Consensus Control Design for Heterogeneous Multi–Agent Systems in Vehicle Platooning Using an Event–Triggering Scheme

1  Introduction

Consensus control has been widely investigated in application domains such as distributed sensor networks [1], autonomous underwater vehicles [2], and cooperative unmanned ground vehicles [3]. In the power-system domain, coordinated protection and control strategies have also been studied to improve grid resilience under the high penetration of renewable energy resources [4]. The consensus problem, a key area of MASs research, seeks to create distributed protocols that, in spite of poor communication and dynamic uncertainty, motivate all agents to agree on specific quantities of interest, such as position, velocity, or heading angle. As a result, a lot of work has been done on this subject, and the literature has revealed many significant findings [5–7]. For example, Ref. [8] used linear consensus protocols to develop a theoretical foundation for average consensus under fixed and switching topologies. Ren and Beard [9] extended consensus analysis to multi–agent systems with dynamically changing directed interaction topologies and established convergence conditions based on directed spanning–tree connectivity. In order to minimize communication frequency while maintaining convergence guarantees, Ref. [10] presented distributed event–triggered consensus techniques. Adaptive consensus controllers for heterogeneous MASs with uncertain nonlinear dynamics were developed by [11]. Sampled–data consensus systems with time–varying sampling intervals and transmission delays were examined by [12]. For second–order MASs that are susceptible to input saturation, Ref. [13] developed fixed–time consensus methods. Consensus-based distributed secondary-control schemes have also been extensively reviewed for DC microgrids [14], demonstrating the broader applicability of graph-based coordination in networked power systems. However, such power-system-oriented methods do not directly address distributed event-triggered consensus co–design for heterogeneous vehicle platoons comprising mixed first-order and second-order agents. Moreover, existing studies on self-triggered consensus, communication-impaired formation control, and stochastic multi–agent systems [15–17] consider different agent dynamics, communication conditions, or control objectives. Therefore, a unified framework combining mixed-order heterogeneous dynamics, neighbor-dependent triggering, and simultaneous LMI–based design of controller gains and triggering matrices remains insufficiently investigated. In [18], researchers combine fixed–time reference generation with a self–structuring neural network that accounts for unknown nonlinear dynamics and external disturbances to create an event–triggered formation-control scheme for underactuated unmanned surface vessels. Conversely, Ref. [19] uses a mixed event-triggered mechanism to study completely distributed leader-following consensus for nonlinear fractional-order multi–agent systems, allowing coordinated behaviour through local information while lowering communication demand, because most existing approaches either concentrate on homogeneous agent dynamics or on first–order and second–order linear models.

The above–mentioned literature is all in the study of isomorphic multi–agent systems, all of which have the same dynamic characteristics. In the actual communication–limited multi–agent system, the competition for bandwidth of heterogeneous agents makes only some agents able to transmit data, and it is necessary to rationally schedule agents to reduce the transmission of information, save network resources, and ensure the optimization of network utilization and control performance. For example, Refs. [20,21] considered cooperative output regulation of heterogeneous linear multi–agent systems based on event–triggered control. In [20], a homogeneous MASs event–triggering mechanism is first established, followed by a new distributed event trigger control scheme is proposed for the internal reference model of each multi–agent to solve the problem of heterogeneous MAS cooperative output regulation. In [21] proposed a basic event–triggered control scheme for output regulation for multi–agent models. Based on this result, a distributed self–triggering control scheme is proposed, which can avoid continuous monitoring and measurement errors. Existing studies [21–24] investigate the consensus control problem of heterogeneous multi–agent systems based on event triggering mechanism under fixed and switching topology. Because the following elements are identified in Table 1 and an unfilled research gap is eliminated, the proposed multi–agent control, in contrast to the conventional event–triggered approaches, can offer a higher level of dependability and resilience.

Although event-triggered consensus for MASs has been documented [10,21,24], the majority of current findings either examine first-order and second-order systems independently or concentrate on homogeneous agent dynamics. Furthermore, traditional event-triggering conditions ignore neighbor information and rely simply on the state error of each agent. On the other hand, this work’s novel contributions are:

1.   This paper studies the consensus control problem of second–order and first-order heterogeneous multi–agent systems, based on a new distributed event–triggering mechanism. The authors propose a co–design method where the stability conditions (using Lyapunov–Krasovskii with Kronecker product) are solved simultaneously to obtain both the controller gains (𝒦1,𝒦2) and the trigger matrices (ρ1,ρ2,ρ3) using LMI tools.

2.   Unlike traditional event–triggered mechanisms that rely only on the agent’s own state error (e.g., Refs. [19,24] mentioned in the References), the proposed trigger condition considers both the agent’s own information update error and the error between its current information and that of its neighbors.

3.   The proposed strategy defines separate event–triggering functions for position and velocity, allowing independent distributed transmission of position and velocity information. The strategy determines the next transmission time for each type of information. In contrast, the agent’s own state error is employed in [19,24]. By including neighbor errors, the trigger condition directly reflects the global consensus error, leading to fewer unnecessary transmissions during transients.

4.   The trigger mechanism contains multiple tunable parameters (ξ1,ξ2,ζ1,ζ2,ζ3), and matrices (σ1,σ2,σ3), allowing researchers to adjust the information transmission times and controller gain according to actual task requirements.

5.   While many existing studies focus on homogeneous systems with identical agent dynamics, our research explicitly addresses a heterogeneous multi-agent system comprising both first-order and second–order agents within a unified event-triggered control framework. This mechanism is designed to automatically reduce transmissions once the system reaches consensus. Specifically, when (μxi) and (μvi) (position/velocity update status) tend to zero (i.e., consensus is achieved), the system transmits information less frequently to maintain the achieved state.

2  Related Work

Due to its numerous applications in distributed sensor models, power sector, wireless communication, and cooperative unmanned ground vehicles, the consensus control problem for multi–agent systems has attracted a lot of attention in the last ten years [25,26]. Early research, mostly focused on homogeneous multi-agent systems, established theoretical foundations for average consensus under fixed and switching topologies by Olfati–Saber and Murray. Later, Ren and Beard [9], extended the consensus analysis to dynamically changing directed interaction topologies and derived convergence conditions based on directed spanning-tree connectivity. Realizing that network communication resources are inherently limited, researchers such as Dimarogonas and Johansson developed event–triggered control algorithms to reduce transmission frequency while maintaining convergence guarantees. However, these early event-triggered methods mostly relied on each agent’s own state error as the triggering condition, without considering interactions with neighboring agents. Researchers have studied cooperative output regulation using event–triggered control based on internal reference models for heterogeneous multi–agent systems [27,28], where agents may have different dynamics (e.g., mixed first–order and second–order dynamics). They have also developed distributed self–triggering schemes that do not require continuous monitoring. Although consensus control for heterogeneous systems under fixed and switching topologies has been studied more recently, the majority of current methods either tackle first–order and second–order linear models independently or concentrate on homogeneous agent dynamics. The relationship between an agent and its neighbors, which contains important information about the consensus status of the entire system, is usually ignored by conventional event–triggered mechanisms, which define trigger conditions only based on the difference between an agent’s current state and its last transmitted state. Additionally, most studies treat controller design and communication scheduling as distinct problems, resulting in suboptimal trade–offs between control performance and resource utilization [29,30]. This is because the literature currently in publication lacks systematic co–design frameworks that simultaneously optimize controller gains and event–trigger parameters. The present study fills these gaps by proposing a novel event–triggered consensus control strategy that incorporates neighbor information into trigger conditions, creates independent event functions for position and velocity, and offers a co–design framework for simultaneously determining trigger parameters and controller gains using linear matrix inequalities. Thus, following are the main innovations of this paper: (i) a neighbor-information-dependent event trigger for mixed first-order/second-order heterogeneous MASs; (ii) independent event functions for position and velocity with guaranteed minimum transmission intervals via parameters ξ1,ξ2; (iii) a co-design LMI framework that simultaneously solves for trigger matrices ρ1,ρ2,ρ3 and controller gains 𝒦1,𝒦2; and (iv) automatic reduction of transmissions as consensus because the trigger condition uses μxi,μvi that disappear at consensus. By including neighbor errors, the trigger condition directly reflects the global consensus error, leading to fewer unnecessary transmissions during transients.

3  The Requirement of Vehicle Platoon Control Design

This research investigates a group of m vehicles traveling in a single–file formation. The vehicles are numbered sequentially from 1,2,…,m, with vehicle 1 serving as the leader. Every following vehicle, denoted as i>1, comes equipped with sensors that measure the distance to the vehicle directly ahead. The inter–vehicle distance is defined as (di(t):=xi−1(t)−xi(t)−d0) for (i=2,3,…,m). Since x˙i−1(t)=vi−1(t), and x˙i(t)=vi(t) differentiating the inter–vehicle distance yields d˙i(t)=x˙i−1(t)−x˙i(t)=vi−1(t)−vi(t), i=2,3,…,m. Thus, the distance dynamics depend on the relative velocity between two consecutive vehicles and do not assume that the preceding vehicle is stationary. Here, xi(t) represents the position of vehicle i at time t, while d0 stands for a minimum safe gap, which accounts for both the length of the vehicle and an additional required clearance between two successive vehicles, as illustrated in Fig. 1. The lead vehicle, i=1, is assumed to maintain a constant reference velocity v0 using a properly designed velocity controller. The entire platoon vehicle is expected to meet the following criteria as outlined in reference [31].

Figure 1: Two vehicles in a platoon configuration.

There are some basic necessary conditions. Every vehicle in the platoon should move at the same speed as the leader. That is, limt→∞|vi(t)−v1(t)|=0, i=2,3,…,m, where vi(t) stands for the speed of vehicle i. The gap between two vehicles ought to grow as their speed increases. More precisely, the distance each follower keeps from the car ahead should eventually match a value proportional to its own speed, for example limt→∞|di(t)−κvi(t)|=0, i=2,3,…,m. Here, κ is what we call the time–headway coefficient. This coefficient determines the following distance as a function of speed. No vehicle should ever move in reverse, meaning vi(t)≥0, t≥0, i=1,2,…,m. Thus, every vehicle moves forward or remains stationary. Finally, the distance di(t)>0 factor should be greater than zero for all i=2,…,m.

Each follower vehicle asymptotically synchronizes its velocity with the preceding vehicle while maintaining the desired inter-vehicle spacing. The lead vehicle and a few following vehicles that have both position and velocity sensors are modelled as second-order agents in this platoon setup (1), whilst other following vehicles that can only measure position are modelled as first-order agents (2). Real-world car fleets, where various cars have varying degrees of automation, are reflected in this heterogeneity. In order to meet the platoon criteria of speed matching and safe distance keeping outlined in Section 3, the control objective is to achieve both position consensus and velocity consensus (as indicated in Section 3) among all vehicles.

4  Heterogeneous Multi–Agent System for Platoon Model Based on Event–Triggered Mechanism

Consider a heterogeneous multi–agent system for a vehicle platoon, consisting of both first–order and second–order agents. Specifically, the first group of agents are second–order agents, and the remaining agents are first-order agents. The dynamic model of the second-order multi-agent system is given by:

{x˙i(t)=vi(t)v˙i(t)=ui(t),i=1,2,3,…,m(1)

In the above equation, xi(t)∈ℛx, vi(t)∈ℛv, and ui(t)∈ℛu represent the position, velocity, and control input of the ith second–order agent, respectively. The dynamic model of the first–order multi–agent system is:

x˙i(t)=ui(t),i=(m+1),(m+2),…,n(2)

Let m be the number of second-order agents and n the total number of agents (n>m). In the above equation, xi(t)∈ℛx, and ui(t)∈ℛu, represent the position and control input of the ith first–order agent, respectively. The structure of the agent node is shown in Fig. 2.

Figure 2: Structure of the ith intelligent agent node.

Definition 1: The closed–loop heterogeneous multi–agent systems (1) and (2) are said to achieve asymptotic consensus if its trajectories satisfy the following limits:

{limt→∞‖vi(t)−vi−1(t)‖=0,i,j∈ℐn={2,…,m}limt→∞‖di(t)−κvi(t)‖=0,i,j∈ℐm={2,…,m}(3)

Remark 1:Eq. (3) defines the desired consensus objective of the heterogeneous multi-agent system. Under the proposed distributed control protocol and event-triggering mechanism, and provided that the LMI conditions derived in Section 5 are satisfied, the closed-loop system asymptotically achieves the consensus behavior specified in Eq. (3). In terms of platooning, this ensures that inter-vehicle distances stabilise to a constant value (the required time-headway κvi(t)), and string stability is attained because the lead vehicle’s disturbances do not intensify throughout the chain.

Assumption 1: The communication topology graph G of the multi–agent system is an undirected connected graph.

Assumption 2: Both the actuator and the controller operate in an event-driven manner, while the sensor operates in a time–driven manner, with sampling period of ℏ, with ℏ > 0, respectively.

Firstly, the following consensus protocols are designed for heterogeneous multi–agent systems (1) and (2):

ui(t)=−𝒦1{∑j∈𝒩i1aij((xi(kℏ)−xj(kℏ))}−𝒦1{∑j∈𝒩i2aij((vi(kℏ)−vj(kℏ))}, i∈ℐmui(t)=−𝒦2{∑j∈𝒩iaij((xi(kℏ)−xj(kℏ))},i=(m+1),(m+2),…,𝒩(4)

Remark 2: 𝒩i1 and 𝒩i2 denote the neighbor sets of the ith second–order and first–order agents, respectively. It is clear that 𝒩i=𝒩i1⋃𝒩i2, 𝒦1 and 𝒦2 are the gain matrices of the consensus controllers to be designed.

Given the limited network communication resources, this research adopts an event–triggered mechanism to reduce the number of transmissions of the state information of agent i, thereby lowering the update frequency of controller i and the amount of information transmitted over the network to neighboring agents. Let tsiℏ denote the ith transmission instant of the position information of agent i, and let tℓiℏ denote the ith (ℓ=0,1,2,…,i∈ℐm;j∈𝒩i1) transmission instant of the velocity information of agent i. Thus, the transmission states of the position and velocity information of agent i can be described as follows:

x^i(kℏ)=xi(tsiℏ),k∈[tsi,ts+1i), i∈ℐnv^i(kℏ)=vi(tℓiℏ),k∈[tℓi,tℓ+1i), i∈ℐm(5)

The next position information transmission time is defined as:

ts+1iℏ=inf{t:t>(ξ1tsi+ℏ), gi(t)>0}(6)

The next speed information transmission moment is defined as:

tℓ+1iℏ=inf{t:t>(ξ2tℓi+ℏ), fi(t)>0}(7)

Remark 3: From Eqs. (6) and (7) that an information transmission interval is always longer than one sampling period. Even when the system satisfies the triggering condition in every sampling period during a certain information transmission phase, the transmission interval can still be made longer than one sampling period by adjusting the parameters ξ1, and ξ2 as appropriate, thereby reducing bandwidth occupancy throughout the entire information transmission process.

Where (ξ1,ξ2)≥1 are tuning parameters and gi(t) and fi(t) are event–triggering functions. The proposed event-triggering mechanism is designed as follows:

For second−order agents (position):gi(t)=exiT(kℏ)ρ1exi(kℏ)−σ1μxiT(kℏ)ρ1μxi(kℏ)>0, i=1,2,…,mFor second−order agents (velocity): fi(t)=eviT(kℏ)ρ2evi(kℏ)−σ2μviT(kℏ)ρ2μvi(kℏ)>0, i=1,2,…,mFor First−order agents: gi(t)=exiT(kℏ)ρ3exi(kℏ)−σ3μxiT(kℏ)ρ3μxi(kℏ)>0, i=(m+1),…,n(8)

where

exi(kℏ)=xi(kℏ)−x^i(kℏ),i∈ℐn,k∈[tsi,ts+1i)evi(kℏ)=vi(kℏ)−v^i(kℏ),i∈ℐm,k∈[tℓi,tℓ+1i)(9)

The parameters σ1,σ2,σ3 are positive, dimensionless trigger-threshold coefficients. The parameter σ1 is associated with the position transmission of the second-order agents, σ2 is associated with their velocity transmission, and σ3 is associated with the position transmission of the first-order agents. A transmission is generated when the weighted local update error exceeds the corresponding weighted neighbor-disagreement threshold eiTρkei>σkμiTρkμi. Therefore, σk regulates the communication–performance trade-off of the corresponding triggering channel. Furthermore, we define:

μxi(kℏ)=∑j∈𝒩iaij[xi(kℏ)−xj(kℏ)]μvi(kℏ)=∑j∈𝒩i1aij[vi(kℏ)−vj(kℏ)](10)

These equations include the agent node, its neighboring agent nodes, and the controller input information for node i at the most recent sampling instant. The scalars σ1, σ2, σ3>0 are the threshold coefficients associated with the second-order position trigger, the second-order velocity trigger, and the first-order position trigger, respectively. They scale the corresponding neighbor-disagreement terms and regulate the communication–performance trade–off. The matrices ρk=ρkT>0, k=1,2,3, are symmetric positive–definite weighting matrices of appropriate dimensions. They define the quadratic metrics used to evaluate the local information-update errors and neighbor-disagreement signals and are determined from the derived LMI conditions. The scalar thresholds σk and weighting matrices ρk are distinct design quantities; σk does not modify or adjust ρk.

Remark 4: Compared with existing literature, the proposed triggering mechanism depends not only on the information update of the agent itself but also to the information of neighboring agents. Here, μxi(kℏ), and μvi(kℏ) describe the position and velocity information update status of the entire multi–agent system. When the whole system approaches toward consensus, i.e., limk→∞μxi(kℏ)=0, and limk→∞μvi(kℏ)=0. The information will be transmitted only when Eqs. (6) and (7) satisfied. This allows the system to be better maintained after it has reached consensus.

Remark 5: The parameters ξ1 and ξ2 determine the minimum transmission intervals and satisfy ξ1,ξ2≥1. In contrast, σk, k=1,2,3, are the threshold parameters of the event-triggering functions. As σk→0+, the corresponding triggering condition approaches eiT(kℏ)ρkei(kℏ)>0. Consequently, any nonzero local update error can activate the associated trigger after the minimum transmission interval imposed by Eqs. (6) and (7) has elapsed. Conversely, increasing σk raises the relative threshold associated with the neighbor-disagreement term and generally reduces the triggering frequency. Therefore, ξ1,ξ2 and σ1,σ2,σ3 provide separate tuning of the minimum transmission interval and trigger sensitivity, respectively.

Based on the above event–triggering scenario, the following consensus control protocol can be designed as follows:

ui(t)=−𝒦1{∑j∈𝒩i1aij((x^i(kℏ)−x^j(kℏ))}−𝒦1{∑j∈𝒩i2aij((v^i(kℏ)−v^j(kℏ))}, i∈ℐmui(t)=−𝒦2{∑j∈𝒩iaij((x^i(kℏ)−x^j(kℏ))},i=(m+1),(m+2),…,𝒩(11)

It is obtained from the Eqs. (10) and (11).

ui(t)=−𝒦1∑j∈𝒩iaij[xi(kℏ)−xj(kℏ)−exi(kℏ)+exj(kℏ)]−𝒦1∑j∈𝒩i2aij[vi(kℏ)−vj(kℏ)−evi(kℏ)+evj(kℏ)⏟], i∈ℐmui(t)=−𝒦2∑j∈𝒩iaij[xi(kℏ)−xj(kℏ)−exi(kℏ)+exj(kℏ)],i=(m+1),(m+2),…,𝒩,t∈[kℏ,(k+1)ℏ)(12)

Then Eqs. (1) and (2) can be converted to:

x˙i(t)=vi(t)v˙i(t)=−𝒦1∑j∈𝒩iaij[xi(kℏ)−xj(kℏ)−exi(kℏ)+exj(kℏ)]−𝒦1∑j∈𝒩i2aij[vi(kℏ)−vj(kℏ)−evi(kℏ)+evj(kℏ)],i∈ℐmx˙i(t)=−𝒦2∑j∈𝒩iaij[xi(kℏ)−xj(kℏ)−exi(kℏ)+exj(kℏ)],i=(m+1),(m+2),…,𝒩,t∈[kℏ,(k+1)ℏ)(13)

Next, we define the augmented matrices as follows:

xT(kℏ)=[x1T(kℏ)x2T(kℏ)…xmT(kℏ)]T∈ℛm×1vT(kℏ)=[v1T(kℏ)v2T(kℏ)…vmT(kℏ)]T∈ℛm×1exiT(kℏ)=[ex1T(kℏ)ex2T(kℏ)…exmT(kℏ)]T;i∈ℐmeviT(kℏ)=[ev1T(kℏ)ev2T(kℏ)…evmT(kℏ)]T;i∈ℐm𝒵(kℏ)=[xT(kℏ)vT(kℏ)𝒳T(kℏ)]T∈ℛ(m+n)×1𝒲(kℏ)=[exT(kℏ)evT(kℏ)ℰT(kℏ)]T∈ℛ(m+n)×1𝒳(kℏ)=[xm+1T(kℏ)xm+2T(kℏ)…xnT(kℏ)]T∈ℛ(n−m)×pℰ(kℏ)=[em+1T(kℏ)em+2T(kℏ)…enT(kℏ)]T∈ℛ(n−m)×p

Now, we present the Laplacian matrix ℒ∈ℛn×n, where ℒ11∈ℛ(n−m)×(n−m), ℒ12∈ℛ(n−m)×m, ℒ21∈ℛm×(n−m), ℒ22∈ℛm×m.

ℒ=[ℒ11ℒ12ℒ21ℒ22]

Substituting (11) into (1) and (2) and using the definitions in (14), we obtain the compact form (14) after collecting terms involving 𝒵(kℏ) and 𝒲(kℏ). From the Eq. (13), it can be obtained:

χ˙(t)=𝒵11𝒳(kℏ)+𝒲11𝒲(kℏ),k∈[tℓi,tℓ+1i)(14)

For notational convenience in the subsequent stability analysis, define χ(t):=𝒵(t), χ(kℏ):=𝒵(kℏ). Thus, χ(t) is not a transformed state; it is the same augmented state vector previously denoted by 𝒵(t), where

𝒵11=[0ℐm×m0k1ℒ22k1ℒ22k1ℒ21k2ℒ120k2ℒ11]𝒲11=[000k1ℒ22k1ℒ22k1ℒ21k2ℒ120k2ℒ11]

To facilitate analysis, let d(t)=t−kℏ, where kℏ≤t≤(k+1)ℏ and k∈𝒩. It is clear that t≠kℏ, d˙(t)=1. When t=kℏ, we have 0≤d(t)≤ℏ. Therefore, Eq. (14) can be transformed as follows:

χ˙(t)=𝒵11χ(kℏ)+𝒲11𝒲(kℏ), t∈[kℏ,(k+1)ℏ)(15)

Then with the definition of d(t)=t−kℏ, our closed–loop (16) model becomes:

χ˙(t)=𝒵11χ(t−d(t))+𝒲11𝒲(t−d(t)), t∈[kℏ,(k+1)ℏ)(16)

Thus, the systems (1) and (2) achieve asymptotic consensus when and only if the closed–loop system (16) is asymptotically stable. For ease of analysis, we define ℰ1=[Im×m00], ℰ2=[0Im×m0], ℰ3=[00Im×m], and ℰ13=[Im×m0000Im×m], ex(kℏ)=ℰ1𝒲(kℏ), ev(kℏ)=ℰ2𝒲(kℏ), ℰx(kℏ)=ℰ3𝒲(kℏ). For subsequent proof, when the trigger conditions are not satisfied, then the Eq. (8) can be converted to:

𝒲T(kℏ)ℰ1Tρ1(ℰ1𝒲(kℏ))<σ1{[[ℒ22ℒ21]⊗ℐp(ℰ13𝒵(kℏ))]T}ρ1×[ℒ22ℒ21]⊗ℐp(ℰ13𝒵(kℏ))(17)

𝒲T(kℏ)ℰ2Tρ2(ℰ2𝒲(kℏ))<σ2{[ℒ22⊗ℐp(ℰ2𝒵(kℏ))]T}ρ2×[ℒ22⊗ℐp(ℰ2𝒵(kℏ))]𝒲T(kℏ)ℰ3Tρ3(ℰ3𝒲(kℏ))<σ3{[ℒ12ℒ11]⊗ℐp(ℰ13𝒵(kℏ))]T}ρ3×[ℒ12ℒ11]⊗ℐp(ℰ11𝒵(kℏ))(18)

Before proceeding to the main stability section, the authors will present some Lemmas, which help in the formation of LMIs.

Lemma 1 ([32]): For any constant matrices ℛ∈ℛn×n, ℛ=ℛT>0, ℋ∈Rn×k, and a time–varying function 0<d(t)≤ℏ satisfying δ˙:[−ℏ,0]→ℛn, let the vector function δ˙ be such that ∫t−d(t)tδ˙(s)=ℱφ(t)d(s), where φ(t)∈ℛk. Then the following inequality holds:

−∫t−d(t)tδ˙(t)ℛδ˙(s)≤−φT(t)(d(t)ℋTℛ−1ℋ−ℱTℋ−ℋTℱ)φ(t)(19)

Lemma 2 ([32]): Assume there exist real numbers ξk¯>0(k¯=1,2,3.) and a Lyapunov function 𝒱(t,χ(t),χ˙(t)). For the corresponding solution of the system (16) χ(t) for t≥t0, the function 𝒱(t,χ(t),χ˙(t)) is continuously differentiable when t≠kℏ and satisfies the following conditions:

ξ1‖χ(t)‖2≤𝒱(t,χ(t),χ˙(t))≤ξ2‖χ(t)‖2(20)

𝒱(t,χ(t),χ˙(t))≤−ξ3‖χ(t)‖2, t≠kℏ(21)

limt→kℏ−𝒱(t,χ(t),χ˙(t))≥𝒱(t,χ(t),χ˙(t))|t=kℏ(22)

Lemma 3 ([33]): Let 𝒳, and 𝒴 with appropriate dimensions. If there exists a constant ν>0, then the following holds:

𝒳T𝒴+𝒴T𝒳≤ν𝒳T𝒳+1ν𝒴T𝒴(23)

5  Co–Design of Distributed Event-Triggering Mechanism For Consensus Control

In this section, delay-dependent sufficient conditions are derived for the co-design of the distributed event-triggering mechanism and consensus controller for the heterogeneous multi-agent system represented by the closed-loop model in Eq. (16). The asymptotic stability of the event-triggered closed-loop system, established under the derived LMI conditions, guarantees convergence to the prescribed platoon-consensus objective. In particular, the vehicle velocities synchronize with the leader’s reference velocity, and the spacing errors converge to zero according to the adopted time-headway policy. We first present the following corollary, which provides general requirements for the resultant model (16), is first presented.

The authors note that the following co-design technique is novel for heterogeneous MASs before presenting the LMI conditions: the controller gains (𝒦1,𝒦2) and the event-triggering thresholds (ρ1,ρ2,ρ3) are derived from a single set of LMIs (33) and (34). The majority of current literature [19,24] construct the controller before the event trigger, which is not ideal.

Corollary 1: Given ℏ>0, under the new event–triggered mechanisms (8), there exist real numbers k1, and k2; appropriately dimensional real matrices 𝒫>0, 𝒳>0, ℛ>0, Ψ>0, Y>0, ρk¯>0, (k¯=1,2,3.) and matrices ℳl>0, (l=1,2.) such that the following matrix inequalities hold:

[Λ11+ℏℵ11ℏℱT(Y+ℛ)ℏℳ1T(∙)−ℏ(Y+ℛ)0(∙)(∙)−ℏℛ]<0(24)

[Λ11ℏℱTℛℏℳ2T(∙)−ℏℛ0(∙)(∙)−ℏ(Y+ℛ)]<0(25)

Then the closed–loop system (2) is globally asymptotically stable. Define:

Λ11=ν1T𝒫Tℱ+ℱT𝒫ν1+ν1TΨν1−ν3TΨν3−ν12T𝒳ν12−ν23Tℳ1−ℳ1Tν23−ν12Tℳ2−ℳ2Tν12−ν4T(ℰ1Tρ1ℰ1)ν4−ν4T(ℰ2Tρ2ℰ2)ν4−ν4T(ℰ3TΦ3ℰ3)ν4+φT(t)ν2T{σ1{[ℒ22ℒ21]⊗ℐpℰ13}Tρ1[ℒ22ℒ21]⊗ℐpℰ13+{ν2[ℒ22⊗ℐpℰ2]Tρ2[ℒ22⊗ℐpℰ2]}+{ν3{[ℒ12ℒ13]⊗ℐpℰ13}Tρ3[ℒ12ℒ13]⊗ℐpℰ13}}ν2φ(t)ℵ11=ν12T𝒳ℱ+ℱT𝒳ν12, ℱ=𝒜ν2+ℬν4,ν1= [I00], ν2= [0I0], ν3= [00I], ν4= [000I]

Proof: The Lyapunov-Krasovskii functional for the closed-loop system is chosen as follows:

𝒱(t,χ(t),χ˙(t))=𝒱1(t,χ(t),χ˙(t))+𝒱2(t,χ(t),χ˙(t)), t∈[kℏ,(k+1)ℏ)(26)

where

𝒱1(t,χ(t),χ˙(t))=χ(t)T𝒫χ(t)+∫t−ℏtχ(α)TΨχ(α)dα+∫−ℏ0∫t+βtχ˙(α)Tℛχ˙(α)dαdβ𝒱2(t,χ(t),χ˙(t))=(ℏ−d(t)){[χ(t)−χ(t−d(t))]T𝒳[χ(t)−χ(t−d(t))]+∫t−d(t)tχ˙(α)TYχ˙(α)dα}, β∈[−ℏ,0]

According to the Lemma 2, the term 𝒱1(t,χ(t),χ˙(t))≥0, and 𝒱2(t,χ(t),χ˙(t))>0. Thus t=kℏ, then second condition in the Lemma 2 is established. Take the time derivative of (26), where t∈[kℏ,(k+1)ℏ), with ℏ˙(t)=1, and t≠kℏ.

𝒱˙(t,χ(t),χ˙(t))=2χ(t)T𝒫χ˙(t)+χ(t)TΨχ(t)−χ(t−ℏ)TΨχ(t−ℏ)+ℏχ˙(t)Tℛχ˙(t)−∫t−ℏtχ˙(α)Tℛχ˙(α)dα−[χ(t)−χ(t−d(t))]T𝒳[χ(t)−χ(t−d(t))]+(ℏ−d(t)){[χ˙(t)−χ˙(t−d(t))]T𝒳[χ(t)−χ(t−d(t))]}+[χ(t)−χ(t−d(t))]T𝒳[χ˙(t)−χ˙(t−d(t))]+(ℏ−d(t)){[χ˙(α)TYχ˙(α)−χ˙(t−d(t))TYχ˙(t−d(t))]}−∫t−d(t)tχ˙(α)TYχ˙(α)dα(27)

Further computing with Lemma 1:

=φT(t)[ν1T𝒫ℱ+ℱT𝒫ν1+ν1TΨν1−ν3TΨν3−ν12T𝒳ν12−ν4T(ℰ1Tρ1ℰ1)ν4−ν4T(ℰ2Tρ2ℰ2)ν4−ν4T(ℰ3Tρ3ℰ3)ν4+(ℏ−d(t))(ν12T𝒳ℱ+ℱT𝒳ν12+ℱT𝒴ℱ)]φ(t)+ψ1+ψ2+𝒲4T(ℰ1Tρ1ℰ1)𝒲4++𝒲4T(ℰ2Tρ2ℰ2)𝒲4+𝒲4T(ℰ3Tρ3ℰ3)𝒲4

So

ψ1=−∫t−ℏtχ˙T(α)ℛχ˙(α)dα,ψ2=−∫t−d(t)tχ˙T(α)Yχ˙(α)dα,

With

ψ1+ψ2=−∫t−ℏt−d(t)χ˙T(α)ℛχ˙(α)dα−∫t−d(t)tχ˙T(α)(ℛ+Y)χ˙(α)dα

The terms ∫t−ℏt−d(t)χ˙T(α)dα=ν23φ(t), ∫t−d(t)tχ˙T(α)dα=ν12φ(t) can be obtained from Lemma 3:

ψ1+ψ2≤φT(t){(ℏ−d(t))ℳ1Tℛ−1ℳ1−ν23Tℳ1−ℳ1Tν23+d(t)ℳ2T(ℛ+𝒴)−1ℳ2−ν12Tℳ2−ℳ2Tν12}φ(t)(28)

Add the left and right sides of the Formulas (17)–(19) to get the result:

⇒𝒲T(ℰ1Tρ1ℰ1)𝒲+𝒲T(ℰ2Tρ2ℰ2)𝒲+𝒲T(ℰ3Tρ3ℰ3)𝒲<𝒵T(kℏ)ℰ13T{σ1{[ℒ22ℒ21]⊗ℐp}Tρ1[ℒ22ℒ21]⊗ℐp}(ℰ13𝒵(kℏ))+(𝒵T(kh)ℰ2T){σ2{ℒ22⊗ℐp}Tρ2[ℒ22⊗ℐp]}(ℰ2𝒵(kℏ))+𝒵T(kℏ)ℰ13T{σ3{[ℒ12ℒ13]⊗ℐp}Tρ3[ℒ12ℒ13]⊗ℐp}(ℰ13𝒵(kℏ))(29)

So

⇒𝒲T(ℰ1Tρ1ℰ1)𝒲+𝒲T(ℰ2Tρ2ℰ2)𝒲+𝒲T(ℰ3Tρ3ℰ3)𝒲<φT(t)ν2Tℰ13T{σ1{[ℒ22ℒ21]⊗ℐp}Tρ1[ℒ22ℒ21]⊗ℐp}(ℰ13ν2φ(t))+φT(t)ν2Tℰ2T{σ2{ℒ22⊗ℐp}Tρ2[ℒ22⊗ℐp]}(ℰ2ν2φ(t))+φT(t)ν2Tℰ13T{σ3{[ℒ12ℒ13]⊗ℐp}Tρ3[ℒ12ℒ13]⊗ℐp}(ℰ13ν2φ(t))(30)