Cooperative Control of Heterogeneous Connected Vehicles with Directed Acyclic Interactions
Abstract
Cooperation of multiple connected vehicles has the potential to benefit the road traffic greatly. In this paper, we consider analysis and synthesis problems of the cooperative control of a platoon of heterogeneous connected vehicles with directed acyclic interactions (characterized by directed acyclic graphs). In contrast to previous works that view heterogeneity as a type of uncertainty, this paper directly takes heterogeneity into account in the problem formulation, allowing us to develop a deeper understanding of the influence of heterogeneity on the collective behavior of a platoon of connected vehicles. Our major strategies include an application of the celebrated internal model principle and an exploitation of lowertriangular structures for platoons with directed acyclic interactions. The major findings include: 1) we explicitly highlight the tracking ability of heterogeneous platoons, showing that the followers can only track the leader’s spacing and velocity; 2) we analytically derive a stability region of feedback gains for platoons with directed acyclic interactions; 3) and consequently we propose a synthesis method based on the solution to an algebraic Riccati equation that shares the dimension of single vehicle dynamics. Numerical experiments are carried out to validate the effectiveness of our results.
28
I Introduction
Connected vehicles have recently received increasing attention from both academia and industry, due to the high potential to significantly benefit road transportation [1, 2, 3, 4]. One important application is to develop cooperative control strategies for multiple connected vehicles based on local information to guarantee a certain global coordination, such that the traffic efficiency and road safety are improved. This is called platoon control of connected vehicles [2], and is also known as cooperative adaptive cruise control (CACC) [5].
One main objective in the cooperative control of multiple connected vehicles is to ensure that all vehicles in a group maintain a desired cruising velocity while keeping a prespecified intervehicle distance. This problem has a long history in control theory dating back to the pioneering work in the 1960s [6], where an optimal control framework was introduced to deal with spacing regulation of multiple vehicles. The earliest practices on platoon control can be traced back to the California Partners for Advanced Transportation Technology (PATH) program in the 1980s [7], where many practical issues were discussed, including control architecture, spacing policies, sensors and actuators, and string stability. Recent advances have emerged in the application of advanced control methods for platooning of connected vehicles, such as control [8], distributed model predictive control [9, 10], and sliding mode control [11]. Also, some proofofconcept demonstrations have been performed in the projects of GCDC [3], SARTRE [12] and EnergyITS [13]. The interested reader is referred to [2] for a recent overview.
With the rapid development of vehicletovehicle (V2V) communications, such as DSRC and VANETs [14], one recent research focus of platooning is on developing scalable analysis and synthesis methods for the cooperation of largescale platoons with various communication topologies [2]. For instance, an explicit stabilizing region of linear feedback gains was derived for homogenous platoons with a large class of communication topologies [15]. Barooah et al. introduced a mistuningbased design method to improve closedloop stability margin of platoons with bidirectional topologies [16], which has recently been extended to cover the inertial time lag of vehicle powertrains in [17]. One tradeoff of the mistuningbased controller was highlighted in [18], where it is shown that the closedloop norm increases exponentially as the platoon size grows. This fact is consistent with a highgain condition in the design of distributed controllers for platoons with undirected topologies [19]. More recently, Qin and Orosz proposed a decomposition method for scalable stability analysis of large connected vehicle systems, where stochastic communication delays were covered [20]. The aforementioned studies have offered efficient methods for performance analysis and controller synthesis of largescale platoons of connected vehicles. However, most of them require a key assumption that the dynamics of each vehicle are homogeneous [15, 16, 17, 18, 19, 20]. This assumption allows one to apply the decomposition results in multiagent systems [21, 22] and greatly simplifies the theoretical analysis and synthesis.
The assumption of homogeneity may be too restrictive and impractical since diverse types of vehicles should be allowed in a platoon formation. This leads to the design of heterogeneous platoons, which actually attracts research attention as early as the practices in the PATH program [7]. For instance, an inclusion principle was applied to decompose a string of interconnected heterogeneous vehicles into a set of subsystems, in which overlapping controllers were designed [23]. String stability analysis for heterogeneous vehicle platoons was discussed in [24] and [25]. Naus et al. derived a necessary and sufficient condition in frequencydomain for string stability, where heterogeneous traffic was taken into account [26]. In [27], Ploeg et al. introduced an string stability of cascade systems using inputoutput properties, which is suitable for heterogeneous platoons with nearest interactions. In [28], an adaptive switched control approach was proposed for heterogeneous platoons with communication losses. Besides, Rödönyi discussed an adaptive spacing policy that is able to guarantee string stability [29]. The results of [24, 25, 26, 27, 28, 29] offer some insights on the design of heterogeneous platoons. However, these methods are only applicable for very limited types of communication topologies, e.g., predecessorfollowing (PF) type and predecessorleader following (PLF) type, since most of them rely on the exploitation of a cascade structure in an implicit or explicit way. Recently, many other types of topologies are emerging, e.g., the multiplePF type [15], thanks to the rapid deployment of V2V techniques. New challenges naturally arise for cooperative control of heterogeneous connected vehicles considering the variety of topologies. Note that there are a few recent works that try to address this issue; see e.g., [30, 31, 32, 33]. In these works, the authors typically consider the heterogeneity as a type of uncertainty and assume that the vehicles share an identical nominal model. It means that the results on homogeneous platoons [19, 20, 21, 22] can be basically applied to the nominal platoon system.
In this paper, we consider cooperative control of a platoon of heterogeneous connected vehicles with directed acyclic interactions (see the precise definition in Section IVB), and directly take the heterogeneity into account. In contrast to previous studies, this treatment allows us to develop a deeper understanding of the effects of heterogeneity on the collective behavior of a platoon, as well as to highlight the influence of topological variety introduced by V2V techniques. Under the notion of directed acyclic interactions, the closedloop heterogeneous platoon system becomes decomposable thanks to a lowertriangular structure. This technique does not rely on the normal eigenvalue decomposition or similarity transformation that is widely used in homogeneous platoons (see [2, 34] for example). More precisely, our contributions are:

The internal model principle, a fundamental result in heterogeneous multiagent systems [35], is applied to the analysis of heterogeneous platoons. The internal model principle presents a necessary and sufficient condition for the synchronization of heterogeneous linear networks. We discuss the implication of internal model principle for cooperative control of heterogeneous connected vehicles, and highlight the tracking ability of each following vehicle. To reach the steady consensus state, the leader should run at a constant speed, indicating that the followers can only track the leader’s spacing and velocity.

We derive an explicit and analytical region for feedback gains that guarantee the asymptotical stability of heterogeneous platoons with directed acyclic interactions. This result not only explicitly highlights the necessity of spacing and velocity information to stabilize a heterogeneous platoon, but also points out that the existence of a spanning tree in the communication graph is essential for stabilization. The influence of the heterogeneity in vehicle dynamics is directly reflected in the stability region. Our result generalizes the stability condition in [15] to heterogeneous platoons.

According to the internal stability result, we propose a synthesis method based on the solution to an algebraic Riccati equation (ARE) relying on the dynamics of each individual vehicle. By exploiting a lowertriangular structure, this design method is decoupled from the communication graph. This makes the computational complexity independent of the platoon size. Besides, the synthesis method has a relatively clear physical interpretation on the convergence rate design, which facilitates its application in practice.
The rest of this paper is organized as follows. Section II presents the problem statement. The tracking ability of a heterogeneous platoon is discussed using the internal model principle in Section III. Section IV presents the stability result and introduces an AREbased controller synthesis method. Numerical simulations are shown in Section V, and we conclude the paper in Section VI.
Notations: The fields of real numbers and real matrices are denoted by and , respectively. The closed righthalf complex plane is denoted by . A matrix is represented by its entries for convenience, i.e., , and its transpose is denoted by . The spectrum of a square matrix is denoted by . A matrix is called Hurwitz (or stable) if and only if all of its eigenvalues have negative real parts. An diagonal matrix, whose diagonal entries are and start from the upper left, is denoted by . For matrices and , the Kronecker product of and is denoted by . For any positive integer , let , and the identity matrix of dimension is denoted by .
Ii Problem Statement: Cooperative Control of Heterogeneous Connected Vehicles
We consider the cooperative control of a platoon of heterogeneous vehicles (nodes) running on a straight flat road, consisting of a leading vehicle indexed by and following vehicles indexed from to (see Fig. 1). The control objective is to make the following vehicles move at the same velocity as the leading vehicle while maintaining a fixed formation geometry. As shown in Fig. 1, from a control perspective, the platoon can be viewed as a combination of four main components: 1) vehicle dynamics; 2) distributed controller; 3) information flow topology; 4) formation geometry, which is known as the fourcomponent framework [4]. A categorization of platoon control can be found in [4] and [34] based on the features of each component.
In this section, we briefly introduce the modeling of the four components and present the problem statement of platooning of connected vehicles.
Iia Modeling of Heterogeneous Platoons
IiA1 Vehicular longitudinal dynamics
Vehicle dynamics describe the behavior of each node, which are inherently nonlinear, consisting of the engine, brake system, and rolling resistance, etc. However, a detailed nonlinear model may not be helpful for platoon level analysis, since it in general cannot lead to analytical results, especially when we consider the effect of different communication topologies. In the literature, to strike a balance between accuracy and conciseness, a typical choice is to derive a linear model by either using a hierarchical control framework [7], or employing a feedback linearization technique [15]. The linear model is then served as a basis for theoretical analysis for the cooperative control of a platoon.
Here, we use the following linearized thirdorder model to describe the longitudinal behavior of each vehicle in a platoon
(1) 
where and denote the position, velocity, and acceleration of the th node, respectively; represents the desired acceleration of the th node, and characterizes the inertial time lag of the powertrain system. This model is widely used as a basis for platoon level analysis; see, e.g., [2, 4, 8, 27]. Here, we note that the third equation in (1) is a firstorder inertial function that approximates the acceleration response of vehicle longitudinal dynamics. The parameter can be different, e.g., is small for passenger cars while it is big for commercial cars. Concisely, (1) can be rewritten as
(2) 
where ,
and or , denotes whether the corresponding state of the vehicle can be measured as an output.
IiA2 Model for information flow topologies
Information flow topologies describe how the vehicles in a platoon exchange information with each other, which exerts a great influence on the collective behavior of a platoon. We employ directed graphs to model the information flow topology in a platoon; please refer to [36] for more details on graph theory.
The information flow between followers is modeled by a directed graph with a set of nodes representing each following vehicle and a set of edges representing the information exchange. The adjacency matrix associated with is defined as with each entry
where means vehicle can obtain the information of vehicle . It is assumed that there is no selfloop, i.e., . The degree matrix is defined as , where
The Laplacian matrix associated with is defined as with each entry:
(3) 
Then, we have
To model the connections between the leader and followers, we define a pinning matrix:
(4) 
where means node can obtain the information from the leader; otherwise . Then, the connections in a platoon are described by the matrices and , which are naturally suitable for different communication topologies.
IiA3 Formation geometry
Formation geometry describes the desired intervehicle spacing between two adjacent nodes, which is the main objective in the cooperative control of a platoon. Mathematically, we require
(5) 
where is the desired gap between vehicle and vehicle . This value can be either velocitydependent (referred to as the constant time headway policy), or velocity independent (called the constant spacing policy). In this paper, we use the constant spacing policy, i.e., , as used in [16, 17, 18].
IiA4 Design of distributed controllers
The distributed controller defines a feedback law using local information that is available for each node, such that the collective behavior of a platoon reaches the global coordination (5). In this paper, we consider a linear feedback of the form
(6) 
where is the local feedback gain and is the desired distance vector. In (6), only local information is used for feedback. We note that the controller (6) and its variants have been widely used in [15, 16, 18, 17].
IiB Problem Statement of Heterogeneous Platoon Design
There are two heterogeneous sources in a practical platoon: 1) heterogeneous dynamics ; 2) heterogeneous feedback gains . In previous studies [30, 31], the dynamical heterogeneity is considered as a type of uncertainty, i.e., , where the nominal system is homogeneous and the uncertainty is bounded. Note that the authors of [30, 31] still assume homogenous feedback gains, i.e., , even in the case of heterogeneous dynamics.
In this paper, we directly consider the heterogeneity in dynamics and allow for heterogeneous feedback gains as well. This treatment not only offers more freedom in controller design, but also gains a deeper understanding of the influence of heterogeneity on the collective behavior of a platoon. Nevertheless, this treatment brings new challenges for the analysis and design since we cannot use the decomposition results in the homogeneous case and new tools are needed. Precisely, we seek to address the following issues:

Stability region: We derive an analytical region of where the closedloop platoon is asymptotically stable.

Controller synthesis: We introduce a method to calculate a particular that stabilizes the closedloop platoon.
These three issues are related to each other. Unlike the homogeneous case, the feasibility of the controller (6) becomes nontrivial due to the heterogeneity. It requires careful discussions on the existence of the controller (6) that can achieve the goal (5). In this paper, we apply the internal model principle of multiagent systems [35] to heterogeneous platoon design, which explicitly highlights tracking ability of each following vehicle. Then, an internal stability theorem is derived using the RouthHurwitz stability criterion by exploiting a lowertriangular structure. We further propose an ARE based design method to calculate the feedback gain for each following vehicle in a heterogeneous platoon.
Iii Tracking Ability of Heterogeneous Platoons
In this section, we formally address the feasibility of the controller (6) by using the internal model principle of heterogeneous multiagent systems.
This result is applicable for general heterogeneous multiagent systems, known as the internal model principle, which is originally proved in [35]. A special form of this theorem appeared in [37]. In principle, the conditions in Lemma 1 indicate that all followers are able to track the leader defined by the dynamical matrix and the output matrix . Also, the dynamics of the followers must embed an internal model of the leader; the interested reader is referred to [35, 37] for more details.
In terms of heterogeneous platoons, we have the following result.
Theorem 1
Consider the cooperative control of a platoon of heterogeneous connected vehicles with dynamics defined in (2) and controller given by (6). If the cooperation objective (5) is satisfied, then the leader must move at a constant speed, i.e., , and the spacing information must be measured for each vehicle, i.e., .
Proof:
According to Lemma 1, the conditions in (7a) and (7b) must be satisfied. Since has full column rank, we know that each eigenvalue of is also an eigenvalue of , i.e.,
(8) 
Then, the eigenvalues of are a subset of the largest common set . According to the dynamics (2), we have
Therefore, the matrix can only have zero eigenvalues. In this case, possible choices for include
The first two choices lead to a trivial situation , where the leader’s velocity is zero. This is a special case where the leader move at a constant speed. Here, we consider a broader case: for heterogeneous platoons, the dynamical matrix of the leader is
(9) 
which indicates . In this case, the leader can have a nonzero constant velocity. Also, we know that the following matrices and satisfy the conditions in (7a) and (7b)
(10) 
The observability matrix of is
(11) 
Then, the observability of requires , implying . In our case, it means the spacing information is available to each vehicle, i.e., .
In a platoon of connected vehicles, the leader’s state actually defines the equilibrium point of each follower, i.e., each follower tries to reach consensus on the equilibrium state defined by the leader (implicitly or explicitly). In Theorem 1, we formally show that to reach the steady consensus state, the leader should run at a constant speed, implying that the followers can only track at most the leader’s spacing and velocity and that there should be no acceleration in the leader. Note that this is the objective defined in (5).
In fact, many previous studies directly assume that the leader’s velocity is constant; see, e.g., [9, 10, 15, 17, 16, 18]. The transient from one constant speed to another is usually modeled as a certain disturbance of the leader, where the response of each follower is typically studied using the notion of string stability [24, 26, 27]. In addition, the conditions in Theorem 1 also explicitly highlight the necessity of spacing information for controller feedback.
Remark 1
For homogenous platoons, a feasible solution to (7a) and (7b) is trivial, i.e., , because homogeneous followers share the same internal model with the leader. We note that the conditions in Theorem 1 only highlight the necessary requirements on the dynamics. For controllability, the communication graph should contain at least one spanning tree rooting at the leader [38], i.e., there should exist a directed path from the leader to every follower (a pinning condition is required). In other words, the leader’s information should be available to every follower explicitly or implicitly. As we shall see below, this requirement is confirmed in Theorem 2.
Iv Stability Region And Controller Synthesis
The last section gives some necessary conditions for the platoon design. With these conditions in mind, this section discusses the stability region of heterogeneous platoons with directed acyclic interactions. Also, a synthesis method is proposed based on the solution to an ARE.
Iva Closedloop Platoon Dynamics
Here, we first formulate the closedloop dynamics of heterogeneous platoons. The desired trajectory of the follower is shown as
As shown in Theorem 1, we assume the leader moves at a constant speed, i.e., We then define the following tracking error for each follower:
(12) 
Further, the lumped tracking error of follower is
(13) 
where and denotes the output and state of tracking errors, respectively. Then, the control law (6) can be rewritten into a compact form
(14) 
The closedloop dynamics of tracking errors are written as
(15) 
Also, we know that
For simplicity, we define
(16)  
where a set of new variables are introduced,
Then, the closedloop platoon dynamics can be compactly rewritten as
(17) 
where , and
denote the lumped states of tracking errors, and
(18) 
collects the effects of heterogeneous inertial time lags of followers, and
(19) 
assembles the effects of heterogeneous feedback gains.
Remark 2
In case of homogeneous platoons (i.e., , is usually used as the state variable. As shown in [15], the closedloop dynamics of homogeneous platoons can be written as
(20) 
where . However, when it comes to heterogeneous platoons, the closedloop dynamics will be more complicated than (20) when choosing as the state variable. In contrast, we collect and as the state variables to construct the closedloop platoon dynamics, leading to the concise formulation (17). In (17), both the heterogeneous inertial time lags and controller gains are collected together into diagonal matrices (see (18) and (19)), making it easier to analyze their effects on the closedloop system. In the homogeneous case, the matrices and become homogeneous as well in a form of , and (17) can be transformed into (20) using a certain state transformation.
Remark 3
From (17), it is easy to see that each of the four components in Fig. 1 exerts a certain influence on the platoon dynamics: the matrix and the structure of the state matrix represent the longitudinal dynamics of vehicles; the matrix represents the influence of information flow topology; the vectors and contain the effect of formation geometry; and the matrix shows the effect of the distributed controller. This is consistent with the homogeneous cases; see (20).
IvB Directed Acyclic Graph
We now introduce the definition of directed acyclic graphs.
Definition 1
A directed acyclic graph (DAG) is a finite directed graph with no directed cycles.
Equivalently, a DAG is a directed graph that has a topological ordering, i.e., a sequence of the vertices, such that every edge is directed unidirectionally from preceding nodes to downstream ones in the sequence. In fact, the DAG is a natural extension of unidirectional topologies (see e.g., [10]), including the common PF [8] and PLF [26] topologies as special cases (see Fig. 3).
For example, as shown in Fig. 2, graphs (a)(c) are DAGs, because they are all directed graphs with no directed cycles. Also, Fig. 2(b) and Fig. 2(c) are two types of topological ordering for Fig. 2(a), where the order of vertices is shifted such that only information flow from preceding nodes (the left side) to downstream nodes (the right side) is allowed. On the contrary, the graph in Fig. 2(d), which can be obtained by reversing the information flow direction between nodes and in Fig. 2(a), is not a DAG, since nodes and nodes form two directed cycles.
Based on Definition 1, the following lemma gives a formulation of the topological ordering.
Lemma 2
For a DAG with a set of nodes indexed as , there exists at least one permutation, denoted by the ordered set , such that
(21) 
where is the entry of the adjacency matrix associated with the permutated graph. Equivalently, there exists an invertible permutation matrix , where is the standard unit vector in the th direction, such that
(22a)  
(22b) 
where and are the adjacency matrices of the original and permuted graphs, and is a lowertriangular matrix.
(23)  
The proof is straightforward, and we omit it for brevity. Take graphs (a) and (b) in Fig. 2 for example. The ordered vertex sets of graphs (a) and (b) are and , respectively, and the adjacency matrices are
respectively. Then we know graph (b) is a permutation of graph (a) with the invertible permutation matrix
which satisfies
and is a lowertriangular matrix.
The key fact of Lemma 2 is that the adjacency matrix of a DAG can be transformed into a lowertriangular matrix, which facilitates the analysis and synthesis of heterogeneous platoons subsequently. Furthermore, for the permutation matrix in Lemma 2, we have the following result.
Lemma 3
Consider a matrix , where is the standard unit vector in the th direction, and is a permutation of . Then, for any diagonal matrix , the matrix is still diagonal with the same diagonal entries as , but the order of the entries is permuted.
The proof is straightforward since represents certain column and row operations on the diagonal matrix . Thus, remains diagonal, and only the order of its diagonal entries is permuted according to the permutation .
IvC Stability Region Analysis
We are ready to present the second theorem of this paper.
Theorem 2
Consider the cooperative control of a platoon of heterogeneous connected vehicles with the closedloop dynamics given by (17). If the information flow topology is a DAG, then the platoon is asymptotically stable if and only if the following statements hold.

The spacing and velocity information are measurable, i.e., ;

Every follower can obtain the information of at least one other node, i.e., ;

The local feedback gains satisfy
(24)
Proof:
For simplicity, in (17), we denote
We first analyze the characteristic equation of the closedloop platoon system (17):
If the closedloop system is asymptotically stable, then there are no zero roots, i.e., . Consequently, we have (23).
If the information flow topology is a DAG, then according to Lemma 3, we know that there exists a permutation matrix such that is a lowertriangular matrix. Then, we denote
which are diagonal matrices and have the same diagonal entries as and according to Lemma 3. These facts lead to the equation shown in (25).
(25)  
Therefore, the stability of the closedloop platoon system (17) is equivalent to the stability of the following characteristic equations:
(26) 
The stability of (26) can then be checked using the RouthHurwitz stability criterion, as shown in (27).
(27) 
Thus, we have
(28) 
Then, it is easy to know that , which indicates . In our case, it requires , i.e., the spacing and velocity information are measurable. Also, from the third inequation of (28), we know
After some simple linear algebra, we arrive at the requirements on the local feedback gains, shown in (24). This completes the proof.
There are a number of points that are worth highlighting for Theorem 2. The first condition in Theorem 2 is consistent with Theorem 1, and it states that both spacing and velocity information are necessary to stabilize a heterogeneous platoon system. Also, this result agrees with the earliest platooning practices [7], where only spacing and velocity information are available since the sensing systems are often radarbased and lack the acceleration information of other vehicles. In this case, the condition (24) in Theorem 2 reduces to
(29) 
which guarantees the internal stability of a heterogeneous platoon without relying on the acceleration information. The second condition in Theorem 2 means there should exist at least one spanning tree rooting at the leader, which agrees with [38] (see Remark 1). It is easy to check that all the unidirectional topologies shown in Fig. 3 satisfy this property. In addition, the condition (24) generalizes the stability condition in [15] to heterogeneous platoons with directed acyclic interactions.
Remark 4
It is shown that the acceleration information via V2V communication helps improve string stability [5, 8]. For condition (24), it is easy to see that the feedback gain actually enlarges the stability region of , whose lower bound is reduced from to
The additional freedom brought by acceleration feedback could then be used to improve other performance indexes, e.g., string stability. From this perspective, our result is consistent with the statement in [5, 8] as well.
IvD AREbased Design of Feedback Gains
Theorem 2 gives analytical results on the stability region, within which all the feedback gains in (24) can guarantee asymptotical stability of the system (17). However, Theorem 2 does not indicate how to choose a proper control gain for a specific platoon system.
In this section, based on Theorem 2, we present a feedback gain design method according to the solution to an ARE. This method has a relatively clear physical interpretation, which is easy to use in practice.
Theorem 3
Consider the cooperative control of a platoon of heterogeneous connected vehicles with closedloop dynamics given by (17) and all states measurable, i.e., . If the information flow topology is a DAG with a spanning tree, i.e., , then the control gain given in (30a), where is the root of the ARE in (30b), guarantees the stability of system (17) if .
(30a)  
(30b) 
Proof:
According to (25), we first analyze the stability of the matrix
(31) 
Consider a Lyapunov equation using the positive definite solution to the ARE in (30b):
(32) 
When , by substituting (30a) into (30b), we have
Then, the matrix is Hurwitz. This means all the eigenvalues of its characteristic equation, shown in (33), have negative real parts:
(33) 
Compared to the condition (24) in Theorem 2, where three gain parameters should lie within an explicit and analytic region, Theorem 3 gives an implicit and univariate way to choose feedback gains. The ARE in (30b) corresponds to the infinite time LQR problem for system (2) with a performance index
Since is controllable, the ARE in (30b) always has a unique positive definite solution for any .
Remark 5
The equation (32) actually defines a Lyapunov function with derivative , for the subsystem , where is given in (31). Then, a larger implies a faster convergence rate of this subsystem. In addition, according to (25) and (33), the relative stability of this subsystem is equivalent to that of the original platoon system. This fact brings much convenience to adjust the convergence rate of the platoon system by properly choosing the parameter . Note that the control gains given by the AREbased method is only a subset of all the stable control gains given in Theorem 2; see (24).
Remark 6
Note that the stability of the closedloop system is equivalent to the stability of the subsystem , where is given in (31). We have used this fact in the proofs of Theorems 1 and 2. In homogenous cases with general information flow topologies, it is shown in [21] that the equivalent system becomes , where is also the eigenvalue of the matrix .
V Numerical Results
This section presents numerical simulations to validate the effectiveness of our findings. In particular, we consider a heterogeneous platoon with eight vehicles (one leading vehicle and seven following vehicles) under multiple types of directed acyclic interaction topologies, including PF, PLF, TPF and TPLF topologies (see Fig. 3). Also, simulations with a realistic nonlinear vehicle model are carried out to show effectiveness of our results in real traffic environments.
Va Validations based on the linear model
In the simulations, the desired spacing was set to . The initial states of the leading vehicle were with the velocity profile given by
(34) 
The initial states of all following vehicles were set as . We assume that the states of each vehicle are all measurable, i.e., .
Vehicle Index  

1  0.40  3.00  3.40  0.06  2.00 
2  0.55  1.30  3.55  0.09  2.62 
3  0.32  2.31  3.32  0.10  2.87 
4  0.44  1.65  3.44  0.08  2.97 
5  0.38  3.83  3.38  0.07  3.07 
6  0.51  2.42  3.51  0.05  3.70 
7  0.29  2.91  3.29  0.04  2.79 
First, we validate the asymptotical stability given in Theorem 2. Gien the heterogeneous inertial time lags in Table I, it is easy to check that the feedback gains satisfy the conditions in Theorem 2 for all the four types of DAGs shown in Fig. 3, while the feedback gains do not. The spacing error profiles corresponding to the two sets of feedback gains are shown in Fig. 4 and Fig. 5, respectively. The numerical results clearly demonstrate that the feedback gains can guarantee the asymptotical stability, while the feedback gains cannot. This fact supports the statements in Theorem 2.
Second, we show that it is impossible to track a leading vehicle with acceleration (see Theorem 1). Here, we extend the acceleration process of in (34), and consider a scenario where the speed profile of the leader is
(35) 
The spacing error profiles using control gain and velocity profile (35) are shown in Fig. 6. It is clear that there exists a constant spacing error for each following vehicle in the platoon, meaning that the followers are not able to track the leader if the leader continues to accelerate. When the leader accelerates from one speed to another speed and maintains that speed afterwards, the followers would be able to reach consensus to the new steady state defined by the leader. The transient behavior of each follower is normally studied under the notion of string stability [8], which is beyond the scope of the current manuscript.
PF  PLF  TPF  TPLF  

1  23.71  18.27  18.71  18.29 
3  21.89  17.42  18.14  17.44 
5  20.94  17.07  17.90  17.09 
7  19.95  16.85  17.73  16.87 
Third, we demonstrate that the stability is guaranteed using the design method in Theorem 3 and the convergence rate can be adjusted by tuning the parameter . Here we define an index to quantify the convergence rate, as used in [17],
(36) 
where is a threshold for admissible position tracking error. This performance index indicates the time instant when all the following vehicles reach consensus with the leading vehicle with respect to an admissible error . In the simulations, we set and
and we used the same velocity profile as (34). The performance index values corresponding to different (we used identical for each vehicle) and information topologies are listed in Table II. The spacing error profiles for TPLF topology are given in Fig. 7, which shows that the spacing error converges to , thus the closedloop system is stable. In Table II, it is obvious that the convergence rate can be improved by increasing . A tradeoff is that a larger generally leads to larger feedback gains, which might result in actuator saturations. In practice, it is necessary to tune to obtain a suitable feedback controller for a particular platoon system.