A behavioral approach to the estimation problem and its applications to state-space models

The observer design problem is here investigated in the context of linear left shift invariant discrete behaviors, whose trajectories have support on the positive axis. Necessary and sufficient conditions for the existence of a (consistent) dead-beat observer of some relevant variables from some measured ones, in the presence of some unmeasured (and irrelevant) variables, are introduced, and a complete parametrization of all dead-beat observers is given. Finally, several classical problems addressed for state-space models, like state estimation, the design of unknown input observers or the design of fault detectors and identifiers (possibly in the presence of disturbances), are casted in this general framework, and the aforementioned equivalent conditions and parametrizations are tailored to all these special instances.