Computation of Time Delay Stability Margin for the Automated Vehicular Platoon

School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China National Experimental Teaching Center for Civil Engineering Virtual Simulation (XAUAT), Xi’an University of Architecture and Technology, Xi’an 710055, China College of Computer and Information Engineering, Tianjin Normal University, Tianjin 300387, China School of Electronic and Control Engineering, Chang’an University, Xi’an 710064, China


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][4][5][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. ird, 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]. erefore, it is of high importance to study the vehicular platoon with time delay. erein, 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.
erein, 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. en, 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. erefore, 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. e exact time delay stability margin is obtained by searching the pure imaginary roots of the characteristic equation for the closedloop 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. erefore, 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. e remainder of this paper is organised as follows. e problem statement is presented in Section 2. e 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.

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.

Model for Interconnected Information Flow.
In our work, we consider the platoon with N + 1 identical interconnected vehicles, including one leader and N followers. ere 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.
For both the undirected and directed topologies, the interconnected information flow among the followers can be modelled by a graph G � 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. e adjacency matrix associated with the graph G is characterized by e nonnegative adjacency weight a ij � 1 means that the node i can get information from the node j; otherwise, a ij � 0. Moreover, we assume that a ii � 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 as G � 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 graphs G and G. erefore, two important matrices (i.e., Laplacian matrix and pinning matrix) are introduced as follows. Firstly, the Laplacian matrix L � [l ij ] N×N , which is associated with the graph G, is defined as 2 Complexity Secondly, the pinning matrix P, which is associated with the augmented graph G characterizing the interconnected information flow from the leader to the followers, is defined as where p i � 1 if the edge i, 0 { } ∈ E; otherwise, p i � 0. e expression p i � 1 means that the ith vehicle can get information from the leader. p i is called pinning gain in the field of complex networks [20]. If p i � 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. e graph G containing a spanning tree means that a subset of its edges forms a spanning tree. erefore, 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.

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. en, the longitudinal dynamics of the ith vehicle are denoted by where r i and v i are the position and velocity of the ith vehicle, respectively. e propelling force u i represents the control input. Once u i 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 u i .

Intervehicle Spacing
Policy. e 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: where d ij is the desired intervehicle spacing between the ith vehicle and the jth vehicle. It is assumed that d ij � − d ji . e formation geometry of the vehicular platoon is determined by d ij . ere 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, d ij is set to be a constant number: where d 0 is the constant spacing between the ith vehicle and its preceding vehicle. For the CTH policy, d i− 1,i is dependent on the vehicle velocity v i : where h is the constant time headway.
In accordance with some geometrical considerations, the spacing policy d i− 1,i can be recast into the spacing with respect to the leader vehicle as us, the objective of the platoon control in (4) can be rewritten as follows: where d i0 is the desired distance from the leader to the ith vehicle.

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: Complexity where k r > 0 and k v > 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. erefore, by considering the time delay into the interconnected information flow, the distributed controller for the vehicular platoon with time delay can be designed as 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.

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.

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: where is the velocity error state. r 0 (t) and v 0 (t) 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 erefore, by using the error dynamics (11), system (3) can be rewritten as follows: For simplicity, we define the error state vector as x � [r T , v T ] T . en, the collective closed-loop dynamics of the vehicular platoon are rewritten into a compact form as follows: with where I N×N and 0 N×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. erein, we define the eigenvalue of the augmented Laplacian matrix L + P as λ i , i ∈ V, which mathematically reflects the important features of the augmented graph G. By utilizing the eigenvalue λ i , 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. en, the characteristic equation of system (13) is given by where Before analysing the stability of the closed-loop dynamics (13), some lemmas are provided as follows. Lemma 1 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 graph G is undirected, all the eigenvalues of L + P are positive real numbers, i.e., Lemma 2 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. e proof of Lemma 1 and 2 can be found in [21]. Based on the abovementioned three lemmas, we obtain the first stability result for the closed-loop dynamics (13). (13) is asymptotically stable, the following inequality holds:

Theorem 1. If the closed-loop dynamics
where |λ i | is the module of λ i . Re(λ i ) and Im(λ i ) are the real part and imaginary part of λ i , respectively.
Proof. e closed-loop dynamics (13) should be primarily stable for the delay-free case [25]. When τ � 0, there exists As λ i � Re(λ i ) + ι · Im(λ i ), ι 2 � − 1 is probably a complex number, (18) is transformed into a fourth-order equation as en, the stability of (19) is examined based on the Routh-Hurwitz stability criterion: e Routh table is listed out in equation (20). Considering the definition that k r > 0, k v > 0 in (9) and the fact that Re(λ i ) > 0 from Lemma 1, it is concluded that (18) is asymptotically stable if and only if Taking all λ i into consideration, if the closed-loop dynamics (13) are asymptotically stable, inequality (17) holds.

Remark 1.
e stability condition (17) of eorem 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. For τ � 0, when the graph G is undirected, the closed-loop dynamics (13) are asymptotically stable, for any k r > 0, k v > 0.
When the graph G is undirected, there exists Im(λ i ) � 0 according to Lemma 2. Hence, inequality (17) holds for any k r > 0, k v > 0.

Computation of the Exact Time Delay Stability Margin for the Automated Vehicular Platoon.
For the delay case τ > 0, we review the characteristic equation (16) as It is obvious that f i (s), i ∈ V is a quasi-polynomial and its characteristic equation f i (s) � 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. (13) satisfying condition (17), let ω i > 0 be the root of the following equation:
Proof. According to Corollary 2.4 in [26], it is known that for a quasi-polynomial g(s, e − τs ) � g 0 (s) + g 1 (s)e − τs , if g(s, e − τs ) is Hurwitz stable for τ � 0 and unstable for τ > τ min , there must exist a root on the imaginary axis for g(s, e − τ min s ). In our work, if the stability condition (17) in eorem 1 is satisfied, f i (s) in (22) is Hurwitz stable for τ � 0. us, it is available to utilize Corollary 2.4 in [26] to find the exact time delay stability margin τ min . Afterwards, we aim to find the root ι · ω i on the imaginary axis for τ i .
With respect to the imaginary root ι · ω i , there exists f i (ι · w i ) � 0, which implies that both the real and imaginary parts of f i (ι · w i ) are zero, shown as By rearranging (25), two following trigonometric functions are obtained: According to the wellknown trigonometric property sin 2 (τ i ω i ) + cos 2 (τ i ω i ) � 1, it is concluded that en, two real-valued roots about ω i , which are positive and negative, respectively, can be solved out. Meanwhile, it is obtained that (26). e delay for i ∈ V is also deduced as τ i � (arctan ξ i + kπ)/ω i , k ∈ 0, 1 { }, where the integer k should be chosen as the minimum value satisfying τ i > 0.
Next, we aim to demonstrate that only the situation ω i > 0 needs to be considered. Two cases for the graph G, i.e., undirected and directed topologies, are taken into account as follows: (i) e undirected graph G: in this case, Im(λ i ) � 0. We assume that the two real-valued roots of (27) are ω i1 , ω i2 , and ω i1 > 0, ω i2 < 0. It is obvious that ω i2 � − ω i1 . Based on this, one can find that . ere must exist arctan ξ i2 /ω i2 � arctan ξ i1 /ω i1 such that we can simply take τ i2 � τ i1 . In a word, considering only the positive root ω i > 0 suffices to obtain τ i , because the imaginary roots for f i (s) form complex conjugate pairs. (ii) e directed graph G: in this case, Im(λ i ) ≠ 0. ere must also exist a conjugate eigenvalue of λ i for the directed graph G, where the conjugate eigenvalue is also a root of (27). We define the conjugate eigenvalue as Let ω i1 > 0, ω i2 < 0, ω l1 > 0, and ω l2 < 0. According to (27), it is easy to see that ω l1 � ω i1 and ω l2 � ω i2 . en, one can find that δ l2 � − δ i1 , θ l2 � θ i1 , and ξ l2 � − ξ i1 . ere must exist arctan ξ l2 /ω l2 � arctan ξ i1 /ω i1 such that we can simply take τ l2 � τ i1 . erefore, for the directed graph G, considering only the positive root ω i > 0 still suffices to obtain τ i . en, by synthesizing the above two cases with the fact that ω i > 0, the positive root of (27) is solved out as follows: At last, we scan all the eigenvalue λ i , i ∈ V to compute all τ i and find the exact time margin τ min � min eorem 2 provides a method for finding the exact time delay stability margin τ min of the closed-loop dynamics (13). eorem 2 is a necessary and sufficient condition for the stability for the delay case.

A Rapid Method for Computing the Exact Time Delay
Stability Margin under the Undirected Topology. A largescale vehicular platoon generally generates plenty of eigenvalues of the graph G. It is time consuming to scan all the eigenvalue λ i , i ∈ V to compute all τ i . 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. erefore, 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. (13), under the undirected graph G, let λ max � max i∈V λ i . en, λ max is the most critical eigenvalue, which determines the exact time delay stability margin τ min .

Theorem 3. For the closed-loop dynamics
Proof. Although it is difficult to directly confirm whether the monotonicity of τ i depends on the monotonicity of λ i , we plan to reveal their monotonous relationship through two steps. By treating ω i as an immediate variable, the first step is to verify that the monotonicity of ω i depends on the monotonicity of λ i , and the second step is to verify that the monotonicity of τ i depends on the monotonicity of ω i .
In the first step, we only consider the case ω i > 0 in accordance with eorem 2. When the graph G is undirected, (28) turns into the following form as follows: According to Lemma 2, λ i is a positive real number for the undirected topology, so it can be seen that ω i gets larger with the increase of λ i . en, it is demonstrated that the monotonicity of the immediate variable ω i depends on the monotonicity of λ i .
In the second step, we aim to verify that the monotonicity of τ i depends on the monotonicity of ω i . with respect to (24), there exists We intend to apply the partial derivative property to conform the monotonicity. e partial derivative (zτ i /zω i ) is calculated as follows: . In (30), the term arctan (k v ω i /k r ) still exists. erefore, 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 Comparing with (31), (30) can be rewritten as en, the partial derivative (zd(ω i )/zω i ) is calculated as With the definition that k r > 0, k v > 0 in (9), it is noted that (zd(ω i )/zω i ) � 0 when ω i � 0, and (zd(ω i )/zω i ) < 0 when ω i > 0. en, it is concluded that (zτ i /zω i ) < 0 in (32) when ω i > 0, which means that τ i gets smaller with the increase of ω i . Until now, we verify that the monotonicity of τ i depends on that of ω i by using the partial derivative property.
By taking the abovementioned two steps into account, it is confirmed that τ i gets smaller with the increase of λ i . Hence, the largest eigenvalue λ max is most critical and determines the exact time delay stability margin τ min . □ Remark 3. eorem 3 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 eorem 3, just computing one delay corresponding to the largest eigenvalue λ max 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 eorem 3 can dramatically reduce the computational burden.

Numerical Simulations
In this section, numerical simulations are conducted to verify the effectiveness of the main results.
erein, 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. e pinning matrix is set as P � diag 1, 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.
e eigenvalues of the augmented Laplacian matrix P + L for the undirected topology are λ 1 � 0.382, λ 2 � 1.000, λ 3 � 2.618, and λ 4 � 4.000. It is seen that all the eigenvalues are positive real numbers such that their distribution coincides with Lemma 2. e control gains are tuned as k r � 1 and k v � 1 satisfying eorem 1. We scan every eigenvalue to get that τ 1 � 0.88 s, τ 2 � 0.71 s, τ 3 � 0.44 s, and τ 4 � 0.32 s. en, the exact time delay stability margin Complexity τ min � 0.32 s is obtained. One can find that the exact time delay stability margin is determined by the largest eigenvalue λ 4 � 4.000, which confirms to the conclusion of eorem 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. us, the simulation results under undirected topology support the main results of eorems 2 and 3.

Directed
e eigenvalues of the augmented Laplacian matrix P + L for directed topology are λ 1 � 1.000, λ 2 � 2.233 + 0.793ι, λ 3 � 2.233 − 0.793ι, and λ 4 � 0.534. It is seen that all the eigenvalues have positive real parts such that their distribution coincides with Lemma 1. e control gains are tuned as k r � 1 and k v � 1 satisfying eorem 1. We scan every eigenvalue to get that τ 1 � 0.71 s, τ 2 � 0.60 s, τ 3 � 0.34 s, and τ 4 � 0.83 s. en, the exact time delay stability margin τ min � 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.
us, the simulation results under directed topology support the main results of eorem 2.
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. is is an interesting phenomenon. erefore, 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. e 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. e 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. erefore, finding exact time delay stability margin for nonlinear dynamical systems is our another work in the future.

Conclusions
is 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. erefore, 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.