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On the duality between filtering and Nevanlinna-Pick interpolation
KTH, Superseded Departments, Mathematics.ORCID iD: 0000-0002-2681-8383
2000 (English)In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 39, no 3, 757-775 p.Article in journal (Refereed) Published
Abstract [en]

Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spectral analysis, and speech processing. For this reason, results about positive real transfer functions and their realizations typically have many applications and manifestations. In this paper, we study certain manifolds and submanifolds of positive real transfer functions, describing a fundamental geometric duality between filtering and Nevanlinna Pick interpolation. Not surprisingly, then, this duality, while interesting in its own right, has several corollaries which provide solutions and insight into some very interesting and intensely researched problems. One of these is the problem of parameterizing all rational solutions of bounded degree of the Nevanlinna-Pick interpolation problem, which plays a central role in robust control, and for which the duality theorem yields a complete solution. In this paper, we shall describe the duality theorem, which we motivate in terms of both the interpolation problem and a fast algorithm for Kalman filtering, viewed as a nonlinear dynamical system on the space of positive real transfer functions. We also outline a new proof of the recent solution to the rational Nevanlinna Pick interpolation problem, using an algebraic topological generalization of Hadamard's global inverse function theorem.

Place, publisher, year, edition, pages
2000. Vol. 39, no 3, 757-775 p.
Keyword [en]
Nevanlinna-Pick interpolation, filtering, positive real functions, foliations, degree constraint, realization
URN: urn:nbn:se:kth:diva-20123ISI: 000090047700006OAI: diva2:338816
QC 20100525Available from: 2010-08-10 Created: 2010-08-10Bibliographically approved

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