  Functional observability and output controllability are properties that
establish the conditions respectively for the partial estimation and partial
control of the system state. In the special case of full-state observability
and controllability, the Popov-Belevitch-Hautus (PBH) tests provide conditions
for the properties to hold based on the system eigenspace. Generalizations of
the Popov-Belevitch-Hautus (PBH) test have been recently proposed for
functional observability and output controllability but were proved to be valid
only for diagonalizable systems thus far. Here, we rigorously establish a more
general class of systems based on their Jordan decomposition under which a
generalized PBH test for functional observability is valid. Likewise, we
determine the class of systems under which the generalized PBH test is
sufficient and necessary for output controllability. These results have
immediate implications for observer and controller design, pole assignment, and
optimal placement of sensors and drivers.
