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.
Integrability for Nonlinear Time-Delay Systems / Kaldmäe, Arvo; Califano, Claudia; Moog, Claude H.. - In: IEEE TRANSACTIONS ON AUTOMATIC CONTROL. - ISSN 0018-9286. - 61:7(2016), pp. 1912-1917. [10.1109/TAC.2015.2482003]
Integrability for Nonlinear Time-Delay Systems
CALIFANO, Claudia;
2016
Abstract
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.File | Dimensione | Formato | |
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