  We propose an analytical construction of observable functions in the extended
dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for
approximating the spectral properties of the Koopman operator. The choice of
observable functions is fundamental for the application of EDMD to nonlinear
problems arising in systems and control. Existing methods either start from a
set of dictionary functions and look for the subset that best fits the
underlying nonlinear dynamics or they rely on machine learning algorithms to
"learn" observable functions. Conversely, in this paper, we start from the
dynamical system model and lift it through the Lie derivatives, rendering it
into a polynomial form. This proposed transformation into a polynomial form is
exact, and it provides an adequate set of observable functions. The strength of
the proposed approach is its applicability to a broader class of nonlinear
dynamical systems, particularly those with nonpolynomial functions and
compositions thereof. Moreover, it retains the physical interpretability of the
underlying dynamical system and can be readily integrated into existing
numerical libraries. The proposed approach is illustrated with an application
to electric power systems. The modeled system consists of a single generator
connected to an infinite bus, where nonlinear terms include sine and cosine
functions. The results demonstrate the effectiveness of the proposed procedure
in off-attractor nonlinear dynamics for estimation and prediction; the
observable functions obtained from the proposed construction outperform methods
that use dictionary functions comprising monomials or radial basis functions.
