State Observation of LTV Systems with Delayed Measurements: A Parameter Estimation-based Approach

The paper’s proposition sets an adaptive observer with delayed measurements and claims exponential convergence under a PE condition stated as ∫ C(s)Φ(s)Φ(s)ᵀC(s)ᵀ ds ≥ δ Iq for all sliding windows. However, the gradient law’s parameter error dynamics require PE of Y∘ϕ in the column sense, i.e., ∫ Φ(ϕ(s))ᵀC(ϕ(s))ᵀC(ϕ(s))Φ(ϕ(s)) ds ≥ α In, and the paper neither establishes the propagation of PE through the time-warp ϕ(t) nor reconciles the row-PE vs column-PE mismatch used in the proof; it only alludes to a derivative-based change of variables in a remark without integrating the needed assumptions into the main statement. The proof concludes with x̂−x = −Φ θ̃ but does not justify that θ̃’s decay dominates any potential growth of Φ, nor discuss how the gain Γ ensures this. By contrast, the candidate solution explicitly adds standard assumptions on ϕ (absolute continuity, monotonicity with bounded derivative, ϕ(t)→∞) and uses a time-change argument to show PE of Y∘ϕ, then invokes the correct column-PE for the gradient algorithm to deduce exponential decay of θ̃ and hence of the state error. These points directly patch the paper’s gaps while following the same PEBO structure. See the paper’s proposition and proof steps, including the stated PE condition and the error dynamics, as well as the later remark on ϕ̇ and change of variables, which highlight the omissions in the main result’s hypotheses .