Automatica 54 (2015) 158–165

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Finite-time consensus for second-order multi-agent systems withdisturbances by integral sliding mode✩

Shuanghe Yu, Xiaojun Long 1

School of Information Science and Technology, Dalian Maritime University, Dalian 116026, China

a r t i c l e i n f o

Article history:Received 3 June 2014Received in revised form16 January 2015Accepted 19 January 2015Available online 20 February 2015

Keywords:Multi-agent systems (MAS)Finite-time consensusIntegral sliding mode (ISM)Leader-followingDisturbance

a b s t r a c t

This paper investigates the distributed finite-time consensus problem of second-order multi-agentsystems (MAS) in the presence of bounded disturbances. Based on the continuous homogeneous finite-time consensus protocol for the nominal MAS, the discontinuous or continuous integral sliding mode(ISM) protocols are respectively developed to achieve accurate finite-time consensus in spite of thedisturbances. Simulation results validate the effectiveness of the proposed scheme.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The distributed cooperative control of multi-agent systems(MAS), such as the formation, flocking or rendezvous tasks of mo-bile robots, unmanned aerial vehicles or autonomous underwa-ter vehicles (Do, 2011; Hartley, Trodden, Richards, & Maciejowski,2012; Oh & Ahn, 2013), has attracted significant attentions in thepast decade (Ghommam & Saad, 2014; Li & Kumar, 2012; Liu& Jiang, 2014; Xargay, Kaminer, Pascoal, & Hovakimyan, 2013).Among these paradigms, the typical issue is known as the consen-sus problem, which aims at designing a distributed consensus pro-tocol such that a teamof agents can reach an agreement (Jadbabaie,Lin, & Morse, 2003; Olfati-Saber & Murray, 2004; Ren, 2008; Yu,Chen, & Cao, 2010).

The consensus problem of MAS can be categorized into theconsensus regulating of a leaderless structure and consensustracking of a leader-following structure (Hong, Hu, & Gao, 2006;Ren & Beard, 2005). The past decade has witnessed the dramatic

✩ This work was supported by the Liaoning BaiQianWan Talents Program(2012921079) in China and the Fundamental Research Funds for the CentralUniversities of China and Excellent Scientific and Technological Innovation TeamProgram of DLMU (3132013334). The material in this paper was not presented atany conference. This paper was recommended for publication in revised form byAssociate Editor Riccardo Scattolini under the direction of Editor Frank Allgöwer.

E-mail addresses: shuanghe@dlmu.edu.cn (S. Yu), longxj1985@aliyun.com(X. Long).1 Tel.: +86 411 84723637; fax: +86 411 84723025.

http://dx.doi.org/10.1016/j.automatica.2015.02.0010005-1098/© 2015 Elsevier Ltd. All rights reserved.

progresses in the asymptotic consensus with infinite settling timefor various agent dynamics and graph topologies (Cao, Yu, Ren,& Chen, 2013). However, the finite-time control demonstrateshigher accuracy, stronger robustness and disturbance rejectionproperties besides finite settling time (Wu, Yu, & Man, 1998; Yu,Yu, Shirinzadeh, & Man, 2005). Therefore, many results relatedto finite-time consensus of MAS, i.e., the agreement is reached infinite time, have been developed in recent years from differentperspectives.

Discontinuous finite-time consensus protocols are mainlyachieved by the sign function of states (Cortes, 2006; Hui, 2011;Qing, Haddad, & Bhat, 2010) or relative states (Chen, Lewis, &Xie, 2011; Franceschelli, Giua, Pisano, & Usai, 2013) or slidingmode control (SMC) (Zhang & Yang, 2013) for first-order MASwithconsidering disturbances or not and terminal sliding mode control(TSMC) (Chen, Yue, & Song, 2013;Ghasemi&Nersesov, 2013; Khoo,Xie, & Man, 2009; Meng, Ren, & You, 2010; Zhao & Hua, 2014) ortwisting second-order SMC (Pilloni, Pisano, Franceschelli, & Usai,2013) for second order MAS with disturbances. Nevertheless, allthese protocols involve discontinuous dynamics, which may causechattering in practice.

Inspired by the continuous but non-Lipschitz finite-timecontrol strategies (Bhat & Bernstein, 1998, 2005; Hong, Xu, &Huang, 2002), the continuous finite-time consensus protocols havebeen formulated by using Lyapunov finite-time stability for first-order MAS (Cao, Ren, Casbeer, & Schumacher, 2013; Hui, Haddad,& Bhat, 2008; Wang & Xiao, 2010; Xiao, Wang, Chen, & Gao,2009; Zoghlami, Beji, Mlayeh, & Abichou, 2013), homogeneous

S. Yu, X. Long / Automatica 54 (2015) 158–165 159

finite-time stability (Cao, Ren, & Meng, 2010; Guan, Sun, Wang,& Li, 2012; Li & Wang, 2013; Lu, Lu, Chen, & Lu, 2013; Ou, Du, &Li, 2014; Sun & Zhu, in press; Wang & Hong, 2008; Zhang, Yang,& Zhao, 2013; Zhao, Duan, & Wen, in press; Zhao, Duan, Wen, &Zhang, 2013) or adding a power integrator technique (Du, Li, & Lin,2013; Ou, Du, & Li, 2012) for second-order MAS. However, noneof these works takes into account the presence of disturbances inthe agents’ dynamics except the works (Du, Li, & Qian, 2011; Li,Du, & Lin, 2011), where only the approximate consensuses can bereached.

When the disturbances are unknown but bounded, an efficientapproach is SMC. Through eliminating the reaching phase, adiscontinuous integral sliding mode control (ISMC) possesses therobustness during the entire system responseprocess (Laghrouche,Plestan, & Glumineau, 2007; Sun, Li, & Sun, 2013; Zong, Zhao, &Zhang, 2010). Furthermore, finite-time ISMC has been formulatedfor a single system with disturbances by combining a continuoushomogeneous finite-time control and a discontinuous control(Defoort, Floquet, Kokosy, & Perruquetti, 2009) or a super-twistingcontrol (Chalanga, Kamal, & Bandyopadhyay, 2013; Defoort, Nollet,Floquet, & Perruqetti, 2009).

Through treating the closed-loop nominal MAS controlled bya continuous homogeneous finite-time consensus protocol as thedesired trajectories, the finite-time consensus problem of second-order MAS subject to bounded disturbances is investigated in theframework of ISM. The main contributions are reflected as fol-lows: (1) Compared with the discontinuous terminal sliding mode(TSM) protocols, although the singularity problem of TSMC can beavoided by adopting nonsingular TSM (Feng, Yu, & Man, 2002), thezero velocity states really decelerate and complicate the reachingphase even though they are not attractors, the proposed discontin-uous finite-time ISM consensus protocol possesses the completerobustness through eliminating the reaching phase and rejectingthe disturbances completely. Furthermore, because the fractional-power items of ISM are wrapped in the integral operation, thefirst time derivative of ISM does not produce any negative frac-tional power, which means the singularity issue obsessing TSMCdoes not exist in this scheme any more. (2) Compared with thecontinuous homogeneous finite-time consensus protocols, whereonly the nominal agent model is considered, through append-ing a continuous super-twisting control (Moreno & Osorio, 2012),finite-time consensus of MAS with disturbances is achieved by acontinuous ISM protocol, and the complete disturbance-rejectingproperty is inherited after finite divergence time. (3) Comparedwith the adding a power integrator technique, where the con-sensus protocols are constructed through a complicated processbecause the finite-time consensus and disturbance-rejecting prob-lems are handled together, the proposed approach is easier to bedesigned and implemented because the two problems are sepa-rately dealt with in the process of protocol design.

The rest of the paper is organized as follows: Section 2introduces some necessary preliminaries and formulates theproblem to be addressed. Section 3 investigates the finite-timeconsensus tracking problemof leader-following second-orderMASwith disturbances under the ISM framework. Numerical examplesare provided to verify the theoretical analysis in Section 4. Finally,Section 5 concludes the paper.Notation: Given a vector x = [x1, x2, . . . , xN ]T , α ∈ R and the signfunction sgn(·), define sgn(x) = [sgn(x1), sgn(x2), . . . , sgn(xN)]T ,

|x|α = [|x1|α , |x2|α , . . . , |xN |α]T , sigα(xi) = |xi|α sgn(xi), i = 1,

2, . . . ,N , sigα(x) = [sigα(x1), sigα(x2), . . . , sigα(xN)]T and diag(x)as the diagonal matrix of a vector x. ∥x∥1 =

Ni=1 |xi| , ∥x∥2 =

√xT x and ∥x∥∞ = maxi=1,...,N{|xi|} denote 1-norm, Euclidean

norm and infinity norm of the vector x respectively, and a basic re-lationship among these norms is ∥x∥∞ ≤ ∥x∥2 ≤ ∥x∥1 , ∀x ∈ RN .

Denote 1N = [1, 1, . . . , 1]T ∈ RN . Given a matrix P , λmax(P)and λmin(P) represent its maximum andminimum eigenvalues re-spectively. If P is symmetric, P > 0 implies that it is positivedefinite.

2. Preliminaries and problem formulation

2.1. Graph theory

Consider the MAS with N agents, the communication topologyamong the agents is modeled by a weighted undirected graph G =

(V, E, A)with a set ofN agentsV = {1, 2, . . . ,N}, a set ofM edgesE ⊆ V × V and a weighted adjacency matrix A =

aij ≥ 0

∈

RN×N . (i, j) ∈ E ⇔ aij = aji = 1, otherwise aij = aji = 0, aii = 0for all i ∈ V because of (i, i) ∈ E . Therefore, A is symmetric.The neighbor set of agent i is denoted by Ni = {j ∈ V : (i, j) ∈

E, j = i}. For the leader-followingMAS, another graphG associatesthe system consisting of N followers with the leader. The leaderadjacency matrix is defined as B = [b1, b2, . . . , bN ]T ∈ RN withthe adjacency element bi = 1 if agent i is a neighbor of the leader,otherwise bi = 0.

2.2. Finite-time stability

Lemma 1 (Bhat & Bernstein, 1998). Consider the following system

x = f (x, t), f (0, t) = 0, x ∈ U0 ⊂ Rn. (1)

Suppose that there is a continuous differentiable positive definitefunction V (x) defined in a neighborhood of the origin, and realnumbers c > 0, α ∈ (0, 1), such that V ≤ −cV α , then the originof the system is finite-time stable, and the upper bound of the settlingtime is satisfied with

T ≤V 1−α(0)c(1 − α)

. (2)

Lemma 2 (Hong et al., 2002). Suppose that the system (1) ishomogeneous of degree κ ∈ R with the dilation (r1, r2, . . . , rn), thefunction f (x, t) is continuous and x = 0 is asymptotically stable. Ifκ < 0, the equilibrium x = 0 is finite-time stable.

2.3. Homogeneous finite-time consensus

For the leader-following MAS of multiple double-integratorswith N followers and one leader,

xi = vi, vi = ui + di, i = 1, 2, . . . ,Nx0 = v0, v0 = 0

(3)

where xi, vi, di, ui ∈ R are the position, velocity, disturbance andcontrol input to be designed of the follower i respectively, andx0, v0 ∈ R are the position and velocity of the leader respectively.The dynamics of N followers can also be written in the vector form

x = v, v = u + d (4)

where x = [x1, x2, . . . , xN ]T , v = [v1, v2, . . . , vN ]T , d = [d1, d2,. . . , dN ]T and u = [u1, u2, . . . , uN ]T are the stack vectors ofxi, vi, di and ui respectively. When d = 0, the following finite-timeconsensus tracking result can be established.

Assumption 1. In the MAS (3), the leader is globally reachable.

Lemma 3 (Guan et al., 2012). Suppose that Assumption 1 holds anddi = 0, the consensus tracking can be achieved in finite time T > 0

160 S. Yu, X. Long / Automatica 54 (2015) 158–165

by the homogeneous protocol

unomi = −

Nj=1

aijsigα1(xi − xj) + bisigα1(xi − x0)

−

Nj=1

aijsigα2(vi − vj) + bisigα2(vi − v0)

(5)

where unomi = ui, α1 ∈ (0, 1) and α2 =

2α11+α1

, i.e.,

limt→T

|xi(t)| = 0, limt→T

|vi(t)| = 0

xi(t) = vi(t) = 0, ∀t ≥ T , i = 1, 2, . . . ,N(6)

where xi(t) = xi(t)−x0(t) and vi(t) = vi(t)−v0(t) are the trackingerrors between the follower i and the leader.

Proof. If we choose ϕk(x) = x for k = 1, 2, 3, 4, pi = 0 and bifor i = 1, 2, . . . ,N guarantee the leader is globally reachable, theproof procedure is similar with Theorem 1 in Guan et al. (2012). Soit is omitted here.

2.4. Problem formulation

This paper aims at utilizing the nominal finite-time consensusprotocol (5) to develop a novel protocol such that the accurateconsensus tracking (6) can also be achieved in finite time, whenthe dynamics of each follower in (3) is affected by the boundeddisturbances satisfying Assumption 2 or 3:

Assumption 2. In the followers’ dynamics (4), there exists aconstant l > 0, such that ∥d∥∞ ≤ l.

Assumption 3. In the followers’ dynamics (4), there exists aconstant η > 0, such that ∥d∥∞ ≤ η.

3. Homogeneous finite-time ISM consensus

Based on the nominal closed-loop finite-time consensusdynamics (3) and (5), the ISMs can be designed as

si = vi − vi(0) +

t

0−unom

i dτ = 0, i = 1, 2, . . . ,N (7)

where vi(0) is the initial value of vi. If the MAS keep sliding on theISM, i.e., si = si = 0, si = 0 means

vi = unomi , i = 1, 2, . . . ,N (8)

which is the nominal finite-time consensus dynamics.Note that (7) and (8) can be written in the vector form

s = v − v(0) +

t

0−unomdτ = 0 (9)

v = unom (10)

respectively, where s = [s1, s2, . . . , sN ]T , v(0) = [v1(0), v2(0),

. . . , vN(0)]T and unom= [unom

1 , unom2 , . . . , unom

N ]T .

3.1. Discontinuous ISM consensus protocol

Theorem 1. Suppose that Assumptions 1 and 2 hold. For the MAS(3) and the ISM (7), the discontinuous protocol

ui = unomi + udis

i

= unomi − ksgn(si), i = 1, 2, . . . ,N (11)

or in the vector form

u = unom+ udis

= unom− ksgn(s) (12)

where unomi is (5), udis

i = −ksgn(si), udis= −ksgn(s), k > l is a

constant, can guarantee that the MAS (3) are kept on the ISM (7) fromthe initial time in spite of the disturbances d, i.e., the desired finite timeconsensus dynamics (8) is invariant.

Proof. Firstly, taking the first time derivative of ISM function in (9)and applying the protocol (12) yield

s = −ksgn(s) + d. (13)

Then choosing Lyapunov function as V =12 s

T s and taking the firsttime derivative of V give

V = sT s = sT (−ksgn(s) + d)= −k∥s∥1 + sTd ≤ −k∥s∥1 + l∥s∥1

= −(k − l)∥s∥1 ≤ −(k − l)∥s∥2

= −(k − l)√2V

12 (14)

where k − l > 0 because of k > l. On the basis of Lemma 1, theISM (9) for the MAS (3) with the protocol (12) is finite-time stable.Because the ISM (9) starts on it at the initial time, this means thatthe MAS will not go away from the ISM in the sequential time, i.e.,s = s = 0 for t ≥ 0. Moreover, according to the ISM (9), s = 0means that the nominal finite-time consensus dynamics (10) ismaintained from the initial time. According to Lemma 3, the sameaccurate finite-time consensus tracking (6) can be achieved in spiteof the disturbances.

Remark 1. From (13), on the ISM (9), s = s = 0 for t ≥ 0 meansthat −ksgn(0) + d = 0, i.e., the disturbance is compensated bythe equivalent value [ksgn(0)]eq of discontinuous control actionksgn(s), which can also been viewed as a disturbance observer, i.e.,ksgn(s) = d.

3.2. Continuous super-twisting ISM consensus protocol

Lemma 4 (Li et al., 2011). For xi ∈ R, i = 1, 2, . . . , n, α ∈ (0, 1],then (

ni=1 |xi|)α ≤

ni=1 |xi|α .

Theorem 2. Suppose that Assumptions 1 and 3 hold, for the MAS(3) and the ISM (7), the continuous protocol

ui = unomi + ust

i

usti = −k1sig

12 (si) + ρi

ρi = −k2sgn(si), i = 1, 2, . . . ,N

(15)

or in the vector form

u = unom+ ust

ust= −k1sig

12 (s) + ρ

ρ = −k2sgn(s)

(16)

where unomi is (5), ρ = [ρ1, ρ2, . . . , ρN ]

T and ust= [ust

1 , ust2 , . . . ,

ustN ]

T is the super-twisting control with k1 > 0, k2 > η, can guaranteethat the ISM (7) is resumed in finite divergence time, and then theMASsequentially slide along it to reach the accurate consensus tracking infinite time.

Proof. Firstly, taking the first time derivative of ISM function in (7)and applying the protocol (15) yield

si = usti + di, i = 1, 2, . . . ,N (17)

S. Yu, X. Long / Automatica 54 (2015) 158–165 161

i.e.,

si = −k1sig12 (si) + ρi + di

ρi = −k2sgn(si), i = 1, 2, . . . ,N.(18)

Let zi = ρi + di, (18) can be rewritten as

si = −k1sig12 (si) + zi

zi = −k2sgn(si) + di, i = 1, 2, . . . ,N.(19)

The Lyapunov function candidate is chosen as V =N

i=1 Vi =Ni=1 ξ T

i Piξi, where Pi ∈ R2×2 > 0 and ξi =

sig

12 (si)zi

, which is

absolutely continuous (AC) but not locally Lipschitz on the set φ =

{(si, zi) ∈ R2|si = 0, i = 1, 2, . . . ,N}, due to the terms includ-

ing sig12 (si), i = 1, 2, . . . ,N . This violates the classical Lyapunov

theoremwhich requires the Lyapunov function is continuously dif-ferentiable, or at least locally Lipschitz. Thanks to Zubov theorem,which only requires a Lyapunov function is continuous, V (ξi) canstill be used for the stability analysis. However, here it is necessaryto carefully check V (ϕ(t, s(0), z(0))) is AC along the trajectoriesϕ(t, s(0), z(0)) = [ϕ1(t, s1(0), z1(0)), . . . , ϕN(t, sN(0), zN(0))]Tof the system (19), because the composition of two AC functionsh◦g is not always AC. The checking process is similar withMorenoand Osorio (2012), so it is omitted here. The first time derivative ofξi can be derived as:

ξi =

12|si|−

12 si

zi

=

12|si|−

12 [−k1sig

12 (si) + zi]

−k2sgn(si) + di

=12|si|−

12

−k1sig

12 (si) + zi

−2[k2 − disgn(si)]sig12 (si)

= |si|−12 Aiξi (20)

where Ai =

−

12k1

12

−[k2 − disgn(si)] 0

. Then the first time derivative of

V along the trajectories of system (19) is

V =

Ni=1

Vi =

Ni=1

|si|−12 ξ T

i (ATi Pi + PiAi)ξi (21)

almost everywhere. Since the matrix Ai is Hurwitz if and only ifk1 > 0, k2 > η (Moreno & Osorio, 2012) and Pi is positive definite,one has

V =

Ni=1

Vi = −

Ni=1

|si|−12 ξ T

i Qiξi < 0 (22)

where Pi and Qi are related by the algebraic Lyapunov equation

ATi Pi + PiAi = −Qi, i = 1, 2, . . . ,N (23)

for everyQi > 0, there exists a unique solution Pi > 0, so thatVi is astrict Lyapunov function. Since Vi(ϕi(t, si(0), zi(0))) is AC and Vi isnegative definite almost everywhere the differential equation (20)is satisfied, it follows that Vi(ϕi(t, si(0), zi(0))) is a monotonicallydecreasing function. In light of |si|

12 = |sig

12 (si)| ≤ ∥ξi∥2 ≤

λ−

12

min(Pi)V12i , the inequality (22) can be further deduced to

V ≤ −

Ni=1

λ12min(Pi)V

−12

iλmin(Qi)

λmax(Pi)Vi

≤ −β

Ni=1

V12i (24)

Fig. 1. The topologies.

where β = mini=1,...,N{λ

12min(Pi)λmin(Qi)

λmax(Pi)} > 0. Based on Lemma 4, the

inequalityN

i=1 V12i ≥

Ni=1 Vi

12is satisfied. Then from the in-

equality (24), we have

V ≤ −βV12 . (25)

It follows that V (ϕ(t, s(0), z(0))) ≤ −βV12 (ϕ(t, s(0), z(0))).

Moreover, based on Lemma 1, from (25), we can get V (ϕ(t, s(0),z(0))) and ϕ(t, s(0), z(0)) converge to zero in finite time smallerthan T =

2βV

12 (s(0), z(0)). Because s(0) = 0 but z(0) = 0 means

V (s(0), z(0)) = 0, the system trajectory will leave the ISM firstlyand slide along it again after finite time. According to Lemma 3, theaccurate finite-time consensus tracking is achieved in spite of thedisturbances.

Remark 2. Note that when z = ρ + d = 0 is reached in finitetime, the−ρ item in the super-twisting control ust (16) can be con-sidered as a finite-time disturbance observer, i.e.,

t0 k2sgn(s)dτ =

d, t > T .

Remark 3. As proved in the Theorem 2 in Guan et al. (2012), thenominal finite-time consensus protocol unom

i in (5) is valid forthe switching topologies if Assumption 1 always holds for everytopology. Therefore, the proposed scheme is also feasible for thiscase when the disturbances are considered.

4. Simulations

The same MAS used in Guan et al. (2012) except di = 0 areselected as given in Fig. 1, where the solid lines represent thelinks among the followers and the dotted lines represent the linksbetween the followers and the leader. In all the simulations, α1 =12 and α2 =

23 are chosen for the nominal protocol unom

i in (5), thedisturbances are chosen as d1 = cos(0.1t), d2 = 0.5sin(0.5t +

π4 ),

d3 = 0.6cos(3t), d4 = 0.8sin(2t +π3 ) and d5 is a time-varying

frequency sinusoidal signal as

d5 =

0.2 sin(ω1t) − 0.2 t < 300.2 sin(ω2t) + 0.2 t ≥ 30

where ω1 = 2π( 5.960 t + 0.1) and ω2 = 2π(− 5.9

60 t + 6). Hence,∥d∥∞ = 1, ∥d∥∞ = 3.83, the dotted lines in Figs. 2–5(a) and (b)represent the position and velocity of the leader respectively, thedotted lines in Fig. 4(e) represent the disturbances to be estimated,and the solid lines in Figs. 2–5 represent the positions, velocities,ISM variables, consensus protocols, disturbance estimations andestimation errors of the followers with the different colorsrespectively, but with the same color for the same follower.

For brevity, we firstly consider the MAS (3) under the topologyG1 only.

Case 1: Only approximate consensus tracking shown in Fig. 2(a)and (b), i.e., the MAS evolve in the neighborhood of ISM (9) asshown in Fig. 2(c), can be achieved by the continuous nominalprotocol (5) shown in Fig. 2(d).

162 S. Yu, X. Long / Automatica 54 (2015) 158–165

Fig. 2. Consensus tracking by the nominal protocol. (For interpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

Fig. 3. Consensus tracking by the discontinuous ISM protocol. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)

S. Yu, X. Long / Automatica 54 (2015) 158–165 163

Fig. 4. Consensus tracking by the continuous ISM protocol. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)

Case 2: The accurate nominal finite-time consensus trackingin Guan et al. (2012) as shown in Fig. 3(a) and (b), i.e., the MASalways evolve on the ISM (9) shown in Fig. 3(c), is achieved by thediscontinuous protocol (12) with k = 1.1 as shown in Fig. 3(d), butthe cost is the undesirable chattering of the protocol.

Case 3: The accurate finite-time consensus tracking is recoveredin finite-time as shown in Fig. 4(a) and (b), i.e., the ISM is resumedin finite divergence time as shown in Fig. 4(c), by the continuousprotocol (15) with k1 = 1, k2 = 4 shown in Fig. 4(d). The protocolis chattering-free and the ρ item can be viewed as the finite-timedisturbance observer as shown in Fig. 4(e) and (f).

Actually, the proposed protocols are also valid for the switchingtopologies.

Case 4: For the space limit, only the case 3 is considered exceptthat the topology is switched among G1, G2, G3, G4 sequentiallywith the switching period τ = 10, which is samewith the Example2 in Guan et al. (2012), the continuous ISM protocol (15) can still

achieve accurate consensus tracking in finite time as shown inFig. 5 in spite of the disturbances and topology switching.

5. Conclusions

Under the ISMC framework, the finite-time consensus prob-lem of second-order MAS with disturbances is investigated by acomposite protocol of homogeneous finite-time consensus proto-col for the nominal performance anddiscontinuous slidingmodeorcontinuous super-twisting protocol for the disturbance-rejection.Further investigations on this approach can be performed on thefollowing aspects: When the velocity information are unavailable,finite-time observer should be designed to reconstruct the infor-mation from the position measurements; When only the relativestate information are available, the ISM (9) should be reformu-lated because the absolute velocity information are unavailable;The above results in second-order MAS are extended to the high-order integrator-chain or general MAS with disturbances.

164 S. Yu, X. Long / Automatica 54 (2015) 158–165

Fig. 5. Same with Fig. 4 except the switching topologies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

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Shuanghe Yu received his Bachelor’s degree in AutomaticControl form Beijing Jiaotong University, and Master’sdegree in Control Theory and Applications and Ph.D.degree in Navigation, Guidance and Control both fromHarbin Institute of Technology, China, in 1990, 1996and 2001, respectively. From 2001 to 2003, he hasa postdoctoral research fellow at Central QueenslandUniversity, Australia. From 2003 to 2004, he has a researchfellow at Monash University, Australia. Since the autumnof 2004, he has been a Professor in the Department ofAutomation, Dalian Maritime University, China. His main

research interests include nonlinear control theory and applications in robot, ACdrive and other industrial process.

Xiaojun Long received his B.S. in Automation Controland M.S. degree in Control Theory and Applicationsfrom Dalian Maritime University, Dalian, China in 2009and 2011, respectively. He is now a Ph.D. candidateof the School of Information Science and Technology,Dalian Maritime University. His research interests includenonlinear control theory and finite-time consensus ofmulti-agent systems.