is decided by the control parameter of the systemand the coupling weights impacted by its neighbors whichhas the same dynamics Similarly the realization of the groupconsensus is not affected by the communication delays

Remark 10 In order to achieve group consensus manycontrol protocols have been proposed generally based onthe cooperative relationship among agents According tothe feature of the bipartite graph that is the informa-tion exchange of agents only occuring between differentsubgroups we design protocols (3) and (4) based on thecompetitive relationship This relaxes the condition thatsystem topology needs to satisfy the in-degree balance whenusing cooperative protocols And this also gives us a newperspective to investigate the group consensus problems

32 Group Consensus for Discrete-Time Heterogeneous MASswith Multiple Time Delays under Digraph with In-DegreeBalance In order to meet the demand that informationexchange of agents exists within the same subgroup next wewill discuss the case of the heterogeneousMASswithmultipletime delays under directed graph with in-degree balance

Assumption 11 sum119895isin119873119894 119886119894119895 = 0 119894 isin 1198711 and sum119895isin119873119897 119886119897119895 = 0 119897 isin1198712 where 1198711 and 1198712 denote two subgroupsRemark 12 As in [3 5 18] Assumption 11 means that theinteraction between the two subgroups is balanced which iscalled the in-degree balance This is necessary when groupconsensus is discussed in most situations

Inspired by the control protocol based on the cooperativerelationship the system dynamic is designed and the closedform is listed as follows119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120572[[ sumV119895isin119873119868119894

119886119894119895 (119909119895 (119896 minus 120591119894119895) minus 119909119894 (119896 minus 120591))+ sum

V119895isin119873119868119868119894

119886119894119895119909119895 (119896 minus 120591119894119895)]] minus 120573V119894 (119896 minus 120591) 119894 isin 1205901

(32)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] (33)

where119873119868119894 and119873119868119868119894 are the neighbor sets of the agent 119894 whichbelong to the same and different subgroups of 119894 respectivelyTheorem 13 For the heterogeneous MASs (32) and (33)under directed graph topology with a spanning tree the groupconsensus can be achieved if the following conditions aresatisfied

(i) Assumption 11 holds(ii) there exists a real number 1205961198940 ge 0 satisfying the equa-

tion 120572119889119894 = 4 sin(12059611989402) cos((1 + 2120591)12059611989402) tan(12059611989402)so that the inequality 2120573 minus 120572119889119894 gt 4 sin(12059611989402) sin((1 +2120591)12059611989402) holds

(iii) 1205732120572(120573 + 2) gt max119889119894 120574 lt 1max119889119894 120591 isin [0min12120573 minus 12 12120574max119889119894 minus 12] where 119889119894 =sum119895isin119873119894 119886119894119895 119894 isin 1205901 and 119889119894 = sum119895isin119873119894 119886119894119895 119894 isin 1205902

Proof Let the heterogeneous systemsrsquo dynamics (32) and (33)do 119911-transformation Assuming 119909119894(0) = 0 we have(119911 minus 1)2 119909119904 (119911) = minus120572 ( 119904 + 119863119904119891) 119909119904 (119911) + 120572119860 119904119891119909119891 (119911)minus (119911 minus 1) 120573119890minus120591119909119904 (119911) (119911 minus 1) 119909119891 (119911) = 120574119860119891119904119909119904 (119911) minus 120574 (119891 + 119863119891119904) 119909119891 (119911) (34)

Define 119909(119911) = (119909119904(119911) 119909119891(119911))119879 then (34) can be rewritten as(119911 minus 1) 119909 (119911) = Γ (119911) 119909 (119911) (35)

Complexity 7

whereΓ (119911)= (minus (119911 minus 1)2 119868 minus 120572 ( 119904 + 119863119904119891)120573119911minus120591 120572119860 119904119891120573119911minus120591120574119860119891119904 minus120574 (119891 + 119863119891119904)) (36)

The characteristic equation of system (35) is

det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)

According to Lyapunov stability theory the consensus of thesystem can be realized if the roots of (37) are either at 119911 = 1 orin the unit circle of the complex plane Next we discuss thesetwo cases separately

When 119911 = 1 det((119911 minus 1)119868 minus Γ(119911)) = (120572120573)119899(minus120574)119898 det()Based on Lemma 3 we know that zero is a simple eigenvalueof Then the roots of (37) are at 119911 = 1

When 119911 = 1 define det((119911 minus 1)119868 minus Γ(119911)) = det(119868 + 119867(119911))where 119867(119911) = minusΓ(119911)(119911 minus 1) 119911 = 119890119895119908 Based on generalNyquist criteria one can know that the roots of (37) can belocated in the unit circle of the complex plane if and only if thepoint (minus1 1198950) is not enclosed by the Nyquist curve of119867(119890119895119908)Hence according to the Gerschgorin circle theorem we have120582 (119867 (119890119895120596)) isin 119867119894 119894 isin 1205901 cup 119867119894 119894 isin 1205902 (38)

When 119894 isin 1205901 it yields that119867119894= 119909 119909 isin | 10038161003816100381610038161003816100381610038161003816100381610038161003816119909 minus 120572(119890119895120596 minus 1) 120573 sum

119895isin119873119894



Noting that this inequation is similar to those in Theo-rem 6 then after some derivation processes we can obtain120591 le 12120573 minus 12 (40)

When 119894 isin 1205902 the following inequality can also beobtained by the Gerschgorin circle theorem


119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)

Then we can get 120591 le 12120574119889119894 minus 12 (42)

Combining the above analysis this finishes the proof ofTheorem 13

Based on (32) and (33) consider the following dynamicsas (43) and (44)119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894

119886119894119895119909119895 (119896 minus 120591119894119895)]]minus 120573V119894 (119896 minus 120591119894) 119894 isin 1205901

(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)

Corollary 14 For the heterogeneous MASs (43) and (44)under directed graph topology with a spanning tree the groupconsensus can be achieved if the following conditions aresatisfied



(iii) 1205732120572(120573 + 2) gt max119889119894 120574 lt 1max119889119894(iv) 120591119894 isin [0 12057322120573 minus 12] 119894 isin 1205901 and 120591119894 isin [0 12120574119889119894 minus12] 119894 isin 1205902 where 119889119894 = sum119895isin119873119894 119886119894119895 119894 isin 1205901 and 119889119894 =sum119895isin119873119894 119886119894119895 119894 isin 1205902 respectivelyWe omit the proof of Corollary 14 as it is similar to

Theorem 13

Remark 15 The dynamics of the agents in our work donot contain the virtual velocity estimation which exists inmany related works [15 17 19] for the sake of analyzingheterogeneous systems In those works with the view ofimitating the dynamics of the second-order agents thevelocity estimation is added to the first-order agents but thisrequires extra computation and wastes systemrsquos resources

Remark 16 In this paper the topology of the systems weconsidered is either a bipartite digraph or a digraph satisfyingin-degree balance both of them has a spanning tree They

8 Complexity

G1

G2

01 0101

01

1 2

3 4 5

Figure 1 The bipartite digraph topology of the heterogeneousMASs

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)

Figure 2The state trajectories of the agents under directed topology in Figure 1 with identical input time delays 120591 = 05 120591119894119895 = 1 (a) Positions(b) Velocities

seem to be more specific But in fact group consensusfor heterogeneous systems usually can hardly be achievedunless adding some stronger conditions As in [15ndash19] thesystemsrsquo topology is undirected or containing a spanning treemeanwhile in those papers in order to achieve the group con-sensus additional assumptions are specified like in-degreebalance and the geometric multiplicity of zero eigenvalues ofthe systemrsquos Laplacian matrix which are at least two It will bea challenge to discuss the problem of the group consensus forheterogeneous MASs under a more general condition

Remark 17 Note that all the results in this paper are theoreti-cally obtained in the case 119909119894(119896) V119894(119896) 119906119894(119896) isin R Utilizing theKronecker product they can be generalized to 119909119894(119896) V119894(119896)119906119894(119896) isin R119899 119899 gt 14 Simulation

In this subsection the effectiveness of the obtained results isdemonstrated by several simulations

Example 18 Figure 1 shows a heterogeneous system with thebipartite digraph topology which has a spanning tree As

known the whole system is divided into two subgroups 1198661and1198662 Hereinto agents 1 and 2 and 3 4 and 5 belong to thesetwo subgroups Without loss of generality we further assumethat agents 1 and 5 are second-order and that the dynamicsof the remaining agents are first-order In our scenario thedynamics of agents in the same subgroup do not need tobe the same which is different from the case in [16ndash19]Obviously those are special cases of ours

For simplicity we assume each edgersquos weight as 119886119894119895 =01 119894 119895 isin [1 5] and it is easy to obtain the in-degree ofeach agent 1198891 = 02 1198892 = 01 1198893 = 01 1198894 = 01and 1198895 = 01 Based on Theorem 6 we set 120572 = 04120573 = 05 and 120574 = 1 then the upper bound of inputtime delays is obtained 120591 isin [0min12 92] In thesimulation we choose 120591 = 05 and all the conditions inTheorem 6 are satisfied now Since the realization of thegroup consensus is independent from the communicationtime delays for convenience we choose their values to bethe same as 120591119894119895 = 1 The trajectories of the agents ofthe systems (5) and (6) are shown in Figure 2 The resultsindicate that the systemsrsquo couple-group consensus is reachedasymptotically

Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)

Figure 3 The state trajectories of the agents under directed topology in Figure 1 with different input time delays 1205911 = 05 1205912 = 45 1205913 = 451205914 = 45 1205915 = 05 and 120591119894119895 = 1 (a) Positions (b) Velocities

minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)

Figure 4 The state trajectories of the agents under directed topology in Figure 1 with different input time delays 1205911 = 45 1205912 = 45 1205913 = 451205914 = 45 1205915 = 05 and 120591119894119895 = 1 (a) Positions (b) VelocitiesFromCorollary 7 and according to the control parameters

given above the upper bound of input delays can be calcu-lated for each node 1205911 le 12 1205912 le 92 1205913 le 92 1205914 le 92and 1205915 le 12 In this case we choose 1205911 = 05 1205912 = 451205913 = 45 1205914 = 45 1205915 = 05 It is clear that the conditions inCorollary 7 are all satisfied here Figure 3 demonstrates thatthe couple-group consensus is realized Furthermore if weset one of the input delays 1205911 = 45 for instance to violate theupper boundwe proposed from the trajectory in Figure 4 weknow that the system is divergent

Next we will testify Theorem 13 and Corollary 14

Example 19 Figure 5 shows an in-degree balance graph and1198891 = 01 1198892 = 01 1198893 = 01 1198894 = 01 and 1198895 = 01Set 120572 = 02 120573 = 03 120574 = 1 and 120591 = 1 Obviously theconditions of Theorem 13 are satisfied The trajectories of theagents are shown in Figure 6 Consider the systems (34) and(35) and choose different input timedelays as 1205911 = 1 1205912 = 451205913 = 45 1205914 = 45 and 1205915 = 1 Here Corollary 14 is satisfiedand the trajectories are shown in Figure 7 From Figures 6

10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01

Figure 5 The directed graph topology of the heterogeneous MASs

50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)


and 7 we know that the couple-group consensus is achievedasymptotically When set 1205915 = 45 exceeds the upper boundthe system diverges (Figure 8)

Remark 20 In this paper we designed a competition-basedcontrol protocol under bipartite graph and a cooperation-based protocol under in-degree balance graph to realizegroup consensus Competitive protocol will lead to a coupleof agents which belong to two different subgroups apartfrom each other In this case agents in the same subgrouphave no interactions and the final states of the system areonly two and opposite which is called the couple-groupconsensus Cooperative protocol will lead the agents in onesubgroup to be close to each other while the differentsubgroups need to satisfy the in-degree balance whichmeansall agents from one subgroup impose a weight of zeroon any node within another subgroup In this case morethan two system states can be achieved Both approacheshave their own limitation of topology Next we plan tocombine the competition and cooperation ideas to studythe group consensus problem under more general systemtopology

5 Conclusion

Under two forms of typical topologies the couple-groupconsensus for the discrete-time heterogeneous MASs withinput and communication time delays is investigated Forthe bipartite graph group consensus algorithms based onthe competitive relationship between agents are designedfor the graph satisfying in-degree balance we propose thealgorithms based on cooperative relationship accordinglyUtilizing matrix theory and general Nyquist criteria wetheoretically propose some sufficient algebraic criteria andthe upper bound of the input time delays for ensuring theconsensus From them we find that the achievement ofthe couple-group consensus for the heterogeneous systemsdepends on the input time delays of agents the controlparameters of the systems and the coupling weights betweenthe agents but it is independent of the communication timedelays The simulations verify the correctness of our theoret-ical results Our future work will extend to more complicatedgroup consensus problems of the heterogeneous multiagentsystems for example considering sampled information orswitching topology which commonly exist in reality

Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)

Figure 8 The state trajectories of the agents under directed topology in Figure 5 with different input time delays 1205911 = 1 1205912 = 45 1205913 = 451205914 = 45 1205915 = 45 and 120591119894119895 = 1 (a) Positions (b) VelocitiesData Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61876200 in

part by the Postgraduate Scientific Research and InnovationProject of Chongqing under Grant no CYS18246 in partby the Natural Science Foundation Project of ChongqingScience and Technology Commission under Grant nocstc2018jcyjAX0112 and in part by the Key Theme SpecialProject of Chongqing Science and Technology Commissionunder Grant nos cstc2017zdcy-zdyfx0091 and cstc2017rgzn-zdyfx0022

References

[1] Y Zheng JMa and LWang ldquoConsensus of hybridmulti-agentsystemsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 29 no 4 pp 1359ndash1365 2018

12 Complexity

[2] Y Zhu S Li J Ma and Y Zheng ldquoBipartite consensus innetworks of agents with antagonistic interactions and quanti-zationrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs 2018

[3] D Xie and T Liang ldquoSecond-order group consensus for multi-agent systems with time delaysrdquo Neurocomputing vol 153 pp133ndash139 2015

[4] H Xia T-Z Huang J-L Shao and J-Y Yu ldquoGroup consensusofmulti-agent systemswith communicationdelaysrdquoNeurocom-puting vol 171 pp 1666ndash1673 2016

[5] Q Cui D Xie and F Jiang ldquoGroup consensus trackingcontrol of second-ordermulti-agent systems with directed fixedtopologyrdquo Neurocomputing vol 218 pp 286ndash295 2016

[6] K Liu Z Ji G Xie and LWang ldquoConsensus for heterogeneousmulti-agent systems under fixed and switching topologiesrdquoJournal of e Franklin Institute vol 352 no 9 pp 3670ndash36832015

[7] J Sun and Z Geng ldquoAdaptive consensus tracking for linearmulti-agent systems with heterogeneous unknown nonlineardynamicsrdquo International Journal of Robust and Nonlinear Con-trol vol 26 no 1 pp 154ndash173 2016

[8] Y Jiang L Ji Q Liu S Yang and X Liao ldquoCouple-groupconsensus for discrete-time heterogeneous multiagent systemswith cooperativendashcompetitive interactions and time delaysrdquoNeurocomputing vol 319 pp 92ndash101 2018

[9] Y-J Sun G-L Zhang S-X Zhang and J Zeng ldquoConsensusequilibrium point analysis for a class of discrete-time hetero-geneous multi-agent systemsrdquo Kongzhi yu JueceControl andDecision vol 30 no 8 pp 1479ndash1484 2015

[10] C Liu and F Liu ldquoStationary consensus of heterogeneousmulti-agent systems with bounded communication delaysrdquoAutomatica vol 47 no 9 pp 2130ndash2133 2011

[11] M Di Bernardo A Salvi and S Santini ldquoDistributed consensusstrategy for platooning of vehicles in the presence of time-varying heterogeneous communication delaysrdquo IEEE Transac-tions on Intelligent Transportation Systems vol 16 no 1 pp 102ndash112 2015

[12] J Yu and Y Shi ldquoScaled group consensus in multiagentsystems with firstsecond-order continuous dynamicsrdquo IEEETransactions on Cybernetics 2017

[13] Y Feng S Xu and B Zhang ldquoGroup consensus controlfor double-integrator dynamic multiagent systems with fixedcommunication topologyrdquo International Journal of Robust andNonlinear Control vol 24 no 3 pp 532ndash547 2014

[14] Y Shang ldquoGroup consensus of multi-agent systems in directednetworks with noises and time delaysrdquo International Journal ofSystems Science vol 46 no 14 pp 2481ndash2492 2015

[15] C-L Liu and F Liu ldquoDynamical consensus seeking of hetero-geneous multi-agent systems under input delaysrdquo InternationalJournal ofCommunication Systems vol 26 no 10 pp 1243ndash12582013

[16] C Liu Q Zhou and XHu ldquoGroup consensus of heterogeneousmulti-agent systems with fixed topologiesrdquo International Jour-nal of Intelligent Computing and Cybernetics vol 8 no 4 pp294ndash311 2015

[17] G Wen J Huang C Wang Z Chen and Z Peng ldquoGroupconsensus control for heterogeneous multi-agent systems withfixed and switching topologiesrdquo International Journal of Controlvol 89 no 2 pp 259ndash269 2016

[18] G Wen Y Yu Z Peng and H Wang ldquoDynamical groupconsensus of heterogenousmulti-agent systems with input timedelaysrdquo Neurocomputing vol 175 pp 278ndash286 2015

[19] Y Zheng and L Wang ldquoA novel group consensus protocol forheterogeneous multi-agent systemsrdquo International Journal ofControl vol 88 no 11 pp 2347ndash2353 2015

[20] L Wang and F Xiao ldquoA new approach to consensus problemsin discrete-time multiagent systems with time-delaysrdquo ScienceChina Information Sciences vol 50 no 4 pp 625ndash635 2007

[21] Q Wang and Y Wang ldquoCluster synchronization of a class ofmulti-agent systems with a bipartite graph topologyrdquo ScienceChina Information Sciences vol 57 no 1 pp 1ndash11 2014

[22] Y Tian and C Liu ldquoConsensus of multi-agent systems withdiverse input and communication delaysrdquo IEEE Transactions onAutomatic Control vol 53 no 9 pp 2122ndash2128 2008

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Submit your manuscripts atwwwhindawicom

2 Complexity

At the same time due to the application demands ofmultitasking the group consensus problem of MASs is nowoccupying research hotspots [12ndash14] Currently methods forachieving group consensus include common cooperation-based strategies [15ndash19] and fewer competition-based strate-gies [20] Since the analysis of the group consensus problemof heterogeneous systems is a more complex work thanthat of homogeneous system there are few related articlesThis incites us to study the group consensus problem forheterogeneous multiagent systems The main contributionsof this paper are as follows First a heterogeneous systemis established which is composed of agents with first-orderand second-order dynamics And the sufficient algebraicconditions are correspondingly deduced to ensure the con-sensus of the system The dynamics of the first-order agentsin our system model does not include the virtual velocitywhich exists in [15 17 19] but reduces the flexibility of thesystem Second utilizing the characteristics of two types ofcommon topologies effective control algorithms to achievegroup consensus are proposed based on competitive andcooperative relationships respectively Finally note that thereexists communication delay when the channel is congestedand the input delaywhen the sensor is aging or the computingpower is insufficient while most of articles like [10 1518] did not consider any or just consider one of themThe two kinds of delays are reasonably considered in theagentsrsquo dynamics of our system Utilizing frequency domainanalysis and Gerschgorin circle theorem we derived theupper bound of input time delay to guarantee the consen-sus of system Meanwhile discrete-time based systems arecommon in the real world and current related research isfocused on continuous systems [10ndash19] Hence it is alsoof practical values to investigate the consensus problem ofthe discrete-time systems The above innovations make oursystem model closer to reality and having a wider scope ofapplication

The remaining of this paper is organized as followsIn Section 2 we first list some preliminaries about graphtheory lemmas and problem statements Section 3 containsthe main analysis of the group consensus of discrete-timeheterogeneous MASs with multiple time delays In Section 4several simulations are demonstrated to validate the theoret-ical results Finally some conclusions are drawn based on ourwork

Note Throughout this paper we use R R119899 R119899times119898 denotingthe one-dimensional Euclidean space the n-dimensional realvector space and the 119899times119898 dimensional real matrices respec-tively 119868119873 represents the identity matrix with 119873-dimensionC indicates complex numbers sets the modulus and the realparts of forall119911 isin C are represented as |119911| and Re(119911) det(Λ)and 120582119894(Λ) indicate the determinant and the 119894th eigenvalue ofmatrix Λ2 Preliminaries and Problems Statements

To assist our work some definitions lemmas and statementsare first introduced

21 Graph eory and Interconnection Topology Based ongraph theory in a MAS with 119899 agents the agents and theinformation exchange between them can be described by aweighted directed graph (digraph) 119866 = (119881 119864 119860) where 119881 =V1 V2 V119899 denotes the node set 119864 = 119881 times 119881 denotes theedge set and 119860 = (119886119894119895)119899times119899 isin R119899times119899 denotes the adjacencymatrix 119890119894119895 = (119894 119895) isin 119864 denotes that there exists a directededge from node 119895 to node 119894 in the digraph 119866 so that node 119894can receive information from node 119895 119886119894119895 = 0 if 119890119894119895 isin 119864 inaddition we assume 119886119894119894 = 0 for all 119894 isin 1 119899

Meanwhile the neighbor set and the in-degree of 119894th nodeare represented as 119873119894 = 119895 isin 119881 119890119894119895 isin 119864 and 119889119894 =degin(119894) = sum119895isin119873119894 119886119894119895 respectively Denote in-degree matrix as119863 = diag1198891 1198892 119889119899 then Laplacian matrix 119871 = 119863 minus 119860

A directed spanning tree consists of nodes with one andonly one directed edge for receiving information from othernodes more than a node called the root A graph which has adirected spanning tree means that there exists such a tree thatis composed of several edges and all the nodes of this graph

Definition 1 There is a graph 119866 = (119881 119864) which can becalled bipartite graph when the following two conditions aresatisfied

(i) The vertex 119881 can be completely divided into twodisjoint subsets 1198811 and 1198812

(ii) The two vertexes 119894 119895 associated with each edge 119890 =(119894 119895) of the graph belong to different subset 119894 isin 1198811119895 isin 1198812Lemma 2 (see [21]) Consider a complex network with bipar-tite digraph topology which has a directed spanning tree it thenhasRe(120582119894(119863+119860)) gt 0when 120582119894(119863+119860) = 0 and 119903119886119899119896(119863+119860) =119899 minus 1Lemma 3 (see [20]) Suppose that 119911 = [1199111 1199112 119911119899] and 119871 isinR119899times119899 is a Laplacianmatrixen the following four conditionsare equivalent

(i) 120582119894(119871) 119894 isin 119899 all have positive real parts except a simplezero eigenvalue

(ii) 119871119911 = 0 indicates that 1199111 = 1199112 = sdot sdot sdot = 119911119899(iii) the consensus is asymptotically achieved for a system119911 = minus119871119911(iv) the directed graph represented by 119871 has at least one

directed spanning tree

22 Discrete-Time Heterogeneous Multiagent System Con-sider a discrete-time heterogeneous MASs including 119899 +119898 agents For the purposes of discussion we assume thesecond-order agents are 1 119899 and the first-order agents arerepresented by the remaining 119899 + 1 119899 + 119898 Hence thedynamics of the system are as follows119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896)

V119894 (119896 + 1) = V119894 (119896) + 119906119894 (119896) 119894 isin 1205901 (1)

Complexity 3

119909119897 (119896 + 1) = 119909119897 (119896) + 119906119897 (119896) 119897 isin 1205902 (2)





119860119891119904 119863119891+119860119891+119863119891119904)




3 Main Results



119906119894 (119896) = minus120572[[sum119895isin119873119894


and

119906119897 (119896) = minus120574[[sum119895isin119873119897

119886119897119895 [119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591)]]] 119897 isin 1205902 (4)


V119894 (119896 + 1) = V119894 (119896)minus 120572[[sum

119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591))]] 119897 isin 1205902

(6)





V119895isin119873119894


4 Complexity

119911119909119897 (119911) = 119909119897 (119911)minus 120574[ sum

V119894isin119873119897

119886119897119895 (119909119895 (119911) 119911minus120591119897119895 + 119909119897 (119911) 119911minus120591)] 119897 isin 1205902

(7)



119911minus120591119894119895119886119894119895 119894 = 119895sum119895isin119873119894

119886119894119895119911minus120591 119894 = 119895(8)


119860119891119904 119891+119860119891+119891119904) Then we





det ((119911 minus 1) 119868 minus Φ (119911)) = 0 (12)








119895isin119873119894




119895isin119873119894

119886119894119895 + 119890119895120596 minus 1120573 119890119895120596120591 (15)


120572119889119894 = 4 sin 12059611989402 cos (1 + 2120591) 12059611989402 tan 12059611989402 (16)




Complexity 5






2120573 minus 120572119889119894 gt 4 sin 12059611989402 sin (1 + 2120591) 12059611989402 (20)








119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (26)


119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (27)



From (28) we get

120574 lt 1max 119889119894

120591 isin [0 12120574max 119889119894 minus 12] (29)



minus 120572[[sum119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591119897))]] 119897 isin 1205902

(31)


6 Complexity











+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(32)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] (33)







Complexity 7



det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)





119895isin119873119894






119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)




+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)





Theorem 13



8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3

119909119897 (119896 + 1) = 119909119897 (119896) + 119906119897 (119896) 119897 isin 1205902 (2)





119860119891119904 119863119891+119860119891+119863119891119904)




3 Main Results



119906119894 (119896) = minus120572[[sum119895isin119873119894


and

119906119897 (119896) = minus120574[[sum119895isin119873119897

119886119897119895 [119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591)]]] 119897 isin 1205902 (4)


V119894 (119896 + 1) = V119894 (119896)minus 120572[[sum

119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591))]] 119897 isin 1205902

(6)





V119895isin119873119894


4 Complexity

119911119909119897 (119911) = 119909119897 (119911)minus 120574[ sum

V119894isin119873119897

119886119897119895 (119909119895 (119911) 119911minus120591119897119895 + 119909119897 (119911) 119911minus120591)] 119897 isin 1205902

(7)



119911minus120591119894119895119886119894119895 119894 = 119895sum119895isin119873119894

119886119894119895119911minus120591 119894 = 119895(8)


119860119891119904 119891+119860119891+119891119904) Then we





det ((119911 minus 1) 119868 minus Φ (119911)) = 0 (12)








119895isin119873119894




119895isin119873119894

119886119894119895 + 119890119895120596 minus 1120573 119890119895120596120591 (15)


120572119889119894 = 4 sin 12059611989402 cos (1 + 2120591) 12059611989402 tan 12059611989402 (16)




Complexity 5






2120573 minus 120572119889119894 gt 4 sin 12059611989402 sin (1 + 2120591) 12059611989402 (20)








119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (26)


119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (27)



From (28) we get

120574 lt 1max 119889119894

120591 isin [0 12120574max 119889119894 minus 12] (29)



minus 120572[[sum119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591119897))]] 119897 isin 1205902

(31)


6 Complexity











+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(32)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] (33)







Complexity 7



det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)





119895isin119873119894






119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)




+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)





Theorem 13



8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

119911119909119897 (119911) = 119909119897 (119911)minus 120574[ sum

V119894isin119873119897

119886119897119895 (119909119895 (119911) 119911minus120591119897119895 + 119909119897 (119911) 119911minus120591)] 119897 isin 1205902

(7)



119911minus120591119894119895119886119894119895 119894 = 119895sum119895isin119873119894

119886119894119895119911minus120591 119894 = 119895(8)


119860119891119904 119891+119860119891+119891119904) Then we





det ((119911 minus 1) 119868 minus Φ (119911)) = 0 (12)








119895isin119873119894




119895isin119873119894

119886119894119895 + 119890119895120596 minus 1120573 119890119895120596120591 (15)


120572119889119894 = 4 sin 12059611989402 cos (1 + 2120591) 12059611989402 tan 12059611989402 (16)




Complexity 5






2120573 minus 120572119889119894 gt 4 sin 12059611989402 sin (1 + 2120591) 12059611989402 (20)








119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (26)


119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (27)



From (28) we get

120574 lt 1max 119889119894

120591 isin [0 12120574max 119889119894 minus 12] (29)



minus 120572[[sum119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591119897))]] 119897 isin 1205902

(31)


6 Complexity











+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(32)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] (33)







Complexity 7



det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)





119895isin119873119894






119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)




+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)





Theorem 13



8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5






2120573 minus 120572119889119894 gt 4 sin 12059611989402 sin (1 + 2120591) 12059611989402 (20)








119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (26)


119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (27)



From (28) we get

120574 lt 1max 119889119894

120591 isin [0 12120574max 119889119894 minus 12] (29)



minus 120572[[sum119895isin119873119894



119886119897119895 (119909119895 (119896 minus 120591119897119895) + 119909119897 (119896 minus 120591119897))]] 119897 isin 1205902

(31)


6 Complexity











+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(32)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] (33)







Complexity 7



det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)





119895isin119873119894






119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)




+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)





Theorem 13



8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity











+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(32)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] (33)







Complexity 7



det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)





119895isin119873119894






119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)




+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)





Theorem 13



8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7



det ((119911 minus 1) 119868 minus Γ (119911)) = 0 (37)





119895isin119873119894






119886119894119895119890minus11989512059612059110038161003816100381610038161003816100381610038161003816100381610038161003816le sum119895isin119873119894

1003816100381610038161003816100381610038161003816 120574119886119894119895119890119895120596 minus 1119890minus119895120596120591119894119895 1003816100381610038161003816100381610038161003816 (41)




+ 120572[[ sumV119895isin119873119868119894


V119895isin119873119868119868119894


(43)

and119909119894 (119896 + 1) = 119909119894 (119896) + V119894 (119896) V119894 (119896 + 1) = V119894 (119896)

+ 120574[[ sumV119895isin119873119868119897


V119895isin119873119868119868119897

119886119897119895119909119895 (119896 minus 120591119897119895)]] 119894 isin 1205902(44)





Theorem 13



8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

G1

G2

01 0101

01

1 2

3 4 5


20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

20 40 60 80 100 120 140 160 180 2000t (s)

Agent1Agent5

(b)








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9

20 40 60 80 100 120 140 160 180 2000t (s)

minus10

minus5

0

5

10

15

20Po

sitio

ns o

f age

nts

Agent1Agent2Agent3

Agent4Agent5

(a)

20 40 60 80 100 120 140 160 180 2000t (s)

minus4

minus2

0

2

4

6

8

10

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


minus3

minus2

minus1

0

1

2

3

4

Posit

ions

of a

gent

s

50 100 150 200 250 3000t (s)

times1015

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus6

minus4

minus2

0

2

4

6

8

10

12

14

Velo

citie

s of a

gent

s

times1014

Agent1Agent5

(b)





10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity

G1 G2

1 2

3 4 5

01

01

01

01

01

01 -01

-01


50 100 150 200 250 3000t (s)

2

4

6

8

10

12

14

16

18

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

50 100 150 200 250 3000t (s)

Agent1Agent5

(b)




5 Conclusion


Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

2

4

6

8

10

12

14

16

18Po

sitio

ns o

f age

nts

50 100 150 200 250 3000t (s)

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus2

minus1

0

1

2

3

4

5

Velo

citie

s of a

gent

s

Agent1Agent5

(b)


50 100 150 200 250 3000t (s)

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Posit

ions

of a

gent

s

Agent1Agent2Agent3

Agent4Agent5

(a)

50 100 150 200 250 3000t (s)

minus30

minus20

minus10

0

10

20

30

Velo

citie

s of a

gent

s

Agent1Agent5

(b)





Acknowledgments



References


12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity





























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








group consensus for discrete-time heterogeneous...

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