1. Initial Work on Suspension Diagonalisation – August 2004 Bryan W Barr

2. 2 Basic Concept… UL UR LR LL Force is applied to the mass by applying a signal to the 4 coils (UL, UR, LL and LR). If the force is not the same from all 4 coils then the mass will not be pushed cleanly in position, but will have some element of pitch and yaw motion. The aim of diagonalisation is to adjust the amplitude of the signals going to each coil such that the total force applied has minimal pitch and yaw coupling.

3. 3 Step by Step Take transfer function of pos->pitch and pos->yaw as a reference Diagonalise at low frequency: ~DC (note coil gains) Diagonalise at high frequency: 25 Hz (note coil gains) Work out the HF/DC ratio for all four coils Generate a filter to convert HF (gain=1) to DC (gain=ratio) and apply them to the Pos Output Filter Take transfer function of pos->pitch and pos->yaw and compare with reference – it should be better

4. 4 Diagonalise at DC Essentially, I went the simple route (possibly not the best though) of applying a DC offset of a few thousand counts to the LSC path of an optic Concentrating on the BS optic since that’s the one that seemed to have the highest priority Before applying the offset – note the values of the OSEM PITCH and YAW (oplevs change more but aren’t available on all optics) Apply offset and change the coil gains stepwise (and symmetrically) until the PITCH and YAW values are back to the start position Check by changing the offset – if the values change then it’s not diagonalised

5. 5 Diagonalise at HF This time, a sine wave is applied to the LSC_EXC point of the optic (at 25Hz) – I found 10000 counts to be OK-ish Take a power spectrum of the signal around the sine wave frequency – at least 10 averages and using a BW of 0.1Hz – can probably play around with this though Note the values of the peaks produced and adjust the coil gains to reduce the height of the PITCH and YAW peaks

6. 6 HF diagonalised No “before” picture here. Only after. Here : Pos=156, Pit=3.5, Yaw=1.1 are the optimised diagonalisation peaks at 25 Hz

7. 7 Matrices DC matrix:1.0041.019 0.7921.005 HF matrix:1.0141.0140.986 Relative gains:1.0100.995 1.0140.981

8. 8 Filter Shape ULPOS DC->HF filter Note the gain difference between low and high freq.

9. 9 Transfer functions Reference plotAfter filters Note the poor YAW performance. Thought alignment may have drifted so reset DC alignment and tried again.

10. 10 After DC Matrix Reset Reference plotAfter filters YAW is much better but PITCH is worse than for previous matrix and both are worse at high frequency. Hmmn… have I done something daft? The answer to that is a resounding yes!

11. 11 Ooops! Had the Silly Thing in Reverse! Rather embarrassingly, it appears I can’t count properly – in order to convert to DC from HF one requires gains of DC/HF instead of HF/DC I quickly threw together some filters and did a transfer function…

12. 12 New Improved Formulae Reference plotWith Proper Filters

13. 13 Status… Not the best we can get? Probably not – the question is how do we get better… Possibly change the filter resonant frequency to adjust the coupling at the pendulum peak – may not be correct Is there a better way of diagonalising? Currently, only the BS is done Could quickly diagonalise at DC for the other suspensions in the interim but the HF part takes longer…

14. 14 The End Needs more work Comments, suggestions and insults all welcome. Back in 6 weeks

15. 15 Details… The files included here can be found in caltech/dvsave/bryan/ Also, the small bit of scripting which I used to diagonalise at DC is in the scripts/test/ directory