Input and State Observability of Network Systems with a Single Unknown Input

This paper studies network systems affected by a single unknown input, possibly representing an attack or a failure, to be estimated. The main result is a characterization of input and state observability, namely the conditions under which both the whole network state and the unknown input can be reconstructed from some measured local states. This characterization is in terms of observability of a suitably-defined subsystem, which allows the use of known graphical charactizations of observability of network systems, leading to structural results (true for almost all interaction weights) or strong structural results (true for all non-zero interaction weights). We apply our results to an illustrative example, finding a full characterization of input and state observability of a path graph, affected by a single unknown input and with measurement of a small number of local states.