Nonlinear continuous-time system identification by linearization around a time-varying setpoint

This article addresses the identification of unknown nonlinear continuous-time systems through a linear time-varying (LTV) approximation as a starting point. The mathematical form of the nonlinear system is unknown and is reconstructed by use of a well-designed experiment, followed by LTV and linear parameter-varying (LPV) estimations, and an integration step. The experiment used allows for a linearization of the unknown nonlinear system around a time-varying operating point (system trajectory), resulting in an LTV approximation. After estimating the LTV model, an LPV model is identified, where the parameter-varying (PV) coefficients represent partial derivatives of the unknown nonlinear system evaluated at the trajectory. We demonstrate a structural relation in the LPV model structure that ensures that the LPV coefficient vector is the gradient of the unknown nonlinear system. The nonlinear model of the system is then reconstructed through symbolic integration of the PV coefficients. This identification method enables the estimation of the unknown nonlinear system and its mathematical form using input-output measurements. The article concludes by illustrating the method on simulation examples.