Generalizing the Markov and covariance interpolation problem using input-to-state filters

In the Markov and covariance interpolation problem a transfer function W is sought that match the first coefficients in the expansion of W around zero and the first coefficients of the Laurent expansion of the corresponding spectral density WW*. Here we solve an interpolation problem where the matched parameters are the coefficients of expansions of W and WW* around various points in the disc. The solution is derived using input-to-state filters and is determined by simple calculations such as solving Lyapunov equations and generalized eigenvalue problems.