The Observability in Unobservable Systems

The paper defines the unobservability index ρ/ε via a constrained maximization over trajectories (its Eq. (3)) and, for the linear case x(k+1)=Ax(k), y(k)=Hx(k), reduces it to a quadratic maximization with G the finite-horizon observability Gramian and F=[1 0 … 0]A^K (its Eqs. (5)–(6)) . For nonlinear systems, it proposes an empirical finite-difference construction of G and F that yields the same quadratic program (its Eqs. (7)–(9)) . The candidate solution reproduces exactly these reductions, adds the standard tightening from ≤ to = (and correctly flags the ∞ case when Null(G) contains directions affecting x1(K)), and solves the program as a generalized Rayleigh quotient to obtain ρ/ε = sqrt(λ_max) of (F^T F, G)—a mathematically correct and natural completion of the paper’s outline. The substance and flow match the paper; the model fills in missing linear-algebra details not explicitly written in the paper. Hence both are correct with substantially the same proof idea.