COIT: Collective Observation-Information Tower

• COIT is a geometric framework in distributed set-membership filtering that characterizes collectively available measurement information across sensor networks.
• It links graph topology, measurement propagation, and sensor estimation, yielding verifiable sufficient conditions for asymptotic boundedness.
• COIT generalizes collective detectability and unifies analyses across deterministic, noise-free, and stochastic estimation methods to guide efficient filter design.

Searching arXiv for the primary paper and closely related work on COIT/OIT. The Collective Observation-Information Tower (COIT) is a concept introduced for the analysis of asymptotic boundedness in Distributed Set-Membership Filtering (DSMFing) for linear discrete-time systems subject to unknown-but-bounded disturbances. It characterizes the cumulative, “commonly known” measurement information that is collectively available to a group of sensors as information propagates through a communication graph over time, and it provides a tight outer bound on the evolution of local set-valued estimates within a source component of the network (Li et al., 17 Sep 2025). In this role, COIT links graph topology, measurement structure, and the geometry of distributed set estimates, and it yields an easily verifiable sufficient condition for boundedness that generalizes the well-known collective detectability condition used in distributed observers and distributed Kalman filters (Li et al., 17 Sep 2025).

1. Origin and problem setting

COIT arises in the study of Distributed Set-Membership Filtering, where each sensor maintains a set-valued estimate of the state that encloses all states consistent with its local information and communicated information from the network (Li et al., 17 Sep 2025). In this setting, process and measurement noises are bounded rather than stochastic, so the principal stability question is not mean-square convergence or covariance boundedness, but whether the estimated sets remain uniformly bounded over time despite the wrapping effect (Li et al., 17 Sep 2025).

The motivating problem is explicitly distinguished from two better-studied neighboring regimes. For noise-free distributed estimation, the relevant benchmark is the convergence of Distributed Observers (DOs); for stochastic-noise estimation, it is the bounded error covariance of Distributed Kalman Filters (DKFs) (Li et al., 17 Sep 2025). COIT is introduced because the corresponding boundedness problem for DSMFing had remained underinvestigated relative to those two cases (Li et al., 17 Sep 2025).

A useful antecedent is the Observation-Information Tower (OIT) developed for classical, centralized set-membership filtering. OIT describes how measurements affect the estimate in a set-intersection manner without relying on the initial condition, and it enables explicit necessary and sufficient conditions for stability with respect to the initial condition in linear time-invariant systems (Cong et al., 2022). COIT extends this observation-information perspective to the distributed, graph-structured setting (Li et al., 17 Sep 2025).

2. Graph-theoretic setting and formal definition

The construction of COIT is tied to the communication graph through the notion of a source component, defined as a strongly connected component of the communication graph with no incoming edge from outside (Li et al., 17 Sep 2025). Let

$$G_t^{\mathrm{s}} = (V_t^{\mathrm{s}}, E_t^{\mathrm{s}})$$

denote a source component. For a sensor $i \in V_t^{\mathrm{s}}$ and time $k$, the COIT associated with source component $t$ is defined as (Li et al., 17 Sep 2025)

$$\boxed{ \mathcal{C}_k^{(t)}:=\bigcap_{r = 0}^{k-\tilde{\rho}+1} \; \bigcap_{l \in V_t^{\mathrm{s}}} \mathcal{O}_{k,r}^l }$$

with

$$\tilde{\rho} = \max\{\bar{\rho}_i : i \in V_t^{\mathrm{s}}\},$$

where $\bar{\rho}_i$ is the eccentricity, i.e., the largest path length from sensor $i$ to any other sensor in $V_t^{\mathrm{s}}$ (Li et al., 17 Sep 2025).

The constituent term $\mathcal{O}_{k,r}^l$ is the observation-information set at time $i \in V_t^{\mathrm{s}}$0 contributed by sensor $i \in V_t^{\mathrm{s}}$1’s measurement at time $i \in V_t^{\mathrm{s}}$2:

$i \in V_t^{\mathrm{s}}$3

with

$i \in V_t^{\mathrm{s}}$4

Here $i \in V_t^{\mathrm{s}}$5 is the bounded range of measurement noise at time $i \in V_t^{\mathrm{s}}$6 for sensor $i \in V_t^{\mathrm{s}}$7 (Li et al., 17 Sep 2025).

This distributed definition mirrors the centralized OIT construction, in which the OIT at time $i \in V_t^{\mathrm{s}}$8 with window $i \in V_t^{\mathrm{s}}$9 is given by the intersection $k$0, where each $k$1 is a single-measurement observation-information set propagated to time $k$2 (Cong et al., 2022). The relation is structural: OIT isolates the measurement-induced constraint geometry in the centralized case, whereas COIT aggregates the measurement-induced constraints that become collectively available in a source component of the network (Li et al., 17 Sep 2025, Cong et al., 2022).

3. Outer-bound interpretation and geometric meaning

The defining significance of COIT is that it yields an outer bound for the local set-valued estimate of every sensor inside a source component. Specifically, if sensor $k$3 belongs to $k$4, then at time $k$5 its local set estimate satisfies (Li et al., 17 Sep 2025)

$k$6

This inclusion means that COIT quantifies the best possible “collectively available” set-valued information for the entire source component at time $k$7 (Li et al., 17 Sep 2025). The bound is “collective” because it reflects not only a sensor’s own measurements, but also all information that can reach it through the communication graph over time (Li et al., 17 Sep 2025).

The underlying geometry is intersection-based. Each observation-information set represents the states consistent with one measurement after propagation through the system dynamics and bounded disturbances; COIT then intersects such sets across sensors and times within the communication reach determined by the source component (Li et al., 17 Sep 2025). This suggests that COIT is not merely a graph summary or an observability surrogate, but a set-valued object encoding both dynamic propagation and measurement consistency.

The comparison with OIT clarifies the conceptual lineage. In the centralized setting, the OIT characterizes observational information provided by measurements independently of the initial set, and the filter estimate is contained in the intersection of the OIT and the state-evolution set (Cong et al., 2022). COIT preserves the same observation-information logic while replacing a single measurement stream with distributed measurement streams constrained by network topology (Li et al., 17 Sep 2025).

4. Role in asymptotic boundedness analysis

COIT is introduced specifically to make the boundedness analysis of DSMFing tractable. By analyzing the evolution of $k$8, the theory establishes a checkable sufficient condition under which all local set estimates remain asymptotically bounded (Li et al., 17 Sep 2025).

The sufficient condition is stated at the level of each source component: for every source component $k$9, a specific condition linking the system dynamics $t$0, the collective output matrix $t$1, and the noise directions must hold (Li et al., 17 Sep 2025). In essence, the condition requires that no unobservable, unstable modes are present in the subsystem seen by any source component, with a precise algebraic characterization given in the theorem of the paper (Li et al., 17 Sep 2025).

Two direct conclusions are highlighted. First, if the pair $t$2 is detectable for each source component, then the estimates remain asymptotically bounded (Li et al., 17 Sep 2025). Second, the exact COIT construction allows the result to generalize and sometimes strictly relax the classic collective detectability requirement (Li et al., 17 Sep 2025).

This boundedness role is the distributed analogue of how OIT is used in centralized set-membership filtering. There, OIT supports explicit necessary and sufficient conditions for stability with respect to the initial condition, and detectability guarantees uniformly bounded estimates and estimation gap for bounded initial sets (Cong et al., 2022). COIT transfers that observation-information methodology to the case where information is fragmented across sensors and constrained by graph structure (Li et al., 17 Sep 2025).

5. Relation to collective detectability

A central theoretical contribution of COIT is its relation to collective detectability, the classic structural condition used in distributed observers and distributed Kalman filters (Li et al., 17 Sep 2025). In those settings, collective detectability means that although individual sensors may not observe the full state, the collective sensor group made available through the communication structure renders the system detectable (Li et al., 17 Sep 2025).

COIT generalizes this notion in two senses. It systematically formalizes what “collective information” every sensor in a source component will eventually know, and the sufficient condition for boundedness via COIT is shown to include collective detectability as a special case (Li et al., 17 Sep 2025). At the same time, the paper states that bounded set-membership estimation can be permitted even for some systems that are “not strictly detectable” in the classic sense, owing to the non-probabilistic, set-based nature of the estimation; this is identified as a uniquely beneficial property of DSMF (Li et al., 17 Sep 2025).

A concise comparison is as follows.

Method	Boundedness/convergence relies on	Role of COIT / detectability
Distributed Observer	Collective detectability	---
Distributed Kalman Filter	Collective detectability (for bounded error covariance)	---
Distributed Set-Membership Filter (DSMF)	COIT: Geometry of collectively available information sets	Generalizes detectability

This comparison indicates continuity rather than rupture. COIT links DSMFs to existing distributed estimation methods and reveals the unique characteristic of DSMFs, namely that boundedness can be certified through a set-valued information geometry that may be less restrictive than classical detectability requirements (Li et al., 17 Sep 2025).

A common misconception is to treat COIT as merely a renaming of collective detectability. The definitions and results do not support that reduction. Detectability is a property of $t$3-type pairs; COIT is a geometric object built from propagated observation-information sets, and detectability enters later as a sufficient condition or special case within the boundedness analysis (Li et al., 17 Sep 2025).

6. Analytical and design implications

COIT has direct implications for the design and verification of distributed filters. The paper gives a practical guideline: verify the relevant structural property, using observability decomposition and network topology, for each source component (Li et al., 17 Sep 2025). This makes the boundedness test local to source components rather than requiring a monolithic global argument (Li et al., 17 Sep 2025).

A further design implication is that COIT can help avoid over-engineering. The paper states that additional sensors are not needed for certain marginally stable but unobservable modes, provided they do not cause wrapping in the set-valued estimate (Li et al., 17 Sep 2025). This suggests that, in DSMF design, sufficiency should be evaluated with respect to bounded set propagation rather than imported uncritically from stochastic or noise-free estimation doctrines.

The analytical framework is also presented as unifying. COIT helps unify the analysis for deterministic (DSMF), noise-free (DO), and stochastic (DKF) cases (Li et al., 17 Sep 2025). In the predecessor OIT framework, a similar unification occurs at the centralized level: OIT supports a stability-guaranteed set-membership filtering framework and an efficient constrained zonotopic SMF that overcomes the wrapping effect while remaining well-posed and uniformly bounded under detectability-type conditions (Cong et al., 2022). The distributed COIT framework extends this line of reasoning from initial-condition dependence to network-mediated information aggregation (Li et al., 17 Sep 2025).

A plausible implication is that COIT provides a principled language for separating three coupled factors that are often conflated in distributed estimation: the dynamics $t$4, the measurement structure encoded by the relevant output matrices, and the information-routing constraints imposed by the communication graph. That interpretation is consistent with the paper’s emphasis on “the interplay between communication, system dynamics, and estimation performance” (Li et al., 17 Sep 2025).

7. Conceptual significance and scope

COIT is best understood as a central geometric object for distributed set-based estimation rather than as a general-purpose label for collective sensing (Li et al., 17 Sep 2025). Its formal content is specific: it is defined over source components of a communication graph, built from propagated observation-information sets, and used to derive asymptotic boundedness guarantees for DSMFs in linear discrete-time systems with unknown-but-bounded disturbances (Li et al., 17 Sep 2025).

Its significance follows from that specificity. First, COIT provides a precise outer-bound description of collectively available information. Second, it converts that description into a verifiable sufficient condition for boundedness. Third, it embeds DSMFing within the broader theory of distributed estimation by recovering collective detectability as a special case while revealing a distinctively set-membership advantage (Li et al., 17 Sep 2025).

The relation to OIT further underscores its place in the literature. OIT had already shown that observation-information structures can yield initial-condition-independent bounds, rigorous stability analysis, and stable set-membership filtering algorithms in the centralized case (Cong et al., 2022). COIT extends that perspective to distributed networks and thereby supplies a theoretical bridge between classical set-membership filtering and graph-based distributed estimation (Li et al., 17 Sep 2025).

In summary, the Collective Observation-Information Tower is a framework for expressing what a source component of a sensor network can collectively infer, in set-valued form, about the evolving system state. Through that framework, asymptotic boundedness of distributed set-membership filters becomes analyzable in terms of graph structure, propagated measurement information, and the presence or absence of unobservable unstable modes (Li et al., 17 Sep 2025).