Not all Vinnicombe metric neighbourhoods are homotopically connected

Brian Anderson; Thomas Brinsmead

doi:10.1109/IDC.2002.995361

Not all Vinnicombe metric neighbourhoods are homotopically connected

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Abstract

We prove by counterexample that even for two transfer functions which are close in the Nu-gap metric of Vinnicombe (1993, 1999), there does not necessarily exist a Vinnicombe metric homotopy from one transfer function to the other, such that intermediate transfer functions in the homotopy remain close to the transfer function at the beginning of the homotopy. This implies that the Vinnicombe metric neighbourhoods of some transfer functions in L-infinity space, are not connected.

Original language	English
Pages	29-34
DOIs	https://doi.org/10.1109/IDC.2002.995361
Publication status	Published - 2002
Event	Information, Decision and Control Conference 2002 - Adelaide Australia, Australia Duration: 11 Feb 2002 → 13 Feb 2002

Conference

Conference	Information, Decision and Control Conference 2002
Country/Territory	Australia
Period	11/02/02 → 13/02/02
Other	February 11 2002

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10.1109/IDC.2002.995361

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title = "Not all Vinnicombe metric neighbourhoods are homotopically connected",

abstract = "We prove by counterexample that even for two transfer functions which are close in the Nu-gap metric of Vinnicombe (1993, 1999), there does not necessarily exist a Vinnicombe metric homotopy from one transfer function to the other, such that intermediate transfer functions in the homotopy remain close to the transfer function at the beginning of the homotopy. This implies that the Vinnicombe metric neighbourhoods of some transfer functions in L-infinity space, are not connected.",

author = "Brian Anderson and Thomas Brinsmead",

year = "2002",

doi = "10.1109/IDC.2002.995361",

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AU - Brinsmead, Thomas

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N2 - We prove by counterexample that even for two transfer functions which are close in the Nu-gap metric of Vinnicombe (1993, 1999), there does not necessarily exist a Vinnicombe metric homotopy from one transfer function to the other, such that intermediate transfer functions in the homotopy remain close to the transfer function at the beginning of the homotopy. This implies that the Vinnicombe metric neighbourhoods of some transfer functions in L-infinity space, are not connected.

AB - We prove by counterexample that even for two transfer functions which are close in the Nu-gap metric of Vinnicombe (1993, 1999), there does not necessarily exist a Vinnicombe metric homotopy from one transfer function to the other, such that intermediate transfer functions in the homotopy remain close to the transfer function at the beginning of the homotopy. This implies that the Vinnicombe metric neighbourhoods of some transfer functions in L-infinity space, are not connected.

U2 - 10.1109/IDC.2002.995361

DO - 10.1109/IDC.2002.995361

M3 - Abstract

SP - 29

EP - 34

T2 - Information, Decision and Control Conference 2002

Y2 - 11 February 2002 through 13 February 2002

ER -

Not all Vinnicombe metric neighbourhoods are homotopically connected

Abstract

Conference

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