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 Tue Jul 25 02:03:59 2017, gautam, Update, Optical Levers, Optical lever tuning thoughts Fri Jul 28 15:36:32 2017, gautam, Update, Optical Levers, Optical lever tuning thoughts Tue Aug 1 16:05:01 2017, gautam, Update, Optical Levers, Optical lever tuning - cost function construction Thu Nov 16 13:57:01 2017, gautam, Update, Optical Levers, Optical lever noise Thu Nov 16 15:43:01 2017, rana, Update, Optical Levers, Optical lever noise Tue Nov 21 11:37:29 2017, gautam, Update, Optical Levers, BS OL calibration updated Tue Nov 21 16:28:23 2017, gautam, Update, Optical Levers, BS OL calibration updated Tue Nov 21 23:04:12 2017, gautam, Update, Optical Levers, Oplev "noise budget" Wed Nov 22 05:41:32 2017, rana, Update, Optical Levers, Oplev "noise budget" Wed Nov 22 15:29:23 2017, gautam, Update, Optical Levers, Oplev "noise budget" Wed Nov 22 16:40:00 2017, Koji, Update, Optical Levers, Oplev "noise budget" Wed Nov 22 19:20:01 2017, rana, Update, Optical Levers, Oplev "noise budget" Wed Nov 22 23:56:14 2017, gautam, Update, Optical Levers, Oplev "noise budget" Thu Nov 23 18:03:52 2017, gautam, Update, Optical Levers, Oplev "noise budget"
Message ID: 13156     Entry time: Tue Aug 1 16:05:01 2017     In reply to: 13147     Reply to this: 13432
 Author: gautam Type: Update Category: Optical Levers Subject: Optical lever tuning - cost function construction

### Summary:

I've been trying to put together the cost-function that will be used to optimize the Oplev loop shape. Here is what I have so far.

### Details:

All of the terms that we want to include in the cost function can be derived from:

1. A measurement of the open-loop error signal [using DTT, calibrated to urad/rtHz]. We may want a breakdown of this in terms of "sensing noises" and "disturbances" (see the previous elog in this thread), but just a spectrum will suffice for the optimal controller given the current noises.
2. A model of the optical plant, P(s) [validated with a DTT swept-sine measurement].
3. A model of the controller, C(s). Some/all of the poles and zeros of this transfer function is what the optimization algorithm will tune to satisfy the design objectives.

From these, we can derive, for a given controller, C(s):

1. Closed-loop stability (i.e. all poles should be in the left-half of the complex plane), and exactly 2 UGFs. We can use MATLAB's allmargin function for this. An unstable controller can be rejected by assigning it an extremely high cost.
2. RMS rrror signal suppression in the frequency band (0.5Hz - 2Hz). We can require this to be >= 15dB (say).
3. Minimize gain peaking and noise injection - this information will be in the sensitivity function, $\left | \frac{1}{1+P(s)C(s)} \right |$. We can require this to be <= 10dB (say).
4. RMS of the control signal between 10 Hz and 200 Hz, multiplied by the digital suspension whitening filter, should be <10% of the DAC range (so that we don't have problems engaging the coil de-whitening).
5. Smallest gain margin (there will be multiple because of the various notches we have) should be > 10dB (say). Phase margin at both UGFs should be >30 degrees.
6. Terms 1-5 should not change by more than 10% for perturbations in the plant model parameters (f0 and Q of the pendulum) at the 10% (?) level.

We can add more terms to the cost function if necessary, but I want to get some minimal set working first. All the "requirements" I've quoted above are just numbers out of my head at the moment, I will refine them once I get some feeling for how feasible a solution is for these requirements.

 Quote: An elog with a first pass at a mathematical formulation of the cost-function for controller optimization to follow shortly.

For a start, I attempted to model the current Oplev loop. The modeling of the plant and open-loop error signal spectrum have been described in the previous elogs in this thread.

I am, however, confused by the controller - the MEDM screen (see Attachment #2) would have me believe that the digital transfer function is FM2*FM5*FM7*FM8*gain(10). However, I get much better agreement between the measured and modelled in-loop error signal if I exclude the overall gain of 10 (see Attachments #1 for the models and #3 for measurements).

What am I missing? Getting this right will be important in specifying Term #4 in the cost function...

GV Edit 2 Aug 0030: As another sanity check, I computed the whitened Oplev control signal given the current loop shape (with sub-optimal high-frequency roll-off). In Attachment #4, I converted the y-axis from urad/rtHz to cts/rtHz using the approximate calibration of 240urad/ct (and the fact that the Oplev error signal is normalized by the QPD sum of ~13000 cts), and divided by 4 to account for the fact that the control signal is sent to 4 coils. It is clear that attempting to whiten the coil driver signals with the present Oplev loop shapes causes DAC saturation. I'm going to use this formulation for Term #4 in the cost function, and to solve a simpler optimization problem first - given the existing loop shape, what is the optimal elliptic low-pass filter to implement such that the cost function is minimized?

There is also the question of how to go about doing the optimization, given that our cost function is a vector rather than a scalar. In the coating optimization code, we converted the vector cost function to a scalar one by taking a weighted sum of the individual components. This worked adequately well.

But there are techniques for vector cost-function optimization as well, which may work better. Specifically, the question is  if we can find the (infinite) solution set for which no one term in the error function can be made better without making another worse (the so-called Pareto front). Then we still have to make a choice as to which point along this curve we want to operate at.

 Attachment 1: loopPerformance.pdf  44 kB
 Attachment 2: OplevLoop.png  24 kB  Uploaded Tue Aug 1 17:12:56 2017
 Attachment 3: OL_errSigs.pdf  36 kB  Uploaded Tue Aug 1 17:13:15 2017
 Attachment 4: DAC_saturation.pdf  40 kB  Uploaded Wed Aug 2 01:30:36 2017
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