The Controlled Center Dynamics

The center manifold theorem is a model reduction technique for determining the local asymptotic stability of an equilibrium of a dynamical system when its linear part is not hyperbolic. The overall system is asymptotically stable if and only if the center manifold dynamics is asymptotically stable....

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Bibliographic Details
Published inMultiscale modeling & simulation Vol. 3; no. 4; pp. 838 - 852
Main Authors Hamzi, Boumediene, Kang, Wei, Krener, Arthur J.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2005
Subjects
Applied mathematics
Approximation
Closed loop systems
Control theory
Dynamical systems
Eigenvalues
Equilibrium
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Summary:The center manifold theorem is a model reduction technique for determining the local asymptotic stability of an equilibrium of a dynamical system when its linear part is not hyperbolic. The overall system is asymptotically stable if and only if the center manifold dynamics is asymptotically stable. This allows for a substantial reduction in the dimension of the system whose asymptotic stability must be checked. Moreover, the center manifold and its dynamics need not be computed exactly; frequently, a low degree approximation is sufficient to determine its stability. The controlled center dynamics plays a similar role in determining local stabilizability of an equilibrium of a control system when its linear part is not stabilizable. It is a reduced order control system with a pseudoinput to be chosen in order to stabilize it. If this is successful, then the overall control system is locally stabilizable to the equilibrium. Again, usually low degree approximation suffices.
ISSN:1540-3459
1540-3467
DOI:10.1137/040603139