the system calibration

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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)!