9th LISA SymposiumParis, 22/05/2012

Optimization of the system calibration

for LISA Pathfinder

Giuseppe Congedo(for the LTPDA team)

Giuseppe Congedo - 9th LISA Symposium, Paris 2

Outline Model of LPF dynamics:

what are the system parameters?

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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?

Giuseppe Congedo - 9th LISA Symposium, Paris 3

Motivation

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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 estimationstat. unc. of

PSD estimation

system calibration

calibrated

uncalibrated

Giuseppe Congedo - 9th LISA Symposium, Paris 4

Model of LPF dynamics

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1TM

21ω

1x12x

1o 12o

2TM

22ω

IFO

dfC susC

21S

SC

i,1o i,12o

dfA susA

SCi,f

i,1f i,2f

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: TM1 free along x, TM2/SC follow

Giuseppe Congedo - 9th LISA Symposium, Paris 5

Framework

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sensed relative motion

o1, o12

system calibration(system identification)

parametersω1

2, ω122, S21,

Adf, Asus

diff. operatorΔ

equivalent acceleration

noise

optimization of system calibration(optimal design)

Giuseppe Congedo - 9th LISA Symposium, Paris 6

System calibration

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LPF system

oi,1

oi,12

...

o1

o12

...

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 loopExp. 2: injection into the elect. suspension loop

Giuseppe Congedo - 9th LISA Symposium, Paris 7

System calibration

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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

Giuseppe Congedo - 9th LISA Symposium, Paris 8

Calibration experiment 1

Exp. 1: injection of sine waves into oi,1

injection into oi,1 produces thruster actuation investigation of the drag-free loop

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1TM

21ω

1x12x

1o 12o

2TM

22ω

IFO

dfC susC

21S

SC

i,1o

dfA susA

black: injection

Standard design

Giuseppe Congedo - 9th LISA Symposium, Paris 9

Calibration experiment 2

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1TM

21ω

1x12x

1o 12o

2TM

22ω

IFO

dfC susC

21S

SC

i,12o

dfA susA

Exp. 2: injection of sine waves into oi,12

injection into oi,12 produces capacitive actuation on TM2

investigation of the elect. suspension loop

black: injection

Standard design

Giuseppe Congedo - 9th LISA Symposium, Paris 10

Optimization of system calibration

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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”)

Giuseppe Congedo - 9th LISA Symposium, Paris 11

Optimization strategy

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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

Giuseppe Congedo - 9th LISA Symposium, Paris 12

Experiment constraints

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Can inject a series of windowed sines

Fix the experiment total duration T ~ 2.5 h

For transitory decay, allow gaps of length δtgap = 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.

Giuseppe Congedo - 9th LISA Symposium, Paris 13

System constraints

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Capacitive authority, 10% of 2.5 nN

Thruster authority, 10% of 100 µN

Interferometer range, 1% of 100 µm

→ 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:

Giuseppe Congedo - 9th LISA Symposium, Paris 14

System constraints

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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

oi,12 inj. (Exp. 2)oi,1 inj. (Exp. 1)

maximum injection amplitude (dashed) VS injection frequency

interferometer interferometer

Giuseppe Congedo - 9th LISA Symposium, Paris 15

Optimization of calibration

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initial-guess parametersω1

2, ω122, S21, Adf, Asus

best-fit parametersω1

2, ω122, S21, Adf, Asus

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 space2) rounding the variables (inj. freq.s)

to the nearest grid node3) using direct algorithms robust to

discontinuities (i.e., patternsearch)

Giuseppe Congedo - 9th LISA Symposium, Paris 16

Parameter Description Nominal value

Standard design

σ

Optimal design

σ

ω12 [s-2] Force (per unit mass) gradient on TM1,

“1st stiffness” -1.4x10-6 4x10-10 2x10-10

ω122 [s-2] Force (per unit mass) gradient between

TM1 and TM2, “differential stiffness” -0.7x10-6 2x10-10 1x10-10

S21 Sensing cross-talk from x1 to x12 1x10-4 4x10-7 1x10-7

Adf Thruster actuation gain 1 7x10-4 1x10-4

Asus Elect. actuation gain 1 1x10-5 2x10-6

Optimization of exp. 1 & 2

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Improvement of factor 2 through 7 in precision, especially for Adf (important for the subtraction of thruster noise)

There are examples for which correlation is mitigated: Corr[S21, ω12

2]=-20%->-3%, Corr[ω122, S21]=9%->2%

Giuseppe Congedo - 9th LISA Symposium, Paris 17

Optimization of exp. 1 & 2

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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)

Giuseppe Congedo - 9th LISA Symposium, Paris 18

Optimization of exp. 1 & 2

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Optimized design:Exp. 1: 4 slots @ 0.83 mHz, 3 slots @ 49 mHzExp. 2: 1 slot @ 0.83 mHz, 6 slots @ 49 mHz

why is it so?the physical interpretation is within the system transfer matrix

11, ooi →

121, ooi →1212, ooi →

The optimization:converges to the maxima of the transfer matrixbalances the information among them

•

•

•

112, ooi →

Exp. 1

Exp. 2

•

Giuseppe Congedo - 9th LISA Symposium, Paris 19

Effect of frequency-dependences

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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:

δ1, δ2 = [1x10-6,1x10-3] s-2

τ1, τ2 = [1x105,1x107] s

11, ooi →

121, ooi → 1212, ooi →However, the biggest contribution is due to gas damping, Cavalleri A. et al., Phys. Rev. Lett. 103, 140601 (2009)

112, ooi →

mHz 1 @ s 10×2< -2-112gω

( )( ) ( )[ ]2/12/12 //328/+1 13/=/= kTmππPLMβMτ

( )( ) ( )[ ]( )( ) ( )[ ] s 10×5~m/s 280cm 6.4 Pa 10×58/kg 96.1~

s 10×4~m/s 250cm 6.4 Pa 10×58/kg 96.1~91-26-

8-12-5

τ

τ

(N2, gas venting directly to space)

(Ar)

-2-10 s 10×1>2ωσ

Giuseppe Congedo - 9th LISA Symposium, Paris 20

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

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Giuseppe Congedo - 9th LISA Symposium, Paris 21

Thanks for your attention!

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... and to the Trento team for the laser pointer(the present for my graduation)!