How to Represent and Identify Affine Time-Invariant Systems?

Affine systems are ubiquitous in modeling and emerge naturally from the linearization of nonlinear dynamics. Despite their relevance in applications, their identification remains largely ad hoc, relying on centering the data before applying linear identification methods. This heuristic approach assumes constant offset and can introduce bias. We develop a dedicated framework for affine system identification, deriving identifiability conditions and identification methods based on difference equation representations. Unlike the classical two-step approach, our method identifies the data-generating system under conditions verifiable from data and system complexity. For noisy data in the errors-in-variables setting, we recast the problem as a structured low-rank approximation, leveraging existing optimization techniques for efficient computation.