Parametric polynomial spectral factorization using the sum of roots and its application to a control design problem

Authors:
Hirokazu Anai;Shinji Hara;Masaaki Kanno;Kazuhiro Yokoyama
Affiliations:
Fujitsu Laboratories Ltd, 4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki, 211-8588, Japan;The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan;CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi-shi, Saitama, 332-0012, Japan;Rikkyo University, 3-34-1 Nishi Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan
Venue:
Journal of Symbolic Computation
Year:
2009

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Abstract

This paper presents an algebraic approach to polynomial spectral factorization, an important mathematical tool in signal processing and control. The approach exploits an intriguing relationship between the theory of Grobner bases and polynomial spectral factorization which can be observed through the sum of roots, and allows us to perform polynomial spectral factorization in the presence of real parameters. It is discussed that parametric polynomial spectral factorization enables us to express quantities such as the optimal cost in terms of parameters and the sum of roots. Furthermore an optimization method over parameters is suggested that makes use of the results from parametric polynomial spectral factorization and also employs two quantifier elimination techniques. This proposed approach is demonstrated in a numerical example of a particular control problem.