# Effect of closing SSFS loop on optical matrix

## Introduction

The coupling of CARM/FREQ in almost all channels is quite large and usually dominant over all other degrees of freedom. The very high gain of the SSFS at the active frequencies of the other loops decouples it very well.

The optical matrix computed by means of a Finesse simulation can be used as the starting point fro simulating the effect of closing any control loop. The finesse output is loaded by a Python script and processed to create a full frequency dependent optical matrix. It has as many rows as the number of channels considered (8: B1_ACp, B1_ACq, B5_ACp, B5_ACq , B2_8MHz_ACp, B2_8MHz_ACq, B7_ACp, B8_ACp) and as many columns as the number of DOFs (4: DARM, CARM, MICH, PRCL).

A control matrix is then defined, which simply connects one channel with one degree of freedom, by means of a suitable corrector filter, which contains also the contribution from the electro-mechanics if needed. The stability of the closed loop system is checked.

The SSFS loop is simulated by a very simple corrector, made of a pure integrator and a zero at 10 to roughly compensate for the optical transfer function. The UGF is set at 10 kHz for reference. The corresponding open loop transfer function has a rough 1/f shape, resulting is quite high gain at low frequency:

Then closing the loop is a simple computation of linear algebra. If M is the optical matrix and F the feedback matrix, the closed loop optical response of the photo-diode signals is given simply by

C = M * inv(1 + F*M)

## Effect of SSFS loop

The following plots compares the open loop optical transfer functions (solid lines) with the closed loop one (dotted lines):