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On the Effects of Modeling Errors on Distributed Continuous-time Filtering

Xiaoxu Lyu, Shilei Li, Dawei Shi, Ling Shi

TL;DR

This work analyzes distributed continuous-time filtering under modeling errors by introducing a nominal performance index $\Sigma_u(t)$ and the estimation error covariance $\Sigma_e(t)$. It derives a nominal distributed Kalman filter using parameters $A_u$, $C_{i,u}$, $Q_u$, and $R_{i,u}$, and obtains coupled Lyapunov-type equations governing the evolution of $\Sigma_u(t)$ and $\Sigma_e(t)$. Key results include bounds on $\Sigma_e(t)$ in terms of $\Sigma_u(t)$ and Frobenius-norm deviations, a divergence condition when the process-noise covariance is incorrect, and a relation framework between the two indices that depends on the consensus gain $\gamma_u$. Numerical simulations on vehicle tracking validate the theoretical bounds and illustrate the potential for divergence under modeling errors, offering guidance for robust nominal-design in sensor networks.

Abstract

This paper offers a comprehensive performance analysis of the distributed continuous-time filtering in the presence of modeling errors. First, we introduce two performance indices, namely the nominal performance index and the estimation error covariance. By leveraging the nominal performance index and the Frobenius norm of the modeling deviations, we derive the bounds of the estimation error covariance and the lower bound of the nominal performance index. Specifically, we reveal the effect of the consensus parameter on both bounds. We demonstrate that, under specific conditions, an incorrect process noise covariance can lead to the divergence of the estimation error covariance. Moreover, we investigate the properties of the eigenvalues of the error dynamical matrix. Furthermore, we explore the magnitude relations between the nominal performance index and the estimation error covariance. Finally, we present some numerical simulations to validate the effectiveness of the theoretical results.

On the Effects of Modeling Errors on Distributed Continuous-time Filtering

TL;DR

This work analyzes distributed continuous-time filtering under modeling errors by introducing a nominal performance index and the estimation error covariance . It derives a nominal distributed Kalman filter using parameters , , , and , and obtains coupled Lyapunov-type equations governing the evolution of and . Key results include bounds on in terms of and Frobenius-norm deviations, a divergence condition when the process-noise covariance is incorrect, and a relation framework between the two indices that depends on the consensus gain . Numerical simulations on vehicle tracking validate the theoretical bounds and illustrate the potential for divergence under modeling errors, offering guidance for robust nominal-design in sensor networks.

Abstract

This paper offers a comprehensive performance analysis of the distributed continuous-time filtering in the presence of modeling errors. First, we introduce two performance indices, namely the nominal performance index and the estimation error covariance. By leveraging the nominal performance index and the Frobenius norm of the modeling deviations, we derive the bounds of the estimation error covariance and the lower bound of the nominal performance index. Specifically, we reveal the effect of the consensus parameter on both bounds. We demonstrate that, under specific conditions, an incorrect process noise covariance can lead to the divergence of the estimation error covariance. Moreover, we investigate the properties of the eigenvalues of the error dynamical matrix. Furthermore, we explore the magnitude relations between the nominal performance index and the estimation error covariance. Finally, we present some numerical simulations to validate the effectiveness of the theoretical results.
Paper Structure (12 sections, 10 theorems, 57 equations, 4 figures)

This paper contains 12 sections, 10 theorems, 57 equations, 4 figures.

Table of Contents

  1. Problem Statement
  2. Performance Indices
  3. Nominal Distributed Kalman Filter
  4. Estimation Error Covariance
  5. Performance Analysis
  6. Bounds of the Estimation Error Covariance
  7. Divergence Analysis
  8. Relations between $\Sigma_u(t)$ and $\Sigma_e(t)$
  9. Simulations
  10. Conclusions
  11. Graph Theory
  12. Supporting Lemmas

Key Result

Proposition 1

Consider the system (eq dynamics) and the nominal distributed filter (eq dot x_iu). The estimation error, the augmented estimation error, and the augmented estimation error covariance can be expressed as follows:

Figures (4)

  • Figure 1: Illustration of the communication topology.
  • Figure 2: $\text{MSE}$, $\text{Tr}(\bar{\Sigma}_e)$, the upper bound 1 of $\text{Tr}(\bar{\Sigma}_e)$, the upper bound 2 of $\text{Tr}(\bar{\Sigma}_e)$, and the lower bound of $\text{Tr}(\bar{\Sigma}_u)$ with the increasing consensus parameter in Case 1.
  • Figure 3: Illustration of the divergence caused by the incorrect covariance with the increasing time in Case 2.
  • Figure 4: Illustration of the relations between $\Sigma_u(t)$ and $\Sigma_e(t)$ with the increasing time in Case 3.

Theorems & Definitions (17)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Theorem 1
  • Remark 3
  • Proposition 4
  • Remark 4
  • Theorem 2
  • ...and 7 more