Fitting a signal to a sum-of-exponentials model is a basic problem in signal processing. It can be posed and solved as a Hankel structured low-rank matrix approximation problem. Subsequently, local optimization, subspace, and convex relaxation methods can be used for the numerical solution. In this paper, we show another approach, based on the recently proposed concept of structured data fusion. Structured data fusion problems are solved in the Tensorlab toolbox by local optimization methods. The approach allows fitting of signals with missing samples and adding constraints on the model, such as fixed exponents and common dynamics in multi-channel estimation problems. These problems are non-trivial to solve by other existing methods.

Keywords: system identification; low-rank approximation; mosaic Hankel matrix; tensorlab; structured data fusion.

The paper considers the class of discrete-time, single-input, single-output, nonlinear dynamical systems described by a polynomial difference equation. This class, call polynomial time-invariant, is a proper generalization of the linear time-invariant model class. The identification data is assumed to be generated in the errors-in-variables setting, where the input and the output noise is zero mean, white, and the noise variances is known up to a scaling factor. The identification problem has two sub-problems

structure selection: find the monomials appearing in the difference equation representation of the system, and parameter estimation: estimate the coefficients of the equation.

The main result shows that the parameter estimation by minimization of the 2-norm of the equation error leads to unstructured low-rank approximation of an extended data matrix. The resulting method is computationally robust and efficient due to the use of the singular value decomposition. However, it requires knowledge of the model structure and even when the correct model structure is used, it leads to biased results. For the structure selection, the use 1-norm regularization is proposed. For the bias removal an adjustment of the ordinary least squares estimator is proposed. The resulting adjusted low-rank approximation methods defines an unbiased estimator for the model parameters of the polynomial time-invariant model.

We are focused on numerical methods for decomposing a multivariate polynomial as a sum of univariate polynomials in linear forms. The main tool is the recent result on correspondence between the Waring rank of a homogeneous polynomial and the rank of a partially known quasi-Hankel matrix constructed from the coefficients of the polynomial. Based on this correspondence, we show that the original decomposition problem can be reformulated as structured low-rank matrix completion (or as structured low-rank approximation in the case of approximate decomposition). We construct algorithms for the polynomial decomposition problem. In the case of bivariate polynomials, we provide an extension of the well-known Sylvester algorithm for binary forms.

This paper focuses on a state-space based approach for the identification of a rather general nonlinear block-structured model. The model has several Single-Input Single-Output (SISO) static polynomial nonlinearities connected to a Multiple-Input Multiple-Output (MIMO) dynamic part. The presented method is an extension and improvement of prior work, where at most two nonlinearities could be identified. The location of the nonlinearities or their relation to other parts of the model does not have to be known beforehand: the method is a black-box approach, in which no states, internal signals or structural properties need to be measured or known. The first step is to estimate a partly structured polynomial (nonlinear) state-space model from input-output measurements. Secondly, an algebraic approach is used to split the dynamics and the nonlinearities by decomposing the multivariate polynomial coefficients.