![Fig. 1. CACC-equiped heterogeneous vehicle platoon [6]](/figures/fig-1-cacc-equiped-heterogeneous-vehicle-platoon-6-1ajoq0qq.webp)
Delft University of Technology
An adaptive switched control approach to heterogeneous platooning with inter-vehicle
communication losses
Abou Harfouch, Youssef; Yuan, Shuai; Baldi, Simone
DOI
10.1109/TCNS.2017.2718359
Publication date
2017
Document Version
Accepted author manuscript
Published in
IEEE Transactions on Control of Network Systems
Citation (APA)
Abou Harfouch, Y., Yuan, S., & Baldi, S. (2017). An adaptive switched control approach to heterogeneous
platooning with inter-vehicle communication losses.
IEEE Transactions on Control of Network Systems
,
5
(2018)
(3), 1434-1444. https://doi.org/10.1109/TCNS.2017.2718359
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1
An Adaptive Switched Control Approach to
Heterogeneous Platooning with Inter-Vehicle
Communication Losses
Youssef Abou Harfouch, Shuai Yuan, and Simone Baldi
Abstract—The advances in distributed inter-vehicle communi-
cation networks have stimulated a fruitful line of research in
Cooperative Adaptive Cruise Control (CACC). In CACC, indi-
vidual vehicles, grouped into platoons, must automatically adjust
their own speed using on-board sensors and communication
with the preceding vehicle so as to maintain a safe inter-vehicle
distance. However, a crucial limitation of the state-of-the-art of
this control scheme is that the string stability of the platoon can
be proven only when the vehicles in the platoon have identical
driveline dynamics and perfect engine performance (homoge-
neous platoon), and possibly an ideal communication channel.
This work proposes a novel CACC strategy that overcomes the
homogeneity assumption and that is able to adapt its action and
achieve string stability even for uncertain heterogeneous platoons.
Furthermore, in order to handle the inevitable communication
losses, we formulate an extended average dwell-time framework
and design an adaptive switched control strategy which activates
an augmented CACC or an augmented Adaptive Cruise Control
strategy depending on communication reliability. Stability is
proven analytically and simulations are conducted to validate
the theoretical analysis.
Index Terms—Cooperative adaptive cruise control, switched
control, heterogeneous platoon, adaptive control, networked con-
trol systems.
I. INTRODUCTION
A
UTOMATED driving is an active area of research striv-
ing to increase road safety, manage traffic congestion,
and reduce vehicles’ emissions by introducing automation into
road traffic [1]. Platooning is an automated driving method in
which vehicles are grouped into platoons, where the speed
of each vehicle (except eventually the speed of the leading
vehicle) is automatically adjusted so as to maintain a safe
inter-vehicle distance [2]. The most celebrated technology
to enable platooning is Cooperative Adaptive Cruise Control
(CACC), an extension of Adaptive Cruise Control (ACC) [3]
where platooning is enabled by inter-vehicle communication
in addition to on-board sensors. CACC systems have overcome
ACC systems in view of their better string stability properties
[4]: string stability implies that disturbances which are intro-
duced into a traffic flow by braking and accelerating vehicles
are not amplified in the upstream direction. In fact, while
The authors are with the Delft Center for Systems and Control, Delft
University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands.
E-mail: youssef.harfoush1@gmail.com, (s.yuan-1, s.baldi)@tudelft.nl.
The research leading to these results has been partially funded by the
European Commission FP7-ICT-2013.3.4, Advanced computing, embedded
and control systems, under contract #611538 (LOCAL4GLOBAL) and by the
China Scholarship Council (CSC), File No.20146160098
string stability in ACC strategies cannot be guaranteed for
inter-vehicle time gaps smaller than 1 second [5], CACC was
shown to guarantee string stability for time gaps significantly
smaller than 1 second [6]. This directly leads to improved road
throughput [7], reduced aerodynamic drag, and reduced fuel
consumption [8] over ACC systems.
Despite this potential, state-of-the-art studies and demon-
strations of CACC crucially rely on the assumption of vehicle-
independent driveline dynamics (homogeneous platoon): under
this assumption, a one-vehicle look-ahead cooperative adaptive
cruise controller was synthesized in [6], by using a perfor-
mance oriented approach to define string stability. An adaptive
bidirectional platoon-control method was derived in [9] which
utilized a coupled sliding mode controller to enhance the string
stability characteristics of the bidirectional platoon topology. A
longitudinal controller based on a constant spacing policy was
developed in [10], showing that string stability can be achieved
by broadcasting the leading vehicle’s acceleration and velocity
to all vehicles in the platoon. In [11], a linear controller
was augmented by a model predictive control strategy to
maintain the platoon’s stability while integrating safety and
physical constraints. In addition, for a platoon composed of
identical agents with different controllers, [12] assessed the
performance and challenges, in terms of string stability, of
unidirectional and asymmetric bidirectional control strategies.
Communication is an important ingredient of CACC sys-
tems: the work [13] reviews the practical challenges of CACC
and highlights the importance of robust wireless communica-
tion. From here a series of studies aiming at addressing the
effect of non-ideal communication on CACC performance: in
order to account for network delays and packet losses caused
by the wireless network, an H
∞
controller was synthesized
in [14], guaranteeing string stability criteria and robustness
for some small parametric uncertainty. The authors in [15]
derived a controller that integrates inter-vehicle communica-
tion over different realistic network conditions which models
time delays, packet losses, and interferences. Random packet
dropouts were modeled as independent Bernoulli processes in
[16] in order to derive a scheduling algorithm and design a
controller for vehicular platoons with inter-vehicle network
capacity limitation that guarantees string stability and zero
steady state spacing errors.
All the aforementioned works rely on the crucial pla-
toon’s homogeneity assumption. However, in practice, having
a homogeneous platoon is not feasible: there will always
be some heterogeneity among the vehicles in the platoon
Accepted Author Manuscript. Link to published article (IEEE): https://doi.org/10.1109/TCNS.2017.2718359
2
(e.g. different driveline dynamics, parametric and networked-
induced uncertainties). A study conducted in [17] assessed the
causes for heterogeneity of vehicles in a platoon and their
effects on string stability. A distributed adaptive sliding mode
controller for a heterogeneous vehicle platoon was derived in
[18] to guarantee string stability and adaptive compensation of
disturbances based on constant spacing policy. While address-
ing heterogeneous platoons to some extent, the aforementioned
work neglects the effect of wireless communication, as pointed
out by [13].
The brief overview of the state-of-the-art reveals the need to
develop CACC with new functionalities, that can handle pla-
toons of heterogeneous vehicles, and guarantee string stability
while adapting to changing conditions and unreliable commu-
nication. The main contribution of this paper is to address
for the first time the problem of CACC for heterogeneous
platoons with unreliable communication. The heterogeneity
of the platoon is represented by different (and uncertain)
time constants for the driveline dynamics and possibly dif-
ferent (and uncertain) engine performance coefficients. Using
a Model Reference Adaptive Control (MRAC) augmentation
method, we prove analytically the asymptotic convergence of
the heterogeneous platoon to an appropriately defined string
stable reference platoon. Furthermore, inter-vehicle commu-
nication losses, which are modeled via an extended average
dwell-time framework, are handled by switching the con-
trol strategy of the vehicle at issue to a string stable ACC
strategy with a different reference model. For this adaptive
switching control scheme, stability with bounded state track-
ing error is proven under realistic switching conditions that
match the Packet Error Rate of the two most widely adopted
vehicular wireless communication standards, namely IEEE
802.11p/wireless access in vehicular environment (WAVE) and
long-term evolution (LTE) [19],[20].
The paper is organized as follows. In Section II, the
system structure of a heterogeneous vehicle platoon with
engine performance losses is presented. Section III presents
a MRAC augmentation of a CACC strategy to stabilize the
platoon. Moreover, Section IV presents an adaptive switched
control strategy to stabilize the platoon in the heterogeneous
scenario with engine performances losses while coping with
inter-vehicle communication losses. Simulation results of the
two controllers are presented in Section V along with some
concluding remarks in Section VI.
Notation: The notation used in this paper is as follows:
R, N, and N
+
represent the set of real numbers, natural num-
bers, and positive natural numbers, respectively. The notation
P = P
T
> 0 indicates a symmetric positive definite matrix P,
where the superscript T represents the transpose of a matrix.
The notation k · k represents the Euclidean norm. The identity
matrix of dimension n is denoted by I
n×n
. The notation sup|·|
represents the least upper bound of a function.
II. SYSTEM STRUCTURE
Consider a heterogeneous platoon with M vehicles. Fig.
1 shows the platoon where v
i
represents the velocity (m/s)
of vehicle i, and d
i
the distance (m) between vehicle i
Fig. 1. CACC-equiped heterogeneous vehicle platoon [6]
and its preceding vehicle i − 1. This distance is measured
using a radar mounted on the front bumper of each vehicle.
Furthermore, each vehicle in the platoon can communicate
with its preceding vehicle via wireless communication. The
main goal of every vehicle in the platoon, except the leading
vehicle, is to maintain a desired distance d
r,i
between itself and
its preceding vehicle. Define the set S
M
= {i ∈ N| 1 ≤ i ≤ M}
with the index i = 0 reserved for the platoon’s leader (leading
vehicle). A constant time headway (CTH) spacing policy will
be adopted to regulate the spacing between the vehicles [21].
The CTH is implemented by defining the desired distance as:
d
r,i
(t) = r
i
+ h
i
v
i
(t) , i ∈ S
M
(1)
where r
i
is the standstill distance (m) and h
i
the time headway
(s) (or time gap). It is now possible to define the spacing error
(m) of the i
th
vehicle as:
e
i
(t) = d
i
(t) − d
r,i
(t)
= (q
i−1
(t) − q
i
(t) − L
i
) − (r
i
+ h
i
v
i
(t))
(2)
with q
i
and L
i
representing the rear-bumper position (m) and
length (m) of vehicle i, respectively.
A desired behavior of the platoon is instantiated when the
effect of disturbances (e.g. emergency braking) introduced
along the platoon is attenuated as they propagate in the
upstream direction [6]. Such behavior is denoted with the
term string stability. A standard definition of string stability
considered in this work is given as follows.
Definition 1 : (String stability [6]) Let the acceleration of
vehicle i be denoted with a
i
(t). Then a platoon is considered
string stable if,
sup
ω
|Γ
i
( jω)| = sup
ω
a
i
( jω)
a
i−1
( jω)
≤ 1, 1 ≤ i ≤ M (3)
where, a
i
(s) is the Laplace transform of the acceleration a
i
(t)
of vehicle i.
The control objective is to regulate e
i
to zero for all i ∈ S
M
,
while ensuring the string stability of the platoon. The following
model is used to represent the vehicles’ dynamics in the
platoon
˙e
i
˙v
i
˙a
i
=
0 −1 −h
i
0 0 1
0 0 −
1
τ
i
e
i
v
i
a
i
+
1
0
0
v
i−1
+
0
0
Λ
i
τ
i
u
i
(4)
where a
i
and u
i
are respectively the acceleration (m/s
2
) and
control input (m/s
2
) of vehicle i. Moreover, τ
i
represents each
vehicle’s unknown driveline time constant (s) and Λ
i
represents
the engine’s performance: for the nominal performance we
3
have Λ
i
= 1, while performance might decrease below 1 due
to wear or wind gusts, or increase above 1 due to wind in the
tail; Λ
i
can also be affected by the slope of the road. Model
(4) was proposed in [6] for the special case of Λ
i
= 1, ∀i ∈ S
M
.
The leading vehicle’s model is defined as:
˙e
0
˙v
0
˙a
0
=
0 0 0
0 0 1
0 0 −
1
τ
0
e
0
v
0
a
0
+
0
0
1
τ
0
u
0
. (5)
Note that, under the assumption of a homogeneous platoon
with perfect engine performance, we have τ
i
= τ
0
and Λ
i
= 1,
∀i ∈ S
M
. In this work, we remove the homogeneous assumption
by considering that ∀i ∈ S
M
, τ
i
can be represented as the sum
of two terms:
τ
i
= τ
0
+ ∆τ
i
(6)
where τ
0
is a known constant representing the driveline dy-
namics of the leading vehicle and ∆τ
i
is an unknown constant
deviation of the driveline dynamics of vehicle i from τ
0
. In
fact, ∆τ
i
acts as an unknown parametric uncertainty. In addi-
tion, we remove the perfect engine performance assumption by
considering Λ
i
as an unknown input uncertainty. Substituting
(6) into the third differential equation of (4) we obtain
τ
i
˙a
i
= −a
i
+ Λ
i
u
i
˙a
i
= −
1
τ
0
a
i
+
1
τ
0
Λ
∗
i
[u
i
+ Ω
∗
i
φ
i
]
(7)
where Λ
∗
i
=
Λ
i
τ
0
τ
i
, Ω
∗
i
= −
∆τ
i
Λ
i
τ
0
, and φ
i
= −a
i
.
Substituting (7) in (4), the vehicle model in a heterogeneous
platoon with engine performance loss under spacing policy (1)
can be defined as the following uncertain linear-time invariant
system ∀i ∈ S
M
˙e
i
˙v
i
˙a
i
=
0 −1 −h
i
0 0 1
0 0 −
1
τ
0
e
i
v
i
a
i
+
1
0
0
v
i−1
+
0
0
1
τ
0
Λ
∗
i
[u
i
+ Ω
∗
i
φ
i
].
(8)
We can now formulate the control objective for the hetero-
geneous platoon as follows:
Problem 1: (Adaptive heterogeneous platooning) Design
an adaptive control input u
i
(t), ∀i ∈ S
M
, such that the het-
erogeneous platoon described by (5) and (8) asymptotically
tracks the behavior of a string stable platoon for any possible
vehicles’ parametric uncertainty under ideal communication
between all consecutive vehicles.
III. ADAPTIVE HETEROGENEOUS PLATOONING
In order to design the control input, Section III-A presents
string stable reference dynamics for the vehicles in the platoon,
and Section III-B defines a stabilizing u
i
(t) through a MRAC
augmentation approach.
A. CACC reference model
Under the baseline conditions of identical vehicles, perfect
engine performance, and no communication losses between
any consecutive vehicles, [6] derived, using a CACC strategy,
a controller and spacing policy which proved to guarantee
the string stability of the platoon. The time headway constant
of the spacing policy (1) is set as h
i
= h
C
, ∀i ∈ S
M
, where
the superscript C indicates that communication is maintained
between the vehicle and its preceding one. Moreover, the
CACC baseline controller is defined as:
h
C
˙u
C
bl,i
= −u
C
bl,i
+ K
C
p
e
i
+ K
C
d
˙e
i
+ u
C
bl,i−1
, i ∈ S
M
(9)
where K
C
p
and K
C
d
are the design parameters of the controller.
Without loss of generality here and in the following all initial
conditions of controllers are set to zero. The initial condition
of (9) is set to zero: u
C
bl,i
(0) = 0, ∀i ∈ S
M
. In addition, the
leading vehicle control input is defined as:
h
0
˙u
0
= −u
0
+ u
r
(10)
where u
r
is the platoon’s input representing the desired accel-
eration (m/s
2
) of the leading vehicle, and h
0
a positive design
parameter denoting the nominal time headway. The initial
condition of (10) is set to zero: u
0
(0) = 0. The cooperative
aspect of (9) resides in u
C
bl,i−1
, which is received over the
wireless communication link between vehicle i and i − 1.
We can now define the reference dynamics for (8) as: the
dynamics of system (8) with Ω
∗
i
= 0, Λ
∗
i
= 1, and control input
u
i,m
= u
C
bl,i
. The reference model can be therefore described by:
˙e
i,m
˙v
i,m
˙a
i,m
˙u
i,m
=
0 −1 −h
C
0
0 0 1 0
0 0 −
1
τ
0
1
τ
0
K
C
p
h
C
−
K
C
d
h
C
−K
C
d
−
1
h
C
| {z }
A
C
m
e
i,m
v
i,m
a
i,m
u
i,m
| {z }
x
i,m
+
1 0
0 0
0 0
K
C
d
h
C
1
h
C
| {z }
B
C
w
v
i−1
u
C
bl,i−1
| {z }
w
i
, ∀i ∈ S
M
(11)
where x
i,m
and w
i
are vehicle i’s reference state vector and
exogenous input vector, respectively. Consequently, (11) is of
the following form:
˙x
i,m
= A
C
m
x
i,m
+ B
C
w
w
i
, ∀i ∈ S
M
. (12)
Furthermore, using (10), the leading vehicle’s model becomes
˙e
0
˙v
0
˙a
0
˙u
0
=
0 0 0 0
0 0 1 0
0 0 −
1
τ
0
1
τ
0
0 0 0 −
1
h
0
| {z }
A
r
e
0
v
0
a
0
u
0
| {z }
x
0
+
0
0
0
1
h
0
| {z }
B
r
u
r
. (13)
Reference model (12) has been proven in [6] to be asymp-
totically stable around the equilibrium point
x
i,m,eq
=
0 v
0
0 0
T
for x
0
= x
i,m,eq
and u
r
= 0 (14)
4
where v
0
is a constant velocity, provided that the following
Routh-Hurwitz conditions are satisfied
h
C
> 0, K
C
p
,K
C
d
> 0, K
C
d
> τ
0
K
C
p
. (15)
To assess the string stability of the reference platoon dy-
namics, it is found that
Γ
i
(s) =
1
h
C
s + 1
, ∀i ∈ S
M
(16)
Therefore, we can conclude that (16) satisfies the string
stability condition (3) of Definition 1 for any choice of h
C
> 0,
and thus the defined reference platoon dynamics (12) are string
stable.
B. MRAC augmentation of a baseline controller
In this Section, reference model (12) will be used to
design the control input u
i
(t) such that the uncertain platoon’s
dynamics described by (5) and (8) converge to string stable
dynamics. With this scope in mind, we will augment a baseline
controller with an adaptive term, using a similar architecture
as proposed in [22]. To include the adaptive augmentation, the
input u
i
(t) is split, ∀i ∈ S
M
, into two different inputs:
u
i
(t) = u
bl,i
(t) + u
ad,i
(t) (17)
where u
bl,i
and u
ad,i
are the baseline controller and the adaptive
augmentation controller (to be constructed), respectively.
First, define the control input of the leading vehicle u
0
(t) as
in (10). Moreover, define u
bl,i
(t) = u
C
bl,i
(t). Substituting (17)
into (8), we get the uncertain vehicle model
˙x
i
= A
C
m
x
i
+ B
C
w
w
i
+ B
u
Λ
∗
i
u
ad,i
+ Θ
∗T
i
Φ
i
, ∀i ∈ S
M
(18)
where x
i
=
e
i
v
i
a
i
u
bl,i
T
, and the matrices A
C
m
and B
C
w
are known and defined in (12), and B
u
=
0 0
1
τ
0
0
T
.
The uncertain ideal parameter vector is defined as Θ
∗
i
=
K
∗
u,i
Ω
∗
i
T
where K
∗
u,i
= 1 − Λ
∗−1
i
. The regressor vector is
defined as Φ
i
=
u
bl,i
φ
i
T
. Therefore, the heterogeneous
platoon with engine performance loss and control input (17)
can be defined as system (13)-(18).
Furthermore, taking (12) as the vehicle reference model, the
adaptive control input is defined as
u
ad,i
= −Θ
T
i
Φ
i
(19)
where Θ
i
is the estimate of Θ
∗
i
. Define the state tracking error
as
˜x
i
= x
i
− x
i,m
, ∀i ∈ S
M
. (20)
Replacing (19) in (18) and subtracting (12) results in the
following state tracking error dynamics
˙
˜x
i
= A
C
m
˜x
i
− B
u
Λ
∗
i
˜
Θ
T
i
Φ
i
(21)
where
˜
Θ
i
= Θ
i
− Θ
∗
i
.
Since A
C
m
is stable, there exists a unique symmetric positive
define matrix P
m
= P
T
m
> 0 such that
(A
C
m
)
T
P
m
+ P
m
A
C
m
+ Q
m
= 0
Fig. 2. Networked switched control system
where Q
m
= Q
T
m
> 0 is a designed matrix. Define the adaptive
law
˙
Θ
i
= Γ
Θ
Φ
i
˜x
T
i
P
m
B
u
(22)
with Γ
Θ
= Γ
T
Θ
> 0 being the adaptive gain. Then the following
stability and convergence results can be stated.
Theorem 1: Consider the heterogeneous platoon model (8)
with reference model (12). Then, the adaptive input (19) with
adaptive law (22) makes the platoon’s dynamics asymptoti-
cally converge to string stable dynamics. Consequently,
lim
t→∞
[x
i
(t) − x
i,m
(t)] = 0, ∀i ∈ S
M
and
lim
t→∞
kΘ
T
i
(t)Φ
i
(t)k = 0, ∀i ∈ S
M
.
Proof: See Appendix A.
The results of Theorem 1 hold under the assumption of
ideal continuous communication between the vehicles in the
platoon. However, communication losses are always present
in practice and coping with them is the subject of the next
section.
IV. ADAPTIVE SWITCHED HETEROGENEOUS PLATOONING
One way of handling the unavoidable communication losses
is by switching between CACC and ACC depending on
the network’s state at each single communication link. This
networked switched control system is outlined in Fig. 2. In
this aim, an adaptive switched control method is presented
for the scenario with joint heterogeneous dynamics and inter-
vehicle communication losses. Note that ACC does not require
inter-vehicle communication, but as a drawback it requires to
increase the time gap in order to guarantee string stability
[6]. So, the switched control system also takes into account
that a different spacing policy might be active in the CACC
case (indicated with h
C
) and in the ACC case (indicated
with h
L
), where the superscript L stands for communication
loss. The adaptive switched controller is based on a Mode-
Dependent Average Dwell Time (MDADT) which is used to
characterize the network switching behavior as a consequence
of communication losses.
Definition 2 (Mode-Dependent Average Dwell Time [23]):
For a switched system with S subsystems, a switching signal
σ(·), taking values in {1,2,3,...,S} = M , and for s ≥ t ≥ 0
and k ∈ M , let N
σk
(t, s) denote the number of times subsystem
k is activated in the interval [t, s), and let T
k
(t, s) be the total
time subsystem k is active in the interval [t, s). The switching