# Consensus of Second-Order Heterogeneous Hybrid Multiagent Systems via Event-Triggered Protocols

Authors: Hong Zhang; Yanhan Li; Ying Zheng
Journal: Complexity
 (2022)

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## Abstract

This paper investigates the event-based consensus problem for the heterogeneous hybrid multiagent system (MAS). First, the heterogeneous hybrid MAS is proposed which contains continuous and discrete-time subsystems with second-order and first-order heterogeneous dynamics. Second, the event-triggered protocols are proposed, which mainly include the event-based control laws and event-triggered conditions for different kinds of agents. Then, the consensus conclusions of fixed topology and switching topologies are obtained based on graph theory and nonnegative matrix theory, which include the constraints on control parameters, coupling gains, and sampling interval to guarantee consensus. Finally, a simulation example is given to verify the efficiency of the proposed protocols.


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## Body

## 1. Introduction

With the popularity of distributed artificial intelligence, multiagent system (MAS) has been widely researched and applied to engineering, military, and other fields. It can accomplish huge and complex tasks in the real world through the mutual communication and coordination among individuals. It can also explain some complex phenomena in nature and human society, such as fish schools, bird flocks [1, 2], and the dynamics of opinion forming in human society [3]. At present, the research on multiagent system is mainly about consensus [4–8], flocking [9–11], formation [12–15], and so on.Consensus, as one of the most fundamental cooperative behaviors of MASs, has attracted extensive interest. It means that the agents can reach the same states from any initial states by a suitable consensus algorithm or control law. Up to now, researchers have proposed many consensus algorithms for MASs through different analysis methods, such as the analysis based on nonnegative matrix theory [16–19], Lyapunov function analysis [20, 21], and frequency domain analysis [22–24]. In 2006, Xiao et al. studied the consensus problem for discrete-time first-order MASs with fixed topology and considered the structural decomposition of the leader-follower model [25]. Then, Xie et al. and Ren et al. gave some sufficient conditions for solving the consensus problem for second-order MASs with fixed and switching topologies [26, 27]. Shi et al. further considered the weighted-average consensus problem for second-order MASs and obtained necessary and sufficient conditions [28]. In recent years, the multiagent networks studied have become more and more complicated with the wide application of MASs. The heterogeneous MASs composed of agents with different dynamics are more suitable for real systems. Taking the multirobots systems into account, heterogeneous systems composed of robots with different perceptual capabilities can complete tasks faster [29]. A number of results about the consensus of heterogeneous MASs have been obtained, including low-order linear systems [30–32] and high-order linear systems [33–35]. In addition, the hybrid MASs including continuous and discrete-time subsystems have also attracted attention. Examples include the refrigeration and heating system. The heat-loss dynamics and the control of air conditioners belong to continuous-time systems, whereas the thermostat is controlled by a discrete-time system [36]. Since 2018, Zheng et al. have studied the consensus problems of first-order and second-order hybrid systems and have proposed a game-theoretic approach to analyze the hybrid systems [37–39]. Su et al. designed an event-triggered consensus strategy for the second-order hybrid system [40] in 2019. Other researchers obtained more results about the hybrid MAS [41, 42].In addition, interest in the event-based consensus problem for MASs has grown in the past decade. Compared with traditional methods, the event-triggered method has certain advantages in studying practical problems, such as minimizing the number of control actions and saving energy by updating the controller only at the trigger time and avoiding continuous communications. However, it also brings new theoretical and practical problems. The main task is the design of distributed event-triggered protocols, including event-triggered control laws and trigger conditions. In 2012, Dimarogonas et al. proposed effective event-triggered consensus algorithms for the first-order agents under undirected connected communication topologies [43]. Then, Fan et al. designed a fully distributed event-triggered strategy for solving the consensus problem of general linear MASs [44]. In recent years, several event-triggered consensus problems based on state feedback, output feedback, and leader-follower models were considered in [45–50].In this paper, we consider the heterogeneous multiagent systems with continuous-time and discrete-time individuals (the heterogeneous hybrid MASs). For example, in the complex system of nature and human interaction, the biological signals are continuous signals, whereas the automated instruments with different functions are mostly discrete-time systems. Compared with the MASs in [42], the consensus problem for the heterogeneous hybrid MASs with a more general form is investigated, and the event-triggered protocols are proposed. First-order and second-order dynamic agents coexist in a system. Some of them belong to the continuous-time system, whereas the other agents are controlled by the discrete-time system. The main contributions of this paper are as follows. First, for the different dynamic characteristics of first-order and second-order agents, two kinds of event-triggered control laws are proposed. The event-triggered conditions are designed, which contain the position of all agents and the velocity of only second-order agents. Second, the sampled-data approach is used to solve the consensus problem for the heterogeneous hybrid MASs. On the one hand, the continuous-time subsystem and discrete-time subsystem can be better analyzed by the overall analysis method. On the other hand, this approach shows that the trigger time interval exists in a lower bound. Hence, the Zeno behavior is avoided. Finally, under the assumption that the fixed topology or the union of switching topologies contains a spanning tree, several results for solving the consensus problem are obtained by using graph theory and nonnegative matrix theory. Some selection conditions of control parameters, as well as the constraints of coupling gains and sampling interval, are given to guarantee consensus.Throughout this paper, assume0<M<N, IM=1,2,…,M and IN/IM=M+1,M+2,…,N. N+ represents the set of positive integers. Consider a vector or a matrix A, A∈Rn means A is a real column vector of length N. Similarly, A is an n×p-dimensional real matrix, which is defined by A∈Rn×p. The symbol AT and A represent the transpose and Euclidean norm of A, respectively. The calculational symbols ⊗ represent the Kronecker product of matrices.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

The communication topology is described by a weighted directed graphGV,E,A, where V=v1,…,vN represents the nodes set and E⊆V×V represents the set of edges between nodes. A=aij∈RN×N denotes the related adjacency matrix. If there is a directed path ϵji∈E, indicating that the j th agent can transmit data to the i th agent, then aij>0; otherwise, aij=0. A diagonal matrix D=diagd1,d2,…,dN is defined as the degree matrix of the directed graph G, where di=∑j=1Naij. The Laplacian matrix is expressed as ℒ=D−A. For a weighted directed graph G, it is said to contain a directed spanning tree, indicating that there is a node, which has directed paths that can lead to all other nodes.A nonnegative matrixC∈Rn×r is also called a row stochastic matrix if the sum of its each row is equal to 1. Furthermore, if it also satisfies limk⟶∞Ck=1n×vT, where v∈Rn, then it is indecomposable and aperiodic (SIA).


### 2.2. Problem Formulation

Considering the position and velocity states of the second-order agents and the position states of the first-order agents, the dynamic models are proposed, respectively. Suppose the number of agents in the entire system isN, where the first MM<N agents are second-order agents, the remaining N−M agents are first-order. The second-order dynamic agents are expressed as follows:(1)x˙ict=vict,i∈IM,v˙ict=uict,i∈IM,xidkh+h=xidkh+hvidkh+h22uidkh,i∈IM,vidkh+h=vidkh+huidkh,i∈IM,where xi∈R, vi∈R, and ui∈R represent the position, velocity, and control input of the second-order agent i, respectively. The superscripts c and d denote that the agent belongs to continuous and discrete-time subsystems. h>0 is the sampling interval of the discrete-time subsystem. The dynamic model of first-order agents is given by(2)x˙lct=ulctl∈INIM,xldkh+h=xldkh+huldkh,l∈INIM.Definition 1.
The heterogeneous hybrid MAS is said to reach consensus if the position of all agents satisfies the following conditions from any initial state.(3)limt⟶∞xict=limkh⟶∞xidkh=C,i∈IN,where C is a constant. To ensure that all agents can keep the same position state, (3) also indicates that the velocity of the second-order agents will tend to zero when t⟶∞ or kh⟶∞.
In order to propose the event-triggered protocols to solve this consensus problem, we first design the following event-triggered control laws for second-order agents:(4)uict=−βvictξi+α∑j=1Naijxjc,dtξi−xictξi,i∈IN,t∈tξi,tξ+1i,uidkh=−βvidkξih+α∑j=1Naijxjc,dkξih−xidkξih,i∈IN,kh∈kξih,kξ+1ih,where α,β>0 are control parameters, and t0i,…,tξi,⋯ and k0ih,…,kξih,⋯ are the trigger time series. The event-triggered control laws of the first-order agents are designed as follows:(5)uict=α∑j=1Naijxjc,dtξi−xictξi,i∈INIM,t∈tξi,tξ+1i,uidkh=α∑j=1Naijxjc,dkξih−xidkξih,i∈INIM,kh∈kξih,kξ+1ih.
Before proposing the event-triggered conditions, the following definitions of the combined measurement and combined measurement error are given. The combined measurement is defined as(6)qict=∑j=1Naijxjc,dt−xict,i∈IN,qidkh=∑j=1Naijxjc,dkh−xidkh,i∈IN.
The combined measurement error is defined as(7)exict=qictξi−qict,i∈IN,exidkh=qidkξih−qidkh,i∈IN,evict=victξi−vict,i∈IM,evidkh=vidkξih−vidkh,i∈IM.
In this paper, the event-triggered conditions are presented as follows:(8)tξ+1i=tξi+minlih,i∈IN,li∈N+,kξ+1ih=kξih+minlih,i∈IN,li∈N+,where minlih is the time interval between two adjacent trigger times, which lih satisfies the following inequalities for the continuous and discrete-time agents:(9)exictξi+lih≥γ1qictξi+γ2δtξi+lih,i∈IN,evictξi+lih≥γ1zictξi+γ2δtξi+lih,i∈IM,exidkξih+lih≥γ1qidkξih+γ2δkξih+lih,i∈IN,evidkξih+lih≥γ1zidkξih+γ2δkξih+lih,i∈IM,where γ1,γ2>0 are the threshold parameters, δ>0 is the exponential component. And zi is a new definition of hybrid state quantity for the second-order agent, which is defined as(10)zi=xi+hvi,i=1,2,…,M
According to the event-triggered conditions proposed above, the Zeno behavior is avoided because there is a lower boundh>0 for the event-trigger interval lih, which ensures that the agent does not trigger infinitely for a limited time. Then, based on the definitions of (6) and (7), the event-triggered control laws (4) and (5) can be written as follows:(11)uict=−βvict+evict+αqict+exict,i∈IM,t∈tξi,tξ+1i,uidkh=−βvidkh+evidkh+αqidkh+exidkh,i∈IM,kh∈kξih,kξ+1ih,ulct=αqlct+exlct,l∈IN/IM,t∈tξl,tξ+1l,uldkh=αqldkh+exldkh,l∈IN/IM,kh∈kξlh,kξ+1lh.
Each controller updates only at its own trigger time, so the energy can be saved. For anyt∈tξi,tξ+1i, uit=uitξi does not change, and the continuous-time dynamics in (1) and (2) can be described as(12)xict=xictξi+∫tξitvicτdτ=xictξi+t−tξivictξi+t−tξi22uictξi,i∈IM,vict=victξi+∫tξituicτdτ=victξi+t−tξiuictξi,i∈IM,xlct=xlctξl+∫tξltulcτdτ=xlctξl+t−tξlulctξl,l∈IN/IM.
To analyze the entire MAS with continuous and discrete-time subsystems by the overall analysis method, the sampled-data method is applied in this paper. Considering the continuous subsystem in the discrete-time scale, a new discrete-time scaleκh=li′h is defined to describe the entire MASs. κξih represents the trigger time tξi and kξih. The unified form of the entire MASs can be described as the following expression:(13)xic,dκh+h=xic,dκh+hvic,dκh+h22−βvic,dκξih+αqic,dκξih,i∈IM,vic,dκh+h=vic,dκh+h−βvic,dκξih+αqic,dκξih,i∈IM,xlc,dκh+h=xlc,dκh+hαqlc,dκξih,l∈IN/IM.
Substituting (7) into (13), we get the following expression:(14)xic,dκh+h=xic,dκh+hvic,dκh−h2β2vic,dκh+evic,dκh+h2α2qic,dκh+exic,dκh,i∈IM,vic,dκh+h=vic,dκh−hβvic,dκh+evic,dκh+hαqic,dκh+exic,dκh,i∈IM,xlc,dκh+h=xlc,dκh+hαqlc,dκh+exlc,dκh,l∈IN/IM.
In order to obtain the relationship between consensus achievement and system parameters, the following definitions are given:(15)xskh=x1Tkh,x2Tkh,…,xMTkhT,xfkh=xM+1Tkh,xM+2Tkh,…,xNTkhT,vskh=v1Tkh,v2Tkh,…,vMTkhT,zskh=z1Tkh,z2Tkh,…,zMTkhT,Qskh=q1Tkh,q2Tkh,…,qMTkhT,Qfkh=qM+1Tkh,qM+2Tkh,…,qNTkhT,Exskh=ex,1Tkh,ex,2Tkh,…,ex,MTkhT,Exfkh=ex,M+1Tkh,ex,M+2Tkh,…,ex,NTkhT,Evkh=ev,1Tkh,ev,2Tkh,…,ev,MTkhT.
Then, (14) can be written as(16)xsκh+h=hβ2xsκh+2−hβ2zsκh+h2α2Qsκh−h2β2Evκh+h2α2Exsκh,zsκh+h=3hβ−22xsκh+4−3hβ2zsκh+3h2α2Qsκh−3h2β2Evκh+3h2α2Exsκh,xfκh+h=xfκh+hαQfκh+hαExfκh.
Considering the communication between different dynamics, the Laplacian matrix representing the communication topology is divided into four parts.(17)ℒkh=ℒ1khℒ2khℒ3khℒ4kh,where ℒ1 represents the directed communication from second-order agents to second-order agents; ℒ2, ℒ3, and ℒ4 represent the directed communication from first-order agents to second-order agents, from first-order agents to first-order agents, and from second-order agents to first-order agents, respectively. (16) can be converted into the following expression:(18)xsκh+h=hβ2IM−h2α2ℒ1κhxsκh+2−hβ2zsκh−h2α2ℒ2κhxfκh+h2α2Exsκhh2β2Evκh,zsκh+h=3hβ−22IM−3h2α2ℒ1κhxsκh+4−3hβ2zsκh−3h2α2ℒ2κhxfκh+3h2α2Exsκh−3h2β2Evκh,xfκh+h=−hαℒ3κhxsκh+IN−M−hαℒ4κhxfκh+hαExfκh.
Then, two large matrices(19)Ykh=xsTkh,zsTkh,xfTkhT,Ekh=ExsTkh,EvTkh,ExfTkhT.are defined. We can get(20)Yκh+h=HκhYκh+PEκh,where(21)Hκh=hβ2IM−h2α2ℒ1κh2−hβ2IM−h2α2ℒ2κh3hβ−22IM−3h2α2ℒ1κh4−3hβ2IM−3h2α2ℒ2κh−hαℒ3κh0IN−M−hαℒ4κh⊗In,P=h2α2IM−h2β2IM03h2α2IM−3h2β2IM000hαIN−M⊗In.
Based on the analysis method in [40], the error system is further constructed. Define(22)ϕskh=x2Tkh−x1Tkh,…,xMTkh−x1TkhT,ψskh=z2Tkh−z1Tkh,…,zMTkh−z1TkhT,ϕfkh=xM+1Tkh−x1Tkh,…,xNTkh−x1TkhT,ℒ=l2,2−l1,2l2,3−l1,3⋯l2,N−l1,Nl3,2−l1,2l3,3−l1,3⋯l3,N−l1,N⋮⋮⋱⋮lN,2−l1,2lN,3−l1,3⋯lN,N−l1,N=ℒ¯1ℒ¯2ℒ¯3ℒ¯4,Y¯kh=ϕsTkhψsTkhϕfTkhT.
Then, (18) can be rewritten as(23)Y¯κh+h=H¯κhY¯κh+P¯Eκh,where(24)H¯κh=hβ2IM−h2α2ℒ¯1κh2−hβ2IM−h2α2ℒ¯2κh3hβ−22IM−3h2α2ℒ¯1κh4−3hβ2IM−3h2α2ℒ¯2κh−hαℒ¯3κh0IN−M−hαℒ¯4κh⊗In,P¯=h2α2IM−h2β2IM03h2α2IM−3h2β2IM000hαIN−M×−1M−1IM−100000−1M−1IM−10−1N−M000IN−M⊗In.


## 2.1. Preliminaries

The communication topology is described by a weighted directed graphGV,E,A, where V=v1,…,vN represents the nodes set and E⊆V×V represents the set of edges between nodes. A=aij∈RN×N denotes the related adjacency matrix. If there is a directed path ϵji∈E, indicating that the j th agent can transmit data to the i th agent, then aij>0; otherwise, aij=0. A diagonal matrix D=diagd1,d2,…,dN is defined as the degree matrix of the directed graph G, where di=∑j=1Naij. The Laplacian matrix is expressed as ℒ=D−A. For a weighted directed graph G, it is said to contain a directed spanning tree, indicating that there is a node, which has directed paths that can lead to all other nodes.A nonnegative matrixC∈Rn×r is also called a row stochastic matrix if the sum of its each row is equal to 1. Furthermore, if it also satisfies limk⟶∞Ck=1n×vT, where v∈Rn, then it is indecomposable and aperiodic (SIA).


## 2.2. Problem Formulation

Considering the position and velocity states of the second-order agents and the position states of the first-order agents, the dynamic models are proposed, respectively. Suppose the number of agents in the entire system isN, where the first MM<N agents are second-order agents, the remaining N−M agents are first-order. The second-order dynamic agents are expressed as follows:(1)x˙ict=vict,i∈IM,v˙ict=uict,i∈IM,xidkh+h=xidkh+hvidkh+h22uidkh,i∈IM,vidkh+h=vidkh+huidkh,i∈IM,where xi∈R, vi∈R, and ui∈R represent the position, velocity, and control input of the second-order agent i, respectively. The superscripts c and d denote that the agent belongs to continuous and discrete-time subsystems. h>0 is the sampling interval of the discrete-time subsystem. The dynamic model of first-order agents is given by(2)x˙lct=ulctl∈INIM,xldkh+h=xldkh+huldkh,l∈INIM.Definition 1.
The heterogeneous hybrid MAS is said to reach consensus if the position of all agents satisfies the following conditions from any initial state.(3)limt⟶∞xict=limkh⟶∞xidkh=C,i∈IN,where C is a constant. To ensure that all agents can keep the same position state, (3) also indicates that the velocity of the second-order agents will tend to zero when t⟶∞ or kh⟶∞.
In order to propose the event-triggered protocols to solve this consensus problem, we first design the following event-triggered control laws for second-order agents:(4)uict=−βvictξi+α∑j=1Naijxjc,dtξi−xictξi,i∈IN,t∈tξi,tξ+1i,uidkh=−βvidkξih+α∑j=1Naijxjc,dkξih−xidkξih,i∈IN,kh∈kξih,kξ+1ih,where α,β>0 are control parameters, and t0i,…,tξi,⋯ and k0ih,…,kξih,⋯ are the trigger time series. The event-triggered control laws of the first-order agents are designed as follows:(5)uict=α∑j=1Naijxjc,dtξi−xictξi,i∈INIM,t∈tξi,tξ+1i,uidkh=α∑j=1Naijxjc,dkξih−xidkξih,i∈INIM,kh∈kξih,kξ+1ih.
Before proposing the event-triggered conditions, the following definitions of the combined measurement and combined measurement error are given. The combined measurement is defined as(6)qict=∑j=1Naijxjc,dt−xict,i∈IN,qidkh=∑j=1Naijxjc,dkh−xidkh,i∈IN.
The combined measurement error is defined as(7)exict=qictξi−qict,i∈IN,exidkh=qidkξih−qidkh,i∈IN,evict=victξi−vict,i∈IM,evidkh=vidkξih−vidkh,i∈IM.
In this paper, the event-triggered conditions are presented as follows:(8)tξ+1i=tξi+minlih,i∈IN,li∈N+,kξ+1ih=kξih+minlih,i∈IN,li∈N+,where minlih is the time interval between two adjacent trigger times, which lih satisfies the following inequalities for the continuous and discrete-time agents:(9)exictξi+lih≥γ1qictξi+γ2δtξi+lih,i∈IN,evictξi+lih≥γ1zictξi+γ2δtξi+lih,i∈IM,exidkξih+lih≥γ1qidkξih+γ2δkξih+lih,i∈IN,evidkξih+lih≥γ1zidkξih+γ2δkξih+lih,i∈IM,where γ1,γ2>0 are the threshold parameters, δ>0 is the exponential component. And zi is a new definition of hybrid state quantity for the second-order agent, which is defined as(10)zi=xi+hvi,i=1,2,…,M
According to the event-triggered conditions proposed above, the Zeno behavior is avoided because there is a lower boundh>0 for the event-trigger interval lih, which ensures that the agent does not trigger infinitely for a limited time. Then, based on the definitions of (6) and (7), the event-triggered control laws (4) and (5) can be written as follows:(11)uict=−βvict+evict+αqict+exict,i∈IM,t∈tξi,tξ+1i,uidkh=−βvidkh+evidkh+αqidkh+exidkh,i∈IM,kh∈kξih,kξ+1ih,ulct=αqlct+exlct,l∈IN/IM,t∈tξl,tξ+1l,uldkh=αqldkh+exldkh,l∈IN/IM,kh∈kξlh,kξ+1lh.
Each controller updates only at its own trigger time, so the energy can be saved. For anyt∈tξi,tξ+1i, uit=uitξi does not change, and the continuous-time dynamics in (1) and (2) can be described as(12)xict=xictξi+∫tξitvicτdτ=xictξi+t−tξivictξi+t−tξi22uictξi,i∈IM,vict=victξi+∫tξituicτdτ=victξi+t−tξiuictξi,i∈IM,xlct=xlctξl+∫tξltulcτdτ=xlctξl+t−tξlulctξl,l∈IN/IM.
To analyze the entire MAS with continuous and discrete-time subsystems by the overall analysis method, the sampled-data method is applied in this paper. Considering the continuous subsystem in the discrete-time scale, a new discrete-time scaleκh=li′h is defined to describe the entire MASs. κξih represents the trigger time tξi and kξih. The unified form of the entire MASs can be described as the following expression:(13)xic,dκh+h=xic,dκh+hvic,dκh+h22−βvic,dκξih+αqic,dκξih,i∈IM,vic,dκh+h=vic,dκh+h−βvic,dκξih+αqic,dκξih,i∈IM,xlc,dκh+h=xlc,dκh+hαqlc,dκξih,l∈IN/IM.
Substituting (7) into (13), we get the following expression:(14)xic,dκh+h=xic,dκh+hvic,dκh−h2β2vic,dκh+evic,dκh+h2α2qic,dκh+exic,dκh,i∈IM,vic,dκh+h=vic,dκh−hβvic,dκh+evic,dκh+hαqic,dκh+exic,dκh,i∈IM,xlc,dκh+h=xlc,dκh+hαqlc,dκh+exlc,dκh,l∈IN/IM.
In order to obtain the relationship between consensus achievement and system parameters, the following definitions are given:(15)xskh=x1Tkh,x2Tkh,…,xMTkhT,xfkh=xM+1Tkh,xM+2Tkh,…,xNTkhT,vskh=v1Tkh,v2Tkh,…,vMTkhT,zskh=z1Tkh,z2Tkh,…,zMTkhT,Qskh=q1Tkh,q2Tkh,…,qMTkhT,Qfkh=qM+1Tkh,qM+2Tkh,…,qNTkhT,Exskh=ex,1Tkh,ex,2Tkh,…,ex,MTkhT,Exfkh=ex,M+1Tkh,ex,M+2Tkh,…,ex,NTkhT,Evkh=ev,1Tkh,ev,2Tkh,…,ev,MTkhT.
Then, (14) can be written as(16)xsκh+h=hβ2xsκh+2−hβ2zsκh+h2α2Qsκh−h2β2Evκh+h2α2Exsκh,zsκh+h=3hβ−22xsκh+4−3hβ2zsκh+3h2α2Qsκh−3h2β2Evκh+3h2α2Exsκh,xfκh+h=xfκh+hαQfκh+hαExfκh.
Considering the communication between different dynamics, the Laplacian matrix representing the communication topology is divided into four parts.(17)ℒkh=ℒ1khℒ2khℒ3khℒ4kh,where ℒ1 represents the directed communication from second-order agents to second-order agents; ℒ2, ℒ3, and ℒ4 represent the directed communication from first-order agents to second-order agents, from first-order agents to first-order agents, and from second-order agents to first-order agents, respectively. (16) can be converted into the following expression:(18)xsκh+h=hβ2IM−h2α2ℒ1κhxsκh+2−hβ2zsκh−h2α2ℒ2κhxfκh+h2α2Exsκhh2β2Evκh,zsκh+h=3hβ−22IM−3h2α2ℒ1κhxsκh+4−3hβ2zsκh−3h2α2ℒ2κhxfκh+3h2α2Exsκh−3h2β2Evκh,xfκh+h=−hαℒ3κhxsκh+IN−M−hαℒ4κhxfκh+hαExfκh.
Then, two large matrices(19)Ykh=xsTkh,zsTkh,xfTkhT,Ekh=ExsTkh,EvTkh,ExfTkhT.are defined. We can get(20)Yκh+h=HκhYκh+PEκh,where(21)Hκh=hβ2IM−h2α2ℒ1κh2−hβ2IM−h2α2ℒ2κh3hβ−22IM−3h2α2ℒ1κh4−3hβ2IM−3h2α2ℒ2κh−hαℒ3κh0IN−M−hαℒ4κh⊗In,P=h2α2IM−h2β2IM03h2α2IM−3h2β2IM000hαIN−M⊗In.
Based on the analysis method in [40], the error system is further constructed. Define(22)ϕskh=x2Tkh−x1Tkh,…,xMTkh−x1TkhT,ψskh=z2Tkh−z1Tkh,…,zMTkh−z1TkhT,ϕfkh=xM+1Tkh−x1Tkh,…,xNTkh−x1TkhT,ℒ=l2,2−l1,2l2,3−l1,3⋯l2,N−l1,Nl3,2−l1,2l3,3−l1,3⋯l3,N−l1,N⋮⋮⋱⋮lN,2−l1,2lN,3−l1,3⋯lN,N−l1,N=ℒ¯1ℒ¯2ℒ¯3ℒ¯4,Y¯kh=ϕsTkhψsTkhϕfTkhT.
Then, (18) can be rewritten as(23)Y¯κh+h=H¯κhY¯κh+P¯Eκh,where(24)H¯κh=hβ2IM−h2α2ℒ¯1κh2−hβ2IM−h2α2ℒ¯2κh3hβ−22IM−3h2α2ℒ¯1κh4−3hβ2IM−3h2α2ℒ¯2κh−hαℒ¯3κh0IN−M−hαℒ¯4κh⊗In,P¯=h2α2IM−h2β2IM03h2α2IM−3h2β2IM000hαIN−M×−1M−1IM−100000−1M−1IM−10−1N−M000IN−M⊗In.


## 3. Main Result

### 3.1. Fixed Communication Topology

Consider the consensus problem of fixed communication topology, which means the directed graphGV,E,A does not change over time and aijt=aijkh=aij.Lemma 1.
[8] If the sum of each row of nonnegative matrix H=hij∈RM+N×M+N is a positive constant μ>0, then μ is an eigenvalue of H corresponding to the eigenvector 1N+M. Furthermore, if the algebraic multiplicity of the eigenvalue μ of H is 1 and hii>0,i=1,2,…,M+N, then, for each eigenvalue λ≠μ, λ<μ is satisfied.Remark 1.
In practical applications, the movement of the agents should be multidimensional. This article only considers motion in a single dimension; in other words,In=1, because motion in a certain direction can be decomposed into motion in several independent directions. Therefore, the conclusion under a single dimension can also be extended to a multidimensional scale.Theorem 1.
The matrixH defined in (18) contains eigenvalue 1, and the remaining eigenvalues satisfy λ<1, and all the eigenvalues of H¯ defined in (20) satisfy λ¯<1, if and only if the fixed communication topology contains a directed spanning tree and coupling gains, and sampling interval and control parameters satisfy as follows:(25)43>hβ>23+h2αmaxi∈IM∑j=1Naij,1>hαmaxi∈IN/IM∑j=1Naij.Proof.
Under condition (21), H is a row stochastic matrix with positive diagonal elements, and λ=1 is one of the eigenvalues of H corresponding to eigenvector 1N+M. The column and row transformations of H−IN+M are performed as follows:(26)H−IN+M=hβ−22IM−h2α2ℒ12−hβ2IM−h2α2ℒ23hβ−22IM−3h2α2ℒ12−3hβ2IM−3h2α2ℒ2−hαℒ30−hαℒ4⟶ℒ10ℒ2IM−IM0ℒ30ℒ4⟶IM000ℒ1ℒ20ℒ3ℒ4.
We obtainrankH−IM+N=M+rankℒ. H has one eigenvalue λ=1 with algebraic multiplicity 1, if and only if rankL=N−1, which is equivalent to the topology containing a directed spanning tree [8]. Based on Lemma 1, we can obtain that all the other eigenvalues of H satisfy λ<1.
Next, it can be proved that all the nonzero eigenvalues ofℒ are also the eigenvalues of ℒ¯. By observing (18) and (20), the mapping of the eigenvalues of H and ℒ is the same as the mapping of the eigenvalues of H¯ and ℒ¯. Thus, the eigenvalues of H and H¯ are equal if ℒ and ℒ¯ have the same eigenvalues. Additionally, the eigenvalue of H corresponding to the zero eigenvalue of ℒ is one. Therefore, all eigenvalues of H are also eigenvalues of H¯ except for eigenvalue 1. In other words, all the eigenvalues of H¯ satisfy |λ¯|<1.Lemma 2.
[40] If a matrix H¯ satisfies that all its eigenvalues are inside the unit circle, then the following inequality is satisfied:(27)H¯k≤a·bk,where a and b are positive constants that satisfy a≥1 and 0<b<1.Theorem 2.
Consider the consensus of the heterogeneous hybrid MASs (1) and (2) with fixed communication topology under the event-triggered control laws (4) and (5) and event-triggered conditions (8), (9), and (10) with γ1∈0,1, γ2∈0,∞, and δ∈b,1. If the fixed communication topology G has a directed spanning tree and conditions (21) in Theorem 1 are satisfied, the heterogeneous hybrid MASs can reach the consensus condition (3).Proof.
Firstly, (20) is written in the following form by iteration.(28)Y¯κh=H¯κY¯0+P¯∑s=0κ−1H¯κ−1−sEsh.
By Theorem1 and Lemma 2, we have(29)Y¯κh≤abκ‖Y¯0+P¯∑s=0κ−1abκ−1−s‖Esh,where a≥1 and 0<b<1. According to the designed event-trigger conditions (9) and (10), Eκh can be expressed as(30)Eκh≤γ1QsκhQfκhzsκh+γ2N+Mδκ.
Two positive constantsC1 and C2 are defined, and then(31)QsκhQfκh≤ℒ·ϕsκhϕfκh=C1ϕsκhϕfκh,zsκh≤101×M−1−1M−1×1IM−1·z1κhψsκh≤101×M−1−1M−1×1IM−1zsκh+ψsκh.
Through (28), we further obtain(32)zsκh≤101×M−1−1M−1×1IM−11−101×M−1−1M−1×1IM−1·ψsκh=C2·ψsκh.
From (27) and (29), (30) can be converted as follows:(33)Eκh≤γ1CY¯κh+γ2N+Mδκ,where C is a bounded positive constant only related to communication topology. Letting ρ=P¯ and Z=maxρaγ2N+M/δ−b−ρaγ1C,aY¯0, ρ and Z are also bounded positive constants related to communication topology. Then, the following inequality will be proved by contradiction:(34)Y¯κh≤Zδκ.
Consider that there is a constantk∗>κ>0 that makes inequality (31) invalid. Thus,(35)Y¯k∗h>Zδk∗.
According to (25) and (30), one has(36)Zδk∗<Y¯k∗h≤abk∗Y¯0+ρ∑s=0k∗−1abk∗−1−sγ1CZ+γ2N+Mδs≤abk∗Y¯0+ρaγ1CZ+γ2N+Mbk∗−δk∗b−δ=aY¯0+ρaγ1CZ+γ2N+Mb−δbk∗+ρaγ1CZ+γ2N+Mδ−bδk∗.
Next, (31) can be proved in three cases.Case 1.
(37)Z=ρaγ2N+Mδ−b−ρaγ1C,which indicates that(38)ρaγ1CZ+γ2N+Mδ−b=Z(39)ρaγ1CZ+γ2N+Mδ−b>aY¯0.
According to (33), we have(40)Zδk∗<aY¯0+ρaγ1CZ+γ2N+Mb−δbk∗+ρaγ1CZ+γ2N+Mδ−bδk∗<ρaγ1CZ+γ2N+Mδ−bδk∗=Zδk∗.Case 2.
Z=aY¯0 and ρaγ2N+M/δ−b−ρaγ1C>0, which indicates δ>b. According to (33), we can obtain that(41)Zδk∗<aY¯0+ρaγ1CZ+γ2N+Mb−δδk∗+ρaγ1CZ+γ2N+Mδ−bδk∗=aY¯0δk∗=Zδk∗.Case 3.
Z=aY¯0 and ρaγ2N+M/δ−b−ρaγ1C>0, which indicates aY¯0<ρaγ1CZ+γ2N+M/δ−b. The proof is similar to case 1 that(42)Zδk∗<ρaγ1CZ+γ2N+Mδ−bδk∗<Zδk∗.
Combining the three cases, the inequality (31) holds, from which we can obtain that(43)limκh⟶∞ϕskhψskhϕfkh=0.
Then, we further consider the consensus of the continuous-time agents in real time. The following inequality is established:(44)vict−vjct≤vict−vickh+vickh−vjckh+vjckh−vjct,i∈IM,xict−xjct≤xict−xickh+xickh−xjckh+xickh−xjct,i∈IN.
Whenκh⟶∞, we can get vic,dkξih⟶0,qic,dkξih⟶0. Combined with (4), (5), (6), and (12), when t∈kh,kh+h, the following inequalities are obtained:(45)limt,kh⟶∞vict−vickh≤limt,kh⟶∞h−βvic,dkξih+αqic,dkξih,i∈IM,limt,kh⟶∞=xict−xickh≤limt,kh⟶∞hvic,dkξih+t−kξih2−kh−kξih22−βvic,dkξih+αqic,dkξih,i∈IM,limt,kh⟶∞θ≤limt,kh⟶∞hαqlc,dkξlh,l∈IN/IM.
Then,(46)limt,kh⟶∞xict−xickh=0,i∈IM,limt,kh⟶∞xict−xickh=0,l∈INIM.
From (44) and (46), continuous-time individuals can also achieve consensus on continuous-time scales. Combined with (43), the consensus conditions (3) are satisfied.


### 3.2. Switching Communication Topologies

According to the above conclusions with the fixed topology, the consensus of the heterogeneous hybrid MASs with switching communication topologies is considered. The system description (18) is rewritten as(47)Ykp+1h=Ωkpkph+QEkp,p=1,2,…,∞,where kp∈κ is the p th event-triggering instant of an agent from its initial state, Ωkp=∏s=kpkp+1−1Hsh is the matrix product of H, corresponding to all switching communication topologies during each interval kph,kp+1h, and(48)QEkp=PEkp+1−1h+P∑s=kpkp+1−2Esh·∏d=s+1kp+1−1Hdhis the accumulation of errors. Similarly, the error system description (20) can also be rewritten as(49)Y¯kp+1h=Ω¯kpY¯kph+Q¯Ekp,p=1,2,…,∞,where(50)Ω¯kp=∏s=kpkp+1−1H¯sh,(51)Q¯Ekp=P¯Ekp+1−1h+P¯∑s=kpkp+1−2Esh·∏d=s+1kp+1−1H¯dh.Remark 2.
To simplify subsequent proofs, it is assumed thatkp+1−kp>1 in the subsection on switching communication topologies, which means there are at least two different communication topologies in the period kp,kp+1. Otherwise, it can be regarded as a fixed communication topology in a short period of time, and the convergence will not be destroyed according to the existing conclusions.Theorem 3.
The matrix productΩkp is SIA, and all the eigenvalues of Ω¯kp satisfy λΩ<1 if the union of communication topologies Gkph,Gkp+1h,…,Gkp+1−1h of each interval kph,kp+1h contains a spanning tree and the coupling gains, and sampling interval and control parameters satisfy conditions (21) in Theorem 1.Proof.
Defineℒ˜=∑s=kpkp+1−1ℒsh/kp+1−kp as the Laplacian matrix of the union of directed graphs during time interval kph,kp+1h, and ℒ˜¯ is the matrix obtained by ℒ˜ through the same transformation as ℒ¯ in (19).(52)H˜=∑s=kpkp+1−1Hshkp+1−kpstill satisfies the sum of each row is 1. Then, we can get rankH˜−IN+M=M+rankℒ˜ by taking the elements column and row transforms as follows:(53)H˜−IN+M=hβ−22IM−h2α2ℒ1˜2−hβ2IM−h2α2ℒ2˜3hβ−22IM−3h2α2ℒ1˜2−3hβ2IM−3h2α2ℒ2˜−hαℒ3˜0−hαℒ4˜⟶IM00ℒ˜.
Thus,H˜ has eigenvalue λ˜=1 with algebraic multiplicity 1, if and only if rankℒ˜=N−1, which means that the union of topologies contains a spanning tree. According to Lemma 3.1 in [18], we can get that(54)∏s=kpkp+1−1Hsh>∑s=kpkp+1−1Hsh/kp+1−kp,which indicates that the graph of the matrix product Ωkp also contains a spanning tree. Besides, Ωkp is a stochastic matrix with positive diagonal elements if conditions (21) are satisfied, because the matrix multiplication among stochastic matrixes with positive diagonal elements is closed. In other words, Ωkp is SIA.
Similar to the proof of Theorem1, for the error system, it can be proved that all eigenvalues of Ωkp except eigenvalue 1 are also eigenvalues of Ω¯kp.Theorem 4.
Consider the consensus of the heterogeneous hybrid MASs (1) and (2) with switching communication topologies under the event-triggered control laws (4) and (5) and event-triggered conditions (8), (9), and (10) with γ1∈0,1, γ2∈0,∞, and δ∈b,1. If the union of switching topologies Gkph,Gkp+1h,…,Gkp+1−1h of each interval kph,kp+1h has a directed spanning tree and conditions (21) in Theorem 1 are satisfied, the heterogeneous hybrid MASs can reach the consensus condition (3).Proof.
Through iteration, the error system description (39) with switching communication topologies can be expressed as(55)Y¯kph=Ω¯k0Ω¯k1⋯Ω¯kp−1Y¯0+Ω¯k1Ωk2⋯Ω¯kp−1Q¯Ek0+⋯+Ω¯kp−1Q¯Ekp−2+Q¯Ekp−1.
Let(56)Ω∗=maxΩ¯k0,Ω¯k1,…,Ω¯kp−1.
Combined with Lemma2, we have(57)Y¯kph≤Ω∗pY¯0+∑s=0p−1Ω∗p−1−sQ¯Eks≤a′BpY¯0+∑s=0p−1a′Bp−1−sQ¯Eks,where 0<B<1 and a′>1 are positive constants. Next, the norm of Q¯Eks is considered as follows:(58)Q¯Eks≤ρH¯ks+1hH¯ks+2h⋯H¯ks+1−1hEksh+⋯+ρH¯ks+1−1hH¯ks+1−2h+ρH¯ks+1−1h<ρηs∑τ=ksks+1−1Eτh,where(59)ηs=maxH¯ks+1h⋯Hks+1h,…,H¯ks+1h,1,is a positive constant. In addition, there are bp and bp′ that satisfy bpkp+1−kp=B and bp′kp+1−kp−1=B for each interval kph,kp+1h. Letting η=maxηs,b′=maxbp,bp′, we can get b′<1 and (42) can be converted into(60)BY¯kph<a′b′kpBY¯0+a′ρηb′kp−1Ek0+Ek0+1+⋯+Ek1−1+⋯+a′ρηb′k0−1Ekp−1+Ekp−1+1+⋯+Ekp−1<a′b′kpBY¯0+a′ρη∑s=0kp−1b′kp−1−sEsh.
Thus, let(61)ρ′=a′pηB,(62)Y¯kph<a′b′kpY¯0+ρ′∑s=0kp−1b′kp−1−sEsh,which has a similar form to (25). The subsequent proof of this theorem is the same as Theorem 2.Remark 3.
The parameters of switching communication topologies, such as the switching rate and the dwell time, have certain impacts on the convergence rate. It mainly depends on the structure of each switching communication topology. In a period of time, more different agents communicating can improve the convergence efficiency. If there are different edges in the switching topologies, increasing the switching rate can reduce the time for the system to reach consensus.


## 3.1. Fixed Communication Topology

Consider the consensus problem of fixed communication topology, which means the directed graphGV,E,A does not change over time and aijt=aijkh=aij.Lemma 1.
[8] If the sum of each row of nonnegative matrix H=hij∈RM+N×M+N is a positive constant μ>0, then μ is an eigenvalue of H corresponding to the eigenvector 1N+M. Furthermore, if the algebraic multiplicity of the eigenvalue μ of H is 1 and hii>0,i=1,2,…,M+N, then, for each eigenvalue λ≠μ, λ<μ is satisfied.Remark 1.
In practical applications, the movement of the agents should be multidimensional. This article only considers motion in a single dimension; in other words,In=1, because motion in a certain direction can be decomposed into motion in several independent directions. Therefore, the conclusion under a single dimension can also be extended to a multidimensional scale.Theorem 1.
The matrixH defined in (18) contains eigenvalue 1, and the remaining eigenvalues satisfy λ<1, and all the eigenvalues of H¯ defined in (20) satisfy λ¯<1, if and only if the fixed communication topology contains a directed spanning tree and coupling gains, and sampling interval and control parameters satisfy as follows:(25)43>hβ>23+h2αmaxi∈IM∑j=1Naij,1>hαmaxi∈IN/IM∑j=1Naij.Proof.
Under condition (21), H is a row stochastic matrix with positive diagonal elements, and λ=1 is one of the eigenvalues of H corresponding to eigenvector 1N+M. The column and row transformations of H−IN+M are performed as follows:(26)H−IN+M=hβ−22IM−h2α2ℒ12−hβ2IM−h2α2ℒ23hβ−22IM−3h2α2ℒ12−3hβ2IM−3h2α2ℒ2−hαℒ30−hαℒ4⟶ℒ10ℒ2IM−IM0ℒ30ℒ4⟶IM000ℒ1ℒ20ℒ3ℒ4.
We obtainrankH−IM+N=M+rankℒ. H has one eigenvalue λ=1 with algebraic multiplicity 1, if and only if rankL=N−1, which is equivalent to the topology containing a directed spanning tree [8]. Based on Lemma 1, we can obtain that all the other eigenvalues of H satisfy λ<1.
Next, it can be proved that all the nonzero eigenvalues ofℒ are also the eigenvalues of ℒ¯. By observing (18) and (20), the mapping of the eigenvalues of H and ℒ is the same as the mapping of the eigenvalues of H¯ and ℒ¯. Thus, the eigenvalues of H and H¯ are equal if ℒ and ℒ¯ have the same eigenvalues. Additionally, the eigenvalue of H corresponding to the zero eigenvalue of ℒ is one. Therefore, all eigenvalues of H are also eigenvalues of H¯ except for eigenvalue 1. In other words, all the eigenvalues of H¯ satisfy |λ¯|<1.Lemma 2.
[40] If a matrix H¯ satisfies that all its eigenvalues are inside the unit circle, then the following inequality is satisfied:(27)H¯k≤a·bk,where a and b are positive constants that satisfy a≥1 and 0<b<1.Theorem 2.
Consider the consensus of the heterogeneous hybrid MASs (1) and (2) with fixed communication topology under the event-triggered control laws (4) and (5) and event-triggered conditions (8), (9), and (10) with γ1∈0,1, γ2∈0,∞, and δ∈b,1. If the fixed communication topology G has a directed spanning tree and conditions (21) in Theorem 1 are satisfied, the heterogeneous hybrid MASs can reach the consensus condition (3).Proof.
Firstly, (20) is written in the following form by iteration.(28)Y¯κh=H¯κY¯0+P¯∑s=0κ−1H¯κ−1−sEsh.
By Theorem1 and Lemma 2, we have(29)Y¯κh≤abκ‖Y¯0+P¯∑s=0κ−1abκ−1−s‖Esh,where a≥1 and 0<b<1. According to the designed event-trigger conditions (9) and (10), Eκh can be expressed as(30)Eκh≤γ1QsκhQfκhzsκh+γ2N+Mδκ.
Two positive constantsC1 and C2 are defined, and then(31)QsκhQfκh≤ℒ·ϕsκhϕfκh=C1ϕsκhϕfκh,zsκh≤101×M−1−1M−1×1IM−1·z1κhψsκh≤101×M−1−1M−1×1IM−1zsκh+ψsκh.
Through (28), we further obtain(32)zsκh≤101×M−1−1M−1×1IM−11−101×M−1−1M−1×1IM−1·ψsκh=C2·ψsκh.
From (27) and (29), (30) can be converted as follows:(33)Eκh≤γ1CY¯κh+γ2N+Mδκ,where C is a bounded positive constant only related to communication topology. Letting ρ=P¯ and Z=maxρaγ2N+M/δ−b−ρaγ1C,aY¯0, ρ and Z are also bounded positive constants related to communication topology. Then, the following inequality will be proved by contradiction:(34)Y¯κh≤Zδκ.
Consider that there is a constantk∗>κ>0 that makes inequality (31) invalid. Thus,(35)Y¯k∗h>Zδk∗.
According to (25) and (30), one has(36)Zδk∗<Y¯k∗h≤abk∗Y¯0+ρ∑s=0k∗−1abk∗−1−sγ1CZ+γ2N+Mδs≤abk∗Y¯0+ρaγ1CZ+γ2N+Mbk∗−δk∗b−δ=aY¯0+ρaγ1CZ+γ2N+Mb−δbk∗+ρaγ1CZ+γ2N+Mδ−bδk∗.
Next, (31) can be proved in three cases.Case 1.
(37)Z=ρaγ2N+Mδ−b−ρaγ1C,which indicates that(38)ρaγ1CZ+γ2N+Mδ−b=Z(39)ρaγ1CZ+γ2N+Mδ−b>aY¯0.
According to (33), we have(40)Zδk∗<aY¯0+ρaγ1CZ+γ2N+Mb−δbk∗+ρaγ1CZ+γ2N+Mδ−bδk∗<ρaγ1CZ+γ2N+Mδ−bδk∗=Zδk∗.Case 2.
Z=aY¯0 and ρaγ2N+M/δ−b−ρaγ1C>0, which indicates δ>b. According to (33), we can obtain that(41)Zδk∗<aY¯0+ρaγ1CZ+γ2N+Mb−δδk∗+ρaγ1CZ+γ2N+Mδ−bδk∗=aY¯0δk∗=Zδk∗.Case 3.
Z=aY¯0 and ρaγ2N+M/δ−b−ρaγ1C>0, which indicates aY¯0<ρaγ1CZ+γ2N+M/δ−b. The proof is similar to case 1 that(42)Zδk∗<ρaγ1CZ+γ2N+Mδ−bδk∗<Zδk∗.
Combining the three cases, the inequality (31) holds, from which we can obtain that(43)limκh⟶∞ϕskhψskhϕfkh=0.
Then, we further consider the consensus of the continuous-time agents in real time. The following inequality is established:(44)vict−vjct≤vict−vickh+vickh−vjckh+vjckh−vjct,i∈IM,xict−xjct≤xict−xickh+xickh−xjckh+xickh−xjct,i∈IN.
Whenκh⟶∞, we can get vic,dkξih⟶0,qic,dkξih⟶0. Combined with (4), (5), (6), and (12), when t∈kh,kh+h, the following inequalities are obtained:(45)limt,kh⟶∞vict−vickh≤limt,kh⟶∞h−βvic,dkξih+αqic,dkξih,i∈IM,limt,kh⟶∞=xict−xickh≤limt,kh⟶∞hvic,dkξih+t−kξih2−kh−kξih22−βvic,dkξih+αqic,dkξih,i∈IM,limt,kh⟶∞θ≤limt,kh⟶∞hαqlc,dkξlh,l∈IN/IM.
Then,(46)limt,kh⟶∞xict−xickh=0,i∈IM,limt,kh⟶∞xict−xickh=0,l∈INIM.
From (44) and (46), continuous-time individuals can also achieve consensus on continuous-time scales. Combined with (43), the consensus conditions (3) are satisfied.


## 3.2. Switching Communication Topologies

According to the above conclusions with the fixed topology, the consensus of the heterogeneous hybrid MASs with switching communication topologies is considered. The system description (18) is rewritten as(47)Ykp+1h=Ωkpkph+QEkp,p=1,2,…,∞,where kp∈κ is the p th event-triggering instant of an agent from its initial state, Ωkp=∏s=kpkp+1−1Hsh is the matrix product of H, corresponding to all switching communication topologies during each interval kph,kp+1h, and(48)QEkp=PEkp+1−1h+P∑s=kpkp+1−2Esh·∏d=s+1kp+1−1Hdhis the accumulation of errors. Similarly, the error system description (20) can also be rewritten as(49)Y¯kp+1h=Ω¯kpY¯kph+Q¯Ekp,p=1,2,…,∞,where(50)Ω¯kp=∏s=kpkp+1−1H¯sh,(51)Q¯Ekp=P¯Ekp+1−1h+P¯∑s=kpkp+1−2Esh·∏d=s+1kp+1−1H¯dh.Remark 2.
To simplify subsequent proofs, it is assumed thatkp+1−kp>1 in the subsection on switching communication topologies, which means there are at least two different communication topologies in the period kp,kp+1. Otherwise, it can be regarded as a fixed communication topology in a short period of time, and the convergence will not be destroyed according to the existing conclusions.Theorem 3.
The matrix productΩkp is SIA, and all the eigenvalues of Ω¯kp satisfy λΩ<1 if the union of communication topologies Gkph,Gkp+1h,…,Gkp+1−1h of each interval kph,kp+1h contains a spanning tree and the coupling gains, and sampling interval and control parameters satisfy conditions (21) in Theorem 1.Proof.
Defineℒ˜=∑s=kpkp+1−1ℒsh/kp+1−kp as the Laplacian matrix of the union of directed graphs during time interval kph,kp+1h, and ℒ˜¯ is the matrix obtained by ℒ˜ through the same transformation as ℒ¯ in (19).(52)H˜=∑s=kpkp+1−1Hshkp+1−kpstill satisfies the sum of each row is 1. Then, we can get rankH˜−IN+M=M+rankℒ˜ by taking the elements column and row transforms as follows:(53)H˜−IN+M=hβ−22IM−h2α2ℒ1˜2−hβ2IM−h2α2ℒ2˜3hβ−22IM−3h2α2ℒ1˜2−3hβ2IM−3h2α2ℒ2˜−hαℒ3˜0−hαℒ4˜⟶IM00ℒ˜.
Thus,H˜ has eigenvalue λ˜=1 with algebraic multiplicity 1, if and only if rankℒ˜=N−1, which means that the union of topologies contains a spanning tree. According to Lemma 3.1 in [18], we can get that(54)∏s=kpkp+1−1Hsh>∑s=kpkp+1−1Hsh/kp+1−kp,which indicates that the graph of the matrix product Ωkp also contains a spanning tree. Besides, Ωkp is a stochastic matrix with positive diagonal elements if conditions (21) are satisfied, because the matrix multiplication among stochastic matrixes with positive diagonal elements is closed. In other words, Ωkp is SIA.
Similar to the proof of Theorem1, for the error system, it can be proved that all eigenvalues of Ωkp except eigenvalue 1 are also eigenvalues of Ω¯kp.Theorem 4.
Consider the consensus of the heterogeneous hybrid MASs (1) and (2) with switching communication topologies under the event-triggered control laws (4) and (5) and event-triggered conditions (8), (9), and (10) with γ1∈0,1, γ2∈0,∞, and δ∈b,1. If the union of switching topologies Gkph,Gkp+1h,…,Gkp+1−1h of each interval kph,kp+1h has a directed spanning tree and conditions (21) in Theorem 1 are satisfied, the heterogeneous hybrid MASs can reach the consensus condition (3).Proof.
Through iteration, the error system description (39) with switching communication topologies can be expressed as(55)Y¯kph=Ω¯k0Ω¯k1⋯Ω¯kp−1Y¯0+Ω¯k1Ωk2⋯Ω¯kp−1Q¯Ek0+⋯+Ω¯kp−1Q¯Ekp−2+Q¯Ekp−1.
Let(56)Ω∗=maxΩ¯k0,Ω¯k1,…,Ω¯kp−1.
Combined with Lemma2, we have(57)Y¯kph≤Ω∗pY¯0+∑s=0p−1Ω∗p−1−sQ¯Eks≤a′BpY¯0+∑s=0p−1a′Bp−1−sQ¯Eks,where 0<B<1 and a′>1 are positive constants. Next, the norm of Q¯Eks is considered as follows:(58)Q¯Eks≤ρH¯ks+1hH¯ks+2h⋯H¯ks+1−1hEksh+⋯+ρH¯ks+1−1hH¯ks+1−2h+ρH¯ks+1−1h<ρηs∑τ=ksks+1−1Eτh,where(59)ηs=maxH¯ks+1h⋯Hks+1h,…,H¯ks+1h,1,is a positive constant. In addition, there are bp and bp′ that satisfy bpkp+1−kp=B and bp′kp+1−kp−1=B for each interval kph,kp+1h. Letting η=maxηs,b′=maxbp,bp′, we can get b′<1 and (42) can be converted into(60)BY¯kph<a′b′kpBY¯0+a′ρηb′kp−1Ek0+Ek0+1+⋯+Ek1−1+⋯+a′ρηb′k0−1Ekp−1+Ekp−1+1+⋯+Ekp−1<a′b′kpBY¯0+a′ρη∑s=0kp−1b′kp−1−sEsh.
Thus, let(61)ρ′=a′pηB,(62)Y¯kph<a′b′kpY¯0+ρ′∑s=0kp−1b′kp−1−sEsh,which has a similar form to (25). The subsequent proof of this theorem is the same as Theorem 2.Remark 3.
The parameters of switching communication topologies, such as the switching rate and the dwell time, have certain impacts on the convergence rate. It mainly depends on the structure of each switching communication topology. In a period of time, more different agents communicating can improve the convergence efficiency. If there are different edges in the switching topologies, increasing the switching rate can reduce the time for the system to reach consensus.


## 4. Simulation Examples

A heterogeneous hybrid MAS is assumed to consist of four second-order (SO) agents and four first-order (FO) agents. They both contain two continuous-time (CT) individuals and two discrete-time (DT) individuals, respectively. Letxs0,xf0T=7,5,3,1,−1,−3,−5,−7T and v0T=4,3,2,−1T.Consider the consensus of heterogeneous hybrid MASs with switching topologies. The communication topology can be switched between topology 1 and topology 2 in Figure1 every step. The union of topology 1 and topology 2 has a spanning tree. Suppose the coupling gains of each edge in topology 1 and topology 2 are 1 and α=1.6, β=1.2. By calculating, we choose h=0.6<0.625 to satisfy conditions (21) in Theorem 1. Then, choosing γ1=0.2 and γ2=0.01, we can obtain the simulation graphics as follows. The position and velocity are shown in Figures 2 and 3. The control inputs are shown in Figure 4. And the triggered instants of each agent are shown in Figure 5.Figure 1
Switching topologies.Figure 2
Position of the dynamic agents.Figure 3
Velocity of the second-order dynamic agents.Figure 4
Control inputs of the dynamic agents.Figure 5
Event-triggered instants of the agents.All agents can achieve consensus, and the controllers are triggered a limited number of times within a finite time, which indicates that the event-triggered protocols algorithm is effective.


## 5. Conclusion

In this paper, the event-triggered consensus was studied for the heterogeneous hybrid MASs, consisting of continuous and discrete-time subsystems with second-order and first-order heterogeneous dynamics. We designed the effective event-triggered protocols, including the event-triggered control laws for the first-order and second-order agents, respectively, and the event-triggered conditions, which can make the controllers only update at their own trigger time and ensure all agents meet consensus. Some criteria were obtained for solving the consensus problems of the heterogeneous hybrid MASs with fixed topology and switching topologies. The main results showed that the MASs can reach consensus if the control parameter, coupling gains, and sampling interval meet certain conditions, and the fixed topology or the union of switching topologies contain a directed spanning tree. Future work may consider the consensus of the heterogeneous hybrid MASs with time delay or communication noise.



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*Source: 2665925-2022-04-30.xml*