# Decoupling noisy multivariate polynomials in nonlinear system identification

In the field of system identification, one special type of nonlinear models are the so-called block-oriented models, and more specifically the Wiener-Hammerstein models. When identifying parallel Wiener-Hammer-stein systems, a multiple-input-multiple-output polynomial should be decoupled, that was obtained from noisy measurements. In this work, an earlier developed decoupling algorithm developed for the noiseless case is extended to the noisy case.

The starting point of the research is a multivariate polynomial function under the influence of noise, whose coefficients are approximated. It is also assumed that the covariance matrix on these coefficients is known at the start of the decoupling process. We wish to decouple this function by finding two transformation matrices and a set of univariate polynomials, such that the given function can be expressed as a linear combination of univariate polynomials in linear forms of the input variables.

The earlier developed algorithm uses first-order derivative information of the given multivariate function and involves the so-called Canonical Polyadic Decomposition (CPD) of a tensor, which is, loosely spoken, a generalization of the singular value decomposition for two-dimensional matrices to multidimensional arrays of numbers.  In this work, a weight matrix based on the covariance matrix is added to the CPD. Three different weight matrices were tested: a diagonal weight matrix, block-diagonal weight matrix and a full weight matrix.

Results vary depending on the added noise. Also the full weight matrix, even though containing more information than the remaining weight matrices, is ill-conditioned, which may produce unexpected results in some cases.

#### References

[1] P. Dreesen and M. Ishteva and J. Schoukens, Decoupling Multivariate Polynomials Using First-Order Information and Tensor Decompositions, SIAM J. Matrix Anal. Appl. (36), 864-879, 2015

[2] G. Hollander and P. Dreesen and M. Ishteva and J. Schoukens,’’An initialization method for nonlinear model reduction using the CP decomposition’’, accpeted at the 13th International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA), 2017