The main benefit of doing radiation pressure compensation is that we only need to handle a single suspension plant. If the plant is unchanged, we can invert it or utilizing its features in the digital control filter banks to improve the loops' noise performance.
For example, we can use optimal control techniques (specifically, the H-infinity design) to optimize the DHARD PIT loop, which enables x10 less sensing noise injection to DARM at 10 Hz yet with the same phase margin (30 deg) and a good enough residual rms level (< 1 nrad).
Please see the attached plots for details:
1. In oltf_Hinf_fit.pdf we compare the oltf using H-inf (the red trace) and the one currently used (the black trace; based on the model in LHO:46179 with a 10 W plant which seemed to match measurements done at both 10 W no RPC and at 20 W w/ RPC). The red trace has a ugf of 2.3 Hz with 30 deg phase margin (same as the black trace; the margin can be further increased if we slightly reduce the DC gain of the loop). On the other hand, the roll-off for the H-inf design is much faster than the one used currently, and it should enable about a factor of 10 more reduction of sensing noise injection to DARM in the 10-20 Hz band. At the same time, there is less phase delay at ~ 1 Hz, so the H-inf design might actually be more robust than the current filter at 1 Hz against cross-couplings.
2. We extract from the overall oltf the control filter should go into the DHARD_P filter module in ctrl_Hinf_fit.pdf. Again a comparison between how the H-inf vs. current should look like is shown in the plot.
In case people were interested in implementing it, we also provide the foton file that can be directly put into the filter in dh_p_ctrl.txt. (Only need to change the ctrl filter bank while leaving the DC gain the same, which was 30 in the model we adopted).
3. The anticipated closed-loop noise performance is shown in cl_noise_hinf_fit.pdf. The low-freq rms is ~ 0.7 nrad < 1 nrad that we need (here we assumed no reduction of input noise from the ISIFF). As a comparison, using the original loop design the rms is 0.4 nrad for the same input noise. Therefore in terms of motion stabilization the two loops should be similar. Also shown is the requirement on roll-off so that it equals to the aLIGO design noise. Here we have assumed a sensing noise of 1e-14 rad/rtHz for DHARD (LHO:46178) and an a2l coupling of 1 mm/rad.
Jenne, Hang
We also show the closed-loop response in the attached plot. The amount of gain peaking is similar for the H-inf design and the original loop.
To check the robustness of the controller, we plot the closed-loop responses with suspension plants corresponding to 8 W (blue) and 15 W of input powers. The 8 W one starts to become marginally stable but the 15 W one still seems fine. Since the RPC gain was adjusted every 2 or 3 W change of input power, and we set the gain such that it would tend to under-subtract the RP (i.e., the sus plant corresponds to input > 10 W) than over-subtract (< 10 W). The step size bounds the |error| < ~ 3 W, and the under-subtraction makes it more likely to be 12 W or 13 W. Thus the controller should be stable given the level of error likely to be happening in the RPC.
Also currently we adjust the RPC gain discretely due to lack of commissioning time. In the future the gain can be continuously/adaptively adjusted by monitoring the TR QPD sum value relative to a reference point. The gain also depends on the ASC optical response converting the digital counts back to physical angle in radians, which can be monitored with diagnostic dithering lines (low amp & @ < 10 Hz). (Also if the TCS is settled, the optical gain was generally quite stable over a lock stretch). So in the future the level of plant fluctuation should be even smaller.