Unknown-state, unknown-input reconstruction in discrete-time systems with invariant zeros: geometric solutions for nonminimal triples

An intrinsic restraint of unknown-state, unknown-input reconstruction in systems with invariant zeros is the fact that for any invariant zero, at least one initial state exists, such that when the mode of the invariant zero is suitably injected into the system the output is identically zero. Nonetheless, the problem of reconstructing a generic, bounded, inaccessible input when the initial state is unknown has attracted considerable research efforts, particularly in connection with applications to fault diagnosis. This paper provides a geometric method for solving this problem in discrete-time systems which are completely observable from the available measurements, not completely reachable from the inaccessible inputs in general.