A series of measurements / calculations for the PRM ASC characterization and servo design

1) Actuator characterization

The actuator responses of the PRM in pitch and yaw were measured (attachment figure 1). I believed the calibration of the oplev QPD to be
1 count/urad. The oplev servo loops were turned off at the FM inputs, and the filter banks were turned off so that the response has the open
loop transfer function except for the servo filter.

The measured transfer functions were fitted with LISO. The LISO results (c.f. the source codes) were shown in the figure. The responses also
include the 60Hz comb filter present in the input filters. The responses are well approximated by the single pendulum with f0 of 0.6-0.8 and q of 3.5 and 6.3.

From this measurement, the actuator responses of the PRM at DC are estimated to be 2.2 urad/cnt and 1.8 urad/cnt in pitch and yaw, respectively.

2) Sensor response of the POP QPD

As we already know how the actuators respond, the QPD optical gain can be characterized by measuring the actuator response of the QPD
(attachment figure 2). The QPD signals are such noisy that the response above 1Hz can't be measured with sufficient coherence. Below 1Hz,
the response is well represented by the actuator response measured with the oplev. From this measurement, the optical gains of the QPD
with respect to the PRM motion are 650 cnt/urad and 350 cnt/urad.

3) Open loop transfer function of the current ASC servo

By combining the above information with the servo setting of the servo filters, the open loop transfer functions of the PRM QPD ASC loops
were estimated (attachment figure 3). Actually the expected suppression of the fluctuation is poor. The yaw loop seems to have
too low gain, but in fact increasing gain is not so beneficial as there is no reasonable phase margin at higher frequency.

With the estimated openloop transfer functions and the measured free-running angular fluctuation, the suppressed angular spectra can be
estimated (attachment figure 4). This tells us that the suppression of the angular noise at around 3Hz is not sufficient in both pitch and yaw.
As there is no mechanical resonance in the actuator response at the frequency, intentional placement of poles and zeros in the servo filter is necessary.

4) Newly designed ASC filter

Here is the new design of the QPD ASC servo (attachment figure 5). The target upper UGF is 10Hz with the phase margin of 50 to 60deg.
The servo is AC coupled so that we still can tweak the alignment of the mirror.

As this servo is conditionally stable, at first we should close the loops with stable filter and then some boosts should be turned on.
Estimated suppressed fluctuation is shown in the attachment figure 6. We can see that the fluctuation was made well white between 0.5Hz to 10Hz.

The filter design is shown as follows:

Pitch
FM1: zero at 0Hz, pole at 2000Hz, gain at 2000Hz = 2000

FM3: (boost)
zero: f: 0.5Hz q: 1  /  4.5Hz, q: 1 / f: 1Hz, q: 3
pole: f: 2Hz q: 3  / f: 2.7Hz, q: 2  / f: 1Hz, q: 15

FM9: (HF Roll-off)
pole: f: 40Hz q: 1.7
 
Servo gain: -0.028

Yaw
FM1: zero at 0Hz, pole at 2000Hz, gain at 2000Hz = 2000

FM3: (boost)
zero: f: 0.7Hz q: 2  /  3Hz, q: 7 / f: 2Hz, q: 6
pole: f: 1.02Hz q: 10  / f: 4.5Hz, q: 0.8  / f: 1.5Hz, q: 10

FM9: (HF Roll-off)
pole: f: 40Hz q: 1.7
 
Servo gain: -0.0132