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IEEE/ASME TRANSACTIONS ON MECHATRONICS 1
A Fully Distributed Antiwindup Control Protocol
for Intelligent-Connected Electric Vehicles
Platooning With Switching Topologies
and Input Saturation
Jingyao Wang , Member, IEEE, Xingming Deng , Jinghua Guo , Yugong Luo, and Keqiang Li
Abstract—This article proposes a fully distributed adap-
tive control framework to tackle the problem of intelligent-
connected electric vehicles platooning with input satura-
tion and topology changes. First, a longitudinal vehicle
platoon system is investigated. A linearized dynamics
model of each vehicle is provided while considering input
saturation. Second, abnormal communication scenarios
are given, which are described by the Markovian randomly
switching topologies with several assumptions provided.
Furthermore, a triple-observer structure consisting of a
local observer, a distributed observer, and an antiwindup
observer, is developed to establish a fully distributed adap-
tive antiwindup controller. The controller enables to ensure
the stability of the vehicle platoon system solely requiring
the local output information of each following vehicle, while
avoiding the acquisition of global information. This control
protocol provides a reliable solution for dealing with the
input saturation constraints and the randomly switching
communication topologies. Finally, the feasibility and the
effectiveness of this adaptive protocol are validated by con-
ducting some numerical simulations.
Index Terms—Adaptive antiwindup controller, connected
vehicles, fully distributed protocol, input saturation con-
straint, Markovian randomly switching communication
graphs.
Manuscript received 13 June 2022; revised 17 October 2022; ac-
cepted 26 November 2022. Recommended by Technical Editor M. De-
foort and Senior Editor X. Chen. This work was supported in part by the
National Nature Science Foundation of China under Grant 61803319
and Grant 61903278, in part by the State Key Laboratory of Auto-
motive Safety and Energy under Grant KFY2206, in part by the Key
Technical Innovation and Industrialization Project of Fujian Province of
China under Grant 2022G047, and in part by the Major Science and
Technology Projects of Xiamen of China under Grant 3502Z20201015.
(Corresponding author: Jinghua Guo.)
Jingyao Wang, Xingming Deng, and Jinghua Guo are with the
School of Aerospace Engineering, Xiamen University, Xiamen 361005,
China (e-mail: wangjingyao1@xmu.edu.cn; 23220201151598@stu.xmu
.edu.cn; guojing_0701@live.cn).
Yugong Luo and Keqiang Li are with the School of Vehicle
and Mobility, Tsinghua University, Beijing 100084, China (e-mail:
lyg@tsinghua.edu.cn; likq@tsinghua.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TMECH.2022.3226208.
Digital Object Identifier 10.1109/TMECH.2022.3226208
I. INTRODUCTION
IN RECENT years, with the development of the automotive
industry, the global car ownership rises rapidly. However,
due to the limitation of land resources, current carriage facilities
are insufficient to meet the rapid growth of vehicle ownership,
resulting in a series of issues, such as traffic congestion, energy
waste, and road accident [1]. Intelligent connected technology
has the potential to improve traffic capacity and reduce fuel
consumption, which is one of the significant research branches
of intelligent transportation system [2], [3], [4], [5], [6]. The
platoon control of electric vehicles is one of the intelligent
connected technologies. Each connected vehicle (CV) in lon-
gitudinal platoon can adjust its own driving state so as to keep
the vehicular platoon in a desired configuration based on the
information of adjacent CVs [7]. Hence, the number of traffic
accidents decreases effectively.
Numerous publications have investigated the platoon control
problem of electrical vehicles and vehicular platoon has become
a recognized area over the past decade. Researchers have pro-
posed many methods for different control objectives. With the
purpose of ensuring the platoon stability, Linsenmayer et al. [8]
and Guo et al. [9] developed a distributed control framework
and a scheduling algorithm-based controller, respectively. In
order to enhance the performance of the vehicle platoon, Ma
et al. [10] designed a distributed control strategy based on model
predictive control and simulated annealing-particle swarm opti-
mization algorithm, and Kwon [11] provided an adaptive control
method, which uses a coupled sliding-mode controller. Martinec
et al. [12] analyzed the platoon stability under asymmetric
bidirectional communication topology.
Nowadays, wireless communication technology is generally
employed to enhance the environment perceiving capability of
CVs. Nevertheless, the platoon system suffers from unstable
factors of network, such as package drop-out and cyber-attacks,
thereby making the topology to change [13], [14]. Besides, the
separation and merging of the vehicular platoon also induce
topology switching. Therefore, communication passages maybe
temporarily unavailable for adjacent CVs, which can destabilize
the platoon system in an intolerable way. Zhang et al. [15]
performed a platoon-optimized clustering algorithm to reduce
the randomness of the topology changes. Zhang and Sun [16] put
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2IEEE/ASME TRANSACTIONS ON MECHATRONICS
forward a distributed topology switching algorithm for hetero-
geneous vehicle platoon to make a tradeoff between the platoon
stability and communication expense. Li et al. [17] provided
a connectivity probability enhancing scheme to mitigate the
impact of dynamic topology variations.
When implementing the aforementioned methods, the control
input cannot exceed a distinct threshold due to the control force
of physical actuators, such as break and accelerator; otherwise
the input signal may not work appropriately [18], [19], [20]. Seri-
ous performance degradation may happen if the input saturation
problem is overlooked in the process of designing control strate-
gies. Based on the neural network and sliding-mode technique,
Guo et al. [21] suggested a distributed adaptive approach for
nonlinear vehicle-following platoon system, which can solve the
input saturation problem with external disturbances by adjusting
only a single parameter. Combined with event-triggered scheme,
Feng et al. [22] introduced a tube-cased discrete controller to
handle the saturation constraints, while restraining heteroge-
neous disturbances with an integrated controller. He et al. [23]
discussed the robust global stabilization issue of the vehicle
platoon system subject to input saturation and input-additive
uncertainties via a low-gain state-feedback control law.
Most of the existing works on topology switching for vehicle
platoons do not consider the problem of actuator saturation,
which makes many research works unsatisfactory at the appli-
cation level. Consequently, it is meaningful to study platoon
control problem in the case of switching topologies and input
saturation.
Motivated by the discussion above, this article aims to de-
velop a fully distributed adaptive platoon controller, so as to
ensure the stability of platoon system under switching topologies
and guarantee tracking performance with input saturation. This
article makes the following contributions. First, we propose a
fully distributed platoon control protocol relying on only the
immediate local output information, which means that the con-
struction of the controller has no need of the global information,
such as certain eigenvalue of the Laplacian matrix (known as
the algebraic connectivity) or the total number of CVs. Second,
different from the related works [22], [23], [24], this article
can solve the platoon control problem that simultaneously take
topology changes and input saturation into account. Third, an
algorithm to construct the proposed control protocol is given, in
which the control parameters are designed without utilizing any
global information, only using a vehicle’sdynamics information.
The rest of this article is organized as follows. Section II
presents the dynamics of CVs as well as the topology switching
model among the vehicle platoon, and formulates the concerned
problem. Section III details the proposed control protocol and the
theoretical results. Section IV provides some simulations results
to illustrate the flexibility of the proposed protocol. Finally,
Section V concludes this article.
Notations: For a matrix A,A≥0(A≤0)means that the
matrix Ais positive semidefinite (negative semidefinite), and
ATdenotes the transpose of A.IfAis symmetric, then let
λmax(A)and λmin (A)be the maximal and minimal eigenvalues
of A, respectively. The set of real numbers is denoted by R.
For a vector x∈Rn, define x2xTx.A⊗Brepresents the
Fig. 1. Structure of the vehicle platoon system. Here, we consider
the vehicle platoon system subject to switching topologies caused by
communication failure. Communication failure happens stochastically
among vehicles and makes them lose their connection to their predeces-
sors and the leading vehicle. For example, vehicle icannot have access
to vehicles 0 and i−1because of the network anomaly and the sensor
fault. A triple-observer based adaptive control protocol is designed to
cope with different driving constraints.
Kronecker product between matrices Aand B.Iexpresses an
identity matrix. E[·]indicates the mathematical expectation.
II. INTELLIGENT-CONNECTED ELECTRIC VEHICLE PLATOON
MODEL
A. Vehicle Longitudinal Dynamics
Consider an intelligent electric vehicle platoon consisting of a
leader and Nfollowers, with labels 0 and 1, ..., N, respectively.
A coupled platoon system is built with all the CVs through
interrelated sensors and wireless communication network. Each
CV uses the driving information of neighboring CVs to adjust
its speed in real time, so that the platoon system can command
the CVs to reach the desired platoon formation, that is to say,
to achieve a consistent speed with leading CV and to keep an
expected distance between successive CVs. Fig. 1shows the
structure of the vehicle platoon system.
It is a primary task of the vehicle platoon control to establish a
concise and accurate vehicle dynamics model. The longitudinal
dynamics of electric vehicles is nonlinear because of factors,
such as the motor, wheel, braking system, and the wind resis-
tance. Since this article only studies the longitudinal driving of
vehicle platoons, we make the following reasonable assumptions
and simplifications about the longitudinal dynamics of electric
vehicles.
1) The car body is supposed to be rigid and symmetric.
2) The effect of vibration of the pitch and yaw motions is
neglected.
3) The influence of the longitudinal slip is ignored.
4) The input and output of the vehicle dynamics system can
be classified as a first-order inertial transfer functions.
When the assumptions are satisfied, we can perform dynamic
force analysis on the simplified vehicle longitudinal dynamics
model. Based on the Newton’s second law, the longitudinal
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WANG et al.: FULLY DISTRIBUTED ANTIWINDUP CONTROL PROTOCOL FOR INTELLIGENT-CONNECTED ELECTRIC VEHICLES 3
dynamics model of vehicle ican be obtained [25]
Fd,i(t)−Fc,i (t)−migμi=miai(t)
Fd,i(t)= Td,i (t)
ra,i
Fc,i(t)= 1
2CcρcSc,iv2
i(t)
τi˙
Td,i(t)+Td,i (t)=Te,i(t)(1)
where Fd,i(t)and Fc,i (t)represent the actual driving force of the
vehicle and the air resistance, respectively; Td,i(t)and Te,i(t)
are the actual driving torque and the desired driving torque of the
vehicle, respectively; μiand Ccrepresent the rolling resistance
coefficient and the air resistance coefficient, respectively; Sc,i,
ra,i,mi, and τiare the windward area of the vehicle, the
tire radius, the vehicle mass, and the time constant of vehicle
dynamics, respectively; grepresents the gravity acceleration
constant and ρcrepresents the air density.
When an electric vehicle is equipped with a dc hub motor, its
wheel dynamics model is [26]
Ja,i ˙ωa,i(t)=ηa,iTa,i (t)−Td,i(t)(2)
where Ja,i is the rotational inertia of the motor and the tire,
ωa,i is the wheel speed, ηa,i is the mechanical efficiency of the
transmission system, and Ta,i is the motor torque.
Using the inverse model compensation technique to give a
feedback linearization of the nonlinear terms in (1), then the
desired torque Te,i(t)can be redesigned as [27]
Te,i(t)=ra,i 1
2CcρcSc,iv2
i(t)+τiCcρcSc,ivi(t)˙vi(t)
+migμi+miui(t).(3)
Substituting (3) and (2) into (1), and assuming that the dynam-
ics of the CVs in the platoon are homogeneous, which means
τi=τ>0, then the linearized longitudinal dynamics model of
each electrical vehicle can be given by
˙ai(t)=−1
τai(t)+ 1
τui(t).(4)
Considering the existence of input saturation, rewrite the lin-
earized longitudinal dynamics as
˙xi(t)=Axi(t)+Bsatθ(ui(t)) ,i=1,...,N
yi=Cxi(5)
with
A=⎡
⎣
01 0
00 1
00−1
τ
⎤
⎦,B=⎡
⎣
0
0
1
τ
⎤
⎦
where xi(t)=[piviai]Tand yi(t)∈Rqare the state and the
measured output of vehicle i, respectively, in which pi,vi,
and aiare the position, the speed, and the acceleration of the
vehicle i, respectively; (A, C)is detectable and the matrix Cis
with appropriate dimension. As saturation is a non-negligible
phenomenon, the control input ui(t)is subject to saturation
nonlinearity, which is described as
satθ(s)=⎧
⎨
⎩
¯
θ, 0<¯
θ≤s
s, θ <s<¯
θ
θ,s≤θ < 0
(6)
where ¯
θand θare constants denoting the saturation constraints
of control input. Throughout the manuscript, uiis regarded as
the upper control input, from which the desired driving torque
of the vehicle can be literally calculated and sent to the lower
torque controller for an actual adjustment of the vehicle speed.
The group of followers are designed to track a leader with the
following dynamics:
˙x0(t)=Ax0(t)
y0=Cx0.(7)
B. Graph Theory and Communication Topology Model
1) Graph Theory: The graph theory is utilized to character-
ize the interaction relationship among the CVs in the platoon.
Within this context, a communication topology consisting of a
leading vehicle and Nfollowing vehicles is represented by a
directed graph G(V,E), where V={0,1,2,...,N}is the
set of vehicles in the platoon, and E∈V×Vdenotes the set of
edges with direct connectivity between CVs. Furthermore, the
derived graph Gis said to contain a directed spanning tree if there
exists a tree-type subgraph, which includes all of the nodes of G.
In adjacency matrix A,αij equals 1 if there exists an information
flow from jto i; equals 0, otherwise. Accordingly, the Laplacian
matrix Lis defined by lij −αij ,ifi=j;lij =N
j=iαij ,
otherwise.
2) Example of the Communication Topology Under the Com-
munication Failure: When the perception abnormalities (mainly
caused by sensor fault) as well as the communication anomaly
(caused by the unstable factors of wireless network) occur at
the same time, the vehicle completely lose data availability, just
like vehicles 1 and N−1inFig.2(c), and vehicles N−2 and
Nin Fig. 2(d). In this situation, it is reasonable to think that
the channels between abnormal CVs and the leading vehicle are
interrupted when the communication anomaly exists. Moreover,
the information connection between a following vehicle and
its predecessor is also blocked if the following vehicle has the
perception abnormalities. Herein, we classify the intermittent
perception abnormalities and the communication anomaly as
the communication failure. It is exactly the problem this article
accounting for.
Fig. 2gives an example of the vehicleplatoon system suffering
from the communication failure. In detail, vehicle 1 as well
as vehicles N−2toNlose their data availability since they
are affected by communication failure. Basically, it makes the
normal leader–predecessor following topology to transform to
some topology switching between graphs I and II, which are,
respectively, presented in Fig. 2(c) and (d).
3) Communication Topology Model: In practice, the commu-
nication failure may happen stochastically that both the time
duration and the occurring time of the failure are randomly
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4IEEE/ASME TRANSACTIONS ON MECHATRONICS
Fig. 2. Illustration of switching topologies under random communi-
cation failure. A limited number of following vehicles [vehicles 1 and
N−1in (c), and vehicles N−2and Nin (d)] lose their connection
to the predecessor and the leading vehicle because of the communi-
cation failure. (a) PF (predecessor following) basic topology. (b) LPF
(leader–predecessor following) basic topology. (c) Switching topology
I under randomly communication failure. (d) Switching topology II under
randomly communication failure.
chosen. This fact results in the randomly switching of the
communication graphs. We establish a Markovian modulated
model to describe the topology changes, which means that the
topology switching process caused by communication failure
is governed by a Markovian stochastic process satisfying the
following assumption.
Assumption 1: The communication graph G(t)switches
stochastically among sdifferent graphs, i.e., G(t)∈
{G1,G2,...,Gs}and G(σ(t)) = Gi, where σ(t):i=
{1,2,...,s}. The switching process follows a continuous-time
Markov process, which is ergodic with a transition rate matrix.
Let πbe the distribution of σ(t). Under Assumption 1, the
distribution of the Markovian modulated switching process σ(t)
can be considered unique and invariant, i.e., π=[π1,...,π
s]T
satisfies s
j=1πj=1 and πj≥0,j =1,...,s.
Remark 1: It is worthy mentioning that under Assumption 1,
the distribution of the Markovian modulated switching process
is invariant, and thus the stability for the vehicle platoon systems
in the mean square sense implies the almost sure stability, based
on [28].
Let Pbe a probability measure on state transition. The in-
finitesimal generator of σ(t)is denoted by Qg=(qij ), which is
defined as
P{σ(t+¯τ)=j|σ(t)=i}
=qij ¯τ+o(¯τ),when σ(t)jumps from ito j
1+qii ¯τ+o(¯τ),otherwise (8)
where qij represents the transition rate of state jumping i→j
with qij ≥0ifi=jand qii =−j=iqij .o(¯τ)is given by
lim¯τ→0o(¯τ)¯τ=0. Qgworks as a transition rate matrix.
Assumption 2: The union graph of switching graphs contains
a directed spanning tree rooted at the leading vehicle.
Remark 2: The Markovian randomly switching graphs are
the best way to describe the communication topology under the
communication failure, since they can simulate the randomness
and the arbitrariness of the exception. Also the Assumptions 1
and 2 on the information connection topology are quite general,
because they have no requirement on the dwell-time of each
switching, and allow each possible graph to be disconnected.
These assumptions are weaker than most of the assumptions
adopted in the published literature. For instance, [29], [30], [31],
[32], [33]. In fact, in [29], [30], and [31], the dwell-time of
each graph is lower bounded, which may not be satisfied in
practice, since the communication failure happens in a random
way. In [32] and [33], at least a part of the possible information
connection graphs is assumed to be connected, and for the same
reason this assumption also limits the application of the control
strategy in [32] and [33].
Without loss of generality, the corresponding Laplacian ma-
trix Lcan be written as L=⎡
⎣
00T
N
Lfl Lf
⎤
⎦where the submatrix
Lf∈RN×Nis a nonsingular M-matrix. Then, inspired by the
property of the nonsingular M-matrix [34], it can be inferred
that there exists a diagonal positive-definite matrix ˆ
Rsuch that
ˆ
L=ˆ
RLf+LT
fˆ
Ris positive-definite.
C. Problem Formulation
Under the Markovian switching topologies, the error of the
distance, speed, and acceleration are required to converge to zero
in the mean square sense so as to guarantee the stability of the
platoon system. When adopting the constant spacing strategy,
the control objective of the vehicle platoon becomes
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
lim
t→∞Evi(t)−v0(t)2=0,i=1,...,N
lim
t→∞Eai(t)−a0(t)2=0
lim
t→∞E
pi−1(t)−pi(t)−d−˜
L
2=0
(9)
where dindicates the constant distance between adjacent CVs,
and ˜
Ldenotes the length of a vehicle. Denote the expected
position offset between CV iand CV jby dij =[(i−j)(d+
˜
L)00]T=[
pi−pj00
]T. Then, the control objective can
be rewritten as
lim
t→∞ E
xi−xj−d
ij
2=0,i,j=0,1,...,N. (10)
Remark 3: As mentioned above, the constant spacing strat-
egy is deployed in this study, which enables the vehicles to
achieve the platooning formation under a weaker assumption
on the communication topology than the cases of constant time
headway strategy in [35] and [36], because vehicles’ velocity is
not involved in the expected headway when using the constant
spacing strategy. Furthermore, compared with [31], the control
strategy shown belowis more suitable for applications with small
distances between CVs. It contributes greatly to the utilization
of road resources and the improvement of traffic throughput.
This article is devoted to design a fully distributed control
protocol such that the following vehicles can track the leading
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WANG et al.: FULLY DISTRIBUTED ANTIWINDUP CONTROL PROTOCOL FOR INTELLIGENT-CONNECTED ELECTRIC VEHICLES 5
vehicle under the influence of switching topologies and input
saturation, i.e., to achieve the control objective given in (10)
solely using its immediate local information.
III. DESIGN OF THE ANTIWINDUP ADAPTIVE CONTROLLER
The purpose of constructing an electric vehicle platoon is to
improve the transportation efficiency when ensuring the driving
safety and driving comfort. Influenced by the instability of
wireless network and the vulnerability of automotive sensors,
communication failure may happen at any time in any part of
the platoon, which can deteriorate the data availability of CVs.
It is a non-negligible issue to handle in the design of platoon
control strategy; otherwise it may paralyse the platoon system
and initiate traffic accidents.
Let ei=N
j=0αij (σ(t))(xi−xj−d
ij )be the platoon error
variable. Then, its dynamics can be given by
˙ei=A
N
j=0
αij (σ(t)) xi−xj−d
ij
−B
N
j=0
lij (σ(t))satθ(ui),i=1,...,N. (11)
To realize the control objective (10), the following distributed
adaptive antiwindup observer is formulated
˙
ˆei=(A+FC)ˆei−F
N
j=0
αij (σ(t)) yi−yj−Cd
ij
+B
N
j=0
αij (σ(t)) (satθ(ui)−satθ(uj))
i=1,...,N
˙
hi=(A+BK)hi−(gi+ζi)FC
⎛
⎝ˆei−
N
j=0
αij (σ(t)) [(hi−hj)+(mi−mj)]⎞
⎠
i=1,...,N,
˙mi=Ami+B(satθ(ui)−Khi),i=1,...,N (12)
where we set m0=h0=ˆe0=0. ˆeiis the local observer estimat-
ing error variable eiwith ˆei(0)=0. hiis the distributed observer
acting to achieve consensus for the local observer estimating
error variables, i.e., to make ϕi=ˆei−N
j=1αij (σ(t))[(hi−
hj)+(mi−mj)] →0ast→∞.miis the antiwindup ob-
server working as a compensator to handle the input saturation.
The abovementioned three observers together form a triple-
observer structure. ζi=ϕT
iQϕiis the quadratic extra gain to
deal with the asymmetry of the Laplacian matrix.
Remark 4: It is worthy mentioning that analyzing the sta-
bility of the vehicle platoon systems under controller (15) is
quite challenging since the Laplacian matrix Lof a directed
graph is asymmetric. In order to overcome the abovementioned
challenge, we define the quadratic extra gain ζito assist the
derivation.
The adaptive gain giis given by
˙gi=ϕT
iCTCϕi(13)
with gi(0)>0. Kis the feedback gain matrix such that A+BK
is Hurwitz, and Fis the gain matrix given by F=−Q−1CT,
where Qis a positive-definite solution of the following linear
matrix inequality (LMI)
QA +ATQ−2CTC<0.(14)
With the observer (12), we can propose the following fully
distributed protocol for each vehicle:
ui=f(mi)+Khi(15)
where f(·)is a Lipschitz nonlinear function derived from the
multilevel saturation feedback control algorithm in [37].
Before proceeding to the main results, we present the follow-
ing lemma, which is key to the development of the theoretical
results.
Lemma 1 [38]: For any given system
˙z=Az +B[satθ(f+w(t)) −w(t)] (16)
satisfying that (A, B)is detectable and stabilizable with A
containing eigenvalues of nonpositive real part, the following
statement holds if the controller f(·)is designed based on
the multilevel saturation feedback control algorithm in [37]:
z(t)asymptotically converges to zero if w(t)asymptotically
converges to zero.
Based on Lemma 1, we can prove that the platoon system
can achieve the control objective under the proposed control
protocol (15). Therefore, we can derive the following theorem.
Theorem 1: Suppose that Assumptions 1 and 2 hold and the
vehicle platoon is influenced by input saturation as well as
random communication failure. Then, under the protocol (15),
the vehicle platoon system can be stable and the adaptive gain gi
can converge to constant values, i.e., the control objective given
in (10) is reached solely using its immediate local information.
Proof: 1) Let ˜ei=ˆei−eibe the estimation error of the local
observer. Taking the expectation of ˙
˜ei, then we can get
E˙
˜ei=
s
p=1
E˙
ˆei−˙ei1{σ(t)=p}
=
s
p=1
E˙
ˆei−A
N
j=0
αij (σ(t)) xi−xj−d
ij
−B
N
j=0
αij (σ(t)) (satθ(ui)−satθ(uj)) 1{σ(t)=p}
.
(17)
With some mathematical calculations, we can get from (12)
and (17) that
E˙
˜ei=
s
p=1
E((A+FC)˜ei)1{σ(t)=p}
=E[(A+FC)˜ei].(18)
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6IEEE/ASME TRANSACTIONS ON MECHATRONICS
It can be derived directly from the LMI (14) that A+FC is
Hurwitz. Thus, E[˜ei]asymptotically converges to zero.
2) When σ(t)=p,wehave
˙ϕi=(A+FC)ˆei−F
N
j=0
αij (p)yi−yj−Cd
ij
+B
N
j=0
lij (p)satθ(uj)−
N
j=0
lij (p)(A+BK)hj
−(gj+ζj)FCϕ
j+Amj+B(satθ(uj)−Khj)
=Aˆei+FC(ˆei−ei)−A
N
j=0
lij (p)(hj+mj)
+FC
N
j=0
lij (p)(gj+ζj)ϕj
=Aϕi+FC⎛
⎝˜ei+
N
j=0
lij (p)(gj+ζj)ϕj⎞
⎠.(19)
Noting that ϕ0=0, then recast ˙ϕiinto a compact form
˙ϕ=[IN⊗A+Lf(p)(G+ζ)⊗FC]ϕ+(IN⊗FC)˜e(20)
where ϕ=[ϕT
1,...,ϕ
T
N]T,˜e=[˜eT
1,...,˜eT
N]T,G=
diag(g1,...,g
N), and ζ=diag(ζ1,...,ζ
N).
Consider the Lyapunov function candidate
V=
s
p=1
Vp(21)
with
Vp=E
N
i=0(2gi+ζi)riζi
2+ri(gi−γ)2
2
+β˜eT
iQ˜ei1{σ(t)=p}(22)
where γand βare constants to be determined. Let R=
diag(r1,...,r
N), satisfying R>0 and ˆ
L=s
p=1(RLf(p)+
LT
f(p)R)>0 with ribeing a positive constant. Denote T=
−(QA +ATQ−2CTC). Taking the derivative of Vpand in-
voking (20) yield
˙
Vp=E
N
i=0ri(gi+ζi)˙
ζi+(ζi+gi−γ)ri˙gi
−β˜eT
iT˜ei1{σ(t)=p}
≤E N
i=12ri(gi+ζi)ϕT
iQ˙ϕi+(ζi+gi−γ)riϕT
i
CTCϕi−β˜eT
iT˜ei1{σ(t)=p}
=E2ϕT[(G+ζ)R⊗Q]˙ϕ+ϕT(G+ζ−γIN)
R⊗CTC−β˜eT[IN⊗T]˜eϕ1{σ(t)=p}
=E
ϕT(G+ζ)R⊗QA +ATQ+CTC−((G
+ζ)ˆ
L(σ(t))(G+ζ)+γR)⊗CTCϕ−2ϕT[(G+ζ)
R⊗CTC]˜e−β˜eT[IN⊗T]˜e1{σ(t)=p}(23)
where ˆ
L(σ(t)) = RLf(σ(t)) + LfT(σ(t))R. Noting that
s
p=1
E−ϕT(G+ζ)¯
L(G+ζ)⊗CTCϕ1{σ(t)=p}
≤E−˘πλ¯
L
minϕT(G+ζ)2⊗CTCϕ(24)
where ¯
Ls
p=1ˆ
L(p)and ˘πminp=1,...,s πpwith λ¯
L
min being
the smallest eigenvalue of ¯
L.
Additionally, we have
s
p=1
E−2ϕT(G+ζ)R⊗CTC˜e1{σ(t)=p}
≤
s
p=1
EϕT˘πλ¯
L
min
2s(G+ζ)2⊗CTCϕ1{σ(t)=p}
+
s
p=1
E2sλ2
max(R)λmax CTC
˘πλ¯
L
min
˜eT˜e1{σ(t)=p}(25)
and
s
p=1
E−ϕT˘πλ¯
L
min
2s(G+ζ)2+γR
⊗CTCϕ1{σ(t)=p}
≤
s
p=1
E−ϕT3(G+ζ)R⊗CTCϕ1{σ(t)=p}(26)
where we choose γ=9sλmax(R)
2πλ¯
L
min
and β=2sλ2
max(R)λmax (CTC)
˘πλ¯
L
minλmin (T).
Substituting (24)–(26) into (23) and considering (21) give
˙
V≤
s
p=1 EϕT(G+ζ)R⊗QA +ATQ+CTC
−γR ⊗CTCϕ−˘πλ¯
L
min
sϕT(G+ζ)2⊗CTCϕ
+ϕT˘πλ¯
L
min
2s(G+ζ)2⊗CTCϕ
+2sλ2
max(R)λmax CTC
˘πλ¯
L
min
˜eT˜e−β˜eT[IN⊗T]˜e1{σ(t)=p}!
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WANG et al.: FULLY DISTRIBUTED ANTIWINDUP CONTROL PROTOCOL FOR INTELLIGENT-CONNECTED ELECTRIC VEHICLES 7
≤
s
p=1 E−ϕT[3(G+ζ)R⊗CTC]ϕ
+ϕT(G+ζ)R⊗(QA +ATQ+CTC)ϕ
+2sλ2
max(R)λmax CTC
˘πλ¯
L
min
˜eT˜e
−2sλ2
max(R)λmax CTC
˘πλ¯
L
minλmin (T)˜eT[IN⊗T]˜e1{σ(t)=p}!
≤EϕT(G+ζ)R⊗QA +ATQ−2CTCϕ
=E−ϕT[(G+ζ)R⊗T]ϕ<0.(27)
Then, we reach the conclusion that E[ϕi]asymptotically
converges to zero, and E[gi]asymptotically converges to a finite
constant.
3) It can be verified that E[ζi]asymptotically converges to zero
because of the fact that ζi=ϕT
iQϕiand limt→∞ E[ϕi]=0.
Thus, E[(gi+ζi)FCϕ
i]also goes to zero as time goes to
infinity. Therefore, we can conclude that limt→∞ E[hi]=0
since A+BK is Hurwitz. Then, by Lemma 1, it yields that
limt→∞ E[mi]=0.
4) Since ˆei=ϕi+N
j=0αij (σ(t))[(hi−hj)+(mi−
mj)], we can draw the conclusion that limt→∞ E[ˆei]=0
following from the fact that E[ϕi],E[hi], and E[mi]
asymptotically converge to zero. Then, by the fact that
E[˜ei]goes to zero as time goes to infinity, we can obtain that
limt→∞ E[ei]=0, which implies
lim
t→∞ Exi−xj−dij 2=0.(28)
Therefore, the control objective given in (10) is achieved, which
completes the proof.
Remark 5: There are two main difficulties in designing a
fully distributed protocol. First, how to take care of the coupling
between the randomly switching communication topologies and
the input saturation. To address this issue, the variable ϕiis
introduced into the proposed controller. Second, how to build the
proposed controller that does not require any global information.
To achieve this goal, the adaptive gain giis defined to estimate
the algebraic connectivity of communication graphs, and then
the fully distributed control protocol is constructed.
Remark 6: The fully distributed control protocol without any
global information is beneficial for the platoon system to defend
against malicious cyber-attacks. In fact, the platoon systems
are vulnerable to cyber-attacks due to the openness of com-
munication networks. When the control protocol of the platoon
system contains global information, the control performance can
be seriously deteriorated because of the impact of the attack.
For example, false data injection (FDI) attacks [39] destroy the
data integrity by manipulating the measurement data and control
commands. Any FDI attack on a single vehicle can propagate and
amplify along the topology of the vehicle platoon, if the global
information is unfortunately fabricated by the attacker. In con-
trast, when the control protocol contains only local information,
even if the FDI attack fabricates the information of a vehicle in
Algorithm 1:
For given constants d>0, ˜
L>0, ¯
θ>0 and θ < 0, setting
m0=h0=ˆe0=0, and gi(0)>0, for i=0,1,..., N,
the controller (15) solving the vehicle platooning problem
can be constructed in the following steps:
1) Solve the inequality (14) to get a positive-definite
solution Q, and let the control gain be F=−Q−1CT
such that A+FC is Hurwitz;
2) Choose the feedback gain matrix Kso that A+BK
is Hurwitz;
3) Design the controller ui=f(mi)+Khiwith f(mi)
chosen by utilizing the Multilevel Saturation Feedback
Control Algorithm.
the platoon, the fabricated information will not be spread to the
entire platoon system, thereby enhancing the system’s ability to
defend against malicious cyber-attacks.
Remark 7: The proposed protocol (15) can also be applied to
the case without actuator saturation by simply making mi(t)≡
0. The results of stability can be confirmed through the similar
theoretical proof steps in Theorem 1.
In the following, we present an algorithm to construct the
controller (15).
Remark 8: In this article, a fully distributed antiwindup
adaptive controller is developed for the vehicle platoon system
under random communication failure. The controller realizes
the stability and security of the platoon system under abnor-
mal communication conditions with control signal being in a
reasonable range. Furthermore, this controller derives control
signals based on only local output information. This reduces the
information processing intensity and the communication burden,
while improving the effective utilization of the information and
enhancing the confidentiality of the information. Meanwhile, the
matrix parameters of the controller can be obtained by solving
an LMI, which only relates to the dynamics of each vehicle.
Compared with the existing works [22], [40], [41], the proposed
controller has more advantages in application.
IV. SIMULATION RESULTS
In this section, we intend to give some simulation examples
to validate the feasibility of the proposed control protocol.
A. Simulation Setting
Consider a vehicle platoon consisting of 10 following vehicles
whose dynamics are described by (5). Meanwhile, the dynamics
of the leading vehicle is modeled as
˙x0=Ax0+Bu0(29)
where u0=⎧
⎨
⎩
0.02t, 0≤t≤5s,
0,5s<t≤30s,Aand Bcan be ob-
tained according to the time constant τ0given in Table I. Assume
that the state information is available to each vehicle. We also
assume that vehicles 1,3,4, and 7 are influenced by the random
communication failure, which gives rise to the topology changes
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8IEEE/ASME TRANSACTIONS ON MECHATRONICS
TABLE I
PARAMETER VALUES IN SIMULATION
Fig. 3. Communication topologies: (a) G1.(b)G2. (c) Union graph of G1
and G2.
Fig. 4. Diagram of the topology switching process.
shown in Fig. 3.Fig.4presents the diagram of the topology
switching process used in the simulation. The switching process
is governed by a continuous-time Markov process satisfying
Assumption 1. Note that the union graph of the possible com-
munication graphs contains a directed spanning tree rooted at
the leading vehicle, which makes Assumption 2 hold. Table I
shows some detailed parameters of the simulation setting.
B. Platoon Formation and Maintenance
We design the proposed controller under Algorithm 1. First,
by solving the inequality (14), we get a positive-definite solution
Q=⎡
⎢
⎢
⎣
3.7337 −4.3698 −0.1072
−4.3698 8.2955 1.3179
−0.1072 1.3179 4.2759
⎤
⎥
⎥
⎦
,and let the control
gain be F=−Q−1CT=⎡
⎢
⎢
⎣
−0.7424 −0.4081 0.1072
−0.4081 −0.3511 0.0980
0.1072 0.0980 −0.2614
⎤
⎥
⎥
⎦
.
Second, choose the feedback gain matrix K=
[−0.2000 −0.5667 0.4000]so that A+BK is Hurwitz.
Third, we design the controller (15) by utilizing the multilevel
saturation feedback control algorithm and obtain
f(mi)=satθf1(mi)+0−2.5−4.0mi(30)
where f1(mi)=satθ([−0.9175 −1.0360 −0.1990]mi).Be-
sides, the initial values of the observers hiand mifor following
vehicles are chosen randomly. The initial value of the adaptive
gain is gi(0)=0.5,i=1,...,10.
Fig. 5shows the sample trajectories of the distance error,
the speed error, velocity, and acceleration of following vehi-
cles under the proposed antiwindup controller (15). It can be
observed from Fig. 5that the distance error and the speed
Fig. 5. Simulation results under the antiwindup controller (15).
Fig. 6. Adaptive gain under the antiwindup controller (15).
error asymptotically tend to zero, proving that the stability of
platoon system is realized under the antiwindup controller. This
implies that the following CVs can adjust their interdistance
to the expected value, meanwhile, tracking the reference speed
of the leading CV accurately, though under the influence of
the randomly switching topologies and control input saturation
constraints.
The saturated control input profile in Fig. 5shows that the
value of control input is limited within the preset range, which
implies that the input saturation constraint is satisfied for the
platoon system. Besides, the trajectory of adaptive gains is
depicted in Fig. 6, indicating that the adaptive gain can converge
to some finite value in the end.
C. Performance Comparison
This section is to compare the proposed antiwindup controller
with the existing algorithms given in [42] and [43], thus illus-
trating the effectiveness of the proposed controller.
1) Example 1: Fan et al. [42] developed a distributed
observer-based protocol to solve the consensus problem of
multiagent system under the input saturation and switching
topologies. The switching of the communication graph is de-
terministic and the duration between graph changes is lower
bounded, which are stronger than the assumptions adopted in
this manuscript. For comparison, this example is to employ the
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WANG et al.: FULLY DISTRIBUTED ANTIWINDUP CONTROL PROTOCOL FOR INTELLIGENT-CONNECTED ELECTRIC VEHICLES 9
Fig. 7. Simulation results under the controller
˜
Ain [42].
distributed observer-based protocol in [42] (named as controller
˜
A) solving the vehicle platooning problem as in Section IV-B,
and thus, reflect the superior performance of the proposed con-
troller in handling random communication failure.
Adopting the same topology switching scenario, we can ob-
tain the controller ˜
Awith input saturation as follows:
˜ui(t)= −B˜
PN
j=1
αij (t)(zi−zj)
+αi0(zi−z0),i=1,...,10 (31)
where the observers ziand z0are defined as
˙zi=Azi+˜
F
N
j=1
αij (t)(C(zi−zj)−(yi−yj))
−BB˜
P
N
j=1
αij (zi−zj)+αi0(zi−z0)
˙z0=Az0−H(y0−Cz0)(32)
where ˜
F=⎡
⎢
⎢
⎣
−2.4608 −1.31610.1248
−1.3161 −1.12720.1141
0.12480.1141 −0.3046
⎤
⎥
⎥
⎦
,˜
P=
⎡
⎢
⎢
⎣
0.8215 3.3240 0.6325
3.3240 26.6731 5.1954
0.6325 5.1954 1.0299
⎤
⎥
⎥
⎦
, and H=
⎡
⎢
⎢
⎣
−2.4608 −1.3161 0.1249
−1.3161 −1.1273 0.1142
0.1249 0.1142 −0.3046
⎤
⎥
⎥
⎦
.
Fig. 7draws the trajectories of the distance error and speed
error of following vehicles under controller ˜
A. It follows from
the distance error curves that the interdistance between the
following CVs gradually increases, implying that the controller
˜
Acannot achieve the stability of the vehicle platoon system
under random communication failure.
2) Example 2: To further show the capability of the proposed
fully distributed antiwindup controller, an additional numerical
simulation is carried out to compare it with another antiwindup
controller (assumed as controller ˜
B) in [43]. Lv et al. [43]
proposed a distributed adaptive observer-based antiwindup con-
troller to achieve the semiglobal output consensus for multiagent
Fig. 8. Simulation results under the controller
˜
Bin [43].
systems. The controller ˜
Bis offered as
ui, ˜
B=f
iˆ
ξ
i+Γvi−K(ˆ
ξ
i−ξ
i)(33)
where K=[−0.1000 −0.5300 −0.4000];Γ=[
000.2];
f
i(ˆ
ξ
i)=sat(sat[−0.9500 −1.0360 −0.2000]∗ˆ
ξ
i+
[0−2.3000 −4.1000]∗ˆ
ξ
i)is constructed by [43, Algorithm
1]; the observers vi,ˆ
ξiand the adaptive gain ϑiare given as
˙vi=Avi−˙
ϑi+ϑiN
j=0
αij (vi−vj)(34)
˙
ϑi=⎡
⎣
N
j=0
αij (vi−vj)⎤
⎦
T
j=0
αij (vi−vj)(35)
˙
ˆ
ξi=Aˆ
ξ
i+Bsatθui, ˜
B−Γvi+Kˆ
ξ
i−ξ
i (36)
the error ξ
iis defined as ξ
i=ˆx
i−Πvi, where
˙
ˆxi=Aˆxi+Bsatθui, ˜
B+M(Cˆxi−yi)(37)
with Π=
⎡
⎢
⎢
⎣
100
010
001
⎤
⎥
⎥
⎦
and M=
⎡
⎢
⎢
⎣
−0.8203 −0.4387 −0.0416
−0.4387 −0.3758 0.0381
0.4160 0.0381 −0.1025
⎤
⎥
⎥
⎦
.
Fig. 8presents the trajectories of the distance error and po-
sition of the following vehicles under the antiwindup controller
˜
B. Although the distance error between the following vehicles
and the leading vehicle gradually converges to zero as the time
prolonging, the ninth and tenth following vehicles collide with
each other at about time t=50 s, shown in the position profile.
Therefore, the antiwindup controller ˜
Bfails to address the
vehicle platoon problem under random communication failure.
D. Cosimulation for Platoon Control
In this section, a cosimulation platform for CVs is designed.
The structure of the cosimulation platform is shown in Fig. 9.
The cosimulation platform is mainly built on the basis of two
software: Prescan and MATLAB. Among them, Prescan is used
to establish the traffic scenarios of vehicle platoon and provides
a realistic vehicle dynamics model; MATLAB is used to build
the platoon control algorithm and establish the information
acquisition constraint mechanism for random communication
failure.
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10 IEEE/ASME TRANSACTIONS ON MECHATRONICS
Fig. 9. Structure of cosimulation platform.
Fig. 10. Communication topologies: (d) G3,(e)G4, (f) union graph of
G3and G4.
Fig. 11. Diagram of the topology switching process.
The vehicle system includes six following vehicles and a
leading vehicle. The leading vehicle runs dynamically with its
acceleration given as a0∈[−1,1.5]m/s2.
During the entire operation of the vehicle platoon, we assume
that the following vehicles 1, 3, and 6 are affected by random
communication failure, which means that the above vehicles are
hindered in the process of obtaining measurement data. Then,
the possible communication topologies of the platoon system
are shown in Fig. 10. In addition, the diagram of the topology
switching process is depicted in Fig. 11. Prescan further illus-
trates the real performance of the vehicle platoon simulation, as
shown in Fig. 12. It can be seen from Fig. 12 that there is no
rear-end collision accident in the vehicle platoon, which reflects
the safety of the system.
The curves depicted in Fig. 13 show the behavior of the
vehicles during the simulation. At t=20 s, the vehicle platoon
has basically formed. At t=42 s, the leading vehicle begins
to decelerate, and then the distance error of the following ve-
hicles complete the convergence within a reasonable interval at
Fig. 12. Simulation results of Prescan, where the red car is the leading
vehicle, and the following vehicles are in white. (a) Initial position of the
platoon, where the vehicles all have different initial values of position,
velocity, and acceleration. (b) Following vehicle accelerates with the aim
of following up with the leading vehicle and forming a vehicle platoon.
(c) Vehicle platoon has been formed. (d) Leading vehicle begins to de-
celerate, and the entire platoon begins to decelerate. (e) Entire platoon
completes deceleration and then stops at the target position.
Fig. 13. Simulation results in Prescan under the proposed antiwindup
controller (15).
t=65 s. Therefore, the proposed fully distributed antiwindup
controller can guarantee the stability of the vehicle platoon
system under random communication failure in the simulation
environment of Prescan.
V. D ISCUSSION
This article studies the vehicle platoon secure control problem
of intelligent transportation system. Due to the sensor fault and
the instability of the communication network, the vehicles in
the platoon system may not be able to obtain the information
of the predecessor and the leading vehicle, i.e., the random
communication failure modeled by the Markovian topology
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WANG et al.: FULLY DISTRIBUTED ANTIWINDUP CONTROL PROTOCOL FOR INTELLIGENT-CONNECTED ELECTRIC VEHICLES 11
switching process occurs. To this end, we proposed the secure
control protocol to maintain the vehicle platoon under abnormal
conditions while considering actuator saturation. The controller
is fully distributed, which reduces the communication burden
and the information processing complexity, thereby improving
the efficiency of the platoon formation. And it can be seen from
the simulation results that the proposed controller ensures the
safety and stability of the platoon when the vehicle platoon
suffers from random communication failure. Thus, we achieved
the objective of this manuscript, i.e., design a fully distributed
control protocol so that the following vehicles can track the
leading vehicle under the influence of switching topologies and
input saturation.
It should be pointed out that this article still has the following
limitations. First, the design complexity of the proposed con-
troller is high, and we will further explore the possibility of
reducing the design complexity of the controller in the future.
Second, the proposed control protocol becomes extremely com-
plex when applied to heterogeneous vehicle platoon systems. It
is a challenging topic to reduce the dependence of the algorithm
and the observer on the dynamics. Our future work will focus
on developing a fully distributed control protocol to tackle
the secure control problem in the presence of the malicious
cyber-attacks and input saturation.
VI. CONCLUSION
In this article, an adaptive antiwindup control protocol for the
vehicle platoon system under random communication failure
is proposed. The protocol is fully distributed with only local
output information being utilized. By exploiting the multilevel
saturation feedback algorithm, the input saturation is well han-
dled. Besides, we adopted the Markovian randomly switching
graphs to build the information connection model in a random
communication failure scenario. A triple-observer structure is
established where the adaptive gain is designed to estimate
the algebraic connectivity of communication graphs. Then, we
presented our controller and used the Lyapunov stability theorem
to obtain the stability conditions of the vehicle platoon system
under input saturation and random communication failure.
The simulation section first shows that the proposed controller
is able to form and maintain the vehicle platoon. Subsequently,
with the same driving constraints, we arranged two simulation
comparisons to demonstrate that the proposed controller is
superior in terms of efficiency, feasibility, riding safety, and
riding comfort. Finally, we used the cosimulation platform of
Prescan and MATLAB to show the performance of the proposed
controller in real scenarios, thereby verifying the feasibility of
the controller in practical application.
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Jingyao Wang (Member, IEEE) received the
Ph.D. degree in engineering from Peking Uni-
versity, Beijing, China, in 2017.
Since July 2017, she has been with the
School of Aerospace Engineering, Xiamen Uni-
versity, Xiamen, China, where she is currently
an Associate Professor. Her current research
interests include intelligent transportation sys-
tems, multiagent systems, and network re-
source management.
Xingming Deng received the B.S. degree in en-
gineering from the Department of Automation,
School of Engineering, South China Agricultural
University, Guangzhou, China, in 2020. He is
currently working toward the M.S. degree in en-
gineering with the Department of Automation,
School of Aerospace Engineering, Xiamen Uni-
versity, Xiamen, China.
His research interests include the secure con-
trol of connected automated vehicles and intelli-
gent transportation systems.
Jinghua Guo received the Ph.D. degree in en-
gineering from the Dalian University of Technol-
ogy, Dalian, China, in 2012.
From 2012 to 2015, he finished his Postdoc-
toral Research with Tsinghua University, Beijing,
China. He is currently an Associate Professor
with Xiamen University, Xiamen, China. He has
authored more than 30 journal papers. He has
engaged in more than five sponsored projects.
His current research interests include intelligent
vehicles, vision system, control theory, and
applications.
Yugong Luo received the B.S. and M.S.
degrees in mechanical engineering from
Chongqing University, Chongqing, China, and
the Ph.D. degree in mechanical engineering
from Tsinghua University, Beijing, China, in
1996, 1999, and 2003, respectively.
He is currently a Professor with the School
of Vehicle and Mobility, Tsinghua University. He
was a Visiting Scientist with the Department of
Mechanical Engineering, University of Michigan
Ann Arbor, Ann Arbor, MI, USA, in 2013 and
2014. He is a coauthor of four books. His research interests include
intelligent vehicles, hybrid electric vehicles, and multiobjective control.
Keqiang Li received the B.S. degree in me-
chanical engineering from Tsinghua Univer-
sity, Beijing, China, in 1985, and the M.E.
and Ph.D. degrees in mechanical engineering
from Chongqing University, Chongqing, China,
in 1988 and 1995, respectively.
He is currently a Professor with the School
of Vehicle and Mobility, Tsinghua University. In
addition, he is also the Chief Scientist of Intelli-
gent and Connected Vehicle Innovation Center
of China, and the Director of State Key Labo-
ratory of Automotive Safety and Energy of China. His current research
interests include intelligent connected vehicles, could-based control for
vehicles, and vehicle dynamics systems.
Prof. Li is currently an Academician of the Chinese Academy of
Engineering.
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