# Second-Order Containment Control of Multiagent Systems in the Presence of Uncertain Topologies with Time-Varying Delays

Authors: Fuyong Wang; Hongyong Yang; Zhongxin Liu; Zengqiang Chen
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

This paper considers the containment control problem of second-order multiagent systems in the presence of time-varying delays and uncertainties with dynamically switching communication topologies. Moreover, the control algorithm is proposed for containment control, and the stability of the proposed containment control algorithm is studied with the aid of Lyapunov-Krasovskii function when the communication topology is jointly connected. Some sufficient conditions in terms of linear matrix inequalities (LMIs) are provided for second-order containment control with multiple stationary leaders. Finally, simulations are given to verify the effectiveness of the obtained theoretical results.


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## Body

## 1. Introduction

Recently, there has been a growing interest towards development of the distributed cooperative control of multiagent systems (MASs). To our knowledge, plenty of theoretical results about consensus [1–5] and containment control [6–9] in distributed cooperative control of MASs have been obtained. However, in practical applications, due to communication delays and uncertain topologies that always emerge, analysis and synthesis of distributed cooperative problem have become more complex and difficult. Meanwhile, it is practically significant to investigate the containment control of MASs with delays and uncertainties.For consensus problems of MASs with uncertain topologies, the average consensus with time-varying communication delays is investigated in [10], and several sufficient conditions for average consensus are derived in terms of linear matrix inequalities (LMIs). In [11], the robust discrete-time consensus problem of MASs with uncertain topologies is addressed, and a necessary and sufficient condition for robust discrete-time consensus is obtained by the Lyapunov stability theory. Considering both fixed and switching directed topologies, the average consensus problem in MASs with uncertain topologies and multiple time-varying delays is studied in [12]. For linear time-invariant MASs over Markovian switching networks, stochastic consensus problems with time-varying delays and uncertain topologies are analyzed in [13]. In [14], the consensus problem of MASs in the presence of uncertain topologies with time-varying delays is analyzed by a new approach, and a condition in terms of linear matrix inequalities is presented for consensus for MASs with switching topologies.As a kind of extended consensus problem, containment control has been paid much attention, which aims to design appropriate control protocols to drive the followers to a target area (convex hull formed by the leaders) asymptotically. For linear MASs, the cooperative containment control problem is discussed in [15], and several necessary and sufficient containment conditions are presented by using spectral analysis and matrix theory. Considering unconnected topologies of MASs, the containment control for both first-order and second-order MASs with jointly connected topologies is studied in [16]. Considering communication delays, containment control for second-order MASs with time-varying delays is studied in [17]; both the case with multiple dynamic leaders and the case with multiple stationary leaders are discussed. In [18], the containment control problem for second-order MASs with time-varying delays and jointly connected topologies is investigated, and the stability of the proposed control algorithm is analyzed by Lyapunov-Krasovskii method. For uncertain linear MASs, the containment control problem with the dynamic agents described by fractional-order differential equations is investigated in [19], and some sufficient conditions are presented by the stability theory of fractional-order systems and matrix theory. The above-mentioned works implicitly assume the link weights or interaction strengths can be exactly measured; however, environmental uncertainties and measurement error cannot be ignored in real systems. Motivated by these considerations, the containment control of first-order MASs with uncertain topologies and communication time delays is studied in [20], and the sufficient condition for containment control of MASs under jointly connected topologies is derived with the aid of Lyapunov-Krasovskii function.Two important factors always emerge in some real situations: one is that some uncertainties may exist in the MASs due to measurement error and environmental uncertainties cannot be avoided in real systems and another is that time delays are usually inevitable because of the possible slow process of interactions among the agents in communication networks. Motivated by these two factors, lots of distributed consensus protocols have been developed by some researchers for MASs with uncertainties and delayed communications. However, different from most of those current literatures, for the MASs with uncertainties and time-varying delays, we aim to analyze and investigate the distributed containment control problem with multiple leaders and jointly connected topologies. The algorithms proposed here can ensure that the uncertain MASs achieve containment control in the presence of switching topologies with multiple stationary leaders. By applying linear matrix inequality method, the convergence of the algorithm for the MASs with dynamically switching topologies is analyzed by Lyapunov-Krasovskii method. Finally, numerical simulations are provided to illustrate the effectiveness of the conclusion in this paper.The paper is organized as follows. In Section2, some basic concepts in algebraic graph theory and some related lemmas are presented. In Section 3, the main results for second-order MASs with uncertain topologies and time-varying delays are obtained. Then the containment control protocol is presented for MASs with multiple stationary leaders and dynamically switching topologies. Finally, numerical simulations and conclusion are given in Sections 4 and 5, respectively.


## 2. Preliminaries

LetG=(W,E,A) be an undirected graph consisting of a note set W={1,2,…,n}, an edge set E∈W×W, and a weighted adjacency matrix A=[aij]∈Rn×n. The elements of A satisfy aij>0 if (j,i)∈E, aij=0; otherwise, where (j,i) represents an edge of G, {j,i}∈W. Here, we assume that (i,i)∉E; that is, aii=0. Let Ni=j∈W∣j,i∈E,j≠i denote the set of the neighbors of node i. TheLaplacian matrix corresponding to the undirected graph G is defined as L=[lij]∈Rn×n, where lij=-aij, j≠i, and lij=∑j∈Niaij, j=i.A path between two distinct notesi and j is a finite ordered sequence of distinct edges of G with the form (i1,i2),(i2,i3),…,(iπ-1,iπ), where i1=i and iπ=j. The undirected graph G is said to be connected if there is a path between any distinct pair of nodes. The union of a collection of graphs G1,G2,…,Gq with the same note set W is defined as the graph G1-q with the note set W and edge set equal to the union of the edge sets of all of the graphs in the collection. Moreover, G1,G2,…,Gq is jointly connected if its union graph G1-q is connected [21].Consider an infinite sequence of nonempty, bounded, and contiguous time intervals[tr,tr+1), r=1,2,…, with t1=0 and tr+1-tr≤Ta, Ta>0. In each interval [tr,tr+1), there is a sequence of subintervals [tr,s,tr,s+1) with tr,1=tr and tr,mr+1=tr+1 satisfying tr,s+1-tr,s≥Tb, where Tb>0 and s=1,2,…,mr, mr≥1. The communication topology described by Gr,s switches at tr,s and it does not change during each subinterval [tr,s,tr,s+1). Suppose that the MASs represented by the communication topology graph G consist of n followers and m leaders. Moreover, when the topology of MASs is dynamically switching, we introduce a piecewise constant switching function σ(t):[0,+∞)→P, P={1,2,…,p}, σ=σ(t), σ∈P, where p denotes the total number of all possible communication topologies. Then the communication topology of MASs described by Gσ switches at t and itsLaplacian matrix is denoted by Lσ, and the communication topology among the n followers described by GσF switches at t and itsLaplacian matrix is denoted by LσF.In MASs, an agent is called a leader if it has no neighbor or a follower if it has at least one neighbor [22]. Considering a MAS with n followers and m leaders, since the leaders have no neighbors, theLaplacian matrix L associated with the communication graph G can be partitioned as (1)L=LFLFR0m×n0m×m,where LF∈Rn×n and LFR∈Rn×m. The following lemma about LF and LFR is brought in, which is useful for deriving the main results.Lemma 1 (see [23]).
If there exists at least a path to a leader for each follower, thenLF in (1) is positive definite, -LF-1LFR is a nonnegative matrix, and the sum of the entries in each row equals 1.Definition 2 (see [24]).
LetX={ξ1,ξ2,…,ξn} be a set in a real vector space Z⊆R. CO(X)=∑i=1nβiξi∣ξi∈X,βi∈R,βi≥0,∑i=1nβi=1 denotes the convex hull of the set X.Lemma 3 (see [25]).
For any real differentiable vector functionz(t)∈Rn and any n×n dimensional constant matrix M=MΤ>0, we have the following inequality:(2)zt-zt-τtΤMzt-zt-τt≤h∫t-τttz˙ΤsMz˙sds,where t≥0, 0≤τ(t)≤h.Lemma 4 (see [26]).
LetQ=QΤ, F1, F2, and S(t) be matrixes with appropriate dimensions, and matrix S(t) satisfies SΤ(t)S(t)≤I. Then(3)Q+F1StF2+F2ΤSΤtF1Τ<0,holds if and only if there exists a positive constant ε>0 such that(4)Q+ε-1F1F1Τ+εF2ΤF2<0.Lemma 5 (see [27]).
Letϕ(t):R≥0→R be a piecewise continuous function with the following properties:(1)
ϕ(t) is continuous and differentiable on each subinterval [tr,s,tr,s+1) and switches at tr,s, r=1,2,…, s=1,2,…,mr. Moreover, ϕ(t) is right continuous and right differentiable function at tr,s.(2)
The derivative (including the right derivative) ofϕ(t) for any t∈[0,+∞) is bounded; that is, ϕ˙t≤ω for any t∈[0,+∞) and some constant ω>0.(3)
limt→+∞⁡∫0tϕ(τ)dτ exists and is finite.Then limt→+∞⁡ϕ(t)=0.


## 3. Main Results

Consider second-order MASs consisting ofn followers and m stationary leaders. The followers’ set and the leaders’ set are denoted by F={1,2,…,n} and R={n+1,n+2,…,n+m}, respectively. The dynamics of each agent can be described by the following equation:(5)q˙it=pit,p˙it=uit,i∈F,R,where qi(t)∈RN, pi(t)∈RN, and ui(t)∈RN are the position vector, the velocity vector, and the control input vector of the ith agent, respectively. For simplicity, we assume that N=1 in this paper and the case of N>1 can be obtained with the Kronecker product.In a network with uncertain topology, we consider the following containment control algorithm for (5):(6)pit=0,i∈R;uit=-k1pit-∑j∈Niaij+Δaijtqit-τt-qjt-τt,i∈F,where k1>0 is the feedback gain, Δaij(t) and Δaii(t)=0 are uncertainties, and τ(t) is the time-varying communication delay. Here the uncertainties Δaij(t) exist in the control algorithm, since measurement error and environmental uncertainties cannot be avoided in real systems.Assumption 6.
The time-varying communication delayτ(t) in (6) is bounded; that is, 0≤τ(t)<h, where h>0, t≥0.Assumption 7.
In each interval[tr,tr+1), r=1,2,…, the communication topologies of MASs are jointly connected.Definition 8.
Δ L ( t ) = [ Δ l i j ( t ) ] is the uncertain matrix of MASs, which is defined as Δlij(t)=∑j∈NiΔaij(t), i=j, and Δlij(t)=-Δaij(t), i≠j. For the MASs with n followers and m leaders, the uncertain matrix ΔL(t) can be partitioned into ΔL(t)=ΔLF(t)ΔLFR(t)0m×n0m×m, where ΔLF(t)∈Rn×n, ΔLFR(t)∈Rn×m.In this paper, we assume that the norm bounded parameter uncertainty satisfies(7)ΔLt=C1StC2(8)ΔLFt=D1StD2,where C1, C2, D1, and D2 are constant matrixes with appropriate dimensions, and the diagonal matrix S(t) satisfies(9)STtSt≤I.It follows that the uncertainties of MASs satisfy(10)ΔLΤtΔLt≤α2In+m(11)ΔLFΤtΔLFt≤α2In,where α>0 is an appropriate constant.LetxF(t)=[qFΤ(t),pFΤ(t)]Τ and xL(t)=[qLΤ(t),pLΤ(t)]Τ, where qF(t)=[q1(t),q2(t),…,qn(t)]Τ, pF(t)=[p1(t),p2(t),…,pn(t)]Τ, qL(t)=[qn+1(t),qn+2(t),…,qn+m(t)]Τ, and pL(t)=[pn+1(t),pn+2(t),…,pn+m(t)]Τ. By Definition 8, it follows that the dynamics of MASs resulting from (5) to (6) can be written in a matrix form as(12)x˙Lt=H1xLtx˙Ft=-H2xFt-τt-H3xLt-τt,where H1=0m×mIm×m0m×m0m×m, H2=0n×n-InEFk1In, and H3=0n×m0n×mEFR0n×m, EF=LF+ΔLF(t), EFR=LFR+ΔLFR(t). Since there are multiple stationary leaders in MASs, that is, pL(t)=pL(0)=0, we get x˙L(t)=H1xL(t)=pL(t)0=0 in (12).Letx~F(t)=xF(t)+H2-1H3xL(t); then (12) can be transformed into the following form:(13)x~˙Ft=-H2x~Ft-τt,where H2-1=k1EF-1EF-1-In0n×n.Lemma 9.
LetxL∗=limt→∞⁡xL(t) and xF∗=limt→∞⁡xF(t). The containment control of the second-order MASs can be achieved if xF∗→-H2-1H3xL∗.Proof.
By Lemma1, it follows that -EF-1EFR is a nonnegative matrix and the sum of the entries in each row equals 1. Furthermore, -EF-1EFRqL∗ is located in the convex hull formed by the set of qL∗ from Definition 2, where qL∗=limt→∞⁡qL(t). Thus, if xF∗→-H2-1H3xL∗, that is, qF∗pF∗→-k1EF-1EF-1-In0n×n×0n×m0n×mEFR0n×m×qL∗pL∗=-EF-1EFRqL∗0, where qF∗=limt→∞⁡qF(t), pF∗=limt→∞⁡pF(t), and pL∗=pL(0)=0, then the containment control of second-order MASs can be achieved.Suppose the communication graphGσ on subinterval [tr,s,tr,s+1) has nσ≥1 connected subgraphs Gσi, and each connected subgraph Gσi has dσi=dσFi+dσRi nodes, where dσFi≥1 represents the number of followers, dσRi≥1 represents the number of leaders, i=1,2,…,nσ. TheLaplacian matrix of subgraph Gσi is denoted by Lσi, and the uncertain matrix of subgraph Gσi is denoted by ΔLσi. Then there exists an orthogonal matrix Uσ with appropriate dimensions such that(14)UσΤLσUσ=diag⁡Lσ1,Lσ2,…,Lσnσ(15)UσΤΔLσtUσ=diag⁡ΔLσ1t,ΔLσ2t,…,ΔLσnσt(16)x~FΤtUσ=x~σF1Τt,x~σF2Τt,…,x~σFnσΤt.Assumption 10.
There exists a connectivity subset for MASs in each nonoverlapping time interval[tr,j,tr,s+1)⊂[tr,tr+1), s=1,2,…,mr. For each follower, there exists at least one leader that has a path to the follower in the connectivity subset.Considering the second-order MASs in the presence of dynamically switching and uncertain topologies with time-varying delays, dynamics (13) can be written as(17)x~˙Ft=-Hσ2x~Ft-τt,where Hσ2=0n×n-InEσFk1In, EσF=LσF+ΔLσF(t).In each subinterval[tr,s,tr,s+1), dynamics (17) can be transformed into the following subsystems:(18)x~˙σFit=-Hσ2ix~σFit-τt,i=1,2,…,nσ,where x~σFi(t)=xσFi(t)+Hσ2i-1Hσ3ixσLi(t), Hσ2i=0dσFi×dσFi-IdσFiEσFik1IdσFi, Hσ3i=0dσFi×dσRi0dσFi×dσRiEσFRi0dσFi×dσRi, EσFi=LσFi+ΔLσFi(t), EσFRi=LσFRi+ΔLσFRi(t), xσFi(t)=[qσFiΤ(t),pσFiΤ(t)]Τ, xσLiΤ(t)=[qσLiΤ(t),pσLiΤ(t)]Τ, qσFi(t)=[qσ1i(t),…,qσdσFii(t)]Τ, qσLi(t)=[qσdσFi+1i(t),…,qσdσii(t)]Τ, pσFi(t)=[pσ1i(t),…,pσdσFii(t)]Τ, and pσLi(t)=[pσdσFi+1i(t),…,pσdσii(t)]Τ, i=1,2,…,nσ.Now, we consider a second-order MAS with uncertain topologies and communication time-varying delays. The stability and the convergence of MASs can be achieved as shown in the following Theorem11.Theorem 11.
Consider a second-order dynamic system with dynamics (5) of n followers and m stationary leaders with the switching topologies, and suppose Assumptions 6, 7, and 10 hold. Control protocol (6) can solve the containment control of second-order MASs in the presence of uncertain topologies with time-varying delays, for each subinterval [tr,s,tr,s+1), if there exists a constant ε>0 and an appropriate constant α>0, such that(19)Πσi=Π11Π12Π12ΤΠ22<0,i=1,2,…,nσ,where Π11=(ε-h-1)I2×dσFi, Π12=h-1I2×dσFi+(εh-1)H-σ2i, and Π22=(ε-1-h-1)I2×dσFi+(h+εh2)H-σ2iΤH-σ2i+hα2Jσi, H-σ2i=0dσFi×dσFi-IdσFiLσFik1IdσFi, Jσi=0dσFi0dσFiIdσFi0dσFi.Proof.
For MASs (17), we choose a Lyapunov-Krasovskii function candidate as(20)Vt=x~FΤtx~Ft+∫t-hts-t+hx~˙FΤsx~˙Fsds.Evidently, V(t) is essentially a distributed Lyapunov function. By (16), V(t) can be rewritten as(21)Vt=∑i=1nσx~σFiΤtx~σFit+∫t-hts-t+hx~˙σFiΤsx~˙σFisds.Calculating the derivative of V(t) along the trajectories of (18) yields(22)V˙t=∑i=1nσ-Hσ2ix~σFit-τtΤx~σFit+x~σFiΤt-Hσ2ix~σFit-τt+hx~˙σFiΤtx~˙σFit-∫t-htx~˙σFiΤsx~˙σFisds.
By Assumption6 and Lemma 3, we can get -∫t-htx~˙σFiΤ(s)x~˙σFi(s)ds≤-∫t-τ(t)tx~˙σFiΤ(s)x~˙σFi(s)ds and -∫t-τ(t)tx~˙σFiΤ(s)x~˙σFi(s)ds≤-h-1[x~σFi(t)-x~σFi(t-τ(t))]Τ[x~σFi(t)-x~σFi(t-τ(t))].
Denoteyσi=[x~σFiΤ(t),x~σFiΤ(t-τ(t))]Τ. It follows that(23)V˙t≤∑i=1nσ-x~σFiΤtHσ2ix~σFit-τt-x~σFiΤt-τtHσ2iΤx~σFit+hx~σFiΤt-τtHσ2iΤHσ2ix~σFit-τt-h-1x~σFit-x~σFit-τtΤx~σFit-x~σFit-τt=∑i=1nσyσiΤΩσiyσi,where Ωσi=-h-1I2×dσFih-1I2×dσFi-Hσ2ih-1I2×dσFi-Hσ2iΤ-h-1I2×dσFi+hHσ2iΤHσ2i.
DenoteH-σ2i=0dσFi×dσFi-IdσFiLσFik1IdσFi and Bσi=0dσFi×dσFi0dσFi×dσFiΔLσFi(t)0dσFi×dσFi. Then we have Hσ2i=H-σ2i+Bσi. Let Φσi=-h-1I2×dσFih-1I2×dσFi-H-σ2ih-1I2×dσFi-H-σ2iΤ-h-1I2×dσFi+hH-σ2iΤH-σ2i+hBσiΤBσi and Ψσi=0BσiBσiΤhH-σ2iΤBσi+hBσiΤH-σ2i; we have Ωσi=Φσi+Ψσi. By Lemma 4, it follows that Ωσi=Φσi+Ψσi=Φσi+0I2×dσFiBσiΤI2×dσFihH-σ2i+I2×dσFihH-σ2iΤBσi0I2×dσFiΤ<0 is equivalent to Ωσi=Φσi+Ξσi=Φσi+εI2×dσFihH-σ2iΤI2×dσFihH-σ2iε-10I2×dσFi0I2×dσFiΤ<0, where Ξσi=εI2×dσFiεhH-σ2iεhH-σ2iΤεh2H-σ2iΤH-σ2i+ε-1I2×dσFi, ε>0. By (11), we have ΔLσFiΤ(t)ΔLσFi(t)≤α2IdσFi and BσiΤBσi≤0dσFi×dσFi0dσFi×dσFiα2IdσFi0dσFi×dσFi. Then Φσi≤Γ11Γ12Γ12ΤΓ22, where Γ11=-h-1I2×dσFi, Γ12=h-1I2×dσFi-H-σ2i, and Γ22=-h-1I2×dσFi+hH-σ2iΤH-σ2i+hα2Jσi, Jσi=0dσFi0dσFiIdσFi0dσFi. Therefore, Ωσi is negative definite if condition (19) is satisfied. Thus V˙(t)<0 and the stability of the dynamically switching system (17) is guaranteed.
Now we know system (17) is asymptotically stable. So x~F(t) and x~F(t-τ(t)) are bounded, and x~˙F(t) is also bounded by (13). Consequently the derivative of ∑i=1nσy~σiΤ[Ωσi+Ψσi]y~σi is bounded by (16) and (18). It follows that ∫0+∞(∑i=1nσyσi(s)ΤΩσiyσi(s))ds exists and is finite since ∑i=1nσyσiΤΩσiyσi<0 and 0≥∫0t∑i=1nσyσi(s)ΤΩσiyσi(s)ds≥∫0tV˙(s)ds=V(t)-V(0). By Lemma 5, we can get limt→+∞⁡∑i=1nσyσiΤΩσiyσi=0 and limt→+∞⁡yσi=0 since Ωσi<0. Then limt→+∞⁡x~σFi(t)=x~σFi(t-τ(t))=0. Thus limt→+∞⁡xσFi(t)=-Hσ2i-1Hσ3ixσLi(t).
Furthermore, in any subinterval[tr,s,tr,s+1)⊂[tr,tr+1), s=1,2,…,mr, limt→+∞⁡xσFi(t)=-Hσ2i-1Hσ3ixσLi(t) and limt→+∞⁡qσFi(t)=-EσFi-1EσFRiqσLi(t), i=1,2,…,nσ. Therefore, in the connected portion of MASs in the subinterval [tr,s,tr,s+1) and [tr,s+1,tr,s+2), that is, in the subinterval [tr,s,tr,s+2), limt→+∞⁡xσFi′(t)=-Hσ2i-1′Hσ3i′xσLi′(t) and limt→+∞⁡qσFi′(t)=-EσFi-1′EσFRi′qσLi′(t) still hold, i=1,2,…,nσ′. Then by induction, according to Assumption 7, limt→+∞⁡xF(t)=-H2-1H3xL(t) and limt→+∞⁡qF(t)=-EF-1EFLqL(t), since all agents are jointly connected in each [tr,tr+1). Then we can conclude that the containment control of the second-order MASs with time-varying delays and uncertain topologies can be realized by Lemma 9.Corollary 12.
Consider the second-order MASs with dynamics (5) and suppose Assumptions 6, 7, and 10 hold, for the case where the communication delay τ(t) is time invariant; that is, τ(t)=τ is a constant in (6). Control protocol (6) can solve the containment control of second-order MASs in the presence of uncertain topologies with communication delays, for each subinterval [tr,s,tr,s+1), if there exists a constant ε>0 and an appropriate constant α>0, such that(24)Λσi=Λ11Λ12∗Λ22<0,i=1,2,…,nσ,where Λ11=(ε-1)I2×dσFi, Λ12=(ετ-1)H-σ2i, and Λ22=(ε-1-1)I2×dσFi+(τ+ετ2)H-σ2iΤH-σ2i+τα2Jσi; H-σ2i and Jσi are the same as H-σ2i and Jσi in (19).


## 4. Simulations

Consider dynamic switching topology with 5 followers and 3 leaders as shown in Figure1. The followers’ set and the leaders’ set are denoted by F={1,2,3,4,5} and R={6,7,8}, respectively. Suppose that the communication topology of the MASs randomly switches in {G1,G2,G3} at t=kT, k=0,1,…, T=0.5s, and the connection weights of each edge are all set to 1 in this paper.Figure 1
Three possible graphs of the communication topology.
(a) (b) (c)By the union topology of graphsG1~G3, the system matrix LσF can be obtained. Solving (19) by the Matlab’s LMI toolbox, we can get k1>4.3278 and h<0.4621. Let k1=4.4 and ΔLσF=αsin⁡(t)∗LσF with α=0.1. The time-varying delays of MASs are taken as τ(t)=0.2+0.2sin⁡t in the simulations. For the MASs with multiple stationary leaders, we consider five agents are guided by three leaders moving in a plane. The initial position of followers and leaders are chosen as q1(0)=(1,1)Τ, q2(0)=(1,3)Τ, q3(0)=(3,1)Τ, q4(0)=(3,5)Τ, q5(0)=(5,3)Τ, q6(0)=(7,10)Τ, q7(0)=(10,7)Τ, and q8(0)=(10,10)Τ. The initial velocities of followers and leaders are chosen as p1(0)=(1,1)Τ, p2(0)=(2,2)Τ, p3(0)=(1,1)Τ, p4(0)=(2,2)Τ, p5(0)=(3,3)Τ, and p6(0)=p7(0)=p8(0)=0. Simulation results are shown in Figure 2, which shows that those followers can asymptotically converge to the triangle formed by three leaders; that is, containment control of second-order MASs in the presence of uncertain topologies with multiple stationary leaders and time-varying delays can be achieved.Figure 2
State trajectories of MASs with uncertain topologies.


## 5. Conclusion

In this paper, the distributed containment control for second-order MASs with multiple stationary leaders and uncertain topologies is investigated. The control algorithm of MASs with time-varying delays and jointly connected topologies is proposed. By applying modern control theory and algebraic graph theory, the convergence of MASs for the proposed containment control algorithm is analyzed via Lyapunov-Krasovskii method. Some sufficient conditions in terms of linear matrix inequalities (LMIs) are given for containment control. The correctness and effectiveness of our theoretical results for uncertain MASs with time-varying delays are demonstrated by some simulation examples. The containment control problem for nonlinear systems will be our future work.



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