Functional Observers in Control Systems

• Functional observers are systems designed to estimate a specified function of a state vector, enabling targeted estimation in high-dimensional or networked models.
• They use algebraic rank tests, PBH-type criteria, and graph-theoretic methods to ascertain target observability and streamline sensor placement.
• Extensions to nonlinear, sample-based, and stochastic settings provide practical solutions for monitoring and controlling complex networks.

A functional observer is an observer system designed to estimate a prescribed function of the state vector—rather than the full state itself—using available system inputs and outputs. Functional observers generalize classical observability and Luenberger observer concepts by targeting the reconstruction of a user-specified linear (or nonlinear) function of the state, $z = F x$ (with $F$ a prescribed map), from output measurements and possibly inputs. This concept is pivotal for large-scale or networked systems, where monitoring or controlling the entire state is infeasible, and interest centers on estimating or controlling a select “target” subset or aggregate quantity.

1. Algebraic Criteria for Functional Observability

The canonical context for functional observers is the linear time-invariant (LTI) system: $\dot x = A x + B u, \quad y = C x, \quad z = F x,$ with $x \in \mathbb{R}^n$, $u \in \mathbb{R}^p$, $y \in \mathbb{R}^q$, $z \in \mathbb{R}^r$. The functional observer seeks to reconstruct $F x(0)$ (or $F x(t)$) from the history of $y(t)$ and $u(t)$.

Algebraic Rank Test: The standard necessary and sufficient condition for functional (or “target”) observability is

$$\operatorname{rank} \begin{bmatrix} \mathcal O \ F \end{bmatrix} = \operatorname{rank} \mathcal O,$$

where $\mathcal O$ is the block observability matrix: $$\mathcal O = \begin{bmatrix} C \ C A \ \vdots \ C A^{n-1} \end{bmatrix}$$ If $F = I_n$, this reduces to standard full-state observability ($\operatorname{rank} \mathcal O = n$).

Popov–Belevitch–Hautus (PBH)–type Test: When $A$ is diagonalizable, $(C,A,F)$ is functionally observable if and only if 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 test generalizes to systems with Jordan block structures by appropriate decomposition. These results are formally established in multiple sources including (Montanari et al., 5 Feb 2024, Zhang et al., 2023, Zhang et al., 2023, Montanari et al., 29 Jan 2024, Montanari et al., 2023).

2. Graph-Theoretic and Structural Characterizations

For structured systems—where the zero/nonzero pattern of $A$, $C$, $F$ is prescribed but the numerical values are free—functional observability admits a direct combinatorial interpretation.

Given the directed inference graph $\mathcal{G}(C, A; F)$, with state nodes $\mathcal{X}$, sensor nodes $\mathcal{Y}$, and target set $\mathcal{T}$ (indices where $F$ is nonzero), $(C,A;F)$ is functionally observable if and only if:

• For every target node $x_i \in \mathcal{T}$ there exists a directed path in $\mathcal{G}$ from $x_i$ to some $y_j \in \mathcal{Y}$.
• $\mathcal{T}$ does not overlap with any minimal contraction set in $\mathcal{G}$.

Minimal contractions are state subsets whose sets of direct successors have smaller cardinality than the subset itself (i.e., informational bottlenecks in the network structure). This structural criterion can be efficiently checked via maximum matching in a bipartite expansion and forms the basis for scalable sensor placement in large-scale networks (Montanari et al., 2023, Montanari et al., 2022, Zhang et al., 2023, Zhang et al., 25 Sep 2024).

3. Observer Synthesis and Sensor Placement

When $(C,A;F)$ is functionally observable, a reduced-order observer (of order $$\operatorname{rank} \mathcal{O} - \operatorname{rank} [\mathcal{O}; F]$$) can be constructed using extensions of the Darouach–Trinh approach. The observer is of the general form: $$\dot w = N w + J y + H u, \qquad \hat z = D w + E y,$$ with design matrices $(N, J, H, D, E)$ chosen to ensure $\lim_{t \to \infty} \|\hat z(t) - F x(t)\| = 0$ and assignable error dynamics. The observer achieves the same estimation performance for $z = F x$ as a full-state observer, using (potentially) dramatically fewer sensors and lower order (Montanari et al., 2022, Montanari et al., 29 Jan 2024).

The sensor placement problem—how to select a minimal set of outputs (rows of $C$) so that $(C,A;F)$ is functionally observable—is, in general, NP-hard (Zhang et al., 2023, Zhang et al., 25 Sep 2024). However, greedy and set-cover algorithms offer polynomial-time approximations with provable logarithmic approximation ratios, and closed-form solutions exist for systems with diagonalizable $A$ (Zhang et al., 2023, Zhang et al., 25 Sep 2024).

4. Extensions: Sample-Based, Nonlinear, and Stochastic Settings

Sample-Based Functional Observability: The framework extends to discrete and irregularly sampled systems. The generalization replaces $\mathcal{O}$ with a sample-based observability matrix $O_s(A,C)$ constructed at sampled times. Functional observability is preserved if adding $O(A,F)$ to $O_s(A,C)$ does not increase the rank (Krauss et al., 30 Jun 2025).

Nonlinear Systems: Functional observability extends to the nonlinear context using Lie derivative algebras. One requires that, for the measured outputs $y = h(x)$ and the functional $z = q(x)$, the mapping $q(x)$ can be expressed as a function of finitely many output Lie derivatives,

$$q(x) = \mathcal{W}(L_f^i h_j(x) : i = 0, \dots, v-1, j),$$

where $L_f^i$ is the $i$-th Lie derivative along the dynamics $f$ (Montanari et al., 2023, Kravaris, 30 Dec 2024). The minimal integer $v$ providing such a relationship is known as the functional observer index and gives the lowest possible observer order for pole-assignment. Observer synthesis then leverages an invariance principle on the observer manifold to assign linear error dynamics to the constructed observer (Kravaris, 30 Dec 2024).

Stochastic Systems: In partially observed Markov processes, functional observability is formulated via the existence, for every bounded continuous function $f(x)$, of an approximating function of the measurement stream, guaranteeing the conditional distributions (filters) merge in a suitable sense. Functional observability under appropriate continuity and dominance conditions implies robust filter stability under weak, total-variation, and relative-entropy distances (McDonald et al., 2018).

5. Duality: Output Controllability and Target Observability

The concept of duality that relates controllability and observability in the full-state setting generalizes to target/or functional notions. For linear systems, functional observability of $(C, A; F)$ implies output controllability of the dual system $(A^T, C^T; F)$ (weak duality), and under conditions such as emptiness of minimal dilations, the converse holds (strong duality). In the strong duality case, algorithms for minimal control actuator placement and minimal output sensor placement can be directly ported between the dual problems (Montanari et al., 2023, Montanari et al., 29 Jan 2024).

6. Structural and Modal Functional Observability

Structural Functional Observability (SFO): In structured system theory, SFO is defined generically—almost all realizations of the parameter-free pattern yield a functionally observable system if and only if the generic rank of the augmented observability matrix equals that of the plain observability matrix. For generically diagonalizable patterns, efficient combinatorial criteria and matching-based algorithms provide minimal sensor sets (Zhang et al., 25 Sep 2024).

Modal Functional Observability: This unifies modal observability analysis: $(A,C,F)$ is functionally observable if, and only if, for every eigenvalue $\lambda_i$ (or Jordan block), appending $F_i$ to the block-structured observability matrix does not increase its rank (Zhang et al., 2023, Montanari et al., 5 Feb 2024). This enables blockwise, modal, and PBH-type analysis for arbitrary target functionals.

7. Applications and Special Cases

Functional observers are relevant across many domains, including network monitoring, power grid state estimation, chemical process control, and distributed/incomplete sampling scenarios. For example, in radial distribution networks, functional observability reduces to a statement about vertex-disjoint paths linking non-metered nodes to metered buses; this can be efficiently checked via max-flow algorithms (Bhela et al., 2016). In partial differential equations, optimal sensor placement can be framed as maximizing a “functional observability” spectral constant—unified via quantum ergodicity results—with implications for sensor layout in wave and Schrödinger systems (Privat et al., 2012).

Table: Key Rank/Graph Criteria for Functional Observability

Context	Criterion	References
LTI system	$$\operatorname{rank}\begin{bmatrix}\mathcal{O} \ F\end{bmatrix} = \operatorname{rank} \mathcal{O}$$	(Montanari et al., 2022, Montanari et al., 29 Jan 2024)
Diagonalizable $A$	PBH-type: $$\operatorname{rank}[A-\lambda I; C; F]=\operatorname{rank}[A-\lambda I; C]$$ $\forall \lambda$	(Zhang et al., 2023, Montanari et al., 5 Feb 2024)
Structured system	Each target node reaches some sensor and avoids minimal contractions/dilations	(Montanari et al., 2023, Zhang et al., 2023, Zhang et al., 25 Sep 2024)

• (Montanari et al., 2023) — Target controllability and target observability of structured network systems.
• (Montanari et al., 29 Jan 2024) — Duality between controllability and observability for target control and estimation in networks.
• (Zhang et al., 2023) — Functional observability, structural functional observability and optimal sensor placement.
• (Montanari et al., 2022) — Functional observability and target state estimation in large-scale networks.
• (Zhang et al., 25 Sep 2024) — Generic diagonalizability, structural functional observability and output controllability.
• (Montanari et al., 5 Feb 2024) — On the Popov-Belevitch-Hautus tests for functional observability and output controllability.
• (Kravaris, 30 Dec 2024) — On Functional Observability of Nonlinear Systems and the Design of Functional Observers with Assignable Error Dynamics.
• (Krauss et al., 30 Jun 2025) — On sample-based functional observability of linear systems.
• (Montanari et al., 2023) — Functional observability and subspace reconstruction in nonlinear systems.
• (Zhang et al., 2023) — Observability blocking for functional privacy of linear dynamic networks.
• (Bhela et al., 2016) — Enhancing observability in distribution grids using smart meter data.
• (Privat et al., 2012) — Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains.
• (McDonald et al., 2018) — Stochastic observability and filter stability under several criteria.
• (Gallaun et al., 2019) — Sufficient criteria and sharp geometric conditions for observability in Banach spaces.