Unknown Input Observer Design for Linear Time-Invariant Systems – A Unifying Framework

The paper proves finite-time exactness of the Σd error for strongly detectable systems by choosing the SCB with the special block-lower-triangular Fdd (equation (24)) so that the error splits into md cascaded integrator chains; for each chain the error dynamics coincide with Levant’s robust exact differentiator with bounded last-equation perturbations, enabling an induction over the chains to conclude ed(t) ≡ 0 after some Tf. This is precisely Theorem 2 and its proof, which rely on the bound for ud in SCB (equation (45)), the Luenberger design for Σb ensuring eb decays exponentially (equations (39)–(40)), the observer structure (equations (46)–(49)), and the chain error forms (equations (51)–(54)) together with Levant’s RED (equation (16)) . The candidate model’s solution mirrors these steps: it invokes the same SCB/Fdd choice to get lower-triangular chain coupling, uses boundedness of eb from the Hurwitz (Ab−LbCb), selects Levant-type gains with αi,qi above the perturbation bound for each chain, and completes the finite-time argument by induction. Hence the two arguments are substantially the same and correct.