On the existence of strong functional observer
Michael Di Loreto, Damien Eberard
TL;DR
This paper addresses the problem of when a strong functional observer exists for linear time-invariant systems with an unknown input, formalized as estimating $z=Ex+Fu$ from measurements $y=Cx+Du$. It develops necessary and sufficient conditions based on strong functional detectability, expressed through invariant zeros and rank criteria of $P(s)=sI_n-A-BCD$ and $P_e(s)=P(s)EF$, together with subspace conditions for the strong$^\star$ case. The results generalize and unify prior work on functional observers, state/input reconstruction, and known-input scenarios, providing numerically tractable tests (via invariant zeros, Smith form decompositions, and subspace inclusions) and clarifying the relationships among various detectability notions. The framework clarifies how existing results emerge as special cases and paves the way for designing minimal-order strong functional observers in future work, with potential impact on robust state-function estimation and reverse-engineering of inputs.
Abstract
For arbitrary linear time-invariant systems, the existence of a strong functional observer is investigated. Such observer determines, from the available measurement on the plant, an estimate of a function of the state and the input. This estimate converges irrespective to initial state and input. This formulation encompass the cases of observer existence for known or unknown inputs and generalizes state-of-art. Necessary and sufficient conditions for such an existence are proposed, in the framework of state-space representation. These conditions are based on functional detectability property and its generalizations for arbitrary input, which include considerations on convergence of the estimation, irrespective to the initial state and the input. Known results on state detectability, input reconstruction or functional detectability are retrieved by particularizing the proposed conditions.