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

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 timevarying communication delay and intermittent communication. Finally, the validity of theoretical results is illustrated by numerical simulations.


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][10][11][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][17][18][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 formationcontainment 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 secondorder 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 delaydependent 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 formationcontainment asymptotically within aperiodic intervals.However, intermittent communications were periodical in [27,33,34].Thirdly, the leaders and followers suffer from the timevarying 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  × 1 column vectors and × matrices, respectively.  and 1  denote × identity matrix and  × 1 column vectors with all entries equivalent to 1. () denotes the eigenvalues of matrix , and () indicates the largest singular value of matrix .⊗ stands for Kronecker product, and   represents the transpose matrix of .

Preliminaries
In this section, the formation-containment control of secondorder multiagent systems is formulated, and some useful lemmas and coordination control algorithms with intermittent communication are presented.
In the digraph  of agents (1),   and   denote the interaction topologies of the followers and leaders separately.
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.

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 [  +   ,  +1 ).The values of   are not constants, which is different from [27,33,34].
The closed-loop forms of agents (1) under algorithms (4) and ( 5) are and

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

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

Lemma 7.
The matrix  1 is Hurwitz if and only if where R(⋅) and I(⋅) represent real and imaginary parts of complex number.
Proof.Characteristic polynomial of  1 is given by Using Generalized Routh Criterion, the eigenvalues of   6) and (7) with Assumption 2 reach the desired formation-containment asymptotically, if the following conditions are satisfied simultaneously. () Proof.
Step 2. Containment convergence analysis of the followers.
If the multiagent systems (13) achieve the desired formation, we obtain lim →∞ q  = 0.Then, the system ( 14) becomes where  1 and  2 are defined in the condition (V).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 (V) 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 formationcontainment asymptotically within aperiodic intervals.

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.
Step 1. Asymptotic convergence of leaders' formation is equivalent to the asymptotical stability of the system (35).
Step 2. The followers' containment convergence analysis.If the multiagent system (31) achieves the desired formation, we obtain lim →∞ q  () = 0.Then, the system (32) becomes where  3 ,  4 , and  5 are defined in the condition (V).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 () and (V) hold.Theorem 11 is proved.
Remark 12.In [25], Dong et al. discussed the formationcontainment 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.

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 as 1, 2, 3, 4 and four leaders labeled as 5, 6, 7, 8.The interconnection topology is shown in Figure 2. Obviously, the topology   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  1 ,  2 and  3 are We assume that the agents move in the two-dimensional space, i.e., () ∈  2 , () ∈    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.
Choose the same interaction topology, formation reference function, and initial state values as those in Example 1.According to condition () in Theorem 11, we choose  11 = −2,  12 = −2,  21 = −2,  22 = −2.By computation,  0 = 0.3727,  3 = 1.5617,  4 = 0.2818,  7 = 1.6387,  = 0.43,  = 0.1, and  1 = 0.0002+0.0002sin(),  2 = 0.0004+0.0004cos() 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.

Conclusion
In this paper, we propose the formation-containment algorithms for second-order multiagent systems with intermittent communication.By variable transformation, the formationcontainment 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.

Data Availability
The simulation data generated during this study have been deposited with figshare (https://figshare.com/s/

Figure 2 :
Figure 2: Network of eight agents.

Figure 3 :
Figure 3: The positions of eight agents.

Figure 4 :
Figure 4: The moving trajectories of eight agents.

Figure 5 :Figure 6 :
Figure 5: The velocity states of eight agents.

Figure 7 :Figure 8 :
Figure 7: The positions of eight agents.