Optimization-based computation of analytic interpolants of bounded complexity

This paper provides a unifying algorithm for computing any analytic interpolant of bounded complexity. Such computation can be performed by solving an optimization problem, due to a theorem by Georgiou and Lindquist. This optimization problem is numerically solvable by a continuation method. The proposed numerical algorithm is useful, among other cases, for designing a low-degree controller for a benchmark problem in robust control. The algorithm unifies previously developed algorithms for the Caratheodory extension and the Nevanlinna–Pick interpolation to one for more general interpolation problems.

[1]  Anders Lindquist,et al.  Geometry of the Kimura-Georgiou parametrization of modelling filters , 1989 .

[2]  Tryphon T. Georgiou,et al.  Realization of power spectra from partial covariance sequences , 1987, IEEE Trans. Acoust. Speech Signal Process..

[3]  Ryozo Nagamune,et al.  An extension of a Nevanlinna-Pick interpolation solver to cases including derivative constraints , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[4]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[5]  Tryphon T. Georgiou,et al.  A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint , 2001, IEEE Trans. Autom. Control..

[6]  R. Langer.  Interpolation and Approximation by Rational Functions in the Complex Domain , 1937 .

[7]  Bruce A. Francis,et al.  Feedback Control Theory , 1992 .

[8]  C. Byrnes,et al.  Generalized interpolation in $H^\infty$ with a complexity constraint , 2004 .

[9]  Anders Lindquist,et al.  Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach , 2003, IEEE Trans. Autom. Control..

[10]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[11]  Ryozo Nagamune,et al.  A robust solver using a continuation method for Nevanlinna-Pick interpolation with degree constraint , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[12]  C. Foias,et al.  The commutant lifting approach to interpolation problems , 1990 .

[13]  Anders Lindquist,et al.  On the Duality between Filtering and Nevanlinna--Pick Interpolation , 2000, SIAM J. Control. Optim..

[14]  C. Byrnes,et al.  A Convex Optimization Approach to the Rational Covariance Extension Problem , 1999 .

[15]  Anders Lindquist,et al.  Interior point solutions of variational problems and global inverse function theorems , 2007 .

[16]  P. Enqvist.  A homotopy approach to rational covariance extension with degree constraint , 2001 .

[17]  Tryphon T. Georgiou,et al.  Spectral estimation via selective harmonic amplification , 2001, IEEE Trans. Autom. Control..

[18]  Ryozo Nagamune,et al.  A shaping limitation of rational sensitivity functions with a degree constraint , 2004, IEEE Transactions on Automatic Control.

[19]  T. Georgiou.  A topological approach to Nevanlinna-pick interpolation , 1987 .

[20]  C. Byrnes,et al.  A complete parameterization of all positive rational extensions of a covariance sequence , 1995, IEEE Trans. Autom. Control..

[21]  A. Lindquist,et al.  GENERALIZED INTERPOLATION IN H∞ WITH A COMPLEXITY CONSTRAINT , 2004 .

[22]  Tryphon T. Georgiou,et al.  Kullback-Leibler approximation of spectral density functions , 2003, IEEE Trans. Inf. Theory.

[23]  H. Dym.  J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation , 1989 .

[24]  Tryphon T. Georgiou,et al.  The interpolation problem with a degree constraint , 1999, IEEE Trans. Autom. Control..

[25]  G. Fanizza,et al.  Identification of rational spectral densities using orthonormal basis functions , 2003 .