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Title: Non-uniform small-gain theorems for systems with unstable invariant sets
Authors: Tyukin, Ivan Yu.
Steur, Erik
Nijmeijer, Henk
van Leeuwen, Cees
First Published: Dec-2008
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Citation: Decision and Control, 2008, CDC 2008, 47th IEEE Conference on, Proceedings of, pp. 5080 - 5085.
Abstract: We consider the problem of small-gain analysis of asymptotic behavior in interconnected nonlinear dynamic systems. Mathematical models of these systems are allowed to be uncertain and time-varying. In contrast to standard small-gain theorems that require global asymptotic stability of each interacting component in the absence of inputs, we consider interconnections of systems that can be critically stable and have infinite input-output Linfin gains. For this class of systems we derive small-gain conditions specifying state boundedness of the interconnection. The estimates of the domain in which the systempsilas state remains are also provided. Conditions that follow from the main results of our paper are non-uniform in space. That is they hold generally only for a set of initial conditions in the systempsilas state space. We show that under some mild continuity restrictions this set has a non-zero volume, hence such bounded yet potentially globally unstable motions are realizable with a non-zero probability. Proposed results can be used for the design and analysis of intermittent, itinerant and meta-stable dynamics which is the case in the domains of control of chemical kinetics, biological and complex physical systems, and non-linear optimization.
DOI Link: 10.1109/CDC.2008.4739503
ISSN: 0191-2216
ISBN: 9781424431236
Type: Conference paper
Rights: This is the author's final draft of the paper published as Decision and Control, 2008, CDC 2008, 47th IEEE Conference on, Proceedings of, pp. 5080 - 5085. Copyright © 2008 IEEE. Doi: 10.1109/CDC.2008.4739503. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Leicester’s products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to By choosing to view this document, you agree to all provisions of the copyright laws protecting it.
Appears in Collections:Conference Papers & Presentations, Dept. of Mathematics

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