9th LISA Symposium
Paris, 22/05/2012
Optimization of
the system calibration
for LISA Pathfinder
Giuseppe Congedo
(for the LTPDA team)
Outline
Model of LPF dynamics:
what are the system parameters
?
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
2
Incidentally, we talk about:
Optimization method
System/experiment constraints
System calibration:
how can we estimate them
?
Optimization of the system calibration:
how can we improve those estimates
?
Motivation
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
3
The reconstructed acc. noise is
parameter
-
dependent
For this, we need to
calibrate
the system
In the end,
better precision
in the measured parameters
→
better confidence
in the reconstructued acc. noise
Differential acceleration noise
to appear in Phys. Rev.
Uncertainties on the
spectrum:
Parameter accuracy:
system calibration
Parameter precision:
optimization of calibration
Statistical uncertainty:
PSD estimation
stat. unc. of
PSD estimation
system
calibration
Model of LPF dynamics
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
4
1
TM
2
1
ω
1
x
12
x
1
o
12
o
2
TM
2
2
ω
IFO
df
C
sus
C
21
S
SC
i,1
o
i,12
o
df
A
sus
A
SC
i,
f
i,1
f
i,2
f
guidance signals
: reference
signals for the drag
-
free and
elect. suspension loops
force gradients (~1x10
-
6
s
-
2
)
sensing cross
-
talk (~1x10
-
4
)
actuation gains (~1)
direct forces on
TMs and SC
Science mode:
TM
1
free along
x
, TM
2
/SC follow
Framework
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
5
sensed
relative motion
o
1
,
o
12
system calibration
(system identification)
parameters
ω
1
2
,
ω
12
2
,
S
21
,
A
df
,
A
sus
diff. operator
Δ
equivalent
acceleration
noise
optimization of
system calibration
(optimal design)
System calibration
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
6
LPF system
o
i,1
o
i,12
...
o
1
o
12
...
LPF is a
multi
-
input/multi
-
output
dynamical system.
The determination of the system parameters can be
performed
with
targeted experiments. We mainly focus on:
Exp. 1
: injection into the drag
-
free loop
Exp. 2
: injection into the elect. suspension loop
System calibration
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
7
residuals
cross
-
PSD matrix
We build the
joint
(multi
-
experiment/multi
-
outputs)
log
-
likelihood
for
the problem
The system response is
simulated
with a
transfer matrix
The calibration is performed
comparing
the modeled response
with both translational IFO readouts
Calibration experiment 1
Exp. 1: injection of sine waves into o
i,1
injection into
o
i,1
produces thruster actuation
investigation of the
drag
-
free loop
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
8
1
TM
2
1
ω
1
x
12
x
1
o
12
o
2
TM
2
2
ω
IFO
df
C
sus
C
21
S
SC
i,1
o
df
A
sus
A
black: injection
Standard design
Calibration experiment 2
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
9
1
TM
2
1
ω
1
x
12
x
1
o
12
o
2
TM
2
2
ω
IFO
df
C
sus
C
21
S
SC
i,12
o
df
A
sus
A
Exp. 2: injection of sine waves into o
i,12
injection into
o
i,12
produces capacitive actuation on TM
2
investigation of the
elect. suspension loop
black: injection
Standard design
Optimization of system calibration
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
10
modeled transfer matrix evaluated
after
system calibration
noise cross PSD matrix
input signals
being optimized
estimated
system parameters
input parameters
(injection frequencies)
Question
: how can we optimize the experiments, to get an improvement
in parameter precision?
gradient w.r.t. system parameters
Answer
: use the
Fisher information matrix
of the system
(method already found in literature and named “theory of optimal design of experiments”)
Optimization strategy
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
11
practically speaking
...
Either way, the optimization seeks
to minimize the “covariance
volume” of the system parameters
Perform a
non
-
linear optimization
(over
a discrete space of design parameter
values) of the
scalar estimator
6 optimization criteria are possible:
information matrix
, maximize:
-
the determinat
-
the minimum eigenvalue
-
the trace
[better results, more robust]
covariance matrix
, minimize:
-
the determinant
-
the maximum eigenvalue
-
the trace
Experiment constraints
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
12
Can inject a
series of windowed sines
Fix the
experiment total duration
T
~
2.5 h
For transitory decay, allow gaps of length
δt
gap
= 150 s
Require that each injected sine must start and end at zero (null boudary
conditions)
→ each sine wave has an
integer number of cycles
→ all possible injection frequencies are
integer multiples
of the
fundamental one
→ the optim. parameter space (space of all inj. frequencies) is
intrinsically
discrete
→ the optimization may be
challenging
Divide the experiment in
injection slots
of duration
δt
= 1200 s each.
This set the
fundamental frequency
, 1/1200
~
0.83 mHz.
System constraints
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
13
Capacitive authority, 10% of 2.5 nN
Thruster authority, 10% of 100
µ
N
Interferometer range, 1% of 100
µ
m
→ as the
injection frequencies vary
during the optimization,
the
injection amplitudes are adjusted
according to the constraints above
For safety reason, choose not to exceed:
System constraints
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
14
for almost the entire frequency band, the maximum amplitude is
limited by the
interferometer range
since the data are sampled at 1 Hz, we conservatively limit the
frequency band
to a 10th of Nyquist, so <0.05 Hz
o
i,12
inj. (Exp. 2)
o
i,1
inj. (Exp. 1)
maximum injection amplitude (dashed) VS injection frequency
interferometer
interferometer
Optimization of calibration
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
15
initial
-
guess parameters
ω
1
2
,
ω
12
2
,
S
21
,
A
df
,
A
sus
best
-
fit parameters
ω
1
2
,
ω
12
2
,
S
21
,
A
df
,
A
sus
system calibration
optimization
of system
calibration
optimized
experimental
designs
Discrete optimization
may be an issue!
Overcome the problem by:
1)
overlapping a grid to a continuous
variable space
2)
rounding the variables (inj. freq.s)
to the
nearest grid node
3)
using direct algorithms
robust to
discontinuities
(i.e., patternsearch)
Parameter
Description
Nominal
value
Standard
design
σ
Optimal
design
σ
ω
1
2
[s
-
2
]
Force (per unit mass) gradient on TM
1
,
“1st stiffness”
-
1.4x10
-
6
4x10
-
10
2x10
-
10
ω
12
2
[s
-
2
]
Force (per unit mass) gradient between
TM
1
and TM
2
, “differential stiffness”
-
0.7x10
-
6
2x10
-
10
1x10
-
10
S
21
Sensing cross
-
talk from x
1
to x
12
1x10
-
4
4x10
-
7
1x10
-
7
A
df
Thruster actuation
gain
1
7x10
-
4
1x10
-
4
A
sus
Elect. actuation gain
1
1x10
-
5
2x10
-
6
Optimization of exp. 1 & 2
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
16
Improvement of
factor 2 through 7
in precision,
especially for
A
df
(important for the subtraction of thruster noise)
There are examples for which correlation is mitigated:
Corr[
S
21
,
ω
12
2
]=
-
20%
-
>
-
3%, Corr[
ω
12
2
,
S
21
]=9%
-
>2%
Optimization of exp. 1 & 2
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
17
The optimization converged to:
Exp. 1: lowest (0.83 mHz) and highest (49 mHz) allowed frequencies
Exp. 2: highest (49 mHz) allowed frequency (plus a slot with 0.83 mHz)
Optimization of exp. 1 & 2
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
18
Optimized design:
E
xp. 1
: 4 slots @ 0.83 mHz, 3 slots @ 49 mHz
Exp. 2
: 1 slot @ 0.83 mHz, 6 slots @ 49 mHz
why is it so?
the physical interpretation is within the system transfer matrix
1
1
,
o
o
i
→
12
1
,
o
o
i
→
12
12
,
o
o
i
→
The optimization:
converges to the maxima
of the transfer matrix
balances the information
among them
•
•
•
1
12
,
o
o
i
→
Exp. 1
Exp. 2
•
Effect of frequency
-
dependences
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
19
loss angle
nominal stiffness,
~
-
1x10
-
6
s
-
2
dielectric loss
gas damping
Simulation of the response of the system
to a
pessimistic
range of values:
δ
1
,
δ
2
= [1x10
-
6
,1x10
-
3
] s
-
2
τ
1
,
τ
2
= [1x10
5
,1x10
7
] s
1
1
,
o
o
i
→
12
1
,
o
o
i
→
12
12
,
o
o
i
→
However, the biggest contribution
is due to
gas damping
,
Cavalleri A. et al.,
Phys. Rev. Lett. 103, 140601
(2009)
1
12
,
o
o
i
→
mHz
1
@
s
10
×
2
<
-2
-11
2
g
ω
(
)
(
)
(
)
[
]
2
/
1
2
/
1
2
/
/
32
8
/
+
1
13
/
=
/
=
kT
m
π
π
PL
M
β
M
τ
(
)
(
)
(
)
[
]
(
)
(
)
(
)
[
]
s
10
×
5
~
m/s
280
cm
6
.
4
Pa
10
×
58
/
kg
96
.
1
~
s
10
×
4
~
m/s
250
cm
6
.
4
Pa
10
×
58
/
kg
96
.
1
~
9
1
-
2
6
-
8
-1
2
-5
τ
τ
(N
2
, gas venting directly to space)
(
Ar
)
-2
-10
s
10
×
1
>
2
ω
σ
Concluding remarks
The optimization of the system calibration shows:
‐
improved parameter precision
‐
improved parameter correlation
The optimization converges to
only two
relevant frequencies which
corresponds to the maxima of the system transfer matrix; this leads
to a simplification of the experimental designs
Possible frequency
-
dependences in the stiffness constants
do not
impact
the optimization of the system calibration
However, we must be open to possible frequency
-
dependences in
the
actuation gains
[to be investigated]
The optimization of the system calibration is model
-
dependent, so
it must be performed once we have
good confidence
on the model
22/05/2012
Giuseppe Congedo
-
9th LISA Symposium, Paris
20
Thanks for your attention!
Giuseppe Congedo
-
9th LISA Symposium, Paris
22/05/2012
21
... and to the Trento team for the laser pointer
(the present for my graduation)!
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