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Adaptive Control of Interconnected Networked Systems with
Application to Heterogeneous Platooning*
Youssef Abou Harfouch, Shuai Yuan, and Simone Baldi1
Abstract— Grouping individual vehicles into platoons with
a defined inter-vehicle spacing policy has been proven to
greatly improve road throughput and reduce vehicles’ en-
ergy consumption. The emerging interest in distributed inter-
vehicle communication networks has provided new tools for
further improvements of the performance of this platoon-based
driving pattern. A leading control strategy of such vehicular
cyber-physical systems is Cooperative Adaptive Cruise Control
(CACC). However, a crucial limitation of the state-of-the-art
is that string stability can be proven only when the vehicles
in the platoon have identical driveline dynamics (homogeneous
platoons). In this paper, we present a novel CACC strategy
that overcomes the homogeneity assumption and that is able to
adapt its action and achieve string stability even with uncertain
heterogeneous platoons. Considering a one-vehicle look-ahead
topology, we propose a Model Reference Adaptive Control
augmentation: the control objective is to augment a baseline
CACC, proven to be string stable in the homogeneous scenario,
with an adaptive control term that compensates for each
vehicles unknown driveline dynamics. Asymptotic convergence
of the heterogeneous platoon to a string stable platoon is shown
analytically for an appropriately designed reference model.
Simulations of the proposed CACC strategy are conducted to
validate the theoretical analysis.
I. INTRODUCTION
Smart traffic is an active area of research striving to
increase road safety, manage traffic congestion, and reduce
vehicles’ emission. Besides traffic light control [1], auto-
mated driving has proved to be a recognized solution for
potentially improving road throughput by grouping vehicles
into platoons controlled by one leading vehicle [2]. CACC
is an extension of Adaptive Cruise Control (ACC) [3] where
platooning is enabled by inter-vehicle communication in
addition to on-board sensors. CACC has been studied to
improve the string stability of this type of vehicular cyber-
physical systems [4]. The notion of string stability implies
that disturbances in the form of sudden velocity changes of
the leading vehicle are attenuated as they propagate upstream
throughout the platoon.
Studies have been conducted to develop CACC strategies
that guarantee the string stability of vehicle platoons. Under
the assumption of vehicle-independent driveline dynamics
(homogeneous platoon), a one-vehicle only look-ahead coop-
erative adaptive cruise controller was synthesized, by using
a performance oriented approach to define string stability, in
1The authors are with the Delft Center for Systems and
Control, Delft University of Technology, The Netherlands (e-mail:
youssef.harfoush1@gmail.com, s.yuan-1@tudelft.nl, s.baldi@tudelft.nl).
The research leading to these results has been partially funded by the
European Commission FP7-ICT-2013.3.4, Advanced computing, embedded
and control systems, under contract #611538 (LOCAL4GLOBAL).
order to stabilize the platoon [5]. Moreover, a longitudinal
controller based on a constant spacing policy (velocity-
independent) was developed and showed that string stabil-
ity can be achieved by broadcasting the leading vehicle’s
acceleration and velocity to all vehicles in the platoon [6].
Furthermore, a linear controller was augmented by a model
predictive control strategy to maintain the platoon’s stability
while integrating safety and physical constraints [7].
A review on the practical challenges of CACC highlights
the importance of robust wireless communication that can
account for highly dynamical environments [8]. In order to
account for network delays and packet loss caused by the
wireless network, an H∞controller was synthesized, after
feedback linearisation of the non-linear model, by integrating
parameter uncertainties and was shown to satisfy the string
stability criteria and robustness both in theory and in sim-
ulations [9]. In addition, the effects of packet loss ratios,
beacon sending frequencies and time headway on string
stability were studied in [10]. A controller that integrates
inter-vehicle communication over different realistic network
conditions which models time delays, packet losses, and
interferences was derived in [11]. Moreover, the extension of
wireless communication between all vehicles in the platoon
was studied in [12]. A controller was derived to integrate all
the information received from neighboring vehicles and to
cope with the network induced uncertainties.
All the aforementioned works rely on platoon’s homogene-
ity assumption: however, in practice, having a homogeneous
platoon is not feasible. A study conducted in [13] assessed
the causes of heterogeneity of vehicles in a platoon and
their effects on string stability. In fact, a longitudinal CACC
tracking controller of a heterogeneous platoon (i.e. different
driveline dynamics) was designed by assuming that the host
vehicle can communicate over a wireless communication
network with its preceding vehicle as well as with the platoon
leader [14]. Furthermore, external disturbances (e.g. vehi-
cle accelerations, wind gust, network induced uncertainties,
parametric uncertainties) are always present in practice: a
distributed adaptive sliding mode controller for a hetero-
geneous vehicle platoon was derived that guarantees string
stability and adaptive compensation of disturbances based on
constant spacing policy [15].
The brief overview of the state-of-the-art reveals the
need to develop CACC with new functionalities, that can
handle platoons of heterogeneous vehicles, while adapting
to changing conditions. The main contribution of this paper
is to address CACC for heterogeneous platoons with unre-
liable communication. The heterogeneity of the platoon is
Fig. 1. CACC-equiped heterogeneous vehicle platoon [5]
represented by different (and uncertain) time constants for
the driveline dynamics. Using a Model Reference Adaptive
Control (MRAC) augmentation method, we prove analyti-
cally the asymptotic convergence of the vehicles output to an
appropriately defined string stable reference models output.
Furthermore, we discuss a method to model inter-vehicle
communication losses and a potential switched control strat-
egy to overcome this loss of information while guaranteeing
bounded stability.
The paper is organized as follows. In Section II, the
system structure of a heterogeneous CACC-equiped vehi-
cle platoon is presented. Moreover, the proposed MRAC
augmentation to stabilize the platoon in the heterogeneous
scenario is studied in Section III. Section IV introduces
inter-vehicle communication losses and discusses a switched
control method to cope with them. Simulation results for the
MRAC augmentation controller are presented in Section V
along with some concluding remarks in Section VI.
II. CACC SYS TEM STRUCTURE
Consider a heterogeneous platoon with Mvehicles. Fig.
1 shows the platoon where viand direpresent the velocity
(m/s) of vehicle i, and the distance (m) between vehicle i
and its preceding vehicle i−1, respectively. Furthermore,
each vehicle in the platoon can only communicate with its
preceding vehicle via wireless communication. The main
goal of every vehicle in the platoon, except the leading
vehicle, is to maintain a desired distance dr,ibetween itself
and its preceding vehicle.
A constant time headway (CTH) spacing policy will be
adopted to regulate the spacing between the vehicles. The
CTH is implemented by defining the desired distance as:
dr,i(t) = ri+hvi(t),i∈SM
where riis the standstill distance (m), hthe time head-away
(s), and SM={i∈N|1≤i≤M}with i=0 reserved for
the platoon’s leader (leading vehicle). A CTH indicates that
the vehicle’s desired velocity must be proportional to the
inter-vehicle spacing which imitates the driving intuition of
slowing down as the inter-vehicle spacings decrease. It has
been shown that such spacing policy improves string stability
[16] and safety [17].
It is now possible to define the spacing error (m) of the
ith vehicle as:
ei(t) = di(t)−dr,i(t)
= (qi−1(t)−qi(t)−Li)−(ri+hvi(t)) (1)
with qiand Lirepresenting vehicle i’s rear-bumper position
(m) and length (m), respectively.
The control objective is to regulate eito zero for all i∈
SM, while ensuring the string stability of the platoon. The
following model, derived by Ploeg et al. (2011), is used to
represent the vehicles in the platoon
˙
di
˙vi
˙ai
=
vi−1−vi
ai
−1
τiai+1
τiui
,i∈SM(2)
where aiand uiare respectively the acceleration (m/s2) and
external input (m/s2) of the ith vehicle. Moreover, τi(s)
represents each vehicle’s driveline time constant. Substituting
(1) in (2) we obtain the linear time-invariant state space
system
˙ei
˙vi
˙ai
=
0−1−h
0 0 1
0 0 −1
τi
ei
vi
ai
+
1
0
0
vi−1+
0
0
1
τi
ui.
(3)
Furthermore, the leading vehicle’s model is defined as
˙e0
˙v0
˙a0
=
0 0 0
0 0 1
0 0 −1
τ0
e0
v0
a0
+
0
0
1
τ0
u0.(4)
Note that, under the assumption of a homogeneous platoon,
we have τi=τ0,∀i∈SM. In this work, we remove the
homogeneous assumption by considering that ∀i∈SM,τican
be represented as the sum of two terms
τi=τ0+∆τi(5)
where τ0is a known constant representing the driveline
dynamics of the leading vehicle and ∆τiis an unknown
constant deviation of vehicle i’s driveline dynamics from τ0.
In fact, ∆τiacts as an unknown parametric uncertainty.
Consequently, the model of a vehicle in a heterogeneous
platoon is obtained by using (5) in the third equation of (3)
τi˙ai=−ai+ui
˙ai=−1
τ0
ai+1
τ0hui+Ω∗
iφii,(6)
where Ω∗
i=−∆τi
τiis an unknown ideal constant scalar
parameter, and φi= (ui−ai)is the known scalar regressor.
Using (6) in (3), we can define the vehicle model as the
uncertain LTI of the following form
˙ei
˙vi
˙ai
=
0−1−h
0 0 1
0 0 −1
τ0
ei
vi
ai
+
1
0
0
vi−1
+
0
0
1
τ0
hui+Ω∗
iφii,∀i∈SM.
(7)
We can now formulate the control objective for the het-
erogeneous platoon as follows:
Problem 1: Design an adaptive control input ui(t),∀i∈SM,
such that the heterogeneous platoon described by (4) and (7)
asymptotically tracks the behavior of a string stable platoon
for any possible vehicles parametric uncertainty.
III. CON TROL STRUCTURE
In order to design the control input, Section III-A presents
string stable reference dynamics for the vehicles in the
platoon, and Section III-B defines a stabilizing ui(t)through
a MRAC augmentation approach.
A. CACC Reference Model
Under the baseline conditions of identical vehicles (Ω∗
i=
τ0,∀i∈SM) and perfect inter-vehicle communication, [5] de-
rived, using a CACC strategy a control input that guarantees
the string stability of the platoon and a zero inter-vehicle
spacing error.
The authors defined a new input ξi,bl such that
h˙ui,bl =−ui,bl +ξi,bl ,∀i∈ {0} ∪ SM
ξi,bl =(Kpei+Kd˙ei+ui−1,bl ,∀i∈SM
uri=0
(8)
where uris the platoon input representing the desired ac-
celeration of the leading vehicle. The cooperative aspect
of (8) resides in ui−1which is received over the wireless
communication between vehicle iand i−1.
Therefore, we can now define string stable reference
dynamics for the heterogeneous platoon (4) and (7) as the
platoon with Ω∗
i=0 and ui,m=ui,bl ,∀i∈SM. Substituting
(8) in (7) and extending the state vector with ui,bl we obtain
the following reference model dynamics
˙ei,m
˙vi,m
˙ai,m
˙ui,m
=
0−1−h0
0 0 1 0
0 0 −1
τ0
1
τ0
Kp
h−Kd
h−Kd−1
h
| {z }
Am
ei,m
vi,m
ai,m
ui,m
| {z }
xi,m
+
1 0
0 0
0 0
Kd
h
1
h
| {z }
Bw
vi−1
ui−1,bl !
| {z }
wi
,∀i∈SM
(9)
where xi,mand wiare vehicle is reference state vector and
exogenous input vector, respectively. Consequently, (9) is of
the following form
˙xi,m=Amxi,m+Bwwi,∀i∈SM.(10)
Furthermore, the leading vehicle model becomes
˙e0
˙v0
˙a0
˙u0
=
0 0 0 0
0 0 1 0
0 0 −1
τ0
1
τ0
0 0 0 −1
h
| {z }
Ar
e0
v0
a0
u0
| {z }
x0
+
0
0
0
1
h
|{z}
Br
ur.(11)
[5] showed that controller (8) asymptotically stabi-
lizes system (9) around the equilibrium point xi,m,eq =
0 ¯v00 0T
for x0=xi,m,eq and ur=0, where ¯v0is a
constant velocity, provided that the following Routh-Hurwitz
conditions are satisfied
h>0,Kp,Kd>0,Kd>τ0Kp.(12)
Furthermore, the authors proved the string stability of the
platoon provided that a set of linear matrix inequalities is
satisfied. Therefore, it can be concluded that reference model
(10) provides string stable reference dynamics characterized
by an asymptotically stable equilibrium point of a zero inter-
vehicle spacing error for the vehicles in a heterogeneous
platoon.
B. MRAC augmentation of a baseline controller
In this Section, reference model (10) will be used to
design the control input ui(t)such that the uncertain platoons
dynamics described by (4) and (7) converge to string stable
dynamics. With this scope in mind, we will augment con-
troller (8) with an adaptive term, using a similar architecture
as proposed in [18].
To include the adaptive augmentation, the input uiof each
vehicle is split into two different inputs:
ui=ui,bl +ui,ad (13)
where ui,bl is the baseline controller defined in (8) and ui,ad
the adaptive augmentation controller (to be constructed).
Replacing (13) into (4) and (7), and augmenting the state
vector with ui,bl results in
˙ei
˙vi
˙ai
˙ui,bl
=
0−1−h0
0 0 1 0
0 0 −1
τ0
1
τ0
Kp
h−Kd
h−Kd−1
h
ei
vi
ai
ui,bl
| {z }
xi
+
1 0
0 0
0 0
Kd
h
1
h
vi−1
ui−1,bl !+
0
0
1
τ0
0
| {z }
Bu
hui,ad +Ω∗
iφii,∀i∈SM
which can be written in the following form:
˙xi=Amxi+Bwwi+Bu[ui,ad +Ω∗
iφi].(14)
Note that the leading vehicle’s model is still as in (11). The
adaptive augmentation controller will then be designed in
such a way to estimate and compensate for the unknown
term Ω∗
iφi. Define the adaptive augmentation control input
as
ui,ad =−ˆ
Ωiφi(15)
where ˆ
Ωiis the estimate of Ω∗
i. Replacing (15) in (14) gives
˙xi=Amxi+Bwwi−Bu(ˆ
Ωi−Ω∗
i
| {z }
∆Ωi
)Tφi(16)
where ∆Ωiis the parameter estimation’s error vector. Define
the state tracking error ˜xias:
˜xi=xi−xi,m.(17)
Using this definition together with (10) and (16), the tracking
error dynamics can be derived as follows:
˙
˜xi=˙xi−˙xi,m
=Am(xi−xi,m)−Bu∆ΩT
i
=Am˜xi−Bu∆Ωiφi
(18)
from which the following stability result can be stated.
Theorem 1: Consider the uncertain system dynamics in
(14), and the reference model dynamics in (10) with bounded
external reference input wi. Then for any positive constant
ΓΩthe adaptive input, ∀i∈SM,
ui,ad =−ˆ
Ωiφi(19)
˙
ˆ
Ωi=ΓΩφi˜xiP
mBu(20)
regulates the tracking error (17) asymptotically to zero,
i.e. limt→∞xi(t) = xi,m(t),∀i∈SM. In (20) P
mrepresents the
unique symmetric positive-definite solution of the algebraic
Lyapunov equation
AT
mP
m+P
mAm=−Qm(21)
with Qm=QT
m>0 a design matrix. Moreover, controller
(19)-(20) regulates the estimation error asymptotically to
zero, i.e. limt→∞k∆Ωiφik=0,∀i∈SM.
Proof: See Appendix A.
Remark 1.We have shown that the augmented control
input
ui=ui,bl +ui,ad
=ui,bl −ˆ
Ωiφi
(22)
guarantees the global asymptotic convergence of the vehicle’s
states to the reference ones ∀i∈SM. Since the reference
model was chosen as the string stable controlled vehicle
model in the homogeneous scenario (10), then the hetero-
geneous platoon converges asymptotically to a string stable
platoon, and ei=0, ∀i∈SMasymptotically. Therefore, it
can be concluded that controller (22) satisfies both control
objectives.
IV. PLATOONING UNDER NON-IDEAL INTER-VEHICLE
COMMUNICATION
In practice, the two most widely adopted vehicular wire-
less communication standards are the IEEE 802.11p/wireless
access in vehicular environment (WAVE) and the long-
term evolution (LTE). Both standards, as described by [19],
suffer from erroneous and lost packets which motivates the
extension of the previously designed MRAC augmentation
controller to cope with this loss of information.
In fact, when communication is lost between at least two
vehicles in the platoon, the reference dynamics (10) fail to
guarantee the string stability of the platoon. This is due to
the fact that uC
i−1,bl is no longer present for measurement for
at least one i∈SMfor a certain dwell time.
Consequently, to overcome this, a new string stable refer-
ence model for the heterogeneous platoon (7) can be derived
based on an ACC strategy, and a switched adaptive control
input can be designed to switch from an augmented CACC
strategy to an augmented ACC strategy when communication
is lost, based on a dwell time approach. To design a string
stable ACC reference model, [20] presents a systematic
way of doing so by deriving conditions on the controller’s
parameters. Moreover, to design such a switched adaptive
control strategy one can refer to the methodologies developed
by [21] and [22] on adaptive control of switched linear
systems, and to the work done by [23] for supervisory
switching logics.
Due to the lack of space, this topic will be addressed in
details in an extended version of this work.
V. IL LUS TRATI VE EXA MPL E
In order to validate the theoretical analysis, this section
presents the simulation results of a CACC-equiped hetero-
geneous vehicle platoon under the MRAC augmentation
controller.
To show the technical feasibility of the MRAC aug-
mentation controller, a heterogeneous vehicle platoon was
simulated in Simulink/MATLAB. A platoon of 5+1 vehicles
is considered with the first one being the platoon leader,
vehicle 0. Table I presents the platoon’s characteristics as
well as the chosen time headway constant.
TABLE I
PLATO ON PAR AM ET ER S,M=5, h=0.7S
i0 1 2 3 4 5
τi(s)0.1 0.5 0.4 0.2 0.5 0.25
Table II shows the true values of the constant parametric
uncertainties Ω∗
i,∀i∈SM, which are unknown to the de-
signer.
TABLE II
IDEAL UNCERTAINTIES
i1 2 3 4 5
Ω∗
i-0.8 -0.75 -0.5 -0.8 -0.6
The reference model (10) for the adaptive laws is char-
acterised by τ0=0.1, h=0.7, and {Kp,Kd}, the gains of
the baseline controller (8). In fact, [5] showed that in the
homogeneous scenario with τ0=0.1 and h=0.7, controller
(8) with Kp=0.2 and Kd=0.7 guaranteed the string stability
of the platoon. Therefore, this motivates the choice of these
gain values for designing the baseline controller (8) in
the heterogeneous scenario, and consequently the reference
model (10). The adaptive input (19)-(20) is designed using
(21) with Qm=diag(10,10,70,50).
We run a simulation of the closed-loop system with
controller (22) for 100 seconds with a desired platoon
acceleration a0(t), shown in Fig. 2. As a results, Fig. 3
Time (s)
0 10 20 30 40 50 60 70 80 90 100
Predefined Acceleration: a0(m/s2)
0
0.5
1
1.5
2
Fig. 2. Desired platoon acceleration a0(t)
displays the estimate ˆ
Ωifor each vehicle. It can be seen that
all estimates converge to their respective true value within
approximately 31 seconds. Furthermore, Fig. 4 shows that
Time (s)
0 10 20 30 40 50 60 70 80 90 100
Uncertainty Estimation ˆ
Ωi(t)
-1
-0.8
-0.6
-0.4
-0.2
0
ˆ
Ω1(t)
ˆ
Ω2(t)
ˆ
Ω3(t)
ˆ
Ω4(t)
ˆ
Ω5(t)
Fig. 3. Vehicle 1-5’s parameter uncertainty estimation ˆ
Ωi(t)
the vehicles’ spacing error eiasymptotically converge to zero
∀i∈ {1,2, ..., 5}. Moreover, for vehicle 1, Fig. 5 demonstrates
Time (s)
0 10 20 30 40 50 60 70 80 90 100
Spacing Error ei(t) (m)
-0.5
0
0.5
e1(t)
e2(t)
e3(t)
e4(t)
e5(t)
Fig. 4. Vehicle 1-5’s spacing error ei(t)
the asymptotic convergence of the vehicle’s acceleration to
the desired reference acceleration. All other vehicles exhibit
similar behavior, which is not shown for better readability
of the plot. Fig. 6 is a plot of the vehicles’ velocities as a
response to the desired acceleration a0(t). It is clear that
string stability is maintained and the velocities reach the
common constant velocity of 40 m/s.
VI. CONCLUSIONS
A novel CACC strategy to stabilize a platoon with non-
identical vehicle dynamics was considered. The proposed
control scheme comprises a baseline controller (string stable
under the homogeneous platoon assumption) augmented with
an adaptive term (to compensate for heterogeneous dynam-
ics). The derivation of the reference model and augmented
Time (s)
0 10 20 30 40 50 60 70 80 90 100
Acceleration (m/s2)
0
0.5
1
1.5
2
Fig. 5. Vehicle 1’s acceleration a1(t)(solid) vs. reference model’s
acceleration a1,m(t)(dashed)
Time (s)
0 10 20 30 40 50 60 70 80 90 100
Velocity (m/s)
0
10
20
30
40
50
v0(t)
v1(t)
v2(t)
v3(t)
v4(t)
v5(t)
Fig. 6. Vehicle 0-5’s velocity response vi(t)
controller were provided and their stability and string stabil-
ity properties were analytically studied. Moreover, a potential
solution to cope inevitable communication losses in practice,
modeled using a dwell time approach, has been sketched
through a switched adaptive control strategy. Numerical re-
sults demonstrated the stability of the heterogeneous platoon
under an augmented CACC strategy, whereas the switched
controller will be worked out in future work.
Future works will also aim at guaranteeing the string
stability of the heterogeneous platoon while enhancing the
control strategy’s robustness to unmodelled dynamics [24]
under both ideal and none ideal communication scenarios
using robust switching control techniques [25].
APPENDIX
PROO F OF THE ORE M 1
Define a radially unbounded quadratic Lyapunov candidate
function as:
Vi(˜xi,∆Ωi) = ˜xT
iP
m˜xi+trace(∆ΩiΓ−1
Ω∆Ωi)
where ΓΩ>0 is the gain matrix containing the rates of adap-
tation, and P
m=PT
m>0 is the unique symmetric positive-
definite solution of the following algebraic Lyapunov equa-
tion (21). Taking the time derivative of Vi(˜xi,∆Ωi)and using
the error dynamics in (18) results in:
˙
Vi(˜xi,∆Ωi) = −˜xT
iQm˜xi−2 ˜xT
iP
mBu∆Ωiφi+2tr(∆ΩiΓ−1
Ω
˙
ˆ
Ωi)
Moreover using the identity
aTb=trace(baT)
results in:
˙
Vi(˜xi,∆Ωi) = −˜xT
iQm˜xi
+2trace(∆ΩT
i{Γ−1
Ω
˙
ˆ
Ωi−φi˜xT
iP
mBu})(23)
Choosing the adaptive law as in (20) reduces (23) to the
following form:
˙
Vi(˜xi,∆Ωi) = −˜xT
iQm˜xi≤0 (24)
which proves the uniform ultimate boundedness of (˜xi,∆Ωi).
Furthermore, it can be concluded from (24) that ˜xi∈L2.
In addition, since wiis bounded, then xi,m∈L∞and con-
sequently, xi∈L∞and ui,bl ∈L∞. Moreover, since Ω∗
iis
constant and ∆Ωiis bounded, then the estimated value is also
bounded, ˆ
Ωi∈L∞. Since (xi,ui,bl )∈L∞and the components
of the regressor vector φiare locally Lipschitz continuous,
then the regressor’s components are bounded. Therefore,
ui∈L∞and ˙xi∈L∞. Hence, ˙
˜xi∈L∞, which implies that
¨
Vi∈L∞. Thus, ˙
Viis a uniformly continuous function of
time. In addition, since Vihas a lower bound, ˙
Vi≤0, and
˙
Viis uniformly continuous, then by Barbalat’s Lemma, Vi
tends to a limit, while its derivative tends to zero. Hence,
the tracking error ˜xitends asymptotically to zero as t→∞.
In fact, since Viis radially unbounded, then ˜xiglobally
asymptotically tends to zero as t→∞. Which means that the
tracking error dynamics are globally asymptotically stable.
From (18), it can be deduced that ¨
˜xi∈L∞which indicates
that ˙
˜xiis uniformly continuous. Moreover, since ˜xi→0 as
t→∞then using Barbalat’s lemma, limt→∞k˙
˜xik=0. Which
leads to:
lim
t→∞k∆ΩT
iφik=0,∀i∈SM
This proves that, for any bounded wi, the closed-loop system
globally asymptotically tracks the reference model as t→
∞.
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