J. Kissel

I've installed the first cut of the inverse actuation filter for the hardware injection team, based on the model shown in LHO aLOG 18050, but originally described in detail, with uncertainties in LHO aLOG 17951 and LHO aLOG 18039. Exactly the same filters have been installed in both the HARDWARE and BLND filter banks.

Design Details:
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I've divided the filter into two banks: 
- FM1 "Inv.Act" = the "main" [ct/m] inverse actuation function described below, and 
- FM10 "m_per_strain" = a gain-only filter that is equivalent to the mean IFO arm length of mean([3994.4704 3994.4692]) =  3994.47 [m]

For the "main" [ct/m] inverse actuation function I
- saw that the model parameters of the past six actuation functions we've reconstructed show results that are virtually identical above 20 [Hz], (see pg 1 of attached .pdf)
- grabbed the latest one of these actuation function models from Apr 15 2015 (parameters from H1DARMparams_1113119652.m),
- inverted it, and
- compared/"fit" it against a simple zpk filter that I knew I could easily hand-implement in foton (see pg 2 of the attached .pdf)

I ended up with the following filter:
Matlab Notation:
    toyModel.poles_Hz = [pair(4e3,51) 6e3];
    toyModel.zeros_Hz = [0.01 0.01];
    toyModel.gain     = 5.98e9;
Foton Design String:
    zpk([0.01;0.01],[2517.28+i*3108.58;2517.28-i*3108.58;6000],1,"n")gain(9.51664e+08)
where the gain has been normalized to match the real model at ~100 [Hz] in matlab, and forced to match the matlab at 99.54 [Hz] in foton (see gain forcing in 2015-04-29_H1DARMINJ_FotonDesign_and_MEDM.png). 
Bode plots of these filters implemented in foton can be found in the last two pages of the attached .pdf.

HOWEVER
You'll notice that the "real" model of the inverse actuation function (a) goes to infinity at high frequency, and (b) has phase behavior equivalent to that of a time *advance*. As such, I've made the following design choices / approximations with this simple model of the inverse actuation:
- Without the ability to make higher-than-Nyquist frequency poles, or to have a filter with more zeros than poles, I've rolled off the function at ~5 [kHz] with the above design (note the pair of poles with some complex phase in between is to tweak up the magnitude to achieve good fidelity reproduction out to 2 [kHz]).
- Of course, we must also, unfortunately have *casual* filters in the front end. As such, I compute the amount of time *advance* we need to reconstruct the phase response reasonably well. This is ~250 [us]. or 4.1 16384 [Hz] cycles.

I'M HOPING that the hardware injection team has this ability, to merely time-shift or "mis"-time-stamp their injection at the sub-millisecond time precision, or some how program the injection to happen 4 clock cycles "in the future." We'll be discussing this on the HW INJ call tomorrow to be sure.

Comments on Accuracy and Precision
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Recall from LHO aLOG 18039, that we are confident in the frequency dependence of the open loop gain to +/- 2.5% in magnitude and 1 [deg] between 15 and 700 [Hz] (and probably out to several [kHz], we just don't have any measurements to that high a frequency yet). However, the uncertainty in the overall scale factor is still ~26% (because we haven't yet discerned the scale factor uncertainty between optical gain fluctuations of the sensing function and ESD actuation strength varying due to charge fluctuations).

As such, I would assign the same uncertainty to the inverse actuation function alone, in addition to the residuals between the real model and this simple fit to it. These residuals are less than 1% and 0.5 [deg] between 15 - 1500 [Hz]. In summary, the 1-sigma, 68% confidence intervals are
Overall Scale: 26%, Frequency Dependence: 2.7%, 1 [deg] between 15 and 700 [Hz] (but likely out to 2 [kHz])