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

• DOI: 10.1016/j.automatica.2025.112122
• Publication Date: 2025-01-24T17:26:42Z
• Authors: Arthur N. Montanari; Chao Duan; Adilson E. Motter
• Keywords: Physics - Physics and Society; Optimization and Control (math.OC); FOS: Mathematics; FOS: Electrical engineering, electronic engineering, information engineering; FOS: Physical sciences; Disordered Systems and Neural Networks (cond-mat.dis-nn); Systems and Control (eess.SY); Dynamical Systems (math.DS); Physics and Society (physics.soc-ph); Condensed Matter - Disordered Systems and Neural Networks; Mathematics - Dynamical Systems; Mathematics - Optimization and Control; Electrical Engineering and Systems Science - Systems and Control

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.