Chapter 19 Nonparametric Analysis and Nonlinear State-Space Identification: A Benchmark Example A. Van Mulders, J. Schoukens, and L. Vanbeylen Abstract This paper deals with the identification of nonlinear models. Such models are particularly useful when a linear model cannot describe the system under test well enough. This is the case in the benchmark example considered here: it is a structure that consists of two facing clamped steel beams connected by a non-linearly behaving flexible element. In this paper, the final goal is to construct a nonlinear state-space model, but first, it is shown how to retrieve a lot of information via one (or few) multisine experiments. Via such excitation signals, one gets a quick impression of the linear system dynamics and the levels of even and odd nonlinearities. After this analysis, an attempt is made to model the system by means of nonlinear (polynomial) state-space models, ranging from a model without structure to a block-structured model. Keywords Nonlinear system identification • State-space models • Block-structured models • Benchmark system • Black-box models 19.1 Introduction This paper is a contribution to the IMAC XXXII benchmark on nonlinear system identification. The benchmark case is a mechanical system for which a simulator is provided. This simulator allows one to apply a user-defined force at any one of eight sensor positions, where the displacements, velocities and accelerations are computed. In the identification approaches presented here, no physical insight in the system is required nor used. It are black-box identification methods, delivering models intended to approximate the system behaviour, not models with physically interpretable parameters. This paper consists of two main parts: a nonparametric identification, based on a frequency domain approach, and a parametric identification. The nonparametric approach is based on the use of multisine excitations. These consist of a sum of harmonically related sines with random phase and a user defined amplitude spectrum. Via these excitation signals, information about linear and nonlinear characteristics of the structure is gathered efficiently. More precisely, the linear system dynamics (or Best Linear Approximation) and levels of odd and even nonlinear distortions are estimated. The next part presents the parametric identification. There exist quite some nonlinear black-box approaches: e.g. NARMAX models [1, 2], neural networks and machine learning methods [3–5], block-structured models [6, 7] and nonlinear state-space models [8, 9]. The black-box approaches presented here belong to the last two categories, ranging from unstructured to very structured. A. Van Mulders ( ) • J. Schoukens • L. Vanbeylen Department of ELEC, Vrije Universiteit Brussel, Belgium e-mail: anne.vanmulders@vub.ac.be; johan.schoukens@vub.ac.be; laurent.vanbeylen@vub.ac.be G. Kerschen (ed.), Nonlinear Dynamics, Volume 2: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04522-1__19, © The Society for Experimental Mechanics, Inc. 2014 203
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