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

Authors: Jianzhong Chen; Dongyang Bai; Huan Liang; Yang Zhou
Journal: Journal of Advanced Transportation
 (2018)

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## Abstract

A third-order consensus approach is proposed for the vehicle platoon. For addressing the platoon problem, a realistic, third-order vehicle dynamics model 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.


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## Body

## 1. 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 [1–3] 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 H∞ 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 second-order 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 time-varying 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 vehicle-to-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 [26–29]. 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 Section2, 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.


## 2. Mathematical Preliminaries

Suppose that the platoon consists of a leader vehicle (labeled with 0) andN following vehicles. The intervehicle communication structure of the followers is described by a directed graph (digraph) G=(V,E,A) in which V={1,2,⋯,N} is the set of nodes, E⊆V×V denotes the set of edges, and A=[αij]N×N represents the adjacency matrix. In this paper, we assume αij=1 if vehicle i can receive the information from vehicle j; otherwise, αij=0. Moreover, we assume that there are no self-loops in the digraph; i.e., αii=0. The degree matrix D=diag{d1,⋯,dn} is diagonal matrix, whose elements are di=∑j=1Nαij. The Laplacian matrix of the directed graph L is defined as L=D-A. We also consider another graph G- to model the information exchange among N followers and the leader. To investigate the leader-following problem, we define a diagonal matrix B=diag{b1,⋯,bn} to be a leader adjacency matrix associated with the platoon consisting of N following vehicles and one leader (labeled with 0), where bi>0 if node 0 is a neighbor of node i; otherwise, bi=0. We suppose that node 0 is globally reachable in G-, which means there is a path in G- from every node i in G to node 0 [30].We next recall some important lemmas and theorems used in studying the stability of the vehicle platoon system.LetC([-r,0],Rn) be a Banach space of continuous functions from [-r,0] into Rn with a norm ϕc=maxθ∈[-r,0]ϕ(θ), where · is Euclidean norm. Consider the following time-delay system:(1)x˙t=ft,xt,t>0,xθ=ϕθ,θ∈-r,0,where xt(θ)=x(t+θ), ∀θ∈[-r,0], f:R×C([-r,0],Rn)→Rn is a continuous function, and f(t,0)=0, ∀t∈R. Then the following result holds.Theorem 1 (Lyapunov-Razumikhin theorem, [31]).
Letφ1, φ2, and φ3 be continuous, nonnegative, nondecreasing functions with φ1(s)>0, φ2(s)>0, and φ3(s)>0 for s>0 and φ1(s)=φ2(s)=0. If there is a continuous function V(t,x) such that(2)φ1x≤Vt,x≤φ2x,t∈R,x∈Rn,if, in addition, there exists a continuous nondecreasing function φ4(s) with φ4(s)>s, s>0 such that(3)V˙t,x≤-φ3x,and if Vt+θ,x(t+θ)≤φ4(V(t,x(t))), θ∈[-r,0], then the solution x=0 is uniformly asymptotically stable.Lemma 2 (Hermite-Hadamard inequality, [32]).
Letf:[a,b]→R be a convex function; then(4)fa+b2≤1b-a∫abfxdx≤fa+fb2.Lemma 3 (see [30]).
For anyc,d∈Rn and any positive-definite matrix E∈Rn×n, it holds that(5)2cTd≤cTEc+dTE-1d.


## 3. 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 ofN following vehicles and a leader moving along a single lane. The ith vehicle’s longitudinal dynamics can be described [33]:(6)x˙it=vit,v˙it=ait,a˙it=1Tiuit-ait,where xi, vi, and ai are, respectively, the position, speed, and acceleration of the ith vehicle, ui denotes the desired acceleration which is the control input, and Ti is the time constant of the drivetrain.The consensus control goal of the platoon can be expressed as(7)xit→x0t-i·s,vit→v0t,ait→a0t,where s is the desired distance between two adjacent vehicles which can be set according to a constant spacing policy studied in [34], and i·s is the desired distance of vehicle i from the leader 0.


### 3.1. 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:(8a)uit=∑j=1Nαijβ1xjt-τj-xit-i-j·s+v0t-τ0τj(8b)+∑j=1Nαijβ2vjt-τj-vit(8c)+biβ1x0t-τ0-xit+v0t-τ0τ0-i·s(8d)+β2v0t-τ0-vit(8e)+β3a0t-τ0-ait(8f)+a0t-τ0,where β1>0, β2>0, and β3>0 are the control parameters; (i-j)·s is the desired spacing errors between vehicles i and j which is set according to the spacing policy; and τ0 and τj are, respectively, the time-varying communication delays from the leader and from the vehicle j to the vehicle i. 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 j.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 i and vehicle j with respect to the desired distance (i-j)·s. The term v0t-τ0τj is introduced as the distance compensation due to the time-delay τj of xj.( 2 ) (8b) represents the velocity error between members i and j.( 3 ) (8c) represent the position error between the distance of vehicle i and the platoon leader 0 with respect to the desired distance i·s. The term v0t-τ0τ0 is added as the distance compensation due to the time-delay τ0 of x0.( 4 ) (8d) and (8e) represent, respectively, the velocity error and acceleration error between member i 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.


### 3.2. 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 signalsx0(t), v0(t), and a0(t), i=1,⋯,N, as(9)x-it=xit-x0t+i·s,v-it=vit-v0t,a-it=ait-a0t.We assume that the variation of the vehicle’s velocity during the delay time can be ignored and the leader’s Jerk a˙0(t) is approximately zero. Based on the assumptions, we have vj(t-τj)≈vj(t) and a˙0(t)=0. Then, we can rewrite the coupling control action ui(t) in terms of the state errors x-i(t), v-i(t), and a-i(t). After performing some algebraic manipulation, we obtain the closed-loop dynamics for the generic ith platoon vehicle:(10)x-˙it=v-it,v-˙it=a-it,a-˙it=a˙it-a˙0t=1Ti∑j=1Nαijβ1x-jt-τj-x-it+∑j=1Nαijβ2v-jt-τj-v-it-biβ1x-it+β2v-it-1+biβ3a-it.By defining x-=[x-0,⋯,x-N]T, v-=[v-0,⋯,v-N]T, a-=[a-0,⋯,a-N]T, and ε=x-Tv-Ta-TT, the closed-loop dynamics of the platoon can be written in a more compact form:(11)ε˙t=Y0εt+∑j=1NYjεt-τj.Here(12)Y0=0N×NIN×N0N×N0N×N0N×NIN×N-β1TD+B-β2TD+B-I+β3BT,and(13)Yj=0N×N0N×N0N×N0N×N0N×N0N×Nβ1TAjβ2TAj0N×N,where(14)Aj=0⋯α1j⋯0⋮⋮⋮⋮⋮0⋯αjj⋯0⋮⋮⋮⋮⋮0⋯αNj⋯⋮andT=diag1T1,…,1TN∈RN×N.Applying the Leibniz-Newton formula leads to(15)εt-τj=εt-∫-τj0ε˙t+sds=εt-∑i=0NYi∫-τj0εt+s-τids,where τ0=0. Substituting (15) into (11), we can obtain(16)ε˙t=Fεt-∑j=1N∑i=0NYjYi∫-τj0εt+s-τids,where(17)F=∑i=0NYi=0N×NIN×N0N×N0N×N0N×NIN×N-β1TH-β2TH-I+β3BT,with(18)H=L+B.From (12) and (13), we have YjYi=0 when j=1,⋯,N and i=1,⋯,N. Hence, (16) can be rewritten as(19)ε˙t=Fεt-∑j=1NYjY0∫-τj0εt+sds.Lemma 4 (see [35]).
The matrixH is positive stable if and only if node 0 is globally reachable in G-.
LetHT be the matrix defined by HT=TH. According to Lemma 4, HT is also positive stable since T>0.Lemma 5.
Let the matrixF be as given in (17) and β2(1+biβ3)>β1Ti; F is Hurwitz stable if and only if HT is a positive stable matrix and(20)Reμi1+biβ3Tiβ21+biβ3Ti-β1-β2Imμi2>0,(21)β1Reμi3β21+biβ3Ti-β12+β1β2ReμiImμi21+biβ3Tiβ21+biβ3Ti-3β1-β1Imμi2β11+biβ3Ti3+β23Reμi2+β23Imμi2>0,where μi is the ith eigenvalue of HT.Proof (Sufficiency).
Letλ be the eigenvalue of F; then(22)det⁡λI3N×3N-F=λIN×N-IN×N0N×N0N×NλIN×N-IN×Nβ1HTβ2HTλIN×N+I+β3BT=det⁡λ3IN×N+λ2I+β3BT+λβ2HT+β1HT=∏iNλ3+1+biβ3Tiλ2+β2μiλ+β1μi=∏iNRiλ,μi.Noting that HT is a positive stable, i.e., Re(μi)>0, and choosing the control gains β1>0, β2>0, and β3>0 such that the conditions (20) and (21) are satisfied, we have that, for the ith polynomial Ri(λ,μi),(23)D1i=1+biβ3Ti,D2i=Reμi1+biβ3Tiβ21+biβ3Ti-β1-β2Imμi2,D3i=β1Reμi3β21+biβ3Ti-β12+β1β2ReμiImμi21+biβ3Tiβ21+biβ3Ti-3β1-β1Imμi2β11+biβ3Ti3+β23Reμi2+β23Imμi2are all positive. According to [36, 37], the roots of Ri(λ,μi) lie in the open left half of the complex plane. Thus, F is Hurwitz stable.
(Necessity). IfHT is not positive stable, there exists Re(μi) which is less than or equal to zero. Then, the corresponding D2i will be less than or equal to zero. This contradicts the fact that the matrix F is Hurwitz stable.Theorem 6.
Consider system (11) and take the control parameters β1, β2, and β3 as in Lemma 5. Then, if and only if node 0 is globally reachable in G-, there exists a constant τ^>0, such that when 0≤τj<τ^(j=1,⋯,N), the consensus is reached asymptotically; i.e.,(24)limt→∞⁡εt=0.Proof (Sufficiency).
Choose appropriate control parameters based on Lemma5. Since node 0 is globally reachable in G-, HT is a positive stable. According to Lemma 5, F is Hurwitz stable. There exists a positive-definite matrix P∈R3N×3N to satisfy(25)PF+FTP=-I3N×3N.Consider the following Razumikhin function for system (19):(26)Vε=εTPε,which satisfies(27)λminPε2≤Vε≤λmaxPε2.From (26), we have(28)V˙ε=ε˙TPε+εTPε˙=εTPF+FTPε-2∑j=1N∑i=0NεTPYjY0∫-τj0εt+sds.According to Lemma 3, let c=-YjTY0TPε, d=ε(t+s), and E=P-1; then (28) becomes(29)V˙ε≤-εTε+∑j=1N∑i=0NτjεTPYjY0P-1Y0TYjTPε+∑j=1N∑i=0N∫-τj0εTt+sPεt+sds.Take φ4(s)=ξs for some constant ξ>1. When(30)Vεt+θ<ξVεt,-τ≤θ≤0,we have(31)V˙ε≤-εTε+∑j=1N∑i=0NτjεTPYjY0P-1Y0TYjTP+ξPε,so, if(32)τj<τ^=1PYjY0P-1Y0TYjTP+ξP,then V˙(ε)≤-ηεTε for some constant η>0. Thus, the conclusion follows from Theorem 1.
(Necessity). Notice that system (11) is asymptotically stable for any τj<τ^, j=1,⋯,N. For the special case τj=0, from (19) the system ε˙(t)=Fε(t) is asymptotically stable. The eigenvalues of F have negative real-parts, which implies that H is positive stable. According to Lemma 4, node 0 is globally reachable in G-.


## 3.1. 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:(8a)uit=∑j=1Nαijβ1xjt-τj-xit-i-j·s+v0t-τ0τj(8b)+∑j=1Nαijβ2vjt-τj-vit(8c)+biβ1x0t-τ0-xit+v0t-τ0τ0-i·s(8d)+β2v0t-τ0-vit(8e)+β3a0t-τ0-ait(8f)+a0t-τ0,where β1>0, β2>0, and β3>0 are the control parameters; (i-j)·s is the desired spacing errors between vehicles i and j which is set according to the spacing policy; and τ0 and τj are, respectively, the time-varying communication delays from the leader and from the vehicle j to the vehicle i. 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 j.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 i and vehicle j with respect to the desired distance (i-j)·s. The term v0t-τ0τj is introduced as the distance compensation due to the time-delay τj of xj.( 2 ) (8b) represents the velocity error between members i and j.( 3 ) (8c) represent the position error between the distance of vehicle i and the platoon leader 0 with respect to the desired distance i·s. The term v0t-τ0τ0 is added as the distance compensation due to the time-delay τ0 of x0.( 4 ) (8d) and (8e) represent, respectively, the velocity error and acceleration error between member i 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.


## 3.2. 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 signalsx0(t), v0(t), and a0(t), i=1,⋯,N, as(9)x-it=xit-x0t+i·s,v-it=vit-v0t,a-it=ait-a0t.We assume that the variation of the vehicle’s velocity during the delay time can be ignored and the leader’s Jerk a˙0(t) is approximately zero. Based on the assumptions, we have vj(t-τj)≈vj(t) and a˙0(t)=0. Then, we can rewrite the coupling control action ui(t) in terms of the state errors x-i(t), v-i(t), and a-i(t). After performing some algebraic manipulation, we obtain the closed-loop dynamics for the generic ith platoon vehicle:(10)x-˙it=v-it,v-˙it=a-it,a-˙it=a˙it-a˙0t=1Ti∑j=1Nαijβ1x-jt-τj-x-it+∑j=1Nαijβ2v-jt-τj-v-it-biβ1x-it+β2v-it-1+biβ3a-it.By defining x-=[x-0,⋯,x-N]T, v-=[v-0,⋯,v-N]T, a-=[a-0,⋯,a-N]T, and ε=x-Tv-Ta-TT, the closed-loop dynamics of the platoon can be written in a more compact form:(11)ε˙t=Y0εt+∑j=1NYjεt-τj.Here(12)Y0=0N×NIN×N0N×N0N×N0N×NIN×N-β1TD+B-β2TD+B-I+β3BT,and(13)Yj=0N×N0N×N0N×N0N×N0N×N0N×Nβ1TAjβ2TAj0N×N,where(14)Aj=0⋯α1j⋯0⋮⋮⋮⋮⋮0⋯αjj⋯0⋮⋮⋮⋮⋮0⋯αNj⋯⋮andT=diag1T1,…,1TN∈RN×N.Applying the Leibniz-Newton formula leads to(15)εt-τj=εt-∫-τj0ε˙t+sds=εt-∑i=0NYi∫-τj0εt+s-τids,where τ0=0. Substituting (15) into (11), we can obtain(16)ε˙t=Fεt-∑j=1N∑i=0NYjYi∫-τj0εt+s-τids,where(17)F=∑i=0NYi=0N×NIN×N0N×N0N×N0N×NIN×N-β1TH-β2TH-I+β3BT,with(18)H=L+B.From (12) and (13), we have YjYi=0 when j=1,⋯,N and i=1,⋯,N. Hence, (16) can be rewritten as(19)ε˙t=Fεt-∑j=1NYjY0∫-τj0εt+sds.Lemma 4 (see [35]).
The matrixH is positive stable if and only if node 0 is globally reachable in G-.
LetHT be the matrix defined by HT=TH. According to Lemma 4, HT is also positive stable since T>0.Lemma 5.
Let the matrixF be as given in (17) and β2(1+biβ3)>β1Ti; F is Hurwitz stable if and only if HT is a positive stable matrix and(20)Reμi1+biβ3Tiβ21+biβ3Ti-β1-β2Imμi2>0,(21)β1Reμi3β21+biβ3Ti-β12+β1β2ReμiImμi21+biβ3Tiβ21+biβ3Ti-3β1-β1Imμi2β11+biβ3Ti3+β23Reμi2+β23Imμi2>0,where μi is the ith eigenvalue of HT.Proof (Sufficiency).
Letλ be the eigenvalue of F; then(22)det⁡λI3N×3N-F=λIN×N-IN×N0N×N0N×NλIN×N-IN×Nβ1HTβ2HTλIN×N+I+β3BT=det⁡λ3IN×N+λ2I+β3BT+λβ2HT+β1HT=∏iNλ3+1+biβ3Tiλ2+β2μiλ+β1μi=∏iNRiλ,μi.Noting that HT is a positive stable, i.e., Re(μi)>0, and choosing the control gains β1>0, β2>0, and β3>0 such that the conditions (20) and (21) are satisfied, we have that, for the ith polynomial Ri(λ,μi),(23)D1i=1+biβ3Ti,D2i=Reμi1+biβ3Tiβ21+biβ3Ti-β1-β2Imμi2,D3i=β1Reμi3β21+biβ3Ti-β12+β1β2ReμiImμi21+biβ3Tiβ21+biβ3Ti-3β1-β1Imμi2β11+biβ3Ti3+β23Reμi2+β23Imμi2are all positive. According to [36, 37], the roots of Ri(λ,μi) lie in the open left half of the complex plane. Thus, F is Hurwitz stable.
(Necessity). IfHT is not positive stable, there exists Re(μi) which is less than or equal to zero. Then, the corresponding D2i will be less than or equal to zero. This contradicts the fact that the matrix F is Hurwitz stable.Theorem 6.
Consider system (11) and take the control parameters β1, β2, and β3 as in Lemma 5. Then, if and only if node 0 is globally reachable in G-, there exists a constant τ^>0, such that when 0≤τj<τ^(j=1,⋯,N), the consensus is reached asymptotically; i.e.,(24)limt→∞⁡εt=0.Proof (Sufficiency).
Choose appropriate control parameters based on Lemma5. Since node 0 is globally reachable in G-, HT is a positive stable. According to Lemma 5, F is Hurwitz stable. There exists a positive-definite matrix P∈R3N×3N to satisfy(25)PF+FTP=-I3N×3N.Consider the following Razumikhin function for system (19):(26)Vε=εTPε,which satisfies(27)λminPε2≤Vε≤λmaxPε2.From (26), we have(28)V˙ε=ε˙TPε+εTPε˙=εTPF+FTPε-2∑j=1N∑i=0NεTPYjY0∫-τj0εt+sds.According to Lemma 3, let c=-YjTY0TPε, d=ε(t+s), and E=P-1; then (28) becomes(29)V˙ε≤-εTε+∑j=1N∑i=0NτjεTPYjY0P-1Y0TYjTPε+∑j=1N∑i=0N∫-τj0εTt+sPεt+sds.Take φ4(s)=ξs for some constant ξ>1. When(30)Vεt+θ<ξVεt,-τ≤θ≤0,we have(31)V˙ε≤-εTε+∑j=1N∑i=0NτjεTPYjY0P-1Y0TYjTP+ξPε,so, if(32)τj<τ^=1PYjY0P-1Y0TYjTP+ξP,then V˙(ε)≤-ηεTε for some constant η>0. Thus, the conclusion follows from Theorem 1.
(Necessity). Notice that system (11) is asymptotically stable for any τj<τ^, j=1,⋯,N. For the special case τj=0, from (19) the system ε˙(t)=Fε(t) is asymptotically stable. The eigenvalues of F have negative real-parts, which implies that H is positive stable. According to Lemma 4, node 0 is globally reachable in G-.


## 4. 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.Table 1
Parameters for the traffic simulation and consensus control algorithm.
Parameter Value Parameter Value Vehicle lengthli 4 m Maximum acceleration 3 m2/s Stable speed 25 m/s Maximum deceleration 5 m2/s Vehicle drivetrainTi 0.5 s Maximum velocity 35 m/s Control parameters β 1 = 2, β2=2, β3=3 Standstill distances 15 m ifbi>0 b i = 10We select a typical communication topology: the leader- and predecessor-following topology considering information from both the preceding vehicle and the leader (see Figure1).Figure 1
The information flow topology used in simulations: leader- and predecessor-following topology.


### 4.2. 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. Figure2 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.Figure 2
Platoon performance for the initial scenario: (a) the position errorxi(t)-x0(t)+i⋅s; (b) the speed error vi(t)-v0(t).
(a) (b)


### 4.3. 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 m2/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.Figure 3
Platoon performance for a single large perturbation scenario: (a) the position errorxi(t)-x0(t)+i⋅s; (b) speed; (c) acceleration.
(a) (b) (c)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:(33)δt=Asin⁡0.2πt,A=2.7m/s.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.Figure 4
Platoon performance for the sinusoidal disturbance scenario: (a) the relative positionxi-1(t)-xi(t)+s; (b) the relative speed vi-1(t)-vi(t); (c) acceleration.
(a) (b) (c)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 t=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.Figure 5
Platoon performance for the initial scenario under spoofing attack: (a) the position errorxi(t)-x0(t)+i⋅s; (b) the speed error vi(t)-v0(t).
(a) (b)


### 4.4. 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. Figure6 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 6
Platoon performance in the presence of the initial scenario under different platoon lengths: (a) the position errorxi(t)-x0(t)+i⋅s; (b) the speed error vi(t)-v0(t).
(a) (b)Figure7 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.Figure 7
Platoon performance in the presence of the sinusoidal disturbance scenario under different platoon lengths: (a) speed; (b) acceleration.
(a) (b)


## 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.Table 1
Parameters for the traffic simulation and consensus control algorithm.
Parameter Value Parameter Value Vehicle lengthli 4 m Maximum acceleration 3 m2/s Stable speed 25 m/s Maximum deceleration 5 m2/s Vehicle drivetrainTi 0.5 s Maximum velocity 35 m/s Control parameters β 1 = 2, β2=2, β3=3 Standstill distances 15 m ifbi>0 b i = 10We select a typical communication topology: the leader- and predecessor-following topology considering information from both the preceding vehicle and the leader (see Figure1).Figure 1
The information flow topology used in simulations: leader- and predecessor-following topology.


## 4.2. 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. Figure2 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.Figure 2
Platoon performance for the initial scenario: (a) the position errorxi(t)-x0(t)+i⋅s; (b) the speed error vi(t)-v0(t).
(a) (b)


## 4.3. 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 m2/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.Figure 3
Platoon performance for a single large perturbation scenario: (a) the position errorxi(t)-x0(t)+i⋅s; (b) speed; (c) acceleration.
(a) (b) (c)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:(33)δt=Asin⁡0.2πt,A=2.7m/s.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.Figure 4
Platoon performance for the sinusoidal disturbance scenario: (a) the relative positionxi-1(t)-xi(t)+s; (b) the relative speed vi-1(t)-vi(t); (c) acceleration.
(a) (b) (c)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 t=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.Figure 5
Platoon performance for the initial scenario under spoofing attack: (a) the position errorxi(t)-x0(t)+i⋅s; (b) the speed error vi(t)-v0(t).
(a) (b)


## 4.4. 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. Figure6 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 6
Platoon performance in the presence of the initial scenario under different platoon lengths: (a) the position errorxi(t)-x0(t)+i⋅s; (b) the speed error vi(t)-v0(t).
(a) (b)Figure7 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.Figure 7
Platoon performance in the presence of the sinusoidal disturbance scenario under different platoon lengths: (a) speed; (b) acceleration.
(a) (b)


## 5. 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.



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*Source: 8963289-2018-07-24.xml*