J. Kissel, S. Dwyer, E. Hall, K. Arai, K. Izumi

I'm embarrassed to say I'm overwhelmed by the wealth of open loop gain transfer functions (OLGTFs) that have been taken recently. But, there's also a ton of stuff to explain, so I summarize comparisons and information from the 3 OLGTFs that were taken prior to this weekend here. More to come on this past weekend's new data.

Summary Points: 
- With a "streamlined" matlab model (similar in content to, but separate from the Noise Budget DARM loop model) we can predict the following DARM OLGTFs (measured between 5 and 300 [Hz]):
2015-03-10, Prior to DARM Loop Shape Modification LHO aLOG 17153
2015-04-02, Post DARM loop modification LHO aLOG 17642
2015-04-06, Post DCPD Analog Electronics Compensation Correction LHO aLOG 17710
to within 10% in magnitude and 5 [deg] in phase (between 5 and 300 [Hz]), with the following uncertainties (determined by the standard deviation of the model parameters for these three comparisons):
     - DC optical gain of 1.1e6 +/- 8%
     - residual, [unknown / unaccounted for] time delay of 93 +/- 30 [us]
- The DCPD Analog Electronics Compensation confusingly did *not* affect the frequency dependence of the OLGTF between 2015-04-02 and 2015-04-06, even to the 10-20% level Koji suggests. As of now, based on these measurements alone, I argue that the lack of discernible change in the OLGTF suggests that we do *not* need to make any correction to our CAL-CS front end calibration.
- Just because I can model the open loop gain TFs to this uncertainty level does *not* mean the overall DARM calibration has this uncertainty. We still have a (potentially) large variation in ESD actuation coefficient due to charge, a (potentially) large uncertainty in the DARM coupled cavity pole frequency, a (potentially) large optical gain fluctuation from lock to lock, and (potentially) large discrepancy between the real actuation strength and what we've measured, and all things of which we haven't realized.

Detail Points:
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- Kiwamu had put together a "simple" model of the DARM loop when he'd updated the DELTA L EXTERNAL calibration in the CAL-CS front-end filters after the second-to-last time the DARM loop shape was changed (see LHO aLOG 17151). As he mentions in his aLOG, this code lives here:
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Scripts/DARM_OLTFGTF_LHOaLOG17153.m
That model had included
    Actuation Function
    - The QUAD SS Model that have the current foton filter file's damping filters engaged with gains that are hard-coded into the generate_QUAD_Model_Production function to gather the necessary damped SUS transfer functions in [m/N], i.e. the output of 
    /ligo/svncommon/SusSVN/sus/trunk/QUAD/Common/MatlabTools/QuadModel_Production/generate_QUAD_Model_Production.m
    using the 
    /opt/rtcds/lho/h1/chans/H1SUSETMX.txt
    foton file.
    - An assumed DAC gain of 2^18 / 20 [V/ct].
    - The UIM analog electronics chain using 
    /ligo/svncommon/SusSVN/sus/trunk/Common/MatlabTools/make_OSEM_filter_model.m
    all parameters of which are "as designed" and not measured. This turns the [N] of drive at the UIM into [ct] of drive at the output of the ETMX_L1_LOCK_L bank.
    - The ESD analog electronics chain using a simple, linearized model of 2*alpha*Vbias*40^2, where 40 is the range of the DAC, Vbias is the bias voltage, in units of the DAC [V], and alpha is the force coefficient. He uses the force coefficient that has been confirmed by measurement to be (2.0 * 1.417)e-10 = 2.834e-10 [N/V^2]. (see LHO aLOG 16843) 
    - The ESD driver's single pole filter
    - The correct hierarchical filters from the current foton file (same as above), with known gains grabbed from conlog.
    The DARM filter
    - Loaded in from the H1OMC foton file from the filter archive,
    /opt/rtcds/lho/h1/chans/filter_archive/h1omc/H1OMC_1110098950.txt,
    with known gains from conlog.

    The Sensing Function
    - Assuming the Actuation and DARM DC gains are correct, the remaining scale factor needed to match the OLGTF model to the measurement is what is used for the interferometer's optical gain, in [m/ct]
    - A single-pole zpk filter, with the pole at 389 [Hz]

    Time Delay
    - An overall true time-delay to account for *all* the high frequency behavior like anti-imaging and anti-aliasing filters.

I've used this model as a template to get started, but I've made the following changes to (a) be able to compare many OLGTFs against the model that best represents the DARM loop at that time, and (b) include the now-better-known high-frequency effects:
(1) Pulled out the hard-coded parameters of the model, and put them in a separate function identified by their GPS time,
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Scripts/
H1DARMparams_1109994128.m
H1DARMparams_1111998876.m
H1DARMparams_1112399129.m
(2) I've modified the filter files chosen such that both the OMC *and* the SUS are pulling from the filter archive, not the live data. This will be come relevant later when we track changes in UIM / TST crossover, or when we begin using all stages like LLO has.
(3) I've now included both the analog and digital anti-aliasing in the sensing function, and the analog and digital anti-imaging in the actuation function. The digital AA and AI filters are the same, from the function
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/S7/Common/MatlabTools/iopdownsamplingfilters.m
and the analog AA and AI filters are the same, taken from .mat file inherited from the 40m, which has been pruned and now lives in
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/S7/Common/MatlabTools/analog_65k_AAAI_filter_response.mat

(4) I've included the known time delays:
Sensing: 1 IOP Computation Cycle (65k) + 1 OMC Computation Cycle (16k)
Actuation:1 SUS Computation Cycle (16k) + 3-Cycle IOP Error Checking (65k) + 1 IOP Computation Cycle (65k) + Zero-order Hold Delay (1/2 a 65k cycle)
(which total 205 [us]) and created a new parameter for the "unknown" time delay, which is fit to make the high-frequency phase flat.

Known remaining flaws in the model:
- Mismatch in digital compensation for analog whitening or pre-amps of the DCPDs -- but this should now be *much* better than before (see LHO aLOG 17650)
- Mismatch in digital compensation for analog whitening of the SUS UIM coil driver and now the ETMY ESD Driver filter 
    -- Measurements have shown that the UIM driver agrees quite well with the model (see LHO aLOG 4495), but this hasn't been confirmed for these SUS individually and/or after their increase in range (Integration Issue 762).
    -- The "temporary" ESD low-noise filters (E1500164) were compensated with their designed poles and zeros, I don't know how well they're matched.
- The model uses 389 [Hz] for the DARM coupled-cavity pole, but this is a modelled value from the L1 IFO's losses (see LHO aLOG 15923). Our losses are larger, so we expect our cavity pole frequency to be higher. More to come from Paul and Daniel on this. The data out to 1 [kHz] from over the weekend should also help with getting a measured confirmation.

This model is a function which takes in the parameter files, is called 
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Scripts/H1DARMmodel_preER7.m,
and returns the modelled and measured open loop gain as well as every single possible thing you could possibly ever want out of the model in a parameter structure (including the things that are independently computed in the model itself, like the full actuation function and the full sensing function).
This model can then be looped over, which I've done with the function
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Scripts/CompareDARMOLGTFs.m

I *hope* that this makes it much easier to compare a new DARM OLGTF against others as we take them.