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Communication-Efficient Distributed Kalman Filtering using ADMM

Muhammad Iqbal, Kundan Kumar, Simo Särkkä

TL;DR

This work tackles scalable state estimation in sensor networks by formulating distributed Kalman filtering (DKF) as a consensus optimization and solving it with a communication-efficient ADMM variant. A novel augmented Lagrangian design enables a fully distributed correction step that exchanges only primal variables, with dual-variable exchange eliminated; parameter bounds are derived in terms of the network Laplacian's maximum eigenvalue $\lambda_{\max}(\mathcal{L})$, yielding faster convergence. The authors prove convergence of the state estimates to the true state and that the local covariances converge to the centralized Riccati solution $P^*$, while maintaining unbiased estimates asymptotically. Simulation on a 100-node network demonstrates accurate tracking and fast consensus with reduced communication overhead, underscoring the method's practicality for large-scale distributed sensing systems.

Abstract

This paper addresses the problem of optimal linear filtering in a network of local estimators, commonly referred to as distributed Kalman filtering (DKF). The DKF problem is formulated within a distributed optimization framework, where coupling constraints require the exchange of local state and covariance updates between neighboring nodes to achieve consensus. To address these constraints, the problem is transformed into an unconstrained optimization form using the augmented Lagrangian method. The distributed alternating direction method of multipliers (ADMM) is then applied to derive update steps that achieve the desired performance while exchanging only the primal variables. Notably, the proposed method enhances communication efficiency by eliminating the need for dual variable exchange. We show that the design parameters depend on the maximum eigenvalue of the network's Laplacian matrix, yielding a significantly tighter bound compared to existing results. A rigorous convergence analysis is provided, proving that the state estimates converge to the true state and that the covariance matrices across all local estimators converge to a globally optimal solution. Numerical results are presented to validate the efficacy of the proposed approach.

Communication-Efficient Distributed Kalman Filtering using ADMM

TL;DR

This work tackles scalable state estimation in sensor networks by formulating distributed Kalman filtering (DKF) as a consensus optimization and solving it with a communication-efficient ADMM variant. A novel augmented Lagrangian design enables a fully distributed correction step that exchanges only primal variables, with dual-variable exchange eliminated; parameter bounds are derived in terms of the network Laplacian's maximum eigenvalue , yielding faster convergence. The authors prove convergence of the state estimates to the true state and that the local covariances converge to the centralized Riccati solution , while maintaining unbiased estimates asymptotically. Simulation on a 100-node network demonstrates accurate tracking and fast consensus with reduced communication overhead, underscoring the method's practicality for large-scale distributed sensing systems.

Abstract

This paper addresses the problem of optimal linear filtering in a network of local estimators, commonly referred to as distributed Kalman filtering (DKF). The DKF problem is formulated within a distributed optimization framework, where coupling constraints require the exchange of local state and covariance updates between neighboring nodes to achieve consensus. To address these constraints, the problem is transformed into an unconstrained optimization form using the augmented Lagrangian method. The distributed alternating direction method of multipliers (ADMM) is then applied to derive update steps that achieve the desired performance while exchanging only the primal variables. Notably, the proposed method enhances communication efficiency by eliminating the need for dual variable exchange. We show that the design parameters depend on the maximum eigenvalue of the network's Laplacian matrix, yielding a significantly tighter bound compared to existing results. A rigorous convergence analysis is provided, proving that the state estimates converge to the true state and that the covariance matrices across all local estimators converge to a globally optimal solution. Numerical results are presented to validate the efficacy of the proposed approach.
Paper Structure (7 sections, 8 theorems, 83 equations, 1 figure, 1 algorithm)

This paper contains 7 sections, 8 theorems, 83 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Distributed Kalman Filtering
  4. Distributed Kalman Filtering using ADMM
  5. Stability Analysis
  6. Simulation Results
  7. Conclusion

Key Result

Lemma 1

Let $P_{i, t \mid t-1}$ be a positive definite and symmetric matrix. Then the solution of problem1, considering the augmented Lagrangian augumented_lag is parameterized as $(\xi^*_t, \lambda^*_t)= ((1_N \otimes I_n)\xi^{\dag}_t,(1_N \otimes I_n) \hat{\lambda}_t+\bar{\lambda}_t)$, where $\xi^{\dag}_t

Figures (1)

  • Figure 1: Position and velocity RMSE of the proposed DKF with different sub-iterations obtained from 50 MC runs.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Lemma 3
  • proof
  • ...and 6 more