ConsensusoftheSecond-orderMulti ... · is small enough, the models tend to be synchronous. Us-ing...

Consensus of the Second-order Multi-agent Systems under AsynchronousSwitching with a Controller FaultDianhao Zheng*, Hongbin Zhang, J. Andrew Zhang and Yang Li

Abstract: Asynchronous switching differing from asynchronous consensus may hinder the system to reach a con-sensus. This receives very limited attention, especially when the multi-agent systems have a controller fault. Inorder to analyze the consensus in this situation, this paper studies the consensus of the second-order multi-agentsystems under asynchronous switching with a controller fault. We convert the consensus problems under asyn-chronous switching into stability problems and obtain important results for consensus with the aid of linear matrixinequalities. An example is given to illustrate the effect of asynchronous switching on the consensus, and to validatethe analytical results in this paper.

Keywords: Consensus, Multi-agent systems, Asynchronous switching, Controller faults

1. INTRODUCTION

Multi-agent systems, as a class of complicated dy-namic systems, are composed of multiple interactive intel-ligences. Such systems appear in many applications, suchas robot communication networks [1, 2], and unmannedvehicles [3]. Therefore multi-agent systems have drawnconsiderable attention in recent years [4–8]. In multi-agent systems, all agents coordinate to solve a global com-mon problem, such as consensus [4], containment control[9, 10], and vehicle formations.

Consensus, as the most basic problem of the coordina-tion in multi-agent systems, is to design or analyze a dis-tributed control law to make all agents reach a commonvalue [11]. Whenever the nodes of a network are all inagreement, this common value is called the group deci-sion value. Reaching the group decision value needs toapply inputs that only depend on the states of every nodeand its own neighbours in a distributed way [4].

The consensus problem has been extensively studied.Some basic concepts on consensus and topologies wereintroduced in [4,12]. In [13], results for second-order con-sensus under fixed topologies are presented. Second-orderconsensus under switched topologies was studied in [14].The paper [11] studied the consensus problem for a classof uncertain multi-agent systems under directed switch-

ing networks with uncertainty. The papers [15,16] studiedthe consensus problem for a class of general second-ordermulti-agent systems and presented some results aboutthe necessary and sufficient condition for the consensus.Fixed-time consensus tracking control for second-ordermulti-agent systems with bounded input uncertainties wasintroduced in [17]. A distributed protocol was proposed in[18] based on the information of second-order neighboursfor the robust consensus problem of fractional-order linearmulti-agent systems with positive real uncertainty under afixed undirected topology.

So far, almost all the research efforts on the con-sensus of multi-agent under switched topologies are de-voted to networks without feedback control or with onlysynchronous switching despite the great importance ofthe consensus under asynchronous switching control inboth theoretical and practical aspects [19]. According tothe definition in [20–22], the consensus is called asyn-chronous consensus if the individuals in systems respondto the new information from their neighbours at differ-ent update times. This is different to the consensus underasynchronous switching, which means that the switchingbetween the candidate controllers and topologies are asyn-chronous [19]. For switched systems, the asynchronousswitching often means that the switching of the controllers

This work is supported by the National Natural Science Foundation of China( Grant No.61374117)

Dianhao Zheng is with the School of Information and Communication Engineering, University of Electronic Science and Technology ofChina, Chengdu 611731, China, is also with the Faculty of Engineering and Information Technology, University of Technology Sydney, Syd-ney, NSW 2007, Australia (e-mail: [email protected]). Hongbin Zhang is with the School of Information and Communication, Universityof Electronic Science and Technology of China, Chengdu 611731, China (e-mail:[email protected]). J. Andrew Zhang is with the GlobalBig Data Technologies Centre, University of Technology Sydney, Sydney, NSW 2007, Australia (e-mail:[email protected]). YangLi is with the School of Information and Communication, University of Electronic Science and Technology of China, Chengdu, Sichuan,611731, China. (e-mail:[email protected]).* Corresponding author.

http://www.springer.com/12555

2 Dianhao Zheng, Hongbin Zhang, J. Andrew Zhang and Yang Li

to be designed has a lag to the switching of the systemmodels [23] and the switching between the candidate con-trollers and system modes is asynchronous [24]. There-fore, although asynchronous consensus and consensus un-der asynchronous switching sound similar in their names,their definitions are different and asynchronous switchingmay hinder the way to a consensus.

Asynchronous switching has been studied for switchedsystems in the past few years and the research results onswitched systems have advanced the study on switchedtopologies [23–25]. Due to link failures or reconnec-tion, the topologies of multi-agent systems often change.When the changing is modelled in a switched way, themulti-agent systems under switching topologies can bedescribed and analyzed by switched systems theorems[24]. However, the consensus problem for asynchronousswitching is yet to be solved. Since asynchronous switch-ing widely exists in switched topologies, the consensus ofmulti-agent systems under asynchronous switching needsto be investigated urgently, especially for the systems witha controller fault.

The main purpose of this paper is to study the consen-sus of the second-order multi-agent systems under asyn-chronous switching with a controller fault. We will tacklethe problem via converting the consensus problems underasynchronous switching into stability problems. The restof the paper is organized as follows. In Section 2, somebasic concepts and algebraic graph theories are given.Main results about asynchronous switching are presentedin Section 3. The simulation results are presented in Sec-tion 4. Finally, conclusions are provided in Section 5.

Notations: The superscript “T" stands for matrix trans-position. The mathematical symbols In−1 and 0n−1 meanan identity matrix and a zero matrix with (n−1)× (n−1)dimension, respectively. The sign diag{· · ·} represents ablock-diagonal matrix with proper dimension. If P is agiven matrix, P > (or <) 0 signifies a symmetric andpositive (or negative) definite matrix P. For a functionγ , it is said to be of a class K∞ function if the functionγ : [0,∞)→ [0,∞), γ(0) = 0, is strictly increasing, contin-uous, and unbounded. Scalar multiplication of matrices isdefined as a regular number( called a "scalar" ) multiply-ing every element in the matrix [26].

2. PRELIMINARIES

In this section, some basic concepts and algebraic graphtheories are introduced.

The network of multi-agent systems is often modelledby graph theories. For the multi-agent systems with nagents and the node indexes set I = {1,2, . . . ,n} , thedigraph can be denoted by G = (V,E ,A), where V ={v1,v2, . . . ,vn}, E ⊆ V ×V , A= [ai j] are the set of nodes,the set of edges and the adjacency matrix. The edge be-tween nodes i and j can be described as ei j = (vi,v j). This

method can represent any edge of the graph G. We assumei 6= j for any edge. The set of neighbours of node vi isdenoted by Ni = {v j ∈ V : (vi,v j) ∈ E , j 6= i}. The Lapla-cian matrix L is defined as: li j = ∑

nk=1,k 6=i aik for i = j, and

li j = −ai j, for i 6= j, i, j ∈ I. A Laplacian-like matrix His defined as H = [hi j], where hi j = li j − ln j. The func-tion σ(t) : [0,+∞)→M = {1,2, · · · ,m} stands for theswitching signal of the switching topologies, where m isthe total amount of topologies. The function σ ′(t) denotesthe asynchronous switching of σ(t).

For the switched topologies, t1, t2, t3, · · · , tl , tl+1, · · ·stand for the switching times of the topologies of themulti-agent systems. Let ∆σ(tl) (tl , tl+1) ( or Oσ(tl) (tl , tl+1)) represent the asynchronous (or synchronous) time be-tween the time slots [tl , tl+1). The symbol Tp(0, t) denotesthe running time of the pth topology between the time slots[0, t).

For a group of n agents systems, every agent is mod-elled by the second-order dynamics as

xi(t) = vi(t),vi(t) = ui(t),

(1)

where xi(t) is the state of the ith agent, ui is the controlinput.

For asynchronous switching, there exists a time lag ∆

between the topologies switching signal σ(t) and the feed-back controller coefficients switching σ ′(t). When the lagis small enough, the models tend to be synchronous. Us-ing one of the many algorithms for synchronous switching[27,28] as an example, the corresponding consensus algo-rithm with asynchronous switching can be represented as

ui(t)

= f β0σ(t−∆) ∑j∈Ni(t)

ai jσ(t)(x j(t)−xi(t))− f β1σ(t−∆)vi(t), (2)

where ai jσ(t) , i, j ∈ I are the elements in the adjacency ma-trix A(G). The topology G and σ(t) change at the switch-ing times t1, t2, t3, · · · , tl , tl+1, · · · . The positive parametersβ0,β1 are the coefficients, and f is defined as

f ,measured value

actual value.

The fault considered in this model is uncertain and{0 < fd ≤ f ≤ fu,

fd ≤ 1≤ fu,(3)

where fd and fu are known constants. In this paper, weonly assume the range of the deviation is known.

Remark 1: When the positive constants fd 6= 1 orfu 6= 1 are known, it is implied that consensus feedbackhas an uncertain parameter f . It may originate from inac-curate coefficient β or other factors and results in a con-troller fault.

consensus of the second-order multi-agent systems under asynchronous switching with a controller fault 3

Remark 2: Because of ∆ in the system (2), the switch-ing time of topologies and controllers is different and thesystem (2) is under asynchronous switching. Since allagents update their states at the same time t, it is also asynchronous system. Therefore, the system (2) is a syn-chronous system with asynchronous switching. For asyn-chronous consensus, every agent has its own update timeti, which is different from agent to agent. For the sys-tem (2), the controller could be at fault and the parameterscould be uncertain and mismatched. How to reach a con-sensus in this situation is challenging.

For all initial conditions, the multi-agent systems (1) aresaid to reach a consensus if

limt→+∞

[xi− x j] = 0, limt→+∞

[vi− v j] = 0,∀i, j ∈M, i 6= j.

For a switching σ(t) at any time tk > tl ≥ 0, the switch-ing number in the pth subsystem is denoted by Nσ p(tl , tk),and the total running time of pth subsystem is denotedby Tp(tl , tk) over the interval [tl , tk). In [29], it is shownthat the system has a mode-dependent average dwell time(MDADT) τap if there exist positive numbers N0p(tl , tk)and τap such that

Nσ p(tl , tk)≤ N0p(tl , tk)+Tp(tl , tk)

τap(4)

Based on the concept on MDADT, we can get the fol-lowing Lemma 1.

Lemma 1: Consider a system zt = Aσ(t−∆)zt , σ(t −∆) ∈M with give constants λp > 0, µp > 1, αp > 0, p ∈M. For ∀p ∈M, ∀t ∈ [tl , tl+1), and ∀(σ(tl) = p,σ(t−l ) =q) ∈ M×M, p 6= q, if there exist symmetric matricesPp > 0, such that

ATq Pp +PpAq ≤ αpPp,∀t ∈ [tl , tl +∆l), (5)

ATp Pp +PpAp ≤−λpPp,∀t ∈ [tl +∆l , tl+1) (6)

and Pp(z(tl))≤ µpPq(z(t−l )), (7)

then the system is globally uniformly asymptotically sta-ble with MDADT

τap > τ∗ap ,

∆p_max(λP +αp)+ ln µp

λp, (8)

where ∆p_max , maxl,σ(tl)=p∆σ(tl) [tl , tl+1), for ∀l ∈ N.

Proof: For any t > 0, ∀t ∈ (tl , tl+1).From (8), one has

ln µp +(λp +αp)∆p_max

τap−λp < 0, (9)

and

maxp∈M{ ln µp +(λp +αp)∆p_max

τap−λp}< 0. (10)

A multiple Lyapunov function is constructed as

Vp(z(t)) = z(t)T Ppz(t). (11)

According to (5), (6) and (7), it holds that

Vσ(t)(x(t))

≤ exp{

ασ(tl)∆σ(tl)(tl , t)−λσ(tl)Oσ(tl)(tl , t)}

Vσ(tl)(x(tl))

≤ exp{

ασ(tl)∆σ(tl)(tl , t)−λσ(tl)Oσ(tl)(tl , t)}

×µσ(tl)Vσ(tl)(x(tl−))

......

≤

{l

∏p=1

µσ(tp)

}× exp

{ασ(tl)∆σ(tl)(tl , t)+ · · ·+ασ(t0)∆σ(t0)(tl−1, tl)

}× exp

{−λσ(tl)Oσ(tl)(tl , t)−·· ·−λσ(t0)Oσ(t0)(t0, t1)

}×Vσ(t0)(x(t0)).

(12)

Because of the total switching numbers N(0, t) =m∑

p=1Nσ p(0, t) and t0 = 0, one can get

Vσ(t)(x(t))

≤

{m

∏p=1

µNσ p(0,t)p

}exp

{m

∑p=1

αp∆p(0, t)−λpOp(0, t)

}×Vσ(0)(x(0)).

(13)

According to (4), one has

m

∏p=1

µNσ p(0,t)p =

m

∏p=1

µN0p+

Tp(0,t)τap

p

=exp

{m

∑p=1

{N0p ln µp +

Tp(0, t)τap

ln µp

}} (14)

and

Vσ(t)(x(t))

≤ exp

{m

∑p=1

{N0p ln µp +

Tp(0, t)τap

ln µp

}}

× exp

{m

∑p=1{αp∆p(0, t)−λpTp(0, t)+λp∆p(0, t)}

}×Vσ(0)(x(0))

≤ exp

{m

∑p=1

N0p ln µp +(ln µp

τap−λp)Tp(0, t)

}

× exp

{m

∑p=1

(λp +αp)∆p(0, t)

}Vσ(0)(x(0)).

(15)

From the definition of ∆p_max one can get

∆p(0, t)≤ Nσ p(0, t)∆p_max ≤ N0p +Tp(0, t)

τap. (16)


Then, based on (4), one can get

Vσ(t)(x(t))

≤ exp

{m

∑p=1{N0p ln µp +(λp +αp)N0p∆p_max)}

}

×exp

{m

∑p=1{( ln µp+(λp+αp)τap

τap−λp)}Tp(0, t)

}×Vσ(0)(x(0)).

(17)

From (9), we can conclude that Vσ(t)(x(t)) conver-gences to zero, as t → +∞, if the dwell time satisfiesτap > (λp +αp)∆p_max + ln µp)/λp.

Therefore, we can see that the asymptotic stability isreduced. This completes the proof. �

In order to solve the problems with an uncertain param-eter, the following Lemma 2 is introduced.

Lemma 2: [30] Given matrices Q, R, E and H withproper dimensions, and Q = QT , R = RT > 0, the inequal-ity

Q+HFE +ET FT HT < 0

holds for all F satisfying FT F ≤ R, if and only if thereexists some ε > 0, such that

Q+ εHHT + ε−1ET RE < 0.

3. MAIN RESULTS

In this section, we mainly complete the following threeobjectives:

Firstly, we design a transformation from consensus un-der asynchronous switching to stability. For asynchronousswitching, the states of agents may go to divergenceswhen some necessary and sufficient conditions cannotbe met [15] in the asynchronous switching time slots.We analyze the relationship between consensus underasynchronous switching and the asynchronous stability ofswitched systems and then present the conversion betweenthem. Based on the conversion and the relationship be-tween energy functions and stability, the divergences willcause the increase of the energy functions. That is, asyn-chronous switching may result in the divergences of states,which are marked by the increasing of the energy function.

Secondly, we analyze the consensus with a controllerfault. We rewrite the formation of f and analyze the con-ditions for reaching a consensus when the multi-agent sys-tems are switching with a controller fault based on themethod of the mode-dependent average dwell time. Usingthis method, we can decrease the overall energy functionby adjusting the dwell time of topologies, and hence forcethe system to reach a consensus.

Thirdly, we extend the results on asynchronous switch-ing to the synchronous switching.

3.1. Transformation from Consensus to Asyn-chronous Switching

By the following Theorem 1, the problem of the consen-sus under asynchronous switching is converted into that ofthe asynchronous stability of switched systems.

Theorem 1: For the multi-agent systems with directednetwork G (t), if the system

z(t) = FKσ(t−∆)Baσ(t)z(t),z(t0) = z(0), (18)

is globally uniformly asymptotically stable, then themulti-agent systems with the dynamic system (1) willreach a consensus with the consensus algorithm (2), where

F =

[In−1 0n−1

0n−1 fdiag

], (19)

Kσ(t−∆) =

[0n−1 In−1

−β0In−1 −β1In−1

]σ(t−∆)

, (20)

Baσ(t) =

[H 0n−1

0n−1 In−1

]σ(t)

, (21)

fdiag = diag{ f , · · · , f︸︷︷︸n−1

}. (22)

Proof: Firstly, we define z(t) , [x1 − xn, . . . ,xn−1 −xn,v1− vn, . . . ,vn−1− vn]

T (t).According to the definition of the consensus, if

limt→+∞

z(t) = 0, the multi-agent systems will reach a con-sensus.

From (2), one can get

ui−un

= f β0σ(t−∆) ∑j∈Ni(t)

ai jσ(t)(x j(t)−xi(t))− f β1σ(t−∆)vi(t)

− f β0σ(t−∆) ∑k∈Nn(t)

ankσ(t)(xk(t)−xn(t))+ f β1σ(t−∆)vn(t)

=− f β0σ(t−∆)Li(t)x(t)−β1σ(t−∆)

f vi(t)

+ f β0σ(t−∆)Ln(t)x(t)+β1σ(t−∆)

f vn(t)

=− f β0σ(t−∆)(Li(t)−Ln(t))x(t)− f β1σ(t−∆)

(vi(t)−vn(t))

=− f β0σ(t−∆)Hi(t)[x1− xn, . . . ,xn−1− xn]

T

− f β1σ(t−∆)(vi(t)− vn(t))

(23)

and

[v1− vn, . . . , vn−1− vn]T (t)

=[u1−un, . . . ,un−1−un]T

=− f β0σ(t−∆)H(t)[x1− xn, . . . ,xn−1− xn]

T (t)

− f β1σ(t−∆)[v1− vn, . . . ,vn−1− vn]

T (t)

=[− f β0σ(t−∆)H(t),− f β1σ(t−∆)

I]z(t).

(24)


where Li ( or Hi ) is the ith row vector of the Laplacian ( orH ) matrix.

From (1), one can get

[x1− xn, . . . , xn−1− xn]T

=[v1− vn, . . . ,vn−1− vn]T

=[0n−1, In−1]z.

(25)

Now, combining (24) and (25), we can get

z(t)=[

0n−1 In−1−f β0σ(t−∆)

H(t) −f β1σ(t−∆)In−1

]z(t),z(t0)=z(0).

(26)

We further have[0n−1 In−1

− f β0σ(t−∆)H(t) − f β1σ(t−∆)

In−1

]=

[In−1 0n−10n−1 f In−1

][0n−1 In−1−β0In−1 −β1In−1

]σ(t−∆)

×[

H 0n−10n−1 In−1

]σ(t)

,FKσ(t−∆)Baσ(t),

(27)

where f is a scalar, and fdiag denotes the diagonal ma-trix with diagonal elements. The equation (18) can nowbe proved.

According to the definition of z(t), the consensus prob-lem can be solved if the system (18) is globally uniformlyasymptotically stable. This completes the proof of thistheorem. �

Remark 3: For the case without a controller fault oran uncertain parameter, it can be denoted by f = 1. Fromthe proof of Theorem 1, we can see that Theorem 1 is alsoapplicable to the situations without a controller fault.

3.2. Consensus with a controller faultFor the multi-agent systems with a controller fault rep-

resented by the uncertain parameter, we only know therange of f . In order to solve the asynchronous switchingwith an uncertain parameter f , we rewrite the formationof f .

We define Fu , diag(1, · · · ,1, fu, · · · fu), Fd ,diag(1, · · · ,1, fd , · · · fd), F0 , 1

2 (Fu+Fd), F1 , 12 (Fu−Fd),

then

F = F0 +EF1, (28)

where E = diag(1 · · ·1,e, · · · ,e), and −16 e6 1.Systems (18) can be replaced with

z(t) = FKσ(t−∆)Baσ(t)z(t)

= (F0 +EF1)Kσ(t−∆)Baσ(t)z(t)

= F0Kσ(t−∆)Baσ(t)z(t)+EF1Kσ(t−∆)Baσ(t)z(t)

= Aσ(t−∆)z(t),

(29)

where Aσ(t−∆) =F0Kσ(t−∆)Baσ(t)z(t)+EF1Kσ(t−∆)Baσ(t)z(t).From (29) and the inequality in Lemma 2, we now have

the following theorem:

Theorem 2: For the given constants λp > 0, µp > 1,and the multi-agent systems with switched topologies Gp,p∈M, ∀(σ(ti) = p,σ(t−i ) = q)∈M×M, p 6= q, if thereexist symmetric matrices Pp > 0, positive constants εp, andεpq, ∀p ∈M, such that Dp εpF1 Pp(KpBap)

T

εpF1T −εpIn−1 0n−1

KpBapPp 0n−1 −εpIn−1

< 0, (30)

where Dp = (F0KpBap)Pp +Pp(F0KpBap)T +λpPp, and

P−1p ≤ µpP−1

q , (31)

Dp εpqF1 Pp(KqBap)T

εpqF1T −εpqIn−1 0n−1

KqBapPp 0n−1 −εpqIn−1

< 0, (32)

where Dp = (F0KqBap)Pp +Pp(F0KqBap)T −αpPp, then

the multi-agent systems (1) under consensus algorithm (2)with the fault tolerance (3) will reach a consensus withMDADT

τap > τ∗ap =

∆p_max(λP +αp)+ ln µp

λp.

Proof: Based on the Schur complement lemma, onecan see that the system (30) is equivalent to

Dp−STp

[−εpIn−1 0n−1

0n−1 −εpIn−1

]−1

Sp < 0, (33)

where Sp =

[εpF1

T

(KpBap)Pp

]. This can be further written

as

Dp−STp

[−εpIn−1 0n−1

0n−1 −εpIn−1

]−1

Sp

= Dp +STp

[εp−1In−1 0n−1

0n−1 εp−1In−1

]Sp

= Dp +[

F1 εp−1Pp(KpBap)

T]

Sp

= Dp + εpF1F1T + εp

−1Pp(KpBap)T (KpBap)Pp

< 0.

(34)

According to Lemma 2, one can get

Dp +F1E(KpBap)Pp +Pp(KpBap)T ET F1

T < 0. (35)

This can also be further represented as

Dp +F1E(KpBap)Pp +Pp(KpBap)T ET F1

T

=(F0KpBap +F1EKpBap)Pp

+Pp(F0KpBap +F1E(KpBap))T +λpPp

<0.

(36)


Using diag{Pp−1} to pre- and post-multiply both sides

of systems (36), one has

Pp−1(F0KpBap +F1EKpBap)

+(F0KpBap +F1EKpBap)T Pp

−1 +λpPp−1

< 0.

(37)

Denote Pp = Pp−1. We can also see that Pp > 0 and

systems (37) are equivalent to

Pp(F0KpBap+F1EKpBap)+(F0KpBap+F1EKpBap)Pp+λpPp < 0. (38)

Based on systems (29) and (38), one has

Pp(FKpBap)+(FKpBap)T Pp <−λpPp. (39)

Similarly, we can have

Pp(FKqBap)+(FKqBap)T Pp < αpPp. (40)

From (31), one has

Pp ≤ µpPq. (41)

Combining the results in (39), (40) and (41) and Lemma1, we can then establish the results in Theorem 2. Theproof is completed. �

Remark 4: Theorem 2 is based on Lemma 1 and onthe idea that the topologies changing replaces subsystemswitching. Hence the results on MDADT and switchedsystems are also applicable to the multi-agent systems un-der switched topologies.

Using a similar process, we can establish and proveCorollary 1 below when using a common P to replace Pp.

Corollary 1: For the multi-agent systems withswitched topologies Gp, p∈M, ∀(σ(ti)= p,σ(t−i )= q)∈M×M, p 6= q, if there exist a symmetric matrix P > 0,positive constants εp, ∀p ∈M, such that Dp εpF1 P(KpBap)

T


(KpBap)P 0n−1 −εpIn−1

< 0, (42)

where Dp = (F0KpBap)P+P(F0KpBap)T +λpP, Dp εpqF1 P(KqBap)

T

εpqF1T −εpqI 0n−1

KqBapP 0n−1 −εpqIn−1

< 0, (43)

where Dp = (F0KqBap)P+P(F0KqBap)T −αpP, then the

multi-agent systems (1) under consensus algorithm (2)with the fault tolerance (3) will reach the consensus finallywith MDADT

τap > τ∗ap =

∆p_max(λP +αp)

λp. (44)

Remark 5: MDADT is used in the proof for reachinga consensus when the multi-agent systems are switching.In order to guarantee the condition of MDADT, the statis-tical information on the dwell time under each topology isrequired to confirm that the average dwell time is longerthan the offset.

3.3. Extension to Synchronous SwitchingOne can see that if the time lag ∆(t) = 0, the asyn-

chronous switching is replaced with synchronous switch-ing in the algorithm (2).

In the synchronous switching situation, systems (2)change to

ui(t) = f β0σ(t) ∑j∈Ni(t)

ai jσ(t)(x j(t)−xi(t))− f β1σ(t)vi(t). (45)

Likewise, the following corollaries can be obtained forthe case of synchronous switching multi-agent systems.

Corollary 2: For the given constants λp > 0, µp > 1,and the switched topologies Gp of the multi-agent systems,p∈M, ∀(σ(ti) = p,σ(t−i ) = q)∈M×M, p 6= q, if thereexist symmetric matrices Pp > 0, positive constants εp,∀p ∈M, such that Dp εpF1 Pp(KpBap)

T


KpBapPp 0n−1 −εpIn−1

< 0, (46)

where Dp = (F0KpBap)Pp +Pp(F0KpBap)T +λpPp,

P−1p ≤ µpP−1

q , (47)

then the multi-agent systems (1) under consensus algo-rithm (45) with the fault tolerance (3) reach the consensuswhen

τap > τ∗ap =

ln µp

λp, (48)

Corollary 3: For the switched topologies Gp of themulti-agent systems, p ∈ M, if there exist a symmetricmatrix P > 0, positive constants εp, ∀p ∈M, such that Dp εpF1 P(KpBap)

T


(KpBap)P 0n−1 −εpIn−1

< 0, (49)

where Dp = (F0KpBap)P+P(F0KpBap)T + λpP, then the

multi-agent systems (1) under consensus algorithm (45)with the fault tolerance (3) will reach the consensus fi-nally.

Remark 6: LMIs toolbox in MATLAB is often usedto find proper matrices P and constants ε . Although theexistence of proper matrices and constants is the sufficientcondition, the results in this paper provide a guarantee onthe consensus under asynchronous switching, especiallywhen the systems are with an uncertain parameter.


Since there are many nonlinear cases and Takagi-Sugeno (T-S) fuzzy model has been popularly utilized inrepresenting nonlinear systems [31–33], one of our futureworks is to investigate the coordination of the nonlinearmulti-agent systems under asynchronous switching.

4. NUMERICAL EXAMPLE

In this section, numerical results are presented todemonstrate the potential and validity of our developedtheoretical results. For this example, the second-ordermulti-agent systems are with 4 agents and the topologiesare switched. Through this numerical example, the asyn-chronous switching’s reaction to a consensus can also beseen clearly. In the same set-up, the corresponding vali-dation can be done for the common P case and the syn-chronous switching cases. This is omitted due to spacelimitations.

Consider the adjacency matrices of the two consideredtopologies described by

0 0 0 01 0 0 00 1 0 01 0 0 0

, and

0 0 1 01 0 0 00 1 0 11 0 0 0

. (50)

The two-level signal values (low and high) of switchingsignals σ(t) stand for these two topologies. The purposehere is to show the influence of asynchronous switchingand present the tendency of states and velocities of agentswhen the dwell times satisfy the corresponding conditions(8). In order to achieve this, the changing of energy func-tion (Lyapunov function) is shown in Fig.1, and statesand velocities are shown in Fig.2., when switching signalsσ(t) and σ ′(t) change.

Selection of the parameters for the system model andthe controller is based on the physical meanings of param-eters. The sign µ stands for the jumping strength whentopologies change. The symbols λ and α denote the con-vergence rate and the divergence speed. Changing theseparameters will affect constrained conditions which re-sults in the changing of the average dwell time. The looserthe constrained conditions are, the lager the average dwelltime is.

The maximal delay of asynchronous switching of thefirst topology is 0, and the maximal delay of asynchronousswitching of the second topology is 6 seconds. In this ex-ample, we just know the range 0.9≤ f ≤ 1.1, i.e., fd = 0.9and fu = 1.1. By providing proper constants λ1 = 0.020,µ1 = 1.88, α1 = 0, λ2 = 0.183, µ2 = 2.47, α2 = 0.845, theLMIs toolbox can find proper solutions and get the properdwell time satisfying conditions (8). The simulation re-sults are shown in Fig.1. and Fig.2.

At the running time slots [0,1.730), [8.04,9.77),[15.65,19.74) and [25.47,34.58), the network is syn-chronous, the energy function in Fig.1. drops down

quickly except for at the topologies switching time spot,the states and velocities of agents in Fig.2. are reach-ing a consensus. At the running time slots [1.73,8.04),[9.77,15.65), [19.74,25.47) and [34.58, 40.37), the net-work is asynchronous, the energy function in Fig.1. in-creases quickly, the states and velocities of agents in Fig.2.tend to diverge. When the dwell times meet the conditions(8), the consensus is reached.

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

time(sec)

Lya

pu

no

v f

un

ctio

n a

nd

sw

itch

ed

sig

na

ls

Vσ

σ(t)

σ‘(t)

Fig. 1. The changing of the Lyapunov function andswitched signals

0 20 40 60 80 100 120 140 160 180−20

−10

0

10

20

30

time(sec)(a)

sta

tes a

nd

sw

itch

ed

sig

na

ls

x1

x2

x3

x4

σ(t)

σ‘(t)

0 20 40 60 80 100 120 140 160 180−20

−10

0

10

20

30

time(sec)(b)

ve

locitie

s a

nd

sw

itch

ed

sig

na

ls

v1

v2

v3

v4

σ(t)

σ‘(t)

Fig. 2. The changing of the states, velocities and switchedsignals

5. CONCLUSIONS

The consensus of the second-order multi-agent systemsunder asynchronous switching with a controller fault is in-


vestigated. We analyzed the relationship between consen-sus and steadiness, and prove the consensus, and providedconditions for the consensus, and also extended the re-sults to the synchronous switching multi-agent systems.Simulation results clearly demonstrate the asynchronousswitching’s reaction to a consensus. With both analyti-cal and numerical results, we show that a consensus underasynchronous switching with a controller fault can alsobe reached under proper conditions despite the reaction ofasynchronous switching. The methods developed in thispaper can also be potentially applied to study other prob-lems such as containment control and vehicle formationsin the presence of asynchronous switching for the under-lying systems.

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Dianhao Zheng received the B.Eng. de-gree in electronic science and technol-ogy from China University of Mining andTechnology, Xuzhou, in 2009, and theM.S. degree in circuits and systems fromUniversity of Electronic Science and Tech-nology of China, Chengdu, in 2012. He iscurrently working towards two Ph.D. de-grees in circuits and systems at the Univer-

sity of Electronic Science and Technology of China and in engi-neering and information technology at the University of Tech-nology Sydney. He was an engineer from 2012 to 2014. Hisresearch interests include cooperative control, multi-agent sys-tems, and switched systems.

Hongbin Zhang received the B.Eng. de-gree in aerocraft design from Northwest-ern Polytechnical University, Xian, China,in 1999, and the MEng and PhD degreesin circuits and systems from the Univer-sity of Electronic Science and Technologyof China, Chengdu, in 2002 and 2006, re-spectively. He has been with the School ofElectrical Engineering, University of Elec-

tronic Science and Technology of China, since 2002, where he iscurrently a professor. From August 2008 to August 2010, he hasserved as a research fellow with the Department of Manufactur-ing Engineering and Engineering Management, City Universityof Hong Kong, Kowloon, Hong Kong. His current research inter-ests include intelligent control, autonomous cooperative controland integrated navigation.

J. Andrew Zhang received the B.Sc.degree from Xi’an JiaoTong University,China, in 1996, the M.Sc. degree fromNanjing University of Posts and Telecom-munications, China, in 1999, and the Ph.D.degree from the Australian National Uni-versity, in 2004. Currently, Dr. Zhang isan associate Professor in School of Com-puting and Communications, University of

Technology Sydney, Australia. He was a researcher with Data61, CSIRO, Australia from 2010 to 2016, the Networked Sys-tems, NICTA, Australia from 2004 to 2010, and ZTE Corp, Nan-jing, China from 1999 to 2001. Dr. Zhang’s research interests arein the area of signal processing for wireless communications andsensing, and autonomous vehicular networks. He has won 4 bestpaper awards for his work. He is a recipient of CSIRO Chair-man’s Medal and the Australian Engineering Innovation Awardin 2012 for exceptional research achievements in multi-gigabitwireless communications.

Yang Li received the B.E. degree in elec-tronic and information engineering andM.S. degree in circuits and systemsfromUniversity of Electronic Science and Tech-nology of China, Chengdu, in 2014 and2017, respectively. He is currently work-ing towards his Ph.D. degree in circuitsand systems at the University of Elec-tronic Science and Technology of China,

Chengdu. His research interests include fuzzy control andswitched systems.

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