Revisiting the necessary condition for the existence of Luenberger observer

In the research field of observer design, one necessary and sufficient condition for the existence of Luenberger observer is widely cited as a fundamental theorem. In 1972, Fortmann and Williamson gave an earliest statement and proof of this theorem. The proof, in essentially unchanged form, was written in all books. We point out that this proof is not rigorous in that it neglected the couple relation between the state to be observed and the input term due to control in the error dynamical equation. A strict proof is given by elegantly choosing control input signal, which deals with the coupling between states and inputs skillfully. Other related problems are also discussed using this skill.