11institutetext: Tamás G. Molnár 22institutetext: Department of Mechanical and
Civil Engineering, California Institute of Technology, Pasadena, CA 91125,
USA, 22email<EMAIL_ADDRESS>33institutetext: Michael Hopka, Devesh
Upadhyay, and Michiel Van Nieuwstadt 44institutetext: Ford Research and
Innovation Center, Dearborn, MI 48124, USA, 44email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>55institutetext: Gábor Orosz
66institutetext: Department of Mechanical Engineering and Department of Civil
and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109,
USA, 66email<EMAIL_ADDRESS>
# Virtual Rings on Highways: Traffic Control by Connected Automated Vehicles

Tamás G. Molnár  Michael Hopka  Devesh Upadhyay  Michiel Van Nieuwstadt  and
Gábor Orosz

###### Abstract

This work gives introduction to traffic control by connected automated
vehicles. The influence of vehicle control on vehicular traffic and traffic
control strategies are discussed and compared. It is highlighted that vehicle-
to-everything connectivity allows connected automated vehicles to access the
state of the traffic behind them such that feedback can be utilized to
mitigate evolving congestions. Numerical simulations demonstrate that such
connectivity-based traffic control is beneficial for smoothness and energy
efficiency of highway traffic. The dynamics and stability of traffic flow,
under the proposed controllers, are analyzed in detail to construct stability
charts that guide the selection of stabilizing control gains.

Figure 1:  Dynamics of traffic flows. (a) A chain of human-driven vehicles
(HVs). If the lead vehicle brakes, the vehicle chain amplifies it and the tail
vehicle reduces its speed more than the lead. (b) Vehicle control on large
penetration of automated vehicles (AVs) which influences traffic. The tail
vehicle reduces its speed about as much as the lead. (c) Traffic control with
a single connected automated vehicle (CAV) where the tail vehicle is a
connected human-driven vehicle (CHV). The tail vehicle reduces its speed less
than the lead.

##  1 Introduction

Traffic congestion is a major factor in reducing the efficiency of road
transportation, as it increases travel times, energy consumption of vehicles
and air pollution. Human-driven traffic often suffers from the onset of stop-
and-go traffic jams on highways. These may occur without any incident and may
be triggered by the human driving behavior Orosz2009 . This is illustrated in
Fig. 1(a), where a chain of human-driven vehicles (HVs) is simulated on a
single lane (with details given later in the paper). When the lead vehicle
(black) decelerates, the following human drivers (gray) tend to overreact and
reduce their speed more: they get slower the farther they are behind the lead
vehicle. This, so-called string unstable driving behavior is typical in human-
driven traffic Molnar2021trc . It eventually leads to low speeds and stop-and-
go motion on highways, giving birth to a traffic congestion Orosz2010 .

This chapter is dedicated to strategies that allow prevention or mitigation of
such driving behavior-induced traffic congestions. Alleviating traffic jams
has substantial benefits for each individual vehicle participating in traffic:
it improves their travel time and energy consumption. The key factor to obtain
these benefits is to ensure that vehicles drive as smoothly as possible
without major deceleration. Smoothness is often analyzed via the notion of
string stability Besselink2017 ; Feng2019 ; Ploeg2014a ; Swaroop1996 , which
also has been used recently to evaluate commercial cruise control systems
Ciuffo2021 ; Gunter2020arecommercially ; Gunter2020model . Smooth driving can
be achieved either at the level of individual vehicles by vehicle control or
at large-scale traffic level by traffic control DelleMonache2019 .

Vehicle control can be realized on automated vehicles (AVs), with various
levels of automation. Control strategies include adaptive cruise control (ACC)
for individual AVs Xiao2010 , connected cruise control (CCC) for connected
automated vehicles Orosz2016 , and cooperative adaptive cruise control (CACC)
for vehicle platoons Wang2018b . Controlling vehicles to drive smoothly
achieves significant benefits in energy efficiency, safety and passenger
comfort Dollar2018 ; Ge2018 ; Zheng2020 . While vehicle control guarantees
benefits for the individual AVs only, a sufficient number of well-designed AVs
may positively influence traffic at large and mitigate traffic jams due to
their smooth driving behavior Ard2020 ; Arem2006 ; Cui2017 ; Spiliopoulou2018
. The influence of vehicle control on traffic is demonstrated in Fig. 1(b),
where AVs (green) are mixed with HVs (gray) in the traffic flow. As the AVs
drive smoothly and reduce their speed less than the vehicle they follow, the
traffic jam is mitigated. This way, vehicle control also controls traffic
indirectly, however, there is no direct measurement or feedback of the traffic
state.

Traffic control, on the other hand, directly uses the state of the traffic
flow as feedback. Traffic state estimation may rely on data from loop
detectors or cameras at fixed locations Mehran2011 ; Orosz2010 ; Yu2019a  or
from vehicles traveling in the traffic flow Bekiaris-Liberis2017 ; Chu2016 ;
Chu2019 ; DelleMonache2019a . State estimators may involve Kalman filtering
techniques Herrera2010a ; Work2008a ; Work2009  and data fusion Duret2017 ;
Yuan2014 ; Yuan2012 . Via traffic state feedback, traffic control aims to
maximize the benefits of large-scale traffic, which eventually leads to
benefits for individual vehicles too. Research in traffic control has seen a
surge in recent years Siri2021 , extending from classical traffic control
DelleMonache2019  to reinforcement learning Kreidlieh2018 ; Wu2018 . Traffic
control has been approached from so-called Eulerian and Lagrangian
perspectives Laval2013 . In Eulerian traffic control, traffic flow is
regulated at specific locations along the highway based on data about the
traffic further down the road Bekiaris-Liberis2020 ; Yu2018 . Traffic control
measures include speed limits Karafyllis2018  and ramp metering Koehler2016 ;
Pasquale2018 ; Yu2019 ; Yu2021 . In Lagrangian traffic control, traffic is
regulated by the driving behavior of certain vehicles in the flow Cicic2018 ;
Piacentini2019 , which directly take into account how they affect traffic, and
utilize the traffic state as feedback in their controllers.

Since Lagrangian traffic controllers have been enabled by vehicle automation,
they have appeared more recently than Eulerian traffic control, and their
literature is still less extensive. Meanwhile, technological advancement in
the field of automated vehicles keeps opening new possibilities for traffic
control. In particular, the occurrence of connected automated vehicles (CAVs)
has significant potential that has not yet been addressed by the literature.
Therefore, this chapter is devoted to establishing the notion of Lagrangian
traffic control by connected automated vehicles, and highlighting its
potential for mitigating traffic congestions.

Namely, automated vehicles must monitor the state of the traffic behind them
while traveling on highways in order to achieve Lagrangian control of the
traffic behind. Monitoring traffic can be realized by vehicle-to-everything
(V2X) connectivity, which becomes the enabling technology of traffic control
by connected automated vehicles. A CAV, communicating with vehicles behind it,
receives traffic state information that can be used as feedback Chen2021a .
This approach is demonstrated in Fig. 1(c), where a CAV communicates with and
responds to a connected human-driven vehicle (CHV) behind it. Information from
the CHV (red) allows the CAV (blue) to mitigate the traffic jam efficiently:
the entire vehicle chain reduces its speed less than for the case with a large
number of AVs in Fig. 1(b). This demonstrates the large potential of CAVs in
traffic control that is yet to be exploited.

The feedback loop in Fig. 1(c) is similar to when vehicles travel on a ring
road: they respond to each other in a loop, the first vehicle responding to
the last one. Ring configurations have rich and interesting dynamics
Giammarino2019 ; Orosz2010 ; Allwoerden2021 ; Wang2020 ; Zheng2020  and there
have even been successful implementations of vehicle controllers on ring roads
to mitigate traffic Stern2018 . While the road geometry may not be a ring in
practice, the response of vehicles in Lagrangian traffic control has a similar
structure, which we refer to as virtual rings. The concept of virtual rings
have been introduced in the experiments of Avedisov2018  where three vehicles
performed car following on a straight highway, with the first vehicle
responding to the last one. This concept was further discussed in the context
of traffic control in Avedisov2021 ; Molnar2020cdc , and also utilized in
Wang2020c  where the so-called leading cruise control was introduced for high-
connectivity penetration scenarios.

Here we further elaborate the concept of virtual rings and Lagrangian traffic
control by CAVs. As opposed to previous works Molnar2020cdc ; Wang2020c , we
put more emphasis on the stability analysis of traffic flow, focus on low-
automation and low-connectivity scenarios, and incorporate time delays
associated with the response of vehicles, their controllers and the human
drivers. We first introduce strategies for vehicle and traffic control, then
highlight the benefits of connectivity-enabled feedback loops via simulations,
and finally give a comprehensive stability analysis.

##  2 Traffic Control by Connected Automated Vehicles

We consider traffic regulation by means of controlling the motions of
automated vehicles (AVs) and connected automated vehicles (CAVs) traveling in
the traffic flow. Since the driving behavior of these AVs and CAVs affects the
motion of other vehicles behind them, including human-driven vehicles (HVs)
and connected human-driven vehicles (CHVs), they regulate the traffic behind.

First, we discuss strategies where AVs and CAVs respond to vehicles ahead of
them only. In this case, AVs and CAVs indirectly control the traffic behind
them, without feedback of the traffic state. We call this case as vehicle
control influencing traffic. Then we propose strategies where CAVs respond to
vehicles behind them as well. This allows CAVs to directly control traffic by
using the traffic state behind as feedback. We refer to this case as traffic
control.

We consider single lane traffic for simplicity of exposition. The results
could be extended to multiple lanes by incorporating the cross-lane dynamics.
While lane changes and overtaking requires further investigation and extensive
future work, we remark that traffic controllers realized on CAVs consider the
interest of the vehicles behind them, which makes it less likely for those
vehicles to overtake the CAVs.

###  2.1 Simplified Models for Longitudinal Vehicle and Traffic Dynamics

Consider the scenario in Fig. 1, in which vehicles follow each other on a
single lane of a straight road. We number the vehicles with indices increasing
in the direction of motion. We denote the set of all vehicle indices by
$\mathcal{N}$ that comprises of the indices ${\mathcal{N}=\\{\mathcal{N}_{\rm
HV},\mathcal{N}_{\rm CHV},\mathcal{N}_{\rm AV},\mathcal{N}_{\rm CAV}\\}}$
representing four vehicle types defined previously. We distinguish a so-called
ego vehicle of interest by index $0$. We denote the length of vehicle $n$ by
$l_{n}$, the position of its rear bumper by $s_{n}$ and its speed by $v_{n}$,
${n\in\mathcal{N}}$.

We model the motion of vehicle $n$ by a delayed double integrator with
saturation:

$\displaystyle\begin{split}\dot{s}_{n}(t)&=v_{n}(t)\,,\\\ \dot{v}_{n}(t)&={\rm
sat}(u_{n}(t-\tau_{n}))\,,\end{split}$ (1)

${\forall n\in\mathcal{N}}$, where $u_{n}$ is the desired acceleration of
vehicle $n$, selected by the human driver for HVs and CHVs
(${n\in\\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\\}}$) and prescribed by
the longitudinal controller of AVs and CAVs (${n\in\\{\mathcal{N}_{\rm
AV},\mathcal{N}_{\rm CAV}\\}}$). We assume that each vehicle realizes the
desired acceleration by the help of human action or low-level controllers,
respectively, unless this acceleration is above the acceleration capability
$a_{\rm max}$ or below the braking limit $-a_{\rm min}$ of the vehicle. This
is captured by the saturation function:

${\rm sat}(u)={\rm min}\\{{\rm max}\\{-a_{\rm min},u\\},a_{\rm max}\\}\,.$ (2)

Furthermore, we incorporate the time delay $\tau_{n}$ into the model, that
involves the response time of the vehicle, as well as the driver reaction time
for HVs and CHVs, and feedback delays for AVs and CAVs. For simplicity, we
assume identical delays for HVs and CHVs throughout this study with
${\tau_{n}=\tau}$, ${n\in\\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\\}}$,
and we distinguish the delay of the ego vehicle $0$ by the notation
${\sigma=\tau_{0}}$.

To capture human driver behavior, we use simple car-following models. Namely,
our human driver model (HDM) assumes that vehicle $n$ responds to vehicle
${n+1}$ ahead considering the headway (range) ${h_{n}=s_{n+1}-s_{n}-l_{n}}$
and the speed difference (range rate) ${\dot{h}_{n}=v_{n+1}-v_{n}}$ between
them:

$u_{n}=f_{\rm H}(h_{n},\dot{h}_{n},v_{n})\,,$ (3)

where the specific expression of $f_{\rm H}$ depends on the choice of the
model. Examples for HDM include the optimal velocity model (OVM) Bando1998
and the intelligent driver model (IDM) Treiber2000 , which were shown to
capture human driver behavior observed in experimental data Avedisov2018 ;
Dollar2021 . For simplicity of exposition, throughout this paper we assume
that human drivers are identical in their driving behaviors and $f_{\rm H}$ is
the same for each ${n\in\\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\\}}$.

###### Example 1

For the numerical examples of this paper, we use the optimal velocity model as
human driver model:

$f_{\rm H}(h_{n},\dot{h}_{n},v_{n})=\alpha_{\rm H}\big{(}V_{\rm
H}(h_{n})-v_{n}\big{)}+\beta_{\rm H}\dot{h}_{n}\,.$ (4)

Here ${\alpha_{\rm H}}$ and ${\beta_{\rm H}}$ are parameters describing the
human driver. The second term with coefficient $\beta_{\rm H}$ captures the
response of human drivers to the speed difference
${\dot{h}_{n}=v_{n+1}-v_{n}}$ relative to the vehicle ahead. The first term
with parameter $\alpha_{\rm H}$ characterizes the response to the headway
${h_{n}=s_{n+1}-s_{n}-l_{n}}$ measured from the vehicle ahead. We assume that
human drivers track a headway-dependent desired speed given by the range
policy:

$V_{\rm H}(h)={\rm min}\\{{\rm max}\\{0,F_{\rm H}(h)\\},v_{\rm max}\\}\,.$ (5)

This desired speed is nonnegative and it saturates at the speed limit $v_{\rm
max}$. The speed increases strictly monotonically between $0$ and $v_{\rm
max}$ at a rate defined by $F_{\rm H}$ that is a function of the headway $h$.

###  2.2 Vehicle Control Influencing Traffic

Figure 2:  Vehicle control strategies where the driving behavior of
(connected) automated vehicles influences the traffic behind them, including
(a) cruise control, (b) adaptive cruise control, (c) connected cruise control,
and (d) these strategies realized on a ring road.

Now we consider vehicle control for (connected) automated vehicles where AVs
and CAVs influence the traffic behind them without explicitly responding to
its state. Consider the scenario in Fig. 2(a)-(c). We assume that the ego
vehicle (vehicle $0$) is automated (an AV in Fig. 2(a)-(b) and a CAV in Fig.
2(c)), and it is followed by $N$ human-driven vehicles (HVs) constituting the
traffic influenced by vehicle control.

Our goal is to design a control input $u_{0}$ for the ego vehicle based on the
traffic ahead and observe its effects on the behavior of the traffic behind.
If traffic conditions are free-flowing and there are no vehicles ahead of the
AV such as in Fig. 2(a), then the AV may use cruise control (CC) to track a
reference speed $v_{\rm ref}(t)$ such that its speed $v_{0}(t)$ approaches
$v_{\rm ref}(t)$:

$u_{0}=f_{\text{CC}}(v_{0},\underbrace{v_{\rm ref}}_{\text{reference}})\,.$
(6)

The specific form of $f_{\text{CC}}$ depends on the controller type. The
reference speed $v_{\rm ref}(t)$ may be constant, or time-varying, or it may
even depend on the position $s_{0}(t)$ — an example for the latter case is
when vehicles optimize their set speed based on road elevation He2016 .

If the traffic is more dense, the AV may need to respond to the vehicle ahead;
see Fig. 2(b). This can be achieved by sensing the position and speed of the
vehicle ahead via on-board sensors (radar, lidar, camera or ultrasonics) and
using adaptive cruise control (ACC) that takes into account the positions and
speeds of the AV and the vehicle ahead:

$u_{0}=f_{\text{ACC}}(s_{0},v_{0},\underbrace{s_{1},v_{1}}_{\text{vehicle
ahead}})\,.$ (7)

Finally, if the ego vehicle is equipped with a communication device, i.e., it
is a CAV, then it may respond to connected human-driven vehicles (CHVs)
farther ahead, as shown by Fig. 2(c). This leads to connected cruise control
(CCC) that may potentially take into account the positions and speeds of $M$
vehicles ahead:

$u_{0}=f_{\text{CCC}}(s_{0},v_{0},\underbrace{s_{1},v_{1},\ldots,s_{M},v_{M}}_{\text{multiple
vehicles ahead}})\,.$ (8)

If a vehicle ahead is not connected to or sensed by the CAV, its state can be
omitted from $f_{\text{CCC}}$. An aggregated response to multiple vehicles
ahead facilitates smooth driving by making the CAV more clairvoyant about
upcoming changes in traffic conditions Ge2018 .

###### Example 2

Throughout this paper, we consider the following simple examples as
controllers for AVs and CAVs that influence the traffic behind. The simplest
CC strategy is a proportional controller for speed tracking with a control
gain $\beta$:

$f_{\text{CC}}(v_{0},v_{\rm ref})=\beta\big{(}v_{\rm ref}-v_{0}\big{)}\,.$ (9)

Indeed, this law could be replaced by other more sophisticated controllers.

For ACC, we consider the following control law:

$f_{\text{ACC}}(s_{0},v_{0},s_{1},v_{1})=\alpha\big{(}V(s_{1}-s_{0}-l_{0})-v_{0}\big{)}+\beta\big{(}W(v_{1})-v_{0}\big{)}\,.$
(10)

This is an analog to the optimal velocity model (4), with the difference that
the control gains $\alpha$, $\beta$ and the range policy $V$ can be selected
as control design parameters. For $V$ we keep the form (5) and prescribe the
desired speed-headway relationship via $F$. Furthermore, in (10) we include
the speed policy $W$ that allows the AV to follow the speed $v_{1}$ of the
vehicle ahead or the speed limit $v_{\rm max}$, whichever is smaller:

$W(v_{1})={\rm min}\\{v_{1},v_{\rm max}\\}\,.$ (11)

Finally, for CCC we extend the controller (10) with response to the speed of
multiple vehicles ahead Zhang2016 :

$f_{\text{CCC}}(s_{0},v_{0},s_{1},v_{1},\ldots,s_{M},v_{M})=\alpha\big{(}V(s_{1}-s_{0}-l_{0})-v_{0}\big{)}+\sum_{m=1}^{M}\beta_{m}\big{(}W(v_{m})-v_{0}\big{)}\,,$
(12)

where $\beta_{m}$ is the control gain associated with the speed $v_{m}$ of
vehicle $m$ ahead. If some vehicle $m$ is not connected and cannot be
perceived by the CAV, we omit the response to it by taking ${\beta_{m}=0}$.

###  2.3 Traffic Control

Figure 3:  Traffic control strategies executed by connected automated vehicles
responding to the traffic behind, including (a) traffic control, (b) adaptive
traffic control, (c) connected traffic control, and (d) the virtual ring
associated with these traffic control strategies.

Now consider the scenario illustrated in Fig. 3(a)-(c), where the ego vehicle
is a CAV and at least one vehicle behind it is connected, e.g., a CHV. Once
the CAV syncs with the CHV, it can actively regulate the traffic behind, since
it uses traffic state feedback when responding to the CHV. When there are no
vehicles ahead of the CAV to respond to, such as in Fig. 3(a), the cruise
control (6) can be extended to traffic control (TC) in response to the
positions and speeds of the connected vehicles behind the ego CAV:

$u_{0}=f_{\text{TC}}(\underbrace{s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}_{\text{traffic
behind}},s_{0},v_{0},\underbrace{v_{\rm ref}}_{\text{reference}})\,.$ (13)

The quantities ${s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}$ represent the state of
the traffic behind the CAV. We assume that not all vehicles behind the ego CAV
are connected, thus, only some of these states (corresponding to connected
vehicles) may be available to the CAV and affect $u_{0}$. Accordingly, $-N$
denotes the index of the farthest connected vehicle behind communicating with
the ego CAV. This scenario is ideal for traffic control in the sense that the
CAV does not have to respond to the traffic ahead in order to minimize
collision risks. Such large control freedom enables the CAV to stabilize long
vehicle chains behind it.

However, when the CAV is traveling in denser traffic environments like the one
in Fig. 3(b), it has to respond to the vehicle ahead and has less freedom to
regulate the traffic behind it. In such a setup, the adaptive cruise control
(7) can be extended with feedback from the traffic behind to adaptive traffic
control (ATC):

$u_{0}=f_{\text{ATC}}(\underbrace{s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}_{\text{traffic
behind}},s_{0},v_{0},\underbrace{s_{1},v_{1}}_{\text{vehicle ahead}})\,.$ (14)

Finally, if vehicles ahead of the CAV are also connected, as illustrated by
Fig. 3(c), we may establish connected traffic control (CTC) that responds to
multiple vehicles ahead while also taking into account its effect on the
traffic behind:

$u_{0}=f_{\text{CTC}}(\underbrace{s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}_{\text{traffic
behind}},s_{0},v_{0},\underbrace{s_{1},v_{1},\ldots,s_{M},v_{M}}_{\text{multiple
vehicles ahead}})\,.$ (15)

###### Remark 1

– Connectivity and automation. We remark that although only CTC includes the
word ”connected” in its name, TC and ATC also rely on connectivity to obtain
information from the traffic behind (unless ${N=1}$ and the vehicle behind the
AV is detected by on-board sensors). At the same time, automation is not
required by any of these strategies for vehicles other than the ego vehicle.
Yet, if other connected vehicles happen to possess sufficient levels of
automation (they are CAVs), that opens the possibility of coordinating CAVs
for traffic control Mahbub2021 ; Mahbub2021a ; Piacentini2018 . In this paper,
we will not discuss details about such coordination.

###### Example 3

Analogously to the CCC strategy in (12), response to the traffic behind can be
achieved from the feedback of the speed of each connected vehicle behind. This
leads to the TC algorithm:

$u_{0}=\beta\big{(}v_{\rm
ref}-v_{0}\big{)}+\sum_{n=-N}^{-1}\beta_{n}\big{(}W(v_{n})-v_{0}\big{)}\,,$
(16)

cf. (9). Again, ${\beta_{n}=0}$ is considered if the CAV does not perceive a
vehicle due to the lack of sensors or connectivity.

The ACC strategy (10) can also be extended to ATC with a feedback term from
the traffic behind:

$u_{0}=\alpha\big{(}V(s_{1}-s_{0})-v_{0}\big{)}+\beta\big{(}W(v_{1})-v_{0}\big{)}+\sum_{n=-N}^{-1}\beta_{n}\big{(}W(v_{n})-v_{0}\big{)}\,.$
(17)

In what follows, we focus on scenarios with lean penetration of connectivity.
If only a single vehicle is connected to the CAV, we obtain

$u_{0}=\alpha\big{(}V(s_{1}-s_{0})-v_{0}\big{)}+\beta\big{(}W(v_{1})-v_{0}\big{)}+\beta_{\rm
B}(W(v_{-N})-v_{0})\,,$ (18)

where ${\beta_{\rm B}=\beta_{-N}}$ is the control gain related to the traffic
behind.

Finally, the CCC (12) can be extended to CTC according to

$u_{0}=\alpha\big{(}V(s_{1}-s_{0})-v_{0}\big{)}+\sum_{m=1}^{M}\beta_{m}\big{(}W(v_{m})-v_{0}\big{)}+\sum_{n=-N}^{-1}\beta_{n}\big{(}W(v_{n})-v_{0}\big{)}\,.$
(19)

Figure 4:  Relationships between control strategies (top) and the associated
stability conditions (bottom).

###### Remark 2

– Relationships between control strategies. All of the above vehicle and
traffic controllers can be regarded as special cases of CTC. Ultimately, the
choice of controller depends on the available information: whether there are
vehicles ahead and behind the ego CAV that are connected to it. The various
scenarios are summarized at the top of Fig. 4. When the ego CAV does not
respond to multiple vehicles ahead, the CTC and CCC strategies reduce to ATC
and ACC. When there are no vehicles ahead of the CAV at all, ATC and ACC
reduce to TC and CC. When the CAV does not respond to the traffic behind, the
CTC, ATC and TC strategies reduce to a CAV executing CCC, ACC and CC,
respectively, that influences the following human-driven traffic. Taking
${N=0}$ we can disregard the influence of the CAV on the traffic behind,
leading to CCC, ACC and CC controllers without considering traffic
environment. Finally, if the CAV behaves as a human driver, we recover the
case of human-driven traffic. We will also see these interconnections in the
formulas describing the stability of traffic flow.

###### Remark 3

– Rings and virtual rings. Longitudinal controllers of AVs and CAVs are often
analyzed on ring roads, as illustrated by Fig. 2(d). Mathematically, the ring
configuration has an additional periodic boundary condition. For example, if
there are $N+1$ vehicles (indexed from $-N$ to $0$) on a ring, we have:

$\displaystyle\begin{split}s_{-N}(t)&=s_{1}(t)\,,\\\
v_{-N}(t)&=v_{1}(t)\,.\end{split}$ (20)

This is a useful property for the stability analysis of traffic, as discussed
below. Apart from analysis Giammarino2019 ; Orosz2010 ; Allwoerden2021 ;
Wang2020 ; Zheng2020 , ring roads have been used in experiments, including
successful mitigation of traffic jams Stern2018 .

The ring configuration is important conceptually for traffic control that
involves feedback of the traffic behind. Namely, the structure of which
vehicles respond to which other vehicles includes a ring (a loop), see Fig.
3(d), without physically traveling on a ring road. Therefore, we refer to the
structure of traffic control as virtual ring. The concept of virtual ring
appeared in the experiments of Avedisov2018  and the analysis of Avedisov2021
; Molnar2020cdc . Below we build on these works and give a more detailed
analysis.

We also remark that traffic control on single-lane ring roads has a
significant advantage compared to single-lane straight roads: the CAV can
affect the equilibrium speed of the ring and can decide to travel slower than
the set speed of the vehicle ahead. Once the CAV slows down, all vehicles
including the vehicle ahead will eventually travel slower on the ring. In
contrast, if the CAV decides to travel slower on a straight road, it will fall
behind the vehicle ahead and break up the car-following scenario. Hence the
virtual ring is more appealing for highway applications than considering a
physical ring road.

###### Remark 4

– Scenario in the rest of the paper. In what follows, we analyze the dynamics
and stability of traffic influenced by vehicle control and traffic control.
For simplicity, we consider ACC, like in Fig. 2(b), and ATC with a single CHV
behind the ego CAV, like in Fig. 3(b). This restricts us to lean penetration
of connectivity and we can compare how vehicle control without connectivity
influences traffic to traffic control allowed by connectivity. We use human
driver model (4), ACC (10) and ATC (18) as example. Vehicle $1$ is the lead
vehicle, vehicle $0$ is the ego CAV, and vehicles $-1,\ldots,-N$ are HVs with
$-N$ being connected in the ATC setup. Furthermore, we analyze ACC on a ring
road and relate it to the virtual ring of ATC.

##  3 Benefits of Connectivity

Now we demonstrate that information from connectivity is highly beneficial for
traffic control. We compare the ACC and ATC setups by investigating the
traffic behavior through numerical simulations. We also evaluate the energy
consumption of vehicles and show that utilizing connectivity leads to energy
savings. The parameters used for numerical case studies throughout this paper
are listed in Table 1 in the Appendix.

Figure 5:  Traffic influenced by vehicle control without connectivity using
adaptive cruise control (left) and regulated ty traffic control with
connectivity using adaptive traffic control (right).

###  3.1 Simulation Results

First, we study a baseline scenario without automation. We simulate (1, 3) for
human driver model (4) in Example 1 with typical human driver parameters
selected from Avedisov2018 . The simulation results are shown in Fig. 1(a). In
the simulation, a lead vehicle (black) sequentially decelerates, accelerates
and cruises at constant speed, followed by 11 HVs (gray). The following HVs
tend to reduce their speed more the farther they are from the lead vehicle as
speed perturbations amplify along the vehicle chain. Consequently, the tail
vehicle brakes noticeably more than the lead (purple highlight). This is
called string unstable behavior and often leads to the formation of stop-and-
go traffic jams for large enough number of string unstable drivers.

String instability can be mitigated by vehicle control using AVs or CAVs. To
demonstrate how vehicle control influences traffic, we simulate (1, 3, 7) for
a heterogeneous chain of HVs and AVs using ACC controller (10) in Example 2.
The simulation results shown by Fig. 1(b) indicate that the AVs can
successfully mitigate the onset of a traffic congestion: the tail vehicle
brakes as much as the lead vehicle and not more (purple highlight). We remark
that congestions are often caused by the reaction time of human drivers: a
delayed reaction forces drivers to overreact and brake or accelerate more. As
such, automated vehicles must be designed carefully to reduce their response
delays, and their controllers must be tuned by considering the presence of
delays. There exist strategies for automated vehicles that specifically aim to
eliminate the effect of delays BekiarisLiberis2018 ; Molnar2018its ; Qi2021 ;
Wang2018a ; Zhang2020 . Still, even with well-designed automated vehicles it
may require a relatively large penetration of AVs in the traffic flow to
mitigate congestions if human drivers are string unstable (the penetration is
25% in our example, i.e., every fourth vehicle is automated). Connectivity and
a CCC strategy could lead to further benefits: some level of connectivity may
reduce the required penetration of automation to stabilize traffic
Avedisov2021 . Yet, with vehicle control it is often hard to achieve stability
for CAV penetrations around or below 10%.

The benefits of connectivity are further exploited by traffic control
strategies. Simulation of (1, 3, 14) with ATC controller (18) in Example 3 is
shown in Fig. 1(c). Here a single CAV responds to the vehicle ahead and to a
single CHV ${N=10}$ vehicles behind. With ATC strategy, the ego CAV is able to
stabilize the traffic flow, and each vehicle including the CHV brakes less
than the lead vehicle. This is achieved with about 8% penetration of
automation and 17% penetration of connectivity (one automated and two
connected vehicles out of 12 vehicles). The required penetration of automation
is reduced (compared to when traffic is influenced by vehicle control) thanks
to the extra connectivity — which is a significant benefit since automation
implies more cost and requirements than communication. Moreover, the CHV has
incentive to be connected to the CAV as it can slow down less.

The performance with and without connectivity is further explored in Fig. 5,
where the influence of a single ACC vehicle on traffic is compared to traffic
control by ATC. These scenarios are illustrated in Fig. 2(b) and Fig. 3(b);
their only difference is that the vehicle $-N$ is connected in the ATC setup
and the ego CAV responds to it. It can be seen that ATC results in less speed
reduction and smoother driving for the whole vehicle chain. The price is that
the headway of the CAV executing ATC fluctuates more than that for ACC.
However, it does not compromise safety, the vehicle is still far from
collision. In the meantime, smooth driving leads to less energy consumption,
which is discussed next.

###  3.2 Energy Efficiency

We evaluate the energy consumption for each vehicle by the following measure
He2020 :

$w_{n}(t)=\int_{t_{0}}^{t}v_{n}(\theta)g\big{(}\dot{v}_{n}(\theta)+p(v_{n}(\theta))\big{)}{\rm
d}\theta\,,$ (21)

that is, the energy over unit mass consumed during the time interval
$[t_{0},t]$. It is an integral of the power over unit mass obtained from the
product of speed and commanded acceleration. The latter involves the vehicle
acceleration and the resistance terms represented by $p$ that vehicles need to
overcome. These can be modeled for example by

$p(v)=a_{\rm r}+c_{\rm r}v^{2}\,,$ (22)

where $a_{\rm r}$ accounts for rolling resistance, while $c_{\rm r}v^{2}$
represents air drag. Furthermore, function $g$ in (21) describes how the
commanded acceleration is related to energy consumption. For example, if we
take

$g(x)=\max\\{0,x\\}\,,$ (23)

then we assume energy consumption during acceleration only and no energy
consumption or regeneration during braking.

The energy measure (21, 22, 23) was evaluated for the simulations in Fig. 5;
see the bottom of the plot. The energy consumption stays constant during
braking, increases linearly during constant speed cruising, and increases at a
higher rate during acceleration. Therefore, it is critical for the vehicles to
drive smoothly and avoid decelerations after which they need to accelerate to
recover the speed.

To highlight the difference between the total energy consumption of ACC and
ATC, in Fig. 6 we plot the final value ${w_{0}(t_{\rm f})}$ of the energy
consumed by the CAV over the simulation interval $[t_{0},t_{\rm f}]$.
Specifically, the simulations were repeated and the energy consumption was
calculated for a grid of $\beta$ and $\beta_{\rm B}$ control gains
representing various control designs. The corresponding energy consumption
levels are shown by the colored contours. It can be seen that increasing
$\beta_{\rm B}$ leads to a lower energy level. One should, however, always
keep safety in mind and choose $\beta_{\rm B}$ small enough to maintain a safe
headway similarly to Fig. 5.

The gains corresponding to Fig. 5 are indicated by point A for ACC
(${\beta_{\rm B}=0}$) and point B for ATC (${\beta_{\rm B}\neq 0}$) in Fig. 6.
These two cases are compared on the right of Fig. 6 for various values of $N$.
It can be seen that ATC consistently saves around 2-3% energy for the ego CAV
compared to ACC if ${N\geq 5}$. The energy consumption $w_{-N}(t_{\rm f})$ of
the CHV is also indicated. By deciding to stay connected, the CHV saves
significant energy, around 6-8% if ${N\geq 14}$. The trend is that the farther
the CHV is from the CAV (i.e., the larger $N$ is), the more energy it saves —
this is a consequence of a longer string unstable chain of HVs being more
sensitive to the CAV’s motion.

We remark that the energy contours in Fig. 6 are specific to the velocity
profile of the lead vehicle and to the choice (21) of the energy measure.
Besides, extensive numerical simulations were needed to obtain these contours
and to identify which controller parameters are energy efficient. Finding
energy-optimal parameters through theoretical analysis is a challenging task
He2020 . On the other hand, energy consumption is associated with the
smoothness of driving, thus it is related to the stability of the traffic
flow. String stable chains of vehicles attenuate speed fluctuations, drive
smoother, and hence tend to consume less energy. Stability analysis is more
straightforward than finding energy optima. Hence, below we analyze the
dynamics and stability of traffic to drive the controller parameter selection.

Figure 6:  Total energy consumption of the ego CAV in the simulation scenario
of Fig. 5 with various $\beta$ and $\beta_{\rm B}$ control gains (left). The
energy consumption is reduced in traffic control (ATC, ${\beta_{\rm B}\neq
0}$) compared to vehicle control (ACC, ${\beta_{\rm B}=0}$). The relative
energy benefits are shown for both the ego CAV and the CHV (right).

##  4 Dynamics of Traffic Flow

In the remaining two sections, we study the dynamics and stability of vehicle
and traffic control in detail. The end result is manifested in so-called
stability charts, which can be used to select the control gains of vehicle and
traffic controllers such that string stability is achieved and traffic
congestions are alleviated. These two sections are algebraically more
involved, thus readers who are less interested in the technical details can
skip these and move to the Conclusions.

###  4.1 Linearized Dynamics

To analyze stability, first we consider traffic flows in equilibrium (often
referred to as uniform flow) where all vehicles drive at the same constant
speed. The equilibrium speed is dictated by how fast the lead vehicle travels
— in practice it can be considered as the average speed of the lead vehicle —
and the following vehicles adjust their speed to the lead. Afterwards, we
consider deviations from this equilibrium to quantify how much speed
perturbations amplify. This will also be used to carry out stability analysis
in the next section. We conduct our analysis in the frequency domain by
linearization and by constructing transfer functions that allow us to
characterize the response of individual vehicles and the overall traffic
behavior.

We start with formalizing the equilibrium solution as

$v_{n}(t)\equiv v^{*}\,,\quad s_{n}(t)=s_{n}^{*}(t)=v^{*}t+s_{n}(0)\,,$ (24)

${n\in\mathcal{N}}$, which is associated with constant speed $v^{*}$ that is
identical for each vehicle, and constant headway
${h_{n}^{*}=s_{n+1}^{*}-s_{n}^{*}-l_{n}}$ that may be different for each
vehicle. This equilibrium can be found by substituting (24) into (1, 3) and
(7) for ACC or (14) for ATC, and solving for $v^{*}$ and $h_{n}^{*}$.

We consider perturbations around the equilibrium in the form

$v_{n}(t)=v^{*}+\tilde{v}_{n}(t)\,,\quad
s_{n}(t)=s_{n}^{*}(t)+\tilde{s}_{n}(t)\,,$ (25)

and collect position and speed perturbations into the state vector
$\mathbf{x}$ given by

$\mathbf{x}_{n}(t)=\begin{bmatrix}\tilde{s}_{n}(t)\\\
\tilde{v}_{n}(t)\end{bmatrix}=\begin{bmatrix}\tilde{v}_{n}(0)+\int_{0}^{t}\tilde{v}_{n}(\theta){\rm
d}\theta\\\ \tilde{v}_{n}(t)\end{bmatrix},$ (26)

from which the speed fluctuations can be recovered by

$\tilde{v}_{n}(t)=\mathbf{c}\mathbf{x}_{n}(t),\quad\mathbf{c}=\begin{bmatrix}0&1\end{bmatrix}.$
(27)

Assuming the accelerations of vehicles do not saturate, the linearized
dynamics of HVs and CHVs can be derived from (1, 3) in the form

$\dot{\mathbf{x}}_{n}(t)=\mathbf{a}\mathbf{x}_{n}(t)+\mathbf{a}_{\rm
H}\mathbf{x}_{n}(t-\tau)+\mathbf{b}_{\rm H}\mathbf{x}_{n+1}(t-\tau)\,,$ (28)

${n\in\\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\\}}$. That is, it contains
the delay-free state of vehicle $n$ and the delayed states of vehicles $n$ and
$n+1$ with coefficient matrices

$\mathbf{a}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm
H}=\begin{bmatrix}0&0\\\ -\left.\dfrac{\partial f_{\rm H}}{\partial
h_{n}}\right|_{*}&\left.\dfrac{\partial f_{\rm H}}{\partial
v_{n}}\right|_{*}-\left.\dfrac{\partial f_{\rm
H}}{\partial\dot{h}_{n}}\right|_{*}\end{bmatrix},\quad\mathbf{b}_{\rm
H}=\begin{bmatrix}0&0\\\ \left.\dfrac{\partial f_{\rm H}}{\partial
h_{n}}\right|_{*}&\left.\dfrac{\partial f_{\rm
H}}{\partial\dot{h}_{n}}\right|_{*}\end{bmatrix},$ (29)

where star indicates that partial derivatives are evaluated at the
equilibrium. For AVs performing ACC the linearized dynamics become

$\dot{\mathbf{x}}_{0}(t)=\mathbf{a}\mathbf{x}_{0}(t)+\mathbf{a}_{\rm
F}\mathbf{x}_{0}(t-\sigma)+\mathbf{b}_{\rm F}\mathbf{x}_{1}(t-\sigma)\,,$ (30)

with

$\mathbf{a}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm
F}=\begin{bmatrix}0&0\\\ \left.\dfrac{\partial f_{\text{ACC}}}{\partial
s_{0}}\right|_{*}&\left.\dfrac{\partial f_{\text{ACC}}}{\partial
v_{0}}\right|_{*}\end{bmatrix},\quad\mathbf{b}_{\rm F}=\begin{bmatrix}0&0\\\
\left.\dfrac{\partial f_{\text{ACC}}}{\partial
s_{1}}\right|_{*}&\left.\dfrac{\partial f_{\text{ACC}}}{\partial
v_{1}}\right|_{*}\end{bmatrix}.$ (31)

whereas CAVs executing ATC are described by

$\dot{\mathbf{x}}_{0}(t)=\mathbf{a}\mathbf{x}_{0}(t)+\mathbf{a}_{\rm
FB}\mathbf{x}_{0}(t-\sigma)+\mathbf{b}_{\rm
F}\mathbf{x}_{1}(t-\sigma)+\mathbf{b}_{\rm B}\mathbf{x}_{-N}(t-\sigma)\,,$
(32)

with

$\displaystyle\begin{split}&\mathbf{a}=\begin{bmatrix}0&1\\\
0&0\end{bmatrix},\quad\mathbf{a}_{\rm FB}=\begin{bmatrix}0&0\\\
\left.\dfrac{\partial f_{\text{ATC}}}{\partial
s_{0}}\right|_{*}&\left.\dfrac{\partial f_{\text{ATC}}}{\partial
v_{0}}\right|_{*}\end{bmatrix},\\\ &\mathbf{b}_{\rm F}=\begin{bmatrix}0&0\\\
\left.\dfrac{\partial f_{\text{ATC}}}{\partial
s_{1}}\right|_{*}&\left.\dfrac{\partial f_{\text{ATC}}}{\partial
v_{1}}\right|_{*}\end{bmatrix},\quad\mathbf{b}_{\rm B}=\begin{bmatrix}0&0\\\
\left.\dfrac{\partial f_{\text{ATC}}}{\partial
s_{-N}}\right|_{*}&\left.\dfrac{\partial f_{\text{ATC}}}{\partial
v_{-N}}\right|_{*}\end{bmatrix}.\end{split}$ (33)

In summary, vehicle control influencing traffic is described by (28, 30) at
the linear level, while traffic control is characterized by (28, 32). For the
ring configuration, one also has periodicity given by (20) that yields

$\mathbf{x}_{-N}(t)=\mathbf{x}_{1}(t)\,.$ (34)

###### Example 4

The coefficient matrices (29) describing human drivers have the following
expressions for the case of the optimal velocity model (4):

$\mathbf{a}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm
H}=\begin{bmatrix}0&0\\\ -\alpha_{\rm H}\kappa_{\rm H}&-(\alpha_{\rm
H}+\beta_{\rm H})\end{bmatrix},\quad\mathbf{b}_{\rm H}=\begin{bmatrix}0&0\\\
\alpha_{\rm H}\kappa_{\rm H}&\beta_{\rm H}\end{bmatrix}.$ (35)

The ACC controller (10) is associated with the coefficient matrices

$\mathbf{a}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm
F}=\begin{bmatrix}0&0\\\
-\alpha\kappa&-(\alpha+\beta)\end{bmatrix},\quad\mathbf{b}_{\rm
F}=\begin{bmatrix}0&0\\\ \alpha\kappa&\beta\end{bmatrix},$ (36)

whereas the ATC controller (18) corresponds to

$\mathbf{a}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm
FB}=\begin{bmatrix}0&0\\\ -\alpha\kappa&-(\alpha+\beta+\beta_{\rm
B})\end{bmatrix},\quad\mathbf{b}_{\rm F}=\begin{bmatrix}0&0\\\
\alpha\kappa&\beta\end{bmatrix},\quad\mathbf{b}_{\rm B}=\begin{bmatrix}0&0\\\
0&\beta_{\rm B}\end{bmatrix}.$ (37)

###  4.2 Transfer Functions

We analyze the linearized dynamics in Laplace domain (assuming zero initial
perturbations). Our ultimate goal is to analyze the stability of the system,
in particular, whether speed fluctuations amplify or decay as they propagate
along the chain of vehicles. This amplification property can be well
represented by means of transfer functions describing the response of vehicles
to the neighboring traffic.

First, we formulate the so-called link transfer functions Zhang2016
associated with vehicle pairs. For human drivers, the link transfer function
$T_{\rm H}(s)$ relates the speed fluctuation of an HV to that of the vehicle
ahead as

$V_{n}(s)=T_{\rm H}(s)V_{n+1}(s)\,,$ (38)

where $T_{\rm H}(s)$ can be obtained from (27, 28) in the form

$T_{\rm H}(s)=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm H}{\rm
e}^{-s\tau}\Big{)}^{-1}\mathbf{b}_{\rm H}{\rm
e}^{-s\tau}\begin{bmatrix}\frac{1}{s}\\\ 1\end{bmatrix}.$ (39)

Similarly, in ACC where the automated vehicle responds to the vehicle
immediately ahead only, the response can be characterized by a single link
transfer function $T(s)$:

$V_{0}(s)=T(s)V_{1}(s)\,,$ (40)

whose expression can be derived from (30) as

$T(s)=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm F}{\rm
e}^{-s\sigma}\Big{)}^{-1}\mathbf{b}_{\rm F}{\rm
e}^{-s\sigma}\begin{bmatrix}\frac{1}{s}\\\ 1\end{bmatrix}.$ (41)

On the other hand, in ATC the ego CAV responds to multiple vehicles,
associated with multiple link transfer functions. For example, when responding
to a single vehicle ahead and a single vehicle behind, we obtain

$V_{0}(s)=T_{\rm F}(s)V_{1}(s)+T_{\rm B}(s)V_{-N}(s)\,,$ (42)

where $T_{\rm F}(s)$ and $T_{\rm B}(s)$ are the link transfer functions
characterizing the response forward and backward. These are obtained from
(32):

$\displaystyle\begin{split}T_{\rm
F}(s)&=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm FB}{\rm
e}^{-s\sigma}\Big{)}^{-1}\mathbf{b}_{\rm F}{\rm
e}^{-s\sigma}\begin{bmatrix}\frac{1}{s}\\\ 1\end{bmatrix},\\\ T_{\rm
B}(s)&=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm FB}{\rm
e}^{-s\sigma}\Big{)}^{-1}\mathbf{b}_{\rm B}{\rm
e}^{-s\sigma}\begin{bmatrix}\frac{1}{s}\\\ 1\end{bmatrix}.\end{split}$ (43)

Notice that the denominators of $T_{\rm F}(s)$ and $T_{\rm B}(s)$ are the
same, ${\rm D}(T_{\rm F}(s))={\rm D}(T_{\rm B}(s))$, since the same matrix is
inverted.

Using the link transfer functions, we can analyze the overall response of
traffic. For a chain of $N$ human drivers, the overall response

$V_{-N}(s)=\Gamma(s)V_{0}(s)\,.$ (44)

is characterized by $\Gamma(s)$ given by

$\Gamma(s)=T_{\rm H}(s)^{N}\,.$ (45)

Note that this corresponds to identical human drivers; for the case of a
heterogeneous chain of HVs it should be replaced with
${\Gamma(s)=\prod_{n=-N}^{-1}T_{{\rm H},n}(s)}$.

Introducing the ego vehicle (indexed $0$) and the vehicle ahead of it (lead
vehicle $1$) into the human-driven traffic, like in Figs. 2(b) and 3(b), we
can describe the overall response of the traffic flow from the head vehicle to
the tail vehicle via the head-to-tail transfer function Zhang2016  given by

$V_{-N}(s)=G(s)V_{1}(s)\,.$ (46)

$G(s)$ depends on the human response $\Gamma(s)$ and the control strategy of
the ego vehicle. For vehicle control with ACC, we write (40, 44) and obtain

$G(s)=T(s)\Gamma(s)\,.$ (47)

For the traffic control setup with ATC, we have (42, 44) that lead to

$G(s)=\frac{T_{\rm F}(s)\Gamma(s)}{1-T_{\rm B}(s)\Gamma(s)}\,.$ (48)

If traffic is considered on a ring road, we have the periodic boundary
condition

$V_{-N}(s)=V_{1}(s)\,,$ (49)

that leads to

$G(s)=1\,,$ (50)

as the characteristic equation of the ring configuration.

###### Remark 5

– Dynamics of rings and virtual rings. If vehicle control (47) is executed on
a ring road, (50) leads to

$1-T(s)\Gamma(s)=0\,.$ (51)

Here the left-hand side corresponds to the expression in the denominator of
$G(s)$ in (48). Therefore, the dynamics of traffic control, that involves a
virtual ring, is closely related to how vehicle control influences traffic
when executed on a ring road. As highlighted in the analysis below, the
stability properties of the ring configuration give the backbone of the
stability analysis of the virtual ring.

###### Example 5

Substituting the coefficient matrices in (35) into the link transfer function
(39) describing human drivers leads to

$T_{\rm H}(s)=\frac{\beta_{\rm H}s+\alpha_{\rm H}\kappa_{\rm H}}{s^{2}{\rm
e}^{s\tau}+(\alpha_{\rm H}+\beta_{\rm H})s+\alpha_{\rm H}\kappa_{\rm H}}\,.$
(52)

Similarly, the link transfer function of ACC given by (36) and (41) is

$T(s)=\frac{\beta s+\alpha\kappa}{s^{2}{\rm
e}^{s\sigma}+(\alpha+\beta)s+\alpha\kappa}\,,$ (53)

whereas for ATC (37) and (43) give

$\displaystyle\begin{split}T_{\rm F}(s)&=\frac{\beta s+\alpha\kappa}{s^{2}{\rm
e}^{s\sigma}+(\alpha+\beta+\beta_{\rm B})s+\alpha\kappa}\,,\\\ T_{\rm
B}(s)&=\frac{\beta_{\rm B}s}{s^{2}{\rm e}^{s\sigma}+(\alpha+\beta+\beta_{\rm
B})s+\alpha\kappa}\,.\end{split}$ (54)

Observe that we have ${T(s)=T_{\rm F}(s)/(1-T_{\rm B}(s))}$ that relates ACC
and ATC.

###### Remark 6

– Special cases. When the ego CAV does not respond to the traffic behind, ATC
reduces to ACC. Mathematically, this means setting ${T_{\rm B}(s)=0}$,
${T_{\rm F}(s)=T(s)}$, e.g. by taking ${\beta_{\rm B}=0}$ in Example 5. This
reduces $G(s)$ in (48) to $G(s)$ in (47). If no traffic is considered behind
the ego CAV, then ${N=0}$ and ${\Gamma(s)=1}$ due to (45). In such a case,
$G(s)$ in (47) and (48) both reduce to $T(s)$ describing the response of the
ACC vehicle to the vehicle ahead only (since ${T(s)=T_{\rm F}(s)/(1-T_{\rm
B}(s))}$ in our example). Consequently, we can analyze the dynamics of ATC and
then take these special cases to capture the dynamics of ACC followed by
human-driven traffic or ACC on its own; recall Fig. 4.

##  5 Stability of Traffic Flow

Herein we formalize the stability conditions of traffic flows influenced by
vehicle control and regulated by traffic control, while linking them to those
of the ring configuration.

###  5.1 Stability Conditions

For a chain of vehicles traveling on a straight road, the system is non-
autonomous, forced by the speed perturbations of the lead vehicle. Thus, we
consider the notions of plant stability and string stability Feng2019 ;
Ploeg2014a ; Zhang2016  to analyze the behavior without speed perturbations
and the response to the perturbations, respectively. For the ring
configuration, the system evolves autonomously, and we apply a single notion
of stability; for more details on stability definitions see Giammarino2019 .

String stability is particularly important in trarffic control. It has various
mathematical definitions; see a comprehensive summary in Feng2019 . Now we
consider linear input-to-output string stability with respect to
$\mathcal{L}_{2}$ norm. This string stability notion is directly related to
the magnitude ($\mathcal{H}_{\infty}$ norm) of the transfer function
describing system, which makes it is easy to analyze and useful for control
design. Other notions are also widely-used, such as $\mathcal{L}_{p}$ string
stability for any value of $p$ Ploeg2014a . $\mathcal{L}_{\infty}$ string
stability is particularly relevant for traffic safety, since it characterizes
the overshoot of the response to perturbations. As this notion is related to
the impulse response, it is more challenging to analyze it explicitly, and we
rather consider $\mathcal{L}_{2}$ string stability.

#### Plant Stability of a Vehicle Chain

Plant stability means that each vehicle in the chain can approach a constant
(equilibrium) speed in a stable manner. If the vehicle chain is associated
with head-to-tail transfer function $G(s)$, the plant stability condition is
given by

${\rm
D}(G(s_{\ell}))=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,.$
(55)

That is, the roots $s_{\ell}$ of the characteristic equation ${{\rm
D}(G(s))=0}$ are located in the left-half of the complex plane, where ${\rm
D}(G(s))$ is the denominator of the head-to-tail transfer function. The system
is at the plant stability boundary if either a real root ${s=0}$ is located on
the imaginary axis:

${\rm D}({G(0)})=0\,,$ (56)

or a complex conjugate pair of roots ${s=\pm{\rm i}\Omega}$, ${\Omega>0}$:

${\rm D}({G({\rm i}\Omega)})=0\,.$ (57)

#### String Stability of a Vehicle Chain

String stability requires, on one hand, plant stability as a necessary
condition, and on the other hand, that the speed perturbations of the head
vehicle are not amplified by the vehicle chain. Otherwise large speed
fluctuations could arise at the end of the chain, ultimately leading to
traffic jams and stop-and-go motion. Specifically, we require that the speed
fluctuations $V_{1}({\rm i}\omega)$ of the head vehicle with any given
frequency ${\omega>0}$ are smaller in amplitude than those $V_{-N}({\rm
i}\omega)$ of the tail vehicle, which, based on (46), can be expressed as

$\left|G({\rm i}\omega)\right|<1\,,\quad\forall\omega>0\,.$ (58)

At the string stability boundary we have

$\left|G({\rm i}\omega)\right|=1\quad\iff\quad G({\rm i}\omega)={\rm e}^{-{\rm
i}K}\,,$ (59)

for some ${\omega>0}$ and ${K\in[0,2\pi)}$, i.e., the speed fluctuations of
the head vehicle are ”repeated” by the tail vehicle with a phase lag $K$ (also
called as the wave number).

Furthermore, string stability can also be studied when ${\omega\to 0}$,
shortly referred to as ${\omega=0}$ string stability. For algebraic
convenience, we study this by introducing

$P(\omega)=\frac{1}{\omega^{2}}\left({\rm D}\left(\left|G({\rm
i}\omega)\right|^{2}\right)-{\rm N}\left(\left|G({\rm
i}\omega)\right|^{2}\right)\right),$ (60)

where ${\rm N}$ and ${\rm D}$ denote numerator and denominator, respectively.
With this definition, (58) can be rewritten as

$P(\omega)>0\,,\quad\forall\omega>0\,,$ (61)

and the ${\omega=0}$ string stability boundary can be written simply as

$P(0)=0\,.$ (62)

#### Stability of the Ring Configuration

Finally, the stability of the ring configuration is determined by the roots of
the characteristic equation (50), and stability requires:

$G(s_{\ell})=1\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,$
(63)

analogously to (55). Recall that $G(s)$ is the head-to-tail transfer function
of the corresponding open chain configuration. Since ${G(0)=1}$ automatically
holds, we consider the stability boundary where ${s=0}$ is a double root of
${G(s)-1}$, and where ${s=\pm{\rm i}\omega}$, $\omega>0$ satisfies the
characteristic equation:

$G({\rm i}\omega)=1\,.$ (64)

###  5.2 Relationship of the Stability Conditions

Now let us take a deeper look into the relationship between the various
vehicle configurations and their stability conditions. These relationships are
summarized at the bottom of Fig. 4 for the ${s={\rm i}\omega}$ stability
boundaries.

For vehicle control with (47), we have ${{\rm D}(G(s))={\rm D}(T(s)){\rm
D}(\Gamma(s))}$, hence the plant stability condition becomes

$\displaystyle\begin{split}{\rm
D}(T(s_{\ell}))&=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,\quad\text{and}\\\
{\rm
D}(\Gamma(s_{\ell}))&=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,\end{split}$
(65)

that is, both the AV and HVs must be plant stable individually. For string
stability, we require

$\left|T({\rm i}\omega)\right|\left|\Gamma({\rm
i}\omega)\right|<1\,,\quad\forall\omega>0\,,$ (66)

and at the stability boundary we have

$T({\rm i}\omega)\Gamma({\rm i}\omega)={\rm e}^{-{\rm i}K}\,,$ (67)

which depend both on the response of the AV and the chain of HVs. If the
human-driven traffic is not considered, i.e., ${N=0}$, ${\Gamma({\rm
i}\omega)=1}$, then we recover the string stability limit ${T({\rm
i}\omega)={\rm e}^{-{\rm i}K}}$ for a single ACC vehicle.

Now consider a nonzero (and potentially large) number $N$ of HVs. This leads
to one of the most interesting research questions, that several recent works
have studied Cui2017 ; Qin2020 ; Stern2018 : how many AVs are needed to
mitigate traffic congestions? If a human-driven vehicle chain behind an AV is
long ($N$ is large), it results in a significant (exponential) increase of
speed perturbations. Thus, the AV has to make considerable effort to dampen
the perturbation in order to prevent or mitigate a congestion. Mathematically,
the magnitude ${\left|T_{\rm H}({\rm i}\omega)\right|}$ of the individual
human responses affects the overall response ${\left|\Gamma({\rm
i}\omega)\right|=\left|T_{\rm H}({\rm i}\omega)\right|^{N}}$ more
significantly for large $N$. Therefore, it may be more challenging to
stabilize a long chain of HVs with a single AV.

To highlight this, consider the limit ${N\to\infty}$. If the human drivers are
string stable, i.e., ${\left|T_{\rm H}({\rm i}\omega)\right|<1}$, then
${\left|\Gamma({\rm i}\omega)\right|=0}$ for all ${\omega>0}$. Unless
${\left|T({\rm i}\omega)\right|\to\infty}$, which can only happen at the plant
stability boundary ${{\rm D}(T({\rm i}\omega))=0}$ of the AV, (66) is
automatically satisfied. Thus string stability is achieved for a long string
stable chain of HVs and a plant stable AV. However, when the human drivers are
string unstable — which is more likely the case in real traffic Molnar2021trc
— we have ${\left|T_{\rm H}({\rm i}\omega)\right|>1}$ and ${\left|\Gamma({\rm
i}\omega)\right|\to\infty}$ for some ${\omega>0}$ and ${N\to\infty}$. Thus,
string stability in (66) cannot be achieved by any AV design (as long as
${T({\rm i}\omega)\neq 0}$, i.e., the AV responds to what is ahead). This
shows the fundamental limitation of influencing traffic by vehicle control:
string stability cannot be achieved with arbitrarily small penetration of AVs
(i.e., in the limit ${N\to\infty}$) if the human drivers are string unstable.

If vehicle control is executed on a ring, the stability limit (64) becomes

$T({\rm i}\omega)\Gamma({\rm i}\omega)=1\,.$ (68)

Therefore, the stability boundary of the ring can be obtained as the ${K=0}$
special case of the string stability boundary (67) of the open chain.
Physically, it means that perturbations propagate through the ring and arrive
back to the same vehicle in the same phase due to the periodic boundary
condition. Another special case to mention is when all vehicles on the ring
are identical, achieved by the substitution ${T(s)=T_{\rm H}(s)}$. Then the
characteristic equation (68) gives

$T_{\rm H}({\rm i}\omega)^{N+1}=1\quad\iff\quad T_{\rm H}({\rm i}\omega)={\rm
e}^{-{\rm i}\frac{2k\pi}{N+1}}\,,\quad k\in\\{0,1,\ldots,N\\}\,.$ (69)

When ${N\to\infty}$, the exponent ${K=2k\pi/(N+1)}$ may take any value on
$[0,2\pi)$ and we recover the string stability condition ${T_{\rm H}({\rm
i}\omega)={\rm e}^{-{\rm i}K}}$ for individual human drivers. Thus, the
stability of an infinitely long homogeneous ring is equivalent to the string
stability of a homogeneous chain of vehicles. Conversely, the substitution of
${K=2k\pi/(N+1)}$ into the string stability limit ${T_{\rm H}({\rm
i}\omega)={\rm e}^{-{\rm i}K}}$ of an individual human driver gives the
stability limit (69) of the homogeneous ring.

Figure 7:  Stability chart of adaptive cruise control without considering its
influence on the traffic behind (${N=0}$). Detailed (left) and simplified view
(center). The light gray region enclosed by red is plant stable, the dark gray
region encircled by blue is string stable. Characteristic roots (right) and
frequency response (bottom) for specific control gains selected from the
stability chart.

Additionally, and most importantly, the traffic control setup as a virtual
ring is related to both the open chain and ring configurations in terms of
stability. Consider formula (48) of $G(s)$. Since ${\rm D}(T_{\rm F}(s))={\rm
D}(T_{\rm B}(s))$, we have

${\rm D}(G(s))={\rm D}(T_{\rm B}(s)\Gamma(s))\big{(}1-T_{\rm
B}(s)\Gamma(s)\big{)}\,.$ (70)

Hence, and the plant stability condition (55) leads to

$\displaystyle\begin{split}{\rm D}(T_{\rm
B}(s_{\ell})\Gamma(s_{\ell}))&=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,\quad\text{and}\\\
T_{\rm
B}(s)\Gamma(s)&=1\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,.\end{split}$
(71)

That is, it requires both the plant stability of the associated vehicle
control setup and the stability of the associated virtual ring configuration.

The string stability boundary of ATC can be written in the form

$\frac{T_{\rm F}({\rm i}\omega)\Gamma({\rm i}\omega)}{1-T_{\rm B}({\rm
i}\omega)\Gamma({\rm i}\omega)}={\rm e}^{-{\rm i}K}\,.$ (72)

This reduces to the string stability boundary of traffic flows influenced by
vehicle control when there is no response to the traffic behind, i.e.,
${T_{\rm B}({\rm i}\omega)=0}$, ${T_{\rm F}({\rm i}\omega)=T({\rm i}\omega)}$,
which is achieved by ${\beta_{\rm B}=0}$ in our example. In this sense, the
string stability of ATC is related to that of ACC while the plant stability of
ATC is related to the stability of its virtual ring.

Figure 8:  Stability charts of vehicle control influencing traffic (ACC, first
row) and traffic control (ATC, second and third rows, and TC, fourth row). The
color scheme is the same as in Fig. 7.

###### Example 6

The stability analysis can be carried out for the specific choice (4) of human
driver model and selection (10, 18) of control laws. The expressions (56, 57)
of the plant stability boundaries, (59, 62) of the string stability boundaries
and (64) of the stability boundaries for the ring can be expanded by using
(52, 53, 54). Then, the stability boundaries can be expressed in terms of the
controller parameters such as $\alpha$, $\beta$ or $\beta_{\rm B}$. This
allows one to select control gains such that they ensure stability. The
detailed formulas of this analysis are derived in the Appendix.

The resulting stability boundaries can be visualized in the space of control
parameters, which we refer to as stability charts. The left panel of Fig. 7
shows the stability chart in the $(\beta,\alpha)$ plane for ACC without
considering the traffic behind, i.e., for ${N=0}$. This special case can also
be found in Zhang2016 . Dashed red line indicates the ${s=0}$ plant stability
boundary, solid red line shows the ${s=\pm{\rm i}\Omega}$ plant stability
boundary, and the plant stable region is light gray. Furthermore, dashed blue
lines show the ${\omega=0}$ string stability boundaries (that overlap with the
dashed red lines at ${\alpha=0}$), and colorful curves denote the ${\omega>0}$
string stability boundaries for various values of ${K\in[0,2\pi)}$ indicated
by color. The envelope of these curves is presented in the center with a solid
blue line, and the string stable region is dark gray. The control gains shall
be selected from this region to achieve stability.

Fig. 7 also illustrates the characteristics of traffic dynamics under
different selections of the control gains $(\beta,\alpha)$. The right panels
show the characteristic roots associated with the plant stability condition
(55) for a plant unstable point with a complex pair of unstable roots (A), a
plant stable point (C), and a plant unstable point with a real unstable root
(E). The bottom panels depict the magnitude of the head-to-tail transfer
function related to the string stability condition (58) for a point with
${\omega=0}$ string instability (B), a string stable point (C) and a point
with ${\omega>0}$ string instability (D). Out of these points, only point C
has desired behavior. This illustrates how stability charts may drive the
selection of controller parameters, so that the end result is a stable and
smooth traffic flow.

The stability charts also given another aspect of comparing vehicle and
traffic control strategies. The top row of Fig. 8 show the stability charts in
the $(\beta,\alpha)$ plane for ACC followed by various numbers of human
drivers (${N=1,2,3,4}$). Since the human driver parameters are selected to be
string unstable, the string stable region shrinks as the number of vehicles
increases. Furthermore, it also shifts towards large gains that may require
large control input and acceleration from the ACC vehicle. In comparison, the
string stable region of ATC in the second row of Fig. 8 shrinks faster, but is
located at lower ${\alpha}$ gains associated with smaller acceleration and
better passenger comfort.

For a fixed, reasonably small gain ${\alpha}$, the stability charts of ATC are
plotted in the $(\beta,\beta_{\rm B})$ plane in the third row of Fig. 8. Note
that the points along ${\beta_{\rm B}=0}$ represent ACC as a special case, and
they are located close to or outside the string stable region. Hence for such
realistic choice of ${\alpha}$, traffic control with ATC (${\beta_{\rm B}\neq
0}$) may achieve string stability while vehicle control with ACC (${\beta_{\rm
B}=0}$) may not. For a large number of string unstable human drivers, the
string stable region disappears (see ${N=4}$). However, ATC may still be
beneficial by reducing instability, improving the smoothness of driving
behavior, and reducing energy consumption; as these were highlighted in Figs.
1 and 6.

Finally, the last row of Fig. 8 shows the stability charts of TC (as the
${\alpha=0}$ special case of ATC). Although the string stable region shrinks
as the chain of HVs gets longer ($N$ gets larger), it is significantly larger
than that of ATC. In fact, the string stable region does not vanish even for
chains as long as ${N=10}$ HVs. This indicates that CAVs can control traffic
significantly more efficiently in situations where they do not have to respond
to the position of the vehicle ahead.

##  6 Conclusions

We have discussed vehicle control strategies influencing traffic and traffic
control strategies regulating traffic by means of injecting connected
automated vehicles (CAVs) into the traffic flow. We have shown that vehicle-
to-everything (V2X) connectivity allows a CAV to receive information about the
traffic behind it, respond to it, and mitigate traffic congestions, while also
responding to vehicles ahead. The response to the traffic behind creates a
feedback loop for traffic control, in a similar structure to a ring road,
termed virtual ring. This strategy achieves smooth driving and energy savings
for both the CAV, and the vehicles behind it that decide to stay connected to
help traffic control. We have also analyzed the stability of the traffic flow
and derived stability charts that guide controller tuning.

To realize the proposed traffic control strategies on real highways, it is
essential for the CAV to access information about the traffic behind it. Thus,
sufficient penetration of connectivity is required so that the CAV can sync
with connected vehicles behind it in its communication range, that may be
limited to a few hundred meters. Furthermore, the controllers proposed in this
paper were discussed for a single-lane scenario — for multiple lanes they
should be extended by considering the cross-lane dynamics, and lane
information should be acquired and incorporated into the control laws.
Finally, we emphasize that connected vehicles behind the CAV do not
necessarily have to be automated. If they possess sufficient level of
automation, however, they can coordinate with the CAV for more efficient
traffic control. Our future work will explore coordination of multiple CAVs
and multi-lane dynamics in traffic control.

###### Acknowledgements.

This work was supported by Ford Motor Co. and the University of Michigan’s
Center for Connected and Automated Transportation through the US DOT grant
69A3551747105.

## Appendix

The parameters used throughout the paper are given in Table 1. The detailed
formulas of the stability boundaries plotted in Sec. 5 are given below.

Table 1: Parameters of the numerical case studies. Field | Description | Parameter | Value
---|---|---|---
vehicle properties | vehicle length | $l$ | 5 m
braking limit | $a_{\rm min}$ | 7 m/s2
acceleration limit | $a_{\rm max}$ | 3 m/s2
human-driven vehicle | time delay | $\tau$ | 0.8 s-1
gain for headway response | $\alpha_{\rm H}$ | 0.1 s-1
gain for speed response | $\beta_{\rm H}$ | 0.6 s-1
range policy | $F_{\rm H}(h)$ | ${v_{\rm max}\left(1-\left(\frac{h_{\rm go}-h}{h_{\rm go}-h_{\rm st}}\right)^{2}\right)}$
automated vehicle | time delay | $\sigma$ | 0.6 s-1
gain for headway response | $\alpha$ | 0.4 s-1
gain for speed response | $\beta$ | 0.5 s-1
gain for traffic control | $\beta_{\rm B}$ | 0.2 s-1
range policy | $F(h)$ | ${v_{\rm max}\frac{h-h_{\rm st}}{h_{\rm go}-h_{\rm st}}}$
range policy | standstill headway | $h_{\rm st}$ | 5 m
free flow headway | $h_{\rm go}$ | 55 m
speed limit | $v_{\rm max}$ | 30 m/s
simulations | time interval | ${[t_{0},t_{\rm f}]}$ | [0,60] s
time step | $\Delta t$ | 0.01 s
lead vehicle braking | $a_{1}(t)$ | -1 m/s2, ${t\in[0,10]}$ s
lead vehicle acceleration | $a_{1}(t)$ | 0.5 m/s2, ${t\in[10,30]}$ s
initial conditions |  $s_{n}(t)$, $v_{n}(t)$ | constant $s_{n}^{*}$, $v_{n}^{*}$
$t\in[-\tau_{n},0]$
energy consumption | constant resistance coefficient | $a_{\rm r}$ | 0.0981 m/s2
quadratic resistance coefficient | $c_{\rm r}$ | 0.0003 m-1
resolution of contour plots | ${\Delta\beta\times\Delta\beta_{\rm B}}$ |  ${0.05\times 0.01}$ s-1
stability charts | range policy gradient for HVs | $\kappa_{\rm H}$ | 0.7 s-1
range policy gradient for AVs | $\kappa$ | 0.6 s-1
frequency range for plant stability | ${[\Omega_{\rm min},\Omega_{\rm max}]}$ |  $[0,2\pi]$ rad/s
frequency range for string stability | ${[\omega_{\rm min},\omega_{\rm max}]}$ |  $[0,2\pi]$ rad/s

### Stability of ATC

As highlighted in Fig. 4, the car-following configurations in our examples can
all be considered as the special cases of traffic control with ATC. Therefore,
we derive the stability charts of ATC and then obtain those of other scenarios
as special cases. Throughout the appendix we use (52, 53, 54) as the
expressions of the link transfer functions $T_{\rm H}(s)$, $T(s)$, $T_{\rm
F}(s)$ and $T_{\rm B}(s)$.

First, consider the plant stability of ATC. Based on (71), it decomposes into
the plant stability of the associated vehicle control setup and virtual ring.
Consider the first line of (71) and assume a plant stable chain of HVs
described by $\Gamma(s)$ (the conditions of this are given at the end of the
Appendix). Then we can take ${{\rm D}(T_{\rm B}(s))=0}$, which leads to the
characteristic equation

$s^{2}{\rm e}^{s\sigma}+(\alpha+\beta+\beta_{\rm B})s+\alpha\kappa=0\,.$ (73)

The case ${s=0}$ gives the plant stability boundary

$\alpha=0\,,$ (74)

whereas substitution of ${s=\pm{\rm i}\Omega}$, separation into real and
imaginary parts and solution for $\alpha$ and $\beta$ lead to the plant
stability boundary

$\displaystyle\begin{split}\alpha&=\frac{\Omega^{2}\cos(\Omega\sigma)}{\kappa}\,,\\\
\beta&=\Omega\sin(\Omega\sigma)-\alpha-\beta_{\rm B}\,.\end{split}$ (75)

This can also be rearranged to

$\beta_{\rm
B}=\Omega^{*}(\alpha)\sin\big{(}\Omega^{*}(\alpha)\sigma\big{)}-\alpha-\beta\,,$
(76)

where $\Omega^{*}(\alpha)$ is the solution of
${\alpha=\Omega^{2}\cos(\Omega\sigma)/\kappa}$ for $\Omega$.

Similarly, the second line of (71) leads to the characteristic equation

$s^{2}{\rm e}^{s\sigma}+(\alpha+\beta+\beta_{\rm B})s+\alpha\kappa-\beta_{\rm
B}s\Gamma(s)=0\,.$ (77)

For ${s=0}$, this yields the plant stability boundary

$\alpha=0\,,$ (78)

while for ${s=\pm{\rm i}\Omega}$ it implies

$\displaystyle\begin{split}\alpha&=\frac{\Omega^{2}\cos(\Omega\sigma)-\Omega\Gamma_{\rm
I}\beta_{\rm B}}{\kappa}\,,\\\
\beta&=\Omega\sin(\Omega\sigma)-\alpha-(1-\Gamma_{\rm R})\beta_{\rm
B}\,,\end{split}$ (79)

or, equivalently,

$\displaystyle\begin{split}\beta_{\rm
B}&=\frac{\Omega^{2}\cos(\Omega\sigma)-\alpha\kappa}{\Omega\Gamma_{\rm
I}}\,,\\\ \beta&=\Omega\sin(\Omega\sigma)-\alpha-(1-\Gamma_{\rm R})\beta_{\rm
B}\,,\end{split}$ (80)

where ${\Gamma_{\rm R}=\Re(\Gamma({\rm i}\Omega))}$ and ${\Gamma_{\rm
I}=\Im(\Gamma({\rm i}\Omega))}$. These equations were used to plot the plant
stability boundaries in the $(\beta,\alpha)$ and $(\beta,\beta_{\rm B})$
planes in Figs. 7 and 8.

String stability can be analyzed by the help of the head-to-tail transfer
function (48), which now reads

$G({\rm i}\omega)=\frac{(\beta{\rm i}\omega+\alpha\kappa)(\Gamma_{\rm R}+{\rm
i}\Gamma_{\rm I})}{-\omega^{2}{\rm e}^{{\rm
i}\omega\sigma}+(\alpha+\beta+\beta_{\rm B}){\rm
i}\omega+\alpha\kappa-\beta_{\rm B}{\rm i}\omega(\Gamma_{\rm R}+{\rm
i}\Gamma_{\rm I})}\,,$ (81)

for ${s={\rm i}\omega}$, where ${\Gamma_{\rm R}=\Re(\Gamma({\rm i}\omega))}$
and ${\Gamma_{\rm I}=\Im(\Gamma({\rm i}\omega))}$. Direct substitution into
(60) gives the following expression for $P(\omega)$:

$P(\omega)=\omega^{2}-2\alpha\kappa\cos(\omega\sigma)-2\beta_{\rm
B}\omega\Gamma_{\rm
I}\cos(\omega\sigma)+\alpha^{2}\kappa^{2}\frac{1-\Gamma_{\rm
R}^{2}-\Gamma_{\rm I}^{2}}{\omega^{2}}+2\alpha\kappa\beta_{\rm
B}\frac{\Gamma_{\rm I}}{\omega}+\beta_{\rm B}^{2}\Gamma_{\rm I}^{2}\\\
-2\big{(}\alpha+\beta+(1-\Gamma_{\rm R})\beta_{\rm
B}\big{)}\omega\sin(\omega\sigma)+\big{(}\alpha+\beta+(1-\Gamma_{\rm
R})\beta_{\rm B}\big{)}^{2}-\beta^{2}(\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2})\,.$ (82)

We can construct the ${\omega=0}$ string stability boundaries via (62). Note
that the following hold for ${\omega\to 0}$:

$\displaystyle\begin{split}\lim_{\omega\to 0}\frac{\Gamma_{\rm
I}}{\omega}&=-\frac{N}{\kappa_{\rm H}}\,,\\\ \lim_{\omega\to
0}\frac{\Gamma_{\rm R}-1}{\omega^{2}}&=N\frac{2\kappa_{\rm H}-2\beta_{\rm
H}-(N+1)\alpha_{\rm H}}{2\alpha_{\rm H}\kappa_{\rm H}^{2}}\,.\end{split}$ (83)

This can be shown by substituting (52) into (45), taking ${s={\rm i}\omega}$,
and calculating the limits above by applying L’Hospital’s rule (once for
$\Gamma_{\rm I}/\omega$ and twice for ${(\Gamma_{\rm R}-1)/\omega^{2}}$). At
the limit ${\omega\to 0}$ we get

$P(0)=\alpha\bigg{(}\alpha+2\beta-2\kappa+N\frac{\alpha\kappa^{2}}{\alpha_{\rm
H}\kappa_{\rm H}^{2}}(\alpha_{\rm H}+2\beta_{\rm H}-2\kappa_{\rm
H})-2N\frac{\kappa}{\kappa_{\rm H}}\beta_{\rm B}\bigg{)}\,.$ (84)

Taking ${P(0)=0}$ finally gives the ${\omega=0}$ string stability boundaries
in the form

$\displaystyle\begin{split}\alpha&=0\,,\\\
\alpha&=\frac{2\Big{(}\kappa-\beta+N\frac{\kappa}{\kappa_{\rm H}}\beta_{\rm
B}\Big{)}}{1+\frac{N\kappa^{2}}{\alpha_{\rm H}\kappa_{\rm H}^{2}}(\alpha_{\rm
H}+2\beta_{\rm H}-2\kappa_{\rm H})}\,,\end{split}$ (85)

which can also be expressed as

$\beta_{\rm B}=\frac{\kappa_{\rm
H}}{2N\kappa}\bigg{(}\alpha+2\beta-2\kappa+N\frac{\alpha\kappa^{2}}{\alpha_{\rm
H}\kappa_{\rm H}^{2}}(\alpha_{\rm H}+2\beta_{\rm H}-2\kappa_{\rm
H})\bigg{)}\,.$ (86)

The ${\omega>0}$ string stability boundaries can be found by substitution of
(48) into (59). Then, separation into real and imaginary parts leads to a
linear system of equations for $\alpha$ and $\beta$, that ultimately gives

$\displaystyle\begin{split}\alpha&=\frac{\omega^{2}\big{(}\Gamma_{\rm
I}\sin(\omega\sigma-K)+\Gamma_{\rm
R}\cos(\omega\sigma-K)-\cos(\omega\sigma)\big{)}}{\omega(\Gamma_{\rm R}\sin
K+\Gamma_{\rm I}\cos K)-\kappa(1+\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2}-2\Gamma_{\rm R}\cos K+2\Gamma_{\rm I}\sin K)}\\\
&\quad+\frac{\beta_{\rm B}\omega\big{(}(\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2})\sin K-\Gamma_{\rm R}\sin K+\Gamma_{\rm I}(1-\cos
K)\big{)}}{\omega(\Gamma_{\rm R}\sin K+\Gamma_{\rm I}\cos
K)-\kappa(1+\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2}-2\Gamma_{\rm R}\cos
K+2\Gamma_{\rm I}\sin K)}\,,\\\
\beta&=\frac{\omega^{2}\cos(\omega\sigma)-\kappa\omega\big{(}\Gamma_{\rm
I}\cos(\omega\sigma-K)-\Gamma_{\rm
R}\sin(\omega\sigma-K)+\sin(\omega\sigma)\big{)}}{\omega(\Gamma_{\rm R}\sin
K+\Gamma_{\rm I}\cos K)-\kappa(1+\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2}-2\Gamma_{\rm R}\cos K+2\Gamma_{\rm I}\sin K)}\\\
&\quad+\frac{\beta_{\rm B}\big{(}-\omega\Gamma_{\rm
I}+\kappa\big{(}1+(\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2})\cos K-\Gamma_{\rm
R}(1+\cos K)+\Gamma_{\rm I}\sin K\big{)}\big{)}}{\omega(\Gamma_{\rm R}\sin
K+\Gamma_{\rm I}\cos K)-\kappa(1+\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2}-2\Gamma_{\rm R}\cos K+2\Gamma_{\rm I}\sin K)}\,.\end{split}$ (87)

This can be re-written also in the form

$\displaystyle\begin{split}\beta_{\rm B}&=\frac{\omega^{2}\big{(}-\Gamma_{\rm
I}\sin(\omega\sigma-K)-\Gamma_{\rm
R}\cos(\omega\sigma-K)+\cos(\omega\sigma))\big{)}}{\omega\big{(}\Gamma_{\rm
I}(1-\cos K)-\Gamma_{\rm R}\sin K+(\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2})\sin
K\big{)}}\\\ &\quad+\frac{\alpha\omega(\Gamma_{\rm R}\sin K+\Gamma_{\rm I}\cos
K)-\alpha\kappa\big{(}1+\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2}-2\Gamma_{\rm
R}\cos K+2\Gamma_{\rm I}\sin K\big{)}}{\omega\big{(}\Gamma_{\rm I}(1-\cos
K)-\Gamma_{\rm R}\sin K+(\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2})\sin
K\big{)}}\,,\\\ \beta&=\frac{\omega^{2}\big{(}\Gamma_{\rm
I}\sin(\omega\sigma)-(1-\Gamma_{\rm
R})\cos(\omega\sigma)\big{)}}{\omega\big{(}\Gamma_{\rm I}(1-\cos
K)-\Gamma_{\rm R}\sin K+(\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2})\sin
K\big{)}}\\\ &\quad+\frac{-\alpha\omega\Gamma_{\rm
I}+\alpha\kappa\big{(}1+(\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2})\cos
K-\Gamma_{\rm R}(1+\cos K)+\Gamma_{\rm I}\sin
K\big{)}}{\omega\big{(}\Gamma_{\rm I}(1-\cos K)-\Gamma_{\rm R}\sin
K+(\Gamma_{\rm R}^{2}+\Gamma_{\rm I}^{2})\sin K\big{)}}\,.\end{split}$ (88)

These formulas allowed us to plot the string stability boundaries in the
$(\beta,\alpha)$ and $(\beta,\beta_{\rm B})$ planes in Figs. 7 and 8.

### Stability of ACC

Now we consider the stability of traffic flows influenced by vehicle control
via ACC.

#### ACC Followed by Human-Driven Traffic

Since ACC is a special case of ATC, all stability boundaries can be obtained
by substituting ${\beta_{\rm B}=0}$. The plant stability boundaries (74)-(79)
reduce to

$\alpha=0\,,$ (89)

and

$\displaystyle\begin{split}\alpha&=\frac{\Omega^{2}\cos(\Omega\sigma)}{\kappa}\,,\\\
\beta&=\Omega\sin(\Omega\sigma)-\alpha\,,\end{split}$ (90)

whereas the string stability boundaries become

$\displaystyle\begin{split}\alpha&=0\,,\\\
\alpha&=\frac{2(\kappa-\beta)}{1+\frac{N\kappa^{2}}{\alpha_{\rm H}\kappa_{\rm
H}^{2}}(\alpha_{\rm H}+2\beta_{\rm H}-2\kappa_{\rm H})}\,,\end{split}$ (91)

and

$\displaystyle\begin{split}\alpha&=\frac{\omega^{2}\big{(}\Gamma_{\rm
I}\sin(\omega\sigma-K)+\Gamma_{\rm
R}\cos(\omega\sigma-K)-\cos(\omega\sigma)\big{)}}{\omega(\Gamma_{\rm R}\sin
K+\Gamma_{\rm I}\cos K)-\kappa(1+\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2}-2\Gamma_{\rm R}\cos K+2\Gamma_{\rm I}\sin K)}\,,\\\
\beta&=\frac{\omega^{2}\cos(\omega\sigma)-\kappa\omega\big{(}\Gamma_{\rm
I}\cos(\omega\sigma-K)-\Gamma_{\rm
R}\sin(\omega\sigma-K)+\sin(\omega\sigma)\big{)}}{\omega(\Gamma_{\rm R}\sin
K+\Gamma_{\rm I}\cos K)-\kappa(1+\Gamma_{\rm R}^{2}+\Gamma_{\rm
I}^{2}-2\Gamma_{\rm R}\cos K+2\Gamma_{\rm I}\sin K)}\,.\end{split}$ (92)

#### ACC Followed by Human-Driven Traffic on a Ring

When the ACC scenario with subsequent human-driven vehicles is driven on a
ring road, the characteristic equation becomes (50). As pointed out in Sec.
5.2, the stability boundary is analogous to the plant stability of the virtual
ring in ATC. Thus, for ${s=0}$ we obtain

$\alpha=0\,,$ (93)

cf. (74, 78). For ${s={\rm i}\omega}$, the stability boundary can be obtained
as the ${K=0}$ special case of the ${\omega>0}$ string stability boundary of
the associated straight-road configuration. Thus, substituting ${K=0}$ into
(92) leads to

$\displaystyle\begin{split}\alpha&=\frac{\omega^{2}\big{(}\Gamma_{\rm
I}\sin(\omega\sigma)-(1-\Gamma_{\rm
R})\cos(\omega\sigma)\big{)}}{\omega\Gamma_{\rm I}-\kappa\big{(}(1-\Gamma_{\rm
R})^{2}+\Gamma_{\rm I}^{2}\big{)}}\,,\\\
\beta&=\frac{\omega^{2}\cos(\omega\sigma)-\kappa\omega\big{(}\Gamma_{\rm
I}\cos(\omega\sigma)+(1-\Gamma_{\rm
R})\sin(\omega\sigma)\big{)}}{\omega\Gamma_{\rm I}-\kappa\big{(}(1-\Gamma_{\rm
R})^{2}+\Gamma_{\rm I}^{2}\big{)}}\,.\end{split}$ (94)

Notice that this gives back the plant stability boundaries (90) if we
substitute ${\Gamma({\rm i}\omega)=0}$, i.e., ${\Gamma_{\rm R}=0}$,
${\Gamma_{\rm I}=0}$ (which is the special case of ${N\to\infty}$ string
stable human drivers with ${\left|T_{\rm H}({\rm i}\omega)\right|<1}$, see
Sec. 5.2).

#### ACC without Considering the Traffic Behind

The influence of ACC controllers on the traffic flow behind them is often
neglected during cruise control design. This can be considered as the ${N=0}$
special case of ACC followed by human-driven traffic discussed above. For
${N=0}$, we have ${\Gamma(s)=1}$ that implies ${\Gamma_{\rm R}=1}$,
${\Gamma_{\rm I}=0}$. The plant stability conditions still yield (89, 90),
since we have already considered a plant stable human-driven chain:

$\alpha=0\,,$ (95)

and

$\displaystyle\begin{split}\alpha&=\frac{\Omega^{2}\cos(\Omega\sigma)}{\kappa}\,,\\\
\beta&=\Omega\sin(\Omega\sigma)-\alpha\,.\end{split}$ (96)

On the other hand, substitution of ${N=0}$ into (91) leads to the ${\omega=0}$
string stability boundaries

$\displaystyle\begin{split}\alpha&=0\,,\\\
\alpha&=2(\kappa-\beta)\,,\end{split}$ (97)

while substitution of ${\Gamma_{\rm R}=1}$ and ${\Gamma_{\rm I}=0}$ into (92)
results in the ${\omega>0}$ string stability boundaries

$\displaystyle\begin{split}\alpha&=\frac{\omega^{2}\big{(}\cos(\omega\sigma-K)-\cos(\omega\sigma)\big{)}}{\omega\sin
K-2\kappa(1-\cos K)}\,,\\\
\beta&=\frac{\omega^{2}\cos(\omega\sigma)+\kappa\omega\big{(}\sin(\omega\sigma-K)-\sin(\omega\sigma)\big{)}}{\omega\sin
K-2\kappa(1-\cos K)}\,.\end{split}$ (98)

### Stability of Human-Driven Traffic

Finally, human-driven traffic can also be analyzed as a special case of ATC or
ACC by considering that the ego vehicle that acts like a human driver. In
terms of formulas, this case is recovered by replacing $\alpha$, $\beta$,
$\kappa$, $\sigma$ with $\alpha_{\rm H}$, $\beta_{\rm H}$, $\kappa_{\rm H}$,
$\tau$.

#### Human-Driven Traffic on a Straight Road

The straightforward replacement of parameters leads to the plant stability
conditions for human drivers in the form

$\alpha_{\rm H}=0\,,$ (99)

and

$\displaystyle\begin{split}\alpha_{\rm
H}&=\frac{\Omega^{2}\cos(\Omega\tau)}{\kappa_{\rm H}}\,,\\\ \beta_{\rm
H}&=\Omega\sin(\Omega\tau)-\alpha_{\rm H}\,,\end{split}$ (100)

as well as the string stability conditions

$\displaystyle\begin{split}\alpha_{\rm H}&=0\,,\\\ \alpha_{\rm
H}&=2(\kappa_{\rm H}-\beta_{\rm H})\,,\end{split}$ (101)

and

$\displaystyle\begin{split}\alpha_{\rm
H}&=\dfrac{\omega^{2}\big{(}\cos(\omega\tau-K)-\cos(\omega\tau)\big{)}}{\omega\sin
K-2\kappa_{\rm H}(1-\cos K)}\,,\\\ \beta_{\rm
H}&=\dfrac{\omega^{2}\cos(\omega\tau)+\kappa_{\rm
H}\omega\big{(}\sin(\omega\tau-K)-\sin(\omega\tau)\big{)}}{\omega\sin
K-2\kappa_{\rm H}(1-\cos K)}\,.\end{split}$ (102)

#### Human-Driven Traffic on a Ring

Human-driven traffic on a ring can be analyzed by the characteristic equation
(69). For ${s=0}$ the stability boundary is still

$\alpha_{\rm H}=0\,,$ (103)

whereas for ${s={\rm i}\omega}$ it becomes

$\displaystyle\begin{split}\alpha_{\rm
H}&=\dfrac{\omega^{2}\big{(}\cos\big{(}\omega\tau-\frac{k2\pi}{N+1}\big{)}-\cos(\omega\tau)\big{)}}{\omega\sin\big{(}\frac{k2\pi}{N+1}\big{)}-2\kappa_{\rm
H}\big{(}1-\cos\big{(}\frac{k2\pi}{N+1}\big{)}\big{)}}\,,\\\ \beta_{\rm
H}&=\dfrac{\omega^{2}\cos(\omega\tau)+\kappa_{\rm
H}\omega\big{(}\sin\big{(}\omega\tau-\frac{k2\pi}{N+1}\big{)}-\sin(\omega\tau)\big{)}}{\omega\sin\big{(}\frac{k2\pi}{N+1}\big{)}-2\kappa_{\rm
H}\big{(}1-\cos\big{(}\frac{k2\pi}{N+1}\big{)}\big{)}}\,.\end{split}$ (104)

This latter result can be obtained by substituting ${K=2k\pi/(N+1)}$ into the
string stability boundaries (102) of the human-driven chain of vehicles, as
explained in Sec. 5.2. Alternatively, the same result can be achieved based on
(68, 69) by considering ${\Gamma({\rm i}\omega)={\rm e}^{{\rm
i}\frac{2k\pi}{N+1}}}$ and substituting ${\Gamma_{\rm
R}=\cos\big{(}\frac{2k\pi}{N+1}\big{)}}$, ${\Gamma_{\rm
I}=\sin\big{(}\frac{2k\pi}{N+1}\big{)}}$ into (92) with ${\alpha=\alpha_{\rm
H}}$, ${\beta=\beta_{\rm H}}$, ${\kappa=\kappa_{\rm H}}$ and ${\sigma=\tau}$.

Finally, we remark that the stability of TC, CC followed by human-driven
traffic on a straight road or ring, and CC without considering the traffic
behind can all be obtained as the special cases of ATC: when there is no
response to the distance from the vehicle ahead, i.e., ${\alpha=0}$. For the
${s=0}$ stability boundaries, however, one needs to take special care since it
is exactly ${\alpha=0}$ in ATC.

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