On the Popov–Belevitch–Hautus tests for functional observability and output controllability

Functional observability and output controllability are properties that establish the conditions for the partial estimation and partial control of the system state, respectively. 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 PBH tests have been recently proposed for functional observability and output controllability, but thus far they have only been proven valid for diagonalizable systems. Here, we rigorously establish the generalized PBH test for functional observability, extending its validity to a broader class of systems using Jordan decomposition. 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.