Abstract
This paper addresses the identification of Nonlinear (NL) systems using
a linearization approach, introducing a combined estimator to tackle
this challenge. We assume that the unknown NL system operates around a
stable, slowly varying operating point. The system trajectory is then
perturbed slightly via small, typically fast, input perturbations. We
demonstrate that the NL system’s response to these small perturbations
can be approximated by a Linear Parameter-Varying (LPV) system model.
Furthermore, we show that this LPV model represents the linearized
version of the unknown NL system around the operating point. A new
parametrization for the LPV model coefficients is introduced,
establishing a structural relationship between the LPV coefficients.
This structural relationship reduces the number of parameters to be
estimated and ensures that the LPV model always corresponds to the
linearized form of the NL system. Additionally, we demonstrate that this
LPV model structure allows for the unique reconstruction of the NL
system model through symbolic integration, resulting in a closed-form
nonlinear Ordinary Differential Equation (ODE). This integration
introduces a second structural relationship, linking the LPV model to
the NL model. By leveraging these two structural relationships, we
reformulate the problem of NL system identification via linearization as
a combined estimation problem, leading to a unified LPV-NL estimation
framework. This approach utilizes all available data, including
perturbation data (linear response) and the varying operating point (NL
response). We propose a combined estimator that jointly estimates the
NL-LPV model, capturing the intrinsic structure of NL system
identification through linearization. Finally, we present a numerical
example to illustrate the performance of the proposed method.