Distributed containment of heterogeneous multi-agent systems with switching topologies

Accepted Manuscript

Distributed containment of heterogeneous multi-agent systems with switching topologies Lei Shi, Jinliang Shao, Mengtao Cao, Hong Xia PII: DOI: Reference:

S0925-2312(18)30630-1 10.1016/j.neucom.2018.05.050 NEUCOM 19608

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Neurocomputing

Received date: Revised date: Accepted date:

17 December 2017 15 May 2018 17 May 2018

Please cite this article as: Lei Shi, Jinliang Shao, Mengtao Cao, Hong Xia, Distributed containment of heterogeneous multi-agent systems with switching topologies, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.05.050

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Distributed containment of heterogeneous multi-agent systems with switching topologies✩ Lei Shia , Jinliang Shaoa,∗, Mengtao Caob , Hong Xiab a School

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of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, PR China b School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, PR China

Abstract

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In this article, we investigate the issue of containment control for a heterogeneous multi-agent system composed of stationary leaders, first-order dynamic followers and second-order dynamic followers under switching communication topologies. For the followers with different order dynamic behavior, two distributed dynamic state feedback algorithms are constructed separately for the followers driven by different dynamics. To achieve the objective of this work, we convert the heterogeneous containment control problem into a convergence problem, which is the product of time-varying non-negative matrices whose sum of each row is less than or equal to 1. With the aid of the matrix product technique and the composite binary relation, a sufficient condition can be established to analyze the convergence problem. At last, some numerical examples are presented to verify the feasibility of the theoretical findings.

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The studies on the issue of consensus/synchronization, with the aim to drive a team of units to realize a common goal by communicating with the neighbors, have received increasing research interests over the past few decades because of its wide range of applications in neural network [1–3], sensor network [4–6] and networked multi-agent systems [7–16], etc. Containment control can often be considered as a subclass of multi-agent consensus, in which a small part of the agents are assumed to be the leaders and other agents are naturally called the followers. The research of containment control issue, with the objective to guide the followers into a convex hull structured by the leaders based on a rationally designed protocol, is motivated by numerous natural phenomena. For example, the female moths intermittently release pheromone called bombykol in order to attract male moths, so that male moths often accumulate in the strict geometry of female moths [17].

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1. Introduction

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Keywords: Distributed containment; Heterogeneous multi-agent systems; Switching topologies.

✩ This research was supported in part by the National Science Foundation of China (61403064), the China Postdoctoral Science Foundation (2017M612944), the Special Postdoctoral Foundation of Sichuan Province, and the Fundamental Research Funds for the Central Universities (ZYGX2015J095). ∗ Corresponding author Email addresses: [email protected] (Lei Shi), [email protected] (Jinliang Shao), [email protected] (Mengtao Cao), [email protected] (Hong Xia)

Preprint submitted to Neurocomputing

May 24, 2018

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By observing the research status of containment control in the most recent decade, the existing literature on containment control topics can be roughly divided into two categories: the case with fixed communication topology and the case with switching communication topologies. For the containment control issues under fixed communication topology, researchers with different backgrounds have obtained a lot of valuable results. The authors in [18] presented reveal necessary and sufficient conditions for the containment control with both stationary and dynamic leaders under a fixed directed topology. The containment control that rely solely on the location measurements of the agents was analyzed in [19]. Wen et al. [20] focused on the containment control for higher-order multi-agent systems. Some containment control results using sampling information were reported in [21, 22]. Recently, asynchronous containment control issues for a team of agents driven by second-order dynamics were discussed in [23] and [24]. For another situation that the communication topology is time-variant, some valuable results for containment control have also been presented in the literatures [25–27]. For the containment control behavior of a team of agents driven by first-order dynamics, Cao et al. [25] and Li et al. [26] exploited, respectively, Laplacian transform method and Lyapunov method. Using trajectory analysis, the authors of [27] investigated the containment control of general linear multi-agent systems with switching topologies. Most of the work described previously with regard to containment control considered the agents with with exactly the same dynamics. In practical applications, the agents may adjust their states through different dynamic behaviors due to various environmental or technical limitations, such as achieving the goal at the lowest cost. Therefore, heterogeneous multi-agent systems composed of agents with different dynamic behaviors are more suitable than homogeneous systems. Based on this consideration, the issue of coordinated control of the agents with different dynamic behaviors has become a hot research topic in recent years. The consensus for heterogeneous multi-agent systems with discrete-time dynamics and continuous-time dynamics were investigated in [28] and [29], respectively. By establishing an error system, the leader-following consensus issues for a team of agents driven by heterogeneous dynamics were solved in [30, 31]. Although many interesting results have been reported so far regarding consensus issues and leader-following consensus issues for heterogeneous multi-agent systems, there has not been extensive attention to the containment control issue for heterogeneous multi-agent systems other than [32, 33], in which the containment control of agents with second-order dynamics and general linear dynamics were discussed, respectively. Motivated by the previous discussions, we consider the containment control problem under both heterogeneous dynamics and switching topologies. The contributions of this article are twofold. (1) Heterogeneous containment control generalizes homogeneous containment control and further contains its homogeneous counterpart as a subcase. Considering the containment control issue under switching topologies is more important than the study of the containment control with a fixed topology because of its wide applicability. In addition, different from the results of containment control with fixed topology [18–23, 32, 33], the followers may enter and keep active in the convex hull formed by the leaders when k → ∞ if the switching topologies are considered. Therefore, it is difficult to derive the corresponding error system and study the stability of the error system equivalently like literature [18–23, 32, 33]. This implies that the technical issues brought by considering the heterogeneous containment control with switching topologies are much more challenging because of the time-varying characteristics of the communication topologies. (2) In order to obtain the main result of this article, the heterogeneous multi-agent system is first transformed into an discrete-time augmented system with nonnegative coefficient matrices. Then the accessibility of heterogeneous containment control problem is equivalent to the convergence of the product of infinite time-varying nonnegative matrices with all row sums less than or equal to 1. So far, the

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2. Preliminaries

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Rn and Rn×m denote, respectively, the set of n dimensional real column vectors and the set of n × m-dimensional real matrices. N and Z+ are used to denote the natural number set and the positive integer set, respectively. 1n ∈ Rn is the vector of all 1’s, 0n×m ∈ Rn×m denotes a matrix of all 0’s and In ∈ Rn×n is an identity matrix. ⊗ stands for the Kronecker product. For a matrix H = [hijP ] ∈ Rn×n , diag(H) presents a diagonal matrix with diagonal elements h11 , h22 , n · · · , hnn ; Λi [H] = j=1 hij denotes the sum of the ith row of matrix H and kHk∞ = maxi Λi [H] represents the infinite norm of matrix H; H(m1 → m2 , :) ∈ R(m2 −m1 +1)×n is given to define a matrix made up of the rows from m1 th row to m2 th row in matrix H. In the absence of ambiguity, the element hij in matrix H can be also described by [H]ij in this article. A nonnegative matrix H ∈QRn×n is row-stochastic if Q1n = 1n , where 1n ∈ Rn×1 is a column vector of all 1’s. Besides, k let i=1 H(i) = H(k)H(k − 1) · · · H(1) present the left product of matrices. Consider a heterogeneous system composed of agents with different dynamic behaviors. Without loss of generality, assume that there are m leaders, n first-order dynamic followers and r secondorder dynamic followers. Let Vl = {v1 , v2 , · · · , vm }, Vf = {vm+1 , vm+2 , · · · , vm+n } and Vs = {vm+n+1 , vm+n+2 , · · · , vm+n+r } denote, respectively, the set of leaders, the set of followers with first-order dynamics and the set of followers with second-order dynamics. Information exchange
between agents is visually described through a directed graph G = V, E, A . (vi , vj ) ∈ E if and only if agent vj can receive the information from agent vi . The weighted adjacency matrix A = [aij ](m+n+r)×(m+n+r) satisfies: aij > 0 ⇔ (vj , vi ) ∈ E. Assume that there are no self-loops in the graph G, i.e. aii = 0, i = 1, 2, · · · , m+n+r. A directed path from vi0 to vir is represented by a finite non-empty sequence of edges, e.g., (vi0 , vi1 ) (vi1 , vi2 ) · · · (vir−1 , vir ), where vi0 , vi1 , · · · , vir ∈ V are all different. A directed graph is called a directed tree if a root vertex is existent so that any non-root vertex of the directed graph is reachable by the sole path beginning at the root vertex. A directed graph is called a directed forest if it is composed of one or more directed trees, in which no two directed trees have common vertices. A directed forest is called a directed spanning forest if it is a spanning subgraph of digraph G. Given a nonnegative matrix Q ∈ Rn×n , let G[Q] denote a digraph with weighted adjacency matrix Q, a set of vertices {1, 2, · · · , n} whose elements are the indexes of the rows of Q, and a set of edges E[Q]. For matrices Q = [qij ]n×n and R = [rij ]n×n , the digraphs G[Q] and G[R] are said to be of the same type, denoted by G[Q] ∼ G[R], if qij > 0 ⇔ rij > 0 and qij = 0 ⇔ rij = 0. Suppose that the leaders have no neighbors, then the adjacency matrix A can be expressed in

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product of an infinite number of row stochastic matrices, where all row sums are strictly equal to 1, has been studied extensively in [34, 35]. However, few literatures have studied the products convergence of an infinite number of time-varying matrices whose all row sums are less than or equal to 1. For overcoming the theoretical difficulties, we develop a new method based on the composite of binary relation in this paper. In order to analyze this issue in depth, the rest of this paper is divided into the following sections. In Section 2, some notations and the basic concepts about graph theory are introduced, and some definitions and supporting lemmas are presented. Section 3 shows the model formulation in detail. In Section 4, a sufficiency criterion is established to ensure the heterogeneous containment control. The effectiveness of the control protocol is verified by a simulation example in Section 5. Finally, the conclusion of this article is shown in Section 6.

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the following form:

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0m×m A =  Af l Asl

0m×n Af f Asf

 0m×r Af s  . Ass

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aij and lij = −aij for i 6= j, and

Ds = diag

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 m+n+r X

 j=1  m+n+r X 

am+1,j ,

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am+n+1,j ,

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Df − Af f −Asf

am+2,j , · · · ,

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−Af s Ds − Ass

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am+n,j

j=1

am+n+2,j , · · · ,



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,

am+n+r,j

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Obviously, matrix Df denotes the in-degrees of the first-order dynamic followers, and matrix Ds represents the in-degrees of second-order dynamic followers. The following definitions concerning the convex hull and the composite of binary relation are introduced for subsequent use. Definition 1 ([36]). A set K ⊂ Rp is said to be convex if (1 − a)x + ay ∈ K for any x ∈ K, y ∈ K and a ∈ [0, 1]. The convex hull of a finite set of points x1 , . . . , xn ∈ Rp is the minimal convex set containing Pn all points xi , iP=n 1, . . . , n, denoted by Co{x1 , . . . , xn }. More specifically, Co{x1 , . . . , xn } = { i=1 ai xi |ai ≥ 0, i=1 ai = 1}. Definition 2 ([37]). Let E1 , E2 be relations on a set V. The composite of E1 and E2 is the relation consisting of ordered pairs (a, c), where a, c ∈ V, and for which there exists an element b ∈ V such that (a, b) ∈ E1 and (b, c) ∈ E2 . The composite of E1 and E2 is denoted by E1 o E2 . 3. Model formulation

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, LF =

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with Df = diag

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where lii =

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It can be seen that Af l and Asl describe the information transmission from the leaders to the followers. Af s and Asf show the communication between the sets Vf and Vs . Af f and Ass describe, respectively, the information exchange of the agents inside the sets Vf and Vs . According to the form of adjacency matrix, the Laplacian matrix is defined as   0m×m 0m×(n+r) L = [lij ] = , LR LF

We first consider the continuous-time containment control  vi   x˙ i (t) = 0, x˙ i (t) = ui (t), vi   x˙ i (t) = ϑ˙ i (t), ϑ˙ i (t) = ui (t), vi 4

case. The agents can be modeled as ∈ Vl ;

∈ Vf ; ∈ Vs ,

(1)

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where xi (t) ∈ Rp×1 , ϑi (t) ∈ Rp×1 and ui (t) present the position, the velocity and the control input of agent vi , respectively. A control protocol consisting of two parts for the heterogeneous multi-agent system is presented as follows,  m+n+r
P   γ aij (t) xj (t) − xi (t) , 1    j=1    m+n+r  X 
    −γ ϑ (t) + γ aij (t) xj (t) − xi (t)  2 i 3  ui (t) =  j=1     m+n+r   X 
     + aij (t) ϑj (t) − ϑi (t) ,     j=1

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where γ1 > 0, γ2 > 0 and γ3 > 0 are the feedback control gains. In a realistic multi-agent network, continuous data exchange among the agents is difficult due to practical limitations, such as environmental interference and communication distance limitations. This fact guides us to consider discrete-time dynamic model formulation in which each agent receives its neighbors’ state information only at discrete time instants. At each discrete time instant kT ,
agents communicate with each other via a time varying digraph G(k) = V, E(k), A(k) with the weighted adjacency matrix A(k) = aij (k) , in which we assume that all the positive weighting factors have uniform lower and upper bound, i.e. aij (k) ∈ [α, α], where 0 < α < α. The agents’ discrete-time heterogeneous dynamic behaviors are usually modeled by the following model:  v i ∈ Vl ;   xi (k + 1) = xi (k), xi (k + 1) = xi (k) + T ui (k), vi ∈ Vf ; (3)   xi (k + 1) = xi (k) + T ϑi (k), ϑi (k + 1) = ϑi (k) + T ui (k), vi ∈ Vs ,

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where xi (k) ∈ Rp×1 , ϑi (k) ∈ Rp×1 and ui (k) denote, respectively, the position, the velocity and the control input of agent vi at time kT , in which T > 0 is the constant step-size. For system (3), the following containment control protocol is proposed

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aij (k) xj (k) − xi (k) ,

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By referring to the definitions of containment control with switching topologies in [25–27], below we present the definition of containment control for system (3) under the protocol (4). Definition 3. The containment control problem for the heterogeneous multi-agent system (3) can be solved by using the protocol (4) if the final position states of all the followers can be always in the convex hull formed by the stationary leaders, such as  lim xi (k) ∈ Co x1 (0), x2 (0), · · · , xm (0) , i = m + 1, m + 2, · · · , n, (5)

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 Pm Pm where Co x1 (0), x2 (0), · · · , xm (0) = { i=1 ai xi |ai ≥ 0, i=1 ai = 1} is the minimal convex set containing all points xi (0), i = 1, . . . , m. 5

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Remark 1. For the containment control under switching topologies, the position states of the followers will be varied with the change of the communication topology even if they can enter the convex hull formed by the stationary leaders. Therefore, the definition of heterogeneous containment control under switching topologies considered in this paper is different from the definition of containment control with fixed topology [18–23, 32, 33], in which all followers’ final states will remain stationary in the convex hull formed by the leaders due to the time-invariant characteristic of fixed topology. In order to transform the heterogeneous multi-agent system (3) under protocol (4) into an augmented system, we first introduce the following model transformations. Let  T Xl (k) = xT1 (k), xT2 (k), · · · , xTm (k) ,  T Xf (k) = xTm+1 (k), xTm+2 (k), · · · , xTm+n (k) ,  T Xs (k) = xTm+n+1 (k), xTm+n+2 (k), · · · , xTm+n+r (k) ,  T Vs (k) = ϑTm+n+1 (k), ϑTm+n+2 (k), · · · , ϑTm+n+r (k) , T  1 T T T Zs (k) = Xs (k), Vs (k) + Xs (k) . γ3

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By substituting the protocol (4) into system (3), we can derive Xl (k + 1) = Xl (k),

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 In − γ1 T Df (k) ⊗ Ip Xf (k) + γ1 T [Af l (k) ⊗ Ip ] Xl (k)

+ γ1 T [Af f (k) ⊗ Ip ] Xf (k) + γ1 T [Af s (k) ⊗ Ip ] Xs (k),

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  Ir − γ3 T Ir γ3 T Ir ⊗ Ip Zs (k) (γ2 − γ3 ) T Ir Ir − (γ2 − γ3 ) T Ir − γ3 T Ds (k)    0r×m + ⊗ Ip Xl (k) γ3 T Asl (k)    (8) 0r×n + ⊗ Ip Xf (k) γ3 T Asf (k)    0r×r + ⊗ Ip Zs (k). γ3 T Ass (k) h iT Let Z(k) = XlT (k), XfT (k), ZsT (k) , then systems (6), (7) and (8) can be merged into an augmented system with matrix-vector form as

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Z(k + 1) = [M (k) ⊗ Ip ] Z(k),

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Ir − (γ2 − γ3 ) T Ir − γ3 T Ds (k) − Ass (k)

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It can be seen that the coefficient matrices M (k), k ∈ N in the augmented system are quite complex and also varied with time because of the complexity of heterogeneous dynamics. Q∞ As a result, it is very difficult to discuss the followers’ state over time step by analyzing k=0 M (k) directly. To derive our main result, we need the following lemma to guarantee that M (k) are nonnegative matrices by choosing the proper system parameters γ1 , γ2 and γ3 . Lemma 1. M (k) is a row-stochastic matrix containing positive diagonal elements if the following conditions hold:

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1 1 1 , γ3 < , γ3 ≤ γ2 < + γ3 (1 + ∆), T∆ T T  Pm+n+r aij (k) : i = 1, 2, · · · , m + n + r, k ∈ N . where ∆ = max j=1 γ1 <

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4. Main result

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then the main work we will do next is to solve the convergence problem limk→∞ N (k)N (k − 1) · · · N (0) → 0 and the achievement of the heterogeneous containment control under switching topologies. However, it can be seen that the row sums of N (k) are less than or equal to 1 under the conditions in (10), which implies that N (k) are not stochastic matrices that have some fine properties and have been widely used in the proof of the convergence performance of multi-agent systems, such as [7, 27]. Recently, we have proposed a new approach based on binary relation theory to study the first-order leader-following system with switching topologies in [16]. In this paper, we will develop this method to the more complex situation, that is, the containment control problem of second-order heterogeneous multi-agent systems with switching topologies.

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In this section, the matrix product technique and the composite of binary relation are explored to prove the realization of heterogeneous containment control under switching topologies. Before moving on, we first divide all discrete time instants 0, T, 2T, 3T, · · · , into an infinite number of uniformly bounded time intervals [kj T, kj+1 T ), j = 1, 2, · · · , starting  at k1 = 0. Similar to [23] and [24], we can obtain from (10) that the union of digraphs G M (kj ) , G M (kj +    1) , · · · , G M (kj+1 − 1) has a spanning forest rooted at the vertices in the set {1, 2, · · · , m} if the union of digraphs G(kj ), G(kj + 1), · · · , G(kj+1 − 1) has a spanning forest rooted at the leaders. Let   In − γ1 T Df (k) + γ1 T Af f (k) γ1 T Af s (k) 0n×r , 0r×n Ir − γ3 T Ir γ3 T Ir N (k) =  T Asf (k) (γ2 − γ3 ) T Ir Ir − (γ2 − γ3 ) T Ir − T Ds (k)

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Proof. If γ1 < T1∆ , then In − γ1 T Df (k) + γ1 T Af f (k) is a nonnegative matrix with positive diagonal elements. From the condition γ3 < T1 , we can obtain that Ir − γ3 T Ir is a nonnegative matrix with positive diagonal elements. The condition γ3 ≤ γ2 <
T1 + γ3 (1 + ∆) yields that matrices (γ2 − γ3 ) T Ir and Ir − (γ2 − γ3 ) T Ir − γ3 T Ds (k) − Ass (k) are also
nonnegative, and there are positive diagonal elements in Ir − (γ2 − γ3 ) T Ir − γ3 T Ds (k) − Ass (k) . In addition, the matrices γ1 T Af l (k), γ1 T Af s (k), γ3 T Asl (k) and γ3 T Asf (k) are nonnegative, and all the row sums of matrix M (k) are 1. Therefore, M (k) is a row-stochastic matrix containing positive diagonal elements.

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where (x0 , x0 ) ∈ E M (kj ) , (x0 , x0 ) ∈ E M (kj + 1) , · · · , (x0 , x0 ) ∈ E M (kj + h0 − 1) , (x0 , j1 ) ∈         E M (kj + h0 ) , (j1 , j1 ) ∈ E M (kj + h0 + 1) , (j1 , j1 ) ∈ E M (kj + h0 + 2) , · · · , (j1 , j1 ) ∈ E M (kj+1 − 1) .   Skj+2 −1 Similarly, there is a directed path x1 → (j + 1)1 → (j + 1)2 → · · · → (j + 1)z1 in i=k G M (i) j+1  for each vertex (j + 1)z1 , where x1 ∈ {1, 2, · · · , m} and (j + 1)1 , (j +
1)2 , · · · , (j + 1)z1 ∈ m + 1, m + 2, m + 3, · · · , m + n + 2r . For each edge (j + 1)g−1 , (j + 1)g , where
g ∈ {2, 3, · · · , z1},  there exists h1 such that kj+1 ≤ kj+1 + h1 < kj+2 and (j + 1)g−1 , (j + 1)g ∈ E M (kj+1 + h1 ) . Since the vertices (j + 1)g−1 and (j + 1)g have self-loops in all digraphs G M (q) , q ∈ N, then we have
      (j + 1)g−1 , (j + 1)g ∈ E M (kj+1 ) oE M (kj+1 + 1) o · · · oE M (kj+2 − 1) ,
 
  (j + 1)g−1 , (j + 1)g−1 ∈ E M (kj+1 ) , (j + 1)g−1 , (j + 1)g−1 ∈ E M (kj+1 + 1) , · · · , (j +
 
 
1)g−1 , (j + 1)g−1 ∈ E M (kj+1 + h1 − 1) , (j + 1)g−1 , (j + 1)g ∈ E M (kj+1 + h1 ) , (j + 1)g , (j + 1)g ∈  
 
  E M (kj+1 + h1 + 1) , (j + 1)g , (j + 1)g ∈ E M (kj+1 + h1 + 2) , · · · , (j + 1)g , (j + 1)g ∈ E M (kj+2 − 1) .

where

From the above discussion, if there exists (j + 1)g0 −1 ∈ {(j + 1)1 , (j + 1)2 , · · · , (j + 1)z1−1 } such that j1 = (j + 1)g0 −1 , then       (x0 , j1 ) ∈ E M (kj ) oE M (kj + 1) o · · · oE M (kj+2 − 1) ,
      x0 , (j + 1)g0 ∈ E M (kj ) oE M (kj + 1) o · · · oE M (kj+2 − 1) ,  and if j1 = (j + 1)z1 or j1 ∈ / (j + 1)1 , (j + 1)2 , · · · , (j + 1)z1 , then       (x0 , j1 ) ∈ E M (kj ) oE M (kj + 1) o · · · oE M (kj+2 − 1) ,
      x1 , (j + 1)1 ∈ E M (kj ) oE M (kj + 1) o · · · oE M (kj+2 − 1) .

This implies that for time interval [kj T, kj+2 T ), there always exist two different vertices s1 , s2 ∈       {m + 1, m + 2, · · · , m + n +2r} such that (x1 , s1 ) ∈ E M (kj ) o E M (kj + 1) o · · · o E M (kj+2 − 1) ,    and (x2 , s2 ) ∈ E M (kj ) o E M (kj + 1) o · · · o E M (kj+2 − 1) , where x1 , x2 ∈ {1, 2, · · · , m}. Suppose for time interval [kj T, kj+j 0 T ), 1 ≤ j 0 < n + 2r, there exist j 0 different vertices    s1 , s2 , ·· · , sj 0 ∈ {m + 1, m + 2, · · · , m + n + 2r} such that (xi , si ) ∈ E M (kj ) o E M (kj + 1) o · · · o E M (kj+j 0 − 1) , where xi ∈ {1, 2, · · · , m}, i = 1, 2, · · · , j 0 . In the following, we consider time interval [kj+j 0 T, kj+j 0 +1 T ). It is known that there is a directed path xj 0 → (j + j 0 )1 → (j + j 0 )2  Skj+j0 +1 −1  G M (i) for each vertex (j + j 0 )zj 0 , where xj 0 ∈ {1, 2, · · · , m} and → · · · → (j + j 0 )zj 0 in i=k j+j 0

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   Lemma 2. Suppose that G M (kj ) , G M (kj +   the conditions in (10) hold. If the union of digraphs
1) , · · · , G M (kj+1 − 1) associated with each time interval kj T, kj+1 T contains a spanning forest
rooted at the vertices in the set {1, 2, · · · , m}, then for the longer time intervals kj T, kj+n+2r T , j ∈ N, we have kN (kj+n+2r − 1)N (kj+n+2r − 2) · · · N (kj )k∞ < 1.  Skj+1 −1  Proof. Because i=k G M (i) contains a spanning forest rooted at the vertices {1, 2, · · · , m}, j  Skj+1 −1  there exists a directed path x0 → j1 → j2 → · · · → jz0 in i=k G M (i) for each vertex jz0 , j  where
x0 ∈ {1, 2, · · · , m} and j1 , j2 , · · · , jz0 ∈ m + 1, m + 2, m + 3, · · · , m + n + 2r . Assume x0 , j1 ∈ E M (kj +h0 ) , where kj ≤ kj + h0 < kj+1 . Since vertices x0 and j1 have self-loops in all digraphs G M (q) , q ∈ N, then we can obtain       (x0 , j1 ) ∈ E M (kj ) oE M (kj + 1) o · · · oE M (kj+1 − 1) ,

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 (j+j 0 )1 , (j+j 0 )2 , · · · , (j+j 0 )zj 0 ∈ m+1, m+2, m+3, · · · , m+n+2r . Since j 0 < n+2r, then we can always find two adjacent vertices (j + j 0 )i and (j + j 0 )i+1 that satisfy (j + j 0 )i ∈ {s1 , s2 , · · · , sj 0 } and
0 0 0 (j+ j 0 )i+1 ∈ / {s1 ,s2 , · · · , sj 0 }, where i ∈ {1,   2, · · · , zj − 1}. Besides, we have (j 0+ j )i , (j + j )i+1 ∈ E M (kj+j 0 ) o E M (kj+j 0 + 1) o · · · o E M (kj+j 0 +1 − 1) . Therefore, let (j + j )i+1 = sj 0 +1 , it can be observed that thereexist j0 +1 different vertices s1 , s2 , · · · , sj 0 +1 ∈ {m+1, m+2, · · · , m+n+2r}   such that (xi , si ) ∈ E M (kj ) o E M (kj + 1) o · · · o E M (kj+j 0 +1 − 1) , where xi ∈ {1, 2, · · · , m},
i = 1, 2, · · · , j 0 + 1. This implies that for time interval kj T, kj+n+2r T , there exist n + 2r dif ferent vertices s1 , s2 , ·· · , sn+2r ∈ {m+ 1, m + 2, · · · , m + n + 2r} such that (xi , si ) ∈ E M (kj ) o   E M (kj + 1) o · · · o E M (kj+n+2r − 1) , where xi ∈ {1, 2, · · · , m}, i = 1, 2, · · · , n + 2r. 
Summarizing the analysis above, for an arbitrary time interval kj T, kj+n+2r T and each s ∈ {m xs ∈ {1, 2, · · · , m} such that (xs , s) ∈ E M (kj ) o  + 1, m +2, · · · , m + n + 2r}, there exists  E M (kj + 1) o· · · o E M (kj+n+2r − 1) . According to Definition  2, it is known that there exist  s1 , s2 , · · · , sz ∈ m+1, m+2, · · · , m+n+2r such that (xs , s1 ) ∈ E M (kj ) , (s1 , s2 ) ∈ E M (kj +1) , · · · , (sz , s) ∈ E M (kj+n+2r − 1) . If sd 6= xs and sd−1  = sd−2 = · ·· = s1 = xs , where d ≥ 1, then it can be seen that Λsd [N (kj + d − 1)] < 1 and Λi N (kj + d − 2) = Λi N (kj + d − 3) = · · · = Λi N (kj ) ≤ 1, i = 1, 2, · · · , n + 2r. For convenience, we let R(kj + d − 1) = N (kj + d − 1) · · · N (kj + 1)N (kj ), then   Λsd R(kj + d − 1) n+2r X y=1

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Theorem 1. Under the conditions in (10), the containment control problem for system (3) with protocol (4) can be solved if the
union of digraphs G(kj ), G(kj + 1), · · · , G(kj+1 − 1) associated with each time interval kj T, kj+1 T has a spanning forest rooted at the leaders. Proof. Let Q(β) = N (k1+(β+1)(n+2r) −1)N (k1+(β+1)(n+2r) −2) ), β = 0, 1, · · · , then

Q· · · N (k1+β(n+2r)

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0m×(n+2r)

0(n+2r)×(n+2r)

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 T where P (k) = γ1 T ATfl (k), 0m×r , γ3 T ATsl (k) and R(k) = P (k) + N (k)P (k − 1) + · · · + N (k)N (k − 1) · · · N (1)P (0). From a well-known result that the product of infinite row stochastic matrices 10

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Q∞ is also a row stochastic matrix, we can know that k=0 M (k) is a row stochastic matrix. This implies that lim R(k) is a row stochastic matrix. Therefore, it can be derived from system (8) k→∞  Q∞   Q∞ that limk→∞Xf (k) = k=0 R(k)(1 → n, :) ⊗ Ip Xl (k) and limk→∞ Xs (k) = k=0 R(k)(n + 1 → n + r, :) ⊗ Ip Xl (k). Therefore, for any i = m + 1, m + 2, · · · , m + n + r, one have  lim xi (k) ∈ Co x1 (0), x2 (0), · · · , xm (0) , i = m + 1, m + 2, · · · , n. (11) k→∞

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By Definition 1, it is clear that the followers’ positions always converge to a stationary convex hull spanned by the stationary leaders. From Definition 3, it can be seen that the heterogeneous containment control can be achieved.

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Remark 2. In Theorem 1, a sufficient condition in terms of the communication topologies is established. For any given switching topologies and step-size T , we can always design appropriate parameters γ1 , γ2 and γ3 to satisfy the conditions in (10).

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Remark 3. For realizing the containment control for a group of agents with heterogeneous dynamics under switching topologies in this paper, we first transform the heterogeneous multi-agent systems into an augmented system with time-varying coefficient matrices. Then, the detailed parameter selection strategy in (10) is given to guarantee the nonnegativity of the coefficient matrices. This implies that the heterogeneous containment control issue is transformed into a convergence issue of the product of infinite time-varying nonnegative matrices in which all row sums are less than or equal to 1. Due to the complexity of heterogeneity dynamics and switching topologies, the methods used in [25–27, 32, 33], such as the construction of error system, Laplacian transform and trajectory analysis etc., are useless for the issues considered in this article. For solving this convergence problem, a new method that is the composite of binary relation is developed in this paper. Moreover, this new proposed method can also be easily extended to deal with the containment control problem of a large class of discrete-time multi-agent systems, including linear multi-agent systems with switching topologies and second-order multi-agent systems without velocity measurements, in which the discrete-time systems can be usually transformed into a matrix-vector form with nonnegative coefficient matrices.

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Under protocol (4), we consider a heterogeneous multi-agent system consisting of three leaders and six followers. Denote Vl = {v1 , v2 , v3 }, Vf = {v4 , v5 , v6 } and Vs = {v7 , v8 , v9 }. Let the communication topology at any times kT , k ∈ N can be randomly chosen in Fig. 1, where the weight of each edge is 1. It is assumed that each communication topology is chosen at least once in each time interval of 5T , then it is clear that the union of digraphs Ga , Gb and Gc in each time interval of 5T has a spanning forest rooted at the leaders. Let T = 0.1, and then we select γ1 = 1, γ2 = 1.5, γ3 = 2 that satisfy the conditions in (10). The trajectories of position states of the agents at different time are shown in Fig. 2. It is obvious that the position states of all the followers can be always in the convex hull formed by the stationary leaders.

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The containment control for discrete-time heterogeneous multi-agent systems with switching topologies has been investigated in this paper. Based on two distributed control protocols and the 11

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appropriate model transformations, the heterogeneous containment control issue under switching topologies has been transformed into the products convergence of infinite time-varying nonnegative matrices whose all row sums are less than or equal to 1. Then by using the composite of binary relation, we have solved the convergence problem and established a sufficient condition in terms of the communication topologies to guarantee that the containment control can be achieved. Moreover, a simulation example is also provided to explain the validity of our theoretical result.

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The author would like to thank the editor and the anonymous reviewers for their valuable suggestions.

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References

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[31] L. Ding, W.X. Zheng, Network-based practical consensus of heterogeneous nonlinear multiagent systems, IEEE Transactions on Cybernetics 47(8) (2017) 1841–1851. [32] Y. Zheng, L. Wang, Containment control of heterogeneous multi-agent systems, International Journal of Control 87(1) (2014) 1–8. [33] H. Haghshenas, M. Badamchizadeh, M. Baradarannia, Containment control of heterogeneous linear multi-agent systems, Automatica 54 (2015) 210–216. [34] F. Xiao, L. Wang, Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays, IEEE Transactions on Automatic Control 53(8) (2008) 1804–1816. [35] J. Qin, C. Yu, S. Hirche, Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topology, IEEE Transactions on Industrial Informatics 8(4) (2012) 986–994. [36] R. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press (1972). [37] K. Rosen, Discrete Mathematics and Its Applications. (7th ed.), New York: McGraw-Hill Companies. (2012).

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Lei Shi received the B.S. degree in mathematics and applied mathematics from Shanxi Datong University in 2014. He is currently working toward the Ph.D. degree in the School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu. Her research interests include networked control systems and multi-agent systems.

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Jinliang Shao received the BSc and PhD degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003 and 2009, respectively. During 2014, he was a visiting scholar in Australian National University, Australia. He is currently an associate professor in the School of Automation Engineering, UESTC. His research interests include multiagent system, robust control, and matrix analysis with applications in control theory.

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Hong Xia received the B.S. degree in Mathematics from Shandong Normal University in 2003, M.S. degree in Mathematics from Wuhan University in 2006, and Ph.D. degree in Mathematics from University of Electronic Science and Technology in 2014. She is currently working in School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China. Her research interests include coordination control of multi-agent systems, matrix analysis with applications.

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Mengtao Cao received his PhD in General Mechanics and Foundation of Mechanics from Peking University in 2015. He was a Visiting Scholar in University of California, Riverside from 2016 to 2017, and currently works in University of Electronic Science and Technology of China. His main research interests include cooperative control of Multi-agent Systems, event-driven control, and social dynamics.

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