  In this study, we present a purely data-driven method that uses the Loewner
framework (LF) along with nonlinear optimization techniques to infer quadratic
with affine control dynamical systems that admit Volterra series (VS)
representations from input-output (i/o) time-domain measurements. The proposed
method extensively employs optimization tools for interpolating the symmetric
generalized frequency response functions (GFRFs) derived in the VS framework.
The GFRF estimations are obtained from the Fourier spectrum (phase and
amplitude) of the quasi-steady state system response under harmonic excitation.
Appropriate treatment of these measurements under the developed framework
allows the identification of low-order quadratic state-space models with
non-trivial stable equilibria, such as in the Lorenz '63 forced system. We thus
can achieve low-order global model identification for systems that can
bifurcate to multiple equilibria after solely collecting measurements from a
local stable operational regime. The developed framework is tested for several
examples of increasing dimension and complexity, up to a test case of the
viscous Burgers' equation with Robin boundary conditions. In the latter case,
this study enforces new directions of employing data-driven reduced model
inference that successfully can provide low-order accurate surrogate predictive
models suitable for control. Future directions and open challenges conclude
this work.
