The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree

Bolotnikov, Vladimir; Cameron, Stephen

doi:10.1016/j.cam.2012.02.009

Cited by 7 publications

(7 citation statements)

References 12 publications

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“…Indeed, with γ = 1, the Pick matrix for the reduced interpolation problem, obtained by avoiding the interpolation conditions on the imaginary axis, is positive definite. The same argument for the existence of a minimum of the interpolation problem has been used in Theorem 1.6 of [2] and Section 5 of [22]. In the opposite case, i.e.…”

Section: H ∞ Optimal Control With Jω Invariant Zerosmentioning

confidence: 91%

A reformulation of augmented basic interpolation problem and an application to H∞ control

Stefanovski

2015

Linear Algebra and its Applications

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MSC:In this paper we present a new formulation of the augmented basic interpolation problem (aBIP) with rational matrices, in terms of the stability of four rational matrices, so that the aBIP transforms into a purely linear-algebraic problem. Actually, the existing interpolation condition, given by an integral, is replaced by stability of a rational matrix. The new condition is applied to the H ∞ optimal control of one-block plants having imaginary axis invariant zeros. A new parameter in the parametrization of H ∞ optimal controllers is revealed, which is given in terms of the derivative of the closed-loop system at the imaginary axis invariant zeros of the plant. The H ∞ optimal control algorithm is illustrated by an example, and compared to the existing algorithms.

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Section: H ∞ Optimal Control With Jω Invariant Zerosmentioning

confidence: 91%

A reformulation of augmented basic interpolation problem and an application to H∞ control

Stefanovski

2015

Linear Algebra and its Applications

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“…The author is grateful to Prof. Dušan Georgijević for pointing out to him the paper Bolotnikov and Cameron (2012), and for the help in the proof of Theorems 5 and 6.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…It is solved for the case of scalar interpolants, with the unit circle as a boundary, and only with sufficient conditions, in Bolotnikov and Cameron (2012). One of the purposes of this paper is to present a solution to this important mathematical problem.…”

Section: Introductionmentioning

confidence: 99%

Optimal boundary interpolation and one-block optimal control problem with invariant zeros on the imaginary axis and infinity

Stefanovski¹

2014

International Journal of Control

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We formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points, using the concept of parametrisation of stabilising controllers. Then, we solve the problem of obtaining all its solutions. If interpolation points at infinity are absent, we show that the introduced problem is equivalent to the existing one. We apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity. We show by an example that the solution of optimal interpolation is directly applicable to the one-block H ∞ optimal control with existent invariant zeros on the imaginary axis and infinity. It is seen from the example that not only the transfer matrix of the closed-loop system is constrained on the extended imaginary axis, but also its derivatives.

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“…If the nodes x j are in C r (with possibly some nodes on iR) the problem is much harder, see e.g. [4, Chapter 21], [6], [7], [11] and [15].…”

Section: Problem Formulationmentioning

confidence: 99%

“Wrong” side interpolation by positive real rational functions

Alpay

Lewkowicz

2018

Linear Algebra and its Applications

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Using polynomial interpolation, along with structural properties of the family of rational positive real functions, we here show that a set of m nodes in the open left half of the complex plane, can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m. Moreover we introduce an easy-to-find parametrization in R 2m+3 of a large subset of these interpolating functions.1991 Mathematics Subject Classification. 26C15, 37F10, 46B70, 47N70, 94C05.

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The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree

Cited by 7 publications

References 12 publications

A reformulation of augmented basic interpolation problem and an application to H∞ control

A reformulation of augmented basic interpolation problem and an application to H∞ control

Optimal boundary interpolation and one-block optimal control problem with invariant zeros on the imaginary axis and infinity

“Wrong” side interpolation by positive real rational functions

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