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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.