We consider a Nehari problem for matrix-valued, positive-real functions, and characterize the class of (generically) minimal-degree solutions. Analytic interpolation problems (such as the one studied herein) for positive-real functions arise in time-series modeling and system identification. The degree of positive-real interpolants relates to the dimension of models and to the degree of matricial power-spectra of vector-valued time-series. The main result of the paper generalizes earlier results in scalar analytic interpolation with a degree constraint, where the class of (generically) minimal-degree solutions is characterized by an arbitrary choice of spectral-zeros. Naturally, in the current matricial setting, there is freedom in assigning the Jordan structure of the spectral-zeros of the power spectrum, i.e., the spectral-zeros as well as their respective invariant subspaces. The characterization utilizes Rosenbrock's theorem on assignability of dynamics via linear state feedback.
Bibliographical noteFunding Information:
Manuscript received August 01, 2008; revised May 07, 2009 and July 29, 2009. First published February 02, 2010; current version published May 12, 2010. This work was supported in part by NSF and AFOSR. Recommended by Associate Editor D. Henrion.
Copyright 2010 Elsevier B.V., All rights reserved.
- Analytic interpolation
- McMillan degree constraint
- Multivariable time-series
- Spectral analysis