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

Abstract

Classical controllability and observability characterise reachability and reconstructibility of the full system state and admit equivalent geometric and eigenvalue-based Popov-Belevitch-Hautus (PBH) tests. Motivated by large-scale and networked systems where only selected linear combinations of the state are of interest, this paper studies functional generalisations of these properties. A PBH-style framework for functional system properties is developed, providing necessary and sufficient spectral characterisations. The results apply uniformly to diagonalizable and non-diagonalizable systems and recover the classical PBH tests as special cases. Two new intrinsic notions are introduced: intrinsic functional controllability, and intrinsic functional stabilizability. These intrinsic properties are formulated directly in terms of invariant subspaces associated with the functional and provide verifiable conditions for the existence of admissible augmentations required for functional controller design and observer-based functional controller design. The intrinsic framework enables the generalized separation principle at the functional level, establishing that functional controllers and functional observers can be designed independently. Illustrative examples demonstrate the theory and highlight situations where functional control and estimation are possible despite lack of full-state controllability or observability.