Lately it's been very difficult to fit an efficient SRCL feedforward filter, as reported many times by Camilla et al. (76993). Here I'm trying to figure out why. Spoiler alert: I don't have an answer yet.

The main problem with the SRCL FF filter (see first plot), is that the transfer function to fit has a large phase rotation, that looks basically like a phase advance (the opposite of a delay) of about 2 ms. This is very large, and being an advance, it can't be realized in a precise and simple wasy digitally.

First observation: we can fit a pretty good SRCL FF if we allow for unstable poles, i.e. poles with positive real parts (see second plot) Of course this is not something that can be implemented in the real time system. The fit ends up having a unstable complex pole at about 420 Hz and about 5 Hz. I have no intepretation for the origin of those poles, and they might very well be only a way to reproduce a phase advance (think of the Padé approximation for phase delays).

So the question is: what is the origin of this large phase rotation? It's not seen at LLO, for example see 70548

Second observation: this phase advance appeared after we switched the LSC FF from ETMX (full chain) to ETMY PUM. The third plot compares the MICH and SRCL feedforward to be fit in two cases: an old measurement when the FF was going to ETMX, and a more recent measurement with the FF going to ETMY PUM only. For both MICH and SRCL the orange traces (FF to ETMY) show a phase advanced with respect to the blue traces (FF to ETMX). For some reasons I don't fully understand, this rotation is more problematic for SRCL than for MICH, although fitting MICH has also been more difficult and the MICH FF is relevant at lower frequencies than SRCL, so maybe the phase advanced isn't that problematic.

Looking at the measurement of the MICHFF to DARM and SRCLFF to DARM, one can see that there seems to be an additional phase delay in the FF path through ETMY PUM with respect to the FF path through ETMX full chain. Since this transfer function is at the denominator when computing the ratio SRCLtoDARM/SRCLFFtoDARM that gives use the LSC FF to fit, this seems to explain the additional phase advance we observe.

The ETMY PUM L2 lock filter bank contains a "QPrime" filter module that compensates partially the additional 1/f^2 due to the actuation from L2 instead of L3. This filter however doesn't seem able to explain this additonal phase delay.

I'm now suspicious that there might be something wrong or mistuned in the ETMY L2 drive, maybe a whitening filter missing or not functioning properly or not properly compensated?

It would be worth doing a quick test in the next commissioning time with full IFO locked: inject some white noise on ETMX L2 L and ETMY L2 L and comapre the two transfer functions to DARM. In theory they should be equal, except for a sign difference. If they're not, then there must be something wrong with ETMY, since we're using ETMX L2 to lock without issues.