# Formation Control of Second-Order Multiagent Systems with Time-Varying Delays

Authors: Hong Xia; Ting-Zhu Huang; Jin-Liang Shao; Jun-Yan Yu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A formation control problem for second-order multiagent
systems with time-varying delays is considered. First, a leader-following consensus
protocol is proposed for theoretical preparation. With the help of
Lyapunov-Krasovskii functional, a sufficient condition under this protocol is
derived for stability of the multiagent systems. Then, the protocol is extended
to the formation control based on a multiple leaders’ architecture. It
is shown that the agents will attain the expected formation. Finally, some
simulations are provided to demonstrate the effectiveness of our theoretical
results.


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## Body

## 1. Introduction

Recent years have witnessed a rapidly growing interest in coordinated control of multiagent systems due to its broad applications in various disciplines [1–12]. As one of the important topics in this field, formation control has attracted great attention. Generally speaking, the main objective in formation control is to design appropriate protocol and algorithm such that the agents can achieve and preserve a predefined geometrical shape, such as a chain or a wedge. Potential applications of formation control include lots of cooperative tasks such as surveillance, exploration, search and rescue, transporting large objects, and control of arrays of satellites.In the literature, researchers have proposed numerous approaches for the multiagent systems to achieve the anticipated formation, roughly categorized as behavior-based strategy [13], virtual structure method [14], and leader-following approach [15–21], to name a few. For instance, Xiao et al. [16] developed a formation framework of multiple leaders and applied a class of nonlinear consensus protocols to the formation control. Under the proposed framework, all the agents in the first-order systems could reach the expected formation. An adaptive formation control approach, in the absence of the velocity information of the leader, was proposed in [17]. Besides, the authors in [18] investigated the leader-following formation control problems for nonlinear systems under fixed and switching topologies. The above works, however, did not take into account the effects of time delays.Owing to the finite speed of information transmission and processing, time delays are inevitable in multiagent systems. In particular, one type of time delays is communication delays, whose effects on multiagent systems have been addressed by many researchers [3–9]. For formation control with time delays, Luo et al. [19] gave a sufficient condition of formation control of multiagent systems by using Lyapunov stability theory. Also, Rezaee and Abdollahi [20] provided a motion synchronization strategy with time delays. Note that the time delays in both papers were assumed to be constant. In reality, it is more practical when the time-varying delays are accounted for. Lu et al. [21] studied the formation control of second-order multiagent systems with time-varying delays, where the time delays existed only in the transmission of position information between neighbors. For second-order multiagent systems, it is worthwhile to mention that the information exchanged between neighbors may include velocity information as well as position information.Motivated by the above analysis, we consider a leader-following formation control problem for second-order multiagent systems, with time-varying delays existing in the transmission of both velocity and position. Here, we adopt the formation framework proposed in [16]. More specifically, the formation information is divided into two parts: the global one and the local one, where the former determines the geometric pattern of the desired formation and the latter decides the relative information of agents with respect to their neighbors. Only a small number of agents called leaders have access to the global information, and the other agents called followers regulate their states according to the local information.An outline of this paper is as follows. Section2 provides some preliminary notions of graph theory and formulates the formation control problem. By utilizing Lyapunov-Krasovskii functional, Section 3 presents the main results under the proposed control protocol. Then several simulations are illustrated in Section 4. Finally, the conclusion is drawn in Section 5.The following notations will be used throughout this paper. Given a matrix, the superscripts “T” and “−1” stand for its transposition and inverse, respectively; Λ(·) and ∥·∥ denote the set of all eigenvalues and the spectral norm of the matrix, respectively. Let In be an n×n identity matrix, 1n=[1⋯1]T∈ℝn, and 0n×n represents an n×n zero matrix. And diag{b1,…,bn} denotes a diagonal matrix with diagonal elements being b1,…,bn. For a complex number μ∈ℂ, Re(μ), Im(μ), and |μ| are its real part, imaginary part, and modulus, respectively. ⊗ denotes the Kronecker product.


## 2. Preliminaries and Problem Formulation

Let𝒢=(𝒱,ℰ,A) be a weighted directed graph with the set of nodes 𝒱={v1,v2,…,vn}, set of edges ℰ⊆𝒱×𝒱, and a weighted adjacency matrix A=[aij] with nonnegative elements aij. The node indexes belong to a finite index set ℐ={1,2,…,n}. An edge in ℰ is denoted by eji=(vj,vi), where eji∈ℰ if and only if aij≠0. In this case, we say node j is a neighbor of node i and denote the neighbors of node i by 𝒩i={j∈𝒱:(vj,vi)∈ℰ}. Moreover, we assume aii=0 for all i∈𝒱. Let the Laplacian matrix L=[lij]∈ℝn×n associated with A be defined as lii=∑j=1,j≠inaij and lij=-aij.A directed path in directed graph𝒢 from vi1 to vik is a sequence of edges of the form (vi1,vi2),(vi2,vi3),…,(vik-1,vik), where vij∈𝒱 for j=1,2,…,k. A directed graph is called strongly connected if any two distinct nodes of the graph can be connected via a directed path that follows the edges of the graph. A directed tree is a directed graph, where every node, except one special node without any parent, which is called the root, has exactly one parent, and the root can be connected to any other nodes through paths. A spanning tree of a digraph is a directed tree formed by graph edges that connect all the nodes of the graph.For then-agent system considered in this paper, suppose that there are m  (m≤n) leaders and n-m followers. For convenience, we use ℛ={1,2,…,m} and ℱ={m+1,m+2,…,n} to denote, respectively, the leader set and the follower set. Further, the interaction topology among agents is modeled by a direct graph 𝒢.The dynamics of then autonomous agents are given by
(1)x˙i=vi,v˙i=ui,i=1,2,…,n,
where xi, vi, ui∈ℝp denote the position, velocity and control input of agent i, respectively.Using the formation framework in [16], the formation information is divided into two independent parts: the global one and the local one. The global information, which determines the geometric pattern of the expected formation, is represented by a time-dependent column vector F=[f1T,f2T,…,fmT]T∈ℝ2mp with fi=[fixT,fivT]T∈ℝ2p  (i∈ℛ). The local formation information is denoted by a time-independent nonnegative matrix W=[Wm+1T,…,WnT]T∈ℝ(n-m)×m with unit entry sum for each row Wi=(wi1,…,wim)∈ℝ1×m   (i∈ℱ).Denotexℛ=(x1T,x2T,…,xmT)T and vℛ=(v1T,v2T,…,vmT)T; we have the following.Definition 1.
We say that system (1) solves the formation problem if there exists a ℝ2p-valued function fc=[fcx(t)T,fcv(t)T] with fcx(t)∈ℝp, fcv(t)∈ℝp, and f˙cx(t)=fcv(t) such that xi→fix(t)+fcx(t), vi→fiv(t)+fcv(t) for i∈ℛ and xi→(Wi⊗Ip)xℛ, vi→(Wi⊗Ip)vℛ for i∈ℱ as t→∞. In particular, the formation problem is called a time-invariant formation (TIF) problem, a time-varying formation (TVF) problem, and a time-varying formation for trajectory tracking (TVFT) problem if f˙ix=f˙cx=fiv=fcv=0, f˙ix=fiv≠0 with f˙cx=fcv=0 and f˙ix=fiv≠0 with f˙cx=fcv≠0, respectively.The column vectorfc(t), which specifies the state of the formation and may be dependent on initial states or may be an external input, used to guide the group of agents to track a prescribed trajectory. The vector F defines the basic frame of the anticipated formation formed by the leaders, and the nonnegative matrix W specifies the local-state restrictions of followers with respect to their leader neighbors. Since each row entry sum of W equals 1, the followers should lie in the convex region covered by the leaders.


## 3. Main Results

### 3.1. Leader-Following Consensus

For the better understanding of the formation control problem, we first consider a leader-following consensus problem. The multiagent systems solve a leader-following consensus asymptotically iflimt→∞∥xi-f0x∥=0 and limt→∞∥vi-f0v∥=0 for all i=1,2,…,n. Under this circumstance, all the agents can obtain the acceleration f˙0v, but only the root agent can obtain the difference between its state and the formation information, and hence it can pin the other leaders to attain the anticipated formation. Define a matrix B∈ℝn×n as B=diag{b1,…,bn}, where bi>0 if the interaction topology 𝒢 has a spanning tree rooted at i, 1≤i≤n and bi=0 otherwise.The time-varying delayτ(t) is the time delay for information communicated between agents at time t. Owing to the communication time delays, each agent cannot instantly get the information from others. With time-varying delays existing in the transmission of both velocity and position, we now provide the following consensus protocol:
(2)ui(t)=f˙0v+∑j∈𝒩i‍aij[(xj(t-τ(t))-xi(t-τ(t)))+k(vj(t-τ(t))-vi(t-τ(t)))]+bi[(f0x(t-τ(t))-xi(t-τ(t)))+k(f0v(t-τ(t))-vi(t-τ(t)))],i=1,2,…,n,
where the control parameter k>0, A=(aij)n×n is the adjacency matrix corresponding to the graph 𝒢, bi>0 if the interaction topology 𝒢 has a spanning tree rooted at i and bi=0 otherwise, and the time-varying delay τ(t) is a continuously differentiable function with
(3)τ(t)<d1,τ˙(t)≤d2<1.Denotex=(x1T,x2T,…,xnT)T, v=(v1T,v2T,…,vnT)T; then, system (1) with protocol (2) can be written in a matrix form:
(4)x˙=v,v˙=f˙0v1n-(L+B)⊗Ipx(t-τ)-k(L+B)⊗Ipv(t-τ)+B[f0x(t-τ)·1n+kf0v(t-τ)·1n],
where L is the Laplacian matrix associated with 𝒢, B=diag{b1,…,bn}, bi>0 if the interaction topology 𝒢 has a spanning tree rooted at i and bi=0 otherwise.Letx¯=x-f0x1n, v¯=v-f0v1n; then, (4) can be rewritten as
(5)x¯˙=v¯,v¯˙=-(L+B)⊗Ipx¯(t-τ)-k(L+B)⊗Ipv¯(t-τ).Takingε=(x¯T,v¯T)T, (5) can be equally expressed as
(6)ε˙=Yε(t)+Zε(t-τ),
where
(7)Y=(0n×nIn0n×n0n×n)⊗Ip,Z=(0n×n0n×n-H-kH)⊗Ip,H=L+B.
To proceed further, we need the following lemmas.Lemma 2 (see [22]).
Given a complex-coefficient polynomial(8)r(z)=z2+(a+ib)z+c+id,
where a,b,c,d∈ℝ, r(z) is Hurwitz stable if and only if a>0 and abd+a2c-d2>0.Lemma 3.
The matrixF=Y+Z=(0n×nIn-H-kH)⊗Ip is Hurwitz stable if and only if H is positive stable and k>maxλ∈Λ(H)(|Im(λ)|/(|Re(λ)|·|λ|)).Proof.
Letz be an eigenvalue of F. Then one has
(9)det(zI2np-F)=det([zIn-InHzIn+kH]⊗Ip)=det((z(zIn+kH)+H)⊗Ip)=∏i=1n‍(z2+kλiz+λi)p.
Clearly, the Hurwitz stability of matrix F is equivalent to that of the polynomial: r(z)=z2+kλiz+λi, where λi=Re(λi)+iIm(λi) is the ith eigenvalue of H. It follows from Lemma 2 that r(z) is Hurwitz stable if and only if Re(λi)>0 and k>|Im(λi)|/(|Re(λi)|·|λi|), λi∈Λ(H).Lemma 4 (see [4]).
The matrixH=L+B is positive stable if 𝒢 has a spanning tree.Now we give the main result of this subsection.Theorem 5.
For system (6), take
(10)k>k0=maxλ∈Λ(H)|Im(λ)||Re(λ)|·|λ|
and suppose that
(11)0<τ<τ0=12∥ZTZ∥/(1-d2)+∥2YTY+PZZTP∥.
Then, the consensus is reached asymptotically if the graph 𝒢 has a spanning tree.Proof.
Since𝒢 has a spanning tree, H is positive stable based on Lemma 4. It follows from Lemma 3 that F is Hurwitz stable. Therefore, there exists a positive definite matrix P∈ℝ2np×2np such that
(12)PF+FTP=-I2np.
To analyze the convergence of system (6), we define a Lyapunov-Krasovskii functional
(13)V(ε)=εT(t)Pε(t)+∫t-τT‍εT(t)Sε(t)ds+∫-τ0‍∫t+θt‍ε˙T(s)Rε˙(s)dsdθ,
where S=βI2np and R=I2np.
CalculatingV˙(ε) along the solution of (6), we have
(14)V˙(ε)=εT(t)(YTP+PY)ε(t)+2εT(t)PZε(t-τ)+εT(t)βε(t)-β(1-τ˙)εT(t-τ)ε(t-τ)-∫t-τt‍ε˙T(θ)ε˙(θ)dθ+τε˙T(t)ε˙(t).
Due to the fact thatε(t-τ)=ε(t)-∫t-τt‍ε˙(s)ds and 2aTb≤aTΨ-1a+bTΨb holds for any appropriate positive definite matrix Ψ, we can obtain that
(15)2εT(t)PZε(t-τ)=εT(t)(ZTP+PZ)ε(t)+2∫t-τt‍(-ZTPε(t))Tε˙(s)ds≤εT(t)(ZTP+PZ)ε(t)+τεT(t)PZZTPε(t)+∫t-τt‍ε˙T(s)ε˙(s)ds.
Similarly,
(16)τε˙T(t)ε˙(t)=τ[Yε(t)+Zε(t-τ)]T[Yε(t)+Zε(t-τ)]=τ[εT(t)YTYε(t)+εT(t-τ)ZTZε(t-τ)]+εT(t)YTZε(t-τ)+εT(t-τ)ZTYε(t)≤2τ[εT(t)YTYε(t)+εT(t-τ)ZTZε(t-τ)].
Substituting (15) and (16) into (14) leads to
(17)V˙(ε)≤εT(t)[-(PF+FTP)+βI2np+τ(2YTY+PZZTP)-(PF+FTP)+βI2np]ε(t)+εT(t-τ)[-(1-τ˙)βI2np+2τZTZ]ε(t-τ)≤εT(t)(-1+β+τ∥2YTY+PZZTP∥)ε(t)+εT(t-τ)[-(1-d2)β+2τ∥ZTZ∥]ε(t-τ).
Consequently, a sufficient condition for V˙(ε)<0 is
(18)-1+β+τ∥2YTY+PZZTP∥<0,-(1-d2)β+2τ∥ZTZ∥<0.
From (18), we can obtain that
(19)τ<τ0=12∥ZTZ∥/(1-d2)+∥2YTY+PZZTP∥.
Therefore, by Lyapunov-Krasovskii Theorem (see [23]), the error system (6) is uniformly asymptotically stable. Namely, the consensus is reached asymptotically.Remark 6.
It can be seen that many zoom techniques are applied during the derivation ofτ0, which result in a conservative estimation of τ0.


### 3.2. Time-Varying Formation for Trajectory Tracking

In this subsection, we consider the case that the agents form a time-varying formation as they track the desired trajectory. More specifically, the desired trajectory of the formation, represented byfc=[fcx(t)T,fcv(t)T], is assumed to be determined by the following equation:
(20)f˙cx=fcv=g(t,fcx).
Also make the following assumptions about the multiagent system:(A1)
the local interaction topology of the leaders has a spanning tree, and the leaders’ dynamic is unaffected by the followers;(A2)
the root agent in the local interaction topology of the leaders is able to access the reference trajectory;(A3)
in addition to the local information from their neighbors, the leaders can also obtain the global informationF;(A4)
in addition to the local information from their follower neighbors, each follower can also get the local formationW directly or indirectly from the leaders.Using the following control protocol:(21)x˙i=vi,i=1,2,…,n,v˙i=f˙iv+g˙(t,fcx)+∑j=1m‍aij{[(xj(t-τ)-fjx(t-τ))-(xi(t-τ)-fix(t-τ))xj(t-τ)-fjx(t-τ)]+k[(vj(t-τ)-fjv(t-τ))-(vi(t-τ)-fiv(t-τ))vj(t-τ)-fjv(t-τ)]}-bi[(xi(t-τ)-fix(t-τ)-fcx(t-τ))+k(vi(t-τ)-fiv(t-τ)-g(t-τ))],i∈ℛ,v˙i=∑k=1m‍wikv˙k+∑j=1n‍aij{[(xj(t-τ)-∑k=1m‍wjkxk(t-τ))-(xi(t-τ)-∑k=1m‍wikxk(t-τ))]+k[(vj(t-τ)-∑k=1m‍wjkvk(t-τ))-(vi(t-τ)-∑k=1m‍wikvk(t-τ))]}-bi[(xi(t-τ)-∑k=1m‍wikxk(t-τ))+k(vi(t-τ)-∑k=1m‍wikvk(t-τ))],i∈ℱ,
where k>0, A=(aij)n×n is the adjacency matrix corresponding to the graph 𝒢, and bi>0 if there is a spanning tree rooted at i in the graph 𝒢 and bi=0 otherwise, we have the following result.Theorem 7.
Suppose that the graph𝒢 has a spanning tree. Take
(22)k>k0=maxλ∈Λ(H)|Im(λ)||Re(λ)|·|λ|,0<τ<τ0=12∥ZTZ∥/(1-d2)+∥2YTY+PZZTP∥.
With protocol (21), the multiagent systems attain a time-varying formation for trajectory tracking (TVFT) under assumptions (A1)–(A4).Proof.
Letx~i=xi-fix-fcx, v~i=vi-fiv-g(t,fcx) for i∈ℛ, and x~i=xi-∑k=1mwikxk, v~i=vi-∑k=1mwikvk for i∈ℱ. Then we can rewrite protocol (21) as
(23)x~˙i=v~i,v~˙i=∑j=1n‍aij[(x~j(t-τ)-x~i(t-τ))+k(v~j(t-τ)-v~i(t-τ))]-bi(x~i(t-τ)+kv~i(t-τ)),i=1,2,…,n.
Denote x~=(x~1T,x~2T,…,x~nT) and v~=(v~1T,v~2T,…,v~nT); (23) can be expressed in a matrix form:
(24)x~˙=v~,v~˙=-(L+B)⊗Ipx~(t-τ)-k(L+B)⊗Ipv~(t-τ).
It follows from Theorem 5 that x~i and v~i converge to zero asymptotically, and equally, xi→fix+fcx, vi→fiv+fcv for i∈ℛ and xi→(Wi⊗Ip)xℛ, vi→(Wi⊗Ip)vℛ for i∈ℱ as t→∞.Remark 8.
Obviously, Theorem7 still holds if f˙ix=fiv=0, or f˙ix=fiv≠0 and f˙cx=fcv=g(t,fcx)=0. In the form case, the multiagent systems attain a time-invariant formation (TIF). In the latter case, the multiagent systems attain a time-varying formation.


## 3.1. Leader-Following Consensus

For the better understanding of the formation control problem, we first consider a leader-following consensus problem. The multiagent systems solve a leader-following consensus asymptotically iflimt→∞∥xi-f0x∥=0 and limt→∞∥vi-f0v∥=0 for all i=1,2,…,n. Under this circumstance, all the agents can obtain the acceleration f˙0v, but only the root agent can obtain the difference between its state and the formation information, and hence it can pin the other leaders to attain the anticipated formation. Define a matrix B∈ℝn×n as B=diag{b1,…,bn}, where bi>0 if the interaction topology 𝒢 has a spanning tree rooted at i, 1≤i≤n and bi=0 otherwise.The time-varying delayτ(t) is the time delay for information communicated between agents at time t. Owing to the communication time delays, each agent cannot instantly get the information from others. With time-varying delays existing in the transmission of both velocity and position, we now provide the following consensus protocol:
(2)ui(t)=f˙0v+∑j∈𝒩i‍aij[(xj(t-τ(t))-xi(t-τ(t)))+k(vj(t-τ(t))-vi(t-τ(t)))]+bi[(f0x(t-τ(t))-xi(t-τ(t)))+k(f0v(t-τ(t))-vi(t-τ(t)))],i=1,2,…,n,
where the control parameter k>0, A=(aij)n×n is the adjacency matrix corresponding to the graph 𝒢, bi>0 if the interaction topology 𝒢 has a spanning tree rooted at i and bi=0 otherwise, and the time-varying delay τ(t) is a continuously differentiable function with
(3)τ(t)<d1,τ˙(t)≤d2<1.Denotex=(x1T,x2T,…,xnT)T, v=(v1T,v2T,…,vnT)T; then, system (1) with protocol (2) can be written in a matrix form:
(4)x˙=v,v˙=f˙0v1n-(L+B)⊗Ipx(t-τ)-k(L+B)⊗Ipv(t-τ)+B[f0x(t-τ)·1n+kf0v(t-τ)·1n],
where L is the Laplacian matrix associated with 𝒢, B=diag{b1,…,bn}, bi>0 if the interaction topology 𝒢 has a spanning tree rooted at i and bi=0 otherwise.Letx¯=x-f0x1n, v¯=v-f0v1n; then, (4) can be rewritten as
(5)x¯˙=v¯,v¯˙=-(L+B)⊗Ipx¯(t-τ)-k(L+B)⊗Ipv¯(t-τ).Takingε=(x¯T,v¯T)T, (5) can be equally expressed as
(6)ε˙=Yε(t)+Zε(t-τ),
where
(7)Y=(0n×nIn0n×n0n×n)⊗Ip,Z=(0n×n0n×n-H-kH)⊗Ip,H=L+B.
To proceed further, we need the following lemmas.Lemma 2 (see [22]).
Given a complex-coefficient polynomial(8)r(z)=z2+(a+ib)z+c+id,
where a,b,c,d∈ℝ, r(z) is Hurwitz stable if and only if a>0 and abd+a2c-d2>0.Lemma 3.
The matrixF=Y+Z=(0n×nIn-H-kH)⊗Ip is Hurwitz stable if and only if H is positive stable and k>maxλ∈Λ(H)(|Im(λ)|/(|Re(λ)|·|λ|)).Proof.
Letz be an eigenvalue of F. Then one has
(9)det(zI2np-F)=det([zIn-InHzIn+kH]⊗Ip)=det((z(zIn+kH)+H)⊗Ip)=∏i=1n‍(z2+kλiz+λi)p.
Clearly, the Hurwitz stability of matrix F is equivalent to that of the polynomial: r(z)=z2+kλiz+λi, where λi=Re(λi)+iIm(λi) is the ith eigenvalue of H. It follows from Lemma 2 that r(z) is Hurwitz stable if and only if Re(λi)>0 and k>|Im(λi)|/(|Re(λi)|·|λi|), λi∈Λ(H).Lemma 4 (see [4]).
The matrixH=L+B is positive stable if 𝒢 has a spanning tree.Now we give the main result of this subsection.Theorem 5.
For system (6), take
(10)k>k0=maxλ∈Λ(H)|Im(λ)||Re(λ)|·|λ|
and suppose that
(11)0<τ<τ0=12∥ZTZ∥/(1-d2)+∥2YTY+PZZTP∥.
Then, the consensus is reached asymptotically if the graph 𝒢 has a spanning tree.Proof.
Since𝒢 has a spanning tree, H is positive stable based on Lemma 4. It follows from Lemma 3 that F is Hurwitz stable. Therefore, there exists a positive definite matrix P∈ℝ2np×2np such that
(12)PF+FTP=-I2np.
To analyze the convergence of system (6), we define a Lyapunov-Krasovskii functional
(13)V(ε)=εT(t)Pε(t)+∫t-τT‍εT(t)Sε(t)ds+∫-τ0‍∫t+θt‍ε˙T(s)Rε˙(s)dsdθ,
where S=βI2np and R=I2np.
CalculatingV˙(ε) along the solution of (6), we have
(14)V˙(ε)=εT(t)(YTP+PY)ε(t)+2εT(t)PZε(t-τ)+εT(t)βε(t)-β(1-τ˙)εT(t-τ)ε(t-τ)-∫t-τt‍ε˙T(θ)ε˙(θ)dθ+τε˙T(t)ε˙(t).
Due to the fact thatε(t-τ)=ε(t)-∫t-τt‍ε˙(s)ds and 2aTb≤aTΨ-1a+bTΨb holds for any appropriate positive definite matrix Ψ, we can obtain that
(15)2εT(t)PZε(t-τ)=εT(t)(ZTP+PZ)ε(t)+2∫t-τt‍(-ZTPε(t))Tε˙(s)ds≤εT(t)(ZTP+PZ)ε(t)+τεT(t)PZZTPε(t)+∫t-τt‍ε˙T(s)ε˙(s)ds.
Similarly,
(16)τε˙T(t)ε˙(t)=τ[Yε(t)+Zε(t-τ)]T[Yε(t)+Zε(t-τ)]=τ[εT(t)YTYε(t)+εT(t-τ)ZTZε(t-τ)]+εT(t)YTZε(t-τ)+εT(t-τ)ZTYε(t)≤2τ[εT(t)YTYε(t)+εT(t-τ)ZTZε(t-τ)].
Substituting (15) and (16) into (14) leads to
(17)V˙(ε)≤εT(t)[-(PF+FTP)+βI2np+τ(2YTY+PZZTP)-(PF+FTP)+βI2np]ε(t)+εT(t-τ)[-(1-τ˙)βI2np+2τZTZ]ε(t-τ)≤εT(t)(-1+β+τ∥2YTY+PZZTP∥)ε(t)+εT(t-τ)[-(1-d2)β+2τ∥ZTZ∥]ε(t-τ).
Consequently, a sufficient condition for V˙(ε)<0 is
(18)-1+β+τ∥2YTY+PZZTP∥<0,-(1-d2)β+2τ∥ZTZ∥<0.
From (18), we can obtain that
(19)τ<τ0=12∥ZTZ∥/(1-d2)+∥2YTY+PZZTP∥.
Therefore, by Lyapunov-Krasovskii Theorem (see [23]), the error system (6) is uniformly asymptotically stable. Namely, the consensus is reached asymptotically.Remark 6.
It can be seen that many zoom techniques are applied during the derivation ofτ0, which result in a conservative estimation of τ0.


## 3.2. Time-Varying Formation for Trajectory Tracking

In this subsection, we consider the case that the agents form a time-varying formation as they track the desired trajectory. More specifically, the desired trajectory of the formation, represented byfc=[fcx(t)T,fcv(t)T], is assumed to be determined by the following equation:
(20)f˙cx=fcv=g(t,fcx).
Also make the following assumptions about the multiagent system:(A1)
the local interaction topology of the leaders has a spanning tree, and the leaders’ dynamic is unaffected by the followers;(A2)
the root agent in the local interaction topology of the leaders is able to access the reference trajectory;(A3)
in addition to the local information from their neighbors, the leaders can also obtain the global informationF;(A4)
in addition to the local information from their follower neighbors, each follower can also get the local formationW directly or indirectly from the leaders.Using the following control protocol:(21)x˙i=vi,i=1,2,…,n,v˙i=f˙iv+g˙(t,fcx)+∑j=1m‍aij{[(xj(t-τ)-fjx(t-τ))-(xi(t-τ)-fix(t-τ))xj(t-τ)-fjx(t-τ)]+k[(vj(t-τ)-fjv(t-τ))-(vi(t-τ)-fiv(t-τ))vj(t-τ)-fjv(t-τ)]}-bi[(xi(t-τ)-fix(t-τ)-fcx(t-τ))+k(vi(t-τ)-fiv(t-τ)-g(t-τ))],i∈ℛ,v˙i=∑k=1m‍wikv˙k+∑j=1n‍aij{[(xj(t-τ)-∑k=1m‍wjkxk(t-τ))-(xi(t-τ)-∑k=1m‍wikxk(t-τ))]+k[(vj(t-τ)-∑k=1m‍wjkvk(t-τ))-(vi(t-τ)-∑k=1m‍wikvk(t-τ))]}-bi[(xi(t-τ)-∑k=1m‍wikxk(t-τ))+k(vi(t-τ)-∑k=1m‍wikvk(t-τ))],i∈ℱ,
where k>0, A=(aij)n×n is the adjacency matrix corresponding to the graph 𝒢, and bi>0 if there is a spanning tree rooted at i in the graph 𝒢 and bi=0 otherwise, we have the following result.Theorem 7.
Suppose that the graph𝒢 has a spanning tree. Take
(22)k>k0=maxλ∈Λ(H)|Im(λ)||Re(λ)|·|λ|,0<τ<τ0=12∥ZTZ∥/(1-d2)+∥2YTY+PZZTP∥.
With protocol (21), the multiagent systems attain a time-varying formation for trajectory tracking (TVFT) under assumptions (A1)–(A4).Proof.
Letx~i=xi-fix-fcx, v~i=vi-fiv-g(t,fcx) for i∈ℛ, and x~i=xi-∑k=1mwikxk, v~i=vi-∑k=1mwikvk for i∈ℱ. Then we can rewrite protocol (21) as
(23)x~˙i=v~i,v~˙i=∑j=1n‍aij[(x~j(t-τ)-x~i(t-τ))+k(v~j(t-τ)-v~i(t-τ))]-bi(x~i(t-τ)+kv~i(t-τ)),i=1,2,…,n.
Denote x~=(x~1T,x~2T,…,x~nT) and v~=(v~1T,v~2T,…,v~nT); (23) can be expressed in a matrix form:
(24)x~˙=v~,v~˙=-(L+B)⊗Ipx~(t-τ)-k(L+B)⊗Ipv~(t-τ).
It follows from Theorem 5 that x~i and v~i converge to zero asymptotically, and equally, xi→fix+fcx, vi→fiv+fcv for i∈ℛ and xi→(Wi⊗Ip)xℛ, vi→(Wi⊗Ip)vℛ for i∈ℱ as t→∞.Remark 8.
Obviously, Theorem7 still holds if f˙ix=fiv=0, or f˙ix=fiv≠0 and f˙cx=fcv=g(t,fcx)=0. In the form case, the multiagent systems attain a time-invariant formation (TIF). In the latter case, the multiagent systems attain a time-varying formation.


## 4. Simulations

In this section, to illustrate our theoretical results derived in the above section, we will provide several examples. Consider a multiagent system consisting of 25 agents moving in a plane (i.e.,p=2), with the direct interaction topology described in Figure 1. The expected formation is a hexagram. Assume that the first 7 agents are leaders with the spanning tree rooted at agent 7. For simplicity, let aij=1 if agent j is a neighbor of agent i and aij=0 otherwise. With simple calculations, we can obtain that k0=0 and τ0=0.007. Take k=2, τ=0.0065|cost|, and d2=0.05.Figure 1
The interaction topology.Consider(25)fix=Ri(t)(cos(π2+2(i-1)3π)sin(π2+2(i-1)3π)),i=1,2,…,6,Ri(t)≡5(TIF),fix=(0,0)T,i=7,f˙ix=fiv,i=1,2,…,25,fcx=4(sin3tsin3t),fcv=f˙cx,Ri(t)=6sint(TVFT),w241=w246=w247=w3ii-1=w3ii-2=w3i7=13,i=3,4,…,7,w81=w251=w3i+1i-1=w3i+2i-1=23,i=3,4,…,7,w3i+1i-2=w3i+2i=w3i-17=w3i+17=16,i=3,4,…,6,w82=w3i+1i-2=w3i-17=w3i+17=16,i=7,8,wij=0,otherwise.Figure2 shows the trajectories of some agents in time-invariant formation (TIF), where the initial states of the agents are randomly generated in a given bounded region. From Figure 2, we can see that the agents attain the vertexes of a hexagram. In other words, the expected time-invariant formation is reached. For the trajectory tracking formation, Figure 3 illustrates the formation states of agents. Since the expected trajectory of the formation fcx satisfies the equation in (25), the formation in Figure 3 changes with time as it moves along a sinusoidal curve. Furthermore, the position and velocity errors of agents in TVFT with time delays existing only in the transmission of position are shown in Figures 4(a) and 5(a), respectively, and the position and velocity errors of agents in TVFT with time delays existing in the transmission of both position and velocity are shown in Figures 4(b) and 5(b), respectively. It can be seen that all of the errors converge to zero ultimately, while the errors in the latter figures converge to zero faster than those in the former figures.Figure 2
The trajectories of some agents in TIF.Figure 3
The formation states of agents in TVFT.The position errors of agents in TVFT.
(a)
Without velocity transmission delays
(b)
With velocity transmission delaysThe velocity errors of agents in TVFT.
(a)
Without velocity transmission delays
(b)
With velocity transmission delays


## 5. Conclusion

This paper investigates the formation control problem for second-order multiagent systems with time-varying delays. We first consider a leader-following consensus problem. By employing Lyapunov-Krasovskii functional, we prove that the multiagent systems can reach consensus. Then, under a special multiple leaders’ framework, we apply the protocol to the formation control, and derive a sufficient condition for the system to achieve prescribed formation. Moreover, several numerical simulations are shown to verify the theoretical analysis.



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