#### Abstract

Time-varying formation-containment problems for a second-order multiagent system (SOMAS) are studied via pulse-modulated intermittent control (PMIC) in this paper. A distributed control framework utilizing the neighborsâ€™ positions and velocities is designed so that leaders in the multiagent system form a formation, and followers move to the convex hull formed by each leader. Different from the traditional formation-containment problems, this paper applies the PMIC framework, which is more common and more in line with the actual control scenarios. Based on the knowledge of matrix theory, algebraic graph theory, and stability theory, some sufficient conditions are given for the time-varying formation-containment problem of the second-order multiagent system. Some numerical simulations are proposed to verify the effectiveness of the results presented in this paper.

#### 1. Introduction

Many scholars start to pay attention to the multiagent systems (MASs) with the rapid development of complex network system theory. In recent decades, many major breakthroughs have been made in this field [1], and those results are also widely used in various fields of production and life, such as UAV cruise system [2], smart grid [3, 4], economic dispatching [5], and multiple underactuated surface vessels [6]. The most studied collaboration problems of MASs include swarm, consensus, formation, and distributed optimization. In very recent years, the clustering behavior of MASs has also attracted widespread interest, including but not limited to consensus [7â€“9], tracking, and formation [10].

In these cooperative control problems, many important advances have been made in the field of formation control and containment control. Huang et al. [11] studied the containment control problem of MASs via intermittent control-based sampled data information. Wang et al. [12] investigated the containment control problem of first-order MASs in the noisy communication environments. Rahimi et al. [13] studied time-varying formation control of collaborative heterogeneous MASs. However, most of the existing works are carried out on formation control and containment control separately. In many applications, both these cooperative behaviors often require simultaneous implementation. For example, in the coordination of multiple tanker airplanes and multiple UAVs, the tanker airplanes will form a specific formation in advance and wait for the UAVs to reach the area they surround. In order to solve this kind of problem, Dong et al. designed a continuous control strategy [14] for second-order MASs with multiple leaders and multiple followers, which can ensure the formation control of the leaders and containment of the followers. It is worth noting that the above work [14] adopted a continuous-time control which may be difficult to implement in some cases due to sampled measurement. The traditional zero-order sampling control adopts the same amplitude control input in the whole sampling interval, which can also lead to application difficulties. For instance, in a driverless system, it is difficult for the sensors on the vehicle to work all the time, allowing for fuel economy and other reasons.

Aiming at solving the aforementioned problems, the formation-containment problem of the second-order multiagent system (SOMAS) is studied in this paper, and the control framework of pulse-modulated intermittent control (PMIC) is adopted. The main contributions are given as follows. (i) The formation and the containment are achieved simultaneously in second-order MASs, where the leaders form a formation and the followers are contained in this formation. (ii) The PIMC is a framework that can unify impulsive control [8, 15] and zero-order sampling control [16]. It can be applied to a wider range of real-world scenarios. (iii) In this paper, some sufficient conditions are given for the parameters of the control strategy under with/without time delay cases.

The remainder sections of this paper are described as follows. Section 2 lists the basic preliminary knowledge and a model of the problem to be studied. Section 3 introduces the formation-containment analysis without time delay. Section 4 obtains and provides the results when MASs contain time delay. Section 5 gives several simulations to verify the theorems are correct. Section 6 draws the conclusion.

#### 2. Preliminaries

##### 2.1. Graph Theory and Some Lemmas

Let be a weighted digraph, where is a vertex set, is a link set, and is a nonnegative weighted adjacency matrix. The information flow from vertex to vertex is represented by a directed link . The elements of matrix are described as follows: if , and , otherwise. Furthermore, for all , and in an undirected topology. Denote as the set of neighbors of node . The Laplacian matrix is described in terms of , where is the in-degree matrix with . A sequence of edges with different nodes of as can be obtained if there is a directed path from the th agent to the th agent.

In a MAS, a SOMAS with agents is considered, and there are followers and leaders. Assume that followers can receive messages from the leaders or followers, while the leaders can only receive messages from the leaders. Let and denote the sets of followers and leaders, respectively. And the Laplacian matrix is described aswhere , , and .

Lemma 1 (see [17]). *A complex characteristic polynomial is Hurwitz stable if and only if and .*

Lemma 2 (see [18]). *If directed graph contains a spanning tree, then the Laplacian matrix of has a simple zero eigenvalue with as the associated eigenvector, and all the other eigenvalues have positive real parts.*

*Assumption 1. *Each follower of graph has at least one directed path from one leader*.*

*Assumption 2. *Let be the graph associated with the leaders in the MAS, and contains a spanning tree*.*

Lemma 3 (see [19]). *By Assumptions 1 and 2, it is obtained that the eigenvalues of have positive real parts, each row of has a sum equal, and each entry of is nonnegative .*

##### 2.2. Model Formulation and Some Definitions

The control input of the th agent is denoted by , position by , and velocity by , respectively. Consider a SOMAS as

In the following, for the sake of description, we assume that . However, more cases such as can be derived by using the Kronecker product.

Let, , and . Then, the dynamic equation of the th agent can be described in a neat form as follows:

*Definition 1. *With the bounded initial state of each agent being chosen arbitrarily, the leaders in MAS (3) are said to realize time-varying formation if there exists a vector function such thatwhere is a piecewise continuously differentiable vector. Then, we can denote the time-varying formation as .

*Definition 2. *Similarly, with the bounded initial state of each agent being chosen arbitrarily, MAS (3) is said to achieve containment if there exist nonnegative constants such that and

#### 3. Formation-Containment Analysis without Time Delay

In this section, we mainly study how to design the PMIC protocol to make MAS (3) achieve the time-varying formation-containment and propose some sufficient and necessary conditions for parameters. We will investigate the time-varying formation-containment problem in two steps. The first step is to transform the formation-containment problem into a stability problem. The second step is to solve the stability problem according to the related theory.

##### 3.1. Problem Transformation

Consider the following PMIC protocols:where and are feedback gain matrices and . The pulse function is described aswhere is a piecewise continuous function. Let and be the control duration within a complete sampling cycle. is the rest interval, and is the control interval [20].

Under the control framework (6), MAS (3) can be described in a compact form as follows:where and .

Let and . Then, system (8) can be written as

The eigenvalues of relating to are denoted by , where with the eigenvector as , and . Let , where is the Jordan canonical form of , , and . By Lemma 2, one has , where consists of the Jordan blocks relating to . Let ; then, system (9) can be rewritten as

Let , , and . Then, system (10) can be divided as

Let , and . Then, the following lemma is used to transform the formation-containment problem.

Lemma 4. *MAS (3) under the PMIC framework (6) can achieve time-varying formation-containment if*

*Proof. *It is able to be proved by a similar way in [14].

*Remark 1. *Lemma 4 is proved because the following results in [14] are worked out:Considering Lemma 3, we can conclude that equation (13) satisfies Definitions 1 and2, respectively. In other words, Lemma 4 can be proved by the above two equations.

##### 3.2. Control Design

By means of Lemma 4, the time-varying formation-containment problem can be transformed into the convergence analysis of and . This section presents the conditions for the parameters in control protocol (6) when and converge to 0.

*Assumption 3. *For ,

Lemma 5. *Let*

Under Assumption 3, is satisfied if

*Proof. *If Assumption 3 holds, one hasSubstitute into (17), and then premultiply both sides of (17) by . One can obtain thatUnder Assumption 2 and Lemma 2, is nonsingular, obviously. By premultiplying both sides of (18) by , we haveConsidering equation (19), Lemma 5 is set up.

Lemma 6. *, equation (16) holds if the following inequalities are satisfied:where and .*

*Proof. *Equation (16) holds if and only if in (15) is asymptotically stable. The solution of (15) can be written aswhere and . Let and in (22); then, we haveAccording to and the theory of matrix function, one can obtain thatNotice that ; then, the system can be discretely expressed asLet and . It yieldsConsidering , then it is focused on the conditions that the eigenvalues of are encircled by the unit circle. By applying a bilinear transformation, , an updated polynomial can be found as in (26) can be obtained if and only if in (27) is Hurwitz stable. Based on Lemma 1, if (27) is stable, then both equations (20) and (21) hold.

Lemma 7. * holds if and only if ,where and .*

*Proof. *It is obtained thatSubstituting (8) into (30), we can obtain thatBy Lemmas 4 and 5, MAS (8) can achieve formation . Considering Definition 1, when the leadersâ€™ formation is achieved, thenSince , one hasThen, consider the following system: means that system (34) is asymptotically stable. Similar to the analysis of Lemma 5, the conditions that make system (34) asymptotically stable are worked out with (28) and (29).

Theorem 1. *MAS (3) under Assumption 3 and PMIC framework (6) realizes formation-containment if the following conditions simultaneously hold:*(i)*,*(ii)*,**where and .*

*Proof. *It is proved naturally by Lemmas 4, 6, and 7.

*Remark 2. *When selecting parameters in the control rules, we generally give , , , and that meet the conditions first. According to the values of these parameters and conditions (35) and (36) in Theorem 1, we can solve parameter as follows:(i),â€‰where .(ii),where .

#### 4. Formation-Containment Analysis with Time Delay

We know that the time delay cannot be ignored due to its widely existence in the real world. Then, a control protocol with considering time delay is proposed as follows:

The control protocol (39) is substituted into MAS (3), and we can obtain

Considering Assumption 3, when there is a time delay in the system, similar to Section 3, we have

It is not hard to see that Lemma 4 also applies to cases with time delay. Therefore, we still transform the formation-containment problem of the SOMAS into the stability problem of the system.

*Remark 3. *Considering that the time delay of the actual system is not long and the intermittent control has the advantage of selectively adjusting the control time interval , we assume that the time delay .

Lemma 8. * holds with time delay if ,where , , and is the time delay.*

*Proof. *Similarly as Lemma 6, differential equation (41) is solved. Since , when , we can obtain thatwhere and . To solve the stability problem, we need to consider the iterative problem of the system. When , one hasLet ; combined with (43), it is obtained thatwhere and . It is easy to get that if the eigenvalues of meet . Let . It yieldswhere and . We can obtain that two eigenvalues of satisfy . Then, the remaining proof process has been omitted, which is similar to the proof of Lemma 6.

Lemma 9. * holds with time delay if ,where , , and is the time delay.*

*Proof. *This proof is analogous to Lemma 8 and is therefore omitted.

Theorem 2. *MAS (3) under Assumption 3 and control framework (39) realizes formation-containment if conditions (42), (43), (48), and (49) hold.*

*Proof. *It is not hard to prove by applying Lemmas 4, 8, and 9.

#### 5. Simulation

In this section, a two-dimensional formation-containment case of the MAS will be shown by numerical simulations. The simulations illustrate the effectiveness of the results in this paper. In this two-dimensional formation-containment case, suppose there are three leaders and two followers in the MAS. The Laplace matrix of the directed topology is given as

Note that , , and , . Then, matrices and in dynamic equation (3) should be extended as

The formation function for leaders is given by

The eigenvalues of are , , , , and . The function is set, and the control duration is chosen; then, and .

When we do not care about time delays, it follows from (20) and (28) that and . The feedback gains are chosen as and . According to inequalities (21) and (29), is obtained, and is chosen. When time delays are taken into account, we assume the time delay . According to Lemmas 8 and 9, we also choose the feedback gains as and and set .

The simulation results are shown in figures. Among them, Figures 1 and 2 are the trajectories of the position and velocity of each agent changing without considering the time delay, in which the initial state is marked with a circle and the final state is marked with a triangular row. Figures 3 and 4 show the trajectories when time delay is taken into account. The same initial state is indicated by a circle, and the final state is indicated by a triangular row. The initial states of all agents are randomly selected.

During the simulation, the five agents begin with arbitrary initial states. As time goes on, the three leaders reach a time-varying circular formation, and two followers move into the convex hull formed by leaders. At the same time, it can be seen that the system with time delay converges more slowly with the same sampling period.

#### 6. Conclusion

Time-varying formation-containment problems of the SOMAS were studied in this paper. Based on the fundamental theorems of graph theory and matrix theory, the formation-containment problem is transformed into the stability problem of the SOMAS. The PMIC protocols are designed, and the sufficient conditions for the time-varying formation-containment of the SOMAS are given by proving three theorems. The simulations show the correctness of the proposed theoretical results. It is expected that the future work will be carried out on the nonlinear MAS, and the conclusion of this paper will be applied to the actual multi-intelligent vehicle experimental platform.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61972170 and 61973133).