Optimization of the system calibration for LISA Pathfinder

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

2. Outline • Model of LPF dynamics: what are the system parameters? • System calibration: how can we estimate them? • Optimization of the system calibration: how can we improve those estimates? Incidentally, we talk about: • Optimization method • System/experiment constraints Giuseppe Congedo - 9th LISA Symposium, Paris

3. Motivation Differential acceleration noise Uncertainties on the spectrum: system calibration • Statistical uncertainty: PSD estimation uncalibrated stat. unc. of PSD estimation • Parameter accuracy: system calibration calibrated • Parameter precision: optimization of calibration to appear in Phys. Rev. • 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 Giuseppe Congedo - 9th LISA Symposium, Paris

4. Model of LPF dynamics Science mode: TM1 free along x, TM2/SC follow direct forces on TMs and SC • force gradients (~1x10-6 s-2) guidance signals: reference signals for the drag-free and elect. suspension loops • sensing cross-talk (~1x10-4) • actuation gains (~1) Giuseppe Congedo - 9th LISA Symposium, Paris

5. Framework system calibration (system identification) optimization of system calibration (optimal design) parameters ω12, ω122, S21, Adf, Asus equivalent acceleration noise sensed relative motion o1, o12 diff. operator Δ Giuseppe Congedo - 9th LISA Symposium, Paris

6. System calibration oi,1 o1 oi,12 LPF system 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 loop Exp. 2: injection into the elect. suspension loop Giuseppe Congedo - 9th LISA Symposium, Paris

7. System calibration • The system response is simulated with a transfer matrix • The calibration is performed comparing the modeled response • with both translational IFO readouts • We build the joint (multi-experiment/multi-outputs) log-likelihood for the problem cross-PSD matrix residuals Giuseppe Congedo - 9th LISA Symposium, Paris

8. Calibration experiment 1 Standard design Exp. 1: injection of sine waves into oi,1 • injection into oi,1 produces thruster actuation • investigation of the drag-free loop black: injection Giuseppe Congedo - 9th LISA Symposium, Paris

9. Calibration experiment 2 Standard design 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 Giuseppe Congedo - 9th LISA Symposium, Paris

10. Optimization of system calibration Question: how can we optimize the experiments, to get an improvement in parameter precision? Answer: use the Fisher information matrix of the system (method already found in literature and named “theory of optimal design of experiments”) noise cross PSD matrix input parameters (injection frequencies) estimated system parameters input signals being optimized modeled transfer matrix evaluated after system calibration gradient w.r.t. system parameters Giuseppe Congedo - 9th LISA Symposium, Paris

11. Optimization strategy 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 practically speaking... Either way, the optimization seeks to minimize the “covariance volume” of the system parameters Giuseppe Congedo - 9th LISA Symposium, Paris

12. Experiment constraints • Can inject a series of windowed sines • Fix the experiment total durationT ~ 2.5 h • Divide the experiment in injection slots of duration δt = 1200 s each. • This set the fundamental frequency, 1/1200 ~ 0.83 mHz. • 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 Giuseppe Congedo - 9th LISA Symposium, Paris

13. System constraints For safety reason, choose not to exceed: • Thruster authority, 10% of 100 µN • Capacitive authority, 10% of 2.5 nN • Interferometer range, 1% of 100 µm → as the injection frequencies vary during the optimization, the injection amplitudes are adjusted according to the constraints above Giuseppe Congedo - 9th LISA Symposium, Paris

14. System constraints maximum injection amplitude (dashed) VS injection frequency oi,1 inj. (Exp. 1) oi,12 inj. (Exp. 2) interferometer interferometer • 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 Giuseppe Congedo - 9th LISA Symposium, Paris

15. Optimization of calibration Discrete optimization may be an issue! Overcome the problem by: overlapping a grid to a continuous variable space rounding the variables (inj. freq.s) to the nearest grid node using direct algorithms robust to discontinuities (i.e., patternsearch) initial-guess parameters ω12, ω122, S21, Adf, Asus optimized experimental designs optimization of system calibration system calibration best-fit parameters ω12, ω122, S21, Adf, Asus Giuseppe Congedo - 9th LISA Symposium, Paris

16. Optimization of exp. 1 & 2 • Improvement of factor 2 through 7 in precision, • especially for Adf (important for the subtraction of thruster noise) • There are examples forwhich correlation is mitigated: • Corr[S21, ω122]=-20%->-3%, Corr[ω122, S21]=9%->2% Giuseppe Congedo - 9th LISA Symposium, Paris

17. Optimization of exp. 1 & 2 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 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 • Exp. 1 • Exp. 2 • • • The optimization: • converges to the maxima of the transfer matrix • balances the information among them Giuseppe Congedo - 9th LISA Symposium, Paris

19. Effect of frequency-dependences nominal stiffness, ~-1x10-6 s-2 dielectric loss loss angle 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 However, the biggest contribution is due to gas damping, Cavalleri A. et al., Phys. Rev. Lett. 103, 140601 (2009) (Ar) (N2, gas venting directly to space) 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 Giuseppe Congedo - 9th LISA Symposium, Paris

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