Extended Lie Brackets for Nonlinear Time-Delay Systems

In this note the Extended Lie bracket operator is introduced for the analysis and control of nonlinear time-delay systems (NLTDS). This tool is used to characterize the integrability conditions of a given submodule. The obtained results have two fundamental outcomes. First, they define the necessary and sufficient conditions under which a given set of nonlinear one-forms in the
n
-dimensional delayed variables
x
(
t
),...,
x
(
t
-
sD
) , with
D
constant but unknown, are integrable, thus generalizing the well known fundamental Frobenius Theorem to delay systems. Secondly, they set the basis for the extension to this context of the geometric approach used for delay-free systems. The effectiveness of the results is shown by solving the problem of the equivalence of a NLTDS to an accessible Linear Time-Delay System (LTDS) by bicausal change of coordinates.