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On the duality between filtering and Nevanlinna-Pick interpolation (2000)
| Content Provider | CiteSeerX |
|---|---|
| Author | Byrnes, Christopher I. Lindquist, Ers |
| Abstract | Abstract. 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. Key words. Nevanlinna–Pick interpolation, filtering, positive real functions, foliations, degree constraint AMS subject classifications. 30E05, 47N70, 58C99, 93B29, 93E11 |
| File Format | |
| Volume Number | 39 |
| Journal | SIAM J. Control and Optimization |
| Language | English |
| Publisher Date | 2000-01-01 |
| Access Restriction | Open |
| Subject Keyword | Nevanlinna-pick Interpolation Positive Real Transfer Function Central Role Duality Theorem Nevanlinna Pick Interpolation Circuit Synthesis Rational Solution Many Application Spectral Analysis Recent Solution Speech Processing Bounded Degree Degree Constraint Am Subject Classification New Proof Fundamental Geometric Duality Algebraic Topological Generalization Nonlinear Dynamical System Fast Algorithm Complete Solution Certain Manifold Robust Control Positive Real Rational Function Rational Nevanlinna Pick Interpolation Problem Positive Real Function Hadamard Global Inverse Function Theorem Nevanlinna Pick Interpolation Problem Key Word Several Corollary Kalman Filtering Stochastic Linear System Theory Interpolation Problem |
| Content Type | Text |
| Resource Type | Article |
