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ENGG2440 — Modelling and Control Viva Voce Major Assignment — Part 2 For Extra Help: E-mail: XXXXXXXXXX HelpDesk: ES204, Fridays, 11am-1pm Learning Outcomes The Viva Voce major assignment will assess...

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ENGG2440 — Modelling and Control
Viva Voce Major Assignment — Part 2
For Extra Help:
E-mail: XXXXXXXXXX
HelpDesk: ES204, Fridays, 11am-1pm
Learning Outcomes
The Viva Voce major assignment will assess your ability to:
1. Design controllers for the cart-pendulum using a linear approximation.
2. Simulate the closed-loop system using Matlab-Simulink.
3. Evaluate the performance of the controllers on the nonlinear model of the cart-
pendulum system.
Key Point:
1 Inverted pendulum on a cart
We use the cart-pendulum system as a benchmark to design linear controllers. Figure 1
shows the idealised model of the system that consists of a pendulum of mass m and length
` attached to a cart of mass Mc. The pendulum moves under the action of the gravity (g)
and the cart moves on the horizontal direction and is actuated by the control force F . The
state-space model can be written in the form1
ẋ1
ẋ2
ẋ3
ẋ4
 = f(x1, x2, x3, x4, F ), (1)
where the states are:
• x1: the position of the cart,
• x2: the angle of the pendulum,
• x3: the velocity of the cart,
• x4: the angular velocity of the pendulum.
1.1 Problem formulation.
In this lab assignment, we consider the stabilisation problem of two equili
iums of the
cart-pendulum system. The equili
iums are
x̄a =

x̄1a
x̄2a
x̄3a
x̄4a
 =

0
0
0
0
 ; x̄b =

x̄1
x̄2
x̄3
x̄4
 =

0
π
0
0
 . (2)
We consider the control structure given in Figure 2. The control objective is to design the
controllers C1(s) and C2(s) that generate the input force to stabilise the equili
iums x̄a
and x̄b.
1The full model of the cart-pendulum system and the values of the model parameters are explicitly given
in the part I of the VIVA assignment.
1
mailto: XXXXXXXXXX
Figure 1: Cart-pendulum system.
Controller System
Cart-pendulum
Figure 2: Control system for the cart-pendulum.
1.2 Stabilisation of the cart-pendulum about the equili
ium x̄a.
The task in this section is to design a controller to stabilise the equili
ium x̄a. To do that,
consider the linearised model of the cart-pendulum about x̄a:
˙̃xa = Aax̃a +BaF, (3)
y = Cax̃a +DaF, (4)
where the matrices Aa, Ba, Ca and Da can be computed using the linearisation of the
nonlinear model of the cart-pendulum about the equili
ium x̄a.
The control structure for the linearised model of the cart-pendulum is shown in Figure 3,
where the system transfer functions are
H1(s) =
X1(s)
F (s)
, (5)
H2(s) =
X2(s)
F (s)
, (6)
with X1(s) = L[x1(t)], X2(s) = L[x2(t)] and F (s) = L[F (t)], and L is the Laplace trasform
operator. The signals x̄1 and x̄2 are the desired equili
ium points.
Control design. Consider the PD controllers C1(s) and C2(s) given in the form
C1(s) = Ka1 +Ka3s, (7)
C2(s) = Ka2 +Ka4s, (8)
where Ka1, Ka2, Ka3 and Ka4 are the constant gains of the controllers.
Design the controller following the next steps
1. Compute the matrices Aa, Ba, Ca and Da of the linearised model of the cart-pendulum
about x̄a.
2. Obtain the transfer functions H1(s) and H2(s) of the linearised model of the cart-
pendulum about x̄a.
2
Controller Linearised system
Figure 3: Control system for the linearised model of the cart-pendulum.
3. Derive the closed-loop transfer function
HCL(s) =
X1(s)
X̄1(s)
, (9)
where X̄1(s) = L[x̄1(t)].
4. Pole placement. Compute the gains Ka1, Ka2, Ka3 and Ka4 such that the poles of
of the closed-loop transfer function HCL(s) are λ1 = −3, λ2 = −4, λ3 = −5, λ4 = −6.
1.3 Simulation of the cart-pendulum control system about the equi-
li
ium x̄a.
Write a script that performs the following tasks:
1. Define the parameters of the model.
2. Define the matrices of the linearised model.
3. Define the gains of the controller.
4. Simulate the linearised and nonlinear closed-loop systems:
i) Create a Simulink model of the linearised system in closed loop with the PD
controllers. Save your model as CP_Control_Lin_a_yourstudentnumber.slx.
An example of the Simulink model is shown in Figure 4 (see Appendix for details).
ii) Create a Simulink model of the nonlinear system in closed loop with the PD
controllers (use the controller gains obtained for the linearised model).
Save your model as CP_Control_NLin_yourstudentnumber.slx. An example of
the Simulink model is shown in Figure 5.
iii) Export the states, the control input and the simulation time from Simulink to
Matlab.
iv) Simulate both the linearised and the nonlinear closed-loop models with the cart
stating at 0.2m and the pendulum at 20deg. Set the initial conditions of the
velocities to zero. That is x(0) =
[
0.2 20π/180 0 0
]
. Suggestion for the
simulation: use the fixed-step solver ode4 and select 0.02 as step time.
5. Plot the results of the simulations in a figure that shows the time histories of the
position of the cart, the angle of the pendulum, the velocity of the cart, the angula
velocity of the pendulum and the control forces for both the linearised and nonlinea
control systems. An example of the simulation results is shown in Figure 6.
6. Save the script as MainFile_Control_Comparison_a_yourstudentnumber.m.
7. Run the script using different initial conditions for the simulation and analyse the
esults. The initial conditions should be defined in the script.
8. (Optional) Use the function Cart_Pendulum_Animation.m to create an animation of
your control system.
Important: The plots and (optional) animation should be produced automatically when
the script is executed without further intervention of the user.
3
Figure 4: Control system for the linearised model.
Figure 5: Control system for the nonlinear model.
4
XXXXXXXXXX XXXXXXXXXX
Time [s]
-0.2
0
0.2
0.4
C
a
t p
os
iti
on
[m
] Time histories of the states using the nonlinear model and linearised model about EPa
x
1
Nonlinea
x
1
Linearised
XXXXXXXXXX XXXXXXXXXX
Time [s]
-20
0
20
P
en
du
lu
m
a
ng
le
[d
eg
]
x
2
Nonlinea
x
2
Linearised
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Time [s]
-2
0
2
C
a
t v
el
oc
ity
[m
s
]
x
3
Nonlinea
x
3
Linearised
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Time [s]
-600
-400
-200
0
P
en
du
lu
m

at
e
[d
eg
s
]
x
4
Nonlinea
x
4
Linearised
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Time [s]
-50
0
50
100
150
In
pu
t F
o
ce
[N
]
F Nonlinea
F Linearised
Figure 6: Time histories of the states and input.
5
1.4 Stabilisation of the cart-pendulum about the equili
ium x̄b.
The task in this section is to design a controller to stabilise the equili
ium x̄b. To do that,
consider the linearised model of the cart-pendulum about x̄b:
˙̃xb = Abx̃b +BbF, (10)
y = Cax̃b +DbF, (11)
where the matrices Ab, Bb, Cb and Db can be computed using the linearisation of the
nonlinear model of the cart-pendulum about the equili
ium x̄b.
Consider the control structure for the linearised model of the cart-pendulum shown in
Figure 3.
Control design. Consider the PD controllers C1(s) and C2(s) given in the form
C1(s) = Kb1 +Kb3s, (12)
C2(s) = Kb2 +Kb4s, (13)
where Kb1, Kb2, Kb3 and Kb4 are the constant gains of the controllers.
Design the controller following the next steps
1. Compute the matrices Ab, Bb, Cb and Db of the linearised model of the cart-pendulum
about x̄b.
2. Obtain the transfer functions H1(s) and H2(s) of the linearised model of the cart-
pendulum about x̄b.
3. Derive the closed-loop transfer function
HCL(s) =
X1(s)
X̄1(s)
, (14)
where X̄1(s) = L[x̄1(t)].
4. Pole placement. Compute the gainsKb1, Kb2, Kb3 and Kb4 such that the poles of of
the closed-loop transfer function HCL(s) are λ1 = −3, λ2 = −4, λ3 = −5, λ4 = −6.
1.5 Simulation of the cart-pendulum control system about the equi-
li
ium x̄b.
Write a script that performs the following tasks:
1. Define the parameters of the model.
2. Define the matrices of the linearised model.
3. Define the gains of the controller.
4. Simulate the linearised and nonlinear closed-loop systems:
i) Create a Simulink model of the linearised system in closed loop with the PD
controllers. Save your model as CP_Control_Lin_b_yourstudentnumber.slx.
ii) Create a Simulink model of the nonlinear system in closed loop with the PD
controllers (use the controller gains obtained for the linearised model).
Save your model as CP_Control_NLin_yourstudentnumber.slx.
iii) Export the states, the control input and the simulation time from Simulink to
Matlab.
iv) Simulate both the linearised and the nonlinear closed-loop models with the cart
stating at 0.2m and the pendulum at 200deg. Set the initial conditions of the
velocities to zero. That is x(0) =
[
0.2 200π/180 0 0
]
. Note that the ini-
tial condition is given for the states of the nonlinear model. Suggestion for the
simulation: use the fixed-step solver ode4 and select 0.02 as step time.
f) Plot the results of the simulations in a figure that shows the time histories of the
position of the cart, the angle of the pendulum, the velocity of the cart, the angula
velocity of the pendulum and the control forces for both the linearised and nonlinea
control systems. An example of the simulation results is shown in Figure 7.
6
5. Save the script as MainFile_Control_Comparison_b_yourstudentnumber.m.
6. Run the script using different initial conditions for the simulation and analyse the
esults. The initial conditions should be defined in the script.
i) (Optional) Use the function Cart_Pendulum_Animation.m to create an animation of
your control system.
Important: The plots and (optional) animation should be produced automatically when
the script is executed without further intervention of the user.
XXXXXXXXXX XXXXXXXXXX
Time [s]
-0.1
0
0.1
0.2
C
a
t p
os
iti
on
[m
] Time histories of the states using the nonlinear model and linearised model about EP
x
1
Nonlinea
x
1
Linearised
XXXXXXXXXX XXXXXXXXXX
Time [s]
160
180
200
P
en
du
lu
m
a
ng
le
[d
eg
]
x
2
Nonlinea
x
2
Linearised
XXXXXXXXXX
2019engg2440cpcontrol-zug0lmrh.pdf 2019engg2440cpmodelling-rmdj4pge.pdf 2019engg2440vivavoceprep-xasvngrj.pdf cartpendulumanimation-zubf4src.m
Answered Same Day Dec 05, 2021 ENGG2440

Solution

Abr Writing answered on Dec 08 2021
142 Votes
plotresults_Modelling_Lin_a_studentnumber.m
%% Clearing the workspace
clear;
close;
clc;
%% Setting up the Model parameters
setupsim_Modelling_studentnumber;
%% Running the model
sim('CP_Modelling_NLin_studentnumber');
%% Retrieving results
velocity = get(yout, 'velocity').Values.Data;
angular_velocity = get(yout, 'angular_velocity').Values.Data;
inclination = get(yout, 'inclination').Values.Data;
position = get(yout, 'position').Values.Data;
%% Plotting results
t = tiledlayout(5,1);
title(t, 'Time histories of the states using the nonlinear model');
nexttile
plot(tout, position)
xlabel('Time (s)');
ylabel('Cart Position [m]');
nexttile
plot(tout, inclination)
xlabel('Time (s)');
ylabel('Pendulum angle [deg]');
nexttile
plot(tout, velocity)
xlabel('Time (s)');
ylabel('Cart velocity [m/s]');
nexttile
plot(tout, angular_velocity)
xlabel('Time (s)');
ylabel('Pendulum rate [deg/s]');
nexttile
plot(tout, zeros(size(tout, 1),1))
xlabel('Time (s)');
ylabel('Input Force [N]');
setupsim_Modelling_Lin_b_studentnumber.m
function setupsim_Modelling_studentnumbe
assignin('base', 'M', 0.4);
assignin('base', 'm', 0.15);
assignin('base', 'l', 0.2);
assignin('base', 'g', 9.81);
assignin('base', 'I', 0.006);
assignin('base', 'b', 0);
end
setupsim_Modelling_Lin_a_studentnumber.m
function setupsim_Modelling_studentnumbe
assignin('base', 'M', 0.4);
assignin('base', 'm', 0.15);
assignin('base', 'l', 0.2);
assignin('base', 'g', 9.81);
assignin('base', 'I', 0.006);
assignin('base', 'b', 0);
end
plotresults_Modelling_Lin_b_studentnumber.m
%% Clearing the workspace
clear;
close;
clc;
%% Setting up the Model parameters
setupsim_Modelling_studentnumber;
%% Running the model
sim('CP_Modelling_NLin_studentnumber');
%% Retrieving results
velocity = get(yout, 'velocity').Values.Data;
angular_velocity = get(yout, 'angular_velocity').Values.Data;
inclination = get(yout, 'inclination').Values.Data;
position = get(yout, 'position').Values.Data;
%% Plotting results
t = tiledlayout(5,1);
title(t, 'Time histories of the states using the nonlinear model');
nexttile
plot(tout, position)
xlabel('Time (s)');
ylabel('Cart Position [m]');
nexttile
plot(tout, inclination)
xlabel('Time (s)');
ylabel('Pendulum angle [deg]');
nexttile
plot(tout, velocity)
xlabel('Time (s)');
ylabel('Cart velocity [m/s]');
nexttile
plot(tout, angular_velocity)
xlabel('Time (s)');
ylabel('Pendulum rate [deg/s]');
nexttile
plot(tout, zeros(size(tout, 1),1))
xlabel('Time (s)');
ylabel('Input Force [N]');
plotresults_Modelling_studentnumber.m
%% Clearing the workspace
clear;
close;
clc;
%% Setting up the Model parameters
setupsim_Modelling_studentnumber;
%% Running the model
sim('CP_Modelling_NLin_studentnumber');
%% Retrieving results
velocity = get(yout, 'velocity').Values.Data;
angular_velocity = get(yout, 'angular_velocity').Values.Data;
inclination = get(yout, 'inclination').Values.Data;
position = get(yout, 'position').Values.Data;
%% Plotting results
t = tiledlayout(5,1);
title(t, 'Time histories of the states using the nonlinear model');
nexttile
plot(tout, position)
xlabel('Time (s)');
ylabel('Cart Position [m]');
nexttile
plot(tout, inclination)
xlabel('Time (s)');
ylabel('Pendulum angle [deg]');
nexttile
plot(tout, velocity)
xlabel('Time (s)');
ylabel('Cart velocity [m/s]');
nexttile
plot(tout, angular_velocity)
xlabel('Time (s)');
ylabel('Pendulum rate [deg/s]');
nexttile
plot(tout, zeros(size(tout, 1),1))
xlabel('Time (s)');
ylabel('Input Force [N]');
cartpendulumanimation.m
%% This function makes an animation of the pendulum on the cart.
%{
This function was created by Dr Alejandro Donaire at UON
as part of ENGGH2440 - Modelling and Control S2 2019
To produce the anumation, the function requires the simulation variables:
'time' is the simulation time
'qa' is the position of the cart
'qu' is the angle of the pendulum
'qar' is the desired equili
ium position of the cart
'qur' is the desired equili
ium angle of the pendulum
%}
function output=Cart_Pendulum_Animation(time,qa,qu,qar,qur)
%%
clc;
l=0.2;
disp('Animation started');
%% Auxiliar variables
lc=0.06; % Cart length
hc=0.04; % Cart height
xp1plane=0.5; %
xp2plane=-0.5;
yp1plane=-1.5*hc;
%% Compute cartesian coordinates of the cart and pendulum
xm = qa + l*sin(qu); % x-coordinate of the pendulum
ym = l*cos(qu); % y-coordinate of the pendulum
xmr = qar; % x-coordinate reference position of the pendulum
ymr = l*cos(qur); % y-coordinate reference position of the pendulum
%% Allocation space movie
mov(1:length(time)) = struct('cdata',[],'colormap',[]);
%% Dimmension of the figure window for the movie
scrsz = get(0,'ScreenSize');
figmovie=figure('Name','Movie: Pendulum on the cart','Position',[0 0 scrsz(3)*2.5/3 scrsz(3)*1.5/2.9]);
%% Plot the cu
ent position of the cart-pendulum and the static ones from the previous declaration to store each frame in variable mov
for k=1:length(time)
%% Set the labels for each frame of the animation
figmovie;clf
axes('NextPlot','replacechildren','tag','plot_axes')
title('Pendulum on the cart','FontSize',18)
xlabel('x [m]','FontSize',18)
ylabel('y [m]','FontSize',18)
text(-0.05,-0.27,sprintf('Time %0.1f sec',time(k)),'BackgroundColor',[1 1 1],'EdgeColor','k','FontSize',18)
hold on;
%% Draw the suporting plane base for the cart
xplane = [xp2plane xp1plane];
yplane = [yp1plane yp1plane];
area(xplane,yplane,'basevalue',-hc,'facecolor',[0.5 .5 0.5]);
%% Initial position of the cart and pendulum
% Pendulum
line([qa(1),xm(1)],[0,ym(1)],'Color','k','LineStyle','--','LineWidth',1); % Pendulum link
plot(qa(1),0,'Marker','o','MarkerEdgeColor','k','LineStyle','--','MarkerFaceColor','w','MarkerSize',20); % Cart joint
plot(xm(1),ym(1),'Marker','o','MarkerEdgeColor','k','LineStyle','--','MarkerFaceColor','w','MarkerSize',30); % Pendulum mass
% Cart body
xp1i = qa(1)+lc;
xp2i = qa(1)-lc;
yp1i = 0;
yp2i = -hc;
area([xp2i;xp1i],[yp1i;yp1i],'basevalue',-hc,'facecolor','w','LineStyle','--');
% Cart wheels
plot(qa(1)+lc*2/3,-hc,'Marker','o','MarkerEdgeColor','k','MarkerFaceColor','w','MarkerSize',30);
plot(qa(1)-lc*2/3,-hc,'Marker','o','MarkerEdgeColor','k','MarkerFaceColor','w','MarkerSize',30);
%% Reference position of the cart and pendulum
% Pendulum
line([qar,xmr],[0,ymr],'Color',[0 .7 0],'LineStyle','--','LineWidth',1); % Pendulum link
plot(qar,0,'Marker','o','MarkerEdgeColor',[0 .7 0],'LineStyle','--','MarkerFaceColor','w','MarkerSize',20); % Cart joint
plot(xmr,ymr,'Marker','o','MarkerEdgeColor',[0 .7 0],'LineStyle','--','MarkerFaceColor','w','MarkerSize',30); % Pendulum mass
% Cart body
xp1r = lc+qar;
xp2r = -lc+qar;
yp1r = 0;
yp2r = -hc;
area([xp2r;xp1r],[yp1r;yp1r],'basevalue',-hc,'EdgeColor',[0 .7 0],'facecolor','w','LineStyle','--');
% Cart wheels
plot(lc*2/3+qar,-hc,'Marker','o','MarkerEdgeColor',[0 .7 0],'MarkerFaceColor','w','MarkerSize',30);
plot(-lc*2/3+qar,-hc,'Marker','o','MarkerEdgeColor',[0 .7 0],'MarkerFaceColor','w','MarkerSize',30);
%% Cu
ent position of the cart and pendulum
% Cart body
xp1 = qa(k)+lc;
xp2 = qa(k)-lc;
yp1 = 0;
yp2 = -hc;
area([xp2;xp1],[yp1;yp1],'basevalue',yp2,'facecolor',[0 0 .7]);
% cart wheels
plot(qa(k)+lc*2/3,-hc,'Marker','o','MarkerEdgeColor','k','MarkerFaceColor',[0 0 0],'MarkerSize',30);
plot(qa(k)-lc*2/3,-hc,'Marker','o','MarkerEdgeColor','k','MarkerFaceColor',[0 0 0],'MarkerSize',30);
% Pendulum
line([qa(k),xm(k)],[0,ym(k)],'Color','k','LineWidth',1); % Pendulum link
plot(qa(k),0,'Marker','o','MarkerEdgeColor','k','MarkerFaceColor',[0 0 .7],'MarkerSize',20); % Cart joint
plot(xm(k),ym(k),'Marker','o','MarkerEdgeColor','k','MarkerFaceColor','r','MarkerSize',30); % Pendulum mass
%% legend
text(-0.32,0.25,{'{\color[rgb]{0,0,0} --- } Initial position','{\color[rgb]{0,0.7,0} --- } Target postion'},'BackgroundColor',[1 1 1],'EdgeColor','k','FontSize',18) ;
%% x-axis and y-axis limits
xlim([-2.5*l 2.5*l])
ylim([-1.5*l 1.5*l])
grid on
hold off
%% Record frame data
mov(k) = getframe(gcf);
end
%% Create AVI file (uncomment this section to create the video file)
vidObj = VideoWriter('Cart_Pendulum_Animation.avi'); % Create a video object
vidObj.FrameRate = 10; % Set frames per second in video object
open(vidObj); % Open video object
writeVideo(vidObj,mov); % Write the frames mov in video object
close(vidObj) % Close video object
disp('Animation finished - Saved in Cart_Pendulum_Animation.avi')
setupsim_Modelling_studentnumber.m
function setupsim_Modelling_studentnumbe
assignin('base', 'M', 0.4);
assignin('base', 'm', 0.15);
assignin('base', 'l', 0.2);
assignin('base', 'g', 9.81);
assignin('base', 'I', 0.006);
assignin('base', 'b', 0);
end
MainFile_Modelling_Comparison_studentnumber.m
%% Clearing the workspace
clear;
close;
clc;
%% Setting up the Model parameters
setupsim_Modelling_studentnumber;
%% Running the model
sim('CP_Modelling_NLin_studentnumber');
CP_Modelling_Lin_a_studentnumber.slx
[Content_Types].xml

_rels/.rels

metadata/coreProperties.xml
model 2019-12-08T05:43:32Z Kanish Kanish 2019-12-08T05:43:48Z 1.0 R2019
metadata/mwcoreProperties.xml
application/vnd.mathworks.simulink.model Simulink Model R2019
metadata/mwcorePropertiesExtension.xml
9.7.0.1243256 7a3a96c2-6da9-459f-9358-afd0c970636a
metadata/mwcorePropertiesReleaseInfo.xml

9.7.0.1247435
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Nov 07 2019
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10.0
auto
on
auto
auto
5
auto
10*128*eps
1000
4
1
auto
auto
1
1e-3
off
off
VariableStepAuto
VariableStepAuto
auto
DisableAll
UseLocalSettings
Nonadaptive
TrustRegion
off
off
Fast
off
off
Unconstrained
Whenever possible
[]
off
off
3


[]

[]
1
[t, u]
xFinal
xInitial
off
1000
off
off
off
off
Dataset
Dataset
on
off
on
on
off
on
off
streamout
on
on
xout
tout
yout
logsout
dsmout
RefineOutputTimes
[]
out
1
off
timeseries
out.mat
[-inf, inf]



BooleansAsBitfields
PassReuseOutputArgsAs
PassReuseOutputArgsThreshold
ZeroExternalMemoryAtStartup
ZeroInternalMemoryAtStartup
OptimizeModelRefInitCode
NoFixptDivByZeroProtection
UseSpecifiedMinMax
EfficientTunableParamExp


[]
on
on
on
Tunable
off
off
off
double
off
off
on
on
off
off
on
off

on
off
uint_T
on
64
Structure reference
12
128
on
5
off
off
Native Intege
on
on
off
off
off
on
auto
Inherit from target
on
off
off
off
on
on
off
off
level2
Balanced
on
off
off
GradualUnderflow
off



UseOnlyExistingSharedCode


[]
e
o
none
none
none
e
o
none
UseLocalSettings
UseLocalSettings
UseLocalSettings
warning
warning
warning
warning
on
Simplified
e
o
off
off
UseLocalSettings
warning
warning
none
e
o
warning
warning
none
warning
e
o
e
o
e
o
none
warning
none
warning
none
warning
warning
e
o
e
o
none
warning
warning
none
none
none
none
none
none
e
o
e
o
...
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