# Formation-Containment Control of Second-Order Multiagent Systems via Intermittent Communication

Authors: Mao-Dong Xia; Cheng-Lin Liu; Fei Liu
Journal: Complexity
 (2018)

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## Abstract

This paper investigates the formation-containment control of second-order multiagent systems with intermittent communication. Distributed coordination control algorithms are proposed under aperiodic intermittent communication, where each agent only communicates with its neighboring agents on some disconnected time intervals. By means of constructing Lyapunov functions, sufficient convergence conditions are obtained for the leaders reaching a prescribed formation asymptotically and the followers converging into the convex hull formed by leaders asymptotically, respectively. Besides, sufficient convergence conditions are also provided for second-order multiagent systems converging to the desired formation-containment under time-varying communication delay and intermittent communication. Finally, the validity of theoretical results is illustrated by numerical simulations.


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## Body

## 1. Introduction

In the past decades, coordination control of multiagent systems has drawn considerable attention due to its wide engineering applications, such as sensor networks [1], railway traffic control [2], formation control of robots [3], and so on. Furthermore, consensus seeking, formation control, and containment control have become the hot issues of coordination control of multiagent systems and have been extensively studied in various research fields, e.g., biology, physics, control theory, etc.Formation control concerned in this paper requires the agents to reach a prescribed formation by cooperating with each other locally, and the traditional formation control strategies include behavior [4], leader-follower [5, 6], virtual structure approach [7, 8], and consensus-based algorithms [9–12]. Up to now, the formation control problem has been widely studied for homogeneous and heterogeneous multiagent systems [13] with time-invariant or time-varying formation [14] and fixed or switching topologies [15].As a special leader-following coordination control of multiagent systems with multiple leaders, containment control requires the following agents to converge into a convex hull generated by the leaders asymptotically. Wang et al. [16] investigated the containment control of second-order multiagent systems with time-varying communication delays and proved that the containment control was achieved asymptotically as long as the interconnection topology was jointly connected. Mu et al. [17] studied the containment control problem of general linear multiagent systems with fixed topology and switching topologies and obtained some sufficient conditions by constructing Lyapunov functions. Hamed et al. [18] proposed the containment control algorithms for linear heterogeneous multiagent systems, and the convergence conditions were obtained for the followers converging into the convex hull formed by leaders based on output regulation techniques. Kan et al. [19] studied containment control of multiagent systems over directed random graphs to deal with random communication failure and got conditions based on a stochastic version of LaSalles Invariance Principle. In addition, Kan et al. [20] proposed a balanced containment control algorithm for driving the followers to the desired region but also equally space the agents in the desired region when driving the agents towards the target. Moreover, containment control problems were studied for second-order nonlinear multiagent systems, and a distributed algorithm was designed and analyzed on the basis of high-frequency feedback robust control [21].In some practical applications, the leaders not only provide a global reference state, but also are required to accomplish some assigned tasks cooperatively. However, there are no cooperation requirements on the leaders in [16–19, 21]. Specially, formation-containment control means that the leaders’ states achieve an expected formation while the followers’ states converge into the convex hull formed by leaders. Ferrari et al. [22] investigated a formation-containment control for first-order multiagent systems, and convergence conditions were obtained by using the partial difference equation on graph. Zheng et al. [23] studied the formation-containment control problem of second-order multiagent systems with only sampled position data, and got sufficient formation-containment conditions based on algebraic graph theory and matrix theory. In addition, Xia et al. [24] proposed the formation-containment control algorithms for second-order multiagent systems under time-varying delays, and two delay-dependent convergence conditions were obtained for the leaders and followers, respectively, based on Lyapunov-Krasovskii functional. Moreover, Dong et al. [25] analyzed the formation-containment problem of a high-order linear multiagent system with identical time-invariant delay and the convergence conditions were gained in the form of linear matrix inequalities. Dong et al. applied algorithms into a UAV platform, simulation and experimental results were shown the effectiveness of theoretical research in [26].In the above-mentioned works, evidently, the neighboring agents communicate with each other continuously. However, some practical situations require that the information can not be transmitted all the time because of instability of communication devices and limitation of technology. In addition, it is useful to save energy and reduce information exchange times, so some intermittent communication algorithms are applied. Therefore, the coordination control of multiagent systems with intermittent communication has received great interest recently. Huang et al. [27] investigated the consensus problem of second-order systems with partly and completely intermittent communications, and the delay-dependent consensus conditions were obtained by using graph theory and Lyapunov functions. Hu et al. [28] studied the leader-following consensus seeking via the intermittent control, and some sufficient consensus conditions were obtained based on the Lyapunov stability theory. Cheng et al. [29] investigated a decentralized formation control under limited and intermittent communication, and the convergence conditions were obtained by using a navigation function framework. Containment control algorithm under the intermittent sampled data was designed for second-order multiagent systems [30], and the necessary and sufficient conditions were dependent on the gain parameters, the sampling period and the communication width.Motivated by the above survey, we will consider the formation-containment problem of second-order multiagent systems with intermittent communication in this paper. The multiagent system consists of the leaders and followers, where each leader updates the states based on its neighboring leaders and the followers receive information from neighboring agents and leaders. The distributed coordination control algorithms are designed for the leaders and followers, respectively, based on the intermittent information. By transforming the formation-containment problem into an asymptotic stability problem and constructing proper Lyapunov functions, the sufficient convergence conditions are established for the leaders and followers achieving the desired formation-containment asymptotically.Compared with the algorithms proposed in this paper, the existing works [27, 31, 32] are seen as a special case. The contributions of this paper can be summarized into three aspects. Firstly, we study the formation-containment problem with intermittent communication, which is different from [25, 26], where the formation-containment problems were under the common assumption that information was transmitted among agents all the time. Secondly, the leaders and followers in this paper reach the desired formation-containment asymptotically within aperiodic intervals. However, intermittent communications were periodical in [27, 33, 34]. Thirdly, the leaders and followers suffer from the time-varying delays in this paper, but the reference [32], which investigated the consensus with intermittent communication, ignored the influence of communication delay.The rest of this paper is organized as follows. In Section2, some preliminaries are given, and the multiagent system with intermittent communication is formulated. In Section 3, sufficient conditions are obtained for the leaders reaching a prescribed formation and the followers converging into the convex hull formed by leaders, respectively. The effectiveness of our proposed control strategies is illustrated by numerical simulations in Section 4. The conclusion is presented in Section 5.Notation:Rm and Rm×m represent m×1 column vectors and m×m matrices, respectively. In and 1n denote n×n identity matrix and n×1 column vectors with all entries equivalent to 1. λ(L) denotes the eigenvalues of matrix L, and σ(L) indicates the largest singular value of matrix L. ⊗ stands for Kronecker product, and QT represents the transpose matrix of Q.


## 2. Preliminaries

In this section, the formation-containment control of second-order multiagent systems is formulated, and some useful lemmas and coordination control algorithms with intermittent communication are presented.


### 2.1. Formation-Containment Control of Second-Order Multiagent Systems

In this paper, the second-order multiagent systems consist ofM followers indexed by 1,2,⋯,M and N-M leaders indexed by M+1,M+2,⋯,N. The agents’ dynamics are given by(1)p˙it=qit,q˙it=uit,i=1,2,⋯,N,where pi(t)∈Rn, qi(t)∈Rn and ui(t)∈Rn are the position, velocity and input of agent i, respectively.Definition 1 (see [23–26]).
The multiagent system (1) is said to achieve the desired formation-containment asymptotically, if(2)limt→∞pit-gi-pjt-gj=0,i,j∈M+1,M+2,⋯,N,and(3)limt→∞pkt-∑j=M+1Nβk,jpjt=0,k=1,2,⋯,M,where gi∈Rn denotes the desired formation reference vector, and βk,j>0 satisfies ∑j=M+1N‍βk,j=1.The interconnection topology ofN agents (1) is usually denoted by a digraph G(V,E,A) with a set of nodes V={1,…,N}, a set of edges E⊆V×V, and a weighted adjacency matrix A=[aij]∈RN×N with aij≥0. eij=(i,j)∈E is a direct edge from node i to node j. aji>0 if and only if eij∈E, and we define aii=0,i=1,⋯,N. A set of neighbors of node i is denoted by Ni={j∈V:(j,i)∈E}. The Laplacian matrix L is defined as L=diag{∑j=1N‍aij,i∈V}-A. A digraph hasa directed spanning tree if there exists at least one node having a directed path to all the other nodes.In the digraphG of agents (1), GF and GL denote the interaction topologies of the followers and leaders separately. Then, the Laplacian matrix L is decomposed as L=L1L20L3, where L1∈RM×M and L3∈R(N-M)×(N-M) can be regarded as the Laplacian matrices of GF and GL, respectively, and L2∈RM×(N-M).In this paper, we will consider the following topology of agents (1).Assumption 2.
The leaders’ interaction topologyGL composed of the leaders has a directed spanning tree, and each follower has at least one leader that has a directed path to it.


### 2.2. Intermittent Coordination Control Algorithms

To solve the formation-containment control problem of agents (1), most of the existing algorithms are implemented on the basis of a common assumption that each agent communicates continuously with neighbors. In practical situations, however, the information may not be transmitted all the time due to instability of communication devices and limitation of technology. Hence, the following algorithms for the leaders and followers with intermittent communication are proposed, respectively.The main idea of intermittent communication is shown in Figure1. The communication periods are divided into work time and rest time, where the inputs are nonzero on [tk,tk+θk) and zero on [tk+θk,tk+1). The values of θk are not constants, which is different from [27, 33, 34].Figure 1
Intermittent communication.For the followers, the coordination control algorithms are designed as(4)uit=k11∑j∈Niaijpit-pjt+k12∑j∈Niaijqit-qjt,t∈tk,tk+θk,0,t∈tk+θk,tk+1,i=1,2,⋯,M,where k11<0 and k12<0, Ni means the neighbors of agent i, aij>0,j∈Ni and 0<θk≤tk+1-tk, k=0,1,⋯,n.The formation control algorithms for the leaders are constructed as(5)uit=k21∑j∈Niaijpit-gi-pjt-gj+k22∑j∈Niaijqit-qjt,t∈tk,tk+θk,0,t∈tk+θk,tk+1,i=M+1,M+2,…,N,where k21<0 and k22<0, Ni means the neighbors of agent i, aij>0,j∈Ni, and 0<θk≤tk+1-tk, k=0,1,⋯,n.The closed-loop forms of agents (1) under algorithms (4) and (5) are(6)p˙it=qit,q˙it=k11∑j∈Niaijpit-pjt+k12∑j∈Niaijqit-qjt,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1,i=1,2,⋯,M,and(7)p˙it=qit,q˙it=k21∑j∈Niaijpit-gi-pjt-gj+k22∑j∈Niaijqit-qjt,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1,i=M+1,⋯,N.


### 2.3. Useful Lemmas

The following lemmas will play important roles in deriving the main results.Lemma 3 (see [35]).
If the Assumption2 holds, L1 is nonsingular matrix, each entry of -L1-1L2 is nonnegative, and all row sums of -L1-1L2 equal to one.Lemma 4 (see [36]).
Suppose thatM∈Rn×n is a symmetric positive-definite matrix and N∈Rn×n is symmetric. Then for any vector x∈Rn, the following inequality holds:(8)λminM-1NxTMx≤xTNx≤λmaxM-1NxTMx.Lemma 5 (see [37]).
Letg:[t0-τ,∞)→[0,∞) be a continuous function such that(9)g˙t≤-agt+bsupt-τ<θ<tgθholds for t>t0, where a∈R and b∈R. If a>b>0, then,(10)gt≤supt-τ<θ<tgθe-ρt-t0,where ρ>0 is the smallest real root of the equation a-ρ-beρτ=0.Lemma 6 (see [38]).
For any vectorsx,y∈Rn and a positive-definite matrix Q∈Rn×n,(11)xTy+yTx≤xTQx+yTQ-1y holds.


## 2.1. Formation-Containment Control of Second-Order Multiagent Systems

In this paper, the second-order multiagent systems consist ofM followers indexed by 1,2,⋯,M and N-M leaders indexed by M+1,M+2,⋯,N. The agents’ dynamics are given by(1)p˙it=qit,q˙it=uit,i=1,2,⋯,N,where pi(t)∈Rn, qi(t)∈Rn and ui(t)∈Rn are the position, velocity and input of agent i, respectively.Definition 1 (see [23–26]).
The multiagent system (1) is said to achieve the desired formation-containment asymptotically, if(2)limt→∞pit-gi-pjt-gj=0,i,j∈M+1,M+2,⋯,N,and(3)limt→∞pkt-∑j=M+1Nβk,jpjt=0,k=1,2,⋯,M,where gi∈Rn denotes the desired formation reference vector, and βk,j>0 satisfies ∑j=M+1N‍βk,j=1.The interconnection topology ofN agents (1) is usually denoted by a digraph G(V,E,A) with a set of nodes V={1,…,N}, a set of edges E⊆V×V, and a weighted adjacency matrix A=[aij]∈RN×N with aij≥0. eij=(i,j)∈E is a direct edge from node i to node j. aji>0 if and only if eij∈E, and we define aii=0,i=1,⋯,N. A set of neighbors of node i is denoted by Ni={j∈V:(j,i)∈E}. The Laplacian matrix L is defined as L=diag{∑j=1N‍aij,i∈V}-A. A digraph hasa directed spanning tree if there exists at least one node having a directed path to all the other nodes.In the digraphG of agents (1), GF and GL denote the interaction topologies of the followers and leaders separately. Then, the Laplacian matrix L is decomposed as L=L1L20L3, where L1∈RM×M and L3∈R(N-M)×(N-M) can be regarded as the Laplacian matrices of GF and GL, respectively, and L2∈RM×(N-M).In this paper, we will consider the following topology of agents (1).Assumption 2.
The leaders’ interaction topologyGL composed of the leaders has a directed spanning tree, and each follower has at least one leader that has a directed path to it.


## 2.2. Intermittent Coordination Control Algorithms

To solve the formation-containment control problem of agents (1), most of the existing algorithms are implemented on the basis of a common assumption that each agent communicates continuously with neighbors. In practical situations, however, the information may not be transmitted all the time due to instability of communication devices and limitation of technology. Hence, the following algorithms for the leaders and followers with intermittent communication are proposed, respectively.The main idea of intermittent communication is shown in Figure1. The communication periods are divided into work time and rest time, where the inputs are nonzero on [tk,tk+θk) and zero on [tk+θk,tk+1). The values of θk are not constants, which is different from [27, 33, 34].Figure 1
Intermittent communication.For the followers, the coordination control algorithms are designed as(4)uit=k11∑j∈Niaijpit-pjt+k12∑j∈Niaijqit-qjt,t∈tk,tk+θk,0,t∈tk+θk,tk+1,i=1,2,⋯,M,where k11<0 and k12<0, Ni means the neighbors of agent i, aij>0,j∈Ni and 0<θk≤tk+1-tk, k=0,1,⋯,n.The formation control algorithms for the leaders are constructed as(5)uit=k21∑j∈Niaijpit-gi-pjt-gj+k22∑j∈Niaijqit-qjt,t∈tk,tk+θk,0,t∈tk+θk,tk+1,i=M+1,M+2,…,N,where k21<0 and k22<0, Ni means the neighbors of agent i, aij>0,j∈Ni, and 0<θk≤tk+1-tk, k=0,1,⋯,n.The closed-loop forms of agents (1) under algorithms (4) and (5) are(6)p˙it=qit,q˙it=k11∑j∈Niaijpit-pjt+k12∑j∈Niaijqit-qjt,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1,i=1,2,⋯,M,and(7)p˙it=qit,q˙it=k21∑j∈Niaijpit-gi-pjt-gj+k22∑j∈Niaijqit-qjt,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1,i=M+1,⋯,N.


## 2.3. Useful Lemmas

The following lemmas will play important roles in deriving the main results.Lemma 3 (see [35]).
If the Assumption2 holds, L1 is nonsingular matrix, each entry of -L1-1L2 is nonnegative, and all row sums of -L1-1L2 equal to one.Lemma 4 (see [36]).
Suppose thatM∈Rn×n is a symmetric positive-definite matrix and N∈Rn×n is symmetric. Then for any vector x∈Rn, the following inequality holds:(8)λminM-1NxTMx≤xTNx≤λmaxM-1NxTMx.Lemma 5 (see [37]).
Letg:[t0-τ,∞)→[0,∞) be a continuous function such that(9)g˙t≤-agt+bsupt-τ<θ<tgθholds for t>t0, where a∈R and b∈R. If a>b>0, then,(10)gt≤supt-τ<θ<tgθe-ρt-t0,where ρ>0 is the smallest real root of the equation a-ρ-beρτ=0.Lemma 6 (see [38]).
For any vectorsx,y∈Rn and a positive-definite matrix Q∈Rn×n,(11)xTy+yTx≤xTQx+yTQ-1y holds.


## 3. Main Results

In this section, the formation-containment problem is analyzed for second-order multiagent systems with aperiodic intermittent communication.


### 3.1. Intermittent Communication without Communication Delay

In this subsection, we will prove that systems (6) and (7) reach the desired formation-containment asymptotically within aperiodic intermittent intervals.Express the multiagent systems (6) and (7) in a multivariable form as(12)p˙Ft=qFt,q˙Ft=k11L1⊗InpFt+k11L2⊗InpLt+k12L1⊗InqFt+k12L2⊗InqLt,t∈tk,tk+θk,q˙Ft=0,t∈tk+θk,tk+1,and(13)p˙Lt=qLt,q˙Lt=k21L3⊗InpLt-k21L3⊗IngL+k22L3⊗InqLt,t∈tk,tk+θk,q˙Lt=0,t∈tk+θk,tk+1,where pF=[p1T,p2T,⋯,pMT]T,pL=[pM+1T,pM+2T,⋯,pNT]T, qF=[q1T,q2T,⋯,qMT]T, qL=[qM+1T,qM+2T,⋯,qNT]T, and gL=[gM+1T,gM+2T,⋯,gNT]T.Definep¯F(t)=pF(t)-((-L1-1L2)⊗In)pL(t) and q¯F(t)=qF(t)-((-L1-1L2)⊗In)qL(t), and we reformulate the multiagent system (12) as(14)p¯˙Ft=q¯Ft,q¯˙Ft=k11L1⊗Inp¯Ft+k12L1⊗Inq¯Ft--L1-1L2⊗Inq˙Lt,t∈tk,tk+θk,q¯˙Ft=0,t∈tk+θk,tk+1,where p¯F=[p¯1T,p¯2T,…,p¯MT]T and q¯F=[q¯1T,q¯2T,…,q¯MT]T.Definep¯i(t)=pi(t)-gi, i=M+1,M+2,⋯,N and p¯L=[p¯M+1T,p¯M+2T,…,p¯NT]T, and we rewrite the multiagent system (13) as(15)p¯˙Lt=qLt,q˙Lt=k21L3⊗Inp¯Lt+k22L3⊗InqLt,t∈tk,tk+θk,q˙Lt=0,t∈tk+θk,tk+1.The eigenvalues of the Laplacian matrix L3 of N-M leaders are defined as λiL,i∈{M+1,M+2,⋯,N}, and the corresponding eigenvectors are ui. We assume that λM+1L=0 is corresponding to associated eigenvector uM+1=1N-M and 0<Re(λM+2L)≤…≤Re(λNL). Then, the Jordan canonical form J3 of the Laplacian matrix L3 is given by(16)J3=UL-1L3UL=diag0,J¯3=diag0,λM+2L,…,λNL,where UL=uM+1,uM+2,…,uN and UL-1=u~M+1,u~M+2,…,u~NT,Definingς(t)=(UL-1⊗In)(p¯LT,qLT)T yields(17)ς˙t=0IN-Mk21J3k22J3⊗Inςt,t∈tk,tk+θk,ς˙t=0IN-M00⊗Inςt,t∈tk+θk,tk+1,where ς=[ςM+1T,ςM+2T,⋯,ςNT]T. Hence, the formation control problem of multiagent system (7) is solved if and only if ςi(t),i=M+2,M+3,⋯,N converge to zero asymptotically.Denotingς~(t)=(ςM+2T(t),⋯,ςNT(t))T yields(18)ς~˙t=E1⊗Inς~t,t∈tk,tk+θk,ς~˙t=E2⊗Inς~t,t∈tk+θk,tk+1,where E1=0IN-M-1k21J¯3k22J¯3 and E2=0IN-M-100.Lemma 7.
The matrixE1 is Hurwitz if and only if(19)k222R3λiL+k222RλiLI2λiL>-k21I2λiL,where R(·) and I(·) represent real and imaginary parts of complex number.Proof.
Characteristic polynomial ofE1 is given by (20)det⁡μI2N-M-1-E1=det⁡μIN-M-1-IN-M-1-k21J¯3μIN-M-1-k22J¯3=det⁡μ2IN-M-1-k22μJ¯3-k21J¯3=∏i=M+2Nμ2-k22μλiL-k21λiL=∏i=M+2Nμ2+-k22μ-k21RλiL+i​∗IλiL.Using Generalized Routh Criterion, the eigenvalues of E1 lie on the open left half complex plane if and only if R(λiL)>0 and (19) hold. If Assumption 2 holds, RλiL>0,i=M+2,M+3,⋯,N. Lemma 7 is proved.Theorem 8.
The multiagent systems (6) and (7) with Assumption 2 reach the desired formation-containment asymptotically, if the following conditions are satisfied simultaneously.
( i )   k 22 2 R 3 ( λ i L ) + k 22 2 R ( λ i L ) I 2 ( λ i L ) > - k 21 I 2 ( λ i L ), i=M+2,M+3,⋯,N.
( i i )   γ 1 / γ 2 > θ ¯ / θ, where θ¯=max{tk+1-tk-θk} and θ=min{θk} with k=0,1,⋯,n, and γ1=λmin(PL-1QL) and γ2=λmax(PL-1(E2PL+PLE2T)) with positive-definite matrices PL>0 and QL>0 satisfying E1PL+PLE1T=-QL.
( i i i )   k 12 2 R 3 ( λ i F ) + k 12 2 R ( λ i F ) I 2 ( λ i F ) > - k 11 I 2 ( λ i F ), where λiF are the eigenvalues of L1, i=1,2,⋯,M.
( i v )   γ 3 / γ 4 > θ ¯ / θ, where γ3=λmin(PF-1QF) and γ4=λmax(PF-1(F2PF+PFF2T)) with positive-definite matrices PF and QF satisfying F1PF+PFF1T=-QF, F1=0IMk11L1k12L1 and F2=0IM00.Proof.
Step 1. Asymptotic convergence of leaders’ formation is equivalent to the asymptotical stability of system (18).
The matrixE1 is Hurwitz from condition (i) and Lemma 7, so there exists a unique PL=PLT>0 satisfying E1PL+PLE1T=-QL with a given positive-definite matrix QL. Choose a Lyapunov function for (18) as follows:(21)Vt=12ς~TtPL⊗Inς~t.It follows from Lemma 4 that the time derivative of V(t) along the trajectory of system (18) with t∈[tk,tk+θk), k=0,1,⋯,n is(22)V˙t=12ς~TtE1PL+PLE1T⊗Inς~t=-12ς~TtQL⊗Inς~t≤-12λminPL-1QLς~TtPL⊗Inς~t=-γ1Vt,where γ1=λmin(PL-1QL). Meanwhile, the time derivative of V(t) along the trajectory of the system (18) with t∈[tk+θk,tk+1), k=0,1,⋯,n is(23)V˙t=12ς~TtE2PL+PLE2T⊗Inς~t≤12λmaxE2PL+PLE2Tς~Ttς~t≤12λmaxPL-1E2PL+PLE2Tς~TtPL⊗Inς~t=γ2Vt,where γ2=λmax(PL-1(E2PL+PLE2T)).
It follows from (22) that(24)Vt≤V0e-γ1θ0+θ1+…+θk-1+t-tk+γ2θ¯0+θ¯1+…+θ¯k-1≤V0e-γ1kθ+γ2kθ¯holds with t∈[tk,tk+θk), where θ and θ¯ are defined in condition (ii) of Theorem 8. In (24), θ and θ¯ are the minimum control time and the maximum communication loss time, respectively. Meanwhile, we obtain from (23) that(25)Vt≤V0e-γ1θ0+θ1+…+θk+γ2θ¯0+θ¯1+…+θ¯k-1+t-tk-θk≤V0e-γ1k+1θ+γ2k+1θ¯holds with t∈[tk+θk,tk+1).
We conclude from the above analysis that(26)Vt≤V0e-γ1kθ+γ2k+1θ¯holds for arbitrary t∈[tk,tk+1), where γ1θ>γ2θ¯>0 for the condition (ii). Hence, the leaders reach the desired formation asymptotically.
Step 2. Containment convergence analysis of the followers.
If the multiagent systems (13) achieve the desired formation, we obtain limt→∞q˙L=0. Then, the system (14) becomes(27)p¯˙Ftq¯˙Ft=F1⊗Inp¯Ftq¯Ft,t∈tk,tk+θk,p¯˙Ftq¯˙Ft=F2⊗Inp¯Ftq¯Ft,t∈tk+θk,tk+1,where F1 and F2 are defined in the condition (iv). Similar to Step 1, we analyze the asymptotic stability of system (27) and prove that the followers converge to the convex hull of the leaders if the condition (iii) and (iv) hold. Theorem 8 is proved.Remark 9.
Different form the references [27, 33, 34] that investigated the multiagent systems with periodical intermittent communication, Theorem 8 provides the conditions for the agents (6) and (7) reaching the desired formation-containment asymptotically within aperiodic intervals.


### 3.2. Intermittent Communication with Time-Varying Delay

In this subsection, the formation-containment control is studied for the second-order multiagent systems with completely intermittent communication and time-varying communication delay.For the followers, the containment control algorithms under time-varying communication delay become(28)uit=k11∑j∈Niaijpit-τ1t-pjt-τ1t+k12∑j∈Niaijqit-τ1t-qjt-τ1t,t∈tk,tk+θk,0,t∈tk+θk,tk+1,where i=1,2,…,M and τ1(t)>0 is the time-varying communication delay. For the leaders, the formation control algorithms under time-varying communication delay turn to be(29)uit=k21∑j∈Niaijpit-τ2t-gi-pjt-τ2t-gj+k22∑j∈Niaijqit-τ2t-qjt-τ2t,t∈tk,tk+θk,0,t∈tk+θk,tk+1,where i=M+1,M+2,…,N and τ2(t)>0 is the time-varying communication delay.The closed-loop forms of agents (1) with (28) and (29) are given by(30)p˙it=qit,q˙it=k11∑j∈Niaijpit-τ1t-pjt-τ1t+k12∑j∈Niaijqit-τ1t-qjt-τ1t,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1,and(31)p˙it=qit,q˙it=k21∑j∈Niaijpit-τ2t-gi-pjt-τ2t-gj+k22∑j∈Niaijqit-τ2t-qjt-τ2t,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1.Similar to the above subsection, we define p¯F(t)=pF(t)-((-L1-1L2)⊗In)pL(t), q¯F(t)=qF(t)-((-L1-1L2)⊗In)qL(t), where pF=[p1T,p2T,⋯,pMT]T,pL=[pM+1T,pM+2T,⋯,pNT]T, qF=[q1T,q2T,⋯,qMT]T, qL=[qM+1T,qM+2T,⋯,qNT]T, and gL=[gM+1T,gM+2T,⋯,gNT]T. Then, we get(32)p¯˙Ft=q¯Ft,q¯˙Ft=k11L1⊗Inp¯Ft-τ1t+k12L1⊗Inq¯Ltt-τ1t--L1-1L2⊗Inq˙Lt-τ1t,t∈tk,tk+θk,q¯˙Ft=0,t∈tk+θk,tk+1,where p¯F=[p¯1T,p¯2T,⋯,p¯MT]T and q¯F=[q¯1T,q¯2T,⋯,q¯MT]T.Definep¯i(t)=pi(t)-gi, i=M+1,M+2,⋯,N, and p¯L=[p¯M+1T,p¯M+2T,⋯,p¯NT]T, and we rewrite multiagent system (31) as(33)p¯˙Lt=qLt,q˙Lt=k21L3⊗Inp¯Lt-τ2t+k22L3⊗InqLt-τ2t,t∈tk,tk+θk,q˙Lt=0,t∈tk+θk,tk+1.Denote ς(t)=(UL-1⊗In)(p¯LT,qLT)T, and we get(34)ς˙t=0IN-M00⊗Inςt+00k21J3k22J3⊗Inςt-τ2t,t∈tk,tk+θk,ς˙t=0IN-M00⊗Inςt,t∈tk+θk,tk+1,where ς=[ςM+1T,ςM+2T,⋯,ςNT]T. Hence, the formation control problem of multiagent system (31) is solved if and only if limt→∞ςi(t)=0,i=M+2,M+3,⋯,N.Use the Newton-Raphson formula:ς(t-τ2(t)))=ς(t)-∫t-τ2(t)t‍ς˙(s)ds, and denoting ς~(t)=(ςM+2T(t),⋯,ςNT(t))T yields(35)ς~˙t=E3⊗Inς~t-∫t-τ2ttE4⊗Inς~˙sds,t∈tk,tk+θk,ς~˙t=E5⊗Inς~t,t∈tk+θk,tk+1,where E3=0IN-M-1k21J¯3k22J¯3, E4=00k21J¯3k22J¯3, and E5=0IN-M-100.Before presenting the result, we make the following assumption on the time-varying delay.Assumption 10.
0 ≤ τ 1 ( t ) ≤ τ ¯ 0 and 0≤τ2(t)≤τ0, where τ¯0 and τ0 are known constants.Theorem 11.
The multiagent systems (30) and (31) with Assumptions 2 and 10 reach the desired formation-containment asymptotically, if the following conditions are satisfied simultaneously:
( i )   k 22 2 R 3 ( λ i L ) + k 22 2 R ( λ i L ) I 2 ( λ i L ) > - k 21 I 2 ( λ i L ), i=M+2,M+3,⋯,N.
( i i ) (36) min ⁡ θ 2 , γ 0 θ - γ 3 θ ¯ 2 γ 0 , h λ m i n Q L - h 2 σ 2 P L E 3 σ 2 E 3 + σ 2 E 4 λ m a x P L λ m a x P L - 1 > τ 0 ,where h>0, θ=min{θk} and θ¯=max{tk+1-tk-θk} with k=0,1,⋯,n, γ0>0 is the unique real root of the equation γ1-γ0-γ2e2γ0τ0=0, γ1=λmin(QL)-h/λmax(PL)>0,γ2=2τ02σ2(PLE3)(σ2(E3)+σ2(E4))λmax(PL-1)/h, and γ3=λmax(PL-1(E5PL+PLE5T)) with positive-definite matrices PL and QL satisfying the Lyapunov equation E1PL+PLE1T=-QL,
( i i i )   k 12 2 R 3 ( λ i F ) + k 12 2 R ( λ i F ) I 2 ( λ i F ) > - k 11 I 2 ( λ i F ), i=1,2,⋯,M.
( i v ) (37) min ⁡ θ 2 , γ 4 θ - γ 7 θ ¯ 2 γ 4 , h λ m i n P F - h 2 σ 2 P F F 3 σ 2 F 3 + σ 2 F 4 λ m a x P F λ m a x P F - 1 > τ ¯ 0 ,where γ4>0 is the unique real root of the equation γ5-γ4-γ6e2γ4τ¯0=0,γ5=λmin(QF)-h/λmax(PF)>0,γ6=2τ02σ2(PFF3)(σ2(F3)+σ2(F4))λmax(PF-1)/h, and γ7=λmax(PF-1(F5PF+PFF5T)), with positive-definite matrix PF and QF satisfying Lyapunov equation(38)F1PF+PFF1T=-QF,F3=0IMk11L1k12L1,F4=00k11L3k12L3andF5=0IM00.Proof.
Step 1. Asymptotic convergence of leaders’ formation is equivalent to the asymptotical stability of the system (35).
The matrixE3 is Hurwitz from condition (i) and Lemma 7, so there exists a unique PL=PLT>0 satisfying E3PL+PLE3T=-QL with a given positive-definite matrix QL>0. Choose a Lyapunov function for (35) as follows:(39)Vt=12ς~TtPL⊗Inς~t.For t∈[tk,tk+θk), k=0,1,⋯,n, the time derivative of V(t) along the trajectory of system (35) is(40)V˙t=12ς~TtE3PL+PLE3T⊗Inς~t-12ς~TtPLE4⊗In∫t-τ2ttς~˙sds-12PLE4⊗In∫t-τ2ttς~˙sdsTς~t≤-12λminQL-hς~Ttς~t+12hPLE4⊗In∫t-τ2ttς~˙sdsTPLE4⊗In∫t-τ2ttς~˙sds≤-12λminQL-hς~Ttς~t+τ0σ2PLE42h∫t-τ2ttς~˙Tsς~˙sds≤-12λminQL-hς~Ttς~t+τ02σ2PLE4σ2E3+σ2E4hsupt-2τ0≤ϑ≤tς~Tϑςϑ≤-γ1Vt+γ2supt-2τ0≤ϑ≤tVϑ,where γ1>γ2>0 holds for the condition (ii). In the light of Lemma 5, we obtain that(41)Vt≤suptk-2τ0≤ϑ≤tkVϑe-γ0t-tkholds with t∈[tk,tk+θk), where γ0>0 is the unique real root of the equation γ1-γ0-γ2e2γ0τ0=0.
Fort∈[tk+θk,tk+1), k=0,1,⋯,N, the time derivative of V(t) along the trajectory of system (35) is(42)V˙t=12ς~TtE5PL+PLE5T⊗Inς~t≤12λmaxE5PL+PLE5Tς~Ttς~t≤12λmaxPL-1E5PL+PLE5Tς~TtPL⊗Inς~t=γ3Vt,where γ3=λmax(PL-1(E5PL+PLE5T)). Then, we get(43)Vt≤Vtk+θkeγ3t-tk-θk≤suptk+θk-2τ0≤ϑk≤tk+θkVϑeγ3t-tk-θk,holds with t∈[tk+θk,tk+1).
Therefore, we come to the conclusion that(44)Vt≤suptk-2τ0≤ϑ≤tkVϑe-γ0t-tk,t∈tk,tk+θk,suptk+θk-2τ0≤ϑ≤tk+θkVϑeγ3t-tk-θk,t∈tk+θk,tk+1.It follows from (44) that(45)Vt≤sup-2τ0≤ϑ≤0Vϑe-γ0θ0-2τ0+θ1-2τ0+…+θk-1-2τ0+t-tk+γ3θ¯0+θ¯1+…+θ¯k-1≤sup-2τ0≤ϑ≤0Vϑe-γ0kθ-2τ0+γ3kθ¯,holds with t∈[tk,tk+θk), where θ and θ¯ are defined in the condition (ii) of Theorem 11. For t∈[tk+θk,tk+1),(46)Vt≤sup-2τ0≤ϑ≤0Vϑe-γ0θ0-2τ0+θ1-2τ0+…+θk-2τ0+γ3θ¯0+θ¯1+…+θ¯k-1+t-tk-θk≤sup-2τ0≤ϑ≤0Vϑe-γ0k+1θ-2τ0+γ3k+1θ¯.In conclusion, we get(47)Vt≤sup-2τ0≤ϑ≤0Vϑe-γ0kθ-2τ0+γ3k+1θ¯,t∈0,+∞,where γ0(θ-2τ0)>γ3θ¯>0 holds for the condition (ii). Therefore, the leaders reach the desired formation.
Step 2. The followers’ containment convergence analysis.
If the multiagent system (31) achieves the desired formation, we obtain limt→∞q˙L(t)=0. Then, the system (32) becomes(48)p¯˙Ftq¯˙Ft=F3⊗Inp¯Ftq¯Ft-∫t-τ1ttF4⊗Inp¯˙Fsq¯˙Fsds,t∈tk,tk+θk,p¯˙Ftq¯˙Ft=F5⊗Inp¯Ftq¯Ft,t∈tk+θk,tk+1.where F3, F4, and F5 are defined in the condition (iv). Similar to Step 1, we analyze the asymptotic stability of system (48) and prove that the followers converge to the convex hull of the leaders if the conditions (iii) and (iv) hold. Theorem 11 is proved.Remark 12.
In [25], Dong et al. discussed the formation-containment control with identical time-invariant delay in which agents received information all the time. In addition, Wang et al. [39] investigated the containment control of second-order multiagent systems with multiple leaders and obtained the conditions for the agents under symmetric topologies and time-varying delay. Different above references, we consider the formation-containment control of second-order multiagent systems with different time-varying delays and gain the convergence conditions under general directed topology.


## 3.1. Intermittent Communication without Communication Delay

In this subsection, we will prove that systems (6) and (7) reach the desired formation-containment asymptotically within aperiodic intermittent intervals.Express the multiagent systems (6) and (7) in a multivariable form as(12)p˙Ft=qFt,q˙Ft=k11L1⊗InpFt+k11L2⊗InpLt+k12L1⊗InqFt+k12L2⊗InqLt,t∈tk,tk+θk,q˙Ft=0,t∈tk+θk,tk+1,and(13)p˙Lt=qLt,q˙Lt=k21L3⊗InpLt-k21L3⊗IngL+k22L3⊗InqLt,t∈tk,tk+θk,q˙Lt=0,t∈tk+θk,tk+1,where pF=[p1T,p2T,⋯,pMT]T,pL=[pM+1T,pM+2T,⋯,pNT]T, qF=[q1T,q2T,⋯,qMT]T, qL=[qM+1T,qM+2T,⋯,qNT]T, and gL=[gM+1T,gM+2T,⋯,gNT]T.Definep¯F(t)=pF(t)-((-L1-1L2)⊗In)pL(t) and q¯F(t)=qF(t)-((-L1-1L2)⊗In)qL(t), and we reformulate the multiagent system (12) as(14)p¯˙Ft=q¯Ft,q¯˙Ft=k11L1⊗Inp¯Ft+k12L1⊗Inq¯Ft--L1-1L2⊗Inq˙Lt,t∈tk,tk+θk,q¯˙Ft=0,t∈tk+θk,tk+1,where p¯F=[p¯1T,p¯2T,…,p¯MT]T and q¯F=[q¯1T,q¯2T,…,q¯MT]T.Definep¯i(t)=pi(t)-gi, i=M+1,M+2,⋯,N and p¯L=[p¯M+1T,p¯M+2T,…,p¯NT]T, and we rewrite the multiagent system (13) as(15)p¯˙Lt=qLt,q˙Lt=k21L3⊗Inp¯Lt+k22L3⊗InqLt,t∈tk,tk+θk,q˙Lt=0,t∈tk+θk,tk+1.The eigenvalues of the Laplacian matrix L3 of N-M leaders are defined as λiL,i∈{M+1,M+2,⋯,N}, and the corresponding eigenvectors are ui. We assume that λM+1L=0 is corresponding to associated eigenvector uM+1=1N-M and 0<Re(λM+2L)≤…≤Re(λNL). Then, the Jordan canonical form J3 of the Laplacian matrix L3 is given by(16)J3=UL-1L3UL=diag0,J¯3=diag0,λM+2L,…,λNL,where UL=uM+1,uM+2,…,uN and UL-1=u~M+1,u~M+2,…,u~NT,Definingς(t)=(UL-1⊗In)(p¯LT,qLT)T yields(17)ς˙t=0IN-Mk21J3k22J3⊗Inςt,t∈tk,tk+θk,ς˙t=0IN-M00⊗Inςt,t∈tk+θk,tk+1,where ς=[ςM+1T,ςM+2T,⋯,ςNT]T. Hence, the formation control problem of multiagent system (7) is solved if and only if ςi(t),i=M+2,M+3,⋯,N converge to zero asymptotically.Denotingς~(t)=(ςM+2T(t),⋯,ςNT(t))T yields(18)ς~˙t=E1⊗Inς~t,t∈tk,tk+θk,ς~˙t=E2⊗Inς~t,t∈tk+θk,tk+1,where E1=0IN-M-1k21J¯3k22J¯3 and E2=0IN-M-100.Lemma 7.
The matrixE1 is Hurwitz if and only if(19)k222R3λiL+k222RλiLI2λiL>-k21I2λiL,where R(·) and I(·) represent real and imaginary parts of complex number.Proof.
Characteristic polynomial ofE1 is given by (20)det⁡μI2N-M-1-E1=det⁡μIN-M-1-IN-M-1-k21J¯3μIN-M-1-k22J¯3=det⁡μ2IN-M-1-k22μJ¯3-k21J¯3=∏i=M+2Nμ2-k22μλiL-k21λiL=∏i=M+2Nμ2+-k22μ-k21RλiL+i​∗IλiL.Using Generalized Routh Criterion, the eigenvalues of E1 lie on the open left half complex plane if and only if R(λiL)>0 and (19) hold. If Assumption 2 holds, RλiL>0,i=M+2,M+3,⋯,N. Lemma 7 is proved.Theorem 8.
The multiagent systems (6) and (7) with Assumption 2 reach the desired formation-containment asymptotically, if the following conditions are satisfied simultaneously.
( i )   k 22 2 R 3 ( λ i L ) + k 22 2 R ( λ i L ) I 2 ( λ i L ) > - k 21 I 2 ( λ i L ), i=M+2,M+3,⋯,N.
( i i )   γ 1 / γ 2 > θ ¯ / θ, where θ¯=max{tk+1-tk-θk} and θ=min{θk} with k=0,1,⋯,n, and γ1=λmin(PL-1QL) and γ2=λmax(PL-1(E2PL+PLE2T)) with positive-definite matrices PL>0 and QL>0 satisfying E1PL+PLE1T=-QL.
( i i i )   k 12 2 R 3 ( λ i F ) + k 12 2 R ( λ i F ) I 2 ( λ i F ) > - k 11 I 2 ( λ i F ), where λiF are the eigenvalues of L1, i=1,2,⋯,M.
( i v )   γ 3 / γ 4 > θ ¯ / θ, where γ3=λmin(PF-1QF) and γ4=λmax(PF-1(F2PF+PFF2T)) with positive-definite matrices PF and QF satisfying F1PF+PFF1T=-QF, F1=0IMk11L1k12L1 and F2=0IM00.Proof.
Step 1. Asymptotic convergence of leaders’ formation is equivalent to the asymptotical stability of system (18).
The matrixE1 is Hurwitz from condition (i) and Lemma 7, so there exists a unique PL=PLT>0 satisfying E1PL+PLE1T=-QL with a given positive-definite matrix QL. Choose a Lyapunov function for (18) as follows:(21)Vt=12ς~TtPL⊗Inς~t.It follows from Lemma 4 that the time derivative of V(t) along the trajectory of system (18) with t∈[tk,tk+θk), k=0,1,⋯,n is(22)V˙t=12ς~TtE1PL+PLE1T⊗Inς~t=-12ς~TtQL⊗Inς~t≤-12λminPL-1QLς~TtPL⊗Inς~t=-γ1Vt,where γ1=λmin(PL-1QL). Meanwhile, the time derivative of V(t) along the trajectory of the system (18) with t∈[tk+θk,tk+1), k=0,1,⋯,n is(23)V˙t=12ς~TtE2PL+PLE2T⊗Inς~t≤12λmaxE2PL+PLE2Tς~Ttς~t≤12λmaxPL-1E2PL+PLE2Tς~TtPL⊗Inς~t=γ2Vt,where γ2=λmax(PL-1(E2PL+PLE2T)).
It follows from (22) that(24)Vt≤V0e-γ1θ0+θ1+…+θk-1+t-tk+γ2θ¯0+θ¯1+…+θ¯k-1≤V0e-γ1kθ+γ2kθ¯holds with t∈[tk,tk+θk), where θ and θ¯ are defined in condition (ii) of Theorem 8. In (24), θ and θ¯ are the minimum control time and the maximum communication loss time, respectively. Meanwhile, we obtain from (23) that(25)Vt≤V0e-γ1θ0+θ1+…+θk+γ2θ¯0+θ¯1+…+θ¯k-1+t-tk-θk≤V0e-γ1k+1θ+γ2k+1θ¯holds with t∈[tk+θk,tk+1).
We conclude from the above analysis that(26)Vt≤V0e-γ1kθ+γ2k+1θ¯holds for arbitrary t∈[tk,tk+1), where γ1θ>γ2θ¯>0 for the condition (ii). Hence, the leaders reach the desired formation asymptotically.
Step 2. Containment convergence analysis of the followers.
If the multiagent systems (13) achieve the desired formation, we obtain limt→∞q˙L=0. Then, the system (14) becomes(27)p¯˙Ftq¯˙Ft=F1⊗Inp¯Ftq¯Ft,t∈tk,tk+θk,p¯˙Ftq¯˙Ft=F2⊗Inp¯Ftq¯Ft,t∈tk+θk,tk+1,where F1 and F2 are defined in the condition (iv). Similar to Step 1, we analyze the asymptotic stability of system (27) and prove that the followers converge to the convex hull of the leaders if the condition (iii) and (iv) hold. Theorem 8 is proved.Remark 9.
Different form the references [27, 33, 34] that investigated the multiagent systems with periodical intermittent communication, Theorem 8 provides the conditions for the agents (6) and (7) reaching the desired formation-containment asymptotically within aperiodic intervals.


## 3.2. Intermittent Communication with Time-Varying Delay

In this subsection, the formation-containment control is studied for the second-order multiagent systems with completely intermittent communication and time-varying communication delay.For the followers, the containment control algorithms under time-varying communication delay become(28)uit=k11∑j∈Niaijpit-τ1t-pjt-τ1t+k12∑j∈Niaijqit-τ1t-qjt-τ1t,t∈tk,tk+θk,0,t∈tk+θk,tk+1,where i=1,2,…,M and τ1(t)>0 is the time-varying communication delay. For the leaders, the formation control algorithms under time-varying communication delay turn to be(29)uit=k21∑j∈Niaijpit-τ2t-gi-pjt-τ2t-gj+k22∑j∈Niaijqit-τ2t-qjt-τ2t,t∈tk,tk+θk,0,t∈tk+θk,tk+1,where i=M+1,M+2,…,N and τ2(t)>0 is the time-varying communication delay.The closed-loop forms of agents (1) with (28) and (29) are given by(30)p˙it=qit,q˙it=k11∑j∈Niaijpit-τ1t-pjt-τ1t+k12∑j∈Niaijqit-τ1t-qjt-τ1t,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1,and(31)p˙it=qit,q˙it=k21∑j∈Niaijpit-τ2t-gi-pjt-τ2t-gj+k22∑j∈Niaijqit-τ2t-qjt-τ2t,t∈tk,tk+θk,q˙it=0,t∈tk+θk,tk+1.Similar to the above subsection, we define p¯F(t)=pF(t)-((-L1-1L2)⊗In)pL(t), q¯F(t)=qF(t)-((-L1-1L2)⊗In)qL(t), where pF=[p1T,p2T,⋯,pMT]T,pL=[pM+1T,pM+2T,⋯,pNT]T, qF=[q1T,q2T,⋯,qMT]T, qL=[qM+1T,qM+2T,⋯,qNT]T, and gL=[gM+1T,gM+2T,⋯,gNT]T. Then, we get(32)p¯˙Ft=q¯Ft,q¯˙Ft=k11L1⊗Inp¯Ft-τ1t+k12L1⊗Inq¯Ltt-τ1t--L1-1L2⊗Inq˙Lt-τ1t,t∈tk,tk+θk,q¯˙Ft=0,t∈tk+θk,tk+1,where p¯F=[p¯1T,p¯2T,⋯,p¯MT]T and q¯F=[q¯1T,q¯2T,⋯,q¯MT]T.Definep¯i(t)=pi(t)-gi, i=M+1,M+2,⋯,N, and p¯L=[p¯M+1T,p¯M+2T,⋯,p¯NT]T, and we rewrite multiagent system (31) as(33)p¯˙Lt=qLt,q˙Lt=k21L3⊗Inp¯Lt-τ2t+k22L3⊗InqLt-τ2t,t∈tk,tk+θk,q˙Lt=0,t∈tk+θk,tk+1.Denote ς(t)=(UL-1⊗In)(p¯LT,qLT)T, and we get(34)ς˙t=0IN-M00⊗Inςt+00k21J3k22J3⊗Inςt-τ2t,t∈tk,tk+θk,ς˙t=0IN-M00⊗Inςt,t∈tk+θk,tk+1,where ς=[ςM+1T,ςM+2T,⋯,ςNT]T. Hence, the formation control problem of multiagent system (31) is solved if and only if limt→∞ςi(t)=0,i=M+2,M+3,⋯,N.Use the Newton-Raphson formula:ς(t-τ2(t)))=ς(t)-∫t-τ2(t)t‍ς˙(s)ds, and denoting ς~(t)=(ςM+2T(t),⋯,ςNT(t))T yields(35)ς~˙t=E3⊗Inς~t-∫t-τ2ttE4⊗Inς~˙sds,t∈tk,tk+θk,ς~˙t=E5⊗Inς~t,t∈tk+θk,tk+1,where E3=0IN-M-1k21J¯3k22J¯3, E4=00k21J¯3k22J¯3, and E5=0IN-M-100.Before presenting the result, we make the following assumption on the time-varying delay.Assumption 10.
0 ≤ τ 1 ( t ) ≤ τ ¯ 0 and 0≤τ2(t)≤τ0, where τ¯0 and τ0 are known constants.Theorem 11.
The multiagent systems (30) and (31) with Assumptions 2 and 10 reach the desired formation-containment asymptotically, if the following conditions are satisfied simultaneously:
( i )   k 22 2 R 3 ( λ i L ) + k 22 2 R ( λ i L ) I 2 ( λ i L ) > - k 21 I 2 ( λ i L ), i=M+2,M+3,⋯,N.
( i i ) (36) min ⁡ θ 2 , γ 0 θ - γ 3 θ ¯ 2 γ 0 , h λ m i n Q L - h 2 σ 2 P L E 3 σ 2 E 3 + σ 2 E 4 λ m a x P L λ m a x P L - 1 > τ 0 ,where h>0, θ=min{θk} and θ¯=max{tk+1-tk-θk} with k=0,1,⋯,n, γ0>0 is the unique real root of the equation γ1-γ0-γ2e2γ0τ0=0, γ1=λmin(QL)-h/λmax(PL)>0,γ2=2τ02σ2(PLE3)(σ2(E3)+σ2(E4))λmax(PL-1)/h, and γ3=λmax(PL-1(E5PL+PLE5T)) with positive-definite matrices PL and QL satisfying the Lyapunov equation E1PL+PLE1T=-QL,
( i i i )   k 12 2 R 3 ( λ i F ) + k 12 2 R ( λ i F ) I 2 ( λ i F ) > - k 11 I 2 ( λ i F ), i=1,2,⋯,M.
( i v ) (37) min ⁡ θ 2 , γ 4 θ - γ 7 θ ¯ 2 γ 4 , h λ m i n P F - h 2 σ 2 P F F 3 σ 2 F 3 + σ 2 F 4 λ m a x P F λ m a x P F - 1 > τ ¯ 0 ,where γ4>0 is the unique real root of the equation γ5-γ4-γ6e2γ4τ¯0=0,γ5=λmin(QF)-h/λmax(PF)>0,γ6=2τ02σ2(PFF3)(σ2(F3)+σ2(F4))λmax(PF-1)/h, and γ7=λmax(PF-1(F5PF+PFF5T)), with positive-definite matrix PF and QF satisfying Lyapunov equation(38)F1PF+PFF1T=-QF,F3=0IMk11L1k12L1,F4=00k11L3k12L3andF5=0IM00.Proof.
Step 1. Asymptotic convergence of leaders’ formation is equivalent to the asymptotical stability of the system (35).
The matrixE3 is Hurwitz from condition (i) and Lemma 7, so there exists a unique PL=PLT>0 satisfying E3PL+PLE3T=-QL with a given positive-definite matrix QL>0. Choose a Lyapunov function for (35) as follows:(39)Vt=12ς~TtPL⊗Inς~t.For t∈[tk,tk+θk), k=0,1,⋯,n, the time derivative of V(t) along the trajectory of system (35) is(40)V˙t=12ς~TtE3PL+PLE3T⊗Inς~t-12ς~TtPLE4⊗In∫t-τ2ttς~˙sds-12PLE4⊗In∫t-τ2ttς~˙sdsTς~t≤-12λminQL-hς~Ttς~t+12hPLE4⊗In∫t-τ2ttς~˙sdsTPLE4⊗In∫t-τ2ttς~˙sds≤-12λminQL-hς~Ttς~t+τ0σ2PLE42h∫t-τ2ttς~˙Tsς~˙sds≤-12λminQL-hς~Ttς~t+τ02σ2PLE4σ2E3+σ2E4hsupt-2τ0≤ϑ≤tς~Tϑςϑ≤-γ1Vt+γ2supt-2τ0≤ϑ≤tVϑ,where γ1>γ2>0 holds for the condition (ii). In the light of Lemma 5, we obtain that(41)Vt≤suptk-2τ0≤ϑ≤tkVϑe-γ0t-tkholds with t∈[tk,tk+θk), where γ0>0 is the unique real root of the equation γ1-γ0-γ2e2γ0τ0=0.
Fort∈[tk+θk,tk+1), k=0,1,⋯,N, the time derivative of V(t) along the trajectory of system (35) is(42)V˙t=12ς~TtE5PL+PLE5T⊗Inς~t≤12λmaxE5PL+PLE5Tς~Ttς~t≤12λmaxPL-1E5PL+PLE5Tς~TtPL⊗Inς~t=γ3Vt,where γ3=λmax(PL-1(E5PL+PLE5T)). Then, we get(43)Vt≤Vtk+θkeγ3t-tk-θk≤suptk+θk-2τ0≤ϑk≤tk+θkVϑeγ3t-tk-θk,holds with t∈[tk+θk,tk+1).
Therefore, we come to the conclusion that(44)Vt≤suptk-2τ0≤ϑ≤tkVϑe-γ0t-tk,t∈tk,tk+θk,suptk+θk-2τ0≤ϑ≤tk+θkVϑeγ3t-tk-θk,t∈tk+θk,tk+1.It follows from (44) that(45)Vt≤sup-2τ0≤ϑ≤0Vϑe-γ0θ0-2τ0+θ1-2τ0+…+θk-1-2τ0+t-tk+γ3θ¯0+θ¯1+…+θ¯k-1≤sup-2τ0≤ϑ≤0Vϑe-γ0kθ-2τ0+γ3kθ¯,holds with t∈[tk,tk+θk), where θ and θ¯ are defined in the condition (ii) of Theorem 11. For t∈[tk+θk,tk+1),(46)Vt≤sup-2τ0≤ϑ≤0Vϑe-γ0θ0-2τ0+θ1-2τ0+…+θk-2τ0+γ3θ¯0+θ¯1+…+θ¯k-1+t-tk-θk≤sup-2τ0≤ϑ≤0Vϑe-γ0k+1θ-2τ0+γ3k+1θ¯.In conclusion, we get(47)Vt≤sup-2τ0≤ϑ≤0Vϑe-γ0kθ-2τ0+γ3k+1θ¯,t∈0,+∞,where γ0(θ-2τ0)>γ3θ¯>0 holds for the condition (ii). Therefore, the leaders reach the desired formation.
Step 2. The followers’ containment convergence analysis.
If the multiagent system (31) achieves the desired formation, we obtain limt→∞q˙L(t)=0. Then, the system (32) becomes(48)p¯˙Ftq¯˙Ft=F3⊗Inp¯Ftq¯Ft-∫t-τ1ttF4⊗Inp¯˙Fsq¯˙Fsds,t∈tk,tk+θk,p¯˙Ftq¯˙Ft=F5⊗Inp¯Ftq¯Ft,t∈tk+θk,tk+1.where F3, F4, and F5 are defined in the condition (iv). Similar to Step 1, we analyze the asymptotic stability of system (48) and prove that the followers converge to the convex hull of the leaders if the conditions (iii) and (iv) hold. Theorem 11 is proved.Remark 12.
In [25], Dong et al. discussed the formation-containment control with identical time-invariant delay in which agents received information all the time. In addition, Wang et al. [39] investigated the containment control of second-order multiagent systems with multiple leaders and obtained the conditions for the agents under symmetric topologies and time-varying delay. Different above references, we consider the formation-containment control of second-order multiagent systems with different time-varying delays and gain the convergence conditions under general directed topology.


## 4. Numerical Simulations

In this section, two examples are given to verify the effectiveness of our theoretical analysis.Example 1.
Formation-containment control without communication delay.Consider a second-order multiagent system composed of four followers labeled as1,2,3,4 and four leaders labeled as 5,6,7,8. The interconnection topology is shown in Figure 2. Obviously, the topology GL of leaders has a spanning tree, and each follower has at least one leader. For simplicity, the coupling weights are all chosen as 1 for the interconnection topology. Hence, the matrices L1, L2 and L3 are (49)L1=2-10002-10002-1-1002,L2=-10000-10000-10000-1,L3=100-1-11000-11000-11.We assume that the agents move in the two-dimensional space, i.e., p(t)∈R2,q(t)∈R2. The initial positions and velocities of leaders and following agents are chosen between -5 and 5 randomly. The formation reference function is shown as follows: g5=1010T,g6=-1010T,g7=-10-10T,g8=10-10T. Based on the above matrices L1 and L3, eigenvalues are λ1F=1,λ2F=2+i,λ3F=2-i,λ4F=3, λ5L=0,λ6L=1+i,λ7L=1-i,λ8L=2. According to condition (i) of Theorem 8, the control gains are chosen as k11=-2,k12=-2,k21=-2,k22=-2. Choose γ1=0.6907,γ2=1.5617, γ3=0.7597,γ4=1.6387, θ=0.23, and θ¯=0.1, and the conditions in Theorem 8 are satisfied. Then, the communication intervals are set from 0.23 to 0.40 and the communication loss intervals are assigned from 0 to 0.1 randomly. The leaders reach the desired formation asymptotically, and the following agents converge to the convex hull generated by the leaders in Figure 3. In Figure 3, the initial states of all agents are described as diamonds, and it is obvious that the agents are not all in the convex hull formed by leaders. The final states of leaders and followers are described by the shape of circles and five-pointed stars, respectively. In addition, Figures 4 and 5 show the agents’ positions and velocities, respectively, and the variation of control inputs is demonstrated in Figure 6. Therefore, the multiagent system (1) under the algorithms (4) and (5) achieve the desired formation-containment control asymptotically.Figure 2
Network of eight agents.Figure 3
The positions of eight agents.Figure 4
The moving trajectories of eight agents.Figure 5
The velocity states of eight agents.Figure 6
The inputs of eight agents.Example 2.
Formation-containment control with time-varying communication delay.Choose the same interaction topology, formation reference function, and initial state values as those in Example1. According to condition (i) in Theorem 11, we choose k11=-2,k12=-2,k21=-2,k22=-2. By computation, γ0=0.3727, γ3=1.5617,γ4=0.2818,γ7=1.6387, θ=0.43,θ¯=0.1, and τ1=0.0002+0.0002sin⁡(t),τ2=0.0004+0.0004cos⁡(t) are set in order to make the conditions in Theorem 11 hold. Therefore, we take the communication intervals from 0.43 to 0.70 and communication loss intervals from 0 to 0.1 randomly. Then, the leaders reach the desired formation asymptotically, and the followers converge to the convex hull generated by the leaders (see Figure 7). Moreover, the trajectories of positions and velocities are shown in Figures 8 and 9, and the variation of control inputs is demonstrated in Figure 10.Figure 7
The positions of eight agents.Figure 8
The moving trajectories of eight agents.Figure 9
The positions of eight agents.Figure 10
The inputs of eight agents.


## 5. Conclusion

In this paper, we propose the formation-containment algorithms for second-order multiagent systems with intermittent communication. By variable transformation, the formation-containment problems are equivalent to the asymptotic stability problems. With the help of Lyapunov function method, sufficient conditions, which depend on the communication intervals and communication loss intervals, are obtained for the leaders’ formation convergence and following agents’ containment convergence, respectively. Besides, the delay-dependent sufficient conditions are also obtained for second-order multiagent systems converging to the desired formation-containment under time-varying communication delay. However, this paper does not take into account the collision problem, which is nonnegligible in practical engineering. Thus, our further investigation will focus on the collision-avoidance strategy, which can be combined with the formation-containment control algorithm in this paper.



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