the system calibration

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)!