Modeling Parallel Wiener-Hammerstein Systems Using Tensor Decomposition of Volterra Kernels

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.

Identification of block-oriented nonlinear systems starting from linear approximations: A survey

“…Palm (1978Palm ( , 1979 shows that a wide class of Volterra systems can be approximated arbitrarily well using a parallel Wiener-Hammerstein model structure. Dreesen et al (2017) present an identification method based on the decomposition of Volterra kernels.…”

Algorithms for Canonical Polyadic Decomposition With Block-Circulant Factors

“…If there are at least two slices of G that do not contain any missing values, a GEVD can be computed of the subtensor consisting of these slices to partially initialize the optimization-based method. From the decomposition of G an estimate G (n) is obtained, which can be used to find the original circulant factors as described above in (6).…”

“…A Wiener-Hammerstein system is defined as a static nonlinear block sandwiched between two linear time-invariant FIR filters and can be identified by first estimating a Volterra kernel from input/output data and then computing its CPD. This idea was extended to parallel Wiener-Hammerstein systems in [6]. It is shown that the estimated Volterra kernels admit a structured tensor decomposition with almost all block-circulant factors.…”

“…It is shown that the estimated Volterra kernels admit a structured tensor decomposition with almost all block-circulant factors. An optimization-based strategy to solve this problem has been developed in [6]. Here, we focus on the special case depicted in Figure 3.…”

“…If Q (1) and Q (2) are simple moving average filters of length L + 1, it immediately follows from [6] that the third-order Volterra kernel K can be written as a CPD with pure block-circulant factors containing the filter coefficients of P (1) and P (2) . Note that for arbitrary filters Q (1) and Q (2) , one has to resort to optimization-based techniques such as [6]. How the parameters appear in the decomposition is explained in [6] as well.…”