A Data Centric Approach to the Design of Unknown Input Functional Observers

Functional observers have been extensively studied in the classical model-based literature for estimating only a portion of the state without reconstructing the full state, thereby providing computational efficiency and flexibility. Despite their extensive development, functional observer design directly from data remains largely unexplored. Motivated by this gap, a data-driven approach to constructing unknown-input functional observers (UIFOs) for discrete-time linear time-invariant (LTI) systems is presented in this paper. We first revisit the model-based solution and reformulate it for the discrete-time LTI case to enable a smoother transition to the data-driven framework. The main contribution is the derivation of sufficient and necessary conditions for having the existence of a UIFO whose matrix parameters are derived from available historical data, along with a proof of their equivalence to the classical model-based conditions. Although we concentrate on UIFOs, our findings equally hold for unknown-input observers (UIOs). Furthermore, a novel data-driven algorithm is proposed for estimating the observer matrices. The efficacy of the proposed approach is illustrated with an example.