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.

sponsorship: This work was supported in part by the Catalan Institution for Research and Advanced Studies (ICREA); in part by the Fond for Scientific Research Vlaanderen (FWO) under Grant G033822N; and in part by the Spanish Ministry of Science under Grant MCIU/AEI/10.13039/501100011033 and Grant PID2023-148952OB-I00. (Catalan Institution for Research and Advanced Studies (ICREA), Fond for Scientific Research Vlaanderen (FWO)|G033822N, Spanish Ministry of Science under Grant MCIU/AEI, PID2023-148952OB-I00)