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Covariance-Intersection-based Distributed Kalman Filtering: Stability Problems Revisited

Zhongyao Hu, Bo Chen, Chao Sun, Li Yu

TL;DR

This work analyzes the stability of covariance-intersection based distributed Kalman filtering over time-varying directed graphs. It establishes that, for general time-varying systems, a node's stability depends only on the nodes that can uniformly reach it, without requiring global network connectivity. In the periodic setting, the paper proves convergence to a periodic steady state characterized by a Riccati-like equation, and provides a closed-loop stability condition via spectral radius bounds. These results enable steady-state CI filtering with reduced communication while guaranteeing mean-square stability, as supported by simulations. Overall, the paper extends CI KF stability analysis beyond fixed graphs to time-varying and periodic networks, with practical implications for large-scale cyber-physical systems.

Abstract

This paper studies the stability of covariance-intersection (CI)-based distributed Kalman filtering in time-varying systems. For the general time-varying case, a relationship between the error covariance and the observability Gramian is established. Utilizing this relationship, we demonstrate an intuition that the stability of a node is only related to the observability of those nodes that can reach it uniformly. For the periodic time-varying case, it is proved by a monotonicity analysis method that CI-based distributed Kalman filtering converges periodically for any initial condition. The convergent point is shown to be the unique positive definite solution to a Riccati-like equation. Additionally, by constructing an intermediate difference equation, the closed-loop transition matrix of the estimation error system is proved to be Schur stable. Notably, all theoretical results are obtained without requiring network connectivity assumptions. Finally, simulations verify the effectiveness of the stability results.

Covariance-Intersection-based Distributed Kalman Filtering: Stability Problems Revisited

TL;DR

This work analyzes the stability of covariance-intersection based distributed Kalman filtering over time-varying directed graphs. It establishes that, for general time-varying systems, a node's stability depends only on the nodes that can uniformly reach it, without requiring global network connectivity. In the periodic setting, the paper proves convergence to a periodic steady state characterized by a Riccati-like equation, and provides a closed-loop stability condition via spectral radius bounds. These results enable steady-state CI filtering with reduced communication while guaranteeing mean-square stability, as supported by simulations. Overall, the paper extends CI KF stability analysis beyond fixed graphs to time-varying and periodic networks, with practical implications for large-scale cyber-physical systems.

Abstract

This paper studies the stability of covariance-intersection (CI)-based distributed Kalman filtering in time-varying systems. For the general time-varying case, a relationship between the error covariance and the observability Gramian is established. Utilizing this relationship, we demonstrate an intuition that the stability of a node is only related to the observability of those nodes that can reach it uniformly. For the periodic time-varying case, it is proved by a monotonicity analysis method that CI-based distributed Kalman filtering converges periodically for any initial condition. The convergent point is shown to be the unique positive definite solution to a Riccati-like equation. Additionally, by constructing an intermediate difference equation, the closed-loop transition matrix of the estimation error system is proved to be Schur stable. Notably, all theoretical results are obtained without requiring network connectivity assumptions. Finally, simulations verify the effectiveness of the stability results.

Paper Structure

This paper contains 12 sections, 9 theorems, 118 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System description
  4. CI-based distributed Kalman filtering
  5. Problems of interest
  6. The General Time-Varying Case
  7. Stability Analysis
  8. The Periodic Time-Varying Case
  9. Convergence analysis
  10. Closed-Loop performance analysis
  11. Simulations
  12. Conclusion

Key Result

Theorem 1

Consider the plant (1), the sensor network (2), and CI-based distributed Kalman filtering (4)-(10). Let Assumption 1 hold. Define If $([C(k)]_{\mathbb{R}_i},A(k))$ is uniformly observable, then for any $0\leq \hat{P}(0|0)<\infty I$, there exists finite constant $p_{u,i}>0$ and $K>0$ independent of $\hat{P}(0|0)$ such that

Figures (4)

  • Figure 1: The topology of the sensor network.
  • Figure 2: The MSE of CI-based distributed Kalman filtering in the general time-varying system.
  • Figure 3: The MSE of CI-based distributed Kalman filtering in the general time-varying system ($C_1$ in (80) is replaced by $C_1=[0,1,0,0;0,0,0,1]$).
  • Figure 4: The trace of the error covariance of CI-based distributed Kalman filtering in the periodic time-varying system.

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2: rugh1996linear
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 3
  • Theorem 2
  • ...and 18 more