Intermittent Sampled Data Control for Time-Varying Formation-Containment of the Multiagent System with/without Time Delay

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 neighbors’ 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.


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]. e 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 [7][8][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. e 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.
e main contributions are given as follows. (i) e 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) e 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. e 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.

Graph eory and Some Lemmas. Let
} is a vertex set, E⊆V × V is a link set, and A � [a ij ] ∈ R N×N is a nonnegative weighted adjacency matrix. e information flow from vertex j to vertex i is represented by a directed link e ij � (j, i). e elements of matrix A are described as follows: a ij > 0 if e ij ∈ E, and a ij � 0, otherwise. Furthermore, a ii � 0 for all i ∈ V, and a ij � a ji in an undirected topology.
In a MAS, a SOMAS with N agents is considered, and there are M followers and N − M leaders. Assume that followers can receive messages from the leaders or followers, while the leaders can only receive messages from the leaders. Let F � 1, 2, . . . , M { } and E � M + 1, M + 2, . . . , N { } denote the sets of followers and leaders, respectively. And the Laplacian matrix L N is described as where Lemma 1 (see [17]). A complex characteristic polynomial R(z) � z 2 + sz + r is Hurwitz stable if and only if Re(s) > 0 and Re(s)Im(s)Im(r) + Re 2 (s)Re(r) − Im 2 (r) > 0.
Lemma 2 (see [18]). If directed graph G contains a spanning tree, then the Laplacian matrix L ∈ R N×N of G has a simple zero eigenvalue with 1 → N as the associated eigenvector, and all the other N − 1 eigenvalues have positive real parts. Assumption 1. Each follower of graph G has at least one directed path from one leader.

Assumption 2.
Let G E be the graph associated with the leaders in the MAS, and G E contains a spanning tree.
Lemma 3 (see [19]). By Assumptions 1 and 2, it is obtained that the eigenvalues of L 1 have positive real parts, each row of − L − 1 1 L 2 has a sum equal, and each entry of − L − 1 1 L 2 is nonnegative.

Model Formulation and Some Definitions.
e control input of the ith agent is denoted by u i ∈ R n , position by p i ∈ R n , and velocity by v i ∈ R n , respectively. Consider a SOMAS as (2) In the following, for the sake of description, we assume that n � 1. However, more cases such as n > 1 can be derived by using the Kronecker product.
. . . , N). en, the dynamic equation of the ith agent can be described in a neat form as follows: 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

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. e first step is to transform the formation-containment problem into a stability problem. e second step is to solve the stability problem according to the related theory.

Problem Transformation.
Consider the following PMIC protocols: where where a(t) is a piecewise continuous function. Let T � t k+1 − t k and d < T be the control duration within a complete sampling cycle. (t k + d, t k+1 ] is the rest interval, and (t k , t k + d] is the control interval [20]. Under the control framework (6), MAS (3) can be described in a compact form as follows: T . en, system (8) can be written as e eigenvalues of L 3 relating to G E are denoted by (9) can be rewritten as Let Let en, 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.

Control
Design. By means of Lemma 4, the time-varying formation-containment problem can be transformed into the convergence analysis of β E (t) and δ F (t). is section presents the conditions for the parameters in control protocol (6) when β E (t) and δ F (t) converge to 0.
Proof. If Assumption 3 holds, one has (17), and then premultiply both sides of (17) by P − 1 E ⊗ I 2 . One can obtain that lim Under Assumption 2 and Lemma 2, J E is nonsingular, obviously. By premultiplying both sides of (18) by J − 1 Considering equation (19), Lemma 5 is set up. where Proof. Equation (16) holds if and only if β E in (15) is asymptotically stable. e solution of (15) can be written as where C � I N− M− 1 ⊗ A and D � J E ⊗ BK 1 . Let t k � t 0 and t k + d � t in (22); then, we have According to C 2 � 0a and the theory of matrix function, one can obtain that (24) Notice that β E (t k+1 ) � β E (t k + T); then, the system can be discretely expressed as Considering λ i ≠ 0, then it is focused on the conditions that the eigenvalues of Γ are encircled by the unit circle. By applying a bilinear transformation, z � ((s + 1)/(s − 1)), an updated polynomial can be found as where Proof. It is obtained that Substituting (8) into (30), we can obtain that By Lemmas 4 and 5, MAS (8) can achieve formation h E (t). Considering Definition 1, when the leaders' formation h E (t) is achieved, then 4 Complexity (32) en, consider the following system: lim t⟶∞ δ F (t) � 0 2M 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: where d 1 � Proof. It is proved naturally by Lemmas 4, 6, and 7.
□ Remark 2. When selecting parameters in the control rules, we generally give K 1 , K 2 , d 1 , and d 2 that meet the conditions first. According to the values of these parameters and conditions (35) and (36) in eorem 1, we can solve parameter T as follows: where M 1 � 2k 21 d 2 − 2k 22 d 1 .

Formation-Containment Analysis with Time Delay
We know that the time delay cannot be ignored due to its widely existence in the real world. en, a control protocol with considering time delay is proposed as follows: Complexity 5 e 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. erefore, we still transform the formationcontainment 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 d, we assume that the time delay τ < min d, Proof. Similarly as Lemma 6, differential equation (41) is where To solve the stability problem, we need to consider the iterative problem of the system. When t � t k , one has Let ; combined with (43), it is obtained that where and . It is easy to get that where a 1 � − λ i k 11 d 1 T + λ i k 11 d 1 τ + λ i k 11 d 2 − λ i k 12 d 1 − 2 and a 0 � − λ i k 11 d 1 τ − λ i k 11 d 2 + λ i k 12 d 1 + 1. We can obtain that two eigenvalues of Γ 2 satisfy s 1 � s 2 � 0. en, the remaining proof process has been omitted, which is similar to the proof of Lemma 6.
is proof is analogous to Lemma 8 and is therefore omitted.

Simulation
In this section, a two-dimensional formation-containment case of the MAS will be shown by numerical simulations. e 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. e Laplace matrix of the directed topology is given as

en, matrices A and B in dynamic equation (3) should be extended as
e formation function for leaders is given by  (29), T < 1.5396 is obtained, and T � 0.6 is chosen. When time delays are taken into account, we assume the time delay τ � 0.1. According to Lemmas 8 and 9, we also choose the feedback gains as K 1 � I 2 ⊗ [− 1, − 2] and K 2 � I 2 ⊗ [− 1, − 2] and set T � 0.6. e 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. e same initial state is indicated by a circle, and the final state is indicated by a triangular row. e 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. Complexity 7

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. e PMIC protocols are designed, and the sufficient conditions for the time-varying formation-containment of the SOMAS are given by proving three theorems. e 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 multiintelligent vehicle experimental platform.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. "Delayed impulsive control for consensus of multiagent systems with switching communication graphs," IEEE   Complexity