linear dynamic model for advanced ligo isolation system wensheng hua, brain lantz, dan debra,...
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Linear Dynamic Model for Advanced LIGO Isolation System
Wensheng Hua, Brain Lantz, Dan Debra, Jonathan How, Corwin Hardham, Sam Richman, Rana Adhikari, Richard
Mittleman
LIGO-G010135-00-Z
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Aim of modeling
• Help to understand the dynamics of the system.
• Help to control the system.
• Help to design future system.
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What are we trying to model?
•LIGO suspension system is a complicated nonlinear dynamic mass spring system.
•We need to build a linear model of it in full degrees of freedom.
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Stanford Active Platform with Pendulums
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Two stage active platform in the BSC
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Double Stage Prototype
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Elements of the systems.
• Stages (Masses)
• Springs
• Sensors
• Actuators
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Big picture of the model
Mechanical Model
Control Model
Sensor Model
Actuator Model
Sensor output
Actuator
control signal
Force & Torque on stages
Stage positions
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General Form of a Dynamic Model
A
D
B C1/S +uX
y+
X=AX+Bu
Y=CX+Du
X
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Matrix A for a Suspension System
X=
0 I
M K M D
-1 -1A =
M: mass/inertia matrix
K: reaction force matrix
D: Damping force matrix
-1 -1
0 I
M K M D
d
V
d
V=
F = Ma
F = Kd+Dv
a = M Kd + M Dv
-1 -1
X=AXd
Vv = d
a = v
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Stiffness Matrix K• The stiffness matrix is defined as the reaction forces
and torques on stages due to small movement of the stages around equilibrium positions.
• In small range of motion, the changes of reaction forces and torques are linear to the perturbations of the positions of stages.
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3 Steps of making stiffness matrix K
• Convert Stage motion into relative motion of two ends of springs around equilibrium positions.
• Calculate each spring’s reaction force and torque.
• Sum up forces and torques from all springs on stages.
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Final position
Static Force Reaction Force from change of direction
Reaction Force From the deformation of the springEquilibriu
m position
Example: Simple Wire Spring
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Example: Loaded Blade
Free blade
K11 K12
K21 K22
x
F
T= +
K11 K12
K21 K22
x
F
T=
0 -F0x
F0x 0
x
F0x is a matrix such that:
F0x x = F0 x
F0
Loaded blade
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Reason
Top View
X
Side View
F0
T = - F0 x
F0
F = F0
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Some features of the model
• The principle is simply based on F=Kx and F=Ma.
• To linearize each spring is simpler than to
linearizie the whole system at ones.
• Make use of Simulink and toolboxes in Matlab.
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Model constructor• There are many systems to be modeled.• There are many different types of springs, actuators
and sensors in the system.– Springs: Stretchable Wire, Blade, more general spring.– Sensors: Optical Sensor, Geophone.
– Actuators: Voice coil. • For each system, we only need to feed its geometry
and physical information to the model constructor.• The information should be in the form of defined data
structures.– Stage, spring, actuator, sensor, control law.
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1
Out1
1
Gain
1
In1
Simulink
Product line of modeling
Stage file
Spring file
Sensor file
Actuator file
Model Constructor
Mech. Model
Sensor Model
Actuator Model
Control Model
dSpace
Stage(1).Positon =[1 0 0]
Stage(1).mass= 10
Stage(2).Position=[2 0 0]
Stage(2).mass= 5
…
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Tilt horizontal coupling
F
Actuator Side Sensor Side
a
Inertial sensor can’t
distinguish
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Predicted Motion of Optics Table
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Views of the Prototype
inner stage(table top removed)
inner stage with outer stage and supports
assembled system
with table top
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Simulink Model Diagram
Model used to simulate the dynamics of the reference design.The controller can be cross-compiled onto dSPACE hardware and used on the real system.
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10-1
100
101
102
10-4
10-3
10-2
10-1
100
101
102
freq Hz
Vo
lt /
Vo
lt
Transfer functions from actuator Horizontal 1 to sensor STS Horizontal 1
H1->H1 modelH1->H1 data
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10-1
100
101
102
10-3
10-2
10-1
100
101
102
freq Hz
Vo
lt /
Vo
lt
Transfer functions of 2 stage prototype from actuator Horizontal 1 to sensor STS Horizontal 1 2 and 3
H1->H1 modelH1->H2 modelH1->H3 modelH1->H1 data H1->H2 data H1->H3 data
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10-1
100
101
102
10-3
10-2
10-1
100
101
102
freq Hz
Vo
lt /
Vo
lt
Transfer functions of 2 stage prototype from actuator Horizontal 1 to sensor STS Horizontal 1 2 and 3
H1->H1 modelH1->H2 modelH1->H3 modelH1->H1 data H1->H2 data H1->H3 data
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Conclusion• A linear dynamic model of advanced LIGO
isolation system directly based on very simple physics principles.
• This model is used to analysis the dynamics of several prototype systems and to design control laws for them.
• This model is also used to design future LIGO isolation systems. Based on our experience, we feel confident of the predictions which the model made.