The parallel projection operators of a nonlinear feedback system

John C. Doyle; Tryphon T. Georgiou; Malcolm C. Smith

doi:10.1016/0167-6911(93)90019-3

The parallel projection operators of a nonlinear feedback system

John C. Doyle, Tryphon T. Georgiou, Malcolm C. Smith

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

Abstract

This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L₂-stability of a feedback system amounts to a 'coordinatization' of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.

Original language	English (US)
Pages (from-to)	79-85
Number of pages	7
Journal	Systems and Control Letters
Volume	20
Issue number	2
DOIs	https://doi.org/10.1016/0167-6911(93)90019-3
State	Published - Feb 1993

Bibliographical note

Funding Information:
Correspondence to: Prof. T.T. Georgiou, Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, USA. * Supported in part by the NSF and the AFOSR, USA, and the Nuffield Foundation, UK.

Keywords

Gap metric
loop-shaping
passivity
robustness

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title = "The parallel projection operators of a nonlinear feedback system",

abstract = "This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L2-stability of a feedback system amounts to a 'coordinatization' of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.",

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author = "Doyle, {John C.} and Georgiou, {Tryphon T.} and Smith, {Malcolm C.}",

note = "Funding Information: Correspondence to: Prof. T.T. Georgiou, Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, USA. * Supported in part by the NSF and the AFOSR, USA, and the Nuffield Foundation, UK.",

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AU - Georgiou, Tryphon T.

AU - Smith, Malcolm C.

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N2 - This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L2-stability of a feedback system amounts to a 'coordinatization' of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.

AB - This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L2-stability of a feedback system amounts to a 'coordinatization' of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.

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