A generalized entropy criterion for Nevanlinna - Pick interpolation with degree constraint

Christopher I. Byrnes; Tryphon T. Georgiou; Anders Lindquist

doi:10.1109/9.928584

A generalized entropy criterion for Nevanlinna - Pick interpolation with degree constraint

Christopher I. Byrnes, Tryphon T. Georgiou, Anders Lindquist

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

159 Scopus citations

Abstract

In this paper, we present a generalized entropy criterion for solving the rational Nevanlinna - Pick problem for n + 1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.

Original language	English (US)
Pages (from-to)	822-839
Number of pages	18
Journal	IEEE Transactions on Automatic Control
Volume	46
Issue number	6
DOIs	https://doi.org/10.1109/9.928584
State	Published - Jun 2001

Bibliographical note

Funding Information:
Manuscript received October 28, 1998; revised January 25, 2000 and November 7, 2000. Recommended by Associate Editor S. Weiland. This work was supported in part by Grants from the Air Force Office of Scientific Research, the National Science Foundation, TFR, the Göran Gustafsson Foundation, and Southwestern Bell. C. I. Byrnes is with the Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130 USA. T. T. Georgiou is with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. A. Lindquist is with the Division of Optimization and Systems Theory, Royal Institute of Technology, 100 44 Stockholm, Sweden. Publisher Item Identifier S 0018-9286(01)03604-2.

Keywords

Duality
Entropy
Interpolation
Power transmission
Robust control
Spectral estimation

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abstract = "In this paper, we present a generalized entropy criterion for solving the rational Nevanlinna - Pick problem for n + 1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.",

keywords = "Duality, Entropy, Interpolation, Power transmission, Robust control, Spectral estimation",

author = "Byrnes, {Christopher I.} and Georgiou, {Tryphon T.} and Anders Lindquist",

note = "Funding Information: Manuscript received October 28, 1998; revised January 25, 2000 and November 7, 2000. Recommended by Associate Editor S. Weiland. This work was supported in part by Grants from the Air Force Office of Scientific Research, the National Science Foundation, TFR, the G{\"o}ran Gustafsson Foundation, and Southwestern Bell. C. I. Byrnes is with the Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130 USA. T. T. Georgiou is with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. A. Lindquist is with the Division of Optimization and Systems Theory, Royal Institute of Technology, 100 44 Stockholm, Sweden. Publisher Item Identifier S 0018-9286(01)03604-2.",

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AU - Lindquist, Anders

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N2 - In this paper, we present a generalized entropy criterion for solving the rational Nevanlinna - Pick problem for n + 1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.

AB - In this paper, we present a generalized entropy criterion for solving the rational Nevanlinna - Pick problem for n + 1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.

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KW - Robust control

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A generalized entropy criterion for Nevanlinna - Pick interpolation with degree constraint

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