# The Couple-Group Consensus of Heterogeneous Multiagent Systems with Asynchronous Switching in Cooperative-Competitive Networks

Authors: Xingcheng Pu; Xia Sun
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

This paper investigates the problem of leader-following or leaderless couple-group consensus of heterogeneous multiagent systems (HMASs) with cooperative-competitive interaction, asynchronous switching, and controller faults. Some novel protocols have been proposed to solve the couple-group consensus of the two kinds of HMASs. Compared with the existing results, asynchronous control, controller faults, leaders, and the competitive-cooperative interaction are all considered in these novel protocols. By using linear matrix inequality, graph theory, Lyapunov functional theory, and model-dependent dwell time technique (MDADT), some sufficient conditions have been gained for the leader-following or leaderless couple-group consensus of these systems. The obtained results show that the couple-group consensus of HMASs can be transformed into determining the stability of the related switching linear system. These results can also be applied to HMASs with weakly connected topology without satisfying the demand of in-degree balance, strong connectivity, or containing a spanning tree. Some simulation experiments are given to prove the validity of the obtained results.


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## Body

## 1. Introduction

In recent years, distributive control of MASs has become a hot issue in artificial intelligence. The distributive control is that agents can communicate with their neighbors in a distributed way and complete simple tasks, respectively, but the whole system can finish very complicated task in a very efficient way. Consensus, as one of the most fundamental issue of distributive controls, refers to design a consensus protocol to make all agents reach a common state.Group consensus, as the extension of consensus, is to design appropriate protocols and algorithms for MASs to accomplish different tasks or reach different states. In some meaning, consensus is a special case of group consensus. According to the existing results, group consensus can be used to solve multitask problems and decompose large-scale problems flexibly, such as robot formation control [1, 2], distributed sensor networks [3], network security control [4, 5], and unmanned air vehicles [6, 7].In the past decades, group consensus or consensus has been widely studied by many scholars [8–45]. In [8], group consensus of continuous MASs with fixed topology and time-invariant delay has been investigated. In [9], group consensus of discrete-time heterogeneous MASs with directed topology has been studied. It should be pointed that many obtained results of group consensus are needed to suppose that the topologies of these systems are to be fixed. However, in many real situations, the topologies of MASs may change with time-varying. Hence, it is necessary for us to investigate consensus or group consensus of MASs with switching topology. Fortunately, some related results have been obtained for MASs with switching topology [10–17]. On the other hand, time delay inevitably exists between the subsystem and the controller in any switching system. This phenomenon will lead to asynchronous switching [18–25]. In many real situations, asynchronous switching will greatly reduce the performance of MASs. Therefore, it is also very important for us to investigate the consensus or group consensus of MASs with asynchronous switching [21–29]. In [21], input-to-state stability (ISS), integral-ISS (iISS), and et-ISS for a class of stochastic systems with time delay and asynchronous switching have been investigated. In [22], the input-to-state stability (ISS), stochastic ISS (SISS), and integral-ISS (iISS) for a class of switched stochastic systems with time delays have been also studied. It is no doubt that asynchronous switching will prevent the MASs from achieving consensus or group consensus [25]. Up to now, some scholars have paid attention to the consensus of MASs with asynchronous switching, and some results have also been obtained [23–30]. In [23], leaderless and leader-following consensus of a kind of second-order MASs with asynchronous communication topology has been discussed. In [24], the consensus of nonlinear multiagent systems with asynchronous switching has been investigated. It is worth pointing out that the obtained results for consensus of MASs with asynchronous switching are needed to suppose that the system is homogeneous [23–29]. This is to say, each agent owns the same dynamic property. However, in many practical cases, it is impossible for each agent to own same dynamic property due to external influence or interrestriction. Therefore, it is necessary for us to investigate the group consensus of HMASs. Up to now, some meaningful achievements have been obtained for the group consensus of HMASs [30–34]. In [30, 31], based on pinning control, delta operator skill and LMI (linear matrix inequality) technique, some sufficient conditions have been obtained for the couple-group consensus of HMASs with cooperative-competitive relation and switching topology. In [32], group consensus of HMASs with the identical or the diverse input time delays has been discussed in competitive networks. In [33], the couple-group consensus of a class of discrete-time HMASs with communication and input time delays has been studied. In [34], based on fixed and switching topologies, group consensus of HMASs which is composed of first-order and second-order agents has been investigated. On the other hand, although many results have been obtained for the group consensus of MASs, the demands of topology of MASs are very strictly, such as in or out-degree balance, having a spanning tree or bipartite network, etc. Moreover, most of these obtained achievements are needed to assume that the relation between agents is competitive or cooperative, whereas in many practical situations, competitive and cooperative relations may coexist [35]. In [35], we have discussed the couple-group consensus of HMASs with Markov switching and cooperative-competitive relation. In fact, cooperative-competitive relation may exist in many community groups. For instance, members in a football team will try their best to win games and championships, and in this situation, they are cooperative. On the other hand, the competitive relation between members is always inevitable due to the limited resources, such as star status, playing time, and public recognition. Moreover, the dynamic properties of any two people are completely different; this is to say, they are heterogeneous. Therefore, it is necessary for us to consider both the cooperative and competitive relation between agents in HMASs, which is more realistic. Up to now, some meaningful results have been obtained for consensus of HMASs with competitive-cooperative relation [30, 31, 33, 35].Furthermore, faults can also affect the performance of the system. For example, they can lead the system to instability or malfunction. Hence, it is very important for us to investigate the dynamics of MASs with faults [36–40]. However, there are few conclusions on consensus of HMASs with unknown faults and competitive-cooperative relation. As another hot topic of MASs, leader-following consensus of MASs has also gained much interest of many scholars [41, 42]. In [41], based on Lyapunov function and Nyquist stability criterion, leaderless and leader-following consensus of MASs with directed topology has been studied. In [42], leader-following consensus of HMASs with input delays has been investigated and a sufficient condition has been gained to achieve consensus of this system. It is noted that there are few reports on the group consensus of HMASs with competitive-cooperative relation and faults.Incited by the aforementioned discussions, this paper aims to investigate the couple-group consensus of HMASs with asynchronous switching and cooperative-competitive relation, especially for the HMASs with faults. There are mainly triple contributions in this paper. Firstly, based on cooperative-competitive relationship, asynchronous switching, and faults, a novel protocol is proposed for the couple-group consensus of HMASs with time delay, faults, cooperative-competitive relationship, and asynchronous switching. Linear matrix inequality (LMI), graph algebra theory, Lyapunov functional theory, and model-dependent dwell time technique (MDADT) are used to solve the couple-group consensus of this kind of system. Some sufficient conditions have been achieved for the couple-group consensus of this system. In our obtained results, the balance of degree or containing a spanning tree can be relaxed, which has extended the existing results [23–27]. Secondly, leader-following consensus of heterogeneous multiagent systems with asynchronous switching is also investigated. Similarly, based on cooperative-competitive interaction, a novel protocol is also proposed for HMASs with asynchronous switching. Similarly, some sufficient conditions have also been obtained for the couple-group consensus of this system, which extend the existing results [42–45]. Thirdly, some simulation examples have been given to show the validity of the obtained results. The simulation examples have also proved that these obtained results are not only fit for MASs with weakly connected topology, but also they are true for MASs without satisfying equilibrium of in or out degree or having a spanning tree. The rest of this paper is organized as follows: some basic definitions, lemmas, and the description of the problem are introduced in Section 2. In Section 3, some results and proofs are given. In Section 4, some simulation examples are listed to verify the validity of the obtained results. In the final section, summary is made for this paper.


## 2. Preliminaries and Problem Statement

In this section, graph theory, some lemmas, and definitions will be introduced.A graph can be used to describe HMASs which is composed of first and second-order agents. In this paper, the graphG=<V,E,A> represents a multiagent system with n agents. V=<v1,v2,⋯,vn> is the vertex set. E⊆V×V is the edge set. A=aijn×n∈Rn×n denotes the adjacent matrix of G. If agent i can receive information from agent j, then aij>0 ; otherwise, aij=0 . Since both first and second-order agents coexist in this HMASs, the adjacent matrix A of G can be divided into the following form:(1)AssAsfAfsAff,Ass denotes the adjacent matrix of the second-order agents. Aff denotes the adjacent matrix of the first-order agents. Asf denotes the adjacent matrix from second-order agents to first-order agents, and Afs denotes the adjacent matrix from first-order agents to second-order agents. Ni=vjeij∈E denotes the neighbor set of node vi . Of course, neighbors of node vi may be second-order or first-order agents, then its neighbor set can be further divided into two parts: Nif and Nis . Nif and Nis represent the first-order and second-order neighbor set of agent i , respectively.We definedi=deginvi=∑j∈Niaij, then D=diagd1,d2,⋯⋯,dn is the in-degree matrix of G . L=D−A is the Laplacian matrix of graph G. Based on the former discussion, L can also be written as the following forms:(2)L=Dss−Ass+Dsf−Asf−AfsDff−Aff+Dfs,or(3)L=Lss+Dsf−Asf−AfsLff+Dfs,where Dss=diag∑j∈Nisaij,i∈η1(4)Dff=diag∑j∈Nifaij,i∈η2,Dfs=diag∑j∈Nifaij,i∈η1,Dsf=diag∑j∈Nsaij,i∈η2,σt:0,∞⟶χ=1,2,⋯,m is the switching function. m is a positive integer, and it stands for the number of all switching topologies. For n≥1,Tσ+tn,tn+1 and Tσ−tn,tn+1, we represent synchronization and asynchronous time during tn,tn+1.In our model, the HMASs are composed ofn second-order agents and m first-order agents. It can be described as the following differential equations:(5)x˙it=vitv˙it=uit,i∈η1,(6)x˙it=vit,i∈η2,η1=1,2,⋯,n,η2=n+1,n+2,⋯,n+m,η=η1∪η2. uit, xit, and vit represent the control input, position, and speed of agent, respectively. For the simplicity of later discussion, a novel matrix is defined as follows:(7)H=hij=lij−lnj∈Rn−1×n−1,lij and lnj are the elements of L. For HMASs (5) and (6), H can be divided into as following two parts Hs and Hf:(8)Hs=hsij=lsij−lsnj∈Rn−1×n−1,Hf=hfij=lfij−lfn+mj∈Rm−1×m−1,lsij and lsnj are the elements of L and i∈η1Λj∈η2; lfiji∈η1Λj∈η2 is the element of L. aΛb means “a and b”.Lemma 1 (see [46]) (Schur complement lemma).
For any symmetric matrixX=X11X12X21X22 and X11 is a square matrix, then the following conditions (i)–(iii) are equivalent:(i)
X<0(ii)
X22<0,X11−X12X22−1X12T<0(iii)
X11<0,X22−X12TX11−1X12<0Lemma 2 (see [31]).
If,R, E, and M are matrices with appropriate dimensions, N=NT, R=RT, FTF≤R, K is a positive real number, then the following inequalities (9) and (10) are equivalent.(9)N+κMMT+κ−1ETRE<0,(10)N+MFE+ETFTMT<0.Definition 1 (see [33]).
The couple-group consensus of HMASs (5) and (6) can be achieved if and only if the following condition (11) is satisfied for any given initial values.(11)limt→∞xit−xjt=0,i,j∈ηk˙,k=1,2,limt→∞vit−vjt=0,i,j∈η1.Definition 2 (see [45]).
The leader-following group consensus of HMASs (5) and (6) can be realized if and only if the following condition is satisfied for any given initial values.(12)limt→∞xit−x0t=0,i,j∈ηk˙,k=1,2,limt→∞vit−v0t=0,i,j∈η1.x0t and v0t are the position and speed of the leader, respectively.Definition 3 (see [25]).
LetTptl,tk and Nσptl,tktk>tl˙>0 be the total running time and total number of subsystem p during the time interval tl,tk, respectively. If there exists τσp>0 satisfying the following inequality for N0ptl,tk>0 and(13)Nσptl,tk≤N0ptl,tk+Tptl,tkτσp,then ταp is called the mode-dependent average dwell time (MADAT), where σt stands for the switching signal.Remark 1.
Based on average dwell time (ADT), many results have been obtained for switching systems or related systems, such as [18–20]. Compared with ADT, more changing parameters on the influence of each subsystem will be considered in MDADT. Hence, MDADT can be used to reduce the conservation, and it has more broad applications.


## 3. Problem Statement

In this section, the couple-group consensus of HMASs (5) and (6) with asynchronous switching and faults will be discussed. Based on competitive-cooperative relation, asynchronous switching signal, and faults, a novel couple-group protocol (14a) and (14b) is designed as follows:(14a)uit=fiλ0σt−Δ∑j∈Nsiaijσtxjt−xit−∑j∈Ndiaijσtxjt+xit−fiλ1σt−Δvit,i∈η1,(14b)uit=fiλ2σt−Δ∑j∈Nsiaijσtxjt−xit−∑j∈Ndiaijσtxjt+xit,i∈η2,where fi represents the unknown control faults and(15)fi=observedvaluerealvalue.The range offi is given as follows:(16)fd≤fi≤fu,0≤fd≤1≤fu,where fd and fu are two given constants. They can be used to represent the low and upper bound of fi, respectively. λ0λ0, λ1, and λ2 represent different control parameters of this system. Δ is the switching time delay between the subsystem and the feedback controller and aijσt denotes the related element of the adjacency matrix.Remark 2.
In fact, cooperative and competitive relations may coexist in many real situations. Different from the proposed asynchronous switching protocols [23–25], cooperative-competitive relation, faults, and heterogeneous are considered in protocols (14a) and (14b). Here, xjt−xit represents that agent j and i are cooperative, and xjt+xit shows that agents j and i are competitive.Remark 3.
According to the range offi, faults are considered to be uncertain values, and no fault or known fault can be regarded as special cases. Therefore, three cases will be investigated in our model: (a) there is no fault in the system; (b) the system has certain faults; (c) the system has uncertain faults. In our later discussion, the system with faults will be discussed firstly and the leader-following consensus of HMASs (5) and (6) with asynchronous switching and no faults will be studied further.


## 4. Main Results

Theorem1 shows that the couple-group consensus of HMASs (5) and (6) with asynchronous switching and faults can be converted into judging the asymptotically stability of a linear switched system (17).Theorem 1.
We suppose HMASs (5) and (6) with switched topology χ=1,2,…,m, σt−Δ∈χ; if system (12) is asymptotically stable,(17)z˙t=C¯σt−Δzt,zt0=z0,then the couple-group consensus of HMASs (5) and (6) can be achieved under asynchronous switching control protocol (14a) and (14b), where(18)C¯σt−Δ=F0Kσt−ΔB+EF1Kσt−ΔB.Proof.
Letzt=x1−xn,…,xn−1−xn,xn+1−xn+m,…,xn+m−1−xn+m,v1−vn,…,vn−1−vnTt
Two situations (a) and (b) will be discussed in the following list:(a)
When∈η1, based on (14a), we have(19)ui−un=fiλ0σt−Δ∑j∈Nsiaijσtxjt−xit−∑j∈Ndiaijσtxjt+xit−fiλ1σt−Δvit−fnλ0σt−Δ∑j∈Nsnanjσtxjt−xnt−∑j∈Ndnanjσtxjt+xnt−fnλ1σt−Δvnt=fiλ0σt−Δ∑j∈Nsiaijσt−∑j∈Ndiaijσtxjt−fiλ0σt−Δ∑j∈Nsiaijσt+∑j∈Ndiaijσtxit−fnλ0σt−Δ∑j∈Nsnanjσt−∑j∈Ndnanjσtxjt+fnλ0σt−Δ∑j∈Nsnanjσt+∑j∈Ndnanjσtxnt−fnλ1σt−Δvit−vnt+∑j∈Ndnanjσtxnt=−fiλ0σt−ΔLssi+Dsfixst−Asfixft−−fnλ0σt−ΔLssn+Dsfnxst−Asfnxft−fnλ1σt−Δvit−vnt.Letxst−τ=x1t−τ,⋯,xnt−τT and xft−τ=xn+1t−τ,⋯,xm+nt−τT.Lssi, Dsfi, and Asfi are the ith row of matrices Lss, Dsf, and Asf; Lssn, Dsfn, and Asfn are the nth row of matrices Lss, Dsf, and Asf, respectively.Combined (2) with (19), equations (20) and (21) can be obtained:(20)v˙1−v˙n,…,v˙n−1−v˙nT=u1−un,…,un−1−unT,(21)=−λ0σt−ΔFη1Hsx1−xn,…,xn−1−xnt−λ1σt−ΔFη1v1−vn,…,vn−1−vnt,=−λ0σt−ΔFη1Hs,0,0zt+0,0,−λ1σt−ΔFη1zt,Fη1=diagf1−fn,f2−fn,⋯,fn−1−fn.(b)
When∈η2, based on (14b), we have(22)uit=fiλ2σt−Δ∑j∈Nsiaijσtxjt−xit−∑j∈Ndiaijσtxjt+xit,i∈η2,ui−un+m=fiλ2σt−Δ∑j∈NSiaijxjt−τ−xit−τ−∑j∈NDiaijxjt−τ+xit−τ−fn+mλ2σt−Δ∑j∈NSn+man+mjxjt−τ−xn+mt−τ−∑j∈NDn+man+mjxjt−τ+xn+mt−τ=−λ2σt−ΔFη2xn+1−xn+m,⋯,xn+m−1−xn+mt−τ,Fη2=diagfn+1−fn+m,fn+2−fn+m,⋯,fn+m−1−fn+m.
Similarly,Lffi, Dfsi, and Afsi are the ith row of matrices Lff, Dfs, and Afs; Lffn+m, Dfsn+m, and Afsn+m are the n+mth row of matrices Lff, Dfs, and Afs, respectively.
Hence, equation (23) can be obtained:(23)x˙n+1−x˙n+m,…,x˙n+m+1−x˙n+mT=un+1−un+m,…,Vn+m−1−un+mT=−λ2σt−ΔFη2Hfxn+1−xn+m,xn+2−xn+m,⋯,xn+m−1−xn+mt=0,−λ2σt−ΔFη2Hf,0zt.
According to equations (20), (21), and (23), equation (24) can be obtained:(24)z˙t=00I0−λ2σt−ΔF1Hf0−λ0σt−ΔF1Hs0−F1λ1σt−ΔHszt,zt0=z0,F1=diagFη1,Fη2.
Equation (24) can also be rewritten as the following matrix form:(25)z˙t=FKσt−ΔBzt,zt0=z0,where =In000Fη1000Fη2, B=Hs000Hf000In,(26)Kσt−Δ=00I0−λ2Im0−λ0In0−λ1Inσt−Δ.
Obviously, based on the definition ofzt and Definition 3, if system (25) is asymptotically stable, then the couple-group consensus of HMASs (5) and (6) can be realized under the protocol (14a) and (14b). Theorem 1 is proved.
Furthermore, some matricesFu,Fd,F0,F1 and E will be defined as follows:(27)Fu=diag1,⋯,1,fu,⋯,fun+m,E=diag1,…,1,ν,…,νn+m,−1≤ν≤−1,Fd=diag1,…,1,fd,…fdn+m,F0=12Fu+Fd,F1=12Fu−Fd,E=diag1,…,1,ν,…,νn+m,−1≤ν≤−1,F=F0+EF1,then differential equation (25) can also be rewritten as the following form (28):(28)z˙t=F0Kσt−ΔB+EF1Kσt−ΔBzt,zt0=z0.
Combining with Definition1 and equations (19)–(28), the couple-group consensus of system (2) and (6) can be achieved under protocol (14a) and (14b) if the system (17) is asymptotically stable.
Based on Theorem1, the couple-group consensus of heterogeneous MASs (3) and (4) with asynchronous switching and faults can be converted into judging the asymptotically stability of a linear switched system (17). In following Theorem 2 and Corollary 1, some directed results can be deduced from Theorem 1.Theorem 2.
For any given positive numbersαi>0,βi>0,μi>1,i,j∈χ, and σtn=i,t∈tn+Δ,tn+1, σtn−=j,t∈tn,tn+Δ, t>0, Pi is a positive definite matrix if inequalities (29) and (30) hold(29)C¯iPi+PiC¯i+αiPi≤0,tn+Δ,tn+t,C¯jPi+PiC¯j−βiPi≤0,t∈tn,tn+Δ,(30)Pi≤μiPj,ταi>lnμi+αi+βiTimax−αi,then system z˙t=C¯σt−Δzt,zt0=z0 is asymptotically stable and(31)Timax−=maxn,σtn=iTσtn−tn,tn+1,σt−Δ∈χ,C¯i=FoKiBi+EF1KiBi,C¯j=F0KjBj+EF1KjBj,t1,t2,…,tn,… are positive real numbers, and they represent the switching time series, n≥1 and limn⟶∞tn=∞.Proof.
We construct a Lyapunov function as follows:(32)Vit=zTdistPizt.
Whent∈tn,tn+Δ, taking the time derivative of Vit along with z˙t=C¯σt−Δzt, then we have(33)Vi⋅t=z˙TtPizt+zTtPiz˙t=zTtC¯σt−ΔTPi+PiC¯σt−Δzt=zTtC¯jTPi+PiC¯jzt.
Based on (29) and (33), one has the following matrix inequalities:(34)zTtC¯jTPi+PiC¯jzt<βizTtPizt,(35)V˙it<βiVit.
Based on inequality (35), the following inequality can be obtained:(36)Vit≤βit−tnVitn.
When∈tn,tn+Δ , taking the time derivative of Vit along with z˙t=C¯σt−Δzt, we also have(37)Vi⋅t=z˙TtPizt+zTtPiz˙t=zTtC¯σt−ΔTPi+PiC¯σt−Δzt.
Similarly, based on (29) and (37), one has(38)Vtt=zTtC¯iTPi+PiC¯izt<−αizTtPizt.(39)V⋅it<−αiVit,then, we can obtain(40)Vit≤e−αit−tn−ΔVitn+Δ.
Hence, whent∈tn,tn+1, from inequalities (36) and (40), we have(41)Vσtnt≤e−ασnTσ+tn,tn+1+βσtnTσ−tn,tn+1Vσtntn≤e−ασtnTσtn,tn+1+βσtnTσ−tn,tn+1μσtnVσtntn−≤e−ασtnTσ+tn,tn+1+βσtnTσtn,tn+1μσtn⋅μσtn+1e−ασtn+1TσTtn,tn+1+βσtn−1TσTtn,tn+1Vσtn+1tn−1≤⋯≤μM⋅eμ1+μ2,where μM=μσtnμσtn+1⋯μσt1Vσt0t0(42)μ1=−ασtnTσrtn,tn+1+βσtnTσrtn,tn+1−ασtn+1Tσrtn,tn+1,μ2=βσtn−1Tσ−tn,tn+1⋯−ασt0Tσ+tn,tn+1+βσt0Tσ−tn,tn+1.
LetN0,t=Nσ10,t+Nσ20,t+⋯Nσm0,t=∑i∈χNσi0,t be the total switching number during time interval span 0,t, then formula (41) can be rewritten as follows:(43)Vσtnt≤e−ασtnTσ+tn,tn+1+βσtnTσ−tntn+1μσtnVσtntn−≤⋯⋯≤∏i∈χμiN0,texp∑i∈χ−αiTi+0,t+βiTi−0,tVσt00.
From Definition3, we can obtain(44)∏i=χμiNσi<∏i∈χμiN0i0,t+τi0,t/τi=μ1N010,t+T10,t/τa1⋅μ2N020,t+T20,t/τa2⋯⋯=eN0,10,t+τ10,t/τα1lnμ1⋅eN0,20,t+T2,0,t/τα2lnμ2⋯⋯=e∑i=χN0,j0,t+Ti0,t/ταilnμi.
Then, based on formulas (43) and (44), we have(45)Vσtnt≤Vσt00e∑i∈χN0i0,tlnμi+Ti0,tlnμi/ταi+∑i∈χ−αiTi+0,t+βiTi−0,t=Vσt00e∑i=xN0,i0,tlnμi+Ti0,tlnμi/τai+∑i=x−αiTi+0,t+βiTi−0,t+αiTi−0,t=Vσt00e∑i=xN0i0,tlnμi+lnμi/ταi−αiTi0,t⋅e∑i=χβiTi−0,t+αiTi−0,t.
From the definition ofTmax− and MADAT, we have the following inequality:(46)Nσi0,tTmax−≤N0,i0,tTmax−+Ti0,tταiTmax.
On the contrary, sinceNσi0,t≥0 and Tmax−≥Ti−0,t, we have(47)Ti−0,t≤Nσi0,tTmax−.
Moreover, according to inequalities (39) and (40), the following inequality can be obtained:(48)Ti−0,t≤Nσi0,tTimax−≤N0,iTimax++Ti0,tταiTimax−.
Finally, according to (46), inequality (45) can be written as the following form:(49)Vσtnt≤e∑i∈χN0,i0,tlnμi+Ti0,t/τailnμi⋅e∑i∈χβiTi−0,t−αiTi+0,tVσt0t=e∑i∈χN0i0,tlnμi+Ti0,t/ταilnμi⋅e∑i∈χβiTi−0,t−αiiTi0,t−Ti−0,tVσt00=e∑i∈χNoi0,tlnμi⋅e∑i∈χlnμi+αi+βiTimax−/ταi−αiTi0,t+αi+βiNoi0,tTimax−Vσt00=e∑i∈χNoi0,tlnμi⋅e∑i∈χlnμi+αi+βiTimax−/ταi−αiTi0,t+αi+βiNoi0,tTimax−Vσt00.
From Theorem1, if system (17) is asymptotically stable, then the couple-group consensus of HMASs (5) and (6) can be realized under protocol (6) and (7). From inequality (49), system (17) is asymptotically stable if(50)ταi>lnμi+αi+βiTimax−αi.
The proof Theorem2 is proved.
By using matrix transformation, Theorem2 can also be written as another form as Corollary 1.Corollary 1.
Suppose positive real numbersαi>0,μi>1,ζi>0,ζj>0,i,j∈χ,σtn=i,σtn−=j, Pi is a positive definite matrix if inequalities (51)–(53) hold.(51)FoKiBτPi+PiF0XiBαi+αiPiζiBαiτKiτPiF1E∗−ζiI0∗∗−ζiI<0,(52)FoKjBαiTPi+PiFoKjBai−βiPiζiBαiTKjTPiF1E∗−ζjI0∗∗−ζjI<0,(53)Pi≤μiPj,and ταi>lnμi+αi+βiTimax−/αi, then the couple-group consensus of HMASs (5) and (6) can be achieved under protocols (14a) and (14b).The proof is obvious, and then the process of proof is omitted.Remark 4.
Based on [23–27], some sufficient conditions have been obtained for the couple-group consensus of HMASs (3) and (4) in Theorems 1 and 2. Comparing with [23–27], heterogeneous, cooperative-competitive relation, faults and asynchronous switching are all considered, which are more related to real situations. Therefore, Theorems 1 and 2 have extended the existing results. Furthermore, there are no special demands for the topology structure, such as equilibrium of in or out degree, strong connectivity, or containing a spanning tree.
In the following Theorem3, the leader-following couple-group consensus of HMASs (5) and (6) with asynchronous switching will be investigated. We suppose that there are two leaders for the n second-order agents and m first-order agents, respectively. The dynamic models of the second-order leader and first-order leader are given as follows:(54)x˙01t=v0t,v0t=k˙1,x02⋅t=k2,where k1 and k2 are given numbers
Based on competitive-cooperative relationship, a novel protocol (55) is designed as follows:(55)uit=λ0σt−Δ∑j∈Nsiaijσtxjt−xit−∑j∈Ndiaijσtxjt+xit−λ1σt−Δvit−hiλ0σt−Δxit−x0it+λ1σt−Δvit−v0t,i∈η1,(56)uit=λ0σt−Δ∑j∈Nsiaijσtxjt−xit−∑j∈Ndiaijσtxjt+xit−hiλ2σi−Δxit−x0it,i∈η2,where λ0, λ1, and λ2 represent different control parameters of the system, Δ is time delay between the topology switching and the feedback controller switching, and aijσt denotes the element of the adjacency matrix in i−th row and j−th column. If the agent i can communicate with the leader, then there exists an edge from the agent i to the leader, then hi>0; otherwise, hi=0.Theorem 3.
Suppose systems (5) and (6) with switched topology χ=1,2,⋯,m, σt−Δ∈χ if the following system (57) is asymptotically stable(57)z˙t=Aσt−Δzt,zt0=z0,then the leader-following couple-group consensus of HMASs (5) and (6) can be achieved under asynchronous switching control protocols (55) and (56) and(58)z˙t=Aσt−Δzt,zt0=z0,Proof.
We definexi¯t=xit−x0it,(59)v¯it=vit−v0t,x¯sTt=x¯1Tt,x¯2Tt,⋯,x¯nTt,x¯fTt=x¯n+1Tt,x¯n+2Tt,x¯n+mTt,v¯sTt=v¯1Tt,v¯2Tt,v¯nTt,Hs=diaghs,hs,…,hn,Hf=diaghn+1,hn+2,…,hn+m,zt=x¯sTt,x¯fTt,v¯sTtTˉ.Case 1.
i∈η1, combining (5) and (55), we have(60)x¯⋅it=v¯it,x¯⋅st=v¯st,(61)v¯˙it=λ0σt−Δ∑j∈Nsiaijσtx¯jt−x¯it−∑j∈Ndiaijσtx¯jt+x¯it−λ1σt−Δv¯it−hiλ0σt−Δx¯it+λ1σt−Δv¯it=−λ0σt−ΔLss+Dsfx¯it−Asfx¯jt−hsλ0σt−Δx¯it−λ1σt−Δ+hsλ1σt−Δv¯it,(62)v¯˙st=−λ0σt−ΔLss+Dsfx¯st−Asfx¯ft−hsλ0σt−Δx¯st+λ1σt−Δ+hsλ1σt−Δv¯st.Case 2.
i∈η2, combining (6) and (56), we similarly have(63)xi¯⋅t=λ2σt−Δ∑j∈Nsiαijσtx¯jt−x¯it−∑j∈Ndiαijσtx¯jt−x¯it−hiλ2σt−Δx¯it=−β2σt−ΔLff+Dfsx¯it−Afsx¯jt−hiβ2σt−Δx¯it,(64)x¯˙ft=−β2σt−ΔLff+Dfsx¯ft−Afsx¯st−hfβ2σt−Δx¯ft.
On basis of (62) and (64), one has(65)z˙t=00In×nλ2Afs−λ2Lff+Dfs−Hfλ20−λ0Lss+Dsf−Hsλ0λ0Asfλ1+λ1Hsσt−Δ,zt=Aσt−Δzt,zt0=z0.
Combining Definition2 and equations (61)–(64), the leader-following couple-group consensus of HMASs (5) and (6) can be achieved under protocols (55) and (56) if the system (65) is asymptotically stable.
Based on Theorem3, a relation has been established for the leader-following couple-group consensus of HMASs (5) and (6) and a linear switched system (57). From Theorems 2 and 3, we can directly obtain the following Theorem 4.Theorem 4.
For any positive real numbersγi>0,ξi>0, ψi>1,i,j∈χ, and σtn=i,t∈tn+Δtn+1, σtn−=j,t∈tn,tn+Δ, t>0, Qi is a positive definite matrix, and the following inequalities (66a)–(66c) hold:(66a)AQi+QiAi+γiQi<0,tn+Δ,tn+1,(66b)AjQi+QiAj−ξiQi<0,t∈tn,tn+Δ,(66c)Qi≤ψiQj,then the system z˙t=Aσt−Δzt asymptotically stable if(67)τγi>lnψi+γi+ξiTimax−γi,where 0≤t1,t2,⋯,tn,⋯ are positive real numbers, they are switching time series, n≥1, limn⟶∞tn=∞, and(68)Timax−=maxn,σtn=iTσtn−tn,tn+1,σt−Δ∈χ,Timax−=maxn,σtn=iTσtn−tn,tn+1,σt−Δ∈χ,Aσt−Δ=00In×nλ2Afs−λ2Lff+Dfs−Hfλ20−λ0Lss+Dsf−Hsλ0λ0Asfλ1+λ1Hsσt−Δ.Furthermore, according to Theorem3, if system (67) is asymptotically stable, then the leader-following couple-group consensus of HMASs (5) and (6) can be achieved under protocols (55) and (56).The proof can be completed similarly as Theorem2.Remark 5.
In Theorem4, some sufficient conditions have been given for the leader-following couple-group consensus of HMASs (5) and (6). In this case, asynchronous switching and cooperative-competitive relations are considered, which has extended the existing results [41–45].


## 5. Simulation Examples

In this section, some examples will be given to show the validity of the obtained results.Example 1.
The topology structure of the HMASs (5) and (6) is drawn as Figure 1. In Figure 1, node 1, node 2, and node 3 are second-order agents, and node 4 and node 5 are first-order agents. Node 1, node 3, and node 4 belong to G1; node 2 and node 5 belong to G2. The agent 1, agent 3, and agent 4 are cooperative in subgroup 1, while agent 2 and agent 5 are cooperative in subgroup 2. The agents in different subgroups are competitive.
In Figure1(a), the in-degree of node 2 is 0, and the out-degree of node 2 is 1; the in-degree of node 5 is 2, and the out-degree of node 5 is 0. Hence, the equilibrium of in-degree or out-degree of node 2 and 5 cannot be satisfied. On the other hand, in Figure 1(a), there is no path from node 2 to node 4 and there is no any path from node 4 to node 2. Hence, there is no spanning tree in Figure 1(a). Obviously, neither of Figures 1(a) and 1(b) is strongly connective. The adjacency matrices of Figures 1(a) and 1(b) are described as follows:(69)0010000000000101010001010,0011000001100100000100000.
In protocols (14a) and (14b), let fd=0.9900, fu=1.1000, 0.9900≤fi≤1.1000, λ0=1.0000, λ1=2.0000, and λ2=1.5000. The initial state values of node 1, node 2, node 3, node 4, and node 5 are set as (1, 10, −2, 5, 3)T; the initial velocities of the second-order agents node 1, node 2, and node 3 are set as (2, 1.5, 2.2)T.
In Theorem2, let α1=0.22, α2=0.183, β1=0.34, β2=0.51, μ1=1.88, and μ2=2.47, by using the LMI toolbox, feasible solutions of P1 and P2 have been solved as follows:(70)P1=0.15310.0467−0.02550.1381−0.07350.06110.1603−0.14160.04670.1379−0.0826−0.0510−0.14880.1892−0.03280.0539−0.0255−0.08260.3333−0.09500.13470.3287−0.2380−0.47690.1381−0.0510−0.09500.50790.68430.20600.3957−0.5456−0.0735−0.14880.13470.68430.0997−1.16551.6108−0.28430.06110.18920.32870.2060−1.16550.09180.1666−0.11340.1603−0.0328−0.23800.39571.61080.16660.6647−0.1253−0.14160.0539−0.4769−0.5456−0.2843−0.1134−0.12531.5617,P2=0.02910.0363−0.0185−0.02970.0249−0.0229−0.0149−0.03320.03631.0725−0.0071−0.02460.3633−0.2329−0.45740.2262−0.0185−0.00710.0422−0.26000.2204−0.5754−1.20781.7380−0.0297−0.0246−0.26000.0886−0.5754−0.07230.1808−0.29190.02490.36330.2204−0.57540.32040.45830.18080.3115−0.0229−0.2329−0.5754−0.07230.45830.5954−0.1171−0.2919−0.0149−0.4574−1.20780.18080.1808−0.11710.36740.1620−0.03320.22621.7380−0.29190.3115−0.29190.16200.1363.
In this case,σ(t) and σ1(t) are asynchronous switching signals. Figure 2(a) shows that the couple-group consensus of HMASs (5) and (6) has been realized under protocols (10) and (11) if τα1 > 7.9603 and τα2 > 12.5149.
According to Figure1, equilibrium of in-degree or out-degree cannot be stratified and there is no any spanning tree. However, if the conditions of Theorems 1 and 2 are satisfied, the couple-group of HMASs (5) and (6) with asynchronous switching can also be realized.Figure 1
The topology of the HMASs (5) and (6) with asynchronous switching and faults.
(a)(b)Figure 2
The state trajectories of agents in systems (5) and (6) with faults and asynchronous switching. (a) Position. (b) Velocity.
(a)(b)Example 2.
The topology structure of the HMASs (5) and (6) is drawn as Figure 3. In Figure 3, node1, node 2, node 3, and node 6 are second-order agents; node 4, node 5, and node 7 are first-order agents. Node 1, node 2, node 3, and node 6 belong to G1; node 4, node 5, and node 7 belong to G2. The agents node 1, node 2, node 3, and node 6 are cooperative in the same subgroup, while the agents node 4, node 5, and node 7 are competitive in different subgroups.
In Figure3(a), the in-degree of node 1 is 0, and the out-degree of node 1 is 2. The in-degree of node 5 is 1, and the out-degree of node 5 is 2. Hence, the equilibrium of in-degree or out-degree of node 1 and 5 cannot be satisfied. On the other hand, in Figure 3(a), there is no path from node 2 to node 4 and there is no any path from node 4 to node 2. Therefore, there is no spanning tree in Figure 3(a). The adjacency matrices of Figures 3(a) and 3(b) are described as follows:(71)0000000101000011000000000100000100001001000001000,0100000000000011000000000101000100010001000001000.
In protocols (14a) and (14b), let fd=0.9900, fu=1.1000, 0.9900≤fi≤1.1000, λ0=1.0000, λ1=2.0000, and λ2=1.5000. The initial state values of node1, node 2, node 3, node 4, node 5, node 6, and node 7 are set as (1, 10, −2, 5, 3, 6, −3)T, and the initial velocities of the second-order agents node 1, node 2, node 3, and node 6 are set as (2, 1.5, 2.2, 3)T. In Theorem 2, let α1 = 0.22, α2 = 0.183, β1 = 0.34, β2 = 0.51, μ1 = 1.88, and μ2 = 2.47. In this case, two signals σ (t) and σ1 (t) are used to represent asynchronous switching. Figure 4 shows that the couple-group consensus of systems (5) and (6) have been realized under protocols (14a) and (14b) if τα1 > 10.0418 and τα2 > 14.3821.
According to Example2, if the conditions of Theorems 1 and 2 are satisfied, the group of heterogeneous MASs with asynchronous switching can also be realized.Figure 3
The topology of the HMASs (5) and (6) with asynchronous switching and faults.
(a)(b)Figure 4
The state trajectories of agents in systems (5) and (6) with faults and asynchronous switching. (a) Position. (b) Velocity.
(a)(b)Example 3.
The topology structure of the HMASs (5) and (6) with leaders (63) and (64) is drawn as Figure 5. In Figure 5, node1, node 2, and node 3 are second-order agents, and node 4 and node 5 are first-order agents. Node1, node2, and node 3 belong to G1; node 4 and node 5 belong to G2. The agent1, agent 2, and agent 3 are cooperative in subgroup 1, while agent 4 and agent 5 are cooperative in subgroup 2. The agents in different subgroups are competitive.
In protocols (65) and (66), let k1=2,k2=0.2,λ0=0.5, λ1=2.5, λ2=3,x01=15,x02=3, v01=4,Hs=10,10,0T, Hf=12,0T, α1=0.12, α2=0.04, β1 = 0.71, β2 = 0.41, μ1 = 2.01, and μ2 = 1.61. The initial positions of node 1, node 2, node 3, node 4, and node 5 are set as (3, 10, 0, −20, 40)T, and the initial velocities of the second-order agent node 1, node 2, and node 3 are set as (10, 30, 20)T.
In this situation,σ (t) and σ1 (t) are the asynchronous switching signals. Figure 6 shows that the leader-following group consensus of systems (5) and (6) has been realized under protocols (55) and (56) if τα1 > 19.6511 and τα2 > 10.6416. Hence, the validity of Theorems 3 and 4 has been proved.Figure 5
The topology of the HMASs (5) and (6) with leaders (55) and (56).Figure 6
The state trajectories of agents in systems (5) and (6) with asynchronous switching and leaders (55) and (56). (a) Position. (b) Velocity curve.
(a)(b)Example 4.
The topology structure of the heterogeneous MASs (5) and (6) is drawn as Figure 7. In Figure 7, node 1 and node 5 are second-order agents, and node 2, node 3, and node 4 are first-order agents. Node1, node 2, and node 4 belong to G1, while node 3 and node 5 belong to G2.
The adjacent matrices of Figures7(a) and 7(b) can be described as follows:(72)0100010000010000000100110,0000010100000010100000100.
In protocols (14a) and (14b), let fd = 0.87, fu = 1.3, 0.87 ≤ fi ≤ 1.30, λ0 = 1.1, λ1 = 3, and λ2 = 1.2. The initial state values of node 1, node 2, node 3, node 4, and node 5 are set as (3, 2, −5, 6, −6)T, and the initial velocities of the second-order agents node 1 and node 5 are set as (3, 2)T. In this case, two signals σt and σ1t are asynchronous switching signals. Figure 8(a) shows that the couple-group consensus of systems (5) and (6) cannot be realized under protocols (14a) and (14b) if mode-dependent average dwell time (MADAT) does not satisfy the conditions in Theorems 1 and 2.
From Examples1–3, the validities of Theorems 1–4 and Corollary 1 have been examined. Especially in Examples 1–3, the topologies of the HMASs (5) and (6) cannot satisfy equilibrium of in-degree or out-degree or contain a spanning tree, which has extended the results in [21–27, 42–46]. Example 4 shows that if the related conditions of Theorems 1–4 and Corollary.1 cannot be satisfied, then the couple-group consensus of HMASs (5) and (6) cannot be realized.Figure 7
The topology of the HMASs (5) and (6) with asynchronous switching and faults.
(a)(b)Figure 8
The state trajectories of agents in HMASs (5) and (6) with faults and asynchronous switching. (a) Position. (b) Velocity.
(a)(b)


## 6. Conclusions

In this article, based on cooperative-competitive relation, couple-group consensus of a class of continuous HMASs with asynchronous switching and faults is studied. Furthermore, leader-following group consensus of this system is also investigated. By using the linear matrix inequality technique, graph, and Lyapunov stability theory, some sufficient conditions have been achieved. Some examples have been given to address the validity of the obtained results. It is worth pointing out that our results can also be applied to HMASs with weakly connected topology, and it breaks the restriction of in-degree balance and spanning tree. In our further work, we will discuss the couple-group consensus of HMASs with stochastic switching or uncertain disturbances.



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