A. Effler, J. Kissel, B. Shapiro

I've reconstructed the filters and gains that have been used to damp the H2/H1SUSETMY for the H2OAT, calibrating them with the appropriate signal chain gains in an attempt to finally merge models with measured reality. 

I attach two sets of plots.
(1) dampingfilters_H1SUSETMY_20121121.pdf: shows the filters, both in their raw uncalibrated (normalized) form (pg 1), as well as in their calibrated form (pg 2) using the EPICs gains that are currently in place.
(2) 2012-11-21_QUADDampedModel_Top2Top_TFs.pdf: shows a comparison between undamped and damped, model and measurement. The measurements used were taken from H1 SUS ETMY's Phase 3b results.

There is now excellent agreement with model and measurement for all degrees of freedom, most importantly with entirely understood frequency responses and gains. The remaining discrepancies arise from the differences in the undamped modeled plant and measured plant, namely the high frequency zero we believe to arise from drive and sensing chain cross coupling.

Now that model and measurement agree to such precision, we can now
- Explore compensation for the high-frequency zero
- See that this current set of filters and gains over damps most degrees of freedom, most likely reinjecting too much sensor noise into the system for a fully operational aLIGO IFO
- Accurately predict sensor and actuator noise at the test mass and design filters that will meet aLIGO requirements accordingly 
- Comfortably design hierarchical control filters which take into account realistic damping loops.

--------------------------------
Details:

Why we hadn't gotten here before today:
There were three pieces missing from the puzzle that had previously resulted in inaccurate models. 
(1) Prior to this update, the model had still been using legacy filters (in frequency response alone), and was using gains fudged by hand to get roughly the desired damping. This was "good enough" for the purposes needed to date, so we hadn't put much brain power into it. Perhaps more accurately I'd been previously stumped by the bug in the connection matrix described here, which is now fixed.
(2) The filter frequency response, along with using the actually-installed-EPICs gains, were were designed in foton "by hand." I've now exported and reproduced them in Matlab. The filters are calibrated into [N/m] or [N.m/rad] using the signal chain described briefly below (and in detail in LHO aLOG 4756 and LHO aLOG 4563). This calibration is needed because the model is (and has always been) calibrated in physical, SI units. The model of the closed loop is therefore

                                                             +--[m/N]--+
Ext. Forces -[N]-->(+)---------------[N]---------------------|  Plant  | ----------------[m]---------------------+-----> 
                    |                                        +[rad/N.m]+                                         |
                   [N]                                                                                          [m]
                    |   +-------------+              +----------+  +---------+                 +-------------+   |
                    +---|    Drive    |<--[drv. ct]--|EPICS GAIN|--|  DAMP   |<-(+)-[sns. ct]--|    Sense    |<--+
                        +-[N/drv. ct]-+              +-[ct/ct]--+  +-[ct/ct]-+   ^             +-[sns. ct/m]-+
                                                                                 |
                                                                                 +-- EXC

where the DAMP filter units, [ct/ct], are the designed, normalized frequency response. Note, 
- This reconstruction was done using the function
${SusSVN}/sus/trunk/QUAD/Common/FilterDesign/reconstructquadfotonfilters_20121121.m, where you can see the details of each filter design. I've saved the filters in the file
${SusSVN}/sus/trunk/QUAD/Common/FilterDesign/dampingfilters_QUAD_20121121.mat

- In practice, thanks to the powers of linear algebra, instead of treating the sensor chain (Sense) and actuator chain (Drive) independently, I merely multiply the normalized filters by the total DC gain of the signal chain,
 
              % <----------- Sensor Chain [sns. ct/m]------------>   <--- Actuator Chain [N/drv. ct] --->;
              %     OSEM                     SatAmp                              Coil Driver Coil/Magnet ;
              %  Sensitivity                TransImp.    ADC            DAC       TransCond. Force Coeff.;
              %(   [mm/uA]    [m/mm]  [uA/A]   [A/V]   [V/sens ct]   [drive ct/V]    [V/A]      [A/N]     )^-1= [(sns. ct/m) . (N/drv. ct)];
calibration = (( (0.7/76.29)*(1/1e3)*(1e6/1)*(1/240e3)*(40/2^16) ) * ( (2^18/20)*(1/0.009943)*(1/1.694) ))^-1;
            = 55.071 % [(sens ct/m) . (N/drive ct)] or [(sens ct/rad) . (N.m/drive ct)] 


(3) Previously, the damping filter's overall gain had been positive, and the feedback negative sign had been taken care of in the connection matrix, where the filters were hooked into the undamped state space model, in
${SusSVN}/sus/trunk/QUAD/Common/MatlabTools/QuadModel_Production/generate_QUAD_Model_Production.m.
Now that I'm using the actual EPICs gains, where the negative sign is explicitly called out, the sign in the connection matrix has been removed. I mention this only because it had stumped my for a day (let it be a less kids: always pay attention to the phase of your transfer functions!). Note this means all future filters input to generate_QUAD_Model_Production.m must have the overall sign be negative (regardless of whether it's included in the filter itself or explicitly in the EPICs gain).

These changes have been committed to the repository, with these 20121121 filters currently set as default; please svn up!