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Nonuniform small-gain theorems for systems with unstable invariant sets

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posted on 2013-03-06, 15:37 authored by Ivan Tyukin, Erik Steur, Henk Nijmeijer, Cees Van Leeuwen
We consider the problem of asymptotic convergence to invariant sets in interconnected nonlinear dynamical systems. Standard approaches often require that the invariant sets be uniformly attracting, e. g., stable in the Lyapunov sense. This, however, is neither a necessary requirement nor is always useful. Systems may, for instance, be inherently unstable ( e. g., intermittent, itinerant, meta-stable) or the problem statement may include requirements that cannot be satisfied with stable solutions. This is often the case in general optimization problems and in nonlinear parameter identification or adaptation. Conventional techniques for these cases either rely on detailed knowledge of the system's vector-fields or require boundedness of its states. The presently proposed method relies only on estimates of the input-output maps and steady-state characteristics. The method requires the possibility of representing the system as an interconnection of a stable and contracting part with an unstable and exploratory part. We illustrate with examples how the method can be applied to problems of analyzing the asymptotic behavior of locally unstable systems as well as to problems of parameter identification and adaptation in the presence of nonlinear parametrizations. The relation of our results to conventional small-gain theorems is discussed.

History

Citation

SIAM Journal on Control and Optimization, 2008, 47 (2), pp. 849-882 (34)

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

  • AM (Accepted Manuscript)

Published in

SIAM Journal on Control and Optimization

Publisher

Society for Industrial and Applied Mathematics (SIAM)

issn

0363-0129

eissn

1095-7138

Copyright date

2008

Available date

2013-03-06

Publisher version

http://epubs.siam.org/doi/abs/10.1137/060672546

Language

en