# Computation of Time Delay Stability Margin for the Automated Vehicular Platoon

Authors: Xunxun Zhang; Li Li; Xu Zhu
Journal: Complexity
 (2020)

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## Abstract

Due to the limited band width and congestion of communication channels in the wireless vehicle-to-vehicle (V2V) communication, time delay inevitably arises and dramatically leads to the disturbances for the automated vehicular platoon. This paper focuses on computing the exact time delay stability margin. In this study, we treat this problem as a stability issue of a consensus system with time delay, where each vehicle in the platoon is recognized as a node, and the interconnected information flow is represented as a graph. Then, the distributed controller is designed by combining the states of the vehicle itself and its neighbouring vehicles. Furthermore, the stability of the entire platoon is analysed according to the characteristic equation of the closed-loop system, and a necessary and sufficient condition for the exact time delay stability margin is obtained. Especially, for the automated vehicular platoon with undirected topology, it is revealed that exact time delay stability margin is determined by the largest eigenvalue of the augmented Laplacian matrix. Furthermore, a rapid method for finding exact time delay stability margin is proposed. Finally, numerical simulations demonstrate that this work generates exact and satisfactory time delay stability margin for the automated vehicular platoon.


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## Body

## 1. Introduction

Vehicular platoon is a vital component of intelligent vehicle infrastructure cooperative systems (IVICS), which is the frontier of intelligent transportation system (ITS) [1]. An automated vehicular platoon consists of a group of coordinated vehicles, which move at an identical speed and maintain a prespecified formation geometry. Due to team cooperation, the automated vehicular platoon is conductive to improving traffic capacity, enhancing highway safety, and reducing exhaust emission and fuel consumption [2].Recently, the automated vehicular platoon technology has attracted considerable attentions [3–6], which can be recognized as a combination of four components, i.e., vehicular longitudinal dynamics, interconnected information flow, distributed controllers, and intervehicle spacing policy [7, 8]. First, the vehicular longitudinal dynamics describes the dominant longitudinal behaviour of each vehicle. Second, the interconnected information flow depicts how the vehicles in the vehicular platoon exchange state information with the others. While its physical implementation depends on wireless vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication [9]. Moreover, the structure of interconnected information flow can be classified into two types in terms of undirected and directed topologies. Third, the distributed controllers are utilized for the specific feedback control of each vehicle. Most controllers are linear [7, 10]. Fourth, the intervehicle spacing policy arranges the desired distance between the two successive vehicles and further determines the formation geometry.In an automated vehicular platoon, the interconnected information flow depends on wireless communication. Due to the limited band width and congestion of the communication channels, it is inevitable to introduce time delay to the vehicular platoon [11, 12]. Meanwhile, the time delay is the inherent phenomenon in the wireless communication networks, which is universally acknowledged as the main factor for the vehicular platoon and leads to string instability [13]. Therefore, it is of high importance to study the vehicular platoon with time delay. Therein, we focus on the computation of time delay stability margin for the automated vehicular platoon under both undirected and directed topologies.To obtain the time delay stability margin, most literatures analyse the closed-loop system of the vehicular platoon and recognize the entire vehicular platoon as a consensus system [14]. For the stability analysis problem of the consensus system with time delay, the majority of the existing studies usually utilize Lyapunov-Krasovskii methodology [15, 16], Lyapunov–Razumikin methodology [12, 17], and generalised Nyquist criterion [18, 19]. However, these studies can provide only sufficient conditions, which are relatively conservative and fail to find the exact time delay stability margin. On the contrary, because these results mainly rely on some linear matrix inequalities, they are usually imprecise and cumbersome to deploy.Motivated by this fact, we focus on offering an analytical method to find the exact time delay stability margin for the automated vehicular platoon under both the directed and undirected topologies. Therein, a second-order consensus system is used for modelling the longitudinal dynamics of the vehicular platoon, designing the control gains, and describing the interconnected information flow. By investigating the distribution of the roots for the closed-loop system’s characteristic equation, we provide a necessary and sufficient condition for the stability of the vehicular platoon. Then, on the basis of the necessary and sufficient condition, the exact time delay stability margin can be found. Specially, for the undirected topology, by analysing the monotonicity relationship between each eigenvalue of the augmented Laplacian matrix and its corresponding time delay, it is revealed that the exact time delay stability margin is determined by the largest eigenvalue. Therefore, a more rapid method for computing the exact time delay stability margin is further proposed for the undirected topology.In a word, the main contributions of this work are twofold. Firstly, to the best of the authors’ knowledge, it is the first time that the exact time delay stability margin is analytically acquired for the second-order platoon. The exact time delay stability margin is obtained by searching the pure imaginary roots of the characteristic equation for the closed-loop system, and it is further demonstrated that only the positive pure imaginary root should be taken into consideration. Secondly, for the undirected topology, we originally propose a simple and rapid method to calculate the exact time delay stability margin. Only one time delay, which corresponds to the largest eigenvalue of the augmented Laplacian matrix, needs to be computed. Therefore, it is unnecessary to compute the delays for the other eigenvalues. For the large-scale vehicular platoon, the second contribution is particularly useful to dramatically reduce the computational burden.The remainder of this paper is organised as follows. The problem statement is presented in Section2. The computation of time delay stability margin for the automated vehicular platoon is elaborated in Section 3. In Section 4, numerical simulations are provided to demonstrate the validity and superiority of the proposed algorithm. Finally, the conclusions are summarized in Section 5.


## 2. Problem Statement

For the automated vehicular platoon, it can be treated as a combination of four components,i.e., vehicular longitudinal dynamics, interconnected information flow, distributed controllers, and intervehicle spacing policy. In this part, the four components will be elaborately described, respectively.


### 2.1. Model for Interconnected Information Flow

In our work, we consider the platoon withN+1 identical interconnected vehicles, including one leader and N followers. There are two typical interconnected information flows in terms of undirected and directed topologies. For the undirected topology (as depicted in Figure 1(a)), except that the leader vehicle transmits to the follower vehicles, the follower vehicles share state information with their neighbouring vehicles. For the directed topology (as depicted in Figure 1(b)), the vehicle receives the state information only from the following and the preceding vehicle. According to graph theory, there are some distinct properties between these two kinds of topologies. Especially, it is easier to analyse the stability of undirected topology.Figure 1
Typical interconnected information flow for vehicular platoons. (a) Undirected topology. (b) Directed topology.
(a)
(b)For both the undirected and directed topologies, the interconnected information flow among the followers can be modelled by a graphG=V,ℰ,A, where each vehicle is recognized as a node. V=1,2,…,N is the set of nodes and ℰ⊆V×V is the set of edges. The adjacency matrix associated with the graph G is characterized by A=aijN×N, where i,j∈V. The nonnegative adjacency weight aij=1 means that the node i can get information from the node j; otherwise, aij=0. Moreover, we assume that aii=0 (no self-loop is allowed unless otherwise indicated).To model the interconnected information flow between the leader and the followers, an augmented graph is defined asG˜=V˜,ℰ˜,A˜, where V˜=0,1,2,…,N is the set of nodes including the leader and followers, and the index 0 represents the leader vehicle. ℰ˜⊆V˜×V˜ is the edge set appended with the information flow from the leader to the followers, and A˜ is the augmented adjacency matrix.After modelling the interconnected information flow, we aim to describe its property with the graphsG and G˜. Therefore, two important matrices (i.e., Laplacian matrix and pinning matrix) are introduced as follows. Firstly, the Laplacian matrix ℒ=lijN×N, which is associated with the graph G, is defined as(1)lij=−aij,i≠j,∑k=1Naik,i=j.Secondly, the pinning matrixP, which is associated with the augmented graph G˜ characterizing the interconnected information flow from the leader to the followers, is defined as(2)P=diagp1,p2,…,pN,where pi=1 if the edge i,0∈ℰ˜; otherwise, pi=0. The expression pi=1 means that the ith vehicle can get information from the leader. pi is called pinning gain in the field of complex networks [20]. If pi=1, the vehicle i is said to be pinned to the leader.According to the algebraic graph theory [21], the augmented graph G˜ should contain at least one spanning tree rooting from the leader vehicle if the vehicular platoon is asymptotically stable. A spanning tree is an algebraic tree formed by some or all the edges of the augmented graph G˜ that links all the nodes. The graph G˜ containing a spanning tree means that a subset of its edges forms a spanning tree. Therefore, for a stable vehicular platoon, there should exist at least one directed path from the leader to each follower. In contrary, each follower can acquire information from the leader directly or indirectly.


### 2.2. Vehicle Longitudinal Dynamics

For the longitudinal dynamics of each vehicle, it is mathematically charactered by a second-order linear model. Moreover, to reduce the model complexity, the input-output feedback linearisation is applied to build the model. Then, the longitudinal dynamics of theith vehicle are denoted by(3)r˙i=vit,v˙i=uit,where ri and vi are the position and velocity of the ith vehicle, respectively. The propelling force ui represents the control input. Once ui is appropriately chosen, the vehicular platoon is capable of achieving the desired spacing, maintaining an identical speed, and performing synchronous braking maneuvers. It is assumed that an inner-loop automatic controller exists in each vehicle for responding to the control input ui.


### 2.3. Intervehicle Spacing Policy

The intervehicle spacing policy plays a vital role in the vehicular platoon control. In contrary, to track the speed of the leader vehicle and realize the predefined formation geometry, the objective of the vehicular platoon control is governed by the intervehicle policy:(4)rit⟶rjt+dij,vit⟶v0t,i∈V,where dij is the desired intervehicle spacing between the ith vehicle and the jth vehicle. It is assumed that dij=−dji. The formation geometry of the vehicular platoon is determined by dij. There are two main kinds of intervehicle spacing policies, i.e., constant distance (CD) policy and constant time head headway (CTH) policy. For the CD policy, dij is set to be a constant number:(5)dij=j−idi−1,i=j−id0,i∈V,where d0 is the constant spacing between the ith vehicle and its preceding vehicle.For the CTH policy,di−1,i is dependent on the vehicle velocity vi:(6)di−1,i=h⋅vi+d0,i∈V,where h is the constant time headway.In accordance with some geometrical considerations, the spacing policydi−1,i can be recast into the spacing with respect to the leader vehicle as dij=di0−dj0. Thus, the objective of the platoon control in (4) can be rewritten as follows:(7)rit⟶r0t+di0,vit⟶v0t,i∈V,where di0 is the desired distance from the leader to the ith vehicle.


### 2.4. Distributed Controller for the Vehicular Platoon with Time Delay

Nowadays, most literatures focus on designing the controller for the vehicular platoon without time delay [21, 22], and the distributed controller is designed as follows:(8)uit=−kr∑j=0Naijrit−rjt−dij−kv∑j=0Naijvit−vjt,where kr>0 and kv>0 are the control gains for the position and velocity, respectively.However, due to the limited band width and congestion of the communication channels, the time delay inevitably appears in the vehicular platoon. Meanwhile, the time delay is the inherent phenomenon in the wireless communication networks, which is universally acknowledged as the main factor for the performance of the vehicular platoon, leading to the string instability. Therefore, by considering the time delay into the interconnected information flow, the distributed controller for the vehicular platoon with time delay can be designed as(9)uit=−kr∑j=0Naijrit−τ−rjt−τ−dij−kv∑j=0Naijvit−τ−vjt−τ,where τ represents the time delay. Generally, clock synchronization is guaranteed throughout the platoon via GPS [23], so the value of τ can be computed out according to the transmission time.


## 2.1. Model for Interconnected Information Flow

In our work, we consider the platoon withN+1 identical interconnected vehicles, including one leader and N followers. There are two typical interconnected information flows in terms of undirected and directed topologies. For the undirected topology (as depicted in Figure 1(a)), except that the leader vehicle transmits to the follower vehicles, the follower vehicles share state information with their neighbouring vehicles. For the directed topology (as depicted in Figure 1(b)), the vehicle receives the state information only from the following and the preceding vehicle. According to graph theory, there are some distinct properties between these two kinds of topologies. Especially, it is easier to analyse the stability of undirected topology.Figure 1
Typical interconnected information flow for vehicular platoons. (a) Undirected topology. (b) Directed topology.
(a)
(b)For both the undirected and directed topologies, the interconnected information flow among the followers can be modelled by a graphG=V,ℰ,A, where each vehicle is recognized as a node. V=1,2,…,N is the set of nodes and ℰ⊆V×V is the set of edges. The adjacency matrix associated with the graph G is characterized by A=aijN×N, where i,j∈V. The nonnegative adjacency weight aij=1 means that the node i can get information from the node j; otherwise, aij=0. Moreover, we assume that aii=0 (no self-loop is allowed unless otherwise indicated).To model the interconnected information flow between the leader and the followers, an augmented graph is defined asG˜=V˜,ℰ˜,A˜, where V˜=0,1,2,…,N is the set of nodes including the leader and followers, and the index 0 represents the leader vehicle. ℰ˜⊆V˜×V˜ is the edge set appended with the information flow from the leader to the followers, and A˜ is the augmented adjacency matrix.After modelling the interconnected information flow, we aim to describe its property with the graphsG and G˜. Therefore, two important matrices (i.e., Laplacian matrix and pinning matrix) are introduced as follows. Firstly, the Laplacian matrix ℒ=lijN×N, which is associated with the graph G, is defined as(1)lij=−aij,i≠j,∑k=1Naik,i=j.Secondly, the pinning matrixP, which is associated with the augmented graph G˜ characterizing the interconnected information flow from the leader to the followers, is defined as(2)P=diagp1,p2,…,pN,where pi=1 if the edge i,0∈ℰ˜; otherwise, pi=0. The expression pi=1 means that the ith vehicle can get information from the leader. pi is called pinning gain in the field of complex networks [20]. If pi=1, the vehicle i is said to be pinned to the leader.According to the algebraic graph theory [21], the augmented graph G˜ should contain at least one spanning tree rooting from the leader vehicle if the vehicular platoon is asymptotically stable. A spanning tree is an algebraic tree formed by some or all the edges of the augmented graph G˜ that links all the nodes. The graph G˜ containing a spanning tree means that a subset of its edges forms a spanning tree. Therefore, for a stable vehicular platoon, there should exist at least one directed path from the leader to each follower. In contrary, each follower can acquire information from the leader directly or indirectly.


## 2.2. Vehicle Longitudinal Dynamics

For the longitudinal dynamics of each vehicle, it is mathematically charactered by a second-order linear model. Moreover, to reduce the model complexity, the input-output feedback linearisation is applied to build the model. Then, the longitudinal dynamics of theith vehicle are denoted by(3)r˙i=vit,v˙i=uit,where ri and vi are the position and velocity of the ith vehicle, respectively. The propelling force ui represents the control input. Once ui is appropriately chosen, the vehicular platoon is capable of achieving the desired spacing, maintaining an identical speed, and performing synchronous braking maneuvers. It is assumed that an inner-loop automatic controller exists in each vehicle for responding to the control input ui.


## 2.3. Intervehicle Spacing Policy

The intervehicle spacing policy plays a vital role in the vehicular platoon control. In contrary, to track the speed of the leader vehicle and realize the predefined formation geometry, the objective of the vehicular platoon control is governed by the intervehicle policy:(4)rit⟶rjt+dij,vit⟶v0t,i∈V,where dij is the desired intervehicle spacing between the ith vehicle and the jth vehicle. It is assumed that dij=−dji. The formation geometry of the vehicular platoon is determined by dij. There are two main kinds of intervehicle spacing policies, i.e., constant distance (CD) policy and constant time head headway (CTH) policy. For the CD policy, dij is set to be a constant number:(5)dij=j−idi−1,i=j−id0,i∈V,where d0 is the constant spacing between the ith vehicle and its preceding vehicle.For the CTH policy,di−1,i is dependent on the vehicle velocity vi:(6)di−1,i=h⋅vi+d0,i∈V,where h is the constant time headway.In accordance with some geometrical considerations, the spacing policydi−1,i can be recast into the spacing with respect to the leader vehicle as dij=di0−dj0. Thus, the objective of the platoon control in (4) can be rewritten as follows:(7)rit⟶r0t+di0,vit⟶v0t,i∈V,where di0 is the desired distance from the leader to the ith vehicle.


## 2.4. Distributed Controller for the Vehicular Platoon with Time Delay

Nowadays, most literatures focus on designing the controller for the vehicular platoon without time delay [21, 22], and the distributed controller is designed as follows:(8)uit=−kr∑j=0Naijrit−rjt−dij−kv∑j=0Naijvit−vjt,where kr>0 and kv>0 are the control gains for the position and velocity, respectively.However, due to the limited band width and congestion of the communication channels, the time delay inevitably appears in the vehicular platoon. Meanwhile, the time delay is the inherent phenomenon in the wireless communication networks, which is universally acknowledged as the main factor for the performance of the vehicular platoon, leading to the string instability. Therefore, by considering the time delay into the interconnected information flow, the distributed controller for the vehicular platoon with time delay can be designed as(9)uit=−kr∑j=0Naijrit−τ−rjt−τ−dij−kv∑j=0Naijvit−τ−vjt−τ,where τ represents the time delay. Generally, clock synchronization is guaranteed throughout the platoon via GPS [23], so the value of τ can be computed out according to the transmission time.


## 3. Time Delay Stability Margin for the Automated Vehicular Platoon

In this part, we aim to compute the time delay stability margin for the automated vehicular platoon under both undirected and directed topologies. To obtain the time delay stability margin, it is necessary to elaborately analyse the closed-loop system of the vehicular platoon. Besides, to compute the exact time delay stability margin, it is essential to obtain the necessary and sufficient condition. Furthermore, it is revealed that the exact time delay stability margin is determined by the largest eigenvalue, and a more rapid method for computing the exact time delay stability margin is proposed for the undirected topology.


### 3.1. Stability Analysis for the Automated Vehicular Platoon

To analyse the stability of system (3) with the distributed controller (9), the error states, which is determined with the comparison of the state information for the leader vehicle, is defined as follows:(10)r˜=r˜1,r˜2,…,r˜NT,v˜=v˜1,v˜2,…,v˜NT,where r˜i=rit−r0t−di0 is the spacing error state and v˜i=vit−v0t is the velocity error state. r0t and v0t represent the position and velocity of the leader vehicle.By substituting the error states into the distributed controller (9), it can be recast in terms of the error states as(11)uit=−kr∑j=0Naijr˜it−τ−r˜jt−τ−kv∑j=0Naijv˜it−τ−v˜jt−τ.Therefore, by using the error dynamics (11), system (3) can be rewritten as follows:(12)r˜˙i=v˜it,v˜˙i=−kr∑j=0Naijr˜it−τ−r˜jt−τ−kv∑j=0Naijv˜it−τ−v˜jt−τ.For simplicity, we define the error state vector asx˜=r˜T,v˜TT. Then, the collective closed-loop dynamics of the vehicular platoon are rewritten into a compact form as follows:(13)x˜˙=Ax˜+Bx˜t−τ,with(14)A=∗20c0N×NIN×N0N×N0N×N,B=∗20c0N×N0N×N−krℒ+P−kvℒ+P,where IN×N and 0N×N denote the N-dimensional identity matrix and zero matrix, respectively.So far, the closed-loop dynamics of the vehicular platoon is build. In the following, the stability analysis will be implemented according to the augmented Laplacian matrix.Therein, we define the eigenvalue of the augmented Laplacian matrixℒ+P as λi,i∈V, which mathematically reflects the important features of the augmented graph G˜. By utilizing the eigenvalue λi, it is convenient to analyse the stability of the closed-loop dynamics (13). Moreover, to facilitate the stability analysis of the closed-loop dynamics (13), we utilize the method in [24] to divide the entire vehicular platoon into some small synchronous subsystems.Then, the characteristic equation of system (13) is given by(15)detsI−A−Be−τs=∏i=1Nfis=0,where(16)fis=s2+kvs+krλie−τs.Before analysing the stability of the closed-loop dynamics (13), some lemmas are provided as follows.Lemma 1.
When the augmented graphG˜ contains a spanning tree, all the eigenvalues of ℒ+P are located in the open right half complex plane, i.e., λi>0,i∈V.
Lemma1 describes the distribution of the eigenvalues of ℒ+P, and it is fundamental to indicate the stability and compute the exact time delay stability margin from the augmented graph G˜.Lemma 2.
When the graphG is undirected, all the eigenvalues of ℒ+P are positive real numbers, i.e., λi∈ℝ+,i∈V.
Lemma2 describes the special character of undirected topologies. When the interconnected information flow among the follower vehicles is undirected, it is useful to simplify the theoretical deduction of verifying the stability. The proof of Lemma 1 and 2 can be found in [21].Lemma 3.
The closed-loop dynamics (13) are asymptotically stable if and only if every equation (16) is Hurwitz stable, i.e., the roots of the closed-loop system’s characteristic equation are all located in the open left half complex plane.
Based on the abovementioned three lemmas, we obtain the first stability result for the closed-loop dynamics (13).Theorem 1.
If the closed-loop dynamics (13) is asymptotically stable, the following inequality holds:(17)kv2kr>maxi∈VIm2λiReλiλi2,where λi is the module of λi. Reλi and Imλi are the real part and imaginary part of λi, respectively.Proof.
The closed-loop dynamics (13) should be primarily stable for the delay-free case [25]. When τ=0, there exists(18)fis=s2+kvs+krλi=0.
Asλi=Reλi+ι⋅Imλi,ι2=−1 is probably a complex number, (18) is transformed into a fourth-order equation as(19)s4+2kvReλis3+kv2λi2+2krReλis2+2krkvλi2s+kr2λi2=0.
Then, the stability of (19) is examined based on the Routh–Hurwitz stability criterion:(20)s41kv2λi2+2krReλikr2λi2s32kvReλi2krkvλi2s2kv2Reλiλi2+krRe2λi−krIm2λiReλikr2λi2s12krkvλi2kv2Reλiλi2−krIm2λikv2Reλiλi2+krRe2λi−krIm2λis0kr2λi2.
The Routh table is listed out in equation (20). Considering the definition that kr>0,kv>0 in (9) and the fact that Reλi>0 from Lemma 1, it is concluded that (18) is asymptotically stable if and only if(21)kv2kr>Im2λiReλiλi2.
Taking allλi into consideration, if the closed-loop dynamics (13) are asymptotically stable, inequality (17) holds.Remark 1.
The stability condition (17) of Theorem 1 is a necessary and sufficient condition of the closed-loop dynamics (13) for the delay-free case, but it is just a necessary condition for the delay case. For τ>0, the stability of the closed-loop dynamics (13) is not only correlated with the control gains and structure of the interconnected information flow but also dependent on the time delay.Corollary 1.
Forτ=0, when the graph G is undirected, the closed-loop dynamics (13) are asymptotically stable, for any kr>0,kv>0.
When the graphG is undirected, there exists Imλi=0 according to Lemma 2. Hence, inequality (17) holds for any kr>0,kv>0.


### 3.2. Computation of the Exact Time Delay Stability Margin for the Automated Vehicular Platoon

For the delay caseτ>0, we review the characteristic equation (16) as(22)fis=s2+kvs+krλie−τs.It is obvious thatfis,i∈V is a quasi-polynomial and its characteristic equation fis=0 is a transcendental equation. Hence, it is complicated to analyse the roots of the characteristic equation. Despite of this, we are still devoted to checking the property of the roots to find the exact time delay stability margin.Theorem 2.
For the closed-loop dynamics (13) satisfying condition (17), let ωi>0 be the root of the following equation:(23)ωi4−kv2λi2ωi2−kr2λi2=0,with ωi=kv2λi2+kv4λi4+4kr2λi2. Take(24)τi=arctanξi+kπωi,k∈0,1,where ξi=−δi/θi with δi=kvωiReλi+krImλi and θi=kvωiImλi−krReλi. Let τmin=mini∈Vτi. Then, system (13) is asymptotically stable if and only if τ∈0,τmin.Proof.
According to Corollary 2.4 in [26], it is known that for a quasi-polynomial gs,e−τs=g0s+g1se−τs, if gs,e−τs is Hurwitz stable for τ=0 and unstable for τ>τmin, there must exist a root on the imaginary axis for gs,e−τmins. In our work, if the stability condition (17) in Theorem 1 is satisfied, fis in (22) is Hurwitz stable for τ=0. Thus, it is available to utilize Corollary 2.4 in [26] to find the exact time delay stability margin τmin. Afterwards, we aim to find the root ι⋅ωi on the imaginary axis for τi.
With respect to the imaginary rootι⋅ωi, there exists fiι⋅wi=0, which implies that both the real and imaginary parts of fiι⋅wi are zero, shown as(25)−ωi2+δisinτiωi−θicosτiωi=0,θisinτiωi+δicosτiωi=0.
By rearranging (25), two following trigonometric functions are obtained:(26)sinτiωi=θiωi2δi2+θi2,cosτiωi=−δiωi2δi2+θi2,where δi2+θi2=kv2ωi2+kr2λi2. According to the well-known trigonometric property sin2τiωi+cos2τiωi=1, it is concluded that(27)ωi4−kv2λi2ωi2−kr2λi2=0.
Then, two real-valued roots aboutωi, which are positive and negative, respectively, can be solved out. Meanwhile, it is obtained that ξi=tanτiωi=sinτiωi/cosτiωi=−θi/δi based on (26). The delay for i∈V is also deduced as τi=arctanξi+kπ/ωi,k∈0,1, where the integer k should be chosen as the minimum value satisfying τi>0.
Next, we aim to demonstrate that only the situationωi>0 needs to be considered. Two cases for the graph G, i.e., undirected and directed topologies, are taken into account as follows:(i)
The undirected graphG: in this case, Imλi=0. We assume that the two real-valued roots of (27) are ωi1,ωi2, and ωi1>0,ωi2<0. It is obvious that ωi2=−ωi1. Based on this, one can find that δi2=−δi1,θi2=θi1,andξi2=−ξi1. There must exist arctanξi2/ωi2=arctanξi1/ωi1 such that we can simply take τi2=τi1. In a word, considering only the positive root ωi>0 suffices to obtain τi, because the imaginary roots for fis form complex conjugate pairs.(ii)
The directed graphG: in this case, Imλi≠0. There must also exist a conjugate eigenvalue of λi for the directed graph G, where the conjugate eigenvalue is also a root of (27). We define the conjugate eigenvalue as λl=Reλi−ι⋅Imλi,l∈V. Let ωi1>0,ωi2<0,ωl1>0,andωl2<0. According to (27), it is easy to see that ωl1=ωi1andωl2=ωi2. Then, one can find that δl2=−δi1,θl2=θi1,andξl2=−ξi1. There must exist arctanξl2/ωl2=arctanξi1/ωi1 such that we can simply take τl2=τi1. Therefore, for the directed graph G, considering only the positive root ωi>0 still suffices to obtain τi.
Then, by synthesizing the above two cases with the fact thatωi>0, the positive root of (27) is solved out as follows:(28)ωi=kv2λi2+kv4λi4+4kr2λi2.
At last, we scan all the eigenvalueλi,i∈V to compute all τi and find the exact time margin τmin=mini∈Vτi.Remark 2.
Theorem2 provides a method for finding the exact time delay stability margin τmin of the closed-loop dynamics (13). Theorem 2 is a necessary and sufficient condition for the stability for the delay case.


### 3.3. A Rapid Method for Computing the Exact Time Delay Stability Margin under the Undirected Topology

A large-scale vehicular platoon generally generates plenty of eigenvalues of the graphG. It is time consuming to scan all the eigenvalue λi,i∈V to compute all τi. However, it is common knowledge that the most critical eigenvalue directly determines the exact time delay stability margin. If the most critical eigenvalue is found, there is no need to scan all the other eigenvalues. Moreover, because it is very fussy to analyse the most critical eigenvalue for the directed graph, we only consider the undirected topology. Therefore, we focus on look for the most critical eigenvalue under the undirected topology, aiming to reduce the computational burden and quickly compute the exact time delay stability margin.Theorem 3.
For the closed-loop dynamics (13), under the undirected graph G, let λmax=maxi∈Vλi. Then, λmax is the most critical eigenvalue, which determines the exact time delay stability margin τmin.Proof.
Although it is difficult to directly confirm whether the monotonicity ofτi depends on the monotonicity of λi, we plan to reveal their monotonous relationship through two steps. By treating ωi as an immediate variable, the first step is to verify that the monotonicity of ωi depends on the monotonicity of λi, and the second step is to verify that the monotonicity of τi depends on the monotonicity of ωi.
In the first step, we only consider the caseωi>0 in accordance with Theorem 2. When the graph G is undirected, (28) turns into the following form as follows:(29)ωi=12kv2λi2+kv4λi4+4kr2λi2.
According to Lemma2, λi is a positive real number for the undirected topology, so it can be seen that ωi gets larger with the increase of λi. Then, it is demonstrated that the monotonicity of the immediate variable ωi depends on the monotonicity of λi.
In the second step, we aim to verify that the monotonicity ofτi depends on the monotonicity of ωi. with respect to (24), there exists τi=ωi−1arctanξi=ωi−1arctankvωi/kr with ωi>0. We intend to apply the partial derivative property to conform the monotonicity. The partial derivative ∂τi/∂ωi is calculated as follows:(30)∂τi∂ωi=−1ωi2arctankvωikr+1ωi⋅krkvkr2+kv2ωi2=1ωi2−arctankvωikr+ωi⋅krkvkr2+kv2ωi2,with ∂arctankvωi/kr/∂ωi=krkv/kr2+kv2ωi2.
In (30), the term arctankvωi/kr still exists. Therefore, it is uneasy to identify whether (30) is positive or negative. To solve this problem, we deduce the partial derivative once again, and an auxiliary function is given by(31)dωi=−arctankvωikr+ωi⋅krkvkr2+kv2ωi2.
Comparing with (31), (30) can be rewritten as(32)∂τi∂ωi=1ωi2dωi.
Then, the partial derivative∂dωi/∂ωi is calculated as(33)∂dωi∂ωi=−krkvkr2+kv2ωi2+kr3kv−krkv3ωi2kr2+kv2ωi22=−2krkv3ωi2kr2+kv2ωi22.
With the definition thatkr>0,kv>0 in (9), it is noted that ∂dωi/∂ωi=0 when ωi=0, and ∂dωi/∂ωi<0 when ωi>0. Then, it is concluded that ∂τi/∂ωi<0 in (32) when ωi>0, which means that τi gets smaller with the increase of ωi. Until now, we verify that the monotonicity of τi depends on that of ωi by using the partial derivative property.
By taking the abovementioned two steps into account, it is confirmed thatτi gets smaller with the increase of λi. Hence, the largest eigenvalue λmax is most critical and determines the exact time delay stability margin τmin.Remark 3.
Theorem3 reveals the inherent relationship between the exact time delay stability margin and the eigenvalues of the augmented Laplacian matrix for the undirected graph G. According to Theorem 3, just computing one delay corresponding to the largest eigenvalue λmax suffices to obtain the exact time delay stability margin, and it is no longer necessary to scan the other eigenvalues. For the large-scale vehicular platoon, utilizing Theorem 3 can dramatically reduce the computational burden.


## 3.1. Stability Analysis for the Automated Vehicular Platoon

To analyse the stability of system (3) with the distributed controller (9), the error states, which is determined with the comparison of the state information for the leader vehicle, is defined as follows:(10)r˜=r˜1,r˜2,…,r˜NT,v˜=v˜1,v˜2,…,v˜NT,where r˜i=rit−r0t−di0 is the spacing error state and v˜i=vit−v0t is the velocity error state. r0t and v0t represent the position and velocity of the leader vehicle.By substituting the error states into the distributed controller (9), it can be recast in terms of the error states as(11)uit=−kr∑j=0Naijr˜it−τ−r˜jt−τ−kv∑j=0Naijv˜it−τ−v˜jt−τ.Therefore, by using the error dynamics (11), system (3) can be rewritten as follows:(12)r˜˙i=v˜it,v˜˙i=−kr∑j=0Naijr˜it−τ−r˜jt−τ−kv∑j=0Naijv˜it−τ−v˜jt−τ.For simplicity, we define the error state vector asx˜=r˜T,v˜TT. Then, the collective closed-loop dynamics of the vehicular platoon are rewritten into a compact form as follows:(13)x˜˙=Ax˜+Bx˜t−τ,with(14)A=∗20c0N×NIN×N0N×N0N×N,B=∗20c0N×N0N×N−krℒ+P−kvℒ+P,where IN×N and 0N×N denote the N-dimensional identity matrix and zero matrix, respectively.So far, the closed-loop dynamics of the vehicular platoon is build. In the following, the stability analysis will be implemented according to the augmented Laplacian matrix.Therein, we define the eigenvalue of the augmented Laplacian matrixℒ+P as λi,i∈V, which mathematically reflects the important features of the augmented graph G˜. By utilizing the eigenvalue λi, it is convenient to analyse the stability of the closed-loop dynamics (13). Moreover, to facilitate the stability analysis of the closed-loop dynamics (13), we utilize the method in [24] to divide the entire vehicular platoon into some small synchronous subsystems.Then, the characteristic equation of system (13) is given by(15)detsI−A−Be−τs=∏i=1Nfis=0,where(16)fis=s2+kvs+krλie−τs.Before analysing the stability of the closed-loop dynamics (13), some lemmas are provided as follows.Lemma 1.
When the augmented graphG˜ contains a spanning tree, all the eigenvalues of ℒ+P are located in the open right half complex plane, i.e., λi>0,i∈V.
Lemma1 describes the distribution of the eigenvalues of ℒ+P, and it is fundamental to indicate the stability and compute the exact time delay stability margin from the augmented graph G˜.Lemma 2.
When the graphG is undirected, all the eigenvalues of ℒ+P are positive real numbers, i.e., λi∈ℝ+,i∈V.
Lemma2 describes the special character of undirected topologies. When the interconnected information flow among the follower vehicles is undirected, it is useful to simplify the theoretical deduction of verifying the stability. The proof of Lemma 1 and 2 can be found in [21].Lemma 3.
The closed-loop dynamics (13) are asymptotically stable if and only if every equation (16) is Hurwitz stable, i.e., the roots of the closed-loop system’s characteristic equation are all located in the open left half complex plane.
Based on the abovementioned three lemmas, we obtain the first stability result for the closed-loop dynamics (13).Theorem 1.
If the closed-loop dynamics (13) is asymptotically stable, the following inequality holds:(17)kv2kr>maxi∈VIm2λiReλiλi2,where λi is the module of λi. Reλi and Imλi are the real part and imaginary part of λi, respectively.Proof.
The closed-loop dynamics (13) should be primarily stable for the delay-free case [25]. When τ=0, there exists(18)fis=s2+kvs+krλi=0.
Asλi=Reλi+ι⋅Imλi,ι2=−1 is probably a complex number, (18) is transformed into a fourth-order equation as(19)s4+2kvReλis3+kv2λi2+2krReλis2+2krkvλi2s+kr2λi2=0.
Then, the stability of (19) is examined based on the Routh–Hurwitz stability criterion:(20)s41kv2λi2+2krReλikr2λi2s32kvReλi2krkvλi2s2kv2Reλiλi2+krRe2λi−krIm2λiReλikr2λi2s12krkvλi2kv2Reλiλi2−krIm2λikv2Reλiλi2+krRe2λi−krIm2λis0kr2λi2.
The Routh table is listed out in equation (20). Considering the definition that kr>0,kv>0 in (9) and the fact that Reλi>0 from Lemma 1, it is concluded that (18) is asymptotically stable if and only if(21)kv2kr>Im2λiReλiλi2.
Taking allλi into consideration, if the closed-loop dynamics (13) are asymptotically stable, inequality (17) holds.Remark 1.
The stability condition (17) of Theorem 1 is a necessary and sufficient condition of the closed-loop dynamics (13) for the delay-free case, but it is just a necessary condition for the delay case. For τ>0, the stability of the closed-loop dynamics (13) is not only correlated with the control gains and structure of the interconnected information flow but also dependent on the time delay.Corollary 1.
Forτ=0, when the graph G is undirected, the closed-loop dynamics (13) are asymptotically stable, for any kr>0,kv>0.
When the graphG is undirected, there exists Imλi=0 according to Lemma 2. Hence, inequality (17) holds for any kr>0,kv>0.


## 3.2. Computation of the Exact Time Delay Stability Margin for the Automated Vehicular Platoon

For the delay caseτ>0, we review the characteristic equation (16) as(22)fis=s2+kvs+krλie−τs.It is obvious thatfis,i∈V is a quasi-polynomial and its characteristic equation fis=0 is a transcendental equation. Hence, it is complicated to analyse the roots of the characteristic equation. Despite of this, we are still devoted to checking the property of the roots to find the exact time delay stability margin.Theorem 2.
For the closed-loop dynamics (13) satisfying condition (17), let ωi>0 be the root of the following equation:(23)ωi4−kv2λi2ωi2−kr2λi2=0,with ωi=kv2λi2+kv4λi4+4kr2λi2. Take(24)τi=arctanξi+kπωi,k∈0,1,where ξi=−δi/θi with δi=kvωiReλi+krImλi and θi=kvωiImλi−krReλi. Let τmin=mini∈Vτi. Then, system (13) is asymptotically stable if and only if τ∈0,τmin.Proof.
According to Corollary 2.4 in [26], it is known that for a quasi-polynomial gs,e−τs=g0s+g1se−τs, if gs,e−τs is Hurwitz stable for τ=0 and unstable for τ>τmin, there must exist a root on the imaginary axis for gs,e−τmins. In our work, if the stability condition (17) in Theorem 1 is satisfied, fis in (22) is Hurwitz stable for τ=0. Thus, it is available to utilize Corollary 2.4 in [26] to find the exact time delay stability margin τmin. Afterwards, we aim to find the root ι⋅ωi on the imaginary axis for τi.
With respect to the imaginary rootι⋅ωi, there exists fiι⋅wi=0, which implies that both the real and imaginary parts of fiι⋅wi are zero, shown as(25)−ωi2+δisinτiωi−θicosτiωi=0,θisinτiωi+δicosτiωi=0.
By rearranging (25), two following trigonometric functions are obtained:(26)sinτiωi=θiωi2δi2+θi2,cosτiωi=−δiωi2δi2+θi2,where δi2+θi2=kv2ωi2+kr2λi2. According to the well-known trigonometric property sin2τiωi+cos2τiωi=1, it is concluded that(27)ωi4−kv2λi2ωi2−kr2λi2=0.
Then, two real-valued roots aboutωi, which are positive and negative, respectively, can be solved out. Meanwhile, it is obtained that ξi=tanτiωi=sinτiωi/cosτiωi=−θi/δi based on (26). The delay for i∈V is also deduced as τi=arctanξi+kπ/ωi,k∈0,1, where the integer k should be chosen as the minimum value satisfying τi>0.
Next, we aim to demonstrate that only the situationωi>0 needs to be considered. Two cases for the graph G, i.e., undirected and directed topologies, are taken into account as follows:(i)
The undirected graphG: in this case, Imλi=0. We assume that the two real-valued roots of (27) are ωi1,ωi2, and ωi1>0,ωi2<0. It is obvious that ωi2=−ωi1. Based on this, one can find that δi2=−δi1,θi2=θi1,andξi2=−ξi1. There must exist arctanξi2/ωi2=arctanξi1/ωi1 such that we can simply take τi2=τi1. In a word, considering only the positive root ωi>0 suffices to obtain τi, because the imaginary roots for fis form complex conjugate pairs.(ii)
The directed graphG: in this case, Imλi≠0. There must also exist a conjugate eigenvalue of λi for the directed graph G, where the conjugate eigenvalue is also a root of (27). We define the conjugate eigenvalue as λl=Reλi−ι⋅Imλi,l∈V. Let ωi1>0,ωi2<0,ωl1>0,andωl2<0. According to (27), it is easy to see that ωl1=ωi1andωl2=ωi2. Then, one can find that δl2=−δi1,θl2=θi1,andξl2=−ξi1. There must exist arctanξl2/ωl2=arctanξi1/ωi1 such that we can simply take τl2=τi1. Therefore, for the directed graph G, considering only the positive root ωi>0 still suffices to obtain τi.
Then, by synthesizing the above two cases with the fact thatωi>0, the positive root of (27) is solved out as follows:(28)ωi=kv2λi2+kv4λi4+4kr2λi2.
At last, we scan all the eigenvalueλi,i∈V to compute all τi and find the exact time margin τmin=mini∈Vτi.Remark 2.
Theorem2 provides a method for finding the exact time delay stability margin τmin of the closed-loop dynamics (13). Theorem 2 is a necessary and sufficient condition for the stability for the delay case.


## 3.3. A Rapid Method for Computing the Exact Time Delay Stability Margin under the Undirected Topology

A large-scale vehicular platoon generally generates plenty of eigenvalues of the graphG. It is time consuming to scan all the eigenvalue λi,i∈V to compute all τi. However, it is common knowledge that the most critical eigenvalue directly determines the exact time delay stability margin. If the most critical eigenvalue is found, there is no need to scan all the other eigenvalues. Moreover, because it is very fussy to analyse the most critical eigenvalue for the directed graph, we only consider the undirected topology. Therefore, we focus on look for the most critical eigenvalue under the undirected topology, aiming to reduce the computational burden and quickly compute the exact time delay stability margin.Theorem 3.
For the closed-loop dynamics (13), under the undirected graph G, let λmax=maxi∈Vλi. Then, λmax is the most critical eigenvalue, which determines the exact time delay stability margin τmin.Proof.
Although it is difficult to directly confirm whether the monotonicity ofτi depends on the monotonicity of λi, we plan to reveal their monotonous relationship through two steps. By treating ωi as an immediate variable, the first step is to verify that the monotonicity of ωi depends on the monotonicity of λi, and the second step is to verify that the monotonicity of τi depends on the monotonicity of ωi.
In the first step, we only consider the caseωi>0 in accordance with Theorem 2. When the graph G is undirected, (28) turns into the following form as follows:(29)ωi=12kv2λi2+kv4λi4+4kr2λi2.
According to Lemma2, λi is a positive real number for the undirected topology, so it can be seen that ωi gets larger with the increase of λi. Then, it is demonstrated that the monotonicity of the immediate variable ωi depends on the monotonicity of λi.
In the second step, we aim to verify that the monotonicity ofτi depends on the monotonicity of ωi. with respect to (24), there exists τi=ωi−1arctanξi=ωi−1arctankvωi/kr with ωi>0. We intend to apply the partial derivative property to conform the monotonicity. The partial derivative ∂τi/∂ωi is calculated as follows:(30)∂τi∂ωi=−1ωi2arctankvωikr+1ωi⋅krkvkr2+kv2ωi2=1ωi2−arctankvωikr+ωi⋅krkvkr2+kv2ωi2,with ∂arctankvωi/kr/∂ωi=krkv/kr2+kv2ωi2.
In (30), the term arctankvωi/kr still exists. Therefore, it is uneasy to identify whether (30) is positive or negative. To solve this problem, we deduce the partial derivative once again, and an auxiliary function is given by(31)dωi=−arctankvωikr+ωi⋅krkvkr2+kv2ωi2.
Comparing with (31), (30) can be rewritten as(32)∂τi∂ωi=1ωi2dωi.
Then, the partial derivative∂dωi/∂ωi is calculated as(33)∂dωi∂ωi=−krkvkr2+kv2ωi2+kr3kv−krkv3ωi2kr2+kv2ωi22=−2krkv3ωi2kr2+kv2ωi22.
With the definition thatkr>0,kv>0 in (9), it is noted that ∂dωi/∂ωi=0 when ωi=0, and ∂dωi/∂ωi<0 when ωi>0. Then, it is concluded that ∂τi/∂ωi<0 in (32) when ωi>0, which means that τi gets smaller with the increase of ωi. Until now, we verify that the monotonicity of τi depends on that of ωi by using the partial derivative property.
By taking the abovementioned two steps into account, it is confirmed thatτi gets smaller with the increase of λi. Hence, the largest eigenvalue λmax is most critical and determines the exact time delay stability margin τmin.Remark 3.
Theorem3 reveals the inherent relationship between the exact time delay stability margin and the eigenvalues of the augmented Laplacian matrix for the undirected graph G. According to Theorem 3, just computing one delay corresponding to the largest eigenvalue λmax suffices to obtain the exact time delay stability margin, and it is no longer necessary to scan the other eigenvalues. For the large-scale vehicular platoon, utilizing Theorem 3 can dramatically reduce the computational burden.


## 4. Numerical Simulations

In this section, numerical simulations are conducted to verify the effectiveness of the main results. Therein, an automated vehicular platoon with five identical vehicles (one leader and four followers) is considered. Simulations are performed for a single-lane road. As the intelligent transportation system scenario described in [27], we take the structure of the interconnected information flow that the leader vehicle transmits to the follower vehicles. The pinning matrix is set as P=diag1,0,1,0. Meanwhile, the follower vehicles share the state information with their neighbouring vehicles and form undirected topology or directed topology (as depicted in Figure 1). It is noted that the configuration of the interconnected information flow contains a spanning tree and satisfies Lemma 1. The leader vehicle maneuvers at a constant velocity of 20 m/s (i.e., 72 km/h). The CD policy is chosen as the intervehicle spacing policy, and the desired spacing is set as dij=15m. The initial error states are defined as r˜0=5,−5,10,−10T m and v˜0=−2,2,−4,4T m/s. In the following, two scenarios, undirected and directed topologies, are simulated for investigating the stability and the exact time delay stability margin.


### 4.1. Undirected Topology

For an undirected topology, the numerical simulations are conducted to verify the effectiveness of Theorem2 and Theorem 3. The adjacency matrix A and the Laplacian matrix ℒ are given by(34)A=0100101001010010,ℒ=1−100−12−100−12−100−11.The eigenvalues of the augmented Laplacian matrixP+ℒ for the undirected topology are λ1=0.382, λ2=1.000, λ3=2.618, and λ4=4.000. It is seen that all the eigenvalues are positive real numbers such that their distribution coincides with Lemma 2. The control gains are tuned as kr=1 and kv=1 satisfying Theorem 1. We scan every eigenvalue to get that τ1=0.88 s, τ2=0.71 s, τ3=0.44 s, and τ4=0.32 s. Then, the exact time delay stability margin τmin=0.32 s is obtained. One can find that the exact time delay stability margin is determined by the largest eigenvalue λ4=4.000, which confirms to the conclusion of Theorem 3. Two extreme cases are verified with the time delays τ=0.31 s and τ=0.33 s, which are depicted in Figures 2 and 3, respectively. It is seen that the stability is guaranteed in Figure 2 and instability occur in Figure 3. Thus, the simulation results under undirected topology support the main results of Theorems 2 and 3.Figure 2
Spacing errors for an automated vehicular platoon under undirected topology withτ=0.31 s.


### 4.2. Directed Topology

For a directed topology, the numerical simulations are conducted to verify the effectiveness of Theorem2. The adjacency matrix A and the Laplacian matrix ℒ are given by(35)A=0001100010000010,ℒ=100−1−1100−101000−11.The eigenvalues of the augmented Laplacian matrixP+ℒ for directed topology are λ1=1.000, λ2=2.233+0.793ι, λ3=2.233−0.793ι, and λ4=0.534. It is seen that all the eigenvalues have positive real parts such that their distribution coincides with Lemma 1. The control gains are tuned as kr=1 and kv=1 satisfying Theorem 1. We scan every eigenvalue to get that τ1=0.71 s, τ2=0.60 s, τ3=0.34 s, and τ4=0.83 s. Then, the exact time delay stability margin τmin=0.34 s is obtained. Two extreme cases are verified with the time delays τ=0.33 s and τ=0.35 s, which are depicted in Figures 4 and 5, respectively. It is seen that the stability is guaranteed in Figure 4 and instability occurs in Figure 5. Thus, the simulation results under directed topology support the main results of Theorem 2.Figure 3
Spacing errors for an automated vehicular platoon under undirected topology withτ=0.33 s.Figure 4
Spacing errors for an automated vehicular platoon under directed topology withτ=0.33 s.Furthermore, it is noted that the exact time delay stability margin for directed topology in this example is associated with one of the eigenvalues with the largest module. This is an interesting phenomenon. Therefore, our further research topic is to figure out whether this phenomenon is general for directed topologies. Meanwhile, this paper focuses on computing the exact time delay stability margin for linear dynamical systems. The proposed method is only suitable for linear dynamical systems because we obtain the exact time delay stability margin according to the characteristic equation of the closed-loop system. The method cannot be directly applied to nonlinear dynamical systems. However, it can be extended to the nonlinear dynamical systems which can be converted into controllable linear systems via dynamic feedback linearisation. Therefore, finding exact time delay stability margin for nonlinear dynamical systems is our another work in the future.Figure 5
Spacing errors for an automated vehicular platoon under directed topology withτ=0.35 s.


## 4.1. Undirected Topology

For an undirected topology, the numerical simulations are conducted to verify the effectiveness of Theorem2 and Theorem 3. The adjacency matrix A and the Laplacian matrix ℒ are given by(34)A=0100101001010010,ℒ=1−100−12−100−12−100−11.The eigenvalues of the augmented Laplacian matrixP+ℒ for the undirected topology are λ1=0.382, λ2=1.000, λ3=2.618, and λ4=4.000. It is seen that all the eigenvalues are positive real numbers such that their distribution coincides with Lemma 2. The control gains are tuned as kr=1 and kv=1 satisfying Theorem 1. We scan every eigenvalue to get that τ1=0.88 s, τ2=0.71 s, τ3=0.44 s, and τ4=0.32 s. Then, the exact time delay stability margin τmin=0.32 s is obtained. One can find that the exact time delay stability margin is determined by the largest eigenvalue λ4=4.000, which confirms to the conclusion of Theorem 3. Two extreme cases are verified with the time delays τ=0.31 s and τ=0.33 s, which are depicted in Figures 2 and 3, respectively. It is seen that the stability is guaranteed in Figure 2 and instability occur in Figure 3. Thus, the simulation results under undirected topology support the main results of Theorems 2 and 3.Figure 2
Spacing errors for an automated vehicular platoon under undirected topology withτ=0.31 s.


## 4.2. Directed Topology

For a directed topology, the numerical simulations are conducted to verify the effectiveness of Theorem2. The adjacency matrix A and the Laplacian matrix ℒ are given by(35)A=0001100010000010,ℒ=100−1−1100−101000−11.The eigenvalues of the augmented Laplacian matrixP+ℒ for directed topology are λ1=1.000, λ2=2.233+0.793ι, λ3=2.233−0.793ι, and λ4=0.534. It is seen that all the eigenvalues have positive real parts such that their distribution coincides with Lemma 1. The control gains are tuned as kr=1 and kv=1 satisfying Theorem 1. We scan every eigenvalue to get that τ1=0.71 s, τ2=0.60 s, τ3=0.34 s, and τ4=0.83 s. Then, the exact time delay stability margin τmin=0.34 s is obtained. Two extreme cases are verified with the time delays τ=0.33 s and τ=0.35 s, which are depicted in Figures 4 and 5, respectively. It is seen that the stability is guaranteed in Figure 4 and instability occurs in Figure 5. Thus, the simulation results under directed topology support the main results of Theorem 2.Figure 3
Spacing errors for an automated vehicular platoon under undirected topology withτ=0.33 s.Figure 4
Spacing errors for an automated vehicular platoon under directed topology withτ=0.33 s.Furthermore, it is noted that the exact time delay stability margin for directed topology in this example is associated with one of the eigenvalues with the largest module. This is an interesting phenomenon. Therefore, our further research topic is to figure out whether this phenomenon is general for directed topologies. Meanwhile, this paper focuses on computing the exact time delay stability margin for linear dynamical systems. The proposed method is only suitable for linear dynamical systems because we obtain the exact time delay stability margin according to the characteristic equation of the closed-loop system. The method cannot be directly applied to nonlinear dynamical systems. However, it can be extended to the nonlinear dynamical systems which can be converted into controllable linear systems via dynamic feedback linearisation. Therefore, finding exact time delay stability margin for nonlinear dynamical systems is our another work in the future.Figure 5
Spacing errors for an automated vehicular platoon under directed topology withτ=0.35 s.


## 5. Conclusions

This paper investigates the problem of finding the exact time delay margin for the automated vehicular platoon. After designing a distributed controller, we focus on analysing the stability of the entire platoon. By investigating the roots’ distribution of the closed-loop system’s characteristic equation, the exact time delay stability margin is acquired. It is a necessary and sufficient condition, which implies that our result outperforms most existing methods at overcoming conservatism. Furthermore, a rapid method for finding the exact time delay stability margin for undirected topologies is proposed by exploring the monotonicity relationship between each eigenvalue of the augmented Laplacian matrix and its corresponding delay. We analytically demonstrate that the largest eigenvalue determines the exact time delay stability margin. Therefore, it is no longer necessary to compute the delays corresponding to the other eigenvalues. For the large-scale vehicular platoon, utilizing our proposed theorem can dramatically reduce the computational burden. Numerical simulations for undirected and directed topologies are both conducted to verify the effectiveness of the theoretical derivation. Simulation results additionally show that the exact time delay stability margin for directed topologies is associated with one of the eigenvalues with the largest module. Figuring out, whether this interesting phenomenon is general, is the subject of our ongoing work.



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