Integrability for Nonlinear Time-Delay Systems

In this note, the notion of integrability is defined for 1-forms defined in the time-delay context. While in the delay-free case, a set of 1-forms defines a vector space, it is shown that 1-forms computed for time-delay systems have to be viewed as elements of a module over a certain non-commutative...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2016-07, Vol.61 (7), p.1912-1917
Main Authors: Kaldmae, Arvo, Califano, Claudia, Moog, Claude H.
Format: Article
Language:English
Subjects:
Accessibility
Aerospace electronics
algebraic methods
Automatic control
Automatic Control Engineering
Computation
Computer Science
Context
Control systems
Delays
Indexes
Modules
Nonlinearity
Polynomials
Rings (mathematics)
Telecommunications
Time-delay systems
Vector spaces
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Summary:In this note, the notion of integrability is defined for 1-forms defined in the time-delay context. While in the delay-free case, a set of 1-forms defines a vector space, it is shown that 1-forms computed for time-delay systems have to be viewed as elements of a module over a certain non-commutative polynomial ring. Two notions of integrability are defined, strong and weak integrability, which coincide in the delay-free case. Necessary and sufficient conditions are given to check if a set of 1-forms is strongly or weakly integrable. To show the importance of the topic, integrability of 1-forms is used to characterize the accessibility property for nonlinear time-delay systems. The possibility of transforming a system into a certain normal form is also considered.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2015.2482003