Data-driven dynamic interpolation and approximation

Abstract

The behavioral system theory and in particular a result that became known as the “fundamental lemma” give the theoretical foundation for nonparametric representations of linear time-invariant systems based on Hankel matrices constructed from data. These “data-driven” representations led in turn to new system identification, signal processing, and control methods. This paper shows how the approach can be used further on for solving interpolation, extrapolation, and smoothing problems. The solution proposed and the resulting method are general – can deal simultaneously with missing, exact, and noisy data of multivariable systems – and simple—require only the solution of a linear system of equations. In the case of exact data, we provide conditions for existence and uniqueness of solution. In the case of noisy data, we propose an approximation procedure based on ℓ1-norm regularization and validate its performance on real-life datasets. The results have application in missing data estimation and trajectory planning. They open a practical computational way of doing system theory and signal processing directly from data without identification of a transfer function or state-space system representation.