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 Sat Sep 19 23:59:22 2020, anchal, Summary, ALS, ALS noise budget update Sun Sep 20 23:12:13 2020, rana, Summary, ALS, ALS noise budget update Tue Sep 22 00:14:43 2020, anchal, Summary, ALS, ALS noise budget update Tue Sep 22 12:14:42 2020, rana, Summary, ALS, ALS noise budget update Wed Sep 23 11:13:49 2020, anchal, Summary, ALS, ALS noise budget update Wed Oct 7 16:56:23 2020, anchal, Summary, ALS, ALS noise budget update - Updated AUX PDH Loop values Thu Oct 8 11:59:52 2020, rana, Summary, ALS, ALS noise budget update - Updated AUX PDH Loop values Fri Oct 9 18:32:14 2020, anchal, Summary, ALS, ALS noise budget update - Updated AUX PDH Loop values Wed Oct 14 17:03:55 2020, anchal, Summary, ALS, ALS noise budget update - Added whitening filter for ADC
Message ID: 15587     Entry time: Sat Sep 19 23:59:22 2020     Reply to this: 15589
 Author: anchal Type: Summary Category: ALS Subject: ALS noise budget update

### Setting the record straight

I found out an error I did in copying some control model values from Kiwamu's matlab code. On fixing those, we get a considerably reduced amount of total noise. However, there was still an unstable region around the unity gain frequency because of a very small phase margin. Attachment 3 shows the noise budget, ALS open-loop transfer function, and AUX PDH open-loop transfer function with ALS disengaged. Attachment 4 is the yaml file containing all required zpk values for the control model used. Note that the noise budget shows out-of-loop residual arm length fluctuations with respect to PSL frequency. The RMS curve on this plot is integrated for the shown frequency region.

### Trying to fix the unstable region

Adding two more poles at 100 Hz in the ALS digital filter seems to work in making the ALS loop stable everywhere and additionally provides a steeper roll-off after 100 Hz. Attachment 1 shows the noise budget, ALS open-loop transfer function, and AUX PDH open-loop transfer function with ALS disengaged. Attachment 2 is the yaml file containing all required zpk values for the control model used. Note that the noise budget shows out-of-loop residual arm length fluctuations with respect to PSL frequency. The RMS curve on this plot is integrated for the shown frequency region.

But is it really more stable?

• I tried to think about it from different aspects. One thing is sure that  $1+G_{OL}$ remains greater than 1 in all of the frequency region plotted for. This is also evident in the common-mode to residual noise transfer function which shows no oscillation peaks and is a clean mirror image of the open-loop transfer function (See Attachment 1, page 2).
• Another way is to look for the phase margin. This is a little controversial way of checking stability. For clarity, the open-loop transfer function I'm plotting does not contain the '-1' feedback in it. So the bad phase value at unity gain frequency is -180 degrees (or 180 degrees) for us. I've taken the difference from the closest side and got 76.2 degrees of phase margin.
• Another way I checked was by plotting a Nyquist plot for the open-loop transfer function. It is said that if the contour does not encircle the point '-1' in the real axis, then the loop would be stable even if the $f_{180} < f_{UGF}$ where $f_{180}$ is the frequency where phase lag becomes -180 degrees at the lowest frequency. For us, $f_{180}$ is at 1 Hz because of the test mass actuator pole. But I have verified that the Nyquist contour of the open-loop transfer function does not encircle '-1' point. I have not uploaded the Nyquist plot as it is not straight forward to plot. Because of large dc gain, it covers a large region and one needs to zoom in and out to properly follow what the contour is really doing. I didn't get time to make insets for it.

### Is this close to reality?

For that, we'll have to take present noise source estimates but Gautum vaguely confirmed that this looked more realistic now 'shape-wise'. If I remember correctly, he mentioned that we currently can achieve 8 pm of residual rms motion in the arm cavity with respect to the PSL frequency. So we might be overestimating our loop's capability or underestimating some noise source. More feedback on this welcome and required.

The code used to calculate the transfer functions and plot them is in the repo 40m/ALS/noiseBudget

Attachment 5 here shows a block diagram for the control loop model used. Output port 'Res_Disp' is used for referring all the noise sources at the residual arm length fluctuation in the noise budget. The open-loop transfer function for ALS is calculated by -(ALS_DAC->ALS_Out1 / ALS_DAC->ALS_Out2) (removing the -1 negative feedback by putting in the negative sign.) While the AUX PDH open-loop transfer function is calculated by python controls package with simple series cascading of all the loop elements.

 Attachment 1: ALS_nb_ExtraPoles.pdf  146 kB  Uploaded Sun Sep 20 01:29:15 2020
 Attachment 2: ALS_controls.yaml  2 kB  Uploaded Sun Sep 20 01:29:36 2020
 Attachment 3: ALS_nb_Kiwamus_Values.pdf  146 kB  Uploaded Sun Sep 20 01:29:54 2020
 Attachment 4: ALS_controls_Kiwamus_Values.yaml  2 kB  Uploaded Sun Sep 20 01:30:17 2020