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Enhanced Analysis of 2-Input, 2-Output Dynamical System Modeling with Cross-Talk Considerations

This technical note presents an updated assessment of a 2-input, 2-output dynamical system model as developed by S. Vitale. The study focuses on effective dynamics, emphasizing the necessity for only two inputs through thrusters or electrostatic means while addressing actuation cross-talk. The analysis explores signal amplitude adjustments, errors in input response, and the implications of imperfection in measurements. Key results highlighted include methodologies to distinguish actuation cross-talk and noise, with suggestions for further improvements and technical notes.

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Enhanced Analysis of 2-Input, 2-Output Dynamical System Modeling with Cross-Talk Considerations

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  1. Master Plan Analyses DA#6 S. Vitale

  2. X-dynamics update S. Vitale

  3. The changes S. Vitale

  4. 2-inputs/2-outputs system (IFO signals x1, x) Multi parameter dynamical model oi,1 o1 oi, o The effective dynamics S. Vitale

  5. Basic dynamics • 2-body difference of force drives the two differential coordinates. • Only two inputs are needed • Force on x1 can be applied either via oi1 (Thrusters) or via gx1 (Electrostatic) • If gx1=-Hdfoi1 formulas stay the same but cross-talk coefficient changes meaning. • Careful: to apply gx1 must increase stiffness. S. Vitale

  6. Signal amplitude adjusted to compensate for gain and to avoid force saturation S. Vitale

  7. Blue: x. Red: x1 S. Vitale

  8. Real displacement S. Vitale

  9. Results S. Vitale

  10. Same approach: response linear in imperfections Imperfections unknown amplitudes to be extracted by Wiener-Kolmogorov (vectorial) theory Stimuli also applied as forces to distinguish actuation cross-talk y-x Cross-Talk S. Vitale

  11. Starting point S. Vitale

  12. Force input “Displacement” input Linearisation S. Vitale

  13. S. Vitale

  14. Dynamics 2: y- nominal Acting on TM for S/C attitude TM attitude irrelevant to first order S. Vitale

  15. Dynamics 2: Solving for S/C acceleration S. Vitale

  16. Dynamics 3: x and imperfections • No inputs on x • No imperfections on x • Nominal response = 0 S. Vitale

  17. S. Vitale

  18. Various Matrices S. Vitale

  19. Dynamics and control imperfections S. Vitale

  20. Signal and actuation imperfections S. Vitale

  21. Two cases: Displacement command within drag-free loop Excites thrusters Probes thrusters cross-talk Electrostatic TM cross-talk only excited through very low frequency attitude control Force command on TM1 and TM2 Probes electrostatics Numerical Calculation S. Vitale

  22. Acceleration noise on x axis. Noise cross-talk neglected Channel 1 (S/C-TM1) Channel  (TM2-TM1) |Cross-spectrum| S. Vitale

  23. Case 1: displacement excitation y1 y2 Force limited < 10 µN S. Vitale

  24. Excitation S. Vitale

  25. Response S. Vitale

  26. Response, absolute attitude S. Vitale

  27. Results (Preliminary): statistically independent combinations Signal cross-talk y-to- measured with high resolution If ifo sensitivity calibrated on ground, thruster cross-talk and cross-stiffness can be disentangled S. Vitale

  28. Case 2: force excitation (<1µm displacement) y1 y2 S. Vitale

  29. Force excitation S. Vitale

  30. Displacement response S. Vitale

  31. Results (preliminary): statistically independent imperfections Actuation cross-talk measured with high resolution Signal cross-talk y-to- also measured scaled for smaller excitation S. Vitale

  32. Actuation cross-talk channel 1 S. Vitale

  33. Actuation cross-talk channel  S. Vitale

  34. Signal cross-talk channel  S. Vitale

  35. Key element: converting data from displacement to acceleration Acceleration PSD S. Vitale

  36. Time domain operation more transparent Double derivative required Non-casual filter preferred Currently in torsion pendulum: 2nd order fit to 5 points Getting the filter S. Vitale

  37. S. Vitale

  38. Next steps Complete technical note Complete analysis for acceleration PSD Get the number for z -x cross-talk -x cross-talk More difficult -x cross-talk Planning S. Vitale

  39. S. Vitale

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