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1
Smoothing Traffic Flow via Control of
Autonomous Vehicles
Yang Zheng, Member, IEEE, Jiawei Wang, Student Member, IEEE, and Keqiang Li
Abstract—The emergence of autonomous vehicles is expected
to revolutionize road transportation in the near future. Although
large-scale numerical simulations and small-scale experiments
have shown promising results, a comprehensive theoretical un-
derstanding to smooth traffic flow via autonomous vehicles is
lacking. In this paper, from a control-theoretic perspective, we
establish analytical results on the controllability, stabilizability,
and reachability of a mixed traffic system consisting of human-
driven vehicles and autonomous vehicles in a ring road. We show
that the mixed traffic system is not completely controllable, but
is stabilizable, indicating that autonomous vehicles can not only
suppress unstable traffic waves but also guide the traffic flow to
a higher speed. Accordingly, we establish the maximum traffic
speed achievable via controlling autonomous vehicles. Numerical
results show that the traffic speed can be increased by over 6%
when there are only 5% autonomous vehicles. We also design
an optimal control strategy for autonomous vehicles to actively
dampen undesirable perturbations. These theoretical findings
validate the high potential of autonomous vehicles to smooth
traffic flow.
Index Terms—Autonomous vehicle, mixed traffic flow, control-
lability, stabilizability.
I. INTRODUCTION
MODERN societies are increasingly relying on complex
road transportation systems to support our daily mobil-
ity needs. In some big cities, the traffic demand is placing a
heavy burden on existing transportation infrastructures, some-
times leading to severely congested road networks [1]. Traffic
congestion not only results in the loss of fuel economy and
travel efficiency, but also increases the potential risk of traffic
accidents and public health [2].
Understanding traffic dynamics is essential if we are to
redesign infrastructures, or to guide/control transportation, to
mitigate road congestions and smooth traffic flow [3], [4].
The subject of traffic dynamics has attracted research interest
from many disciplines, including mathematics, physics, and
engineering. Since the 1930s, a wide range of models at both
the macroscopic and microscopic levels have been proposed
Copyright (c) 20xx IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
This work is supported by National Key R&D Program of China with
2016YFB0100906. The authors also acknowledge the support from TOYOTA.
Yang Zheng was with the Department of Engineering Science, University
of Oxford, Oxford, OX1 2JD, U.K. He is now with the School of Engineering
and Applied Sciences, and the Harvard Center for Green Buildings and Cities,
Harvard University, Cambridge, MA 02138 (zhengy@g.harvard.edu).
J. Wang and K. Li are with the School of Vehicle and Mobility, Tsinghua
University, Beijing, China, and with the Center for Intelligent Connected
Vehicles & Transportation, Tsinghua University, Beijing China. (e-mail: wang-
jw18@mails.tsinghua.edu.cn, likq@tsinghua.edu.cn). Corresponding author:
Keqiang Li.
to describe traffic behavior [3]. Based on these traffic models,
many control methods have been introduced and implemented
to improve the performance of road transportation systems [4].
Currently, most control strategies rely on actuators at fixed
locations. For example, variable speed advisory or variable
speed limits [5] are commonly implemented through traffic
signs on roadside infrastructure, and ramp metering [6] typi-
cally relies on traffic signals located at the freeway entrances.
These strategies are essentially external regulation methods
imposed on traffic flow.
As a key ingredient of traffic flow, the motion of vehicles
plays a fundamental role in road transportation systems. In the
past decades, major car-manufacturers and technology com-
panies have invested in developing vehicles with high levels
of automation, and some prototypes of self-driving cars have
been tested in real traffic environments [8]. The emergence of
autonomous vehicles (AVs) and vehicular network is expected
to revolutionize road transportation [8]–[10]. In particular, the
advancements of autonomous vehicles offer new opportunities
for traffic control, where autonomous vehicles can receive
information from other traffic participants and act as mobile
actuators to influence traffic flow internally. Most research
on the control of traffic flow via autonomous vehicles has
focused on platooning of a series of adjacent vehicles or
cooperative adaptive cruise control (CACC) [11], [12]. In the
context of platoon control, all involved vehicles are assumed
to be autonomous and can be controlled to maintain a string
stable platoon, such that disturbances along the platoon are
dissipated. Significant theoretical and practical advances have
been made in designing sophisticated controllers at the platoon
level [13]–[15].
While traffic systems with fully autonomous vehicles may
be of great interest in the far future, the near future will
have to meet a mixed traffic scenario where both autonomous
and human-driven vehicles (HDVs) coexist. In fact, early
autonomous vehicles need to cooperate in traffic systems
where most vehicles are human-driven. This situation is more
challenging in terms of theoretical modeling and stability
analysis, and many existing studies are based on numerical
simulations [16]–[18]. One recent concept is the connected
cruise control that considers mixed traffic scenarios where
autonomous vehicles can use the information from multiple
HDVs ahead to make control decisions [19], [20]. More
recently, Cui et al. first pointed out the potential of a single
autonomous vehicle in stabilizing mixed traffic flow [21], and
they also implemented simple control strategies to demonstrate
the dissipation of stop-and-go waves via a single autonomous
vehicle in real-world experiments [22]. The control principle is
arXiv:1812.09544v2 [math.OC] 10 May 2020
2
(a) (b) (c)
Fig. 1. Response profiles to an impulse perturbation in traffic systems on a ring road. Vehicle no.2 has an initial perturbation, and the parameters of human-
driven vehicles are chosen to resemble the wave behavior in the real-world experiment [7]. (a) All the vehicles are human-driven, where the perturbation is
amplified and stop-and go waves will appear accordingly. (b) Vehicle no.1 is a CACC-equipped vehicle which adjusts its behavior passively according to its
direct preceding vehicle. In this case, the perturbation is not amplified but small traffic waves still persist for a long period. (c) Vehicle no.1 adopts an optimal
control strategy considering the global behavior of the entire mixed traffic flow to mitigate undesirable perturbations actively. In this case, the perturbation is
attenuated, and the traffic flow becomes smooth quickly.
essentially a slow-in fast-out approach, which is an intuitive
method to dampen traffic jams [23]. This idea was further
studied by analyzing head-to-tail string stability [24]. More
sophisticated strategies, such as deep reinforcement learning,
have also recently been investigated to improve traffic flow in
mixed traffic scenarios via numerical simulations [25], [26].
Although the potential of autonomous vehicles has been
recognized and demonstrated [21], [22], [24]–[26], a compre-
hensive theoretical understanding is still lacking. In principle,
the behavior of traffic flow emerges from the collective dynam-
ics of many individual human-driven and/or autonomous vehi-
cles [3], where autonomous vehicles can serve as controllable
nodes. In this paper, we introduce a control-theoretic frame-
work for analyzing the mixed traffic system with multiple
autonomous vehicles and human-driven vehicles. By viewing
the autonomous vehicles as controllable nodes, we analyze the
controllability and stabilizability of the mixed traffic system.
Moreover, we formulate the problem of designing a control
strategy for autonomous vehicles to smooth mixed traffic flow
as standard H2optimal control, and discuss the reachability
of desired traffic state. Specifically, the contributions of this
paper are:
1) We prove for the first time that a mixed traffic sys-
tem consisting of one autonomous vehicle and multiple
human-driven vehicles is not completely controllable, but
is stabilizable. This result confirms the high potential
of autonomous vehicles in mixed traffic systems: the
global traffic velocity can be guided to a desired value by
controlling autonomous vehicles properly. Our theoretical
results validate the empirical experiments in [22] that a
single autonomous vehicle is able to stabilize traffic flow.
Note that unlike [21], we prove that there always exists
an uncontrollable mode.
2) We propose an optimal strategy which utilizes a system-
level objective for autonomous vehicles: instead of re-
sponding to traffic perturbations passively, it considers the
global behavior of the entire mixed traffic flow to mitigate
undesirable perturbations actively (see Fig.1 for illustra-
tion). The feedback control of autonomous vehicles is
formulated into the standard H2optimal synthesis prob-
lem [27], where the optimal controller can be computed
efficiently via convex optimization. Numerical results
show a better performance of the proposed controller in
smoothing traffic flow and improving fuel economy than
existing strategies [22].
3) Based on reachability analysis, we show an explicit range
for the desired traffic state in the mixed traffic system.
Moreover, an upper bound of reachable traffic velocity
is derived, indicating that a single autonomous vehicle
can not only smooth traffic flow but also increase traffic
speed. In the numerical experiments, we observed 6%
traffic velocity improvement using only 5% autonomous
vehicles.
4) Finally, we extend our theoretical framework to the case
where multiple autonomous vehicles coexist. We prove
that the results on controllability, stabilizability and reach-
ability remain similar. Unlike previous works focusing
on one single autonomous vehicle only [21], [22], we
show that autonomous vehicles can cooperate with each
other to reduce the time and energy for attenuating
perturbations and smoothing traffic flow. As expected, we
confirm that controlling multiple autonomous vehicles has
a better performance for large-scale mixed traffic systems.
The rest of this paper is organized as follows. Section
II introduces the theoretical modeling of a mixed traffic
system with one autonomous vehicle. The controllability and
stabilizability result is presented in Section III, and Section
IV describes the proposed system-level optimal controller and
analyzes the reachability of the desired system state. The case
of multiple autonomous vehicles is presented in Section V.
Numerical simulations are presented in Section VI, and we
conclude the paper in Section VII.
II. THEORETICAL MODELLING FR AM EWORK OF MIXED
TRA FFIC SY ST EM S
In this section, we introduce the modeling of a mixed traffic
system with one single autonomous vehicle. We consider a
single-lane ring road of length Land with nvehicles. As
discussed in Cui et al. [21], the ring road setting has several
theoretical advantages for modeling a traffic system, including
3
1) the existence of experimental results that can be used to
calibrate model parameters [7], 2) perfect control of average
traffic density, and 3) correspondence with an infinite straight
road with periodic traffic dynamics.
We denote the position of the i-th vehicle as pi(t)along
the ring road, and its velocity is denoted as vi(t) = ˙pi(t). The
spacing of vehicle i,i.e., the distance between two adjacent
vehicles, is defined as si(t) = pi−1(t)−pi(t). Note that
we ignore the vehicle length without loss of generality. For
simplicity, we assume that there is one autonomous vehicle
and the rest are HDVs in this section. The autonomous vehicle
is indexed as no.1. The case of multiple autonomous vehicles
will be discussed in Section V.
A. Modeling Human-driven Vehicles
Human car-following dynamics are typically modeled by
nonlinear processes [3], [4]
˙vi(t) = F(si(t),˙si(t), vi(t)),(1)
meaning that the acceleration of an HDV is a function of its
spacing si(t), the relative velocity between its own and its
preceding vehicle ˙si(t), and its velocity vi(t). Denote s∗, v∗
as the equilibrium spacing and velocity of each HDV, and then
(s∗, v∗)satisfies the following equilibrium equation
F(s∗,0, v∗)=0,(2)
which implies a certain relationship between the equilibrium
spacing and equilibrium velocity for HDVs. We define the
error state of the i-th HDV as
(˜si(t) = si(t)−s∗,
˜vi(t) = vi(t)−v∗.
Applying the first-order Taylor expansion to (1) at (s∗, v∗)
yields a linearized model of each HDV
(˙
˜si(t) = ˜vi−1(t)−˜vi(t),
˙
˜vi(t) = α1˜si(t)−α2˜vi(t) + α3˜vi−1(t),(3)
with α1=∂F
∂s , α2=∂ F
∂˙s−∂F
∂v , α3=∂ F
∂˙sevaluated at the
equilibrium state (s∗, v∗). These three coefficients reflect the
driver’s sensitivity to the error state. Considering the real driver
behavior, the acceleration should increase when the spacing
increases, the velocity of the ego vehicle drops, or the velocity
of the preceding vehicle increases. Hence, we assume that
α1>0,α2> α3>0[21], [28].
In the following, we choose the optimal velocity model
(OVM) [4], [29] to derive explicit expressions of (2) and (3).
The OVM model is given by
F(si(t),˙si(t), vi(t)) = α(V(si(t)) −vi(t)) + β˙si(t),(4)
where α > 0reflects the driver’s sensitivity to the differ-
ence between the current velocity and the spacing-dependent
desired velocity V(si(t)), and β > 0reflects the driver’s
sensitivity to the difference between the velocities of the ego
Fig. 2. Nonlinear spacing-dependent desired velocity function V(s)in the
OVM model. vmax = 30m/s,sst = 5mand sgo = 35m. The specific
mathematical expression is shown in (5) and (6). Note that this figure also
illustrates the relationship between equilibrium spacing s∗and equilibrium
velocity v∗, as shown in (7).
vehicle and the preceding vehicle. V(si(t)) is usually modeled
by a continuous piecewise function
V(s) =
0, s ≤sst,
fv(s), sst < s < sgo,
vmax, s ≥sgo,
(5)
where the desired velocity V(s)is zero for small spacing sst,
and reaches a maximum value vmax for large spacing sgo.
fv(s)is a monotonically increasing function and defines the
desired velocity when the spacing sis between sst and sgo.
There are many choices of fv(s), either in a linear or nonlinear
form. A typical one is of the following nonlinear form
fv(s) = vmax
21−cos(πs−sst
sgo −sst
).(6)
Fig.2 demonstrates a typical example of V(s).
For the general OVM model (4), it is easy to obtain the
following specific equilibrium state (s∗, v∗)that satisfies (2)
v∗=V(s∗).(7)
Furthermore, we can calculate the values of the coefficients in
linearized model (3) as follows
α1=α˙
V(s∗), α2=α+β, α3=β, (8)
where ˙
V(s∗)denotes the derivative of V(s)with respect to s
evaluated at the equilibrium spacing s∗.
B. Modeling Mixed Traffic Systems
For the autonomous vehicle, indexed as i= 1, the acceler-
ation signal is directly used as the control input u(t), and its
car-following model is
(˙
˜s1(t) = ˜vn(t)−˜v1(t),
˙
˜v1(t) = u(t),(9)
where ˜s1(t) = s1(t)−s∗
c,˜v1(t) = vi(t)−v∗with s∗
cbeing
a tunable spacing for the autonomous vehicle at velocity v∗.
Note that s∗
cis a design parameter for the autonomous vehicle,
and v∗is the desired traffic velocity, which the autonomous
vehicle attempts to steer the traffic flow towards. How to
choose a suitable s∗
cis discussed in Section IV-B.
4
……
……
Motion
Direction
Traffic Flow
Direction
Vehicle 1
Vehicle
AV
Vehicle 2
Vehicle − 1
Vehicle
Vehicle + 1
HDV
Fig. 3. Model establishment schematic. AV: autonomous vehicle; HDV: human-driven vehicle. From left to right, we have (a) the ring road traffic scenario
that includes one autonomous vehicle (blue) and n−1HDVs (green); (b) a simplified network system schematic. Purple arrows indicate the interaction
between adjacent vehicles, meaning that each HDV considers the state of its preceding vehicle only. Orange arrows show the information flow of the whole
system, assuming that the traffic state is observable to the autonomous vehicle; (c) the system matrix Aof the mixed traffic dynamics, as shown in (11).
Upon combining the error states of all the vehicles as the
mixed traffic system state,
x(t) = ˜s1(t),˜v1(t),...,˜sn(t),˜vn(t)T,
we arrive at the following canonical linear dynamics from a
global system viewpoint
˙x(t) = Ax(t) + Bu(t),(10)
where
A=
C10. . . . . . 0C2
A2A10. . . . . . 0
0A2A10. . . 0
.
.
..............
.
.
0. . . 0A2A10
0. . . . . . 0A2A1
, B =
B1
B2
B2
.
.
.
B2
,(11)
with each block matrix given by
A1=0−1
α1−α2, A2=0 1
0α3,
C1=0−1
0 0 , C2=0 1
0 0, B1=0
1, B2=0
0.
In (10), the evolution of each vehicle’s state is determined
by its own state and the state of its direct preceding vehicle
only; see Fig.3 for an illustration. In the following, we provide
a theoretical analysis on the potential of the autonomous
vehicle on smoothing the mixed traffic flow and design an
optimal control input u(t)for the autonomous vehicle.
III. CONTROLLABILITY AND STABILIZABILITY
Several real-world field experiments have shown that traffic
waves may easily happen and lead to severe congestions in a
ring road traffic system with human-driven vehicles [7], [22].
Theoretical results were also established that the linearized
traffic system is stable if the following condition holds [21]
α2
2−α2
3−2α1≥0; (12)
otherwise, the traffic system may become unstable and small
perturbations would cause stop-and-go waves. The condi-
tion (12) was first derived in [21] using frequency analysis;
the interested reader can refer to Appendix B for an alternative
proof based on eigenvalue analysis. For the OVM model (4),
the criterion (12) is reduced to
α+ 2β≥2˙
V(s∗),(13)
which indicates that to guarantee traffic stability, human
drivers should have a quicker response to velocity deviations
than the sensitivity of the optimal velocity function with
respect to the equilibrium spacing; otherwise stop-and-go
waves may happen.
Instead of modifying human drivers’ behavior, we reveal in
this section that a mixed traffic system can be always stabilized
by controlling one single autonomous vehicle. Specifically,
we discuss two fundamental concepts, controllability and
stabilizability, of the mixed traffic system (10).
A. Controllability Analysis
The controllability of a dynamical system captures the
ability to guide the system’s behavior towards a desired state
using appropriate control inputs, and the system is stabilizable
if all uncontrollable modes are stable [27]. We first present
three useful lemmas [27], [30].
Lemma 1 (Controllability): The linear system (A, B)in (10)
is controllable if and only if
rank B, AB , . . . , A2n−1B= 2n.
Lemma 1 is the well-known Kalman’s controllability rank
test [30], which provides a necessary and sufficient mathe-
matical condition for controllability. However, computing the
rank requires all the elements of (A, B)to be known, and it
might be numerically unreliable to calculate the rank for large-
scale systems. To facilitate analysis, we can apply a certain
linear transformation and represent the linear system under a
different basis, thus simplifying the system dynamics.
In particular, given a nonsingular T, we define a new state
˜x=T−1x, leading to the following dynamics
˙
˜x(t) = T−1AT ˜x(t) + T−1Bu(t).
Then, we obtain an equivalent linear system (T−1AT, T −1B).
5
Lemma 2 (Invariance under linear transformation): The
linear system (A, B)is controllable if and only if
(T−1AT , T −1B)is controllable for every nonsingular T.
If one can diagonalize the system matrix Avia T−1AT ,
then the controllability of (A, B)will be easier to derive. This
diagonalization may be nontrivial for its original form (A, B).
In this case, we can first apply a certain state feedback to
simplify the system dynamics. Specifically, consider a control
law v(t) = u(t) + Kx(t), and we arrive at
˙x(t) = (A−BK )x(t) + Bv(t).
Lemma 3 (Invariance under state feedback): The linear
system (A, B)is controllable if and only if (A−B K, B)
is controllable for every Kwith compatible dimension.
Now, we are ready to present the result on the controllability
of the mixed traffic system (10).
Theorem 1: Consider the mixed traffic system in a ring road
with one AV and n−1HDVs given by (10). We have
1) The system is not completely controllable.
2) There exists one uncontrollable mode corresponding to a
zero eigenvalue, and this uncontrollable mode is stable.
Proof: Our main idea is to exploit the invariance of
controllability under linear transformation and state feedback.
Using a sequence of state feedback and linear transformation,
we diagonalize the system, leading to an analytical conclusion
on the controllability of (10). Our procedure is as follows.
(A, B)state feedback
−−−−−−−→ (ˆ
A, B)linear transformation
−−−−−−−−−−−→ (e
A, e
B)
First, we transform system (A, B)into (ˆ
A, B)by introduc-
ing a virtual input ˆu(t), defined as
ˆu(t) = u(t)−(α1˜s1(t)−α2˜v1(t) + α3˜vn(t)),
which is the difference between the actual control value and
the acceleration value when the vehicle is controlled by a
human driver. Then, the state space model of (ˆ
A, B)becomes
˙x(t) = ˆ
Ax(t) + Bˆu(t),
where
ˆ
A=
A10. . . . . . 0A2
A2A10. . . . . . 0
0A2A10. . . 0
.
.
..............
.
.
0. . . 0A2A10
0. . . . . . 0A2A1
.(14)
According to Lemma 3, the controllability remains the same
between system (ˆ
A, B)and the original system (A, B ). Note
that ˆ
Ais block circulant, and it can be diagonalized by the
Fourier matrix Fn[31], [32]. We refer the interested reader
to Appendix A for the Fourier matrix Fnas well as precise
definitions and properties of block circulant matrices.
Define ω=e2πj
nwhere j=√−1denotes the imaginary
unit, and define Fngiven by (44) in Appendix A. Then, we
use the transformation matrix F∗
n⊗I2to transform (ˆ
A, B)
into (e
A, e
B), where F∗
ndenotes the conjugate transpose of Fn.
The new system matrix is
e
A= (F∗
n⊗I2)−1ˆ
A(F∗
n⊗I2) = diag(D1, D2, . . . , Dn),(15)
where ⊗denotes the Kronecker product, and
Di=A1+A2ω(n−1)(i−1)
=0−1 + ω(n−1)(i−1)
α1−α2+α3ω(n−1)(i−1), i = 1,2, . . . , n, (16)
and diag(D1, D2, . . . , Dn)denotes a block-diagonal matrix
with D1, D2, . . . , Dnon its diagonal. Using the fact that Fnis
a unitary matrix, the new state variable ˜xafter transformation
becomes
˜x= (F∗
n⊗I2)−1x= (Fn⊗I2)x, (17)
and the new control matrix e
Bis
e
B= (F∗
n⊗I2)−1B=1
√n
B1
B1
.
.
.
B1
.
Therefore, the dynamics of ˜xare
˙
˜x=e
A˜x(t) + e
Bˆu(t)
=
D1
D2
...
Dn
˜x(t) + 1
√n
B1
B1
.
.
.
B1
ˆu(t).(18)
Upon denoting ˜x(t) = ˜x11 ,˜x12,˜x21,˜x22,...,˜xn1,˜xn2T,
(e
A, e
B)is decoupled into nindependent subsystems
d
dt˜xi1
˜xi2=Di˜xi1
˜xi2+0
1
√nˆu(t)
=0−1 + ω(n−1)(i−1)
α1−α2+α3ω(n−1)(i−1)˜xi1
˜xi2+0
1
√nˆu(t).
It is easy to verify that ˙
˜x11 = 0, which means that ˜x11 is
an uncontrollable component, but remains constant during the
dynamic evolution. According to ex= (F∗
n⊗I2)−1x, we know
˜x11 =1
√n (s1(t)−s∗
c) +
n
X
i=2
(si(t)−s∗)!.(19)
Note that system (˜
A, ˜
B)is equivalent to system (ˆ
A, B)due
to the linear transformation. Also, system (ˆ
A, B)has the same
controllability characteristic as the original system (A, B).
Therefore, we conclude that the original system (A, B)is not
completely controllable and has at least one uncontrollable
component which remains constant, as shown in (19).
Furthermore, the uncontrollable component ˜x11 corresponds
to a zero eigenvalue. Since this zero eigenvalue only appears in
D1(see the expression of Diin (16)), its algebraic multiplicity
in ˆ
Ais one. Hence, we conclude that the uncontrollable mode
is stable.
We remark that the uncontrollable mode (19) has a clear
physical interpretation: the sum of each vehicle’s spacing
should remain constant due to the ring road structure of the
mixed traffic system. Theorem 1 differs from the results of [21]
in two aspects: 1) we explicitly point out the existence of
the uncontrollable mode (19); 2) our proof exploits the block
circulant property of the mixed traffic system.
6
B. Stabilizability Analysis
After revealing the uncontrollable component (19), we next
prove that the mixed traffic system is stabilizable. We need
the following PBH test for our stabilizability analysis.
Lemma 4 (PBH controllability criterion): The linear system
(A, B)is controllable if and only if rank(λI −A, B )=2nfor
every eigenvalue λof A. In addition, (A, B)is uncontrollable
if and only if there exists ω6= 0, such that
ωTA=λωT, ωTB= 0,
where ωis a left eigenvector of Acorresponding to λ, and ω
corresponds to an uncontrollable mode.
Theorem 2: Consider the mixed traffic system in a ring road
with one AV and n−1HDVs given by (10). We have
1) The controllability matrix Qc=B , AB, . . . , A2n−1B
satisfies
rank(Qc) = (2n−1,if α1−α2α3+α2
36= 0,
n, if α1−α2α3+α2
3= 0.(20)
2) The mixed traffic system (10) is stabilizable.
Proof: As proved in Theorem 1, systems (e
A, e
B)and
(A, B)share the same controllability characteristics. Here, we
focus on (e
A, e
B), and the main idea is to characterize all the
uncontrollable modes and prove that they are all stable.
•Case 1: α1−α2α3+α2
36= 0.
Since e
Ain (18) is block diagonal, we have
det(λI −e
A) =
n
Y
i=1
det(λI −Di)=0,(21)
where λdenotes an eigenvalue of e
A. Substituting (16) into
(21) leads to the following equation (i= 1,2, . . . , n)
λ2+α2−α3ω(n−1)(i−1)λ+α11−ω(n−1)(i−1) = 0.
(22)
Note that (22) is a second-order complex equation and it is
non-trivial to directly get its analytical roots. Instead, we use
this equation to analyze the properties of the eigenvalues. The
following proof is divided into two steps.
Step 1: We prove that Diand Dj(i6=j) share no common
eigenvalues. Assume there exists a λsatisfying det(λI−Di) =
0and det(λI −Dj)=0,i6=j, which means
(λ2+α2λ+α1= (α3λ+α1)ω(n−1)(i−1),
λ2+α2λ+α1= (α3λ+α1)ω(n−1)(j−1).
Since ω(n−1)(i−1) 6=ω(n−1)(j−1), we obtain α3λ+α1= 0
and λ2+α2λ+α1= 0, leading to
α1−α2α3+α2
3= 0, λ =α3−α2,
which contradicts the condition that α1−α2α3+α2
36= 0.
Therefore, Diand Dj(j6=i)have different eigenvalues.
Step 2: We prove that all the system modes corresponding
to non-zero eigenvalues are controllable. Denote λk6= 0 as the
eigenvalue of Dkand ρas its corresponding left eigenvector.
According to Lemma 4, we need to show ρTe
B6= 0.
Upon denoting ρ=ρT
1, ρT
2, . . . , ρT
nTwhere ρi=
ρi1, ρi2T∈R2×1, i = 1,2, . . . , n, the condition ρTe
A=
λkρTleads to
ρT
iDi=λkρT
i, i = 1,2, . . . , n. (23)
Since λkis not an eigenvalue of Di, i 6=k, we obtain ρi=
0, i 6=k. Hence, ρTe
B=ρT
kB1=ρk2. Assume ρk2= 0, then
substituting (16) into (23) yields
ρk100−1 + ω(n−1)(k−1)
α1−α2+α3ω(n−1)(k−1)=λkρk10.
(24)
The only solution to (24) is ρk1= 0, indicating that
the left eigenvector ρ= 0, which is false. Accordingly, the
assumption that ρk2= 0 does not hold. Therefore, we have
ρTe
B=ρk26= 0, meaning that the mode corresponding to λkis
controllable. In other words, the system modes corresponding
to nonzero eigenvalues are all controllable. Because the only
zero eigenvalue λ= 0 appears in det(λI −D1)=0and
the corresponding mode is uncontrollable, we conclude that if
α1−α2α3+α2
36= 0, there are 2n−1controllable modes in
the system (e
A, e
B), meaning that rank(Qc) = 2n−1.
•Case 2: α1−α2α3+α2
3= 0.
Substituting this condition into (22) yields
(λ+α2−α3)λ+α3−α3ω(n−1)(i−1)= 0, i = 1,2, . . . , n,
which gives the eigenvalues of Dias follows
λi1=α3−α2, λi2=α3ω(n−1)(i−1) −1.
The rest proof is organized into two steps.
Step 1: we prove that there are n−1uncontrollable modes
corresponding to α3−α2. It is easy to see that α3−α2is the
common eigenvalue for each block Di, i = 1,2, . . . , n,i.e.,
the algebraic multiplicity of α3−α2is n. We consider its left
eigenvector ρ=ρT
1, ρT
2, . . . , ρT
nT. Similar to (23), we obtain
ρT
iDi= (α3−α2)ρT
i, i = 1,2, . . . , n.
Expanding this equation leads to
ρi1ρi20−1 + ω(n−1)(i−1)
α1−α2+α3ω(n−1)(i−1)= (α3−α2)ρi1ρi2,
from which we have ρi1=−α3ρi2, i = 1,2, . . . , n. There-
fore, we can choose nlinearly independent left eigenvectors
corresponding to α3−α2as
ρ(1) =−α3,1,0,0,0,0,...,0,0,
ρ(2) =−α3,1, α3,−1,0,0,...,0,0,
ρ(3) =−α3,1,0,0, α3,−1,...,0,0,
.
.
.
ρ(n)=−α3,1,0,0,0,0...,α3,−1.
(25)
From these left eigenvectors, it is easy to verify that
ρ(1)Te
B6= 0 and ρ(i)Te
B= 0, i = 2,3, . . . , n, meaning
that for α3−α2, there are n−1uncontrollable modes.
Step 2: we consider the rest of eigenvalues, i.e.
λi2=α3ω(n−1)(i−1) −1, i = 1,2, . . . , n.
7
The zero eigenvalue λ12 = 0 still corresponds to an uncon-
trollable mode, as shown in (19). We prove that the modes
associated with λi2=α3ω(n−1)(i−1) −1, i = 2,3, . . . , n
are controllable. The proof is similar to the case of α1−α2α3+
α2
36= 0. For λk2=α3ω(n−1)(k−1) −1,(k6= 1), denote
its left eigenvector as
ρ=ρT
1, ρT
2, . . . , ρT
nT,
where ρi=ρi1, ρi2T∈R2×1, i = 1,2, . . . , n. Then we have
ρi= 0, i 6=k, since λk2is not an eigenvalue of other blocks
Di, i 6=k. For ρk=ρk1, ρk2T, we have ρk26= 0, which is
similar to the argument in (24). Therefore, ρTe
B6= 0, meaning
that the mode corresponding to λk2(k6= 1) is controllable.
In summary, the eigenvalue λ=α3−α2is associ-
ated with n−1uncontrollable modes and one controllable
mode. Since λ=α3−α2<0, the uncontrollable modes
are all stable. The n−1modes associated with λi2=
α3ω(n−1)(i−1) −1, i = 2,3, . . . , n are controllable, and
the zero eigenvalue corresponds to an uncontrollable mode.
In total, there are ncontrollable modes in the system (e
A, e
B),
meaning that rank(Qc) = n. Finally, the system (A, B)is
stabilizable since all its uncontrollable mode are stable.
Theorem 2 shows that the mixed traffic system (10) always
has one uncontrollable mode corresponding to a zero eigen-
value, and the rest of modes are either controllable or stable.
This result has no requirement on the car-following behavior of
other human-driven vehicles or the scale nof the mixed traffic
system. By choosing an appropriate control input, one single
autonomous vehicle can always stabilize the global traffic flow
at an equilibrium traffic velocity.
IV. OPTIMAL CONTROL AND REACHABILITY ANALYS IS
We have shown that a mixed traffic system with one single
autonomous vehicle is always stabilizable. In this section,
we proceed to design an optimal control strategy to reject
perturbations in the mixed traffic system using standard control
theory [27]. Moreover, we discuss the reachability of the
equilibrium traffic state and show that the autonomous vehicle
can increase the traffic equilibrium velocity.
A. Optimal Control Formulation and its Solution
To reflect traffic perturbations, we assume that there exists a
disturbance signal wi(t)in each vehicle’s acceleration signal,
i.e.,˙
˜vi=α1˜si(t)−α2˜vi(t)+α3˜vi−1(t)+wi(t).The linearized
dynamics of HDVs in (10) become
˙xi(t) = A1xi(t) + A2xi−1(t) + H1wi(t),
with H1=0,1T.
Then, we design an optimal control input u(t) = −Kx(t)
to minimize the influence of disturbances wi(t)on the traf-
fic system, where K∈R1×2ndenotes the feedback gain.
Mathematically, this can be formulated into the following
optimization problem
min
KkGzw k2
subject to u=−Kx, (26)
where Gzw denotes the transfer function from disturbance
signal w(t) = w1(t), . . . , wn(t)to the performance state
z(t) = γs˜s1(t), γv˜v1(t), . . . , γs˜sn(t), γv˜vn(t), γuu(t)T,
with positive weights γs>0, γv>0, γu>0, and k·kdenotes
the H2norm of a transfer function that captures the influence
of disturbances. Note that the performance state can also be
written into
z(t) = Q1
2
0x(t) + 0
R1
2u(t),(27)
where Q1
2=diag(γs, γv, . . . , γs, γv)and R1
2=γudenote
the square roots of state and control performance weights,
respectively.
The optimization problem (26) is in the standard form of the
H2optimal controller synthesis [27]. Here, we briefly present
the steps to obtain a convex formulation for (26).
Lemma 5 (H2norm of a transfer function [27]): Given a
stable linear system ˙x(t) = Ax(t) + Hw(t), z(t) = Cx(t),
the H2norm of the transfer function from disturbance w(t)
to performance signal z(t)can be computed by
kGzw k2= inf
X0{TraceCXCT|AX +XAT+H H T0},
where Trace(·)denotes the trace of a symmetric matrix.
When applying state-feedback u=−Kx, the dynamics of
the closed-loop traffic system become
˙x(t)=(A−BK )x(t) + Hw(t),
z(t) = Q1
2
−R1
2Kx(t).(28)
Using Lemma 5 and a standard change of variables Z=KX,
the optimal control problem (26) can be equivalently reformu-
lated as
min
X,Z Trace(QX) + Trace RZX−1ZT
subject to (AX −BZ)+(AX −BZ)T+H HT0,
X0.
By introducing YZX −1ZTand using the Schur com-
plement, a convex reformulation to (26) is derived as follows.
min
X,Y,Z Trace(QX) + Trace(RY )
subject to (AX −BZ)+(AX −BZ)T+H HT0,
Y Z
ZTX0, X 0.
(29)
Problem (29) is convex and ready to be solved using general
conic solvers, e.g., Mosek [33], and the optimal controller is
recovered as K=ZX−1.
Remark 1 (Active response to traffic perturbations): Some
traditional strategies for autonomous vehicles, e.g., CACC
[13], [14], mainly focus on the performance of the autonomous
vehicles themselves, corresponding to a local-level considera-
tion, and they typically respond to external perturbations in a
passive way. Although they can improve traffic stability in
mixed traffic flow [16], [17], there always exists a certain
requirement on the penetration rate of autonomous vehicles.
8
Instead, our formulation directly considers the global traffic
behavior, i.e., the state and behavior of all the involved vehi-
cles, and aims at minimizing the influence of disturbances on
the entire traffic system via controlling autonomous vehicles.
In this way, the resulting system-level strategy enables au-
tonomous vehicles to respond to traffic perturbations actively;
see Fig. 1 for an illustration.
Remark 2 (Parameter selection of controllers): There al-
ready exist several control strategies for autonomous vehicles
to stabilize traffic flow, e.g., FollowerStopper and PI with
Saturation in [22]. We note that there are many parameters that
need to be pre-determined in these strategies. Different choices
of parameter values may lead to different performance, which
are not completely predictable. Instead, only three parameters
need to be designed in (26), i.e., the weight coefficients in
the cost function, γs,γv, and γu. Moreover, we can adjust
their values to achieve different and predictable results. For
example, setting a larger value to γsand γvtypically allows
to stabilize the traffic in a shorter time, and setting a larger
value to γunormally helps to keep a lower control energy for
the autonomous vehicle.
Remark 3 (Feasibility of the proposed strategy): Prob-
lem (29) can be formulated into a standard semidefinite
program (SDP), for which there exist efficient algorithms to
get a solution of arbitral accuracy in polynomial time [34].
In particular, the well-established interior-point method has
a computational complexity of O(max(m, n)mn2.5), where
nis the dimension of the positive semidefinite constraint
and mis the number of equality constraints in a standard
SDP [34]. In this paper, we use the conic solver Mosek [33]
to solve the resulting SDP from (29). In addition, computing
the optimal control strategy from (29) requires explicit car-
following models of HDVs; see (1). Lyapunov-type meth-
ods [35] and data-driven methods [36] have been proposed
for practical estimation of car-following behaviors based on
vehicle trajectories. It would be interesting for future work to
incorporate model identification in the design of robust optimal
control strategies.
B. Reachability and Maximum Traffic Velocity
We have shown that the mixed traffic system is always
stabilizable and upon choosing the weight coefficients γs,
γv, and γu, we can solve (29) to obtain an optimal control
strategy u(t) = −Kx(t)to reject the influence of disturbances.
Considering the definition of state x(t)and denoting K=
k1,1, k1,2, k2,1, k2,2,, . . . , kn,1, kn,2∈R1×2n, the optimal
control strategy is implemented as
u(t) = −(k1,1(s1(t)−s∗
c) + k1,2(v1(t)−v∗))
−
n
X
i=2
(ki,1(si(t)−s∗) + ki,2(vi(t)−v∗)) .(30)
Recall that (s∗, v∗)is the traffic equilibrium state of HDVs
satisfying (2), and s∗
c>0is the desired spacing for the
autonomous vehicle that is free to choose. We observe that an
improper choice of s∗
cmay cause the mixed traffic system to
fail to reach the desired velocity v∗due to the uncontrollable
mode (19). In fact, we can further predict the system final
state, which sheds some insight on the reachability of the
equilibrium traffic state.
The result of reachability is stated as follows.
Theorem 3: Consider the mixed traffic system in a ring
road with one AV and n−1HDVs given by (10). Suppose a
stabilizing feedback gain is found by (26) and the matrix (34)
is non-singular. Then the traffic system is stabilized at v∗if
and only if the desired spacing s∗
cof the AV satisfies
s∗
c=L−(n−1)s∗,(31)
with s∗given by the equilibrium equation (2).
Proof: With a stabilizing controller for the autonomous
vehicle, the mixed traffic system (10) is stable and the state
x(t)will approach an equilibrium point, where ˙x(t)=0. We
analyze the dynamics of each vehicle in (3) and (9) separately,
leading to
˜s1(tf) = se,˜vi(tf) = ve,˜si(tf) = α2−α3
α1
ve, i = 2,3, . . . , n,
where se, veare constant values, and tfis the time when the
system reaches its equilibrium point. Considering the desired
state in the controller xdes =s∗
c, v∗, s∗, v∗, . . . , s∗, v ∗T, the
final state of the system (10) must be in the following form
xf=sf,AV, vf, sf,HDV, vf, . . . , sf,HDV , vfT,(32)
where sf,AV =s∗
c+se,sf,HDV =s∗+α2−α3
α1
ve,vf=v∗+ve.
We next calculate the exact value of seand ve. In the final
state, all the vehicles have zero acceleration, indicating that
the control input u(t)must be zero, i.e.,u(t) = −Kx(t)=0.
Besides, according to the controllability analysis, we know
that there exists an uncontrollable mode (s1(t)−s∗
c) +
Pn
i=2 (si(t)−s∗)remaining constant; see (19). Combining
these two conditions leads to the following linear equations
(α2−α3
α1Σn
i=2ki,1+ Σn
i=1ki,2ve+k1,1se= 0,
(n−1)α2−α3
α1ve+se=L−(n−1)s∗−s∗
c,(33)
and its solution offers the exact value of seand ve,i.e.,
se
ve=M−10
L−(n−1)s∗−s∗
c,
where Mis a non-singular matrix as
M=α2−α3
α1Σn
i=2ki,1+ Σn
i=1ki,2k1,1
(n−1)α2−α3
α11.(34)
To reach the desired equilibrium state (s∗, v∗), we should
have se= 0 and ve= 0, which is equivalent to L−(n−
1)s∗−s∗
c= 0.This completes our proof.
Since the mixed traffic system in (10) is stabilizable, it can
be guided to reach any equilibrium state with traffic speed v∗
via controlling the autonomous vehicle properly. In practice,
however, the spacing of the autonomous vehicle cannot be
negative, i.e.,s∗
c>0, which is equivalent to
s∗
max <L
n−1.(35)
Recall that (s∗, v∗)should satisfy the HDV equilibrium equa-
tion F(s∗,0, v∗), and v∗usually increases as s∗grows up
9
according to the real driving behavior, as illustrated in Fig.2 for
the OVM model. Therefore, the requirement of the equilibrium
spacing in (35) sets up a maximum equilibrium traffic velocity
v∗
max. This leads to the following result.
Corollary 1 (Range of reachable traffic velocity): There
exists a reachable range for the traffic velocity in the mixed
traffic system (10):
06v∗< v∗
max,(36)
where v∗
max satisfies
FL
n−1,0, v∗
max= 0.
This result reveals the upper bound of reachable traffic ve-
locity, indicating that mixed traffic flow with one autonomous
vehicle can be steered exactly towards a velocity within the
range of (0, v∗
max). Note that v∗
max is higher than the equi-
librium traffic speed with HDVs only, where the equilibrium
spacing is s∗=L/n. A physical interpretation is that the
autonomous vehicle can follow its preceding vehicle at a
shorter distance and leave more space for its following HDVs,
which in turn triggers the HDVs to travel at a higher speed in
the equilibrium; see Fig.4 for illustration.
Remark 4: In the case of vehicle platoons where all involved
vehicles have autonomous capabilities, the vehicles can be
controlled to reach the same desired velocity with separate
desired spacings [13]–[15]. In a mixed traffic system, although
only the autonomous vehicles are under direct control, all
the other HDVs can be influenced indirectly. One distinction
is that the desired state (s∗, v∗)for HDVs should satisfy
their corresponding car-following behaviors, while the desired
state (s∗
c, v∗)for the autonomous vehicle can be designed
separately.
V. TR AFFI C SYS TE MS W IT H MULTI PL E AUTONOMOUS
VEHICLES
The mixed traffic system with a single autonomous vehicle
is stabilizable, which is independent of the number of vehicles
n. Also, an optimal control strategy u(t)can be obtained by
solving an optimization problem. However, it might be not
practical to control a mixed traffic system consisting of many
HDVs and a single autonomous vehicle. In this section, we
extend our previous analysis to a mixed traffic system with
multiple autonomous vehicles.
A. Theoretical Framework
Assume that there are nvehicles in the traffic flow with k
autonomous vehicles (k < n). The indices of the autonomous
vehicles are i1,i2, ..., ik, for which we define a set SAV =
{i1, i2, . . . , ik}. The error state of an HDV indexed as i(i /∈
SAV) is still defined as ˜si(t) = si(t)−s∗,˜vi(t) = vi(t)−v∗,
where (s∗, v∗) satisfies (2) and its linearized model remains
the same as (3).
For an autonomous vehicle indexed as ir(r= 1,2...,k),
the acceleration signal is directly used as its control input
uir(t), and its car-following model is
(˙
˜sir(t) = ˜vir−1(t)−˜vir(t),
˙
˜vir(t) = uir(t),(37)
Traffic Flow
Direction
Vehicle 1
Vehicle Vehicle 3
…
…
…
…
Vehicle 2
Vehicle Vehicle 3
Vehicle 1
Vehicle 2
Traffic Flow
Direction
(a)
Traffic Flow
Direction
Vehicle 1
Vehicle Vehicle 3
…
…
…
…
Vehicle 2
Vehicle Vehicle 3
Vehicle 1
Vehicle 2
Traffic Flow
Direction
(b)
Fig. 4. Illustration of the scenario where the autonomous vehicle increases the
traffic speed. (a) When all vehicles are human-driven, the spacing between
two vehicles is equal for homogeneous car-following dynamics. (b) In the
case of mixed traffic systems, the autonomous vehicle can be controlled to
follow its preceding vehicle in a shorter distance, and the other HDVs have
a larger spacing at the equilibrium state. According to F(s∗,0, v∗)=0, the
equilibrium velocity v∗increases as s∗grows up. Hence, the entire traffic
flow speed can be increased via controlling the autonomous vehicle.
where ˜sir(t) = sir(t)−s∗
ir,c,˜vir(t) = vir(t)−v∗with s∗
ir,c
being a tunable desired spacing for the autonomous vehicle at
velocity v∗.
To derive the global dynamics, we lump the error states of
all the vehicles as the mixed traffic system state,
x(t) = ˜s1(t),˜v1(t),...,˜sn(t),˜vn(t)T,
and lump all the control inputs as
u(t) = ui1(t), ui2(t), . . . , uik(t)T.
Then the state-space model of the entire mixed traffic system
can be given by
˙x(t) = Akx(t) + Bku(t),(38)
where
Ak=
A11 0. . . . . . 0A12
A22 A21 0. . . . . . 0
0A32 A31 0. . . 0
.
.
..............
.
.
0. . . 0A(n−1)2 A(n−1)1 0
0. . . . . . 0An2An1
,
Bk=P1, P2, . . . , Pk.
In the system matrix Ak, we have
(Ar2=C2, Ar1=C1,if r∈SAV,
Ar2=A2, Ar1=A1,if r /∈SAV,
and the other blocks are zero, where A1,A2,C1and C2are
the same as that in (11). In Bk, each column Pris a 2n×1
vector, in which only the (2ir)-th entry is one and the others
are zero.
10
B. Controllability and Stabilizability
When there exist multiple autonomous vehicles, we observe
that the controllability and stabilizability of the mixed traffic
system remain unchanged compared to the case discussed in
Section III, where there is only one autonomous vehicle. This
result is summarized as follows.
Theorem 4: Consider a mixed traffic system with multiple
autonomous vehicles in (38). We have
1) The mixed traffic system (38) is not completely control-
lable, and there still exists an uncontrollable mode.
2) The mixed traffic system (38) is stabilizable.
Proof: To analyze the system controllability, we first
define a virtual control input as
ˆuk(t) = u(t)−¯ui1,¯ui2. . . , ¯uikT,
with ¯uir=α1˜sir(t)−α2˜vir(t) + α3˜vir−1(t),ir∈SAV.
Then (38) becomes
˙x(t) = ˆ
Ax(t) + Bkˆuk(t),
with ˆ
Adefined by (14). Using F∗
n⊗I2as the transformation
matrix, (ˆ
A, Bk)can be transformed into (e
A, e
Bk), given by
˙
˜x=e
A˜x(t) + e
Bkˆuk(t)(39)
with e
A=diag(D1, D2, . . . , Dn)being the same as (15) and
e
Bkdefined as
e
Bk= (F∗
n⊗I2)−1Bk=he
P1,e
P2,..., e
Pki,
where e
Pr=1
√n0,1,0,¯ωir−1,...,0,¯ω(n−1)(ir−1)T, r =
1, . . . , k. After the transformation, the new state variable ˜x
is the same as (17).
Note that the dynamics (39) shall be reduced to the case
with a single autonomous vehicle (18) when k= 1 and i1=
1. Upon denoting ˜x(t) = ˜x11 ,˜x12,˜x21 ,˜x22,...,˜xn1,˜xn2T,
(e
A, e
Bk)can be decoupled into nindependent subsystems (q=
1,2, . . . , n)
d
dt˜xq1
˜xq2=Di˜xq1
˜xq2+Fqˆu(t),
where
Fq=1
√n0 0 ·· · 0
¯ω(q−1)(i1−1) ¯ω(q−1)(i2−1) ··· ¯ω(q−1)(ik−1).
It is not difficult to see that ˙
˜x11 = 0, which means that ˜x11 is
an uncontrollable mode. Thus, the mixed traffic system (38) is
not completely controllable. Note that ˜x11 corresponds to the
zero eigenvalue and remains constant. Similar to the analysis in
Section III-A, the algebraic multiplicity of the zero eigenvalue
is one and hence the uncontrollable mode is stable.
As has been shown in Section III, system ( e
A, e
B) in (18) is
stabilizable. Note that the first column in e
Bk,i.e.,e
P1, is equal
to e
B, which indicates the stabilizability of system ( e
A, e
P1).
Then it is easy to observe that system ( e
A, e
Bk) is also stabiliz-
able. According to Lemmas 2 and 3, we can conclude that the
mixed traffic system with multiple autonomous vehicles given
by (38) is stabilizable.
The specific expression of the uncontrollable mode is
˜x11 =Pi∈SAV si(t)−s∗
i,c+Pi∈{1,2,...,n}\SAV (si(t)−s∗)
√n,
which remains unchanged during the system evolution. The
physical interpretation is the same as the case where there
is one single autonomous vehicle: the sum of each vehicle’s
spacing should remain constant due to the ring road structure.
C. Optimal Control and Reachability Analysis
Our controller formulation proposed in Section IV-A can
be easily applied to the mixed traffic system with multiple
autonomous vehicles in (38). Define the feedback controller
as u(t) = −Kx(t), where
K=KT
1, KT
2, . . . , KT
kT,(40)
with Kr∈R1×2ndenoting the feedback gain for the au-
tonomous vehicle indexed as ir,i.e.,uir(t) = −Krx(t).
Then following the process in Section IV-A, we can obtain an
optimal control strategy. We remark that the specific feedback
gain Krfor each autonomous vehicle may differ from each
other depending on their positions in the traffic system. The
resulting controller is a cooperation strategy for multiple AVs
to achieve an optimal performance of the entire traffic system.
The reachability analysis is as follows.
Theorem 5: Consider the mixed traffic system in a ring road
with multiple AVs in (38). Suppose a stabilizing feedback gain
(40) is found by (26) and the coefficient matrix of (42) is non-
singular. Then the traffic system can be stabilized at velocity
v∗, if and only if the sum of the desired spacing s∗
i,c (i∈SAV)
of each AV satisfies
X
i∈SAV
s∗
i,c=L−(n−k)s∗,(41)
with s∗given by the equilibrium equation (2).
Proof: Denote (r= 1,2, . . . , n)
Kr=hk(r)
1,1, k(r)
1,2, k(r)
2,1, k(r)
2,2,, . . . , k(r)
n,1, k(r)
n,2i.
Similar to the reachability analysis in Section IV, the final
state of stable system in (38) can be obtained via
˜vi(tf) = ve, i ∈ {1,2, . . . , n},
˜si(tf) = α2−α3
α1
ve, i ∈ {1,2, . . . , n}\SAV ,
˜si(tf) = se,i, i ∈SAV.
where veand se,i, i ∈SAV should satisfy
η(1)ve+Pi∈SAV k(1)
i,1se,i = 0,
.
.
.
η(k)ve+Pi∈SAV k(k)
i,1se,i = 0,
(n−k)α2−α3
α1ve+Pi∈SAV se,i =Lk,
(42)
with (r= 1,2, . . . , n)
η(r)=α2−α3
α1X
i∈{1,2,...,n}\SAV
k(r)
i,1+X
i∈{1,2,...,n}
k(r)
i,2,
Lk=L−(n−k)s∗−X
i∈SAV
s∗
i,c.
11
(a) (b) (c)
Fig. 5. Stabilizing traffic flow and increasing traffic speed. (a) The traffic system with human-driven vehicles only is unstable when α= 0.6, β = 0.9in the
OVM model. (b) The mixed traffic system becomes stable after introducing an autonomous vehicle with an appropriate control strategy. (c) The traffic flow
can be guided to a higher stable velocity (6% improvement) via controlling the autonomous vehicle.
The condition that the traffic flow reaches the desired
equilibrium velocity v∗is equivalent to ve= 0, se,i =
0, i ∈SAV. The solution to (42) is all zeros, i.e.,ve= 0
and se,i = 0, i ∈SAV, if and only if the constant vector is
zero, i.e.,Lk= 0, which leads to (41).
Since the spacing of each autonomous vehicle cannot be
negative, i.e.,s∗
i,c >0(i∈SAV), we have Pi∈SAV s∗
i,c =
L−(n−k)s∗>0.In this case, the maximum spacing for
each HDV in the equilibrium can be increased to
s∗
max,k <L
n−k,
which sets up a new maximum equilibrium traffic velocity
(v∗)max,k.
Corollary 2: There exists a reachable range for the traffic
velocity in the mixed traffic system with kautonomous vehi-
cles given by (38), i.e.,06v∗< v∗
max,k,where
FL
n−k,0, v∗
max,k= 0.
This result generalizes the statement in Corollary 1, and it
can be observed clearly that a larger proportion of autonomous
vehicles leads to a higher reachable traffic velocity.
VI. NUMERICAL EX PE RI ME NT S
Our theoretical results are obtained using a linearized model
of the mixed traffic system. We evaluate their effectiveness
in the presence of nonlinearities arising in the car-following
dynamics. In this section, we conduct multiple simulation ex-
periments to validate our results based on a realistic nonlinear
OVM model, as shown in (4). All the experiments are carried
out in MATLAB.
A. Experimental Setup
Similarly to [37], we set the parameters in the OVM model
(4) as follows: α= 0.6,β= 0.9,sgo = 35,vmax = 30,sst =
5. One can verify that this set of values violates the stability
condition (13), which means that a ring-road traffic system
with such HDVs only is unstable and stop-and-go waves may
happen in case of any perturbations.
For the parameters in the performance output (27), we
choose γs= 0.03,γv= 0.15,γu= 1. Based on the approach
in Section IV, an optimal linear feedback gain Kfor the
autonomous vehicle is obtained using Mosek [33]. To avoid
crashes, we also assume that all the vehicles are equipped with
a standard automatic emergency braking system
˙v(t) = amin ,if v2
i(t)−v2
i−1(t)
2si(t)≥ |amin|,
where the maximum deceleration rate of each vehicle is set to
amin =−5m/s2.
B. Stabilizing Traffic Flow and Increasing Traffic Velocity
Our first experiment aims to show that a typical nonlinear
mixed traffic flow can be stabilized by one single autonomous
vehicle, as proved in Theorem 2 after linearization. We con-
sider the case where n= 20 and L= 400. Each vehicle
has a weak perturbation around its equilibrium state at initial
time, in the sense that the position and the velocity of the
i-th vehicle are iL/n +δs, and vini +δv, where vini is the
equilibrium velocity corresponding to the equilibrium spacing
L/n,δs ∼U[−4,4] and δv ∼U[−2,2] with U[a, b]denoting
a uniform distribution between aand b.
When all the vehicles are under human control, it is clearly
observed in Fig.5(a) that the initial perturbations inside the
traffic flow are amplified gradually, and each vehicle’s velocity
keeps fluctuating, finally inducing a stop-and-go wave. In
contrast, if there is one autonomous vehicle using the proposed
control method, the traffic flow can be stabilized to the
original average velocity 15m/s within a short time (Fig.5(b)).
Moreover, by adjusting the equilibrium velocity v∗(as stated
in Section IV-B) and the corresponding equilibrium spacing
s∗and s∗
caccording to (2) and (31) respectively, the AV can
steer the entire traffic flow towards a higher velocity, from
15m/s to 16m/s; see Fig.5(c). In this case we observed 6%
improvement of traffic velocity when there exist only 5% AVs
(one out of 20) in the mixed traffic system.
C. Smoothing Traffic Flow via Multiple Autonomous Vehicles
Fig.1 and Fig.5 have demonstrated the ability of a single
autonomous vehicle to smooth the traffic flow where there
exist certain perturbations. We proceed to conduct numerical
experiments for the scenario where the traffic system has two
autonomous vehicles. We consider different values of nand
12
(a) (b)
Fig. 6. Simulation results at different system scales. We ran 2000 random
simulations for each value of n. The parameters are as follows: γs= 0.03,
γv= 0.15,γu= 1. (a) The control energy R∞
0uTudt needed to stabilize
the traffic flow for each autonomous vehicle. (b) The time required to stabilize
the traffic system.
let L= 20n. The experimental setup at initial time is the same
as that in the previous experiment. The results are shown in
Fig.6. It is clear that both the settling time and the control
energy of each autonomous vehicle decrease by a factor of
two approximately, when there are two autonomous vehicles
in the traffic system uniformly. Based on the results, we may
estimate the market penetration rate of autonomous vehicles
to control traffic flow effectively when adopting the optimal
control strategy. In the scenario of Fig.6, if one wants to reject
the influence of the perturbation on traffic flow within 30
seconds, a single autonomous vehicle can control the traffic
flow consisting of around 20 HDVs. This number agrees with
the results from real-world experiments [22].
D. Dampening Traffic Waves and Comparison with Existing
Strategies
As our last experiment, we consider a scenario with the
presence of infrastructure bottlenecks or lane changing [22],
where one vehicle has a rapid deceleration representing a
strong perturbation. In the beginning, the traffic flow is at
the equilibrium state with the velocity 15m/s. And then
at t= 20s, the i-th vehicle decelerates to 5m/s in two
seconds. We observe that if all the vehicles are human-driven,
the perturbation may grow stronger during the propagation
process (Fig.7(a)), while the autonomous vehicle with an
optimal control strategy can respond actively to attenuate the
perturbation and stabilize the traffic flow (Fig.7(b)). Here we
only show the case where the 6th vehicle is under the strong
perturbation. Indeed, the experiment results confirm that our
strategy allows one autonomous vehicle to dampen strong
traffic waves wherever they come from.
Next, we compare our proposed strategy with existing ones.
As shown in Fig.1, our strategy allows the autonomous vehicle
to mitigate undesired perturbations in an active way instead
of responding passively as CACC-type controllers. Here, we
proceed to make comparisons with two heuristic controllers
that aim at dampening traffic waves: FollowerStopper and
PI with Saturation [22]. Note that the FollowerStopper and
PI with Saturation use a command velocity vcmd as the
control input, and we add a proportional controller, i.e.,
u(t) = kp(vcmd(t)−v(t)), with kp= 0.6, to serve as a
lower-level controller. In all tested scenarios, the perturbation
was successfully dampened using the three methods. However,
0 20 40 60 80
t[s]
0
10
20
30
Velocity [m/s]
OVM
Average velocity
(a)
0 20 40 60 80
t[s]
0
10
20
30
Velocity [m/s]
OVM
Autonomous vehicle
Average velocity
(b)
Fig. 7. Numerical results for the scenario with a rapid and strong perturbation
in the 6th vehicle. (a) The traffic system consists of HDVs only. (b) The
mixed traffic system has an autonomous vehicle that adopts the optimal control
strategy. In each panel, the right figure shows the vehicles’ trajectories, where
the red zone represents the traffic wave; the left figure shows the vehicles’
velocities, where the red line denotes the perturbation and the black line is
the average velocity of all vehicles.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Position of the perturbation (perturbed vehicle ID)
30
40
50
60
70
80
Maximum spacing [m]
Optimal control strategy
FollowerStopper
PI with Saturation
Fig. 8. Comparison between three strategies of the maximum spacing of the
autonomous vehicle, i.e.,maxts1(t), during the experiment with a rapid and
strong perturbation.
since FollowerStopper and PI with Saturation are essentially
slow-in fast-out strategies, they tend to leave a long gap from
the preceding vehicle, which may cause vehicles from adjacent
lanes to cut in. In contrast, our optimal strategy avoids this
problem and keeps the spacing within a moderate range. As
shown in Fig.8, this finding holds irrespectively of the position
of the perturbation, and hence confirms the advantage of our
strategy.
Finally, we consider the comparison of fuel consumption for
the three control methods. It has been demonstrated in [22]
that applying FollowerStopper or PI with Saturation to one
autonomous vehicle can result in a 40% reduction of the total
fuel consumption of the entire traffic flow. Here we are inter-
ested in whether our strategy can achieve further improvement.
An instantaneous fuel consumption model in [38] is utilized
to estimate the fuel consumption rate fi(mL/s) of the i-th
vehicle, which is given by
fi=(0.444 + 0.090Rivi+ [0.054a2
ivi]ai>0,if Ri>0,
0.444,if Ri≤0,
13
Fig. 9. Comparison between three strategies of the total fuel consumption of
the entire traffic flow, i.e.,F C in (43), during the experiment with a rapid
and strong perturbation.
where Ri= 0.333 + 0.00108v2
i+ 1.200ai. Then, the total fuel
consumption of the entire traffic flow F C is calculated as
F C =
N
X
i=1 Ztf
t=0
fidt,(43)
where tf= 100sdenotes the end time of the simulation.
The simulation setup is the same as that in Fig.7, and the
results are shown in Fig.9. It is observed that our proposed
strategy achieves evidently lower fuel consumption than Fol-
lowerStopper and PI with Saturation when the perturbation
happens within the range from the 1st to the 10th vehicle. This
result validates the great potential of our strategy in improving
fuel economy. We note that, when the perturbation happens
within the range from the 11th to the 20th vehicle, which is
ahead of the autonomous vehicle in a small distance, all of the
three strategies require the autonomous vehicle to brake hard
to guarantee safety. In these cases, the three control strategies
have similar performance in terms of fuel consumption.
VII. CONCLUSION
Unlike the traditional control methods that regulate traffic
flow externally at fixed positions, autonomous vehicles can
be used as mobile actuators to control traffic flow internally.
In this paper, we have introduced a comprehensive theoretical
analysis to address the potential of autonomous vehicles on
smoothing mixed traffic flow. Specifically, we have analyzed
the controllability, stabilizability, and reachability of mixed
traffic systems. Also, an optimal control strategy has been
introduced to actively smooth mixed traffic flow.
A few other topics are worth further investigations. First,
we have assumed autonomous vehicles have access to the
global traffic state, i.e., the information of all other human-
driven vehicles. Due to the limit of communication ranges,
autonomous vehicles may be only able to obtain the infor-
mation of its neighboring vehicles. It is interesting to design
a localized optimal controller, and this leads to the notion
of structured controller synthesis [39], [40]. Second, we have
assumed homogeneous dynamics for human-driven vehicles,
and potential time delays are ignored. One interesting direction
is to consider heterogeneity and time delay in controlling
mixed traffic systems. We note that some recent work has
considered the effect of heterogeneity and time delays at the
level of platoon control [37], [41], [42], which may offer some
insights for controller design in mixed traffic systems. Finally,
our current analysis focuses on the single-lane ring road
setting, and it would be interesting to extend our analysis to
the scenarios with multiple lanes and lane-changing behavior.
APPENDIX A
DIAGONALIZATION OF BLOCK CIRCULANT MATRICES
Define ω=e2πj
n, where j=√−1denotes the imaginary
unit, and the Fourier matrix Fnis defined as [31], [32]
F∗
n=1
√n
1 1 1 . . . 1
1ω ω2. . . ωn−1
1ω2ω4. . . ω2(n−1)
.
.
..
.
..
.
..
.
.
1ωn−1ω2(n−1) . . . ω(n−1)(n−1)
,(44)
where F∗
ndenotes the conjugate transpose matrix of Fn. By
the definition of Fourier matrix, we know that Fnand F∗
nare
symmetric, i.e.,F∗
n= (F∗
n)T,Fn=FT
n, and that Fnis a
unitary matrix, i.e.,FnF∗
n=In, where Indenotes the n×n
identity matrix.
Given M1, M2, . . . , Mn∈Rm×m, a block circulant matrix
is of the following form
M=
A1A2. . . An
AnA1. . . An−1
.
.
..
.
..
.
.
A2A3. . . A1
∈Rmn×mn.(45)
As shown in [31], [32], a block circulant matrix given by (45)
can be diagonalized as follows
diag(D1, D2, . . . , Dn)=(F∗
n⊗Im)−1ˆ
A(F∗
n⊗Im),
where
D1
D2
.
.
.
Dn
= (√nF ∗
n⊗Im)
M1
M2
.
.
.
Mn
.
APPENDIX B
PROO F OF T HE STABILITY CONDITION (12)
When all the vehicles are controlled by human drivers, the
system dynamics is given by
˙x=ˆ
Ax, (46)
where ˆ
Ais the same as (14). To analyze the stability of
matrix ˆ
A, it is necessary and sufficient to study its eigenvalues’
distribution. Since ˆ
Ais a block circulant matrix, it can be
diagonalized to simplify the eigenvalue calculation. As shown
in Appendix A, ˆ
Acan be diagonalized into
ˆ
A= (F∗
n⊗I2)·diag(D1, D2, . . . , Dn)·(Fn⊗I2),
where Di=A1+A2ω(n−1)(i−1), i = 1,2, . . . , n. Since
det(λI −A) = det(λI −diag(D1, D2, . . . , Dn))
=
n
Y
i=1
det(λI −Di),(47)
14
then the eigenvalues λof ˆ
Acan be calculated by
λ2+α2−α3ω(n−1)(i−1)λ+α11−ω(n−1)(i−1) = 0,
(48)
where i= 1,2, . . . , n. Note that (48) is a second-order com-
plex equation, which makes it non-trivial to get the analytical
roots. Instead, we transform this equation to continue our
analysis. Substituting the expression of ωinto (48) leads to
ei−1
n·2πj =α1+α3λ
α1+α2λ+λ2=H(λ), i = 1,2, . . . , n, (49)
which means that the eigenvalues of ˆ
Acorrespond to the
solutions of (49). Note that ei−1
n·2πj is the i-th complex root
of zn= 1, indicating that for all the eigenvalues λof A, the
values of H(λ)constitute nunit roots. As nchanges, H(λ)
corresponds to different unit roots. Therefore, if all the roots
of |H(λ)|= 1 have negative real parts, then the solutions of
equation (49), i.e., the eigenvalues of matrix ˆ
A, have negative
real parts. We conclude that the condition that all the roots of
|H(λ)|= 1 have negative real parts is sufficient to guarantee
that ˆ
Ais stable. Note that this condition becomes sufficient
and necessary for the case where the system is stable for any
n. This is because that H(λ)can be any unit root eθj , for
θ∈[0,2π).
The rest of analysis is the same as [21]. Since all rational
functions are meromorphic, H(λ)is a meromorphic function.
Because α1and α2are positive real numbers, the poles of
H(λ)are in the left half plane, indicating that H(λ)is holo-
morphic in the right half plane. Meanwhile, |H(λ)| → 0when
Re(λ)→ ∞. According to Maximum Modulus Principle [43],
the extreme value of |H(λ)|in the right half plane can only
be obtained on the imaginary axis. To avoid eigenvalues with
positive real parts, |H(λ)|should not be more than 1on the
imaginary axis. Therefore, that the roots of |H(λ)|= 1 have
negative real part is equivalent to |H(jv)| ≤ 1,∀v∈R.This
inequality leads to the stability criterion α2
2−α2
3−2α1≥0.
ACK NOW LE DG ME NT
The authors thank Prof. Antonis Papachristodoulou,
Prof. Jianqiang Wang, Dr. Qing Xu, Mr. Licio Romao,
Mr. Suhao Yan and Mr. Chaoyi Chen for comments and dis-
cussions. We would also like to thank Dr. Ross Drummond at
the University of Oxford for providing constructive feedback.
REFERENCES
[1] M. Batty, “The size, scale, and shape of cities,” science, vol. 319, no.
5864, pp. 769–771, 2008.
[2] J. I. Levy, J. J. Buonocore, and K. Von Stackelberg, “Evaluation of the
public health impacts of traffic congestion: a health risk assessment,”
Environmental health, vol. 9, no. 1, p. 65, 2010.
[3] D. Helbing, “Traffic and related self-driven many-particle systems,”
Reviews of modern physics, vol. 73, no. 4, p. 1067, 2001.
[4] G. Orosz and et al., “Traffic jams: dynamics and control,” 2010.
[5] S. Smulders, “Control of freeway traffic flow by variable speed signs,”
Trans. Research Part B: Methodological, vol. 24, pp. 111–132, 1990.
[6] M. Papageorgiou, E. Kosmatopoulos, and I. Papamichail, “Effects of
variable speed limits on motorway traffic flow,” Transportation Research
Record, no. 2047, pp. 37–48, 2008.
[7] Y. Sugiyama, M. Fukui, and et al, “Traffic jams without bottlenecks-
experimental evidence for the physical mechanism of the formation of
a jam,” New journal of physics, vol. 10, no. 3, p. 033001, 2008.
[8] D. J. Fagnant and K. Kockelman, “Preparing a nation for autonomous
vehicles: opportunities, barriers and policy recommendations,” Trans-
portation Research Part A: Policy and Practice, pp. 167–181, 2015.
[9] J. Contreras-Castillo, S. Zeadally, and J. A. Guerrero-Iba˜
nez, “Internet of
vehicles: Architecture, protocols, and security,” IEEE internet of things
Journal, vol. 5, no. 5, pp. 3701–3709, 2017.
[10] C. Chen, L. Liu, T. Qiu, and et al., “Drivers intention identification and
risk evaluation at intersections in the internet of vehicles,” IEEE Internet
of Things Journal, vol. 5, no. 3, pp. 1575–1587, 2018.
[11] S. E. Shladover, C. A. Desoer, J. K. Hedrick, and et al., “Automated
vehicle control developments in the path program,” IEEE Transactions
on vehicular technology, vol. 40, no. 1, pp. 114–130, 1991.
[12] S. E. Li, Y. Zheng, K. Li, Y. Wu, J. K. Hedrick, F. Gao, and
H. Zhang, “Dynamical modeling and distributed control of connected
and automated vehicles: Challenges and opportunities,” IEEE Intelligent
Transportation Systems Magazine, vol. 9, no. 3, pp. 46–58, 2017.
[13] G. J. Naus, R. P. Vugts, J. Ploeg, M. J. van de Molengraft, and
M. Steinbuch, “String-stable cacc design and experimental validation:
A frequency-domain approach,” IEEE Transactions on vehicular tech-
nology, vol. 59, no. 9, pp. 4268–4279, 2010.
[14] V. Milan´
es, S. E. Shladover, J. Spring, and et al., “Cooperative adaptive
cruise control in real traffic situations.” IEEE Trans. Intelligent Trans-
portation Systems, vol. 15, no. 1, pp. 296–305, 2014.
[15] Y. Zheng, S. E. Li, K. Li, F. Borrelli, and J. K. Hedrick, “Dis-
tributed model predictive control for heterogeneous vehicle platoons
under unidirectional topologies,” IEEE Transactions on Control Systems
Technology, vol. 25, no. 3, pp. 899–910, 2017.
[16] B. Van Arem, C. J. Van Driel, and R. Visser, “The impact of cooperative
adaptive cruise control on traffic-flow characteristics,” IEEE Transac-
tions on Intelligent Transportation Systems, vol. 7, pp. 429–436, 2006.
[17] S. E. Shladover, D. Su, and X.-Y. Lu, “Impacts of cooperative adaptive
cruise control on freeway traffic flow,” Transportation Research Record,
vol. 2324, no. 1, pp. 63–70, 2012.
[18] H. Liu, X. D. Kan, S. E. Shladover, X.-Y. Lu, and R. E. Ferlis,
“Modeling impacts of cooperative adaptive cruise control on mixed
traffic flow in multi-lane freeway facilities,” Transportation Research
Part C: Emerging Technologies, vol. 95, pp. 261–279, 2018.
[19] G. Orosz, “Connected cruise control: modelling, delay effects, and
nonlinear behaviour,” Vehicle System Dynamics, vol. 54, no. 8, pp. 1147–
1176, 2016.
[20] I. G. Jin and G. Orosz, “Connected cruise control among human-driven
vehicles: Experiment-based parameter estimation and optimal control de-
sign,” Transportation Research Part C: Emerging Technologies, vol. 95,
pp. 445–459, 2018.
[21] S. Cui, B. Seibold, R. Stern, and D. B. Work, “Stabilizing traffic flow via
a single autonomous vehicle: Possibilities and limitations,” in Intelligent
Vehicles Symposium (IV), 2017 IEEE. IEEE, 2017, pp. 1336–1341.
[22] R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting,
M. Churchill, N. Hamilton, H. Pohlmann, F. Wu, B. Piccoli et al., “Dis-
sipation of stop-and-go waves via control of autonomous vehicles: Field
experiments,” Transportation Research Part C: Emerging Technologies,
vol. 89, pp. 205–221, 2018.
[23] R. Nishi, A. Tomoeda, K. Shimura, and K. Nishinari, “Theory of jam-
absorption driving,” Transportation Research Part B: Methodological,
vol. 50, pp. 116–129, 2013.
[24] C. Wu, A. M. Bayen, and A. Mehta, “Stabilizing traffic with autonomous
vehicles,” in 2018 IEEE International Conference on Robotics and
Automation (ICRA). IEEE, 2018, pp. 1–7.
[25] C. Wu, A. Kreidieh, K. Parvate, E. Vinitsky, and A. M. Bayen, “Flow:
Architecture and benchmarking for reinforcement learning in traffic
control,” arXiv preprint arXiv:1710.05465, 2017.
[26] C. Wu, A. Kreidieh, and et al., “Emergent behaviors in mixed-autonomy
traffic,” in Conference on Robot Learning, 2017, pp. 398–407.
[27] S. Skogestad and I. Postlethwaite, Multivariable feedback control:
analysis and design. Wiley New York, 2007, vol. 2.
[28] R. E. Wilson and J. A. Ward, “Car-following models: fifty years of linear
stability analysis–a mathematical perspective,” Transportation Planning
and Technology, vol. 34, no. 1, pp. 3–18, 2011.
[29] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama,
“Dynamical model of traffic congestion and numerical simulation,”
Physical review E, vol. 51, no. 2, p. 1035, 1995.
[30] R. E. Kalman, “Mathematical description of linear dynamical systems,”
Journal of the Society for Industrial and Applied Mathematics, Series
A: Control, vol. 1, no. 2, pp. 152–192, 1963.
[31] J. A. Marshall, M. E. Broucke, and B. A. Francis, “Formations of
vehicles in cyclic pursuit,” IEEE Transactions on automatic control,
vol. 49, no. 11, pp. 1963–1974, 2004.
15
[32] B. J. Olson, S. W. Shaw, C. Shi, C. Pierre, and R. G. Parker, “Circulant
matrices and their application to vibration analysis,” Applied Mechanics
Reviews, vol. 66, no. 4, p. 040803, 2014.
[33] A. Mosek, “The mosek optimization software,” Online at http://www.
mosek. com, vol. 54, no. 2-1, p. 5, 2010.
[34] A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization:
analysis, algorithms, and engineering applications. Siam, 2001, vol. 2.
[35] O. Gomez, Y. Orlov, and I. V. Kolmanovsky, “On-line identification
of siso linear time-invariant delay systems from output measurements,”
Automatica, vol. 43, no. 12, pp. 2060–2069, 2007.
[36] Q. Lin, Y. Zhang, S. Verwer, and J. Wang, “Moha: a multi-mode
hybrid automaton model for learning car-following behaviors,” IEEE
Transactions on Intelligent Transportation Systems, vol. 20, no. 2, pp.
790–796, 2018.
[37] I. G. Jin and G. Orosz, “Optimal control of connected vehicle systems
with communication delay and driver reaction time,” IEEE Transactions
on Intelligent Transportation Systems, no. 8, pp. 2056–2070, 2017.
[38] D. P. Bowyer, R. Akccelik, and D. Biggs, Guide to fuel consumption
analyses for urban traffic management, 1985, no. 32.
[39] Y. Zheng, R. P. Mason, and A. Papachristodoulou, “Scalable design of
structured controllers using chordal decomposition,” IEEE Transactions
on Automatic Control, vol. 63, no. 3, pp. 752–767, 2018.
[40] M. R. Jovanovi´
c and N. K. Dhingra, “Controller architectures: Tradeoffs
between performance and structure,” European Journal of Control,
vol. 30, pp. 76–91, 2016.
[41] M. Di Bernardo, A. Salvi, and S. Santini, “Distributed consensus strategy
for platooning of vehicles in the presence of time-varying heterogeneous
communication delays,” IEEE Transactions on Intelligent Transportation
Systems, vol. 16, no. 1, pp. 102–112, 2015.
[42] F. Gao, S. E. Li, Y. Zheng, and D. Kum, “Robust control of hetero-
geneous vehicular platoon with uncertain dynamics and communication
delay,” IET Intelligent Transport Systems, vol. 10, pp. 503–513, 2016.
[43] R. B. Burckel, An introduction to classical complex analysis. Academic
Press, 1980, vol. 1.
