Existence and Design of Functional Observers for Time-Delay Systems with Delayed Output Measurements

This paper investigates the problem of functional state estimation for linear time-delay systems in which the delay affecting the state evolution differs from the delay affecting the output measurements. While existing observer designs typically assume instantaneous output availability, practical systems often exhibit measurement delays that are distinct from and not aligned with the intrinsic state delay. We explicitly distinguish between the state delay $\tau$ and the measurement delay $h$ and address the problem of estimating a desired functional $z(t)=Fx(t)$ under such mismatched delay conditions. Three functional observer structures are proposed to accommodate different delay configurations, each capable of realizing functional observers of different orders. This flexibility is important since a functional observer whose order equals the number of estimated functionals may not always exist. For each structure, algebraic existence conditions are established together with constructive synthesis procedures. A functional augmentation framework is developed to derive verifiable rank-based conditions for observers of various orders. In addition, the notion of generalized functionals, defined over an augmented delayed state vector, is introduced to provide greater flexibility in satisfying observer existence conditions and facilitating systematic design. Numerical examples illustrate the proposed theory.