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Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems pp 222–241Cite as

Topological approaches to robustness

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 185))

Abstract

This paper takes an input-output point of view to infinite-dimensional systems, namely, a system is considered to be a (possibly unbounded) operator on L 2. Any such linear, shift-invariant system with closed graph has a natural, frequency domain representation. Such a system also has a transfer function expressed in terms of quotients of H ∞ functions. We will characterize basic properties such as stability, causality and stabilizability in terms of the frequency domain representation. We will also discuss the issue of uncertainty. The framework for the discussion will be the gap metric and the graph topology. The gap metric is a measure of “angle” between the graphs of two systems which is defined in terms of projection operators. The computation of the gap metric in terms of an H ∞ optimization problem will be described. We discuss the L 2-gap metric and the v-gap metric—an indexed L 2-gap metric which induces the graph topology. The robustness of feedback systems under gap-ball uncertainty will be considered, as well as the optimization of robustness. A class of infinite dimensional systems will be described which can be approximated by finite dimensional ones and which admit rational suboptimal controllers. A new approach to the computation of rational suboptimal controllers will be presented for the case of a rational transfer function times a delay. The approach will be illustrated by a worked example.

Supported by the National Science Foundation, Air Force Office of Scientific Research and the Nuffield Foundation

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, University of Minnesota, 55455, Minneapolis, MN, U.S.A.

    Tryphon T. Georgiou

  2. Department of Engineering, University of Cambridge, CB2 1PZ, Cambridge, U.K.

    Malcolm C. Smith

Authors
  1. Tryphon T. Georgiou
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  2. Malcolm C. Smith
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Editor information

R. F. Curtain A. Bensoussan J. L. Lions

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© 1993 Springer-Verlag

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Georgiou, T.T., Smith, M.C. (1993). Topological approaches to robustness. In: Curtain, R.F., Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems. Lecture Notes in Control and Information Sciences, vol 185. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0115027

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  • DOI: https://doi.org/10.1007/BFb0115027

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56155-2

  • Online ISBN: 978-3-540-47480-7

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