Analytic interpolation and the degree constraint

• Autorzy: Tryphon T. Georgiou
• Języki publikacji: EN

Abstrakty

EN

Analytic interpolation problems arise quite naturally in a variety of engineering applications. This is due to the fact that analyticity of a (transfer) function relates to the stability of a corresponding dynamical system, while positive realness and contractiveness relate to passivity. On the other hand, the degree of an interpolant relates to the dimension of the pertinent system, and this motivates our interest in constraining the degree of interpolants. The purpose of the present paper is to make an overview of recent developments on the subject as well as to highlight an application of the theory.

Słowa kluczowe

EN

analytic interpolation; spectralanalysis of time-series; uniformly optimal control

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2001
• Tom: 11
• Numer: 1
• Strony: 271-279

Daty

wydano

2001

otrzymano

2000-09-01

poprawiono

2001-01-01

Twórcy

autor

Tryphon T. Georgiou

• Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street S.E., Minneapolis, MN 55455, USA

Bibliografia

• Akhiezer N.I. (1965): The Classical Moment Problem, Hafner Publishing, Translation 1965, New York: Oliver and Boyd.
• Ball J.A. and Helton J.W. (1983): A Beurling-Lax theoremfor the Liegroup U(m, n) which contains most classical interpolation theory. -J. Op. Theory, Vol.9, No.1, pp.107-142.
• Byrnes C.I., Georgiou T.T. and Lindquist A.(2000a): A generalized entropy criterion for Nevanlinna-Pick interpolationwith degree constraint. -IEEE Trans. Automat. Contr., (to appear).
• Byrnes C.I., Georgiou T.T. and Lindquist A.(2000b): A new approach to spectral estimation: A tunable high-resolutionspectral estimator. -IEEE Trans. Signal Process., Vol.48, No.11, pp.3189-3206.
• Byrnes C.I., Georgiou T.T. and Lindquist A.(2000c): Generalized interpolation in cH^∞: Solutions of boundedcomplexity. - (in preparation).
• Byrnes C.I., Gusev S.V. and Lindquist A. (1999): A convex optimization approach to the rational covariance extensionproblem. -SIAM J. Contr. Optim., Vol.37, pp.211-229.
• Byrnes C.I., Landau H.J. and Lindquist A. (1997): On the well-posedness of the rational covariance extensionproblem, In: Current and Future Directions in Applied Mathematics, (M. Alber, B. Hu, J. Rosenthal, Eds.). - Boston: Birkhauser, pp.83-106.
• Byrnes C.I., Lindquist A., Gusev S.V. and Matveev A.S.(1995): A complete parametrization of all positive rational extensions of acovariancesequence. - IEEE Trans. Automat. Contr., Vol.AC-40, No.11, pp.1841-1857.
• Delsarte Ph., Genin Y. and Kamp Y. (1981): On therole of the Nevanlinna-Pick problem in circuits and systemtheory. - Circuit Theory Applics., Vol.9, No.1, pp.177-187.
• Francis B.A. (1987): A Course in H_infty, Control Theory. -New York: Springer-Verlag.
• Gantmacher F.R. (1959): The Theory of Matrices. - New York: Chelsea Publishing Company.
• Georgiou T.T. (1983): Partial Realization of Covariance Sequences. - Ph.D. Thesis, CMST, University of Florida, Gainesville.
• Georgiou T.T. (1987a): Realization of power spectra from partialcovariance sequences. -IEEE Trans. Acoust. Speech Signal Process., Vol.ASSP-35, No.4, pp.438-449.
• Georgiou T.T. (1987b): A topological approach to Nevanlinna-Pickinterpolation. -SIAM J. Math. Anal., Vol.18, No.5, pp.1248-1260.
• Georgiou T.T. (1999): The interpolation problem with a degreeconstraint. -IEEE Trans. Automat. Contr., Vol.44, No.3, pp.631-635.
• Georgiou T.T. (2000a): Signal estimation via selectiveharmonicamplification: MUSIC, Redux. - IEEE Trans. Signal Process., Vol.48, No.3, pp.780-790.
• Georgiou T.T. (2000b): Subspace analysis of state covariances. - Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, Istanbul, June 2000 (CD-ROM).
• Georgiou T.T. (2001): Spectral estimation via selectiveharmonic amplification. -IEEE Trans. Automat. Contr., Vol.46, No.1, pp.29-42.
• Geronimus Ya.L.(1961): Orthogonal Polynomials. - English translation from Russian, New York: Consultants Bureau Inc.
• Grenander U. and Szeg G. (1958): Toeplitz Forms and Their Applications. -Berkeley: Univ. California Press.
• Helton J.W. (1982): Non-Euclidean analysis and electronics. -Bull. Amer. Math. Soc., Vol.7, No.1, pp.1-64.
• Kalman R.E. (1982): Realization of covariance sequences, In: Toeplitz Centennial (I. Goh-berg, Ed.). - Boston: Birkhauser, pp.331-342.
• Stoica P. and Moses R. (1997): Introduction to Spectral Analysis. -New Jersey: Prentice Hall.
• Sarason D. (1967): Generalized interpolation in H_infty. -Trans. Amer. Math. Soc., Vol.127, No.2, pp.179-203.
• Tannenbaum A.R. (1982): Modified Nevanlinna-Pick interpolation of linear plants withuncertainty in the gain factor. - Int. J. Contr., Vol.36, No.2, pp.331-336.
• Walsh J.L. (1956): Interpolation and Approximationby Rational Functions in the Complex Domain. - Providence, R. I.: Amer. Math. Soc.
• Youla D.C. and Saito M. (1967): Interpolationwith positive-real functions.- J. Franklin Institute, Vol.284, No.1, pp.77-108.
• Zames G. (1981): Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. - IEEE Trans. Automat. Contr., Vol.26, No.2, pp.301-320.