# Fully Distributed Event-Triggered Containment Control of Uncertain Multiagent Systems

Authors: Zhaodong Liu; Zhi Liu; Xuewu Qian; Ancai Zhang; Zhenxing Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates the event-triggered containment control problem of a class of uncertain nonlinear multiagent systems (MASs). By employing the local relative information, we design an adaptive event-triggered containment algorithm. The proposed containment algorithm can cope with the unavailability of global topology information and uncertain dynamics of follower agents. Therefore, the presented containment algorithm is free of global topology information, i.e., the designed algorithm is fully distributed. In addition, it is proved that Zeno behavior will not occur. At last, a numerical example is given to verify our event-triggered containment algorithm.


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## Body

## 1. Introduction

During past two decades, cooperative control of MASs has been a hot research field for the reason of broad engineering applications, such as formation control [1], attitude alignment [2], mobile sensor networks [3], multirobot systems [4], and so forth [5, 6]. In practice, MASs are composed of mobile units, such as UAVs and robots, which are equipped with embedded control chips and limited energy supply. To extend the working hours of MASs, researchers developed various cooperative algorithms. Event-triggered cooperative algorithms can effectively reduce communications among neighbor agents and have gained extensive attention.MASs can be divided into three cases, i.e., leaderless MASs, leader-follower MASs, and multileader MASs. The control objectives of those three kinds of MASs are consensus control, tracking control, and containment control, respectively. Consensus control and tracking control have been widely studied, from linear MASs [1, 7, 8] to nonlinear MASs [9–11]. Event-triggered control technique is also employed to design consensus and tracking controllers for various kinds of MASs [12–16]. Unlike consensus and tracking control, the objective of containment control is that each follower agent enters into the convex hull spanned by leader agents. In papers [2, 17–19], containment control problems of different kinds of MASs were studied. Event-triggered containment problems are also considered by researchers [20–24]. Miao studied the event-triggered containment control for first/second-order MASs with constant time delay in [20]. Event-triggered containment control of second-order MASs with sampled position data and time-varying input delays are considered in [21, 22]. Rong and her coauthors studied the event-triggered containment control problem for general MIMO linear MASs in [23]. Xu studied the event-triggered containment problem of Euler–Lagrange MASs in [24].Cooperative algorithms of MASs use the local information to achieve global control tasks. However, most of the existing control algorithms of MASs need the information of Laplacian which is a global information. Control algorithms that do not need any global information are said to be fully distributed. Many researchers devoted to develop the fully distributed algorithms for MASs. Papers [25–28] presented some typical works of fully distributed algorithms. Recently, some researchers studied the fully distributed control problems of MASs via event-triggered approach as well. Zhu and Cheng studied the fully distributed event-triggered control problems for linear MASs in [29, 30]. Li and his coauthors investigated both event-triggered consensus and tracking problems of second-order nonlinear MASs in [31]. Apart from Euler–Lagrange MASs in [24], event-triggered containment problem of multiple leaders MASs was seldom considered.Inspired by the above papers, we study the fully distributed event-triggered containment problem of a class of uncertain MASs. Not only the Laplacian of MASs but also the dynamics of follower agents are unavailable to design the containment controllers. To deal with these two unknown data, adaptive techniques are used to design event-triggered containment controllers. The main contributions of this paper are listed as follows: (i) our containment algorithm is fully distributed and does not use any global information of MASs; (ii) our containment algorithm is robust and still works when the dynamics of follower agent is uncertain; and (iii) our presented event-triggered algorithm is free of Zeno behavior.


## 2. Problem Statement and Preliminaries

Notations. Let IN∈RN×N be the N-dimensional identity matrix, 1N∈RN be a vector with each entry being 1, ⋅ be the Euclidean norm (2-norm), ⊗ be the Kronecker product, and diaga1,…,aN be a diagonal matrix. We use λ1A≤λ2A≤…≤λNA, to denote N eigenvalues of a symmetric matrix A∈RN×N. AF=trATA denotes the Frobenius norm of matrix A.


### 2.1. Preliminaries

In this paper, there areN+M agents in the considered MASs. The communication topology among those N+M agents is depicted by a graph G=V,ℰ,A, where V=1,…,N+M, ℰ=V×V, and A=aij∈ℝN+M×N+M are the agent set, edge set, and adjacency matrix, respectively. i,j∈ℰ denotes a directed edge from agent j to agent i and the associated weighing factor aij>0; otherwise, aij=0. If aij=aji>0, then there is an undirected edge between agent i and agent j. We assume that there is no self-loop edge, i.e., aii=0,∀i∈V. Edge sequence i2,i1i3,i2,…,ik,ik−1 denotes a directed path from agent i1 to agent ik. L=lij∈ℝN+M×N+M denotes the Laplacian matrix of graph G, with lii=∑j=1N+Maij and lij=−aij. Suppose that there are N follower agents and M leader agents in the considered MASs. For simplicity, we assume that agents indexed by 1,…,N are the followers, and agents indexed by N+1,…,N+M are the leaders. In practice, the desired trajectories of followers are generated by leaders. Hence, we assume that each leader is not influenced by other agents, that is, aij=0,i=N+1,…,N+M. For brevity, we use F=1,…,N and L=N+1,…,N+M to denote the follower set and leader set, respectively.For the topology of MASs, we have the following assumption.Assumption 1.
For every follower agenti∈F, there exists at least a leader agent j∈L that has a path to follower agent i.Assumption 2.
The induced subgraph of follower agents is an undirected graph.
Since each leader agent is not influenced by other agents, the associated Laplacian matrix can be partitioned as follows:(1)L=L1L20M×N0M×M,L1∈ℝN×N,L2∈ℝN×M.Lemma 1 (see [19]).
If Assumptions 1 and 2 hold,L1 is positive definite, each entry of −L1−1L2 is nonnegative and the sum of each row of −L1−1L2 equals one.Definition 1 (see [32]).
A setC⊆ℝn is convex if 1−λx+λy∈C, for any x,y∈C and any λ∈0,1. The convex hull CoX of a finite set of points X=x1,x2,…,xq is the minimal convex set containing all points in X. That is, CoX=∑i=1qλixixi∈X,λi≥0,∑i=1qλi=1.


### 2.2. Problem Statement

In this paper, we study event-triggered containment control problem of uncertain MASs without using any global topology information.Consider a network system ofN+M agents with the following dynamics:(2)x˙it=Axit+Bfixit+uit,i∈F,(3)x˙it=Axit+Buit,i∈L,where A∈ℝn×n,B∈ℝn×m,xit∈ℝn, and uit∈ℝm are the system matrix, input matrix, state vector, and control input of agent i, respectively; input nonlinearity fixit is an unknown smooth function.Based on the fact thatfixit is smooth and Stone–Weierstrass approximation theory [33], the unknown nonlinear function fixit can be approximated by(4)fixit=WiTϕixit+εi,∀xit∈Π,with ϕi⋅:ℝn⟶ℝs being a known basis function, Wi∈ℝs×m being the neural network (NN) weight matrix, Π⊂ℝm being a compact set, and εi∈ℝm being the approximation error such that(5)εi∞≤εM,∀i∈F.Suppose that bothmaxxit∈Πϕixit and WiF are bounded, i.e., there exist nonnegative numbers ϕM,WM≥0 such that(6)maxxit∈Πϕixit≤ϕM,WiF≤WM,i∈F.For the control input of each leader, we assume thatuit satisfies the following assumption, i∈L.Assumption 3.
uit is bounded by a nonnegative constant τ, that is, uit≤τ,∀i∈L.
The control objective of MASs (2) and (3) is to design event-triggered controllers ui,i∈F for follower agents such that the states of follower agents enter into the convex hull spanned by leaders’ states, that is, limt⟶∞distxit,CoL=0, where CoL=CoxN+1t,…,xN+Mt.


## 2.1. Preliminaries

In this paper, there areN+M agents in the considered MASs. The communication topology among those N+M agents is depicted by a graph G=V,ℰ,A, where V=1,…,N+M, ℰ=V×V, and A=aij∈ℝN+M×N+M are the agent set, edge set, and adjacency matrix, respectively. i,j∈ℰ denotes a directed edge from agent j to agent i and the associated weighing factor aij>0; otherwise, aij=0. If aij=aji>0, then there is an undirected edge between agent i and agent j. We assume that there is no self-loop edge, i.e., aii=0,∀i∈V. Edge sequence i2,i1i3,i2,…,ik,ik−1 denotes a directed path from agent i1 to agent ik. L=lij∈ℝN+M×N+M denotes the Laplacian matrix of graph G, with lii=∑j=1N+Maij and lij=−aij. Suppose that there are N follower agents and M leader agents in the considered MASs. For simplicity, we assume that agents indexed by 1,…,N are the followers, and agents indexed by N+1,…,N+M are the leaders. In practice, the desired trajectories of followers are generated by leaders. Hence, we assume that each leader is not influenced by other agents, that is, aij=0,i=N+1,…,N+M. For brevity, we use F=1,…,N and L=N+1,…,N+M to denote the follower set and leader set, respectively.For the topology of MASs, we have the following assumption.Assumption 1.
For every follower agenti∈F, there exists at least a leader agent j∈L that has a path to follower agent i.Assumption 2.
The induced subgraph of follower agents is an undirected graph.
Since each leader agent is not influenced by other agents, the associated Laplacian matrix can be partitioned as follows:(1)L=L1L20M×N0M×M,L1∈ℝN×N,L2∈ℝN×M.Lemma 1 (see [19]).
If Assumptions 1 and 2 hold,L1 is positive definite, each entry of −L1−1L2 is nonnegative and the sum of each row of −L1−1L2 equals one.Definition 1 (see [32]).
A setC⊆ℝn is convex if 1−λx+λy∈C, for any x,y∈C and any λ∈0,1. The convex hull CoX of a finite set of points X=x1,x2,…,xq is the minimal convex set containing all points in X. That is, CoX=∑i=1qλixixi∈X,λi≥0,∑i=1qλi=1.


## 2.2. Problem Statement

In this paper, we study event-triggered containment control problem of uncertain MASs without using any global topology information.Consider a network system ofN+M agents with the following dynamics:(2)x˙it=Axit+Bfixit+uit,i∈F,(3)x˙it=Axit+Buit,i∈L,where A∈ℝn×n,B∈ℝn×m,xit∈ℝn, and uit∈ℝm are the system matrix, input matrix, state vector, and control input of agent i, respectively; input nonlinearity fixit is an unknown smooth function.Based on the fact thatfixit is smooth and Stone–Weierstrass approximation theory [33], the unknown nonlinear function fixit can be approximated by(4)fixit=WiTϕixit+εi,∀xit∈Π,with ϕi⋅:ℝn⟶ℝs being a known basis function, Wi∈ℝs×m being the neural network (NN) weight matrix, Π⊂ℝm being a compact set, and εi∈ℝm being the approximation error such that(5)εi∞≤εM,∀i∈F.Suppose that bothmaxxit∈Πϕixit and WiF are bounded, i.e., there exist nonnegative numbers ϕM,WM≥0 such that(6)maxxit∈Πϕixit≤ϕM,WiF≤WM,i∈F.For the control input of each leader, we assume thatuit satisfies the following assumption, i∈L.Assumption 3.
uit is bounded by a nonnegative constant τ, that is, uit≤τ,∀i∈L.
The control objective of MASs (2) and (3) is to design event-triggered controllers ui,i∈F for follower agents such that the states of follower agents enter into the convex hull spanned by leaders’ states, that is, limt⟶∞distxit,CoL=0, where CoL=CoxN+1t,…,xN+Mt.


## 3. Adaptive Event-Triggered Containment Algorithm

In this section, we will give an adaptive event-triggered algorithm to solve the containment problem of MASs (2) and (3).Let(7)xft=x1Tt,…,xNTtT,xlt=xN+1Tt,…,xN+MTtT,γt=γ1Tt,…,γNTtT=−L1−1L2⊗Inxlt.From Lemma1, we know that γit,i∈F, is a convex combination of states of leader agents. If follower agent i can track γit, then xit enters into CoL, i∈F. We call γit the target state trajectory of follower agent i,i∈F.Denoteδit=xit−γit as the tracking error of follower agent i and δt=δ1Tt,…,δNTtT. Then, we have(8)δ˙t=x˙ft+L1−1L2⊗Inx˙lt=IN⊗Axft+IN⊗BFxft+uft+L1−1L2⊗InIM⊗Axlt+IM⊗Bult=IN⊗Aδt+IN⊗BFxft+uft+L1−1L2⊗Bult,where(9)Fxft=f1Tx1t,…,fNTxNtT,uft=u1Tt,…,uNTtT,ult=uN+1Tt,…,uN+MTtT.Letξit=∑j=1N+Maijxit−xjt be the local relative state of follower agent i,i∈F. ξitki denotes the sample value of ξit at sample instant tki, and eit denotes sample error of ξitki, i.e.,(10)eit=ξitki−ξit,t∈tki,tk+1i,i∈F.By usingξitki, we give the following dynamical containment controller for each follower agent:(11)uit=−ciBTPξitki−W^iTϕixit−αhBTPξitki,c˙it=1η1−β12ξiTtkiPBBTPξitki−σcit−1,W^˙it=ϕixitξiTtkiPB−ρW^it,t∈tki,tk+1i,i∈F,where cit is the dynamical gain of uit with ci0≥1, α≥εM+τ is a positive constant, β1∈0,2 is a constant, W^it is the dynamical estimation of Wi with W^i0=0, σ>0 and ρ>0 are the turning scalars, and P>0 is the unique solution of the following Riccati equation:(12)PA+ATP−PBBTP+In=0,and hi⋅ is a discontinuous function such that, for z∈ℝm,(13)hz=zz,ifz≠0,0,ifz=0.The trigger function for adaptive controller (11) is given as(14)fit=ki1eiTtPBBTPeit−ki2ξiTtkiPBBTPξitki−r,where(15)ki1=μcit+ϕM+α+2−β11−β4β52β4,r>0ki2=2−β11−β52,μ>0,β4∈0,1,β5∈0,1.Remark 1.
Usually, the feedback gain of a consensus controller depends on the eigenvalues of the Laplacian matrix, which are the global topology information. Since adaptive gainscit can be adjusted by ξitki automatically, adaptive controller (11) does not rely on any global topology information.Remark 2.
Trigger function (14) is a hybrid trigger function. Unlike trigger functions in other papers, we use a positive constant r to replace the exponential decay term. The exponential decay term only guarantees that there is no Zeno behavior for finite time t. By choosing a relative small r, a satisfactory tracking error δt and a lower bound of tk+1i−tki can be guaranteed.Remark 3.
In the design of dynamical controller (11), we use the σ-modification technique. Containment controller (11) and trigger function (14) only guarantee that tracking error δt is uniformly ultimately bounded. If we do not employ the σ-modification technique, the dynamical gains cit and estimation parameters W^it may increase unboundedly. To ensure the boundedness of cit and W^it, we use σ-modification technique to adjust the dynamical parameters.
Denote(16)ξt=ξ1Tt,…,ξNTtT,ξtk=ξ1Ttk1,…,ξNTtkNT,ct=diagc1t,…,cNt,ε=ε1T,…,εNTT,W˜it=Wi−W^it,W˜=diagW˜1t,…,W˜Nt,ϕxft=ϕ1Tx1t,…,ϕNTxNtT,Hξtk=hTBTPξ1tk1,…,hTBTPξNtkNT.
Then, combining (4), (8), and (11), we have, for t∈tki,tk+1i,(17)δ˙t=IN⊗Aδt−ct⊗BBTPξtk+IN⊗BW˜Ttϕxft+IN⊗Bε−αIN⊗BHξtk−−L1−1L2⊗Bult.Theorem 1.
Suppose that Assumptions 1 and 2 hold, containment problem of MASs (2) and (3) will be solved by dynamical controller (11). The tracking errors δit and adaptive parameters cit and W^it are uniformly ultimately bounded (UUB). Furthermore, controller (11) does not exist Zeno behavior.Proof.
Take Lyapunov function candidate as(18)V=12δTtL1⊗Pδt+∑i=1Nη2c˜i2t+∑i=1N12trW˜iTtW˜it,where c˜it=cit−c¯ with c¯ being a constant to be determined later.
Along with (11 and 17), we compute V˙ as follows:(19)V˙=δTtL1⊗PAδt−δTtL1ct⊗PBBTPξtk+δTtL1⊗PBW˜Ttϕxft+δTtL1⊗PBε−αδTtL1⊗PBHξtk+∑i=1Nηc˜itc˜˙it−δTt−L1L1−1L2⊗PBult+∑i=1NtrW˜iTtW˜˙it.
From the definition ofξit, one obtains(20)ξt=L1⊗Inxft+L2⊗Inxlt=L1⊗Inxft−−L1−1L2⊗Inxlt=L1⊗Inδt.
Hence, we get(21)V˙=δTtL1⊗PAδt−ξTtct⊗PBBTPξtk+ξTtIN⊗PBW˜Ttϕxft+ξTtIN⊗PB×ε−αξTtIN⊗PBHξtk+∑i=1Nηc˜itc˜˙it−ξTt.−L1−1L2⊗PBult+∑i=1NtrW˜iTtW˜˙it−∑i=1NαξiTtPBhBTPξitki−∑i=1NξiTtPBuilt=δTtL1⊗PAδt−ξTtct⊗PBBTPξtk+∑i=1NξiTtPBW˜iTtϕixit+∑i=1NξiTtPBεi+∑i=1Nηc˜itc˜˙it+∑i=1NtrW˜iTtW˜˙it,where uilt are convex combinations of leaders’ inputs with u1lTt,…,uNlTtT=−L1−1L2⊗Imult. Since the sum of each row of −L1−1L2 is equal to 1, one has(22)uilt≤τ,i∈F.
Denoteet=e1Tt,…,eNTtT. Since ξt=ξtk−et, one gets(23)−ξTtct⊗PBBTPξtk=−ξTtkct⊗PBBTPξtk+eTtct⊗PBBTPξtk≤−1−β12ξTtkct⊗PBBTPξtk+∑i=1Ncit2β1eiTtPBBTPeit,where β1∈0,2.
For the third term ofV˙ (21), one has(24)ξiTtPBW˜iTtϕixit=ξiTtkiPBW˜iTtϕixit−eiTtPBW˜iTtϕixit.
Note that(25)ξiTtkiPBW˜iTtϕixit=trW˜iTtϕixitξiTtkiPB,(26)−eiTtPBW˜iTtϕixit=−trW˜iTtϕixiteiTtPB≤ϕMW˜itFBTPeit≤β2ϕM2W˜itF2+ϕM2β2eiTtPBBTPeit,where β2∈0,ρ/2ϕM.
For the fourth, fifth, and sixth terms, one obtains(27)ξiTtPBεi=ξiTtkiPBεi−eiTtPBεi≤εMBTPξitki+εMBTPeit,−αξiTtPBhBTPξitki≤−αBTPξitki+αBTPeit,(28)−ξiTtPBuilt≤τBTPξitki+τBTPeit.
Hence, we have(29)∑i=1NξiTtPBεi−αξiTtPBhBTPξitki−ξiTtPBuilt≤∑i=1N2αBTPeit≤∑i=1Nαβ3+αβ3eiTtPBBTPeit,where β3>0 is a real constant. From (11), we have(30)∑i=1Nηc˜itc˙it=1−β12ξTtkct⊗PBBTPξtk−c¯1−β12ξTtkIN⊗PBBTPξtk−∑i=1Nσc˜itcit−1,(31)trW˜iTtW˜˙it=−trW˜iTtϕixitξiTtkiPB+ρtrW˜iTtW^it.
Substituting (23)–(31) into (21), we obtain(32)V˙≤δTtL1⊗PAδt+∑i=1Nβ2ϕM2W˜itF2−c¯1−β12ξTtkIN⊗PBBTPξtk−∑i=1Nσc˜itcit−1+∑i=1NρtrW˜iTtW^it+∑i=1Ncit2β1+ϕM2β2+αβ3eiTtPBBTPeit+Nαβ3.
Note that(33)−ξTtkIN⊗PBBTPξtk=−ξTtIN⊗PBBTPξt−2ξTtIN⊗PBBTPet−eTtIN⊗PBBTPet,(34)−2ξTtIN⊗PBBTPet≤β4ξTtIN⊗PBBTPξt+∑i=1N1β4eiTtPBBTPeit,where β4∈0,1. Since ξt=L1⊗Inδt and L1 is positive definite, we have the following inequalities:(35)ξTtIN⊗PBBTPξt=δTtL12⊗PBBTPδt,≥λ¯L12λ¯L1δTtL1⊗PBBTPδt,where λ¯L1 and λ¯L1 are the smallest and largest eigenvalues of L1, respectively.
Combining (32)–(35), one has(36)V˙≤δTtL1⊗PAδt+∑i=1Nβ2ϕM2W˜itF2−c¯1−β121−β4β5λ¯L12λ¯L1δTtL1⊗PBBTP×δt−∑i=1Nσc˜itcit−1+∑i=1NρtrW˜iTtW^it+∑i=1Ncit2β1+ϕM2β2+αβ3+c¯2−β121−β4β4β5×eiTtPBBTPeit−∑i=1Nc¯1−β121−β5×ξiTtkiPBBTPξitki+Nαβ3,where β5∈0,1 is a constant.
Choosing enough largec¯ such that(37)c¯≥λ¯L12−β11−β4β5λ¯L12,c¯≥12β1μ,c¯≥12β2μ,c¯≥1β3μ.
Then, Riccati equation (12) and trigger function (14) guarantee that(38)V˙≤−12δTtL1⊗Inδt+∑i=1Nβ2ϕM2W˜itF2+Nr−∑i=1Nσc˜itcit−1+∑i=1NρtrW˜iTtW^it+Nαβ3.
Sincecit=c˜it+c¯ and W^it=Wi−W˜it, one gets(39)−σc˜itcit−1=−σc˜itc˜it+c¯−1=−σc˜i2t−σc˜itc¯−1≤−σ2c˜i2t+σ2c¯−12,(40)trW˜iTtW^it=trW˜iTtWi−W˜it≤WMW˜itF−W˜itF2≤−1−β62WMW˜itF2+WM2β6,where β6∈0,1/2WM is a constant.
Combining (38)–(40) yields(41)V˙≤−12δTtL1⊗Inδt−∑i=1Nσ2c˜i2t−∑i=1Nρ1−β62WM−β22ϕMW˜itF2+ς,where ς=Nαβ3+Nσ/2c¯−12+NWM/2β6+Nr. Since β2∈0,ρ/2ϕM and β6∈0,1/2WM, we get(42)V˙≤−12δTtL1⊗Inδt−∑i=1Nσ2c˜i2t−∑i=1Nρ2W˜itF2+ς≤−κV+ς,where κ=min1/λ¯p,σ/η,ρ with λ¯p being the largest eigenvalue of P.
After simple calculations, we have(43)Vt≤V0−ςκe−κt+ςκ,which indicates that tracking error δt, dynamical gains cit, and W^it are UUB.
In the last step, we prove that event-triggered controller (11) does not exist Zeno behavior.
Whent=tki, sample error eit will be set to zero. Dini derivative will be used to analyze eit, and we get(44)D+eit=ddteiTteit≤e˙it.
Sinceeit=ξitki−ξit, for t∈tki,tk+1i, we have(45)e˙it=−ξ˙it=−Aξit−∑j=1NaijBfixit−fjxjt−∑j=1N+Maijuit−ujt=Aeit−Aξitki−∑j=1NaijBfixit−fjxjt−∑j=1N+Maijuit−ujt.
Denote(46)φki=maxt∈tki,tk+1iAξitki+∑j=1NaijBfixit−fjxjt+∑j=1N+Maijuit−ujt.
And, one gets(47)D+eit≤A⋅eit+φki.
Notice thateitki=0. Hence, for t∈tki,tk+1i, eit satisfies following inequality:(48)eit≤1AφkieAt−tki−1.
Letψkit=ki2/ki1ξiTtkiPBBTPξitki+r/ki1. From trigger function (14), we know that the k+1th event of agent i will not be triggered unless BTPeit exceeds ψkit, i.e.,(49)BTPeit≤ψkit,∀t∈tki,tk+1i,limt⟶tk+1iBTPeit=ψki.
BecauseBTPeit≤BTP⋅eit, one has(50)limt⟶tk+1i‖eit‖≥ψkit‖BTP‖.
By (48) and (50), one obtains the interinterval satisfying(51)tk+1i−tki≥1AlnAψkitBTPφki+1.
Due to the existence of positive realr>0, ψkit is greater than zero for any given t>0. That means tk+1i−tki is strictly positive.


## 4. Simulations

One numerical example is given in this section to verify the adaptive event-triggered containment controller (11) developed in this paper.Consider a leader-follower multiagent system with four follower agents and two leader agents. The Laplacian matrix associated with topology is given as follows:(52)L=20−1−10001−1000−1−130−10−10020−1000000000000.The dynamics of leaders is depicted by the linearized model of the longitudinal dynamics of an aircraft [34], described by (3) with(53)A=−0.27701.0000−0.0002−17.1000−0.1780−12.200000−6.6700,B=006.67.xit=xi1t,xi2t,xi3tT denotes the state of each agent with xi1t being the angle of attack, xi2t being the pitch rate, and xi3t being the elevator angle. From Stone–Weierstrass approximation theory, the approximation error εi can be arbitrarily selected. We set ϕixit=4xi1sinxi1t,2cosxi2tT and εM=0.01. The initial conditions of follower agents are given as x10=1,−1,0T,x20=2,0,1.5T,x30=1.5,0.5,−1.5T,andx40=−0.9,−0.5,1.5. We choose control inputs of leader agents as u5t=−0.5andu6t=0.5 and initial conditions as x50=−1,0.1,0.7Tandx60=0.5,0,−0.8T.We set the initial values of adaptive gainscit and W^it as ci0=1 and W^i0=0,0T. By solving Riccati equation (12), we get BTP=1.2566,−0.9116,1.3097T. According to controller design (11), we set β1=0.2,β4=0.5,β5=0.2,η=1,σ=0.2,ρ=0.1,μ=0.2, and α=0.51.The simulation results are presented in Figures1–6. Figures 1 and 3 show the trajectories of follower agents and leader agents. The black lines denote the trajectories of leader agents which indicate that the states of follower agents enter into the convex hull spanned by leaders’ states. Figure 4 displays the triggering instants of each agent. Figure 5 displays the adaptive gains cit. Figure 6 presents the norms of estimation coefficients W^it.Figure 1
xi1t trajectories of each agent.Figure 2
xi2t trajectories of each agent.Figure 3
xi3t trajectories of each agent.Figure 4
Triggering instants of each follower agent.Figure 5
Adaptive gainscit.Figure 6
Norms ofW^it.


## 5. Conclusions

In this paper, we have studied the event-triggered containment problem of a class of uncertain MASs. The global topology information is unavailable, and parameters of follower agents are uncertain. Based on adaptive control technique, we have proposed an event-triggered containment controller which does not rely on any global topology information and can estimate the uncertain parameters. Under this containment algorithm, follower agents enter into the convex hull of leader agents. Moreover, the proposed algorithm has no Zeno behavior. In the future, we will concentrate our study on event-triggered consensus problem of MAS with unknown control directions.



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