# Identifying parallel Wiener-Hammerstein systems by decomposing Volterra kernels

To capture the nonlinear effects of the real world, the focus of system identification is shifting from linear to nonlinear dynamical models. Every nonlinear dynamic system with fading memory can be approximated with arbitrary precision by its Volterra kernel description, which generalizes the concept of the finite impulse response to the nonlinear case, in much the same way as a Taylor series expansion for function approximation. However, the Volterra series provides a non-parametric representation, which lacks physical and intuitive interpretation. To take advantage of the Volterra representation while aiming for an interpretable block-oriented model, we study ways to impose the desired structure using tensor techniques.

We successfully generalized [1] an existing tensor technique [2,3] for identifying Wiener-Hammerstein systems to identifying the more challenging but more powerful parallel Wiener-Hammerstein systems. Tensor techniques, and in particular well-designed structured canonical polyadic decompositions, thus once again proved useful in the system identification community. The next step is to make the algorithm numerically stable and robust to noise.

#### References

- P. Dreesen, D. T. Westwick, J. Schoukens, and M. Ishteva. Modeling parallel Wiener-Hammerstein systems using tensor decomposition of Volterra kernels. The 13th International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA 2017), Grenoble, France, February 2017. Volume 10169 of the series Lecture Notes in Computer Science, pp. 16–25.
- D. Westwick, M. Ishteva, P. Dreesen, and J. Schoukens. Tensor factorization based estimates of parallel Wiener Hammerstein models. 20th IFAC World Congress, Toulouse, France, July 2017.
- de M. Goulart, J.H., Boizard, M., Boyer, R., Favier, G., Comon, P.: Tensor CP decomposition with structured factor matrices: Algorithms and performance. IEEE J. Sel. Top. Signal Process. 10(4), 757—769, 2016.
- Favier, G., Kibangou, A.Y.: Tensor-based methods for system identification -- part 2: Three examples of tensor-based system identifcation methods. Int. J. Sci. Techn. Autom. Control & Comp. Eng. (IJ-STA) 3(1), 870—889, 2006