Physica A 388 (2009) 193–208

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Physica A

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Leader-following consensus problem with a varying-velocity leader andtime-varying delaysKe Peng ∗, Yupu YangDepartment of Automation, Shanghai Jiao Tong University, 800# Dongchuan road, Shanghai, 200240, People’s Republic of China

a r t i c l e i n f o

Article history:Received 23 April 2008Received in revised form 9 July 2008Available online 17 October 2008

Keywords:Multi-agent systemConsensusLeader-followingTime-varying delayLyapunov–Krasovskii functional

a b s t r a c t

In this paper, we study a leader-following consensus problem for a multi-agent systemwith a varying-velocity leader and time-varying delays. Here, the interaction graph amongthe followers is switching and balanced. At first, we propose a neighbor-based rule forevery agent to track a leader whose states may not be measured. In addition, we considerthe convergence analysis of this multi-agent system under two different conditions: theconnection between the followers and the leader is time-invariant and time-varying. Forthe first case, a novel decomposition method is introduced to facilitate the convergenceanalysis. By utilizing a Lyapunov–Krasovskii functional, we obtain sufficient conditions foruniformly ultimately boundedness of the tracking errors. Finally, two simulations are alsopresented to illustrate our theoretical results.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Recently, themulti-agent distributed coordination problemhas attractedmany researchers of a broad range of disciplinesincluding biology, physics, computer science and control engineering. This is partly because of broad applications in manyareas such as aggregation behavior of animals in Refs. [1–3], collective motion of particles in Refs. [4–6], distributedcomputations in Ref. [7] and formation control in Refs. [8–15].In the research of multi-agent systems[16], the main challenge is how to design simply control rules including neighbor-

based rules [4–6,8–11,15,17] for simple agents (with limited computing power and information interaction capability) toachieve a prescribed group behavior. In 1986, Reynolds first proposed a computer animation model to simulate collectivebehaviors of multiple agents in Ref. [17]. Vicsek et al. proposed and analyzed a neighbor-based swarm model in Ref. [4].In addition, there has been a great amount of renewed interest in self-organized group with leaders. Mu et al. studied thecollective dynamics of a group of motile particles with a leader in Ref. [18]. For leader–follower networks, a simple yetgeneralmethod is described in Ref. [19]. Jadbabaie et al. considered the coordination of a group ofmobile autonomous agentsfollowing an a actual leader in Ref. [20]. Olfati-Saber introduced a theoretical framework to design a flocking algorithmwithvirtual leader/following architecture in Ref. [9]. In most of the current relevant works, the states of leader are known. In fact,some variables of the leader in a multi-agent system may not be measurable. Hong et al. considered a consensus problemwith an active leader and switching undirected topologies in Ref. [10]. In Ref. [21], Ren proposed and analyzed consensustracking algorithms under a directed fixed information exchange topology and only a few agents can obtain a time-varyingconsensus reference state. Moreover, in Ref. [22], Ren also extended the results on consensus tracking algorithms in Ref. [21]to the case where there appears switching interaction topology and actuator saturation.On the other hand, besides the variation of information interaction topology, time-delay is another key factor influencing

the stability of the multi-agent system. Olfati-Saber first considered consensus problem with fixed undirected networks

∗ Corresponding author. Tel.: +86 21 34204261; fax: +86 21 34204427.E-mail address: Kpeng1979@yahoo.com.cn (K. Peng).

0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.10.009

194 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

and time-delays in Ref. [8]. Based on a reduced-order Lyapunov–Krasovskii functional, Lin et al. studied average consensusproblem with switching directed topology in Ref. [15]. Hu et al. considered a leader-following consensus problem withswitching directed topologies in Ref. [11]. However, for a fraction of agents which are sometimes connected with the leader,the states of the leader with a constant velocity are entirely known.In this paper, we consider a leader-following consensus problem for a multi-agent systemwith a varying-velocity leader

and time-varying delays. Here, the velocity of the leader cannot be measured and every follower can obtain the position ofthe leader only when they are connected with the leader. However, since the velocity of the leader cannot be measured, thereduced-order Lyapunov–Krasovskii functional in Ref. [15] cannot be extended to our case. We will consider two differentcases: The leader adjacency matrix which is used to represent the connection between the followers and the leader istime-invariant and time-varying. For the first one, we introduce a novel decomposition method which decomposes thegroup tracking dynamics into two subsystems. Then by utilizing a Lyapunov–Krasovskii functional, we derive a sufficientcondition for uniformly ultimately boundedness of the tracking errors; While for the second case, we directly use theLyapunov–Krasovskii functional and get a more conservative sufficient condition for uniformly ultimately boundedness.The rest of this paper is organized as follows: in Section 2, we provide some preliminaries and present the leader-

following multi-agent model. In Section 3, we analyze the stability of this model and present two sufficient conditionsuniformly ultimately boundedness of the tracking errors. To illustrate our theoretical results, some simulations are providedin Section 4. Finally, some conclusions are drawn in Section 5.By convention, R and Z+ represent the real number set and the positive integer set, respectively; In is a n × n identity

matrix. For any vector x, xT denotes its transpose and its norm ‖x‖ is the Euclidean norm. For a matrix P ∈ Rn×n, the norm‖P‖ = max

{√λ : λ is an eigenvalue of PTP

}.

2. Problem statement

In general, information exchange between agents in amulti-agent system can bemodeled by directed/undirected graphs[8,23]. Before we proceed, therefore, some basic concepts on graph theory are provided as below.Let G = (V, E,A) be a weighted digraph (or directed graph) of order n with the set of vertices V = {v1, v2, . . . , vn},

set of directed edges E ⊆ V × V , and a weighted adjacency matrix A = [aij]. The node indices belong to a finite setI = {1, 2, . . . , n}(n ∈ Z+). An edge of G is denoted by eij = (vi, vj). The adjacency elements associated with the edges ofthe graph are positive, i.e. eij ∈ E ⇔ aij > 0. In this case, we say node j is a neighbor of node i and define the set of labels ofthose agents by Ni(t)which are out-neighbors of agent i (i = 1, . . . , n) at time t , that is

Ni = {vj ∈ V : (vi, vj) ∈ E}. (1)

The in-degree and out-degree of node vi are, respectively, defined as

din(vi) =n∑j=1

aji, dout(vi) =n∑j=1

aij. (2)

We say a node vi of a digraph G = (V, E, A) is balanced if and only if its in-degree and out-degree are equal, i.e.dout(vi) = din(vi). A digraph G = (V, E, A) is called balanced if and only if all of its nodes are balanced, or

∑j aij =

∑j aji,

for i = 1, . . . , n. The degree matrix of G is D = diag{d1, d2, . . . , dn} ∈ Rn×n in which di =∑j∈Niaij for i = 1, . . . , n, then

the Laplacian of the weighted digraph G is defined as L = D− Awhich is generally not symmetric. By definition, every rowsum of L is 0 and the Laplacian matrix has a zero eigenvalue corresponding to a right eigenvector 1 = (1, 1, . . . , 1)T ∈ Rn.A digraph G is called strongly connected (SC) if and only if any two distinct nodes of the graph can be connected via a pathwhich follows the direction of the edges of the digraph G. A directed path is a sequence of edges in a directed graph of theform (i1, i2), (i2, i3), . . . , (if−1, if ), where j = 1, . . . , f ∈ Z+, ij ∈ V and (ij, ik) ∈ E . There exists a directed path from node ito node j, then node j is said to be reachable from node i. For any node i, if there is a path from every node in digraph G to it,then node i is called a globally reachable node in G.Here, we consider a leader-following consensus problem for a multi-agent system in which there are n followers and

a leader. we denote the state of follower i by xi for i = 1, . . . , n and the information exchange between all followerscan be described by directed graph G. In addition, directed graph G is used to model information exchange among nfollowers and the leader. In fact, G includes n followers (related to graph G) and a leader (labeled n + 1) with directededges from some agents to the leader. Moreover, the information topology G is time-varying. Suppose that there is aninfinite sequence of bounded, nonoverlapping, contiguous time-intervals [ti, ti+1), i = 0, 1, . . . , starting at t0 = 0. DenoteG = {G1,G2, . . . ,GN} as a set of the graphs with all possible topologies, which includes all possible interconnection graphs(involving n followers and a leader) and denoteP = 1, 2, . . . ,N as its index set.Since the interconnection topology is variable, we define a piecewise-constant switching signal σ : [0,∞) → P.

Therefore, Lp(p ∈ P) is time-varying (switched at ti, i = 0, 1, . . .), but it is time-invariant in any interval [ti, ti+1). In thefollowing, we suppose that there exist fixed positive constants aij, bi (i, j = 1, . . . , n) such that:

aij(t) ={aij, follower j is a neighbor of follower i at t,0, otherwise (3)

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 195

and

bi(t) ={bi, follower i is connected to the leader at t,0, otherwise.

(4)

All the considered followers in this paper move in anm-dimensional space

xi = ui ∈ Rm, i = 1, . . . , n (5)

where ui is the control input. A leader is described by a double integrator of the form:{x0 = v0v0 = a(t) = ao(t)+ δ(t), x0, v0, δ ∈ Rmy = x0

(6)

where y(t) = x0(t) is the measured output and a(t) is the input. In addition, its velocity keeps changing and cannot bemeasured.In our problem description, the input a0(t) is known and δ(t) is unknown but bounded with a given upper bound δ

(that is, ‖δ(t)‖ ≤ δ). The input a(t) is known if and only if δ = 0. On the other hand, y = x0 is the only variable that canbe obtained directly by the followers when they are connected to the leader. Since v0(t) cannot be measured, we have toestimate v0(t) only by using the information obtained from its neighbors in a decentralized way. The estimate of v0(t) byfollower i is denoted by vi(t) (i = 1, 2, . . . , n). In addition, due to time-varying delays, each follower cannot instantly get theinformation from others or the leader. Thus, a neighbor-based coupling rule which consists of two parts can be expressedas follows:

• a neighbor-based rule:

ui = k

[ ∑j∈Ni(t)

aij(t)(xi(t − τ)− xj(t − τ))+ bi(t)(xi(t − τ)− x0(t − τ))

]+ vi(t)

−αγ k

[ ∑j∈Ni(t)

aij(t)(vi(t − τ)− vj(t − τ))

], i = 1, . . . , n (7)

where 0 < γ < 1,α > 0, k > 0.Ni(t) denotes the set of out-neighbors of follower i. The time-delay τ(t) is a continuouslydifferentiable function with

τ(t) < d1, τ ≤ d2 < 1; (8)

• a neighbor-based dynamics system to estimate v0

vi = a0 − γ k

[ ∑j∈Ni(t)

aij(t)(xi(t − τ)− xj(t − τ))+ bi(t)(xi(t − τ)− x0(t − τ))

]

−αk

[ ∑j∈Ni(t)

aij(t)(vi(t − τ)− vj(t − τ))

], i = 1, . . . , n. (9)

Remark 1. Note that ui in (8) and vi in (9) both depend on the information from its out-neighbors and itself. In the specialcase that time-delay τ = 0 and α = 0, algorithms (8) and (9) are equivalent to algorithms (3) and (4) in Ref. [10].

By taking x = (x1, . . . , xn)T, v = (v1, . . . , vn)T, (5) and (9) can be written in a matrix form:{x = −k(Lσ + Bσ )

⊗Imx(t − τ)+ kBσ1

⊗x0(t − τ)+ v(t)− αγ kLσ

⊗Imv(t − τ)

v = 1⊗a0 − γ k(Lσ + Bσ )

⊗Imx(t − τ)+ γ kBσ1

⊗x0(t − τ)− αkLσ

⊗Imv(t − τ)

(10)

where⊗denotes the Kronecker product, σ : [0,∞)→ P = {1, 2, . . . ,N} is a piecewise-constant switching signal with

successive switching times, Lσ is the Laplacian for the n followers, the leader adjacency matrix Bσ is an n × n diagonalmatrix whose ith diagonal element is bi(t) at time t and is utilized to represent the connections between the followers andthe leader. Furthermore, denote x = x− 1

⊗x0 and v = v − 1

⊗v0. Due to Lσ

⊗Imx = Lσ

⊗Imx, we can obtain an error

dynamics of (9) as follows:

ε(t) = Mε(t)+ Eσ ε(t − τ)+ g, (11)

196 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

where

ε =

[xv

]∈ R2mn, Eσ =

[−k(Lσ + Bσ ) −αγ kLσ−γ k(Lσ + Bσ ) −αkLσ

]⊗ Im

g =[

0mn×1−1

⊗δ

]∈ R2mn, M =

[0n In0n 0n

]⊗ Im.

(12)

When every follower can be connected with the leader for all time, which means that bi(t) = b, t < 0, i = 1, . . . , n, that is,Bσ = bIn, we can obtain that the error dynamics which has the same form as (11) and (12), except that

Eσ =[−k(Lσ + bIn) −αγ kLσ−γ k(Lσ + bIn) −αkLσ

]⊗ Im. (13)

Remark 2. Although we also consider the case when the coupling topology is balanced, the reduced-order Lyapunov–Krasovskii functional cannot be employed here. This is because both

∑i∈I x(t) = 0 and

∑i∈I v(t) = 0 do not hold for

all t ≥ 0.

3. The stability analysis

In this section, the convergence analysis of system (11) is given for the consensus problem of multi-agent system (10). Ifthe information of the input a(t) can be used in local control design,we canprove that although the leader keeps changing, allthe agents can follow the leader, that is, consensus is achieved. If not, we can also get some estimation of the tracking errors.We need to consider two cases: (1) every follower can be connected to the leader for all time, that is, the leader adjacencymatrix Bσ = bIn is time-invariant; (2) only some followers can obtain the states of the leader, that is, Bσ is time-varying.For the first case, we propose a novel decomposition method and decompose the error dynamics denoted by (11)

into two subsystems: center tracking dynamics describing the tracking error dynamics of the center of all agents andcohesion dynamics representing relative dynamics of the tracking errors of individual agents. However, for the second case,this decomposition method cannot make the resulting two subsystems entirely decoupled and will result in additionalconservation of the upper bound of the time-delay. So, we will directly use Lyapunov–Krasovskii functional to analyze theconvergence property of error dynamics in (11) when only some followers can be connected to the leader.

3.1. The leader adjacency matrix is time-invariant

We first define a coordinate transformation

z = I2 ⊗ S ⊗ Imε (14)

where S ∈ Rn×n is the (full-rank) transformation matrix defined by

S :=

1n

1n

1n· · ·

1n

1 −1 0 · · · 01 0 −1 · · · 0...

. . .. . .

. . ....

1 · · · · · · 0 −1

(15)

and z := [z1, z2, . . . , z2n]T ∈ R2mn is the transformed coordinate. Define zox := z1 ∈ Rm, zex := [z2, . . . , zn] ∈Rm(n−1), zov := zn+1 ∈ Rm, zev = [zn+2, . . . , z2n] ∈ Rm(n−1). We can show that zox, zov , zex and zev are given by

zox =1n

n∑i=1

xi =1n

n∑i=1

xi − x0,

zov =1n

n∑i=1

vi =1n

n∑i=1

vi − v0,

(16)

{zex =

[x1 − x2, x1 − x3, . . . , x1 − xn

]T,

zev =[v1 − v2, v1 − v3, . . . , v1 − vn

]T.

(17)

Using (15), we can rewrite the error dynamics (11) as follows:

I2 ⊗ (S−TS−1)⊗ Imz = (I2 ⊗ S−1 ⊗ Im)TM(I2 ⊗ S−1 ⊗ Im)z(t)

+ (I2 ⊗ S−1 ⊗ Im)TEσ (I2 ⊗ S−1 ⊗ Im)z(t − τ)+ (I2 ⊗ S−1 ⊗ Im)Tg (18)

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 197

where the inverse of S in (15) is given as

S−1 =

11n

1n

· · ·1n

11n− 1

1n

· · ·1n

11n

1n− 1 · · ·

1n

.... . .

. . .. . .

...

11n

1n

· · ·1n− 1

. (19)

Then, using (19),we can show that

S−TS−1 =[

n 01×(n−1)0(n−1)×1 I

](20)

where I ∈ R(n−1)×(n−1) is a symmetric and positive-definitive matrix. In addition, since (2), the transformed informationgraph has the following structure:

S−TLσ S−1 =[

0 01×(n−1)0(n−1)×1 Lσ

](21)

where the ijth element of the matrix Lσ ∈ R(n−1)×(n−1) is

Lijσ = L(i+1),(j+1)σ , for all i, j = 1, . . . , n− 1. (22)

Consequently, the error dynamics (11) can be decomposed as two subsystems: center tracking dynamics denoted asfollows:

zo = Mozo(t)+ Eozo(t − τ)+ go, go =[0m×1δ(t)

]∈ R2m (23)

where

Mo =[0 10 0

]⊗ Im, Eo =

[−kb 0−γ kb 0

]⊗ Im (24)

and cohesion dynamics

ze = Meze + Eeσ ze(t − τ) (25)

where

Me =[0n−1 In−10n−1 0n−1

]⊗ Im,

Eeσ =[−k(bIn−1 + I−1Lσ ) −αγ kI−1Lσ−γ k(bIn−1 + I−1Lσ ) −αkI−1Lσ

]⊗ Im.

(26)

Thus, the error dynamics (11) is decomposed as center tracking dynamics and cohesion dynamics which are entirelydecoupled. Moreover, center tracking dynamics does not depend on the switches among Gp(p ∈ P).In the following, we will consider the convergence analysis of two subsystems (23) and (25). We first assume that the

interconnection graph is always balanced and SC, though the interconnection topology keeps changing. To analyze thesetwo subsystems, we first need to derive some properties of many matrices. For p ∈ P, denote Lsym,p = 1

2 (Lp + LTp) and

Mp1 = I−1Lsym,p. The next Lemma is given forMp1.

Lemma 1. If digraph Gp is balanced and SC, then the symmetric matrix Mp1 associated with Gp is positive definite.

Proof. We can show that S−TLsym,pS−1 = diag[0, Lsym,p], where Lsym,p = 12 (Lp + L

Tp). Since S

−1 is full-rank, which is acongruence transform form, thus, it preserves the signs of eigenvalues [24]. Also, if Gp is balanced and SC, Lsym,p has only oneeigenvalue at zero with all the others being strictly positive, since, in this case, Lsym,p is the Laplacian matrix of the (SC andundirected) mirror graph of Gp [8]. Therefore, Lsym,p is positive definite. Also since I is symmetric and positive definite,Mp1is positive definite. The proof is completed. �

198 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

Based on Lemma 1 and the fact that the setP is finite,

λ1 = min{eigenvalues ofMp1 ∈ R(n−1)×(n−1),∀Gp is connected} > 0.

λ2 = max{‖I−1Lp‖ ∈ R(n−1)×(n−1),∀Gp is connected} > 0.

(27)

Lemma 2. Assume that Qep = −[Pe(Eep+Me)+ (Eep+Me)TPe] and k > 14γ b(1−γ 2)

, then the smallest eigenvalue of Qep(p ∈ P)

is

µe = (1− γ 2)k[(1+ α)λ1 + b] + γ −√{(1− γ 2)k[(1+ α)λ1 + b] − γ

}2+ 1 > 0. (28)

Proof. By the definition of Qep, we can obtain that

Qep =[2k(1− γ 2)(bIn−1 +Mp) −In−1

−In−1 2γ In−1 + 2αk(1− γ 2)Mp

]⊗ Im. (29)

Let µij, i = 1, . . . , n − 1, j = 1, 2 denote the (at most) 2(n − 1) different eigenvalues of Qep ∈ R2m(n−1)×2m(n−1). Based onλi(Mp1), the eigenvalue ofMp1, we have 2(n− 1) eigenvalues in the following forms:

µi,1 = (1− γ 2)k[(1+ α)λi(Mp1)+ b] + γ +√∆e

µi,2 = (1− γ 2)k[(1+ α)λi(Mp1)+ b] + γ −√∆e

(30)

for i = 1, . . . , n − 1 and ∆e ={(1− γ 2)k[(1+ α)λi(Mp1)+ b] + γ

}− 4kγ (1 − γ 2)[(1 + α)λi(Mp1) + b] + 1. Since

4γ kb(1 − γ 2) > 1, we know that µi,2 > 0 and it increases as k or λi(Mp1) increases. So the smallest eigenvalue of Qp hasthe form in (28). This completes the proof. �

Remark 3. By taking the derivative ofµe with respect to α, it is easily obtained thatµe is a strictly increasing function of α.Therefore, when time-delay is not taken into consideration, increasing α can improve the stability of cohesion dynamics.

Lemma 3. For p ∈ P, we have

‖PeEe‖ = (1− γ 2)kω1, ‖Ee‖ =√1+ γ 2kω1 (31)

where ω1 = max{b+ λ2, αλ2

}.

Proof. By the definition of matrix norm, these results are easily obtained. �

To study the boundedness of center tracking dynamics, We first need to introduce the following results on theboundedness of solutions for general functional differential equations [25]. Let (C, |||.|||) denote the Banach space ofcontinuous functions ϕ : [−r, 0] → Rn with supremum norm: |||ϕ||| = sup−r≤θ≤0 ‖ϕ(θ)‖. We consider the followingsystem:{

x = f (t, xt)xt0 = ϕ

(32)

where xt(θ) = x(t + θ), θ ∈ [−r, 0], r > 0, f : R × C([−r, 0],Rn)→ Rn is a continuous function and f (t, 0) = 0 for allt ∈ R.

Definition 1. Let x(t) = x(t, t0, ϕ) be any solution of system (32), system (32) is said to be(1) uniformly bounded (U.B.), if for any α > 0, there exists β = β(α) > 0 such that for any t0 ∈ R, ‖ϕ‖ < α implies‖x(t)‖ < β for all t ≥ t0;

(2) uniformly ultimately bounded (U.U.B.), if there exists a positive constant B, for any α > 0 there exists a T = T (α) > 0such that for any t0 ∈ R, ‖ϕ‖ < α implies ‖x(t)‖ < B for all t ≥ t0 + T .

Lemma 4 ([26]). Assume that there exists a functional V (t, φ) : [0,∞) × C[−τ , 0] → R which is continuous with respect to(t, φ), together with continuous strictly increasing functions: Wi(r) : [0,∞) → [0,∞)(i = 1, . . . , 4) satisfying Wi(0) = 0and Wi(r)→∞ as r →∞. If

W1(‖x(t)‖) ≤ V (t, xt)

≤ W2(‖x(t)‖)+W3

(∫ t

t−τW4(‖x(s)‖)ds

)(33)

V(32)(t, xt) ≤ −W4(‖x(t)‖)+M (34)

for some constant M > 0, then (32) is uniformly bounded (U.B.) and uniformly ultimately bounded (U.U.B.).

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 199

Lemma 5. Assume that system (32) with x = [x1, . . . , xn] ∈ Rn is linear with respect to x(t) and x(t − τ) (τ is a boundedtime-delay), that is

x(t) = A1(t)x(t)+ A2(t)x(t − τ)+ I(t) (35)

where A1(t) and A2(t) are n×n time-varyingmatrices, ‖I(t)‖ ≤ Q (Q is a positive constant), then there exists a positive constantK such that∫ 0

−τ

∫ t

t+θxT(s)x(s)dsdθ ≤ K

∫ t

t−τxT(s)x(s)ds. (36)

Proof. For the given system (35) and any t ≥ 0, if∫ 0−τ

∫ tt+θ x

T(s)x(s)dsdθ > 0 and is bounded, then there at least exist somexi(ts)(i = 1, . . . , n) and some time-interval [tj, tk] ⊆ [t, t + τ ] such that

xi(ts) 6= 0, for ts ∈ [tj, tk]. (37)

Without loss of generality, assume that xi(ts) > 0 for ts ∈ [tj, tk], that is, xi(ts) is a strictly increasing function with respectto ts ∈ [tj, tk]. In this case, therefore,

x2i (ts) =

(xi(tj)+

∫ ts

tjxi(tθ )dθ

)2≥ 0 for ts ∈ [tj, tk] (38)

where the last equality holds at most for some xi(tp), tp ∈ [tj, tk]. Consequently, for any t ≥ 0,∫ t

t−τxT(s)x(s)ds ≥

∫ t

t−τx2i (s)ds ≥

∫ tk

tjx2i (ts)dts > 0. (39)

Hence, there exists a positive constant K satisfying

K := maxt≥0 and

∫ tt−τ x

T(s)x(s)ds>0

∫ 0−τ

∫ tt+θ x

T(s)x(s)dsdθ∫ tt−τ x

T(s)x(s)ds. (40)

In addition, there is a special case to consider: for t ≥ 0, if∫ tt−τ x

T(s)x(s)ds = 0, then∫ 0−τ

∫ tt+θ x

T(s)x(s)dsdθ = 0. To prove

this proposition, we first assume that if∫ tt−τ x

T(s)x(s)ds = 0, then∫ 0−τ

∫ tt+θ x

T(s)x(s)dsdθ > 0 holds. In this case, accordingto the previous proof, we can immediately obtain that

∫ tt−τ x

T(s)x(s)ds > 0 which leads to a contradiction. Hence, (36) alsoholds for this special case. This completes the proof. �

Theorem 1. Consider these two subsystems (23) and (25), when every follower can be connected to the leader for all time, forany fixed 0 < γ < 1, α > 0, λ1 defined in (27), if the switching interconnection graph keeps balanced and SC, we take a constant

k >1

4γ b(1− γ 2)(41)

and the time-varying delay τ is sufficiently small, then

limt→∞‖zo(t)‖ ≤ Co, lim

t→∞‖ze(t)‖ = 0 (42)

for some constant Co depending on δ, d1 and d2 (d1 and d2 defined in (8)). Moreover, if a(t) is known (δ = 0), thenlimt→∞ ‖zo(t)‖ = 0.

Proof. This proof consists of two parts.(1) To analyze the uniformly ultimately boundedness of center tracking dynamics (23), we define a Lyapunov–Krasovskii

functional as follows:

Vo = V1o + V2o + V3o, (43)

where

V1o = zTo (t)Pozo(t),

V2o =∫ t

t−τzTo (t)Sozo(t)ds,

V3o =∫ 0

−τ

∫ t

t+θzo(s)Rozo(s)dsdθ

(44)

200 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

where Po =[1 −γ−γ 1

]⊗ Im, So = βoI2m and Ro = I2m. Consider the time derivative of Vio (i = 1, 2, 3) along the solution of

(23) respectively:

V1o = zTo (t)(MToPo + PoMo)zo(t)+ 2z

To (t)Po[Eozo(t − τ)+ go] (45)

V2o = zTo (t)Sozo(t)− (1− τ )zTo (t − τ)Sozo(t − τ) (46)

V3o = −∫ t

t−τzTo (θ)Rozo(θ)dθ + τ z

To (t)Rozo(t). (47)

For any a, b ∈ Rn and any symmetric positive-definite matrixΦ ∈ Rn×n,

2aTb ≤ aTΦ−1a+ bTΦb (48)

and from the Leibniz–Newton formula zo(t − τ) = z(t)−∫ tt−τ z(s)ds, we can obtain that

2zTo (t)PoEozo(t − τ) = zTo (t)(E

ToPo + PoEo)zo(t)+ 2(E

ToPzo(t))

T∫ t

t−τzo(s)ds

≤ zTo (t)(EToPo + PoEo)zo(t)+ τ z

To (t)PoEoR

−1o E

ToPozo(t)+

∫ t

t−τzo(s)Rozo(s)ds. (49)

Consequently,

V1o ≤ zTo (t)[−Qo + τPoEoR−1o E

ToPo]zo(t)+

∫ t

t−τzTo (θ)Rozo(θ)dθ + 2zo(t)

TPogo (50)

where

Qo = −((Mo + Eo)TPo + Po(Mo + Eo)

)=

[2kb(1− γ 2) −1−1 2γ

]⊗ Im (51)

and its smallest eigenvalue is µo = kb(1− γ 2)+ γ −√[kb(1− γ 2)− γ ]2 + 1 > 0 (by virtue of (41)). Similarly,

V3o = −∫ t

t−τzTo (θ)Rozo(θ)dθ + τ(Mozo(t)+ Eozo(t − τ)+ go)

TRo(Mozo(t)+ Eozo(t − τ)+ go)

= 2τ[zTo (t)M

ToRoMozo(t)+ z

To (t − τ)E

ToRoEozo(t − τ)

]+ gToRogo

−

∫ t

t−τzTo (θ)Rozo(θ)dθ + 2τ [z

To (t)M

To + z

To (t − τ)E

To ]Rogo. (52)

From (46), (50) and (52), we can obtain that

Vo ≤ zTo (t)[−Qo + So + τ(2MToRoMo + PoEoR

−1o E

ToPo)

]zo(t)

+ zTo (t − τ)[−(1− τ )So + 2τEToRoEo

]zo(t − τ)

+2zTo (t)(Po + τMToRo)go + τ z

To (t − τ)E

ToRogo + g

ToRogo. (53)

Since ‖Mo‖ = 1, ‖PoEoR−1o EToPo‖ = k

2b2(1− γ 2)2 and ‖Eo‖ = kb√1+ γ 2, we have

zTo (t)[−Qo + So + τ(2MToRoMo + PoEoR

−1o E

ToPo)

]zo(t)+ 2zTo (t)(Po + τM

ToRo)go

≤[−µo + βo + τ(2+ k2b2(1− γ 2)2)

]‖zo(t)‖2 + 2 (1+ γ + d1) ‖zo(t)‖δ

≤ −[(1− ζ1o)µo − βo − τ(2+ k2b2(1− γ 2)2)

]‖zo(t)‖2 + C1o (54)

where 0 < ζ1o < 1, C1o =2(1+γ+d1)2

ζ1oµoδ2is a constant. If we choose time-delay τ such that

µo(1− ζ1o)− βo − τ(2+ k2b2(1− γ 2)2) > 0 (55)

then

zTo (t)[−Qo + So + τ(2MToRoMo + PoEoR

−1o E

ToPo)

]zo(t)+ 2zTo (t)(Po + τM

ToRo)go ≤ −η1o‖zo‖

2+ C1o (56)

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 201

for some η1o > 0. Similarly,

zTo (t − τ)[−(1− τ )So + 2τEToRoEo

]zo(t − τ)+ τ zTo (t − τ)E

ToRogo

≤[−(1− d2)βo + 2τk2b2(1+ γ 2)

]‖zo(t − τ)‖2 + d1

√1+ γ 2‖zo(t − τ)‖δ

≤[−(1− d2)(1− ζ2o)βo + 2τk2b2(1+ γ 2)

]‖zo(t − τ)‖2 + C2o

where 0 < ζ2o < 1, C2o =(1+γ 2)d1(1−d2)ζ2oβo

δ2is a constant. If we choose time-delay τ such that

(1− d2)(1− ζ2o)βo − 2τk2b2(1+ γ 2) > 0 (57)

then

zTo (t − τ)[−(1− τ )So + 2τEToRoEo

]zo(t − τ)+ τ zTo (t − τ)E

ToRogo

≤ −η2o‖zo(t − τ)‖2 + C2o (58)

for some η2o > 0. To solve the upper and lower bounds of βo, we submit (41) into (55) and obtain that

(1− ζ1o)µo − βo − τk2b2(32γ 2 + 1)(1− γ 2)2 > 0. (59)

From (57) and (59), it is easily obtained that

4τk2b2(1+ γ 2)(1− d2)(1− ζ2o)

< βo <2(1+ γ 2)(1− ζ1o)µo

4(1+ γ 2)+ (1− d2)(32γ 2 + 1)(1− γ 2)2. (60)

After multiplying (55) by (1− d2)(1− ζ2o)/2 and submitting it into (57), we have τ < d1o, where

d1o := limζ1o→0

limζ2o→0

(1− d2)(1− ζ1o)(1− ζ2o)µo2k2b2(1+ γ 2)+ 2(1− d2)(1− ζ2o)[2+ k2b2(1− γ 2)2]

=(1− d2)µo

2k2b2(1+ γ 2)+ 2(1− d2)[2+ k2b2(1− γ 2)2]. (61)

When both (60) and (61) hold, then

Vo ≤ −ηo‖zo(t)‖2 + C3o (62)

where ηo = η1o and C3o =[1+ 2(1+γ+d1)2

ζ1oµo+

2(1+γ 2)d1(1−d2)ζ2oβo

]δ2. Since the eigenvalues of Po are either 1− γ or 1+ γ , we have

‖Po‖ = 1+ γ . In addition, we replace xwith zo in (40) and obtain a positive constant K1. It follows that

(1− γ )‖zo‖2 ≤ Vo(t, zot) ≤ (1+ γ )‖zo‖2 + K1

(∫ t

t−τ‖zo(s)‖2ds

)(63)

where the last inequality holds according to Lemma 5. We further choose the functionWi(r) : R+ → R+(i = 1, . . . , 4) asfollows.W1(r) = (1− γ )r2,W2(r) = (1+ γ )r2,W3(r) = K 1r = (βo +

K1ηo)r andW4(r) = ηor2, then

W1(‖zo(t)‖) ≤ Vo(t, zot)

≤ W2(‖zo(t)‖)+W3

(∫ t

t−τW4(‖zo(s)‖)ds

)(64)

for all t ≥ 0. This shows that Vo(t, zo(t)) satisfies the condition (33) of Lemma 4. At the same time, (62) can be rewritten as

Vo(t, zot) ≤ −W4(‖zo(t)‖)+ C3o. (65)

This shows that condition (34) of Lemma 4 holds.Therefore, from Lemma 4, we get center tracking dynamics (23) is U.U.B., which implies (42) with taking

Co =

√1+

2(1+ γ + d1)2

ζ1oµo+2(1+ γ 2)d1(1− d2)ζ2oβo

δ. (66)

Furthermore, if δ = 0, then limt→∞ zo(t) = 0.(2) On the other hand, we apply a similar Lyapunov–Krasovskii functional to analyze the convergence of cohesion

dynamics

Ve = V1e + V2e + V3e, (67)

202 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

where

V1e = zTe (t)Poeze(t),

V2e =∫ t

t−τzTe (t)Seze(t)ds,

V3e =∫ 0

−τ

∫ t

t+θze(s)Reze(s)dsdθ

(68)

where Pe =[In−1 −γ In−1−γ In−1 In−1

]⊗ Im, Se = βeI2m(n−1) and Re = I2m(n−1). Consider the time derivative of Vie (i = 1, 2, 3) along

the solution of (25) respectively:

V1e = zTe (t)(MTe Pe + PeMe)ze(t)+ 2z

Te (t)PeEepze(t − τ) (69)

V2e = zTe (t)Sezoe(t)− (1− τ )zTe (t − τ)Seze(t − τ) (70)

V3e = −∫ t

t−τzTe (θ)Reze(θ)dθ + τ z

Te (t)Reze(t). (71)

Following the lines of the analysis of center dynamics, we can obtain that

Ve = V1e + V2e + V3e≤ zTe (t)

[−Qep + Se + τ(2MTe ReMe + PeEepR

−1e E

TepPe)

]ze(t)

+ zTe (t − τ)[−(1− τ )Se + 2τETepReEep

]ze(t − τ) (72)

where

Qep = −[Pe(Eep +Me)+ (Eep +Me)TPe]

=

[2k(1− γ 2)(bIn−1 +Mp) −1

−1 2αk(1− γ 2)Mp

]⊗ Im (73)

and its smallest eigenvalue is µe > 0.Therefore, we get a sufficient condition for Ve < 0 as

−Qep + Se + τ(2MTe ReMe + PeEepR−1e E

TepPe) < 0 (74)

−(1− τ )Se + 2τETepReEep < 0. (75)

From Lemma 3, these two matrix inequalities can be rewritten as

−µe + βe + [2+ (1− γ 2)2k2ω21]τ < 0

−(1− d2)βe + 2(1+ γ 2)k2ω21τ < 0(76)

where

ω1 = max{b+ λ2, αλ2

}. (77)

After some manipulations, it is easily obtained that

2τ(1+ γ 2)k2ω211− d2

< βe <2(1+ γ 2)µe

4(1+ γ 2)+ (32γ 2 + 1)(1− γ 2)2(1− d2)(78)

τ < d1e :=(1− d2)µe

2(1+ γ 2)k2ω21 + (1− d2)[2+ k2ω21(1− γ 2)2

] . (79)

Therefore, if both (79) and (80) hold, then Ve < −ηe‖ξ(t)‖2 for some positive ηe, where ξ T(t) = [zTe (t) zTe (t−τ)] ∈ R2m(n−1).

This completes the proof. �

Hence, we get a sufficient condition for U.U.B. of center tracking dynamics (22) and asymptotic convergence of cohesiondynamics (23), that is, all of (60) and (78) and

τ < d1 = min {d1o, d1e} (80)

hold. Furthermore, this condition guarantees that error dynamics (11) is U.U.B. when every follower can be connected tothe leader for all time.

Remark 4. The above result holds if σ(t) is switching arbitrarily fast in a small time interval.

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 203

Remark 5. Obviously, Theorem 1 still holds if the time-delay τ is constant.

Remark 6. As stated in Remark 3, α plays an important role in improving the stability of cohesion dynamics. Here, it shouldbe pointed out that when

α ≤ α := 1+b

maxp∈P‖I−1Lp‖

, (81)

gain α may not only enhance the stability of cohesion dynamics but also increase the upper bound of allowed time-delay.

3.2. Only some followers can be connected to the leader

To analyze the convergence property of error dynamics (11) when the connections between the followers and the leaderare time-varying, we still need some properties of some important matrices. For p ∈ P, denote Lsym,p = 1

2 (Lp + LTp) and

Mp2 = Lsym,p + Bp. The next Lemma is given forMp2.

Lemma 6 (Lemma 5, [11]). Suppose that digraph Gp is balanced, then the symmetric matrix Mp2 is positive definite if and only ifnode (n+ 1) is a globally reachable node in Gp.

Note that node n+1 is a globally reachable node in Gp which is a more weaker condition than strong connectedness (SC)and implies that when time-delay need not be considered, some states of leaders can be directly or indirectly obtained byall followers. Based on Lemma 1 and the fact that the setP is finite, when node n+ 1 is a globally reachable node in Gp, wedefine

λ3 = min{eigenvalues ofMp2 ∈ Rn×n} > 0,λ4 = max{‖Lp + Bp‖, p ∈ P} > 0,λ5 = max{‖Lp‖, p ∈ P} > 0.

(82)

Lemma 7. Assume that Qp = −[P(Ep + M)+ (Ep + M)TP] and k > 14γ (1−γ 2)λ3

, then the smallest eigenvalue of Qp(p ∈ P) islarger than

µ = (1− γ 2)kλ3 + γ −√[(1− γ 2)kλ3 − γ

]2+ 1 > 0. (83)

Proof. By the definition of Qp, we can obtain that

Qp =[2k(1− γ 2)Mp2 −In

−In 2γ In + 2αk(1− γ 2)Lsym,p

]⊗ Im

≥ Q p :=[2k(1− γ 2)Mp2 −In

−In 2γ In

]⊗ Im (84)

where the relation≥means that Qp − Q p is a nonnegative matrix and this is because for any column vector x ∈ Rn×16= 0,

xTLsym,px ≥ 0 holds. Then, according to the method used in Lemma 2, we immediately obtain (83). This completes theproof. �

Lemma 8. For p ∈ P, we have

‖PEp‖ = (1− γ 2)kω2, ‖Ep‖ =√1+ γ 2kω2 (85)

where ω2 = max{λ4, αλ5

}.

Theorem 2. Consider the error dynamics (11) with time-varying delay τ , when only some followers can be connected to theleader, for any fixed 0 < γ < 1, α > 0, λ3 defined in (82), if the switching interconnection graph keeps balanced and node n+ 1is a globally reachable node in Gp for all time, we take a constant

k >1

4γ (1− γ 2)λ3(86)

and the time-varying delay τ is sufficient small, then

limt→∞‖ε(t)‖ ≤ C (87)

for some constant C depending on δ, d1 and d2. Moreover, if a(t) is known (δ = 0), then limt→∞ ‖zo(t)‖ = 0.

204 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

Proof. Similarly to the proof of Theorem 1, we also define a Lyapunov–Krasovskii functional as follows:

V = V1 + V2 + V3, (88)

where

V1 = εT(t)Pε(t),

V2 =∫ t

t−τεT(t)Sε(t)ds,

V3 =∫ 0

−τ

∫ t

t+θε(s)Rε(s)dsdθ

(89)

where P =[In −γ In−γ In In

]⊗ Im, S = βI2mn and R = I2mn. Consider the time derivative of Vi (i = 1, 2, 3) along the solution of

(11) respectively:

V1 = εT(t)(MTP + PM)ε(t)+ 2εT(t)P[Epε(t − τ)+ g] (90)

V2 = εT(t)Sε(t)− (1− τ )εT(t − τ)Sε(t − τ) (91)

V3 = −∫ t

t−τεT(θ)Rε(θ)dθ + τ εT(t)Rε(t). (92)

From the Leibniz–Newton formula, we can obtain that

2εT(t)PEpε(t − τ) = εT(t)(ETpP + PEp)ε(t)+ 2(ETpPε(t))

T∫ t

t−τε(s)ds

≤ εT(t)(ETpP + PEp)ε(t)+ τεT(t)PEpR−1ETpPε(t)+

∫ t

t−τε(s)Rε(s)ds.

Consequently,

V1 ≤ εT(t)[−Qp + τPEpR−1ETpP

]ε(t)+

∫ t

t−τεT(θ)Rε(θ)dθ + 2ε(t)TPg (93)

where Qp is defined in Lemma 7. Similarly,

V3 = −∫ t

t−τεT(θ)Rε(θ)dθ + τ(Mε(t)+ Epε(t − τ)+ g)TR(Mε(t)+ Epzo(t − τ)+ g)

= 2τ[εT(t)MTRMε(t)+ εT(t − τ)ETpREpε(t − τ)

]+ gTRg

−

∫ t

t−τεT(θ)Rε(θ)dθ + 2τ [εT(t)MT + εT(t − τ)ETp ]Rg. (94)

From (91), (93) and (94), we can obtain that

V ≤ εT(t)[−Qp + S + τ(2MTRM + PEpR−1ETpP)

]ε(t)

+ εT(t − τ)[−(1− τ )S + 2τETpREp

]ε(t − τ)

+ 2εT(t)(P + τMTR)g + τεT(t − τ)ETpRg + gTRg. (95)

Since ‖M‖ = 1, Lemmas 7 and 8, we have

εT(t)[−Qp + S + τ(2MTRM + PEpR−1ETpP)

]ε(t)+ 2εT(t)(P + τMTR)g

≤[−µ+ β + τ(2+ k2ω22(1− γ

2)2)]‖ε(t)‖2 + 2 (1+ γ + d1) ‖ε(t)‖δ

≤ −[(1− ζ1)µ− β − τ(2+ k2ω22(1− γ

2)2)]‖ε(t)‖2 + C1 (96)

where 0 < ζ1 < 1, C1 =2(1+γ+d1)2

ζ1µδ2is a constant. If we choose time-delay τ such that

(1− ζ1)µ− β − τ(2+ k2ω22(1− γ2)2) > 0 (97)

then

εT(t)[−Qp + S + τ(2MTRM + PEpR−1ETpP)

]ε(t)+ 2εT(t)(P + τMTR)g

≤ −η1‖ε‖2+ C1 (98)

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 205

for some η1 > 0. Similarly,

εT(t − τ)[−(1− τ )S + 2τETpREp

]ε(t − τ)+ τεT(t − τ)ETpRg

≤[−(1− d2)β + 2τk2ω22(1+ γ

2)]‖ε(t − τ)‖2 + d1

√1+ γ 2‖ε(t − τ)‖δ

≤[−(1− d2)(1− ζ2)β + 2τk2ω22(1+ γ

2)]‖ε(t − τ)‖2 + C2

where 0 < ζ2 < 1, C2 =2(1+γ 2)d1(1−d2)ζ2β

δ2is a constant. If we choose time-delay τ such that

(1− d2)(1− ζ2)β − 2τk2ω22(1+ γ2) > 0 (99)

then

εT(t − τ)[−(1− τ )S + 2τETpREp

]ε(t − τ)+ τεT(t − τ)ETpRg

≤ −η2‖ε(t − τ)‖2 + C2 (100)

for some η2 > 0. To solve the upper and lower bounds of β , we submit (86) into (97) and obtain that

(1− ζ1)µ− β − τk2(32γ 2λ23 + ω

22)(1− γ

2)2 > 0. (101)

From (99) and (101), it is easily obtained that

4τk2ω22(1+ γ2)

(1− d2)(1− ζ2)< β <

2(1+ γ 2)(1− ζ1)µ

4(1+ γ 2)+ (1− d2)(32γ 2λ23/ω

22 + 1)(1− γ 2)2

. (102)

After multiplying (97) by (1− d2)(1− ζ2)/2 and submitting it into (99), we have τ < d1, where

d1 := limζ1→0

limζ2→0

(1− d2)(1− ζ1)(1− ζ2)µ8k2ω22(1+ γ 2)+ 2(1− d2)(1− ζ2)[2+ k2ω

22(1− γ 2)2]

=(1− d2)µ

8k2ω22(1+ γ 2)+ 2(1− d2)[2+ k2ω22(1− γ 2)2]

(103)

where ω2 = max{λ4, αλ5

}. When both (102) and (103) hold, then

V ≤ −η‖ε(t)‖2 + C (104)

where η = η1 and C =[1+ 2(1+γ+d1)2

µ+2(1+γ 2)d1(1−d2)β

]δ2. Since the eigenvalues of P are either 1 − γ or 1 + γ , we have

‖P‖ = 1+ γ . In addition, we replace xwith ε in (40) and obtain a positive constant K2. It follows that

(1− γ )‖ε‖2 ≤ V (ε) ≤ (1+ γ )‖ε‖2 + K2

(∫ t

t−τ‖ε(s)‖2ds

)(105)

where the last inequality holds according to Lemma 5. We further choose the functionWi(r) : R+ → R+(i = 1, . . . , 4) asfollows.W1(r) = (1− γ )r2,W2(r) = (1+ γ )r2,W3(r) = K 2r=(β +

K2η)r andW4(r) = ηr2, then

W1(‖ε(t)‖) ≤ V (t, εt)

≤ W2(‖ε(t)‖)+W3

(∫ t

t−τW4(‖ε(s)‖)ds

)(106)

for all t ≥ 0. This shows that V (t, ε(t)) satisfies condition (33) of Lemma 4. At the same time, (104) can be rewritten as

V (t, εt) ≤ −W4(‖ε(t)‖)+ C . (107)

This shows that condition (34) of Lemma 4 holds. �

Therefore, from Lemma 4, we get that error dynamics (11) is U.U.B., which implies (87) with taking

C =

√1+

2(1+ γ + d1)2

ζ1µ+2(1+ γ 2)d1(1− d2)ζ2β

δ. (108)

Furthermore, if δ = 0, then limt→∞ ε(t) = 0.

206 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

Fig. 1. Four different balanced and SC directed graphs.

Fig. 2. The leader-following errors with α = 0 and τ(t) = 0.011| cos(9t)|.

Remark 7. Compared to Theorem 1, Theorem 2 can be suitable to more general cases, that is, only some followers canconnected to the leader.

Remark 8. Since during the proof of Theorem 2, parameter α is neglected, this leads to possible conservation. In addition,from Theorem 2, we cannot obtain some results of α similar to those in Remarks 3 and 6.

4. Simulation results

In this section, to illustrate our theoretical results derived in the above section, we will provide two numericalsimulations. Without loss of generality, we takem = 1 in numerical simulations.

4.1. The first simulation

In Fig. 1. there are four different balanced and SC directed graphs. We assume that ai,j = 0.6, for all i, j ∈ I andb = 2 which means that all followers can be connected to the leader for all time. Here we consider the case in whicharbitrary switching among these four graphs in Fig. 1. happens every 0.001 second. In addition, take k = 0.5, γ = 0.5,δ(t) = 0.04 sin(t)+0.01 cos(t) and themaximumof the derivative of time-varying delay d2 = 0.1.With simple calculations,we can obtain that α = 1.66. In the following, we consider two cases:

(1) α = 0. At this time, we can obtain that τ ≤ 0.011(the maximum delay bound d1) and take τ(t) = 0.011| cos(9t)|;(2) α = 1.6. Similarly, we can obtain that τ ≤ 0.013 and take τ(t) = 0.013| cos(7.5t)|.

The simulation results are shown in Figs. 2 and 3. respectively. From Figs. 2 and 3, we can see that when 0 < α < α, positionerrors and velocity errors converge together more quickly and the allowed time-delay is larger than the case when α = 0.

4.2. The second simulation

Herewe consider the case inwhich the interaction topology is arbitrarily switched at every 0.001 s among three digraphsGi (i = 1, 2, 3). When it is assumed that ai,j = 0.6 and bi = 2 for all i, j = 1, . . . , 4, the Laplacian Li (i = 1, 2, 3) of

K. Peng, Y. Yang / Physica A 388 (2009) 193–208 207

Fig. 3. The leader-following errors with α = 1.6 and τ(t) = 0.013| cos(7.5t)|.

Fig. 4. The leader-following errors with α = 0 and τ(t) = 0.0025| cos(20t)|.

Gi (i = 1, 2, 3) as well as the leader adjacency matrices Bi (i = 1, 2, 3) have the following forms:

L1 =

1.2 −0.6 0 −0.6−0.6 1.2 −0.6 00 −0.6 0.6 0−0.6 0 0 0.6

, L2 =

1.2 0 −0.6 −0.6−0.6 0.6 0 0−0.6 −0.6 1.2 00 0 −0.6 0.6

L3 =

0.6 0 −0.6 00 0.6 −0.6 00 −0.6 1.2 −0.6−0.6 0 0 0.6

, {B1 = diag{1 0 1 1},B2 = diag{0 1 1 1},B3 = diag{1 1 0 1}.

In addition, we take k = 1.5, γ = 0.5. δ(t) = 0.04 sin(t) + 0.01 cos(t) and d2 = 0.05. With simple calculations, we canobtain that τ < 0.0027s. Therefore, we take τ(t) = 0.0025| cos(20t)|. The simulation results are shown in Figs. 4 and 5.From the following figures, we can easily obtain a similar result to Remark 3 that when α < λ4/λ5−1 (λ4 and λ5 are definedin (82)), increasing α can make the tracking errors converge more quickly.

5. Conclusion

In this paper, we consider a leader-following coordination problem for a multi-agent system with a varying-velocityleader and time-varying delays. Although the states of the leader keeps changing and some state may not be measured, weproposed a neighbor-based rule for each agent to follow such leader. We also present sufficient conditions for uniformlyultimately boundedness of the tracking errors even when the coupling topology is switching and balanced. Finally, two

208 K. Peng, Y. Yang / Physica A 388 (2009) 193–208

Fig. 5. The leader-following errors with α = 1.2 and τ(t) = 0.0025| cos(20t)|.

numerical simulations are presented to illustrate our theoretical results. Since we only investigate the case when time-varying delays are uniform, how to consider the nonuniform time-delay is our future research.

Acknowledgements

We would like to thank the associated editor and the anonymous reviewers for their insightful comments andsuggestions.

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