Robust stability of feedback systems: A geometric approach using the gap metric

Ciprian Foias, Tryphon T. Georgiou, Malcolm C. Smith

Research output: Contribution to journalArticlepeer-review

52 Scopus citations


A geometric framework for robust stabilization of infinite-dimensional time-varying linear systems is presented. The uncertainty of a system is described by perturbations of its graph and is measured in the gap metric. Necessary and sufficient conditions for robust stability are generalized from the time-invariant case. An example is given to highlight an important difference between the obstructions, which limit the size of a stabilizable gap ball, in the time-varying and time-invariant cases. Several results on the gap metric and the gap topology are established that are central in a geometric treatment of the robust stabilizability problem in the gap. In particular, the concept of a `graphable' subspace is introduced in the paper. Subspaces that fail to be graphable are characterized by an index condition on a certain semi-Fredholm operator.

Original languageEnglish (US)
Pages (from-to)1518-1537
Number of pages20
JournalSIAM Journal on Control and Optimization
Issue number6
StatePublished - 1993

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