A Dynamic Generalized Kalman Consensus Filter for Switching Sensor Networks

I. INTRODUCTION

• Centralized Kalman filtering requires agents to send observations to a fusion center, which becomes impractical as network size grows under communication and processing constraints.The introduction also notes that routing all data through a single node introduces a vulnerability to failure.Source: §I · p. 1
• Earlier approaches include an all-to-all decentralized filter, a scalable filter using embedded average consensus, and methods addressing oblivious agents or consensus-iteration requirements.The all-to-all method did not address scalability; later work reported inadequate performance for the embedded-consensus filter and a need for sufficient iterations in discrete-time systems.Source: §I · p. 1
• KCF’s unweighted average consensus gives equal weight to neighboring agents, including oblivious agents, and can lead to divergence.GKCF, ICF, and other methods use weighted average consensus to address this deficiency.Source: §I · p. 1
• Most consensus-based distributed filters assume fixed communication topologies, although mobile agents can move in and out of communication range.The paper identifies switching links, oblivious agents, and intermittent observations as relevant conditions for maintaining estimation accuracy.Source: §I · pp. 1–2
• DGKCF uses locally computed information-based consensus weights for target tracking under switching communication topologies, without requiring a global network parameter.Simulations report lower RMSE and MAE and faster convergence than the distributed filters considered, particularly with oblivious agents and switching topologies.Source: §I · p. 2

II. PROBLEM FORMULATION

• The system is a discrete-time LTI dynamical system monitored by N sensor agents, with process noise and agent-specific measurements.The process and measurement noises are zero-mean white Gaussian processes with covariances Q and R_i, respectively.Source: §II · p. 2
• The initial state, measurement noise, and process noise are mutually independent, and the full system state is observable across all agents collectively.Individual agents are not required to observe the entire state on their own.Source: §II · p. 2
• Communication is represented by an undirected graph whose active topology is selected from a finite admissible set by a piecewise-constant switching signal.Every admissible graph is assumed connected.Source: §II · p. 2
• Each agent seeks a local estimator that produces an accurate estimate of the system state, using a Kalman-filter posterior update.The Kalman gain is K_i(k); the formulation distinguishes prior and posterior estimates and their error covariances.Source: §II · p. 2
• The notation x+ denotes a variable’s value at the next time step.For example, x+ = f(x, w) represents x(k + 1) = f(x(k), w(k)).Source: §II · p. 2

III. ALGORITHM DEVELOPMENT

• Oblivious agents may lag several iterations, and KCF's equal weighting can let their inaccurate estimates degrade the fused estimate.These agents lack direct target observations and depend on estimates from neighbors that may already be outdated.Source: §III · p. 3
• Each agent collects a measurement, forms local information, exchanges estimates and information with neighbors, fuses those inputs, updates its estimate, and propagates it forward.The information-form filter uses a fused measurement information vector and matrix.Source: §III · pp. 2–3
• Because changing graph degree can make fixed ϵ slow or divergent, DGKCF replaces it with information-based weights that require no global convergence parameter.The fixed step size depends on the graph's maximum degree and is typically chosen before filtering.Source: §III · pp. 3–4
• Symmetrically combined inverse covariance traces measure link reliability; normalized weights favor lower-uncertainty neighbors and become uniform when covariances match.The trace is used as a computationally efficient, well-conditioned scalar uncertainty measure.Source: §III · p. 4
• Nonnegative weights summing to one preserve convexity in the consensus update, while additional iterations can move local estimates toward the global information-weighted average.Multiple weighted consensus iterations may be performed between measurement updates when communication allows.Source: §III · p. 4
• Weights are recomputed from local covariance and neighbor information at each time step, adapting consensus to connectivity changes and estimate quality while accommodating oblivious agents.The scheme assigns larger weights to more reliable estimates and smaller weights to less reliable ones.Source: §III · p. 4

IV. SIMULATION RESULTS

• The simulation compares the proposed filter with CKF, KCF, and GKCF for a target tracked by ten agents, including at least one oblivious agent.The LTI state vector has four position and velocity components across two dimensions; process covariance is diag(10, 10, 1, 1) and measurement covariance is 100I2.Source: §IV · p. 4
• The communication graph switches from G1 to G2 at k = 150, then to G3 and G4 at k = 300 and k = 450; only a single agent directly observes the target at each time step.The consensus step size is 0.65/∆0, with ∆0 = ∆(G1).Source: §IV · p. 4
• After each graph switch, KCF and GKCF perform worse than DGKCF because their initially chosen consensus step size is no longer optimal for the updated topology.The resulting slow convergence affects oblivious-agent estimates, which depend on a properly tuned consensus step size.Source: §IV · pp. 5–6
• DGKCF converges faster than GKCF at computational cost comparable to GKCF and approaches the CKF benchmark, while KCF does not converge during the simulation.Figure 6 reports RMSE of estimated state vectors averaged across agents.Source: §IV · p. 5, Figure 6 · p. 6

• DGKCF replaces fixed consensus step sizes with information-based weights computed from neighboring agents’ prior error covariance matrices.The weights use locally available information, so the method requires no global network parameters and can adapt to changing connectivity.Source: §V · p. 6
• DGKCF achieves lower RMSE and MAE and faster convergence than KCF and GKCF under switching topologies.Source: Table I · p. 6, §V · p. 6
• Future work will establish formal convergence guarantees and extend the framework to multi-target tracking.Source: §V · p. 6