  Modern data-driven control applications call for flexible nonlinear models
that are amenable to principled controller synthesis and realtime feedback.
Many nonlinear dynamical systems of interest are control affine. We propose two
novel classes of nonlinear feature representations which capture control affine
structure while allowing for arbitrary complexity in the state dependence. Our
methods make use of random features (RF) approximations, inheriting the
expressiveness of kernel methods at a lower computational cost. We formalize
the representational capabilities of our methods by showing their relationship
to the Affine Dot Product (ADP) kernel proposed by Casta\~neda et al. (2021)
and a novel Affine Dense (AD) kernel that we introduce. We further illustrate
the utility by presenting a case study of data-driven optimization-based
control using control certificate functions (CCF). Simulation experiments on a
double pendulum empirically demonstrate the advantages of our methods.
