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A Note on Optimal Distributed State Estimation for Linear Time-Varying Systems

Irene Perez-Salesa, Rodrigo Aldana-Lopez, Carlos Sagues

TL;DR

The paper tackles distributed state estimation for continuous-time, linear time-varying systems under stochastic noise in sensor networks. It demonstrates that the ODEFTC algorithm yields estimates at each node whose error covariance asymptotically matches the centralized Kalman-Bucy filter as the consensus gain $\kappa$ increases, while establishing a computable bound on $\kappa$ that guarantees stability. Extending prior results from the LTI setting to LTV systems, it shows that the covariance mismatch between the distributed and centralized filters vanishes in the limit of large $\kappa$, even in the presence of transitory mismatches in the local covariances. Simulation results corroborate the theoretical claims and reveal that the derived stability bound can be considerably tighter than previous bounds in the LTI case, enabling more efficient distributed estimation in practice.

Abstract

In this technical note, we prove that the ODEFTC algorithm constitutes the first optimal distributed state estimator for continuous-time linear time-varying systems subject to stochastic disturbances. Particularly, we formally show that it is able to asymptotically recover the performance, in terms of error covariance of the estimates at each node, of the centralized Kalman-Bucy filter, which is known to be the optimal filter for the considered class of systems. Moreover, we provide a simple sufficient value for the consensus gain to guarantee the stability of the distributed estimator.

A Note on Optimal Distributed State Estimation for Linear Time-Varying Systems

TL;DR

The paper tackles distributed state estimation for continuous-time, linear time-varying systems under stochastic noise in sensor networks. It demonstrates that the ODEFTC algorithm yields estimates at each node whose error covariance asymptotically matches the centralized Kalman-Bucy filter as the consensus gain increases, while establishing a computable bound on that guarantees stability. Extending prior results from the LTI setting to LTV systems, it shows that the covariance mismatch between the distributed and centralized filters vanishes in the limit of large , even in the presence of transitory mismatches in the local covariances. Simulation results corroborate the theoretical claims and reveal that the derived stability bound can be considerably tighter than previous bounds in the LTI case, enabling more efficient distributed estimation in practice.

Abstract

In this technical note, we prove that the ODEFTC algorithm constitutes the first optimal distributed state estimator for continuous-time linear time-varying systems subject to stochastic disturbances. Particularly, we formally show that it is able to asymptotically recover the performance, in terms of error covariance of the estimates at each node, of the centralized Kalman-Bucy filter, which is known to be the optimal filter for the considered class of systems. Moreover, we provide a simple sufficient value for the consensus gain to guarantee the stability of the distributed estimator.
Paper Structure (15 sections, 9 theorems, 70 equations, 2 figures)

This paper contains 15 sections, 9 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The ODEFTC Algorithm
  5. Assumptions
  6. Optimal Centralized Filter
  7. Auxiliary Results
  8. Optimality of ODEFTC for LTV Systems
  9. Error and Covariance Dynamics
  10. Choice of Consensus Gain for Stability
  11. Analysis under Consensus Assumption for $\mathbf{P}_i(t)$
  12. Effect of the Transitory in Convergence of $\mathbf{P}_i(t)$
  13. Proof of Theorem \ref{['th:main']}
  14. Simulation Example
  15. Conclusions

Key Result

Lemma 1

Let Assumption assum:bounds-Z hold. Consider a graph $\mathcal{G}$ with algebraic connectivity $\lambda_\mathcal{G}$, $N$ nodes and $\ell$ edges. For given design parameters $\alpha > 0$ and $\gamma \in (0,1)$, set $\xi \geq 2L/(\alpha \sqrt{\lambda_{\mathcal{G}}})$ and $T_{\max}=\ell \pi/(\alpha \g

Figures (2)

  • Figure 1: Communication graph $\mathcal{G}$.
  • Figure 2: MSE over 100 noise realizations for ODEFTC (all nodes are plotted, continuous line) vs. for the centralized Kalman-Bucy filter (dashed line).

Theorems & Definitions (19)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 2
  • proof
  • ...and 9 more