Application of low-rank approximation for nonlinear system identification

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 1) structure selection: find the monomials appearing in the difference equation representation of the system, and 2) 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.