Abstract
Providing flexibility and user-interpretability in nonlinear system identification can be achieved by means of block-oriented methods. One of such block-oriented system structures is the parallel Wiener-Hammerstein system, which is a sum of Wiener-Hammerstein branches, consisting of static nonlinearities sandwiched between linear dynamical blocks. Parallel Wiener-Hammerstein models have more descriptive power than their single-branch counterparts, but their identification is a non-trivial task that requires tailored system identification methods. In this work, we will tackle the identification problem by performing a tensor decomposition of the Volterra kernels obtained from the nonlinear system. We illustrate how the parallel Wiener-Hammerstein block-structure gives rise to a joint tensor decomposition of the Volterra kernels with block-circulant structured factors. The combination of Volterra kernels and tensor methods is a fruitful way to tackle the parallel Wiener-Hammerstein system identification task. In simulation experiments, we were able to reconstruct very accurately the underlying blocks under noisy conditions.
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Notes
- 1.
Remark that the introduction of the extra mode
is similar to the extraction of the weights \(\lambda _i\) in the notation \(\left[ \!\left[ \mathbf {\lambda }; \mathbf {A}, \mathbf {B}, \mathbf {C} \right] \!\right] \) of [10] where the columns of the factor matrices \(\mathbf {A}\), \(\mathbf {B}\) and \(\mathbf {C}\) are scaled to have unit norm. Our notation is intentionally different in the sense that we have normalized the first elements of the columns of \(\mathbf {P}\) and \(\mathbf {q}\) equal to one, for practical purposes.
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Acknowledgments
This work was supported in part by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem), the Belgian Government through the Inter-university Poles of Attraction (IAP VII) Program, by the ERC Advanced Grant SNLSID under contract 320378 and by FWO projects G.0280.15N and G.0901.17N. The authors want to thank Otto Debals and Nico Vervliet for help with the use of Tensorlab/SDF and the suggestion to extract the vector \(\mathbf {q}\) into an additional tensor mode.
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Dreesen, P., Westwick, D.T., Schoukens, J., Ishteva, M. (2017). Modeling Parallel Wiener-Hammerstein Systems Using Tensor Decomposition of Volterra Kernels. In: Tichavský, P., Babaie-Zadeh, M., Michel, O., Thirion-Moreau, N. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2017. Lecture Notes in Computer Science(), vol 10169. Springer, Cham. https://doi.org/10.1007/978-3-319-53547-0_2
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