In LHO:46179 we reported an anomalous phase advance in the DHARD PIT OLTF at ~ 0.8 Hz. 

It turns out that such a feature can be modeled by including cross-couplings in both actuators and sensors. Please see the attached plot. 

In the model we considered the single-arm case which is equivalent and the X and Y arms are balanced. On the other hand, we assumed the ETM has an actuation strength 10% stronger than the ITM, and each soft mode angle we sensed was actually one ideal soft mode minus one ideal hard mode. Without any fine-tuning of the parameters, the model taking into account the cross-couplings almost perfectly matched to the measurement data. 

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The mathematical details were also attached (actImba.pdf). The basic idea is to project all the quantities with imperfections onto the ideal hard-soft basis (also note that the hard/soft radiation feedback is independent of the strength of actuation magnets), and then extend the regular SISO loop analysis to the MIMO case. The latter is easily achieved by replacing the vector transfer functions into transfer matrices. The effective oltf is then obtained by first finding the close-loop response and then use oltf= 1/cltf - 1.