Identification of discontinuous nonlinear systems via a multivariate Padé approach

Abstract

We present a nonlinear system identification technique based on multi-dimensional rational polynomials. A multi-dimensional Padé–Legendre approximation is developed to circumvent challenges in dealing with sharp shocks. The purpose of this paper is to investigate the accuracy of such approximations for identification of various nonlinear systems, particularly systems with a non-smooth response surface. This identification approach utilizes the generalized form of a Padé–Legendre approximation for studying multivariable functions. In the studied problems, the nonlinearity is a function of state variables (displacement and velocity), which requires multi-dimensional formulation. Furthermore, a spatial filter is applied to minimize the effects of the singular points in the applicable rational function of the response surface. This study presents different types of nonlinearities including smooth, irregular, and hysteretic functions, in order to demonstrate the performance of the approach under different conditions. In order to study the robustness of the method in comparison to other identification techniques based on plain polynomial representation, a nonlinear system with a sharp discontinuous restoring force surface is considered. The performance of both approaches is investigated for different degrees of “sharpness”. In addition, the accuracy of the identified models to represent the nonlinear system is verified by comparing the output of the system (computed on the basis of the identified model) from data sets corresponding to different excitations than those used for identification purposes. It is shown that the proposed approach provides a robust identification technique for a broad class of highly-nonlinear systems, and it is particularly advantageous to use when dealing with systems incorporating discontinuous properties.