The Effect of Sensor Fusion on Data-Driven Learning of Koopman Operators

The paper’s Theorem 2 asserts the existence of a similarity transform T that yields the block-triangular OC-KO structure (16) and shows that an appropriately projected, time-delay-stacked output z_k is diffeomorphically conjugate to the observable lifted coordinates ψ_o(x_k). The proof relies on the standard observable decomposition for finite-dimensional LTI systems and the full-column-rank property of a stacked observability operator O_N, after which choosing W_ψ = O_N^T gives an invertible map S = O_N^T O_N and linear conjugacy z_k = S ψ_{o,k} (see Theorem 2 and equations (16)–(17) in the paper ). The candidate solution follows the same essential route, but with a more explicit construction: it factors out the unobservable subspace via a quotient, builds T to achieve the block form (16), proves O_{n_o-1} has full column rank under observability, and then selects W_ψ so that S := W_ψ O_N is invertible, yielding linear conjugacy between z_k and ψ_{o,k}. Aside from minor notational/dimensional slips in the paper (e.g., writing O ∈ R^{Np×n_o} when the stack has N+1 blocks), the arguments align. Hence both are correct and substantially the same proof.