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.
Classical controllability and observability are central concepts in linear systems theory, characterising state reachability and reconstructibility. They admit geometric interpretations via invariant subspaces and are commonly verified using controllability and observability matrices. Equivalent Popov-Belevitch-Hautus (PBH) tests provide eigenvalue-based rank conditions that expose modal structure and enable verification without constructing large matrix powers.
Motivated by modern applications in large-scale and networked systems, increasing attention has been devoted to functional generalisations of classical system properties [1]- [9]. Rather than controlling or estimating the full state vector, T. Fernando is with the Department of Electrical, Electronic and Computer Engineering, University of Western Australia (UWA), 35 Stirling Highway, Crawley, WA 6009, Australia. (email: tyrone.fernando@uwa.edu.au) the objective is to control or reconstruct a prescribed linear functional z(t) = F x(t), representing quantities of practical relevance such as aggregate power, population totals, or reduced-order performance variables. This shift leads naturally to functional analogues of controllability, stabilizability, observability, and detectability. Existing work introduces these functional notions using subspace-based definitions that generalise Kalman’s geometric framework [10], [11]. Subspace-based characterisations are available for functional observability [12], [13], functional detectability [14], functional controllability and functional stabilizability [9], and target output controllability [15]. These formulations preserve classical dualities, provide conceptual clarity, and reduce to standard notions when F is the identity matrix. Extensions of subspace-based methods for functional observability have been explored for sample-based linear systems [16], nonlinear systems [17]- [19], networked systems [2]- [6], and diverse applications including power systems [20] and event-triggered estimation [21], [22].
However, subspace characterisations do not directly expose the role of individual eigenmodes and are often inconvenient for large-scale or structured systems, where local, eigenvalue-based tests are preferable. Corresponding PBHstyle eigenvalue-eigenspace based tests for functional properties remain limited in scope and have largely been confined to restricted system classes [1], [4], [7] while other frequency-domain strategies for functional state estimation appear in [23]. This paper develops a unified PBH-style framework [24] for established functional system properties, including functional controllability, functional stabilizability, functional observability, functional detectability, and target output controllability. By working directly with eigenvectors and generalized eigenvectors arranged along Jordan chains, we obtain necessary and sufficient spectral characterisations that avoid the construction of full controllability or observability matrices. The framework recovers the classical PBH test as a special case and applies uniformly to both diagonalizable and non-diagonalizable systems. Within this framework, these functional properties are not separate phenomena, but different manifestations of the same underlying eigenspace geometry, from which a generalised separation principle follows naturally.
Moreover, for the design of functional controllers and functional observers, augmentation-based formulations [9], [25] remain practically useful. Once suitable augmentation matrices are available, they provide explicit realizations and constructive synthesis procedures for both functional controller design and functional observer design. However, unlike in the case of functional observer design, for a given quadruple (A, B, C, F ) it is not known a priori whether augmentation matrices exist that satisfy the required rank conditions for functional controller design. This paper addresses precisely this issue by introducing two intrinsic functional properties: intrinsic functional controllability and intrinsic functional stabilizability. For these intrinsic notions, both subspace-based and eigenvalue-eigenspace-based criteria are developed.
In this sense, the intrinsic notions elevate augmentationbased designs from realization-dependent constructions to verifiable structural properties of the quadruple (A, B, C, F ), and play a key role in establishing the Generalized Separation Principle by certifying the existence of augmentation matrices required to realise observer-based functional controllers. Contributions. This paper develops a unified PBH-style framework yielding necessary and sufficient spectral characterisations of seven functional system properties: functional controllability (FC), intrinsic functional controllability (IFC), functional stabilizability (FS), intrinsic functional stabilizability (IFS), functional observability (FO), functional detectability (FD), and target output controllabi
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