From Nonlinear Identification to Linear Parameter Varying Models: Benchmark Examples

Abstract

Linear parameter-varying (LPV) models form a powerful model class to analyze and control a (nonlinear) system of interest. Identifying a LPV model of a nonlinear system can be challenging due to the difficulty of selecting the scheduling variable(s) a priori, which is quite challenging in case a first principles based understanding of the system is unavailable. This paper presents a systematic LPV embedding approach starting from nonlinear fractional representation models. A nonlinear system is identified first using a nonlinear block-oriented linear fractional representation (LFR) model. This nonlinear LFR model class is embedded into the LPV model class by factorization of the static nonlinear block present in the model. As a result of the factorization a LPV-LFR or a LPV state-space model with an affine dependency results. This approach facilitates the selection of the scheduling variable from a data-driven perspective. Furthermore the estimation is not affected by measurement noise on the scheduling variables, which is often left untreated by LPV model identification methods. The proposed approach is illustrated on two well-established nonlinear modeling benchmark examples.

A Latent Restoring Force Approach to Nonlinear System Identification

“…The resulting errors are 18.47% for the linear model and 0.17% for the nonlinear model, the normalised error of 0.17% corresponds to a root-mean-squared-error of 0.0021 in the testing data which is competitive with results seen in the literature, e.g. Schoukens and Tóth [41]. In summary, the proposed approach for inference over the nonlinear restoring force in the system and the linear system parameters has proven to be very effective in identification of the Silverbox.…”

“…The benchmark ( [34]) is an experimental dataset produced by an electrical system which implements a nonlinearity similar in behaviour to the theoretical Duffing oscillator. It has been investigated previously in the literature as a benchmark dataset for nonlinear system identification, for example see [40][41][42][43]. The benchmark itself aims to replicate, electronically, the behaviour of a Duffing oscillator seen previously in Equation (9).…”

Linear parameter-varying model for a refuellable zinc–air battery

“…However, it is acknowledged that nonlinear models are less flexible than comparable linear models and the mathematical tools are lacking for nonlinear systems. Alternatively, nonlinear behaviour can be captured via a linear parameter-varying (LPV) model, which approximates a nonlinear system with high accuracy [ 35 , 36 ].…”

Linear Parameter Varying Representation of a class of MIMO Nonlinear Systems

“…LPV embedding of NLFR structures, sometimes called Lur'e systems, has been studied in prior works (Seron and De Doná, 2015;Hanafi et al, 2018;Schoukens and Tóth, 2018). However, these works are limited to single-inputsingle-output static nonlinearities.…”