Research Article Event-Based Consensus for General Linear Multiagent Systems under Switching Topologies Yinshuang Sun , 1 Zhijian Ji , 1 and Kaien Liu 2 1 Institute of Complexity Science, College of Automation, Qingdao University, Qingdao 266071, China 2 School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China Correspondence should be addressed to Zhijian Ji; [email protected] Received 14 December 2019; Revised 21 January 2020; Accepted 3 February 2020; Published 28 March 2020 Academic Editor: Lucia Valentina Gambuzza Copyright©2020YinshuangSunetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, event-triggered leader-following consensus of general linear multiagent systems under both fixed topology and switching topologies is studied. First, centralised and decentralised event-triggered control strategies based on neighbors’ state estimation are proposed under fixed topology, in which the controller is only updated at the time of triggering. Obviously, compared with the continuous time control algorithms, the event-triggered control strategies can reduce the communication frequency among agents effectively. Meanwhile, event-triggering conditions are derived for systems to achieve consensus by using the Lyapunov stability theory and model transformation method. en, the theoretical results obtained under the fixed topology are extended to the switching topologies, and the sufficient conditions for the system to achieve leader-following consensus under the switching topologies are given. However, different from fixed topology, the control input of each agent is updated both at event-triggering and topology switching time. Finally, Zeno behaviors can be excluded by proving that the minimum triggering interval of each agent is strictly positive, and the effectiveness of the event-triggered protocol is verified by simulation experiments. 1. Introduction In recent years, multi-agent systems have great potential of application in the fields of biology, engineering, physics, society and so on. Also its distributed cooperative control [1, 2] has attracted more and more researchers' attention. For example, the controllability [3–11] and the consensus prob- lem [12–19] are widely studied in multi-agent systems. Among them, as the basic problem of multi-agent cooperative control, consensus is widely used in formation control [20], cluster control, sensor network and other aspects, which is a research hot issue in the control discipline at present. In practical applications, the information needed for cooperative control among agents is transmitted through the network. It is necessary to design a reasonable controller to ensure the control performance of the system due to the finite energy of the agent and the limited network band- width. It is well known that periodic sampling control [21–23] can save resources, but when the system runs in an ideal environment or the system state tends to be consensus gradually, it will cause unnecessary resource waste if the control task is executed periodically. In order to reduce this unnecessary waste of resources, a new simple event-triggered control strategy based on feedback mechanism was proposed in [24]. In short, the event-triggered control strategy means that the control task is executed as required. On the premise of ensuring the closed-loop system has certain performance, only can the task be executed once when a specific event occurs (such as the state error exceeds the preset threshold value). e advantage of event-triggered control strategies is that it can not only guarantee the performance of the system but also save the network and computing resources. At present, event-triggered mechanism has been applied to the research of consensus for multiagent systems effectively. For example, Dimarogonas and Frazoli and Dimarogonas et al. [25, 26] studied the consensus of a first-order multiagent system in undirected topology, and Yan et al. [27] investigated the consensus of a second-order multiagent system based on event-triggered mechanism in directed topology. Event- triggering conditions based on composite measurements were Hindawi Complexity Volume 2020, Article ID 5972749, 14 pages https://doi.org/10.1155/2020/5972749
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Research ArticleEvent-Based Consensus for General Linear MultiagentSystems under Switching Topologies
Yinshuang Sun 1 Zhijian Ji 1 and Kaien Liu 2
1Institute of Complexity Science College of Automation Qingdao University Qingdao 266071 China2School of Mathematics and Statistics Qingdao University Qingdao 266071 China
Correspondence should be addressed to Zhijian Ji jizhijianpkuorgcn
Received 14 December 2019 Revised 21 January 2020 Accepted 3 February 2020 Published 28 March 2020
Academic Editor Lucia Valentina Gambuzza
Copyright copy 2020Yinshuang Sun et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this paper event-triggered leader-following consensus of general linear multiagent systems under both fixed topology andswitching topologies is studied First centralised and decentralised event-triggered control strategies based on neighborsrsquo stateestimation are proposed under fixed topology in which the controller is only updated at the time of triggering Obviouslycompared with the continuous time control algorithms the event-triggered control strategies can reduce the communicationfrequency among agents effectively Meanwhile event-triggering conditions are derived for systems to achieve consensus by usingthe Lyapunov stability theory and model transformation method en the theoretical results obtained under the fixed topologyare extended to the switching topologies and the sufficient conditions for the system to achieve leader-following consensus underthe switching topologies are given However different from fixed topology the control input of each agent is updated both atevent-triggering and topology switching time Finally Zeno behaviors can be excluded by proving that the minimum triggeringinterval of each agent is strictly positive and the effectiveness of the event-triggered protocol is verified by simulation experiments
1 Introduction
In recent years multi-agent systems have great potential ofapplication in the fields of biology engineering physicssociety and so on Also its distributed cooperative control [12] has attracted more and more researchers attention Forexample the controllability [3ndash11] and the consensus prob-lem [12ndash19] are widely studied in multi-agent systemsAmong them as the basic problem of multi-agent cooperativecontrol consensus is widely used in formation control [20]cluster control sensor network and other aspects which is aresearch hot issue in the control discipline at present
In practical applications the information needed forcooperative control among agents is transmitted through thenetwork It is necessary to design a reasonable controller toensure the control performance of the system due to thefinite energy of the agent and the limited network band-width It is well known that periodic sampling control[21ndash23] can save resources but when the system runs in anideal environment or the system state tends to be consensus
gradually it will cause unnecessary resource waste if thecontrol task is executed periodically In order to reduce thisunnecessary waste of resources a new simple event-triggeredcontrol strategy based on feedback mechanism was proposedin [24] In short the event-triggered control strategy meansthat the control task is executed as required On the premise ofensuring the closed-loop system has certain performanceonly can the task be executed once when a specific eventoccurs (such as the state error exceeds the preset thresholdvalue) e advantage of event-triggered control strategies isthat it can not only guarantee the performance of the systembut also save the network and computing resources Atpresent event-triggered mechanism has been applied to theresearch of consensus for multiagent systems effectively Forexample Dimarogonas and Frazoli and Dimarogonas et al[25 26] studied the consensus of a first-order multiagentsystem in undirected topology and Yan et al [27] investigatedthe consensus of a second-order multiagent system based onevent-triggered mechanism in directed topology Event-triggering conditions based on composite measurements were
HindawiComplexityVolume 2020 Article ID 5972749 14 pageshttpsdoiorg10115520205972749
designed in [28] to address the consensus of a multiagentsystem Hu et al and Li et al [29 30] discussed the leader-following consensus of the second-order multiagent systeme centralized and decentralized event-triggered strategieswere proposed in [31] which made the state of all agentsconverge to the same value gradually In [32] a consensuscontrol algorithm based on event-triggered mechanism wasproposed for the nonlinear multiagent system In [33] a newaverage consensus problem under event-triggered controlstrategy was set up which limits the measurement error ofeach agent to a threshold that changes with time Meng andChen [34] studied the average consensus of the event-trig-gered multi-integrator systems under fixed and switchingtopologies and designed an event-triggered scheme based onquadratic Lyapunov function which made each agentrsquos stateconverge to itrsquos initial average eventually Xiao et al [35]discussed the average consensus of the network of the inte-grator system with unidirectional information link In orderto reduce the communication cost a distributed state con-sensus sampling data control scheme based on edge-event wasproposed In recent years more and more attention has beenpaid to general linear systems In [36] the consensus ofgeneral linear multiagent systems under integral event-trig-gered strategy has been considered In [37] the consensus ofgeneral linear systems under fixed topology and switchingtopologies has been investigated Hu et al [38] studied theleader-following consensus of general linear multiagent sys-tem under fixed topology and Zhu et al [39] studied theconsensus of general linear multiagent system under mixedevent-triggering conditions
With this background we consider the event-triggeredleader-following consensus of the general linear multiagentsystem under fixed topology and switching topologiesEvent-triggered control mechanism is designed for eachagent respectively Under the mechanism multiagent sys-tems can achieve leader-following consensus graduallyMoreover a continuous event-triggering condition is pro-posed which uses the state error between the follower andleader to design the triggering conditions of each agentunder the fixed topology and switching topologies re-spectively e consensus problem is transformed into thestability problem by the method of model transformationand the sufficient conditions for the system to achieveleader-following consensus are obtained by using Lyapunovstability theory In addition all the proposed event-triggeredmechanism can exclude Zeno behavior Finally the accuracyof the conclusion is verified by simulation experiment
e structure of this paper is as follows Section 2 in-troduces some concepts of the graph theory and systemmodel e leader-following consensus of the systems underthe fixedswitching topologies is considered in Sections 3and 4 In addition the effectiveness of the results is shownthrough simulation experiment in Section 5 Section 6summarizes this paper
e following notations are used in this paper otimes denotesthe Kronecker product IN denotes the N-dimensionalidentity matrix Rn and Rmtimesn indicate the set of n dimen-sional real vectors and m times n dimensional real matricesrespectively
2 Preliminaries
21eory ofGraph For a multiagent system composed of aleader and follower agents its communication topology canbe represented by an undirected graph G (V E) whereV 0 1 N 0 denotes the leader and 1 N denotethe followers EsubeV times V denotes the edges set e con-nection matrix between the follower agent i(i 1 N)
and leader 0 is D diag a10 aN01113864 1113865 where ai0 is theconnection weight between the leader 0 and follower i Ifai0 gt 0 the follower agent i can receive state information ofleader 0 otherwise ai0 0
e communication network among the followers isdenoted by G (V E A) where V 1 2 N andEsubeV times V are obtained from E by removing all edgesamong the leader 0 and followers in V andA (aij) isin RNtimesN is the weighted adjacency matrix of graphG where aij gt 0 for (i j) isin E if agent i obtains informationfrom agent j We assume that (i i) notin E and hence aii 0For a given graph G with the adjacency matrix A theLaplacian matrix used in this paper is L D minus A where D isa diagonal matrix its diagonal elements are dii 1113936jisinNi
|aij|and define Ni as the neighbor set of agent i inV ereforethe elements of L are
Lik 1113936
jisinNi
aij
11138681113868111386811138681113868
11138681113868111386811138681113868 k i
minus aik kne i
⎧⎪⎨
⎪⎩(1)
A path from the vertex i to vertex k is a sequence ofadjacent edges in the form (i i + 1) (i + 1 i + 2)
(k minus 1 k) e undirected graph is said to be connected ifthere exists a path between any two distinct vertices
22 SystemModel Consider a multiagent system composedof the leader 0 and N followers e dynamics of leader 0 is
_x0(t) Ax0(t) (2)
where x0(t) isin Rn is the state and A isin Rntimesn is constantmatrix
Accordingly each follower has the following lineardynamic equation
_xi(t) Axi(t) + Bui(t) i 1 N (3)
where xi(t) isin Rn and ui(t) isin Rp are the state and input ofthe ith follower agent respectively A isin Rntimesn andB isin Rntimesp
are constant matrices Denote the initial state of the ithfollower as xi(0)
Definition 1 If there is a control input ui(t) the leader 0 andfollower i for any initial state satisfy the following conditions
limt⟶infin
xi(t) minus x0(t)
0 i 1 2 N (4)
en the leader (2) is said to be successfully tracked byfollower (3)
Assumption 1 e communication network topology G
among followers is connected
2 Complexity
Assumption 2 e pair (A B) is stabilizableBased on Assumption 2 there is a symmetric positive
definite matrix P that satisfies the following algebraic Riccatiand Lyapunov inequality with βgt 0
ATP + PA minus 2βPBBTP + βIlt 0 (5)
ATP + PAlt 0 (6)
Lemma 1 (see [37]) For an undirected and connected graphG the eigenvalues of L are real and can be labelled as
0 λ1(L)lt λ2(L)le middot middot middot le λN(L) (7)
Lemma 2 (see [40]) For any x y isin R and βgt 0 one has thefollowing property
xyleβ2x2
+12β
y2 (8)
Lemma 3 (see [41]) (Comparison Principle) Consider adifferential equation (dudt) f(t u) u(t0) u0 wheretgt 0 f(t u) is continuous and satisfies the local Lipschitzcondition in t Let [t0 T) be the maximum existence intervalof the solution u where Tcan be infinite If for any t isin [t0 T)
satisfiesdv
dtlef(t v)
v t0( 1113857le u0
(9)
then v(t)le u(t) t isin [t0 T)
3 Leader-Following Control of MultiagentSystems under Fixed Topology
In this part we consider the leader-following control ofmultiagent systems (2) and (3) under the event-triggeredstrategy Based on the general event-triggered control lawwe put forward two kinds of piecewise continuous controlmechanisms which are centralized event-triggered mech-anism and decentralized event-triggered mechanism withstate estimation in order to minimize the frequency ofcontroller updating e analysis shows that under the twocontrol mechanisms multiagent system (3) can track thesystem (2) successfully with appropriate event-triggeringfunction e minimum interval between any two consec-utive event-triggering instants under the two controlmechanisms is greater than 0 and Zeno behavior can beexcluded
31 Centralized Event-Triggered Control Strategy Under thecentralized event-triggered strategy all agents i in system (3)are triggered synchronously at the time tk(k 0 1 ) Atthe triggering instants all agents send their states infor-mation to neighbours and update the control law with thereceived state information Compared with the control
protocol in continuous time each agent i only updates thecontrol input at the event instants under the event-triggeredmechanism So ui is a piecewise continuous function andthe updating frequency can be reduced
We consider the following control input for the ithfollower
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tk( 1113857 minus xj tk( 11138571113872 1113873
minus Kai0(t) xi tk( 1113857 minus x0 tk( 1113857( 1113857
(10)
where t isin [tk tk+1) K isin Rptimesn is the control gain matrix to bedesigned and xi(tk) is the sampling state of agent i at the kthtriggering instant Since there does not exist control inputfor leader 0 we take x0(tk) x0(t) t isin [tk tk+1) For con-venience we make t0 0
e event-triggering time sequence tk1113864 1113865 is determined bythe following triggering functions
f(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2λmax(D)λmax(W)
1113944
N
i11113954x
Ti 1113954xi
+ 1113944
N
i1e
Ti ei ge 0
(11)
at is tk+1 inf tgt tk | f(t)ge 01113864 1113865 where 0lt κ lt 1 0ltαlt (ai0min
λmin(W)λmax(P)) W PBBTPe state error between the follower and leader is defined
as 1113954xi(t) xi(t) minus x0(t) 1113954x(t) [1113954xT1 1113954xT
N]T For agent ithe measurement error is defined as ei(t) xi(tk) minus
xi(t) t isin [tk tk+1) en formula (10) is converted to
ui(t) minus K 1113944N
j0aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873⎛⎝ ⎞⎠
minus Kai0 1113954xi(t) + ei(t)( 1113857
(12)
Combining (2) (3) and (12) we get_1113954x(t) IN otimesA( 11138571113954x(t) minus IN otimesBK( 1113857 (L + D)otimes In( 1113857(1113954x(t) + e(t))
IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t))
(13)
Remark 1 rough the model transformation the leader-following control problem between systems (2) and (3) canbe interpreted by the stability problem of system (13)
Next we will give the following consensus conditionsunder the centralized event-triggering protocol (10)
Theorem 1 Under Assumptions 1 and 2 centralized event-triggered control strategy (10) can make multiagent system (3)track system (2) successfully under event-triggering condition(4) where feedback gain matrix K satisfies K BTP andW PBK
Complexity 3
Proof We consider a candidate Lyapunov function asfollows
V1 eαt 1113936N
i11113954xT
i P1113954xi (14)
Along with the trajectories of the state as described in (8)the time derivative of Lyapunov function is
_V1 αeαt
1113944
N
i11113954x
Ti P1113954xi + 2e
αt1113944
N
i11113954x
Ti P _1113954xi
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti P A1113954xi + Bui( 1113857
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi + 2e
αt1113944
N
i11113954x
Ti PBui
(15)
where
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti PBK ai0 1113954xi + ei( 1113857 + 1113944
N
j1aij 1113954xi minus 1113954xj + ei minus ej1113872 1113873⎛⎝ ⎞⎠
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
(16)
where W PBKAccording to the property of L LT in undirected graph
G we can deduce
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aji1113954x
Tj W 1113954xj minus 1113954xi1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Tj W 1113954xi minus 1113954xj1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(17)
Similarly
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(18)
Hence
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(19)
Combining equality (15) yields
_V1 eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(20)
In the light of Lemma 2 we have
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
le12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873 +
12eαt
1113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
(21)
By substituting the abovementioned formula intoequation (20) we obtain
4 Complexity
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(22)
From Lemma 2 we have
eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873le 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
(23)
Together with (22) we can get that
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(24)
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi minus e
αt1113944
N
i11113954x
Ti Wai0ei
le minus eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i11113954x
Ti Wai01113954xi
+12eαt
1113944
N
i1e
Ti Wai0ei
le minus12eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i1e
Ti Wai0ei
(25)
Combining (24) and (25) we arrive at
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi + e
αt1113944
N
i1e
Ti Wai0ei
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
le eαt
1113954xT
IN otimes αP( 11138571113954x + eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minus eαt
1113954xT(LotimesW)1113954x minus e
αt1113954x
T(DotimesW)1113954x
+ eαt
eT(DotimesW)e + 2e
αte
T(DotimesW)e
(26)
Under Assumption 1 by using Lemma 1 1113954xT(LotimesW)1113954xgeλ2(L)1113954xT(IN otimesW)1113954x holds Consequently
_V1 le eαt
1113954xT
IN otimes PA + ATP minus λ2(L)W1113872 11138731113872 11138731113954x
+ eαt αλmax(P) minus ai0min
λmin(W)1113872 11138731113954x2
+ eαt
ai0maxλmax(W) + 2λmax(D)λmax(W)1113872 1113873e
2
(27)
Using inequality (5) and event-triggering condition (11)we claim that the following inequality holds
Complexity 5
_V1 le (κ minus 1)eαt
ai0minλmin(W) minus αλmax(P)1113872 11138731113954x
2
minus eαtλ2(L)
21113954x
T1113954x le minus e
αtλ2(L)
21113954x
T1113954x
(28)
It can be seen from (28) that V1 is not increasingtherefore
V1(0)geV1(t) eαt 1113936N
i11113954xi(t)TP1113954xi(i)ge eαtλmin(P)1113954x(t)2
(29)
at is to say 1113954x(t)le(V1(0)λmin(P))
1113968eminus (α2)t ie
limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0 whichmeans limt⟶infin||xi(t) minus x0(t)|| 0 i 1 2 N
holds
Theorem 2 Under the conditions of eorem 1 system (13)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
IN otimesA
+||(L + D)otimesBK||1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (30)
Proof From the mechanism of event-triggering strategythe event interval between tk and tk+1 is the timethat (||e(t)||1113954x(t)) grows from 0 to
(κ(ai0min
λmin(W)minus αλmax(P))(ai0maxλmax(W)+2λmax(D)λmax(W)))
1113969
e time derivative of (e(t)||1113954x(t)||) has
ddt
e(t)
1113954x(t)
ddt
e(t)Te(t)1113872 111387312
1113954x(t)T1113954x(t)1113872 111387312
e(t)Te(t)1113872 1113873
(12)prime1113954x(t) minus e(t)Te(t)1113872 1113873
121113954x(t)T1113954x(t)1113872 1113873
(12)prime
1113954x(t)2
e(t)T _e(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
minus e(t)T _1113954x(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
le _1113954x(t)
1113954x(t)+
_1113954x(t)e(t)
1113954x(t)2
_1113954x(t)
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+(L + D)otimesBKe
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+IN otimesA
+(L + D)otimesBK1113872 1113873e
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
2
(31)
6 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
designed in [28] to address the consensus of a multiagentsystem Hu et al and Li et al [29 30] discussed the leader-following consensus of the second-order multiagent systeme centralized and decentralized event-triggered strategieswere proposed in [31] which made the state of all agentsconverge to the same value gradually In [32] a consensuscontrol algorithm based on event-triggered mechanism wasproposed for the nonlinear multiagent system In [33] a newaverage consensus problem under event-triggered controlstrategy was set up which limits the measurement error ofeach agent to a threshold that changes with time Meng andChen [34] studied the average consensus of the event-trig-gered multi-integrator systems under fixed and switchingtopologies and designed an event-triggered scheme based onquadratic Lyapunov function which made each agentrsquos stateconverge to itrsquos initial average eventually Xiao et al [35]discussed the average consensus of the network of the inte-grator system with unidirectional information link In orderto reduce the communication cost a distributed state con-sensus sampling data control scheme based on edge-event wasproposed In recent years more and more attention has beenpaid to general linear systems In [36] the consensus ofgeneral linear multiagent systems under integral event-trig-gered strategy has been considered In [37] the consensus ofgeneral linear systems under fixed topology and switchingtopologies has been investigated Hu et al [38] studied theleader-following consensus of general linear multiagent sys-tem under fixed topology and Zhu et al [39] studied theconsensus of general linear multiagent system under mixedevent-triggering conditions
With this background we consider the event-triggeredleader-following consensus of the general linear multiagentsystem under fixed topology and switching topologiesEvent-triggered control mechanism is designed for eachagent respectively Under the mechanism multiagent sys-tems can achieve leader-following consensus graduallyMoreover a continuous event-triggering condition is pro-posed which uses the state error between the follower andleader to design the triggering conditions of each agentunder the fixed topology and switching topologies re-spectively e consensus problem is transformed into thestability problem by the method of model transformationand the sufficient conditions for the system to achieveleader-following consensus are obtained by using Lyapunovstability theory In addition all the proposed event-triggeredmechanism can exclude Zeno behavior Finally the accuracyof the conclusion is verified by simulation experiment
e structure of this paper is as follows Section 2 in-troduces some concepts of the graph theory and systemmodel e leader-following consensus of the systems underthe fixedswitching topologies is considered in Sections 3and 4 In addition the effectiveness of the results is shownthrough simulation experiment in Section 5 Section 6summarizes this paper
e following notations are used in this paper otimes denotesthe Kronecker product IN denotes the N-dimensionalidentity matrix Rn and Rmtimesn indicate the set of n dimen-sional real vectors and m times n dimensional real matricesrespectively
2 Preliminaries
21eory ofGraph For a multiagent system composed of aleader and follower agents its communication topology canbe represented by an undirected graph G (V E) whereV 0 1 N 0 denotes the leader and 1 N denotethe followers EsubeV times V denotes the edges set e con-nection matrix between the follower agent i(i 1 N)
and leader 0 is D diag a10 aN01113864 1113865 where ai0 is theconnection weight between the leader 0 and follower i Ifai0 gt 0 the follower agent i can receive state information ofleader 0 otherwise ai0 0
e communication network among the followers isdenoted by G (V E A) where V 1 2 N andEsubeV times V are obtained from E by removing all edgesamong the leader 0 and followers in V andA (aij) isin RNtimesN is the weighted adjacency matrix of graphG where aij gt 0 for (i j) isin E if agent i obtains informationfrom agent j We assume that (i i) notin E and hence aii 0For a given graph G with the adjacency matrix A theLaplacian matrix used in this paper is L D minus A where D isa diagonal matrix its diagonal elements are dii 1113936jisinNi
|aij|and define Ni as the neighbor set of agent i inV ereforethe elements of L are
Lik 1113936
jisinNi
aij
11138681113868111386811138681113868
11138681113868111386811138681113868 k i
minus aik kne i
⎧⎪⎨
⎪⎩(1)
A path from the vertex i to vertex k is a sequence ofadjacent edges in the form (i i + 1) (i + 1 i + 2)
(k minus 1 k) e undirected graph is said to be connected ifthere exists a path between any two distinct vertices
22 SystemModel Consider a multiagent system composedof the leader 0 and N followers e dynamics of leader 0 is
_x0(t) Ax0(t) (2)
where x0(t) isin Rn is the state and A isin Rntimesn is constantmatrix
Accordingly each follower has the following lineardynamic equation
_xi(t) Axi(t) + Bui(t) i 1 N (3)
where xi(t) isin Rn and ui(t) isin Rp are the state and input ofthe ith follower agent respectively A isin Rntimesn andB isin Rntimesp
are constant matrices Denote the initial state of the ithfollower as xi(0)
Definition 1 If there is a control input ui(t) the leader 0 andfollower i for any initial state satisfy the following conditions
limt⟶infin
xi(t) minus x0(t)
0 i 1 2 N (4)
en the leader (2) is said to be successfully tracked byfollower (3)
Assumption 1 e communication network topology G
among followers is connected
2 Complexity
Assumption 2 e pair (A B) is stabilizableBased on Assumption 2 there is a symmetric positive
definite matrix P that satisfies the following algebraic Riccatiand Lyapunov inequality with βgt 0
ATP + PA minus 2βPBBTP + βIlt 0 (5)
ATP + PAlt 0 (6)
Lemma 1 (see [37]) For an undirected and connected graphG the eigenvalues of L are real and can be labelled as
0 λ1(L)lt λ2(L)le middot middot middot le λN(L) (7)
Lemma 2 (see [40]) For any x y isin R and βgt 0 one has thefollowing property
xyleβ2x2
+12β
y2 (8)
Lemma 3 (see [41]) (Comparison Principle) Consider adifferential equation (dudt) f(t u) u(t0) u0 wheretgt 0 f(t u) is continuous and satisfies the local Lipschitzcondition in t Let [t0 T) be the maximum existence intervalof the solution u where Tcan be infinite If for any t isin [t0 T)
satisfiesdv
dtlef(t v)
v t0( 1113857le u0
(9)
then v(t)le u(t) t isin [t0 T)
3 Leader-Following Control of MultiagentSystems under Fixed Topology
In this part we consider the leader-following control ofmultiagent systems (2) and (3) under the event-triggeredstrategy Based on the general event-triggered control lawwe put forward two kinds of piecewise continuous controlmechanisms which are centralized event-triggered mech-anism and decentralized event-triggered mechanism withstate estimation in order to minimize the frequency ofcontroller updating e analysis shows that under the twocontrol mechanisms multiagent system (3) can track thesystem (2) successfully with appropriate event-triggeringfunction e minimum interval between any two consec-utive event-triggering instants under the two controlmechanisms is greater than 0 and Zeno behavior can beexcluded
31 Centralized Event-Triggered Control Strategy Under thecentralized event-triggered strategy all agents i in system (3)are triggered synchronously at the time tk(k 0 1 ) Atthe triggering instants all agents send their states infor-mation to neighbours and update the control law with thereceived state information Compared with the control
protocol in continuous time each agent i only updates thecontrol input at the event instants under the event-triggeredmechanism So ui is a piecewise continuous function andthe updating frequency can be reduced
We consider the following control input for the ithfollower
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tk( 1113857 minus xj tk( 11138571113872 1113873
minus Kai0(t) xi tk( 1113857 minus x0 tk( 1113857( 1113857
(10)
where t isin [tk tk+1) K isin Rptimesn is the control gain matrix to bedesigned and xi(tk) is the sampling state of agent i at the kthtriggering instant Since there does not exist control inputfor leader 0 we take x0(tk) x0(t) t isin [tk tk+1) For con-venience we make t0 0
e event-triggering time sequence tk1113864 1113865 is determined bythe following triggering functions
f(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2λmax(D)λmax(W)
1113944
N
i11113954x
Ti 1113954xi
+ 1113944
N
i1e
Ti ei ge 0
(11)
at is tk+1 inf tgt tk | f(t)ge 01113864 1113865 where 0lt κ lt 1 0ltαlt (ai0min
λmin(W)λmax(P)) W PBBTPe state error between the follower and leader is defined
as 1113954xi(t) xi(t) minus x0(t) 1113954x(t) [1113954xT1 1113954xT
N]T For agent ithe measurement error is defined as ei(t) xi(tk) minus
xi(t) t isin [tk tk+1) en formula (10) is converted to
ui(t) minus K 1113944N
j0aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873⎛⎝ ⎞⎠
minus Kai0 1113954xi(t) + ei(t)( 1113857
(12)
Combining (2) (3) and (12) we get_1113954x(t) IN otimesA( 11138571113954x(t) minus IN otimesBK( 1113857 (L + D)otimes In( 1113857(1113954x(t) + e(t))
IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t))
(13)
Remark 1 rough the model transformation the leader-following control problem between systems (2) and (3) canbe interpreted by the stability problem of system (13)
Next we will give the following consensus conditionsunder the centralized event-triggering protocol (10)
Theorem 1 Under Assumptions 1 and 2 centralized event-triggered control strategy (10) can make multiagent system (3)track system (2) successfully under event-triggering condition(4) where feedback gain matrix K satisfies K BTP andW PBK
Complexity 3
Proof We consider a candidate Lyapunov function asfollows
V1 eαt 1113936N
i11113954xT
i P1113954xi (14)
Along with the trajectories of the state as described in (8)the time derivative of Lyapunov function is
_V1 αeαt
1113944
N
i11113954x
Ti P1113954xi + 2e
αt1113944
N
i11113954x
Ti P _1113954xi
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti P A1113954xi + Bui( 1113857
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi + 2e
αt1113944
N
i11113954x
Ti PBui
(15)
where
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti PBK ai0 1113954xi + ei( 1113857 + 1113944
N
j1aij 1113954xi minus 1113954xj + ei minus ej1113872 1113873⎛⎝ ⎞⎠
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
(16)
where W PBKAccording to the property of L LT in undirected graph
G we can deduce
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aji1113954x
Tj W 1113954xj minus 1113954xi1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Tj W 1113954xi minus 1113954xj1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(17)
Similarly
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(18)
Hence
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(19)
Combining equality (15) yields
_V1 eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(20)
In the light of Lemma 2 we have
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
le12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873 +
12eαt
1113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
(21)
By substituting the abovementioned formula intoequation (20) we obtain
4 Complexity
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(22)
From Lemma 2 we have
eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873le 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
(23)
Together with (22) we can get that
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(24)
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi minus e
αt1113944
N
i11113954x
Ti Wai0ei
le minus eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i11113954x
Ti Wai01113954xi
+12eαt
1113944
N
i1e
Ti Wai0ei
le minus12eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i1e
Ti Wai0ei
(25)
Combining (24) and (25) we arrive at
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi + e
αt1113944
N
i1e
Ti Wai0ei
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
le eαt
1113954xT
IN otimes αP( 11138571113954x + eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minus eαt
1113954xT(LotimesW)1113954x minus e
αt1113954x
T(DotimesW)1113954x
+ eαt
eT(DotimesW)e + 2e
αte
T(DotimesW)e
(26)
Under Assumption 1 by using Lemma 1 1113954xT(LotimesW)1113954xgeλ2(L)1113954xT(IN otimesW)1113954x holds Consequently
_V1 le eαt
1113954xT
IN otimes PA + ATP minus λ2(L)W1113872 11138731113872 11138731113954x
+ eαt αλmax(P) minus ai0min
λmin(W)1113872 11138731113954x2
+ eαt
ai0maxλmax(W) + 2λmax(D)λmax(W)1113872 1113873e
2
(27)
Using inequality (5) and event-triggering condition (11)we claim that the following inequality holds
Complexity 5
_V1 le (κ minus 1)eαt
ai0minλmin(W) minus αλmax(P)1113872 11138731113954x
2
minus eαtλ2(L)
21113954x
T1113954x le minus e
αtλ2(L)
21113954x
T1113954x
(28)
It can be seen from (28) that V1 is not increasingtherefore
V1(0)geV1(t) eαt 1113936N
i11113954xi(t)TP1113954xi(i)ge eαtλmin(P)1113954x(t)2
(29)
at is to say 1113954x(t)le(V1(0)λmin(P))
1113968eminus (α2)t ie
limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0 whichmeans limt⟶infin||xi(t) minus x0(t)|| 0 i 1 2 N
holds
Theorem 2 Under the conditions of eorem 1 system (13)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
IN otimesA
+||(L + D)otimesBK||1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (30)
Proof From the mechanism of event-triggering strategythe event interval between tk and tk+1 is the timethat (||e(t)||1113954x(t)) grows from 0 to
(κ(ai0min
λmin(W)minus αλmax(P))(ai0maxλmax(W)+2λmax(D)λmax(W)))
1113969
e time derivative of (e(t)||1113954x(t)||) has
ddt
e(t)
1113954x(t)
ddt
e(t)Te(t)1113872 111387312
1113954x(t)T1113954x(t)1113872 111387312
e(t)Te(t)1113872 1113873
(12)prime1113954x(t) minus e(t)Te(t)1113872 1113873
121113954x(t)T1113954x(t)1113872 1113873
(12)prime
1113954x(t)2
e(t)T _e(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
minus e(t)T _1113954x(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
le _1113954x(t)
1113954x(t)+
_1113954x(t)e(t)
1113954x(t)2
_1113954x(t)
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+(L + D)otimesBKe
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+IN otimesA
+(L + D)otimesBK1113872 1113873e
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
2
(31)
6 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
Assumption 2 e pair (A B) is stabilizableBased on Assumption 2 there is a symmetric positive
definite matrix P that satisfies the following algebraic Riccatiand Lyapunov inequality with βgt 0
ATP + PA minus 2βPBBTP + βIlt 0 (5)
ATP + PAlt 0 (6)
Lemma 1 (see [37]) For an undirected and connected graphG the eigenvalues of L are real and can be labelled as
0 λ1(L)lt λ2(L)le middot middot middot le λN(L) (7)
Lemma 2 (see [40]) For any x y isin R and βgt 0 one has thefollowing property
xyleβ2x2
+12β
y2 (8)
Lemma 3 (see [41]) (Comparison Principle) Consider adifferential equation (dudt) f(t u) u(t0) u0 wheretgt 0 f(t u) is continuous and satisfies the local Lipschitzcondition in t Let [t0 T) be the maximum existence intervalof the solution u where Tcan be infinite If for any t isin [t0 T)
satisfiesdv
dtlef(t v)
v t0( 1113857le u0
(9)
then v(t)le u(t) t isin [t0 T)
3 Leader-Following Control of MultiagentSystems under Fixed Topology
In this part we consider the leader-following control ofmultiagent systems (2) and (3) under the event-triggeredstrategy Based on the general event-triggered control lawwe put forward two kinds of piecewise continuous controlmechanisms which are centralized event-triggered mech-anism and decentralized event-triggered mechanism withstate estimation in order to minimize the frequency ofcontroller updating e analysis shows that under the twocontrol mechanisms multiagent system (3) can track thesystem (2) successfully with appropriate event-triggeringfunction e minimum interval between any two consec-utive event-triggering instants under the two controlmechanisms is greater than 0 and Zeno behavior can beexcluded
31 Centralized Event-Triggered Control Strategy Under thecentralized event-triggered strategy all agents i in system (3)are triggered synchronously at the time tk(k 0 1 ) Atthe triggering instants all agents send their states infor-mation to neighbours and update the control law with thereceived state information Compared with the control
protocol in continuous time each agent i only updates thecontrol input at the event instants under the event-triggeredmechanism So ui is a piecewise continuous function andthe updating frequency can be reduced
We consider the following control input for the ithfollower
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tk( 1113857 minus xj tk( 11138571113872 1113873
minus Kai0(t) xi tk( 1113857 minus x0 tk( 1113857( 1113857
(10)
where t isin [tk tk+1) K isin Rptimesn is the control gain matrix to bedesigned and xi(tk) is the sampling state of agent i at the kthtriggering instant Since there does not exist control inputfor leader 0 we take x0(tk) x0(t) t isin [tk tk+1) For con-venience we make t0 0
e event-triggering time sequence tk1113864 1113865 is determined bythe following triggering functions
f(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2λmax(D)λmax(W)
1113944
N
i11113954x
Ti 1113954xi
+ 1113944
N
i1e
Ti ei ge 0
(11)
at is tk+1 inf tgt tk | f(t)ge 01113864 1113865 where 0lt κ lt 1 0ltαlt (ai0min
λmin(W)λmax(P)) W PBBTPe state error between the follower and leader is defined
as 1113954xi(t) xi(t) minus x0(t) 1113954x(t) [1113954xT1 1113954xT
N]T For agent ithe measurement error is defined as ei(t) xi(tk) minus
xi(t) t isin [tk tk+1) en formula (10) is converted to
ui(t) minus K 1113944N
j0aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873⎛⎝ ⎞⎠
minus Kai0 1113954xi(t) + ei(t)( 1113857
(12)
Combining (2) (3) and (12) we get_1113954x(t) IN otimesA( 11138571113954x(t) minus IN otimesBK( 1113857 (L + D)otimes In( 1113857(1113954x(t) + e(t))
IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t))
(13)
Remark 1 rough the model transformation the leader-following control problem between systems (2) and (3) canbe interpreted by the stability problem of system (13)
Next we will give the following consensus conditionsunder the centralized event-triggering protocol (10)
Theorem 1 Under Assumptions 1 and 2 centralized event-triggered control strategy (10) can make multiagent system (3)track system (2) successfully under event-triggering condition(4) where feedback gain matrix K satisfies K BTP andW PBK
Complexity 3
Proof We consider a candidate Lyapunov function asfollows
V1 eαt 1113936N
i11113954xT
i P1113954xi (14)
Along with the trajectories of the state as described in (8)the time derivative of Lyapunov function is
_V1 αeαt
1113944
N
i11113954x
Ti P1113954xi + 2e
αt1113944
N
i11113954x
Ti P _1113954xi
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti P A1113954xi + Bui( 1113857
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi + 2e
αt1113944
N
i11113954x
Ti PBui
(15)
where
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti PBK ai0 1113954xi + ei( 1113857 + 1113944
N
j1aij 1113954xi minus 1113954xj + ei minus ej1113872 1113873⎛⎝ ⎞⎠
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
(16)
where W PBKAccording to the property of L LT in undirected graph
G we can deduce
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aji1113954x
Tj W 1113954xj minus 1113954xi1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Tj W 1113954xi minus 1113954xj1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(17)
Similarly
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(18)
Hence
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(19)
Combining equality (15) yields
_V1 eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(20)
In the light of Lemma 2 we have
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
le12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873 +
12eαt
1113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
(21)
By substituting the abovementioned formula intoequation (20) we obtain
4 Complexity
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(22)
From Lemma 2 we have
eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873le 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
(23)
Together with (22) we can get that
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(24)
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi minus e
αt1113944
N
i11113954x
Ti Wai0ei
le minus eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i11113954x
Ti Wai01113954xi
+12eαt
1113944
N
i1e
Ti Wai0ei
le minus12eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i1e
Ti Wai0ei
(25)
Combining (24) and (25) we arrive at
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi + e
αt1113944
N
i1e
Ti Wai0ei
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
le eαt
1113954xT
IN otimes αP( 11138571113954x + eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minus eαt
1113954xT(LotimesW)1113954x minus e
αt1113954x
T(DotimesW)1113954x
+ eαt
eT(DotimesW)e + 2e
αte
T(DotimesW)e
(26)
Under Assumption 1 by using Lemma 1 1113954xT(LotimesW)1113954xgeλ2(L)1113954xT(IN otimesW)1113954x holds Consequently
_V1 le eαt
1113954xT
IN otimes PA + ATP minus λ2(L)W1113872 11138731113872 11138731113954x
+ eαt αλmax(P) minus ai0min
λmin(W)1113872 11138731113954x2
+ eαt
ai0maxλmax(W) + 2λmax(D)λmax(W)1113872 1113873e
2
(27)
Using inequality (5) and event-triggering condition (11)we claim that the following inequality holds
Complexity 5
_V1 le (κ minus 1)eαt
ai0minλmin(W) minus αλmax(P)1113872 11138731113954x
2
minus eαtλ2(L)
21113954x
T1113954x le minus e
αtλ2(L)
21113954x
T1113954x
(28)
It can be seen from (28) that V1 is not increasingtherefore
V1(0)geV1(t) eαt 1113936N
i11113954xi(t)TP1113954xi(i)ge eαtλmin(P)1113954x(t)2
(29)
at is to say 1113954x(t)le(V1(0)λmin(P))
1113968eminus (α2)t ie
limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0 whichmeans limt⟶infin||xi(t) minus x0(t)|| 0 i 1 2 N
holds
Theorem 2 Under the conditions of eorem 1 system (13)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
IN otimesA
+||(L + D)otimesBK||1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (30)
Proof From the mechanism of event-triggering strategythe event interval between tk and tk+1 is the timethat (||e(t)||1113954x(t)) grows from 0 to
(κ(ai0min
λmin(W)minus αλmax(P))(ai0maxλmax(W)+2λmax(D)λmax(W)))
1113969
e time derivative of (e(t)||1113954x(t)||) has
ddt
e(t)
1113954x(t)
ddt
e(t)Te(t)1113872 111387312
1113954x(t)T1113954x(t)1113872 111387312
e(t)Te(t)1113872 1113873
(12)prime1113954x(t) minus e(t)Te(t)1113872 1113873
121113954x(t)T1113954x(t)1113872 1113873
(12)prime
1113954x(t)2
e(t)T _e(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
minus e(t)T _1113954x(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
le _1113954x(t)
1113954x(t)+
_1113954x(t)e(t)
1113954x(t)2
_1113954x(t)
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+(L + D)otimesBKe
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+IN otimesA
+(L + D)otimesBK1113872 1113873e
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
2
(31)
6 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
Proof We consider a candidate Lyapunov function asfollows
V1 eαt 1113936N
i11113954xT
i P1113954xi (14)
Along with the trajectories of the state as described in (8)the time derivative of Lyapunov function is
_V1 αeαt
1113944
N
i11113954x
Ti P1113954xi + 2e
αt1113944
N
i11113954x
Ti P _1113954xi
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti P A1113954xi + Bui( 1113857
eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi + 2e
αt1113944
N
i11113954x
Ti PBui
(15)
where
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti PBK ai0 1113954xi + ei( 1113857 + 1113944
N
j1aij 1113954xi minus 1113954xj + ei minus ej1113872 1113873⎛⎝ ⎞⎠
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
(16)
where W PBKAccording to the property of L LT in undirected graph
G we can deduce
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
eαt
1113944
N
i11113944
N
j1aji1113954x
Tj W 1113954xj minus 1113954xi1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Tj W 1113954xi minus 1113954xj1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(17)
Similarly
eαt
1113944
N
i11113954x
Ti W 1113944
N
j1aij ei minus ej1113872 1113873
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(18)
Hence
eαt
1113944
N
i11113954x
Ti PBui
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus
12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857 minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
(19)
Combining equality (15) yields
_V1 eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(20)
In the light of Lemma 2 we have
minus eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW ei minus ej1113872 1113873
le12eαt
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873 +
12eαt
1113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
(21)
By substituting the abovementioned formula intoequation (20) we obtain
4 Complexity
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(22)
From Lemma 2 we have
eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873le 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
(23)
Together with (22) we can get that
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(24)
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi minus e
αt1113944
N
i11113954x
Ti Wai0ei
le minus eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i11113954x
Ti Wai01113954xi
+12eαt
1113944
N
i1e
Ti Wai0ei
le minus12eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i1e
Ti Wai0ei
(25)
Combining (24) and (25) we arrive at
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi + e
αt1113944
N
i1e
Ti Wai0ei
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
le eαt
1113954xT
IN otimes αP( 11138571113954x + eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minus eαt
1113954xT(LotimesW)1113954x minus e
αt1113954x
T(DotimesW)1113954x
+ eαt
eT(DotimesW)e + 2e
αte
T(DotimesW)e
(26)
Under Assumption 1 by using Lemma 1 1113954xT(LotimesW)1113954xgeλ2(L)1113954xT(IN otimesW)1113954x holds Consequently
_V1 le eαt
1113954xT
IN otimes PA + ATP minus λ2(L)W1113872 11138731113872 11138731113954x
+ eαt αλmax(P) minus ai0min
λmin(W)1113872 11138731113954x2
+ eαt
ai0maxλmax(W) + 2λmax(D)λmax(W)1113872 1113873e
2
(27)
Using inequality (5) and event-triggering condition (11)we claim that the following inequality holds
Complexity 5
_V1 le (κ minus 1)eαt
ai0minλmin(W) minus αλmax(P)1113872 11138731113954x
2
minus eαtλ2(L)
21113954x
T1113954x le minus e
αtλ2(L)
21113954x
T1113954x
(28)
It can be seen from (28) that V1 is not increasingtherefore
V1(0)geV1(t) eαt 1113936N
i11113954xi(t)TP1113954xi(i)ge eαtλmin(P)1113954x(t)2
(29)
at is to say 1113954x(t)le(V1(0)λmin(P))
1113968eminus (α2)t ie
limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0 whichmeans limt⟶infin||xi(t) minus x0(t)|| 0 i 1 2 N
holds
Theorem 2 Under the conditions of eorem 1 system (13)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
IN otimesA
+||(L + D)otimesBK||1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (30)
Proof From the mechanism of event-triggering strategythe event interval between tk and tk+1 is the timethat (||e(t)||1113954x(t)) grows from 0 to
(κ(ai0min
λmin(W)minus αλmax(P))(ai0maxλmax(W)+2λmax(D)λmax(W)))
1113969
e time derivative of (e(t)||1113954x(t)||) has
ddt
e(t)
1113954x(t)
ddt
e(t)Te(t)1113872 111387312
1113954x(t)T1113954x(t)1113872 111387312
e(t)Te(t)1113872 1113873
(12)prime1113954x(t) minus e(t)Te(t)1113872 1113873
121113954x(t)T1113954x(t)1113872 1113873
(12)prime
1113954x(t)2
e(t)T _e(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
minus e(t)T _1113954x(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
le _1113954x(t)
1113954x(t)+
_1113954x(t)e(t)
1113954x(t)2
_1113954x(t)
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+(L + D)otimesBKe
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+IN otimesA
+(L + D)otimesBK1113872 1113873e
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
2
(31)
6 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus 2eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(22)
From Lemma 2 we have
eαt
21113944
N
i11113944
N
j1aij ei minus ej1113872 1113873
TW ei minus ej1113872 1113873le 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
(23)
Together with (22) we can get that
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
minus 2eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
(24)
minus eαt
1113944
N
i11113954x
Ti Wai0 1113954xi + ei( 1113857
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi minus e
αt1113944
N
i11113954x
Ti Wai0ei
le minus eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i11113954x
Ti Wai01113954xi
+12eαt
1113944
N
i1e
Ti Wai0ei
le minus12eαt
1113944
N
i11113954x
Ti Wai01113954xi +
12eαt
1113944
N
i1e
Ti Wai0ei
(25)
Combining (24) and (25) we arrive at
_V1 le eαt
1113944
N
i11113954x
Ti αP1113954xi + 2e
αt1113944
N
i11113954x
Ti PA1113954xi
minus eαt
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus eαt
1113944
N
i11113954x
Ti Wai01113954xi + e
αt1113944
N
i1e
Ti Wai0ei
+ 2eαt
1113944
N
i11113944
N
j1aije
Ti Wei
le eαt
1113954xT
IN otimes αP( 11138571113954x + eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minus eαt
1113954xT(LotimesW)1113954x minus e
αt1113954x
T(DotimesW)1113954x
+ eαt
eT(DotimesW)e + 2e
αte
T(DotimesW)e
(26)
Under Assumption 1 by using Lemma 1 1113954xT(LotimesW)1113954xgeλ2(L)1113954xT(IN otimesW)1113954x holds Consequently
_V1 le eαt
1113954xT
IN otimes PA + ATP minus λ2(L)W1113872 11138731113872 11138731113954x
+ eαt αλmax(P) minus ai0min
λmin(W)1113872 11138731113954x2
+ eαt
ai0maxλmax(W) + 2λmax(D)λmax(W)1113872 1113873e
2
(27)
Using inequality (5) and event-triggering condition (11)we claim that the following inequality holds
Complexity 5
_V1 le (κ minus 1)eαt
ai0minλmin(W) minus αλmax(P)1113872 11138731113954x
2
minus eαtλ2(L)
21113954x
T1113954x le minus e
αtλ2(L)
21113954x
T1113954x
(28)
It can be seen from (28) that V1 is not increasingtherefore
V1(0)geV1(t) eαt 1113936N
i11113954xi(t)TP1113954xi(i)ge eαtλmin(P)1113954x(t)2
(29)
at is to say 1113954x(t)le(V1(0)λmin(P))
1113968eminus (α2)t ie
limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0 whichmeans limt⟶infin||xi(t) minus x0(t)|| 0 i 1 2 N
holds
Theorem 2 Under the conditions of eorem 1 system (13)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
IN otimesA
+||(L + D)otimesBK||1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (30)
Proof From the mechanism of event-triggering strategythe event interval between tk and tk+1 is the timethat (||e(t)||1113954x(t)) grows from 0 to
(κ(ai0min
λmin(W)minus αλmax(P))(ai0maxλmax(W)+2λmax(D)λmax(W)))
1113969
e time derivative of (e(t)||1113954x(t)||) has
ddt
e(t)
1113954x(t)
ddt
e(t)Te(t)1113872 111387312
1113954x(t)T1113954x(t)1113872 111387312
e(t)Te(t)1113872 1113873
(12)prime1113954x(t) minus e(t)Te(t)1113872 1113873
121113954x(t)T1113954x(t)1113872 1113873
(12)prime
1113954x(t)2
e(t)T _e(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
minus e(t)T _1113954x(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
le _1113954x(t)
1113954x(t)+
_1113954x(t)e(t)
1113954x(t)2
_1113954x(t)
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+(L + D)otimesBKe
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+IN otimesA
+(L + D)otimesBK1113872 1113873e
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
2
(31)
6 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
_V1 le (κ minus 1)eαt
ai0minλmin(W) minus αλmax(P)1113872 11138731113954x
2
minus eαtλ2(L)
21113954x
T1113954x le minus e
αtλ2(L)
21113954x
T1113954x
(28)
It can be seen from (28) that V1 is not increasingtherefore
V1(0)geV1(t) eαt 1113936N
i11113954xi(t)TP1113954xi(i)ge eαtλmin(P)1113954x(t)2
(29)
at is to say 1113954x(t)le(V1(0)λmin(P))
1113968eminus (α2)t ie
limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0 whichmeans limt⟶infin||xi(t) minus x0(t)|| 0 i 1 2 N
holds
Theorem 2 Under the conditions of eorem 1 system (13)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
IN otimesA
+||(L + D)otimesBK||1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (30)
Proof From the mechanism of event-triggering strategythe event interval between tk and tk+1 is the timethat (||e(t)||1113954x(t)) grows from 0 to
(κ(ai0min
λmin(W)minus αλmax(P))(ai0maxλmax(W)+2λmax(D)λmax(W)))
1113969
e time derivative of (e(t)||1113954x(t)||) has
ddt
e(t)
1113954x(t)
ddt
e(t)Te(t)1113872 111387312
1113954x(t)T1113954x(t)1113872 111387312
e(t)Te(t)1113872 1113873
(12)prime1113954x(t) minus e(t)Te(t)1113872 1113873
121113954x(t)T1113954x(t)1113872 1113873
(12)prime
1113954x(t)2
e(t)T _e(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
minus e(t)T _1113954x(t)
1113954x(t)e(t)minus
e(t)1113954x(t)T _1113954x(t)
1113954x(t)21113954x(t)
le _1113954x(t)
1113954x(t)+
_1113954x(t)e(t)
1113954x(t)2
_1113954x(t)
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+(L + D)otimesBKe
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
le IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
+IN otimesA
+(L + D)otimesBK1113872 1113873e
1113954x(t)1 +
e(t)
1113954x(t)1113888 1113889
IN otimesA
+(L + D)otimesBK1113872 1113873 1 +e(t)
1113954x(t)1113888 1113889
2
(31)
6 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
Denote z (||e(t)||1113954x(t)) then
_zle IN otimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873(1 + z)2 (32)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (IN otimesA + ||(L + D)otimesBK||)(1 + ψ)2 and ψ0 0en from Lemma 3 zleψ(t 0) It can be seen from (11)that
ψ(τ 0)
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
(33)
erefore
τ IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
IN otimesA
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868 +(L + D)otimesBK1113872 1113873
3times 1 +
κ ai0minλmin(W) minus αλmax(P)1113872 1113873
ai0maxλmax(W) + 2λmax(D)λmax(W)
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(34)
Obviously τ gt 0It is assumed that the Zeno behavior occurs which
means that there exists a positive constant tlowast such thatlimk⟶infintk tlowast Let ε0 (12)τ ere exists a positive in-teger N0 such that tlowast minus ε0 le tk le tlowast for the abovementionedε0 gt 0 according to the definition of sequence limit wherekgeN0 erefore tlowast + ε0 le tk + 2ε0 le tk+1 holds whenkgeN0 is contradicts with tlowast ge tk+1 for kgeN0 us Zenobehavior is strictly excluded
32 Decentralized Event-Triggered Control Strategy ecentralized event-triggered mechanism given in the previoussection sets a global state error threshold for all agents Oncethe system error reaches the threshold all agents in thesystem perform control tasks at the same time In thissection an error threshold based on the state of its neighbornode is set for each agent When the state error of the agentreaches the set threshold the agent triggers the event in-dependently and executes the control task
e triggering time of the kth event of the i agent isdefined as ti
k(k 0 1 ) In the design of this section itshould be noted that the agent triggers asynchronously thatis each agent has its own event-triggering sequence emeasurement error of agent i is defined as ei(t)
xi(tik) minus xi(t) t isin [ti
k tik+1) It is clear that ei(ti
k) 0 whent ti
kFor a multiagent system composed of (2) and (3) we
consider the following decentralized event-triggered controlprotocol
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(35)
where t isin [tik ti
k+1) tj
kprime argmin
lisinNtgetj
l
t minus tj
l1113966 1113967 representsthe latest event-triggering time before t for agent jAccording to (35) agent iwill update control input ui at both
its triggering instants (ti0 ti
1 ) and neighbor agent j eventinstants (t
j0 t
j1 ) e event-triggering instant sequence
tik1113864 1113865 for agent i is determined by the following decentralized
event-triggering function
fi(t) minus (1 minus κ) 1113944N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
+ λmax(W) ei
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
1113944
N
j14aij + 2ai01113872 1113873ge 0
(36)
where 0lt κlt 1 W PBBTP According to the definition ofmeasurement error and 1113954xi(t) xi(t) minus x0(t) (35) can berewritten as
ui(t) minus K 1113944N
j0aij xi(t) minus xj(t) + ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j0aij xi(t) minus x0(t) minus xj(t) minus x0(t)1113872 11138731113872 1113873
minus K 1113944
N
j0aij ei(t) minus ej(t)1113872 1113873
minus K 1113944N
j1aij 1113954xi(t) minus 1113954xj(t) + ei(t) minus ej(t)1113872 1113873
minus Kai0 1113954xi(t) + ei(t)( 1113857
(37)
Combining (2) (3) with (37) yields_1113954x(t) IN otimesA( 11138571113954x(t) minus ((L + D)otimesBK)(1113954x(t) + e(t)) (38)
Theorem 3 Under Assumption 1 the multiagent systems (3)with protocol (35) can track system (2) successfully under theevent-triggering condition (36) where K BTP andW PBBTP
Complexity 7
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
Proof Define the Lyapunov function
V2 12
1113944
N
i11113954x
Ti P1113954xi (39)
Following the same proof as that of eorem 1 the timederivation of V2 along the trajectory of system (38) isobtained
_V2 le 1113936N
i11113954xT
i PA1113954xi minus12
1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi +
12
1113944
N
i1e
Ti Wai0ei
+ 1113944N
i11113944
N
j1aije
Ti Wei
le 1113954xT
IN otimesPA( 11138571113954x minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
minus12
1113944
N
i11113954x
Ti Wai01113954xi + 1113944
N
i11113944
N
j1aije
Ti Wei +
12e
Ti Wai0ei
⎛⎝ ⎞⎠
le 1113954xT
IN otimesPA( 11138571113954x +14
1113944
N
i11113944
N
j1e
Ti 4aij + 2ai01113872 1113873Wei
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le 1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+14
1113944
N
i1λmax(W) ei
2
1113944
N
j14aij + 2ai01113872 1113873
minus14
1113944
N
i11113944
N
j1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
(40)
According to (6) and event-triggering condition (36) wecan find that
_V2 le12
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
minusκ4
1113944
N
j11113944
N
i1aij 1113954xi minus 1113954xj1113872 1113873
TW 1113954xi minus 1113954xj1113872 1113873
le minusκ2
1113954xT(LotimesW)1113954x
le minusκ2λmax(L)λmax(W)1113954x
2
le 0
(41)
It can be seen from the abovementioned formula that V2is not increasing therefore
V2(0)geV2(t) 12
1113944
N
i11113954xi(t)
TP1113954xi(t)ge
12λmin(P)1113954x(t)
2
(42)
at is to say ||1113954x(t)||le2(V2(0)λmin(P))
1113968 0
According to LaSallersquos invariance principle we canobtain that system (38) can achieve consensus that islimt⟶infin1113954xi 0 which is equivalent tolimt⟶infinxi(t) minus x0(t) 0 i 1 2 N e proof iscompleted
Theorem 4 Under the conditions of eorem 3 system (38)does not exhibit Zeno behavior e interval between any twoconsecutive event-triggering instants of the system is not lessthan
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3times 1 + (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠ (43)
Proof It is similar to the proof of eorem 2 e eventinterval between ti
k and tik+1 is (ei(t)1113954xi(t)) which grows
from 0 to ((1 minus κ)(ai0λmin(W)(2dii + ai0)λmax(W)))12 etime derivative of (||ei(t)||1113954xi(t)) is
8 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
d
dt
ei
1113954xi
le_1113954xi(t)
1113954xi(t)
+
_1113954xi(t)
ei(t)
1113954xi(t)
2
_1113954xi(t)
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+BK Li + ai0( 1113857otimes In( 1113857
ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
+A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 ei
1113954xi(t)
1 +
ei(t)
1113954xi(t)
1113888 1113889
A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 +ei(t)
1113954xi(t)
1113888 1113889
2
(44)
where Li is the row i of the Laplace matrix LLet zi (ei(t)||1113954xi(t)||) then
_zi le A + BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873 1 + zi( 11138572 (45)
Consider that a nonnegative function ψ(tψ0) satisfies_ψ (||A|| + BK((Li + ai0)otimes In))(1 + ψ)2 and ψ0 0according to Lemma 3 zi leψ(t 0) It can be seen from (36)
ψ τik 01113872 1113873 (1 minus κ)
ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12
(46)
Hence
τik
A+ BK Li + ai0( 1113857otimes In( 11138571113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113872 1113873
3(1 + ψ(τ))
3minus 11113872 1113873
A+ BK Li + ai0( 1113857otimes In( 1113857
1113872 1113873
3
times 1 + (1 minus κ)ai0λmin(W)
2dii + ai0( 1113857λmax(W)1113888 1113889
12⎛⎝ ⎞⎠
3
minus 1⎛⎝ ⎞⎠
(47)
Similar to eorem 2 that Zeno behavior that does notoccur can be proved by contradiction which is omittedhere
4 Leader-Following Control of MultiagentSystems under Switching Topologies
In this part we consider the extended case that the inter-connection network switches according to signal σ(t) and isnot connected all the time It is worth noting that unlike thefixed topology the controller updates only when the event istriggered In the switching topologies the controller updatesin the following two cases (1) event-triggering instant (2)Communication topology switching instant
e control input of the ith agent is defined as follows
ui(t) minus K 1113944jisinNi(t)
aij(t) xi tik1113872 1113873 minus xj t
ikprime1113872 11138731113872 1113873
minus Kai0(t) xi tik1113872 1113873 minus x0(t)1113872 1113873
(48)
where t isin [tik ti
k+1) Different from control protocols (10) and(35) Ni(t) and aij(t) in (48) are changed under the switchingtopologies Matrices Lσ(t) and Dσ(t) in Gσ(t) represent Lap-lacian matrix and connection matrix between leader andagent respectively Switching signal σ(t) [0infin)⟶ P is apiecewise continuous constant function which is used todescribe the switching law of communication topology AlsoGσ(t) p isin P1113966 1113967 is a set of graphs that are switched within afinite setP 1 2 in any finite time interval Consider anonempty and continuous infinite sequence [ts ts+1) wherek 0 1 and t0 0 Suppose thatGσ(t) is switched only atand remains unchanged in t isin [ts ts+1)
Remark 2 It should be noted that graph Gσ(t) may beconnected or unconnected in interval [ts ts+1)
By replacing the similar variables in Section 32 we canderive that
_1113954x(t) IN otimesA( 11138571113954x(t) minus Lσ(t) + Dσ(t)1113872 1113873otimesBK1113872 1113873(1113954x(t) + e(t))
(49)
Theorem 5 Under Assumptions 1 and 2 if feedback gainmatrix K satisfies K BTP and W PBK then the protocol(48) still makes the multiagent system with (3) track thesystem (2) successfully if the event-triggering conditionsatisfies
fi(t) minus κai0min
λmin(W) minus αλmax(P)
ai0maxλmax(W) + 2diimax
λmax(W)1113944
N
i11113954x
Ti 1113954xi
+ 1113944N
i1e
Ti ei ge 0
(50)
where 0lt κlt 1 0lt αle (ai0σ(t)λmin(W)λmax(P))
Proof Construct the Lyapunov function for system (49) asfollows
V3 eαt 1113936N
i11113954xT
i P1113954xi (51)
Similar to Section 32 taking the derivative of V3 alongthe trajectory of system (49) yields
Complexity 9
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
_V3 le2eαt
1113944
N
i11113954x
Ti PA1113954xi minus e
αt1113944
N
i11113944
N
j1aij1113954x
Ti W 1113954xi minus 1113954xj1113872 1113873
+ eαt
1113944
N
i1e
Ti Wai0ei + 2e
αt1113944
N
i11113944
N
j1aije
Ti Wei
+ eαt
1113944
N
i11113954x
Ti αP1113954xi minus e
αt1113944
N
i11113954x
Ti Wai01113954xi
le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x minus e
αt1113954x
TLσ(t) otimesW1113872 11138731113954x
+ eαt
1113944
N
i1e
Ti ai0σ(t)
W + 2diiσ(t)W1113874 1113875ei
+ eαt
1113944
N
i11113954x
Ti αP minus ai0σ(t)
W1113874 11138751113954xi
(52)
(i) If the graph Gp is not connected during t isin [ts ts+1)according to the event-triggering condition (50) andequation (6) one has
_V3 le eαt
1113954xT
IN otimes PA + ATP1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
le 0
(53)
It can be seen from the abovementioned formula that V3is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(54)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
(ii) If the graph Gp is connected during t isin [ts ts+1)then
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
+ eαt
1113944
N
i1αλmax(P) minus ai0σ(t)
λmin(W)1113874 1113875 1113954xi
2
+ eαt
1113944
N
i1ai0σ(t)
λmax(W) + 2diiσ(t)λmax(W)1113874 1113875 ei
2
(55)
According to event-triggering condition (50) andequation (5)
_V3 le eαt
1113954xT
IN otimes PA + ATP minus λ2 Lσ(t)1113872 1113873W1113872 11138731113872 11138731113954x
le minus eα(t)
λ2 Lσ(t)1113872 1113873
21113954x
T1113954x
(56)
It can be seen from (56) that V3 is not increasing hence
V3(t)geV3 ts+1( 1113857 eαts+1 1113944
N
i11113954xi ts+1( 1113857
TP1113954xi ts+1( 1113857
ge eαts+1λmin(P) 1113954x ts+1( 1113857
11138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138682
(57)
ie 1113954x(ts+1)le(V3(t)λmin(P))
1113968eminus (α2)ts+1 le
(V3(0)λmin(P))1113968
eminus (α2)ts+1
1 2
3 4 0
Figure 1 Communication topology G
1 12 2
3 34 40 0
Figure 2 Communication topology G1 and G2
0 5 10 15 20t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 3e 1st state error trajectory of each agent under protocol(10)
10 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
In summary ||1113954x(ts+n)||le(V3(ts+(nminus 1))λmin(P))
1113969
eminus (α2)ts+n le middot middot middot le(V3(0)λmin(P))
1113968eminus (α2)ts+n ie 1113954x(t)le
(V3(t)λmin(P))
1113968eminus (α2)t le middot middot middot le
(V3(0)λmin(P))
1113968eminus (α2)t
so limt⟶infin1113954x(t) 0 is equivalent to limt⟶infin1113954xi(t) 0and accordingly limt⟶infinxi(t) minus x0(t) 0 i 1 2 N
is established
Remark 3 Index (α2) can be approximated as the con-vergence rate of multiagent system (49) and the conver-gence rate can be changed by adjusting α
Theorem 6 Under the conditions of eorem 5 system (49)does not have Zeno behavior e interval between any two
consecutive event-triggering instants of the system is not lessthan
INotimesA1113868111386811138681113868
11138681113868111386811138681113868111386811138681113868
1113868111386811138681113868 +||(L + D)otimesBK||1113872 1113873
3
times 1 +κ ai0σ(t)
λmin(W) minus αλmax(P)1113874 1113875
ai0σ(t)λmax(W) + 2diiσ(t)
λmax(W)
⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠
12
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
3
minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(58)
Proof e proof is similar to that of eorem 2
ndash10
ndash5
0
5
10
x i2(t)
0 5 10 15 20t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 4e 2st state error trajectory of each agent under protocol(10)
0 02 04 06 08 1t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 5 Event times instants for four agents in eorem 1
0 2 4 6 8 10t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 6e 1st state error trajectory of each agent under protocol(35)
x i2(t)
ndash10
ndash5
0
5
10
0 2 4 6 8 10t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 7 e 2nd state error trajectory of each agent underprotocol (35)
Complexity 11
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
5 Simulation
In this part we consider the trajectories of the state errorsbetween the follower and leader under the fixed topologyand the switching topology respectively where the dynamicequations of the leader and the follower are given by (2) and(3) respectively and the communication network topologyamong agents is shown in Figures 1 and 2 Assume thatxi [xi1 xi2]
T and A and B are chosen as follows
A 0 05
minus 48 01113890 1113891 B 0
minus 051113890 1113891 it is easy to prove that the
Assumption 2 is satisfied By solving Riccati equation byMATLAB we know that feedback gain matrix K BTP
[minus 04995 minus 11343]T Let the leaderrsquos initial state be x0(0)
[2 3]T and the followerrsquos initial state be x1(0) [minus 1 1]T
x2(0) [minus 2 minus 3]T x3(0) [5 minus 6]T x4(0) [4 2]T
Example 1 Under the centralized event-triggering pro-tocol (10) the leader-following consensus of the multi-agent system composed of (2) and (3) is considered ecommunication network among agents is shown in Fig-ure 1 and the corresponding weights are all 1 It can beseen from Figures 3 and 4 that followers can successfullyfollow the leader Figure 5 shows the event instants of eachfollower with the centralized event-triggering protocol(10) It can be seen that protocol (10) can effectively reducethe number of communications among agents thus re-ducing the waste of resources Also there is no Zenobehavior
Example 2 In this example we illustrate the leader-fol-lowing consensus of the multiagent system under the dis-tributed event-triggering protocol (35) e communicationnetwork among agents is shown in Figure 1 It can be seenfrom Figures 6 and 7 that followers can successfully followthe leader Figure 8 shows the event triggering time of eachfollower under the decentralized event triggering protocol
(35) and Zeno behavior is excluded e simulation resultsverify eorems 3 and 4
Example 3 Finally the leader-following consensus of themultiagent system under the control protocol (48) isconsidered e communication network among agentswill randomly switch between G1 and G2 as shown inFigure 2 where G1 is a connected graph and G2 is anunconnected graph e state errors between the followeragent i and leader 0 are shown in Figures 9 and 10 re-spectively It indicates that all followers can successfullyfollow the leader Figure 11 shows the event-triggeringinstants of each follower under (48) and there is no Zenobehavior
x1x2
x3x4
0 02 04 06 08 10
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
t (s)
Figure 8 Event times instants for four agents in eorem 3
0 20 40 60 80t (s)
ndash4
ndash2
0
2
4
x i1(t)
x11(t) ndash x01(t)x21(t) ndash x01(t)
x31(t) ndash x01(t)x41(t) ndash x01(t)
Figure 9e 1st state error trajectory of each agent under protocol(48)
x i2(t)
ndash15
ndash10
ndash5
0
5
10
15
0 20 40 60 80t (s)
x12(t) ndash x02(t)x22(t) ndash x02(t)
x32(t) ndash x02(t)x42(t) ndash x02(t)
Figure 10 e 2nd state error trajectory of each agent underprotocol (48)
12 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
6 Conclusions and Future Work
In this paper the leader-following control of general linearmultiagent systems based on event-triggering mechanismunder both fixed topology and switching topologies havebeen studied Under the fixed topology two different controlprotocols are designed in order to reduce waste of resourcesBased on these two control protocols we propose twodifferent triggering functions ie centralized event-trig-gering function and decentralized event-triggering functionwith state error between the follower and leader When thetriggering function exceeds 0 the agent will update thecontrol input at the triggering instants rough theoreticalanalysis the sufficient conditions are derived for the systemto achieve leader-following consensus under two controlprotocols and event-triggering conditions e conditionsobtained under fixed topology are extended to switchingtopologies (different from the fixed topology the controllerupdate at the triggering time and also the switching time)e results show that the conditions to achieve leader-fol-lowing are also valid under switching topologies Finally it isproved that the system can realize leader-following controlwithout Zeno behavior e simulation results verify theeffectiveness of the theoretical analysis In the future we willfurther study the leader-following control of the linearmultiagent system with interference delay and otherfactors
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China (Grantno 61873136 6190321061374062 and 61603288) Science Foundation of ShandongProvince for Distinguished Young Scholars (GrantnoJQ201419) and Shandong Provincial Natural ScienceFoundation China (Grantno ZR201709260010)
References
[1] D Meng ldquoDynamic distributed control for networks withcooperative-antagonistic interactionsrdquo IEEE Transactions onAutomatic Control vol 63 no 8 pp 2311ndash2326 2018
[2] D Meng ldquoBipartite containment tracking of signed net-worksrdquo Automatica vol 79 pp 282ndash289 2017
[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 42202 2019
[4] Y Chao and Z Ji ldquoNecessary and sufficient conditions formulti-agent controllability of path and star topologies byexploring the information of second-order neighboursrdquo IMAJournal of Mathematical Control and Information 2016
[5] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018
[6] Z Ji H Lin and H Yu ldquoProtocols design and uncontrollabletopologies construction for multi-agent networksrdquo IEEETransactions on Automatic Control vol 60 no 3 pp 781ndash7862015
[7] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017
[8] N Cai M He Q Wu and M J Khan ldquoOn almost con-trollability of dynamical complex networks with noisesrdquoJournal of Systems Science and Complexity vol 32 no 4pp 1125ndash1139 2019
[9] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofmulti-agent systems under directed topologyrdquo InternationalJournal of Robust and Nonlinear Control vol 27 no 18pp 4333ndash4347 2017
[10] Y Guan Z Ji L Zhang and L Wang ldquoControllability ofheterogeneous multi-agent systems under directed andweighted topologyrdquo International Journal of Control vol 89no 5 pp 1009ndash1024 2016
[11] Z Lu L Zhang Z Ji and L Wang ldquoControllability of dis-crete-time multiagent systems with directed topology andinput delayrdquo International Journal of Control vol 89 no 1pp 179ndash192 2016
[12] X Liu Z Ji and T Hou ldquoStabilization of heterogeneousmulti-agent systems via harmonic controlrdquo Complexityvol 2018 Article ID 8265637 9 pages 2018
[13] K Liu Z Ji andW Ren ldquoNecessary and sufficient conditionsfor consensus of second-order multi-agent systems underdirected topologies without global gain dependencyrdquo IEEETransactions on Cybernetics vol 47 no 8 pp 2089ndash20982017
[14] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018
0 05 1 15t (s)
0
1
2
3
4
Even
t-trig
gerin
g tim
e ins
tant
s
x1x2
x3x4
Figure 11 Event times instants for four agents in eorem 5
Complexity 13
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
[15] J Xi C Wang X Yang and B Yang ldquoLimited-budget outputconsensus for descriptor multiagent systems with energyconstraintsrdquo IEEE Transactions on Cybernetics pp 2168ndash2275 2020 httparxivorgabs190908345
[16] Q Qi H Zhang and Z Wu ldquoStabilization control for linearcontinuous-time mean-field systemsrdquo IEEE Transactions onAutomatic Control vol 64 no 9 pp 3461ndash3468 2019
[17] L Tian Z Ji T Hou and K Liu ldquoBipartite consensus oncoopetition networks with time-varying delaysrdquo IEEE Accessvol 6 no 1 pp 10169ndash10178 2018
[18] J Xi M He H Liu and J Zheng ldquoAdmissible outputconsensualization control for singular multi-agent systemswith time delaysrdquo Journal of the Franklin Institute vol 353no 16 pp 4074ndash4090 2016
[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004
[20] L Wang J Xi M He and G Liu ldquoRobust time-varyingformation design for multi-agent systems with disturbancesextended-state-observer methodrdquo International Journal ofRobust and Nonlinear Control httparxivorgabs190908974 2019
[21] K Liu and Z Ji ldquoConsensus of multi-agent systems with timedelay based on periodic sample and event hybrid controlrdquoNeurocomputing vol 270 pp 11ndash17 2017
[22] H Liu G Xie and L Wang ldquoNecessary and sufficientconditions for solving consensus problems of double-inte-grator dynamics via sampled controlrdquo International Journal ofRobust and Nonlinear Control vol 20 no 15 pp 1706ndash17222010
[23] Y Gao B Liu J Yu J Ma and T Jiang ldquoConsensus of first-order multi-agent systems with intermittent interactionrdquoNeurocomputing vol 129 pp 273ndash278 2014
[24] P Tabuada ldquoEvent-triggered real-time scheduling of stabi-lizing control tasksrdquo IEEE Transactions on Automatic Controlvol 52 no 9 pp 1680ndash1685 2007
[25] D V Dimarogonas and E Frazzoli ldquoDistributed event-triggered control strategies for multi-agent systemsrdquo inProceeings of the 2009 47th Annual Allerton Conference onCommunication Control and Computing IEEE MonticelloIL USA October 2009
[26] D V Dimarogonas E Frazzoli and K H JohanssonldquoDistributed event-triggered control for multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 57 no 5pp 1291ndash1297 2012
[27] H Yan Y Shen H Zhang and H Shi ldquoDecentralized event-triggered consensus control for second-order multi-agentsystemsrdquo Neurocomputing vol 133 no 8 pp 18ndash24 2014
[28] Y Fan G Feng Y Wang and C Song ldquoDistributed event-triggered control of multi-agent systems with combinationalmeasurementsrdquo Automatica vol 49 no 2 pp 671ndash675 2013
[29] J Hu G Chen and H Li ldquoDistributed event-triggeredtracking control of second-order leader-follower multi-agentsystemsrdquo in Proceeedings of the 30th Chinese ControlConference Yantai China July 2011
[30] H Li X Liao T Huang and W Zhu ldquoEvent-Triggeringsampling based leader-following consensus in second-ordermulti-agent systemsrdquo IEEE Transactions on AutomaticControl vol 60 no 7 pp 1998ndash2003 2015
[31] D Xie S Xu Y Zou and Z Li ldquoEvent-triggered consensuscontrol for second-order multi-agent systemsrdquo IET Controleory amp Applications vol 9 no 5 pp 667ndash680 2015
[32] D Xie S Xu Y Chu and Y Zou ldquoEvent-triggered averageconsensus for multi-agent systems with nonlinear dynamicsand switching topologyrdquo Journal of the Franklin Institutevol 352 no 3 pp 1080ndash1098 2015
[33] G S Seyboth D V Dimarogonas and K H JohanssonldquoEvent-based broadcasting for multi-agent average consen-susrdquo Automatica vol 49 no 1 pp 245ndash252 2013
[34] X Meng and T Chen ldquoEvent based agreement protocols formulti-agent networksrdquo Automatica vol 49 no 7 pp 2125ndash2132 2013
[35] F Xiao X Meng and T Chen ldquoAverage sampled-dataconsensus driven by edge eventsrdquo in Proceedings of theChinese Control Conference (CCC) pp 6239ndash6244 HefeiChina July 2012
[36] Z Zhang and L Wang ldquoDistributed integral-type event-triggered synchronization of multi-agent systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 28 no 14pp 4175ndash4187 2018
[37] Z Zhang F Hao L Zhang and L Wang ldquoConsensus oflinear multi-agent systems via event-triggered controlrdquo In-ternational Journal of Control vol 87 no 6 pp 1243ndash12512014
[38] W Hu L Liu and G Feng ldquoLeader-following consensus oflinear multi-agent systems by distributed event-triggeredcontrolrdquo in Proceedings of the 34th Chinese ControlConference Hangzhou China July 2015
[39] W Zhu Z-P Jiang and G Feng ldquoEvent-based consensus ofmulti-agent Systems with general linear modelsrdquo Automaticavol 50 no 2 pp 552ndash558 2014
[40] C Nowzari and J Cortes ldquoDistributed event-triggered co-ordination for average consensus on weight-balanced di-graphsrdquo Automatica vol 68 no 4 pp 237ndash244 2016
[41] Z Li and Z Duan Hinfin Cooperative Control of Multi-AgentSystems A Consensus Region Approach CRC Press Bocaraton FL USA 2014
14 Complexity
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