This paper addresses the identification and model reduction of a class of continuous-time Nonlinear (NL) systems through a linearization approach. Although the NL system is considered unknown, we demonstrate that its response to perturbations around a varying operating point can be approximated by a Linear Time-Varying (LTV) system model. We show that this LTV model is the linearization of the NL system around the system trajectory. Then, we demonstrate that the vector of LTV coefficients is the gradient of the NL system, evaluated at the system trajectory. This vector field represents the sensitivity of the NL function (describing the system dynamics) with respect to the input space. By estimating this vector field, we identify a low-dimensional subspace within the input space where the NL function exhibits the most variability. This is achieved by employing the Active Subspace (AS) method, which transforms the high-dimensional input space of the NL system into a lower-dimensional space using the estimated LTV coefficients (gradient vector). Subsequently, the LTV/LPV (Linear Parameter-Varying) approximation of the unknown reduced NL model in the new low-dimensional space is recovered. A new parametrization for the LPV coefficients is introduced to ensure that the LPV model remains the linearized version of an unknown NL system. This LTV/LPV model is then utilized to reconstruct the closed form of the NL system in the new (reduced) coordinates. Finally, the performance of the proposed method is illustrated through numerical examples.

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.

This paper handles the identification of nonlinear systems through linear time-varying (LTV) approximation. The mathematical form of the nonlinear system is unknown and regenerated through an experiment followed by LTV and linear parameter-varying (LPV) estimation and integration. By employing a well-designed experiment the linearized model of the nonlinear system around a time-varying trajectory is obtained. The result is an LTV approximation of the nonlinear system around that trajectory. Having estimated the LTV model, an LPV model is identified. It is shown that the parameter-varying (PV) coefficients of this LPV model are partial derivatives of the nonlinear system evaluated at the trajectory. In this paper, we will show that there exists a relation between the LPV coefficients. This structural relation in the LPV model ensures the integrability of PV coefficients for nonlinear reconstruction. Indeed, the vector of the LPV coefficients is the gradient of the nonlinear system evaluated at the trajectory. Then, the nonlinear system is reconstructed through symbolic integration of the coefficients. The proposed method is a data-driven scheme that can reconstruct an estimate of the nonlinear system and its mathematical form using input-output measurements. Finally, the use of the proposed method is illustrated via a simulation example.