  The Koopman framework is a popular approach to transform a finite dimensional
nonlinear system into an infinite dimensional, but linear model through a
lifting process, using so-called observable functions. While there is an
extensive theory on infinite dimensional representations in the operator sense,
there are few constructive results on how to select the observables to realize
them. When it comes to the possibility of finite Koopman representations, which
are highly important form a practical point of view, there is no constructive
theory. Hence, in practice, often a data-based method and ad-hoc choice of the
observable functions is used. When truncating to a finite number of basis,
there is also no clear indication of the introduced approximation error. In
this paper, we propose a systematic method to compute the finite dimensional
Koopman embedding of a specific class of polynomial nonlinear systems in
continuous-time such that, the embedding, without approximation, can fully
represent the dynamics of the nonlinear system.
