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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