Comments on "Realization and reduction of Markovian models from nonstationary data"

In the above paper, Baram determines a basis in which to construct realizations from measurement data and for model reduction of these realizations. The basis selected by Baram is similar to the "cost-decoupled" basis introduced earlier [1], [2], with one exception. Both have diagonal state covariances, but the second matrix diagonalized is different. Of course, this makes the two algorithms different and the nature of these differences is pointed out in this correspondence. The principal difference is that Baram's algorithm for model reduction approximates (in a least squares sense) all of the covariance sequences, whereas the model reduction of [1], [2] matches the first two covariance sequences exactly. The cost decoupled basis guarantees several different properties: 1) output covariance matching, and 2) an equivalent quadratic performance metric (i.e., "cost-equivalent" realizations).