A Third-Order Consensus Approach for Vehicle Platoon with Intervehicle Communication

A third-order consensus approach is proposed for the vehicle platoon. For addressing the platoon problem, a realistic, third-order vehicle dynamicsmodel is used, and the spacing policy and the vehicle acceleration error are embedded into the consensus protocol. A sufficient and necessary condition of asymptotically stability is obtained for the vehicle platooning system.Numerical simulations for several traffic scenarios are carried out. The results demonstrate the effectiveness and the robustness of the presented approach.


Introduction
Traffic congestion is a serious problem and considerable challenge in many parts of the world.How to alleviate traffic congestion has attracted great concern in recent years.Platoon based cooperative driving is one of the promising approaches to improve traffic flow, enhance traffic capacity, and reduce fuel consumption (see [1][2][3] and the references therein).The main goal of vehicle platoon control is to ensure that all vehicles keep the consensus speed and maintain the desired intervehicle distance prespecified by the spacing policy.The platoon studies can date back to the Partners for Advanced Transportation Technology (PATH) program [4].Since then, researchers have introduced and implemented various control strategies such as the consensus control, model predictive control [5], optimal control [6], sliding-mode control [7], and  ∞ control [8].
This paper is concerned with the consensus control strategy for platooning of vehicles.Consensus control is an active research field in multivehicle cooperative control.The pioneering work has been reported by Fax and Murray [9].They have developed a theoretical framework of consensus for cooperative control of multiple vehicles.They focus on vehicles with first order dynamics and consider fixed time delays and different communication topologies.Later, Ren [10] has studied cooperative control of vehicles modeled by secondorder dynamics and introduced consensus strategies under directed information topologies.Wang et al. [11] have proposed a weighted and constrained consensus control strategy for platoon coordination.They have studied the consensus control under a stochastic framework.The communication noises are considered while time-varying delays are not taken into account.
di Bernardo et al. [12] have investigated the vehicle platooning problem in the presence of heterogeneous timevarying delays, introduced a distributed control protocol to guarantee second-order consensus in vehicles platoon, and proved the stability of platoon based on Lyapunov-Razumikhin theorem.di Bernardo et al. [13] have modified the spacing policy in the control strategy and extended the approach in [12].The proposed algorithm is validated by experiments performed on a three-vehicle platoon.Santini et al. [14,15] have also proposed a second-order consensus algorithm for the vehicle platoon with intervehicle communications.The constant time headway (CTH) spacing policy and the time-varying delays are embedded in the algorithm.The performance of the algorithm is compared with a well-known Cooperative Adaptive Cruise Control (CACC) algorithm and is validated in the realistic scenario.In [16], the platoon problem in the presence of malicious attacks is studied, and a new second-order consensus strategy has been proposed to enhance the protection level of platoons.The designed strategy is validated by analytical and experimental results.Yan et al. [17] have presented a control strategy for vehicle platoon to deal with the actuator saturation and absent velocity measurement.
In [18,19], the authors have suggested a distributed control strategy to achieve third-order consensus of a dynamic network in the presence of time-varying heterogeneous delays.Saeednia and Menendez [20] have discussed the truck platooning problem and presented a distributed algorithm based on the average consensus algorithm.They have compared the distributed algorithm with a centralized optimization-based algorithm by simulating multiple scenarios.Wang et al. [21] have suggested the distributed consensus algorithm and protocol for CACC system.Zegers et al. [22] have adopted a realistic longitudinal vehicle dynamics model and the CTH spacing policy for the consensus problem.A three-vehicle platoon test is used to validate the performance of the control approach.
Jia and Ngoduy [23] have considered the packet loss and transmission delay and developed consensus control algorithms for the multiple platoons cooperative driving.Jia and Ngoduy [24] have further studied the cooperative model considering vehicle-to-vehicle (V2V) and vehicleto-infrastructure (V2I) communication and suggested an improved consensus control strategy.In [25], the authors have suggested a control algorithm of mixed vehicle platoon based on a unified model, in which the connected and autonomous vehicle and the human-driven vehicle are described by the different control models.
In this paper, we propose a novel third-order consensus strategy for the vehicle platooning system.Comparing with the studies [18,19], the paper has two contributions.Firstly, the leader's acceleration is not assumed to be zero in our consensus algorithm, which is more adaptable to complex driving environments.Secondly, the speed errors between following vehicles are taken into account, whereas this factor has not been considered in the works [18,19].In the cooperative driving environment, the introduction of the speed error information can further improve the stability of traffic flow [26][27][28][29].The asymptotic platoon stability is investigated by using Lyapunov-Razumikhin theorem.The effectiveness of the proposed approach is evaluated by simulations for several traffic scenarios.
The rest of the paper is organized as follows.In Section 2, the mathematical preliminaries are introduced.In Section 3, we present a third-order consensus control algorithm and carry out the stability analysis.Numerical simulations can be found in Section 4. Some conclusions are drawn in Section 5.

Mathematical Preliminaries
Suppose that the platoon consists of a leader vehicle (labeled with 0) and  following vehicles.The intervehicle communication structure of the followers is described by a directed graph (digraph)  = (, , ) in which  = {1, 2, ⋅ ⋅ ⋅ , } is the set of nodes,  ⊆  ×  denotes the set of edges, and  = [  ] × represents the adjacency matrix.In this paper, we assume   = 1 if vehicle  can receive the information from vehicle ; otherwise,   = 0.Moreover, we assume that there are no self-loops in the digraph; i.e.,   = 0.The degree matrix  = diag{ 1 , ⋅ ⋅ ⋅ ,   } is diagonal matrix, whose elements are   = ∑  =1   .The Laplacian matrix of the directed graph  is defined as  =  − .We also consider another graph  to model the information exchange among  followers and the leader.To investigate the leader-following problem, we define a diagonal matrix  = diag{ 1 , ⋅ ⋅ ⋅ ,   } to be a leader adjacency matrix associated with the platoon consisting of  following vehicles and one leader (labeled with 0), where   > 0 if node 0 is a neighbor of node ; otherwise,   = 0. We suppose that node 0 is globally reachable in , which means there is a path in  from every node  in  to node 0 [30].
We next recall some important lemmas and theorems used in studying the stability of the vehicle platoon system.

Platooning Control
The cooperative driving strategy of the platoon is to make each member of the platoon follow the leader's behavior and maintain the desired small intervehicle spacing.Consider a platoon consisting of  following vehicles and a leader moving along a single lane.The th vehicle's longitudinal dynamics can be described [33]: where   , V  , and   are, respectively, the position, speed, and acceleration of the th vehicle,   denotes the desired acceleration which is the control input, and   is the time constant of the drivetrain.
The consensus control goal of the platoon can be expressed as where  is the desired distance between two adjacent vehicles which can be set according to a constant spacing policy studied in [34], and  ⋅  is the desired distance of vehicle  from the leader 0.

Consensus Control Algorithm.
To achieve the control goal that the platoon members follow the leader's state, we design the following consensus control algorithm embedding the spacing policy information and the time-varying communication delays: where  1 > 0,  2 > 0, and  3 > 0 are the control parameters; ( − ) ⋅  is the desired spacing errors between vehicles  and  which is set according to the spacing policy; and  0 and   are, respectively, the time-varying communication delays from the leader and from the vehicle  to the vehicle .
Here, the effect of position difference is ignored and it is assumed that all neighboring vehicles can receive the beacon simultaneously from the leader and the vehicle .
The algorithm (8a), (8b), (8c), (8d), (8e), and (8f) can be described in detail as follows: (1) (8a) is the position error between the distance of vehicle  and vehicle  with respect to the desired distance ( − ) ⋅ .The term V 0 ( −  0 )  is introduced as the distance compensation due to the time-delay   of   .
(3) (8c) represent the position error between the distance of vehicle  and the platoon leader 0 with respect to the desired distance  ⋅ .The term V 0 ( −  0 ) 0 is added as the distance compensation due to the time-delay  0 of  0 .
(4) (8d) and (8e) represent, respectively, the velocity error and acceleration error between member  and the platoon leader 0. The leader acceleration is introduced in (8f).
According to (8a), (8b), (8c), (8d), (8e), and (8f), the consensus algorithm is designed based on state errors between the vehicle itself and the delayed state information of its neighboring vehicles obtained via wireless communication.The acceleration error is embedded into the proposed algorithm.The control algorithm using acceleration information has some advantages such as improving control reactivity and avoiding vehicle falling too far behind the vehicle ahead [19].The leader's acceleration is not assumed to be zero in the algorithm (8a), (8b), (8c), (8d), (8e), and (8f) to adapt to more complex driving environments.

Stability Analysis.
To prove asymptotic stability of the closed-loop dynamics driven by the control action, we first define position, velocity, and acceleration errors with respect to the reference signals  0 (), V 0 (), and  0 (),  = 1, ⋅ ⋅ ⋅ , , as We assume that the variation of the vehicle's velocity during the delay time can be ignored and the leader's Jerk ȧ 0 () is approximately zero.Based on the assumptions, we have V  ( −   ) ≈ V  () and ȧ 0 () = 0.Then, we can rewrite the coupling control action   () in terms of the state errors   (), V  (), and   ().After performing some algebraic manipulation, we obtain the closed-loop dynamics for the generic th platoon vehicle: , the closed-loop dynamics of the platoon can be written in a more compact form: Here and where ] Applying the Leibniz-Newton formula leads to where  0 = 0. Substituting ( 15) into (11), we can obtain where with From ( 12) and ( 13), we have     = 0 when  = 1, ⋅ ⋅ ⋅ ,  and  = 1, ⋅ ⋅ ⋅ , .Hence, ( 16) can be rewritten as Lemma 4 (see [35]).The matrix  is positive stable if and only if node 0 is globally reachable in .
Let   be the matrix defined by   = .According to Lemma 4,   is also positive stable since  > 0.
Lemma 5. Let the matrix  be as given in (17) where   is the th eigenvalue of   .
Proof (Sufficiency).Let  be the eigenvalue of ; then Noting that   is a positive stable, i.e., Re(  ) > 0, and choosing the control gains  1 > 0,  2 > 0, and  3 > 0 such that the conditions ( 20) and ( 21) are satisfied, we have that, for the th polynomial   (,   ), are all positive.According to [36,37], the roots of   (,   ) lie in the open left half of the complex plane.Thus,  is Hurwitz stable.(Necessity).If   is not positive stable, there exists Re(  ) which is less than or equal to zero.Then, the corresponding  2 will be less than or equal to zero.This contradicts the fact that the matrix  is Hurwitz stable.Theorem 6.Consider system (11) and take the control parameters  1 ,  2 , and  3 as in Lemma 5.Then, if and only if node 0 is globally reachable in , there exists a constant τ > 0, such that when 0 ≤   < τ ( = 1, ⋅ ⋅ ⋅ , ), the consensus is reached asymptotically; i.e., lim →∞  () = 0. ( Proof (Sufficiency).Choose appropriate control parameters based on Lemma 5. Since node 0 is globally reachable in ,   is a positive stable.According to Lemma 5,  is Hurwitz stable.There exists a positive-definite matrix  ∈  3×3 to satisfy Consider the following Razumikhin function for system (19): which satisfies From ( 26), we have According to Lemma 3, let  = −     0 ,  = ( + ), and  =  −1 ; then (28) becomes Take  4 () =  for some constant  > 1.When we have then V() ≤ −   for some constant  > 0. Thus, the conclusion follows from Theorem 1. (Necessity).Notice that system (11) is asymptotically stable for any   < τ,  = 1, ⋅ ⋅ ⋅ , .For the special case   = 0, from (19) the system ε () = () is asymptotically stable.The eigenvalues of  have negative real-parts, which implies that  is positive stable.According to Lemma 4, node 0 is globally reachable in .

Simulations
4.1.Simulation Setting.We adopt PLEXE simulator [38] in our simulation.PLEXE integrates the network simulator OMNeT++/MiXiM and the road traffic simulator SUMO, which are used to simulate V2V communication based on the 802.11p standard and the vehicle dynamics with the consensus algorithm, respectively.The parameters for the traffic simulation and consensus control algorithm are specified in Table 1.Control parameters are selected to guarantee consensus according to Theorem 6.The parameters of delay are not set since they are implemented in PLEXE to simulate more realistic vehicle dynamics.
We select a typical communication topology: the leaderand predecessor-following topology considering information from both the preceding vehicle and the leader (see Figure 1).

Platoon Formation and Maintenance.
We first consider the platoon composed of seven following vehicles and a leader initially starting from different positions with different speeds.It is assumed that there are no packet losses to avoid the effect of the communication on the system performance.Figure 2 shows the results for this initial scenario.The results illustrate that all vehicles converge toward the desired positions satisfying the spacing policy requirements (see Figure 2(a)) and reach the leader speed (see Figure 2(b)).The platoon forms and maintains the behavior imposed by the leader.The results confirm the ability of the proposed consensus approach to create and maintain the platoon.

Performance for Perturbations.
We consider a single large perturbation scenario similarly reported in [24], where the leader decelerates from 25 m/s to 10 m/s, then keeps the speed at 10 m/s for some time, and accelerates with 2 m 2 /s from 10 m/s to 25 m/s.The test is used to evaluate the ability of the approach in tracking the leader motion.The simulation results are shown in Figure 3.We can see that all vehicles almost simultaneously start to decelerate/accelerate (see Figure 3(c)) and the following vehicles can be fast and correctly track the leader speed (see Figure 3(b)), which confirms the tracking performance of the proposed algorithm.
To further verify the efficiency of the presented algorithm, we consider a periodic disturbance, where the following sinusoidal disturbance is added onto the leading vehicle speed: The goal is to investigate if the errors are amplified along the vehicle string (string stability).Figure 4 illustrates the simulation results.We can see that the relative position (see Figure 4(a)) and relative speed (see Figure 4(b)) are attenuated along the platoon, which illustrates the proposed approach can counteract the influence of the periodic disturbance and maintain the string stability.We next study another kind of perturbation coming from security risks discussed by [16].We consider one of the attacks, i.e., spoofing, and adopt similar scenario presented in [16].For the initial scenario in Section 4.2, an internal adversary controls the third vehicle and injects fraudulent information by means of setting its acceleration to the maximum value at  = 30s.This control lasts for two seconds.The simulation results are shown in Figure 5.It can be seen that after about 10 s transient stage, the members' states could be recovered.If we extend the voting technique developed in [16] to our algorithm, it will be possible to obtain better results, which will be left in our future work.

Impact of Platoon Length.
In this subsection, we study the performance of the proposed consensus algorithm for different platoon lengths.We consider the platoons with 4, 7, 10, 13, and 16 vehicles.Figure 6 shows the errors of the last vehicle of the platoon with respect to the leader 0 under different platoon lengths for the initial scenario.We can see that the platoons reach the consensus for different platoon lengths.The longer the platoon length is, the longer the convergence time is.From Figure 6, it also can be seen that the longer the platoon length is, the larger the speed error is.This is because the last vehicle of the platoon minimizes the larger position error with respect to the leader.
Figure 7 illustrates the state errors of the last vehicle of the platoon with respect to the leader under different platoon lengths for the sinusoidal disturbance scenario.From Figure 7, we can see that the strategy is able to track the leader's motion for different platoon lengths.

Conclusions
In this paper, we have proposed a novel third-order consensus strategy for the vehicle platoon and have proven the asymptotically stability of the platooning algorithm in presence of time-varying delays.We have tested several traffic scenarios including the initial case, the large perturbation, the sinusoidal disturbance, and the perturbation coming from security risks.The simulation results illustrate the effectiveness of the approach and confirm the robustness of the proposed strategy in the presence of perturbations.In addition, the proposed consensus control approach shows the effectiveness for different platoon lengths.Future work will    be devoted to investigating the effect of switching communication network topologies.Furthermore, more sophisticated spacing strategy should be introduced to determine the desired distance of vehicle.In addition, how to choose the optimal control parameter values will be studied in future work.

Figure 1 :Figure 2 :
Figure 1: The information flow topology used in simulations: leader-and predecessor-following topology.

Figure 7 :
Figure 7: Platoon performance in the presence of the sinusoidal disturbance scenario under different platoon lengths: (a) speed; (b) acceleration.
and  2 (1 +    3 ) >  1   ;  is Hurwitz stable if and only if   is a positive stable matrix and

Table 1 :
Parameters for the traffic simulation and consensus control algorithm.