IIR Filtering Ideas

This post is going to be in pseudo-formal speak since I'm fleshing out ideas for my dissertation. Although that's tough without adding any equations.

The problem of constructing the optimal IIR controller for a given closed-loop plant and disturbance model can be solved as an LQR problem with a particular state-space system. However, the condition that the plant transfer matrix is commutable with the filter turns out to be overly restrictive. Suppose the transfer matrix can be factored into the product of minimum and non-minimnum phase matrices. We can define a new filter which is the product of the minimum phase component and the optimal filter F, this leaving the non-minimum phase component to be compensated. The requirement is now that this non-minimum phase component is commutable, which happens if its equal to a scalar transfer function times the identify matrix. If this is satisfied, we can perform the LQR problem to identify the controller, then multiply it by the inverse of the minimum phase component to recover the actual optimal filter that is implemented in the software.

For the adaptive optics experiment, the lack of significant DM dynamics simplifies the problem. If the non-minimum phase transfer matrix consists solely of n-step delays on the diagonal, then the optimal IIR filter is simply the n-step Kalman predictor for the disturbance model. The disturbance model is itself in innovations form, thus the Kalman predictor can be constructed directly from the state-space model generated by the subspace identification algorithm.

...some math showing how this is done....

The result is a filter which predicts the disturbance wavefronts n-steps ahead on the basis of the current wavefront measurement.

Beautiful.