Observability-Constrained Updating (OC)

• Observability-constrained updating is a methodology that maintains unobservable system characteristics by enforcing strict uncertainty quantification.
• It uses techniques like projected Jacobians and covariance adjustments in EKF, VINS, and LINS to prevent artificial reduction in uncertainty along unobservable directions.
• OC methods extend to epistemic models by synthesizing privatized action updates, ensuring leak-free belief revisions in multi-agent systems.

Observability-constrained updating (OC) is a principled methodology for preserving the inherent unobservable features of a dynamical or epistemic system during state (or belief) estimation. OC techniques enforce correct uncertainty quantification and consistency by guaranteeing that estimation algorithms do not introduce spurious information or overconfidence about directions or facts that are fundamentally unobservable given the available data and sensing setup. This article reviews the theoretical foundations, mathematical constructions, and key implementations of OC updating, including both state estimation in robotics/localization and knowledge update in epistemic multi-agent systems.

1. Observability in State Estimation and Epistemic Models

In nonlinear state estimation, particularly for inertial navigation systems (INS) and sensor fusion frameworks, observability characterizes which components of the system state can be inferred uniquely from the observed data under the given sensing and motion model. Mathematically, for a discrete-time system

$$x_{k+1} = f(x_k, u_k, w_k), \quad y_k = h(x_k) + e_k,$$

with process noise $w_k \sim \mathcal{N}(0, Q_k)$ and observation noise $e_k \sim \mathcal{N}(0,R_k)$, the (local) observability matrix encodes the directions in state space that can or cannot be recovered from the output history. A vector $v$ belongs to the unobservable subspace $\mathcal{N}_k = \operatorname{Null}(O_k)$ if $O_k v = 0$, meaning $v$ cannot be inferred from outputs $y_k$ over future times.

Analogously, in epistemic logic models such as Kripke structures for multi-agent knowledge, observability-constrained updates synthesize model transformations ensuring that the effect of a communication or belief update is limited exactly to the epistemically accessible content; facts not observable (or not meant to be revealed) to a given agent remain so, and private knowledge does not leak to irrelevant parties.

2. Motivation: Consistency and Filter Overconfidence

Naive application of state estimators, e.g., the Extended Kalman Filter (EKF), to partially observable systems often leads to inconsistent uncertainty estimates and artificial reduction in the variance along unobservable directions. This is particularly acute for yaw (heading) estimation in magnetic field–aided or visual/lidar–aided INS, where rotation about the gravity axis is not observable from purely relative sensors (magnetometers constrain only the local magnetic field model, not absolute heading) (Huang et al., 2024, Li, 2024).

The standard EKF, by linearizing about the time-varying current estimate, may locally “see” some unobservable directions as being weakly observed, causing the posterior covariance $P_k^\phi$ to shrink below its information-theoretic lower bound $P_0^\phi$. This overconfidence can be quantified by metrics such as normalized estimation error squared (NEES). In epistemic updates, improperly applied update mechanisms (full product update in DEL) may leak information to agents not intended to receive it, violating information privacy.

3. Mathematical Formulation of OC Updating

3.1 OC-EKF: Projecting onto Observable Subspace

In the OC-EKF framework, at each step, the observability matrix $O_k$ is constructed (locally or over a sliding window) to determine a basis $N_k$ for the unobservable subspace. The key step is to form the orthogonal projector $W = I - N_k(N_k^\top N_k)^{-1}N_k^\top$ onto the observable subspace.

The OC updating then replaces the Jacobians in the EKF with projected versions (e.g., $H_k$ replaced by $H_k W$) or, equivalently, projects the posterior covariance after each update:

$P_k^{+,OC} = W P_k^+ W^\top + P_N$

where $P_N = N_k(N_k^\top N_k)^{-1}N_k^\top$ is the projector onto the unobservable directions. This ensures that no spurious reduction of uncertainty can occur along unobservable modes, maintaining $P_k^\phi \geq P_0^\phi$ (Huang et al., 2024).

3.2 OC Constraints via Invariant Functionals

For general nonlinear systems, OC constraints may be expressed via invariant covectors $n_4(x)$ whose evolution is constant under the system flow and parametrization. In visual-inertial (VINS) or lidar-inertial (LINS) systems, such an $n_4(x)$ encodes the unobservable yaw–translation manifold and is kept constant by treating $h_{OC}(x) = n_4(x) - b_4$ as a zero-innovation pseudo-measurement, enforced at each filter update (Li, 2024).

3.3 OC in Epistemic Logic: Privatized, Pointed Updates

In multi-agent epistemic models, observability-constrained updating is realized by synthesizing action models that stratify the Kripke structure into fully privatized agent-clusters, so that only targeted agents acquire new beliefs and all other agents' epistemic states remain unaffected (Schlögl et al., 2024). The pointed update operation $(M,w) \odot (U, \alpha_0)$ applies the synthesized action model only to the reachable world–event pairs from the actual world, preventing model blow-up and information leakage outside the targeted agent cluster.

4. Representative OC Algorithms and Implementations

4.1 OC-EKF in Magnetic Field–Aided INS

Huang et al. extend the observability-constrained EKF to magnetic field–aided inertial navigation:

• State: $x = [p^n; v^n; q; \theta]$, with $\theta$ magnetometer field coefficients.
• Prediction: Standard inertial kinematics.
• Magnetometer measurements are linear in $\theta$, but do not observe yaw.
• Unobservable direction: rotation about gravity axis.
• OC-EKF projects update Jacobians/covariances to preserve this structure; transition Jacobian $F_k$ is minimally modified (using basis-alignment and block optimization) to ensure $F_k N_k = N_{k+1}$ at each step (Huang et al., 2024).

4.2 OC-VINS and OC-LINS

Li et al. prove that in VINS (with two non-collinear features) and LINS (with one feature), the global translation (3 DOF) and rotation about gravity (1 DOF) are unobservable. They define an invariant covector $n_4(x)$ and enforce the OC constraint $n_4(x(t)) = n_4(x(0))$ by incorporating it as a pseudo-measurement within EKF or ESKF frameworks (Li, 2024).

4.3 OC Updates in Distributed Epistemic Systems

OC update synthesis in Kripke models for multi-agent systems is achieved by recursively constructing privatized action models for deterministic belief increase (DBI) goal formulas and applying the pointed update operator. The resulting update is leak-free, preserves consistency if possible, and achieves minimal change—even under repeated applications—without exponential model growth (Schlögl et al., 2024).

5. Practical Consequences and Comparative Performance

Empirical evaluations demonstrate clear benefits of OC updating:

• In magnetic field–aided INS, OC-EKF yields lower position and yaw RMSE than standard EKF, and NEES values consistent with the true uncertainty (NEES for OC-EKF: 1.0 vs. EKF: up to 3). The perceived yaw uncertainty from standard EKF can arbitrarily sink below the initial level, while OC-EKF never violates the lower prior bound (Huang et al., 2024).
• In VINS/LINS, OC updating avoids artificial information gain along the unobservable yaw–translation manifold and consistently maintains filter reliability under reasonable feature-observability conditions (Li, 2024).
• In epistemic multi-agent systems, OC updates guarantee that only targeted agents' beliefs are affected, preserving privacy and avoiding redundant model expansion (Schlögl et al., 2024).

6. Limitations, Generalizations, and Domain-Specific Variants

OC updating methods assume the (local) unobservable subspace is correctly identified and remains consistent during operation. For nonstationary environments, moving feature sets, or degenerate motion, the unobservable directions may shift or lose their invariance; enforcement of the OC constraint may need to be suspended or adapted (Li, 2024).

Current OC epistemic update frameworks (DBI goals) do not directly handle negative beliefs, group announcements, or disjunctive information, which would require more advanced action models or new update semantics (Schlögl et al., 2024).

In online convex optimization (OCO), observability constraints manifest not via estimation nullspaces but as periodic feedback and delayed aggregation, restricting the granularity of available gradient/constraint information. Algorithms such as Periodic Queueing and Gradient Aggregation (PQGA) address these constraints by designing update rules compatible with information-limited regimes, but are not directly instances of state-space nullspace constraints (Wang et al., 2021).

7. Summary Table: Key Properties of OC Updating in Major Domains

Domain	Unobservable Features	OC Enforcement Mechanism	Notable Effect
INS, VINS, LINS	Yaw (gravity axis), translation	Jacobian/covariance projection, invariant covector as constraint	Consistent uncertainty, improved accuracy
Multi-agent epistemic	Private facts, unrelated beliefs	Privatized action model synth., pointed update	Leak-free, minimal-change, efficient model
OCO (periodic)	Slot-level gradients, constraints	Periodic virtual queues, delayed aggregation	Sublinear regret/violation under info constraints

Observability-constrained updating provides a rigorous solution to the fundamental problem of preserving physically and epistemically required indeterminacies in estimation and inference, ensuring no artificial certainty or information leakage is introduced in the estimation process across a wide range of applied domains (Huang et al., 2024, Li, 2024, Schlögl et al., 2024, Wang et al., 2021).