Summary: At the 40m meeting yesterday, Eric Q. gave the suggestion that we accept the input matrix weirdness and adjust the output matrix by driving each coil individually so that it refers to the same degrees of freedom. After testing this strategy, I don't think it will work.
Yesterday evening I tested this idea by driving one ITMY coil at a time, and measuring the response of each of the free swing modes at the drive frequency. I followed more or less the same procedure as the standard diagonalization: responses to each of the possible stimuli are compared to build a matrix, which is inverted to describe the responses given the stimuli. For the input matrix, the sensor readings are the responses and the free swing peaks are the stimuli. For the output matrix, the sensors transformed by the diagonalized input matrix as the responses of the dofs which are compared, and the drive frequency peak associated with a coil output is the stimulus. However, the normalization still happens to each dof independently, not to each coil independently.
The output matrix I got had good agreement with the ITMY input matrix in the previous elog: for each dof/osem the elements had the same sign in both input and output matrices, so there are no positive feedback loops. The relative magnitude of the elements also corresponded well within rows of the input matrix. So the input and output matrices, while radically different from the ideal, were consistent with each other and referred to the same dof basis. So, I applied these new matrices (both input and output) to the damping loops to test whether this approach would work.
drive-generated output matrix:
UL UR LR LL SD
pit 1.701 -0.188 -2.000 -0.111 0.452
yaw 0.219 -1.424 0.356 2.000 0.370
pos 1.260 1.097 0.740 0.903 -0.763
sid 0.348 0.511 0.416 0.252 1.000
but 0.988 -1.052 0.978 -0.981 0.060
However, when Gautam attempted to lock the Y arm, we noticed that this change significantly impacted alignment. The alignment biases were adjusted accordingly and the arm was locking. But when the dither was run, the lock was consistently destroyed. This indicates that the dither alignment signals pass through the SUS screen output matrix. If the output matrix pitch and yaw columns refer instead to the free swing eigenmodes, anything that uses the output matrix and attempts to align pitch and yaw will fail. So, the ITMY matrices were restored to their previous values: a close to ideal input matrix and naive output matrix. We could try to change everything that is affected by the output matrices to be independent of a transformation to the free swing dof basis, and then implement this strategy. But to me, that seems like an unneccessary amount of changes with unpredictable consequences in order to fix something that isn't really broken. The damping works fine, maybe even better, when the input matrix is set by the output matrix: we define pitch, for example, to be "The mode of motion produced by a signal to the coils proportional to the pitch row of the naieve output matrix," and the same for the other dofs. Then you can drive one of these "idealized" dofs at a time and measure the sensor responses to find the input matrix. (That is how the input matrix currently in use for ITMY was found, and it seems to work well.)