EDIT, JCD, 17Feb2015: Updated loop diagram and calculation: http://131.215.115.52:8080/40m/11043
Okay, Koji and I talked (after he talked to Rana), and I relooked at the original cartoon from when Rana and I were thinking about this the other day.
The original idea was to be able to actuate on the MC frequency (using REFL as the sensor), without affecting the ALS loop. Since actuating on the MC will move the PSL frequency around, we need to tell the ALS error signal how much the PSL moved in order to subtract away this effect. (In reality, it doesn't matter if we're actuating on the MC or the ETMs, but it's easier for me to think about this way around). This means that we want to be able to actuate from point 10 in the diagram, and not feel anything at point 4 in the diagram (diagram from http://131.215.115.52:8080/40m/11011)
This is the same as saying that we wanted the green trace in http://131.215.115.52:8080/40m/11009 to be zero.
So. What is the total TF from 10 to 4?
.
So, to set this equal to zero (ALS is immune to any REFL loop actuation), we need .
Next up, we want to see what this means for the closed loop gain of the whole system. For simplicity, let's let , where * can be either REFL or ALS.
Recall that the closed loop gain of the system (from point 1 to point 2) is
, so if we let and simplify, we get
This seems a little scary, in that maybe we have to be careful about keeping the system stable. Hmmmm. Note to self: more brain energy here.
Also, this means that I cannot explain why the filter wasn't working last night, with the guess of a complex pole pair at 1Hz for the MC actuator. The ALS plant has a cavity pole at ~80kHz, so for our purposes is totally flat. The only other thing that comes to mind is the delays that exist because the ALS signals have to hop from computer to computer. But, as Rana points out, this isn't really all that much phase delay below 100Hz where we want the cancellation to be awesome.
I propose that we just measure and vectfit the transfer function that we need, since that seems less time consuming than iteratively tweaking and checking.
Also, I just now looked at the wiki, and the MC2 suspension resonance for pos is at 0.97Hz, although I don't suspect that that will have changed anything significantly above a few Hz. Maybe it makes the cancellation right near 1Hz a little worse, but not well above the resonance.
