  The purpose of this paper is to propose a novel perspective, based on
Willems' "behavior theory", on the design of an unknown-input observer for a
given linear time-invariant discrete-time state-space model, with unknown
disturbances affecting both the state and the output equations. The problem is
first addressed assuming that the original system model is known, and later
assuming that the model is unknown but historical data satisfying a certain
assumption are available. In both cases, fundamental concepts in behavior
theory, as the projection of a behavior, the inclusion of a behavior in another
one, and the use of kernel and image representations, provide quite powerful
tools to determine necessary and sufficient conditions for the existence of an
unknown-input observer (UIO), as well as algorithms to design one of them, if
it exists.
