This technical note is devoted to the observers and observer-based control for linear descriptor systems. First, the complete proof of necessary and sufficient conditions for the existence of Luenberger observers for an arbitrary order is given, then the existence conditions for the controller based on these observers are presented in the LMI formulation. The design approach is simple and based on the new study of the constrained generalized Sylvester equations. It unifies all the observers and observer-based controllers of different orders. A numerical example is presented to illustrate our approach.