Behaviors, trajectories and data: A novel perspective on the design of unknown-input observers

The paper’s Theorem 12 gives a correct necessary-and-sufficient data-driven criterion for the existence of a UIO and an explicit construction (via Ω and the blocks S3–S6), with A* Schur, and matches the behavioral/data-compatibility framework built under the full-row-rank data assumption (Definition 10, Proposition 11) . The candidate’s necessity part agrees with the paper: any UIO induces a left-kernel L_UIO with [Vp Vf] mapped to [−A* I] through some Ω . However, in the sufficiency direction the candidate makes a sign error in mapping from the window identity x(t+1)=A* x(t)+Pu u(t)+Qu u(t+1)+Py y(t)+Qy y(t+1) to the observer matrices: they set Bu_UIO=Pu−A*Qu and By_UIO=Py−A*Qy, whereas the paper (and the direct check of the acceptor identity) require Bu_UIO=Pu+ A*Qu and By_UIO=Py+ A*Qy (equivalently, Bu_UIO=S3+ A* S4 and By_UIO=S5+ A* S6) . With the candidate’s signs, the acceptor identity fails in general, so the proposed observer does not guarantee inclusion nor convergence. The rest of the argument (error dynamics: Schur A* implies asymptotic decay) is standard and correct when the mapping is fixed .