# Decoupling Multivariate Nonlinearities in System Identification

#### Problem statement

Nonlinear system identification often makes use of coupled multiple-input-multiple-output static nonlinear functions to represent nonlinearities [5, 6]. For the sake of model interpretability and to limit the rapid increase of parameters it is desirable to find a parsimonious description. A simplified description may also serve as a good approximation.

We assume that a linear transformation is present at the input as well as at the output, concealing the internal variables between which univariate polynomial mappings exist as in the figure below.

Figure: Decoupling scheme. A given multivariate function f(u) is decoupled into a linear combination of univariate functions in linear forms of the input variables.

#### First-order Approach

The proposed solution method [2] is based on a first-order information approach. By evaluating the Jacobian of the function f(u) in several operating points, a tensor decomposition [4]  problem is obtained from which the unknown linear transformations can be determined, as well as the internal univariate mappings. The canonical polyadic decomposition [4] provides a numerical procedure to determine the unknown linear basis transformations, providing access to the internal variables such that the full structure can be recovered.

#### Applications and extensions

An application to the identification of a parallel Wiener-Hammerstein system was described in [2]. The method can easily be generalized to the question of partially decoupling a given multivariate function in which a ‘block-decoupling’ is obtained. In this case the function f(u) is decoupled in mutually decoupled internal multiple-input-multiple-output nonlinearities in which each has a small number of outputs depending on a small number of inputs. The computational core is in this case the block-term tensor decomposition and is described in [3]. A related method to unravel the Wiener-Hammerstein structure in a given state-space model using the full decoupling approach and linear algebra tools is described in [4]

#### References

[1] P. Dreesen, M. Ishteva, and J. Schoukens. Decoupling multivariate polynomials using first-order information and tensor decompositions. SIAM J. Matrix Anal. Appl., 36(2):864-879, 2015.

[2] P. Dreesen, M. Schoukens, K. Tiels, and J. Schoukens. Decoupling static nonlinearities in a parallel Wiener-Hammerstein system: A first-order approach. In Proc. 2015 IEEE International Instrumentation and Measurement Technology Conference (I2MTC 2015), pages 987-992, Pisa, Italy, 2015.

[3] P. Dreesen, T. Goossens, M. Ishteva, L. De Lathauwer, and J. Schoukens. Block-decoupling multivariate polynomials using the tensor block-term decomposition. In E. Vincent, A. Yeredor, Z. Koldovsky, and P. Tichavsky, editors, Proc. 12th International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA 2015), volume 9237 of Lecture Notes on Computer Science (LNCS), pages 14-21, 2015.

[4] P. Dreesen, M. Ishteva, and J. Schoukens. Recovering Wiener-Hammerstein nonlinear state-space models using linear algebra. Proc. 17th IFAC Symp. Syst. Ident. (SYSID 2015), Beijing, China, pp. 951-956, 2015.