Access to this full-text is provided by MDPI.
Content available from Symmetry
This content is subject to copyright.
Citation: Kong, W.; Cai, T.; Luo, Y.;
Wang, X.; Jiang, F. Cooperative
Multi-Objective Control of
Heterogeneous Vehicle Platoons on
Highway with Varying Slopes.
Symmetry 2022,14, 2647. https://
doi.org/10.3390/sym14122647
Academic Editors: Savin Treanta
and Octav Olteanu
Received: 20 November 2022
Accepted: 5 December 2022
Published: 14 December 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
symmetry
S
S
Article
Cooperative Multi-Objective Control of Heterogeneous Vehicle
Platoons on Highway with Varying Slopes
Weiwei Kong 1,* , Tianmao Cai 1, Yugong Luo 2, Xuetong Wang 3and Fachao Jiang 1
1College of Engineering, China Agricultural University, Beijing 100083, China
2School of Vehicle and Mobility, Tsinghua University, Beijing 100084, China
3Intelligent System and Software Engineering Center, Pan Asia Technical Automotive Center Co., Ltd.,
Shanghai 200120, China
*Correspondence: kongweiwei@cau.edu.cn
Abstract:
Stability, vehicle safety, energy saving, and passenger comfort are the major objectives
of vehicle platooning control. These objectives are coupled, interrelated, and even conflicting, so
integrated optimization of multiple objectives is quite challenging. Particularly for heterogeneous
platoons, the difficulties are intensified for the differences in vehicle dynamics. In this paper, the
concept of symmetry is utilized in the platooning control, that is, the design method of each vehicle’s
controller is the same. For each controller, it is to solve the optimal solution of multi-objective collab-
orative optimization. The concept of asymmetry is meanwhile embodied in the parameter setting
of each controller, for the vehicle heterogeneity. The contents of this study are as follows. First, a
mathematical model is established, in which the differences in vehicle dynamic characteristics of
heterogeneous platoon, road slope, and aerodynamics are all taken into account. Then, based on dis-
tributed nonlinear model predictive control (DNMPC) method, multi-objective control strategies are
proposed for the leader and followers, cooperatively. Furthermore, a weight coefficient optimization
method is presented, to further improve the platoon’s multi-objective synthesis performance. Finally,
comparative experiments are carried out. Results demonstrate that, compared with the classic cruise
control method of vehicle platoons, the proposed approach can reduce energy consumption by more
than 5% and improve tracking performance on the premise of passenger comfort. Real-road experi-
ments verify that the proposed control system can function effectively and satisfy the computational
requirements in real applications.
Keywords:
heterogeneous vehicle platoon; multi-objective control; nonlinear model predictive
control; distributed control; energy saving
1. Introduction
Currently, there is a widespread concern over vehicle platoon studies due to their
considerable potential to enhance road safety, reduce energy consumption and improve
traffic efficiency. Early research mainly focused on homogenous platoons. In recent years,
more and more studies on heterogeneous platoons have been carried out. Stability, vehicle
safety, energy saving, and passenger comfort are the major objectives in the control of
autonomous vehicles. Previous studies on heterogeneous platoons have probably focused
on one or two objectives, especially stability control. It is of great significance to study the
multi-objective control method of heterogeneous platoons, taking all the four objectives
into account.
The study on vehicle platoons can trace back to the California PATH project in the
1980s, which proposed the concept of “Platoon” for the first time [
1
,
2
]. Since then, vehicle
platooning control has been a topic of wide concern. The existing studies on platooning
control could be classified according to different control objectives, as shown in Table 1.
The early studies mainly focused on tracking control and stability control [
3
–
10
],
which are the basis of vehicle platoons. Tracking control is usually achieved by the cruise
Symmetry 2022,14, 2647. https://doi.org/10.3390/sym14122647 https://www.mdpi.com/journal/symmetry
Symmetry 2022,14, 2647 2 of 20
control system. A Swedish scholar, Alam, proposed a control structure for the truck
platoon, in which the leader was controlled by cruise control and the followers were
controlled by adaptive cruise control (ACC) [
11
]. With the development of vehicle-to-
vehicle (V2V) communication technology, the cooperative adaptive cruise control (CACC)
system gradually attracted more and more attention [
12
]. Chiedu, N.M. and Keyvan,
H.Z. studied a stability analysis of CACC-based platoons [
13
]. Rakkesh et al. studied a
homogenous platoon composed of eight vehicles to compare CACC and ACC systems,
and it was proven that the CACC system had better vehicle tracking and energy saving
performance [
14
]. Zegers et al. designed a multi-layer control architecture based on
CACC, in order to achieve the stability of and the expected spacing between vehicles of the
platoon [15].
Most of the above research was conducted on homogenous platoons. The studies on
heterogeneous platoons are more meaningful in practical applications, and, in the mean-
while, more difficult due to the significant differences in dynamic characteristics between
vehicles [
16
,
17
]. In recent years, many scholars have been committed to studies on heteroge-
neous platoons, mainly on stability control. Reference [
18
] analyzed the stability control of
a heterogeneous platoon with switched interaction topology, time-varying communication
delay, and lag of actuators. Delft University of Technology proposed a novel CACC method
for heterogeneous platoons, which effectively achieved stability control [
19
]. Scholars at
the University of Manchester proposed a two-layer distributed control scheme to maintain
the stability of a heterogeneous vehicle platoon moving with a constant spacing policy
assuming constant velocity of the leading vehicle [
20
]. In reference [
21
], stability control for
a heterogeneous vehicle platoon was studied, subject to external bounded unknown accel-
eration disturbances. Reference [
22
] presented an integrated platoon control framework
for heterogeneous vehicles on curved roads with varying slopes and wireless communi-
cation delays, in order to guarantee that the perturbations did not grow unbounded as
they propagated through the platoon. Zheng et al. at Tsinghua University introduced a
distributed model predictive control algorithm for heterogeneous vehicle platoons, which
could guarantee internal stability for any unidirectional topology [
23
]. In 2019, Li et al.
further studied the distributed platoon control with more generic topologies [
24
]. All of
these studies aim at the stability control of heterogeneous platoons.
In addition to stability control, studies on energy-saving control of platoons have
attracted much attention from scholars. The existing research on energy-saving control of
platoons can be categorized into three approaches, shown as follows:
(1) Energy-saving control based on decreasing the air resistance of vehicles: Refer-
ences [
25
,
26
] analyzed the aerodynamics of vehicle platoons, and studies in reference [
26
]
have shown that vehicles in different positions of a platoon faced different air resistance.
Swedish scholars have designed a small distance between vehicles of a platoon in order to
increase the fuel efficiency [
27
]. Chalmers University of Technology utilized a stochastic
optimization method to optimize the speed curve of the leading vehicle, and this method
was proven to be more energy efficient than cruise control [28].
(2) Energy-saving control by avoiding unnecessary rapid accelerations or decelerations
of platoons based on road information and predicted information of the surrounding
vehicles [
5
]: Turri et al. at the Royal Swedish Institute of Technology proposed a two-layer
control architecture for heavy-duty truck platoons [
29
]. The upper layer obtained and
predicted the road geometry information, and utilized a dynamic programming method to
calculate the optimal speed curve of platoons. The lower layer achieved vehicle safety and
energy saving control based on the MPC method. Assad Alam et al. studied the influence of
different road slopes on the fuel consumption of heavy-duty truck platoons, and proposed a
method to calculate the optimal energy-saving speed curve by predicting the information of
the road ahead [
30
]. Zhang et al. at Tsinghua University designed an energy management
strategy based on predicting the behavior of the preceding vehicles [31].
Symmetry 2022,14, 2647 3 of 20
(3) Energy-saving control based on reducing frequent gear shifts: Valerio Turri et al.
discussed a control architecture that could calculate the optimal sequence of gear shifts for a
given reference speed profile, and this could realize energy saving and smooth tracking [
32
].
In the above studies, the majority focused on one single performance factor as the
experimental objective, and only a few concerned two objectives, mainly for homogeneous
platoons, such as refs. [
5
,
6
,
11
,
14
]. Recently, some scholars have gradually become concerned
about the multi-objective control of heterogeneous platoons. Zhai et al. [
33
] proposed a
switched control strategy of heterogeneous vehicle platoons for multiple objectives with
state constraints. In this study, although multiple objectives were taken into account, only
fuel economy was designed as an objective function, while vehicle safety and passenger
comfort were designed as state constraints. In other words, this method could optimize
the single performance of energy saving, and did not actually achieve the integrated
optimization of energy saving, safety, and passenger comfort.
Table 1. Classification of existing vehicle platooning control studies.
Tracking/Safety Performance Stability Performace Energy-Saving Performance Comfort Performance
[3–7,11–14,16,33] [8–10,15,17–24] [5,11,14,17,25–33] [6,33]
In summary, fruitful results have been achieved on the stability control or energy-saving
control of vehicle platoons. However, the existing studies mainly focus on one or two objectives,
and mostly for homogeneous platoons. There still lack systematic studies on multi-objective
control of heterogeneous platoons. The major challenge is to achieve the integrated optimization
of the four major objectives, for the reason that these objectives are coupled, interrelated, and
sometimes even conflicting and contradictory. Furthermore, the vehicle dynamics differences
for heterogeneous platoons exacerbate the difficulty. The motivation of this work is to solve
this problem. The main work and contributions are as follows:
(1)
A two-layer architecture of the heterogeneous platoon control system is presented
,
consisting of a control layer and a dynamic layer, with a distributed controller for each
vehicle. This hierarchical and distributed structure is especially suitable for hetero-
geneous platoon. For dynamic layer, a nonlinear dynamic model of a heterogeneous
platoon is presented, characterizing the differences in dynamic properties between
vehicles and the influence of the road slope and wind resistance. For the control layer,
a wealth of information is utilized for multi-objective solving, including not only the
current states of the vehicles, but also their predicted states over a period of time, as
well as the expected control signals.
(2)
A cooperative multi-objective control strategy of a heterogeneous platoon is pro-
posed, based on distributed nonlinear model predictive control (DNMPC) method.
Multi-objective DNMPC controllers are designed for the leading vehicle and the fol-
lowing vehicles, cooperatively. For each controller, objective function integrates
multiple sub-objective functions, each of which depicts one targeted performance.
With this method, the optimization of multiple targets of heterogeneous platoons can
be achieved.
(3)
A weight coefficient optimization method based on a non-dominated sorting ge-
netic algorithm (NSGA-II) is presented
, to obtain the optimal weight coefficient set
of multiple targets. Instead of the common empirical method in the existing studies,
this proposed method is able to achieve coordinated adjustment between multiple
targets, which can effectively improve the multi-objective collaborative optimization
capability of the heterogeneous platoons.
The remainder of this paper is organized as follows. Section 2describes the multi-
objective control system architecture, and demonstrates the dynamic model of the hetero-
geneous platoon. In Section 3, the cooperative multi-objective control strategy based on
the DNMPC method is presented. The stability analysis based on the Lyapunov theory is
Symmetry 2022,14, 2647 4 of 20
introduced as well. Section 4elaborates on the NSGA-II-based weight coefficient optimiza-
tion method. Section 5describes the simulation experiments and real-road tests. Section 6
presents the main conclusions of this investigation.
2. Cooperative Multi-Objective Control System of a Heterogeneous Platoon
2.1. Architecture of the Multi-Objective Control System of a Heterogeneous Platoon
The architecture of the multi-objective control system of a heterogeneous platoon is
shown in Figure 1, where
α
represents the road slope, and
u∗
i(:
t)
(i= 1,
. . .
,n) is the optimal
control variable of vehicle i, calculated by its controller at time t.
u∗
i(:
t)
is a sequence,
composed of N
p
control variables during a predicted time domain [t,t+N
p∆t
], and there
are N
p
time steps in one predicted time domain.
u∗
i(1
t)
is the optimal control variable for
the present moment, and y
i
refers to the state information of vehicle i, specifically including
vehicle position, speed, and the torque.
Symmetry 2022, 14, x FOR PEER REVIEW 4 of 20
the DNMPC method is presented. The stability analysis based on the Lyapunov theory is
introduced as well. Section 4 elaborates on the NSGA-II-based weight coefficient optimi-
zation method. Section 5 describes the simulation experiments and real-road tests. Section
6 presents the main conclusions of this investigation.
2. Cooperative Multi-Objective Control System of a Heterogeneous Platoon
2.1. Architecture of the Multi-Objective Control System of a Heterogeneous Platoon
The architecture of the multi-objective control system of a heterogeneous platoon is
shown in Figure 1, where
α
represents the road slope, and *(:| )
i
ut
(i = 1,…, n) is the
optimal control variable of vehicle i, calculated by its controller at time t. *(:| )
i
ut
is a se-
quence, composed of Np control variables during a predicted time domain [t, t + NptΔ],
and there are Np time steps in one predicted time domain. *(1 | )
i
ut
is the optimal control
variable for the present moment, and yi refers to the state information of vehicle i, specif-
ically including vehicle position, speed, and the torque.
As shown in Figure 1, the architecture is a two-layer one. The control layer is com-
posed of the DNMPC controllers of each vehicle. In this study, the concept of symmetry
is utilized in the platooning control, that is, the design method of each vehicle’s controller
is the same. Each controller obtains road information, and sends the optimal control signal
to the dynamic layer, from which the state information of each vehicle can be acquired.
Dynamic layer is the dynamic model of the heterogeneous platoon.
The proposed architecture and control method in this paper are applicable for vari-
ous types of communication topologies. In this paper, a predecessor-following leader
(PFL) type of communication topology is selected for illustration, which has been reflected
by the dashed lines in Figure 1.
Road Information
DNMPC
controller of
the leader
DNMPC
controller for
follower 1
DNMPC
controller for
follower N-1
······
······
Dynamic
Layer
Control
Layer
*
1
(:| )ut
*
2
(:| )ut
*
1
(:| )
N
ut
−
α
*
1
(1 | )ut
*
2
(1 | )ut
*
(1 | )
N
ut
y1
y1y2yN-1
[y1,y2][y1,yN-1,yN]
Figure 1. Architecture of the multi-objective control system of a heterogeneous platoon.
The architecture possesses the following advantages:
(1) With a distributed control method for the control layer, each controller is designed
with full consideration of the differences in dynamic characteristics between the vehicles.
This makes it possible to achieve the best overall performance of the heterogeneous pla-
toon. Moreover, compared with centralized control, it better computational real-time per-
formance, which is more conducive to practical application.
(2) Rich information is supplied to the controller for the calculation of the optimal
control variables. The predicted information is fully utilized, which can effectively im-
prove the stability and energy-saving performance of the platoon. As shown in Figure 1,
for vehicle i, its controller could obtain the vehicle’s current state yi, control signals *(:| )
i
ut
, the leading vehicle’s state and control signals, y1 and *
1(:| )ut
, and the neighboring vehi-
cles’ state and control signals, yi−1 and *
1(:| )
i
ut
−. It is important to note that these signals
Figure 1. Architecture of the multi-objective control system of a heterogeneous platoon.
As shown in Figure 1, the architecture is a two-layer one. The control layer is composed
of the DNMPC controllers of each vehicle. In this study, the concept of symmetry is utilized
in the platooning control, that is, the design method of each vehicle’s controller is the
same. Each controller obtains road information, and sends the optimal control signal to the
dynamic layer, from which the state information of each vehicle can be acquired. Dynamic
layer is the dynamic model of the heterogeneous platoon.
The proposed architecture and control method in this paper are applicable for various
types of communication topologies. In this paper, a predecessor-following leader (PFL)
type of communication topology is selected for illustration, which has been reflected by the
dashed lines in Figure 1.
The architecture possesses the following advantages:
(1) With a distributed control method for the control layer, each controller is designed
with full consideration of the differences in dynamic characteristics between the vehi-
cles. This makes it possible to achieve the best overall performance of the heterogeneous
platoon. Moreover, compared with centralized control, it better computational real-time
performance, which is more conducive to practical application.
(2) Rich information is supplied to the controller for the calculation of the optimal
control variables. The predicted information is fully utilized, which can effectively improve
the stability and energy-saving performance of the platoon. As shown in Figure 1, for
vehicle i, its controller could obtain the vehicle’s current state y
i
, control signals
u∗
i(:
t)
, the
leading vehicle’s state and control signals, y
1
and
u∗
1(:
t)
, and the neighboring vehicles’
state and control signals, y
i−1
and
u∗
i−1(:
t)
. It is important to note that these signals include
not only the current signal, but also the expected one during a predicted time domain.
(3) Based on the vehicle state from dynamic layer, each controller uses a feedforward–
feedback control structure, which is beneficial to improve the control effect.
Symmetry 2022,14, 2647 5 of 20
(4) In the dynamic layer, road information and wind resistance are taken into account
in the dynamic model of the platoon, which could further improve the platoon’s energy-
saving performance.
2.2. Dynamic Model of a Heterogeneous Platoon
Force analysis of one vehicle on the ramp is shown in Figure 2, where
α
represents
road slope, F
T
for driving force, F
W
for air resistance, F
g
for slope resistance, and F
f
for
rolling resistance.
Symmetry 2022, 14, x FOR PEER REVIEW 5 of 20
include not only the current signal, but also the expected one during a predicted time
domain.
(3) Based on the vehicle state from dynamic layer, each controller uses a feedfor-
ward–feedback control structure, which is beneficial to improve the control effect.
(4) In the dynamic layer, road information and wind resistance are taken into account
in the dynamic model of the platoon, which could further improve the platoon’s energy-
saving performance.
2.2. Dynamic Model of a Heterogeneous Platoon
Force analysis of one vehicle on the ramp is shown in Figure 2, where α represents
road slope, FT for driving force, FW for air resistance, Fg for slope resistance, and Ff for
rolling resistance.
f
F
T
F
α
w
F
g
F
Figure 2. Force analysis of a vehicle on the ramp.
FT, FW, Fg, and Ff can be calculated according to Equations (1)–(4), described as fol-
lows:
0
4()
m
Tq
w
i
F
Tt
r
η
= (1)
2
1()
2
wd
FCAvt
ρ
= (2)
cos
f
Ffmg
α
= (3)
sin
g
Fmg
α
= (4)
where m
η
, i0, rw and Tq represent the transmission efficiency, transmission ratio, rolling
radius of the wheel, and the motor torque, respectively. Cd, A,
ρ
and v represent the
aerodynamic drag coefficient, frontal area, air density, and the vehicle speed. f represents
the coefficient of rolling resistance, which of the highway is usually 0.012.
Integrating Equations (1)–(4), the longitudinal dynamics model of one vehicle can be
obtained, shown as follows:
2
0
()
41
() () cos sin .
2
Twf g
m
qd
w
mv t F F F F
iTt CAvt fmg mg
r
η
ρ
αα
=−−−
=− −−
(5)
Dynamic characteristics between vehicles of a heterogeneous platoon are different.
The dynamic model of a heterogeneous platoon is given as follows:
,0,, 2
,
,,
() ()
4() ()
() () sin cos , N
2
() () ()
ii
qi i mi Di i
ii
iwi i
iqi qi i
St vt
Tti CdA
vt vt g fg i
mr m
Tt Tt ut
ηραα
τ
=
=− −−∈
+=
(6)
where Si(t), i
τ
and ui(t) represent the position, delay coefficient of driving system, and
the expected torque for vehicle i.
In addition to the road slope, the aerodynamics is taken into account when establish-
ing the dynamic model. Reference [34] revealed that a vehicle’s air resistance varied with
Figure 2. Force analysis of a vehicle on the ramp.
FT,FW,Fg, and Ffcan be calculated according to Equations (1)–(4), described as follows:
FT=4i0ηm
rw
Tq(t)(1)
Fw=1
2CdAρv2(t)(2)
Ff=f mg cos α(3)
Fg=mg sin α(4)
where
ηm
,i
0
,r
w
and T
q
represent the transmission efficiency, transmission ratio, rolling
radius of the wheel, and the motor torque, respectively. C
d
,A,
ρ
and vrepresent the
aerodynamic drag coefficient, frontal area, air density, and the vehicle speed. frepresents
the coefficient of rolling resistance, which of the highway is usually 0.012.
Integrating Equations (1)–(4), the longitudinal dynamics model of one vehicle can be
obtained, shown as follows:
m.
v(t) = FT−Fw−Ff−Fg
=4i0ηm
rwTq(t)−1
2CdAρv2(t)−f mg cos α−m g sin α.(5)
Dynamic characteristics between vehicles of a heterogeneous platoon are different.
The dynamic model of a heterogeneous platoon is given as follows:
.
Si(t) = vi(t)
.
vi(t) = 4Tq,i(t)i0,iηm,i
mirw,i−CD(di)Aiρ
2mivi(t)2−gsin α−f g cos α,i∈N
τi
.
Tq,i(t) + Tq,i(t) = ui(t)
(6)
where Si(t), τiand ui(t) represent the position, delay coefficient of driving system, and the
expected torque for vehicle i.
In addition to the road slope, the aerodynamics is taken into account when establishing
the dynamic model. Reference [
34
] revealed that a vehicle’s air resistance varied with the
distance between vehicles. In the existing research, the aerodynamic drag coefficient is
usually a constant, which is inconsistent with the aerodynamics characteristics. In this
paper, the mathematical formula between aerodynamic drag coefficient and the vehicle
spacing is established, given as follows:
CD(di) = C0
D,i(1−alsq
blsq +di
), (7)
Symmetry 2022,14, 2647 6 of 20
where
C0
D,i
is the nominal air drag coefficient for a single vehicle, a
lsq
and b
lsq
are the
empirical coefficients [
35
], and d
i
refers to the distance between the ego vehicle and the
preceding one. According to Formula (7), the relationship between the air drag coefficient
and the spacing is shown in Figure 3. As shown in Figure 3, the air drag coefficient is
greatly affected by the distance between vehicles, when the distance is within 50 m.
Symmetry 2022, 14, x FOR PEER REVIEW 6 of 20
the distance between vehicles. In the existing research, the aerodynamic drag coefficient
is usually a constant, which is inconsistent with the aerodynamics characteristics. In this
paper, the mathematical formula between aerodynamic drag coefficient and the vehicle
spacing is established, given as follows:
0
,
() (1 ),
lsq
Di Di
lsq i
a
Cd C bd
=−
+ (7)
where 0
,
D
i
C is the nominal air drag coefficient for a single vehicle, alsq and blsq are the em-
pirical coefficients [35], and di refers to the distance between the ego vehicle and the pre-
ceding one. According to Formula (7), the relationship between the air drag coefficient
and the spacing is shown in Figure 3. As shown in Figure 3, the air drag coefficient is
greatly affected by the distance between vehicles, when the distance is within 50 m.
Figure 3. Relationship between air drag coefficient and the spacing.
Equations (6) and (7) form the dynamic model of the heterogeneous platoon, which
possesses advantages as follows:
(1) Different properties of each vehicle is presented, such as, m
η
, i0, rw, and other pa-
rameters of each vehicle, can be different.
(2) Road slope and aerodynamics are considered, and the mathematical relationship
between air resistance and vehicle spacing is taken into account as well.
3. Cooperative Multi-Objective Control Strategy Based on the DNMPC Method
In this section, a cooperative multi-objective control strategy of heterogeneous pla-
toons based on DNMPC method is presented. DNMPC is the improvement based on
MPC, whose important advantage is that multi-objective collaboration can be achieved.
Further considering dynamic differences in heterogeneous platoons, the DNMPC method
is presented based on MPC. DNMPC controllers are designed for the leader and the fol-
lowers, respectively and cooperatively.
First, a sub-objective function is designed for each performance. Then, multi-objec-
tive function and constraints are established. Finally, for the entire control system of the
heterogeneous platoon, stability analysis is conducted based on Lyapunov theory.
3.1. Multi-Objective DNMPC Controller of the Leader
3.1.1. Sub-Objective Function for Energy Saving
The energy-saving objective function J1 (k|t) is expressed as follows:
1112
(|)|| (|) ||,Jkt WPkt t=⋅Δ
(8)
where W1 represents the weight coefficient of the energy consumption for the leader, and
tΔ is a time step. P1 (k|t) represents the motor power, and the energy consumption during
the predicted time domain is calculated by accumulating the energy power of the motor
for Np time steps.
Figure 3. Relationship between air drag coefficient and the spacing.
Equations (6) and (7) form the dynamic model of the heterogeneous platoon, which
possesses advantages as follows:
(1) Different properties of each vehicle is presented, such as,
ηm
,i
0
,r
w
, and other
parameters of each vehicle, can be different.
(2) Road slope and aerodynamics are considered, and the mathematical relationship
between air resistance and vehicle spacing is taken into account as well.
3. Cooperative Multi-Objective Control Strategy Based on the DNMPC Method
In this section, a cooperative multi-objective control strategy of heterogeneous pla-
toons based on DNMPC method is presented. DNMPC is the improvement based on
MPC, whose important advantage is that multi-objective collaboration can be achieved.
Further considering dynamic differences in heterogeneous platoons, the DNMPC method is
presented based on MPC. DNMPC controllers are designed for the leader and the followers,
respectively and cooperatively.
First, a sub-objective function is designed for each performance. Then, multi-objective
function and constraints are established. Finally, for the entire control system of the
heterogeneous platoon, stability analysis is conducted based on Lyapunov theory.
3.1. Multi-Objective DNMPC Controller of the Leader
3.1.1. Sub-Objective Function for Energy Saving
The energy-saving objective function J1(k|t) is expressed as follows:
J1(k|t) =||W1P1(k|t)·∆t||2, (8)
where W
1
represents the weight coefficient of the energy consumption for the leader, and
∆t
is a time step. P
1
(k|t) represents the motor power, and the energy consumption during
the predicted time domain is calculated by accumulating the energy power of the motor
for Nptime steps.
The motor power of one vehicle i, that is P
i
(k|t), can be calculated separately according
to the braking and driving conditions, given by Equation (9). T
q,i
,r
w,i
,i
g,i
,
ηd
,
ηb
denote
the motor torque, rolling radius of the wheel, transmission ratio, driving efficiency and
braking efficiency of the motor for vehicle i.
Pi(k|t) =
4Tq,i(k|t)vi(k|t)ig,i
rw,iηd,Tq,i(k|t)≥0
4Tq,i(k|t)vi(k|t)ig,i
rw,iηb,Tq,i(k|t)<0
(9)
Symmetry 2022,14, 2647 7 of 20
3.1.2. Sub-Objective Function for Stability and Passenger Comfort
For the leading vehicle, its speed should keep as constant as possible in order to ensure
the platoon stability and passenger comfort. Thus the stability and passenger comfort
objective function J2(k|t) is expressed as follows:
J2(k
t) = ||R1(up
1(k
t)−u0(vp
1(k
t)))||2, (10)
where R
1
represents the weight coefficient,
up
1(k
t)
represents the expected torque of the
leading vehicle, and
u0(v1p(k|t))
is the torque when the vehicle is driving at a constant
speed, which is given as Equation (11). In order to improve comfort, the rate of torque
change should be kept as low as possible.
u0(vp
i(Np
t)) = rw,i
4i0,iηm,i(1
2CD,iAiρvip(Np
t)2
+mig f cos α+migsin α),i=1, . . . , N (11)
3.1.3. Multi-Objective Function and Constraints of the Leader
Considering stability, passenger comfort and energy saving targets, the objective
function and constraint for the leader’s DNMPC controller is designed, shown as follows:
minJ1(t) =
Np−1
∑
k=0
(J1(k|t) + J2(k|t))
s.t.vmin ≤v1p(k|t)≤vmax
Tmin ≤u∗
1(k
t)≤Tmax
v1p(Np
t) = veco
Tq,1p(Np
t) = u0(v1p(Np
t))
(12)
where J
1
(t) represents the comprehensive objective function for the leading vehicle, N
p
is
the quantity of time steps during a predictive time domain, v
min
is the minimum speed
for a vehicle on the highway, v
max
is the maximum speed, T
min
is the minimum torque
for the motor, Tmax is the maximum torque, veco is the vehicle’s economic speed set by the
experience, and u∗
1(k
t)is the optimal control sequence to be solved.
3.2. Multi-Objective DNMPC Controller of the Followers
3.2.1. Sub-Objective Function for Vehicle Tracking Performance
The vehicle tracking performance of the followers represents the driving safety of
a platoon, and, meanwhile, has a significant impact on the platoon’s stability. In this
study, according to the selected PFL communication topology, the tracking performance is
described by the tracking error between the ego vehicle and the leader, and then between
the ego vehicle and the preceding one.
As shown in Figure 1,y
i
refers to the state information of vehicle i. The real state of
the ego vehicle iis expressed as Equation (13). The desired state of vehicle iis calculated
according to the state of the leader, expressed by Equation (14). The desired state of vehicle
icalculated according to the preceding vehicle i−1 is expressed as Equation (15).
yp
i= [Sp
ivp
iTp
q,i]T(13)
yi,des = [Sa
1−(i−1)d va
1Ta
q,1]T(14)
yi,i−1,des = [Sa
i−1−d va
i−1Ta
q,i−1]T(15)
As shown in Equations (13)–(15), the vehicle state set is composed of the position S,
the speed v, and the torque T
q
. Superscript pdenotes that this state is obtained by in-vehicle
Symmetry 2022,14, 2647 8 of 20
sensors, and adenotes that the state is obtained by V2V communication. The state may
vary due to the communication delay. ddenotes the desired spacing between vehicles. y
i,des
and y
i,i−1,des
represent the desired state set of vehicle icalculated according to the leader,
and the preceding vehicle, respectively.
Then, the tracking objective function for vehicle iis expressed as Equation (16), where
Qiand Giare the weight coefficients.
J1,i(k|t) =
Qi(yi,des(k
t)−yp
i(k
t))
2
+
Gi(yi,i−1,des(k
t)−yp
i(k
t))
2
(16)
3.2.2. Sub-Objective Function for Energy Saving
Similar to the energy-saving sub-objective function for the leader, that for the following
vehicle iis expressed as follows:
J2,i(k|t) =||WiPi(k|t)·∆t||2, (17)
where W
i
represents the weight coefficient. The calculation method of the vehicle’s energy
consumption is the same as that of the leader, described as Equation (9).
3.2.3. Sub-Objective Function for Passenger Comfort
Similar to the passenger comfort sub-objective function for the leader, that for the
following vehicle iis expressed as follows:
J3,i(k
t) = ||Ri(uip(k
t)−u0(vp
i(k
t)))||2, (18)
where R
i
,
up
i(k
t)
, and
u0(vip(k|t))
represent the weight coefficient, the expected torque of
vehicle i, and the torque when the vehicle is driving at a constant speed, respectively. The
calculation of u0(vip(k|t)) is the same as that of the leader, expressed as Equation (11).
3.2.4. Sub-Objective Function for Communication Stability
In order to further improve the stability performance of the platoon, the accuracy
of information transmission should be ensured. For this, the communication stability
sub-objective function is designed, given as follows:
J4,i(k
t) =
Fi((yp
i(k
t)−ya
i(k
t)))
2, (19)
where F
i
is the weight coefficient,
yp
i(k
t)
is the expected state set of vehicle i, and
ya
i(k
t)
is the state set sent to other vehicles of the platoon by V2V communication.
3.2.5. Multi-Objective Function and Constraints of the Followers
Taking all these targets into consideration at the same time, the objective function and
constraints for the follower’s DNMPC controller is designed, shown as follows:
minJi(t) =
Np−1
∑
k=0
(J1,i(k|t) + J2,i(k|t) + J3,i(k|t) + J4,i(k|t))
s.t.vmin ≤vip(k|t)≤vmax
Tmin ≤u∗
i(k
t)≤Tmax
vip(Np
t) = v1p(Np
t)
Sip(Np
t) = S1p(Np
t)−(i−1)d
Tq,ip(Np
t) = u0(vip(Np
t))
(20)
Symmetry 2022,14, 2647 9 of 20
where J
i
(t) represents the objective function for the following vehicle i(i= 2,
. . .
,n). The
terminal constraints are designed to ensure that the vehicle state could be the desired one
calculated according to the state of the leader.
3.3. Stability Analysis Based on Lyapunov Theory
The stability of the proposed control system is analyzed based on the Lyapunov theory.
The control system can be expressed as follows:
x(t) = f(x(t),u∗(t)). (21)
Assume that x= 0 is a balance point. Based on the framework of Lyapunov theory, the
stability is defined as follows:
Definition 1.
For any
e
> 0, there exists
δ
(
e
) > 0, satisfying
||x(0)||<δ(ε)⇒||x(t)||<ε,∀t≥0
.
It is controllable and stable near the initial point x = 0.
Definition 2.
If the closed loop system is stable at the balance point x = 0, and there exists
δ
that
satisfies
||x(0)|| <δ⇒lim
t→∞x(t)→0
, the closed loop system is asymptotically stable nearby the
balance point.
At random moment t, the comprehensive multi-objective function for the vehicle i’ s
controller is shown as follows:
J∗
∑(t) =
N
∑
i=1
J∗
i(xi(t),u∗
i(·|t)) (22)
where irepresents the number of the vehicle, and Nrepresents the quantity of the vehicles
in the platoon. At the moment t, the cost function for vehicle iis given as follows:
J∗
i,∑(x(t)) = Li(x∗
i(k
t),u∗
i(k
t),xa
i(k
t),xa
j(k
t),xa
1(k
t))
=
Np
∑
k=1
Qi(yi,des(k
t)−yp
i(k
t))
2
+
Gi(yi,i−1,des(k
t)−yp
i(k
t))
2
+
WiPi(k
t)·∆t
2+||Ri(uip(k
t)−u0(vp
i(k
t)))||2
+
Fi((yp
i(k
t)−ya
i(k
t)))
2
(23)
At the moment t+ 1, the value to be optimized is given as follows:
J∗
i,∑(t+1)≤
Li(x∗
i(:
t+1),u∗
i(:
t+1),xa
i(:
t+1),xa
j(:
t+1),xa
1(:
t+1))
=
Np−1
∑
k=0
li(x∗
i(k
t+1),u∗
i(k
t+1),
xa
i(k
t+1),xa
j(k
t+1),xa
1(k
t+1))
(24)
Symmetry 2022,14, 2647 10 of 20
then,
J∗
i,∑(t+1)−J∗
i,∑(t)≤
−
Np−1
∑
k=0
Li(x∗
i(k
t),u∗
i(k
t),xa
i(k
t),xa
j(k
t),xa
1(k
t))
+
Np−1
∑
k=1
Li(x∗
i(k
t),u∗
i(k
t),x∗
i(k
t),x∗
j(k
t),x∗
1(k
t))
=−li(x∗
i(0
t),u∗
i(0
t),xa
i(1
t),xa
j(1
t),xa
1(1
t))
+
Np−1
∑
k=1n[li(x∗
i(k
t),u∗
i(k
t),x∗
i(k
t),x∗
j(k
t),x∗
1(k
t)]
−li(x∗
i(k
t),u∗
i(k
t),xa
i(k
t),xa
j(k
t),xa
1(k
t))o
(25)
Analyzing Equation (25), the formula could be obtained, as follows:
li(x∗
i(k
t),u∗
i(k
t),x∗
i(k
t),x∗
j(k
t),x∗
1(k
t))
−li(x∗
i(k
t),u∗
i(k
t),xa
i(k
t),xa
j(k
t),xa
1(k
t))
=N
∑
i=1
n
Gi(yj∗(k
t)−y∗
i(k
t))
2−
Gi(yja(k
t)−y∗
i(k
t))
2o
−
Fi(y∗
i(k
t)−ya
i(k
t))
2
(26)
According to the norm triangle inequality, Equation (26) could be expressed as follows:
N
∑
i=1
n
Gi(yj∗(k
t)−y∗
i(k
t))
2−
Gi(yja(k
t)−y∗
i(k
t))
2o
−
Fi(y∗
i(k
t)−ya
i(k
t))
2
≤N
∑
i=1
Gi(yj∗(k
t)−ya
j(k
t))
2−
Fi(y∗
i(k|t)−ya
i(k|t))
2
(27)
One single step iteration of each controller is given as follows:
J∗
∑(x(t+1)) −J∗
∑(x(t))
≤ − N
∑
i=1
Li(x∗
i(1
t),u∗
i(0
t),xa
i(1
t),xa
j(1
t),xa
1(1
t)) +
Np−1
∑
k=1
ε∑(k)(28)
where,
ε∑(k) =
N
∑
i=1
[∑
j
Gi(yj(k
t)−yp
i(k
t))
2−
Fi(yp
i(k
t)−ya
i(k
t)
2](29)
Only if
Np−1
∑
k=1
ε∑(k)≤
0 can the stability of the platoons’ control system can be achieved.
In the formula, G
i
and F
i
are the weight coefficients, set by the designers. Therefore, as
long as artificially set coefficients satisfy
ε∑(k)≤
0, the asymptotic stability of the platoon’s
control system can be guaranteed based on the Lyapunov theory.
4. NSGA-II-Based Weight Coefficient Optimization
The empirical method is commonly utilized to determine the weight coefficient in the
existing research. In this study, the NSGA-II-based weight coefficient optimization method
is presented, to obtain the optimal weight coefficient set for each following vehicle. This
proposed method takes into account the differences in dynamic characteristics between
vehicles, and is able to effectively improve the multi-objective integrated performance of
the heterogeneous platoon.
For each follower of a heterogeneous platoon, the control block diagram with the
NSGA-II-based weight coefficient optimization method is shown in Figure 4.
Symmetry 2022,14, 2647 11 of 20
Symmetry 2022, 14, x FOR PEER REVIEW 11 of 20
Dynamic model
of the vehicle
DNMP
controller
NSGA-II-based weight
coefficient optimization
L
δ
Δ
*
(1 | )uk
Q
i
G
i
W
i
p
δ
Δ
E
Figure 4. Control block diagram with the NSGA-II-based weight coefficient optimization method.
As shown in Figure 4, the weight coefficient optimization calculation is executed of-
fline. After one complete control cycle, this optimization calculation is performed. L
δ
Δ
refers to the root mean square of the tracking error between the ego vehicle and the lead-
ing vehicle, p
δ
Δ refers to the root mean square of the tracking error with the preceding
vehicle, and E represents the energy consumption of the ego vehicle. Qi, Gi, and Wi are
weight coefficients of the multi-objective function to be optimized. For the weight coeffi-
cient optimization module, the objective function is designed as follows:
min [ () () ()],
LP
L
XXEX
δδ
=Δ Δ (30)
where X represents the state variable set at any moment through the control cycle.
The optimization solution for the weight coefficients is carried out based on the func-
tion shown as Equation (30). The optimization solution process of genetic algorithm (GA)
includes selection, crossover and mutation. On the basis of classic GA, the NSGA-II algo-
rithm introduces an elite strategy to further expand the sampling space, which is able to
prevent the loss of the optimal solution during the update of the population. The specific
solution can be solved simply by using MATLAB, and therefore the solution process will
not be described.
5. Simulation and Analysis
In order to verify the effectiveness of the proposed method, a simulation platform is
developed, and comparative experiments are carried out. The approach for comparison is
the classic cruise control method. Comparison simulation tests have been conducted on
the road with designed slope curve, and the actual highway with varying slopes, sepa-
rately. Moreover, a real-road experiment is conducted to verify the effectiveness and real-
time computational performance in real applications.
5.1. Simulation Platform and Simulation Setting
Based on Matlab/Simulink, the dynamic model, the DNMPC controller for each ve-
hicle, the cooperative multi-objective control algorithm, and the off-line weight coefficient
optimization algorithm were built for a heterogeneous platoon, which consisted of five
trucks. Parameters of the driving road and vehicles were set based on PreScan.
The parameters of the five trucks were designed according to the actual vehicle’s pa-
rameters of the FAW Jiefang vehicles. Two types of vehicles with different dynamic char-
acteristics were chosen to form the platoon, as shown in Table 2.
Table 2. Dynamic parameters of two types of vehicles.
No. Mass Rolling Radius of the Wheel Frontal Area of the Vehicle
1 3900 kg 0.364 m 2.4 m2
2 6100 kg 0.497 m 4.8 m2
For the simulation tests on the road with designed slope curve and tests on the actual
highway with varying slopes, the simulation settings are exactly the same. The initial speed of
each vehicle is 22 m/s, and the desired speed is 23.5 m/s, which is the average economic-speed
on the highway for this platoon. The initial spacing is just the desired one, which is 15 m.
Figure 4. Control block diagram with the NSGA-II-based weight coefficient optimization method.
As shown in Figure 4, the weight coefficient optimization calculation is executed
offline. After one complete control cycle, this optimization calculation is performed.
∆δL
refers to the root mean square of the tracking error between the ego vehicle and
the leading vehicle,
∆δp
refers to the root mean square of the tracking error with the pre-
ceding vehicle, and Erepresents the energy consumption of the ego vehicle. Q
i
,G
i
, and
W
i
are weight coefficients of the multi-objective function to be optimized. For the weight
coefficient optimization module, the objective function is designed as follows:
minL= [∆δL(X)∆δP(X)E(X)], (30)
where Xrepresents the state variable set at any moment through the control cycle.
The optimization solution for the weight coefficients is carried out based on the
function shown as Equation (30). The optimization solution process of genetic algorithm
(GA) includes selection, crossover and mutation. On the basis of classic GA, the NSGA-II
algorithm introduces an elite strategy to further expand the sampling space, which is able
to prevent the loss of the optimal solution during the update of the population. The specific
solution can be solved simply by using MATLAB, and therefore the solution process will
not be described.
5. Simulation and Analysis
In order to verify the effectiveness of the proposed method, a simulation platform is
developed, and comparative experiments are carried out. The approach for comparison is
the classic cruise control method. Comparison simulation tests have been conducted on the
road with designed slope curve, and the actual highway with varying slopes, separately.
Moreover, a real-road experiment is conducted to verify the effectiveness and real-time
computational performance in real applications.
5.1. Simulation Platform and Simulation Setting
Based on Matlab/Simulink, the dynamic model, the DNMPC controller for each
vehicle, the cooperative multi-objective control algorithm, and the off-line weight coefficient
optimization algorithm were built for a heterogeneous platoon, which consisted of five
trucks. Parameters of the driving road and vehicles were set based on PreScan.
The parameters of the five trucks were designed according to the actual vehicle’s
parameters of the FAW Jiefang vehicles. Two types of vehicles with different dynamic
characteristics were chosen to form the platoon, as shown in Table 2.
Table 2. Dynamic parameters of two types of vehicles.
No. Mass Rolling Radius of the Wheel Frontal Area of the Vehicle
1 3900 kg 0.364 m 2.4 m2
2 6100 kg 0.497 m 4.8 m2
For the simulation tests on the road with designed slope curve and tests on the actual
highway with varying slopes, the simulation settings are exactly the same. The initial
speed of each vehicle is 22 m/s, and the desired speed is 23.5 m/s, which is the average
economic-speed on the highway for this platoon. The initial spacing is just the desired one,
which is 15 m.
Symmetry 2022,14, 2647 12 of 20
In order to purely verify the effectiveness of the multi-objective control method,
in the comparative tests on the two types of roads, the weight coefficients are set as
empirical values, according to the traditional way. In this case, the comparison results
show completely the differences between these two control methods, without the impact
of weight coefficient optimization. Then, the effectiveness of the NSGA-II-based weight
coefficient optimization algorithm is verified and comparative test is described in detail in
Section 5.2.3.
5.2. Simulation Result and Discussion
5.2.1. Test on the Road with Designed Slope Curve
The comparative simulation test is carried out on the road with designed slope curve
for the heterogeneous platoon with five trucks. The road is designed as shown in Figure 5.
According to the standard of the highway, the slope range is set as (−0.066, 0.066) rad.
Symmetry 2022, 14, x FOR PEER REVIEW 12 of 20
In order to purely verify the effectiveness of the multi-objective control method, in the
comparative tests on the two types of roads, the weight coefficients are set as empirical val-
ues, according to the traditional way. In this case, the comparison results show completely
the differences between these two control methods, without the impact of weight coefficient
optimization. Then, the effectiveness of the NSGA-II-based weight coefficient optimization
algorithm is verified and comparative test is described in detail in Section 5.2.3.
5.2. Simulation Result and Discussion
5.2.1. Test on the Road with Designed Slope Curve
The comparative simulation test is carried out on the road with designed slope curve
for the heterogeneous platoon with five trucks. The road is designed as shown in Figure
5. According to the standard of the highway, the slope range is set as (−0.066, 0.066) rad.
Position (m)
Slope (rad )
020 40 60 80 100
-0.1
-0.05
0
0.05
0.1
Figure 5. Slope curve of the designed road.
As shown in Table 2, there are two types of vehicles with different masses in the
platoon. The performance of the platoon may vary greatly with different mass distribu-
tions. In this paper, three platoons with different mass distributions have been tested sep-
arately, and their vehicle tracking and energy saving performances have been analyzed,
as shown in Table 3.
Table 3. Tracking and energy-saving performances of platoons with different mass distributions.
Performances
Mass Distribution
Speed Tracking
Error (m/s)
Distance Tracking Error
(m)
Energy Consuming
(kW·h)
6100, 6100, 6100, 3900, 3900 0.1087 0.3164 2.1364
3900, 3900, 6100, 6100, 6100 0.1136 0.3348 2.0421
6100, 3900, 6100, 3900, 6100 0.1246 0.3442 2.0267
3900, 3900, 3900, 3900, 3900 0.0419 0.0816 1.5269
It is obvious from the test results that the control performance of homogeneous pla-
toon is better than that of heterogeneous platoon. It also confirms a consensus that heter-
ogeneity of vehicle dynamics makes platooning control more difficult.
Different types of heterogeneous platoons are tested, as shown in Table 3. Speed
tracking error and distance tracking error are typical indicators for tracking/safety perfor-
mance of platooning control, representing the deviation between actual speed/spacing
and the expected one. For platoon 3, in which two types of vehicles are arranged in alter-
nating order, every two adjacent vehicles affect each other, so vehicle tracking perfor-
mance of this platoon is the worst. With the alternating order, the vehicle with a larger
frontal area could withstand wind resistance for the following vehicle, and energy con-
suming of the following one could be effectively reduced. Therefore, energy saving per-
formance of platoon 3 is the best, as shown in Table 3.
Figure 5. Slope curve of the designed road.
As shown in Table 2, there are two types of vehicles with different masses in the
platoon. The performance of the platoon may vary greatly with different mass distributions.
In this paper, three platoons with different mass distributions have been tested separately,
and their vehicle tracking and energy saving performances have been analyzed, as shown
in Table 3.
Table 3. Tracking and energy-saving performances of platoons with different mass distributions.
Mass Distribution
Performances Speed Tracking Error
(m/s)
Distance Tracking
Error (m)
Energy Consuming
(kW·h)
6100, 6100, 6100, 3900, 3900 0.1087 0.3164 2.1364
3900, 3900, 6100, 6100, 6100 0.1136 0.3348 2.0421
6100, 3900, 6100, 3900, 6100 0.1246 0.3442 2.0267
3900, 3900, 3900, 3900, 3900 0.0419 0.0816 1.5269
It is obvious from the test results that the control performance of homogeneous platoon
is better than that of heterogeneous platoon. It also confirms a consensus that heterogeneity
of vehicle dynamics makes platooning control more difficult.
Different types of heterogeneous platoons are tested, as shown in Table 3. Speed track-
ing error and distance tracking error are typical indicators for tracking/safety performance
of platooning control, representing the deviation between actual speed/spacing and the
expected one. For platoon 3, in which two types of vehicles are arranged in alternating
order, every two adjacent vehicles affect each other, so vehicle tracking performance of this
platoon is the worst. With the alternating order, the vehicle with a larger frontal area could
withstand wind resistance for the following vehicle, and energy consuming of the following
one could be effectively reduced. Therefore, energy saving performance of platoon 3 is the
best, as shown in Table 3.
The mass distribution of platoon 1 is the most popular, so platoon 1 is selected in
this paper to make the comparative analysis between the proposed DNMPC-based multi-
objective control method and the classic cruise control method. For the test on the road
Symmetry 2022,14, 2647 13 of 20
with designed slope curve, vehicle speed, speed tracking error, distance tracking error, and
energy consumption are shown in Figure 6.
Symmetry 2022, 14, x FOR PEER REVIEW 13 of 20
The mass distribution of platoon 1 is the most popular, so platoon 1 is selected in this
paper to make the comparative analysis between the proposed DNMPC-based multi-ob-
jective control method and the classic cruise control method. For the test on the road with
designed slope curve, vehicle speed, speed tracking error, distance tracking error, and
energy consumption are shown in Figure 6.
(a) Speed curve of each vehicle with the proposed method
−
−
(b) Speed tracking error with the proposed method
−
−
−
(c) Distance tracking error with the proposed method
(d) Comparison result of energy consumption
Figure 6. Test results of the heterogeneous platoon driving on the designed road.
Figure 6. Test results of the heterogeneous platoon driving on the designed road.
According to Figure 6, several points could be drawn, shown as follows.
Symmetry 2022,14, 2647 14 of 20
(1) As shown in Figure 6b, when driving on a sloped road, the proposed DNMPC-
based cooperative multi-objective control method can ensure that the tracking error fluctu-
ates in a small range, and the platoon stability could be quickly achieved.
(2) As shown in Figure 6a, the vehicles accelerate a little earlier before going uphill, in
order to avoid sudden acceleration when reaching the uphill. During the climbing process,
vehicles reduce the speed to ensure sufficient torque. During the downhill process, motors
of vehicles do not generate torque, and in the meanwhile, the energy is recovered. As
the red line shows in Figure 6d, the energy consumption of the platoon decreases during
downhill due to the energy recovery. Thus, it can be seen that the proposed DNMPC-based
cooperative multi-objective control method takes into account the impact of road slope
on the energy consumption, and calculates an optimized speed curve according to the
road slope.
(3) The detailed comparison results of the simulation test on the designed road are
shown in Table 4. As can be observed from Table 4, compared with the classic cruise control
method, the proposed DNMPC-based multi-objective control method can effectively reduce
energy consumption by 5.14%, while maintaining a good vehicle tracking performance.
Table 4.
Comparison results of the heterogeneous platoon on the designed road with different
control methods.
Performances
Control Method The Proposed DNMPC-Based Multi-Objective
Control Method The Cruise Control Method
Average speed tracking error (m/s) 0.1087 0.5103
Average distance tracking error (m) 0.3164 0.3157
Total energy consumption (kW·h) 2.1364 2.2523
5.2.2. Test on the Actual Highway with Varying Slopes
In order to further verify the effectiveness of the proposed multi-objective control
method, an actual highway is chosen, which is a road section of the highway from Beijing
to Tianjin. The slope of the chosen road section is shown as Figure 7, and simulation tests
haven been carried out. In order to clearly show the impact of road slope on the test results,
a space-time conversion is made and therefore the horizontal axis in Figure 7is time.
Symmetry 2022, 14, x FOR PEER REVIEW 14 of 20
According to Figure 6, several points could be drawn, shown as follows.
(1) As shown in Figure 6b, when driving on a sloped road, the proposed DNMPC-
based cooperative multi-objective control method can ensure that the tracking error fluc-
tuates in a small range, and the platoon stability could be quickly achieved.
(2) As shown in Figure 6a, the vehicles accelerate a little earlier before going uphill,
in order to avoid sudden acceleration when reaching the uphill. During the climbing pro-
cess, vehicles reduce the speed to ensure sufficient torque. During the downhill process,
motors of vehicles do not generate torque, and in the meanwhile, the energy is recovered.
As the red line shows in Figure 6d, the energy consumption of the platoon decreases dur-
ing downhill due to the energy recovery. Thus, it can be seen that the proposed DNMPC-
based cooperative multi-objective control method takes into account the impact of road
slope on the energy consumption, and calculates an optimized speed curve according to
the road slope.
(3) The detailed comparison results of the simulation test on the designed road are
shown in Table 4. As can be observed from Table 4, compared with the classic cruise con-
trol method, the proposed DNMPC-based multi-objective control method can effectively
reduce energy consumption by 5.14%, while maintaining a good vehicle tracking perfor-
mance.
Table 4. Comparison results of the heterogeneous platoon on the designed road with different con-
trol methods.
Control Method
Performances
The Proposed DNMPC-Based Multi-Objective
Control Method The Cruise Control Method
Average speed tracking error (m/s) 0.1087 0.5103
Average distance tracking error (m) 0.3164 0.3157
Total energy consumption (kW·h) 2.1364 2.2523
5.2.2. Test on the Actual Highway with Varying Slopes
In order to further verify the effectiveness of the proposed multi-objective control
method, an actual highway is chosen, which is a road section of the highway from Beijing
to Tianjin. The slope of the chosen road section is shown as Figure 7, and simulation tests
haven been carried out. In order to clearly show the impact of road slope on the test re-
sults, a space-time conversion is made and therefore the horizontal axis in Figure 7 is time.
−
−
−
−
Figure 7. Road slope of the chosen highway section.
Still taking platoon 1 in Table 3 as an example, analyze the platoon’s vehicle tracking
performance and the energy saving performance, as shown in Figure 8.
Figure 7. Road slope of the chosen highway section.
Still taking platoon 1 in Table 3as an example, analyze the platoon’s vehicle tracking
performance and the energy saving performance, as shown in Figure 8.
The platoon adjusted its speed from 22 m/s to the economic speed 23.5 m/s during the
initial 10 s, and then drove at a constant speed of 23.5 m/s. As shown in Figure 8a,b, when
driving at a constant speed, both the speed tracking error and distance tracking error can
be kept within a quite small range. Even when the platoon adjusted its speed, the proposed
DNMPC-based cooperative multi-objective control method could ensure that the tracking
error fluctuated in a small range, and the platoon stability could be quickly achieved.
Symmetry 2022,14, 2647 15 of 20
Symmetry 2022, 14, x FOR PEER REVIEW 15 of 20
−
−
−
(a) Speed tracking error with the proposed method
−
(b) Distance tracking error with the proposed method
Energy cost (kW.h)
Time (s)
Multi-objective control
PID cruise control
(c) Comparison result of energy consumption
Figure 8. Test results of the heterogeneous platoon driving on the actual highway with varying slopes.
The platoon adjusted its speed from 22 m/s to the economic speed 23.5 m/s during
the initial 10 s, and then drove at a constant speed of 23.5 m/s. As shown in Figure 8a,b,
when driving at a constant speed, both the speed tracking error and distance tracking
error can be kept within a quite small range. Even when the platoon adjusted its speed,
the proposed DNMPC-based cooperative multi-objective control method could ensure
that the tracking error fluctuated in a small range, and the platoon stability could be
quickly achieved.
The detailed comparison of the results of the simulation test on the actual highway
with varying slopes are shown in Table 5. As shown in Figure 8c and Table 5, compared
with the cruise control method, the proposed multi-objective control method shows better
energy saving performance and vehicle tracking performance. The energy consumption
can be saved by 5.66%, while reducing vehicle tracking error.
Table 5. Comparison of the results of the heterogeneous platoon on the actual sloped road with
different control method.
Control Method
Performances
The Proposed DNMPC-Based Multi-Objective
Control Method
The Cruise Control Method
Average speed tracking error (m/s)
0.1203
0.2764
Average distance tracking error (m)
0.2810
0.2908
Total energy consumption (kW·h)
3.2785
3.4752
Figure 8.
Test results of the heterogeneous platoon driving on the actual highway with varying slopes.
The detailed comparison of the results of the simulation test on the actual highway
with varying slopes are shown in Table 5. As shown in Figure 8c and Table 5, compared
with the cruise control method, the proposed multi-objective control method shows better
energy saving performance and vehicle tracking performance. The energy consumption
can be saved by 5.66%, while reducing vehicle tracking error.
Table 5.
Comparison of the results of the heterogeneous platoon on the actual sloped road with
different control method.
Performances
Control Method The Proposed DNMPC-Based Multi-Objective
Control Method The Cruise Control Method
Average speed tracking error (m/s) 0.1203 0.2764
Average distance tracking error (m) 0.2810 0.2908
Total energy consumption (kW·h) 3.2785 3.4752
5.2.3. Test of NSGA-II-Based Weight Coefficient Optimization Method
In order to verify the effectiveness of the proposed weight coefficient optimization
method, the comparative simulation test is carried out.
According to reference [
23
], the weight coefficients are designed with the empirical
method, as shown in Table 6. In addition, the optimal weight coefficients are calculated
for each follower with the proposed NSGA-II-based optimization method, as described in
Symmetry 2022,14, 2647 16 of 20
Section 4. With different weight coefficients, the performances of the heterogeneous platoon
based on the multi-objective control method are shown in Figure 9. The detailed comparison
results are shown in Table 7. As shown in Figure 9and Table 7, with the proposed NSGA-
II-based weight coefficient optimization method, the vehicle tracking performance and
energy saving performance of the platoon can be improved simultaneously.
Table 6. Weight coefficients with the empirical method and the optimization method.
No. Weight Coefficients with the Empirical Method Weight Coefficients with the Optimization Method
1Q1 = 5, G1 = 5, W1 = 5 /
2Q2 = 5, G2 = 5, W2=5 Q2 = 3.2102, G2 = 1.8589, W2 = 4.3012
3Q3 = 5, G3 = 5, W3=5 Q3 = 0.4688, G3 = 5.8102, W3 = 1.7082
4Q4 = 5, G4 = 5, W4=5 Q4 = 0.1540, G4 = 3.7788, W4 = 1.2393
5Q5 = 5, G5 = 5, W5=5 Q5 = 0.5830, G5 = 1.8529, W5 = 22.2115
Symmetry 2022, 14, x FOR PEER REVIEW 16 of 20
5.2.3. Test of NSGA-II-Based Weight Coefficient Optimization Method
In order to verify the effectiveness of the proposed weight coefficient optimization
method, the comparative simulation test is carried out.
According to reference [23], the weight coefficients are designed with the empirical
method, as shown in Table 6. In addition, the optimal weight coefficients are calculated
for each follower with the proposed NSGA-II-based optimization method, as described in
Section 4. With different weight coefficients, the performances of the heterogeneous pla-
toon based on the multi-objective control method are shown in Figure 9. The detailed
comparison results are shown in Table 7. As shown in Figure 9 and Table 7, with the pro-
posed NSGA-II-based weight coefficient optimization method, the vehicle tracking per-
formance and energy saving performance of the platoon can be improved simultaneously.
Table 6. Weight coefficients with the empirical method and the optimization method.
No. Weight Coefficients with the Empirical Method Weight Coefficients with the Optimization Method
1 Q1 = 5, G1 = 5, W1 = 5 /
2 Q2 = 5, G2 = 5, W2 = 5 Q2 = 3.2102, G2 = 1.8589, W2 = 4.3012
3 Q3 = 5, G3 = 5, W3 = 5 Q3 = 0.4688, G3 = 5.8102, W3 = 1.7082
4 Q4 = 5
,
G4 = 5
,
W4 = 5 Q4 = 0.1540, G4 = 3.7788, W4 = 1.2393
5 Q5 = 5, G5 = 5, W5 = 5 Q5 = 0.5830, G5 = 1.8529, W5 = 22.2115
Table 7. Comparison results of the heterogeneous platoon with different weight coefficients.
Performances The Typical Empirical
Method
The Proposed Optimization
Method
Percentage of the
Improvement
Average speed tracking error (m/s) 0.2272 0.1234 45.7%
Average distance tracking error (m) 0.3226 0.1996 38.1%
Total energy consumption (kW·h) 0.7903 0.7884 0.24%
(a) Average speed tracking error of the platoon
(b) Average distance tracking error of the platoon
0 10203040
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
原模型
NSGA- II 优化模型
Average speed tra cking error (m/s)
Time (s)
empirical method
with optimization
010203040
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Average distanc e trac king error (m)
Time
(
s
)
empirical method
with optimization
Symmetry 2022, 14, x FOR PEER REVIEW 17 of 20
(c) Total energy consumption of the platoon
Figure 9. Test results of the heterogeneous platoon with different weight coefficients.
5.3. Real Road Experiment Based on Micro-Vehicle Platoon
In order to verify the effectiveness and real-time performance of the proposed control
system, a real-road experiment is conducted based on three micro-vehicles, which are
manufactured by JROBOT, as shown in Figure 10. Vehicle 1 is a wheeled one, WART-
HOG01, which acts as the leader in the platoon, and vehicle 2 and 3, Komodo, are the
followers.
Figure 10. Three micro-vehicles for the experiment.
This experiment aims to verify the real-time computational performance of the pro-
posed control system, and therefore an ordinary vehicle control unit (VCU) is chosen as
the core controller, as shown in Figure 11a. The test field consists of a straight section and
curved section, and the snapshot of road experiment is shown in Figure 11b.
(a) VCU (b) Snapshot of the road experiment
Figure 11. Photos of the real-road experiment.
The speed trajectories of three vehicles, and the tracking error of two followers, are
shown in Figure 12. The blue line represents vehicle 1, the leader, the orange dashed line
for vehicle 2, and the green dotted line for vehicle 3. The platoon entered the curved sec-
tion at about 28 s, so the speed of vehicle 1 decreased suddenly (blue line of Figure 12a),
and speed tracking error of two followers increased (Figure 12b) but soon was regulated
010203040
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Total en ergy consumptio n (kW.h)
Time (s)
empirical meth od
with optimization
Figure 9. Test results of the heterogeneous platoon with different weight coefficients.
Symmetry 2022,14, 2647 17 of 20
Table 7. Comparison results of the heterogeneous platoon with different weight coefficients.
Performances The Typical Empirical
Method
The Proposed
Optimization Method
Percentage of the
Improvement
Average speed tracking error (m/s)
0.2272 0.1234 45.7%
Average distance tracking error (m)
0.3226 0.1996 38.1%
Total energy consumption (kW·h) 0.7903 0.7884 0.24%
5.3. Real Road Experiment Based on Micro-Vehicle Platoon
In order to verify the effectiveness and real-time performance of the proposed control
system, a real-road experiment is conducted based on three micro-vehicles, which are
manufactured by JROBOT, as shown in Figure 10. Vehicle 1 is a wheeled one, WARTHOG01,
which acts as the leader in the platoon, and vehicle 2 and 3, Komodo, are the followers.
Symmetry 2022, 14, x FOR PEER REVIEW 17 of 20
(c) Total energy consumption of the platoon
Figure 9. Test results of the heterogeneous platoon with different weight coefficients.
5.3. Real Road Experiment Based on Micro-Vehicle Platoon
In order to verify the effectiveness and real-time performance of the proposed control
system, a real-road experiment is conducted based on three micro-vehicles, which are
manufactured by JROBOT, as shown in Figure 10. Vehicle 1 is a wheeled one, WART-
HOG01, which acts as the leader in the platoon, and vehicle 2 and 3, Komodo, are the
followers.
Figure 10. Three micro-vehicles for the experiment.
This experiment aims to verify the real-time computational performance of the pro-
posed control system, and therefore an ordinary vehicle control unit (VCU) is chosen as
the core controller, as shown in Figure 11a. The test field consists of a straight section and
curved section, and the snapshot of road experiment is shown in Figure 11b.
(a) VCU (b) Snapshot of the road experiment
Figure 11. Photos of the real-road experiment.
The speed trajectories of three vehicles, and the tracking error of two followers, are
shown in Figure 12. The blue line represents vehicle 1, the leader, the orange dashed line
for vehicle 2, and the green dotted line for vehicle 3. The platoon entered the curved sec-
tion at about 28 s, so the speed of vehicle 1 decreased suddenly (blue line of Figure 12a),
and speed tracking error of two followers increased (Figure 12b) but soon was regulated
010203040
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Total en ergy consumption (kW.h)
Time (s)
empirical method
with optimization
Figure 10. Three micro-vehicles for the experiment.
This experiment aims to verify the real-time computational performance of the pro-
posed control system, and therefore an ordinary vehicle control unit (VCU) is chosen as
the core controller, as shown in Figure 11a. The test field consists of a straight section and
curved section, and the snapshot of road experiment is shown in Figure 11b.
Symmetry 2022, 14, x FOR PEER REVIEW 17 of 20
(c) Total energy consumption of the platoon
Figure 9. Test results of the heterogeneous platoon with different weight coefficients.
5.3. Real Road Experiment Based on Micro-Vehicle Platoon
In order to verify the effectiveness and real-time performance of the proposed control
system, a real-road experiment is conducted based on three micro-vehicles, which are
manufactured by JROBOT, as shown in Figure 10. Vehicle 1 is a wheeled one, WART-
HOG01, which acts as the leader in the platoon, and vehicle 2 and 3, Komodo, are the
followers.
Figure 10. Three micro-vehicles for the experiment.
This experiment aims to verify the real-time computational performance of the pro-
posed control system, and therefore an ordinary vehicle control unit (VCU) is chosen as
the core controller, as shown in Figure 11a. The test field consists of a straight section and
curved section, and the snapshot of road experiment is shown in Figure 11b.
(a) VCU (b) Snapshot of the road experiment
Figure 11. Photos of the real-road experiment.
The speed trajectories of three vehicles, and the tracking error of two followers, are
shown in Figure 12. The blue line represents vehicle 1, the leader, the orange dashed line
for vehicle 2, and the green dotted line for vehicle 3. The platoon entered the curved sec-
tion at about 28 s, so the speed of vehicle 1 decreased suddenly (blue line of Figure 12a),
and speed tracking error of two followers increased (Figure 12b) but soon was regulated
010203040
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Total en ergy consumption (kW.h)
Time (s)
empirical method
with optimization
Figure 11. Photos of the real-road experiment.
The speed trajectories of three vehicles, and the tracking error of two followers, are
shown in Figure 12. The blue line represents vehicle 1, the leader, the orange dashed line for
vehicle 2, and the green dotted line for vehicle 3. The platoon entered the curved section at
about 28 s, so the speed of vehicle 1 decreased suddenly (blue line of Figure 12a), and speed
tracking error of two followers increased (Figure 12b) but soon was regulated to within
0.5 m/s. During the whole process, the platoon was able to drive safely and stably, speed
tracking error was controlled within
±
0.8 m/s, and distance tracking error was controlled
within ±1 m.
Symmetry 2022,14, 2647 18 of 20
Symmetry 2022, 14, x FOR PEER REVIEW 18 of 20
to within 0.5 m/s. During the whole process, the platoon was able to drive safely and sta-
bly, speed tracking error was controlled within ±0.8 m/s, and distance tracking error was
controlled within ±1 m.
0 102030405060
Time (s)
0
0.5
1
1.5
2
2.5
Speed (m/s)
vehicle 1
vehicle 2
vehicle 3
Time (s)
Speed (m/s)
(a) Speed trajectories of three vehicles
0 102030405060
Time (s)
−1.5
−1
−0.5
0
0.5
1
1.5
Speed tracking error (m/s)
vehicl e 2
vehicl e 3
Time (s)
Speed tr acking error (m/s)
(b) Speed tracking error of two followers
0 1020 30405060
Time (s)
−2
−1
0
1
2
Distance tacking error (m)
vehicl e 2
vehicl e 3
Time (s)
Distance tracking err or (m)
(c) Distance tracking error of two followers
Figure 12. Results of the real-road experiment.
The experiment verifies that the proposed control method can work on the VCUs of
micro-vehicles, and ensure the stability of the platoon. Thus, the real-time computational
requirements can be satisfied in real applications.
6. Conclusions
Aiming to improve the overall performance of a heterogeneous platoon on the high-
way, this paper presents a cooperative multi-objective control system, which takes four
major objectives into consideration, as well as the road slope. The following conclusions
can be drawn:
(1) A two-layer architecture of the multi-objective control system for heterogeneous
platoons is presented. For the dynamic layer, a nonlinear model of a heterogeneous pla-
toon is established, depicting various dynamic characteristics of vehicles and the influence
of road slope and wind resistance. For the control layer, rich information is provided to
distributed controllers for the calculation of the optimal control variables. The proposed
architecture is the basic of multi-objective control of heterogeneous platoons.
(2) A cooperative multi-objective control strategy based on the DNMPC method is
proposed, and controllers for the leader and followers are designed cooperatively. Com-
prehensive objective functions with multiple targets are built up, achieving integrated
optimization of safety, stability, energy saving, and passenger comfort. Through compar-
ative simulation tests on the highway with slopes, it is verified that, compared with the
Figure 12. Results of the real-road experiment.
The experiment verifies that the proposed control method can work on the VCUs of
micro-vehicles, and ensure the stability of the platoon. Thus, the real-time computational
requirements can be satisfied in real applications.
6. Conclusions
Aiming to improve the overall performance of a heterogeneous platoon on the high-
way, this paper presents a cooperative multi-objective control system, which takes four
major objectives into consideration, as well as the road slope. The following conclusions
can be drawn:
(1) A two-layer architecture of the multi-objective control system for heterogeneous
platoons is presented. For the dynamic layer, a nonlinear model of a heterogeneous platoon
is established, depicting various dynamic characteristics of vehicles and the influence of
road slope and wind resistance. For the control layer, rich information is provided to
distributed controllers for the calculation of the optimal control variables. The proposed
architecture is the basic of multi-objective control of heterogeneous platoons.
(2) A cooperative multi-objective control strategy based on the DNMPC method is
proposed, and controllers for the leader and followers are designed cooperatively. Com-
prehensive objective functions with multiple targets are built up, achieving integrated
optimization of safety, stability, energy saving, and passenger comfort. Through compar-
ative simulation tests on the highway with slopes, it is verified that, compared with the
classic cruise control method of vehicle platoons, the proposed approach can improve the
fuel economy by more than 5% and reduce tracking error simultaneously, on the premise
of ensuring safety and passenger comfort.
(3) The NSGA-II-based weight coefficient optimization method is presented, to obtain
the optimal weight coefficient set for each vehicle. Through comparative simulation tests,
Symmetry 2022,14, 2647 19 of 20
it is shown that, compared with the commonly used empirical method, multi-objective
collaborative optimization capability of the heterogeneous platoon can be further improved.
(4) In the simulation tests, three types of heterogeneous platoons with different struc-
tural parameters have been tested, and the performances have been analyzed.
(5) The proposed control system was developed and equipped on three micro-vehicles.
Real-road experiments show that the proposed control system can effectively work, and
real-time computational requirements can be satisfied in real applications.
The quality of information transmission between controllers will greatly affect the
performances of platooning control. There is an assumption in this study, which is that a
V2V (vehicle-vehicle) communication network is ideal. We will further study the platooning
control method with non-ideal communication in the future.
Author Contributions:
Conceptualization, W.K. and Y.L.; methodology, Y.L.; software, X.W.; valida-
tion, W.K., X.W. and Y.L.; writing—original draft preparation, W.K. and T.C.; writing—review and
editing, F.J. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by National Natural Science Foundation of China grant number
[52002209], and the State Key Laboratory of Automotive Safety and Energy grant number [KFY2210].
And the APC was funded by [KFY2210].
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Shladover, S.E.; Desoer, C.A.; Hedrick, J.K.; Tomizuka, M.; Walrand, J.; Zhang, W.-B.; McMahon, D.H.; Peng, H.; Sheikholeslam, S.;
McKeown, N. Automated vehicle control developments in the PATH program. IEEE Trans. Veh. Technol.
1991
,40, 114–130. [CrossRef]
2. Shladover, S.E. PATH at 20-history and major milestones. IEEE Trans. Intell. Transp. Syst. 2007,8, 584–592. [CrossRef]
3.
Liu, Y.; Yao, D.; Li, H.; Lu, R. Distributed cooperative compound tracking control for a platoon of vehicles with adaptive NN.
IEEE Trans. Cybern. 2022,52, 7039–7048. [CrossRef]
4.
Guo, G.; Dandan, L. Adaptive sliding mode control of vehicular platoons with prescribed tracking performance. IEEE Trans. Veh.
Technol. 2019,68, 7511–7520. [CrossRef]
5.
Guo, G.; Wang, Q. Fuel-efficient en route speed planning and tracking control of truck platoons. IEEE Trans. Intell. Transp. Syst.
2019,20, 1–13. [CrossRef]
6. Peter, A.C. Stable control of vehicle convoys for safety and comfort. IEEE Trans. Autom. Control 2007,52, 526–531.
7.
Zuo, L.; Wang, P.; Yan, M.; Zhu, X. Platoon tracking control with road-friction based spacing policy for nonlinear vehicles. IEEE
Trans. Intell. Transp. Syst. 2022,23, 20810–20819. [CrossRef]
8.
Besselink, B.; Johansson, K.H. String Stability and a Delay-Based Spacing Policy for Vehicle Platoons Subject to Disturbances.
IEEE Trans. Autom. Control 2017,62, 4376–4391. [CrossRef]
9.
Zheng, Y.; Li, S.E.; Li, K.; Wang, L.-Y. Stability Margin Improvement of Vehicular Platoon Considering Undirected Topology and
Asymmetric Control. IEEE Trans. Control Syst. Technol. 2016,24, 1253–1265. [CrossRef]
10.
Zheng, Y.; Li, S.E.; Wang, J.; Cao, D.; Li, K. Stability and Scalability of Homogeneous Vehicular Platoon: Study on the Influence of
Information Flow Topologies. IEEE Trans. Intell. Transp. Syst. 2016,17, 14–26. [CrossRef]
11.
Alam, A.A.; Gattami, A.; Johansson, K.H. An experimental study on the fuel reduction potential of heavy duty vehicle platooning.
In Proceedings of the 13th International IEEE Conference on Intelligent Transportation Systems, Madeira, Portugal, 19–22
September 2010; pp. 306–311.
12.
Dey, K.C.; Yan, L.; Wang, X.; Wang, Y.; Shen, H.; Chowdhury, M.; Yu, L.; Qiu, C.; Soundararaj, V. A Review of Communication,
Driver Characteristics, and Controls Aspects of Cooperative Adaptive Cruise Control (CACC). IEEE Trans. Intell. Transp. Syst.
2016,17, 491–509. [CrossRef]
13.
Chiedu, N.M.; Keyvan, H.Z. Energy-based analysis of string stability in vehicle platoons. IEEE Trans. Veh. Technol.
2022
,71, 5915–5929.
14.
Rakkesh, S.T.; Weerasinghe, A.R.; Ranasinghe, R.A.C. An intelligent highway traffic model using cooperative vehicle platooning techniques.
In Proceedings of the Moratuwa Engineering Research Conference (MERCon), Moratuwa, Sri Lanka, 29–31 May 2017; pp. 170–175.
15.
Zegers, J.C.; Semsar-Kazerooni, E.; Fusco, M.; Ploeg, J. A multi-layer control approach to truck platooning: Platoon cohesion
subject to dynamical limitations. In Proceedings of the IEEE International Conference on Models and Technologies for Intelligent
Transportation Systems (MT-ITS), Naples, Italy, 26–28 June 2017; pp. 128–133.
Symmetry 2022,14, 2647 20 of 20
16.
Chehardoli, H.; Ghasemi, A. Adaptive Centralized/Decentralized Control and Identification of 1-D Heterogeneous Vehicular
Platoons Based on Constant Time Headway Policy. IEEE Trans. Intell. Transp. Syst. 2018,19, 3376–3386. [CrossRef]
17.
He, Z.C.; Kang, H.; Li, E.; Zhou, E.L.; Cheng, H.T.; Huang, Y.Y. Coordinated control of heterogeneous vehicle platoon stability
and energy-saving control strategies. Phys. A Stat. Mech. Its Appl. 2022,606, 128155. [CrossRef]
18.
Chehardoli, H.; Homaeinezhad, M.R. Stable control of a heterogeneous platoon of vehicles with switched interaction topology, time-
varying communication delay and lag of actuator. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2017,231, 4197–4208. [CrossRef]
19.
Harfouch, Y.A.; Yuan, S.; Baldi, S. An Adaptive Switched Control Approach to Heterogeneous Platooning with Intervehicle
Communication Losses. IEEE Trans. Control Netw. Syst. 2018,5, 1434–1444. [CrossRef]
20.
Hu, J.; Bhowmick, P.; Arvin, F.; Lanzon, A.; Lennox, B. Cooperative control of heterogeneous connected vehicle platoons: An
adaptive leader-following approach. IEEE Robot. Autom. Lett. 2020,5, 977–984. [CrossRef]
21.
Guo, X.G.; Wang, J.L.; Liao, F.; Teo, R.S.H. Sting stability of heterogeneous leader-following vehicle platoons based on constant
spacing policy. In Proceedings of the 2016 IEEE Vehicles Symposium (IV), Gothenburg, Sweden, 19–22 June 2016; pp. 761–766.
22.
Xu, L.; Zhuang, W.; Yin, G.; Bian, C.; Wu, H. Modeling and robust control of heterogeneous vehicle platoons on curved roads
subject to disturbances and delays. IEEE Trans. Veh. Technol. 2019,68, 11551–11564. [CrossRef]
23.
Zheng, Y.; Li, S.E.; Li, K.; Borrelli, F.; Hedrick, J.K. Distributed Model Predictive Control for Heterogeneous Vehicle Platoons
Under Unidirectional Topologies. IEEE Trans. Control Syst. Technol. 2017,25, 899–910. [CrossRef]
24.
Li, S.E.; Qin, X.; Zheng, Y.; Wang, J.; Li, K.; Zhang, H. Distributed platoon control under topologies with complex eigenvalues:
Stability analysis and controller synthesis. IEEE Trans. Control Syst. Technol. 2019,27, 206–220. [CrossRef]
25.
Ebrahim, H.M.; Dominy, R.G.; Leung, P.S. Evaluation of vehicle platooning aerodynamics using bluff body wake generators and
CFD. In Proceedings of the International Conference for Students on Applied Engineering (ICSAE), Newcastle upon Tyne, UK,
20–21 October 2016; pp. 218–223.
26.
Fu, L.; He, B.; Wu, Y.; Hu, X.; Lai, C. The influence of inter-vehicle distance on aerodynamic characteristics of vehicle platoon.
Automot. Eng. 2007,29, 365–368.
27.
Norrby, D. A CFD Study of the Aerodynamic Effects of Platooning Trucks. Master’s Thesis, KTH Royal Institute Technology,
Stockholm, Sweden, 2014.
28.
Caltagirone, L.; Torabi, S.; Wahde, M. Truck Platooning Based on Lead Vehicle Speed Profile Optimization and Artificial
Physics. In Proceedings of the IEEE 18th International Conference on Intelligent Transportation Systems, Gran Canaria, Spain,
15–18 September 2015; pp. 394–399.
29.
Turri, V.; Besselink, B.; Johansson, K.H. Cooperative look-ahead control for fuel-efficient and safe heavy-duty vehicle platooning.
IEEE Trans. Control Syst. Technol. 2017,25, 12–28. [CrossRef]
30.
Alam, A.; Besselink, B.; Turri, V.; MåRtensson, J.; Johansson, K.H. Heavy-Duty Vehicle Platooning for Sustainable Freight
Transportation: A Cooperative Method to Enhance Safety and Efficiency. IEEE Control Syst. Mag. 2015,35, 34–56.
31.
Zhang, S.; Luo, Y.; Li, K.; Li, V. Real-Time Energy-Efficient Control for Fully Electric Vehicles Based on Explicit Model Predictive
Control Method. IEEE Trans. Veh. Technol. 2018,67, 4693–4701. [CrossRef]
32.
Turri, V.; Besselink, B.; Johansson, K.H. Gear management for fuel-efficient heavy-duty vehicle platooning. In Proceedings of the
IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV, USA, 12–14 December 2016; pp. 1687–1694.
33.
Zhai, C.; Liu, Y.; Luo, F. A switched control strategy of heterogeneous vehicle platoon for multiple objectives with state constraints.
IEEE Trans. Intell. Transp. Syst. 2019,20, 1883–1896. [CrossRef]
34.
He, B. Research on Automotive Aerodynamic Characteristics of Platoon. Ph.D. Thesis, Jilin University, Changchun, China, 2009.
35. Hucho, W.-H. Aerodynamics of Road Vehicles. Annu. Rev. Fluid Mech. 1993,25, 485–537. [CrossRef]
Available via license: CC BY 4.0
Content may be subject to copyright.
