I show here the matrix formalism to calculate analytically the loop TF relationships for the IMC w/ both FSS actuators so that it would be easier to interperet the results.

The attached PDF shows the Mathematica notebook and the associated block diagram.

In the notebook, I have written the single hop connection gains into the K matrix. P is the optical plant, C is the Common electronic gain, F is the 'fast' NPRO PZT path, and M is the phase Modulator.

G is the closed loop gain matrix. The notation is similar to matlab SS systems; the first index is the row and the second index is the column. If you want to find the TF from node 2 to node 3, you would ask for G[[3,2]].

As examples, I've shown how to get the FAST gain TF that I recently made with the Koji filter box as well as the usual OLG measurement that we make from the MC servo board front panel.