Dynamic Event-Triggered Scheduling and Platooning Control Co-Design for Automated Vehicles Over Vehicular Ad-Hoc Networks

Xiaohua Ge,, Shunyuan Xiao, Qing-Long Han,,Xian-Ming Zhang,, and Derui Ding,

Abstract—This paper deals with the co-design problem of event-triggered communication scheduling and platooning control over vehicular ad-hoc networks (VANETs) subject to finite communication resource. First, a unified model is presented to describe the coordinated platoon behavior of leader-follower vehicles in the simultaneous presence of unknown external disturbances and an unknown leader control input. Under such a platoon model, the central aim is to achieve robust platoon formation tracking with desired inter-vehicle spacing and same velocities and accelerations guided by the leader, while attaining improved communication efficiency. Toward this aim, a novel bandwidth-aware dynamic event-triggered scheduling mechanism is developed. One salient feature of the scheduling mechanism is that the threshold parameter in the triggering law is dynamically adjusted over time based on both vehicular state variations and bandwidth status. Then, a sufficient condition for platoon control system stability and performance analysis as well as a co-design criterion of the admissible event-triggered platooning control law and the desired scheduling mechanism are derived. Finally,simulation results are provided to substantiate the effectiveness and merits of the proposed co-design approach for guaranteeing a trade-off between robust platooning control performance and communication efficiency.

I. INTRODUCTION

VEHICLE platooning has been regarded as a promising intelligent transportation system technology for achieving an automated highway system due to its promising benefits, including reduced fuel consumption, improved road safety, highway capacity and traffic congestion relief [1]. A key control objective of platooning is to maneuver a group of automated vehicles to establish and maintain a platoon formation with desired longitudinal spacing and cruise speed.For this purpose, cooperative adaptive cruise control, which serves as an extension of adaptive cruise control, has been extensively developed for vehicle platooning via employing wireless vehicle-to-vehicle and/or vehicle-to-infrastructure communication, as demonstrated in Fig. 1. In this sense,platoon vehicles can not only measure their relative positions and velocities, but also share useful data (e.g., absolute position, velocity and/or acceleration) among their neighboring vehicles or roadside infrastructure via integrating wireless communication interface on board. In practice,vehicles with integrated wireless communication capabilities form a mobile ad hoc network on a road, commonly known as VANET [2], which constitutes the cornerstone of modern transportation systems.

Fig. 1. VANET-enabled vehicle platooning in a highway scenario under wireless vehicle-to-vehicle (V2V) communication and vehicle-toinfrastructure (V2I) communication.

The advent of wireless VANET technologies for vehicular platoons greatly facilitates inter-vehicle communication and cooperation, which, however, poses several challenging issues that need to be carefully addressed for platooning control.First, due to the digital nature of wireless communication,onlypacket-baseddata transmissions are permitted among interacting vehicles. More specifically, measured vehicle data is required to be firstly sampled and digitized before it is shared over VANETs. Besides, virtually all control protocols and algorithms are implemented on digital modules/computers, which means that platoon controllers operate only at discrete sampling times. In this sense, conventional data communication mechanisms for vehicular platoons [3]–[5],which utilize continuous-time vehicular data (e.g.,xi(t) withtdenoting the continuous time) for achieving platoon controller design, are no longer valid in a sampling setting of VANETs.Therefore, newsampled-data platooning controlapproaches are demanded such that only sampled sensor data packets at discretized instants of time are necessary for computing desired platoon control commands for automated vehicles.Second, the wireless bandwidth and data rates of VANETs are inherently limited because of the shared and broadcast nature of a wireless communication protocol. For example, the VANET under an IEEE 802.11p-standard dedicated shortrange communication protocol offers a maximum data rate at 27 Mb/s in 10-MHz channels [6]. The bandwidth and data rate issues become particularly conspicuous when platoon scale becomes large and all vehicles broadcast their data through a shared networked medium in an instantaneous way.Therefore, it is of great significance to efficiently schedule data transmissions between intercommunicating vehicles such that scarce communication resource can be occupied economically during implementation of platooning control strategies. However, how to reliably schedule and disseminate data among surrounding vehicles in a resource-efficient manner remains a challenge. This is because unilaterally reducing the frequency of inter-vehicle data transmissions ineluctably degrades desired platoon control performance.Unfortunately, the existing continuous and periodic communication mechanisms [3]–[5], [7]–[11] do not account for the resource constraint during platoon controller design.This thus calls for novelresource-efficient platooning controlstrategies which promise both satisfactory communication efficiency and platoon control performance.

An event-triggered scheduling mechanism (ESM) has attracted considerable attention over the past decade due to its prominent advantage in sustaining desired system performance while keeping reasonable communication expenditure [12]. The rationale behind such an ESM is that the decisions to perform sampling and/or data transmission actions are judged by a well-designed triggering mechanism,and thus the precious communication resource is only occupied when “needed” or “necessary”. A considerable amount of research on ESMs has yet been carried out for various networked systems [13]–[17]. In the context of eventtriggered platooning control, however, a few results have been reported [18]–[21]. One of the main difficulties arises from designing a suitable Zeno-free ESM to guarantee the desired platoon control performance in the presence of unknown inputs. Although the ESM-based platooning control strategies above are effective in specific configurations, there are several significant issues that remain open. Thefirstone is that the ESMs in [19], [20] essentially fall into a category of static ESMs, where the threshold parameter in the corresponding triggering law is persistently fixed [17]. While such static ESMs offer benefits in terms of computation and implementation, they naturally represent a conservative solution that still leads to unnecessary data transmissions and broadcasts, which has been well justified in the literature [17].On the other hand, although dynamic ESMs are studied in[18], [21], the triggering laws therein are designed merely on the basis of vehicular state changes but irrespective of variable bandwidth status. This means that when actual network traffic load is low and bandwidth status is idle, the proposed ESMs in[18], [21] might still prevent data packets from being transmitted over a network. Therefore, an intelligent dynamic ESM that is aware of changing bandwidth is highly preferable in VANET-enabled platooning control systems. Thesecondissue is that the performance between the ESMs in [18]–[21]and existing ones lacks comprehensive comparison and formal theoretical justification. As a matter of fact, most of the existing ESMs have been largely evaluated via numerical validation. There is thus a clear need to present theoretical ways of evaluating different dynamic and static ESMs that successfully achieve the same control objective in order to understand which one is advantageous and under what cost.Third, since ESMs aim to reduce data transmissions, platoon control performance is inevitably compromised to certain extent. Therefore, it is necessary to conduct an insightful trade-off investigation between desired platoon control performance and expected communication resource expenditure, which, however, has not been adequately addressed in the literature. How to tackle the significant issues above in the context of VANET-enabled platooning control motivates this study.

In this paper, we investigate the problem of event-triggered platooning control for a convoy of wirelessly connected vehicles. We begin with a general model of vehicle longitudinal dynamics, which incorporates the simultaneous presence of an unknown external disturbance input on each follower vehicle and an unknown leader control input. The novelty of the paper lies in the development of a novel dynamic event-triggered scheduling and control co-design approach to solving the platooning control problem regardless of the effects of the unknown external disturbance and leader control input.

The main contributions of the paper are summarized as follows.1) A new bandwidth-aware dynamic event-triggered scheduling mechanism (DESM)is developed to efficiently alleviate communication resource expenditure. This is done by dynamically adjusting the threshold parameters in the desired triggering laws based on a bandwidth acknowledgement parameter such that the number of transmitted data packets over VANET can be significantly reduced when bandwidth is busy or slightly decreased when bandwidth is idle. Via suitably prescribing the bandwidth acknowledgement parameter, it is shown that the proposed DESM straightforwardly degenerates into two typical DESMs: one accounts for a consistent low network traffic load scenario and the other accounts for a consistent high network traffic load scenario;

2) The theoretical relationship between the proposed DESMs and some existing ESMs is explicitly disclosed. It is demonstrated that the proposed DESMs are more flexible for achieving a trade-off between communication efficiency and platoon control performance; and3) A scalable co-designcriterionon the existence of desired event-triggered platoon control law and DESMs is presented. Both controller gain matrix and trigger weighting matrix can be obtained from the derived criterion that is independent of platoon scale.

The remainder of the paper is organized as follows.Preliminaries are provided in Section II. Formulation of the problem to be addressed is presented in Section III. The new bandwidth-aware DESM is elaborated in Section IV. The internal stability and performance analysis of the resulting closed-loop platooning control system is provided in Section V.The main co-design result for desired platoon formation control protocol and DESM is stated in Section VI. Simulation results under various scheduling mechanisms and information flow topologies are shown in Section VII. Concluding remarks are drawn in Section VIII.

II. PRELIMINARIES

A. Notation

Denote by Rmthem-dimensional Euclidean space and Rm×nthe set ofm×nreal matrices. A symmetric matrix Φ>0(Φ ≥0) means that it is positive definite (positive semidefinite). The set of nonnegative integers is denoted by N={0,1,2,...} . Denote by 1 =[1,1,...,1]Ta column vector of an appropriate dimension, I an identity matrix of an appropriate size, and 0 a zero vector or matrix of an appropriate size. diag{Φ1,Φ2,...,Φm} represents anm-block diagonal matrix with Φi, ?i=1,2,...,m, being thei-th diagonal element. The Euclidean norm is denoted by ‖·‖.Denote byL2[0,∞) the space of square-integrable vector functions over [0,∞). For a measurable matrix-valued functionw(t):R →Rn∫satisfyingw(t)∈L2[0,∞), its norm is given bydenotes the Kronecker product. The transpose of a matrix (vector) Φ is denoted by ΦT. If a matrix is invertible, the superscript “–1”represents the matrix inverse. Matrices, if not explicitly stated,are assumed to have compatible dimensions.

B. Algebraic Graph Theory

Let V={1,2,...,N} denote an index set ofNnodes,E ?V×V denote an edge set of paired nodes with (i,j)representing an edge (or a link) of nodeiand nodej, and A=[aij]∈RN×Ndenote a weighted adjacency matrix withai jbeing the adjacency element (or coupling weight) between nodeiand nodej. Then, a weighted and undirected graph G=(V,E,A) models the information flow topology among theNnodes. Moreover,ai j>0 ?(i,j)∈E if and only if there is a bidirectional information link between nodeiand nodej.Self-loops are excluded in the graph, i.e.,aii=0 ,i∈V. A path of G is a concatenation of edges (i1,i2), (i2,i3),...,,(ip?1,ip)∈E , in which all nodesi1,i2,...,ipare distinct. G is a connected graph if for anyi,j∈V there is a path from nodeito nodej. The Laplacian matrix of graph G is defined as L=D?A , where D=diag{d1,d2,...,dN} withfor anyi∈V.

III. PROBLEM FORMULATION

A. Vehicle Longitudinal Dynamics

Consider a platoon ofN+1 automated vehicles moving along a straight and flat road. The vehicular platoon is composed of one leader, indexed by 0, andNfollowers,indexed byi=1,2,...,N, whose information flow topology is modeled by a digraphThe linearized thirdorder model, which has been widely adopted to describe the longitudinal dynamics of a platoon vehicle [3], [4], [7],[9]–[11], can be described by

wherepi(t)∈R ,denote the absolute position, velocity and acceleration of thei-th vehicle, respectively; τp>0 represents the inertial time lag in the powertrain;ui(t)∈R denotes the desired control input acting on thei-th vehicle;wi(t)∈R denotes the generally unknown but bounded input (e.g., disturbance or modeling error) imposed on thei-th vehicle. Without loss of generality,it is assumed thatw0(t)=0 on the leader vehicle. However,different from the existing vehicle models [4], [5] where the leader’s control inputu0(t) is assumed to be zero, the control inputu0(t)∈L2[0,∞) of the leader in (1) allows to be timevarying and not accessible to any follower vehicle, which is more involved than the zero leader control input case.

B. A Platoon Formation Tracking Objective

Remark 1:The aim of platooning control is to design a suitable distributed coordinated control lawui(t) for anyi∈V such that theNfollower vehicles can achieve the same speed and acceleration with the leader, while simultaneously maintaining the desired longitudinal spacing. From Definition 1, it is clear that ifxi(t)?x0(t)?di→0 ast→∞, then one has thatxi(t)?di?(xj(t)?dj)→0. Therefore, the spacing errors of the follower vehicles satisfy that ψi(t)=pi?1(t)?pi(t)?denoting the constant spacing (CS) between vehicleiand its preceding vehiclei?1,meanwhile, the velocities and accelerations satisfy thatvi(t)?v0(t)→0 andai(t)?a0(t)→0. Apart from the CS policy [4], [9], [10], [19], [21], another type of spacing policy that has been commonly adopted in the literature is referred to as the time headway policy (also known as speed-dependent spacing policy). For example, under a constant time headway(CTH) policy, the vectordiis refined aswheresidenotes a constant distance that f orms the gap between vehicleiand its preceding vehiclei?1 at standstill anddenotes the desired time headway in seconds for vehicleiarriving at its preceding vehicle’s positionpi?1(t). It is clear that the CTH policy regulates intervehicle distance based on the real-time speed of each platoon vehicle. Hence, the automated platoon under a CTH policy enjoys a safety guarantee during high-speed driving owe to the resultant large spacings. Nevertheless, large inter-vehicle distances imply a loose platoon of vehicles, which inevitably decreases traffic throughput on highways and may further induce traffic congestion. In contrast, platooning under a CS policy normally yields a tighter vehicle convoy with relatively small spacings, thereby offering high traffic efficiency.However, how to guarantee driving safety at a high speed is regarded as a potential challenge of platooning control under a CS policy, especially in the presence of unknown disturbances. In this study, we are interested in addressing the resource-efficient platooning control problem under a CS policy as well as unknown disturbance and leader inputs,which poses a great challenge to the platoon safety and platoon control design. This is because much fewer vehicular data packets can be utilized for platoon controller design and implementation.

C. An Event-Based Distributed Platooning Control Law

The traditional continuous distributed platooning control law often takes the following form

whereK=[kp,kv,ka] represents the controller gain matrix to be designed. Clearly, to achieve the desired platoon tracking control objective, each vehicleiin the platoon needs to gather the continuous vehicle statexj(t) from its underlying neighbors over the topologyat anyt∈R, which may be impractical for VANET-enabled platoons. In this paper, for each platoon memberi∈V, we are interested in constructing and designing the following event-based distributed platoon formation tracking control law

D. The Problem to Be Addressed

For each follower vehiclei∈V, denote the formation tracking error by

as a performance output of the resulting closed-loop system(25). The main problem to be addressed is stated as follows.

Problem 1:For the vehicular platoon (1), the control objective is to design a suitable event-based distributed coordination control lawui(t) of the form (4) under a suitable event-triggered scheduling mechanism such that 1) The resulting closed-loop platoon tracking error system in terms of

TABLE I ACRONYMS AND MATHEMATICAL NOTATIONS ADOPTED FOR EVENT-TRIGGERED MECHANISMS

IV. A NOVEL DYNAMIC EVENT-TRIGGERED SCHEDULING MECHANISM

The vehicular platoon in (1) is maneuvered over a VANET,where each follower vehicleiequips an onboard sensor,allowing it to measure the continuous state information, to sample the measurement at discretized instants of time, and to further transmit the sampled data to its neighboring vehicles in accord with a suitable information flow topology. However, as VANET is a typical shared digital communication network medium, a significant yet challenging issue is to develop a resource-aware scheduling mechanism for better allocating the limited communication resource. In this section, we propose a novel DESM such that the desired distributed platooning control task over the VANET can be successfully accomplished while data transmissions between wirelessly connected vehicles can be regulated in a resource-aware manner.

The acronyms and mathematical notations used in the subsequent development are provided in Table I. It is noteworthy that the sensors deployed on the follower vehicles are time-driven and can periodically sample the continuous vehicle statexi(t) at discrete instants {kh}. Whether or not the current sampled data packet (k,xi(kh)) should be transmitted over the VANET is decided by a suitable ESM. For simplicity, it is assumed that data transmissions over the VANET are carried out in a single packet manner and each transmitted data packet can successfully arrive at its destination. This allows an emphasis to be placed on the development of the ESMs.

Before detailing the key elements of the DESM, we shall start with a brief description of the traditional SESMs; see,e.g., [13], [19], [20], [25]–[29]. By “static” in this paper, it refers to a scenario that the threshold parameter in a triggering law isfixedduring the whole implementation of the ESM. In contrast, the threshold parameter in a DESM isdynamically adjustedover time. Specifically, an SESM recursively computes the sequence of event releasing instantsfor each vehiclei∈V according to the following triggering law

which naturally excludes the Zeno phenomenon.

It is noted that the threshold parameterin (8)characterizes the frequency of the invoked data transmissions and thus the number of total transmitted data packets over the VANET. For example, a smallin SESM (8) often leads to a high frequency of data packet transmissions and vice versa[25], [26]. Therefore, the threshold parametercan be regarded as a communication scheduling parameter that relates to the data transmission rate over the VANET.Nevertheless, from a resource-efficient perspective, it is undesirable to consistently fix the threshold parameter during the implementation of the SESM since the real-time data transmission rate over a practical communication network often varies over time. For example, the IEEE 802.11p standard WLAN possesses various data transmission rates at 3, 4.5, 6, 9, 12, 18, 24, and 27 Mbps [21], due to variable interference and wireless fading conditions. In such a practical scenario, intelligent ESMs are demanded to schedule data transmissions less frequently over the network when the current transmission network experiences a high network traffic load and more often when the network experiences a low network traffic load. This thus motivates the following DESM which provides more intelligent and flexible communication scheduling.

The schematic of the DESM, hereafter known as DESMbis illustrated in Fig. 2, where Storeiat the local controller side works in an event-driven manner and retains only the latest arrived data packets from vehiclei’s neighbors, and ZOHikeeps the control input unchanged fromuntil the next updating instant. More specifically, DESMbdetermines the sequence of event releasing instants for each vehiclei∈V via the following triggering law

Fig. 2. Schematic of dynamic event-triggered platooning control for vehicle i ∈V under DESMb over a VANET.

where α ∈[0,1] is a bandwidth status acknowledgement parameter with α=0 indicating ‘busy’, α=1 indicating ‘idle’and 0<α<1 indicating ‘moderate’; σ1i(kh) and σ2i(kh)represent two dynamic threshold parameters whose values are severally adjusted online at each sampling time according to the following laws

Remark 3:Note that the minimum inter-event time represents an important property of an ESM as it depicts how‘far’ the event releasing intervals are away from the Zeno phenomenon. Lemma 5 (or Lemma 6) states that the minimum inter-event time of DESM2cannot be less than that of SESM(or DESMb) and the minimum inter-event time of SESM (or DESMb) cannot be less than that of DESM1, which can be described as

V. STABILITY AND PERFORMANCE ANALYSIS

VI. SCHEDULING AND CONTROL CO-DESIGN

Theorem 1 establishes a sufficient condition under which the closed-loop system (25) is asymptotically stable under the prescribed performance requirement. However, it cannot be used directly to solve out the control gainKdue to the coupling termUT BK. Apart from this, the linear matrix inequality (LMI) (27) has a dimension of 1 7N×17N, whereNdenotes the number of follower vehicles in the platoon. As the platoon scale goes larger, the computational complexity of verifying the LMI will increase dramatically, which is unfavorable for practical platooning control design. In what follows, let λi,i∈V , be the eigenvalues of the matrix H, and further denote bymini∈V{λi} andmaxi∈V{λi}.

We are now in a position to present a simple and efficient criterion for co-designing an admissible distributed platoon formation control protocol (4) and desired ESMs, including SESM (8), DESMb(11), DESM1(15) and DESM2(16).

Remark 4:Some key features of Theorem 2 are highlighted as follows.First, Theorem 2 is numerically efficient for large platoons owing to the dimension-reduced LMI conditions compared with Theorem 1.Second, Theorem 2 can be tailored to suit the different scheduling scenarios. To be more specific,by settingin (28), Theorem 2 reduces to the codesign criterion under SESM (8) or DESM1(15) which corresponds to the case that the network traffic load is low and the bandwidth can be allocated to transmit more data packets.Similarly, the co-design criterion under DESM2(16) can be retrieved by lettingin (28), which relates to the scenario that the bandwidth status is busy and vehicular data packets should be transmitted less frequently over the VANET.Third, Theorem 2 establishes the explicit relationship between the dynamic communication scheduling parameter and the information flow topology connectivity.More specifically, from the (5,5)- th block of, it can be clearly observed thatimplies, which further leads tobecause>0. Notice thatcharacterizes the topology connectivity andTherefore, by exposing the constraint, the system designers can carefully configure the information flow topology (e.g., by pinning suitable follower vehicles or choosing appropriate coupling weights) such that the network resource can be used at prescribed levels. This also demonstrates that the proposed criterion enables the co-design of vehicular networking and platoon control in a unified framework.

Before closing this section, Algorithm 1 is presented to achieve the simultaneous dynamic event-triggered communication scheduling and platoon formation tracking control.

Algorithm 1 Scheduling and Control Co-Design 1: For a prescribed V2V communication topology configuration,compute , from the graph matrix and determine the bounds λii ∈V H λ,λ 2: For a suitable , if , then goto Step 1; otherwise continue σ σM >0 σM ≥1/λ2 3: For given scalars h, β, γ, η, α, , , , , solve Theorem 2 to determine a feasible solution . If Theorem 2 is infeasible,then reset the scalars and repeat Step 3; otherwise continue tsim ti0=0 i ∈V 4: Set the simulation time and for all m=0:tsim/h?1 M ?1i?2ii ∈V(K,Φ)5: for do i=1:N 6: for do j=1:N 7: for do ?xjs(t) ?G 8: Collect from each vehicle j’s neighbors over 9: end for ui(t) ?xi(t) ?xj(t)10: Compute in (4) under K, and from its nei- ghbors 11: Derive the longitudinal vehicle dynamics (2) under (sim,xi(simh))12: Obtain the current sampled data packet σ1i(simh) σ2i(simh)σαi(simh)ui(t)13: Calculate in (14a) and in (14b) for de- termining in (13)14: for do j=1:N j (sjmh)>0 ?15: if then verify DESMb (11)fσm m+1h ←sjmh 16: Update tj ?xj(t)←xj(tj m+1h)17: Update 18: end if 19: end for 20: end for 21: end for

VII. SIMULATION RESULTS

To verify the control performance and communication efficiency of the proposed dynamic event-triggered scheduling and control co-design approach for vehicle platooning, we perform comprehensive comparative analysis involving timevarying external disturbance and leader control (acceleration)input, different ESMs and various information flow topologies. Specifically, we consider an automated platoon of one leader vehicle and ten follower vehicles. The longitudinal dynamics of thei-th vehicle,={0,1,2,...,10}, is described by system (1) with τp=0.5 s. The desired spacing between any consecutive platoon members during platoon driving is set as=10 m. The platoon driving starts withp0=100m, an initial cruise speed 5 m/s and zero initial tracking errors. In what follows, we examine four different topologies for the vehicular platoon, as shown in Fig. 3, where t he adjacency elements of the graphare taken asaij=1/Nfor (i,j)∈. The subsequent simulation will be performed fortsim=100s underh=2m s and β=0.6. During platoon driving, a sinusoidal external disturbance:20))m/s2,t∈[20,25]s andwi(t)=0 m/s2, otherwise is imposed on each follower vehiclei∈V and a time-varying control commandu0(t) of the following form

Fig. 3. Four information flow topologies for the simulated platoon: (a) Bidirectional (BD); (b) Leader-two-bidirectional (LTBD); (c) Leaderbidirectional (LBD); (d) Leader-two-predecessor-bidirectional (LPBD).

acts on the leader vehicle, allowing it to accelerate the convoy gradually for 35 s, then decelerate it to a new speed 7.5 m/s and maintain this cruise speed for the convoy afterwards. Such a leader command could be used to simulate overtaking and lane changing maneuvers, where the new lane adopts a higher speed. The central aim is to ensure that the platoon attains the same time-varying leader speed/acceleration and maintains the desired spacing guided by the leader regardless of external disturbances and leader acceleration variations, while at the same time preserving satisfactory communication efficiency.

To quantitatively evaluate the communication scheduling efficiency of an ESM, we define an average transmission rateon a finite time interval [ 0,Tmh)s as

A. Platooning Under Different Event-Triggered Mechanisms

First, we investigate the platoon behavior and communication efficiency under different ESMs over the LBD topology in the simultaneous presence of the concerned external disturbancewi(t) and time-varying leader acceleration inputu0(t). Specifically, the following four different ESMs are examined:

Solving Algorithm 1 with η=0.5, it is found that the concerned platoon formation tracking problem is feasible under the proposed event-triggered platoon tracking control law (4). The simulation results are provided in Fig. 4.Specifically, Fig. 4 (a) depicts the relationship between the minimalH∞performance levels γminand the average data transmission ratesunder the above four ESMs. Note that under DESMb, four additional cases of the bandwidth indicationαare considered. Fig. 4 (b) shows the trajectories of the average spacing errorsof the controlled platoon under different ESMs. From Fig. 4, it can be clearly seen that as the value ofαincreases, more sampled vehicular data packets allow to be broadcast over the network, thereby leading to better platoon control performance. This is reasonable because a largerαindicates more spare bandwidth resource and thus permits a higher data transmission rate. As a result, the platoon control performance in terms of the optimalH∞attenuation level and the resulting average spacing error can be significantly improved. On the other hand, although DESM2achieves the minimal communication resource expenditure compared with the other ESMs, the platoon control performance is compromised, which can be seen from the largestH∞performance level γminin Fig. 4 (a) and the worst average spacing error in Fig. 4 (b). A possible cause of such control performance degradation is that some data packets that are not triggered by DESM2may contain useful information to achieve and maintain the desired platoon behavior. On the contrary, DESM1ensures the best platoon control performance amongst the six DESMs, while resulting in the highest average transmission rate. Hence, at the cost of the platoon control performance, the consumption of the finite bandwidth resource over VANET can be reduced significantly. It thus can be concluded that the proposed DESMbprovides a more flexible scheduling mechanism for the platoon to achieve a trade-off between satisfactory communication efficiency and desired platoon control performance.

Fig. 4. Comparisons of platoon control performance and communication efficiency under different ESMs over the LBD topology: (a) Optimal H∞performance levels γmin versus average data transmission rates (%);(b) Average spacing errors (t).

B. Platooning Under Different Information Flow Topologies

We next carry out the simulation by testing the developed co-design approach for the vehicular platoon over four different information flow topologies depicted in Fig. 3. The comparative results in terms of the minimalH∞performance levels γmin, average data transmission ratesand average spacing errorsunder the developed DESMb(α =0.45)over the BD topology (η =0.048), the LTBD topology(η =0.048) , the LBD topology (η =0.23) and the LPBD topology (η =0.23) are presented in Fig. 5.

Fig. 5. Comparisons of platoon control performance and communication efficiency under DESMb (α =0.45) over different topologies: (a) Optimal H∞performance levels γmin versus average data transmission rates (%);

Based on the simulation results in Fig. 5, we note the following two points:1)Comparing the BD and LTBD topologies in the same parameter setting, better platoon control performance (smaller γminandcan be warranted over LTBD at the cost of a slightly higher average transmission rateof the whole convoy. An important reason for such a control performance improvement is that LTBD allows two platoon members (vehicles 1 and 2) to be informed by the leader, whereas BD enables only the first platoon member to be informed. Generally speaking, to achieve the collective platoon formation tracking objective,the more platoon members are informed by the leader, the better global tracking behavior of the platoon will be. This observation can be further confirmed by the LBD case, where the best platoon control performance (the smallest γminandis preserved because all the platoon members are informed by the leader over LBD; and2)Comparing the LBD and LPBD topologies in the same parameter setting, it can be observed that the communication efficiency under LPBD is significantly boosted at the expense of degraded platoon control performance (larger γminandA possible cause of such control performance degradation is that the DESMbover LPBD employs more neighboring vehicles' information than that over LBD to verify the triggering condition, which can be seen fromin (11). This eventually may result in less released data packets but worse platoon control performance. Note that the comparative studies between the other ESMs, including DESM2, DESM1, SESM and DESMb(under differentα), and the four topologies in Fig. 3 can be conducted in a similar way. The results and discussions are omitted for brevity.

To further demonstrate that how the vehicular platoon over different topologies behaves under the DESMb-based eventtriggered platooning control law, we only present the simulation results over the BD topology and the LBD topology. The time responses of the vehicle positionspi(t),relative positions to the leaderp0(t)?pi(t), velocitiesvi(t) and accelerationsai(t) of the controlled platoon over BD and LBD are shown in Fig. 6 and Fig. 7, respectively. It can be observed from Fig. 6 (c) and (d) that the imposed external disturbancewi(t) on each follower vehicle and leader acceleration changesu0(t) can both cause certain fluctuations of the vehicles'accelerations and velocities during the relevant periods of time. Their adverse effects, however, are then effectively attenuated by the proposed platooning control law within an acceptable time span. This is particular the case under the LBD topology, where the controlled platoon shows a remarkably stable behavior with short settling times that allow rapid platoon maneuvering, as clearly demonstrated in Fig. 7 (c) and (d).

Fig. 6. Time responses of the controlled platoon under DESMb (α =0.45)over BD topology: (a) Vehicles' positions pi(t), ; (v) Vehicles' relative positions to the leader p0(t)?pi(t), i ∈V ; (c) Vehicles' velocities vi(t), ;(d) Vehicles' accelerations a i(t),.

Fig. 7. Time responses of the controlled platoon under DESMb ( α=0.45)over LBD topology: (a) Vehicles’ positions pi(t), ; (v) Vehicles’ relative positions to the leader p0(t)?pi(t), i ∈V ; (c) Vehicles’ velocities vi(t), ;(d) Vehicles’ accelerations a (t), .

Finally, we provide an intuitionistic evaluation of the resultant vehicular platoon behavior via developing a platoon driving maneuver environment, which simulates and visualizes the motions of the platoon vehicles on a straight and flat road. The snapshots of the simulated platoon driving maneuvers over the BD and LBD topologies at different times are shown in Fig. 8 and Fig. 9, respectively. Obviously, it can be seen that the simulated platoon well maintains the desired spacing, stability, and resilience performance requirements within the simulation time.

Fig. 8. Snapshots of the simulated platoon driving scenario (with real-time vehicular position pi , velocity vi and acceleration ai) over the BD topology at different times.

Fig. 9. Snapshots of the simulated platoon driving scenario (with real-time vehicular position pi , velocity vi and acceleration ai) over the LBD topology at different times.

The simulation results above substantiate that the proposed dynamic event-triggered scheduling and platooning control co-design approach can not only drive a group of automated vehicles to perform the desired platoon formation tracking task against the simultaneous unknown disturbances and leader control input changes, but also serve as an effective means for guaranteeing the resource-efficient requirement of vehicular platoons when the communication resource is limited. The approach is well-suited for networked vehicular platoon systems under finite resource budget, and, as compared with the traditional ESMs and time-triggered scheduling mechanisms, the newly developed DESMboffers a meaningful design freedom for seeking a preferable trade-off between desired platoon control performance and satisfactory communication efficiency over VANET.

VIII. CONCLUSIONS

The problem of event-triggered platooning control over resource-constrained VANET has been studied. Novel bandwidth-aware DESMs have been developed to alleviate the bandwidth occupancy in an intelligent and flexible manner. Formal theoretical relationship between the proposed DESMs and the widely-adopted ESMs have been established.To preserve the stability and robustness of the resulting closed-loop platooning control system against simultaneous unknown disturbances on follow vehicles and control input changes on the leader, an admissible event-triggered distributed platoon formation tracking control law has been constructed and designed. Numerically efficient co-design criterion and algorithm have been established to determine both desired platooning control law and relevant DESMs.Finally, simulation results under different information flow topologies and DESMs have shown that the proposed dynamic event-triggered scheduling and platooning control co-design approach can achieve a meaningful trade-off between desired platoon control performance and expected communication efficiency over VANETs.

Albeit the proposed co-design approach shows its promising advantages in simulations, there are still several challenging issues for future research. For example, we have employed a linearized state-space model for describing vehicle longitudinal dynamics. Nevertheless, a real vehicle longitudinal control system mainly consists of engine, torque converter, transmission, drive train, throttle actuator, and brake actuator, which makes the vehicular longitudinal dynamics inherently nonlinear and uncertain. It would be more practical to reevaluate the proposed platooning control approach for nonlinear and uncertain vehicle longitudinal dynamics. On the other hand, the main focus of this study has been placed on achieving desired resource efficiency and platoon control performance (including individual vehicle stability and prescribedH∞attenuation performance). How to initiate novel resource-efficient platooning control strategies that further preserve satisfactory operational performance,such as driving safety, fuel efficiency, and riding comfort,constitutes an interesting and significant topic.

APPENDIX