Distributed node‐to‐node consensus of multi‐agent systems with stochastic sampling

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2016; 26:110–124Published online 28 January 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3302

Distributed node-to-node consensus of multi-agent systems withstochastic sampling

Ying Wan1, Guanghui Wen1, Jinde Cao1,2,*,† and Wenwu Yu1,3

1Department of Mathematics, Southeast University, Nanjing 210096, China2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

3Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

SUMMARY

This paper is concerned with the mean square node-to-node consensus tracking problem for multi-agentsystems with nonidentical nonlinear dynamics and directed topologies. The randomly occurred uncertaintiesin the sampling devices may result in stochastically varied sampling periods, which lead to the investigationof node-to-node consensus problem under stochastic sampling. By employing the input-delay method anddiscontinuous Lyapunov functional approach, it arrives at some sufficient conditions under which the stateof each follower can track that of the corresponding leader asymptotically in the mean square sense. Finally,some numerical simulations are provided to verify the effectiveness of the theoretical results. Copyright ©2015 John Wiley & Sons, Ltd.

Received 2 October 2014; Revised 3 December 2014; Accepted 24 December 2014

KEY WORDS: multi-agent system; distributed control; node-to-node consensus; stochastic sampling

1. INTRODUCTION

Distributed coordination and cooperative control has been a hot topic in recent years, mainly becauseof its extensive applications including the formation of UAV, attitude synchronization of spacecraft,and distributed sensor networks [1–5], only to name a few. The requirement of cooperation andcoordination stems from some natural and social phenomena that one single agent may be incom-petent for a specific challenging task. Among the diverse cooperative control objectives, reaching acommon state while utilizing only the local relative information, which is also known as distributedconsensus, has become one of the prerequisite and significant issues [6]. Considerable efforts havebeen devoted to the investigation of consensus problems and numerous fruitful results concerningthe different consensus protocols have been established up to date.

Consensus problems can be roughly classified into two categories: leader-following consensusand leaderless consensus. As for leaderless consensus, the ultimate agreement state of all the agentsis usually unknown in advance, whereas in the case of leader-following consensus, the states ofagents can track a prescribed trajectory, which is provided by a real or virtual leader. From thispoint, the leader-following consensus deserves more studies for future practical utilities [7]. Par-ticulary, recent years have witnessed many excellent works concerning the consensus problem ofsecond-order multi-agent systems [8–13]. In addition, Li et al. [14] and Yu et al. [15] studied theleader-following consensus problems of multi-agent systems with general linear node and higher-order node dynamics over fixed topology, respectively. Qin et al. [16] studied the leader-followingconsensus problem for multiple agents with general linear system dynamics under switching topolo-

*Correspondence to: Jinde Cao, Department of Mathematics, Southeast University, Nanjing 210096, Jiangsu, China.†E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

CONSENSUS OF LINEAR MULTI-AGENT SYSTEMS 111

gies. If there exist more than one leader in the network and the dynamics of each leader isindependent of each other, then the states of the followers will approach to the convex hull spannedby these leaders. This issue is often referred as containment control problem, which is addressedin [17–20], and so on. However, noting that the interactions between the leaders widely exist andeach follower needs only to track the state of the corresponding leader in many real practice, insteadof the convex combination of the states of all the leaders, thus, it is of great interest to study thisnew kind of consensus, namely, node-to-node consensus [21, 22]. Node-to-node consensus withlinear and Lur’s node dynamics is investigated in [21] and [23], respectively, both with continuouscommunication exchanges between the agents. Constant sampling communication between the twolayers is further considered in [22] for the distributed node-to-node consensus. The node-to-nodeconsensus will play a crucial role in the state estimation and reconstruction of multilayer networks.

With the fast development of computer and communication technology, the increasing demandsfor more advanced information processing in real applications have made the theory of digital signalprocessing more complete and powerful. To deal with a continuous signal with the implementa-tion of digital sensors, filters, or controllers, the first task is to transform the continuous signalsinto discrete signals by sampling, thus, the communication and sensing interactions among theagents are occurred periodically rather than continuously. The sampled-data coordination algorithmsfor double-integrator dynamics are studied in [24, 25]. Appropriate sampling period is critical forachieving specific performance. According to the different sampling rates and sampling intervals,the sampling control can be divided into three classes: the single-rate digital control, the multi-rate digital control, and the digital control with time-varying sampling periods [26]. Single-ratedigital control refers to sampling all signals with one constant rate, whereas different signals aresampled at diverse but constant rate, one speaks of multi-rate digital control. The time-varyingsampling, in which sampling rate varies with time, has been attracting researchers’ attentionsfor its practical applicability. For the purpose of utilizing the network bandwidth efficientlyand avoiding network congestions simultaneously, the stochastic sampled-data control scheme isadopted [27, 28]. When the randomly occurring uncertainties are taken into account, robust syn-chronization for chaotic systems using stochastic sampled-data control is studied by Lee et al. [29].Furthermore, in real networked control systems, the sampling period is actually stochastically vary-ing because of the randomly occurring disturbances of sampling devices thus making the samplingperiod vary stochastically.

So motivated by the aforementioned investigations, this paper studies the mean square node-to-node consensus for multi-agent system with stochastic sampling. The objective of this paper isto present some sufficient conditions under which the states of the followers can track that of thecorresponding leaders asymptotically in the mean square sense. Specifically, we consider a systemof two layers: the leaders’ layer and the followers’ layer. For simplicity, it is assumed that thesetwo layers have the same topology. The leader’s dynamics are only affected by its neighbors inthe leaders’ layer, whereas the follower’s dynamics are not only affected by its neighbors in thefollowers’ layer but also affected by the leaders. The information among the same layer and betweenthe different layers are exchanged only at stochastic sampling instants. From a practical point ofview, it is often impossible to let all the followers directly sense the states of their correspondingleaders, so only a small fraction of followers are allowed to receive the information of the leadersdirectly. By employing the delayed-input approach [30], the original sampled-data system can beconverted into the delayed continuous system equivalently. By introducing stochastic variables withrespect to possible sampling intervals, the retarded continuous system can be expressed by thesestochastic variables. The discontinuous Lyapunov–Krasovskii functionals are properly constructedfor our main results. With free-weight matrix method, reciprocally combination method [31] anddelay-decomposition method, some criteria in form of LMIs are provided subsequently.

The rest of the paper is organized as follows. Some preliminaries and the formulation of the node-to-node consensus problem with stochastic sampling are presented in Section 2. In Section 3, themain theoretical results and analysis are discussed. In Section 4, some simulation examples are givento verify the validity of the obtained theoretical results. Conclusions are finally drawn in Section 5.

Throughout the paper, let R and N be the sets of real and natural numbers, respectively. Rn is then-dimension real vector space and Rn�n is the set of real matrices. In indicates the n � n identity

Copyright © 2015 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2016; 26:110–124DOI: 10.1002/rnc

112 Y. WAN ET AL.

matrix. The superscript T represents the transpose for real matrices. Notation diag¹ı1; ı2; � � � ; ınºis a diagonal matrix with diagonal elements ı1; ı2; � � � ; ın. Symbol ˝ denotes the Kroneckerproduct. k � k represents the Euclidean vector norm. P r¹Xº is the occurrence probability of eventX . E¹˛º is the mathematical expectation of stochastic variable ˛, and E¹˛jˇº refers to conditionalexpectation of stochastic variable ˛ in terms of stochastic variable ˇ.

2. PRELIMINARIES AND FORMULATION OF THE MODEL

In this section, some preliminaries and model formulation of mean square node-to-node consensusare provided.

2.1. Preliminaries

The underlying communication topology of the considered multi-agent system is represented by adirected graph. Let G.V; E ;A/ be a directed graph with the set of nodes V D ¹ �1; �2; � � � ; �N º, theset of directed edges E � V � V , and a weighted adjacency matrix A D Œaij �N�N with elementsaij > 0; i; j D 1; 2; � � � ; N . An edge eij in G is denoted by the ordered pair of nodes .�j ; �i /, where�j and �i are called the parent and child nodes, respectively, and eij 2 E if and only if aij > 0. Adirected path from node �i to �j is a sequence of edges, .�i ; �k1/, .�k1 , �k2/; � � � ; .�kl ; �j /, withdistinct nodes �km , m D 1; 2; � � � ; l . Furthermore, self-loops are not allowed, which indicate thateach agent cannot obtain its absolute state information, that is, ai i D 0, for all i D 1; 2; � � � ; N .The Laplacian matrix L D

�lij�N�N

is defined by lij D �aij ; i ¤ j and li i DPNkD1 aik for

i D 1; 2; � � � ; N .

2.2. Formulation of the model

Suppose the multi-agent system under consideration consists of two layers: the leaders’ layer andthe followers’ layer. Each layer contains N coupled agents, and the two layers have the identicaltopology structure. It is further assumed that the relative states between each pair of neighboringnodes are measured only at discrete stochastic sampling times tk; k 2 N. Thus, the dynamics of thei th agent in the leaders’ layer are given by

Pxi .t/ D fi .xi .t/; t/C c

NXjD1

aij Œxj .tk/ � xi .tk/�; t 2 Œtk; tkC1/: (1)

According to the definition of Laplacian matrix, system (1) can be equivalently written as

Pxi .t/ D fi .xi .t/; t/ � c

NXjD1

lijxj .tk/; t 2 Œtk; tkC1/; (2)

for i D 1; 2; � � � ; N , where xi .t/ D .xi1.t/; � � � ; xin.t//T 2 Rn represents the state variable of thei th leader, c > 0 is the coupling strength, L D

�lij�N�N

is the Laplacian matrix of the commu-nication topology G, and fi .xi .t/; t/ D .fi1.xi .t/; t/; � � � ; fin.xi .t/; t//

T describes the nonlineardynamics of the i th agent.

The dynamics of the i th agent in the followers’ layer are given as

PQxi .t/ D fi . Qxi .t/; t/ � c

NXjD1

lij Qxj .tk/C cpi .xi .tk/ � Qxi .tk//; t 2 Œtk; tkC1/; (3)

for i D 1; 2; � � � ; N , where Qxi .t/ D . Qxi1.t/; � � � ; Qxin.t//T 2 Rn is the state variable of the i th

follower, and the dynamic nonlinear function of the i th follower is assumed to be the same as thei th leader. pi 2 ¹0º

SRC and pi ¤ 0 if and only if the i th follower can directly sense the state of

the i th leader, that is, there exists a directed link from the i th leader to the i th follower.



Assumption 1Suppose the nonlinear intrinsic dynamic function of each agent fi W RN � R 7! RN in (1) and (3)satisfies the following conditions: there exists positive definite matrix �i D diag¹�i1; � � � ; �inº >0, i D 1; 2; � � � ; N such that

0 6 fij .b; t/ � fij .a; t/b � a

6 �ij ; fij .0; t/ D 0

for all t > t0 and a; b 2 R; a ¤ b, i D 1; 2; � � � ; N , j D 1; 2; � � � ; n.

Assumption 2For each follower i , i D 1; 2; � � � ; N , there exists at least one leader that has a directed path fromitself to that follower.

Remark 1A detailed algorithm has been formulated in [21] to underline the issue of the kinds of followersand at least how many followers should be pinned to directly sense their corresponding leaders suchthat Assumption 2 holds. In this paper, only a small fraction of followers are assumed to directlysense the states of the corresponding leaders. Additionally, noting that if there exists no informationexchanges between the two layers, the node-to-node consensus of multi-agent systems (1) and (3)cannot be achieved.

The sampled-data systems (2) and (3) can be equivalently transformed into the following retardedfunctional differential equations by using the delay-input approach. Letting �.t/ D t � tk , t 2Œtk; tkC1/, one obtains P�.t/ D 1 and

Pxi .t/ D fi .xi .t/; t/ � c

NXjD1

lijxj .t � �.t//; (4)

PQxi .t/ D fi . Qxi .t/; t/ � c

NXjD1

lij Qxj .t � �.t//C cpi .xi .t � �.t// � Qxi .t � �.t///: (5)

Letting ei .t/ D xi .t/ � Qxi .t/ gives

Pei .t/ D fi .xi .t/; t/ � fi . Qxi .t/; t/ � c

NXjD1

lij ej .t � �.t// � cpiei .t � �.t//: (6)

The following error dynamical system can be thus obtained as

Pe.t/ D F.x.t/; Qx.t/; t/ � c. OL˝ In/e.t � �.t//; (7)

where e.t/ D .eT1 .t/; � � � ; eTN .t//

T ,F.x.t/; Qx.t/; t/ D .f T1 .x1.t/; t/ � fT1 . Qx1.t/; t/; � � � ;

f TN .xN .t/; t/ � fTN . QxN .t/; t//

T , and OL D LC diag¹p1; p2; � � � ; pN º.For notational simplicity, denote A D �c. OL˝ In/. Then, one has

Pe.t/ D F.e.t/; t/C Ae.t � �.t//; t 2 Œtk; tkC1/: (8)

Because the sampling instants are stochastically selected, a stochastic variable d.t/ D tkC1 � tkis introduced here to indicate the stochastically varying sampling interval. Suppose that there existm discrete sampling intervals, 0 D d0 < d1 < � � � < dm, and d.t/ takes the value of di withprobability ıi , that is,

P r ¹d.t/ D diº D ıi ; i D 1; 2; � � � ; m; (9)


114 Y. WAN ET AL.

where ıi are known constants satisfyingmPiD1

ıi D 1. Then, one can derive that E¹d.t/º DPmiD1 ıi .t/di .Before moving on, two stochastic variable �i .t/ and ıi .t/ are defined as

�i .t/ D

²1; di�1 6 �.t/ < di ;0; otherwise;

ıi .t/ D

²1; d.t/ D di ;

0; otherwise;

(10)

for i D 1; 2; � � � ; m. Then, the probabilities of �i .t/ and ıi .t/ are calculated as

P r¹ıi .t/ D 1º D P r¹d.t/ D diº D ıi ;

P r¹�i .t/ D 1º D P r¹di�1 6 �.t/ < diº DmXjDi

ıjdi � di�1

djD �i :

Remark 2Note that the stochastic variables �i .t/ and ıi .t/ both satisfy the Bernoulli distribution, so itfollows that

E¹�i .t/º D �i ; E¹.�i .t/ � �i /2º D �i .1 � �i /;

E¹ıi .t/º D ıi ; E¹.ıi .t/ � ıi /2º D ıi .1 � ıi /:

Furthermore, �i .t/.ıi .t// and �j .t/.ıj .t// are independent when i ¤ j , that is,

P r¹¹ıi .t/ D 1º \ ¹ıj .t/ D 1ºº D 0;

P r¹¹�i .t/ D 1º \ ¹�j .t/ D 1ºº D 0:

Based on the analysis of the stochastic variables �i .t/ and ıi .t/, the error dynamic system (8)with stochastic sampling can be formulated as

Pe.t/ D F.e.t/; t/C A

mXiD1

�i .t/e.t � �i .t//; t 2 Œtk; tkC1/; (11)

where di�1 6 �i .t/ < di , i D 1; 2; � � � ; m.

Definition 1The distributed mean square node-to-node consensus of multi-agent system with leaders’ dynamicsgiven in (1) and followers’ dynamics given in (3) is said to be achieved if, for any initial conditions,

limt!1

E¹kxi .t/ � Qxi .t/kº D 0; 8i D 1; 2; � � � ; N;

or equivalently, the error system (11) is globally mean square asymptotically stable, that is,

limt!1

E¹ke.t/kº D 0; 8i D 1; 2; � � � ; N:

Lemma 1 ([32])For any matrix P > 0, scalars a, and b with b > a, a vector function u.t/ W Œa; b� 7! Rn such thatthe integrations concerned are well-defined, then Z b

a

u.s/ds

!TP

Z b

a

u.s/ds

!6 .b � a/

Z b

a

uT .s/Pu.s/ds:



Lemma 2 ([33])Let u.t/ 2 W Œa; b/ and u.a/ D 0. Then for any matrix M > 0, the following inequality holds

Z b

a

uT .s/Mu.s/ds 6 4.b � a/2

�2

Z b

a

PuT .s/M Pu.s/ds:

Lemma 3 ([31])Let ˛1; ˛2; � � � ; ˛N W Rm 7! R have positive values in an open subset D of Rm. Then, thereciprocally convex combination of ˛i over D satisfies

min´�i j�i>0;

Pi

�iD1

μXi

1

�i˛i .t/ D

Xi

˛i .t/C maxˇij .t/

Xi¤j

ˇij .t/

subject to

²ˇij W Rm 7! R; ˇij .t/ D ˇj i .t/;

�˛i .t/ ˇij .t/

ˇij .t/ ˛j .t/

�> 0

³:

Remark 3According to Definition 1, the distributed mean square node-to-node consensus proposed in thispaper requires the states of leaders and those of the corresponding followers to achieve consensusin the sense of mean square, whereas the states of the agents in the same layer may fail to reachthe same value. This definition is of important significance in the distributed monitoring and signalreconstruction in networks.

Remark 4The objective of distributed node-to-node consensus is to reconstruct the states of N leaders, soit is not necessary to employ redundant (more than N ) followers to achieve this goal if it can berealized by involvingN followers. Redundant followers may result in unnecessary facility and com-munication cost in real applications. Additionally, some useful algorithms in graph theory such aswidth-first or depth-first algorithm can be used to identify the underlying topology of the leaders’layer. In order to reconstruct the states of leader in real applications, the topology of the followers’layer can be set to be identical as that of the leaders’ layer. If the topologies of these two layers aredifferent, the neighbors of agent i may not be the corresponding leaders of the neighbors of fol-lower i , thus, leader i and follower i would then be influenced by different nonlinear dynamics fromtheir neighbors, this would make the accurate node-to-node consensus become rather difficult andeven impossible.

3. MAIN RESULTS

In this section, the distributed node-to-node consensus problem under stochastic sample-data controlis investigated.

Theorem 1Under Assumptions 1 and 2, for given positive constants �i ; ıi ; hi ; di and constant matrices A,�, distributed mean square node-to-node consensus can be achieved if there exist positive definitematrices Mi , Qi , iD1; 2; � � � ; m, positive definite diagonal matrix W , and constant matrix N1, N2,Si ; i D 1; 2; � � � ; m with appropriate dimensions such that the following LMIs hold:�

Mi Si� Mi

�> 0; i D 1; 2; � � � ; m; (12)


116 Y. WAN ET AL.

D

0BB@11 12 13 �

TW CN1� 22 23 0

� � 33 0

0 0 0 �W

1CCA < 0; (13)

where

11 D ��1M1 �

mXiD1

OQi ;

12 W

8̂<:̂.1; 1/ D �1M1 � �1S1 C OQ1 C �1N1A;

.1; 2/ D �1S1;

.1; i/ D OQ iC12C � iC1

2N1A; i D 3; 5; � � � ; 2m � 1;

13 D P �N1;

22 W

8̂̂ˆ̂̂̂̂̂ˆ̂<ˆ̂̂̂̂ˆ̂̂̂̂:

.i; i/ D � iC12.�2M iC1

2C S iC1

2C STiC1

2

/ � OQ iC12; i D 1; 3; � � � ; 2m � 1;

.i; i/ D �� i2M i

2� � i

2C1M i

2C1; i D 2; 4; � � � ; 2m � 2;

.2m; 2m/ D ��mMm;

.i; i C 1/ D � iC12.M iC1

2� S iC1

2/; i D 1; 3; � � � ; 2m � 1;

.i; i C 1/ D � i2C1

.M i2C1� S i

2C1/; i D 2; 4; � � � ; 2m � 2;

.i; i C 2/ D � i2C1

S i2; i D 2; 4; � � � ; 2m � 2;

23.i; 1/ D � iC12ATN T

2 ; i D 1; 3; � � � ; 2m � 1;

33 D

mXiD1

.�ih2iMi C ıid

2i Qi / �N2 �N

T2 ;

OQi D

mXjDi

ıjhi�

2

4djQj ; hi D di � di�1; i D 1; 2; � � � ; m;

� D diag¹�1; �2; � � � ; �N º 2 RnN�nN :

ProofConsider the following discontinuous Lyapunov–Krasovskii functional of the error dynamicalsystem (11)

V.t/ , V1.t/C V2.t/C V3.t/; t 2 Œtk; tkC1/; (14)

where

V1.t/ D eT .t/Pe.t/;

V2.t/ D

mXiD1

�i .t/hi

Z �di�1�di

Z t

tC�

PeT .s/Mi Pe.s/dsd;

V3.t/ D

mXiD1

ıi .t/

d2i

Z t

tk

PeT .s/Qi Pe.s/ds ��2

4

Z t

tk

.e.s/ � e.tk//TQi .e.s/ � e.tk//ds

!

,mXiD1

V3i .t/;

P D P T > 0, Mi D M Ti > 0, and Qi D QT

i > 0, i D 1; 2; � � � ; m. According to Lemma 2, onecan verify that V3.t/ > 0. Additionally, V3i .t/ vanishes at the sampling instant tk . Thus, one haslimt!t�

k

V.t/ > V.tk/. �



The infinitesimal operator L of V.et / is defined as

LV.et / D limh!0C

1

h¹E¹V.etCh/jetº � V.et /º: (15)

According to (14) and (15), one obtains

E¹LV1.t/º D E¹2eT .t/P Pe.t/º;

E¹LV2.t/º D E

´mXiD1

�i .t/h

2i PeT .t/Mi Pe.t/ � �i .t/hi

Z t�di�1

t�di

PeT .s/Mi Pe.s/ds

!μ

D E

´mXiD1

"�i .t/h

2i PeT .t/Mi Pe.t/ � �i .t/hi

Z t�di�1

t��i .t/

PeT .s/Mi Pe.s/ds

C

Z t��i .t/

t�di

PeT .s/Mi Pe.s/ds

!#μ:

Invoking Lemmas 1 and 3, one has that the following two inequalities hold for the integral term ofLV2.t/,

hi

Z t�di�1

t��i .t/

PeT .s/Mi Pe.s/ds >hi

�i � di�1

Z t�di�1

t��i .t/

Pe.s/ds

!TMi

Z t�di�1

t��i .t/

Pe.s/ds

!

D1

1 � �i

Z t�di�1

t��i .t/

Pe.s/ds

!TMi

Z t��i .t/

t�di

Pe.s/ds

!;

hi

Z t��i .t/

t�di

PeT .s/Mi Pe.s/ds >1

�i

Z t��i .t/

t�di

Pe.s/ds

!TMi

Z t��i .t/

t�di

Pe.s/ds

!;

(16)

where �i Ddi � �i .t/

di � di�1. Subsequently, it yields that

hi

Z t�di�1

t��i .t/

PeT .s/Mi Pe.s/ds C

Z t��i .t/

t�di

PeT .s/Mi Pe.s/ds

!

>�´1i .t/

´1i .t/

�T 0B@1

1 � �iMi 0

01

�iMi

1CA� ´1i .t/ ´2i .t/ �

>�´1i .t/

´2i .t/

�T �Mi Si� Mi

� �´1i .t/ ´2i .t/

�;

(17)

where ´1i .t/ DR t�di�1t��i .t/

Pe.s/ds, ´2i .t/ DR t��i .t/t�di

PeT .s/ds and

�Mi Si� Mi

�> 0; i D 1; 2; � � � ; m:

For the purpose of utilizing the information of delay �.t/ more sufficiently, the stochastic vari-able �ij .t/ is defined as follows: �ij D 1 when ıi .t/�j .t/ D 1, otherwise, �ij D 0, j 6 i D1; 2; � � � ; m. Thus, one has


118 Y. WAN ET AL.

P r¹�ij .t/ D 1º D ıiP r¹�i .t/ D 1º D ıidj � dj�1

di, �ij ; (18)

wheremPiD1

iPjD1

�ij D 1.

Then, it follows from (18) that V3i .t/ can be transformed to

V3i .t/ D ıi .t/d2i

Z t

t��.t/

PeT .s/Qi Pe.s/ds

�

iXjD1

�ij .t/

�2

4

Z t

t��j .t/

.e.s/ � e.t � �j .t///TQi .e.s/ � e.�j .t///ds

!:

(19)

Furthermore, the expectation of LV3i .t/ can be obtained as

E¹LV3i .t/º D ıi .t/d2i PeT .t/Qi Pe.t/

�

iXjD1

´�ij

�2

4

�e.t/

e.t � �j .t//

�T �Qi �Qi

� Qi

��e.t/

e.t � �j .t//

�μ:

The free-weight matrix method can be applied by using the following equality:

E

´2.eT .t/N1 C Pe

T .t/N2/

�Pe.t/C F.e.t/; t/C A

mXiD1

�i .t/e.t � �i .t//

!μD 0; (20)

where N1 and N2 are two arbitrarily given matrices with appropriate dimensions.Based on Assumption 1 and the S-procedure, one can obtain that E¹LV.t/º < 0 if there exists

positive definite diagonal matrix W D diag¹w1; w2; � � � ; wnº 2 RnN�nN such that

�T .t/��.t/ � 2

NXjD1

.wjFj .ej .t/; t//T .Fj .ej .t/; t/ ��j ej .t// < 0; (21)

where �.t/ D�eT .t/; eT .t � �1.t//; e

T .t � d1/; � � � ; eT .t � �m.t//; e

T .t � dm/; PeT .t/

�Tand

� D

0@11 12 13� 22 23� � 33

1A : (22)

Denote .t/ D ŒeT .t/; eT .t � �1.t//; e

T .t � d1/; � � � ; eT .t � �m.t//; e

T .t � dm/; PeT .t/; F T .e.t/; t/�T ,

then (21) can be written as

T .t/ .t/ < 0: (23)

Thus, under the conditions of Theorem 1, it can be concluded that E¹jje.t/jjº will approach tozero asymptotically, which indicates that node-to-node consensus in the mean square sense with theleader agents’ dynamics given in (1) and the follower agents’ dynamics given in (3) can be achieved.The proof is thus completed.

When m D 1, 0 D d0 < d1 D d , and ı1 D 1, the previously investigated node-to-nodeconsensus problem with stochastic sampled-data control is then reduced to the normal node-to-nodeconsensus problem with constant sampling period d . Thus, the following corollary can be derived.



Corollary 1Under Assumptions 1 and 2, for given constant matrices A, �, distributed node-to-node consensuswith constant sampling period d can be achieved if there exist positive definite matrices P , M ,and Q; positive definite diagonal matrix W ; and constant matrices N1, N2, and S with appropriatedimensions such that the following LMIs hold:�

M S

� M

�> 0; (24)

and

D

0BB@11 12 13 �

TW CN1� 22 23 0

� � 33 0

0 0 0 �W

1CCA < 0; (25)

where

11 D �M ��2

4Q;

12.1; 1/ DM � S C�2

4QCN1A;

12.1; 2/ D S;

13 D P �N1;

22.1; 1/ D �2M C S C ST �

�2

4Q;

22.1; 2/ DM � S;

22.2; 2/ D �M;

23.1; 1/ D ATN T

2 ;

33 D d2.M CQ/ �N2 �N

T2 ;

� D diag¹�1; �2; � � � ; �N º 2 RnN�nN :

Remark 5Previous works of the distributed node-to-node consensus under constant sampling include [22].Reciprocally combination method and delay-decomposition approach are further utilized in thispaper, which will be helpful for deriving less conservative results when compared with the resultsin [22].

4. NUMERICAL SIMULATIONS

In this section, some numerical simulations are presented to demonstrate the effectiveness of ourtheoretical results.

Example 1Suppose there are four agents in each layer with the topology structure G illustrated in Figure 1,where the weights are marked on the corresponding edges. The solid lines indicate the neighboringinteractions between the agents in the same layer, and the communication links between the twolayers are represented in dashed lines. Let fi .xi .t/; t/ D .sin.xi1.t//; cos.xi2.t///, i D 1; 2; � � � ; 8.The leaders’ states are xi .t/ D .xi1.t/; xi2.t//, i D 1; 2; 3; 4, and the corresponding followers’states are xi .t/ D .xi1.t/; xi2.t//, i D 5; 6; 7; 8.

Choose three possible sampling intervals as d1 D 0:1, d2 D 0:2, and d3 D 0:3 with probability

ı1 D1

3, ı2 D

1

3, and ı3 D

1

3, respectively. The coupling strength c is selected as 0.6. Nodes 6 and


120 Y. WAN ET AL.

Figure 1. Communication topology G for the multi-agent system with two layers.

8 in the follower’s layer are selected to directly sense the states of their corresponding leaders. InFigure 1, one obtains

OL D

0B@3:5 �1:5 �2 0

0 2 0 0

0 �2 4:5 �2:50 0 0 2

1CA : (26)

Solving the LMIs in Theorem 1 gives that P D QP ˝ I2, where

QP D

0B@0:2892 0:0333 �0:0349 0:0076

� 0:3532 �0:0473 0:0045

� � 0:2780 �0:0611� � � 0:3532

1CA ; (27)

The state trajectories of eight agents are shown in Figures 2 and 3, respectively. The consensuserrors are illustrated in Figure 4, where e1j .t/ D x1j .t/�x5j .t/, e2j .t/ D x2j .t/�x6j .t/, e3j .t/ Dx3j .t/ � x7j .t/, e4j .t/ D x4j .t/ � x8j .t/, and j D 1; 2. It can be seen that the states of allthe followers can track the states of their corresponding leaders asymptotically, which verify theanalytical results very well.

0 2 4 6 8 10−4

−3

−2

−1

0

1

2

3

4

t

Sta

tes

of a

gent

s in

two

leve

ls

x

11

x21

x31

x41

x51

x61

x71

x81

Figure 2. State trajectories of xi1.t/ in two layers, i D 1; 2; � � � ; 8.



0 2 4 6 8 10−5

−4

−3

−2

−1

0

1

2

3

t

Sta

tes

of a

gent

s in

two

leve

ls

x

12

x22

x32

x42

x52

x62

x72

x82

Figure 3. State trajectories of xi2.t/ in two layers, i D 1; 2; � � � ; 8.

0 2 4 6 8 10−6

−5

−4

−3

−2

−1

0

1

2

t

Con

sens

us e

rror

s

Figure 4. Consensus errors eij , i D 1; 2; 3; 4; j D 1; 2 (Case 1).

Figure 5 illustrates the relationship between the states’ convergence time and the expectation ofthe stochastic sampling period d.t/. The simulation results are with the same initial values, couplingstrength, and communication topology as Case 1, but the values of E¹d.t/º vary from 0 to 0.8 s, withwhich the mean square node-to-node consensus can all be achieved by solving the correspondingLMIs of Theorem 1. Fifty independent experiments are conducted, and the mean convergence timesare calculated. It can be seen from Figure 5 that the convergence time on the E¹d.t/º first decreasesand then increases. It reveals the fact that the appropriate value of E¹d.t/º is beneficial for the fastconvergence rate.

Example 2The agents’ dynamics are taken the same as those in Example 1. However, in this example, the sam-pling period is set as a constant d > 0. Comparisons between the results given Corollary 1 and thoseprovided in Theorem 2 of [22] in terms of maximum allowable sampling interval dmax are per-formed. Table. I shows the corresponding maximum allowable sampling interval dmax of these twomethods under different values of coupling strength c. It can be seen that the sufficient conditions in


122 Y. WAN ET AL.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.84

6

8

10

12

14

16

18

20

E(d(t)) (s)

Sta

tes

conv

erge

nce

time

(s)

Figure 5. The errors’ convergence time as function of the expectation of di , i D 1; 2; 3, where the accuracyof convergence is set as 10�4 around the origin.

Table I. Maximum allowable value ofconstant sampling period dmax .

c Corollary 1 Theorem 2 of [22]

0.6 0.887 0.3631 0.530 0.2221.5 0.350 0.1482 0.266 0.111

Corollary 1 contribute to larger dmax , which indicates that the results given in Corollary 1 are lessconservative than those in Theorem 2 of [22]. Additionally, Table I reveals that with the increase ofthe coupling strength c, the corresponding maximum allowable sampling interval dmax decreases.This fact can also be induced from 12 and 33 of (25), because in order to ensure the solvabilityof (25), there is a trade-off between the constant sampling period d and the coupling strength c.

5. CONCLUSIONS

In this paper, we have studied the distributed node-to-node consensus in the sense of mean squarewith stochastic sampled-data communications and under a directed topology. Unlike the contain-ment problem, the agents in the leaders’ layer also have information exchanges at sampling instants.Delay-input method has been employed, and several stochastic variables have been defined totransform the original sampled-data systems into delayed continuous ones with detailed expres-sions relating to the stochastic sampling instants. We have derived some criteria to ensure thateach follower can track the states of its corresponding leaders asymptotically. A discontinuousLyapunov–Krasovskii functional has been appropriately constructed, and the LMIs in Theorem 1have been derived using the delay-decomposition method, reciprocally combination method, andthe free-weight matrix approach, which are beneficial for reducing the conservation of the obtainedconditions. Lastly, two examples have provided in Section 4, and the simulation results verify themain results very well. Future research will focus on the case in which the agents can only sense theoutput measurements between neighboring agents. Another potential topic of future research is theH1 node-to-node consensus of multi-agent systems in the presence of external disturbances.



ACKNOWLEDGEMENTS

This work was funded by the National Natural Science Foundation of China under grants 61272530,11072059, 61304168, and 61322302; the Natural Science Foundation of Jiangsu Province of Chinaunder grants BK2012741 and BK20130595; and the Specialized Research Fund for the DoctoralProgram of Higher Education under grants 20110092110017 and 20130092110017.

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Distributed node‐to‐node consensus of multi‐agent systems with stochastic sampling

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