Analytic interpolation with a degree constraint for matrix-valued functions

Mir Shahrouz Takyar, Tryphon T. Georgiou

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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.

Original languageEnglish (US)
Article number5404367
Pages (from-to)1075-1088
Number of pages14
JournalIEEE Transactions on Automatic Control
Issue number5
StatePublished - May 2010

Bibliographical note

Funding 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


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