Multi-agent containment control with random link failures over dynamic cooperative networks

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 9665–9676 www.elsevier.com/locate/jfranklin

Multi-agent containment control with random link failures over dynamic cooperative networks ✩ Yan Xu a, Yuhua Cheng b, Libing Bai b, Xilin Zhang b,∗ a School

of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, PR China b School of Automation Engineering, University of Electronic Science and Technology of China, Sichuan 611731, PR China Received 26 February 2019; received in revised form 16 September 2019; accepted 16 September 2019 Available online 20 September 2019

Abstract In this contribution, the containment control problem for a team of agents with discrete-time secondorder dynamics over dynamic cooperative networks is discussed, where data from the controller to the actuator may be lost randomly and it is described by a random variable obeying Bernoulli distribution. A random-based distributed controller is designed by using the information from neighbors. Based on the hybrid tools of graph theory and nonnegative matrix, it is shown that the implementation of containment control is related to the dynamic cooperative networks and the successful rate of information transmission. Finally, a simulation is carried out to demonstrate the result in this paper. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Over the past several decades, a large number of researchers have been interested in the application of coordinated control of multi-agent systems in various fields such as physics and computer science. Consensus and leader-following consensus are the fundamental issues in multi-agent coordination, and their purposes are to ensure that the states of all agents ✩

This research was supported by the National Science Foundation of China (U1830207, 61772003). Corresponding author. E-mail addresses: [email protected] (Y. Xu), [email protected] (Y. Cheng), [email protected] (L. Bai), [email protected] (X. Zhang). ∗

https://doi.org/10.1016/j.jfranklin.2019.09.022 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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in the system can realize consistent. To date, many related results about consensus and leader-following consensus with different system settings such as time delay [1,2], switching topologies [3–6], noise [7], hybrid dynamics [8,9], quantization [10], event-trigger control [11,12], adaptive control [13], random link failures [14–17], cooperation-competition interactions [18], and asynchronous setting [19,20] have been studied in detail. In the reference [21], Ji et al. introduced an interesting potential application of cooperative control of autonomous vehicle route guidance systems. In this type of system, a small group of autopilot vehicles are referred to as leaders, which are furnished with sensors that can detect hazards on the road, and the rest of the autopilot vehicles that are not equipped with sensors are called followers. Taking advantage of the distributed information interaction, the leaders successfully guide the followers to move to a secure area formed by the leaders. Recently, the authors of [22] presented another similar engineering example in which a group of unmanned aerial vehicles need to be controlled during the migrating in a dangerous environment. By constantly sending information to the followers, the leaders successfully guide the followers to an area in which all the aerial vehicles fly at a lower cost. We believe that this work is inspired by the wild geese flying in a herringbone. Obviously, if there are a large number of no-sensor followers in both cases, in order to lead the entire fleet of the multi-agent system away from danger as quickly as possible, it is best to have multiple leaders. This special type of multi-agent cooperative control problem is called distributed containment control. Generally speaking, the purpose of containment control is eventually leading all followers into an area surrounded by the leaders, and theoretically, the area is called the leaders’ convex hull. In the past ten years, a great number of studies about multi-agent containment control over a fixed topology have been performed [23–28]. Cao et al. discussed this problem by some experiments and algorithms in a multiple intelligent vehicle system expressed as a second-order dynamical model in [23]. Liu et al. [24] established some necessary and sufficient conditions for realizing containment control. Moreover, the problem of containment control with dynamic leaders, which only depends on the position measurement of agents, was investigated in [25]. Communication time delay as a common phenomenon, has been analyzed in [26] in detail. Shi et al. [27–29] studied the multi-agent containment control problem under asynchronous setting. Afterwards, the situation that researchers consider is becoming more and more realistic, like the limited scope of communication. In that circumstance, the communication links between any two agents may be unreliable, in the sense that the neighbors of each agent may shift over time. Which is why some results of multi-agent containment control over dynamic cooperative topologies have been published such as [30–32]. The systems expressed in different order bring different complexities. The containment control problem of directed switching topologies in first-order systems was studied in [30], and in second-order systems in [31]. Along this research line, the paper [32] explored multi-agent containment control to a heterogeneous system with both first- and second-order dynamic agents. The containment control results with random switching topologies have been obtained in [33,34]. In all the references mentioned above, all the results are established on one assumption, that is each agent can successfully receive information from its neighbors at all time. In fact, after the agents receive the neighbors’ information, the loss of control packets may occur during the information transmission from the controller to the actuator due to communication noise, interference, or congestion. Therefore, it is reasonable and significant to consider multiagent systems under random environment caused by unreliable wireless communication links or other practical reasons. Recently, some valuable consensus results with random link failures have been obtained in [14–17]. Nevertheless, so far as we know, very few work has studied the containment control problem with random link failures. On this account, this paper aims

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to analyze the containment control behavior in the environment of random link failures, where the transmission of information from the controller to the actuator is likely to fail. A secondorder multi-agent system is considered, and the influence of dynamic cooperative networks on the system’s behavior is also taken into account. In this piece of research, the random information transmission is modeled as a Bernoulli stochastic process. In general, it is a challenge to solve the containment control problem with random link failures, especially when dynamic cooperative networks are also considered. In order to overcome this challenge, this paper constructs an augmented system to simulate the phenomenon of random link failures, and then the correlation theories of graph theory and nonnegative matrix are used to investigate this augmented system. Finally, it is shown that the followers can still be guaranteed to move into the leaders’ convex hull gradually under the circumstance of random link failures, and the followers after entering the convex hull will always keep moving due to the consideration of dynamic cooperative networks. The remaining of this paper is arranged as follows. Section 2 introduces some notations and basic concepts of matrix and graph theory. The derivation of the model is presented in Section 3. Next, Section 4 performs the model transformations, and then analyzes the transformed augmented system in detail. In Section 5, the effectiveness of the theoretical result is demonstrated by a simulation example. At last, Section 6 concludes this paper. 2. Preparatory work 2.1. Some notations In denotes an n-order identity matrix and 0n × n is an n-order matrix whose all elements are 0. N expresses the set of natural numbers. A real matrix Q is nonnegative, denoted by Q ≥ 0, if its all elements are equal or greater than 0. Let 1n be an n-dimensional column vector where all elements are equal to 1. If Q1n = 1n , then the nonnegative matrix Q is row-stochastic, further, if limk→∞ Qk = 1n yT , where y ∈ Rn×1 , then we say Q is stochastic indecomposable and aperiodic (SIA). For a nonnegative matrix Q, we use G [Q] to present a digraph with a nonegative weighted adjacency matrix Q ∈ Rn×n , E [Q] to present an edge set and {1, 2, . . . , n} to present a vertex set in which the elements are associated with the row indexes of matrix Q. For two nonnegative matrices Q and H, it is said that digraphs G {Q} and G {H } are the same type, which is written as G {Q} ∼ G {H }, if [Q]ij > 0⇔[H]ij > 0 and [Q]i j = 0 ⇔ [H ]i j = 0. 2.2. Directed graph G = (V , E , A ) indicates a digraph consisting of a set of vertices V = {v1 , v2 , . . . , vn }, a set of edges E ∈ V × V , and a nonnegative adjacency matrix A = [ai j ]n×n . A directed edge in E is presented by an ordered pair (vi , v j ), and we supposed that there are no selfloops (vi , vi ), i = 1, 2, . . . , n in the digraph G . In the weighted adjacency matrix A , aij > 0 if and only if (v j , vi ) ∈ E , otherwise ai j = 0. One path from vi0 to vir in digraph G is defined as vi0 → vi1 → · · · → vir , where vi0 , vi1 , · · · , viy ∈ V are all distinct. The distance between vertices vi and v j , denoted by d (vi , v j ), is the number of edges along the shortest path from vi to v j . In a group of n agents, a leader is the agent which has no neighbors, and there is no restriction on the number of each follower’s neighbours. Suppose that there are m leaders and n − m followers, and leaders set and the followers set are indicated by Vl = {v1 , v2 , . . . , vm }

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and V f = {vm+1 , vm+2 , . . . , vn }, respectively. Furthermore, the digraph G = (E , V , A ) describes the information exchange in the system formed by the n agents. Edge (vi , v j ) ∈ E if agent v j can receive the information from agent vi . According to the division of leaders and followers, the adjacency matrix A can be partitioned as the following form:   0m×(n−m) 0 . A = m×m Afl Af f  n  n D f = diag j=1 am+1, j , . . . , j=1 an, j denotes the degree matrix of the followers. 2.3. Definition and lemmas In this section, a few definitions and lemmas will be introduced, which will be used in the investigation of the containment control problem with random link failures. Definition 1 [35]. The convex hull of a finite set of points x1 , . . . , xn ∈ R p×1 is the minimal convex set containing by Co{x1 , . . . , xn }. More specifically, nall points xi , i =1n, . . . , n, denoted  Co{x1 , . . . , xn } = i=1 ai xi | ai ≥ 0, i=1 ai = 1 . Lemma 1 [27]. A row-stochastic matrix Q is SIA if the digraph G [Q] has a directed spanning tree in which the root vertex has a self-loop. Lemma 2 [36]. For each infinite sequence of n × n SIA matrices Q1 , Q2 , . . . , Qk , . . . , there exists a n-dimensional column vector ξ , such that limk→∞ Qk Qk−1 · · · Q1 = 1n ξ T . 3. Model formulation In practical applications, because of some practical limitations such as the unreliable communication channels and the limited sensory scope of agents, the information exchange between any two agents may be not continuous. For above-mentioned reasons, we mainly consider about the discrete time multi-agent system in which each agent receives the information from its neighbors only at discrete time. Thus, we suppose that at each discrete time instant kτ , where k = 0, 1, 2, . . . and τ is the step size,  agents exchange  information with each other according to a time-varying digraph G (k) = V , E (k), A (k) in which it is assumed that all the positive weighting factors aij have uniform lower and upper bounds, i.e. ai j (k) ∈ [α, α], where 0 < α < α. Then, the discrete-time dynamical systems are given by xi (k + 1) = xi (k), i = 1, . . . , m,

(1)

and xi (k + 1) = xi (k) + τ ϑi (k), ϑi (k + 1) = ϑi (k) + τ ui (k), i = m + 1, . . . , n,

(2)

where xi (k), ϑi (k), ui (k) ∈ R p represent the position, velocity and control protocol of agent vi at instant kτ , respectively. Consider the following distributed control protocol:     ui (k) = −γ1 ϑi (k) + γ2 ai j (k ) x j (k ) − xi (k) + γ3 ai j (k ) ϑ j (k ) − ϑi (k) , (3) j∈Ni (k)

j∈Ni (k)

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where γ 1 > 0, γ 2 > 0, γ 3 > 0 are fixed gain parameters, Ni (k) = {v j | (v j , vi ) ∈ E (k), j = 1, . . . , n} denotes the neighbor set of agent vi . According to the protocol, it is shown that each agent always can receive the information of position and velocity from its neighbors at each time instant, and obviously, there is no data loss. Whereas the fact we mentioned before, that is the actuator may not be able to receive accurate state information from the controller because of the instability of the network environment, such as data loss, channel fading and inefficient noise measurement. Hence, it is strongly necessary to consider the network link failures during the study to design a more appropriate control protocol. Then we consider to add the link failures at instant kτ , and the control protocol is designed as:     ui (k) = −γ1 ϑi (k) + γ2 φ(k )ai j (k ) x j (k ) − xi (k) + γ3 φ(k )ai j (k ) ϑ j (k ) − ϑi (k) , j∈Ni (k)

j∈Ni (k)

(4) where the stochastic variable φ(k) ∈ {0, 1} is a Bernoulli binary distributed white sequence. In particular, φ(k) = 1 means that the states information are successfully transmitted to the actuator at time instant kτ , and φ(k) = 0 indicates that the information failed to be transmitted. Let the Bernoulli stochastic variable satisfies: ¯ Prob{φ(k) = 1} = φ, ¯ Prob{φ(k) = 0} = 1 − φ,

(5)

where φ¯ is the success probability of the information transmission. Remark 1. So far, the consensus problem of multi-agent systems without leader with random link failures has been studied in literature [14,15], where a continuous packet loss process is considered. Unlike these literatures, this paper considers a discrete-time multi-agent system with multiple leaders, and its purpose is to propose a distributed control protocol (4) to implement containment control in the environment of random link failures. From protocol (4), we can observe that when data does not lose, the agents can use the neighbors’ information to update the states. Conversely, when data is lost, the agents cannot update the states by using the neighbors’ information. Definition 2. Using protocol (5), the problem of containment control with random link failures can be solved if all followers gradually enter the convex hull formed by the leaders in expectation, namely,

   lim E xi (k) ∈ Co x1 (0), x2 (0), . . . , xm (0) , vi ∈ V f . k→∞

4. Main result In this section, the target is to transform the problem of containment control with random link failures and dynamic cooperative topologies into a products convergence problem of infinite nonnegative random matrices. And next, by the combined using of graph theory and SIA matrices, it can be proved that the products convergence problem can be solved. The following gives the proof of the above problem. We begin with performing the following model transformations, which assist us in transforming the containment control system into an equivalent compact augmented system. Let

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 xl (k) = x1 (k), . . . , xm (k) T ,

 x f (k) = xm+1 (k), . . . , xn (k) T ,

 ϑ f (k) = ϑm+1 (k), . . . , ϑn (k) T ,

 Zl (k) = xlT (k), xlT (k) T ,

 γ3 Z f (k) = x Tf (k), ϑ Tf (k) + x Tf (k) T , γ2

 Z (k) = ZlT (k), Z Tf (k) T .

(6)

Substitute the protocol (4) into the systems (1) and (2), then we can obtain the following system with random coefficient matrix, Z (k + 1) = (k )Z (k ), where



I2m (k) = G f l (k)

(7)

 02m×2(n−m) , E (k) + G f f (k)

in which the random matrices E(k), Gfl (k) and Gff (k) are described as

γ2 τ In−m − γγ23τ In−m I γ3 n−m   E (k) = (γ1 γ3 −γ2 )τ , 2 )τ 1 − (γ1 γ3γ−γ In−m − φ(k)γ3 τ D f (k) In−m γ3 3   0(n−m)×m 0(n−m)×m , G f l (k) = 0(n−m)×m φ(k)γ3 τ A f l (k)   0(n−m)×(n−m) 0(n−m)×(n−m) , G f f (k) = (8) 0(n−m)×(n−m) φ(k)γ3 τ A f f (k)  n  n with D f (k) = diag j=1 am+1, j (k ), . . . , j=1 an, j (k ) . For each matrix (k), we define a new random matrix   01×2(n−m) 1 . Q(k) = (9) G f l (k)12m E (k) + G f f (k) Below we show a lemma that plays a crucial role in obtaining results. Lemma 3. There is a series of continuous, non-empty, uniformly bounded time intervals [k j τ, k j+1 τ ), j ∈ N, starting at k0 = 0, with the property that the union of digraphs G (k j ), . . . , G (k j+1 − 1) contains a spanning forest rooted at the leaders. If the success probability of the information transmission satisfies φ¯ > 0 and the parameters γ 1 , γ 2 , γ 3 satisfy the following inequalities:  γ2 < τ1 , γ3 (10) ¯ 3 − γ2 < 1 , γ1 + φγ γ3

τ

where = (n − 1)α, then E[Q(k j+1 − 1)] · · · E[Q(k j + 1)]E[Q(k j )] are SIA matrices for any j ∈ N. Proof. When γγ23 < τ1 , we can calculate that the matrices In−m − γγ23τ In−m and γγ23τ In−m are ¯ 3 dmax − γ2 < nonnegative and contain positive diagonal elements. Under the inequality γ1 + φγ γ3 γ2 τ 1 , we can derive that (1 − ) I is a nonnegative matrix with positive diagonal elements n−m τ γ3

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2 )τ and the matrix (γ1 γ3γ−γ In−m is nonnegative. Thus, the matrix E(E (k)) is nonnegative and 3 contains positive diagonal elements, which together with the fact that the matrices E[G f f (k)] and E[G f l (k)] are nonnegative, guarantees that the matrix E[Q(k)] is a row-stochastic matrix with positive diagonal elements. For convenience, let the matrix E[Q(k)] be partitioned as follows   01×2(n−m) 1 , E[Q(k)] = E[Q1 (k)] E[Q2 (k)]

where Q1 (k) = G f l (k)12m and Q2 (k) = E (k) + G f f (k). Let E[M(k j )] = E[Q(k j+1 − 1)] · · · E[Q(k j )]. According to a known fact that the product of row-stochastic matrices is a row-stochastic matrix, it can be obtained E[M(k j )] is a row-stochastic matrix, which can be presented as   01×4(n−m) 1 E[M(k j )] = , E[N (k j )] E[Q2 (k j+1 − 1)] · · · E[Q2 (k j )] where E[N (k j )] = E[Q1 (k j+1 − 1)] + E[Q2 (k j+1 − 1)]E[Q1 (k j+1 − 2)] + · · · + E[Q2 (k j+1 − 1)] · · · E[Q2 (k j − 1)]E[Q1 (k j )]. Denote E[Q2 (k)] = E[Q21 (k)] + E[Q22 (k)], (11)     where E[Q21 (k)] = diag E[E (k)] and E[Q22 (k)] = E[E (k)] − diag E[E (k)] + E[G f f (k)]. Then we have E[Q2 (k j+1 − 1)] · · · E[Q2 (k j )] = E[Q21 (k j+1 − 1) + Q22 (k j+1 − 1)] · · · E[Q21 (k j ) + Q22 (k j )] k j+1 −1

≥ E[Q21 (k j+1 − 1)] · · · E[Q21 (k j )] +



f (i)

(12)

i=k j

where f (i) = E[Q21 (k j+1 −1)] · · · E[Q21 (i+1)]E[Q22 (i)]E[Q21 (i−1)] · · · E[Q21 (k j )], i = k j , . . . , k j+1 −1. Furthermore, we can calculate that   In−m 0 , i = k j , . . . , k j+1 −1. f (i) ∼ E[Q22 (i)] ∼ (n−m)×(n−m) In−m A f f (i) Thus ∃ δ > 0 such that ⎛ 0(n−m)×(n−m) k j+1 −1 ⎜ f (i) ≥ δ ⎝ I i=k j

In−m

k j+1 −1

n−m

⎞

⎟ . A f f (i)⎠

(13)

i=k j

Since the union of digraphs G (k j ), . . . , G (k j+1 − 1) contains a spanning forest in which the  k j+1 −1  leaders are the root vertices, digraph G A f f (i) contains a spanning forest. Assume i=k j that the leaders vl1 , . . . , vls are the root vertices in spanning forest, and the vertices directly connected to these root vertices are the followers vm+w1 , . . . , vm+wh . Then it can be observed  k j+1 −1  that the root vertices of the spanning forest in G A f f (i) are w1 , . . . , wh . It follows i=k j  k j+1 −1  from Eq. (13) that digraph G f (i) contains a spanning forest with the root veri=k j  tices w1 + n − m, . . . , wh + n − m. By (12), it can be obtained that digraph G E[Q2 (k j+1 − 1)] · · · E[Q2 (k j )] contains a spanning forest with the root vertices w1 + n − m, . . . , wh + n − m.

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Now we analyze the vector E[N (k j )]. It is assumed that the vertex set of digraph G E[M(k j )] is {0, 1, . . . , 2(n − m)} whose elements are associated with the row indexes of matrix E[M(k j )]. According to the definition of Gfl (k), we have [N(kj )]i > 0,  i = w1 + n − M(k means that (0, w + n − m) ∈ G ) m, w2 + n −m, . . . ,wh + n − m. This 1 j , . . . , (0, wh +   n − m) ∈ G M(k j ) . Therefore, G E[M(k j )] contains a spanning tree in which 0 is the root vertex, which together with the fact that the vertex 0 has self-loop, guarantees that the matrices E[M(k j )] = E[Q(k j+1 − 1)] · · · E[Q(k j )], j ∈ N are SIA.  Based on Lemma 3, we now present the main result of this paper. Theorem 1. There is a series of continuous, non-empty, uniformly bounded time intervals [k j τ, k j+1 τ ), j ∈ N, starting at k0 = 0. Using protocol (4), the containment control problem for systems (1) and (2) can be solved if the following conditions hold: (i) the ers (ii) the (iii) the

union of digraphs G (k j ), . . . , G (k j+1 − 1) contains a spanning forest with the leadas the root vertices; success probability of the information transmission satisfies φ¯ > 0; parameters γ 1 , γ 2 , γ 3 satisfy the inequalities in Eq. (10).

Proof. It is known from Lemma 3 that matrices E[M(k j )], j ∈ N are SIA. It follows from Lemma 2 that lim E[Q(k)] · · · E[Q(0)] = lim E[M (k j )] · · · E[M (k0 )] j→∞ ⎛ ⎞ 1 0 ··· 0 ⎜ ⎟ = ⎝ ... ... . . . ... ⎠.

k→∞

1

0

···

(14)

0

This implies that limk→∞ E[Q2 (k)] · · · E[Q2 (0)] = 02(n−m)×2(n−m) . Consequently, for system (7), we have lim E[Z (k + 1)] = lim E[(k)] · · · E[(0)]Z (0) k→∞   02m×2(n−m) I2m Z (0), = lim E[Q2 (k)] · · · E[Q2 (0)] k→∞ E[(k)]   02m×2(n−m) I2m Z (0), = limk→∞ E[(k)] 02(n−m)×2(n−m)

k→∞

where limk→∞ E[(k)] is a row-stochastic matrix. And then lim E[Z f (k)] = lim E[ (k)]Zl (0),

k→∞

k→∞

Fig. 1. Switching communication topologies.

(15)

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5 4 3 2

Y-axis

1 0 -1 -2 -3 -4 -5 -4

-3

-2

-1

0

1

2

3

4

X-axis

(a) the position trajectories of the agents 8 7

i 1

|| || , i=5,6,...,10

6 5 4 3 2 1 0

0

10

20

30

40

50

60

70

80

90

100

Time/

(b) the velocity norm trajectories of the followers Fig. 2. The state trajectories of the agents. In (a), the red diamonds represent the fixed positions of the leaders, and the blue solid lines represent the position trajectories of the followers at different instants. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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where limk→∞ E[ (k)] is a row-stochastic matrix. Furthermore, lim E[x f (k)] = lim E[ (k)]xl (0),

k→∞

k→∞

where limk→∞ E[ (k)] is a row-stochastic matrix. That is to say, the containment control with dynamic cooperative networks and random link failures is achieved.  Remark 2. Compared with the existing works on the containment control without considering random packet losses from controller to actuator [23–34], the main contribution of this paper is to extend the containment control result to an environment of random link failures for the first time. In addition, this paper also considers the effect of dynamic cooperative networks on the result, i.e., the followers who enter the convex hull formed by the leaders will keep moving instead of stationary. 5. Numerical simulation Consider four leaders and six followers. The random variable φ obeying Bernoulli distribution satisfies  0.6, if φ = 1, Prob{φ} = 0.4, if φ = 0. The possible communication topologies are shown in Fig. 1, where all edge weights are assumed to be 0.2, and hence = 1.8. To satisfy the inequalities in Eq. (10), we let γ1 = 3, γ2 = 2, γ3 = 1 and τ = 0.2. All discrete instants are divided into intervals [3sτ, 3(s + 1)τ ), s ∈ N, which are uniformly bounded. Assume that the communication topologies in each time interval [3sτ, 3(s + 1)τ ) obey the following rule: from Ga to Gb , Gb to Gc , and then, Gc to Ga . Obviously, the union of topologies in each time interval has a spanning forest with the leaders as the root vertices. Finally, Figs. 2a and b shows, respectively, the position and velocity norm trajectories of the agents at different instants, from which one can observe that all the followers gradually merge into the leaders’ convex hull and keep moving. 6. Conclusion This piece of research investigated the problem of multi-agent containment control, and the main highlight is that an environment of random link failures was considered. Firstly, appropriate model transformation vectors were designed to transform the initial system into a compact augmented system with nonnegative random coefficient matrices. In addition, the properties of nonnegative matrices were used to solve the stability of the augmented system. Finally, it was shown that the containment control under the environment of random link failures can be achieved if the union of dynamic cooperative networks contains a spanning forest rooted in the leaders. A simulation was used to demonstrate the theoretical result. Acknowledgment The authors would like to thank the Associate Editor and all the reviewers for their valuable comments and suggestions.

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