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Non-uniform small-gain theorems for systems with unstable invariant sets

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conference contribution
posted on 01.12.2009, 14:44 by Ivan Yu. Tyukin, Erik Steur, Henk Nijmeijer, Cees van Leeuwen
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.

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Citation

Decision and Control, 2008, CDC 2008, 47th IEEE Conference on, Proceedings of, pp. 5080 - 5085.

Published in

Decision and Control

Publisher

Institute of Electrical and Electronics Engineers (IEEE)

issn

0191-2216

isbn

9781424431236

Copyright date

2008

Available date

01/12/2009

Publisher version

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4739503

Language

en

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