# Distributed Continuous-Time Containment Control of Heterogeneous Multiagent Systems with Nonconvex Control Input Constraints

Authors: Xue Li; Lulu Wang; Yinsen Zhang
Journal: Complexity
 (2022)

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## Abstract

This paper focuses on studying containment control problem with switching communication graphs of continuous-time heterogeneous multiagent systems where the control inputs are constrained in a nonconvex set. A nonlinear projection algorithm is proposed to address the problem. We discuss the stability and containment control of the system with switching topologies and nonconvex control input constraints under three different conditions. It is shown that all agents converge to the convex hull of the given leaders ultimately while staying in the nonconvex set under the premise that at least one directed path from leaders to the agents exists in each bounded time interval. Finally, the validity of the results obtained in this paper is verified by simulation.


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## Body

## 1. Introduction

With the in-depth research of autonomous control of multiagent systems in recent years, many cooperative control problems of multiagent systems have become the focus of many fields, such as control theory, biology, robotic systems, and spacecraft systems [1–5]. The study and exploration of multiagent systems provided a unified framework and theoretical basis for various practical problems such as unmanned aerial vehicles, formation aircrafts, multiple robots, and other practical applications. A Lyapunov-based approach was proposed to address the consensus problem in [6]. Lin et al. [7] emphasized on the consensus control problem in consideration of nonconvex control input constraints. The consensus control problem of multiagent systems with time delays was studied in consideration of external interference in [8]. And the work [9] was centered on the containment control problem. Lin et al. [10] studied distributed optimization problems for continuous-time and discrete-time multiagent systems with different constraints.As an important branch of control theory, there were many articles focusing on the study of containment control [11–20]. Ji et al. and Li et al. [11, 13] solved the containment control problem with fixed topology. The output formation-containment problem of heterogeneous system was investigated in [14, 15]. For double-integrator multiagent systems, containment control problem with fixed communication topology and position measurements was addressed in [16, 17]. Two distributed algorithms were proposed for containment control in the case of only using absolute position measurements and relative position measurements, respectively. However, the ‘sign’ function was employed in [17], which may lead to the chattering phenomenon. Cheng et al. [18] avoided employing the ‘sign’ function, and both disturbance and measurement noise were also taken into consideration. Notarstefano et al. and Zhang et al. [19, 20] investigated containment control with switching topologies. Cao et al. [12] studied the solution of containment control in consideration of switching topologies and fixed simultaneously.In many practical systems, the control inputs were generally constrained in a convex or a nonconvex hull, while most of existing works studied the multiagent problem without considering the constraint of control inputs. The authors of [21–24] studied the consensus problem with position constraints on the basis of the property of stochastic matrices. Nevertheless, these approaches could not be applied to the realization of containment control with the nonconvex control inputs’ constraints. The containment control problem with input saturation of the second-order agent system was studied in [25, 26]. However, control input of each agent was supposed to be in a hypercube. In reality, on account of the physical limitation, the control inputs were often constrained in a convex or a nonconvex region. For instance, the maximum driving forces of quadrotors, which formed a nonconvex region, were in the direction of the diagonal axis. A multiagent system model and a projection consensus algorithm were introduced in [23], and it placed emphasis on the effects of control input constraints. The algorithm was executed locally by each agent and its relationship with the alternate projection method was discussed in [23]. Yang et al. and Lin et al. [27, 28] took nonconvex velocity and control inputs’ constraints into consideration, but what they emphasized on was the consensus control problem which meant all the followers had to reach a consensus.In [29], all agents were assumed to be in the form of the second-order dynamics. In practical applications, the multiagent systems might contain different kinds of agents. Our focus of this paper is to study the containment control for heterogeneous multiagent systems with nonconvex input constraints. Since the agents in this paper have different dynamics, the analysis for the case with all identical agents in [29] cannot be applied to this paper directly. In this study, we expand the results of [29] and mainly focus on the containment control problem of continuous-time heterogeneous multiagent systems, given the nonconvex control input constraints and switching communication graphs. Li et al. [30] studied the containment control of heterogeneous multiagent systems. However, they did not consider the constraints of control inputs and heterogeneous multiagent systems at the same time. To analyze the stability and convergence of the system, a nonlinear projection algorithm is proposed to address the problem. Then, a model transformation is introduced and estimates the distance from the followers to the nonconvex hull by using the Lyapunov function we construct. We prove that the distance decreases and ultimately all followers converge to the nonconvex hull.


### 1.1. Notations

Assume thatℝm×n represents the set of m×n-dimensional real matrix. xT is the transposing matrix of x. x is the Euclidean norm of x. Wχx denotes the projection of a vector x onto a closed convex set χ, i.e., Wχx=argminx¯∈χx−x¯.


## 1.1. Notations

Assume thatℝm×n represents the set of m×n-dimensional real matrix. xT is the transposing matrix of x. x is the Euclidean norm of x. Wχx denotes the projection of a vector x onto a closed convex set χ, i.e., Wχx=argminx¯∈χx−x¯.


## 2. Preliminaries

### 2.1. Graph Theory

GV,E,A is a directed weighted graph representing the communication graphs among the agents, where V=v1,v2,…vn is the set of node representing the agents and E⊆V×V is the set of directed edges representing the communication between the agents. A=aij∈Rn×n is a weighted adjacent matrix. Laplacian is a very important matrix in graph theory, which is defined as L=D−A, where(1)D=∑j=1,i≠jnaij,i=j,0,i≠j.Then, the Laplacian matrix can be expressed as follows:(2)L=lijn×n,lij=∑j=1,i≠jnaij,i=j,−aij,i≠j.For heterogeneous multiagent systems consisted ofn first-order and m second-order agents, the adjacent agents of second-order agent i1 can be denoted as Ni1=Ni1s∪Ni1f. Similarly, the adjacent agents of first-order agent i2 can be denoted as Ni2=Ni2s∪Ni2f.s and f represent the second-order and the first-order agents, respectively. Partition the matrix A and D as(3)A=AsAsfAfsAf,D=Ds+Dsf00Df+Dfs,where Dsf=diag∑j∈Ni1fai1j,i1=1,2…m and Asf represents the adjacent relationship of the second-order and the first-order agents. Dfs=diag∑j∈Ni2sai2j,i2=1,2…n and Afs represents the adjacent relationship of the first-order and the second-order agents. Then, the Laplacian matrix can be expressed as follows:(4)L=D−A=L¯s−Asf−AfsL¯f,where L¯s=Dsf+Ls, L¯f=Dfs+Lf, and Ls and Lf are the Laplacian matrix of second-order agents and first-order agents, respectively.


### 2.2. Nonconvex Constraints

The control inputs are generally subject to nonconvex constraints in many practical cases. Hence, we have the following assumption.Assumption 1.
(see [7]). E∈ℝr is a nonempty bounded closed set, 0∈Ei and maxx∈EiQEix=η¯,minx∉EiQEix=η¯>0. QEi is a nonconvex constraints’ operator and is defined as follows:(5)QEix=xxmax0≤ε≤xε|δεx‖x‖∈Ei,∀0≤δ≤1,x≠0,0,x=0.
In this study, the operatorQEi. is used to convert x to QEix which has the alignment with the direction of x and satisfies QEix≤x and δQEix∈Ei, for all 0≤δ≤1. maxx⊆EiQEix=η¯ means that the value cannot be arbitrarily large. We do not have any requirement for Ei to be convex or nonconvex. What we require is that the distance from any point outside Ei to the origin is lower bounded by a positive constant.


## 2.1. Graph Theory

GV,E,A is a directed weighted graph representing the communication graphs among the agents, where V=v1,v2,…vn is the set of node representing the agents and E⊆V×V is the set of directed edges representing the communication between the agents. A=aij∈Rn×n is a weighted adjacent matrix. Laplacian is a very important matrix in graph theory, which is defined as L=D−A, where(1)D=∑j=1,i≠jnaij,i=j,0,i≠j.Then, the Laplacian matrix can be expressed as follows:(2)L=lijn×n,lij=∑j=1,i≠jnaij,i=j,−aij,i≠j.For heterogeneous multiagent systems consisted ofn first-order and m second-order agents, the adjacent agents of second-order agent i1 can be denoted as Ni1=Ni1s∪Ni1f. Similarly, the adjacent agents of first-order agent i2 can be denoted as Ni2=Ni2s∪Ni2f.s and f represent the second-order and the first-order agents, respectively. Partition the matrix A and D as(3)A=AsAsfAfsAf,D=Ds+Dsf00Df+Dfs,where Dsf=diag∑j∈Ni1fai1j,i1=1,2…m and Asf represents the adjacent relationship of the second-order and the first-order agents. Dfs=diag∑j∈Ni2sai2j,i2=1,2…n and Afs represents the adjacent relationship of the first-order and the second-order agents. Then, the Laplacian matrix can be expressed as follows:(4)L=D−A=L¯s−Asf−AfsL¯f,where L¯s=Dsf+Ls, L¯f=Dfs+Lf, and Ls and Lf are the Laplacian matrix of second-order agents and first-order agents, respectively.


## 2.2. Nonconvex Constraints

The control inputs are generally subject to nonconvex constraints in many practical cases. Hence, we have the following assumption.Assumption 1.
(see [7]). E∈ℝr is a nonempty bounded closed set, 0∈Ei and maxx∈EiQEix=η¯,minx∉EiQEix=η¯>0. QEi is a nonconvex constraints’ operator and is defined as follows:(5)QEix=xxmax0≤ε≤xε|δεx‖x‖∈Ei,∀0≤δ≤1,x≠0,0,x=0.
In this study, the operatorQEi. is used to convert x to QEix which has the alignment with the direction of x and satisfies QEix≤x and δQEix∈Ei, for all 0≤δ≤1. maxx⊆EiQEix=η¯ means that the value cannot be arbitrarily large. We do not have any requirement for Ei to be convex or nonconvex. What we require is that the distance from any point outside Ei to the origin is lower bounded by a positive constant.


## 3. System Modeling

Consider a continuous-time multiagent system consisted ofl leaders, n first-order followers, and m second-order followers. The first-order agents have the following dynamic equation:(6)x˙ft=uft.The dynamic equations of the second-order agents can be denoted as follows:(7)x˙st=vst,v˙st=ust,where s,f∈Γ represent the second-order and the first-order agents, respectively, and xst, vst, and ust represent the position information, speed information, and control input of the second-order agents separately. xft and uft represent the position information and control input of the first-order agents separately. The set of all followers is expressed by Γ=1,2,…,n+m. We assume all leaders are stationary and the position states are denoted by Y=y1,y2,…yl.In this study, if all the followers can converge into the convex set of leaders under the control protocol for any initial value given, that is,limt⟶∞xit−Wχxit=0, the control protocol can realize the containment control.


## 4. Model Transformation

To analyze the containment control of continuous-time heterogeneous multiagent system, a control protocol is proposed in this study as follows:(8)us=QEi−qsvs+∑i∈Γasitxit−xst−gst(xst−Wχst(xst)),uf=QEi∑i∈Γafitxit−xft−gftxft−Wχftxft,where s,f∈Γ and qs is speed decay factor of followers. asit and afit are the weight of edge i,s,i,f at time t. And, we assume the weight of edge is always positive and lower bounded by a constant εd. If follower i can get the information from one or more agent directly, then git=g¯i, otherwise, git=0. Assume that χt is the convex set of the leaders. The Laplacian matrix of the heterogeneous multiagent system can be denoted by the following matrix:(9)L=L¯s−Asf−AfsLf.Then, algorithm (8) can be converted into(10)ust=QEi−qsvst−L¯sxst+Afsxft−gst(xst−Wχst(xst)),uft=QEi−L¯fxft+Asfxst−gft(xft−Wχft(xft)).The following model transformation is introduced for further derivation:(11)zit=QEiΦtΦt,when Φt=0,zit=1. For the first-order agents, (11) can be denoted as follows:(12)zft=QEiωftωft,where ωft=−L¯fxft+Asfxst−gft(xft−Wχft(xft)). For the second-order agents, (11) can be transformed as follows:(13)zst=QEi−qsvst+ωst−qsvst+ωst,where ωst=−L¯sxst+Afsxft−gst(xst−Wχst(xst)). From the above definition, it is easy to know the range of zit, that is, 0<zit<1. We assume that zit is lower bounded by a positive constant πi=minzit>0, for all i. According to the above definition, system (8) can be converted into the following form.For the first-order agents,(14)x˙ft=zftωft.For the second-order agents,(15)x˙st=vst,v˙st=−qszstvst+zstωst.Denoteds=max∑j∈Γasjt,v¯st=xst+vst/cs, where cs=ds+g¯s is a positive constant. Then, (15) can be expressed in the following form:(16)x˙st=csv¯st−csxst,v¯˙st=cs−qszstv¯st+qszst−cs−zstL¯s+gstcsxst+zstAfscsxft+zstgstcsWχstxst.


## 5. Main Result

Prior to the main theorems, we need to give the definition of switching communication graphs. Denote an infinite sequence of nonempty bounded continuous intervals asts,ts+1, s=0,1,2… and 0≤ts+1−ts≤T, where T is a positive constant. Divide the above interval into a series of subintervals which are represented as ts0,ts1),[ts1,ts2)…[tsn,tsn+1 with ts0=ts and tsn+1=ts+1. We assume there exists a constant ρ≥0 that makes tsn+1−tn≥ρ. The communication topologies change at t=tsn and do not change in the subinterval tsn,tsn+1.Assumption 2.
Suppose that there exist at least one path between leaders and followers in the intervalts,ts+1 for every agent. In other words, each follower can receive information from leaders directly or indirectly in the every interval ts,ts+1.Assumption 3 (see [29]).
∑j∈Γaijtxjt−xit≤Ni/2 and gitxit−Wχitxit≤Ni/2 for some constant Ni.Lemma 1 (see [29]).
Suppose thatU⊆ℝr is a nonempty closed set. For any vector xi∈ℝr,i∈1,2…,n, and ∑i=1nai=1, we have(17)∑i=1naixi−WU∑i=1naixi≤∑i=1naixi−WUxi.
To analyze the stability and convergence of the system, we construct a Lyapunov function as follows:(18)Ft=maxkζkt−Wχktζkt,where t≥T0, k∈1,2…2n+m.Theorem 1.
Ifqs≥cs+1/πi and η¯/2<Ni<η¯, under assumption 1 and 2, Ft is a nonincreasing function relative to time t and Ft≥0, namely, the limit of Ft exits. And the following inequalities hold:(19)dxst−Wχstxstdt≤csv¯st−Wχstv¯st−csxst−Wχstxst,(20)dv¯st−Wχstv¯stdt≤cs−qszstv¯st−Wχstv¯st+qszst−cs−zstL¯s+gstcsxst−Wχstxst+zstAsfcsxft−Wχstxft,(21)dxft−Wχftxftdt≤−L¯fzft+gftxft−Wχftxft+Asfzftxst−Wχstxst.Proof.
For the second-order agents, we can deduce(22)limΔt⟶0xst+Δt=limΔt⟶01−csΔtxst+limΔt⟶0csΔtv¯st,where Δt is time increment and Δt⟶0. cs is a positive constant. Then, it is evident that csΔt>0, 1−csΔt>0, and 1−csΔt+csΔt=1. According to Lemma 1, (22) is converted to(23)xst+Δt−Wχstxst+Δt≤1−csΔtxst+Δt−Wχstxst+Δt+csΔtv¯st+Δt−Wχstv¯st+Δt.
Analogously, we can do the same conversion forv¯st:(24)v¯st+Δt=1+cs−qszstΔtv¯st+qszst−cs−zstL¯s+gstcsΔtxst+zstAsfcsΔtxft+zstgstcsΔtWχstxst.
To facilitate later calculations, we have simplified the upper formula to the following form:(25)v¯st+Δt=β1iv¯st+β2ixst+β3ixft+β4iWχstxst,where(26)β1i=1+cs−qszstΔt,β2i=qszst−cs−zstL¯s+gstcsΔt,β3i=zstAsfcsΔt,β4i=zstgstcsΔt.
For allt≥T0, we have(27)β1i>0,β3i≥0,β4i≥0.
Whenqs≥cs+1/πi, we obtain(28)β2i=qszst−cs−zstL¯s+gstcs≥qsπi−cs−1≥0.
Andβ1i+β2i+β3i+β4i=1. Then, (24) can be converted into the following forms:(29)v¯st+Δt−Wχstv¯st+Δt≤β1iv¯st−Wχstv¯st+β2ixst−Wχstxst+β3ixft−Wχftxft+β4iWχstxst−Wχst(Wχst(xst)).
Combined with the following conditions,χst∈χ, χft∈χ, and WχstWχstxst=Wχstxst; it follows that(30)v¯st+Δt−Wχstv¯st+Δt≤β1iv¯st−Wχstv¯st+β2ixst−Wχstxst+β3ixft−Wχftxft.
Similarly, for the first-order agents, we can deduce(31)limΔt⟶0xft+Δt=limΔt⟶0x˙ftΔt+limΔt⟶0xftΔt,where Δt is time increment and Δt⟶0. According to (31) and Lemma 1, we have(32)xft+Δt=1−zftL¯f+gftΔtxft+AsfzftΔtxst+gftzftΔtWχftv¯ft.
To facilitate later calculations, we have simplified the upper formula to the following form:(33)xft+Δt=α1ixft+α2ixst+α3iWχftxft,where α1i=1−zftL¯f+gftΔt, α2i=AsfzftΔt, and α3i=gftzftΔt. We can deduce from (33)(34)xft+Δt−Wχftxft+Δt≤α1ixft−Wχftxft+α2ixst−Wχstxst+α3iWχftxft−Wχft(Wχft(xft)).
Similar to the second-order agents, according to the conditionχft∈χ and WχstWχstxst=Wχstxst, we have WχftWχftxft=Wχftxft. Then,(35)xft−Wχftxft≤α1ixft−Wχftxft+α2ixst−Wχstxst.
It is evident that everyζkt−Wχζkt is a convex hull that is composed of ζjt−Wχζjt. From (23), (30), and (35), we can obtain that(36)ζjt+Δt−Wχjtζjt+Δt≤maxjζjt−Wχjtζjt.
Thus,Ft+Δt≤Ft can be deduced which indicate that Ft is a nonincreasing function relative to time t. It is obvious that Ft≥0. Thus, it is easy to know that the limit of Ft exists.
For the second-order agents, according to the definition of derivative and (23), we have(37)limΔt⟶0xst+Δt−Wχstxst+Δt−xst−WχstxstΔt≤csv¯st−Wχstv¯st−csxst−Wχstxst.
That is,(38)dxst−Wχstxstdt≤csv¯st−Wχstv¯st−csxst−Wχstxst.
Similarly, according to (30), we have(39)limΔt⟶0v¯st+Δt−Wχstv¯st+Δt−v¯st−Wχstv¯stΔt≤λ1iv¯st−Wχstv¯st+λ2ixst−Wχstxst+λ3ixft−Wχftxft.
That is,(40)dv¯st−Wχstv¯stdt≤λ1iv¯st−Wχstv¯st+λ2ixst−Wχstxst+λ3ixft−Wχftxft,where(41)λ1i=cs−qszst,λ3i=zstAsfcs,λ2i=qszst−cs−zstL¯s+gstcs.
For the first-order agents, similarly, according to (35), we have(42)limΔt⟶0xft+Δt−Wχftxft+Δt−xft−WχftxftΔt≤μ1ixft−Wχftxft+μ2ixst−Wχstxst.
That is,(43)dxft−Wχftxftdt≤μ1ixft−Wχftxft+μ2ixst−Wχstxst,where μ1i=−zftL¯f+gft,μ2i=Asfzft.Theorem 2.
Under assumption1 and 2, all the followers can move into the convex hull spanned by static leaders with algorithm (3), i.e., limt⟶∞xit−Wχitxit=0.Proof.
We will discuss the convergence of the heterogeneous multiagent system with switching graphs and nonconvex control input constraints at intervals in three cases.Case 1
: suppose that there exists a agentin can receive the information from leaders at time t=tsn. At this point, git=g¯i. First of all, we assume in is a second-order agent. From (40), we have(44)dv¯sint−Wχsintv¯sintdt≤λ1inv¯sint−Wχsintv¯sint+−λ1in−zsintg¯sincsinFtsn≤csin−qsinxsint−Wχsintxsint+qsin−csin−πing¯sincsinFtsn.
According to the definition of calculus, for allt∈[tsn,tsn+1), we have(45)v¯sint−Wχsintv¯sint≤ecsin−qsint−tsnFtsn+∫tsntecsin−qsint−τqsin−csin−πing¯sincsinFtsn≤ecsin−qsint−tsnFtsn+qsin−csin−πing¯sin/csinFtsnqsin−csin1−ecsin−qsint−tsn≤1+πing¯sinecsin−qsint−tsn−1qsinqsin−csinFtsn.
For allt∈[tsn,tsn+1), thus, we have(46)v¯sint−Wχsintv¯sint≤m1Ftsn,where m1=1+πing¯sinecsin−qsint−tsn−1/qsinqsin−csin. It is obvious that 0<m1<1. According to (38) and (46), we obtain(47)xsint−Wχsintxsint≤m1Ftsn+1−m1e−qsint−tsnFtsn.
For allt∈[tsn,tsn+1), we have(48)xsint−Wχsinxsint≤m2Ftsn,where m2=1+1−m1eqsint-tsn. Obviously, 0<m2<1.
Ifin is a first-order agent, we have(49)dxfint−Wχfintxfintdt≤−g¯finxfint−Wχfintxfint≤−g¯finFtsn≤e−g¯fint−tsnFtsn.
Then,(50)xfint−Wχfintxfint≤n1Ftsn,where n1=e−g¯fint−tsn. Obviously, 0<n1<1.Case 2
: If there exists a followerin such that xint−Wχintxint≤binFtsn, for the second-order agents, from (23), we obtain(51)xsint−Wχsintxsint≤e−csint−tsnbinFtsn+1−e−csint−tsnFtsn.
For allt∈[tsn,tsn+1), we have(52)xsint−Wχsintxsint≤m3Ftsn,where m3=1+bin−1e−csint-tsn. Obviously, 0<m3<1. Then, we can obtain that(53)dv¯sint−Wχsintv¯sintdt≤(qsinzsint−csin)m3+1−m3Ftsn−v¯sint−Wχsintv¯sint+1−m3csin−qsinzsint+1×v¯sint−Wχsintv¯sint≤qsin−csm3+1−m3qsin−csπin×v¯st−Wχsintv¯sint+qsin−csm3+1−m3Ftsn.
And(54)v¯sint−Wχsintv¯sint≤1+1−m3csπin−qsin−1×e−Bst−tsn−1BsFtsn,where(55)Bs=1+1−m3csinπin−qsin.
Hence,(56)v¯sint−Wχsintv¯sint≤m4Ftsn,where(57)m4=1+1−m3csinπin−qsin−1×e−Bst−tsn−1Bs,0<m4<1.
For the first-second agents, similarly, we have(58)dxfint−Wχfintxfintdt≤g¯inbinFtsn.
Then,(59)xfint−Wχfintxft≤n2Ftsn,where n2=e−g¯int-tsnbin,0<n2<1.Case 3
: suppose that there exists a followerin such that ainj0tsn>0, xj0tsn−Wχj0txj0t≤ej0Ftsn, where 0<ej0<1. According to the above assumption, for the second-order agents, we have(60)xsintsn−Wχsintxsintsn≤1+ej0−1e−g¯sint−tsnFtsn.
Hence,(61)dv¯sint−Wχsintv¯sintdt≤csin−qsinv¯sint−Wχsintv¯sint+qsin−csin+πincsinqsinej0−1e−csinTFtsn.
For allt∈[tsn,tsn+1), we have(62)v¯sint−Wχsintv¯sint≤m5Ftsn,where m5=1+πincsin/qsin1−ej0e−csint-tsnecsin−qsint-tsn−1/qsincsin−qsin, 0≤m5≤1. Furthermore,(63)xsint−Wχsintxsint≤m6Ftsn,where m6=1+m5−11−ecsint-tsn, 0<m6<1. And similarly, for the first-order agents, there exists 0<n3<1 that makes(64)xfint−Wχfintxfint≤n3Ftsn.
Under assumption2, we know that there exists at least one follower which can receive the information directly from leaders. From the analysis of the above cases, in the time interval ts,ts+1, we have the following conclusions for all agents:(65)xin1ts+1−Wχxin1ts+1≤γ1Fts,v¯in1ts+1−Wχv¯in1ts+1≤γ2Fts,where the constant γ1,γ2∈0,1. And there exists another follower in2≠in1 which can receive the information from leaders or in1 in the interval ts+1,ts+2. Hence,(66)xin1ts+2−Wχxin1ts+2≤γ3Fts+1,v¯in1ts+2−Wχv¯in1ts+2≤γ4Fts+1,xin2ts+2−Wχxin2ts+2≤γ3Fts+1,v¯in2ts+2−Wχv¯in2ts+2≤γ4Fts+1,where the constant γ3,γ4∈0,1. Then, we can draw the following conclusions:(67)xits+n−Wχxits+n≤γFts,v¯its+n−Wχv¯its+n≤γFts,where γ∈0,1. Hence, there exists a constant ε∈0,1 that makes Fts+nr≤εrFts. Then, we can know the limit of Ft exists and limt⟶∞maxζkt−Wχζkt=0. Therefore, we can draw the conclusion that the control protocol can realize the containment control.


## 6. Simulation

Consider a continuous-time heterogeneous multiagent system with 6 followers and 4 leaders. According to the topology assumptions, we design four different communication topologies which are shown in Figure1. And they switch every 1.5s. The control inputs are limited in the following set: χ=x|x≤1∪x|x−0,3/2T≤0.5∪x|x−0,−3/2T≤0.5. Suppose the weight of every edge is 0.6. Initial position values of followers are x10=−3,3T, x20=−3,0T, x30=−3,−3T, x40=3,3T, x50=3,0T, and x60=3,−3T. Initial position values of leaders are y10=−2,2T, y20=2,2T, y30=−2,−2T, and y40=2,−2T. Initial velocity of agents are v10=v30=1,1T, v20=1,0T, v40=−1,−1T, v50=−1,0T, and v60=−1,0T. We suppose each parameter qi is taken as qi=10. Figure 2 shows the trajectory of followers. As you can see in Figures 2 and 3, all the followers finally converge to the set composed of multiple stationary leaders while the control input of each followers remains in their corresponding constraint set. The simulation result is consistent with theorem.Figure 1
Four directed graphs.
(a)(b)(c)(d)Figure 2
Position of followers.Figure 3
Nonconvex constraints of control input.


## 7. Conclusion

This paper focuses on distributed containment control with nonconvex control input constraints of heterogeneous multiagent systems. For this system, a protocol with nonconvex control input is proposed. Based on the Lyapunov function and the definition of the derivative, we can theoretically infer the distance between each follower and the set consisted of leaders in each time interval. Then, we can obtain that each follower is driven to converge to a convex set composed of multiple stationary leaders. Future work could be directed to the containment control of the system which consisted of more different order multiagent.



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*Source: 7081091-2022-04-25.xml*