On the existence of strong functional observer

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.