Identifiability in the Behavioral Setting

Abstract

Identifiability, i.e., uniqueness of a solution of the identification problem, is a fundamental issue in system identification and data-driven control. Necessary and sufficient identifiability conditions for deterministic linear time-invariant systems that do not require a priori given input/output partitioning of the variables nor controllability of the true system are derived in the article. The prior knowledge needed for identifiability is the number of inputs, lag, and order of the true system. Our results are based on a modification of the notion of a most powerful unfalsified model for finite data and a novel algorithm for its computation. We provide a generalization of a result that became known as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fundamental lemma</i> and a novel nonparametric data-driven representation of the system behavior based on general data matrix structures. The results assume exact data, however, low-rank approximation allows their application in the case of noisy data. We compare empirically low-rank approximation of the Hankel, Page, and trajectory matrices in the errors-in-variables setting. Although the Page and trajectory matrices are unstructured, the parameter estimates obtained are less accurate than the one obtained from the Hankel matrix.