The parallel projection operators of a nonlinear feedback system

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

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


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 languageEnglish (US)
Pages (from-to)79-85
Number of pages7
JournalSystems and Control Letters
Issue number2
StatePublished - Feb 1993
Externally publishedYes

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.


  • Gap metric
  • loop-shaping
  • passivity
  • robustness


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