Unified Eigenvalue-Eigenspace Criteria for Functional Properties of Linear Systems and the Generalized Separation Principle

Abstract

Classical controllability and observability admit equivalent Popov-Belevitch-Hautus (PBH) tests based on eigenvalue-wise rank conditions. This paper extends this framework to functional properties of linear systems, establishing necessary and sufficient PBH-style conditions for Functional Controllability, Functional Stabilizability, Functional Observability, and Functional Detectability in terms of generalized eigenspaces. By contrast, Target Output Controllability is shown not to admit an independent eigenvalue-wise characterization in general; a necessary and sufficient eigenstructure condition is derived instead. We further introduce Intrinsic Functional Controllability and Intrinsic Functional Stabilizability, which give necessary and sufficient conditions for the existence of the augmentation matrices required for functional controller and observer synthesis. These intrinsic properties yield a Generalized Separation Principle that recovers the classical separation principle as a special case.

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