ODEFTC: Distributed Optimal Estimation

• The ODEFTC algorithm is a distributed estimation framework that combines local Kalman–Bucy filtering with consensus protocols to match centralized performance as consensus gain increases.
• It employs Riccati-type evolution of local error covariances and Lyapunov-based analysis to ensure exponential decay of estimation errors in a network of sensor nodes.
• By tuning the consensus gain above a defined threshold, the method achieves robust network synchronization and optimal state estimation in continuous-time LTV stochastic systems.

The ODEFTC (Optimal Distributed Estimation based on Fixed-Time Consensus) algorithm is a distributed estimation framework for continuous-time linear time-varying (LTV) stochastic systems. It targets scenarios where a network of sensor nodes collectively seeks to estimate the evolving state of a dynamical system under stochastic disturbances, such as process and measurement noise. Each node maintains a local estimator combining the classical Kalman–Bucy structure with an inter-node consensus protocol, ensuring both stability and asymptotic optimality: as consensus gain increases, individual node estimates (including error covariances) converge to those of an idealized, centralized Kalman–Bucy filter.

1. Structural Design of the ODEFTC Algorithm

At its core, ODEFTC equips each network node with a local estimator driven by the following principles:

• Each node $i$ uses its own measurement sequence to update a local estimate $\hat{x}_i(t)$ of the system state $x(t)$, along with a local error covariance estimate $P_i(t)$. The state evolution conforms to a continuous-time LTV system subject to stochastic noise.
• The state-update equation augments the standard Kalman–Bucy innovation term with a consensus term. Formally, the dynamics for node $i$ are of the type:

$$\frac{d\hat{x}_i}{dt} = A(t)\hat{x}_i(t) + K_i(t)\left[y_i(t) - C_i(t)\hat{x}_i(t)\right] + \kappa P_i(t) \sum_{j \in \mathcal{N}_i} \left[\hat{x}_j(t) - \hat{x}_i(t)\right]$$

where $\kappa$ is the consensus gain and $K_i(t)$ is the local Kalman gain.

• The error covariance $P_i(t)$ evolves independently via a Riccati-type equation, closely following the centralized filter's Riccati equation, but computed locally.
• Auxiliary consensus variables, specifically the consensus matrices $Q_i$, are incorporated to synchronize the information matrices across the network via a fixed-time consensus protocol.

This structure yields algorithmic simplicity and modularity: local innovations act as in the centralized filter, while consensus terms ensure global alignment across the distributed architecture.

2. Proof of Asymptotic Optimality

The theoretical foundation of ODEFTC rests on its guarantee to attain centralized filter performance as consensus gain increases:

• Define the local estimation error at node $i$ as $e_i(t) = x(t) - \hat{x}_i(t)$.
• The collective network error $e(t)$ can be decomposed into the consensus (average) component $\bar{\epsilon}(t)$ and the disagreement component $\tilde{e}(t)$ such that $$e(t) = \mathbf{1}_N \otimes \bar{\epsilon}(t) + \tilde{e}(t)$$.
• The average component $\bar{\epsilon}(t)$ evolves according to the centralized Kalman–Bucy error update:

$$\dot{\bar{\epsilon}}(t) = \left[A(t) - K(t) C(t)\right] \bar{\epsilon}(t)$$

which converges to zero due to centralized filter properties.

• The disagreement component $\tilde{e}(t)$ is driven by a Lyapunov function $$V(\tilde{e}(t)) = \tilde{e}(t)^\top (I_N \otimes P(t)^{-1}) \tilde{e}(t)$$, whose time derivative contains a negative term $-2\kappa \lambda_\mathcal{G} \|\tilde{e}(t)\|^2$ (with $\lambda_\mathcal{G}$ the algebraic connectivity of the network), ensuring exponential decay if $\kappa$ is chosen large enough:

$$\kappa > \kappa_0 = \frac{c^2}{2 r_1 \lambda_\mathcal{G}}$$

where $c$ bounds the measurement matrix and $r_1$ the noise covariance.

• For the error covariance mismatch $X(t) = \mathbb{P}(t) - (U_N \otimes P(t))$, a vectorized Lyapunov argument shows:

$$\lim_{t\to\infty} \|\mathbb{P}_i(t) - P(t)\| \leq b(\kappa),\quad \text{with}\quad \lim_{\kappa\to\infty} b(\kappa) = 0$$

Thus, ODEFTC aligns both state estimate and covariance of each node asymptotically with the centralized solution, under appropriate consensus gain conditions.

3. Error Covariance Convergence and Network Synchronization

A central claim is that each $P_i(t)$ (local error covariance at node $i$) evolves uniformly and reaches the centralized Riccati solution after consensus is attained:

• For all $t \geq 0$, $P_i(t)$ remains bounded.
• Once auxiliary consensus variables synchronize, $P_i(t)$ strictly follows:

$$\dot{P}(t) = A(t)P(t) + P(t)A(t)^\top + W(t) - P(t)C(t)^\top R(t)^{-1} C(t) P(t)$$

matching the centralized Kalman–Bucy covariance evolution.

• This ensures that, after an initial consensus transient, the entire filtering network operates with error-covariance performance indistinguishable from a centralized optimal estimator.

4. Role of Consensus Gain and Lyapunov-Based Stability

The consensus gain $\kappa$ is fundamental:

• It appears directly in the state update’s consensus term as an amplification of discrepancies between neighbors.
• The threshold

$$\kappa > \kappa_0 = \frac{c^2}{2 r_1 \lambda_\mathcal{G}}$$

ensures exponential decay of disagreement and stability in state and covariance dynamics.

• The ability to tune $\kappa$ directly controls the speed of network synchronization and the tightness of the covariance bound $b(\kappa)$; increasing $\kappa$ improves performance up to practical communication and computation limits.

The value $\kappa_0$ is often less conservative than prior results for LTI systems, meaning smaller gains can guarantee stability in practical networks.

5. Applications and Practical Implications

ODEFTC provides significant utility in various domains:

Application Area	Objective	ODEFTC Feature
Distributed sensor networks	Target tracking, env. monitoring	Decentralized processing, scalability
Navigation in large-scale networks	Distributed data fusion	Robust consensus with minimal communication
Systems with communication constraints	Optimal estimation under limited bandwidth	Asymptotic covariance matching

• In all cases, each node requires only local computations and limited neighbor-to-neighbor communication (state and consensus variables), enabling scalability and inherent robustness to failures.
• The algorithm bridges centralized and decentralized paradigms: even in the absence of global data aggregation, optimal state estimation performance (in error covariance sense) is retained as $\kappa$ increases.
• The explicit analytic condition for stability allows distributed, adaptive setting of consensus gain based solely on known measurement and network graph parameters.

6. Theoretical and Methodological Impact

ODEFTC provides a rigorous framework for distributed optimal state estimation in continuous-time, time-varying contexts:

• It constitutes the first distributed estimator for continuous-time LTV systems that provably recovers centralized Kalman–Bucy filter performance (Perez-Salesa et al., 21 Oct 2025).
• The Lyapunov-based dual error decomposition (network mean and disagreement) clarifies the role of consensus in distributed estimation, extending stability analysis beyond previous LTI results.
• The results enable less conservative designs, broadening the applicability of distributed filtering to scenarios where previous gain requirements would be impractical.

A plausible implication is that ODEFTC lays the foundation for further extensions, such as time-varying graph topologies or asynchronous updates, while preserving near-optimal estimation properties. Its clear separation of consensus and filtering dynamics also increases transparency for hardware or embedded implementations focused on large-scale sensor arrays.