Distributed Set-Membership Filtering

• Distributed Set-Membership Filtering (DSMFing) is a family of distributed estimation techniques that use set intersections to manage unknown-but-bounded uncertainties in sensor networks.
• The methodology integrates local prediction, update, and fusion steps to refine feasible belief sets, ensuring convergence and asymptotic boundedness under convex constraints.
• DSMFing employs various set representations such as zonotopes, ellipsoids, and constrained zonotopes, making it suitable for robust multi-agent system applications and control scenarios.

to=arxiv_search ￣奇米影视json {"query":"Distributed Set-Membership Filtering DSMFing distributed set-membership filtering arXiv (Farina et al., 2018, Li et al., 17 Sep 2025, Ding et al., 2023, Ding et al., 2023)", "max_results": 10} to=arxiv_search 北京赛车前json {"query":"(Farina et al., 2018)", "max_results": 5} to=arxiv_search 凤凰大参考 to=arxiv_search аанацҳауеитjson {"query":"(Li et al., 17 Sep 2025)", "max_results": 5} Distributed Set-Membership Filtering (DSMFing) denotes a class of distributed estimation methods for networks of sensors or agents subject to unknown-but-bounded uncertainties, in which each node propagates and refines a local feasible set or belief set rather than a covariance matrix or a single probabilistic posterior. In the static linear-regression setting, distributed projection algorithms converge asymptotically to an element of the global feasible set under convex measurement sets and a nonempty global intersection (Farina et al., 2018). In linear discrete-time dynamic systems, a classical DSMFing architecture consists of local prediction, local update, and intersection-based fusion of posterior beliefs, and its asymptotic boundedness is studied through graph-dependent collective observation information (Li et al., 17 Sep 2025). In multi-agent systems with absolute and relative measurements, DSMFing has been formulated through uncertain ranges, constrained zonotopes, ellipsoids, and related convex set descriptions, with local communication replacing centralized fusion (Ding et al., 2023).

1. Core formulation and mathematical objects

In the dynamic setting, a representative DSMFing model is the linear discrete-time sampled-data system

${\mathbf{x}_{k + 1} = A{\mathbf{x}_k} + B{\mathbf{w}_k}, \qquad {\mathbf{y}_k^i = {C_i}{\mathbf{x}_k} + {\mathbf{v}_k^i,$

where $A$ is nonsingular, $C_i$ may be null if sensor $i$ has no measurements, and the realizations $x_k$, $w_k$, $y_k^i$, and $v_k^i$ are unknown but bounded. The range of an uncertain variable $\mathbf{x}$ is written as $$\llbracket\mathbf{x}\rrbracket := \{\mathbf{x}(\omega):\omega\in\Omega\}$$, and set size is measured by the diameter $A$0 (Li et al., 17 Sep 2025).

A central object is the local belief $A$1, defined from the perspective of sensor $A$2 as an outer bound of the posterior range $A$3, where $A$4 collects all measurements in the network (Li et al., 17 Sep 2025). This formulation is deliberately nonstochastic: the objective is not a minimum-variance estimate but a guaranteed bounding set consistent with the dynamics, measurements, and assumed error bounds.

In the static linear-regression setting, the unknown parameter $A$5 is observed through

$A$6

which induces the local feasible strip

$A$7

Each node accumulates measurements through the nonincreasing local feasible sets

$A$8

and the global asymptotic feasible parameter set is

$A$9

The standing assumption is $C_i$0, which holds when the bounded-error assumptions are not violated (Farina et al., 2018).

The network model is a directed graph $C_i$1. In the projection-based regression algorithms, strong connectivity and row-stochastic weights are assumed for the consensus step. In the dynamic boundedness analysis, the graph is decomposed into source components, reflecting strongly connected components with no incoming edges from outside; this topology enters the boundedness condition through the propagation of collective observation information (Farina et al., 2018, Li et al., 17 Sep 2025).

2. Interpolatory distributed estimation in static settings

A rigorous foundation for DSMFing in static linear regression is provided by distributed interpolatory algorithms based on projections onto local feasible sets (Farina et al., 2018). Two algorithms are considered.

The first is an incremental, cyclic, sequential projection method: $C_i$2 with the essential requirement that each agent projects infinitely often on its local feasible set. The second is a distributed consensus–then–projection scheme: $C_i$3 where only the current estimates $C_i$4 are communicated; measurements and feasible sets remain local (Farina et al., 2018).

The projection operator onto a closed set $C_i$5 is

$C_i$6

For the strip

$C_i$7

the Euclidean projection has the closed form

$C_i$8

If $C_i$9, the denominator disappears. This operation is $i$0 per projection, and for strips it modifies only the component of $i$1 along $i$2 (Farina et al., 2018).

The key theoretical statement is asymptotic interpolation. Under convex measurement sets and nonempty $i$3, both the cyclic incremental projections and the consensus–projection iterates converge to a common point $i$4; if $i$5, then all node estimates converge to $i$6 (Farina et al., 2018). The term “interpolatory” is precise: the limit belongs to the true global feasible set itself, not merely near it.

The proof mechanism is Fejér-type monotonicity with respect to $i$7. In the incremental case, the projection inequality

$i$8

for $i$9 yields bounded nonincreasing distance sequences and vanishing disagreement. In the distributed case, nonexpansiveness of projections plus Jensen’s inequality produces a weighted descent in $x_k$0, and the vanishing projection residuals imply asymptotic agreement through input-to-state stability of consensus (Farina et al., 2018).

This static theory is narrower than full dynamic DSMFing, but it establishes a foundational point: under convexity and nonempty feasibility, distributed set-membership updates can be both communication-efficient and provably interpolatory.

3. Dynamic DSMFing architectures and asymptotic boundedness

For linear discrete-time systems with unknown-but-bounded process and measurement noises, a “classical DSMFing” architecture abstracts many zonotopic and ellipsoidal implementations (Li et al., 17 Sep 2025). At each sensor $x_k$1, the recursion has three stages.

The local prediction is

$x_k$2

The local update is

$x_k$3

where

$x_k$4

The fusion step is intersection-based: $x_k$5 where $x_k$6 collects $x_k$7 and its 1-hop in-neighbors (Li et al., 17 Sep 2025).

This formulation is set-agnostic, but tractable implementations typically use zonotopes, constrained zonotopes, or ellipsoids. The analysis emphasizes the wrapping effect: Minkowski sums, linear transformations, intersections, and outerbounding with tractable shapes cause accumulation of conservatism and geometric inflation of the estimated sets. Asymptotic boundedness is therefore a central stability notion. A DSMF is asymptotically bounded if, for all $x_k$8,

$x_k$9

This limsup diameter condition is the formal metric used in the paper (Li et al., 17 Sep 2025).

The 2025 boundedness analysis introduces the Collective Observation-Information Tower (COIT), defined for a source component $w_k$0 by

$w_k$1

where $w_k$2 is the observation-information set contributed by sensor $w_k$3’s measurement at time $w_k$4, propagated to time $w_k$5. COIT captures exactly the observation-information sets that are commonly known across the source component by time $w_k$6 and links graph reachability to the evolution of belief sets (Li et al., 17 Sep 2025).

The main sufficient condition for asymptotic boundedness is stated after an observability decomposition of the lumped source-component pair $w_k$7. DSMFing is asymptotically bounded if, for each source component, the unobservable substate matrix $w_k$8 is marginally stable and, for every eigenvalue $w_k$9 with $y_k^i$0,

$y_k^i$1

Corollary 1 states that if $y_k^i$2 is detectable for every source component, then DSMFing is asymptotically bounded; in that case $y_k^i$3 is Schur and the rank condition holds trivially (Li et al., 17 Sep 2025).

A recurrent misconception is that DSMF boundedness is equivalent to the collective detectability condition known from distributed observers and distributed Kalman filters. The 2025 result explicitly states a broader sufficient condition: boundedness can still hold when collective detectability fails, provided marginally stable unobservable modes are not excited by process noise and observable-part coupling through the stated rank condition (Li et al., 17 Sep 2025).

4. Absolute and relative measurements in multi-agent systems

A major technical difficulty in distributed SMFing is the treatment of relative measurements. The 2023 framework for multi-agent systems with absolute and relative measurements addresses this by introducing a set description based on uncertain variables, termed the uncertain range, and proving that the accurate description requires only a single calculation step rather than multiple iterations (Ding et al., 2023).

For agent $y_k^i$4, the general nonlinear model is

$y_k^i$5

Given a known neighbor range $y_k^i$6, the feasible range induced on $y_k^i$7 by a relative measurement is

$y_k^i$8

The paper shows that repeated mutual refinement is unnecessary: only the first iteration reduces uncertainty; subsequent iterations do not improve (Ding et al., 2023).

On this basis, two distributed DSMFing frameworks are proposed. One computes the joint uncertain range of the agent itself and its neighbors and then projects onto the local subspace. The other computes only the marginal uncertain range of each local system. A rigorous set analysis yields the inclusion relation

$y_k^i$9

or equivalently that the distributed SMF can be considered as the process of computing the marginal uncertain range to outer bound the projection of the uncertain range obtained by the centralized SMF in the corresponding subspace (Ding et al., 2023).

A related cooperative-state-estimation formulation uses discrete-time linear agent dynamics, absolute measurements, and relative measurements, with uncertainty sets represented as zonotopes or constrained zonotopes (Ding et al., 2023). In that setting, the distributed constrained zonotopic algorithm operates by local prediction, formation of a joint prior with neighbors, joint measurement update with absolute and relative constraints, projection onto the agent’s own coordinates, and an interval-hull step for complexity control. The paper also proposes an OIT-Inspired centralized constrained zonotopic algorithm as a finite-horizon benchmark and reports that, at each time, both centralized OIT-inspired and distributed SMF sets contain the true positions of UAVs, while the standard centralized constrained-zonotope SMF produces the tightest sets (Ding et al., 2023).

This part of the literature clarifies a second common misconception: relative measurements do not force DSMFing into iterative message passing over repeatedly refined pairwise sets. In the uncertain-range framework, the exact relative-measurement description is obtained in a single step (Ding et al., 2023).

5. Set representations, computational structure, and communication

DSMFing is not tied to a single set class. In the static regression setting, the natural local measurement set is a strip; with constant noise bounds and normalized regressors, each node can maintain its cumulative feasible set through running maxima and minima of measurements, and projection onto a strip is $v_k^i$0 (Farina et al., 2018). This is the most explicit closed-form case.

In nonlinear multisensor systems, Wang, Shen, and Zhu formulate centralized and distributed set-membership information fusion with ellipsoidal uncertainty sets and derive analytical fusion formulae that are similar in form to the classic information filter (Wang et al., 2017). Their distributed fusion formula is

$v_k^i$1

$v_k^i$2

A notable feature is that the knowledge of the correlation among measurement noises across sensors is not required (Wang et al., 2017).

Constrained zonotopes are used when intersections with linear measurement relations must remain tractable. In the cooperative multi-agent estimator, affine maps, Minkowski sums, joint-update constraints, and projection/fusion are all expressed in constrained-zonotope form, while interval hulls and generator truncation are introduced to limit complexity growth (Ding et al., 2023). In the asymptotic boundedness framework, the analysis remains representation-agnostic, but the simulations use constrained zonotopes and explicitly identify outerbounding as a source of wrapping (Li et al., 17 Sep 2025).

Ellipsoidal DSMFing is prominent in leader–follower synchronization and fuzzy filtering. In the leader–follower setting, each agent runs a correction–prediction set-membership filter, with one correction SDP and one prediction SDP per time step, and shares only its corrected point estimate with neighbors (Bhattacharjee et al., 2020). In the nonlinear fuzzy attack-detection setting, each agent maintains prediction and estimation ellipsoids, computes them through LMIs derived from the S-procedure and Finsler’s lemma, and uses ellipsoid intersection tests for attack detection (Rahimifard et al., 2022).

Communication structure follows the set representation. The projection-based static algorithms transmit only current estimates, not measurements or raw feasible sets (Farina et al., 2018). The constrained-zonotopic cooperative estimators exchange compact descriptors such as centers, generators, and optionally constraint matrices; interval hulls reduce payload at the cost of conservatism (Ding et al., 2023). Ellipsoidal leader–follower DSMFing transmits only corrected state estimates, keeping ellipsoid parameters local (Bhattacharjee et al., 2020).

6. Relation to neighboring paradigms and representative application domains

DSMFing differs from Bayesian and Kalman-type estimation in both assumptions and guarantees. Under unknown-but-bounded noise, probabilistic filters require noise models such as Gaussian assumptions and yield point estimates not guaranteed to be feasible, whereas set-membership estimators produce estimates that respect hard bounds, with guaranteed interpolation when the global feasible set is nonempty (Farina et al., 2018). In the distributed-observer and distributed-Kalman-filter literature, the central questions are convergence of state estimates or boundedness of error covariance; in DSMFing, the object of interest is the size and evolution of the belief set itself (Li et al., 17 Sep 2025).

The approach has been specialized to several distributed control and monitoring problems. In leader–follower synchronization of discrete-time linear multi-agent systems, each agent is equipped with a set-membership filter in correction–prediction form, and the corrected state estimates are used in the local control law; under the stated graph and Riccati-based conditions, the global disagreement error is bounded and its upper bound is monotonically decreasing (Bhattacharjee et al., 2020). In nonlinear leader-following systems under replay attacks and false data injection attacks, a distributed fuzzy set-membership filtering method computes a prediction ellipsoid set and an estimation ellipsoid set for each agent; if the two sets do not intersect at the current time, a sensor attack is declared, while inconsistency between the current prediction set and the previous updated set indicates a control-signal or communication-signal attack (Rahimifard et al., 2022).

Another strand extends DSMFing to adaptive parameter estimation over diffusion networks. In robust set-membership diffusion normalization subband adaptive filtering, the parameter of interest is a common global vector $v_k^i$3, the local constraint is $v_k^i$4, and a MAD-based robust boundary is used to improve robustness to impulsive noise (Han et al., 3 Jun 2026). This is a different algorithmic lineage from state-set propagation, but it still fits the broad DSMFing theme of distributed set-membership estimation with data-selective updates.

A plausible implication of the present literature is that DSMFing is best understood not as one algorithm but as a family of distributed set-based estimators unified by three structural commitments: unknown-but-bounded uncertainty, local communication, and explicit set propagation or set intersection. Within that family, the decisive technical issues are convexity or tractable outer-bounding, topology-dependent information aggregation, and the extent to which distributed marginals can approximate or outer bound the projection of a centralized feasible set (Farina et al., 2018, Li et al., 17 Sep 2025, Ding et al., 2023).