  Recently, the sinosoidal output response in power series (SORPS) formalism
was presented for system identification and simulation. Based on the concept of
characteristic curves (CCs), it establishes a mathematical connection between
power series and Fourier series for a first-order nonlinear system [F. J.
Gonzalez, Sci. Rep. 13, 1955, (2023)]. However, the system identification
procedure discussed there, based on fast Fourier transform (FFT), presents the
limitations of requiring a sinusoidal single tone for the dynamical variable
and equally spaced time steps for the input-output dataset (DS). These
limitations are here addressed by introducing a different approach: we use a
power series-based model for system modeling using two hyperparameters
$\hat{A}_0$ and $\hat{A}_1$ optimally defined depending on the DS. Overall,
this work expands the applicability of the SORPS formalism to arbitrary DSs and
represents a groundbreaking contribution for the concept of CCs. The method of
CCs is complementary to the commonly used approaches of NARMAX-models and
sparse regression techniques, which emphasize the estimation of the individual
parameter values of the model. However, the CCs-based methods emphasize the
computation of the CCs as a whole, thus presenting the advantages that the
system identification is uniquely defined, and that it can be applied for any
system without additional algebraic operations. Thus, the parsimonious
principle defined by the NARMAX-philosophy is extended from the concept of a
model with as few parameters as possible to the concept of finding the lowest
model order that correctly describe the input-output DS. This opens up a wide
variety of potential applications in vibration analysis, structural dynamics,
viscoelastic materials, nonlinear electric circuits, voltammetry techniques,
structural health monitoring, and fault diagnosis.
