Functional Observability Criterion

• Functional observability is a criterion that defines conditions for reconstructing specific state functions from outputs without measuring the complete state.
• It employs algebraic rank tests, modal and graph-theoretic methods to ensure unique reconstruction and guide optimal sensor placement.
• The framework enables efficient observer design and privacy preservation across linear, nonlinear, sample-based, and stochastic systems.

The Functional Observability Criterion provides necessary and sufficient conditions for reconstructing prescribed functions—linear or nonlinear—of the state trajectory of a dynamical system from available system outputs, without requiring reconstruction of the entire state. Functional observability is foundational for the estimation, monitoring, and control of large-scale, networked, structured, or uncertain systems where full-state measurement is infeasible or undesirable. This criterion underpins optimal sensor placement, observer design, privacy-blocking, and the analysis of structural and sample-based identifiability.

1. Algebraic and Modal Characterizations in Linear Systems

In continuous-time LTI systems

$$\dot{x}(t) = A x(t) + B u(t),\qquad y(t) = C x(t),\qquad z(t) = F x(t)$$

functional observability for a triple $(C, A; F)$—or target observability—requires that knowledge of $u(\cdot)$ and $y(\cdot)$ over a finite interval uniquely determines $z(0) = F x(0)$. The definitive algebraic criterion is the Jennings–Fernando–Trinh rank test: $$\operatorname{rank}\begin{bmatrix}O \ F\end{bmatrix} = \operatorname{rank}(O)$$ where $O = [C; CA; \ldots; CA^{n-1}]$ is the classical observability matrix. This is necessary and sufficient for uniquely reconstructing $F x(0)$ independently of auxiliary states (Montanari et al., 2023, Montanari et al., 2022, Montanari et al., 29 Jan 2024).

The modal (or PBH-type) characterization, extended to general (including non-diagonalizable) $A$ via Jordan decomposition, asserts for all $\lambda \in \mathbb{C}$: $$\operatorname{rank}\begin{bmatrix}A - \lambda I \ C \ F \end{bmatrix} = \operatorname{rank}\begin{bmatrix}A - \lambda I \ C \end{bmatrix}$$ This ensures that every eigenvector of $A$ not distinguishable by $C$ is annihilated by $F$; no unobservable mode leaks into the functional of interest (Montanari et al., 5 Feb 2024, Zhang et al., 2023, Zhang et al., 2023).

2. Graph-Theoretic and Structural Criteria

For structured networked systems or large-scale settings, functional observability reduces to combinatorial connectivity properties in a suitably defined system graph. For $(C, A; F)$, define the inference digraph $G(C, A; F)$ over state nodes $X$, sensor nodes $Y$, and

• edges $x_j \to x_i$ if $A_{ij} \neq 0$
• edges $x_j \to y_i$ if $C_{ij} \neq 0$
• targets $T \subset X$ defined by nonzero rows in $F$

The graph-theoretic criterion asserts (Montanari et al., 2023, Montanari et al., 2022):

• Reachability: Every $x \in T$ has a directed path to some $y \in Y$
• No target contraction: For any subset $T' \subset T$, its set of direct successors $S(T')$ satisfies $|S(T')| \ge |T'|$ (i.e., no set of targets is "bottlenecked")

This test can be implemented via maximum matching in an associated bipartite graph in $O(E \sqrt{V})$ time, enabling polynomial-time verification and near-optimal sensor placement via set cover approximations.

Structural Functional Observability (SFO): For structured triples $(\bar{A},\bar{C},\bar{F})$, SFO requires that almost every consistent numerical realization is functionally observable. In generically-diagonalizable systems, SFO admits closed-form and matching-theoretic sensor placement criteria (Zhang et al., 25 Sep 2024, Zhang et al., 2023). For general systems, SFO reduces to inclusion criteria for outputs of all maximum linkings or dilation-freeness in the dynamic graph.

3. Duality, Observer Design, and Minimality

Duality

Functional observability is tightly connected to output controllability by duality. Weak duality states that functional observability of $(C, A; F)$ implies output controllability of the dual $(A^\top, C^\top; F)$; under an additional geometric (orthogonality) condition on the controllability/observability Gramian, strong duality holds, reversing the implication (Montanari et al., 2023, Montanari et al., 29 Jan 2024). This yields direct algorithmic equivalence between minimal sensor and minimal actuator placement for function estimation/control.

Observer Design

Given the algebraic criterion or its structural versions, it is possible to synthesize minimal-order functional observers of the form

$$\dot{w} = N w + J y + H u,\qquad \hat{z} = D w + E y$$

where $(N, J, H, D, E)$ are constructed so that $\hat{z}(t) \to z(t)$ asymptotically. The existence of such an observer is equivalently certified by the functional rank test (Montanari et al., 29 Jan 2024, Montanari et al., 2022). The order of the observer is typically lower (sometimes dramatically) than that required for full-state estimation.

Sensor Placement and Complexity

• Exact minimal-sensor selection is NP-hard in general but admits $(1+\ln r)$-approximation via greedy set cover (Zhang et al., 2023).
• In generically-diagonalizable or special cases, closed-form or matching-based minimal sensor placement is possible (Zhang et al., 25 Sep 2024).
• For targets with self-loops, set cover greedy methods suffice; otherwise, contraction checks must be included, increasing computational complexity (Montanari et al., 2023, Montanari et al., 2022).

4. Extensions: Sample-Based, Nonlinear, Infinite-Dimensional, and Stochastic Systems

Sample-Based Functional Observability: For irregular/intermittent sampling, $k$ output samples at times $\{t_i\}$ suffice to reconstruct $F x(0)$ if

$$\operatorname{rank}\begin{bmatrix} O_s(A,C) \ O(A,F) \end{bmatrix} = \operatorname{rank} O_s(A,C)$$

where $O_s(A,C)$ is the sample-based observability matrix. Sampling times must ensure that observable A-modes are adequately excited; fewer than $n$ samples may suffice if only a reduced functional is targeted (Krauss et al., 30 Jun 2025).

Nonlinear Systems:

• For smooth dynamical systems $\dot{x} = f(x),~ y = h(x),~ z = g(x)$,
  • Local functional observability at $x_0$ is determined by the Lie-derivative rank test:

  $$\operatorname{dim} \{\nabla O(x_0)\} = \operatorname{dim} \{\nabla O(x_0), \nabla g(x_0)\}$$

where $O(x)$ is the span of all Lie-derivatives of $h_j(x)$. If adding $\nabla g(x_0)$ does not increase the rank, $z$ is functionally observable from the output flow (Montanari et al., 2023, Kravaris, 30 Dec 2024).

• For observer synthesis, sufficient conditions involve expressing all Lie-derivatives of $q(x)$ as smooth functions of Lie-derivatives of the measured outputs, enabling functional observer design with assignable error dynamics.

Infinite-Dimensional Systems:

• In Banach-space or PDE settings, functional observability (e.g., final-state observability, optimal observability) is characterized via uncertainty relations and dissipation estimates (semigroup decay), with geometric set properties (thick sets) dictating sensor domain optimality (Gallaun et al., 2019, Privat et al., 2012).

Stochastic Systems:

• For partially observed Markov processes (POMP), the criterion requires that for any continuous bounded $f(x)$, there exists a measurable function of future outputs $g(Y_{1:N})$ approximating $f(x)$ under all initializations. Observability in this sense is both necessary and sufficient for filter stability under weak, total-variation, and relative-entropy merging of filtering distributions (McDonald et al., 2018).

5. Applications: Large-Scale, Power Grids, Privacy, and Quantum Systems

Functional observability has been applied in diverse settings:

• Large-scale\,/\,networked systems: Fast polynomial algorithms for minimal sensor deployment and functional observer design reduce sensing and computational load relative to full-state estimators, with performance validated on synthetic and real network models (Montanari et al., 2022, Montanari et al., 2023).
• Power systems: For radial distribution feeders with smart meters, functional observability is equivalent to the existence of vertex-disjoint paths from unknown to metered buses—checkable by matching in linear time (Bhela et al., 2016).
• Privacy-blocking: In network privacy, the functional PBH-criterion quantifies the (NP-hard) problem of blocking particular state variables to prevent adversarial inference of sensitive functionals (Zhang et al., 2023).
• Quantum ergodic domains: The spectral observability functional $J(\omega)=\inf_j \int_\omega \phi_j^2$ governs optimal sensor placement for wave and Schrödinger equations, with quantum ergodicity properties guaranteeing asymptotic optimality (Privat et al., 2012).

6. Comparative Summary and Key Algorithmic Procedures

The following table summarizes core aspects across main settings:

System Type	Algebraic Criterion	Structural/Graph Criterion	Observer Design
LTI (finite)	$$\operatorname{rank}\begin{bmatrix}O \ F\end{bmatrix} = \operatorname{rank} O$$	Reachability + no contraction/dilation	Luenberger/Kalman, reduced order
Sample-based	$$\operatorname{rank}\begin{bmatrix}O_s \ O_F\end{bmatrix} = \operatorname{rank} O_s$$	Matching in sample-excitation bipartite graph	Least-squares with assigned samples
Structured/SFO	Generic rank / contraction-free paths	Paths/linkings/dilations in graph $\mathcal{G}(\bar A, \bar C)$	Matching-based algorithms
Nonlinear	Lie-derivative rank test	Symbolic reachability in derivatives	Input–output/differential observers
Stochastic	Approximation for all $f$ by $g(Y_{1:N})$	Notion of filter information propagation	Nonlinear Bayesian filters

Key algorithmic techniques include maximum matching (Hopcroft-Karp or minimum-weight in bipartite graphs), greedy set cover for sensor placement, and pole-assignment in observer synthesis. The duality with target controllability allows direct transfer of methods between minimal control/estimation problems under strong conditions.

Functional observability provides an analytically rigorous and computationally tractable foundation for targeted estimation in high-dimensional and complex dynamical systems, spanning finite, infinite, nonlinear, sampled, and stochastic domains (Montanari et al., 2023, Montanari et al., 29 Jan 2024, Montanari et al., 2022, Zhang et al., 25 Sep 2024, Zhang et al., 2023, Kravaris, 30 Dec 2024, Privat et al., 2012, Bhela et al., 2016, Krauss et al., 30 Jun 2025, McDonald et al., 2018, Montanari et al., 2023, Zhang et al., 2023, Gallaun et al., 2019, Montanari et al., 5 Feb 2024).