- Seminar: BLIND AND OFF-GRID ACOUSTIC ECHOES RETRIEVAL USING MULTICHANNEL ANNIHILATING FILTERS
Decoupling Multivariate Nonlinearities in System Identification
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.
The proposed solution method  is based on a first-order information approach. By evaluating the Jacobian of the function f(u) in several operating points, a tensor decomposition  problem is obtained from which the unknown linear transformations can be determined, as well as the internal univariate mappings. The canonical polyadic decomposition  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 . 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 . 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 
 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.
 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.
 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.
 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.