#### 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.

#### 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 Section 2, 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: and represent column vectors and matrices, respectively. and denote identity matrix and column vectors with all entries equivalent to . denotes the eigenvalues of matrix , and indicates the largest singular value of matrix . stands for Kronecker product, and represents the transpose matrix of .

#### 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 of followers indexed by and leaders indexed by . The agents’ dynamics are given bywhere , and are the position, velocity and input of agent , respectively.

*Definition 1 (see [23–26]). *The multiagent system (1) is said to achieve the desired formation-containment asymptotically, ifandwhere denotes the desired formation reference vector, and satisfies .

The interconnection topology of agents (1) is usually denoted by a digraph with a set of nodes , a set of edges , and a weighted adjacency matrix with . is a direct edge from node to node . if and only if , and we define . A set of neighbors of node is denoted by . The Laplacian matrix is defined as . A digraph has* a directed spanning tree* if there exists at least one node having a directed path to all the other nodes.

In the digraph of agents (1), and denote the interaction topologies of the followers and leaders separately. Then, the Laplacian matrix is decomposed as , where and can be regarded as the Laplacian matrices of and , respectively, and .

In this paper, we will consider the following topology of agents (1).

*Assumption 2. *The leaders’ interaction topology 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 Figure 1. The communication periods are divided into work time and rest time, where the inputs are nonzero on and zero on . The values of are not constants, which is different from [27, 33, 34].

For the followers, the coordination control algorithms are designed aswhere and , means the neighbors of agent , and , .

The formation control algorithms for the leaders are constructed aswhere and , means the neighbors of agent , , and , .

The closed-loop forms of agents (1) under algorithms (4) and (5) areand

##### 2.3. Useful Lemmas

The following lemmas will play important roles in deriving the main results.

Lemma 3 (see [35]). *If the Assumption 2 holds, is nonsingular matrix, each entry of is nonnegative, and all row sums of equal to one.*

Lemma 4 (see [36]). *Suppose that is a symmetric positive-definite matrix and is symmetric. Then for any vector , the following inequality holds:*

Lemma 5 (see [37]). *Let be a continuous function such thatholds for , where and . If , then,where is the smallest real root of the equation *

Lemma 6 (see [38]). *For any vectors and a positive-definite matrix , 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 asandwhere , , , and .

Define and , and we reformulate the multiagent system (12) aswhere and .

Define , and , and we rewrite the multiagent system (13) asThe eigenvalues of the Laplacian matrix of leaders are defined as , and the corresponding eigenvectors are . We assume that is corresponding to associated eigenvector and . Then, the Jordan canonical form of the Laplacian matrix is given bywhere and ,

Defining yieldswhere . Hence, the formation control problem of multiagent system (7) is solved if and only if converge to zero asymptotically.

Denoting yieldswhere and .

Lemma 7. *The matrix is Hurwitz if and only ifwhere and represent real and imaginary parts of complex number.*

*Proof. *Characteristic polynomial of is given by Using Generalized Routh Criterion, the eigenvalues of lie on the open left half complex plane if and only if and (19) hold. If Assumption 2 holds, . 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.**, .**, where and with , and and with positive-definite matrices and satisfying .**, where are the eigenvalues of , .**, where and with positive-definite matrices and satisfying , and *

*Proof. *Step 1. Asymptotic convergence of leaders’ formation is equivalent to the asymptotical stability of system (18).

The matrix is Hurwitz from condition and Lemma 7, so there exists a unique satisfying with a given positive-definite matrix . Choose a Lyapunov function for (18) as follows:It follows from Lemma 4 that the time derivative of along the trajectory of system (18) with , iswhere . Meanwhile, the time derivative of along the trajectory of the system (18) with , iswhere .

It follows from (22) thatholds with , where and are defined in condition of Theorem 8. In (24), and are the minimum control time and the maximum communication loss time, respectively. Meanwhile, we obtain from (23) thatholds with .

We conclude from the above analysis thatholds for arbitrary , where for the condition . 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 . Then, the system (14) becomeswhere and are defined in the condition . 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 and 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 becomewhere and is the time-varying communication delay. For the leaders, the formation control algorithms under time-varying communication delay turn to bewhere and is the time-varying communication delay.

The closed-loop forms of agents (1) with (28) and (29) are given byandSimilar to the above subsection, we define , , where , , , and . Then, we getwhere and .

Define , , and , and we rewrite multiagent system (31) asDenote , and we getwhere . Hence, the formation control problem of multiagent system (31) is solved if and only if .

Use the Newton-Raphson formula: , and denoting yieldswhere , , and .

Before presenting the result, we make the following assumption on the time-varying delay.

*Assumption 10. * and , where and 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:**, .**where , and with , is the unique real root of the equation , , and with positive-definite matrices and satisfying the Lyapunov equation ,**, .**where is the unique real root of the equation , and , with positive-definite matrix and satisfying Lyapunov equation*

*Proof. *Step 1. Asymptotic convergence of leaders’ formation is equivalent to the asymptotical stability of the system (35).

The matrix is Hurwitz from condition and Lemma 7, so there exists a unique satisfying with a given positive-definite matrix . Choose a Lyapunov function for (35) as follows:For , , the time derivative of along the trajectory of system (35) iswhere holds for the condition . In the light of Lemma 5, we obtain thatholds with , where is the unique real root of the equation

For , , the time derivative of along the trajectory of system (35) iswhere . Then, we getholds with .

Therefore, we come to the conclusion thatIt follows from (44) thatholds with , where and are defined in the condition of Theorem 11. For ,In conclusion, we getwhere holds for the condition . 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 . Then, the system (32) becomes