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.
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Notes
- 1.
No steady-state response could be obtained after 20 simulated periods.
- 2.
In the table, the root-mean-square error (RMSE) of the output is used as a measure for the model quality. It is defined as \(\sqrt{\frac{\sum _{l=1 }^{n_{y } }\sum _{ k=1}^{N}{e}^{2}(l,k)} {n_{y}N}}\), with e(l, k) the output error at time index k and output position (sensor position index) l: \(e(:,k) = y_{model}(k) - y_{meas}(k)\). Hence, the errors on the 8 output positions are averaged. The errors are either based on the entire frequency band (“overall RMSE”) or based on frequencies up to 379. 5 Hz (“in-band RMSE”).
- 3.
Not higher, since the nonlinear model should not be extrapolated.
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Acknowledgements
This work is sponsored by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish Government (Methusalem Grant METH-1), the Belgian Program on Inter-university Poles of Attraction (IAP VII/19 - Dysco), and by the ERC advanced grant SNLSID, under contract 320378.
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Van Mulders, A., Schoukens, J., Vanbeylen, L. (2014). Nonparametric Analysis and Nonlinear State-Space Identification: A Benchmark Example. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 2. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-04522-1_19
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