1. 9th LISA Symposium Paris, 22/05/2012 Optimization of the system calibration for LISA Pathfinder Giuseppe Congedo (for the LTPDA team)

2. Outline  Model of LPF dynamics: what are the system parameters? 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris2 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?

3. Motivation 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris3  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 calibrated uncalibrated

4. Model of LPF dynamics 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris4 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

5. Framework 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris5 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)

6. System calibration 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris6 LPF system o i,1 o i,12... o1o1 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

7. System calibration 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris7 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

8. 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 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris8 black: injection Standard design

9. Calibration experiment 2 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris9 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

10. Optimization of system calibration 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris10 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”)

11. Optimization strategy 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris11 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

12. Experiment constraints 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris12  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.

13. System constraints 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris13  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:

14. System constraints 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris14  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

15. Optimization of calibration 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris15 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)

16. ParameterDescription 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 17x10 -4 1x10 -4 A sus Elect. actuation gain 11x10 -5 2x10 -6 Optimization of exp. 1 & 2 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris16  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%

17. Optimization of exp. 1 & 2 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris17 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)

18. Optimization of exp. 1 & 2 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris18 Optimized design: Exp. 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 The optimization:  converges to the maxima of the transfer matrix  balances the information among them Exp. 1 Exp. 2

19. Effect of frequency-dependences 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris19 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 However, the biggest contribution is due to gas damping, Cavalleri A. et al., Phys. Rev. Lett. 103, 140601 (2009) (N 2, gas venting directly to space) (Ar)

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 22/05/2012Giuseppe Congedo - 9th LISA Symposium, Paris20

21. Thanks for your attention! Giuseppe Congedo - 9th LISA Symposium, Paris22/05/201221... and to the Trento team for the laser pointer (the present for my graduation)!