A Combined Estimator for Nonlinear System Identification via LPV Approximations

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 (nominal) operating point. The system trajectory is then perturbed slightly via small input perturbations. While the system's operating point evolves slowly over time, the small (broadband) input signal excites the system dynamics. 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, referred to as “gradient-parameterized” LPV coefficients, is introduced, establishing a structural relationship between the 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). The proposed approach employs a combined estimator to locally identify the joint LPV-NL model within the neighborhood of the system's operating point. Although the estimated NL model is localized around this nominal trajectory, an experimental approach is proposed to reduce the linear approximation error, thereby enlarging the validity region of the final model. Finally, we present a numerical example to illustrate the performance of the proposed method.