5.30.10

I've been sidetracked the last couple weeks working on this idea of quantifying frozen flow layers. The problem is to identify how many layers are moving in a phase profile, and estimate their velocities. A common idea among some "predictive" AO controllers is to then use this information to generate control commands some number of steps in the future.

Of course, this only works when the velocities are constant and pretty well known, but it seems to be a common approach among certain fields. Based on some comments I've heard, they like it because it incorporates some knowledge about the physics behind the problem. I think deep down some of them just don't trust the completely black box methods that's common in hard core controls applications.

We typically use one of these feared methods to identify a state space model for the turbulence. One question that's bothered us though, is how can we extract the velocity and layer information? Since the controller developed from the state space is optimal, the velocity info has to be embedded in there, but since the states are a product of the ID it isn't clear how.

One approach people in the AO community have tried is to generate a bunch of image correlations from the data, and look for peaks. If the phase is composed of a finite number of layers moving with distinct velocities, the correlations between images separated by enough delay should develop peaks corresponding to each layer. In our case, we have a state space model. And while we could just generate a sequence of data and use these methods, I'd be cooler if we could identify the velocity straight from the system matrices directly.

Coincidentally, I've been reading this book on subspace identification, and it has a good review about calculating the state and output covariance matrices for a state space system. Its very easy to compute the covariance matrices for any number of time steps, so I started to wonder if you could compute the covariance function directly from these matrices. After much, much head banging, it turns out you can.

I don't want to reveal the exact details, but I'll just say even though its not theoretically complicated, its pretty cumbersome in the 2D case, and required many many cups of coffee and a nontrivial amount of cursing to figure out. I still haven't tried it out on real data, but in all the simple 2 layer, integer velocity cases I've developed it works swimmingly, and seems relatively robust to random similarity transformations to the state space.