Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

The paper’s Theorem 4.3 proves uniform exponential detectability by: (i) transforming the error dynamics via continuous QR into a block upper-triangular form, (ii) using negativity of the upper Bohl exponents on the lower block to obtain uniform exponential stability there, (iii) solving a reduced-order Riccati equation on the upper block under uniform complete observability, and (iv) invoking a standard triangular/cascade stability result to conclude uniform exponential stability of the full error system. The candidate solution follows the same structure, adds a Lyapunov/information-form derivation for the reduced block, and reaches the identical gain L(t)=Q̄P Q̄^T C^T and conclusion. Minor notation issues appear in both sources (transpose in the Riccati drift term and a missing trailing P in one printed equation), but the underlying argument is consistent with the paper’s proof and standard results. Key steps are explicitly present in the paper: the detectability statement and proof sketch (Theorem 4.3) , the detectability definition and UCO-to-RDE facts , the QR machinery and block-triangularization relations , and the Bohl-exponent link used to infer stability for the lower block ; the block-triangular proof step appears verbatim in the paper’s sketch .